PHYS-L: is Circulation theory "wrong" ????

[William Beaty <billb@ESKIMO.COM>]

In PHYS-L on Sat, 16 Jan 1999, JACK L. URETSKY (C)1998; HEP DIVISION,              ARGONNE NATIONAL LAB ARGONNE, IL 60439 wrote:

>         It's a quantitative question that depends upon the "angle of
> attack"  of the airfoil.  If the angle is sufficient to give lift, then
> the downstream deflection is greater than the upstream deflection,
> giving a net lift.  This is apparently illustrated in the
> Anderson-Eberhart paper.

I totally agree with this reasoning as far as the 3D case is concerned.
The Anderson-Eberhart paper (http://www.aa.washington.edu/faculty/eberhardt/lift.htm)
argues that 3D aircraft must impose a net downwards deflection upon the
air being encountered by the wings, and therefor derive an upwards F=MA
lifting force.  I agree because there is no way for the aircraft to push
upon the earth directly, and because the forces created by the wing must
obviously deflect mass (with no "earth's surface" being required.)  I
agree because the downstream "wake" of a 3D aircraft is totally different
than the upstream patterns of air motion: it contains a pair of large
wake-vortices which presumably are carrying air downwards, as
illustrated in my "disk balloons" example at
http://www.amasci.com/wing/rotbal.html.  Clearly a 3D aircraft applies a
reaction force between its wings and the oncoming parcels of air, and
consequently the air accelerates downwards, and the craft accelerates
upwards and therefor opposes gravity and flys horizontally.

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.

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.  The wing *TRIES* to deflect air
downwards, but it only succeeds in creating a circular flow; where every
parcel of downward-moving downstream air is match by an upstream parcel
which is forced to move upwards.  Isn't that what "circulation" means,
after all?  A tilted airfoil tries to deflect air downwards, but in the 2D
world, this air encounters a sort of "solid floor", and as a result, it
ends up causing an equal upwash ahead of the wing.  Upwash equals
downwash, therefor there is no *NET* change in the vertical velocity or
vertical momentum of the oncoming air.  By conservation of momentum, there
must be zero lifting force!  Yet the wing does fly.  Why?  I suspect it
works like so: the wing essentially pushes downwards upon a (horizontally
wide) "rigid column" of 2D air which is resting upon the earth and
prevented from moving downwards.  Therefor there is an instantaneous
contact-force pair between the wing and the earth; as if the wing was
resting upon the earth itself (or was using a pressure-bubble like a
hovercraft; in other words, flight by "ground effect.")


Another way to think about the problem.  It appears to me that the
following flow pattern is impossible in the two dimensional world:


       upstream of               downstream of
       the airfoil               the airfoil

  ---------------------      ___
                                ---___
  ---------------------      ___      ---___
                                ---___
  ---------------------      ___      ---___
                                ---___
  ---------------------      ___      ---___
                                ---___
  ---------------------      ___      ---___
                                ---___
                                      ---___


In order to permanently deflect one streamline, the airfoil would have to
deflect ALL the downstream streamlines, agreed?  This phenomena only
arises in a 2D world.  The tilted flow in the right-hand diagram would
crash into the earth.  Or, if the earth was removed and the atmosphere was
infinitely tall, then the change from horizontal-flow to diagonal-flow
would require an infinite momentum change.  The airfoil cannot create the
infinite force needed to permanently deflect the infinite mass of
two-dimensional air.  With no deflection, there is no F=MA force. However,
since the airfoil TRIES to deflect the air downards, it is applying a
force to the air and therefor to the earth's surface.  The earth's surface
answers: it applies an upward 3rd-law force to the air and therefor to the
airfoil.  As a consequence, the two-dimensional airfoil rides indirectly
upon the two-dimensional earth: it flies entirely by "ground effect."

The airfoil in the 2D world cannot deflect air, however it is allowed to
move air around by creating a CIRCULAR flow.  The circular flow doesn't
impose a permanent net downwards deflection upon oncoming parcels of air
in the way that a 3D aircraft does.  The circular flow requires that the
"upwash" equal the "downwash".  If upwash equals downwash, there is no net
downwash, and no net F=MA lifting force. In the 2D world, the airfoil
cannot fly by deflecting air downwards. Instead there is a contact-force
with the earth, and the airfoil "rides on a bubble."   This bubble is its
circulation pattern which has higher pressure below, and which pushes upon
the earth like some vast automobile tire.

I conclude that, in the two-dimensional world, an airfoil "flys" by
contact forces with the distant earth's surface (or equivalently, by
attempting to deflect the entire 2D atmosphere).  On the other hand, a 3D
aircraft flys by directing a stream of air downwards in three dimensions,
and the air far below the aircraft can move aside in order to allow the
downwards flow to pass by.  Such a thing is not possible in 2D.  Hence,
aircraft in "flatland" rely on the unique physics of the two-dimensional
world, and they essentially are hovercrafts, no matter how high they may
fly above the ground.

Hey everybody on PHYS-L!  Please feel free to find the huge flaw in my
reasoning above.  I cannot find it.  I've cc:'d TAP-L to attract more
"opponents."  :)  I seriously want to get to the bottom of my confusion.
Either that, or I want to contribute to a "phase change" where a bunch of
people leap onto my bandwagon and stop using 2D circulation theory to
explain real-world flight.


What I'd REALLY love to see is the graph of net vertical velocity of
vertical slices of the atmosphere surrounding a wing in a 2D world.  My
intuition is that this graph is flat except near the airfoil, and near the
airfoil every upward part is matched elsewhere by an equal downward part.
If true, then isn't my intuition correct, and the wing causes circulation
but no net deflection of oncoming air, and therefor no net change in
momentum?

"Bill Beaty (and Dr. Weltner, etc.) CANNOT be right, because it would mean
that any explanation of flight which is based on 2D circulation is
profoundly misleading when used to explain how 3D airplanes fly.  That
many textbooks and experts cannot possibly be wrong!"

My experiences with "educator misconceptions" shows that the above
argument is invalid.  That many textbooks CAN be wrong, and frequently
are. That's why I'm on such a personal crusade!

I usually limit myself to K-6 textbook issues.  In the case of flight, the
(apparent) conceptual garbage from the university level is reaching down
and distorting explanations of flight in grade school.  Therefor I have no
inhibitions about attacking circulation theory.  It might after all be
"wrong."  If I can find the flaw in 2D circulation explanations of flight,
then I can weaken their hold on K-6 textbook explanations of flight, and
defeat the typical arguments from authority. ( Don't try to reason it out
for yourself, instead listen to expert opinions contained in advanced
textbooks? NOT! ) The flaw appears to be located in the "flatland physics"
which does not apply to any 3D aircraft.  My reasoning is obviously
oversimplified, and it is possible to argue about all sorts of little
issues.  However, if my reasoning has a *fundamental* flaw, I need
somebody would set me straight.

In conclusion, it appears to me that 2D circulation theory is fine when
its use is confined to calculations of lifting force of airfoil
cross-sections.  Local to the airfoil surface, 3D air flows are probably
identical to air flows in a 2D world, and so any lifting-force
calculations that are based on 2D circulation will match the real-world 3D
results.  However, since 2D circulation theory only *EXPLAINS* the flight
of "flatland aircraft", we should never unthinkingly apply it to 3D craft
which fly by an entirely different physical mechanism: by creating
downwards-moving 3D trailing vortices.

As always, more stuff and old message-threads are available here:

  Airfoil Lifting Force Misconception in K-6 Textbooks
  http://www.amasci.com/wing/airfoil.html

  "Science Myths" in K-6 Textbooks and Popular culture
  http://www.amasci.com/miscon/miscon.html

(note the new domain name, but the old will always work)

   "It is a good morning exercise for a research scientist to discard a
    pet hypothesis every day before breakfast. It keeps him young."
       -- Konrad Lorenz

   "If you make people think they're thinking, they'll love you.  If you
    REALLY make them think, they'll hate you" - D. Marquis

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William J. Beaty                                  SCIENCE HOBBYIST website
billb@eskimo.com                                  http://www.amasci.com
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