Steady supersonic combustion flows with a contact discontinuity in two-dimensional finitely long nozzles

TL;DR

This paper develops a mathematical framework for analyzing steady supersonic combustion flows with contact discontinuities in finite nozzles, establishing existence, uniqueness, and error estimates for solutions of the governing equations.

Contribution

It introduces a novel fixed point iteration scheme for the free boundary problem and compares the full 2D model with a quasi-one-dimensional approximation.

Findings

Existence of a fixed point solution for the flow equations.

Uniqueness of the steady combustion flow with contact discontinuity.

Quantitative error estimates between 2D and quasi-1D models.

Abstract

In this paper, we are concerned with the two-dimensional steady supersonic combustion flows with a contact discontinuity moving through a nozzle of finite length. Mathematically, it can be formulated as a free boundary value problem governed by the two -dimensional steady combustion Euler equations with a contact discontinuity as the free boundary. The main mathematical difficulties are that the contact discontinuity is a characteristic free boundary and the equations for all states are coupled with each other due to the combustion process. We first employ the Lagrangian coordinate transformation to fix the free boundary. Then by introducing the flow slope and Bernoulli function, we further reduce the fixed boundary value problem into an initial boundary value problem for a first order hyperbolic system coupled with several ordinary differential equations. A new iteration scheme is developed near the background states by employing the intrinsic structure of the equation for the mass fraction of the non-combustion gas. We show that there is a fixed point for the iteration by deriving some novel $C^{1,\alpha}$-estimates of the solutions and applying the fixed point theorem, and then the uniqueness of the fixed point is proved by a contraction argument. On the other hand, a quasi-one-dimensional approximate system is often used to simplify the two-dimensional steady supersonic combustion model. The error between these two systems is estimated. Finally, given a piece-wise $C^{1,\alpha}$-solution containing a contact discontinuity with piece-wise constant states on the entrance of the nozzle, we can show that the solution is the piece-wise constant states with a straight contact discontinuity.