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An airfoil's shape causes air above the wing to move faster than below the wing to allow for the difference in air pressure as dictated by Bernoulli's principle. This is what many sources say, but I have yet to see any explain what about the airfoil's shape or why exactly air speeds up above but doesn't below the airfoil. If I were to assume, maybe it would have something to do with the law of conservation of momentum or energy?

If you could send any sources, that would be much appreciated too!

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    $\begingroup$ Have you red this already?And for compressibility effects, this could also help. Please let me know if that leaves questions open. $\endgroup$ – Peter Kämpf Oct 31 '20 at 10:47
  • $\begingroup$ The air on top has to travel further. $\endgroup$ – Michael Hall Nov 1 '20 at 15:44
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The flow of a viscous fluid can be described using the Navier-Stokes equations. This includes describing flow of air around an airfoil. Your assumption about conservation of momentum or energy is correct, both of these enter in the derivation of Navier-Stokes, together with conservation of mass:

[The Navier-Stokes Equations] describe how the velocity, pressure, temperature, and density of a moving fluid are related. [...]

The Navier-Stokes equations consists of a time-dependent continuity equation for conservation of mass, three time-dependent conservation of momentum equations and a time-dependent conservation of energy equation.

(NASA, emphasis mine)

Since there is no generic way to analytically solve Navier-Stokes, numerical approximations are typically used to derive a solution (see Computational Fluid Dynamics). When solving for the flow around a wing, the solutions show faster flow at lower pressure above the wing and slower flow at higher pressure below the wing. This might not be the simple answer you are looking for, but any answer that does not involve complex maths usually oversimplifies something.

While you are right that Bernoulli's principle relates the speed of the flow and its pressure, you have to be careful about implying causality from this:

Bernoulli-only explanations imply that a speed difference arises from causes other than a pressure difference, and that the speed difference then leads to a pressure difference by Bernoulli's principle. This implied one-way causation is a misconception. The real cause-and-effect relationship between pressure and velocity is reciprocal.

(Wikipedia)

To understand why airfoils are shaped the way they are, I recommend you have a look at the NASA page on Aerodynamic Forces. They have a good description of how the total force is generated from the pressure difference:

Aerodynamic Forces

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  • $\begingroup$ Thanks so much! So just to clarify, while many people say that the reason for pressure difference is Bernoulli's principle (as in velocity causes the variation in pressure), it's actually the pressure that causes the variation in velocity? Or is it a mix of both? $\endgroup$ – SharkyPanda Oct 31 '20 at 10:52
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    $\begingroup$ No, it's not like one is causing the other. Pressure and velocity are not independent, but that does not imply one is causing the other. Both pressure and velocity distribution are a result of how the air flows. And this flow can be described with Navier-Stokes (although not analytically solved). $\endgroup$ – Bianfable Oct 31 '20 at 10:54

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