Smashis a common project between INRIA-Sophia Antipolis–Méditerranée and Aix-Marseille University. Its main topic is related to both the mathematical and numerical modelling of heterogeneous and complex flows in heterogeneous media, such as (inert or reactive) granular materials, interface problems or polluant propagation in urban environments.This project team was previously located at both locations (till 2008) and is now uniquely located at Marseille. While two INRIA members left Smashto join another INRIA project team PUMASin 2009, two associated professors have been hired at Marseille both in September 2009: Fabien Petitpas who has got a University-CNES chair , and Nicolas Favrie, who has got an associate professor position at the Civil Engineering Department of Polytech'Marseille School, Aix-Marseille I University.

Smashis a common project between INRIA Sophia Antipolis – Méditerranée and Aix-Marseille University. Its main topic is related to both the mathematical and numerical modelling of heterogeneous flows such as multiphase media, granular materials and interface problems.

The first issue deals with the
*design*and
*improvements*of
*theoretical models*for multiphase and interfacial
flows. Particular attention is paid to
*well posedness*issues and
*system's hyperbolicity*.

The second issue deals with the
*design*of
*appropriate numerical schemes*. These models are not
known as well as conventional single fluid models and pose
numerical challenges such as, for example, the numerical
approximation of
*non–conservative terms*. These numerical issues pose
*theoretical*questions such as,
*shock wave existence in multiphase mixture*,
*cell averages of non–conservative variables*,
*Chapman–Jouguet detonation conditions for heterogenous
explosives*and so on.

The final aim is to
*implement*the resulting algorithms on
*parallel machines*for solving
*large scale problems*for the design of advanced
technology systems in Space, Defense and Nuclear energy, but
also in security problems such as the propagation of
polluants in urban atmosphere and environment.

One of the main original features of the
Smashresearches on
heterogeneous flows lies in the way we deal with multiphase
mixtures. Our aim is to solve the
*same equations*everywhere with the
*same numerical*method :

in pure fluid,

in multi-velocity mixtures,

in artificial smearing zones at material interfaces or in mixture cells,

in shocks, phase transition fronts, detonation waves,

in elastic-plastic materials.

An example of such computations is given in the figure .

There are some advantages with this approach :

the most obvious relies in the
*coding simplicity*and
*robustness*as a
*unique algorithm*is used;

*conservation principles*are guarantied
*for the mixture*. Conventional algorithms are able
to preserve mass conservation only when dealing with
interfaces;

*interface conditions are perfectly matched*even for
the coupling of complex media (granular flows, capillary
fluids, transition fronts) even in the presence of
*shocks*;

this approach is the
*only one able to deal with dynamic appearance of
interfaces*(cavitation, spallation);

our methods allow the
*coupling of multi-velocities, multi-temperatures
mixtures*to macroscopic interfaces where a single
velocity must be present. To illustrate this capability
consider the example of a cloud of bubbles rising up in a
liquid to the surface, where a free boundary is present.
Two velocities have to be considered for the bubbles
rising, while a single velocity must be present just
after their crossing through the interface. This is also
the only method able to deal with such situations.

Our approach rises increasing attention from the
*scientific community*as well as
*the industry*. As will be detailed further, many
projects are currently under development with french oriented
research centers (DGA, CNES, SNECMA) as well as foreign ones
(Idaho National Laboratory - USA, ADD, Korea).

Richard Saurel has received the Prix Edmond Brun award from the French Academy of Sciences for his research work in multiphase flows in the spatial sciences area, last October 2010.

Conventional one-dimensional models of two-phase mixtures having two velocities form a system of six partial differential equations: two mass, two momentum and two energy equations. Those models are not hyperbolic and are consequently ill posed. It means that there is no continuous dependance from initial data and boundary conditions to the solution. In other words, wave propagation may have no physical sense.

This issue has been understood by and subtle remedy was given by . They proposed an extended model with seven equations. The extra differential equation replaced the pressure equilibrium assumption in the mixture. Thanks to this new equation, the model was correctly posed, unconditionally hyperbolic.

This model had little diffusion as it was presented in the context of a specific problem of detonation physics. Also, the model was difficult to solve at the numerical level, in particular with modern algorithms based on the Riemann problem solution. In we developed the first Godunov type method for this model and derived accurate approximation formulas for the non-conservative terms. Moreover, a specific relaxation method was built in order to solve these equations in the presence of stiff relaxation terms. This issue was particularly important as,

this model was involving two pressure and two velocities,

at an interface the jump condition corresponds to continuous normal velocities and continuous pressures,

in order to fulfill this condition it was necessary to relax the two pressures and velocities to unique equilibrium variables.

Such an issue was reached by using specific relaxation solvers, with infinite relaxation parameters like in . With this solver, the model was able to solve interface problems (air/water for example) and multiphase mixtures with two velocities. Important applications of fundamental and applied physics were possible to solve. Financial supports from DGA and CEA helped us to pursue the investigations.

Denoting
p_{r}=
p_{1}-
p_{2},
u_{r}=
u_{1}-
u_{2}, the two-phase flow model presents under
the form (
) :

Only the equations for phase 1 are written, since those of phase 2 are symmetric. General closure relations for this system need the determination of :

the interface velocity
u_{I}and pressure
p_{I}that respectively represent the velocity and
pressure that exert at the boundary of a cloud of bubbles
or droplets,

the average interface velocity
u_{I}^{'}and pressure
p_{I}^{'}that exert in the bulk of a two-phase control
volume,

the relaxation parameters and that control the rate at which velocities and pressures relax to mechanical equilibrium respectively.

These relations were unknown, either estimated in limit cases only, or determined by experimental means. In order to determine these closure relations a new homogenization method has been built in .

This new averaging method considers the mixture at the discrete level, with a stencil composed of three computational cells. In each cell, at each cell boundary and at each internal boundary separating the phases, the Riemann problem of the pure fluid equations (RP) is solved. The RP solution provides all local interfacial information. These RP solutions are then averaged in the computational cell as done originally with the first version of the Godunov method, derived originally for the Euler equations. In our context, extra difficulties appear, due to the presence of internal material interfaces, material discontinuities at cell boundaries and variable sub-volumes, due to the phase presence in the cells. But the philosophy was the same as with the Godunov method : we are dealing with average RP solutions and not with discretized partial differential equations.

The resulting system of this averaging procedure is a quite complicated discrete system in algebraic form. It corresponds to the result of the Discrete Equations Method (DEM). The closure relations for the various interface variables have been obtained by reaching the continuous limit of these discrete equations , that provide information easier to interprete than discrete formulas.

With this strong modelling foundations, it was possible to consider problems with extended physics : turbulence, phase transition, ions and electrons in plasma mixtures, granular materials, chemical reactions, continuum media with elastic-plastic effects. An example is shown in the figures - .

Most of these extensions are done with the help of the Hamilton principle of least action , to develop appropriate single phase material models that are then coupled with the DEM to form a multiphase flow model.

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
,
: 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
, nor interface reconstruction
and so on. The same equations are solved everywhere
,
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 Smashteam :

a model involving
*capillary*,
*compressibility*and
*viscous effects*
. 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
;

*phase transition*in
*metastable liquids*. 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
,
and
. 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
, and implemented in a
parallel, three-dimensional code
. This model is able to
reproduce phase separation and other complex phenomena
;

extending the approach to deal with
*fluid-structure interactions*. A non-linear elastic
model for compressible materials has been built
. 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
, that has been persued last
year
;

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
, closed by appropriate shock
conditions
, 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
.

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.

All the mathematical models considered and studied in Smashconsist in hyperbolic systems of PDE's. Most of the attention is focused on the 7 equation model for non-equilibrium mixtures and the 5 equation model for mechanical equilibrium mixtures. The main difficulty with these models is that they cannot be written under divergence form. Obviously, the conservation principles and the entropy inequality are fulfilled, but some equations (the volume fraction equation in particular) cannot be cast under conservative form. From a theoretical point of view, it is known since the works of Schwartz that the product of two distributions is not defined. Therefore, the question of giving a sense to this product arises and as a consequence, the numerical approximation of non-conservative terms is unclear , . Aware of this difficulty, we have developed two specific methods to solve such systems.

The first one is the
*discrete equations method*(DEM) presented previously as
a new homogenization method. It is moreover a numerical
method that solves non-conservative products for the 7
equation model in the presence of shocks. With this method,
Riemann problem solutions are averaged in each sub-volume
corresponding to the phase volumes in a given computational
cell. When a shock propagates inside a cell, each interaction
with an interface, corresponds to the location where
non-conservative products are undefined. However, at each
interaction, a diffraction process appears. The shock
discontinuity splits in several waves : a left facing
reflected wave, a right facing transmitted wave and a contact
wave. The interface position now corresponds to the one of
the contact wave. Along its trajectory, the velocity and
pressure are now continuous : this is a direct
consequence of the diffraction process. The non-conservative
products that appear in these equations are precisely those
that involve velocity, pressure and characteristic function
gradient. The characteristic function gradient remains
discontinuous at each interface (it corresponds to the
normal) but the other variables are now continuous.
Corresponding non-conservative products are consequently
perfectly defined : they correspond to the local
solution of the Riemann problem with an incoming shock as
initial data. This method has been successfully developed and
validated in many applications
,
,
,
.

The second numerical method deals with the numerical
approximation of the
*five equation model*. Thanks to the shock relations
previously determined, there is no difficulty to solve the
Riemann problem. However, the next step is to average (or to
project) the solution on the computational cell. Such a
projection is not trivial when dealing with a
non-conservative variable. For example, it is well known that
pressure or temperature volume average has no physical
meaning. The same remark holds for the
*cell average*of volume fraction and internal energy. To
circumvent this difficulty a new relaxation method has been
built
. This method uses
*two main ideas*.

The first one is to
*transform*one of the
*non-conservative products*into a
*relaxation term*. This is possible with the volume
fraction equation, where the non conservative term
corresponds to the asymptotic limit of a pressure relaxation
term. Then, a splitting method is used to solve the
corresponding volume fraction equation. During the hyperbolic
step, there is no difficulty to derive a positivity
preserving transport scheme. During the stiff relaxation
step, following preceding analysis of pressure relaxation
solvers
, there is no difficulty neither
to derive entropy preserving nor positive relaxation
solvers.

The second idea deals with the
*management of the phase's energy equations*, which are
also present under
*non-conservative form*. These equations are able to
compute regular/smooth solutions, such as expansion waves,
but are inaccurate for shocks. Thus they are only used at
shocks to predict the solution. With the predicted internal
energies, phase's pressures are computed and then
*relaxed to equilibrium*. It results in an
*approximation*of the volume fraction at shocks. This
approximation is then used in the
*mixture equation of state*, that is unambiguously
determined. This equation of state is based on the
*mixture energy*, a supplementary equation. This
equation, apparently redundant, has to be fulfiled however.
Its numerical approximation is obvious even in the presence
of shocks since it is a conservation law. With the help of
the mixture energy and predicted volume fraction, the
*mixture pressure*is now computed, therefore closing the
system. This treatment guarantees
*correct*,
*convergent*and
*conservative wave transmission*across material
interfaces separating pure media. When the interface
separates a fluid and a mixture of materials, the correct
partition of energies among phases is fulfiled by replacing
at the shock front the internal energy equations by their
corresponding jumps
. To ensure the numerical
solution strictly follows the phase's Hugoniot curves, the
poles of these curves are transported
. With this treatment, the method
also converges for multiphase shocks.

This method is very
*efficient*and
*simple to implement*. This also helped us considerably
to solve very large systems of hyperbolic equations, like
those arising for elastic materials in large deformations.
The fluid-solid coupling via diffuse interfaces with extreme
density ratios was done efficiently, as shown in figure
.

Another difficulty encountered in solving two-phase flow problems comes from the high disparity between the wave speeds of each existing fluid material. In particular, one of the fluids may be very close to the incompressibility limit. In that case, we face up the problem of very low Mach number flows. The numerical treatment of these flows is still a problem and involves non trivial modifications of the original upwind schemes , . Our investigations in that domain concern both acoustic and incompressible aspects in methodologies for setting up suitable numerical methods.

About 15 years ago, working on the physics of detonation waves in highly energetic materials, we discovered a domain where flow conditions were extreme. Numerical simulations in detonation conditions were a true challenge. The mathematical models as well as numerical methods must be particularly well built. The presence of material interfaces was posing considerable difficulties.

During the years 90–95, we have investigated open and
classified litterature in the domain of multimaterial
shock-detonation physics codes. We came to the conclusion
that nothing was clear regarding
*mixture cells*. These
*mixture cells*are a consequence of the numerical
diffusion or cell projection of flow variables at contact
discontinuites.

Thus, we have developed our own approach. On the basis of multiphase flow theory, revisited for a correct treatment of wave dynamics, we have proposed to solve mixture cells as true multiphase mixtures. These mixtures, initially out of equilibrium, were going to relax to mechanical equilibrium with a single pressure and velocity.

From this starting point, many extensions have been done, most times initiated by applications connected to the Defense domain. Collaborations have never stopped with these specialized laboratories since 1993. Applications have also been done with Space, Automotive, Oil, Nuclear engineering domains. International projects have started with the US and Korea.

From the technology developed in the Defense area, important applications are now coming for Space industry (CNES and SNECMA). The aim is to restart the Ariane cryogenic engine several times, for orbit change. Restarting a cryogenic engine is very challenging as the temperature difference between cryogenic liquid and walls is about 300K. Stiff phase change, cavitation, flashing in ducts and turbopumps are expected. These phenomena have to be particularly well computed as it is very important to determine the state of the fluids at the injection chamber. This is crucial for the engine ignition and combustion stability.

From a modelling point of view, our models and methods are aimed to replace the technology owned by space laboratories, taken 10 years ago from nuclear laboratories.

To deal with these industrial relations, the startup RS2N has been created in 2004 on the basis of the Innovation Law of the Minister Claude Allègre.

Four contracts with the Gramat Research Center (DGA) are under realization for the modelling of explosions with liquid tanks, granular materials, combustion of particle clouds, phase change etc. The total amount is 2M€ for 6 years work. They will end in 2013.

A contract with DGA (REI) is under realization for the modelling of solid-fluid coupling in extreme conditions. The diffuse interface theory is under extension to build equations which will be valid in pure solids, pure fluids as well as interfaces. The total amount is 300 K€ for 3 years work. It will end in 2011.

A contract with Chungnam National University (South Korea) is starting in order to model supercavitation around a high velocity topedo, propulsed by underwater solid rocket motor. The total amount is 100 K$ for one year work. It will end in 2010, but will possibly continue.

CNES and SNECMA have joined their support and efforts to ask Smashfor the development of flow solvers to study the restart stage of cryogenic engine under microgravity.

A three year project plan has been validated during 2009 for the total amount of 650 K€. In addition, Richard Saurel has been asked by SNECMA to make different expertise works in other areas, such as cavitation and system codes.

This work deals with the building of a discrete model able to describe and to predict the evolution of complex gas flows in heterogeneous media. In many physical applications, large scales numerical simulation is no longer possible because of a lack of computing resources. Indeed the medium topology may be complex due to the presence of many obstacles (walls, pipes, equipments, geometric singularities, ...). Aircraft powerplant compartments are examples where topology is complex due to the presence of pipes, ducts, coolers and other equipments. Other important examples are gas explosions and large scale dispersion of hazardous materials in urban places, cities or undergrounds involving obstacles such as buildings and various infrastructures. In all cases efficient safety responses are required. Then a new discrete model is built and solved in reasonable execution times for large cells volumes including such obstacles. Quantitative comparisons between experimental and numerical results are shown for different significant test cases, showing excellent agreement .

Conventional modelling of two-phase dilute suspensions is achieved with the Euler equations for the gas phase and gas dynamics pressureless equations for the dispersed phase, the two systems being coupled by various relaxation terms. The gas phase equations form a hyperbolic system but the particle phase corresponds to a hyperbolic degenerated one. Numerical difficulties are thus present when dealing with the particles system. In the present work, we consider the addition of turbulent effects in both phases in a thermodynamically consistent manner. It results in two strictly hyperbolic systems describing phases dynamics. Another important feature is that the new model has improved physical capabilities. It is able, for example, to predict particle dispersion, while the conventional approach fails. These features are highlighted on several test problems involving particles jets dispersion and are compared against experimental data. With the help of a single parameter (a turbulent viscosity), excellent agreement is obtained for various experimental configurations studied by different authors .

To investigate the effects of explosive composition on Al combustion, in particular regarding its oxygen balance, several liquid mixtures are experimentally studied with varying oxygen balance. They are then loaded with Al particles and the velocity of detonation (VOD) is recorded. Computational results with the help of conventional Chapman Jouguet (CJ) codes are compared but fail to reproduce experimental observations. A new multiphase flow model out of thermal equilibrium is then considered. Two options are considered as limiting cases : stiff thermal relaxation and vanishing heat exchange between Al and detonation products. With this last option, predictions are in excellent agreement with the experiments. This suggests that temperature disequilibrium plays a major role in heterogeneous explosives detonation dynamics .

For the simulation of light water nuclear reactor coolant flows, general two-phase models (valid for all volume fractions) have been generally used which, while allowing for velocity disequilibrium, normally force pressure equilibrium between the phases (see, for example, the numerous models of this type described by H. Städtke in ). These equations are not hyperbolic, their physical wave dynamics are incorrect, and their solution algorithms rely on dubious truncation error induced artificial viscosity to render them numerically well posed over a portion of the computational spectrum. The inherent problems of the traditional approach to multiphase modelling, which begins with an averaged system of (ill-posed) partial differential equations (PDEs) which are then discretized to form a numerical scheme, are avoided by employing a new homogenization method known as the Discrete Equation Method (DEM), . This method results in well-posed hyperbolic systems, this property being important for transient flows. This also allows a clear treatment of non-conservative terms (terms involving interfacial variables and volume fraction gradients) permitting the solution of interface problems without conservation errors, this feature being important for the direct numerical simulation of two-phase flows.

Unlike conventional methods, the averaged system of PDEs for the mixture are not used, and the DEM method directly obtains a well-posed discrete equation system from the single-phase conservation laws, producing a numerical scheme which accurately computes fluxes for arbitrary number of phases and solves non-conservative products. The method effectively uses a sequence of single phase Riemann problem solutions. Phase interactions are accounted for by Riemann solvers at each interface. Non-conservative terms are correctly approximated. Some of the closure relations missing from the traditional approach are automatically obtained. Lastly, the continuous equation system resulting from the discrete equations can be identified by taking the continuous limit with weak-wave assumptions. In this work, this approach is tested by constructing a DEM model for the flow of two compressible phases in one-dimensional ducts of spatially varying cross-section with explicit time integration. An analytical equation of state is included for both water vapor and liquid phases, and a realistic interphase mass transfer model is developed based on interphase heat transfer. A robust compliment of boundary conditions are developed and discussed. Though originally conceived as a first step toward implicit time integration of the DEM method (to relieve time step size restrictions due to stiffness and to achieve tighter coupling of equations) in multidimensions, this model offers some unique capabilities for incorporation into next generation light water reactor safety analysis codes. We demonstrate, on a converging-diverging two-phase nozzle, that this well-posed, 2-pressure, 2-velocity DEM model can be integrated to a realistic and meaningful steady-state with both phases treated as compressible .

We consider the flow of an ideal fluid in a
two-dimensional bounded domain, admitting flows through the
boundary of this domain. The flow is described by the Euler
equations with
*non-homogeneous*Navier slip boundary conditions. We
establish the solvability of this problem in the class of
solutions with
L_{p}-bounded vorticity,
p(1,
]. To prove the solvability we
realize the passage to the limit in Navier-Stokes equations
with vanishing viscosity
.

We find a sufficient condition of hyperbolicity for a system of partial differential equations governing the motion of a one-dimensional porous medium, so ensuring the well posedness of a solution for the associated Cauchy problem. We study propagation of linear waves in presence of a pure-fluid/porous-medium interface and we deduce novel expressions for the reflection and transmission coefficients in terms of the spectral properties of the governing differential system. We also propose an indirect method for measuring Biot s parameters when the measurement of the reflection and transmission coefficients associated to acoustic waves is possible .

A macroscopic model describing elastic-plastic solids is derived in a special case of the internal specific energy taken in separate form: it is the sum of a hydrodynamic part depending only on the density and entropy, and a shear part depending on other invariants of the Finger tensor. In particular, the relaxation terms are constructed compatible with the von Mises yield criteria. Also, the Maxwell type material behavior is shown up: the deviatoric part of the stress tensor is decaying during plastic deformations. Numerical examples show the ability of this model to deal with real physical phenomena .

A multiphase hyperbolic model for dynamic and irreversible powder compaction is built. Three major issues have to be addressed in this aim. The first one is related to the irreversible character of powder compaction. When a granular media is subjected to a loading-unloading cycle the final volume is lower than the initial one. To deal with this hysteresis phenomenon a multiphase model with relaxation is built. During loading, mechanical equilibrium is assumed corresponding to stiff mechanical relaxation, while during unloading non-equilibrium mechanical transformation is assumed. Consequently, the sound speeds of the limit models are very different during loading and unloading. These differences in acoustic properties are responsible for irreversibility in the compaction process. The second issue is related to dynamic effects where pressure and shock waves play important role. Wave dynamics is guaranteed by the hyperbolic character of the equations. Each phase compressibility is considered, as well as configuration pressure and energy. The third issue is related to multidimensional situations that involve material interfaces. Indeed, most processes with powder compaction entail free surfaces. Consequently the model has to be able to solve interfaces separating pure fluids and granular mixtures. These various issues are solved by a unique model fitting the frame of multiphase theory of diffuse interfaces , , . Model's ability to deal with these various effects is validated on basic situations, where each phenomenon is considered separately. Special attention is paid to the validation of the hysteresis phenomenon that occurs during powder compaction. Basic experiments on energetic material (granular HMX) and granular NaCl compaction are considered and are perfectly reproduced by the model. Excepting the materials equations of state (hydrodynamic and granular pressures and energies) that are determined on the basis of separate experiments found in the literature, the model is free of adjustable parameter. Its ability to reproduce the hysteresis phenomenon is due to a relaxation parameter that tends either to infinity in the loading regime, or to zero in the unloading stage. Discontinuous evolution of this relaxation parameter is explained , .

For this work, a hybrid numerical method using a Godunov
type scheme is proposed to solve the Green-Naghdi model
describing dispersive
*shallow water*waves. The corresponding equations are
rewritten in terms of new variables adapted for numerical
studies. In particular, the numerical scheme preserves the
dynamics of solitary waves. Some numerical results are
shown and compared to exact and/or experimental ones in
different and significant configurations. A dam break
problem and an impact problem where a liquid cylinder is
falling to a rigid wall are solved numerically. This last
configuration is also compared with experiments leading to
a good qualitative agreement
.

This study realized under DGA grant deals with the development of models and computational tools for nano-structured explosives. Comparative experiments are done at Nuclear Federal Center, Sarov, Russia.

This study realized under DGA grant, deals with the development of multiphase algorithms to compute the dispersion of a multiphase mixture in air and its interaction with detonation products.

This study realized under DGA grant, deals with the development of a conservative elastic-plastic-fluid flow model to deal with fluid-solid coupling in extreme deformations. A collaboration with Prof. S.K. Godunov is also active in this area.

Modelling and simulation of two-phase flows in cryogenic engine of space launchers (Ariane V) is the aim of this contract. A first contract is under realization with CNES. Another one with SNECMA. These two supports are aimed to continue during 4 years.

Scientific collaboration with the Lavrentyev Institute of Hydrodynamics in Russia: Nikolaï Makarenko has been invited for one month staying in 2010.

In the academic year 2009–2010, project members have taught the following courses :

Aix-Marseille I University : 138 h (partial discharge of education),

first and second year in
*fluid mechanics*and
*numerical methods in fluid dynamics*;

*two-phase dilute flows*.

Aix-Marseille I University : 192 h,

First, second and third year in
*Mathematics, Structure modelling, Programming
languages, Material resistance and concrete structure
study in civil engineering*.

Aix-Marseille III University : 192 h,

*mathematical methods for physicists*,
*continuum mechanics*;

*two-phase flows modelling*.

Aix-Marseille I University : 140h (in half delegation at CNRS),

First and second year in
*mathematics*,
*fluid mechanics*and
*thermics*.

Aix-Marseille I University : 192 h,

first, second and third year in
*programming languages for scientific
computing*,
*fluid mechanics*and
*numerical methods for fluid dynamics*;

*high performance computing*.

Aix-Marseille I University : 64 h,

First year in
*Thermodynamics*.

Aix-Marseille I University : 16h (in delegation at INRIA),

*Heterogeneous flows*;

Fifth year in
*Heterogeneous flows*.

is still this year the Director of the Mechanical Engineering Department of Polytech Engineering School of Marseille with a partial discharge of education (96 hours).

is Director of the Master M2
*Diphasic flows, Energetics and Combustion*.

is Director of the Doctoral School in
Engineering Sciences, including all research units of
Marseilles, Aix and Toulon in
*Mechanics*,
*Acoustics*,
*Energetics*,
*Macroscopic Physics*,
*Micro and Nanoelectronics*. The laboratories are
CNRS UMR and UPR units: LMA, IUSTI, IRPHE, M2P2, IM2NP.
The doctoral school involves more than 300 researchers
and 200 PhD students.

This year, the project has harbored the following Ph. D Students :

Aix-Marseille University, RS2N-Région PACA grant ,

*Modelling irreversible and dynamic compaction of
powders*, since 2008.

Aix-Marseille University and DGA Gramat, DGA financial support,

*Experimental and numerical study of liquid and solid
dispersion under explosion conditions*.

Aix-Marseille University, DGA grant ,

*Modelling multiphase explosions and dispersion
phenomena*, since 2007.

Aix-Marseille University, MRE support,

*Modélisation et simulation numérique de la dispersion
de fluides dans un milieu fortement hétérogène*, since
2009.

Aix-Marseille University, salaried, (professor at the CPGE of "Lycée Notre Dame de Sion", Marseille,

*Écoulements des eaux peu profondes avec effet de
cisaillement*, since 2009.

Aix-Marseille University, CNES-SNECMA support,

*Simulation numérique directe de la crise d'ébullition
dans les systèmes spatiaux*, since November 1rst
2010.

Members of the project team Smashhave delivered invited lectures and/or have been coorganizers in the following conferences , schools and/or seminars :

*ICMS, Mathematical challenges and modelling in
hydroelasticity*, Edinburgh, United Kingdom, 21-24
Juin, 2010,
;

(
http://

*CANUM 2010*, Carcans-Maubuisson, May 31–June 4,2010
(S. Gavrilyuk only),
;

(
http://

*Advances in Continuum Mechanics*, Bohum, Germany,
June 30–July 2, 2010 (S. Gavrilyuk only),
;

(
http://

*International school "Variational methods in solid and
fluid mechanics", CISM*, July 2010 , Udine, Italy (S.
Gavrilyuk has also co-organized that school),
;

(
http://

*CIRM Workshop "Méthodes de domaine fictif pour des
conditions aux limites immergées "*, August
30–September 3, Marseille, 2010;

*École d'été franco-allemande de l'UFA (Université
Franco-Allemande) 2010 : Modélisation, simulation
numérique et optimisation en mécanique des fluides :
théorie et pratique*, September 6–9 2010 (Nicolas
Favrie has given an invited lecture on
*Solid-fluid elasto-plastic diffuse interface model in
cases of extreme deformations*,
);