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Atmospheric Models 1.

0 Aerodynamic Variables
Four of the most frequently used words in aerodynamics: Pressure (p) Density () Temperature (T) Flow Velocity (V)

1.1

Pressure

The pressure is defined at a point in the fluid or a point on a solid surface. The pressure can vary from one point to another. Definition: Pressure is the normal force per unit area exerted on a surface due to the time rate of change of momentum of the gas molecules impacting on (or crossing) that surface (J. Anderson, Fundamentals of Aerodynamics, 2001, pg. 13) 1 = lim

dA = Elemental Area at B (m2) dF = Force on 1 side of dA due to pressure (N)

1.2

Density
The density of a material is a measure of the amount of material contained in a given volume. 2 = lim

From Source:

Laurent Dala, UWE

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Atmospheric Models
<<<<< Note: 1.3 & 1.4 (up to viscosity) were missing from original handout! >>>>>

1.3

Temperature
The temperature T of a gas is directly proportional to the average kinetic energy of the molecules of the fluid. In fact, if KE is the mean molecular kinetic energy, then temperature is given by: 3 = 2 Where k is the Boltzmann constant (k = 1.38 x 10-23 J/K) Temperature is a point property, which can vary from point to point in the gas & has an important role in high-speed aerodynamics Source: J. Anderson, Fundamentals of Aerodynamics, 2001, pg. 14

1.4

Flow Velocity
A velocity is a vector value & as such must contain both scalar value & direction. In a flowing fluid at each region (or point) in the fluid there is not necessarily the same velocity. Hence this is also a point property which can vary from point to point in the flow. The velocity of a flowing gas at any fixed point B in space is the velocity of an infinitesimally small fluid element as it sweeps through B. (J. Anderson, Fundamentals of Aerodynamics, 2001, pg. 14) Viscosity of oil > Viscosity of air Sutherlands Law: 3 =
/

is reference data T0 = 288.16 K Where: 0 = 1.7894 x 10-5 kg/ms

0

Dynamic Viscosity: 4 =

Newtonian Fluids For many simple fluids, such as air and water, is a thermodynamic property which depends only on Temperature & Pressure, but not on the shear rate Note: Newtonian Non Newtonian Fluids =

2.0

Equation of State
This may also be referred to as the Ideal Gas Law or the Perfect Gas Law. 5 = Using SI units R = Specific Gas Constant = 287.05287 J/kg.K = & =

From Source:

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Atmospheric Models 3.0 The Hydrostatic Equation

For a fluid at rest, the pressure is constant over any horizontal surface but decreases with altitude.

If the pressure is assumed to change from p at one end of the cylinder to p + p at the other end, then the net force acting on the cylinder is: . + = .

This net force must be 0 unless the fluid cylinder is accelerating to the right (p < 0) or to the left (p > 0). Hence for a fluid at rest, p = 0 & the pressure is constant along any horizontal line (or surface). The net downwards force acting on the fluid cylinder is: . + + = . . + . + . = . OR 7 = . . + . + . =0 =0 =0

Where: . + . .

4.0

An Energy Balance Equation 4.1 Some Definitions

Adiabatic Process One in which no heat is added to or taken away from the system Isothermal Process One in which the temperature remains constant Reversible Process One in which no dissipative phenomena occur, that is, where the effects of viscosity, thermal conductivity & mass diffusion are absent Isentropic Process One that is both Adiabatic & Reversible OR If there is no heat transfer or friction in the process Page 3

From Source:

Atmospheric Models 4.2 Isothermal Case

If: T = T0 Then the equation of state becomes: = 8 =

Where p0 occurs at z = 0 Substitute (10) into (8): = 11 , = ,

Where 0 occurs at z = 0

4.3

12 =0 For an isentropic process, ds = 0, where s is the entropy of the system. Note: Entropy & the Second Law of Thermodynamics: 13 =

First Law of Thermodynamics: The change in energy in a system is: 14 For a reversible process: 15 = = = +

Where dv is an elemental change in the volume due to a displacement of the boundary of the system By deriving (13) it can be shown that From Source: Laurent Dala, UWE = =

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Atmospheric Models
16 = + . +

= .

Substitute this into (16): 18 = .

For a Perfect Gas: = . & = . Therefore by substituting into (16) & (18): . + 19 = 20 = = = = . ln + + . ln = = = 21 22 = = . ln + . ln .

Therefore:

. ln ln =

+ . ln . ln

& &

0= ln

. ln =

+ . ln . ln

Therefore:

=
Where:

23 = 1 & =

Therefore:

24

25

From Source:

Laurent Dala, UWE

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Atmospheric Models
Where: = =

4.4

Constant Lapse Rate

In the troposphere, the temperature decreases with altitude z, according to the relation: 29 = Where T0 = 288.16 K at z = 0 & is the lapse rate = .

30

Note: In isothermal conditions T = T0 = Constant. Therefore = 0 Using the hydrostatic equation derived from (9): = By substituting equation (29) into it we find: 31 = = 32 =

. =

.

From Source:

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