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Time-periodic transonic shock solution in divergent nozzles

Xiaomin Zhang, Peng Qu, Huimin Yu

TL;DR

The paper proves the global existence and dynamical stability of time-periodic transonic shock solutions for a quasi-one-dimensional isothermal compressible Euler system under periodic boundary forcing. It develops a two-step iterative scheme that decouples the moving shock (a free boundary) from the subsonic flow by leveraging Rankine-Hugoniot conditions and a reformulation using Riemann invariants. The authors establish existence and periodicity of a weak solution, provide convergence and continuity estimates for the iterative scheme, and show stability against small perturbations, including exponential decay in the supersonic region and controlled shock motion. These results mathematically justify engineering observations of periodically moving shocks in divergent nozzles and inform boundary-control strategies for shock dynamics in duct flows.

Abstract

We demonstrate that it is possible to control a normal transonic shock to move periodically by adjusting the boundary conditions at the entrance or the exit of the tube, for which, the phenomena has been observed in engineering. In this paper, we describe the gas by a quasi-one-dimensional compressible Euler equations with temporal periodic boundary conditions and prove the global existence and dynamical stability of the time-periodic transonic shock solution with an iteration method. The major difficulty is to determine the position of the moving shock front, which can be obtained by a free boundary problem in the subsonic domain. We decouple this free boundary problem by the $Rankine-Hugoniot$ conditions and a two-step iteration process.

Time-periodic transonic shock solution in divergent nozzles

TL;DR

The paper proves the global existence and dynamical stability of time-periodic transonic shock solutions for a quasi-one-dimensional isothermal compressible Euler system under periodic boundary forcing. It develops a two-step iterative scheme that decouples the moving shock (a free boundary) from the subsonic flow by leveraging Rankine-Hugoniot conditions and a reformulation using Riemann invariants. The authors establish existence and periodicity of a weak solution, provide convergence and continuity estimates for the iterative scheme, and show stability against small perturbations, including exponential decay in the supersonic region and controlled shock motion. These results mathematically justify engineering observations of periodically moving shocks in divergent nozzles and inform boundary-control strategies for shock dynamics in duct flows.

Abstract

We demonstrate that it is possible to control a normal transonic shock to move periodically by adjusting the boundary conditions at the entrance or the exit of the tube, for which, the phenomena has been observed in engineering. In this paper, we describe the gas by a quasi-one-dimensional compressible Euler equations with temporal periodic boundary conditions and prove the global existence and dynamical stability of the time-periodic transonic shock solution with an iteration method. The major difficulty is to determine the position of the moving shock front, which can be obtained by a free boundary problem in the subsonic domain. We decouple this free boundary problem by the conditions and a two-step iteration process.
Paper Structure (10 sections, 7 theorems, 211 equations)

This paper contains 10 sections, 7 theorems, 211 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The existence of steady transonic shock solution
  4. The derivation and properties of equivalent systems
  5. Periodic solutions to a kind of ODE system
  6. Existence of the time-periodic weak solution
  7. boundedness and time-periodicity
  8. convergence
  9. Continuity modulus estimate and convergence
  10. Stability around the time-periodic weak solution

Key Result

Theorem 1.1

(Existence of the time-periodic weak solution) Under the assumptions a2-a7, we have 1) There exists a constant $\epsilon_{1}>0$, such that for any $0<\epsilon\leq\epsilon_{1}$, any given $T\in\mathbb{R}_{+}$, and any given functions $\bar{\rho}_{l}(t), \bar{u}_{l}(t)$ and $\bar{\rho}_{r}(t)$ satisf there exists an initial value $(\rho^{(T)}, ~u^{(T)})(0,x)$ such that the initial-boundary value pr

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 9 more