A splitting method adapted to the simulation of mixed flows in pipes with a compressible two-layer model

EDF R&D, Dept. Mécanique des Fluides, Énergetique et Environnement, 6 quai Watier, 78401 Chatou cedex, France
Laboratoire de Mathématiques, UMR 5127-CNRS and Université Savoie Mont Blanc, 73376 Le Bourget-du-Lac Cedex, France
Institut de Mathématiques de Marseille, UMR 7373-CNRS, Université Aix-Marseille and Centrale Marseille, 13453, Marseille Cedex, France

^* Corresponding author: charles.demay@outlook.fr

Received: 5 January 2018
Accepted: 27 August 2018

Abstract

The numerical resolution of the Compressible Two-Layer model proposed in [27] is addressed in this work with the aim of simulating mixed flows and entrapped air pockets in pipes. This five-equation model provides a unified two-phase description of such flows which involve transitions between stratified regimes (air–water herein) and pressurized or dry regimes (pipe full of water or air). In particular, strong interactions between both phases and entrapped air pockets are accounted for. At the discrete level, the coexistence of slow gravity waves in the stratified regime with fast acoustic waves in the pressurized regime is difficult to approximate. Furthermore, the two-phase description requires to deal with vanishing phases in pressurized and dry regimes. In that context, a robust splitting method combined with an implicit-explicit time discretization is derived. The overall strategy relies on the fast pressure relaxation in addition to a mimetic approach with the shallow water equations for the slow dynamics of the water phase. It results in a three-step scheme which ensures the positivity of heights and densities under a CFL condition based on the celerity of material and gravity waves. In that framework, an implicit relaxation-like approach provides stabilization terms which are activated according to the flow regime. Numerical experiments are performed beginning with a Riemann problem for the convective part. The overall approach is then assessed considering relevant mixed flow configurations involving regime transitions, vanishing phases and entrapped air pockets.