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 B V ∩ L 1 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 L 1 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 ) ( U 0 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 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.