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Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

Myoungjean Bae, Gui-Qiang G. Chen, Mikhail Feldman

Abstract

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak and strong shock solutions, when a steady supersonic flow impinges upon the wedge whose angle is less than the detachment angle, and then conjectured that the steady weak shock solution is physically admissible. The fundamental issue of whether one or both of the steady weak/strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled definitively. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the 2-D Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions. In this sense, we have to understand the reflection configurations to understand fully the global entropy solutions of 2-D hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory.

Prandtl-Meyer Reflection Configurations, Transonic Shocks, and Free Boundary Problems

Abstract

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak and strong shock solutions, when a steady supersonic flow impinges upon the wedge whose angle is less than the detachment angle, and then conjectured that the steady weak shock solution is physically admissible. The fundamental issue of whether one or both of the steady weak/strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled definitively. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the 2-D Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions. In this sense, we have to understand the reflection configurations to understand fully the global entropy solutions of 2-D hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory.

Paper Structure

This paper contains 54 sections, 109 theorems, 1195 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Mathematical Problems and Main Theorems
  3. Mathematical Problems
  4. Structure of Solutions of Problem 2.9
  5. Near the origin
  6. Away from the origin
  7. Global configurations of the solutions of Problem \ref{['problem-2']}
  8. Main Theorems
  9. Change of the Parameters and Basic Properties
  10. Straight oblique shocks in the self-similar plane
  11. New parameters $(v_{\infty}, \beta)$
  12. Main Theorems in the $(v_{\infty},\beta)$--Parameters
  13. Further Features of Problem 2.34
  14. Uniform Estimates of Admissible Solutions
  15. Directional Monotonicity Properties of Admissible Solutions
  16. ...and 39 more sections

Key Result

Lemma 2.4

Given any $\gamma\ge 1$ and $(\rho_{\infty}, u_{\infty})$ with $u_{\infty}>c_\infty=\rho_{\infty}^{(\gamma-1)/2}>0$, there exist unique $\underline{u}^{(\rho_{\infty}, u_{\infty})}\in (0, u_{\infty})$ and $\theta_{\rm d}^{(\rho_{\infty}, u_{\infty})}\in(0,\frac{\pi}{2})$ such that the following prop

Figures (16)

  • Figure 1.1: Admissible solutions in the $(v_{\infty}, \beta)$--parameters in the rotated coordinates $(\xi_1,\xi_2)$ by angle $\theta_{\rm w}$ counterclockwise (Left: $0<\beta<\beta_{\rm s}^{(v_{\infty})}$; Right: $\beta_{\rm s}^{(v_{\infty})}\le\beta<\beta_{\rm d}^{(v_{\infty})}$).
  • Figure 1.1: The shock polar for potential flow
  • Figure 1.2: The cone of monotonicity
  • Figure 2.1: Supersonic flow impinging upon a solid wedge
  • Figure 2.2: Shock polars in the $(u,v)$--plane
  • ...and 11 more figures

Theorems & Definitions (226)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: The steady Prandtl-Meyer reflection solution
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Remark 2.7
  • Definition 2.8
  • Definition 2.10
  • Definition 2.11
  • ...and 216 more