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 $$BV\cap L^1$$ 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 $$\text {M}_{\infty }\rightarrow \infty $$ 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. 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 $$O(\tau ^{-1})$$ , that is, the $$L^1$$ convergence rate between the two solutions is of order $$O(\tau ^{\frac{3}{2}})$$ under the assumption that the initial perturbation has compact support.