Why airplanes fly: the final truth Questions about the reasons aircrafts fly are frequent among scientist. Since the time I was in high school, even if I now work on the other side of  the fluid world (Low $Re$ regime), I've kept asking my professors, advisors, colleagues, what was their own explanation of 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 overall and simplest 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 we have this just 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.                  
 A: People have been flying airplanes for the last century, so it's not news.
Let me just point out a few things.
Here's one of my favorite pictures, of a Cessna 172 Skyhawk. It's the same model I fly. I'd like to draw your attention to a few things about it.



*

*Notice the pilot (in the white shirt) is looking right at the camera. That's because he's flying in formation with the photographer's airplane, and he really doesn't want to get too close. It's a long way down.

*The airplane is not level. It is banked toward the photographer. That means it is turning left because its lift vector is not vertical, exactly as in a bicycle or motorcycle. (It also means the photographer's plane is in a left turn, so they stay in formation.)

*The air is flowing past the airplane at somewhere around 100 miles per hour. Even though the plane is nicely streamlined, the air makes a pretty loud hissing sound.

*Notice the propeller, it is being cranked at around 30 revolutions per second. It is creating a forward pull of around 200 lbs, which is overcoming the drag of that 100 mph hurricane blowing past the entire structure.

*Notice the shape of the wings. They are more curved on top, of course, but you can also see that they work to deflect that wind downward. If you could see the wind, you would see that it forms a wake, just like the wake of a planing speedboat. It takes a lot of downward force to deflect the air down into that wake. In fact, the downward force is just the weight of the plane. It is surfing, plain and simple.

*What you can't see is the weight of the engine up front. In fact, if it weren't for the tail pushing down and holding the nose up, the plane would just drop straight to the ground. The reason for that is, if it slows down the nose drops, causing it to speed up, causing the nose to rise, causing it to slow down, etc. It's for stability. The speed controls itself. To set the speed, you turn a little wheel that applies some forward or backward pressure on the stick.

*Notice the sides of the airplane. They are broad and flat. This, with the vertical stabilizer, keeps the plane pointed into the wind, like a weathervane. If you're having trouble getting down and need more drag, you can push one of the rudder pedals. That causes the plane to get a bit sideways to the wind, so the wind drags a lot on the side of the plane. If you're sitting in the seat, you feel a strong sideways force, just like a car skidding.

*Notice the wing is not straight. It bends up on the left and right (called dihedral). That's another stability feature. If something causes the plane to get a little crosswise to the wind, one of the wings will project forward. Because of its dihedral angle, the wind will get "under it" more than the other one, and lift it. That causes the plane to bank, which causes it to start turning, so it is no longer crossways to the wind. (The Spirit of St. Louis had no dihedral, and a small tail. It kept Lindbergh awake for a day and a half using the rudder just trying to keep it straight.)

*Notice the wheels. There are two on the sides, and the one in the front can steer. The two side wheels are set back of the center of gravity so when it is on the ground it doesn't fall over backward. This is better than the old style, where the third wheel was at the back and the side wheels were set forward. The reason was, with the old style if the plane got the least bit sideways on the runway while landing, it could have a vicious tendency to turn even more sideways, or "ground loop", badly damaging the plane and possibly killing the occupants. Pilots of "tail-draggers" are skilled at preventing this.
So I hope you get from that a bit of what is second-nature to pilots.
A: My favorite explanation for the lift generated by airfoils is this:
One side of the wing does the pushing. It accelerates a very large number of air molecules in a direction roughly 90 degrees from the direction of the wing's apparent direction of travel. The other side of the wing creates a free space for the air molecules to occupy. Air has mass, and therefore inertia, so there is a short delay before the air in the newly formed space reaches its previous density. By the time it does (at flying speeds), the wing has passed, or nearly passed that location.
Here is an example that illustrates what I mean:
Most people who have traveled in an elevator have noticed what happens when it begins to descend. Although their feet don't leave the floor of the elevator, they feel lighter for a short time. If the person were to be standing on a spring-type bathroom scale, I expect it would show decreased "weight" or pressure against the scale. The "upper" surface of a wing experiences the same thing, caused more or less the same way. The curved and downward angled upper wing surface causes the "floor" to be pulled out from under the air column above, reducing its pressure against the wing.
If I have made my explanation clear enough, it should be easy to understand. It's an attempt to explain what really happens, as opposed to using math, which can not be used to completely explain any real object or occurrence. That's a problem with math that tends to be forgotten, especially among mathematicians. Math can not provide a completely correct answer even when adding one apple to one apple, because an "apple" can not be fully defined using math or measurements. Apples are too complex for that, and no two are the same.
A: My apologies, I won't be reading your entire question.  
But still I will provide an answer.  Why is that?  Because flight does not require any of the things you talk about.  
You could build an airplane that would fly with no "airfoil" shapes.  You could build an airplane that would fly with completely flat rectangular wings made out of plywood.  The important thing would be the angle of attack of the wings to the air.  Consider a flat piece of wood, like plywood.  Push it through the air in a direction exactly parallel to its flat dimensions and it develops no lift.  Tilt the wood so the leading edge is "up" compared to the direction it is moving and you can feel the lift.  
The lift can be thought of a few ways.  Think of the air molecules hitting the surface of the wood.  They bounce off, in a downward direction.  Well if we are pushing air downward, we must have an equal and opposite force, which is the lift.  Or another way: we are gathering air under the board, it gets a little pressurized.  The pressure is pushing up on the wood.  This is really the same picture as the first if you think about it.  
All the rest with airfoils and so on, all this has to do with developing lift efficiently, developing lift while minimizing drag.  An airplane with flat plywood wings would fly, but it would have a lot of drag and would therefore be very inefficient.  
A: Lift is caused by air that is deflected downwards, through Newton's law of action and reaction, and not by something like the Bernoulli effect. This article goes to a lot of detail to explain that. The full book, Understanding Flight, is available from Amazon.
A: So the first thing to note here is that we genuinely don’t have it all thrashed out. Lift is a complicated and well studied force and the result is that we have multiple different models that all belong to different time periods and people. However, we do know some things. I should also say that you basically have it all there, it’s just that it’s interconnected. Now let’s dig into it all…

*

*To start with, yes, there is a downturning of the flow and it isn’t a fallacy, there is a Newtonian element to lift as explained above. However, it is not enough to explain all the particular signs of lift nor it’s strength. This is caused (if I understand it correctly) by the ‘circulation’ around the airfoil

*‘Circulation’ as per the K-J airfoil theory is a VERY useful mathematical model of how air flows about an airfoil. But it is a model. It works by creating potential flow streamlines that flow straight around an airfoil and that do not turn. It then calculates the circulation around the airfoil and adds that to the streamlines, the result is quite an accurate and easy to use model for 2D infinite-span airfoil theory. However. Circulation is not a ‘real’ thing. It is useful in that it accounts for the slower moving air on the bottom of the airfoil and the faster moving air on top in the model, and as a result factors Bernoulli’s theorem into the equations.

*Bernoulli’s theorem accounts for the missing strength and observations around lift. The pressure difference between top and bottom of airfoil, and the extra strength that we see in real life that isn’t seen with pure Newtonian calculations.
To wrap it up I want to direct you to NACA report No 116 ‘Applications of modern Hydrodynamics to aeronautics’. This was a paper written in the early 1920s by Ludwig Prandtl (the one and only ) It has a very very helpful and thorough derivation of lift which, simply put into words, attributes half the lift to pressure differences (Bernoulli) and half the lift to impulse (Newtonian). The resulting equation is Lift=(density)x(length of chord)x(velocity)x(circulation coefficient).
I hope this helps.

