The generation of lift occurs because of the pressure variation around the airfoil. When you integrate the pressure times the area times the local normal base vector around the entire surface, you obtain a single aerodynamic force. The component of that force perpendicular to the flight direction is the lift.
Now why does the pressure vary? Because the velocity around the airfoil varies and pressure and velocity are related along a streamline by Bernoulli's equation.
Why does the velocity vary? Because the flow can't go through the surface of the airfoil. In an ideal flow situation, the flow is tangent to the surface of the airfoil. In reality, the flow at the surface sticks to the surface and the velocity is zero, but the external flow reacts to the edge of the boundary layer from the surface to free stream. The free stream flow must simultaneously conserve mass, momentum and energy (that's where the Euler equations come in)
Solutions of the Euler equations introduce the theory of bound vorticity within a lifting airfoil
Extensions of the bound vorticity theory to three dimensions leads to the Prandtl lifting line theory and the vortex lattice method.
None of this is classified; it is just complex. Undergraduate aero engineering students learn all of this and how to derive the necessary equations that describe flow around an airfoil.
Tom Benson, Editor.
GRC/TB: No ... saying that the answer is complex means just that ... it is complex, it isn't simple. But we know what it is.
Many people look for simple answers to questions, when the answer may not be simple. When the answer is really complex, people make simplifying assumptions so that they can get a simple answer. Unfortunately, with fluid mechanics, when you make simplifying assumptions you can get the wrong answer. That's what has happened with the incorrect theories and their inability to produce meaningful results.
Fluid mechanics comes down to satisfying the conservation laws for mass, momentum and energy. That's what the Euler equations (and the more complete Navier-Stokes equations) express. There are some limited solutions to the Euler equations and the bound vortex / Prandtl lifting line theory are some of them. These solutions correctly predict observed flow phenomenon.
CJ: I hear you say that you know, but I don't think you do. If you want to know, take a look at Why It Is Possible to Fly showing in particular that Kutta-Zhukovsky's circulation theory for lift, which you refer to, is un-physical: An airplane does not take off by shedding transversal vorticity as indicated in GRCs picture. This is a common misconception which GRC unfortunately contributes to transmit to the young generation. Often a 2d bathtub foam surface experiment is used to support the circulation theory, but this is misleading since an airplane wing acts in 3d. In fact, 2d flight is impossible. There are no pictures of shed transversal vorticity from airplane wings, because they don't exist. 3d streamwise vorticity: Yes! 2d transversal vorticity: No! Right?