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Re: a couple of points on flight

On Fri, 15 Jan 1999, John Mallinckrodt wrote:

On Fri, 15 Jan 1999, William Beaty wrote:

Here is a central problem: in 2D, the air far upstream from the wing is
flowing horizontally, right? And the air far downstream is also flowing
horizontally, correct? These flows come about through the superposition
of circulation and constant horizontal flight. It seems to me that this
guarantees a ZERO change in momentum of the air. It looks as if the 2D
flow diagram is symmetrical about a vertical line, and upwash exactly
equals downwash.

Maybe we need to remind ourselves that "horizontal" here means "parallel
to the earth's surface." The downward flowing air must eventually meet
the earth in which case it pushes on and imparts "downward" momentum to
the earth.

Yes, but if the upstream/downstream pattern are mirror images of each
other, then the upstream pattern would pull upwards on the earth while the
downstream pattern pushes the earth downwards, and the forces and momenta
subtract.

If somebody standing on the ground was shooting a machine-gun up at a
solid object, and the object was reflecting the bullets, the object would
experience a lifting force. This situation resembles that with the
airfoil in a 2D world. Yet the airfoil seems to be *pulling* its own
bullets upwards, then reflecting them downwards to create lift, then
*pulling* on the bullets to stop their downward motion. The total
momentum change would be zero. What am I missing? Maybe I should try to
simulate this with a large spreadsheet.

Or, to approach the problem from a different angle, what would happen if
we took a horizontal streamline in a 2D world and deflected it downwards?
Imagine that the earth is 10000KM below.

To deflect one horizontal streamline, won't we also have to deflect each
adjacent streamline as well? That takes more momentum. How far up and
down must we reach to deflect the streamlines? All the way down to the
earth, no? ANd so the earth will push on the bottom streamline, and so
all the streamlines will feel this occur. Since the gas is
incompressible, it cannot be deflected at all. If the earth was removed,
we still would need to change the vertical component of momentum for all
of the deflected streamlines (an infinite change in momentum), even though
we only want to deflect a single streamline. In a 3D world, this problem
does not arise, because deflecting a small sheaf of streamlines would
simply cause the air underneath to move in the 3rd dimension to get out of
the way.


If you stay in the earth's frame it may look like there has
been no change in momentum, but you aren't doing your analysis in an
inertial frame.

Bouyancy forces should eliminate the non-inertial frame's effects upon the
air parcels, no? Suppose that instead of an earth-frame, we remove
gravity and instead somehow apply a "downwards" force to the airfoil but
not to the air. I would think that this would not change the physics,
since the airfoil would still have "weight", yet the gas would be
"weightless," same as when a 2D airfoil flies over a 2D planet.


And while 3D may offer another way around the apparent
dilemma, I don't think it is necessary in order to resolve the paradox.

If a 3D aircraft flys because it flings its wingtip-vortices downwards
like I've describe here:

http://www.amasci.com/wing/rotbal.html

...then "flight" in a 2D world is distinct from the "flight" of a real 3D
aircraft, and while the 2D model would be convenient for calculating the
lifting force, it would be wrong to use it in explanations of the lifting
force associated with a real aircraft.


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