Originally Posted by HazelNuts39
Your theory is fine for a given "geometry", which term includes a given angle of attack. For an airplane at a given weight, AoA (1g) at constant Mach (free stream) increases with altitude, and hence Cp (absolute).
It's not really my theory, I am just recounting the thoughts of others.
What do you think of the following? The need to maintain a constant weight in equilibrium (vertically-unaccelerated flight) means constant lift (of magnitude = weight). And this total lift derives from the coefficient of lift c_L (along the chord line). And this c_L derives from integrating the difference between C_p on the upper surface and C_p on the lower surface along the chord line, and then dividing by the chord. So c_L is a function of C_p. So, so is the total lift. All at a given AoA, as you observed. Different AoA, different c_L, different lift; you're accelerating up or down now.
This doesn't seem to fit with what you are suggesting.
Originally Posted by HazelNuts39
BTW, according to Wikipedia, Prandtl-Glauert "is derived from linearized equations".
I find Wikipedia a pretty poor source for understanding almost anything mathematical. To me, a "linearized equation" would mean an equation with only linear terms of its variables. And that is misleading.
From what I understand, Prandtl used the approximation in his lectures in Göttingen as early as 1922. Glauert "derived" the compressibility correction in 1928, and it was used until 1939 when there were better. (Ref: Anderson, Fundamentals of Aerodynamics. It may well be in his history, also, but I don't have that.)
Glauert's derivation starts from a linear partial differential equation (of second order, called the "linearised velocity potential equation), which is an approximation of the velocity potential equation, which is not a linear PDE. He does it by formally transforming the independent variables so that the linear PDE transforms into Laplace's equation (the sum of the second partials on each independent variable, aka the Laplacian, = 0), which is easy to solve; and then one transforms back. But remember that this derivation starts from an approximation in the first place (a linear PDE that "corresponds" to the velocity potential PDE).
The Prandtl-Glauert formula is a "reasonable" approximation for thin airfoils, low AoA and M.freestream < 0.7 or thereabouts.
How do we know all this? Curve-fitting. You have the data. You try a simple quadratic fit, and it does pretty well. Then along comes someone else, who starts from another well-known approximation (which is also necessary for mathematicians to derive anything useful from the original equation), and derives yours from it. Now you can all be even more convinced that it does a good job. Within the region in which it does a good job, of course. Outside that region, it doesn't do a good job (this is a tautology, obviously). That is what curve-fitting is all about. But if someone else wants to use a different phrase for this process, that's OK with me.
PBL