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Hypersonic flow onto a large curved wedge and the dissipation of shock wave

Dian Hu, Aifang Qu

Abstract

The flow field with a Mach number larger than 5 is named hypersonic flow. In this paper, we explore the existence of smooth flow field after shock for hypersonic potential flow past a curved smooth wedge with neither smallness assumption on the height of the wedge nor that it is a BV perturbation of a line. The asymptotic behaviour of the shock is also analysed. We prove that for given Bernoulli constant of the incoming flow, there exists a sufficient large constant such that if the Mach number of the incoming flow is larger than it, then there exists a global shock wave attached to the tip of the wedge together with a smooth flow field between it and the wedge. The state of the flow after shock is in a neighbourhood of a curve that is determined by the wedge and the density of the incoming flow. If the slope of the wedge has a positive limit as $x$ goes to infinity, then the slope of the shock tends to that of the self-similar case that the same incoming flow past a straight wedge with slope of the limit. Specifically, we demonstrate that if the slope of the wedge is parallel to the incoming flow at infinity, the strength of the shock will diminish to zero at infinity. The restrictions on the surface of a wedge have been greatly relaxed compared to the previous works on supersonic flow past wedges. The method employed in this paper is characteristic decomposition, and the existence of the solution is obtained by finding an invariant domain of the solution based on geometry structures of the governing equations. The ideas and methods presented here may be applicable to other problems.

Hypersonic flow onto a large curved wedge and the dissipation of shock wave

Abstract

The flow field with a Mach number larger than 5 is named hypersonic flow. In this paper, we explore the existence of smooth flow field after shock for hypersonic potential flow past a curved smooth wedge with neither smallness assumption on the height of the wedge nor that it is a BV perturbation of a line. The asymptotic behaviour of the shock is also analysed. We prove that for given Bernoulli constant of the incoming flow, there exists a sufficient large constant such that if the Mach number of the incoming flow is larger than it, then there exists a global shock wave attached to the tip of the wedge together with a smooth flow field between it and the wedge. The state of the flow after shock is in a neighbourhood of a curve that is determined by the wedge and the density of the incoming flow. If the slope of the wedge has a positive limit as goes to infinity, then the slope of the shock tends to that of the self-similar case that the same incoming flow past a straight wedge with slope of the limit. Specifically, we demonstrate that if the slope of the wedge is parallel to the incoming flow at infinity, the strength of the shock will diminish to zero at infinity. The restrictions on the surface of a wedge have been greatly relaxed compared to the previous works on supersonic flow past wedges. The method employed in this paper is characteristic decomposition, and the existence of the solution is obtained by finding an invariant domain of the solution based on geometry structures of the governing equations. The ideas and methods presented here may be applicable to other problems.
Paper Structure (25 sections, 38 theorems, 368 equations, 14 figures)

This paper contains 25 sections, 38 theorems, 368 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Basic differential relations
  3. Preliminary
  4. Characteristic decomposition
  5. Differential relations on the wedge boundary.
  6. Differential relations on the shock fronts.
  7. A limit case
  8. The limit solution
  9. Differential relations for the limit solution
  10. Global attached shock to a curved wedge without convexity
  11. Some a priori estimates
  12. $(\xi, \eta)$ Coordinates
  13. Thickness of the shock layer
  14. A priori estimates related to the characteristics in the shock layer
  15. Extension of the local solution and uniform estimates
  16. ...and 10 more sections

Key Result

Theorem 1.1

For the supersonic incoming flow with fixed Bernoulli's constant $\mathbf{B}$, there exists $\mathbf{M}_1>1$ only depending on the wall and $\mathbf{B}$, such that, for $\underline{\mathbf{M}}>\mathbf{M}_1$, the problem eq:EulerEquations, eq:initialdata and eq:VelocityWallCondition admits a global p where $(u_\infty, v_\infty)$ is the constant supersonic solution for corresponding problem involvin

Figures (14)

  • Figure 1: supersonic flow onto a curved wedge
  • Figure 2: supersonic flow onto a bullet wedge
  • Figure 3: detached shock
  • Figure 4: flow with limit speed onto a curved wedge
  • Figure 5: conservation of mass
  • ...and 9 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Lemma 2.2: Characteristic expressions
  • Lemma 2.3: Characteristic decompositions
  • Lemma 2.4
  • Lemma 2.5
  • Definition 3.1
  • Remark 3.1
  • Lemma 3.1
  • ...and 57 more