Features of flow in the region of the stagnation point in supersonic flow past power-law bodies

Abstract

Bodies of power-law shape with a generator of the form y=x^α, 0.5<α<1, have at the apex a vertical tangent (like blunt bodies) and infinite curvature (like sharp ones). Supersonic flow past axisymmetric bodies with power-law longitudinal profile has been studied by Radvogin in [1, 2], where on the basis of the analysis of a large quantity of numerical calculations certain empirical laws of similarity were formulated. It follows from these relations that the position and form of the subsonic sector of the shock wave are not determined by the singularity in the body's profile, but by the position on the body of the sonic point and its “bluntness” in relation to the cone of the “critical” opening angle. In O. V. Titov's work the results obtained in [1, 2] are confirmed analytically, but here it is assumed that the curvature of the shock wave and the second derivative of the curvature with respect to the longitudinal curvilinear coordinate are finite at the apex. This assumption imposes a limitation on the contour of the body; if it is satisfied, the curvature of the profile past which the flow takes place is also finite. Therefore it is natural to consider the case when this assumption is not made. In this paper we study the flow past bodies of power-law shape with shock waves of infinite curvature at the apex and finite curvature but an infinite second derivative of the curvature with respect to the longitudinal coordinate.