It seems that Abbott and Von Guenhoff went through the exact derivation as James Dwinnell did in my old aerodynamics textbook. We are in complete agreement concerning both the origin and constraints of the Bernoulli equation. I also found the Roger Long article quite interesting, especially the mention of the Coanda effect. That is something not predicted by theory but since it is an observed effect we just name it and drive on.

Now lets look at some implications.

First, the air doesn’t “flow”, it just sits there waiting for something to happen. Then a wing passes by supporting an airplane in straight and level flight. After the wing has past, the air that once was resting is now moving downward, and slightly forward. There was obviously some force acting on the now moving mass of air and there must have also been an equal and opposite force acting on the airplane. The force on the mass of air put it in motion because there was nothing other than the inertia of the mass to resist it. The airplane remained steady on its course because the force on it was resisted by an equal and opposite force due to gravity. So far Newton’s laws of motion are satisfied and it is obvious that the force maintaining the aircraft in flight is a reaction to the wing deflecting a mass of air downward. The wing in its essence is just an air deflector.

Now because Mr. Euler, Bernoulli, and company are so smart, they figured out how the wing does it, and here is how.

The wing moving through the air parts the air with some of it passing above and some below. The air moving over the wing is bounded below by the surface of the wing, and above by the air that is far enough away to not be immediately effected by the wing’s passage. This forms the stream tube in which Bernoulli’s equation applies. Pressure is reduced above the wing and across the entire cross section of the stream tube according to the relationship he defined. Whatever pressure distribution then exists along the chord of the wing, a mirror image of that distribution also exists at the upper boundary (Neither can exist alone. Where one is, the other is there also). Granted that boundary isn’t as well defined as the solid surface of the wing, but it is real nontheless. The free stream air at the boundary moves downward toward the area of reduced pressure (and having been set in motion, continues this motion after the wing passes). The wing doesn’t move upward since the net upward force is balanced by the weight of the airplane. A similar thing happens to the air going below the wing but the pressures are less negative or even slightly positive depending on the angle of attack. The final net effect of the wing on the air is to accelerate it downward, and slightly forward.

A thing that is heavier than air cannot fly except it moves air downward. Helicopters do it with rotors, birds and bees do it with the complex flapping motion of their wings, and a fixed wing does it by causing a pressure field to be formed when it is in motion that “pulls” the air downward by its’ passage. For all of its complicated formulas, and mathematics, in the final analysis, aerodynamics is the study of how to accelerate air to produce a force to sustain, propel, or maneuver a flying machine.

Ben, I enjoyed the joke. LOL