MIXED CONVECTION IN A LID-DRIVEN SQUARE CAVITY FILLED WITH POROUS MEDIUM IN THE PRESENCE OF THERMAL...

MIXED CONVECTION IN A LID-DRIVEN SQUARE CAVITY FILLEDWITH POROUS MEDIUM IN THE PRESENCE OF THERMAL

RADIATION AND NON-UNIFORM HEATING

T. R. Mahapatra1, D. Pal2 and S. Mondal3∗

1Department of Mathematics, Visva-Bharati (A Central University), Santiniketan-731 235,West-Bengal, India

Email: [email protected] of Mathematics, Visva-Bharati (A Central University), Santiniketan-731 235,

West-Bengal, IndiaEmail: [email protected]

3∗Department of Mathematics, Bengal Institute of Technology and Management,Santiniketan-731 236, West-Bengal, IndiaEmail: [email protected]

Received 16 December 2012; accepted 19 May 2013

ABSTRACT

Influence of thermal radiation and heat generation on mixed-convection flow in a lid-drivensquare cavity filled with porous medium in the presence of non-uniformly heated walls is stud-ied numerically by using finite-difference technique with staggered grid distribution. Rightvertical wall is considered to be adiabatic and top wall is maintained at a constant cold temper-ature, whereas left vertical and bottom walls are non-uniformly heated. The upper horizontalwall moves on its own plane from left to right at a constant speed by keeping all other wallsfixed. The simulations are performed for a wide range of physical parameters in order to obtainnumerical solutions in terms of streamlines, isotherms, local Nusselt numbers and the averageNusselt number. Non-uniform heating causes wavy type of local Nusselt number at the bottomand left vertical walls. Average Nusselt number increases with decrease in the thermal radiationparameter. As velocity parameter increases the local Nusselt number at the left vertical wallincreases upto a certain value and thereafter opposite trend is observed.

Keywords: Lid-driven cavity; porous medium; mixed convection; non-uniform heating; ther-mal radiation.

1 INTRODUCTION

Studies on natural-convection flow in a lid-driven square cavity filled with a porous mediumhave received considerable attention in recent years because of its relevance to many industrialand technological applications such as crude oil production, storage of nuclear waste, compactbeds for the chemical industry and thermal insulation etc. Cheng (Cheng 1978) provided a com-prehensive review on free convection in fluid-saturated porous medium focusing on geothermalsystem. Prasad and Koseff (Prasad and Koseff 1996) performed an experimental investigationon mixed convection flow in a lid-driven cavity flow considering hot moving bottom wall for

Int. J. of Appl. Math. and Mech. 9(13): 23-51, 2013.

24 T. R. Mahapatra, D. Pal and S. Mondal

high Reynolds and Grashof numbers. Shankar and Deshpande (Shankar and Deshpande 2000)reviewed and showed its applications from a scientific and industrial point of view. In the lastfew decades, a number of experimental and numerical studies have been performed to analyzethe flow field and heat transfer characteristics of lid-driven cavity flow such as heat exchang-ers, solar power collectors, packed bed catalytic reactors and so on (Nield and Bejan 2006).Khanafer and Chamkha (Khanafer and Chamkha 1999) studied the unsteady mixed convectionflow in a lid-driven enclosure filled with Darcian fluid-saturated uniform porous medium withinternal heat generation. They reported that the Darcy number is most important parameter inconvective flows. Moreover, the presence of internal heat generation provides an additionaldynamic in overall convective flow system, which has significant influence on the features ofisotherms and streamlines for small values of the Richardson number. Al-Amiri (Al-Amiri2000) investigated the momentum and heat transfer in a lid-driven square cavity filled with aporous medium which was heated from a driving wall by giving a general formulation of the mo-mentum equation in the presence of inertial and viscous effects. Khanafer and Vafai (Khanaferand Vafai 2002) analyzed the double-diffusive mixed convection in a lid-driven porous enclo-sure with horizontal walls having different but constant temperature and concentration.It is well known that Darcy’s law is based on an empirical formula relating the pressure gra-dient, the gravitational force and the bulk viscous resistance in a porous medium. Hence, themathematical formulations based on the Darcy’s law neglect the effects of a solid boundary orthe inertia forces on fluid flow and heat transfer through a porous medium. But in general, theinertia and boundary effects become significant when the fluid velocity is high and the heattransfer is taken in the near-wall region. Pal (Pal 2010) studied non-Darcy mixed convectionheat transfer in a porous medium with variable porosity. A model developed by Vafai and Tien(Vafai and Tien 1981) bridged the gap between the Darcy and Navier-Stokes equations by intro-ducing the Darcy-Forchheimer model, which describes the effect of inertia as well as viscousforces in porous media. Medeiros et al. (Medeiros et al. 1999) studied numerically (finite-volume method) heat transfer by natural convection in a porous cavity under a non-Darcianapproach for uniform porosity using Darcy-Brinkman-Forchheimer model. They compared thenumerical results with those works that considered uniform porosity and they found excellentaccuracy. Nithiarasu et al. (Nithiarasu et al. 1999) examined the effect of buoyancy force onthe cavity flow and heat transfer in a fluid-saturated non-Darcian porous medium. Later, Basaket al. (Basak et al. 2006) investigated a natural-convection flow in a square cavity filled witha porous medium. They considered both uniform and non-uniform heating from below by us-ing the Darcy-Forchheimer model. Oztop (Oztop 2006) investigated combined convection heattransfer and fluid flow numerically in a partially heated porous lid-driven cavity. Kandaswamyet al. (Kandaswamy et al. 2008) analyzed the effects of Prandtl number on mixed convectionin lid-driven square cavity filled with a porous medium. They found that the dominance effectsof conduction for low Prandtl numbers and as Prandtl numbers are increased then the mixedconvection and forced convection dominate the temperature field. Oztop and Varol (Oztop andVarol 2009) investigated the velocity and temperature distributions in a lid-driven cavity filledwith porous media with non-uniformly heated bottom wall. Oztop et al. (Oztop et al. 2009)performed numerical simulations of the conduction-combined forced and natural convection(mixed convection) heat transfer and fluid flow for 2-D lid-driven square enclosure. They foundthat the heat transfer is a decreasing function of thermal conductivity ratio for all the values ofRichardson number. Basak et al. (Basak et al. 2010) analyzed mixed convection in a lid drivenporous square cavity with linearly heated side wall(s). Muthtamilselvan et al. (Muthtamilselvanet al. 2010) studied the characteristics of a lid-driven flow in a two-dimensional square cavityfilled with heat-generating porous medium.


Mixed Convection In A Lid-Driven Square Cavity Filled With Porous Medium 25

New dimensions are added in the study of natural convection flow in a cavity filled with porousmedium in the presence of thermal radiation and internal heat source/sink. It is well knownthat the effect of thermal radiation is important in space technology and high temperature pro-cesses. Thermal radiation also plays an important role in controlling heat transfer process inpolymer processing industry. The effect of radiation on heat transfer problems have been stud-ied by many authors (Hossain and Takhar 1996, Badruddin et al. 2007 and Salman Ahmed etal. 2009). There are few works in the field of mixed convection cavity flow in presence ofthermal radiation. This concept of convecting flow is utilized in critical technology applicationlike in nuclear reactor cooling, the reactor bed can be modelled as a heat generating porousmedium are very important from the point of view of convective flows. Seth et al. (Seth et al.2011) analyzed the effect of rotation on unsteady hydromagnetic natural convection flow pastin impulsively moving vertical plate with ramped temperature in a porous medium with ther-mal diffusion and heat absorption. Zahmatkesh (Zahmatkesh 2007) studied free convection ina cavity filled with porous medium in the presence of thermal radiation. Devika et al. (Devikaet al. 2011) investigated the effect of radiation on non-Darcy convection heat transfer flow ofa viscous electrically conducting fluid through a porous medium in a vertical channel. Rajputand Sahu (Rajput and Sahu 2012) studied natural convection in unsteady hydromagnetic cou-ette flow through a vertical channel in the presence of thermal radiation. Reddaiah and PrasadaRao (Reddaiah and Prasada Rao 2012) performed the effect of radiation on hydromagnetic con-vective heat transfer flow of a viscous electrically conducting fluid in a non-uniformly heatedvertical channel. Mahapatra et al. (Mahapatra et al. 2011) analyzed natural convection in alid-driven square cavity filled with porous medium in the presence of thermal radiation withuniform heating of walls of the cavity. Again, Mahapatra et al. (Mahapatra et al. 2012) inves-tigated Combined effects of thermal radiation and heat generation on natural convection in asquare cavity filled with Darcy-Forchheimer porous medium and then Mahapatra et al. (Maha-patra et al. 2012) studied influence of thermal radiation on non-Darcian natural convection in asquare cavity filled with fluid saturated porous medium of uniform porosity.Thus the motivation of the present study is to examine the influence of thermal radiation and in-ternal heat generation on natural convection in a lid-driven square cavity having fluid-saturatedporous medium using finite-difference technique with staggered grid distribution. The rightvertical wall is considered to be adiabatic and the top wall is maintained at a constant coldtemperature. Both the left vertical and bottom walls are heated non-uniformly. The upper hori-zontal wall moves on its own plane from left to right at a constant speed keeping all other wallsfixed. The poisson equation for pressure is derived using momentum and continuity equationswhich are then solved by Bi-CG-Stab method. The numerical results for streamlines, isotherms,velocity profiles and temperature profiles, the local Nusselt numbers and the average Nusseltnumber are presented graphically and in tabular form.

2 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS

The physical configuration considered in the present investigation involves a square cavity filledwith a fluid-saturated porous medium. It is assumed that the fluid obeys non-Darcian and theBoussinesq approximations. The physical model of the problem is shown in Fig. 1. It con-sists of a square porous cavity of side length L and thermal conductivity k. The right verticalwall is considered to be adiabatic and the top wall is maintained at constant cold temperature.Both the left vertical and bottom walls are heated non-uniformly. The upper horizontal wallmoves in its own plane from left to right at a constant speed keeping all other walls fixed. TheRosseland approximation (Rohsenow et al. 1998) has been used to describe the radiative heat



V = 0, T=Tc ,

U = V = 0,

T = (Th - Tc) sin ( Y/L) + Tc

L

L

U = V = 0, T = (Th - Tc) sin ( X/L) +Tc

Adiabatic

Wall

Y

X

U=U0

Porous medium

Figure 1: Schematic diagram of the physical system.

flux in the energy equation. The radiative heat flux in the x-direction is considered negligible incomparison to the y-direction. The fluid inside the enclosure is considered to be Newtonian andincompressible in nature. It is assumed to possess constant fluid properties. The thermophysicalproperties of the fluid at a reference temperature are taken as constant except in the buoyancyterm in V− momentum equation. In the Cartesian coordinate system, the basic equations areas follows:(i) Continuity equation:

∂U

∂X+∂V

∂Y= 0, (1)

(ii) U− momentum equation:

∂U

∂t′+U

ϵ

∂U

∂X+V

ϵ

∂U

∂Y= −1

ρ

∂P

∂X+ ν

(∂2U

∂X2+∂2U

∂Y 2

)− νϵ

KU − 1.75 | V⃗ |√

150KϵU, (2)



(iii) V− momentum equation:

∂V

∂t′+U

ϵ

∂V

∂X+V

ϵ

∂V

∂Y= −1

ρ

∂P

∂Y+ν

(∂2V

∂X2+∂2V

∂Y 2

)− νϵ

KV − 1.75 | V⃗ |√

150KϵV +gβ(T −Tc), (3)

(iv) Energy equation:

∂T

∂t′+ U

∂T

∂X+ V

∂T

∂Y= α

(∂2T

∂X2+∂2T

∂Y 2

)− 1

ρCp

∂qr∂Y

+Q

ρCp

(T − Tc). (4)

with boundary conditions

U = U0, V = 0 and T = Tc at Y = L, (5)

U = V = 0 and T = (Th − Tc) sin(πX

L) + Tc at Y = 0, (6)

U = V = 0 and∂T

∂X= 0 at X = L, (7)

U = V = 0 and T = (Th − Tc) sin(πY

L) + Tc at X = 0. (8)

Now we define the following non-dimensional variables:

x =X

L, y =

Y

L, u =

UL

α, v =

V L

α, θ =

T − TcTh − Tc

, p =PL2

ρα2, t =

αt′

L2, (9)

where g, Cp, t′, T , Th and Tc indicate acceleration due to gravity, specific heat at constant pres-sure, time, fluid temperature, temperature of the hot wall and temperature of the cold wallrespectively. Here α, ρ, ν, ϵ and K denote thermal diffusivity, density, kinematic viscos-ity, the porosity of the porous medium and permeability of the porous medium respectively.Here the radiation heat flux qr is considered according to Rosseland approximation such thatqr = − 4σ

3k∗∂T 4

∂Y, where σ and k∗ are the Stefan-Boltzmann constant and the mean absorption co-

efficient, respectively. Following Raptis (Raptis 1998), the fluid-phase temperature differenceswithin the flow are assumed to be sufficiently small so that T 4 may be expressed as a linearfunction of temperature. This is done by expanding T 4 in a Taylor series about the temperatureTc and neglecting higher order terms to yield T 4 = 4T 3

c T − 3T 4c (Zahmatkesh 2007).

Using the non-dimensional variables, the dimensional equations reduce to following non- di-mensional form :

∂u

∂x+∂v

∂y= 0, (10)

∂u

∂t= −∂p

∂x+ Pr

(∂2u

∂x2+∂2u

∂y2

)− 1

ϵ

(∂u2

∂x+∂uv

∂y

)− Prϵ

Dau− 1.75

√u2 + v2√

150Da ϵu, (11)

∂v

∂t= −∂p

∂y+Pr

(∂2v

∂x2+∂2v

∂y2

)− 1

ϵ

(∂v2

∂y+∂uv

∂x

)−Prϵ

Dav− 1.75

√u2 + v2√

150Da ϵv+Pr Ra ϵ θ, (12)

∂θ

∂t=

(∂2θ

∂x2+∂2θ

∂y2

)−(∂uθ

∂x+∂vθ

∂y

)+

4 NR

3

∂2θ

∂y2+He θ. (13)

The dimensionless boundary conditions are as follows:

u = A, v = 0 and θ = 0 at y = 1, (14)



u = v = 0 and θ = sin(πx) at y = 0, (15)

u = v = 0 and∂θ

∂x= 0 at x = 1, (16)

u = v = 0 and θ = sin(πy) at x = 0. (17)

A =LU0

α, Pr =

ν

α,Da =

K

L2, Ra =

gβ(Th − Tc)

ναL3, NR =

4σT 3c

kk∗, He =

QL2

k, (18)

where A, k, Q, Da, Ra, NR and He denote a velocity parameter, thermal conductivity, heatgeneration constant, the Darcy number, Rayleigh number, thermal radiation parameter and heatgeneration parameter, respectively. The heat transfer coefficient in terms of the local Nusseltnumber (Nu) is defined by

Nu = −∂θ

∂n, (19)

where n denotes the normal direction on a plane.The local Nusselt number at the bottom wall (Nub) and left vertical wall (Nul) are defined as

Nub = −∂θ∂y

y=0

and Nul = −∂θ∂x

x=0

. (20)

The average Nusselt number at the hot walls is given by,

NuH

y=0

=

∫ 1

0

Nub dx and NuH

x=0

=

∫ 1

0

Nul dy. (21)

The fluid motion in terms of stream function ψ is obtained from velocity components u andv. The relationships between stream function, ψ (Batchelor 1993) and velocity components fortwo-dimensional flows are

u =∂ψ

∂y, v = −∂ψ

∂x. (22)

From the definition of stream function, the positive sign of ψ denotes anti-clockwise circulationwhereas the negative sign of ψ represents clockwise circulation.

3 METHOD OF SOLUTION

3.1 Finite-difference formulation

Control-volume based finite-difference discretization of the continuity, momentum and tem-perature equations are carried out using staggered grid, popularly known as MAC cell methodwhich was also studied by Mahapatra et al. (Mahapatra et al. 2002). Here, the u and v veloc-ity components are evaluated at different locations of the control volume whereas the pressureand temperature are evaluated at same location of the control volume as shown in Fig. 2. Itcan be noted from the figure 2 that u, v velocity components are stored at the mid point of thevertical and horizontal faces respectively whereas pressure and temperature values are stored atthe centre of the cells. In MAC method we use different cells to discretized different equations.While discretizing continuity and temperature equation we use the cell ABCD as can be seenin Fig. 2. The cells EFGH and A′B′C ′D′ are used to discretize u-momentum and v-momentum



equations. Now, the non-dimensional u-momentum, v-momentum and temperature equationscan be written in the following form:

∂u

∂t= −∂p

∂x+ UDPC, (23)

∂v

∂t= −∂p

∂y+ V DPC, (24)

∂θ

∂t= TDPC, (25)

here,

UDPC = Pr

(∂2u

∂x2+∂2u

∂y2

)− 1

ϵ

(∂u2

∂x+∂uv

∂y

)− Prϵ

Dau− 1.75

√u2 + v2√

150Da ϵu, (26)

V DPC = Pr

(∂2v

∂x2+∂2v

∂y2

)− 1

ϵ

(∂v2

∂y+∂uv

∂x

)− Prϵ

Dav − 1.75

√u2 + v2√

150Da ϵv + PrRaϵθ, (27)

TDPC =

(∂2θ

∂x2+∂2θ

∂y2

)−(∂uθ

∂x+∂vθ

∂y

)+

4

3 NR

∂2θ

∂y2+He θ. (28)

The convective terms in the momentum equations are differenced with a hybrid formula con-sisting of central differencing and second-order upwind differencing. The diffusive terms aredifferenced by second order accurate three point central difference formula. The source termsare evaluated at the centre of the corresponding control volumes. The finite difference forms ofp, u, v, θ with t = nδt, x = iδx, y = jδy are

p(x, y, t) = p(iδx, jδy, nδt) = pnij,

u(x, y, t) = u(iδx, jδy, nδt) = unij,

v(x, y, t) = v(iδx, jδy, nδt) = vnij,

θ(x, y, t) = θ(iδx, jδy, nδt) = θnij, (29)

where superscript n refers to the time direction, subscripts i and j refer to the spatial directions,δt is the time increment and δx, δy are the length and width of the control volume. Now dis-cretization of the continuity equation at (i, j) cell takes the following form,

unij − uni−1j

δx+vnij − vnij−1

δy= 0. (30)

The finite-difference formulation of the momentum equation in x−direction using uniform grid-spacing is

un+1ij − unijδt

= −pni+1j − pnij

δx+ (UDPC)nij, (31)



where

(UDPC)nij = Pr Diffunij −

1

ϵConunij −

Prϵ

Daunij −

1.75√150Daϵ

(√(unij)

2 +(vt + vb)2

4

)unij,(32)

where, Diffunij , Conunij are diffusive and convective terms of the u-momentum equation at the

nth time level at (i, j)th cell and vt = 0.5(vnij + vni+1j), vb = 0.5(vnij−1 + vni+1j−1).

Table 1: Grid independence test for different grids.

No. of Grid points Iteration |ψmin|20× 20 10075 17.4840× 40 31009 17.1180× 80 123083 16.98160× 160 491051 16.96

Table 2: Comparison of average Nusselt number NuH |x=0 for different values of Ra whenall the cavity walls are fixed, the left vertical wall is uniformly heated and Pr = 1, Da =10−6, ϵ = 0.4, A = 0.0, NR = 0.0, He = 0.0.

NuH |x=0

Ra Nithiarasu et al. Medeiros et al. Present results(Nithiarasuetal. 1997) (Medeirosetal. 1999)

108 2.97 3.05 3.07109 11.46 12.10 12.23

5× 109 23.09 24.74 24.43

In the y−direction, the finite-difference formulation of the momentum equation is

vn+1ij − vnijδt

= −pnij+1 − pnij

δy+ (V DPC)nij, (33)

where,

(V DPC)nij = Pr Diffvnij −

1

ϵConvnij −

Prϵ

Davnij −

1.75√150Daϵ

(√(ut + ub)2

4+ (vnij)

2

)vnij

+0.5RaϵPr(θnij + θnij+1),(34)

where, Diffvnij , Convnij are diffusive and convective terms of the v-momentum equation at the

nth time level at (i, j)th cell and ut = 0.5(unij+1 + uni−1j+1), ub = 0.5(unij + uni−1j). Now, thediscretized form of energy equation is

θn+1ij − θnijδt

= (TDPC)nij, (35)



Table 3: Computed values of NuH |y=0 when Da = 10−3 and A = 1.0, for various values ofRa, Pr, NR and He.

Ra He NR Pr NuH |y=0

0.7 2.1452106 1.0 1.0 1.0 2.2208

7.0 2.43112.0 1.8457

106 1.0 1.0 0.7 2.14520.5 2.3309

1.0 2.1452106 3.0 1.0 0.7 1.9894

5.0 1.7986104 0.2697105 1.0 1.0 0.7 0.7224106 2.1452

where

(TDPC)nij = Diffθnij − Conθnij +4

3 NR

(θnij+1 − 2θnij + θnij−1

∂y2

)+Heθnij. (36)

Here, Diffθnij , Conθnij are diffusive and convective terms of the energy equation at the nth time

level at (i, j)th cell. The diffusive terms in u−momentum equation are discretized as

∂2u

∂x2=uni+1j − 2unij + uni−1j

δx2+O(δx2), (37)

∂2u

∂y2=unij+1 − 2unij + unij−1

δy2+O(δy2), (38)

and convective terms in u−momentum equation are discretized as

∂u2

∂x= (1− β)

unrunr − unl u

nl

δx+ β

unrϕnur − unl ϕ

nul

δx, (39)

∂uv

∂y= (1− β)

unt vnt − unb v

nb

δy+ β

vnt ϕnut − vnb ϕ

nub

δy. (40)

Here β, the combination factor the central-differencing and second order upwind-differencing,is determined from the numerical stability criteria. When β = 0, the convective terms arediscretized by the central-difference whereas it is second-order upwind scheme when β = 1. Inthe second-order upwind scheme, the choice of taking the momentum flux ϕ passing throughthe interface of the control volume depends on the sign of convecting velocities at that interface.Thus the momentum fluxes for u-momentum are given by

if unr ≥ 0, ϕnur = unij, (41)

if unr < 0, ϕnur = uni+1j, (42)

if unl ≥ 0, ϕnul = uni−1j, (43)

if unl < 0, ϕnul = unij, (44)



where unr = 0.5(unij + uni+1j), unl = 0.5(uni−1j + unij). Suffix u of ϕ represents the quantity for

u-momentum. Similarly,

if vnt ≥ 0, ϕnut = unij, (45)

if vnt < 0, ϕnut = unij+1, (46)

if vnb ≥ 0, ϕnub = unij−1, (47)

if vnb < 0, ϕnub = unij, (48)

where unt = 0.5(unij +unij+1), u

nb = 0.5(unij−1+u

nij), v

nt = 0.5(vnij +v

ni+1j) and vnb = 0.5(vnij−1+

vni+1j−1).The finite-difference discretization of the diffusive and convective terms of the v−momentumequation are similar to those of the u−momentum equation.The diffusive terms in temperature equation are discretized as

∂2θ

∂x2=θni+1j − 2θnij + θni−1j

δx2+O(δx2), (49)

∂2θ

∂y2=θnij+1 − 2θnij + θnij−1

δy2+O(δy2), (50)

and convective terms in temperature equation are discretized as

∂uθ

∂x= (1− β)

unr θnr − unl θ

nl

δx+ β

unrϕnθr − unl ϕ

nθl

δx, (51)

∂vθ

∂y= (1− β)

vnt θnt − vnb θ

nb

δy+ β

vnt ϕnθt − vnb ϕ

nθb

δy. (52)

Thus the momentum fluxes for temperature equation are given by

if unr ≥ 0, ϕnθr = θnij, (53)

if unr < 0, ϕnθr = θni+1j, (54)

if unl ≥ 0, ϕnθl = θni−1j, (55)

if unl < 0, ϕnθl = θnij. (56)

Suffix θ of ϕ represents the quantity for temperature equation. Similarly,

if vnt ≥ 0, ϕnθt = θnij, (57)

if vnt < 0, ϕnθt = θnij+1, (58)

if vnb ≥ 0, ϕnθb = θnij−1, (59)

if vnb < 0, ϕnθb = θnij. (60)

The convected momentum fluxes (unt , unb , u

nr , u

nl , v

nt , v

nb , v

nr , v

nl ) at the interfaces are linearly

interpolated for both u−momentum, v−momentum and temperature equations. Here suffixes,t, b, r, l denote corresponding value at top, bottom, right and left middle positions of the suitablecell faces.A pressure-Poisson equation is derived combining the discretized continuity and momentumequations. To deduce the pressure-Poisson equation, the local dialation term at (n + 1)th levelDiln+1

ij is set equal to zero. The detailed derivation of pressure-Poisson equation is presented



belowThe finite-difference discretization of u-momentum equation at (i, j) cell is

un+1ij − unijδt

= −(pni+1j − pnij

δx

)+ (UDPC)nij. (61)

The finite-difference discretization of u-momentum equation at (i− 1, j) cell is

un+1i−1j − uni−1j

δt= −

(pnij − pni−1j

δx

)+ (UDPC)ni−1j. (62)

Subtracting (62) from (61) and multiplying by 1/δx we get,

un+1ij − un+1

i−1j

δtδx−unij − uni−1j

δtδx= −

(pni+1j − 2pnij + pni−1j

δx2

)+

(UDPC)nij − (UDPC)ni−1j

δx. (63)

The finite-difference discretization of v-momentum equation at (i, j) cell is

vn+1ij − vnijδt

= −(pnij+1 − pnij

δy

)+ (V DPC)nij. (64)

The finite-difference discretization of v-momentum equation at (i, j − 1) cell is

vn+1ij−1 − vnij−1

δt= −

(pnij − pnij−1

δy

)+ (V DPC)nij−1. (65)

Subtracting equation (65) from equation (64) and multiplying by 1/δy we get,

vn+1ij − vn+1

ij−1

δtδy−vnij − vnij−1

δtδy= −

(pnij+1 − 2pnij + pnij−1

δy2

)+

(V DPC)nij − (V DPC)nij−1

δy. (66)

Addition of equations (63)and(66) we get,

2(A+B)pnij − Apni+1j − Apni−1j −Bpnij+1 −Bpnij−1 =

(Diln+1

ij −Dilnijδt

)−[(UDPC)nij − (UDPC)ni−1j

δx+

(V DPC)nij − (V DPC)ni−1j

δy

]. (67)

where, for A,B are presented below.

A =1

δx2, B =

1

δy2. (68)



The terms Dilnij and Diln+1ij are called local dilatation terms at n-th and (n+1)th time level

respectively and are defined as

Dilnij =

(unij − uni−1j

δx+vnij − vnij−1

δy

), (69)

Diln+1ij =

(un+1ij − un+1

i−1j

δx+vn+1ij − vn+1

ij−1

δy

). (70)

Assuming Diln+1ij = 0 we get the required Poisson equation for pressure equation (67) as

2(A+B)pnij − Apni+1j − Apni−1j −Bpnij+1 −Bpnij−1 =

−[Dilnijδt

+1

δx{(UDPC)nij − (UDPC)ni−1j}+

1

δy{(V DPC)nij − (V DPC)ni−1j}

]. (71)

The solution procedure starts with the initializing the velocity and temperature fields. Thisis done either from the results of previous cycle or from the prescribed initial and boundaryconditions. Using these velocity and temperature fields, pressure-Poisson equation is solvedusing Bi-CG-Stab (Bi-Conjugate Gradient Stabilized) method. Knowing pressure field, theu−momentum, v−momentum and temperature equations are solved and the values of u, v, θare updated to get the values at (n + 1)th time level. Using the values of u and v at (n + 1)thtime level, the value of the divergence of velocity field is checked for its limit. If its absolutevalue is less than 0.5 × 10−5 and steady state reaches then iteration process stops, otherwisepressure-Poisson equation is solved again for pressure.

3.2 Numerical Stability Criteria

Linear stability of fluid flow gives δt1 ≤ Min[δx|u| ,

δy|v|

], which is related to convection of fluid,

i.e., fluid should not move more than one cell width per time step (Courant, Friedrichs and Lewycondition). Also, from the Hirt’s stability analysis, δt2 ≤ Min

[1

2Pr. δx2.δy2

(δx2+δy2)

]. This condition

roughly states that momentum cannot diffuse more than one cell width per time step. The timestep actually used in the computations is determined from δt = FCT ×

[Min(δt1, δt2)

], where

the factor FCT varies from 0.2 to 0.4. The upwinding parameter β is governed by the inequalitycondition 1 ≥ β ≥ Max

[|uδtδx|, |vδt

δy|]. As a rule of thumb, β is taken approximately 1.2 times

larger than what is found from the above inequality condition.

3.3 Numerical algorithm

One complete calculation cycle comprises of the following steps:Stage 1a) Velocities unij , v

nij and temperature θnij are initialized at each cell (i, j). This is done either

from the result of previous cycle or from the prescribed initial and boundary conditions.



b) Time step (δt) is calculated from stability criteria.c) The Poisson equation for pressure is solved to get the intermediate pressure field (pnij) usingvelocities unij , v

nij and temperature θnij at the nth time step by Bi-CG-Stab Method.

d)The momentum and temperature equations are solved to get velocities un+1ij , vn+1

ij and tem-perature θn+1

ij at the (n+1)th time step in an explicit manner using pressure (pnij) as found fromsolution of Poisson equation for pressure and the velocities (unij, v

nij) as well as temperature (θnij).

Stage 2e)The maximum cell divergence of velocity field is calculated and is checked for its limit. If sat-isfied, steady state convergence or limiting non-dimensional time criteria is checked for whetherto stop calculation. If maximum divergence is not satisfactorily low it becomes necessary to goto step (a).This completes the necessary calculations for advancing the flow field through one cycle intime. The process is to be repeated until steady state convergence or limiting non-dimensionaltime is achieved. In the Fig. 3, the flow chart describing the above steps is presented.

4 RESULTS AND DISCUSSIONS

Numerical results have been obtained for the study of natural convection flow in a square cavityfilled with fluid-saturated porous media having uniform porosity (ϵ = 0.4), Rayleigh numberRa (104 ≤ Ra ≤ 5 × 109), Darcy number Da (10−4 ≤ Da ≤ 10−2), thermal radiation pa-rameter NR(0.2 ≤ NR ≤ 2.0) (Pal and Mondal 2007, Shanker et al. 2010), heat generatingparameter He (1.0 ≤ He ≤ 8.0) (Olajuwon 2009, Shanker et al. 2010), Prandtl numberPr (0.7 ≤ Pr ≤ 7.0) and the parameter A (1.0 ≤ A ≤ 1000), which have been illustratedthrough several tables and figures. Grid independence test is provided in Table 1 for variousgrid size. It is important to note that as the number of grid points are increased, the number ofiterations for obtaining the converged results for | ψmin | drastically increased. But when thenumber of grid points is increased from 80× 80 to 160× 160, no significant change is found inthe value of | ψmin |. Hence, all the results are computed taking 80× 80 grid points.When all the cavity walls are fixed, the left vertical wall is uniformly heated and Pr = 1, Da =10−6, ϵ = 0.4, NR = 0.0, He = 0.0, A = 0, the average Nusselt number obtained fromthe present study and those of Nithiarasu et al. (Nithiarasu et al. 1997) and Medeiros et al.(Medeiros et al.1999) have been compared for various values of Ra. This comparison is shownin Table 2. From this study it is observed that there exists a good agreement among the resultobtained in the present study and previously published works. It is noted from this table that anaverage Nusselt number increases with increase in the value of Rayleigh number Ra. Table 3presents the computed values of the average Nusselt number at the bottom wall (NuH |y=0) withϵ = 0.4, A = 1.0 and Da = 10−3 for various values of Pr, NR, He and Ra. From this table, itis observed that with increase in the values of Pr and Ra, the average Nusselt number increases,whereas average Nusselt number increases with decrease in the value of NR. It is also evidentfrom this table that the average Nusselt number decreases with increase in the values of He.The variations on the streamlines and isotherms for different values of Ra(= 104 to 106) andDa(= 10−4 to 10−2) with non-uniform heating of the bottom and left vertical walls are pre-sented in Figs. 4 to 13. Due to heating of the bottom and left vertical walls, the fluid rises upalong the side of the hot vertical wall which flows towards the cold vertical wall, forming aroll with clockwise rotation inside the cavity. Fig. 4 to 6 depict the streamlines and isothermsfor Da = 10−3 and Ra = 104 to Ra = 106. Distinctive isotherms and streamlines for threedifferent values of Ra can be noted. It is observed from Figs. 4(a), 5(a) and 6(a) that as Ra

increases from 104 to 106, the values of the stream function increase i.e. the flow rate increases.



Figs. 4(b), 5(b) and 6(b) depict that the isothermal lines change its value smoothly from one hotvertical and bottom walls to cold vertical and top walls with the isothermal lines having greatervalue (θ > 0.37) which cover almost the entire cavity for both the values of Ra = 104, 106

due to enhanced convection from the hot vertical and bottom wall sides to the cold wall sides ofthe cavity. Results also indicate that the non-uniform heating cause formation of small stagnantfluid region attached to the corners of the heated bottom and vertical walls.Figs. 6 to 8 depict the effects of variation of Da on the streamlines and isotherms for fixedvalues of Ra = 106, Pr = 1, A = 1.0, NR = 1.0 and He = 1.0. These three figures indicatethat for Da = 10−4 to Da = 10−2, the values of stream functions are very similar to thoseobtained with variations in Ra = 104 to 106 (as seen in Figs. 4 to 6) except the changes seento the corners of the heated bottom and one vertical plate of the cavity for non-uniform heatingas seen from Figs. 6(a), 7(a) and 8(a). Further, Figs. 6(b), 7(b) and 8(b) show the greatervalues of Da, the isotherm contours are highly dense at the center of the bottom and left walls,which indicates higher local heating rate due to non-uniform heating at bottom and left verticalwalls. Further, as the Darcy number is increased from 10−4 to 10−2, the isotherms get denser tothe heated walls. Similar behavior of streamlines and isotherms are observed by increasing thevalues of Pr, He and decreasing the values of NR as can be seen from Figs. 8 to 13.Fig. 14 shows that as velocity parameter A increases from 1 to 1000, the local Nusselt numberat the left vertical wall (Nul) increases upto a certain value of y and thereafter opposite trend isobserved for heat transfer rate at left vertical wall. The effects of the Darcy number Da on thelocal Nusselt number at the bottom wall and the left vertical wall (Nub, Nul) in the presence ofthermal radiation and heat generation are depicted in Figs. 15 and 16 when Pr = 0.7, Ra = 106.It is observed from these figures that for non-uniform heating the local Nusselt number at thebottom wall and the left vertical wall produced a wavy type of local heat transfer rate with itsminimum values at both the edges. The physical reason for this type of behavior may be due tothe choice of sinusoidal temperature boundary conditions at the heated walls. It is interestingto note that for all the values of NR, the non-uniform heating enhances the heat transfer rate atcentral regime of the cavity walls which is due to the fact that non-uniform heating causes thetemperature contours to compress toward the boundary of the cavity. Thus, the heat transferrate at the left vertical wall is very high when 0.6 ≤ y ≤ 0.8 of the left vertical wall, due tothe presence of discontinuity in the temperature boundary conditions at this edge. Further, thelocal heat transfer rate at the cold right vertical wall has its minimum value. It is also observedthat the local Nusselt number decreases exponentially to the asymptotic value of 1.0 at the rightwall as seen in Fig.15(a), 16(a) and at the top wall as seen from Fig. 15(b).



H G

F E

B A

C D

B' A'

C' D'

pij-1

ij-1

pi-1j

i-1j

pij+1

ij+1

pij

ij

!x

!y

ui-1j-1

vi-1j-1 vij-1 vi+1j-1

ui-1j uij

uij+1ui-1j+1

vij-2

uij-1

vij vi+1j

ui+1j ui-2j

vi-1j

vij+1

pi+1j

i+1j

Figure 2: Control volume for u- momentum, v- momentum and temperature equations.



Yes

START

Read input parameters

Initialize with un, v

n,

n

Set boundary conditions

Calculate Time- Step

Solve pressure Poisson equation by Bi-CG-Stab method using

un, v

n&

nto get p

n

Solve momentum and temperature equations using un, v

n,

n& p

nfor u

n+1, v

n+1&

n+1

Calculate Max. Divergence and Max ( !" #$ !" #$ %" #)

Is Divergence, limit and

steady-state criteria

attained?

Time to stop?

Print Results

STOP

Yes

No

No

Figure 3: A flow chart for solving governing equations using MAC method.



0.62

-0.21

-0.17

-0.14 -0.11-0.07

-0.04

(a) (b)

0.25

0.87

0.75

0.25

0.37

0.50

0.62

0.87

0.75

0.12

0.25

0.37

0..50

Figure 4: (a) Streamlines and (b) Isotherms for Ra = 104, Da = 10−3, Pr = 1.0, NR = 1.0 andHe = 1.0.

-0.2-0.7

-1.2-1.6-2.1

-2.6

(a)

-3.1

0.25 0.12

0.37

0.5

0.620.75

0.87

0.87

0.75 0.62

0.500.37

0.25

0.12

(b)




-3.6

-15.6-13.2

-6.0-8.4-10.8

(a)

-1.2

0.120.25

0.37

0.50

0.87

0.75

0.62

0.12

0.870.75

0.62

0.50

0.37

0.25

(b)


-2.3-6.1

-21.3

-9.9-13.7

-17.5

-25.1

(b)

0.120.25

0.37

0.50

0.87

0.75 0.62

0.120.25

0.37

0.62

0.50

0.750.87

(a)




-0.2-0.8

-1.5-2.1

-2.8-3.5

(a)

-4.1

0.120.37

0.50

0.62

0.87

0.75

0.87

0.75

0.12

0.25

0.37

0.50

0.62

(b)


-0.1

-0.8 -1.4

-2.1-2.7

-3.4

-4.0

(a)

0.120.25

0.37

0.5

0.62

0.87

0.75

0.87

0.75

0.62

0.5

0.37

0.25

0.12

(b)

Figure 9: (a) Streamlines and (b) Isotherms for NR = 1.0, He = 1.0, Pr = 0.7, Da = 10−4

and Ra = 106.



-0.8-1.5

-2.1-2.8

-3.4-4.1

(a)

-0.2

0.120.25

0.37

0.5

0.62

0.87

0.75

0.87

0.75

0.12

0.250.37

0.5

0.62

(b)

Figure 10: (a) Streamlines and (b) Isotherms for NR = 0.5, He = 1.0, Pr = 0.7, Da = 10−4

and Ra = 106.

-0.7-2.5

-4.3-6.1-7.9-9.7

(a)

-11.5

0.12

0.25

0.25

0.370.50

0.870.75

0.62

0.87

0.75 0.62

0.12

0.250.370.50

(b)

Figure 11: (a)Streamlines and (b)Isotherms for NR = 0.2, He = 1.0, Pr = 0.7, Da = 10−4

and Ra = 106.



-0.1

-0.8

-1.5-2.1

-2.8

-3.5-4.1

(a)

0.120.25

0.37

0.5

0.62

0.87

0.75

(b)

0.87

0.75

0.62 0.5

0.37

0.25

0.12

Figure 12: (a) Streamlines and (b) Isotherms for He = 2.0, NR = 1.0, Pr = 0.7, Da = 10−4

and Ra = 106.



-0.1-0.8

-1.5-2.2

-2.8 -3.5

(a)

-4.2

0.120.25

0.37

0.5

0.62

0.75

0.87

0.87

0.75

0.62

0.5

0.37

0.25

0.12

(b)

Figure 13: (a) Streamlines and (b) Isotherms for He = 5.0, NR = 1.0, Pr = 0.7, Da = 10−4

and Ra = 106.



0.0 0.2 0.4 0.6 0.8 1.0-8

-6

-4

-2

0

2

4

6

8

A= 1

, 200

, 500

, 100

0

loca

l Nus

selt

num

ber a

t lef

t ver

tical

wal

l

y

A= 1, 200, 500, 1000

Figure 14: Local Nusselt number at the left vertical wall for different values of velocity param-eter A.



-6

-4

-2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

-6

-4

-2

0

2

4

Ra=106, Pr=0.7, He=1

Da=10 -3, NR =2,1.25,0.5

Da=10-3, NR=2,1.25,0.5

Nu b

-6-4-20246

0.0 0.2 0.4 0.6 0.8 1.0

-6

-4

-2

0

2

4

(b)

Ra=106, Pr=0.7, He=1Da=10 -3

, NR =2,1.25,0.5

Da=10-3, NR= 2,1.25,0.5

Nul

Ra=106, Pr=0.7, He=1

Da=10 -4

, NR =2,1.25,0.5

Da=10-4, NR=2,1.25,0.5

Nul

y

Ra=106, Pr=0.7, He=1

Da=10 -4, N

R = 2, 1.25, 0.5

Da=10-4, NR= 2,1.25,0.5

Nu b

X(a)

Figure 15: Local Nusselt number at (a) the bottom wall and (b) the left vertical wall for differentvalues of Da and NR.



-6

-4

-2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

-6

-4

-2

0

2

Ra=106, Pr=0.7, NR=1

Da=10-3, He=1, 5, 8

Nu b

-2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

-3

-2

-1

0

1

2

3

Da=10-3,He=1,5,8

Da=10-3, He=1, 5, 8

Nul

Ra=106, Pr=0.7, NR=1

(b)

Ra=106, Pr=0.7, NR=1

Da=10-4, He=1, 5, 8

Nul

y(a)

Ra=106, Pr=0.7, NR=1

Da=10-4, He=1, 5, 8

x

Nu b

Figure 16: Local Nusselt number at (a) the bottom wall and (b) the left vertical wall for differentvalues of Da and He.



5 CONCLUSIONS

The effects of thermal radiation and heat generation on natural-convection flow in a lid-drivensquare cavity filled with fluid-saturated porous medium in the presence of non-uniformly heat-ing of side walls has been studied numerically by using finite-difference technique with stag-gered grid distribution. Numerical results for the velocity and temperature fields, local Nusseltnumbers and average Nusselt number are obtained for representative governing physical param-eters. Streamlines and isotherms for various values of Darcy number and Rayleigh number areshown graphically. Thus, follwing conditions can be drawn from the present study:1) Non-uniform heating of the bottom and left vertical walls causes the fluid to rise upwardsalong the side of the hot vertical wall which later flows towards the cold vertical wall, forminga roll with clockwise rotation inside the cavity.2)Non-uniform heating causes wavy type of local Nusselt number at the bottom and left verticalwalls.3)Average Nusselt number increases with decrease in the thermal radiation parameter whereasreverse effect is observed by increasing the value of heat generating parameter.

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