Expert: Paul Soderman - 3/25/2008

Question
Hello Sir:
I hope this question does not seem obscure.

When an object moves thru the atmosphere, such as an airplane, the disturbance that it creates propagates thru out the whole atmosphere, spreading out as it goes to infinity.  This is a simple consequence of conservation of momentum, action/reaction, however one wants to label it.

When an airplane generates lift, how much mass of air is initially involved in the momentum changes around the wing, and at what point does it become this same momentum just spreading out into a larger and larger mass?  The answer must be very subtle, not clearly defined.  

From a circulation standpoint, the circulation generated by the wing has a tangential velocity that varies inversely with radius.  As you go further from the wing, the slower tangential velocity is associated with a "longer" streamline, so the circulation is the same.  I believe that this is the same as saying the wing causes a certain amount of momentum change, and this same momentum spreads out into more and more mass, until it is so spread out as to be undetectable (by humans).

Newton originally thought that the amount of mass deflected was equal to the chord of the wing times the sine of the angle.  But this was in error.  In Von Karman's book, he says that the mass involved (2-D) is equal to 3.14 times the chord.  I have seen other books that use an "apparent mass" concept that basically says that most of the lift is generated within a few chord lengths.  However, this apparent mass is not an identifiable region in the flow field.  

Answer
Michael
I am not sure of your question, but let me try to clarify a few things.  The lift on a wing can be explained by pressure distributions on the surface or by a change in momentum of the fluid by the wing.  For the momentum method we fix the coordinate system to the wing and assume a mass of air is approaching the wing and is deflected downward.  The choice of momentum volume affected by the wing is somewhat arbitrary, but the usual method is to calculate the deflection of a tube of air with diameter equal to wing span.  We could use a larger volume, but the effect of the wing goes to zero at infinity, and the calculation becomes tedious.  The finite tube calculation is not difficult but involves estimating the air deflection angle.  See the following for the complete equations:  

http://www.onemetre.net/Design/Downwash/Momentum/Momentum.htm

As you can see, the method works best for an elliptical wing with uniform downwash.  Since most wings are more complex, the surface pressure analysis is the method usually used by aerodynamicists to calculate wing lift.  For quick approximations, they might use a vortex lifting-line analysis.

Paul