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Transonic shock solutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindrical nozzle

Beixiang Fang, Xin Gao, Wei Xiang, Qin Zhao

TL;DR

This work proves the existence and location of three-dimensional axisymmetric transonic shocks for steady full Euler flows with large swirl in a finite cylindrical nozzle under prescribed exit pressure. The authors develop a two-step strategy: first constructing special radial shock backgrounds with nonzero swirl, then proving the existence of perturbed shock solutions for small boundary data by solving a coupled elliptic–hyperbolic system behind the shock. A key novelty is the handling of strong elliptic–hyperbolic coupling induced by swirl, including new decompositions, higher-dimensional reformulations to address axis singularities, and solvability conditions that determine the shock position. The results establish a rigorous pathway to determine shock locations via boundary data perturbations, offering new insight into swirl-dominated transonic nozzle flows and expanding the mathematical theory of multi-dimensional Euler shocks under Courant–Friedrichs-type prescriptions.

Abstract

This paper concerns the existence and location of three-dimensional axisymmetric transonic shocks with large swirl velocity for shock solutions of the steady compressible full Euler system in a cylindrical nozzle with prescribed receiver pressure. As far as we know, it is the first mathematical result on the three-dimensional transonic shock with either large vorticity or large swirl velocity. One of the key difficulties is the fact that the Euler system is elliptic-hyperbolic composite for the flow behind the shock front, and its elliptic part and hyperbolic part are strongly coupled in the lower order terms because of the large swirl velocity, such that they cannot be simply decoupled in the principal parts as the case for the flow without swirls or with small swirl velocity. New decomposition techniques for the elliptic-hyperbolic composite system are developed to deal with this difficulty, and the solvability condition for the boundary value problem is deduced to determine the location of the shock front. It finally turns out that the non-zero swirl velocity, which brings new challenging difficulties in the analysis, plays an essential and fundamental role in determining the location of the shock front. Another key difficulty in the analysis is that there are no readily established shock solutions available. Non-trivial special shock solutions are first constructed as background solutions under the assumption that the flow parameters depend only on the radial distance to the symmetric axis. Necessary and sufficient conditions for the existence of such solutions are also found. Even though the shock location can be arbitrarily shifted for the special shock solutions, it will be shown that the shock solution with a determined shock position can be established as the boundary data are perturbations of one of the established special shock solutions under certain conditions.

Transonic shock solutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindrical nozzle

TL;DR

This work proves the existence and location of three-dimensional axisymmetric transonic shocks for steady full Euler flows with large swirl in a finite cylindrical nozzle under prescribed exit pressure. The authors develop a two-step strategy: first constructing special radial shock backgrounds with nonzero swirl, then proving the existence of perturbed shock solutions for small boundary data by solving a coupled elliptic–hyperbolic system behind the shock. A key novelty is the handling of strong elliptic–hyperbolic coupling induced by swirl, including new decompositions, higher-dimensional reformulations to address axis singularities, and solvability conditions that determine the shock position. The results establish a rigorous pathway to determine shock locations via boundary data perturbations, offering new insight into swirl-dominated transonic nozzle flows and expanding the mathematical theory of multi-dimensional Euler shocks under Courant–Friedrichs-type prescriptions.

Abstract

This paper concerns the existence and location of three-dimensional axisymmetric transonic shocks with large swirl velocity for shock solutions of the steady compressible full Euler system in a cylindrical nozzle with prescribed receiver pressure. As far as we know, it is the first mathematical result on the three-dimensional transonic shock with either large vorticity or large swirl velocity. One of the key difficulties is the fact that the Euler system is elliptic-hyperbolic composite for the flow behind the shock front, and its elliptic part and hyperbolic part are strongly coupled in the lower order terms because of the large swirl velocity, such that they cannot be simply decoupled in the principal parts as the case for the flow without swirls or with small swirl velocity. New decomposition techniques for the elliptic-hyperbolic composite system are developed to deal with this difficulty, and the solvability condition for the boundary value problem is deduced to determine the location of the shock front. It finally turns out that the non-zero swirl velocity, which brings new challenging difficulties in the analysis, plays an essential and fundamental role in determining the location of the shock front. Another key difficulty in the analysis is that there are no readily established shock solutions available. Non-trivial special shock solutions are first constructed as background solutions under the assumption that the flow parameters depend only on the radial distance to the symmetric axis. Necessary and sufficient conditions for the existence of such solutions are also found. Even though the shock location can be arbitrarily shifted for the special shock solutions, it will be shown that the shock solution with a determined shock position can be established as the boundary data are perturbations of one of the established special shock solutions under certain conditions.
Paper Structure (20 sections, 19 theorems, 440 equations, 1 figure)

This paper contains 20 sections, 19 theorems, 440 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical formulation and main theorems
  3. Free boundary value problem for 3-d steady axisymmetric full Euler flows in a finite cylinder
  4. Step 1: Existence of special shock solutions with large non-zero swirl
  5. Step 2: Existence of small perturbed shock solutions with prescribed receiver pressure.
  6. Existence of the quasi-supersonic solution.
  7. Approximate shock position, and the quasi-subsonic solution behind the free shock front.
  8. Special steady normal shock solutions with non-zero swirl
  9. Existence of steady special shock solutions
  10. Properties of the special shock solutions
  11. Quasi-supersonic solutions in the cylinder
  12. Linearized problem in the quasi-subsonic domain
  13. Reformulation of the nonlinear free boundary value problem in the quasi-subsonic domain behind the shock front
  14. Linearized boundary value problems for the two elliptic equations
  15. Well-posedness of $\llbracket \textit{Problem I}\rrbracket$
  16. ...and 5 more sections

Key Result

Theorem 2.2

Either for given $C^7$-functions $(\bar{w}_-(r), \bar{q}_-(r))$ with $\bar{w}_-(0)=0$ and $\bar{q}_-(r)>0$ and for given boundary values $\overline{M}_-(0)$ and $\bar{p}_-(0)$ on the symmetry axis with $\overline{M}_-(0)>1$ and $\bar{p}_-(0)> 0$, or for given $C^7$-functions $(\bar{p}_-(r), \bar{s}_

Figures (1)

  • Figure 1.1: Transonic shock flow with large swirl velocity in a cylinder.

Theorems & Definitions (43)

  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Theorem 2.9
  • Remark 2.10
  • ...and 33 more