Transonic shocks for steady Euler flows with rotating effect in two-dimensional almost flat nozzles

TL;DR

This paper proves the existence and stability of transonic shocks in a rotating Euler flow within almost flat nozzles, considering the effects of Coriolis force and small perturbations in flow conditions.

Contribution

It introduces a new analysis of transonic shocks in rotating Euler flows, including special solutions and a method for determining shock positions under perturbations.

Findings

Existence of special transonic shock solutions depending on upstream Mach number.

Shock position can be determined and shown to exist under small perturbations.

A nonlinear iteration scheme is developed to construct solutions near initial approximations.

Abstract

We address the existence and stability of transonic shocks for the two-dimensional steady rotating Euler system in an almost flat nozzle. Under the influence of the Coriolis force, we first establish a class of special transonic shock solutions in a flat nozzle, whose states depend on the vertical variable. It is shown that these solutions exist if and only if the upstream Mach number satisfies certain conditions, while the shock position is arbitrary. We then determine the shock position and establish the existence of the transonic shock solution under small perturbations of the incoming supersonic flow, the exit pressure, and the upper nozzle wall. The problem is formulated as a free boundary problem for a hyperbolic-elliptic mixed nonlinear system. We decompose the hyperbolic and elliptic modes in terms of the deformation and vorticity, and analyze the solvability condition to determine the admissible shock positions. Starting from the obtained initial approximation of the shock solution, a nonlinear iteration scheme can be constructed to derive a transonic shock solution in which the shock front is close to the initial approximating position.