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The present paper reports on an analytical and numerical study of combined Soret and Dufour effects on thermosolutal convection in a horizontal porous cavity saturated with an electrically conducting binary fluid under a magnetic field. The horizontal walls of the system are subject to vertical uniform fluxes of heat and mass, whereas the vertical walls are assumed to be adiabatic and impermeable. The main governing parameters of the problem are the Rayleigh, the Hartmann, the Soret, the Dufour and the Lewis numbers, the buoyancy ratio, the enclosure aspect ratio, and the normalized porosity of the porous medium. An asymptotic parallel flow approximation is applied to determine the onset of subcritical nonlinear convection. In addition, a linear stability analysis is performed to predict explicitly the thresholds for the onset of stationary, overstable and oscillatory convection, and the Hopf bifurcation as functions of the governing parameters. The combined effect of a magnetic field, Soret and Dufour parameters have a noticeable influence on the intensity of the convective flow, the heat and mass transfer rates, and the thresholds of linear convection. It is found that the imposition of a magnetic field delays the onset of convection and its intensification can lead to the total suppression of the convective currents. The heat transfer rate increases with the Dufour number and decreases with the Soret number and vice versa for the mass transfer rate.

Different from other alloys, the observation in this work on the dendritic mushy zone shows that the freckles are formed in two different regions before and after peritectic reaction in directional solidification of Sn–Ni peritectic alloys. In addition, the experimental results demonstrate that the dendritic morphology is influenced by the temperature gradient zone melting and Gibbs–Thomson effects. A new Rayleigh number (RaP) is proposed in consideration of both effects and peritectic reaction. The prediction of RaP confirms the freckle formation in two regions during peritectic solidification. Besides, heavier thermosolutal convection in samples with larger diameter is also demonstrated.

International audience; Experiments of directional solidification of TiAl cylindrical samples were conducted within the frame of the ESA GRADECET project. The experiments were performed in the ESA "Large Diameter Centrifuge" using a furnace with a well defined thermal protocol. The furnace was mounted in the centrifuge and free to tilt in such a way that the total apparent gravity (sum of terrestrial gravity and centrifugal acceleration) was aligned to the cylinder centerline. Several centrifugation levels were investigated besides to one reference case out of the centrifuge. In this work, we present 3D numerical simulations of these experiments paying special attention in the liquid thermosolutal buoyancy convection and aluminum macrosegregation. The numerical model accounts for the non-inertial accelerations that appear in the rotating reference frame (centrifugal and Coriolis), motionless solid, the-mosolutal Boussinesq approximation and an infinitely fast microscopic diffusion model between the phases to depict the solid growth. The results showed that the Coriolis acceleration entirely modifies the liquid flow regime during solidification leading to a 3D aluminum segregation pattern with respect to the case solidified under normal terrestrial gravity conditions. Additionally the magnitude of aluminum segregation increases with the level of centrifu-* Corresponding author

In this paper, three-dimensional thermosolutal natural convection and entropy generation in a cubic cavity filled with a non-Newtonian Carreau-Yasuda fluid has been simulated by Lattice Boltzmann Method (LBM). This study has been conducted for certain pertinent parameters of Rayleigh number (Ra = 104 and 105), the Buoyancy ratio (N = -1, 0.1, 1), the Lewis number (Le = 0.5, 2.5, 10, 100, and 1000), power-law indexes (n = 0.5, 1, and 1.5), Carreau number (Cu = 0.01, 0.1, 1, 10 and 100), and Carreau-Yasuda parameter (m = 0.1, 0.5, and 1). Results indicate that the rise of Rayleigh number enhances heat and mass transfer for various studied parameters. The increase in power-law index provokes heat and mass transfer to drop gradually. However, the effect of power-law index on heat and mass transfer rises steadily as Rayleigh number rises. The enhancement of Carreau number decreases heat and mass transfer, but; the increases in the Carreau-Yasuda parameter augments heat and mass transfer significantly. The augmentation of the buoyancy ratio number enhances heat and mass transfer. It was found that mass transfer increases as Lewis number augments. The augmentation of Rayleigh number enhances different entropy generations and declines the average Bejan number. The increase in the power-law index provokes various irreversibilities to drop significantly. The enhancement of the buoyancy ratio causes the summation entropy generations to increase considerably. The rise of Carreau number and Carreau-Yasuda parameter decreases and increases the total irreversibilities; respectively.

Embedded the thermal and solutal buoyancy into the momentum conservation equation as an additional force term using the Boussinesq approximation, a 2D CA-FVM model is extended to simulate the dendritic growth with thermosolutal convection. The model is firstly validated by comparison of numerical predictions with the benchmark test of Rayleigh – Benard convection and the analytical solutions of the stagnant film model for the free dendritic growth with thermosolutal convection, and good agreements between the numerical results with analytical solutions are obtained. Later, numerical simulations for both the equiaxed and columnar dendritic growth of Fe-0.82wt%C binary alloy with thermosolutal convection are performed. The results show that, for the equiaxed dendritic growth in an undercooled melt, the dendrite tip growth rapidly decreases from the high velocity to a relative low steady-state value. With the further growth of dendrite, the thermosolutal convection induced by the solute rejection and latent heat release is enhanced and four vortexes are developed between the dendrite arms. Thus, the asymmetries of the dendrite morphology, temperature and solute profiles are intensified. For the columnar dendritic growth with thermosolutal convection under the unidirectional solidification process, the thermosolutal convection transports the rejected solute downward and makes the solute enrich at the interdendritic region. The thermosolutal convection facilitates the upstream dendritic growth, but inhibits the downstream dendritic growth. Moreover, with the increase of deflection angle of gravity, the advection on the top region and the clock-wise vortex flow at the interdendritic region intensified, and finally the columnar dendrite morphology becomes more asymmetrical.

A numerical investigation of Soret and Dufour effects on heat and mass transfer in a Salt Gradient Solar Pond (SGSP) and its storage efficiency under meteorological conditions of Tangier (Morocco) is presented. This SGSP consists of an open square cavity filled with a saline solution. Its vertical and bottom walls are adiabatic and impermeable. Transfers are described by the Navier-Stokes, thermal energy and diffusion equations which are solved with a finite volume scheme and the Gauss method. The link between the pressure and velocity fields is carried out by the SIMPLE algorithm. The numerical model, developed in Fortran 95 language, is satisfactorily validated by comparisons with previously published numerical and experimental studies. Results show that the Dufour effect on thermosolutal convection in a SGSP is more relevant than the Soret effect. Nevertheless, increasing the value of the Dufour coefficient from 0 to 0.8 improves heat and mass transfer in the SGSP. Therefore, the amount of the thermal energy accumulated in the storage zone is reduced by about 5.56 %. In addition, the heat losses through the SGSP free surface are improved by around 8.53 %.

In this study, the bifurcation phenomenon and the existence of dual asymmetry solutions of double-diffusive convection driven by an internal thermal and solutal source in a horizontal cavity have been investigated systematically. The problem is solved by the SIMPLE algorithm with the QUICK scheme in a non-uniform staggered grid system. The Rayleigh numbers, 200 ≤ Ra ≤ 106, buoyancy ratios, −20 ≤ Nc ≤ 20, Soret numbers, 0 ≤ Sr ≤ 0.8, and Dufour numbers, 0 ≤ Df ≤ 0.8, are considered in this study. Different streamlines for the onset and evolution of static bifurcation are tracked. The paper aims to determine the critical Rayleigh numbers and buoyancy ratios for the onset of symmetry-breaking through bifurcation diagrams based on the vortex length. The results indicate that increasing buoyancy can destabilize the symmetric system. The strong couple diffusion effect delays the onset of bifurcation flow. A pair of asymmetric modes can be obtained under different initial conditions.

The present study simulates the 3D unsteady double-diffusive natural convection subject to opposing thermal and solutal buoyancy forces (N

The unsteady double-diffusive natural convection flow in a rectangular enclosure filled with soft rocks which are subjected to humid air, including water vapor absorption and thermoelasticity, is studied numerically. The high order compact finite difference schemes are adopted for better simulation of this problem, and a Darcy–Brinkman–Forchheimer extended model combined with the absorption and thermoelasticity effects is introduced. The effects of the thermal and solutal Rayleigh numbers, hygroscopicity, Darcy numbers and thermoelasticity on the flow are investigated in detail. Results are presented in the form of streamlines and isolines of temperature and mass fraction as well as the plots of average Nusselt and Sherwood numbers over the walls under different conditions. The flow structure develops from conduction-dominated to steady convection-dominated as increasing Rayleigh number or increasing buoyancy ratio as well as increasing Darcy numbers. The thermoelasticity effects of soft rocks play an important role in the analysis of thermal convection flow by the change of temperature and permeability, while the water vapor absorption effects may have a slight effect on the flow in the beginning and vanish with the absorption effects becoming weak. Nevertheless, the effect may significant reduce the mechanical strength properties of soft rocks.

Thermosolutal convection with reaction in a porous media of Darcy type is studied using the energy method. The effect of the reaction terms on the temperature and salt Rayleigh numbers are presented graphically. Furthermore, nonlinear energy stability boundaries for different values of the reaction terms are compared with the linear instability boundaries obtained by Pritchard and Richardson (2007).

Thermosolutal convection in a square cavity filled with a binary perfect gas mixture and submitted to an oriented magnetic field taking into account the effect of radiation heat transfer is numerically investigated. The cavity is heated and cooled along the active walls whereas the two other walls are adiabatic and insulated. Entropy generation due to heat and mass transfer, fluid friction and magnetic effect has been determined for laminar flow by solving numerically: The continuity, momentum energy and mass balance equations, using a Control Volume Finite-Element Method. The structure of the studied flows depends on five dimensionless parameters which are: The Grashof number, the buoyancy ratio, the Hartman number, the inclination angle of the magnetic field and the radiation parameter.

We describe a mathematical model of double-diffusive (thermosolutal) convection in a saturated porous layer, when the solubility of the solute depends on the temperature, and the porosity and permeability of the porous medium evolve through dissolution and precipitation. We present the results of linear and weakly nonlinear stability analyses and explore the longer-term development of the system numerically. When the solutal concentration gradient is destabilising, the dynamics are somewhat similar to those previously found for single-species convection (Ritchie & Pritchard, J. Fluid Mech., vol. 673, 2011, pp. 286–317), including the occurrence of subcritical instabilities driven by a reaction–diffusion mechanism. However, when the solutal concentration gradient is stabilising and the thermal gradient is destabilising, novel dynamics emerge. These include a vertical segregation of circulation cells and porosity perturbations near the onset of convection, and over longer time scales the formation of a low-permeability region in the middle of the layer, pierced by occasional high-permeability channels. Under these conditions, convection may die away to nearly zero for extended periods before resuming vigorously in localised regions at later times.

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.

This study attempts to examine convection caused by the combined effects of temperature and concentration gradients in an annular enclosure with steps. An experimental study of thermosolutal convection of a CuSO4+ H2SO4+ H2O solution in a horizontal annular enclosure with steps is conducted using an electrochemical system that enables either opposing or cooperating in temperature and concentration gradients to be imposed. The shadowgraph recording technique is utilized to visualize and analyze the flow field phenomenon. The large difference between the thermal and solutal diffusion rates causes the flow to exhibit double-diffusive characteristics. In both cooperating and opposing cases, some finger-type flow can be visualized in the layered flow structure. Furthermore, Sh (mass transfer rate) increased with increasing Grt for fixed Grm and S*. Sh is larger in the cooperating case than in the opposing case for given Grt and Grm values.

An experimental study of thermosolutal convection in an inclined rectangular enclosure with a partition is presented in this article. Aspect ratio, partition ratio, and inclination angle were kept constant at Ar =0.5, Ap = 0.25 and φ = 30°, respectively. The convective flow is generated by both inclined temperature and concentration gradients under limiting current condition. Both the thermal and solutal buoyancies, which either cooperated or opposed one another, were induced from the copper plates. The temperature gradient was maintained and controlled using two separate constant temperature baths that circulated heated or cooled water through a heat exchanger. We used copper sulphate-sulfuric acid solution as both the working fluid and the electrolyte. An electrochemical method based on a diffusion-controlled electrode reaction was employed to create the concentration gradient. We used the shadowgraph recording technique to visualize and analyze the flow field phenomenon. Thermal Grashof numbers ranging from 8.16 × 105 to 16.32 × 105 and a solutal Grashof number Grm = 4.36 × 106 were investigated. It is demonstrated that the mass transfer rate increases with the increasing thermal Grashof numbers within our experimental ranges. Multilayer structures are found in the cooperating case or the opposing case.

We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting.

The onset of thermosolutal convection due to thermal diffusion in a fluid saturated isotropic porous layer, subject to a gravity gradient is investigated using the Galerkin technique. Results reveal that the Soret parameter affects the pattern of convection only when its magnitude is large, both in the presence and absence of variable gravity field.

The solidification of a dilute alloy (bismuth-tin) under Bridgman crystal growth conditions is investigated. Computations are performed in two dimensions with a uniform grid. The simulation includes the species concentration, temperature and flow fields, as well as conduction in the ampoule. Fully transient simulations have been performed, with no simplifying steady state approximations. Results are obtained under microgravity conditions for pure bismuth, and for Bi-0.1 at.% Sn and Bi-1.0 at.% Sn alloys, and compared with experimental results obtained from crystals grown in the microgravity environment of space. For the Bi-1.0 at.% Sn case the results indicate that a secondary convective cell, driven by solutal gradients, forms near the interface. The magnitude of the velocities in this cell increases with time, causing increasing solute segregation at the solid/liquid interface. Finally, a comparison between model predictions and results obtained from a space experiment is reported. The concentration-dependence of the alloy melting temperature is incorporated in the model for this case. Satisfactory correspondence is obtained between the predicted and experimental results in terms of solute concentrations in the solidified crystal.

The purpose of this paper is to investigate the influence of the Soret effect on natural convection in a vertical cavity filled with a binary liquid. The two vertical walls of the cavity are subject to constant fluxes of heat while the two horizontal ones are adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity. In the limit of tall enclosures an analytical model is derived on the basis of a parallel flow approximation. The study is completed by a numerical solution of the full governing equations.

A numerical study is carried out on double-diffusive natural convection within a vertical circular porous annulus. Motions are driven by the externally applied constant temperature and concentration differences imposed across the vertical walls of the enclosure. In the formulation of the problem, use is made of the Brinkman extended Darcy model which allows the no-slip boundary condition on a solid wall, to be satisfied. The flow is assumed to be laminar and two dimensional. The density variation is taken into account by the Boussinesq approximation. The control-volume approach is used for solving the governing equations. Solutions for the flow fields, temperature and concentration distributions and Nusselt, Nui and Sherwood, Shi numbers are obtained in terms of the governing parameters of the problem. The effect of both the Darcy number, Da, and the radius ratio, κ, on Nui and Shi is found to be significant. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present model as limiting cases.

Pb-2.2 and 5.8 wt pct Sb alloys were directionally solidified with a positive thermal gradient of 140 K cm−1 at growth speeds ranging from 0.8 to 30 µm s−1, and then quenched to retain the mushyzone morphology. Chemical analysis along the length of the directionally solidified portion and in the quenched melt ahead of the dendritic array showed extensive longitudinal macrosegregation. Cellular morphologies growing at smaller growth speeds are associated with larger amounts of macrosegregation as compared with the dendrities growing at higher growth speeds. Convection is caused, mainly, by the density inversion in the overlying melt ahead of the cellular/dendritic array because of the antimony enrichment at the array tip. Mixing of the interdendritic and bulk melt during directional solidification is responsible for the observed longitudinal macrosegregation.

The onset of double diffusive convective flows in an inclined fluid layer, when constant fluxes of heat and mass are applied on the two oposing boundaries of the layer, is investigated. The case of equal and opposing buoyancy forces is considered. A numerical linear stability theory is used to determine the critical Rayleigh number for the onset of convection. The existence of a subcritical Rayleigh number, for the onset of finite amplitude convection, is demonstrated on the basis of the parallel flow approximation.

Two dimensional, thermosolutal convection in molten Pb-Sn housed within a square enclosure, is simulated using a spectral method. The predictions illustrate two regimes of liquid convection which develop in response to simultaneous horizontal temperature and species concentration gradients applied to the system. Sherwood numbers for the high Schmidt number fluid are quite high, and change markedly depending upon the regime of operation. Heat transfer rates are only moderately affected by convection due to the low Prandtl number of the melt. The predictions at relatively large Rayleigh number and buoyancy ratio display highly oscillatory behavior in a manner consistent with previous experimental observations.

The essential principles of thermosolutal convection are outlined, and how convection provides a transport mechanism between the mushy region of a casting and the open bulk liquid is illustrated. The convective flow patterns which develop assist in heat exchange and macroscopic solute segregation during solidification; they also provide a mechanism for the transport of dendritic fragments from the mushy region into the bulk liquid. Surviving fragments become nuclei for equiaxed grains and so lead to blocking of the parental columnar, dendritic growth front from which they originated. The physical steps in such a sequence are considered and some experimental data are provided to support the argument.

The rate of layer growth during thermosolutal convection in a square enclosure is studied both experimentally and analytically. The thermosolutal convection is induced by the combined thermal and solutal buoyancies which either augment or oppose each other. The layer growth process is attributable to the vertical solutal boundary flow which accumulates and stratifies along the horizontal wall. A mathematical model, based on the filling box process used to predict the layer growth rate, is developed. The solutal boundary flow that enters or exits the stratified layer is calculated by using an integral solution for natural convection due to combined thermal and solutal buoyancies along a vertical plate. Comparison of layer growth rate between the prediction and the data is made, and the agreement is excellent if the actual thickness of the solutal boundary layer flow at the inlet and the outlet of the stratified layer can be accurately determined. The actual thickness of the solutal boundary layer flow at the inlet and the outlet is found to increase with increasing buoyancy ratio.

This paper reports the numerical simulations of double diffusive natural convection flows in a binary fluid contained in a two-dimensional enclosure, with imposed horizontal temperature and concentration differences between the vertical walls. The analysis concerns the influence of the different parameters governing the problem in the heat transfer characteristics and on the flow structure. The study is focused on steady-state solutions in the cooperating situation. At high Lewis numbers, numerical simulations show that heat transfer decreases with increasing buoyancy ratios. This phenemenon is analysed on the basis of the flow structure, and a scale analysis is provided to estimate the decrease in heat transfer. Then, the influence of the parameters on the formation of a multicellular flow structure is discussed.

This paper reports the analytical and numerical results on double diffusive natural convection in a binary fluid contained in a two-dimensional enclosure where horizontal temperature and concentration differences are specified. A numerical code, based on a finite volume procedure, and a scaling law approach are used to analyse the influence of the different parameters which characterize those thermosolutal flows: in this first paper, the mass transfer problem is studied in the steady-state. A general mass transfer correlation is proposed, which is valid over a wide range of parameters.

This paper focuses on the effect of natural convection on the accuracy of diffusion coefficient measurements in concentrated liquid alloys. This convective effect is modelled using the concept of effective diffusivity, introduced in a former work for the case of dilute systems. Our computer simulations support the validity of this approach and show that it is very difficult in practice to reach conditions of negligible convective solute transport. The possibility of realizing the measurements of diffusion coefficients in microgravity is also discussed.

International audience; In this paper, we study the effect of a homogeneous longitudinal through flow on the onset of convection in a horizontal porous layer saturated by a binary fluid and heated from below or above. The layer boundaries are subjected to a constant heat flux. The investigation is made by taking the Soret effect into account. It is found that in the case of positive separation ratio when the denser component moves toward the cooler wall, through flow has no effect on the stability threshold but exerts an orientating effect on the convective patterns. For negative separation ratio, a strong destabilization occurs of the spatially homogeneous state with respect to long-wave disturbances. The stability range for long-wavelength convective rolls is defined.

International audience; Heat transfer by natural convection and surface condensation in two-dimensional enclosures in contact with a cold external ambient through a wall of finite thickness was studied numerically. Special attention was given on the modeling of the flow of a binary mixture consisting of humid air. Low-Mach number assumption was introduced in order to account for decreases in mixture mass and average pressure within the enclosure between the initial and steady states. The computations show that thermodynamic balances are satisfied within the accuracy of the numerical procedure. The heat and fluid flows with and without condensation are compared for various operating conditions. It is shown that vapor condensation increases the heat transfer rate at the cold wall at the early stage of the transient regime. The decrease in the average density of the mixture leads to significant variable's reductions at steady-state which, in turn, causes lower overall heat transfer rate than for dry air.

The influence of laminar thermosolutal convection on solute segregation and heat transfer in the floating zone crystal growth of a dilute binary alloy is considered. The solute gradient is assumed to be destabilizing, and the thermal gradient is assumed perpendicular to it. The evolution of non-linear states from the critical points predicted by linear theory is studied. For vanishing thermal Rayleigh numbers, two solutions, corresponding to upflow and downflow at the meniscus, are obtained. The presence of side heating establishes upflow as the preferential direction; the downflow solution is found to persist for low thermal Rayleigh numbers. Flow and heat transfer for high thermal Rayleigh numbers are also computed.

When a solution containing a non-linear profile of solute concentration is heated from below, convection can occur in layers. Observations of this phenomenon are presented, and these are compared with calculations based on a simple model. It is concluded that the model contains the essential features of the process.

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored.

Les mouvements 2D engendres par l'instabilite de circulation de Langmuir de couches stratifiees d'eau de profondeur finie sont etudies avec une hypothese simplificatrice qui les rendent analogues mathematiquement a la convection thermosolutale a double diffusion avec une concentration en solute et un flux thermique constants aux frontieres

International audience; Freckles are common defects in industrial casting. They result from thermosolutal convection due to buoyancy forces generated from density variations in the liquid. The present paper proposes a numerical analysis for the formation of channel segregation using the three-dimensional (3D) cellular automaton (CA)—finite element (FE) model. The model integrates kinetics laws for the nucleation and growth of a microstructure with the solution of the conservation equations for the casting, while introducing an intermediate modeling scale for a direct representation of the envelope of the dendritic grains. Directional solidification of a cuboid cell is studied. Its geometry, the alloy chosen as well as the process parameters are inspired from experimental observations recently reported in the literature. Snapshots of the convective pattern, the solute distribution, and the morphology of the growth front are qualitatively compared. Similitudes are found when considering the coupled 3D CAFE simulations. Limitations of the model to reach direct simulation of the experiments are discussed.

Corrigendum to this publication: http://hal.archives-ouvertes.fr/hal-00528029; International audience; This call describes a numerical comparison exercise for the simulation of ingot solidification of binary metallic alloys. Two main steps are proposed, which may be treated independently: 1. The simulation of the full solidification process. First a specified 'minimal' solidification model is used and the contributors are provided with the corresponding sets of equations. The objective is to verify the agreement of the numerical solutions obtained by different contributors. Then different physical solidification models may be compared to check the features that allow for the best possible prediction of the physical phenomena. 2. A separate preliminary exercise is also proposed to the contributors, only concerned with the convective problem in the absence of solidification, in conditions close to those met in solidification processes. Two problems are considered for the case of laminar natural convection: transient thermal convection for a pure liquid metal with a Prandtl number on the order of 10(-2), and double-diffusive convection in an enclosure for a liquid binary metallic mixture with a Prandtl number on the order of 10(-2) and a Lewis number on the order of 10(4).

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.

Double-diffusive systems, such as thermosolutal convection, in which the density depends on two components that diffuse at different rates, are prone to both steady and oscillatory instabilities. Such systems can evolve into layered states, in which both components, and also the density, adopt a “staircase” profile. Turbulent transport is enhanced significantly in the layered state. Here we exploit an analogy between magnetic buoyancy and thermosolutal convection in order to demonstrate the phenomenon of magnetic layering. We examine the long-term nonlinear evolution of a vertically stratified horizontal magnetic field in the so-called “diffusive regime,” where an oscillatory linear instability operates. Motivated astrophysically, we consider the case where the viscous and magnetic diffusivities are much smaller than the thermal diffusivity. We demonstrate that diffusive layering can occur even for subadiabatic temperature gradients. Magnetic layering may be relevant for stellar radiative zones, with implications for the turbulent transport of heat, magnetic field, and chemical elements.

International audience; A comparative study among solidification experiments for three selected alloys (Sn-3 wt.%Pb, Sn-6.5 wt.%Pb and Sn-10 wt.%Pb) was conducted on a benchmark experiment model under the same experimental conditions. The goal of this paper is to analyze the effect of variation in concentration on the solidification process with respect to different aspects: thermal, dynamic, structure, and morphology of segregation. Experimental results consist of instantaneous temperature maps provided by a lattice of 50 thermocouples welded on the large crucible side and post-mortem characterizations of the samples, such as X-ray imaging, solute local composition and metallography. Measurement of the instantaneous temperature field and numerical computation of liquid solid interface evolution allows us to evaluate the effect of variation in concentration on thermosolutal convection behavior. Experimental results show that an increase in concentration greatly enhances the mechanism of the columnar-to-equiaxed transition (CET) and leads to refinement of the equiaxed structure. However, a significant effect of solutal element (lead) stratification is observed, which can slow down thermosolutal convection, in particular for large concentrations. Especially, without any stirring lead segregation which likely occurs during the melting phase may suppress natural convection. Furthermore, lead stratification is significantly reduced when electromagnetic stirring is opposed to natural convection before the solidification phase begins.

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Thermosolutal convection of a water–ethanol binary mixture, with a 50% mass fraction, at a high pressure of 750 bar, in a three-dimensional horizontal cavity, filled with an aluminum oxide (Al2O3) porous medium, is numerically investigated. Such investigation is examined in terms of the compositional separation versus the thermal conductivity and permeability of Al2O3 porous medium. The thermal diffusion, or Soret effect, is analyzed globally with a separation ratio and locally with the distributions of ethanol mole fraction on the horizontal and vertical lines in the center of the porous cavity. The Soret coefficient of ethanol at 30 °C was found to vary between −0.00699 and −0.00610 when pressure increases from 1 to 750 bar, leading to a ± 13.74 % relative variation around the mean value. On the other hand, the Soret coefficient of ethanol at 750 bar will vary from −0.00697 to −0.00534 when the temperature increases from 10 to 50 °C, leading to ± 26.48 % relative variation around the mean value. The details of the compositional separation at the steady state of thermosolutal convection are also applied to study the characteristics of Soret effect. When the value of permeability is less than 104 md, the compositional separation in the cavity is evident. However, when the value of permeability is larger than 104 md, the thermosolutal convection creates a mixing of the substances and the separation is diminished. Recommendations are made for the experimental design based on the results of numerical analysis.

Abstract The influence of the melt flow on the solidification structure is bilateral. The flow plays an important role in the solidification pattern, via the heat transfer, grain distribution, and segregations. On the other hand, the crystal structure, columnar or equiaxed, impacts the flow, via the thermosolutal convection, the drag force applied by the crystals on the melt flow, etc. As the aim of this research was to further explore the solidification–flow interaction, experiments were conducted in a cast cell (95 * 95 * 30 mm3), in which an ammonium chloride–water solution (between 27 and 31 wt pct NH4Cl) was observed as it solidified. The kinetic energy (KE) of the flow and the average flow velocity were calculated throughout the process. Measurements of the volume extension of the mush in the cell and the velocity of the solid front were also taken during the solidification experiment. During the mainly columnar experiments (8 cm liquid height) the flow KE continuously decreased over time. However, during the later series of experiments at higher liquid height (9.5 cm), the flow KE evolution presented a strong peak shortly after the start of solidification. This increase in the total flow KE correlated with the presence of falling equiaxed crystals. Generally, a clear correlation between the strength of the flow and the occurrence of equiaxed crystals was evident. The analysis of the results strongly suggests a fragmentation origin of equiaxed crystals appearing in the melt. The transition from purely columnar growth to a strongly equiaxed rain (CET) was found to be triggered by (a) the magnitude of the coupling between the flow intensity driven by the equiaxed crystals, and (b) the release and transport of the fragments by the same flow recirculating within the mushy zone. Graphical Abstract Coupling mechanism at the origin of CET: 1-2 strong flow running through the mush transporting out dendrite fragments (white dots); 3-4 equiaxed growth and drag of the downward flow. If the vortex is sufficiently stable, the horizontal configuration can lead to freckle appearing; a) vertical solidification front; b) horizontal solidification front.

A transient two-dimensional ISPH method based on the time-fractional derivative was applied for emulating thermosolutal convection of the nano-encapsulated phase change material (NEPCM) embedded in an annulus between an inner wavy shape and outer hexagonal-shaped cavity. The impacts of a magnetic field and double rotations amongst an inner wavy shape and outer hexagonal-shaped cavity on the heat and mass transmission of NEPCM in an annulus have been conducted. Effects of a time parameter τ (0.01 − 1), frequency parameter ω (1 − 7), fractional time derivative α (0.95 − 1), Darcy parameter Da (10−2 − 10−4), Hartmann number Ha (0 − 100), fusion temperature θf (0.05 − 0.8), and Rayleigh number Ra (103 − 105) on the contours of temperature, heat capacity, concentration, and velocity field as well as profiles of Nu ¯ and Sh ¯ are investigated. The main findings signaled that the double rotations plays effectively in speed up the nanofluid movements, and changing the features of temperature, concentration, and heat capacity inside an annulus. An augmentation in a frequency parameter boosts the nanofluid speed by 128.57%. A decline in α from 1 to 0.95 enhances the maximum nanofluid velocity by 13.73%. The nanofluid movements within an annulus are reduced according to an increase in Ha and a decrease in Da. The power in the Rayleigh number enhances the nanofluid movements within an annulus.

International audience; A special case that stands out in thermosolutal convection is that for which the thermal and solutal convection forces are equal and generate opposite effects (N = -1). This antagonistic effect between the two buoyancy forces is expected to generate unpredictable and unique behaviors when combined with thermodiffusion and internal heating phenomena. The present numerical study is dedicated to this particular case considering a binary mixture confined in a vertical rectangular cavity with an aspect ratio A = 2. The lattice Boltzmann method with multiple relaxation time is used to analyze the effect of the control parameters which are the Soret parameter (Sr = -0.5, 0 and 0.5), and the internal to external Rayleigh numbers ratio (0

An analysis of thermosolutal convection and entropy generation considering Soret and Dufour effects inside an inclined rectangular enclosure attached with porous wall has been investigated. The physical parameters are in the ranges such as buoyancy ratio (-10 ≤ N ≤ 10), permeability (10−9 ≤ Da ≤ 10−1) and thickness (0 ≤ d ≤ 1.0) of porous wall, enclosure inclination angle (0° ≤ Φ ≤ 90°), Soret number (0 ≤ Sr ≤ 1.5) and Dufour number (0 ≤ Du ≤ 1.5). Heatlines and masslines taking Soret and Dufour effects into account are obtained as effective tools that visualizes the heat and species transported paths. Results demonstrate that Soret number has mild impact on heat transfer rate and promotes moisture transportation, while increasing Dufour number could boot heat transfer rate and reduce moisture transfer rate. Furthermore, increasing Soret numbers enhances entropy generation induced by heat transfer and fluid friction, while entropy generation induced by fluid friction, heat and moisture transfer is dropped with stronger Dufour effect. Present work can benefit the heat and moisture transfer performance and entropy generation minimization of thermal storage wall system.

This paper deals with thermosolutal convection in an inclined enclosure inserted with heat-generating porous blocks under the influence of magnetic field. The general Brinkman-extended Darcy model is adopted to formulate the fluid flow in the enclosure. An extensive series of numerical simulations is investigated in the range of parameters 0 ≤ Ha ≤ 150, 103 ≤ Ra ≤ 107, −10.0 ≤ N ≤ 10.0, 10−9 ≤ Da ≤ 10−1, 0.1 ≤ Kr ≤ 10 and −90° ≤ Φ ≤ 90°. Streamline, isotherms, isoconcentrations and masslines are produced to illustrate the fluid, heat and moisture flow structures. It is founded that overall Nusselt number is an increasing function of Ra, N, Da and Kr in the vertical enclosure, while decreasing with Ha. The permeability and thermal conductivity of porous blocks have no significant effect on moisture transfer rate. The Nusselt and Sherwood curves for different inclination angle are presented to be parabolic, and the maximum are near at Ф = −10° and 30°, respectively. In addition, correlations of the overall Nusselt and Sherwood numbers depending on thermal Rayleigh number, buoyancy ratio, Hartmann number and inclination angle have been obtained, which are beneficial to determine heat and moisture transfer rates in electrical devices.

International audience; Solidification of alloys in a thermal gradient usually involves the generation of flows by thermal or thermosolutal convection. We experimentally study their effects on the dynamics of a solidification interface by inducing a controlled sweeping flow in a directional solidification device. Flow is induced in the sample from an external thermosiphon. Downstream inclination of microstructures and downstream sidebranch development are observed. However, the major outcome is the evidence of large scale travelling waves on the solidification interface. They are induced by the coupling between solidification and flow and yield repetitive striations of the solid phase. Two waves are observed and characterized.