Section: Scientific Foundations

Modelling of Interface and Multi-Fluid Problems

In order to solve interfaces separating pure fluids or pure materials, two approaches have been developed. The first one has been described previously. It consits in solving a non-equilibrium flow model with two pressures and two velocities, and then in relaxing instantaneously these variables to equilibrium ones. Such a method allows a perfect fulfillment of interface conditions in mixture cells, that appear as a result of numerical diffusion at material interfaces.

The second option consists in determining the asymptotic model that results from stiff mechanical relation. In the context of two fluids, it consists in a set of five partial differential equations [35] , [32]  : two masses, one mixture momentum, one mixture energy and one volume fraction equations. Such a system is obviously less general than the previous non-equilibrium system, but it is particularly interesting in solving interface problems, where a single velocity is present. More precisely, it is more appropriate and simpler, when considering extra physics extensions such as, phase transition, capillary effects, elastic-plastic effects.

Contrarily to conventional methods, there is no need to use a front tracking method, nor level set [28] , nor interface reconstruction and so on. The same equations are solved everywhere [36] , [37] and the interface is captured with the 5 equation model. This model provides correct thermodynamic variables in artificial mixture zones. Although seemingly artificial, this model can handle huge density ratio, and materials governed by very different equations of state, in multi-dimensions. It is also able to describe multiphase mixtures where stiff mechanical relaxation effects are present, such as, for example, reactive powders, solid alloys, composite materials etc.

Several extentions have been done during these recent years by the Smash team :

• a model involving capillary, compressibility and viscous effects [40] . This is the first time such effects are introduced in a hyperbolic model. Validations with experiments done at IUSTI (the laboratory where the group of Marseille is located at) have shown its excellent accuracy, as shown in the figure 4 ;

  Figure 4. Comparison of the drop shape during formation (experiment in grey area, computations in lines). No interface tracking nor interface reconstruction method are used. The same equations are solved at each mesh point. The model accounts for compressible, viscous and capillary effects. The compressible effects are negligible in the present situation, but they become fundamental in other situations (phase transition for example) where the full thermodynamics of each fluid is mandatory. The method treats in a routinely manner both merging and fragmentation phenomena.

• phase transition in metastable liquids [9] . This is the first time a model solves the ill-posedness problem of spinodal zone in van der Waals fluids.

The combination of capillary and phase transition effects is under study in order to build a model to perform direct numerical simulation (DNS) of phase transition at interfaces, to study explosive evaporation of liquid drops, or bubble growth in severe heat flux conditions. This topic has important applications in nuclear engineering and future reactors (ITER for example). A collaboration has been started with the Idaho National Laboratory, General Electrics, and MIT (USA) in order to build codes and experiments on the basis of our models and numerical methods. In another application domain, several contracts with CNES and SNECMA have been concluded to model phase transition and multiphase flows in the Ariane VI space launcher cryogenic engine.

In the presence of shocks, fundamental difficulties appear with multiphase flow modelling. Indeed, the volume fraction equation (or its variants) cannot be written under divergence form. It is thus necessary to determine appropriate jump relations.

In the limit of weak shocks, such relations have been determined by analysing the dispersive character of the shock structure in [44] , [30] and [29] . Opposite to single phase shocks, backward information is able to cross over the shock front in multiphase flows. Such phenomenon renders the shocks smooth enough so that analytical integration of the energy equations is possible. Consequently, they provide the missing jump condition.

These shock conditions have been validated against all experimental data available in the various American and Russian databases, for both weak and very strong shocks.

At this point, the theory of multiphase mixtures with single velocity was closed. Thanks to these ingredients we have done important extensions recently :

• restoration of drift effects : a dissipative one-pressure, one-velocity model has been studied in [39] , and implemented in a parallel, three-dimensional code [38] . This model is able to reproduce phase separation and other complex phenomena [31] ;

• extending the approach to deal with fluid-structure interactions. A non-linear elastic model for compressible materials has been built [2] . It extends the preceding approach of Godunov to describe continuum media with conservative hyperbolic models. When embedded in our multiphase framework, fluid solid interactions are possible to solve in highly non-linear conditions with a single system of partial differential equations and a single algorithm. This was the aim of Nicolas Favrie's PhD thesis [26] , that has been persued last year [27] ;

• determining the Chapman–Jouguet conditions for the detonation of multiphase explosives. The single velocity - single pressure model involves several temperatures and can be used to describe the non-equilibrium detonation reaction zone of condensed heterogenous energetic materials. Since the work of Zeldovich-Neumann and Doering (ZND model), the detonation dynamics of gaseous and condensed energetic materials is described by the ZND approach, assuming mixtures in thermal equilibrium. However, in condensed energetic materials, the mixture is not of molecular type and the thermal equilibrium assumption fails. With the help of the same model used for phase transition [9] , closed by appropriate shock conditions [44] , it is now possible to develop a ZND type model with temperature disequilibrium. This opens a new theory for the detonation of condensed materials. Successful computations of multidimensional detonation waves in heterogenous explosives have been done with an appropriate algorithm in [41] .

Obviously, all these models are very different from the well studied gas dynamics equations and hyperbolic systems of conservation laws. The building of numerical schemes requires special attention as detailed hereafter.