Local supersonic region on a body moving at subsonic speeds

Summary

A simple graphical proof is presented which gives a clear physical explanation of why a local supersonic region enclosed by a sonic line and a convex profile must always have an inflection point on any characteristic Mach line at the point where the local velocity corresponds to $M* = \sqrt 2$. This inflection point is shown to be directly related to the fact that the velocity ellipse is tangent to the characteristic epicycloid in the hodograph plane at $M* = \sqrt 2$

The physical insight gained by the previous geometric proof then leads one to investigate the corresponding shock polar case in the hodograph plane. Then it is found that a straight line can be simultaneously tangent to the sonic circle and any shock polar (or characteristic) epicycloid only at $M = \sqrt {\frac{{\gamma + 3}}{2}}$. This condition is then shown to correspond to the minimum Mach number at which any shock wave can form without producing an inflection point in a curved stream line, which is in agreement with the analytical analyses of Emmons (1946) and Lin and Rubinov (1948), it is also pointed out that $M = \sqrt {\frac{{\gamma + 3}}{2}}$ Corresponds to the most stable normal shock that can terminate a local supersonic region on a convex profile since it yields the maximum absolute value of static pressure, and because it can be consistent with any upstream radius of curvature.

Finally it is shown that in the local supersonic flow over a thin convex profile the isentropic flow is limited to $M* \approx \sqrt 2$ This velocity limitation is shown to be somewhat analogous to the sonic velocity limit in a Laval nozzle because of the fact that pressure signals cannot be propagated sufficiently far upstream by a smooth convex profile.