From: "JACK L. URETSKY (C)1998; HEP DIVISION, ARGONNE NATIONAL LAB ARGONNE, IL 60439" <JLU@HEP.ANL.GOV>
Date: Sun, 17 Jan 1999 15:01:25 -0600
Hi William Beaty-
Well, I'm willing to give you some hints.
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My conceptual confusion lies with the 2D case. In the 2D "world", it
appears to me that the air-flow patterns upstream and downstream of the 2D
airfoil are mirror images of each other, and so the airfoil on average
does not throw air downwards (nor does it leave air moving downwards.) It
appears to me that the wing is directly reacting against an infinite mass
of downstream air (or possibly is directly reacting against the distant
surface of the earth which blocks the vertical motion of the 2D air). Why
do I think this? See below.
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Your assumption that upstream and downstream are mirror images is
false.
The statement that the wing is directly reacting against an "infinite
mass of downstream air" (at any instant, at least) is not supportable by
any legitimate calculation.
The surface of the earth roughly analogous to an infinite conducting
plane in 2-D electrostatics. The potential of a 2-D point charge near such
a plane is a logarithm. The 2-D wing is like a 2-D charge distribution
(plus circulation).
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Since the air in the 2D world is treated as incompressible, then whenever
a downward force is applied to a long horizontal downstream parcel of air,
this downward force will be instantly communicated all the way down to the
earth's surface, and the earth's surface will instantly prevent that chunk
of 2D atmosphere from moving downwards.
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Also in the subsonic 3-D world is air incompressible.
Circulation is, I believe, still accounted for by the Kutta-Jokowski
hypothesis, which I will encourage you to learn about for yourself.
As I bow out of this discussion, I will remind the list that
Aristotelean argument does not replace a successful mathematical description
of nature.
Regards,
Jack
"I scored the next great triumph for science myself,
to wit, how the milk gets into the cow. Both of us
had marveled over that mystery a long time. We had
followed the cows around for years - that is, in the
daytime - but had never caught them drinking fluid of
that color."
Mark Twain, Extract from Eve's
Autobiography