Asymptotic Behaviour of the Location of a Detached Shock Wave in a Nearly Sonic Flow

Abstract

This paper gives an example of physical applications of our results on some singular solutions of the Tricomi equation.
Making use of the solution subject to appropriate boundary conditions in the hodograph plane, we have some informations about the asymptotic behaviour of a detached shock wave in a nearly sonic flow, as follows.
The distance, b, of a detached shock wave from an obstacle and the curvature of a shock at its nose, 1/R, vary, respectively, with such powers of (M_∞−1) as
1⁄b∝(M_∞−1)², 1⁄R∝(M_∞−1)³,
M_∞ being the Mach number at infinity.
In axisymmetrical cases, we can obtain similar results by assuming that the flow behind a detached shock wave may be expressed by the asymptotic solution of the sonic flow due to Guderley and Barish. With the same notations as before, the results are
1⁄b∝(M_∞−1)^2⁄3, 1⁄R∝(M_∞−1)^5⁄3.