{
  "id": "2107.11725",
  "version": "v1",
  "published": "2021-07-25T04:53:36.000Z",
  "updated": "2021-07-25T04:53:36.000Z",
  "title": "Convergence Rate of Hypersonic Similarity for Steady Potential Flows Over Two-Dimensional Lipschitz Wedge",
  "authors": [
    "Jie Kuang",
    "Wei Xiang",
    "Yongqian Zhang"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "physics.flu-dyn"
  ],
  "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 \\cite[Page 67]{anderson} for more details)as the incoming Mach number $\\textrm{M}_{\\infty}\\rightarrow\\infty$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-25T04:53:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convergence rate",
      "steady potential flows",
      "two-dimensional lipschitz wedge",
      "initial-boundary value problem",
      "steady irrotational euler flow"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}