Stability of transonic contact discontinuity for two-dimensional steady compressible Euler flows in a finitely long nozzle

Feimin Huang; Jie Kuang; Dehua Wang; Wei Xiang

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arXiv · 2012.10039

Stability of transonic contact discontinuity for two-dimensional steady compressible Euler flows in a finitely long nozzle

Feimin Huang·Jie Kuang·Dehua Wang·Wei Xiang

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Abstract

We consider the stability of transonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle. This is the first work on the mixed-type problem of transonic flows across a contact discontinuity as a free boundary in nozzles. We start with the Euler-Lagrangian transformation to straighten the contact discontinuity in the new coordinates. However, the upper nozzle wall in the subsonic region depending on the mass flux becomes a free boundary after the transformation. Then we develop new ideas and techniques to solve the free-boundary problem in three steps: (1) we fix the free boundary and generate a new iteration scheme to solve the corresponding fixed boundary value problem of the hyperbolic-elliptic mixed type by building some powerful estimates for both the first-order hyperbolic equation and a second-order nonlinear elliptic equation in a Lipschitz domain; (2) we update the new free boundary by constructing a mapping that has a fixed point; (3) we establish via the inverse Lagrangian coordinates transformation that the original free interface problem admits a unique piecewise smooth transonic solution near the background state, which consists of a smooth subsonic flow and a smooth supersonic flow with a contact discontinuity.

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Feimin Huang, Jie Kuang, Dehua Wang, Wei Xiang. 2020-12-18. Stability of transonic contact discontinuity for two-dimensional steady compressible Euler flows in a finitely long nozzle. https://arxiv.org/abs/2012.10039

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