Transonic shocks for 3-D axisymmetric compressible inviscid flows in cylinders

TL;DR

This paper proves the existence of axisymmetric weak solutions with transonic shocks, vorticity, and swirl in 3D cylinders for the steady Euler system, using Helmholtz decomposition and iterative methods.

Contribution

It introduces a novel approach to construct solutions with transonic shocks and swirl in 3D flows by combining Helmholtz decomposition with an iterative scheme.

Findings

Existence of axisymmetric weak solutions with transonic shocks in 3D cylinders.

Development of an iterative method based on Helmholtz decomposition.

Handling of nonzero vorticity and swirl in steady Euler flows.

Abstract

We establish the existence of an axisymmetric weak solution to the steady Euler system with a transonic shock, nonzero vorticity, and nonzero swirl in a three-dimensional cylinder. When prescribing the supersonic solution in the upstream region by axisymmetric functions with variable entropy and variable angular momentum density(=swirl), we construct such a solution by using a Helmholtz decomposition of the velocity field and the method of iteration. An iteration scheme is developed using a delicate decomposition of the Rankine-Hugoniot conditions on the transonic shock via Helmholtz decomposition.