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

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 ∩L 1 space. The rate we established is the same as the one predicted by Newtonian-Busemann law (see (3.29) in [2, Page 67] for more details) as the incoming Mach number M∞ → ∞ 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 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 (τ ) h,ν (x, •) to the initial-boundary value problem for the scaled equations and the trajectories P h (x, 0)(U ν 0 ) by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of P h and U (τ ) h,ν , we can further establish the L 1 estimates of order τ 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. Based on it, we can further establish a better convergence rate by considering the hypersonic flow past a two-dimensional Lipschitz slender wing and show that for the length of the wing with the effect scale order O(τ −1 ), that is, the L 1 convergence rate between the two solutions is of order O(τ 3 2 ) under the assumption that the initial perturbation has compact support.