On the instability problem of a 3-D transonic oblique shock wave

TL;DR

This paper investigates the stability of 3-D transonic oblique shock waves in steady supersonic flow past a wedge, demonstrating that such shocks are generally unstable due to overdetermined conditions.

Contribution

It provides a rigorous proof that 3-D transonic shocks are unstable, addressing a longstanding open question in shock wave theory.

Findings

3-D transonic shocks are generally unstable.

The shock problem is overdetermined, leading to instability.

Supports the idea that only weak shocks are physically stable.

Abstract

In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated in pages 317 of [9] that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge, then it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there possibly appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined, which implies that the 3-D transonic shock is unstable in general.