Numerical validation of homogeneous multi-fluid models

We consider a 1D periodic system containing a large number of layers. Each layer is com-posed 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 equa-tions that govern the multi-layer system, and two isentropic homogeneous (average) mod-els effectively describing such a complex two-fluid mixture.The first homogeneous model is a 2 x 2 isentropic one-pressure two-fluid model. The sec-ond one is described by a 3 x 3 system, which additionally takes into account some turbu-lent 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 x 2 model is recovered from the 3 x 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 ve-locity. 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 mono-tone (non-monotone) behaviour of the corresponding sound speed in the 2 x 2 model as a function of the mass fraction of the phases.(c) 2022 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates a one-dimensional periodic system consisting of multiple layers, each composed of compressible fluids with different densities and equations of state. A detailed numerical solution of the Euler equations governing the multi-layer system is compared with two isentropic homogeneous models that effectively describe the complex two-fluid mixture. The first model is a 2x2 isentropic one-pressure two-fluid model, while the second model is a 3x3 system that takes into account turbulent effects. The numerical solution and the two models show good agreement for smooth solutions, but differ for shock waves. The second model provides a better description for large density ratios, while the first model is more accurate for moderate density ratios.