Vertical Porous Enclosure Research Articles

Some Insights on Unsteady Double Diffusive Natural Convection of Porous Medium in Vertical Open-Ended Cylinder

The work presented in this paper deals with combined heat and mass transfer by natural convection in porous media. The aim is to investigate numerically the effect of control parameters on the flow behavior as well as the enhancement of heat transfer in vertical porous enclosure. The side wall temperature is periodic function of time which is the case in several physical problems. The Darcy’s flow model coupled with the energy and mass equations is considered. The numerical results show that the flow’s behavior is strongly dependent on the buoyancy ratio values. Three types of flows take place: chimney, reversal and top aspiration flow. Further, the effect of buoyancy ratio (N) at which the different flow types occur is significantly influenced by the values of control parameters. The relative heat transfer enhancement between constant and periodical wall temperature is profoundly affected by thermal Rayleigh number (Ra), buoyancy ratio and dimensionless amplitude (XA). In the case of opposing double diffusive flow, the relative difference becomes to the favor of the stationary case for high Ra and XA.

Entropy generation for unsteady natural convection and thermal radiation inside a porous enclosure

Entropy generation problem for unsteady natural convection and thermal radiation in a vertical porous enclosure, submitted to a uniform hot walls temperature and filled with a fluid saturated porous medium, has been performed. A time–space variations study is presented for fluid flow, heat transfer and radiative transfer irreversibilities, local entropy generation Sgen and modified Bejan number Bem. A sensitivity study shows that Bem increases, particularly near the boundaries, due to an increase in N or Br values and it decreases as τD or R values increase. For the cases of ε and ω, an inversion phenomenon is observed. Furthermore, Sgen and the total entropy generation Egen values rise when there is an increase in ε, τD, R or Br values. In opposition, Sgen and Egen values decrease with the increase of ω or N values.

Thermosolutal convection in saturated porous enclosure with concentrated energy and solute sources

Double diffusive natural convection within a vertical porous enclosure with localized heating and salting from one side is numerically studied by the finite element based finite volume method. In the formulation of the problem, use is made of the Darcy model, which allows the slip boundary condition on a solid wall to be satisfied. Comparisons with benchmark solutions for natural convection in fluid saturated porous enclosures are first presented to validate the code. Following that, an extensive series of numerical simulations is conducted in the range of −55 ⩽ N ⩽ + 55 and 0.125 ⩽ L ⩽ 0.875, where N and L are the buoyancy ratio and the element location, respectively. Streamlines, heatlines, masslines, isotherms and iso-concentrations in the system are produced to illustrate the flow structure transition from solutal dominated opposing to thermal dominated and solutal dominated aiding flows, respectively. The computed average Nusselt and Sherwood numbers provide guidance for locating the heating and salting element.

Free convection from one thermal and solute source in a confined porous medium

This paper reports a numerical study of double diffusive natural convection in a vertical porous enclosure with localized heating and salting from one side. The physical model for the momentum conservation equation makes use of the Darcy equation, and the set of coupled equations is solved using the finite-volume methodology together with the deferred central difference scheme. An extensive series of numerical simulations is conducted in the range of −10 ⩽ N ⩽ + 10, 0 ⩽ R t ⩽ 200, 10−2 ⩽ Le ⩽ 200, and 0.125 ⩽ L ⩽ 0.875, where N, R t , Le, and L are the buoyancy ratio, Darcy-modified thermal Rayleigh number, Lewis number, and the segment location. Streamlines, heatlines, masslines, isotherms, and iso-concentrations are produced for several segment locations to illustrate the flow structure transition from solutal-dominated opposing to thermal dominated and solutal-dominated aiding flows, respectively. The segment location combining with thermal Rayleigh number and Lewis number is found to influence the buoyancy ratio at which flow transition and flow reversal occurs. The computed average Nusselt and Sherwood numbers provide guidance for locating the heating and salting segment.

Soret effect on the boundary layer flow regime in a vertical porous enclosure subject to horizontal heat and mass fluxes

The combined thermo- and double-diffusive convection in a vertical tall porous cavity subject to horizontal heat and mass fluxes was investigated analytically and numerically using the Darcy model with the Boussinesq approximation. The investigation focused on the effect of Soret diffusion on the boundary layer flow regime. The governing parameters were the thermal Rayleigh number, R T, the Lewis number, Le, the buoyancy ratio, N, the Soret parameter, M, which characterized the Soret effect, and the aspect ratio of the enclosure, A r . The results demonstrated the existence of a boundary layer flow solution for which the Soret parameter had a strong effect on the heat and mass transfer characteristics. For M ≠ 1 and M ≠ −1/ Le, the profiles of the vertical velocity component, v, temperature, T, and solute concentration, S, exhibited boundary layer behaviors at high Rayleigh numbers. Furthermore, as R T increased, the temperature and solute concentration became vertically and linearly stratified in the core region of the enclosure. The thermo-diffusion effect on the boundary layer thickness, δ, was discussed for a wide range of the governing parameters. It was demonstrated analytically that the thickness of the boundary layer could either increase or decrease when the Soret parameter was varied, depending on the sign of the buoyancy ratio. The effect of R T on the fluid flow properties and heat and mass transfer characteristics was also investigated.

Solidification of a binary alloy: Finite-element, single-domain simulation and new benchmark solutions

A finite-element simulation of binary alloy solidification based on a single-domain formulation is presented and tested. Resolution of phase change is first checked by comparison with the analytical results of Worster [M.G. Worster, Solidification of an alloy from a cooled boundary, J. Fluid Mech. 167 (1986) 481–501] for purely diffusive solidification. Fluid dynamical processes without phase change are then tested by comparison with previous numerical studies of thermal convection in a pure fluid [G. de Vahl Davis, Natural convection of air in a square cavity: a bench mark numerical solution, Int. J. Numer. Meth. Fluids 3 (1983) 249–264; D.A. Mayne, A.S. Usmani, M. Crapper, h-adaptive finite element solution of high Rayleigh number thermally driven cavity problem, Int. J. Numer. Meth. Heat Fluid Flow 10 (2000) 598–615; D.C. Wan, B.S.V. Patnaik, G.W. Wei, A new benchmark quality solution for the buoyancy driven cavity by discrete singular convolution, Numer. Heat Transf. 40 (2001) 199–228], in a porous medium with a constant porosity [G. Lauriat, V. Prasad, Non-darcian effects on natural convection in a vertical porous enclosure, Int. J. Heat Mass Transf. 32 (1989) 2135–2148; P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transf. 40 (1997) 3955–3967] and in a mixed liquid–porous medium with a spatially variable porosity [P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Natural convective heat transfer in an enclosure filled with fluid saturated variable porosity medium, Int. J. Heat Mass Transf. 40 (1997) 3955–3967; N. Zabaras, D. Samanta, A stabilized volume-averaging finite element method for flow in porous media and binary alloy solidification processes, Int. J. Numer. Meth. Eng. 60 (2004) 1103–1138]. Finally, new benchmark solutions for simultaneous flow through both fluid and porous domains and for convective solidification processes are presented, based on the similarity solutions in corner-flow geometries recently obtained by Le Bars and Worster [M. Le Bars, M.G. Worster, Interfacial conditions between a pure fluid and a porous medium: implications for binary alloy solidification, J. Fluid Mech. (in press)]. Good agreement is found for all tests, hence validating our physical and numerical methods. More generally, the computations presented here could now be considered as standard and reliable analytical benchmarks for numerical simulations, specifically and independently testing the different processes underlying binary alloy solidification.

Natural convection in a vertical porous slot heated from below and with horizontal concentration gradients

Double-diffusive natural convection in a slender vertical porous enclosure is studied both analytically and numerically. The buoyancy forces that induce the fluid motion result from the imposition of both a vertical temperature gradient and a horizontal solutal gradient. The first part of the study contains an analytical solution valid for stratified flows in enclosures with relatively high aspect ratios. The second part of the study contains a numerical study of the full governing equations that validate the analytical model. Comparison between the numerical and analytical solutions covers the thermal Rayleigh number range −6×10 2⩽ R T ⩽10 4, the buoyancy range 0⩽ N⩽10 3 and the Lewis number range 10 −2⩽ Le⩽10 2. In the absence of a horizontal solutal gradient ( N=0), the solution takes the form of standard Bénard bifurcation. The asymmetry resulting from the imposition of a small ( N⪡1) horizontal solutal gradient is investigated. The existence of multiple solutions, for a given range of the governing parameters, is demonstrated.

NATURAL CONVECTION IN A THERMALLY STRATIFIED WAVY VERTICAL POROUS ENCLOSURE

The effect of surface undulations on natural convection in a thermally stratified vertical porous enclosure has been analyzed numerically by the finite-element method. The influence of increasing number of surface undulations ( N ) with different wave phases (φ) and different wave amplitudes ( a ) has been analyzed. Increasing either number of waves per unit length ( N ) or wave amplitudes ( a ) or thermal stratification ( S ) is seen to decrease Nusselt number (Nu). Interestingly, the periodic variation in Nu with wave phase in the presence of thermal stratification is seen to be sensitive to the number of waves ( N ) per unit length. Maximum heat flux is obtained when the wave phase 210° h φ h 240° for 0 h S h 1. Features such as secondary circulations near the wavy wall and multicellular circulations in the core of the flow field have been noticed.

Stability analysis of thermosolutal convection in a vertical packed porous enclosure

A linear stability theory is performed to investigate the stability of the quiescent state and fully developed thermosolutal convection within a vertical porous enclosure subject to horizontal opposing gradients of temperature and solute. The fluid motion is modeled using the unsteady form of Darcy’s law coupled with energy and species conservation equations. The effect of different thermal and solutal boundary conditions is considered. The linearized governing equations are solved numerically using a finite element method. The thresholds for oscillatory and stationary convection are determined as functions of the governing parameters. It is concluded that the porosity and the acceleration parameter of the porous medium have a strong effect on the onset of overstability for a confined enclosure and on the wave number for an infinite enclosure. The stability analysis of fully developed flows within a slender enclosure reveals that an increase in the porosity and the acceleration parameter of the porous media delays the appearance of oscillatory finite amplitude flows. A nonlinear numerical solution is also computed by solving the full governing equations using a finite element method. Within the overstable regime, nonlinear traveling waves exist within slender enclosures, subject to Dirichlet thermal and solutal boundary conditions.

Boundary layer flows in a vertical porous enclosure induced by opposing buoyancy forces

Double diffusive natural convection induced in a vertical porous layer subject to opposing horizontal gradients of heat and mass is studied analytically and numerically using the Darcy model with the Boussinesq approximation. The governing parameters for the problem are the thermal Rayleigh number, R T, the Lewis number, Le, the buoyancy ratio, N and the aspect ratio, A, of the enclosure. The analytical solution is developed on the basis of the parallel flow approximation. A numerical study is performed to validate the results of the analytical predictions. It is demonstrated in this study that there exists a domain in ( Le, N) plane where, at large R T, boundary layer profiles are obtained for the velocity and density but not for the temperature and concentration. The boundary layer regimes obtained for this domain ( N<0) are extremely different from those found in the previous studies for the case of aiding buoyancy forces ( N>0).

Double-diffusive convection instability in a vertical porous enclosure

The Galerkin and the finite element methods are used to study the onset of the double-diffusive convective regime in a rectangular porous cavity. The two vertical walls of the cavity are subject to constant fluxes of heat and solute while the two horizontal ones are impermeable and adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity. For this situation, a steady rest state solution corresponding to a purely diffusive regime is possible. To demonstrate whether the solution is stable or unstable, a linear stability analysis is carried out to describe the oscillatory and the stationary instability in terms of the Lewis number, Le, normalized porosity, ε, and the enclosure aspect ratio, A. Using the Galerkin finite element method, it is shown that there exists a supercritical Rayleigh number, RsupTC, for the onset of the supercritical convection and an overstable Rayleigh number, RoverTC, at which overstability may arise. Furthermore, the overstable regime is shown to exist up to a critical Rayleigh number, RoscTC, at which the transition from the oscillatory to direct mode convection occurs. By using an analytical method based on the parallel flow approximation, the convective heat and mass transfer is studied. It is found that, below the supercritical Rayleigh number, RsupTC, there exists a subcritical Rayleigh number, RsubTC, at which a stable convective solution bifurcates from the rest state through finite-amplitude convection. In the range of the governing parameters considered in this study, a good agreement is observed between the analytical predictions and the finite element solution of the full governing equations. In addition, it is found that, for a given value of the governing parameters, the converged solution can be permanent or oscillatory, depending on the porous-medium porosity value, ε.

Multiple solutions for double-diffusive convection in a vertical porous enclosure

A numerical study is made of double-diffusive natural convection in a rectangular fluid-saturated vertical porous enclosure. The flows are driven by conditions of uniform heat and mass fluxes imposed along the two vertical side walls of the cavity where the two buoyancy effects can either augment or counteract each other. An extensive series of numerical simulations is conducted in the range 1 ⩽ R T ⩽ 165, 1 ⩽ Le ⩽ 10 3, −20 ⩽ N ⩽ 20 and A = 1, where R T , Le, N and A are the Darcy-modified Rayleigh number, Lewis number, buoyancy ratio and aspect ratio of the enclosure, respectively. For aiding flows ( N > 0) the behaviour of the resulting double-diffusive convection is in qualitative agreement with the available numerical results. For opposing flows ( N < 0) the existence of multiple steady states is demonstrated. It is determined that, for a given value of N, both Lewis and Rayleigh numbers have an influence on the domain of existence of these multiple steady states. Comprehensive Nusselt and Sherwood number data are presented as functions of the governing parameters mentioned above. The effects of the buoyancy ratio are found to be rather significant on the flow pattern and heat and mass transfer, especially for the opposing flows.

On natural convection in vertical porous enclosures due to opposing fluxes of heat and mass prescribed at the vertical walls

A two-dimensional mathematical model based on Darcy's law with Boussinesq approximation has been used to study double-diffusive natural convection in a rectangular fluid-saturated vertical porous enclosure subject to opposing and horizontal gradients of heat and solute. Results are presented for 50 ⩽ R c ⩽ 250 , 0.01 ⩽ N ⩽ 10, 10 ⩽ Le ⩽ 40 and 1 ⩽ A ⩽ 10, where R c , N, Le and A correspond to the solutal Rayleigh-Darcy number, inverse of buoyancy ratio, Lewis number and enclosure aspect ratio, respectively. The numerical integration of the full problem reveals that for sufficiently large R c , Le and A, there is a domain of N in which one obtains oscillating convection. Outside this domain, the solution approaches steady-state convection, for which analytical solutions are developed and presented. The agreement between the analytical and the numerical solutions is shown to be satisfactory.

On natural convection in vertical porous enclosures due to prescribed fluxes of heat and mass at the vertical boundaries

Unsteady and steady convection in a fluid-saturated, vertical and homogeneous porous enclosure has been studied numerically on the basis of a two-dimensional mathematical model. The buoyancy forces that induce the fluid motion are due to cooperative and constant fluxes of heat and mass on the vertical walls. For the steady state, an analytical solution, valid for stratified flow in slender enclosures, is presented. Scale analysis is applied to the two extreme cases of heat-driven and solute-driven natural convection. Comparisons between the fully numerical and analytical solutions are presented for 0.1⩽ R c⩽500, 2 ⩽ Le ⩽ 10 2 10 2 ⩽ N ⩽ 10 4 and 1 ⩽ A ⩽ 10, where R c, Le, N and A denote the solutal Rayleigh-Darcy number, Lewis number, inverse of buoyancy ratio and enclosure aspect ratio, respectively. The numerical results show that for any value of Le > 1, there exists a minimum A below which the concentration field in the core region is rather uniform and above which it is linearly stratified in the vertical direction. For sufficiently high aspect ratios, the agreement between the numerical and analytical solutions is good. The results of the scale analysis agree well with approximations of the analytical solution in the heat-driven and solute-driven limits. The numerical results indicate that for Le > 1 the thermal layers at the top and the bottom of enclosure are thinner than their solutal counterparts. In the boundary layer regime, and for sufficiently high A, the thicknesses of the vertical boundary layers of velocity, concentration and temperature are shown to be equal, regardless of the value of Le.

Non-Darcian effects on natural convection in a vertical porous enclosure

The relative importance of inertia and viscous forces on natural convection in porous media is examined via the Darcy-Brinkman-Forchheimer solutions for a differentially heated vertical cavity. These results indicate that there exists an asymptotic convection regime where the solution is independent of the permeability of the porous matrix, or the Darcy ( Da) and Forchheimer ( Fs) numbers. The higher the fluid Rayleigh number, Ra, the earlier this regime starts in terms of Da and Fs. In this convective state, not only the average Nusselt number but also the temperature and flow fields for a fixed Ra match with the fluid results provided the conductivity ratio for the porous medium is unity. Otherwise, the Nusselt number always increases with the fluid Rayleigh number, the Darcy number and the conductivity ratio while decreases when Fs increases. The effect of Prandtl number is not straightforward, and depends on the flow regime and other parameters. The numerical solutions further indicate that it is possible that a porous medium can transport more energy than a saturating fluid alone if the porous matrix is highly permeable and the thermal conductivity of solid particles is greater than that for the fluid. Several criteria to delimit the various convective flow regimes are also presented for a square cavity.

Natural convection in a vertical porous enclosure with internal heat generation

A theoretical study is performed on steady natural convection inside a rectangular porous enclosure with uniform internal heat generation and cooling from the side walls. For low Rayleigh numbers, the problem is tackled by regular expansion in the Rayleigh number and the resulting Poisson equations are sequentially solved by the modified Galerkin method. For large Rayleigh numbers, the boundary layer equations valid near the vertical walls along with the core region equations are examined using the Gill linearization method. The analytical predictions agree well with the numerical solutions based on the power-law difference representation of the full governing equations.

Numerical Study of High Rayleigh Number Convection in a Vertical Porous Enclosure

Numerical Study of High Rayleigh Number Convection in a Vertical Porous Enclosure