This is the kind of confusion I expected...
The confusion is over semantics. There is
nothing wrong with either the Bernoulli or Newton perspective. See
here for a discussion at the High School senior level. Both Bernoulli and Newton allow you to correctly calculate the forces on the wing, they just do it from different perspectives. Newton considers forces, Bernoulli considers energy. Both are correct.
Now, having said that, neither Newton nor Bernoulli allows you to compute the
inputs to the equation. You need to know the fluid speeds in order to use either method, and neither approach will give it to you. That's where the confusion sets in.
The most common error is the "Equal Path" argument. It specifies that you can compute the fluid velocity above and below the wings as follows: Consider two parcels of air, one just above the other, that strike the front stagnation point of the wing. The top parcel goes over the top of the wing, the bottom parcel goes underneath,
[this part is wrong]and they meet again at the rear stagnation point. Since the wing is curved, the top parcel has to travel faster, so its dynamic pressure on the wing is lower, says Bernoulli.
[/end of error]
You all correctly note that the above cannot be true because, if so, paper planes wouldn't work and aircraft couldn't fly upside down. The reason the above argument is false is because (a) the two parcels absolutely do not have to meet up again, and (b) we can't be sure where the stagnation points are -- they in fact move depending on angle of attack and wing shape.
Finding the fluid speeds and the stagnation points requires much heavier duty mathematics. Or we can just build a model and find them with smoke wands and pitot probes. The latter is what was actually done in the early days of flight. But whatever method you use, once you find these properties, you can plug the values back into Bernoulli, and you
will get the right answer.
What actually happens is, yes indeed, the aerfoil accelerates the flow. The flow above is faster, and the entire flow is deflected, both of which contribute to lift.
What is less obvious is
how the aerfoil accelerates the flow. It has to do so by starting the fluid, creating shear and vorticity that remains attached to the wing surfaces, and the flow has to remain attached to the aerfoil or else it stalls -- and when it stalls, lift disappears. This part is much, much more complicated. But, if you're willing to just take a snapshot in steady-state, you can compute the answer almost instantly.
The steady-state answer is good enough for 99 questions out of 100. For the really weird behavior, what happens in separated flow, high alpha, transsonic flow, etc., you're better off doing CFD, wind tunnel and flight tests and finding out experimentally.
The starting flow problem above can be verified mathematically. In Euler flow, i.e. in simulations where viscosity is set to zero, there is no lift produced. We can create it by setting an additional boundary condition on the total circulation, and if we do so we can use the Euler equations to get a pretty good lift estimate. But this is cheating. Physically, what this means is that in a truly inviscid flow, the flow does not start properly and our wing remains stalled no matter what. We need at least
some viscosity to create the boundary layer, and we need the boundary layer to keep the flow attached. Fortunately, while air is relatively inviscid, it is not totally inviscid. Flying through superfluid helium, on the other hand, is impossible, so don't try it.
