Drela,

I would expect to be able to say whether an airplane actually does or does not impart "vertical momentum" to the flow. The fact that the answer depends upon the choice of control volume aspect ratio isn't very satisfying. Suppose I take a REALLY big control volume that is still far from the ground, and I make it a tall vertical slab. The width of the control volume might be many spans, but the height is enough that the aspect ratio is still very large (effectively infinite). As I make this control volume bigger, I might expect to capture all of the momentum imparted by the airplane. Once I'm satisfied that I have captured all of the momentum, I sum it up and find that the rate of momentum addition is equal in magnitude to the airplane lift.

Then, while keeping the height of my control volume fixed, I imagine increasing the width until the control volume becomes very short and wide. If I captured all of the momentum with my original (REALLY big) control volume then I should get the same answer when I sum all of the momentum. But I don't, I get zero. The conclusion has to be that you can never make the Trefftz plane big enough to capture all of the momentum imparted by the airplane. (Not too surprising given the nature of a 2D vortex).

One difference between the real world and one in which you can imagine arbitrarily large control volumes is that real world flows have viscosity. With viscosity included, I would expect to be able to truly capture all of the momentum imparted by an airplane inside a finite control volume. What is the rate of momentum transfer in this case?

One other question about your reply. Your case a) with a cubic control volume would have a square Trefftz plane. You suggested that delta(M) would be zero in this case. I get something like half the lift. I only get zero when I stretch the Trefftz plane to be infinitely short and wide. Am I doing something wrong?

Thanks for your reply!