Alan Baker wrote:
Jim Logajan wrote:
Because the airplane and the Earth have zero relative vertical
velocity during straight and level flight, conservation of momentum
requires the net vertical flow of air to also be zero.

Therefore in subsonic flows where the fluid is assumed
incompressible, to the extent any fluid is moving downward,
conservation of mass requires an equal amount of mass must be moving
upward (the continuity requirement.)

Hence airplanes must cause air to move in circles.

Nope. Wrong.

The aircraft is experience an force upward the entire time it is in
flight. That force means there must be an equal force acting on the
air, and since the air was not moving vertically (in our idealized
case for this discussion) before the aircraft arrived, the force
exerted on it must mean that it is moving downward afterward it has
passed.

Nothing you wrote in your paragraph contradicts anything in my
paragraphs. So I'm at a loss therefore as to what specific statements you
claim are wrong. So how about I break it down into smaller claims and you
tell me which of these statements you agree with and which you disagree
with:

1) Conservation of momentum requires that at all times during flight that
net vertical momentum of the total system must be zero. Agree or
disagree?

2) We can treat air as incompressible, so conservation of mass means the
net vertical mass flow of the system during level flight must be zero.
Agree or disagree?

3) Therefore if, say, the downwash is 1 kg/s at any given instant due to
the wing, somewhere else in the fluid there must be an upwash at that
same instant of 1 kg/s. Agree or disagree?

4) Because upwash mass rate equals downwash mass rate, at some point the
downward flow reverses direction and becomes the upwash. Agree or
disagree?

Yes some deflection downward occurs. But I don't know that it could
be said to "diffuse" in any sense due to conservation of mass and
momentum requirements.

As the air the plane has forced downward encounters more air, the
momentum is diffused so that a greater and greater mass of air moves
downward at smaller and smaller velocities (net)...

...until it encounters the ground.

Keep in mind that balloons need no downwash to stay aloft. Yet we know
from conservation laws that the *static* pressure on the surface of the
earth must be increase due to their presence. Nothing you've written
rules out the possibility that the *dynamic* pressure of the downwash
translates into a *static* pressure increase well before the downwash
reaches the surface of the earth. The physics of the situation do not
seem to rule out that a priori.