The amplification of weak shock waves in axisymmetric supersonic flow and their reflection from an axis of symmetry

Abstract

The problem of the amplification of weak shock waves when a supersonic flow approaches the axis of symmetry and they are reflected from this axis is considered within the framework of an ideal (non-viscous and non-heat-conducting) gas model. A non-linear theory is developed to investigate the amplification of shock waves, and Euler's equations are integrated numerically with an explicit construction of the head shock wave – the boundary of the unperturbed flow. In the simplest linear theory, in contradiction with numerical results, the amplification of weak shock waves is independent of the Mach number M0 of the flow in front of the shock wave and of the adiabatic exponent of the gas. The non-linear theory is free from this drawback. In this theory, obtaining the dependence of the intensity of the shock wave on the distance to the axis of symmetry reduces to the numerical solution of several unconnected Cauchy problems for two ordinary differential equations. Here the limit of applicability of the theory is also determined. In addition to investigating the amplification of weak shock waves, by numerical integration of Euler's equations on grids that are finer towards the axis of symmetry, irregular reflection is calculated and the dimensions of Mach discs are determined for different M0 for low initial intensities of the shock waves. These results confirm the well-known assertions that the size of the Mach disc is negligibly small for a shock wave of low initial intensity. A non-linear theory is constructed and a number of features of conical flows and related proofs of the impossibility of regular reflection of stationary shock waves of any initial intensity from the axis of symmetry, that are of independent interest, are considered, as well as the supersonic flow at the trailing edges of solids of revolution with finite vertex angles.