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Introduction

## Publications

Publications (77)

The present paper addresses the convergence of a first-order in time incremental projection scheme for the time-dependent incompressible Navier–Stokes equations to a weak solution. We prove the convergence of the approximate solutions obtained by a semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non unifor...

In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite vol...

We prove in this paper the convergence of the Marker-and-Cell scheme for the discretization of the steady-state and time-dependent incompressible Navier–Stokes equations in primitive variables, on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the...

In this paper we present a pressure correction scheme for the compressible Navier-Stokes equations. The space discretization is staggered, using either the Marker-And-Cell (MAC) scheme for structured grids, or a nonconforming low-order finite element approximation for general quandrangular, hexahedral or simplicial meshes. For the energy balance eq...

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the app...

The explosivity of dust clouds is greatly influenced by several parameters which depend on the operating conditions, such as the initial turbulence, temperature or ignition energy, but obviously also on the materials composition. In the peculiar case of a mixture of two combustible powders, the physical and chemical properties of both dusts have an...

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids.

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids.

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids.

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the app...

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the app...

During decommissioning of UNGG (Uranium Natural Graphite Gas) nuclear reactors and wastes reconditioning operations, mixtures of graphite and metal dusts can be encountered. In this perspective, an extensive experimentation has been realized on graphite/magnesium and graphite/iron mixtures to determine their ignition sensitivity and explosivity. Th...

We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven ; they yie...

We address in this paper a non-linear parabolic system, which is built to retain the main mathematical difficulties of the P
\(_1\) radiative diffusion physical model. We propose a finite volume fractional-step scheme for this problem enjoying the following properties. First, we show that each discrete solution satisfies a priori \({ L}^\infty \)-e...

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hex...

We prove in this paper the
convergence of the Marker and cell (MAC) scheme for the discretization of the steady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the existence of d...

We propose a numerical scheme
for the incompressible Navier-Stokes equations. The pressure is approximated at the cell centers while the vector valued velocity degrees of freedom are localized at the faces of the cells. The scheme is able to cope with unstructured non-conforming meshes, involving hanging nodes. The discrete convection operator, of...

In the context of LES of turbulent flows, the control of kinetic energy seems to be an essential requirement for a numerical scheme. Designing such an algorithm, that is, as less dissipative as possible while being simple, for the resolution of variable density Navier–Stokes equations is the aim of the present work. The developed numerical scheme,...

We assess in this paper the capability of a pressure correction scheme to compute shock solutions of the homogeneous model for barotropic two-phase flows. This scheme is designed to inherit the stability properties of the continuous problem: the unknowns (in particular the density and the dispersed phase mass fraction y) are kept within their physi...

We review in this paper an explicit scheme for the numerical simulation of inviscid
compressible flows; we analyze it for both the barotropic Euler equations and the full
Euler equations for an ideal gas. In each case, we summarize the theoretical results that
were recently obtained concerning the stability and consistency of the schemes and presen...

In this paper, we build and analyze the stability and consistency of an explicit scheme for the compressible barotropic Euler equations. This scheme is based on a staggered space discretization, with an upwinding performed with respect to the material velocity only (so that, in particular, the pressure gradient term is centered). The velocity conve...

In this paper, we build and analyze the stability and consistency of an explicit scheme for the Euler equations. This scheme is based on staggered space discretizations, with an upwinding performed with respect to the material velocity only. The pressure gradient is defined as the transpose of the natural velocity divergence, and is thus centered....

In this paper, we analyze the stability and consistency of a time-implicit scheme and a pressure correction scheme, based on staggered space discretizations, for the compressible barotropic Euler equations. We first show that the solutions to these schemes satisfy a discrete kinetic energy and a discrete elastic potential balance equations. Integra...

We propose in this paper a finite volume scheme to compute the solution of convection-diffusion equation on unstructured and non–conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second order spatial convergence rate for the Laplace equation on any unstructu...

SUMMARY We propose, in this paper, a finite volume scheme to compute the solution of the convection–diffusion equation on unstructured and possibly non-conforming grids. The diffusive fluxes are approximated using the recently published SUSHI scheme in its cell centred version, that reaches a second-order spatial convergence rate for the Laplace eq...

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low-degree
nonconforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure.
The numerical method falls into the class of algebraic projection methods. We provide an error...

Nous présentons dans cet article une nouvelle méthode de correction de pression pour les écoulements dilatables. Nommée « méthode de pénalité-projection », cette technique diffère des schémas de projection usuels par l’ajout dans l’étape de prédiction d’un terme de pénalisation, construit pour contraindre la vitesse à satisfaire le bilan de masse....

We prove in this paper the continuity of the natural projection operator from W 01,q (Ω) d, q ∈ [1, + ∞), d = 2 or d = 3, to the MAC discrete space of piecewise constant functions over the dual cells, endowed with the finite volume W 01,q-discrete norm. Since this projection operator is also a Fortin operator (that is an operator which "preserves"...

We present in this paper a class of schemes for the numerical simulation of compressible flows. In order to ensure the stability of the discretizations in a wide range of Mach numbers and introduce sufficient decoupling for the numerical resolution, we choose to implement and study pressure correction schemes on staggered meshes. The implicit versi...

We develop in this paper a discretization for the convection term in variable density unstationary Navier–Stokes equations, which applies to low-order non-conforming finite element approximations (the so-called Crouzeix–Raviart or Rannacher–Turek elements). This discretization is built by a finite volume technique based on a dual mesh. It is shown...

We present in this paper a class of schemes for the solution of the barotropic Navier- Stokes equations. These schemes work on general meshes, preserve the stability properties of the continuous problem, irrespectively of the space and time steps, and boil down, when the Mach number vanishes, to discretizations which are standard (and stable) in th...

In this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function u (n) of the sequence with respect...

The 1D Burgers equation is used as a toy model to mimick the resulting behaviour of numerical schemes when replacing a conservation
law by a form which is equivalent for smooth solutions, such as the total energy by the internal energy balance in the Euler
equations. If the initial Burgers equation is replaced by a balance equation for one of its e...

We propose a discretization for the MAC scheme of the viscous dissipation term τ(u) : Δu (where τ(u) stands for the shear stress tensor associated to the velocity field u), which is suitable to obtain an unconditionally stable scheme for the compressible Navier-Stokes equations. It is also shown,
in some model cases, to ensure the strong convergenc...

In the context of Large Eddy Simulation, the use of a turbulence model brings the question of the implementation of the eddy–viscosity.
In this communication, we propose to assess the discretization of the diffusive term based on a low–order non–conforming finite
element. For this, we build a manufactured solution of the incompressible steady Stoke...

We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree nonconforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error an...

In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with an equation of state of the form p=rho^gamma (where p stands for the pressure and rho for the density). This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element...

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift ve...

We present in this short note a simple construction of the convection op-erator in the variable density Navier-Stokes equations (i.e. the discrete analogue of ∂ t (ρu) + div(ρu × u), where ρ stands for the density and u for the velocity) for MAC discretizations, which ensures the control of the kinetic energy. We thereby extend a similar constructi...

We prove in this paper the convergence of the Marker-and-Cell (MAC) scheme for the discretization of the steady state compressible Stokes equations on two- or three-dimensional Cartesian grids. The existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the approximate solution...

We prove that, under a cfl condition, the explicit upwind finite vol-ume discretization of the convection operator C(u) = ∂ t (ρu) + div(uq), with a given density ρ and momentum q, satisfies a discrete kinetic energy decrease property, provided that the convection operator satis-fies a "consistency-with-the-mass-balance property", which can be simp...

We construct a discretization of the nonlinear compressible Navier-Stokes equations on a staggered grid such that the discrete kinetic energy is preserved.

In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state rho=p , based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a f...

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obta...

We address in this paper a fractional-step scheme for the simulation of incompressible flows falling in the class of penalty-projection methods. The velocity prediction is similar to a penalty method prediction step, or, equivalently, differs from the incremental projection method one by the introduction of a penalty term built to enforce the diver...

We address in this paper a parabolic equation used to model the phase mass balance in two-phase flows, which differs from
the mass balance for chemical species in compressible multicomponent flows by the addition of a nonlinear term of the form
∇·ρφ(y)ur, where y is the unknown mass fraction, ρ stands for the density, φ is a regular function such t...

. In this paper, we build a L2-stable discretization of the non-linear convection term in Navier-Stokes equations for non-divergence-free flows, for non-conforming low order Stokes finite elements. This discrete operator is obtained by a finite volume technique, and its stability relies on a result interesting for its own sake: the L 2-stability of...

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift ve...

Nous presentons dans cet article une nouvelle methode de correction de pression pour les ecoulements dilatables. Nommee « methode de penalite-projection », cette technique differe des schemas de projection usuels par l’ajout dans l’etape de prediction d’un terme de penalisation, construit pour contraindre la vitesse a satisfaire le bilan de masse....

In this paper, we give a new (and simpler) stability proof for a cell-centered colocated finite volume scheme for the 2D Stokes problem, which may be seen as a particular case of a wider class of methods analyzed in [10]. The definition of this scheme involves two grids. The coarsest is a triangulation of the computational domain by acute-angled si...

We present in this paper a pressure correction scheme for barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solu-tion. The stability proof is based on two independent resul...

Bubbly flows appear in a large variety of engineering applications from the petroleum to the nuclear industry. A common model
used in these contexts is the so-called drift–flux model where the slip velocity (the difference between the velocities of
the gas and of the liquid) is expressed on the basis of empirical correlations. However, depending on...

This paper is devoted to a review of the analysis tools which have been developed for thethe mathematical study of cell centred finite volume schemes in the past years.We first recall the general principle of the method and give some simple examples.We then explain how the analysis is performed for elliptic equations and relate it to the analysis o...

We present and analyse in this paper a novel cell-centered colocated finite volume scheme for incompressible flows. Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes. Under a simple geometrical assumption for the clusters, we obtai...

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, wh...

The penalty–projection method for the solution of Navier–Stokes equations may be viewed as a projection scheme where an augmentation term is added in the first stage, namely the solution of the momentum balance equation, to constrain the divergence of the predicted velocity field. After a presentation of the scheme in the time semi-discrete formula...

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.
It has been developed following two main ideas.
On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volum...

We study a colocated cell centered finite volume method for the approximation of the incompressible Navier-Stokes equations posed on a 2D or 3D finite domain. The discrete unknowns are the components of the velocity and the pressures, all of them colocated at the center of the cells of a unique mesh; hence the need for a stabilization technique, wh...

We address in this paper a class of physical problems which can be set under the form of the momentum and mass balance equations, supplemented by the balance equation of an independent unknown field z. The fluid density is supposed to be given as a nonlinear function of this latter unknown. In particular, governing equations of some reactive low Ma...

This paper is devoted to the presentation of a numerical scheme for the simulation of gravity currents of non-Newtonian fluids. The two dimensional computational grid is fixed and the free-surface is described as a polygonal interface independent from the grid and advanced in time by a Lagrangian technique. Navier–Stokes equations are semi-discreti...

In this paper we propose and analyze a finite element scheme for a class of variational nonlinear and nondifferentiable mixed inequalities including balance equations governing incompressible creeping flows of Bingham fluids. For numerical efficiency reasons, equal-order piecewise linear approximations are used for both velocity and pressure, and t...

We propose a numerical method to calculate unsteady flows of Bingham fluids without any regularization of the constitutive law. The strategy is based on the combination of the characteristic/Galerkin method to cope with convection and of the Fortin–Glowinsky decomposition/coordination method to deal with the non-differentiable and non-linear terms...

This paper addresses a sub-problem of low Mach number reactive flows modelling: the solution of the mass and momentum balance equations for a pressure-independent variable density flow. For this objective, we develop a novel numerical scheme: the time discretisation is performed by an incremental projection method, using the original form of the ma...

We address the finite element solution of the Cahn-Hilliard/Navier-Stokes model proposed in (2) with artificial boundary conditions. Our aim is to perform simulations focussing on a particular zone of the flow, to be further exploited for upscaling purposes in a two-level physical modelling. To obtain as less as possible perturbating boundary condi...

Fire is a major concern for the nuclear safety due to potential severe consequences of an uncontrolled fire on the surroundings of a nuclear plant. Since more a twenty years, a research program addressing this topic is in progress at the french "Institut de Radioprotection et de Sûreté Nucléaire" (IRSN). Within this framework, a computational code,...

In this paper, we design a finite-volume based numerical scheme for the solution of the nonlinear balance equations of RNG variants of the well-known k − ε model. In this class of models, the description of the turbulence relies on two variables, the turbulent kinetic energy k and its dissipation rate ε, which, for physical reasons, must remain pos...

This paper is concerned with a pressure correction scheme for the homogeneous model of a two-phase flow with two barotropic phases. This scheme combines finite element and finite volume discretizations and is based on an original pressure correction step coupling the mixture mass balance and the mass balance of one of the phases. It respects the es...