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

TL;DR

This paper proves the existence and determines the location of three-dimensional transonic shocks with large swirl velocity in a cylindrical nozzle, introducing new techniques to handle the complex elliptic-hyperbolic coupled system.

Contribution

It is the first mathematical analysis of 3D transonic shocks with large swirl velocity, developing novel decomposition methods for the elliptic-hyperbolic system and establishing shock existence conditions.

Findings

Established existence of 3D transonic shocks with large swirl velocity.

Developed new decomposition techniques for elliptic-hyperbolic systems.

Determined conditions for shock location and constructed background shock solutions.

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.