A previous question by David Zaslavsky was a request for a broad, "how things work" type of explanation of the lift of an airfoil. The answers given there are enlightening, but don't address a more specific question I have.
First a summary of my current understanding. There is no logical reason to assume that parcels of air take equal time to travel above and below the wing. If we did make such an assumption, Bernoulli's principle would make all kinds of incorrect predictions. "How things work" explanations (e.g., this one) often speak as though Bernoulli's principle is one effect, and then there are some other effects, such as conservation of momentum; it's implied that these effects are additive, and it's often stated that the Bernoulli effect is negligible. But in fact Bernoulli's principle is, under certain hypotheses, equivalent to conservation of energy. Conservation of energy and conservation of momentum are not effects to be added. They're two physical laws, both of which need to be obeyed.
It's only under certain hypotheses that Bernoulli's principle is equivalent to conservation of energy. It requires nonviscous flow, and the version given in freshman physics textbooks also assumes incompressible flow, although there is also a version for compressible flow. The WP article on Bernoulli's principle states that it's usually valid for Mach numbers lower than about 0.3, which I guess might be valid for birds and some small planes (cruise speed for a Cessna 172 is Mach 0.25), but not passenger jets.
Based on this, I can envision two logical possibilities:
(1) The hypotheses of Bernoulli's principle hold for birds and some small planes. The equation gives at least an approximately correct result for the net force on the wing, but there is no obvious, simple way to know what velocities to assume.
(2) The hypotheses of Bernoulli's principle fail in these cases.
Which of these is correct? If #2, which hypothesis is it that fails? Is the incompressible-flow version inapplicable but the compressible-flow version OK?