Impulsive Motion of a Sphere at Supersonic Speeds

Abstract

This paper presents an analytical transient solution to the subsonic flow near the stagnation region of a sphere which starts impulsively at a constant supersonic speed. The analysis is based upon a series expansion in time of the flow variables and of the shape of the moving shock. The coefficients of the series are determined analytically by substituting the series into the differential equations of motion and the standard Rankine-Hugoniot jump conditions. The series is extended over 30 terms at stagnation point and up to nine terms near the sonic point. The first four terms are in agreement with the known solutions. By recasting them in Euler’s transformation, the series is analytical beyond their natural region of convergence. The results match the experiments and are in agreement with the known steady-state numerical solutions.