Aerodynamic Analysis of Sports Equipment

Items where aerodynamics directly affect performance, such as golf balls, soccer balls, tennis balls, bicycle helmets, and ski jumping suits.

Especially for balls, surface texture (dimples, panel seams) has a dramatic impact on drag. Compared to a smooth sphere's $C_D \approx 0.47$, the dimples on a golf ball reduce $C_D$ to about 0.25, nearly halving it.

The aerodynamic drag on a sphere is expressed by the following equation.

Here $A = \pi d^2/4$ is the sphere's frontal projected area.

Sphere drag exhibits a dramatic phenomenon called the "drag crisis." When the Reynolds number exceeds a certain critical value, the boundary layer transitions from laminar to turbulent, causing the separation point to move downstream and the wake to shrink. As a result, $C_D$ drops sharply.

Exactly. A golf ball's initial velocity is about 70 m/s, $Re \approx 2 \times 10^5$. A smooth sphere would still be in the high-drag regime, but thanks to the dimples, it enters the low-drag regime. This nearly doubles the flight distance.

A rotating sphere experiences a Magnus force.

Spin parameter:

Here $\omega$ is the angular velocity, $d$ is the sphere's diameter. Larger $S$ results in greater deflection.

Yes. With no spin ($S \approx 0$), the Kármán vortex street behind the sphere becomes unstable, causing lateral forces to fluctuate randomly over time. This is the cause of the irregular trajectory change known as the "knuckle effect." LES is essential for reproducing this in CFD.

• Surface texture resolution: 10--20 cells per dimple required, tens of millions of cells for an entire golf ball
• Handling rotating bodies: Sliding Mesh or Overset Mesh
• Low-Re transition: Transition model ($\gamma$-$Re_\theta$) is essential
• Unsteady vortices: LES or DDES required
• Flight trajectory: Coupling of 6DOF motion model with CFD

Unsteady aerodynamic analysis of rotating spheres is computationally expensive. Proper selection of methods is crucial.

LES for a golf ball (diameter 42.7mm, 300--500 dimples) requires 100--300 million cells. However, approaches that substitute the dimple effect with an equivalent roughness model are also being researched. This is a method that sets an equivalent sand-grain roughness $K_s$ in the wall function.

There are two approaches for CFD meshing of rotating spheres.

1. Sliding Mesh Method

• Set a rotating region around the sphere and physically rotate it
• High accuracy but computationally expensive
• Time step: $\Delta t \cdot \omega \cdot d < 1°$ (less than 1 degree rotation per step)

2. MRF + Wall Rotation Speed

• Steady approximation. Apply rotating wall condition to the sphere surface
• Only the steady component of the Magnus force can be predicted
• Cannot capture the dynamics of unsteady vortices

$y^+ < 1$ is essential for predicting the drag crisis. Since the transition location in the boundary layer governs $C_D$, resolving the viscous sublayer determines accuracy.

Soccer ball panel seams affect aerodynamics.

• 32-panel (traditional): Many seams, strong surface roughness effect. Drag crisis occurs at low Re
• 6-panel (Jabulani type): Fewer seams, closer to smooth. Unstable behavior with no spin
• Analysis key point: Model seam grooves geometrically and capture unsteady vortices with LES

Yes. Having fewer panels and shallower seams made the drag crisis transition sharper, destabilizing vortex shedding with no spin and generating large lateral force fluctuations.

Bicycle aerodynamic analysis has also become popular recently. The rider's body accounts for 70--80% of the total drag.

• Helmet: Vent placement can cause a 5--10% difference in $C_D$
• Rider posture: Upper body angle causes large variation in $C_DA$, from 0.20--0.35 m^2
• Drafting: Entering the wake of a rider ahead reduces drag by 30--40%