Inviscid, Compressible Flow Past Thin Airfoils

4.1 Introduction

Compressibility effects are significant when the incoming Mach numbers or the local Mach numbers take values that are no longer small compared to unity. Indeed, one often considers that if the Mach number is larger than 0.3 the flow can no longer be treated as incompressible. Such a situation can occur at very low incoming Mach number, if the geometry is such that the incompressible solution reaches infinite velocities at some point. This is the case, as we have seen in Chap. 3, at a sharp leading edge, Fig.3.12a. This would also be the case at the shoulder of a double wedge profile, Fig. 3.30, even at zero incidence. In both cases, the incompressible flow solution admits an infinite velocity. If an inviscid, compressible flow model were used, the velocity would not be infinite, but the flow would accelerate beyond sonic speeds and a supersonic bubble terminated by a shock would appear in the midst of the overall subsonic flow field. Although these are interesting theoretical problems, we will not focus our attention on this type of compressibility effects induced by discontinuities in the geometry, that can be eliminated by introducing a blunt nose or a rounded shoulder. In the framework of thin airfoil theory, compressibility effects will be the result of a combination of thickness, camber, angle of incidence and Mach number such that, as the Mach number is progressively increased from low subsonic to supersonic values, all the other parameters being held fixed, the solution deviates progressively more and more from the incompressible flow solution, exhibiting in general shock waves, such as the detached shock in Fig.4.1.