Hypersonic Similarity for Steady Compressible Full Euler Flows Over Two-Dimensional Lipschitz Wedges

TL;DR

This paper rigorously proves the hypersonic similarity law for steady Euler flows over Lipschitz wedges, establishing optimal convergence rates and justifying Van Dyke's theory through mathematical analysis.

Contribution

It provides the first rigorous mathematical proof of hypersonic similarity for Euler flows with characteristic boundary conditions, including optimal convergence rates.

Findings

Established the optimal convergence rate to hypersonic similarity law.

Proved $L^1$--stability and global existence of entropy solutions.

Validated Van Dyke's similarity theory mathematically.

Abstract

We establish the optimal convergence rate to the hypersonic similarity law, which is also called the Mach number independence principle, for steady compressible full Euler flows over two-dimensional slender Lipschitz wedges. The problem can be formulated as the comparison of the entropy solutions in $BV\cap L^{1}$ between the two initial-boundary value problems for the compressible full Euler equations with parameter $\tau>0$ and the hypersonic small-disturbance equations with curved characteristic boundaries. We establish the $L^1$--convergence estimate of these two solutions with the optimal convergence rate, which justifies Van Dyke's similarity theory rigorously for the compressible full Euler flows. This is the first mathematical result on the comparison of two solutions of the compressible Euler equations with characteristic boundary conditions. To achieve this, we first employ the special structures of the two systems and establish the global existence and the $L^1$--stability of the entropy solutions under the smallness assumptions on the total variation of both the initial data and the tangential slope function of the wedge boundary. Based on the $L^1$--stability properties of the approximate solutions to the scaled equations with parameter $\tau$, a uniform Lipschtiz continuous map with respect to the initial data and the wedge boundary is obtained. Next, we compare the solutions given by the Riemann solvers of the two systems by taking the boundary perturbations into account case by case. Then, for a given fixed hypersonic similarity parameter, as the Mach number tends to infinity, we establish the desired $L^1$--convergence estimate with the optimal convergence rate. Finally, we show the optimality of the convergence rate by investigating a special solution.