On the thermal anisotropy affecting transfers in a multilayer porous mediumSur l'effet de l'anisotropie thermique sur les transferts dans un milieu poreux multicouches

An analytical model for studying double-diffusive natural convection within a multilayer anisotropic porous medium is developed and validated with respect to a direct numerical silmulation. The studied domain is composed of two horizontal porous layers where the lower one is thermally anisotropic and is submitted to a uniform horizontal heat flux and a vertical mass flux. The assumption of parallel flow is validated and the effect of anisotropy on dynamic transitions is investigated. To cite this article: R. Bennacer et al., C. R. Mecanique 332 (2004).     

Convection in anisotropic inclined porous layers

Two-dimensional convective flow in a system of fluid-saturated porous layers is considered for small fluid velocities. The layers are inclined at an angle to the horizontal and there is a temperature gradient across them. A mathematical model of the system is constructed and the resulting series solution is numerically summed. The unique features of this model include its ability to represent non-symmetric multi-layered systems and materials whose permeabilities are anisotropic.The model is applied to various physical configurations. System aspect ratios, system symmetry and permeablity anisotropies are found to influence the effect that layers of different materials have on the convective flow.     

Upscaling Forchheimer law

We investigate the high velocity flow in heterogeneous porous media. The model is obtained by upscaling the flow at the heterogeneity scale where the Forchheimer law is assumed to be valid. We use the method of multiple scale expansions, which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. We show that Forchheimer law does not generally survive upscaling. The macroscopic flow law is strongly non-linear and anisotropic. A 2-point Padé approximation of the flow law in the form of a Forchheimer law is given. However, this approximation is generally poor. These results are illustrated in two particular cases: a layered composite porous media and a composite constituted by a square array of circular porous inclusions embedded in a porous matrix. We show that non-linearities are sources of anisotropy.     

Buoyant flow in long vertical enclosures in the presence of a strong horizontal magnetic field. Part 2. Finite enclosures

Numerical computations and experiments were carried out for a buoyant flow of liquid metal (mercury in the experiments) in a long vertical enclosure of square cross-section, in the presence of a uniform horizontal magnetic field. A strong emphasis is put on the case of a magnetic field perpendicular to the applied temperature gradient for two reasons: (1) the MHD damping is smaller than with any other orientation, and (2) the quasi-two-dimensionality of the flow in this case yields a quite efficient velocity measurement technique. The enclosure is heated by a thermally controlled flow of water from one of the vertical walls and cooled by a similar technique from the facing wall. Those two walls are good thermal conductors (thick copper plates in the experiments), whereas the four other walls are thermally insulating. All walls are electrically insulated from the fluid. In this paper, as well as in the companion paper by Tagawa et al. (Eur. J. Mech. B Fluids 21 (4) (2002) 383–398), we model analytically the Hartmann layers present along the walls perpendicular to the magnetic field. This modeling, which yields boundary conditions for the core flow without any meshing of the thin layers, is quite accurate when Hartmann layers are stable. The numerical results are in fairly good agreement with the experimental data. They namely reveal how the heat flux and the fluid flow organization depend on the magnetic field.     

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

This article presents a new nonlinear finite‐volume scheme for the nonisothermal two‐phase two‐component flow equations in porous media. The face fluxes are approximated by a nonlinear two‐point flux approximation, where transmissibilities nonlinearly depend on primary variables. Thereby, we mainly follow the ideas proposed by Le Potier combined with a harmonic averaging point interpolation strategy for the approximation of arbitrary heterogeneous permeability fields on polygonal grids. The behavior of this interpolation strategy is analyzed, and its limitation for highly anisotropic permeability tensors is demonstrated. Moreover, the condition numbers of occurring matrices are compared with linear finite‐volume schemes. Additionally, the convergence behavior of iterative solvers is investigated. Finally, it is shown that the nonlinear scheme is more efficient than its linear counterpart. Copyright © 2016 John Wiley & Sons, Ltd.     

Steady seepage flow under aprons in anisotropic ground

Seepage under a flat apron in a two-layer ground consisting of homogeneous isotropic layers of equal thickness was investigated by means of the analytic theory of linear differential equations by Polubarinova-Kochina [1]. An approximate solution of the same problem for layers of different thickness by the methods of the variational calculus was given in [2]. The solution to such a problem for homogeneous anisotropic layers of different thickness by means of a special complex variable representation and linear conjugation was given in [3] by Prusov and the author of the present paper. The method of [3] is now used to investigate two-dimensional steady seepage flow under two aprons in a multilayer ground consisting of n homogeneous anisotropic layers of different thickness, the principal directions of anistropy of each layer having an arbitrary position relative to the horizontal.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 155–158, October–December, 1981.     

Buoyant plume above a line source of heat in an anisotropic porous medium

Buoyancy-induced convection arising from a horizontal line heat source embedded in an anisotropic porous medium is investigated analytically. The porous medium is anisotropic is permeability with its principal axes oriented in a direction that is oblique to the gravity vector. Assuming the boundary layer approximation, closed-form exact similarity solutions for both flow and temperature fields are presented and compared with those of isotropic case. Scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime for which the conditions of validity are obtained. Effects of both anisotropic parameters (K* and %) and Rayleigh number Ra_L are observed to be strongly significant. It is demonstrated that a minimum (maximum) intensity of the thermal convective plume above the line source of heat can be obtained if the porous matrix is oriented with its principal axis with higher permeability parallel (perpendicular) to the vertical direction.     

Solutions for axial and transverse boundary layers in the case of steady laminar flow past a horizontal plate embedded in a saturated porous medium

The present paper is concerned with the interesting situation of the existence of transverse, velocity and thermal, boundary layers in the case of flow past a horizontal plate embedded in a saturated porous medium. The analysis shows that these layers are thicker than the axial boundary layers. However, the increase in the porous parameterB reduces the thickness. The expressions for the boundary layer thickness, the shear-stress at the wall and the specific heat flux are obtained and the behaviour of the above quantities whenB takes different values are discussed in detail.     

Vibration effect on convection onset in a system consisting of a horizontal pure liquid layer and a layer of liquid-saturated porous medium

The onset of convection in a system of two horizontal layers (a pure liquid and a porous medium saturated with the same liquid) heated from below under the action of vertical vibration is investigated. For describing the free thermal convection, in the liquid layer the Boussinesq approximation and in the porous layer the Darcy-Boussinesq approximation are used. In the limiting case of a thin liquid layer, effective boundary conditions on the upper boundary of the porous layer with account for convection in the liquid layer are obtained and it is shown that vibration has a stabilizing effect, whereas the presence of a liquid layer leads to destabilization. For an arbitrary liquid to porous layer thickness ratio the onset of convection is investigated numerically. In the case of a thin liquid layer there are two (short-and long-wave) unstable modes. In the case of thick layers the neutral curves are unimodal. Vibration has a stabilizing effect on perturbations with any wave number but affects short-wave perturbations much more strongly than long-wave ones.     

Radiative Effect on Natural Convection Flows in Porous Media

A regular two-parameter perturbation analysis based upon the boundary layer approximation is presented here to study the radiative effects of both first- and second-order resistances due to a solid matrix on the natural convection flows in porous media. Four different flows have been studied, those adjacent to an isothermal surface, a uniform heat flux surface, a plane plume and the flow generated from a horizontal line energy source on a vertical adiabatic surface. The first-order perturbation quantities are presented for all these flows. Numerical results for the four conditions with various radiation parameters are tabulated.     

层状复合材料的弹塑性模型

通过将宏微观应力及应变分成面内及面外两部分,利用理想界面上的面外应力与面内应变连续条件,应用微观应力及应变均匀化方法得到了两层复合材料的弹塑性分析模型,并在此基础上构造了多层复合材料的全量型三维显式本构模型.各向同性层组成的复合材料算例表明了该模型的可靠性,同时该模型完全适用于各向异性层组成的复合材料弹塑性分析.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Effects of Anisotropy and Drying Air Parameters on Drying of Deformable Porous Media Hydro-Dynamically and Thermally Anisotropic

The purpose of this work was to investigate numerically the drying of saturated deformable porous media. The considered sample is a rectangular porous plate which assumed to be both hydro-dynamically and thermally anisotropic, while the mechanical behavior of the sample is supposed to be isotropic. All walls of the plate are subjected to a convective heat flux. Moreover, the top and bottom walls are allowed the mass transfer. The Darcy–Brinkman extended model was used as the momentum balance equation for the liquid and solid phases. The energy balance equation is based on the local thermodynamic equilibrium assumption between the both phases. The lattice Boltzmann method is used to solve the governing differential equation system. A comprehensive analysis of the effect of anisotropy and the drying air parameters on macroscopic fields is investigated throughout this work.     

Mixed convection in the presence of horizontal impermeable surfaces in saturated porous media with variable permeability

Boundary layer approximation is applied for mixed convection about a horizontal flat plate in a saturated porous medium with aiding external flows. Similarity solutions are obtained, incorporating the variation of permeabilty, for 1) horizontal flat plate at zero angle of attack with constant heat flux; 2) stagnation point flows about horizontal flat plates with wall temperature varying asT _w ∝x ². The temperature and velocity profiles for different values of Ra/(RePr)^3/2 and the parameters governing the flow are obtained. The heat transfer rate is calculated and its implications in a geothermal application is discussed. Further, the criteria for pure mixed convection about horizontal flat plates in a porous media are established.     

Weak 2D Convective Plumes in a Sloping Permeable Layer

This is a study of thermal plumes in a permeable fluid-saturated slab of a porous medium (of finite uniform thickness but otherwise infinite extent) that is heated by either an instantaneous or a steady line source embedded in the medium. The slab, which may be horizontal or sloping, is initially at ambient conditions; the impervious upper surface remains at the ambient temperature, while the impervious base is thermally insulated. The transient temperature distribution, the surface heat flux and the convective flow velocity are calculated for small instantaneous line heat energy sources. They show how the flow develops, spreads and slows as time progresses, and an estimate of the time to decay is given. Steady-state temperature and velocity profiles are calculated for embedded line sources that provide heat energy at a small constant rate, and the surface heat flux distribution is calculated.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium

A local thermal non-equilibrium model has been considered for the case of thermally fully developed flow within a constant heat flux tube filled with a porous medium. Exact temperature profiles for the fluid and solid phases are found after combining the two individual energy equations and then transforming them into a single ordinary differential equation with respect to the temperature difference between the solid phase and the wall subject to constant heat flux. The exact solutions for the case of metal-foam and air combination reveal that the local thermal equilibrium assumption may fail for the case of constant heat flux wall. The Nusselt number is presented as a function of the Peclet number, which shows a significant increase due to both high stagnant thermal conductivity and thermal dispersion resulting from the presence of the metal-foam.     

Influence of anisotropy of the permeability on the convection and heat transfer in a porous annular layer

Investigations into the convective transport of heat in porous materials are of interest for many applications in connection with the problem of increasing the efficiency of thermal insulation. In [1–5], convection in Isotropic porous media was considered. However, in many cases porous materials have an essential anisotropy of their permeability. Convective heat transfer has been inadequately studied for this case. In [6], the linearized equations were used to study the convection between infinite horizontal planes with a filling of an anisotropic material; the value of the critical Rayleigh number was found, and this agreed satisfactorily with experimental data. In the present paper, we investigate numerically convection between two infinite coaxial cylinders with an anisotropic porous filling, using the equations of convection in the Darcy—Boussinesq approximation [1–3]. The permeability tensor in the annular region is constructed from its principal values, which can be found experimentally. A method of calculation is developed and a parametric study made of the structure of the flow and of the local and averaged characteristics of the heat transfer, which are of interest for the design of thermal insulation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–64, January–February, 1980.     

Parallel folding in multilayered structures

A recent two-layer nonlinear model for parallel folding is extended to a full multilayer formulation for n identical layers. The general behaviour of the multilayer is characterized using a sinusoidal Galerkin approximation substituted into the linearized potential energy function and equations for the wavelength and critical load are found. Comparisons with experiments conducted on layers of paper constrained between two sheets of foam rubber show that the trend of the load-amplitude plots is in good agreement with that predicted. Excellent correlation is also achieved between the wavelengths calculated by the linearized buckling analysis and those seen experimentally.     

Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.