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Diffuse interfaces modelling

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Diffuse interfaces are a consequence of numerical diffusion at contact discontinuities separating various materials. They appear with any Eulerian hyperbolic solver and result in computational mixture cells. This has serious consequences on the thermodynamic state computation as the equations of state of the fluids in contact are discontinuous. To circumvent this difficulty artificial mixture cells were considered as true multiphase mixtures with stiff mechanical relaxation effects (Saurel and Abgrall, 1999). This method was simplified by Kapila et al. (2001) with the help of asymptotic analysis, resulting in a single velocity, single pressure but multi-temperature flow model. This model present serious difficulties for its numerical resolution, as one of the equations is non-conservative, but is an excellent candidate to solve mixture cells as well as pure fluids.

In the presence of shocks, jump conditions have to be provided. They have been determined in Saurel et al. (2007) in the weak shock limit. When compared against experiments for both weak and strong shocks, excellent agreement was observed. These relations are accepted as closure relations for the Kapila et al. (2001) model in the presence of shocks.

Mass transfer modeling in this model was addressed in Saurel et al. (2008), in the context of evaporation and flashing fronts. With the help of corresponding heat and mass transfer terms, it was possible to deal with high speed cavitating flows.

Oppositely to the previous example of endothermic phase transition, when exothermic effects are considered as for example with high energetic materials, detonation waves appear. With the help of the shock relations and governing equations inside the reaction zone, generalized Chapman-Jouguet conditions are obtained as well as detonation wave structure of heterogenous explosives (Petitpas et al., 2009).

With the same multiphase flow model, solved at each mesh point with the same numerical scheme is it thus possible to deal with:

- material interfaces dynamics, eventually in the presence of surface tension (Perigaud and Saurel, 2005) and hyper-elastic solids (Favrie et al., 2009),

- shocks and detonation waves in heterogeneous energetic materials,

- phase transition fronts.

More recently, dynamic powders compaction including irreversible effects has been considered (Saurel et al., 2010) in the same theoretical frame. In addition, gas permeation effects have been restored, resulting in velocity drift effects in the Kapila et al. (2001) model. Slight velocity disequilibrium effects can thus be considered, extending diffuse interface modeling capabilities to fluids mixing and extra physics.

Kapila A., Menikoff R., Bdzil J., Son S., Stewart D. (2001) Two-phase modeling of DDT in granular materials: reduced equations, Physics of Fluids, 13, pp. 3002-3024

Perigaud G., Saurel R. (2005) A compressible flow model with capillary effects, Journal of Computational Physics, 209, pp. 139-178

Saurel R. and Abgrall R. (1999) A multiphase Godunov method for compressible multifluid and multiphase flows. Journal of Computational Physics, 150, pp 425-467

Saurel R., Petitpas F., Abgrall R. (2008), Modelling phase transition in metastable liquids. Application to cavitating and flashing flows, Journal of Fluid Mechanics, 607: 313-350

Favrie N., Gavrilyuk S. and Saurel R. (2009) Solid-fluid diffuse interface model in cases of extreme deformations. Journal of Computational Physics, vol. 228, Issue 16(1), pp 6037-6077

Petitpas F., Saurel R., Franquet E. and Chinnayya A. (2009) Modelling detonation waves in condensed energetic materials: Multiphase CJ conditions and multidimensional computations. Shock Waves, Vol. 19, Number 5, pp. 377-401

Saurel R., Petitpas F. and Berry R.A. (2009) Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. Journal of Computational Physics 228, pp 1678-1712

Saurel R., Favrie N., Petitpas F., Lallemand M.H. and Gavrilyuk S. (2010) Modelling irreversible dynamic compaction of powders. Journal of Fluid Mechanics, in press

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