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A new measure of roughness for defining the aerodynamic performance of sports balls S J Haake, S R Goodwill and M J Carre Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2007 221: 789 DOI: 10.1243/0954406JMES414 The online version of this article can be found at: http://pic.sagepub.com/content/221/7/789

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A new measure of roughness for deﬁning the aerodynamic performance of sports balls S J Haake1∗ , S R Goodwill1 , and M J Carre2 1 The Centre for Sport and Exercise Science, Shefﬁeld Hallam University, Shefﬁeld, UK 2 Department of Mechanical Engineering, The University of Shefﬁeld, Shefﬁeld, UK The manuscript was received on 16 June 2006 and was accepted after revision for publication on 27 March 2007. DOI: 10.1243/0954406JMES414

Abstract: A new analysis is presented of the major ﬁndings in sports ball aerodynamics over the last 20 years, leading to a new method for deﬁning surface roughness and its effects on the aerodynamic performance of sports balls. It was shown that the performance of balls in soccer, tennis, and golf are characterized by the position of the separation points on the surface of the ball, and that these are directly inﬂuenced by the roughness of the surface at a given Reynolds number and spin rate. The traditional measure of roughness k/D (the ratio of surface asperity dimension to diameter) was unable to predict the transition from laminar to turbulent ﬂow for different sports balls. However, statistical measures of roughness commonly used in tribology were found to correlate well with the Reynolds number at transition and the minimum Cd after transition. It was concluded that this new measure and a further one of dimension should allow the complete characterization of the aerodynamic performance of sports balls. The effects of surface roughness on spin rate decay were also considered, and it was found that tennis balls had spin decay over six times that of golf balls and was due to the increased skin friction of the nap.

1

INTRODUCTION

People’s love of sport is reﬂected in the current wave of research being carried out on sports equipment and technology. The idea that sport should be a legitimate area of research is not new and attempts to explain the deviations of sports balls in ﬂight were made by such illustrious scientists as Newton, Rayleigh, and Thompson [1–3]. In more recent years, impending tournaments such as Wimbledon, the soccer world cup, and the Olympics have provoked a rash of popular articles on the interaction between aerodynamics and sport. In particular, ball sports receive much attention from both researchers and the media alike.

∗ Corresponding author: The Centre for Sport and Exercise Science,

The Faculty of Health and Wellbeing, Shefﬁeld Hallan University, Collegiate Hall, Collegiate Crescent, Shefﬁeld S10 2BP, UK. email: [email protected]

JMES414 © IMechE 2007

It was not until 1985, however, that Mehta carried out the ﬁrst review of sports ball aerodynamics (in the Annual Review of Fluid Mechanics) by covering the sports of cricket, golf, and baseball. Mehta pulled together both his and other researchers’ work in a succinct way and showed that the drag on a ball was dominated by the size and deﬂection of the wake. The drag was inﬂuenced by the position of separation of the boundary layer from the ball’s surface which was, in turn, affected by the condition of the ﬂow within the boundary layer. Transition of the boundary layer from laminar to turbulent ﬂow was seen to be affected by a number of factors, such as Reynolds number, surface roughness, and spin rate. Since 1985, research in this area has continued apace, particularly in the sports of golf, tennis and soccer. It is the intention of this paper to outline the more important pieces of work in sports ball aerodynamics that have emerged in the last 20 years, and to use them to formulate a new methodology for determining the effect of roughness on performance. Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

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2 THE FUNDAMENTAL MECHANICS OF SPORTS BALL AERODYNAMICS The drag and lift forces on a rotating sphere of crosssectional area A, travelling at velocity U∞ through a ﬂuid of density ρ are given as 1 2 ρACD U∞ 2 1 2 FL = ρACL U∞ 2

FD =

Figure 1(a) shows the classical results found by Achenbach, which have been used in almost all studies of sports balls ever since. It can be seen that a drag coefﬁcient CD of approximately 0.52 exists

(1) (2)

where CD and CL are the drag and lift coefﬁcients, respectively. Both CD and CL are expected to be functions of the Reynolds number Re and the spin parameter α (deﬁned as the ratio of the peripheral velocity to the translational velocity) given by ρU∞ D U∞ D = µ ν ωD α= 2U∞

Re =

(3) (4)

where ν is the kinematic viscosity of the air, α the spin parameter, and D and ω are the diameter and spin of the sphere, respectively. Most researchers in sports ball aerodynamics start with the work carried out by Achenbach in the 1970s which was actually nothing to do with sport at all. Achenbach [4] was interested with the transportation of spherical fuel elements in high-pressure pipes and carried out a number of carefully controlled experiments on a polished metal sphere in a high-pressure wind tunnel. Table 1 shows that the turbulence of the wind tunnel used was less than half a per cent and that the blockage ratio (the ratio of the areas of the model to the wind tunnel section) was one per cent.

Table 1

Fig. 1

(a) Drag coefﬁcient as a function of Reynolds number for a smooth metal sphere and (b) schematics and graphs showing the position of the boundary layer separation and transition points for smooth spheres with respect to stagnation point [5]

Experimental details for wind tunnel tests on roughened spheres and sports balls (N/A = not applicable)

Sport Reynolds number Spin parameter, α Turbulence level Diameter of sphere Ratio of diameter of sting to sphere Blockage (ratio of area of model to tunnel) CD of smooth sphere (subcritical region)

[4, 5]

[6]

[7]

[8]

[9, 10]

[11, 12]

[13]

General 40 000–6 000 000 N/A 0.45% 200 mm

Golf 40 000–260 000 0–0.3 0.2% 103 mm

Golf 40 000–250 000 0–1.4 0.4–0.8% 42.6 mm

Tennis 80 000–300 000 N/A 88 500 that the values collapse onto a single line. Bearman and Harvey, on the other hand only showed the independence of CL for Re > 126 000 and ignored data away from this line. It can be seen that the CL for Smits and Smith are higher than those of Bearman and Harvey, which may be due to a different dimple pattern analysed or to a systematic shift due to the side supports that was accounted for in their calculation of CD but not in CL . Figure 4(c) shows lift and drag coefﬁcients versus spin rates for two values of Reynolds number. It can be seen that for Re = 39 805 CL tends towards negative

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Fig. 4

(a) Drag coefﬁcient versus Reynold number; (b) lift coefﬁcient versus spin parameter for different Reynolds numbers (×10−3 ); (c) lift and drag coefﬁcients bar different Reynolds numbers (×10−3 ); (d) spin rate decay versus spin parameter [7]

values at a spin rate of 0.4, and that this coincides with a turning point in CD . The authors considered that this was caused by the boundary layer on one side of the ball remaining laminar whereas the other was turbulent. The laminar side separated early whereas the turbulent side separated late such that the wake was deﬂected in the opposite direction to scenarios at much higher spin parameters [23]. Smits and Smith [7] also published for the ﬁrst time signiﬁcant amount of data on the spin rate decay of golf balls. Lieberman [17] had previously indicated that spin rate decayed to two-thirds to three-quarters of its initial value over its ﬂight and was exponential in nature. Smits and Smith found that ω˙ ∝ U∞ ω so that for constant velocity the spin decayed exponentially with time. This implied 2 ∝ ωR 2 /U∞ = αR and a non-dimensional that ωR ˙ 2 /U∞ 2 which increases linearly with spin parameter ωR ˙ 2 /U∞ rate for a given radius (Fig. 4(d)). This feature was subsequently incorporated by Smits and Smith into their trajectory models to simulate the ﬂight of golf balls.

Tavares et al. [24] used radar measurements to determine the spin rate decay of golf balls with different moments of inertia and different dimple patterns. They showed that spin rate decay was increased with deeper dimples and with lower moments of inertia, but was only signiﬁcant for spin rate greater than about 0.5. Other researchers [9, 10] studied the effect of the number of dimples on a golf ball by manufacturing balls with different numbers and geometries of dimples. Table 2 shows that Aoki et al. [9, 23] increased Table 2

Speciﬁcations for dimple shapes used by Aoki et al. [9, 10]

No. of dimples

Distance between dimples (mm)

Width of dimples (mm)

Depth of dimples (mm)

Dimple depth to width ratio

104 184 328 328 504

3.897 2.043 0.650 0.650 0.297

3.528 3.528 3.528 3.528 3.046

0.338 0.338 0.338 0.646 0.292

0.096 0.096 0.096 0.183 0.096

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A new measure of roughness for deﬁning the aerodynamic performance

the number of dimples from 104 to 504 while keeping the width to depth ratio constant. They also looked at the effect of doubling the dimple depth to width ratio with 328 dimples. Table 1 shows that their measured value for the drag coefﬁcient on a smooth sphere was low. It can be seen from Fig. 5(a), however, that their values for CD of a golf ball with 328 dimples (a typical number for a golf ball) compares very well with Bearman and Harvey [6]. It can be seen in Fig. 5(a) that increasing the number of dimples decreases the critical Reynolds number and increases the minimum CD in the supercritical region. Figure 5(b) shows that deepening the dimples causes a further decrease in the critical Reynolds number while retaining the low minimum CD . Deepening the dimples, however, causes a rise in CD with Re, probably caused by a thickening of the turbulent

Fig. 5

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boundary layer and earlier separation. Figure 5(b) also shows that increasing the number of similar dimples from 328 to 504 does not effect the critical Re, but increases the minimum CD by a small amount (about 0.02). What can be concluded is that transition occurs earlier if the number of dimples is increased, or if the dimples are made deeper. Thus, increasing the number of dimples while decreasing their depth appears to more or less cancel each other out. A slight increase in number of dimples from 328 to 504 with a decrease in depth from 0.34 to 0.29 mm more or less cancel each other out. Achenbach [5] and all subsequent authors have used the parameter k/D to characterize roughness. In this case, however, the balls with 104, 184, and 328 dimples would all have the same

(a) Drag coefﬁcient versus Reynolds number for golf balls with different number of dimples of the same depth; (b) drag coefﬁcient versus Reynolds number for golf balls with dimples of different depth; (c) drag and lift coefﬁcient versus spin parameter for golf balls with 104 and 184 dimples of the same depth; (d) drag and lift coefﬁcient versus spin parameter for a golf ball with 328 dimples of different depth (all data from Aoki et al. [9, 10, 23])

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Fig. 6

Comparison of (a) drag coefﬁcient and (b) lift coefﬁcient with spin parameter for different researchers

roughness parameter (k/D = 0.008) but very different performance. Equally, the balls with 328 and 504 dimples would have different roughnesses but would perform almost identically. This issue of an adequate deﬁnition of roughness will be discussed in a later section. The effect of increasing the number of dimples on a spinning golf ball in the supercritical region is shown in Fig. 5(c). It can be seen that both the drag and lift coefﬁcients increase with spin parameter and are thus very similar to the trends of Bearman and Harvey. Increasing the number of dimples from 104 to 184 increases the drag by approximately 0.05, and the lift by about 0.07 at α = 0.15. This implies that the extra roughness associated with the more densely packed dimples not only encourages the asymmetry of the wake but also thickens the boundary layer. This induces early separation and produces a larger drag coefﬁcient. Figure 5(d) compares the golf balls with 328 dimples but different depths. It shows that increasing the depth of the dimple both increases the drag and decreases the lift coefﬁcient, implying that too much roughness associated with dimple depth causes early separation of the boundary layer. Figures 6(a) and (b) summarize the drag and lift coefﬁcients found by all the researchers discussed in this section. All researchers agree that increasing the spin parameter increases both the lift and drag coefﬁcients. The data shows that the actual magnitudes depend upon the number of dimples, the spin parameter, Reynolds number, and the critical transition. It also depends, to some extent, on the validity of the experiment performed.

method of Davies [15] to drop tennis balls across the working section of a wind tunnel in order to determine lift and drag coefﬁcients up to spin parameters of approximately 0.8 (Fig. 7). He found that the drag coefﬁcient of a non-spinning ball was approximately 0.51 and that it increased to about 0.76 with increasing spin parameter, whereas the lift coefﬁcient increased with spin rate to a maximum of about 0.31. Stepanek found, therefore, that the drag coefﬁcient for a non-spinning tennis ball was equivalent to that for a smooth ball but increased rapidly when the ball was spinning. Stepanek found that unlike golf balls, the lift and drag coefﬁcients were independent of Reynolds number, and did not ﬁnd any evidence of critical transition. Further work on tennis balls was carried out by Chadwick in the late 1990s [26–29] to try to quantify the effect of surface condition and ball size on lift

3.2 Tennis Stepanek [25] produced the ﬁrst signiﬁcant piece of work on tennis ball aerodynamics. He used the

Fig. 7

Drag and lift coefﬁcient versus spin parameter for spinning tennis balls [26]

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and drag. Chadwick found that the drag coefﬁcient was higher for a tennis ball than that for the smooth or a rough spheres analysed by Achenbach and that, like Stepanek, CD was more or less independent of Reynolds number. Furthermore, they found that the state of the covering inﬂuenced the drag coefﬁcient with a raised nap giving a value about 10 per cent higher than a shaved nap. Most interestingly, it was seen that transition did not occur with a tennis ball and that separation almost always occurred at 80◦ (with respect to the stagnation point). They concluded that either the boundary layer ﬂowing over a tennis ball was in the supercritical or transcritical regime, or that the nap acted merely as a trip regardless of the condition of the boundary layer. It was also noted that the nap tended to lie ﬂat at high-wind speeds for a non-spinning ball and became raised at high-spin parameters. Mehta and Pallis [30] responded by carrying out wind tunnel tests on a small selection of tennis balls and subsequently suggested that the boundary layer was indeed in the transcritical regime, with drag coefﬁcients of around 0.6–0.65. They conﬁrmed the work of Chadwick and found that the height of the nap affected the drag coefﬁcient. They postulated that the increase in drag over and above the transcritical value for a sphere was due to the nap ﬁbres acting as a series of bent cylinder-like ﬁlaments, which was signiﬁcant at low Re and lessened at higher Re. To look at this hypothesis in more detail, Goodwill et al. [13] carried out a detailed survey of the aerodynamic performance of tennis balls. Tennis balls were supported in the centre of the test section of a highspeed wind tunnel using two horizontal stings 3.5 mm in diameter giving a ratio of support to ball diameter of around 5 per cent. The stings allowed the ball to be rotated up to 2750 r/min (289 rad s−1 ) with the simultaneous measurement of lift force. Ten brands of ball were mounted in the wind tunnel and the variability between balls of the same brand and between balls of different brands was studied. Figure 8(a) shows the drag coefﬁcient of four of the tennis balls studied by Goodwill et al. to show the range between brands. It can be seen that the drag coefﬁcient drops initially from around 0.65 to about 0.60 at Re ≈ 150 000 before increasing marginally again. The range between the slowest and fastest ball brands is about 0.05 whereas the range for different balls of the same brand was found to be ±0.02. It was concluded, therefore, that the balls could not be distinguished between each other within errors and that all new tennis balls play roughly the same. Mehta and Pallis [30] and Chadwick [28] explained the initial drop in CD by the laying down of the ﬁbres of the felt. Individual ﬁbres have a high drag coefﬁcient at low Re which contributes to the overall drag coefﬁcient. Once the ﬁbres have laid ﬂat, then a further JMES414 © IMechE 2007

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increase in Re causes increased thickening of the turbulent boundary layer which causes the separation points to move upstream. If the ball is in the transcritical regime, as advocated by Mehta and Pallis then the drag coefﬁcient is at least 0.2 higher than the largest transcritical value found by Achenbach [5] of 0.4. This implies that the extra drag due to the felt contributes approximately 30 per cent towards the total drag coefﬁcient. A second experiment was carried out in which tennis balls were artiﬁcially worn in a wear rig to simulate play under controlled conditions. Figure 8(b) shows that worn tennis balls act in a similar manner to new tennis balls, and that the most worn ball (1500 impacts) has a drag coefﬁcient that is approximately 0.04 lower than a new ball. Thus, a worn ball in which the felt has been abraded and removed has a 6 per cent lower drag coefﬁcient. Goodwill et al. showed that a difference of this magnitude would cause a typical serve with the same initial conditions to travel 40 cm further horizontally than a standard ball at the receiver’s baseline. Also shown in Figure 8(b) is data from Mehta and Pallis [30] who analysed a used tournament ball. The plot appears to indicate transition with a critical Reynolds number of about 100 000 and a minimum drag coefﬁcient of about 0.46. Tennis balls are changed during tournaments after either 7 or 9 sets and it can be estimated that a single ball may only be hit around 30 times before it is changed. It is surprising, therefore, that the tournament ball exhibits transition when the balls subjected to over 50 times normal wear do not. Without observing the surface of the tournament ball ﬁrst hand, it is difﬁcult to hypothesize the reasons for such different behaviour in comparison to all the other tennis balls tested. Figures 8(c) and (d) show the lift and drag coefﬁcient for spinning tennis balls at two different Reynolds numbers equivalent to a ball speed of 25 and 50 ms−1 for a number of different ball brands. It can be seen that the drag coefﬁcients at α = 0 compare very well with Fig. 8(a) with the drag value at Re = 105 000 being higher than that at Re = 210 000. The range of CD values between brands is similar to that of the non-spinning balls in Fig. 8(a). It can be seen that CD increases with spin parameter by about 6 per cent over the range studied and that the results are much higher than those measured by Stepanek [25] and Chadwick [28], particularly for low spin rates. The values for nonspinning balls, however, do match the new balls tested by Mehta and Pallis [30]. Chadwick attempted to take into account the fact that the ﬁbres of the felt stand on end due to centrifugal forces as the ball spins and used an effective ball diameter of 73 mm. Using a diameter of 65 mm would have increased the lift coefﬁcient to coincide with the experimental data of Goodwill et al. The lift coefﬁcients increase with the spin parameter at both Reynolds numbers (Figs 8(c) and (d)). It is Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

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Fig. 8

(a) Drag coefﬁcient versus Re for new tennis balls; (b) drag coefﬁcient versus Re for worn tennis balls; (c) drag and lift coefﬁcients versus spin parameter for new spinning tennis balls at Re = 105 000; (d) drag and lift coefﬁcients versus spin parameter for new spinning tennis balls at Re = 210 000; (e) drag and lift coefﬁcients versus spin parameter for worn spinning tennis spinning tennis balls at Re = 105 000; (f) drag and lift coefﬁcients versus spin parameter for worn spinning tennis balls at Re = 210 000 (additional data from Goodwill et al. [13])

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A new measure of roughness for deﬁning the aerodynamic performance

interesting that the lift coefﬁcient appears to show a dip at Re = 210 000 between α = 0.05 and α = 0.15 and that the drag coefﬁcient does not increase quite as much as at the lower Re in this range either. Similarities are seen in Figs 8(e) and (f ) for the worn balls. It can be seen that the new ball (zero impacts) follows the same lines as in Figs 8(c) and (d), i.e. there is a dip in the CL data at the higher Re. There is an even more marked dip in the data for the worn balls (particularly at 1500 impacts) which exists in all the lift and drag curves in Figs 8(e) and (f). These characteristics can be explained if the nature of the felt on a spinning tennis ball is discussed. Chadwick [28] showed that the height of the nap on the leading surface of a non-spinning ball decreased with wind speed from a natural length of 5 mm to around 2 mm at Re ≈ 150 000. The nap then compressed to around 1 mm as the speed was increased further to Re ≈ 300 000. Chadwick also showed that the nap of a non-spinning ball extended to over 8 mm in length as the balls was spun up to 290 rad s−1 . These features of the nap can be used to explain the different scenarios in Fig. 8, as follows. 3.2.1

Non-spinning ball (Figs 8(a) and (b))

As the Reynolds number increases to around 150 000, the height of the nap at the front of the ball decreases and is laid ﬂat around the sides of the ball. Thus, the ball changes from a ball with high nap to one with low nap and the skin friction decreases because of the reduced ﬁbre height. Once the ﬁbres are ﬂat, the boundary layer is probably in the transcritical regime and any further increase in the Reynolds number causes the separation points to move upstream, giving a further increase in the drag coefﬁcient. 3.2.2

Unused spinning balls at low speed (Fig. 8(c))

When a ball is spun, the nap height depends upon the spin rate and the Reynolds number. At low Reynolds number, the nap of an unused tennis ball remains at its original height of around 5 mm and any spin increases its height all around the sphere. The extra surface friction causes the drag coefﬁcient to increase as the spin parameter increases and the separation points move equally on each side to give a steady rise in the lift coefﬁcient. 3.2.3

New spinning balls at high speed (Fig. 8(d))

At high Reynolds number, the nap of a new tennis ball depresses to around 2 mm at the front of the ball and lays ﬂat on its upper and lower surfaces. Low spin rates do not raise the height of the nap much and therefore the skin friction is quite low. This means that for spins up to α ≈ 0.15, ﬂow is less dominated by skin friction. It would be expected that the boundary layer JMES414 © IMechE 2007

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of the advancing side of the spinning ball would be transcritical and that an increase in the spin would not affect the position of the separation point. On the retreating side, however, the boundary layer might be supercritical and the separation point would retreat downstream. This would have the effect of decreasing the drag coefﬁcient and giving a slightly smaller lift coefﬁcient than expected. 3.2.4 Worn spinning balls at low speed (Fig. 8(e)) As the nap is worn away (1500 impacts), the importance of skin friction decreases. The advancing side of the ball remains transcritical whereas the retreating side of the ball becomes supercritical. The latter allows the separation point on that side of the ball to move downstream giving a lower CD and CL than for a new ball. The effect starts to disappear as the spin parameter increases when the nap is raised by the centrifugal forces of the spinning ball. 3.2.5 Worn spinning balls at high speed (Fig. 8(f )) At high speed, the nap is depressed for both the new and worn balls. The results show that the balls display the dip in CD and CL for α < 0.15 seen in the previous graphs, and explained by the supercritical nature of the retreating side of the ball. For α > 0.15, the nap stands up and the friction forces increase which overwhelms any tendency of the boundary layer to be supercritical and gives a higher lift and drag coefﬁcient for the new ball than the worn ball. In conclusion, therefore, the aerodynamics of a tennis ball is dominated by the condition of the felt. Raising or lowering the nap, either naturally during play or through artiﬁcial wear causes the drag coefﬁcient to increase or decrease. The height of the nap is also affected by the speed of the ball through the air and its spin, and it is likely that the boundary layer can be in the supercritical or transcritical regimes, depending upon the combination of speed, spin, and wear. 3.3

Soccer

Soccer balls have only recently been analysed aerodynamically, probably because the balls require a much larger wind tunnel if the experiments are not to suffer from blockage. Carre et al. [11] overcame this problem by manufacturing a one-third scale model of football, complete with seams and with the roughness parameter k/D scaled accordingly. A further analysis was carried out with a mini football. Figure 9(a) shows the variation of drag coefﬁcient with Reynolds number for the football model in comparison to a smooth sphere and a golf ball. It can be seen that transition for a football lies somewhere between the smooth Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

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Fig. 9

Aerodynamic coefﬁcients for soccer balls; (a) drag coefﬁcient versus Re for a non-spinning football model (closed symbols) and a mini-football (open symbols); (b) drag coefﬁcient versus spin parameter at different Re for a model football; (c) lift coefﬁcient versus spin parameter at different Re for a model football; (d) simulated ratio of side to drag force (CL /CD ) for an 18.4 ms−1 free kick [11, 12]

sphere and a golf ball, suggesting that the seams of a football are effective at tripping the boundary layer, but are not as efﬁcient as dimples. It is possible that the drag coefﬁcient increases faster for footballs in the supercritical and transcritical region than for golf balls due to the relatively large areas of ‘smooth’ panel between seams causing earlier separation. Both the football model and the mini-football lie approximately on the same line, although the mini-football may just be undergoing transition at Re ≈ 110 000. Figures 9(b) and (c) show the variation of drag and lift coefﬁcient with spin parameter for the model ball only. The drag coefﬁcients for a non-spinning ball at α = 0 in Fig. 9(b) match the values seen in

Fig. 9(a). It can be seen that for Reynolds numbers well above transition, the drag coefﬁcient is independent of Reynolds number. This implies that the separation points remain roughly in the same position as more spin is placed on the ball and that extra spin does not cause the boundary layer to thicken very much. For Reynolds numbers at or below transition (i.e. data at Re = 85 000 and 110 000), it can be seen that the drag coefﬁcient decreases with spin parameter before levelling off to a more or less constant value. This is explained if the lift coefﬁcients are considered in Fig. 9(c). It can be seen that the lift coefﬁcient increases with spin parameter and tends to lie along the same line when above transition,

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A new measure of roughness for deﬁning the aerodynamic performance

much like the golf ball data in Fig. 4(b). At Reynolds numbers approaching transition (Re = 110 000) the lift coefﬁcient is lower and below transition negative lift coefﬁcients are found. This is due to the boundary layer on the retreating side of the spinning ball remaining laminar whereas the advancing side is turbulent. The separation point on the laminar side advances upstream causing the wake to deﬂect in the opposite direction to that when both sides of the ball have turbulent boundary layers. The reversal of the lift coefﬁcient at α ≈ 0.26 in Fig. 9(c) for Re = 85 000 coincides with the turning point in the drag coefﬁcient in Fig. 9(b). Again, this agrees with the work in golf ball carried out by Smits and Smith [7] shown in Fig. 4(c). Carre et al. used these results to analyse several free kicks that were notable for their curvature in the air, and their ability to deceive goal keepers. If the spin is assumed to remain constant throughout the trajectory, then the spin parameter actually rises as the ball slows, also causing the lift coefﬁcient to rise. Figure 9(d) shows the ratio of side to drag force (effectively CL /CD ) for a free kick at 18.4 ms−1 with different amounts of spin, simulated to land in the top corner of the goal from 18 m if 64 rad s−1 of sidespin is applied. In this case the ‘lift’ force has been applied as a side force since the spin axis is in the vertical direction and the resultant force in the horizontal direction. The ball slows from Re ≈ 270 000 on launch to Re ≈ 190 000 at the goal, before increasing again at the end of the shot due to the effects of gravity. It can be seen that the ratio of side to drag force rises signiﬁcantly at the end of ﬂight as the ball nears the goal and could be enough to signiﬁcantly fool a goalkeeper. The ratio of side to drag force drops slightly over the full trajectory at low

Fig. 10

801

spins, and rises at high spins. There appears to be an optimum value of spin between 48 and 64 rad s−1 where the ratio remains approximately constant and still rises signiﬁcantly at the end. Above this value of spin, the ratio rises continuously over the trajectory and shows a smaller rise at the end. 3.4

Spin decay

Lieberman [17] ﬁrst quoted rates of decay for golf balls in ﬂight to be from a quarter to two-thirds for trajectories lasting between 6 and 8 s. Smits and Smith [7] suggested that spin rate decay is exponential with time for a constant velocity such that dω = −cU∞ ω dt

(5)

where c is a constant. The non-dimensional parame2 is proportional to αR. ter ωR ˙ 2 /U∞ 2 for a golf ball is Figure 4(d) shows that ωR ˙ 2 /U∞ indeed linear with the spin parameter and was found to be applicable for all Reynolds numbers. This gave them all the information they needed to describe a golf ball trajectory, i.e. the variation of the lift and drag coefﬁcients with velocity and spin, and the decay of the spin over the trajectory. Tarnowksi [8] investigated the spin decay of tennis balls by projecting them in a sports hall at approximately 28 and 32 ms−1 and with spins between −48 and +427 rad s−1 . The angle was altered so that the ball landed in approximately the same position at the other end of a tennis court. Two high-speed videos were used to measure the speed, spin, and angle on launch and on landing. Figure 10(a) shows that the

(a) Landing spin rate versus launch spin rate for tennis balls projected at 28 ms−1 for a given horizontal distance; (b) a comparison of spin rate decay parameter between tennis and golf balls

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ﬁnal spin was proportional to the initial spin and that approximately 14 per cent of spin was lost during the trajectory, regardless of initial spin rate or velocity. It appears, therefore, that the spin decay for tennis balls acts in the same way as that for golf balls found by Smits and Smith. Although the time of ﬂight was not recorded in this experiment, it is possible to predict the time of ﬂight using a trajectory programme [28], which allows an indication of the constant c in equation (4) to be 2 versus spin rate found. Figure 10(b) shows ωR ˙ 2 /U∞ for tennis balls compared to data for golf balls. It can 2 for tennis balls is approximately be seen that ωR ˙ 2 /U∞ ten times that of golf balls. Given that the radius of a tennis ball is approximately 1.55 times that of a golf ball, then the data shows that the spin decay ω˙ for a tennis ball (or the coefﬁcient of moment as deﬁned by Tavares et al. [24]) is approximately 6.5 times that of a golf ball at the same spin rate and velocity. This is probably due to the individual ﬁbres in the felt penetrating through the boundary layer and into the ﬂow of the free stream. Achenbach [4] showed that the proportion of the CD due to friction for a non-spinning smooth ball was approximately 5 per cent of the transcritical value. The work here implies that the proportion of CD due to friction for a spinning tennis ball could be at least 20 per cent of the total (i.e. 0.13). This agrees with the work of Chadwick and Haake [27], Chadwick [28], and Mehta and Pallis [30] who showed that the drag coefﬁcient could be altered by at least 10 per cent by artiﬁcially varying the length of the nap, and with the earlier conclusion that the extra friction of the nap contributes about 30 per cent to the drag coefﬁcient.

Fig. 11

It would be interesting to carry out a similar experiment to determine the spin rate decay of soccer balls, with the expectation that they would behave more like golf balls than tennis balls because of their relatively smooth surface.

4

MEASURES OF ROUGHNESS

Figures 2(c) and (d) showed that there was a high correlation between the critical Reynolds number, the minimum drag coefﬁcient at transition, and the roughness parameter k/D for spheres covered in glass beads. It was found that a larger roughness k/D caused transition to occur at low Re but with relatively high supercritical values of CD . Figures 11(a) and (b) show similar graphs to those in Fig. 2 with the inclusion of soccer and golf balls. For the soccer ball model, the depth of the seams was measured from the original CAD drawing and the value k taken as the maximum depth of seam found. The dimensions of the golf balls were given by Aoki et al. in their original paper. It can be seen that the golf balls do not ﬁt well to the lines through the rough sphere data, primarily because three of the golf balls had the same dimple depth even though they had a different number of dimples and showed clear aerodynamic differences. The soccer ball is even further away from the best ﬁt line and it is evident that the parameter k/D is not enough on its own to give an indication of the aerodynamic performance of sports balls. An alternative method is to look at statistical techniques used by tribologists to characterize surfaces

A comparison of aerodynamic performance of sports balls of different roughness k/D; (a) critical Reynolds number at transition; (b) minimum CD value at transition. The lines show the best ﬁt to the data of Achenbach [5] shown in Fig. 2

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[31]. The characterization of the roughness of a surface requires that the proﬁle be measured in some manner and digitized such that the surface height zi is known at distance xi from a given datum, and is given by zi = Z (xi )

at

xi = x1 + (i − 1) × x

Table 3 indicates the values of the critical Reynolds numbers, the minimum CD values at transition, the k/D values, and the statistical measures of roughness calculated using equations (6) to (9) for the spheres studied in this paper. Figure 12 shows schematics through the various spheres (with the surface proﬁles exaggerated) and their calculated skewness values. The CAD drawing of the model soccer ball was analysed through three orthogonal axes and it was found that the ball was dominated by ﬂat peaks with an average skewness of −2.5. The golf balls of Aoki et al. range from a ball dominated by peaks for 104 dimples (skewness = −0.70) to a ball dominated by valleys for 504 dimples. Finally, the skewness of one of Achenbach’s rough spheres was analysed for k/D = 1.25 per cent with 2.5 mm dimples on a 200 mm diameter sphere. It was assumed that the glass beads used in his experiment were adhered so that they were one layer thick and were packed so that there were no gaps between them. It was thought that these assumptions could not be applied to the other spheres in Achenbach’s experiment since smaller beads would be likely to have more than one layer, making a geometric analysis difﬁcult. It was found that Achenbach’s roughest sphere had a skewness value of approximately −0.5 and was thus dominated by peaks. Figure 13(a) shows the critical Re plotted against skewness for all the spheres analysed. It can be seen that there is a strong relationship between Re and skewness for all the balls and that roughness dominated by valleys produces a lower critical Reynolds number than roughness dominated by peaks.

(6)

A mean can be determined across the length examined such that n zi = 0 (7) i=1

when the heights of n points are measured. The following parameters can then be used to characterize the roughness of the surface Standard deviation n 1 Rq = z2 (8) n i=1 i Skewness Rsk =

n n z3 (n − 1)(n − 2)Rq3 i=1 i

(9)

The skewness can be used to give an idea of the shape of the surface proﬁles; a negative skewness indicates a proﬁle where wide peaks are separated by narrow valleys, whereas a positive skewness indicates a proﬁle with wide valleys separated by narrow peaks. For a sphere, the surface heights zi are taken with reference to its mean radius. Table 3

Roughness measures and aerodynamic parameters for sports balls and rough spheres

Rough spheres [5] D = 200 mm

k/D

k/D (×1000)

Rq

Rsk

0 0.05 0.3 0.5 1.0 2.5

0 0.000 25 0.001 50 0.002 50 0.005 00 0.012 50

0 0.25 1.5 2.5 5.0 12.5

0

0

D 0

0.39

−0.49

1.9

No. of dimples

k (mm)

k/D (×1000)

Rq

Rsk

104 184 328 328 504

0.338 0.338 0.338 0.646 0.292

7.9 7.9 7.9 15.2 6.9

0.13 0.13 0.13 0.24 0.10

−0.70 −0.17 0.34 0.39 0.54

Orientation

k maximum

k/D (×1000)

Rq

Rsk

x-axis y-axis z-axis Average

2.21 2.12 2.16 2.16

33.5 32.1 32.7 32.8

0.56 0.38 0.33 0.42

−1.9 −2.7 −2.9 −2.5

Soccer balls [11] D = 66 mm

Rq

k (mm)

Golf balls [9] D = 42.6 mm

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Rq D 3.1 3.2 3.0 3.7 2.4 Rq D 8.5 5.7 4.9 6.4

(×1000)

(×1000)

(×1000)

Recritical (/1000)

CD minimum

381 274 189 145 118 81

0.07 0.07 0.09 0.10 0.14 0.20

Recritical (/1000)

CD minimum

113 94 60 40 60

0.19 0.20 0.235 0.26 0.26

Recritical (/1000)

CD minimum

153

0.20

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Figure 13(b) shows the minimum value of CD at transition versus skewness for the different sports balls. It can be seen that the skewness can be used to predict the minimum CD value for all the balls except the soccer ball, which lies some way away from the best ﬁt line for the other balls. This can be explained if the soccer ball is compared with the roughened sphere having 2.5 mm beads and the golf ball having 104 dimples, which have a very similar minimum CD . The roughness of these two balls are likely to cause continual mixing of the boundary layer so that slowing of the ﬂuid close to the surface of the sphere is minimized. In contrast, the seams on a football are likely to trip the boundary layer so that it becomes turbulent, but then the large expanse of ﬂat panel following the seam does not mix the boundary layer further but thickens it so that separation occurs relatively early in comparison to a roughened sphere or a golf ball. It is evident, therefore, that both the shape of the roughness elements and their absolute size inﬂuences the aerodynamic parameters and it might be that contours of skewness versus k/D may be more useful. However, further analysis of this kind requires a selection of carefully chosen roughnesses to give a good range of skewnesses and depths, coupled with wind tunnel testing to measure the required critical Reynolds numbers and drag coefﬁcients. Fig. 12

A schematic of sections through sports ball with different roughnesses (with surface geometries exaggerated) indicating values of skewness; (a) a model of a soccer ball across three orthogonal axes [11]; (b) golf balls with different dimple designs [9]; (c) a 200 mm sphere covered with a monolayer of 2.5 mm glass beads [5]

Fig. 13

5

CONCLUSIONS

It is interesting to note the similarities of the published results of the different sports balls since the publication of work by Mehta [14]. Golf, tennis, and soccer all show evidence of transition characteristics in the

A comparison of aerodynamic performance of sports balls using skewness to deﬁne roughness; (a) critical Reynolds number at transition (/103 ); (b) minimum CD at transition

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A new measure of roughness for deﬁning the aerodynamic performance

lift and drag coefﬁcients with spin. However, while the boundary layers of golf and soccer balls lie in the laminar and supercritical regimes, the boundary layer for tennis balls probably lies in the supercritical and transcritical regimes. Spin decay is an area which does not have a lot of reported information, presumably because of the difﬁculty in carrying out the experiments. However, tennis balls experience spin decay ten times higher than golf balls, partly due to their different moment of inertia but also due to the higher skin friction of the nap. The performance of sports balls is dictated by the position of the separation points of the boundary layer. These points are directly inﬂuenced by the roughness of the ball’s surface and it has been shown in this paper that more useful measures of roughness are needed to properly characterize the aerodynamics of a sports ball. One such measure is skewness which identiﬁes the surface as one dominated by peaks or by valleys. It seems that a surface with valleys give a lower critical Reynolds number and a higher drag coefﬁcient at transition than a surface with peaks. However, since skewness gives a normalized measure of surface shape, it is clear that a measure of dimension such as k/D is also needed and would require a set of wind tunnel experiments to be carried out with carefully characterized rough spheres.

REFERENCES

11

12

13

14 15 16

17

18

1 Newton, I. New theory of light and colours. Philos. Trans. R. Soc. Lond., 1672, 1, 678–688. 2 Rayleigh, L. On the irregular ﬂight of a tennis ball. Mess. Math., 1877, 7, 14–16. Reprinted in Scientiﬁc Papers (Cambridge, UK, 1899) 1, 344–346. 3 Thompson, J. J. The dynamics of a golf ball. Nature, 1910, 85, 2151–2157. 4 Achenbach, E. Experiments on the ﬂow past spheres at very high Reynolds numbers. J. Fluid Mech., 1972, 54(3), 565–575. 5 Achenbach, E. The effects of surface roughness and tunnel blockage on the ﬂow past spheres. J. Fluid Mech., 1974, 65(1), 113–125. 6 Bearman, P. W. and Harvey, J. K. Golf ball aerodynamics. Aeronaut. Q., 1976, 27, 112–122. 7 Smits, A. J. and Smith, D. R. A new aerodynamical model of a golf ball in ﬂight. Science and Golf II: In Proceedings of the World scientiﬁc congress of golf (Ed. A. J. Cochran and M. R. Farrally), 1994, pp. 340–346 (E&FN Spon, London). 8 Tarnowksi, K. Analysis of tennis impacts on a realistically supported racket. M.Eng Dissertation, University of Shefﬁeld, 2004. 9 Aoki, E., Oike, A., Nonaka, M., and Yamaguchi, K. Effect of dimple number on the ﬂying characteristics and ﬂow patterns of a golf ball. In The engineering of sport 4 (Eds S. Ujihashi and S. J. Haake), 2002, pp. 330–336 (The

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International Sports Engineering Association, Shefﬁeld, UK). Aoki, E., Nonaka, M., Goto, T., Miyamoto, M., and Sugiura, M. Effect of dimple structure on the ﬂying characteristics and ﬂow patterns of a golf ball. In The engineering of sport 5 (Eds M. Hubbard, R. D. Mehta, and J. M. Pallis), 2004, pp. 49–55 (The International Sports Engineering Association, Shefﬁeld, UK). Carre, M. J., Hanna, R. K., Wilms, J., Goodwill, S. R., and Haake, S. J. Understanding the aerodynamics of a spinning soccer ball. In The engineering of sport 5 (Eds M. Hubbard, R. D. Mehta and J. M. Pallis), 2004, vol. 1, pp. 70–76 (The International Sports Engineering Association, UK). Carre, M. J., Haake, S. J., and Goodwill, S. R. Understanding the effect of seams on the aerodynamics of an association football. Proc. IMechE, 2005, 219(C7), 657–666. Goodwill, S. R., Chin, S. B., and Haake, S. J. Aerodynamics of spinning and non-spinning tennis balls. J. Wind Eng. Ind. Aerodyn., 2004, 92, 935–958. Mehta, R. Aerodynamics of sports balls. Annu. Rev. Fluid Mech., 17, 151–189. Davies, J. M. The aerodynamics of golf balls. J. Appl. Phys., 1949, 20, 821–828. Williams, D. Drag force on a golf ball in ﬂight and its practical signiﬁcance. Q. J. Mech. Appl. Mech., 1959, XII, 387. Lieberman, B. Estimating lift and drag coefﬁcients from golf ball trajectories. Science and Golf: In Proceedings of the World scientiﬁc congress of golf (Ed. A. J. Cochran), 1990, pp. 187–192 (E&FN Spon, London, UK). Chikaraishi, T., Alaki, Y., Maehara, K., Shimosaka, H., and Fukazawa, F. A new method on measurement of trajectories of a golf ball. Science and Golf: In Proceedings of the World scientiﬁc congress of golf (Ed. A. J. Cochran), 1990, pp. 193–198 (E&FN Spon, London, UK). Zagarola, M. V., Lieberman, B., and Smits, A. J. An indoor testing range to measure the aerodynamics performance of golf balls. Science and Golf II: In Proceedings of the World scientiﬁc congress of golf (Eds A. J. Cochran and M. R. Farally), 1994, pp. 348–354 (E&FN Spon, London, UK). Smits, A. J. and Ogg, S. Golf ball aerodynamics. In The engineering of sport 5 (Eds M. Hubbard, R. D. Mehta, and J. M. Pallis), 2004b, pp. 3–12 (The International Sports Engineering Association, Shefﬁeld, UK). Aoyama, S. A modern method for the measurement of aerodynamic lift and drag on golf balls. Science and Golf: In Proceedings of the World scientiﬁc congress of golf (Ed. A. J. Cochran), 1990, pp. 199–204 (E&FN Spon, London, UK). Maccoll, J. Aerodynamics of a spinning sphere. J. R. Aeronaut. Soc., 1928, 32, 777–798. Aoki, E., Nakayama, Y., Hayasida, T., Yamaguti, N., and Sugiura, M. Flow characteristics of a golf ball using visualisation techniques. Science and Golf III: In Proceedings of the World scientiﬁc congress of golf (Eds M. R. Farrally and A. J. Cochran), 1998, pp. 446–456 (E&FN Spon, London, UK).

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24 Tavares, G., Shannon, K., and Melvin, T. Golf ball spin decay model based upon radar measurements. Science and Golf III: In Proceedings of the World scientiﬁc congress of golf (Eds M. R. Farrally and A. J. Cochran), 1998, pp. 464–473 (E&FN Spon, London, UK). 25 Stepanek, A. The aerodynamics of tennis balls – the top spin lob. Am. J. Phys., 1988, 56, 138–142. 26 Chadwick, S. G. and Haake, S. J. The drag coefﬁcient of tennis balls. In The engineering of sport (Eds A. Subic and S. J. Haake), 2000a, pp. 169–176 (Blackwell Science, Oxford, UK). 27 Chadwick, S. G. and Haake, S. J. Methods to determine the aerodynamic forces acting on tennis balls in ﬂight. In Tennis science and technology (Eds S. J. Haake and

28 29

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A. Coe), 2000b, pp. 127–134 (Blackwell Science, Oxford, UK). Chadwick, S. G. The aerodynamics of tennis balls. PhD Thesis, The University of Shefﬁeld, UK, 2003. Haake, S. J., Chadwick, S. G., Dignall, R. J., Goodwill, S. R., and Rose, P. Engineering tennis – slowing the game down. Sports Eng., 2000, 3(2), 131–143. Mehta, R. and Pallis, J. The aerodynamics of a tennis ball, In The engineering of sport 4 (Eds S. Ujihashi and S. J. Haake), 2001, pp. 1–13 (The International Sports Engineering Association, Shefﬁeld, UK). Whitehouse, D. J. (Ed) Handbook of surface metrology, 1994, pp. 45–49 (Rank Taylor Hobson Ltd., IOP Publishing, London, UK).

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A new measure of roughness for defining the aerodynamic performance of sports balls S J Haake, S R Goodwill and M J Carre Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2007 221: 789 DOI: 10.1243/0954406JMES414 The online version of this article can be found at: http://pic.sagepub.com/content/221/7/789

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789

A new measure of roughness for deﬁning the aerodynamic performance of sports balls S J Haake1∗ , S R Goodwill1 , and M J Carre2 1 The Centre for Sport and Exercise Science, Shefﬁeld Hallam University, Shefﬁeld, UK 2 Department of Mechanical Engineering, The University of Shefﬁeld, Shefﬁeld, UK The manuscript was received on 16 June 2006 and was accepted after revision for publication on 27 March 2007. DOI: 10.1243/0954406JMES414

Abstract: A new analysis is presented of the major ﬁndings in sports ball aerodynamics over the last 20 years, leading to a new method for deﬁning surface roughness and its effects on the aerodynamic performance of sports balls. It was shown that the performance of balls in soccer, tennis, and golf are characterized by the position of the separation points on the surface of the ball, and that these are directly inﬂuenced by the roughness of the surface at a given Reynolds number and spin rate. The traditional measure of roughness k/D (the ratio of surface asperity dimension to diameter) was unable to predict the transition from laminar to turbulent ﬂow for different sports balls. However, statistical measures of roughness commonly used in tribology were found to correlate well with the Reynolds number at transition and the minimum Cd after transition. It was concluded that this new measure and a further one of dimension should allow the complete characterization of the aerodynamic performance of sports balls. The effects of surface roughness on spin rate decay were also considered, and it was found that tennis balls had spin decay over six times that of golf balls and was due to the increased skin friction of the nap.

1

INTRODUCTION

People’s love of sport is reﬂected in the current wave of research being carried out on sports equipment and technology. The idea that sport should be a legitimate area of research is not new and attempts to explain the deviations of sports balls in ﬂight were made by such illustrious scientists as Newton, Rayleigh, and Thompson [1–3]. In more recent years, impending tournaments such as Wimbledon, the soccer world cup, and the Olympics have provoked a rash of popular articles on the interaction between aerodynamics and sport. In particular, ball sports receive much attention from both researchers and the media alike.

∗ Corresponding author: The Centre for Sport and Exercise Science,

The Faculty of Health and Wellbeing, Shefﬁeld Hallan University, Collegiate Hall, Collegiate Crescent, Shefﬁeld S10 2BP, UK. email: [email protected]

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It was not until 1985, however, that Mehta carried out the ﬁrst review of sports ball aerodynamics (in the Annual Review of Fluid Mechanics) by covering the sports of cricket, golf, and baseball. Mehta pulled together both his and other researchers’ work in a succinct way and showed that the drag on a ball was dominated by the size and deﬂection of the wake. The drag was inﬂuenced by the position of separation of the boundary layer from the ball’s surface which was, in turn, affected by the condition of the ﬂow within the boundary layer. Transition of the boundary layer from laminar to turbulent ﬂow was seen to be affected by a number of factors, such as Reynolds number, surface roughness, and spin rate. Since 1985, research in this area has continued apace, particularly in the sports of golf, tennis and soccer. It is the intention of this paper to outline the more important pieces of work in sports ball aerodynamics that have emerged in the last 20 years, and to use them to formulate a new methodology for determining the effect of roughness on performance. Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

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2 THE FUNDAMENTAL MECHANICS OF SPORTS BALL AERODYNAMICS The drag and lift forces on a rotating sphere of crosssectional area A, travelling at velocity U∞ through a ﬂuid of density ρ are given as 1 2 ρACD U∞ 2 1 2 FL = ρACL U∞ 2

FD =

Figure 1(a) shows the classical results found by Achenbach, which have been used in almost all studies of sports balls ever since. It can be seen that a drag coefﬁcient CD of approximately 0.52 exists

(1) (2)

where CD and CL are the drag and lift coefﬁcients, respectively. Both CD and CL are expected to be functions of the Reynolds number Re and the spin parameter α (deﬁned as the ratio of the peripheral velocity to the translational velocity) given by ρU∞ D U∞ D = µ ν ωD α= 2U∞

Re =

(3) (4)

where ν is the kinematic viscosity of the air, α the spin parameter, and D and ω are the diameter and spin of the sphere, respectively. Most researchers in sports ball aerodynamics start with the work carried out by Achenbach in the 1970s which was actually nothing to do with sport at all. Achenbach [4] was interested with the transportation of spherical fuel elements in high-pressure pipes and carried out a number of carefully controlled experiments on a polished metal sphere in a high-pressure wind tunnel. Table 1 shows that the turbulence of the wind tunnel used was less than half a per cent and that the blockage ratio (the ratio of the areas of the model to the wind tunnel section) was one per cent.

Table 1

Fig. 1

(a) Drag coefﬁcient as a function of Reynolds number for a smooth metal sphere and (b) schematics and graphs showing the position of the boundary layer separation and transition points for smooth spheres with respect to stagnation point [5]

Experimental details for wind tunnel tests on roughened spheres and sports balls (N/A = not applicable)

Sport Reynolds number Spin parameter, α Turbulence level Diameter of sphere Ratio of diameter of sting to sphere Blockage (ratio of area of model to tunnel) CD of smooth sphere (subcritical region)

[4, 5]

[6]

[7]

[8]

[9, 10]

[11, 12]

[13]

General 40 000–6 000 000 N/A 0.45% 200 mm

Golf 40 000–260 000 0–0.3 0.2% 103 mm

Golf 40 000–250 000 0–1.4 0.4–0.8% 42.6 mm

Tennis 80 000–300 000 N/A 88 500 that the values collapse onto a single line. Bearman and Harvey, on the other hand only showed the independence of CL for Re > 126 000 and ignored data away from this line. It can be seen that the CL for Smits and Smith are higher than those of Bearman and Harvey, which may be due to a different dimple pattern analysed or to a systematic shift due to the side supports that was accounted for in their calculation of CD but not in CL . Figure 4(c) shows lift and drag coefﬁcients versus spin rates for two values of Reynolds number. It can be seen that for Re = 39 805 CL tends towards negative

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Fig. 4

(a) Drag coefﬁcient versus Reynold number; (b) lift coefﬁcient versus spin parameter for different Reynolds numbers (×10−3 ); (c) lift and drag coefﬁcients bar different Reynolds numbers (×10−3 ); (d) spin rate decay versus spin parameter [7]

values at a spin rate of 0.4, and that this coincides with a turning point in CD . The authors considered that this was caused by the boundary layer on one side of the ball remaining laminar whereas the other was turbulent. The laminar side separated early whereas the turbulent side separated late such that the wake was deﬂected in the opposite direction to scenarios at much higher spin parameters [23]. Smits and Smith [7] also published for the ﬁrst time signiﬁcant amount of data on the spin rate decay of golf balls. Lieberman [17] had previously indicated that spin rate decayed to two-thirds to three-quarters of its initial value over its ﬂight and was exponential in nature. Smits and Smith found that ω˙ ∝ U∞ ω so that for constant velocity the spin decayed exponentially with time. This implied 2 ∝ ωR 2 /U∞ = αR and a non-dimensional that ωR ˙ 2 /U∞ 2 which increases linearly with spin parameter ωR ˙ 2 /U∞ rate for a given radius (Fig. 4(d)). This feature was subsequently incorporated by Smits and Smith into their trajectory models to simulate the ﬂight of golf balls.

Tavares et al. [24] used radar measurements to determine the spin rate decay of golf balls with different moments of inertia and different dimple patterns. They showed that spin rate decay was increased with deeper dimples and with lower moments of inertia, but was only signiﬁcant for spin rate greater than about 0.5. Other researchers [9, 10] studied the effect of the number of dimples on a golf ball by manufacturing balls with different numbers and geometries of dimples. Table 2 shows that Aoki et al. [9, 23] increased Table 2

Speciﬁcations for dimple shapes used by Aoki et al. [9, 10]

No. of dimples

Distance between dimples (mm)

Width of dimples (mm)

Depth of dimples (mm)

Dimple depth to width ratio

104 184 328 328 504

3.897 2.043 0.650 0.650 0.297

3.528 3.528 3.528 3.528 3.046

0.338 0.338 0.338 0.646 0.292

0.096 0.096 0.096 0.183 0.096

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the number of dimples from 104 to 504 while keeping the width to depth ratio constant. They also looked at the effect of doubling the dimple depth to width ratio with 328 dimples. Table 1 shows that their measured value for the drag coefﬁcient on a smooth sphere was low. It can be seen from Fig. 5(a), however, that their values for CD of a golf ball with 328 dimples (a typical number for a golf ball) compares very well with Bearman and Harvey [6]. It can be seen in Fig. 5(a) that increasing the number of dimples decreases the critical Reynolds number and increases the minimum CD in the supercritical region. Figure 5(b) shows that deepening the dimples causes a further decrease in the critical Reynolds number while retaining the low minimum CD . Deepening the dimples, however, causes a rise in CD with Re, probably caused by a thickening of the turbulent

Fig. 5

795

boundary layer and earlier separation. Figure 5(b) also shows that increasing the number of similar dimples from 328 to 504 does not effect the critical Re, but increases the minimum CD by a small amount (about 0.02). What can be concluded is that transition occurs earlier if the number of dimples is increased, or if the dimples are made deeper. Thus, increasing the number of dimples while decreasing their depth appears to more or less cancel each other out. A slight increase in number of dimples from 328 to 504 with a decrease in depth from 0.34 to 0.29 mm more or less cancel each other out. Achenbach [5] and all subsequent authors have used the parameter k/D to characterize roughness. In this case, however, the balls with 104, 184, and 328 dimples would all have the same

(a) Drag coefﬁcient versus Reynolds number for golf balls with different number of dimples of the same depth; (b) drag coefﬁcient versus Reynolds number for golf balls with dimples of different depth; (c) drag and lift coefﬁcient versus spin parameter for golf balls with 104 and 184 dimples of the same depth; (d) drag and lift coefﬁcient versus spin parameter for a golf ball with 328 dimples of different depth (all data from Aoki et al. [9, 10, 23])

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S J Haake, S R Goodwill, and M J Carre

Fig. 6

Comparison of (a) drag coefﬁcient and (b) lift coefﬁcient with spin parameter for different researchers

roughness parameter (k/D = 0.008) but very different performance. Equally, the balls with 328 and 504 dimples would have different roughnesses but would perform almost identically. This issue of an adequate deﬁnition of roughness will be discussed in a later section. The effect of increasing the number of dimples on a spinning golf ball in the supercritical region is shown in Fig. 5(c). It can be seen that both the drag and lift coefﬁcients increase with spin parameter and are thus very similar to the trends of Bearman and Harvey. Increasing the number of dimples from 104 to 184 increases the drag by approximately 0.05, and the lift by about 0.07 at α = 0.15. This implies that the extra roughness associated with the more densely packed dimples not only encourages the asymmetry of the wake but also thickens the boundary layer. This induces early separation and produces a larger drag coefﬁcient. Figure 5(d) compares the golf balls with 328 dimples but different depths. It shows that increasing the depth of the dimple both increases the drag and decreases the lift coefﬁcient, implying that too much roughness associated with dimple depth causes early separation of the boundary layer. Figures 6(a) and (b) summarize the drag and lift coefﬁcients found by all the researchers discussed in this section. All researchers agree that increasing the spin parameter increases both the lift and drag coefﬁcients. The data shows that the actual magnitudes depend upon the number of dimples, the spin parameter, Reynolds number, and the critical transition. It also depends, to some extent, on the validity of the experiment performed.

method of Davies [15] to drop tennis balls across the working section of a wind tunnel in order to determine lift and drag coefﬁcients up to spin parameters of approximately 0.8 (Fig. 7). He found that the drag coefﬁcient of a non-spinning ball was approximately 0.51 and that it increased to about 0.76 with increasing spin parameter, whereas the lift coefﬁcient increased with spin rate to a maximum of about 0.31. Stepanek found, therefore, that the drag coefﬁcient for a non-spinning tennis ball was equivalent to that for a smooth ball but increased rapidly when the ball was spinning. Stepanek found that unlike golf balls, the lift and drag coefﬁcients were independent of Reynolds number, and did not ﬁnd any evidence of critical transition. Further work on tennis balls was carried out by Chadwick in the late 1990s [26–29] to try to quantify the effect of surface condition and ball size on lift

3.2 Tennis Stepanek [25] produced the ﬁrst signiﬁcant piece of work on tennis ball aerodynamics. He used the

Fig. 7

Drag and lift coefﬁcient versus spin parameter for spinning tennis balls [26]

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A new measure of roughness for deﬁning the aerodynamic performance

and drag. Chadwick found that the drag coefﬁcient was higher for a tennis ball than that for the smooth or a rough spheres analysed by Achenbach and that, like Stepanek, CD was more or less independent of Reynolds number. Furthermore, they found that the state of the covering inﬂuenced the drag coefﬁcient with a raised nap giving a value about 10 per cent higher than a shaved nap. Most interestingly, it was seen that transition did not occur with a tennis ball and that separation almost always occurred at 80◦ (with respect to the stagnation point). They concluded that either the boundary layer ﬂowing over a tennis ball was in the supercritical or transcritical regime, or that the nap acted merely as a trip regardless of the condition of the boundary layer. It was also noted that the nap tended to lie ﬂat at high-wind speeds for a non-spinning ball and became raised at high-spin parameters. Mehta and Pallis [30] responded by carrying out wind tunnel tests on a small selection of tennis balls and subsequently suggested that the boundary layer was indeed in the transcritical regime, with drag coefﬁcients of around 0.6–0.65. They conﬁrmed the work of Chadwick and found that the height of the nap affected the drag coefﬁcient. They postulated that the increase in drag over and above the transcritical value for a sphere was due to the nap ﬁbres acting as a series of bent cylinder-like ﬁlaments, which was signiﬁcant at low Re and lessened at higher Re. To look at this hypothesis in more detail, Goodwill et al. [13] carried out a detailed survey of the aerodynamic performance of tennis balls. Tennis balls were supported in the centre of the test section of a highspeed wind tunnel using two horizontal stings 3.5 mm in diameter giving a ratio of support to ball diameter of around 5 per cent. The stings allowed the ball to be rotated up to 2750 r/min (289 rad s−1 ) with the simultaneous measurement of lift force. Ten brands of ball were mounted in the wind tunnel and the variability between balls of the same brand and between balls of different brands was studied. Figure 8(a) shows the drag coefﬁcient of four of the tennis balls studied by Goodwill et al. to show the range between brands. It can be seen that the drag coefﬁcient drops initially from around 0.65 to about 0.60 at Re ≈ 150 000 before increasing marginally again. The range between the slowest and fastest ball brands is about 0.05 whereas the range for different balls of the same brand was found to be ±0.02. It was concluded, therefore, that the balls could not be distinguished between each other within errors and that all new tennis balls play roughly the same. Mehta and Pallis [30] and Chadwick [28] explained the initial drop in CD by the laying down of the ﬁbres of the felt. Individual ﬁbres have a high drag coefﬁcient at low Re which contributes to the overall drag coefﬁcient. Once the ﬁbres have laid ﬂat, then a further JMES414 © IMechE 2007

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increase in Re causes increased thickening of the turbulent boundary layer which causes the separation points to move upstream. If the ball is in the transcritical regime, as advocated by Mehta and Pallis then the drag coefﬁcient is at least 0.2 higher than the largest transcritical value found by Achenbach [5] of 0.4. This implies that the extra drag due to the felt contributes approximately 30 per cent towards the total drag coefﬁcient. A second experiment was carried out in which tennis balls were artiﬁcially worn in a wear rig to simulate play under controlled conditions. Figure 8(b) shows that worn tennis balls act in a similar manner to new tennis balls, and that the most worn ball (1500 impacts) has a drag coefﬁcient that is approximately 0.04 lower than a new ball. Thus, a worn ball in which the felt has been abraded and removed has a 6 per cent lower drag coefﬁcient. Goodwill et al. showed that a difference of this magnitude would cause a typical serve with the same initial conditions to travel 40 cm further horizontally than a standard ball at the receiver’s baseline. Also shown in Figure 8(b) is data from Mehta and Pallis [30] who analysed a used tournament ball. The plot appears to indicate transition with a critical Reynolds number of about 100 000 and a minimum drag coefﬁcient of about 0.46. Tennis balls are changed during tournaments after either 7 or 9 sets and it can be estimated that a single ball may only be hit around 30 times before it is changed. It is surprising, therefore, that the tournament ball exhibits transition when the balls subjected to over 50 times normal wear do not. Without observing the surface of the tournament ball ﬁrst hand, it is difﬁcult to hypothesize the reasons for such different behaviour in comparison to all the other tennis balls tested. Figures 8(c) and (d) show the lift and drag coefﬁcient for spinning tennis balls at two different Reynolds numbers equivalent to a ball speed of 25 and 50 ms−1 for a number of different ball brands. It can be seen that the drag coefﬁcients at α = 0 compare very well with Fig. 8(a) with the drag value at Re = 105 000 being higher than that at Re = 210 000. The range of CD values between brands is similar to that of the non-spinning balls in Fig. 8(a). It can be seen that CD increases with spin parameter by about 6 per cent over the range studied and that the results are much higher than those measured by Stepanek [25] and Chadwick [28], particularly for low spin rates. The values for nonspinning balls, however, do match the new balls tested by Mehta and Pallis [30]. Chadwick attempted to take into account the fact that the ﬁbres of the felt stand on end due to centrifugal forces as the ball spins and used an effective ball diameter of 73 mm. Using a diameter of 65 mm would have increased the lift coefﬁcient to coincide with the experimental data of Goodwill et al. The lift coefﬁcients increase with the spin parameter at both Reynolds numbers (Figs 8(c) and (d)). It is Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

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Fig. 8

(a) Drag coefﬁcient versus Re for new tennis balls; (b) drag coefﬁcient versus Re for worn tennis balls; (c) drag and lift coefﬁcients versus spin parameter for new spinning tennis balls at Re = 105 000; (d) drag and lift coefﬁcients versus spin parameter for new spinning tennis balls at Re = 210 000; (e) drag and lift coefﬁcients versus spin parameter for worn spinning tennis spinning tennis balls at Re = 105 000; (f) drag and lift coefﬁcients versus spin parameter for worn spinning tennis balls at Re = 210 000 (additional data from Goodwill et al. [13])

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interesting that the lift coefﬁcient appears to show a dip at Re = 210 000 between α = 0.05 and α = 0.15 and that the drag coefﬁcient does not increase quite as much as at the lower Re in this range either. Similarities are seen in Figs 8(e) and (f ) for the worn balls. It can be seen that the new ball (zero impacts) follows the same lines as in Figs 8(c) and (d), i.e. there is a dip in the CL data at the higher Re. There is an even more marked dip in the data for the worn balls (particularly at 1500 impacts) which exists in all the lift and drag curves in Figs 8(e) and (f). These characteristics can be explained if the nature of the felt on a spinning tennis ball is discussed. Chadwick [28] showed that the height of the nap on the leading surface of a non-spinning ball decreased with wind speed from a natural length of 5 mm to around 2 mm at Re ≈ 150 000. The nap then compressed to around 1 mm as the speed was increased further to Re ≈ 300 000. Chadwick also showed that the nap of a non-spinning ball extended to over 8 mm in length as the balls was spun up to 290 rad s−1 . These features of the nap can be used to explain the different scenarios in Fig. 8, as follows. 3.2.1

Non-spinning ball (Figs 8(a) and (b))

As the Reynolds number increases to around 150 000, the height of the nap at the front of the ball decreases and is laid ﬂat around the sides of the ball. Thus, the ball changes from a ball with high nap to one with low nap and the skin friction decreases because of the reduced ﬁbre height. Once the ﬁbres are ﬂat, the boundary layer is probably in the transcritical regime and any further increase in the Reynolds number causes the separation points to move upstream, giving a further increase in the drag coefﬁcient. 3.2.2

Unused spinning balls at low speed (Fig. 8(c))

When a ball is spun, the nap height depends upon the spin rate and the Reynolds number. At low Reynolds number, the nap of an unused tennis ball remains at its original height of around 5 mm and any spin increases its height all around the sphere. The extra surface friction causes the drag coefﬁcient to increase as the spin parameter increases and the separation points move equally on each side to give a steady rise in the lift coefﬁcient. 3.2.3

New spinning balls at high speed (Fig. 8(d))

At high Reynolds number, the nap of a new tennis ball depresses to around 2 mm at the front of the ball and lays ﬂat on its upper and lower surfaces. Low spin rates do not raise the height of the nap much and therefore the skin friction is quite low. This means that for spins up to α ≈ 0.15, ﬂow is less dominated by skin friction. It would be expected that the boundary layer JMES414 © IMechE 2007

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of the advancing side of the spinning ball would be transcritical and that an increase in the spin would not affect the position of the separation point. On the retreating side, however, the boundary layer might be supercritical and the separation point would retreat downstream. This would have the effect of decreasing the drag coefﬁcient and giving a slightly smaller lift coefﬁcient than expected. 3.2.4 Worn spinning balls at low speed (Fig. 8(e)) As the nap is worn away (1500 impacts), the importance of skin friction decreases. The advancing side of the ball remains transcritical whereas the retreating side of the ball becomes supercritical. The latter allows the separation point on that side of the ball to move downstream giving a lower CD and CL than for a new ball. The effect starts to disappear as the spin parameter increases when the nap is raised by the centrifugal forces of the spinning ball. 3.2.5 Worn spinning balls at high speed (Fig. 8(f )) At high speed, the nap is depressed for both the new and worn balls. The results show that the balls display the dip in CD and CL for α < 0.15 seen in the previous graphs, and explained by the supercritical nature of the retreating side of the ball. For α > 0.15, the nap stands up and the friction forces increase which overwhelms any tendency of the boundary layer to be supercritical and gives a higher lift and drag coefﬁcient for the new ball than the worn ball. In conclusion, therefore, the aerodynamics of a tennis ball is dominated by the condition of the felt. Raising or lowering the nap, either naturally during play or through artiﬁcial wear causes the drag coefﬁcient to increase or decrease. The height of the nap is also affected by the speed of the ball through the air and its spin, and it is likely that the boundary layer can be in the supercritical or transcritical regimes, depending upon the combination of speed, spin, and wear. 3.3

Soccer

Soccer balls have only recently been analysed aerodynamically, probably because the balls require a much larger wind tunnel if the experiments are not to suffer from blockage. Carre et al. [11] overcame this problem by manufacturing a one-third scale model of football, complete with seams and with the roughness parameter k/D scaled accordingly. A further analysis was carried out with a mini football. Figure 9(a) shows the variation of drag coefﬁcient with Reynolds number for the football model in comparison to a smooth sphere and a golf ball. It can be seen that transition for a football lies somewhere between the smooth Proc. IMechE Vol. 221 Part C: J. Mechanical Engineering Science

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Fig. 9

Aerodynamic coefﬁcients for soccer balls; (a) drag coefﬁcient versus Re for a non-spinning football model (closed symbols) and a mini-football (open symbols); (b) drag coefﬁcient versus spin parameter at different Re for a model football; (c) lift coefﬁcient versus spin parameter at different Re for a model football; (d) simulated ratio of side to drag force (CL /CD ) for an 18.4 ms−1 free kick [11, 12]

sphere and a golf ball, suggesting that the seams of a football are effective at tripping the boundary layer, but are not as efﬁcient as dimples. It is possible that the drag coefﬁcient increases faster for footballs in the supercritical and transcritical region than for golf balls due to the relatively large areas of ‘smooth’ panel between seams causing earlier separation. Both the football model and the mini-football lie approximately on the same line, although the mini-football may just be undergoing transition at Re ≈ 110 000. Figures 9(b) and (c) show the variation of drag and lift coefﬁcient with spin parameter for the model ball only. The drag coefﬁcients for a non-spinning ball at α = 0 in Fig. 9(b) match the values seen in

Fig. 9(a). It can be seen that for Reynolds numbers well above transition, the drag coefﬁcient is independent of Reynolds number. This implies that the separation points remain roughly in the same position as more spin is placed on the ball and that extra spin does not cause the boundary layer to thicken very much. For Reynolds numbers at or below transition (i.e. data at Re = 85 000 and 110 000), it can be seen that the drag coefﬁcient decreases with spin parameter before levelling off to a more or less constant value. This is explained if the lift coefﬁcients are considered in Fig. 9(c). It can be seen that the lift coefﬁcient increases with spin parameter and tends to lie along the same line when above transition,

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much like the golf ball data in Fig. 4(b). At Reynolds numbers approaching transition (Re = 110 000) the lift coefﬁcient is lower and below transition negative lift coefﬁcients are found. This is due to the boundary layer on the retreating side of the spinning ball remaining laminar whereas the advancing side is turbulent. The separation point on the laminar side advances upstream causing the wake to deﬂect in the opposite direction to that when both sides of the ball have turbulent boundary layers. The reversal of the lift coefﬁcient at α ≈ 0.26 in Fig. 9(c) for Re = 85 000 coincides with the turning point in the drag coefﬁcient in Fig. 9(b). Again, this agrees with the work in golf ball carried out by Smits and Smith [7] shown in Fig. 4(c). Carre et al. used these results to analyse several free kicks that were notable for their curvature in the air, and their ability to deceive goal keepers. If the spin is assumed to remain constant throughout the trajectory, then the spin parameter actually rises as the ball slows, also causing the lift coefﬁcient to rise. Figure 9(d) shows the ratio of side to drag force (effectively CL /CD ) for a free kick at 18.4 ms−1 with different amounts of spin, simulated to land in the top corner of the goal from 18 m if 64 rad s−1 of sidespin is applied. In this case the ‘lift’ force has been applied as a side force since the spin axis is in the vertical direction and the resultant force in the horizontal direction. The ball slows from Re ≈ 270 000 on launch to Re ≈ 190 000 at the goal, before increasing again at the end of the shot due to the effects of gravity. It can be seen that the ratio of side to drag force rises signiﬁcantly at the end of ﬂight as the ball nears the goal and could be enough to signiﬁcantly fool a goalkeeper. The ratio of side to drag force drops slightly over the full trajectory at low

Fig. 10

801

spins, and rises at high spins. There appears to be an optimum value of spin between 48 and 64 rad s−1 where the ratio remains approximately constant and still rises signiﬁcantly at the end. Above this value of spin, the ratio rises continuously over the trajectory and shows a smaller rise at the end. 3.4

Spin decay

Lieberman [17] ﬁrst quoted rates of decay for golf balls in ﬂight to be from a quarter to two-thirds for trajectories lasting between 6 and 8 s. Smits and Smith [7] suggested that spin rate decay is exponential with time for a constant velocity such that dω = −cU∞ ω dt

(5)

where c is a constant. The non-dimensional parame2 is proportional to αR. ter ωR ˙ 2 /U∞ 2 for a golf ball is Figure 4(d) shows that ωR ˙ 2 /U∞ indeed linear with the spin parameter and was found to be applicable for all Reynolds numbers. This gave them all the information they needed to describe a golf ball trajectory, i.e. the variation of the lift and drag coefﬁcients with velocity and spin, and the decay of the spin over the trajectory. Tarnowksi [8] investigated the spin decay of tennis balls by projecting them in a sports hall at approximately 28 and 32 ms−1 and with spins between −48 and +427 rad s−1 . The angle was altered so that the ball landed in approximately the same position at the other end of a tennis court. Two high-speed videos were used to measure the speed, spin, and angle on launch and on landing. Figure 10(a) shows that the

(a) Landing spin rate versus launch spin rate for tennis balls projected at 28 ms−1 for a given horizontal distance; (b) a comparison of spin rate decay parameter between tennis and golf balls

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ﬁnal spin was proportional to the initial spin and that approximately 14 per cent of spin was lost during the trajectory, regardless of initial spin rate or velocity. It appears, therefore, that the spin decay for tennis balls acts in the same way as that for golf balls found by Smits and Smith. Although the time of ﬂight was not recorded in this experiment, it is possible to predict the time of ﬂight using a trajectory programme [28], which allows an indication of the constant c in equation (4) to be 2 versus spin rate found. Figure 10(b) shows ωR ˙ 2 /U∞ for tennis balls compared to data for golf balls. It can 2 for tennis balls is approximately be seen that ωR ˙ 2 /U∞ ten times that of golf balls. Given that the radius of a tennis ball is approximately 1.55 times that of a golf ball, then the data shows that the spin decay ω˙ for a tennis ball (or the coefﬁcient of moment as deﬁned by Tavares et al. [24]) is approximately 6.5 times that of a golf ball at the same spin rate and velocity. This is probably due to the individual ﬁbres in the felt penetrating through the boundary layer and into the ﬂow of the free stream. Achenbach [4] showed that the proportion of the CD due to friction for a non-spinning smooth ball was approximately 5 per cent of the transcritical value. The work here implies that the proportion of CD due to friction for a spinning tennis ball could be at least 20 per cent of the total (i.e. 0.13). This agrees with the work of Chadwick and Haake [27], Chadwick [28], and Mehta and Pallis [30] who showed that the drag coefﬁcient could be altered by at least 10 per cent by artiﬁcially varying the length of the nap, and with the earlier conclusion that the extra friction of the nap contributes about 30 per cent to the drag coefﬁcient.

Fig. 11

It would be interesting to carry out a similar experiment to determine the spin rate decay of soccer balls, with the expectation that they would behave more like golf balls than tennis balls because of their relatively smooth surface.

4

MEASURES OF ROUGHNESS

Figures 2(c) and (d) showed that there was a high correlation between the critical Reynolds number, the minimum drag coefﬁcient at transition, and the roughness parameter k/D for spheres covered in glass beads. It was found that a larger roughness k/D caused transition to occur at low Re but with relatively high supercritical values of CD . Figures 11(a) and (b) show similar graphs to those in Fig. 2 with the inclusion of soccer and golf balls. For the soccer ball model, the depth of the seams was measured from the original CAD drawing and the value k taken as the maximum depth of seam found. The dimensions of the golf balls were given by Aoki et al. in their original paper. It can be seen that the golf balls do not ﬁt well to the lines through the rough sphere data, primarily because three of the golf balls had the same dimple depth even though they had a different number of dimples and showed clear aerodynamic differences. The soccer ball is even further away from the best ﬁt line and it is evident that the parameter k/D is not enough on its own to give an indication of the aerodynamic performance of sports balls. An alternative method is to look at statistical techniques used by tribologists to characterize surfaces

A comparison of aerodynamic performance of sports balls of different roughness k/D; (a) critical Reynolds number at transition; (b) minimum CD value at transition. The lines show the best ﬁt to the data of Achenbach [5] shown in Fig. 2

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[31]. The characterization of the roughness of a surface requires that the proﬁle be measured in some manner and digitized such that the surface height zi is known at distance xi from a given datum, and is given by zi = Z (xi )

at

xi = x1 + (i − 1) × x

Table 3 indicates the values of the critical Reynolds numbers, the minimum CD values at transition, the k/D values, and the statistical measures of roughness calculated using equations (6) to (9) for the spheres studied in this paper. Figure 12 shows schematics through the various spheres (with the surface proﬁles exaggerated) and their calculated skewness values. The CAD drawing of the model soccer ball was analysed through three orthogonal axes and it was found that the ball was dominated by ﬂat peaks with an average skewness of −2.5. The golf balls of Aoki et al. range from a ball dominated by peaks for 104 dimples (skewness = −0.70) to a ball dominated by valleys for 504 dimples. Finally, the skewness of one of Achenbach’s rough spheres was analysed for k/D = 1.25 per cent with 2.5 mm dimples on a 200 mm diameter sphere. It was assumed that the glass beads used in his experiment were adhered so that they were one layer thick and were packed so that there were no gaps between them. It was thought that these assumptions could not be applied to the other spheres in Achenbach’s experiment since smaller beads would be likely to have more than one layer, making a geometric analysis difﬁcult. It was found that Achenbach’s roughest sphere had a skewness value of approximately −0.5 and was thus dominated by peaks. Figure 13(a) shows the critical Re plotted against skewness for all the spheres analysed. It can be seen that there is a strong relationship between Re and skewness for all the balls and that roughness dominated by valleys produces a lower critical Reynolds number than roughness dominated by peaks.

(6)

A mean can be determined across the length examined such that n zi = 0 (7) i=1

when the heights of n points are measured. The following parameters can then be used to characterize the roughness of the surface Standard deviation n 1 Rq = z2 (8) n i=1 i Skewness Rsk =

n n z3 (n − 1)(n − 2)Rq3 i=1 i

(9)

The skewness can be used to give an idea of the shape of the surface proﬁles; a negative skewness indicates a proﬁle where wide peaks are separated by narrow valleys, whereas a positive skewness indicates a proﬁle with wide valleys separated by narrow peaks. For a sphere, the surface heights zi are taken with reference to its mean radius. Table 3

Roughness measures and aerodynamic parameters for sports balls and rough spheres

Rough spheres [5] D = 200 mm

k/D

k/D (×1000)

Rq

Rsk

0 0.05 0.3 0.5 1.0 2.5

0 0.000 25 0.001 50 0.002 50 0.005 00 0.012 50

0 0.25 1.5 2.5 5.0 12.5

0

0

D 0

0.39

−0.49

1.9

No. of dimples

k (mm)

k/D (×1000)

Rq

Rsk

104 184 328 328 504

0.338 0.338 0.338 0.646 0.292

7.9 7.9 7.9 15.2 6.9

0.13 0.13 0.13 0.24 0.10

−0.70 −0.17 0.34 0.39 0.54

Orientation

k maximum

k/D (×1000)

Rq

Rsk

x-axis y-axis z-axis Average

2.21 2.12 2.16 2.16

33.5 32.1 32.7 32.8

0.56 0.38 0.33 0.42

−1.9 −2.7 −2.9 −2.5

Soccer balls [11] D = 66 mm

Rq

k (mm)

Golf balls [9] D = 42.6 mm

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Rq D 3.1 3.2 3.0 3.7 2.4 Rq D 8.5 5.7 4.9 6.4

(×1000)

(×1000)

(×1000)

Recritical (/1000)

CD minimum

381 274 189 145 118 81

0.07 0.07 0.09 0.10 0.14 0.20

Recritical (/1000)

CD minimum

113 94 60 40 60

0.19 0.20 0.235 0.26 0.26

Recritical (/1000)

CD minimum

153

0.20

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Figure 13(b) shows the minimum value of CD at transition versus skewness for the different sports balls. It can be seen that the skewness can be used to predict the minimum CD value for all the balls except the soccer ball, which lies some way away from the best ﬁt line for the other balls. This can be explained if the soccer ball is compared with the roughened sphere having 2.5 mm beads and the golf ball having 104 dimples, which have a very similar minimum CD . The roughness of these two balls are likely to cause continual mixing of the boundary layer so that slowing of the ﬂuid close to the surface of the sphere is minimized. In contrast, the seams on a football are likely to trip the boundary layer so that it becomes turbulent, but then the large expanse of ﬂat panel following the seam does not mix the boundary layer further but thickens it so that separation occurs relatively early in comparison to a roughened sphere or a golf ball. It is evident, therefore, that both the shape of the roughness elements and their absolute size inﬂuences the aerodynamic parameters and it might be that contours of skewness versus k/D may be more useful. However, further analysis of this kind requires a selection of carefully chosen roughnesses to give a good range of skewnesses and depths, coupled with wind tunnel testing to measure the required critical Reynolds numbers and drag coefﬁcients. Fig. 12

A schematic of sections through sports ball with different roughnesses (with surface geometries exaggerated) indicating values of skewness; (a) a model of a soccer ball across three orthogonal axes [11]; (b) golf balls with different dimple designs [9]; (c) a 200 mm sphere covered with a monolayer of 2.5 mm glass beads [5]

Fig. 13

5

CONCLUSIONS

It is interesting to note the similarities of the published results of the different sports balls since the publication of work by Mehta [14]. Golf, tennis, and soccer all show evidence of transition characteristics in the

A comparison of aerodynamic performance of sports balls using skewness to deﬁne roughness; (a) critical Reynolds number at transition (/103 ); (b) minimum CD at transition

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A new measure of roughness for deﬁning the aerodynamic performance

lift and drag coefﬁcients with spin. However, while the boundary layers of golf and soccer balls lie in the laminar and supercritical regimes, the boundary layer for tennis balls probably lies in the supercritical and transcritical regimes. Spin decay is an area which does not have a lot of reported information, presumably because of the difﬁculty in carrying out the experiments. However, tennis balls experience spin decay ten times higher than golf balls, partly due to their different moment of inertia but also due to the higher skin friction of the nap. The performance of sports balls is dictated by the position of the separation points of the boundary layer. These points are directly inﬂuenced by the roughness of the ball’s surface and it has been shown in this paper that more useful measures of roughness are needed to properly characterize the aerodynamics of a sports ball. One such measure is skewness which identiﬁes the surface as one dominated by peaks or by valleys. It seems that a surface with valleys give a lower critical Reynolds number and a higher drag coefﬁcient at transition than a surface with peaks. However, since skewness gives a normalized measure of surface shape, it is clear that a measure of dimension such as k/D is also needed and would require a set of wind tunnel experiments to be carried out with carefully characterized rough spheres.

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