Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness of Transonic Shock Solutions for Two-Dimensional Steady Compressible Euler Flows in an Expanding Nozzle

Beixiang Fang, Xin Gao, Wei Xiang

TL;DR

This work addresses the uniqueness of transonic shock solutions for the 2D steady Euler equations in a strictly expanding nozzle with a prescribed exit pressure. The authors develop a priori estimates for the subsonic region behind the shock that do not require the post-shock state to be a small perturbation of a uniform state, then reformulate the Rankine–Hugoniot conditions on a fixed domain using a Lagrangian transform. By proving the monotonicity of the solvability condition and applying a contraction mapping, they establish that two transonic shock configurations must coincide when nozzle perturbations are sufficiently small. The results advance rigorous understanding of shock uniqueness in multidimensional nozzle flows and hinge on precise control of the elliptic subproblem behind the shock and the shock-wall intersection point. The approach combines elliptic PDE theory, RH condition reformulations, and fixed-domain techniques to obtain a sharp, quantitative uniqueness result.

Abstract

In this paper, we are trying to show the uniqueness of transonic shock solutions in an expanding nozzle under certain conditions and assumptions on the boundary data and the shock solution. The idea is to compare two transonic shock solutions and show that they should coincide if the perturbation of the nozzle is sufficiently small. To this end, a condition on the pressure of the flow across the shock front is proposed, such that a priori estimates for the subsonic flow behind the shock front could be established without the assumption that it is a small perturbation of the unperturbed uniform subsonic state. With the help of these estimates, the uniqueness of the position of the intersection point between the shock front and the nozzle boundary could be further established by demonstrating the monotonicity of the solvability condition for the elliptic sub-problem of the subsonic flow behind the shock front. Then, via contraction arguments, two transonic shock solutions could be verified to coincide as the perturbation is small, which leads to the uniqueness of the transonic shock solution.

Uniqueness of Transonic Shock Solutions for Two-Dimensional Steady Compressible Euler Flows in an Expanding Nozzle

TL;DR

This work addresses the uniqueness of transonic shock solutions for the 2D steady Euler equations in a strictly expanding nozzle with a prescribed exit pressure. The authors develop a priori estimates for the subsonic region behind the shock that do not require the post-shock state to be a small perturbation of a uniform state, then reformulate the Rankine–Hugoniot conditions on a fixed domain using a Lagrangian transform. By proving the monotonicity of the solvability condition and applying a contraction mapping, they establish that two transonic shock configurations must coincide when nozzle perturbations are sufficiently small. The results advance rigorous understanding of shock uniqueness in multidimensional nozzle flows and hinge on precise control of the elliptic subproblem behind the shock and the shock-wall intersection point. The approach combines elliptic PDE theory, RH condition reformulations, and fixed-domain techniques to obtain a sharp, quantitative uniqueness result.

Abstract

In this paper, we are trying to show the uniqueness of transonic shock solutions in an expanding nozzle under certain conditions and assumptions on the boundary data and the shock solution. The idea is to compare two transonic shock solutions and show that they should coincide if the perturbation of the nozzle is sufficiently small. To this end, a condition on the pressure of the flow across the shock front is proposed, such that a priori estimates for the subsonic flow behind the shock front could be established without the assumption that it is a small perturbation of the unperturbed uniform subsonic state. With the help of these estimates, the uniqueness of the position of the intersection point between the shock front and the nozzle boundary could be further established by demonstrating the monotonicity of the solvability condition for the elliptic sub-problem of the subsonic flow behind the shock front. Then, via contraction arguments, two transonic shock solutions could be verified to coincide as the perturbation is small, which leads to the uniqueness of the transonic shock solution.

Paper Structure

This paper contains 11 sections, 7 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. The transonic shock problem in a nozzle with a receiver pressure condition at the exit
  3. Existence of transonic shock solutions
  4. Uniqueness of transonic shock solutions for the problem $\llbracket\textit{TSPL}\rrbracket$
  5. Outline of the paper.
  6. A priori estimates for subsonic solutions behind the shock front
  7. Uniqueness of the transonic shock solutions
  8. Some properties of supersonic solution $U_-$ ahead of the shock front
  9. Reformulation of the R-H conditions \ref{['tilgi']}-\ref{['tilg4']}
  10. Uniqueness of $\psi(\eta_0)$
  11. Uniqueness of the transonic shock solutions

Key Result

Theorem 1.3

Suppose that specialcondition-Peinterval hold. Then there exists a sufficiently small positive constant $\sigma_0$, only depends on $\overline{U}_{\pm}$ and $L$, such that for any $0< \sigma \leq \sigma_0$, there exists a solution $(U_-^{\sharp}, U_+^{\sharp} ; \varphi_{\sharp})$ to the problem $\ll where the constant $C_-^{\sharp}$ depends on $\overline{U}_-$, $U_0$, $\gamma$ and $\alpha$, and th

Figures (3)

  • Figure 1.1: The steady Euler flow with a transonic shock front in an expanding nozzle.
  • Figure 1.2: The shock polar on $\theta$-$p$ plane.
  • Figure 1.3: The steady Euler flow with a transonic shock front in the Lagrangian coordinates.

Theorems & Definitions (16)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • ...and 6 more