Modelling of an Homogeneous Equilibrium Mixture Model (HEM)
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Modelling of an Homogeneous Equilibrium Mixture Model (HEM) A. Bernard-Champmartin · O. Poujade · J. Mathiaud · J.-M. Ghidaglia
Received: 17 October 2012 / Accepted: 19 April 2013 / Published online: 29 May 2013 © Springer Science+Business Media Dordrecht 2013
Abstract We present here a model for two phase flows which is simpler than the 6equations models (with two densities, two velocities, two temperatures) but more accurate than the standard mixture models with 4 equations (with two densities, one velocity and one temperature). We are interested in the case when the two-phases have been interacting long enough for the drag force to be small but still not negligible. The so-called Homogeneous Equilibrium Mixture Model (HEM) that we present is dealing with both mixture and relative quantities, allowing in particular to follow both a mixture velocity and a relative velocity. This relative velocity is not tracked by a conservation law but by a closure law (drift relation), whose expression is related to the drag force terms of the two-phase flow. After the derivation of the model, a stability analysis and numerical experiments are presented. Keywords Mixture model · Drift closure · Two phase flows · Darcy law Mathematics Subject Classification (2010) 76T10 · 76N99 · 65Z05 · 65M06 · 41A60
A. Bernard-Champmartin () · O. Poujade · J. Mathiaud CEA, DAM, DIF, 91297 Arpajon, France e-mail: [email protected] A. Bernard-Champmartin · J. Mathiaud · J.-M. Ghidaglia CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue du Président Wilson, 94230 Cachan, France Present address: A. Bernard-Champmartin INRIA Sophia Antipolis Méditerranée, 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France Present address: A. Bernard-Champmartin LRC MESO, ENS Cachan/CEA, DAM, DIF, 61 avenue du Président Wilson, 94235 Cachan Cedex, France
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A. Bernard-Champmartin et al.
1 Introduction Dispersed flows (i.e. droplets in a carrying fluid) as described in [21] are ubiquitous in nature and in industrial applications (see e.g. [3, 12, 13, 15, 23, 27, 30, 34]). During the past fifteen years, the continuous increase of computational power has triggered the interest of applying computational fluid dynamics to flows involving several phases. However a full description of these flows requires such a tremendous amount of computational power that one must resort to simplified models. These flows are governed by the exchanges between the two phases due to temperature, pressure or velocity differences. Depending on the level of equilibrium between the two phases and the degree of complexity that one wants to conserve, different types of model are available (cf. [37] or [9] for a complete review). A first possibility consists in describing all fields for each phase. This leads to a set of 6 (or 7) equations depending on whether or not equilibrium in pressure is assumed (see e.g. [11, 32, 36]). Due to the numerical complexity of such models (including the presence of stiff source terms, high number of equations, loss of hyperbolicity in so
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