Effects of Viscous Dissipation on Unsteady Free Convection in a Fluid Past a Vertical Plate Immersed in a Porous Medium

The effects of viscous dissipation on unsteady free convection from an isothermal vertical flat plate in a fluid saturated porous medium are examined numerically. The Darcy–Brinkman–Forchheimer model is employed to describe the flow field. A new model of viscous dissipation is used for the Darcy–Brinkman–Forchheimer model of porous media. The simultaneous development of the momentum and thermal boundary layers are obtained by using a finite difference method. Boundary layer and Boussinesq approximation have been incorporated. Numerical calculations are carried out for various parameters entering into the problem. Velocity and temperature profiles as well as local friction factor and local Nusselt number are shown graphically. It is found that as time approaches infinity, the values of friction factor and heat transfer coefficient approach steady state.     

Flow near the permeable boundary of a porous medium: An experimental investigation using LDA

 Fluid flow at the interface of a porous medium and an open channel is the governing phenomenon in a number of processes of industrial importance. Traditionally, this has been modeled by applying the Brinkman’s modification of Darcy’s law to obtain the velocity profile in terms of an additional parameter known as the “apparent viscosity” or the “slip coefficient”. To test this ad hoc approach, a detailed experimental investigation of the flow was conducted using Laser Doppler Anemometry (LDA) in the close vicinity of the permeable boundary of a porous medium. The porous medium used in the experiments consisted of a network of continuous glass strands woven together in a random fashion. A Hele–Shaw cell was partially filled with a fibrous preform such that an open channel flow is coupled with the Darcy flow inside the preform through the permeable interface of the preform. The open channel portion of the Hele–Shaw cell also acts as an ideal porous medium of known in-plane permeability which is much higher than the permeability of the fibrous porous medium. A viscous fluid is injected at a constant flow rate through the above arrangement and a saturated and steady flow is established through the cell. Using LDA, steady state velocity profiles are accurately measured by traversing across the cell in the direction perpendicular to the flow. A series of experiments were conducted in which fluid viscosity, flow rate, solid volume fraction of the porous medium and depth of the Hele–Shaw cell were varied. For each and every case in which the conditions for Hele–Shaw approximation were valid, the depth of the boundary layer zone or the screening length inside the fibrous preform was found to be of the order of the channel depth. This is much larger as compared to the Brinkman’s prediction of the screening length which is of the order of √K, where K is the permeability of the fibrous porous medium. Based on this finding, we modified the boundary condition in the Brinkman’s solution and found that the velocity profile results compared well with the experimental data for the planar geometry and the fibrous preforms for volume fractions of 7%, 14% and 21% for Hele–Shaw cell depths of 1.6 and 3.175 mm. For a cell depth of 4.8 cm, in which the Hele–Shaw approximation was not valid, the boundary layer thickness or the screening length was found to be less than the mold or channel depth but was still much larger than the Brinkman’s prediction. Received: 10 May 1996 / Accepted: 26 August 1996     

The Onset of Double-Diffusive Nanofluid Convection in a Layer of a Saturated Porous Medium

The onset of convection in a horizontal layer of a porous medium saturated by a nanofluid is studied analytically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. For the porous medium, the Brinkman model is employed. Three cases of free–free, rigid–rigid, and rigid–free boundaries are considered. The analysis reveals that for a typical nanofluid (with large Lewis number), the prime effect of the nanofluids is via a buoyancy effect coupled with the conservation of nanoparticles, whereas the contribution of nanoparticles to the thermal energy equation is a second-order effect. It is found that the critical thermal Rayleigh number can be reduced or increased by a substantial amount, depending on whether the basic nanoparticle distribution is top-heavy or bottom-heavy, by the presence of the nanoparticles. Oscillatory instability is possible in the case of a bottom-heavy nanoparticle distribution.     

Mixed Convection in a Vertical Porous Channel

A numerical study of mixed convection in a vertical channel filled with a porous medium including the effect of inertial forces is studied by taking into account the effect of viscous and Darcy dissipations. The flow is modeled using the Brinkman–Forchheimer-extended Darcy equations. The two boundaries are considered as isothermal–isothermal, isoflux–isothermal and isothermal–isoflux for the left and right walls of the channel and kept either at equal or at different temperatures. The governing equations are solved numerically by finite difference method with Southwell–Over–Relaxation technique for extended Darcy model and analytically using perturbation series method for Darcian model. The velocity and temperature fields are obtained for various porous parameter, inertia effect, product of Brinkman number and Grashof number and the ratio of Grashof number and Reynolds number for equal and different wall temperatures. Nusselt number at the walls is also determined for three types of thermal boundary conditions. The viscous dissipation enhances the flow reversal in the case of downward flow while it counters the flow in the case of upward flow. The Darcy and inertial drag terms suppress the flow. It is found that analytical and numerical solutions agree very well for the Darcian model. An erratum to this article is available at .     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Effect of distributed gas injection on aerodynamic characteristics of a body of revolution in a supersonic flow

Results of experimental and numerical investigations of the effect of gas injection through a permeable porous surface on the drag coefficient of a cone-cylinder body of revolution in a supersonic flow with the Mach number range M _h = 3–6 are presented. It is demonstrated that gas injection through a porous nose cone with gas flow rates being 6–8% of the free-stream flow rate in the mid-section leads to a decrease in the drag coefficient approximately by 5–7%. The contributions of the decrease in the drag force acting on the model forebody and of the increase in the base pressure to the total drag reduction are approximately identical. Gas injection through a porous base surface with the flow rate approximately equal to 1% leads to a threefold increase in the base pressure and to a decrease in the drag coefficient. Gas injection through a porous base surface with the flow rate approximately equal to 5% gives rise to a supersonic flow zone in the base region.     

The Onset of Convection in a Layer of a Porous Medium Saturated by a Nanofluid: Effects of Conductivity and Viscosity Variation and Cross-Diffusion

The linear stability theory for the Horton–Rogers–Lapwood problem is extended to the case where the porous medium is saturated by a nanofluid with thermal conductivity and viscosity dependent on the nanoparticle volume fraction. The effects of Brownian motion and thermophoresis are considered. In conjunction with the Brownian motion, the nanoparticle fraction becomes stratified, and hence the viscosity and the conductivity are stratified. The nanofluid is assumed to be dilute and this enables the porous medium to be treated as a weakly heterogeneous medium with variation, in the vertical direction, of conductivity and viscosity. In turn this allows an approximate analytical solution to be obtained.     

Numerical simulation of stabilization of the boundary layer on a surface with a porous coating in a supersonic separated flow

Stability of a supersonic (M _∞ = 5.373) boundary layer with local separation in a compression corner with a passive porous coating partly absorbing flow perturbations is considered by solving two-dimensional Navier-Stokes equations numerically. The second mode of disturbances of a supersonic boundary layer is demonstrated to be the most important one behind the boundary-layer reattachment point. The possibility of effective stabilization of these disturbances behind the reattachment point with the use of porous coatings is confirmed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 39–47, March–April, 2007.     

Relaxation Effects of a Saturated Porous Media Using the Two-Dimensional Generalized Thermoelastic Theory

A model of the equations of a generalized thermoelasticity (GT) with relaxation times for a saturated porous medium is given in this article. The formulation can be applied to the GT theories: Lord–Shulman theory, Green–Lindsay theory, and Coupled theory for the porous medium. A two-dimensional thermoelastic problem that is subjected to a time-dependent thermal/mechanical source is investigated with the model of the generalized porous thermoelasticity. By using the Laplace transform and the Fourier transform technique, solutions for the displacement, temperature, pore pressure, and stresses are obtained with a semi-analytical approach in the transform domain. Numerical results are also performed for portraying the nature of variations of the field variables. In addition, comparisons are presented with the corresponding four theories.     

Fully Developed Mixed Convection Flow in a Vertical Channel Containing Porous and Fluid Layer with Isothermal or Isoflux Boundaries

An analysis of fully developed combined free and forced convective flow in a fluid saturated porous medium channel bounded by two vertical parallel plates is presented. The flow is modeled using Brinkman equation model. The viscous and Darcy dissipation terms are also included in the energy equation. Three types of thermal boundary conditions such as isothermal–isothermal, isoflux–isothermal, and isothermal–isoflux for the left–right walls of the channel are considered. Analytical solutions for the governing ordinary differential equations are obtained by perturbation series method. In addition, closed form expressions for the Nusselt number at both the left and right channel walls are derived. Results have been presented for a wide range of governing parameters such as porous parameter, ratio of Grashof number and Reynolds number, viscosity ratio, width ratio, and conductivity ratio on velocity, and temperature fields. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature.     

Natural convection from a cylinder in a narrow gap and in a porous medium

The problem of natural convection from a horizontal cylinder in a narrow gap and in a porous medium is solved both theoretically and experimentally. An integral method for calculating heat transfer from the cylinder for constant flux on its surface was suggested. Numerical analysis clarified the role of regime and geometrical factors. It is shown that natural convection from a cylinder in a porous medium can be modeled by a Hele Shaw cell. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 140–150. January–February, 1999.     

Statistical characteristics of the motion of a fluid particle in a medium with random porosity

In the study of flow of a neutral admixture in a porous medium, it is most often assumed in the stochastic formulation that the porosity is constant and a determinate quantity, and the velocity is a random function [1–4]. The velocity distribution is usually regarded as known. Flow in a porous medium with random porosity has been studied to a far lesser extent. We note [5], which studies the averaged equations obtained within the framework of the correlation approximation. We consider the model problem of one-dimensional motion of a fluid particle (position of the front for flow of a neutral admixture in a porous medium) in a medium with random porosity. For a particular form of random porosity field, expressions are obtained for the one- and two-point densities of the distribution of the position of the particle. A study is made of the dependences of the first four moments and the correlation function of the position of the particle as functions of the time. It is shown that for large values of the time the motion of the particle is asymptotically similar to Brownian motion. It is shown by means of numerical modeling that the results obtained transfer to the case of an arbitrary random porosity field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 59–65, November–December, 1986.     

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles

A numerical study is performed to analyze steady laminar forced convection in a channel in which discrete heat sources covered with porous material are placed on the bottom wall. Hydrodynamic and heat transfer results are reported. The flow in the porous medium is modeled using the Darcy–Brinkman–Forchheimer model. A computer program based on control volume method with appropriate averaging for diffusion coefficient is developed to solve the coupling between solid, fluid, and porous region. The effects of parameters such as Reynolds number, Prandtl number, inertia coefficient, and thermal conductivity ratio are considered. The results reveal that the porous cover with high thermal conductivity enhances the heat transfer from the solid blocks significantly and decreases the maximum temperature on the heated solid blocks. The mean Nusselt number increases with increase of Reynolds number and Prandtl number, and decrease of inertia coefficient. The pressure drop along the channel increases rapidly with the increase of Reynolds number.     

On solving the problem of reflection of linear waves in a fluid from a porous half-space saturated by this fluid

Solutions of the problem of reflection of a stepwise pressure wave in a linearly compressed fluid from a flat boundary of a porous medium of infinite length saturated by the same fluid are obtained in the acoustic approximation. Based on analytical solutions, a numerical analysis is performed to reveal the specific features of the reflected and incident waves, depending on porosity and permeability of the porous half-space. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 16–26, September–October, 2006.     

Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.     

Convective stability of a binary mixture in a fractured porous medium

The effect of the gravity field and the concentration and thermal gradients on the convective stability of a mixture in a nonhomogeneous porous medium containing a system of thin highly permeable fractures separating relatively impermeable blocks is analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 88–97, January–February, 1994.     

The effect of a transition layer between a fluid and a porous medium: shear flow in a channel

Flow in a three-layer channel is modeled analytically. The channel consists of a transition layer sandwiched between a porous medium and a fluid clear of solid material. Within the transition layer, the reciprocal of the permeability varies linearly across the channel. The Brinkman model is used for the momentum equations for the porous medium layer and the transition layer. The velocity profile is obtained in closed form in terms of Airy, exponential, and polynomial functions. The overall volume flux and boundary friction factors are calculated and compared with values obtained with a two-layer model employing the Beavers–Joseph condition at the interface between a Darcy porous medium and a clear fluid.     

Analysis of hydrodynamic and thermal dispersion in porous media by means of a local approach

A pore scale analysis is implemented in this numerical study to investigate the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure. The macroscopic characteristics of the transport phenomena are evaluated with an averaging technique of the controlling variables at a pore scale level in an elementary cell of the porous structure. The Darcy–Forchheimer model describes the fluid motion through the porous medium while the continuity and Navier–Stokes equations are applied within the unit cell. An average energy equation is employed for the thermal part of the porous medium. The macroscopic pressure loss is computed in order to evaluate the dominant microscopic inertial effects. Local fluctuations of velocity and temperature at the pore scale are instrumental in the quantification of the thermal dispersion through the total effective thermal diffusivity. The numerical results demonstrate that microscopic inertia contributes significantly to the magnitude of the macroscopic pressure loss, in some instances with as much as 70%. Depending on the nature of the porous medium, the thermal dispersion may have a marked bearing on the heat transfer, particularly in the streamwise direction for a highly conducting fluid and certain values of the Peclet number.     

A Conceptual Study of Cavity Aeroacoustics Control Using Porous Media Inserts

In this study, an integrated flow simulation and aeroacoustics prediction methodology is applied to testing a sound control technique using porous inserts in an open cavity. Large eddy simulation (LES) combined with a three-dimensional Ffowcs Williams–Hawkings (FW–H) acoustic analogy is employed to predict the flow field, the acoustic sources and the sound radiation. The Darcy pressure – velocity law is applied to conceptually mimic the effect of porous media placed on the cavity floor and/or rear wall. Consequently, flow in the cavity could locally move in or out through these porous walls, depending on the local pressure differences. LES with “standard” subgrid-scale models for compressible flow is carried out to simulate the flow field covering the sound source and near fields, and the fully three-dimensional FW–H acoustic analogy is used to predict the sound field. The numerical results show that applying the conceptual porous media on cavity floor and/or rear wall could decrease the pressure fluctuations in the cavity and the sound pressure level in the far field. The amplitudes of the dominant oscillations (Rossiter modes) are suppressed and their frequencies are slightly modified. The dominant sound source is the transverse dipole term, which is significantly reduced due to the porous walls. As a result, the sound pressure in the far field is also suppressed. The preliminary study reveals that using porous-inserts is a promising technology for flow and sound radiation control.     

Law of flow of a viscoplastic fluid through a porous medium with allowance for inertial losses

A macroscopic law of flow of a viscoplastic Schwedoff-Bingham fluid through a porous medium is obtained on the basis of percolation theory with allowance for viscous and inertial losses. The asymptotics of the flow law are estimated and expressions for determining the limiting pressure gradient as a function of the microinhomogeneity parameters are given. Satisfactory qualitative agreement between the theoretical and known experimental data is observed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 68–73, January–February, 1999.