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On Admissible Locations of Transonic Shock Fronts for Steady Euler Flows in an Almost Flat Finite Nozzle with Prescribed Receiver Pressure

This paper concerns the existence of transonic shock solutions to the 2‐D steady compressible Euler system in an almost flat finite nozzle (in the sense that it is a generic small perturbation of a flat one), under physical boundary conditions proposed by Courant‐Friedrichs in [14], in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock front is some of the most desirable information one would like to determine. However, the location of the normal shock front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. Once an initial approximation is obtained, a further nonlinear iteration can be constructed and proved to lead to a transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived that yields the existence of the solutions to the proposed free boundary problem. By this condition, it will be shown that, for the proposed linear free boundary problem, if the nozzle is either strictly expanding or contracting, there exists only one solution as long as the given pressure at the exit lies within a certain interval; while if the nozzle is generic with both expanding and contracting portions, there may exist more than one solution for the same given receiver pressure, which implies the existence of more than one initial approximating shock front. This nonuniqueness of the initial approximations will lead to the nonuniqueness of the transonic shock solutions for generic nozzles with the same given receiver pressure, which shows the instability of the unperturbed normal shock solution under general perturbations of the nozzle boundary and how this instability may behave. © 2020 Wiley Periodicals LLC