Mechanics of porous elastic materials containing multiphase fluid

A continuum theory of mixtures for a porous elastic solid saturated by immiscible viscous fluids is presented. The theory includes micro-inertial effects for the local fluctuation in volume fractions of the solid and fluid constituents. Gradients of volume fraction of both the elastic solid and fluid constituents are included in the constitutive variables. Equations governing the macroscopic motion are developed and show that the present theory contains both Biot's equations and multiphase Darcy flow through porous media as special cases.     

A stabilized volume‐averaging finite element method for flow in porous media and binary alloy solidification processes

A stabilized equal‐order velocity–pressure finite element algorithm is presented for the analysis of flow in porous media and in the solidification of binary alloys. The adopted governing macroscopic conservation equations of momentum, energy and species transport are derived from their microscopic counterparts using the volume‐averaging method. The analysis is performed in a single domain with a fixed numerical grid. The fluid flow scheme developed includes SUPG (streamline‐upwind/Petrov–Galerkin), PSPG (pressure stabilizing/Petrov–Galerkin) and DSPG (Darcy stabilizing/Petrov–Galerkin) stabilization terms in a variable porosity medium. For the energy and species equations a classical SUPG‐based finite element method is employed. The developed algorithms were tested extensively with bilinear elements and were shown to perform stably and with nearly quadratic convergence in high Rayleigh number flows in varying porosity media. Examples are shown in natural and double diffusive convection in porous media and in the directional solidification of a binary‐alloy. Copyright © 2004 John Wiley & Sons, Ltd.     

Modeling of ammonothermal growth of gallium nitride single crystals

Ammonothermal growth of GaN crystals with a retrograde solubility has been modeled and simulated here using fluid dynamics, thermodynamics and heat transfer models. The nutrient is considered as a porous media bed and the flow in the porous charge is simulated using the Darcy-Brinkman-Forchheimer model. The resulting governing equations are solved using the finite volume method. For the case of retrograde solubility, the charge is put above the baffle. The temperature difference between the dissolving zone and growth zone is found smaller than that applied on the sidewall of autoclave. The baffle opening has a strong effect on the nutrient transport and supersaturation of GaN species in the growth zone.     

Analysis of the vacuum infusion moulding process: I. Analytical formulation

The present work is primarily concerned with the analytical formulation of governing equations for flow of incompressible fluids through compacting porous media and their application to vacuum infusion (VI) of composite materials. The literature on VI and the effects of compacting media on permeability and flow is reviewed. A complete development of the proposed governing equation is shown along with a suggested numerical solution. The proposed model is subsequently used to quantify the effect of process parameters such as inlet and outlet pressures, fibre architecture and lay-up. Implications for industrial production are discussed.     

An efficient solution method for incompressible N‐S equations using non‐orthogonal collocated grid

An efficient strategy for the solution of N‐S Equations using collocated, non‐orthogonal grids is presented. The governing equations have been discretized in the physical plane itself without co‐ordinate transformation, thereby retaining the lucidity of the basic finite volume method. The non‐orthogonal terms and QUICK type corrections for the convective terms in the momentum equations are treated explicitly, while the other terms are taken in implicit form. In the pressure correction equation, the non‐orthogonal terms have been dropped altogether. The discretized equations have been solved by the preconditioned conjugate gradient square method. The specific combination of above steps has resulted in better convergence properties as compared to those of existing algorithms, even for highly skewed grids. The scheme has been validated against benchmark solutions such as lid‐driven flow in square and skewed cavities and experi mental results of flow over a single cylinder. Its applicability has also been illustrated for flow through a bank of staggered cylinders, with anti‐symmetric inlet and outlet boundary conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

Accelerating iterative solution methods using reduced‐order models as solution predictors

We propose the use of reduced‐order models to accelerate the solution of systems of equations using iterative solvers in time stepping schemes for large‐scale numerical simulation. The acceleration is achieved by determining an improved initial guess for the iterative process based on information in the solution vectors from previous time steps. The algorithm basically consists of two projection steps: (1) projecting the governing equations onto a subspace spanned by a low number of global empirical basis functions extracted from previous time step solutions, and (2) solving the governing equations in this reduced space and projecting the solution back on the original, high dimensional one. We applied the algorithm to numerical models for simulation of two‐phase flow through heterogeneous porous media. In particular we considered implicit‐pressure explicit‐saturation (IMPES) schemes and investigated the scope to accelerate the iterative solution of the pressure equation, which is by far the most time‐consuming part of any IMPES scheme. We achieved a substantial reduction in the number of iterations and an associated acceleration of the solution. Our largest test problem involved 93 500 variables, in which case we obtained a maximum reduction in computing time of 67%. The method is particularly attractive for problems with time‐varying parameters or source terms. Copyright © 2006 John Wiley & Sons, Ltd.     

A scalable and implicit meshless RBF method for the 3D unsteady nonlinear Richards equation with single and multi‐zone domains

The Local Hermitian Interpolation (LHI) method is a strong‐form meshless numerical technique which uses Radial Basis Function (RBF) interpolants to satisfy linear and nonlinear governing equations and boundary operators. Recent developments have shown that, for linear transport problems, applying the PDE governing equation directly to the basis functions can greatly improve the accuracy and stability of the resulting solutions. In this work, the LHI formulation with local PDE‐interpolation is extended to the nonlinear gravity‐driven Richards equation, in order to solve unsteady problems involving flow in unsaturated porous media. The application of the linearised PDE‐operator to the basis functions incorporates information, such as the effective velocity field, directly into the solution construction. This results in a form of ‘analytical upwinding’ which helps to stabilise the solution. In addition, the local interpolation itself satisfies the linearised governing equation, allowing for more accurate reconstruction of partial derivatives, and hence a more accurate solution. The procedure is tested using a 3D infiltration problem with a known analytical solution. The performance of the LHI method, both with and without local PDE interpolation, is compared to the Finite Element method via the FEMWATER software. The LHI formulation with PDE data centres shows consistent improvement over the FEMWATER solutions, reducing errors by several orders of magnitude. In addition, a procedure is introduced to model strongly heterogeneous and layered soils. The physically correct matching conditions are applied over layer interfaces, i.e. continuity of pressure and mass flux. The ‘double collocation’ property of the Hermitian RBF method is exploited to enforce both matching conditions at the same set of locations on the layer interface. This procedure allows the accurate capture of solutions across such interfaces, replicating the required discontinuities in the first derivatives of the pressure profile. The multi‐layer formulation is validated using a transient two‐layer infiltration problem, with the analytical solution replicated to a high precision in a variety of configurations. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis of Bénard convection in rectangular cavities filled with a porous medium

Summary Bénard convection in fluid-saturated porous cavities is studied in this research. The unsteady Navier-Stokes equations and energy equation describing the transient heat and fluid flow are expressed in primitive variables. A semi-implicit splitting finite element method is adopted to solve the coupled governing equations. The phenomena are discussed for a series of Rayleigh numbers, aspect ratios and different porous media. The results show that the strength of Bénard convection and heat-transfer rate become weaker due to the existence of more flow restriction in the porous media. Furthermore, the progress of flow evolution is also retarded compared with the non-porous medium. In some conditions, the heat transfer phenomenon is obviously altered because of the dramatic change in the flow pattern for different porous media.     

Investigation of turbulent heat transfer and nanofluid flow in a double pipe heat exchanger

In present study, heat transfer and turbulent flow of water/alumina nanofluid in a parallel as well as counter flow double pipe heat exchanger have been investigated. The governing equations have been solved using an in-house FORTRAN code, based on finite volume method. Single-phase and standard k-ε models have been used for nanofluid and turbulent modeling, respectively. The internal fluid has been considered as hot fluid (nanofluid) and the external fluid, cold fluid (base fluid). The effects of nanoparticles volume fraction, flow direction and Reynolds number on base fluid, nanofluid and wall temperatures, thermal efficiency, Nusselt number and convection heat transfer coefficient have been studied. The results indicated that increasing the nanoparticles volume fraction or Reynolds number causes enhancement of Nusselt number and convection heat transfer coefficient. Maximum rate of average Nusselt number and thermal efficiency enhancement are 32.7% and 30%, respectively. Also, by nanoparticles volume fraction increment, the outlet temperature of fluid and wall temperature increase. Study the minimum temperature in the solid wall of heat exchangers, it can be observed that the minimum temperature in counter flow has significantly reduced, compared to parallel flow. However, by increasing Reynolds number, the slope of thermal efficiency enhancement of heat exchanger gradually tends to a constant amount. This behavior is more obvious in parallel flow heat exchangers. Therefore, using of counter flow heat exchangers is recommended in higher Reynolds numbers.     

An effective stress based numerical model for hydro-mechanical analysis in unsaturated porous media

 A fully coupled flow-deformation model is presented for the behaviour of unsaturated porous media. The governing equations are derived based on the equations of equilibrium, effective stress concept, Darcy's law, Henry's law, and the conservation of fluid mass. Macroscopic coupling between the flow and deformation fields is established through the effective stress parameters. The microscopic link between the volumetric deformations of the two pore system (i.e. the pore-air and the pore-water) is established using Betti's reciprocal theorem. Both links are essential for a proper modelling of flow and deformation in unsaturated porous media. The discretised form of the governing equations is obtained using the finite element technique. As application of the model, experimental results from several laboratory tests reported in the literature are modelled numerically. Good agreement is obtained between the numerical and the experimental results in all cases.     

Topology optimization for unsteady flow

A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier–Stokes equations. A Brinkman penalization is used to enforce zero‐velocities in solid material. The resulting parameter optimization problem is solved by a non‐linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design‐dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non‐physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady‐state counterparts. Copyright © 2011 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.     

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.     

An artificial compressibility algorithm for modelling natural convection in saturated packed pebble beds: A heterogeneous approach

This work is concerned with the modelling of heat and fluid flow through saturated packed pebble beds. A volume‐averaged set of local thermal disequilibrium governing equations is employed to describe the latter as a heterogeneous porous medium with porosity varying from 0.39 to 0.99. The thermal disequilibrium approach, together with stated porosity upper limit, allows for the modelling of wall effects such as wall channeling and wall–bed radiative heat transfer. The resulting set of coupled non‐linear partial differential equations is solved via a locally preconditioned artificial compressibility method, where spatial discretization is effected with a compact finite volume edge‐based discretization scheme. The latter was done in the interest of accuracy. Stabilization is effected via JST scalar‐valued artificial dissipation. This is the first instance in which an artificial compressibility algorithm is applied to modelling heat and fluid flow through heterogeneous porous materials. For this reason, special attention was given to the calculation of acoustic velocities, stabilization scaling factors, as well as allowable time‐step sizes. The developed technology is validated by application to the modelling of a number of benchmark test cases. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulation of thermogravitational energy transport of a hybrid nanoliquid within a porous triangular chamber using the two-phase mixture approach

This study is a computational investigation of transient thermogravitational energy transport in H₂O/Al₂O₃ nanoliquid and water-based copper/aluminum oxide hybrid nanofluid (water/Al₂O₃-Cu) inside a horizontal isosceles triangular enclosure with porous medium. The governing equations for two-phase mixture flow have been derived by the use of Darcy-Brinkman model for porous media without the Forchheimer term (inertia loss). The control equations have been discretized using the finite volume technique. The effects of porosity factor, Rayleigh number, and Darcy number on the liquid motion and transient energy transport have been studied. The results have shown that convective thermal transmission in the nanofluid inside the triangular cavity generally consists of three phases: initial, transient, and quasi-steady, all of which are described in detail. It has been found that a rise of the porosity factor, Rayleigh number, or Darcy number always leads to an increment of the average Nusselt number and energy transport intensity. It has been also observed that with a rise of the Darcy number and strengthening of flow motion (convection), the instability in both flow and temperature fields increases and the distribution of isotherms and streamlines becomes completely asymmetric.     

Magnetohydrodynamic poiseuille flow through a porous medium

Abstract

This paper is concerned with formulating equations for the flow of an electrically conducting fluid through a non‐conducting porous medium with non‐porous and non‐conducting boundaries. Equations are developed for the general case of the flow of a solid‐fluid suspensions; flow through a porous medium is treated as a special case by letting the velocity of the particle phase goes to zero. Two cases are considered. Exact solution is obtained for the case of flow between parallel plates, but for flow in pipes of square and circular cross sections, the equations have to be solved numerically. The numerical technique developed can treat elliptical cross sections as well. The flow in all cases is assumed to be steady, laminar, incompressible, viscous, and fully developed. The results are presented in terms of a parameter which measures the resistance of the porous medium.     

Partitioned formulation and stability analysis of a fluid interacting with a saturated porous medium by localised Lagrange multipliers

In this work, a partitioned scheme for the numerical simulation of the surface‐coupled problem of a fluid interacting with a saturated porous medium (fluid‐porous‐media interaction) is proposed by adopting the method of localised Lagrange multipliers, which facilitates an automatic spatial partitioning of the problem and a parallel treatment of the interacting components, and allows for using tailored solvers optimised for each subproblem. Moreover, proceeding from the interaction between an incompressible bulk fluid with a saturated biphasic porous medium with intrinsically incompressible and inert constituents, the characteristics of the governing equations are scrutinised, and the various constraints within the subsystems are identified. Following this, the method of perturbed Lagrange multipliers is used to replace the constrained equation systems within each subdomain by unconstrained ones. Furthermore, considering the one‐dimensional (1D) version of the equations, a stability analysis of the proposed solution method is performed, and the unconditional stability of the partitioned solution scheme is shown. Solving 1D and 2D numerical benchmark examples, the applicability of the proposed scheme is demonstrated. Copyright © 2015 John Wiley & Sons, Ltd.     

Object‐oriented finite element analysis of thermo‐hydro‐mechanical (THM) problems in porous media

The design, implementation and application of a concept for object‐oriented in finite element analysis of multi‐field problems is presented in this paper. The basic idea of this concept is that the underlying governing equations of porous media mechanics can be classified into different types of partial differential equations (PDEs). In principle, similar types of PDEs for diverse physical problems differ only in material coefficients. Local element matrices and vectors arising from the finite element discretization of the PDEs are categorized into several types, regardless of which physical problem they belong to (i.e. fluid flow, mass and heat transport or deformation processes). Element (ELE) objects are introduced to carry out the local assembly of the algebraic equations. The object‐orientation includes a strict encapsulation of geometrical (GEO), topological (MSH), process‐related (FEM) data and methods of element objects. Geometric entities of an element such as nodes, edges, faces and neighbours are abstracted into corresponding geometric element objects (ELE–GEO). The relationships among these geometric entities form the topology of element meshes (ELE–MSH). Finite element objects (ELE–FEM) are presented for the local element calculations, in which each classification type of the matrices and vectors is computed by a unique function. These element functions are able to deal with different element types (lines, triangles, quadrilaterals, tetrahedra, prisms, hexahedra) by automatically choosing the related element interpolation functions. For each process of a multi‐field problem, only a single instance of the finite element object is required. The element objects provide a flexible coding environment for multi‐field problems with different element types. Here, the C++ implementations of the objects are given and described in detail. The efficiency of the new element objects is demonstrated by several test cases dealing with thermo‐hydro‐mechanical (THM) coupled problems for geotechnical applications. Copyright © 2006 John Wiley & Sons, Ltd.