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International audience; In this work, we consider the momentum transport of a incompressible fluid in a like Beavers and Joseph (1967) system. For this purpose, in the context of the volume averaging method, we use a one-domain approach (ODA). Thus, the momentum generalized transport equations (GTE), which are written in terms of position-dependent effective medium coefficients, are valid everywhere in the system and contains two Brinkman corrections in addition to a Darcy’s term. The ODA predictions are tested against the results obtained from averaging the local profiles resulting from pore-scale simulations. One of the key points for solving the ODA remains on the prediction of the permeability, which in this work is obtained either by solving the associated local closure problem or from pore-scale profiles. Our analysis shows that the GTE for momentum transport accurately predicts the average velocity profiles everywhere in the system. To this end, the first and the second Brinkman’s corrections, as well as a position-dependent intrinsic permeability tensor in Darcy’s term must be included.

International audience; Heterogeneous porous structures are very often encountered in natural environments, bioremediation processes among many others. Reliable models for momentum transport are crucial whenever mass transport or convective heat occurs in these systems. In this work, we derive a large-scale average model for incompressible single-phase flow in heterogeneous and hierarchical soil porous media composed of two distinct porous regions embedding a solid impermeable structure. The model, based on the local mechanical equilibrium assumption between the porous regions, results in a unique momentum transport equation where the global effective permeability naturally depends on the permeabilities at the intermediate mesoscopic scales and therefore includes the complex hierarchical structure of the soil. The associated closure problem is numerically solved for various configurations and properties of the heterogeneous medium. The results clearly show that the effective permeability increases with the volume fraction of the most permeable porous region. It is also shown that the effective permeability is sensitive to the dimensionality spatial arrangement of the porous regions and in particular depends on the contact between the impermeable solid and the two porous regions.

International audience; This paper describes the first step of a study dedicated to the development of a macroscopic model of casting of a metal foam by infiltration and solidification of a liquid metal in a porous mould. The first stage presented here describes a local model of injection of the metallic melt in a capillary tube and subsequent solidification of the metal by heat transfer to the duct walls.The model is intended to account for the air/liquid interface displacement during the infiltration phase, for the heat transfer to the wall and for the growth of the solid phase in the presence of the fluid flow. The objective is to determine the influence of the operating conditions on the penetration depth and on the solidification time in a simplified geometry before using this local information in a macroscopic homogenized model presently under development.

We develop asymptotic modeling for two- or three-dimensional viscous fluid flow and convective transfer at the interface between a fluid and a porous layer. The asymptotic model is based on the fact that the thickness d of the interfacial transition region Ω_{fp} of the one-domain representation is very small compared to the macroscopic length scale L. The analysis leads to an equivalent two-domain representation where transport phenomena in the transition layer of the one-domain approach are represented by algebraic jump boundary conditions at a fictive dividing interface Σ between the homogeneous fluid and porous regions. These jump conditions are thus stated up to first-order in O(d/L) with d/L≪1. The originality and relevance of this asymptotic model lies in its general and multidimensional character. Indeed, it is shown that all the jump interface conditions derived for the commonly used 1D-shear flow are recovered by taking the tangential component of the asymptotic model. In that case, the comparison between the present model and the different models available in the literature gives explicit expressions of the effective jump coefficients and their associated scaling. In addition for multi-dimensional flows, the general asymptotic model yields the different components of the jump conditions including a new specific equation for the cross-flow pressure jump on Σ.

In this work we solve effective-medium equations for modeling momentum and heat transfer in a parallel-plate channel partially filled with a porous insert. In order to avoid specifying the boundary conditions at the fluideporous boundary, we solve equations involving position-dependent coefficients (i.e., a one-domain approach). The solution of the momentum-transport problem is carried out using implicit integral equation formulations based on Green’s functions, whereas the heat transfer problem was solved numerically using the finite element method. The simulations were performed in terms of several values of the porosity, the Peclet number, the size of porous insert and the thermal conductivities ratio. In agreement with previous works, it was found that the thermal performance is improved by either increasing the size of the porous insert or by favoring mixing inside the channel. A drawback of this approach is the high computational demand associated to modeling transport in the vicinity of the porous medium and the fluid. In this way, the extents and limitations about the use of a one-domain formulation are exposed in a practical application. The results from this work should serve as motivation for more experimental and theoretical research; in particular, the derivation and application of jump boundary conditions.

International audience; The purpose of this work is to investigate the extents of the local thermal equilibrium (LTE) assumption at the fluid-porous medium boundary (i.e., in a heterogeneous region). The analysis is performed in terms of the one-domain approach. Therefore, we posed and solved the associated closure problems in order to compute the spatial dependence of the effective coefficients at the fluid-porous medium boundary. Steady-state comparisons with direct numerical simulations evidence that the LTE is, in general, justifiable everywhere in the system, i.e., in both homogeneous and heterogeneous regions.

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; Stability analyses of thermal and/or solutal natural convection in a configuration composed by a fluid layer overlying a homogeneous porous medium have been performed using different modeling approaches, especially for the treatment of the interfacial region. Comparisons between the one-domain approach and the two-domain formulation have shown important discrepancies of the marginal stability curves. This note shows that, according to Kataoka [I. Kataoka, Local instant formulation of two-phase flow, Int. J. Multiphase Flow 12(5) (1986) 745–758.], the differentiation of the macroscopic properties of the porous layer at the interface (porosity, permeability, thermal effective diffusivity) must be considered in the meaning of distributions. In that case, the one- and the two-domain approaches are shown to be equivalent and very good agreement is indeed found when comparing the results obtained with both approaches.

This study deals with the onset of thermal natural convection in a system consisting of a fluid layer overlying a homogeneous porous medium. A linear stability analysis is carried out, using the so-called two-domain approach and including the Brinkman term in the porous region (2ΩDB). Results are systematically compared to those obtained using the one-domain approach (1Ω) and the classical Darcy formulation of the two-domain approach (2ΩD). A better agreement is found between the 2 Ω DB and 2ΩD neutral curves, than with the 1Ω curves, indicating that the inclusion of the Brinkman term plays a secondary role on the stability results. The different treatment of the interfacial region is discussed on the basis of these results.

A stability analysis of thermal natural convection in superposed fluid and porous layers is carried out. The two-layer system is described using a one-domain formulation, and the eigenvalue problem resulting from the stability analysis is solved using the generalized integral transform technique (GITT). The numerical results confirm that the onset of convection can have a bimodal nature depending on the depth ratio. The influence of the dimensionless permeability and thermal diffusivity ratio are investigated.

This paper deals with natural convection driven by combined thermal and solutal buoyancy forces in a binary fluid. The configuration under study is a confined enclosure partially filled with a vertical porous layer. The mathematical description of the problem is based on a one-domain formulation of the conservation equations. The set of numerical results presented here quantitatively shows the influence of the porous layer on the flow structure and on heat and species transfer in the enclosure. The paper is focused on the analysis of the influence of the characteristic parameters governing double diffusive convection, namely the ratios of solutal and thermal parameters: the diffusivities and the buoyancy forces. Heat and mass transfer is analyzed as a function of the permeability of the porous layer. It is shown that the coupling of the flow penetration in the porous layer with the combined buoyancy forces induces a specific behavior of the flow structure and average heat transfer in the enclosure.

This paper describes the solution of a steady natural convection problem in porous media by the dual reciprocity boundary element method. The boundary element method for the coupled set of mass, momentum, and energy equations in two-dimensions is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on augmented scaled thin plate splines. Numerical examples include convergence studies with different mesh size, uniform and non-uniform mesh arrangement, and constant, linear, and quadratic boundary field discretisations for differentially heated rectangular cavity problems at filtration with Rayleigh number of Ra p ¼ 25; 50, and 100, Darcy numbers Da ¼ 10 23 ; and 10 25 , and aspect ratios A ¼ 1=2; 1, and 2. The solution is assessed by comparison with reference results of the fine-mesh finite volume method. q 2003 Elsevier Ltd. All rights reserved.

The momentum balance at the interface between a liquid and a porous substrate is investigated for a configuration with forced flow parallel to the interface. An heterogeneous continuously varying transition layer between the two outer bulk regions is introduced. The stress jump coefficient earlier introduced in the jump interface condition is here derived as an explicit function of the variations of the velocity and effective properties of the transition layer. Agreement is found between our numerical results based on the single-domain approach and the existing ad hoc estimates in the literature. Further advantages of this non-homogeneous analysis are also provided.

This paper deals with the derivation of the macroscopic momentum transport equation in a non-homogeneous solidifying columnar dendritic mushy zone using the method of volume averaging. One of the originalities of this study lies in the derivation of an associated closure problem for the determination of the spatial evolution of the effective transport properties in such a complex situation. In this analysis—where the phase change has been included at the different stages of the derivation—all the terms arising from the averaging procedure (geometrical moments, phase interactions, interfacial momentum transport due to phase change, porosity gradients, etc.) are systematically estimated and compared on the basis of the characteristic length-scale constraints associated with the porous structures presenting evolving heterogeneities. For dendritic structures with “moderate” (but not small) evolving heterogeneities, we show that phase change and non local effects could hardly affect the determination of the permeability and inertia tensors. Finally, a closed form of the macroscopic momentum equation is proposed and a discussion is presented about the need to consider inertia terms and the second Brinkman correction (explicitly involving gradients of the liquid volume fraction) in such non-homogeneous systems.

In order to improve mass transport description in solidification modeling, this study deals with the determination of the solute dispersion tensor in columnar dendritic mushy zone. The closure problem associated with the derivation of the macroscopic species conservation equation, using the volume averaging method, is solved numerically. Using schematic structures and digitized images of real dendritic structures observed experimentally during solidification of succinonitrile-4 wt.% acetone, the influence of tortuosity and microscopic dispersion on the determination of the effective solute dispersion coefficients is represented in terms of the Peclet number.

This paper describes the solution of a steady state natural convection problem in porous media by the dual reciprocity boundary element method (DRBEM). The boundary element method (BEM) for the coupled set of mass, momentum, and energy equations in two dimensions is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on augmented scaled thin plate splines. Numerical examples include convergence studies with different mesh size, uniform and non-uniform mesh arrangement, and constant and linear boundary field discretizations for differentially heated rectangular cavity problems at filtration with Rayleigh numbers of Ra*=25, 50, and 100 and aspect ratios of A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM). Copyright © 2000 John Wiley & Sons, Ltd.

This paper addresses the derivation of the macroscopic momentum equation for flow through a nonhomogeneous porous matrix, with reference to dendritic structures characterized by evolving heterogeneities. A weighted averaging procedure, applied to the local Stokes' equations, shows that the heterogeneous form of the Darcy's law explicitly involves the porosity gradients. These extra terms have to be considered under particular conditions, depending on the rate of geometry variations. In these cases, the local closure problem becomes extremely complex and the full solution is still out of reach. Using a simplified two-phase system with continuous porosity variations, we numerically analyze the limits where the usual closure problem can be retained to estimate the permeability of the structure.

Turbulent heat transfer in a channel partially filled by a porous medium is investigated using a direct numerical simulation of an incompressible flow. The porous medium consists of a three-dimensional Cartesian grid of cubes, which has a relatively high permeability. The energy equation is not solved in the cubes. Three different heating configurations are studied. The simulation is performed for a bulk Reynolds number Reb = 5500 and a Prandtl number Pr = 0.1. The turbulent flow quantities are compared with the results of Breugem and Boersma [“Direct numerical simulations of turbulent flow over a permeable wall using a direct and a continuum approach,” Phys. Fluids 17, 025103 (2005)] to validate the numerical approach and macroscopic turbulent quantities are analyzed. Regarding the temperature fields, original results are obtained. The temperature fields show an enhanced turbulent heat transfer just above the porous region compared to the solid top wall, which can be related to the large vortical structu...

Modeling transport phenomena in hierarchical systems can be carried out by either a one domain approach or a two domain approach. The first one involves assuming the system as a pseudo-continuum and is expressed in terms of position-dependent effective medium coefficients. In the two domain approach, the differential equations have position-independent coefficients but require accounting for the corresponding boundary conditions that couple the equations between each homogeneous region. For momentum transport between a porous medium and a fluid, stress boundary conditions have been derived in terms of a jump coefficient that needs to be predicted within a two-domain approach formulation. However, continuity of the velocity is postulated at the dividing surface. In this work, we propose a methodology for the derivation of boundary conditions for both the velocity and the stress. These conditions are expressed in terms of jump coefficients that are computed from the solution of an ancillary macroscopic closure problem. This problem accounts for the deviations from the one and two domain approaches. From the closure problem solution we were also able to determine the position at which the jump conditions should be applied, i . e . , the dividing surface position. In addition, we used this methodology adopting the assumptions proposed by Ochoa-Tapia and Whitaker as well as those by Beavers and Joseph. We found that any version of the two domain approach was in agreement with the one domain approach in the bulk of the porous medium and the fluid. However, the same is not true for the process of capturing the essential information of the inter-region.

International audience; We investigate the role of the inertia of the flow through the dendritic mushy zone in the numerical prediction of channel segregations during columnar solidification. The contribution of inertia is included in the momentum transport equation through the quadratic Forchheimer correction term. The study reveals a significant influence of the Forchheimer term in the vicinity of the liquidus front, i.e. at high liquid fractions. The natural convective flow field in this region is modified due to the additional inertial drag. This strongly influences the convective transport of solute and thereby incurs a modification of the dynamics of the advancement of the mushy zone. The most notable consequence is a significant decrease in the predicted channel segregation.

This study deals with macroscopic modeling of heat transfer in porous media subjected to high temperature. The derivation of the macroscopic model, based on thermal non-equilibrium, includes coupling of radiation with the other heat transfer modes. In order to account for non-Beerian homogenized phases, the radiation model is based on the generalized radiation transfer equation and, under some conditions, on the radiative Fourier law. The originality of the present upscaling procedure lies in the application of the volume averaging method to local energy conservation equations in which radiation transfer is included. This coupled homogenization mainly raises three challenges. First, the physical natures of the coupled heat transfer modes are different. We have to deal with the coexistence of both the material system (where heat conduction and/or convection take place) and the non-material radiation field composed of photons. This radiation field is homogenized using a statistical approach leading to the definition of radiation properties characterized by statistical functions continuously defined in the whole volume of the porous medium. The second difficulty concerns the different scales involved in the upscaling procedure. Scale separation, required by the volume averaging method, must be compatible with the characteristic length scale of the statistical approach. The third challenge lies in radiation emission modeling, which depends on the temperature of the material system. For a semi-transparent phase, this temperature is obtained by averaging the local-scale temperature using a radiation intrinsic average while a radiation interface average is used for an opaque phase. This coupled upscaling procedure is applied to different combinations of opaque, transparent, or semi-transparent phases. The resulting macroscopic models involve several effective transport properties which are obtained by solving closure problems derived from the local-scale physics.

This paper deals with natural convection in confined porous media, driven by cooperating thermal and solutal buoyancy forces. The physical model for the momentum conservation equation makes use of the B:rinkman extension of the classical Darcy equation, and the set of coupled equations is solved using a finite volume approach. The numerical simulations presented here span a wide range of the main parameters (the Rayleigh and Darcy numbers) in the domain of positive buoyancy numbers and for Le > 1. When possible, the results are compared with previous numerical data or existing scaling laws. The results are mainly analyzed in terms of the average heat and mass transfers at the walls of the enclosure. Although the mass transfer characteristics are fairly well predicted by the scale analysis, it is shown that convective heat transfer has a specific behavior in given ranges of the governing parameters.

International audience; This study proposes a downscaling methodology consisting of describing the convective heat transfer at the pore scale of the porous media from the knowledge of the velocity and temperature fields at the Darcy scale. The analysis is based on the derivation of local deviation equations on a local unit cell. A penalization technique is used to deal with the macroscopic source terms involved in these equations. The partial differential equations system is closed assuming periodic conditions at the boundaries of the unit cell. Numerical simulations illustrate the relevance and the limitations of the downscaling procedure applied to two-dimensional periodic arrays of cylinders.

A gravity-driven falling film on a saturated porous inclined plane is studied via a continuum approach, where the liquid and porous layers are considered as a single composite layer. Using a weighted residual technique, a two-equation model is derived in terms of the local flow rate $q(x, t)$ and the entire layer thickness $H(x, t)$. Its linear stability analysis has been satisfactorily compared to the results of the Orr–Sommerfeld problem. The principal effect of the porous substrate on the film flow is to displace the liquid–porous interface to an effective liquid–solid interface located at the lower boundary of the upper momentum boundary layer in the porous medium. The stability and dynamics of the film is thus only weakly affected by the presence of a permeable substrate. In both the linear and the nonlinear regimes, the spatial response of a falling film on a porous medium is not very different from that observed on an impermeable inclined wall. However, the wavy motion of the film triggers a significant exchange of mass at the liquid–porous interface.

International audience

International audience; During the solidification of metal alloys, chemical heterogeneities at the scale of the product develop. It is referred to as "macrosegregation". Numerical simulation tools exist in the industry. However, their predictive capabilities are not validated and are still limited. A 2D numerical benchmark is presented, based on the solidification of metallic Pb-Sn alloys. Concerning the numerical benchmark, a "minimal" common model of solidification is assumed, including columnar growth without undercooling, fixed solid, isotropic permeability of the mushy region, local thermodynamic equilibrium, lever-rule assumption for the local average composition. We focus our attention on the numerical method used to solve the average conservation equations: Finite Volume, Finite Element, Velocity-Pressure coupling treatment, scheme for convective terms, etc. At this stage of the work, we cannot exhibit a reference solution. However we draw some conclusions on the effects of the grid dependency, in particular on the location and sizes of the segregate channels. The development of both thermally and solutal driven convections in the first stage of the process (cf. low Prandtl and high Lewis numbers) and the relative independency of the convective scheme are also discussed. This presentation also have the goal to call other contributors to join this benchmark [1] in order to enrich the exercise and to reach a reference solution for this important problem in metallurgy.

International audience; The numerical prediction of channel segregations during two-dimensional columnar solidification of Sn-Pb alloy is analyzed when inertia is considered in the dendritic mushy zone. This contribution is included in the momentum transport equation through the quadratic Forchheimer correction term. A significant influence of the Forchheimer term in the vicinity of the liquidus region is found. The natural convective flow field and therefore the global shape of the mushy zone are modified giving rise to significant decrease of channel segregation, due to additional drag in these regions.

International audience; We study a convective heat transfer problem in a fluid-porous domain in the case of the local thermal non-equilibrium assumption (LTNE). The issue of this study is to determine appropriate boundary conditions to model heat transfer, while using models with a different number of equations: a two-temperature model in the homogeneous porous region versus a one-temperature model in the free region. To proceed, a two-step up-scaling approach is used, which has the particularity to provide closed jump relations depending on intrinsic characteristic of the interface. Thus, the use of jump or continuity conditions depend only on the interface location inside the fluid-porous transition region. The pertinence of the approach is illustrated on a 2D convective heat transfer problem considering a solid heat source in the porous medium.

International audience; The onset of double-diffusive natural convection in under-ice melt ponds is investigated through a linear stability analysis. The three-layer configuration is composed by a fluid layer (melt pond) overlying a saturated porous medium (ice matrix), which in turn overlies another fluid layer (under-ice melt pond). Water density inversion is taken into account by adopting a density profile with a quadratic temperature dependence and a linear concentration dependence. We show that the key parameter affecting stability is the depth of the ice matrix, while the depths of the upper and lower fluid layers play a marginal role. A Hopf bifurcation is observed in the whole range of parameters studied, and the size of the convection cells depends on ice permeability. The influence of the external temperature gradient is investigated by means of the definition of an extra thermal parameter accounting for the relative position of the density maximum. It is shown that convection is favored by larger temperature gradients, which occur during Arctic summer.

In many industrial processes or natural phenomena coupled heat and mass transfer and fluid flow take place in configurations combining a clear fluid and a porous medium. Since the pioneering work by Beavers and Joseph (1967), the modelling of such systems has been a controversial issue, essentially due to the description of the interface between the fluid and the porous domains. The validity of the so-called one-domain approach — more intuitive and numerically simpler to implement — compared to a two-domain description where the interface is explicitly accounted for, is now clearly assessed. This paper reports recent developments and the current state of the art on this topic, concerning the numerical simulation of such flows as well as the stability studies. The continuity of the conservation equations between a fluid and a porous medium are examined and the conditions for a correct handling of the discontinuity of the macroscopic properties are analyzed. A particular class of problems dealing with thermal and double diffusive natural convection mechanisms in partially porous enclosures is presented, and it is shown that this configuration exhibits specific features in terms of the heat and mass transfer characteristics, depending on the properties of the porous domain. From the viewpoint of the stability of convection in a horizontal layer where a fluid layer lies on top of a porous medium, the analysis shows that the onset of convection is strongly influenced by the presence of the porous medium. The case of thermal convection is fully detailed and many open problems arise in the field of double diffusive convection.Copyright © 2010 by ASME

A gravity-driven film flow on a slippery inclined plane is considered within the framework of long-wave and boundary layer approximations. Two coupled depth-averaged equations are derived in terms of the local flow rate $q(x, t)$ and the film thickness $h(x, t)$. Linear stability analysis of the averaged equations shows good agreement with the Orr–Sommerfeld analysis. The effect of a slip at the wall on the primary instability has been found to be non-trivial. Close to the instability onset, the effect is destabilising whereas it becomes stabilising at larger values of the Reynolds number. Nonlinear travelling waves are amplified by the presence of the slip. Comparisons to direct numerical simulations show a remarkable agreement for all tested values of parameters. The averaged equations capture satisfactorily the speed, shape and velocity distribution in the waves. The Navier slip condition is observed to significantly enhance the backflow phenomenon in the capillary region of the solitary waves with a possible effect on heat and mass transfer.

International audience; During the solidification of metal alloys, chemical heterogeneities at the product scale (macrosegregation) develop. Numerical simulation tools are beginning to appear in the industry, however their predictive capabilities are still limited. We present a numerical benchmark exercise treating the performance of models in the prediction of macrosegregation. In a first stage we defined a "minimal" (i.e. maximally simplified) solidification model, describing the coupling of the solidification of a binary alloy and of the transport phenomena (heat, solute transport and fluid flow) that lead to macrosegregation in a fully columnar ingot with a fixed solid phase. This model is solved by four different numerical codes, employing different numerical methods (FVM and FEM) and various solution schemes. We compare the predictions of the evolution of macrosegregation in a small (10×6 cm) ingot of Sn-10wt%Pb alloys. Further, we present the sensitivities concerning the prediction of instabilities leading to banded channel mesosegregations.

International audience; Refers to: Call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys International Journal of Thermal Sciences, Volume 48, Issue 11, November 2009, Pages 2013-2016, M. Bellet, H. Combeau, Y. Fautrelle, D. Gobin, M. Rady, E. Arquis, O. Budenkova, B. Dussoubs, Y. Duterrail, A. Kumar, C.A. Gandin, B. Goyeau, S. Mosbah, M. Založnik

PurposeThis paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.Design/methodology/approachIn the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.FindingsNumerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.Research limitations/implicationsAccurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.Practical implicationsThe present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.Originality/valueThe paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

The momentum transfer between a homogeneous fluid and a porous medium in a system analogous to the one used by Beavers and Joseph (J Fluid Mech 30:197-207, 1967) is studied using volume averaging techniques. In this article, we present a closed generalized momentum transport equation (GTE) that is valid everywhere and is expressed in terms of position-dependent effective transport coefficients, which are computed from the solution of associated closure problems previously reported. A combination of the velocity profiles from the GTE in the definition of the excess terms that define the jump coefficients allows their computation using numerical techniques. The calculations are in concordance with those resulting from the work of Goyeau et al. (Int J Heat Mass Transf. 46:4071-4081, 2003), showing a strong dependence with the porosity. In addition, the effects of the roughness of the boundary on the computation of the position-dependent permeability tensor in the inter-region are also analyzed.

In the quest (i) to determine the form of the boundary conditions that must be applied at a fluid/porous interface and (ii) to determine the value of the jump parameters that appear in the expression for these boundary conditions, two different approaches are commonly considered: the so-called one-domain and two-domain approaches. These approaches are commonly thought to be different, and they are thus sometimes compared to each other to determine the value of jump parameters. In this article, we show that the two-domain and discontinuous one-domain approaches are actually strictly equivalent, provided that the latter is mathematically interpreted in the sense of distributions. This equivalence is shown in details for a heat conduction problem and for the more classical Darcy-Brinkman problem. We show in particular that interfacial jumps are introduced in the discontinuous one-domain approach through Dirac delta functions. Numerical issues are then discussed that show that subtle discretization truncation errors give rise to large variations that can be mis-interpreted as the sign of the existence of jump parameters.

In this paper, mass transfer at the fluid-porous medium boundaries is studied. The problem considers both diffusive and convective transport, along with adsorption and reac- tion effects in the porous medium. The result is a mass flux jump condition that is expressed in terms of effective transport coefficients. Such coefficients (a total dispersion tensor and effective reaction and adsorption coefficients) may be computed from the solution of the corresponding closure problem here stated and solved as a function of the Peclet number (Pe), the porosity and a local Thiele modulus. For the case of negligible convective transport (i.e., Pe � 1), the closure problem reduces to the one recently solved by the authors for diffusion and reaction between a fluid and a microporous medium.

This article deals with the onset of thermosolutal natural convection in horizontal superposed fluid and porous layers. A linear stability analysis is performed using the one-domain approach. As in the thermal convection case, the results show a bimodal nature of the marginal stability curves where each mode corresponds to a different convective instability. At small wave numbers, the convective flow occurs in the whole cavity (“porous mode”) while perturbations of large wave numbers lead to a convective flow mainly confined in the fluid layer (“fluid mode”). Furthermore, it is shown that the onset of thermosolutal natural convection is characterized by a multi-cellular flow in the fluid region for negative thermal Rayleigh numbers. For positive thermal Rayleigh numbers, the convective flow takes place both in the fluid and porous regions. The influence of the depth ratio and thermal diffusivity ratio is also investigated for a wide range of the thermal Rayleigh numbers.

The time evolution of a thin liquid film flowing down a heated solid porous substrate is investigated. Using the Navier–Stokes and Darcy–Brinkman equations in the film and the porous layer, respectively, the problem is reduced to the study of the evolution equation for the free surface of the liquid film derived through a long-wave approximation. A linear stability analysis of the base flow is performed and the critical Reynolds and Marangoni numbers are obtained. A nonlinear analysis using continuation techniques shows that the base flow yields to stationary surface structures ranging from surface waves to large amplitude structures resembling sliding drops or ridges. It is also shown under what conditions the porous layer can be replaced by an effective slip boundary condition at the liquid-solid interface. Then, the corresponding slip length is calculated from the porous layer characteristics (thickness, porosity, and Darcy number).

The aim here is to describe macroscopic models of conductive heat transfer within systems comprising two solid phases, using the method of volume averaging. The presentation of this technique largely stems from work by Carbonell, Quintard, and Whitaker [1–3]. The macroscopic conservation equations are set up under the assumption of local thermal equilibrium, leading to a model governed by a single equation. The effective thermal conductivity of the equivalent medium is obtained by solving the associated closure problems. The case where thermal equilibrium does not pertain, leading to a model with two energy conservation equations, is discussed briefly.

A macroscopic modeling of diffusion and chemical reaction in double emulsion systems using the method of volume-averaging is presented. In this three-phase system, chemical reaction takes place in the drops and membrane phases (ω-region) while passive diffusion is considered in the continuous external phase (η-region). First, a generalized one-equation model, free of the usual length scale constraints, is derived in order to describe the solute transfer in both homogeneous regions and in the ω–η inter-region. The up-scaling in the ω-region is based in the local mass equilibrium assumption between the two phases. Equations in both homogeneous regions are deduced from the generalized one-equation model. Then, the jump boundary condition at the dividing surface is derived and associated closure problems are established in order to calculate the jump coefficients.

Large particle transport in porous media plays an important role in chemical and mechanical engineering but also in the medical field, especially in cancer treatment. The major difficulty in large particle transport results from the fact that the particles themselves influence the effective transport properties by pore plugging due to particle entrapment. In this study, we used a random walk model describing the particle transport inside the pores. The effective macroscopic transport properties are determined using the results of the random walk model. It is shown that the permeability tensor strongly depends on the particle size and the injection point location, whereas the dispersion coefficients remain independent. We also determined the maximal particle radius for which particle transport can be described by the convection-diffusion equation. Another important point in the chemical and medical field is the final particle distribution, and particularly the distribution of the quantity of liquid transported by the particles. Our results show that large particles do not lead to a homogeneous liquid distribution. Hence when a large quantity of liquid should be homogeneously distributed in the porous medium, the use of smaller particles is recommended.

This paper deals with the macroscopic modeling and numerical simulation of columnar dendritic solidification of binary alloys. The macroscopic governing equations and associated effective transport properties were previously derived using a volume averaging technique with local closure. The macroscopic model takes into account the spatial variation of the pore-scale geometry within the mushy zone, which leads to additional terms involving porosity gradients. The second important feature concerns solute mass conservation, which is described by considering a macro-scale non-equilibrium accounting for chemical exchanges at the solid–liquid interface. A simplified version of the model is validated through a comparison of the numerical solution to three experiments available in the literature. Porosity extra terms are systematically estimated on the basis of these numerical simulations, and the influence on solidification of effective transport properties such as permeability and interfacial solute exchange coefficients is investigated.

International audience; This paper deals with the derivation of a macroscopic model for columnar dendritic solidification of binary mixtures using the volume averaging method with closure. The main originalities of the model are first related to the explicit description of evolving heterogeneities of the dendritic structures and their consequences on the derivation of averaged conservation equations, where additional terms involving porosity gradients are present, and on the determination of effective transport properties. These average properties are defined by the associated closure problems taking into account the geometry of the dendrites and the local intensity of the flow. The macroscopic solute transport is obtained by considering macroscale non-equilibrium giving rise to macroscopic dispersion and interfacial exchange phenomena. Mass exchange coefficients are accurately explicited as a function of the local geometry. © 2004 SBMAC.

International audience; Pressure-driven membrane processes have strongly gained importance in industrial separations over the past three decades. Numerous improvements in the technology – for instance, development of highly selective and permeable membranes, improvement in peripheral technology – have led to widespread adaptation of this process in chemical, environmental, pharmaceutical and biomedical applications. Membrane fouling and subsequent permeate flux decline are inevitably associated with the pressure-driven membrane processes. Despite the various studies on membrane fouling and related phenomena, the fundamental mechanisms and processes involved are still not fully understood. Typical observations of permeate flux over time reveal a rapid initial decline followed by a more gradual long-term decline. This indicates the increase of the membrane resistance with time. The resistance of membrane is modeled via different mechanisms (pore blockage, concentration polarization, and cake formation). These models depend on the experimental parameters and do not describe the fouling process completely. It should be noted that the characteristics of the system change with time because of the evolution of molecules (such as proteins) or their interactions with environment (a porous medium and a fluid). Furthermore the problem is multiphysics (hydrodynamics, mass transport, physical chemistry) and multiscale (molecule, membrane pores and membrane). The fouling of membrane during the filtration of complex media containing microorganisms is frequent (wastewater, cleaning solutions, microorganisms cultures etc). Microorganisms can attach, grow, multiply, and relocate on the membrane surfaces they can excrete extracellular polymer substances (EPS). EPS are primarily composed of polysaccharides and proteins. The fouling can be located on the surface and in the volume of the membrane. The fouling of membranes by proteins is also present in many post treatment of food products (extraction of proteins from milk, concentration of cooking juice for example). The objective of this work was to focus on the fouling of ultrafiltration and microfiltration membranes by proteins. In contrast of small molecules that behave like rigid molecules, most proteins do not simply attach or detach from an interface with certain adsorption and desorption probabilities. Instead, the complex composition and structure of proteins causes more complex phenomena such as structural re-arrangements, changing surface affinities during adsorption, positive cooperative effects, overshooting adsorption kinetic or surface aggregation.Firstly we were interested in better understanding the interactions between the membranes and proteins and the mechanisms of fouling. Secondly the modeling of the membrane process was developed in order to describe and predict the performance of the filtration system. The proposed model could then be validated by new filtration experiments.BSA (Bovine Serum Albumin) solutions in Milli-Q water was used as a model of proteins in the project. The commercial PES UF/MF membranes were used. The membrane nominal cut-offs obtained from manufacturer were 0.1 and 0.01 µm. Ultrafiltration experiments performed on a plate module (Ray flow X100, Orelis-Novasep) allowing the use of 100 cm² flat membrane. Fouling with 2 L of the BSa solution was performed over night at ambient temperature at constant TMP of 1.5 bar. Adsorption isotherms in batch mode were carried out to characterize the interactions. The mechanisms (pore blocking, cake formation, concentration polarization) occurring in the fouling membranes were identified with filtration experiments. The membrane properties (permeability, porosity, pore diameter, thickness of the different parts of the membrane) were analyzed by means of experiments (filtration, SEM, AFM etc).The method of volume averaging is used for the modeling approach. It is one of the techniques for modeling of the porous media. It provides a rigorous foundation for the analysis of a porous system. The development is based on classical continuum physics, and it provides both the spatially smoothed equations and a method to predict the effective transport coefficients that appear in those equations. In this project, local equations in the porous medium (membrane) were identified and described at the local scale thanks to experimental results of measurements of interactions and identification of mechanisms. The model of membrane fouling is based on diffusion of species and adsorption at the fluid-solid interface. The method of volume averaging was then applied for upscaling the local equations and obtaining the macroscopic equations with effective coefficients.

International audience; The study of gas slip-flow in porous media is relevant in many applications ranging from nanotechnology to enhanced oil recovery and in any situation involving low-pressure gas-transport through structures having sufficiently small pores. In this paper, we use the method of volume averaging for deriving effective-medium equations in the framework of a slightly compressible gas flow. The result of the upscaling process is an effective-medium model subjected to time- and length-scale constraints, which are clearly identified in our derivation. At the first order in theKnudsen number, the macroscopic momentum transport equation corresponds to a Darcy-like model involving the classical intrinsic permeability tensor and a slip-flow correction tensor that is also intrinsic. It generalizes the Darcy-Klinkenberg equation for ideal gas flow, and exhibits a more complex form for dense gas. The component values of the two intrinsic tensors were computed by solving the associated closure problems on two- and three-dimensional periodic unit cells. Furthermore, the dependence of the slip-flow correction with the porosity was also verified to agree with approximate analytical results. Our predictions show a power-law relationship between the permeability and the slip-flow correction that is consistent with other works. Nevertheless, the generalization of such a relationship to any configuration requires more analysis.; The study of gas slip-flow in porous media is relevant in many applications ranging from nanotechnology to enhanced oil recovery and in any situation involving low-pressure gas-transport through structures having sufficiently small pores. In this paper, we use the method of volume averaging for deriving effective-medium equations in the framework of a slightly compressible gas flow. The result of the upscaling process is an effective-medium model subjected to time- and length-scale constraints, which are clearly identified in our derivation. At the first order in the Knudsen number, the macroscopic momentum transport equation corresponds to a Darcy-like model involving the classical intrinsic permeability tensor and a slip-flow correction tensor that is also intrinsic. It generalizes the Darcy-Klinkenberg equation for ideal gas flow, and exhibits a more complex form for dense gas. The component values of the two intrinsic tensors were computed by solving the associated closure problems on two- and three-dimensional periodic unit cells. Furthermore, the dependence of the slip-flow correction with the porosity was also verified to agree with approximate analytical results. Our predictions show a power-law relationship between the permeability and the slip-flow correction that is consistent with other works. Nevertheless, the generalization of such a relationship to any configuration requires more analysis.