From: spgreg@mars.lerc.nasa.gov (Greg Macrae)
Subject: spinning balls was: misconceptions
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Organization: NASA Lewis Research Center
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Date:  7 Apr 1992 18:34 EDT  

From: spgreg@mars.lerc.nasa.gov (Greg Macrae)
Newsgroups: sci.aeronautics
Subject: floating balls was Re: Aeronautical misconceptions
Summary:  
Date:  7 Apr 1992 16:38 EDT  
Distribution: usa
Keywords:  

Ah, it has been a while since I attempted an explanation of lift, let's 
see if I can do it justice...

In article <1992Apr6.173315.480@ualr.edu>, rjvictor@acs.harding.edu writes...
>  I have observed: That a ping-pong ball, when placed upon an upward directed
>stream of air (such as that produced by a hair dryer) will "hover" at the
>approximate center of this stream.

The key word here is approximate.  The ball is not stable at the exact center.
Most of the comments that followed this were very wrong.  I will not include 
them in an attempt to minimize confusion.

As suggested, the air in the center of the stream moves faster than the 
air near the edge.  For a laminar case, one can say that the air has concentric
layers (hence the name laminar which means layered).  This is neither necessary 
nor appropriate for most cases.  The general velocity difference should hold 
for turbulent cases.  We can treat the turbulence as perturbations, I will 
neglect those terms.  

The short answer is that the ball remains at the center of the stream because 
it is spinning.  This is called the Kutta-Joukouski effect (I think).  This 
has absolutely nothing to do with the Bernoulli principle.  In fact, Bernoulli
describes the free stream (or static) pressure and the ball sees a pressure 
closer to the stagnation pressure.  The stagnation pressure from the faster 
air near the center of the stream is higher than the sagnation pressure from 
the slower air near the edge.  This tends to push the ball away from the 
center.  The viscous effects of the air cause the ball to spin in a direction 
appropriate to the effect of the center of the stream on one side of the ball. 
Furthermore,they allow the ball to impart a circulation to the boundary layer 
air.  

Now the Kutta-Joukouski theorem gets involved.  A circulating flow generates 
lift.  The simplest case to evaluate is cylindrical flow.  The lift is:

  l = - Integral from 0 to 2pi [p (sin theta) R dtheta] = rho v Gamma

where p is pressure, theta is the angle measured from the linear flow stream 
in the direction of the circulation and R is the radius of the cylinder, or 
rho is the density, v is the free stream velocity, and Gamma is a measure of 
the strength (speed) of the vortex (or circulation).  The pressure 
distribution can be generated assuming an inviscid, incompressible,
irrotational flow.  Because we are assuming velocities for the flow (by 
choosing a circulation constant and a free stream velocity), we can use 
Bernoulli's principal to get the pressure, thus:

  p = P + rho v^2 / 2 - rho / 2 [- (2v (sin theta) + Gamma / 2pi / R) ]^2

where P is the free stream pressure and all other terms are defined above.
Oddly, there is no drag produced by the circulation component of the flow
(neglecting viscous effects of course).  All drag is produced by the free 
stream component and is the same as for a non-rotating cylinder.  A similar
analysis can be made for a sphere with similar results, ie. lift from the
circulation component and drag from the linear component.  

The definitions I am using have positive lift directed radially outward from
the flow stream and drag axial in the direction of the flow stream.  Lift will 
be negative because of the definition of Gamma so it will tend to draw the 
ball towards the center of the flow.  The effect is most obvious when the 
stream is not vertical.  Then the ball spins below the center of flow stream 
at a point where the pressure exerted by the flow plus the lift equal the 
weight in the vertical direction and the horizontal position is determined by 
the balance of the pressure from the flow and the horzontal component of lift.
The previous assumption that turbulent effects can be treated as a negligible
pertubation can be verified by observing the small fluctuations in the balls 
position.  

General Lift:
Aerodynamic lift is always due to pressure.  Several statements can be 
made that are always true.  Lift is always the integral of the component of 
the pressure vector in the lift direction across the surface of the airfoil.  
Sometimes it is easier to integrate the fluid velocity across a boundary that 
is well separated from the surface of the airfoil.  In this case, a momentum 
balance will give you the lift.  Newton's laws still hold and all lifting 
bodies produce a momentum change in the surounding fluid that exactly 
balances the lift.  The momentum balance equation may not be easy to use 
when there is a thrust term.

I hope it helps,
Greg
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     MacRae                  |      Bright red pepper pod...
                             | It needs but shiny wings and look...
  spgreg@mars.lerc.nasa.gov  |        Darting dragonfly!
                             |                      -Basho
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