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Effects of buoyancy ratio on double-diffusive natural convection in a lid-driven cavity
International Journal of Heat and Mass Transfer 57 (2013) 771–785
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effects of buoyancy ratio on double-diffusive natural convection in a lid-driven cavity Tapas Ray Mahapatra a, Dulal Pal a, Sabyasachi Mondal b,⇑ a b
Department of Mathematics, Visva-Bharati (A Central University), Santiniketan 731235, West Bengal, India Department of Mathematics, Bengal Institute of Technology and Management, Santiniketan 731236, West Bengal, India
a r t i c l e
i n f o
Article history: Received 20 June 2012 Received in revised form 4 September 2012 Accepted 7 October 2012 Available online 5 December 2012 Keywords: Double-diffusive Lid-driven cavity flow Finite-difference method Non-uniformly heated Non-uniformly concentrated
a b s t r a c t The effects of uniform and non-uniform heating of wall(s) on double-diffusive natural convection in a liddriven square enclosure are analyzed. It is assumed that the left vertical wall and bottom wall are heated and concentrated (uniformly or non-uniformly), while the other vertical wall is maintained at a constant cold temperature and the top wall is well insulated which moves with a constant speed. The governing equations are solved numerically using staggered grid finite-difference method. Streamlines, isotherms, local Nusselt number, local Sherwood number, average Nusselt number and average Sherwood number for various values of buoyancy ratio and thermal Rayleigh number are obtained. The results are compared with previously published work and excellent agreement has been obtained. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Fluid flow, heat and mass transfer induced by double-diffusive natural convection in fluid saturated porous media have practical importance in many engineering applications. This aspect of fluid dynamics has gained considerable attention in recent years among the researchers. Migration of moisture in fibrous insulation, drying processes, chemical reactors, transport of contaminants in saturated soil and electro-chemical processes are some examples of double-diffusive natural convection phenomena. Double-diffusion occurs in a wide range of scientific field, such as oceanography, astrophysics, geology, biology and chemical processes. So, the researchers have keen interest in the study of heat and mass transfer in enclosure and cavity. Double-diffusive natural convection in cavities has been subject of an intensive research due to its importance in various engineering and geophysical problems. This includes nuclear reactors, solar ponds, geothermal reservoirs, solar collectors, crystal growth in liquids, electronic cooling and chemical processing equipments. Wee et al. [1] and Beghein et al. [2] presented comprehensive reviews on the study of heat and moisture transfer by natural convection in a rectangular cavity. Thereafter, Ghorayeb et al. [3] analyzed the onset of oscillatory flow in double diffusive convection in a square cavity with equal but opposing horizontal temperature
⇑ Corresponding author. E-mail addresses: [email protected] (T.R. Mahapatra), [email protected] (D. Pal), [email protected] (S. Mondal). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.028
and concentration gradients numerically. The results presented the influence of the Lewis number on the transition of steady state convective and oscillatory flow structures occurring beyond this transition. Deng et al. [4] investigated fluid, heat and contaminant transport structures of laminar double diffusive mixed convection in a two-dimensional ventilated enclosure numerically. The effect of thermal modulation on the onset of double-diffusion natural convection in a horizontal fluid layer was studied analytically and numerically using a linear stability analysis by Malashetty et al. [5]. They found that the frequency symmetric modulation was destabilizing while high frequency symmetric modulation was always stabilizing. The effects of combined thermal and solutal buoyancy induced by temperature and concentration gradients have, however, not been widely studied. Yan [6,7] and Lee et al. [8] analyzed the transport phenomena of developing laminar mixed convection heat and mass transfer in rectangular ducts. Later, Alimi et al. [9] studied the buoyancy effects on mixed convection heat and mass transfer in an inclined duct preceded with a double step expansion. Brown and Lai [10] numerically examined combined heat and mass transfer from a horizontal channel with an open cavity heated from below numerically. Teamah [11] studied double-diffusive convective flow in a rectangular enclosure with the upper and lower surfaces being insulated and impermeable by imposing constant temperature and concentration along the left and right walls of the enclosure and a uniform magnetic field was applied in a horizontal direction. Saha et al. [12] investigated the new characteristics of the airflow and heat/contaminant transport mechanism inside a vented cavity in terms of streamlines, isotherms and isoconcentration lines.
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wycz et al. [17] discovered that the discontinuity can be avoided by choosing a non-uniform temperature distribution along the walls (i.e. non-uniformly heated walls), in an investigation on mixed convection flow over a vertical plate, which is non-uniformly heated/cooled. Roy and Basak [18] solved the nonlinear coupled partial differential equations for flow and temperature fields with both uniform and non-uniform temperature distributions prescribed at the bottom wall and at one vertical wall. Teamah et al. [19] studied the effect of the heater length, Rayleigh number, Prandtl number and buoyancy ratio on both average Nusselt and Sherwood number with uniform heating at left vertical wall. Heat and mass transfer in lid-driven enclosures have received less attention in the above literature. In drying technology, better understanding of the drying process is vital for optimum performance of the drying chamber. Alleborn et al. [20] investigated a two-dimensional flow accompanied by heat and mass transport in a shallow lid-driven cavity with a moving heated lid and a moving cooled lid. Their results showed that the drying rates were enhanced by increasing the velocity and become increasingly independent of the gravity orientation because of the dominance of forced convection. Wan and Kuznetsov [21] investigated the Fig. 1. Schematic diagram of the physical system.
Kuznetsov and Nield [13] studied the double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate analytically. Later, Nield and Kuznetsov [14] presented an analytical treatment of double-diffusive nanofluid convection in a porous medium. Teamah et al. [15] studied the effects for a wide range of thermal Grashof number and aspect ratio coupled with the inclination angle. The obtained results for average Nusselt and Sherwood numbers were correlated. Trevisan and Bejan [16] investigated the phenomenon of natural convection caused by combined temperature and concentration buoyancy effects in a rectangular enclosure with uniform heat and mass fluxes applied along the vertical walls numerically and analytically. They found that in the case of uniformly heated walls, the finite discontinuity in the temperature distribution appeared at the right edge of the bottom wall. Minko-
Table 1 Grid independence test when Pr = 0.7, Le = 2.0, N = 1, A = 1 and Ra⁄ = 103. No. of Grid points
20 20 40 40 80 80 160 160
Uniformly heated and uniformly concentrated
Non-uniformly heated and non-uniformly concentrated
Iteration
jwminj
Iteration
jwminj
19,493 78,004 312,806 1,251,253
2.1149 2.0906 2.0856 2.0846
19,856 79,449 318,646 1,274,605
1.8134 1.7924 1.7887 1.7878
Table 2 Computed values of NuH jy¼0 and NuH jx¼0 when Pr = 0.7, Le = 2.0, A = 1 and N = 1.0 for various values of Ra⁄. Ra⁄
N
103 104 2 104 5 104 105 2 105 5 105 8 105 106
1.0
Uniformly heated and uniformly concentrated
Non-uniformly heated and non-uniformly concentrated
NuH jy¼0
NuH jx¼0
NuH jy¼0
NuH jx¼0
4.293 5.424 5.891 6.663 7.421 8.351 9.859 10.783 11.265
0.409 0.779 0.980 1.318 1.663 2.120 2.918 3.416 3.676
1.325 2.263 2.659 3.299 3.863 4.555 5.670 6.340 6.684
0.186 0.391 0.524 0.739 0.969 1.269 1.765 2.063 2.217
Table 3 Computed values of ShH jy¼0 and ShH jx¼0 when Pr = 0.7, Le = 2.0, A = 1 and N = 1.0 for various values of Ra⁄.
Fig. 2. Control volume for u-momentum, v-momentum, temperature and concentration equations.
Ra⁄
N
103 104 2 104 5 104 105 2 105 5 105 8 105 106
1.0
Uniformly heated and uniformly concentrated
Non-uniformly heated and non-uniformly concentrated
ShH jy¼0
ShH jx¼0
ShH jy¼0
ShH jx¼0
4.671 6.162 6.790 7.813 8.799 10.010 11.947 13.13 13.75
0.515 1.176 1.468 1.993 2.529 3.20 4.349 5.075 5.459
1.613 2.849 3.377 4.221 4.963 5.845 7.251 8.087 8.514
0.204 0.672 0.861 1.176 1.512 1.929 2.601 3.019 3.245
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fluid flow in a rectangular cavity whose lid vibrates in one standing wavelength. The studies of double-diffusive natural convection have centered their analysis on the limit cases of dominating thermal buoyancy force or concentration buoyancy force. The aim of the present investigation is to study the effects of buoyancy ratio, circulations, heat transfer rate at the heated walls in terms of local Nusselt number, average Nusselt number and local Sherwood number, average Sherwood number when the bottom wall and left vertical wall are heated and concentrated (uniformly and non-uniformly) and right vertical wall is cooled by means of a constant temperature while the top wall is well insulated with a constant speed from left to right. The numerical work presented in this paper focuses on an entirely new class of flows in this kind of cavities. The thermal and mass exchanges generated in the case of co-operating thermal and concentration buoyancy effects with uniform and non-uniform boundary conditions have been analyzed.
2. Governing equations and boundary conditions An unsteady-state two-dimensional square cavity of height L as shown in Fig. 1 is considered. It is assumed that the top wall is moving from left to right at a constant speed U0 and is considered to be adiabatic. The bottom wall and left vertical wall are heated and concentrated (uniformly or non-uniformly) and right vertical wall is cooled by means of a constant temperature. The thermophysical properties of the fluid are assumed to be constant except the density variation in the buoyancy force, which is approximated according to the Boussinesq approximation. This variation, due to
(a)
both temperature and concentration gradients, can be described by the following equation:
q ¼ q0 ½1 bT ðT T c Þ bS ðC C c Þ;
ð1Þ
where bT and bS are the thermal and concentration expansion coefficients, respectively. In the Cartesian coordinate system, the fundamental governing equations are as follows:
@U @V þ ¼ 0; @X @Y
ð2Þ !
@U @U @U 1 @P @2U @2U ; þ þV ¼ þm 0 þU @t @X @Y q @X @X 2 @Y 2 ! @V @V @V 1 @P @2V @2V þ U m þ þ V ¼ þ @t 0 @X @Y q @Y @X 2 @Y 2
@T @T @T þU þV @t 0 @X @Y @C @C @C þV þU @t0 @X @Y
þ gbT ðT T c Þ þ gbS ðC C c Þ; ! @2T @2T ; þ ¼a @X 2 @Y 2 ! @2C @2C : ¼D þ @X 2 @Y 2
ð3Þ
ð4Þ ð5Þ
ð6Þ
The associated boundary conditions are when t0 = 0 for 0 6 X, Y 6 L:
UðX; YÞ ¼ 0 ¼ VðX; YÞ;
ð7Þ
TðX; YÞ ¼ T c ;
ð8Þ
CðX; YÞ ¼ C c ;
(b)
Fig. 3. Comparison of present local Nusselt number at the bottom and left vertical walls for uniform heating (—) and non-uniform heating (- - -) with Roy and Basak [18].
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when t0 > 0 for 0 6 X, Y 6 L:
UðX; LÞ ¼ U 0 ;
UðX; 0Þ ¼ Uð0; YÞ ¼ UðL; YÞ ¼ 0;
ð11Þ
Here x and y are dimensionless coordinates along the horizontal and vertical directions respectively; u and v are dimensionless velocity components in the x- and y-directions respectively; h and S denote the dimensionless temperature and concentration respectively; p is the dimensionless pressure parameter. Using these dimensionless variables, we obtain the following dimensionless governing equations from Eqs. (2)–(6):
ð12Þ
@u @ v þ ¼ 0; @x @y
ð13Þ
! @u @p @2u @2u @u2 @uv ¼ þ Pr þ þ ; @t @x @x2 @y2 @x @y
ð9Þ
VðX; 0Þ ¼ VðX; LÞ ¼ Vð0; YÞ ¼ VðL; YÞ ¼ 0; TðX; 0Þ ¼ T h or TðX; 0Þ ¼ ðT h T c Þ sinðpX=LÞ þ T c ; @T ðX; LÞ ¼ 0; @Y Tð0; YÞ ¼ T h or Tð0; YÞ ¼ ðT h T c Þ sinðpY=LÞ þ T c ; TðL; YÞ ¼ T c ; CðX; 0Þ ¼ C h or CðX; 0Þ ¼ ðC h C c Þ sinðpX=LÞ þ C c ; @C ðX; LÞ ¼ 0; @Y Cð0; YÞ ¼ C h or Cð0; YÞ ¼ ðC h C c Þ sinðpY=LÞ þ C c ; CðL; YÞ ¼ C c ;
ð10Þ
ð14Þ
where, X and Y are the distances measured along the horizontal and vertical directions respectively; U and V are velocity components in the X- and Y-directions respectively; T and C denote the temperature and concentration respectively; m, a and D are kinematic viscosity, thermal diffusivity and mass diffusivity respectively; P is the pressure and q is the density; Th and Tc are the temperatures at the hot and cold walls respectively; Ch and Cc are the concentrations at the hot and cold walls respectively; L is the side of the square cavity. We now introduce dimensionless variables given as follows:
t¼
at
0
2
;
x¼
L T Tc ; h¼ Th Tc
X ; L
Y UL ; u¼ ; L a C Cc S¼ : Ch Cc y¼
v¼
VL
a
2
;
p¼
PL
qa2
;
ð15Þ ð16Þ
ð17Þ
@v @p @2v @2v þ ¼ þ Pr @y @t @x2 @y2 @h ¼ @t
!
2 @v @uv þ þ Pr Ra ðh þ NSÞ; ð19Þ @y @x
! @2h @2h @uh @ v h ; þ þ @x2 @y2 @x @y
@S 1 @2S @2S ¼ þ @t Le @x2 @y2
!
ð18Þ
@uS @ v S þ : @x @y
ð20Þ
ð21Þ
The dimensionless boundary conditions are when t = 0 for 0 6 x, y 6 1:
uðx; yÞ ¼ 0 ¼ v ðx; yÞ;
ð22Þ
hðx; yÞ ¼ 0;
ð23Þ
Sðx; yÞ ¼ 0;
(a)
(b)
(c)
Fig. 4. Contour plots for Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and (a) N = 50, (b) N = 1, (c) N = 50 (uniformly heated and uniformly concentrated).
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The average Nusselt numbers at the bottom wall and left vertical wall are given by,
when t > 0 for 0 6 x, y 6 1:
uðx; 1Þ ¼ A;
uðx; 0Þ ¼ uð0; yÞ ¼ uð1; yÞ ¼ 0;
ð24Þ
v ðx; 0Þ ¼ v ðx; 1Þ ¼ v ð0; yÞ ¼ v ð1; yÞ ¼ 0;
ð25Þ
hðx; 0Þ ¼ 1 or hðx; 0Þ ¼ sinðpxÞ;
@h ðx; 1Þ ¼ 0; @y
ð26Þ
hð0; yÞ ¼ 1 or hð0; yÞ ¼ sinðpyÞ;
hð1; yÞ ¼ 0;
ð27Þ
Sðx; 0Þ ¼ 1 or Sðx; 0Þ ¼ sinðpxÞ;
@S ðx; 1Þ ¼ 0; @y
ð28Þ
Sð0; yÞ ¼ 1 or Sð0; yÞ ¼ sinðpyÞ;
Sð1; yÞ ¼ 0;
ð29Þ
NuH jy¼0 ¼
Z
Z
1
0
Nub dx and NuH jx¼0 ¼
1
Nul dy:
ð32Þ
0
The local Sherwood number at the left vertical wall and bottom walls are defined as Shl = (@S)/(@x)jx=0 and Shb = (@S)/(@y)jy=0. The average Sherwood numbers at he bottom wall and left vertical wall are given by,
ShH jy¼0 ¼
Z
1 0
Shb dx and ShH jx¼0 ¼
Z
1
Shl dy:
ð33Þ
0
where 3. Method of solution 3
2
GrT ¼ gbT ðT h T c ÞL =m ; GrC ; N¼ GrT
3
2
GrC ¼ gbS ðC h C c ÞL =m ;
m Pr ¼ ; Ra ¼ GrT Pr ; A ¼ U 0 L=a: a
ð30Þ ð31Þ
Here A, N, Le, Pr, GrC, GrT and Ra are a velocity parameter, buoyancy ratio, Lewis number, Prandtl number, solutal Grashof number, thermal Grashof number and thermal Rayleigh number respectively. The heat transfer coefficient in terms of the local Nusselt number (Nu) is Nu = (@h)/(@n), where n denotes the normal direction to a plane. The local Nusselt number at the left vertical wall and bottom walls are defined as Nul = (@h)/(@x)jx=0 and Nub = (@h)/(@y)jy=0.
Control-volume based finite-difference discretization of Eqs. (17)–(21) are carried out using staggered grid, popularly known as MAC cell method. The grid alignment of the velocities are evaluated at different locations of the control volume whereas the pressure and temperature are evaluated at same location of control volume as shown in Fig. 2. The non-dimensional equations have been derived in distinct types of cells for four equations, viz., (i) continuity cell, (ii) u-momentum cell, (iii) v-momentum cell and (iv) temperature cell. Now, the non-dimensional Eqs. (18)–(21) can be written in the following form:
Fig. 5. Contour plots for Pr = 0.7, Le = 2.0, A = 1, N = 1 and (a) Ra⁄ = 103, (b) Ra⁄ = 104, (c) Ra⁄ = 105 (uniformly heated and uniformly concentrated).
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Fig. 6. Contour plots for Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and (a) N = 50, (b) N = 1, (c) N = 50 (non-uniformly heated and non-uniformly concentrated).
Fig. 7. Contour plots for Pr = 0.7, Le = 2.0, A = 1, N = 1 and (a) Ra⁄ = 103, (b) Ra⁄ = 104, (c) Ra⁄ = 105 (non-uniformly heated and non-uniformly concentrated).
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785
@u @p ¼ þ UDPC; @t @x @v @p ¼ þ VDPC; @y @t @h ¼ TDPC; @t @S ¼ CDPC: @t here;
" # @2u @2u @u2 @uv ; þ 2 þ UDPC ¼ Pr 2 @x @y @x @y " # @2v @2v @ v 2 @uv þ Pr Ra ðh þ NSÞ VDPC ¼ Pr þ 2 þ 2 @x @y @x @y " # @2h @2h @uh @ v h ; þ þ TDPC ¼ @x2 @y2 @x @y " # 1 @2S @2S @uS @ v S : CDPC ¼ þ þ Le @x2 @y2 @x @y
777
ð34Þ
and the source terms are centrally differenced without changing the position of the cavity at the center of the control volumes. Let
ð35Þ
t ¼ ndt;
ð36Þ ð37Þ
x ¼ idx;
hðx; y; tÞ ¼ hðidx; jdy; ndtÞ ¼ hnij ; Sðx; y; tÞ ¼ Sðidx; jdy; ndtÞ ¼ Snij , where superscript n refers to the time direction, subscripts i an, j refer to the spatial directions, dt is the time increment and dx, dy are the length and width of the control volume. Now pressure gradient are discritized as n
ð38Þ ð39Þ ð40Þ ð41Þ
The convective terms in the momentum equations are differenced with a hybrid formula consisting of central differencing and secondorder upwinding. The diffusive terms are differenced by second-order-accurate three-point central difference formula in both the cases
y ¼ jdy and pðx; y; tÞ ¼ pðidx; jdy; ndtÞ ¼ pnij ;
n
@p piþ1j pij ¼ þ Oðdx2 Þ; @x dx n n @p pijþ1 pij ¼ þ Oðdy2 Þ: @y dy
ð42Þ ð43Þ
Discretization of the continuity equation at (i, j) cell takes the following form,
unij uni1j dx
þ
v nij v nij1 dy
¼ 0:
ð44Þ
The finite-difference formulation of the momentum equation in xdirection using uniform grid-spacing is
unþ1 unij ij dt
¼
pniþ1j pnij dx
þ ðUDPCÞnij ;
ð45Þ
Fig. 8. Contour plots for Pr = 0.7, Ra⁄ = 103, Le = 2.0, A = 1, N = 1 and (a) t = 0.0112, (b) t = 0.0335, (c) t = 0.0558, (d) t = 1.7779 (non-uniformly heated and non-uniformly concentrated).
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where
ðUDPCÞnij ¼ Pr Diff unij Con unij :
ð46Þ
Here, Diff unij , Con unij are diffusive and convective terms of the umomentum equation at the nth time level at (i, j)th cell. In the y-direction, the finite-difference formulation of the momentum equation is
v nþ1 v nij ij dt
¼
pnijþ1 pnij dy
þ ðVDPCÞnij ;
ð47Þ
where,
!
ðVDPCÞnij ¼ Pr Diff v nij Con v nij þ Pr Ra
hnij þ hnijþ1 Snij þ Snijþ1 þN : 2 2
hnþ1 hnij ij ¼ ðTDPCÞnij ; dt
ð49Þ
where
ðTDPCÞnij ¼ Diff hnij Con hnij : Diff hnij ;
ð50Þ
Con hnij
Here, are diffusive and convective terms of the energy equation at the nth time level at (i, j)th cell. Now, we have discritized form of concentration equation (37)
Snþ1 Snij ij ¼ ðCDPCÞnij ; dt
ð51Þ
where
ð48Þ Here, Diff v nij , Con v nij are diffusive and convective terms of the v-momentum equation at the nth time level at (i, j)th cell. Now, we have discritized form of energy equation (36)
ðCDPCÞnij ¼
1 Diff Snij Con Snij : Le
ð52Þ
Here, Diff Snij and Con Snij denote diffusive and convective terms of the concentration equation at the nth time level at (i, j)th cell. The diffusive terms in u-momentum equation are discretized as
(a)
(b)
(c)
(d)
Fig. 9. Local Nusselt number and local Sherwood number for uniformly heated and uniformly concentrated for Pr = 0.7, Le = 2.0, A = 1 and Ra⁄ = 103.
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785 n
n
n
@ 2 u uiþ1j 2uij þ ui1j ¼ þ Oðdx2 Þ; @x2 dx2 n n n @ 2 u uijþ1 2uij þ uij1 ¼ þ Oðdy2 Þ; 2 2 @y dy
ð53Þ ð54Þ
and convective terms in u-momentum equation are discretized as
Here b is the upwinding parameter which is determined from the numerical stability criteria which is described in Section 3.1. The finite-difference formulation for the diffusive and convective terms in the v-momentum equation are similar to that in umomentum formulation. The diffusive terms in temperature equation are discretized as n
un un unl unl un /n unl /nul @u2 ¼ ð1 bÞ r r þ b r ur ; @x dx dx un v n unb v nb v n /n v nb /nub @uv ¼ ð1 bÞ t t þ b t ut ; @y dy dy
ð55Þ ð56Þ
779
n
n
@ 2 h hiþ1j 2hij þ hi1j ¼ ; @x2 dx2 n n n 2 @ h hijþ1 2hij þ hij1 ¼ ; @y2 dy2
ð57Þ ð58Þ
and convective terms in temperature equation are discretized as when unr P 0 then /nur ¼ unij ; when unl P 0 then /nul ¼ uni1j ; when unr < 0 then /nur ¼ uniþ1j ; when unl < 0 then /nul ¼ unij ; when v nt P 0 then /nut ¼ unij ; when v nb P 0 then /nub ¼ unij1 ; when v nt < 0 then /nut ¼ unijþ1 ; when v nb < 0 then /nub ¼ unij .
un hn unl hnl un /n unl /nhl @uh ¼ ð1 bÞ r r þ b r hr ; @x dx dx n n n n n n n n v h v b hb v t /ht v b /hb @v h ¼ ð1 bÞ t t þb ; @y dy dy
(a)
(b)
(c)
(d)
Fig. 10. Local Nusselt number and local Sherwood number for non-uniformly heated and non-uniformly concentrated for Pr = 0.7, Le = 2.0, A = 1 and Ra⁄ = 103.
ð59Þ ð60Þ
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T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785
when unr P 0 then /nhr ¼ hnij ; when unl P 0 then /nhl ¼ hni1j ; when unr < 0 then /nhr ¼ hniþ1j ; when unl < 0 then /nhl ¼ hnij ; when v nt P 0 then /nht ¼ hnij ; when v nb P 0 then /nhb ¼ hnij1 ; when v nt < 0 then /nht ¼ hnijþ1 ; when v nb < 0 then /nhb ¼ hnij .
4. Results and discussion
The finite-difference formulation for the diffusive and convective terms in the concentration equation are similar to that in temequation formulation. The convected momentum fluxes perature unt ; unb ; unr ; unl ; v nt ; v nb ; v nr ; v nl at the interfaces are linearly interpolated for both u-momentum and v-momentum fluxes. In the second-order upwinding scheme, the choice of taking the momentum flux / passing through the interface of the control volume depends on the sign of convecting velocities at that interface. Here b is the combination factor between the central-differencing scheme and second order upwind-differentiating scheme. When b = 0, the convective terms are represented by the central-difference whereas it is second-order upwinding when b = 1. A pressure-Poisson equation is derived combing the discretized momentum and continuity equations. Here the local dilation term nþ1 at (n + 1)th level Div ij is set equal to zero. Thus the final form of the pressure-Poisson equation is n
n
2ða þ bÞpnij apniþ1j apni1j bpijþ1 bpij1 " n # o 1n o Dij 1 n ðUDPCÞnij ðUDPCÞni1j þ ðVDPCÞnij ðVDPCÞni1j ; ¼ þ dy dt dx
Grid independence test is provided in Table 1 for various grid sizes when Pr = 0.7, Le = 2.0, N = 1, A = 1 and Ra⁄ = 103. It is important to note that as the number of grid points are increased, the number of iterations required for getting the converged results for jwminj increases. But when the number of grid points increases from 80 80 to 160 160, no significant change found in the value of jwminj. Hence, all the results are computed taking 80 80 grid points. Tables 2 and 3 display the computed values of NuH jy¼0 , NuH jx¼0 , ShH jy¼0 and ShH jx¼0 when Pr = 0.7, Le = 2.0, A = 1 and N = 1.0 for various values of Ra⁄ on the heated walls. It is seen from these two tables that NuH jy¼0 , NuH jx¼0 , ShH jy¼0 and ShH jx¼0 increase with increase in Ra⁄. Fig. 3 shows the comparison of present results with that of Roy and Basak [18] for local Nusselt number at bottom wall and left vertical wall with uniform and non-uniform heating when A = 0 and other parameters are same as in [18]. From this graph it is observed that a very good agreement has been obtained with the previously published results. The numerical results for streamlines, isotherms, isoconcentrations, local Nusselt number, average Nusselt number, local Sherwood number and average Sherwood number at the heated walls are presented in Figs. 4–12 for uniform and non-uniform heating.
ð61Þ where Dnij is the divergence of the velocity field at the cell (i, j) at n-th time level, which is to be zero for the convergence of the flow in successive iterations of the method and
a¼
1 ; dx2
b¼
1 ; dy2
Dnij ¼
unij uni1j dx
þ
v nij v nij1 dy
:
ð62Þ
3.1. Solution procedure and numerical stability criteria Now the iteration process is described to obtain the solutions of the basic equations with appropriate boundary conditions. In the derivation of pressure-Poisson equation, the divergence term at n-th time level ðDnij Þ is retained and evaluated in the pressure-Poisson iteration. It is done because the discretized form of divergence of velocity field, i.e., Dnij is not guaranteed to be zero initially. The solution procedure starts with the initializing the velocity field. This is done either from the result of previous cycle or from the prescribed initial and boundary conditions. Using this velocity field pressure-Poisson equation is solved using Bi-CG-Stab method [22]. Knowing the pressure field, equation for u-momentum, v-momentum, temperature and concentration are updated to get u,v,h and S at (n + 1)th time level. Then using the values of u and v at (n + 1)th time level, the value of the divergence of velocity field is calculated and checked for its limit. If its absolute value is less than 0.5 105 and steady state is reached then iteration process stops, otherwise pressure-Poisson equation is solved again for pressure. h i dx dy Linear stability of fluid flow gives dt 1 6 Min juj ; jv j , which is related to the convection of fluid, i.e., fluid should not move more than one cell width per time step (Courant, Friedrichs and Lewy condition). Also, from the Hirt’s stability analysis, we have h i 2 dy2 dt2 6 Min 2P1 r ðdxdx2 þdy 2 Þ . This condition roughly states that momentum cannot diffuse more than one cell width per time step. The time step is determined from dt = FCT [Min (dt1,dt2)], where the factor FCT varies from 0.2 to 0.4. The upwinding parameter b is h i ; v dt . As governed by the inequality condition 1 P b P Max udt dx dy a rule of thumb, b is taken approximately 1.2 times larger than what is found from the above inequality condition.
(a)
(b) Fig. 11. Local Nusselt number for different values of A (a) uniformly heated and uniformly concentrated (b) non-uniformly heated and non-uniformly concentrated Pr = 0.7, Le = 2.0, N = 1 and Ra⁄ = 103.
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The effects of N and Ra⁄ on streamline, isotherm and isoconcentration contours for A = 1, Le = 2 and Pr = 0.7 are displayed in Figs. 4–7. Fig. 4 shows the effect of N on the streamlines, isotherms as well as on the isoconcentration for wide range of variations in the buoyancy ratio (N) with uniform heating at left vertical wall and the bottom walls. As N decreases from 50 to 1, the values of stream function decrease i.e., strength of flow circulation decreases. But when the value of N is further decreased from 1 to 50 then the values of stream function increase, i.e., strength of flow circulation increase with decrease in N. It is clear from Fig. 4(a) that when N = 50 the contours of both the isotherms and isoconcentrations are mainly concentrated near the cold vertical wall but when the value of N is decreased to 1, they are dis-
781
persed. Again when the value of N is further decreased to 50, the contours are concentrated near the lower half of the cold vertical wall and upper half of the hot vertical wall. Fig. 4(c) shows that the isotherm and isoconcentration lines are almost parallel to the horizontal wall in the middle part of the cavity for N = 50, indicating that most of the heat transfer is carried out by heat conduction. This is due to an increase in the thermal boundary layer thickness. At low buoyancy ratio, the entire enclosure is influenced by the flow structure and as the buoyancy ratio increases the boundary layer thickness becomes thinner. This change in the flow structure for high buoyancy ratio has a significant influence on the concentration field, which builds up a vertical stratification of the enclosure. Again it is interesting to note that for
(a)
(b)
(c)
(d)
Fig. 12. Average Nusselt number and average Sherwood number for different values of N: (a,b) uniformly heated and non-uniformly concentrated and (c,d) uniformly heated and uniformly concentrated when Pr = 0.7, Le = 2.0, A = 1 and Ra⁄ = 103.
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N = 50, the effect of solutal buoyancy force is in the opposite direction of thermal buoyancy force. When N = 1, the isothermal and isoconcentration countours tend toward the left wall, as both thermal buoyancy force and solutal buoyancy force are equal. When N = 50, the effect of solutal buoyancy force is in the same direction of thermal buoyancy force. In such case the isothermal and isoconcentration contours tend toward the right wall. Streamline, isotherm and isoconcentration contours for various Ra ¼ 103 to 105 with uniform heating of a vertical wall and the bottom wall are displayed in Fig. 5. As expected due to a hot vertical wall, fluids rise up along the side of the hot vertical wall and flow down along the cold vertical wall forming a roll with clockwise rotation inside the cavity. As Ra increases from 103 to 105, the values of the stream function increase. The streamlines are crowded near the cavity wall and the cavity core is empty. It is clear from
Fig. 5(c) that when Ra ¼ 105 , the contours of both the isotherms and isoconcentrations are mainly concentrated near the cold vertical wall but when Ra decreases to 104, they are dispersed. Again when Ra is further decreased to 103, the contours are more dispersed towards the left vertical wall. As seen in Figs. 4 and 5, uniform heating of the bottom wall and left vertical wall cause a finite discontinuity in Dirichlet type of boundary conditions for the temperature distribution at one edge of the bottom wall. In contrast, the non-uniform heating removes the singularity at the edge of the bottom wall and provides a smooth temperature distribution in the entire cavity. Figs. 6 and 7 present the streamline, isotherm and isoconcentration contours for various values of N and Ra when the bottom wall and left vertical wall are non-uniformly heated. Comparing Figs. 4 and 6 or Figs. 5 and 7 it can be said that the patterns of streamlines are almost similar for uniformly and non-uniformly heated and concentrated cases. But significant differences
Fig. 13. Least square curve fitting of average Nusselt number with thermal Rayleigh number at bottom and left vertical walls for (a,c) uniformly heated and uniformly concentrated and (b,d) non-uniformly heated and non-uniformly concentrated when Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and N = 1.
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are observed in the patterns of isotherms and isoconcentrations for uniformly and non-uniformly heated and concentrated cases. Fig. 8 shows the changes of the patterns of the streamlines, isotherms and isoconcentrations with time when bottom and left walls are non-uniformly heated and concentrated with Ra ¼ 103 , Pr ¼ 0:7, Le ¼ 2:0, N ¼ 1 and A = 1. The pattern of the stream lines show large eddy covering almost whole of the cavity but does not show any eddy in the left lower corner of the cavity when nondimensional time t = 0.0112. A small eddy is formed at the left lower corner at time t = 0.0335 and the large eddy covering almost whole of the cavity increases. After that, size of both the eddies increase when t = 0.0558. Ultimately the flow inside the cavity reaches steady state and the eddies stop increasing when t = 1.7779. The pattern of isotherms changes significantly with time. Initially when t = 0.0112, the isotherm contours are concen-
783
trated near the left and lower walls. After that, as time progresses isotherms spread through out the cavity. The similar thing happens to the pattern of isoconcentrations also. Fig. 9 displays the local Nusselt number and local Sherwood number at the heated walls for various values of N due to uniform temperature and uniform concentration boundary conditions. It is interesting to note that in all the figures, the local Nusselt number and the local Sherwood number is zero at the left edge of bottom wall. Fig. 9(a) and (c) show that heat transfer rate and mass transfer rate are very high at the right edge of the bottom wall due to discontinuity present in temperature and concentration boundary conditions at this edge. On the other hand, the local Nusselt number and the local Sherwood number at the left vertical wall smoothly increase from bottom edge and attain a maximum value near the mid point of the left vertical wall which then decrease to a
Fig. 14. Least square curve fitting of average Sherwood number with thermal Rayleigh number at bottom and left vertical walls for (a,c) uniformly heated and uniformly concentrated and (b,d) non-uniformly heated and non-uniformly concentrated when Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and N = 1.
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Table 4 Computed values of ‘‘coefficient’’ and ‘‘power’’ with standard error of least square curve fitting formula of average Nusselt number with thermal Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions when Pr = 0.7, Le = 2.0, A = 1 and N = 1. Boundary conditions
Value (bottom wall)
Standard error (bottom wall)
Value (left vertical wall)
Standard error (left vertical wall)
Uniform
Coefficient Power
1.2804 0.1558
0.1146 0.0071
0.0342 0.3385
0.0015 0.0032
Non-uniform
Coefficient Power
0.2575 0.2356
0.002 0.0006
0.0143 0.3655
0.0009 0.0048
number at bottom and left vertical walls for uniform and non-uniform boundary conditions. The fitted model is of the form,
NuH
or ShH ¼ coefficient Rpower : a
ð63Þ
The values of ‘‘coefficient’’ and ‘‘power’’ with standard error of least square curve fitting formula of average Nusselt number or Sherwood number with Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions when Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and N = 1 is found from Tables 4 and 5. The following relation are obtained for cases a and b (uniform boundary conditions) and c and d (non-uniform boundary conditions) as follows: Case a
Table 5 Computed values of ‘‘coefficient’’ and ‘‘power’’ with standard error of least square curve fitting formula of average Sherwood number with thermal Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions when Pr = 0.7, Le = 2.0, A = 1 and N = 1. Boundary conditions
Value (bottom wall)
Standard error (bottom wall)
Value (left vertical wall)
Standard error (left vertical wall)
Uniform
Coefficient Power
1.2476 0.1724
0.0997 0.0063
0.0530 0.3356
0.0008 0.0012
Non-uniform
Coefficient Power
0.3227 0.2371
0.0044 0.0011
0.0291 0.3419
0.0027 0.0069
NuH ¼ 1:2804 R0:1558 ðat bottom wallÞ; a NuH ¼ 0:0342 R0:3385 ðat left vertical wallÞ: a
ð64Þ
Case b
ShH ¼ 1:2476 R0:1724 ðat bottom wallÞ; a ShH ¼ 0:0530 R0:3356 ðat left vertical wallÞ: a
ð65Þ
Case c
NuH ¼ 0:2575 R0:2356 ðat bottom wallÞ; a NuH ¼ 0:0143 R0:3655 ðat left vertical wallÞ: a
ð66Þ
Case d non-zero value as depicted in Fig. 9(b) and (d). The physical reason of this behavior is due to high flow rate. Also, it is seen from these two figures that the local Nusselt number and the local Sherwood number increase with increase in N. Fig. 10 shows the local Nusselt number and local Sherwood number at the heated walls for various values of N due to non-uniform temperature and non-uniform concentration boundary conditions. It is interesting to note from this figure that for all values of N, the non-uniform heating enhances the heat transfer at the central region of the walls. It is also observed from this figure that the local Nusselt number and the local Sherwood number increase with increase in N which attains a maximum value near the right side of the bottom wall, but the local Nusselt number and the local Sherwood number are very high near the middle region of the left vertical wall forming wavy type of heat and mass transfer rate. Fig. 11 shows the effect of A on heat transfer rate when Ra ¼ 103 , Pr ¼ 0:7, Le ¼ 2:0 and N = 1 for uniform and non-uniform heating. From Fig. 11(a) it is seen that for uniform heating, the local Nusselt number at the left vertical wall is minimum at the bottom edge of that wall for all the values of A. It is also seen that the local Nusselt number increases with increase in A. But from Fig. 11(b) it is seen that for non-uniformly heating, the local Nusselt number at the left vertical wall is minimum at the top edge of that wall. Again it is seen that near the bottom edge of left vertical wall the local Nusselt number increases with increase in A and opposite trend is seen near the top edge of the wall. Fig. 12 shows the effect of buoyancy ratio N on the average Nusselt and the average Sherwood number with uniform and non-uniform boundary conditions. It is seen from this figure that when N = 1 and Ra ¼ 103 , Pr ¼ 0:7, Le ¼ 2:0 and A = 1, the average Nusselt number and the average Sherwood number take same minimum value for uniform and non-uniform boundary conditions at the left vertical wall (Fig. 12(b) and (d)). At the bottom wall, uniform and non-uniform boundary conditions produce S-type of the average Nusselt number and the average Sherwood number with their maximum value at right edge of the bottom wall (Fig. 12(a) and (c)). Figs. 13 and 14 show a least square curve fitting of average Nusselt number and average Sherwood number with thermal Rayleigh
ShH ¼ 0:3227 R0:2371 ðat bottom wallÞ; a ShH ¼ 0:0291 R0:3419 ðat left vertical wallÞ: a
ð67Þ
It is observed from Tables 4 and 5 that the standard error of estimation is very less, so the relation obtained for average Nusselt number and average Sherwood number as function of thermal Rayleigh number would result into almost same values as obtained from numerical computation of the set of differential equations with appropriate boundary conditions. Thus it is interesting to found that without wasting much time on computation, relations from Eqs. (64)–(67) can well be used for estimating average Nusselt and average Sherwood number for various values of thermal Rayleigh number. 5. Conclusion The main objective of the current investigation is to study the influence of buoyancy ratio on double-diffusive natural convection in a lid-driven square cavity with uniform and non-uniform thermal and concentration boundary conditions at the bottom wall and at the left vertical wall. It is observed that in the case of uniform heating and uniform concentration, the heat and mass transfer rate are zero for both left edge of the bottom wall and bottom edge of left vertical wall. Further, it is observed that the isotherm and isoconcentration lines are almost parallel to the horizontal wall in the middle part of the cavity when N = 50, indicating that most of the heat transfer is carried out by heat conduction. This is due to the increase in thermal boundary layer thickness. As, the buoyancy ratio increases the boundary layer thickness becomes thinner. The change in the flow structure for high buoyancy ratio has a significant influence on the concentration field, which builds up a vertical stratification of the enclosure. When buoyancy ratio N = 50, the effect of solutal buoyancy force is in the opposite direction of the thermal buoyancy force. The magnitude of the thermal buoyancy force is very small compared with the solutal buoyancy force due to which isotherm and isoconcen-
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tration tend to move to left wall whereas, when the buoyancy ratio N = 1 both thermal and solutal buoyancy forces are equal for which isothermal and isoconcentration tend to move to the right wall. The change of patterns with respect to time is described here. It is concluded that the heat and mass transfer rate are very high at the right edge of the bottom wall. In contrast, for the case of non-uniform heating and non-uniform concentration the heat and mass transfer rate are minimum at the left edge of bottom wall and top corner of left vertical wall. Also, it is seen that average Nusselt number and the average Sherwood number attains their minimum value when the value of buoyancy ratio is 1 at left vertical wall for both uniform and non-uniform boundary conditions. Finally, least square curves are fitted for average Nusselt number and average Sherwood number with thermal Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions and the standard errors are calculated. Acknowledgement The work of two of the authors (T.R.M and D.P) is supported under SAP (DRS PHASE II) program of UGC, New Delhi, India. The authors wish to thank the referees for helpful comments and suggestions which improved the quality of the paper considerably. References [1] H.K. Wee, R.B. Keey, M.J. Cunningham, Heat and moisture transfer by natural convection in a rectangular cavity, Int. J. Heat Mass Transfer 32 (1) (1989) 765– 1778. [2] C. Beghein, F. Haghighat, F. Allard, Numerical study of double-diffusive natural convection in a square cavity, Int. J. Heat Mass Transfer 35 (1992) 833–846. [3] K. Ghorayeb, H. Khallouf, A. Mojtabi, Onset of oscillatory flows in double diffusive convection, Int. J. Heat Mass Transfer 42 (1999) 629–643. [4] Qi-Hong Deng, Jiemin Zhou, Chi Mei, Yong-Ming Shen, Fluid, heat and contaminant transport structures of laminar double diffusive mixed convection in a two dimensional ventilated enclosure, Int. J. Heat Mass Transfer 47 (2004) 5257–5269.
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[5] M.S. Malashetty, D. Bassavaraja, Effect of thermal modulation on the onset of double diffusive convection in a horizontal fluid layer, Int. J. Therm. Sci. 44 (2005) 323–332. [6] W.M. Yan, Turbulent mixed convection heat and mass transfer in a wetted channel, J. Heat Transfer 117 (1) (1995) 229–233. [7] W. Yan, Combined buoyancy effects of thermal and mass diffusion on laminar forced convection in horizontal rectangular ducts, Int. J. Heat Mass Transfer 39 (7) (1996) 1479–1488. [8] K. Lee, H. Tsai, W. Yan, Mixed convection heat and mass transfer in vertical rectangular ducts, Int. J. Heat Mass Transfer 40 (7) (1997) 1621–1631. [9] S.E. Alimi, J. Orfi, S.B. Nasrallah, Buoyancy effects on mixed convection heat and mass transfer in a duct with sudden expansions, Heat Mass Transfer 41 (6) (2005) 559–567. [10] N. Brown, F. Lai, Correlations for combined heat and mass transfer from an open cavity in a horizontal channel, Int. Commun. Heat Mass Transfer 32 (8) (2005) 1000–1008. [11] M.A. Teamah, Numerical simulation of double diffusive natural convection in rectangular enclosure in the presences of magnetic field and heat source, Int. J. Therm. Sci. 47 (2008) 237–248. [12] S. Saha, M.N. Hasan, I.A. Khan, Double diffusive mixed convection heat transfer inside a vented square cavity, Chem. Eng. Res. Bull. 13 (2009) 17–24. [13] A.V. Kuznetsov, D.A. Nield, Double-diffusive natural convective boundarylayer flow of a nanofluid past a vertical plate, Int. J. Therm. Sci. 50 (2011) 712– 717. [14] D.A. Nield, A.V. Kuznetsov, The Cheng–Minkowycz problem for the doublediffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid, Int. J. Heat Mass Transfer 54 (2011) 374–378. [15] M.A. Teamah, S.M. ElSherbiny, R.A. Saleh, Effect of Grashof number and aspect ratio on double-diffusive laminar natural convection in a symmetrical trapezoidal enclosure, Alexandria Eng. J. 49 (2) (2010) 89–100. [16] O.V. Trevisan, A. Bejan, Combined heat and mass transfer by natural convection in a vertical enclosure, J. Heat Transfer 109 (1987) 104–112. [17] W.J. Minkowycz, E.M. Sparrow, G.E. Schneider, R. H Fletcher, Handbook of Numerical Heat Transfer, Wiley, New York, 1988. [18] S. Roy, T. Basak, Finite element analysis of natural convection flows in a square cavity with non-uniformly heated wall(s), Int. J. Eng. Sci. 43 (2005) 668–680. [19] M.A. Teamah, M.M. Khairat Dawood, W.M. El-Maghlany, Double diffusive natural convection in a square cavity with segmental heat sources, Eur. J. Sci. Res. 54 (2) (2011) 287–301. [20] N. Alleborn, H. Raszillier, F. Durst, Lid-driven cavity with heat and mass transport, Int. J. Heat Mass Transfer 42 (1999) 833–853. [21] Q. Wan, A.V. Kuznetsov, Investigation of the acoustic streaming in a rectangular cavity induced by the vibration of its lid, Int. Commun. Heat Mass Transfer 31 (4) (2004) 467–476. [22] [22] C.T. Kelley, Iterative methods for linear and nonlinear equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1995.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effects of buoyancy ratio on double-diffusive natural convection in a lid-driven cavity Tapas Ray Mahapatra a, Dulal Pal a, Sabyasachi Mondal b,⇑ a b
Department of Mathematics, Visva-Bharati (A Central University), Santiniketan 731235, West Bengal, India Department of Mathematics, Bengal Institute of Technology and Management, Santiniketan 731236, West Bengal, India
a r t i c l e
i n f o
Article history: Received 20 June 2012 Received in revised form 4 September 2012 Accepted 7 October 2012 Available online 5 December 2012 Keywords: Double-diffusive Lid-driven cavity flow Finite-difference method Non-uniformly heated Non-uniformly concentrated
a b s t r a c t The effects of uniform and non-uniform heating of wall(s) on double-diffusive natural convection in a liddriven square enclosure are analyzed. It is assumed that the left vertical wall and bottom wall are heated and concentrated (uniformly or non-uniformly), while the other vertical wall is maintained at a constant cold temperature and the top wall is well insulated which moves with a constant speed. The governing equations are solved numerically using staggered grid finite-difference method. Streamlines, isotherms, local Nusselt number, local Sherwood number, average Nusselt number and average Sherwood number for various values of buoyancy ratio and thermal Rayleigh number are obtained. The results are compared with previously published work and excellent agreement has been obtained. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Fluid flow, heat and mass transfer induced by double-diffusive natural convection in fluid saturated porous media have practical importance in many engineering applications. This aspect of fluid dynamics has gained considerable attention in recent years among the researchers. Migration of moisture in fibrous insulation, drying processes, chemical reactors, transport of contaminants in saturated soil and electro-chemical processes are some examples of double-diffusive natural convection phenomena. Double-diffusion occurs in a wide range of scientific field, such as oceanography, astrophysics, geology, biology and chemical processes. So, the researchers have keen interest in the study of heat and mass transfer in enclosure and cavity. Double-diffusive natural convection in cavities has been subject of an intensive research due to its importance in various engineering and geophysical problems. This includes nuclear reactors, solar ponds, geothermal reservoirs, solar collectors, crystal growth in liquids, electronic cooling and chemical processing equipments. Wee et al. [1] and Beghein et al. [2] presented comprehensive reviews on the study of heat and moisture transfer by natural convection in a rectangular cavity. Thereafter, Ghorayeb et al. [3] analyzed the onset of oscillatory flow in double diffusive convection in a square cavity with equal but opposing horizontal temperature
⇑ Corresponding author. E-mail addresses: [email protected] (T.R. Mahapatra), [email protected] (D. Pal), [email protected] (S. Mondal). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.028
and concentration gradients numerically. The results presented the influence of the Lewis number on the transition of steady state convective and oscillatory flow structures occurring beyond this transition. Deng et al. [4] investigated fluid, heat and contaminant transport structures of laminar double diffusive mixed convection in a two-dimensional ventilated enclosure numerically. The effect of thermal modulation on the onset of double-diffusion natural convection in a horizontal fluid layer was studied analytically and numerically using a linear stability analysis by Malashetty et al. [5]. They found that the frequency symmetric modulation was destabilizing while high frequency symmetric modulation was always stabilizing. The effects of combined thermal and solutal buoyancy induced by temperature and concentration gradients have, however, not been widely studied. Yan [6,7] and Lee et al. [8] analyzed the transport phenomena of developing laminar mixed convection heat and mass transfer in rectangular ducts. Later, Alimi et al. [9] studied the buoyancy effects on mixed convection heat and mass transfer in an inclined duct preceded with a double step expansion. Brown and Lai [10] numerically examined combined heat and mass transfer from a horizontal channel with an open cavity heated from below numerically. Teamah [11] studied double-diffusive convective flow in a rectangular enclosure with the upper and lower surfaces being insulated and impermeable by imposing constant temperature and concentration along the left and right walls of the enclosure and a uniform magnetic field was applied in a horizontal direction. Saha et al. [12] investigated the new characteristics of the airflow and heat/contaminant transport mechanism inside a vented cavity in terms of streamlines, isotherms and isoconcentration lines.
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wycz et al. [17] discovered that the discontinuity can be avoided by choosing a non-uniform temperature distribution along the walls (i.e. non-uniformly heated walls), in an investigation on mixed convection flow over a vertical plate, which is non-uniformly heated/cooled. Roy and Basak [18] solved the nonlinear coupled partial differential equations for flow and temperature fields with both uniform and non-uniform temperature distributions prescribed at the bottom wall and at one vertical wall. Teamah et al. [19] studied the effect of the heater length, Rayleigh number, Prandtl number and buoyancy ratio on both average Nusselt and Sherwood number with uniform heating at left vertical wall. Heat and mass transfer in lid-driven enclosures have received less attention in the above literature. In drying technology, better understanding of the drying process is vital for optimum performance of the drying chamber. Alleborn et al. [20] investigated a two-dimensional flow accompanied by heat and mass transport in a shallow lid-driven cavity with a moving heated lid and a moving cooled lid. Their results showed that the drying rates were enhanced by increasing the velocity and become increasingly independent of the gravity orientation because of the dominance of forced convection. Wan and Kuznetsov [21] investigated the Fig. 1. Schematic diagram of the physical system.
Kuznetsov and Nield [13] studied the double-diffusive natural convective boundary-layer flow of a nanofluid past a vertical plate analytically. Later, Nield and Kuznetsov [14] presented an analytical treatment of double-diffusive nanofluid convection in a porous medium. Teamah et al. [15] studied the effects for a wide range of thermal Grashof number and aspect ratio coupled with the inclination angle. The obtained results for average Nusselt and Sherwood numbers were correlated. Trevisan and Bejan [16] investigated the phenomenon of natural convection caused by combined temperature and concentration buoyancy effects in a rectangular enclosure with uniform heat and mass fluxes applied along the vertical walls numerically and analytically. They found that in the case of uniformly heated walls, the finite discontinuity in the temperature distribution appeared at the right edge of the bottom wall. Minko-
Table 1 Grid independence test when Pr = 0.7, Le = 2.0, N = 1, A = 1 and Ra⁄ = 103. No. of Grid points
20 20 40 40 80 80 160 160
Uniformly heated and uniformly concentrated
Non-uniformly heated and non-uniformly concentrated
Iteration
jwminj
Iteration
jwminj
19,493 78,004 312,806 1,251,253
2.1149 2.0906 2.0856 2.0846
19,856 79,449 318,646 1,274,605
1.8134 1.7924 1.7887 1.7878
Table 2 Computed values of NuH jy¼0 and NuH jx¼0 when Pr = 0.7, Le = 2.0, A = 1 and N = 1.0 for various values of Ra⁄. Ra⁄
N
103 104 2 104 5 104 105 2 105 5 105 8 105 106
1.0
Uniformly heated and uniformly concentrated
Non-uniformly heated and non-uniformly concentrated
NuH jy¼0
NuH jx¼0
NuH jy¼0
NuH jx¼0
4.293 5.424 5.891 6.663 7.421 8.351 9.859 10.783 11.265
0.409 0.779 0.980 1.318 1.663 2.120 2.918 3.416 3.676
1.325 2.263 2.659 3.299 3.863 4.555 5.670 6.340 6.684
0.186 0.391 0.524 0.739 0.969 1.269 1.765 2.063 2.217
Table 3 Computed values of ShH jy¼0 and ShH jx¼0 when Pr = 0.7, Le = 2.0, A = 1 and N = 1.0 for various values of Ra⁄.
Fig. 2. Control volume for u-momentum, v-momentum, temperature and concentration equations.
Ra⁄
N
103 104 2 104 5 104 105 2 105 5 105 8 105 106
1.0
Uniformly heated and uniformly concentrated
Non-uniformly heated and non-uniformly concentrated
ShH jy¼0
ShH jx¼0
ShH jy¼0
ShH jx¼0
4.671 6.162 6.790 7.813 8.799 10.010 11.947 13.13 13.75
0.515 1.176 1.468 1.993 2.529 3.20 4.349 5.075 5.459
1.613 2.849 3.377 4.221 4.963 5.845 7.251 8.087 8.514
0.204 0.672 0.861 1.176 1.512 1.929 2.601 3.019 3.245
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fluid flow in a rectangular cavity whose lid vibrates in one standing wavelength. The studies of double-diffusive natural convection have centered their analysis on the limit cases of dominating thermal buoyancy force or concentration buoyancy force. The aim of the present investigation is to study the effects of buoyancy ratio, circulations, heat transfer rate at the heated walls in terms of local Nusselt number, average Nusselt number and local Sherwood number, average Sherwood number when the bottom wall and left vertical wall are heated and concentrated (uniformly and non-uniformly) and right vertical wall is cooled by means of a constant temperature while the top wall is well insulated with a constant speed from left to right. The numerical work presented in this paper focuses on an entirely new class of flows in this kind of cavities. The thermal and mass exchanges generated in the case of co-operating thermal and concentration buoyancy effects with uniform and non-uniform boundary conditions have been analyzed.
2. Governing equations and boundary conditions An unsteady-state two-dimensional square cavity of height L as shown in Fig. 1 is considered. It is assumed that the top wall is moving from left to right at a constant speed U0 and is considered to be adiabatic. The bottom wall and left vertical wall are heated and concentrated (uniformly or non-uniformly) and right vertical wall is cooled by means of a constant temperature. The thermophysical properties of the fluid are assumed to be constant except the density variation in the buoyancy force, which is approximated according to the Boussinesq approximation. This variation, due to
(a)
both temperature and concentration gradients, can be described by the following equation:
q ¼ q0 ½1 bT ðT T c Þ bS ðC C c Þ;
ð1Þ
where bT and bS are the thermal and concentration expansion coefficients, respectively. In the Cartesian coordinate system, the fundamental governing equations are as follows:
@U @V þ ¼ 0; @X @Y
ð2Þ !
@U @U @U 1 @P @2U @2U ; þ þV ¼ þm 0 þU @t @X @Y q @X @X 2 @Y 2 ! @V @V @V 1 @P @2V @2V þ U m þ þ V ¼ þ @t 0 @X @Y q @Y @X 2 @Y 2
@T @T @T þU þV @t 0 @X @Y @C @C @C þV þU @t0 @X @Y
þ gbT ðT T c Þ þ gbS ðC C c Þ; ! @2T @2T ; þ ¼a @X 2 @Y 2 ! @2C @2C : ¼D þ @X 2 @Y 2
ð3Þ
ð4Þ ð5Þ
ð6Þ
The associated boundary conditions are when t0 = 0 for 0 6 X, Y 6 L:
UðX; YÞ ¼ 0 ¼ VðX; YÞ;
ð7Þ
TðX; YÞ ¼ T c ;
ð8Þ
CðX; YÞ ¼ C c ;
(b)
Fig. 3. Comparison of present local Nusselt number at the bottom and left vertical walls for uniform heating (—) and non-uniform heating (- - -) with Roy and Basak [18].
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when t0 > 0 for 0 6 X, Y 6 L:
UðX; LÞ ¼ U 0 ;
UðX; 0Þ ¼ Uð0; YÞ ¼ UðL; YÞ ¼ 0;
ð11Þ
Here x and y are dimensionless coordinates along the horizontal and vertical directions respectively; u and v are dimensionless velocity components in the x- and y-directions respectively; h and S denote the dimensionless temperature and concentration respectively; p is the dimensionless pressure parameter. Using these dimensionless variables, we obtain the following dimensionless governing equations from Eqs. (2)–(6):
ð12Þ
@u @ v þ ¼ 0; @x @y
ð13Þ
! @u @p @2u @2u @u2 @uv ¼ þ Pr þ þ ; @t @x @x2 @y2 @x @y
ð9Þ
VðX; 0Þ ¼ VðX; LÞ ¼ Vð0; YÞ ¼ VðL; YÞ ¼ 0; TðX; 0Þ ¼ T h or TðX; 0Þ ¼ ðT h T c Þ sinðpX=LÞ þ T c ; @T ðX; LÞ ¼ 0; @Y Tð0; YÞ ¼ T h or Tð0; YÞ ¼ ðT h T c Þ sinðpY=LÞ þ T c ; TðL; YÞ ¼ T c ; CðX; 0Þ ¼ C h or CðX; 0Þ ¼ ðC h C c Þ sinðpX=LÞ þ C c ; @C ðX; LÞ ¼ 0; @Y Cð0; YÞ ¼ C h or Cð0; YÞ ¼ ðC h C c Þ sinðpY=LÞ þ C c ; CðL; YÞ ¼ C c ;
ð10Þ
ð14Þ
where, X and Y are the distances measured along the horizontal and vertical directions respectively; U and V are velocity components in the X- and Y-directions respectively; T and C denote the temperature and concentration respectively; m, a and D are kinematic viscosity, thermal diffusivity and mass diffusivity respectively; P is the pressure and q is the density; Th and Tc are the temperatures at the hot and cold walls respectively; Ch and Cc are the concentrations at the hot and cold walls respectively; L is the side of the square cavity. We now introduce dimensionless variables given as follows:
t¼
at
0
2
;
x¼
L T Tc ; h¼ Th Tc
X ; L
Y UL ; u¼ ; L a C Cc S¼ : Ch Cc y¼
v¼
VL
a
2
;
p¼
PL
qa2
;
ð15Þ ð16Þ
ð17Þ
@v @p @2v @2v þ ¼ þ Pr @y @t @x2 @y2 @h ¼ @t
!
2 @v @uv þ þ Pr Ra ðh þ NSÞ; ð19Þ @y @x
! @2h @2h @uh @ v h ; þ þ @x2 @y2 @x @y
@S 1 @2S @2S ¼ þ @t Le @x2 @y2
!
ð18Þ
@uS @ v S þ : @x @y
ð20Þ
ð21Þ
The dimensionless boundary conditions are when t = 0 for 0 6 x, y 6 1:
uðx; yÞ ¼ 0 ¼ v ðx; yÞ;
ð22Þ
hðx; yÞ ¼ 0;
ð23Þ
Sðx; yÞ ¼ 0;
(a)
(b)
(c)
Fig. 4. Contour plots for Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and (a) N = 50, (b) N = 1, (c) N = 50 (uniformly heated and uniformly concentrated).
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The average Nusselt numbers at the bottom wall and left vertical wall are given by,
when t > 0 for 0 6 x, y 6 1:
uðx; 1Þ ¼ A;
uðx; 0Þ ¼ uð0; yÞ ¼ uð1; yÞ ¼ 0;
ð24Þ
v ðx; 0Þ ¼ v ðx; 1Þ ¼ v ð0; yÞ ¼ v ð1; yÞ ¼ 0;
ð25Þ
hðx; 0Þ ¼ 1 or hðx; 0Þ ¼ sinðpxÞ;
@h ðx; 1Þ ¼ 0; @y
ð26Þ
hð0; yÞ ¼ 1 or hð0; yÞ ¼ sinðpyÞ;
hð1; yÞ ¼ 0;
ð27Þ
Sðx; 0Þ ¼ 1 or Sðx; 0Þ ¼ sinðpxÞ;
@S ðx; 1Þ ¼ 0; @y
ð28Þ
Sð0; yÞ ¼ 1 or Sð0; yÞ ¼ sinðpyÞ;
Sð1; yÞ ¼ 0;
ð29Þ
NuH jy¼0 ¼
Z
Z
1
0
Nub dx and NuH jx¼0 ¼
1
Nul dy:
ð32Þ
0
The local Sherwood number at the left vertical wall and bottom walls are defined as Shl = (@S)/(@x)jx=0 and Shb = (@S)/(@y)jy=0. The average Sherwood numbers at he bottom wall and left vertical wall are given by,
ShH jy¼0 ¼
Z
1 0
Shb dx and ShH jx¼0 ¼
Z
1
Shl dy:
ð33Þ
0
where 3. Method of solution 3
2
GrT ¼ gbT ðT h T c ÞL =m ; GrC ; N¼ GrT
3
2
GrC ¼ gbS ðC h C c ÞL =m ;
m Pr ¼ ; Ra ¼ GrT Pr ; A ¼ U 0 L=a: a
ð30Þ ð31Þ
Here A, N, Le, Pr, GrC, GrT and Ra are a velocity parameter, buoyancy ratio, Lewis number, Prandtl number, solutal Grashof number, thermal Grashof number and thermal Rayleigh number respectively. The heat transfer coefficient in terms of the local Nusselt number (Nu) is Nu = (@h)/(@n), where n denotes the normal direction to a plane. The local Nusselt number at the left vertical wall and bottom walls are defined as Nul = (@h)/(@x)jx=0 and Nub = (@h)/(@y)jy=0.
Control-volume based finite-difference discretization of Eqs. (17)–(21) are carried out using staggered grid, popularly known as MAC cell method. The grid alignment of the velocities are evaluated at different locations of the control volume whereas the pressure and temperature are evaluated at same location of control volume as shown in Fig. 2. The non-dimensional equations have been derived in distinct types of cells for four equations, viz., (i) continuity cell, (ii) u-momentum cell, (iii) v-momentum cell and (iv) temperature cell. Now, the non-dimensional Eqs. (18)–(21) can be written in the following form:
Fig. 5. Contour plots for Pr = 0.7, Le = 2.0, A = 1, N = 1 and (a) Ra⁄ = 103, (b) Ra⁄ = 104, (c) Ra⁄ = 105 (uniformly heated and uniformly concentrated).
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Fig. 6. Contour plots for Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and (a) N = 50, (b) N = 1, (c) N = 50 (non-uniformly heated and non-uniformly concentrated).
Fig. 7. Contour plots for Pr = 0.7, Le = 2.0, A = 1, N = 1 and (a) Ra⁄ = 103, (b) Ra⁄ = 104, (c) Ra⁄ = 105 (non-uniformly heated and non-uniformly concentrated).
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785
@u @p ¼ þ UDPC; @t @x @v @p ¼ þ VDPC; @y @t @h ¼ TDPC; @t @S ¼ CDPC: @t here;
" # @2u @2u @u2 @uv ; þ 2 þ UDPC ¼ Pr 2 @x @y @x @y " # @2v @2v @ v 2 @uv þ Pr Ra ðh þ NSÞ VDPC ¼ Pr þ 2 þ 2 @x @y @x @y " # @2h @2h @uh @ v h ; þ þ TDPC ¼ @x2 @y2 @x @y " # 1 @2S @2S @uS @ v S : CDPC ¼ þ þ Le @x2 @y2 @x @y
777
ð34Þ
and the source terms are centrally differenced without changing the position of the cavity at the center of the control volumes. Let
ð35Þ
t ¼ ndt;
ð36Þ ð37Þ
x ¼ idx;
hðx; y; tÞ ¼ hðidx; jdy; ndtÞ ¼ hnij ; Sðx; y; tÞ ¼ Sðidx; jdy; ndtÞ ¼ Snij , where superscript n refers to the time direction, subscripts i an, j refer to the spatial directions, dt is the time increment and dx, dy are the length and width of the control volume. Now pressure gradient are discritized as n
ð38Þ ð39Þ ð40Þ ð41Þ
The convective terms in the momentum equations are differenced with a hybrid formula consisting of central differencing and secondorder upwinding. The diffusive terms are differenced by second-order-accurate three-point central difference formula in both the cases
y ¼ jdy and pðx; y; tÞ ¼ pðidx; jdy; ndtÞ ¼ pnij ;
n
@p piþ1j pij ¼ þ Oðdx2 Þ; @x dx n n @p pijþ1 pij ¼ þ Oðdy2 Þ: @y dy
ð42Þ ð43Þ
Discretization of the continuity equation at (i, j) cell takes the following form,
unij uni1j dx
þ
v nij v nij1 dy
¼ 0:
ð44Þ
The finite-difference formulation of the momentum equation in xdirection using uniform grid-spacing is
unþ1 unij ij dt
¼
pniþ1j pnij dx
þ ðUDPCÞnij ;
ð45Þ
Fig. 8. Contour plots for Pr = 0.7, Ra⁄ = 103, Le = 2.0, A = 1, N = 1 and (a) t = 0.0112, (b) t = 0.0335, (c) t = 0.0558, (d) t = 1.7779 (non-uniformly heated and non-uniformly concentrated).
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where
ðUDPCÞnij ¼ Pr Diff unij Con unij :
ð46Þ
Here, Diff unij , Con unij are diffusive and convective terms of the umomentum equation at the nth time level at (i, j)th cell. In the y-direction, the finite-difference formulation of the momentum equation is
v nþ1 v nij ij dt
¼
pnijþ1 pnij dy
þ ðVDPCÞnij ;
ð47Þ
where,
!
ðVDPCÞnij ¼ Pr Diff v nij Con v nij þ Pr Ra
hnij þ hnijþ1 Snij þ Snijþ1 þN : 2 2
hnþ1 hnij ij ¼ ðTDPCÞnij ; dt
ð49Þ
where
ðTDPCÞnij ¼ Diff hnij Con hnij : Diff hnij ;
ð50Þ
Con hnij
Here, are diffusive and convective terms of the energy equation at the nth time level at (i, j)th cell. Now, we have discritized form of concentration equation (37)
Snþ1 Snij ij ¼ ðCDPCÞnij ; dt
ð51Þ
where
ð48Þ Here, Diff v nij , Con v nij are diffusive and convective terms of the v-momentum equation at the nth time level at (i, j)th cell. Now, we have discritized form of energy equation (36)
ðCDPCÞnij ¼
1 Diff Snij Con Snij : Le
ð52Þ
Here, Diff Snij and Con Snij denote diffusive and convective terms of the concentration equation at the nth time level at (i, j)th cell. The diffusive terms in u-momentum equation are discretized as
(a)
(b)
(c)
(d)
Fig. 9. Local Nusselt number and local Sherwood number for uniformly heated and uniformly concentrated for Pr = 0.7, Le = 2.0, A = 1 and Ra⁄ = 103.
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785 n
n
n
@ 2 u uiþ1j 2uij þ ui1j ¼ þ Oðdx2 Þ; @x2 dx2 n n n @ 2 u uijþ1 2uij þ uij1 ¼ þ Oðdy2 Þ; 2 2 @y dy
ð53Þ ð54Þ
and convective terms in u-momentum equation are discretized as
Here b is the upwinding parameter which is determined from the numerical stability criteria which is described in Section 3.1. The finite-difference formulation for the diffusive and convective terms in the v-momentum equation are similar to that in umomentum formulation. The diffusive terms in temperature equation are discretized as n
un un unl unl un /n unl /nul @u2 ¼ ð1 bÞ r r þ b r ur ; @x dx dx un v n unb v nb v n /n v nb /nub @uv ¼ ð1 bÞ t t þ b t ut ; @y dy dy
ð55Þ ð56Þ
779
n
n
@ 2 h hiþ1j 2hij þ hi1j ¼ ; @x2 dx2 n n n 2 @ h hijþ1 2hij þ hij1 ¼ ; @y2 dy2
ð57Þ ð58Þ
and convective terms in temperature equation are discretized as when unr P 0 then /nur ¼ unij ; when unl P 0 then /nul ¼ uni1j ; when unr < 0 then /nur ¼ uniþ1j ; when unl < 0 then /nul ¼ unij ; when v nt P 0 then /nut ¼ unij ; when v nb P 0 then /nub ¼ unij1 ; when v nt < 0 then /nut ¼ unijþ1 ; when v nb < 0 then /nub ¼ unij .
un hn unl hnl un /n unl /nhl @uh ¼ ð1 bÞ r r þ b r hr ; @x dx dx n n n n n n n n v h v b hb v t /ht v b /hb @v h ¼ ð1 bÞ t t þb ; @y dy dy
(a)
(b)
(c)
(d)
Fig. 10. Local Nusselt number and local Sherwood number for non-uniformly heated and non-uniformly concentrated for Pr = 0.7, Le = 2.0, A = 1 and Ra⁄ = 103.
ð59Þ ð60Þ
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when unr P 0 then /nhr ¼ hnij ; when unl P 0 then /nhl ¼ hni1j ; when unr < 0 then /nhr ¼ hniþ1j ; when unl < 0 then /nhl ¼ hnij ; when v nt P 0 then /nht ¼ hnij ; when v nb P 0 then /nhb ¼ hnij1 ; when v nt < 0 then /nht ¼ hnijþ1 ; when v nb < 0 then /nhb ¼ hnij .
4. Results and discussion
The finite-difference formulation for the diffusive and convective terms in the concentration equation are similar to that in temequation formulation. The convected momentum fluxes perature unt ; unb ; unr ; unl ; v nt ; v nb ; v nr ; v nl at the interfaces are linearly interpolated for both u-momentum and v-momentum fluxes. In the second-order upwinding scheme, the choice of taking the momentum flux / passing through the interface of the control volume depends on the sign of convecting velocities at that interface. Here b is the combination factor between the central-differencing scheme and second order upwind-differentiating scheme. When b = 0, the convective terms are represented by the central-difference whereas it is second-order upwinding when b = 1. A pressure-Poisson equation is derived combing the discretized momentum and continuity equations. Here the local dilation term nþ1 at (n + 1)th level Div ij is set equal to zero. Thus the final form of the pressure-Poisson equation is n
n
2ða þ bÞpnij apniþ1j apni1j bpijþ1 bpij1 " n # o 1n o Dij 1 n ðUDPCÞnij ðUDPCÞni1j þ ðVDPCÞnij ðVDPCÞni1j ; ¼ þ dy dt dx
Grid independence test is provided in Table 1 for various grid sizes when Pr = 0.7, Le = 2.0, N = 1, A = 1 and Ra⁄ = 103. It is important to note that as the number of grid points are increased, the number of iterations required for getting the converged results for jwminj increases. But when the number of grid points increases from 80 80 to 160 160, no significant change found in the value of jwminj. Hence, all the results are computed taking 80 80 grid points. Tables 2 and 3 display the computed values of NuH jy¼0 , NuH jx¼0 , ShH jy¼0 and ShH jx¼0 when Pr = 0.7, Le = 2.0, A = 1 and N = 1.0 for various values of Ra⁄ on the heated walls. It is seen from these two tables that NuH jy¼0 , NuH jx¼0 , ShH jy¼0 and ShH jx¼0 increase with increase in Ra⁄. Fig. 3 shows the comparison of present results with that of Roy and Basak [18] for local Nusselt number at bottom wall and left vertical wall with uniform and non-uniform heating when A = 0 and other parameters are same as in [18]. From this graph it is observed that a very good agreement has been obtained with the previously published results. The numerical results for streamlines, isotherms, isoconcentrations, local Nusselt number, average Nusselt number, local Sherwood number and average Sherwood number at the heated walls are presented in Figs. 4–12 for uniform and non-uniform heating.
ð61Þ where Dnij is the divergence of the velocity field at the cell (i, j) at n-th time level, which is to be zero for the convergence of the flow in successive iterations of the method and
a¼
1 ; dx2
b¼
1 ; dy2
Dnij ¼
unij uni1j dx
þ
v nij v nij1 dy
:
ð62Þ
3.1. Solution procedure and numerical stability criteria Now the iteration process is described to obtain the solutions of the basic equations with appropriate boundary conditions. In the derivation of pressure-Poisson equation, the divergence term at n-th time level ðDnij Þ is retained and evaluated in the pressure-Poisson iteration. It is done because the discretized form of divergence of velocity field, i.e., Dnij is not guaranteed to be zero initially. The solution procedure starts with the initializing the velocity field. This is done either from the result of previous cycle or from the prescribed initial and boundary conditions. Using this velocity field pressure-Poisson equation is solved using Bi-CG-Stab method [22]. Knowing the pressure field, equation for u-momentum, v-momentum, temperature and concentration are updated to get u,v,h and S at (n + 1)th time level. Then using the values of u and v at (n + 1)th time level, the value of the divergence of velocity field is calculated and checked for its limit. If its absolute value is less than 0.5 105 and steady state is reached then iteration process stops, otherwise pressure-Poisson equation is solved again for pressure. h i dx dy Linear stability of fluid flow gives dt 1 6 Min juj ; jv j , which is related to the convection of fluid, i.e., fluid should not move more than one cell width per time step (Courant, Friedrichs and Lewy condition). Also, from the Hirt’s stability analysis, we have h i 2 dy2 dt2 6 Min 2P1 r ðdxdx2 þdy 2 Þ . This condition roughly states that momentum cannot diffuse more than one cell width per time step. The time step is determined from dt = FCT [Min (dt1,dt2)], where the factor FCT varies from 0.2 to 0.4. The upwinding parameter b is h i ; v dt . As governed by the inequality condition 1 P b P Max udt dx dy a rule of thumb, b is taken approximately 1.2 times larger than what is found from the above inequality condition.
(a)
(b) Fig. 11. Local Nusselt number for different values of A (a) uniformly heated and uniformly concentrated (b) non-uniformly heated and non-uniformly concentrated Pr = 0.7, Le = 2.0, N = 1 and Ra⁄ = 103.
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785
The effects of N and Ra⁄ on streamline, isotherm and isoconcentration contours for A = 1, Le = 2 and Pr = 0.7 are displayed in Figs. 4–7. Fig. 4 shows the effect of N on the streamlines, isotherms as well as on the isoconcentration for wide range of variations in the buoyancy ratio (N) with uniform heating at left vertical wall and the bottom walls. As N decreases from 50 to 1, the values of stream function decrease i.e., strength of flow circulation decreases. But when the value of N is further decreased from 1 to 50 then the values of stream function increase, i.e., strength of flow circulation increase with decrease in N. It is clear from Fig. 4(a) that when N = 50 the contours of both the isotherms and isoconcentrations are mainly concentrated near the cold vertical wall but when the value of N is decreased to 1, they are dis-
781
persed. Again when the value of N is further decreased to 50, the contours are concentrated near the lower half of the cold vertical wall and upper half of the hot vertical wall. Fig. 4(c) shows that the isotherm and isoconcentration lines are almost parallel to the horizontal wall in the middle part of the cavity for N = 50, indicating that most of the heat transfer is carried out by heat conduction. This is due to an increase in the thermal boundary layer thickness. At low buoyancy ratio, the entire enclosure is influenced by the flow structure and as the buoyancy ratio increases the boundary layer thickness becomes thinner. This change in the flow structure for high buoyancy ratio has a significant influence on the concentration field, which builds up a vertical stratification of the enclosure. Again it is interesting to note that for
(a)
(b)
(c)
(d)
Fig. 12. Average Nusselt number and average Sherwood number for different values of N: (a,b) uniformly heated and non-uniformly concentrated and (c,d) uniformly heated and uniformly concentrated when Pr = 0.7, Le = 2.0, A = 1 and Ra⁄ = 103.
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N = 50, the effect of solutal buoyancy force is in the opposite direction of thermal buoyancy force. When N = 1, the isothermal and isoconcentration countours tend toward the left wall, as both thermal buoyancy force and solutal buoyancy force are equal. When N = 50, the effect of solutal buoyancy force is in the same direction of thermal buoyancy force. In such case the isothermal and isoconcentration contours tend toward the right wall. Streamline, isotherm and isoconcentration contours for various Ra ¼ 103 to 105 with uniform heating of a vertical wall and the bottom wall are displayed in Fig. 5. As expected due to a hot vertical wall, fluids rise up along the side of the hot vertical wall and flow down along the cold vertical wall forming a roll with clockwise rotation inside the cavity. As Ra increases from 103 to 105, the values of the stream function increase. The streamlines are crowded near the cavity wall and the cavity core is empty. It is clear from
Fig. 5(c) that when Ra ¼ 105 , the contours of both the isotherms and isoconcentrations are mainly concentrated near the cold vertical wall but when Ra decreases to 104, they are dispersed. Again when Ra is further decreased to 103, the contours are more dispersed towards the left vertical wall. As seen in Figs. 4 and 5, uniform heating of the bottom wall and left vertical wall cause a finite discontinuity in Dirichlet type of boundary conditions for the temperature distribution at one edge of the bottom wall. In contrast, the non-uniform heating removes the singularity at the edge of the bottom wall and provides a smooth temperature distribution in the entire cavity. Figs. 6 and 7 present the streamline, isotherm and isoconcentration contours for various values of N and Ra when the bottom wall and left vertical wall are non-uniformly heated. Comparing Figs. 4 and 6 or Figs. 5 and 7 it can be said that the patterns of streamlines are almost similar for uniformly and non-uniformly heated and concentrated cases. But significant differences
Fig. 13. Least square curve fitting of average Nusselt number with thermal Rayleigh number at bottom and left vertical walls for (a,c) uniformly heated and uniformly concentrated and (b,d) non-uniformly heated and non-uniformly concentrated when Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and N = 1.
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785
are observed in the patterns of isotherms and isoconcentrations for uniformly and non-uniformly heated and concentrated cases. Fig. 8 shows the changes of the patterns of the streamlines, isotherms and isoconcentrations with time when bottom and left walls are non-uniformly heated and concentrated with Ra ¼ 103 , Pr ¼ 0:7, Le ¼ 2:0, N ¼ 1 and A = 1. The pattern of the stream lines show large eddy covering almost whole of the cavity but does not show any eddy in the left lower corner of the cavity when nondimensional time t = 0.0112. A small eddy is formed at the left lower corner at time t = 0.0335 and the large eddy covering almost whole of the cavity increases. After that, size of both the eddies increase when t = 0.0558. Ultimately the flow inside the cavity reaches steady state and the eddies stop increasing when t = 1.7779. The pattern of isotherms changes significantly with time. Initially when t = 0.0112, the isotherm contours are concen-
783
trated near the left and lower walls. After that, as time progresses isotherms spread through out the cavity. The similar thing happens to the pattern of isoconcentrations also. Fig. 9 displays the local Nusselt number and local Sherwood number at the heated walls for various values of N due to uniform temperature and uniform concentration boundary conditions. It is interesting to note that in all the figures, the local Nusselt number and the local Sherwood number is zero at the left edge of bottom wall. Fig. 9(a) and (c) show that heat transfer rate and mass transfer rate are very high at the right edge of the bottom wall due to discontinuity present in temperature and concentration boundary conditions at this edge. On the other hand, the local Nusselt number and the local Sherwood number at the left vertical wall smoothly increase from bottom edge and attain a maximum value near the mid point of the left vertical wall which then decrease to a
Fig. 14. Least square curve fitting of average Sherwood number with thermal Rayleigh number at bottom and left vertical walls for (a,c) uniformly heated and uniformly concentrated and (b,d) non-uniformly heated and non-uniformly concentrated when Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and N = 1.
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Table 4 Computed values of ‘‘coefficient’’ and ‘‘power’’ with standard error of least square curve fitting formula of average Nusselt number with thermal Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions when Pr = 0.7, Le = 2.0, A = 1 and N = 1. Boundary conditions
Value (bottom wall)
Standard error (bottom wall)
Value (left vertical wall)
Standard error (left vertical wall)
Uniform
Coefficient Power
1.2804 0.1558
0.1146 0.0071
0.0342 0.3385
0.0015 0.0032
Non-uniform
Coefficient Power
0.2575 0.2356
0.002 0.0006
0.0143 0.3655
0.0009 0.0048
number at bottom and left vertical walls for uniform and non-uniform boundary conditions. The fitted model is of the form,
NuH
or ShH ¼ coefficient Rpower : a
ð63Þ
The values of ‘‘coefficient’’ and ‘‘power’’ with standard error of least square curve fitting formula of average Nusselt number or Sherwood number with Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions when Pr = 0.7, Le = 2.0, A = 1, Ra⁄ = 103 and N = 1 is found from Tables 4 and 5. The following relation are obtained for cases a and b (uniform boundary conditions) and c and d (non-uniform boundary conditions) as follows: Case a
Table 5 Computed values of ‘‘coefficient’’ and ‘‘power’’ with standard error of least square curve fitting formula of average Sherwood number with thermal Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions when Pr = 0.7, Le = 2.0, A = 1 and N = 1. Boundary conditions
Value (bottom wall)
Standard error (bottom wall)
Value (left vertical wall)
Standard error (left vertical wall)
Uniform
Coefficient Power
1.2476 0.1724
0.0997 0.0063
0.0530 0.3356
0.0008 0.0012
Non-uniform
Coefficient Power
0.3227 0.2371
0.0044 0.0011
0.0291 0.3419
0.0027 0.0069
NuH ¼ 1:2804 R0:1558 ðat bottom wallÞ; a NuH ¼ 0:0342 R0:3385 ðat left vertical wallÞ: a
ð64Þ
Case b
ShH ¼ 1:2476 R0:1724 ðat bottom wallÞ; a ShH ¼ 0:0530 R0:3356 ðat left vertical wallÞ: a
ð65Þ
Case c
NuH ¼ 0:2575 R0:2356 ðat bottom wallÞ; a NuH ¼ 0:0143 R0:3655 ðat left vertical wallÞ: a
ð66Þ
Case d non-zero value as depicted in Fig. 9(b) and (d). The physical reason of this behavior is due to high flow rate. Also, it is seen from these two figures that the local Nusselt number and the local Sherwood number increase with increase in N. Fig. 10 shows the local Nusselt number and local Sherwood number at the heated walls for various values of N due to non-uniform temperature and non-uniform concentration boundary conditions. It is interesting to note from this figure that for all values of N, the non-uniform heating enhances the heat transfer at the central region of the walls. It is also observed from this figure that the local Nusselt number and the local Sherwood number increase with increase in N which attains a maximum value near the right side of the bottom wall, but the local Nusselt number and the local Sherwood number are very high near the middle region of the left vertical wall forming wavy type of heat and mass transfer rate. Fig. 11 shows the effect of A on heat transfer rate when Ra ¼ 103 , Pr ¼ 0:7, Le ¼ 2:0 and N = 1 for uniform and non-uniform heating. From Fig. 11(a) it is seen that for uniform heating, the local Nusselt number at the left vertical wall is minimum at the bottom edge of that wall for all the values of A. It is also seen that the local Nusselt number increases with increase in A. But from Fig. 11(b) it is seen that for non-uniformly heating, the local Nusselt number at the left vertical wall is minimum at the top edge of that wall. Again it is seen that near the bottom edge of left vertical wall the local Nusselt number increases with increase in A and opposite trend is seen near the top edge of the wall. Fig. 12 shows the effect of buoyancy ratio N on the average Nusselt and the average Sherwood number with uniform and non-uniform boundary conditions. It is seen from this figure that when N = 1 and Ra ¼ 103 , Pr ¼ 0:7, Le ¼ 2:0 and A = 1, the average Nusselt number and the average Sherwood number take same minimum value for uniform and non-uniform boundary conditions at the left vertical wall (Fig. 12(b) and (d)). At the bottom wall, uniform and non-uniform boundary conditions produce S-type of the average Nusselt number and the average Sherwood number with their maximum value at right edge of the bottom wall (Fig. 12(a) and (c)). Figs. 13 and 14 show a least square curve fitting of average Nusselt number and average Sherwood number with thermal Rayleigh
ShH ¼ 0:3227 R0:2371 ðat bottom wallÞ; a ShH ¼ 0:0291 R0:3419 ðat left vertical wallÞ: a
ð67Þ
It is observed from Tables 4 and 5 that the standard error of estimation is very less, so the relation obtained for average Nusselt number and average Sherwood number as function of thermal Rayleigh number would result into almost same values as obtained from numerical computation of the set of differential equations with appropriate boundary conditions. Thus it is interesting to found that without wasting much time on computation, relations from Eqs. (64)–(67) can well be used for estimating average Nusselt and average Sherwood number for various values of thermal Rayleigh number. 5. Conclusion The main objective of the current investigation is to study the influence of buoyancy ratio on double-diffusive natural convection in a lid-driven square cavity with uniform and non-uniform thermal and concentration boundary conditions at the bottom wall and at the left vertical wall. It is observed that in the case of uniform heating and uniform concentration, the heat and mass transfer rate are zero for both left edge of the bottom wall and bottom edge of left vertical wall. Further, it is observed that the isotherm and isoconcentration lines are almost parallel to the horizontal wall in the middle part of the cavity when N = 50, indicating that most of the heat transfer is carried out by heat conduction. This is due to the increase in thermal boundary layer thickness. As, the buoyancy ratio increases the boundary layer thickness becomes thinner. The change in the flow structure for high buoyancy ratio has a significant influence on the concentration field, which builds up a vertical stratification of the enclosure. When buoyancy ratio N = 50, the effect of solutal buoyancy force is in the opposite direction of the thermal buoyancy force. The magnitude of the thermal buoyancy force is very small compared with the solutal buoyancy force due to which isotherm and isoconcen-
T.R. Mahapatra et al. / International Journal of Heat and Mass Transfer 57 (2013) 771–785
tration tend to move to left wall whereas, when the buoyancy ratio N = 1 both thermal and solutal buoyancy forces are equal for which isothermal and isoconcentration tend to move to the right wall. The change of patterns with respect to time is described here. It is concluded that the heat and mass transfer rate are very high at the right edge of the bottom wall. In contrast, for the case of non-uniform heating and non-uniform concentration the heat and mass transfer rate are minimum at the left edge of bottom wall and top corner of left vertical wall. Also, it is seen that average Nusselt number and the average Sherwood number attains their minimum value when the value of buoyancy ratio is 1 at left vertical wall for both uniform and non-uniform boundary conditions. Finally, least square curves are fitted for average Nusselt number and average Sherwood number with thermal Rayleigh number at bottom and left vertical walls for uniform and non-uniform boundary conditions and the standard errors are calculated. Acknowledgement The work of two of the authors (T.R.M and D.P) is supported under SAP (DRS PHASE II) program of UGC, New Delhi, India. The authors wish to thank the referees for helpful comments and suggestions which improved the quality of the paper considerably. References [1] H.K. Wee, R.B. Keey, M.J. Cunningham, Heat and moisture transfer by natural convection in a rectangular cavity, Int. J. Heat Mass Transfer 32 (1) (1989) 765– 1778. [2] C. Beghein, F. Haghighat, F. Allard, Numerical study of double-diffusive natural convection in a square cavity, Int. J. Heat Mass Transfer 35 (1992) 833–846. [3] K. Ghorayeb, H. Khallouf, A. Mojtabi, Onset of oscillatory flows in double diffusive convection, Int. J. Heat Mass Transfer 42 (1999) 629–643. [4] Qi-Hong Deng, Jiemin Zhou, Chi Mei, Yong-Ming Shen, Fluid, heat and contaminant transport structures of laminar double diffusive mixed convection in a two dimensional ventilated enclosure, Int. J. Heat Mass Transfer 47 (2004) 5257–5269.
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