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Modelling of natural convection flows with large temperature differences : a benchmark problem for low Mach number solvers. Part 2. Contributions to the June 2004 conference

Henri Paillère, Patrick Le Quéré, Catherine Weisman, Jan Vierendeels, Erik Dick, Malte Braack, Frédéric Dabbene, Alberto Beccantini, Etienne Studer, Thibaud Kloczko, Christophe Corre, Vincent Heuveline, Masoud Darbandi, Seyed Farid Hosseinizadeh (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In the second part of the paper, we compare the solutions produced in the framework of the conference “Mathematical and numerical aspects of low Mach number flows” organized by INRIA and MAB in Porquerolles, June 2004, to the reference solutions described in Part 1. We make some recommendations on how to produce good quality solutions, and list a number of pitfalls to be avoided.

Modelling of Natural Convection Flows with Large Temperature Differences: A Benchmark Problem for Low Mach Number Solvers. Part 2. Contributions to the June 2004 conference

Henri Paillère, Patrick Le Quéré, Catherine Weisman, Jan Vierendeels, Erik Dick, Malte Braack, Frédéric Dabbene, Alberto Beccantini, Etienne Studer, Thibaud Kloczko, Christophe Corre, Vincent Heuveline, Masoud Darbandi, Seyed Farid Hosseinizadeh (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In the second part of the paper, we compare the solutions produced in the framework of the conference “Mathematical and numerical aspects of low Mach number flows” organized by INRIA and MAB in Porquerolles, June 2004, to the reference solutions described in Part 1. We make some recommendations on how to produce good quality solutions, and list a number of pitfalls to be avoided.

Modelling of Natural Convection Flows with Large Temperature Differences: A Benchmark Problem for Low Mach Number Solvers. Part 1. Reference Solutions

Patrick Le Quéré, Catherine Weisman, Henri Paillère, Jan Vierendeels, Erik Dick, Roland Becker, Malte Braack, James Locke (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

There are very few reference solutions in the literature on non-Boussinesq natural convection flows. We propose here a test case problem which extends the well-known De Vahl Davis differentially heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference...

More pressure in the finite element discretization of the Stokes problem

Christine Bernardi, Frédéric Hecht (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

For the Stokes problem in a two- or three-dimensional bounded domain, we propose a new mixed finite element discretization which relies on a nonconforming approximation of the velocity and a more accurate approximation of the pressure. We prove that the velocity and pressure discrete spaces are compatible, in the sense that they satisfy an inf-sup condition of Babuška and Brezzi type, and we derive some error estimates.

Mortar finite element discretization of a model coupling Darcy and Stokes equations

Christine Bernardi, Tomás Chacón Rebollo, Frédéric Hecht, Zoubida Mghazli (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

As a first draft of a model for a river flowing on a homogeneous porous ground, we consider a system where the Darcy and Stokes equations are coupled via appropriate matching conditions on the interface. We propose a discretization of this problem which combines the mortar method with standard finite elements, in order to handle separately the flow inside and outside the porous medium. We prove a priori and a posteriori error estimates for the resulting discrete problem. Some numerical experiments...

Multimodels for incompressible flows : iterative solutions for the Navier-Stokes / Oseen coupling

L. Fatone, P. Gervasio, A. Quarteroni (2001)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In a recent paper [4] we have proposed and analysed a suitable mathematical model which describes the coupling of the Navier-Stokes with the Oseen equations. In this paper we propose a numerical solution of the coupled problem by subdomain splitting. After a preliminary analysis, we prove a convergence result for an iterative algorithm that alternates the solution of the Navier-Stokes problem to the one of the Oseen problem.

Multimodels for incompressible flows: iterative solutions for the Navier-Stokes/Oseen coupling

L. Fatone, P. Gervasio, A. Quarteroni (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In a recent paper [4] we have proposed and analysed a suitable mathematical model which describes the coupling of the Navier-Stokes with the Oseen equations. In this paper we propose a numerical solution of the coupled problem by subdomain splitting. After a preliminary analysis, we prove a convergence result for an iterative algorithm that alternates the solution of the Navier-Stokes problem to the one of the Oseen problem.

Multi-parameter asymptotic error resolution of the mixed finite element method for the Stokes problem

Aihui Zhou (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper, a multi-parameter error resolution technique is applied into a mixed finite element method for the Stokes problem. By using this technique and establishing a multi-parameter asymptotic error expansion for the mixed finite element method, an approximation of higher accuracy is obtained by multi-processor computers in parallel.

Multiple spatial scales in engineering and atmospheric low Mach number flows

Rupert Klein (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this...

Multiple spatial scales in engineering and atmospheric low Mach number flows

Rupert Klein (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of...

Multiscale stochastic homogenization of convection-diffusion equations

Nils Svanstedt (2008)

Applications of Mathematics

Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂ u ε ω / ∂ t + 1 / ϵ 3 𝒞 T 3 ( x / ε 3 ) ω 3 · ∇ u ε ω - div α T 1 ( x / ε 1 ) ω 1 , T 2 ( x / ε 2 ) ω 2 , t ∇ u ε ω = f . It is shown, under certain structure assumptions on the random vector field 𝒞 ( ω 3 ) and the random map α ( ω 1 , ω 2 , t ) , that the sequence { u ϵ ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂ u / ∂ t - div ( ℬ ( t ) ∇ u ) = f .

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