SMASH is a common project between INRIA and Université de Provence. The main topic of our project focuses on problems related to both mathematical and numerical modelization of full heterogeneous flows such as multiphase media, granular materials or reactive flows with mass transfer. The scientific themes in concern are the setting up and perfecting of models of these flow problems including both definition and analysis of discretization methods and schemes for their numerical simulation. The aim is to implement the resulting algorithms using either multigrid or domain decomposition techniques on parallel machines.

One of the main original feature of the SMASH research on that topic lies in the way we deal with multifluid flows (interface problems). We use an eulerian approach with a diffuse interface model. The two different media are not modelled separately (using adequate scheme for each phase) nor the interface needs to be defined explicitely. On the opposite, the diffuse interface zone is considered as a true multiphase region and described with a two-phase model. With such an eulerian approach, numerical problems related to the discontinuities of the state equations through the interface which occur when using separate models for each phase, do not need to be treated anymore.

The domains in which two-phase flows are of interest are widely spread through the industry (nuclear industry, oil company industry, car engine technology, food industry ...) but also in forest fire, biomedical engineering, detonation or astrophysics research areas.

The microcospic description of an heterogeneous medium has to take
into account the specific physical properties of each material
component. For both pratical and numerical reasons, it may be
impossible to take into account all those microscopic features and for
each material. As an illustration, one

The flows we are interested in are characterized by the existence at the microscopic scale of interfaces between each fluid medium which is clearly distinct and identified. There are mainly two ways of considering the description model of such media.

In the first approach, namely the multifluid flow model, the
description of the fluid is defined at the heterogeneity scale. The
resulting mathematical model is in fact very simple and consists in
the set of either Euler or Navier-Stokes equations if the material
under consideration is either an inviscid or viscous fluid
respectively, or in the set of equations derived from solid mechanics
(elasticity for example) if the material is considered as an elastic
solid medium. It remains the main difficult problem of describing and
defining interfaces separating each component of that heterogeneous
medium together with the definition of the interaction model involving
the different physical exchange parameters of each material.

Such a microstructure approach may be considered as long as the
number of interfaces is not too large. When it is not the case, it
should be prefered to adopt what we call the multiphase flow
approach.

A typical multiphase flow contains a range of about

The definition and setting of such multiphase models may be based on different techniques.

The first technique, the full description of which may be found in

Another technique is based on the Hamilton principle. We indeed know
that the set of equations of the inviscid fluid mechanics may be
derived from a variational principle a priori both the kinetic and
potential energies of the multiphase flow. Writing down the
variational principal

Using either the first or the second technique results in a set of non linear partial differential equations (PDE) that we need to discretize by some numerical scheme in order to get a set of algebraic equations that in turn will furnish the discrete solution. To summarize, we are therefore faced on mainly two types of difficulties: the first class of difficuties consists in finding the good set of PDE's to model the multiphase flow we want to study; the second class of difficulties is based on the choice of the discretization scheme.

At last, there is another way of defining the multiphase flow model:
it is therefore possible to end up with the set of algebraic equations
without considering the preliminary step of defining the set of non
linear PDE's. It is what it is done in

Concerning the simulation of interface flow problems (with a microscale description of the interface) SMASH adopts the ``front capturing'' approach, that is an eulerian approach. As a matter of fact, the interface is not defined explicitely but intrinsically in the domain where some variable function has high variation values. If there is no need to explicitely define those interfaces, the drawback of the front capturing approach in opposition to the ``front tracking'' approach (where the interface has to be fully described and followed in space and time) is that the resulting discrete interface may be quite diffuse, creating a numerical artificial mixing fluid zone of the two media. The main originality of the methods developed by SMASH and based on the front capturing approach lies in that this artificial numerical zone is considered as a true two-phase flow region. Although seemingly artificial, this permits to consider a model for which we may define a numerical method that may be applied spacewise everywhere and therefore to ensure and justify the equations of state to be satisfied by the well defined corresponding variables.

The front capturing - diffuse interface technique has proven its
ability of solving interface flow problems more simply than usually
done

All the mathematical models considered and studied by SMASH consist
in either a hyperbolic or parabolic system of PDE's.
The approximation step consists in replacing the original mathematical
system by some discrete equivalent system of algebraic
equations. Those discretization techniques should respect and satisfy
the properties of the continuous model such as conservativity,
positiveness preserving (e.g. pressure, density), jump conditions,
entropy inequality etc. They generally use the finite volume
approximation techniques in which the resulting discrete equation at the cell
interface consists in solving some Riemann problem

Most of our numerical methods are based on finite volume
discretizations Riemann problems corresponding to the
evolution of the medium variables through a plane interface separating two
physical states. SMASH focuses its efforts into the development of
approximation methods using either exact or approximate Riemann
solvers. The complexity of the flows we are considering implies the
use of other state equations than the classical equation of state used
for perfect fluids. In such cases we generally have to use approximate Riemann
solvers.

When the resulting model under study is non conservative, which occurs
as a ``mechanical'' consequence when using the averaging process or
when introducing some turbulence model to the physics of the medium,
additional mathematical difficulties arise (as the question of giving
a sense to the distribution product to which we do not know a sensible
answer since Schwartz works

Another difficulty encountered in solving two-phase flow problems
comes from the high disparity of the wave speeds of each existing
fluid material. In particular, one of the fluid may be very close to
the incompressibility limit. In that case, we face up the problem of very
low Mach number flows for which its numerical treatment is still not
fully resolved and envolves non trivial modifications of the original
numerical scheme. Our investigations in that domain concerns both
acoustic and incompressible aspects in methodologies for setting
suitable numerical methods

Once getting the fully discrete system resulting from some discretization scheme to the full set of PDE's defining the mathematical model, we face up now the issue of choosing the adequate discrete solver for large algebraic systems which are generally not linear. We usually have to use iterative solvers and we focus on the particular and interesting class of hierarchical techniques such as multilevel or multigrid solvers.

In a multigrid
method smoother which may be any
classical iterative method having the property of efficiently and
rapidly damping the high frequency modes of the error associated with
the finest level discretization; the residual equation, written on a
coarser level may therefore be approximately solved by the same
smoother in the sense that high frequency modes of the coarse level
error are in turn rapidly damped by that smoother. When suitably
projected back to the finest level, both high frequency (considered
on the fine level) and low frequency (which are also high frequency at
the coarser level) modes of the original error have been damped
resulting in a better approximation of the discrete exact solution. In
fact, when more than two grid levels are considered, each residual
equation is solved using the same smoother before being projected onto a
coarser level, and so on, until reaching the coarsest level where a
direct method may be used to exactly solve the coarsest residual
equation. Getting the final solution consists in projecting back from
coarse to fine level, combined with or without post smoothing, the
corrected error until reaching the finest discretization level on
which the discrete solution has to be defined. In multigrid theory, it
can be shown that the complexity of these methods is proportional to
the discretization node number when applied to elliptic type problems.
That therefore means that the solution cost of the finest level
discrete system of algebraic equations up to the order of finest level
discretization error is directly proportional to the
total number of degrees of freedom of that finest level.

Concerning the definition of the grid levels, we focus on construction methods based on the finest level ingredients as initial data, where the finest level discretization grid is supposed to be fully unstructured. This means that we are interested in either agglomerating or reconstruction techniques (geometrical definition of the grid levels) or algebraic techniques (the algebraic definition of the discrete coarser level residual problem is directly defined from the discrete fine level algebraic operators).

With a large experience in working on numerical fluid mechanics problems, SMASH studies focuses its particular interests on compressible multiphase or mutilfuid flows; the application domains envolves energy and transport industries: aeronautics, car engine combustion research, space research, oil company research, research on electricity and nuclear centers hazards, but envolves also other various domains such as astrophysics and detonation matters.

The main needs in that aera concerns external aerodynamics, where turbulence has to be taken into account to model external complex flows around a complete geometry profile since the numerical simulations are still too costly to be a true part of the optimization process. SMASH is interested in defining suitable multigrid solvers for the Navier Stokes equations using unstructured finite volume discretization grids. In addition, the unsteady feature of turbulent flow problems encountered e.g. in vibration of some material structure and damaged material problems requires the need of LES type models (large Eddy Simulation techniques).

Another application aera we are concerned with is the behaviour of space launcher engine using liquid fuel or powder propellents. Since the media envolves very heterogeneous ingredients, they necessitate two-phase modelizations.

Diesel engine car industry, due to the new generation injection system technologies, uses high pressure spray injection techniques. Our eulerian modelization techniques may be used to describe the first steps of atomization of the spray. These phenomena are preceded by a dynamic liquid-vapor transition phase during which the compressibility feature of the flow appears to be crucial.

The needs in numerical modelization are concerned with thermohydraulics for nuclear energy production centers (as CEA and EDF), flow problems in pipe lines for oil company industry (extraction, oil transport and refining) and all other industries related to energy production or chemical engineering.

In these industrial domains, turbines, boilers and pipes envolve very complex flows which often evolve either violently and/or at high speed, or on the contrary evolve very slowly with a large stabilization time. Describing those phenomena which are intrinsically two-phase type flow problems appears to be fundamental in studies of security issues against hazards (nuclear industry, oil-chemical industry or chemical engineering industry). The ability of a CFD code to treat efficiently and accurately the different phase flow regimes (gaz/liquid flows, bubbling flows, packet flows, liquid film and droplet flows etc.) is still problematic. In addition, when dealing with very low Mach number flows, the numerical complexity and difficulty are still increased.

Multiphase modelization techniques may also find a place in other various application domains. That is for instance the case in astrophysics for modelling keplerian type flows in a proto-planetary system. This type of study have been done in the project in collaboration with astrophysicists to validate some scenarii of planet formation based on the accretion disk assumption.

Another interesting domain of application concerns multiphase flow problems in highly energy-giving granular media. Most of the existing CFD codes that aim to simulate those types of multiphase flows uses the Euler equation model closed by an equation of state defined for the fluid mixture. That necessarily assumes that the phase temperatures (or phase density ...) equilibrium holds without making sure that such assumptions remain pertinent to this type of flow problems. We can therefore enhance our efforts to define models in which the physics is better and widely described for such media.

Changes of phase can appears in two different ways : the first one
(boiling) is governed by thermal diffusion and occurs at constant
pressure. On the opposite, the second one (cavitation) is due to a
fast pressure drop at constant temperature.
This work have considered the modelling of evaporation waves in
overheated fluids where cavitation occurs. The propagation speed of
these evaporation fronts (sometimes named cavitation fronts) exceeds
largely the thermal diffusion velocity and one can guess that the
dynamics of these front does not depend on diffusion processes.
Endothermic kinetic reactions occurs in these fronts leading to a
decrease of the pressure. For this reason, these front can be
considered as negative shock waves. If one adopt this point of
view, an additional relation is needed to close the
Rankine-Hugoniot relationships that govern the propagation of shock
waves. The assumption done in this work is that this relationship is
given by the Chapman-Jouget condition for deflagrations. From a
physical point of view, this means that we assume that the speed of
propagation of the front corresponds to the maximal rate of vapor
production.

The numerical part of this work have been devoted to the construction
of a reactive Riemann solver including evaporation waves.
The Ph. D. thesis of Olivier LeMetayer on this subject have been
defended september 19, 2003. An article in the Int. J. of Thermal
Sciences have been accepted

This work deals with the construction of a five equation two-phase
flow model. Using asymptotic analysis, this model has been derived from
a more general seven equation model proposed in

This work deals with the modelling of surface tension for compressible flows. The CSF approach of J. Brackbill originally set up for incompressible flows have been extended to compressible ones. In this approach, the surface tension is replaced by a volume force proportional to the gradient of the volume fraction. A conservative form of the equations have been derived and its numerical approximation have been studied. The numerical experiments shows that this approach is computationally robust and allows to consider the physics of surface tension phenomena even with large density differences between the two fluids. The thesis of Guillaume Perigaud have been defended on this subject november 27.

The simulation of disperse bubbly flows is very often done with the use of the so-called ``drift flux'' model where the slip velocity between the dispersed and the continuous phases is specified by an algebraic relationship. This simple approach suffers from a severe drawback : Depending on the flow regime, the slip relationship can change and with it, the mathematical nature of the model that can switch from hyperbolic to a non-hyperbolic system. Using asymptotic analysis, we have derived a model for bubbly flow where the slip velocity is expressed by a Darcy type relation. This model is always hyperbolic with a clear mathematical structure. The present work aims to define a numerical approximation for this model.

A first study of the 3-D modelling of a wastewater treatment plant have been performed during a 4 month visit of René Bwemba. Special attention have been paid to the modelling of dissolved oxygen in a three phase flow model.

The masters thesis of Erwin Franquet have been devoted to the modelling of evaporation fronts. In the framework of a one velocity, one pressure model, a relaxation method have been studied. Numerical tests on condensation and evaporation fronts have been performed.

The simulation of the steady flow of a viscous compressible fluid was mastered during the last decade and answered rather well to the need of tools for efficient devices and products (airplanes,...). The new needs in terms of low pollution and high safety of our society push forward the problem of accurately predicting unsteady flows. Unsteadiness is the cause of noise and structure vibration. It is at the heart of environment events.

Then slower flows involving vortices are in the center of new investigations. Turbulence phenomena have to be simulated not only in the mean, as in the past where statistical models were applied. New models have to predict a part of unsteady behavior, and this is progressively affordable due to the progress in computer and numerical schemes performances.

A three-year cooperation with the university of Pisa
has produced a set of new models and schemes for the
Large Eddy Simulation (LES) of compressible flows.
An original feature is the development of a sixth order
numerical filter in a scheme applying to unstructured meshes.
LES calculation could be applied to an airplane geometry
at high angle of attack (

The emission and propagation of acoustic waves in a compressible flow is one of the important subject of the decade, due to new standards in acoustical pollution. In rather fast flows, the model has to be a costly one, derived from the Euler equations. The basic numerical tool of the team is a variation of the one used for unsteady flow. A high accuracy is obtained on regions where the mesh is regular. Then the accuracy of the prediction depends on three concurrent factors, model accuracy, scheme accuracy, round off errors. We have proposed a new version of the Non Linear Disturbance Equation (NLDE) that saves, much more surely than the standard one, the overall accuracy. A paper on this novelty is in under refereeing process.

Study of instantaneous pressure relaxation methods

From a theoretical point of view, we have studied during the master
thesis of Zulfukar Arslan the detailed discrete system obtained with
upwind schemes in the isentropic case and showed that this discrete
system cannot be an accurate approximation of the incompressible Euler
equations. From a numerical point of view,
we have extended the preconditioning techniques studied in

A particular feature of compressible flows
is the arising of small details with high gradient.
In that case, affordable meshes may not produce
second-order numerical convergence.
The basic principles of our study are the
following ones (

- to design mesh adaptation methods relying on optimality,

- to show/check that these methods offer second order accuracy for stiff and discontinuous flows,

- to derive methods for certification of the numerical accuracy
(

The progress of the ACI-GRID "MECAGRID" throughout this year can be summarized in three points: The first objective was to study the project's needs and the solutions available to install a computing grid between the different partners, and then to agree on a roadmap for the deployment of the selected middleware: Globus. In a second step, the first tests validating the chosen solution (globus+MPI) were carried out with the cluster of the INRIA-Sophia and a cluster of the INSA-Lyon, using the application code AERO3D. Those tests were conclusive. Finally, the current main objective for now is the update of the iusti and cemef clusters in order to be able to launch parallel programs using MPI. For this purpose, the establishment of a VPN (Virtual Private Network) based on protected tunnels is being finalized.

The goal of this work done during the master thesis of Rodolphe Lanrivain was to develop a mesh partition tool taking into account heterogeneous processors speed and communication time. A mesh partition algorithm have been devised that achieves load balancing with heterogeneous CPU while trying to minimize the communication time on an non-homogeneous network. This new tool have been tested on a Finite Element application solving the Stokes equation and shows an improvement of 30% with respect to the use of an homogeneous mesh partitioner.

In the context of the MecaGRID initiative, the Smash team develops a
set of computational tools for demonstrations in Grid Computing.
The central kernel is a CFD code (AERO) derived from the synthesis of
research developed by the SMASH Team in collaboration with the
University of Colorado at Boulder. This code
executes in parallel using MPI and is natually suited for MPI Grid
Computing.
A new kernel, AEDIF (derived from AERO), computes two-phase flows
using a new diphasic models developed by Smash (

The goal of this CIFRE funding with IFP is the direct numerical simulation of flow instabilities in oil pipes. The long term objective is to built a direct simulation tool that can be used to derive closure relationships for one-dimensional averaged codes.

In association with the STH/LTA departement of CEA Cadarache, an initiative for the numerical simulation of low Mach number multiphase flows in the nuclear industry by 7 equation models have been launched. This initiative have supported the Ph. D Thesis of Angelo Murrone.

In the framework of a preliminary program to study convective pattern in bubbly flows, IRSN and smash have engaged an initiative for the numerical modelling of bubbly flows.

In the framework of ``Action de Recherche Concertée''
``ARC Diphasique'' of CNRS, and in association with CORIA,
ECL and PARIS VI from the academic side and Renault
and Peugeot from the industrial side, the project works on the
modelling of cavitating sprays for Diesel engines. This initiative
have supported the Ph. D thesis of Olivier Le Metayer.

This initiative launched in November 2002 associates the CEMEF of Ecole de Mines de Paris, IUSTI of the University of Provence and INRIA in the construction of a computational grid from three PC clusters located in the PACA region. This grid will be devoted to massively parallel applications in multimaterial fluid mechanics.

With the help of a ``CARI'' grant, the project has harboured for a 4 month visit, René Bwemba, head of the computer sciences departement of the University of Douala, Cameroun. This visit has been devoted to the study of the modelling of wastewater treatment.

Project members have teached the following courses :

: Cours du DEA de dynamique non-linéaire et applications, University de Nice-Sophia Antipolis, 12 h (A. Dervieux).

: cours de Maîtrise d'Ingénieurie Mathématique, University de Nice-Sophia Antipolis, 20 h (H. Guillard).

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

University de Provence, ``Modélisation et simulation de la transition de phase dynamique. Application aux injecteurs de carburant à haute pression.'', September 19, 2003.

University de Provence, ``Modèles bifluides à 6 et 7 équations pour les écoulements diphasiques à faible nombre de Mach'', December 4, 2003.

University de Provence, ``Elaboration et Résolution de modèles d'écoulements compressibles à interface en présence de capillarité et de viscosité'', November 27, 2003.

University of Nice-Sophia-Antipolis, "Méthodes de volumes-Eléments finis pour les écoulements diphasiques à faible nombre de Mach"

University of Nice-Sophia-Antipolis, "Méhodes numériques pour les écoulements diphasiques"

ENSMP, "Simulation numérique de mousses sur grille de calcul"

The following masters thesis have taken place :

``Ecoulements isentropique à faible nombre de Mach'', DEA de Mathématiques Appliquées, University of Montpellier.

``Partitionnement de maillage sur une grille de calcul'', DEA de Mathématiques Appliquées, University of Bordeaux I

``Simulation numérique directe de la propagation d'un front de changement de phase'', DEA de Mécanique-Energétique, University of Provence.

The following seminars have been organized by the project :

SMASH Seminar on Cavitation, february 17, with as lecturers: M.V. Salvetti, F. Beux, E. Sinibaldi, L. D'Agostino (Pise), R. Saurel (Marseille), Ph. Helluy et Th. Barberon (Toulon)

SMASH Seminar on LES, novembre 14, with as lecturers: C. Le Ribault (Lyon), S. Camarri (Pise)

Hervé Guillard have been member of the scientific committee of CFM2003.

Project members have participated to the following conferences and seminars:

15

eSéminaire de Mécanique des fluides numérique, January 28-30, Institut des Sciences et Techniques Nucléaires, Saclay, (Angelo Murrone).CFM2003, Congrès Français de Mécanique, September 22-26, Nice, (Hervé Guillard, Angelo Murrone, Guillaume Perrigaud)

International Workshop on Multiphase and Complex Flow Simulation for Industry, Cargese, October 20-24, (Hervé Guillard, Angelo Murrone, Phillippe Helluy)

Alain Dervieux, april 23, ``Upwind mixed element-volume'', Charles University, Prague

Alain Dervieux, november 27, at ``Mesh adaption and validation in CFD'', Inauguration of Espace Jacques-Louis Lions, Saint-Cloud.