Abstract
Air rifles and air pistols find widespread use in formal and recreational sports events. Despite their widespread use in sport, they have rarely been studied scientifically. The influence of air rifle pellet geometry on aerodynamic drag was investigated experimentally and theoretically Results indicate that the overall drag of non-spherical pellets was dominated by the drag on their front face, with the face contributing approximately 65 % of the overall pellet drag, while base drag contributed almost all of the remaining 35 %. The net drag contribution of the pellet side-slopes was close to zero. The geometry of the front face had a weak influence on the drag acting on the pellet base. This influence was exercised through the behaviour of the free shear layer separating from the pellet head rim. It was apparent that the presence of the tail in a dome-head pellet enabled flow reattachment and a rise in base pressure, which reduced the base drag. In contrast, at Re~54,000, flow reattachment on the rear surface of the spherical pellet was not possible. For this reason, its base drag was higher than that for the dome-head pellet. Flat-, cone-, and cavity-head pellets had higher overall drag coefficient
Introduction
Major international and national sports competitions (Olympic Games, European and World Championships) include air rifle and air pistol events. Air rifle and pistol target practice is also a popular recreational sport worldwide and in the UK alone, up to five million individuals participate. [1] Aerodynamic drag is a major influence on air rifle pellet trajectory and shot accuracy. Once the pellet leaves the rifle barrel and assumes free flight, aerodynamic drag opposes the motion of the pellet, slowing it down. This prolongs the pellet flight period, during which time gravity and aerodynamic forces acting on the pellet cause it to deviate in a complex manner from a linear path to the target. Despite the key role of aerodynamic drag on pellet trajectory, few scientific studies have been published on this topic. [2] [3] [4] This contrasts with a plethora of aerodynamic studies for balls and other projectiles in sports such as golf, [5] badminton, [6] football/soccer, [7] [8] baseball, [9] tennis, [10] table tennis, [11] [12] cricket, [13] and frisbee. [14] Aerodynamic drag of ordnance projectiles has been extensively researched for many decades, however, little data has been published in scientific literature.
Introduction
This paper explores the relationship between aerodynamic drag and air rifle pellet geometry for several commercially available pellets of 4.5 mm (0.177 in) calibre. The great majority of commercially available pellets are manufactured from soft pure or alloyed lead, while a few are made from harder metals such as tin, or partially from polymers or steel. Pellets come in a wide variety of shapes, which range from that of a sphere to the common diabolo shape shown in Fig. 1 . Air rifle and pistol pellets fly through the atmosphere usually at subsonic speeds, which range from Mach number of approximately 0.5 to 0.8. During pellet free flight, flow separation often occurs at the head rim as shown in Fig. 1 , with the flow often re-attaching close to the pellet tail rim near point C, but only to detach once again at the tail rim. The detaching boundary layers at the head and tail rims form free shear layers which engulf low pressure recirculation zones labelled A and B in Fig. 1(b) . Separation of the flow from the pellet head and tail rims is an important contributory factor to pellet drag, as it causes lower pressures within the separation regions A and B, which contribute substantially to pellet drag. For example, the low pressure recirculation region B, downstream of the pellet base, contributes a significant proportion of the total pellet drag (at least 35 % at M a~0 .6, see later). The pellet shown in Fig. 1 has a fairly rounded front face so that the flow separating close to the head rim does so at an angle of approximately 45 o to the pellet axis. In the case of pellets with a blunt flat face, the separating flow will leave the pellet rim at a greater angle of up to 90 o , resulting in wider separation zones and wake and generally higher pellet drag.
Introduction
Pellets almost invariably have a hollow blunt base, and when they are in free flight, their large base area is exposed to the low pressure recirculation region B (Fig. 1 ). The design of B A C the hollow base is deliberate. When the pellet is inside the rifle barrel, propelled by high pressure gas, the gas pressure expands the base rim and forces it into the barrel rifling grooves, so as to seal the propelling gases. The barrel helical grooves force the pellet to spin, so that it leaves the barrel gyroscopically stabilised, similar to firearm bullets. The deep hollowing of the pellet shown in Fig. 1(a) moves the pellet centre of mass (CM) forward so that it lies ahead of the centre of pressure (CP). This provides aerodynamic stabilisation, so that when the pellet leaves the barrel and assumes free flight, it is stabilised aerodynamically and gyroscopically.
Introduction
In contrast to air rifle pellets, which have blunt tails, firearm bullets generally have gradually tapered "boat" tails, with the tail diameter gradually reducing toward their tail end to avoid separation of the flow from the side surfaces of the bullet tail. The tail tapering decreases the base area of the bullet that is exposed to the low pressure recirculation region downstream of the base, thus reducing the base drag. The tapered tail also encourages pressure recovery along the sides of the tail, which raises the base pressure and reduces drag further. [20] [21]
Introduction
[22] Mair [20] studied the reductions that could be obtained by adding boat tails to truncated blunt-base projectiles. Reductions of more than 50 % in base drag were obtained by adding to the blunt base a short (~0.5 calibre long) boat tail afterbody having gradual downstream taper of 22 o . Figure 2 shows the five pellets investigated, all of which had 4.5 mm head diameters (0.177 calibre). All of the pellets were made of lead and are currently available commercially. The first pellet was spherical, the second had a domed-faced head, the third had a flat-faced head with a head rim chamfer, and the fourth had a cone-faced head. The fifth pellet had a truncated cone face, with the apex replaced by a deep cavity, but was otherwise almost identical to the fourth pellet with the cone-shaped head.
Experimental approach and systems

Pellets investigated
For the measurement of drag coefficient, all pellets were mounted on identical stings, as
Pellets investigated
shown in the lower part of Fig. 2 . The stings were made of ground high-carbon steel rod of 1 mm diameter, attached securely inside the pellet base cavity using epoxy adhesive. During the curing of the adhesive, a custom-made jig ensured that the sting and pellet axes were effect on sphere wake of sting/sphere sectional area ratio. An open jet wind tunnel and associated drag force measuring system were constructed for these investigations and are shown in Fig. 3 . Two electrically driven, high-speed, two-stage, centrifugal compressors discharged pressurised air into a 110 mm diameter pipe in which a series of honeycomb and mesh screens reduced swirl and velocity mal-distributions, with very small pressure drop penalty due to the large pipe diameter.
Pellets investigated
The air flow then passed through a contraction into a 50 mm diameter pipe, where an additional set of honeycomb and mesh screens conditioned the flow and further reduced turbulence intensity. Finally, the air flow passed through a second contraction and emerged through a 17 mm diameter nozzle as a free jet. Both contractions had a curved internal profile where ρ is the jet free stream density. In subsequent discussions in this paper, the value of the free jet velocity, U, was quoted and this was calculated from the following expression shown in Eq. (3):
Pellets investigated
The free jet static temperature, T, was calculated for isentropic flow using the measured stagnation pressure and stagnation temperature (p oo , T oo ) in the wind tunnel, just upstream of the jet nozzle contraction as shown in Eq. (4):
Pellets investigated
ఊିଵ ఊ Figure 4 shows the radial profile of the free jet velocity measured at 0, 10 and 20 mm downstream of the nozzle exit plane. At 10 mm, the radial velocity profile was flat at~205 m/s up to ±8 mm from the jet centreline. Therefore, the ±2.25 mm head radii of the 4.5 mm pellets were readily encompassed within this flat profile. All subsequent measurements of drag and pressure distributions were made with the front of the pellet nose placed at 10 mm from the nozzle exit plane. 
Blockage ratio
The blockage ratio, defined as the pellet maximum cross sectional area divided by the jet nozzle cross sectional area, was~7. [31] Maskel [32] provides correlations which can be used to correct for blockage effects in [33] suggested limiting the blockage ratio below 7.5 % in closed tunnels, while they also stated that blockage effects in open jet tunnels are smaller than those in closed tunnels. In open wind tunnels, as the one used here, the unconstrained jet is free to expand radially around the pellet, in a manner similar to that of the unconstrained atmosphere being displaced around a flying pellet.
Turbulence intensity
The turbulence intensity of the free jet was measured using a custom-build hot wire anemometer probe constructed with 5 μm tungsten wire, 1.24 mm long, which was spotwelded at the tips of~0.5 mm stainless steel needle prongs. The wire was operated in constant temperature mode using a custom-build servo-amplifier, constructed along the lines described by Itsweire and Helland [34] and Osorio et al. [35] At 204 m/s, the square wave test response of the probe/amplifier system was high, at 17 kHz (bridge ratio 5, overheat ratio 0.5, and jet temperature compensation was as suggested by Bruun [36] and Benjamin and Roberts [37] ).
Turbulence intensity
Measurements of turbulence intensity were made radially by traversing across the free jet at a distance of 10 mm downstream of the nozzle exit plane when the jet velocity was at 204 m/s (M a~0 .58). Figure 4 shows that the turbulence intensity was 0.35 % at the free jet centreline and it stayed at approximately that level as the jet was traversed radially up to a radius of approximately 6 mm from the jet centreline. The intensity then rose gradually to~2 % as the radius of 8 mm was approached, and then it rose sharply when the radius was increased beyond 8 mm, indicating the presence of a highly turbulent shear layer at the interface of the jet and the stationary atmosphere (the nozzle radius was ±8.5 mm). 
Measurement of pellet drag
The pellet drag was measured using the load-cell system shown in Fig When the pellet was mounted, the air flow imposed drag forces, not just on the pellet but also on the pellet support sting. Therefore, the force measured by the load cell, was the gross drag value, which included the air drag force on the pellet itself plus the "tare" drag due to the air jet impinging on the sting. In order to evaluate the net drag on the pellet, the tare drag on the sting was measured (after each measurement of the gross drag), using the additional arrangement shown in Fig. 6 . An identical pellet was mounted, using a transverse sting (item D in Fig. 6 ), to a support system (items E, F, G) which was independent of the load cell. An axial "dummy" sting (item C), not connected to the pellet, was placed in position and attached to the load cell of Fig. 5 . Thus, the arrangement of The calculated axial drag force F dx from band "d" was a result of the mean pressure of band
Measurement of pellet drag
, deviating from the atmospheric pressure p a .
Measurement of pellet drag
The axial force contributions of all 27 area bands around the pellet were then summed algebraically, which provided the overall pellet net drag force, F D . This overall net force was then entered as the drag force in Eq. In this paper, values of pressure coefficient were evaluated using the following general expression for pressure coefficient as shown in Eq. (7), ௦ ଶ where is the pressure coefficient, p s , is a pressure on the surface of a pellet, and p is the free stream static pressure measured with the pitot tube (invariably, the measured free stream static pressure was found to be equal to the atmospheric pressure, p a , as the stream was a free jet).
Measurement of pellet drag
In the case of pellets numbered 2, 3, 4 and 5, the pressure coefficient was calculated for each surface area band. For each surface area band, the value for p s that was used in Eq. (7) The pressure measurements at each taping azimuthal angle, θ o , were used to calculate the surface pressure coefficient at each angle, θ o , using Eq. (7) above. In the case of this pellet, the value of p s in Eq. (7) was the pressure measured at the tapping hole when the hole was An implicit assumption was that the pressure measured at the tapping was uniform around the pellet circumference in the plane normal to the flow. This assumption was considered to be reasonable because the velocity profile across the free stream jet was found to be uniform and also because care was taken to ensure that the pellet axis of rotation was accurately positioned perpendicularly to the flow. The estimate of pellet overall C d obtained from Eq. In total, 18 repeat measurements of C d were made for the five pellets. The maximum % deviation from a pellet mean value was calculated to be 4.2 % and the minimum was 0.04 %.
Measurement of pellet drag
The standard deviation of the 18 values for % deviation was 1.3 %. These results therefore provided an estimate of 1.3 % for the repeatability of any individual C d measurement (standard deviation from the mean).
Accuracy of C d measurements with load cell system
Error propagation analysis [42] was carried out in order to estimate the accuracy of the value of C d measured with the load cell system. Uncertainties in the measurements of individual derived variables were estimated to be: drag force, 2.9 %; pellet head cross sectional area, 0.3 %; density, 2.3 %; velocity squared, 2.1 %. Therefore, the overall error in the measurement of the pellet C d was greenanthony@example.org %.
Accuracy of C d measurements with load cell system
The following factors were found to be crucial in achieving high repeatability and accuracy of C d measurements: precise alignment of the pellet axis of symmetry with the axis of the jet; minimising pellet tare drag force, by shielding the load cell and pellet sting from the air jet;
Accuracy of C d measurements with load cell system
checking daily the calibrations of the data acquisition and load cell measuring systems; measurement of local atmospheric pressure hourly, using a calibrated pressure transducer.
Accuracy of C d calculated from surface pressure distributions
Error propagation analysis was also used to estimate the error in overall pellet C d obtained from surface pressure distribution measurements. The measurement of the radial location of each of the 28 pressure tappings was subject to an estimated random error of 0.05 mm. As can be deduced from Fig. 7 , an error in the radial location of a shared tapping almost always This error is not straightforward to estimate, but it is believed that its contribution to the overall error in C d was small and was thus neglected.
Accuracy of C d calculated from surface pressure distributions
The propagation analysis gave an error estimate of 12.2 % in the value of C d obtained from surface pressure distribution for each of the four pellets numbered 2 to 5 in Fig. 2 . The measurement of pressure at each tapping had an uncertainly of 2.2 % and when this was accumulated over 28 tappings, it accounted for almost all of the overall error of 12.2 % in the value of C d . The rest of the error resulted from uncertainties in the measurement of pellet head cross sectional area, jet velocity, and jet density.
Results
This section presents mostly experimental results of pressure distributions around several pellets and drag coefficients for these pellets obtained using two independent approaches, the load cell system and pressure-area integration. The section also includes results of potential flow analysis in sub-section 3.3, which help in the interpretation of experimental results, in particular how the pellet nose shape affects pressure distribution on the front face of pellets.
Results
Because potential flow analysis cannot predict flow separation, this aspect is addressed mainly by the experimental results.
Pressure distribution and drag for spherical pellet and systems validation
The Table 1 , which also provides additional information about the small spheres.
Pressure distribution and drag for spherical pellet and systems validation
The value of C d was measured at a constant value of M a = 0.58 for all the spheres, while the value of Re for each sphere varied depending on the sphere diameter. For the smallest sphere of 3.08 mm diameter, the Re value was 37,500, while the Re for the largest sphere of 4.75 mm diameter was 57,000. Overall, good agreement was found between the measured values of C d for the nine spheres and those from published sources. Figure 9 shows the distribution of the pressure coefficient around the surface of the 4.5 mm spherical lead pellet (No. 1 in Fig. 2 ) that was measured as described in the previous section using the arrangement shown in Fig. 8 . The values of pressure coefficient shown in Fig. 9 were calculated using the pressures measured at the pressure tapping hole, when the hole was Aoki et al. [5] found that the minimum value of pressure coefficient for several smooth spheres occurred at~74 o .
Pressure distribution and drag for spherical pellet and systems validation
The pressure coefficient distribution of Fig. 9 for the 4.5 mm spherical lead pellet was integrated numerically using Eq. (8), as described above, which gave a value of C d = 0.539.
Pressure distribution and drag for spherical pellet and systems validation
The numerical integration was also carried out using Eq. (6) given by Eq. (9) 3.2 Pressure and drag force distributions around the dome-head pellet Figure 10 shows the measured distributions of pressure coefficient and drag force around the dome-head pellet. Pressure coefficients are shown for 27 surface area bands labelled "a" to "aa" (see Fig. 7 ) around the dome-head pellet at Re~54,000 and M a~0 .58. For information, the data of Fig. 10 have also been re-plotted in Fig. 11 but this time, following convention, the x-axis shows the axial distance (streamwise direction) from the pellet nose tip. Figure 10 shows that the flow was brought to rest at the stagnation point at the centre of the pellet nose and as a result, the pressure coefficient for band "a" rose to just over 1.0. Further away from the stagnation point, close to the pellet head rim (area bands "g" and "h"), the pressure coefficient was close to its lowest value of approximately -0.6.
Pressure distribution and drag for spherical pellet and systems validation
At the head rim, the flow velocity attained a high value and, as a result, the surface pressure reduced to a minimum (a more detailed discussion of the influence of nose shape on flow velocity follows in the next section). Between area bands "a" and "h", on the pellet front face, the pressure coefficient reduced progressively, as the flow velocity rose from zero at the centre of the nose to a high value at the pellet rim.
Pressure distribution and drag for spherical pellet and systems validation
Downstream of area band "h", the pressure coefficient remained approximately constant at approximately -0.6 for six downstream area bands "i" to "n". This suggests that the flow boundary layer had separated from the surface of the pellet, somewhere close to the head rim (between surface area bands "g" and "h"), and had become a free shear layer. As the free stream flow was not in contact anymore with the pellet surface, it had no significant influence on the surface pressure between bands "h" to "n". This was reflected in the pressure coefficient remaining approximately constant at -0.6.
Pressure distribution and drag for spherical pellet and systems validation
The free shear layer that left the pellet at the head rim formed an interface between the high velocity free stream and a low-pressure recirculation region near the pellet surface between bands "h" and "n". Entrainment of fluid from this recirculation region into the layer maintained this region at low pressure.
Pressure distribution and drag for spherical pellet and systems validation
Close to the pellet tail at band "n", the surface pressure begun to rise substantially. The pressure coefficient rose rapidly from approximately -0.6 at band "n" to -0.1 at band "r" on the pellet tail rim. This rise was caused by the shear layer impinging and re-attaching onto the pellet surface just downstream of area band "n". Figure 10 also shows the distribution of 27 local forces acting on the area bands "a" to "aa".
Pressure distribution and drag for spherical pellet and systems validation
All 27 forces shown in the figure were pressure forces on the bands and these forces were calculated and resolved axially as described previously. The sign convention for the forces, outlined in Fig. 10 , is as follows: a positive local force (e.g. on band "d" on the front face) was a drag force that retarded the pellet (i.e. contributed to the overall drag force on the pellet), while a negative force (e.g. on band "m" on the rear slope) accelerated the pellet and thus local drag force was based on measured local gauge pressure; +ve force ൙ĚƌĂŐŝŶƐƚƌĞĂŵǁ ŝƐĞĚŝƌĞĐƟŽŶ͖ -ve force ≡ force in counter-streamwise direction; 1 g force = 0.0098 N Note: the empty and filled data markers (o and •) correspond to area "bands" a, b, c, etc, not to the pressure tappings Concluding this section it can be said that: (a) the dome-head pellet drag is dominated by the drag on its front face and its base, which contribute~62 % and~38 % respectively of the overall pellet drag; and, (b) the detailed behaviour of the separated boundary layers can affect the drag contributions made by the two side-slopes and by the base of the pellet.
Relationship between pellet nose shape and surface pressure
The preceding section concluded that the majority of overall drag on the dome-head pellet emanates from its front face (~62 %). This section considers in more detail how the geometry of the front face influences the pressure distribution on the face and thereby pellet Because it is a simplified analysis, it will be seen from the following discussion that potential flow provides useful physical insights into how the shape of pellet nose can influence surface flow velocities and surface pressures and, thereby, pellet drag. Potential flow theory is therefore used here as an aid to understanding but not as a predictive tool of complex flow behaviour. Computational fluid dynamics would be a more appropriate predictive tool, as it does not include major simplifying assumptions such as inviscid and irrotational flow and it can incorporate compressibility.
Relationship between pellet nose shape and surface pressure
In the following discussion, a key relationship of potential flow theory is as shown in Eq. (10) [45]:
Relationship between pellet nose shape and surface pressure
where u and v are the components in the x and y directions of the surface flow velocity, V.
Relationship between pellet nose shape and surface pressure
Once the Cartesian velocity components u and v have been calculated from potential flow, Equation (10) implies that the surface velocity follows the body surface slope. This is true only in theory, as in real flows there is the presence of the boundary layer and surface no-slip condition which modify further the surface velocity. Furthermore, the flow may separate from the surface, in which case potential flow theory is no longer applicable. In physical terms, the pressures on the surfaces of the five shapes in Fig. 13(a) with the oval-nose shape, which retains considerable proportion of its horizontal velocity component, u. Because the oval-nose shape has a higher velocity, V, near its axis (y < 1.1 mm), it will experience a lower pressure than the flat-nose shape and therefore lower drag force near the axis. This is illustrated well in Fig. 14(a) , where the pressure coefficients were calculated using the surface velocity and the Bernoulli Eq. (12) . The pressure coefficient for the oval shape can be seen to be significantly lower for y < 1.1 mm. Returning to Fig. 13(b) and (c), consider now the surface velocity for the two shapes, further away from their axes, close to their rim (y > 1.75 mm). Figure 13 (b) and (c) show that the situation has been reversed, and that the surface velocity for the oval-nose shape is now smaller than that for the flat-nose shape. It can be seen from the figure that this is because the oval-nose shape, with its much smaller surface slope, accelerates the flow vertically much less than the flat-nose shape does (vertical velocity component, v, smaller in the case of oval shape for y > 1.75 mm). Because the oval-nose shape has a lower resultant velocity, V, near its rim (y > 1.75 mm), it will also experience a higher pressure near its rim than the flat-nose shape. This is also shown in Fig. 14(a) , where the pressure coefficient for Returning briefly to the five shapes of Fig. 13(a) , with real flows the likelihood of flow separation upstream of the nose rim would be significant in bodies having profiles such as those of the flat-nose shape (iii) and some of the other shapes, such as the circular-nose one
Relationship between pellet nose shape and surface pressure
(ii) and cavity-nose shape (iv).
Relationship between pellet nose shape and surface pressure
Setting aside for a moment flow separation, the pressure coefficient curves of Fig. 14(a) provide some useful physical insights of how nose geometry affects surface pressure. Figure   14 (a) shows that the flat nose body has a much lower theoretical value of pressure coefficient at the body rim (y = 2.25 mm) than the oval-nose shape. This low value of the coefficient for the flat-nose shape is due to greater curvature of the streamlines around the rim of the flatnose, when compared with the curvature for the oval-nose shape, as can be seen from Fig.   13 (a)(i) and (iii). Close to the rim, the flow has to turn over a greater angle in the case of the flat-nose shape. This requires substantially higher centripetal acceleration at the rim of the flat-nose shape in order to ensure radial equilibrium of the fluid elements as they flow past its rim. In turn, this higher centripetal acceleration needs to be provided by a high pressure Although not discussed in great detail, the physical principles discussed above apply also to the other shapes (ii), (iv) and (v) shown in Fig. 13(a) . shows an inflexion in the curve for the cavity-head pellet between area bands "d" and "f"
Comparison of pressure distribution and drag for four pellets
(reproduced in 3 repeated experiments). It is suggested that this was due to reversed flow leaving the cavity and detaching temporarily over the cavity lip, but subsequently reattaching a short distance downstream onto the sloped pellet surface. Next, considering the dome-head pellet, the pressure coefficient level very close to the pellet axis (bands "a" and "b", r = 0 to 0.64 mm) was high, due to the nearly flat face there. Away from the axis (beyond band "d")
Comparison of pressure distribution and drag for four pellets
toward the head rim, the coefficient declined rapidly.
Comparison of pressure distribution and drag for four pellets
Returning to Fig. 15 , it can be seen that for all four pellets, the flow separation occurred somewhere between bands "g" and "h". After flow separation, the value of the pressure coefficient for all four pellets remained almost constant for a long distance downstream, from band "h" all the way to band "n". Pressure recovery begun around band "n" due to impingement and re-attachment of the shear layer onto the rear slope of the pellet. Figure 15 suggests that the value of the pressure coefficient at the detachment point (between bands "g"
Comparison of pressure distribution and drag for four pellets
and "h") affected the downstream location where the flow re-attached onto the pellet tail surface. For example, Fig. 15 shows that in the case of the dome-head pellet, re-attachment occurred relatively early at area band "n", while for the cavity-head pellet, re-attachment occurred two area bands downstream at "p". Therefore, Fig. 15 suggests that re-attachment and pressure recovery occurred earlier when the value of the coefficient at the upstream detachment point was low. A physical explanation for this observation can be proposed with the help of Fig. 12 . The free shear layer, detaching from the pellet head rim, is subjected to a radial pressure gradient because the free stream is at atmospheric pressure, while the forward recirculation zone is at lower, sub-atmospheric pressure. This causes the shear layer to curve toward the pellet axis and move closer to the pellet surface. As a result, the impingement point onto the rear pellet slope is also further upstream. This is what appears with the domehead pellet, where the lower pressure in the recirculation zone intensified the radial pressure gradient on the shear layer, increasing its curvature and causing it to impinge earlier onto the pellet sloped tail, around band "n". In the case of the cavity-head pellet, for which Fig. 15 shows a higher pressure within the recirculation region, the converse is true and shear layer re-attachment occurred later, around band "p" (see Fig. 15 ).
Comparison of pressure distribution and drag for four pellets
Figures 12 and 15 show that flow detached again at the tails of the four pellets, between area bands "q" and "r". Close examination of Fig. 15 suggests a link between the pressure coefficient at the two flow detachment points, at the pellet head and tail rims. The value of the pressure coefficient at the head rim appears to relate to the value of the coefficient at the tail rim. For example, the dome-head pellet had the lowest value for the coefficient at bands "g" and "h", but at the tail band "r" its coefficient value was highest. Conversely, the cavityhead pellet had the highest coefficient value at bands "g" and "h" but the lowest value at band "r". There was therefore an inverse relationship between the values of the pressure coefficient at the two detachment points. This is confirmed by an almost perfect Pearson correlation coefficient of -0.976 between the pressure coefficients at the head bands "h" and the tail bands "r" (Pearson correlation coefficient values are quoted in the rest of this paper). The boundary layer detaching from the pellet tail (between bands "q" and "r") became a free shear layer, which engulfed a large, low pressure, recirculation region behind the pellet base.
Comparison of pressure distribution and drag for four pellets
As shown in Fig. 15 , the pressure within the recirculation region was sub-atmospheric, while that of the free stream was atmospheric. The free shear layer was therefore subjected to a radial pressure gradient, which caused it to curve toward the pellet axis and enclose the rear recirculation region, as shown in Fig. 12 . Figure 15 suggests that the pressure within this recirculation region had a positive relationship with the pressure at the tail detachment point.
Comparison of pressure distribution and drag for four pellets
For example, the dome-head pellet had the highest coefficient value at area bands "q" and "r" and the highest average coefficient value for area bands "s" to "aa". Similarly, the cavityhead pellet had the lowest coefficient value at area bands "q" and "r" and lowest average value for area bands "s" to "aa". In fact, for the four pellets, a high positive correlation coefficient of 0.924 was calculated between the averaged pressure coefficient at bands "q"
Comparison of pressure distribution and drag for four pellets
and "r" and the averaged coefficient at bands "s" to "aa". Equation (13) provides the local drag coefficient for any pellet surface area band, calculated using the local drag force and the pellet cross sectional area at the head rim. As mentioned previously, if the local drag forces and the local drag coefficients for the 27 surface area bands are summed up algebraically, the overall drag force and overall drag coefficient for a pellet as a whole are obtained. Table 2 was constructed using the local drag coefficients. For each pellet, Table 2 breaks down the pellet overall drag coefficient into four pellet regions: the front face, forward and rear slopes, and pellet base. It is more convenient to first consider the drag coefficient contributions of the forward and rear pellet slopes. For all four pellets, Table 2 shows that the drag force on the forward slope was offset almost entirely by the drag force on the rear slope. Quantitatively, the net contribution of the two slopes toward the overall drag coefficient for each pellet was -2 %, -4 %, -8 %, and -4 % for the dome-head, flat-head, cone-head and cavity-head pellets, respectively.
Comparison of pressure distribution and drag for four pellets
The net influence of the two side slopes on the overall drag was small due to several (interrelated) factors. Firstly, despite the rear slope having a greater projected area (normal to the pellet axis) than the forward slope, some of this larger area was rendered less important due to pressure recovery as a result of shear layer re-attachment close to the pellet tail (see Fig. 15 ). Secondly, a lower pressure on the rear slope did not automatically lead to a reduction in drag because the lower pressure also encouraged earlier pressure recovery, which reduced rear slope area on which the low pressure acted. Table 2 shows that the overall pellet drag coefficient was made up almost entirely of the drag forces on the face of the pellet and on its base, with the former contributing 61 -71 % of the pellet overall drag and the base contributing 32 -42 %, depending on individual pellet geometry. Examining the values of C d for the front face of the four pellets, it can be seen that the greatest value of C d was 0.50 for the flat-head pellet, followed by 0.44 for the cavityhead, 0.40 for the cone-head and 0.31 for the dome-head pellet. Table 2 shows also the variation in C d for the pellet bases. The greatest value of 0.27 was for the cavity-head, followed by 0.25 for the cone-head, 0.23 for the flat-head pellet and 0.20 for the dome-head pellet. These results correlate positively and highly with the pressure coefficient at the head rim; in fact there is a +0.99 correlation coefficient between the base C d for the four pellets and their pressure coefficients at the head rim separation point (i.e., average for bands "g" and "h"). However, only a fairly weak relationship was found between the value of C d for the front face and the C d for the pellet base (correlation coefficient 0.55). Table 2 Notes: (a) pellet serial number in Fig. 2; (b) all values of C d in this Table were calculated from pressure-area integration using information in Fig. 7 and the data plotted in Fig. 15 ; 
Comparison of pellet overall drag coefficient
The load cell system of Figs. 5 and 6 was used to measure the drag coefficient of the five pellets at Re~54,000 and M a~0 .58. The values obtained are shown below in Table 3 : with the data here, and they show that the cone-head pellet in the previous study [2] had a larger value of overall pellet C d than the flat-head pellet, by approximately 8 %. However, this reversed at higher values of M a greater than 0.65, where the flat-head pellet in the study [2] had a larger value of C d than the cone-head pellet. Recently, Salimipour et al. [4] reported an extensive analysis of aerodynamic and dynamic behaviour in flight of four 4.5 mm calibre pellets, with the flow analysis being carried out using compressible three-dimensional computational fluid dynamics. Of the four pellets, two had similar head shapes to the two of the pellets being discussed here. One pellet had a flat-head shape similar to pellet 3, and a second pellet had a cone-head shape similar to pellet 4. The analysis in Salimipour et al. 
Conclusions
A low turbulence jet with a nominal diameter of 17 mm proved to be a suitable facility for investigation at M a~0 .6 of drag forces on small spheres and air rifle pellets having diameters in the range of 3 to 4.5 mm. The small drag forces involved required careful calibration and system set-up to achieve satisfactory repeatability and accuracy of the measured values of drag coefficient.
Conclusions
At M a~0 .6, the behaviour of a small 4.5 mm spherical pellet proved to be very similar to widely published behaviour for spheres being a value of C d~0 .5, which was independent of Re in the range 37,000 to 57,000, and flow separation at azimuthal angle of~78 o .
Conclusions
The overall drag of four non-spherical pellets was dominated by the drag on their front face.
Conclusions
The face contributed approximately 65 % of the overall pellet drag, while base drag contributed almost all of the remaining 35 %. The net drag contribution of the two side slopes of the pellets was close to zero.
Conclusions
The theoretical analysis of 2-D nose shapes and the measured drag distributions for five pellets showed that a streamlined pellet nose could, in theory, reduce the drag generated on the front face. However, this effect would mostly be limited to a central region of the front face, close to the pellet axis. The results of the theoretical potential flow analysis can only be indicative, as they assume incompressible and inviscid flow and do not allow the prediction of flow separation. In real pellets, flow separation near the rim of the front face can substantially affect the drag on the pellet face and also the base drag, as the pellet base is engulfed in the wake formed as a result of flow separation.
Conclusions
The geometry of the front face has a weak influence on the drag acting on the pellet side slopes and its base. This influence was exercised through the behaviour of the free shear layer separating from the pellet head rim. However, only a weak relationship was found between the value of C d for the front face and the value of C d for the pellet base (correlation coefficient 0.55).
Conclusions
Detailed theoretical and experimental studies suggested that for low pellet face drag, the pellet nose shape should be such as to minimise rotation of the approaching free stream away from the pellet axis. That is, it is necessary to minimise flow deceleration axially and also minimise flow acceleration transversely. Therefore, pellet nose shapes with spherical, ogive, or ellipsoidal nose geometries are likely to have smaller face drag than flat-head or cavityhead pellets.
Conclusions
The dome-head pellet was found to have 17 % lower overall C d than the spherical pellet, likely due to the much lower base drag in the case of the dome-head pellet, which more than compensated for higher dome-head face drag. It was apparent that the presence of the tail in the dome-head pellet enabled flow reattachment and a rise in base pressure, thus reducing its base drag. In contrast, at Re~54,000, flow reattachment on the rear surface of the spherical pellet was not possible, and it is likely that because of this reason its base drag was higher than that for the dome-head pellet.
Conclusions
The flat, cone, and cavity-head pellets had higher overall C d values than the spherical pellet, by 10 %, 31 % and 40 %, respectively. It was shown that this was due to their having substantially higher face drag, which was not compensated sufficiently by their lower base drag.
Conclusions
When the value of C d for a pellet was measured with the load cell system, the repeatability of the value was 1.3 % and the error in the value of C d was estimated by means of uncertainty analysis to be 4.2 %. On the other hand, when the value of C d was calculated from pressure distribution around the pellet, the error estimated by means of uncertainty analysis was considerably greenanthony@example.org %.
Declaration of conflicting interest
The author declared no potential conflicts of interest with respect of the research, authorship, and/or publication of this article.
Funding
This research received no specific financial support from funding agencies in the public, commercial or not-for-profit sectors. 
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{'paper_id': '131768755', 'title': 'Influence of air rifle pellet geometryon aerodynamic drag', 's2_url': 'https://api.semanticscholar.org/CorpusID:131768755', 'source': 'S2ORC'}