Examples of non-uniqueness of almost-unidirectional gas flow

Abstract

Modeling of non-linear phenomena always brings the question of uniqueness of the exact as well as numerical solution. The paper is devoted to the non-unique behavior of gases, which has been observed in special axisymmetric nozzles at transonic flow regimes. We describe the flow by means of the compressible Euler equations. Surprisingly, the non-uniqueness is recorded already by the simplest stationary quasi-one-dimensional model. In this case, we prove the non-uniqueness analytically. The non-unique behavior is present also in the three-dimensional model, which we illustrate numerically using an original mass-conserving axisymmetric finite volume scheme. Numerical example of non-unique solutions corresponding to an axisymmetric nozzle of a complicated geometry shows a good agreement between the quasi-one-dimensional analytical and axisymmetric three-dimensional numerical results.