Hypersonic Similarity for the Two Dimensional Steady Potential Flow with Large Data

TL;DR

This paper rigorously proves the hypersonic similarity principle for 2D steady potential flow, showing that solutions at large Mach numbers approximate scaled solutions, validating Van Dyke's theory mathematically.

Contribution

It provides the first rigorous mathematical validation of hypersonic similarity for large data flows in 2D steady potential flow, using a modified Glimm scheme.

Findings

Validated hypersonic similarity principle mathematically.

Constructed global entropy solutions for large data flows.

Showed solutions approach hypersonic small-disturbance solutions as Mach number increases.

Abstract

In this paper, we establish the first rigorous mathematical global result on the validation of the hypersonic similarity, which is also called the Mach-number independence principle, for the two dimensional steady potential flow. The hypersonic similarity is equivalent to the Van Dyke's similarity theory, that if the hypersonic similarity parameter $K$ is fixed, the shock solution structures (after scaling) are consistent, when the Mach number of the flow is sufficiently large. One of the difficulty is that after scaling, the solutions are usually of large data since the perturbation of the hypersonic flow is usually not small related to the sonic speed. In order to make it, we first employ the modified Glimm scheme to construct the approximate solutions with large data and find fine structure of the elementary wave curves to obtain the global existence of entropy solutions with large data, for fixed $K$ and sufficiently large Mach number of the incoming flow $M_{\infty}$. Finally, we further show that for a fixed hypersonic similarity parameter $K$, if the Mach number $M_{\infty}\rightarrow\infty$, the solutions obtained above approach to the solution of the corresponding initial-boundary value problem of the hypersonic small-disturbance equations. Therefore, the Van Dyke's similarity theory is first verified rigorously.