In this paper natural convection flows in a square cavity filled with a porous matrix has been investigated numerically when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Darcy-Forchheimer model without the inertia term is used to simulate the momentum transfer in the porous medium. Penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for a range of parameters (Rayleigh number Ra, 103 ≤ Ra ≤ 106, Darcy number Da, 10-5 ≤ Da ≤ 10-3, and Prandtl number Pr, 0.2 ≤ Pr ≤ 100) in terms of stream functions and isotherm contours, and local and average Nusselt numbers. © 2006 Elsevier Ltd. All rights reserved.

Tanmay Basak

Department Of Chemical Engineering

Satyajit Roy

Department of Mathematics

finite element method

Mathematical models

Numerical methods

Nusselt number

Porous materials

Prandtl number

DARCY FORCHHEIMER MODEL

NON UNIFORM HEATING

PENALTY FINITE ELEMENT METHOD

Square cavity

Natural convection

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International Journal of Heat and Mass Transfer

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. © 2012 Elsevier Ltd. All rights reserved.

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

The present numerical study deals with mixed convection flows within square enclosures filled with porous media. The influence of various thermal boundary conditions on bottom and side walls based on thermal aspect ratio (A) is investigated for a wide range of parameters (1 ≤ Re ≤ 100, 0.015 ≤ Pr ≤ 7.2, 10 -5 ≤ Da ≤ 10 -3 and 10 3 ≤ Gr ≤ 10 5). A penalty finite element method with bi-quadratic elements has been used to investigate the results in terms of streamlines, isotherms and heatlines and average Nusselt numbers. Lid driven effect is dominant at low Darcy number (Da = 10 -5), whereas buoyancy driven effect is dominant at high Darcy numbers (Da = 10 -4 and Da = 10 -3) for Re = 1. Asymmetric pattern is observed in isotherms and heatlines for Re = 100. It is found that thermal gradient is high at the center of the bottom wall for A = 0.1 due to large dense heatlines at that zone and that is low for A = 0.9 irrespective of Re, Pr and Gr. Overall heat transfer rates are higher for A = 0.1 compared to other thermal aspect ratios (A = 0.5, A = 0.9) irrespective of Darcy number, Prandtl number and Reynolds number. © 2012 Elsevier Ltd. All rights reserved.

Numerical study of mixed convection within porous square cavities using Bejan's heatlines: Effects of thermal aspect ratio and thermal boundary conditions

Mixed convection flows in a lid-driven square cavity filled with porous medium are studied numerically using penalty finite element analysis for uniformly heated bottom wall, linearly heated side walls or cooled right wall. The top wall is well insulated and set with uniform velocity. The relevant parameters in the present study are Darcy number (Da = 10- 5 s(-) 10- 3), Grashof number (Gr = 103 s(-) 105), Prandtl number (Pr = 0.015-10) and Reynolds number (Re = 1-102). The isotherms are generally symmetric at smaller Pr irrespective of Da and Re at Gr = 105 for linearly heated side walls. The isotherms are also almost symmetric at small Re with higher Gr(Gr = 105) and Da(Da = 10- 3) and natural convection is found to be dominant whereas the isotherms are compressed near the left and bottom walls at higher Re for linearly heated side walls. Compression of isotherms and dominance of forced convection is also observed at Re = 102 for linearly heated left wall and cooled right wall. The local Nusselt number of the bottom wall (Nub) for low Da is almost constant whereas Nub for higher Da and Pr shows non-monotonic variation at Re = 10 and an overall decreasing trend is observed at Re = 102 for linearly heated side walls. The local Nusselt numbers of left and right walls (Nul and Nur) increase for Da = 10-3 and Pr = 0.7 whereas oscillatory trend is observed at Re = 10-102 and Pr = 10. For linearly heated left wall and cooled right wall, Nub increases from the left edge towards the right edge of the bottom wall for both cases Re = 10 and 102 irrespective to Pr and Da. It is also observed that Nur for higher Pr and Da is found to increase monotonically at both Re = 10 and 102 and Nul show non-monotonic trend for higher Pr and Da at Re = 10 whereas for Re = 102, Nul increases monotonically. Average Nusselt numbers fenced(over(Nub, -), over(Nul, -), over(Nur, -)) are found almost invariant with Gr for low Pr with all Da for linearly heated side walls or cooled right wall. On the other hand, over(Nub, -) is found to vary exponentially at higher Pr and Da and oscillations inover(Nul, -) and over(Nur, -) are observed for Re = 10 whereas increasing trend is observed for over(Nul, -) and over(Nur, -) for Re = 102 for both linearly heated side walls or cooled right wall. © 2010 Elsevier Ltd. All rights reserved.

Analysis of mixed convection in a lid-driven porous square cavity with linearly heated side wall(s)

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy-Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 103 ≤ Ra ≤ 106, Darcy number Da, 10-5 ≤ Da ≤ 10-3, and Prandtl number Pr, 0.71 ≤ Pr ≤ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ≤ 10 -5 irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ≥ 10-4. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented. © 2005 Elsevier Ltd. All rights reserved.

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Convection Heat Transfer

Handbook of Porous Media

Transport Phenomena in Porous Media III

Preface

International Communications in Heat and Mass Transfer

VISCOUS DISSIPATION EFFECTS ON FREE CONVECTION IN A POROUS CAVITY

Steady natural convection flow in a square cavity filled with a porous medium for linearly heated side wall(s)

Journal	Data powered by TypesetInternational Journal of Heat and Mass Transfer
Publisher	Data powered by TypesetElsevier Ltd
ISSN	00179310
Open Access	No