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Review of tennis ball aerodynamics

Review DOI: 10.1002/jst.11

Review of tennis ball aerodynamics Rabindra Mehta1,, Firoz Alam 2 and Aleksandar Subic2 1 2

Sports Aerodynamics Consultant, U.S.A. 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 (CDffi0.6 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

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: [email protected] 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. Sports Technol. 2008, 1, No. 1, 7–16

Keywords: . . . .

tennis tennis balls aerodynamics coefficient of drag

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

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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 (CL and CD ) 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 CD , and between 0.075 and 0.275 for CL, depending on the

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spin parameter (S), which was varied between about 0.05 and 0.6. The extrapolated CD 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 Chadwick and Haake [15] obtained tennis ball CD measurements using a force balance mounted in a wind tunnel. The initial measurements gave a CD 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 CD 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 CD 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 CD 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.

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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. Sports Technol. 2008, 1, No. 1, 7–16

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

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Table 1. Physical dimensions for some widely used tennis balls [13]. Ball name

Mass (gm)

Diameter (mm)

Bartlett Wilson Rally 2 Wilson US Open 3 Wilson DC 2 Slazenger 1 Slazenger 4 Dunlop 3 TI Kennex Pro Tretorn Micro X Penn Tennis Master series Tretorn Plus Dunlop 2 Grand Prix

57 57 58 59 57 57 57 57 58 58 58 57

65.0 69.0 64.5 64.5 65.5 65.5 65.5 64.0 65.0 63.5 64.5 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 CD 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).

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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 CD 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 CD, as long as other parameters,

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Sports Technol. 2008, 1, No. 1, 7–16

Review of tennis ball aerodynamics

This trajectory plot illustrates the significance of the reduction in CD , 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. Sports Technol. 2008, 1, No. 1, 7–16

Table 2a. Average speed and spin rate for some male tennis players [20].

Player name

Average speed (km/h)

Average spin (rpm)

No. serve

Andre Agassi Mark Philippoussis Pete Sampras Tomas Muster Michael Chang Tim Henman

164 198 193 169 180 193

2249 2198 2699 2754 1677 1548

9 3 11 8 7 2

Table 2b. Average speed and spin rate for some female tennis players [20].

Player Name Venus Williams Anna Kournikova Monica Seles Lindsay Davenport Mary Jo Fernandez Martina Hingis

Average speed (km/h)

Average spin (rpm)

No. serve

151 146 153 145 138

2598 2250 1287 2678 601 2103

8 12 9 9 9 5

–

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].

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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 CD 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 CD 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 CD on the worn balls is still evident with the maximum difference apparent at S 5 0.15 with the new ball CD 5 0.67 versus 0.61 for one with 1...