# Why a Topspin Tennis Ball Curves Down

· fluid mechanics
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### Why Explanation by Circulation is Unphysical and thus Incorrect

The explanation why a topspin/backspin tennis ball curves down/up presented in fluid mechanics textbooks (and on Wikipedia) is based on the Magnus effect according to the following picture showing potential flow around a section of a rotating circular cylinder augmented by circulation:

As shown in the picture below this flow generates lift as a reaction to redirecting the incoming flow (left) by adding circulation (middle) to make the flow turn around the cylinder section (or ball):

The problem with this explanation is that it is non-physical. Why? Because the circulation visibly changes the direction of the incoming flow. The real flow around a spinning tennis ball has instead the following flow pattern (left) with a non-rotating ball as comparison (right):

We see to the right the flow around a ball  without spin with the incoming flow horizontal and symmetric separation without force up or down.

We see to the left a topspin ball with still incoming flow horizontal with unsymmetric separation redirecting the air up with a resulting force downand thus giving a force down making a topspin tennis ball curve down.

### Correct Explanation by Unsymmetric Separation by Different Effective Reynolds Number

Why is then the separation unsymmetric for a spinning tennis ball? Because the friction below is smaller than on top because of different relative velocities with thus a larger effective Reynolds number on top than below, combined with the observation that the separation point moves from the crest at lower Reynolds number (laminar separation) further back at higher (turbulent separation).

In a laminar boundary layer, the fluid particles gradually change velocity from zero at the boundary to the free stream value away from the boundary, while the fluid particles in a turbulent boundary layer can have near free stream speed also very close to the boundary. A turbulent boundary layer acts like a small friction on the free stream flow, with friction coefficient tending to zero with the viscosity, in the limit like  slip boundary condition allowing fluid particles to tangentially slide without friction along the boundary. On the other hand, fluid particles in a laminar boundary layer satisfy a no-slip boundary conditionwith vanishing tangential velocity on the boubary. In both casesthe velocity normal to the boundary vanishes on the boundary. We can thus make the association:

• laminar boundary layer — no-slip boundary condition
• turbulent boundary layer — slip boundary condition.

We learn that the text-book explanation of the Magnus effect by large scale circulation is unphysical and thus incorrect.

Why a Golf Ball has Dimples
The dimples of a golf ball make the boundary layer turbulent which delays separation and reduces drag from about 0.5 for laminar separation with no-slip (above right) to about 0.2 for turbulent separation with slip:

We see a pattern of four (counter-rotating) rolls of streamwise vorticity forming the smaller wake of turbulent separation with smaller drag.

### Why a Backspin Table Tennis Ball Curves Down

A table tennis ball with a strong backspin also curves down, which depends on delayed turbulent separation: A table tennis ball has a smooth surface and the boundary layer is turbulent (with slip) only below because the higher relative velocity below triggers turbulence, while the low relative velocity on top is not large enough to trigger turbulence. A backspin chop is difficult to return.