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Acta Appl Math (2014) 129:121
DOI 10.1007/s10440-013-9827-2
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 ows which is simpler than the 6-equations 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 ow. After the derivation of the model, a stability analysis and numerical experiments are presented.
Keywords Mixture model Drift closure Two phase ows Darcy law
Mathematics Subject Classication (2010) 76T10 76N99 65Z05 65M06 41A60
A. Bernard-Champmartin ( ) O. Poujade J. Mathiaud
CEA, DAM, DIF, 91297 Arpajon, France e-mail: mailto:[email protected]
Web End [email protected]
A. Bernard-Champmartin J. Mathiaud J.-M. Ghidaglia
CMLA, ENS Cachan, CNRS, UniverSud, 61 Avenue du Prsident Wilson, 94230 Cachan, France
Present address:A. Bernard-Champmartin
INRIA Sophia Antipolis Mditerrane, 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 Prsident Wilson, 94235 Cachan Cedex, France
Modelling of an Homogeneous Equilibrium Mixture Model (HEM)
2 A. Bernard-Champmartin et al.
1 Introduction
Dispersed ows (i.e. droplets in a carrying uid) 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 fteen years, the continuous increase of computational power has triggered the interest of applying computational uid dynamics to ows involving several phases. However a full description of these ows requires such a tremendous amount of computational power that one must resort to simplied models.
These ows are governed by the exchanges between the two phases due...