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

作者：Kuang, Jie^*; Xiang, Wei; Zhang, Yongqian

来源：Calculus of Variations and Partial Differential Equations, 2023, 62(3): 106.

DOI：10.1007/s00526-023-02449-y

• 35B07
• 35B20
• 35D30
• 76J20
• 76L99
• 76N10

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 n L1 space. The rate we established is the same as the one predicted by NewtonianBusemann law(see (3.29) in [2, p 67] formore details) as the incoming Mach numberM8. 8 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 Ph 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 (t) h,. (x, center dot) to the initial-boundary value problem for the scaled equations and the trajectories Ph(x, 0)(U. 0) by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of Ph andU (t) h,., we can further establish the L1 estimates of order t 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 flowpast a two-dimensional Lipschitz slender wing and show that for the length of the wing with the effect scale order O( t -1), that is, the L1 convergence rate between the two solutions is of order O( t 3 2) under the assumption that the initial perturbation has compact support.

• 单位
  复旦大学; 中国科学院