Supersonic flow onto a solid wedge

TL;DR

This paper analyzes the behavior of supersonic flow onto a wedge, proving that weak shocks form at the wedge tip under acceleration and developing new theoretical tools for elliptic regions in potential flow.

Contribution

It provides the first rigorous proof that sharp wedges produce weak shocks at the tip during acceleration, with detailed analysis of self-similar solutions and elliptic regions for supersonic flows.

Findings

Weak shocks form at the wedge tip during acceleration.

Self-similar solutions exist for a range of flow conditions.

Theoretical tools for analyzing elliptic regions in potential flow are developed.

Abstract

We consider the problem of 2D supersonic flow onto a solid wedge, or equivalently in a concave corner formed by two solid walls. For mild corners, there are two possible steady state solutions, one with a strong and one with a weak shock emanating from the corner. The weak shock is observed in supersonic flights. A long-standing natural conjecture is that the strong shock is unstable in some sense. We resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed. More precisely we prove that for upstream state as initial data in the entire domain, the time-dependent solution is self-similar, with a weak shock at the tip of the wedge. We construct analytic solutions for self-similar potential flow, both isothermal and isentropic with arbitrary $\gamma\geq 1$. In the process of constructing the self-similar solution, we develop a large number of theoretical tools for these elliptic regions. These tools allow us to establish large-data results rather than a small perturbation. We show that the wave pattern persists as long as the weak shock is supersonic-supersonic; when this is no longer true, numerics show a physical change of behaviour. In addition we obtain rather detailed information about the elliptic region, including analyticity as well as bounds for velocity components and shock tangents.