Incorrect Non-Physical Explanation of the Magnus Effect

1. Incorrect Non-Physical Explanation of the Magnus Effect Björn Borg's revolutionary open stance topspin forehand with body facing forward. The explanation why a topspin tennis ball curves down presented in fluid mechanics text books (and on Wikipedia) is based on the Magnus effect according to the following figures:  Potential flow with no lift  Potential flow + circulation with lift Following the ball with our eyes we have a motionless ball surrounded by a horisontal flow of air from left to right, say. The left figure shows a fully symmetric so called potential flow with zero lift force acting perpendicular to the flow direction, because of vertical symmetry. The right figure shows the vertically non symmetric flow obtained by superimposing a large scale circulation around the ball, with a resulting downward non-zero lift force, referred to as the Magnus effect, which can be understood as a reaction from the visible change of direction of the flowfrom incoming to outgoing, supposedly caused by the presence of the (rotating) ball. The problem with this explanation is that it is non-physical, and thus is not the correct explanation why a topspin tennis ball curves down. Potential flow with circulation satisfies the Euler equations for inviscid flow, but is non-physical  because, as we show in more detail below, a tennis ball cannot, by rotation, cause large scale change of direction of the flow or circulation around itself real flow has a turbulent wake behind the ball and is not horisontally symmetric. This article is closely related to the Knols  Why It Is Possible to FlyThe Spell of Prandtl's Boundary Layer TheoryThe Spell of Kutta-Zhukovsky's Circulation Theory and New Theory of Drag and Lift, showing that the classical lift theory for a wing based on large scale circulation around the wing, is non-physical by the same reason that the above explanation of the Magnus effect is non-physical, because there is no large scale circulation in real flow. See also the Knol on d'Alembert's paradox[1] and videos from wind tunnel tests.

2. Potential flow with no lift  Potential flow + circulation with lift

3. Navier-Stokes Simulation We simulate the flow around a tennis ball by solving the incompressible Navier Stokes equations using a finite element method [1] and obtain the following turbulent solution (vorticity left and pressure right) for a non spinning ball: We see a flow with vertical symmetry with low pressure on top and below and in a turbulent wake behind the ball resulting in zero lift and substantial drag.  For a spinning ball we obtain: We see that now the pressure distribution is also vertically non-symmetric with a lower pressure (blue) below than on top resulting in a downward lift force. The computation thus shows that a topspin tennis ball curves down, which we know fits with reality. Björn Borg was a specialist on topspin forhand giving him 5 Wimbledon titles in a row.  Evidently, Björn Borg's secret of the topspin lies in the non-symmetric separation for a  topspin tennis ball. We now seek a correct explanation  of this effect, understanding that the standard explanation based on large scale circulation is wrong. We shall also explain the surprising fact that a back-spin table tennis ball also curves down.

5. Analysis We can imagine fluid particles flowing past the ball first deviating and attaching (coming close) to the front of the ball,  then flowing around the ball and finally separating to form a turbulent wake in the back.  A correct explanation of the Magnus effect presented in [2] is based on flow separation in slightly viscous flow with an important distinction between  laminar separation (boundary layer laminar before separation) turbulent separation  (boundary layer turbulent before 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 a 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 condition with vanishing tangential velocity on the boubary. In both cases the 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 note that potential flow satisfies a slip boundary condition.  We also note that fluid particles can only separate from the boundary tangentially, since the velocity normal to the boundary is zero. More precisely, potential flow can only separate at a stagnation point where the tangential velocity vanishes, because in potential flow fluid particles tend to follow the curvature of the boundary as long as the tangential velocity is not small (which reflects a property of harmonic functions [1]). Since potential flow has a slip boundary condition there is no retardation of the  tangential velocity with the result that potential flow has a tendency to stick to the boundary as long as there is no opposing flows causing stagnation. 

7. Correct Physical Explanation of the Magnus Effect We can now understand why a top-spin tennis ball curves down as a result of non-symmetric separation: separation at the crest on top because of large friction with no-slip boundary condition delayed separation below  because of small friction with slip boundary condition. The boundary layer can be turbulent for a fast ball with strong topspin and laminar for slow ball with little topspin non-symmetric  with non-symmetric separation in both cases causing the ball the curve down. With the same argument we can understand why a back-spin tennis ball curves up. We can thus explain the observations made in The Physics of Topspin on a Tennis Ball. We can also explain, by a partly different argument, why a slow tabletennis ball with strong backspin curves down as the result of a different form of non-symmetric separation, laminar on top and turbulent below:  laminar separation at the crest on top because of low relative velocity delayed turbulent separation below with slip because of high relative velocity. The difference is the explanation of delayed separation: A tabletennis 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.