3-D Axisymmetric Transonic Shock Solutions of the Full Euler System in Divergent Nozzles

Abstract

We establish the stability of 3-D axisymmetric transonic shock solutions of the steady full Euler system in divergent nozzles under small perturbations of an incoming radial supersonic flow and a constant pressure at the exit of the nozzles. To study 3-D axisymmetric transonic shock solutions of the full Euler system, we use a stream function formulation of the full Euler system for a 3-D axisymmetric flow. We resolve the singularity issue arising in stream function formulations of the full Euler system for a 3-D axisymmetric flow. We develop a new scheme to determine a shock location of a transonic shock solution of the steady full Euler system based on the stream function formulation.

On Admissible Positions of Transonic Shocks for Steady Isothermal Euler Flows in a Horizontal Flat Nozzle under Vertical Gravity

“…See also, for instance, [10,31] and references therein for literatures on quasi-one-dimensional nozzle flows. As to the steady multi-dimensional models such as potential equations or the Euler system, thanks to continuous efforts of many mathematicians, there have been substantial progresses in the past two decades, for instance, see [1,2,3,4,5,6,7,8,9,11,15,16,17,20,21,22,23,24,25,27,28,32,33,34,35,36,37,38,39,40,41]. Two typical kinds of nozzles are studied.…”

“…Based on this shock solution, in [9] by Chen and in a series of papers [20,21,25] by Li-Xin-Yin, the well-posedness of shock solutions in an expanding nozzle has been established, with prescribed pressure at the exit as suggested by Courant and Friedrichs. See also [22,23,24,33,34] for related studies on transonic shocks in a 3-D axisymmetric conic nozzle. The other is a flat nozzle with two parallel walls.…”