The questions about the reasons the aircraft fly are frequent among scientist, since the high school until now, even if I work on the other side of fluid mysterious world (the Low Reynolds numbers regime), I've always asked to my professors, advisors, colleagues, what is their own explanation of the flight. I know about the controversy about the push downward, considered a common fallacy by the NASA website, and about the Anderson argument, denying the erroneous principles of equal times, and the overstimated role of the Bernoulli theorem. My best "global" and simple explanation is taken from Anderson, and consists in the following:
Somehow the air reaching the first edge of the wing, after the interaction with it, is going donward. This must be the result of some kind of force, and therefore, for the third Newton's Law, there must be an opposite force of equal strenght in the opposite direction, which pushes the aircraft up.
First: why does the air go down? Answer: the angle of attack and shape of the airfoil, together with simple pressure and stagnation arguments. Second: which is the role of the Bernoulli theorem here? If the air is pushed down by means of the "geometry", we don't need the difference of velocity between the upper and the lower part of the wing, but just we have this as a consequence of the change of pressure (due to the shape). Is that right?
My second question, actually, is about the most common and sophisticated explanation: the Starting Vortex Balance. The main argument is: due to the Kutta condition (a body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge) the vorticity "injected"by means of the viscous diffusion by the boundary layer generated near the airfoil, to the surrounding flow, transforms in a continuum of mini-starting-vortexes. This leaves the airfoil, and remains (nearly) stationary in the flow.It rapidly decays through the action of viscosity.
By means of the Kelvin Theorem, which in the 2D case is nothing but the statement that the vorticity is constant along every particle's path, this vorticity must be balanced by the formation of an equal but opposite "bound vortex" around the airfoil. This vortex, being causes an higher velocity on the top of the wing, and a lower velocity under it, causing the arising of the lift by means of the Bernoulli phenomenon.
Now, I wonder:
1) We are assuming that the vorticity leaving the wall (the no-slip condition make the airfoil wall a sheet of infinite vorticity) by diffusion, leaves the boundary layer and enters in the region where the Reynolds number is high enough to allow us to apply the Euler Equation, then the Kelvin Theorem (valid only for inviscid fluids). I usually explain this using the vorticity equation, that is a local version (in 2D) of the Kelvin Theorem: $$\partial_t\omega+\boldsymbol u \cdot \nabla \omega=\nu \nabla^2 \omega$$ in the boundary layer the viscosity terme dominates, whereas in the outer layer it can be neglected. When the vorticity arrives in the outer layer, vorticity is conserved and we can say that the structures arriving from the boundary layer must be balanced (in terms of vorticity) by the fluid in this region. And we can do this only because the circulation, which is the actual constant, is a line-integral, and if we don't cross the airfoil/BL region we don't have problem of any sort. Is this correct?
2) In this explanation the Bernoulli theorem seems to be a cause, generating the lift through the difference of velocity. Is this right?
Thanks in advance. I feel always a kid, asking about that issue. And at the same time, I feel an actual ignorant but curious scientist.