Supersonic flows with a contact discontinuity to the two-dimensional steady rotating Euler system

TL;DR

This paper investigates the stability of supersonic flows with contact discontinuities in a rotating Euler system, establishing existence and stability results under perturbations using a Lagrangian approach.

Contribution

It introduces a novel analysis of the stability of supersonic contact discontinuities in a rotating Euler system with a curved nozzle, including the effect of Coriolis force.

Findings

Existence of stable supersonic shear flows with contact discontinuity

Development of an iteration scheme for hyperbolic systems

Proof of smooth flow separation by a contact discontinuity

Abstract

This paper concerns the structural stability of supersonic flows with a contact discontinuity in a finitely long curved nozzle for the two-dimensional steady compressible rotating Euler system. Concerning the effect of Coriolis force, we first establish the existence of supersonic shear flows with a contact discontinuity in the flat nozzle. Then we consider the stability of these background supersonic shear flows with a contact discontinuity when the incoming supersonic flow and the upper and lower nozzle walls are suitably perturbed. The problem can be formulated as an initial boundary value problem with a contact discontinuity as a free boundary. To deal with the free boundary value problem, the Lagrangian transformation is introduced to straighten and fix the contact discontinuity. The rotating Euler system is reduced to a first order hyperbolic system for the Riemann invariants. We design an iteration scheme and derive some estimates for the solution to the hyperbolic system. Finally, by using the inverse Lagrangian transformation, we prove the original free boundary problem admits two layers of smooth supersonic flows separated by a smooth contact discontinuity.