Brinkman-extended Darcy Model Research Articles

Effect of uniform inclined magnetic field on natural convection and entropy generation in an open cavity having a horizontal porous layer saturated with a ferrofluid

ABSTRACTNumerical analysis of natural convection combined with entropy generation in a square open cavity partially filled with a porous medium has been performed for a ferrofluid under the effect of inclined uniform magnetic field. Governing equations with corresponding boundary conditions formulated in dimensionless stream function and vorticity using Brinkman–extended Darcy model for porous layer have been solved numerically using finite difference method. An influence of key parameters on ferrofluid flow and heat transfer has been analyzed. It has been found that an inclusion of spherical ferric oxide nanoparticles can lead to a diminution of entropy generation in the case of similar flow and heat transfer structures.

Effect of uniform inclined magnetic field on mixed convection in a lid-driven cavity having a horizontal porous layer saturated with a ferrofluid

MHD mixed convection in a lid-driven cavity with partially filled with a porous medium saturated with a ferrofluid has been analyzed numerically. The domain of interest consists of a bottom porous layer and a nanofluid layer over the porous one with a heated motionless bottom wall and cooled upper moved wall. The governing partial differential equations formulated on the basis of a single-phase model for nanofluid, Brinkman-extended Darcy model for porous layer and Boussinesq approximation for buoyancy force have been solved by finite difference method of the second-order accuracy. Analysis has been carried out for a wide range of Hartmann number, magnetic field inclination angle, Darcy number, porous layer height, and nanoparticles volume fraction. It has been revealed that average Nusselt number is a non-monotonic function of Darcy number and porous layer height, while a growth of Hartmann number illustrates the heat transfer rate reduction.

Slip–Brinkman Flow Through Corrugated Microannulus with Stationary Random Roughness

This paper presents a boundary perturbation method of the Brinkman-extended Darcy model to investigate the flow in corrugated microannuli cylindrical tubes with slip surfaces. The stationary random model is used to mimic the surface roughness of the cylindrical walls. The tube is filled with a porous medium. We shall consider the two cases where corrugations are either perpendicular or parallel to the flow, and particular attention is given to the effect of the phase shift. The effects of the corrugations on the flow rate and pressure gradient are investigated as functions of wavelength, the permeability of the medium, the radius ratio and the slip parameter. Particular surface roughnesses are examined as special cases of stationary random surface. It is found that the effect of the partial slip is significant on the corrugation functions. The limiting cases of Stokes and Darcy’s flows and no-slip case are discussed.

Numerical simulations and optimization of porous pin fins in a rectangular channel

ABSTRACTIn this study, the multiparameter constrained optimization procedure integrating the design of experiments (DOE), genetic algorithm (GA), and computational fluid dynamics (CFD) is proposed to design three-dimensional porous pin fins in a rectangular channel. The Forchheimer–Brinkman extended Darcy model and the two-equation energy model are adopted to describe the fluid flow and heat transfer characteristics in the porous media. The elliptical, coupled, steady-state, and three-dimensional governing partial differential equations for laminar forced convection with porous pin fins in a rectangular channel are solved numerically using the finite volume approach. The numerical optimization provides a reliable and economic mean of designing a heat transfer channel with porous pin fin arrays.

A Semi-analytical Solution for Start-Up Flow in an Annulus Partially Filled with Porous Material

This study investigates the start-up flow in a horizontal annulus partially filled with clear fluid and partially with a fluid-saturated porous material (composite channel) as a result of the sudden application of constant pressure gradient in horizontal direction. Momentum transfer in the porous medium is simulated using the Brinkman-extended Darcy model. The fluid and porous regions are linked together by equating their velocities and incorporating the shear stress jump conditions at the interface. With the application of Laplace transform technique, the governing partial differential equations are transformed into ordinary differential equations which are solved in the Laplace domain to obtain exact solutions. Inversion of the exact solution in Laplace domain to time domain is done using the Riemann-sum approximation approach. Validation of the Riemann-sum approximation method is achieved by comparing numerical values obtained with those of exact solution obtained for steady state flow and transient solution obtained by implicit finite difference method. At large values of time, transient solution obtained using Riemann-sum approximation approach and implicit finite difference method coincide with closed form solution obtain exactly for steady state flow showing excellent agreement between these methods. Line graphs are used to discuss the effect of the different flow parameters involved in the flow formation.

A fourth order compact scheme for heat transfer problem in porous media

In this paper, we have extended the recently proposed fourth order compact scheme by Pandit et al. (2007) which was used earlier only for the convection–diffusion equations without considering reaction and nonhomogeneous source terms, and designed to compute flow in a two sided lid-driven differentially heated square cavity filled with a fluid saturated porous medium. The governing Navier–Stokes (N–S) equations in streamfunction–vorticity (ψ−ζ) form of Brinkman-extended Darcy model including the energy transport equation are all solved as a coupled system of equations for the five field variables consisting of streamfunction, vorticity, two velocities and temperature. Moreover, local entropy generation distributions are determined based on the obtained dimensionless velocity and temperature values. The derivative source term present in vorticity equation has been treated as fourth order compact using Padé scheme. The details for the derivation of difference relations at boundaries to generate accurate and stable solutions are also given. We have computed the results for three different cases depending on the direction of moving walls. To assess the numerical accuracy of our proposed scheme, one pertinent test problem with known exact solution is used. The higher order compact scheme adopted in the present study yields consistent performance for variation of key parameters e.g. Richardson number (Ri), Darcy number (Da), Grashof number (Gr) and fixed Prandtl number (Pr)=0.7. The present results are compared with numerical results available in the literature and excellent match is observed in all the cases, establishing the efficiency and accuracy of the extended combined formulations of our fourth order compact scheme and Padé schemes.

Effect of anisotropic permeability on fluid flow through composite porous channel

This paper presents a theoretical model for a fully developed flow through a composite porous channel. The channel consists of a fluid layer sandwiched between two porous layers. A generalized Brinkman-extended Darcy model for the porous layers and Navier–Stokes equations for the fluid layer are used to investigate the flow in detail. The continuity of stress and velocity are used at the interface and no slip at the impermeable walls. We assume that the porous layers are anisotropic. Accordingly, \(\varphi \) is taken as the angle between the horizontal direction and the principal axes with permeability \(K_{2}\) or \(K_{4}\) for the lower or upper porous layers considering the anisotropic nature of the porous medium. It is shown that the anisotropic permeability and orientation angle \(\varphi \) have a strong effect on the fluid flow and skin friction. We present some important findings based on the response to variations in the anisotropy.

Hydrodynamic Anisotropy Effects on Radiation-Mixed Convection Interaction in a Vertical Porous Channel

The effects of hydrodynamic anisotropy on the mixed-convection in a vertical porous channel heated on its plates with a thermal radiation are investigated analytically for fully developed flow regime. The porous medium is anisotropic in permeability whose principal axes are oriented in a direction that is oblique to the gravity. The generalized Brinkman-extended Darcy model which allows the no-slip boundary-condition on solid wall is used in the formulation of the problem. The flow reversal, the thermal radiation influence for natural, and forced convection are considered in the limiting cases for low and high porosity media. It was found that the anisotropic permeability ratio, the orientation angle of the principal axes of permeability and the radiation parameter affected significantly the flow regime and the heat transfer.

Three-Dimensional Simulation of Pulsatile Flow Through a Porous Bulge

The present research deals with simulation of pulsatile flow through a tubular bulge filled with homogeneous, isotropic, saturated porous medium. For comparison, flow fields are also computed for the flow through clear medium in the same geometry. Both cases involve three-dimensional, unsteady, laminar, incompressible flow of Newtonian fluid. Flow through the porous medium is modeled via Forchheimer–Brinkman-extended Darcy model. While semi-analytical solutions are presented for the asymptotic cases, numerical techniques are used for the general solutions of the conservation equations. The governing equations, in both porous and clear media, are discretized by an implicit finite volume technique over an unstructured tetrahedral mesh. The resulting algebraic equations are solved using stabilized biconjugate gradient technique. Simulations include Darcy numbers of $$10^{-4}$$ and $$10^{-2}$$ at Reynolds numbers of 500 and 2000 based on the peak incoming average velocity and main tube diameter. The Womersley number is chosen to be 11.3 to mimic biomedical application. Results show that the porous medium naturalizes the recirculations and secondary flows while experiencing higher-pressure drop and wall shear compared to that of the clear medium.

Unsteady Couette Flow in a Composite Channel Partially Filled with Porous Material: A Semi-analytical Approach

This study is devoted to investigate the unsteady fully developed time-dependent Couette flow in a composite channel partially filled with porous material. The Brinkman-extended Darcy model is used to simulate momentum transfer in the porous medium. The fluid and porous regions are interlinked by equating the velocity and by considering shear stress jump conditions in the interface. The solutions of the governing equations are obtained using a Laplace transform technique. However, the Riemann-sum approximation method is used to invert from Laplace domain to time domain. The solution obtained is validated by presenting comparisons with closed form solution obtained for steady flow which has been derived separately and also by implicit finite difference method. During the course of numerical comparison, an excellent agreement was found between steady-state solution obtained exactly and unsteady solution obtained by implicit finite difference method or Riemann-sum approximation method at large values of time. The effect of various flow parameters entering into the problem is discussed with the aid of line graphs. The results obtained here may be further used to verify the validity of obtained numerical solutions for more complicated time-dependent Couette flow in composite channel.

Unsteady Conjugate Natural Convection in a Vertical Cylinder Containing a Horizontal Porous Layer: Darcy Model and Brinkman-Extended Darcy Model

Transient natural convection in a vertical cylinder partially filled with a porous media with heat-conducting solid walls of finite thickness in conditions of convective heat exchange with an environment has been studied numerically. The Darcy and Brinkman-extended Darcy models with Boussinesq approximation have been used to solve the flow and heat transfer in the porous region. The Oberbeck–Boussinesq equations have been used to describe the flow and heat transfer in the pure fluid region. The Beavers–Joseph empirical boundary condition is considered at the fluid–porous layer interface with the Darcy model. In the case of the Brinkman-extended Darcy model, the two regions are coupled by equating the velocity and stress components at the interface. The governing equations formulated in terms of the dimensionless stream function, vorticity, and temperature have been solved using the finite difference method. The main objective was to investigate the influence of the Darcy number \(10^{-5}\le \hbox {Da}\le 10^{-3}\), porous layer height ratio \(0\le d/L\le 1\), thermal conductivity ratio \(1\le k_{1,3}\le 20\), and dimensionless time \(0\le \tau \le 1000\) on the fluid flow and heat transfer on the basis of the Darcy and non-Darcy models. Comprehensive analysis of an effect of these key parameters on the Nusselt number at the bottom wall, average temperature in the cylindrical cavity, and maximum absolute value of the stream function has been conducted.

Numerical optimization of gas diffusion layer with a wavy channel

In this study, the multi-parameter constrained optimization procedure integrating the design of experiments (DOE), full factorial experimental design (FFED), genetic algorithm (GA) and computational fluid dynamics (CFD) are proposed to design a gas diffusion layer (GDL) linked to a wavy channel. The Forchheimer–Brinkman extended Darcy model and two-equation energy model are adopted to describe the fluid flow and heat transfer characteristics in the porous media. Numerical computations are performed with a wavy channel for the parameters studied. That is, Reynolds number (Re), wave amplitude (α), wave number (β) and gas diffusion layer porosity (ε) in the range of 200≤Re≤1000, 0.1≤α≤0.3, 2≤β≤10, 0.4≤ε≤0.5, respectively. The interactive effects of Re, α, β, and ε on the local Nusselt number (Nu) and the average Nusselt number (Nu¯) are discussed in this study. The results show that computations of local and average Nusselt numbers within the wavy channel are enhanced compared to the straight channel.In addition, the optimization of this problem is also presented by using a full factorial experimental design and the genetic algorithm (GA) method. The objective function defined as thermal performance factor (TPF) has developed a correlation function with three design parameters. That is, wave amplitude α, wave number β and gas diffusion layer porosity ε. The results show that the thermal performance factor (TPF) prediction can be approached under the maximum error of 6.2% from the present regression model, which is compared with CFD results.

Rotation effects on non-Darcy convection in an enclosure filled with porous medium

Natural convection heat transfer in a rotating, differentially heated enclosure is studied numerically in this paper taking into consideration the Forchheimer–Brinkman-extended Darcy model. The enclosure is filled with a fluid-saturated porous medium and executes a steady counterclockwise angular velocity about its longitudinal axis. The staggered grid arrangement together with the Marker and Cell (MAC) method was employed to solve the governing equations. The governing parameters considered are the porosity, 0.4≤ϵ≤0.99, the Darcy number, 0.005≤Da≤0.01 and the Taylor number, 8.9×104≤Ta≤3.8×105, and the centrifugal force is assumed weaker than the Coriolis force. It is found that higher porosities have weaker flow circulation when the Coriolis effect is smaller than the buoyancy effect. The global quantity of the heat transfer rate increases by increasing the porosity and the Darcy number and decreases by increasing the Taylor number.

Effect of anisotropic permeability on convective heat transfer through a porous river bed underlying a fluid layer

A study is made of convective heat transfer through packed porous beds which consist of a horizontal fluid layer (river bed) and a porous zone with anisotropic permeability and underlaid by a surface heated to a constant temperature \(T_1\). The free surface of the fluid layer overlying the horizontal porous layer is heated isothermally at temperature \(T_2\) (\(>T_1\)). Using the Navier–Stokes model for the fluid layer and the generalized Brinkman-extended Darcy model for the porous zone, an exact solution is found for a fully developed system of forced convective flow through the superposed layers. The Beavers–Joseph condition is applied at the interface between the two layers. The influence of hydrodynamic anisotropy on the convective phenomenon is investigated analytically. It is shown that the anisotropic permeability ratio \(K^*\), the inclination angle of the principal axes \(\varphi \) of the porous medium, and the thickness of the porous lining \(\epsilon \) have a strong influence on the convective flow and the heat transfer rate. This analysis helps to predict environmental aquatic behavior.

On multiple steady states for natural convection (low Prandtl number fluid) within porous square enclosures: Effect of nonuniformity of wall temperatures

A Brinkman extended Darcy model has been used to study the effect of spatial nonuniformity of wall temperatures on the multiplicity of steady convective flows within a square porous enclosure saturated with low Prandtl fluid (Pr=0.026). The convection is assumed to be driven by the hotter bottom wall in conjunction with colder side walls, where bottom-side wall junctions maintain continuity of temperature to represent a more realistic situation. This configuration enforces nonuniformity on either bottom wall or side walls or both bottom and side wall temperatures. The degree of nonuniformity of wall temperatures are varied parametrically in terms of thermal aspect ratio (Θ) to simulate various possible scenarios of nonuniform bottom/side wall temperatures and steady solution branches are traced in the parameter space of Θ to investigate the variation of multiplicity of the flows. A perturbation technique has been used to initiate the steady solution branches, which are then traced by numerical continuation scheme. A penalty finite element approximation in conjunction with Newton–Raphson method and parameter continuation scheme has been used to construct the bifurcation diagrams of various steady branches. This work reveals three steady symmetric branches and four steady asymmetric branches with exhibition of symmetry breaking bifurcation. This work also shows that the nonuniformity of wall temperatures play a crucial role for the existence of multiple solutions as well as the multiplicity of convective flows.

Effects of Variable Fluid Properties and MHD on Mixed Convection Heat Transfer from a Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium

The effects of variable Fluid Properties like variation of permeability, porosity, thermal conductivity and magnetic field on Mixed Convection Heat transfer from Vertical Heated Plate Embedded in a Sparsely Packed Porous Medium have been approached numerically. The boundary layer flow in the porous medium is governed by Lapwood - Forchheimer - Brinkman extended Darcy model and the Lorentz force. The natures of these equations are highly non-linear and coupled each other. The non-linear differential equations are non-dimensionalised using the non-dimensional parameter involving Grashoff number Gr, Prandtl number Pr, Hartmann number M, Eckert number E and so on. Similarity transformations are employed and the resulting ordinary differential equations are solved numerically by using shooting algorithm with Runge - Kutta and Newton - Raphson method to obtain velocity and temperature distributions. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in decreasing the flow field, whereas its influence is reversed on the rate of heat transfer for all values of permeability considered. The effect of Magnetic field is diminution with velocity of the fluid flow. Further, the obtained results under the two limiting conditions were found to be in good agreement with the existing results.

MOMENTUM INTEGRAL METHOD FOR FORCED CONVECTION IN THERMAL NONEQUILIBRIUM POWER-LAW FLUID-SATURATED POROUS MEDIA

Forced convection heat transfer for power-law fluid flow in porous media was studied analytically. The analytical solutions were obtained based on the Brinkman-extended Darcy model for fluid flow and the two-equation model for forced convection heat transfer. As a closed-form exact velocity profile is unobtainable for the general power-law index, an approximate velocity profile based on the parabolic model is proposed by subscribing to the momentum boundary layer integral method. Heat transfer analysis is based on the two-equation model by considering local thermal nonequilibrium between fluid and solid phases and constant heat flux boundary conditions. The velocity and temperature distributions obtained based on the parabolic model were verified to be reasonably accurate and improvement is justified compared to the linear model. The expression for the overall Nusselt number was derived based on the proposed parabolic model. The effects of the governing parameters of engineering importance such as Darcy number, power-law index, nondimensional interfacial heat transfer coefficient, and effective thermal conductivity ratio on the convective heat transfer characteristics of non-Newtonian fluids in porous media are analyzed and discussed.

Numerical simulation of mixed convection in a porous medium filled with water/Al2 O3 nanofluid

AbstractThe present work encloses the application of a Brinkman–extended Darcy model in a problem concerning mixed convection ina lid–driven porous cavity using nanofluids. The transport equations are solved numerically by the finite volume method on a co–located grid arrangement using the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme. The effects of governing parameters, namely, Grashof number (Gr), Darcy number (Da), and solid volume fraction $(\chi )$, on the streamlines and the isotherms are studied. The present results are validated by favorable comparisons with previously published results and are in good agreement with them. The present numerical results show that the addition of nanoparticles to a base fluid has produced an augmentation of the heat transfer coefficient and it is found to increase significantly with an increase of the particle volume concentration. It is observed from the results that at the higher value of the Grashof number (Gr = 104 ), the average Nusselt number increases with an increase in the Darcy number for a constant solid volume fraction. The detailed results are reported by means of streamlines, isotherms, and Nusselt numbers. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21029

Transient behavior of fluid flow and heat transfer in vertical channels partially filled with porous medium: Effects of inertial term and viscous dissipation

In this article, transient hydrodynamic and heat-transfer behavior of Newtonian fluid flow in vertical parallel-plate channels partially filled with a porous medium has been investigated numerically. In this regard, the influences of macroscopic local inertial term and the viscous heating due to the viscous dissipation were taken into account in the momentum equations of porous region and the thermal energy equations, respectively. Moreover, Forchheimer–Brinkman extended Darcy model was used to model fluid flow in the porous region. In addition, an analytical solution was obtained in the case of negligible Brinkman and Forchheimer number values at the steady-state conditions. The predicted results were compared with those predicted by a two-parameter perturbation technique developed by the present authors at the steady-state conditions and good agreement was obtained. The predicted results clearly indicate that neglecting the inertial effect in high permeability porous media or high velocity flows can alter substantially the flow and heat transfer characteristics.

Onset of Vibrational Convection in a Binary Fluid Saturated Non-Darcy Porous Layer Heated from Above

A linear stability analysis is used to investigate the influence of mechanical vibration on the onset of thermosolutal convection in a horizontal porous layer heated and salted from above. Vibrations are considered with arbitrary amplitude and frequency. The Brinkman extended Darcy model is used to describe the flow and the Oberbeck-Boussinesq approximation is employed. Continued fraction method and Floquet theory are used to determine the convective instability threshold. It is found that the solutal Rayleigh number has the stabilizing effect. The existence of a closed disconnected loop of synchronous mode is predicted in the marginal curve for moderate values of solutal Rayleigh number and vibration amplitude.