Detached Shock Past a Blunt Body

Abstract

In \(\mathbb{R}^{2}\), a symmetric blunt body \(W_{b}\) is fixed by smoothing out the tip of a symmetric wedge \(W_{0}\) with the half-wedge angle \(\theta _{w}\in (0, \frac{\pi }{2})\). We first show that if a horizontal supersonic flow of uniform state moves toward \(W_{0}\) with a Mach number \(M_{\infty }>1\) being sufficiently large depending on \(\theta _{w}\), then the half-wedge angle \(\theta _{w}\) is less than the detachment angle so that there exist two shock solutions, a weak shock solution and a strong shock solution, with the shocks being straight and attached to the vertex of the wedge \(W_{0}\). Such shock solutions are given by a shock polar analysis, and they satisfy entropy conditions. The main goal of this work is to construct a detached shock solution of the steady Euler system for inviscid compressible irrotational flow in \(\mathbb{R}^{2}\setminus W_{b}\). Especially, we seek a shock solution with the far-field state given as the strong shock solution obtained from the shock polar analysis. Furthermore, we prove that the detached shock forms a convex curve around the blunt body \(W_{b}\) if the Mach number of the incoming supersonic flow is sufficiently large, and if the boundary of \(W_{b}\) is convex.