THE GAME WITHOUT DIMPLES
Figure 2A shows a side elevation view of the average trajectory of PGA Tour players. The initial ball speed is 165 miles per hour, and the initial backspin rate is2,685 revolutions per minute. It is launched at an initial angle of 11.2 degrees and continues to climb at that rate, defying gravity, for about the first 130 yards of its flight. The flight path appears to be a straight line upward over this range, although closer examination shows that it actually curves upward. This allows the ball to rise to a height of 90 feet at around 170 yards or at about two-thirds of the flight distance. From this height, the descent takes a further 100 yards for a carry distance of 269 yards. In contrast, the lower dashed curve is the flight distance of a smooth golf ball given the same launch conditions, that is, struck by a driver with the same loft and at the same impact speed of 112 miles per hour.
Of course, if the game was played with smooth golf balls, then the launch conditions to obtain maximum distance would be different. In fact, the likely range of optimum launch conditions would be as shown in the upper dashed trajectories. The lower curve is for a launch of 23 degrees with the same 165 miles per hour speed and 2,685 revolutions per minute backspin. The upper one is for a launch at 27 degrees with the backspin rate decreased to only 1,000 revolutions per minute. This one carries 167 yards, still 100 yards less than the average PGA Tour distance.
To take this comparison one historical step further, we might imagine what the game might have become if the first golf balls had been smooth, molded ones, and the smooth ball had become the required standard. Without further regulation of the ball, except the rule that the ball should be smooth, they would likely have improved in springiness to get the higher ball speeds of today. However, to achieve the maximum drive distance of approximately 170 yards, the ball would
Figure 2A Average PGA Tour player performance with smooth and regular golf balls.
Have to be teed up about 6 inches off the ground and struck outside of the front foot to launch at a high angle using a driver having only a small loft to obtain a very low backspin rate. The average player would drive the ball only about 125 yards, somewhere around the distance of the average amateur 8- or 9-iron in the modern game. I think we get the picture that the game would be a whole lot less exciting.
So how does the modern ball with its precisely optimized dimple pattern make such a difference? To answer this question, we will first consider the golf ball as simply a passive, non-spinning projectile passing through the air. After this, the effect of giving the ball backspin to become an active flier will be considered. The effect of the latter is analogous to winding up the propeller on a rubber band powered model aircraft.
THE GOLF BALL AS A PROJECTILE
The fundamental effect of dimples is illustrated in Figure 2B—which gives the relative drag force acting on a smooth ball, a bullet-shaped projectile, and a golf ball at speeds up to 250 miles per hour. This is strictly a comparison of the relative forces for the different speeds and hides the fact that the drag force increases with the square of the ball speed. So, if the speed through the air doubles from 75 miles per hour to 150 miles per hour, the actual drag force on these objects increases fourfold. This is the main reason why, with all of their athleticism and raw power, the longest drivers of the ball only manage to carry the ball about twice as far as the newest beginner of the game. The relative drag force of Figure 2B simply takes out the effects of speed squared.
As seen in the top curve of Figure 2B, the relative drag force on a smooth ball has an approximately constant value near 1.0 over the range of ball speeds involved in most golf shots: approximately 50 to 200 miles per hour. In contrast, over the same speed range, a dimpled golf ball is subjected to a relative drag force less than half of that for the smooth ball. These differences are caused by the differences in the flow of the air around the ball and particularly the manner in which it separates from the ball as the ball passes by.
The upper illustration in Figure 2C shows how the air would flow around a smooth golf ball at these speeds. As the ball pushes forward, the air is forced to speed up sideways around the ball. The layer of air near to the ball on the leading surface gains energy of motion, so-called kinetic energy, and so it must lose energy in a different form to conform to the law of energy conservation. It suffers a net loss of internal energy. Of interest here is the drop in the pressure of the moving
Figure 2B Relative drag of a golf ball, smooth sphere, and bullet with the same diameter.
Air. It is easy to imagine the molecules of air separating further as they speed up around the outside of the ball, therefore becoming less dense with lowered pressure. Maximum air speed and minimum pressure occur as the air reaches the full diameter of the ball. Here the lower-pressure air must rejoin the higher-pressure air behind the ball. This causes a chaotic mixing of the lower- and higher-pressure air, resulting in turbulence and the formation of a wake.
The process thus dumps lower-pressure air behind the ball while the ball is pushing against higher-pressure air in front. This pressure difference between the front and back is what is referred to as drag. So to reduce drag, we need the high-velocity air passing over the ball to travel down the back surface of the ball. If it does this, it will lose speed and by conservation of energy regain some of the pressure. This will
Figure 2C Separation of air flow behind smooth and dimpled non-spinning balls.
Result in a smaller wake, with pressure nearer to the air ahead of the ball, therefore reducing drag. As shown in the lower illustration of Figure 2C, the dimples do this by “tripping” the layer of air passing over the ball, giving rise to a thin turbulent layer. This layer mixes with adjacent thin layers of higher-pressure air and is able to pass down over the rear surface of the ball before separating in a much-reduced wake. This behavior, of the airflow remaining in a thin layer around to the back of the ball, starts at a critical airspeed, which depends on the ball diameter and the surface roughness. For the golf ball diameter, the critical speed with a dimpled surface can be seen on the lower curve in Figure 2B to be approximately 40 miles per hour. For a smooth sphere, following the upper curve to its smallest value in the lower-right corner, it can be seen that the critical speed is approximately 225 miles per hour. So now we can see that dimples do not universally lower drag. It would seem very unusual that a rougher surface should always slip more easily through the air than a smooth one. Instead, the dimples simply move the critical speed lower so that golf is played in the so-called supercritical region of ball flight. Notice finally that in the supercritical region for a smooth ball, starting at 225 miles per hour, the relative drag force is less than half of that for the dimpled ball, which puts things back into common-sense order. For a game involving a more effective launch device, such as the catapult firing balls at speeds above 225 miles per hour, smooth balls would fly much further than golf balls of the same diameter and weight.
As for the bullet shape, the reduced-pressure layer formed around the nose can pass smoothly along the sides, increasing in pressure before joining the air behind the bullet. It is only included in the diagram (Figure 2B) to indicate the surprising efficiency of the dimpled golf ball. The effect of surface roughness on golf ball flight is so dramatic that it is no surprise that the smooth, molded gutta-percha balls, introduced in the 1850s, did not fly as far as the stitched leather case balls they were intended to replace. However, the “guttie” balls were found to carry much further when they became roughened in play. It was a small step from this realization to the molding of marks into the ball, eventually becoming a pronounced dimple pattern, and, in the past few decades, finely optimized dimple forms using sophisticated fluid dynamics software.