Benoît Goyeau

Numerical study of double-diffusive natural convection in a porous cavity using the Darcy-Brinkman formulation

Abstract 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.

Average momentum equation for interdendritic flow in a solidifying columnar mushy zone

Abstract 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.

Stability of Natural Convection in Superposed Fluid and Porous Layers Using Integral Transforms

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.

Direct numerical simulation of turbulent heat transfer in a fluid-porous domain

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...

Natural convection in porous media—dual reciprocity boundary element method solution of the Darcy model

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

Dual reciprocity boundary element method solution of natural convection in Darcy -Brinkman porous media

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.

Stability of natural convection in superposed fluid and porous layers: Influence of the interfacial jump boundary condition

Macroscopic modeling of momentum transport at a fluid-porous interface has been improved by the derivation of a stress jump boundary condition related to the spatial variations of the effective properties of the porous medium at the interfacial region. This Communication concerns the influence of this jump condition on the onset of thermal natural convection in fluid/porous stratified layers. A linear stability analysis shows that, for small depth ratio, the effective jump coefficient strongly influences the bimodal marginal stability curves. At large wave numbers, the “fluid mode” (the convective flow is confined in the fluid layer) is found to be more unstable while the porous mode, corresponding to small wave numbers, remains unchanged. The influence of the thickness and the dimensionless permeability of the porous layer is also presented.

Coupled Upscaling Approaches For Conduction, Convection, and Radiation in Porous Media: Theoretical Developments

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.

Natural convection in partially porous media: a brief overview

Purpose – This paper aims to provide a limited, but selective bibliography on modelling heat and mass transfer in composite fluid‐porous domains.Design/methodology/approach – Since the pioneer study by Beavers and Joseph, the problem of interface continuity and/or jump conditions at a fluid‐porous interface has been of interest to the fluid mechanics and heat and mass transfer community. The paper is concerned both with numerical simulations of heat and fluid flow in such systems, and with the linear stability problems.Findings – The one‐ and two‐domain formulations are equivalent. Using the Darcy‐Brinkman extension instead of the Darcy model reduces the number of ad hoc parameters in this configuration.Research limitations/implications – The problem of double diffusive convection has still to be solved and analyzed.Practical implications – The discussion on the interface conditions is of great relevance to many industrial and practical situations.Originality/value – The important question of the macrosco...

A numerical simulation of columnar solidification: influence of inertia on channel segregation

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.