Convergence Rate of Hypersonic Similarity for Steady Potential Flows Over Two-Dimensional Lipschitz Wedge

TL;DR

This paper establishes the convergence rate of hypersonic similarity in steady potential flows over 2D Lipschitz wedges, confirming the Newtonian-Busemann law prediction as Mach number increases.

Contribution

It provides a rigorous proof of the convergence rate for hypersonic similarity in steady Euler flows over Lipschitz wedges, using boundary approximation and Riemann solutions.

Findings

Convergence rate matches the Newtonian-Busemann law prediction.

Established $L^1$ difference estimates with order $ au^2$.

Validated hypersonic small-disturbance approximation for large Mach numbers.

Abstract

This paper is devoted to establishing the convergence rate of the hypersonic similarity for the inviscid steady irrotational Euler flow over a two-dimensional Lipschitz slender wedge in $BV\cap L^1$ space. The rate we established is the same as the one predicted by Newtonian-Busemann law (see (3.29) in \cite[Page 67]{anderson} for more details)as the incoming Mach number $\textrm{M}_{\infty}\rightarrow\infty$ for a fixed hypersonic similarity parameter $K$. The hypersonic similarity, which is also called the Mach-number independence principle, is equivalent to the following Van Dyke's similarity theory: For a given hypersonic similarity parameter $K$, when the Mach number of the flow is sufficiently large, the governing equations after the scaling are approximated by a simpler equation, that is called the hypersonic small-disturbance equation. To achieve the convergence rate, we approximate the curved boundary by piecewisely straight lines and find a new Lipschitz continuous map $\mathcal{P}_{h}$ such that the trajectory can be obtained by piecing together the Riemann solutions near the approximated boundary. Next, we derive the $L^1$ difference estimates between the approximate solutions $U^{(\tau)}_{h,\nu}(x,\cdot)$ to the initial-boundary value problem for the scaled equations and the trajectories $\mathcal{P}_{h}(x,0)(U^{\nu}_{0})$ by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of $\mathcal{P}_{h}$ and $U^{(\tau)}_{h,\nu}$, we can further establish the $L^1$ estimates of order $\tau^2$ between the solutions to the initial-boundary value problem for the scaled equations and the solutions to the initial-boundary value problem for the hypersonic small-disturbance equations, if the total variations of the initial data and the tangential derivative of the boundary are sufficiently small.