Hypersonic similarity law for steady compressible Euler flows past slender bodies within the framework of Radon measure solutions

TL;DR

This paper develops a mathematical framework for hypersonic similarity laws in steady compressible Euler flows past slender bodies, showing flow structures depend only on a combined parameter as slenderness and Mach number grow large.

Contribution

It introduces a new hypersonic small-disturbance system and proves Radon measure solutions converge under the limit of slenderness and high Mach number.

Findings

Flow structures depend only on parameter K and gamma in the hypersonic limit.

Radon measure solutions of original flows converge to small-disturbance solutions as slenderness tends to zero.

Explicit Radon measure solutions are derived for both 2D and 3D scenarios.

Abstract

In this paper, we establish a mathematical theory on statement and validation of the hypersonic similarity law within the framework of Radon measure solutions of steady compressible Euler equations. We consider two scenarios: (1) two-dimensional steady non-isentropic compressible Euler flows past an infinitely long slender curved wedge; (2) three-dimensional steady non-isentropic compressible Euler flows past an infinitely long axisymmetric cone. It turns out that, for the hypersonic flow passing through a slender body with tiny slenderness $\tau$, if the parameter $K\doteq M_{\infty}\tau$ is fixed, by taking $\tau \to 0$ (i.e., the Mach number of the upcoming flow $M_{\infty} \to \infty$), the flow field structures (after scaling) no longer depend on the body's shape and the Mach number $M_{\infty}$ independently, but only on $K$ and adiabatic index $\gamma$ of the polytropic gas. Mathematically, for non-isentropic Euler flows, we find a new system of hypersonic small-disturbance equations to describe steady compressible hypersonic flows passing slender bodies. We demonstrate that as $ \tau \to0$, under suitable non-dimensional scalings, the Radon measure solutions of the original problems of hypersonic flow passing bodies converge to those of corresponding hypersonic small-disturbance problems. The explicit forms of the Radon measure solutions derived for the two scenarios effectively simplify the convergence analysis.