First thing, that's a common misconception that the plane flies due to the Bernoulli effect.
See the these questions: What really allows airplanes to fly? Why does the air flow faster over the top of an airfoil? for the correct explanation.
Assuming that the bernoulli effect does explain flight, the answer to your question is:
Pressure of a fluid is force exerted by a fluid on a unit area of a neighboring body (the force exists inside the fluid as well, but it is balanced). So, if there is more pressure underneath, the upward force is greater (area if top and bottom of wings are approximately the same)
If the upward force is greater, the net force due to pressure is upwards. This force is called 'lift', and it balances gravity, helping the plane fly.
Read http://enwp.org/Pressure for more info.
First to Address the Equivalence of Bernoulli's Equation to Conservation of Momentum:
There are (at least) three popular explanations for lift on an airfoil:
Faster air on the top has lower static pressure than slow moving air on the bottom. The resulting pressure difference multiplied by the area is equal to the lift.
The airfoil deflects air downward and by Newton's 3rd law an equal and opposite force (lift) is applied to the wing.
Bound circulation on the wing generates lift due to the Kutta--Joukowski theorem.
All three are equivalent.
Bernoulli's equation is derived from conservation of momentum (Navier-Stokes equations) with the assumption that the velocity has a potential function. Bernoulli's principle is merely the mechanism for the equal and opposite force to be applied to the wing in explanation #2.
Circulation is required for there to be any downward deflection of air. Without circulation, the flow would not exit smoothly at the trailing edge. Bound circulation is a result of boundary layers forming on the top and bottom surfaces. (Also a result of conservation of momentum)
To get a decent estimate of the lift curve of an airfoil at small angles-of-attack, panel methods are used to solve for the tangential velocity at the surface of an airfoil. This tangential velocity is then fed into Bernoulli's equation to get the pressures. Integrating the pressure over the surface of the airfoil we can find the lift.
These panel method's are inviscid but add just the right amount of circulation to get a realistic solution. (satisfying the Kutta condition)
While all three of the explanations above are valid, they are just rewording the unfulfilling explanation of the reason for lift: Conservation of Momentum.
Range of Applicability of Bernoulli's Equation
The incompressible version of Bernoulli's equation, $1/2 \mathbf{U}\cdot\mathbf{U} + p/\rho = const$, is valid:
- along streamlines
- along vortex lines (lines parallel to $\omega = \nabla \times \mathbf{U}$)
- and everywhere in irrotational flow ($\omega = 0$)
(for details see Ch. 2 of Viscous Fluid Flows by Frank M. White)
For airplanes, you can trace a streamline far in front of the aircraft into a region where $\mathbf{U} = const$ and hence $\omega = 0$. This means that the $const$ in Bernoulli's equation is the same everywhere and the equation can be used to find pressures anywhere on the surface of the wing.
In practice, if the Bernoulli equation is used, the velocities are found by panel methods. (If you are doing a viscous simulation you probably are calculating the pressures directly without using Bernoulli's equation.) The velocities from the panel method are analogous to the edge velocity at the top of the boundary layer near the surface. Any good viscous flow book will show that for boundary layers $dp/dy \approx 0$, where $y$ is oriented normal to the surface. This is also true for compressible flows up to about $M \approx 5$.
This method of using Bernoulli's equation to find the pressures (and by extension, forces) over the surface is a good approximation in many cases. When the results are not accurate, it is typically a result of inaccurate velocity information. For example, due to their inviscid nature, panel methods are not very good at describing separated flow such as an airfoil at high angle-of-attack.
Compressible vs. Incompressible
$M < 0.3$ as the limit of incompressible flow is thrown around a lot but typically without any justification. Consider gas at rest with density $\rho_0$ that is then accelerated isentropically to Mach number $M$. The density of the gas will change in this new state and is given by:
$$\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{1/(\gamma-1)}$$
As it turns out, in air $M \approx 0.3 \rightarrow \rho_0/\rho = 0.95$ and it is then assumed for practical purposes that if the density changes by no more than 5% the flow can be assumed to be incompressible.
You are correct that there are compressible versions of Bernoulli's equation. The results of calculations using the compressible equation should be a good approximation as long as the inputted velocities are accurate and the streamlines can be traced into the freestream (i.e. not separated flow).
Best Answer
See: https://aviation.stackexchange.com/questions/39146/how-do-symmetrical-airfoils-generate-lift and Bernoulli and Newton.
An oversimplified and short explanation is that the upper surface has a longer path whether the airfoil is symmetrical or not, that provides the lift. If the angle of attack was zero, like symmetrical rocket fins, there would be no lift; only straight travel, accounting for weight and force / vector.