Convergence rate of hypersonic similarity for steady potential flows over two-dimensional Lipschitz wedge

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 ∩ L¹ space. The rate we established is the same as the one predicted by Newtonian-Busemann law (see (3.29) in [2, p 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 L¹ difference estimates between the approximate solutions U_h,ν^(τ)(x, ·) to the initial-boundary value problem for the scaled equations and the trajectories P_h(x, 0)(U₀^ν) by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of P_h and U_h,ν^(τ), we can further establish the L¹ estimates of order τ² 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 L¹ convergence rate between the two solutions is of order O(τ^3/2) under the assumption that the initial perturbation has compact support. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.