Unsteady compressible flow in ducts with varying cross-section : comparison between the nonconservative Euler system and the axisymmetric flow model

Abstract

Most of nonconservative hyperbolic systems corresponds to a reduction of an initial three-dimensional problem deriving from a homogenization procedure. Unfortunately, the reduced model gives rise to two new difficulties : the resonance problem corresponding to a splitting or a merging of the genuinely nonlinear waves and the non uniqueness of the Riemann problem solution. The question arises to check whether the two problems correspond and provide similar solutions, at least numerically. In this paper, we propose a comparison between the one-dimensional nonconservative Euler equations modelling the duct with variable cross-sectional area with its original three-dimensional conservative Euler system. Based on the classification of the Riemann problems proposed in \cite{ClRo}, we compare the numerical results of the two models for a large series of representative configurations. We also propose a new example of non uniqueness for the Riemann problem involving the resonance phenomena.