ThatsMaths

Golf balls fly further today, thanks to new materials and mathematical design. They are a triumph of chemical engineering and aerodynamics. They are also big business, and close to a billion balls are sold every year. [TM081: search for “thatsmaths” at Irish Times ].

Simulation of flow around the dimples of a golf ball. Image from http://www.bioe.umd.edu/~balaras/html/topics.shtml

The golfer controls the direction and spin of the ball by variations in his swing. A pro can swing his driver at up to 200 km/h, driving the ball 50% faster than this as it leaves the tee, on a trajectory about 10º above the horizon. By elementary mechanics the vertical motion is decelerated by gravity, and the ball should bounce after about 200 metres and stop a few tens of metres further on.

How come then that professional golfers regularly reach over 300 metres? Gravity is not the only force acting on the ball, and aerodynamic forces can extend the range. In addition to drag, which slows down the flight, there is a lifting force that extends it. Drag is the force you feel when walking against a strong wind. Lift, the force that enables air-planes to fly, results from the back spin of the ball. Air passing over the top of the ball is speeded up while air below is slowed. This results – by way of Bernoulli’s principle – in lower pressure above and an upward force on the ball. Lift allows golfers to achieve greater distances.

The character of fluid flow depends on a dimensionless quantity, the Reynolds number. Named for Belfast-born scientist Osborne Reynolds, it determines whether the flow is laminar (smooth) or turbulent (rough). Normally the drag force increases with speed. In 1912, Gustave Eiffel – of Eiffel Tower fame – made a remarkable discovery, known as the drag crisis. Studying flow around a smooth sphere, he found a drop in the drag force as the flow speed increased above a critical Reynolds number.

The drag crisis extends the range of a rapidly travelling ball, but a smooth ball needs speeds in excess of 300km/h, unattainable in golf. So how can the drag crisis help? The answer lies in the dimples. The aerodynamics of a ball are determined by its mass and shape, in particular the nature of the surface. A complex pattern of dimples of varying sizes, shapes and depths influence the air flow around the ball. By roughening the surface, the critical Reynolds number is reduced to speeds within the golfer’s range. The dimples cause a transition to turbulence at a lower Reynolds number. The resulting reduction in drag can double the distance flown compared to what is possible with a smooth ball.

Dimples give golf balls their aerodynamic properties. Most balls have about 300 dimples. Manufacturers promise greater control, stability and velocity on longer shots. Hundreds of dimple patterns have been devised and patented but the optimal design remains a challenge. How many dimples should there be and what shape and pattern should they have?

Simulation of flow over dimples (image from PCAM)

The Curious Drag Crisis

The Reynolds number is defined as Re = VL/ν where V is the flow speed, L the ball size and ν is the viscosity coefficient. The transition from laminar to turbulent flow occurs at a critical value of Re which depends on details of the system, such as surface roughness.

Normally the drag force increases with speed. In 1912, Gustave Eiffel – of Eiffel Tower fame – made a remarkable discovery, known as the drag crisis. Studying flow around a smooth sphere, he found a drop in the drag force as the flow speed increased above Reynolds number 200 000 and continued to drop until about Re = 300 000 (see figure above).

The drag crisis was explained by Ludwig Prandtl in terms of his boundary layer theory. Reynolds had found that found that as the speed increases, the flow changes from laminar to turbulent. Prandtl argued that the turbulence mixes rapidly-moving external air into the boundary layer, increasing the range over which it adheres to the surface, making the trailing low pressure wake smaller and thereby reducing the drag force.

The drag crisis extends the range of a rapidly travelling ball, but a smooth ball needs speeds in excess of 300km/h, unattainable in golf. So how can the drag crisis help?

Figure from NASA: https://www.grc.nasa.gov/www/K-12/airplane/dragsphere.html

The answer lies in the dimples. By roughening the surface, the critical Reynolds number is reduced to about 50 000, with speeds within the golfer’s range. The dimples cause a transition to turbulence at a lower Reynolds number. The resulting reduction in drag can double the distance flown compared to what is possible with a smooth ball. The figure below shows that there is a range of Reynolds numbers for which the drag on a rough ball is substantially less than that on a smooth ball for the same flow conditions.

Sources

Arnold, Douglas N., 2015: The Flight of a Golf Ball. In Princeton Companion to Applied Mathematics, Ed. N. J. Higham et al., Princeton Univ. Press, ISBN: 9-780-691-15039-0

Elias Balaras: Research Topics. http://www.bioe.umd.edu/~balaras/html/topics.shtml

NASA: Drag on a Sphere. https://www.grc.nasa.gov/www/K-12/airplane/dragsphere.html

Smith, C. E., N. Beratlis, E. Balaras, K. Squires, and M. Tsunoda. 2012. Numerical investigation of the flow over a golf ball in the subcritical and supercritical regimes. International Journal of Heat and Fluid Flow 31:262–73.