Detached shock past a blunt body

TL;DR

This paper investigates the formation and shape of detached shocks around a blunt body in supersonic flow, proving the existence of convex detached shocks for high Mach numbers and convex bodies.

Contribution

It constructs a detached shock solution for the steady Euler system around a blunt body and proves its convexity under high Mach number conditions.

Findings

Existence of detached shock solutions for high Mach numbers.

Detached shocks form convex curves around the blunt body.

Strong shock solutions satisfy entropy conditions.

Abstract

In $\R^2$, 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 $M_{\infty}>1$ sufficiently large, %depending on $\theta_w$, then there exist two shock solutions, {\emph{a weak shock solution and a strong shock solution}}, with the shocks being straight and attached to the tip 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 $\R^2\setminus W_b$. In particular, we seek a shock solution with the far-field state being 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.