Approximate transonic shock solution for hypersonic flow past a large curved convex wedge

TL;DR

This paper develops an approximate analytical method for solving the transonic shock problem in hypersonic flow over a large curved convex wedge, using transformations and elliptic estimates to handle the free boundary.

Contribution

It introduces a novel approach combining hodograph transformation and asymptotic boundary construction to approximate the free boundary in hypersonic flow problems.

Findings

Constructed an approximate boundary based on asymptotic states.

Solved a narrow-region elliptic boundary value problem.

Provided error estimates for the free boundary approximation.

Abstract

In this paper, we study the existence of a global transonic shock generated by hypersonic potential flow over a large curved convex wedge. Modeling 2-dimensional steady potential flow leads to a free boundary value problem of quasilinear equation. By applying the hodograph transformation, in which one of the coordinate is the unknown function, this problem is reduced to a free boundary value problem of linear equation. For hypersonic incoming flow with adiabatic index $\gamma=2$, we construct an approximate boundary based on the asymptotic state. Then, within the region defined by this approximate boundary, we solve a narrow-region elliptic boundary value problem. This solution serves as the approximate solution to the free boundary problem that we seek. Utilizing uniform weighted Schauder estimates for the elliptic mixed boundary value problem in the narrow region, we obtain an error estimate for the free boundary.