Free convection and radiation characteristics for a vertical plate embedded in a porous medium

Steady two‐dimensional free convection flow due to combined effect of radiation and convection through a porous medium bounded by a vertical infinite plate is considered. The behaviour of Darcy and non‐Darcy flow is investigated. The flow of water through different porous media under different environmental conditions is discussed. Effect of four non‐dimensional parameters, i.e. Prandtl number (Pr), modified Grashof number (G), permeability parameter (K) and radiation parameter (N) has been studied. Effect of these parameters on Nusselt number is analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Topology optimization of creeping fluid flows using a Darcy–Stokes finite element

A new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no‐slip condition along the solid–fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy's law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy–Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no‐slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid–fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy–Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0–1 (void–solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed finite element method for coupled thermo‐hydro‐mechanical process in poro‐elasto‐plastic media at large strains

A mixed finite element for coupled thermo‐hydro‐mechanical (THM) analysis in unsaturated porous media is proposed. Displacements, strains, the net stresses for the solid phase; pressures, pressure gradients, Darcy velocities for pore water and pore air phases; temperature, temperature gradients, the total heat flux are interpolated as independent variables. The weak form of the governing equations of coupled THM problems in porous media within the element is given on the basis of the Hu–Washizu three‐filed variational principle. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of the THM constitutive model for unsaturated porous media based on the CAP model. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elasto‐plastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is utilized. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization and the softening behaviours caused by thermal and chemical effects. Copyright © 2005 John Wiley & Sons, Ltd.     

Natural convection between coaxial vertical cylinders partially filled with a porous material

This paper presents laminar fully developed free convection flow between two coaxial vertical cylinders partially filled with a porous matrix when the cylinders are kept at different temperatures. The Brinkman-extended Darcy model is used to simulate momentum transfer in the porous region. The two regions are coupled by equating the velocity and by considering shear stress jump condition at the interface. The effect of Darcy number on the velocity throughout porous domain and adjustable coefficient in the stress jump condition at the interface is shown graphically. It is observed that velocity is influenced by the shear stress jump condition at the interface.     

Discontinuous finite volume methods for the stationary Stokes–Darcy problem

In this paper, we analyze discontinuous finite volume methods for the stationary Stokes–Darcy problem that models coupled fluid flow and porous media flow. The discontinuous finite volume methods are combinations of finite volume method and discontinuous Galerkin method with three interior penalty types (incomplete symmetric, nonsymmetric, and symmetric), briefly, using discontinuous functions as trial functions in the finite volume method. Optimal error estimates in broken H¹ norm are obtained for the three discontinuous finite volume methods. Optimal error estimates in the standard L² norm are derived for the symmetric interior penalty discontinuous finite volume method. Numerical experiments are presented to confirm the theoretical results with non‐matching meshes across the common interface of Stokes region and Darcy region. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonisothermal cross-flow around a cylinder with a square cross section and an impermeable core covered with a porous layer

Nonisothermal cross flow of a viscous incompressible fluid around a porous cylinder with a square cross section is considered. Main attention is paid when an impermeable core of the cylinder is surrounded with a porous layer. The full system of Navier–Stokes and energy equations is integrated numerically by the finite-volume method. The hydrodynamic interaction between the flow and the matrix of the porous layer is described by Darcy’s law. At moderate Reynolds numbers, the influence of the permeability of the porous layer on the nature of the flow and the heat exchange between the cylinder and the flow is studied. It is shown that, with increasing permeability, heat transfer from the cylinder increases mainly on its front side. From the analysis of the data obtained, an approximate formula for the mean Nusselt number as a function of the Reynolds and Darcy numbers is derived. The results of the calculation of hydrodynamic and thermal characteristics of the cross-flow around an impermeable and a fully permeable cylinder are also presented.     

On bounded approximations of periodicity for computational homogenization of Stokes flow in porous media

By separation of scales and the homogenization of a flow through porous media, a two‐scale problem arises where a Darcy‐type flow is present on the macroscale and a Stokes flow on the subscale. In this paper, the problem is given as the minimization of a potential. Additional constraints imposing periodicity in a weak sense are added using Lagrange multipliers. In particular, the upper and lower energy bounds for the corresponding strongly periodic problem are produced, quantifying the accuracy of the weakly periodic boundary conditions. A numerical example demonstrates the evaluation of energy bounds and the performance of weakly periodic boundary conditions on a representative volume element. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons Ltd     

Estimation of the dynamic permeability of an assembly of permeable spherical porous particles using the cell model

This paper is concerned with the theoretical estimation of the dynamic permeability of an assembly of permeable porous particles subject to an oscillatory Stokes flow. A cell model is used to approximate the hydrodynamic interactions of particles. The flow field inside the representative porous sphere is governed by Darcy’s law and that within the fluid region by the unsteady Stokes equations. Faxén’s laws for drag and torque exerted on the representative particle are derived, and the results are compared in special cases with the existing literature. The dynamic permeability of the bed of particles is obtained and the variation in dynamic permeability with various parameters such as porosity, Darcy number and, frequency is analyzed.     

Two‐scale computational modelling of water flow in unsaturated soils containing irregular‐shaped inclusions

The focus of this paper is two‐dimensional computational modelling of water flow in unsaturated soils consisting of weakly conductive disconnected inclusions embedded in a highly conductive connected matrix. When the inclusions are small, a two‐scale Richards’ equation‐based model has been proposed in the literature taking the form of an equation with effective parameters governing the macroscopic flow coupled with a microscopic equation, defined at each point in the macroscopic domain, governing the flow in the inclusions. This paper is devoted to a number of advances in the numerical implementation of this model. Namely, by treating the micro‐scale as a two‐dimensional problem, our solution approach based on a control volume finite element method can be applied to irregular inclusion geometries, and, if necessary, modified to account for additional phenomena (e.g. imposing the macroscopic gradient on the micro‐scale via a linear approximation of the macroscopic variable along the microscopic boundary). This is achieved with the help of an exponential integrator for advancing the solution in time. This time integration method completely avoids generation of the Jacobian matrix of the system and hence eases the computation when solving the two‐scale model in a completely coupled manner. Numerical simulations are presented for a two‐dimensional infiltration problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Mixed finite element method for saturated poroelastoplastic media at large strains

A mixed finite element for hydro‐dynamic analysis in saturated porous media in the frame of the Biot theory is proposed. Displacements, effective stresses, strains for the solid phase and pressure, pressure gradients, and Darcy velocities for the fluid phase are interpolated as independent variables. The weak form of the governing equations of coupled hydro‐dynamic problems in saturated porous media within the element are given on the basis of the Hu–Washizu three‐field variational principle. In light of the stabilized one point quadrature super‐convergent element developed in solid continuum, the interpolation approximation modes for the primary unknowns and their spatial derivatives of the solid and the fluid phases within the element are assumed independently. The proposed mixed finite element formulation is derived. The non‐linear version of the element formulation is further derived with particular consideration of pressure‐dependent non‐associated plasticity. The return mapping algorithm for the integration of the rate constitutive equation, the consistent elastoplastic tangent modulus matrix and the element tangent stiffness matrix are developed. For geometrical non‐linearity, the co‐rotational formulation approach is used. Numerical results demonstrate the capability and the performance of the proposed element in modelling progressive failure characterized by strain localization due to strain softening in poroelastoplastic media subjected to dynamic loading at large strain. Copyright © 2003 John Wiley & Sons, Ltd.     

Quasi-static deformation of a multilayered poroelastic half-space by two-dimensional buried sources

Summary The fully coupled Biot quasi-static theory of fluid-infiltrated porous materials is used to study the two-dimensional plane strain deformation of a multilayered poroelastic half-space by internal sources. Pure compliance formulation, in which the stresses and the pore pressure are taken as the basic state variables, is used. Displacements are then obtained by integrating the coupled constitutive relations and the fluid flux from the Darcy law. The problem is formulated in terms of the propagator matrices. Simplified explicit expressions for the elements of the 6 × 6 propagator matrix are obtained. The propagator matrix is also used to obtain explicit expressions for the surface displacements and fluid flow due to a line force and a fluid injection source buried in a poroelastic half-space.     

Transient Couette flow in a rotating non-Darcian porous medium parallel plate configuration: network simulation method solutions

The transient, viscous, incompressible, hydrodynamic Couette flow in a rotating porous medium channel is studied in this paper. The channel comprises a pair of infinitely long parallel plates which rotate with uniform angular velocity about an axis normal to the plates. The porous medium is simulated using a Darcy–Forchheimer drag force model which includes both bulk matrix porous drag (dominant at low Reynolds numbers) and second order inertial impedance (dominant at higher Reynolds numbers). The two-dimensional Navier–Stokes equations are reduced to a (z*, t*) coordinate system incorporating Coriolis terms, and appropriate initial and boundary conditions are prescribed. Separate porous drag body force terms are incorporated in both the primary and secondary flow momentum equations. Using a set of transformations, the model is rendered dimensionless and shown to be dictated by the Ekman number, Forchheimer number, Darcy number and Reynolds number in a (z, t) coordinate system. Numerical solutions are obtained for the transformed model using the Network Simulation Method. The influence of the hydrodynamic parameters are computed graphically and also the interaction of parameters on the velocity fields is discussed at length. Excellent agreement is found with earlier non-porous flow studies. The analysis has important applications in geophysics and also chemical engineering systems.     

泡沫型多孔介质中非达西流动特性的研究

本文通过将泡沫材料复杂结构进行几何简化，形成立方框架结构，利用简化的等效单元流道，提出一种数理模型，它既考虑固体网架表面对流体的摩擦阻力作用，也考虑网架迎着流向的迎风阻力（形状阻力），得到了在多孔介质中非达西流的压力降与流速的关系式。并用简化的框架结构导出了预估泡沫材料渗透率ｋ和Ｆ－Ｗ关系式中流速平方项的系数Ｆ。另外，还通过专用的实验装置测定了四种不同孔径泡沫陶瓷的ｋ和Ｆ。结果表明，根据所给模型理论预测的结果与实验值一致性较好，在泡沫型多孔材料的应用中能够提供流动特性的相关数据。     

再生骨料影响混凝土渗透性的数值模拟

出于环境保护及经济方面的考虑,再生骨料在混凝土生产中得到越来越多的应用。混凝土可认为是由天然或再生骨料、多孔基体和界面过渡区三相组成的复合材料。基于随机骨料结构生成算法,建立了三相混凝土数值模型并应用于气体渗透性的分析。对于混凝土二维数值模型的稳态渗透问题,应用有限单元法求解压力场,并基于压力-流量的宏观等效关系计算混凝土材料的总体渗透率。数值分析结果表明:1) 总体渗透率随过渡区厚度及其相对渗透率的增加而非线性地增长;2) 由于骨料的存在而导致的稀释效应及曲折度效应是影响混凝土总体渗透性的两个重要因素;3) 再生骨料渗透率对混凝土宏观渗透率的影响呈“S”形曲线的关系。出于控制回收骨料混凝土渗透性能的考虑,回收骨料的渗透率最好比砂浆基体的渗透率低一个数量级,同时回收骨料的掺量也需要合理选择确定。     

A numerical contact algorithm in saturated porous media with the extended finite element method

In this paper, a coupled hydro-mechanical formulation is developed for deformable porous media subjected to crack interfaces in the framework of extended finite element method. Governing equations of the porous medium consist of the momentum balance of the bulk together with the momentum balance and continuity equations of the fluid phase, known as formulation. The discontinuity in fractured porous medium is modeled for both opening and closing modes that results in the fluid flow within the fracture, and/or contact behavior at the crack edges. The fluid flow through the fracture is assumed to be viscous and is modeled by employing the Darcy law in which the permeability of fracture is obtained using the cubic law. The contact condition in fractured porous medium is handled by taking the advantage from two different algorithms of LATIN method and penalty algorithm. The effect of contact on fluid phase is employed by considering no leak-off from/into the porous medium. The nonlinearity of coupled equations produced due to opening and closing modes is carried out using an iterative algorithm in the Newton–Raphson procedure. Finally, several numerical examples are solved to illustrate the performance of proposed X-FEM method for hydro-mechanical behavior of fractured porous media with opening and closing modes.     

A numerical study of filling process through multilayer preforms in resin injection/compression molding

In resin injection/compression molding (RI/CM), a preform often comprises layers of different fiber reinforcements. Each fiber reinforcement has unique through thickness and in-plane permeabilities as well as compressibility, creating a heterogeneous porous medium in the mold cavity. In the present article, numerical simulation is utilized to investigate the filling process of RI/CM in such a heterogeneous porous medium. The filling stage is simulated in a full three-dimensional space by using control volume/finite element method and based upon an appropriate filling algorithm. The flow in the open gap which may be present in the mold cavity is modeled by Darcy’s law using an equivalent permeability. Numerical simulations of filling process for preforms containing two and three layers of different reinforcements in various stacking sequences are conducted with the aid of computer code developed in this study. Results show that the injection time as well as flow front progression depends on fiber types in the whole preform, fiber stacking sequence and open gap provided in the mold cavity. Simulated results also suggest that the presence of open gap at top of reinforcement can lead to both low injection time and uniform flow pattern.     

Mixed finite elements and Newton‐type linearizations for the solution of Richards' equation

We present the development of a two‐dimensional Mixed‐Hybrid Finite Element (MHFE) model for the solution of the non‐linear equation of variably saturated flow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest‐order Raviart–Thomas (RT₀) elements and is ‘exactly’ mass conserving. Hybridization is used to overcome the ill‐conditioning of the mixed system. The scheme is globally first‐order in space. Nevertheless, numerical results employing non‐uniform meshes show second‐order accuracy of the pressure head and normal fluxes on specific grid points. The non‐linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE‐Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation techniques. Copyright © 1999 John Wiley & Sons, Ltd.     

Higher‐resolution convection schemes for flow in porous media on highly distorted unstructured grids

Higher‐resolution schemes are presented for convective flow approximation on highly distorted unstructured grids. The schemes are coupled with continuous full‐tensor Darcy‐flux approximations. A sequence of non‐uniform and distorted grid formulations are developed and compared for a range of unstructured meshes with variable grid spacing. The higher‐order schemes are constructed using non‐uniform grid slope limiters such that they are stable with a local maximum principle, ensuring that solutions are free of spurious oscillations. Benefits of the resulting schemes are demonstrated for classical test problems in reservoir simulation including cases with full‐tensor permeability fields. The test cases involve a range of unstructured grids with variations in grid spacing, orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher‐order formulations are shown to effectively reduce numerical diffusion, leading to improved resolution of concentration and saturation fronts. Copyright © 2008 John Wiley & Sons, Ltd.     

Simulation of 3D flow in porous media by boundary element method

In the present paper problem of natural convection in a cubic porous cavity is studied numerically, using an algorithm based on a combination of single domain and subdomain boundary element method (BEM). The modified Navier–Stokes equations (Brinkman-extended Darcy formulation with inertial term included) were adopted to model fluid flow in porous media, coupled with the energy equation using the Boussinesq approximation. The governing equations are transformed by the velocity–vorticity variables formulation which separates the computation scheme into kinematic and kinetic parts. The kinematics equation, vorticity transport equation and energy equation are solved by the subdomain BEM, while the boundary vorticity values, needed as a boundary conditions for the vorticity transport equation, are calculated by single domain BEM solution of the kinematics equation. Computations are performed for steady state cases, for a range of Darcy numbers from 10^?6 to 10^?1, and porous Rayleigh numbers ranging from 50 to 1000. The heat flux through the cavity and the flow fields are analyzed for different cases of governing parameters and compared to the results in some published studies.