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

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.