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At its simplest a solution comprises one liquid component is vast excess, the solvent, and another, the solute, which is dispersed in the solvent. The solvent is more than just a useful medium in which to disperse the solute although one might argue that a key role is to inhibit associations of the solute molecules. In an even cursory examination of the properties of solutions, a key consideration is the distance between solute molecules. An interesting calculation offers insight into the dependence of solute-solute distances on solute concentration [1]. For a simple non-ionic solute ( e.g. urea) in aqueous solution at concentration $\mathrm{c}_{j} \mathrm{~mol dm}^{-3}$, the average solute-solute distance $\mathrm{d}$ is given by equation (a) where $\mathrm{N}_{\mathrm{A}}$ is the Avogadro number.
$d=\left(N_{A} \, c_{j}\right)^{-1 / 3}$
At $\mathrm{c} = 10^{-2} \mathrm{~mol dm}^{-3}$, $\mathrm{d} = 5.5 \mathrm{~nm}$. If the solute is a 1:1 salt where 1 mole of salt yields two moles of solute ions, $\mathrm{d} = 4.4 \mathrm{~nm}$. With increase in solute concentration, the mean distance between solute molecules decreases.
An interesting feature of aqueous solutions is worthy of comment. If a given water molecule is hydrogen bonded (indicating strong cohesion) to four nearest neighbour water molecules, that water molecule exists in a state of low density-high molar volume. In other words cohesion is linked to low density, a pattern contrary to that encountered in most systems. Nevertheless in reviewing the properties of aqueous solutions and water, one must be wary of overstressing the importance of hydrogen bonding. Indeed liquid water has a modest viscosity which is not the conclusion would draw from some models for liquid water which emphasize the role of water-water hydrogen bonding.
Footnote
[1] R. A. Robinson and R. H. Stokes, Electrolyte Solutions, Butterworths, London 2nd. Edn. Revised, 1965.
1.14.71: Time and Thermodynamics (Timenote)
We note two comments in monographs dealing with thermodynamics.
One comment [1] states that ‘… thermodynamics deals with systems at equilibrium, time is not a thermodynamic co-ordinate.’
The reference here is in the context of systems at equilibrium [1].
A stronger statement with a different view is made by McGlashan [2].
Thus
‘We shall be using time $\mathrm{t}$ as one of our variables in this chapter. There are those who say that time has no place in thermodynamics. They are wrong.’
Some history sets the scene.
Once upon a time chemists used the calorie as a unit of energy. In fact there were three different units named calorie: thermochemical calorie, international calorie and 150 C calorie. In common they defined energy in terms of the amount of energy required to raise by one Kelvin, the temperature of one gram of pure liquid water under specified conditions of temperature and pressure. Time is not mentioned, directly or indirectly, in this definition. Then Joule showed there is an equivalence between heat and mechanical energy. It is just a small step to relate thermal energy to kinetic energy and, hence, to time. If a calorimetric definition of energy had been adopted, then its unit would be a base unit. In practice, this would be a regression to the situation before Joule determined the mechanical equivalent of heat.
Wood and Battino [1] and McGlashan[2] are both right. Time is an important thermodynamic variable for formulating the conditions under which systems approach an equilibrium state. However, time is not used to describe the properties of these systems after equilibrium is attained.
Footnotes
[1] S. E. Wood and R. Battino, Thermodynamics of Chemical Systems, Cambridge Univeristy Press, Cambridge,1990, page 2.
[2] M. L. McGlashan, Chemical Thermodynamics, Academic Press, London, 1979, page 102; the footnotes in this text are often provocative.
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A colleague has filled a flask with water and asks us by phone to estimate the volume of water in the flask. Clearly this is an impossible task but our colleague offers further information. In answer to the first question, our colleague informs us that there are 2 moles of water in the flask. Immediately we suggest that the volume of water is $36 \mathrm{cm}^{3}$. Not good enough! Our colleague demands a more precise estimate. We know that the volume of water($\ell$) depends on temperature and pressure and so request new this information. We are told that the temperature is $298.2 \mathrm{~K}$ and the pressure is $101325 \mathrm{~N m}^{-2}$. We summarize this information in the following form.
$\mathrm{V}=\mathrm{V}\left[298.2 \mathrm{~K} ; 101325 \mathrm{~N} \mathrm{~m} \mathrm{~m}^{-2} ;(\ell) ; 2 \text { moles }\right]$
Our colleague offers further information such as the vapor pressure and heat capacity of water($\ell$) under these conditions. But we decline this offer on the grounds that no further information is required. We know that having defined the variables in the square brackets [.....], a unique volume $\mathrm{V}$ is defined. We may not immediately know the actual volume but given a little time in a scientific library we will be in a position to report volume $\mathrm{V}$.
The variables in the square brackets are the INDEPENDENT VARIABLES [1]. For a system containing one chemical substance we define the volume as follows.
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}\right]$
The term independent means that within limits [1] we can change $\mathrm{T}$ independently of the pressure and $\mathrm{n}_{1}$; change $\mathrm{p}$ independently of $\mathrm{T}$ and $\mathrm{n}_{1}$; change $\mathrm{n}_{1}$ independently of $\mathrm{T}$ and $\mathrm{p}$. There are some restrictions in our choice of independent variables. At least one variable must define the amount of all chemical substances in the system and one variable must define the 'hotness' of the system.
The molar volume of liquid chemical substance 1 at the specified temperature and pressure, $V_{1}^{*}(\ell)$ is obtained from equation (b) by fixing $\mathrm{n}_{1}$ at 1 mol. Thus
$\mathrm{V}_{1}^{*}(\ell)=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}=1 \mathrm{~mol}\right]$
If the composition of a given closed system is specified in terms of the amounts of two chemical substances, 1 and 2, four independent variables $\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]$ define the independent variable $\mathrm{V}$ [2]. Thus
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]$
Actually there is merit in writing equation (d) in terms of three intensive variables which in turn defines the molar volume $\mathrm{V}_{\mathrm{m}}$ of the binary system at given mole fraction $x_{1}=1-x_{2}$. Thus
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right]$
For a system containing i - chemical substances where the amounts can be independently varied, the dependent extensive variable $\mathrm{V}$ is defined by equation (f).
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots, \mathrm{n}_{\mathrm{i}}\right]$
Similarly the dependent intensive variable $\mathrm{V}_{\mathrm{m}}$ is defined by equation (g).
$\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}, \mathrm{x}_{2} \ldots \ldots \ldots, \mathrm{x}_{\mathrm{i}-1}\right]$
Footnotes
[1] The phrase 'independent variable' is important. With reference to the properties of an aqueous solution containing ethanoic acid, the number of components for such a solution is 2, amount of water and amount of ethanoic acid. The actual amounts of ethanoic acid, water, ethanoate and hydrogen ions are determined by an equilibrium constant which is an intrinsic property of this system at given $\mathrm{T}$ and $\mathrm{p}$. From the point of the Phase Rule [2], the number of components equals 2. For the same reason when we consider the volume of a system containing only $\mathrm{n}_{j}$ moles of water we disregard evidence that water partly self-dissociates into $\mathrm{H}^{+} (\mathrm{aq})$ and $\mathrm{OH}^{-} (\mathrm{aq})$.
[2] In terms of the Phase Rule, for two components ($\mathrm{C} = 2$) and one phase ($\mathrm{P} = 1$), the number of degrees of freedom $\mathrm{F}$ equals 3. These degrees of freedom refer to a set of intensive variables. Hence, for a solution where substance 1 is the solvent and substance 2 is the solute, the system is defined by specifying the three (intensive) degrees of freedom, $\mathrm{T}, \mathrm{~p}$ and, for example, solute molality.
1.14.73: Variables: Gibbsian and Non-Gibbsian
Experience shows that the thermodynamic state of a closed single phase system can be defined by a minimum set of independent variables where at least one variable is a measure of the ‘hotness’ of the system; e.g. temperature. The volume of an aqueous solution containing $\mathrm{n}_{1}$ moles of water and $\mathrm{n}_{j}$ moles of urea is defined by the set of independent variables, $\mathrm{T}, \mathrm{~p}, \mathrm{~n}_{1} \text { and } \mathrm{n}_{j}$.
$\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{\mathrm{j}}\right]$
Having defined the parameters set out in the brackets [...] the volume of the system, the dependent variable, is uniquely defined. In fact we can replace $\mathrm{V}$ in this equation by $\mathrm{G}, \mathrm{~H} \text { and } \mathrm{~S}$ in order to define unique Gibbs energy, enthalpy and entropy respectively.
The set of independent variables in equation (a) is called Gibbsian because the set comprises the intensive variables $\mathrm{T}$ and $\mathrm{p}$ together with the extensive composition variables [1]. The general form of equation (a) defining the thermodynamic potential function, Gibbs energy $\mathrm{G}$ is as follows where $\xi$ is the extensive composition variable.
$\mathrm{G}=\mathrm{G}[\mathrm{T}, \mathrm{p}, \xi]$
Other sets of independent variables are used in conjunction of the thermodynamic potential functions, enthalpy $\mathrm{H}$, energy $\mathrm{U}$ and Helmholtz energy $\mathrm{F}$.
$\mathrm{F}=\mathrm{F}[\mathrm{T}, \mathrm{V}, \xi]$
$\mathrm{U}=\mathrm{U}[\mathrm{S}, \mathrm{V}, \xi]$
$\mathrm{H}=\mathrm{H}[\mathrm{S}, \mathrm{p}, \xi]$
In equations (c) and (d), $\mathrm{V}$ is an extensive variable and in equations (d) and (e) S is an extensive variable. The sets of independent variables in equations (c), (d) and (e) are called non-Gibbsian [1].
Footnote
[1] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douhéret, Phys. Chem. Chem. Phys., 2001, 3, 1465.
1.14.74: Vaporization
A given chemical substance $j$ can exist in phases I and II. For phase I,
$\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{I})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{I})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I}) \label{a}$
$\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{I}), \mathrm{~H}_{\mathrm{j}}^{*}(\mathrm{I}) \text { and } \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{I})$ are the molar thermodynamic energy, enthalpy and volume respectively of chemical substance $j$ in phase I at pressure $\mathrm{p}$. Chemical substance $j$ can also exist in phase II at the same pressure $\mathrm{p}$.
$\mathrm{U}_{\mathrm{j}}^{*}(\mathrm{II})=\mathrm{H}_{\mathrm{j}}^{*}(\mathrm{II})-\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{II}) \label{b}$
Equations \ref{a} and \ref{b} are quite general. In an important application we identify phase II as the vapor phase which we assume to have the properties of a perfect gas. Phase I is the liquid state. For the process `liquid → vapor' ( i.e. vaporization) at temperature $\mathrm{T}$,
$\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})=\Delta_{\mathrm{vap}} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T})-\mathrm{p} \,\left[\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~g})-\mathrm{V}_{\mathrm{j}}^{*}(\ell)\right]$
But at temperature $\mathrm{T}$, $\mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~g})-\mathrm{V}_{\mathrm{j}}^{*}(\ell) \gg 0$ Also for one mole of a perfect gas, $\mathrm{p} \, \mathrm{V}_{\mathrm{j}}^{*}(\mathrm{~g})=\mathrm{R} \, \mathrm{T}$.
Hence,
$\Delta_{\text {vap }} \mathrm{U}_{\mathrm{j}}^{*}(\mathrm{~T})=\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{*}(\mathrm{~T})-\mathrm{R} \, \mathrm{T}$
$\Delta_{\text {vap }} \mathrm{H}_{\mathrm{j}}^{\mathrm{N}}(\mathrm{T})$ is obtained from the dependence of vapour pressure on temperature; see Clausius - Clapeyron Equation. Hence we obtain the molar thermodynamic energy of vaporisation.
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The (shear) viscosities of salt solutions have been extensively studied [1,2]. A frequently cited paper (always worth reading in terms of the care taken in the experimental study) was published by Jones and Dole [3]. The dependence of the viscosity of an aqueous salt solution $\eta(\mathrm{aq})$ (at fixed temperature and pressure) on concentration of salt $j$ is described by the Jones-Dole equation; equation (a) where $\eta_{1}^{*}(\ell)$ is the viscosity of water($\ell$) at the same $\mathrm{T}$ and $\mathrm{p}$ [3,4].
$\eta(\mathrm{aq}) / \eta_{1}^{*}(\ell)=1+\mathrm{A} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}+\mathrm{B} \, \mathrm{c}_{\mathrm{j}}$
Equation (a) is re-expressed as follows. By definition;
$\psi=\left[\eta(\mathrm{aq})-\eta_{1}^{*}(\ell)\right] /\left[\eta_{1}^{*}(\ell) \, \mathrm{c}_{\mathrm{j}}^{1 / 2}\right]$
Hence
$\psi=\mathrm{A}+\mathrm{B} \, \mathrm{c}_{\mathrm{j}}^{1 / 2}$
A plot of $\psi$ against $\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}$ has intercept $\mathrm{A}$ and slope $\mathrm{B}$, the Jones-Dole $\mathrm{B}$-viscosity coefficient [5]. The $\mathrm{A}$ coefficient describes the impact of charge-charge interactions on the viscosity of a solution, being generally positive and estimated using the Falkenhagen equation [6-8]. The $\mathrm{B}$ coefficient characterizes ion-solvent interactions at defined $\mathrm{T}$ and $\mathrm{p}$. For a 1:1 salt $j$, the $\mathrm{B}_{j}$ coefficient for salt $j$ is expressed as the sum of ionic $\mathrm{B}$ coefficients.
$\mathrm{B}_{\mathrm{j}}=\mathrm{B}_{+}+\mathrm{B}$
For example, $\mathrm{B}_{j}$ for a series of salts with a common anion, the changes in $\mathrm{B}_{j}$ reflect changes in $\mathrm{B}_{+}$ for the cations. The pattern in $\mathrm{B}\left(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~I}^{-};\mathrm{aq}; 298.2 \mathrm{~K}\right)$ reflects the changes in $\mathrm{B}\left(\mathrm{R}_{4}\mathrm{N}^{+} \mathrm{~I}^{-};\mathrm{aq}; 298.2 \mathrm{~K}\right)$ through the series from $\mathrm{R}$ = methyl to $\mathrm{R}$ = n-butyl. In fact the change in this case is indicative of the change in character from ‘structure breaking’ $\mathrm{Me}_{4}\mathrm{N}^{+}$ to hydrophobic ‘structure forming’ $\mathrm{Bu}_{4}\mathrm{N}^{+}$ ions [9]. In broad terms a positive $\mathrm{B}$ coefficient indicates a tendency for the solute to enhance water-water interactions and thus raise the viscosity whereas a negative coefficient indicates a tendency to induce disorder [10]. Ionic B-viscosity coefficients are linked to the hydration properties of ions. An important link was suggested by Gurney [11]. On the grounds that $\mathrm{K}^{+}$ and $\mathrm{Cl}^{-}$ ions are roughly the same size, Gurney argued that for aqueous solutions, $\mathrm{B}\left(\mathrm{K}^{+}\right)=\mathrm{B}\left(\mathrm{Cl}^{-}\right)$. Hence one can estimate single-ion $\mathrm{B}$-viscosity coefficients. $\mathrm{A}$ negative $\mathrm{B}$-ionic coefficient indicates that the ion is a ‘structure breaker’ and a positive $\mathrm{B}$-ionic viscosity coefficient indicates that the ion is a ‘structure former’.
The terms ‘structure breaker’ and ‘structure former’ were extensively used in the decades from 1950 to 1990. However their popularity waned towards the end of the century as more precise descriptions of ionic hydration were sought.
Footnotes
[1] We confine attention to shear viscosities (i.e. resistance to shear). The related bulk viscosities (i.e. resistance to compression) are rarely discussed in the present context.
[2] Units; dynamic viscosity; traditional unit = poise, symbol $\mathrm{P}$
$\mathrm{P}=10^{-1} \text { Pas}$. But $\mathrm{Pa} = \mathrm{~kg m}^{-1} \mathrm{~s}^{-2}$ Then $\mathrm{P} = 10^{-1} \mathrm{~kg m}^{-1} \mathrm{~s}^{-1} SI unit; η = [kg m-1 s-1 ] [3] G. Jones and M. Dole, J. Am. Chem. Soc., 1929,51,2950. [4] Other equations have been suggested as alternatives to the Jones-Dole equation. 1. The concentration cj is replaced by the ionic strength of the salt solution; C. Wu, J. Phys Chem.,1968,72,2663. 2. For non-electrolytes, the coefficient A is zero but a new term is usually added linear in the concentration of solute. 1. R. Robinson and R. Mills, Viscosity of Electrolytes and Related Properties, Pergamon, London, 1965. 2. W. Devine and B. M. Lowe, J.Chem.Soc.,A,1971,2113. 3. The ratio \(\eta(\mathrm{aq}) / \eta_{1}^{*}(\ell)$ is expressed as a function of the volume fraction of the solute.
1. D. G. Thomas, J. Colloid. Interface Sci.,1965,20,267.
2. B. R. Breslau and I. F. Mikker, J. Phys. Chem.,1970,74,1056.
3. J. Vand, J.Phys.Chem.,1948,52,277.
4. D. Eagland and G. Pilling, J. Phys. Chem., 1 972,76,1902.
4. Add a term linear in $\left(c_{j}\right)^{2}$ to equation (a). N. Martinus, C. D. Sinclair and C. A. Vincent, Electrochim. Acta, 1977,22,1183.
5. Somewhat outside the terms of reference of the account developed here is a treatment of viscosity in terms of kinetic phenomena; e.g. in terms of Transition State Theory. S. Glasstone, K. J. Laidler and H. Eyring, Theory of Rate Processes, McGraw-Hill, New York,1941. For application of this model see for example,
1. W. Good, Electrochim. Acta, 1964, 9, 203; 1965, 10,1; 1966, 11, 759,767; 1967, 12,103 1.
2. J. C. MacDonald, Electrochim. Acta, 1972,17,1965.
[5] See for example,
1. R. L. Kay, K. T. Vituccio, C. Zawoyski and D. F. Evans. J. Phys. Chem.,1966, 70, 2336.
2. Y. Tamaki, Y. Ohara, M. Inabe, T. Mori and F. Numata, Bull. Chem. Soc. Jpn.,1983,56,1930.
[6] H. S. Harned and B. B. Owen, The Physical Chemistry of Electrolyte Solutions, Reinhold, New York, 3rd. edition, 1958,p.240.
[7] See for example;
1. A. Kacperska, S. Taniewska-Osinska, A. Bald and S. Szejgis, J. Chem. Soc. Faraday Trans.1,1989,85,4147.
2. A. Sacco, G. Petrella, A. Dell’Atti and A. de Giglio, J. Chem. Soc. Faraday Trans.1, 1982,78,1507.
[8] An alternative approach involves calculating the A coefficient using the Falkenhagen theory and hence writing the Jones-Dole equation as follows.
$\eta(\mathrm{aq}) / \eta_{\mathrm{I}}^{*}(\ell)-1-\mathrm{A} \,\left(\mathrm{c}_{\mathrm{j}}\right)^{1 / 2}=\mathrm{B} \, \mathrm{c}_{\mathrm{j}}$
In a plot of the LHS of this equation against $\mathrm{c}_{j}$, the slope equals B. See for example, K. Tamaki, K. Suga and E. Tanihara, Bull. Chem. Soc. Jpn.,1987.60,1225.
[9] B. M. Lowe and G. A. Rubienski, Electrochem. Acta, 1974,19,393.
[10] E. R. Nightingale, J. Phys. Chem.,1959,63,1381;1962,66,894. See also, D. T. Burns, Electrochim Acta, 1965,10,985.
[11] R. W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
1.14.76: Work
This term ‘work’ makes it first key appearance (at least in thermodynamics) in the context of the statement that if work is done on a closed system the thermodynamic energy of the system increases given that heat q is zero. This simple statement understates the complexity of the term ‘work’ in thermodynamics.
In general terms work done on a closed thermally insulated system raises the energy of that system and is given by the product of intensive and capacity factors [1]. Three examples make the point.
The analysis is complicated by the fact that changes in a given system can take one of two limiting forms; e.g. frozen and equilibrium. In the case of surface tension, frozen ( plastic) surface tension describes the case where the intermolecular distances in the surface increase. The equilibrium case describes the case where molecules in the bulk phase and in the surface exchange to hold the change in the surface as a reversible (equilibrium) process.
Footnote
[1] E. F. Caldin, Chemical Thermodynamics, Clarendon Press, Oxford, 1958.
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Chapter 1. Introduction to Aerodynamics
1.1 Aerodynamics
Aerodynamics is probably the first subject that comes to mind when most people think of Aeronautical or Aerospace Engineering. Aerodynamics is essentially the application of classical theories of “fluid mechanics” to external flows or flows around bodies, and the main application which comes to mind for most aero engineers is flow around wings.
The wing is the most important part of an airplane because without it there would be no lift and no aircraft. Most people have some idea of how a wing works; that is, by making the flow over the top of the wing go faster than the flow over the bottom we get a lower pressure on the top than on the bottom and, as a result, get lift. The aero engineer needs to know something more than this. The aero engineer needs to know how to shape the wing to get the optimum combination of lift and drag and pitching moment for a particular airplane mission. In addition he or she needs to understand how the vehicle’s aerodynamics interacts with other aspects of its design and performance. It would also be nice if the forces on the wing did not exceed the load limit of the wing structure.
If one looks at enough airplanes, past and present, he or she will find a wide variety of wing shapes. Some aircraft have short, stubby wings (small wing span), while others have long, narrow wings. Some wings are swept and others are straight. Wings may have odd shapes at their tips or even attachments and extensions such as winglets. All of these shapes are related to the purpose and design of the aircraft.
In order to look at why wings are shaped like they are we need to start by looking at the terms that are used to define the shape of a wing.
A two dimensional slice of a wing cut parallel to the centerline of the aircraft fuselage or body is called the airfoil section. A straight line from the airfoil section leading edge to its trailing edge is called the chord line. The length of the chord line is referred to as the chord. A line drawn half way between the airfoil section’s upper and lower surfaces is called the camber line. The maximum distance between the camber line and chord line is referred to as the airfoil’s camber and is usually enumerated as a percent of chord. We will see that the amount of airfoil camber and the location of the point of maximum camber are important numbers in defining the shape of an airfoil and predicting its performance. For most airfoils the maximum camber is on the order of zero to five percent and the location of the point of maximum camber is between 25% and 50% of the chord from the airfoil leading edge.
When viewed from above the aircraft the wing shape or planform is defined by other terms.
Note that the planform area is not the actual surface area of the wing but is “projected area” or the area of the wing’s shadow. Also note that some of the abbreviations used are not intuitive; the span, the distance from wing tip to wing tip (including any fuselage width) is denoted by b and the planform area is given a symbol of “S” rather than perhaps “A”. Sweep angles are usually given a symbol of lambda (λ).
Another definition that is based on the planform shape of a wing is the Aspect Ratio (AR).
AR = b2/S.
Aspect ratio is also the span divided by the “mean” or average chord. We will later find that aspect ratio is a measure of the wing’s efficiency in long range flight.
Wing planform shapes may vary considerably from one type of aircraft to another. Fighter aircraft tend to have low aspect ratio or short, stubby wings, while long range transport aircraft have higher aspect ratio wing shapes, and sailplanes have yet higher wing spans. Some wings are swept while others are not. Some wings have triangular or “delta” planforms. If one looks at the past 100 years of wing design he or she will see an almost infinite variety of shapes. Some of the shapes come from aerodynamic optimization while others are shaped for structural benefit. Some are shaped the way they are for stealth, others for maneuverability in aerobatic flight, and yet others just to satisfy their designer’s desire for a good looking airplane.
In general, high aspect ratio wings are desirable for long range aircraft while lower aspect ratio wings allow more rapid roll response when maneuverability is a requirement. Sweeping a wing either forward or aft will reduce its drag as the plane’s speed approaches the speed of sound but will also reduce its efficiency at lower speeds. Delta wings represent a way to get a combination of high sweep and a large area. Tapering a wing to give it lower chord at the wing tips usually gives somewhat better performance than an untapered wing and a non-linear taper which gives a “parabolic” planform will theoretically give the best performance.
In the following material we will take a closer look at some of the things mentioned above and at their consequences related to the flight capability of an airplane.
Before we take a more detailed look at wing aerodynamics we will first examine the atmosphere in which aircraft must operate and look at a few of the basic relationships we encounter in “doing” aerodynamics.
1.2 Air, Our Flight Environment
Airplanes operate in air, a gas made up of nitrogen, oxygen, and several other constituents. The behavior of air, that is the way its properties like temperature, pressure, and density relate to each other, can be described by the Ideal or Perfect Gas Equation of State:
where P is the barometric or hydrostatic pressure, ρ is the density, and T is the temperature. R is the gas constant for air. In this equation the temperature and pressure must be given in absolute values; in other words, temperature must be in Kelvin or Rankine, not Celsius (Centigrade) or Fahrenheit. Of course the units must all be consistent with those used in the gas constant:
1.3 Units
This brings us to the subject of units. It is important that all the units in the perfect gas equation be compatible; i.e., all English units or all SI units, and that we be careful if solving for, for example, pressure, to make sure that the units of pressure come out as they should (pounds per square foot in the English system or Pascals in SI). Unfortunately many of us don’t have a clue as to how to work with units.
It is popular in U. S. scientific circles to try to convince everyone that Americans are the only people in the world who use “English” units and the only people in the world who don’t know how to use SI units properly. Nothing could be further from the truth. No one in the world actually uses SI units correctly in everyday life. For example, the rest of the world commonly uses the Kilogram as a unit of weight when it is actually a unit of mass. They buy produce in the grocery store in Kilograms, not Newtons. You would also be hard pressed to find anyone in the world, even in France, who knows that a Pascal is a unit of pressure. Newtons and Pascals are simply not used in many places outside of textbooks. In England the distances on highways are still given in miles and speeds are given in mph even as the people measure shorter distances in meters (or metres), and the government is still trying to get people to stop weighing vegetables in pounds. There are many people in England who still give their weight in “stones”.
As aerospace engineers we will find that, despite what many of our textbooks say, most work in the industry is done in the English system, not SI, and some of it is not even done in proper English units. Airplane speeds are measured in miles per hour or in knots, and distances are often quoted in nautical miles. Pressures are given to pilots in inches of mercury or in millibars. Pressures inside jet and rocket engines are normally measured in pounds-per-square-inch (psi). Airplane altitudes are most often quoted in feet. Engine power is given in horsepower and thrust in pounds. We must be able to work in the real world, as well as in the politically correct world of the high school or college physics or chemistry or even engineering text.
It should be noted that what we in America refer to as the “English” unit system, people in England call “Imperial” units. This can get really confusing because “imperial” liquid measures are different from “American” liquid measures. An “imperial” gallon is slightly larger than an American gallon and a “pint” of beer in Britain is not the same size as a “pint” of beer in the U. S.
So, there are many possible systems of units in use in our world. These include the SI system, the pound-mass based English system, the “slug” based English system, the cgs-metric system, and others. We can discuss all of these in terms of a very familiar equation, Isaac Newton’s good old F = ma. Newton’s law relates units as well as physical properties and we can use it to look at several common unit systems.
Force = mass x acceleration
1 Newton = 1 kg x 1 meter/sec2
1 pound-force = 1 pound-mass x 32.17 ft/sec2
1 Dyne = 1 gram x 1 cm/sec2
1 pound-force = 1 slug x 1 ft/sec2
The first and last of the above are the systems with which we need to be thoroughly familiar; the first because it is the “ideal” system according to most in the scientific world, and the last because it is the semi-official system of the world of aerospace engineering.
In using any unit system there are three basic requirements:
1. Always write units with any number that has units.
2. Always work through the units in equations at the same time that you work out the numbers.
3. Always reduce the final units to their simplest form and verify that they are the appropriate units for that number.
Following the above suggestions would eliminate about half of the wrong answers found on most student homework and test papers.
In doing engineering problems one should carry through the units as described above and make sure that the units make sense for the answer and that the magnitude of the answer is reasonable. Good students do this all the time while poor ones leave everything to chance.
The first part of this is simple. If the units in an answer don’t make sense, for example, if the speed for an airplane is calculated to be 345 feet per pound or if we calculate a weight to be 1500 kilograms per second, it should be easy to recognize that something is wrong. A fundamental error has been made in following through the problem with the units and this must be corrected.
The more difficult task is to recognize when the magnitude of an answer is wrong; i.e., is not “in the right ballpark”. If we are told that the speed of a car is 92 meters/sec. or is 125 ft/sec. do we have any “feel” for whether these are reasonable or not? Is this car speeding or not? Most of us don’t have a clue without doing some quick calculations (these are 205 mph and 85 mph, respectively). Do any of us know our weight in Newtons? What is a reasonable barometric pressure in the atmosphere in any unit system?
So our second unit related task is to develop some appreciation for the “normal” range of magnitudes for the things we want to calculate in our chosen units system(s). What is a reasonable range for a wing’s lift coefficient or drag coefficient? Is it reasonable for cars to have 10 times the drag coefficient of airplanes?
With these cautions in mind let’s go back and look at our “working medium”, the standard atmosphere.
1.4 The Standard Atmosphere
We said we were starting with the Ideal Gas Equation of State, P=RT. We will also make use of the Hydrostatic Equation, another relationship you have seen before in chemistry and physics:
This tells us how pressure changes with height in a column of fluid. This tells us how pressure changes as we move up or down through the atmosphere.
These two equations, the Perfect Gas Equation of State and the Hydrostatic Equation, have three variables in them; pressure, density, and temperature. To solve for these properties at any point in the atmosphere requires us to have one more equation, one involving temperature. This is going to require our first assumption. We must have some relationship that can tell us how temperature should vary with altitude in the atmosphere.
Many years of measurement and observation have shown that, in general, the lower portion of the atmosphere, where most airplanes fly, can be modeled in two segments, the Troposphere and the Stratosphere. The temperature in the troposphere is found to drop fairly linearly as altitude increases. This linear decrease in temperature continues up to about 36,000 feet (about 11,000 meters). Above this altitude the temperature is found to hold constant up to altitudes over 100,000 ft. This constant temperature region is the lower part of the Stratosphere. The troposphere and stratosphere are where airplanes operate, so we need to look at these in detail.
1.5 The Troposphere
We model the linear temperature drop with altitude in the troposphere with a simple equation:
Talt = Tsealevel Lh
where “L” is called the “lapse rate”. From over a hundred years of measurements it has been found that a normal, average lapse rate is:
L = 3.56oR / 1000 ft = 6.5oK / 1000 meters .
This is often taught to pilots in a strange mixture of units as 1.98 degrees Centigrade per thousand feet!
The other thing we need is a value for the sea level temperature. Our model, also based on averages from years of measurement, uses the following sea level values for pressure, density, and temperature.
TSL = 288 oK = 520 oR
So, to find temperature at any point in the troposphere we use:
T (oR) = 520 – 3.56(h),
where h is the altitude in thousands of feet, or
T (oK) = 288 – 6.5 (h)
where h is the altitude in thousands of meters.
We need to stress at this point that this temperature model for the Troposphere is merely a model, but it is the model that everyone in the aviation and aerospace community has agreed to accept and use. The chance of ever going to the seashore and measuring a temperature of 59oF is slim and even if we find that temperature it will surely change within a few minutes. Likewise, if we were to send a thermometer up in a balloon on any given day the chance of finding a “lapse rate” equal to the one defined as “standard” is slim to none, and, during the passage of a weather front, we may even find that temperature increases rather than drops as we move to higher altitudes. Nonetheless, we will work with this model and perhaps later learn to make corrections for non-standard days.
Now, if we are willing to accept the model above for temperature change in the Troposphere, all we have to do is find relationships to tell how the other properties, pressure and density, change with altitude in the Troposphere. We start with the differential form of the hydrostatic equation and combine it with the Perfect Gas equation to eliminate the density term.
or,
which is rearranged to give
dP/P = – (g/RT)dh.
Now we substitute in the lapse rate relationship for the temperature to get
dP/P = {g/[R(TSL-Lh)]}dh.
This is now a relationship with only one variable (P) on the left and only one (h) on the right. It can be integrated to give
In a similar manner we can get a relationship to find the density at any altitude in the troposphere
So now we have equations to find pressure, density, and temperature at any altitude in the troposphere. Care has to be taken with units when using these equations. All temperatures must be in absolute values (Kelvin or Rankine instead of Celsius or Fahrenheit). The exponents in the pressure and density ratio equations must be unitless. Exponents cannot have units!
We can use these equations up to the top of the Troposphere, that is, up to 11,000 meters or 36,100 feet in altitude. Above that altitude is the Stratosphere where temperature is modeled as being constant up to roughly 100,000 feet.
1.6 The Stratosphere
We can use the temperature lapse rate equation result at 11,000 meters altitude to find the temperature in this part of the Stratosphere.
Tstratosphere = 216.5oK = 389.99oR = constant
The equations for determining the pressure and density in the constant temperature part of the stratosphere are different from those in the troposphere since temperature is constant. And, since temperature is constant both pressure and density vary in the same manner.
The term on the right in the equation is “e” or 2.718, evaluated to the power shown, where h1 is the 11,000 meters or 36,100 ft (depending on the unit system used) and h2 is the altitude where the pressure or density is to be calculated. T is the temperature in the stratosphere.
Using the above equations we can find the pressure, temperature, or density anywhere an airplane might fly. It is common to tabulate this information into a standard atmosphere table. Most such tables also include the speed of sound and the air viscosity, both of which are functions of temperature. Tables in both SI and English units are given below.
Table 1.1: Standard Atmosphere in SI Units
h (km) T (degrees C) a (m / sec) Px10^(-4)(N/m^2 ) (pascals) P (kg/m^3) u x10^5 (kg/m sec)
0 15 340 10.132 1.226 1.78
1 8.5 336 8.987 1.112 1.749
2 2 332 7.948 1.007 1.717
3 -4.5 329 7.01 0.909 1.684
4 -11 325 6.163 0.82 1.652
5 -17.5 320 5.4 0.737 1.619
6 -24 316 4.717 0.66 1.586
7 -30.5 312 4.104 0.589 1.552
8 -37 308 3.558 0.526 1.517
9 -43.5 304 3.073 0.467 1.482
10 -50 299 2.642 0.413 1.447
11 -56.5 295 2.261 0.364 1.418
12 -56.5 295 1.932 0.311 1.418
13 -56.5 295 1.65 0.265 1.418
14 -56.5 295 1.409 0.227 1.418
15 -56.5 295 1.203 0.194 1.418
16 -56.5 295 1.027 0.163 1.418
17 -56.5 295 0.785 0.141 1.418
18 -56.5 295 0.749 0.121 1.418
19 -56.5 295 0.64 0.103 1.418
20 -56.5 295 0.546 0.088 1.418
30 -56.5 295 0.117 0.019 1.418
45 40 355 0.017 0.002 1.912
60 70.8 372 0.003 0.00039 2.047
75 -10 325 0.0006 0.00008 1.667
Table 1.2: Standard Atmosphere in English Units
h (ft) T (degrees F) a (ft/sec) p (lb/ft^2) p (slugs/ft^3) u x 10^7 (sl/ft-sec)
0 59 1117 2116.2 0.002378 3.719
1,000 57.44 1113 2040.9 0.00231 3.699
2,000 51.87 1109 1967.7 0.002242 3.679
3,000 48.31 ll05 1896.7 0.002177 3.659
4,000 44.74 ll02 1827.7 0.002112 3.639
5,000 41.18 1098 1760.8 0.002049 3.618
6,000 37.62 1094 1696 0.001988 3.598
7,000 34.05 1090 1633 0.001928 3.577
8,000 30.49 1086 1571.9 0.001869 3.557
9,000 26.92 1082 1512.9 0.001812 3.536
10,000 23.36 1078 1455.4 0.001756 3.515
ll,000 19.8 1074 1399.8 0.001702 3.495
12,000 16.23 1070 1345.9 0.001649 3.474
13,000 12.67 1066 1293.7 0.001597 3.453
14,000 9.1 1062 1243.2 0.001546 3.432
15,000 5.54 1058 1194.3 0.001497 3.411
16,000 1.98 1054 1147 0.001448 3.39
17,000 -1.59 1050 1101.1 0.001401 3.369
18,000 -5.15 1046 1056.9 0.001355 3.347
19,000 -8.72 1041 1014 0.001311 3.326
20,000 -12.28 1037 972.6 0.001267 3.305
21,000 -15.84 1033 932.5 0.001225 3.283
22,000 -19.41 1029 893.8 0.001183 3.262
23,000 -22.97 1025 856.4 0.001143 3.24
24,000 -26.54 1021 820.3 0.001104 3.218
25,000 -30.1 1017 785.3 0.001066 3.196
26,000 -33.66 1012 751.7 0.001029 3.174
27,000 -37.23 1008 719.2 0.000993 3.153
28,000 -40.79 1004 687.9 0.000957 3.13
29,000 -44.36 999 657.6 0.000923 3.108
30,000 -47.92 995 628.5 0.00089 3.086
31,000 -51.48 991 600.4 0.000858 3.064
32,000 -55.05 987 573.3 0.000826 3.041
33,000 -58.61 982 547.3 0.000796 3.019
34,000 -62.18 978 522.2 0.000766 2.997
35,000 -65.74 973 498 0.000737 2.974
40,000 -67.6 971 391.8 0.0005857 2.961
45,000 -67.6 971 308 0.0004605 2.961
50,000 -67.6 971 242.2 0.0003622 2.961
Table 1.2: Standard Atmosphere in English Units (con’t)
h (ft) T (degrees F) a (ft/sec) p (lb/ft^2) p (slugs/ft^3) u x 10^7(sl/ft-sec)
60,000 -67.6 971 150.9 0.000224 2.961
70,000 -67.6 971 93.5 0.0001389 2.961
80,000 -67.6 971 58 0.0000861 2.961
90,000 -67.6 971 36 0.0000535 2.961
100,000 -67.6 971 22.4 0.0000331 2.961
150,000 113.5 1174 3.003 0.00000305 4.032
200,000 159.4 1220 0.6645 0.00000062 4.277
250,000 -8.2 1042 0.1139 0.00000015 3.333
A look at these tables will show a couple of terms that we have not discussed. These are the speed of sound “a”, and viscosity “μ”. The speed of sound is a function of temperature and decreases as temperature decreases in the Troposphere. Viscosity is also a function of temperature.
The speed of sound is a measure of the “compressibility” of a fluid. Water is fairly incompressible but air can be compressed as it might be in a piston/cylinder system. The speed of sound is essentially a measure of how fast a sound or compression wave can move through a fluid. We often talk about the speeds of high speed aircraft in terms of Mach number where Mach number is the relationship between the speed of flight and the speed of sound. As we get closer to the speed of sound (Mach One) the air becomes more compressible and it becomes more meaningful to write many equations that describe the flow in terms of Mach number rather than in terms of speed.
Viscosity is a measure of the degree to which molecules of the fluid bump into each other and transfer forces on a microscopic level. This becomes a measure of “friction” within a fluid and is an important term when looking at friction drag, the drag due to shear forces that occur when a fluid (air in our case) moves over the surface of a wing or body in the flow.
Two things should be noted in these tables about viscosity. First, the units look sort of strange. Second, the viscosity column is headed with μ X 10x. The units are the proper ones for viscosity in the SI and English systems respectively; however, if you talk to a chemist or physicist about viscosity they will probably quote numbers with units of “poise”. The 10xnumber in the column heading means that the number shown in the column has been multiplied by 10xto give it the value shown. This is, to most of us, not intuitive. What this means is that in the English unit version of the Standard Atmosphere table, the viscosity at sea level has a value of 3.719 times ten to the minus 7.
So now we can find the properties of air at any altitude in our model or “standard” atmosphere. However, this is just a model, and it would be rare indeed to find a day when the atmosphere actually matches our model. Just how useful is this?
In reality this model is pretty good when it comes to pressure variation in the atmosphere because it is based on the hydrostatic equation which is physically correct. On the other hand, pressure at sea level does vary from day to day with weather changes, as the area of concern comes under the various high or low pressure systems often noted on weather maps. Temperature represents the greatest opportunity for variation between the model and the real atmosphere, after all, how many days a year is the temperature at the beach 59oF (520oR)? Density, of course, is a function of pressure and temperature, so its “correctness” is dependent on that of P and T.
On the face of things, it appears that the Standard Atmosphere is somewhat of a fantasy. On the other hand, it does give us a pretty good idea of how these properties of air should normally change with altitude. And, we can possibly make corrections to answers found when using this model by correcting for actual sea level pressure and temperature if needed. Further, we could define other “standard” atmospheres if we are looking at flight conditions where conditions are exceptionally different from this model. This is done to give “Arctic Minimum” and “Tropical Maximum” atmosphere models.
In the end, we do all aircraft performance and aerodynamic calculations based on the normal standard atmosphere and all flight testing is done at standard atmosphere pressure conditions to define altitudes. The standard atmosphere is our model and it turns out that this model serves us well.
One way we use this model is to determine our altitude in flight.
1.7 Altitude Measurement
The pilot of an aircraft needs to know its altitude and there are several ways we could measure the altitude of an airplane. Radar might be used to measure the plane’s distance above the ground. Global Positioning System satellite signals can determine the plane’s position, including its altitude, in three dimensional space. These and some other possible methods of altitude determination depend on the operation of one or more electrical systems, and while we may want to have such an instrument on our airplane, we also are required to have an “altimeter” that does not depend on batteries or generators for its operation. Further, the altitude that the pilot needs to know is the height above sea level. The obvious solution is to use our knowledge that the pressure varies in a fairly dependable fashion with altitude.
If we know how pressure varies with altitude then we can measure that pressure and determine the altitude above a sea level reference point. In other words, if we measure the pressure as 836 pounds per square foot we can look in the standard atmosphere table and find that we should be at an altitude of 23,000 ft. So all we need to do is build a simple mechanical barometer and calibrate its dial so it reads in units of altitude rather than pressure. As the measured pressure decreases, the indicated altitude increases in accord with the standard atmosphere model. This is, in fact, how “simple” altimeters such as those sometimes used in cars or bikes or even “ultra-lite” aircraft. A barometer measures the air pressure and on some type of dial or scale, instead of pressure units, the equivalent altitudes are indicated.
The “simple” altimeter, however, might not be quite accurate enough for most flying because of the variations in atmospheric pressure with weather system changes. The simple altimeter would base its reading on the assumption that the pressure at sea level is 2116 psf . If, however, we are in an area of “high” pressure, the altimeter of an airplane sitting at sea level would sense the higher than standard pressure and indicate an altitude somewhat below sea level. Conversely, in the vicinity of a low pressure atmospheric system the altimeter would read an altitude higher than the actual value. If this error was only a few feet it might not matter, but in reality it could result in errors of several hundred feet in altitude readings. This could lead to disaster in bad weather when a pilot has to rely on the altimeter to ensure that the plane clears mountain peaks or approaches the runway at the right altitude. Hence, all aircraft today use “sensitive” altimeters that allow the pilot to adjust the instrument for changes in pressure due to atmospheric weather patterns.
The sensitive altimeter, shown in the next figure, has a knob that can be turned to adjust the readout of the instrument for non-standard sea level pressures. This can be used in two different ways in flight. When the aircraft is sitting on the ground at an airport the pilot can simply adjust the knob until the altimeter reads the known altitude of the airport. In flight, the pilot can listen to weather report updates from nearby airports, reports that will include the current sea level equivalent barometric pressure, and turn the knob until the numbers in a small window on the altimeter face agree with the stated pressure. These readings are usually given in units of millimeters of mercury where 29.92 is sea level standard. Adjusting the reading in the window to a higher pressure will result in a decrease in the altimeter reading and adjusting it lower will increase the altitude indication. With proper and timely use of this adjustment a good altimeter should be accurate within about 50 feet.
It should be noted that we could also use density to define our altitude and, in fact, this might prove more meaningful in terms of relating to changes in an airplane’s performance at various flight altitudes because engine thrust and power are known to be functions of density and the aerodynamic lift and drag are also functions of density. However, to “measure” density would require measurement of both pressure and temperature. This could introduce more error into our use of the standard atmosphere for altitude determination than the use of pressure alone because temperature variation is much more subject to non-standard behavior than that of pressure. On the other hand, we do sometimes find it valuable to calculate our “density altitude” when looking at a plane’s ability to take-off in a given ground distance.
If we are at an airport which is at an altitude of, lets say, 4000 ft and the temperature is higher than the 44.74oF predicted by the standard atmosphere (as it probably would be in the summer) we would find that the airplane behaves as if it is at a higher altitude and will take a longer distance to become airborne than it should at 4000 ft. Pilots use either a circular slide rule type calculator or a special electronic calculator to take the measured real temperature and combine it with the pressure altitude to find the “density altitude”, and this can be used to estimate the extra takeoff distance needed relative to standard conditions.
Some may wonder why we can’t simply use temperature to find our altitude. After all, wasn’t one of our basic assumptions that in the Troposphere, temperature dropped linearly with altitude? Wouldn’t it be really easy to stick a thermometer out the window and compare its reading with a standard atmosphere chart to find our altitude?
Of course, once we are above the Troposphere this wouldn’t do any good since the temperature becomes constant over thousands of feet of altitude, but why wouldn’t it work in the Troposphere?
Thought Exercise
• Think about and discuss why using temperature to find altitude is not a good idea.
• Why is pressure the best property to measure to find our altitude?
• Perhaps using density to find altitude would be a better idea since density has a direct effect on flight performance. Think of one reason why we don’t have altimeters that measure air density.
1.8 Bernoulli’s Equation
You have undoubtedly been introduced to a relationship called Bernoulli’s Equation or the Bernoulli Principle somewhere in a previous Physics or Chemistry course. This is the principle that relates the pressure to the velocity in any fluid, essentially showing that as the speed of a fluid increases its pressure decreases and visa versa. This principle can take several different mathematical forms depending on the fluid and its speed. For an incompressible fluid such as water or for air below about 75% of the speed of sound this relationship takes the following form:
P + ½ρV2 = P0
(hydro)static pressure + dynamic pressure = total pressure
[internal energy + kinetic energy = total energy]
This relationship can be thought of as either a measure of the balance of pressure forces in a flow, or as an energy balance (first law of thermodynamics) when there is no change in potential energy or heat transfer.
Bernoulli’s equation says that along any continuous path (“streamline”) in a flow the total pressure, P0, (or total energy) is conserved (constant) and is a sum of the static pressure and the dynamic pressure in the flow. Static pressure and dynamic pressure can both change, but they must change in such a way that their sum is constant; i.e., as the flow speeds up the pressure decreases.
* ASSUMPTIONS: It is very important that we know and understand the assumptions that limit the use of this form of Bernoulli’s equation. The equation can be derived from either the first law of thermodynamics (energy conservation) or from a balance of forces in a fluid through what is known as Euler’s Equation. In deriving the form of the equation above some assumptions are made in order to make some of the math simpler. These involve things like assuming that density is a constant, making it a constant in an integration step in the derivation and making the integration easier. It is also assumed that mass is conserved, a seemingly logical assumption, but one that has certain consequences in the use of the equation. It is also assumed that the flow is “steady”, that is, the speed at any point in the flow is not varying with time. Another way to put this is that the speed can vary with position in the flow (that’s really what the equation is all about) but cannot vary with time.
The assumption of constant density, which we usually call an assumption of incompressible flow, means that we have to observe a speed limit. As air speeds up and the speed approaches the speed of sound its density changes; i.e., it becomes compressible. So when our flow speeds get too near the speed of sound, the incompressible flow assumption is violated and we can no longer use this form of Bernoulli’s equation. When does that become a problem?
Some fluid mechanics textbooks use a mathematical series relationship to look at the relationship between speed or Mach number (Mach number, the speed divided by the speed of sound, is really a better measure of compressibility than speed alone) and they use this to show that the incompressible flow assumption is not valid above a Mach number of about 0.3 or 0.3 times the speed of sound. This is good math but not so good physics. The important thing is not how the math works but how the relationship between the two pressures in Bernoulli’s equation changes as speed or Mach number increases. We will examine this in a later example to show that we are actually pretty safe in using the incompressible form of Bernoulli’s equation up to something like 75% of the speed of sound.
The other important assumptions in this form of Bernoulli’s equation are those of steady flow and mass conservation. Steady flow means pretty much what it sounds like; the equation is only able to account for changes in speed and pressure with position in a flow field. It was assumed that the flow is exactly the same at any time.
The mass conservation assumption really relates to looking at what are called “streamlines” in a flow. These can be thought of at a basic level as flow paths or highways that follow or outline the movement of the flow. Mass conservation implies that at any point along those paths or between any two streamlines the mass flow between the streamlines (in the path) is the same as it is at any other point between the same two streamlines (or along the same path).
The end result of this mass conservation assumption is that Bernoulli’s equation is only guaranteed to hold true along a streamline or path in a flow. However, we can extend the use of the relationship to any point in the flow if all the flow along all the streamlines (or paths) at some reference point upstream (at “∞”) has the same total energy or total pressure.
So, we can use Bernoulli’s equation to explain how a wing can produce lift. If the flow over the top of the wing is faster than that over the bottom, the pressure on the top will be less than that on the bottom and the resulting pressure difference will produce a lift. The study of aerodynamics is really all about predicting such changes in velocity and pressure around various shapes of wings and bodies. Aerodynamicists write equations to describe the way air speeds change around prescribed shapes and then combine these with Bernoulli’s equation to find the resulting pressures and forces.
Let’s look at the use of Bernoulli’s equation for the case shown below of a wing moving through the air at 100 meters/sec. at an altitude of 1km.
We want to find the pressure at the leading edge of the wing where the flow comes to rest (the stagnation point) and at a point over the wing where the speed has accelerated to 150 m/s.
First, note that the case of the wing moving through the air has been portrayed as one of a stationary wing with the air moving past it at the desired speed. This is standard procedure in working aerodynamics problems and it can be shown that the answers one finds using this method are the correct ones. Essentially, since the process of using Bernoulli’s equation is one of looking at conservation of energy, it doesn’t matter whether we are analyzing the motion (kinetic energy) involved as being motion of the body or motion of the fluid.
Now let’s think about the problem presented above. We know something about the flow at three points:
Well in front of the wing we have what is called “free stream” or undisturbed, uniform flow. We designate properties in this flow with an infinity [∞] subscript. We can write Bernoulli’s equation here as:
Note that it is at this point, the “free stream” where all the flow is uniform and has the same total energy. If at this point the flow was not uniform, perhaps because it was near the ground and the speed increases with distance up from the ground, we could not assume that each “streamline” had a different value of total pressure (energy).
At the front of the wing we will have a point where the flow will come to rest. We call this point the “stagnation point” if we can assume that the flow slowed down and stopped without significant losses. Here the flow speed would be zero. We can write Bernoulli’s equation here as:
Pstagnation + 0 = P0
At this point the flow has accelerated to 150 m/s and we can write Bernoulli’s equation as:
Now we know that since the flow over the wing is continuous (mass is conserved) the total pressure (P0) is the same at all three points and this is what we use to find the missing information. To do this we must understand which of these pressures (if any) are known to us as atmospheric hydrostatic pressures and understand that we can assume that the density is constant as long as we are safely below the speed of sound.
Initially we know that the pressure in the atmosphere is that in the standard atmosphere table for an altitude of 1 km or 89870 Pascals and that the density at this altitude is 1.112 kg/m3. Looking at the problem, the most logical place for standard atmosphere conditions to apply is in the “free stream” location because this is where the undisturbed flow exists. Hence
And, using these in Bernoulli’s equation at the free stream location we calculate a total pressure
P0 = 95430 Pa
Now that we have found the total pressure we can use it at any other location in the flow to find the other unknown properties.
At the stagnation point
Pstagnation = P0 = 95430 Pa
At the point where the speed is 150 m/s we can rearrange Bernoulli’s equation to find
As a check we should confirm that the static pressure (P3) at this point is less than the free stream static pressure (P) since the speed is higher here and also confirm that the static pressures everywhere else in the flow are lower than the stagnation pressure.
Now let’s review the steps in working any problem with Bernoulli’s equation. First we must sketch the flow and write down everything we know at various points in that flow. Second we must write Bernoulli’s equation at every point in the flow where we either know information or want to know something. Third we must carefully assess which pressure, if any, can be obtained from the standard atmosphere table. Fourth we must look at all these points in the flow and see which point gives us enough information to solve for the total pressure (P0). Finally we use this value of P0 in Bernoulli’s equation at other points in the flow to find the other missing terms. Attempting to skip any of the above steps can lead to mistakes for most of us.
One of the most common problems that people have in working with Bernoulli’s equation in a problem like the one above is to assume that the stagnation point is the place to start the solution of the problem. They look at the three points in the flow and assume that the stagnation point must be the place where everything is known. After all, isn’t the velocity at the stagnation point equal to zero? Doesn’t this mean that the static pressure and the total pressure are the same here? And what other conclusion can be drawn than to assume that this pressure must then be the atmospheric pressure?
Well, the answer to the first two questions is “yes” but a third “yes” does not follow. What is known at the stagnation point is that the static pressure term in the equation is now the static pressure at a stagnation point and is therefore called the stagnation pressure. And, since the speed is zero, the stagnation pressure is equal to the total pressure in the flow. Neither of these pressures, however, is the atmospheric pressure.
Why is the pressure at the stagnation point not the pressure in the atmosphere? Well, this is where our substitution of a moving flow and a stationary wing for a moving wing in a stationary fluid ends up causing us some confusion. In reality, this stagnation point is where the wing is colliding head-on with the air that it is rushing through. The pressure here, the stagnation pressure, must be equal to the pressure in the atmosphere plus the pressure caused by the collision between wing and fluid; i.e., it must be higher than the atmospheric pressure.
Our approach of modeling the flow of a wing moving through the stationary atmosphere as a moving flow around a stationary wing makes it easier to work with Bernoulli’s equation in general; however, we must keep in mind that it is a substitute model and alter our way of looking at it appropriately. In this model the hydrostatic pressure is not the pressure where the air is “static”, it is, rather, the pressure where the flow is “undisturbed”. This is at the “free stream” conditions, the point upstream of the body (wing, in this case) where the flow has not yet felt the presence of the wing. This is where the undisturbed atmosphere exists. Between that point and the wing itself the flow has to change direction and speed as it moves around the body, so nowhere else in the flow field will the pressure be the same as in the undisturbed atmosphere.
1.9 Airspeed Measurement
Now that we know something about Bernoulli’s equation we can look at another use of the relationship, the measurement of airspeed. Rearranging the equation we can write:
So, if we know the total and static pressures at a point and the density at that point we can easily find the speed at that point. All we need is some way to measure or otherwise find these quantities.
We can find the total pressure (P0) by simply inserting an open tube of some kind into the flow so that it is pointed into the oncoming flow and then connected to a pressure gage of some sort.
The static pressure can be found in a similar manner but the flow must be going parallel to the openings in the tube or surface.
On an airplane we usually mount a pitot tube somewhere on the wing or nose of the aircraft where it will generally point into the undisturbed flow and not be behind a propeller. The static pressure reading on an airplane is normally taken via a hole placed at some point on the side of the airplane where the flow will have the same static pressure as the freestream flow instead of using a separate static probe. This point is usually determined in flight testing. There is usually a static port on both sides of the plane connected to a single tube through a “T” connection. The static port looks like a small, circular plate with a hole in its center. One of the jobs required of the pilot in his or her preflight inspection of the aircraft is to make sure that both the pitot tube and static ports are free of obstruction, a particularly important task in the Spring of the year when insects like to crawl into small holes and build nests.
In a wind tunnel and in other experimental applications we often use a single instrument to measure both total and static pressures. This instrument is called a pitot-static tube and it is merely a combination of the two probes shown above.
In both the lab case and the aircraft case it is the difference in the two pressures, P0 – P, that we want to know and this can be measured with several different types of devices ranging from a “U-tube” liquid manometer to a sophisticated electronic gage. In an aircraft, where we don’t want our knowledge of airspeed to depend on a source of electricity and where a liquid manometer would be cumbersome, the pressure difference is measured by a mechanical device called an aneroid barometer.
But let’s go back and look at the equation used to find the velocity and see if this causes any problem.
This shows that we also need to know the density if we wish to find the speed. In the lab we find the density easily enough by measuring the barometric pressure and the temperature and calculating density using the Ideal Gas Law,
or,
and, using this we can find the exact or “true” airspeed.
In an airplane we want simplicity and reliability, and while we could ask the pilot or some flight computer to measure pressure and temperature, then calculate density, then put it into Bernoulli’s equation to calculate airspeed, this seems a little burdensome and, of course, the use of computers or calculators might depend on electricity. Hence, we do not usually have an instrument on an aircraft that displays the true airspeed; instead we choose to simply measure the difference in the two above pressures using a mechanical instrument and then calibrate that instrument to display what we call the indicated airspeed, a measurement of speed based on the assumption of sea level density.
Another name for the indicated airspeed is the “sea level equivalent airspeed”, the speed which would exist for the measured difference in static and total pressure if the aircraft was at sea level.
The true and indicated airspeeds are directly related by the square root of the ratio of sea level and true densities.
The airspeed indicator on an aircraft then measures the indicated airspeed and not the true airspeed. It is a sealed instrument with the static pressure going to the instrument container and the total pressure connected to an aneroid barometer inside the container. As the difference in these two pressures changes, the indicator needles on the instrument face move over a dial marked off, not for a range of pressures, but for a range of speeds. Each such instrument is carefully calibrated to ensure accurate measurement of indicated airspeed.
So, just as we found that the altimeter on an airplane measures the wrong altitude unless we are able to adjust it properly, the airspeed indicator does not measure the real airspeed. Is this a problem for us?
It turns out that, as far as the performance of the aircraft is concerned; i.e., its ability to take off in a certain distance, to climb at a certain rate, etc., is actually dependent on the indicated airspeed rather than the true airspeed. Yes, we want to know the true airspeed to know how fast we are really going and for related flight planning purposes, but as far as knowing the speed at which to rotate on takeoff, the best speed at which to climb or glide, and so on, we are better off using the indicated airspeed.
The indicated airspeed, since the density is assumed to always be sea level conditions, is really a function only of the difference in total and static pressures, P0 – P, which we know from Bernoulli’s equation is equal to:
and we are going to find that the terms on the right, the dynamic pressure, is a very important term in accounting for the forces on a body in a fluid. In other words, the plane’s behavior in flight is much more dependent on the dynamic pressure than on the airspeed alone.
Example: Let’s look at the difference between true and indicated airspeed just to get some idea of how big this difference might be. Lets pick an altitude of 15,000 feet and see what the two values of airspeed would be if the pitot-static system is exposed to a pressure difference of 300 pounds-per-square-foot (psf). The density in the standard atmosphere for 15,000 feet is 0.001497 sl/ft3 while that at sea level is 0.002378 sl/ft3.
So the difference in these two readings can be significant, but that is OK. We use the indicated airspeed to fly the airplane and use the true airspeed when finding the time for the trip. Note that when working Bernoulli’s equation problems, such as in finding the variations in pressures and velocity around a wing, you always want to use the true airspeed and the real pressures and density at altitude.
Finally, while on the subject of airspeed, we should note that even though we often calculate the speed of an aircraft or wing in units of feet/sec. or meters/sec., most airspeed indicators will show the airspeed in units of either miles-per-hour or knots. The knot is a rather ancient unit of speed used for centuries by sailors and once measured by timing a knotted rope as it was lowered over the side of a ship into the flowing sea.
A knot is a nautical-mile-per-hour and a nautical mile is a set fraction of the earth’s circumference. In relationship to more familiar English units:
1 knot (kt) = 1.15 mph
1 nautical mile (nm) = 1.15 “statute” miles (mi).
It is common practice in all parts of the world for our politically correct unit systems to be totally ignored and to do all flight planning and flying using units of knots and nautical miles for speed and distance.
1.10 Bernoulli’s Equation for Compressible Flow
The form of Bernoulli’s equation that we have been using is for incompressible flow as has been noted several times. What if the flow isn’t incompressible?
If Bernoulli’s equation was derived without making the assumption of constant air density it would come out in a different form and would be a relationship between pressures and Mach number. The relationship would also have another parameter in it, a term called gamma (γ). Gamma is simply a number for a given gas and the number depends on the number of atoms in the gas molecule, whether it is monatomic or diatomic, etc. Air is really a mixture of gasses but, in general, it is considered a diatomic gas. Its value of gamma is 1.4.
[Another name for gamma is the “ratio of specific heats” or the specific heat at constant pressure divided by the specific heat at constant density. These specific heats are a measure of the way heat is transferred in a gas under certain constraints (constant pressure or density) and this is, in turn, dependent on the molecular composition of the gas. In some other fields, Thermodynamics for example, the letter “k” is used for this ratio instead of γ.]
When a flow must be considered compressible this relationship between pressures and speed or Mach number takes the form below:
(P0/P) = {1 + [(γ-1)/2]M2 }[(γ-1)/(γ)]
If you use both this equation and the incompressible form of Bernoulli’s equation to solve for total pressure for given speeds from zero to 1000 ft/sec., using sea level conditions and the speed of sound at sea level to find the Mach number associated with each speed, and then compare the compressible and incompressible values of the total pressure (P0) you will find just over 2% difference at 700 ft/sec. and 5% at 900 ft/sec. In other words, the use of Bernoulli’s incompressible equation to find pressure and speed relationships is pretty reasonable up to speeds of about 75% of the speed of sound!
1.11 Forces in a Fluid
Above it was noted that the behavior of an airplane in flight is dependent on the dynamic pressure rather than on speed or velocity alone. In other words, it is a certain combination of density and velocity and not just density or velocity alone that is important to the way an airplane or a rocket flies. A question that might be asked is if there are other combinations of fluid properties that also have a major influence on aerodynamic forces.
We have already looked at one of these, Mach number, a combination of the speed and the speed of sound. Why is Mach number a “unique” combination of properties? Are there others that are just as important?
There is a fairly simple way we can take a look at how such combinations of fluid flow parameters group together to influence the forces and moments on a body in that flow. In more sophisticated texts this is found through a process known as “dimensional analysis”, and in books where the author was more intent on demonstrating his mathematical prowess than in teaching an understanding of physical reality, the process uses something called the “Buckingham-Pi Theorem”. Here, we will just be content with a description of the simplest process.
If we look at the properties in a fluid and elsewhere that cause forces on a body like an airplane in flight we could easily name several things like density, pressure, the size of the body, gravity, the “stickiness” or “viscosity” of the fluid and so on. As it turns out we could fairly easily say that most forces on an aircraft or rocket in flight are in some way functions of the following things:
where,
ρ = density
V = velocity
l = a representative length or size of the body
μ = viscosity
P = pressure
g = gravity (weight)
a = speed of sound
Viscosity must be considered to account for friction between the flow and the body and the speed of sound is included because somewhere we have heard that there are things like large drag increases at speeds near the speed of sound.
We really don’t know at this point exactly how general to be in looking at these terms. For example, we already know from Bernoulli’s equation that it is velocity squared that is important and not just velocity, at least in some cases. And, we might expect that instead of length it is length squared (area) that is important in the production of forces since we know forces come from a pressure acting on an area. So let’s be completely general and say the following:
Our simple analysis does not seek to find exact relationships or numbers but only the correct functional dependencies or combination of parameters. The analysis is really just a matter of balancing the units on the two sides of the equation. On the left side we have units of force (pounds or Newtons) where we know that one pound is equal to one slug times one ft/sec.2 or that one Newton is a kilogram-meter per second2. So the combination of all the units inherent in all the terms on the right side of the equation must also come out in these exact same unit combination as found on the left side of the equation. In other words, when all the units are accounted for on the right hand side of the equation the must combine to have units of force;
(sl)1(ft)1(sec)-2 or (mass)1(length)1(time)-2.
So, in this game of dimensional analysis the procedure is to replace each physical term on both sides of the equation with its proper units. Then we can simply add up all the exponents on both sides and write equations relating unit powers. For example, on the left we have units of mass (slugs or kg) to the first power. On the right there are several terms that also have units of mass in them and their exponents must add up to match the one on the left.
sl1 = (slA)(slD)(slG)
or since exponents add:
1 = A + D + G
We can do the same math for the other units of length and time and get two more relationships among the exponents:
1 = -3A + B + C – D + E – G + H
-2 = -B – D – 2E – 2G – H
These three equations of unit exponents can then be solved in terms of three of the “unknowns”, A, B, and C.
A = 1 – D – G
B = 2 – D – 2E – 2G – H
C = 2 – D + E
So, where we had density to the A power, or units of (sl/ft3)A, we now have:
(sl/ft3)A = (sl)(sl)-D(sl)-G(ft)-3(ft)3D(ft)3G.
We do this with every term on the right of the functional relationship and then rearrange the terms, grouping all terms with the same letter exponent and looking at the resulting groupings. We will get:
So, what does this tell us? It tells us that it is the groupings of flow and body parameters on the right that are important instead of the individual parameters in determining how a body behaves in a flow. Let’s examine each one.
The first term on the right is the only one that has no undefined exponent. The equation essentially says that one of the physical quantities that influences the production of forces on a body in a fluid is this combination of density, velocity squared, and some area (length squared).
If the force is a function of this combination of terms it is just as easily a function of this group divided by two; i.e.,
Note that we have used the letter “S” for the area. This may seem an odd choice since in other fields it is common to use S for a distance; however, it is conventional in aerodynamics to use S for a “representative area”. The area actually used is, as its name implies, one representative of the aerodynamics of the body. On an airplane, the dominant area for lift and drag is the wing, and S becomes the “planform area” of the wing. On a missile the frontal area is commonly used for S as is the case for automobiles and many other objects.
The second thing we note is that the term on the right is now the dynamic pressure times the representative area. So we have verified that the dynamic pressure is indeed very important in influencing the performance of a vehicle in a fluid.
If we look at this grouping of terms:
we note that it has units of force (pressure times an area). This means that all the units on the right hand side of our equation are in this one term. The other combinations of parameters on the right side of our equation must be unitless. This is immediately obvious in one case, V/a, where both numerator and denominator are speeds and it can be verified in all the others by looking at their units. We should recognize V/a as the Mach number!
We now rewrite the equation:
This says that the unitless combination of terms on the left is somehow a function of the four combinations of terms on the right. What are these terms and what role do they play in the production of forces on a body in a fluid?
1.12 Force Coefficients
First let’s look at the term on the left. This unitless term tells us the proper way to “non-dimensionalize” fluid forces. Instead of talking about lift we will talk about a unitless lift coefficient, CL:
We will also talk about a non-dimensional drag coefficient, CD:
We use these unitless “coefficients” instead of the forces themselves for two reasons. First, they are nice because they are unitless and we don’t need to worry about what unit system we are working in. If a wing has a lift coefficient of, say, 1.5, it will be 1.5 in either the English system or in SI or in any other system. Second, our analysis of units, this “dimensional analysis” business, has told us that it is more appropriate to the understanding of what happens to a body in a flow to look at lift coefficient and drag coefficient than it is to look just at lift and drag.
1.13 “Similarity Parameters”
Now, what about the terms on the right? Our analysis tells us that these groupings of parameters play an important role in the way force coefficients are produced in a fluid. Lets look at the simplest first, V/a.
V/a, by now, should be a familiar term to us. It is the ratio of the speed of the body in the fluid to the speed of sound in the fluid and it is called the Mach Number.
M = V/a .
If we are flying at the speed of sound we are at “Mach One” where V = a. But what is magic about Mach 1? There must be something important about it because back in the middle of the last century aerodynamicists were making a big deal about breaking the sound barrier; i.e., going faster than Mach 1. To see what the fuss was and continues to be all about lets look at what happens on a wing as it approaches the speed of sound.
As air moves over a wing it accelerates to speeds higher than the “free stream” speed. In other words, at a speed somewhat less than the speed of sound, the speed on top of the wing may have reached speeds greater than the speed of sound. This acceleration to supersonic speed does not cause any problem. It is slowing the flow down again that is problematic. Supersonic flow does not like to slow down and often when it does so it does it quite suddenly, through a “shock wave”. A shock wave is a sudden deceleration of a flow from supersonic to subsonic speed with an accompanying increase in pressure (remember Bernoulli’s equation). This sudden pressure change can easily cause the flow over the wing to break away or separate, resulting in a large wake behind the wing and an accompanying high drag.
So at some high but subsonic speed (Mach number) supersonic flow over the wing has developed to the extent that a shock forms, and drag increases due to a separated wake and losses across the shock. The point where this begins to occur is called the critical Mach number, Mcrit. Mcrit will be different for each airfoil and wing shape. The result of all this is a drag coefficient behavior something like that shown in the plot below:
Actually the theory for subsonic, compressible flow says that the drag rise that begins at the critical Mach number climbs asymptotically at Mach 1; hence, the myth of the “sound barrier”. Unfortunately many people, particularly theoreticians, seemed to believe that reality had to fit their theory rather than the reverse, and thought that drag coefficient actually did become infinite at Mach 1. They had their beliefs reinforced when some high powered fighter aircraft in WWII had structural and other failures as they approached the speed of sound in dives. When the shock wave caused flow separation it changed the way lift and drag were produced by the wing, sometimes leading to structural failure on wings and tail surfaces that weren’t designed for those distributions of forces. This flow separation could also make control surfaces on the tail and wings useless or even cause them to “reverse” in their effectiveness. The pilot was left with an airplane which, if it stayed together structurally, often became impossible to control, leading to a crash. Sometimes, if the plane held together and the pilot could retain consciousness, the plane’s Mach number would decrease sufficiently as it reached lower altitude (the speed of sound is a function of temperature and is higher at lower altitude) and the problem would go away, allowing the pilot to live to tell the story.
At any rate, experimentalists came to the rescue, noting that bullets had for years gone faster than the speed of sound (“you’ll never hear the shot that kills you”) and designing a bullet shaped airplane, the Bell X-1, with enough thrust to get it to and past Mach 1.
Once the plane is actually supersonic, there are actually two shocks on the wing, one at the leading edge where the flow decelerates suddenly from supersonic freestream speed to subsonic as it reaches the stagnation point, and one at the rear where the supersonic flow over the wing decelerates again. As a result, the “sonic boom” one hears from an airplane at supersonic speeds is really two successive booms instead of a single bang.
So it is important that we be aware of the Mach number of a flow because the forces like drag can change dramatically as Mach number changes. We certainly don’t want to try to predict the forces on a supersonic aircraft from test results at subsonic speeds or vice-versa. On the other hand, as long as everything we are considering happens below the critical Mach number we may not need to worry about these “compressibility effects”. In general, below Mcrit we can consider the flow to be “incompressible” and assume that density is a constant. Above Mcrit we can’t do this and we must use a compressible form of equations like Bernoulli’s equation.
Based on the above, one name we often give to the Mach number is a “similarity parameter”. Similarity parameters are things we must check in making sure our experimental measurements and calculations properly account for things such as the drag rise that starts at Mcrit. We don’t want to try to predict compressible flow effects using incompressible equations or test results or vice-versa.
The other three terms in our force relationship are also similarity parameters. Let’s look at the first term on the right hand side of that relationship. This grouping of terms is known as the Reynolds Number. Reynolds Number may be seen in different texts abbreviated in different ways [Re, Rn, R, Rex, etc.] depending on the convention in the field of use and on its application. Here we will use Re.
Reynolds number is really a ratio of the inertial forces and viscous forces in a fluid and is, in a very real way, a measure of the ability of a flow to follow a surface without separating.
Reynolds number is also an indicator of the behavior of a flow in a thin region next to a body in a flow where viscous forces are dominant, determining whether that flow is well behaved (laminar) or randomly messy (turbulent). This thin region is called the boundary layer.
Inertial forces are those forces that cause a body or a molecule in a flow to continue to move at constant speed and direction. Viscous forces are the result of collisions between molecules in a flow that force the flow, at least on a microscopic scale, to change direction. The combination of these forces, as reflected in the Reynolds Number, can lead a flow to be smooth and orderly and easily break away from a surface or to be random and turbulent and more likely to follow the curvature of a body. They also govern the magnitude of friction or viscous drag in between a body and a flow.
In general, a laminar boundary layer , which occurs at lower Reynolds Numbers, results in low friction drag (skin friction) but isn’t very good at resisting separation and may promote a large “wake” drag. A turbulent boundary layer , which occurs at higher Reynolds Numbers, has higher friction drag but resists flow separation better leading to lower “wake” drag. So which do we want? This all depends on the shape of the body and the relative magnitudes of wake and friction drag.
A classic case to examine is the flow around a circular cylinder or, in three dimensions, over a sphere.
Flow over a circular cylinder or sphere will generally follow its surface about half way around the body and then break away or separate, leaving a “wake” of “dead” air. This wake causes a lot of drag. This wake is similar to that seen when driving in the rain behind cars and especially large trucks. On a truck the point where the flow separates is about at the rear corners of the trailer body or tailgate. On a car it is often less well defined with separation occurring somewhere between the top of the rear window and the rear of the vehicle. On an aerodynamically well designed car the separation point would be right at the rear deck corner or at the “spoiler” if the car has one. A small wake gives lower drag than a large wake.
On a circular cylinder or sphere the separation point will largely depend on the Reynolds Number of the flow. At lower values of Re the flow next to the body surface in the “boundary layer tends to be “laminar”, a flow that is not very good at resisting breaking away from the surface or separating. Lower Re will usually result in separation somewhat before the flow has reached the half-way point (90 degrees) around the shape, giving a large wake and high “wake drag”. At higher Re the flow in the boundary layer tends to be turbulent and is able to resist separation, resulting in separation at some point beyond the half-way point and a smaller wake and wake drag.
Because the wake drag is the predominant form of drag on a shape like a sphere or circular cylinder, far greater in magnitude than the friction drag, the point of flow separation is the dominant factor in determining the value of the body’s drag coefficient.
In the above graph the Reynolds Number is based on the diameter of the sphere and the drag coefficient drops from a value of about 1.2 to about 0.3 as the “critical” Reynolds Number is passed at a value of about 385,000. This is a huge decrease in drag coefficient and illustrates how powerful an influence Re can have on a flow and the resulting forces on a body.
Note that a flow around a cylinder or sphere could fall in the high drag portion of the above curve because of several things since Re depends on density, speed, a representative length, and viscosity. Density and viscosity are properties of the atmosphere which, in the Troposphere, both decrease with altitude. The major things affecting Re and the flow behavior at any given altitude will be the flow speed and the body’s “characteristic length” or dimension. Low flow speed and/or a small dimension will result in a low Re and consequently in high drag coefficient. A small wire (a circular cylinder) will have a much larger drag coefficient than a large cylinder at a given speed.
Of course drag coefficient isn’t drag. Drag also depends on the body projected area, air density, and the square of the speed:
Hence, it cannot be stated in general that a small sphere will have more drag than a large one because the drag will depend on speed squared and area as well as the value of the drag coefficient. On the other hand, many common spherically shaped objects will defy our intuition in their drag behavior because of this phenomenon. A bowling ball placed in a wind tunnel will exhibit a pronounced decrease in drag as the tunnel speed increases and a Reynolds Number around 385,000 is passed. A sphere the size of a golf ball or a baseball will be at a sub-critical Reynolds Number even at speeds well over 100 mph. This is the reason we cover golf balls with “dimples” and baseball pitchers like to scuff up baseballs before throwing them. Having a rough surface can make the boundary layer flow turbulent even at Reynolds numbers where the flow would normally be laminar.
The CD behavior shown in the plot above is for a smooth sphere or circular cylinder. The same shape with a rough surface will experience “transition” from high drag coefficient behavior to lower CD values at much lower Reynolds Numbers. A rough surface creates its own turbulence in the boundary layer which influences flow separation in much the same way as the “natural” turbulence that results from the forces in the flow that determine the value of Re. Early golfers, playing with smooth golf balls, probably found out that, once their ball had a few scuffs or cuts, it actually drove farther. They probably then started experimenting with groove patterns cut into the surface of the balls. This led to the dimple patterns we see today which effectively lower the drag of the golf ball (this is not all good since the same dimples make a golfer’s hook or slice worse). The stitches on a baseball have the same effect, and baseball pitchers have found in their own somewhat unscientific experiments that further scuffing the cover of a new ball can make it go faster just as spitting on it can make it do other strange things.
The sphere or cylinder, as mentioned earlier, is a classic shape where there are large Re effects on drag. Other shapes, particularly “streamlined” or low drag shapes like airfoils and wings, will not exhibit such dramatic drag dependencies on Re, but the effect is still there and needs to be considered.
Like Mach number, Re is considered a “similarity” parameter, meaning that it we want to know what is happening to things like lift and drag coefficient on a body we must know its Reynolds Number and know which side of any “critical Re” we might be on. Flow over a wing could be quite different at sub-critical values of Re than at higher values, primarily in terms of the location of flow separation, stall, and in the magnitude of the friction drag that depends on the extent of laminar and turbulent flow on the wing surface.
So in doing calculations and wind tunnel tests we need to look at the magnitude of Re and its consequences. In doing so, we can get into some real quandaries when we test scale models in a wind tunnel. If we, for example, test a one-tenth scale model of an airplane in a wind tunnel our “characteristic dimension” in the Reynolds Number will be 1/10 of full scale. So, if we want to match the test Re with that of the full scale plane, one of the other terms must be changed by a factor of ten. Changing velocity by an order of magnitude to ten times the full scale speed will obviously get us in trouble with the other similarity parameter, Mach number, so that won’t do any good. We must change something else or “cheat” by using artificial roughness to create a turbulent boundary layer when the value of Reynolds number is really too low for a turbulent boundary layer.
One of the biggest breakthroughs in aeronautics came in the 1920s when the National Advisory Committee for Aeronautics (the NACA) at Langley Field, VA built what they called a “Variable Density Wind Tunnel”. This was a test facility where the air density could be increased by a factor of 20, allowing the testing of 1/20 scale models at full scale Reynolds Numbers. The VDWT, now a National Historical Monument, was a subsonic wind tunnel built inside of a egg-shaped pressurized steel shell. Quite sophisticated for its time, the tunnel was pressurized to 20 atmospheres after the wing model was installed. The tunnel was operated and the test model was moved through a range of test angles of attack by operators who observed their tests through pressure tight windows that resembled ship portholes. The model wings, each with a five inch chord and a thirty inch span, were machined to very tight tolerances based on dimensions worked out by NACA scientists and engineers. For the first time the world of aeronautics had reliable, full scale aerodynamic measurements of wings with a wide range of airfoil shapes.
Today in wind tunnel testing we usually “cheat” on Reynolds Number by using “trip strips”, small sandpaper like lines placed near the leading edges of wings and fuselages to force the flow to transition from laminar to turbulent flow at locations calculated prior to the tests. Although the Variable Density Wind Tunnel was retired long ago, we still have a unique capability at what is now NASA-Langley Field with the National Transonic Facility, a wind tunnel in which the working gas is nitrogen at temperatures very near its point of liquification. Since density, viscosity, and speed of sound all change with temperature it is possible in the NTF to simulate full scale Reynolds Numbers and Mach numbers at the same time. While testing is not easy at these low temperatures, properly run investigations in the NTF can yield aerodynamic data that can be obtained in no other manner.
In concluding this discussion of Reynolds Number it should be noted that the “characteristic length” in Re may take several different values depending on convention and that the value of Re at which transition from laminar to turbulent flow occurs can also vary with application. As shown earlier, we base the Reynolds Number for a circular cylinder or sphere on its diameter. The transition Re is just under 400,000. However, if we were mechanical or civil engineers working with flows through pipes, we would use the pipe diameter as our dimension and we would find that transition takes place at Re of about 5000, some two orders of magnitude different from transition on the sphere. We need to be aware of this different perspective on the important magnitudes of Re when talking with our friends in other fields about flows.
When we are talking about the value of Reynolds Number on a wing or an airplane the characteristic dimension used is the mean or average chord of the wing, the average distance from the wing’s leading to trailing edge. When we are doing detailed calculations on the behavior of the flow in the boundary layer we will use yet another dimension, the distance from the stagnation point over the surface of the body to the point where we are doing the calculations.
OK, we have found our something about the importance of Mach Number and Reynolds Number, what about the other two groupings of parameters in the force dependency equation we had earlier?
There were two other terms, gl/V2, and P/ρV . The first of these is a ratio of gravitational and inertial forces and relates to forces which arise as a consequence of a body being close to an “interface” such as the ground or the surface of the ocean. This term is usually inverted and its square root is known as the Froude Number.
Froude Number = V / [gl]1/2 ,
where the length in the equation is the distance between the body and the fluid interface; i.e., the height of the airplane above the runway or the depth of a submarine below the surface. Unless that distance is less than about twice the diameter of the submarine or the chord of the wing, Froude Number can be generally ignored; but within this range, it can help account for increases in drag or lift that may occur when a body is near a surface.
The final term, the Euler Number, P/ρV, is really not important in air, but is used when looking for the likelyhood of cavitation effects on a body in water. When the pressure in water is lowered to the vapor point of water at a given temperature, the water will boil even though temperature is lower than its usual boiling point. When flow accelerates around a ship propeller or hull the pressure can become low enough locally to boil the water, and this boiling or cavitation can cause serious increases in drag, noise, loss of propeller thrust, and damage to the surface itself. The term used to examine this is a modification of the one found in our analysis, rewritten slightly to account for the importance of the vapor pressure.
Note that the denominator is a familiar grouping of parameters, twice the dynamic pressure.
1.14 Airfoils and Their Aerodynamic Coefficients
At the beginning of this text we looked at the terminology that is used to define the shape of the airfoil, things like chord, camber, and thickness, as well as the things like span, chord, sweep, and aspect ratio which define the shape of a complete wing. In the section above we found that the best way to talk about the forces acting on an airfoil (2-D) or a wing (3-D) is in terms of non-dimensional coefficients. Let’s take a look again at those coefficients and how they are defined in both two and three dimensions and look at the way these change as a function of the wing or airfoil angle of attack to the flow.
A few pages back we defined the lift and drag coefficients for a wing. These are repeated below and are extended to define a pitching moment coefficient:
Note again that the area “S” is the “planform” area of the wing, the “projected” area one would see looking down on the wing and not its actual surface area. The moment coefficient has a slightly different denominator which includes the mean or average chord along with the planform area in order to make the coefficient unitless, since the pitching moment has units of force times distance. A little later we will look at the moment in more detail and see what reference point(s) should be used in measuring or calculating that moment.
The coefficients above are in three-dimensional form and are as we would define them for a full wing. If we are just looking at the airfoil section, a two-dimensional slice of the wing, the area in the denominator would simply become the chord.
It would be helpful to know something about the typical ranges of these coefficients in either two or three dimensions so we will have some idea how valid our calculations may be.
-2.0 < CL < +2.0 for a normal wing or airfoil with no flaps
-3.0 < CL < +3.0 for a wing or airfoil with flaps
0.005 < CD < 0.025 for a wing or airfoil (more for a whole airplane)
-0.1 < CM < 0.0 for a wing or airfoil when measured at c/4
It should also be noted that the pitching moment is defined as positive when it is “nose up”. Depending on the direction of the airfoil or wing within an axis system this may be counter to mathematical custom. In fact, most texts commonly treat aerodynamics problems as if the freestream velocity is coming from the left in the positive x direction and the wing or airfoil is facing toward the left in a negative x direction. In such a depiction a positive pitching moment would be clockwise even though customary mathematical treatment would assign this a negative sign.
1.15 Angle of Attack
We also need to define the way we relate the orientation of the wing to the flow; i,e., the wing or airfoil angle of attack, α. At the same time we can look at the definition for the direction of lift and drag.
Note that the angle of attack, α, is defined as the angle between the wing or airfoil chord and the freestream velocity vector, not between the velocity and the horizontal. Also note that Lift and Drag are defined as perpendicular and parallel, respectively, to the freestream velocity vector, not “up and back” or perpendicular and parallel to the chord. These definitions often require care because they are not always intuitive. In many situations the force we define as lift will have a “forward” component to it along the wing chord line. This is, in fact, the reason that helicopter and gyrocopter rotor blades can rotate without power in what is known as the “autorotation” mode, giving lift with an unpowered rotor.
Given these definitions lets look at the way lift and drag coefficient typically change with angle of attack.
Let’s look at a few things in the above plots:
1. The lift coefficient varies linearly with angle of attack until the wing approaches stall. Stall begins when the flow begins to break away or separate from the wing’s upper surface and may progress quickly or gradually over the surface.
2. When the lift coefficient no longer increases with angle of attack, in the middle of stall progression, we have CLmax the largest value of lift coefficient we can get for a particular airfoil or wing at a given Reynolds number. The value of CLmax will, in general, increase with increasing Re.
3. The lift coefficient is not shown as zero in value at a zero angle of attack. On a “symmetrical” airfoil; i.e., one with no camber, the lift will be zero at zero angle of attack. On a cambered airfoil the lift will be zero at some negative angle of attack which we call the “zero lift angle of attack”, αL0.
4. Drag increases rapidly in stall.
We should explore this idea of stall further because something seems amiss here. The plot above shows that stall is where we have our maximum lift coefficient. How can that be true when we’ve always been told that stall is where an airplane looses lift?
Our dilemma here stems from the meanings of two different terms, lift and lift coefficient. Let’s look at the relationship between lift and lift coefficient.
Lift=CL[1/2ρV2S]
To have lift we must have two things, a good lift coefficient and the speed necessary to turn that lift coefficient into lift. So what is the problem here? The problem is drag.
When an airplane reaches CLmax it has its highest possible lifting capability or lift coefficient. But looking at the other plot, the one of drag coefficient, we see that it also has a rapidly increasing drag coefficient. The resulting drag causes a reduction in speed and, since speed is squared in the relationship above it has a much more powerful influence on lift than does the lift coefficient. Lift decreases as the speed drops and, indeed, even though we have a very high lift coefficient we can’t make enough lift to overcome the plane’s weight. And, as we can see from the plot on the left, if we go past the stall angle of attack we also get a decrease in lift coefficient.
As noted above up until stall the lift coefficient is shown to vary linearly with angle of attack and to become zero at the zero lift angle of attack L0. Given this we can write a simple equation for the value of CL.
Theory will show and experiment will verify that for an airfoil section (a two dimensional slice of a wing) this slope, dCL /dα, is equal to 2π, where the angles of attack are expressed in radians. So theoretically for an airfoil section this equation becomes:
For a three dimensional wing the slope of the curve may be less and both theory and experiment can be used to determine that 3-D slope.
So how does the shape of the airfoil section influence this relationship? First, the shape of the camber line will determine the value of the zero lift angle of attack. We will find in a later course in aerodynamics that there are ways to calculate L0 from the shape of the airfoil camber line.
It is harder and may even be impossible to calculate the value of CLmax and the angle of attack for stall. These are functions of the thickness of the airfoil and the shape of its surface as well as of things like the Reynolds Number. We have to look at friction effects and Boundary Layer Theory to even begin to deal with the stall region successfully. Our only alternative is to use wind tunnel testing. And this is what the people at the National Advisory Committee for Aeronautics (NACA) which was mentioned earlier did in their Variable Density Wind Tunnel.
1.16 NACA Airfoils
In the 1920s the scientists and engineers at the NACA set out to for the first time systematically examine how things like camber line shape and thickness distributions influenced the behavior of airfoils. They began their quest by deciding how to define an airfoil shape using only three numbers and four digits. They looked at the best airfoil shapes of their time and found a good way to vary the thickness of the airfoil from its leading edge to the trailing edge and wrote an equation for this distribution that allowed everything to be defined in terms of one, two-digit number. Then they wrote two equations for the airfoil camber line, one for the front part of the wing up to the point of maximum camber and the other for the aft part of the wing behind the maximum camber location. These two equations depended on two single-digit numbers, one giving the location of the maximum camber point in tenths of the chord and the other giving the maximum distance between the camber and chord lines in hundredths of the chord. Then they used these four digits to identify the resulting airfoil.
For example, the NACA 2412 airfoil is an airfoil where each of the numbers means something about the shape:
2 The maximum distance between the chord and camber lines is 2% of the chord.
4 – The location of the point of maximum camber is at 40% chord.
12 – The maximum thickness of the airfoil is 12% of the chord.
These four digits (three numbers) could then be used with related equations to draw the airfoil shape.
Using this method the NACA could systematically look at the effects of just the amount of maximum camber by testing a series of shapes such as 1412, 2412, 3412, etc. , or they could hold the maximum camber steady and look at the effect of placement of maximum camber (2212, 2312, 2412, 2512, etc.) , or fix the shape of the camber line and look at the effect of thickness variation (0006, 0009, 0012, 0015, 0021, etc.). Note that this last set of airfoils are all symmetrical; i.e., zero camber. And this is precisely what was done with tests on hundreds of airfoils. A wide variety of airfoils were tested in the Variable Density Wind Tunnel and the results plotted. Sample plots are presented in Appendix A.
The first set, Figures A-1 (A&B), is for an NACA 0012 airfoil section. Note that this is a symmetrical airfoil section (the first zero indicates that there is no camber and thus the second number is meaningless) and has a 12% thickness. We first see that there are several plots on each of the two graphs and that there are multiple definitions of the vertical axis for each plot. On the left hand plot the horizontal axis is the angle of attack given in degrees and the primary vertical axis is the section or two-dimensional lift coefficient. There is also a secondary vertical axis for the pitching moment coefficient (with the moment measured relative to c/4, the quarter chord ).
There is a lot of information given on plot A. The first thing we note is that there are two sets of lift coefficient curves and that each of these seems to have a couple of different plots of the stall region. Note that the linear portion of one set of these curves goes through the plot axis showing that the lift coefficient is zero at an angle of attack of zero. These curves are for the basic 0012 airfoil which is a symmetrical airfoil and will have no loft at zero angle of attack.
There are, as noted above, at least two different stall region plots for this symmetrical airfoil. To understand these we need to look at the inset information about Reynolds number (noted as simply “R” on these plots). There are four different symbols shown in this inset. The “diamond” symbol denoted data for the wing tested at a Reynolds number of 9 x 106, the square for Re = 6 x 106, and the circle for Re = 3 x 106. The other symbol, a triangle, is for the airfoil tested with a “standard roughness” or a roughened surface at a Re = 6 x 106. Usually the higher the value of Reynolds number the higher the value of maximum lift coefficient but in this particular case there is only a slight increase with CLmax moving up from about 1.5 to 1.6 as Reynolds number on the smooth airfoil increases from 3 to 6 x 106. For the smooth surface stall always occurs at 16 degrees angle of attack. Note also that the airfoil also stalls at minus 16 degrees angle of attack, verifying its symmetrical behavior. The rough surface causes a much earlier stall and lower value of CLmax.
So, what is the other set of “lift curves” that are displaced up and to the left on this plot? These are the data for the 0012 airfoil with a trailing edge flap. Other inset information on the plot tells us that this is a split flap with a chord of 20% of the airfoil chord and it is deflected 60 degrees. Note that the basic airfoil is drawn to scale on the right hand plot and the flap and its deflection are noted in dashed lines on the drawing.
This data shows that for this particular flap the lift curve shifts to the left such that it has zero lift at minus 12 degrees, in other words αLO, the zero lift angle of attack, is minus 12 degrees for the flapped airfoil. The curves show that the deflection of the flap has increased CLmax from about 1.6 up to about 2.4, a 50% increase. CLmax for the rough surface airfoil has gone from 1.0 to 1.9.
Now, there are two other curves on this plot. They are curves for the pitching moment coefficient about the quarter chord, with and without flap deflection. First note that the scale to the left of the plot is different for moment coefficient than it was for lift coefficient. Do not confuse these two scales. Next, note that the upper of these two curves is for the wing without flaps and that it has a value of zero over the range of angle of attack where the lift curve is linear. This means that the pitching moment is zero at the quarter chord for this airfoil. This is true for all symmetrical airfoils. We define the place where the pitching moment is zero as the center of pressure (sometimes called the center of lift) and we define the place where the pitching moment coefficient is constant with changing angle of attack as the aerodynamic center. We will look at these further a little later. For a symmetrical airfoil the center of pressure and the aerodynamic center are both at c/4.
Now let’s look at the plot B for the 0012 airfoil. This plot displays test results for drag coefficient and moment coefficient and plots them versus the lift coefficient, not the angle of attack. So to find the drag coefficient at a certain angle of attack requires us to first find the value of CL at that angle of attack and then use it to find CD.
Again on this plot there are different sets of data for different values of Reynolds number and roughness. Note that CD is smallest at zero angle of attack for this symmetrical airfoil as would be expected, and that as angle of attack is increased CD increases. The moment curve on this plot is for the moment coefficient at the aerodynamic center and not for c/4 as in the plot on the left. However, as noted above, for a symmetrical airfoil the aerodynamic center is at the quarter chord so these are the same here.
The next plot in Appendix A, Figure A-2 (A&B), shows similar data for the NACA 2412 airfoil. Like the 0012 airfoil this shape has 12% thickness but now it has a slight (one percent) camber with the maximum camber located at 40% of the chord. So we can compare these plots with the first ones to see the effect of added camber. Looking at all the same things mentioned above we see that the airfoil now has a positive lift coefficient (0.15) at zero angle of attack and that the zero lift angle of attack is now minus-one degree. CLmax is still about 1.6 for the unflapped wing but with flap deflection it has increased slightly from the symmetrical case. Pitching moment coefficient is no longer zero at c/4 but it is still constant, now at a value of about – 0.025 ; i.e., a slightly nose down pitching moment. Since the pitching moment at c/4 is still constant this point is still the aerodynamic center but it is no longer the center of pressure. The minimum drag coefficient is about the same as for the symmetrical wing and it is still minimum at about zero angle of attack but this is no longer at zero lift coefficient.
We can see the result of further increases in camber to two percent in the plots for the NACA 2412 airfoil. There is no data for flap deflection included in these plots. In this plot we can see that the pitching moment continues to increase negatively (nose down) as camber is increased and that the minimum drag coefficient has also begun to increase. The zero lift angle of attack and the lift at zero angle of attack are both seen to continue to increase in magnitude as camber is added as theory would suggest.
The next plots, Figure A-3 (A&B), for the NACA 2415 show very slight changes in the airfoil behavior with increased thickness from 12% to 15% but for the NACA 2421 section we see that the added thickness is resulting in earlier stall, decreased CLmax, increased drag coefficient, and in a non-linear moment coefficient variation with angle of attack about the quarter chord. This might suggest to us that 21% thickness is a little too thick.
The data (Figure A-4 (A&B), for the NACA 4412 airfoil continues to illustrate the trends discussed above, looking at the effect of adding more camber.
The last two plot sets, Figures A-5 and A-6, are for a different design of airfoil, the NACA 6-series airfoil, a shape we will discuss first before looking further at the graphs.
The NACA tested a wide variety of airfoil shapes in its four digit series investigations. However there were some limits to the shape variations one could get using the four digit designation. They couldn’t, for example, look at an airfoil with maximum camber at 25% of the chord, only 20% or 30%. Hence they went on to develop a 5 digit airfoil series. Later they looked at an airfoil series designed to optimize the use of laminar and turbulent flow in the boundary layer to minimize drag and developed the 6-series airfoils. Let’s look at both of these.
The NACA 5-digit series used the same thickness distribution as the four digit series but allowed more flexibility in defining the position of maximum camber. It also attempted to relate the amount of camber and the first digit in the airfoil designation to a “design CL”. The design CL is the lift coefficient viewed as optimum for the performance objectives of a given airplane. A long range transport may have a design lift coefficient of around 0.3 while a fighter aircraft might have a higher design CL. An example of an NACA 5-digit airfoil is given below with an explanation of the numbering system:
NACA 23021
2 – the maximum camber is approximately 0.02 and the design CL is 2 x 0.15 = 3.0
30 – the position of the maximum camber is (0.30/2) times the chord or 0.15c
21 – the maximum thickness is 0.21c
The NACA 6-series (note that this is simply called the “six” series and not the six digit series although most of the designation numbers do have six digits) was developed in the 1930s in an attempt to design a series of airfoil shapes which optimized the areas of laminar and turbulent flow in the boundary layer on the wing. As we discussed earlier, laminar flow in the boundary layer is a low friction flow and that is good; however, laminar flow is poor at resisting flow separation and separation results in high drag and low lift. A turbulent boundary layer is much better at resisting flow separation than a laminar one but it has higher friction drag.
Flow separation is much more likely when the flow is decelerating (where the pressure is increasing, known as an “adverse” pressure gradient because separation is likely). The flow over an airfoil will usually be accelerating over the front of the shape up to the point of its minimum pressure which us usually at about the point of maximum thickness. Here it is safe to have laminar flow because the flow is not likely to try to separate. If we want more laminar flow and, hence, a larger part of the airfoil with low friction drag, we can move the maximum thickness point more toward the rear of the airfoil. The idea is to get as large as possible an area of laminar flow and then let the boundary layer transition to turbulent flow before the thickness starts to decrease so the turbulent boundary layer will resist separation. This results in an airfoil with a smaller leading edge radius than the older designs and with the maximum thickness further back.
Two such 6-series shapes are sketched as part of their NACA data plots in Appendix A, the NACA 651-212 and 651-412 airfoils. One thing that is immediately obvious when comparing these two data plots to those of the NACA 4-digit airfoil data is the “bucket” in the center of the drag coefficient curves in the right hand plots. This so-called “drag bucket” is characteristic of the 6-series airfoils. All of these airfoils have a region of angle of attack or lift coefficient over which the drag is considerably lower than at other angles of attack. This is the range of angle of attack where laminar flow can exist over a large part the forward portion of the airfoil, giving a reduction in friction drag.
The 6-series airfoil numbering system is devised to help the designer/aerodynamicist select the best airfoil for the job by telling where the center of the “drag bucket” is located and telling the extent (width) of that drag bucket. In other words, if a designer wants a wing with an airfoil section that gives a design lift coefficient of 0.2 he or she wants the center of the drag bucket to be at a CL of 0.2 so the airplane will be able to do its design mission at the lowest possible drag conditions. Hence, one of the numbers in the 6-series designation tells the CL for the center of the drag bucket. Lets look at one of these airfoil designations and see what the numbers mean.
NACA 651-212
6 – this is merely the “series” designation
5 – the minimum pressure location at zero lift is at 50% chord
1 – this is a subscript which may or may not appear in a 6 series designation. It means that the width of the “drag bucket” extends for a range of CL of 0.1 above and below the design CL
2 – the design CL is 0.2
12 – as always, the maximum thickness as a % chord
Now if we again look at the data for the above airfoil we see that the drag bucket is indeed centered at a lift coefficient of about 0.2 and the bucket extends for at least a CL range of 0.1 on both sides of its center. Similarly if we look at the plots for the NACA 651-412 we find that the drag bucket is further to the right and centered at about a CL of 0.4.
A closer comparison of these 6-series aerodynamic data with those of the 4-digit series cases will show that not everything is better. The 6-series airfoils often stall a little earlier due to their smaller leading edge radii and hence have slightly lower CLmax values than their earlier counterparts. Also their drag coefficients outside of the range of the drag bucket may be higher than conventional airfoils. As in all things in real life improvements in one area are often accompanied by penalties in others. Nonetheless, the 6-series airfoils are excellent designs and are still used today on many aircraft.
1.17 Pitching Moment
Before we look further at airfoil development we should step back a bit and look further at two things mentioned earlier, the aerodynamic center and the center of pressure. These are two important points on the airfoil that depend on the behavior of the pitching moment. A moment must be referenced to a point and these happen to be very meaningful points about which to reference the pitching moment on an airfoil or wing.
The moment on an airfoil is related primarily to the way lift is produced on the shape. Drag also contributes to the moment but to a much smaller extent. At subsonic speeds the lift on most airfoils is higher on the front part of the shape than at the rear and might look something like this:
If we choose to talk about the pitching moment about the airfoil leading edge, the moment would always be nose down or counterclockwise. If we sum up the moments at some point about half way back along the chord the moment would be nose up or clockwise since the lift forces to the left are greater than those on the right. Obviously there is some point between the airfoil leading edge and its center where the moments would sum to zero. This would be the center of pressure. It might be interesting and useful to know where this place is since it seems to be a natural balance point or sorts. The only problem is that this position may move as the airfoil changes angle of attack. For example, at higher angle of attack even more of the lift might be produced near the front of the airfoil and the center of pressure would move closer to the nose of the airfoil.
As it turns out, according to aerodynamic theory which you will examine in a later course, there is another point which is of even more interest, the aerodynamic center. This is the point where the moment coefficient (not the moment itself) is constant over a wide range of angle of attack. Basic aerodynamic theory will tell us that this is approximately at the quarter-chord of the airfoil in subsonic flow. This was seen to be the case for the airfoils whose data are shown in Appendix A.
Theory will also show and experiment will verify that for a symmetrical airfoil such as the NACA 0012 shape in Appendix A the aerodynamic center and the center of pressure are in the same place. In other words, for symmetrical airfoils at the aerodynamic center the pitching moment is not only constant but is also zero. This makes the aerodynamic center a very convenient place to locate major structural elements or to use as the balance point for control surfaces.
1.18 Camber and Flaps, and Flight at Reduced Speeds
We have seen in the data of Appendix A that as the camber of a wing increases its aerodynamic performance changes. Looking at the data for the NACA 0012, 1412, and 2412 airfoils we saw that as camber (indicated by the first term in the NACA four digit numbering system) is increased the lift curve shifts to the left giving more lift coefficient at zero angle of attack, an increasingly negative angle of attack for zero lift coefficient, and a slowly increasing value of maximum lift coefficient. This is accompanied by a slight increase in drag coefficient and a negative increase in pitching moment coefficient at the aerodynamic center as camber increases. In later aerodynamics courses you will learn how to predict these changes which result from modification of the airfoil camber line shape. It is evident now, however, that increasing camber can give higher lift coefficients and both the airplane designer and the pilot may wish to take advantage of this. One type of flight where this becomes very useful is low speed flight, especially in takeoff and landing.
To fly the lift of the airplane must equal its weight.
This relationship says that lift comes from four things, the lift coefficient, the density, the velocity, and the wing planform area. There is nothing much we can do about the density, it comes with the altitude, and while there are ways to change the wing area while a plane is in flight, these are often impractical. We note that speed is a powerful factor since it is squared.
The equation above essentially tells us that if we want to fly at a lower speed with a given wing and altitude we must increase the lift coefficient. We can do this to a certain extent as we increase the angle of attack from the zero lift angle of attack to the angle for stall, but stall defines our limit.
We, of course, don’t want to try to fly at CLmax because we will stall, but the stall speed does define the minimum limit for our possible range of flight speed at a given altitude. If we want to fly at lower speed we need to increase the value of CLmax. The graphs in Appendix A show us that this can be done with flaps. On the plot for the NACA 1412 airfoil we see that by deflecting a split flap with a length of 20% of the airfoil chord to an angle of 60 degrees we can increase the maximum lift coefficient from 1.6 to 2.5, a change which would lower our stall speed by 20 percent. This means that we can fly at a 20% lower speed.
This is a powerful effect but it isn’t free. It is accompanied by a large increase in drag coefficient and a huge change in pitching moment. This may mean a need for a larger horizontal stabilizer or canard to counter the pitch change and we will need to deal with the drag.
In the early days of flight there was no real need for flaps. Aircraft flew at speeds very slow by today’s standards and their stall speeds were often very low due to large wing areas. But as research showed how to reduce airplane drag and better engines gave increasingly more power and thrust not as much wing area was needed to cruise at the resulting higher speeds. Airplane designers found that making a plane cruise at over 200 mph and still land at something like 60 mph or less was a problem. Landing speed is important because it is directly related to stopping distance and the needed runway length. Higher landing speeds also may increase the risk of landing accidents.
Most airplanes cruise at a lift coefficient of about 0.2 to 0.3 and the best subsonic airfoils will have a CLmax no higher than 1.8. These two factors define the problem of high speed cruise and low speed landing.
In 1933 the Boeing Company brought out the Boeing 247 airliner, a thoroughly modern plane for its time which took advantages of all the accumulated progress in engine and airframe design. Its cruise speed was 188 mph at an altitude of 8000 feet. The 247 weighed 13,650 pounds and had a wing area of 836 ft2. A quick calculation of its cruise lift coefficient gives CL = 0.23, a reasonable value. At that time existing runways required a landing speed of around 60 mph and a calculation of the 247’s lift coefficient at sea level at 60 mph gives 1.77, pretty near the maximum for a conventional wing.
The Douglas Aircraft Company (now a part of Boeing) decided to build a bigger and more comfortable airliner and came out with the DC-1, the prototype airplane, and the production model DC-2. The DC-2 weighed 36% more than the 247 at 18,560 pounds and had a slightly higher wing area of 939 ft2, but it cruised at about the same speed and altitude as its competitor. The cruise CL for the DC-2 comes out to be about 0.27, higher than the 247 because of its significantly larger weight. This higher “wing loading” gave the DC-2 passengers a more comfortable ride than the 247.
To land the DC-2 at 60 mph requires a CL of about 2.15, too high for a normal wing. The solution was to add flaps, giving extra camber and a higher CLmax when needed for landing and takeoff. This allowed the larger, more comfortable DC-2 to fly as fast as the 247 and still land and takeoff at all the commercial airports of its day. The DC-2 and its even larger and more comfortable sibling, the DC-3, went on to revolutionize the airline industry.
It should be noted that large flap deflections are used in landing where the added drag may actually be advantageous and smaller deflections are used in takeoff where lower drag and rapid acceleration are a must.
There are many types of flaps. There are both leading edge and trailing edge flaps and an array of variations on both. Aerodynamic theory tells us that a camber increase is most effective when it is made near the trailing edge of an airfoil, hence, the trailing edge flap is the primary type of flap used on wings. The deflection of virtually any type of trailing edge flap from a simple flat plate to a complex, multi-element flap system will shift the “lift curve” to the left and increase CLmax. Some trailing edge flaps have slots and multiple elements to help control the flow over the flaps and prevent separation to give even higher lift coefficient and often these flaps deploy in such a way as to temporarily add extra wing area.
Leading edge flaps do little to move the lift curve to the left but can do a lot to allow the airfoil to go to a higher angle of attack before stalling, by controlling the flow over the “nose” of the airfoil and delaying separation. Leading edge flaps are often used in takeoff and landing in conjunction with trailing edge flaps. A few aircraft have been designed with leading edge flaps or slots fixed permanently into the wing to give them lower stall speeds.
The effects of both leading and trailing edge flaps are shown in the following figure.
The following table lists typical magnitudes of lift coefficient with and with out both leading and trailing edge flaps for a “Clark Y” airfoil. The Clark Y airfoil is a famous non-NACA airfoil shape developed by Virginius Clark. Clark had served on the same commission as many of the founders of the NACA, a commission charged with studying European airfoil sections after World War I and, using much of the same information that the NACA used to develop its original 4 digit airfoils, Clark developed the Clark Y and other airfoils as part of his Ph.D research at MIT. In the Clark Y, Virginius Clark designed an airfoil shape with a flat bottom that made it easy to manufacture, and because of this and its excellent aerodynamic performance, it was used widely for everything from airplane wings to propeller blades.
Table 1.3: Leading edge and trailing edge flap and slot effects on a Clark Y airfoil
Configuration C sub Lmax alpha sub stall
Basic Clark Y airfoil 1.29 15
with 30% plain flap at 45° 1.95 12
with "fixed" slot and no flap 1.77 24
with slot and plain flap 2.18 19
with 40% Fowler flap at 40° 3.09 14
with deployable slot and Fowler flap 3.36 16
To summarize, a trailing edge flap can have a significant “camber effect”; i.e., can shift the “lift curve” to the left, increasing the zero lift angle of attack and the value of CLmax. It can also be deployed in such a way as to temporarily increase wing area. A leading edge flap or slot will probably not produce a “camber effect” (if it does it is likely to be a “negative” one, slightly shifting the “lift curve” to the right) but will help retard stall to a higher angle of attack whether used on a wing with or without flaps.
1.19 Transonic and Supersonic Airfoils and Wings
Earlier we looked at the way flow can accelerate to supersonic speeds over an airfoil or wing as the free stream speed approaches the speed of sound and how, at Mach numbers higher than some “critical Mach number” the deceleration of that supersonic flow back to subsonic speeds can result in sudden flow separation and a drag increase. There are two ways to reduce this drag. One is by sweeping the wing and the other is by designing a special airfoil section.
1.20 Wing Sweep
The first way found to successfully decrease the drag rise that occurs in the transonic flight regime is by sweeping the wing. Theory and experiment showed that both the onset and the magnitude of the drag increase were functions of the “normal component” of the free stream Mach number, M∞. In other words, if a wing is swept 450 the normal component of the free stream Mach number is M∞ cosine θ or 0.707
As with almost everything this benefit of sweep comes at a cost. Sweeping the wing also reduces its lift coefficient at a given angle of attack (reduces the slope of the lift curve) and the curved flow over the wing itself can lead to premature stall near the wing tips and a phenomenon known as “pitch up”
While most swept wings are angled toward the rear of the plane or swept back, theoretically it doesn’t matter whether the wing is swept forward or aft. Several early swept wing designs were drawn with forward swept wings which allowed the wing spar or structure to pass through the fuselage aft of the cockpit and presented fewer internal design problems but it was soon discovered that there was indeed a problem with forward swept wings. This problem was caused by the same type of curving flow that caused tip stall on the aft swept wings except that the result on the forward swept wing was an added lift at the wing tips that tended to twist the wings to their breaking point at fairly low speeds. There was little point in sweeping a wing to lower the transonic drag rise when the wing would break off long before such speeds were reached and the added weight needed to stiffen the wing to prevent failure made the airplanes too heavy. This problem was finally solved in the 1970s with the use of fabric based composite structures which could be designed in such a way that the wing got stronger as it tried to twist off. The X-29 experimental aircraft successfully proved that swept forward wings could indeed be used in transonic flows.
1.21 Supercritical Airfoils
The other method used to reduce the transonic drag rise on a wing was developed by Richard Whitcomb at NASA-Langley Research Center in the 1960’s. Dr. Whitcomb essentially reshaped the conventional airfoil section to do three things. He increased the “roundness” of the airfoil leading edge to give a more gradual acceleration of the flow to a speed lower than conventional airfoil shapes so when supersonic flow resulted on the surface it was weaker. He reduced the wing camber at its mid chord area to flatten the upper surface and allow a longer region of this weaker supersonic flow before allowing it to decelerate, giving less separation and drag. Finally, to make up for the lift that was lost by designing for slower upper surface flow, Whitcomb designed his airfoil with significant “aft camber” on its lower surface, noting that camber has a very powerful effect neat an airfoil trailing edge. The result was an airfoil somewhat like the one below which gave excellent aerodynamic performance with a reduced transonic drag rise. This type of airfoil is called a Whitcomb airfoil or a “supercritical” airfoil.
As it turned out, this airfoil was an excellent design for all ranges of flight with its only drawback being a tendency toward a large pitching moment due to the aft lower camber. Subsequent designs have reduced this problem and variants of this design are used on almost every type of aircraft today.
While discussing Richard Whitcomb and transonic flow, his “coke bottle” fuselage design should also be mentioned. In the early 1950’s as jet fighters were approaching Mach 1 in capability, the transonic drag rise of the whole airplane continued to be a problem in “breaking the so-called sound barrier”. Convair and the Air Force hoped that a new, highly swept, delta wing fighter designated the F-102 would be able to routinely fly at supersonic speeds; however two prototype aircraft failed to reach Mach 1. Whitcomb realized that, at the speed of sound, the air cannot be compressed any further and needs some place to go or it will simply push outward from the plane, displacing other airflow and causing drag. He suggested a redesign of the F-102 fuselage with a reduced cross section in the vicinity of the wing to allow this supersonic air a place to go without pushing away outer flows. The “wasp-waist” or “coke-bottle” fuselage was the result and the design which could formerly not make it to the speed of sound reached Mach 1.22 on its first flight.
1.22 Three Dimensional Aerodynamics
Most of the things discussed in the previous sections were two dimensional phenomenon such as airfoil section design, the effects of camber, etc. Wing sweep was the exception. But there are many other variations in wing planform and shape that will influence the aerodynamic performance of a wing. Let’s start by looking at the Aspect Ratio.
A wing which is producing lift must have a lower pressure on its upper surface than on the lower surface. At the wing tips there is nothing to prevent the air from the lower surface from trying to go around the wing tip to the upper surface where the lower pressure acts like a vacuum. The result is some loss of lift near the wing tip. An ideal wing would have the same lift from wing tip to wing tip but a real wing doesn’t.
This loss of lift is felt for some distance inboard of the wing tips. The question is what percentage of the wing area this affects and this will depend on the wing Aspect Ratio, AR.
Aspect Ratio is a measure of the wing’s span divided by its “mean” or average chord. It can also be expressed in terms of the square of the span and the planform area.
AR = b2/S = b/cavg
where b = wing span, S = planform area, c = chord
To see why aspect ratio is important we can look at two different wing planforms of the same area but different aspect ratios.
This flow around the wing tip results in two other problems, the production of a drag called the induced drag and the creation of a tornado-like swirling flow, called a wing-tip vortex behind the wing tip that can be a hazard to following aircraft.
The trailing vortices (one vortex from each wing tip) can continue for miles behind an aircraft and the “strength” of the vortices will depend on the weight of the generating aircraft. A following airplane, particularly a small one, may find itself suddenly turned upside down (or worse) if it encounters one of these vortices. This is particularly dangerous near the ground and is one of the reasons for required separation times between landing and taking off airplanes at airports.
The 3-D problem of concern here is the added drag which comes from this flow around the wing tips and the decreased lift. Because of this flow the lift coefficient on a 3-D wing will be lower at a given angle of attack than a 2-D airfoil section of the same shape. Aerodynamic theory can be used to calculate the 3-D effect as a function of the planform shape of the wing and that effect can be characterized as an aspect ratio effect. A 2-D airfoil section is said to have an infinite aspect ratio and for the 2-D case theory gives a slope for the “lift curve” (dCL/dα) of 2π. For the 3-D case where aspect ratio is finite the slope of the lift curve will be found to decrease with decreasing aspect ratio.
The added drag in 3-D comes from the same phenomenon that causes lift. We define life as a force perpendicular to the free stream velocity. Induced drag is a force that is perpendicular to the “downwash” velocity caused by the flow around the wingtip. This downwash velocity is small compared to the freestream flow and the induced drag is correspondingly small but it is still a force we need to understand and deal with.
Theory will show the induced drag coefficient CDi to be:
From this it can be seen that as AR increases the induced drag decreases and that in the 2-D case where the theoretical aspect ratio is infinity, the induced drag coefficient is zero. But, what about the other term in the equation, e?
e is called the “Oswald efficiency factor”. The value of e will be somewhere between zero and one with one being the best or “minimum induced drag” case. Theory will show that e is a function of the way lift acts along the span of the wing which is a function of several things, including wing planform shape, wing sweep, wing twist, taper, the various airfoil sections used across the span, etc. A study of this theory will show that the best case, when e = 1.0 occurs when the lift is distributed along the wing span in an elliptical manner. This minimum induced drag case is often called the elliptical lift distribution case.
There are many ways to get an elliptical lift distribution. The easiest to visualize is that where the wing planform is shaped like an ellipse, that is where the wing chord varies elliptically along its span. In the 1940’s many World War II fighters were built with elliptical planforms to try to minimize induced drag. The most famous of these was the British Spitfire aircraft.
It is possible to get an elliptical or near-elliptical lift distribution in other ways with the right combination of wing taper, twist, and sweep or by varying the airfoil section used as you go out the span. Some of these combinations can give values of induced drag coefficient at or near a minimum while minimizing the difficulty of building the wing. In some cases a complicated planform is required for other reasons such as “stealth” or minimizing the radar return of the aircraft. The B-2, “stealth” bomber, despite its “saw-tooth” shaped wing planform, has a nearly elliptical lift distribution.
Often twist is part of this scheme because it may also be needed for control in stall. Because the control surfaces used for roll control, the ailerons, are near the wing tips we want to design the wing so the outboard portion of the wing will not stall when the inboard section begins to stall. For this reason the part of the wing near the tip is often twisted to give it a smaller angle of attack than the rest of the wing. Some airplanes use different airfoil sections near the tip than on the rest of the wing for this reason.
It should be stressed that while the elliptical lift distribution is ideal aerodynamically there are other factors which must be considered when designing the wing. For example, another lift distribution may well be the optimum one for structural strength or for control responsiveness. The British Spitfire was a very efficient airplane aerodynamically because of its elliptical wing planform but pilots found that they could not roll the fighter as quickly as their German opponents in dogfights. As a result the beautiful Spitfire wing was “clipped” in later versions to allow greater roll rates because in the real world of aerial warfare maneuverability was found to be more important than aerodynamic efficiency and drag.
It should also be stressed that induced drag is only a portion of the drag. This “drag due to lift” is independent of other sources of drag such as friction between the air and the “skin” or surface of the aircraft or the “pressure drag” which comes from the normal variation of pressures around the airfoil. These other types of drag must be calculated from aerodynamic theory and “boundary layer” theory, the subjects of two courses later in the curriculum.
Homework 1
1. Write a calculator or computer program to find standard atmosphere conditions (pressure, temperature and density) for any altitude in the troposphere and stratosphere in both SI and English units. Turn in a listing of the program and a print-out for conditions every 1,000 meters (SI units) and every 1000 feet (English units) up to 100,000 feet or 30,000 meters.
2. A compressed air tank is fitted with a window of 150 mm diameter. A U-tube manometer using mercury as its operating fluid is connected between the tank and the atmosphere and reads 1.80 meters. What is the total load acting on the bolts securing the window? The relative density of mercury is 13.6.
3. On a certain day the sea level pressure and temperature are 101,500 N/m2 and 25°C, respectively. The temperature is found to fall linearly with altitude to -55°C at 11,300 meters and be constant above that altitude.
4. An aircraft with no instrument errors and with an altimeter calibrated to ISA specifications has an altimeter reading of 5000 meters. What is the actual altitude of the aircraft? What altitude would the altimeter show when the plane lands at sea level?
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textbooks/eng/Aerospace_Engineering/Aerodynamics_and_Aircraft_Performance_(Marchman)/01%3A_Introduction_to_Aerodynamics.txt
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Introduction
In Chapter 1 we looked at the Standard Atmosphere, the environment in which aircrafts operate, at Bernoulli’s equation and its relationship to airplane (or more specifically, wing) aerodynamics, and at some basic parameters that influence the aerodynamic performance of an airplane. In this chapter we will look at the way that we account for airplane propulsion; i.e., jet or propeller engines. This means we will be looking at the factors that affect things like airplane thrust and power. We will also find that the same factors that explain thrust can also be used to account for some of the drag on an airplane.
In looking at thrust, power, and drag we are interested in how these may vary with airplane speed and with altitude. We must have a basic understanding of these dependencies if we are to eventually use these in determining the performance of an airplane.
Airplane engines are, of course, the subject of entire engineering courses dealing with things such as internal combustion engines and air-breathing jet engines. We want to confine ourselves to a very simple approach to understanding how propulsion works without going into any more of the details than are absolutely necessary. Fortunately, we will be able to do this.
2.1 Jet Engines
Jet engines come in a wide range of designs. Most are considered “turbine” engines because turbines are used to extract energy from the high speed exhaust flow to drive a “compressor” to compress the flow into the engine prior to fuel addition and combustion, but at very high speeds (hypersonic flow) it is possible to get compression through shock waves and a non-turbine jet engine called a “ramjet” is the result. However, we are going to restrict ourselves to subsonic, incompressible flight where turbines and compressors are always needed.
The most basic type of jet engine is called a “turbojet” and it consists basically of an inlet, followed by a compressor that increases the pressure (and lowers the speed) of the air before it enters the combustion chamber where fuel is added and ignited. After combustion a turbine extracts enough energy from the high energy (high speed) exhaust products to drive the compressor, and the flow then exits through the engine exhaust at high speed to provide thrust.
It turns out that a pure turbojet isn’t a very efficient way to make thrust. It creates thrust through a very high speed exhaust and this is both very noisy and very loss prone. The high speed exhaust jet essentially rips its way through the surrounding air and this violent interaction between exhaust and atmosphere results in a lot of friction-like losses and makes a lot of noise.
We will look at jet propulsion in terms of momentum changes (energy per unit time) with the difference between the momentum in the exhaust and the engine inlet accounting for the thrust and, at first glance, it will seem that any way we can get a change in momentum is just as good as any other way, but that is not the case.
Momentum is essentially the mass multiplied by speed (velocity). This means that there are two ways to get a momentum change. One is to take a small amount of mass and accelerate it to a very high speed as is done in a turbojet engine. Another is to take a large amount of mass and accelerate it by a lesser amount. As it turns out, the latter way is the most efficient way to get thrust. It is sort of like comparing the effects of a large ceiling fan rotating slowly with those of a little “personal” fan. If you made two propeller driven air boats of the type used in swamps, one boat with a small propeller and the other with a large one, you would find that the boat with the larger prop would need less power to move at a given speed than the one with the small prop.
Fan-jets rely on this principle to provide more efficient thrust to an airplane than turbo-jets. In a fan-jet, the engine turbine or turbines drive both the compressor that works on the air going into the combustion chamber and a large fan that adds momentum to a large mass of air going around the engine core without being used to burn fuel. This fan or “bypass” air then mixes with the higher speed core, combustion products to give a high momentum total engine exhaust that derives its momentum from the large mass of the bypass flow and the high speed of the core flow.
So, for a given amount of thrust we will need a given amount of momentum change of the air going through the engine between its entrance and exit. We will look at how this momentum change mathematically accounts for the thrust a little later. The point here is that the most efficient way to get this momentum change with minimal losses is to accelerate a large mass of air by a small amount (small change in speed). This means that the bigger the bypass ratio (the ratio of the bypass air mass to the mass of air going through the engine core) the more efficient the engine is. But there is a limit to this.
As the fan jet engine gets larger due to higher bypass ratio design, the engine enclosure (nacelle) also gets larger and it produces more drag. So at some point it makes more sense to replace the bypass “fan” with a large propeller. The result is the “turboprop” engine.
In the turboprop engine the flow through the engine core is really not used to produce any significant thrust. The exhaust turbine is designed to take all of the energy it can from the exhaust to drive the propeller and all of the engine’s thrust comes from the flow through the propeller. In a turboprop engine the amount of thrust that comes from the core flow is so negligible that, in some engine designs, the core flow actually goes “backwards”.
We might wonder, if the turboprop engine is more efficient than the fan-jet, which is in turn more efficient than the turbojet, why fan-jets are the engine of choice for most airplanes today? The answer is in the desired speed of flight.
Just as there is a big drag rise on a wing as it approaches the speed of sound, there are drag type losses on a propeller blade when its speed approaches Mach one. In fact, for a given propeller rotation speed, the limit on practical diameter for the prop is determined by the radius at which the propeller blade section reaches its critical Mach number. And, since the airspeed seen by the propeller blades is a function of both their rotational speed and the speed of the airplane, this limits the speed of the aircraft. Propeller design can extend this speed range somewhat with things like swept blade tips but the turboprop will always impose limits on aircraft cruise speeds.
Also, it turns out that the rotational speeds needed for a turboprop propeller are an order of magnitude below those of those in an efficient turbine core and this necessitates speed reduction gears between the turbine and the prop and this introduces both noise and vibrations that are not found in the fan-jet.
2.2 Propeller Engines
So what is the difference between a turboprop and a propeller driven by an internal combustion engine? From the point of view of the thrust provided by the propeller, there isn’t much difference. The difference is in the engine and the gearing that drives the propeller.
The turboprop is driven by a small turbine (jet) engine that sends as much of its energy as possible to the propeller through a driveshaft and reduction gear system. The IC engine propeller is attached to the driveshaft of an internal combustion engine that, like most automobile engines, uses the burning of gasoline or diesel fuel in a piston/cylinder type motor to turn the shaft.
Today most internal combustion driven propeller engines are found on smaller general aviation aircraft. This type of engine has provided reliable, affordable power for airplanes since the first flight of the Wright brothers in 1903. Over the years there have been many fascinating variations of IC engine used in airplanes, from the “rotary” engines of World War I in which the driveshaft was attached to the airplane and the propeller and engine actually rotated together around the shaft, to the massive piston engines of the 1940s and 1950s with dozens of cylinders arranged around the driveshaft like kernels on an ear of corn, to the four and six cylinder, car type but air cooled, engines usually found on today’s GA airplanes. These many varieties of IC engines would make an interesting and exhausting study in themselves, but that is beyond the scope of this text.
As far as we will be concerned, a propeller engine is a propeller engine, whether driven by a turbine or an IC engine or a rubber band. We will merely be concerned with the “power” output by the engine and we will call this the “shaft power” regardless of the type of engine that drives the shaft.
2.3 Thrust and Power
This brings us to the main difference in the way we will talk about propulsion for jet and prop engines. For jet powered aircraft, whether turbojets or fanjets, we will characterize the propulsion properties of the airplane in terms of thrust. For propeller powered airplanes, whether the propeller is attached to an IC engine or a turbine, we will talk about performance in terms of power.
Power and thrust are merely two different ways of looking at aircraft propulsion and performance. They are directly related to each other through speed.
Power = (Thrust)(Velocity)
While we normally talk about jet propulsion in terms of thrust and propeller propulsion in terms of power, there is little reason beyond convention that we must do so. We could talk about the power of a jet engine and the thrust of a propeller and we sometimes do so. Perhaps one reason for this distinction is that we will later find it convenient to look at the variation of both power and thrust with velocity and we will find that it is common to assume that thrust is fairly constant with speed for a jet and power is fairly constant with speed for a propeller driven plane.
The units normally associated with power and thrust, respectively, are pounds and horsepower. Yes, these are “politically incorrect” units; nonetheless, they are far more widely used than Newtons and Watts, their SI equivalents. [Have you ever heard anyone talk about the power of their car engine in watts?] This, of course, means we need to learn how the unit of horsepower relates to basic units in the “English” system.
1 horsepower = 550 foot-pounds / second.
[ A bit of engineering trivia: this conversion was used so often in the days of slide rule calculations that most slide rules had a special mark on them at the 550 location on the slide.]
2.4 Thrust and Conservation Laws
To find out how things like altitude and airspeed affect thrust and power we need to take a look at how the air goes through the propeller or the jet engine when an airplane is in flight and how the momentum of the air changes as it follows that path. To do this we will need to look at two “conservation laws”, conservation of mass and conservation of momentum.
2.4.1 Mass Conservation
In its simplest concept mass conservation is often stated something like “mass cannot be either created or destroyed; i.e., it is constant or conserved”. This is often accompanied by a qualifier noting that, in an atomic reaction, mass can actually be created (fusion reaction) or destroyed (fission reaction). This is an interesting way to look at mass if one is looking at the mass in the universe or in a closed container but it doesn’t help us when talking about engines. We need to look at the conservation of mass in a flow; that is, in the air going through a room or a pipe or a propeller or a jet engine.
If we had a sealed room filled with air it would be simple to state that the amount of air in the room is a constant. We could have people and plants in the room with the chemical reactions that are part of human breathing and plant chemistry continually altering the chemical constituencies in the “air”; nonetheless, the total mass of the “air” would remain constant.
The picture changes when we add ventilation to the room, either by using a forced ventilation system such as an air conditioning or heating system or by simply opening windows and doors. With either system there would be new air coming into the windows, doors, or intake vents and old air going out of other windows, doors, or exhaust vents. If we had a room with only a forced air inlet and no exhaust, the mass of air in the room would increase as air came in through the inlet. To accommodate this increasing mass the room would either have to expand like a balloon or the pressure and density of the air in the room would have to increase. Note that, if we assumed that the air was “incompressible” it would be impossible to pump new air into the room without providing an exhaust for an equal amount of air to escape. This would require conservation in the mass of the air in the room.
So in the example of room ventilation, conservation of mass for the air in the room would simply mean that, as a mass of new air enters the room, the same amount of mass of air must leave the room. Room or window air conditioners work this way, taking a given mass flow rate from the room, sending it through cooling coils, and returning that same mass flow rate to the room after some heat was removed from the air.
This brings us to the subject of mass flow rate, often called “m-dot” and given the symbol of a lower case “m” with a dot on top of the letter to represent a time derivative of the mass; i.e., mass per unit time, dm/dt.
When we speak of a room with vents or doors and windows we must talk about mass flow rates, and we say that in order to have mass conservation we must have no change in the mass within the room per unit time, simply another way of saying that the amount of mass that goes in during a given time period must equal the amount of mass that goes out in that same time. This is stated as:
dm/dt = 0 .
In other words, the amount of mass in the room does not change with time.
We often put this in equation form, saying that
dm/dt = Σ ρVA = 0.
Here, we are saying that the mass flow rate is equal to the density of the air, multiplied by its speed, as it passes through an area of size “A”. In other words, if air at sea level density is blowing through a window at a speed of 20 feet-per-second and if that window has an opening of 2 feet by 4 feet, we can calculate the mass of air per unit time that is passing through the window.
dm/dt = ρVA = (0.002378 sl/ft3)(20 ft/sec.)(2 ft x 4 ft) = 0.3805 sl/sec.
[Note here that the units of mass rate of flow have been found to be slugs per second. In the SI system they would be found in kilograms per second and in a version of the “English” system often used in fields such as Mechanical Engineering the units of mass flow rate would be pounds-mass per second.]
Now, if conservation of mass is met for the air in the room, the same mass of air per unit time must be going out of another opening or openings.
(dm/dt)in + (dm/dt)out = 0
or
(ρVA)in + (ρVA)out = 0
So, if there is a single window letting in the air flow found above and the exit is through a door, we can use conservation of mass to determine the speed of the air going out the door.
(ρVA)out = – (ρVA)in .
Just as three factors, the size (area) of the window, the speed of the air flow through the window, and the density of the air, determined the “mass flow rate” of the air coming into the room, the same three things determine the exit mass flow rate. In reality, all three of these things could be different at the exit (door, in this case), so, if we want to find the speed of the exiting air we must know both the area of the door and the density of the air at the door. However, there is no reason why the air flowing through the room would have changed density so we are safe in assuming “incompressible” flow, that is, density is constant. This gives us a simple equation:
(VA)out = – (VA)in .
So, if the door is 3 feet wide by 7 feet high, giving an area of 21 ft2, while the window had an area of 8 ft2, the speed of the air going out the door is:
Vout/Vin = – Ain/Aout
or
Vout = – Vin(Ain/Aout)
or in this case,
Vout = – 20 ft/sec. (8 ft2 / 21 ft2) = – 7.62 ft/sec.
Now, why is there a minus sign with the exit velocity? This is because we, for no real reason, chose to give a positive sense to the velocity going in the window and since velocity is a vector; i.e., it has a direction, we have designated the flow of air into the room as positive. This means that the negative sign on the exit air velocity tells us it is going out of the room. While this may seem like an un-needed complication here, there are cases where it can help us figure out what is happening.
For example, suppose there are five windows and two doors in our room and we are told that air is coming into all five windows at a certain speed and is going out one door at a given speed, what is happening at the other door? Is the flow through that second door going into or out of the room?
We would have to write the complete equation for mass flow conservation to find both the amount and direction of the flow through the second door.
(ρVA)w1 + (ρVA)w2 + (ρVA)w3 + (ρVA)w4 + (ρVA)w5 – (ρVA)D1 + (ρVA)D2 = 0 .
Note that we have assigned positive values to the flow through all the windows since we were told that the flow was coming into all of them. We have also assigned a negative value to the flow out the first door since the flow was said to be out of that door. Also note that we did not assign a sign (direction) to the flow through the second door because we have no idea which way it is going. Now, if we put all the needed information for the five windows and first door into the terms in the equation and if we know the area of the second door and assume that density is the same everywhere (incompressible flow), we can solve for the speed (velocity) of the flow using the mass conservation relationship above and find both the magnitude of the speed and its direction (sign).
Exercise 2.1
Try doing the above problem assuming that all five windows are 2 ft X 4 ft in size and that air is blowing in at 20 ft/sec.
Assume that the two doors are both 3 ft X 7 ft in size and that the flow out of the first door is measured at 50 ft/sec.
Find the speed and direction of the flow through the second door.
NOTE: Here we considered all flow INTO our “system” or “control volume” as POSITIVE, and all flow OUT of the system as NEGATIVE. If we do not know its direction, we assume it is positive in value and the solution of the equation will give us a negative answer if we assumed the wrong direction. Later, when we look at the Momentum Equation we will use a unit vector, n, to assign a positive direction within our chosen axis system for flow through an opening, and that unit vector will always point OUT of the system.
OK, that was simple enough, but how do we deal with mass conservation when we are looking at flow through a jet engine or a propeller?
Mass conservation through a propeller or a jet engine works just like mass conservation in a flow going through a room. In fact, for the jet engine it is even simpler than the average room because there is only one well defined entrance and exit, or is that really the case?
Technically, there is a second source of incoming mass in any jet engine and that is the mass flow of fuel coming into the engine. There is air coming into the engine inlet of a known area at (supposedly) a known speed and density, but the flow going out of the engine isn’t really just air, it is the gas that comes from combustion of the incoming air and the incoming fuel. The mass rate of flow coming out of the exit must account for both the mass of the entering air and the entering fuel, so our mass conservation relationship must recognize this.
(dm/dt)inlet + (dm/dt)fuel + (dm/dt)exhaust = 0
We would normally write this as:
(ρAV)inlet + (dm/dt)fuel = – (ρAV)exhaust .
So we must know the mass flow rate of the fuel. Usually the mass flow rate of the fuel is very small compared to that of the inlet air so perhaps that term can be neglected. So what’s the big deal? If we can neglect the fuel flow rate we are back to the one window, one door example and life is easy. Unfortunately there is another factor that we must not forget and that is density. Usually the flow through the exhaust of a jet engine is going pretty fast, near or greater than the speed of sound; i.e., we can no longer assume that density is constant as we did in the room ventilation example.
To solve this problem we have to know either the exit flow density or its speed in order to solve the equation for the “other” parameter (exit speed of density), and since the fuel mass flow contributes to this exit density we probably should not assume it to be negligible even if its velocity is almost negligible.
Making mass conservation for a jet even more complex is the fact that most of today’s jets are “fan jets” where there are essentially two entrance flows, one that goes through the engine core, mixing with the fuel to form a high speed exhaust, and another, larger, flow that is accelerated through the fan. We might analyze this problem by accounting for two separate entrance flows and two separate exit flows, or by assuming (correctly in most cases) that the two exit flows mix before leaving the engine covering or “nacelle” to form a single, mixed exhaust.
In any case, the jet engine flow problem is a little simpler for many people to understand than the propeller flow problem because the entrance and exit areas are normally pretty well defined. How do we define entrance and exit flows when we draw the flow through a propeller?
When a flow is going through a propeller, just what are the entrance and exit areas? There really is no physical entrance or exit. Of course, we know the flow goes through the propeller itself, so, is the propeller area used for both the flow “entrance” and its “exit”? This hardly makes sense. How can we talk about the changes in the flow between the entrance and exit when there is no physical distance between the entrance and exit?
Let’s look at what we know intuitively about the flow through a propeller (or a fan). We know that the flow behind the propeller or fan is moving faster than the flow in front of it. We know that in some way, a way that can be analyzed in detail by looking at each propeller or fan blade as a little rotating wing that does work on the air, the propeller essentially adds energy to the flow. We also know, if we think about it a bit, that we cannot use Bernoulli’s equation to compare the flow upstream and downstream of the prop or fan because energy is added at the prop or fan and Bernoulli’s equation assumes that energy is constant through the flow. We also know that there are limits to what a fan or propeller can do to accelerate a flow due to tip speed limits on the blades themselves and these limits essentially mean that we can pretty safely assume incompressible flow through the system.
Putting all these facts together, we can draw a picture that looks something like the flow should appear through a propeller or fan. We know that somewhere upstream of the propeller the flow is undisturbed; i.e., it is at “free-stream” or atmospheric conditions. We know that somewhere downstream of the prop the static pressure in the mass of air that went through the propeller must return to its free-stream value.
We will imagine a “stream-tube”, or three-dimensional path of constant mass flow, that starts out in the undisturbed flow upstream of the prop, goes through the prop (becoming the same diameter as the prop at that location, and then continues downstream until the point we mentioned above where the static pressure has returned to the atmospheric value. What must that “stream-tube” look like?
A stream-tube is defined as a three-dimensional flow path in which the mass flow rate is the same at every point along its journey. Essentially, as shown in the following figure, the upstream cross sectional area of the stream-tube (its “capture” area) must have the same amount of mass flow rate through it as goes through the prop itself. Likewise, the “exit” area for our stream-tube must also allow passage of the same mass flow as went through the capture area and the prop “disk” area.
So why is the “stream-tube” in the figure above getting progressively smaller as the flow goes from the atmospheric pressure, free-stream capture area to the atmospheric pressure exit area somewhere downstream? First, we know the velocity in the exit area must be larger than in the capture (inlet) area; hence, if mass flow rate is the same and the flow is incompressible, the area must decrease in inverse proportion to the speed increases. But why do we assume that this area decrease (and speed increase) is smooth and continuous? Isn’t there simply a big jump in speed across the propeller disk?
Well, we probably could analyze everything in terms of some kind of instantaneous jump in flow speed at the propeller disk based on an energy balance, assuming that the energy added by the prop produces a sudden increase in flow kinetic energy and speed. However, we know from real world measurements that this speed increase is not instantaneous and that part of the increase is seen in front of the propeller as the flow speeds up from its “free-stream” velocity to the velocity right at the front of the prop disk. We also know that it takes a couple of propeller diameters downstream before the flow in the “propwash” reaches top speed. Based on this combination of reality and convenience, we choose to model the speed increase as a continuous one within a “stream-tube” shaped like a converging nozzle of circular cross section, as shown in the figure above.
This ideal picture, of course, ignores a lot of things such as the losses due to turbulence and rotational flow effects; nonetheless, it is one that works fairly well. So, what do we propose to do with this model and with the model of the flow through a jet engine? What we want to do is use these to determine how thrust is produced and find the properties that determine how thrust varies with speed and altitude.
2.4.2 Thrust
Our goal is to take a look at propulsion. How do we account for thrust or power in aircraft performance evaluations?
There are two ways to do this. One would look at energy additions to the flow and a conservation of energy. But, as noted in the propeller discussion above, this would be very tedious, requiring us to do aerodynamic analyses of each propeller blade, accounting for losses due to compressibility effects near the blade tips and for the interference between the flow over one blade and the following blade. There are books on how to do this, the oldest of which went under titles such as “Airscrew Theory”, and this is the type of analysis that companies making propellers must use. The problem would be even more interesting in a jet engine with us having to account for energy gains and losses due to flow around compressor blades and turbine blades, combustion of fuel, and flow though internal nozzles.
It turns out that the simplest way to look at thrust is to look at momentum conservation.
2.4.3 Momentum conservation
Momentum conservation, like mass conservation and energy conservation, is one of the “big three” conservation “laws” that we all saw somewhere back in some Physics course. On the face of it, conservation of momentum is a simple concept. Just as in mass conservation of a flow we must account for all mass flows that enter or leave the flow-field under consideration, in looking at momentum conservation we must consider all things that could possibly account for momentum changes and, ultimately, in forces.
Essentially, the concept we are looking at is one that says that the change of momentum in a body or “system” with time must equal the forces on that body or system. The idea is that either forces on a body or system will cause its momentum to change or a momentum change within the system or body will result in a force.
(d/dt)(momentum) = (d/dt)(mv) = Force
This is a simple idea that is often made to look very complicated when derived in most textbooks on fluid mechanics. If, for example, you kick a soccer ball, the force you impart to the ball will result in a change in momentum in the ball. If the ball was standing still before it was kicked, the force will change its momentum from zero to a value related to the force of the impact and the mass of the ball. If the ball was already moving, the kick may send moving in another direction, so this concept is directional; i.e., it is a vector concept, as would be expected when a force is involved.
In looking at aircraft propulsion we are interested in the reverse action; that is, creating a momentum change in order to get a force, changing the momentum of the flow through the engine or propeller to create thrust.
Just as in working with Bernoulli’s equation we had a choice of modeling the flow as a moving fluid going past a wing or body, or as a body moving through still air, we have to make a similar choice here. We will, for example, choose to look at the flow through a jet engine or a propeller as if the engine (prop) is standing still and the flow is moving past it. This is really a choice between having to consider the momentum of the moving engine or the momentum of the moving air. Either view will give the same answer for the thrust, but the moving air model is usually a little easier to work with. Either way, we must be very careful to account for all possible momentum changes in both the engine and the flow.
We first need to look at what kinds of momentum changes might be present as well as what kinds of forces might be involved. To do this, let’s look at one of the simplest of “jet” engines, but one of the hardest to analyze, a rubber balloon that is inflated and released.
Let’s look at the illustration above and list all of the ways that momentum might play a role as well as all the forces involved. There will be at least two sources of change of momentum for the balloon and at least three forces that might be involved.
Momentum change sources:
1. The change in momentum of the balloon (the “system”) with time because of the change in mass of air inside the balloon with time and due to any changes in velocity of that mass. [As the balloon expels air through its inlet/outlet, the mass of the “system” itself is changing and, even if its speed was constant, the momentum of the system would change.]
2. The momentum of the flow exiting the “system” (balloon); i.e., the mass flow of air through the inlet/outlet (jet) multiplied by its velocity.
Both of these terms above are directional because of the velocities associated with them. The momentum of the balloon itself is related to the balloon’s velocity and the momentum of the flow through the exit is obviously related to the direction of the flow through the exit.
Forces on the balloon:
1. The major force on the balloon will be the one we choose to call thrust. This is essentially what we are trying to find.
2. nother force on the balloon that we might not think of at first is that due to gravity; i.e., its weight.
3. Finally, there would be any pressure forces caused by pressures acting on areas. These might include pressure drag on the balloon itself or differences in pressure across system boundaries. Often we find that pressure forces tend to balance out or sum to zero but there are some cases where these must be considered.
4. We could also consider friction forces or even electromagnetic or other forces if we wished but we will limit ourselves to the first three forces mentioned above.
How do we describe each of these sources of momentum change or forces in a very general way? Let’s look at each of these listed above.
1. The change in momentum of the “system” with time involves the changes in both mass and velocity of the system:
d/dt [(mass)(velocity)] ,
and, since the system mass can be written as its density times its volume, we might look at this as
d/dt [(density)(volume)(velocity)]
2. The change in momentum due to the flow out of (and in general) into the system with time is essentially the mass rate of flow (dm/dt) across any entrances or exits multiplied by the speed at which that mass is passing through the entrance or exit areas. We know that the mass rate of flow is the density multiplied by both the velocity and the flow cross sectional area, so this term is expressed as:
(dm/dt)(velocity) = (density)(velocity)(area)x(velocity) .
3. The weight is just the mass (density x volume) multiplied by the acceleration of gravity.
(density)(volume)(g) .
4. The pressure forces are just pressures acting on an area:
(pressure)(area)
Now, to work with all these we need to put them together in the form of some kind of equation. The equation must essentially say that the momentum changes must be balanced by the forces involved. This can be thought of as forces causing momentum change (the soccer player’s foot kicks the ball) or momentum changes causing forces (the thrust from a released balloon). The equation that usually results from a much more formal derivation is a complicated looking, vector relationship called the momentum equation.
2.4.4 The Momentum Equation
Before you panic at the vector notation and the double and triple integrals, take a deep breath and see how these terms relate to the ones presented above.
A triple integral over “R” (the mathematical “region” or the “system”) is nothing but the volume. If the density and velocity of everything contained in the region or system is the same; i.e., if it is a homogeneous system, then this term is nothing but the time derivative of the density times the volume times the velocity; i.e., of the system mass times its speed as it was stated in the section above.
So why do we make it so complicated looking? One reason might be just to impress our friends in liberal arts or to show our parents how hard our courses are. A better reason is to allow the momentum equation to account for non-homogenous system effects. Suppose, for example, that our “system” was not a balloon filled with nice homogenous air, but a baseball or golf ball with a solid filling made of several layers, each with different densities, and further, that someone had made the ball with its heavier core somewhat “off center”. You can buy such “trick” golf balls at novelty shops and when you hit them with a golf club (impart a force to the system!), instead of traveling in a straight line they wobble around as if they were drunk. Because the momentum equation can account for this “non-homogeneity” it can account for the wobbly motion of the trick golf ball. In a similar way the last term on the right, the gravity or weight term, can account for gravitational effects on a non-homogeneous mass.
Two of the terms in the equation have double integrals. You might have guessed by now that the double integral over a distance “S” must relate to some kind of area, and looking at the terms would confirm that. The double integral term on the left relates to the momentum carried with a flow into or out of the system over an entrance or exit area. This term is written in this complex way to be able to account for non-uniform velocities over the entrance or exit and even for non-uniform densities over these areas. If we assume that all of the entrance or exit flow is the same fluid moving at the same speed then the density and velocity terms can come outside the integral and the integral itself becomes nothing but the entrance or exit area. So, again, why make it look this complicated? Well, in many cases the flow out of an opening is not uniform because friction forces cause it to move more slowly near the edges of the opening than at the center, and this comprehensive form of the momentum equation can account for that if we want it to do so. Similarly, the pressure term on the right hand side of the equation can account for pressure variation over a surface.
What about the vector notation, the Vn term, in the double integral term on the left? First, the momentum equation is a vector equation, meaning that each of the terms has a direction and the solution of the equation for a force such as thrust or drag will give both a magnitude and a direction for that force. Second, for one of the terms on each side of the equation, it is only the parameter “normal” to a defined surface or boundary that will cause a force and the “unit vector” n is used to designate that normal direction. We will always define the direction for this unit vector as pointing out of the system, even where the flow is coming into the system.
What then do these vector quantities mean? Each of the velocities can have up to three terms in them, one associated with each direction in a selected axis system. In the case of velocity in a conventional x, y, z axis system, we normally use the terms u, v, and w to designate the x, y, and z components of velocity, respectively. So we would write a velocity vector as:
V = ui + vj + wk ,
where i, j, and k are the unit vectors in the positive x, y, and z directions. In a similar manner, the gravity vector could have up to three components; however, we sometimes try to define our coordinate system so one axis is in the direction of gravitational acceleration to eliminate two of these components.
The Vn term is then the “dot product” of two vectors where both the V and the n vectors may have x, y, and z components, but only the like directed components multiply with each other, then sum to give a “scalar” quantity with a magnitude but no direction. So, if the velocity is in the same direction as the normal vector (as is often the case for flows into or out of a system) the result is simply the magnitude of the velocity. At the other extreme, if the velocity is at a 90 degree angle to the normal vector the dot product gives zero.
Again one might ask, why make things so complicated with all these integrals and vectors and dot products and the like? It is done this way because it is a very versatile equation that can account for fully three dimensional motion. For example, should that soccer player kick the ball at a 90 degree angle to its existing direction of motion, this relationship would, provided we knew the force of the kick and the mass and velocity of the ball, tell us the ball’s new direction and speed even though the direction would be in neither the original direction of motion or in the direction of the kick. Similarly, if there is a bend in a pipe we can use the equation to find the magnitude and direction of the force that will occur when water flows through that bend in the pipe.
The trick to using the momentum equation is to follow the rule of thumb that often distinguishes an engineer from a pure scientist or mathematician; that is to use proper alignment of axis systems and to set system boundaries and to make good assumptions that will eliminate as much of the complexity as possible. Fortunately we can do a lot of this as we use the momentum equation to look at thrust.
2.4.5 Thrust (again)
Let’s look at the flow through a jet engine in terms of the momentum equation.
In the illustration above we have aligned the engine with the “x” axis and we have flow coming into the engine inlet in the x direction and another flow coming out of the engine, also in the x direction. We want to know the thrust as a function of this information. Let’s look at what we can say about the various terms in the momentum equation.
The first term on the left hand side of the equation is a “time dependent” term to account for changes in momentum of the “system” itself with time. Here our system is the entire jet engine, and, if we assume that the engine (airplane) is in “steady” or constant speed flight, there is nothing in the term (density, velocity, or volume) that is changing with time. So, this term is zero.
The second term on the left accounts for the momentum carried into or out of the system as flow enters or leaves. Obviously, this term will not go away since we have air coming into the engine and combustion products going out the other end. First we need to ask if these two flows are “uniform” across their respective entrance or exit areas. If we can assume that they are uniform and can assume that all of the flow has the same density, then this term (actually two terms, one for the entrance and one for the exit) becomes:
ρ1V1(V1•n1)A1 + ρ2V2(V2•n2)A2 .
Now, what do we do with the vector business? The flows are both in the positive x direction. The first normal unit vector is in the negative x direction while the second is in the positive x direction. The result is:
ρ1V12A1 + ρ2V22A2 , (all in the x direction)
Ok, that takes care of the left hand side of the momentum equation. What happens to the terms on the right? The first term on the right is the “external” force which, in this case, is the thrust we want to find. The second term on the right is perhaps the hardest to understand physically so we will come back to this.
The third term on the right is the gravity term, really the weight. If we assume that this is acting at a 90 degree angle to the x axis or the direction of flight and thus is perpendicular to all the other forces and momentum changes in which we have an interest we might simply neglect this term. Actually it would be more proper to say that its component in the x direction is zero. In reality, this term would tell us that there must be a force to oppose the weight and this would be the aerodynamic lift which, in turn, would be related in the momentum equation to a vertical change in momentum of the flow as it moved around the wing and the corresponding pressure distribution around the wing. In essence we are choosing to ignore the vertical components of the forces and momentum changes.
Now let’s go back to the second term on the right, the only term with pressures in it. This term looks at forces caused by pressures acting on areas. If we were looking at the lift force we would use this term to integrate the pressure distribution around a wing. On the engine we will assume that the flows over the outside of the engine casing or nacelle are symmetrical, that is that the same pressure distribution exists on the top as on the bottom of the nacelle, and that the net effect of these pressures (at least in the x direction) is zero. But what about pressures across the entrance and exit?
Pressures across the entrance and exit?!! How can this mean anything when there are no real surfaces here, just flows going in or out? This is where the concept of a “system” boundary gets interesting. When there is a real boundary such as the engine nacelle the bounds of the system are easy to understand. But these “open” ends of the “system” are also boundaries over which we must account for all the terms in the equation. In other words, just as we had to account for the flow through these somewhat imaginary boundaries, we must also account for pressure changes across them. But how can these pressures cause real forces when there are no “real” surfaces for them to act on? This becomes one of those “leaps of faith” that we often must take in applying equations to physical situations.
No, there are no surfaces at the entrance and exit where the pressure differences across the surface cause a force; however, we must account for them anyway if there is a pressure differential between the surrounding atmosphere and the flow into the entrance and out of the exit. This is probably the easiest to understand when we look at the exhaust flow.
Coming out of the exhaust is a flow of the combustion products of air and fuel that has been heated and pressurized in the engine combustor. After combustion we want to turn that added energy into as high a momentum (the second term on the left hand side of the momentum equation) as possible. This means that we want to “expand” the gas in a exit nozzle, lowering its pressure with a corresponding increase in speed (ala Bernoulli’s equation) as much as possible to get a high momentum. The ideal situation is to expand it to the point where the exiting gas has the same pressure as the atmosphere into which it will exit. If it expands too much or too little there will be losses as the flow pressure comes to equilibrium with the atmosphere. It turns out (and the momentum equation essentially tells us this) that the losses from over or under-expansion are equivalent to the pressure force that would be on a surface with the same area as the exit with a pressure difference equal to that under or over-expansion delta-P. This is why some high performance jets have variable area exit nozzles on their engines.
The same problem can occur to a lesser degree at the engine inlet but a properly designed engine inlet and compressor section can eliminate most of the loss.
So, how do we deal with this pressure term? We either must know the differences between the atmospheric pressure and those of the entrance and exit flows and compute values for these terms, being careful to account properly for the unit vector signs, or we must assume that these losses are negligible. Let us take the easy way out and assume that these terms are of little consequence because we have a properly designed engine.
OK, where does this leave us? We have ended up with a relatively simple equation:
ρ1V12A1 + ρ2V22A2 = – Fe
rearranging this gives:
Thrust = Fe = ρ1V12A1 ρ2V22A2 .
Looking at this we see that the second term on the right will be much greater than the first term, so, the thrust will have a negative sign. Is this ok? Sure it is. It just says the thrust force is in the negative x direction, toward the left, just as we want it to be.
Wow! That sure was a lot of work to get a fairly obvious answer; the thrust is equal to the momentum change from engine inlet to exit! Isn’t this somewhat intuitive? Yes, it sortof is intuitive to many of us. On the other handit does keep the mathematicians and theoreticians in our midst happy, and more importantly, it tells us that in arriving at this “intuitive” answer we have made some important assumptionsabout pressure behavior and axis system selection, etc.
Ok, now that we have all that under our belts what important facts about propulsion can be drawn from this solution? To see this, let’s play around with the equation above a little by accounting for conservation of mass.
Now, recognizing that V1 is our “free stream” speed, V, and that the entering air density is also that of the atmosphere, ρ, we can write this as
Thrust = ρV2A1 ρ2V22A2
And looking only at the magnitude of the thrust (as said above, the relationship above gives a negative thrust, signifying simply that it is to the left in our original illustration of the engine moving from right to left)
Thrust = ρ2V22A2 ρV2A1
We now define the “static thrust” as T0, the thrust when the engine is standing still (V = 0). This is the amount of thrust that would be measured on an engine test stand and is a standard piece of information that would exist for any engine.
T0 = ρ2V22A2
This allows us to rewrite the general thrust relationship as:
Thrust = T0 ρV2A1 ,
or simply as:
T = T0 – a V2 ,
where:
a = ρA1
What does all this tell us? First, all the thrust equations tell us that thrust is a function of the atmospheric density. Unlike velocity, which we earlier found to vary with the square root of density, thrust decreases in direct proportion to the decrease of density in the atmosphere. Thus, we write:
Talt = Tsl(ρalt/ρSL)
This is an important relationship between thrust and altitude that we will use in all performance calculations.
Second, we learn that, in general, the thrust of an engine varies with speed according to the relationship:
T = T0 – a V2
It should be noted, as always, that these equations involve important assumptions, such as the assumption that engine exit pressure and entrance pressure are both equal to the pressure in the free stream atmosphere. Pilots of jet aircraft will tell you that the thrust to static thrust relationship shown above doesn’t, for example, account for an engine surge on initial acceleration down the runway as a “ram effect” into the engine inlet occurs. This “ram effect” is essentially one of these pressure effects that we chose to ignore.
2.4.6 Propeller Thrust
It should be noted that we would get essentially the exact same thrust equation looking at the flow through a propeller as we do with a jet engine. Keeping in mind an earlier discussion, we would draw our “system” as shown below using boundaries that represent a “stream tube” of constant mass flow. In this case we have no easy way of knowing the exact values for the entering flow area or the exit area but we would get exactly the same equation as we found for the jet and we would still find
T = T0 – aV2.
In this chapter we have looked at the relatively simple models of aircraft propulsion that we will use in examining aircraft performance. In doing this we have used some basic physical concepts of conservation (mass and momentum), both of which can provide very powerful tools for evaluating forces and motions in fluid flows and other areas. We made a lot of simplifying assumptions along the way in order to understand some very basic concepts related to jet and propeller propulsion; in particular, to give a basis for modeling the way both thrust and power vary with speed and altitude. We will find these concepts very useful in later chapters.
Homework 2
1. In a wind tunnel the speed changes as the cross sectional area of the tunnel changes. If the speed in a 6′ x 6′ square test section in 100 mph, what was the speed upstream of the test section where the tunnel measures 20′ x 20′? Use conservation of mass and assume incompressible flow. Conservation of mass requires that as the flow moves through a path or a duct the product of the density, velocity and cross sectional area must remain constant; i.e. that ρVA = constant.
2. A model is being tested in a wind tunnel at a speed of 100 mph.
(a) If the flow in the test section is at sea level standard conditions, what is the pressure at the model’s stagnation point?
(b) The tunnel speed is being measured by a pitot-static tube connected to a U-tube manometer. What is the reading on that manometer in inches of water.
(c) At one point on the model a pressure of 2058 psf is measured. What is the local airspeed at that point?
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textbooks/eng/Aerospace_Engineering/Aerodynamics_and_Aircraft_Performance_(Marchman)/02%3A_Propulsion.txt
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Chapter 3. Additional Aerodynamics Tools
Introduction
All of the introductory aerodynamic concepts needed for the aircraft performance material to be covered in the following chapters were presented in Chapter 1. In general, it will be left up to the readers’ choice of many excellent texts on the subject of aerodynamics to provide in-depth coverage of the field, including complete derivations of aerodynamic theories and discussions of their usage. The field of aerodynamics is often separated into five or more segments based on the flow characteristics and the assumptions that can or must be made to analyze those flows in detail. These divisions would normally include:
• Incompressible or subsonic aerodynamics
• Transonic aerodynamics
• Supersonic aerodynamics
• Hypersonic aerodynamics
• Boundary layer theory
Other, even more specialized, segments of the aerodynamics field might include such topics as rarefied gas dynamics and magneto-hydrodynamics.
In chapter one we looked at a few basic concepts relevant to the first three topics above with an emphasis on the incompressible flow regime and, hopefully, enough discussion of the assumptions involved for the reader to recognize when he or she is in danger of the need to account for transonic or supersonic flow effects. We looked at some of the basic conclusions that come from an analysis of two and three dimensional flow around airfoils and wings. We learned, for example, that the camber of an airfoil will determine the angle of attack at which the airfoil lift coefficient is zero and that we can temporarily change camber with things like wing warping or its modern equivalent, “morphing”, or, in a more conventional manner with flaps. We might be curious as to how much of a change camber can give in the zero lift angle of attack.
We learned that there is a certain type of spanwise lift distribution on a three dimensional wing that will give “optimum” aerodynamic performance by giving “minimum induced drag” and we found that higher aspect ratio wing planforms also give better performance than wings with low AR.
In this chapter we are going to take a very elementary look at these two fundamental wing and airfoil configuration effects, just enough of a look so we will have at least one or two basic tools that might help us find out something about the effects of wing design on aircraft performance should we need to do so. We will do this, not through the type of thorough analysis that would be found in most good aerodynamic textbooks, but with a couple of somewhat over simplified approaches that are, nonetheless, often useful.
In order to develop the desired “back of the envelope” methods of looking at some basic influences of airfoil and wing shape on aerodynamics and performance we need to first take a quick look at how an aerodynamicist would make a mathematical model of a wing or airfoil. Let’s begin by looking at the flow around a lifting airfoil.
3.1 Airfoils (2-D Aerodynamics)
For an airfoil or wing to product lift the flow over its upper surface must move faster than the flow over its lower surface. If this occurs, Bernoulli’s equation would tell us that the faster flow over the upper surface will give a lower pressure than the slower flow over the lower surface and this pressure differential will produce lift. If we look at the flow at a point some distance behind the leading edge of an airfoil we will find that we could represent it somewhat as shown in the figure below with a large velocity vector on top of the airfoil and a smaller one on the bottom.
Another way to represent this same flow would be with a combination of a “uniform” flow and a circular flow, such that the velocities add on top of the wing and subtract on the bottom as shown below.
If these two flows can be said to end up giving the same result, the flow shown in Figure 3.2 can be said to be a way to model the flow around an airfoil. This, in fact, works very well and is the basic idea behind the way an aerodynamicist would model the flow around an airfoil.
We could look at the upper and lower surface velocities at several points along the chord and find that at any point we can model that local flow by a “uniform” flow and a circular type flow as shown in Figure 3.3.
In this model, the “uniform” flows on the upper and lower surface would be exactly the same as the upstream or “freestream” velocity, V. The only thing that would change at each position along the chord line would be the circular type flow, which would get more powerful where the speed differential between upper and lower surfaces gets larger and less as it gets smaller.
These “circular” flows are known as vortices (a single circular flow is a vortex) and they are the mathematical equivalent of a little tornado. It turns out that vortices are very important flows in aerodynamics and they occur physically in many places. Of course, there are no vortices in the middle of a wing. Here, we are just using a real physical flow, a vortex, to create a mathematical model that gives us the result that we know actually exists.
These purely circular flows or vortices can be described mathematically in terms of their circulation. Circulation is technically related to the integration of a velocity around a closed path, a concept that we need not go into here and it is given a symbol gamma [γ or Γ]. The lower case form of the Greek letter, γ, is used when we are combining the effects of a lot of small vortices and the upper case form, Γ, is used either when there is only a single vortex or when we are looking at the combined effect of a lot of small vortices. The circulation, gamma, is essentially the measurement of the rotational speed of the spinning flow in a vortex and it is sometimes called the strength of a vortex.
So, the basic concept here is that to get a lifting flow (higher speed on the top and lower speed on the bottom) we combine a vortex flow with a uniform flow. If we do this and mathematically analyze the result we will find a very important and very simple relationship between the lift produced and the circulation and free stream speed that says:
Lift = ρVΓ
The lift is equal to the flow density times the free stream velocity times the circulation. You can find a well done derivation of this basic principle in any good aerodynamics textbook and there you will also find that this concept is so important that it is given a name, the Kutta-Joukowski Theorem.
As always, we should look at the units involved with this parameter, the circulation. We should keep in mind that we looked at this concept of a circular type flow as a way to look at lift on a two-dimensional slice of a wing, an airfoil section. So the lift we are talking about is the lift per unit span of the wing; i.e., the lift per foot or lift per meter. Knowing that, we can see what the units of gamma must be.
Lift (pounds per foot) = density (slugs per foot cubed) x velocity (feet per second) x gamma
Equating the units in this relationship tells us that the units for gamma must be:
(lb/ft) ÷ [(sl/ft3)(ft/sec)] = (lb ft sec)/sl
and knowing that F = ma; i.e., 1 lb = 1 sl ft/sec2 , we will find the units for gamma to be
units for Γ = ft2 / sec .
We may find these units a little strange. They are neither units for speed or acceleration. However, that’s ok, because circulation is not speed or acceleration, it is circulation!
So, what do we do with this? If this were a rigorous aerodynamics course or text we would take a few thousand of these little tornados (vortices) and lay them side by side along the chord or camber line of our airfoil section and do a complicated calculation to see exactly what value of gamma each must have to give the correct lift for an airfoil of a given shape at a given angle of attack to the free stream flow. We would base this on the shape we wanted our airfoil camber line to have. Finally, we would add up or “integrate” the combined circulations (gammas) for all our little vortices and find the total lift on the airfoil.
We would also look at the “distribution” of those circulations or vortex strengths and find what we could call their “centroid” of vorticity; i.e., the place on the airfoil chord where the total lift could be said to act if all the little vortices could be replaced with a single big vortex. We would call this place the “center of lift”.
If we did all this (and, again, we can find this derivation in its full glory in any good aerodynamics text), we will find a few interesting and very useful results that we will list below:
For a “symmetrical” airfoil (no camber)
• The “center of lift” is at the “quarter chord”; i.e., one fourth of the distance along the chord line from the leading edge.
• The two dimensional lift coefficient will be CL = 2πα where α is the angle of attack (angle between the chord line and the free stream velocity vector).
For a cambered airfoil (non-symmetrical)
• The “center of lift” is not at the quarter chord and it, in fact, moves as angle of attack changes.
• The two dimensional lift coefficient will now be CL = 2π(α – αL0), where αL0 is called the “zero lift angle of attack” and is a negative angle for a positively cambered airfoil. This means that the lift curve shifts to the left as camber increases and that, at a given angle of attack, the cambered airfoil will produce a larger lift coefficient than the symmetrical airfoil (provided the airfoil is at an angle of attack below that for stall).
We talked about these same results in Chapter One without saying much about their origins. This shift in lift curve and increase in lift coefficient with increasing camber is the basis for the use of flaps as a temporary way to increase the lift coefficient when a boost in lifting capability is needed such as on landing.
The aerodynamic theory that would predict all this is called “thin airfoil theory” and it does this precisely as described above, by assuming that thousands of tiny vortices are laid side by side in what is called a vortex sheet along the airfoil’s camber line. Knowing the mathematical description of the shape of the desired camber line, the free stream velocity and the angle of attack, and requiring that there be no flow through the camber line and that the flow does not go from one surface to the other around the trailing edge, thin airfoil theory can tell the needed distribution of circulation along the camber line from leading edge to trailing edge to give a simulation of the real flow around the airfoil.
This method can’t predict stall. To do that we would need to consider the effects of shear or viscosity in the flow and this would require us to look at the “boundary layer” or viscous flow region around the actual wing surface. This too, is far beyond the scope of the material we want to investigate in this text.
So, is there a simple way to predict the effects of camber without resorting to thin airfoil theory of some equally messy procedure? It turns out that there is a “back of the envelope” method called “Weissinger’s Approximation”
3.2 Weissinger’s Approximation
Weissenger’s Approximation is based on the symmetrical airfoil results of thin airfoil theory that were listed in a section above. These results say that for a symmetrical airfoil (essentially a flat plate) the lift acts at the quarter chord and the lift coefficient is 2-pi times the angle of attack.
We also have the Kutta-Joukowski Theorem which says that lift is equal to the flow density multiplied by the circulation and the freestream velocity.
We can, therefore, combine these two results by saying that we can model the lift on a flat plate by placing a single vortex at the quarter chord of the flat plate (since this is where theory says the net lift acts). All of this gives us the picture shown below.
Basically, we are going to combine these two ideas that describe lift by selecting a place where we would impose a condition of no flow through the flat plate that will equate the lift from the two theoretical results. To do this we need to know something about a vortex that we have not previously introduced. That is, the velocity found in or introduced by a vortex at various distances from its center. Aerodynamic theory would tell us that the circular or tangential velocity in a vortex varies inversely with its radius and is a function of the circulation, Γ, in the vortex. Theoretically, a vortex has an infinite circumferential velocity at its center and this velocity gets smaller as we move further away from the center. This will give
Vvortex = Γ / (2πr)
In using this definition of the circumferential velocity in a vortex it is conventional in the field of aerodynamics to define the positive direction for Γ as clockwise. This defies conventional mathematical practice and care has to be taken in being consistent in using the convention.
This is illustrated below.
Going back to our combination of thin airfoil theory results and the Kutta-Joukowski Theorem we have the following task. We want to find some point on the flat plate where the vortex that we have placed at the quarter chord will give just the right amount of velocity to counteract the component of the freestream velocity normal to the plate such that our two looks at lift or lift coefficient will give the same answers. One theory, the Kutta-Joukowski Theorem tells us that L = ρVΓ and the other tells us that the lift coefficient CL = 2πα.
Realizing that the lift on a two dimensional flat plate is equal to the lift coefficient times the dynamic pressure, multiplied by the length of the plate (a “one dimensional area”), we can write:
L = ρVΓ = (2πα)(½ρV2)(c) ,
where “c” is the chord (length) of the flat plate.
It is easily seen that what is being sought here is a relationship between the angle of attack and the circulation in the vortex. We are relating the physical reality of lift increasing with angle of attack to the mathematical model that says lift increases with circulation.
Γ = παVc .
It is this relationship that we use to create our “back of the envelope model” called Weissinger’s Approximation. To see this we need to look again at our flat plate with the vortex at the quarter chord and ask ourselves at what radius from the vortex will the velocity from the vortex be just enough to balance the normal component of the freestream velocity.
If we look at this illustration and use the last equation above to define the circulation, Γ, in terms of the freestream velocity and the angle of attack and then equate the velocity from the vortex and the normal component of the freestream velocity, we find
Vθ = Γ / (2πr) = (παVc) / (2πr) = V∞n = V sin(α) ≈ V α
or,
(αVc) / (2r) = V α , for small angles of attack.
The final outcome of this is that the distance, r, at which we must solve for “no flow through the flat plate” to make the two theoretical models compatible is:
r = c/2 .
We must solve for no flow through the plate at a point three-fourths the way back (at the three-quarter chord point) to make this “approximation” work. We call this point the “control point”.
OK, so what’sthe big deal? We’ve found a new way to get a result we already knowand it is only for a flat plate! How can this “approximation” tell us anything we don’talready know?
The reason this is so useful is that we can “build” approximate models of cambered or flapped airfoils from flat plates. Consider the simple case of a symmetrical airfoil with a plain flap taking up the final 20% of its chord. We can model this as two sequential flat plates with one flat plate of length of 80% of the airfoil chord and the other 20% of the chord and we can deflect the “flap” and see what it does to the lift coefficient. The way this is done is illustrated in the figure below:
In the above case we will end up with two equations and two unknowns, the two values of circulation. To get these two equations we look at the flows normal to the two plates at their respective “control points” that have been placed at the ¾ chord points on each plate or panel. Lets look first at the control point on the first panel.
There will be three velocities that must be accounted for at this point; the component of the freestream velocity normal to the panel, the velocity induced by the vortex on the first panel, and the velocity induced by the vortex on the second panel. We can see in the figure below that the freestream velocity is directed upward while the velocities from the two vortices will be in opposite directions (that due to the first vortex is “down” and the velocity from the second vortex is “up”, if we assume the vortices are “positive” or clockwise). We can account for these directions by either mental bookkeeping and noting that they all must add to zero, or by conscientiously accounting for signs on the vector quantities, noting that the radii (r11 and r21) have signs related to their direction.
Note that the velocity induced at control point 1 by the vortex on panel 2 is not exactly perpendicular to panel 1. We could be very precise and use a little trigonometry to figure out the exact angle and account for it to find the true normal component or we could just assume it is close enough to normal to ignore the angle. Since this is an approximate method anyway, except in the case of very large angles (say ten degrees or more) we will usually ignore the error. Doing this, we can write an equation adding the two vortex components of velocity and equating their sum to the free stream component.
V11 + V21 = V∞n1 = Vsin1) ≈ V α1
where
V11 = Γ1 /(2πr11)
V21 = Γ /(2πr21)
and where
r11 = 0.4c
r21 ≈ – 0.25c
Now, we would do exactly the same thing at the control point on the second panel.
Here our equation will be similar to the one at the control point on panel one except that we will probably have to account for the fact that the flap deflection angle will probably be too large to simply ignore in finding the normal component of the freestream velocity.
V12 + V22 = V∞n2 = Vsin2) = Vsin(α + δ)
where
δ = flap deflection angle
V12 = Γ1 /(2πr12)
V22 = Γ2 /(2πr22)
r12 ≈ 0.75c
r22 = 0.1c
We would then simultaneously solve these two equations, each having two unknowns (Γ1 and Γ2) for those unknown values. We would then use those two values of circulation to find the total lift and lift coefficient where the lift would simply be found from the Kutta-Joukowski Theorem, summing the lifts on each panel.
L = ΣρVΓ = ρV [Γ1 + Γ2]
and the lift coefficient would be
CL = L / (½ρV2c) = 2[Γ1 + Γ2] / (Vc)
Exercise:
Solve the problem above for a freestream speed of 100 mph, a flap deflection of 20 degrees, and a 5 foot wing chord at sea level conditions at an angle of attack of 5 degrees. How does this lift coefficient compare with that for a symmetrical airfoil at the same angle of attack?
Note that in the above approximate solution for a flapped, symmetrical airfoil we have a very crude “lift distribution” over the airfoil chord, that is, we know the way lift can be divided to act at the two locations above. If we solved this for several different angles of attack we would find that the relative values of the two vortex strengths (circulations) would change. If we looked closely at this we would find that, unlike for the symmetrical airfoil where the lift can always be said to act at the airfoil quarter chord, for the cambered airfoil the point where the lift will act (center of lift or center of pressure) will move with angle of attack and will be a function of camber. If we wanted to find this more precisely we could use more panels and more vortices and control points (along with more equations). For the above example it might be natural to simply divide the airfoil into five panels of equal lengths, each 20% of the original chord, looking something like the sketch below.
And, it is not too hard to imagine extending this method to a truly cambered airfoil as shown in Figure 3.12.
This simple approximation based on the results of flat plate aerodynamic behavior has, in this way, been stretched into a true “numerical solution” for an airfoil of any camber shape with as many panels as we wish to use. We can use it to find the total lift or lift coefficient for an airfoil with any camber shape and also to find the way lift is distributed over the airfoil chord.
Another thing we can find from this is the “pitching moment” of the airfoil, its tendency to rotate nose up or nose down about any desired reference point on the chord. The pitching moment is just found by taking each “lift” force found from the individual circulations and multiplying it by the distance or “moment arm” between that vortex and the desired reference point. If, for example, we want to use the leading edge of the wing as our moment reference (a common choice in theoretical or numerical calculations) we only need to take each value of gamma times the distance from the airfoil leading edge and that vortex and sum these to get the total pitching moment about the leading edge.
For the symmetrical airfoil (flat plate) where the center of lift is always at the quarter chord, this pitching moment around the leading edge will always be the lift multiplied by the quarter chord distance. The pitching moment coefficient for this case would be the lift multiplied by the quarter chord distance, divided by the dynamic pressure times the square of the chord. Do a quick calculation to see what this would be?
3.3 Pitching Moment
While we are talking about finding the pitching moment let’s take a look at some special cases. Pitching moment can be calculated or measured about any point we wish to use. Often in analytical or numerical calculations it is convenient to find the pitching moment around the wing leading edge. On the other hand, in wind tunnel testing it might be more convenient to measure the moment at some point between 20% and 50% back from the leading edge because of the ease of attaching the force and moment balance system there.
In Chapter One we mentioned two significant locations where the pitching moment or its coefficient has special meanings. These were the center of pressure (center of lift) and the aerodynamic center; points where the moment coefficient is zero or where it remains constant as the lift coefficient and angle of attack change. According to thin airfoil theory these are coincident at the quarter chord for a symmetrical airfoil but for the cambered airfoil, the center of pressure will usually move as angle of attack changes while the aerodynamic center remains at the quarter chord. We can verify that this theoretical result is a pretty good match for reality by looking at the aerodynamic data plots presented in Appendix A (discussed earlier in Chapter One). Often these plots present two graphs for pitching moment coefficient with the first one (on the left in most figures) a plot of CMc/4 (the moment coefficient at the quarter chord) and the other plot (usually on the right) of CMAC (moment coefficient at the aerodynamic center). In the plots in Appendix A for cambered airfoils it can be seen that CMc/4 is non-zero in value and changes with angle of attack while CMAC is relatively constant prior to the onset of stall. In the plots of symmetrical airfoil data both of these quantities are zero (or near zero) in value and constant prior to stall. In stall, the lift or pressure distribution over the airfoil is changing drastically as the flow begins to separate over progressively larger portions of the airfoil and the approximations of thin airfoil theory are far from valid.
As noted in the previous section, the pitching moment and moment coefficient can be calculated along with the lift when using the Weissinger Approximation. To find the location of the center of pressure we need only use the definition of that point as being where the moment is zero to find its location. Based on the illustration below, we can assume the center of pressure is located at some point Xcp and sum the moments about that unknown point due to the various circulation induced lift forces to equal zero.
ρV[Γ1(x1-xcp) + Γ2(x2-xcp) + • • • • + Γn(xn-xcp)] = 0
In this equation x1, x2, through xn are the locations along the chord of the various vortices (each at the quarter chord of its panel) and xcp is the unknown location of the center of pressure. This is solved for xcp, the location of the center of pressure. Note that for a symmetrical airfoil (flat plate) this position should not change with angle of attack while for a cambered airfoil xcp will be different for every angle of attack.
In a similar, but slightly more complicated fashion, we could find the location of the aerodynamic center from Weissinger Approximation lift results. This would involve finding the point where dCM/dCL = 0. We will leave this calculation for a text or course in aerodynamics.
3.4 Wings (3-D Aerodynamics)
As was pointed out in Chapter One, the main difference between two-dimensional flow around an airfoil and three-dimensional flow around a wing is the flow around the wing tip from the bottom to the top of the wing. This results in several things:
• An outward flow along the bottom of the wing near the tip.
• An inward flow along the top of the wing near the tip.
• A trailing vortex system.
• A “downwash” on the wing caused by the trailing vortex system.
• An “induced drag” caused by the downwash.
This vortex flow off of each wing tip is a very real flow that can be seen in the wind tunnel with either smoke or by simply sticking a string in the flow behind the wing tip. It can also be seen on airplanes in flight when atmospheric conditions are right. If there is sufficient moisture in the air in the form of high relative humidity or due to water vapor in the jet engine exhaust being pulled into the trailing vortices, the low pressure in the vortex core will cause the water vapor to condense, making it visible as a pair of “white tornados” trailing behind each wing tip. It is interesting to watch these when they are visible from high flying jets and to see just how long they persist. They can exist for many miles behind the generating aircraft, illustrating both the amount of energy in the vortices and the danger to other aircraft that might encounter them.
These “trailing vortices” are the key factor in trying to create any kind of mathematical model of the 3-D flow around and behind a real wing. In essence, this is done by bending the vortices that were used to model the flow over an airfoil at a 90 degree angle and allowing them to trail behind the wing. And, since we know that the lift on the wing varies along its span and doesn’t just stay the same all the way out to the wing tip, we allow vortices to turn right or left and come off the wing all along its span. This produces what is called a “horseshoe vortex system”.
While this theoretical derivation is beyond the intended scope of this material, its essence is found in the following facts or assumptions:
• The total circulation on the wing is assumed to vary along the span and to go to zero at the wing tips.
• The “trailing vortices” induce a downward flow or “downwash” that creates a small but significant downward velocity on the wing itself. (Actually, this is a way to account for the downward momentum of the flow that results from the lift force. This could also be found from the momentum equation if we had sufficient information.)
• In the same way that the interaction of the freestream velocity, V , with the vortex circulations on the wing results in lift (Kutta-Joukowski Theorem), the interaction of this “downwash” velocity with the vortex circulations causes a drag. We call this the “induced drag”.
Using this horseshoe vortex model and assuming that all the vortices on the wing are bundled into a single, rope-like, core at the quarter chord of the wing and also assuming that the wing quarter chord is unswept, a method referred to as “lifting line theory” can be used to find the lift variation along the span and the induced drag for any unswept wing of moderate to high aspect ratio. Any good text on aerodynamics will have a full development of this approach to 3-D aerodynamics and will present its results.
One of the results of the use of lifting line theory will be an “optimum” case, that is, a spanwise lift distribution on the wing where the induced drag is a minimum. This turns out to be a solution where the spanwise distribution of circulation over the wing is elliptical in shape and mathematical form. This special solution will give the equation for induced drag coefficient that we have already cited in Chapter One,
CDi = CL2 / (πAR)
where “AR” is the aspect ratio,
AR = b2/S = b/cavg .
The more general (non-optimum) form of this equation for induced drag coefficient includes another term , “e” , known as Oswald’s efficiency factor, that accounts for non-elliptical spanwise variations of circulation.
CDi = CL2 / (πARe)
This Oswald’s efficiency factor can be calculated from lifting line theory in the form of a Fourier series.
While it is not the purpose of this text to go into much detail examining lifting line theory, I would like to briefly look at what it says about lift for the special, minimum drag, case of the elliptical lift distribution because we often use this special case to take a first look at the influences of wing design on aerodynamics and performance. Without deriving them, we will simply look at some of the most important results of lifting line theory related to lift:
L = (π/4)ρVcenter
where
b = wing span
Γcenter = the value of the circulation at the center of the wing span
and
Γcenter = (2CLVS) / (πb)
Note that this makes the lift coefficient a function of the aspect ratio:
CL = L /(½ρV2S) = (πcenter) / (2VS) = (π AR Γcenter) / (2bV)
This says that for any given angle of attack (or value of circulation since Γcenter must increase with angle of attack) a higher aspect ratio wing design will give a higher lift coefficient than a low AR wing.
If we plot lift coefficient versus angle of attack for two wings of different aspect ratio we would find that the “slope” of the lift curve would decrease with decreasing aspect ratio. The maximum value of the slope is 2-pi, the two dimensional airfoil value, equivalent to an infinite aspect ratio wing.
Another way to look at this same effect is to look at the angle of attack at which two wings of different aspect ratios must be flown to get the same value of lift coefficient:
α2 = α1 + (CL/π) [(1/AR2) – (1/AR1)]
The above relationships give us valuable tools for examining the effects of design variables like wing aspect ratio on the aerodynamics and performance of an aircraft. Should we wish to use a moderate aspect ratio wing instead of a high aspect ratio design for, say, structural reasons or for improved roll dynamics, we can see what penalty we will pay. Often the penalty may not be as great as we might at first believe based on our basic knowledge that high aspect ratio is usually desirable.
These equations are, as mentioned earlier, for the ideal case, the elliptical lift distribution or minimum induced drag case. Nonetheless, they can give us a pretty good idea of how things like changes in aspect ratio will influence the performance of any three-dimensional wing.
One final thing I would like to mention in this chapter relates to the assumptions mentioned earlier for lifting line theory. It was noted that this theory assumes that the quarter chord of the wing is unswept. Another assumption inherent in the use lifting line theory that was not mentioned is that the theory is not very good for low aspect ratio wings. Even with these two important limitations, lifting line theory gives us some important insights into 3-D aerodynamics. But, what would we do if we are looking at a wing with sweep or low aspect ratio?
For swept wings, wings with low aspect ratio, or any other wing beyond the bounds of the assumptions of lifting line theory, the usual approach is to go to some variation of what is called “vortex lattice” theory. The wing (or even the entire airplane) is divided into panels and a single “horseshoe” vortex is placed at the quarter chord of each panel in what is essentially a three-dimensional version of Weissinger’s Approximation. If the wing is broken into N panels, there are N horseshoe vortices and these must be solved by setting up N equations. The equations are solved for the condition of no flow through the panels (the same idea as with Weissinger’s Approximation) at the center span of the three-quarter chord of the panel. This solution is three times as complex as the 2-D Weissinger method because each horseshoe vortex has three segments (one spanning the panel’s quarter chord and the other two acting as “trailing vortices” from the first vortex at the panel’s edges. Just as in the 2-D method, the equations solved at the “control points” had to account for all the velocities induced by all the vortices on the airfoil as well as the freestream velocity, in the 3-D vortex lattice approach the equation at each panel’s control point must account for the velocities induced by all three vortex segments from all N panels as well as for the freestream velocity.
Vortex lattice methods can be intimidating at first; however, like the Weissinger Approximation in 2-D, they merely use N equations to solve for N unknowns where the unknown is the strength of the horseshoe vortex on each 3-D panel. Everything else boils down to modeling the geometry of the 3-D wing. The result will be the 3-D velocity vector parallel to each panel control point and these can be used to find the pressures and forces and moments all around the wing.
3.5 Vortex Aerodynamics: Winglets
We will look at one final subject in this chapter, that of “vortex aerodynamics”. As we have already seen, vortices play a major role in the way we mathematically model airfoil and wing aerodynamics. And, as we have seen in the case of three dimensional flows around wing tips, vortices actually play a very real role in creating things like lift and drag and the math models we construct to explain aerodynamics also do an excellent job of modeling these real vortices and their effects.
There are many situations in addition to the wing tip vortices where these tornado-like flows exist in “real life” and play a major role in creating forces and moments on airplanes and wings. One of the jobs of the aerodynamicist is to correctly model these vortices and their effects and another job may be to determine if there are ways to use these swirling flows to our advantage in ways that will improve an airplane’s aerodynamics and flight performance.
We will look at two very important and interesting cases where vortex aerodynamics plays important roles. These are in the use of “winglets” and “leading edge extensions” (sometimes known as “strakes” or “wing gloves”).
The winglet is, by now, a fairly well known addition to the wing tips of many airplanes but its purpose is widely misunderstood. For many, many years engineers and scientists tried many approaches that might reduce the strength or effects of wing tip vortices or that would eliminate them altogether. Unfortunately, the laws of Physics are hard to overcome, and this rotational energy that we quantify as “circulation” has to go somewhere at the wingtip and there is really no way to eliminate it in flight other than to let it slowly dissipate as a trailing vortex pair somewhere in the atmosphere far downstream of the aircraft. We can put big plates on the wingtip or create interesting wingtip shapes iin attempts to eliminate or hasten the dissipation of the trailing vortices but the usual effect is simply to create a slightly different circular of tangential velocity distribution within the vortex itself, a pattern that may or may not make the vortices less dangerous to trailing aircraft and may or may not result in any reduction in the wing’s induced drag. Usually the effects of such “fixes” are greater in their inventor’s imagination than in real life.
The winglet, more properly known as the Whitcomb winglet after their inventor, Richard Whitcomb of NASA-Langley, is, contrary to conventional wisdom, not designed to eliminate the wingtip vortices. Rather than try to eliminate something that can’t be eliminated without eliminating the wing’s lift, Whitcomb decided to use the trailing vortices to create a positive effect. In a conversation with the author, Mr. Whitcomb explained that he saw the winglets as working much like the keel on a sailboat that is sailing or “tacking” into the wind. A sailboat keel is a kind of wing on the bottom of the boat, and when a sailboat is sailing into the wind, the forward force is not coming from the sail at all. Instead, the sail is creating a sideward force that pushes the keel through the water in such a way as to create a forward directed force. The forward force pushing a sailboat into the wind is coming from a small wing in the water rather than from the large wing we call a sail.
In creating the winglet, Richard Whitcomb reasoned that if the large wing on a sailboat could create a flow over a smaller wing (the keel) that would produce a thrust, there ought to be a way to take the flow created by a large wing on an airplane (the tip vortex) and use it to create a thrust on a smaller wing. That smaller wing ended up being the winglet. The Whitcomb winglet is placed at what would appear to be a negative angle of attack on the wingtip such that the combination of the freestream velocity and the wingtip vortex flow velocities creates a “lift” on the winglet that is actually pointed forward, giving a thrust. And it turns out that this thrust is significant enough to improve the lift-to-drag ratios on a wing by fifteen to twenty percent. This can result in very significant improvements in airplane flight performance and economics.
3.6 Vortex Aerodynamics: Leading Edge Vortex
The second type of real vortex that we want to examine very briefly is the “leading edge vortex”. Such a vortex forms when a wing is swept to angles of about 50 degrees or more. These vortices, illustrated in figure 3.18 below, result from a combination of a three-dimensional, spanwise flow on the wing and from the normal flow around the leading edge of the wing. This combined flow actually separates from the wing surface at the leading edge but, due to the rotational flow in the vortex, reattaches to the wing surface in such a way that the wing does not stall at the usual fifteen to twenty degree angle of attack but has attached flow and lift up to much larger angles of attack.
Wings are usually swept to delay the onset of the transonic drag rise near Mach One and to reduce the magnitude of that drag rise. On the other hand, swept wings produce less conventional lift at a given angle of attack than an unswept wing, in much the same manner as lower aspect ratio wings give less lift at a given angle of attack than high aspect ratio wings. The effect of the leading edge vortex is two-fold. First, by keeping the flow attached to much higher than normal angles of attack, they enable the wing to produce lift at these higher than normal angles of attack. Added to this increased angle of attack advantage is an extra lift, called “vortex lift” that is created by the very low pressure at the vortex core. Unfortunately, this low pressure in the vortex also adds some drag, but the net effect can be very useful in allowing airplanes that need highly swept wings to operate efficiently at transonic and supersonic speeds to still get the lift they need at lower subsonic speeds. They also can allow military fighter airplanes to “fly” at very unusual attitudes (angles of attack) that can be very useful in air-to-air combat.
Sometimes we want to create this same capability on wings that aren’t swept that much. This can be done by adding highly swept leading edge extensions or strakes. These strakes create their own leading edge vortices which continue over the wing behind them, giving many of the benefits mentioned above and allowing airplanes with relatively low wing sweep to fly at much higher than normal angles of attack when needed. One airplane that makes good use of this effect is the Navy’s F-18. Its very long and highly swept strakes enable it to operate at the high angles or attack and low speeds needed for landing and takeoff on aircraft carriers while also giving it very useful high angle of attack maneuverability that is very useful in fighter combat.
As was mentioned at the beginning of this chapter, the coverage in the chapter is not really needed to be able to understand or work with the material on aircraft performance that will follow. It has been included simply to fill in some of the blanks that might have been left open in Chapter One. On the other hand, it can be used to enhance the following aircraft performance coverage and to better relate it to some of the basic concepts in aerodynamics.
Homework 3
1. The Airbus 380-100 is designed to cruise at 35,000 ft at a Mach number of 0.85. Its weight in cruise is approximately one million pounds. It has a wing area of 9100 ft2 and a wing span of 262 ft. Assuming that lift equals weight and thrust equals drag and that in cruise at this altitude 50,000 pounds of thrust is needed, find:
a. The flight speed in miles per hour and in knots
b. The lift coefficient
c. The drag coefficient
d. Its Reynolds number based on mean chord
2. If an airplane is taking off by simply accelerating down the runway until it has sufficient speed for its lift to equal its weight and its wing is at a five-degree angle of attack, what speeds are required for takeoff at sea level and at 5000 feel altitude assuming that its wing has a lift curve slope (dCL/dα) of 0.08 per degree and a zero-lift angle of attack (αL0) of minus one degree? Also find the indicated airspeed at both altitudes. Assume the airplane weighs 11,250 pounds and has a wing area of 150 ft2.
Note: Normally most aircraft would accelerate at a low value of lift coefficient and then “rotate” to increase their angle of attack to a value that will give lift + weight at the defined take-off speed. This would allow take-off in a shorter distance than the method above. We will look at this in detail later in the course.
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textbooks/eng/Aerospace_Engineering/Aerodynamics_and_Aircraft_Performance_(Marchman)/03%3A_Additional_Aerodynamics_Tools.txt
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Introduction
Now that we have examined the origins of the forces which act on an aircraft in the atmosphere, we need to begin to examine the way these forces interact to determine the performance of the vehicle. We know that the forces are dependent on things like atmospheric pressure, density, temperature and viscosity in combinations that become “similarity parameters” such as Reynolds number and Mach number. We also know that these parameters will vary as functions of altitude within the atmosphere and we have a model of a standard atmosphere to describe those variations. It is also obvious that the forces on an aircraft will be functions of speed and that this is part of both Reynolds number and Mach number.
Many of the questions we will have about aircraft performance are related to speed. How fast can the plane fly or how slow can it go? How quickly can the aircraft climb? What speed is necessary for lift‑off from the runway?
In the previous section on dimensional analysis and flow similarity we found that the forces on an aircraft are not functions of speed alone but of a combination of velocity and density which acts as a pressure that we called dynamic pressure. This combination appears as one of the three terms in Bernoulli’s equation
$P+\frac{1}{2} \rho V^{2}=P_{0}$
which can be rearranged to solve for velocity
$V=\sqrt{2\left(P_{0}-P\right) / \rho}$
In chapter two we learned how a Pitot‑static tube can be used to measure the difference between the static and total pressure to find the airspeed if the density is either known or assumed. We discussed both the sea level equivalent airspeed which assumes sea level standard density in finding velocity and the true airspeed which uses the actual atmospheric density. In dealing with aircraft it is customary to refer to the sea level equivalent airspeed as the indicated airspeed if any instrument calibration or placement error can be neglected. In this text we will assume that such errors can indeed be neglected and the term indicated airspeed will be used interchangeably with sea level equivalent airspeed.
$V_{I N D}=V_{e}=V_{S L}=\sqrt{\frac{2\left(P_{0}-P\right)}{\rho_{S L}}}$
It should be noted that the equations above assume incompressible flow and are not accurate at speeds where compressibility effects are significant. In theory, compressibility effects must be considered at Mach numbers above 0.3; however, in reality, the above equations can be used without significant error to Mach numbers of 0.6 to 0.7.
The airspeed indication system of high speed aircraft must be calibrated on a more complicated basis which includes the speed of sound:
$V_{\mathrm{IND}}=\sqrt{\frac{2 a_{S L}^{2}}{\gamma-1}\left[\left(\frac{P_{0}-P}{\rho_{S L}}+1\right)^{\frac{\gamma-1}{\gamma}}-1\right]}$
where $a_{sl}$ = speed of sound at sea level and ρSL = pressure at sea level. Gamma is the ratio of specific heats (Cp/Cv) for air.
Very high speed aircraft will also be equipped with a Mach indicator since Mach number is a more relevant measure of aircraft speed at and above the speed of sound.
In the rest of this text it will be assumed that compressibility effects are negligible and the incompressible form of the equations can be used for all speed related calculations. Indicated airspeed (the speed which would be read by the aircraft pilot from the airspeed indicator) will be assumed equal to the sea level equivalent airspeed. Thus the true airspeed can be found by correcting for the difference in sea level and actual density. The correction is based on the knowledge that the relevant dynamic pressure at altitude will be equal to the dynamic pressure at sea level as found from the sea level equivalent airspeed:
An important result of this equivalency is that, since the forces on the aircraft depend on dynamic pressure rather than airspeed, if we know the sea level equivalent conditions of flight and calculate the forces from those conditions, those forces (and hence the performance of the airplane) will be correctly predicted based on indicated airspeed and sea level conditions. This also means that the airplane pilot need not continually convert the indicated airspeed readings to true airspeeds in order to gauge the performance of the aircraft. The aircraft will always behave in the same manner at the same indicated airspeed regardless of altitude (within the assumption of incompressible flow). This is especially nice to know in take‑off and landing situations!
4.1 Static Balance of Forces
Many of the important performance parameters of an aircraft can be determined using only statics; ie., assuming flight in an equilibrium condition such that there are no accelerations. This means that the flight is at constant altitude with no acceleration or deceleration. This gives the general arrangement of forces shown below.
In this text we will consider the very simplest case where the thrust is aligned with the aircraft’s velocity vector. We will also normally assume that the velocity vector is aligned with the direction of flight or flight path. For this most basic case the equations of motion become:
T – D = 0
L – W = 0
Note that this is consistent with the definition of lift and drag as being perpendicular and parallel to the velocity vector or relative wind.
Now we make a simple but very basic assumption that in straight and level flight lift is equal to weight,
L = W
We will use this so often that it will be easy to forget that it does assume that flight is indeed straight and level. Later we will cheat a little and use this in shallow climbs and glides, covering ourselves by assuming “quasi‑straight and level” flight. In the final part of this text we will finally go beyond this assumption when we consider turning flight.
Using the definition of the lift coefficient
$C_{L}=\frac{L}{\frac{1}{2} \rho V_{\infty}^{2} S}$
and the assumption that lift equals weight, the speed in straight and level flight becomes:
$V=\sqrt{\frac{2 W}{\rho S C_{L}}}$
The thrust needed to maintain this speed in straight and level flight is also a function of the aircraft weight. Since T = D and L = W we can write
D/L = T/W
or
Therefore, for straight and level flight we find this relation between thrust and weight:
The above equations for thrust and velocity become our first very basic relations which can be used to ascertain the performance of an aircraft.
4.2 Aerodynamic Stall
Earlier we discussed aerodynamic stall. For an airfoil (2‑D) or wing (3‑D), as the angle of attack is increased a point is reached where the increase in lift coefficient, which accompanies the increase in angle of attack, diminishes. When this occurs the lift coefficient versus angle of attack curve becomes non‑linear as the flow over the upper surface of the wing begins to break away from the surface. This separation of flow may be gradual, usually progressing from the aft edge of the airfoil or wing and moving forward; sudden, as flow breaks away from large portions of the wing at the same time; or some combination of the two. The actual nature of stall will depend on the shape of the airfoil section, the wing planform and the Reynolds number of the flow.
We define the stall angle of attack as the angle where the lift coefficient reaches a maximum, CLmax, and use this value of lift coefficient to calculate a stall speed for straight and level flight.
Note that the stall speed will depend on a number of factors including altitude. If we look at a sea level equivalent stall speed we have
It should be emphasized that stall speed as defined above is based on lift equal to weight or straight and level flight. This is the stall speed quoted in all aircraft operating manuals and used as a reference by pilots. It must be remembered that stall is only a function of angle of attack and can occur at any speed. The definition of stall speed used above results from limiting the flight to straight and level conditions where lift equals weight. This stall speed is not applicable for other flight conditions. For example, in a turn lift will normally exceed weight and stall will occur at a higher flight speed. The same is true in accelerated flight conditions such as climb. For this reason pilots are taught to handle stall in climbing and turning flight as well as in straight and level flight.
For most of this text we will deal with flight which is assumed straight and level and therefore will assume that the straight and level stall speed shown above is relevant. This speed usually represents the lowest practical straight and level flight speed for an aircraft and is thus an important aircraft performance parameter.
We will normally define the stall speed for an aircraft in terms of the maximum gross takeoff weight but it should be noted that the weight of any aircraft will change in flight as fuel is used. For a given altitude, as weight changes the stall speed variation with weight can be found as follows:
It is obvious that as a flight progresses and the aircraft weight decreases, the stall speed also decreases. Since stall speed represents a lower limit of straight and level flight speed it is an indication that an aircraft can usually land at a lower speed than the minimum takeoff speed.
For many large transport aircraft the stall speed of the fully loaded aircraft is too high to allow a safe landing within the same distance as needed for takeoff. In cases where an aircraft must return to its takeoff field for landing due to some emergency situation (such as failure of the landing gear to retract), it must dump or burn off fuel before landing in order to reduce its weight, stall speed and landing speed. Takeoff and landing will be discussed in a later chapter in much more detail.
4.3 Perspectives on Stall
While discussing stall it is worthwhile to consider some of the physical aspects of stall and the many misconceptions that both pilots and the public have concerning stall.
To the aerospace engineer, stall is CLmax, the highest possible lifting capability of the aircraft; but, to most pilots and the public, stall is where the airplane looses all lift! How can it be both? And, if one of these views is wrong, why?
The key to understanding both perspectives of stall is understanding the difference between lift and lift coefficient. Lift is the product of the lift coefficient, the dynamic pressure and the wing planform area. For a given altitude and airplane (wing area) lift then depends on lift coefficient and velocity. It is possible to have a very high lift coefficient CL and a very low lift if velocity is low.
When an airplane is at an angle of attack such that CLmax is reached, the high angle of attack also results in high drag coefficient. The resulting high drag normally leads to a reduction in airspeed which then results in a loss of lift. In a conventionally designed airplane this will be followed by a drop of the nose of the aircraft into a nose down attitude and a loss of altitude as speed is recovered and lift regained. If the pilot tries to hold the nose of the plane up, the airplane will merely drop in a nose up attitude. Pilots are taught to let the nose drop as soon as they sense stall so lift and altitude recovery can begin as rapidly as possible. A good flight instructor will teach a pilot to sense stall at its onset such that recovery can begin before altitude and lift is lost.
It should be noted that if an aircraft has sufficient power or thrust and the high drag present at CLmax can be matched by thrust, flight can be continued into the stall and post‑stall region. This is possible on many fighter aircraft and the post‑stall flight realm offers many interesting possibilities for maneuver in a “dog-fight”.
The general public tends to think of stall as when the airplane drops out of the sky. This can be seen in almost any newspaper report of an airplane accident where the story line will read “the airplane stalled and fell from the sky, nosediving into the ground after the engine failed”. This kind of report has several errors. Stall has nothing to do with engines and an engine loss does not cause stall. Sailplanes can stall without having an engine and every pilot is taught how to fly an airplane to a safe landing when an engine is lost. Stall also doesn’t cause a plane to go into a dive. It is, however, possible for a pilot to panic at the loss of an engine, inadvertently enter a stall, fail to take proper stall recovery actions and perhaps “nosedive” into the ground.
4.4 Drag and Thrust Required
As seen above, for straight and level flight, thrust must be equal to drag. Drag is a function of the drag coefficient CD which is, in turn, a function of a base drag and an induced drag.
CD = CD0 + CDi
We assume that this relationship has a parabolic form and that the induced drag coefficient has the form
CDi = KCL2
We therefore write
CD = CD0 + KCL2
K is found from inviscid aerodynamic theory to be a function of the aspect ratio and planform shape of the wing
where e is unity for an ideal elliptical form of the lift distribution along the wing’s span and less than one for non‑ideal spanwise lift distributions.
The drag coefficient relationship shown above is termed a parabolic drag “polar” because of its mathematical form. It is actually only valid for inviscid wing theory not the whole airplane. In this text we will use this equation as a first approximation to the drag behavior of an entire airplane. While this is only an approximation, it is a fairly good one for an introductory level performance course. It can, however, result in some unrealistic performance estimates when used with some real aircraft data.
The drag of the aircraft is found from the drag coefficient, the dynamic pressure and the wing planform area:
Therefore,
Realizing that for straight and level flight, lift is equal to weight and lift is a function of the wing’s lift coefficient, we can write:
giving:
The above equation is only valid for straight and level flight for an aircraft in incompressible flow with a parabolic drag polar.
Let’s look at the form of this equation and examine its physical meaning. For a given aircraft at a given altitude most of the terms in the equation are constants and we can write
where
The first term in the equation shows that part of the drag increases with the square of the velocity. This is the base drag term and it is logical that for the basic airplane shape the drag will increase as the dynamic pressure increases. To most observers this is somewhat intuitive.
The second term represents a drag which decreases as the square of the velocity increases. It gives an infinite drag at zero speed, however, this is an unreachable limit for normally defined, fixed wing (as opposed to vertical lift) aircraft. It should be noted that this term includes the influence of lift or lift coefficient on drag. The faster an aircraft flies, the lower the value of lift coefficient needed to give a lift equal to weight. Lift coefficient, it is recalled, is a linear function of angle of attack (until stall). If an aircraft is flying straight and level and the pilot maintains level flight while decreasing the speed of the plane, the wing angle of attack must increase in order to provide the lift coefficient and lift needed to equal the weight. As angle of attack increases it is somewhat intuitive that the drag of the wing will increase. As speed is decreased in straight and level flight, this part of the drag will continue to increase exponentially until the stall speed is reached.
Adding the two drag terms together gives the following figure which shows the complete drag variation with velocity for an aircraft with a parabolic drag polar in straight and level flight.
4.5 Minimum Drag
One obvious point of interest on the previous drag plot is the velocity for minimum drag. This can, of course, be found graphically from the plot. We can also take a simple look at the equations to find some other information about conditions for minimum drag.
The requirements for minimum drag are intuitively of interest because it seems that they ought to relate to economy of flight in some way. Later we will find that there are certain performance optima which do depend directly on flight at minimum drag conditions.
At this point we are talking about finding the velocity at which the airplane is flying at minimum drag conditions in straight and level flight. It is important to keep this assumption in mind. We will later find that certain climb and glide optima occur at these same conditions and we will stretch our straight and level assumption to one of “quasi”‑level flight.
We can begin with a very simple look at what our lift, drag, thrust and weight balances for straight and level flight tells us about minimum drag conditions and then we will move on to a more sophisticated look at how the wing shape dependent terms in the drag polar equation (CD0 and K) are related at the minimum drag condition. Ultimately, the most important thing to determine is the speed for flight at minimum drag because the pilot can then use this to fly at minimum drag conditions.
Let’s look at our simple static force relationships:
L = W, T = D
to write
D = W x D/L
which says that minimum drag occurs when the drag divided by lift is a minimum or, inversely, when lift divided by drag is a maximum.
This combination of parameters, L/D, occurs often in looking at aircraft performance. In general, it is usually intuitive that the higher the lift and the lower the drag, the better an airplane. It is not as intuitive that the maximum lift‑to drag ratio occurs at the same flight conditions as minimum drag. This simple analysis, however, shows that
MINIMUM DRAG OCCURS WHEN L/D IS MAXIMUM.
Note that since CL / CD = L/D we can also say that minimum drag occurs when CL/CD is maximum. It is very important to note that minimum drag does not connote minimum drag coefficient.
Minimum drag occurs at a single value of angle of attack where the lift coefficient divided by the drag coefficient is a maximum:
Dmin occurs when (CL/CD)max
As noted above, this is not at the same angle of attack at which CDis at a minimum. It is also not the same angle of attack where lift coefficient is maximum. This should be rather obvious since CLmax occurs at stall and drag is very high at stall.
Since minimum drag is a function only of the ratio of the lift and drag coefficients and not of altitude (density), the actual value of the minimum drag for a given aircraft at a given weight will be invariant with altitude. The actual velocity at which minimum drag occurs is a function of altitude and will generally increase as altitude increases.
If we assume a parabolic drag polar and plot the drag equation
for drag versus velocity at different altitudes the resulting curves will look somewhat like the following:
Note that the minimum drag will be the same at every altitude as mentioned earlier and the velocity for minimum drag will increase with altitude.
We discussed in an earlier section the fact that because of the relationship between dynamic pressure at sea level with that at altitude, the aircraft would always perform the same at the same indicated or sea level equivalent airspeed. Indeed, if one writes the drag equation as a function of sea level density and sea level equivalent velocity a single curve will result.
To find the drag versus velocity behavior of an aircraft it is then only necessary to do calculations or plots at sea level conditions and then convert to the true airspeeds for flight at any altitude by using the velocity relationship below.
4.6 Minimum Drag Summary
We know that minimum drag occurs when the lift to drag ratio is at a maximum, but when does that occur; at what value of CL or CD or at what speed?
One way to find CL and CD at minimum drag is to plot one versus the other as shown below. The maximum value of the ratio of lift coefficient to drag coefficient will be where a line from the origin just tangent to the curve touches the curve. At this point are the values of CL and CD for minimum drag. This graphical method of finding the minimum drag parameters works for any aircraft even if it does not have a parabolic drag polar.
Once CLmd and CDmd are found, the velocity for minimum drag is found from the equation below, provided the aircraft is in straight and level flight
As we already know, the velocity for minimum drag can be found for sea level conditions (the sea level equivalent velocity) and from that it is easy to find the minimum drag speed at altitude.
It should also be noted that when the lift and drag coefficients for minimum drag are known and the weight of the aircraft is known the minimum drag itself can be found from
It is common to assume that the relationship between drag and lift is the one we found earlier, the so called parabolic drag polar. For the parabolic drag polar
it is easy to take the derivative with respect to the lift coefficient and set it equal to zero to determine the conditions for the minimum ratio of drag coefficient to lift coefficient, which was a condition for minimum drag.
Hence,
This gives
or
and
The above is the condition required for minimum drag with a parabolic drag polar.
Now, we return to the drag polar
and for minimum drag we can write
which, with the above, gives
or
From this we can find the value of the maximum lift‑to‑drag ratio in terms of basic drag parameters
And the speed at which this occurs in straight and level flight is
So we can write the minimum drag velocity as
or the sea level equivalent minimum drag speed as
4.7 Review: Minimum Drag Conditions for a Parabolic Drag Polar
At this point we know a lot about minimum drag conditions for an aircraft with a parabolic drag polar in straight and level flight. The following equations may be useful in the solution of many different performance problems to be considered later in this text. There will be several flight conditions which will be found to be optimized when flown at minimum drag conditions. It is therefore suggested that the student write the following equations on a separate page in her or his class notes for easy reference.
EXAMPLE 4.1
An aircraft which weighs 3000 pounds has a wing area of 175 square feet and an aspect ratio of seven with a wing aerodynamic efficiency factor (e) of 0.95. If the base drag coefficient, CDO, is 0.028, find the minimum drag at sea level and at 10,000 feet altitude, the maximum lift‑to-drag ratio and the values of lift and drag coefficient for minimum drag. Also find the velocities for minimum drag in straight and level flight at both sea level and 10,000 feet. We need to first find the term K in the drag equation.
K = 1 / (πARe) = 0.048
Now we can find
We can check this with
The velocity for minimum drag is the first of these that depends on altitude.
At sea level
To find the velocity for minimum drag at 10,000 feet we an recalculate using the density at that altitude or we can use
It is suggested that at this point the student use the drag equation
and make graphs of drag versus velocity for both sea level and 10,000 foot altitude conditions, plotting drag values at 20 fps increments. The plots would confirm the above values of minimum drag velocity and minimum drag.
4.8 Flying at Minimum Drag
One question which should be asked at this point but is usually not answered in a text on aircraft performance is “Just how the heck does the pilot make that airplane fly at minimum drag conditions anyway?”
The answer, quite simply, is to fly at the sea level equivalent speed for minimum drag conditions. The pilot sets up or “trims” the aircraft to fly at constant altitude (straight and level) at the indicated airspeed (sea level equivalent speed) for minimum drag as given in the aircraft operations manual. All the pilot need do is hold the speed and altitude constant.
4.9 Drag in Compressible Flow
For the purposes of an introductory course in aircraft performance we have limited ourselves to the discussion of lower speed aircraft; ie, airplanes operating in incompressible flow. As discussed earlier, analytically, this would restrict us to consideration of flight speeds of Mach 0.3 or less (less than 300 fps at sea level), however, physical realities of the onset of drag rise due to compressibility effects allow us to extend our use of the incompressible theory to Mach numbers of around 0.6 to 0.7. This is the range of Mach number where supersonic flow over places such as the upper surface of the wing has reached the magnitude that shock waves may occur during flow deceleration resulting in energy losses through the shock and in drag rises due to shock‑induced flow separation over the wing surface. This drag rise was discussed in Chapter 3.
As speeds rise to the region where compressiblility effects must be considered we must take into account the speed of sound a and the ratio of specific heats of air, gamma.
Gamma for air at normal lower atmospheric temperatures has a value of 1.4.
Starting again with the relation for a parabolic drag polar, we can multiply and divide by the speed of sound to rewrite the relation in terms of Mach number.
where
or
The resulting equation above is very similar in form to the original drag polar relation and can be used in a similar fashion. For example, to find the Mach number for minimum drag in straight and level flight we would take the derivative with respect to Mach number and set the result equal to zero. The complication is that some terms which we considered constant under incompressible conditions such as K and CDO may now be functions of Mach number and must be so evaluated.
Often the equation above must be solved itteratively.
4.10 Review
To this point we have examined the drag of an aircraft based primarily on a simple model using a parabolic drag representation in incompressible flow. We have further restricted our analysis to straight and level flight where lift is equal to weight and thrust equals drag.
The aircraft can fly straight and level at a wide range of speeds, provided there is sufficient power or thrust to equal or overcome the drag at those speeds. The student needs to understand the physical aspects of this flight.
We looked at the speed for straight and level flight at minimum drag conditions. One could, of course, always cruise at that speed and it might, in fact, be a very economical way to fly (we will examine this later in a discussion of range and endurance). However, since “time is money” there may be reason to cruise at higher speeds. It also might just be more fun to fly faster. Flight at higher than minimum-drag speeds will require less angle of attack to produce the needed lift (to equal weight) and the upper speed limit will be determined by the maximum thrust or power available from the engine.
Cruise at lower than minimum drag speeds may be desired when flying approaches to landing or when flying in holding patterns or when flying other special purpose missions. This will require a higher than minimum-drag angle of attack and the use of more thrust or power to overcome the resulting increase in drag. The lower limit in speed could then be the result of the drag reaching the magnitude of the power or the thrust available from the engine; however, it will normally result from the angle of attack reaching the stall angle. Hence, stall speed normally represents the lower limit on straight and level cruise speed.
It must be remembered that all of the preceding is based on an assumption of straight and level flight. If an aircraft is flying straight and level at a given speed and power or thrust is added, the plane will initially both accelerate and climb until a new straight and level equilibrium is reached at a higher altitude. The pilot can control this addition of energy by changing the plane’s attitude (angle of attack) to direct the added energy into the desired combination of speed increase and/or altitude increase. If the engine output is decreased, one would normally expect a decrease in altitude and/or speed, depending on pilot control input.
We must now add the factor of engine output, either thrust or power, to our consideration of performance. It is normal to refer to the output of a jet engine as thrust and of a propeller engine as power. We will first consider the simpler of the two cases, thrust.
4.11 Thrust
We have said that for an aircraft in straight and level flight, thrust must equal drag. If the thrust of the aircraft’s engine exceeds the drag for straight and level flight at a given speed, the airplane will either climb or accelerate or do both. It could also be used to make turns or other maneuvers. The drag encountered in straight and level flight could therefore be called the thrust required (for straight and level flight). The thrust actually produced by the engine will be referred to as the thrust available.
Although we can speak of the output of any aircraft engine in terms of thrust, it is conventional to refer to the thrust of jet engines and the power of prop engines. A propeller, of course, produces thrust just as does the flow from a jet engine; however, for an engine powering a propeller (either piston or turbine), the output of the engine itself is power to a shaft. Thus when speaking of such a propulsion system most references are to its power. When speaking of the propeller itself, thrust terminology may be used.
The units employed for discussions of thrust are Newtons in the SI system and pounds in the English system. Since the English units of pounds are still almost universally used when speaking of thrust, they will normally be used here.
Thrust is a function of many variables including efficiencies in various parts of the engine, throttle setting, altitude, Mach number and velocity. A complete study of engine thrust will be left to a later propulsion course. For our purposes very simple models of thrust will suffice with assumptions that thrust varies with density (altitude) and throttle setting and possibly, velocity. We already found one such relationship in Chapter two with the momentum equation. Often we will simplify things even further and assume that thrust is invariant with velocity for a simple jet engine.
If we know the thrust variation with velocity and altitude for a given aircraft we can add the engine thrust curves to the drag curves for straight and level flight for that aircraft as shown below. We will normally assume that since we are interested in the limits of performance for the aircraft we are only interested in the case of 100% throttle setting. It is obvious that other throttle settings will give thrusts at any point below the 100% curves for thrust.
In the figure above it should be noted that, although the terminology used is thrust and drag, it may be more meaningful to call these curves thrust available and thrust required when referring to the engine output and the aircraft drag, respectively.
4.12 Minimum and Maximum Speeds
The intersections of the thrust and drag curves in the figure above obviously represent the minimum and maximum flight speeds in straight and level flight. Above the maximum speed there is insufficient thrust available from the engine to overcome the drag (thrust required) of the aircraft at those speeds. The same is true below the lower speed intersection of the two curves.
The true lower speed limitation for the aircraft is usually imposed by stall rather than the intersection of the thrust and drag curves. Stall speed may be added to the graph as shown below:
The area between the thrust available and the drag or thrust required curves can be called the flight envelope. The aircraft can fly straight and level at any speed between these upper and lower speed intersection points. Between these speed limits there is excess thrust available which can be used for flight other than straight and level flight. This excess thrust can be used to climb or turn or maneuver in other ways. We will look at some of these maneuvers in a later chapter. For now we will limit our investigation to the realm of straight and level flight.
Note that at the higher altitude, the decrease in thrust available has reduced the “flight envelope”, bringing the upper and lower speed limits closer together and reducing the excess thrust between the curves. As thrust is continually reduced with increasing altitude, the flight envelope will continue to shrink until the upper and lower speeds become equal and the two curves just touch. This can be seen more clearly in the figure below where all data is plotted in terms of sea level equivalent velocity. In the example shown, the thrust available at h6 falls entirely below the drag or thrust required curve. This means that the aircraft can not fly straight and level at that altitude. That altitude is said to be above the “ceiling” for the aircraft. At some altitude between h5 and h6 feet there will be a thrust available curve which will just touch the drag curve. That altitude will be the ceiling altitude of the airplane, the altitude at which the plane can only fly at a single speed. We will have more to say about ceiling definitions in a later section.
Another way to look at these same speed and altitude limits is to plot the intersections of the thrust and drag curves on the above figure against altitude as shown below. This shows another version of a flight envelope in terms of altitude and velocity. This type of plot is more meaningful to the pilot and to the flight test engineer since speed and altitude are two parameters shown on the standard aircraft instruments and thrust is not.
It may also be meaningful to add to the figure above a plot of the same data using actual airspeed rather than the indicated or sea level equivalent airspeeds. This can be done rather simply by using the square root of the density ratio (sea level to altitude) as discussed earlier to convert the equivalent speeds to actual speeds. This is shown on the graph below. Note that at sea level V = Ve and also there will be some altitude where there is a maximum true airspeed.
4.13 Special Case of Constant Thrust
A very simple model is often employed for thrust from a jet engine. The assumption is made that thrust is constant at a given altitude. We will use this assumption as our standard model for all jet aircraft unless otherwise noted in examples or problems. Later we will discuss models for variation of thrust with altitude.
The above model (constant thrust at altitude) obviously makes it possible to find a rather simple analytical solution for the intersections of the thrust available and drag (thrust required) curves. We will let thrust equal a constant
T = T0
therefore, in straight and level flight where thrust equals drag, we can write
where q is a commonly used abbreviation for the dynamic pressure.
or
and rearranging as a quadratic equation
Solving the above equation gives
or
In terms of the sea level equivalent speed
These solutions are, of course, double valued. The higher velocity is the maximum straight and level flight speed at the altitude under consideration and the lower solution is the nominal minimum straight and level flight speed (the stall speed will probably be a higher speed, representing the true minimum flight speed).
There are, of course, other ways to solve for the intersection of the thrust and drag curves. Sometimes it is convenient to solve the equations for the lift coefficients at the minimum and maximum speeds. To set up such a solution we first return to the basic straight and level flight equations T = T0 = D and L = W.
or
solving for CL
This solution will give two values of the lift coefficient. The larger of the two values represents the minimum flight speed for straight and level flight while the smaller CL is for the maximum flight speed. The matching speed is found from the relation
4.14 Review for Constant Thrust
The figure below shows graphically the case discussed above. From the solution of the thrust equals drag relation we obtain two values of either lift coefficient or speed, one for the maximum straight and level flight speed at the chosen altitude and the other for the minimum flight speed. The stall speed will probably exceed the minimum straight and level flight speed found from the thrust equals drag solution, making it the true minimum flight speed.
As altitude increases T0 will normally decrease and VMIN and VMAX will move together until at a ceiling altitude they merge to become a single point.
It is normally assumed that the thrust of a jet engine will vary with altitude in direct proportion to the variation in density. This assumption is supported by the thrust equations for a jet engine as they are derived from the momentum equations introduced in chapter two of this text. We can therefore write:
EXAMPLE 4.2
Earlier in this chapter we looked at a 3000 pound aircraft with a 175 square foot wing area, aspect ratio of seven and CDO of 0.028 with e = 0.95. Let us say that the aircraft is fitted with a small jet engine which has a constant thrust at sea level of 400 pounds. Find the maximum and minimum straight and level flight speeds for this aircraft at sea level and at 10,000 feet assuming that thrust available varies proportionally to density.
If, as earlier suggested, the student, plotted the drag curves for this aircraft, a graphical solution is simple. One need only add a straight line representing 400 pounds to the sea level plot and the intersections of this line with the sea level drag curve give the answer. The same can be done with the 10,000 foot altitude data, using a constant thrust reduced in proportion to the density.
Given a standard atmosphere density of 0.001756 sl/ft3, the thrust at 10,000 feet will be 0.739 times the sea level thrust or 296 pounds. Using the two values of thrust available we can solve for the velocity limits at sea level and at l0,000 ft.
= 63053 or 5661
VSL = 251 ft/sec (max)
or = 75 ft/sec (min)
Thus the equation gives maximum and minimum straight and level flight speeds as 251 and 75 feet per second respectively.
It is suggested that the student do similar calculations for the 10,000 foot altitude case. Note that one cannot simply take the sea level velocity solutions above and convert them to velocities at altitude by using the square root of the density ratio. The equations must be solved again using the new thrust at altitude. The student should also compare the analytical solution results with the graphical results.
As mentioned earlier, the stall speed is usually the actual minimum flight speed. If the maximum lift coefficient has a value of 1.2, find the stall speeds at sea level and add them to your graphs.
4.15 Performance in Terms of Power
The engine output of all propeller powered aircraft is expressed in terms of power. Power is really energy per unit time. While the propeller output itself may be expressed as thrust if desired, it is common to also express it in terms of power.
While at first glance it may seem that power and thrust are very different parameters, they are related in a very simple manner through velocity. Power is thrust multiplied by velocity. The units for power are Newton‑meters per second or watts in the SI system and horsepower in the English system. As before, we will use primarily the English system. The reason is rather obvious. The author challenges anyone to find any pilot, mechanic or even any automobile driver anywhere in the world who can state the power rating for their engine in watts! Watts are for light bulbs: horsepower is for engines!
Actually, our equations will result in English system power units of foot‑pounds per second. The conversion is
one HP = 550 foot-pounds/second.
We will speak of two types of power; power available and power required. Power required is the power needed to overcome the drag of the aircraft
Preq = D x V
Power available is equal to the thrust multiplied by the velocity.
Pav = T x V
It should be noted that we can start with power and find thrust by dividing by velocity, or we can multiply thrust by velocity to find power. There is no reason for not talking about the thrust of a propeller propulsion system or about the power of a jet engine. The use of power for propeller systems and thrust for jets merely follows convention and also recognizes that for a jet, thrust is relatively constant with speed and for a prop, power is relatively invariant with speed.
Power available is the power which can be obtained from the propeller. Recognizing that there are losses between the engine and propeller we will distinguish between power available and shaft horsepower. Shaft horsepower is the power transmitted through the crank or drive shaft to the propeller from the engine. The engine may be piston or turbine or even electric or steam. The propeller turns this shaft power (Ps) into propulsive power with a certain propulsive efficiency, ηp.
The propulsive efficiency is a function of propeller speed, flight speed, propeller design and other factors.
It is obvious that both power available and power required are functions of speed, both because of the velocity term in the relation and from the variation of both drag and thrust with speed. For the ideal jet engine which we assume to have a constant thrust, the variation in power available is simply a linear increase with speed.
It is interesting that if we are working with a jet where thrust is constant with respect to speed, the equations above give zero power at zero speed. This is not intuitive but is nonetheless true and will have interesting consequences when we later examine rates of climb.
Another consequence of this relationship between thrust and power is that if power is assumed constant with respect to speed (as we will do for prop aircraft) thrust becomes infinite as speed approaches zero. This means that a Cessna 152 when standing still with the engine running has infinitely more thrust than a Boeing 747 with engines running full blast. It also has more power! What an ego boost for the private pilot!
In using the concept of power to examine aircraft performance we will do much the same thing as we did using thrust. We will speak of the intersection of the power required and power available curves determining the maximum and minimum speeds. We will find the speed for minimum power required. We will look at the variation of these with altitude. The graphs we plot will look like that below.
While the maximum and minimum straight and level flight speeds we determine from the power curves will be identical to those found from the thrust data, there will be some differences. One difference can be noted from the figure above. Unlike minimum drag, which was the same magnitude at every altitude, minimum power will be different at every altitude. This means it will be more complicated to collapse the data at all altitudes into a single curve.
4.16 Power Required
The power required plot will look very similar to that seen earlier for thrust required (drag). It is simply the drag multiplied by the velocity. If we continue to assume a parabolic drag polar with constant values of CDO and K we have the following relationship for power required:
We can plot this for given values of CDO, K, W and S (for a given aircraft) for various altitudes as shown in the following example.
We will note that the minimum values of power will not be the same at each altitude. Recalling that the minimum values of drag were the same at all altitudes and that power required is drag times velocity, it is logical that the minimum value of power increases linearly with velocity. We should be able to draw a straight line from the origin through the minimum power required points at each altitude.
The minimum power required in straight and level flight can, of course be taken from plots like the one above. We would also like to determine the values of lift and drag coefficient which result in minimum power required just as we did for minimum drag.
One might assume at first that minimum power for a given aircraft occurs at the same conditions as those for minimum drag. This is, of course, not true because of the added dependency of power on velocity. We can begin to understand the parameters which influence minimum required power by again returning to our simple force balance equations for straight and level flight:
Thus, for a given aircraft (weight and wing area) and altitude (density) the minimum required power for straight and level flight occurs when the drag coefficient divided by the lift coefficient to the two‑thirds power is at a minimum.
Assuming a parabolic drag polar, we can write an equation for the above ratio of coefficients and take its derivative with respect to the lift coefficient (since CL is linear with angle of attack this is the same as looking for a maximum over the range of angle of attack) and set it equal to zero to find a maximum.
Note that
The lift coefficient for minimum required power is higher (1.732 times) than that for minimum drag conditions.
Knowing the lift coefficient for minimum required power it is easy to find the speed at which this will occur.
Note that the velocity for minimum required power is lower than that for minimum drag.
The minimum power required and minimum drag velocities can both be found graphically from the power required plot. Minimum power is obviously at the bottom of the curve. Realizing that drag is power divided by velocity and that a line drawn from the origin to any point on the power curve is at an angle to the velocity axis whose tangent is power divided by velocity, then the line which touches the curve with the smallest angle must touch it at the minimum drag condition. From this we can graphically determine the power and velocity at minimum drag and then divide the former by the latter to get the minimum drag. Note that this graphical method works even for nonparabolic drag cases. Since we know that all altitudes give the same minimum drag, all power required curves for the various altitudes will be tangent to this same line with the point of tangency being the minimum drag point.
One further item to consider in looking at the graphical representation of power required is the condition needed to collapse the data for all altitudes to a single curve. In the case of the thrust required or drag this was accomplished by merely plotting the drag in terms of sea level equivalent velocity. That will not work in this case since the power required curve for each altitude has a different minimum. Plotting all data in terms of Ve would compress the curves with respect to velocity but not with respect to power. The result would be a plot like the following:
Knowing that power required is drag times velocity we can relate the power required at sea level to that at any altitude.
or
The result is that in order to collapse all power required data to a single curve we must plot power multiplied by the square root of sigma versus sea level equivalent velocity. This, therefore, will be our convention in plotting power data.
4.17 Review
In the preceding we found the following equations for the determination of minimum power required conditions:
We can also write
Thus, the drag coefficient for minimum power required conditions is twice that for minimum drag. We also can write
Since minimum power required conditions are important and will be used later to find other performance parameters it is suggested that the student write the above relationships on a special page in his or her notes for easy reference.
Later we will take a complete look at dealing with the power available. If we know the power available we can, of course, write an equation with power required equated to power available and solve for the maximum and minimum straight and level flight speeds much as we did with the thrust equations. The power equations are, however not as simple as the thrust equations because of their dependence on the cube of the velocity. Often the best solution is an itterative one.
If the power available from an engine is constant (as is usually assumed for a prop engine) the relation equating power available and power required is
For a jet engine where the thrust is modeled as a constant the equation reduces to that used in the earlier section on Thrust based performance calculations.
EXAMPLE 4.3
For the same 3000 lb airplane used in earlier examples calculate the velocity for minimum power.
• It is suggested that the student make plots of the power required for straight and level flight at sea level and at 10,000 feet altitude and graphically verify the above calculated values.
• It is also suggested that from these plots the student find the speeds for minimum drag and compare them with those found earlier.
4.18 Summary
This chapter has looked at several elements of performance in straight and level flight. A simple model for drag variation with velocity was proposed (the parabolic drag polar) and this was used to develop equations for the calculations of minimum drag flight conditions and to find maximum and minimum flight speeds at various altitudes. Graphical methods were also stressed and it should be noted again that these graphical methods will work regardless of the drag model used.
It is strongly suggested that the student get into the habit of sketching a graph of the thrust and or power versus velocity curves as a visualization aid for every problem, even if the solution used is entirely analytical. Such sketches can be a valuable tool in developing a physical feel for the problem and its solution.
Homework 4
1. Use the momentum theorem to find the thrust for a jet engine where the following conditions are known:
inlet velocity 300 fps
inlet flow density 0.0023 sl/ft^3
inlet area 4 ft^2
exit flow velocity 1800 fps
exit flow density unknown
exit area 2 ft^2
fuel flow rate 5lb_m/sec
Assume steady flow and that the inlet and exit pressures are atmospheric.
2. We found that the thrust from a propeller could be described by the equation T = T0 – aV2. Based on this equation, describe how you would set up a simple wind tunnel experiment to determine values for T0 and a for a model airplane engine. Assume you have access to a wind tunnel, a pitot-static tube, a u-tube manometer, and a load cell which will measure thrust. Draw a sketch of your experiment.
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textbooks/eng/Aerospace_Engineering/Aerodynamics_and_Aircraft_Performance_(Marchman)/04%3A_Performance_in_Straight_and_Level_Flight.txt
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Chapter 5. Altitude Change: Climb and Guide
Introduction
Through the basic power and thrust performance curves considered in the last chapter we have been able to investigate the straight and level flight performance of an aircraft. We must now add another dimension to our study of performance, that of changes in altitude. We know that from the straight and level data we can determine the theoretical maximum altitude, or ceiling, for a given aircraft. The question to be answered now is how do we get the aircraft from one altitude to the other? This discussion must include the investigation of possible rates of climb and descent, the distance over the ground needed to climb a given altitude and the range of the aircraft in a glide. How fast can I get from altitude A to altitude B? How far can I glide after my engine fails? If I take off 600 feet from the end of the runway, can I clear the trees ahead?
To look at altitude changes we need to think in terms of energy changes. In climb we are turning kinetic and internal (engine) energy into an increase in potential energy. In a glide we are converting potential energy into velocity (kinetic energy) which will give us needed lift for flight.
One of the questions above involved the rate of climb. In climbing, the aircraft is increasing its potential energy. Rate of climb then involves the change of potential energy in a given time. The engine provides the needed energy for climb and the engine energy output per unit time is power (work per unit time). We are aware that a certain amount of power is required for straight and level flight at a given speed. To climb at that same speed then requires extra power and the amount of that extra power will determine the rate at which climb will occur. The maximum rate of climb at a given speed will then depend on the difference between the power available from the engine at that speed and the power required for straight and level flight. This can be determined from the power performance information studied in the last chapter.
The concept of adding power to increase altitude (climb) is usually not intuitive. Most of us are conditioned by experience with cars, boats and bicycles to think of speed increase as a consequence of adding power. These, of course, are vehicles limited to the altitude of the road or water surface. If we think about a car going over a hill, however, the process is not hard to understand. If a car is traveling at, say, 55 mph (since none of us would think of driving at speeds over the limit!) and we start up a hill holding the accelerator (throttle) steady, the car will decelerate as it climbs the hill, trading kinetic energy for potential energy. To maintain our 55 mph (keeping kinetic energy constant) as we move up the hill we must add power. The same is true in an aircraft.
One of the most difficult things for a flight instructor to teach a new pilot is that the throttle controls the altitude and the control stick or yoke controls the speed. This is, of course, not entirely true since the two controls are used simultaneously; however, this is the analogy that will best serve the pilot in a difficult situation. For example, on an approach to landing the pilot is attempting to hold a steady descent toward the runway. If a sudden downdraft causes a loss of altitude the pilot must take immediate action to regain the lost altitude or run the risk of an unplanned encounter with the ground short of the runway! Pulling back on the control to bring the nose of the aircraft up is the most common instinctive response since the aircraft is descending with the nose down. This will, however, merely increase the angle of attack and result in a reduction in speed, possibly leading to stall and certainly leading to further loss of lift and altitude. The proper response, adding power, will result in a climb to recover from the altitude loss. The ultimate control of the aircraft in such a circumstance will require the coordinated use of both controls to regulate both speed and altitude during this most difficult phase of flight.
The pilot in the above situation is not going to stop think about her or his aircraft’s power available or power required performance curves. This is the job of the engineer who designs the airplane to be able to meet the pilot’s needs in such a situation. This is our job in the sections that follow. In this study we must add an angle to our previous illustration of the balance of forces on the airplane. This will be the angle of climb, θ, which will be considered positive in a climb and negative in a glide or descent.
Summing the forces in the above figure along the thrust axis we find:
But this equation is a static relationship which does not allow acceleration; i.e., does not allow a change in kinetic energy. To consider all the forces which may be involved in climb we must also consider acceleration, so the above equation becomes:
If we rearrange this equation and divide by weight (mg) we get:
Then, multiplying by velocity, we have:
or
Now, V sinθ turns out to be the vertical velocity or the rate of climb as shown in Figure 5.2.
So we have (Pav – Preq) / W = dh/dt + (V/g)(dV/dt).
And, we can rearrange this to give
dh/dt = (Pav – Preq) / W – (V/g)(dV/dt).
It is common to refer to the first term in parentheses on the right in this equation as the excess power. When the excess power is divided by the weight as in the above equation it becomes the specific excess power, Ps.
Ps = (Pav – Preq) / W .
Now, going back to an earlier form of the equation, we can write
Ps = dh/dt + (V/g)(dV/dt)
This is a very important relationship which tells us that we can use our excess power (the power over and above that needed for straight and level flight) to either climb (dh/dt) or to change our speed (accelerate or decelerate) or to do both at the same time. We can also convert speed to altitude or altitude into speed.
In real situations a pilot will initiate a climb by both increasing the throttle (adding engine power) and slowing down, meaning that both engine power and kinetic energy are being converted into rate of climb. In descent the pilot will often reduce the plane’s power setting (throttle) so that not all of the decrease in potential energy goes into increasing the speed but some goes into the energy needed to maintain lifting flight.
In reality, how much gain can be realized by converting kinetic energy to potential energy without changing the engine power setting? We can find this fairly easily if we look directly at such an exchange
or
giving
Using this we can find that for an aircraft flying 200 mph and slowing to 160 mph during an initial climb, the altitude gain from swapping kinetic energy for potential energy comes to 483 feet. This is pretty small if one is thinking of making a 5000 ft climb but it may be useful in terms of emergency avoidance maneuvers. On the other hand, the same equation will show that an airplane flying 500 mph can gain over 5000 feet in altitude by slowing to 300 mph and if one is looking at a supersonic speed aircraft this kinetic for potential energy swap becomes very significant in accounting for climb and descent capabilities. Note that the mass of the aircraft is not in the equation above.
In a later chapter we will return to the concepts of specific excess energy and of trading speed for altitude or visa-versa. For the present we will look at the simpler case of un-accelerated flight and assume that all climbs and descents are done at constant speed and that the rate of change of altitude is only a function of the use of excess power. This will essentially assume that the altitude to be gained or lost by changing our airspeed is negligible.
We will start our “static” look at altitude change by looking at gliding flight with zero power available.
5.1 Gliding Flight
The first case we will consider will be the simple case of non-powered descent, or glide. This is a very important performance situation for an aircraft since all aircraft are susceptible to engine failure. One of the first things a student pilot is taught to do is to properly handle an “engine-out” in his or her aircraft; how to set up the best speed to optimize the rate of descent in order to allow maximum time to call for help, restart the engine, prepare for an emergency landing, etc. For some aircraft of course, the unpowered glide is normal. Sailplanes and hang gliders come to mind immediately but one should also consider that the Space Shuttle is nothing but an airplane with an “engine-out” in its descent from orbit to landing!
In an unpowered glide there are only three forces acting on the aircraft, lift, drag and weight. These forces must reach an equilibrium state in the glide. It is up to the pilot to make sure that the equilibrium reached is optimum for survival and, in most aircraft, it is up to the aircraft designer to make the airplane so that it will seek a reasonable equilibrium position on its own. The airplane which stalls and goes into a spin upon loss of an engine will not be very popular with most pilots! We must now determine what those optimum conditions are.
Using Figure 5.1 and using a thrust of zero we can write the following two simple force balance relations in the lift and drag directions:
Dividing the second equation by the first gives
There’s that term again, L/D!
This tells us a very simple and very important fact: the glide angle depends only on the lift-to-drag ratio.
The glide angle is:
and
Something seems wrong here. Does this mean that glide angle has nothing to do with the weight of the aircraft? It sure seems like a heavy airplane wouldn’t glide like a light one. Will a Boeing 747 glide just like a Cessna 152? What about the Space Shuttle?
Yes, the equation doesn’t contain the weight of the aircraft even though it was in the original force balance equations. The glide angle depends only on the lift-to-drag ratio and that ratio depends on parameters such as CD0, K and e as discussed in the last chapter.
But doesn’t the fact that there must be sufficient lift to support the weight (or at least most of it) mean that weight really is a factor? Not really, since drag is also seen to be a function of weight and in the ratio of lift-to-drag, the weight “divides out” of the relationship. A Boeing 747 can indeed glide as well as a Cessna 152.
So, what are our concerns in a glide? Essentially we want to know how far the aircraft can glide (range) and how long will it take to reach the ground (endurance).
5.1.1 Range in a Glide
We will look at the range assuming an absence of any natural wind. This is, of course, rarely the case in real life but is the easiest case for us to examine. We will also assume a steady glide, meaning that the pilot has set (or trimmed) the aircraft such that it will hold the selected indicated airspeed and angle of glide during the entire descent. The geometry of the situation is rather simple as shown below.
From the figure it is clear that the glide angle is the arc-tangent of the change in altitude divided by the range.
This gives a range of:
and since the tangent of the glide angle is simply the lift‑to‑drag ratio we have
Maximum range in a glide occurs at the maximumto-drag ratio; i.e., at minimum drag conditions! We already know how to find anything we may wish to know about minimum drag conditions so we know how to determine conditions for maximum range in an un-powered glide.
To the pilot this means that he or she must, upon loss of engine power, trim the aircraft to glide at the indicated (sea level equivalent) airspeed for minimum drag, a speed which the engineer has provided in the aircraft owner’s handbook, if maximum range is desired. The pilot would then fly the aircraft to hold the desired speed in the glide.
Usually, maximum range is not the most desirable goal in an “engine-out” situation. The best solution is usually to optimize the time before “ground encounter” (hopefully a landing!). This means going for minimum rate of descent rather than maximum range. This is, again, one of those things which may not be intuitive to most people, even to pilots, and there are many cases where planes have crashed as pilots tried unsuccessfully to stretch their range after an engine fails. Wind, which wasn’t included in the above calculations, can cut range to zero or can enhance it. Distances are hard to judge from the air. Student pilots are taught that in an “engine-out” situation it is time that should be optimized rather than range. The pilot needs to know the airspeed for minimum rate of descent rather than for maximum range in order to trim the aircraft for a descent which will allow the maximum time to try to restart the engine, to prepare for an emergency landing, to radio for help, etc. This means we are interested in the rate of descent.
Looking at rate of descent is a little more complicated than looking at range. We will consider two cases, the small glide angle case where we can make some simplifying assumptions and the general or large angle case. In looking at the small angle case we will use the usual mathematical assumption that the cosine of an angle is close enough to unity that we can approximate it as one. The usual limit of this assumption is about 5 degrees since a check on our calculator will show that cos 5º = 0.99619. However, our angle of interest is the glide angle which we already know is equal to the arc‑tangent of D/L. We would also like to assume that the sine of that angle is approximately equal to its tangent. Because of this we will stretch the applicability of the small angle rationalization to include glide angles up to about 15 degrees.
(At fifteen degrees the cosine is 0.9659, so we are still within 5% to of our goal of cosine = 1.0. This is usually pretty good in the real world. Also the tangent of fifteen degrees is 0.2679 while the sine is 0.2588, making our sine = tangent assumption good with less than 4% error.)
A useful result of the small angle assumption is that it will allow us to further assume that lift is approximately equal to weight. Since we had
and if
then
This might be referred to as “quasi‑level” flight. The main advantage of this assumption is that it allows us to continue to relate velocity to weight through
even though flight isn’t really straight and level.
Now we want to begin to look at rate of change of altitude, dh/dt or h. This is the rate of climb when defined in terms of a positive change of altitude as was shown in Figure 5.2.
From Figure 5.2 we see that the rate of climb is equal to the plane’s airspeed multiplied by the sine of the angle of climb. Referring to our earlier force balance equations for the glide case (no thrust) we can write
or
and using the small angle assumption that weight is approximately equal to lift gives
Changing to a form which uses the force coefficients
Now use the other small angle assumption for velocity
we have
or finally
Note that this is a negative rate of climb since we are looking at the case of glide or descent (we assumed no thrust).
From the above it is obvious that for the minimum rate of descent for a given aircraft and altitude will occur when CD/CL3/2 is at a minimum. Looking back at our study of power in the previous chapter we find that this is the same condition found for minimum power required.
In review, we have found the conditions needed for flight in an un-powered glide for two optimum cases, minimum rate of descent and maximum range with no wind. These are found to occur when the descending aircraft is trimmed to hold an indicated airspeed for minimum power required conditions and for minimum drag, respectively. We know everything about both of these conditions from the previous chapter’s discussion.
We have found that for any glide, the range with no wind is
and for glide angles of fifteen degrees or less the rate of descent is
These can be used to find the range and rate of descent for any glide condition where we know the appropriate lift and drag coefficients (angle of attack) and are not limited to the optimum cases. In addition, we know that to optimize range we need to fly at minimum drag conditions while for a minimum rate of descent, we need to fly at the conditions for minimum power required.
Most aircraft in a glide will satisfy the fifteen degree small angle assumption used in the above. A few, such as the Space Shuttle, will not. It is therefore worthwhile to back up and briefly consider the case of steep glide angles. This is, of course, the general case without the small angle assumption. We must use the force balance equations as developed without the approximations. These become
and
The velocity equation cannot assume straight and level flight and the first of the above two equations must be used to insert aircraft weight into the relationship.
or
The glide angle definition is unchanged
and we can use this relation with some simple trigonometry to find a relationship between the cosine of the glide or climb angle and the lift and drag coefficients.
The rate of climb (rate of descent) equation now becomes
or
This is a relation which will determine the rate of descent for any glide angle. It is noted that this equation is not really any more complicated mathematically than that found using the small glide angle approximation. The difference is that there is now no correlation between the minimum rate of descent and the condition for minimum power required.
5.2 Time to Descend
Using the rate of descent and the altitude change
it is possible to determine the time required for that descent.
dt = dh / (dh/dt)
If the rate of descent is constant this can become
In reality we have already shown that for both the general and the small angle cases the rate of descent is not constant but depends on altitude since it is a function of density. The complete equation would therefore be
and by using the equations for density variation in the standard atmosphere one could insert density as a function of h to give a general equation for time of descent. However, to get a simpler picture of the time to descend problem we will assume that an incremental approach can be used where the density, and thus rate of descent, can be assumed constant over reasonably small increments of altitude during descent. For example, over an increment of altitude of 1000 feet we can base our calculations on the density (rate of descent) half-way between the upper and lower altitude without introducing much error. This can be repeated incrementally to find the time of descent over larger altitude changes. A few simple examples might help illustrate this process.
EXAMPLE 5.1
A sailplane weighs one-thousand pounds and has a wing loading (W/S) of 12.5 pounds per square foot with a drag polar given by
Find the time to glide from 1000 feet to sea level at minimum rate of descent (minimum sink rate).
Solution: Minimum sink rate occurs at conditions for minimum power required
We can check the resulting lift-to-drag ratio to determine if the small angle approximations are valid
Thus we can find the velocity from the “quasi-level” equation
and using the density for a 500 foot altitude we have
and the rate of descent becomes
giving a time to descend the 1000 feet
EXAMPLE 5.2
Consider descent of the same sailplane from a much higher altitude. We can use a descent from 20,000 feet to investigate the inaccuracies of using the incremental approach to the time to descend problem. Suppose that in order to get a first guess for the time to descend we assumed a single increment using the density at 10,000 feet. We will first find an airspeed
then a rate of descent
giving a time for descent of
We should expect improved accuracy if we use four increments of 5000 feet each, calculating velocities and rates of descent at 17,500; 12,500; 7,500; and 2,500 foot altitudes as shown in the following table.
Table 5.1: Example 2
h(ft) h(mean)(ft) σ V(fps) dh/dt=Vsinθ(fps)
20,000-15,000 17,500 0.5793 124.49 4.258
15,000-10,000 12,500 0.6820 114.74 3.924
10,000-5,000 7,500 0.7982 106.05 33.627
5,000-SEA LEVEL 2,500 0.9288 98.32 3.363
VsL = 94.752f
The total time to descend is found by summing the incremental times from each of the 5000 foot descents.
giving t = 5313.8 sec = 88.6 min
This gives a time to descend from 20,000 feet of 88.6 minutes, a difference of only 0.2 minutes or 10.8 seconds from the gross, single increment solution.
Does the above show that there is little point in breaking the glide into increments to find the time of descent or simply that the increments chosen were too large to make much difference? A solution of the “exact” integral equation for the 20,000 foot descent will result in a time of descent of 5426.5 seconds or 90.4 minutes. There is only a two minute difference between the “exact” solution and the worst possible approximation; a 2% error!
5.3 Climbing Flight
As discussed earlier, the addition of power above that required for straight and level flight at a given speed will make possible either an increase in altitude or a change in speed or both. If speed is held constant while power (or thrust) is added the result will be a climb. Since climb is best thought of as an increase in potential energy we can best analyse it on an energy usage basis as reflected in power or energy addition per unit time. To begin our look at climb we can return to the figure used earlier and again write force balance equations in the lift and drag directions, this time adding the thrust vector.
It should be emphasized that we are assuming that climb occurs at constant speed. This means physically that climb is a straight exchange of energy from the engine for a gain in potential energy. It also means that our force balance equations sum to zero; ie, are static equations with no acceleration. We will not, however, restrict ourselves too much. As every good engineer should we will fudge a little by saying that we are flying at “quasi-steady: conditions and tolerate very small accelerations which are inevitable in real flight.
The rate of climb relation is still
From the Thrust/Drag force balance above we can write the angle of climb
The rate of climb is then
Note from the above that the angle of climb depends on the amount of excess thrust while the rate of climb depends on the amount of excess power. Not surprisingly, this is the same kind of dependence we found in the gliding case except there we spoke of drag instead of thrust.
Since the angle of climb and rate of climb both can be directly related to previously discussed performance curves for an aircraft, we can take a look at these parameters as they relate to these graphs. A typical plot of thrust and drag (thrust required) is shown below. At any given velocity the difference between the thrust and drag curves can be divided by the aircraft weight to determine the maximum possible angle of climb at that speed using the relationship defined earlier. Of course, at any given speed, not all of the excess available thrust need be used for climb if a lower angle of climb is desired. As the thrust and drag curves move together to the left and right, the possible angle of climb narrows toward zero at the velocities where thrust equals drag.
The velocity where the maximum possible angle of climb occurs is that for which the vertical distance between the thrust and drag curves is maximum. This could be found from an actual data plot by simply using a ruler or a pair of dividers to find this maximum. It could also be found analytically if functional relationships are known for the thrust and drag curves by taking the derivative of the difference in thrust and drag with respect to the velocity and setting that equal to zero to determine the maximum.
5.3.1 Case when Thrust is Constant
A simple case occurs when it can be assumed that the thrust available from an engine is constant, an assumption often made for jet engines. If the thrust is constant the maximum difference between thrust and drag and, hence, the maximum angle of climb, must occur when the drag is minimum. Once again minimum drag conditions become the optimum for a performance parameter. It should also be obvious that when thrust is not a constant, minimum drag is probably not the condition needed for maximum angle of climb.
The reader should note that no reference has been made in the above to a parabolic drag polar and the conclusions reached are not restricted to such a case. In the case of the parabolic drag polar we know how to determine the lift and drag coefficients and the speed for minimum drag from our previous study.
A typical plot of power versus velocity is shown below. We know from above that the rate of climb is equal to the difference in the power available and that required, at a given speed, divided by the aircraft weight. Thus the power available / power required graph can be used to graphically determine the rate of climb at any speed in the same manner as the thrust curves were used above. In cases where the power available is assumed constant, as is often the case in a simple representation of a propeller powered aircraft, the maximum rate of climb will occur at the speed where power required is a minimum. We know from the previous chapter how to determine the conditions for minimum power required. If power available is not constant, maximum rate of climb will not necessarily occur at the speed for minimum power required.
Note that the graph shown below plots P√σ versus Ve since this allows the power required data at all altitudes to collapse to a single curve as derived in Chapter 4.
It should also be noted that maximum rate of climb and maximum angle of climb do not occur at the same speed.
It is interesting to compare the power performance curves, and hence the rate of climb, for the two simple models we have chosen for jet and prop aircraft. In the plot which follows, the prop aircraft is assumed to have constant power available and the jet to have constant thrust. Since power available equals thrust multiplied by velocity, the jet power available data lies in a diagonal line starting at the origin. The power required curve assumes a common aircraft. In other words this is a comparison of the same airplane with two different types of engine. It is obvious that at lower speeds the rate of climb for the prop exceeds that for the jet while at higher speeds the jet can outclimb the prop. This comparison, while fictional, is typical of the differences between similar jet and prop aircraft. It shows one reason why one would not design a jet powered crop duster since such an aircraft needs a high rate of climb at very low speeds.
5.3.1.1 Special Case: Constant Thrust
In the case mentioned above as a simple model for a jet aircraft, finding the maximum angle of climb is easy since it must occur at the speed for minimum drag or maximum lift-to-drag ratio. The conditions for maximum rate of climb are not as simple. Looking at rate of climb again we recall
and assuming quasi-level flight we can write
Thus, we have a relationship which has the lift coefficient as the variable.
Again rising the quasi-level assumption which assumes that lift is essentially equal to weight
We now have a relation which includes both lift and drag coefficients as variables.
However, we know that drag coefficient depends on the lift coefficient in the drag polar.
This gives
or
The above equation is for the constant thrust case and shows the rate of climb as a function of only one variable, the lift coefficient. To determine the optimum rate of climb it is then necessary to take the derivative of this equation with respect to the lift coefficient. Only the terms in the brackets need be included in the derivative since it will be set equal to zero.
This gives
which can be solved via the quadratic equation to find the value of the lift coefficient which will give the highest rate of climb for this special case of constant thrust.
EXAMPLE 5.3
A given aircraft has CD0=0.013, K = 0.157, W = 35,000 lb, S = 530 sqft, T/W = 0.429 and thrust is constant with speed. Find the best rate of climb and the associated angle of climb.
Before starting our solution we should make sure we understand what is being asked. Note that the best angle of climb was not requested. The angle of climb sought was that for the best rate of climb case. Students sometimes assume that the answer sought is always for some optimum case.
To find the maximum rate of climb we use the relation found above to solve for the lift coefficient.
This can then be used to find the associated speed of flight for maximum rate of climb.
The angle of climb for maximum rate of climb (not maximum angle of climb) can then be found as follows:
Finally, these are used together to find the rate of climb itself.
Note: units of feet per minute are traditional.
Now let’s look at the other optimum, that of maximum angle of climb for this same aircraft. Maximum angle of climb occurs at conditions for minimum drag or maximum L/D .
We can then find the lift coefficient associated with maximum angle of climb and the airspeed at which that occurs.
Finally, the rate of climb for the maximum angle of climb
Lets look at the answers above and make sure they are logical.
• The maximum rate of climb should be higher than the rate of climb for maximum angle of climb. Is that true?
• The climb angle for the maximum rate of climb case should be less than the maximum angle of climb. Is that true?
• Maximum angle of climb should occur at a lower airspeed than that for maximum rate of climb. Is that the case?
In all cases the above questions are satisfied. These are some of the questions that the student should ask in reviewing the solutions to a problem. Often asking questions such as these can catch errors which might otherwise be ignored.
One situation in which all pilots are interested in both rate of climb and angle of climb is on takeoff. In a normal takeoff the pilot wants to initially climb at the speed which will give the maximum rate of climb. This will allow the aircraft to gain altitude in as short a time as possible, an important goal as a precaution against engine or other problems in takeoff. Should an engine fail on takeoff, maximum altitude is desired to allow time to recover and make an emergency landing. There are, however, some situations in which it is in the pilot’s best interest to forgo best rate of climb and go for best angle of climb. An obvious case is when the aircraft must clear an obstacle at the end of the runway such as a tree or tower. The figure below illustrates both cases.
The airplane which flew at maximum rate of climb would have reached the desired altitude faster than the plane which flew at maximum angle of climb if that darn tree hadn’t been in the way!
5.4 Time to Climb
To find the time to climb from one altitude to another we must integrate over the time differential
To integrate this expression we must know how V sin θ varies as a function of altitude. We are usually going to be interested in the minimum time to climb as a limiting case. This will, of course, occur at the speed for maximum rate of climb. This speed will be a function of altitude.
If we can find the rate of climb at each altitude we can plot rate of climb versus altitude as shown below. The area under the curve between the two desired altitudes represents the time to climb between those two altitudes.
Either of the above methods can be used to find the time to climb. In reality they are the same. The analytical method may not be as simple as it appears at first since the equations must account for the velocity and climb angle variation with altitude, necessitating the incorporation of the standard altitude density equations into the integral. The equations could be simplified by the assumption of a constant velocity climb or a constant angle climb.
5.5 Power Variation with Altitude
We dealt earlier with the variation of power required (to overcome drag) with altitude and how the power required curves could be merged into one by plotting power multiplied by the square root of the density ratio. The power available must also be multiplied by the square root of the density ratio to be included on the same performance plot. In addition to this we must be aware of how the power available actually varies with altitude.
For both jet engines (turbojet, fan-jet and turbo-prop) and piston engines the power produced by the engine drops in proportion to the decrease in density with increased altitude.
For a turbocharged piston engine the turbocharger is designed to maintain sea level intake conditions up to some design altitude. A simple model of power variation with altitude for a turbocharged engine will have power constant at its sea level value up to about 20,000 feet and dropping in direct proportion to decreasing density at higher altitudes.
20K=P20KρALTρ20K” title=”Ph>20K=P20KρALTρ20K” class=”mathml mathjax”>
More complicated situations are possible with multiple stages of turbocharging.
It must be remembered that in plotting power data versus the sea level equivalent velocity we must both account for the real variation in power available as just discussed and multiply that result by the square root of the density ratio to make the power available curves compatible with the power required curves. This is not redundant. The first change is made to account for the real altitude effects and the second for a plotting scheme needed to collapse all power required data to a single curve.
5.6 Ceiling Altitudes
In earlier discussion we spoke of the ceiling altitude as that at which climb was no longer possible. This would be the altitude where the power available curve just touches the power required curve, indicating that the aircraft can fly straight and level at only one speed at that altitude. Here the maximum rate of climb is zero. We define this altitude as the absolute ceiling. This definition is, however, somewhat misleading.
Theoretically, based on our previous study, it would take an infinite amount of time to reach the ceiling altitude. One could look at the rate of climb possible for an aircraft which is, say, 500 feet below its absolute ceiling. A very low rate of climb would be found, resulting in a very large amount of time required to climb that last 500 feet to reach the absolute ceiling. Because of this we define a more practical ceiling called the service ceiling. The definition of the service ceiling is based on the rate of climb; ie, at what altitude is the maximum rate of climb so low as to make further climb impractical. This is different for jet and piston powered aircraft. For the piston aircraft, the service ceiling is the altitude at which the rate of climb is 100 feet per minute (or 0.5 meters per second). For the jet aircraft, the service ceiling is the altitude at which the rate of climb is 500 feet per minute (or 2.5 meters per second).
It should be noted that many fighter and high performance aircraft may, in reality, be able to exceed even their absolute ceiling through the use of energy management approaches. An aircraft may climb to its service ceiling, for example, and then go into a dive, building up excess kinetic energy, and then resume a climb, using both the excess power and the excess kinetic energy to climb to altitudes higher than that found as “absolute”. Also, at very high altitude it may be necessary to include orbital dynamics in the consideration for climb and ceiling capabilities.
Homework 5
1. An aircraft weighs 3000 lb and has a 175 ft2 wing area, an aspect ratio of 7, and an Oswald Efficiency Factor, e, of 0.95. If CD0 is 0.028, plot drag versus velocity for sea level and 10,000 feet altitudes, plotting drag in 20 fps intervals. Also calculate the values of minimum drag and the velocity for minimum drag at both altitudes and compare them with the results on your graph. Use the graph paper provided; do not plot by computer.
2. Using a sea level value of thrust of 400 lb and assuming that thrust is constant with velocity but varies with density (altitude), calculate the maximum and minimum true airspeeds at sea level and at 10,000 ft altitudes and confirm these answers graphically.
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textbooks/eng/Aerospace_Engineering/Aerodynamics_and_Aircraft_Performance_(Marchman)/05%3A_Altitude_Change-_Climb_and_Guide.txt
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Introduction
A Little Background
In the earliest days of powered flight the primary concern was getting the aircraft into the air and back down safely (with safely meaning the ability to limp away after the “landing”). The Wright’s famous first flight was shorter than a football field and even for a couple of years after December of 1903 they were content to circle around the family farm The Wright’s home built engines couldn’t run for long periods of time and they simply didn’t envision the need or desire for flights over distances of over a few miles. In 1908, however, Scientific American magazine challenged aviation experimenters to produce an aircraft or “aero-plane”[1] which could fly, in public view, over a distance of one mile! While the Wright’s claimed to be able to make such a flight, their obsession with secrecy as they sought military sales and their egotistical belief that no one else could approach their expertise in aviation led them to ignore the prize offered by Scientific American for the one mile public flight.
It was Glenn Curtiss, a builder of motorcycle engines and holder of numerous world speed records in motorcycle racing, who, in July of 1908, made the first public one-mile flight. Curtiss, who had worked with Alexander Bell and others to develop their own airplanes, made the flight with newspaper reporters watching and with movie cameras recording the flight. Curtiss became the top aviator in America and the Wrights were furious, leading to numerous legal suits as Wilbur and Orville sought to prove in the courts that Curtiss and Bell had infringed on their patents. Curtiss went on to outperform the Wrights and others in aviation meets in America and Europe. The Wright’s subsequent patent suits aimed at reserving for themselves the sole rights to design and build airplanes in the United States stagnated aircraft development in America and shifted the scene of aeronautical progress to Europe where it remained until after World War I.
As aircraft and aviation continued to develop, range and endurance became the primary objective in aircraft design. In war, bombers needed long ranges to reach enemy targets beyond the front lines and by the end of World War I huge bombers had been developed in several countries. After the war, European governments subsidized the conversion of these giants into passenger aircraft. Some larger planes had even been built as passenger carrying vehicles before the conflict. Sikorski’s early designs are good examples. In the United States, however, with little government interest in promoting air travel for the public until the late 1920’s, long range aircraft development was more fantasy than fact. In 1927 Lindberg’s trans-Atlantic flight captured the public’s imagination and interest in long range flight increased. Lindberg’s flight, like that of Curtiss, was prompted by a prize from the printed media, illustrating the role of newspapers and magazines in spurring technological progress.
By the late 1930’s the public began to see flight as a way to travel long distances in short times, national and international airline routes had developed, and planes like the “China Clipper” set standards for range and endurance. World War II forced people and governments to think in global terms leading to wartime development of bombers capable of non‑stop flight over thousands of miles and to post‑war trans‑continental and trans‑ocean aircraft. Since 1903 we have seen aircraft ranges go from feet to non‑stop circling the globe and endurances go from minutes to days!
6.1 Fuel Usage and Weight
In studying range and endurance we must, for the first time in this course, consider fuel usage. In the aircraft of Curtiss and the Wrights, it was not uncommon for the engine to quit from mechanical problems or overheating before the fuel ran out. In today’s aircraft, range and endurance depend on the amount of fuel on board. When the last drop of fuel is gone the plane has reached its limit for range and endurance. One could, of course, include the glide range and endurance after the aircraft runs out of fuel, but an airline that operated that way would attract few passengers!
Fuel usage depends on engine design, throttle settings, altitude and a number of other factors. It is, however, not the purpose of this text to study engine fuel efficiency or the pilot’s use of the throttle. We will assume that we are given an engine with certain specifications for efficiency and fuel use and that the throttle setting is that specified in the aircraft handbook or manual for optimum range or endurance at the chosen altitude. It is assumed that the student will take a separate course on propulsion to study the origins of the figures used here for these parameters.
Our primary concern in fuel usage will be the change in the weight of the aircraft with time. Many of our performance equations used in previous chapters include the aircraft weight. In those chapters we treated weight as a constant. Weight is, in reality, constant only for the glider or sailplane. For other aircraft the weight is always changing, always decreasing as the fuel is burned. This means that the aerodynamic performance of the airplane changes during the flight. This does not, however, negate the value of the methods used earlier to study cruise and climb. Those calculations will normally be done using the maximum gross weight of the airplane which will lead to a conservative or “worst case” analysis of those performance parameters. We can also use the methods developed earlier to look at the “instantaneous” capabilities of the aircraft at a given weight, realizing that at a later time in flight and at a lower weight, the performance may be different.
In considering range and endurance it is imperative that we consider weight as a variable, changing from maximum gross weight at take‑off to an empty fuel tank weight at the end of the flight. To do this we will deal with fuel usage in terms of the weight of the fuel (as opposed to fuel volume, in gallons, which we normally use for automobiles). When our concern is endurance we are interested in the change in weight of the fuel per unit time
$\dfrac{dW_f}{dt}$
and, when range is the concern we want to know how the weight of fuel decreases with distance traveled.
$\dfrac{dW_f}{dS}$
Aircraft engine manufacturers like to specify engine fuel usage in terms of specific fuel consumption. For jet engines this becomes a thrust specific fuel consumption and for prop aircraft, a power specific fuel consumption. Since thrust and power bring different units into the equations we must consider the two cases separately.
6.2 Range and Endurance: Jet
We speak of the engine output of a jet engine in terms of thrust; therefore, we speak of the fuel usage of the jet engine in terms of a thrust specific fuel consumption, Ct. Ct is the mass of fuel consumed per unit time per unit thrust. The unit of time should be seconds and the unit for thrust should be in pounds or Newtons of thrust.
[Ct] = (sl/sec)/lbthrust , or = (kg/sec)/Nthrust
The above is the proper definition of thrust specific fuel consumption, however, it is not really exactly what we need for our calculations. We would prefer a definition based on weight of fuel consumed instead of the mass. We will thus define a weight specific fuel consumption, t, as the weight of fuel used per unit time per unit thrust. This gives units of (time)-1.
$\left[\gamma_{\mathrm{t}}=\mathrm{gC}_{\mathrm{t}}\right]=\left(\mathrm{lb}_{\text {fuel }} / \mathrm{sec}\right) / \mathrm{lb}_{\text {thrust }}, \text { or }=\left(\mathrm{N}_{\text {fuel }} / \mathrm{sec}\right) / \mathrm{N}_{\text {thrust }}=(\mathrm{sec})^{-1}$
The reader should be aware that many aircraft performance texts and propulsion texts are very vague regarding the units of specific fuel consumption. Some even define it in terms of mass and give it units of 1 / sec., making it dimensionally incorrect. Part of the confusion, particularly in older American propulsion texts, lies in the use of the pound‑mass as the unit of mass. This gives a combination of pounds‑mass divided by pounds‑force, which, in reality gives sec2 / ft. The situation is then further complicated by the author seemingly throwing in a term called gc which is supposed to resolve the lbm /lbf issue. At any rate it is very important that the engineer using specific fuel consumption carefully consider the units involved before beginning the solution of a range or endurance problem. A correctly specified weight specific fuel consumption will have units of sec-1 and will do so without the use of anything called gc.
It should be added that one might also encounter specific fuel consumption which has been calculated using mass of fuel (kg) and thrust in kilograms, a non-SI unit which is used in much of the world (virtually no one in the world knows what a Newton is or how to use it). If this is done the numerical value should be the same as obtained using the weight specific fuel consumption definition above.
Some sources of specific fuel consumption data use units of (hours)-1 since the hour is a more convenient unit of time. The use of seconds is, however, correct in any standard unit system and the student may be well advised to convert hours to seconds before beginning calculation even though this will ultimately result in the calculation of endurance in seconds, giving rather large numbers for answers.
To find endurance we want the rate of fuel weight (Wf) change per unit time which can be written in terms of the thrust specific fuel consumption
$\mathrm{dW}_{\mathrm{f}} / \mathrm{dt}=\gamma_{\mathrm{t}} \mathrm{T}$
And, in straight and level flight where thrust equals drag
$\mathrm{d} \mathrm{W}_{\mathrm{f}} / \mathrm{dt}=\gamma_{\mathrm{t}} \mathrm{D}$
For maximum endurance we want to minimize the above term. This clearly shows that for maximum endurance the jet plane must be flown at minimum drag conditions. We will look at how to find that endurance after taking a brief look at range.
To find the flight range we must look at the rate of change of fuel weight with distance of flight. We might pause a little at this point to realize that this may be more complicated than endurance because range will depend on more than simply the aerodynamic performance of the airplane. It will also require consideration of the wind speed. An airplane can fly forever at a speed of 100 mph into a 100 mph head‑wind and still have a range of zero! For now, however, we will put those worries aside and look at the simple mathmatics with which we begin consideration of the problem. We looked above at the rate of weight change with time. We can combine this with the change of distance with time (speed) to get the rate of change of weight with distance.
$\mathrm{dW}_{\mathrm{f}} / \mathrm{dS}=\left(\mathrm{dW}_{\mathrm{f}} / \mathrm{dt}\right) /(\mathrm{dS} / \mathrm{dt})=\gamma_{\mathrm{t}}(\mathrm{T} / \mathrm{V})=\gamma_{\mathrm{t}}(\mathrm{D} / \mathrm{V})$
Note that we still assume D = T or straight and level flight.
From the above it is obvious that maximum range will occur when the drag divided by velocity (D/V) is a minimum. This is not a condition which we have studied earlier but we can get some idea of where this occurs by looking at the plot of drag versus velocity for an aircraft.
On this plot a line drawn from the origin to intersect the drag curve at any point has a tangent equal to the drag at the point of intersection divided by the velocity at that point. The minimum possible value of D/V for the aircraft represented by the drag curve must then be found when the line is just tangent to the drag curve. This point will give the velocity for maximum range. Note that it is a higher speed than that for minimum drag (which, in turn was higher than the speed for minimum power).
In the above we have found the conditions needed to achieve maximum range and endurance for a jet aircraft. We have not yet found equations for the actual range or endurance. To find these we need to return to the time and distance differentials and integrate them. For time we have
We now wish to put the equations in a form which includes the weight of the aircraft instead of the weight of the fuel. Since the change in weight of the aircraft in flight is equal and opposite the weight of fuel consumed
dW = – dWfuel
we have
Finally, integrating over time to find the endurance gives
In a similar manner the range is found from the distance differential
or
In the above equations we must know how the aircraft velocity, thrust and specific fuel consumption vary with aircraft weight. At this point we need to make some assumptions about the way the flight is to be conducted. This is sometimes called the “flight schedule”.
6.3 Approximate Solutions for Range and Endurance for a Jet
The first assumption to be made in finding range and endurance equations is that the flight will be essentially straight and level. In order to give ourselves some leeway we will call this “quasi‑level” flight, our desire being merely to use the L = W. T = D relations.
T = D = W(D/L) = W(CD/CL) ,
and we can also use
Substituting these into the range and endurance relationships above give
At this point we need to make some further assumptions about the flight schedule in order to simplify the integration of these equations. For example, in the endurance equation, if we assume that the flight is made at constant angle of attack, we are assuming
that the lift and drag coefficients are constant for the entire flight. If we also assume that the specific fuel consumption is constant for the flight the only variable left in the integral is weight itself and the integral becomes:
The range integral contains an additional variable, the density of the atmosphere. It is still possible to make a couple of combinations of assumptions which will result in simple integration and realistic flight conditions. The first case will be to assume cruise at both constant altitude and constant angle of attack giving both density and the lift and drag coefficients as constants in the integration.
A second simple case combines the assumptions of constant angle of attack and constant speed, which can be used with the earlier form of the range equation.
giving
Note that the last equation above is simply the endurance equation multiplied by the velocity. This should not be surprising since this is the case where velocity is constant.
In the final equations above for range and endurance we should note that if standard units are used with specific fuel consumptions in sec.-1, range will be given in feet or meters and endurance in seconds. We may find it easier to ascertain the degree to which our answers are realistic if we convert these answers to miles or kilometers and hours.
In finding the above equations for range and endurance we have looked only at special cases which would result in simple integrations. If we know more complicated flight schedules we can determine the functional relationships between the lift and drag coefficients, velocity, density, etc. and weight loss during flight and insert them into the original integrals to solve for range and endurance. The above cases are, however, very close to actual operational cruise conditions for long range aircraft and will probably suffice for an introductory study of aircraft performance. Let’s take a look at those simple cases.
Both range cases included our endurance assumption of constant angle of attack and specific fuel consumption. The first case combined these assumptions with specification of constant altitude. This appears to be the simplest case to actually fly but to see what it actually means we need to go back to the straight and level flight velocity relation
V = [2W/(ρSCL)]1/2
or V∝W1/2,when ρ and CL are both constant.
If altitude (density) and angle of attack (lift coefficient) are both constant it is obvious that the velocity must change as the weight changes. In other words, for this flight schedule as fuel is burned and the weight of the aircraft decreases, the flight speed must decrease in proportion to the square root of the weight.
The other case, constant speed combined with constant angle of attack, is seen from the velocity relation above to require that density decrease in proportion to the weight.
W/ρ = const. when V and CL are constant.
This means that as the aircraft burns off fuel, the aircraft will slowly move to higher altitudes where the density is lower. This is commonly known as the drift‑up flight schedule. This is actually very similar to the way that commercial airliners fly long distance routes. Those of you who have been on such flights will recall the pilot announcing that “we are now cruising at 35,000 feet and will climb to 39,000 feet after crossing the Mississippi” or some such plan. While the FAA will not allow aircraft to simply “drift‑up” as they fly from coast‑to‑coast, they will allow schedules which incrementally approximate the drift‑up technique.
It must be noted that the two range equations above will give two different answers for the same amount of fuel. Also note that the equations are based on only the cruise portion of the flight. An actual flight will include take‑off, climb to cruise altitude, descent and landing in addition to cruise. Allowance also must be made for reserve fuel to handle emergency situations and “holds” imposed by air traffic controllers.
The biggest assumption used in all the integrations above is that of constant angle of attack. While this fits our conditions for optimum cases such as maximum endurance
occurring at maximum lift‑to drag ratio (minimum drag), it may not fit real flight very well. While the pilot can easily monitor his or her airspeed and altitude, the airplane’s angle of attack is not as easily monitored and directly controlled.
The equations above for range and endurance are valid for any flight condition which falls within the assumptions made in their derivation. If we have a Boeing 747 flying at an angle of attack of eight degrees and a speed of 250 miles per hour these equations can be used to find the range and endurance even though this is obviously not an optimum speed and angle of attack. Should we wish to determine the optimum range or endurance we must use the values of lift and drag coefficient and the velocity which we found earlier to be needed for these optimums.
Earlier we found that for maximum endurance the aircraft needs to fly at minimum drag conditions. Our actual endurance equation confirms this, showing endurance as a function of the lift‑to‑drag coefficient ratio which will be a maximum if drag is a minimum.
We also found that range would be optimum if the drag divided by velocity was a minimum. The correlation between this condition and the range equations derived is not as obvious as that of minimum drag with the endurance equation. Using the straight and level flight force relations which can be manipulated to show
D = W[D/L] = W[CD/CL]
the quantity V/D can be written
V/D = [V/W][CL/CD] .
Now using the velocity relation for straight and level flight
we find
Therefore, we find that the maximum range occurs when, for a given weight and altitude
CL1/2 /CD is a maximum.
If we assume a parabolic drag polar with constant CD0 and K we can write
CL1/2 /CD = CL1/2 / [CD0 + KCL2]
To find when this combination of terms is at a maximum we can take its derivative with respect to its variable (CL) and set it equal to zero.
Solving this gives
½(CD0 + KCL2)CL1/2 – (2KCL2) / CL1/2 = 0
or
CD0 + KCL2 -4KCL2 = 0
then
CD0 = 3KCL2
and, finally
CL = [CD0 / 3K]1/2 .
Thus, for maximum range
Using this in the drag polar gives the value of drag coefficient for maximum range
CDmaxR = CD0 + KCLmaxR2 = CD0 + KCD0/(3K) = (4/3)CD0 .
These are referred to as the conditions for “instantaneous” maximum range. The term instantaneous is used because the calculations are for a given weight and we know that weight is changing during the flight. In other words, at any point during the flight, at the weight and altitude at that point, the lift and drag coefficients found above will give the best range.
6.4 Range and Endurance: Prop
We will now look at range and endurance for propeller driven aircraft in which the engine performance is normally expressed in terms of power instead of thrust. An examination of range and endurance for aircraft which have performance measured in terms of power (propeller aircraft) is made by defining a power specific fuel consumption similar to the thrust specific fuel consumption used for jets. The power specific fuel consumption is defined as the mass of fuel consumed per unit time per unit shaft power. The units are slugs per unit power per second in the English system or kilogram per unit power per second in SI units.
[Cp] = sl/(power-sec) or kg/(power-sec)
The power units used are horsepower in the English system and watts in SI units.
Just as we did in the jet (thrust) case, we will often find an alternate definition of specific fuel consumption given in terms of the weight of fuel consumed instead of the mass.
While the proper time unit is seconds we will often find such data given for an engine in terms of hours. We will develop our equations in terms of the fundamental units (seconds for time) and, as in the jet case, assume “quasi‑level” flight which has
Pavail = Preq = DV
In dealing with prop engines we must consider the propulsive efficiency, ηp, which relates the shaft power, Ps, coming from the engine itself to the power effectively used by the prop to transfer momentum to the air.
As for the jet, to find endurance we must consider
and for range are interested in
From the above equations it is obvious that, for a given specific fuel consumption and efficiency, the rate of fuel use is a minimum (instantaneous endurance is a maximum) when the power required (DV) is a minimum. It is also obvious that the fuel use per amount of distance traveled is a minimum (instantaneous range is a maximum) when the drag is a minimum.
So we again run into our old friends minimum power required and minimum drag as conditions needed for optimum flight. We already know how to find these graphically from power versus velocity plots as shown below. This graphical determination of minimum power and minimum drag speeds is valid for any drag polar, even if not parabolic.
At this point we should pause and say: “Hey, wait just a minute! It was only a couple of pages back that you said that maximum endurance occurred at minimum drag conditions. Now you say it’s maximum range that I get at minimum drag conditions. Make up your mind, for Pete’s sake!”
The problem is that in one case we are talking about jets and the other, prop aircraft. This means that we must be very careful to see which type of plane we are dealing with before starting any calculations. It is very easy to get into a big rush and get the two cases mixed up (especially in the heat of battle on a test or exam!).
Now, as we did for the jet, we can develop integrals to determine the range or endurance for any flight situation. For endurance we have
and for range
6.5 Approximate Solutions for Range and Endurance for a Prop Aircraft
Once more we will assume “quasi‑level” flight and manipulate the terms in our force balance relations to give
$D = W\left[\dfrac{D}{L}\right] = W\left[\dfrac{C_D}{C_L}\right]$
This makes the endurance integral
$\mathrm{E}=-\int_{W_{1}}^{\mathrm{W} 2} \frac{\eta_{\mathrm{p}}}{\gamma_{\mathrm{p}}} \frac{1}{\mathrm{~V}} \frac{\mathrm{C}_{\mathrm{L}}}{\mathrm{C}_{\mathrm{D}}} \frac{\mathrm{d} \mathrm{W}}{\mathrm{W}}$
Using the straight and level velocity relation
$\mathrm{V}=\left[2 \mathrm{~W} /\left(\rho \mathrm{SC}_{\mathrm{L}}\right)\right]^{1 / 2}$
we get
$\mathbf{E}=-\int_{\mathbf{W} 1}^{\mathrm{W} 2} \frac{\eta_{\mathrm{p}}}{\gamma_{\mathrm{p}}}[\rho \mathrm{S} / 2]^{1 / 2} \frac{\mathrm{C}_{\mathrm{L}}^{3 / 2}}{\mathrm{C}_{\mathrm{D}}} \frac{\mathrm{dW}}{\mathrm{W}^{3 / 2}}$
The range integral can be written in a similar fashion as
$R=-\int_{W_{1}}^{W_{2}} \frac{\eta_{P}}{\gamma_{P}} \frac{C_{L}}{C_{D}} \frac{d W}{W}$
Now we need to consider the same flight schedules examined in the jet case. Constant angle of attack flight will give constant lift and drag coefficients and constant altitude will give constant density. We will also assume constant specific fuel consumption.
For range we need only to use the constant angle of attack assumption to give a simple integral. The resulting range is
$R=\frac{\eta_{P}}{\gamma_{P}} \frac{C_{L}}{C_{D}} \ln \left(\frac{W_{1}}{W_{2}}\right), \quad(\text { const } \alpha)$
For endurance we will consider two cases. The first holds both altitude and angle of attack constant, giving
$E=-\frac{\eta_{P}}{\gamma_{P}} \sqrt{\frac{\rho S}{2}} \frac{C_{L}^{3 / 2}}{C_{D}} \int_{W_{1}}^{W_{2}} \frac{d W}{W^{3 / 2}}$
which integrates to
$E=\frac{\eta_{P}}{\gamma_{P}} \sqrt{2 \rho S} \frac{C_{L}^{3 / 2}}{C_{D}}\left(\frac{1}{\sqrt{W_{2}}}-\frac{1}{\sqrt{W_{1}}}\right),\left(\begin{array}{l} \text { const } \alpha \ \text { const } \rho \end{array}\right)$
The second case has angle of attack and velocity constant
$E=-\frac{\eta_{P}}{\gamma_{P}} \frac{1}{V} \frac{C_{L}}{C_{D}} \int_{W_{1}}^{W_{2}} \frac{d W}{W}$
or
$E=\frac{\eta_{P}}{\gamma_{P}} \frac{1}{V} \frac{C_{L}}{C_{D}} \ln \left(\frac{W_{1}}{W_{2}}\right), \quad\left(\begin{array}{l} \text { const } \alpha \ \text { const } V \end{array}\right)$
This is the “drift up” flight schedule.
6.6 Wind Effects
The above range and endurance equations for both jet or prop aircraft were derived assuming no atmospheric winds. The speeds in the equations are the airspeeds, not speeds over the ground. If there is a wind the airspeed is, of course, not equal to the speed over the ground.
Endurance calculations are not altered by the presence of an atmospheric wind. If our concern is how long the aircraft can stay in the air at a given airspeed and altitude and we don’t particularly care if it is making progress over the ground we need not worry about winds. We are doing endurance calculations based only on the aerodynamic behavior of the airplane at a given speed and altitude in a mass of air.
Range is related to speed across the ground rather than the airspeed; thus, if there is a wind our range equation results need to be re-evaluated to account for the wind. The logic of this is simple: a headwind will slow progress over the ground and reduce range while a tailwind will increase range. What is not so obvious is how to correct the calculations to account for this wind. Since our usual concern is to find the maximum range, we will examine the correction for wind effects only for this optimum situation.
Maximum range for a jet was found to occur when D / V was a minimum while, for a prop, maximum range occurred at minimum drag conditions. The velocities for both cases can be determined graphically by finding the point of tangency for a line drawn from the zero velocity origin on either the drag versus velocity curve in the jet case or the power required versus velocity curve for the prop plane. We can use an extension of this graphical approach to find the speed for best range with either a head wind or a tail wind.
The important first step in determining optimum range in the presence of an atmospheric wind is to find a new airspeed for best range with a wind. This new speed will then be used to calculate a new value of the optimum range. The new value of best range airspeed is found as illustrated in the figures below. The first task is to draw a conventional drag versus velocity (for a jet) or power required versus velocity (for a prop) plot. To this plot is added a new origin, displaced to the left by the value of a tailwind or to the right by the magnitude of the tailwind. A line is then drawn from the displaced origin, tangent to the drag or power curve and the point of tangency locates the new velocity for optimum range with a wind. The magnitude of this new optimum range velocity is read with respect to the original origin (not the displaced origin). This speed is an airspeed, not a ground speed.
This new optimum range velocity is then used to find a new range value from the same equations developed previously. Using the new velocity, new values of lift and drag coefficients are first calculated and these new coefficients and velocity are used to find the optimum range with the wind. To this new range must be added another range which results purely from the aircraft’s time of exposure (endurance) to the wind. This endurance is also found using the newly found optimum range velocity and associated coefficients. The final corrected range for maximum range in a wind is
Rwith wind = RCorrected + VwECorrected for a tailwind
or
Rwith wind = RCorrected VwECorrected for a headwind
6.7 Let the Buyer Beware
Airplane manufacturers, like those of automobiles and other products, like to do anything they can to make their product look good and sometimes they hope that the buyer doesn’t look too closely at the contradictions in their specifications and advertising. A car may be advertised as having seating for five, an EPA fuel economy rating of 38 mpg, the ability to go 542 miles on a single tank of gas and a top speed of 120 miles per hour. Most of us, however, know not to expect that car to go 542 miles on a single tank of gas while carrying 5 people at a speed of 120 mph! Those who believe it will would also probably be dumb enough to pay sticker price.
What about airplanes? Is this product of an industry which is regulated at every step by the FAA just as subject to contradiction in specifications as a car?
Let’s look at a few simple examples taken from a general aviation Aircraft Fleet Directory of a few years back. A Cessna 150, the most widely used two place aircraft in the country, quotes a range of 815 nautical miles on 32 gallons (210 pounds) of fuel. The plane has an empty weight (no pilot, passenger, baggage or fuel) of 1104 pounds and a maximum gross takeoff weight of 1600 pounds. This means that with the full fuel tanks needed for maximum range there is only a 286 pound allowance for both pilot and passenger, hardly enough for two adults and luggage! This is why one of the favorite questions of flight examiners who are preparing for a private pilot check‑ride in a Cessna 150 involves weight and balance of the aircraft and why sometimes pilots may have to actually pump fuel out of an airplane before takeoff.
A Cessna 172, the most popular four place aircraft in the world, is a little better than the 150 cited above. It has an empty weight of 1387 pounds and to reach its advertised range of 742 miles it has a fuel tank which holds 288 pounds of gas. This gives a total weight for airplane and fuel of 1675 pounds. The maximum gross takeoff weight of the 172 is 2300 pounds, leaving 625 pounds allowance for four passengers and their stuff; an average of 156 pounds each! It is beginning to look like airplanes are designed like those “four-place” cars which have a rear seat about large enough to seat two small poodles!
With another Cessna product, the all around best of their 4 seat line , the Skylane, things are a little better. Its listed empty weight of 1707 pounds, range of 979 nautical miles on 474 pounds of fuel and max gross weight of 2950 pounds leave 769 pounds for pilot, passengers and accessories (192 pounds each). Finally an airplane for real people!
Lest the naive get the idea that this is only a problem for small single engine airplanes, let’s look at one more example, the eight place Learjet 25C. It claims a range of 2472 miles, just the ticket for the rich young business tycoon to fill with seven of her closest friends for a transcontinental weekend jaunt. The listed fuel capacity of 7393 pounds, adds to the quoted “zero fuel” weight of 11,400 pounds to give a 18,793 pound airplane. So how much is left for those 8 passengers? The listed max gross weight of the Learjet 25C is 15,000 pounds! With a full tank of gas the airplane is over its maximum allowable takeoff weight! With a 160 pound pilot and no other passengers or payload this airplane can carry enough fuel for a real range of about 1150 miles, less than half that advertised. Why claim a range of almost 2500 miles? Well, the fuel tanks are big enough to carry the needed fuel. If only the airplane could get off the ground!
Homework 6
1. An aircraft weighs 56,000 pounds and has 900ft2 wing area. Its drag polar equation is given by CD= 0.016 + 0.04CL2. The plane has a turbojet engine with constant thrust at any given altitude as shown below:
Table 6.1: Question 1
altitude (ft) 0 5000 10,000 15,000 20,000 25,000 30,000
thrust (lb) 6420 5810 5200 4590 4000 3360 2700
a. Find the minimum thrust required for straight and level flight and the corresponding true airspeeds at sea level and at 30,000 ft.
b. Find the minimum power required and the corresponding true airspeeds at sea level and 30,000 ft.
2. For the aircraft above:
a. plot thrust and drag vs Vefor straight and level flight.
b. plot altitude vs Vemax and Vmax for straight and level flight.
Figure 6.6: Maximum (& Min) Speed for Straight and Level Flight Versus Altitude
c. find the altitude for maximum true airspeed.
d. find the maximum obtainable altitude.
e. compare V at minimum drag from the plot and the calculation.
f. calculate (L/D)max.
References
Figure 6.1: Kindred Grey (2021). “Finding Velocity for Maximum Range .” CC BY 4.0. Adapted from James F. Marchman (2004). CC BY 4.0. Available from https://archive.org/details/6.1-updated
Figure 6.2: Kindred Grey (2021). “Velocities for Minimum Power and Drag.” CC BY 4.0. Adapted from James F. Marchman (2004). CC BY 4.0. Available from https://archive.org/details/6.2-updated
Figure 6.3: Kindred Grey (2021). “Speed for Best Range with Wind (Jet).” CC BY 4.0. Adapted from James F. Marchman (2004). CC BY 4.0. Available from https://archive.org/details/6.3-updated
Figure 6.4: Kindred Grey (2021). “Speed for Best Range with Wind (Prop).” CC BY 4.0. Adapted from James F. Marchman (2004). CC BY 4.0. Available from https://archive.org/details/6.4-updated
Figure 6.5: Kindred Grey (2021). “Thrust and Drag Versus V_e For Straight and Level Flight.” CC BY 4.0. Adapted from James F. Marchman (2004). CC BY 4.0. Available from https://archive.org/details/hw-6-part-1
Figure 6.6: Kindred Grey (2021). “Maximum (& Min) Speed for Straight and Level Flight Versus Altitude.” CC BY 4.0. Adapted from James F. Marchman (2004). CC BY 4.0. Available from https://archive.org/details/hw-6-part-2
1. The term “aero-plane” originally referred to a wing, a geometrically planar surface meant to support a vehicle in flight through the air. By 1903 the term had become associated with the entire flying vehicle. By the 1920s the American press and magazines had changed the word to “airplanes”; however, it is still common in Britain to see “aeroplane” used in books and papers.
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Chapter 7. Accelerated Performance: Takeoff and Landing
Introduction
To this point, all of our discussion has related to static or unaccelerated flight where F = ma = 0. Even in climb and descent we assumed “quasi-level” conditions where the forces on the aircraft summed to zero. If we are to look at the performance of an airplane during take‑off and landing we must, for the first time, consider acceleration (during takeoff) and deceleration (during landing). We will also have a couple of new forces to consider in the ground reaction force and ground friction.
In take-off, the airplane accelerates from zero groundspeed (but not necessarily zero airspeed!) to a speed at which it can lift itself from the ground. The thrust must exceed drag for acceleration to take place and the lift won’t equal weight until the moment of liftoff. The plane may accelerate along the ground at a given angle of attack (or lift coefficient) until the speed reaches the point where the dynamic pressure combines with the lift coefficient to give lift equal to the weight or it may accelerate at some angle of attack determined by its landing gear height until it reaches a speed which will give lift equals to weight when the aircraft is then rotated (tail down, nose up) to a higher angle of attack and lift coefficient.
Any pilot will tell you that take‑off and landing are what flight is all about. The thrill of full throttle and maximum acceleration as the plane roars down the runway, followed by the freeing of the soul which comes from cheating gravity and breaking the bond with the earth is incomparable. Of course the pilot hopes this occurs before the end of the runway is reached and in such a way as to allow clearance of the water tower at the end of the strip!
In landing, deceleration must be provided through braking, aerodynamic drag, ground friction and possibly reverse thrust to slow the plane to zero speed; hopefully before it reaches the end of the runway!
Landing is the ultimate challenge of person against nature as the pilot once again attempts to remain in control of a planned encounter with the ground in a vehicle moving at speeds which can result in instant mutilation and death if there is the slightest miscalculation of crosswind or downdraft. Of course, all of this must be done in such a manner as to assure the passenger that every move is as safe and natural and controlled as a Sunday afternoon drive to the golf course.
Wind will be a factor in take‑off and landing and one would think it would be obvious that the pilot should position the aircraft at the end of the runway which will result in operation into the wind. This will result in a reduction in the length of the ground roll in either take‑off or landing. To some, however this may not be obvious.
The author once sat on a graduate committee of a student in Transportation Engineering who had taken several courses in airport design. When asked what role the prevailing winds played in the design of airports the student appeared puzzled. Given a hint that it had something to do with the way the runways were aligned, he still drew a blank. Finally, when asked to draw a runway and show an airplane getting ready to take‑off at one end and to explain which way the wind would be blowing, the student’s eyes lit up in an apparent revelation of truth. He drew the runway horizontal across the center of the blackboard with the airplane at the right end, ready to begin a take‑off roll toward the left. Then he triumphantly drew an arrow to indicate a wind moving from right‑to‑left, the same direction as the motion of the aircraft!
As despair and gloom settled over the faculty in the room I, rather reluctantly, asked him why the airplane would take‑off in the same direction as the wind blew. He replied that the answer was obvious, “So the wind will carry the pollution away with the airplane!” Watch out for environmentalists who design airports!
To study aircraft performance in take‑off and landing we must make sure we have proper definitions of what these phases of flight entail. Then we must consider the forces acting on the airplane. We will begin this study by looking at take‑off.
7.1 Takeoff Performance
The definition used by the Federal Aviation Administration for take‑off includes the ground run from zero ground speed to the point where the wheels leave the ground, plus the distance required to clear a 50 foot obstacle. The distance over the ground for all of the above is computed at maximum gross weight at sea level standard conditions. The “worst case” condition is often also calculated for a hot day at high altitude (100ºF in Denver).
We will concern ourselves only with the ground run portion of the take‑off run, knowing that we can find the distance to clear a 50 foot obstacle from our climb equations. That climb would be calculated for maximum angle of climb conditions.
The first step in the calculation of the ground run needed for take‑off is an examination of the forces on the aircraft. In addition to the lift, drag, thrust and weight, we must now consider the ground friction and the “Resultant” force of the ground in supporting all or part of the weight of the aircraft. These are shown in the figure below. The coefficient of friction will depend on the ground surface and braking friction.
A summation of the vertical forces in Figure 7.1 gives
L + R ‑ W = 0
or
R = W ‑ L
Summing the horizontal forces gives
Note that in the above relation we have, for the first time, an acceleration. These forces change as the aircraft accelerates from rest to take‑off speed.
Combining the two equations above we have a single relation
which can be rearranged to give
Our desire is to integrate this or a related equation to get the time and distance needed for the take‑off ground run. To do this we must first account for the dependence of both lift and drag on velocity. This gives
where CLg denotes the lift coefficient during the takeoff or landing ground run and not that in flight or at the point of takeoff or touchdown itself.
The above equation still contains thrust and weight, both of which may well change during the take‑off ground run. Thrust is known to be a function of velocity, however, weight will be a function of the rate of fuel use (specific fuel consumption) and will be a function of time rather than speed. In order to keep our analysis relatively simple we will consider the weight change during the take‑off roll to be negligible and treat weight as a constant in the equation. We will use the thrust model which we derived from the Momentum Equation in Chapter 2,
T = T0 -aV2
In this equation T0 is the thrust at zero velocity or the “static thrust”, a is a constant (which could be zero) and T is the thrust at any speed. Substituting this model for thrust into our acceleration equation gives:
It should be noted that the velocity in this equation is the airspeed and not the speed relative to the ground. When we look at the take‑off distance we will have to be concerned with both the ground speed and the airspeed. The simplest case will be when there is no ground wind; ie, when the airspeed and ground speed are equal.
In the above relation all of the terms in brackets and parentheses are essentially constant for a given aircraft at a given runway altitude and for a given runway surface. The lift coefficient is given the special designation of CLg to denote that it is the value for the ground run only. In a normal take‑off roll the airplane accelerates to a pre-determined speed and then “rotates” to a higher angle of attack which will produce enough lift to result in lift‑off at that speed. Hence, the ground run lift coefficient will probably not be the same as the take‑off lift coefficient. The drag coefficient could be similarly subscripted; however, since CD is a function of CL and will subsequently be written in that manner, this will not be done at this point.
Since most of the terms in the equation can be treated as constants the equation can be simplified as follows:
dV/dt = A – BV2 ,
where
and
This acceleration relationship can be integrated to obtain the time for the ground run of take‑off.
Assuming that the airplane starts the take‑off run from rest and that there is no ground wind and that the upper limit is the take‑off velocity VTO, we have
NOTE: This may be the first time that the reader has ever seen an inverse hyperbolic tangent. What should follow is a frantic search of your calculator to see if there is any such key or combination of keys along with an equally harried check of the indices of high school and college trig and calculus texts to see just what the heck this thing is. Waiting to figure this out during a test could result in considerable embarrassment.
A question which should be considered here is “What is a good value for the take‑off speed?”
The very lowest speed at which the airplane can possibly lift off of the ground is the stall speed for straight and level flight at the runway altitude. It is, however, not safe to attempt takeoff at this minimum speed with the airplane right on the verge of stall. A somewhat higher than stall speed will give a margin of safety which will allow take‑off at a fairly low speed without risk of stall due to unexpected gusts or similar problems. Commonly used values for take-off speed range from 10 to 20 percent higher than straight and level stall speed.
1.1 Vstall < VTO < 1.2 Vstall .
We will assume the higher value,
VTO = 1.2 Vstall ,
unless told otherwise.
Far more important than the time required for the take-off ground run is the distance required. It is always nice to know that the pilot can get the airplane into the air before it reaches the end of the runway! To find the take‑off distance we must integrate over distance instead of time.
dV/dS = (dV/dt)/dS/dt) = (A – BV2) / V
Rearranging this gives
which is integrated to get
Finally
Now, assuming that the airplane starts from rest, no wind and lift off at VTO we have
We will later investigate the case of take‑off in a wind.
Before going further with an analytical analysis of the takeoff ground run it is worthwhile to pause and examine the physical aspects of the problem. These are too often lost in the equations, especially when we have hidden a lot of terms behind convenient terms like A and B. Let’s first write the last equation for take‑off distance in its full glory.
It is obvious from the above equation that many factors influence the take-off distance.
It is, for example, intuitive that the ground friction will retard take‑off. The retarding force due to friction will decrease as the lift increases during the take‑off run. So, it appears that it might be to our advantage to move down the runway at a high angle of attack such that high lift is generated which will result in a reduction in the friction force and enhance the airplane’s acceleration to take‑off speed. On the other hand, a high angle of attack will also give a high drag coefficient, retarding acceleration. At some point in the take‑off run the drag force will exceed the friction force. Does this mean the pilot should begin the take‑off run at a high angle of attack and then lower it to reduce drag so as to hold some friction/drag ratio at an optimal value?
What about the value of the friction coefficient? Do we use one type of ground run on a concrete runway and another on a grass strip? What about soft dirt? Typical values of friction coefficient are:
Table 7.1: Typical Values of Friction Coefficients
Concrete, asphalt 0.02 - 0.05
Hard Turf 0.04 - 0.05
Normal turf, short grass 0.05
Normal turf, long grass 0.07 - 0.10
Soft ground 0.10 - 0.30
One should probably use the lower of the above values for a particular surface unless instructed to do otherwise.
For a “soft field” take‑off such as on long grass or soft ground, pilots are taught to do several things to reduce the role of ground friction on the take‑off roll. Usually, the use of flaps is recommended to increase the lift coefficient and, if the airplane has a tricycle type of landing gear (nose wheel and two main wheels), the pilot is taught to keep the nose up, which will both reduce the friction on that wheel and give a higher angle of attack and lift coefficient. One reason for the popularity of the “tail dragger” style of aircraft in the early days of aviation was it’s natural superiority in soft field takeoffs, which were common at the airfields of the day.
7.2 Minimum take-off ground run:
In a normal take‑off, as mentioned earlier, the aircraft accelerates along the runway at a fairly constant angle of attack until the desired take‑off speed is reached. The plane is then rotated to give an increased angle of attack and lift coefficient such that lift equals or exceeds the weight, allowing lift‑off. The angle of attack during that ground roll and, hence the lift and drag coefficients, is largely determined by the relative lengths of the landing gear and the angle at which the wing is attached to the fuselage.
Many factors influence the size and placement of the landing gear. It is nice if the gear struts are long enough to keep the propeller from hitting the runway (this can be a real problem with a tail mounted prop) and it is also good if the center of gravity of the aircraft is between the main and auxillary gear. The main gear should be close to the CG to allow ease of rotation but far enough away to prevent inadvertent rotation. There is also the question of where the gear are stored in a retractable system
The wing angle of placement on the fuselage will primarily be a function of optimal cruise considerations such that things like the fuselage drag is minimized and pilot visibility is satisfactory when the wing is at the best combination of lift and drag coefficient for cruise as determined by using relations of previous chapters. It is also nice if, in cruise conditions, the aisle in a commercial aircraft is relatively level.
An important task for the designer is to find the wing angle of attack which will minimize the take‑off ground run and then to design the landing gear such that under normal conditions the plane sits on its gear with the wing at that angle. Let’s try to find that angle or, more precisely, the lift and drag coefficients at that angle of attack.
We first return to the equation for acceleration in the ground run.
Our desire is to maximize this acceleration at all times during the run. Assuming that the only variable we have is angle of attack, i.e., CL and CD, assuming that we have a parabolic drag polar, and further assuming that the take‑off speed VTO is independent of CLg, we can find the maximum acceleration by taking the derivative with respect to CLg and equating the result to zero. The assumption that VTO is independent of CLg means that the plane will be rotated at VTO to achieve lift off rather than allowed to continue to accelerate until lift off occurs at CLg.
or
This gives the best value of the ground run lift coefficient for minimum ground run length.
7.2.1 Airplane design note
This tells us that if we want to take-off in the shortest possible ground run distance we will design the airplane so that with a normal load distribution on the runway its wing will be at the angle of attack which will give the above value of lift coefficient. We may be able to do this by making the wheel struts or supports the right length. In other words, a well designed airplane will have its wing attached to the fuselage at an angle so the fuselage is level at cruise conditions and will have its landing gear height set to put the wing at the optimum take-off angle of attack at maximum gross weight conditions when sitting on the ground.
It might be interesting to see just how much difference having the optimum ground run lift coefficient makes by finding the best CLg for takeoff and then calculating the resulting takeoff distance as well as the distance at somewhat higher and lower values of CL.
7.2.2 Power based engine performance
A factor not previously noted in this discussion is that we have accounted for the output of the aircraft’s propulsion system in terms of thrust and not power. This was natural because we were dealing with force equations. What do we do when we have an aircraft which has a power based propulsion system (propeller)? We know that thrust is equal to power divided by velocity but how do we use that in the equations? Perhaps an example will provide an answer:
EXAMPLE 7.1
For an aircraft with the following properties find the minimum ground run distance at sea level standard conditions.
W = 56,000 lb
VTO = 1.15 Vstall
ηp = 0.75
S = 1000 sq. ft.
CD = 0.024 + 0.04CL2
TO= 13000 lb
CLmax = 2.2
μ = 0.025
Ps = 4800 hp
Let’s first find the stall speed and then the take‑off speed.
VTO = 1.15 Vstall = 168 fps.
Now we must face the problem of having power information and equations that demand thrust data. We have been given the static thrust and we can assume that the power available which was given will be the power in use at the moment of take‑off. We then have to determine how thrust varies and how to fit it to our assumed thrust versus velocity relation used in the take‑off acceleration equation.
At take-off speed
so, the thrust at take off is
TTO = Pav/VTO = 3600 hp / 168 fps = (1980000 ft-lb/sec) / 168 fps = 11786 lb
Our thrust versus velocity relationship is
T = T0 -aV2.
Substituting the takeoff speed and thrust and the static thrust we can find the value for a.
11786 lb = 13000 lb – a (168 fps)2
a = 0.0430 lb-sec2/ft2
Our thrust relationship to be used in the take‑off equations is then
T = 1300 – 0.0430 V2.
Now we need to determine the lift coefficient for minimum ground run.
The drag coefficient at the minimum ground run lift coefficient is:
CD = CD0 + KCL2 = 0.024 + 0.04(0.3125)2 = 0.0279
Finally we can use all of the above to determine the take‑off ground run.
S = 2314 ft.
7.3 Takeoff Without Rotation
As described previously, a conventional take‑off run would be made at the angle of attack dictated by the airplane configuration and the landing gear geometry, all of which has probably been designed to give near optimal ground run acceleration. When a predetermined take‑off speed is reached the pilot raises the nose of the aircraft to increase the angle of attack and give the lift needed for lift‑off. But, what would happen if, instead of rotating, the airplane was allowed to simply continue to accelerate until it gained enough speed to lift off without rotation?
Continued ground run acceleration to take‑off without rotation is not an optimum way to achieve flight. It will always require more runway than a conventional take‑off. There are, however, a limited number of aircraft which are designed for this type of lift-off. One well known example is the B-52 bomber. This aircraft has what might be described as “bicycle” type landing gear with the gear located entirely in the long fuselage and placed well fore and aft the center of gravity. This placement and the long, low fuselage make rotation virtually impossible. The result is the need for very long runways and very long, shallow approaches to landing.
Optimizing the takeoff run for an aircraft like the B-52 is different from the maximum acceleration optimum for a conventional take-off. Since the plane cannot rotate, the gear design and wing placement on the fuselage must be arranged such that the wing angle of attack is that desired for a safe and efficient lift-off. Too high an angle of attack might result in take-off at conditions too near stall and too low an angle might require too much runway. In the following example we look at such an aircraft where the design is such that the ground run angle of attack of the wing is set to give take-off at a speed 20% above stall speed.
EXAMPLE 7.2
The aircraft defined below is designed for take-off with no rotation, thus the ground run angle of attack (and, therefore, CL and CD) is the same as that at take-off. Find the take-off distance at sea level standard conditions.
W = 75,000 lb
CD = 0.02 + 0.05CL2
μ = 0.02
S = 2500 sq. ft.
CLmax = 1.5
T = T0 = 12,000 lb
VTO = 1.2 VSTALL
We must first find the stall speed, the take-off speed, and the related take-off (and, thus, ground run) lift coefficient.
VTO = 1.2 Vstall = 155.7 fps.
We can then find the lift coefficient for this take-off speed.
Actually we could have skipped some of the above as we realized the following.:
CLg = CLTO = [Vstall/VTO]2 CLmax = [1/1.2]2 (1.5) = 1.042
Using the above lift coefficient we find the drag coefficient.
CD = CD0 + KCL2 = 0.02 + 0.05(1.042)2 = 0.0742
We are now ready to find the takeoff distance.
S = 3324 ft.
7.4 Thrust Augmented Take-off
Although not commonly seen today, a technique once regularly used by military cargo aircraft and bombers like the B‑52 to reduce the take‑off distance involved the augmentation of ground run thrust through the use of strap‑on or built in solid rockets. This system was often referred to as JATO for jet assisted take‑off even though it used rockets and not jets. Calculation of ground runs for this type of take‑off require breaking the ground run distance integral into two parts to account for the two different levels of thrust used in the run. The resulting equation is as follows:
Let us return to the last example and see what happens if we try to shorten the ground run of this airplane by the use of 15,000 pounds of extra thrust obtained from JATO units which are fired for the first ten seconds of the ground run to boost the plane’s initial acceleration.
The total thrust during the first ten seconds of the ground run will be 27,000 pounds. Thus, for that portion of the run the A term in the ground run distance equation will be
The B term will not be changed.
Now we must determine the velocity of the aircraft at the end of this first ten seconds of acceleration since the limits on the distance equation are velocities. To find this we go to the relationship for take‑off ground run time
Since the initial velocity is zero and t1 – t2 = 10 sec we have
Solving gives the speed at the end of the augmented thrust portion of the take‑off run.
V1 = 107 fps.
At this point it wouldn’t hurt to check the units in the equations above and make sure that we really did end up with units of feet per second.
The entire distance for take‑off can now be found as follows:
STOTAL = 2480 ft.
The JATO boost in this example gave a 25% reduction in the ground run needed for takeoff. This could be important for such an aircraft if it is operating out of short, remote airfields often found in “third world” countries or in military operations.
7.5 Ground Wind Effects
Earlier we mentioned the importance of ground wind in the take‑off of aircraft. It is rare that a ground wind does not exist, thus, our “no‑wind” equations are, hopefully, worst case predictions since taking off into the wind will reduce the distance for the ground run. Finding the distance required for take‑off into a ground wind (assuming the pilot has the good sense to fly into the wind and not attempt a “downwind” take‑off) requires another look at the equations. Note: There are sometimes conditions such as “downhill” runways or end‑of‑runway obstacles which may at times necessitate a downwind takeoff.
In the take‑off equations it is important to realize that, as noted when first presented, the distance and acceleration are measured relative to the ground; however, the aerodynamic forces in the equations are obviously dependent on airspeed and not ground speed. We must consider this in our equations. In doing so we will use the following designations for different speeds:
VG = GROUND SPEED
VA = AIRSPEED
VW = WIND SPEED (PARALLEL TO RUNWAY)
This gives
VA = VG ± VW (+ if a head wind, – if a tail wind).
Returning to the basic equation of motion we have
dVG/dt = A – BVA2
However,
Thus
dVG/dt = dVA/dt = A – BVA2 .
So, to determine the time for take‑off we use
dt = dVA /(A – BVA2) .
To find take‑off distance we use
or
This becomes
Finally we have a differential which includes wind effects. We will write it only for the case of the headwind since this would be the normal situation.
Now, we must also note that the take‑off speed of the aircraft is airspeed and not ground speed. The time and distance equations above may be integrated above to give
Finally, realizing that take‑off usually starts from rest at zero ground speed (at t = 0), we obtain
Note that the take‑off speed in these equations is the airspeed for takeoff and not the ground speed.
7.6 Landing
Landing, like take‑off, is properly defined as having at least two parts; an “approach” over a 50 foot obstacle to touchdown and the landing ground run. These, in turn, might be divided into several other segments. The approach will usually not be a non‑powered glide as studied earlier. The normal approach to landing for most aircraft is a powered descent. The FAA definition of the landing terminal glide over an obstacle is, however based on an unpowered glide as the limiting case. We have already considered gliding flight and should be able to deal with this portion of flight. A real descent can be the most interesting portion of the flight for a pilot as he of she corrects for side‑winds, updrafts, and downdrafts while aiming for a hoped‑for touchdown point on the runway. All of this is done at a descent rate of about 500 feet per minute (about 8 mph).
Here we will concern ourselves with only the touchdown through full stop portion of the landing. Again our primary concern will be ground run distance with the hope that full stop occurs before the end of the runway.
The equations of motion for the landing ground run are identical to those for takeoff, however, the terms in the equations can assume very different magnitudes from those in take‑off. To slow the aircraft in its landing ground run high drag is desirable, negative or “reverse” thrust may be used, and brakes will be used during much of the run to greatly increase the friction term. The boundary conditions on the integrals are essentially reversed with the initial speed being the touchdown or “contact” speed and the final ground speed being zero; however the solution may need to be broken into several segments to account for a sequence of events as part of the landing ground roll.
Before we look at the equations let’s look at a typical landing as seen by a small plane, general aviation pilot. The approach‑to‑landing descent will probably be made using full flaps, at least in its final “glide” (this will be true for almost any aircraft). This will lower the stall speed and allow approach and touchdown at a lower flight speed. It will also steepen the approach glide and, on the ground, add to the drag to help slow the aircraft.
As soon as the pilot feels that the aircraft is under full control after touchdown he or she may raise the flaps. While this reduces the drag and contributes to a longer ground roll, it also reduces the lift, increasing ground friction forces and allowing better directional control of the aircraft in a crosswind. After this is done the brakes will be applied to further slow the aircraft to a stop. Larger, jet aircraft may apply reverse thrust very soon after touchdown and before use of brakes to improve deceleration.
Now, let’s look again at the equations of motion for an aircraft on the ground. We can still use
dS = VdV / (A – BV2) .
We define VC as the speed of initial ground contact on landing at some point defined as S1 and conditions at the next point in the ground roll sequence as S2 and V2, giving the following integrated equation:
or
In the rare case where none of the parameters in the equation (T, μ, CLg, etc.) change during the ground run; i.e., where the airplane simply touches down and coasts to a stop, our final speed V2 = 0, giving
For a first estimate of a minimum landing ground run one could assume that the pilot is able to apply the brakes almost instantly after touchdown and that thrust is simply zero during the entire roll and, thus, use the above equation to calculate a landing ground roll distance. In reality, as mentioned previously, the ground roll would have to be determined by adding a series of the previous ΔS equations, each with its appropriate starting and ending speeds and values for thrust and friction and the like.
The time for the landing ground roll is found from
dV/dt = A – BV2
or
Integration of this equation can take several different forms depending on the relative magnitudes and signs of A and B. Looking again at these terms
note that A will almost always be negative since thrust will always be zero or negative, if not at touchdown, then very quickly thereafter. Braking forces could also be large enough to make B negative, depending on the relative magnitudes of the lift and drag coefficients. In various landing situations it may be possible to have any combination of negative or positive terms and this affects the form of the integral. The difficulty arises in the fact that integration gives a square root of the product of A and B as well as other terms with square roots of A and B individually or ratios of A and B. The result can be an imaginary answer if the correct solution is not chosen.
The time of landing ground roll solution is given for the four possible combinations of A and B below.
0, B>0 :t2−t1=12⋅ABlnA+V⋅BA−VB” title=”1. A>0, B>0 :t2−t1=12⋅ABlnA+V⋅BA−VB” class=”mathml mathjax”> 0, B<0 :t2−t1=1−ABtan−1(V−BA)” title=”2. A>0, B<0 :t2−t1=1−ABtan−1(V−BA)” class=”mathml mathjax”> 0 :t2−t1=1−ABtan−1(VB−A)” title=”3. A<0, B>0 :t2−t1=1−ABtan−1(VB−A)” class=”mathml mathjax”> <img src=”https://pressbooks.lib.vt.edu/app/up...f3594162d9.png” alt=”4. A<0, B<0 :t2−t1=12ABln[V−B−−AV−B+−A]” title=”4. A<0, B
7.7 Effect of Wind on Landing Ground Roll
As in the case of taking off, all landings should be made into the wind (with the same exceptions noted for take-off). The equations must then be written to account for the different velocity terms. This is done exactly as it was for the take-off case.
dVg/dS = (dVg/dt)/(dS/dt) = [A – BVA2] / Vg
or
Vg(dVg/dS) = A – BVA2 = Vg(dVg/dS)
For the headwind case this gives:
(VA Vw)(dVA/dS) = A – BVA2
and
Integrating and noting that when the aircraft has come to rest on the ground the velocity will equal that of the wind component along the runway VW,
The last term is evaluated using the time equation already discussed.
EXAMPLE 7.3
The following aircraft touches down in landing at a speed 30% above its stall speed. The pilot applies the brakes when the plane has slowed to 80% of its touchdown speed. If there is no wind, find the distance required for the aircraft to come to a complete stop on the runway.
W = 30,000 lb
μB = 0.5
μ= 0.02
S = 750 sq.ft
CLmax = 2.2 (with flaps)
Assume that the lift‑to drag ratio at 1.3 times the stall speed has a value of eight and is constant throughout the ground roll and that thrust is zero at touchdown and throughout the ground roll.
Since everything is related to the stall speed we will first find its value.
giving a touchdown speed of
Vc = 1.3Vstall = 160.7 fps
This speed gives a lift coefficient of
We will assume this lift coefficient is constant through the ground run.
We were not given a drag polar equation or its constants but we do know the lift-to-drag ratio and can find the drag and drag coefficient as follows:
DVC = W/(L/D) = 3750 lb, CDg = D/[½ρVC2S] = 0.1627 .
Now we can find the A and B terms for the distance solution. We must solve for the distance in two parts, the distance between touchdown and application of the brakes and the remaining distance to full stop.
Before braking
giving a distance of
After braking
giving the rest of the ground roll distance as:
The total ground roll in landing is the sum of the two distances above:
STOT = S1 + S2 = 2075.4 ft.
7.8 FAA AND OTHER DEFINITIONS OF TAKEOFF AND LANDING PARAMETERS
7.8.1 Takeoff
As discussed earlier, there are many components which may be included in the calculations of takeoff and landing distances. In the previous calculations only the actual ground run distances were considered and these, especially during landing, may be composed of multiple segments where different values of friction coefficient and thrust apply. A complete look at takeoff must also include the distance between the initiation of rotation and the establishment of a constant rate of climb and the distance needed to clear a defined obstacle height as shown in the figure below.
Several different terms may be used in a complete discussion of takeoff. These include the following:
Ground Roll: The distance from the start of the ground run or release of brakes until the point where the wheels leave the ground. This includes the distance needed to achieve the needed lift to equal the weight during rotation. The takeoff velocity must be at least 1.1 times the stall speed and is normally specified as between 1.1 and 1.2 times that speed.
Obstacle Clearance Distance: The distance between the point of brake release and that where a specified altitude is reached. This altitude is usually defined as 50 feet for military or smaller civil aviation aircraft and 35 feet for commercial aircraft.
Balanced Field Length: The length of the field required for safe completion of takeoff should one engine on a multi‑engine aircraft fail at the worst possible time during takeoff ground run. This distance includes the obstacle clearance distance. The balanced field length is sometimes also called the FAR Takeoff Field Length because it is a requirement for FAA certification in FAR 25 for commercial aircraft and includes the 35 foot obstacle clearance minimum. In the early part of the takeoff ground run the loss of one engine would usually lead to a decision to abort the takeoff, apply brakes and come to a stop. The “worst possible time” for engine failure would be when it is no longer possible to stop the aircraft before reaching the end of the runway and the decision must be made to continue the takeoff with one engine out.
Decision Speed (V1): The speed at which the distance to stop after the failure of one engine exactly equals the distance to continue takeoff on the remaining engines and to clear the FAA defined obstacles. In calculating this speed one cannot assume the possibility of using reverse thrust as part of the braking process.
7.8.2 Landing
As in takeoff, landing includes several possible segments as shown in Figure 7.3. Our previous calculations included only the actual ground roll distance but a complete definition may also include the portion of the approach needed to clear a defined obstacle and that needed to transition from a steady approach glide to touchdown (the “flare distance”). Note that the landing ground run could also include portions with reversed thrust used alone or with the brakes.
The weight of the aircraft at landing is normally less than that at takeoff due to the use of fuel during the flight, however it is common to calculate the landing distance of trainer aircraft and of most propeller driven aircraft at takeoff weight. For non‑trainer jets, landing weight is normally assumed to be 85% of the takeoff weight. Military requirements usually assume landing with a full payload and about half of the fuel.
As in takeoff, there are several definitions associated with landing which should be familiar to the performance engineer:
FAR 23 Landing Field Length: This distance includes that needed to clear a 50 foot obstacle at approach speed flying down a defined approach glidepath (normally about 3 degrees). Touchdown is usually at about 1.15 times the stall speed. This total distance is usually about twice that of the calculated ground roll distance. This distance is normally about the same as that specified in requests for proposals for military aircraft.
FAR25 Landing Field Length: This distance adds to that of FAR 23 above an arbitrary two-thirds as a safety margin.
Homework 7
1. An aircraft has the following specifications:
W = 24,000 lb
S = 600 ft2
CD0 = 0.02
K = 0.056
This aircraft has run out of fuel at an altitude of 30,000 ft. Find the initial and final values of its airspeed for best range, the glide angle for best range, its rate of descent at this speed, and the time taken to descend to sea level at this speed.
2. For the aircraft above, assume a sea level thrust of 6,000 pounds and assume that thrust at altitude is equal to the sea level thrust times the density ratio (sigma). Find the true airspeeds for best rate of climb at sea level, at 20,000 ft, 30,000 ft and 40,000 ft. Also find the ceiling altitude.
3. For an aircraft where:
W = 10,000 lb
W/S = 50 psf
CD0 = 0.015
K = 0.02
Find the best rate of climb and the velocity for best rate of climb at sea level where T = constant = 4,000 lb and at an altitude of 40,000 ft where T = 2,000 lb.
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textbooks/eng/Aerospace_Engineering/Aerodynamics_and_Aircraft_Performance_(Marchman)/07%3A_Accelerated_Performance-_Takeoff_and_Landing.txt
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Introduction
Historical Introduction
Thus far all of our performance study has involved straight line flight. Unfortunately, unless our airplane is flying from a runway that is exactly in line with our destination runway and there is no wind on the route, straight line flight isn’t very practical! We need to be able to turn.
While the need to be able to turn is fairly obvious to us, a look at early aviation will show that it often was the last thing on the mind of many aviation pioneers. The uniqueness of the Wright Flyer was not its ability to fly a few feet in a straight line over the sand at Kitty Hawk. It was unique in its ability to turn and maneuver. There are claims that earlier experimenters in England, France, Russia, and the United States may indeed have made short, uncontrolled “hops” or even legitimate straight line “flights” in powered vehicles before December 17, 1903 but there are no claims for “controlled” flight of a powered, heavier than air, man (or woman) carrying vehicle prior to this date.
There are several ways to turn a vehicle in flight. Early experimenters such as Otto Lilienthal in Germany and Octave Chanute in this country knew that shifting the weight of the “pilot” suspended beneath their early “hang gliders” would tilt or “bank the wings to allow turns. Others such as Samuel P. Langley, the turn of the century director of the Smithsonian who had government funding to build and fly the first airplane, designed their craft to be steered with a rudder like a ship. Neither method of turning was very efficient. Langley’s heavier than air powered models, for example, flew very well but couldn’t adjust for winds and flew in long circles instead of a straight line as they had been designed to do.
Banking the wings (called the “aero‑planes” in the 1890’s) tilts the lift force to the side and the sideward component of the lift results in a turn, but since part of the lift is now being used to turn the vehicle the remaining lift may not be enough to oppose the weight unless additional power is added. Using a rudder alone results in a side force on the fuselage of the aircraft and, hence, a turning force. The resulting turn, by causing one wing to move forward faster than the other, usually leads to a bank. Neither method, employed alone, provides a very satisfactory means of turning and the result is usually a very large radius turn. Many early experimenters, in fact, appeared to want to turn without banking since they were “flying” very close to the ground and a banked wing might touch the ground and cause a crash.
The Wright brothers designed a complex mechanism involving coordinated rudders and twisting of wings to combine both roll and yaw in a “coordinated”, efficient turn. Their original purpose may have actually been to try to bank the airplane opposite the turn to prevent ground contact by the wing but they found that a properly coordnated turn could make their vehicle quite maneuverable. When the Wrights took their aircraft to Europe in 1908 they amazed European aviators with their craft’s ability to turn and maneuver. French airplanes, which were the most sophisticated in Europe, used only rudders to turn. The Wright Flyer, with its “wing warping” system and coordinated rudder, was literally able to fly circles around the French aircraft.
The Wrights had made this system of ropes and pulleys which connected rudder to twisting wing tips to a cradle under the pilot’s body, the central focus of their patent on the airplane. When world motorcycle high speed record holder and engine designer Glenn Curtiss, with funding from Alexander Graham Bell and others, built and flew an airplane with performance as good as or better than the Wright Flyer, the Wrights sued for patent violation. The Curtiss planes, which used either small, separate wings near the wing tips or wingtip mounted triangular flaps (later to be called ailerons) and which relied on pilot operation of separate controls like today’s stick and rudder system, were able to achieve the same turning performance as the Wrights Flyer. Curtiss, a much more flamboyant and public figure than either of the Wrights, quickly captured the attention and imagination of the American public, infuriating the Wrights who had shunned public attention while convincing themselves that no one else was capable of duplicating their aerial feats.
The decade long court battle between Curtiss and the Wright family over patent rights to devices capable of efficiently turning an airplane is credited by most historians as allowing European aviators and designers to forge far ahead of Americans. The Wrights were so absorbed with protecting their patent that they made no further efforts to improve the airplane and the threat of a Wright lawsuit kept all American airplane designers but Curtiss out of the business. Curtiss, whose lack of respect for caution had earlier enabled him to set the world motorized speed record on a motorcycle with a V‑8 engine, with moral and financial support from Bell and Henry Ford and others, kept the patent suit in court through appeal after appeal and continued to build and sell airplanes. To get around the Wright patent, Curtiss, at one time built his aircraft without ailerons or other roll controls and then shipped them to nearby Canada where one of Bell’s companies added the ailerons before the planes were shipped to customers in Europe! Meanwhile, Curtiss continued to experiment and innovate and it is no accident that when the first World War drew American participation it was Curtiss and not Wright aircraft that went to war. After the war it was the famed Curtiss “Jenny” that brought the “barnstorming” age of aviation to all America.
I hope the reader will pardon the above slip into historical fascination. By now some of you are asking what the heck all of this has to do with aircraft performance in turns? The facts are, however, that the first ten to fifteen years of American flight really were dominated by the airplane’s ability to turn.
8.1 The Mechanics of a Turn
To keep the physics of our discussion as simple as possible, lets consider only turns at constant radius in a horizontal plane. This is the ideal turn with no loss or gain of altitude which every student pilot practices by flying in circles around some farmer’s silo or other prominent landmark.
Our objectives in looking at turning performance will be to find things like the maximum rate of turn and the minimum turning radius and to determine the power or thrust needed to maintain such turns. We will begin by looking at two types of turns.
Today’s airplanes, in general, make turns using the same techniques pioneered by the Wrights and improved by Curtiss; coordinated turns using rudder and aileron controls to combine roll and yaw. The primary exception would be found in some evasive turning maneuvers made by military aircraft and in the everyday turns of most student pilots!
Non‑winged vehicles such as missiles, airships, and submarines still make turns like those of early French aviators and of Langleys “Aerodrome”, using rudder and body or fuselage sideforce to generate a “skiding” turn. We will examine this technique before looking at the more sophisticated coordinated turn.
The acceleration in any turn of radius $R$ is given by the following relation:
$a_r = \dfrac{V^2}{R}. \nonumber$
This acceleration is directed radially inward toward the center of the circle and is properly termed the centripital acceleration.
We can also consider the acceleration from the perspective of the rate of change of the “heading angle”, ψ, as shown in the figure below.
The “skid to turn” technique is illustrated below for a constant radius, horizontal turn. A rudder (or even vectored thrust) is used to angle the vehicle and the sideforce created by the flow over the yawed body creates the desired acceleration.
The equations of motion become
L – W = 0
Y = mV(dψ/dt) = m(V2/R)
For the skid turn examined above, the turning rate and radius depend on the amount of side force which can be generated on the body of the vehicle. Note that the lift (or buoyancy in the cases of submarines and airships) does not enter into the problem.
R = (WV2)/(gY) , dψ/dt = (gY)/(WV)
With the above relationships one can determine the radius and rate for a horizontal turn where body force, not wing lift, carries the vehicle through a turn. This is how non-winged rockets and missiles turn. Airplanes use a much more powerful force, the lift, to turn. By using the lift to provide the needed sideforce and to counteract the weight in a coordinated fashion an airplane can make a much more efficient turn than missiles or dirigibles.
Let’s look at the coordinated turn. In the ideal coordinated turn as illustrated in Figure 8.3, the aerodynamic lift is used to balance the weight such that horizontal flight is maintained and to provide a side force which produces the desired turning acceleration. No actual side force is generated on the fuselage of the aircraft. This type of turn requires the pilot to use the rudder and the ailerons and the throttle to give the ideal balance of bank angle and forces which will create a constant radius turn and maintain altitude.
If this turn is properly coordinated the resulting combined acceleration and gravitational force felt by both airplane and pilot will be directed “down” along the vertical axis of the aircraft and will be felt by the pilot as an increased force into the seat. The improperly coordinated will be felt as including a side force pushing the pilot left or right in the seat. These same forces act on the “ball” in the aircraft’s “turn‑slip” indicator, moving the ball off center in an uncoordinated turn. We will look at the turn-slip indicator later.
If a turn is not coordinated several results may occur. The turning radius will not be constant and the airplane will either “skid” outward to a larger radius turn or “slip” inward to a smaller radius. There could also be a gain or loss of altitude.
In the coordinated turn, part of the lift produced by the wing is used to create the turning acceleration. The remainder of the lift must still counteract the weight to maintain horizontal flight.
We now find ourselves looking for the first time at a situation where lift is not equal to weight. In a coordinated turn the lift must be greater than the weight and we define a “load factor”, n, to account for this inequality.
$L = nW$
This load factor can then be related to the bank angle used in the turn, to the turn radius, and to the rate of turn. Returning to the vertical force balance equation we have
$\mathbf{L} \cos \varphi-\mathbf{W}=\mathbf{n} \mathbf{W} \cos \varphi-\mathbf{W}=\mathbf{0}$
which gives:
$\cos \varphi=1 / \mathrm{n}$
Using the other equation of motion we can find the turn radius
$\mathbf{R}=\left(\mathbf{m} \mathbf{V}^{2}\right) /(\mathbf{L} \sin \varphi)=\left(\mathbf{m} \mathbf{V}^{2}\right) /(\mathbf{n} \mathbf{W} \sin \varphi)=\left(\mathbf{V}^{2} / \mathbf{n g}\right)(\mathbf{1} / \sin \varphi)$
Knowing that the cosine of the bank angle is equal to 1/n we can find the value of the sine of the bank angle by constructing a right triangle
hence,
and the turning radius becomes
R = (V2/g){ 1 / [n2 – 1]1/2} .
In dealing with turns we must remember that lift is no longer equal to weight. The lift coefficient is then
therefore
The above allows us to write the turning radius in another manner,
It should be noted here that if a small turning radius is desired a high load factor and lift coefficient are needed and low altitude will help. High wing loading (W/S) will also allow a tighter turn.
The rate of turn in a coordinated turn is
or
Alternatively,
The same factors which contribute to small turning radii give high rates of turn.
8.2 Load factor (n)
From the above equations it is obvious that the load factor plays an important role in turns. In straight and level flight the load factor, n, is 1. In maneuvers of any kind the load factor will be different than 1. In a turn such as those described it is obvious that n will exceed 1. The same is true in maneuvers such as “pull ups”.
The load factor is simply a function of the amount of lift needed to perform a given maneuver. If the required bank angle for a coordinated turn is 60° the load factor must equal 2. This means that the lift is equal to twice the weight of the aircraft and that the structure of the aircraft must be sufficient to carry that load. It also means that the pilot and passengers must be able to tolerate the loading imposed on them by this turn, a load which is forcing their body into their seat with an effect twice that of normal gravity. This “2g” load or acceleration is also forcing their blood from their heads to their feet and having other interesting effects on the human body.
If we look at the lift relation
L = nmaxW = CLmaxρV2S]
we see that the maximum lift and therefore the maximum load factor that may be generated aerodynamically is a function of the maximum lift coefficient (stall conditions).
One must realize that the aircraft, or, more precisely, its wings, may be capable of generating far higher load factors than either the pilot and passengers or the aircraft structure may be able to tolerate. It is not hard to design aircraft which can tolerate far higher “g‑loads” than the human body, even when the body is in a prone position in a specially designed seat and uniform. Engineers in the industry will tell you that they could design far more agile fighters at much lower cost if the military didn’t insist on having pilots in the cockpit!
All aircraft, from a Cessna 152 to the X‑31, are designed to tolerate certain load factors. The aerobatic version of the Cessna 152 is certified to tolerate a higher load factor than the “commuter” version of that aircraft. An aerobatic aircraft must be designed for a load factor of 6.
The FAA also imposes certain flight restrictions on commercial aircraft based on passenger comfort. It is possible to do aerobatics in a Boeing 777 but most of the passengers wouldn’t like it. Passenger carrying commercial flight is therefore normally restricted to “g‑loads” of 1.5 or less even though the aircraft themselves are capable of much more.
8.3 The two-minute turn
General aviation pilots are usually familiar with the “standard rate” or “two‑minute” turn. This turn, at a rate of three‑degrees per second (0.05236 rad/sec), is used in maneuvers under controlled instrument flight conditions. To make such a turn the pilot uses an instrument called a “turn‑slip” indicator. This instrument, illustrated below, consists of a gyroscope which is partially restrained and attached to a needle indicator, and a curved tube containing a ball in white kerosene. As the airplane turns, the gyroscope deflects the indicator needle as it attempts to remain fixed in orientation. The “precession” force of the gyroscope and the resulting needle displacement is proportional to the turn rate. The accuracy of this indication is not dependent on the degree to which the turn is coordinated. The ball in the curved tube will stay centered if the turn is coordinated while it will move to the side (right or left) if it is not coordinated. One of the sets of markings on the face of the instrument indicates a two minute turn.
8.4 The Turn-Slip Indicator
To make a two‑minute turn the pilot need only place the aircraft in a turn such that the needle is at the standard turn indication in the desired direction. To turn 90º the turn rate is maintained for 30 seconds, one minute for 180º, etc. The vertical speed indicator (rate of climb) is used to maintain altitude and the ball is kept centered to coordinate the turn.
Many pilots are taught, incorrectly, that the two‑minute turn mark on the turn-slip indicator is an indication of a 15 degree bank angle, with the next mark being 30º and so on. Some pilots even refer to the turn‑slip indicator as the “turn‑bank” indicator when the instrument has absolutely no way to detect bank. It is possible, using a “cross control” technique, to turn the aircraft via yaw with no bank (much like a missile turns) and see that the instrument indicates the correct rate of turn even though there is no bank and, similarly, the aircraft may be placed in roll without turning and the indicator will remain centered.
Why would this error in flight instruction occur? The answer lies partly in the difficulty in eradicating longstanding lore and partly in the fact that, for a small general aviation trainer airplane, a coordinated two‑minute turn does occur at about a 15 degree bank angle. Let’s look at the numbers.
From our previous equations we have
Inserting fifteen degrees as the bank angle and a two minute turn rate (0.05236 rad/sec) gives a velocity of 165 ft/sec or 112 mph. This is indeed close to the speed at which such an airplane would fly in a turn. If we, however, look at a faster aircraft, lets say one that is operating at 350 miles per hour, and use the two‑minute rate of turn we get a very different bank angle of 30 degrees!
Suppose you are a passenger in a Boeing 737 traveling at 600 mph and the pilot set up a two minute turn. This would give a bank angle of 55 degrees. It would also give a load factor of 1.75! This is higher than the FAA allows for airline operations. For this reason airliners use turn rates slower than the two minute turn in flight and only make two minute turns at low speeds, perhaps when operating in the “pattern” around airports.
8.5 Instantaneous versus sustained turn conditions
The previously derived relations will give the instantaneous turn rate and radius for a given set of flight conditions. In other words, for a given set of initial flight conditions we can determine the turn rate and radius, etc. Another question which must be asked is “Can the airplane sustain that turn rate?” The pilot may be able to, for example, place the plane in a 60 degree bank at 250 mph but may find that there is not enough engine thrust to hold that speed and bank angle while maintaining altitude.
Example $1$
For the airplane with the specifications below find the maximum turn rate and minimum radius of turn and the speeds at which they occur. Also determine if this turn can be sustained at sea level standard conditions.
W/S = 59.88 lb/ft2
S = 167 ft2
CLmax = 1.5
nmax = 6
CD0 = 0.018
K = 0.064
Tmax = 5000 lb
The maximum turning rate is
= 0.424 rad/sec = 24.29 o/sec .
The velocity for this turn rate is
The minimum turning radius is
Now we must see if the plane has enough thrust to operate at these conditions. The drag coefficient at maximum lift coefficient is
CD = 0.018 + 0.064 CLmax2 = 0.162 .
At the speed found above the drag is then
This drag exceeds the thrust available from the aircraft engine!
If the above aircraft enters a coordinated turn at the maximum turn rate it will quickly slow to a lower speed and turning rate with a larger turn radius or it will lose altitude.
8.6 The V‑n or V‑g Diagram
A plot which is sometimes used to examine the combination of aircraft structural and aerodynamic limitations related to load factor is the V‑n or V‑g diagram. This is a plot of load factor n versus velocity.
We know that when lift exceeds weight
We know that one limit is imposed by stall
Rearranging this we can write
and we can rearrange this as
Plotting n versus V will then give a curve like that shown below.
We can also consider negative load factors which will relate to “inverted” stall; ie, stall at negative angle of attack. At negative angle of attack, unless the wing is untwisted and constructed of symmetrical airfoil sections, CLmax will be different from that at positive angle of attack. This will give a different but similar curve below the axis. Combining this with the plot above gives the following plot.
To the left of this curve is the post‑stall flight region which, with the exception of high performance military aircraft, represents a out‑of‑bounds area for flight.
Other limits must also be considered. There will obviously be an upper speed limit such as that found earlier for straight and level flight. There will also be limits imposed by the structural design of the aircraft. Depending on the aircraft’s structural category (utility, aerobatic, etc.) it will be designed to structurally absorb load factors up to a given limit at positive angle of attack and another limit at negative angle of attack. Once these are defined, the complete V‑n diagram denotes an operating envelope in terms of load factor limits.
The point where the structural limit line and the stall limit intersect is termed a “corner point”. The velocity at this point is limited by both maximum structural load factor and CLmax. The velocity at that point is
At speeds below the “corner velocity” it is impossible to structurally damage the airplane aerodynamically because the plane will stall before damage can occur. At speeds above this value it is possible to place the aircraft in a maneuver which will result in structural damage, provided the plane has sufficient thrust to reach that speed and load.
It is possible for a wind “gust” to cause loads which exceed the above limits. Such gusts may be part of what is referred to as wind shear and are common around thunderstorms or mountain ridges. Gusts can be in either the vertical or horizontal direction. The primary effect of a horizontal gust is to increase or decrease the likelihood of stall due to the change in speed relative to the wing. This is often the cause of wind shear accidents around airports where the aircraft is operating at near-stall conditions.
If a gust is vertical, we can look at its effect in terms of change of angle of attack. Suppose we have a vertical gust of magnitude wg. Its effect on the angle of attack and CL is seen below.
so the change in lift is
Thus
If, for example, an aircraft in straight and level flight encounters a vertical gust of magnitude wg the new load factor is
(for straight and level flight n = 1)
The effect of the gust on the load factor is therefore amplified by the flight speed V. This effect can be plotted on the V‑n diagram to see if it results in stall or structural failure.
For the case illustrated above, the gust will cause stall if it occurs at a flight speed below Va and can cause structural failure if it occurs at speeds above Vb.
Homework 8
We wish to compare the performance of two different types of “General Aviation” aircraft; the popular Cessna Citation III business jet and the best all-around, four place, single engine, piston plane in the business, the Cessna 182. Approximate aerodynamic and performance characteristics are given in the table below:
Table 8.1: Aerodynamic and Performance Characteristics
CITATION III CESSNA 182
Wingspan 53.3 ft 35.8ft
Wing area 318 ft^2 174 ft^2
Normal gross weight 19,815 lb 2,950 lb
Total thrust at sea level 7300 lb ---------
Usable power at sea level --------- 230 hp
C_D0 0.02 0.025
Oswald Efficiency Factor (e) 0.81 0.80
1. Calculate and tabulate the thrust required (drag) versus Ve data for both aircrafts and plot the results on the same graph [Figure 8.11]. Plot the sea level thrust available curves for both aircrafts on the same graph [Figure 8.11].
2. Calculate the maximum velocity at sea level for both aircraft and compare with that indicated on the graph [Figure 8.11].
3. Calculate and tabulate the power required versus Ve data for both aircraft and plot each on a separate graph [Figures 8.12 and 8.13]. Plot the sea level power available on the same graphs.
Figure 8.12: Power at Sea Level For Prop
8.13 Power at Sea Level for Jet
4. Calculate and tabulate the rate of climb (in ft/min) versus velocity data at sea level for both aircraft for normal gross weight and plot the data on the same graph [Figure 8.14].
Figure 8.14: Rate of Climb at Sea Level for Citation III and C-182
5. Calculate and tabulate the maximum rate of climb versus altitude data for both aircraft and plot it on the same graph. [Figure 8.15]. Determine the absolute ceilings of both aircraft.
6. Calculate the time required to climb from sea level to 20,000 ft for both aircraft. Assume that the curves (in 5) are close enough to linear to use a linear approximation for the calculation.
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Chapter 9. The Role of Performance in Aircraft Design: Constraint Analysis
Introduction
In the proceeding chapters we have looked at many aspects of basic aircraft performance. These included takeoff and landing, turns, straight and level flight in cruise, and climb. If we were to look at the relationships we found for any of these we could see how we might design an airplane to best accomplish the task at hand. In other words, if we wanted to design an aircraft that could takeoff and land in a very short distance we can look at the takeoff and landing distance equations and identify the factors that would minimize these distances. To takeoff in a short distance we might want a high maximum lift coefficient to get a low takeoff speed, a large wing area to give a lot of lift at low speed, and a lot of thrust to accelerate to takeoff distance in as short a ground run as possible. To land in a short distance we might want to also design a plane with a large wing and high maximum lift coefficient but now the thrust isn’t as important as the amount of braking friction available unless it is reverse thrust that we are talking about.
Of course there are limits to be considered. High thrust will minimize the takeoff ground run but once thrust becomes as high as the weight of the plane we might as well take off vertically! And a big wing area gives us high drag along with high lift. Nonetheless, we can see that three parameters; thrust, weight, and wing area, are important factors to consider in takeoff.
We would find, if we looked at the equations we derived for the other types of flight mentioned above that these same three parameters pop up everywhere. The only problem is that we would find that their relationships in cruise aren’t necessarily the same as they are in takeoff and landing. And they may be different still in climb.
If we want an airplane that only does one thing well we need only look at that one thing. If, for example, we went all out to create a plane that could takeoff in a very short distance and then look at its performance in straight and level cruise we would probably find that it isn’t very good. And its climbing performance may be even worse!
This isn’t really much different from designing any other product that is capable of more than one task. A car can be designed to go really fast or to get really good gas mileage, but probably not both. Car tires can be designed to have high traction in mud and snow or to give great mileage at highway speeds but any attempt to design an “all weather touring” tire will result in a compromise with less traction than a mud and snow tire and poorer performance at high speeds than the high speed highway tire design.
The question with the design of an airplane as with a car or a tire, is how do we arrive at the best compromise that will result in a good all around design while still being better than average in one or two desired areas?
One way to approach this would be to go back to the equations in earlier chapters and iterate among them, trying to find wing areas, weights, and engine sizes that would accomplish our design objectives. We might start with cruise since a certain minimum range is often a design objective. We know that in cruise since lift must equal weight, we can select a design value of cruise lift coefficient (commonly around 0.2 to 0.3) and a desired cruise speed and altitude and solve for the needed wing area.
S = 2W/(ρV2CL)
If our desire is to look at an optimum range we might want to find the ratio of lift to drag that will maximize range (for example, for a propeller driven plane Rmax occurs with flight at [L/D]max or at minimum drag conditions). Finding this value of drag would set the thrust we need for cruise. Within all this we could look at the effects of aspect ratio and Oswald’s efficiency factor to find how wing planform shape will affect our results.
In this manner, we can find values of weight, wing area, and thrust that match our desired cruise capability. However, these may not represent the best combination of these parameters if another of our goals is to achieve a certain climb rate.
Another factor to consider would be the desired maximum speed at the cruise altitude. This can be put into the drag equation with the numbers found above to get the thrust or power needed to reach that maximum speed.
The cruise based calculations mentioned above would give us valuable design information for our airplane based on a desired cruise speed and altitude for a design weight and would tell us the wing area and thrust needed for that cruise condition and the thrust needed to cruise at a desired maximum speed. It would not, however tell us if this would result in a good ability to climb or the ability to takeoff and land in a reasonable distance. We would also need to look at these requirements and our design objectives.
To see if we can climb at the desired rate over a reasonable range of altitudes we would need to look at the climb relationship:
dh/dt = [Pavail – Preq] / W
This would give us another value of thrust needed to reach the target rate of climb for a given weight and, since the equation contains power required, which is drag times speed, the wing area would also be a factor.
Finally, we would need to look at the takeoff and landing relationships and at our target values for ground run or for the total takeoff or landing distance. These relationships also involve thrust, weight, and wing area.
The essence of all this is that if we even have only three primary design objectives; a cruise specification, a climb requirement, and a takeoff or landing constraint, we can end up with three different values for wing area and thrust required for a given aircraft weight. We would then have to decide which of these three requirements was most important and which was least important and then start varying design parameters in an iterative manner until we got all three objectives to result in the same weight, wing area, and engine thrust.
This process would become even more cumbersome as we added other design objectives such as a minimum turning radius or a minimum stall speed.
The question is; is there some way to analyze all of these at the same time and come to a decision about optimum or reasonable compromise values of weight, wing area, and engine thrust without having to go through iteration after iteration? Fortunately, the answer is yes. The method normally used is called “constraint analysis”.
9.1 Constraint Analysis
Constraint analysis is essentially a way to look at aircraft weight, wing area, and engine thrust for various phases of flight and come to a decision about meaningful starting values of all three parameters for a given set of design objectives. It does this by looking at two important ratios, the thrust-to-weight ratio (T/W), the wing loading or ratio of weight-to-planform area (W/S). Or in some cases the power-to-weight ratio (P/W) is used instead of T/W. These two ratios are both very reflective of the design philosophy and objectives of any particular airplane.
As one of my students once put it, the thrust-to-weight ratio (T/W) is a measure of how much of a rocket your plane is. The more efficient a plane is in things like cruise the lower its value of T/W. The limit is a sailplane with T/W = 0 and at the other extreme we have fighter aircraft where T/W approaches unity. If T/W = 1.0 or greater we need no wing. The vehicle can get into the air with no lift at all. The value of T/W will depend on the desired flight speed, the wing area, and the efficiency (L/D) of the wing. In cruise where lift = weight and thrust = drag, T/W = 1 / [L/D], meaning that the high value of L/D that is needed for a large range goes hand in hand with a low thrust-to-weight ratio.
The other parameter, W/S, or wing loading, is also generally low for sailplanes and high for fighters. Wing loading for sailplanes is usually in the range of 5-8 pounds per square foot, around 17 lb/ft2 for general aviation planes, and over 100 lb/ft2 for fighters. This ratio is a measure of aerodynamic efficiency as well as a measure of the way the structure is designed.
These two ratios are tied together in aircraft performance through the same power relationship that we looked at when we first examined climb and glide.
9.2 Specific Excess Power
In an earlier chapter on climb and glide we looked at something called specific excess power and defined it as:
Ps = [Pavail Preq] / W = [(T – D)V] / W
We may, hopefully, remember using this relationship to find the rate of climb but we may not recall that it was only the correct rate of climb in a special case, where speed (V) was constant; i.e., the static rate of climb:
[dh/dt]static = [(T – D)V] / W
If we go back to that earlier chapter we will find that in a more general relationship we had:
Ps = [Pavail Preq] / W = [(T – D)V] / W = [dh/dt] + (V/g)(dV/dt).
In other words, only when velocity (V) is constant is this relationship strictly equal to the rate of climb.
In reality, the specific excess power relationship tells us how the excess engine power, Pavail – Preq , can be used to increase the aircraft’s potential energy (climb) or its kinetic energy (speed). In other words this equation is really an energy balance. Power required is the power needed to maintain straight and level flight, i.e., to overcome drag and to go fast enough to give enough lift to equal the weight. If the engine is capable of producing more power than the power required, that excess power can be used to make the plane accelerate to a faster speed (increasing kinetic energy) or to climb to a higher altitude (increasing potential energy), or to give some combination of both. It can tell us how much speed we can gain by descending to a lower altitude, converting potential energy to kinetic energy, or how we can perhaps climb above the static ceiling of the aircraft by converting excess speed (kinetic energy) into extra altitude (potential energy).
In essence this is a pretty powerful relationship and it can be used to analyze many flight situations and to determine an airplane’s performance capabilities. Let’s look at how the equation can be rearranged to help us examine the performance needs in various types of flight.
Rearranging the equation we have:
(T/W)V = (D/W)V +dh/dt + (V/g)(dV/dt)
Now we can expand the first term on the right hand side by realizing that
D = (CD0 + kCL2) ½ ρV2S.
Substituting this for drag in the equation and dividing the entire equation by V we can get:
(T/W) = [(CD0 + kCL2) ½ ρV2S) /W] + (1/V)dh/dt + (1/g)(dV/dt)
Now, to simplify things a little we are going to use a common substitution for the dynamic pressure:
q = ½ ρV2 .
We will also define the lift coefficient in terms of lift and weight using the most general form where in a turn or other maneuver lift may be equal to the load factor n times the weight.
CL = L / qS = nW / qS
Putting all of this together will give:
T/W = (qCD0)/(W/S) + (kn2/q)(W/S) + (1/V)dh/dt + (1/g)dV/dt
In the equation above we have a very general performance equation that can deal with changes in both speed and altitude and we find that these changes are functions of the thrust-to-weight ratio, T/W, and the wing loading W/S. Note that just as the drag equation is a function of both V and 1/V, this is a function of both W/S and 1/(W/S).
9.3 Straight and level flight
We can use the above relationship to make plots of the thrust-to-weight ratio versus the wing loading for various types of flight. If, for example, we want to look at conditions for straight and level flight we can simplify the equation knowing that:
Straight and level flight: n = 1, dh/dt = 0, dV/dt = 0, giving:
T/W = (qCD0)/(W/S) + (k/q)(W/S)
So for a given estimate of our design’s profile drag coefficient, aspect ratio, and Oswald efficiency factor [ k = 1/(πARe)] we can plot T/W versus W/S for any selected altitude (density) and cruise speed.
We could get a different curve for different cruise speeds and altitudes but at any given combination of these this will tell us all the combinations of thrust-to-weight values and wing loadings that will allow straight and level flight at that altitude and speed.
In understanding what this really tells us we perhaps need to step back and look at the same situation another way. In straight and level flight we know:
L = W = qSCL
and T = D = qS (CD0 + kCL2)
And if we simply combine these two equations we will get the same relationship we plotted above.
T/W = (qCD0)/(W/S) + (k/q)(W/S)
Hence, what we have done through the specific excess power relationship is nothing but a different way to get a familiar result. We just end up writing that result in a different form, in terms of the thrust-to-weight ratio and the wing loading.
We need to note that to make the plot above we had to choose a cruise speed. We need to keep in mind that there are limits to that cruise speed. We can’t fly straight and level at speeds below the stall speed or above the maximum speed where the drag equals the maximum thrust from the engine. We could put these limits on the same plot if we wish. For example, let’s look at stall.
At stall Vstall = [2W/(ρSCLmax)]1/2
And this can be written [W/S] = ½ ρVstall2CLmax
On the plot above this would be a vertical line, looking something like this
Here we should note that the space to the right of the dashed line for stall is “out of bounds” since to fly here would require a higher maximum lift coefficient.
9.4 Climb
We could return to the reorganized excess power relationship
T/W = (qCD0)/(W/S) + (kn2/q)(W/S) + (1/V)dh/dt + (1/g)dV/dt
and look at steady state climb. For climb at constant speed dV/dt = 0 and our equation becomes
T/W = (qCD0)/(W/S) + (kn2/q)(W/S) + (1/V)dh/dt
and we can plot T/W versus W/S just as we did in the cruise case, this time specifying a desired rate of climb along with the flight speed and other parameters. Doing this will add another curve to our plot and it might look like the figure below.
This addition to the plot tells us the obvious in a way. It says that we need a higher thrust-to-weight ratio to climb than to fly straight and level.
Each plot of the specific power equation that we add to this gives us a better definition of our “design space”. It tells us that to make the airplane do what we want it to do we are restricted to certain combinations of T/W and W/S.
9.4.1 Caution
It should be noted that in plotting curves for cruise and climb a flight speed must be selected for each. It is, for example, a common mistake for students to look at the performance goals for an aircraft design and just plug in the numbers given without thinking about them. Design goals might include a maximum speed in cruise of 400 mph and a maximum range goal of 800 miles, however these do not occur at the same flight conditions. Just as a car cannot get its best gas mileage when the car is moving at top speed, an airplane isn’t going to get maximum range at its top cruise speed. In fact, the equations used to find the maximum range for either a jet or a prop aircraft assume flight at very low speeds, speeds that one would never really use in cruise unless desperate to extend range in some emergency situation. When plotting the cruise curve in a constraint analysis plot it should be assumed that the aircraft is cruising at a desired “normal” cruise speed, which will be neither the top speed at that altitude nor the speed for maximum range. As an example, most piston engine aircraft will cruise at an engine power setting somewhere between 55% and 75% of maximum engine power. On the other hand, the climb curve should be plotted for optimum conditions; i.e., maximum rate of climb (minimum power required conditions for a prop aircraft) since that is the design target in climb.
9.5 Altitude effects
Obviously altitude is a factor in plotting these curves. The cruise curve will normally be plotted at the desired design cruise altitude. The climb curve would probably be plotted at sea level conditions since that is where the target maximum rate of climb is normally specified. This presents somewhat of a problem since we are plotting the relationships in terms of thrust and weight and thrust is a function of altitude while weight is undoubtedly less in cruise than at takeoff and initial climb-out. One way to resolve this issue is to write our equations in terms of ratios of thrust at altitude divided by thrust at sea level and weight at altitude divided by weight at takeoff.
Talt / Tsl and Walt / WTO .
Going back to our main equation:
T/W = (qCD0)/(W/S) + (kn2/q)(W/S) + (1/V)dh/dt + (1/g)dV/dt ,
we rewrite this in terms of the ratios above to allow us to make our constraint analysis plots functions of TSL and WTO.
TSL / WTO = [(Walt/WTO) / (Talt/TSL)] {[q/(Walt/WSL)](CD0)/(WTO/S)
+ (kn2/q)(WTO/S)(Walt/WTO) + (1/V)dh/dt + (1/g)dV/dt} .
Some references give these ratios, which have been italicized above, symbols such as α and β to make the equation look simpler.
Note that the thrust ratio above is normally just the ratio of density since it is normally assumed that
Talt / Tsl = ρalt / ρSL .
9.6 Other Design Objectives Including Take-off
What other design objectives can be added to the constraint analysis plot to further define our design space? One that is fairly easy to deal with is turning.
Often a set of design objectives will include a minimum turn radius or minimum turn rate. If we assume a coordinated turn we find that once again the last two terms in the constraint analysis relationship go to zero since a coordinated turn is made at constant altitude and airspeed. All we need to do is go to the turn equations and find the desired airspeed and load factor (n), put these into the equation and plot it. Normally we would look at turns at sea level conditions and at takeoff weight. This would give a curve that looks similar to the plots for cruise and climb.
The plot that will be different from all of these is that for takeoff. The takeoff equation seen in an earlier chapter is somewhat complex because takeoff ground distances depend on many things, from drag coefficients to ground friction.
STO = (1/2B) ln [A /(A – BVTO2)]
where A = g[(T0/W) – μ]
and B = (g/W) [ ½ ρS(CD μCLg) + a] .
It should be recalled that CLg is the value of lift coefficient during the ground roll, not at takeoff, and its value is μ/2k for the theoretically minimum ground run. The last parameter in the “B” equation above is “a”, a term that appears in the thrust equation:
T = T0 –aV2 ,
a relationship that comes from the momentum equation where T0 is the “static thrust” or the thrust when the airplane is standing still.
It can be noted that in the A and B terms respectively we have the thrust-to-weight ratio and the inverse of the wing loading (W/S); hence, for a given set of takeoff parameters and a desired ground run distance (STO) a plot can be made of T/W versus W/S. This relationship proves to be a little messy with both ratios buried in a natural log term and the wing loading in a separate term. An iterative solution may be necessary.
An alternative approach often proposed in books on aircraft design is based on statistical takeoff data collected on different types of aircraft. The figure below (Raymer, 1992) is based on a method commonly used in industry.
In this approach a “Take-Off-Parameter”, TOP, is proposed to be a function of W/S, T/W, CLTO, and the density ratio sigma (σ) where:
W/S = (TOP)σCLTO(T/W). [σ = ρalt/ρSL]
The value of “TOP” is found from the chart above. One finds the desired takeoff distance in feet on the vertical axis and projects over to the plot for the type of aircraft desired, then drops a vertical line to the TOP axis to find a value for that term. Once the value of TOP has been found the relationship above is plotted to give a straight line from the origin of the constraint analysis graph.
Two things should be noted at this point. First is that the figure from Raymer on the preceding page has two types of plots on it, one for ground run only and the other for ground run plus the distance required to clear a 50 ft obstacle. Either can be used depending on the performance parameter which is most important to meeting the design specifications. The second is that the takeoff parameter (TOP) defined for propeller aircraft is based on power requirements (specifically, horsepower requirements) rather than thrust. For the prop aircraft Raymer defines TOP as follows:
W/S = (TOP)σCLTO(hp/W) .
It should be noted here that it is often common when conducting a constraint analysis for a propeller type aircraft to plot the power-to-weight ratio versus wing loading rather than using the thrust-to-weight ratio. This can be done fairly easily by going back to the constraint analysis equations and substituting P/V everywhere that a thrust term appears.
9.7 Landing
In reality, the landing distance is pretty much determined by the stall speed (the plane must touch down at a speed higher than stall speed, often about 1.2 VStall) and the glide slope (where obstacle clearance is part of the defined target distance). Again it is common for aircraft design texts to propose approximate or semi-empirical relationships to describe this and those relationships show landing distance to depend only on the wing loading. This makes sense when one realizes that, unless reverse thrust is used in the landing ground run, thrust does not play a major role in landing. Raymer proposed the relationship below:
Slanding = 80(W/S)[1/(σCLmax)] + Sa
where
Sa = 1000 for an airliner with a 3 degree glideslope
600 for a general aviation type power off approach
450 for a STOL 7 degree glideslope
Raymer also suggests multiplying the first term on the right in the distance equation above by 0.66 if thrust reversers are to be used and by 1.67 when accounting for the safety margin required for commercial aircraft operating under FAR part 25.
Note here that the weight in the equation is the landing weight but that in calculating this landing distance for design purposes the takeoff weight is usually used for general aviation aircraft and trainers and is assumed to be 0.85 times the takeoff weight for jet transports.
The relationship above, since it does not depend on the thrust, will plot on our constraint analysis chart as a vertical line in much the same way the stall case did, but it will be just to the left of the stall line.
9.8 Optimum design points
In this final plot the space above the climb and takeoff curves and to the left of the landing line is our acceptable design space. Any combination of W/S and T/W within that space will meet our design goals. What we want, however, is the “best” combination of these parameters for our design goals. The optimum will be found at the intersections of these curves. In the figure above this will be either where the takeoff and climb curves intersect or where the takeoff and landing curves intersect.
By “optimum” we mean that we are looking for the minimum thrust-to-weight ratio that will enable the airplane to meet its performance goals and we would like to have the highest possible wing loading. The desire for minimum thrust is obvious, based on the need to minimize fuel consumption and engine cost. The goal of maximum wing loading may not be as obvious to the novice designer but this means the wing area is kept to a minimum which gives lower drag. It also gives a better “ride” to the airplane passengers. As wing loading increases the effects of turbulence and gusts in flight are minimized, smoothing out the “bumps” in flight.
9.9 The design process
Constraint analysis is an important element in a larger process called aircraft design. There are many good textbooks available on aircraft design and the Raymer text referenced earlier is one of the best. Another good text that combines an examination of the design process with a look as several design case studies is Aircraft Design Projects for Engineering Students, by Jenkinson and Marchman, published by the AIAA.
The design process usually begins with a set of design objectives such as these we have examined, a desired range, payload weight, rate of climb, takeoff and landing distances, top speed, ceiling, etc. The first step in the process is usually to look for what are called “comparator” aircraft, existing or past aircraft that can meet most or all of our design objectives. This data can give us a place to start by suggesting starting values of things like takeoff weight, wing area, aspect ratio, etc. that can be used in the constraint analysis equations above. These are then plotted to find “optimum” values of wing loading and thrust-to-weight ratio.
The constraint analysis may be performed several times, looking at the effects of varying things like wing aspect ratio on the outcome. The analysis may suggest that some of the “constraints” (i.e., the performance targets) need to be relaxed. What can be gained by accepting a lower cruise speed or a longer takeoff distance. We might find, for example, that by accepting an additional 500 feet in our takeoff ground run we can get by with a significantly smaller engine.
Design is a process of compromise and no one design is ever best at everything. But through good use of things like constraint analysis methods we can turn those compromises into optimum solutions.
Acknowledgment: Thanks to Dustin Grissom for reviewing the above and developing examples to go with it.
Homework 9
Looking again at the aircraft in Homework 8 with some additional information:
Cessna 182 Cessna Citation III
b = 35.8 ft b = 53.3 ft
S = 174 ft^2 S = 318 ft^2
W_max = 2950 lb W_max = 19,815 lb
Fuel = 65 gal (@ 6 lb/gal) Fuel = 1119 gal (@ 6.67 lb/gal)
Single engine 230 hp at SL 2 engines with 3650 lb thrust each at SL
eta_p = 0.8
gamma_p = 0.45 lb/hp-hr gamma_T = 0.6 lb/lb_thrust-hr
C_D0 = 0.025 C_Do = 0.02
e = 0.8 e = 0.81
1. Find the maximum range and the maximum endurance for both aircraft.
a. What altitude gives the best range for the C-182? Do you think this is a reasonable speed for flight?
b. What altitude gives the best endurance for the C-182? Is this a reasonable flight speed?
c. Endurance for the C-182 can be found two ways (constant altitude or constant velocity). Which gives the best endurance?
2. Find the range for the C-182 assuming the flight starts at 150 mph and an altitude of 7500 feet and stays at constant angle of attack.
3. How sensitive is the maximum range for the Cessna 182 to aspect ratio and the Oswald efficiency factor, i.e. to the wing planform shape? To answer this, plot range versus aspect ratio using e.= 0.8 and varying AR from 4 through 10, and plot range versus e for an aspect ratio of 7.366 with e varying from 0.6 through 1.0.
Figure 9.7: Effect of R & e Variation on max Range Cessna 182
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Appendix A: Airfoil Data
In Chapter 3 of this text we discussed many of the aspects of airfoil design as well as the NACA designations for several series of airfoils. Lift, drag, and pitching moment data for hundreds of such airfoil shapes was determined in wind tunnel tests by the National Advisory Committee for Aeronautics (NACA) and later by NASA, the National Aeronautics and Space Administration. This data is most conveniently presented in plots of lift coefficient versus angle of attack, pitching moment coefficient versus angle of attack, drag coefficient versus lift coefficient, and pitching moment coefficient versus lift coefficient and is found in literally hundreds of NACA and NASA Reports, Notes, and Memoranda published since the 1920s.
Many of the more important airfoil shapes have their test results summarized in the Theory of Wing Sections, a Dover paperback publication authored by Ira Abbott and Albert Von Doenhoff and first published in 1949. While the date of original publication might lead one to think this material must be out of date, that is simply not true and the Theory of Wing Sections is one of the most valuable references in any aerospace engineer’s personal library.
In the following appendix material a selection of airfoil graphical data is presented which can be found in the Theory of Wing Sections and in the non-copyrighted NACA publications which are the source of the Dover publication’s data. The airfoils presented represent a cross section of airfoil shapes selected to illustrate why one would select one airfoil over another for any given aircraft design or performance requirement.
Figure A-1 shows data for the NACA 0012 airfoil, a classic symmetrical shape that is used for everything from airplane stabilizers and canards to helicopter rotors to submarine “sails”. Note that for the symmetrical shape the lift coefficient is zero at zero angle of attack. These graphs show test results for several different Reynolds numbers and for “standard roughness” on the surface. They also show what happens when a 20% chord flap is deflected 40 degrees. Note that the flap deflection shifts the lift curve far to the left giving a zero lift angle of attack of roughly minus 12 degrees while it increases the maximum lift coefficient (Re= 6 x 106) from just under 1.6 to 2.4, a huge increase in lifting capability that can contribute to large decreases in takeoff and landing distances. Also note that the pitching moment coefficient at c/4 (in the left hand graph) is essentially zero from -12 degrees to+ 14 degrees angle of attack and then goes negative in stall at positive angle of attack. In the right hand graph the moment curve shown is for the moment at the “aerodynamic center” rather than the quarter chord but since it is also zero in this plot it confirms the theoretical prediction that for a symmetrical airfoil the center of pressure (where the moment is zero) coincides with the aerodynamic center.
Figure A-2 gives similar data for the NACA 2412 airfoil, another 12% thick shape but one with camber. Note that the lift coefficient at zero angle of attack is no longer zero but is approximately 0.25 and the zero lift angle of attack is now minus two degrees, showing the effects of adding 2% camber to a 12% thick airfoil. Also note that the moment coefficient at the quarter chord is no longer zero but is still relatively constant between the onset of positive and negative stall. The moment coefficient is negative over most of the range of angle of attack indicating a nose down pitching moment and positive stability. Adding 2% camber has also resulted in a slight increase in CLmax from about 1.6 to 1.7 when compared to the 0012 airfoil.
When Figure A-3 is compared with A-I and A-2 one can see the effect of added thickness as the percent thickness increases from 12 to 15 percent. This shows up primarily as a slight increase in drag coefficient and a slight reduction in CLmax compared to the 12% thick equally cambered airfoil in A-2.
Figure A-4 returns to a 12% thick airfoil but one with 4% camber and a comparison with the previous figures will show how the increase in camber increases the lift at zero angle of attack, takes the zero lift angle of attack down to minus four degrees and increases the nose down pitching moment which is still constant between stall angles when measured at the quarter chord (aerodynamic center).
Figures A-5 and A-6 are for “6-series” airfoils, the so-called “laminar flow” airfoil series developed in the 1930s and used extensively in wing designs well into the late 1900s. Both figures show 12% thick airfoils. The distinguishing features of these graphs are the pronounced “drag buckets” in the right hand plots. Note that the first number to the right of the hyphen in the airfoil designation tells the location of the center of the drag bucket; i.e., the center of the bucket is at a CL of 0.1 for the 641-112 and at 0.4 for the 641-412. In this manner the airfoil designation in the “6-series” is a handy tool for the designer, allowing easy selection of an airfoil that has its “drag bucket” centered at perhaps the cruise lift coefficient for a transport aircraft or at the lift coefficient which is best for climb or maneuver in a fighter. Also note that the difference in camber produces the same kind of shifts in the lift curve as noted in the 4 digit series airfoils in the earlier plots.
These 6 plots are just the tip of the iceberg when exploring the many airfoil shapes which have been investigated by the NACA, NASA, and others over the years but the general features noted above will hold true for almost any variations in airfoil shape.
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The aim of this chapter is to give a broad overview of the activities related to the field referred to as aerospace engineering. More precisely, it aims at summarizing briefly the main scope in which the student will develop his or her professional career in the future as an aerospace engineer. First, a rough overview of what engineering is and what engineers do is given, with particular focus on aerospace engineering. Also, a rough taxonomy of the capabilities that an engineer is supposed to have is provided. Second, the focus is on describing the different aerospace activities, i.e., the industry, the airlines, the military air forces, the infrastructures on earth, the research institutions, the space agencies, and the international organizations. Last but not least, in the believe that research, development, and innovation is the key element towards the future, an overview of the current aviation research agenda is presented.
01: The Scope
Following Wikipedia [7], engineering can be defined as:
The application of scientific, economic, social, and practical knowledge in order to design, build, and maintain structures, machines, devices, systems, materials, and processes. It may encompass using insights to conceive, model, and scale an appropriate solution to a problem or objective. The discipline of engineering is extremely broad, and encompasses a range of more specialized fields of engineering, each with a more specific emphasis on particular areas of technology and types of application.
The foundations of engineering lays on mathematics and physics, but more important, it is reinforced with additional study in the natural sciences and the humanities. Therefore, attending to the previously given definition, engineering might be briefly summarized with the following six statements:
• to adapt scientific discovery for useful purposes;
• to create useful devices for the service of society;
• to invent solutions to meet society’s needs;
• to come up with solutions to technical problems;
• to utilize forces of nature for society’s purposes;
• to convert energetic resources into useful work.
On top of this, according to current social sensitivities, one should add: in an environmentally friendly manner.
Following Wikipedia [6], aerospace engineering can be defined as:
a primary branch of engineering concerned with the research, design, development, construction, testing, and science and technology of aircraft and spacecraft. It is divided into two major and overlapping branches: aeronautical engineering and astronautical engineering. The former deals with aircraft that operate in Earth’s atmosphere, and the latter with spacecraft that operate outside it.
Therefore, an aerospace engineering education attempts to introduce the following capabilities Newman [3, Chap. 2]:
• Engineering fundamentals (maths and physics); innovative ideas conception and problem solving skills; the vision of high-technology approaches to engineering complex systems; and the idea of technical system integration and operation.
• knowledge in the technical areas of aerospace engineering including mechanics and physics of fluids, aerodynamics, structures and materials, instrumentation, control and estimation, humans and automation, propulsion and energy conversion, aeronautical and astronautical systems, infrastructures on earth, the air navigation system, legislation, air transportation, etc.
• The methodology and experience of analysis, modeling, and synthesis.
• Finally, an engineering goal of addressing socio-humanistic problems.
As a corollary, an aerospace engineering education should produce engineers capable of the following Newman [3, Chap. 2]:
• Conceive: conceptualize technical problems and solutions.
• Design: study and comprehend processes that lead to solutions to a particular problem including verbal, written, and visual communications.
• Development: extend the outputs of research.
• Testing: determine performance of the output of research, development, or design.
• Research: solve new problems and gain new knowledge.
• Manufacturing: produce a safe, effective, economic final product.
• Operation and maintenance: keep the products working effectively.
• Marketing and sales: look for good ideas for new products or improving current products in order to sell.
• Administration (management): coordinate all the above.
Thus, the student as a future aerospace engineer, will develop his or her professional career accomplishing some of the above listed capabilities in any of the activities that arise within the aerospace industry.
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It seems to be under common agreement that the aerospace activities (in which aerospace engineers work) can be divided into seven groups Franchini et al. [2]:
• the industry, manufacturer of products;
• the airlines, transporters of goods and people;
• the military air forces, demanders of high-level technologies;
• the space agencies, explorers of the space;
• the infrastructures on earth, supporter of air operations;
• the research institutions, guarantors of technological progress;
• the international organizations, providers of jurisprudence.
The aerospace industry
The aerospace industry is considered as an strategic activity given that it is a high technology sector with an important economic impact. The Aerospace sector is an important contributor to economic growth everywhere in the world. The european aerospace sector represents a pinnacle of manufacturing which employed almost half a million highly skilled people directly in 2010 and it continuously spins-out technology to other sectors. About 2.6 million indirect jobs can be attributed to air transport related activities and a contribution of around €250 billion1 (around 2.5%) to european gross domestic product in 2010. Therefore, the aerospace industry is an important asset for Europe economically, being a sector that invests heavily in Research and Development (R & D) compared with other industrial sectors. The aerospace sector is also an important pole for innovation.
The aerospace industry accomplish three kind of activities: aeronautics (integrated by airships, propulsion systems, and infrastructures and equipments); space; and missiles. Grosso modo, the aeronautical industry constitutes around the 80-90% of the total activity.
The fundamental characteristics of the aerospace industry are:
• Great dynamism in the cycle research-project-manufacture-commercialization.
• Specific technologies in the vanguard which spin-out to other sectors.
• High-skilled people.
• Limited series (non mass production) and difficult automation of manufacturing processes.
• Long term development of new projects.
• Need for huge amount of capital funding.
• Governmental intervention and international cooperation.
The linkage between research and project-manufacture is essential because the market is very competitive and the product must fulfill severe safety and reliability requirements in order to be certified. Thus, it is necessary to continuously promote the technological advance to take advantage in such a competitive market.
The quantity of units produced a year is rather small if we compare it with other manufacture sectors (automobile manufacturing, for instance). An airship factory only produces tens of units a year; in the case of space vehicles the common practice is to produce a unique unit. These facts give a qualitative measure of the difficulties in automating manufacturing processes in order to reduce variable costs.
The governmental intervention comes from different sources. First, directly participating from the capital of the companies (many of the industries in Spain and Europe are state owned). Indirectly, throughout research subsides. Also, as a direct client, as it is the case for military aviation. The fact that many companies do not have the critical size to absorb the costs and the risks of such projects makes common the creation of long-term alliances for determined aircrafts (Airbus) or jet engines (International Aero Engines or Eurojet).
Airlines
Among the diverse elements that conform the air transportation industry, airlines represent the most visible ones and the most interactive with the consumer, i.e., the passenger. An airline provides air transport services for traveling passengers and/or freight. Airlines lease or own their aircraft with which to supply these services and may form partnerships or alliances with other airlines for mutual benefit, e.g., Oneworld, Skyteam, and Star alliance. Airlines vary from those with a single aircraft carrying mail or cargo, through full-service international airlines operating hundreds of aircraft. Airline services can be categorized as being intercontinental, intra-continental, domestic, regional, or international, and may be operated as scheduled services or charters.
The first airlines were based on dirigibles. DELAG (Deutsche Luftschiffahrts-Aktiengesellschaft) was the world’s first airline. It was founded on November 16, 1909, and operated airships manufactured by the zeppelin corporation. The four oldest non- dirigible airlines that still exist are Netherlands’ KLM, Colombia’s Avianca, Australia’s Qantas, and the Czech Republic’s Czech Airlines. From those first years, going on to the elite passenger of the fifties and ultimately to the current mass use of air transport, the world airline companies have evolved significantly.
Flag companies Low Cost companies
Operate hubs and spoke Operate point to point
Hubs in primary international airports Mostly regional airports
Long rotation times (50 min) Short rotation times (25 min)
Mixed fleets Standardized fleets
Selling: agencies and internet Selling: internet
Extras included (Business, VIP louges, catering) No extras included in the tickets
Table 1.1: Comparison between flag companies and low cost companies.
Traditional airlines were state-owned. They were called flag companies and used to have a strong strategic influence. It was not until 1978, with the United States Deregulation Act, when the market started to be liberalized. The main purpose of the act was to remove government control over fares, routes, and market entry of new airlines in the commercial aviation sector. Up on that law, private companies started to emerge in the 80’s and 90’s, specially in USA. Very recently, a new phenomena have arisen within the last 10-15 years: the so called low cost companies, which have favored the mass transportation of people. A comparison between low cost companies and traditional flag companies is presented in Table 1.1. It provides a first understanding of the main issues involved in the direct operating costs of an airline, which will be studied in Chapter 8. The competition has been so fierce that many traditional companies have been pushed to create their own low cost filial companies, as it the case of lberia and its filial lberia Express. See Figure 1.1.
Figure 1.1: Flag companies (e.g. lberia) and low cost companies (e.g., lberia Express and Ryanair).
Military air forces
The military air forces are linked to the defense of each country. In that sense, they play a strategic role in security, heavily depending on the economical potential of the country and its geopolitical situation. Historically, it has been an encouraging sector for technology and innovation towards military supremacy. Think for instance in the advances due to World War II and the Cold War. Nowadays, it is mostly based on cooperation and alliances. However, inherent threats in nations still make this sector a strategic sector whose demand in high technology will be maintained. An instance of this is the encouraging trend of the USA towards the development of Unmanned Air Vehicles (UAV) in the last 20 years in order to maintain the supremacy in the middle east minimizing the risk of soldiers life.
Space agencies
Figure 1.2: International Space Station (ISS).
There are many government agencies engaged in activities related to outer space exploration. Just to mention a few, the China National Space Administration (CNSA), the Indian Space Research Organization (ISRO), the Russian Federal Space Agency (RFSA) (successor of the Soviet space program), the European Space Agency (ESA), and the National Aeronautics and Space Administration (NASA). For their interest, the focus will be on these last two.
The European Space Agency (ESA) was established in 1975, it is an intergovernmental organization dedicated to the exploration of space. It counts currently with 20 member states: Austria, Belgium, Check Republic, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, the Nederland, Norway, Poland, Portugal, Romania, Spain, Sweden, Switzerland, and United Kingdom. Moreover, Hungary and Canada have a special status and cooperate in certain projects.
In addition to coordinating the individual activities of the member states, ESA’s space flight program includes human spaceflight, mainly through the participation in the International Space Station (ISS) program (Columbus lab, Node-3, Cupola), the launch and operation of unmanned exploration missions to other planets, asteroids, and the Moon (probe Giotto to observe Halley’s comet; Cassine-Hyugens, joint mission with NASA, to observe Saturn and its moons; Mars Express, to explore mars; Rosetta to perform a detailed study of comet 67P/Churyumov?Gerasimenko), Earth observation (Meteosat), science (Spacelab), telecommunication (Eutelsat), as well as maintaining a major spaceport, the Guiana Space Centre at Kourou, French Guiana, and designing launch vehicles (Ariane).
The National Aeronautics and Space Administration (NASA) is the agency of the United States government that is responsible for the civilian space program and for aeronautics and aerospace research. NASA was established by the National Aeronautics and Space Act on July 29, 1958, replacing its predecessor, the National Advisory Committee for Aeronautics (NACA). NASA science is focused on better understanding of Earth through the Earth Observing System, advancing heliophysics through the efforts of the Science Mission Directorate’s Heliophysics Research Program, exploring bodies throughout the Solar System with advanced robotic missions such as New Horizons, and researching astrophysics topics, such as the Big Bang, through the Great Observatories and associated programs.
United States space exploration efforts have since 1958 been led by NASA, including the Apollo moon-landing missions, the Skylab space station, the Space Shuttle, a reusable space vehicles program whose last mission took place in 2011 (see Figure 2.7), the probes (Pioner, Viking, etc.) which explore the outer space. Currently, NASA is supporting the ISS, and the Mars Science Laboratory unmanned mission known as curiosity. NASA not only focuses on space, but conducts fundamental research in aeronautics, such in aerodynamics, propulsion, materials, or air navigation.
Infrastructures on earth
In order to perform safe operations either for airliners, military aircraft, or space missions, a set of infrastructures and human resources is needed. The necessary infrastructures on earth to assist flight operations and space missions are: airports and air navigation services on the one hand (referring to atmospheric flights); launch bases and control and surveillance centers on the other (referring to space missions).
The airport is the localized infrastructure where flights depart and land, and it is also a multi-modal node where interaction between flight transportation and other transportation modes (rail and road) takes place. It consists of a number of conjoined buildings, flight field installations, and equipments that enable: the safe landing, take-off, and ground movements of aircrafts, together with the provision of hangars for parking, service, and maintenance; the multi-modal (earth-air) transition of passengers, baggage, and cargo.
The air navigation is the process of steering an aircraft in flight from an initial position to a final position, following a determined route and fulfilling certain requirements of safety and efficiency. The navigation is performed by each aircraft independently, using diverse external sources of information and proper on-board equipment. The fundamental goals are to avoid getting lost, to avoid collisions with other aircraft or obstacles, and to minimize the influence of adverse meteorological conditions. Air navigation demands juridic, organizational, operative, and technical framework to assist aircraft on air fulfilling safe operations. The different Air Navigation Service Providers (ANSP) (AENA in Spain, FAA in USA, Eurocontrol in Europe, etc.) provide these frameworks, comprising three main components:
• Communication, Navigation, and Surveillance (CNS).
• Meteorological services.
• Air Traffic Management (ATM).
• Air Space Management (ASM).
• Air Traffic Flow Management (ATFM).
• Air Traffic Services (ATS) such traffic control and information.
A detailed insight on these concepts will be given in Chapter 10 and Chapter 11.
A launch base is an earth-based infrastructure from where space vehicles are launched to outer space. The situation of launch bases depends up on different factors, including latitudes close to the ecuador, proximity to areas inhabited or to the sea to avoid danger in the first stages of the launch, etc. The most well known bases are: Cape Kennedy in Florida (NASA); Kourou in the French Guyana (ESA); Baikonur en Kazakhstan (ex Soviet Union space program). Together with the launch base, the different space agencies have control centers to monitor the evolution of the space vehicles, control their evolution, and communicate with the crew (in case there is crew).
Aerospace research institutions
The research institutions fulfill a key role within the aerospace activities because the development of aviation and space missions is based on a continuos technological progress affecting a variety of disciplines such as aerodynamics, propulsion, materials, avionics, communication, airports, air navigation, etc. The research activity is fundamentally fulfilled at universities, aerospace companies, and public institutions.
Some relevant aerospace research institutions are: the National Aeronautics and Space Administration (NASA) in the United States of America (USA),; the French Aerospace Lab (ONERA), the German Aerospace Center (DLR), or Spanish the Instituto Nacional de Técnica Aeroespacial (INTA) in Europe; the Japan Aerospace Exploration Agency (JAXA) and the China National Space Administration (CNSA)) in Asia; and the (Roscosmos) State Corporation for Space Activities
International organizations
In order to promote a reliable, efficient, and safe air transportation, many regulations are needed. This regulatory framework arises individually in each country but always under the regulatory core of two fundamental supranational organizations: The International Civil Aviation Organization (ICAO) and the International Air Transport Association (IATA) .
ICAO was created as a result of the Chicago Convention. ICAO was created as a specialized agency of the United Nations charged with coordinating and regulating international air travel. The Convention establishes rules of airspace, aircraft registration and safety, and details the rights of the signatories in relation to air travel. In the successive revisions ICAO has agreed certain criteria about the freedom of overflying and landing in countries, to develop the safe and ordered development of civil aviation world wide, to encourage the design and use techniques of airships, to stimulate the development of the necessary infrastructures for air navigation. Overall, ICAO has encourage the evolution of civil aviation.
The modern IATA is the successor to the International Air Traffic Association founded in the Hague in 1919. IATA was founded in Havana, Cuba, in April 1945. It is the prime vehicle for inter-airline cooperation in promoting safe, reliable, secure, and economical air services. IATA seeks to improve understanding of the industry among decision makers and increase awareness of the benefits that aviation brings to national and global economies. IATA ensures that people and goods can move around the global airline network as easily as if they were on a single airline in a single country.
In addition to the cited organizations, it is convenient to mention the two most important organization with responsibility in safety laws and regulations, including the airship project and airship certification, maintenance labour, crew training, etc.: The European Aviation Safety Agency (EASA) in the European Union and the Federal Aviation Administration (FAA). Spain counts with the Agencia Estatal de Seguridad Aérea (AESA), dependent on the ministry of infrastructures (fomento). AESA is also responsible for safety legislation in civil aviation, airships, airports, air navigation, passengers rights, general aviation, etc.
1. one billion herein refers to 1.000.000.000 monetary units.
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Aviation has dramatically transformed society over the past 100 years. The economic and social benefits throughout the world have been immense in shrinking the planet with the efficient and fast transportation of people and goods. However, encouraging challenges must be faced to cope with the expected demand, but also to meet social sensitivities.
These challenges have led to the formation of the Advisory Council for Aeronautics Research in Europe (ACARE) to define a Strategic Research Agenda (SRA) for aeronautics and air transport in Europe ACARE [1]. The goals set by the SRA have had a clear influence on current aeronautical research, delivering important initiatives and benefits for the aviation industry, including among others the Clean Sky Joint Technology Initiative, which pursues a greener aviation, and the Single European Sky ATM Research (SESAR) Joint Undertaking, which pursues a more efficient ATM system. Initiatives in the same direction have been also driven in USA within the Next Generation of air transportation system (NextGen).
1.03: Aviation research agenda
The growth of air traffic in the past 20 years has been spectacular, and forecasts indicate that there will continue in the future. According to Eurocontrol, in 2010 there was 9.493 million IFR2 flights in Europe, around 26000 flights a day. Traffic demand will nearly triple, and airlines will more than double their fleets of passenger aircraft within 20 years time. This continuous growth in demand will bring increased challenges for dealing with mass transportation and congestion of ATM and airport infrastructure.
Aviation is directly impacted by energy trends. The oil price peaks in the last period (2008-2011) due to diverse geopolitical crisis are not isolated events, the cost of oil will continue to increase. Dependence on fuel availability will continue to be a risk for air transport, specially if energy sources are held in a few hands. Aviation will have to develop long-term strategies for energy supply, such as alternative fuels, that will be technically suitable and commercially scaleable as well as environmentally sustainable.
Climate change is a major societal and political issue and is becoming more. Globally civil aviation is responsible for 2% of CO2 of man made global emissions according to the United Nations’ Intergovernmental Panel on Climate Change (IPCC). As aviation grows to meet increasing demand, the IPCC forecasts that its share of global man made CO2 emissions will increase to around 3% to 5% in 2050 Penner [4]. Non-CO2 emissions including oxides of nitrogen and condensation trails which may lead to the formation of cirrus clouds, also have impacts but require better scientific understanding. Thus, reducing emissions represents a major challenge, maybe the biggest ever. Reducing disturbance around airports is also a challenge with the need to ensure that noise levels and air quality around airports remain acceptable.
2. IFR stands for Instrumental Flight Rules and refers to instrumental flights
1.3.02: Clean Sky
Clean Sky is a Joint Technology Initiative (JTI) that aims at developing a mature breakthrough clean technologies for air transport. By accelerating their deployment, the JTI will contribute to Europe’s strategic environmental and social priorities, and simultaneously promote competitiveness and sustainable economic growth.
Joint Technology Initiatives are specific large scale research projects created by the European Commission within the 7th Framework Programme (FP7) in order to allow the achievement of ambitious and complex research goals, set up as a public-private partnership between the European Commission and the European aeronautical industry.
Clean Sky will speed up technological breakthrough developments and shorten the time to market for new and cleaner solutions tested on full scale demonstrators, thus contributing significantly to reducing the environmental footprint of aviation (i.e. emissions and noise reduction but also green life cycle) for our future generations. The purpose of Clean Sky is to demonstrate and validate the technology breakthroughs that are necessary to make major steps towards the environmental goals set by ACARE and to be reached in 2020 when compared to 2000 levels:
• 50% reduction of CO2 emissions;
• 80% reduction of NOx emissions;
• 50% reduction of external noise; and
• a green product life cycle.
Clean Sky JTI is articulated around a series of the integrated technology demonstrators:
• Eco Design.
• Smart Fixed Wing Aircraft.
• Green Regional Aircraft.
• Green Rotorcraft.
• Systems for Green Operations.
• Sustainable and Green Engines.
Figure 1.3: Contributors to reducing emissions. Adapted from Clean Sky JTI.
Therefore, the reduction in fuel burn and CO2 will require contributions from new technologies in aircraft design (engines, airframe materials, and aerodynamics), alternative fuels (bio fuels), and improved ATM and operational efficiency (mission and trajectory management). See Figure 1.3. ACARE has identified the main contributors to achieving the above targets. The predicted contributions to the 50% CO2 emissions reduction target are: efficient aircraft: 20-25%; efficient engines: 15-20%; improved air traffic management: 5-10%; bio fuels: 45-60%.
1.3.03: SESAR
ATM, which is responsible for sustainable, efficient, and safe operations in civil aviation, is still nowadays a very complex and highly regulated system. A substantial change in the current ATM paradigm is needed because this system is reaching the limit of its capabilities. Its capacity, efficiency, environmental impact, and flexibility should be improved to accommodate airspace users’ requirements. The Single European Sky ATM Research (SESAR) Program aims at developing a new generation of ATM system.
The SESAR program is one of the most ambitious research and development projects ever launched by the European Community. The program is the technological and operational dimension of the Single European Sky (SES) initiative to meet future capacity and air safety needs. Contrary to the United States, Europe does not have a single sky, one in which air navigation is managed at the European level. Furthermore, European airspace is among the busiest in the world with over 33,000 flights on busy days and high airport density. This makes air traffic control even more complex. The Single European Sky is the only way to provide an uniform and high level of safety and efficiency over Europe’s skies. The major elements of this new institutional and organizational framework for ATM in Europe consist of: separating regulatory activities from service provision and the possibility of cross-border ATM services; reorganizing European airspace that is no longer constrained by national borders; setting common rules and standards, covering a wide range of issues, such as flight data exchanges and telecommunications.
The mission of the SESAR Joint Undertaking is to develop a modernized air traffic management system for Europe. This future system will ensure the safety and fluidity of air transport over the next thirty years, will make flying more environmentally friendly, and reduce the costs of air traffic management system. Indeed, the main goals of SESAR are SESAR Consortium [5]:
• 3-fold increase the air traffic movements whilst reducing delays;
• improvement the safety performance by a factor of 10;
• 10% reduction in the effects aircraft have on the environment;
• provide ATM services at a cost to airspace users with at least 50% less.
1.04: References
[1] ACARE (2010). Beyond Vision 2020 (Towards 2050). Technical report, European Commission. The Advisory Council for Aeronautics Research in Europe.
[2] Franchini, S., López, O., Antoín, J., Bezdenejnykh, N., and Cuerva, A. (2011). Apuntes de Tecnología Aeroespacial. Escuela de Ingeniería Aeronáutica y del Espacio. Universidad Politécnica de Madrid.
[3] Newman, D. (2002). Interactive aerospace engineering and design. McGraw-Hill.
[4] Penner, J. (1999). Aviation and the global atmosphere: a special report of IPCC working groups I and III in collaboration with the scientific assessment panel to the Montreal protocol on substances that deplete the ozone layer. Technical report, International Panel of Climate Change (IPCC).
[5] SESAR Consortium (April 2008). SESAR Master Plan, SESAR Definition Phase Milestone Deliverable 5.
[6] Wikipedia. Aerospace Engineering. http://en.wikipedia.org/wiki/Aerospace_engineering. Last accesed 30 sept. 2013.
[7] Wikipedia. Engineering. http://en.wikipedia.org/wiki/Engineering. Last accesed 30 sept. 2013.
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The aim of this chapter is to present the student some generalities focusing on the atmospheric flight of airplanes. First, a classification of aerospace vehicles is given. Then, focusing on airplanes (which will be herein also referred to as aircraft), the main parts of an aircraft will be described. Third, the focus is on characterizing the atmosphere, in which atmospheric flight takes place. Finally, in order to be able to describe the movement of an aircraft, different system references will be presented. For a more detailed description of an aircraft, please refer for instance to any of the following books in aircraft design: Torenbeek [8], Howe [5], Jenkinson et al. [6], and Raymer et al. [7].
Figure 2.1: Classification of air vehicles. Adapted from Franchini et al. [3].
02: Generalities
An aircraft, in a wide sense, is a vehicle capable to navigate in the air (in general, in the atmosphere of a planet) by means of a lift force. This lift appears due to two different physical phenomena:
• aeroestatic lift, which gives name to the aerostats (lighter than the air vehicles), and
• dynamic effects generating lift forces, which gives name to the aerodynes (heavier than the air vehicles).
Figure 2.2: Aerostats.
An aerostat is a craft that remains aloft primarily through the use of lighter than air gases, which produce lift to the vehicle with nearly the same overall density as air. Aerostats include airships and aeroestatic balloons. Aerostats stay aloft by having a large "envelope" filled with a gas which is less dense than the surrounding atmosphere. See Figure 2.2 as illustration.
Aerodynes produce lift by moving a wing through the air. Aerodynes include fixed-wing aircraft and rotorcraft, and are heavier-than-the-air aircraft. The first group is the one nowadays know as airplanes (also known simply as aircraft). Rotorcraft include helicopters or autogyros (Invented by the Spanish engineer Juan de la Cierva in 1923).
A special category can also be considered: ground effect aircraft. Ground effect refers to the increased lift and decreased drag that an aircraft airfoil or wing generates when an aircraft is close the ground or a surface. Missiles and space vehicles will be also analyzed as classes of aerospace vehicles.
2.01: Classification of aerospace vehicles
A first division arises if we distinguish those fixed-wing aircraft with engines from those without engines.
Figure 2.3: Gliders.
A glider is an aircraft whose flight does not depend on an engine. The most common varieties use the component of their weight to descent while they exploit meteorological phenomena (such thermal gradients and wind deflections) to maintain or even gain height. Other gliders use a tow powered aircraft to ascent. Gliders are principally used for the air sports of gliding, hang gliding and paragliding, or simply as leisure time for private pilots. See Figure 2.3.
Aerodynes with fixed-wing and provided with a power plant are known as airplanes1. An exhaustive taxonomy of airplanes will not be given, since there exist many particularities. Instead, a brief sketch of the fundamentals which determine the design of an aircraft will be drawn. The fundamental variables that must be taken into account for airplane design are: mission, velocity range, and technological solution to satisfy the needs of the mission.
The configuration of the aircraft depends on the aerodynamic properties to fly in a determined regime (low subsonic, high subsonic, supersonic). In fact, the general configuration of the aircraft depends upon the layout of the wing, the fuselage, stabilizers, and power plant. This four elements, which are enough to distinguish, grosso modo, one configuration from another, are designed according to the aerodynamic properties.
Then one possible classification is according to its configuration. However, due to different technological solutions that might have been adopted, airplanes with the same mission, could have different configurations. This is the reason why it seems more appropriate to classify airplanes attending at its mission.
Two fundamental branches exist: military airplanes and civilian airplanes.
Figure 2.4: Military aircraft types.
The most usual military missions are: surveillance, recognition, bombing, combat, transportation, or training. For instance, a combat airplane must flight in supersonic regime and perform sharp maneuvers. Figure 2.4 shows some examples of military aircraft.
Figure 2.5: Types of civilian transportation aircraft.
In the civil framework, the most common airplanes are those dedicated to the transportation of people in different segments (business jets, regional transportation, medium-haul transportation, and long-haul transportation). Other civil uses are also derived to civil aviation such fire extinction, photogrametric activities, etc. Figure 2.5 shows some examples of civilian aircraft.
1. Also referred to as aircraft. From now on, when we referred to an aircraft, we mean an aerodyne with fixed-wing and provided with a power plant.
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A rotorcraft (or rotary wing aircraft) is a heavier-than-air aircraft that uses lift generated by wings, called rotor blades, that revolve around a mast. Several rotor blades mounted to a single mast are referred to as a rotor. The International Civil Aviation Organization (ICAO) defines a rotorcraft as supported in flight by the reactions of the air on one or more rotors. Rotorcraft include:
• Helicopters.
• Autogyros.
• Gyrodinos.
• Combined.
• Convertibles.
Figure 2.6: Helicopter.
A helicopter is a rotorcraft whose rotors are driven by the engine (or engines) during the flight, to allow the helicopter to take off vertically, hover, fly forwards, backwards, and laterally, as well as to land vertically. Helicopters have several different configurations of one or more main rotors. Helicopters with one driven main rotor require some sort of anti-torque device such as a tail rotor. See Figure 2.6 as illustration of an helicopter.
An autogyro uses an unpowered rotor driven by aerodynamic forces in a state of autorotation to generate lift, and an engine-powered propeller, similar to that of a fixed- wing aircraft, to provide thrust and fly forward. While similar to a helicopter rotor in appearance, the autogyro’s rotor must have air flowing up and through the rotor disk in order to generate rotation.
The rotor of a gyrodyne is normally driven by its engine for takeoff and landing (hovering like a helicopter) with anti-torque and propulsion for forward flight provided by one or more propellers mounted on short or stub wings.
The combined is an aircraft that can be either helicopter or autogyro. The power of the engine can be applied to the rotor (helicopter mode) or to the propeller (autogyro mode). In helicopter mode, the propeller assumes the function of anti-torque rotor.
The convertible can be either helicopter or airplane. The propoller-rotor (proprotor) changes its attitude 90 [deg] with respect to the fuselage so that the proprotor can act as a rotor (helicopter) or as a propeller with fixed wings (airplane).
2.1.03: Missiles
A missile can be defined as an unmanned self-propelled guided weapon system.
Missiles can be classified attending at different concepts: attending at the trajectory, missiles can be cruise, ballistic, or semi-ballistic. A ballistic missile is a missile that follows a sub-orbital ballistic flightpath with the objective to a predetermined target. The missile is only guided during the relatively brief initial powered phase of flight and its course is subsequently governed by the laws of orbital mechanics and ballistics. Attending at the target, missiles can be classified as anti-submarines, anti-aircraft, anti-missile, anti-tank, anti-radar, etc. If we look at the military function, missiles can be classified as strategic and tactical. However, the most extended criteria is as follows:
• Air-to-air: launched from an airplane against an arial target.
• Surface-to-air: design as defense against enemy airplanes or missiles.
• Air-to-surface: dropped from airplanes.
• Surface-to-surface: supports infantry in surface operations.
The general configuration of a missile consists in a cylindrical body with an ogival warhead and surfaces with aerodynamic control. Missiles also have a guiding system and are powered by an engine, generally either a type of rocket or jet engine.
2.1.04: Space vehicles
A space vehicle (also referred to as spacecraft or spaceship) is a vehicle designed for spaceflight. Space vehicles are used for a variety of purposes, including communications, earth observation, meteorology, navigation, planetary exploration, and transportation of humans and cargo. The main particularity is that such vehicles operate without any atmosphere (or in regions with very low density). However, they must scape the Earth’s atmosphere. Therefore, we can identify different kinds of space vehicles:
• Artificial satellites.
• Space probes.
• Manned spacecrafts.
• Space launchers.
A satellite is an object which has been placed into orbit by human endeavor, which goal is to endure for a long time. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon. They can carry on board diverse equipment and subsystems to fulfill with the commended mission, generally to transmit data to Earth. A taxonomy can be given attending at the mission (scientific, telecommunications, defense, etc), or attending at the orbit (equatorial, geostationary, etc).
A space probe is a scientific space exploration mission in which a spacecraft leaves Earth and explores space. It may approach the Moon, enter interplanetary, flyby or orbit other bodies, or approach interstellar space. Space probes are a form of robotic spacecraft. Space probes are aimed for research activities.
The manned spacecraft are space vehicles with crew (at least one). We can distinguish space flight spacecrafts and orbital stations (such ISS). Those missions are also aimed for research and observation activities.
Figure 2.7: Space shuttle: Discovery.
Space launchers are vehicles which mission is to place another space vehicles, typically satellites, in orbit. Generally, they are not recoverable, with the exception of the the American space shuttles (Columbia, Challenger, Discovery, Atlantis, and Endevour). The space shuttle was a manned orbital rocket and spacecraft system operated by NASA on 135 missions from 1981 to 2011. This system combined rocket launch, orbital spacecraft, and re-entry spaceplane. See Figure 2.7, where the Discovery is sketched. Major missions included launching numerous satellites and interplanetary probes, conducting space science experiments, and 37 missions constructing and servicing the ISS.
The configuration of space vehicles varies depending on the mission and can be unique. As a general characteristic, just mention that launchers have similar configuration as missiles.
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Figure 2.8: Parts of an aircraft.
Before going into the fundamentals of atmospheric flight, it is interesting to identify the fundamental elements of the aircraft2. As pointed out before, there are several configurations. The focus will be on commercial airplanes flying in high subsonic regimes, the most common ones. Figure 2.8 shows the main parts of a typical commercial aircraft.
The central body of the airplane, which hosts the crew and the payload (passengers, luggage, and cargo), is the fuselage. The wing is the main contributor to lift force. The surfaces situated at the tail or empennage of the aircraft are referred to as horizontal stabilizer and vertical stabilizer. The engine is typically located under the wing protected by the so-called gondolas (some configurations with three engines locate one engine in the tail).
2. Again, in the sense of a fixed-wing aircraft provided with a power plant.
2.02: Parts of the aircraft
The fuselage is the aircraft’s central body that accommodates the crew and the payload (passengers and cargo) and protect them from the exterior conditions. The fuselage also gives room for the pilot’s cabin and its equipments, and serves as main structure to which the rest of structures (wing, stabilizers, etc.) are attached. Its form is a trade off between an aerodynamic geometry (with minimum drag) and enough volume to fulfill its mission.
Figure 2.9: Types of fuselages. © Adrián Hermida / Wikimedia Commons / CC-BY-SA-3.0.
Most of the usable volume of the fuselage is derived to passenger transportation in the passenger cabin. The layout of the passenger cabin must fulfill IATA regulations (dimensions of corridors, dimensions of seats, distance between lines, emergency doors), and differs depending on the segment of the aircraft (short and long-haul), the passenger type (economic, business, first class, etc.), or company policies (low cost companies Vs. flag companies). Cargo is transported in the deck (in big commercial transportation aircraft generally situated bellow the passenger cabin). Some standardized types of fuselage are depicted in Figure 2.9.
2.2.02: Wing
A wing is an airfoil that has an aerodynamic cross-sectional shape producing a useful lift to drag ratio. A wing’s aerodynamic quality is expressed as its lift-to-drag ratio. The lift that a wing generates at a given speed and angle of attack can be one to two orders of magnitude greater than the total drag on the wing. A high lift-to-drag ratio requires a significantly smaller thrust to propel the wings through the air at sufficient lift.
Figure 2.10: Aircraft’s plant-form types. © Guy Inchbald / Wikimedia Commons / CC-BY-SA-3.0.
The wing can be classified attending at the plant-form. The elliptic plant-form is the best in terms of aerodynamic efficiency (lift-to-drag ration), but it is rather complex to manufacture. The rectangular plant-form is much easier to manufacture but the efficiency drops significantly. An intermediate solution is the wing with narrowing (also referred to as trapezoidal wing or tapered wing). As the airspeed increases and gets closer to the speed of sound, it is interesting to design swept wings with the objective of retarding the effects of sharpen increase of aerodynamic drag associated to transonic regimens, the so-called compressibility effects. The delta wing is less common, typical of supersonic flights. An evolution of the delta plant-form is the ogival plant-form. See Figure 2.10.
Figure 2.11: Wing vertical position. © Guy Inchbald / Wikimedia Commons / CC-BY-SA-3.0.
Attending at the vertical position, the wing can also be classified as high, medium, and low. High wings are typical of cargo aircraft. It allows the fuselage to be nearer the floor, and it is easier to execute load and download tasks. On the contrary, it is difficult to locate space for the retractile landing gear (also referred to as undercarriage). The low wing is the typical one in commercial aviation. It does not interfere in the passenger cabin, diving the deck into two spaces. It is also useful to locate the retractile landing gear. The medium wing is not typical in commercial aircraft, but it is very common to see it in combat aircraft with the weapons bellow the wing to be dropped. See Figure 2.11.
Figure 2.12: Wing and empennage devices. Wikimedia Commons / Public Domain.
Usually, aircraft’s wings have various devices, such as flaps or slats, that the pilot uses to modify the shape and surface area of the wing to change its aerodynamic characteristics in flight, or ailerons, which are used as control surfaces to make the aircraft roll around its longitudinal axis. Another kind of devices are the spoilers which typically used to help braking the aircraft after touching down. Spoilers are deflected so that the lift gets reduced in the semi-wing they are acting, and thus they can be also useful to help the aircraft rolling. If both are deflected at the same time, the total lift of the aircraft drops and can be used to descent quickly or to brake after touching down. See Figure 2.12.
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Figure 2.13: Aircraft’s empennage types. © Guy Inchbald / Wikimedia Commons / CC-BY-SA-3.0.
The empennage, also referred to as tail or tail assembly, gives stability to the aircraft. Most aircraft feature empennage incorporating vertical and horizontal stabilizing surfaces which stabilize the flight dynamics of pitch and yaw as well as housing control surfaces. Different configurations for the empennage can be identified (See Figure 2.13):
The conventional tail (also referred to as low tail) configuration, in which the horizontal stabilizers are placed in the fuselage. It is the conventional configuration for aircraft with the engines under the wings. It is structurally more compact and aerodynamically more efficient.
The cruciform tail, in which the horizontal stabilizers are placed midway up the vertical stabilizer, giving the appearance of a cross when viewed from the front. Cruciform tails are often used to keep the horizontal stabilizers out of the engine wake, while avoiding many of the disadvantages of a T-tail.
The T-tail configuration, in which the horizontal stabilizer is mounted on top of the fin, creating a "T" shape when viewed from the front. T-tails keep the stabilizers out of the engine wake, and give better pitch control. T-tails have a good glide ratio, and are more efficient on low speed aircraft. However, T-tails are more likely to enter a deep stall, and is more difficult to recover from a spin. T-tails must be stronger, and therefore heavier than conventional tails. T-tails also have a larger cross section.
Twin tail (also referred to as H-tail) or V-tail are other configuration of interest although much less common.
2.2.04: Main control surfaces
The main control surfaces of a fixed-wing aircraft are attached to the airframe on hinges or tracks so they may move and thus deflect the air stream passing over them. This redirection of the air stream generates an unbalanced force to rotate the plane about the associated axis.
The main control surfaces are: ailerons, elevator, and rudder.
Ailerons are mounted on the trailing edge of each wing near the wingtips and move in opposite directions. When the pilot moves the stick left, the left aileron goes up and the right aileron goes down. A raised aileron reduces lift on that wing and a lowered one increases lift, so moving the stick left causes the left wing to drop and the right wing to rise. This causes the aircraft to roll to the left and begin to turn to the left. Centering the stick returns the ailerons to neutral maintaining the bank angle. The aircraft will continue to turn until opposite aileron motion returns the bank angle to zero to fly straight.
Figure 2.14: Actions on the control surfaces. © Ignacio Icke / Wikimedia Commons / CC-BY-SA-3.0.
An elevator is mounted on the trailing edge of the horizontal stabilizer on each side of the fin in the tail. They move up and down together. When the pilot pulls the stick backward, the elevators go up. Pushing the stick forward causes the elevators to go down. Raised elevators push down on the tail and cause the nose to pitch up. This makes the wings fly at a higher angle of attack, which generates more lift and more drag. Centering the stick returns the elevators to neutral position and stops the change of pitch.
The rudder is typically mounted on the trailing edge of the fin, part of the empennage. When the pilot pushes the left pedal, the rudder deflects left. Pushing the right pedal causes the rudder to deflect right. Deflecting the rudder right pushes the tail left and causes the nose to yaw to the right. Centering the rudder pedals returns the rudder to neutral position and stops the yaw.
2.2.05: Propulsion plant
The propulsion in aircraft is made by engines that compress air taken from the exterior, mix it with fuel, burn the mixture, and get energy from the resulting high-pressure gases.
There are two main groups: propellers and jets.
A propeller is a type of fan that transmits power by converting rotational motion into thrust. The first aircraft were propelled using a piston engine. Nowadays, piston engines are limited to light aircraft due to its weight and its inefficient performance at high altitudes. Another kind of propelled engine is the turbopropoller (also referred to as turboprop) engine, a type of turbine engine which drives an aircraft propeller using a reduction gear. Turboprop are efficient in low subsonic regimes.
A jet engine is a reaction engine that discharges a fast moving jet to generate thrust by jet propulsion in accordance with the third Newton’s laws of motion (action-reaction).
Figure 2.15: Propulsion plant.
It typically consists of an engine with a rotating air compressor powered by a turbine (the so-called Brayton cycle), with the leftover power providing thrust via a propelling nozzle. This broad definition of jet engines includes turbojets, turbofans, rockets, ramjets, pulse jets. These types of jet engines are primarily used by jet aircraft for long-haul travel. Early jet aircraft used turbojet engines which were relatively inefficient for subsonic flight. Modern subsonic jet aircraft usually use high-bypass turbofan engines which provide high speeds at a reasonable fuel efficiency (almost as good as turboprops for low subsonic regimes).
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The International Standard Atmosphere (ISA) is an atmospheric model of how the pressure, temperature, density, and viscosity of the Earth’s atmosphere change over a wide range of altitudes. It has been established to provide a common reference for the atmosphere consider standard (with an average solar activity and in latitudes around 45N). This model of atmosphere is the standard used in aviation and weather studies. The temperature of air is a function of the altitude, given by the profiles established by the International Standard Atmosphere (ISA) in the different layers of the atmosphere. The reader is referred, for instance, to Anderson [1] and Franchini and García [2] for a deeper insight.
2.03: Standard atmosphere
Hypothesis 2.1 (Standard atmosphere)
The basic typotheses of ISA are:
• Complies with the perfect gas equation:
$p = \rho RT, \label{eq2.3.1.1}$
where $R$ is the perfect gas constant for air ($R = 287.053 [J/kg\ K]$), $p$ is the pressure, $\rho$ is the density, and $T$ the temperature.
• In the troposphere the temperature gradient is constant.
$\begin{array} {c} {\text{Troposphere } (0 \le h < 11000 [m])} \ {T = T_0 - \alpha h,} \end{array}$
where $T_0 = 288.15 [K], \alpha = 6.5 [K/km]$.
• In the tropopause and the inferior stratosphere the temperature is constant.
$\begin{array} {c} {\text{Tropopause and inferior stratosphere } (11000 [m] \le h < 20000 [m])} \ {T = T_{11}} \end{array}$
where $T_{11} = 216.65[K]$.
• The air pressure at sea level ($h = 0$) is $p_0 = 101325 [Pa]$. In Equation ($\ref{eq2.3.1.1}$), the air density at sea level yields $\rho_0 = 1.225 [kg/m^3]$.
• The acceleration due to gravity is constant ($g = 9.80665 [m/s^2]$).
• The atmosphere is in calm with respect to Earth.
2.3.02: Fluid-static equation
Fluid statics (also called hydrostatics) is the science of fluids at rest, and is a sub-field within fluid mechanics. It embraces the study of the conditions under which fluids are at rest in stable equilibrium.
Figure 2.16: Differential cylinder of air. Adapted from Franchini et al. [3].
If we assume the air at rest as in Hypothesis (2.1), we can formulate the equilibrium of a differential cylindrical element where only gravitational volume forces and pressure surface forces act (see Figure 2.16):
$pdS - (p + dp) dS = \rho g d Sdh,$
which gives rise to the equation of the fluid statics:
$\dfrac{d p}{dh} = -\rho g.$
2.3.03: ISA equations
Considering Equation (2.3.1.1), Equations (2.3.1.2)-(2.3.1.3), and Equation (2.3.2.2), the variations of $\rho$ and $p$ within altitude can be obtained for the different layers of the atmosphere that affect atmospheric flight:
Troposphere ($0 \le h < 11000 [m]$): Introducing Equation (2.3.1.1) and Equation (2.3.1.2) in Equation (2.3.2.2), it yields:
$\dfrac{d p}{dh} = -\dfrac{p}{R (T_0 - \alpha h)} g.$
Integrating between a generic value of altitude $h$ and the altitude at sea level ($h = 0$), the variation of pressure with altitude yields:
$\dfrac{p}{p_0} = (1 - \dfrac{\alpha}{T_0} h)^{\tfrac{g}{R\alpha}}.\label{eq2.3.3.2}$
With the value of pressure given by Equation ($\ref{eq2.3.3.2}$), and entering in the equation of perfect gas (2.3.1.1), the variation of density with altitude yields:
$\dfrac{\rho}{\rho_0} = (1 - \dfrac{\alpha}{T_0} h)^{\tfrac{g}{R\alpha} - 1}.$
Introducing now the numerical values, it yields:
$T[k] = 288.15 - 0.0065 h [m];$
$\rho [kg/m^3] = 1.225 (1 - 22.558 \times 10^{-6} \times h [m])^{4.2559};$
$p [P a] = 101325 (1 - 22.558 \times 10^{-6} \times h[m])^{5.2559}$
Tropopause and Inferior part of the stratosphere ($11000 [m] \le h < 20000 [m]$): Introducing Equation (2.3.1.1) and Equation (2.3.1.3) in Equation (2.3.2.2), and integrating between a generic altitude ($h > 11000 [m]$) and the altitude at the tropopause ($h_{11} = 11000 [m]$):
$\dfrac{p}{p_{11}} = \dfrac{\rho}{\rho_{11}} = e^{-\tfrac{g}{RT_{11}} (h - h_{11})}$
Figure 2.17: ISA atmosphere. © Cmglee / Wikimedia Commons / CC-BY-SA-3.0.
Introducing now the numerical values, it yields:
$T[k] = 216.65;$
$\rho [kg/m^3] = 0.3639e^{-157.69 \cdot 10^{-6} (h[m] - 11000)};$
$p [P a] = 22632 e^{-157.69 \cdot 10^{-6} (h[m] - 11000)}.$
2.3.04: Warm and cold atmospheres
For warm and cold days, it is used the so called warm (ISA+5, ISA+10, ISA+15, etc.) and cold (ISA-5, ISA-10, ISA-15, etc.), where the increments (decrements) represent the difference with respect to the 288.15 [K] of an average day.
Given the new \(T_0\), and given the same pressure at sea level (\(p_0\) does not change), the new density at sea level can be calculated. Then the ISA equation are obtained proceeding in the same manner.
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The atmospheric flight mechanics uses different coordinates references to express the positions, velocities, accelerations, forces, and torques. Therefore, before going into the fundamentals of flight mechanics, it is useful to define some of the most important ones:
Definition 2.1 (Inertial Reference Frame)
According to classical mechanics, an inertial reference from $F_I (O_I, x_I, y_I, z_I)$ is either a non accelerated frame with respect to a quasifixed reference star, or either a system which for a punctual mass is possible to apply the second Newton's law:
$\sum \vec{F}_I = \dfrac{d (m \cdot \vec{V}_I)}{dt} \nonumber$
Definition 2.2 (Earth Reference Frame)
An earth reference frame $F_e (O_e, x_e, y_e, z_e)$ is a rotating topocentric (measured from the surface of the earth) system. The origin Oe is any point on the surface of earth defined by its latitude $\theta_e$ and longitude $\lambda_e$. Axis ze points to the center of earth; $x_e$ lays in the horizontal plane and points to a fixed direction (typically north); $y_e$ forms a right-handed thrihedral (typically east).
Such system is sometimes referred to as navigational system since it is very useful to represent the trajectory of an aircraft from the departure airport.
Theorem 2.2 Flat earth
The earth can be considered flat, non rotating, and approximate inertial reference frame. Consider $F_I$ and $F_e$. Consider the center of mass of the aircraft denoted by $CG$. The acceleration of $CG$ with respect to $F_I$ can be written using the well-known formula of acceleration composition from the classical mechanics:
$\vec{a}_I^{CG} = \vec{a}_e^{CG} + \vec{\Omega} \wedge (\vec{\Omega} \wedge \vec{r}_{OICG}) + 2 \vec{\Omega} \wedge \vec{V}_e^{CG},$
where the centripetal acceleration $(\vec{\Omega} \wedge (\vec{\Omega} \wedge \vec{r}_{OICG}))$ and the Coriolis acceleration $(2 \vec{\Omega} \wedge \vec{V}_e^{CG})$ are neglectable if we consider typical values: $\vec{\Omega}$ (the earth angular velocity) is one revolution per day; $\vec{r}$ is the radius of earth plus the altitude (around 6380 [km]); $\vec{V}_e^{CG}$ is the velocity of aircraft in flight (200-300 [m/s]). This means $\vec{a}_I^{CG} \approx \vec{a}_e^{CG}$ and therefore $F_e$ can be considered inertial reference frame.
Definition 2.3 (Local Horizon Frame)
A local horizon frame $F_h (O_h, x_h, y_h, z_h)$ is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axes $(x_h, y_h, z_h)$ are defined parallel to axes $(x_e, y_e, z_e)$.
In atmospheric flight, this system can be considered as quasi-inertial.
Definition 2.4 (Body Axes Frame)
A body axes frame $F_b (O_b, x_b, y_b, z_b)$ represents the aircraft as a rigid solid model. It is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis $x_b$ lays in to the plane of symmetry and it is parallel to a reference line in the aircraft (for instance, the zero-lift line), pointing forwards according to the movement of the aircraft. Axis $z_b$ also lays in to the plane of symmetry, perpendicular to $x_b$ and pointing down according to regular aircraft performance. Axis $y_b$ is perpendicular to the plane of symmetry forming a right-handed thrihedral ($y_b$ points then to the right wing side of the aircraft).
Definition 2.5 (Wind Axes Frame)
A wind axes frame $F_w (O_w, x_w, y_w, z_w)$ is linked to the instantaneous aerodynamic velocity of the aircraft. It is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis $x_w$ points at each instant to the direction of the aerodynamic velocity of the aircraft $\vec{V}$. Axis $z_w$ lays in to the plane of symmetry, perpendicular to $x_w$ and pointing down according to regular aircraft performance. Axis $y_b$ forms a right-handed thrihedral.
Notice that if the aerodynamic velocity lays in to the plane of symmetry, $y_w \equiv y_b$.
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Exercise $1$ International Standard Atmosphere
After the launch of a spatial probe into a planetary atmosphere, data about the temperature of the atmosphere have been collected. Its variation with altitude ($h$) can be approximated as follows:
$T = \dfrac{A}{1 + e^{\tfrac{h}{B}}},\label{eq2.5.1}$
where $A$ and $B$ are constants to be determined.
Assuming the gas behaves as a perfect gas and the atmosphere is at rest, using the following data:
• Temperature at $h = 1000$, $T_{1000} = 250\ K$;
• $p_0 = 100000 \dfrac{N}{m^2}$;
• $\rho_0 = 1 \dfrac{Kg}{m^3}$;
• $T_0 = 300\ K$;
• $g = 10 \dfrac{m}{s^2}$.
determine:
1. The values of $A$ and $B$, including their unities.
2. Variation law of density and pressure with altitude, respectively $\rho (h)$ and $p (h)$ (do not substitute any value).
3. The value of density and pressure at $h = 1000 m$.
Answer
We assume the following hypotheses:
(a) The gas is a perfect gas.
(b) It fulfills the fluidostatic equation.
Based on hypothesis (a):
$P = \rho RT.\label{eq2.5.2}$
Based on hypothesis (b):
$dP = -\rho gdh.\label{eq2.5.3}$
Based on the data given in the statement, and using Equation ($\ref{eq2.5.2}$):
$R = \dfrac{P_0}{\rho_0 T_0} = 333.3 \dfrac{J}{(Kg \cdot K)}$
1. The values of $A$ and $B$:
Using the given temperature at an altitude $h = 0$ $(T_0 = 300\ K)$, and Equation ($\ref{eq2.5.1}$):
$300 = \dfrac{A}{1 + e^0} = \dfrac{A}{2} \to A = 600 \ K.$
Using the given temperature at an altitude $h = 1000$ ($T_{1000} = 250\ K$), and Equation ($\ref{eq2.5.1}$):
$250 = \dfrac{A}{1 + e^{\tfrac{1000}{B}}} = \dfrac{600}{1 + e^{\tfrac{1000}{B}}} \to B = 2972\ m.$
2. Variation law of density and pressure with altitude:
Using Equation ($\ref{eq2.5.2}$) and Equation ($\ref{eq2.5.3}$):
$dP = -\dfrac{P}{RT} gdh.\label{eq2.5.7}$
Integrating the differential Equation ($\ref{eq2.5.7}$) between $P(h = 0)$ and $P, h = 0$ and $h$:
$\int_{P_0}^{P} \dfrac{dP}{P} = \int_{h = 0}^{h} -\dfrac{g}{RT} dh.\label{eq2.5.8}$
Introducing Equation ($\ref{eq2.5.1}$) in Equation ($\ref{eq2.5.8}$):
$\int_{P_0}^{P} \dfrac{dP}{P} = \int_{h = 0}^{h} -\dfrac{g(1 + e^{\tfrac{h}{B}})}{RA} dh.\label{eq2.5.9}$
Integrating Equation ($\ref{eq2.5.9}$):
$Ln \dfrac{P}{P_0} = -\dfrac{g}{RA} (h + Be^{\tfrac{h}{B}} - B) \to P = P_0 e^{-\tfrac{g}{RA} (h + Be^{\tfrac{h}{B}} - B)}.\label{eq2.5.10}$
Using Equation ($\ref{eq2.5.2}$), Equation ($\ref{eq2.5.1}$), and Equation ($\ref{eq2.5.10}$):
$\rho = \dfrac{P}{RT} = \dfrac{P_0 e^{-\tfrac{g}{RA} (h + Be^{\tfrac{h}{B}} - B)}}{R \tfrac{A}{1 + e^{\tfrac{h}{B}}}}\label{eq2.5.11}$
3. Pressure and density at an altitude of 1000 m:
Using Equation ($\ref{eq2.5.10}$) and Equation ($\ref{eq2.5.11}$), the given data for $P_0$ and $g$, and the values obtained for $R, A$, and $B$:
• $\rho (h = 1000) = 1.0756 \tfrac{kg}{m^3}.$
• $P(h = 1000) = 89632.5 Pa.$
2.06: References
[1] Anderson, J. (2012). Introduction to flight, seventh edition. McGraw-Hill.
[2] Franchini, S. and García, O. (2008). Introducción a la ingeniería aeroespacial. Escuela Universitaria de Ingeniería Técnica Aeronáutica, Universidad Politécnica de Madrid.
[3] Franchini, S., López, O., Antoín, J., Bezdenejnykh, N., and Cuerva, A. (2011). Apuntes de Tecnología Aeroespacial. Escuela de Ingeniería Aeronáutica y del Espacio. Universidad Politécnica de Madrid.
[4] Gómez-Tierno, M., Pérez-Cortés, M., and Puentes-Márquez, C. (2009). Mecánica de vuelo. Escuela Técnica Superior de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid.
[5] Howe, D. (2000). Aircraft conceptual design synthesis, volume 5. Wiley.
[6] Jenkinson, L. R., Simpkin, P., Rhodes, D., Jenkison, L. R., and Royce, R. (1999). Civil jet aircraft design, volume 7. Arnold London.
[7] Raymer, D. P. et al. (1999). Aircraft design: a conceptual approach, volume 3. American Institute of Aeronautics and Astronautics.
[8] Torenbeek, E. (1982). Synthesis of subsonic airplane design: an introduction to the preliminary design of subsonic general aviation and transport aircraft, with emphasis on layout, aerodynamic design, propulsion and performance. Springer.
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Aerodynamics is the discipline that studies the forces and the resulting motion of objects in the air. Therefore, the basis of atmospheric flight is found on the study of aerodynamics (a branch of fluid mechanics). We start then giving the fundamentals of fluid mechanics in Section 3.1. Section 3.2 and Section 3.3 are devoted to the study of aerodynamics of airfoils and wings, respectively. Finally, Section 3.4 analyzes high-lift devices. Thorough references are Anderson [2] and Franchini and García [3, Chap. 3].
03: Aerodynamics
A fluid is a substance, such a liquid or a gas, that changes its shape rapidly and continuously when acted on by forces. Fluid mechanics is the science that study how the fluid qualities respond to such forces and what forces the fluid applies to solids in contact with the fluid. Readers are referred to Franchini et al. [4]. More detailed information can be found in Franchini and García [3, Chap. 3].
3.01: Fundamentals of fluid mechanics
Many of the fundamental laws of fluid mechanics apply to both liquid and gases. Liquids are nearly incompressible. Unlike a gas, the volume of a given mass of the liquid remains almost constant when a pressure is applied to the fluid. The interest herein is centered in gases since the atmosphere in which operate is a gas commonly know as air. Air is a viscous, compressible fluid composed mostly by nitrogen (78%) and oxygen (21%). Under some conditions (for instance, at low flight velocities), it can be considered incompressible.
Gas: A gas consists of a large number of molecules in random motion, each molecule having a particular velocity, position, and energy, varying because of collisions between molecules. The force per unit created on a surface by the time rate of change of momentum of the rebounding molecules is called the pressure. As long as the molecules are sufficiently apart so that the intermolecular magnetic forces are negligible, the gas acts as a continuos material in which the properties are determined by a statistical average of the particle effects. Such a gas is called a perfect gas.
Figure 3.1: Stream line.
Stream line: The air as a continuos fluid flows under determined patterns1 confined into a finite space (the atmosphere). The curves tangent to the velocity vector of the flow at each point of the fluid in an instant of time are referred to as streamlines. Two streamlines can not cross each other except in points with null velocity, otherwise will mean that one point has two different velocities.
Figure 3.2: Stream tube.
Stream tube: A stream tube is the locus2 of the streamlines which pass through a closed curve in a given instant. The stream tube can be though as a pipe inside the fluid, through its walls there is no flow.
1. the movement of a fluid is governed by the Navier-Stokes partial differential equations. The scope of this course does not cover the study of Navier-Stokes.
2. In geometry, a locus is a collection of points that share a property.
3.1.02: Continuity equation
One of the fundamentals of physics stays that the matter in the interior of an isolated system is not created nor destroyed, it is only transformed. If one thinks in open systems (not isolated), such as human beings or airplanes in flight, its mass is constantly varying.
In a fluid is not easy to identify particles or fluid volumes since they are moving and deforming constantly within time. That is way the conservation of mass must be understood in a different way:
Recall the concept of stream tube. Assuming through its walls there is no flow, and that the flow is stationary across any section area (the velocity is constant), the mass that enters per unit of time in Section $A_1 (\rho_1 V_1 A_1)$ will be equal to the mass that exits Section $A_2 (\rho_2 V_2 A_2)$, where $\rho$ is the density, $V$ is the velocity, and $A$ is the area. Therefore, the continuity of mass stays:
$\rho_1 V_1 A_1 = \rho_2 V_2 A_2.$
Since Section $A_1$ and Section $A_2$ are generic, one can claim that the product $\rho VA$ is constant along the stream tube. The product $\rho VA$ is referred to as mass flow $\dot{m}$ (with dimensions [kg/s]).
Figure 3.3: Continuity equation.
Compressible and incompressible flow
In many occasions occurs that the density of a fluid does not change due to the fact that it is moving. This happens in liquids and, in some circumstances, in gases (think in the air confined in a room). Notice that one can not say that the air is incompressible, but an air flow is incompressible.
The movement of air in which the velocity is inferior to 100 [m/s] can be considered incompressible. When the air moves faster, as is the case in a jet airplane, the flow is compressible and the studies become more complicated as it will be seen in posterior courses.
3.1.03: Quantity of movement equation
The quantity of movement equation in a fluid (also referred to as momentum equation) is the second Newton law expression applied to a fluid:
$\sum \vec{F} = m \dfrac{d(\vec{V})}{dt}.$
The forces exerted over the matter, solid or fluid, can be of two types:
• Distance-exerted, e.g., gravitational, electric, magnetic; related to mass or volume.
• Contact-exerted, e..g, pressure or friction, related to the surface of contact.
Euler equation
We assume herein that the fluid is inviscous and, therefore, we do not consider frictional forces. The only sources of forces are pressure and gravity. We consider one-dimensional flow along the longitudinal axis.
Figure 3.4: Quantity of movement. Adapted from $F_{\text{RANCHINI}}$ et al. [4].
Assume there is a fluid particle with circular section $A$ and longitude $dx$ moving with velocity $u$ along direction $x$. Apply second Newton law ($m \dot{u} = \sum F$; $m = \rho Adx$):
$(\rho A dx) \dfrac{du}{dt} = -Adp - (\rho A dx) g \dfrac{dz}{dx}.\label{eq3.1.3.2}$
The force due to gravity only affects in the direction of axis $x$.
Considering a reference frame attached to the particle in which $dt = \dfrac{dx}{u}$ (stationary flow), the acceleration can be written as $\tfrac{du}{dt} = \dfrac{du}{dx/u}$ and Equation ($\ref{eq3.1.3.2}$), dividing by $Adx$. yields:
$\rho u \dfrac{du}{dx} = - \dfrac{dp}{dx} - \rho g \dfrac{dz}{dx}.\label{eq3.1.3.3}$
Equation ($\ref{eq3.1.3.3}$) is referred to as Euler equation. This is the unidimensional case. The Euler equation is more complex, considering the tridimensional motion. The complete equation will be seen is posterior courses of fluid mechanics.
Bernoulli equation
Consider Equation ($\ref{eq3.1.3.3}$) and notice that the particle moves on the direction of the streamline. If the flow is incompressible (or there exist a relation between pressure and density, relation called barotropy), the equation can be integrated along the streamline:
$\rho \dfrac{u^2}{2} + p + \rho gz = C.\label{eq3.1.3.4}$
The value of the constant of integration, $C$, should be calculated with the known conditions of a point. Bernouilli equation expresses that the sum of the dynamic pressure ($\rho \tfrac{u^2}{2}$), the static pressure ($p$), and the piezometric pressure ($\rho g z$) is constant along a stream tube.
In the case of the air moving around an airplane, the term $\rho g z$ does not vary significantly between the different points of the streamline and can be neglected. Equation ($\ref{eq3.1.3.4}$) is then simplified to:
$\rho \dfrac{u^2}{2} + p = C.$
This does not occur in liquids, where the density is an order of thousands higher than in gases and the piezometric term is always as important as the rest of terms (except if the movement is horizontal).
If at any point of the streamline the velocity is null, the point will be referred to as stagnation point. At that point, the pressure takes a value known as stagnation pressure ($p_T$), so that at any other point holds:
$u = \sqrt{2 \dfrac{p_T - p}{\rho}}.$
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Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. Every single fluid inherently has viscosity. The less viscous the fluid is, the easier is the movement (fluidity). For instance, it is well known that the honey has low fluidity (high viscosity), while water presents, compared to honey, high fluidity (low viscosity). Viscosity is a property of the fluid, but affects only if the fluid is under movement. Viscosity describes a fluid’s internal resistance to flow and may be thought as a measure of fluid friction. In general, in any flow, layers move at different velocities and the fluid’s viscosity arises from the shear stress between the layers that ultimately opposes any applied force.
Viscosity stress
The viscosity force is a friction force and therefore the shear stress $\tau$ (viscosity stress) is a force over a unity of surface.
Figure 3.5: Viscosity. Adapted from $F_{\text{RANCHINI}}$ et al. [4]
The relationship between the shear stress and the velocity gradient can be obtained considering two plates closely spaced at a distance $z$, and separated by a homogeneous fluid, e.g., water or oil. Assuming that the plates are very large, with a large area $A$, and that the lower plate is fixed, let a force $F$ be applied to the upper plate. Thus, the force causes the substance between the plates to undergo shear flow with a velocity gradient $du/dz$. The applied force is proportional to the area and velocity gradient in the fluid:
$F = \mu A \dfrac{du}{dz},$
where coefficient $\mu$ is the dynamic viscosity.
This equation can be expressed in terms of shear stress, $\tau = F/A$. Thus expressed in differential form for straight, parallel, and uniform flow, the shear stress between layers is proportional to the velocity gradient in the direction perpendicular to the layers:
$\tau = \mu \dfrac{du}{dz}.$
Some typical values of $\mu$ in regular conditions are: $0.26 [N \cdot s/m^2]$ for oil; $0.001 [N \cdot s/m^2]$ for water; and $0.000018 [N \cdot s/m^2]$ for air.
Boundary layer
If one observes the flow around an airfoil, it will be seen that fluid particles in contact with the airfoil have null relative velocity. However, the velocity of the particles at a (relatively low) distance is approximately the velocity of the exterior stream. This thin layer, in which the velocity perpendicular to the airfoil varies dramatically, is known as boundary layer and plays a very important role.
Figure 3.6: Airfoil with boundary layer. The boundary layer has been overemphasized for clarity. Adapted from Franchini et al. [4].
The aerodynamic boundary layer was first defined by Ludwig Prandtl in 1904 Prandtl [5]. It allows to simplify the equations of fluid flow by dividing the flow field into two areas: one inside the boundary layer, where viscosity is dominant and the majority of the drag experienced by a body immersed in a fluid is created, and one outside the boundary layer where viscosity can be neglected without significant effects on the solution. This allows a closed-form solution for the flow in both areas, which is a significant simplification over the solution of the full Navier-Stokes equations. The majority of the heat transfer to and from a body also takes place within the boundary layer, again allowing the equations to be simplified in the flow field outside the boundary layer.
In high-performance designs, such as commercial transport aircraft, much attention is paid to controlling the behavior of the boundary layer to minimize drag. Two effects have to be considered. First, the boundary layer adds to the effective thickness of the body through the displacement thickness, hence increasing the pressure drag. Secondly, the shear forces at the surface of the wing create skin friction drag.
Reynolds number
The dimensionless Reynolds number is due to the studies of Professor Osborne Reynolds (1842-1912) about the conditions in which the flow of fluid in pipes transitioned from laminar flow to turbulent flow. Reynolds used a fluid made of a mixture of water and glycerin, so that varying the mixture the viscosity of the fluid could be modified. When the proportion of glycerin was high, the flow was smooth; by injecting a thread of ink in a pipe with the fluid the thread of ink was flowing smoothly. When the proportion of water was high, by injecting the ink in the pipe a spinning movement was noticed (vortexes) and soon the ink was blurred into the fluid. Reynolds called laminar flow the smooth flow and turbulent flow the chaotic one. He also proved that the character of the flow depended on an dimensionless parameter:
$\text{Re} = \rho V D/\mu,$
where $V$ is the mean velocity of the fluid and $D$ is the diameter of the pipe. Posterior research named this number the Reynolds number.
Figure 3.7: Boundary layer transition. Adapted from $F_{\text{RANCHINI}}$ et al. [4].
More precisely, the Reynolds number $\text{Re}$ is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions:
$\text{Re} = \dfrac{\rho V^2 D^2}{\mu VD}.$
Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion; turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices, and other flow instabilities.
In the movement of air around the wing, instead of $D$ it is used the chord of the airfoil, $c$, the most adequate characteristic longitude. In flight, the Reynolds number is high, of an order of millions, which means the viscosity effects are low and the boundary layer is thin. In this case, the Euler Equation (3.1.3.3) can be used to determine the exterior flow around the airfoil. However, the boundary layer can thicken and the boundary layer drops off along the body, resulting in turbulent flow which increases drag.
Therefore, at high Reynolds numbers, such as typical full-sized aircraft, it is desirable to have a laminar boundary layer. This results in a lower skin friction due to the characteristic velocity profile of laminar flow. However, the boundary layer inevitably thickens and becomes less stable as the flow drops off along the body, and eventually becomes turbulent, the process known as boundary layer transition. One way of dealing with this problem is to suck the boundary layer away through a porous surface (Boundary layer suction). This can result in a reduction in drag, but is usually impractical due to the mechanical complexity involved and the power required to move the air and dispose of it. Natural laminar flow is the name for techniques pushing the boundary layer transition aft by shaping of an airfoil or a fuselage so that their thickest point is aft and less thick. This reduces the velocities in the leading part and the same Reynolds number is achieved with a greater length.
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The speed of sound in a perfect gas is:
$a = \sqrt{\gamma RT},$
where $R$ is the constant of the gas, $T$ the absolute temperature, and $\gamma$ the adiabatic coefficient which depends on the gas. In the air $\gamma = 1.4$ and $R = 287.05\ [J/KgK]$. Therefore, the speed of sound in the air is 340.3 [m/s] at sea level in regular conditions.
Mach number
Mach number is the quotient between the speed of an object moving in the air (or any other fluid substance), typically an aircraft or a fluid particle, and the speed of sound of the air (or substance) for its particular physical conditions, that is:
$M = \dfrac{V}{a}.$
Depending on the Mach number of an air vehicle (airplane, space vehicle, or missile, for instance), five different regimes can be considered:
1. Incompressible: $M< 0.3$, approximately. In this case, the variation of the density with respect to the density at rest can be neglected.
2. Subsonic (compressible subsonic): $0.3 \le M < 0.8$, approximately. The variations in density must be included due to compressibility effects. Two different regimes can be distinguished: low subsonic ($0.3 \le M < 0.6$, approximately) and high subsonic ($0.6 \le M < 0.8$, approximately). While regional aircraft typically fly in low subsonic regimes, commercial jet aircraft typically fly in high subsonic regimes (trying to be the closest to transonic regimes while avoiding its negative effects in terms of aerodynamic drag).
3. Transonic: $0.8 \le M < 1$, approximately. This is complex situation since around the aircraft coexist both subsonic flows and supersonic flows (for instance, in the extrados of the airfoil the flow accelerates and can be supersonic while the flow entering through the leading edge was subsonic).
4. Supersonic: $M \ge 1$, and then the flow around the aircraft is also at $M \ge 1$. Notice that the flow at $M = 1$ is known as sonic.
5. Hipersonic: $M \gg 1$ (in practice, $M > 5$). In these cases phenomena such as the kinetic heat or molecules dissociation appears.
In order to understand the importance of the Mach number it is important to notice that the speed of sound is the velocity at which the pressure waves or perturbations are transmitted in the fluid.
Figure 3.8: Effect of the speed of sound in airfoils ($M_a$ corresponds to Mach number).
Imagine a compressible air flow with no obstacles. In this case, the pressure will be constant along the whole flow, there are no perturbations. If we introduce an airplane moving in the air, immediately appears a perturbation in the field of pressures near the airplane. Moreover, this perturbation will travel in the form of a wave at the speed of sound throughout the whole fluid field. This wave represents some kind of information emitted to the rest of fluid particles, so that the fluid adapts its physical conditions (trajectory, pressure, temperature) to the upcoming object.
If the airplane flies very slow ($M = 0.2$), the waves will travel fast relative to the airplane ($M = 1$ versus $M = 0.2$) in all directions. In this form the particles approaching the airplane are well informed of what is coming and can modify smoothly its conditions. If the velocity is higher, however still below $M = 1$, the modification of the fluid field is not so smooth. If the airplane flies above the speed of sound (say $M = 2$), then in this case the airplane flies twice faster than the perturbation waves, so that waves can not progress forwards to inform the fluid field. The consequence is that the fluid particles must adapt its velocity and position in a sudden way, resulting in a phenomena called shock wave.
Therefore, when an aircraft exceeds the sound barrier, a large pressure difference is created just in front of the aircraft resulting in a shock wave. The shock wave spreads backwards and outwards from the aircraft in a cone shape (a so-called Mach cone). It is this shock wave that causes the sonic boom heard as a fast moving aircraft travels overhead. At fully supersonic speed, the shock wave starts to take its cone shape and the flow is either completely supersonic, or only a very small subsonic flow area remains between the object’s nose and the shock wave. As the Mach number increases, so does the strength of the shock wave and the Mach cone becomes increasingly narrow. As the fluid flow crosses the shock wave, its speed is reduced and temperature, pressure, and density increase. The stronger the shock, the greater the changes. At high enough Mach numbers the temperature increases so much over the shock that ionization and dissociation of gas molecules behind the shock wave begins.
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Figure 3.9: Aerodynamic forces and moments.
The geometric figure obtained as a cross section of an airplane wing is referred to as airfoil. An airfoil-shaped body moved through a fluid produces an aerodynamic force. The component of this force perpendicular to the direction of motion is called lift. The component parallel to the direction of motion is called drag. In order to be able to calculate the movement of an airplane, an important issue is to determine the forces and torques around the center of gravity produced by the effects of air. Readers are referred to $F_{\text{RANCHINI}}$ et al. [4]. Other introductory references on airfoil aerodynamics are Franchini and García [3, Chap. 3] and Anderson [1, Chap. 4-5].
The aerodynamic forces are: The aerodynamic torques are:
• $L$: Lift force.
• $D$: Drag force.
• $Q$: Lateral force.
• $M_x$: Roll torque.
• $M_y$: Pitch torque.
• $M_z$: Yaw torque.
3.02: Airfoils shapes
Figure 3.10: Description of an airfoil.
The main parts of an aerodynamic airfoil are: the chord, $c$, which is the segment joining the leading edge of the airfoil, $x_{le}$, and the trailing edge, $x_{te}$. To describe the airfoil, one only needs to know the functions $z_e (x)$ and $z_i (x)$, the extrados and the intrados of the airfoil, respectively. See Figure 3.10.
Another form of describing airfoils is to consider them as the result of two different contributors:
• $C(x)$, representing the camber of the airfoil:
$C(x) = \dfrac{z_e (x) + z_i (x)}{2}.\label{eq3.2.1.1}$
• $E(x)$, which gives the thickness of the airfoil:
$E(x) = z_e (x) - z_i (x).\label{eq3.2.1.2}$
According to Equation ($\ref{eq3.2.1.1}$) and Equation ($\ref{eq3.2.1.2}$), it yields:
$z_e (x) = C(x) + \dfrac{1}{2} E(x).$
$z_i (x) = C(x) - \dfrac{1}{2} E(x).$
A symmetric airfoil is that in which $C(x) = 0 \forall x$.
The thickness of the airfoil is the maximum of $E(x)$. The relative thickness of the airfoil is the quotient between thickness and camber, and it is usually expressed in percentage.
Figure 3.11: Description of an airfoil with angle of attack.
The angle of attack $\alpha$ is the angle formed by the direction of the velocity ($u_{\infty}$) of the air current with no perturbation, that is, sufficiently far away, with a reference line in the airfoil, typically the chord line. See Figure 3.11. Therefore, to give an angle of attack to the airfoil it is only necessary to rotate it around an arbitrary point $x_0$. In the hypothesis of a low relative thickness ($\le$ 15 - 18%), a mathematical approximation of this rotation is obtained adding to both $z_e (x)$ and $z_i (x)$ the straight line $z = (x_0 - x) \alpha$, so that it yields:
$z_e (x) = C(x) + \dfrac{1}{2} E(x) + (x_0 - x) \alpha.$
$z_i (x) = C(x) - \dfrac{1}{2} E(x) + (x_0 - x) \alpha.$
This breaking down analysis into different contributors will be very useful in the future to analyze the contributors of an airfoil to aerodynamic lift: camber, thickness, and angle of attack.
3.2.02: Generation of aerodynamic forces
Figure 3.12: Pressure and friction stress over an airfoil.
Figure 3.13: Aerodynamic forces and torques over an airfoil.
Figure 3.14: Aerodynamic forces and torques over an airfoil with angle of attack.
The actions of the air over a body which moves with respect to it give rise, at each point of the body’s surface, to a shear stress tangent to the surface due to viscosity and a perpendicular stress due to the pressure. Thus, a pressure distribution and a shear stress distribution are obtained over the surface of the body.
Integrating the distribution over the surface of the body, one obtains the aerodynamic forces:
$f_{aero} = \int (p(x) - p_{\infty}) \cdot dx + \int \tau (x) \cdot dx.$
Taking the resultant of the distribution and multiplying by the distance to a fixed point (typically the aerodynamic center, located approximately at $c/4$), one obtains the aerodynamic torques:
$m_{ca} = \int (p(x) - p_{\infty}) (x - x_{c/4}) \cdot dx + \int \tau (x - x_{c/4}) \cdot dx.$
Figure 3.12 , Figure 3.13, and Figure 3.14 illustrate it.
The lift in an airfoil comes, basically, from the pressure forces. The drag in an airfoil
comes from both the friction forces (shear stress) and the pressure forces.
Regarding drag forces, it is important to remember the already mentioned about the boundary layer. The thicker the boundary layer is, the greater the drag due to pressure effects is. In particular, when the flow drops off along the airfoil and becomes turbulent (boundary layer transition), the drag due to pressure effects increases dramatically. Furthermore, friction forces exist, which are greater in turbulent flow rather than in laminar flow. Typically, the contribution to drag of friction forces is lower than the contribution of pressure forces. Therefore, a smart design of an airfoil regarding the behavior of the boundary layer is key to minimize drag forces.
The lift forces are due to the camber, the angle of attack, and the thickness of the airfoil, which conform an airfoil shape so that the pressures in the extrados are lower that pressures in the intrados. The generation of lift can be summarized as follows:
Figure 3.15: Lift generation. Modified from Wikimedia Commons / Public Domain.
• Because of the law of mass continuity (Equation (3.1.2.1)) the flow velocity increases over the top surface of the airfoil more than it does over the bottom surface. This is illustrated in Figure 3.15.
• As a consequence of Bernoulli effect3 (Equation (3.1.3.5)), the pressure over the top surface of the airfoil is less than the pressure over the bottom surface.
• Because of the lower pressure over the top surface of the airfoil is less than the pressure over the bottom surface, the airfoil experiences a lift force upwards.
Therefore, this simplified statement of the equations of fluid mechanics give a qualitative idea of the aerodynamic forces. However, the resolution of the equations of fluid mechanics (Navier-Stokes equations) is extremely difficult, even though counting with the most powerful numerical tools. From the theoretical point of view they are studied using simplifications. From the experimental point of view, it is common practice to test scale-models in wind tunnels. The wind tunnels is an experimental equipment able to produce a controlled air flow into a testing chamber.
3. For an incompressible flow, from Bernoulli Equation (3.1.3.5), where the velocity increases, the static pressure decreases.
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The fundamental curves of an aerodynamic airfoil are: lift curve, drag curve, and momentum curve. These curves represent certain dimensionless coefficients related to lift, drag, and momentum.
The interest first focuses on determining the pressure distribution over airfoil’s intrados and extrados so that, integrating such distributions, the global loads can be calculated. Again, instead of using the distribution of pressures $p(x)$, the distribution of the coefficient of pressures $c_p(x)$ will be used.
The coefficient of pressures is defined as the pressure in the considered point minus the reference pressure, typically the static pressure of the incoming current $p_{\infty}$, over the
dynamic pressure of the incoming current, $q = \rho_{\infty} u_{\infty}^2 /2$, that is:
$c_p = \dfrac{p - p_{\infty}}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2}.$
Using Equation (3.1.3.5) and considering constant density, it yields:
$c_p = 1 - \left ( \dfrac{V}{u_{\infty}} \right )^2,$
being $V$ the velocity of the air flow at the considered point.
Figure 3.16. Distribution of Coefficient of Pressures. Adapted from $F_{\text{RANCHINI}}$ et al. [4]
Figure 3.16 shows a typical distribution of coefficient of pressures over an airfoil. Notice that z-axis shows negative $c_p$ and the direction of arrows means the sign of $c_p$. An arrow which exits the airfoil implies $c_p$ is negative, which means the air current accelerates in that area (airfoil’s extrados) and the pressure decreases (suction). On the other hand, where arrows enter the airfoil there exist overpressure, that is, decelerated current and positive $c_p$. Notice that if there exist a stagnation point ($V = 0$), which is by the way typical, $c_p = 1$.
The dimensionless coefficients are:
$c_l = \dfrac{l}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2 c};$
$c_d = \dfrac{d}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2 c};$
$c_m = \dfrac{m}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2 c^2};$
The criteria of signs is as follows: for $c_l$, positive if lift goes upwards; for $c_d$, positive if drag goes backwards; for $c_m$, positive if the moment makes the airfcraft pitch up.
The equation that allow $c_l$ and $c_m$ to be obtained from the distributions of coefficients of pressure in the extrados, $c_{pe} (x)$, and the intrados, $c_{pe} (x)$, are:
$c_l = \dfrac{1}{c} \int_{x_{le}}^{x_{te}} (c_{pi} (x) - c_{pe} (x)) dx = \dfrac{1}{c} \int_{x_{le}}^{x_{te}} c_l (x) dx,$
$\begin{array} {rcl} {c_m} & = & {\dfrac{1}{c^2} \int_{x_{le}}^{x_{te}} (x_0 - x) (c_{pi} (x) - c_{pe} (x)) dx = \ ...} \ {...} & = & {\dfrac{1}{c^2} \int_{x_{le}}^{x_{te}} (x_0 - x) c_l (x) dx.} \end{array}$
If $x_0$ is chosen so that the moment is null, $x_0$ will coincide with the center of pressures of the airfoil (also referred to as aerodynamic center), $x_{cp}$, which is the point of application of the vector lift:
$x_{cp} = \dfrac{\int_{x_{le}}^{x_{te}} x c_l (x) dx}{\int_{x_{le}}^{x_{te}} c_l (x) dx}.$
Notice that in the case of a wing or a full aircraft, lift and drag forces have unities of force [N], and the pitch torque has unities of momentum [Nm]. To represent them it is common agreement to use $L, D$, and $M$. In the case of airfoils, due to its bi-dimensional character, typically one talks about force and momentum per unity of distance. In order to notice the difference, they are represented as $l, d$, and $m$.
Characteristic curves
The characteristic curves of an airfoil are expressed as a function of the dimensionless coefficients. These characteristic curves are, given a Mach number, a Reynolds number, and the geometry of the airfoil, as follows:
• The lift curve given by $c_l (\alpha)$.
• The drag curve, $c_d (c_l)$, also referred to as polar curve.
• The momentum curve, $c_m (\alpha)$.
Moreover, there is another typical curve which represents the aerodynamic efficiency as a function of $c_l$ given $\text{Re}$ and $M$. The aerodynamic efficiency is $E = \tfrac{c_l}{c_d} (E = \tfrac{L}{D})$, and measures the ratio between lift generated and drag generated. The designer aims at maximizing this ratio.
Lets focus on the curve of lift. Typically this curve presents a linear zone, which can be approximated by:
$c_l (\alpha) = c_{l0} + c_{l\alpha} \alpha = c_{l \alpha} (\alpha - \alpha_0),$
Figure 3.17: Lift and drag characteristic curves.
where $c_{l \alpha} = d c_l /d \alpha$ is the slope of the lift curve, $c_{l0}$ is the value of $c_l$ for $\alpha = 0$ and $\alpha_0$ is the value of $\alpha$ for $c_l = 0$. The linear theory of thin airfoils in incompressible regime gives a value to $c_{l \alpha} = 2\pi$, while $c_{l0}$, which depends on the airfoil’s camber, is null for symmetric airfoils. There is an angle of attack, referred to as stall angle, at which this linear behavior does not hold anymore. At this point the curve presents a maximum. One this point is past lift decreases dramatically. This effect is due to the boundary layer dropping of the airfoil when we increase too much the angle of attack, reducing dramatically lift and increasing drag due to pressure effects. Figure 3.17.b illustrates it.
The drag polar can be approximated (under the same hypothesis of incompressible flow) to a parabolic curve of the form:
$c_d (c_l) = c_{d_0} + c_{d_i} c_l^2,$
where $c_{d_0}$ is the parasite drag coefficient (the one that exist when $c_l = 0$) and $c_{d_i}$ is the induced coefficient (drag induced by lift). This curve is referred to as parabolic drag polar.
The momentum curve can be approximately constant (under the same hypothesis of incompressible flow) if one choses adequately the point $x_0$. This point is the aerodynamic center of the airfoil. Under incompressible regime, this point is near $0.25c$.
Lift and drag curves are illustrated in Figure 3.17.a for a typical airfoil.
3.2.04: Compressibility and drag-divergence Mach number
Given an airfoil with a specific angle of attack, if the speed of flight increases, the velocities of the air flow over the airfoil also increase. In that case, the coefficient of pressures increases and also the coefficient of lift does so. For Mach number close to $M = 1$, $c_p$ and $c_l$ can be approximated by the Prandtl-Glauert transformation:
$c_p = \dfrac{c_{p, inc}}{\sqrt{1 - M^2}};$
$c_l = \dfrac{c_{l, inc}}{\sqrt{1 - M^2}};$
where subindex $inc$ refers to the value of the coefficient in incompressible flow.
Figure 3.18: Divergence Mach.
Figure 3.19: Supercritical airfoils.
On the other hand, the coefficient of drag remains practically constant until the airplane reaches the so called critic velocity, a subsonic velocity for which a point of extrados reaches the sonic velocity. It appears a supersonic region and waves shocks are created, giving rise to an important increase of drag. This phenomena is referred to as drag divergence.
The velocity at which this phenomena appears is refereed to as drag divergence Mach number, $M_{DD}$. Commercial aircraft can not typically overpass this velocity. There is not a unique definition on how to calculate this velocity. Two of the most used conditions are:
$\dfrac{\partial c_d}{\partial M} = 0.1; \text{ and }$
$\Delta c_d = 0.002.$
Airlines seek to fly faster if the consumption does not raise too much. For that reason, it is interesting to increase $M_{DD}$. In transonic regimes, airfoils can be designed with thin relative thickness. Another design is the so-called supercritical airfoils, whose shape permits reducing the intensity of the shock wave. In supersonic regimes, it appears another contributor to drag, the wave drag. Supersonic airfoils are designed very thin with very sharp leading edges.
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After having studied the aerodynamics fundamentals in bi-dimensional airfoils, we proceed on studying the aerodynamic fundamentals in three-dimensional wings. Readers are referred to Franchini et al. [4]. Other introductory references on wing aerodynamics are Franchini and García [3, Chap. 3] and Anderson [1, Chap. 4-5].
3.03: Wing aerodynamics
Figure 3.20: Wing geometry
In order to characterize the geometry and nomenclature of a typical commercial aircraft wing, the following wing elements are illustrated in Figure 3.20:
• Wingspan $b$.
• Chords: root chord $c_r$ and tip chord $c_t$.
• Leading and trailing edges, and the line corresponding to the locus of $c/4$ points.
• $c/4$ swept $\wedge_{c/4}$.
The area enclosed into the leading and trailing edge and the marginal borders (the section with $c_r$) view in a plant-form is referred to as wet wing surface $S_w$. The quotient between the wet wing surface and the wingspan is referred to as the geometric mean chord $\bar{c}$. which represents the mean chord that a rectangular wing with the same $b$ and $S_w$ would have.
The enlargement, $A$, is defined as:
$A = \dfrac{b}{\bar{c}} = \dfrac{b^2}{S_w}.$
There is also a parameter measuring the narrowing of the wing: $\lambda = c_t / c_r$.
3.3.02: Flow over a finite wing
Figure 3.21: Sketch of the coefficient of lift along a wingspan.
Figure 3.21 shows the distribution of coefficient of lift along the wingspan of four rectangular wings flying in incompressible flow with an attack angle of 10 [deg]. The four wings use the same aerodynamic airfoil and differ in the enlargement (8,10,12, and infinity). It can be observed that if the enlargement is infinity the wing behaves as the bi-dimensional airfoil $y$ ($c_l (y)$ constant). On the other hand, if the enlargement is finite, $c_l (y)$ shows a maximum in the root of the wing ($y = 0$) and goes to zero in the tip of the wing ($y/c = A/2$). As the enlargement decreases, the maximum $c_l (y)$ also decreases.
Figure 3.22: Whirlwind trail
The explanation behind this behavior is due to the difference of pressures between extrados and intrados. In particular, in the region close to the marginal border, there is an air current surrounding the marginal border which passes from the intrados, where the pressure is higher, to extrados, where the pressure is lower, giving rise to two vortexes, one in each border rotating clockwise and counterclockwise. This phenomena produces downstream a whirlwind trail. Figure 3.22 illustrates it.
Figure 3.23: Effective angle of attack.
The presence of this trail modifies the fluid field and, in particular, modifies the velocity each wing airfoil sees. In addition to the freestream velocity, $u_{\infty}$, a vertical induced velocity, $u_i$, must be added (See Figure 3.23). The closer to the marginal border, the higher the induced velocity is. Therefore, the effective angle of attack of the airfoil is lower that the geometric angle, which explains both the reduction in the coefficient of lift (with respect to the bi-dimensional coefficient) and the fact that this reduction is higher when one gets closer to the marginal border.
3.3.03: Lift and induced drag in wings
In order to represent the lift curve, a dimensionless coefficient ($C_L$) will be used. $C_L$ is defined as:
$C_L = \dfrac{L}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2 S_w},$
which can also be expressed as:
$C_L = \dfrac{L}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2 S_w} = {1}{\tfrac{1}{2} \rho_{\infty} u_{\infty}^2 S_w} \int_{-b/2}^{b/2} \dfrac{1}{2} \rho_{\infty} u_{\infty}^2 c(y) c_l (y) dy = \dfrac{1}{S_w} \int_{-b/2}^{b/2} c(y) c_l (y) dy.$
Figure 3.24: Induced drag.
Another consequence of the induced velocity is the appearance of a new component of drag (see Figure 3.24), the induced drag. This occurs because the lift is perpendicular to the effective velocity and therefore it has a component in the direction of the freestream (the direction used to measure the aerodynamic drag).
3.3.04: Characteristic curves in wings
The curve of lift and the drag polar permit knowing the aerodynamic characteristics of the aircraft.
Lift curve
The coefficient of lift depends, in general, on the angle of attack, Mach and Reynolds number, and the aircraft configuration (flaps, see Section 3.4). The most general expression is:
$C_L = f(\alpha , M, \text{Re}, configuration).$
As in airfoils (under the same hypothesis of incompressible flow), in wings typically the lift curve presents a linear zone, which can be approximated by:
$C_L (\alpha) = C_{L0} + C_{L\alpha} \alpha = C_{L \alpha} (\alpha - \alpha_0),$
where $C_{L \alpha} = d C_L/d \alpha$ is the slope of the lift curve, $C_{L0}$ is the value of $C_L$ for $\alpha = 0$ and $\alpha_0$ is the value of $\alpha$ for $C_L = 0$. There is a point at which the linear behavior does not hold anymore, whose angle is referred to as stall angle. At this angle the curve presents a maximum. Once this angle is past, lift decreases dramatically.
According to Prandtl theory of large wings, the slope of the curve is:
$C_{L \alpha} = \dfrac{d C_L}{d\alpha} = \dfrac{c_{l\alpha} e}{1 + \tfrac{c_{l\alpha}}{\pi A} },$
where $e \le 1$ is an efficiency form factor of the wing, also referred to as Oswald factor. In elliptic plantform $e = 1$.
Drag polar
The aircraft’s drag polar is the function relating the coefficient of drag with the coefficient of lift, as mentioned for airfoils.
The coefficient of drag depends, in general, on the coefficient of lift, Mach, and Reynolds number, and the aircraft configuration (flaps, see Section 3.4). The most general expression is:
$C_D = f(C_L, M, \text{Re}, configuration).$
The polar can be approximated to a parabolic curve of the form:
$C_D (C_L) = C_{D_0} + C_{D_i} C_L^2,$
where $C_{D_0}$ is the parasite drag coefficient (the one that exists when $C_L = 0$) due to friction
and pressure effects in the wing, fuselage, etc., and $C_{D_i} = \tfrac{1}{\pi Ae}$ is the induced coefficient (drag induced by lift) fundamentally due to the induced velocity and the whirlwind trail. This curve is referred to as parabolic drag polar. The typical values of $C_{D_0}$ depend on the aircraft but are approximately 0.015 - 0.030 and the parameter of aerodynamic efficiency e can be approximately 0.75 0.85.
Figure 3.25: Characteristic curves in wings.
The lift curve ($C_L - \alpha$) and the drag ploar ($C_D - C_L$) are represented in Figure 3.25 for a wing with four different enlargements. Both the slope and the maximum value of the lift curves increase when the enlargement increases. For the polar case, it can be observed how drag reduces as the enlargement increases.
3.3.05: Aerodynamics of wings in compressible and supersonic regimes
The evolution of the coefficients of lift and drag for a wing presents similarities with which has already been exposed for airfoils. However, instead of reducing the relative thickness and the use of supercritical airfoil to aft the divergency, in the case of wings there exist an additional resource: wing swept $\Delta_{c/4}$.
The use of swept for the design of the wing permits reducing the effective Mach number (the reduction factor is approximately $\cos \Delta_{c/4}$). Then the behavior of the airfoils is as they were flying slower and consequently the divergence Mach can be seen as higher. However, the use of swept makes the aircraft structurally complicated and, moreover, both $C_{L_{\alpha}}$ and $C_{L_{\max}}$ decrease. That is why commercial aviation tends to develop supercritical airfoils to minimize the swept.
For supersonic flight, the wings are typically designed with great swept and small enlargement. The extreme case is the delta wing.
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High-lift devices are designed to increase the maximum coefficient of lift. A first classification differences active and passive devices. Active high-lift devices require energy to be applied directly to the air (typically provided by the engine). Their use has been limited to experimental applications. The passive high-lift devices are, on the other hand, extensively used. Passive high-lift devices are normally hinged surfaces mounted on trailing edges and leading edges of the wing. By their deployment, they increase the aerodynamic chord and the camber of the airfoil, modifying thus the geometry of the airfoil so that the stall speed during specific phases of flight such as landing or take-off is reduced significantly, allowing to fly slower than in cruise. There also exist other types of high-lift devices that are not explicitly flaps, but devices to control the boundary layer. Readers are referred to $F_{\text{RANCHINI}}$ et al. [4].
3.04: High-lift devices
As shown in previous sections, there is a maximum coefficient of lift, $C_{L_{\max}}$, that can not be exceeded by increasing the angle of attack. Consider the uniform horizontal flight, where the weight of the aircraft ($W = mg$) must be balanced by the lift force, i.e.:
$W = L = \dfrac{1}{2} \rho V^2 S_w C_L.$
Therefore, the existence of the maximum coefficient of lift, $C_{L_{\max}}$, implies that the aircraft can not fly below a minimum velocity, the stall speed, $V_S$:
$V_S = \sqrt{\dfrac{W}{\tfrac{1}{2} \rho S_w C_{L_{\max}}}}.\label{eq3.4.1.2}$
Looking at equation ($\ref{eq3.4.1.2}$), it can be deduced that increasing the area ($S_w$) and the maximum coefficient of lift ($C_{L_{\max}}$) allows to fly at a lower airspeed since the minimum speed ($V_S$) decreases.
Deploying high-lift devices also increases the drag coefficient of the aircraft. Therefore, for any given weight and airspeed, deflected flaps increase the drag force. Flaps increase the drag coefficient of an aircraft because of higher induced drag caused by the distorted span-wise lift distribution on the wing with flaps extended. Some devices increase the planform area of the wing and, for any given speed, this also increases the parasitic drag component of total drag.
By decreasing operating speed and increasing drag, high-lift device shorten takeoff and landing distances as well as improve climb rate. Therefore, these devices are fundamental during take-off (reduce the velocity at which the aircraft’s lifts equals aircraft’s weight), during the initial phase of climb (increases the rate of climb so that obstacles can be avoided) and landing (decrease the impact velocity and help braking the aircraft).
3.4.02: Types of high-lift devices
The passive high-lift devices, commonly referred to as flaps, are based on the following three principles:
• Increase of camber.
• Increase of wet surface (typically by increasing the chord).
• Control of the boundary layer.
There are many different types of flaps depending on the size, speed, and complexity of the aircraft they are to be used on, as well as the era in which the aircraft was designed. Plain flaps, slotted flaps, and Fowler flaps are the most common trailing edge flaps. Flaps used on the leading edge of the wings of many jet airliners are Krueger flaps, slats, and slots (Notice that slots are not explicitly flaps, but more precisely boundary layer control devices).
The plain flap is the simplest flap and it is used in light . The basic idea is to design the airfoil so that the trailing edge can rotate around an axis. The angle of that deflexion is the flap deflexion $\delta_f$. The effect is an increase in the camber of the airfoil, resulting in an increase in the coefficient of lift.
Another kind of trailing edge high-lift device is the slotted flap. The only difference with the plain flap is that it includes a slot which allows the extrados and intrados to be communicated. By this mean, the flap deflexion is higher without the boundary layer dropping off.
The last basic trailing edge high-lift device is the flap Fowler. This kind of flap combines the increase of camber with the increase in the chord of the airfoil (and therefore the wet surface). This fact increases also the slope of the lift curve. Combining the different types, there exist double and triple slotted Fowler flaps, combining also the control of the boundary layer. The Fairey-Youngman, Gouge, and Junkers flaps combine some of the exposed properties.
The last trailing edge high-lift device is the split flap (also refereed to as intrados flap). This flap provides, for the same increase of lift coefficient, more drag but with less torque.
The most important leading edge high devices are: slot, the leading edge drop flap, and the flap Krueger.
The slot is a slot in the leading edge. It avoid the dropping off of the boundary layer by communicating extrados and intrados. The leading edge drop has the same philosophy as the plain flap, but applied in the leading edge instead of the trailing edge. The Kruger flaps works modifying the camber of the airfoil but also acting in the control of the boundary layer.
See Figure 3.26 and Figure 3.27.
Figure 3.26: Types of high-lift devices. © NiD.29 / Wikimedia Commons / CC-BY-SA-3.0.
Figure 3.27: Effects of high lift devices in airfoil flow, showing configurations for normal, take-off, landing, and braking. © Andrew Fry / Wikimedia Commons / CC-BY-SA-3.0.
3.4.03: Increase in CLmax
Table 3.1 shows the typical values for the increase of coefficient of lift in airfoils.
High-lift devices $\Delta c_{L_{\max}}$
Trailing edge devices
Plain flap and intrados flap
Slotted flap
Fowler flap
Doble slotted Fowler flap
Tripple slotted Fowler flap
0.9
1.3
$1.3 c'/c^*$
$1.6 c'/c$
$1.9 c'/c$
Leading edge devices
Slot
Krueger and drop flap
Slat
0.2
0.3
$0.4 c'/c$
* $c'$ is the extended chord and $c$ to the nominal chord.
Table 3.1: Increase in $C_{l_{\max}}$ of airfoils with high lift devices. Data retrieved from $F_{\text{RANCHINI}}$ et al. [4].
The increase in the maximum coefficient of lift of the wing ($\Delta C_{L_{\max}}$) can be related with the increase of the maximum coefficient of lift of an airfoil ($\Delta c_{L_{\max}}$). For slotted and Fowler flaps, the expression is:
$\Delta C_{L_{\max}} = 0.92 \Delta c_{l_{\max}} \dfrac{S_{fw}}{S_w} \cos \wedge_{1/4},$
where $\wedge_{1/4}$ refers to the swept measured from the locus of the $c/4$ of all airfoils and $S_{fw}$ refers to the surface of the wing between the two extremes of the flap. If the flap is a plain flap, the expression is:
$\Delta C_{L_{\max}} = 0.92 \Delta c_{L_{\max}} \dfrac{S_{fw}}{S_w} \cos^3 \wedge_{1/4}.$
In the Table 3.2 the typical values of $C_{L_{\max}}$ and flap deflections in different configurations are given.
High-lift device $\delta_f\ TO^*$ $\delta_f\ LD$ $\tfrac{C_{L_{\max}}}{\cos \wedge_{1/4}}\ TO$ $\tfrac{C_{L_{\max}}}{\cos \wedge_{1/4}}\ LD$
Plain flap
Slotted flap
Fowler flap
Doble slotted** flap
Tripple slotted flap and slat
$20^{\circ}$
$20^{\circ}$
$15^{\circ}$
$20^{\circ}$
$20^{\circ}$
$60^{\circ}$
$40^{\circ}$
$40^{\circ}$
$50^{\circ}$
$40^{\circ}$
1.4-1.6
1.5-1.7
2-2.2
1.7-1.95
2.4-2.7
1.7-2
1.8-2.2
2.5-2.9
2.3-2.7
3.2-3.5
* $TO$ and $LD$ refers to take off and landing, respectively.
** Double and triple slotted flaps have always Fowler effects increasing the chord.
Table 3.2: Typical values for $C_{L_{\max}}$ in wings with high-lift devices. Data retrieved from $F_{\text{RANCHINI}}$ et al. [4].
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Exercise $1$ Airfoils
1. In a wind tunnel experiment it has been measured the distribution of pressures over a symmetric airfoil for an angle of attack of $14^{\circ}$. The distribution of coefficient of pressures at the intrados, $C_{pl}$, and extrados, $C_{pE}$, of the airfoil can be respectively approximated by the following functions:
$C_{pl} (x) = \begin{cases} 1 - 2 \tfrac{x}{c}, \ \ \ \ \ \ & 0 \le x \le \tfrac{c}{4}, \ \tfrac{2}{3} (1 - \tfrac{x}{c} ) \ \ \ \ \ \ & \tfrac{c}{4} \le x \le c; \end{cases}\nonumber$
$C_{pE} (x) = \begin{cases} - 12 \tfrac{x}{c}, \ \ \ \ \ \ & 0 \le x \le \tfrac{c}{4}, \ 4 (-1 + \tfrac{x}{c} ) \ \ \ \ \ \ & \tfrac{c}{4} \le x \le c. \end{cases}\nonumber$
(a) Draw the curve that represents the distribution of pressures.
(b) Considering a chord $c = 1$ m, obtain the coefficient of lift of the airfoil.
(c) Calculate the slope of the characteristic curve $c_l (\alpha)$.
2. Based on such airfoil as cross section, we build a rectangular wing with a wingspan of $b = 20$ m and constant chord $c = 1$ m. The distribution of the coefficient of lift along the wingspan of the wing ($y$ axis) for an anglee of attack $\alpha = 14^{\circ}$ is approximated by the following parabolic function:
$c_l (y) = 1.25 - 5 (\dfrac{y}{b})^2, - \dfrac{b}{2} \le y \le \dfrac{b}{2}.\nonumber$
(a) Draw the curve $c_l (y)$.
(b) Calculate the coefficient of lift of the wing.
Answer
1. (a) The curve is as follows:
Figure 3.28: Distribution of the coefficient of pressures.
(b) The coefficient of lift for the airfoil for $\alpha = 14^{\circ}$ can be calculated as follows:
$c_l = \dfrac{1}{c} \int_{x_{le}}^{x_{te}} (c_{pl} (x) - c_{pE} (x)) dx,\label{eq3.5.1}$
In this case, with $c = 1$ and the given distributions of pressures of Intrados and extrados, Equation ($\ref{eq3.5.1}$) becomes:
$c_l = \dfrac{1}{c} \left [ \int_{0}^{1/4} ((1 - 2x) - (-12x)) dx + \int_{1/4}^{1} (2/3 (1 - x) - 4(-1 + x)) dx \right ] = 1.875. \nonumber$
(c) The characteristic curve is given by:
$c_l = c_{l_0} + c_{l_{\alpha}} \alpha.\nonumber$
Since the airfoil is symmetric: $c_{l_0} = 0$. Therefore $c_{l_{\alpha}} = \tfrac{c_l}{\alpha} = \tfrac{1.875 \cdot 360}{14 \cdot 2\pi} = 7.16 \ 1/rad$.
2. (a) The curve is as follows:
Figure 3.29: Coefficient of lift along the wingspan.
(b) The coefficient of lift for the wing for $\alpha = 14^{\circ}$ can be calculated as follows:
$C_L = \dfrac{1}{S_w} \int_{-b/2}^{b/2} c(y) c_l (y) dy.\label{eq3.5.2}$
Substituting in Equation ($\ref{eq3.5.2}$) considering $c(y) = 1$ and $b = 20$:
$C_L = \dfrac{1}{20} \int_{-10}^{10} \left (1.25 - 5 (\dfrac{y}{20})^2 \right ) dy = 0.83.\nonumber$
Exercise $2$ Airfoils
Figure 3.30: Characteristic curves of a NACA 4410 airfoil.
We want to know the aerodynamic characteristics of a NACA-4410 airfoil for a Reynolds number $\text{Re} = 100000$. Experimental results gave the characteristic curves shown in Figure 3.30.
Calculate:
1. The expression of the lift curve in the linear range in the form: $c_l = c_{l0} + c_{l\alpha} \alpha$.
2. The expression of the parabolic polar of the airfoil in the form: $c_d = c_{d0} + b_{c_l} + kc_l^2.$
3. the angle of attack and the coefficient of lift corresponding to the minimum coefficient of drag.
4. The angle of attack, the coefficient of lift and the coefficient of drag corresponding to the maximum aerodynamic efficiency.
5. The values of the aerodynamic forces per unity of longitude that the model with chord $c= 2$ m would produce in the wind tunnel experiments with an angle of attack of $\alpha = 3^{\circ}$ and incident current with Mach number $M = 0.3$. Consider ISA conditions at an altitude of $h = 1000$ m.
Answer
We want to approximate the experimental data given in Figure 3.30, respectively to a straight line and a parabolic curve. Therefore, to univocally define such curves, we must choose:
• Two pair of points ($c_l, \alpha$) of the $c_l (\alpha)$ curve in Figure 3.30.a.
• Three pair of points ($c_l, c_d$) of the $c_l (c_d)$ curve in Figure 3.30.b.
According to Figure 3.30 For $\text{Re} = 100000$ we choose (any other combination properly chosen must work):
$c_l$ $\alpha$
0.5 $0^{\circ}$
1 $4^{\circ}$
Table 3.3: Date obtained from Figure 3.30.a.
$c_l$ $c_d$
0 0.03
1 0.0175
1.4 0.0325
Table 3.4: Data obtained from Figure 3.30.b.
1. The expression ofthe lift curve in the linear range in the form: $c_l = c_{l0} + c_{l\alpha} \alpha$:
With the data in Table 3.3:
$c_{l0} = 0.5;$
$c_{l\alpha} = 7.16 \cdot 1/rad.$
The required curve yields then:
$c_l = 0.5 + 7.16 \alpha [\alpha \ in \ rad]\label{eq3.5.5}$
2. The expression of the parabolic polar of the airfoil in the form: $c_d = c_{d0} + bc_l + kc_l^2$:
With the data in Table 3.4 we have a system of three equations with three unkowns that is to be solved. It yields:
$c_{d0} = 0.03;$
$b = -0.048;$
$k = 0.0357.$
The expression of the parabolic polar yields:
$c_d = 0.03 - 0.048 c_l + 0.0357 c_l^2.\label{eq3.5.9}$
3. The angle of attack and the coefficient of lift corresponding to the minimum coefficient of drag:
In order to do so, we seek the minimum of the parabolic curve:
$\dfrac{dc_l}{dc_d} = 0 = b + 2 \cdot kc_l.$
Substituting in Equation ($\ref{eq3.5.9}$):
$\dfrac{dc_l}{dc_d} = 0 = -0.048 + 2 \cdot 0.0357 c_l \to (c_l)_{c_{d_{\min}}} = 0.672.$
Substituting $(c_l)_{c_{d_{\min}}}$ in Equation ($\ref{eq3.5.5}$), we obtain:
$(\alpha)_{c_{d_{\min}}} = 0.024\ rad (1.378^{\circ}).\nonumber$
4. The angle of attack, the coefficient of lift and the coefficient of drag corresponding to the maximum aerodynamic efficiency:
The aerodynamic efficiency is defined as:
$E = \dfrac{l}{d} = \dfrac{c_l}{c_d}.\label{eq3.5.12}$
Substituting the parabolic polar curve in Equation ($\ref{eq3.5.12}$), we obtain:
$E = \dfrac{c_l}{c_{d0} + bc_l + kc_l^2}.$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dc_l} = 0 = \dfrac{c_{d0} - kc_l^2}{(c_{d0} + bc_l + kc_l^2)^2} \to (c_l)_{E_{\max}} = \sqrt{\dfrac{c_{d0}}{k}}.$
Substituting according to the values previously obtained ($c_{d0} = 0.03, k = 0.0357$): $(c_l)_{E_{\max}} = 0.91$. Substituting in Equation ($\ref{eq3.5.5}$) and Equation ($\ref{eq3.5.9}$), we obtain:
$\bullet (\alpha)_{E_{\max}} = 0.058\ rad\ (3.33^{\circ});$
$\bullet (c_d)_{E_{\max}} = 0.01588.$
5. The values of the aerodynamic forces per unity of longitude that the model with chord $c = 2$ m would produce in the wind tunnel experiments with an angle of attack of $\alpha = 3^{\circ}$ and incident current with Mach number $M = 0.3$:
According to ISA:
$\bullet \rho (h = 1000) = 0.907\ kg/m^3$;
$\bullet a(h = 1000) = \sqrt{\gamma_{air} R (T_0 - \lambda h)} = 336.4\ m/s$;
where a corresponds to the speed of sound, $\gamma_{air} = 1.4, R = 287\ J/KgK, T_0 = 288.15\ k$ and $\lambda = 6.5 \cdot 10^{-3}$.
Since the experiment is intended to be at $M = 0.3$:
$V = M \cdot a = 100.92\ m/s.$
Since the experiment is intended to be at $\alpha = 3^{\circ}$, using Equation ($\ref{eq3.5.5}$) and Equation ($\ref{eq3.5.9}$):
$c_l = 0.87;$
$c_d = 0.01526.$
Finally:
$l = c_l \dfrac{1}{2} \rho c V^2 = 8036.77\ N/m;$
$d = c_d \dfrac{1}{2} \rho c V^2 = 140.96\ N/m.$
Exercise $3$ Wings
Figure 3.31: Plant-form of the wing (Dimensions in meters)
We want to analyze the aerodynamic performances of a trapezoidal wing with a plant-form as in Figure 3.31 and an efficiency factor of the wing of $e = 0.96$. Moreover, we will employ a NACA 4415 airfoil with the following characteristics:
• $c_l = 0.2 + 5.92 \alpha$.
• $c_d = 6.4 \cdot 10^{-3} - 1.2 \cdot 10^{-3} c_l + 3.5 \cdot 10^{-3} c_l^2$.
Calculate:
1. The following parameters of the wing4: chord at the root; chord at the tip; mean chord; wing-spanl wet surface; enlargement.
2. The lift curve of the wing in the linear range.
3. The polar of the wing assuming that it can be calculated as $C_D = C_{D_0} + C_{D_i} C_L^2$.
4. Calculate the optimal coefficient of lift, $C_{L_{opt}}$, for the wing. Compare it with the airfoil's one.
5. Calculate the optimal coefficient of drag, $C_{D_{opt}}$, for the wing. Compare it with the airfoil's one.
6. Maximum aerodynamic efficiency, $E_{\max}$, for the wing. Compare it with the airfoils's one.
7. Discuss the differences observed in $C_{L_{opt}}$, $C_{D_{opt}}$, and $E_{\max}$ between the wing and the airfoil.
Answer
1. Chord at the root; chord at the tip; mean chord; wing-span; wet surface; enlargement:
According the Figure 3.31:
The wing-span, $b$, is $b = 16\ m$. The chord at the ip, $c_t$, is $c_t = 0.75\ m$. The chord at the root, $c_r$, is $c_r = 2\ m$.
$C_l = C_{l_0} + C_{l_{\alpha}} \alpha.$
Therefore, with Equation (3.67) and Equation (3.68) in Equation (3.69), we have that:
We can also calculate the wet surface of the wing calculating twice the area of a tropezoid as follows:
$S_w = 2 \left (\dfrac{(c_r + c_t)}{2} \dfrac{b}{2} \right ) = 22\ m^2.\nonumber$
The mean chord, $\bar{c}$, can be calculated as $\bar{c} = \tfrac{S_w}{b} = 1.375\ m$; and the enlargement, $A$, as $A = \tfrac{b}{\bar{c}} = 11.63$.
2. Wing's lift curve:
The lift curve of a wing can be expressed as follows:
$C_L = C_{L_0} + C_{L_{\alpha}} \alpha,\label{eq3.5.21}$
and the slope of the wing's lift curve can be expressed related to the slope of the airfoil's lift curve as:
$C_{L_{\alpha}} = \dfrac{C_{l_{\alpha}}}{1 + \tfrac{C_{t_{\alpha}}}{\pi A}} e = 4.89 \cdot 1/rad.\nonumber$
In order to calculate the independent term of the wing’s lift curve, we must consider the fact that the zero-lift angle of attack of the wing coincides with the zero-lift angle of attack of the airfoil, that is:
$\alpha (L = 0) = \alpha (l = 0).\label{eq3.5.22}$
First, notice that the lift curve of an airfoil can be expressed as follows
$C_l = C_{l_0} + C_{l_{\alpha}} \alpha.\label{eq3.5.23}$
Therefore, with Equation ($\ref{eq3.5.21}$) and Equation ($\ref{eq3.5.22}$) in Equation ($\ref{eq3.5.23}$), we have that:
$C_{L_0} = C_{l_0} \dfrac{C_{L_{\alpha}}}{C_{l_{\alpha}}} = 0.165.\nonumber$
The required curve yields then:
$C_L = 0.165 + 4.89 \alpha [\alpha \ in \ rad].\nonumber$
3. The expression of the parabolic polar of the wing:
Notice first that the statement of the problem indicates that the polar should be in the following form:
$C_D = C_{D0} + C_{D_i} C_L^2.\label{eq3.5.24}$
For the calculation of the parabolic drag of the wing we can consider the parasite term approximately equal to the parasite term of the airfoil, that is, $C_{D_0} = C_{d_0}$.
$C_{D_i} = \dfrac{1}{\pi Ae} = 0.028.\nonumber\] The expression of the parabolic polar yields then: $C_D = 0.0064 + 0.028 C_L^2.\label{eq3.5.25}$ 4. The optimal coefficient of lift, \(C_{L_{opt}}$, for the wing. Compare it with the airfoils's one.
The optimal coefficient of lift is that making the aerodynamic efficiency maximum. The aerodynamic efficiency is defined as:
$E = \dfrac{L}{D} = \dfrac{C_L}{C_D}.\label{eq3.5.26}$
Substituting the parabolic polar given in Equation ($\ref{eq3.5.24}$) in Equation ($\ref{eq3.5.26}$), we obtain:
$E = \dfrac{C_L}{C_{D_0} + C_{D_i} C_L^2}.$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_L} = 0 = \dfrac{C_{D_0} - C_{D_i} C_L^2}{(C_{D_0} + C_{D_i} C_L^2)^2} \to (C_L)_{E_{\max}} = C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}}.\label{eq3.5.28}$
For the case of an airfoil, the aerodynamic efficiency is defined as:
$E = \dfrac{1}{d} = \dfrac{c_l}{c_d}.\label{eq3.5.29}$
Substituting the parabolic polar curve given in the statement in the form $c_{d_0} + bc_l + kc_l^2$ in Equation ($\ref{eq3.5.29}$), we obtain:
$E = \dfrac{c_l}{c_{d_0} + bc_l + kc_l^2}.$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_l} = 0 = \dfrac{c_{d_0} - kc_l^2}{(c_{d0} + bc_l + kc_l^2)^2} \to (C_l)_{E_{\max}} = (c_l)_{opt} = \sqrt{\dfrac{c_{d_0}}{k}}.\label{eq3.5.31}$
According to the values previously obtained ($C_{D_0} = 0.0064$ and $C_{D_i} = 0.028$) and the values given in the statement for the airfoil's polar ($c_{d_0} = 0.0064$, $k = 0.0035$), substituting them in Equation ($\ref{eq3.5.28}$) and Equation ($\ref{eq3.5.31}$), respectively, we obtain:
• $(C_L)_{opt} = 0.478;$
• $(c_l)_{opt} = 1.35.$
5. The optimal coefficient of drag, $C_{D_{opt}}$ for the wing. Compare it with the airfoils's one:
Once the optimal coefficient of lift has been obtained for both airfoil and wing, simply by substituting their values into both parabolic curves given respectively in the statement and in Equation ($\ref{eq3.5.25}$), we obtain:
$C_{D_{opt}} = 0.0064 + 0.028 C_{l_{opt}}^2 = 0.01279.\nonumber$
$c_{d_{opt}} = 6.4 \cdot 10^{-3} - 1.2 \cdot 10^{-3} c_{l_{opt}} + 3.5 \cdot 10^{-3} c_{l_{opt}}^2 = 0.01115.\nonumber$
6. Maximum aerodynamic efficiency $E_{\max}$ for the wing. Compare it the the airfoils's one:
The maximum aerodynamic efficiency can be obtained as:
$E_{\max_{wing}} = \dfrac{C_{L_{opt}}}{C_{D_{opt}}} = 37.\nonumber$
$E_{\max_{airfoil}} = \dfrac{c_{l_{opt}}}{c_{d_{opt}}} = 121.\nonumber$
7. Discuss the differences observed in $C_{L_{opt}}$, $C_{D_{opt}}$ and $E_{\max}$ between the wing and the airfoil.
According to the results it is straightforward to see that meanwhile the optimum coefficient of drag is similar for both airfoil and wing, the optimum coefficient of lift is approximately three times lower that the airfoil’s one. Obviously this results in an approximately three time lower efficiency for the wing when compared to the airfoil’s one.
What does it mean? A three dimensional aircraft made of $2D$ airfoils generates much more drag than the $2D$ airfoil in order to achieve a required lift. Therefore, we can not simply extrapolate the analysis of an airfoil to the wing.
Such loss of efficiency is due to the so-called induced drag by lift. The explanation behind this behavior is due to the difference of pressures between extrados and intrados. In particular, in the region close to the marginal border, there is an air current surrounding the marginal border which passes from the intrados, where the pressure is higher, to extrados, where the pressure is lower, giving rise to two vortexes, one in each border rotating clockwise and counterclockwise. This phenomena produces downstream a whirlwind trail.
The presence of this trail modifies the fluid filed and, in particular, modifies the velocity each wing airfoil "sees". In addition to the freestream velocity $u_{\infty}$, a vertical induced velocity ui must be added (See Figure 3.23). The closer to the marginal border, the higher the induced velocity is. Therefore, the effective angle of attack of the airfoil is lower that the geometric angle, which explains both the reduction in the coefficient of lift (with respect to the bi-dimensional coefficient) and the appearance of and induced drag (See Figure 3.24).
Exercise $4$ Airfoils and Wings
1. In a wind tunnel experiment we have measured teh distribution of pressures over a symmetric airfoil with angle of attack $6^{\circ}$. The distributions of the coefficient of pressures for intrados $(C_{pl})$ and extrados $(C_{pE})$ can be approximated by the following functions:
$C_{pl} (x) = \begin{cases} 10 \tfrac{x}{c}, \ \ \ \ \ \ \ & 0 \le \tfrac{x}{c} \le \tfrac{1}{10}, \ 2 - 10 \tfrac{x}{c}, \ \ \ \ \ & \tfrac{1}{10} \le \tfrac{x}{c} \le \tfrac{1}{5}; \ 0, \ \ \ \ \ \ & \tfrac{1}{5} \le \tfrac{x}{c} \le 1. \end{cases}\nonumber$
$C_{pE} (x) = \begin{cases} -15 \tfrac{x}{c}, \ \ \ \ \ \ \ & 0 \le \tfrac{x}{c} \le \tfrac{1}{5}, \ \tfrac{-15}{4} (1 - \tfrac{x}{c}, \ \ \ \ \ & \tfrac{1}{5} \le \tfrac{x}{c} \le 1. \end{cases}\nonumber$
(a) Draw the curve $-C_p (\tfrac{x}{c})$.
(b) Considering $c = 1 [m]$, calculate the coefficient of lift of the airfoil.
2. Based on the previous airfoil as transversal section, we want to design a rectangular wing with wing-span $b$ and constant chord $c = 1\ m$. The distribution of the coefficient of lift along the wing-span for angle of attack $6^{\circ}$ can be approximated by the following function:
$c_l (y) = c_{l_{airfoil}} \cdot (1 - \dfrac{4}{A} \cdot (\dfrac{y}{b})^2), -\dfrac{b}{2} \le y \le \dfrac{b}{2},\nonumber$
being $c_{l_{airfoil}}$ the coefficient of lift of the airfoil previously calculated and A the enlargement of the wing.
(a) Calculate the coefficient of lift of the wing as a function of the enlargement $A$.
(b) Calculate the coefficient of lift of the wing for $A = 1$, $A = 8yA =\infty$.
(c) Draw the distribtution of the coefficient of lift along the wing-span for $A = 1, A = 8yA = \infty$. Discuss the results.
Answer
1. Airfoil:
(a) The curve is as follows:
Figure 3.32: Distribution of the coefficient of pressures.
(b) The coefficient of lift of the airfoil for $\alpha = 6^{\circ}$ can be calculate as follows:
$c_l = \dfrac{1}{c} \int_{x_{le}}^{x_{te}} (c_{pl} (x) - c_{pE} (x)) dx.\label{eq3.5.32}$
In this case, with $c = 1$ and the given distributions of pressures of intrados and extrados, Equation ($\ref{eq3.5.32}$) becomes:
$c_l = \dfrac{1}{c} \left [\int_{0}^{1/10} (10x) dx + \int_{1/10}^{1/5} (2 - 10x) dx + \int_{1/5}^{1} (0) dx - \int_{0}^{1/5} (-15x) dx - \int_{1/5}^{1} (-15/4 (1 - x)) dx \right ] = 1.6.\nonumber$
2. Wing:
(a) The coefficient of lift for the wing for $\alpha = 6^{\circ}$ can be calculated as follows:
$C_L = \dfrac{1}{S_w} \int_{-b/2}^{b/2} c(y) c_l (y) dy.\label{eq3.5.33}$
Substituting in Equation ($\ref{eq3.5.33}$) considering $c(y) = 1$:
$C_L = \dfrac{1}{b} \int_{-b/2}^{b/2} 1.6 \left (1 - \dfrac{4}{A} (\dfrac{y}{b})^2 \right ) dy = 1.6(1 - \dfrac{1}{3A}).\nonumber$
(b) The values of $C_L$ for the different enlargements are:
• $A = 1 \to C_L = 1.06.$
• $A = 8 \to C_L = 1.53.$
• $A = \infty \to C_L = 1.6.$
Considering, for instance, a wing-span $b = 20\ m$, the curve is as follows:
Figure 3.33: Coefficient of lift along the wingspan
(c) The discussion has to do with the differences in lift generation between finite and infinity wing.
It can be observed that if the enlargement is infinity the wing behaves as the bi-dimensional airfoil $y (c_l(y)$ constant). On the other hand, if the enlargement is finite, $c_l (y)$ shows a maximum in the root of the wing $(y = 0)$ and goes to zero in the tip of the wing $(y/c = A/2)$. As the enlargement decreases, the maximum $c_l(y)$ also decreases.
The explanation behind this behavior is due to the difference of pressures between extrados and intrados. In particular, in the region close to the marginal border, there is an air current surrounding the marginal border which passes from the intrados, where the pressure is higher, to extrados, where the pressure is lower, giving rise to two vortexes, one in each border rotating clockwise and counterclockwise. This phenomena produces downstream a whirlwind trail.
The presence of this trail modifies the fluid field and, in particular, modifies the velocity each wing airfoil "sees". In addition to the freestream velocity, $u_{\infty}$, a vertical induced velocity, $u_i$, must be added (See Figure 3.23). The closer to the marginal border, the higher the induced velocity is. Therefore, the effective angle of attack of the airfoil is lower that the geometric angle, which explains both the reduction in the coefficient of lift (with respect to the bi-dimensional coefficient) and the fact that this reduction is higher when one gets closer to the marginal border.
Exercise $5$ High-Lift devices
Figure 3.34: Plant-form of the wing (dimensions in meters).
We want to analyze the aerodynamic performances of a trapezoided wing with a plant-form as in Figure 3.34. The wing mounts two triple slotted Fowler flaps. The efficiency factor (Oswald factor) of the wing is $e = 0.96$. The wing is build employing $NACA$ 4415 airfoils with the following characteristics:
• $c_l = 0.2 + 5.92 \alpha$. ($\alpha$ en radianes)
• $c_d = 6.4 \cdot 10^{-3} - 1.2 \cdot 10^{-3} c_l + 3.5 \cdot 10^{-3} c_l^2$.
On regard of the effects of the Fowler flaps in the maximum coefficient of lift, it is known that:
• The increase of the maximum coefficient of lift of the airfoil ($\Delta c_{l_{\max}}$) can be approximated by the following expression:
$\Delta c_{l_{\max}} = 1.9 \dfrac{c'}{c},\label{eq3.5.34}$
being $c$ the chord in the root and $c'$ the extended chord (consider $c' = 3\ [m]$).
• The increase of the maximum coefficient of lift of the wing ($\Delta C_{L_{\max}}$) can be related to the increase of the maximum coefficient of lift of the airfoil ($\Delta c_{l_{\max}}$) by means of the following expression:
$\Delta C_{L_{\max}} = 0.92 \Delta c_{l_{\max}} \dfrac{S_{fw}}{S_w} \cos \wedge.\label{eq3.5.35}$
Based on the data given in Figure 3.34, calculate:
(a) Chord in the root and tip of the wing. wing-span and enlargement. Wet wing surface ($S_w$) and surface wet by the flaps ($S_{fw}$). Aircraft swept ($\wedge$) measured from the leading edge.
Assuming a clean configuration (no flap deflection), typical of cruise conditions, and knowing also that the stall of the airfoil takes place at an angle of attack of $15^{\circ}$:
(b) calculate the maximum coefficient of lift of the airfoil.
(c) calculate the expression of the lift curve of the wing in its linear range.
(d) calculate the maximum coefficient of lift of the wing (assume that the aircraft (wing) stalls also at an angle of attack of $15^{\circ}$)
Assuming a configuration with flaps fully deflected, typical of a final approach, calculate:
(e) the maximum coefficient of lift of the wing.
It is known that the mass of the aircraft is $4500\ kg$. For sea level $ISA$ conditions and force due to gravity equal to $9.81\ m/s^2$:
(f) calculate the stall speeds of the aircraft for both configurations (clean and full).
(g) compare and discuss the results.
Answer
(a). Chord at the root; chord at the tip; wing-span and enlargement; wing wet surface and flap wet surface; swept:
According to Figure 3.34:
• The wing-span, $b$, is $b = 18\ m$.
• The chord at the tip, $c_t$, is $c_t = 0.75\ m$.
• The chord at the root, $c_r$, is $c_r = 2\ m$.
We can also calculate the wet surface of the wing calculating twice the area of a trapezoid as follows:
$S_w = 2 \left ( \dfrac{(c_r + c_t)}{2} \dfrac{b}{2} \right ) = 24.75\ m^2.\nonumber$
In the same way, the flap wet surface $(S_{fw})$ can be calculated as follows:
$S_{fw} = 2 \left ( 1 \cdot 1.5 + \dfrac{1}{2} 1 \cdot 0.25 \right ) = 3.25 \ m^2.\nonumber$
The mean chord, $\bar{c}$, can be calculated as $\bar{c} = \tfrac{S_w}{b} = 1.375\ m$; and the enlargement, $A$, as $A = \tfrac{b}{\bar{c}} = 13.09$.
Finally, the swept of the wing ($\wedge$) measured from the leading edge is:
$\wedge = \arctan (\dfrac{0.25}{1}) = 14^{\circ}.\nonumber$
(b). $c_{l_{\max}}$
According to the expression given in the statement for the airfoil's lift curve: $c_l = 0.2 + 5.92 \alpha$, and given that the airfoils stalls at $\alpha = 15^{\circ}$, the maximum coefficient of lift will be given by the value of the coefficient of lift at the stall angle:
$c_{l_{\max}} = 0.2 + 5.92 \cdot 15 \dfrac{2\pi}{360} = 1.74.$
(c). Wing's lift curve:
The lift curve of a wing can be expressed as follows:
$C_L = C_{L_0} + C_{L_{\alpha}} \alpha,\label{eq3.5.37}$
and the slope of the wing’s lift curve can be expressed related to the slope of the airfoil’s lift curve as:
$C_{L_{\alpha}} = \dfrac{C_{l_{\alpha}}}{1 + \tfrac{C_{l_{\alpha}}}{\pi A}} e = 4.96 \cdot 1/rad.\nonumber$
In order to calculate the independent term of the wing’s lift curve, we must consider the fact that the zero-lift angle of attack of the wing coincides with the zero-lift angle of attack of the airfoil, that is:
$\alpha (L = 0) = \alpha (l = 0).\label{eq3.5.38}$
First, notice that the lift curve of an airfoil can be expressed as follows
$C_l = C_{l_0} + C_{l_{\alpha}} \alpha.\label{eq3.5.39}$
Therefore, with Equation ($\ref{eq3.5.37}$) and Equation ($\ref{eq3.5.38}$) in Equation ($\ref{eq3.5.39}$), we have that:
$C_{L_0} = C_{l_0} \dfrac{C_{L_{\alpha}}}{C_{l_{\alpha}}} = 0.1678.\label{eq3.5.40}$
The required curve yields then:
$C_L = 0.1678 + 4.96 \alpha \ [\alpha \ in \ rad].\nonumber$
(d). $C_{L_{\max}}$ in clean configuration:
Given the expression in Equation ($\ref{eq3.5.40}$), and given that the aircraft (wing) stalls at $\alpha = 15^{\circ}$, the maximum coefficient of lift will be given by:
$C_{L_{\max}} = 0.1678 + 4.96 \cdot 15 \dfrac{2\pi}{360} = 1.466.$
(e). $C_{L_{\max}}$ in full configuration (with all flaps deflected):
In order to obtain the maximum coefficient of lift for full configuration $(C_{L_{\max}})$ we have that:
$C_{L_{\max_f}} = C_{L_{\max}} + \Delta C_{L_{\max}}.$
As it was given in the Equation ($\ref{eq3.5.35}$), $\Delta C_{L_{\max}}$ can be expressed as:
$\Delta C_{L_{\max}} = 0.92 \Delta c_{l_{\max}} \dfrac{S_{fw}}{S_w} \cos \wedge, \nonumber$
where $\wedge$, $S_{fw}$, and $S_w$ are already known and $c_{l_{\max}}$ was given in Equation ($\ref{eq3.5.34}$). Therefore, $\Delta C_{L_{\max}} = 0.334$.
$C_{L_{\max_f}}$ yields then 1.8.
f. $V_{stall}$:
Knowing that $L = C_L \cdot \tfrac{1}{2} \rho S_w V^2$, that the flight can be considered to be equilibrated, i.e., $L = m \cdot g$, and that the stall speed takes place when the coefficient of lift is maximum, we have that:
$V_{stall} = \sqrt{\dfrac{m \cdot g}{\tfrac{1}{2} \rho S_w C_{L_{\max}}}} = 44.56 \ m/s.\nonumber$
For the case of full configuration, we use $C_{L_{\max_f}}$ and consider the wing-and-flap wet surface (notice that we are not including the chord extension in the wing wet surface). The stall velocity yields:
$V_{stall_f} = \sqrt{\dfrac{m \cdot g}{\tfrac{1}{2} \rho (S_w + S_{fw}) C_{L_{\max_f}}}} = 37.8 \ m/s. \nonumber$
g. Discussion:
High-lift devices are designed to increase the maximum coefficient of lift. By their deployment, they increase the aerodynamic chord and the camber of the airfoil, modifying thus the geometry of the airfoil so that the stall speed during specific phases of flight such as landing or take-off is reduced significantly, allowing to flight slower than in cruise.
Deploying high-lift devices also increases the drag coefficient of the aircraft. Therefore, for any given weight and airspeed, deflected flaps increase the drag force. Flaps increase the drag coefficient of an aircraft because of higher induced drag caused by the distorted span-wise lift distribution on the wing with flaps extended. Some devices increase the planform area of the wing and, for any given speed, this also increases the parasitic drag component of total drag.
By decreasing operating speed and increasing drag, high-lift device shorten takeoff and landing distances as well as improve climb rate. Therefore, these devices are fundamental during take-off (reduce the velocity at which the aircraft’s lifts equals aircraft’s weight), during the initial phase of climb (increases the rate of climb so that obstacles can be avoided) and landing (decrease the impact velocity and help braking the aircraft).
4. Based on the given data in Figure 3.31.
3.06: References
[1] Anderson, J. (2012). Introduction to flight, seventh edition. McGraw-Hill.
[2] Anderson, J. D. (2001). Fundamentals of aerodynamics, volume 2. McGraw-Hill New York.
[3] Franchini, S. and García, O. (2008). Introducción a la ingeniería aeroespacial. Escuela Universitaria de Ingeniería Técnica Aeronáutica, Universidad Politécnica de Madrid.
[4] Franchini, S., López, O., Antoín, J., Bezdenejnykh, N., and Cuerva, A. (2011). Apuntes de Tecnología Aeroespacial. Escuela de Ingeniería Aeronáutica y del Espacio. Universidad Politécnica de Madrid.
[5] Prandtl, L. (1904). Über Flüssigkeitsbewegung bei sehr kleiner Reibung. In Proceedings of 3rd International Mathematics Congress, Heidelberg.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/03%3A_Aerodynamics/3.05%3A_Problems.txt
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A Structure holds things together, carries loads, and provides integrity. Structural engineering is the application of statics and solid mechanics to devise structures with sufficient strength, stiffness, and useful life to fulfill a mission without failure with a minimum amount of weight. Aerospace engineers pay particular attention to designing light structures due to the strong dependence of weight on operational costs. The aim of this chapter is to give an overview of aircraft structures. The chapter starts with some generalities in Section 4.1. Then material properties are analyzed in Section 4.2 focusing on aircraft materials. The Loads that appear in an aircraft structure will be described in Section 4.3. Finally, Section 4.4 will be devoted to describe the fundamental structural components of an aircraft. Thorough references on the matter are, for instance, Megson [3] and Cantor et al. [1].
04: Aircraft structures
Figure 4.1: Normal stress. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
It is the task of the designer to consider all possible loads. The combination of materials and design of the structure must be such that it can support loads without failure. In order to estimate such loads one can take measurements during the flight, take measurements of a scale-model in a wind tunnel, do aerodynamic calculations, and/or perform test-flights with a prototype. Aircraft structures must be able to withstand all flight conditions and be able to operate under all payload conditions.
A force applied lengthwise to a piece of structure will cause normal stress, being either tension (also refereed to as traction) or compression stress. See Figure 4.1. With tensile loads, all that matters is the area which is under stress. With compressive loads, also the shape is important, since buckling may occur. Stress is defined as load per area, being $\sigma = F/A$.
Figure 4.2: Bending. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Figure 4.3: Torsion. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Figure 4.4: Shear stress due to bending. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Figure 4.5: Shear stress due to torsion. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Figure 4.6: Stresses in a plate. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
If a force is applied at right angles (say perpendicular to the lengthwise of a beam), it will apply shear stress and a bending moment. See Figure 4.2. If a force is offset from the line of a beam, it will also cause torsion. See Figure 4.3. Both bending and torsion causes shear stresses. Shear is a form of loading which tries to tear the material, causing the atoms or molecules to slide over one another. See Figure 4.4 and Figure 4.5. Overall, a prototypical structure suffers from both normal ($\sigma$) and shear ($\tau$) stresses. See Figure 4.6 in which an illustrative example of the stresses over a plate is shown.
Figure 4.7: Normal deformation. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Figure 4.8: Tangential deformation. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Structures subject to normal or shear stresses may also be deformed. See Figure 4.7 and Figure 4.8.
Figure 4.9: Behavior of an isotropic material. Adapted from $F_{\text{RANCHINI}}$ et al. [2].
Strain, $\epsilon = \tfrac{\Delta l}{l_I} = \tfrac{l - l_I}{l_I}$ is the proportional deflection within a material as a result of an applied stress. It is impossible to be subjected to stress without experiencing strain. For elastic deformation, which is present below the elastic limit, Hooke’s law applies: $\sigma = E \epsilon$, where $E$ is refereed to as the modulus of Young, and it is a property of the material. The stresses within a structure must be kept below a defined permitted level, depending of the requirements of the structure (in general, stresses must no exceed the elastic limit, $\sigma_y$). See Figure 4.9.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/04%3A_Aircraft_structures/4.01%3A_Generalities.txt
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As a preliminary to the analysis of loads and basic aircraft structural elements presented in subsequent sections, we shall discuss some of the properties of materials and the materials themselves that are used in aircraft construction. Readers are referred to Megson [3].
4.02: Materials
Several factors influence the selection of the structural materials for an aircraft. The most important one is the combination of strength and lightness. Other properties with different importance (sometimes critical) are stiffness, toughness, resistance to corrosion and fatigue, ease of fabrication, availability and consistency of supply, and cost (also very important). A brief description of some of the most important properties is given in the sequel:
Ductility: Ductility refers to a solid material’s ability to deform under tensile stress, withstanding large strains before fracture occurs. These large strains are accompanied by a visible change in cross-sectional dimensions and therefore give warning of impending failure.
Strength: The strength of a material is its ability to withstand an applied stress without failure. The applied stress may be tensile, compressive, or shear. Strength of materials is a subject which deals with loads, deformations, and the forces acting on a material. Looking at Figure 4.9, it is associated to $\sigma_B$ (breaking stress); the greater $\sigma_B$ is the more strengthless is the material.
Toughness: Toughness is the ability of a material to absorb energy and plastically deform without fracturing. Toughness requires a balance of strength and ductility. Strength indicates how much force the material can support, while toughness indicates how much energy a material can absorb before fracturing. Looking at Figure 4.9, it is associated to the difference between $\sigma_B$ and $\sigma_y$; the greater this difference is the more capacity the material ha to absorb impact energy by plastic deformation.
Brittleness: A brittle material exhibits little deformation before fracture, the strain normally being below 5%. Brittle materials therefore may fail suddenly without visible warning. Brittleness and toughness are antonyms.
Elasticity: A material is said to be elastic if deformations disappear completely on removal of the load. Looking at Figure 4.9, this property is associated to $\sigma_y$ (elastic limit); the greater $\sigma_y$ is the more elastic the material. Notice that, within the elastic zone, stress and strain are linearly related with the Young Modulus (E), i.e, $\sigma = E \cdot \epsilon$.
Stiffness: Stiffness is the resistance of an elastic body to deformation by an applied force. Looking at Figure 4.9, this property is associated to $\sigma_y$ (elastic limit); the lower $\sigma_y$ is the more stiff the material. Elasticity and Stiffness are antonyms.
Plasticity: A material is perfectly plastic if no strain disappears after the removal of load. Ductile materials are elastoplastic and behave in an elastic manner until the elastic limit is reached after which they behave plastically. When the stress is relieved the elastic component of the strain is recovered but the plastic strain remains as a permanent set.
Fatigue: Mechanical fatigue occurs due to the application of a very large number of relatively small cyclic forces (always below the breaking stress $\sigma_B$) which results in material failure. For instance, every single flight of an aircraft can be considered as a cycle. In this manner, the aircraft can regularly withstand the nominal loads (always below the breaking stress $\sigma_B$), but after a large amount of cycles some parts of the structure might fail due to mechanical fatigue. For these reasons, aircraft may be tested for three times its life-cycle. In order to be able to withstand such testing, many aircraft components may be made stronger than is strictly necessary to meet the static strength requirements. The parts that might suffer from mechanical fatigue are termed fatigue-critical.
Corrosion: Corrosion is the gradual destruction of materials (usually metals) by chemical reaction with its environment. Roughly speaking, it has to do with the oxidation of the material and thus the loss of some of its properties. Corrosion resistance is an important factor to consider during material selection. Methods to prevent corrosion include: painting, which however incorporates an important amount of weight; anodizing, in which the aircraft is treated with a stable protective oxide layer; cladding, which basically consists of adding a layer of pure aluminum to the surface material (essentially, to attach a less noble material to a more noble material); and finally cadmium plating, which consists of covering the surface material with a more noble material (assuming the structure is made of a less noble material). These ideas are based on having two different materials with very different properties in terms of oxidation, so that if one suffers corrosion, the other does not.
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The main groups of materials used in aircraft construction nowadays are steel, aluminum alloys, titanium alloys, and fibre-reinforced composites.
Titanium alloys
Titanium alloys possess high specific properties, have a good fatigue strength/tensile strength ratio with a high fatigue limit, and some retain considerable strength at temperatures up to 400-500C. Generally, there is also a good resistance to corrosion and corrosion fatigue although properties are adversely affected by exposure to temperature and stress in a salty environment. The latter poses particular problems in the engines of carrier operated aircraft. Further disadvantages are a relatively high density so that weight penalties are imposed if the alloy is extensively used, coupled with high costs (of the material itself and due to its fabrication), approximately seven times those of aluminum and steel. Therefore, due its very particular characteristics (good fatigue strength/tensile strength at very high temperatures), titanium alloys are typically used in the most demanding elements of jet engines, e.g., the turbine blades.
Steels
Steels result of alloying Iron (Fe) with Carbon (C). Steels were the materials of the primary and secondary structural elements in the 30s. However, they were substituted by aluminum alloys as it will be described later on. Its high specific density prevents its widespread use in aircraft construction, but it has retained some value as a material for castings of small components demanding high tensile strengths, high stiffness, and high resistance to damage. Such components include landing gear pivot brackets, wing-root attachments, and fasteners.
Aluminum alloys
If one thinks in pure aluminum, the first thought is that it has virtually no structural application. It has a relatively low strength and it is extremely flexible. Nevertheless, when alloyed with other metals its mechanical properties are improved significantly, preserving its low specific weight (a key factor for the aviation industry). The typical alloying elements are copper, magnesium, manganese, silicon, zinc, and lithium. Aluminum alloys substituted steel as primary and secondary structural elements of the aircraft after World War II and thereafter. Four groups of aluminum alloy have been used in the aircraft industry for many years and still play a major role in aircraft construction: Al-Cu (2000 series); Al-Mg (5000 series); Al-Mg-Si (6000 series); Al-Zn-Mg (7000 series)1. The latest aluminum alloys to find general use in the aerospace industry are the aluminum-lithium (Al-Li, 8000 series) alloys.
Figure 4.10: Sketch of a fibre-reinforced composite materials. © PerOX / Wikimedia Commons / Public Domain.
Alloys from each of the above groups have been used extensively for airframes, skins, and other stressed components. Fundamentally, because all of them have a very low specific weight. Regarding the mechanical properties of the different alloys, the choice has been influenced by factors such as strength (proof and ultimate stress), ductility, easy of manufacture (e.g. in extrusion and forging), resistance to corrosion and suitability for protective treatments (e.g., anodizing), fatigue strength, freedom from liability to sudden cracking due to internal stresses, and resistance to fast crack propagation under load.
Unfortunately, as one particular property of aluminum alloys is improved, other desirable properties are sacrificed. Since the alloying mechanisms/process are complicated (basically micro-structural/chemical processes), finding the best trade-off is a challenging engineering problem. In the last 10 years, aluminum alloys are being systematically substituted by fibre-reinforced composite materials, first in the secondary structures, and very recently also in the primary structural elements (as it is the case of A350 or B787 Dreamliner).
Fibre-reinforced composite materials
Composite materials are materials made from two or more constituent materials with significantly different physical or chemical properties, that when combined produce a material with characteristics different from the individual components. In particular, the aircraft manufacturing industry uses the so-called fibre-reinforced composite materials, which consist of strong fibers such as glass or carbon set in a matrix of plastic or epoxy resin, which is mechanically and chemically protective.
A sheet of fibre-reinforced material is anisotropic, i.e. its properties depend on the direction of the fibers working at traction-compression. Therefore, in structural form two or more sheets are sandwiched together to form a lay-up so that the fibre directions match those of the major loads. This lay-up is embedded into a matrix of plastic or epoxy resin that fits things together and provides structural integrity to support both bending and shear stresses.
In the early stages of the development of fibre-reinforced composite materials, glass fibers were used in a matrix of epoxy resin. This glass-reinforced plastic (GRP) was used for helicopter blades but with limited use in components of fixed wing aircraft due to its low stiffness. In the 1960s, new fibre-reinforcements were introduced; Kevlar, for example, is an aramid material with the same strength as glass but is stiffer. Kevlar composites are tough but poor in compression and difficult to manufacture, so they were used in secondary structures. Another composite, using boron fibre, was the first to possess sufficient strength and stiffness for primary structures. These composites have now been replaced by carbon- fibre-reinforced plastics (CFRP), which have similar properties to boron composites but are very much inexpensive.
Typically, CFRP has a Young modulus of the order of three times that of GRP, one and a half times that of a Kevlar composite and twice that of aluminum alloy. Its strength is three times that of aluminum alloy, approximately the same as that of GRP, and slightly less than that of Kevlar composites. Nevertheless, CFRP does suffer from some disadvantages. It is a brittle material and therefore does not yield plastically in regions of high stress concentration. Its strength is reduced by impact damage which may not be visible and the epoxy resin matrices can absorb moisture over a long period which reduces its matrix- dependent properties, such as its compressive strength; this effect increases with increase of temperature. On the contrary, the stiffness of CFRP is much less affected than its strength by the absorption of moisture and it is less likely to fatigue damage than metals.
Replacing 40% of an aluminum alloy structure by CFRP results, roughly, in a 12% saving in total structural weight. Indeed, nowadays the use of composites has been extended up to 50% of the total weight of the aircraft, covering most of the secondary structures of the aircraft and also some primary structures. For instance, in the case of the Airbus A350XWB, the empennage and the wing are manufactured essentially based on CRPF. Also, some parts of the nose and the fuselage are manufactured on CRPF. The A350XWB material breakdown is as follows (in percentage of its structural weight) according to Airbus:
• 52% fiber-reinforced composites.
• 20% aluminum alloys.
• 14% titanium.
• 7% steel.
• 7% miscellaneous.
1. The following aluminum alloys are commonly used in aircraft and other aerospace structures: 7075 aluminum; 6061 aluminum; 6063 aluminum; 2024 aluminum; 5052 aluminum.
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The structure of a typical commercial aircraft is required to support two distinct classes of loads: the first, termed ground loads, include all loads encountered by the aircraft during movement or transportation on the ground such as taxiing, landing, or towing; while the second, air loads, comprise loads imposed on the structure during flight operations2.
The two above mentioned classes of loads may be further divided into surface forces which act upon the surface of the structure, e.g., aerodynamic forces and hydrostatic pressure, and body forces which act over the volume of the structure and are produced by gravitational and inertial effects, e.g., force due to gravity. Calculation of the distribution of aerodynamic pressure over the various surfaces of an aircraft’s wing was presented in Chapter 3.
Basically, all air loads are the different resultants of the corresponding pressure distributions over the surfaces of the skin produced during air operations. Generally, these resultants cause direct loads, bending, shear, and torsion in all parts of the structure.
2. In Chapter 7 we will examine in detail the calculation of ground and air loads for a variety of cases.
4.03: Loads
The fuselage will experience a wide range of loads from a variety of sources.
The weight of both structure of the fuselage and payload will cause the fuselage to bend downwards from its support at the wing, putting the top in tension and the bottom in compression. In maneuvering flight, the loads on the fuselage will usually be greater than for steady flight. Also landing loads may be significant. The structure must be designed to withstand all loads cases in all circumstances, in particular in critical situations.
Most of the fuselage of typical commercial aircraft is usually pressurized (this also applies for other types of aircraft). The pressure inside the cabin corresponds, during the cruise phase, to that at an altitude of 2000-2500 [m] (when climbing/descending below/above that altitude, it is usually changed slowly to adapt it to terrain pressure). Internal pressure will generate large bending loads in fuselage frames. The structure in these areas must be reinforced to withstand these loads. Also, for safety, the designer must consider what would happen if the pressurization is lost. The damage due to depressurization depends on the rate of pressure loss. For very high rates, far higher loads would occur than during normal operation.
Doors and hatches are a major challenge when designing an aircraft. Depending on their design, doors will or will not carry some of the load of the fuselage structure. Windows, since they are very small, do not create a severe problem. On the floor of the fuselage also very high localized loads can occur, especially from small-heeled shoes. Therefore floors need a strong upper surface to withstand high local stresses.
4.3.02: Wing and tail loads
The lift produced by the wing creates a shear force and a bending moment, both of which are at their highest values at the root of the wing. Indeed, the root of the wing is one (if not the most) structurally demanding elements of the aircraft. The structure at this point needs to be very strong (high strenght) to resist the loads and moments, but also quite stiff to reduce wing bending. Thus, the wing is quite thick at the root.
Another important load supported by the wing is, in the case of wing-mounted engines, that of the power plant. Moreover, the jet fuel is typically located inside the wing. Therefore, an appropriate location of the power plant weight together with a correct distribution of the jet fuel (note that it is being consumed during the flight) contribute to compensate the lift forces during the flight, reducing the shear force and bending moment at the wing root. Fuel load close to the tips reduces this moment. Therefore the order in which the tanks are emptied is from the root to the tip. Nevertheless, when the aircraft is on the ground the lift is always lower than weight (when the aircraft is stopped, there is no lift), and all three forces, i.e., its structural weight, fuel, and power plant, can not be compensated by upwards lift. Therefore, the wing must also be design to withstand these loads which requires a design compromise.
The tailplane, rudder, and ailerons also create lift, causing a torsion in the fuselage. Since the fuselage is cylindrical, it can withstand torsion very effectively.
4.3.03: Landing gear loads
The main force caused by the landing gear is an upward shock during landing. Thus, shock-absorbers are present, absorbing the landing energy and thus reducing the force done on the structure. The extra work generated during a hard landing results in a very large increase in the force on the structure.
4.3.04: Other loads
Other loads include engine thrust on the wings or fuselage which acts in the plane of symmetry but may, in the case of engine failure, cause severe fuselage bending moments; concentrated shock loads during a catapult launch for fighters; and hydrodynamic pressure on the fuselages or floats of seaplanes.
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Aircraft are generally built up from the basic components of wings, fuselages, tail units and control surfaces, landing gear, and power plant.
Figure 4.11: Aircraft monocoque skeleton.
The structure of an airplane is the set of those elements whose mission is to transmit and resist the applied loads; to provide an aerodynamic shape and to protect passengers, payload, systems, etc. from the environmental conditions encountered during the flight. These requirements, in most aircraft, result in thin shell structures where the outer surface or skin of the shell is usually supported by longitudinal stiffening elements and transverse frames to enable it to resist bending, compressive, and torsional loads without buckling. Such structures are known as semi-monocoque, while thin shells which rely entirely on their skins for their capacity to resist loads are referred to as monocoque.
4.04: Structural components of an aircraft
The fuselage should carry the payload, and is the main body to which all parts are connected. It must be able to resist bending moments (caused by weight and lift from the tail), torsional loads (caused by fin and rudder), and cabin pressurization. The structural strength and stiffness of the fuselage must be high enough to withstand these loads. At the same time, the structural weight must be kept to a minimum.
Figure 4.12: Semimonocoque Airbus A340 rear fuselage, seen from inside. © Sovxx / Wikimedia Commons / CC-BY-SA-3.0.
In transport aircraft, the majority of the fuselage is cylindrical or near-cylindrical, with tapered nose and tail sections. The semi-monocoque construction, which is virtually standard in all modern aircraft, consists of a stressed skin with added stringers to prevent buckling, attached to hoop-shaped frames. See Figure 4.12.
The fuselage has also elements perpendicular to the skin that support it and help keep its shape. These supports are called frames if they are open or ring-shaped, or bulkheads if they are closed.
Disturbances in the perfect cylindrical shell, such as doors and windows, are called cutouts. They are usually unsuitable to carry many of the loads that are present on the surrounding structure. The direct load paths are interrupted and as a result the structure around the cut-out must be reinforced to maintain the required strength.
In aircraft with pressurized fuselages, the fuselage volume both above and below the floor is pressurized, so no pressurization loads exist on the floor. If the fuselage is suddenly de-pressurized, the floor will be loaded because of the pressure difference. The load will persist until the pressure in the plane has equalized, usually via floor-level side wall vents. Sometimes different parts of the fuselage have different radii. This is termed a double-bubble fuselage. Pressurization can lead to tension or compression of the floor-supports, depending on the design.
Frames give the fuselage its cross-sectional shape and prevent it from buckling when it is subjected to bending loads. Stringers give a large increase in the stiffness of the skin under torsion and bending loads, with minimal increase in weight. Frames and stringers make up the basic skeleton of the fuselage. Pressure bulkheads close the pressure cabin at both ends of the fuselage, and thus carry the loads imposed by pressurization. They may take the form of flat discs or curved bowls. Fatigue-critical areas are at the fuselage upper part and at the joints of the fuselage frames to the wing spars.
Figure 4.13: Structural wing sketch.
Figure 4.14: Structural wing torsion box. Adapted from © User Kadellar / Wikimedia Commons / / CC-BY-SA-3.0.
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Providing lift is the main function of the wing of an aircraft. A wing consists of two essential parts. The internal wing structure, consisting of spars, ribs, and stringers, and the external wing, which is the skin.
Ribs give the shape to the wing section, support the skin (prevent buckling), and act to prevent the fuel flowing around as the aircraft maneuvers. Its primary structural function is to withstand bending moments (the moment resultant of aerodynamic forces) and shear stresses (due to the vertical and horizontal resultant of forces). They serve as attachment points for the control surfaces, flaps, landing gear, and engines. They also separate the individual fuel tanks within the wing.
The wing stringers (also referred to as stiffeners) are thin strips of material (a beam) to which the skin of the wing is fastened. They run spanwise and are attached between the ribs. Their job is to stiffen the skin so that it does not buckle when subjected to compression loads caused by wing bending and twisting, and by loads from the aerodynamic effects of lift and control-surface movements.
The ribs also need to be supported, which is done by the spars. These are simple beams that usually have a cross-section similar to an I-beam. The spars are the most heavily loaded parts of an aircraft. They carry much more force at its root, than at the tip. Since wings will bend upwards, spars usually carry shear forces and bending moments.
Aerodynamic forces not only bend the wing, they also twist it. To prevent this, a second spar is introduced. Torsion now induces bending of the two spars. Modern commercial aircraft often use two-spar wings where the spars are joined by a strengthened section of skin, forming the so-called torsion-box structure. The skin in the torsion-box structure serves both as a spar-cap (to resist bending), as part of the torsion box (to resist torsion) and to transmit aerodynamic forces.
4.4.03: Tail
For the structural components of the stabilizers of the tail, fundamentally all exposed for the wing holds.
4.4.04: Landing gear
The landing gear (also referred to as undercarriage) of an aircraft supports the aircraft on the ground, provide smooth taxiing, and absorb shocks of taxiing and landings. It has no function during flight, so it must be as small and light as possible, and preferably easily retractable.
Due to the weight of the front (containing cabin and equipment) and rear parts (where the empennage is located) of the aircraft, large bending moments occur on the centre section of the fuselage. Therefore, to withstand these bending moments, a strong beam is located. This reduces the space in which the landing gear can be retracted.
When an aircraft lands, a large force is generated on the landing gear as it touches the ground. To prevent damage to the structure, this shock must be absorbed and dissipated as heat by the landing gear. If the energy is not dissipated, the spring system might just make the aircraft bounce up again.
After touchdown, the aircraft needs to brake. Disc brakes are primarily used. The braking of an aircraft can be supplemented by other forms of braking, such as air brakes, causing a large increase in drag, or reverse thrust, thrusting air forward.
4.05: References
[1] Cantor, B., Assender, H., and Grant, P. (2010). Aerospace materials. CRC Press.
[2] Franchini, S., López, O., Antoín, J., Bezdenejnykh, N., and Cuerva, A. (2011). Apuntes de Tecnología Aeroespacial. Escuela de Ingeniería Aeronáutica y del Espacio. Universidad Politécnica de Madrid.
[3] Megson, T. (2007). Aircraft structures for engineering students. A Butterworth- Heinemann Title.
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In this chapter the goal is to give a brief overview of the different instruments, systems, and subsystems that one can find in a typical aircraft. First, in Section 5.1, the focus will be on instruments. Notice that modern aircraft are becoming more sophisticated and classical instruments are being substituted by electronic displays. Aircraft systems will be briefly analyzed in Section 5.2. Again, many elements of classical mechanical (pneumatic, hydraulic) systems are being substituted by electronics. Thus, in modern terminology, the discipline that encompasses instruments (as electronic displays) and electronic systems is referred to as avionics. An introductory reference is Franchini et al. [1], in which this chapter is inspired. Thorough references on aircraft systems are Moir and Seabridge [4], Kossiakoff et al. [2], Tooley and Wyatt [5], and Langton et al. [3].
05: Aircraft instruments and systems
Flight instruments are specifically referred to as those instruments located in the cockpit of an aircraft that provide the pilot with the information about the flight situation of the aircraft, such as position, speed, and attitude. The flight instruments are of particular use in conditions of poor visibility, such as in clouds, when such information is not available from visual reference outside the aircraft. The term is sometimes used loosely as a synonym for cockpit instruments as a whole, in which context it can include engine instruments, navigational instruments, and communication equipment.
Historically, the first instruments needed on board were the magnetic compass and a clock in order to calculate directions of flight and times of flight. To calculate the remaining fuel in the tanks, a glass pipe showing the level of fuel was presented on the cockpit. Before World War I, cockpits begin to present altimeters, anemometers, tachometers, etc. In the period between wars (1919-1939), the era of the pioneers, more and more sophisticated instruments were demanded to fulfill longer and longer trips: the directional gyro (heading indicator) and the artificial horizon (attitude indicator) appeared, and the panel of instruments started to have a standard layout.
Nowadays, in the era of electronics and information technologies, the cockpits present the information in on-board computers, using digital indicators and computerized elements of measure. Since instruments play a major role in controlling the aircraft and performing safe operations in compliance with air navigation requirements, it is necessary to present data in a clean and standard layout, so that the pilot can interpret them rapidly and clearly. The design of on board instruments requires knowing the physical variables one wants to measure, and the concepts and principles within each instrument.
5.01: Aircraft instruments
Different sources of information are needed for the navigation of an aircraft in the air.
Certain data come by measuring physical magnitudes of the air surrounding the aircraft, such as the pressure (barometric altimeter) or the velocity of air (pitot tube). Other data are obtained by measuring the accelerations of the aircraft using accelerometers. Also the angular changes (changes in attitude) and changes in the angular velocity can be measured using gyroscopes. The course of the aircraft is calculated through the measure of the direction of the magnetic field of the Earth.
Figure 5.1: Barometric altimeter.
A barometric altimeter is an instrument used to calculate the altitude based on pressure measurements. Figure 5.1 illustrates how a barometric altimeter works and how it looks like. Details on how the altimeter indicator works will be given later on when analyzing the altimeter. Still nowadays, most of the aircraft use the barometric altimeters to determine the altitude of the aircraft. An altimeter cannot, however, be adjusted for variations in air temperature. As already studied in Chapter 2, ISA relates pressure and altitude.
Figure 5.2: Diagram of barometric settings.
Differences in temperature from the ISA model will, therefore, cause errors in the indicated altitude. An aneroid or mercury barometer measures the atmospheric pressure from a static port outside the aircraft and based on a reference pressure. The aneroid altimeter can be calibrated in three manners (QNE, QNH, QFE) to show the pressure directly as an altitude above a reference (101225 [Pa] level, sea level, the airport, respectively). Please, refer to Figure 5.2. Recall Equation (2.3.3.2):
$\dfrac{p}{p_0} = \left (1 - \dfrac{\alpha}{T_0} h \right )^{\tfrac{g}{R\alpha}}; \ \ \ and\ thus$
$\dfrac{p_{ref}}{p_0} = \left (1 - \dfrac{\alpha}{T_0} h_{ref} \right )^{\tfrac{g}{R\alpha}}.$
Isolating $h$ and $h_{ref}$, respectively, and subtracting, it yields:
$h - h_{ref} [m] = \dfrac{T_0}{\alpha} \left [ \left (\dfrac{p_{ref}}{p_0}\right )^{\tfrac{g}{R\alpha}} - \left (\dfrac{p}{p_0}\right )^{\tfrac{g}{R\alpha}} \right ].$
The reference values can be adjusted, and there exist three main standards:
• QNE setting: the baseline pressure is 101325 Pa. This setting is equivalent to the air pressure at mean sea level (MSL) in the ISA.
• QNH setting: the baseline pressure is the real pressure at sea level (not necessarily 101325 [Pa]). In order to estimate the real pressure at sea level, the pressure is measured at the airfield and then, using equation (2.3.3.3), the real pressure at mean sea level is estimated (notice that now $p_0 \ne 101325$). It captures better the deviations from the ISA.
• QFE setting: where $p_{ref}$ is the pressure in the airport, so that $h - h_{ref}$ reflects the altitude above the airport.
Figure 5.3: Pitot tube.
A pitot tube is an instrument used to measure fluid flow velocity. A basic pitot tube consists of a tube pointing directly into the fluid flow, in which the fluid enters (at aircraft’s airspeed). The fluid is brought to rest (stagnation). This pressure is the stagnation pressure of the fluid, which can be measured by an aneroid. The measured stagnation pressure cannot itself be used to determine the fluid velocity (airspeed in aviation). Using Bernoulli’s equation (see Equation (3.1.3.6)), the velocity of the incoming flow (thus the airspeed of the aircraft, since the pitot tube is attached to the aircraft) can be calculated. Figure 5.3 illustrates how a pitot tube works and how it looks like. Please, refer to Exercise 5.3.1 as an illustration of pitot tube equations. Details on how the airspeed indicator works will be given later on in this chapter.
Figure 5.4: Gyroscope and accelerometeres.
Figure 5.5: Diagram of ST-124 gimbals with accelerometers and gyroscopes (conforming the basic elements of a Inertial Measurement Unit). Author NASA/MSFC / Wikimedia Commons / Public Domain.
A gyroscope is a mechanical (also exist electronic) device based on the conservation of the kinetic momentum, i.e., a spinning cylinder with high inertia rotating at high angular velocity, so that the kinetic momentum is very high and it is not affected by external actions. Thus, the longitudinal axis of the cylinder points always in the same direction. Figure 5.4.a illustrates it. An accelerometer is a device that calculates accelerations based on displacement measurements. It is typically composed by a mass-damper system attached to a spring as illustrated in Figure 5.4.b. When the accelerometer experiences an acceleration, the mass is displaced. The displacement is then measured to give the acceleration (applying basic physics and the Second Newton Law). A typical accelerometer works in a single direction, so that a set of three is needed to cover the three directions of the space. The duple gyroscopes and accelerometer conforms the basis of an Inertial Measurement Unit (IMU), an element used for inertial navigation (to be studied in Chapter 10), i.e., three accelerometers measure the acceleration in the three directions and three gyroscopes measure the angular acceleration in the three axis; with an initial value of position and attitude and via integration, current position, velocity, attitude, and angular velocity can be calculated1. Figure 5.5 illustrates it. See also Exercises 5.3.1-5.3.2 for an insight on IMU usage in the context of Inertial Navigation.
The aircraft can also send electromagnetic waves to the exterior to know, for instance, the altitude with respect to the ground (radio-altimeter), or the presence of clouds in the intended trajectory (meteorologic radar). It can also receive electromagnetic waves from specific aeronautical radio-infrastructures, both for en-route navigation (VOR,2 NDB, etc.), and for approach and landing phases (ILS, MLS, etc). Also, the new systems of satellite navigation (GPS, GLONASS, and the future GALILEO) will be key in the future for more precise and reliable navigation. Aircraft have on-board instruments (the so-called navigation instruments) to receive, process, and present this information to the pilot.
1. Notice that these calculations are complicated, since the values need to be projected in the adequate reference frames, and also the gravity, which is always accounted by the accelerometer in the vertical direction, needs to by considered. This is not covered in this course and will be studied in more advanced courses of navigation.
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ICAO establishes the criteria (some are rules, other recommendations) to design, manufacture, and install the instruments. Some of these recommendations are:
• All instruments should be located in a way that can be read clearly and easily by the pilot (or the corresponding member of the crew).
• The illumination should be enough to be able to read without disturbance nor reflection at dark.
• The flight instruments, navigation instruments, and engines instruments to be used by the pilot must be located in front of his/her view.
• All flight instruments must be grouped together in the instrument panel.
• All engine instruments should be conveniently grouped to be readable by the appropriate member of the crew.
• The multiengine aircraft must have identical instruments for each engine, and be located in a way that avoids any possible confusion.
• The instruments should be installed so that are subject to minimal vibrations.
5.1.03: Instruments to be installed in an aircraft
The instruments to be installed in an aircraft are, on the one hand, flight and navigation instruments, and, on the other, instruments of the power plant.3
Flight and navigation instruments
ICAO establishes that the minimum required flight and navigation instruments are:
Airspeed indicator: The airspeed indicator presents the aircraft’s speed (usually in knots) relative to the surrounding air. It works by measuring the pressure (static and dynamic) in the aircraft’s pitot tube. The indicated airspeed must be corrected for air density (using barometric and temperature data) in order to obtain the true airspeed, and for wind conditions in order to obtain the ground speed.
Attitude indicator (artificial horizon): The attitude indicator (also known as an artificial horizon) presents the aircraft’s attitude relative to the horizon. This instrument provides information to the pilot on, for instance, whether the wings are leveled or whether the aircraft nose is pointing above or below the horizon.
Figure 5.6: Airspeed indicator: © Mysid / Wikimedia Commons / CC-BY-SA-3.0; Attitude indicator: © El Grafo / Wikimedia Commons / GNU-3.0; Altimeter: © Bsayusd / Wikimedia Commons / Public Domain; Heading indicator: © Oona RŁisŁnen / Wikimedia Commons / CC-BY-SA-3.0.
Altimeter: The altimeter presents the altitude of the aircraft (in feet) above a certain reference (typically sea-level, destination airport, or 101325 isobar according to the three different barometric settings studied in Chapter 2) by measuring the difference between the pressure in aneroid capsules inside the barometric altimeter and the atmospheric pressure obtained through the static ports. The variations in volume of the aneroid capsule, which contains a gas, due to pressure differences are traduced into altitude by a transducer. If the aircraft ascends, the capsule expands as the static pressure drops causing the altimeter to indicate a higher altitude. The opposite occurs when descending.
Heading indicator (directional gyro): The heading indicator (also known as the directional gyro) displays the aircraft’s heading with respect to the magnetic north. The principle of operation is based on a gyroscope.
Magnetic compass: The compass shows the aircraft’s heading relative to magnetic north. It a very reliable instrument in steady level flight, but it does not work well when turning, climbing, descending, or accelerating due to the inclination of the Earth’s magnetic field. The heading indicator is used instead (based on gyroscopes, more reliable instruments).
Figure 5.7: Turn and slip: Author User:Dhaluza / Wikimedia Commons / Public Domain; Variometer: © User:The High Fin Sperm Whale / Wikimedia Commons / CC-BY-SA-3.0; Magnetic Compass: © User:Chopper / Wikimedia Commons / CC-BY-SA-3.0.
Turn indicator (turn and slip): The turn indicator (also known as turn and slip) displays direction of turn and rate of turn. The direction of turn displays the rate that the aircraft’s heading is changing. The internally mounted inclinometer (some short of balance indicator or ball) displays quality of turn, i.e. whether the turn is correctly coordinated, as opposed to an uncoordinated turn, wherein the aircraft would be in either a slip or a skid.
Vertical speed indicator (variometer): The vertical speed indicator (also referred to as variometer) displays the rate of climb or descent typically in feet per minute. This is done by sensing the change in air pressure.
Additionally, an indicator of exterior air temperature and a clock are also required.
Additional panel instruments: Obviously, most aircraft have more than the minimum required instruments. Additional panel instruments that may not be found in smaller aircraft are:
The Course Deviation Indicator (CDI): is an instrument used in aircraft navigation to determine an aircraft’s lateral position in relation to a track, which can be provided, for instance, by a VOR or an ILS. This instrument can also be integrated with the heading indicator in a horizontal situation indicator.
A Radio Magnetic Indicator (RMI): is generally coupled to an Automatic Direction Finder (ADF), which provides bearing for a tuned NDB. While simple ADF displays may have only one needle, a typical RMI has two, coupling two different ADF receivers, allowing the pilot to determine the position by bearing interception.
ILS Instrumental Landing System: This system is nowadays fundamental for the phases of final approach and landing in instrumental conditions. The on-board ILS instrumental system indicates a path angle and an alignment with the axis of the runway, i.e., it assists pilots in vertical and lateral navigation.
Figure 5.8: Navigation instruments: Course Deviation Indicator (CDI) and Radio Magnetic Indicator (RMI). Notice that the reading of these instruments is not covered in this course. Nevertheless, in Chapter 10 an interpretation is provided.
Power plant instruments
ICAO also establishes a minimum required set of instruments for the power plant. We will just mention a few, not going into details:
• Tachometer for measuring the velocity of turn of the crankshaft (or the compressor in a jet).
• Indicator of the temperature of air entering the carburetor (just for piston aircraft).
• Indicator of the temperature of oil at the entrance and exit.
• Indicator of the temperature at the entrance of the turbine and the exit gases (just for jet aircraft).
• Indicator of fuel pressure and oil pressure.
• Indicator of tank level.
• Indicator of thrust (jets) and motor-torque (propellers).
Figure 5.9: Six basic instruments in a light twin-engine airplane arranged in a basic- T. From top left: airspeed indicator, attitude indicator, altimeter, turn coordinator, heading indicator, and vertical speed indicator. © User:Meggar / Wikimedia Commons / CC-BY-SA-3.0.
3. the content of this subsection has been partially based on Wikipedia flight instruments.
5.1.04: Instruments layout
Flight and navigation instruments layout
Most aircraft are equipped with a standard set of flight instruments which provide the pilot with information about the aircraft’s attitude, airspeed, and altitude.
Most aircraft built since the 50s have four of the flight instruments located in a standardized pattern called the T-arrangement, which has become throughout the years a standard. The attitude indicator is at the top center, airspeed indicator to the left, altimeter to the right, and heading indicator below the attitude indicator. The other two, turn indicator and vertical speed indicator, are usually found below the airspeed indicator and altimeter, respectively, but for these two there is no common standard. The magnetic compass will be above the instrument panel. In newer aircraft with electronic displays substituting conventional instruments, the layout of the displays conform to the basic T-arrangement. The basic T-arrangement can be observed in Figure 5.6 and Figure 5.9.
Power plant instruments layout
This instruments layout is less standardized and we will not go into detail.
Figure 5.10: Aircraft cockpit.
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The content of this section is inspired by Wikipedia [6].
A cockpit or flight deck is the area, usually in the nose of an aircraft, from which the cabin crew (pilot and co-pilots) commands the aircraft. Except for some small aircraft, modern cockpits are physically separated from the cabin. The cockpit contains the flight instruments on an instrument panel, and the controls which enable the pilot to fly the aircraft, i.e., the control yoke (also known as a control column) that actuates on the elevator and ailerons4, the pedals that actuates on the rudder, and the throttle level position to adjust thrust.
The layout of cockpits in modern airliners has become largely unified across the industry. The majority of the systems-related actuators (typically some short of switch), are usually located in the ceiling on an overhead panel. These are for instance, actuators for the electric system, fuel system, hydraulic system, and pressurization system. Radio communication systems are generally placed on a panel between the pilot’s seats known as the pedestal. The instrument panel or instrument display is located in front of the pilots, so that all displays are visible. In modern electronic cockpits, the block displays usually regarded as essential are Mode Control Panel (MCP), Primary Flight Display (PFD), Navigation Display (ND), Engine Indicator and Crew Alerting System (EICAS), Flight Management System (FMS), and back-up instruments. Thus, these five elements (together with the back-ups) compose the instrument panel (containing all flight and navigation instruments as electronic displays) in a modern airliner. Notice that mechanical instruments have been substituted by electronic displays, and this is why this discipline is now referred to as avionics systems (the electronics on board the aircraft).
Figure 5.11: Aircraft glass cockpit displays: MCP, PFD, and ND.
Mode Control Panel (MCP): A MCP is an instrument panel that permits cabin crew to control the autopilot and related systems. It is a long narrow panel located centrally in front of the pilot, just above the PFD and rest of displays. The panel covers a long but narrow area usually referred to as the glareshield panel as illustrated in Figure 5.11.a. The MPC contains the elements (mechanical or digital) that allow the cabin crew to select the autopilot mode, i.e., to specify the autopilot to hold a specific altitude, to change altitude at a specific rate, to maintain a specific heading, to turn to a new heading, to follow a route of waypoints, etc., and to engage or disengage the auto-throttle. Thus, it permits activating different levels of automation in flight (from fully automated to fully manual). Notice that MCP is a Boeing designation (that has been informally adopted as a generic name); the same unit with the same functionalities on an Airbus aircraft is referred to as the FCU (Flight Control Unit).
Figure 5.12: EICAS/ECAM cockpit displays.
Primary Flight Display (PFD): The PFD is a modern, electronic based aircraft instrument dedicated to flight information. It combines the older instruments arrangement (T- arrangement or T-arrangement plus turn and slip and variometer) into one compact display, simplifying pilot tasks. It is located in a prominent position, typically centered in the cockpit for direct view. It includes in most cases a digitized presentation of the attitude indicator (artificial horizon), air speed indicator, altitude indicator, and the vertical speed indicator (variometer). Also, it might include some form of heading indicator (directional gyro) and ILS/VOR deviation indicators (CDI). Figure 5.11.b illustrates it.
Navigation Display (ND): The ND is an electronic based aircraft instrument showing the route, information on the next waypoint, current wind speed and wind direction. It can also show meteorological data such as incoming storms, navaids5 located on earth. This electronic display is sometimes referred to as MFD (multi-function display). Figure 5.11.c illustrates how it looks like.
Engine Indication and Crew Alerting System (EICAS) (used by Boeing) or Electronic Centralized Aircraft Monitor (ECAM) (by Airbus): The EICAS/ECAM displays information about the aircraft’s systems, including its fuel, electrical, and propulsion systems (engines). It allows the cabin crew to monitor the following information: values for the different engines, fuel temperature, fuel flow, the electrical system, cockpit or cabin temperature and pressure, control surfaces and so on. The pilot may select display of information by means of button press. The EICAS/ECAM display improves situational awareness by allowing the cabin crew to view complex information in a graphical format and also by alerting the crew to unusual or hazardous situations. For instance, for the EICAS display, if an engine begins to lose oil pressure, an alert sounds, the display switches to the page with the oil system information and outline the low oil pressure data with a red box.
Figure 5.13: Flight Management System (FMS) Control Display Unit.
Flight Management System (FMS): The FMS is a specialized computer system that automates a wide variety of in-flight tasks, reducing the workload on the flight crew. Its primary function is in-flight management of the flight plan6, which is uploaded before departure and updated via data-link communications. Another function of the FMS is to guide the aircraft along the flight plan. This is done by measuring the current state (position, velocity, heading angle, etc.) of the aircraft, comparing them with the desired one, and finally setting a guidance law. From the cockpit, the FMS is normally controlled through a Control Display Unit (CDU) which incorporates a small screen and keyboard or touchscreen. The FMS sends the flight plan for display to the Navigation Display (ND) and other electronic displays in order them to present the following flight plan information: waypoints, altitudes, speeds, bearings, navaids, etc.
Figure 5.14: Aircraft electrical generation sources: ground unit and APU.
4. An alternative to the yoke in most modern aircraft is the centre stick or side-stick (colloquially known as joystick).
5. navaids refers to navigational aids and will be studied in Chapter 10. It includes VORs, DMEs, ILS, NDB, etc.
6. the flight plan will be studied in Chapter 10.
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In order an aircraft to fulfill its mission, e.g., to transport passengers from one city to another in a safe, comfortable manner, many systems and subsystems are needed. These must be fully integrated since most of them are interdependent. In this section we present some of the main systems that can be found in an aircraft, e.g., electrical system, fuel system, hydraulic, flight control system, etc. More detailed information can be consulted, for instance, in Moir and Seabridge [4] and Langton et al. [3].
5.02: Aircraft systems
The electrical system is of great importance. Many elements run with electric energy, e.g: Indicator instruments, navigation and communication equipments, electro-actuators, electro-pneumatic mechanisms, illumination, passenger comfort (meals, entertainment, etc).
The electrical system is formed by the unities and basic components which generate, store, and distribute the electric energy to all systems that need it. Generally, in aircraft the primary source is Alternating Current (AC), and the secondary source in Direct Current (DC). The typical values of the AC are 115 V and 400 Hz, while the typical value of DC is 28 V. Due to safety reasons, the principal elements of the systems must be redundant (back-up systems), at least be double. Therefore, we can distinguish:
• Power generation elements (AC generation).
• Primary power distribution and protection (AC distribution).
• Power conversion and energy storage (AC to DC and storage).
• Secondary power distribution and protection elements (DC distribution).
Figure 5.15: A380 power system components.
There are different power generation sources for aircraft. They can be either for nominal conditions, for redundancy, or to handle emergency situations. These power sources include:
• Engine driven AC generators.
• Auxiliary Power Units (APU).
• External power, also referred to as Ground Power Unit (GPU).
• Ram Air Turbines (RAT).
The engine driven AC generators are the primary source of electrical energy. Each of the engines on an aircraft drives an AC generator. The produced power using the rotation of the turbine in nominal flight is used to supply the entire aircraft with electrical energy.
When the aircraft is on the ground, the main generators do not work, but still electrical energy is mandatory for handling operation, maintenance actions, or engine starting. Therefore, it is necessary to extract the energy from other sources. Typically, the aircraft might use an external source such a GPU, or the so-called Auxiliary Power Unit (APU). The APU is a turbine engine situated in the rear part of the aircraft body which
produces electrical energy. The APU is typically used on the ground as primary source of electrical energy, while on air is a back-up power source.
Some aircraft are equipped with Ram Air Turbines (RAT). The RAT is an air-driven turbine, normally stowed in the aircraft ventral or nose section. The RAT is used as an emergency back-up element, which is deployed when the conventional power generation elements are unavailable, i.e., in case of failure of the main generator or the APU when on air or ground, respectively.
Engine driven generators, GPU, APU, and RAT produce AC current. This AC current is distributed throughout the system to feed the elements of the aircraft that require electrical input, e.g., lighting, heating, communication systems, etc. However, it is also necessary to store energy in case of emergency. The energy must be stored in batteries working in DC. Therefore, the energy must be converted (AC to DC), for which one needs transformation units. The most frequently used method of power conversion in modern aircraft electrical system is the Transformer Rectifier Unit (TRU), which converts a three- phase 115V AC current into 28V DC current. Then the DC current stored in batteries can be distributed by means of a secondary DC distribution system, used also to feed certain elements of the aircraft that require electrical input. Notice that both the generation and the distribution need protection elements (for the case of AC current: under/over-voltage protection, under/over frequency protection, differential current protection, current phase protection) and control elements (in order to regulate voltage).
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The main purpose of an aircraft fuel system is to provide a reliable supply of fuel to the power plant. Given that an aircraft with no fuel (or with no properly supplied fuel) can not fly (unless gliding), this system is key to ensure safe operations. The commonly used fuel is high octane index gasoline for piston aircraft, and some type of kerosene for jet aircraft. Even though fuel systems differ greatly due to the type of fuel and the type of mission, one can distinguish the following needs: refuel and defuel; storage; fuel pressurization; fuel transfer; engine feed; etc. Thus, the system is fundamentally composed by:
• tanks;
• fuel hydrants;
• feeding pumps;
• pipes and conducts;
• valves and filters;
• sensors, indicators, and control elements.
Figure 5.16: Diagrammatic representation of the Boeing 737-300 fuel system. © User:RosarioVanTulpe / Wikimedia Commons / Public Domain.
Tanks are used to storage fuel. Three main types can be distinguished: independent tanks; integrated tanks; interchangeable tanks. The independent tanks (concept similar to car tanks) are nowadays obsolete, just present in regional aircraft. The most extended in commercial aviation are integrated tanks, meaning that the tank is also part of the structure of (typically) the wing. The integral tanks are painted internally with a anti-corrosion substance and sealing all union and holes. The interchangeable tanks are those installed for determined missions.
The filling up and emptying process is centralized in a unique point, the fuel hydrant, which supplies fuel to all tanks thanks to feeding pumps which pump fuel throughout the pipes and conducts conforming the distribution network of the system. To be more precise, there are two fundamental types of pumps: the fuel transfer pumps, which perform the task of transferring fuel between the aircraft tanks, and the fuel booster pumps (also referred to as engine feed pumps), which are used to boost (preventing from flameouts and other inconveniences) the fuel flow form the fuel system to the engine.
The system is completed with valves, filters, sensors, indicators, and control elements. Valves can be simply transfer valves or non-return valves (to preserve the logic direction of fuel flow) or vent valves (to eliminate air during refueling). Filters are used to remove contaminants in the system. Last, different sensors are located within the system to measure different performance parameters (fuel quantity, fuel properties, fuel level, etc). The measurements are displayed in several indicators, some of them shown directly to the pilot, some others analyzed in a control unit. Both pilot and control unit (the later automatically) might actuate on the system to modify some of the performances. Notice that the subsystems that encompass the indicators, displays, and control unit might be also seen as part of an electronic or avionics system.7
Figure 5.16 shows a diagrammatic representation of the Boeing 737-300 fuel system.
7. 1 Engine Driven Fuel Pump - Left Engine; 2 Engine Driven Fuel Pump - Right Engine; 3 Crossfeed Valve; 4 Left Engine Fuel Shutoff Valve; 5 Right Engine Fuel Shutoff Valve; 6 Manual Defuling Valve; 7 Fueling Station; 8 Tank No. 2 (Right); 9 Forward Fuel Pump (Tank No. 2); 10 Aft Fuel Pump (Tank No. 2); 11 Left Fuel Pump (Center Tank); 12 Right Fuel Pump (Center Tank); 13 Center Tank; 14 Bypass Valve; 15 Aft Fuel Pump (Tank No. 1); 16 Forward Fuel Pump (Tank No. 1); 17 Tank No. 1 (Left); 18 Fuel Scavenge Shutoff Valve; 20 APU Fuel Shutoff Valve; 21 APU; 22 Fuel Temperature Sensor; 23-36 Indicators.
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Hydraulic systems have been used since the early 30s and still nowadays play an important role in modern airliners. The basic function of the aircraft hydraulic system is to provide the required power to hydraulic consumers, such for instance: primary flight controls (ailerons, rudder, and elevator); secondary flight controls (flaps, slats, and spoilers); other systems, such landing gear system (extension and retraction, braking, steering, etc.), or door opening, etc.
The main advantages of hydraulic systems are:
• relative low weight in comparison with the required force to apply; • simplicity in the installation;
• low maintenance;
• high efficiency with low losses, just due to liquid friction.
Figure 5.17: Aircraft hydraulic system: aileron actuator and landing gear actuator.
The main components of a hydraulic system are:
• a source of energy (any of the sources of the electrical system, i.e., engine driven alternator, APU, RAT);
• a reservoir or tanks to store the hydraulic fluid;
• a filter to maintain clean the hydraulic fluid;
• a mean of storing energy such as an accumulator (high density fluid tank);
• pipeline manifold (pipe or chamber branching into several openings);
• pumps (engine driven or electric), pipes, and valves;
• a mechanism for hydraulic oil cooling;
• pressure and temperature sensors;
• actuators (actuate mechanically on the device).
5.2.04: Flight control systems- Fly-By-Wire
Fly-by-wire Wikipedia [7] is a system that replaces the conventional manual flight controls of an aircraft with an electronic interface. The movements of flight controls are converted to electronic signals transmitted by wires (hence the fly-by-wire term), and flight control computers determine how to move the actuators at each control surface to provide the adequate response. The fly-by-wire system also allows automatic signals sent by the aircraft’s computers to perform functions without the pilot’s input, as in systems that automatically help stabilize the aircraft.
Figure 5.18: Flight control system: conventional and flight by wire.
Mechanical and hydro-mechanical flight control systems are relatively heavy and require careful routing of flight control cables through the aircraft by systems of pulleys, cranks, tension cables, and hydraulic pipes. Both systems often require redundant backups to deal with failures, which again increases weight. Also, both have limited ability to compensate for changing aerodynamic conditions. The term fly-by-wire implies a purely electrically-signaled control system. However, it is used in the general sense of computer- configured controls, where a computer system is interposed between the operator and the final control actuators or surfaces. This modifies the manual inputs of the pilot in accordance with control parameters.
Command: Fly-by wire systems are quite complex; however their operation can be explained in relatively simple terms. When a pilot moves the control column (also referred to as sidestick or joystick), a signal is sent to a computer through multiple wires or channels (a triplex is when there are three channels). The computer receives the signals, which are then sent to the control surface actuator, resulting in surface motion. Potentiometers in the actuator send a signal back to the computer reporting the position of the actuator. When the actuator reaches the desired position, the two signals (incoming and outgoing) cancel each other out and the actuator stops moving.
Automatic Stability Systems: Fly-by-wire control systems allow aircraft computers to perform tasks without pilot input. Automatic stability systems operate in this way. Gyroscopes fitted with sensors are mounted in an aircraft to sense movement changes in the pitch, roll, and yaw axes. Any movement results in signals to the computer, which automatically moves control actuators to stabilize the aircraft to nominal conditions.
Digital Fly-By-Wire: A digital fly-by-wire flight control system is similar to its analog counterpart. However, the signal processing is done by digital computers and the pilot literally can "fly-via-computer". This also increases the flexibility of the flight control system, since the digital computers can receive input from any aircraft sensor, e.g., altimeters and pitot tube. This also increases the electronic stability, because the system is less dependent on the values of critical electrical components in an analog controller. The computers sense position and force inputs from pilot controls and aircraft sensors. They solve differential equations to determine the appropriate command signals that move the flight controls to execute the intentions of the pilot. The Airbus Industries Airbus A320 became the first airliner to fly with an all-digital fly-by-wire control system.
Main advantages: Summing up, the main advantages of fly-by-wire systems are:
• decrease in weight, which results in fuel savings;
• reduction in maintenance time (instead of adjusting the system, pieces are simply changed by new ones, so that maintenance is made more agile);
• better response to air gusts, which results in more comfort for passengers;
• automatic control of maneuvers (the systems avoid the pilot executing maneuvers with exceed of force in the controls).
5.2.05: Air conditioning and pressurisation system
The cabin air conditioning seeks keeping the temperature and humidity of the air in the cabin within certain range of values, avoidance ice and steam formation, the air currents and bad smells. The flight at high altitudes also force to pressurize the cabin, so that passengers can breath sufficient oxygen (remember that human being rarely can reach 8.000 m mountains in the Himalayan, only after proper natural conditioning not to suffer from altitude disease). That is why, cabins have an apparent atmosphere bellow 2500 m. Both systems can be built independently.
5.2.06: Other systems
There are many other systems in the aircraft, some are just in case of emergency. For the sake of brevity, we just mention some of them providing a brief description. Notice also that this taxonomy is not standard and thus the reader might encounter it in a different way in other textbooks.
Pneumatic system: Pursues the same function as hydraulic systems, actuating also in control surfaces, landing gears, doors, etc. The only difference is that the fluid is air.
Oxygen system: Emergency system in case the cabin is depressurized.
Ice and rain protection system: In certain atmospheric conditions, ice can be formed rapidly with influence in aerodynamic surfaces. There exist preventive systems which heat determined zones, and also corrective systems that meld ice once formed.
Fire protection system: Detection and extinction system in case of fire.
Information and communication system: Provides information and permits both internal communication with the passengers (musical wire), and external communication (radio, radar, etc). External communication includes VHF and HF communication equipment. Also, flight-deck audio systems might be included. Information system refers also to data- link communications, including all kind of data broadcasted from the ATM units, but also all aircraft performance data (speed, pressure, altitude, etc.) that can be recorded using Flight Data Recorder (FDR), Automatic Dependent Surveillance Broadcast (ADSB), and track-radar. Please, refer to Chapter 11.
Air navigation system: It includes all equipment needed for safe navigation. All navigation instruments and electronic displays already described in Section 5.1 might be seen as part of this system. Inertial navigation systems (IMUs) can also be categorized as air navigation equipment. The Traffic Alert and Collision Avoidance System (TCAS) can be also included in this system. Please, refer to Chapter 11.
Avionics system: This term is somehow confuse, since avionics refers to the electronics on board the aircraft. As it has been exposed throughout the chapter, electronics is becoming more and more important in modern aircraft. Practically every single system in an aircraft has electronic elements (digital signals, displays, controllers, etc.) to some extent. Therefore, is not clear whether avionics should be a system by itself, but is becoming more and more popular to use the term avionics systems to embrace all electronics on board the aircraft (and sometimes also the earth-based equipment that interrogates the aircraft). Characteristic elements of avionics are microelectronic devices (microcontrollers), data buses, fibre optic buses, etc. It is also important to pay attention to system design and integration since the discipline is transversal to all elements in the aircraft.
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Exercise $1$ Pitot tube
Figure 5.19L Pitot Tube.
Aircraft use pitot tubes to measure airspeed. They consists of a tube pointing directly into the fluid flow, such that the moving fluid is brought to rest (stagnation pressure of the air, $p_T$). Typically, pitot tubes include also a static port to measure the static pressure of the air ($p_{\infty}$). See Figure 5.19 as illustration considering the air as a compressible flow.
Consider the following measurements on board the aircraft:
• The Pitot tube measures a stagnation pressure $p_T = 36975\ Pa$.
• The static part measures a static pressure of $p_{\infty} = 22500\ Pa$.
Assume also that:
• the air can be considered an ideal gas.
• the air should be considered a compressible fluid. For compressible flow, one has that
$P_T = P_{\infty} \cdot \left (1 + \dfrac{\gamma - 1}{2} M^2 \right )^{\tfrac{\gamma}{\gamma - 1}},$
with $\gamma = 1.4$ the adiabatic coefficient of air, and $M$ the Mach number:
$M = \sqrt{V_{TAS}}{\sqrt{\gamma RT}}$
Calculate the calibrated airspeed of the aircraft (CAS).8
Answer
Notice that one can apply Bernoulli’s equation to fluid’s stream line within the Pitot tube. Assuming compressible flow, one has:
$P_T = P_{\infty} \cdot \left (1 + \dfrac{\gamma - 1}{2} M^2 \right )^{\tfrac{\gamma}{\gamma - 1}}.$
Assuming also the air can be considered an ideal gas, one has:
$P = \rho \cdot R \cdot T.$
In addition, one has the following relation:
$M = \dfrac{V_{TAS}}{\sqrt{\gamma RT}}.$
All in all, elaborating with this three equations, one has:
$P_T = P_{\infty} \cdot \left (1 + \dfrac{\gamma - 1}{2} \dfrac{\rho_{\infty} \cdot V_{TAS}^2}{\gamma \cdot P_{\infty}} \right )^{\tfrac{\gamma}{\gamma - 1}}.$
Considering $P_T - P_{\infty} = \Delta P$ and isolating $V_{TAS}^2$:
$V_{TAS}^2 = \dfrac{2\gamma}{\gamma - 1} \cdot \dfrac{P_{\infty}}{\rho_{\infty}} \left (\left (\dfrac{\Delta P}{P_{\infty}} + 1 \right )^{\tfrac{\gamma - 1}{\gamma}} - 1 \right ).$
However, one should notice that only with the pitot tube and the static port, neither temperature nor density can be measured. Thus, $V_{TAS}$ can not be directly calculated (we would need additional instruments/sensors). This is the reason behind the Calibrated Airspeed (CAS) concept: the true airspeed an aircraft would have if flying with standard mean sea level conditions. Thus, CAS is defined as follows:
$V_{CAS}^2 = \dfrac{2\gamma}{\gamma - 1} \cdot \dfrac{P_{MSL}}{\rho_{MSL}} \left (\left (\dfrac{\Delta P}{P_{MSL}} + 1 \right )^{\tfrac{\gamma - 1}{\gamma}} - 1 \right ).\label{eq5.3.8}$
Indeed, the airspeed that is displayed in the cockpit to the pilot (referred to as Indicated Airspeed) is the CAS speed corrected with instrument errors.
Now, entering in Eq. ($\ref{eq5.3.8}$) with the values given in the statement, one has the solution to the problem:
$V_{CAS} = 150\ m/s.\nonumber$
8. assume mean sea level conditions are the standard ones according to ISA, $P_{MSL} = 101325\ Pa$ and $\rho_{MSL} = 1.225\ kg/m^3$
5.04: References
[1] Franchini, S., López, O., Antoín, J., Bezdenejnykh, N., and Cuerva, A. (2011). Apuntes de Tecnología Aeroespacial. Escuela de Ingeniería Aeronáutica y del Espacio. Universidad Politécnica de Madrid.
[2] Kossiakoff, A., Sweet, W. N., Seymour, S., and Biemer, S. M. (2011). Systems engineering principles and practice, volume 83. Wiley.com.
[3] Langton, R., Clark, C., Hewitt, M., and Richards, L. (2009). Aircraft fuel systems. Wiley Online Library.
[4] Moir, I. and Seabridge, A. (2008). Aircraft systems: mechanical, electrical and avionics subsystems integration, volume 21. Wiley. com.
[5] Tooley, M. and Wyatt, D. (2007). Aircraft communication and navigation systems (principles, maintenance and operation).
[6] Wikipedia (2013a). Cockpit. http://en.wikipedia.org/wiki/Cockpit. Last accesed 25 feb. 2013.
[7] Wikipedia (2013b). Fly-by-wire. http://en.wikipedia.org/wiki/Fly-by-wire. Last accesed 25 feb. 2013.
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Aircraft require to thrust themselves to accelerate and thus counteract drag forces. In this chapter, we look at the way aircraft engines work. All aircraft propulsion systems are based on the principle of reaction of airflow through a power plant system. The two means for accelerating the airflow surrounding the aircraft that are presented in this chapter are through propellers and jet expansion, which give rise to the so-called propeller engines and jet engines to be studied in Section 6.1 and Section 6.2, respectively. In Section 6.3 the different types of jet engines will be studied. A third type of propulsion systems are the rocket engines, but they are used in spacecrafts and lay beyond the scope of this course. An introductory reference on the topic is Newman [3, Chapter 6]. Thorough references are, for instance, Mattingly et al. [2] and Jenkinson et al. [1].
The design of an aircraft engine must satisfy diverse needs. The first one is to provide sufficient thrust to counteract the aerodynamic drag of the aircraft, but also to exceed it in order to accelerate. Moreover, it must provide enough thrust to fulfill with the operational requirements in all circumstances (climbs, turns, etc.). Moreover, commercial aircraft focus also on high engine efficiency and low fuel consumption rates. On the contrary, fighter aircraft might require an important excess of thrust to perform sharp, aggressive maneuvers in combat.
06: Aircraft propulsion
Typically, general aviation aircraft are powered by propellers and internal combustion piston engines (similar to those used in the automobile industry). The basic working principles are as follows: the air in the surroundings enters the engine, it is mixed with fuel and burned, thereby releasing a tremendous amount of energy in the mix (air and fuel) that is employed in increasing its energy (heat and molecular movement). This mix at high speed is exhausted to move a piston that is attached to a crankshaft, which in turn acts rotating a propeller.
The process of combustion in the engine provides very little thrust. Rather, the thrust is produced by the propeller due to aerodynamics. Propellers have various (two, three, or four) blades with an airfoil shape. The propeller acts as a rotating wing, creating a lift force due to its motion in the air. The aerodynamics of blades, i.e., the aerodynamics of helicopters, are slightly different than those studied in Chapter 3, and lay beyond the scope of this course. Nevertheless, the same principles apply: the engine rotates the propeller, causing a significant change in pressure across the propeller blades, and finally producing a net balance of forwards lift force.
6.01: The propeller
Figure 6.1: Propeller schematic.
A schematic of a propeller propulsion system is shown in Figure 6.1. As the reader would notice, this illustration has strong similarities with the continuity equation illustrated in Figure 3.3. The thrust force is generated due to the change in velocity as the air moves across the propeller between the inlet (0) and outlet (e). As studied in Chapter 3, the mass flow into the propulsion system (considered as a stream tube) is a constant.
The fundamentals of propelled aircraft flights are based on Newton’s equations of motion and the conservation of energy and momentum:
Attending at conservation of momentum principle, the force or thrust is equal to the mass flow times the difference between the exit and inlet velocities, expressed as:
$F = \dot{m} \cdot (u_e - u_0),\label{eq6.1.1}$
where $u_0$ is the inlet velocity, $u_e$ is the exit velocity, and $\dot{m}$ is the the mass flow. The exit velocity is higher than the inlet velocity because the air is accelerated within the propeller.
Attending at the conservation of energy principle for ideal systems, the output power of the propeller is equal to the kinetic energy flow across the propeller. This is expressed as follows:
$P = \dot{m} \cdot (\dfrac{u_e^2}{2} - \dfrac{u_0^2}{2} = \dfrac{\dot{m}}{2} \cdot (u_e - u_0) \cdot (u_e + u_0).\label{eq6.1.2}$
where $P$ denotes propeller power.
As real systems do not behave ideally, the propeller efficiency can be defined as:
$\eta_{prop} = \dfrac{F \cdot u_0}{P}.\label{eq6.1.3}$
where $F \cdot u_0$ is the useful work, and $P$ refers to the input power, i.e., the power that goes into the engine. In other words, the efficiency $\eta$ is a ratio between the real output power generated to move the aircraft and the input power demanded by the engine to generate it. In an ideal system $F \cdot u_0 = P$. In real systems $F \cdot u_0 < P$ due to, for instance, mechanical losses in transmissions, etc.
Operating with Equation ($\ref{eq6.1.1}$) and Equation ($\ref{eq6.1.2}$) and substituting in Equation ($\ref{eq6.1.3}$) it yields:
$\eta_{prop} = \dfrac{2 \cdot u_0}{u_e + u_0}.$
In order to obtain a high efficiency ($\eta_{prop} \sim 1$), one wants to have $u_e$ as close as possible to $u_0$. However, looking at Equation ($\ref{eq6.1.1}$), at very close values for input and input velocities, one would need a much larger mass flow to achieve a desired thrust. Therefore, there are find a compromise. Rewriting Equation ($\ref{eq6.1.1}$) as:
$\dfrac{F}{\dot{m} u_0} = \dfrac{u_e}{u_0} - 1,$
leads to a relation for propulsive efficiency. Notice that, if $u_e = u_0$, there is no thrust. For higher values of thrust, the efficiency drops dramatically.
Besides the propeller efficiency, other effects contribute to decrease the efficiency of the system. This is the case of the thermal effects in the engine. The thermal efficiency can be defined as:
$\eta_t = \dfrac{P}{\dot{m}_f \cdot Q},$
where $P$ is power, $\dot{m} f$ is the mass flow of fuel, and $Q$ is the characteristic heating value of the fuel.
Finally, the overall efficiency can be defined combining both as follows:
$\eta_{overall} = \eta_t \cdot \eta_{prop} = \dfrac{F \cdot u_0}{\dot{m}_f \cdot Q}.$
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Figure 6.2: Jet engine: Core elements and station numbers. Adapted from: © Jeff Dahl / Wikimedia Commons / CC-BY-SA-3.0.
Even though there are various types of jet engines (also referred to as gas turbine engines) as it is to be studied in Section 6.3, all of them share the same core elements, i.e., inlet, compressor, burner, turbine, and nozzle. Figure 6.2 illustrates schematically a jet engine with its core elements and the canonical engine station numbers, which are typically used to notate the airflow characteristics ($T, p, \rho$, etc.) through the different components. In this Figure, the station 0 represent the free-stream air flow; 1 represents the entrance of the inlet; 2 and 3 represent the entrance and exit of the compressor, respectively; 4 and 5 represent the entrance and exit of the turbine, respectively; 6 and 7 represent the entrance and exit of the after-burner1 (in case there is one, which is not generally the case), respectively; and finally 8 represents the exit of nozzle.
Roughly speaking, the inlet brings free-stream air into the engine; the compressor increases its pressure; in the burner fuel is injected and combined with high-pressure air, and finally burned; the resulting high-temperature exhaust gas goes into the power turbine generating mechanical work to move the compressor and producing thrust when passed through a nozzle (due to action-reaction Newton’s principle). Details of these engine core components are given in the sequel.
1. Notice that in the figure there is not after-burner, but however the station numbers 6 and 7 have been added for the sake of generalizing.
6.02: The jet engine
Before analyzing the characteristics and equations of the elements of the jet engine, let us briefly explain some basic concepts regarding thermodynamics (useful to understand what follows in the section). For more insight, the reader is referred to any undergraduate text book on thermodynamics.
The first law of the thermodynamics can be stated as follows:
$\Delta E = Q + W.\label{eq6.2.1.1}$
where $E$ denotes de energy of the system, $Q$ denotes de heat, and $W$ denotes the work. In other words, an increase (decrease) in the energy of the system results in heat and work.
The energy of the system can be expressed as:
$E = U + \dfrac{mV^2}{2} + mgz,$
where $U$ denotes the internal energy, the term $\tfrac{mV^2}{2}$ denotes the kinetic energy, and the term $mgz$ denotes the patential energy (with $z$ being the altitude). In the case of a jet engine, $z$ can be considered nearly constant, and the potential term thus neglected. Also, it is typical to use stagnation values for pressure and temperature of the gas, i.e., the values that the gas would have considering $V = 0$ as already presented in Chapter 3. Under this assumption, the kinetic terms can be also neglected. Therefore, Equation ($\ref{eq6.2.1.1}$) can be expressed as:
Figure 6.3: Sketch of an adiabatic process. © Yuta Aoki / Wikimedia Commons / CC-BY-SA-3.0.
$\Delta U = Q + W.$
Now, the work can be divided into two terms: mechanical work ($W_{mech}$) and work needed to expand/contract the gas ($\Delta (PV)$), i.e.,
$W = W_{mech} + \Delta (PV).$
Also, the enthalpy ($h$) of the system can be defined as: $h = U + PV$. In sum, the energy equation (Equation ($\ref{eq6.2.1.1}$)) can be expressed as follows:
$\Delta h = Q + W_{mech}.$
An increase of enthalpy can be expressed as follows:
$\Delta h = c \cdot \Delta T;$
where $c$ is the specific heat of the gas and $T$ is the temperature.
Moreover, we state now how the stagnation values are related to the real values:
$h_t = h + \dfrac{V^2}{2};$
$p_t = p + \dfrac{1}{2} \cdot \rho \cdot V^2.$
Notice that, if the process is adiabatic then $Q = 0$, and thus the increase (decrease) in enthalpy is all turned into mechanical work. Adiabatic processes will be assumed for the stages at the compressor and turbine. Moreover, for an adiabatic process there are some relations between pressure and temperature for an ideal gas, i.e.,
$P \cdot V^{\gamma} = constant \to \dfrac{p_a}{p_b} = \left (\dfrac{T_a}{T_b} \right )^{\tfrac{\gamma}{\gamma - 1}},$
being $a$ and $b$ state conditions within the adiabatic process and $\gamma$ the ratio of specific heats.2
2. This ratio is also referred to as heat capacity ratio or adiabatic index.
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Figure 6.4: Types of inlets.
The free-stream air enters the jet engine at the inlet (also referred to as intake). There exist a variety of shapes and sizes dependent on the speed regime of the aircraft. For subsonic regimes, the inlet design in typically simple and short (e.g., for most commercial and cargo aircraft). The surface front is called the inlet lip, which is typically thick in subsonic aircraft. See Figure 6.4.a.
On the contrary, supersonic aircraft inlets have a relatively sharp lip as illustrated in Figure 6.4.b. This sharpened lip minimizes performance losses from shock waves due to supersonic regimes. In this case, the inlet must slow the flow down to subsonic speeds before the air reaches the compressor.
An inlet must operate efficiently under all flight conditions, either at very low or very high speeds. At low speeds the free-stream air must be pulled into the engine by the compressor. At high speeds, it must allow the aircraft to properly maneuver without disrupting flow to the compressor.
Given that the inlet does no thermodynamic work, the total temperature through the inlet is maintained constant, i.e.:
$\dfrac{T_{2t}}{T_{1t}} = \dfrac{T_{2t}}{T_0} = 1.$
The total pressure through the inlet changes due to aerodynamic flow effects. The ratio of change is typically characterized by the inlet pressure recovery (IPR), which measures how much of the free-stream flow conditions are recovered and can be expressed as follows:
$IPR = \dfrac{p_{2t}}{p_{0t}} = \dfrac{p_{2t}}{p_0}; M < 1.\nonumber$
As pointed out before, the shape of the inlet, the speed of the aircraft, the airflow characteristics that the engine demands, and aircraft maneuvers are key factors to obtain a high pressure recovery, which is also related to the efficiency of the inlet expressed as:
$\eta_i = \dfrac{p_{2t}}{p_{1t}} = \dfrac{p_{2t}}{p_0}.\nonumber$
6.2.03: Compressor
Figure 6.5: Types of jet compressors.
In the compressor, the pressure of the incoming air is increased by mechanical work. There are two fundamental types of compressors: axial and centrifugal. See Figure 6.5 as illustration of these two types.
Figure 6.6: Axial compressor.
In axial compressors the flow goes parallel to the rotation axis, i.e., parallel to the axial direction. In a centrifugal compressor the airflow goes perpendicular to the axis of rotation. The very first jet engines used centrifugal compressors, and they are still used on small turbojets. Modern turbojets and turbofans typically employ axial compressors. An axial compressor is composed by a duple rotor-stator (if the compressor is multistage, then there will one duple per stage). In short, the rotor increases the absolute velocity of the fluid and the stator converts this into pressure increase as Figure 6.6 illustrates.
A typical, single-stage, centrifugal compressor increases the airflow pressure by a factor of 4. A similar single stage axial compressor will produce a pressure increase of between 15% and 60%, i.e., pressure ratios of 1.15-1.6 (small when compared to the centrifugal one). The fundamental advantage of axial compressor is that several stages can be easily linked together, giving rise to a multistage axial compressor, which can supply air with a pressure ratio of 40. It is much more difficult to produce an efficient multistage centrifugal compressor and therefore most high-compression jet engines incorporate multistage axial compressors. If only a moderate amount of compression is required, the best choice would be a centrifugal compressor.
Let us now focus on the equations that govern the evolution of the airflow over the compressor.
The pressure increase is quantified in terms of the so-called compressor pressure ratio (CPR), which is the ratio between exiting and entering air pressure. Using the station numbers of Figure 6.2, the CPR can be expressed as the stagnation pressure at stage 3 ($p_{3t}$) divided by the stagnation pressure at stage 2 ($p_{2t}$):
$CPR = \dfrac{p_{3t}}{p_{2t}}.\nonumber$
The process can be considered adiabatic. Thus, according to the thermodynamic relation between pressure and temperature given in Equation (6.2.1.9), CPR can be also expressed as follows:
$CPR = \dfrac{p_{3t}}{p_{2t}} = \left (\dfrac{T_{3t}}{T_{2t}} \right )^{\tfrac{\gamma}{(\gamma - 1)}},\nonumber$
where $\gamma$ is the ratio of specific heats ($\gamma \approx 1.4$ for air).
Referring the reader to Section 6.2.1 and doing some algebraic operations, the mechanical work consumed by the compressor can be expressed:
$W_{comp} = \dfrac{cT_{2t}}{\eta_c} (CPR^{\tfrac{(\gamma - 1)}{\gamma}} - 1),$
where $c$ is the specific heat of the gas and $\eta_c$ is the compressor efficiency. The efficiency factor is included to account for the real performance as opposed to the ideal one. Notice that the needed mechanical work is provided by the power turbine, which is connected to the compressor by a central shaft.
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The combustion chamber (also referred to as burner or combustor) is where combustion occurs. Fuel is mixed with the high-pressure air coming out of the compressor, and combustion occurs. The resulting high-temperature exhaust gas is used to turn the power turbine, producing the mechanical work to move the compressor and eventually producing thrust after passing through the nozzle.
The burner is located between the compressor and the power turbine. The burner is arranged as some short of annulus so that the central engine shaft connecting turbine and compressor can be allocated in the hole. The three main types of combustors are annular; can; and hybrid can-annular.
Figure 6.7: Combustion chamber or combustor.
Can combustors are self-contained cylindrical combustion chambers. Each can has its own fuel injector. Each can get an air source from individual opening. Like the can type combustor, can-annular combustors have discrete combustion zones contained in separate liners with their own fuel injectors. Unlike the can combustor, all the combustion zones share a common air casing. Annular combustors do not use separate combustion zones and simply have a continuous liner and casing in a ring (the annulus).
Many modern burners incorporate annular designs, whereas the can design is older, but offers the flexibility of modular cans. The advantages of the can-annular burner design are that the individual cans are more easily designed and tested, and the casing is annular. All three designs are found in modern gas turbines.
The details of mixing and burning the fuel are very complicated and therefore the equations that govern the combustion process will not be studied in this course. For the purposes of this course, the combustion chamber can be considered as the place where the air temperature is increased with a slight decrease in pressure. The pressure in the combustor can be considered nearly constant during burning. Using the station numbers from Figure 6.2, the combustor pressure ratio (CbPR) is equal to the stagnation pressure at stage 4 ($p_{4t}$) divided by the stagnation pressure at stage 3 ($p_{3t}$), i.e.:
$BPR = \dfrac{p_{4t}}{p_{3t}} \sim 1.\nonumber$
The thermodynamics in the combustion chamber are different from those of the compressor and turbine because in the combustion chamber heat is released during the combustion process. In the compressor and turbine, the processes are adiabatic (there is no heat involved): pressure and temperature are related, and the temperature change is determined by the energy equation.
In the case of the combustion chamber, the process is not adiabatic anymore. Fuel is added in the chamber. The added mass of the fuel can be accounted by using a ratio $f$ of fuel flow to air mass flow, which can be quantified as:
$f = \dfrac{\dot{m}_f}{\dot{m}} = \dfrac{\tfrac{T_{4t}}{T_{3T}} - 1}{\tfrac{\eta_b Q}{cT_{3t}} - \tfrac{T_{4t}}{T_{3T}}},$
where $\dot{m}_f$ denotes the mass flow of fuel, $Q$ is the heating constant (which depends on the fuel type), $c$ represents the average specific heat, $T_{t3}$ is the stagnation temperature at the combustor entrance, $T_{4t}$ is the stagnation temperature at the combustor exit, and $\eta_b$ is the combustor efficiency. This ratio is very important for determining overall aircraft performance because it provides a measure of the amount of fuel needed to burn a determined amount of air flow (at the conditions of pressure and temperature downstream the compressor) and subsequently generate the corresponding thrust.
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The turbine is located downstream the combustor and transforms the energy from the hot flow into mechanical work to move the compressor (remember that turbine and compressor are linked by a shaft). The turbine is composed of two rows of small blades, one that rotates at very high speeds (the rotor) and the other that remains stationary (the stator). The blades experience flow temperatures of around $1400^{\circ} K$ and must, therefore, be either made of special metals (typically titanium alloys) that can withstand the heat.
Depending on the engine type, there may be multiple turbine stages present in the engine. Turbofan and turboprop engines usually employ a separate turbine and shaft to power the fan and gearbox, respectively, and are referred to as two-spool engines. Three- spool configurations exist for some high-performance engines where an additional turbine and shaft power separate parts of the compressor.
The derivation of equations that govern the evolution of the air flow over the turbine are similar to those already exposed for the compressor. As the flow passes through the turbine, pressure and temperature decrease. The decrease in pressure through the turbine is quantified with the so-called turbine pressure ratio (TPR), i.e., the ratio of the exiting to the entering air pressure in the turbine. Using the station numbers of Figure 6.2, the TPR is equal to the stagnation pressure at point 5 ($p_{5t}$) divided by the stagnation pressure at point 4 ($p_{4t}$), i.e.,
$TPR = \dfrac{p_{5t}}{p_{4t}}.\nonumber$
Figure 6.8: Turbine and schematic blade.
Given that the process can be considered adiabatic, pressure and temperature are related as in Equation (6.2.1.9) so that:
$TPR = \dfrac{p_{5t}}{p_{4t}} = \left (\dfrac{T_{5t}}{T_{4t}} \right )^{\tfrac{\gamma}{\gamma - 1}}.$
Again, referring the reader to Section 6.2.1 and doing some algebraic operations, the turbine mechanical work $W_{turb}$ can be expressed as follows:
$W_{turb} = (\eta_t c T_{4t}) (1 - TPR^{\tfrac{(\gamma - 1)}{\gamma}}),\label{eq6.2.5.2}$
where $c$ is the specific heat of the gas and $\eta_t$ is the turbine efficiency.
Compressor and turbine stages work attached one to the other. This relationship can be expressed by setting the work done by the compressor equal to the work done by the turbine, i.e., $W_{turb} = W_{comp}$. Hence, the conservation of energy is ensured. Equating Equation (6.2.3.1) and Equation ($\ref{eq6.2.5.2}$) yields:
$\dfrac{cT_{2t}}{\eta_c} (CPR^{\tfrac{(\gamma - 1)}{\gamma}} - 1) = (\eta_t c T_{4t}) (1 - TPR^{\tfrac{(\gamma - 1)}{\gamma}}).$
Figure 6.9: Variable extension nozzle.
6.2.06: Nozzles
The final stage of the jet engine is the nozzle. The nozzle has three functions, namely: a) to generate thrust; b) to conduct the exhaust gases back to the free-stream conditions; and c) to establish the mass flow rate through the engine by setting the exhaust area. The nozzle lays downstream the turbine.3
There are different shapes and sizes depending on the type of aircraft performance. Simple turbojets and turboprops typically have fixed-geometry convergent nozzles. Turbofan engines sometimes employ a coannular nozzle where the core flow exits the center nozzle while the fan flow exits the annular nozzle. After-burning turbojets and some turbofans often incorporate variable-geometry convergent-divergent nozzles (also referred to as de Laval nozzles), where the flow is first compress to flow through the convergent throat, and then is expanded (typically to supersonic velocities) through the divergent section.
Figure 6.10: Convergent-divergent nozzle. In the left-land side, the figure shows approximate flow velovity ($v$), together with the effect on temperature ($T$) and pressure ($p$).
Let us now move on analyzing in brief the equations governing the evolution of the flow in the nozzle. The nozzle exerts no work on the flow, and thus both the stagnation temperature and the stagnation pressure can be considered constant. Recalling the station number from Figure 6.2, we write:
$\dfrac{p_{8t}}{p_{5t}} = \left (\dfrac{T_{8t}}{T_{5t}} \right )^{\tfrac{\gamma}{(\gamma - 1)}} = 1,\nonumber$
where 5 corresponds to the turbine exit and 8 to the nozzle throat.
The stagnation pressure at the exit of the nozzle is equal to the free-stream static pressure, unless the exiting flow is expanded to supersonic conditions (a convergent- divergent nozzle). The nozzle pressure ratio (NPR) is defined as:
$NPR = \dfrac{p_{8t}}{p_8} = \dfrac{p_0}{p_8},$
where $p_{8t}$ is the stagnation nozzle pressure or the free-stream static pressure. In order to determine the total pressure at the nozzle throat $p_8$, a term referred to as overall engine pressure ratio (EPR) is used. The EPR is defined to be the total pressure ratio across the engine, and can be expressed as follows:
$EPR = \dfrac{p_{8t}}{p_{2t}} = \dfrac{p_{3t}}{p_{2t}} \dfrac{p_{4t}}{p_{3t}} \dfrac{p_{5t}}{p_{4t}} \dfrac{p_{8t}}{p_{5t}},$
where the compressor, combustor, turbine, and nozzle stages are all represented.
Similarly, the Engine Temperature Ration (ETR) can be expressed as:
$ETR = \dfrac{T_{8t}}{T_{2t}} = \dfrac{T_{3t}}{T_{2t}} \dfrac{T_{4t}}{T_{3t}} \dfrac{T_{5t}}{T_{4t}} \dfrac{T_{8t}}{T_{5t}},$
from which the nozzle stagnation temperature ($T_{8t}$) can be calculated.
Considering Equation (6.2.1.7), isolating the exit velocity and doing some algebra, it yields:
$u_e = u_8 = \sqrt{2c \eta_n T_{8t} \left [1 - (\dfrac{1}{NPR})^{\tfrac{\gamma - 1}{\gamma}} \right ]},$
where $\eta_n$ is the nozzle efficiency, which is normally very close to 1.
The nozzle performance equations work just as well for rocket engines except that rocket nozzles always expand the flow to some supersonic exit velocity.
Summing up, all the necessary relations between jet engine components have been stated in order to obtain the thrust developed by the jet engine. Notice that, as already pointed out in Equation (6.1.1.1), the thrust would be:
$Thrust = \dot{m} \cdot (u_e - u_0).$
3. Notice that in this description of the core elements of a jet engine the after-burner has been omitted. It there is one (fundamentally, for supersonic aircraft), it would located downstream the turbine and upstream the nozzle.
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Figure 6.11: Relative suitability of the turboprop, turbofans, and ordinary turbojects for the flight at the 10 km attitude in various speeds. Adapted from Wikimedia Commons / CC-BY-SA-3.0.
Some of the most important types of jet engines will be now discussed. Specifically, turbojets, turbofans, turboprops, and after-burning turbojets. As a first touch, Figure 6.11 illustrate a sketch of the relative suitability of some of these types of jets. It can be observed that turboprop are more efficient in low subsonic regimes; turbofans are more efficient in high subsonic regimes; and turbojets (also after-burning turbojets) are more efficient for supersonic regimes. If one looks at higher mach numbers (\(M > 3-4\)), ramjet, scramjets or rockets will be needed. However these last types are beyond the scope of this course.
6.03: Types of jet engines
Figure 6.12: Turbojet with centrifugal compressor.
Figure 6.13: Turbojet with axial compressor.
A turbojet is basically what has been already exposed in Section 6.2. It is composed by an inlet, a compressor, a combustion chamber, a turbine, and a nozzle. The reader is referred back to Figure 6.2 as illustration. As already mentioned, there are two main types of turbojets depending on the type of compressor: axial or centrifugal. Figures 6.12-6.13 show schematic and real jet engines with centrifugal and axial flow, respectively.
6.3.02: Turbofans
Most modern commercial aircraft use turbofan engines because of their high thrust and good fuel efficiency at high subsonic regimes. A turbofan engine is similar to a basic jet engine. The only difference is that the core engine is surrounded by a fan in the front and an additional fan turbine at the rear. The fan and fan turbine are connected by an additional shaft. This type of arrangement is called a two-spool engine (one spool for the fan, one spool for the core). Some turbofans might have additional spools for even higher efficiency.
The working principles are very similar to basic jet engines: the incoming air is pulled in by the engine inlet. Some of it passes through the fan and continues on throughout compressor, combustor, turbine, and nozzle, identical to the process in a basic turbojet. The fan causes additional air to flow around (bypass) the engine. This produces greater thrust and reduces specific fuel consumption. Therefore, a turbofan gets some of its thrust from the core jet engine and some from the fan. The ratio between the air mass that flows around the engine and the air mass that goes through the core is called the bypass ratio.
Figure 6.14: Turbofan.
There are two types of turbofans: high bypass and low bypass, as illustrated in Figure 6.14. High bypass turbofans have large fans in front of the engine and are driven by a fan turbine located behind the primary turbine that drives the main compressor. Low bypass turbofans permit a smaller area and thus are more suitable for supersonic regime. A turbofan is very fuel efficient. Indeed, high bypass turbofans are nearly as fuel efficient as turboprops at low speeds. Moreover, because the fan is embedded in the inlet, it operates more efficiently at high subsonic speeds than a propeller. That is why turbofans are found on high-subsonic transportation (typical commercial aircraft) and propellers are used on low-speed transports (regional aircraft).
6.3.03: Turboprops
Figure 6.15: Turboprop engines.
Figure 6.16: A statically mounted Pratt & Whitney J58 engine with full after-burner. Wikimedia Commons / Public Domain.
Many regional aircraft use turboprop engines. There are two main parts in a turboprop engine: the core engine and the propeller. The core engine is very similar to a basic turbojet except that instead of expanding all the hot exhaust gases through the nozzle to produce thrust, most of this energy is used to turn the turbine. The shaft drives the propeller through gear connections and produces most of the thrust (similarly to a propeller). Figure 6.15 illustrates a turboprop.
The thrust of a turboprop is the sum of the thrust of the propeller and the thrust of the core, which is very small. Propellers become less efficient as the speed of the aircraft increases. Thus, turboprops are only used for low subsonic speed regimes aircraft. A variation of the turboprop engine is the turboshaft engine. In a turboshaft engine, the gearbox is not connected to a propeller but to some other drive device. Many helicopters use turboshaft engines.
6.3.04: After-burning turbojet
Modern fighter aircraft typically mount an after-burner. Other alternatives are either a low bypass turbofan or a turbojet. The explanation behind this is that fighters typically need extra thrust to perform sharp maneuvers and fulfill its mission. The after-burner is essentially a long tailpipe into which additional fuel is sprayed directly into the hot exhaust and burned to provide extra thrust. When the after-burner is turned off, the engine performs as a basic turbojet. The exhaust velocity is increased compared to that with after-burner off because higher temperatures are involved.
6.4: References
[1] Jenkinson, L. R., Simpkin, P., Rhodes, D., Jenkison, L. R., and Royce, R. (1999). Civil jet aircraft design, volume 7. Arnold London.
[2] Mattingly, J. D., Heiser, W. H., and Pratt, D. T. (2002). Aircraft engine design. AIAA.
[3] Newman, D. (2002). Interactive aerospace engineering and design. McGraw-Hill.
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Consider the following reference frames:1
Definition 7.1 (Earth Reference Frame)
An Earth reference frame $F_e(O_e, x_e, y_e, z_e)$ is a rotating topocentric (measured from the surface of the Earth) system. The origin Oe is any point on the surface of Earth defined by its latitude θe and longitude λe. Axis ze points to the center of Earth; xe lays in the horizontal plane and points to a fixed direction (typically north); ye forms a right-handed thrihedral (typically east).
Such system it is sometimes referred to as navigational system since it is very useful to represent the trajectory of an aircraft from the departure airport.
Definition 7.2 (Wind Axes Frame)
A wind axes frame $F_w (O_w, x_w, y_w, z_w)$ is linked to the instantaneous aerodynamic velocity of the aircraft. It is a system of axes centered in any point of the symeetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis $x_w$ points at each instant to the plane of symmetry, perpendicular to $x_w$ and pointing down according to regular aircraft performance. Axis $y_w$ forms a right-handed thrihedral.
Orientation angles
There exist several angles used in flight mechanics to orientate the aircraft with respect to a determined reference. The most important ones are:
• Sideslip angle, $\beta$, and angle of attack, $\alpha$: The angles of the aerodynamic velocity, $\vec{V}$, (wind axes reference frame) with respect the body axes reference frame.
• Roll, $\mu$, pitch, $\gamma$, and yaw, $\chi$, velocity angles: The angles of the wind axes reference frame with respect of the Earth reference frame. This angles are also referred to as bank angle, flight path angle, and heading angle.
1. Please, refer to Section 2.4 and/or Appendix A for a more detailed definition of the different reference frames.
7.1.02: Reference frames
Consider also the following hypotheses:
Hypothesis 7.1 Flat Earth model:
The Earth can be considered flat, non rotating, and approximate inertial reference frame.
Figure 7.1: Wind axes reference frame.
Hypothesis 7.2 Constant gravity
The acceleration due to gravity in atmosphere flight of an aircraft can be considered constant (\(g = 9.81[m/s^2]\)) and perpendicular to the surface of Earth.
Hypothesis 7.3 Moving Atmosphere
Wind is taken into account. Vertical component is neglected due its low influence. Only kinematic effects are considered, i.e., dynamic effects of wind are also neglected due its low influence.
Hypothesis 7.4 6-DOF model
The aircraft is considered as a rigid solid with six degrees of freedom, i.e., all dynamic effects associated to elastic deformations, to degrees of freedom of articulated subsystems (flaps, ailerons, etc.), or to the kinetic momentum of rotating subsystems (fans, compressors, etc ), are neglected.
Hypothesis 7.5 Point mass model
The translational equations are uncoupled from the rotational equations by assuming that the airplane rotational rates are small and that control surface deflections do not affect forces. This leads to consider a 3 Degree Of Freedom (DOF) dynamic model that describes the point variable-mass motion of the aircraft.
Hypothesis 7.6 Fixed engines
We assume the aircraft is a conventional jet airplane with fixed engines.
Hypothesis 7.7 Variable mass
The aircraft is modeled as variable mass particle.
Hypothesis 7.8 Forces acting on an aircraft
The external actions acting on an aircraft can be decomposed, without loss of generality, into propulsive, aerodynamic, and gravitational.
Hypothesis 7.9 Symmetric flight
We assume the aircraft has a plane of symmetry, and that the aircraft flies in symmetric flight, i.e., all forces act on the center of gravity and the thrust and the aerodynamic forces lay on the plane of symmetry.
Hypothesis 7.10 Small thrust angle of attack
We assume the thrust angle of attack is small.
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3D motion3
Under Hypotheses 7.1-7.10, the 3DOF equations governing the translational 3D motion of an airplane are the following:
• 3 dynamic equations relating forces to translational acceleration.
• 3 kinematic equations giving the translational position relative to an Earth reference frame.
• 1 equation defining the variable-mass characteristics of the airplane versus time.
The equation of motion is hence defined by the following Ordinary Differential Equations (ODE) system:
Definition 7.3 (3DOF equations of 3D motion)
$m \dot{V} = T - D - mg \sin \gamma;$
$m V \dot{\chi} \cos \gamma = L \sin \mu;$
$m V \dot{\gamma} = L \cos \mu - mg \cos \gamma;$
$\dot{x}_e = V \cos \gamma \cos \chi + W_x;$
$\dot{y}_e = V \cos \gamma \sin \chi + W_y;$
$\dot{h}_e = V \sin \gamma;$
$\dot{m} = -T \eta.$
Figure 7.2: Aircraft forces.
Where in the above:
• the three dynamics equaitions are expressed in an aircraft based reference frame, the wind axes system $F_w (O, x_w, y_w, z_w)$, usually $x_w$ coincident with the velocity vector.
• the three kinematic equations are expressed in a ground based reference frame, the Earth reference frame $F_e (O_e, x_e, y_e, z_e)$ and are usually referred to as down range (or longitude), cross range (or latitude), and altitude, respectively.
• $x_e, y_e$ and $h_e$ denote the components of the center of gravity of the aircraft, the radio vector $\vec{r}$, expressed in an Earth reference frame $F_e (O_e, x_e, y_e, z_e)$.
• $W_x$, and $W_y$ denote the components of the wind, $\vec{W} = (W_x, W_y, 0)$, expressed in an Earth reference frame $F_e (O_e, x_e, y_e, z_e)$.
• $\mu, \chi$, and $\gamma$ are the bank angle, the heading angle, and the flight-path angle, respectively.
• $m$ is the mass of the aircraft and $\eta$ is the specific fuel consumption.
• $g$ is the acceleration due to gravity.
• $V$ is the true air speed of the aircraft.
• $T$ is the engines' thrust, the force generated by the aircraft's engines. It depends on the altitude $h$, Mach number $M$, and throttle $\pi$ by an assumedly known relationship $T = T(h, M, \pi)$.
• lift, $L = C_L S \hat{q}$, and drag, $D = C_D S \hat{q}$ are the components of the aerodynamic force, where $C_L$ is the dimensionless coefficient of lift and $C_D$ is the dimensionless coefficient of drag, $\hat{q} = \tfrac{1}{2} \rho V^2$ is referred to as dynamic pressure, $\rho$ is the air density, and $S$ is the wet wing surface. $C_L$ is, in general, a function of the angle of attack, Mach and Reynolds number: $C_L = C_L (\alpha, M, \text{Re})$. $C_D$ is, in general, a function of the coefficient of lift: $C_D = C_D (C_L (\alpha, M, \text{Re}))$.
Additional assumptions are:
Hypothesis 7.11 Parabolic drag polar
A parabolic drag polar is assumed, $C_D = C_{D_0} + C_{D_i} C_L^2$.
Hypothesis 7.12 Standard atmosphere model
A standard atmosphere is defined with $\Delta_{ISA} = 0$.
Vertical motion
Considerer the additional hypothesis for a symmetric flight in the vertical plane:
Hypothesis 7.13 Vertical motion
• $\chi$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $\mu = 0$.
• There are no actions out of the vertical plane, i.e., $W_y = 0$.
Definition 7.4 (3DOF equations of vertical motion)
The 3DOF equations governing the translational vertical motion of an airplane is given by the following ODE system:
$m \dot{V} = T - D - mg \sin \gamma,$
$m V \dot{\gamma} = L - mg \cos \gamma,$
$\dot{x}_e = V \cos \gamma \cos \chi + W_x,$
$\dot{h}_e = V \sin \gamma,$
$\dot{m} = -T \eta.$
Horizontal motion
Considerer the additional hypothesis for a symmetric flight in the horizontal plane:
Hypothesis 7.14 Horizontal motion
We consider flight in the horizontal plane, i.e., $\dot{h}_e = 0$ and $\gamma = 0$.
Definition 7.5 (3DOF equations of horizontal motion)
The 3DOF equations governing the translational horizontal motion of an airplane is given by the following ODE system:
$m \dot{V} = T - D$
$m V \dot{\chi} = L\sin \mu,$
$0 = V \cos \mu - mg,$
$\dot{x}_e = V \cos \chi + W_x,$
$\dot{y}_e = V \sin \chi + W_y,$
$\dot{m} = -T \eta.$
3. The reader is encouraged to read Appendix A for a better understanding.
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Consider the additional hypotheses:
• Consider a symmetric flight in the horizontal plane.
• $\chi$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $\mu = 0$.
• There is no wind.
• The mass and the velocity of the aircraft are constant.
The 3DOF equations governing the motion of the airplane are:4
$T = D,\label{eq7.1.4.1}$
$L = mg, (which\ implies \ n = 1),\label{eq7.1.4.2}$
$\dot{x}_e = V,\label{eq7.1.4.3}$
Recall the following expressions already exposed in Chapter 3:
• $L = \tfrac{1}{2} \rho SV^2 C_L (\alpha ); C_L = C_{L_0} + C_{L_{\alpha}} \alpha,$,
• $D = \tfrac{1}{2} \rho SV^2 C_D (\alpha ); C_D = C_{D_0} + k C_L^2$,
• $E = \tfrac{L}{D} = \tfrac{C_L}{C_D} = \tfrac{C_L}{C_{D_0} + k C_L^2}$, with $E_{\max} = \tfrac{1}{2\sqrt{C_{D_0} k}}$.5
Considering these expressions, System of equations ($\ref{eq7.1.4.1}$), ($\ref{eq7.1.4.2}$) and ($\ref{eq7.1.4.3}$) can be expressed as:
$T = \dfrac{1}{2} \rho S V^2 C_{D_0} + \dfrac{2k(mg)^2}{\rho SV^2},\label{eq7.1.4.4}$
$mg = \dfrac{1}{2} \rho SV^2 (C_{L_0} + C_{L_{\alpha}} \alpha ),\label{eq7.1.4.5}$
$\dot{x}_e = V.$
Expression ($\ref{eq7.1.4.5}$) says that in order to increase velocity it is necessary to reduce the angle of attack and vice-versa. Expression ($\ref{eq7.1.4.4}$) gives the two velocities at which an aircraft can fly for a given thrust.
4. $n = \tfrac{L}{mg}$ is referred to as load factor
5. remember that $E_{\max}$ refers to the maximum efficiency.
7.1.05: Performances in a steady linear flight
Consider the additional hypotheses:
• Consider a symmetric flight in the vertical plane.
• $\chi$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $\mu = 0$.
• There is no wind.
• The mass, the velocity, and the flight path angle of the aircraft are constant.
The 3DOF equations governing the motion of the airplane are:
$T = D + mg \sin \gamma , \label{eq7.1.5.1}$
$L = mg \cos \gamma,\label{eq7.1.5.2}$
$\dot{x}_e = V \cos \gamma \cos \chi ,$
$\dot{h}_e = V \sin \gamma,\label{eq7.1.5.4}$
Typically, commercial and general aviation aircraft have a relation $T/(mg)$ so that flight path angles are small $(\gamma \ll 1)$. Therefore, Expression ($\ref{eq7.1.5.1}$) can be expressed as
$\gamma \cong \dfrac{T - D}{mg},$
and Expression ($\ref{eq7.1.5.2}$) can be expressed as
$L \cong mg, \to n \cong 1.$
Therefore the flight path angle can be controlled by means of the power plant thrust.
Another important characteristic in ascent (descent) flight is the Rate Of Climb (ROC), which is given by Expression ($\ref{eq7.1.5.4}$) as:
$V_{ROC} = \dfrac{dh_e}{dt} = V \sin \gamma .$
7.1.06: Performances in steady ascent and descent filght
In all generality, a glider is an aircraft with no thrust. In stationary linear motion in vertical plane, the equations are as follows:
$D = mg \sin \gamma,$
$L = mg \cos \gamma,$
and dividing:
$\tan \gamma_d = \dfrac{D}{L} = \dfrac{C_D}{C_L} = \dfrac{1}{E(\alpha)},$
Figure 7.3: Aircraft forces in a horizontal loop.
where $\gamma_d$ is the descent path angle $(\gamma_d = -\gamma)$. As in stationary linear-horizontal flight, in order to increase the velocity of a glider it is necessary to reduce the angle of attack. Moreover, the minimum gliding path angle will be obtained flying with the maximum aerodynamic efficiency. The descent velocity of a glider $(V_d)$ can be defined as the loss of altitude with time, that is:
$V_d = V\sin \gamma_d \cong V \gamma_d.$
7.1.07: Performances in gliding
Horizontal stationary turn
Consider the additional hypotheses:
• Consider a symmetric flight in the horizontal plane.
• There is no wind.
• The mass and the velocity of the aircraft are constant.
The 3DOF equations governing the motion of the airplane are:
$T = D,\label{eq7.1.7.1}$
$m V \dot{\chi} = L \sin \mu ,\label{eq7.1.7.2}$
$L \cos \mu = mg,\label{eq7.1.7.3}$
$\dot{x}_e = V \cos \chi, \label{eq7.1.7.4}$
$\dot{y}_e = V \sin \chi.\label{eq7.1.7.5}$
In a uniform (stationary) circular movement, it is well known that the tangential velocity is equal to the angular velocity ($\dot{\chi}$) multiplied by the radius of turn $(R)$:
$V = \dot{\chi} R.$
Therefore, System ($ref{eq7.1.7.1}$, $ref{eq7.1.7.2}$, $ref{eq7.1.7.3}$, $ref{eq7.1.7.4}$, $ref{eq7.1.7.5}$) can be rewritten as:
$T = \dfrac{1}{2} \rho SC_{D_0} + \dfrac{2kn^2 (mg)^2}{\rho V^2 S},$
$n \sin \mu = \dfrac{V^2}{gR},$
$n = \dfrac{1}{\cos \mu} \to n > 1,$
$\dot{x}_e = V \cos \chi,$
$\dot{y}_e = V \sin \chi.$
where $n = \dfrac{L}{mg}$ is the load factor. Notice that the load factor and the bank angle are $mg$ inversely proportional, that is, if one increases the other reduces and vice versa, until the bank angle reaches $90^{\circ}$, where the load factor is infinity.
The stall speed in horizontal turn is defined as:
$V_S = \sqrt{\dfrac{2mg}{\rho S C_{L_{\max}}} \dfrac{1}{\cos \mu}}$
Ideal looping
The ideal looping is a circumference of radius R into a vertical plane performed at constant velocity. Consider then the following additional hypotheses:
• Consider a symmetric flight in the vertical plane.
• $\chi$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $\mu = 0$.
• There is no wind.
• The mass and the velocity of the aircraft are constant.
The 3DOF equations governing the motion of the airplane are:
$T = D + mg \sin \gamma,$
$L = mg \cos \gamma + mV \dot{\gamma},$
$\dot{x}_e = V \cos \gamma,$
$\dot{h}_e = V \sin \gamma,$
Figure 7.4: Aircraft forces in a vertical loop.
In a uniform (stationary) circular movement, it is well known that the tangential velocity is equal to the angular velocity ($\dot{\gamma}$ in this case) multiplied by the radius of turn ($R$):
$V = \dot{\gamma} R.$
The load factor and the coefficient of lift in this case are:
$n = \cos \gamma + \dfrac{V^2}{gR},$
$C_L = \dfrac{2mg}{\rho V^2 S} (\cos \gamma + \dfrac{V^2}{gR}).$
Notice that the load factor vaires in a sinusoidal way along the loop, reaching a maximum value at the superior point ($n_{\max} = 1 + \tfrac{V^2}{gR})$ and a minimum value at the inferior point ($n_{\min} = \tfrac{V^2}{gR} - 1$).
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After analyzing the performances of an aircraft in the air, we will analyze the performances of an aircraft while taking off and landing.
Figure 7.5: Take off distances and velocities.
Take off
The take off is defined as the maneuver covering those phases from the initial acceleration at the runway’s head until the aircraft reaches a prescribed altitude and velocity (defined by the aeronavegability norms). This maneuver is performed with maximum thrust, deflected flaps, and landing gear down.
We can divide the maneuver in two main phases:
1. Rolling in the ground ($0 \le V \le V_{LOF}$): From the initial acceleration to the velocity of take off ($V_{LOF}$), when the aircraft does not touch the runway.
(a) Rolling with all the wheels in the ground ($0 \le V \le V_R$): The aircraft takes off rolling with all the wheels in the ground until it reaches a velocity called rotational velocity, $V_R$.
(b) Rolling with the aft wheels in the ground ($V_R \le V \le V_{LOF}$): At $V_R$ the nose rotates upwards and the aircraft keeps rolling but now just with the aft wheels in the ground.
2. Path in the air ($V_{LOF} \le V \le V_2$): From the instant in which the aircraft does not touch the runway to the instant in which the aircraft reaches a velocity $V_2$ at a given altitude $h$ (such altitude is usually defined as $h = 35\ ft(10.7\ m)$).
(a) Track of curve transition $(V \approx V_{LOF})$: The aircraft needs a transition until it reaches the desired ascent flight path angle.
(b) Straight accelerated track $(V_{LOF} \le V \le V_2)$: The aircraft accelerates with constant flight path angle until it reaches $V_2$ at a given altitude $h$.
Figure 7.6: Forces during taking off
Of all the above, we are interested on analyzing sub-phase 1.(a). In order to get approximate numbers of taking off distances and times, let us assume the aircraft performs a uniform accelerated movement and the only force is thrust $T$. According to Newton's second law:
$ma = T.$
The acceleration is $a = \tfrac{dV}{dt}$. Then:
$V = \int \dfrac{T}{m} dt = \dfrac{T}{m} t.$
If we make the integral defined between $t = 0$ and $t = t_{TO}$, and $V = 0$ and $V = V_{TO}$:
$V_{TO} = \int_{0}^{t_{TO}} \dfrac{T}{m} dt = \dfrac{T}{m} t_{TO} \to t_{TO} = \dfrac{V_{TO} m}{T}.$
The velocity is $V = \tfrac{dx}{dt}$. Then:
$x = \int \dfrac{T}{m} t dt = \dfrac{T}{m} \dfrac{t^2}{2}.$
If we make the integral defined between $t = 0$ and $t = t_{TO}$ and $x = 0$ and $x = x_{TO}$:
$x_{TO} = \int_{0}^{t_{TO}} \dfrac{T}{m} t dt = \dfrac{T}{m} \dfrac{t_{TO}^2}{2} \to x_{TO} = \dfrac{V_{TO}^2 m}{2T}.$
Landing
Figure 7.7: Landing distances and velocities.
The landing is defined as the maneuver covering those phases starting from a prescribed altitude (defined by the aeronavegability norms, typically $h = 50\ ft$ and $V_A = 1.3 V_S$) until the aircraft stops (to be more precise, when the aircraft reaches a constant taxiing velocity). This maneuver is performed with minimum thrust, deflected flaps, and landing gear down.
We can divide the maneuver in two main phases:
1. Path in the air ($V_A \ge V \le V_{Touch}$): From the instant in which the aircraft reaches a prescribed altitude performing a steady descent to the instant the aircraft touches down.
(a) Final approach: it consist in a steady straight trajectory at a velocity typically 1.3 the stall velocity of the aircraft in the landing configuration.
(b) Transition: The aircraft performs a transition between the straight trajectory to the horizontal plane of the runway. It can be supposed as a circumference. This transition is performed at a touchdown velocity $V_{touch} \approx 1.15 V_S$.
2. Rolling in the ground ($V_{Touch} \ge V \ge 0$): From the instant of touchdown to the instant in which stops:
(a) Rolling with the aft wheels in the ground: the nose rotates downwards and the aircraft rolls but now just with the aft wheels in the ground.
(b) Rolling with all the wheels in the ground: the aircraft keeps rolling with all the wheels in the ground until it stops.
The equations of sub-phase 2.(b) are basically the same as the equations for sub-phase 1.(a) in taking off. The only differences are that thrust is minimum, zero, o even negative (in reverse gear aircrafts), drag is maximized deflecting the spoiler; the coefficient of friction is much higher due to break and downforce effects. The kinematic analysis for distances and times follow the same patterns as for take off: the movement can be considered herein uniformly decelerated.
7.1.09: Performances in the runway
In this section, we study the range and endurance for an aircraft flying a steady, linear-horizontal flight.
• The range is defined as the maximum distance the aircraft can fly given a quantity of fuel.
• The endurance is defined as the maximum time the aircraft can be flying given a quantity of fuel.
Considerer the additional hypotheses:
• Consider a symmetric flight in the horizontal plane.
• $\chi$ can be considered constant.
• The aircraft performs a leveled wing flight, i.e., $\mu = 0$.
• There is no wind.
• The velocity of the aircraft is constant.
The 3DOF equations governing the motion of the airplane are:
$T = D,\label{eq7.1.9.1}$
$L = mg,\label{eq7.1.9.2}$
$\dot{x}_e = V,\label{eq7.1.9.3}$
$\dot{m} = -\eta T.\label{eq7.1.9.4}$
Equation ($\ref{eq7.1.9.4}$) means that the aircraft losses weight as the fuel is burt, where $\eta$ is the specific fuel consumption. Notice that Equation ($\ref{eq7.1.9.4}$) is just valid for jets.
The specific fuel consumption is defined in different ways depending of the type of engines:
• jets: $\eta_j = \tfrac{-dm/dt}{T}$.
• Propellers: $\eta_p = \tfrac{-dm/dt}{P_m} = \tfrac{-dm/dt}{TV}$, where $P_m$ is the mechanical power.
Focusing on jet engines, operating with Equation ($\ref{eq7.1.9.1}$, $\ref{eq7.1.9.2}$, $\ref{eq7.1.9.3}$, $\ref{eq7.1.9.4}$), considering $E = L/D$, and taking into account the initial state $(\cdot)_i$, and the final state $(\cdot)_f$ we obtain the distance and time flown as:
$x_e = -\int_{m_i}^{m_f} \dfrac{V}{\eta_j T} dm = -\int_{m_i}^{m_f} \dfrac{1}{\eta_j g} VE \dfrac{dm}{m},$
$t = -\int_{m_i}^{m_f} \dfrac{1}{\eta_j T} dm = -\int_{m_i}^{m_f} \dfrac{1}{\eta_j g} E \dfrac{dm}{m},$
In order to integrate such equations we need to make additional assumptions, such for instance consider constant specific fuel consumption and constant aerodynamic efficiency (remember that the velocity has been already assumed to be constant).
Range and endurance (maximum distance and time, respectively) are obtained assuming the aircraft flies with the maximum aerodynamic efficiency (given the weights of the aircraft and given also that for a weight there exists an optimal speed):
$x_{e\ \max} = \dfrac{1}{\eta_j g} VE_{\max} \ln \dfrac{m_i}{m_f},$
$t_{\max} = \dfrac{1}{\eta_j g} E_{\max} \ln \dfrac{m_i}{m_f},$
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Weights of the aircraft
Let us start defining the different weights of the aircraft:
• $OEW$: The Operating Empty Weight is the basic weight of an aircraft including the crew, all fluids necessary for operation such as engine oil, engine coolant, water, unusable fuel and all operator items and equipment required for flight but excluding usable fuel and the payload. Also included are certain standard items, personnel, equipment, and supplies necessary for full operation.
• $PL$: The Payload is the load for what the company charges a fee. In transportation aircraft it corresponds to the passenger and its luggage, together with the cargo.
• $FW$: The Fuel Weight of an aircraft is the total weight of fuel carried at take off and it is calculated adding the following two weights:
1. $TF$: Trip Fuel is the total amount of fuel estimated to be consumed in the trip.
2. $RF$: Reserve Fuel is the weight of fuel to allow for unforeseen circumstances, such as an inaccurate weather forecast, alternative arrival airports, etc.
• $TOW$: The TakeOff weight of an aircraft is the weight at which the aircraft takes off. $TOW = OEW + PL + FW$.
• $LW$: The Landing Weight of an airplane is the total weight of the airplane at destination with no use of reserve fuel. $LW = OEW + PL + RF$.
• $ZFW$: the Zero Fuel Weight of an airplane is the total weight of the airplane and all its contents, minus the total weight of the fuel on board. $ZFW = OEW + PL$.
Figure 7.8: Take-off weight components. © Mohsen Alshayef / Wikimedia Commons / CC-BY-SA-3.0.
Limitation on the weight of an aircraft
Due to different features, such structural limits, capacity of tanks, or capacity of passengers and cargo, some of the weights have limitations:
1. $MPL$: The Maximum PayLoad of an aircraft is limited due to structural limits and capacity constraints.
2. $MFW$: The Maximum Fuel Weight is the maximum weight of fuel to be carried and it is limited by the capacity of tanks.
3. $MZFW$: The Maximum Zero Fuel Weight is the maximum weight allowed before usable fuel and other specified usable agents (engine injection fluid, and other consumable propulsion agents) must be loaded in defined sections of the aircraft as limited by strength and airworthiness requirements. It may include usable fuel in specified tanks when carried instead of payload. The addition of usable and consumable items to the zero fuel weight must be in accordance with the applicable government regulations so that airplane structure and airworthiness requirements are not exceeded.
4. $MTOW$: The Maximum Takeoff Weight of an aircraft is the maximum weight at which the pilot of the aircraft is allowed to attempt to take off due to structural or other limits.
5. $MLW$: The Maximum Landing Weight of an aircraft is the maximum weight at which the pilot of the aircraft is allowed to attempt to land due to structural or other limits. In particular, due to structural limits in the landing gear.
Payload-range diagram
A payload range diagram (also known as the elbow chart) illustrates the trade-off between payload and range. The top horizontal line represents the maximum payload. It is limited structurally by maximum zero fuel weight ($MZFW$) of the aircraft. Maximum payload is the difference between maximum zero-fuel Weight and operational empty weight ($OEW$). Moving left-to-right along the line shows the constant maximum payload as the range increases. More fuel needs to be added for more range.
Weight in the fuel tanks in the wings does not contribute as significantly to the bending moment in the wing as does weight in the fuselage. So even when the airplane has been loaded with its maximum payload that the wings can support, it can still carry a significant amount of fuel.
Figure 7.9: Payload-range diagram.
The vertical line represents the range at which the combined weight of the aircraft, maximum payload and needed fuel reaches the maximum take-off weight ($MTOW$) of the aircraft. See point A in Figure 7.9. If the range is increased beyond that point, payload has to be sacrificed for fuel.
The maximum take-off weight is limited by a combination of the maximum net power of the engines and the lift/drag ratio of the wings. The diagonal line after the range-at- maximum-payload point shows how reducing the payload allows increasing the fuel (and range) when taking off with the maximum take-off weight. See point B in Figure 7.9.
The second kink in the curve represents the point at which the maximum fuel capacity is reached. Flying further than that point means that the payload has to be reduced further, for an even lesser increase in range. See point C in Figure 7.9. The absolute range is thus the range at which an aircraft can fly with maximum possible fuel without carrying any payload.
In order to relate the ranges with weights we can use to so-called Breguet equation:
$R = \dfrac{1}{g\eta_j} VE \ln \dfrac{TOW}{LW},$
For the three marked points, respecitvely $A$, $B$ and $C$:
$R_A = \dfrac{1}{g\eta_j} VE \ln \dfrac{MTOW}{OEW + MPL + RF},$
$R_B = \dfrac{1}{g\eta_j} VE \ln \dfrac{MTOW}{MTOW - MFW + RF},$
$R_C = \dfrac{1}{g\eta_j} VE \ln \dfrac{OEW + MFW}{OEW + RF},$
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Mechanics of atmospheric flight studies aircraft performances, that is, the movement of the aircraft in the air in response to external forces and torques, and the stability and control of the aircraft’s movement, analyzing thus the rotational movement of the aircraft. The study of performances, stability, and control plays a major role in verifying the design requirements. For instance, one must be able to analyze the required power for cruise flight, the required power settings and structural design for climbing with a desired angle at a desired velocity, the range and autonomy of the aircraft, the distances for taking off and landing, the design for making the aircraft stable under disturbances, and so on and so forth. The reader is referred to Franchini and García [3] and Anderson [1] as introductory references. Appendix A complements the contents of this chapter.
07: Mechanics of flight
In Section 7.1 we have studied the performances of the aircraft, modeling the aircraft as a 3DOF solid and studying the point mass model movement due to external actions. In this section we study the fundamentals of stability and control, considering the aircraft as a 6DOF6 model, so that we must take into consideration the geometric dimensions, the distribution of mass, and thus we study the external forces and torques which define the movement of the center of gravity and the orientation and angular velocity of the aircraft.
Figure 7.10: Diagram showing the three main cases for aircraft pitch static stability, following a pitch disturbance: Aircraft is statically stable (corrects attitude); Aircraft is statically neutral (does not correct attitude); Aircraft is statically unstable (exacerbates attitude disturbance). © User:Ariadacapo / Wikimedia Commons / CC0-1.0.
6. Please refer to Appendix A for the deduction of the 6-DOF equations.
7.02: Stability and control
A vehicle is said to be in equilibrium when remains constant or the movement is uniform, that is, both the linear and angular quantity of movement are constant. In the case of an aircraft, the state of equilibrium typically refers to an uniform movement in which the angular velocities are null, and then the movement is simply a translation.
The stability is a property related with the state of equilibrium, which studies the behavior of the aircraft when any of the variables describing its state of equilibrium suffers from a variation (for instance a wind gust). The variation in one of those variables is typically referred to as perturbation.
The stability can be studied in two different ways depending on the time scale:
• Static stability.
• Dynamic stability.
Static stability
The interest lies on the instant of time immediately after the perturbation takes place. In the static stability, the forces and torques which appear immediately after the perturbation are studied. If the value of the state variables describing the equilibrium tends to increase or amplify, the state of equilibrium is statically unstable. On the contrary, it is statically stable. Figure 7.10 shows a sketch of an aircraft with three different options for the static stability after a pitch down disturbance, e.g., a downwards wind gust.
Dynamic stability
Figure 7.11: Diagram showing the three main cases for aircraft pitch dynamic stability. Here all three cases are for a statically stable aircraft. Following a pitch disturbance, three cases are shown: Aircraft is dynamically unstable (although statically stable); Aircraft is dynamically damped (and statically stable); Aircraft is dynamically overdamped (and statically stable). © User:Ariadacapo / Wikimedia Commons / CC0-1.0.
The dynamic stability studies the evolution with time of the different variables of flight (yaw, pitch, and roll angles, velocity, angular velocities, altitude, etc.) when the condition of equilibrium is perturbed. Figure 7.11 shows a sketch of an aircraft with different dynamic-stability behaviors after a pitch down disturbance, e.g., a downwards wind gust. In order to study this evolution we need to solve the system of equation describing the 6DOF movement of the aircraft. As illustrated in Appendix A, the equations are:
$\vec{F} = m \dfrac{d\vec{V}}{dt},$
$\vec{G} = l \dfrac{d\vec{\omega}}{dt}.$
An aircraft is said to be dynamically stable when after a perturbation the variables describing the movement of the aircraft tend to a stationary value (either the same point of equilibrium or a new one). On the contrary, if the aircraft does not reach an equilibrium it is said to be dynamically unstable.
Figure 7.12: Feedback loop to control the dynamic behavior of the system: The sensed value is subtracted from the desired value to create the error signal, which is amplified by the controller. © User:Myself / Wikimedia Commons / CC-BY-SA-3.0.
As the reader might imagine, aircraft are designed to be dynamically stable. Indeed, advanced flight mechanics studies (out of the scope of this book) will focus on the analysis of aircraft stability and control. A typical analysis would consist in: the linearization of the 6DoF equations (for which one needs the so-called stability derivatives); the characterization of the flight modes (for which one would uncouple longitudinal and lateral dynamics and calculate eigenvalues and eigenvectors), i.e., short period, phugoid, spiral, rolling convergence and Dutch roll; the analysis of open loop response (uncontrolled aircraft motion subject to a control actions -a perturbation-, e.g., a throttle step), which would end up activation some of the flight modes; and finally the closed-loop control response (controlled motion of the aircraft), which consists in the fundamentals of an autopliot design (one sets a reference value to track and the control system is designed to correct measured/estimated deviations from it). The reader is referred to Etkin and Reid [2] for a thorough course on this matter.
7.2.02: Fundamentals of control
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The external input of a system is called the reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. The usual objective of control theory is to calculate solutions for the proper corrective action from the controller that result in system stability, that is, the system will hold the reference state values and not oscillate around them.
The input and output of the system are related to each other by what is known as a transfer function (also known as the system function or network function). The transfer function is a mathematical representation, in terms of spatial or temporal frequency, of the relation between the input and output of a linear time-invariant system.
Example
Consider an aircraft’s autopilot, with one of its functionalities being to maintain altitude at a reference value provided by the pilot. The controller is the autopilot, the plant is the aircraft (the equations of motion), and the system is together the aircraft and the autopilot.
The system output is the aircraft’s altitude, and the control itself is the pitching which determines the deflection of the elevator needed.
In a closed-loop control system, a sensor monitors the system output (the aircraft’s altitude, in this case the barometric altitude) and feeds back to a controller that adjusts the control (the elevator) to maintain the desired system output (the reference altitude). Now when the aircraft flies above the desired altitude (there is an error measured/estimated by the aircraft), the elevator position changes to pitch down, speeding the vehicle, and descending to the desired value. Feedback from measuring the aircraft’s altitude has allowed the controller to dynamically compensate for changes to the altitude.
7.2.03: Longitudinal balancing
The longitudinal balancing is the problem of determining the state of equilibrium of a longitudinal movement in which the lateral and directional variables are considered uncoupled. For the longitudinal analysis, one must consider forces on $z$-axis ($F_z$) and torques around $y$-axis ($M_y$). Generally, it is necessary to consider external actions coming from aerodynamics, propulsion, and gravity. However, it is common to consider only the gravity and the lift forces in wing and horizontal stabilizer. Additional hypotheses include: no wind; mass and velocity are constant.
The equations to be fulfilled are:
$\sum F_z = 0,$
$\sum M_y = 0.$
Which results in
$mg - L - L_t = 0,$
$-M_{ca} + Lx_{cg} - L_t l = 0,$
where $L_t$ is the lift generated by the horizontal stabilizer, $M_{ca}$ is the pitch torque with respect to the aerodynamic center, $x_{cg}$ is the distance between the center of gravity and the aerodynamic center, and $l$ is the distance between the center of gravity and the aerodynamic center of the horizontal stabilizer.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/07%3A_Mechanics_of_flight/7.02%3A_Stability_and_control/7.2.01%3A_Fundamental_of_stability.txt
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Longitudinal static stability
Consider an aircraft in horizontal, steady, linear flight. The aircraft is in equilibrium under equations (7.2.3.3)-(7..2.3.4).
Figure 7.13: Longitudinal equilibrium. Adapted from Franchini and García [3].
Consider now that such equilibrium is perturbed by a vertical wind gust, so that the angle of attack increases, that is, there is a perturbation in the angle of attack. In this case, both $L$ and $L_t$ increase according to the lift-angle of attack curves (we assume that the behavior for the horizontal stabilizer is similar to the one for the wing7) so that we have $L + \Delta L$ and $L_t + \Delta L_t$. If $\Delta L_t l > \Delta L$, then the angle of attack tends to decrease and the aircraft is statically stable. On the contrary, the aircraft is statically unstable. In other words, the torque generated by the horizontal stabilizer is stabilizer (so the name). Obviously, this also depends on the relative position of the aerodynamic centers with respect to the center of gravity of the aircraft8. Therefore, the aerodynamic design is also key to determine the static stability.
The external longitudinal moments acting on the center of gravity can be made dimensionless as follows:
$\dfrac{\sum M_{cg, y}}{\tfrac{1}{2} \rho SV_{\infty}^2 \bar{c}} = c_{M, cg},$
where $\sum M_{cg, y} = -M_{ca} + Lx_{cg} - L_t l$ and $\bar{c}$ is the mean chord of the aircraft. The dimensionless equation is:
$c_{M, cg} = c_{M0} + c_{M\alpha} \alpha + c_{M \delta_e} \delta_e,\label{eq7.2.4.2}$
where $c_{M, cg}$ is the coefficient of moments of the aircraft with respect to its center of gravity, $c_{M0}$ is the coefficient of moments independently of the angle of attack and the deflection of the elevator, $c_{M\alpha}$ is the derivative of the coefficient of moments of the aircraft with respect to the angle of attack, and $c_{M\delta_e}$ is the derivative of the coefficient of moments of the aircraft with respect to the deflection of the elevator. The coefficients of Equation ($\ref{eq7.2.4.2}$) $c_{M0}$, $c_{M\alpha}$, and $c_{M\delta}$ depend on the geometry and the aerodynamics of the aircraft.
Figure 7.14: Longitudinal stability. Adapted from Franchini and García [3].
In Figure 7.14 the coefficient of moments of two aircraft as a function of the angle of attack is shown for a given center of gravity, a given aerodynamic center for wing and horizontal stabilizer, and a given deflection of the elevator. The intersection of the curves with the abscissa axis determine the angle of attack of equilibrium $\alpha_e$. Imagine that a perturbation appears, for instance a wind gust which decreases the angle of attack so that both aircrafts have an angle of attack $\alpha_1 < \alpha_e$. In the case or aircraft (a), the $c_{M,cg,b} > 0$ (the curve for $\alpha_1$ is above the abscissa axis), which means the moment tends to pitch up the aircraft so that it returns to the initial state of equilibrium: the aircraft is statically stable. In the case or aircraft (b), $c_{M,cg,b} < 0$ (the curve for $\alpha_1$ is below the abscissa axis) and the moment tends to pitch down the aircraft, so that it is statically unstable.
From this reasoning, we can conclude that for an aircraft to be statically stable it must be fulfilled:
$\dfrac{dc_{M,cg}}{d\alpha} = c_{M\alpha} < 0.$
$c_{M\alpha}$ depends, among other, of the center of gravity of the aircraft. Therefore, one of the key issues in the design of an aircraft is to determine the center of gravity to make the aircraft statically stable. This is not trivial, since the center of gravity varies during the flight (depends on the payload, varies as the fuel is burnt, etc.). Therefore, during a flight the pilot (or the autopilot in control systems) must modify the angle of attack to maintain the flight in the equilibrium (since $\alpha_e$ varies). As the center of gravity makes its way aft the angle of attack of equilibrium increases. There is a point for which $c_{M\alpha} = 0$, the neutral point. The center of gravity can not go back beyond this point by any means because $c_{M\alpha} > 0$ and the aircraft becomes statically unstable.
Longitudinal control
The coefficient $c_{M\delta_e}$ is referred to as the power of the longitudinal control and represents a measure of the capacity that the elevator has to generate a moment and, therefore, to change the angle of attack at which the aircraft can fly in equilibrium ($\alpha_e$).
An elevator's positive deflection $(\delta_e > 0)$ generates an increase in the horizontal stabilizer's lift ($\Delta L_t$), which gives rise to a negative pitch moment $M_{cg} < 0$ ($c_{M\delta_e} < 0$). In the condition of equilibrium, the sum of moments around the center of gravity is null, and therefore it must be fulfilled that
$c_{M, cg} = c_{M0} + c_{M\alpha} \alpha + c_{M\delta_e} \delta_e = 0.$
Two main problems can be derived in the longitudinal control:
1. Determine the deflection angle of the elevator, $\delta_e$, to be able to fly in equilibrium at a given angle of attack, $\alpha_e$:
$\delta_e = \dfrac{-c_{M0} - c_{M\alpha} \alpha_e}{c_{M\delta_e}}$
2. Determine the angle of attack to fly in equilibrium, $\alpha_e$, for a known deflection of the elevator, $\alpha_e$:
$\alpha_e = \dfrac{-c_{M0} - c_{M\delta_e} \delta_e}{c_{M\alpha}}$
Figure 7.15: Effects of elevator on moments coefficient. Adapted from Franchini and García [3].
Figure 7.15 shows the effects of the elevator’s deflection in the angle of attack of equilibrium. Simplifying, for a $\delta_e < 0$, the angle of attack of equilibrium at which the aircraft flies increases and so does the coefficient of lift. Since the lift must be equal to weight, the aircraft must fly slower. In other words, the elevator is used to modify the velocity of a steady horizontal flight.
As we have pointed out before, the geometric condition of the aircraft vary during the flight. Therefore it is necessary to re-calculate this conditions and modify the variables continuously. This is made using control systems.
7. To be more precise, it is necessary to derive a model which gives us the effective angle of attack of the stabilizer since the wing modifies the incident current. However, this will be studied in posterior courses.
8. remember that the aerodynamic center is the point at with the pitching moment does not vary with respect the increase in $C_L$.
7.2.05: Lateral-directional stability and control
Consider again an aircraft in horizontal, steady, linear flight. Suppose in this case that the lateral-directional elements (vertical stabilizer, rudder, ailerons) do not produce forces nor moments, so that there not exists a primary problem of balancing (as there was in the longitudinal case) since we have a longitudinal plane of symmetry.
In this case, the lateral-direction control surfaces (rudder and ailerons) fulfill a mission of secondary balancing since they are used when there exists an asymmetry (propulsive or aerodynamic). For instance, aircraft must be able to fly under engine failure, and thus the asymmetry must be compensated with the rudder. Another instance could be the landing operation under lateral wind, which must be also compensated with the rudder deflection. Notice that the center of gravity lays on the plane of symmetry, so that its position does not affect the lateral-directional control. Further mathematical analysis will be studied in posterior courses.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/07%3A_Mechanics_of_flight/7.02%3A_Stability_and_control/7.2.04%3A_Longitudinal_stability_and_control.txt
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Exercise $1$ Performances
Consider on Airbus A-320 with the following characteristics:
• $m = 64\ tonnes.$
• $S_w = 122.6\ m^2.$
• $C_D = 0.024 + 0.0375 C_L^2$.
1. The aircraft starts an ascent maneuver with uniform velocity at 10.000 feet of altitue (3048 meters). At that flight level, the typical performances of the aircraft indicate a velocity with respect to air of 289 knots ($148.67\ m/s$) and a rate of climb (vertical velocity) of $2760\ feet/min$ ($14\ m/s$). Assuming that $\gamma \ll 1$, calculate:
(a) The angle of ascent, $\gamma$.
(b) Required thrust at those conditions.
2. The aircraft reaches an altitude of $11000\ m$, and performs a horizontal, steady, straight flight. Determine:
(a) The velocity corresponding to the maximum aerodynamic efficiency.
3. The pilot switches off the engines and starts gliding at an altitude of $11000\ m$. Calculate:
(a) The minimum descent velocity (vertical velocity), and the corresponding angle of descent, $\gamma_d$:
Answer
Besides the data given in the statement, the following data have been used:
• $g = 9.81\ m/s^2$.
• $R = 287\ J/(kgK)$.
• $\alpha_T = 6.5 \cdot 10^{-3}\ K/m$.
• $\rho_0 = 1.225\ kg/m^3$.
• $T_0 = 288.15\ K$.
• $ISA: \rho = \rho_0 (1 - \tfrac{\alpha_T h}{T_0})^{\tfrac{gR}{\alpha_T} - 1}$.
1. Uniform-ascent under the following flight conditions:
$\bullet$ $h = 3048\ m$. Using $ISA \to \rho = 0.904\ kg/m^3$.
$\bullet$ $V = 148.67\ m/s$.
$\bullet$ $h_e = 14\ m/s$.
The system that governs the motion of the aircraft is:
$T = D + mg \sin \gamma;$
$L = mg \cos \gamma;$
$\dot{x}_e = V\cos \gamma;$
$\dot{h}_e = V \sin \gamma.$
Assuming that $\gamma \ll 1$, and thus that $\cos \gamma \sim 1$ and $\sin \gamma \sim \gamma$, System (B.1) becomes:
$T = D + mg \gamma;\label{eq7.3.5}$
$L = mg;\label{eq7.3.6}$
$\dot{x}_e = V;$
$\dot{h}_e = V_{\gamma}.\label{eq7.3.8}$
(a) From Equation ($\ref{eq7.3.8}$), $\gamma = \tfrac{\dot{h}_e}{V} = 0.094\ rad\ (5.39^{\circ})$.
(b) From Equation ($\ref{eq7.3.5}$), $T = D + mg\gamma$.
$D = C_D \dfrac{1}{2} \rho S_w V^2,\label{eq7.3.9}$
where $C_D = 0.024 + 0.0375 C_L^2$, and $\rho, S_w, V^2$ are known.
$C_L = \dfrac{L}{\tfrac{1}{2} \rho S_w V^2} = 0.512,\label{eq7.3.10}$
where, according to Equation ($\ref{eq7.3.6}$), $L = mg$. With Equation ($\ref{eq7.3.10}$) in Equation ($\ref{eq7.3.8}$), $D = 41398\ N$.
Finally:
$T = D + mg\gamma = 100\ kN.\nonumber$
2. Horizontal, steady, straight flight under the following flight conditions:
$\bullet$ $h = 11000\ m$. Using $ISA \to \rho = 0.3636\ kg/m^3$.
$\bullet$ The aerodynamic efficiency is maximum.
The system that governs the motion of the aircraft is:
$T = D;$
$L = mg \cos \gamma.\label{eq7.3.12}$
The maximum Efficiency is $E_{\max} = \tfrac{1}{2\sqrt{C_{D_0} C_{D_i}}} = 16.66$.
The optimal coefficient of lift is $C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}} = 0.8$.
$C_L = \dfrac{L}{\tfrac{1}{2} \rho S_w V^2} \to V = \dfrac{L}{\tfrac{1}{2} \rho S_w C_L} = 187\ m/s,\nonumber$
where, according to Equation ($\ref{eq7.3.12}$), $L = mg$, and in order to fly with maximum efficiency: $C_L = C_{L_{opt}}$.
3. Gliding under the following flight conditions:
$\bullet$ $h = 11000\ m$. Using $ISA \to \rho = 0.3636\ kg/m^3$.
$\bullet$ At the minimum descent velocity.
The system that governs the motion of the aircraft is:
$D = mg\sin \gamma_d;\label{eq7.3.13}$
$L = mg \cos \gamma_d;\label{eq7.3.14}$
$\dot{x}_e = V \cos \gamma_d;\label{eq7.3.15}$
$\dot{h}_{e_{des}} = V \sin \gamma_d.\label{eq7.3.16}$
Notice that $\gamma_d = -\gamma$.
Assuming that $\gamma_d \ll 1$, and thus that $\cos \gamma_d \sim 1$ and $\sin \gamma_d \sim \gamma_d$, System ($\ref{eq7.3.13}$, $\ref{eq7.3.14}$, $\ref{eq7.3.15}$, $\ref{eq7.3.16}$) becomes:
$D = mg \gamma_d;\label{eq7.3.17}$
$L = mg;\label{eq7.3.18}$
$\dot{x}_e = V;$
$\dot{h}_{e_{des}} = V\gamma_d.$
In order to fly with maximum descent velocity $\dot{h}_{e_{des}}$ must be maximum. Operating with Equation ($\ref{eq7.3.17}$) and Equation ($\ref{eq7.3.18}$), $\gamma_d = \tfrac{D}{L}$.
$\dot{h}_{e_{des}} = V\gamma = V \dfrac{D}{L} = V \dfrac{(0.024 + 0.0375 C_L^2) \tfrac{1}{2} \rho S_w V^2}{C_L \tfrac{1}{2} \rho S_w V^2}.\label{eq7.3.21}$
knowing that $C_L = \tfrac{L}{\tfrac{1}{2} \rho S_w V^2}$, where, according to Equation ($\ref{eq7.3.18}$), $L = mg$, Equation ($\ref{eq7.3.21}$) becomes:
$\dot{h}_{e_{des}} = \dfrac{V}{mg} \left (0.024 \dfrac{1}{2} \rho S_w V^2 + \dfrac{0.0375 (mg)^2}{\tfrac{1}{2} \rho S_w V^2} \right ).\label{eq7.3.22}$
make $\tfrac{\partial \dot{h}_{e_{des}}}{\partial V} = 0$.
The velocity with respect to air so that the vertical velocity is minimum is:
$V = \sqrt[4]{\dfrac{4}{3} \dfrac{C_{D_i}}{C_{D_0}}} \sqrt{\dfrac{mg}{\rho S_w}} = 142.57\ m/s.$
Substituting $V = 142.57\ m/s$ in Equation ($\ref{eq7.3.22}$), $\dot{h}_{e_{des}} = 9.87\ m/s$.
Exercise $2$ Runway performances
We want to estimate the take-off distance of an Airbus A-320 taking off at Madrid-Barajas airport. Such aircraft mounts two turbojets, which thrust can be estimated as: $T = T_0 (1 - k\cdot V^2)$, where $T$ is the thrust, $T_0$ is the nominal thrust, $k$ is a constant and $V$ is the true airspeed.
Considering that:
• $g \cdot (\tfrac{T_0}{m\cdot g} - \mu_r) = 1.31725 \tfrac{m}{s^2};$
• $\tfrac{\rho S(C_D - \mu_r C_L) + 2 \cdot T_0 \cdot k}{2 \cdot m} = 3.69 \cdot 19^{-5} \tfrac{m}{s^2};$
• The velocity of take off is $V_{TO} = 70\ m/s;$
where $g$ is the force due to gravity, $m$ is the mass of the aircraft, $\mu_r$ is the friction coefficient, $\rho$ is the density of air, $S$ is the wet surface area of the aircraft, $C_D$ is the coefficient of lift9.
Figure 7.16: Forces during taking off
Calculate:
1. Take-off distance.
Answer
We apply the 2nd Newton's Law:
$\sum F_z = 0;\label{eq7.3.24}$
$\sum F_x = m \dot{V}.\label{eq7.3.25}$
Regarding Equation ($\ref{eq7.3.24}$), notice that while rolling on the ground, the aircraft is assumed to be under equilibrium along the vertical axis.
Looking at Figure 7.16, Equation ($\ref{eq7.3.24}$)-($\ref{eq7.3.25}$) become:
$L + N - mg = 0;\label{eq7.3.26}$
$T - D - F_F = m \dot{V}.$
being $L$ the lift, $N$ the normal force, $mg$ the weight; $T$ the trust, $D$ the drag and $F_F$ the total friction force.
It is well known that:
$L = C_L \dfrac{1}{2} \rho S V^2;\label{eq7.3.28}$
$D = C_D \dfrac{1}{2} \rho S V^2.\label{eq7.3.29}$
It is also well known that:
$F_F = \mu_r N.$
Equation ($\ref{eq7.3.26}$) states that: $N = mg - L$. Therefore:
$F_F = \mu_r (mg - L).\label{eq7.3.31}$
Given that $T = T_0 (1 - kV^2)$, with Equation ($\ref{eq7.3.31}$) and Equations ($\ref{eq7.3.28}$)-($\ref{eq7.3.29}$), Equation (9.5.9) becomes:
$\right (\dfrac{T_0}{m} -\mu_r g \right) + \dfrac{(\rho S (\mu_r C_L - C_D) - 2T_0 k)}{2m} V^2 = V.\label{eq7.3.32}$
Now, we have to integrate Equation ($\ref{eq7.3.32}$).
In order to do so, as referred in the statement: $T_0, m, \mu_r, g, \rho, S, C_L, C_D$ and $k$ can be considered constant along the take off phase.
We have that:
$\dfrac{dV}{dt} = \dfrac{dV}{dx} \dfrac{dx}{dt},$
and knowing that $\tfrac{dx}{dt} = V$, Equation ($\ref{eq7.3.32}$) becomes:
$\dfrac{(\tfrac{T_0}{m} - \mu_r g) + \tfrac{(\rho S (\mu_r C_L - C_D) - 2T_0 k)}{2m} V^2}{V} = \dfrac{dV}{dx}.\label{eq7.3.34}$
In order to simplify Equation ($\ref{eq7.3.34}$):
• $(\tfrac{T_0}{m} - \mu_r g) = g(\tfrac{T_0}{mg} - \mu_r) = A;$
• $\tfrac{(\rho S (\mu_r C_L - C_D) - 2T_0 k)}{2m} = B.$
We proceed on integrating Equation ($\ref{eq7.3.34}$) between $x = 0$ and $x_{T0}$ (the distance of take off); $V = 0$ (Assuming the maneuver starts with the aircraft at rest) and the velocity of take off that was given in the statement: $V_{TO} = 70\ m/s$. It holds that:
$\int_{0}^{x_{TO}} dx = \int_{0}^{V_{TO}} \dfrac{VdV}{A+ BV^2}.$
Integrating:
$\left [ x \right ]_{0}^{x_{TO}} = \left [ \dfrac{1}{2B} \ln (A + BV^2) \right ]_{0}^{V_{TO}}.$
Substituting the upper and lower limits:
$x_{TO} = \dfrac{1}{2B} \ln (1 + \dfrac{B}{A} V_{TO}^2).$
Substituting the data given in the statement:
• $A = 1.31725 \tfrac{m}{s^2};$
• $B = -3.69 \cdot 10^{-5} \tfrac{m}{s^2};$
• $V_{TO} = 70\ m/s.$
The distance to take off is $x_{TO} = 2000\ m$.
Exercise $3$ Performances
An aircraft has the following characteristics:
• $S_w = 130\ m^2$.
• $b = 40\ m$.
• $m = 70000\ kg$.
• $T_{\max, av} (h = 0) = 120000\ N$ (Maximum available thrust at sea level).
• $C_{D_0} = 0.02$.
• Oswald coefficient (wing efficiency coefficient): $e = 0.9$.
• $C_{L_{\max}} = 1.5$.
We can consider that the maximum thrust only varies with altitude as follows: $T_{\max, av} (h) = T_{\max, av} (h = 0) \tfrac{\rho}{\rho_0}$. Consider standard atmosphere $ISA$ and constant gravity $g = 9.8\ m/s^2$. Determine:
1. The required thrust to fly at an altitude of $h = 11000\ m$ with $M_{\infty} = 0.7$ in horizontal, steady, straight flight.
2. The maximum velocity due to propulsive limitations of the aircraft and the corresponding Mach number in horizontal, steady, straight flight at $h = 11000$.
3. The minimum velocity due to aerodynamic limitations (stall speed) in horizontal, steady, straight flight at an altitude of $h = 11000$.
4. The theoretical ceiling (maximum altitude) in horizontal, steady, straight flight.
We want to perform a horizontal turn at an altitude of $h = 11000$ with a load factor $n = 2$, and with the velocity corresponding to the maximum aerodynamic efficiency in horizontal, steady, straight flight. Determine:
5. The required bank angle.
6. The radius of turn.
7. The required thrust. Can the aircraft perform the complete turn?
We want to perform a horizontal turn with the same load factor and the same radius as in the previous case, but at an altitude corresponding to the theoretical ceiling of the aircraft.
8. Can the aircraft perform such turn?
Answer
Besides that data given in the statement, the following data have been used:
• $R = 287\ J/(kgK)$.
• $\gamma_{air} = 1.4.$
• $\alpha_T = 6.5 \cdot 10^{-3} \ K/m$.
• $\rho_0 = 1.225 \ kg/m^3$.
• $T_0 = 288.15\ K$.
• $ISA: \rho = \rho_0 (1 - \tfrac{\alpha_T h}{T_0})^{\tfrac{gR}{\alpha_T} - 1}.$
1. Required Thrust to fly a horizontal, steady, straight flight under the following flight conditions:
$\bullet$ $h = 11.000\ m$;
$\bullet$ $M_{\infty} = 0.7.$
According to $ISA$:
$\bullet$ $\rho (h = 11000) = 0.364\ Kg/m^3$;
$\bullet$ $a (h = 11000) = \sqrt{\gamma_{air} R (T_0 - \alpha_T h)} = 295.04\ m/s$.
where a corresponds to the speed of sound.
The system that governs the dynamics of the aircraft is:
$T = D;\label{eq7.3.37}$
$L = mg;\label{eq7.3.38}$
being $L$ the lift, $mg$ the weight; $T$ the trust and $D$ the drag.
It is well known that:
$L = C_L \dfrac{1}{2} \rho S_w V^2;\label{eq7.3.39}$
$D = C_D \dfrac{1}{2} \rho S_w V^2.\label{eq7.3.40}$
It is also well known that the coefficient of drag can be expressed in a parabolic form as follows:
$C_D = C_{D_0} + C_{D_i} C_L^2,\label{eq7.3.41}$
where $C_{D_0}$ is given in the statement and $C_{D_i} = \tfrac{1}{\pi A e}$. The enlargement, $A$, can be calculated as $A = \tfrac{b^2}{S_w} = 12.30$ and therefore: $C_{D_i} = 0.0287$.
According to Equation ($\ref{eq7.3.38}$): $L = 686000\ N$. The velocity of flight can be calculated as $V = M_{\infty} a = 206.5\ m/s$. Once these values are obtained, with the values of density and wet surface, and entering in Equation ($\ref{eq7.3.39}$), we obtain that $C_L = 0.68$.
With the values of $C_L$, $C_{D_i}$, and $C_{D_0}$, entering in Equation ($\ref{eq7.3.41}$) we obtain that $C_D = 0.0332$.
Looking now at Equation ($\ref{eq7.3.37}$) and using Equation ($\ref{eq7.3.40}$), we can state that the required thrust is as follows:
$T = C_D \dfrac{1}{2} \rho S_w V^2.\nonumber$
Since all values are known, the required thrust yields:
$T = 33567\ N.\nonumber$
Before moveing on, we should look wether the required thrust exceeds or not the maximum available thrust at the given altitude. In order to do that, it has been given that the maximum thrust only varies with altitude as follows:
$T_{\max, av} (h) = T_{\max, av} (h = 0) \dfrac{\rho}{\rho_0}.$
The maximum available thrust at $h = 11000$ yields:
$T_{\max, av} (h = 11000) = 35657.14\ N.\label{eq7.3.43}$
Since $T < T_{\max, av}$, the flight condition is flyable.
2. The maximum velocity due to propulsive limitations of the aircraft and the corresponding Mach number is horizontal, steady, straight flight at $h = 11000$:
The maximum velocity due to propulsive limitation at the given altitude implies flying at the maximum available thrust that was obtained in Equation ($\ref{eq7.3.43}$).
Looking again at Equation ($\ref{eq7.3.37}$) and using Equation ($\ref{eq7.3.40}$), we can state that:
$T_{\max, av} = C_D \dfrac{1}{2} \rho S_w V^2.\label{eq7.3.44}$
Using Equation ($\ref{eq7.3.41}$) and Equation ($\ref{eq7.3.39}$), and entering in Equation ($\ref{eq7.3.44}$) we have that:
$T_{\max, av} = \left ( C_{D_0} + C_{D_i} \left (\dfrac{L}{\tfrac{1}{2} \rho S_w V^2} \right )^2 \right ) \dfrac{1}{2} \rho S_w V^2.\label{eq7.3.45}$
Multiplying Equation ($\ref{eq7.3.45}$) by $V^2$ we obtain a quadratic equation of the form:
$ax^2 + bx + c = 0.$
where $x = V^2$, $a = \tfrac{1}{2} \rho S_w C_{D_0}$, $b = -T_{\max, av}$, and $c = \dfrac{C_{D_i} L^2}{\tfrac{1}{2} \rho S_w}$.
Solving the quadratic equation we obtain two different speeds at which the aircraft can fly given the maximum available thrust10. Those velocities yield:
$V_1 = 228\ m/s;\nonumber$
$V_2 = 151\ m/s.\nonumber$
The maximum corresponds, obviously, to $V_1$.
3. The minimum velocity due to aerodynamic limitations (stall speed) in horizontal, steady, straight flight at al altitde of $h = 11000$.
The stall speed takes place when the coefficient of lift is maximum, therefore, using Equation ($\ref{eq7.3.39}$), we have that:
$V_{stall} = \sqrt{\dfrac{L}{\tfrac{1}{2} \rho S_w C_{L_{\max}}}} = 139\ m/s.\nonumber$
4. The theoretical ceiling (maximum altitude) in horizontal, steady, straight flight.
In order to obtain the theoretical ceiling of the aircraft the maximum available thrust at that maximum altitude must coincide with the minimum required thrust to fly horizontal, steady, straight flight at that maximum altitude, that is:
$T_{\max, av} = T_{\min}.\label{eq7.3.47}$
Let us first obtain the minimum required thrust to fly horizontal, steady, straight flight. Multiplying and dividing by $L$ in the second term of Equation ($\ref{eq7.3.37}$), and given that the aerodynamic efficiency is $E = \tfrac{L}{D}$, we have that:
$T = \dfrac{D}{L} L = \dfrac{L}{E}.$
Since $L$ is constant at those conditions of flight, the minimum required thrust occurs when the efficiency is maximum: $T_{\min} \Leftrightarrow E_{\max}$.
Let us now proceed deducing the maximum aerodynamic efficiency:
The aerodynamic efficiency is defined as:
$E = \dfrac{L}{D} = \dfrac{C_L}{C_D}.\label{eq7.3.49}$
Substituting the parabolic polar curve given in Equation ($\ref{eq7.3.41}$) in Equation ($\ref{eq7.3.49}$), we obtain:
$E = \dfrac{C_L}{C_{D_0} + C_{D_i} C_L^2}.\label{eq7.3.50}$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_L} = 0 \dfrac{C_{D_0} - C_{D_i} C_L^2}{(C_{D_0} + C_{D_i} C_L^2)^2} \to (C_L)_{E_{\max}} = C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}}.\label{eq7.3.51}$
Substituting the value of $C_{L_{opt}}$ into Equation ($\ref{eq7.3.50}$) and simplifying we obtain that:
$E_{\max} = \dfrac{1}{2\sqrt{C_{D_0} C_{D_i}}}.\nonumber$
$E_{\max}$ yields 20.86, and $T_{\min} = 32870\ N$.
According to Equation ($\ref{eq7.3.43}$) and based on Equation ($\ref{eq7.3.47}$) with $T_{\min} = 32870\ N$, we have that:
$32870 = T_{\max, av} (h = 0) \dfrac{\rho}{\rho_0}.\nonumber$
Given that $T_{\max, av} (h = 0)$ was given in the statement and $\rho_0$ is known according to $ISA$, we have that $\rho = 0.335\ kg/m^3$.
Since $\rho_{h_{\max}} < \rho_{11000}$ we can easily deduce that the eiling belongs to the stratosphere. Using the $ISA$ equation corresponding to the stratosphere we have that:
$\rho_{h_{\max}} = \rho_{11} \exp^{-\tfrac{g}{TR_{11}} (h_{\max} - h_{11})}.\label{eq7.3.52}$
where the subindex 11 corresponds to the values at the tropopause $(h = 11000\ m)$. Operating in Equation ($\ref{eq7.3.52}$), the ceiling yields $h_{\max} = 11526\ m$.
5. The required bank angle:
The equations governing the dynamics of the airplane in an horizontal turn are:
$T = D;\label{eq7.3.53}$
$m V \dot{\chi} = L \sin \mu;\label{eq7.3.54}$
$L \cos \mu = mg.\label{eq7.3.55}$
In a uniform (stationary) circular movement, it is well known that the tangential velocity is equal to the angular velocity ($\dot{\chi}$) multiplied by the radius o turn ($R$):
$V = \dot{\chi} R.$
Therefore, Sytem ($\ref{eq7.3.53}$, $\ref{eq7.3.54}$ and $\ref{eq7.3.55}$) can be rewritten as:
$T = D;\label{eq7.3.57}$
$n \sin \mu = \dfrac{V^2}{gR};\label{eq7.3.58}$
$n = \dfrac{1}{\cos \mu};\label{eq7.3.59}$
where $n = \tfrac{L}{mg}$ is the load factor.
Therefore, looking at Equation ($\ref{eq7.3.59}$), it is straightforward to determine that the bank angle of turn is $\mu = 60^{\circ}$.
6. The radius of turn.
First, we need to calculate the velocity corresponding to the maximum efficiency. As we have calculated before in Equation ($\ref{eq7.3.51}$), the coefficient of lift that generates maximum efficiency is the so-called optimal coefficient of lift, that is, $C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}} = 0.834$. The velocity yields then:
$V = \sqrt{\dfrac{L}{\tfrac{1}{2} \rho S_w C_{L_{opt}}}} = 186.4\ m/s.\nonumber$
Entering the Equation ($\ref{eq7.3.58}$) with $\mu = 60\ [deg]$ and $V = 186.4\ m/s$; $R = 4093.8\ m$.
7. The required thrust.
As exposed in Question 3, Equation ($\ref{eq7.3.57}$) can be expressed as:
$T = \dfrac{1}{2} \rho S_w V^2 C_{D_0} + \dfrac{L^2}{\tfrac{1}{2} \rho V^2 S_w} C_{D_i}.\nonumber$
Since all values are known: $T = 32451\ N$
Since $T \le T_{\max, av} (h = 110000)$, the aircraft can perform the turn.
8. We want to perform a horizontal turn with the same load factor and the same radius as the pervious case, but at an altitude corresponding to the thearetical ceiling of the aircraft. Can the aircraft perform the turn?
If the load factor is the same, $n = 2$, necessarily (according to Equation ($\ref{eq7.3.59}$) the bank angle is the same, $\mu = 60^{\circ}$. Also, if the radius of turn is the same, $R = 4093.8\ m$, necessarily (according to Equation ($\ref{eq7.3.58}$), the velocity of the turn must be the same as in the previous case, $V = 186.4\ m/s$. Obviously, since the density will change according to the new altitude ($\rho = 0.335\ kg/m^3$), the turn will not be performed under maximum efficiency conditions.
In order to know wheter the turn can be performed or not, we must compare the required thrust with the maximum available thrust at the ceiling altitude:
$T = \dfrac{1}{2} \rho S_w V^2 C_{D_0} + \dfrac{L^2}{\tfrac{1}{2} \rho V^2 S_w} C_{D_i} = 32983\ N.\nonumber$
$T_{\max, av} (h = 11526) = T_{\max, av} (h = 0) \dfrac{\rho}{\rho_0} = 32816\ N.\nonumber$
Since $T > T_{\max, av}$ at the ceiling, the turn can not be performed.
Exercise $4$ Weights
Consider an Airbus A-320. Simplifying, we assume the wing of the aircraft is rectangle and it is composed on NACA 4415 airfoils. The characteristics of the NACA 4415 airfoils as follows:
• $c_l = 0.2 + 5.92 \alpha$. ($\alpha$ in radians).
• $c_d = 6.4 \cdot 10^{-3} - 1.2 \cdot 10^{-3} c_l + 3.5 \cdot 10^{-3} c_l^2$.
Regarding the aircraft, the following data are known:
• Wing wet surface of 122.6 $[m^2]$.
• Wing-span of 34.1 $[m]$.
• Oswald efficiency factor of 0.95.
• Speific consumption per unity of thrust and time: $\eta_j = 6.8 \cdot 10^{-5} [\tfrac{Kg}{N \cdot s}]$.
Calculate:
1. The lift of the wing in its linear range.
2. The drag polar curve, assuming it can be expressed as: $C_D = C_{D_0} + C_{D_i} C_L^2$.
3. The optimal coefficient of lift of the wing, $C_{L_{opt}}$. Compare it with the airfoil's one.
On regard of the characteristic weights of the aircraft, the following data are known:
• Operating empty weight $OEW = 42.4\ [Ton]$.
• Maximum take-off weight $MTOW = 77000 \cdot g\ [N]$11.
• Maximum fuel weight $MFW = 29680 \cdot g\ [N]$.
• Maximum zero fuel weight $MZFW = 59000 \cdot g\ [N]$.
• Moreoveer, the reserve fuel ($RF$) can be calculated as the 5% of the Trip Fuel ($TF$).
Calculate:
4. The Payload and Trip Fuel in the following cases:
(a) Initial weight equal to $MTOW$ with the Maximum Payload $(MPL)$.
(b) Initial weight equal to $MTOW$ with the Maximum Fuel Weight $(MFW)$.
(c) Initial weight equal to $OEW$ plus $MFW$.
Assuming that in cruise conditions the aircraft flies at a constant altitude of $h = 11500\ [m]$, constant Mach number $M = 0.78$, and maximum aerodynamic efficiency, considering $ISA$ standard atmosphere, calculate:
5. Range and autonomy of the aircraft for the three initial weights pointed out above.
6. According to the obtained results, draw the payload-range diagram.
Answer
1. Wing's lift curve:
The lift curve of a wing can be expressed as follows:
$C_L = C_{L_0} + C_{L_{\alpha}} \alpha,\label{eq7.3.60}$
and the slope of the wing's lift curve can be expressed related to the slope of the airfoil's lift curve as:
$C_{L_{\alpha}} = \dfrac{C_{l_{\alpha}}}{1 + \tfrac{C_{l_{\alpha}}}{\pi A}} e = 4.69 \cdot 1/rad.\nonumber$
In order to calculate the independent term of the wing's lift curve, we must consider the fact that the zero-lift angle of attack of the wing coincides with the zero-lift angle of attack of the airfoil, that is:
$\alpha (L = 0) = \alpha (l = 0).\label{eq7.3.61}$
First, notice that the lift curve of an airfoil can be expressed as follows
$C_l = C_{l_0} + C_{l_{\alpha}} \alpha.\label{eq7.3.62}$
Therefore, with Equation ($\ref{eq7.3.60}$) and Equation ($\ref{eq7.3.61}$) in Equation ($\ref{eq7.3.62}$), we have that:
$C_{L_0} = C_{l_0} \dfrac{C_{L_{\alpha}}}{C_{l_{\alpha}}} = 0.158.\nonumber$
The required curve yields then:
$C_L = 0.158 + 4.69 \alpha \ [\alpha \ in\ rad].\nonumber$
2. The expression of the parabolic polar of the wing:
Notice first that the statement of the problem indicates that the polar should be in the following form:
$C_D = C_{D_0} + C_{D_i} C_L^2.\label{eq7.3.63}$
For the calculation of the parabolic drag of the wing we can consider the parasite term approximately equal to the parasite term of the airfoil, that is, $C_{D_0} = C_{d_0}$.
The induced coefficient of drag can be calculated as follows:
$C_{D_i} = \dfrac{1}{\pi Ae} = 0.035.\nonumber$
The expression of the parabolic polar yields then:
$C_D = 0.0064 + 0.035 C_L^2.$
3. The optimal coefficient of lift, $C_{L_{opt}}$, for the wing. Compare it with the airfoils's one.
The optimal coefficient of lift is that making the aerodynamic efficiency maximum. The aerodynamic efficiency is defined as:
$E = \dfrac{L}{D} = \dfrac{C_L}{C_D}.\label{eq7.3.65}$
Substituting the parabolic polar curve given in Equation ($\ref{eq7.3.63}$) in Equation ($\ref{eq7.3.65}$), we obtain:
$E = \dfrac{C_L}{C_{D_0} + C_{D_i} C_L^2}.\label{eq7.3.66}$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_L} = 0 = \dfrac{C_{D_0} - C_{D_i} C_L^2}{(C_{D_0} - C_{D_i} C_L^2)^2} \to (C_L)_{E_{\max}} = C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}}.\label{eq7.3.67}$
For the case of an airfoil, the aerodynamic efficiency is defined as:
$E = \dfrac{l}{d} = \dfrac{c_l}{c_d}.\label{eq7.3.68}$
Substituting the parabolic polar curve given in the statement in the form $c_{d_0} + bc_l + kc_l^2$ in Equation ($\ref{eq7.3.68}$), we obtain:
$E = \dfrac{c_l}{c_{d_0} + bc_l + kc_l^2}.$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_l} = 0 = \dfrac{c_{d_0} - kc_l^2}{(c_{d_0} + bc_l + kc_l^2)^2} \to (c_l)_{E_{\max}} = (c_l)_{opt} = \sqrt{\dfrac{c_{d_0}}{k}}.\label{eq7.3.70}$
According to the values previously obtained ($C_{D_0} = 0.0064$ and $C_{D_i} = 0.035$) and the values given in the statement for the airfoil's polar ($c_{d_0} = 0.0064$, $k = 0.0035$), substituting them in Equation ($\ref{eq7.3.67}$) and Equation ($\ref{eq7.3.70}$), respectively, we obtain:
$\bullet$ $(C_L)_{opt} = 0.42;$
$\bullet$ $(c_l)_{opt} = 1.35.$
4. Payload and Trip Fuel for case (a), (b), and (c):
Before starting we the particular cases, it is necessary to point out that:
$TOW = OEW + PL + FW,\label{eq7.3.71}$
$FW = TF + RF,\label{eq7.3.72}$
$MZFW = OEW + MPL.\label{eq7.3.73}$
Moreover, according to the statement,
$RF = 0.05 \cdot TF.$
Based on the data given in the statement, and using Equation ($\ref{eq7.3.73}$):
$MPL = 16.6\ [Ton.].$
Case (a) Initial weight12 $\to$ $MTOW$ with $MPL$.
Equation ($\ref{eq7.3.71}$) becomes:
$MTOW = OEW + MPL + TF + 0.05 \cdot TF,\label{eq7.3.76}$
Isolating in Equation ($\ref{eq7.3.76}$): $TF = 17.14\ [Ton]$. The payload is equal to the maximum payload $MPL$.
Case (b) Initial weight $\to$ $MTOW$ with $MFW$.
Equation ($\ref{eq7.3.71}$) becomes:
$MTOW = OEW + PL + MFW,\label{eq7.3.77}$
Isolating in Equation ($\ref{eq7.3.77}$): $PL = 4.92\ [Ton]$. In order to calculate the trip fuel, looking at Equation ($\ref{eq7.3.72}$), we have that
$MFW = TF + RF = TF + 0.05 \cdot TF.$
Isolating, $TF = 28.266\ [Ton.]$.
Case (c) Initial weight $\to$ $OEW + MFW$.
Equation ($\ref{eq7.3.71}$) becomes:
$TOW = OEW + MFW,$
That means $PL = 0$, in order to calculate the trip fuel, we proceed exactly as in Case (b). Looking at Equation ($\ref{eq7.3.72}$), we have that
$MFW = TF + RF = TF + 0.05 \cdot TF.$
Isolating, $TF = 28.266\ [Ton.]$
5. Range and Endurance for cases (a), (b) and (c): Considering that the aircraft performs a linear, horizontal steady flight, we have that:
$L = mg;\label{eq7.3.81}$
$T = D;\label{eq7.3.82}$
$\dot{x} = V;\label{eq7.3.83}$
$\dot{m} = -\eta T.\label{eq7.3.84}$
Since $\dot{x} = \tfrac{dx}{dt}$, it is clear that the Range, $R$, looking at Equation ($\ref{eq7.3.83}$), can be expressed as:
$R = \int_{t_i}^{t_f} V dt.\label{eq7.3.85}$
Now, since $\dot{m} = \tfrac{dm}{dt} = -\eta T$, Equation ($\ref{eq7.3.85}$):
$R = \int_{m_i}^{m_f} -\dfrac{V}{\eta T} dm,\label{eq7.3.86}$
where $m_i$ is the initial mass and $m_f$ is the final mass.
Using Equation ($\ref{eq7.3.81}$) and Equation ($\ref{eq7.3.82}$), Equation ($\ref{eq7.3.86}$) yields:
$R = \int_{m_i}^{m_f} - \dfrac{VE}{\eta g} \dfrac{dm}{m}.$
with $E = \tfrac{L}{D}, V, \eta$, and $g$ are constant values.
Integrating:
$R = \dfrac{VE}{\eta g} \ln (\dfrac{m_i}{m_f}).\label{eq7.3.88}$
For the endurance, we operate analogously, integrating Equation ($\ref{eq7.3.84}$), which yields
$t = \int_{m_i}^{m_f} - \dfrac{1}{\eta T} dm.\label{eq7.3.89}$
Using Equation ($\ref{eq7.3.81}$) and ($\ref{eq7.3.82}$), ($\ref{eq7.3.89}$) yields:
$t = \int_{m_i}^{m_f} - \dfrac{E}{\eta g} \dfrac{dm}{m}.$
where again $E = \tfrac{L}{D}$, $\eta$ and $g$ are constant values. Integrating:
$t = \dfrac{E}{\eta g} \ln (\dfrac{m_i}{m_f}).\label{eq7.3.91}$
Before calculating Range and Endurance for the three cases, we need to calculate the values of velocity and aerodynamic efficiency, which is maximum.
The velocity can be expressed as $V = M \cdot a$, where $a$ is the speed of sound, that can be expressed as
$a = \sqrt{\gamma RT},$
where $\gamma = 1.4$ is adiabatic coefficient of air, and $R = 287\ J/KgK$ is the perfect gas constant. Notice that, using $ISA$, the temperature at the stratosphere is constant, and thus $T = 216.6\ K$.
Substituting all terms, it yields $v = 230.1\ [m/s]$.
The aeerodynamic efficiency is maximum. Thus, substituting $C_{L_{opt}}$ obtained in Equation ($\ref{eq7.3.67}$) into Equation ($\ref{eq7.3.66}$), it yields:
$E = \dfrac{1}{2\sqrt{C_{D_0} C_{D_i}}} = 33.4.$
Now, we should calculate the initial and final mass for of the three cases a), b), c). Notice that the final mass results from subtracting the trip fuel from the take-off mass:
$m_f = m_i - TF.$
Thus,
(a) $m_i = \dfrac{MTOW}{g} [Kg]$ and $m_f = 59860\ [Kg]$.
(b) $m_i = \dfrac{MTOW}{g} [Kg]$ and $m_f = 48734\ [Kg]$.
(c) $m_i = 72080 [Kg]$ and $m_f = 43814\ [Kg]$.
Substituting in Equation ($\ref{eq7.3.88}$) and Equation ($\ref{eq7.3.91}$), it yields:
(a) $R_a = 2900\ [Km]$ and $t_a = 12600\ [s]$.
(b) $R_b = 5270\ [Km]$ and $t_b = 22900\ [s]$.
(c) $R_c = 5736\ [Km]$ and $t_c = 24928\ [s]$.
6. Payload-Range diagram cases (a), (b) and (c):
Figure 7.17: Payload-range diagram
Exercise $5$
An aircraft has the following characteristics:
• $S_w = 130\ m^2$.
• $b = 40\ m$.
• $m = 70000\ kg$.
• $T_{\max, av, 0} = 130000\ N$ (Maximum available thrust at sea level).
• $C_{D_0} = 0.02$.
• Oswald efficiency factor: $e = 0.9$.
The maximum available thrust can be considered to vary according to the following law:
$T_{\max, av} (h) = T_{\max, av, 0} \dfrac{\rho}{\rho_0}.\nonumber$
Consider $ISA$ atmosphere and constant gravity $g = 9.8\ m/s^2$. Determine:
1. The required thrust to fly at an altitude of $h = 11250\ m$ at $M_{\infty} = 0.78$ in steady linear-horizontal flight.
2. The maximum speed of the aircraft due to propulsive limitations in steady linear-horizontal flight at an altitude of $h = 11250\ m$.
3. The theoretical ceiling of the aircraft in steady linear-horizontal flight.
We want now to deteremine the surface of the horizontal stabilizer and as a design criterion we take the flight conditions of equilibrium at an altitude of $h = 11250\ m$ & $M_{\infty} = 0.78$. In those conditions, the coefficient of lift of the stabilizer is equal to 1.4. Moreover, we can assume that the distribution of lift of the wing can be reduced to a resultant force in the aerodynamic center (lift) and a pitching down moment with respect to the aerodynamic center equal to $10000\ N \cdot m$. The distance between the aerodynamic center and the center of gravity (note that the aerodynamic center is closer to the nose of the aircraft) is $x_{cg} = 2\ m$. The distance between the stabilizer and the center of gravity is $l = 20\ m$.
4. Determine the surface of the horizontal stabilizer for those flight conditions.
Answer
Besides the data given in the statement, the following data have been used:
• $R = 287\ J/(kgK)$.
• $\gamma_{air} = 1.4$.
• $T_{11} = 216.6\ K$.
• $\rho_{11} = 0.36\ kg/m^3$.
• $T_0 = 288.15\ K$.
• $ISA: \rho = \rho_{11} \cdot e^{\tfrac{-gR}{T_{11}}} (h - 11000)$.
1. Required Thrust to fly a horizontal, steady, straight flight under the following flight conditions:
$\bullet$ $h = 11.250\ m$.
$\bullet$ $M_{\infty} = 0.78$.
According to $ISA$:
$\bullet$ $\rho (h = 11250) = 0.3461\ Kg/m^3$.
$\bullet$ $a (h = 11250) = \sqrt{\gamma_{air} R(T_{11})} = 295\ m/s$.
where a corresponds to the speed of sound.
The system that governs the dynamics of the aircraft is:
$T = D,\label{eq7.3.95}$
$L = mg,\label{eq7.3.96}$
being $L$ the lift, $mg$ the weight; $T$ the trust and $D$ the drag.
It is well known that:
$L = C_L \dfrac{1}{2} \rho S_w V^2;\label{eq7.3.97}$
$D = C_D \dfrac{1}{2} \rho S_w V^2.\label{eq7.3.98}$
It is also well known that the coefficient of drag can be expressed in a prabolic form as follows:
$C_D = C_{D_0} + C_{D_i} C_L^2,\label{eq7.3.99}$
where $C_{D_0}$ is given in the statement and $C_{D_i} = \tfrac{1}{\pi Ae}$. The enlargement $A$ can be calculated as $A = \tfrac{b^2}{S_w} = 12.30$ and therefore: $C_{D_i} = 0.0287$.
According to Equation ($\ref{eq7.3.96}$): $L = 686000\ N$. The velocity of flight can be calculated as $V = M_{\infty} a = 230.1\ m/s$. Once these values are obtained, with the values of density and wet surface, and entering in Equation ($\ref{eq7.3.97}$), we obtain that $C_L = 0.5795$.
With the values of $C_L, C_{D_i}$ and $C_{D_0}$, entering in Equation ($\ref{eq7.3.99}$) we obtain that $C_D = 0.0295$.
Looking now at Equation ($\ref{eq7.3.95}$) and using Equation ($\ref{eq7.3.98}$), we can state that the required thrust is as follows:
$T = C_D \dfrac{1}{2} \rho S_w V^2.\nonumber$
Since all values are known, the required thrust yields:
$T = 35185\ N.\nonumber$
Before moving on, we should look wether the required thrust exceeds or not the maximum available thrust at the given altitude. In order to do that, it has been given that the maximum thrust only varies with altitude as follows:
$T_{\max, av} (h) = T_{\max, av, 0} \dfrac{\rho}{\rho_0}.\label{eq7.3.100}$
The maximum available thrust at $h = 11250$ yields:
$T_{\max, av} (h = 11250) = 36729\ N.\label{eq7.3.101}$
Since $T < T_{\max, av}$, the flight condition is flyable.
2. The maximum velocity due to propulsive limitations of the aircraft in horizontal, steady, straight flight at $h = 11250$:
The maximum velocity due to propulsive limitation at the given altitude implies flying at the maximum available thrust that was obtained in Equation ($\ref{eq7.3.101}$).
Looking agian at Equation ($\ref{eq7.3.95}$) and using Equation ($\ref{eq7.3.98}$), we can state that:
$T_{\max, av} = C_D \dfrac{1}{2} \rho S_w V^2.\label{eq7.3.102}$
Using Equation ($\ref{eq7.3.99}$) and Equation ($\ref{eq7.3.97}$), and entering in Equation ($\ref{eq7.3.102}$) we have that:
$T_{\max, av} = \left (C_{D_0} + C_{D_i} \left (\dfrac{L}{\tfrac{1}{2} \rho S_w V^2} \right )^2 \right ) \dfrac{1}{2} \rho S_w V^2.\label{eq7.3.103}$
Multiplying Equation ($\ref{eq7.3.103}$) by $V^2$ we obtain a quadratic equation of the form:
$ax^2 + bx + c = 0.$
where $x = V^2$, $a = \dfrac{1}{2} \rho S_w C_{D_0}, b = - T_{\max, av}$ and $c = \dfrac{C_{D_i} L^2}{\tfrac{1}{2} \rho S_w}$.
Solving the quadratic equation we obtain two different speeds at which the aircraft can fly given the maximum available thrust13. Those velocities yield:
$V_1 = 218\ m/s;\nonumber$
$V_2 = 184.06\ m/s.\nonumber$
The maximum speeds corresponds, obviously, to $V_1$.
3. The theoretical ceiling (maximum altitude) in horizontal, steady, straight flight.
In order to obtain the theoretical ceiling of the aircraft the maximum available thrust at that maximum altitude must coincide with the minimum required thrust to fly horizontal, steady, straight flight at that maximum altitude, that is:
$T_{\max, av} = T_{\min}.\label{eq7.3.105}$
Let us first obtain the minimum required thrust to fly horizontal, steady, straight flight. Multiplying and dividing by $L$ in the second term of Equation ($\ref{eq7.3.95}$), and given that the aerodynami efficiency is $E = \tfrac{L}{D}$, we have that:
$T = \dfrac{D}{L} L = \dfrac{L}{E}.$
Since $L$ is constant at those conditions of flight, the minimum required thrust occurs when the efficiency is maximum: $T_{\min} \Leftrightarrow E_{\max}$.
Let us now proceed deducing the maximum aerodynamic efficiency:
The aerodynamic efficiency is defined as:
$E = \dfrac{L}{D} = \dfrac{C_L}{C_D}.\label{eq7.3.107}$
Substituting the parabolic polar curve gicen in Equation polar curve given in Equation ($\ref{eq7.3.99}$) in Equation ($\ref{eq7.3.107}$), we obtain:
$E = \dfrac{C_L}{C_{D_0} + C_{D_i} C_L^2}.\label{eq7.3.108}$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_L} = 0 = \dfrac{C_{D_0} - C_{D_i} C_L^2}{(C_{D_0} + C_{D_i} C_L^2)^2} \to (C_L)_{E_{\max}} = C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}}.$
Substituting the value of $C_{L_{opt}}$ into Equation ($\ref{eq7.3.108}$) and simplifying we obtain that:
$E_{\max} = \dfrac{1}{2\sqrt{C_{D_0} C_{D_i}}}.\nonumber$
$E_{\max}$ yields 20.86, and $T_{\min} = 32904\ N$.
According to Equation ($\ref{eq7.3.101}$) and based on Equation ($\ref{eq7.3.105}$) with $T_{\min} = 32904\ N$, we have that:
$32904 = T_{\max, av, 0} \dfrac{\rho}{\rho_0}.\nonumber$
Given that $T_{\max, av, 0}$ was given in the statement and $\rho_0$ is known according to $ISA$, we have that $\rho = 0.3101\ kg/m^3$.
Since $\rho_{h_{\max}} < \rho_{11000}$ we can easily deduce that the ceiling belongs to the stratosphere. Using the $ISA$ equation corresponding to the stratosphere we have that:
$\rho_{h_{\max}} = \rho_{11} \exp^{-\tfrac{g}{RT_{11}} (h_{\max} - h_{11})}.\label{eq7.3.110}$
where the subindex 11 corresponds to the values at the tropopause $(h = 11000\ m)$. Operating in Equation ($\ref{eq7.3.110}$), the ceiling yields $h_{\max} = 11946\ m$.
4. We want to determine the surface of the horizontal stabilizer:
In order to do that, we state the equations for the longitudinal balancing problem:
$mg - L - L_t = 0;\label{eq7.3.111}$
$-M_{ca} + Lx_{cg} - L_t l = 0;\label{eq7.3.112}$
where $L_t$ is the lift generated by the horizontal stabilizer, $M_{ca} = 10000\ Nm$ is the pitch torque with respect to the aerodynamic center, $x_{cg} = 2\ m$ is the distance between the center of gravity and the aerodynamic center, and $l = 20\ m$ is the distance between the center of gravity and the aerodynamic center of the horizontal stabilizer.
Entering in Equation ($\ref{eq7.3.111}$), we have that $L = mg - L_t$. Knowing that $L_t = 0.5 \rho S_t V^2 C_L$, and substituting in Equation ($\ref{eq7.3.112}$), we have that:
$L_t = \dfrac{mg \cdot x_{cg} - M_{ca}}{l + x_{cg}} \to S_t = \dfrac{1}{0.5 \rho V^2 C_{L_t}} \dfrac{mg \cdot x_{cg} - M_{ca}}{l + x_{cg}} = 4.78\ m^2.$
Figure 7.18: Longitudinal equilibrium.
Exercise $6$ Performances
Consider an Airbus A-320 with the following characteristics:
• $m = 64$ tonnes.
• $S_w = 122.6\ m^2$
• $T_{\max, av, 0} = 130000\ N$ (maximum available thrust at sea level).
• $C_D = 0.024 + 0.0375 C_L^2$.
The maximum available thrust can be considered to vary according to the following law:
$T_{\max, av} (h) = T_{\max, av, 0} \dfrac{\rho}{\rho_0}.\nonumber$
Consider $ISA$ atmosphere and constant gravity $g = 9.8\ m/s^2$.
1. The aircraft starts an uniform ascent maneuver (constant speed and flight path angle) at an altitude of 10.000 feet ($3048\ m$). At this flight level, the aerodynamic speed is $150\ m/s$ and the vertical speed is $12\ m/s$. Assuming a small angle of attack $\gamma \ll 1$, calculate:
(a) The ascent flight path angle, $\gamma$.
(b) The required thrust at these conditions.
(c) Thust relation14 with respect to the maximum available thrust at that altitude.
2. Calculate the maximum angle of ascent (flight path angle) in uniform ascent at an altitude of 10000 feet. In these conditions, determine:
(a) The aerodynamic speed and the vertical speed.
3. We want now to analyze the turn performances in the horizontal plane. Consider sea level conditions, maximum aerodynamic efficiency, and structural limitations characterized by a maximum load factor of $n_{\max} = 2.5$. In these conditions, calculate:
(a) The required bank angle.
(b) Speed an radius of turn.
(c) The required thrust.
(d) Can the aircraft perform the complete turn?
Answer
Besides the data given in the statement, the following data have been used:
• $g = 9.81\ m/s^2$.
• $R = 287\ J/(kgK)$.
• $\alpha_T = 6.5 \cdot 10^{-3}\ K/m$.
• $\rho_0 = 1.225\ kg/m^3$.
• $T_0 = 288.15\ K$.
• $ISA: \rho = \rho_0 (1 - \dfrac{\alpha_T h}{T_0})^{\tfrac{gR}{\alpha_T} - 1}$.
1. Uniform-ascent under the following flight conditions:
$\bullet$ $h = 3048\ m$. Using $ISA \to \rho = 0.904\ kg/m^3$.
$\bullet$ $V = 150\ m/s$.
$\bullet$ $\dot{h}_e = 12\ m/s$.
The system that governs the motion of the aircraft is:
$T = D + mg \sin \gamma;\label{eq7.3.114}$
$L = mg \cos \gamma;\label{eq7.3.115}$
$\dot{x}_e = V \cos \gamma;\label{eq7.3.116}$
$\dot{h}_e = V \sin \gamma.\label{eq7.3.117}$
Assuming that $\gamma \ll 1$, and thus that $\cos \gamma \sim 1$ and $\sin \gamma \sim \gamma$, System ($\ref{eq7.3.114}$, $\ref{eq7.3.115}$, $\ref{eq7.3.116}$, $\ref{eq7.3.117}$) becomes:
$T = D + mg \gamma;\label{eq7.3.118}$
$L = mg;\label{eq7.3.119}$
$\dot{x}_e = V;\label{eq7.3.120}$
$\dot{h}_e = V_{\gamma}. \label{eq7.3.121}$
(a) From Equation $\ref{eq7.3.121}$, $\gamma = \tfrac{\dot{h}_e}{V} = 0.08\ rad (4.58^{\circ})$.
(b) From Equation $\ref{eq7.3.118}$, $T = D + mg\gamma$.
$D = C_D \dfrac{1}{2} \rho S_w V^2,\label{eq7.3.122}$
where $C_D = 0.024 + 0.0375 C_L^2$, and $\rho, S_w, V^2$ are known.
$C_L = \dfrac{L}{\tfrac{1}{2} \rho S_w V^2} = 0.5057.\label{eq7.3.123}$
where, according to Equation ($\ref{eq7.3.119}$), $L = mg$. With Equation ($\ref{eq7.3.123}$) in Equation ($\ref{eq7.3.122}$), $D = 41398\ N$.
Finally:
$T = D + mg \gamma = 91886\ N.\nonumber$
(c) The maximum thrust at those conditions is
$T_{\max, av} (h = 3048) = T_{\max, av, 0} \dfrac{\rho}{\rho_0} = 95510\ N.\nonumber$
The relation is $\sqcap = \tfrac{T}{T_{\max, av}} = 0.962$.
2. Maximum angle of ascent.
We consider again the set of equations ($\ref{eq7.3.118}$, $\ref{eq7.3.119}$, $\ref{eq7.3.120}$, $\ref{eq7.3.121}$). Diving Equation ($\ref{eq7.3.118}$) by $mg$, we have that:
$\gamma = \dfrac{T}{mg} - \dfrac{1}{E}.$
In order $\gamma$ to be maximum:
$\bullet$ $T = T_{\max, av} = 95510\ N$.
$\bullet$ $E = E_{\max}$.
Let us now proceed deducing the maximum aerodynamic efficiency:
The aerodynamic efficiency is defined as:
$E = \dfrac{L}{D} = \dfrac{C_L}{C_D}.\label{eq7.3.125}$
Substituting the parabolic polar curve given in the statement in Equation ($\ref{eq7.3.125}$), we obtain:
$E = \dfrac{C_L}{C_{D_0} + C_{D_i} C_L^2}.\label{eq7.3.126}$
In order to seek the values corresponding to the maximum aerodynamic efficiency, one must derivate and make it equal to zero, that is:
$\dfrac{dE}{dC_L} = 0 = \dfrac{C_{D_0} - C_{D_i} C_L^2}{(C_{D_0} - C_{D_i} C_L^2)^2} \to (C_L)_{E_{\max}} = C_{L_{opt}} = \sqrt{\dfrac{C_{D_0}}{C_{D_i}}}.\label{eq7.3.127}$
Substituting the value of $C_{L_{opt}}$ into Equation ($\ref{eq7.3.126}$) and simplifying we obtain that:
$E_{\max} = \dfrac{1}{2\sqrt{C_{D_0} C_{D_i}}}.\nonumber$
Substituting: $C_{L_{opt}} = 0.8, E_{\max} = 16.66$, and $\gamma_{\max} = 5.27^{\circ}$.
The aerodynamic velocity will be given by:
$V = \sqrt{\dfrac{m \cdot g}{\tfrac{1}{2} \rho S_w C_{L_{opt}}}} = 119\ m/s.\nonumber$
From Equation ($\ref{eq7.3.117}$), $\dot{h}_e = V \cdot \gamma = 10.95\ m/s$.
3. We want to perform a horizontal turn at an altitude of $h = 0$ with a load factor $n = 2.5 = n_{\max}$, and with the velocity corresponding to the maximum aerodynamic efficiency in horizontal, steady, straight flight.
The equations governing the dynamics of the airplane in an horizontal turn are:
$T = D;\label{eq7.3.128}$
$mV \dot{\chi} = L \sin \mu;\label{eq7.3.129}$
$L \cos \mu = mg.\label{eq7.3.130}$
In a uniform (stationary) circular movement, it is well known that the tangential velocity is equal to the angular velocity $\dot{\chi}$ multiplied by the radius of turn $R$:
$V = \dot{\chi} R.$
Therefore, System ($\ref{eq7.3.128}$, $\ref{eq7.3.129}$, $\ref{eq7.3.130}$) can be rewritten as:
$T = D;\label{eq7.3.132}$
$n \sin \mu = \dfrac{V^2}{gR};\label{eq7.3.133}$
$n = \dfrac{1}{\cos \mu};\label{eq7.3.134}$
where $n = \tfrac{L}{mg}$ is the load factor.
(a). The required bank angle:
Therefore, looking at Equation ($\ref{eq7.3.134}$), it is straightforward to determine that the bank angle of turn is $\mu = 66.4^{\circ}$.
(b). Velocity and radius of turn.
First, we need to calculate the velocity corresponding to the maximum efficiency. As we have calculated before in Equation ($\ref{eq7.3.127}$), the coefficient of lift that generates maximum efficiency is the so-called optimal coefficient of lift, that is, $C_{L_{opt}} = \sqrt{\tfrac{C_{D_0}}{C_{D_i}}} = 0.8$. The velocity yields then:
$V = \sqrt{\dfrac{L}{\tfrac{1}{2} \rho_0 S_w C_{L_{opt}}}} = 161.57\ m/s.\nonumber$
Entering in Equation ($\ref{eq7.3.133}$) with $\mu = 66.4$ [deg] and $V = 161.57\ m/s$; $R = 1167\ m$.
(c) The required thrust.
Equation ($\ref{eq7.3.132}$) can be expressed as:
$T = \dfrac{1}{2} \rho_0 S_w V^2 C_{D_0} + \dfrac{L^2}{\tfrac{1}{2} \rho_0 V^2 S_w} C_{D_i}.\nonumber$
Since all values are known: $T = 94093\ N$
(d). Since $T \le T_{\max, av} (h = 0)$, the aircraft can perform the turn.
9. All this variables can be considered constant during take off.
10. Notice that given an altitude and a thrust setting, the aircraft can theoretically fly at two different velocities meanwhile those velocities lay between the minimum velocity (stall) and a maximum velocity (typically near the divergence velocity).
11. $g$ represents the force due to gravity.
12. Notice that we can convert mass in weight by simply multiplying by the force due to gravity.
13. Notice that given an altitude and a thrust setting, the aircraft can theoretically fly at two different velocities meanwhile those velocities lay between the minimum velocity (stall) and a maximum velocity (typically near the divergence velocity).
14. It refers to a value between 0 & 1 associated to the throttle level position.
7.4: References
[1] Anderson, J. (2012). Introduction to flight, seventh edition. McGraw-Hill.
[2] Etkin, B. and Reid, L. D. (1996). Dynamics of flight: stability and control. Wiley, New York.
[3] Franchini, S. and García, O. (2008). Introducción a la ingeniería aeroespacial. Escuela Universitaria de Ingeniería Técnica Aeronáutica, Universidad Politécnica de Madrid.
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Let us first try to roughly describe transport as that economical activity aimed at transporting people from one place to another. If one thinks on his/her daily activity, time dedicated to travel consumes an important share within a day (estimated in 10-15% on average). In selecting the transportation mean, one would think first on availability and then in a trade-off between time and cost (other metrics might come into play, such as confort, safety perception, environmental perception, etc.).
Air transportation has become paramount since it plays an integral role in our way of life. Commercial airlines allow millions of people every year to attend business conventions or take vacations around the globe. Air transportation also represents the fastest way to ship most types of cargo over long distances. Air transportation must be seen both as a business and as a technical and operational activity, in which many stakeholders are involved. Thus, air transport can be defined as follows:
Definition 8. 1: Definiton of air transport
Multi-stakeholder industrial added value chain, whose ultimate goal is to provide of the service of air travel from one point to another to an end user: the passenger.
Notice that typically the air transportation is seen as a multi-modal transportation between one point (home; office) to the origin airport and from the destination airport to another point (hotel; business center). Notice also that a value chain is a set of activities that firm operating in a specific industry performs in order to deliver a valuable product or service to the market. In particular the air transportation activity is participated by many stakeholders, including: airports, air navigation services providers, manufacturers (and its providers), airlines (and its service providers), etc.1 Thus, when one pays for a flight ticket, the price is distributed among the different stakeholders that contribute adding value to the product (service of flying), i.e., the airline as operator of the aircraft, the manufacturer as producer of the aircraft (in turn, paying all the engineering needed for the design, development and manufacturing of an aircraft), the airport as provider of take-off, landing, and processing service, the air navigation service provider as provider of air navigation services (communications, navigation, surveillance, ATM services, etc.), etc.
1. Notice that airports and air navigation service providers will be analysed in Chapter 9 and Chapter 10, respectively. In this chapter we will focus on manufacturers and airlines.
8.1.02: History
Providing a thorough history of air transport is out of the scope of this book. On the contrary, we will focus on briefly stating some relevant periods and associated milestones for the development of air transport. These periods include: Pioneer Period (with the first flight by Wright Brothers), the Interwar’s Periods (with the first airlines and the first north Atlantic Crossing), the Post War period (with the foundation of ICAO), the Jet Period (with the first Jet Aircraft), the Liberalization Period (with the de-regulation act in the United States), the Modern Aircraft Period (with the A320 as the first fly-by-wire aircraft and A350 and B787 including more and more composite materials), the Crisis Period (driving an model change, including low cost and mergings), and the Green transportation Period (a new period that seems to be starting, including more efficient engines and biofuels). Please, refer to Table 8.1 and Figures 8.1-8.3.
Figure 8.1: Air transport History: Pioneer and Interwar's Periods.
Figure 8.2: Air transport History: PostWar and Jet periods.
Figure 8.3: Air Transport: Liberalization and economic mature.
8.1.03: Facts and Figures
As it was introduced at the beginning of the Chapter, Air transportation plays an integral role in today’s society. Let us analyse some numbers that support this affirmation.
Socioeconomic importance of air transport
First, it should be highlighted its socioeconomic relevance. The following aspects should be considered:
• The air transportation industry (in some forums generally refereed to as aviation industry) is a high-technological industry with high economical impact, including high skilled jobs and an important contribution to richness and wellness (increases productivity by encouraging innovation).
• The air transportation industry is essential for globalisation, facilitating international meetings and the shipping of goods.
• The air transportation industry is an important instrument for regional integration, e.g., Islands, yet also a touristic catalyst.
• Due its inherent characteristics, the air transportation industry needs for huge amount of capital. This make it a key strategic asset for countries, including subsidizes and Government intervention.
• Linked to the later, it is used as international political instrument.
Quantitative Figures
We provide herein some numbers for Europe and US:
Europe: Roughly 30,000 flights fly over European skies on a daily basis; representing 11 million flights and 1,600 million passengers per year. From an economic perspective, the aviation industry is considered a strategic activity given its economic and societal impact. The air transport industry in Europe directly employs between 1.4 and 2 million jobs and contributes €110 billion (roughly 0.8%) to European gross domestic product (GDP). The total impacts (direct + indirect + induced) mean the air transport sector supports between 4.8 and 5.5 million jobs and contributed €510 billion (roughly 3.6%) to GDP in Europe.2 The aviation industry is an important economic asset for Europe. Moreover, as a sector, it invests heavily in research, development, and innovation (RDI).
United States: The aviation sector in EU today3 employs between 1.4 and 2 million people (desegregated numbers: air traffic management 65.000, airports 156.000, airlines 579.800, and aerospace 379.600: total employment in air transport was 800.800 vs. 379.600 in aerospace) and indirectly supports between 4.8 and 5.5 million jobs.
Needless to say, other sources to check this information (updated and extended to other regions if desired by the reader) include:
• Air Transport Action Group (ATAG), which provides a global vision of Air Transport;
• IATA, which also provides a global vision shifted towards the airline industry. IATA includes the WATS (World Air Transport Statistics) and the Economic Performance of the Airline Industry with data on fleets, airline rankings, demand, etc.
• Other sources (in this case pure data) include for instance: the ICAO data+ database; the USDOT (US Department of Transportation); and the MIT Airline Data Program.
2. Steer Davies Gleave: Study on employment and working conditions in air transport and airports, Final report 2015 & Aviation: Benefits Beyond Borders, Report prepared by Oxford Economics for ATAG, April 2014. EU’s GDP in 2014 was €14.000 bi.
3. Air Transport and Aerospace Education Synergies and Differences. A. Kazda. Workshop on education and training needs for aviation engineers and researchers in Europe; September 23 2015; Brussels. URL: http://www.airtn.eu/downloads/air-tr...pace-education—synerg.pdf
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First in Section 8.1, we will introduce the concept air transportation by defining it, briefly describing its history, and finally presenting some facts and figures aimed at providing a quantitative measure of its importance. Second, in Section 8.2, we will analyse the complex regulatory framework needed for reliable and safe air transportation. ICAO and IATA will be studied in Section 8.2.1 and Section 8.2.2, respectively. Third, for air transport economy we need to consider the performances of the aircraft studied in Chapter 7 and the particular characteristics of air transportation. Thus, this chapter will briefly focus on the types of aircraft and manufactures in Section 8.3, on the structure of costs of a typical airline in Section 8.4, and on aviation’s environmental fingerprint in Section 8.5. A good introductory reference is Navarro [7]. Thorough overviews are given, for instance, in Pindado Carrión [10] and Belobaba et al. [2].
08: Air transportation
With the advent of the aerial transportation mode, in the 20s-30s of the last century, new challenges were to be faced. It was a new mean, the Air, which introduced a mind change in the concept of borders and sovereignty. The vehicle, the aircraft, was also new at that time. Moreover, the transport (the flight) was international, something that added complexity. All in all, these revolutionary transportation mode brought in new modes of crime (hijacking), redefinition of the concept of sovereignty, intricate insurances, and the need to define the contract between the air carrier and the passenger.
Then, the question is how the first international flights were possible. Well, thanks to bilateral agreements: a vis-a-vis agreement dealing with commercial aspects. These agreements were referred to as Air Service Agreements (ASAs): Bilateral air transport agreement between two nations to allow international commercial transit and traffic between and over their territories.. Examples of ASAs include the first one, signed in 1913 between Germany and France, and the Bermudas ASAs signed in 1946 between United Kingdom and the United Stated of America.
Other agreements had a multilateral character, typically signed by a minimum number of countries and dealing with more general aspects. The first multilateral agreement is that of Paris (Paris Convention) in 1919, with a jurisdiccional/political character. It established: the sovereignty of nations over the air; the nationality of the aircraft; universal airworthiness rules; and the rights of the state to take measures on safety. Other multilateral agreements include the Warsaw Treaty in 1929 (related to Air Transport), and the Chicago Convention in 1944 (the sucesor of Paris Convention). Indeed, Chicago’s Convention is a fundamental milestone for the development of civil aviation: Among other things, it gave birth to the International Civil Aviation Organization (ICAO) (See Section 8.2.1).
Originally, airlines were state owned (flag carriers), operating as monopolists in domestic, highly protected markets. This started to change in 1978, with the Deregulation act signed by President Nixon in the United States. It modified the regulation of the US domestic market in the following directions: US airlines had freedom to enter/exit any US domestic market; each airline could determine its frequencies and number of seats in any domestic market; each airline could determine the airfares and number of seats per airfare class. This was revolutionary, the reader should notice that until then the origin- destination pairs to serve, the number of frequencies and seats, and even the fair were regulated. This milestone triggered modifications in the ASAs, moving from traditional ASAs to Open Market ASAs (earliest in 1978-1979). Other landmarks in international airline regulation include: the three-stage liberalization of the intra-European Union market (1988-1993); the Asia-Pacific Economic Community (APEC) multilateral ASA; and the Open Skies Agreement between the EU and the whole USA in 2007. The extent (in the sense the signatory countries allow the others certain rights) of these agreements are typically based on what is known as the freedoms of the air (see Figure 8.4 and Table 8.2).
8.02: Regulatory framework
8.2.1 ICAO5
The International Civil Aviation Organization (ICAO) is an agency of the United Nations, created in 1944 to promote the safe and orderly development of international civil aviation. It sets standards and regulations necessary for aviation safety, security, efficiency, and regularity, as well as for aviation environmental protection.
Why are Standards Necessary?
Air transportation (and in turn the whole aviation industry) is made possible by the existence of universally accepted standards known as Standards and Recommended Practices, or SARPs. SARPs cover all technical and operational aspects of international civil aviation, such as safety, personnel licensing, operation of aircraft, aerodromes, air traffic services, accident investigation, and the environment. Without SARPs, our aviation system would be at best chaotic and at worst unsafe.
How it works
The constitution of ICAO is the Convention on International Civil Aviation, drawn up by a conference in Chicago in December 1944, and to which each ICAO Contracting State is a party. The Organization is made up of an Assembly, a Council of limited membership with various subordinate bodies and a Secretariat. The Assembly, composed of representatives from all Contracting States, is the sovereign body of ICAO. It meets every three years, reviewing in detail the work of the Organization and setting policy for the coming years.
Foundation of ICAO
The consequence of the studies initiated by the US and subsequent consultations between the Major Allies was that the US government extended an invitation to 55 states or authorities to attend, in November 1944, an International Civil Aviation Conference in Chicago. Fifty four states attended this conference end of which a Convention on International Civil Aviation was signed by 52 States set up the permanent International Civil Aviation Organization (ICAO) as a mean to secure international cooperation an highest possible degree of uniformity in regulations and standards, procedures, and organization regarding civil aviation matters.
The most important work accomplished by the Chicago Conference was in the technical field because the Conference laid the foundation for a set of rules and regulations regarding air navigation as a whole which brought safety in flying a great step forward and paved the way for the application of a common air navigation system throughout the world.
From the very assumption of activities of ICAO, it was realized that the work of the Secretariat, especially in the technical field, would have to cover the following major activities: those which covered generally applicable rules and regulations concerning training and licensing of aeronautical personnel both in the air and on the ground, communication systems and procedures, rules for the air and air traffic control systems and practices, airworthiness requirements for aircraft engaged in international air navigation as well as their registration and identification, aeronautical meteorology, and maps and charts. For obvious reasons, these aspects required uniformity on a world-wide scale if truly international air navigation was to become a possibility.
Chicagos’s convention
In response to the invitation of the United States Government, representatives of 54 nations met at Chicago from November 1 to December 7, 1944, to make arrangements for the immediate establishment of provisional world air routes and services and to set up an interim council to collect, record and study data concerning international aviation and to make recommendations for its improvement. The Conference was also invited to discuss the principles and methods to be followed in the adoption of a new aviation convention.
Convention on International Civil Aviation (also known as Chicago Convention), was signed on 7 December 1944 by 52 States. Pending ratification of the Convention by 26 States, the Provisional International Civil Aviation Organization (PICAO) was established. It functioned from 6 June 1945 until 4 April 1947. By 5 March 1947 the 26th ratification was received. ICAO came into being on 4 April 1947. In October of the same year, ICAO became a specialized agency of the United Nations linked to Economic and Social Council (ECOSOC). The Convention on International Civil Aviation set forth the purpose of ICAO:
WHEREAS the future development of international civil aviation can greatly help to create and preserve friendship and understanding among the nations and peoples of the world, yet its abuse can become a threat to the general security; and
WHEREAS it is desirable to avoid friction and to promote that co- operation between nations and peoples upon which the peace of the world depends;
THEREFORE, the undersigned governments having agreed on certain principles and arrangements in order that international civil aviation may be developed in a safe and orderly manner and that international air transport services may be established on the basis of equality of opportunity and operated soundly and economically;
Have accordingly concluded this Convention to that end.
The Convention has since been revised eight times (in 1959, 1963, 1969, 1975, 1980, 1997, 2000 and 2006). It is constituted by a preamble and 4 parts:
• Air navigation.
• Organization of the international civil aviation.
• International air transport.
• Final dispositions.
Some important articles are:
• Article 1: Every state has complete and exclusive sovereignty over airspace above its territory.
• Article 5: (Non-scheduled flights over state’s territory): The aircraft of states, other than scheduled international air services, have the right to make flights across state’s territories and to make stops without obtaining prior permission. However, the state may require the aircraft to make a landing.
• Article 6: (Scheduled air services) No scheduled international air service may be operated over or into the territory of a contracting State, except with the special permission or other authorization of that State.
• Article 10: (Landing at customs airports): The state can require that landing to be at a designated customs airport and similarly departure from the territory can be required to be from a designated customs airport.
• Article 13: (Entry and clearance regulations) A state’s laws and regulations regarding the admission and departure of passengers, crew or cargo from aircraft shall be complied with on arrival, upon departure and whilst within the territory of that state.
• Article 24: Aircraft flying to, from or across, the territory of a state shall be admitted temporarily free of duty. Fuel, oil, spare parts, regular equipment and aircraft stores retained on board are also exempt custom duty, inspection fees or similar charges.
• Article 29: Before an international flight, the pilot in command must ensure that the aircraft is airworthy, duly registered and that the relevant certificates are on board the aircraft. The required documents are: certificate of registration; certificate of airworthiness; passenger names; place of boarding and destination; crew licenses ; journey logbook; radio license; cargo manifest.
• Article 30: The aircraft of a state flying in or over the territory of another state shall only carry radios licensed and used in accordance with the regulations of the state in which the aircraft is registered. The radios may only be used by members of the flight crew suitably licensed by the state in which the aircraft is registered.
• Article 32: the pilot and crew of every aircraft engaged in international aviation must have certificates of competency and licenses issued or validated by the state in which the aircraft is registered.
• Article 33: (Recognition of certificates and licenses) Certificates of Airworthiness, certificates of competency and licenses issued or validated by the state in which the aircraft is registered, shall be recognized as valid by other states. The requirements for issue of those Certificates or Airworthiness, certificates of competency or licenses must be equal to or above the minimum standards established by the Convention.
• Article 40: No aircraft or personnel with endorsed licenses or certificate will engage in international navigation except with the permission of the state or states whose territory is entered. Any license holder who does not satisfy international standard relating to that license or certificate shall have attached to or endorsed on that license information regarding the particulars in which he does not satisfy those standards.
The Convention is supported by eighteen annexes containing standards and recommended practices (SARPs). The annexes are amended regularly by ICAO and are as follows:
• Annex 1: Personnel Licensing
• Annex 2: Rules of the Air
• Annex 3: Meteorological Service for International Air Navigation
– Vol I: Core SARPs
– Vol II: Appendices and Attachments
• Annex 4: Aeronautical Charts
• Annex 5: Units of Measurement to be used in Air and Ground Operations
• Annex 6: Operation of Aircraft
– Part I: International Commercial Air Transport: Aeroplanes – Part II: International General Aviation: Aeroplanes
– Part III: International Operations: Helicopters
• Annex 7: Aircraft Nationality and Registration Marks
• Annex 8: Airworthiness of Aircraft
• Annex 9: Facilitation
• Annex 10: Aeronautical Telecommunications
– Vol I: Radio Navigation Aids
– Vol II: Communication Procedures including those with PANS status – Vol III: Communication Systems
∗ Part I: Digital Data Communication Systems
∗ Part II: Voice Communication Systems
– Vol IV: Surveillance Radar and Collision Avoidance Systems – Vol V: Aeronautical Radio Frequency Spectrum Utilization
• Annex 11: Air Traffic Services: Air Traffic Control Service, Flight Information Service and Alerting Service
• Annex 12: Search and Rescue
• Annex 13: Aircraft Accident and Incident Investigation
• Annex 14: Aerodromes
– Vol I: Aerodrome Design and Operations – Vol II: Heliports
• Annex 15: Aeronautical Information Services
• Annex 16: Environmental Protection
– Vol I: Aircraft Noise
– Vol II: Aircraft Engine Emissions
• Annex 17: Security: Safeguarding International Civil Aviation Against Acts of
Unlawful Interference
• Annex 18: The Safe Transport of Dangerous Goods by Air
• Annex 19: Safety Management
Figure 8.4: Freedoms of the Air.
Freedoms of the air
Table 8.2 presents the freedoms of the air.6 Chicago Convention signing states recognize each other the 1st and the 2nd. ICAO officially recognizes the first five "freedoms" as such. Then, the 6th to the 9th (also referred to as full cabotage) are included in some ASAs. For instance full cabotage applies for European Countries within European airspace.
5. The information included in this section has been retrieved from ICAO’swebsite @ http://www.icao.int/Pages/icao-in-brief.aspx.
6. According to the Manual on the Regulation of International Air Transport (Doc 9626, Part 4)
8.2.02: IATA
8.2.2 IATA8
Air transport is one of the most dynamic industries in the world. The International Air Transport Association (IATA) is its global trade organization. Over 60 years, IATA has developed the commercial standards that built a global industry. Today, IATA’s mission is to represent, lead, and serve the airline industry. Its members comprise some 240 airlines representing 84% of total air traffic.
IATA seeks to improve understanding of the industry among decision makers and increase awareness of the benefits that aviation brings to national and global economies. It ensures that people and goods can move around the global airline network as easily as if they were on a single airline in a single country. In addition, it provides essential professional support to all industry stakeholders with a wide range of products and expert services, such as publications, training and consulting. IATA’s financial systems also help carriers and the travel industry maximize revenues.
For consumers, IATA simplifies the travel and shipping processes, while keeping costs down. Passengers can make one telephone call to reserve a ticket, pay in one currency and then use the ticket on several airlines in several countries. IATA allows airlines to operate safely, securely, efficiently and economically under clearly defined rules.
History
IATA was founded in Havana, Cuba, in April 1945. It was the prime vehicle for inter- airline cooperation in promoting safe, reliable, secure, and economical air services. At the founding, its main goals were:
• to promote safe, regular, and economical air transport for the benefit of the peoples of the world, to foster air commerce, and to study the problems connected therewith;
• to provide means for collaboration among the air transport enterprises engaged directly or indirectly in international air transport service; and
• to cooperate with the newly created International Civil Aviation Organization (ICAO - the specialized United Nations agency for civil aviation) and other international organizations.
The international scheduled air transport industry is now more than 100 times larger than it was in 1945. Few industries can match the dynamism of that growth, which would have been much less spectacular without the standards, practices, and procedures developed within IATA.
8. The information included in this section has been retrieved from IATA’swebsite @ http://www.iata.org/about/Pages/index.aspx.
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As it has been exposed in the two previous sections, ICAO and IATA are the two fundamental international organizations that provide a juridic framework to all stakeholders in order them to carry out the air transportation business under reliable, safe, and efficient standards. Intuitively, one can think straightforward that one of the key elements of air transportation are aircraft. Therefore, we move on to analyze the market of aircraft for commercial air transportation. We will analyze the main manufactures, the fundamental aircraft, its prices, and future trends in the industry.
8.03: The market of aircraft for commercial air transportation
In a wide sense, the current market of commercial aircraft is dominated by four major manufacturers of civil transportation aircraft:
• Airbus, based in Europe.
• Boeing, based in the United States.
• Bombardier, based in Canada.
• Embraer, based in Brazil.
Airbus: Airbus9 is an aircraft manufacturing subsidiary of EADS, a European aerospace company. Based in Toulouse, and with significant activity across Europe, the company produces more than half of the world’s jet airliners. Airbus began as a consortium of aerospace manufacturers, Airbus Industrie. The consolidation of European defence and aerospace companies in 1999 and 2000 allowed the establishment of a simplified joint- stock company in 2001, owned by EADS (80%) and British Aerospace (BAE) Systems (20%). In 1006, BAE sold its shareholding to EADS. Airbus employs around 52,000 people at sixteen sites in four European Union countries: France, Germany, the United Kingdom, and Spain. Final assembly production is at Toulouse (France), Hamburg (Germany), Seville (Spain) and, since 2009, Tianjin (People’s Republic of China). The company produced and markets the first commercially viable fly-by-wire airliner, the Airbus A320, and the world’s largest airliner, the A380.
Figure 8.5: Aircraft manufacturers.
Boeing: The Boeing Company10 is an American multinational aerospace and defense corporation, founded in 1916 in Seattle, Washington. Boeing has expanded over the years, merging with McDonnell Douglas in 1997. Boeing is made up of multiple business units, which are Boeing Commercial Airplanes (BCA); Boeing Defense, Space & Security (BDS); Engineering, Operations & Technology; Boeing Capital; and Boeing Shared Services Group. Boeing is among the largest global aircraft manufacturers by revenue, orders and deliveries, and the third largest aerospace and defense contractor in the world based on defense-related revenue. Boeing is the largest exporter by value in the United States. Its Boeing 737 has resulted (counting with the different versions and evolutions) the all times most sold aircraft type.
Embraer: Embraer11 is a Brazilian aerospace conglomerate that produces commercial, military, and executive aircraft and provides aeronautical services. Embraer is the third-largest commercial aircraft manufacturing in the world, and the forth-largest aircraft manufacturing when including business jets into account, and it is Brazil’s top exporter of industrial products.
Bombardier: Bombardier12 is a Canadian conglomerate. It is a large manufacturer of regional aircraft and business jets. Its headquarters are in Montreal, Canada.
9. Information retrieved from http://en.wikipedia.org/wiki/Airbus.
10. Information retrieved from http://en.wikipedia.org/wiki/Boeing.
11. Information retrieved from http://en.wikipedia.org/wiki/Embraer.
12. Information retrieved from http://en.wikipedia.org/wiki/Bombardier_Inc..
8.3.02: Types of aircraft
Boeing and Airbus concentrate on wide-body and narrow-body jet airliners, while Bombardier and Embraer concentrate on regional airliners. Large networks of specialized parts suppliers from around the world support these manufacturers, who sometimes provide only the initial design and final assembly in their own plants.
Jet aircraft can be generally divided into:
• Medium/long-haul (> 100 seats)
– Wide body (two decks): A380; B787; A350; B747; A340 family.
– Narrow body (one deck): B737 family and A320 family.
• Short haul (< 100 seats)
– Bombardier CRJ700/900.
– Embraer 170-175-190-195.
Table 8.3: Long-haul aircraft specifications. MTOW in tones, Range in kilometers, longitude and wing-span in meters. The Mach number corresponds to long-range operating Mach and max
Table 8.4: Medium-haul aircraft specifications. MTOW in tones, Range in kilometers, longitude and wing-span in meters. The Mach number corresponds to long-range operating Mach and maximum operating Mach, respectively. Data retrieved from http://www.airliners.net.
Table 8.5: Regional aircraft specifications. MTOW in tones, range in kilometers with typical pax, longitude and wing-span in meters. Mach number corresponds to maximum cruising speed. The velocity is given in km/h and corresponds to the maximum cruising velocity. Data retrieved from http://www.airliners.net.
Regional propellers are also short range with typically 30-80 seats. Table 8.3, Table 8.4, and Table 8.5 show the specifications of different aircraft.
Table 8.6: Airbus medium-haul aircraft 2012 average prices list (million USD). Data retrieved from Airbus.
Table 8.7: Airbus long-haul aircraft 2012 average prices list (million USD). Data retrieved from Airbus.
Table 8.6 and Table 8.7 show the average prices of Airbus manufactured aircraft. As it can be observed, medium-haul aircraft have a price of an order of 100 million USD, while the cost of the new A380 is around 400 million USD. Obviously, these prices depend upon many factors, such for instance, the configuration selected by the airline, commercial agreements, the currency exchange rates (note that aircraft are sold in USD currency), etc. Therefore, due to such high value, the policies of aircraft acquisition are key for the viability of the airline. In particular, airlines might acquire the aircraft or simply use it within a leasing or renting formula. Also, after some years of use, old aircraft are sold in the second hand market. In this way, airlines might rotate quickly their fleet in order to count with the newest technological advances.
Figure 8.6: Airbus A320 family.
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Short-term trends point towards a next generation of medium-haul aircraft. Both Boeing and Airbus have recently relaunched evolved versions of their successful families A320 and B737, respectively. Also, China is trying to get into this niche with the design of its first medium-haul airliner.
Table 8.8: A320neo family specifications. max refers to the maximum capacity layout; typ refers to the typical seats layout. Maximum range refers to fully loaded. Data retrieved from Wikipedia A320neo.
A320neo family: The Airbus A320neo13 is a series of enhanced versions of A320 family under production by Airbus. It entered in service with Lufthansa in January 2015. The letters "neo" stand for "New Engine Option". The main change is the use of the larger and more efficient engines which results in 15% less fuel consumption, 8% lower operating costs, less noise production, and a reduction of NOx by at least 10% compared to the A320 series according to Airbus. Two power plants will be available: either the CFM International LEAP-X or the Pratt & Whitney PW1000G. The airframe will also receive some modifications, including the addition of "Sharklet" wingtips to reduce drag and interior modifications for the passengers comfort such as larger luggage spaces and an improved air purification system. The A320neo family specifications can be consulted in Table 8.8.
Table 8.9: B737 MAX family specifications. Data retrieved from Wikipedia B737 MAX.
B737 MAX: The Boeing 737 MAX14 is a new family of aircraft being developed by Boeing in order to replace the current Boeing 737 generation family. Its first flew was on January 29, 2016, and the first delivery in May 2017. The primary change will be the use of the larger and more efficient CFM International LEAP-1B engines. The airframe is to receive some modifications as well. The 737 MAX first delivery in 2017 was 50 years after the 737 first flew. The original three variants of the new family are the 737 MAX 7, the 737 MAX 8 and the 737 MAX 9, which are based on the 737-700, 737-800 and 737-900ER, respectively, which are the best selling versions of the current 737 generation family. A fourth variant, the 737Max 10 was later on proposed to compete with the A321neo. Boeing claims the 737 MAX will provide a 16% lower fuel burn than the current Airbus A320, and 4% lower than the Airbus A320neo. The B737 MAX family specifications can be consulted in Table 8.9.
Comac C919: The Comac C91915 is a planned family of 168-190 seat narrow-body airliners to be built by the Commercial Aircraft Corporation of China (Comac). It will be the largest commercial airliner designed and built in China since the defunct Shanghai Y-10. Its first flight took place in May 2017, with first deliveries scheduled for 2019-2020. The C919 forms part of China’s long-term goal to break Airbus and Boeing’s duopoly, and is intended to compete against Airbus A320neo and the Boeing 737 MAX.
Dimensions of the C919 are very similar to the Airbus A320. Its fuselage will be 3.96 meters wide, and 4.166 meters high. The wingspan will be 33.6 meters. Its cruise speed will be Mach 0.785 and it will have a maximum altitude of 12,100 meters. There will be two variants. The standard version will have a range of 4,075 km, with the extended-range version able to fly 5,555 km. The capacity will go from 156 passengers (with two classes) to 174 passengers (1 class and maximum density of seats).
13. INformation retrieved from http://en.wikipedia.org/wiki/A320_neo
14. Data retrieved from http://en.wikipedia.org/wiki/Boeing_737_MAX
15. Data retrieved from http://en.wikipedia.org/wiki/Comac_C919
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The calculus of economic costs constitutes a necessity within every single enterprise. It is valuable for measuring the efficiency of the different areas, deciding on new investments, and obviously to set the prices for the supplied products (in the case of airlines, services) is based on desired profits and estimated forecasts. Two important references in airlines’ economics are Doganis [5] and Doganis [4].
Focusing on costs, the breaking down taxonomy varies depending on the company. However they are typically adjusted to the cost classification stablished by ICAO. A fundamental division arises when dividing operational costs and non operational costs (also refereed to as operative and non-operative costs):
• Operational costs: Expenses associated with administering a business on a daily basis. Operating costs include both fixed costs and variable costs. According to a somehow canonical definition, fixed costs, such as infrastructures or advertising, remain the same regardless of the number of products produced; variable costs, such as materials or labour, can vary according to how much product is produced. In airlines terminology, they are referred to as Direct Operational Costs (DOC) and Indirect Operational Costs (IOC):
– DOC are related to the operation of the aircraft
– IOC are related to the running of the airline company and, therefore, regardless of the aircraft operation.
• Non-operational costs: associated to expenses not related to day to day operations, typically financial costs.
Table 8.10: Cost structure of a typical airline.
We could keep breaking down the different costs, however with this general framework we can expose a typical taxonomy of the cost structure of an airline company as illustrated in Table 8.10.
The non-operational costs are also refereed to as capital costs or simply financial costs. As pointed out above, they can be divided into:
• Loans amortization.
• Capital interests.
The concept loans amortizations refers to the distribution of an acquisition in different payment periods. This fact implies typically interests.
8.04: Airlines' cost structure
As pointed out above, it is under common agreement to establish a classification based in fixed costs ($c_f$) and variable costs ($c_v$):
$c = c_f + c_v BT,$
where c are the operational costs, and $BT$ refers to Block Time.16 The difference between $BT$ and flight time is small in long flights, but it can be important in short flights. BT can be expressed as a function of the aircraft range, $R$, as follows:
$BT = A + B \cdot R + C \cdot \ln R.$
where $A$, $B$ and $C$ are parameters set by the airline. A linear approximation is usually adopted:
$BT = A + B \cdot R.$
Therefore the operation costs can be expressed either as a function of block time or range:
$c = c_f + c_v BT = c_f + c_v (A + B \cdot R) = c_f' + c_v' \cdot R.$
Using these expressions one can generate the so called cost indicators: cost per unity of time, range, payload, or any other parameter to be taken into account at accounting level. Three of these cost indicators are:
The hourly costs:
$\dfrac{c}{BT} = c_v + \dfrac{c_f}{A + B \cdot R}.$
The kilometric costs:
$\dfrac{c}{R} = \dfrac{c_f'}{R} + c_v' = \dfrac{c_f'}{R} + Bc_v.$
The cost per offered ton and kilometer ($OTK$):
$\dfrac{c}{(MPL) \cdot R} = \dfrac{c_f'}{(MPL) \cdot R} + \dfrac{c_v B}{MPL},$
where $MPL$ refers to maximum payload.
Besides this cost ratios, there are also some other ratios in which airlines generally measure their operations. These are Passenger Kilometer Carried ($PKC$), the passenger carried per kilometer flown; Seat Kilometer Offered ($SKO$), seats offered per kilometer flown; the Factor of Occupancy ($FO=PKC/SKO$):
$PKC = OTK \cdot FO.$
These three ratios, together with the above described cost ratios represent the best metric to analyze the competitiveness of an airline.
Structure of the operational costs
We can identify four main groups:
• Labour;
• Fuel;
• Aircraft rentals and depreciation & amortization; and
• Others: Maintenance, landing and air navigation fees, handling, ticketing, etc.
Table 8.11: Evolution of airlines’ operational costs 2001-2008. Source IATA.
Figure 8.7: European % share of airline operational costs in 2008. Data retrieved from IATA.
Labour costs Labour has been traditionally the main budget item not only for airline companies, but also for any other company. However, the world has shifted to one more deregulated and globalized. This fact has lead to a less stable labour market, in which airlines companies are not a exception.
Therefore, meanwhile traditional flag companies had a very consolidated labour staff, the airline deregulation in the 70s brought the appearance of fierce competitors as the low cost companies were (and still are). In that sense all companies have been making big efforts in reducing their labour cost as we can see in Table 8.11 and Figure 8.7 and this tendency will continue growing.
Nowadays one can find staff with the same qualification and responsibilities in very different contractual conditions, or even people that works for free (as it is the case of many pilots pursuing the aircraft type habilitation).
Figure 8.8: Evolution of the price of petroleum 1987-2012. © TomTheHand / Wikimedia Commons / CC-BY-SA-3.0.
Fuel costs: The propellant used in aviation are typically kerosenes, which it is produced derived from the crude oil (or petroleum). The crude oil is a natural, limited resource which is under high demand. It is also a focus for geo-politic conflicts. Therefore its price is highly volatile in the short term and this logically affects airline companies in their operating cost structures. However, as Figure 8.8 illustrates, the long-term evolution of petroleum price has shown since 2000 a clear increasing tendency in both real and nominal value. This increase in the price of fuel has modified substantially the operational cost structure of airline companies; in some cases fuel expenses represent 30-40% of the total operating costs. Notice that in Europe the impact was mitigated due to the strength of Euro with respect to the Dollar, but most likely today in European companies the weight of fuel costs is as well in the 30-40% of the total operating costs.
Since the crude oil is limited as a natural resource and day after day the extractions are more expensive, and since the demand is increasing due to the rapid evolution of countries such China, India, Brazil, etc., forecasts predict that this tendency or price increase will continue. Therefore, airlines will have to either increase tickets (as they have done); reduce other costs items; and encourage research on the direction of alternative fuels.
Maintenance: The maintenance costs depend on the maintenance program approved by the company, the complexity of the aircraft design (number of pieces), the reliability of the aircraft, and the price of the spare pieces. The maintenance is carried out at different levels after the dictation of an inspection:
• Routine check: At the gate after every single flight (before first flight or at each stop when in transit). It consists of visual inspection; fluid levels; tyres and brakes; and emergency equipment. The standard duration is around 45 to 1 hour, however the minimum required time to perform it is 20 minutes.
• Check A: At the gate every 500 flight hours. It consists of routine light maintenance and engine inspection. The standard duration is around 8-10 hours (a night).
• Check B: At the gate every 1500 flight hours. It is similar to A check but with different tasks and may occur between consecutive A checks. The duration is between 10 hours and 1 day.
• Check C: In the hangar every 15-18 moths on service. It consists of a structural inspection of airframe and opening access panels; routine and non routine maintenance; and run-in tests. The duration ranges between 3 days and 1 week.
• Check D: In the hangar after some years (around 8) of service. This last level requires a complete revision (overhaul). It consists of major structural inspection of airframe after paint removal; engines, landing gear, and flaps removed; instruments, electronic, and electrical equipment removed; interior fittings (seats and panels) removed; hydraulic and pneumatic components removed. The duration is around 1 month with the aircraft out of service.
The traditional companies used to have internal maintenance services, typically hosted in their hubs. However, this strategy is shifting due to different flight strategies (point to point) and also to take advantage of reduced cost in determined geographic zones. Therefore, nowadays many flight companies externalize these services. This is the last aspect in which the low cost strategies have modified the air transportation industry.
Regarding spare pieces, meanwhile in other industries do not exist a clear regulation in regard of spare pieces, the aeronautical industry and the american government have been pioneers regulating the market of spare pieces. Both original manufactures and spare companies can provide spare pieces. In order private companies to be allowed to commerce sparse pieces they need an habilitation named PMA (Parts Manufacturer Approval), while the original pieces referred to as OEMs (Original Equipment Manufactures).
Handling: The handling services consist of the assistance on the ground given to aircraft, passenger, and freight, so that a stopover in any airport is carried out properly. Handling include devices, vehicles, and services such fuel refilling, aircraft guidance, luggage management, or cabin cleaning.
The handling services can be grouped into:
• Aircraft handling.
• Operational handling.
• Payload handling.
Aircraft handling defines the servicing of an aircraft while it is on the ground, usually parked at a terminal gate of an airport. It includes:
• Operations: includes communications, download and load of cargo, passenger transportation, assistance for turn on, pushback, etc.;
• Cleaning: exterior cleaning, cabin clean up, restrooms, ice or snow, etc.; and
• Fill in and out: Fuel, oil, electricity, air conditioning, etc.;
Operational handling assits in:
• Administrative assistance.
• Flight operations: includes dispatch preparation and modifications on the flight plan;
• Line maintenance: includes the maintenance prior departure, spare services and reservation of parking lot or hangar;
• Catering: includes the unloading of unused food and drink from the aircraft, and the loading of fresh food and drink for passengers and crew. Airline meals are typically delivered in trolleys. Empty or trash-filled trolley from the previous flight are replaced with fresh ones. Meals are prepared mostly on the ground in order to minimize the amount of preparation (apart from chilling or reheating) required in the air.
Payload handling refers to:
• Passenger handling: includes assistance in departure, arrival and transit, tickets and passport control, check-in, and luggage transportation towards the classification area;
• Classification, load, and download of luggage;
• Freight and main services;
• Transportation of passengers, crew and payload between different airport terminals.
As in the case of maintenance, the handling services used to be handled by flag companies. However, this activity was liberalized (in Europe, in the 90s) and it is being increasingly outsourced. As a result, many independent company have arisen in past 10-15 years.
Landing and air navigation fees: According to $IATA$, the landing fees and air navigation costs represent around 10% of the operating cost for airline companies. Each nation establishes navigation fees due to services provided when overflying an airspace region under its sovereignty. Moreover, each airport establishes landing fees for the services provided to the aircraft when approaching and landing. In Spain, $AENA$ charges for approaching (notice that can be interpreted as a landing fee).
The landing fee is established taking into consideration the MTOW of the aircraft and the type of flight (Schengen, International, etc.). $AENA$ gives therefore a unitary fee that must be multiplied by the aircraft $MTOW$. The formula is as follows:
$L_{fee} = u_l \cdot (\dfrac{MTOW}{50} )^{0.9},$
where $L_{fee}$ is the total landing fee, $u_l$ is the unitary landing fee and $MTOW$ is the maximum take off weight (in tons) of the aircraft. For instance, this unitary fee depends on the airport and ranges 12 to 17 €.
On the other hand, the air navigation fees in Europe are invoiced and charged by Eurocontrol by means of the following formula.
$\text{Navigation fee} = \text{unit rate} \cdot \text{distance coef} \cdot \text{weight coef}$
where the unit rate is established in the different European $FIR/UIR$17. For instance, in Spain, the unit rates of $FIR$ Madrid, $FIR$ Barcelona and $FIR$ Canarias, respectively, 71.84 €, 71.84 € and 58.52 €. The distance coefficient is the orthodromic distance (in nautical miles) over 100. The weight coefficient is $\sqrt{MTOW/50}$. Therefore, the navigation fee results in:
$\text{Navigation fee} = \text{unit rate} \cdot \dfrac{d}{100} \cdot \sqrt{\dfrac{MTOW}{50}}.$
Depreciation: The depreciation of an aircraft (the most important good airline companies have in their accounting) is typically imposed by the national (or international) accounting regulations, and it is typically associated to the following factors:
• Use.
• The course of time.
• Technological obsolescence.
The Use, corresponding to flight hour (also referred to as block hour), could be included somehow into the DOC. The depreciation due to the course of time is due to the loss of efficiency with respect to more modern aircraft. Last but not least, the incorporation by the competitors of a technological breakthrough (such, for instance, Airbus with the fly-by-wire) implies that the aircraft get depreciated immediately in the market.
The estimation of depreciation set the pace for fleet renovation, and it is obviously association to the utility life of the aircraft. A wide-body jet is typically depreciated to a residual value of (0-10%) in 14-20 years. The utility life of an aircraft is set to 30 years.
Aircraft acquisition: The acquisition of an aircraft is a costly financial operation, indeed the companies typically acquire several aircraft at the same time, not only one. These investments must financed by means of bank loans (also by increases of capital, emissions of bonds and obligations, etc.), which imply interests.
Other forms of aircraft disposition are the operational leasing (a simply renting) and the financial leasing (renting with the right of formal acquisition). Among the operational leasing there exist different types:
• Dry leasing: the aircraft is all set to be operated, but it does not include crew, maintenance, nor fuel. Sometimes the insurance is neither included.
• Wet leasing: like dry leasing but including crew.
• $ACMI$ leasing: includes Aircraft, Crew, Maintenance and Insurance.
• Charter leasing: Includes everything, even airport and air navigation fees.
There exist important leasing companies, such $GECAS$, $ILFC$, Boeing Capital Corp or $CIT$ group. Notice that approximately half of the total orders made to the manufactures correspond to leasing companies. The financial leasing is also very extended. The only difference with operational leasing is that they include a policy with the operator’s right to acquire the aircraft at a preset date and price.
Insurances: An insurance is a practice or arrangement by which a company or government agency provides a guarantee of compensation for specified loss, damage, illness, or death in return for payment of a premium. The characteristic of an insurance contract is the displacement of a risk by means of paying a price.
The aeronautical insurance, when compared to maritime or terrestrial, have some peculiarities: The reality of air traffic proofs that air accidents occur with rather low regularity, which makes difficult to apply the rules of big numbers. Moreover, exceptionally an accident produces partial damage, but, on the other hand, catastrophic damages including death of crew and passengers, resulting in high compensatory payments. In these circumstances, the insurance companies have agreed to subscribe common insurances so that the risk is hold by a pool of insurance companies.
16. The time in block hours is the time between the instant in which the aircraft is pulled out in the platform and the instant in which the aircraft parks at the destination airport. It includes, therefore, taxi out and taxi in.
17. Flying Information Region and Upper Information Region. The meanings of these regions will be studied in Chapter 10.
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A simple description of air transportation could be based on three elements: the aircraft, the airport, and the air navigation system. The first transports people and goods; airports allow passengers and goods to change transportation mode; the later provides services to ensure air operations are performed in a safe way. The activities of each of these three elements have a characteristic environmental impact, including construction, life-cycle, and reposition. A introductory reference on the topic is BENITO and BENITO [3]. Even though the different environmental impact sources will be briefly described, the focus will be on aircraft operations’ environmental fingerprint, in particular to its contribution to climate change. The reader is referred to Schumann [14] for a recent, thorough overview.
8.05: Environmental impact
Attending to its geographical range, the different impact sources can be classified into Benito and Benito [3]:
• Local effects
– Noise.
– Local air pollution.
– Use of surrounding areas.
• Global effects
– Consumption of non-recyclable materials.
– Use of airspace and radio-electric spectrum.
– Contribution to climate change.
Local effects referred to those effects that are only perceptible in the vicinity of airports. This includes noise nuisance due to aircraft operations (mainly take off and landing), the air pollution due to the airport activity, and also the use of areas for the purpose of airport activity, e.g, areas with bird colonies or natural interest.
On the other hand, global effects refer to those effects that affect the sustainability of the planet worldwide. Among this, we can cite the consumption of non-recyclable materials that are finite and, moreover, need to be stored somewhere after the life-cycle, the use of airspace by aircraft and electromagnetic waves emitted by navigation services and aircraft to provide communication, navigation, and surveillance services, and the contribution of the industry to global warming.
In the sequel, the focus will be on aircraft operations’ environmental impact, namely noise and emissions (\(CO_2\), \(NO_x\), etc.) that contribute to climate change.
8.5.02: Aircraft operations' environmental fingerprint
Noise
Noise nuisance is an important environmental impact in the vicinity of airports. The problem is not related to an isolated take-off or landing operation, but as a consequence of the total set of departures and arrivals taking place in the airport daily. In order to understand the problem, try to empathize with a neighborhood (including hospitals, schools, houses, etc.) that has to bear systematically with an important amount of noise. To quantify it, the decibel [Db] is used. In order to provide a qualitative reference, it is worth mentioning that a typical commercial aircraft during take off emits 130 dB; the pain threshold is 140 dB; a launcher during take off is 180 dB; a concert is 110 dB; a train 80 dB; a conversation 40 dB; etc.
The most important noise emission sources within an aircraft are due to the engines, which work at high power settings during take off and initial climb. The fundamental contribution to this noise is due to rotatory elements (compressor, turbine, fans, etc.). The second fundamental contribution is due to the exhausted jet in case of turbojets. Moreover, there is also a so-called aerodynamic noise, coming form the wing, the fuselage, the empennage, and the landing gear as the aircraft flies. Sound waves propagate in the air at the speed of sound. The intensity that an observer suffers is proportional to the intensity in the source (the aircraft in this case) and inversely proportional to the square distance between source and receiver, i.e., the closer the aircraft is, the more intense the noise suffered by the observer.
Noise mitigation strategies: There are four fundamental strategies to mitigate noise:
• Reduction of noise in the source (airframe and engines).
• Urban management and planning.
• Take-off and landing noise abatement procedures.
• Operative restrictions.
The continuous development of more and more modern aircraft and jet engines has led in the past to substantial reduction of noise (among other improvements) emissions. This is expected to continue in the future, since noise emissions are regulated by authorities. It obviously requires the application of new technologies coming from research and innovation.
Urban managing and planning refers to limiting the urban areas next to current limits of the airport, but also to potential future enlargements.
If it happens that there is a neighborhood next to an airport, and the neighborhood is suffering from noise, an interesting strategy is to design the so-called noise abatement procedures both for departure and arrival. These are typically continuous climb or continuous descent procedures that modify the flight path to avoid overflying certain areas. A good reference on this issue is Prats-Menéndez [11].
Last, if any of the previous strategies has not been developed, one can always restrict operations, for instance, at night hours. This is not desirable in terms of the economy of the industry, but it might be mandatory due to local legislation.
Climate change impact.
Aviation is one of the transport sectors with currently moderate climate impact. Air transportation contributes a small but growing share of global anthropogenic climate change impact. As aviation grows to meet increasing demand, the United Nations Intergovernmental Panel on Climate Change (IPCC) forecasted in 1999 that its share of global man made $CO_2$ emissions will increase to around 3% to 5% in 2050 (in 1999 it was estimated to be 2%) PENNER [9]. Moreover, the Royal Commission of Environmental Pollution (RCEP) has estimated that the aviation sector will be responsible for 6% or the total anthropogenic radiative forcing by 2050 Royal-Commission [12]. The development of mitigation methods for this purpose is in line with aviation visions and research programs, such as ACARE [1], the European aeronautics projects Clean Sky18 and SESAR CONSORTIUM [15], and the U.S. Next Generation19 strategy.
The climate impact of aviation results from $CO_2$ and non-$CO_2$ emissions PENNER [9]. While $CO_2$ is the most widely perceived greenhouse gas agent in aviation, mainly because its long lifetimes in the atmosphere and because of its considerable contribution to radiative forcing, emissions from aircraft engines include other constituents that contribute, via the formation or destruction of atmospheric constituents, to climate change. The non-$CO_2$ emissions (nitrogen oxides, water vapor, aerosols, etc.) have shorter lifetimes but contribute a large share to aviation climate impact, having a higher climate impact when emitted at cruise than at ground levels. One of these non-$CO_2$ contributors to climate change is the formation of contrails, which have received significative attention PENNER [9]. The relative importance of $CO_2$ and non-$CO_2$ depends strongly on the time horizon for evaluation of climate impacts and scenarios, e.g., future air traffic development. The non-$CO_2$ effects are more important for short time horizons than for long horizons.
Aviation $NO_x$ Climate Impact: Nitrogen oxides ($NO_x$, i.e., $NO$ and $NO_2$) are one of the major non-$CO_2$ emissions. $NO_x$ emissions in the troposphere and lower stratosphere contribute to ozone ($O_3$) formation and methane ($CH_4$) reduction. Both are important greenhouse gases. On average, the $O_3$ impact of aviation $NO_x$ is expected to be stronger than the impact on $CH_4$, which increases the greenhouse effect, though the precise amounts are uncertain. The amount of $NO_x$ emissions depends on fuel consumption and the engine’s type-specific emission index. The emission index for $NO_x$ depends on the engine and combustor architecture, power setting, flight speed, ambient pressure, temperature, and humidity. This dependence has to be taken into account when considering changes in aircraft design and operations.
Figure 8.9: $CO_2$ and global warming emissions.
Figure 8.10: Aircraft emissions contributing to global warming. Data retrieved from BENITO and BENITO [3].
Aviation Water Vapor Climate Impact: The climate impact of water vapor emissions without contrail formation is relatively small for subsonic aviation. The relative impact increases with altitude because of longer lifetimes and lower background concentrations at higher altitudes in the stratosphere, and would be more important for supersonic aircraft; water vapor would also become more important when using hydrogen-powered aircraft. The total route time in the stratosphere can be used as an indicator for water vapor climate impact.
Figure 8.11: Contrails.
Contrails: Contrails (short for condensation trails) are thin, linear ice particle clouds often visible behind cruising aircraft. They form because, under appropriate atmospheric conditions, the exhausted water vapor resulting from combustion inside aircraft engines mixes with cold ambient air, leading to local liquid saturation, condensation of water vapor, and subsequent freezing. A comprehensive analysis of the conditions for persistent contrail formation from aircraft exhausts is given in SCHUMANN [13].
Linear contrails may persist for hours and may eventually evolve into diffuse cirrus clouds, modifying thus the natural cloudiness. As a consequence persistent contrails modify the radiation balance of the Earth-Atmosphere system, resulting into a net increase of earth’s surface warming.
Contrails form when a mixture of warm engine exhaust gases and cold ambient air reaches saturation with respect to water, forming liquid drops that quickly freeze. Contrails form in the regions of airspace that have ambient relative humidity with respect to water ($RH_w$) greater than a critical value $r_{contr}$. Regions with $RH_w$ greater or equal than 100% are excluded because clouds are already present. Contrails can persist when the environmental relative humidity with respect to ice ($RH_i$) is greater than 100%. Thus, persistent contrail favorable regions are defined as the regions of airspace that have: $r_{contr} \le RH_w < 100%$ and $RH_i \ge 100%$.
The estimated critical relative humidity for contrail formation at a given temperature $T$ (in degrees Celsius) can be calculated as:
$r_{contr} = \dfrac{G (T - T_{contr}) + e_{sat}^{liq} (T_{contr})}{e_{sat}^{liq} (T)}\label{eq8.5.2.1}$
where $e_{sat}^{liq} (T)$ is the saturation vapor pressure over water at a given temperature. The estimated threshold temperature (in degrees Celsius) for contrail formation at liquid saturation is:
$T_{contr} = -46.46 + 9.43 \log (G - 0.053) + 0.72 \log^2 (G - 0.053),\label{eq8.5.2.2}$
where
$G = \dfrac{EI_{H_2O} C_p P}{\epsilon Q (1 - \eta)}.\label{eq8.5.2.3}$
In equation ($\ref{eq8.5.2.3}$), $EI_{H_2O}$ is the emission index of water vapor, $C_p$ is the isobaric heat capacity of air, $P$ is the ambient air pressure, $\epsilon$ is the ratio of molecular masses of water and dry air, $Q$ is the specific heat combustion, and $\eta$ is the average propulsion efficiency of the jet engine.
$RH_i$ is calculated by temperature and relative humidity using the following formula:
$RH_i = RH_w \dfrac{6.0612 \exp \tfrac{18.102T}{249.52 + T} }{6.1162 \exp \tfrac{22.577 T}{237.78 + T}},\label{eq8.5.2.4}$
where $T$ is the temperature in degrees Celsius.
Climate impact mitigation Options
Strategies for minimizing the climate impact of air traffic include identifying the most efficient options for airframe and propulsion technology, air traffic management, and alternative route network concepts. Economic measures and market-based incentives may also contribute, but these are out of the scope of this chapter. Minimizing the climate impact of aviation would require addressing all climate impact components. In the following, due to its relative importance, only the reduction of emissions of $CO_2$ and a contrail mitigation strategy are considered.
Minimizing $CO_2$ Emissions: Minimum fuel consumption is of primary interest for the aviation industry because it reduces costs. However, fuel is not the only cost driver and various constraints cause fuel penalties. Although fuel reduction below the current state is challenging, further reduction of fuel consumption and hence of fossil $CO_2$ climate impact is nevertheless feasible. As exposed in Section 1.3, the aim is to reduced the $CO_2$ emissions by 50% due to 2050 when compared to 2010 emissions. The reduction in $CO_2$ will require contributions from new technologies in aircraft design (engines, airframe materials, and aerodynamics), alternative fuels (bio fuels), and improved ATM and operational efficiency (mission and trajectory management). See Figure 1.3.
Contrail impact mitigation strategies: Several strategies for persistent contrail mitigation have been studied. See for instance Gierens et al. [6]. As illustration, a flight planning contrail mitigation strategy is herein presented. Further mitigation potential can be achieved by developing optimized aircraft and jets for these alternative trajectories.
Example $1$
In this example the aim is at showing a contrail mitigation strategy based on modifying the vertical profile of the flight. More information about this example can be consulted in Soler et al. [16].
More specifically, we optimize the trajectory of a B757-200 BADA 3.6 Nuic [8] model aircraft performing the en-route part of a flight San Francisco ($SFO$) - New York ($JFK$) between the waypoint20 Peons as initial fix and the waypoint Magio as final fix. The route is composed by waypoints given in Table 8.12.
Table 8.12: Route’s waypoints, navaids, and fixes
We assume all pairs formed by two consecutive waypoints are connected by bi-directional airways. On an airway, aircraft fly at different flight levels to avoid collisions. The different flight levels are vertically separated 1000 feet. On a bi-directional airway, each direction has its own set of flight levels according to the course. In east direction flights, aircraft are assigned odd flight levels separated 2000 feet. We then assume the aircraft can flight the route in any (if only one) of the following flight levels:
$\{FL270, FL290, FL310, FL330, FL350, FL370, FL390, FL410\}\label{eq8.5.2.5}$
Figure 8.12: Longitude-latitude grid points that present favorable conditions for persistent contrail formation for different barometric altitudes.
The flight we are analyzing is inspired in $DAL30$, with scheduled departure from $SFO$ at 06:30 a.m. on June the 30th, 2012. Data of air temperature and relative humidity for June the 30th, 2012 at time 18.00 Z21 (10.00 a.m. $PST$) have been retrieved from the $NCEP/DOE\ AMIP-II$ Reanalysis data provided by the System Research Laboratory at the National Oceanic & Atmospheric Administration ($NOAA$)22. The data have a global spatial coverage with different grid resolutions. Our data have a global longitude-latitude grid resolution of $2.5^{\circ} \times 2.5^{\circ}$. Regarding the vertical resolution, the data are provided in 17 pressure levels (hPa): 1000, 925, 850, 700, 600, 500, 400, 300, 250, 200, 150, 100, 70, 50, 30, 20, 10.
According to what has been exposed in Section 8.5.2, we compute the latitude- longitude grid points that are favorable to persistent contrail formation at different barometric altitudes (which defines the pressure). We do so based on gathered data of air temperature and relative humidity, and using equations (8.5.2.1)-(8.5.2.4) with the following: $EI_{H_2O} = 1.25$; $C_p = 1004[J/KgK]$; $\epsilon = 0.6222$; $Q = 43 \cdot 10^{6} [J/Kg]$; and $\eta = 0.15$. The longitude-latitude grid points with favorable conditions for persistent contrail formation are represented as red dots in Figure 8.12 for different barometric altitudes.
Figure 8.13: Favorable regions of contrail formation over USA at different flight levels. The horizontal route is depicted to illustrate how the same horizontal route under different flight levels might increase/reduce potential persistent contrail generation.
In order to analyze the regions of persistent contrail formation in our case study, we first need to estimate the values of temperature and relative humidity for the flight levels given in set (8.5.2.5). In order to do that, we use the International Standard Atmosphere ($ISA$) equations to convert altitude into barometric altitude, and then run a linear interpolation between the data of air temperature and relative humidity corresponding to the 17 pressure levels and the desired flight levels (already converted into barometric altitude). Once we have the values of temperature and relative humidity at the desired flight levels, we proceed on using equations (8.5.2.1)-(8.5.2.4) as exposed above. The favorable regions of persistent contrail formation over the USA at the different flight level can be consulted in Figure 8.13.
It can be observed that flying at high flight levels, e.g., $FL370, FL390$, and $FL410$, implies overflying regions of persistent contrail generation. On the contrary, flying at low flight levels, e.g., $FL270, FL290$, and $FL310$, implies non overflying regions of persistent contrail generation and thus minimizes environmental impact. However, it is obviously more efficient in terms of fuel burned to fly higher, which is actually what airlines do. Indeed, the flight in which this example is based on flow a flight plan at $FL390$ and $FL410$. Concluding, trade-off strategies (fuel-environmental impact) must be found.
20. Waypoints may be a simple named point in space or may be associated with existing navigational aids, intersections, or fixes.
21. Z-hour corresponds to Universal Time Coordinates (UTC). The Pacific Standard Time (PST) is given by UTC - 8 hours.
22. The data have been downloaded from NOAA website @ http://www.esrl.noaa.gov/psd/
8.6: References
[1] ACARE (2010). Beyond Vision 2020 (Towards 2050). Technical report, European Commission. The Advisory Council for Aeronautics Research in Europe.
[2] Belobaba, P., Odoni, A., and Barnhart, C. (2009). The global airline industry. Wiley. [3] Benito, A. and Benito, E. (2012). Descubrir el transporte aéreo y el medio ambiente. Centro de Documentación y Publicaciones Aena.
[4] Doganis, R. (2002). Flying off course: The economics of international airlines. Psychology Press.
[5] Doganis, R. (2006). The airline business. Psychology Press.
[6] Gierens, K., Lim, L., and Eleftheratos, K. (2008). The Open Atmospheric Science Journal 2, 1–7.
[7] Navarro, L. U. (2003). Descubrir el transporte aéreo. Centro de Documentación y Publicaciones de AENA.
[8] Nuic, A. (2005). User Manual for the base of Aircraft Data (BADA) Revision 3.6. Eurocontrol Experimental Center.
[9] Penner, J. (1999). Aviation and the global atmosphere: a special report of IPCC working groups I and III in collaboration with the scientific assessment panel to the Montreal protocol on substances that deplete the ozone layer. Technical report, International Panel of Climate Change (IPCC).
[10] Pindado Carrión, S. (2006). ETSI Aeronáuticos. Universidad Politécnica de Madrid . [11] Prats-Menéndez, X. (2010). Contributions to the Optimisation of aircraft noise abatement procedures. PhD thesis, Universitat Politècnica de Catalunya.
[12] Royal-Commission (2002). The environmental effects of civil aircraft in flight. Technical report, Royal Commission of Environmental Pollution, TR, London, England, UK.
[13] Schumann, U. (1996). Meteorologische Zeitschrift-Berlin- 5, 4–23.
[14] Schumann, U., editor (2012). Atmospheric Physics: Background—Methods—Trends. Research Topics in Aerospace. Springer.
[15] SESAR Consortium (April 2008). SESAR Master Plan, SESAR Definition Phase Milestone Deliverable 5.
[16] Soler, M., Zou, B., and Hansen, M. (2014). Transportation Research Part C: Emerging Technologies 48, 172–194.
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The aim of this chapter is to give a brief overview of airports, a fundamental infrastructure to facilitate intermodal transportation. Section 9.1 is devoted to provide a brief overview of airports’ history, introducing their naming nomenclature, describing the variables that potentially affect the demand of air transportation. The Master Plan, the set of official documents for the design and development of an airport, is described in Section 9.2. Section 9.3 is devoted to provide a description of the configuration of a modern airport, including air-side and land-side elements. Finally, Section 9.4 analyzes airport operations. Some introductory aspects suitable for this type of course can be consulted in Franchini et al. [2]. Two thorough references on the matter are De Neufville and Odoni [1], García Cruzado [3].
09: Airports
Over 100 years ago, it arose the necessity of using existing terrains to carry out the first flights. Those terrains were named airfields. Later on, airfields evolved to what is referred to as aerodrome. ICAO, in its Annex 14 ICAO [4] defines aerodrome as:
Definition 9.1: Aerodrome
A defined area on land or water (including any buildings, installations, and equipment) intended to be used either wholly or in part for the arrival, departure, and surface movement of aircraft.
After World War II, when commercial aviation reached its maturity, the term airport was generalized. The term airport refers to an aerodrome that is licensed by the responsible government organization (FAA in the United States; AESA in Spain). Airports have to be maintained to high safety patterns according to ICAO standards.
An airport is an intermodal transportation facility where passengers connect from/to ground transportation to/from air transportation. As it will be described in detail in Section 9.3, airports can be divided into land-side and air-side. The land-side embraces all facilities of the airport in which passengers arrive/depart the airport terminal building and move through the terminal building to clear security controls. Air-side embraces those infrastructures devised for movement of the airplanes on the airports surface, but also the boarding lounges. Roughly speaking, land-side corresponds to those facilities in which both passengers and companions (not passengers such as family, friends, etc.) cohabit. On the contrary, air-side infrastructure include all those areas in which only passengers with tickets (and obviously also airport employees) are allowed to be, including those infrastructures made for aircraft parking, taxiing, and landing/taking-off.
The most simple airport consists of one runway (or helipad), but other common components are hangars and terminal buildings. Apart from these, an airport may have a variety of facilities and infrastructures, including airline’s services, e.g., hangars; air traffic control infrastructures and services, e.g, the control tower; passenger facilities, e.g., restaurants and lounges; and emergency services, e.g., fire extinction unit. A more general definition could be as follows:
Definition 9.2: Airport
A localized infrastructure where flights depart and land, acting also as a multi-modal node where the interaction between flight transportation and other transportation modes (rail and road) takes place. It consists of a number of conjoined buildings, flight field installations, and equipments that enable: the safe landing, take- off, and ground movements of aircraft, together with the provision of hangars for parking, service, and maintenance; the multi-modal (ground-air) transition of passengers, baggage, and cargo. From a socioeconomic perspective, airports can be also considered a pole for economic growth, a door-gate of a country-region, and an entertainment area (shopping, eating & drinking).
9.01: Introduction
Airports are uniquely represented by their IATA airport code and ICAO airport code. IATA 3-letter airport codes are typically abbreviated of their names, such as MAD for Madrid Barajas International Airport. Exceptions to this rule are, for instance, O’Hare International Airport in Chicago (retains the code ORD from its former name of Orchard Field) and some named after a prominent national celebrity, e.g., John F. Kennedy, Paris Charles de Gaulle, Istanbul Atartuk , etc. The ICAO 4-letter airport identifier codes uniquely identify individual airports worldwide. Usually, the first letter of ICAO codes identify the country. In the continental USA, the first letter is \(K\). In Europe, the first letters is either \(L\) or \(E\).
9.1.02: The demand of air transportation
The variables that influence in the potential demand of air transportation in a determined airport can be itemized as follows:
• Historical tendency of geographical related airports.
• Demographic variables of the population under the region of influence of the airport.
• The economical character (industrial, technological, financial, touristic) of the region.
• Intermodal transportation network.
• Urban and regional strategic development plan.
• Competitors prices.
These items can be reduced to one: the Gross Domestic Product (GDP) per capita of the region. GDP per capita and demand of air transportation are strongly correlated.
Table 9.1: List of biggest airports in 2015 in volume of passengers. % of change refers to the increase of traffic with respect to 2014. Data retrieved from Wikipedia.
Table 9.2: List of biggest airports in 2015 by aircraft movements. % of change refers to the increase of movements with respect to 2014. Data retrieved from Wikipedia.
Therefore, according to the long-term estimation of GDP growth, the demand of air transportation is also expected to increase as a worldwide average rate of 5%. Thus, existing airports should be enlarged to absorb increasing demand, but also new airports should be opened in the future. Table 9.1 and Table 9.2 give a quantitative measure of the busiest airports worldwide.
Analyzing the biggest airports by number of passenger, obviously the big cities appear in the first positions, i.e., Beijing, London, Tokio, Chicago, Los Angles, Paris, etc. Notice however that Atlanta, not such an important city, is the world’s busiest airport. This is due to the fact that Atlanta is Delta’s hub. Also Dallas, American Airlines’ hub, appears in the first positions. Other important issues to notice are: the increasing presence of east and south east asian airports, with very important inter-annual growths; and the fact that Madrid Barajas dropped 12% in 2013 (also 9% in 2012). The later can be explained due to two different phenomena: the very important crisis that Europe, particularly the mediterranean countries, are suffering; and the acquisition of Iberia, Spanish flag company whose hub was Barajas, by British Airways, which has shifted a little bit the South America’s connexion demand towards United Kingdom. In 2015, Madrid Barajas recovered traffic growing 12%. The privatisation of AENA in 2014 might partially explain it.
In terms of movements, USA’s airports cope the first positions. This is due to the fact that many cities in the United States act as hubs. Many connections between american cities are done on a daily basis with medium-haul aircraft types (transporting less people). Also, oversees flights that arrive at the United States typically go first to the airline’s hub and then transit to a domestic flight. On the contrary, asian companies have recently started an strategy towards buying big airplanes (A380), transporting thus more people with less movements.
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Different types of studies are performed in airport planning, including facility planning, financial planning, traffic and markets, economics, environment, etc. Three different planning levels can be identified: system planning; master planning; project planning.
System planning: An airport system plan is a representation of the aviation facilities required to meet the immediate and future needs of a metropolitan area, region, state, or country. Therefore, it is a political planning level. Its overall purpose is to determine the extent, type, nature, location, and timing of airport development to establish a viable, balanced, and integrated system of airports to meet the transportation needs of a region/country.
Master planning: It is related to strategic planning of infrastructures for a single airport, and it is closely related to the Master plan. Detailed information will be given in Section 9.2.1.
Project planning: Project planning refers to a particular project in an airport, e.g., a new runway, a new taxiway, extension of the apron, extension of the terminal building, etc., which has been specified (obviously, only in the case of big projects) in the master plan. Notice that the administrative documents that are compulsory vary from one country to another (even among different regional entities within one country). Typically, one would find: a descriptive report; drawings; technical prescriptions; and a budget.
9.02: Airport Planning
Definition 9.3: The Master Plan
The Master Plan is an official set of documents with information, studies, methodologies and performances to be carried out in the design and construction of a new airport or an important enlargement into an existing one.
In other words, a Master Plan is a guide for:
• Developing the physical facilities of an airport.
• Developing land adjacent to the airport and establishing access requirements.
• Determining the environmental effects of airport construction and operations.
• Proving the feasibility of the proposed developments through a thorough investigation of alternative options.
• Establishing a timeline for the improvements proposed in the plan.
• Establishing an achievable financial plan to support the implementation schedule.
More specifically, a Master Plan should include the following studies/documents:
• Study of the existing situation: physical medium data (topography, meteorology, etc.), socioeconomic data (demography, GPD, etc.), comparative studies with proximal airports, physical assets, etc.
• Demand forecast: flights and types of aircraft, including a complete long-term demand forecast.
• Demand/capacity analysis and facility requirements.
• Alternatives development.
• Preferred development plan
• Implementation plan
• Environmental impact assessment
• Stakeholder and public involvement
Figure 9.1: Master plan flowchart.
The master plan encompasses first a study of the current situation, including physical data (topography, meteorology, etc.), socioeconomic data (demography, GPD, etc.), comparative studies with nearby airports, etc. Aeronautical data such forecasted flights and types of aircraft, including a complete long-term demand forecast, are also needed. The forecasted demand is thereafter casted against the existing capacity (for which one should consider aircraft operations and an specific level of service for passengers). The capacity-demand imbalances raise the future necessities in terms of runways, taxiways, platform positions, terminal buildings, etc. Whether a new airport is needed or an enlargement of the existing one suffices should be analyzed. In either cases, different layout alternatives must be proposed. In case of the necessity of building a new airport, the key decision is to select the best emplacement. The Master Plan flowchart in Figure 9.1 illustrates the different processes that a typical Master Plan involves.
Traffic forcast
The traffic prognosis is the key element within airport planning. It constitutes the baseline to define the facilities that are to be required together with the times at which those facilities will be necessary. It is provided in three different time horizons (short [5 years], medium [10-15 years], and long term [20-30 years]) and for three different scenarios (pessimistic, nominal, optimistic). Please refer to Exercise 1.1 as an illustrative instance.
The principal items for which estimates are usually needed include:
• The volume and peaking characteristics of passengers, aircraft, vehicles, and cargo.
• Number and types of aircraft needed to serve the above traffic.
• Number of general aviation aircraft and the number of movements generated.
• The performance and operating characteristics of ground access systems.
There are several forecasting methods or techniques available to airport planners ranging from subjective judgment to sophisticated mathematical modeling:
• Time series method.
• Market share method.
• Econometric modeling.
• Simulation modeling.
• Delphi method.
Econometric modeling represents the most sophisticated and complex technique in airport demand forecasting. Simple and multiple regression analysis techniques (linear and nonlinear) are often applied.
Multiple regression analysis can be regarded as an extension of simple linear regression analysis (which involves only one independent variable) to the situation where two or more independent variables are considered. The general form of a polynomial regression model for $m$ independent variables is
$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots + \beta_m X_m + \varepsilon,$
where $\beta_0, \beta_1, ..., \beta_m$ are the regression coefficients that need to be estimated. The independent variables $X_1, X_2, ..., X_m$ may all be separate basic variables, or some of them may be functions of a few basic variables. $Y$ represents an individual observation and $\varepsilon$ is the error component reflecting the difference between an individual’s observed response $Y$ and the true average response $\mu_{Y|X_1, X_2, ..., X_m}$.
Demand and capacity analysis
The analysis of capacity in an airport must be done for each of its different components, since the bottleneck could be in any of them:
• Capacity of parking lots (parking positions).
• Capacity of the passenger terminal (pax/h).
• Capacity of ramp and apron (parking position).
• Capacity of taxiways (mov/h).
• Capacity of the runway (mov/h).
Selection of the emplacement
The emphasis in airport planning is normally on the expansion and improvement of existing airports. However, if an existing airport cannot be expanded to meet the future demand or the need for new airport is identified in an airport system plan, a process to select a new airport emplacement may be required. For the selection of the emplacement one must take into account:
• The climatology (wind, fog, temperature, etc.).
• The topography (unevenness and slopes of the terrain).
• Obstacles in the surroundings (for safe taking off and landing).
• Intermodal connexions.
• Availability of terrains.
• Environmental impact.
Indeed, the master plan includes an environmental impact report of the operations in the selected emplacement. Economic studies in terms of operations and future development are also included.
Wind speed and direction: On the airport surface, the speed and direction of winds directly affect aircraft runway utilization. The best operational configuration is headwind: It allows an aircraft to achieve lift at slower ground speeds (with obviously greater true airspeeds) and shorter runway lengths. However, most of the times one would be affected by crosswinds to some extent. Lateral wind might be dangerous and sometimes operations would be cancelled if it exceeds determined safety thresholds (which depend on the aircraft type). ICAO provides requirements to runway design so that 95% of the annual wind conditions at the airport allow safe operations, which can be measured in terms of maximum permitted crosswind. According to ICAO, the crosswind component must not exceed:
• 37 km/h (20 kt) in the case of aeroplanes whose reference field length1 is 1500 m or over, except that when poor runway braking action owing to an insufficient longitudinal coefficient of friction is experienced with some frequency, a crosswind component not exceeding 24 km/h (13 kt) should be assumed;
• 24 km/h (13 kt) in the case of aeroplanes whose reference field length is 1200 m or up to but not including 1500 m; and
• 19 km/h (10 kt) in the case of aeroplanes whose reference field length is less than 1200 m.
Therefore, finding a location with low wind intensities (fulfilling ICAO restrictions) or with a clearly dominant wind direction is key. Indeed, as it will be described in Section 9.3.2, the orientation of the runway (or runways) will be partially driven by the blowing direction of the dominant winds. Therefore, starting 10-15 years prior the construction of the airport, a detailed study on the wind patterns is carried out to statistically select the most appropriate runway’s configuration. Wind intensities and directions are used to complete a so-called wind rose diagram. Please, refer to Exercise 1.2 as an illustrative instance.
Orography: The orography of the site is also key for two fundamental reasons: first of all, ICAO establishes requirements in terms of maximum slopes for runways and taxiways; second, ICAO also establishes requirements in terms of obstacles to facilitate the design of more efficient and safer departure and arrival procedures, for which ICAO specifies a set of obstacle limitation surfaces that must not be violated by orographic accidents (e.g., mountains) or human-made buildings:
• Outer horizontal surface and inner horizontal surface.
• Conical surface.
• Approach surface and Inner approach surface.
• Transitional surface and Inner transitional surface.
• Balked landing surface and take-off climb surface.
Therefore, selection a relatively flat emplacement would reduce the cost of the construction of the airport (since less terrain movement would be needed to make it flat). Also, selecting an emplacement with relatively few obstacles would not limit the possible directions of the runway.
Environmental issues: The construction of a new airport (also the enlargement) of an existing one implies a tremendous environmental impact, associated to: social factors such as land development, displacement and relocation, parks, recreational ares, historical places; ecological factors such as wildlife, waterfowl, flora, fauna, endangered species, and wetlands and coastal zones; and pollution factors such as air quality, water quality, and noise.
1. please, refer to Definition 9.5.
9.2.02: Physical environment of the airport
Site data
A set of data referring to the aerodrome emplacement (determined by the geographical coordinates referred to the World Geodesic System) must be supplied to the authorities to be ultimately published in the corresponding Aeronautical Information Service (AIS):
• Aerodrome’s Reference Point.
• Aerodrome’s Elevation.
• Coordinates of the runway’s thresholds.
• Coordinates of the parking positions.
• Mean elevation of each of the thresholds.
• Elevation of the runway’s heads.
• Maximum elevation of the touchdown zone.
Please, refer to Exercise 1.3 as an illustrative instance.
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Figure 9.2: Schematic configuration of an airport. Adapted from Franchini et al. [2].
Figure 9.3: Typical airport infrastructure. © Robert Aehnelt. / Wikimedia Commons / CC-BY-SA-3.0.
Airports are divided into land-side and air-side areas. Figure 9.19 illustrates an schematic flow in an airport. Figure 9.3 shows a layout of a medium size airport.
land-side areas
Land-side areas include parking lots, fuel tank farms, and access roads. Access from land-side areas to air-side areas is controlled at most airports by security systems and personal. Passengers on commercial flights access air-side areas through terminals, where they can purchase tickets, check luggage in, and clear security. One security has been cleared, the passenger is in the air-side areas.
Air-side areas
The air-side is partially composed by a set of infrastructures formed by the runway (or runways), taxiway (or taxiways) and high-speed taxiways, together with the ramp and the apron. Also the waiting areas, which provide passenger access to aircraft and typically include duty free shops and restaurants, are considered air-side and referred to as concourses.2 Due to their high capacity and busy airspace, most international airports have air traffic control located on site. This is also considered air-side infrastructure. Notice that minor airports might not necessarily have a control tower, instead some air traffic control services would be allocated within the airport facilities.
The area where aircraft park next to a terminal to load passengers and baggage is known as a ramp or platform. Parking areas for aircraft away from terminals are generally called aprons. The difference between ramp and apron is that the ramp is typically connected to the terminal with fingers.
A taxiway is a path on an airport connecting runways with ramps, hangars, terminals, and other facilities. They are typically build on asphalt (or more rarely concrete). Busy airports typically construct high-speed or rapid-exit taxiways in order to allow aircraft to leave the runway at higher speeds. This allows the aircraft to exit the runway quicker, permitting another one to land or depart in a shorter space of time, increasing thus the capacity of the airport as it will be mentioned later on.
2. this term is often used interchangeably with terminal waiting lounges.
9.3.02: The runway
According to ICAO [4] a runway is:
Definition 9.4: Runway
defined as a rectangular area on a land aerodrome prepared for the landing and takeoff of aircraft.
Runways are typically build based on asphalt or concrete over a previously leveled and compacted surface. Runways are defined together with safety areas, that might be of compacted natural terrain. On both sides of the runway, there is the strip. In the heads of the runway we find a Stop-Way (SWY) area, a Clear-Way (CWY) area, and a Runway End Safety Area (RESA). All these areas are due to safety reasons and their specifications are stated by ICAO and can be consulted in ICAO [4]. Related to these safety areas, ICAO defines de following declared distances:
• take-off run available (TORA): The length of runway declared available and suitable for the ground run of an aeroplane taking off.
• take-off distance available (TODA): The length of the take-off run available plus the length of the clearway, if provided.
• accelerate-stop distance available (ASDA): The length of the take-off run available plus the length of the stopway, if provided.
• landing distance available (LDA): The length of runway which is declared available and suitable for the ground run of an aeroplane landing.
Figure 9.4: Runway declared distances. Adapted from © User:Mormegil / Wikimedia Commons / CC-BY-SA-3.0.
Figure 9.4 sketches them. Notice that in this figure the threshold has been displaced (limiting thus landings) and there is a pre-threshold area only allowed to be used as stopway for 06R’s take-offs. Please refer to Section 9.4.2 for more information on visual aids and markings. Please, refer to Exercise 1.4 as an illustrative instance.
Table 9.3: Runway ICAO categories (alph. code refers to the type of aircraft). Data retrieved from ICAO [4].
Table 9.4: Minimum runway’s width [m] ICAO identifiers. Data retrieved from ICAO [4].
Runway categories
ICAO has established different categories for the runways according to the size of the aircraft that can operate in such runways. The identifiers are two: a letter associated to the wing-span of the aircraft; and a number designating the longitude of the runway, measured in terms of reference field length as defined in Definition 9.5. Table 9.3 shows these categories. The width of the runways is obviously related with the category. It is shown in Table 9.4.
Definition 9.5: Reference field length
The minimum field length required for take-off at maximum certificated take-off mass, sea level, standard atmospheric conditions, still air and zero runway slope, as shown in the appropriate aeroplane flight manual prescribed by the certificating authority or equivalent data from the aeroplane manufacturer. Field length means balanced field length for aeroplanes, if applicable, or take-off distance in other cases.
For instance, Category 4 establishes a reference field length greater that 1800 m, however runways can be much longer up to 4500 m. Therefore, at the real operation conditions, the distances that aircraft actually need must be compensated, e.g., with the temperature of the airport, elevation of the airport, and slope of the runway. As an illustration, the same aircraft would need much less distance to take off in an airport located at sea level that in an airport located at 3000 [m]. Further insight on this will be given in Section 9.4.3. Please, refer to Exercise 1.5 as an illustrative instance.
Runway identifiers
Figure 9.5: Runway designators.
Runways are identified attending at its geographical orientation, starting from the north and running clockwise, rounding to the closest tens of grades. Therefore, a runway which is being approach with a course of \(89^{\circ}\) (that is, approaching from West to East) will be designated 09. Obviously, in the other head of the runway there is a difference of \(180^{\circ}\), that is, the designator will be 27 (see Figure 9.3). Notice that the heads in courses near the North are identified as 36 instead of using 0. Figure 9.5.b illustrates an example.
When there exist parallel runways, as is the case of big airports such Madrid Barajas, a letter L (left) or right (R)3 is added prior to the number. L or R is added attending at what the pilot is seeing when approaching on his/her right and left. See for instance Figure 9.5.a, which shows an airport layout with three parallel runways.
Runways configuration
The most commonly used configurations are:
• Unique runway configuration.
• Cross runways configuration.
• V runways configuration.
• Parallel runways configuration.
• Double parallel runways configuration.
Figure 9.6: Adolfo Suarez Madrid Barajas layout chart. © AENA. / AIP AENA.
Figure 9.7: FAA airport diagram of O’Hare International Airport. © FAA. / Wikimedia Commons / Public domain.
Notice that these configurations are made attending at the design requirements. One key indicator is the forecasted demand, which determined whether is enough with one runway to cope with all expected demand. Another factor could be related to the dominant winds. If there are two dominant directions, we might be forced to design two runways in different directions not to cancel operation on a regular basis. An overview of different spanish airports layout can be consulted in AIP AENA. See Figure 9.6 and Figure 9.7.
Capacity of the runway
Table 9.5: ICAO minimum distance in airport operations. Heavy refers to aircraft with \(MTOW\) > 136000 [kg]. Medium refers to aircraft with 7000 < \(MTOW\) < 136000 [kg]. Any other aircraft with \(MTOW\) < 7000 [kg] are light. Data retrieved from ICAO [4].
As studied in Chapter 3, an aircraft generates two vortexes in the tips of the wing that travel backwards behind the aircraft. Such trails remain a long distance behind the aircraft and can disturb aircraft flying behind, becoming a potential danger. In order to prevent such danger, ICAO has established a required minimum separation. Table 9.5 shows these distances in airport operations, being aircraft 1 the preceding aircraft. These separations are the key factor that determines the capacity of a runway in nominal conditions. A single runway configuration might have a maximum capacity of approximately 50-60 movements per hour. In configurations with more than one runway, the capacity increases.
In general, the capacity of a runways depends on different factors, some based on the available infrastructure, and other related with airport operations:
• The conditions of Air Traffic Control (ATC) in approach and take-off.
• Longitude, orientation, and number of runways.
• The use of the system of runways for different operations (take-off and landing).
• The number, location, and characteristics of the rapid exit taxiways.
• The number of taxiways and the waiting points to runways heads.
• Mix of aircraft.
• Atmospheric conditions (wind, rain, fog, etc.)
• Conditions of the pavement.
• Type of visual aids.
• Approach and take off procedures.
• Interferences of the Terminal Maneuvering Area (TMA) with nearby airports or other flights (military, training, general aviation, etc.)
3. There might also exist C (center) in case of three parallel runways.
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In general, the terminal area designates the set of infrastructures inside the airport different from the aircraft movement area (apron, taxiways, runways). We can distinguish:
• Auxiliary aeronautical buildings (control tower, fire extinction building, etc.).
• Freight processing areas (freight terminals).
• Aircraft processing areas (hangars, etc.).
• Industrial and commercial areas (pilot schools, catering services, mail services, etc.).
• Passenger processing and attention infrastructures (referred to as passenger terminal).
We will focus in what follows on the passenger terminal. An airport passenger terminal is a building at an airport whose main functions are:
Figure 9.8: Aircraft fed by a finger.
• The interchange of transportation mode terrestrial-aerial.
• The processing of the passenger before boarding: check-in, security controls, shopping, etc.
• The processing of the passenger after disembarking: luggage claim, customs and security, facilities (car rental, for instance), etc.
• It also fulfills a function of distributing the flows of passengers. Typically passenger reach check in in a bunch, but then they walk alone in small groups, and they reach the gate again as a bunch. Therefore, big lounges and long decks or walkways are needed to distribute the flows.
• Give room to aircraft parking positions fed by fingers, as illustrated in Figure 9.8.
Terminal layout
The terminal layout depends on many factors and typically differs from one airport to another. However, there are some patterns that are typically followed:
• Arrival and departure flows are separated, typically in different levels.
• Domestic and international flow are separated, typically in the same level.
Figure 9.9: Typical design of a terminal, showing departure (right half of page) and arrival levels (left half): 1. Departures lounge; 2. Gates and fingers; 3. Security clearance gates; 4 Baggage check-in; 5. Baggage carousels. © Ohyeh. / Wikimedia Commons / CC-BY-SA-3.0.
Figure 9.9 shows a typical layout of a medium size airport. We can observe how arrivals and departure flows are separated (typically in two different floors). We can observe that the departure passenger process starts by queuing to check-in, clearing security, waiting in the concourse and proceed to gate, clear the boarding security control, and finally embarking. Notice that aircraft are fed by fingers. On the other hand, the arrival passenger will disembark and go directly to claim luggage (notice that there is no customs on arrivals, so we assume this is the domestic part of the airport).
Terminal configuration
Figure 9.10: Typical terminal configurations. © Robert Aehnelt. / Wikimedia Commons / CC-BY-SA-3.0.
The configuration of the terminal is determined by the number and type of aircraft we want to directly feed with fingers. Figure 9.10 shows some of the typical configurations.
The standard configuration allows many aircraft, but the terminal must be long and therefore the distance to be walked by passenger. Another strategy is to use piers. A pier design uses a long, narrow building with aircraft parked on both sides. Piers offer high aircraft capacity and simplicity of design, but often result in a long distance from the check-in counter to the gate and might create problems of capacity due to aircraft parking manoeuvres.
Another typical configuration is to build one or more satellite associated to a main passenger processor terminal. The main difference between a satellite and a passenger processor terminal is that the satellite does not allow check-in, nor security controls, is just to give access to gate with walkways, concourses, and maybe duty free shops. The main advantage is that aircraft can park around its entire perimeter. The main disadvantage is that they are expensive: a subway transportation infrastructure in typically needed, also luggages must be transported to the main terminal building. Think for instance in Madrid Barajas with a standard linear terminal (T1 - T2 - T3) and a Terminal + Satellite (T4 + T4S), in which the satellite is not a processor. In this case a subway transportation infrastructure together with an automated system for luggages were needed and thus constructed.
9.3.04: Airport services
International customs
Any international airport must necessarily have customs facilities, and often require a more perceptible level of physical security. This includes national police and custom agents, drug inspections, and, in general, any inspection to ensure migration and commerce regulations.
Security
Airports are required to have security services in most countries. These services might be sublet to a private security company or carried out by the national security services of the country (sometimes, one would find a mixture between these two). Airport security normally requires baggage checks, metal screenings of individual persons, and rules against any object that could be used as a weapon. Since the September 11, 2001 attacks, airport security has been dramatically increased worldwide.
Intermodal connections
Airports, specially the largest international airports located in big cities, are often located next to highways or are served by their own highways. Traffic is fed into two access roads (loops) one sitting on top of the other to feed both departures and arrivals (typically in two different levels). Also, many airports have the urban rail system directly connecting the main terminals with the inner city. Very recently, to facilitate connections with medium distance cities (up to 500-600 [km]), there are projects (if is not a reality already) to incorporate the high speed train in the airport facility, connecting big capitals with other important cities.
Shop and food services
Every single airport, even the smallest ones, have shops and food courts (at least one little shop to buy a snack and soda). These services provide passengers food and drinks before they board their flights. If we move to large international airports, these resemble more like a shopping mall, with many franchise food places and the most well-known retail branches (specially clothes stores). International areas usually have a duty-free shop where travelers are not required to pay the usual duty fees on items. Larger airlines often operate member-only lounges for premium passengers (VIP lounges). The key of this business is that airports have a captive audience, sometimes with hours of layover in connections, and consequently the prices charged for food are generally much higher than elsewhere in the region.
Cargo and freight services
Airports are also facilities where large volumes of cargo are continuously moved throughout the entire globe. Cargo airlines carry out this business, and often have their own adjacent infrastructure to rapidly transfer freight items between ground and air modes of transportation.
Support Services
Other services that provide support to airlines are aircraft maintenance, pilot services, aircraft rental, and hangar rental. At major airports, particularly those used as hubs by major airlines, airlines may operate their own support facilities. If this is not the case, every single company operating an airport must have access to the above mentioned services, which are typically rented on demand.
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The main function of an airport, besides facilitating the passenger intermodal connection, is to ensure that aircraft can land, take off, and move around in an efficient and safe manner. Thus, many systems and subsystems are needed to facilitate achieving this end, encompassing many protocols and processes. These processes are most of the times hardly visible to passengers, but have extraordinary complexity, specially at large international airports. We will focus herein on the airport operations duties of Air Traffic Management (ATM) and, fundamentally, on the airport navigational aids that must be available to ensure safe operations. Last, we will briefly point out some issues regarding safety management and environmental concerns in airport operations.
9.04: Airport operations
ATM will be deeply described in Chapter 10. As a rough definition, we can say that ATM is about the processes, procedures, and resources which come into play to make sure that aircraft are safely guided in the skies and on the ground. Therefore, it plays an important role in airport operations.
Air traffic control: ATC (to be also studied in Chapter 10) is the tactical part within the Air Traffic Management (ATM) system. It is on charge of separating aircraft safely in the sky as flying at the airports where arriving and departing. These duties are carried out by air traffic controllers, who direct aircraft movements, usually via VHF radio. Air traffic control responsibilities at airports are usually divided into two main areas: ground control and tower control.
Ground control is responsible for directing all ground traffic in designated movement areas, except for the case of traffic on runways, i.e, ground control in on charge of aircraft movements in aprons and taxiways, but also all service vehicles movements (fuel trucks, push-back vehicles, luggage trollies, etc.). Ground Control commands these vehicles on which taxiways to use, which runway to proceed (only for aircraft in this case), where to park, when to cross runways, etc. When a plane is ready to take off, it must wait in the runway head and turned over to tower control, who is responsible to authorize take-off and surveil the operation thereafter. After a plane has landed, it exits the runway and then is automatically turned over to ground control.
Tower Control controls aircraft on the runway and in the controlled airspace immediately surrounding the airport, the so-called Control Zone (CTR) or Terminal Maneuvering Area (TMA).4 They coordinate the sequencing and spacing of aircraft and direct aircraft on how to safely join and leave the CTR/TMA circuit of arrivals and departures.
Communication services: Together with ATC services, the ATM provides an information system that apply both for airport operations and en-route operations. In regard of airport operations, pilots check before take off the so-called Automatic Terminal Information Service (ATIS), which provides information about airport conditions. The ATIS contains information about weather, which runway and traffic patterns are in use, and other information that pilots should be aware of before boarding the aircraft and entering the movement area and the airspace.
4. The difference between CTR and TMA can be consulted in Chapter 10.
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The maneuvers of approach and landing are assisted from the airport by means of radio- electric and visual aids. The flight that is assisted with radio-electric aids is said to be under Instrumental Flight Rules (IFR flight); the flight that is assisted only with visual rules is said to be under Visual Flight Rules (VFR flight).
Visual aids
When flying, there are a number of visual aids available to pilots:
• Signaling devices (such for instance, windsock indicator).
• Guidance signs (information, compulsory instructions, etc.).
• Signs painted over the pavement (runway, taxiways, aprons).
• Lights (runway, taxiway).
Windsock: Planes take-off and land in the presence of head/tail wind in order to achieve maximum performance. Wind speed and direction information is available through the ATIS or ATC, but pilots need instantaneous information during landing. For this purpose, a windsock is kept in view of the runway. As already pointed out before in order to justify the fact that airports might have runways in different directions, the presence of wind is dangerous and limiting in terms of operations. The aeronautical authorities have established a maximum crosswind of 15-40 knots depending of the aircraft (for instance, a B777 has a limit of 38 knots) for landing and take-off. If these values are exceeded the runway can not be operated.
Guidance signs: Airport guidance signs provide moving directions and information to aircraft operating in the airport, but also to airport vehicles. There are two classes of guidance signs at airports, with several types of each:
Figure 9.11: Airport visual aids.
• Location signs (yellow colored on black background): Identifies the runway or taxiway in which the aircraft is or is about to enter.
• Direction/runway exit signs (black colored on yellow background): Identifies the intersecting taxiways the aircraft is approaching when rolling on the runway right after having landed. They also have an arrow indicating the direction to turn.
• Other: Many airports use conventional traffic signs such for instance stop signs throughout the airport.
Mandatory instruction signs: They show entrances to runways or critical areas. Vehicles and aircraft are required to stop at these signs until the control tower provides clearance to proceed on.
• Runway signs (white on red): These signs simply identify a runway intersection ahead of the aircraft.
• Frequency change signs: Typically consists of a stop sign and an instruction to change to another frequency. These signs are used at airports with different areas of ground control, where different communication frequencies might be used.
• Holding position signs: A single solid yellow bar across a taxiway (painted over the pavement) indicates a position where ground control may require a stop. If a two solid yellow bars and two dashed yellow bars are encountered (painted over the pavement), it indicates a holding position for a runway intersection ahead. Runway holding lines must never be crossed without ATC permission.
Signs painted over the pavement: There are three main sets of signs painted over the pavement:
• Runway signs (white).
• Taxiway signs (yellow).
• Apron signs (red).
Figure 9.12: Runway pavement signs. © User:Mormegil / Wikimedia Commons / CC-BY-SA-3.0.
The runway signs, white colored, can be consulted in Figure 9.3 and Figure 9.12, namely: threshold, aiming point, touchdown zone, center line, runway designator, edge lines, etc. The taxiway signs, yellow colored, can also be consulted in Figure 9.3, namely: strip and axis lines, holding positions, crossing points, etc. The apron signs, red colored, are typically the so-called envelopes where aircraft park.
Lighting: Airports have lighting devices that help guide planes using the runways and taxiways at night or in rain/fog. There are two main sets of lighting in the movement area:
• Runway lights (green, red, and white).
• Taxiway signs (blue and green).
Figure 9.13: Runway lighting. © Hansueli Krapf / Wikimedia Commons / CC-BY-SA-3.0.
On runways, green lights indicate the beginning of the runway for landing, while red lights indicate the end of the runway. Runway edge lights are white spaced out on both sides of the runway. Some airports have more sophisticated lighting on the runways including embedded lights that run down the centerline of the runway and lights that help indicating the approach. Along taxiways, blue lights indicate the taxiway’s edge, and some airports have embedded green lights that indicate the centerline. See Figure 9.13 as illustration.
Other light devices also help pilots approaching, as in the case of the Precision Approach Path Indicator (PAPI). The PAPI is a visual aid that provides guidance information to help pilots acquire and maintain the correct approach (in the vertical plane) to an airport. It consists of 4 lights display in a row located on the right side of the runway, approximately 300 meters beyond the landing threshold. These lights emit in the red spectrum below the gliding path and in the white spectrum above it. In order to follow the correct glide slope, a pilot would maneuver the aircraft to obtain an equal number of red and white lights, i.e, 2 red lights on the left part and 2 white lights on the right part.
Instrumental Aids
Besides visual aids, that all airport have, a majority of big airports have also a number of radio-electric aids to assist aircraft and pilots:
Figure 9.14: PAPI: The greater number of red lights visible compared with the number of white lights visible in the picture means that the aircraft is flying below the glide slope. Wikimedia Commons / Public domain.
A typical instrumental aid located in airport is the so-called VHF omnidirectional range (VOR), which help pilots finding a desired flying course. VORs are often installed together with a Distance Measuring Equipment (DME), which provides the distance between the aircraft and air (located in the airport). In this way, a pilot can use course and distance to proceed safely towards the runway head. These two equipments are also for en-route navigation and will be described in detail in Chapter 10. There is one instrumental aid that is only used in airport operations: the Instrument Landing System (ILS).
Figure 9.15: ILS: The emission patterns of the localizer and glide slope signals. Note that the glide slope beams are partly formed by the reflection of the glide slope aerial in the ground plane. © User:treesmill / Wikimedia Commons / CC-BY-SA-3.0.
Instrument Landing System (ILS): An ILS is a ground-based instrumental approach system that provides precision guidance to an aircraft approaching and landing on a runway. In poor visibility conditions (rain, fog, dark, etc.), pilots will be aided by an ILS to instrumentally find the runway and fly the correct approach and land safely, even if they cannot see the ground at some point of the approach procedure (or even during the whole approach procedure).
An ILS consists of two independent subsystems, one providing lateral guidance (localizer), and the other providing vertical guidance (glide slope or glide path). Aircraft guidance is provided by the ILS receivers in the aircraft by performing a modulation depth comparison. The localizer receiver on the aircraft measures the Difference in the Depth of Modulation5 (DDM) of two signals, one of 90 Hz and the other of 150 Hz. The difference between the two signals varies depending on the position of the approaching aircraft from the centerline. If there is a predominance of either 90 Hz or 150 Hz modulation, the aircraft is off the centerline. In the cockpit, the needle on the Horizontal Situation Indicator (HSI, the instrument part of the ILS), or Course Deviation Indicator (CDI), will show that the aircraft needs to fly left or right to correct the error to fly down the center of the runway. If the DDM is zero, the aircraft is on the centerline of the localizer coinciding with the physical runway centerline.
Figure 9.16: ILS: Localizer array and approach lighting. Wikimedia Commons / Public Domain.
A glide slope (GS) or glide path (GP) antenna array is sited to one side of the runway touchdown zone. The GP signal is transmitted on a carrier frequency between 328.6 and 335.4 MHz. The centerline of the glide slope signal is arranged to define a glide slope of approximately \(3^{\circ}\) above horizontal (ground level). The pilot controls the aircraft so that the indications on the instrument (i.e., the course deviation indicator (see Chapter 5)) remain centered on the display. This ensures the aircraft is following the ILS centerline (i.e., it provides lateral guidance). The vertical guidance is shown on the instrument panel by the glide slope indicator, and aids the pilot in reaching the runway at the proper touchdown point. Many modern aircraft are able to embed these signals into the autopilot, allowing the approach to be flown automatically.
Table 9.6: ILS categories. Data retrieved from ICAO [4].
According to ICAO, there are different ILS categories attending at the visual range and the decision altitude: the visual range is the longitudinal distance at which the pilot is able to clearly distinguish the signs painted over the pavement or the lights if flying at dark of fog (generally speaking at low visibility conditions); the decision altitude is the minimum vertical altitude at which the pilot must abort the approach in case of not seeing any of the visual aids. See Table 9.6.
5. It is based on the concept of space modulation, a radio amplitude modulation technique specifically used in ILS that incorporates the use of multiple antennas fed with various radio frequency powers and phases to create different depths of modulation within various volumes of three-dimensional airspace.
9.4.03: Aircraft characteristics related to airport planning
Every single aircraft type must provide a document named aircraft characteristics related to airport planning to the authorities pertaining a set of data related to airport planning and aircraft operations. This document provides, in a standardized format, airplane characteristics data for general airport planning and operations. Data include airplane characteristics and performances, ground maneuvering, terminal servicing, operating conditions, and pavement data. Within aircraft performance, take-off/landing distances are provided for different altitudes and temperatures. A typical document would include:
• Airplane description;
• Airplane performance;
• Ground maneuvering;
• Terminal servicing;
• Jet engine wake and noise data;
• Pavement data;
• Scaled drawings.
The reader is referred to check the document for different aircraft types (notice that the documents are public). For instance, Boeing provides access to all its aircraft’s airport planning documents through Boeing’s aircraft characteristics related to airport planning.6 Please, refer to Exercise 1.5 and Exercise 1.6 as illustrative instances.
9.4.04: Safety management and environment
Safety: Safety is the most important concern in airport operations. Thus, every airfield includes equipment and procedures for handling emergency situations. Commercial airfields include at least one emergency vehicle (with the corresponding crew) and a fire extinction unit specially equipped for dealing with airfield incidents and accidents.
Potential airfield hazards to aircraft include scattered fragments of any kind, nesting birds, and environmental conditions such as ice or snow. The field must be kept clear of any scattered fragment using cleaning equipment so that doesn’t become a projectile and enter an engine duct. Similar concerns apply to birds nesting near an airfield which might endanger aircraft operations due to impact. To threaten birds, falconry is practiced within the airport boundaries. In adverse weather conditions, ice and snow clearing equipment can be used to improve traction on the landing strip. For waiting aircraft, special equipment and fluids are used to melt the ice on the wings.
Environmental concerns: As already exposed, the construction of new airports (or the enlargement of an existing one) has also a tremendous environmental impact, affecting on the countryside, historical sites, local flora and fauna. In addition, airport operations also cause important environmental impact: vehicles operating in airports (aircraft but also surface vehicles) represent a major source of noise and air pollution which can be very disturbing and damaging for nearby residents and users. Moreover, operating aircraft have a dramatic impact on inhabiting birds colonies and affect entire neighborhoods generating noise (please refer to Chapter 8).
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Exercise $1$ Traffic Demand
The exercise is related to Master planning for airports. Students will team up in groups of four people each. The exercise is to be completed during the class. Students are allowed to use any mean, e.g., books, laptops, internet, to find a solution to the problem.
Table 9.7: Historical data [2004-2015 period] about number of passenger and number of operations in the Adolfo-Suarez Madrid Barajas airport.
The historical data contained in Table 9.7 with the number of passengers and the number of operations in the Adolfo-Suarez Madrid Barajas airport for the period 2004-2015 is made available.
Table 9.8: Historical data [2004-2015 period] of GDP growth.
Table 9.9: Forecast [2016-2030] of GDP growth.
Historical data [period 2004-2015] of the Spanish Gross Domestic Product (GDP) [in nominal terms, i.e., not considering inflation] are given in Table 9.8. Also, forecasts 2016-2030 for the World GDP forecast [in nominal terms] from both IMF (International Monetary Found) and CEPREDE7 are given in Table 9.9.
1. With these data (Pax and GDP), do a traffic forecast (number of pax. 2016-2030) using econometric models. Find a solution (traffic forecast) for three different scenarios (pessimistic, nominal, optimistic). Depict a sketch figure of the evolution of the number of passenger in the period 2016-2030 for the three scenarios.
The maximum capacity of the Airport Adolfo-Suarez Madrid Barajas has been declared to be 70 Million passenger a year.
1. According to the different forecasts, when is it estimated the airport to be non capable of coping with the passengers’ demand?
2. What other metrics should we look at (besides that of total amount of passengers) in order to analyze the capacity of the airport? Why?
Answer
Econometric models represent one of the most sophisticated and complex technique in airport demand forecasting. Simple and multiple regression analysis techniques (linear and nonlinear) are often applied.
Multiple regression analysis can be regarded as an extension of simple linear regression analysis (which involves only one independent variable) to the situation where two or more independent variables are considered. The general form of a polynomial regression model for m independent variables is
$Y = \beta_0 + \beta_1 X_1+ \beta_2 X_2 + ... + \beta_m X_m + \varepsilon,$
where $\beta_0, \beta_1, ..., \beta_m$ are the regression coefficients that need to be estimated. The independent variables $X_1, X_2,... , X_m$ may all be separate basic variables, or some of them may be functions of a few basic variables. $Y$ represents an individual observation and $\varepsilon$ is the error component reflecting the difference between an individual’s observed response $Y$ and the true average response $\mu_{Y|X_1,X_2,...,X_m}$.
In this particular case, for the sake of simplicity, we run a linear regression analysis as follows:
$Y = \beta_0 + \beta_1 X_1 + \varepsilon,$
where $Y$ represents individual observations, i.e., values of traffic demand (pax) between years 2004-2015; and $X_1$ represents the explanatory variables, i.e., the gross domestic product growth in the same period. Therefore, we want to estimate the coefficients $\beta_0$ and $\beta_1$ that best fit the data; in other words, we want to estimate the line that minimizes the errors $\varepsilon$ of the observations. We will use minimum squares method.
Taking years 2004-2015, the solution is:
$Y = -0.1307 + 2.7182 X,\label{eq9.5.3}$
Figure 9.17: Linear regression analysis.
where $Y$ represent de % of passenger growth and X represents the percentage of GDP growth. See Figure 9.17. The goodness of fit measured in term of R2 (coefficient of determination) is rather low (0.6080), meaning that the curve does not fit very well the data (as it can be observed from the graph). Other regression might be done in which some observations could be considered outliers or even considering nonlinear functions.
Given Eq. ($\ref{eq9.5.3}$), we build three different scenarios for GDP growth: nominal (the given one in Table 9.8), optimistic (in this case the one in Table 9.8 + 0.5%), and pessimistic (in this case the one in Table 9.8 - 0.5%). Again, notice that other scenarios could have been built yielding different results.
Figure 9.18: Scenarios of traffic forecast.
2. When is it estimated the airport to be non capable of coping with the passengers’ demand?
• Pessimistic: 70M Pax exceeded in 2026.
• Nominal: 70M Pax exceeded in 2023.
• Optimistic: 70M Pax exceeded in 2021.
3. What other metrics should we look at (besides that of total amount of passengers) in order to analyze the capacity of the airport? Why?
One should also look at the operations/design hour; Pax/design hour. Also, number of vehicles accessing the airport; number of people using the subway; etc.
Exercise $2$ Wind Rose
Table 9.10: Example of historical wind data.
For a given site located in Somewhere, historical wind data have been already collected for the last 15 years as illustrated in Table 9.10.
1. Fill in the wind rose diagram sketched bellow and propose the most suitable direction (or directions) for a new runway (or runways).
Figure 9.19: Wind Rose coordinate system and template with cross wind component limits of 13 knots.
Answer
According to ICAO’s Annex 14 (Recommendation 3.1):
The number and orientation of runways at an aerodrome should be such that the usability factor of the aerodrome is not less than 95 per cent for the airplanes that the aerodrome is intended to serve.
Figure 9.20: Wind coverage for runways 9-27 and 3-21.
Then, taking that recommendation into consideration, we first calculate the runway orientation the yields the maximum percentage of wind between parallel lines. This is the runway 9-27 (see Figure 9.20). It will permit around 90-91% of the operations. If one wants to follow the recommendations, we must build another runway. We do that trying to maximize the % of operations. The solution is the 3-21 runway (see Figure 9.20), which would permit around 97.5% of the operations.
Exercise $3$ Aerodrome data
The exercise is related to identifying the main aerodrome data for a particular airport. For the airport Adolfo Suarez Madrid Barajas, identify:
1. The following data about the site:
• Aerodrome’s Reference Point;
• Aerodrome’s Elevation;
• Coordinates of the runway’s thresholds;
• Coordinates of the parking positions;
• Mean elevation of each of the thresholds;
• Elevation of the runway’s heads;
• Maximum elevation of the touchdown zone;
Answer
All data can be consulted at AENA’s AIS Adolfo Suarez Madrid Barajas. In particular, one should consult:
• Aerodrome data
• Aerodrome chart
• Aerodrome ground movement chart
• Aircraft parking/docking chart
• Aerodrome obstacle chart
Figure 9.21: Aerodrome data.
The solution is sketched in Figure 9.21. Please, notice that the solution to Exercise 9.4 is also given in this Figure.
Exercise $4$ Airfield data
The exercise is related to identifying the main aerodrome data for a particular airport. For the airport Adolfo Suarez Madrid Barajas (For the runway 36L/14R), identify the following data about the movement area:
• Dimensions (length and width);
• Usability of both 36L and 18R for take-offs and landings.
• Safety areas (strip and runway end safety areas for 36L/18R);
• Displacement of the threshold for 36L and 18R;
• Declared distances for 36L and 18R;
• Identify the localizer and the glide path for the ILS of the runway 18R. Write down their frequencies.
• Pavement Classification Number of runways 36L and 18R. What kind of pavement is it? How can we know it?
All this information can be consulted at Adolfo Suarez Madrid Barajas Aerodrome’s chart given in the appendix.
Answer
All data can be consulted at AENA’s AIS Adolfo Suarez Madrid Barajas consulting the same documents as in Exercise 1.3. The solution is sketched in Figure 9.21.
Exercise $5$ Airfield Design
A regional airport to be designed has the following emplacement characteristics:
• Located at sea level;
• Emplaced in flat terrain;
• Located in a standard latitude of 40 deg ($\Delta T = 0$).
The critical aircraft has the following characteristics:
• Reference Field length $\to\0 1100 m. • Wingspan \(\to$ 28 m.
Do a preliminary design of the runway according to ICAO’s regulations in App. 14. Follow the following steps (use sketches if needed):
Table 9.11: Runway ICAO categories (alph. code refers to the type of aircraft).
Table 9.12: Minimum runway’s width [m] ICAO identifiers.
1. Identify the reference code of the aerodrome (see Table 9.11);
2. Select length and width of the runway (see Table 9.12);
3. Dimension the safety areas (strip and runway end safety area);
4. Identify the designators of the two runway heads according to the runway direction selected in the previous exercise.
5. Choose one of the runway’s heads and displace the threshold 100 m. Publish the declared distances of this runway head (notice that the stopway and the clearway need to be considered; if needed dimension them following ICAO recommendations).
The needed information can be found in the tables below and in the appendix containing ICAO’s appendix 14 (3.10-3.17).
Answer
Figure 9.22: Runway design.
1. The reference code of the aerodrome would be 2C.
2. Given that the altitude of the aerodrome is 0, conditions are ISA standard, and there is no slope, the distance should be the given reference field of the critical aircraft (1100 m). Any larger runway would also work but implies over-investment. Width should also be the minimum (30 m) for the same reason.
3. Dimension must be at least: Strip (1220 m $\times$ 180 m) and RESA (60 m $\times$ 90 m) starting at the edge of the runway head. Authorities recommend RESA length of 120 m for instrumental aerodrome.
4. The designators would be 05 and 23.
5. There must be a Clearway (length less than have of the take-off run $\times$ 75 m width from the center line); Stopway is not mandatory, but it must have same width as the runway. Please check Figure 9.22 for the solution.
6. Declared distances can be also checked in Figure 9.22. Notice that in case of a SWY, the length of the runway should be 1100 + SWY. ASDA would then be 1100 + SWY.
Exercise $6$ Visual aids data
The exercise is related to identifying the main aerodrome visual aids for a particular airport. For the airport Adolfo Suarez Madrid Barajas, identify:
1. the main markings of any of the runways, i.e.:
$\bullet$ Designator
$\bullet$ Threshold and pre-threshold
$\bullet$ Center lines and side line
$\bullet$ Aiming point
$\bullet$ Touchdown zone
2. the main lights of any of the runways.
3. the instrumental aids in the airfield, i.e, VOR, DME, ILS. Write down the location and the frequency.
4. the main markings in the taxiways.
5. the main markings in apron and ramp.
Answer
All data can be consulted at AENA’s AIS Adolfo Suarez Madrid Barajas. In particular, you should consult:
• Aerodrome data
• Aerodrome chart
• Aerodrome ground movement chart
• Aircraft parking/docking chart
Exercise $7$ Visual aids design
The exercise is a continuation of Exercise 9.5.5 , thus related to a preliminary runway design. Assume you have done a preliminary design of the runway, including dimensions, safety areas, etc. (i.e., assume you have assessed Exercise 9.5.5 ). Do a preliminary design of the main markings of the runway, i.e.:
• Designator
• Threshold and pre-threshold
• Center lines and side line
• Aiming point
• Touchdown zone
and the main lights of the runway.
Answer
All data can be consulted at ICAO’s Annex 14 and ICAO’s Runway design manual (Part I and IV).
Exercise $8$ Take-off length calculation
We want to estimate the take-off distance of a typical commercial jet aircraft. Such aircraft mounts two turbojets, which thrust can be estimated as: $T = T_0 (1 − k \cdot V^2)$, where $T$ is the thrust, $T_0$ is the nominal thrust, $k$ is a constant and $V$ is the true airspeed.
Figure 9.23: Forces during taking off run.
Consider Figure 9.23, where $g$ is the force due to gravity, $m$ is the mass of the aircraft, $F_F$ corresponds to the friction force (being $\mu_r$ the friction coefficient of the pavement), and $L$ and $D$ are lift and drag force, respectively, which can be expressed as:
$L = C_L \dfrac{1}{2} \rho SV^2;$
$D = C_D \dfrac{1}{2} \rho SV^2;$
where $\rho$ is the density of air, $S$ is the wet surface area of the aircraft, $C_D$ is the coefficient of drag (which can be approximated to the parasite coefficient of drag, i.e., $C_D = C_{D_0}$) and $C_L$ is the coefficient of lift.8
Find:
1. An analytic expression for the take-off distance of this generic aircraft.
Consider a B-737-800, which values can be approximated to:
• $T_0 = 149000\ [N]$ and $k = 1 \cdot 10^{-5}$.
• $C_{D_0} = 0.0357$ (with flap configuration for take-off)
• $S = 124.65\ [m^2]$;
• $m = 78300\ [kg] (MTOW)$;
• $V_{TO} = 1.2 V_{Stall}$ (with flop configuration for take-off). We can consider $V_{LOF} = 1.1 V_{Stall}$;
• $V_{stall} = \sqrt{\tfrac{2mg}{\rho S C_{L_{\max}}}}$;
• $C_{L_{\max}} = 2$ and $C_L = 0.8 C_{L_{\max}}$.
Moreover, we can consider $\mu_r = 0.025$.
According to the previously selected numbers:
2. Take-off distance for different altitudes (sea level, 2000 ft, 4000 ft, 6000 ft, 8000 ft, 10000 ft) under calm conditions and maximum take-off weight. Compare these results with the figures published in the 737 Airplane Characteristics for Airport Planning document. Discuss them.
Answer
[1] We apply the 2nd Newton's Law:
$\sum F_z = 0;\label{eq9.5.6}$
$\sum F_x = m \dot{V}.\label{eq9.5.7}$
Regarding Equation ($\ref{eq9.5.6}$), notice that while rolling on the ground, the aircraft is assumed to be under equilibrium along the vertical axis.
Looking at Figure 9.23, Equations ($\ref{eq9.5.6}$)-($\ref{eq9.5.7}$) become:
$L + N - mg = 0;\label{eq9.5.8}$
$T - D - F_F = m \dot{V}.\label{eq9.5.9}$
being $L$ the lift, $N$ the normal force, mg the weight; $T$ the trust, $D$ the drag and $F_F$ the total friction force.
It is well known that:
$L = C_L \dfrac{1}{2} \rho S V^2;\label{eq9.5.10}$
$D = C_D \dfrac{1}{2} \rho SV^2.\label{eq9.5.11}$
It is also well known that:
$F_F = \mu_r N.$
Equation ($\ref{eq9.5.8}$) state that: $N = mg - L$. Therefore:
$F_F = \mu_r (mg - L).\label{eq9.5.13}$
Given that $T = T_0 (1 - kV^2)$, with Equation ($\ref{eq9.5.13}$) and Equations ($\ref{eq9.5.10}$)-($\ref{eq9.5.11}$). Equation ($\ref{eq9.5.9}$) becomes:
$(\dfrac{T_0}{m} - \mu_r g) + \dfrac{(\rho S(\mu_r - C_D) - 2T_0 k)}{2m}V^2 = \dot{V}.\label{eq9.5.14}$
Now, we have to integrate Equation ($\ref{eq9.5.14}$).
In order to do so, we know, as it was stated in the statement, that: $T_0, m, \mu_r, g, \rho, S, C_L, C_D$ and $k$ can be considered constant along the take off phase.
We have that:
$\dfrac{dV}{dt} = \dfrac{dV}{dx} \dfrac{dx}{dt},$
and knowing that $\tfrac{dx}{dt} = V$, Equation ($\ref{eq9.5.14}$) becomes:
$\dfrac{(\tfrac{T_0}{m} - \mu_r g) + \tfrac{(\rho S (\mu_r C_L - C_D) - 2T_0k)}{2m} V^2}{V} = \dfrac{dV}{dx}.\label{eq9.5.16}\) In order the simplify Equation ($\ref{eq9.5.16}$): • $(\tfrac{T_0}{m} - \mu_r g) = A$; • $\tfrac{(\rho S(\mu_r C_L - C_D) - 2 T_0k)}{2m} = B$. We proceed on integrating Equation ($\ref{eq9.5.16}$) between $x = 0$ and $x_{LOF}$ (the distance of lift off); $V = 0$ (assuming the maneuver starts with the aircraft at rest) and the lift off speed: $V_{LOF}$. It holds that: \[\int_{0}^{x_{LOF}} dx = \int_{0}^{V_{LOF}} \dfrac{VdV}{A + BV^2}.$
Intergrating:
$\langle x \rangle_0^{x_{LOF}} = \langle \dfrac{1}{2B} Ln (A + BV^2) \rangle_0^{V_{LOF}}$
Substituting the upper and lower limits:
$x_{LOF} = \dfrac{1}{2B} Ln (1 + \dfrac{B}{A} V_{LOF}^2).\label{eq9.5.19}$
[2] With the values given in the statement and using Eq ($\ref{eq9.5.19}$) and substituting it yields:
• $x_{LOF} (h = 0) = 2605.5\ [m]$
• $x_{LOF} (h = 2000\ ft) = 2764.4\ [m]$
• $x_{LOF} (h = 4000\ ft) = 2935.4\ [m]$
• $x_{LOF} (h = 6000\ ft) = 3119.7\ [m]$
• $x_{LOF} (h = 8000\ ft) = 33186\ [m]$
• $x_{LOF} (h = 10000\ ft) = 3533.4\ [m]$
Figure 9.24 illustrates it. If we look at the official documents, for sea level conditions it can be observed that both values are similar. According to the Figure, the aircraft could not take-off with $MTOW$ for altitude above $2000\ ft$. Repeating the analysis for a mass of 60 tons, results present more similarities with tables.
Exercise $9$ B-737-800 Aircraft Characteristics related to Airport Planning
For the B-737-800, obtain:
• General characteristics (weights);
• General dimensions;
• Payload diagram;
• Take-off runway requirements (sea level; $4000\ ft; 8000\ ft$);
• Landing requirements (sea level; $4000\ ft; 8000\ ft$);
• Turning radii requirements.
Answer
For the solution, place refer to the B-737-800 airport manual that can be accessed at Boeing’s aircraft characteristics related to airport planning.9
8. Notice that $T_0, k, g, m, \mu_r, S, C_{D_0}$ and $C_L$ can be considered constant during take off.
9.6: References
[1] De Neufville, R. and Odoni, A. (2003). Airport systems: planning design, and management, volume 1. McGraw-Hill New York.
[2] Franchini, S., López, O., Antoín, J., Bezdenejnykh, N., and Cuerva, A. (2011). Apuntes de Tecnología Aeroespacial. Escuela de Ingeniería Aeronáutica y del Espacio, universidad politécnica de madrid edition.
[3] García Cruzado, M. (2000). ETSI Aeronáuticos. Madrid .
[4] ICAO (1999). Aerodrome Design and Operations, volume Annex 14, Volume 1. International Civil Aviation Organization, third edition edition.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/09%3A_Airports/9.05%3A_Exercises.txt
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Definition 10.1: Air navigation
The air navigation is the process of steering an aircraft in flight from an initial position to a final position, following a determined route, and fulfilling certain requirements of safety and efficiency. The navigation is performed by each aircraft independently, using diverse external sources of information and proper on- board equipment.
Besides the primary goal above mentioned (safe and efficient controlled flight towards destination), three important additional goals can be mentioned:
• Avoid getting lost.
• Avoid collisions with other aircraft or obstacles.
• Minimize the influence of adverse meteorological conditions.
Definition 10.2: Air navigation system wide perspective
In addition, from a system wide perspective, the main goal of air navigation is to make possible air transportation day after day by means of providing the required services to perform operations safely and efficiently.
Figure 10.1: Dual vision of Air navigation: single flight vs system-wide perspective.
Figure 10.1 illustrates these dual vision in the definition of air navigation.
10.1.02: History
The first navigation techniques at the beginning of the 20th century were rudimentary. The navigation was performed via terrain observation and pilots were provided with maps and compasses to locate the aircraft.
Dead reckoning
Some years later the dead reckoning was introduced. The dead reckoning consists in estimating the future position of the aircraft based on the current position, velocity, and course. Pilots were already provided with anemometers to calculate the airspeed of the aircraft and clock to measure time. The flights were designed based on points (typically references in the terrain), and pilots had to follow the established track.
Obviously, when trying to fly a track from one point to another using dead reckoning, the errors were tremendous. This was due to three main reasons:
• Errors in the used instruments (anemometer, compass, and clock).
• Piloting errors.
• Wind effects.
Figure 10.2: Triangle of velocities.
The fist two are inherent to all types of navigation and they will always be to some extent. Nevertheless, errors in instruments are being reduced. Also piloting errors are being minimized due to automatic control systems. One can think that these two errors will converge to some bearable values. On the contrary, wind effects are more relevant, and still play a key role in the uncertainty of aircraft trajectories. Based on these errors, but in particular on wind effects, one can define the triangle of velocities and the track and course angles. See Figure 10.2.
Track angle (TR) (also referred to as course angle): Is the angle between the North (typically magnetic, but the geographic North can be also used) and the absolute velocity of the aircraft and it corresponds with the real track or course the aircraft is flying. The absolute velocity of the aircraft is the sum of the aerodynamic velocity and wind speed: $\vec{V}_{abs} = \vec{V}_{aer} + \vec{V}_w$. This is called triangle of velocities.
The heading angle (HDG): It is the angle between the North (typically magnetic, but the geographic North can be also used) and the aerodynamic velocity vector. Notice that if we assume symmetric flight, it also coincides with the longitudinal axis of the aircraft. Typically it does not coincide with the track angle since the aircraft might have to compensate cross wind.
For instance, looking at Figure 10.2.b, the aircraft heading is the vector joining A and B, but the real track or course is represented by the vector joining A and C. The corresponding angles would be calculated establishing a reference (typically the magnetic North). Notice that the difference between heading and track is referred to as drift angle.
Some other elements that are important in defining the flight orientation are: the desired track, the cross-track error, and the bearing:
Desired Track (DTR) angle: Is the angle between the north (typically magnetic) and the straight line joining two consecutive waypoints in the flight. Is the track we want to fly, which in ideal conditions would coincide with the track we are actually flying. Unfortunately, this rarely happens.
Cross-Track Error (XTE): Is the distance between the position of the aircraft and the line that represents the desired track. Notice that the distance between a point and a line is the perpendicular to the line passing through the point. Thus XTE = d(DTR,TR), where d can be defined as the norm 2 (the euclidean distance).
Bearing: It is defined as the angle between the north (typically magnetic) and the straight line (in a sphere or ellipsoid would not be exactly straight) connecting the aircraft with a reference point. Note that the bearing depends on the selected reference point. Nowadays, these points typically coincide either with navigational aids located on earth or waypoints calculated based on the information of at least two navigational aids located on earth.
Astronomic navigation
Besides the errors caused by wind effects, dead reckoning navigation had one fundamental drawback: It was required that the selected points acting as a reference were visible by the pilots in any circumstance. As the reader can intuitively imagine, these points were sometimes difficult to identify in case of adverse meteorological conditions (rain, fog, etc.) or at dark during night flights. Moreover, it was really difficult to obtain references over monotone landscapes as it is the case for oceans.
Figure 10.3: Astronomic navigation: sextant and astrolab.
Therefore, pioneer aviators started to use astronomic devices. Devices such as the astrolabe1 and the sextant2 had been used since centuries for maritime navigation. Using these devices, pilots (helped by a man on board that was term navigator) were able to periodically determine the position and minimize errors.
Thanks to this combined type of navigation: the astronomic navigation used together with the dead reckoning navigation, the most important feats among the pioneers were given birth. Thus, in light of history, one can claim that the first oceanic flights in 1919 (Alcock and Brown) and 1929 (Linderbergh) were, in part, thanks to the implementation of the astronomic navigation, which allowed pilots to reach destination without getting lost.
Navigation aids
All the previously described navigation techniques did not require any infrastructure support on the ground, and therefore they can be considered as autonomous navigation techniques. However, such navigation was complicated and required lot of calculations on board. The navigator had to be continuously doing very complicated calculations and this was not operative at all.
As a consequence, there was a necessity of some type of earth-based navigation aids. The first to appear, in 1918, were the so-called aerial beacon (light beacon). This allowed night flight over networked areas, such the USA. However, these aids were limited. In 1919, the radio communications were started to be used. First, installing transmitters in the cockpits to communicate. Afterwards, using the radio-goniometry.3 Radio-goniometers were installed on board and the navigation was performed determining determining the orientation of the aircraft with respect to two transmitter ground-stations which position was known.
Later on, in 1932, the Low Frequency Radio-Range (LFR) appeared, which was the main navigation system used by aircraft for instrument flying in the 1930s and 1940s until the advent of the VHF omnidirectional range (VOR) in the late 1940s. It was used for en route navigation as well as instrumental approaches. Based on a network of radio towers which transmitted directional radio signals, the LFR defined specific airways in the sky. Pilots navigated the LFR by listening to a stream of automated Morse codes. It was some sort of binary codification: hearing a specified tone meant to turn left (in analogy, 1 to turn left) and hearing a different specified tone meant to turn right (in analogy, 0 to turn right).
Since the 40s towards our days, the navigational aids have evolved significantly. To cite a few evolutions, the appearance of VOR and DME in the late 40s-early 50s, the concept of Area Navigation (RNAV) in the late 60s-early 70s, the fully automated ILS approach system in the late 60s, or even the satellite navigation (still to be fully operative) contributed to improve navigation performances. We will not describe them now, since all these types of navigation will be studied later on. As a consequence of the appearance of all these navigation aids throughout the years, nowadays the navigation is mainly performed using instrumental navigation techniques.
Navigation in the presence of other aircraft
Being able to fly, not getting lost, and avoiding terrain obstacles was at the beginning already a big challenge. At the beginning, due to the limited number of aircraft, the navigation did not consider the possibility of encountering other aircraft that might cause a collision. With the appearance of airports, attracting many aircraft to the same physical volume, the concept of navigation shifted immediately to that one of circulation.
Circulation can be defined as the movement to and from or around something. In air navigation, it appeared the necessity of making the aircraft circulate throughout certain structures defined in the air space or following certain rules. To avoid collisions, there were defined some rules based on the capacity of being able to see and to be seen. In the cases of approaches and departures in airports, it appeared the necessity of existence of someone with capability to assign aircraft a sequence to take off. These were the precursors of what we know today as air traffic controllers. In 1935, the first control center for air routes was created in the USA.
The air navigation as a system
As a result, juridic, operative, and technical support frameworks to regulate the air navigation were necessary. The technical and operative frameworks should supply:
• Information system prior departure: related to meteorology, operative limitations, and limitations in the navigation aids.
• Tactical support to pilots: related to possible modifications in the conditions of the flight, specially to avoid potential conflicts with other aircraft or within regions under bad weather conditions.
• Radio-electric infrastructures: to provide aircraft navigation aids.
These three items have constituted the basic pillars in what is referred to as system of air navigation throughout its whole development. The technical and operative framework that conform the system (based on the so-called CNS-ATM4 concept) will be studied in the forthcoming sections.
The juridic framework should take into account the following aspects: formation and licenses of the aeronautical personal; communication systems and procedures; rules about systems and performances on air, and air traffic control; air navigation requirements for aircraft certification, registration and identification for aircraft that carry out international flights; and aeronautical meteorology, maps, and navigation charts; among others. We give know a brief overview of the juridic framework. More in depth analysis will be undertaken in posterior courses regarding air law.
The International regulation (juridic) framework
The very first concern, at the beginning of the 20th century, was that of being able to fly. Within the following 30 years the concern changed to that of being able to fly anywhere (within the possibilities of the aircraft), even though in the presence of adverse navigation conditions, and avoiding collisions with other aircraft and the terrain.
The necessity of flying anywhere (including international flights) encouraged the development of an international regulation framework which could establish the rights and obligations when going beyond the domestic borders. Navigation systems and equipments should be also uniform, so that the crew could maintain the same modus operandi when trespassing borders.
In 1919, the International Commission for Air Navigation (ICAN) was created to provide international regulations. In practice, it was the principal organ of an international arrangement requiring administrative, legislative and judicial agents. At the end World War II, in November 1944, 55 states were invited to Chicago to celebrate a conference the Chicago Convention was signed. The Chicago Convention, as studied in Chapter 8, promoted the safe and orderly development of international civil aviation throughout the world. It set standards and regulations necessary for aviation safety, security, efficiency and regularity, as well as for aviation environmental protection. This obviously included air navigation regulations. The Chicago Convention gave birth to ICAO, successor of ICAN. The development of ICAO’s regulation (see contents of Chicago convention in Chapter 8) has led to the creation of international juridic framework.
In Spain we count with AENA (Aeropuertos Españoles y Navegación Aérea), divided into two main directions: Airports and Air Navigation. Regarding the Air Navigation direction, its main function is to provide aircraft flying in what is termed as civil traffic (commercial flights and general aviation flights) all means so that aircraft are able to navigate and circulate with safety, fluidity, and efficiency over the air space under Spanish responsibility. AENA is therefore the Air Navigation Service Provider (ANSP) in Spain. Also in Spain, the regulator organ is AESA (Agencia Estatal de Seguridad Aérea), which depends of the Dir. General de Aviación Civil, Ministerio de Fomento. AESA is the state body that ensures civil aviation standards in all aeronautical activity in Spain.
If we draw a parallelism, the European Civil Aviation Conference (ECAC) is the European regulator organ and the European Organization for the Safety of Air Navigation (EUROCONTROL) is the ANSP in Europe.5 In the USA, both the function of regulator and ANSP is held by the FAA.
Technical and operative framework
The main goal of air navigation is to make possible air transportation day after day by means of providing the required services to perform operations safely and efficiently. These services are provided based on an organization, human resources, technical means, and a defined modus operandi.
The so constituted system is referred to as CNS-ATM (Communications, Navigation & Surveillance-Air Traffic Management). Therefore, CNS corresponds to the required technical means to fulfill the above mentioned air navigation’s main goal, while ATM refers to the organizational scope and the definition of operational procedures. The CNS will be studied in Chapter 11. ATM is a fundamental part of the so called air navigation services, which is to be studied in the forthcoming sections.
1. An astrolabe is an elaborate inclinometer, historically used by astronomers, navigators, and astrologers. Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars, determining local time given local latitude and vice-versa.
2. The sextant is an instrument that permits measuring the angles between two objects, such for instance a star or planet and the horizon. Knowing the elevation of the sun the hour of the day, the latitude at which the observer is located can be determined.
3. A goniometer is an instrument that either measures an angle or allows an object to be rotated to a precise angular position.
4. As it will be introduced later on, CNS stands for Communication, Navigation and Surveillance; ATM stands for Air Traffic Management.
5. To be more precise, it is the ANSP in Belgium, Netherlands, Luxembourg, and north-west Germany
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In this chapter we analyze air navigation as a whole, including an introduction and historical perspective in Section 11.1, the the different Air Navigation Services in Section 10.1.2, focusing on ASM in Section 10.3, ATFM in Section 10.4, and ATS in Section 10.5. Section 10.6 is devoted to analysing the flight plan. Finally, in Section 10.7, we analyze the project SESAR, giving an overview of future trends in the air navigation system. A good introduction is given in Sáez and Portillo [6]. In depth studies that can be consulted include Sáez et al. [5], Sáez Nieto [7], Pérez et al. [4], and Nolan [3]. The reader is also referred to Lloret [2].
10: Air navigation- ATM
Figure 10.4: The air navigation services.
The Air Navigation Services (ANS) conform all the services that are provided to airspace users (i.e., aircraft) such that all flight operations are safely (and efficiently) performed. These services are provided by each country within the volumes of air under its responsibility. The provider of these services, which can be either government departments, state-owned companies, or privatised organisations, are referred to as Air Navigation Service Providers as introduced in the previous section. Figure 10.4 presents an schematic of the different services.
10.02: Air Navigation Services
The Aeronautical Information Services (AIS) can be defined as:
The AIS provides the necessary information to ensure aeronautical operations develop with safety, regularity, economy and efficiency. All the information is made public and distributed air navigation central services.
This information included in the AIS is composed of:
• Aeronautical Information Publication (AIP).
• AIP Amendments (AMDT) and Supplements (SUP).
• Notice to Airmen (NOTAM)-SNOWTAMs.
• Aeronautical Information Circulars (AIC).
The AIP is a basic aeronautical information manual. It contains permanent information and long-term changes and it is used essentially for air navigation and airport operations. It contains information on availability of routes, navigation charts, etc. both for airports and en-route areas.
Regular amendments (AMDT) include small changes and editorial corrections in the AIP. The most important one is the AIRAC cycle, an AIP revision issued every 28 days with information on the airspace, including routes. The SUP complement or vary the information contained in the AIP. They contain temporary information that requires extensive texts and/or explanatory graphics.
NOTAM are notices distributed by means of telecommunication containing information concerning the establishment, condition or change in any aeronautical facility, service, procedure or hazard, the timely knowledge of which is essential to personnel concerned with flight operations. NOTAMs are issued by national authorities for a number of reasons, such as:
• Hazards such as air-shows, parachute jumps and glider or micro-light flying;
• Flights by important people such as heads of state;
• Closed runways, taxiways, etc;
• Unserviceable radio navigational aids, lights, etc.;
• Military exercises with resulting airspace restrictions;
An Aeronautical Information Circular (AIC) is a notice containing information that does not qualify for the origination of a NOTAM or for inclusion in the AIP, but which relates to flight safety, air navigation, technical, administrative or legislative matters.
Further information can be consulted in ICAO’s Annex 15 [1].
10.2.02: Meteorological Services (MET)
Meteorological service for international aviation is provided by meteorological authorities designated by states through the use of standardized MET products and services delivered in accordance with ICAO Annex 3 regulations. Each State also establishes a suitable number of meteorological offices, i.e. aerodrome meteorological offices, meteorological watch offices (MWOs) and aeronautical meteorological stations. Those services have been established on the prevailing state-of-the-art available in the 1960’s, and consist mainly in coded information, which is composed of:
• METAR/TAF: Aerodrome MET conditions/forecast;
• SIGMET: En-route significant weather advisory;
• AIRMET/GAMET: En-route weather phenomena (less significant).
Moreover, weather forecasts (including wind and convective areas, both very relevant) of en-route conditions, except forecasts for low-level flights issued by meteorological offices, are prepared by world area forecast centres (WAFCs). This ensures the provision of high-quality and uniform forecasts for flight planning and flight operations.
Figure 10.5: Meteorological effects.
Figure 10.5 presents a Cumulonimbus: a highly convective region that should be avoided when encountered. On the right-hand side, winds over the North-Atlantic region are presented, where a Jet Stream6 can be seen. Airlines would plan its flight plane taking advantage of favourable winds. In the Figure, the Great-Circle Distance (also referred to as orthodromic or minimum distance path) is compared with wind optimal trajectories. The difference is significative. Indeed, anyone who would have flown over the North Atlantic would have noticed that flying eastwards is much faster than flying westwards. The main reason is the Jet Stream.
6. Jet Stream refers to fast flowing, narrow air currents found in the upper atmosphere or in troposphere of Earth. There is typically one in the North Atlantic.
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Figure 10.6: ATM levels.
The Air Navigation Service Providers (ANSPs), i.e., AENA in Spain, FAA in the USA, Eurocontrol in central Europe, must have technical capabilities to develop and support a technical CNS infrastructure, but on the other hand, it is also needed a highly structured organization with high skilled people, forming the operational support needed to provide transit, communication, and surveillance services. This operational infrastructure is referred to as Air traffic Management (ATM).
ATM is about the process, procedures, and resources which come into play to make sure that aircraft are safely guided in the skies and on the ground. If we consider the time-horizon between the management activity and the aircraft operation, we can identify three levels of systems:
• AirSpace management (ASM). (Strategic level).
• Air Traffic Flow and capacity Management (ATFM). (Pre tactical level).
• Air Traffic Control (ATC).7 (Tactical level)
Airspace management (ASM)
The first layer of the ATM system, the so-called Airspace Management (ASM), is performed at strategic level before aircraft departure, within months/years look-ahead time. The ASM is an activity which includes airspace modeling and design. As aircraft fly in the sky, they follow pre-planned routes conformed by waypoints, airways, departure and arrival procedures, etc. The route followed by an aircraft is selected by the company before departure based on the airspace design previously made by the ASM. The ASM activity includes, among others, the definition of the network or routes (referred to as ATS routes), the organization of the airspace in regions and control sectors, the classification (determined airspace is only flyable by aircraft fulfilling determined conditions) and the delimitation (some regions of the airspace might be limited/restrung/prohibited for civil traffic) of the airspace. All this information is in turn published in the AIP. Section 10.3 will be devoted to airspace management and organization, and all these issues will be tackled in detail.
Air traffic flow and capacity management (ATFM)
The second layer of the ATM system is the so called Air Traffic Flow and Capacity Management (ATFM8). It is performed at pre-tactical level before aircraft departure, within weeks up to three hours look ahead time. The idea is the following: Once the flight plan has been determined by the company according to its individual preferences and fulfilling the ASM airspace design and organization, the next step is to match the flight plan with all flights to be operating at the same time windows in the same areas in order to check whether the available capacity is exceeded. This is an important step as only a certain number of flights can be safely handled at the same time by each air traffic controller in the designated volumes of airspace under his/her responsibility. All flight plans for flights into, out of, and around a region, e.g., Europe, must be submitted to an air traffic flow and capacity management unit (the Eurocontrol’s Central Flow Management Unit (CFMU) in Europe), where they are analyzed and processed. Matching the requested fights against available capacity is first done far in advance for planning purposes, then on the day before the flight, and finally, in real-time, on the day of the flight itself. If the available capacity is exceeded, the flight plans are modified, resulting in reroutings, ground delays, airborne delays, etc. Section 10.4 will be devoted to study the ATFM.
Air traffic control (ATC)
The third layer of the the ATM system is the so-called Air Traffic Control (ATC9). It is performed at tactical level, typically during the operation of the aircraft or instants before departure. The idea is the following: once the flight plan has been approved by ATFM, it has to be flown. Unfortunately, there are many elements that introduce uncertainty in the system (atmospheric conditions, measurement errors, piloting errors, modeling errors, etc.) and the flight intentions, i.e., the flight plan, is rarely fully fulfilled. Thus, there must be a unit to ensure that all flight evolve safely, detecting and avoiding any potential hazard, e.g., a potential conflict, adverse meteorological conditions, by modifying the routes. This task is fulfilled by ATC. ATC is executed over different volumes of airspace (route, approximation, surface) in different dependences (Area Control Centers (ACC) and Control Tower) by different types of controllers. Controllers use communication services to advise pilots. Also, pilots are aided by the FIS and ALS. All these systems together increase the situational awareness of the pilot to circumvent any potential danger. Section 10.5 will be devoted to study the ATS, in particular the ATC.
7. To be more precise, ATC belongs, together with the Flying Information Service (FIS) and the alert service (ALS), to the so-called Air Transit Services (ATS). Due to its importance, we restring ourselves to ATC.
8. The acronym ATFCM might be used as well. ATFM and ATFCM can be used interchangeably
9. As already mentioned before, to be more precise, the third layer is referred to as Air Transit Services (ATS), which is composed of the Flying Information Services (FIS), the Alert Services (ALS), and, fundamentally, the Air Traffic Control (ATC). Since ATC is by far the most important one, it has been typically used as the third layer by itself. We adopt the same criterion, but it is convenient not to forget the FIS and ALS.
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The surrounding air is the fundamental mean in which aircraft fly and navigate. In aeronautics, the airspace is considered as the volume of air above the earth surface in which aircraft carry out their activity. The development of aviation has encouraged the organization of the airspace:
First, in the sense of sovereignty and responsibility of the different states. As a state has perfectly defined its territory and its territorial waters, where its laws apply, there must also exist a sovereign airspace. Notice that the international organizations (ICAO) define also the responsibility over the ocean.10
Second, with the airspace politically delimited, the airspace must also be organized to allow the efficient and safe development of aircraft operations. It has been established a network of routes, equipped with navigation aids, so that the aircraft can navigate following the routes. Communication and surveillance services are also provided. This network of routes is referred to as ATS (Air Traffic Services) routes. These routes go through regions that determine the volumes of responsibility over which the functions of surveillance and control are executed. These regions are referred to as FIR/UIR (Flying Information Region/Upper Information Region). Inside each region, there are defined different areas of control depending on the phase of the flight.
Figure 10.7: Air Traffic Control.
10. In aviation there is international air, in analogy with the international waters. The whole airspace is under responsibility of one or more states according to ICAO.
10.03: Airspace Management (ASM)
A route is a description of the path followed by an aircraft when flying between airports. A complete route between airports often uses several airways connected by waypoints. However, airways can not be directly connected to airports. The transition from/to airports and airways is defined in a different way as we will see later on. Thus, the network of ATS routes refer only to the en-route part of the flight (excluding operations near airports).
Airways
Figure 10.8: Air navigation chart: VOR stations as black hexagons (PPM, DQO, MXE), NDB station as brown spot (APG), VORNAV intersection fixes as black triangles (FEGOZ, BELAY, SAVVY), RNAV fixes as blue stars (SISSI, BUZIE, WINGO), VORNAV airways in black (V166, V499), RNAV airways in blue (TK502, T295), and other data. Author: User:Orion 8 / Wikimedia Commons / Public Domain.
An airway has no physical existence, but can be thought of as a motorway in the sky. In Europe, airways are corridors 10 nautical miles (19 km) wide. On an airway, aircraft fly at different flight levels to avoid collisions. The different flight levels are vertically separated 1000 feet.11 On a bi-directional airway, each direction has its own set of flight levels according to the course of the aircraft:
• Course to the route between $0^{\circ}$ and $179^{\circ}$: east direction $\to$ Odd flight levels.
• Course to the route between $180^{\circ}$ and $359^{\circ}$: west direction $\to$ Even flight levels.
Each airway starts and finishes at a waypoint, and may contain some intermediate waypoints as well. Airways may cross or join at a waypoint, so an aircraft can change from one airway to another at such points. A waypoint is thus most often used to indicate a change in direction, speed, or altitude along the desired path. Where there is no suitable airway between two waypoints, ATC may allow a direct waypoint to waypoint routing which does not use an airway. Additionally, there exist special tracks known as ocean tracks, which are used across some oceans. Free routing is also permitted in some areas over the oceans.
Waypoints
A waypoint is a predetermined geographical position that is defined in terms of latitude/longitude coordinates (altitude is ignored). Waypoints may be a simple named point in space or may be associated with existing navigational aids, intersections, or fixes. Recently, it was typical that airways were laid out according to navigational aids such as VORs, NDBs, and therefore the position of the VORs or NDBs gave the coordinates of the waypoint (in this case, simply referred to as navaids). Nowadays, the concept of area navigation (RNAV) allows also to calculate a waypoint within the coverage of station-referenced navigation aids (VORs, NDBs) or within the limits of the capability of self contained aids, or a combination of these. Waypoints used in aviation are given five-letter names. These names are meant to be pronounceable or have a mnemonic value, so that they may easily be conveyed by voice.
ATS routes network
Summing up, the complete network of routes formed by airways and waypoints is referred to as ATS routes. The ATS routes are published in the basic manual for aeronautical information referred to as Aeronautical Information Publication (AIP). AIP publishes information for en-route and aerodromes in different charts (the so-called navigation charts), which are usually updated once a month coinciding with the Aeronautical Information Regulation and Control (AIRAC) cycle. Ocean tracks might change twice a day to take advantage of any favorable wind.
11. This only applies up to 41000 ft., where the separation increases to 2000 ft.
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As mentioned in the introduction of the section, ATS routes go through regions that determine the volumes of responsibility over which the functions of surveillance and control are executed. These regions are referred to as FIR/UIR (Flying Information Region/Upper Information Region).
Figure 10.9: UIRs in the North Atlantic region and western Europe.
FIR regions cover an area of responsibility of a state up to 24500 feet (FL245). Over a FIR, it is defined an UIR, which covers flight levels above FL245. The UIR had to be defined when jets appeared. Notice that jets, opposite to propellers, flight more efficiently in upper flying levels. Typically the geographical area (surface on earth) for both FIRs and UIRs coincide. Figure 10.9 shows the FIR/UIR structure for the North Atlantic region and western Europe in the upper level. In the case of Spain, there are three FIR/UIR regions of responsibility:
• Barcelona FIR/UIR.
• Madrid FIR/UIR.
• Canarias FIR/UIR.
The need for surveillance and control inside FIR/UIR regions differs depending on the phase of the flight. Specifically, there are different needs when the aircraft is cruising (typically with stationary behavior) rather that when the aircraft is in the vicinity of an airport (either departing or arriving). Therefore, we can define different areas inside a FIR/UIR:
• En-Route airspace: Volumes of airspace containing airways connecting airports’ TMA (Terminal Maneuvering Areas), i.e., volumes containing the network of ATS routes.
• TMA: Volumes of airspace situated above one or more airports. In the TMA are located the SIDs12 and STARs procedures that allow aircraft to connect airways with the runway when departing or arriving.
• CTR (Control zone, also referred to as transit zone): The CTR is inside a TMA, it is a volume of controlled airspace around an airport where air traffic is operating to and from that airport. Aircraft can only fly in it after receiving a specific clearance from air traffic control. CTRs contains ATZs.
• ATZ (Aerodrome Traffic Zone): ATZ are zones around an airport with a radius of 2 nm or 2.5 nm, extending from the surface to 2,000 ft (600 m) above aerodrome level. Aircraft within an ATZ must obey the instructions of the tower controller.
The labour of surveillance and control can be also divided in three levels as follows:
• En-Route control (also termed area control): executed over the En-Route airspace.
• Approximation control: executed over the CTR and TMA.
• Aerodrome control: executed over the ATZ.
Figure 10.10: Control dependences, type of control and volumes of airspace under control.
This labour is carried out in different dependences:
• Control towers (TWR), where the aerodrome control is executed.
• Approximation offices (APP), where the approximation control is executed.
• Area Control Centers (ACC), where the en-route (area) control is executed.
Every airport has a control tower to host controllers and equipment so that surveillance, communications, and control is provided to aircraft when departing and arriving. Typically, the aerodrome control includes taxiing operations, take-off and landing, and its dependences are on the control tower (with the controller on the top with complete vision of the aerodrome). The initial climb and the approach phases are controlled within the approximation control level, which dependences are typically located in the base of the control tower. The area control level is provided from the so called ACC, which are not typically located in airports. For instance, in Spain there are five ACCs: Barcelona and Palma de Mallorca in FIR/UIR Barcelona; Madrid and Sevilla in FIR/UIR Madrid; Canarias in FIR/UIR Canarias.
Therefore, the airspace is divided and assigned to control dependences so that controllers can provide surveillance, communications, and control services (ATC services), and thus aircraft operate safely. When the traffic within an airspace portion assigned to a control dependence exceeds the capacities of the controller to carry out his/her duty with safety, such dependencies are divided in what is known as ATC sectors. In this way, the separation between all aircraft within an ATC sector is responsibility of a controller.
He/She gives the proper instructions to avoid any potential conflict, and also coordinates the aircraft transfer between his/her sector and adjoining sectors. Obviously, the dimensions of the sector are determined by both the volume of traffic and the characteristics of it.
12. SID and STAR will be studied in Section 10.6.3.
10.3.03: Restrictions in the airspace
The use of the airspace by ATS routes is limited to some areas of special use due to military operations, environmental policies, or simply security reasons. Therefore, different special use airspaces are defined and designated as:
• Prohibited (P).
• Restricted (R).
• Dangerous (D).
The prohibited zones contain a defined volume of airspace over a sovereign state in which the flight of any aircraft in prohibited, with the exception of those aircraft that are authorized by the ministry of defense. These areas are established due to national security reasons and are published in the navigation charts. The restricted zones contain a defined volume of airspace over a sovereign state in which the flight of any aircraft in restricted. In order to enter such zones, the aircraft must be authorized by ATC. The dangerous zones contain a define volume of airspace over a sovereign state in which there might be operations or activities considered of risk to other aircraft.
10.3.04: Classification of the airspace according to ICAO
There are two different rules under which aircraft can operate, both based on the instruments on board and the qualification of the crew:
• Visual Flight Rules (VFR).
• Instrumental Flight Rules (IFR).
In order operations to be carried out under VFR, the meteorological conditions must be good enough to allow pilots identify the visual references in the terrain and other aircraft. Such meteorological conditions are referred to as Visual Meteorological Conditions (VMC). VFR require a pilot to be able to see outside the cockpit, to control the aircraft’s altitude, navigate, to maintain distance to surrounding clouds, and avoid obstacles and other aircraft. Essentially, pilots in VFR are required to see and avoid.
If the meteorological conditions are below the VMC threshold, the flight must be performed under IFR. Such meteorological condition are referred to as Instrumental Meteorological Conditions (IMC). Notice that IFR flights are under control by ATC services.
Figure 10.11: Classes of Airspace in the USA (altitudes AGL in feet).
Since sometimes VFR and IFR flights must share the same airspace, it was necessary to regulate the operations. With that aim, ICAO has defined seven different classes of airspaces: A, B, C, D, E, F, and G. The most restrictive one is Class A, where only IFR flights are permitted. The least restrictive is Class G, where both IFR and VFR flights are permitted. In any of the other airspace classes, sovereign authorities derive additional rules (based on the ICAO definitions) for VFR cloud clearance, visibility, and equipment requirements. Classes A-E are referred to as controlled airspace. Classes F and G are uncontrolled airspace.Figure 10.11 sketches the classes of airspace in the US (notice that class F in not use in the US). Table 10.1 include some of the main features by ICAO.
Table 10.1: Airspace classification.
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The Air Traffic Flow (and Capacity) management can be defined as follows [ICAO Ann. 11]
A service established with the objective of contributing to a safe, orderly and expeditious flow of air traffic by ensuring that ATC capacity is utilized to the maximum extent possible and that the traffic volume is compatible with the capacities declared by the appropriate ATS authority.
Figure 10.12: ATFM sketch.
In other words, the main issue of ATFM is to maintain an equilibrium between capacity and demand. The demand corresponds to the amount of flights expected for a particular time window in a particular region of the airspace (to be more precise, in the ATC sectors). The capacity is driven by the physical characteristics of the element of the system (ATC sectors for en-route or within TMAs) and operational factors (aircraft and human). Figure 10.12 sketches it; for instance if the capacity of Sector V would be 4, there would be and extra aircraft within the sector for a particular time window.13 Actions such as delaying or re-routing some of the aircraft would be required.
In Europe, it is carried out by Eurocontrol’s CFMU (Central Flow Management Unit), which acts as the coordinator at European level, and the FMP (Flow Management Positions), which are unities stablished in the control centers to guarantee the fulfilment of ATFM regulations and to monitor the execution of the operation (reporting to CFMU in case of congestion). The overall objective is to optimise traffic flows according to air traffic control capacity while enabling airlines to operate safe and efficient flights. Eurocontrol starts planning operations as early as possible (sometimes more than one year in advance), consolidating the air traffic forecasts issued by the aviation industry and the capacity plans issued by the Air Traffic Control Centres and airports. It also defines operational scenarios to anticipate specific events which may cause congestion (such as sporting events, Christmas skiing or summer holiday traffic).
This ATFM function can be divided into three temporal phases:
Strategic ATFM, carried out from some months (even one year) to roughly 2 days before the operation. It is based on traffic forecasts (typically coming from repetitive flight plans). Under expected congestion, CFMU might publish a Route Availability Document (RAD). The RAD exposes the set of routes that will not be permitted to the aircraft in order to avoid determined areas to be congested. The company could present the most efficient flight plan taking only into account the limitations in RAD.
Pre-tactical ATFM, carried out from roughly 2 days to roughly 3 hours before the operation. It is also based on traffic forecasts (obviously more up to date), yet considers information from FMPs on the short-term expected capacity. Also, weather forecasts are taken into account to for, instance, reorganise traffic in case of expected convective weather in a particular region of the airspace. The regulations that might apply are published in the so called ATFM Notification Messages (ANMs) & Regulations. They inform about areas where the regulation is going to be applied (airports, sectors, etc.), type of traffic affected by the regulation, flight levels affected, period in which the regulation is expected, procedure of assignation of CTOT. The regulations are issued to the airlines so that they can finally complete their flight plans during the day of operation.
Tactical ATFM, which takes place the day of operation. It is based on the actual flight plans submitted by all companies (typically, flights plans must be submitted to CFMU at least three hours before the operation) and it consists on real-time coordination and balancing of capacity and demand. CMFU uses real time information on the capacity of the sectors based on the information of the FMPs. Under any unexpected situation, CFMU will re-route or delay (the latest is the most typical practise) the flight assigning a Calculated Take-Off Time (CTOT):
\[CTOT = EOBT = TAXI\ TIME + Delay,\]
where EOBT stands for Estimated Off Block Time (stated in the original flight plan). The rule for assignation of CTOT is as follows:
In ordering the flows, it will be only taken into account the estimated time of entry in the regulation area. That is, the priority will be given to the first aircraft estimated to enter the regulated area.
13. notice that the figure illustrate a predicted scenario once CFMU has gathered the information of all flights
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The Air Traffic Services can be defined as follows [ICAO Annex. 11]:
Air traffic service (ATS) is a generic term meaning variously, flight information service, alerting service, air traffic advisory service, and air traffic control service (area control service, approach control service or aerodrome control service).
They represent the tactical layer of the Air Traffic Management and consist of the Alerting service (ALS), the Flying Information Service (FIS), and the Air Traffic Control (ATC) service.
10.05: Air Traffic Services (ATS)
Alerting service (ALS) can be defined as [ICAO Annex. 11]:
a service provided to notify appropriate organizations regarding aircraft in need of search and rescue aid, and assist such organizations as required.
In other words, it is a protocol established to deploy all necessary resources for search and rescue in case of any situation of emergency notified by the crew of the aircraft. Indeed, there is an emergency code (squawk 7700 code) that pilots must tune in the transponder to notify an emergency.
The Flying Information Service (FIS) can be defined as [ICAO Annex. 11]:
Flight information service is a service provided for the purpose of giving advice and information useful for the safe and efficient conduct of flights.
10.5.02: Air Traffic Control
Air Traffic Control (ATC) Service is probably the most well known service within the ATS services. It is provided for the purpose of preventing collisions (between aircraft, and between aircraft and obstacles); and expediting and maintaining an orderly flow of air traffic.
Figure 10.13: A typical minimum required separation for the en-route phase.
Before analysing the rol of air traffic controllers, it should be pointed out that aircraft have a safety volumen that should not be violated. A loss of separation minima), also referred to as mid-air conflict, can be defined as any situation in which a pair of aircraft are situated within a horizontal and vertical separation, \(D\) and \(H\), equal or lower than the minimum established by the regulators, \(D_{\min}\) and \(H_{\min}\), respectively. Figure 10.13 illustrates it for the typical values for en-route phase. When this cylinder is violated, a Mid-Air Conflict (MAC) occurs.
Figure 10.14: Vertical advisories through climb/descend maneuver.
Figure 10.15: Horizontal advisories through a turn maneuver (also referred to as vectoring) or a speed advisory.
Air traffic controllers (aided with safety alarms, human support tools, and other layers that apply such as the TCAS) are in charge of identifying potential Mid-air-Conflicts and advising aircraft different manoeuvres to avoid the conflict. Figure 10.14 and Figure 10.15 sketch vertical and horizontal advisories, respectively. If two aircraft are already in conflict, different back-up layers will get activated, e.g., TCAS, to avoid a potential collision.
Types of ATC
There are different types of ATC services, associated to different phases of the flight. These are: area control service, approach control service, aerodrome control service. Each of them acts on different volumes of airspace as introduced in Section 10.3.2 and sketched in Figure 10.10. These three types of ATC services give rise to three types of control (also sketched in Figure 10.10), namely:
• Aerodrome control (also referred to as Tower control) and labeled TWR control.
• Approximation control (also referred to as TMA control) and labeled APP control.
• Route control, labeled ACC control.
Aerodrome control: The volume of airspace in which aircraft separations are responsibility of the aerodrome control function is the so called Air Traffic Zone (ATZ), and includes the manoeuvres area and the aerodrome circuit. The aerodrome control is executed from the top of the control tower (in order to have complete visibility of the airfield). In a control tower, we typically find three types of controllers:
• Aerodrome controller, who assigns runways.
• Taxiway controller (surface controller), who manages surface movements.
• Authorisation controller, who authorises take-off.
Figure 10.16: Multidirectional flow of incoming and outgoing aircraft. Star indicate points of conflicting flows. Blue areas illustrate areas of sequencing.
Approximation control: Besides the ATZ (Airspace under responsibility of the tower control), the controlled airspace through which aircraft transit from landing/take-off to en-route is refereed to as Control Zone (CTR) (Also Terminal Maneuvering Area (TMA) or simply Control Area (CTA)). Approximation control is executed either from dependences inside the tower control or from dependences within an ACC (no direct vision is needed in this case). Approximation controllers have to lead with: take-off traffic, i.e., diverging aircraft (diverging flow); and landing traffic, i.e., converging aircraft (converging flow). Moreover, they have to manage separation between take-off and arriving flows, following a four corner post strategy as illustrated in Figure 10.16.
Approximation controllers’ main task are threefold:
• Sequencing, which is the action that establishes the time-order to access the point in which the common track starts.
• Merging, which is the action that allows aircraft coming from different routes to access that point in the given sequence.
• Metering, is the action that provides the required separation.
En-Route control: the controlled airspace through which aircraft transit the en-route phase is refereed to as en-route airspace. En-route control is executed from dependences referred to as Area Control Centres (ACC). Typically there is 1 to 2 ACCs per FIR/UIR. Two types of en-route controllers can be found in ACCs: executive and planners.
Executive controllers are in charge of handling potential conflicts (identifying and eventually solving) within the assigned ATC sector. Planner controllers are in charge of transmitting aircraft from one adjacent sector to another. It should be noticed that this two roles also appear in approximation control.
En-route aircraft are typically established at a constant speed and flight level. Nevertheless, we might also encounter aircraft ascending/descending: this type of aircraft are referred to as evolution traffic.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/10%3A_Air_navigation-_ATM/10.05%3A_Air_Traffic_Services_%28ATS%29/10.5.01%3A_ALS_and_FIS.txt
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A flight plan is an aviation term defined by the International Civil Aviation Organization (ICAO) as:
Specified information provided to air traffic services units, relative to an intended flight or portion of a flight of an aircraft.14
A flight plan is prepared on the ground and specified in three different manners: as a document carried by the flight crew, as a digital document to be uploaded into the Flight Management System (FMS), and as a summary plan provided to the Air Transit Services (ATS). It gives information on route, flight levels, speeds, times, and fuel for various flight segments, alternative airports, and other relevant data for the flight, so that the aircraft properly receives support from ATS in order to execute safe operations. Two safety critical aspects must be fulfilled: fuel calculation, to ensure that the aircraft can safely reach the destination, and compliance with Air Traffic Control (ATC) requirements, to minimize the risk of collision.
Flight planning is the process of producing a flight plan to describe a proposed aircraft flight. Flight planning requires accurate weather forecasts so that fuel consumption calculations can account for the fuel consumption effects of head or tail winds and air temperature. Furthermore, due to ATC supervision requirements, aircraft flying in controlled airspace must follow predetermined routes.
An effective flight plan can reduce fuel costs, time-based costs, overflight costs, and lost revenue from payload that can not be carried, simply by efficiently modifying the route and altitudes, speeds, or the amount of departure fuel.
From the point of view of ATS provision, the flight planning process starts when the company declares its intention in operating a particular route and finishes when the aircraft takes off. The importance of planning can be seen from two different perspectives: on the one hand, airlines have to assign its (limited) resources; on the other ANSPs need to know the demand in advance and adjust its capacity to it (if impossible, to modify the demand).
The process to follow has there main milestones:
• Coordination of slots.
• Presentation of the flight plan.
• Assignation of departure time.
14. ICAO Document 4444.
10.06: Flight plan
The process of coordination of slots essentially consists of meeting the demand of flights and airport capacity A slot is the administrative authorisation received by an airline company to make use of time window for take-off in a particular airport. This process is based on (European) Regulation CEE 93/95 and the recommendations of IATA.
Figure 10.17: Process of coordination of slots.
The coordination activities are carried out (at international level) in two seasons independently: summer season (March to October); and winter season (October to March). Figure 10.17 presents a timeline sketch of the different activities in the process of coordination of slots. These activities are: first, initial assignation of slots (based on historical analysis and petitions for the actual season); second, an assignation in the IATA conference; third, the post-conference coordination (assignation, re-assignation, cancelation, modification, etc.), which takes place since 2 months before starting the season; fourth, the coordination of slots during the season.
10.6.02: Flight Plan Document
The flight plan is the confirmation by the airline company of the use of an assigned slot. The flight plan represents the linkage item between the agreed planning and the execution of the flight so that Air Traffic Services (ATS) can be provided during the flight in order to ensure safe operations. Typical strategy in regular flights is to submit a repetitive flight plan to the authorities.
Figure 10.18: Flight Plan FAA International Form 7233-4.
Regulation establish that the flight plan must be presented/submitted prior departure (except for some cases of VFR in non-controlled airspace class E, F, G) via two procedures: presentation at the Airport Reservation Office in the origin airport, or the presentation at the Integrated Flight Plan Service (dependent of Eurocontrol in Europe). The flight plan must be submitted within a minimum time prior departure (Estimated Off-Block Time [EOBT]). For IFR subject to ATFM, 3 hours before EOBT. Figure 10.18 presents a FAA’s flight plan form.
10.6.03: Navigation charts
Figure 10.19: Phases in a flight.
Table 10.2: Navigation charts. Data retrieved from Annex 4 ICAO.
The navigation charts are essential for air navigation. Their characteristics are internationally standardized in different ICAO documents, such for instance, the Annex 4: Aeronautical charts. According to the flight rules applied (IFR or VFR), the phase and level of flight, the characteristics of such charts differ. A complete route of an aircraft flying between two airports can be divided in three main parts: origin, en-route, and destination. Also, origin can be divided into take-off and initial climb, and destination can be divided into approach and landing. Figure 10.19 shows an schematic representation of the phases of a typical flight. Table 10.2 show the existing navigation charts according to ICAO’s Annex 4.
Figure 10.20: En-Route upper navigation chart of the Iberic Peninsula.
The en-route phase is defined by a series of waypoints and airways (the already presented ATS routes). The upper en-route navigation chart of the Iberic Peninsula is given in Figure 10.20. However, airports can not be directly connected by airways. Terminal Maneuvering Areas (TMA) are defined to describe a designated area of controlled airspace surrounding an airport due to high volume of traffic. Operational constraints and arrival and departure procedures are defined inside an airport TMA.
Figure 10.21: Instrumental approximation chart: APP to Adolfo Suarez Madrid Barajas, Runway 32L.
A flight departing from an airport must follow a Standard Instrument Departure (SID) which defines a pathway from the runway to a waypoint or airway, so that the aircraft can join the en-route sector in a controlled manner. Before landing, an aircraft must follow two different procedures. It must follow first a Standard Terminal Arrival Route (STAR), which defines a pathway from a waypoint or airway to the Initial Approach Fix (IAF). Then, proceed from the IAF to runway following a final approach procedure. Figure 10.21 shows a final approach chart.
The reader is referred to the ANSP providers’ AIP for more information on navigation charts.15 For instance, AENA publishes its AIP in AIP AENA.16 The en route charts can be consulted at AIP AENA EN-Route.17 Airport charts can be consulted at AIP AENA Aerodromes.18
15. Notice that these documents are published in public access in the internet, but the ANSP holds the copyright on them, so they can not be published in this text book unless explicitly permitted.
10.07: SESAR concept
Single European Sky
Contrary to the United States, Europe does not have a single sky, one in which air navigation is managed at the European level. Furthermore, European airspace is among the busiest in the world with over 33,000 flights on busy days and high airport density. This makes air traffic control even more complex. The EU Single European Sky is an ambitious initiative launched by the European Commission in 2004 to reform the architecture of European air traffic management. It proposes a legislative approach to meet future capacity and safety needs at a European rather than a local level. The Single European Sky is the only way to provide a uniform and high level of safety and efficiency over Europe’s skies. The key objectives are to :
• Restructure European airspace as a function of air traffic flows;
• Create additional capacity; and
• Increase the overall efficiency of the air traffic management system.
The major elements of this new institutional and organizational framework for ATM in Europe consist of: separating regulatory activities from service provision, and the possibility of cross-border ATM services; reorganizing European airspace that is no longer constrained by national borders; setting common rules and standards, covering a wide range of issues, such as flight data exchanges and telecommunications.
SESAR
As part of the Single European Sky initiative, SESAR (Single European Sky ATM Research) represents its technological dimension. It will help create a paradigm shift, supported by state-of-the-art and innovative technology. The SESAR program will give Europe a high-performance air traffic control infrastructure which will enable the safe and environmentally friendly development of air transport. The goals of SESAR are:
• triple capacity of the system;
• increase safety by a factor of 10;
• reduce environmental impact by 50%; and
• reduce the overall cost of the system by 50%.
We will not cover more details of SESAR in this course. The reader is referred to SESAR,19 and the master plan in SESAR Consortium [8] for more information on SESAR.
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Exercise $1$ Navigation
• For the Approximation Chart given in Figure 10.21, identify the Navaids and Associated Frequencies. Identify the different final approach paths for both the lateral and vertical profiles (write down the initial approach fix, initial fix, and the final approach fix/point. Indicate altitudes and distances to the relevant fixes). Identify the go-around procedure.
• For the En-route upper airspace chart in Figure 10.20, identify the UIRs. Identify P,D,R volumes of airspace. Select an airway between two arbitrary waypoints/navaids. Identify the name of the airway, altitude limits, radial, and distance between waypoints/navaids. Select an arbitrary navaid and right down the type, coordinates, and frequency.
• Access your country’s AIP, download a SID and a STAR corresponding to the capital city airport and try to identify all relevant elements.
Answer
An schematic solution is provided below. The reader should notice that the charts analysed below correspond to the Spanish AIP in 2016. For different parts of the world, some differences in charts might be encountered. For the US, see for instance (Citar FAA)
Figure 10.22: Exercise: En-Route Chart.
1) En route Chart: Information is provided in Figure 10.22. The delimitation of UIR regions is marked therein. Yet, a track of airway UN870 between PISUS and PONEM is analysed. Information on that particular track is boxed. The name of the airway (UN870), the magnetic direction ($70^{\circ}$), the minimum FL (FL245, corresponding to upper airspace), and the distance between waypoints (42 NM) can be readily identified. Moreover the > marker indicates that the airway can be only flown in that particular direction. In order to know the flight levels that are permitted, one has to check the ATS basic RNAV routes in the AIP (ENR 3 - ATS routes). In this particular case, EVEN FLs are only permitted.
Figure 10.23: Instrumental app. chart: APP to Adolfo Suarez Madrid Barajas, RWY 32L.
2) Final approximation chart: Notice that this is a non-precision approach (a non ILS approach). Check both the text with descriptions and Figure 10.23.
1) Frequencies for the APP control (e.g., 134.950), TWR control, and AFIS service. These should be tuned on-board to communicate with ATC.
2) Terrain map with altitude in ft and P,D,R areas in pink and altitude restrictions. E.g., marked area LED41 in Figure 10.23 denotes a Dangerous (D) airspace between altitude 5000 ft and Ground.
3) Navaids: For the sake of simplicity, just Colmenar Viejo (labeled CNR) is herein highlighted in Figure 10.23. It can be observed the frequency to tune it (117.30), its geographic location in the map, its coordinates, and the typology (DVOR/DME).
4) Approach procedure: horizontal profile. Only the procedure starting at IAF (Initial Approach Fix) TOBEK is exposed. Aircraft take radial $048^{\circ}$ (with respect to Perales (PDT)); reach IAF TOBEK (4.9 NM from DME Perales); then turn to reach Radial 326 (with respect to Perales), reach the Initial Fix (IF) at 10.2 NM and 5.9 NM with respect to DME Barajas (labeled BRA) and DME Perales, respectively; follow Radial 145 with respect to BRA; reach Final Approach Fix (FAF) at 5.1 NM from DME Barajas; proceed to runway.
5) Approach procedure: vertical profile. Similarly, TOBEK procedure is exposed. Aircraft level off at 5000 ft, overflying TOBEK; then descent following Radial 326 with respect to PDT to 4000 ft (2070 with respect to Ground), level-off and intercept the IF; then proceed down following Radial 145 with respect to BRA, level-off at 3400 ft and intercept FAF; from FAF onwards proceed down with a 5.1% slope.
6) The go-around procedure is highlighted in Figure 10.23. Notice that the Go-Around procedure starts at the MAPT (Mixed Approach Point).
7) Last but not least, in the Tables marked below in the chart one can observe times between FAF and MAPT (3.6 NM) and the rate of descent (ROD) required to maintain 5.1% slope at different Ground Speeds.
3) SID and STAR Route: These questions are left as an open exercise to students. Notice that finding the information in the AIP is not straight forward, thus leaving the exercise open should help the student going through the process of finding the needed information in the AIP. For the interpretation of both SIDs and STAR, one can check for instance IVAO’s SID explanation20 and IVAO’s STAR explanation,21 respectively.
Exercise $2$ ANS Services
Consider an intended flight between airport A and B (e.g., Continental Europe) at time $H$ of day $D$. According to current operations, analyze:
• How the sketched air navigation services in Figure 10.4 would affect your intended flight plan at strategic level, i.e., months before Time H and Day D. What is the status of your flight plan? Which role do Communications, Navigation, and Surveillance play at this stage?
• How the sketched air navigation services in Figure 10.4 would affect your intended flight plan at pre-tactical, i.e., 1-2 days before operation up to three hours before Time H and Day D. What is the status of your flight plan? Which role do Communications, Navigation, and Surveillance play at this stage?
• How the sketched air navigation services in in Figure 10.4 would affect your intended flight plan during the tactical phase (pre-flight and execution), i.e., from 3 hours before departure to real time execution of the flight. What is the status of your flight plan? Which role do Communications, Navigation, and Surveillance play at this stage?
Answer
Figure 10.24L Flight Plan processes.
Please, refer to the air navigation services sketched in Figure 10.4. Aditionally, all proceses in which a flight plan is involved are sketched in Figure 10.24. The following paragraphs try to assess (schematically) the proposes questions:
1) Strategic level
• First one should go into the process of coordination of slots and request one for Day D and Hour H. This should be done according to the season calendar (at least 6 months before departure). Please, refer to Figure 10.17 as an example.
• Assume you have got a slot. If your flight has certain periodicity, you will produce a repetitive flight plan. In order to do so, you must rely on the ASM, in particular on the availability of routes: check AIP for charts. Notice also that ATFM will publish a Route Availability Document (RAD) and your should also comply with it.
• The flight plan should be sent to ATFM (strategic layer) in order them to check that you flight plan is compliant with the structure of routes and the RAD document (consistency check). It could be rejected and thus returned to the flight operations center for further flight planning (also, suggestions of re-routing are typically issued) or accepted. Notice that the Flight Plan submission is a continuous process (you can submit as many time as you will) that might take place at strategic level, however it also exists (probably more intensively) at pre-tactical level.
• Notice that at this point Met Services are not very relevant, since forecast are only able to predict weather within one weak (with a lot of uncertainty though). Much more precise forecast will be made available at later stages of the process.
• CNS do not play an important role here: only notice that the network on routes has been (originally) built based on the existence of navaids located in certain geographical locations. Notice also that the whole structure of ATS routes is designed such that, later during the flight, CNS can be provided.
2) Pre-tactical level
• At this point, 1-2 days, the process is accelerated as time gets closer to departure. Notice that assuming IFR flight (typical of commercial aviation), three hours before departure one must send the flight plan to ATFM dependences. The Flight Plan will include a EOBT (Estimated off Block Time) that ATFM will use for its tactical planning (ATFM allocation of slots in following steps).
• At this stage, Met Service is very important, since much more accurate forecasts are available for both the flight operation center (for fine tuning of the flight plan, e.g., taking into consideration favourable winds) and the ANSP (e.g., identification of potential hazards that might affect capacity of the system).
• ATFM pre-tactical service will be in charge of analysing the capacity of the different sectors in the Network. It will gather information from the Flow Management Positions (1 per sector), Met information, any issue that might appear as a NOTAM, etc. An ATFM Notification Message will be produced including all regulations (capacity restrictions) that apply for the next day. This information might be relevant to flight dispatching to re-do the flight plan if needed.
• CNS: Do not play any important role at this time (just as before)
3) Tactical phase (pre-flight and execution)
• ATFM would at this stage (2-3 Hours before departure) balance capacity and demand. Demand will be given by all submitted flight plans (notice that 3 hours before departure no more flight plans will be admitted) and capacity regulations (included in the ATFM notification messages). ATFM will approve or delay the flight. In any case, ATFM will provide a Calculated Take Off Time (CTOT) (EOBT + taxi + possibly a delay).
• As for MET, short time forecasts and real time information is very relevant at this stage. It is important to distribute Information for pilots and controllers (e.g., METARs) to take demissions before departing and during the flight. In particular, wind direction and intensity will be the driver for the operational configuration (operational runway heads) of both departure and destination airport. This might be helpful to choose SID and STAR (Check AIP). Also, en-route weather hazards must be avoided during the flight.
• Regarding ATS, they are of course very relevant before departure and during the flight: agreeing changes (Pilot-controller) in the intended flight plan; advising manoeuvres in case of potential conflict or weather hazard; using FIS in case any relevant information; ALS might get activated in case of emergency.
• Finally, CNS do play a very important role during execution. Regarding communications, one should consider both fixed service (flight plan submissions to all ATC sectors) and mobile service (voice and data link communications). Regarding navigation systems, they are fundamental to know where the aircraft is and how to follow the route (VOR, DME, etc. are relevant both in Conventional Nav. and RNAV; the information is presented to the pilot using on-board cockpit instruments). As for surveillance systems, they are aimed at providing controllers with the information of the aircraft (position, altitude, velocity, etc.) in their screens (controller working position) to monitor its evolution inflight and, if needed, instruct them the appropriate manoeuvre.
Exercise $3$ ATFM Exercise
Consider 8 aircraft that all depart at the same time T from different airports. All Aircraft have sent its intended flight plans before departure (at T-3H). Tactical ATFM is in charge of analysing whether the routes are compliant with the network of routes (they are) and asses any potential imbalances during the execution of the flight. Conditions of the problem are:
• According to the different FMP, capacity in Sectors I, II, III, IV, and V is 2 at any time.
• Stars correspond to the intended position of $Acj (j = 1,...,8)$ at a future time $T+3h$. Note that the size of the aircraft has been artificially overemphasised.
• Squares correspond to the entry/exit waypoints to sectors.
• Distances are (all units in km):
Airway a) $A - AC1 = 5; Ac1 - C = 45; C - Ac2 = 5; Ac2 - J = 50$;
Airway b) $M - B = 50; B - Ac6 = 10; Ac6 - D = 35; D - Ac7 = 10; Ac7 - E = 25$;
Airway c) $G - Ac3 = 5; Ac3 - Ac4 = 20; Ac4 - H = 25; H - Ac5 = 10; Ac 5 - I = 50$;
Airway d) $F - L = 35; L - Ac8 = 5; Ac8 - K = 50$;
• All aircraft fly at the same speed: $200\ km/h$.
The questions are the following:
Figure 10.25: ATFM layout
• Is demand balanced with capacity (consider not only $T+3H$, but all the potential times in which the aircraft would be overflying Sectors I to V)? If yes, identify capacity imbalances.
• According to the given layout in Figure 10.25 (and acting in the same way as the tactical ATFM faces this issue today), quantitatively assess the measures that should be taken into account (come up with a solution trying to minimise disruptions in the intended flight plans).
Answer
We should first check whether there exist capacity imbalances. In order to do that, we should analyse the status of the aircraft at different time instants both before and after the snapshot time ($T+3H$). The reader should note that this is an ATFM related exercise, i.e., the picture showed represents a simulation of what is supposed to happen (according to submitted flight plans) at time T+3H. What ATFM does (among other issues) is to simulate all submitted flight plans and check for capacity-demand imbalances.
We thus simulate at three different instants of time, namely: $T+3H-3$ min; $T+3H-1.5$ min; $T+3H+7.5$ min; and $T+3H+15$ min. We evaluate the demand (number of aircraft) in each Sector (notice that an aircraft at interSector position is considered to belong to both sectors). The count is as follows:
• $T + 3H - 3$ min (notice that aircraft fly 10 km in 3 min):
- Ac2 in S.I; Ac6 @ B (SECTOR I)
- Ac2 @ B; Ac7 @ D (SECTOR II)
- Ac7 @ D; Ac 8 IN S.III (SECTOR III)
- Ac5 @ H (SECTOR IV)
- Ac 4 in S.V; Ac5 @ H (SECTOR V)
• $T + 3H - 1.5$ min (notice that aircraft fly 5 km in 1.5 min):
- Ac1 @ A; Ac2 @ C (SECTOR I)
- Ac6 in S.II (SECTOR II)
- Ac7 in S.III; Ac8 @ L (SECTOR III)
- Ac5 in S.IV; Ac8 @ L (SECTOR IV)
- Ac3 @ G; Ac2 @ C; Ac4 in S.V (SECTOR V)
• $T + 3H + 7.5$ min (notice that aircraft fly 25 km in 7.5 min):
- Ac1 in S.I (SECTOR I)
- Ac6 in S.II (SECTOR II)
- Ac7 @ E (SECTOR III)
- Ac5 in S.IV; Ac8 in S.IV; Ac4 @ H (SECTOR IV)
- Ac3 in S.V; Ac2 in S.V (SECTOR V)
• $T + 3H + 15$ min (notice that aircraft fly 50 km in 15 min):
- (SECTOR I)
- (SECTOR II)
- Ac6 in S.III (SECTOR III)
- Ac5 @ I; Ac4 in S.IV; Ac3 in S.IV; Ac8 @ K; Ac2 @ J (SECTOR IV)
- Ac1 in S.V; Ac2 @ J (SECTOR V)
Given that the capacity of each sector at any time is 2, it can be readily observed that it is exceeded. Current ATFM studies run the so-called CASA (computer assisted slot allocation) software to balance capacity and demand, essentially a first come-first serve algorithm, that computes the CTOT time imposing ground delays to the "last come" aircraft. The purpose herein is not to replicate the CASA algorithm, but provide a solution that balances demand with capacity. The following is proposed: to delay on ground Ac 4 more that 7.5 min (e.g., 8 min); to delay Ac 3 more that 16.5 min (e.g., 17 min); to delay Ac 2 less that 1.5 min (e.g., 1 min); and to delay Ac 1 more that 1.5 min (e.g., 2 min).
Let us now analyze demand with this new CTOTs (after imposing the proposed delays) at the problematic times and thereafter:
• $T + 3H + 15$ min $\to$
- Ac1 in S.I (SECTOR I)
- (SECTOR II)
- Ac6 in S.III (SECTOR III)
- Ac5 @ I; Ac8 @ K (SECTOR IV)
- Ac2 in S.V; Ac 4 in S.V (SECTOR V)
• $T + 3H + 18$ min $\to$
- (SECTOR I)
- (SECTOR II)
- Ac6 in S.III (SECTOR III)
- Ac2 in S.IV; Ac4 in S.IV (SECTOR IV)
- Ac1 in S.V; Ac3 in S.V (SECTOR V)
It can be seen that now demand does not go beyond the capacity (2 aircraft per sector at any time). The reader should notice that this solution is not intended to be exhaustive, computer assisted simulations are required to test whether this statement holds for all times. In any case, it should serve as example to understand how ATFM works.
Exercise $4$ ATC Exercise
Consider 2 aircraft flying in the configuration sketched in Figure 10.26. ATC is in charge of avoiding any potential conflict during the flight. Conditions of the problem are:
• Ac.1 and Ac. 2 are stablished at constant FL.
• The executive controller in charge of the sector can advise aircraft to modify the speed (speed advisory). Note that no vertical manoeuvres nor vectoring (turn advisories) are considered.
• Stars correspond to the position of Ac 1 and Ac 2 at time t (t correspond to real time, i.e., the sketch is what the controller is seeing in her/his screen). Note that the size of the aircraft has been overemphasised.
• Distances are (all units in km): d1=60 km; d2= 40 km
• True Airspeeds are:22 $V_{TAS_1}$ = 300 km/h; $V_{TAS_2}$ = 200 Km/h. Unless ATC advisory, aircraft are supposed to keep constant speed and track.
• Loss of separation minima can be approximated to a distance of 10 Km.
The questions are the following:
• Is there any potential conflict envisioned?
• Assuming the controller wants to resolve the conflict with only one advisory (either to aircraft 1 or 2) and this advisory can be provided instantaneously at time t: Which speed advisory could be given?
Answer
The equations of motion (taking the waypoint as the origin of coordinates) can be stated as follows:
$x_1 = 60 - V_1 \cdot t \label{eq10.8.1}$
$y_1 = 0 \label{eq10.8.2}$
$x_2 = 0 \label{eq10.8.3}$
$y_2 = 40 - V_2 \cdot t \label{eq10.8.4}$
Let us calculate first the time to reach WP I for each of the aircraft:
$t_{airc_1} = \dfrac{60}{300} = \dfrac{1}{5}\nonumber$
$t_{airc_2} = \dfrac{40}{200} = \dfrac{1}{5}\nonumber$
Therefore, it is straightforward to see that there is potential conflict. Recall that a conflict will exist if the minimum distance is violated (in this case, we assume $d_{\min}$ = 10 km).
The distance can be calculated as follows:
$d^2 = x_1^2 + y_2^2\nonumber$
By substituting $x_1$ and $y_2$ in Eq. ($\ref{eq10.8.1}$) - ($\ref{eq10.8.4}$) and setting $d = 10$, one has a quadratic equation on $t$:
$5.1 + t^2 \cdot 130 - t \cdot 520 = 0\nonumber$
Solving it one gets the time window in which we have the conflict:
$t \in [0.1722, 0.2277]\nonumber$
In order to avoid the conflict, we decide for instance to modify the airspeed of aircraft 1. In order to avoid any potential conflict, $d \ge 10$; $\forall t$, in particular $d \ge 10$; $\forall t \in [0.1722, 0.2277]$.
We have four options:
• accelerate or decelerate aircraft 1.
• accelerate or decelerate aircraft 2.
If we decide to decelerate any of the aircraft, we must impose that at the maximum time of the conflict interval, i.e., $t = 0.2277$, $d = 10$. Since we are reducing speed of one of the aircraft, it will fly slower and arrive later to the conflicting points. We must ensure that at the latest time it has not arrived yet. If we were to accelerate one of the aircraft, we would have to impose the minimum separation criteria at the soonest time of conflict.
The question that arises is: what is the correct strategy? In principle, any of them is valid. Notice however that, we might have problems related to a converging relative velocity.
Figure 10.27: Solution to ATC exercise.
For instance, if we decide to decelerate aircraft 1 and impose that at the maximum time of the conflict interval, i.e., $t = 0.2277$, $d = 10$, substituting we get a quadratic equation on V1, which solution yields 226.94 km/h (the other solution is 300 km/h, i.e., not touching the aircraft). In this case, since aircraft 1 is still flying faster than aircraft 2, we can not ensure that there is no conflict afterwards. Indeed there is. Figure 10.27.a illustrates the solution.
If we lower the velocity to 225 km/h, then there is no conflict anymore. Figure 10.27.b illustrates the solution.
22. note that wind can be neglected
10.9: References
[1] Annex, ICAO (2010). ICAO, Annex 15. Aeronautical Information Services. International Civil Aviation Organization.
[2] Lloret, J. (2017). Introduction to Air Navigation: A technical and operational approach. Javier Lloret [Ed], third edition.
[3] Nolan, M. S. (2010). Fundamentals of air traffic control. Cengage Learning.
[4] Pérez, L., Arnaldo, R., Saéz, F., Blanco, J., and Gómez, F. (2013). Introducción al sistema de navegación aérea. Garceta.
[5] Sáez, F., Pérez, L., and Gómez, V. (2002). La navegación aérea y el aeropuerto. Fundación AENA.
[6] Sáez, F. and Portillo, Y. (2003). Descubrir la Navegación Aérea. Aeropuertos Españoles y Navegación Aérea (AENA).
[7] Sáez Nieto, F. J. (2012). Navegación Aérea: Posicionamiento, Guiado y Gestión del Tráfico Aéreo. Garceta.
[8] SESAR Consortium (April 2008). SESAR Master Plan, SESAR Definition Phase Milestone Deliverable 5.
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In this chapter we analyze the technical enablers that are needed to support aerial operations. There are communications, navigation, and surveillance. Communications systems are studied in Section 11.2, focusing on both fixed and mobile services. Navigation systems are covered in Section 11.3, including autonomous navigation systems and non-autonomous ones, e.g., VOR, DME, NDB, GNSS. Lastly, surveillance systems (radar, TCAS, ADSB) are studied in Section 11.4. In depth studies that can be consulted include Britting [1], Kayton and Fried [2], and Tooley and Wyatt [4]. The reader is also referred to Lloret [3].
11: Air navigation- CNS
Communications, navigation, and surveillance are essential technological systems for pilots in the air and air traffic controllers on the ground. They facilitate the process of establishing where the aircraft is and when and how it plans to arrive at its destination. It also facilitates the process of identifying and avoiding potential threats, e.g., potential conflicts with other aircraft or incoming storms. In order an aircraft to fly from one point to another in a safe way, it must keep continuos contact with the control services on earth by means of communication systems, it must use the navigation systems to continuously determine its position and address to the desired destination. In this whole process, the control services must use the surveillance systems to monitor aircraft and avoid any potential hazard.
Communication
The communications are utilized to issue aeronautical information and provide flying aircraft with air transit services. The air transit services are provided form the different control centers (in which air traffic controllers operate), which communicate with aircraft to give instructions, or simply to inform about potential danger. On the other hand, aircraft must use the proper communication equipment (radios, datalink) to receive this service (by receiving this service it is meant to maintain bidirectional communication with control centers). Besides the communication aircraft-control center (the so-called mobile communications), there must be a communication network between ground stations, i.e., control centers, flight plan dispatchers, meteorological centers, etc. More details about the communication service will be given in Section 11.2.
Navigation
The navigation services refer to ground or orbital (satellites) infrastructures aimed at providing aircraft in flight with information to determine their positions and be able to navigate to the desired destination in the airspace. As already described in Chapter 5, the aircraft will have the required on-board equipment (navigation instruments and displays) to receive this service. More details about the navigation systems will be given in Section 11.3.
Surveillance
The objective of the surveillance infrastructure is to enable a safe, efficient, and cost- effective air navigation service. In airspaces with medium/high traffic density, the function of surveillance requires the use of specific systems that allow controllers to know the position of all aircraft that are flying under their responsibility1 airspace. This service has been typically provided by radar stations in the ground. In this way the evolution of aircraft is monitored and potential threats can be identified and avoided. An instance of this would be two aircraft evolving in such a way that a potential conflict2 is expected in the mid-term. The controller would advise instructions (using the above mentioned communication system) to the involved aircraft to avoid this threat. Automatic Dependent Surveillance-Broadcast (ADS-B) will be replacing radar as the primary surveillance method for controlling aircraft worldwide. There are also airborne systems that fulfill a surveillance function. That is the case of the Traffic Collision Avoidance System or Traffic alert and Collision Avoidance System (both abbreviated as TCAS). More details about the surveillance systems will be given in Section 11.4.
1. Notice that each control center has the responsibility over a volume of airspace.
2. In air navigation, a conflict is defined by a loss of separation minima. This separation minima is typically defined by a circle of 5 NM in the horizontal plane and a vertical distance of 1000 ft.
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The technical means included under the term aeronautical communication fulfill a mission of spreading any information of interest to aircraft operations. This information is real- time information,3 and therefore is not included in the official documents published by the authorities. According to the provided service, ICAO has classified the communications in two main groups:
• Aeronautical Fixed Service (AFS): between terrestrial stations, i.e., fixed stations.
• Aeronautical Mobile Service (RR S1.32):4 between terrestrial stations and aircraft (mobile stations).
3. information of two types: communications with pilots on real time, or information prior departure delivered in what is termed as NOTAM (NOtice To AirMen), which deals with information about the flight plan, meteorological conditions, operative conditions of navaids and/or ATS routes, etc.
4. notation according to ICAO Annex 10-Vol II.
11.02: Communication systems
As defined by ICAO Standards documents in Annex 10 Vol II:
The AFS is a telecommunication service between specified fixed points provided primarily for the safety of air navigation and for the regular, efficient, and economical operation of air services.
This service is typically on charge of spreading all the information prior departure (in NOTAMs), related with flight plans, meteorological information, operative state of the air space, etc. Such information must be transmitted to all fixed point or stations, e.g., control
centers, that might need it to provide support to the aircraft in flight. This information is typically generated in one point and it is distributed using specific terrestrial networks. It is provided by voice and data networks and circuits, including:
• the Aeronautical Fixed Telecommunication Network (AFTN);
• the Common ICAO Data Interchange Network (CIDIN);
• the Air Traffic Services (ATS) Message Handling System (AMHS);
• the meteorological operational circuits, networks, and broadcast systems;
• the ATS direct speech networks and circuits;
• the inter-centre communications (ICC).
The major part of data message interchange in the AFS is performed by the Aeronautical Fixed Telecommunications Network, AFTN. This is a message handling network running according to ICAO Standards documented in Annex 10 to the ICAO Convention, in which it is defined as:
A worldwide system of aeronautical fixed circuits provided, as part of the aeronautical fixed service, for the exchange of messages and/or digital data between aeronautical fixed stations having the same or compatible communications characteristics.
ATFN exchanges vital information for aircraft operations such as distress messages, urgency messages, flight safety messages, meteorological messages, flight regularity messages, and aeronautical administrative messages. One example could be the the spreading of the different flight plans that airlines must submit to authorities prior departure, which must be necessarily transmitted to the different control centers. The technology on which the AFTN is based is referred to as messages commutation. It transmits messages at low speed and therefore the network has low capacity. As a consequence, AFTN is completely outdated (however still widely used).
In order to create a technological upgrade to cope with the increasing volume of information, the CIDIN was conceived in the 1980’s to replace the core of the AFTN. The technology on which the CIDIN is based is referred to as packages commutation and it is considered as a high speed, high capacity transmission network. Typically, most nodes which are part of the AFTN have also CIDIN capability, and thus the CIDIN can be considered as a data transport network which supports the AFTN.
Nevertheless, the volume of information needed is increasing more and more and CIDIN is about to be obsolete (if is not already). The equipment and protocols upon which CIDIN (supporting also ATFN) is based need to be replaced by more modern technology with new messaging requirements. To meet these requirements, the ICAO has specified the ATS Message Handling System (AMHS), a standard for ground-ground communications not fully deployed yet. The AMHS is an integral part of the CNS/ATM concept, and it is associated to the Aeronautical Telecommunication Network (ATN) environment.
The goal of ATN is to be the aeronautical internet, a worldwide telecommunications network that allow any aeronautical actor (ATS services, airlines, private aircraft, meteorological services, airport services, etc.), exchange information in a safe way (control instructions, meteorological messages, flight parameters, position information, etc.), under standard message formats and standard communication protocols.5
The European AMHS makes use of a TCP/IP network infrastructure, in line with the recent evolution of the ATN concept for ground communications. In addition to being the replacement for AFTN/CIDIN technology, the AMHS also provides increased functionality, in support of more message exchanges than those traditionally conveyed by the AFTN and/or CIDIN. This includes, for example, the capability to exchange binary data messages or to secure message exchanges by authentication mechanisms.6
5. The standards of the ATN can be consulted in the ICAO DOC 9705-AN/956: Manual of Technical Provisions for the ATN.
6. The standards of the AMHS can be consulted in the ICAO Doc 9880-AN/466: Manual on Detailed Technical Specifications for the Aeronautical Telecommunication Network (ATN).
11.2.02: Aeronautical mobile service
On the other hand, the aeronautical mobile service includes all technical means required to support the communications between the aircraft and the ATS services (information, surveillance, and control) based on earth. These communications are typically pilot- controller.
As defined by ICAO Standards documents in Annex 10 Vol II:
The aeronautical mobile service is a mobile service between aeronautical stations and aircraft stations, or between aircraft stations, in which survival craft stations may participate; emergency position-indicating radio-beacon stations may also participate in this service on designated distress and emergency frequencies.
The ultimate goal of this service is to allow communications between pilot and controller. In particular, in one control sector, the controller must be able to communicate with all aircraft inside the sector using only one of these radio channels (each sector has a unique frequency assigned). Therefore, the number and dimension of the sectors condition the location of the communication centres. The frequency assigned to each sector establish a double direction channel: pilot-controller; controller-pilot. That is the fundamental instrument in the functions of information, surveillance, and control of aircraft in flight.
Figure 11.1: Radio communications
The categories of messages handled by the aeronautical mobile service and the order of priority in the establishment of communications and the transmission of messages shall be as follows:
1. Distress calls, distress messages, and distress traffic (emergency messages).
2. Urgency messages.
3. Communications relating to direction finding (to modify the course).
4. Flight safety messages (movement and control).
5. Meteorological messages (meteorological information).
6. Flight regularity messages.
There are two types of aircraft-controller communications:
• Controller-pilot voice communications.
• Controller-pilot data-link communications (CPDLC).
Voice communications These services are provided wireless, using radio channels. In the case of aeronautical communications, it is used the VHF (Very High Frequencies) band and HF (High Frequency) band. The channels in HF are only used for long-distance communications, when it is impossible to establish communication using VHF. VHF radio communications (for civil aviation) operate in the frequency range extending from 118MHz to137MHz.7 HF radio communications utilize practically the whole HF spectrum (3MHz to 30MHz), depending on times of the day, seasonal variations, solar activity, etc.
Figure 11.2: Datalink control and display unit (DCDU) on an Airbus A330. © User:SempreVolando / Wikimedia Commons / CC-BY-3.0.
CPDLC communications: A mean of communication between controller and pilot, using data link for ATC communication. Messages can be transmitted using both VHF bands or satellite bands. The way it works is rather simple: when any of either the pilot or the controller wants to establish contact, a message containing the request/instructions is sent. Figure 11.2 illustrates a pilot interface for sending and receiving CPDLC messages.
The first data link ground-air communication were due to ACARS (Aircraft Communication Addressing and Reporting System) in 1978. This service is provided via Inmarsat satellite. Its main drawback is that is not compatible with the ATN. CPDLC was later generalised under FANS (Boeing’s avionics equipment), which has evolved to FANSB, a system with advanced capabilities, e.g., radar mode S, RNP.
7. Notice that the VHF range is 30MHz to 300MHz.
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Navigation systems allow aircraft to know their positions at any time. It is important to distinguish between the systems that assist pilots (navigational aids) to steer their aircraft, and the techniques that pilots use to navigate. The navigational aids constitute infrastructures capable to provide pilots all needed information in terms of position and guidance. On the other hand, the navigation techniques refer to the way in which pilots use these data about the position of the aircraft to navigate. In what follows, we are going to focus on the navigational aids systems.
The navigational aids systems can be classified in two main groups:
• Autonomous systems: Those systems that make only use of the means available in the aircraft to obtain information about its position.
• Non autonomous systems: Those external systems that provide the aircraft with the information about its position.
Table 11.1: Navigational aids systems.
Table 11.1 provides a (non necessarily exhaustive) taxonomy.
11.03: Navigation systems
Using only autonomous navigation systems, the most advanced navigation technique to be used is dead reckoning. As shown in Section 11.1, the dead reckoning consist in predicting the future position of the aircraft based on the current position, velocity, and course. Obviously, a reference (or initial) position of the aircraft must be known. In order to determine this reference position, different means can be utilized, e.g., observing a point near the aircraft which position is known (very rudimentary), the observation of celestial bodies (also rudimentary), or the use of the so-called autonomous systems, which are also able to determine the velocity and course of the aircraft.
The two principal autonomous systems are:
• The Doppler radar.
• The Inertial Navigation System (INS).
Doppler radar: A Doppler radar is a specific radar that makes use of the Doppler effect to calculate the velocity of a moving object at some distance. It does so by beaming a microwave signal towards the target, e.g., a flying aircraft, and listening its reflection. Once the reflection has been listened, it is treated analyzing how the frequency of the signal has been modified by the object’s motion. This variation gives direct and highly accurate measurements of the radial component of a target’s velocity relative to the radar.
Figure 11.3: Doppler effect: change of wavelength caused by motion of the source. © User:Tkarcher / Wikimedia Commons / CC-BY-SA-3.0.
Figure 11.4: Scheme of an Inertial Navigation System (INS). The output refers to position, attitude, and velocity.
Inertial navigation system (INS): An inertial navigation system (INS) includes at least a computer and a platform or module containing accelerometers, gyroscopes, or other motion-sensing devices. The later is referred to as Inertial Measurement Unit (IMU). The computer performs the navigation calculations. The INS is initially provided with its position and velocity from another source (a human operator, a GPS satellite receiver, etc.), and thereafter computes its own updated position and velocity by integrating information received from the motion sensors. Figure 11.4 illustrates it schematically. The advantage of an INS is that it requires no external references in order to determine its position, orientation, or velocity once it has been initialized. On the contrary, the precision is limited, specially for long distances. There are two fundamental inertial navigation systems:
• stable platform systems (aligned with the global reference frame)
• and strap-down systems (aligned with the body frame).
Gyroscopes measure the angular velocity of the aircraft in the inertial reference frame (for instance, the earth-based reference frame). By using the original orientation of the aircraft in the inertial reference frame as the initial condition and integrating the angular velocity, the aircraft’s orientation (attitude) can be known.
Accelerometers measure the linear acceleration of the aircraft, but in directions that can only be measured relative to the moving system (since the accelerometers are attached to the aircraft and rotate with it, but are not aware of their own orientation). Based on this information alone, it is known how the aircraft is accelerating relative to itself, i.e., in a non-inertial reference frame such as the wind reference frame, that is, whether it is accelerating forward, backward, left, right, upwards, or downwards measured relative to the aircraft, but not the direction (attitude) relative to the Earth. The attitude will be an input provided by the gyroscopes.
By tracking both the angular velocity of the aircraft and the linear acceleration of the aircraft measured relative to itself, it is possible to determine the linear acceleration of the aircraft in the inertial reference frame. Performing integration on the inertial accelerations (using the original velocity as the initial conditions) using the correct kinematic equations yields the inertial velocities of the system, and integration again (using the original position as the initial condition) yields the inertial position. These calculations are out of the scope of this course since one needs to take into account relative movement, which is to be studied in advance courses of mechanics. However, some insight is given in Chapter 7 and Appendix A. Two exercises (see Exercises 11.1-11.2) have been proposed for interested readers.
Figure 11.5: Accuracy of navigation systems in 2d. © Johannes Rössel / Wikimedia Commons / CC-BY-SA-3.0
Errors in the inertial navigation system: All inertial navigation systems suffer from integration drift: small errors in the measurement of acceleration and angular velocity are integrated into progressively larger errors in velocity, which are compounded into still greater errors in position. Since the new position is calculated from the previous calculated position and the measured acceleration and angular velocity, these errors are cumulative and increase at a rate roughly proportional to the time since the initial position was input. Therefore the position must be periodically corrected by input from some other type of navigation system. The inaccuracy of a good-quality navigational system is normally less than 0.6 nautical miles per hour in position and on the order of tenths of a degree per hour in orientation. Figure 11.5 illustrates it in relation with other on-autonomous navigation systems (to be studied in what follows).
Accordingly, inertial navigation is usually supplemented with other navigation systems (typically non-autonomous systems), providing a higher degree of accuracy. The idea is that the position (in general, the state of the aircraft) is measured with some sensor, e.g., the GPS, and then, using filtering techniques (Kalman filtering, for instance), estimate the position based on a weighted sum of both measured position and predicted position (the one resulting from inertial navigation). The weighting factors are related to the magnitude of the errors in both measured and predicted position. By properly combining both sources the errors in position and velocity are nearly stable over time. The equation of the Kalman filter are not covered in this course and will be studied in more advanced courses.
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Non autonomous systems require the information generated by terrestrial stations or satellites to determine the position, course, and/or velocity of the aircraft. In this manner, the transmission station (transmitter) produces electromagnetic waves that are received in the reception point (receptor).
The supplied information is referred to as observables (it can be a distance, a course, etc.). Such information locates the aircraft inside a so-called situation surface, i.e., if the observable is the distance, one knows that the aircraft is located at some point on the surface of a sphere with center the transmission center.
More precisely, a situation surface is the geometric locus of the space which is compatible with the observables. The types of situation surfaces are:
• Plane perpendicular to the surface of earth where the aircraft is located. The observable is the course.
• Spherical surface with the center in the transmission station. The observable is the distance.
• Hyperboloid of revolution, being the focuses two external transmission centers exchanging information. The observable is the distance.
In general, using information coming only from one transmitter, it is impossible to locate an aircraft; more sources of information are needed. If two surfaces are intersected,
one obtains a curve. If three (or more) surfaces are intersected, one obtains a point (if more than three, ideally also a point). Therefore, in order to locate an aircraft one needs either two transmitter (generating two surfaces) plus the altitude given by the altimeter, or three transmitters (generating three surfaces).
We turn now the discussion to analyze how such observables are obtained, i.e., how we are able to determine a distance or a course using terrestrial stations or satellites. They are obtained using different techniques based on electromagnetic fields, namely:
• Radiotelemetry.
• Radiogoniometry.
• Scanning beam.
• Spatial modulation.
• Doppler effect.
Radiotelemetry: It is based on the consideration that the electromagnetic waves travel at constant velocity, the velocity of light (c = 300000 km/s), and in straight line8. Under these assumptions, if we measure the time that the waves take from the instant in which the transmitter emits the wave and it receives it back after being rebooted by the receptor (the aircraft), by simple kinematic analysis one can obtain the distance.
Radiogoniometry: It is based on the consideration that the electric and magnetic field that constitute an electromagnetic wave are both perpendicular to the direction of propagation of the wave. Using this technique, by measuring the phases of the electric and magnetic fields of the wave, one can determine the angle that forms the longitudinal axis of the aircraft with the direction of the transmitted wave.
Scanning beam: It is based on the fact that the electromagnetic wave emitted by the transmitter has a dynamic radiation diagram9, with a narrow principal lobe and very small secondary lobes. The receptor (aircraft) is only illuminated (radiated) if the principal lobe points to the aircraft. In this way, knowing the movement law of the radiation diagram, when the aircraft is illuminated one can obtain the direction between the transmitter and the aircraft.
Figure 11.6: Scanning beam radiation diagram and a radar antenna that produces a directional radiation. Notice that the radar is a surveillance system.
Spatial modulation: This technique is original from air navigation. Two different electromagnetic waves are used. The first one is the reference signal, generating an omnidirectional magnetic field so that all points of the region receive the same information. These kind of antennas are referred to as isotropic antennas, and their radiation is referred to as isotropic or omnidirectional radiation. The second signal generates a (either static or dynamic) directional magnetic field. The comparison between the phases of the reference signal and the directional signal determines the direction of the aircraft.
Doppler effect: It is based on the change of frequency of a wave produced by the relative movement of the generating source (transmitter) with respect to the receiver (aircraft). In this way, one can obtain the distance between transmitter and aircraft.
Table 11.2: Navigation aids based on situation surface and the technique.
Table 11.3: Classification of the navigation aids based on the flight phase.
Table 11.2 and Table 11.3 show a classification of the different navigation aids as a function of the different techniques and the different situation surfaces, and a classification of the different navigation aids as a function of the different flight phases, respectively.
The most important ones using the technique of radiotelemetry are of two kinds: those that locate the aircraft in spheres; those that locate the aircraft in revolution hyperboloids. The most important ones among the first ones are: DME (Distance Measurement Equipment); TACAN (TACtical Air Navigation equipment)10, typically used in military aviation; GNSS (Global Navigation Satellites Systems); Radar11 (Radio detection and ranging)12. Due to its importance, we will just analyze more in depth the DME and the GNSS systems. Radiotelemetry is also used in another way by the so called hyperbolic systems (those systems that locate the aircraft in revolution hyperboloids): LORAN-C, Omega, DECCA. None of them is being used nowadays. Due to its historial importance, we will focus on LORAN-C.
Regarding the navigational aids that use the technique of radiogoniometry, the most important one is: NDB (Non-Directional (radio) Beacon).13 For those using Spatial modulation, the most important ones are: VOR (VHF Omnidirectional Radio range); ILS (Instrument Landing System). The ILS has been already studied in Chapter 9. We will focus on the VOR14. Finally, we will analyze the navigational aids that use Scanning beam. This technique is based on concentrating the radiation of electromagnetic waves in a particular direction. Big antennas and high frequencies must be used. The most important systems are: MLS (Microwave landing system); Radar (radio detection and ranging).
Figure 11.7: VOR-DME.
8. Notice that the fact that the light travels in a straight line was proven false in Einstein’s theory of general relativity. Inside the atmosphere (using terrestrial transmitter with aircraft as receptors) one can assume as hypothesis a straight line. When using satellites, the trajectory is a curve and the straight line must be corrected.
9. A diagram of radiation is a graphic representation of the intensity of a radiated signal in each direction. In some cases, there exist a principle lob and secondary lobs of less intensity.
10. Equivalent to the use together of a VOR and a DME.
11. This system uses both radiotelemetry and scanning beam techniques.
12. this system is specific of the surveillance and it will be analyzed later on.
13. The on-board equipment that captures the information is called Automatic Direction Finder (ADF). Thus, sometimes, these equipments are named as a whole as NDB-ADF.
14. Source: Wikipedia: VOR
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Distance Measurement Equipment (DME) is a transponder-based radio navigation system that measures slant range distance by timing the propagation delay of radio signals. The DME system is composed of a transmitter/receiver (interrogator) in the aircraft and a receiver/transmitter (transponder) on the ground. Aircraft interrogate and the DME ground station responds.
Aircraft use DMEs to determine their distance from a land-based transponder (terrestrial station) by sending and receiving pulse pulses of fixed duration and separation. The ground stations are typically co-located with VORs. A low-power DME can also be co-located with an ILS glide slope antenna installation where it provides an accurate distance to touchdown. The simultaneous syntonization of two terrestrial DME stations by the aircraft allows us to locate the aircraft in two dimensions: latitude and longitude. By means of an altimeter, the aircraft is located in the 3D space. When co-located with a VOR, it also provides the direction of flight (read at the on-board ADF equipment). Therefore, the duple VOR-DME provides both position and course. It’s important to understand that DME provides the physical distance from the aircraft to the DME transponder. This distance is often referred to as slant range and depends trigonometrically upon both the altitude above the transponder and the ground distance from it. DME operation will continue and possibly expand as an alternate navigation source to space-based navigational systems such as GPS and Galileo.
Figure 11.8: GNSS systems
11.3.04: Distance Measurement Equipment (DME)
Global Navigation Satellite Systems (GNSS): GNSS are global systems that use a medium high constellation of satellites describing quasi-circular orbits inclined with respect to the terrestrial equator. Currently, just two systems are in practice active: The american GPS and the Russian GLONASS. Some other constellations are in development, such as the European Union Galileo positioning system, the Chinese Compass navigation system, and Indian Regional Navigational Satellite System.
GNSS systems are based on the transmission of an electromagnetic wave by the satellite that is captured and de-codified by the receiver (the aircraft). The basic information that we can obtain is the time that the signal takes while traveling. This time provides a so-called pseudo-distance. This is due to the fact that there exists a synchrony error between the time of the aircraft and the time of the satellites, and therefore, we can not know with certainty the real distance. These systems provide location and time information anywhere on or near the Earth where there is a line of sight to four or more GPS satellites.15
Figure 11.9: Service Areas of Satellite Based Augmentation Systems (SBAS). © User:Persimplex / Wikimedia Commons / CC-BY-SA-3.0.
This system is intended to offer higher precision (with an error of about 10m in determining the position), global coverage, and continuos navigation. However, one of the fundamental drawbacks that have made so far these systems impractical for air navigation is their strategic character in terms of national security. GLONASS was only open to limited civilian use in 2007. The GPS is maintained by the United States government and is freely accessible by anyone with a GPS receiver. However, its reliability is not complete in terms of precision and continuity in the coverage, i.e., the signal has no integrity due to its military character.
Therefore, in order GNSS to be used (still in a limited way and always with back-up systems) in some phases of the flight, a first generation of GNSS (the so-called GNSS-1) was conceived as a combination between the existing satellite navigation systems, i.e., GPS and GLONASS, and some type of augmentation system. Augmentation of a global navigation satellite system (GNSS) is a method of improving the navigation system’s attributes, such as accuracy, reliability, and availability, through the integration of external information into the calculation process. These additional information can be for example about sources of satellite error (such as clock drift, ephemeris, or ionospheric delay), or about additional aircraft information to be integrated in the calculation process. There are three types of augmentation systems, namely:
• The Satellite-Based Augmentation System (SBAS): supports wide-area or regional augmentation through the use of additional satellite-broadcast messages. SBAS systems are composed of multiple, strategically located ground stations. The ground stations take measurements of GNSS satellite signals are used to generate information messages that are sent back to the satellite constellation, which finally broadcasts the messages to the end users (aircraft). Regional SBAS include WAAS (US), EGNOS (EU), SDCM (Russia), MSAS (Japan), and GAGAN (India).
• The Ground-Based Augmentation System (GBAS): a system that supports augmentation through the use of terrestrial radio messages. As for SBAS, terrestrial stations take GNSS signal measurements and generate information messages, but in this case these messages are directly transmitted to the end user (the aircraft). GBAS include, for instance, the LAAS (US).
• The Airborne Based Augmentation System (ABAS): in this augmentation system the ground stations analyze only information coming from the aircraft. This information is transmitted back to the aircraft.
GNSS-2 is the second generation of satellite systems that will provide a full civilian satellite navigation system. These systems will provide the accuracy and integrity necessary for air navigation. These fully civil satellite systems include the European Galileo, which is expected to be fully operative in 2020. Also a civil GPS version is under development.
15. Notice that in order to determine the position of the receptor (the aircraft) we need in this case 4 satellites. This is because an extra satellite is needed to determine the synchrony error between the time of the aircraft and the time of the satellites.
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The LOng RAnge Navigation system is constituted by a chain of stations that allow a wide coverage range using low frequency radio signals. It is an evolution of its precursor: Loran-A, developed during World War II. LORAN is based on measuring the time difference between the receipt of signals from a pair of radio transmitters. A given constant time difference between the signals from the two stations can be represented by a hyperbolic line of position. If the positions of the two synchronized stations are known, then the position of the raircraft can be determined as being somewhere on a particular hyperbolic curve where the time difference between the received signals is constant. In ideal conditions, this is proportionally equivalent to the difference of the distances from the aircraft to each of the two stations.
An aircraft which only receives signals form a pair of LORAN stations cannot fully fix its position. The aircraft must receive and calculate the time difference between a second pair of stations. This allows to be calculated a second hyperbolic line on which the aircraft is located. In practice, one of the stations in the second pair also may be (and frequently is) in the first pair. This means signals must be received from at least three LORAN transmitters to locate exactly the aircraft. By determining the intersection of the two hyperbolic curves, the location of the aircraft can be determined.
LORAN has been widely used to navigate when overflying oceans, where DME and VOR coverage ranges are insufficient. In recent decades LORAN use has been in steep decline, with the GNSS systems as primary replacement. However, there have been attempts to enhance LORAN, mainly to serve as a backup to GNSS systems.
Figure 11.10: LORAN.
11.3.06: LORAN-C
A NDB is a radio transmitter at a terrestrial location that is used to obtain the course or position of an aircraft. Due to the fact that NDB uses radiogoniometry, its signals are affected (more than other aids) by atmospheric conditions, mountainous terrain, coastal refraction, and electrical storms, particularly at long range. The navigation based on NDB aids consists of two fundamental parts: the automatic direction finder (ADF), which is the equipment on-board the aircraft that detects the NDB’s signal, and the NDB transmitter. ADF equipment determines the direction to the NDB station relative to the aircraft, which is presented to the pilot on a Radio Magnetic Indicator (RMI). In this way, in a simple, intuitive manner pilots know if the aircraft is addressing towards an NDB; if not, they now de deviation and can correct the course.
Figure 11.11: Non Directional Beacon.
NDBs are also used to determine airways of fixes. NDB bearings16 provide a method for defining a network of routes aircraft can fly. In this way, the network of terrestrial NDB stations (also VORs) can uniquely define a network of fixes (connected by airways, i.e., the bearings) in the sky. Indeed, 20-30 years ago, the routes aircraft followed to complete a flight plan were only based on NDB/VOR stations. In a navigation chart a NDB is designated by a symbol as in Figure 11.11.a. More recently, another way of navigation has arisen: the so-called RNAV. It is based on calculating fixes based on the information provided by two aids. For instance, using the information coming from two NDBs, fixes are computed by extending lines through known navigational reference points until they intersect. In this manner, many fictitious (in the sense that are not related to an existing terrestrial station) fixes or waypoints have been defined, increasing the network of routes and thus the capacity and efficiency of the system. See Figure 11.11.b.
16. A bearing is a line passing through the station that points in a specific direction.
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VOR It is a type of short-range radio navigation system, enabling aircraft to determine their position and/or course by receiving VHF radio signals transmitted by a network of fixed ground radio stations. The VOR was developed in the US during World War II and finally deployed by 1946. VORs can be considered all fashioned, but they have played a key role in the development of the modern air navigation. As we pointed out in the case of NDBs, VORs have been traditionally used as intersections along airways, and thus, to configure airways. Many people have been claiming throughout years that the GNSS system will sooner rather than later substitute them (as well as NDBs, DMEs, etc.), but however VORs still play a fundamental role in air navigation. Indeed, VOR is the standard air navigational system in the world, used by both commercial and general aviation.
The way a fix or a direction can be obtained based on VOR information is identical as what have been exposed for NDBs. However, VOR’s signals provide considerably greater accuracy (90 meters approx.) and reliability than NDBs due to a combination of factors. VHF radio is less vulnerable to diffraction (course bending) around terrain features and coastlines. Phase encoding suffers less interference from thunderstorms.
Figure 11.12: VOR (Author: Denelson83 / Wikimedia Commons / Public Domain), VOR- DME (Author: User:mamayer / Wikimedia Commons / / CC0 1.0), and VORTAC (Author: User:Denelson83 / Wikimedia Commons / Public Domain) symbols on a navigation chart.
Typically, VOR stations have co-located DME or military TACAN. A co-located VOR and TACAN is called a VORTAC. A VOR co-located only with DME is called a VOR-DME. A VOR radial with a DME distance allows a one-station position fix. VOR-DMEs and TACANs share the same DME system. The different symbols that identify this co-inhabiting systems are illustrated in Figure 11.12.
Figure 11.13: Spatial modulation in VORs. A radio beam sweeps (30 times per sec.). When the beam is at the local magnetic north, the station transmits a second, omni-directional signal. The time between the omni-directional signal and instant in which the aircraft receives the directional beam gives the angle from the VOR station (105 deg in this case). © User:Orion 8 / Wikimedia Commons / CC-BY-SA-3.0.
A VOR ground station emits an omnidirectional signal, and a highly directional second signal that varies in phase 30 times a second compared to the omnidirectional one. By comparing the phase of the directional signal to the omnidirectional one, the angle (bearing) formed by the aircraft and the station can be determined. Figure 11.13 illustrates it. This line of position is called the "radial" from the VOR. This bearing is then displayed in the cockpit of the aircraft in one of the following four common types of indicators:
Figure 11.14: VOR displays interpretation.
1. Omni-Bearing Indicator (OBI): is the typical light-airplane VOR indicator. It consists of a knob to rotate an "Omni Bearing Selector" (OBS), and the OBS scale around the outside of the instrument, used to set the desired course. A "course deviation indicator" (CDI) is centered when the aircraft is on the selected course, or gives left/right steering commands to return to the course. An ambiguity (TO-FROM) indicator shows whether following the selected course would take the aircraft to, or away from the VOR station. A thorough explanation on how this instrument works is given in Figure 11.14.
2. Radio Magnetic Indicator (RMI): features a course arrow superimposed on a rotating card which shows the aircraft’s current heading at the top of the dial. The "tail" of the course arrow points at the current radial from the station, and the "head" of the arrow points at the inverse (180 deg different) course to the station.
3. Horizontal Situation Indicator (HSI): is considerably more expensive and complex than a standard VOR indicator, but combines heading information with the navigation display in a much more user-friendly format, approximating a simplified moving map.
4. An Area Navigation (RNAV) system is an onboard computer with display and up-to-date navigation database. At least two VOR stations (or one VOR/DME station) is required for the computer to plot aircraft position on a moving map, displaying the course deviation relative to a VOR station or waypoint.
Figure 11.15: MLS coverage.
11.3.08: VOR
A microwave landing system (MLS) is a precision landing system originally intended to replace or supplement instrument landing systems (ILS). MLS has a number of operational advantages when compared to ILS, for instance, including a wide selection of channels to avoid interference with other nearby airports, excellent performance in all weather conditions, less influence of the orography in the quality of the signal, and more flexible range of vertical and horizontal descent angles, which in principle would allow for efficient descents. The system may be divided into five functions: approach azimuth, back azimuth, approach elevation, range and data communications.
MLS systems became operational in the 1990s. However, it has not been used much. This is due to two main reasons: first, the ILS has evolved and it is now more robust; second, and more important, GNSS systems allowed the expectation of the same level of positioning detail with no equipment needed at the airport. However, the GNSS navigation is still not a reality and therefore, and MLS continues to be of some interest in Europe.
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The technical means included under the term aeronautical surveillance fulfill a mission of providing real-time information over the position of the aircraft to ATC function, i.e., controllers, with the aim of ensuring safety by properly separating them and avoiding thus any potential conflict. The surveillance has been traditionally (and still is nowadays) carried out in the different control dependences as shown in Figure 10.10, i.e., Area Control Centers (ACC) centers, Approximation (APP) dependences, and Control Tower (TWR), using the radar. However, within the use of satellite communications, most likely Automatic Dependent Surveillance Broadcast (ADSB) will be replacing (sooner rather than later) radar as the primary surveillance method for controlling aircraft worldwide. ADSB will increase the situational awareness of pilots with cockpit displays, enabling them more autonomy in self-separation. There are also airborne systems that fulfill a surveillance function. That is the case of the Traffic Collision Avoidance System or Traffic alert and Collision Avoidance System (both abbreviated as TCAS), which acts as an automatic advisory back-up system in case of imminent threat, i.e., when the human-based ATC layer has failed.
11.04: Surveillance systems
Radar was developed before and during World War II, where it played a key role in all aerial battles. The term RADAR was coined in 1941 by the United States Navy as an acronym for radio detection and ranging.
A radar system has a transmitter that emits electromagnetic radio signals in predetermined directions. When these come into contact with an object they are usually reflected back towards the receiver. A radar receiver is usually in the same location as the transmitter. By using using radiotelemetry techniques, the position of the radiated object can be determined and displayed. If the object is moving, there is a slight change in the frequency of the radio waves due to the Doppler effect.
Figure 11.16: Radar antenna and ATC display.
In aviation, two radar techniques are applied:
• The original technique described above, that detects the objects due to its finite magnitude. This kind of radars are referred to as simply primary radars (PSR). In this case the aircraft is a passive object.
• The secondary radar (SSR): in this case, the radar requires the aircraft to carry an on board equipment called transponder. The transponder is interrogated from earth, responding with coded values such as flight level, flight code, direction, or velocity. This version was standardized by ICAO in the 80s with the aim at supporting air traffic control and surveillance.
The presentation of data in the screen that use controllers is very different in both cases. In the primary radar, only points (called targets) are presented with no identification, nor any information. Fixed targets can be mountains or any other orographic accident, while mobile targets can be identified with aircraft. Thus, the PSR is more interpretative. In the case of the secondary radar, the targets that are presented in the screen have a identification code, and provide also data such as flight level or velocity. Obviously, this information is much more useful for a controller to fulfill the surveillance function since each aircraft has a unique transponder.
The information is supplied in three different scenarios with three different types of equipment: Long range secondary radar for En-route control; primary radar and short range secondary radar for approach; surface radar (primary) at the airport.
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Figure 11.17: TCAS protection volume and traffic/resolution advisories
Traffic Collision Avoidance System (TCAS)17 is an airborne aircraft collision avoidance system designed to reduce the incidence of Mid-Air Collision (MAC) between aircraft. It acts as the last safety back-up layer. Based on SSR transponder signals, it monitors the airspace around an aircraft for other aircraft equipped with a corresponding active transponder (independently of air traffic control) and advises instructions to pilots in case of the presence of other aircraft which may present a threat.
A TCAS installation consists of the following components: Telecommunication systems (antennas, transponder, etc.), TCAS computer unit, and cockpit presentation. In modern aircraft, the TCAS cockpit display may be integrated in the Navigation Display (ND).
TCAS issues the following types of advisories: Traffic advisory (TA) and Resolution advisory (RA). Traffic advisory is a situational awareness advisory, i.e., pilots must be aware of conflicting aircraft to either maintain separation in visual rules or coordinate with ATC to avoid the thread in instrumental rules. The RA is the last safety layer. It is advised when a mid-air collision is to occur within less than 25 to 35 seconds (depending on the TCAS generation). In this case, pilots are expected to respond immediately and the controller is no longer responsible for separation of the aircraft involved in the RA until the conflict has been resolved. Typically the RA will involve coordinated instructions to the two aircraft involved, e.g., flight level up and flight level down.
17. also refereed to as Aircraft Collision Avoidance System (ACAS).
11.4.03: ADSB
Automatic Dependent Surveillance-Broadcast (ADS-B) is a GNSS based surveillance technology for tracking aircraft. It is still under development and most likely will replace radar as main surveillance system.
ADS-B technology consists of two different services: ADS-B Out and ADS-B In. ADS-B Out periodically broadcasts information about aircraft, such as identification code, position, course, and velocity, through an onboard transmitter. ADS-B In is the reception by aircraft of traffic information, flight information, and weather information, as well as other ADS-B data such as direct communication from nearby aircraft. The system relies on two fundamental components: a satellite navigation system (GPS nowadays; in the future a GNSS system with more integrity is desirable) and a datalink (ADS-B unit). With all this information, two fundamental issues will be acquired: first, controllers will be able to position and separate aircraft with improved precision and timing (since the information is more accurate); second, pilots will increase their situational awareness.
The potential benefits of ADS-B are:
• Improve situational awareness: Pilots in an ADS-B equipped cockpit will have the ability to see, on their in-cockpit flight display, other traffic operating in the airspace as well as access to clear and detailed weather information. They will also be able to receive pertinent updates ranging from temporary flight restrictions (TFR’s) to runway closings.
• Improve visibility: aircraft will be benefited by air traffic controllers ability to more accurately and reliably monitor their position. Fully equipped aircraft using the airspace around them will be able to more easily identify and avoid conflict with ADS-B out equipped aircraft. ADS-B provides better surveillance in fringe areas of radar coverage.
• Others such as: Reduce environmental impact (more efficient trajectories), increase safety (by increasing situational awareness and visibility as mentioned above), increase capacity and efficiency of the system (enhance visual approaches, closely spaced parallel approaches, reduced spacing on final approach, reduce aircraft separations, improve ATC services in non-radar airspace (such oceans, enabling free routes), etc.
Figure 11.18: ADS-B sketch. Author: User:AuburnADS-B / Wikimedia Commons / Public Domain.
Nowadays, most airliners are equipped with ADS-B. However, since the equipment is very expensive, most regional aircraft do not have it. Therefore, still ADS-B can not be used as primary surveillance system due to its low degree of implantation. Nevertheless, there is a road map both in Europe and the US to increasingly equip all aircraft with ADSB by 2020. Another issue is the low integrity of GPS as main satellite system. The implementation of the GNSS-2 will circumvent this problem.
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Exercise $1$ Inertial Navigation Systems
Consider an aircraft flying constant altitude $h$ as illustrated in Figure 11.21. Assume the Earth is rotating at angular rate $\vec{\Omega}_E = \Omega \cdot \vec{k}_E$, and there is no wind (calm conditions). Assume also the aircraft is equipped with a strap-down inertial navigation system. At time $t$, accelerometers and gyroscopes are providing the following measurements:
• Angular velocity: $\vec{W} = W_x \cdot \vec{i}_b + W_y \cdot \vec{j}_b + W_z \cdot \vec{k}_b \approx 0$.
• Body-Frame accelerations:18 $\vec{a}_b = a_x \cdot \vec{i}_b + a_y \cdot \vec{j}_b + a_z \cdot \vec{k}_b$.
Figure 11.19: Inertial Navigation System (Exercise 3.1)
Consider:
• the aircraft is flying at speed $\vec{V} = V \cdot \vec{i}_b$ at time $t$.
• the angle $\theta$ can be considered approximately constant.
Obtain:
1. a symbolic expression of the acceleration terms $a_x$, $a_y$ and $a_z$, identifying those that are due to the motion of the aircraft with respect to Earth and those that are due to inertial and gravitational effects.
2. Substitute using the values below and provide the value (and direction) of the acceleration of the aircraft with respect to Earth:
$\bullet$ $\Omega = 1\ rev./day$.
$\bullet$ $W_x \approx 0; W_y \approx 0$
$\bullet$ $a_x = 0.9831\ m/s^2; a_y = 0.0257\ m/s^2, a_z = -9.8269\ m/s^2$.
$\bullet$ $V = 250\ m/s; h = 11.000\ m$.
$\bullet$ $R_E = 6378\ km; g = 9.81\ m/s^2$.
$\bullet$ $\theta = 45\ deg$.
Answer
Let us start saying that the absolute acceleration of a body is $\vec{a}_i = \tfrac{d^2 \vec{r}}{dt^2}|_i$, and the velocity of a body is $\vec{V}_i = \tfrac{d\vec{r}}{dt}|_i$, being frame $i$ a inertial reference frame (e.g., a fixed star) and $\vec{r}$ the radio vector and $t$ the time.
Then, according the Newton's second law: $m \cdot \tfrac{d^2 \vec{r}}{dt^2}|_i = \sum \vec{F}_{ext}$, being $m$ the mass of the body and $F_{ext}$ the external forces.
Using Coriolis formula, one has that the derivative on a generic vectorial magnitude ($\vec{A}$) in absolute terms (with respect to an inertial reference frame $i$) is equal to its derivative in relative terms (with respect to a non-inertial reference frame $e$) plus the vectorial product of the relative angular velocity of the two frames ($\vec{w}_{ei}$) and the generic vectorial magnitude $\vec{A}$. In order words:
$\dfrac{d\vec{A}}{dt}|_i = \dfrac{d\vec{A}}{dt}|_e + \vec{w}_{ei} \wedge \vec{A}\label{eq11.5.1}$
Thus, we can say that:
$\dfrac{d\vec{r}}{dt}|_i = \dfrac{d\vec{r}}{dt}|_e + \vec{w}_{ei} \wedge \vec{r}$
Taking derivatives:
$\dfrac{d^2\vec{r}}{dt^2}|_i = \dfrac{d^2\vec{r}}{dt^2}|_e + \dfrac{d}{dt}[\vec{w}_{ei} \wedge \vec{r}]$
This results in:
$\dfrac{d\vec{V}}{dt}|_i = \dfrac{d\vec{V}}{dt}|_e + \vec{w}_{ei} \wedge \vec{V}\label{eq11.5.4}$
In other words, absolute terms are equal to relative terms plus Coriolis terms. We can also say that:
$\dfrac{d^2\vec{r}}{dt^2}|_i = \dfrac{d}{dt} (\dfrac{d\vec{r}}{dt}|_i)|_i = \dfrac{d}{dt} (V + \vec{w}_{ei} \wedge \vec{r})|_i,$
which elaborating yields:
$\dfrac{d^2\vec{r}}{dt^2}|_i = \dfrac{d\vec{V}}{dt}|_i + \dfrac{d}{dt}(\vec{w}_{ei} \wedge \vec{r})|_i\label{eq11.5.6}$
Equation ($\ref{eq11.5.6}$) is also referred to as Navigation equation. We can apply Coriolis formula in ($\ref{eq11.5.1}$) to the first term of the right hand side in Equation ($\ref{eq11.5.6}$), which yields $\tfrac{d\vec{V}}{dt}|_e + \vec{w}_{ei} \wedge \vec{V}|_e$ as in Eq. ($\ref{eq11.5.4}$), and to the second term of the right hand side in Equation ($\ref{eq11.5.6}$), which yields $\tfrac{d(\vec{w}_{ei} \wedge \vec{r})}{dt} + \vec{w}_{ei} \wedge (\vec{w}_{ei} \wedge \vec{r})$. Thus, Eq. ($\ref{eq11.5.6}$) results in:
$\vec{a} |_i = \vec{a}|_e + 2 \cdot \vec{w}_{ei} \wedge \vec{V} + \vec{w}_{ei} \wedge (\vec{w}_{ei} \wedge \vec{r}),\label{eq11.5.7}$
where $\vec{V}$ and $\vec{r}$ are magnitudes refereed to the non-inertial reference frame $e$.
Equation ($\ref{eq11.5.7}$) is the well known composition of accelerations equation. In the context of navigation, it states that the absolute acceleration (with respect to an inertial reference frame i) is equal to the relative acceleration (with respect to a non-inertial reference frame e, typically the Earth) plus Coriolis effects (second term in the right hand side of ($\ref{eq11.5.7}$)) and centrifugal effects (third term in the right hand side of ($\ref{eq11.5.7}$)).
In the context of the problem under analysis, acceleration measured by the Inertial Measurement Unit are absolute ($\vec{a}_b$), including gravitational effects and inertial (Coriolis and centrifugal) terms. However, in order to further compute the position of the aircraft with respect to Earth ($vec{a}_e$), we are interested in relative acceleration, i.e., the acceleration of the body (the aircraft) with respect to Earth. In other words:
$\vec{a}_b = \vec{a}_e + \vec{g} + Coriolis + Centrifugal$
Let’s now work on each of these terms. First, define the following vectors:
$\vec{g} = -g \cdot \vec{k}_b$
$\vec{w}_{ei} = \Omega \cdot \vec{k}_e$
$\vec{r} = (R_e + h) \cdot \vec{k}_b$
$\vec{V} = V \cdot \vec{i}_b$
Then, operating, one has:
$Coriolis: 2 \Omega V \sin \theta \cdot \vec{j}_b$
$Centrifugal: -\Omega^2 (R_e + h) \cos \theta \cdot \vec{i}_e$
$Gravity: -g \cdot \vec{k}_b.$
Figure 11.20: Inertial Navigation System, including gravitational acceleration, Coriolis acceleration ($2\Omega V$), and centrifugal acceleration ($\Omega^2 (R_e + h) \cos (\theta)$).
Please, see Figure 11.20.
We should project the centrifugal term into body-frame axis:
$Centrifugal: -\Omega^2 (R_e + h) \cos \theta \cdot (\sin \theta \cdot \vec{i}_b + \cos \theta \cdot \vec{k}_b)$
Then, we can say:
$a_x \cdot \vec{i}_b = (a_{ex} - \Omega^2 (R_e + h) \cos \theta \sin \theta) \cdot \vec{i}_b,$
$a_y \cdot \vec{j}_b = (a_{ey} + 2\Omega V \sin \theta) \cdot \vec{j}_b,$
$a_z \cdot \vec{k}_b = (a_{ez} - g - \Omega^2 (R_e + h) \cos^2 \theta) \cdot \vec{k}_b,$
where $a_{ex}$, $a_{ey}$, and $a_{ez}$ are the accelerations of the aircraft with respect to Earth and $a_x$, $a_y$, and $a_z$ are the absolute accelerations measured by the IMU.
Let us know particularize to the given values, resulting:
$a_x \cdot \vec{i}_b = (a_{ex} - 0.0168941032554) \cdot \vec{i}_b,$
$a_y \cdot \vec{j}_b = (a_{ey} + 0.0257111281143) \cdot \vec{j}_b,$
$a_z \cdot \vec{k}_b = (a_{ez} - -9.82689410326) \cdot \vec{k}_b.$
With the measurements of the accelerometers being $a_x = 0.9831\ m/s^2$, $a_y = 0.0257\ m/s^2$, and $a_z = -9.8269\ m/s^2$, respectively, one has: Let us know particularize to the given values, resulting:
$a_{ex} \cdot \vec{i}_b \approx 1 \cdot \vec{i}_b$
$a_{ey} \cdot \vec{j}_b \approx 0 \cdot \vec{j}_b$
$a_{ez} \cdot \vec{k}_b \approx 0 \cdot \vec{k}_b$
Thus, the acceleration of the aircraft with respect to Earth, $\vec{a}_e = 1 \cdot \vec{i}_b \ m/s^2$. This acceleration is the one that should be used to obtain the position of the aircraft via double integration (given certain initial conditions).
Exercise $2$ Inertial Navigation Systems II
Figure 11.21: INS Sketch.
Consider an aircraft flying at constant altitude as illustrated in Figure 11.21. Assume the Earth can be considered flat, non-rotating,19 and there is no wind (calm conditions). Assume also the aircraft is equipped with a strap-down inertial navigation system. At time $t$, accelerometers and gyroscopes are providing the following measurements:
• Angular velocity: $\vec{w}_b = w \cdot \vec{k} \approx 0$.
• Body-Frame forces: $\vec{f}_b = (f_{bx} \cdot \vec{i}_b, f_{by} \cdot \vec{j}_b)$.
Given the following initial conditions:
• initial heading/track angle: $\theta_0$;
• initial time: $t_0$;
• initial position: $\vec{r}_0 = (x_0 \cdot \vec{i}_e, y_0 \cdot \vec{j}_e)$;
• initial velocity: $\vec{v}_0 = (v_0 \cdot \vec{i}_b, 0 \cdot \vec{j}_b)$;
Calculate:
• Position of the aircraft at time $t$.
Answer
First of all, the reader should notice that by assuming a non-rotating Earth, we are not considering Inertial (Coriolis and Centrifugal) terms. Also, because the movement is considered horizontal, gravity does not play any role in this problem. Of course, a realistic Inertial Navigation problem would require to conduct the analysis in the previous exercise. For the sake of simplicity, we focus herein on integrating the accelerations to obtain the position (something missing in the previous exercise).
Notice that $\vec{w} = \dot{\theta}$, being $\theta$ an arbitrary angle between a fixed direction, e.g., $\vec{i}_e$, and the track of the aircraft, i.e., $\vec{i}_b$.
Thus we can obtain the variation of $\theta$ over time $\theta (t)$ by simply integrating the flowing equation:
$\int_{\theta_0}^{\theta} d\theta = \int_{t_0}^{t} w(t) dt \to \theta (t).\label{eq11.5.26}$
Now, since the measurement of the gyroscope can be approximated to zero, i.e., $\vec{w} \approx 0$, Equation $\ref{eq11.5.26}$ yields:
$\theta = \theta_0.$
The strap-down accelerometers provide measurements of absolute forces in the body frame axis that can be readily (under the conditions herein assumed) transformed in accelerations, i.e., $\vec{a}_b = (a_{bx} \cdot \vec{i}_b, a_{by} \cdot \vec{j}_b)$.
Now, in order to expressed the absolute acceleration in the Earth reference frame, we have to simply apply a rotation:
$\begin{bmatrix} \vec{a}_{ex} \ \vec{a}_{ey} \end{bmatrix} = \begin{bmatrix} \cos \theta_0 & -\sin \theta_0 \ \sin \theta_0 & \cos \theta_0 \end{bmatrix} \cdot \begin{bmatrix} \vec{a}_{bx} \ \vec{a}_{by} \end{bmatrix}$
We now that:
$\dfrac{d\vec{V}}{dt} = \vec{a};$
$\dfrac{d\vec{r}}{dt} = d \vec{V}.$
By integrating once, we obtain:
$v_x = v_0 + (a_{xb} \cdot \cos \theta_0 - a_{yb} \cdot \sin \theta_0) \cdot t$
$v_y = (a_{yb} \cdot \sin \theta_0 + a_{xb} \cdot \cos \theta_0) \cdot t$
By integrating twice, we obtain:
$x = x_0 + v_0 \cdot t + (a_{xb} \cdot \cos \theta_0 - a_{yb} \cdot \sin \theta_0 ) \cdot \dfrac{t^2}{2}$
$y = y_0 + (a_{yb} \cdot \sin \theta_0 + a_{xb} \cdot \cos \theta_0) \cdot \dfrac{t^2}{2}$
Notice that $t$ is supposed to be sufficiently small (indeed, equivalent to the frequency of measurement) such that measurements can be considered constant along the time interval.
18. for the sake of simplicity, we assume accelerometers directly provide accelerations after having measured forces and having done the appropriate transformations.
19. one can assume both centrifugal and Coriolis terms are neglectable in the formula that relates absolute and relative acceleration. In other words, absolute and relative accelerations can be consider identical. Notice that this is true herein because we have considered the Earth non-rotating.
11.6: References
[1] Britting, K. R. (2010). Inertial navigation systems analysis.
[2] Kayton, M. and Fried, W. R. (1997). Avionics navigation systems. John Wiley & Sons.
[3] Lloret, J. (2017). Introduction to Air Navigation: A technical and operational approach. Javier Lloret [Ed], third edition.
[4] Tooley, M. and Wyatt, D. (2007). Aircraft communication and navigation systems (principles, maintenance and operation).
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This appendix is devoted to the deduction of the 6DOF general equations of motion of the aircraft. The reader is referred to Gómez-Tierno et al. [1] for a thorough and comprehensive overview. Other references on mechanics of flight are, for instance, Hull [2] and Yechout et al. [3].
12: 6-DOF Equations of Motion
Definition 12.1: Inertial Reference Frame
According to classical mechanics, a inertial reference frame $F_I (O_I , x_I , y_I , z_I)$ is either a non accelerated frame with respect to a quasi-fixed reference star, or either a system which for a punctual mass is possible to apply the second Newton’s law:
$\sum \vec{F}_1 = \dfrac{d(m \cdot \vec{V}_I)}{dt}\nonumber$
Definition 12.2: Earth Reference Frame
An earth reference frame $F_e(O_e, x_e, y_e, z_e)$ is a rotating topocentric (measured from the surface of the earth) system. The origin Oe is any point on the surface of earth defined by its latitude $\theta_e$ and longitude $\lambda_e$. Axis $z_e$ points to the center of earth; $x_e$ lays in the horizontal plane and points to a fixed direction (typically north); $y_e$ forms a right-handed thrihedral (typically east).
Such system it is sometimes referred to as navigational system since it is very useful to represent the trajectory of an aircraft from the departure airport.
Hypothesis 12.1: Flat earth
The earth can be considered flat, non rotating and approximate inertial reference frame. Consider $F_I$ and $F_e$. Consider the center of mass of the aircraft denoted by $CG$. The acceleration of $CG$ with respect to $F_1$ can be written using the well-known formula of acceleration composition from the classical mechanics:
$\vec{a}_I^{CG} = \vec{a}_e^{CG} + \vec{\Omega} \wedge (\vec{\Omega} \wedge \vec{r}_{O_I CG}) + 2 \vec{\Omega} \wedge \vec{V}_e^{CG},$
where the centripetal acceleration and the Coriolis acceleration are neglectable if we consider typical values: $\vec{\Omega}$, the earth angular velocity is one revolution per day; $\vec{r}$ is the radius of earth plus the altitude (around 6380 [km]); $\vec{V}_e^{CG}$ is the velocity of the aircraft in flight (200-300 [m/s]). This means $\vec{a}_I^{CG} = \vec{a}_e^{CG}$ and therefore $F_e$ can be considered inertial reference frame.
Definition 12.3: Local Horizon Frame
A local horizon frame $F_h(O_h, x_h, y_h, z_h)$ is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axes $(x_h, y_h, z_h)$ are defined parallel to axes $(x_h, y_h, z_h)$.
In atmospheric flight, this system can be considered as quasi-inertial.
Definition 12.4: Body Axes Frame
A body axes frame $F_b(O_b,x_b,y_b,z_b)$ represents the aircraft as a rigid solid model. It is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis xb lays in to the plane of symmetry and it is parallel to a reference line in the aircraft (for instance, the zero-lift line), pointing forwards according to the movement of the aircraft. Axis $z_b$ also lays in to the plane of symmetry, perpendicular to xb and pointing down according to regular aircraft performance. Axis $y_b$ is perpendicular to the plane of symmetry forming a right-handed thrihedral ($y_b$ points then the right wing side of the aircraft).
Definition 12.5: Wind Axes Frame
A wind axes frame $F_w (O_w , x_w , y_w , z_w )$ is linked to the instantaneous aerodynamic velocity of the aircraft . It is a system of axes centered in any point of the symmetry plane (assuming there is one) of the aircraft, typically the center of gravity. Axis $x_w$ points at each instant to the direction of the aerodynamic velocity of the aircraft $\vec{V}$, Axis $z_w$ lays in to the plane of symmetry, perpendicular to $x_w$ and pointing down according to regular aircraft performance. Axis $y_b$ forms a right-handed thrihedral.
Notice that if the aerodynamic velocity lays in to the plane of symmetry, $y_w \equiv y_b$.
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According to the classical mechanics, to orientate without loss of generality a reference frame system $F_I$ with respect to another $F_F$: if both have common origin, it is necessary to perform a generic rotation until axis coincide; if the origin differs, it is necessary, together with the mentioned rotation, a translation to make origins coincide.
There are different methods to orientate two systems with common origin, such for instance, directors cosinos, quaternions or the Euler angles, which indeed will be used in this dissertation.
Definition 12.6 Euler angles
Euler angles represent three composed and finite rotations given in a pre-establish order that move a reference frame to a given referred frame. This is equivalent to saying that any orientation can be achieved by composing three elemental and finite rotations (rotations around a single axis of a basis), and also equivalent to saying that any rotation matrix can be decomposed as a product of three elemental rotation matrices.
Remark 12.1
The pre-establish order of elemental rotations is usually referred to as Convention. In aeronautics and space vehicles it is universally utilized the Tailt-Bryan Convention. Such convention is also referred to as Convention 321.
Definition 12.7 Transformation or rotation matrix
If the three components of a vector $\vec{A}$ in $F_I$ are known, the transformation or rotation matrixx $L_{F_I}$ expresses a vector $\vec{A}$ in the reference system $F_F$ as follows:
$\vec{A}_F = L_{F_I} \vec{A}_I$
Remark 12.2
$L_{F_I}$ can be obtained by simply obtaining the three individual rotation matrixes and properly multiplying them.
Example 12.1 Convention 321
Given two reference systems, $F_I$ and $F_F$, with common origin, we want to make $F_I$ coincide with $F_F$: first we rotate $F_I$ around axis $z_I$ an angle $\delta_3$, obtaining the first intermediate reference systems $F_1$. Second, we rotate system $F_1$ around axis $y_1$ an angle $\delta_2$, obtaining the second intermediate reference system $F_2$. Third, we rotate the system $F_2$ around axis $x_2$ an angle $\delta_1$, obtaining the final reference system $F_F$.
First, we express the unit vector of $F_1$ as a function of unit vector of $F_I$:
$\begin{bmatrix} \vec{i}_1 \ \vec{j}_1 \ \vec{k}_1 \end{bmatrix} = \begin{bmatrix} \cos \delta_3 & \sin \delta_3 & 0 \ -\sin \delta_3 & \cos \delta_3 & 0 \ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \vec{i}_I \ \vec{j}_I \ \vec{k}_I \end{bmatrix}$
The rotation matrix will be:
$L_{1I} = R_3 (\delta_3) = \begin{bmatrix} \cos \delta_3 & \sin \delta_3 & 0 \ -\sin \delta_3 & \cos \delta_3 & 0 \ 0 & 0 & 1 \end{bmatrix}.$
where $R_3 (\delta_3)$ is the notation of the individual matrix of rotation of an angle $\delta_3$ around the third axis (axis $z$).
Therefore, the vector $\vec{A}$ expressed in the first intermediate reference frame $F_1$ (Notated $\vec{A}_1$) will be:
$\vec{A}_1 = L_{1I} \vec{A}_I.$
Operating analogously for the second individual rotation:
$\begin{bmatrix} \vec{i}_1 \ \vec{j}_1 \ \vec{k}_1 \end{bmatrix} = \begin{bmatrix} \cos \delta_2 & 0 & -\sin \delta_2 \ 0 & 1 & 0 \ \sin \delta_2 & 0 & \cos \delta_2 \end{bmatrix} \begin{bmatrix} \vec{i}_1 \ \vec{j}_1 \ \vec{k}_1 \end{bmatrix}$
$L_{21} = R_2 (\delta_2) = \begin{bmatrix} \cos \delta_2 & 0 & -\sin \delta_2 \ 0 & 1 & 0 \ \sin \delta_2 & 0 & \cos \delta_2 \end{bmatrix}$
$\vec{A}_2 = L_{21} \vec{A}_1.$
Finally, for the third individual rotation:
$\begin{bmatrix} \vec{i}_F \ \vec{j}_F \ \vec{k}_F \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \ 0 & \cos \delta_1 & \sin \delta_1 \ 0 & -\sin \delta_1 & \cos \delta_1 \end{bmatrix} \begin{bmatrix} \vec{i}_2 \ \vec{j}_2 \ \vec{k}_2 \end{bmatrix}$
Figure 12.1: Fuler angles
$L_{F2} = R_1 (\delta_1) = \begin{bmatrix} 1 & 0 & 0 \ 0 & \cos \delta_1 & \sin \delta_1 \ 0 & -\sin \delta_1 & \cos \delta_1 \end{bmatrix}$
$\vec{A}_F = L_{F2} \vec{A}_2.$
Composing:
$\vec{A}_F = L_{F2} L_{21} L_{1I} \vec{A}_I,$
and the global rotation matrix will be:
$L_{FI} = \begin{bmatrix} \cos \delta_2 \cos \delta_3 & \cos \delta_2 \sin \delta_3 & -\sin \delta_2 \ \sin \delta_1 \sin \delta_2 \cos \delta_3 - \cos \delta_1 \sin \delta_3 & \sin \delta_1 \sin \delta_2 \sin \delta_3 + \cos \delta_1 \cos \delta_3 & \sin \delta_1 \cos \delta_2 \ \cos \delta_1 \sin \delta_2 \cos \delta_3 + \sin \delta_1 \sin \delta_3 & \cos \delta_1 \sin \delta_2 \sin \delta_3 - \sin \delta_1 \cos \delta_3 & \cos \delta_1 \cos \delta_2 \end{bmatrix}$
12.02: Orientation between reference frames
To situate the wind axis reference frame with respect to the local horizon reference frame, the general form given in Example (12.2) is particularized for:
• $F_I \equiv F_h; F_F \equiv F_w,$
• $\delta_3 \equiv \chi \to \text{Yaw angle,}$
• $\delta_2 \equiv \gamma \to \text{Flight path angle,}$
• $\delta_1 = \equiv \mu \to \text{Bank angle}.$
The transformation matrix will be:
$L_{wh} = \begin{bmatrix} \cos \gamma \cos \chi & \cos \gamma \sin \chi & - \sin \gamma \ \sin \mu \sin \gamma \cos \chi - \cos \mu \sin \chi & \sin \mu \sin \gamma \sin \chi + \cos \mu \cos \chi & \sin \mu \cos \gamma \ \cos \mu \sin \gamma \cos \chi + \sin \mu \sin \chi & \cos \mu \sin \gamma \sin \chi - \sin \mu \cos \chi & \cos \mu \cos \gamma \end{bmatrix}.$
12.2.02: Body axed-Wind axes orientation
To situate the wind axis reference frame with respect to the local horizon reference frame, the general form given in Example (12.2) is particularized for:
• $F_I \equiv F_w; F_F \equiv F_b.$
• $\delta_3 \equiv -\beta \to \text{Sideslip angle,}$
• $\delta_2 \equiv \alpha \to \text{Angle of attack,}$
• $\delta_1 = 0.$
The transformation matrix will be:
$L_{wh} = \begin{bmatrix} \cos \alpha \cos \beta & -\cos \alpha \sin \beta & - \sin \alpha \ \sin \beta & \cos \beta & 0 \ \sin \alpha \cos \beta & -\sin \alpha \sin \beta & \cos \alpha \end{bmatrix}$
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The physic-mathematical model governing the movement of the aircraft in the atmosphere are the so-called general equations of motion: three equations of translation and three equations of rotation. The fundamental simplifying hypothesis is:
Theorem 12.2 6-DOF model
The aircraft is considered as a rigid solid with six degrees of freedom, i.e., all dynamic effects associated to elastic deformations, to degrees of freedom of articulated subsystems (flaps, ailerons, etc.), or to the kinetic momentum of rotating subsystems (fans, compressors, etc ), are neglected.
12.03: General equations of motion
The dynamic model governing the movement of the aircraft is based on two fundamental theorems of the classical mechanics: the theorem of the quantity of movement and the theorem of the kinetic momentum:
Theorem 12.1 Quantity of movement
The theorem of quantity of movement establishes that:
$\vec{F} = \dfrac{d(m \vec{V})}{dt},\label{eq12.3.1.1}$
where $\vec{F}$ is the resulting of the external forces, $\vec{V}$ is the absolute velocity of the aircraft (respect to a inertial reference frame), $m$ is the mass of the aircraft, and $t$ is the time.
Remark 12.3
For a conventional aircraft holds that the variation of its mass with respect to time is sufficiently slow so that the term $\dot{m} \vec{V}$ Equation ($\ref{eq12.3.1.1}$) could be neglected.
Theorem 12.2 Kinematic momentum
The theorem of the kinematic momentum establishes that:
$\vec{G} = \dfrac{d\vec{h}}{dt},$
$\vec{h} = I \vec{\omega},$
where $\vec{G}$ is the resulting of the external momentum around the center of gravity of the aircraft, $\vec{h}$ is the absolute kinematic momentum of the aircraft, $I$ is the tensor of inertia, and $\vec{\omega}$ is the absolute angular velocity of the aircraft.
Definition 12.8 Tensor of Inertia
The tensor of inertia is defined as:
$I = \begin{bmatrix} I_x & -I_{xy} & -J_{xz} \ -J_{xy} & I_y & -J_{yz} \ -J_{xz} & -J_{yz} & I_z \end{bmatrix}.$
where $I_x , I_y, I_z$ are the inertial momentums around the three axes of the reference system, and $J_{xy}, J_{xz}, J_{yz}$ are the corresponding inertia products.
The resulting equations from both theorems can be projected in any reference system. In particular, projecting them into a body-axes reference frame (also to a wind-axes reference frame) have important advantages.
Theorem 12.3 Field of velocities
Given a inertial reference frame denoted by $F_0$ and a non-inertial reference frame $F_1$ whose related angular velocity is given by $\vec{\omega}_{01}$, and given also a generic vector $\vec{A}$, it holds:
$\{ \dfrac{\partial \vec{A}}{\partial t} \}_1 = \{ \dfrac{\partial \vec{A}}{\partial t} \}_0 + \vec{\omega}_{01} \wedge \vec{A}_1$
The three components expressed in a body-axes reference frame of the total force, the total momentum, the absolute velocity, and the absolute angular velocity are denoted by:
$\vec{F} = (F_x, F_y, F_z)^T,$
$\vec{G} = (L, M, N)^T,$
$\vec{V} = (u, v, w)^T,$
$\vec{\omega} = (p, q, r)^T.$
Therefore, the equations governing the motion of the aircraft are:
$F_x = m(\dot{u} - rv + qw),\label{eq12.3.1.10}$
$F_y = m(\dot{v} + ru - pw),\label{eq12.3.1.11}$
$F_z = m(\dot{w} - qu + pv),\label{eq12.3.1.12}$
$L = I_x \dot{p} - J_{xz} \dot{r} + (I_z - I_y) qr - J_{xz} pq,\label{eq12.3.1.13}$
$M = I_y \dot{q} - (I_z - I_x) pr - J_{xz} (p^2 - r^2),\label{eq12.3.1.14}$
$N = I_z \dot{r} - J_{xz} \dot{p} + (I_x - I_y) pq - J_{xz} qr,\label{eq12.3.1.15}$
System ($\ref{eq12.3.1.10}$ - $\ref{eq12.3.1.15}$) is referred to as Euler equations of the movement of an aircraft.
12.3.02: Forces acting on an aircraft
Hypothesis 12.3 Forces acting on an aircraft
The external actions acting on an aircraft can be decomposed, without loss of generality, into propulsive, aerodynamic and gravitational, notated respectively with subindexes ($(\cdot)_T, (\cdot)_A, (\cdot)_G$):
$\vec{F} = \vec{F}_T + \vec{F}_A + \vec{F}_G,$
$\vec{G} = \vec{G}_T + \vec{G}_A,$
The gravitational force can be easily expressed in local horizon axes as:
$(\vec{F}_G)_h = \begin{bmatrix} 0 \ 0 \ mg \end{bmatrix},$
where $g$ is the acceleration due to gravity.
Theorem 12.4 Constant gravity
The acceleration due to gravity in atmospheric flight of an aircraft can be considered constant ($g = 9.81[m/s^2]$), due to a small altitude of flight when compared to the radius of earth. Therefore, the little variations of $g$ as a function of $h$ are neglectable.
To project the force due to gravity into wind-axes reference frame:
$(\vec{F}_G)_w = L_{wh} (\vec{F}_G)_h = \begin{bmatrix} -mg \sin \gamma \ mg \cos \gamma \sin \mu \ mg \cos \gamma \cos \mu \end{bmatrix}.$
Introducing the propulsive, aerodynamic and gravitational actions in System (12.3.1.10-12.3.1.15):
$-mg \sin \gamma + F_{T_x} + F_{A_x} = m (\dot{u} - rv + qw),$
$mg \cos \gamma \sin \mu + F_{T_y} + F_{A_y} = m (\dot{v} + ru - pw),$
$mg \cos \gamma \cos \mu + F_{T_z} + F_{A_z} = m (\dot{w} - qu + pv),$
$L_T + L_A = I_x \dot{p} - J_{xz} \dot{r} + (I_z - I_y) qr - J_{xz} pq,$
$M_T + M_A = I_y \dot{q} - (I_z - I_x) pr - J_{xz} (p^2 - r^2),$
$N_T + N_A = I_z \dot{r} - J_{xz} \dot{p} + (I_x - I_y) pq - J_{xz} qr.$
The three aerodynamic momentum of roll, pitch and yaw $(L_A,M_A,N_A)$ can be controlled by the pilot through the three command surfaces, ailerons, elevator and rudder, whose deflections can be respectively notated by $\delta_a, \delta_e, \delta_r$. Notice that such deflection have also influence in the three components of aerodynamic force, and therefore the 6 equations are coupled and must be solved simultaneously.
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Theorem 12.5 Point mass model
The translational equations (12.3.1.10)-(12.3.1.12) are uncoupled from the rotational equations (12.3.1.13)-(12.3.1.15) by assuming that the airplane rotational rates are small and that control surface deflections do not affect forces. This leads to consider a 3 Degree Of Freedom (DOF) dynamic model that describes the point variable-mass motion of the aircraft.
Under this hypothesis, the translational problem (performances) can be studied separately from the rotational problem (control and stability).
12.04: Point mass model
Therefore, the dynamic equations governing the translational motion of the aircraft are uncoupled:
$-mg \sin \gamma + F_{T_x} + F_{A_x} = m (\dot{u} - rv + qw),$
$mg \cos \gamma \sin \mu + F_{T_y} + F_{A_y} = m (\dot{v} + ru - pw),$
$mg \cos \gamma \cos \mu + F_{T_z} + F_{A_z} = m (\dot{w} - qu + pv),$
The aerodynamic forces, expressed in wind axes, are as follows:
$(\vec{F}_A)_w = \begin{bmatrix} -D \ -Q \ -L \end{bmatrix},$
where $D$ is the aerodynamic drag, $Q$ is the aerodynamic lateral force, and $L$ is the aerodynamic lift.
The propulsive forces, expressed in wind axes, are as follows:
$(\vec{F}_T)_w = \begin{bmatrix} T \cos \epsilon \cos v \ T \cos \epsilon \sin v \ -T \sin \epsilon \end{bmatrix},$
where $T$ is the thrust, $\epsilon$ is the thrust angle of attack, and $ν$ is the thrust sideslip.
Theorem 12.6 Fixed engines
We assume the aircraft is a conventional jet airplane with fixed engines. Almost all existing aircrafts worldwide have their engines rigidly attached to their structure.
12.4.02: Mass relations
Hypothesis 12.7 Variable mass
The aircraft is modeled as variable mass particle.
The variation of mass is given by the consumed fuel during the flight:
$\dot{m} + \phi = 0.$
12.4.03: Kinematic relations
Hypothesis 12.8 Moving Atmosphere
The atmosphere is considered moving, i.e., wind is taken into consideration. Vertical component is neglected due its low influence. Only kinematic effects are considered, i.e., dynamic effects of wind are also neglected due its low influence. The wind velocity $\vec{W}$ can be expressed in local horizon axes as:
$\vec{W}_h = \begin{bmatrix} W_x \ W_y \ 0 \end{bmatrix}$
Considering the earth axes reference system as a inertial system, and assuming that earth axes are parallel to local horizon axes, the absolute velocity $\vec{V}^G = \vec{V}^A + \vec{W}$ can be expressed referred to a wind axes reference as follows:
$\vec{V}_e^G = \begin{bmatrix} \dot{x}_e \ \dot{y}_e \ \dot{z}_e \end{bmatrix} = L_{hw} \vec{V}_w^A + \vec{W}_h = L_{hw} \begin{bmatrix} u \ v \ w \end{bmatrix} + \begin{bmatrix} W_x \ W_y \ 0 \end{bmatrix} = L_{wh}^T \begin{bmatrix} V \ 0 \ 0\end{bmatrix} + \begin{bmatrix} W_x \ W_y \ 0\end{bmatrix}.$
Remark 12.4
Notice that the absolute aerodynamic velocity $\vec{V}^A$ expressed in a wind axes reference frame is $(V, 0, 0)$.
Therefore, the kinematic relations are as follows:
$\dot{x}_e = V \cos \gamma \cos \chi + W_x,\label{eq12.4.3.3}$
$\dot{y}_e = V \cos \gamma \sin \chi + W_y,$
$\dot{z}_e = -V \sin \gamma,\label{eq12.4.3.5}$
Equations ($\ref{eq12.4.3.3}$)-($\ref{eq12.4.3.5}$) provide the movement law and the trajectory of the aircraft can be determined.
Notice that Equation ($\ref{eq12.4.3.5}$) is usually rewritten as
$\dot{h}_e = V \sin \gamma,$
Remark 12.5
If one wants to model a flight over a spherical earth, since the radius of earth is sufficiently big and the angular velocity of earth is sufficiently small, it holds that the rotation of earth has very low influence in the centripetal acceleration and it is thus neglectable. Therefore, the hypothesis of flat earth holds in the dynamics of an aircraft moving over an spherical earthwith the following kinematic relations:
$\dot{\lambda}_e = \dfrac{V \cos \gamma \cos \chi + W_x}{(R + h) \cos \theta},$
$\dot{\theta}_e = \dfrac{V \cos \gamma \sin \chi + W_y}{R + h},$
$\dot{h}_e = V \sin \gamma,$
where $\lambda$ and $\theta$ are respectively the longitude and latitude and $R$ is the radius of earth.
12.4.04: Angular kinematic relations
In what follows the three components of absolute angular velocity of the aircraft are related with the orientation angles of the aircraft with respect to a local horizon reference system:
$\vec{\omega}_I \approx \vec{\omega}_h = \begin{bmatrix} p \ q \ r \end{bmatrix} = \dot{\mu} \vec{i}_w + \dot{\gamma} \vec{j}_1 + \dot{\chi} \vec{k}_h.$
Projecting the unit vectors in wind axes using the appropriate transformation matrices:
$p = \dot{\mu} - \dot{\chi} \sin \gamma$
$q = \dot{\gamma} \cos \mu + \dot{\chi} \cos \gamma \sin \mu$
$r = -\dot{\gamma} \sin \mu + \dot{\chi} \cos \gamma \cos \mu$
12.4.05: General differential equations system
For a point mass model, the general differential equations system governing the motion of an aircraft is stated as follows:
$-mg \sin \gamma + T \cos \epsilon \cos v - D = m (\dot{V}),\label{eq12.4.4.1}$
$mg \cos \gamma \sin \mu + T \cos \epsilon \sin v - Q = - mV (\dot{\gamma} \sin \mu + \dot{\chi} \cos \gamma \cos \mu),$
$mg \cos \gamma \cos \mu - T \sin \epsilon - L = - mV (\dot{\gamma} \cos \mu - \dot{\chi} \cos \gamma \sin \mu),$
$\dot{x}_e = V \cos \gamma \cos \chi + W_x,$
$\dot{y}_e = V \cos \gamma \sin \chi + W_y,$
$\dot{z}_e = - V \sin \gamma,$
$\dot{m} + \phi = 0.\label{eq12.4.4.7}$
If we assume the following hypothesis:
Hypothesis 12.9 Symmetric flight
We assume the aircraft has a plane of symmetry, and that the aircraft flies in symmetric flight, i.e., all forces act on the center of gravity and the thrust and the aerodynamic forces lay on the plane of symmetry. This leads to non sideslip, i.e., $\beta = ν = 0$, and non lateral aerodynamic force, i.e., $Q = 0$, assumptions.
Hypothesis 12.10 Small thrust angle of attack
We assume the thrust angle of attack is small $\epsilon \ll 1$, i.e., $\cos \epsilon \approx 1$ and $\sin \epsilon \approx 0$. For commercial aircrafts, typical performances do not exceed $\epsilon = \pm 2.5 [deg] (\cos 2.5 = 0.999)$; in taking off rarely can go up to $\epsilon = 5 - 10 [deg]$, but still $\cos 10 = 0.98$
ODE system ($\ref{eq12.4.4.1}$-$\ref{eq12.4.4.7}$) is as follows:
$-mg \sin \gamma + T - D = m (\dot{V}),$
$mg \cos \gamma \sin \mu = -m V (\dot{\gamma} \sin \mu - \dot{\chi} \cos \gamma \cos \mu),\label{eq12.4.4.9}$
$mg \cos \gamma \cos \mu - L = -m V (\dot{\gamma} \cos \mu + \dot{\chi} \cos \gamma \sin \mu),\label{eq12.4.4.10}$
$\dot{x}_e = V \cos \gamma \cos \chi + W_x,$
$\dot{y}_e = V \cos \gamma \sin \chi + W_y,$
$\dot{z}_e = -V \sin \gamma,$
$\dot{m} + \phi = 0.$
Operating Equation ($\ref{eq12.4.4.9}$) $\cdot \cos \mu$ - Equation ($\ref{eq12.4.4.10}$) $\cdot \sin \mu$ it yields:
$L \sin \mu = m V\dot{\chi} \cos \gamma.$
Operating Equation ($\ref{eq12.4.4.9}$) $\cdot \sin \mu$ + Equation ($\ref{eq12.4.4.10}$) $\cdot \cos \mu$ it yields:
$L \cos \mu - mg \cos \gamma = m V \dot{\gamma}.$
12.5: References
[1] Gómez-Tierno, M., Pérez-Cortés, M., and Puentes-Márquez, C. (2009). Mecánica de vuelo. Escuela Técnica Superior de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid.
[2] Hull, D. G. (2007). Fundamentals of Aiplane Flight Mechanics. Springer.
[3] Yechout, T., Morris, S., and Bossert, D. (2003). Introduction to Aircraft Flight Mechanics: Performance, Static Stability, Dynamic Stability, and Classical Feedback Control. AIAA Education Series.
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The aim of this chapter is to provide the student with a series of laboratory exercises such that theoretical content and exercises can be contextualised. Different open-source (or academic free-basis licensing) softwares will be used: namely, XLFR5, for solving an airfoil design exercise in Section 13.1; Python to integrate the equations of motion of an aircraft in Section 13.2; and Eurocontrol’s Nest to analyse Air Traffic Management after producing an operational flight plan in Section 13.3.
13: Hands-on Laboratories
In this laboratory exercise, the goal is to combine theory and practice aided by a popular aerodynamic design software called XFLR5, an open-source software developed by MIT’s Prof. Drela.
Figure B.1: XLFR5
13.01: Aerodynamics - Airfoil design
http://www.xflr5.com1 is an analysis tool for airfoils, wings and planes operating at low Reynolds Numbers. It includes:
• XFoil’s Direct and Inverse analysis capabilities.
• Wing design and analysis capabilities based on the Lifting Line Theory, on the Vortex Lattice Method, and on a 3D Panel Method (details on such methods are out of the scope of this book).
13.1.02: Airfoil design exercise
1. Design an airfoil based on thickness, camber as function of the longitudinal dimension $x$.
2. Choose 4 airfoils among the NACA 4-digit series. Plot them, and identify its main characteristics.2
Consider incompressible flow. Consider an aircraft flying at $h = 11000\ m$ in a determined incompressible velocity regime (Consider $M < 0.3$). The characteristic chord can be considered as $c = 5\ m$. The viscosity of air can be considered as $\mu = 0, 000018\ Ns/m^2$.
3. Calculate the Reynolds number for those conditions and proceed on with the subsequent analysis.
4. Plot the main characteristic curves of the 5 airfoils (the 4 NACA ones and the one you have defined before). Compare them and discuss the effects of thickness, camber and angle of attack in the generation of lift and drag forces. Check the pressure distribution, the coefficient of pressures and the boundary layer thickness for different angles of attack. Compare all five airfoils by plotting them together.
5. Repeat the previous analysis for different Reynolds numbers. How it affects the aerodynamics of the airfoil?
6. Discuss on the aerodynamic center and the center of pressures of the airfoils.
7. Analyse the stall of the airfoils.
2. Try to combine different cambers and thickness. In particular, choose at least one symmetric airfoil.
13.1.03: Proposed solution
Figure 13.2: XLFR5 analysis
The solution to this exercise is left open. The software is very user-friendly. Find here a video tutorial. Extension of this exercise could be the 3D analysis (wing analysis) and the Aircraft analysis. In the sequel of this Section, a very brief and schematic overview of NACA airfoils is given. Also, some XLFR5 plots are presented in Figure 13.2.
On NACA Airfoils
The NACA airfoils are a series of airfoils created by NACA (Nacional Advisory Committee for Aeronautics), including the following series: Four-digit-series; Five-digit-series; Modifications in four and five-digit series; 1-series; 6-series; 7-series; 8-series.
4-digit series:
• First digit: describes the maximum camber as a percentage of the chord (% c).
• Second digit: describes the maximum camber’s distance measured from the leading edge in 1/10 of the percentage of the chord (% c).
• Third and fourth digit: describing the maximum thickness as a percentage of the chord (% c). [By default the maximum thickness is 30% of the chord]
Some examples include:
• NACA 2412
- Maximum camber is 2% of c. (0.02c)
- Maximum camber located at 40% (0.4c) of the leading edge.
- Maximum thickness of 12% of the chord (0.12c)
• NACA 0015
- Symmetric airfoil (00)
- Maximum thickness of 15% of the chord (0.15c)
5-digit series:
• First digit: describes the $C_l$ multiplying the digit by 0.15.
• Second and third digits: dividing them by 2, describes the maximum camber’s distance measured from the leading edge as percentage of the chord (% c).
• Fourth and fifth digits: describing the maximum camber as a percentage of the chord (% c).
• By default the maximum thickness is 30% of the chord.
The following example illustrates it:
• NACA 12345
- $C_l = 0.15.$
- Maximum camber located at 11.5% (0.115c) of the leading edge. This implies $x_{mc} = 0.15$.
- Maximum camber of 45% of the chord (0.45c)
Notice that camber line is defined as follows [$y$ and $x$ normalized with the chord]:
$y = \begin{cases} \tfrac{k_1}{6} \{ x^3 - 3mx^2 + m^2 (3 - m)x \} & 0 \le x \le m, \ \tfrac{k_1 m^3}{6} (1 - x) & m \le x \le 1; \end{cases}\nonumber$
with $m$ is chosen so that the maximum camber takes place in $x = c_{mc}$.
Modifications in four and five-digit series: The fourth and fifth digit series can be modifies by adding two digits with a dash.
• First digit after the dash: describes how rounded is the shape, being 0 very sharp and 6 exactly as the original airfoil, and 9 more rounded that the original.
• Second digit after the dash: describing the maximum thickness distance measured from the leading edge in 1/10 as a percentage of the chord (% c)
The following example illustrates it:
• NACA 1234-05
- NACA 1234 with sharp leading edge shape.
- Maximum thickness located at 50% c (0.5c) measured from the leading edge.
1 series:
• First digit: describes the series.
• Second digit: describes the minimum pressure’s distance measured from the leading edge in 1/10 as a percentage of the chord (% c).
• Third digit [after a dash line]: describes $C_l$ in 1/10.
• Fourth and fifth digits [after a dash line]: describe the maximum thickness in 1/10 as a percentage of the chord.
Consider the following example as illustration:
• NACA 16-123
- Minimum pressure located at 60% of the chord.
- $C_l = 0.1.$
- $E_{\max} = 0.23c$ measured from the leading edge.
6 series: It is essentially an improvement of 1-series to maximize the laminar flow:
• First digit: describes the series.
• Second digit: describes the minimum pressure’s distance measured from the leading edge in 1/10 as a percentage of the chord (% c).
• Third digit [typically as a subindex]: describes the fact that drag remains low a number of tenths below of $C_l$.
• Fourth digit [after a dash line]: describes $C_l$ in 1/10.
• Fifth and sixth digits [after a dash line]: describe the maximum thickness in 1/10 as a percentage of the chord.
• "a=. . . " [followed by a decimal number]: describes the fraction of chord in which the laminar flow remains. By default a=1.
Please find below an example:
• NACA 61 - 345 a = 0.5
- Minimum pressure located at 10% of the chord.
- $C_l = 0.3$. What this means is that the airfoil was designed for maximum efficiency at a lift coefficient of approximately 0.3
- $E_{\max} = 0.45c$ measured from the leading edge.
- The laminar flow is maintained over 50% of the chord.
7 and 8 series: Correspond to additional improvements to maximize the laminar flow both in extrados and intrados.
• First digit: describes the series.
• Second digit: describes the minimum pressure’s distance in the extrados measured from the leading edge in 1/10 as a percentage of the chord (% c).
• Third digit: describes the minimum pressure’s distance in the intrados measured from the leading edge in 1/10 as a percentage of the chord (% c).
• Letter Letter referring to an standard airfoil of previous NACA series
• Fourth digit [after a dash line]: describes Cl in 1/10.
• Fifth and sixth digits [after a dash line]: describe the maximum thickness in 1/10 as a percentage of the chord.
The following example illustrates it:
• NACA 712A345
- Minimum pressure located at 10% of the chord in the extrados.
- Minimum pressure located at 20% of the chord in the intrados.
- $C_l = 0.3.$
- $E_{\max} = 0.45c$ measured from the leading edge.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/13%3A_Hands-on_Laboratories/13.01%3A_Aerodynamics_-_Airfoil_design/13.1.01%3A_Overview_of_XFLR5.txt
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This exercise consists in obtaining and analyzing the 3DOF performances of a BADA model aircraft. The evolution of aircraft state variables over time $(V(t), \gamma (t), \chi (t), x_e(t), y_e(t), h_e(t), m(t))$, including the 4D trajectory $(x_e, y_e, h_e, t)$, in a typical flight of a commercial transportation aircraft are to be analysed. An aircraft type among those in BADA database (See [1]) will be chosen. Python programming Language will be used to integrate the resulting set of differential equations and plot the results.
13.02: Flight Mechanics - Aircraft motion
BADA is a collection of ASCII files, which specifies operation performance parameters and operating procedure parameters for 295 aircraft types. These parameters of aircraft performance and the model is designed for use in trajectory simulation and prediction algorithms within the domain of Air Traffic Management (ATM). All files are maintained within a configuration management system at the Eurocontrol Experimental Centre (EEC) at Bretigny-sur-Orge, France. A complete description of BADA is available in the BADA 3.6 User Manual [1]. User Manual for Revision 3.6 of BADA provides definitions of each of the coefficients and then explains the file formats.
Figure 13.3: Flight mechanics lab using Python.
13.2.02: Overview of Python
Python is a widely used general-purpose, open-source, high-level programming language. Python supports multiple programming paradigms, including object-oriented, imperative and functional programming or procedural styles. It features a dynamic type system and automatic memory management and has a large and comprehensive standard library. Please, visit Python.3
Installation
It is recommended to Install the Anaconda Distribution.4 You can either download Python 2 or 3. For the sake of compatibility with the packages to be used throughout the course, Python 2.7 is recommended. Anaconda is a completely free Python distribution (including for commercial use and redistribution). It includes more than 300 of the most popular Python packages for science, math, engineering, and data analysis.
Installing packages
Python capabilities are being built up based on continuous contributions of the community. These contributions are typically encapsulated in packages. For instance, We can start by Poliastro5 library, a space engineering library.
To install this or any other package, we can make use of different package managers. There are two main package managers for python (both of them come with anaconda):
Just invoke the any of the following sentences in your terminal/cmd window:
conda install poliastro -c poliastro
pip install poliastro
Integrated Development Environment (IDE) environment
We can use Python either invoking it form the terminal, or using ad-hoc IDE environments. In particular, Anaconda distribution comes with:
• Spyder $\to$ Desktop script.
• i-Python $\to$ html development environment
Getting started with Python
Write your first script in Python and run it using Spyder:
# @author: manuelsolerarnedo (# to comment)
print "hello world"
a = 2
b = 3
c = a + b
print c
If you want to further learn about python (with aeronautical applications), the AeroPython course is strongly recommended (AeroPython Course).6 Notice however that the course has been prepared as a i-python notebook. Should you want to continue, just download the notebooks and start coding!
5. https://poliastro.github.io/ by Juan Luis Cano
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/13%3A_Hands-on_Laboratories/13.02%3A_Flight_Mechanics_-_Aircraft_motion/13.2.01%3A_Overview_of_BADA.txt
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In order to obtain the 4D trajectory, the flight will be divided into three phases: climb, cruise, and descent.
1. The climb phase will be assumed to be a symmetric flight into the vertical plane non considering any wind and assuming the heading angle to be zero. Therefore, the ODE system to be used is as follows:
$m \dot{V} = T - D - m \cdot g \cdot \sin \gamma, \nonumber$
$m V \dot{\gamma} = L - m \cdot g \cdot \cos \gamma,\nonumber$
$\dot{x}_e = V \cdot \cos \gamma,\nonumber$
$\dot{h}_e = V \cdot \sin \gamma,\nonumber$
$\dot{m} = -T \cdot \eta.\nonumber$
where according to BADA, $\eta = ( \tfrac{C_{f1}}{1000*60} ) \cdot (1 + \tfrac{V}{C_{f2}} )$, with $V$ in knots.
In order to integrate the system, one should:
1.1 Set the initial conditions for all the state variables, initial and final time. This conditions must be selected according to typical values of aircraft performance.
1.2 Set the control variables $(T(t), C_L (t))$,7 for instance to the following values:
* $T = 0.8 \cdot T_{\max}$, where $T_{\max} = C_{tc1} \cdot (1 - \tfrac{h_e}{C_{tc2}} + C_{tc3} \cdot h_e^2)$, $h_e$ in feet,
* $C_L = C_{L_{opt}}$.
1.3 Use a suitable numerical method to solve the resulting system.
2. The cruise phase will be assumed to be a symmetric flight into the horizontal plane not considering any wind. Therefore, the ODE system to be used is as follows:
$m \dot{V} = T - D,\nonumber$
$m V \dot{\chi} = L \sin \mu, \nonumber$
$\dot{x}_e = V \cos \chi,\nonumber$
$\dot{y}_e = V \sin \chi, \nonumber$
$\dot{m} = -T \eta,\nonumber$
being $L = \tfrac{mg}{\cos \mu}$.
In order to solve the system, one should:
2.1 Set the initial conditions for all the state variables, initial and final time. Initial conditions and initial time will coincide with the final conditions of the previous phase. Set the final time according to typical values of aircraft performance.
2.2 Set the control variables $(T(t), \mu (t))$ to the following values:
* $T = 0.5 * T_{\max}$,
* $\mu = 0$.
2.3 Use a suitable numerical method to solve the resulting system.
3. The landing phase will be assumed to be gliding performance not considering any wind. Therefore, the ODE system to be used is as follows:
$m \dot{V} = -D - mg \sin \gamma, \nonumber$
$m V \dot{\gamma} = L - mg \cos \gamma, \nonumber$
$\dot{x}_e = V \cos \gamma,\nonumber$
$\dot{h}_e = V \sin \gamma, \nonumber$
3.1 Set the initial conditions for all the state variables, initial and final time. Initial conditions and initial time will coincide with the final conditions of the previous phase. Set the final time according to typical values of aircraft performance.
3.2 Set the control variable $(C_L(t))$ to the following value:
* $C_L = C_{L_{opt}}.$
3.3 Use a suitable numerical method to solve the resulting system.
7. Notice that $C_L$ acts as control.
13.2.04: Proposed solution
Hereby, an schematic solution for the aircraft trajectory is proposed. Notice that, for the sake of conciseness, only the climb phase is presented. Both Cruise and descent phases are left as exercises to the students:
• An Airbus A320 is selected. A Python based code (A320.py), including A320 BADA parameter values, is provided along the text.
• A Python based code (file ODE_Aircraft.py), including values and description of the different steps, is also provided.
• A 4th order Runge-Kutta method is used to solve the set of differential equations. Notice that numerical methods are out of the scope of this course; they are to be studied within a numerical calculus course.
Figure 13.4: Aircraft climb motion solution.
• The evolution mass, true airspeed, altitude, and flight path angle, is presented in Figure 13.4. Notice that the oscillations in Flight Path Angle and True airspeed are natural with open-loop, fixed controls (as it is our case): they correspond to the activation of the so-called phugoid mode. Real aircraft include close-loop control systems that counteract this oscillating behaviour. Aircraft engine modes and aircraft response to both open-loop and close loop control are out of the scope of this course. They should be studied in advanced courses of mechanics of flight and aircraft dynamic stability and control.
A320.py data file
Main file ODE_Aircraft.py
Figure 13.4 presents the solution obtained after running the given code in Python. Notice that this solution corresponds only to the climb phase. Recall what was mentioned before on the oscillations.
Students are challenged to complete the exercise by integrating a cruise phase and then a descent phase. Students are also challenged to solve the problem in a more operational manner, that is, instead of ascending at constant Thrust and optimum coefficient of lift, something that aircraft do not do in real operation, one could set two constraints to close up the degrees of freedom of the problem:
• ascent at a rate of climb of 2000 ft/min;
• follow a constant CAS procedure (say 250 kts), reach the transition Mach (say M = 0.78), and then a constant Mach.
The student should observe how oscillations disappear.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/13%3A_Hands-on_Laboratories/13.02%3A_Flight_Mechanics_-_Aircraft_motion/13.2.03%3A_Aircraft_motion_exercise.txt
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The laboratory consists of two parts: design a flight plan; and the analyse of the air navigation network capacity and airspace design using NEST modelling tool. It is aimed at providing the student with a first overview of flight plan design and Nest capabilities.
13.03: Flight Plan analysis
NEST modelling tool8 is a scenario-based modelling tool used by the EUROCONTROL Network Manager and the Air Navigation Service Providers (ANSPs) for:
• designing and developing the airspace structure,
• planning the capacity and performing related post operations analyses,
• organising the traffic flows in the ATFCM strategic phase,
• preparing scenarios to support fast and real-time simulations,
• and for ad-hoc studies at local and network level.
NEST is used to optimise the available resources and improve performance at network level. Upon request to Eurocontrol,9 access can be granted at no cost. Installation is straight forward (only under Windows though). First of all, remark that one should select an available airac cycle. Up to date airac cycles are available to be downloaded within the NEST settings. Nest includes three blocks with data to visualise, namely: airspace data, network data, and flight data:
Figure 13.5: Nest: Airspace design.
• Airspace data: Nest includes capabilities to visualise many elements conforming the airspace structure, including for instance the FIR/UIR regions; elemental ATC sectors; airports; navaids and waypoints. Figure 13.5 illustrate these features.
• Network data: Nest also includes information on the network of routes, including different types of airspaces (e.g., free routing airspaces), regulations that apply (e.g., RAD regulations).
Figure 13.6: Nest Flights in Europe for January 25th 2014.
• Flights data: Furthermore Nest includes detailed information on flights (historical by default; however one can also download and display forecasted traffic or even create an scenario of trafffic). For instance, Figure 13.6.a presents all flights in a day in Europe. In this particular case, in January 25th 2014, we had 19359 flight in Europe. These flights can be analysed in depth, e.g., by filtering all those going/coming to/from the US (642 flights) as illustrated in Figure 13.6.b; all those that are flown using a B-738 (3204 in total) as illustrated in Figure 13.6.c; and all those that depart from London Heathrow (EGLL in ICAO’s terminology) as illustrated in Figure 13.6.d.
Figure 13.7: Nest Flight analysis capabilities.
For each particular flight, one can analyse the following information: the horizontal route, including origin and destination airport, overflying sectors, waypoints, airways, times, etc, for which one should use the Flight Route Viewer (see Figure 13.7.a); the vertical profile, for which one should use the 4D Vertical Profile Viewer (see Figure 13.7.b). Note the reader that Nest allows to compare planned flights with actual (really flown) flights as in Figure 13.7.b. Also, allows to compare regulated flights with both planned and actual flights. This can be observed in Figure 13.7.c, where a flight has been regulated: the difference between ETOT (Estimated Take-Off Time) and CTOT (Calculated Take-Off Time), 36 minutes in this case, results in the delay imposed by ATFM. Another feature is the analysis of airspace sectors, in which one can analyse the entry flights and occupancy counts among other issues. An example with the flight list in LECM UIR is given in Figure 13.7.d.
9. Access EUROCONTROL One Sky Online (Extranet - https://ext.eurocontrol.int/); select "Subscribe for online services" and select DDR2. The request will be processed and access might be granted; access DDR2 home page and select Tools Download tab Select NEST, download the package and install it.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/13%3A_Hands-on_Laboratories/13.03%3A_Flight_Plan_analysis/13.3.01%3A_Overview_of_Nest.txt
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You are in charge of preparing the most efficient route for a flight between LEMD (Adolfo Suarez Madrid Barajas) and LEBL (Barcelona). Assume your aircraft type is an B737-800 and your requested flight level (RFL) will be always above FL245 (upper airspace). Assume also your flight is RNAV. Flight is with Estimated Off-Block Date (EOBD) YYYYMMDD and Estimated Off-Block Time (EOBT) HHMMZ.
I.1. Establish a route between the given airports. You must notice that routes start at the end of a SID and end at the beginning of a STAR. The route will be composed by Waypoints/Navaids joined by Airways. En route chart should be consulted in the AIP at En-Aire AIP10. Note the reader that Route Availability Document (RAD) and the ATFM regulations should be consulted to find the route. However for the sake of simplification, we can consider fully route availability (in other words, no RAD nor ATFM regulations apply for this exercise).
I.2. Assume that the flight plan should include also SID and STAR. Select the appropriate SID and STAR, assuming that the operative runways heads will be: LEMD RWY 14R (south configuration); LEBL RWY 07R (east configuration) according to current wind conditions.11
I.3. Choose a precision (ILS based) approximation chart.
I.5. Establish a required flight level and a cruising speed.
I.6. Fill in a Flight Plan international Form. Note that for many fields there is no information in the exercise statement. Fill them at your discretion. Check for instance the following document for some information IVAO Flight Plan Doc.12
11. Notice that a METAR should be given in a real scenario.
13.3.03: Part II- Nest analysis
II.1 Select the latest available AIRAC cycle, analyse different flights (at different days) flying between LEMD and LEBL. Choose one of those flights between LEMD and LEBL and:
II.1.1 Indicate the EOBT, EOBT, COBT, CTOT, AOBT, ATOT. What is the difference between them? Could you find one flight (with the same or a different city pair) with different ETOT and CTOT. What is the reason behind this difference?
II.1.2 Indicate Route length and requested flight level for you flight LEMD to LEBL.
II.1.3 Analyze its horizontal profile throughout the Flight Route Viewer. The analysis should include the Waypoints and ATC sector the aircraft overflies, together with times (overfly/entry) and altitudes. A table is required.
II.1.4 Analyze its vertical profile throughout the 4D vertical profile Viewer. Compare Initial/Regulated/Actual vertical profile. Include plots.
II.2 Now, it is time to analyse the ACCs/Traffic Volumes in which your flight flies. For the sake of simplicity, focus on ATC Sector Teruel (LECM TER):
II.2.1 Analyze (include plot and discuss) the sector daily entry counts.
II.2.2 Analyze (include plot and discuss) the occupancy counts.
II.2.3 Analyze (include plot and discuss) the AIRAC summary.
13.3.04: Proposed solution
Hereby, an schematic solution for the Flight Plan is proposed:
Figure 13.8: SID-14R: PINAR2B.
Figure 13.9: En-Route Chart.
Figure 13.10: STAR.
Figure 13.11: Precision Approach.
• The selected route is depicted within different Navigation Charts, including SID (PINAR2B), En-ROUTE, STAR (CASPE 1U), and Precision Approach. See Figure 13.8, Figure 13.9, Figure 13.10, Figure 13.11. Charts were accessed at AIP En-AIRE.
• Requested Flight Level could be, for instance, FL300; and Cruising Speed M078.
Figure 13.12: Flight Plan Form (FAA Form 7233-4)
• A Flight Plan Form has been (partially) filled in and is presented in Figure 13.12. Please, check for instance the document referred in the statement for a deeper understanding. In particular, try to figure out what is the equipment of the aircraft (e.g., S refers to standard equipment such as VHF, VOR ILS; D refers to DME equipment; etc.)
Similarly, an schematic solution for the NEST analysis is given below:
Figure 13.13: Nest: Madrid (LEMD) - Barcelona (LEBL) flights
• Figure 13.13 includes top and lateral view of all flights between Madrid (LEMD) and Barcelona (LEBL) in a day.
• For a finer analysis, we choose Flight IBE2770 (AIRAC 1602) on February 21 2016: EOBT was 07:10 and ETOT was 07:24; COBT and CTOT were identical to the estimated ones (in other words, no ATFM delay was enforced); AOBT was 07:08 and ATOT was 07:24 (in other words, ATC authorised Off-Block 2 minutes earlier than expected, however take-off took place exactly when expected). A difference between estimated and calculated would imply an ATFM delay. This is typically associated to a regulation (one can find flights falling in this category by simply filtering). Please, refer to Figure 13.7.c.
• Aircraft was an A333, RFL was FL300, and total route length was 300,90 NM.
Figure 13.14: IBE 2770 analysis: Route Description.
Figure 13.15: IBE 2770 analysis: Vertical profile.
Route Description and Vertical Profile are presented in Figure 13.14 and Figure 13.15, respectively. Waypoints, Sectors, Altitudes, and Overflying times can be checked. Additionally, a comparison between the planned flight and the actually flown flight can be checked.
Figure B.16: LECMTER Sector Analysis.
Figure B.16 includes an analysis over SECTOR TERUEL (LECMTER). The analysis includes entry counts and occupancies.
13.4: References
[1] Nuic, A. (2005). User Manual for the base of Aircraft Data (BADA) Revision 3.6. Eurocontrol Experimental Center.
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textbooks/eng/Aerospace_Engineering/Fundamentals_of_Aerospace_Engineering_(Arnedo)/13%3A_Hands-on_Laboratories/13.03%3A_Flight_Plan_analysis/13.3.02%3A_Part_I-_Flight_Planning.txt
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Part of the purpose of the course is to help you to understand why biofuels are needed and how to make them, at the current state-of-the-art.
Why biofuels? To look at the situation a little more broadly, the question then becomes: why alternative fuels?
As climate change becomes an issue of ever-stronger concern in the world, stronger efforts are being devoted to tackling this issue. The International Energy Agency (IEA) has recently proposed the 2 °C scenario (2DS) as a way to handle the climate change issue. The 2DS has become a largely used quote for many policymakers and scientists. The 2DS scenario requires that carbon dioxide (CO2) emissions in 2060 should be reduced by 70% in comparison to the 2014 level. The transport sector plays an important role to achieve this goal considering that the transportation sector is responsible for about 23% of total CO2 emissions. Although electricity has been considered as a promising option for reducing CO2 emissions in transportation (Yabe, Shinoda, Seki, Tanaka, & Akisawa, 2012), transport biofuel is estimated to be the key alternative energy in the transport sector (Ahlgren, Börjesson Hagberg, & Grahn, 2017). The share of biofuels in total transportation-fuel consumption by 2060 is predicted to be 31%, followed by electricity at 27% based on the mobility model results of IEA for the 2DS. Biofuels production must be increased by a factor of 10 to achieve this goal (Oh, Hwang, Kim, Kim, & Lee, 2018). In addition to the need for climate change adaptation, the increasing concerns over energy security is another main driver for the policy-makers belonging to the Organisation for Economic Co-operation and Development (OECD) to promote the production of renewable energy (Ho, Ngo, & Guo, 2014). Last but not least, world energy demand will continue increasing. The world energy demand was 5.5 x 1020 J in 2010. The studies predict an increase of a factor of 1.6 to reach a value of 8.6 x 1020J in 2040. The bioenergy delivery potential of the world's total land area excluding cropland, infrastructure, wilderness, and denser forests is estimated at 190 x 1018 J yr-1, 35% of the current global energy demand (Guo, Song, & Buhain, 2015).
In short, there are three main reasons to develop biofuels (Fig 1.1):
1. to meet the needs of increasing energy demand;
2. dependence on foreign fuel sources can be problematic, depending on US domestic fuel production;
3. to reduce greenhouse gas (GHG) emissions.
We will explore each of these reasons in more depth in the following sections.
References
Ahlgren, E. O., Börjesson Hagberg, M., & Grahn, M. (2017). Transport biofuels in global energy–economy modeling–a review of comprehensive energy systems assessment approaches. Gcb Bioenergy, 9(7), 1168-1180.
Guo, M. X., Song, W. P., & Buhain, J. (2015). Bioenergy and biofuels: History, status, and perspective. Renewable & Sustainable Energy Reviews, 42, 712-725. doi:10.1016/j.rser.2014.10.013
Ho, D. P., Ngo, H. H., & Guo, W. (2014). A mini review on renewable sources for biofuel. Bioresource Technology, 169, 742-749. doi:10.1016/j.biortech.2014.07.022
Oh, Y. K., Hwang, K. R., Kim, C., Kim, J. R., & Lee, J. S. (2018). Recent developments and key barriers to advanced biofuels: A short review. Bioresource Technology, 257, 320-333. doi:10.1016/j.biortech.2018.02.089
Yabe, K., Shinoda, Y., Seki, T., Tanaka, H., & Akisawa, A. (2012). Market penetration speed and effects on CO2 reduction of electric vehicles and plug-in hybrid electric vehicles in Japan. Energy Policy, 45, 529-540.
1.02: Increasing Energy Demand
The energy needs of most of the developed countries in the western world are increasing at a modest level. However, in underdeveloped countries, where the economy is booming, energy demands are increasing dramatically, e.g., in China and India. Figure 1.2 shows the gross domestic product (GDP) growth rate of the countries of the world in 2009. Most of the growth occurred in Asia. Africa and South America experienced some growth in areas, while the US did not experience much growth. However, if many of the third world countries were to dramatically increase their standard of living, there are estimates that worldwide energy consumption would double (the world uses ~13 TW). But where would that energy come from, particularly since there aren't huge stockpiles of crude oil sitting around? Petroleum cannot supply it all, and neither can natural gas or coal.
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textbooks/eng/Biological_Engineering/Alternative_Fuels_from_Biomass_Sources_(Toraman)/01%3A_Why_Alternative_Fuels_from_Biomass/1.01%3A_Why_Biofuels.txt
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