European Journal of Physics PAPER Effects of turbulence on the drag force on a golf ball To cite this article: Rod Cross 2016 Eur. J. Phys. 37 054001 View the article online for updates and enhancements. Related content - Measurements of drag and lift on smooth balls in flight Rod Cross and Crawford Lindsey - The effect of spin in swing bowling in cricket: model trajectories for spin alone Garry Robinson and Ian Robinson - Physics of knuckleballs Baptiste Darbois Texier, Caroline Cohen, David Quéré et al. Recent citations - Resources for sports engineering education Tom Allen and John Eric Goff This content was downloaded from IP address 37.44.203.148 on 07/02/2018 at 18:10
European Journal of Physics Eur. J. Phys. 37 (2016) 054001 (9pp) doi:10.1088/0143-0807/37/5/054001 Effects of turbulence on the drag force on a golf ball Rod Cross Physics Department, University of Sydney, Sydney, NSW, Australia E-mail: rodney.cross@sydney.edu.au Received 16 May 2016, revised 16 June 2016 Accepted for publication 5 July 2016 Published 10 August 2016 Abstract Measurements are presented of the drag force on a golf ball dropped vertically into a tank of water. As observed previously in air, the drag coefficient drops sharply when the flow becomes turbulent. The experiment would be suitable for undergraduate students since it can be undertaken at low ball speeds and since the effects of turbulence are easily observed on video film. A modified golf ball was used to show how a ball with a smooth and a rough side, such as a cricket ball, is subject to a side force when the ball surface itself is asymmetrical in the transverse direction. Keywords: dimples, Reynolds number, drag crisis, separation point (Some figures may appear in colour only in the online journal) 1. Introduction It is well known that the aerodynamic drag force on a ball depends on its diameter, speed and surface roughness. A surprising result is that the drag force can be smaller on a rough ball than a smooth ball, especially at speeds of interest in ball sports. It is for that reason that golf balls are dimpled. The dimples act to generate turbulent flow around the ball at relatively low ball speeds, with the result that the air pressure immediately behind the ball is higher than it would otherwise be if the flow was laminar. Consequently, the drag force on the ball is reduced. The effect is commonly studied in a wind tunnel, a device that is not normally available in an undergraduate physics laboratory. Nevertheless, the effect can easily be observed in a student laboratory, using water instead of air. Differences between golf and smooth balls can be found in many books and articles on aerodynamics [1 5], and are summarised in figure 1 in terms of the variation in the drag coefficient, C D, with Reynolds number, Re. With a smooth ball, the onset of turbulence and a corresponding drop in the drag coefficient occurs at Re» 200 000, where Re = rvd h and 0143-0807/16/054001+09$33.00 2016 IOP Publishing Ltd Printed in the UK 1
Figure 1. Typical variation of the drag coefficient, C D versus Re for a smooth ball and a golf ball. Figure 2. Arrangement used to measure the speed of a golf ball dropped vertically into a fish tank. The average ball speed over the 10 cm travel distance is v = 0.5( v + v ). A 1 2 where ρ is the fluid density, v is the ball speed, D is the ball diameter and η is the fluid viscosity. For air, r = 1.21 kg m 3 and h = 1.8 10-5 Pa s. For a golf ball, D = 42.7 mm. For a smooth golf ball in air, turbulence would commence at about v = 70 m s 1. The longest drives in golf are struck at ball speeds around 80 m s 1 but the ball slows down through the air and the dimples act to carry the ball further. For a dimpled golf ball, turbulence commences at Re» 50 000 or at a ball speed about 17 ms 1. For a dimpled golf ball in water, where r = 1000 kg m 3 and h = 8.9 10-4 Pa s, turbulence commences at a ball speed of only about 1.0 ms 1. In this paper, measurements are presented of the drag force on a golf ball dropped vertically at low speed into a fish tank filled with water, as indicated in figure 2 and as described previously [6]. The velocity of the ball through the water was measured by filming the ball with a video camera, at drop heights varying from zero up to 70 cm. Motion of the ball was analysed using Tracker motion analysis software. The advantages of this technique are that (a) the experiment can easily be conducted at low ball speeds in an undergraduate laboratory and (b) the onset of turbulence can be observed visually. More commonly, this type of experiment is undertaken in undergraduate laboratories using steel balls in a viscous 2
Figure 3. Ball height, y, versus time for a drop height of 1 cm using (a) a quadratic fit and (b) a cubic fit to the data points. fluid, in order to measure the terminal velocity [7]. In the present experiment, the ball speed did not remain constant and the drag coefficient varied with ball speed, thereby leading to a more challenging problem in data analysis. 2. Data analysis A potential disadvantage of the technique described in this paper is that the drag force is not measured at a fixed ball speed. In a wind tunnel, the drag force is measured at a fixed air speed, while the ball remains at rest in the tunnel. The drag coefficient is calculated from the measured force on the ball and can be determined over a range of different air speeds simply by changing the air speed. Similarly, when measuring the terminal velocity of a ball in a fluid, the ball speed remains constant. The ball speed can be changed by varying the mass or diameter of the ball. In the present experiment, the force on the ball is measured in terms of the deceleration of the ball as it falls through the water. The drag force, F D, is conventionally described by the relation 1 FD = CDrAv2, ( 1) 2 where C D is the drag coefficient, ρ is the fluid density, A = pr 2 is the cross-sectional area of the ball, R is the ball radius and v is the relative speed of the ball and the fluid. The drag coefficient can therefore be calculated in terms of the measured drag force and the measured value of v. Therein lies a problem with the present technique. If the velocity decreases by a factor of say two as the ball falls through the water, then v 2 changes by a factor of four. The calculated value of C D could therefore vary by a large factor, depending on the assumed value of v. The uncertainty in the quoted value of C D can be reduced by using average values of v and F D during the fall and by measuring the deceleration of the ball over a relatively small fall distance. However, the uncertainty in the measured deceleration increases when the change in velocity decreases, especially when the deceleration is determined by differentiating the displacement versus time data twice. Nevertheless, values of C D were obtained with a 3
Figure 4. Drag force F D versus average ball speed v A for 42 different ball drops. The solid curve is a best fit curve assuming that F D is proportional to v A 2. maximum estimated error of about ±10% by filming the fall at 300 frames s 1 with a Casio EX-F1 video camera and by restricting the measured fall distance to about 10 cm. When a ball of mass M is falling vertically through water at speed v, the equation of motion has the form d M v dt = Mg - FB - FD, ( 2) where FB = ( 4 3) pr3rg is the buoyant force. For the golf ball used in this experiment, M = 45.44 0.01 g, D = 42.70 0.05 mm, FB = 0.400 N and Mg = 0.445 N. The buoyant force is only slightly less than Mg, and the dominant force on the ball at high ball speeds is the drag force. The terminal velocity is about 0.4 ms 1 and can be measured by releasing the ball from rest when it is partly submerged in the upper surface of the water. When dropped from a height of 1 cm or more, the ball decelerated over the whole 23 cm of the fall distance to the bottom of the fish tank. At a drop height of 1 cm above the water surface, the ball speed decreased from about v1 = 0.7 ms 1 to about v2 = 0.42 ms 1 over the 10 cm fall distance shown in figure 2. A typical result is shown in figure 3, where the vertical height of the bottom of the ball above the bottom of the tank, y, is shown as a function of time. All y(t) data for all drop heights could be fitted accurately by a quadratic with a correlation coefficient greater than 0.9999. In figure 2(a), y = 0.146-0.693t + 0.746t 2, indicating that v =- 0.693 + 1.492t and dv dt = 1.492 ms 2. In theory, the deceleration should not remain constant in time since the drag force is proportional to v 2. A cubic fit to the experimental data solves this problem, as indicated in figure 3(b). For the cubic fit, dv dt = 1.80 3.408t, indicating that the acceleration decreases with time since the drag force decreases with time. In terms of the correlation coefficient, the cubic fit was found to be indistinguishable from the excellent quadratic fit shown in figure 3(a), and in all other cases. Furthermore, the mean value of the deceleration obtained from a cubic fit was the same, within experimental error, as the constant value of the deceleration obtained from the simple quadratic fits. Consequently, the deceleration value obtained from the quadratic fit in figure 3(a) was used to calculate the drag force 4
Figure 5. Drag coefficient C D versus average ball speed v A for the 42 different ball drops. corresponding to the average ball speed, va = ( v1 + v2) 2 = 0.56 ms 1,infigure 3(a). From equation (2), FD = 0.114 N, giving CD = 0.51 from equation (1), consistent with other measurements of C D for a golf ball at low ball speeds [1 5]. The same procedure was used to calculate C D for all other drop heights. The procedure was checked by solving equation (2) numerically to find F D, v and y as functions of time for a given value of C D. A quadratic fit toy(t) was then used to estimate the average deceleration and the average ball speed, giving a value for C D consistent with the assumed value to within 10% or better. 3. Drag force results Measurements of the drag force are shown in figure 4 as a function of average ball speed for 42 individual ball drops. For comparison, a smooth curve fit is shown, assuming that F D is proportional to v A squared and that C D is independent of v A. The fitted curve implies from equation (1) that C D has a constant value of 0.335 at all ball speeds. In fact, C D varies with v A. The calculated values of C D, obtained from the measured values of F D and v A, are shown for each ball drop in figure 5. The value of C D averaged over all 42 drops was 0.38. The result is consistent with previous measurements of C D for a golf ball in air [1 5], showing a sudden drop in C D from about 0.5 to about 0.25 when Re is about 50 000 or when v A is about 1.0 ms 1 in water. 4. Onset of turbulence Several video images of the falling ball are presented in figure 6, showing the onset of turbulence behind the ball when the ball speed exceeds about 1.2 m s 1. At lower ball speeds, the flow of water around the ball is laminar and is not visible in the video film. Furthermore, the ball enters the water silently at low ball speeds and there is no splash. At ball speeds above about 1.2 ms 1, there is a loud splash since water is ejected from the rear of the ball into the air. The onset of turbulence coincides with a significant decrease in the drag coefficient, as 5
Figure 6. Video images at three different average ball speeds, v A, showing the onset of turbulence at higher ball speeds. The ball enters the water at the top of each image. indicated in figure 5. The sudden decrease in the drag coefficient as the ball speed increases is commonly known as the drag crisis. A more dramatic effect is observed at higher ball speeds, as shown in figure 7. As water travels from the front end of the ball vertically upwards, it separates from the ball at points near the front end, creating a relatively wide and deep air cavity behind the ball, plus a wine glass shape fountain of water extending up into the air. As the ball slows down, the separation points move towards the rear of the ball. As a result, water flows inwards towards the rear of the ball, rather than vertically, causing the air cavity to pinch inwards and eventually to separate completely from the ball. By the time the ball speed has decreased to 1.2 ms 1, the turbulent flow pattern immediately behind the ball in figure 7 is qualitatively similar to the flow pattern observed at 1.2 ms 1 in figure 6. In both cases, the separation points are near the back end of the ball rather than being near the front end. 6
Figure 7. Result when the golf ball was dropped from a height of 70 cm into the fish tank, at three selected times after the ball enters the water. As the ball slows down, the separation point moves from the front half of the ball to the rear half. 5. Simulation of side force on a cricket ball It is well known that the location of the separation points plays a critical role in fluid flows [1]. At low Reynolds numbers, the separation points on a sphere are close to the equator, meaning that water or air flows from the front of the ball to the equator and then separates from the ball, leaving a region of low pressure immediately behind the ball. Consequently, the drag force is relatively high and the drag coefficient is typically about 0.5. If the flow becomes turbulent, air or water can flow along the surface of the sphere past the equator before it separates, resulting in a higher pressure behind the ball and a reduction in the drag coefficient. At even higher speeds, the flow becomes turbulent sooner, the separation points move back towards the equator and the drag coefficient increases again. At the higher speeds, the separation point can even be well in front of the equator, as indicated at early times in figure 7. These effects were observed clearly in the present experiment and were investigated further using a modified golf ball where half the dimples were removed by sanding one half of the ball. Such a golf ball is illegal, but it is a well-established practice in cricket to polish one side of the ball during play and to allow the other side to roughen during play. The left-toright asymmetry introduces a left-to-right side force on the ball due to delayed separation on the rough side of the ball. As shown in [8] and [9], an equivalent effect can be obtained by tilting the spin axis so that the ball seam is aligned at an angle to the direction of motion of the ball. In that case the ball surface itself can remain smooth on both sides of the ball but the 7
Figure 8. Golf ball with dimples removed on the left half of ball. (a) The separation point is further behind the ball on the dimpled (right hand) side. (b) As the ball slows down, the separation point on the smooth side moves towards the rear and the air cavity separates from the ball. raised seam on one side generates turbulence when air encounters the seam. In other words, one side of the ball is rougher than the other because it contains a raised seam. The modified golf ball was dropped from a height of about 30 cm into the fish tank so that the smooth half was on one side of the ball and the dimpled half faced the other side. The result is shown in figure 8. Initially, the separation points were close to the front of the ball on the smooth (left hand) side of the ball, and close to the rear of the ball on the dimpled (right hand) side. As a result, the water was deflected to the left behind the ball. Since the force of the ball on the water acted from right to left, the force of the water on the ball acted from left to right, resulting in a sideways deflection of the ball from left to right. The magnitude of the sideways deflection in figure 8 can be estimated in terms of the sideways deflection of the ball compared with the point of entry into the water. As the ball slowed down, the separation point on the smooth side of the ball moved towards the rear of the ball, as indicated in figure 8(b), since the onset of turbulence was delayed at the lower ball speed. The separation point on the dimpled side of the ball is closer to the rear of the ball than in figures 6 and 7, probably because the ball is moving left to right as well as downwards. In terms of the left to right motion alone, the separation point is near the black line equator drawn around the ball rather than the rear (lhs) of the ball. In this situation, the location of the separation point should therefore be specified in terms of the actual direction of motion of the ball rather than the vertical direction. 8
6. Conclusions There have been many studies of the impact and fall of smooth spheres through water [10, 11] but the present study appears to be the first to examine the fall of a golf ball through water. The results show clearly the significance of turbulence in changing the location of the separation point on the ball and reducing the drag coefficient. These effects are often described in the literature without clear experimental evidence to illustrate those effects or rely on relatively old wind tunnel studies for experimental support. The experiment described in this paper can easily be performed by undergraduate students and would be of interest especially to those keen on ball sports such as soccer, baseball, cricket and golf where the drag crisis plays a major role in determining the ball speed and trajectory through the air. The experimental technique could also benefit researchers and ball manufacturers interested in the aerodynamics of sports balls. If the ball is spinning as it enters the water, it is possible to measure the lift (or Magnus) force as well as the drag force [12]. References [1] Fox R W, Pritchard P J and McDonald A T 2009 Introduction to Fluid Mechanics 7th edn (New York: Wiley) [2] MacDonald W M and Hanzely S 1991 The physics of the drive in golf Am. J. Phys. 59 213 8 [3] Libii J N 2007 Dimples and drag: experimental demonstration of the aerodynamics of golf balls Am. J. Phys. 75 760 3 [4] Choi J, Jeon W-P and Choi H 2006 Mechanism of drag reduction by dimples on a sphere Phys. Fluids 18 041702 [5] https://en.wikipedia.org/wiki/golf_ball [6] Cross R 2016 Vertical impact of a sphere falling into water Phys. Teach. 54 153 5 [7] Owen J and Ryu W 2005 The effects of linear and quadratic drag on falling spheres: an undergraduate laboratory Eur. J. Phys. 26 1085 91 [8] Mehta R D 2005 An overview of cricket ball swing Sports Eng. 8 181 92 [9] Cross R 2012 Aerodynamics in the classroom and at the ball park Am. J. Phys. 80 289 97 [10] Aristoff J, Truscott T, Techet A and Bush J 2010 The water entry of decelerating spheres Phys. Fluids 22 032102 [11] Duez C, Ybert C, Clanet C and Bocquet L 2007 Making a splash with water repellency Nat. Phys. 3 180 3 [12] Truscott T and Techet A 2009 Water entry of spinning spheres J. Fluid Mech. 625 135 65 9