Supersonic flow onto a solid wedge

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.