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A 3D microscale solidification model, resolving complex interaction of solute partition and diffusion, interdendritic thermosolutal convection, dendrite formation and remelting, was used to study freckle formation for unidirectional solidified Ga-In alloys. The simulated results are validated against real time, in situ X-ray radiographic experiments, highlighting both similarities and differences between model and experiment. The redistribution of solute concentration and the formation of segregated solutal channels were both captured by the numerical model. Effects of thermo-solutal convection due to temperature and solute concentration variations were investigated and the formation of segregated solutal channels (freckles) was also examined in detail. With the assistance of the numerical solution, an improved understanding of the freckle channel formation at the microstructural level was provided.

International audience; Heat and mass transfer by natural convection coupled to wall surface condensation or evaporation in a two-dimensional cavity subjected to uniform, but time-dependent wall temperatures is investigated numerically. At initial state, the cavity is filled with quiescent humid air at uniform temperature and density. By decreasing the wall temperature, condensation occurs at the four wall surfaces until an equilibrium thermodynamic state is reached. The walls are then heated and evaporation of the liquid film water is considered. Various time-variations of the wall temperature were investigated. Since the mass of humid air and average pressure experience large changes during transient regimes, a weakly compressible formulation has been used. The model considers only condensation/evaporation under the thin film approximation. The computations were carried out for temperatures of humid air varying between 300 K and 350 K, and pressure variations around atmospheric pressure. The typical width of the cavities is L = 0.1 m. The results show that very different transient flow structures occur during condensation and evaporation processes. The thickness distributions of the the water films condensed at the walls are discussed, and it is shown that the thicknesses reflect the flow structures. The effect of the cavity aspect ratio reveals more complicated results than for convection without phase change at the wall surfaces.

The combined effects of the double-diffusion and of the viscous dissipation on the convective instability in a fluid-saturated porous medium with a basic horizontal throughflow are investigated. A horizontal porous layer with an impermeable adiabatic lower wall and an impermeable isothermal upper wall is considered. The parallel boundary walls are assumed to have uniform, but unequal, concentrations of the solute. A linear stability analysis is carried out both numerically and by a first-order perturbation method. General disturbances having the form of oblique rolls are considered, reducing either to longitudinal rolls or to transverse rolls in the special cases of roll axes parallel or orthogonal to the basic flow direction, respectively. It is shown that the combined effects of viscous dissipation and mass diffusion may lead to the instability of the basic horizontal flow. Either the longitudinal rolls or the transverse rolls may be the preferred modes of instability depending on the value of the viscous dissipation parameter Xi. The longitudinal rolls are the most unstable when Xi < 61.86657.

The aim of this dissertation is at modeling and numerically simulating thermosolutal convection within cavities filled by binary gas mixtures of ideal gases. A new problem formulation, based on the weakly compressible approximation, has been derived in order to account for the changes in density as well as thermodynamic pressure. The ideal gas law and global conservation of mass are invoked for the predictions of density field and thermodynamic pressure. The first part of the manuscript is devoted to the mathematical derivation of the governing equations and boundary conditions, numerical procedure implemented and, checks of the accuracy of the results through deeply comparisons with updated benchmark solutions. The emphasis is put on the efficiency of algorithm used for solving the pressure-velocity coupling for unsteady, transitional flow regimes. Thermosolutal convection without phase changes at the cavity surfaces is first considered in the second part of the manuscript. We investigated the case of vertical cavities with horizontal gradients of concentration and temperature. The results clearly show that the dilute approximation fails to be valid as soon as the maximum concentration difference exceeds a critical value function of the flow parameters. Surface condensation of water vapor or evaporation of liquid water film in vertical cavities is then considered. The specific thermal boundary conditions of uniform but time-varying temperatures of the four walls are considered. Periodic variations of the flow, temperature and relative humidity fields are discussed in detail. The evolutions of thicknesses of the water film over the four walls are also predicted, L'objectif de c e mémoire est d'apporter une contribution à la modélisation et la simulation numérique de la convection thermosolutale de mélanges binaires de gaz parfaits contenus dans des cavités. Un modèle a été élaboré en se basant sur l'approximation de faible compressibilité. Le premier chapitre précise la démarche suivie dans la modélisation et une formulation originale en est déduite afin de traiter les différents types de conditions aux limites et de conditions de références hydrostatiques analysés dans le mémoire. Les variations de masse volumique sont déduites de la loi des gaz parfaits et la pression thermodynamique est calculée à partir de la conservation de la masse totale. La méthode numérique repose sur la méthode des volumes finis mise en uvre sur des maillages décalés. Le couplage vitesse-pression est traité par un nouvel algorithme dont l'efficacité est discutée en détail. La démarche numérique est validée via des comparaisons avec des solutions de références, en régime stationnaire comme en régime transitoire pour des écoulements transitionnels. Dans la seconde partie du mémoire, on considère d'abord la convection thermosolutale dans une cavité rectangulaire verticale dans le cas où les écoulements sont induits par des gradients horizontaux de température et de concentration. On discute en particulier les limites de l'approximation d'extrême dilution. La condensation de vapeur d'eau et l'évaporation d'un film d'eau liquide sur les parois d'une cavité sont ensuite étudiées en régime transitoire. Ces changements de phase surfaciques sont associés à la convection naturelle dans une cavité dont les températures des quatre parois varient au cours du temps

International audience; The present work deals with an analytical solution based on the parallel flow assumption obtained in the case of a vertical porous layer heated and salted from the long vertical sides with uniform fluxes of heat and mass, respectively. The study concerns a specific case for which the buoyancy ratio and the separation coefficient are identical. For this particular situation, the external mass flux is compensated by the Soret effect, which leads to zero concentration gradient on the vertical walls. The problem is first analyzed by solving numerically the full governing equations and the aspect ratio required to satisfy numerically the parallel flow conditions is determined. Analytical solutions for the pseudo-conductive and boundary layer regimes are proposed and discussed. The N-Le plane is divided into regions with specific behaviors and the results obtained are presented in terms of boundary layer thickness, heat transfer (Nusselt number), and mass transfer (Sherwood number) versus the main governing parameters.

Thermosolutal natural convection induced in a vertical porous layer heated and salted with uniform fluxes is studied analytically and numerically. The study is focused on a specific case where the separation coefficient is identical to the ratio of buoyancy forces. Analytical results, describing both pseudo-conductive and boundary layer regimes, are discussed. Specific behaviour, corresponding to this particular situation, is presented. To cite this article: M. Er-Raki et al., C. R. Mecanique 336 (2008).

International audience; The linear stability of a plane layer with horizontal temperature and concentration stratification corresponding to gradient zone of a solar pond is investigated. The problem is described by Navier-Stokes equations with Boussinesq-Oberbeck approximation. Two source terms are introduced in the energy equations: the absorption of solar energy characterized by the extinction radiative coefficient μe and by the parameter f defined as the ratio of extracted heat flux to absorbed heat flux in the lower convective zone. The influence of the parameters μe and f on the onset of thermosolutal convection in the case of confined and infinite layers is analyzed. It is found that convection starts in an oscillatory state, independently of the RaS value. Different convection solutions were found for marginal stability and steady state.

Numerical simulation is used in order to analyse the influence of the thermosolutal convection on the chemical segregation during the Bridgman solidification of GaInSb concentrated semiconductor alloys under external fields. The results, in agreement with the experimental data, show a strong solutal damping effect on the thermally driven convection when the solidification of the alloys occurs at fast pulling rates of the crucible. Consequently, the chemical segregation increases and the solid-liquid interface becomes morphologically unstable. The analytical calculations which describe the solutal effect on the melt convection and the interface curvature show that the damping solutal effect can be avoided and the quality of crystals improved in two cases : a) when the growth occurs at low and variable pulling rates ; b) when the solidification occurs under alternating magnetic fields. The numerical simulation and the experimental results show that the crystals obtained at low pulling rates are more homogeneous than the alloys solidified at fast pulling rates. The modelling of the solidication under alternating magnetic fields show that the chemical segregations and the interface curvatures are maintained at lower values, but it is difficult to achieve the experimental conditions required by the simulation., La simulation numérique a été utilisée pour étudier l'influence de la convection thermo-solutale sur les ségrégations chimiques lors de la solidification Bridgman d'alliages GaInSb concentrés. Les résultats, en accord avec les résultats expérimentaux, montrent un fort amortissement solutal de la convection thermique pendant la solidification des alliages solidifiés rapidement. A la suite de cet amortissement solutal, les ségrégations chimiques augmentent et l'interface devient morphologiquement instale. A partir d'un calcul analytique qui prend en compte l'effet solutal sur la convection et la courbure de l'interface, deux méthodes ont été proposées pour éviter l'amortissement de l'écoulement et pour améliorer la qualité des cristaux : a) la croissance à vitesses faibles et variables ; b) la croissance sous champ magnétique alternatif. La simulation numérique et les expériences montrent que les cristaux obtenus en utilisant des vitesses de croissance faibles sont plus homogènes et leur qualité structurale est améliorée par rapport aux cristaux obtenus à grandes vitesses. La modélisation de la solidification d'alliages concentrés sous champ magnétique alternatif montre des ségrégations chimiques et courbures de l'interfaces faibles. Cependant, il est difficile d'obtenir expérimentalement les conditions optimales prédites.

Numerical simulation is used in order to analyse the influence of the thermosolutal convection on the chemical segregation during the Bridgman solidification of GaInSb concentrated semiconductor alloys under external fields. The results, in agreement with the experimental data, show a strong solutal damping effect on the thermally driven convection when the solidification of the alloys occurs at fast pulling rates of the crucible. Consequently, the chemical segregation increases and the solid-liquid interface becomes morphologically unstable. The analytical calculations which describe the solutal effect on the melt convection and the interface curvature show that the damping solutal effect can be avoided and the quality of crystals improved in two cases : a) when the growth occurs at low and variable pulling rates ; b) when the solidification occurs under alternating magnetic fields. The numerical simulation and the experimental results show that the crystals obtained at low pulling rates are more homogeneous than the alloys solidified at fast pulling rates. The modelling of the solidication under alternating magnetic fields show that the chemical segregations and the interface curvatures are maintained at lower values, but it is difficult to achieve the experimental conditions required by the simulation.; La simulation numérique a été utilisée pour étudier l'influence de la convection thermo-solutale sur les ségrégations chimiques lors de la solidification Bridgman d'alliages GaInSb concentrés. Les résultats, en accord avec les résultats expérimentaux, montrent un fort amortissement solutal de la convection thermique pendant la solidification des alliages solidifiés rapidement. A la suite de cet amortissement solutal, les ségrégations chimiques augmentent et l'interface devient morphologiquement instale. A partir d'un calcul analytique qui prend en compte l'effet solutal sur la convection et la courbure de l'interface, deux méthodes ont été proposées pour éviter l'amortissement de l'écoulement et pour améliorer la qualité des cristaux : a) la croissance à vitesses faibles et variables ; b) la croissance sous champ magnétique alternatif. La simulation numérique et les expériences montrent que les cristaux obtenus en utilisant des vitesses de croissance faibles sont plus homogènes et leur qualité structurale est améliorée par rapport aux cristaux obtenus à grandes vitesses. La modélisation de la solidification d'alliages concentrés sous champ magnétique alternatif montre des ségrégations chimiques et courbures de l'interfaces faibles. Cependant, il est difficile d'obtenir expérimentalement les conditions optimales prédites.

We provide an overview of current techniques and typical applications of numerical bifurcation analysis in fluid dynamical problems. Many of these problems are characterized by high-dimensional dynamical systems which undergo transitions as parameters are changed. The computation of the critical conditions associated with these transitions, popularly referred to as ‘tipping points’, is important for understanding the transition mechanisms. We describe the two basic classes of methods of numerical bifurcation analysis, which differ in the explicit or implicit use of the Jacobian matrix of the dynamical system. The numerical challenges involved in both methods arementioned and possible solutions to current bottlenecks are given. To demonstrate that numerical bifurcation techniques are not restricted to relatively low-dimensional dynamical systems, we provide several examples of the application of the modern techniques to a diverse set of fluid mechanical problems.

A vertical enclosure, submitted to horizontal temperature and concentration gradients, is characterized by complex bifurcation phenomena. We are concerned by the situation where the opposing buoyancy forces due to horizontal thermal and concentration gradients are equal. In this work, the linear stability of equilibrium solution is studied in the case of infinite and rectangular cell. The structure of subcritical steady solutions has been studied numerically for several values of the aspect ratio of the enclosure.

An incompressible Oldroydian viscoelastic fluid layer heated and soluted from below in the presence of suspended (dust) particles and uniform vertical magnetic field to include the effect of Hall currents in porous medium is considered. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, Oldroydian viscoelastic fluid behaves like an ordinary Newtonian fluid. Dust particles and Hall currents are found to have a destabilizing effect on the thermosolutal convection, whereas magnetic field is found to have a stabilizing effect on the thermosolutal convection. Medium permeability has both stabilizing and destabilizing effect on the thermosolutal convection under certain conditions. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The case of overstability is also considered.