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Transonic Shocks for 2-D Steady Euler Flows with Large Gravity in a Nozzle for Polytropic Gases

Beixiang Fang, Xin Gao, Wei Xiang, Qin Zhao

TL;DR

The paper tackles the existence of transonic shocks in 2-D steady Euler flows of polytropic gases under large vertical gravity in a nozzle, with exit-pressure boundary data. It constructs special normal transonic shock solutions, analyzes gravity-driven coupling, and reformulates the nonlinear free boundary problem behind the shock in Lagrangian coordinates to obtain a fixed-shock, subsonic boundary-value problem. A linearized well-posedness theory in weighted Hölder spaces is developed, and a nonlinear iteration scheme is designed to determine the shock location and full post-shock state, showing that gravity plays a dominant role in fixing the shock position. The results provide a rigorous framework for the existence and stability of transonic shocks under large external forcing and small boundary perturbations, with implications for nozzle flow design under gravity.

Abstract

In this paper, we are concerned with the existence of transonic shock solutions for two-dimensional (2-d) steady Euler flows of polytropic gases with the vertical gravity in a horizontal nozzle under a pressure condition imposed at the exit of the nozzle. The acceleration of the gravity g is assumed to take a generic value. We first show that the existence of special transonic shock solutions with the flow states depending only on the variable in the gravity direction can be established if and only if the Mach number of the incoming flow satisfies certain conditions. However, the shock position of the special solutions is arbitrary in the nozzle. We determine the shock position and establish the existence of transonic shock solution when the boundary data are small perturbations of the special shock solutions under certain conditions. Mathematically, the perturbation problem can be formulated as a free boundary problem of a nonlinear system of hyperbolic-elliptic mixed type and composite. Key difficulties in the analysis mainly comes from the vertical gravity. Methods and techniques are developed in this paper to deal with these key difficulties. Finally, it turns out that the vertical gravity plays a dominant role in the mechanism determining the shock position.

Transonic Shocks for 2-D Steady Euler Flows with Large Gravity in a Nozzle for Polytropic Gases

TL;DR

The paper tackles the existence of transonic shocks in 2-D steady Euler flows of polytropic gases under large vertical gravity in a nozzle, with exit-pressure boundary data. It constructs special normal transonic shock solutions, analyzes gravity-driven coupling, and reformulates the nonlinear free boundary problem behind the shock in Lagrangian coordinates to obtain a fixed-shock, subsonic boundary-value problem. A linearized well-posedness theory in weighted Hölder spaces is developed, and a nonlinear iteration scheme is designed to determine the shock location and full post-shock state, showing that gravity plays a dominant role in fixing the shock position. The results provide a rigorous framework for the existence and stability of transonic shocks under large external forcing and small boundary perturbations, with implications for nozzle flow design under gravity.

Abstract

In this paper, we are concerned with the existence of transonic shock solutions for two-dimensional (2-d) steady Euler flows of polytropic gases with the vertical gravity in a horizontal nozzle under a pressure condition imposed at the exit of the nozzle. The acceleration of the gravity g is assumed to take a generic value. We first show that the existence of special transonic shock solutions with the flow states depending only on the variable in the gravity direction can be established if and only if the Mach number of the incoming flow satisfies certain conditions. However, the shock position of the special solutions is arbitrary in the nozzle. We determine the shock position and establish the existence of transonic shock solution when the boundary data are small perturbations of the special shock solutions under certain conditions. Mathematically, the perturbation problem can be formulated as a free boundary problem of a nonlinear system of hyperbolic-elliptic mixed type and composite. Key difficulties in the analysis mainly comes from the vertical gravity. Methods and techniques are developed in this paper to deal with these key difficulties. Finally, it turns out that the vertical gravity plays a dominant role in the mechanism determining the shock position.

Paper Structure

This paper contains 12 sections, 8 theorems, 174 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Free boundary problem and main theorem
  3. Transonic shock solution
  4. Special normal transonic shock solutions
  5. Main theorem
  6. Reformulation of the free boundary value problem behind the shock front
  7. Flatten the nozzle wall and fix the shock front
  8. Reformulated problem in the subsonic domain
  9. Linearized problem in the subsonic domain
  10. Nonlinear free boundary problem
  11. Linearized problem and approximate shock position
  12. Nonlinear iteration scheme

Key Result

Theorem 2.1

For a given $C^3$-function $\bar{q}_-(y)>0$ in $\mathcal{D}_-^{\bar{x}_s}$, the special normal transonic shock solutions of the form $(\overline{U}_- (y);\ \overline{U}_+ (y);\ \bar{x}_s)$ as described in BS--thm-BS++thm with $\overline{M}_-(1)>1$ and $\bar{p}_-(1)>0$, satisfy E1-E4 and RH1-RH4, if and where $\overline{M}_-^2(y)=\frac{\bar{q}_-^2(y)\bar{\rho}_-(y)}{\gamma \bar{p}_-(y)}$. Then su

Figures (1)

  • Figure 1.1: Transonic flow with a vertical gravity in a nozzle.

Theorems & Definitions (19)

  • Definition 2.1: Transonic shock solution
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Remark 2.3: Definition of the weighted Hölder norms
  • Definition 3.1: Subsonic solution behind the shock front
  • Theorem 4.1
  • proof
  • ...and 9 more