Review. Review of tennis ball aerodynamics. Rabindra Mehta 1,, Firoz Alam 2 and Aleksandar Subic 2

Review of tennis ball aerodynamics Review DOI: 10.1002/jst.11 Review of tennis ball aerodynamics Rabindra Mehta 1,, Firoz Alam 2 and Aleksandar Subic 2 1 Sports Aerodynamics Consultant, U.S.A. 2 School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Australia The aerodynamics of a tennis ball are reviewed here with reference to several wind tunnel measurement efforts. Measurements for a wide variety of tennis balls, including the oversized balls, are presented. Flow visualization results have shown that the separation location on a non-spinning tennis ball occurred relatively early, near the apex, and appeared very similar to a laminar separation in the subcritical Reynolds number regime. The flow regime (boundary layer separation location) appears to be independent of Reynolds number in the range, 167,000oReo284,000. Asymmetric boundary layer separation and a deflected wake flow, depicting the Magnus effect, have been observed for the spinning ball. Aerodynamic force (drag and lift) measurements for nonspinning and spinning balls are reviewed for a wide range of Reynolds numbers and spin rates. Relatively high drag coefficients (C D ffi0.6 to 0.7), have been measured for new nonspinning tennis balls. The observed (unexpected) behavior of the tennis ball drag coefficient is explained in terms of a flow model that includes the drag contribution of the fuzz elements. & 2008 John Wiley and Sons Asia Pte Ltd Keywords:. tennis. tennis balls. aerodynamics. coefficient of drag 1. HISTORICAL BACKGROUNDy The game of tennis originated in France some time during the 12th century and was referred to as jes de paume, the game of the palm played with the bare hand. As early as the 12th century, a glove was used to protect the hand. Starting in the 16th century and continuing until the middle of the 18th century, rackets of various shapes and sizes were introduced. Around 1750, the present configuration of a lopsided head, thick gut and longer handle emerged. The original game known as real tennis, was played on a stone surface surrounded by four high walls and covered by a sloping roof. The shape of the new racket enabled players to scoop balls out of the corners and to put cut or spin on the ball. The rackets were usually made of hickory or ash and heavy sheep gut was *209 Orchard Glen Court, Mountain View, CA 9404, U.S.A. E-mail: rabi44@aol.com ya substantial part of this section 1. Historical Background has been reproduced from Balls and Ballistics, In: Materials in Sports Equipment, ed: Mike Jenkins, ISBN: 1 85573 599 7, by kind permission of Woodhead Publishing. used for the strings. The old way of stringing a racket was to loop the side strings round the main strings. This produced a rough and smooth effect in the strings, hence the practice of calling rough or smooth to win the toss at the start of a tennis match. Only royalty and the very wealthy played the game. The oldest surviving real tennis court, located at Hampton Court Palace, was built by King Henry VIII in approximately 1530. The present day game of lawn tennis was derived from real tennis in 1873 by a Welsh army officer, Major Walter Wingfield. Balls used in the early days of real tennis were made of leather stuffed with wool or hair. They were hard enough to cause injury or even death. Starting from the 18th century, strips of wool were wound tightly around a nucleus of strips rolled into a small ball. String was then tied in different directions around the ball and a white cloth covering was sewn around it. The original lawn tennis ball was made of India rubber, the result of a vulcanisation process invented by Charles Goodyear in the 1850s. Today, the size, bounce, deformation and colour of the ball must be approved by the world governing body for tennis, the International Tennis Federation (ITF). Ball performance characteristics are based on varying dynamic and aerodynamic Sports Technol. 2008, 1, No. 1, 7 16 & 2008 John Wiley and Sons Asia Pte Ltd 7

Review properties. Tennis balls are classified as Type 1 (fast speed), Type 2 (medium speed), Type 3 (slow speed) and high altitude. Type 1 balls are intended for slow pace court surfaces, such as clay. Type 2 balls, the traditional standard tennis balls, are meant for medium paced courts, such as a hard court. Type 3 balls are intended for fast courts, such as grass. High altitude balls are designed for play above 1219 m (4000 ft). Tennis balls may be pressurised or pressureless. Today s pressurised ball design consists of a hollow rubber-compound core, containing a slightly pressurized gas and covered by a felt fabric cover. The hourglass seam on the ball is a result of the adhesive drying during the curing process. Once removed from its pressurised container, the gases within a pressurised ball begin to leak through the core and fabric and the ball eventually loses bounce. Pressureless balls are filled with microcellular material. Subsequently, pressureless balls wear from play, but do not lose bounce through gas leakage. As a costsaving measure, pressureless balls are often recommended for people who play infrequently. The tennis ball must have a uniform outer surface consisting of a fabric cover and be white or yellow in colour. Ball seams must be free of stitches. All balls must weigh more than 56.0 g and less than 59.4 g. Types 1 and 2 ball diameters must be between 6.541 cm and 6.858 cm; Type 3 balls must be between 6.985 cm and 7.302 cm in diameter. It was in fact the flight of a tennis ball that first inspired scientists to think and write about sports ball aerodynamics. Newton [1] noted how the flight of a tennis ball was affected by spin and he wrote I remembered that I had often seen a tennis ball y describe such a curveline. For, a circular as well as a progressive motion being communicated to it by that stroke, its part on that side, where the motions conspire, must press and beat the contiguous air more violently than on the other, and there excite a reluctancy and reaction of the air proportionably greater. Over 200 years later, Rayleigh [2] in a paper entitled On the Irregular Flight of a Tennis Ball, commented that y a rapidly rotating ball moving through the air will often deviate considerably from the vertical plane. He added the following interesting thoughts: y if the ball rotate, the friction between the solid surface and the adjacent air will generate a sort of whirlpool of rotating air, whose effect may be to modify the force due to the stream. Despite all this early attention when the first review article on sports ball aerodynamics was published [3], no detailed scientific studies on tennis balls had been reported in the open literature. 2. TENNIS BALL AERODYNAMICS STUDIES TO DATE The first published study of tennis ball aerodynamics was written by Stepanek [4] who measured the lift and drag coefficients on a spinning tennis ball simulating the topspin lob. The aerodynamic forces were determined by projecting spinning tennis balls into a wind tunnel test section. Empirical correlations for the lift and drag coefficients (C L and C D ) were derived in terms of the spin parameter (S) only; it was concluded that a Reynolds number dependence could be neglected. Stepanek measured values of between 0.55 and 0.75 for C D, and between 0.075 and 0.275 for C L, depending on the spin parameter (S), which was varied between about 0.05 and 0.6. The extrapolated C D for the non-spinning case was found to be approximately 0.51. Some work on the aeromechanical and aerodynamic behaviour of tennis balls was conducted in the Engineering Department at Cambridge University in the late 1990 s [5,6]. One of the more significant conclusions of these investigations was that the tennis ball would reach a quasi-steady aerodynamic state very soon after leaving the racket, in approximately 10 ball diameters, which is equivalent to only 3% of its trajectory [5]. So the initial transient stage, when the ball is still deformed and the flow around it is still developing, will not generally make a significant contribution to the overall flight path. Based on comparisons with Achenbach s [7,8] drag measurements on rough spheres, it was estimated that the critical Reynolds number for a tennis ball would be about 85 000, based on a nap or fuzz height of about 1 mm. It was therefore deduced that for Reynolds numbers normally encountered during a serve, 100 000oReo200 000 (corresponding to a serving velocity range of 26oUo46 m/s [93.6oUo165.5 km/h]), the ball would be in the supercritical regime giving a drag coefficient of approximately 0.3 to 0.4. However, recent measurements on nonspinning tennis balls [9 14] showed that the drag coefficient was higher and appeared to be independent of Reynolds number. 2.1 Effects of Fuzz R. Mehta, F. Alam and A. Subic Chadwick and Haake [15] obtained tennis ball C D measurements using a force balance mounted in a wind tunnel. The initial measurements gave a C D of approximately 0.52 for a standard tennis ball and it was found to be independent of Re over the range, 200 000oReo270 000. Chadwick and Haake [16] and Haake et al. [9] reported that C D ffi0.55 over the same Re range for a standard tennis ball, a pressureless ball and a larger ball. The difference between the two reported C D levels is attributed to the technique used to measure the ball diameter [5]. Chadwick and Haake [15] used an outer (projected) diameter, which included the nap or fuzz height. Their results also showed that the tennis ball C D could be increased (by raising the fuzz) or decreased (by shaving off the fuzz) by up to 10% [9,15,16]. More recently, Alam et al. [12,17,18] conducted a series of experimental investigations on more than 12 different tennis balls used in various tournaments around the world, shown in Figure 1. The objectives of these studies were to verify previously published results and to quantify the spin effects on tennis ball aerodynamics. Physical dimensions of these balls are shown in Table 1. Alam et al. [12,17] reported that the average drag coefficient for non-spinning new tennis balls varies between 0.55 and 0.65 (see Figure 2). These values are slightly higher compared to previous studies [4,9,15]. However, recent measurements conducted by Mehta strongly support the findings of Alam et al. A detailed explanation is given below in the discussion section. In addition, both investigations attempted to quantify the effects of seam orientation (as a tennis ball possesses complex seam) on drag coefficients. 8 www.sportstechjournal.com & 2008 John Wiley and Sons Asia Pte Ltd Sports Technol. 2008, 1, No. 1, 7 16

Review of tennis ball aerodynamics Figure 1. Balls used for experimental measurements [13]. 2.2 Effects of Seam Unlike cricket balls and baseballs [3], the seam on a tennis ball is indented and the cover surface is very rough, thus obscuring or overwhelming any seam effects. Although ball seam orientation can affect the flight and trajectory of other sports balls, these effects were not significant on the tennis ball. A study conducted by Mehta and Pallis [11] at Reynolds numbers between 46 000 and 161 000 on two Wilson US Open tennis balls using quantitative measurements and flow visualisation concluded that there were no significant effects of the seam on the aerodynamic properties of tennis balls. They also reported that for Re4150 000, the data in the transcritical regime for each ball can be averaged to give a single value for Sports Technol. 2008, 1, No. 1, 7 16 & 2008 John Wiley and Sons Asia Pte Ltd www.sportstechjournal.com 9

Review R. Mehta, F. Alam and A. Subic Table 1. Physical dimensions for some widely used tennis balls [13]. Ball name Mass (gm) Diameter (mm) Bartlett 57 65.0 Wilson Rally 2 57 69.0 Wilson US Open 3 58 64.5 Wilson DC 2 59 64.5 Slazenger 1 57 65.5 Slazenger 4 57 65.5 Dunlop 3 TI 57 65.5 Kennex Pro 57 64.0 Tretorn Micro X 58 65.0 Penn Tennis Master series 58 63.5 Tretorn Plus 58 64.5 Dunlop 2 Grand Prix 57 65.5 Figure 4. Orientation of seam towards wind direction, Wilson US Open 3 [12]. 2.3 Effects of Larger Diameter Figure 2. Drag coefficients as function of Reynolds number for a series of new tennis balls [13]. Figure 3. Drag coefficient versus Reynolds number for new tennis balls in transcritical regime [11]. the C D and these data are presented in Figure 3. These findings have been confirmed by Alam et al. [12 14]. The study was based on six new tennis balls with four different seam orientations as shown in Figure 4. The study showed that the seam orientation has minimal effect at high Reynolds numbers (over 92 000 or 80 km/h speed). However, an average of 8% drag coefficient variation due to seam orientation was found at lower Reynolds numbers (below 80 km/h speed). Serve has become a dominant factor influencing the outcomes of tennis games as the ball travels very fast and the returning player and spectators cannot follow the flight of the ball. To slow down the serve, the ITF decided in the 1990 s to start field testing of a slightly larger oversized tennis ball. This decision was instigated by a concern that the serving speed in (men s) tennis had increased to the point where the serve dominates the game. The fastest recorded serve was produced by Greg Ruzedski in March 1998, measured at 66.6 m/s or 240 km/h [19]. The main evidence for the domination of the serve in men s tennis has been the increase in the number of sets decided by tie breaks at the major tournaments [9]. This is particularly noticeable on the faster grass courts, such as those used at Wimbledon. Today, players not only can serve at very high speeds but also can impart high spin rates. Alam and Subic [20] compiled data from one of the major tournaments held in the U.S.A. The entire tournament was filmed using a high speed camera. The average speed and spin introduced by some of the renowned tennis players are shown in Table 2a. Generally, if the diameter of the ball is larger, the drag force will be greater due to the larger projected frontal area. A larger diameter ball, such as the Wilson Rally 2 (69 mm diameter compared to a regular diameter of 64.5 mm) was developed (see Figure 5). Tests conducted by Mehta and Pallis [10], Pallis and Mehta [21], Haake et al. [9] and Alam et al. [12 14] indicated that there is no significant variation in drag coefficient of larger diameter ball compared to regular sized balls. The important point to note is that the C D values for the larger balls are comparable to those for the regular balls. Of course, this is not all that surprising because a simple scaling of the size should not affect the C D, as long as other parameters, 10 www.sportstechjournal.com & 2008 John Wiley and Sons Asia Pte Ltd Sports Technol. 2008, 1, No. 1, 7 16

Review of tennis ball aerodynamics This trajectory plot illustrates the significance of the reduction in C D, on a typical tennis stroke. It was shown that if the drag coefficient of the worn ball was reduced even further, then the ball would travel faster through the air, and give the receiver a significantly shorter time to react to the shot. Mehta and Pallis [11] and Haake et al. [21] also studied the effects of spinning balls. They initially measured aerodynamic properties at spin rate of 1 4 revs/sec for a larger diameter Figure 5. Comparison of a normal sized tennis ball (a) to the oversized ball (b) with a 6.5% larger diameter; [13]. such as the surface characteristics (e.g. the fuzz), are not altered significantly. As mentioned earlier, the drag on the oversized ball will increase by an amount proportional to the projected frontal (cross-section) area, and the desired effect of increasing the flight time for a given serve velocity will be attained. However, Alam et al. [12] found that the Bartlett ball with a diameter of 65 mm has the highest drag coefficient (over 15%). A close visual inspection revealed that the Bartlett ball has a prominent seam (width and depth) compared to other regular balls. 3. AERODYNAMICS OF SPINNING TENNIS BALLS Modern day tennis players not only serve very fast but also spin the ball at a high rate (see Table 2). Spinning can affect the aerodynamic drag and lift of a tennis ball, and thus the motion and trajectory of the ball. The so-called Magnus effect on a sphere is well-known in fluid mechanics. In tennis, apart from the flat serve where there is zero or very little spin imparted to the ball, almost all other shots involve the ball rotating around some axis. In addition to Stepanek s [4] earlier work, the aerodynamics of spinning tennis balls was recently studied by Chadwick [22], Goodwill and Haake [23], Alam et al. [13,14,18] and Mehta and Pallis [10,11]. In this case, apart from the drag and gravitational forces, the lift (or side) force also come into play because a Magnus force is generated due to the spin. Goodwill and Haake [23] measured the aerodynamic forces of new balls, as well as some worn balls (60, 500, 1000 and 1500 impacts, which approximately corresponds to two to 50 games if only one ball is used). For the non-spinning tests, the measurements were conducted in the Reynolds number range of 85 000oReo250 000, which corresponds to a velocity range of 20oUo60 m/s. Tests for the spinning conditions (250 2750 rpm) were conducted at wind speeds of 25 and 50 m/ s. The data for the new tennis balls revealed that all balls have similar drag coefficients (0.6 0.7). However, a heavily worn ball exhibits a slight decrease in drag coefficient (Figure 6a). The study also found that a worn ball produces slightly lower lift coefficient compared to a new tennis ball. However, the authors also noted that the differences in lift and drag coefficients of new and worn balls are negligible at high Reynolds numbers. Based on these findings, the authors estimated flight trajectory for a new ball and a worn ball, shown in Figure 6b. Table 2a. Average speed and spin rate for some male tennis players [20]. Player name Table 2b. Average speed and spin rate for some female tennis players [20]. Player Name Average speed (km/h) Average speed (km/h) Average spin (rpm) Andre Agassi 164 2249 9 Mark Philippoussis 198 2198 3 Pete Sampras 193 2699 11 Tomas Muster 169 2754 8 Michael Chang 180 1677 7 Tim Henman 193 1548 2 Average spin (rpm) Venus Williams 151 2598 8 Anna Kournikova 146 2250 12 Monica Seles 153 1287 9 Lindsay Davenport 145 2678 9 Mary Jo Fernandez 138 601 9 Martina Hingis 2103 5 No. serve No. serve Figure 6. (a) Drag coefficient of ten worn balls (two of each category) versus Reynolds number; (b) predicted trajectory for new and worn standard size balls and an oversize ball; [23]. Sports Technol. 2008, 1, No. 1, 7 16 & 2008 John Wiley and Sons Asia Pte Ltd www.sportstechjournal.com 11

Review model ball (280 mm) at Reynolds numbers between 167 000 284 000 and standard tennis ball at 18 72 revs/sec under range of speeds (39 66 m/s). The C D for the spinning balls are shown in Figure 7 as a function of the spin parameter (S) for Reynolds number 5 105 000 and 210 000, respectively. For balls subjected to 0 and 60 impacts, the C D increases with S, presumably due to the fuzz elements standing up when the ball is rotated [22]. Also, note that with lift generated on spinning balls, there will be an additional contribution of induced drag. The lower C D on the worn balls is still evident with the maximum difference apparent at S 5 0.15 with the new ball C D 5 0.67 versus 0.61 for one with 1500 impacts. For the higher Reynolds number of 210 000, the C D for the new ball is about 0.03 0.04 higher than that of the heavily worn ball for all values of S. The data for the lift coefficient, C L, are shown in Figure 8, again for Reynolds number 5 105 000 and 210 000, respectively. In general, the C L increases with S for all the balls, as would be expected, with almost linear relations at both values of Reynolds number. For the lower Reynolds number, there is some effect of wear on the C L, especially at R. Mehta, F. Alam and A. Subic the lower values of S, but this trend is not repeated at the higher Reynolds number. In general, there does not appear to be any strong effect of wear on the ball lift coefficient. Alam et al. [13, 18] conducted a series of experimental studies using a six component force sensor in a wind tunnel with a test section 3 m wide, 2 m high and 9 m long (Figure 9). Twelve balls were used for spinning tests under a range of Reynolds numbers (46 000 and 161 000; speeds of 40 140 km/ h) at a spin rate of 500 3000 rpm (8.33 50 rev/s). The results indicate that with an increase of spin rate, the lift coefficient or down force coefficient depending on topspin or back spin increases. However, with an increase of Reynolds numbers, the Figure 10. Wind tunnel set-up for smoke flow visualization studies over a 28 cm (11 in) diameter tennis ball model at NASA Ames Research Center, flow in the wind tunnel is from left to right [11] (Image courtesy of NASA Ames Research Center and Cislunar Aerospace Inc.). Figure 7. Drag coefficient for spinning balls. (a) U 5 25 m/s (Re 5 105 000); (b) U 5 50 m/s (Re 5 210 000); [23]. Figure 8. Lift coefficient for spinning balls. (a) U 5 25 m/s (Re 5 105 000); (b) U 5 50 m/s (Re 5 210 000); [23]. Figure 11. Flow visualization of 28 cm diameter tennis ball model with no spin (Re 5 167 000) at NASA Ames Research Center. Flow is from left to right [11] (Image courtesy of NASA Ames Research Center and Cislunar Aerospace Inc.). Figure 9. Test section of RMIT University Industrial Wind Tunnel with tennis ball experimental set-up [18]. Figure 12. Flow visualization on ball with topspin (counter-clockwise rotation at 4 revs/sec, Re 5 167 000) at NASA Ames Research Center, flow is from left to right [11] (Image courtesy of NASA Ames Research Center and Cislunar Aerospace Inc.). 12 www.sportstechjournal.com & 2008 John Wiley and Sons Asia Pte Ltd Sports Technol. 2008, 1, No. 1, 7 16

Review of tennis ball aerodynamics Figure 13. Flow visualization on ball with underspin (clockwise rotation at 4 revs/sec, Re 5 167 000) at NASA Ames Research Center. Flow is from left to right [11] (Image courtesy of NASA Ames Research Center and Cislunar Aerospace Inc.). Figure 15. Flow regimes on a sphere [11, based on 24]. Figure 14. Flow pattern around a used tennis ball, side view. (a) Airflow around a worn ball (non-spinning); (b) airflow around a worn ball (spinning); [18]. lift force coefficient decreases. The reduction of lift force coefficients at high Reynolds numbers (over 120 km/h) is minimal. The studies also found that with an increase of spin rate, the drag coefficient also increases. The average C D value varies between 0.6 and 0.8, which is significantly higher compared to non-spinning balls. One of the reasons for higher drag coefficients of a tennis ball when spun is believed to be the effect of fuzz elements. A close visual inspection of each ball after the spin revealed that the hairy stuff (fuzz) comes outward from the surface (but not up-rooted) and the surface becomes very rough. As a result, it is believed that the fuzz element generates additional drag. However, as Reynolds number increases, the rough surface (fuzz elements) becomes streamlined and reduces the drag. Flow visualization photos of non-spinning and spinning new and worn tennis balls are shown in Figures 10 14. The relatively early boundary layer separation on the non-spinning ball and the asymmetric separation on the spinning balls are clearly observed in these photographs. The asymmetric separation leads to the generation of the Magnus force which makes the ball deviate from a straight flight path. 4. DISCUSSION The value of C D for tennis balls is relatively high for the following reasons. The turbulent boundary layer separation location for the tennis ball appears to be comparable to that seen for laminar separation at relatively low Reynolds number. Total drag on a bluff body, such as a sphere or tennis ball, consists mainly of pressure drag (due to the pressure difference Figure 16. Drag coefficient versus Reynolds number for smooth and rough spheres [11]. between the front and back of the ball) and a very small contribution due to viscous or skin friction drag (due to the no slip condition). Achenbach [24] showed that the viscous drag for a smooth sphere approaching the transcritical regime was about 2% of the total drag. So the bulk of the total drag is accounted for by pressure drag, which in turn is determined solely by the boundary layer separation location on the ball. Because the separation location for the tennis ball is comparable to that for laminar separation at low Reynolds number, one would expect the C D for the tennis ball to be around 0.5. Previously measured C D values were 0.51 [4] and 0.55 [9,16]. Alam et al. [12 14] determined C D values between 0.55 and 0.65, except for the Bartlett ball, which has a C D value of 0.70 at high Reynolds numbers. As mentioned earlier, Alam et al. s studies were based on 12 brand new (unused) tennis balls and the tests were conducted using six-component force sensor in an industrial wind tunnel. Mehta and Pallis [11] found the C D value of around 0.62 for new tennis balls. The results agreed well with the findings of Alam et al. [12 14]. However, some of the differences in the measured C D values between different investigations can be attributed to the different techniques used to measure the ball diameter. It is believed that higher C D values (over 0.5) are generated in the transcritical regime (Figure 15). Once transition has occurred, the transition and separation locations start to creep upstream and so the C D starts to increase. At some Sports Technol. 2008, 1, No. 1, 7 16 & 2008 John Wiley and Sons Asia Pte Ltd www.sportstechjournal.com 13

Review point the transition location moves all the way up to the stagnation location and the separation location is then totally determined by the development of the turbulent boundary layer. With increasing roughness, the boundary layer growth rate is increased, thus resulting in earlier separation and higher C D. The constant level achieved by the C D in the transcritical regime is also expected to increase with increasing roughness, as evidenced in Achenbach s [8] measurements (data for two roughness levels are shown in Figure 16). However, Achenbach s data show an upper limit of C D ffi0.4 on spheres with increasing roughness (figures 2 and 4 in Achenbach [8] show this limit for a k/d range of 0.0025 0.0125). The measured separation location for this value of C D was about hsffi1001. This is still in the region of the adverse pressure gradient and so one would expect the boundary layer separation location to continue moving upstream with increasing surface roughness. However, the point to note is that while the boundary layer growth (rate of thickening) increases with increasing roughness, so does the skin friction coefficient, and the behaviour of the separation location is then determined by the behaviour of these competing effects. The increasing skin friction coefficient makes the boundary layer more resilient to separation, thus opposing the tendency of a boundary layer to separate as it thickens. So it is entirely possible that for certain types of roughness, such as the round glass beads investigated by Achenbach for example, a limit is reached for the C D level in the transcritical regime because the effects of the boundary layer thickening are offset by those due to the increasing skin friction coefficient. In principle, though, there is no reason why the separation location cannot continue to creep forward for other types of roughness elements, which may be more effective at thickening the boundary layer than increasing the skin friction coefficient. Therefore, it is believed that the absolute limit for the turbulent boundary layer separation location in the transcritical regime is the same as that for laminar boundary layer separation in the subcritical regime (h s ffi801). Laminar boundary layer separation occurs upstream of the sphere apex because of the presence of an adverse pressure gradient in this region. The adverse pressure gradient is generated in this region due to an upstream influence of the separated near wake. One effect which occurs is that initially, when the flow is first turned on, the laminar boundary layer separates at the apex and immediately a pressure minimum is generated upstream of it due to streamline curvature effects, much in the same way as that generated near the exit region of a contraction [25]. Once this adverse pressure gradient is generated, the laminar boundary layer separation tends to move to that location. This is probably the most upstream location that the adverse pressure gradient can move up to. Assuming that a very thick (weak) turbulent boundary layer can become as prone to separation as a laminar layer, it will separate as soon as it encounters an adverse pressure gradient (at about h s ffi801), just like the laminar layer. Therefore, if the location for turbulent separation in the transcritical regime is similar to that of laminar separation in the subcritical regime, the pressure drag should also be comparable, thus giving a total drag of C D ffi0.5. R. Mehta, F. Alam and A. Subic On examining the tennis ball, the relatively rough surface on the felt is readily apparent. The roughness actually results from the junctions of the fuzz elements, where they are embedded within the fabric covering on the ball. However, in addition, the fuzz elements have a finite thickness and length and this forms an additional porous coating on the ball through which air can still flow. So the tennis ball can be thought of as a very rough sphere with a porous coating. Subsequently, each fuzz element will also experience pressure drag and when this is summed-up for all the fuzz elements on the ball s surface, the additional drag contribution is obtained and this is herein termed the fuzz drag. Therefore the present data suggest that the contribution of the fuzz drag to the total drag on the tennis ball is between 20 and 40%, depending on the Reynolds number. The other trend in the tennis ball C D measurements, which was initially puzzling, was the higher values of C D at the lower Reynolds number (Figure 17). At first it was tempting to discard the trend by attributing it to experimental error because both the tunnel reference pressure and drag force (drag count), become harder to measure accurately as the wind tunnel flow speed reduces (the percentage error increases as the mean values are lower). Compared to the smooth sphere, the overall drag count error for the tennis balls would be lower because the drag is higher. The first effect, which is perhaps not too surprising, is the change in orientation of some of the Figure 17. Drag coefficient versus Reynolds number for new tennis balls [11]. Figure 18. Effect of flow velocity on fuzz element orientation, flow is from left to right. (a) U 5 20 m/s (45 mph, Re 5 100 000); (b) U 5 60 m/s (135 mph, Re 5 260 000); [11] (Image courtesy of NASA Ames Research Center and Cislunar Aerospace Inc.). 14 www.sportstechjournal.com & 2008 John Wiley and Sons Asia Pte Ltd Sports Technol. 2008, 1, No. 1, 7 16

Review of tennis ball aerodynamics 5. CONCLUDING REMARKS A comprehensive review of the published research to date has led to the following conclusions regarding tennis ball aerodynamics: Figure 19. Drag coefficient versus Reynolds number for used Wilson US Open balls [11]. filaments. As the flow velocity is increased, many of the filaments that are initially standing almost perpendicular to the ball s surface are forced to lay down due to aerodynamic drag effects. Note how in Figure 18 the fuzz filaments, particularly over the front face of the ball and up to the apex region, tend to lay down at the higher flow speed. Hence, the contribution of the fuzz drag is reduced at the higher flow speeds or Reynolds numbers. Also, the fuzz element Reynolds number (based on filament diameter) is estimated to be of order 20, and this puts it in a range where the C D (for a circular cylinder) is much higher (C D ffi3) and a strong function of the Reynolds number, with the C D decreasing with increasing Reynolds number [26, figure 10.12]. Therefore the higher C D level at the lower ball Reynolds number is attributed to the combined effect of fuzz filament orientation and Reynolds number effects on the individual filaments. The critical role of fuzz in determining tennis ball drag was borne out succinctly in the results for the used balls (Figure 19). One of the balls was used in the 1997 US Open for nine games and the other used balls were played with by recreational players for the noted number of games, using only two balls at a time. The baseline data for the new Wilson US Open ball are also shown. The 1997 US Open ball indicates a supercritical behaviour with the critical Reynolds number of approximately 100 000 and then a gradual approach towards the transcritical regime. For the balls used by the recreational players, after three games the C D behaviour is comparable to that of the new ball. However, after six games the C D is clearly higher and to confirm this increase, both the used balls were tested and showed a very consistent trend with excellent repeatability. This initial increase in C D is known to tennis players who often refer to the felt as having fluffed-up (fuzz is raised), which would obviously account for the higher drag. The fluffing-up was not apparent when these two balls were examined visually, but the present C D measurements strongly suggest that the felt texture, perhaps the internal structure, must have been affected. The nine game ball clearly exhibited a worn felt and therefore it was not too surprising to see a lower measured C D. The fact that a fluffed-up felt results in an increased C D and a worn felt in a reduced C D was not too surprising because the effect had been previously observed [9,15,16]. The flow over a new tennis ball is generally in the transcritical regime, where the separation location does not move significantly with Reynolds numbers. This in turn implies that the C D is independent of Reynolds numbers because the total drag on a bluff body, such as a round ball, is almost completely accounted for by the pressure drag. The fuzz on a tennis ball causes early transition of the laminar boundary layer and rapid thickening of the turbulent boundary layer. This results in the separation location moving up to the apex region, comparable to that for laminar boundary layer separation at subcritical Reynolds number. The high C D value at low Reynolds numbers is believed to be the combined effect of fuzz filament orientation and Reynolds number effects on the individual filaments. The average drag coefficient varies between 0.55 and 0.65 for new tennis balls in the Reynolds number range of 69 000oReo161 000 (60 140 km/h). However, the C D value for worn (used) tennis balls is slightly lower compared to new tennis balls. The seam orientation has negligible effect on drag coefficient at high Reynolds numbers. However, some effects have been noted at lower Reynolds numbers (8% increase of C D value). The average drag coefficient of a recently approved oversized tennis ball is comparable to that of the standard-sized balls. However, the drag on the oversized balls is higher by virtue of the larger cross-sectional area and so the desired effect of slowing down the game (increased tennis ball flight time) will be achieved. However, slowing down the game can also be achieved by using a regular size Bartlett tennis ball as it has the highest C D value of 0.7. Spin has significant effects on the drag and lift force coefficients. The C D increases with an increase of spin rate. The average C D value varies between 0.6 and 0.8 for the spin rate of 8.33 50 rev/s in the Reynolds number range, 69 000 161 000 (60 140 km/h). The lift or down force coefficient (depending on spinning direction) also increases with an increase of spin rate, as expected. However, the rate of increase of C L reduces at higher Reynolds numbers (161 000). REFERENCES 1. Newton I. New theory of light and colours. Philosophical Transactions of the Royal Society London 1672; 1: 678 688. 2. Rayleigh, Lord. On the irregular flight of a tennis ball. Messenger of Mathematics 1877; 7: 14 16. 3. Mehta RD. Aerodynamics of sports balls. Annual Review of Fluid Mechanics 1985; 17: 151 189. 4. Stepanek A. The aerodynamics of tennis balls the topspin lob. American Journal of Physics 1988; 56: 138 141. Sports Technol. 2008, 1, No. 1, 7 16 & 2008 John Wiley and Sons Asia Pte Ltd www.sportstechjournal.com 15

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