Numerical validation of homogeneous multi-fluid models

Abstract

We consider a 1D periodic system containing a large number of layers. Each layer is composed of two sub-layers of compressible fluids having different densities and equations of state. We present a comparison between a detailed numerical solution of the Euler equations that govern the multi-layer system, and two isentropic homogeneous (average) models effectively describing such a complex two-fluid mixture.The first homogeneous model is a 2×2 isentropic one–pressure two-fluid model. The second one is described by a 3×3 system, which additionally takes into account some turbulent effects in terms of the corresponding energy. An effective entropy can be introduced in the second model, which is constant along material lines for smooth solutions. The 2×2 model is recovered from the 3×3 model, in the limit of vanishing turbulent energy.The detailed numerical solution of the multi-layer system is carried out by a suitably designed second order finite-volume shock-capturing scheme in Lagrangian coordinates, which makes use of a new Roe-type numerical flux function.For smooth solutions the two models are both in very good agreement with the detailed numerical solution of multi-layer Euler equations.When a shock develops, the multi-layer solution becomes highly oscillatory and transforms into a dispersive shock for large amplitude shocks. It is found that, in the case of moderate density ratio, the first model shows a better agreement with the corresponding shock velocity. However, for large density ratio, the second model provides a better description of the multi-layer system. The case of moderate (large) density ratio is associated to a monotone (non-monotone) behaviour of the corresponding sound speed in the 2×2 model as a function of the mass fraction of the phases.