Exploring the Confusing Impact of Compressibility on CL Measurements

[greg_rack]

Homework Statement

Given a geometric angle of attack of 3deg, and the lift-curve below(measured compressible CL, M>0.3), use the Prandtl-Glauert correction to calculate the lift coefficient of the wing assuming incompressible flow.

DISCLAIMER: with Cl I mean the lift of the wing profile, the airfoil, and with the CL the lift of the whole wing

Relevant Equations

Induced AOA: $\alpha_i=\frac{57.3C_L}{\pi e_1 A}$

This situation is really confusing me. First of all, does accounting compressibility affect the measurements for CL? If yes, how does CL change as a function of the increasing mach number?

Back to the problem, above is the "in-flight" lift curve for a plane traveling at high speeds, so the CLs are supposed to be for compressible flow.
To find the related CL for incompressible flow, since we can apply the Prandtl-Glauert(PG) correction only to profile Cl, we must first switch to the lift-curve of the airfoil(Cl versus alpha).
We find that a Cl of 0.5 is obtained for alpha=1.8, so the slope changes and is higher, as expected.

Now, we can apply the "reverse" PG correction to this Cl coefficient to get the airfoil incompressible curve: a Cl of 0.433 is obtained for alpha=1.8.
And here is where I got stuck... how do I get from the Cl curve, to the CL curve(to find incomp. CL at alpha is 3), if I don 't know the induced angle of attack?

 

[Valkarie]

Or, is there another way?The way to get from the Cl curve to the CL curve is to use a relationship between the lift coefficient and the angle of attack called the lift curve slope. This relationship is given by: CL = CL0 + K*alpha, where CL0 is the lift coefficient at zero angle of attack, and K is the lift curve slope (the rate of change of lift coefficient with angle of attack). For any given angle of attack, you can find the corresponding lift coefficient by plugging it into the equation and solving for CL. Using this equation, you can then calculate the incompressible CL at any given angle of attack.