On the Equivalence of the Discontinuous One- and Two-Domain Approaches for the Modeling of Transport Phenomena at a Fluid/Porous Interface

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.     

An Apparent Interface Location as a Tool to Solve the Porous Interface Flow Problem

A new approach for solving the laminar flow problem above a porous medium is presented here, using an apparent interface for which both superficial velocity and intrinsic shear stress are continuous. The derivation of this approach is based on a detailed investigation of the Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38:2635–2646, 1995a) jump condition and its sensitivity to the value of β (the jump condition coefficient) and to an error in the interface location. While the value of the jump condition coefficient is highly sensitive to the interface location, the new apparent interface approach does not require an a priori information about the location of the interface. This approach can be easily used knowing only one measurable parameter—the maximum velocity or the flow rate. Validation of the apparent interface approach against measurements from other works shows that it can be successfully used to predict the velocity profile for different geometrical models.     

A numerical investigation of droplet impact on a heated wall in the film boiling regime

In this research, a novel approach is proposed to simulate the impingement of a water droplet on a superheated wall by solving conservation equations of mass, momentum and energy. Due to high temperature of the surface, a vapor layer is formed between droplet and surface. The vapor layer is captured using a very fine mesh near the surface. The level set method is applied for interface tracking and appropriate jump conditions are imposed at the interface by the ghost fluid method. The proposed algorithm is validated by comparing numerical results to the available experimental and analytical solutions. The effect of impact velocity, surfactant and polymer additives on the heat removal is studied. Simulations show that an increase in impact velocity enhances the heat removal. On the other hand, polymer additive has no significant effect while surfactant enhances dissipated heat by increasing the contact time or even by sticking the droplet on the surface.     

Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium

An analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier–Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated.     

Slow Motion of a Porous Sphere Translating Along the Axis of a Circular Cylindrical Pore Subject to a Stress Jump Condition

The coupled flow problem of an incompressible axisymmetrical quasisteady motion of a porous sphere translating in a viscous fluid along the axis of a circular cylindrical pore is discussed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stress along with continuity of normal stress and velocity components are employed. The flow through the porous particle is governed by the Brinkman model and the flow in the outside porous region is governed by Stokes equations. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are satisfied first at the cylindrical pore wall by the Fourier transforms and then on the surface of the porous particle by a collocation method. The collocation solutions for the normalized hydrodynamic drag force exerted by the clear fluid on the porous particle is calculated with good convergence for various values of the ratio of radii of the porous sphere and pore, the stress jump coefficient, and a coefficient that is proportional to the permeability. The shape effect of the cylindrical pore on the axial translation of the porous sphere is compared with that of the particle in a spherical cavity; it found that the porous particle in a circular cylindrical pore in general attains a lower hydrodynamic drag than in a spherical envelope.     

Mathematical model and numerical method for spontaneous potential log in heterogeneous formations

This paper introduces a new spontaneous potential log model for the case in which formation resistivity is not piecewise constant. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on the interfaces. It has beer/ shown that the elliptic interface problem has a unique weak solution. Furthermore, a jump condition capturing finite difference scheme is proposed and applied to solve such elliptic problems. Numerical results show validity and effectiveness of the proposed method.     

A numerical method for forced convection in porous and homogeneous fluid domains coupled at interface by stress jump

A numerical method was developed for flows involving an interface between a homogeneous fluid and a porous medium. It is based on the finite volume method with body‐fitted and multi‐block grids. The Brinkman–Forcheimmer extended model was used to govern the flow in the porous medium region. At its interface, the flow boundary condition imposed is a shear stress jump, which includes the inertial effect, together with a continuity of normal stress. The thermal boundary condition is continuity of temperature and heat flux. The forced convection through a porous insert over a backward‐facing step is investigated. The results are presented with flow configurations for different Darcy numbers, 10^?2 to 10^?5, porosity from 0.2 to 0.8, Reynolds number from 10 to 800, and the ratio of insert length to channel height from 0.1 to 0.3. The heat transfer is improved by using porous insert. To enhance the heat transfer with minimal frictional losses, it is preferable to have a medium length of insert with medium Darcy number, and larger Reynolds number. The interfacial stress jump coefficients β and β₁ were varied from ?1 to 1, and within this range the average and local lower‐wall Nusselt numbers are not sensitive to the parameters. Copyright © 2007 John Wiley & Sons, Ltd.     

多孔介质——自由流界面应力跳跃条件下流动特性解析解

对非对称多孔介质--自由流复合通道内多孔介质内部及多孔介质与自由流体界面处复杂质量、动量输运特性进行研究. 在多孔介质区采用Brinkman-extended Darcy模型并结合速度连续,剪切应力跳跃的界面条件对此复合通道内流体的传递现象进行求解,提出了考虑界面应力跳跃时非对称复合通道各区域流体运动速度及摩擦系数的解析式,分析了界面应力跳跃系数,达西数及无量纲多孔层偏心厚度对流体速度及摩擦系数的影响. 结果表明:改变界面性质可在一定条件下明显控制各区域流体速度分布;在达西数、多孔层偏心厚度一定情况下,界面应力系数的增大会使界面流速减小,而使流体摩擦系数增大,特别是界面应力系数小于0的情况下变化更明显,此时若不考虑界面应力系数则会造成较大误差. 当界面应力系数及多孔层偏心厚度均为较小负数值时,改变多孔层偏心厚度对界面速度的影响要大于改变界面应力系数的情况;而当界面应力系数及多孔层偏心厚度为较大正数值时,情况则相反. 较大达西数下,界面应力系数及多孔层偏心厚度对流体摩擦系数的影响均较大,继续减小达西数至一定程度时,界面应力系数对流体摩擦系数的影响可忽略不计而认为只与多孔层偏心厚度相关,且对较大多孔层偏心厚度更敏感.      

Analytical investigation of the fluid flow in the interface region between a porous medium and a clear fluid in channels partially filled with a porous medium

In this paper analytical solutions for the steady fully developed laminar fluid flow in the parallel-plate and cylindrical channels partially filled with a porous medium and partially with a clear fluid are presented. The Brinkman-extended Darcy equation is utilized to model the flow in a porous region. The solutions account for the boundary effects and for the stress jump boundary condition at the interface recently suggested by Ochoa-Tapia and Whitaker. The dependence of the velocity on the Darcy number and on the adjustable coefficient in the stress jump boundary condition is investigated. It is shown that accounting for a jump in the shear stress at the interface essentially influences velocity profiles.     

A study of internal heat transfer in nonuniform porous structures

The results of theoretical and experimental studies of heat transfer and pressure drop in nonuniform porous materials and systems are presented. In experiments, measurements were made of the air flow rate, inlet and outlet air pressures, and air and porous sample temperatures. Experimental determination of the heat transfer coefficient in porous structures is associated with certain difficulties. The problem of determining a temperature difference between coolant and porous skeleton is the most complex. As a rule, under laboratory conditions this difference is small and cannot be found with sufficient accuracy. In the present work, the method of determination of the internal heat transfer coefficient is based on solving the inverse unsteady heat transfer problem for porous structures. Using this approach, the heat transfer coefficient is calculated indirectly or on the basis of the porous material temperature variation over time.     

Investigation on Heat and Mass Transfer in a Evaporator of a Capillary-Pumped Loop with the Lattice Boltzmann Method: Pore Scale Simulation

A two-dimensional transient numerical model based on the lattice Boltzmann method (LBM) for the global evaporator of a capillary-pumped loop (CPL) is proposed to describe heat and mass transfer with evaporation in the porous wick, heat conduction in the cover plate, and heat transfer in the vapor groove. To indicate the stochastic phase distribution characteristics of most porous wick, the quartet structure generation set (QSGS) is introduced for generating more realistic microstructures of porous media. By using the present lattice Boltzmann algorithm along with the porous structure, the heat and mass transfer of an evaporator on pore scale can be predicted without resorting to any empirical parameters determined case by case. The energy equations for entire evaporator are solved as a conjugate problem, which are solved by means of a spatially varying relaxation time in the lattice Boltzmann model and the liquid flow is driven via the interfacial mass flux. A convective boundary condition considering the latent heat during the evaporation on the interface is introduced into the lattice Boltzmann model based on the nonequilibrium extrapolation rule. Especially, the bounce-back rule and the equilibrium rule of the LBM are, respectively, introduced to deal with the momentum boundary conditions inside the porous wick and on the evaporation interface in order to ensure the stability and the efficiency of the LBM model. Numerical results corresponding to different working conditions and different working fluids are presented, which provide guidance for the evaporator design of a CPL system.     

Turbulent Flow Around Fluid–Porous Interfaces Computed with a Diffusion-Jump Model for <Emphasis Type="Italic">k</Emphasis> and <Emphasis Type="Italic">ε</Emphasis> Transport Equations

Flow over vegetation and bottom of rivers can be characterized by some sort of porous structure of irregular surface through which a fluid permeates. Also, in engineering systems, one can have components that make use of a working fluid flowing over irregular layers of porous material. This article presents numerical solutions for such hybrid medium, considering here a channel partially filled with a flat porous layer saturated by a fluid flowing in turbulent regime. One unique set of transport equations is applied to both the regions. A diffusion-jump model for both the turbulent kinetic energy and its dissipation rate, across the interface, is presented and discussed upon. The discretization steps taken for numerically accommodating such model in the system of algebraic equations are presented. Numerical results show the effects of Reynolds number, porosity, and permeability on mean and turbulence fields. Results indicate that when negative values for the stress jump coefficient are applied, the peak of the turbulent kinetic energy distribution occurs at the macroscopic interface.     

Conjugate natural convection between horizontal concentric cylinders filled with a porous medium

A conjugate problem of natural convection between two horizontal concentric, isothermal cylinders filled with a fluid-saturated porous medium is studied. The flow field and energy equations are solved under the conditions of equality in temperature but with jump in heat flux at the fluid-solid interface. Numerical results by a finite-difference technique are presented for a large variation in the parameters entering the problem. A comparison of the solution with non-conjugate problem is also given.Das konjugierte Problem der freien Konvektion zwischen zwei horizontalen, konzentrischen, isothermen Zylindern, die mit einem fluid-gesättigten porösen Medium gefüllt sind, wurde hier untersucht.Das Strömungsfeld und die Energiegleichung sind unter den Bedingungen gelöst worden, daß die Temperatur gleich ist, aber ein Wärmeflußsprung am Übergang von der Flüssigkeits- zur Festphase stattfindet. Die numerischen Ergebnisse werden mit dem Finite-Differenzen-Verfahren für viele Parametervariationen dargestellt. Zudem wurde ein Lösungsvergleich mit einem nicht konjugierten Problem gegeben.     

An efficient second‐order accurate cut‐cell method for solving the variable coefficient Poisson equation with jump conditions on irregular domains

A numerical method is presented for solving the variable coefficient Poisson equation on a two‐dimensional domain in the presence of irregular interfaces across which both the variable coefficients and the solution itself may be discontinuous. The approach involves using piecewise cubic splines to represent the irregular interface, and applying this representation to calculate the volume and area of each cut cell. The fluxes across the cut‐cell faces and the interface faces are evaluated using a second‐order accurate scheme. The deferred correction approach is used, resulting in a computational stencil for the discretized Poisson equation on an irregular (complex) domain that is identical to that obtained on a regular (simple) domain. In consequence, a highly efficient multigrid solver based on the additive correction multigrid (ACM) method can be applied to solve the current discretized equation system. Several test cases (for which exact solutions to the variable coefficient Poisson equation with and without jump conditions are known) have been used to evaluate the new methodology for discretization on an irregular domain. The numerical solutions show that the new algorithm is second‐order accurate as claimed, even in the presence of jump conditions across an interface. Copyright © 2006 John Wiley & Sons, Ltd.     

Thermomechanics of solids with general imperfect coherent interfaces

The objective of this contribution is to develop a thermodynamically consistent theory for general imperfect coherent interfaces in view of their thermomechanical behavior and to establish a unified computational framework to model all classes of such interfaces using the finite element method. Conventionally, imperfect interfaces with respect to their thermal behavior are often restricted to being either highly conducting (HC) or lowly conducting (LC) also known as Kapitza. The interface model here is general imperfect in the sense that it allows for a jump of the temperature as well as for a jump of the normal heat flux across the interface. Clearly, in extreme cases, the current model simplifies to HC and LC interfaces. A new characteristic of the general imperfect interface is that the interface temperature is an independent degree of freedom and, in general, is not a function of only temperatures across the interface. The interface temperature, however, must be computed using a new interface material parameter, i.e., the sensitivity. It is shown that according to the second law, the interface temperature may not necessarily be the average of (or even between) the temperatures across the interface. In particular, even if the temperature jump at the interface vanishes, the interface temperature may be different from the temperatures across the interface. This finding allows for a better, and somewhat novel, understanding of HC interfaces. That is, a HC interface implies, but is not implied by, the vanishing temperature jump across the interface. The problem is formulated such that all types of interfaces are derived from a general imperfect interface model, and therefore, we establish a unified finite element framework to model all classes of interfaces for general transient problems. Full details of the novel numerical scheme are provided. Key features of the problem are then elucidated via a series of three-dimensional numerical examples. Finally, we recall since the influence of interfaces on the overall response of a body increases as the scale of the problem decreases, this contribution has certain applications to nano-composites and also thermal interface materials.     

Drying of porous oil shales

Rate processes including change of phase are modelled analytically for a half-space porous substance exposed to a jump in external temperature. The model predicts, in a closed form, the pressure build-up and the rate of evaporation of volatiles from a porous matrix. It assumes two distinct regions separated by a moving interface where the change of phase takes place. One region maintains its initial concentration of volatiles while the second is devoid of volatiles. Different thermophysical properties are considered for the two regions. The model was applied for the evaporation of moisture from oil-shale. Results are given in a parametric form     

Fluid penetration effects in porous media contact

The contact boundary conditions at the interface between two fluid-saturated porous bodies are derived. The general derivation is performed within the well-founded framework of the Theory of Porous Media (TPM) based on the constituent balance relations of mass, momentum, and energy accounting for finite discontinuities at the contact surface. Particular attention is drawn to the effects associated with the interstitial fluid flux across the interface. The derived contact conditions include two kinematic continuity conditions for the solid velocity and the fluid seepage velocity as well as two jump conditions for the effective solid stress and the pore-fluid pressure. As an application, the common case of biphasic porous media contact proceeding from materially incompressible constituents and inviscid fluid properties is discussed in detail.      

波浪与外圆弧开孔壁双圆筒柱的相互作用

应用比例边界有限元法(SBFEM)研究了短峰波与圆筒外接圆弧开孔结构物的相互作用. 求解时将外接圆弧延伸构建一个虚拟圆, 该圆的孔隙影响系数可由矩阵G_0统一进行表达. 整个流场可划分成一个有限域和一个无限域. SBFEM只需对虚拟圆边界进行离散, 使空间维数降低一阶, 在圆的半径方向保持解析, 并且无限域处的辐射边界条件能自动满足. 利用变分原理推导SBFEM方程, 有限域和无限域分别采用贝塞尔函数和汉克尔函数作为基函数来求得对应域的解. 将计算结果与解析解和其他数值方法进行了比较, 验证了该方法是一种用很少单元便能得到精确结果的高效算法. 进一步研究了诸如短峰波波向、结构的几何、材料参数等因素对结构所受波浪载荷及绕射波轮廓的影响, 并进行了分析.      

A simple finite element method for Stokes flows with surface tension using unfitted meshes

This paper presents a simple finite element method for Stokes flows with surface tension. The method uses an unfitted mesh that is independent of the interface. Due to the surface force, the pressure has a jump across the interface. Based on the properties of the level set function that implicitly represents the interface, the jump of the pressure is removed, and a new problem without discontinuities is formulated. Then, classical stable finite element methods are applied to solve the new problem. Some techniques are used to show that the method is equivalent to an easy‐to‐implement method that can be regarded as a traditional method with a modified pressure space. However, the matrix of the resulting linear system of equations is the same as that of the traditional method. Optimal error estimates are derived for the proposed method. Finally, some numerical tests are presented to confirm the theoretical results. Copyright © 2015 John Wiley & Sons, Ltd.     

A numerical method for flows in porous and homogenous fluid domains coupled at the interface by stress jump

A numerical method was developed for flows involving an interface between a homogenous fluid and a porous medium. The numerical method is based on the finite volume method with body‐fitted and multi‐block grids. A generalized model, which includes Brinkman term, Forcheimmer term and non‐linear convective term, was used to govern the flow in the porous medium region. At its interface, a shear stress jump that includes the inertial effect was imposed, together with a continuity of normal stress. Furthermore, the effect of the jump condition on the diffusive flux was considered, additional to that on the convective part which has been usually considered. Numerical results of three flow configurations are presented. The method is suitable for coupled problems with regions of homogeneous fluid and porous medium, which have complex geometries. Copyright © 2006 John Wiley & Sons, Ltd.