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The Seventh Asian Congress of Fluid Mechanics Dec 8 12, 1997, Chennai (Madras) ,
NATURAL CONVECTION IN A RECTANGULAR POROUS CAVITY
M.Venkatachalappa 1.5.Shivakumara M.Sankar B.M.R.Prasanna UGC-DSA Centrein FluidMechanics,Departmentof Mathematics,BangaloreUniversity,Bangalore 560001,India. ABSTRACT The present numerical study based on ADI (Alternate Direction Implicit) method combined with upwind differencing scheme in the presence of non-Darcy effects on natural convection in a cavity with high porosity porous media show that the inertia effects, viscosity and aspect ratios have significant influence on the streamline pattern and temperature distributions. These effects reduce the heat transfer rate from the rectangular cavity with side walls maintained at different temperatures and the horizontal walls kept adiabatic and become more noticeable near the walls. The results also show that as resistance due to solid matrix increases the temperature profile show channelling effects. 1. Introduction Because of its relevance in variety of situations like geothermal system, thermal insulation, coal and grain storage, solid matrix heat exchangers, nuclear waste disposal and so on, natural convection in a porous medium is a well developed field of investigation. Most works on convection in porous media are concerned with the channel ( either vertical or horizontal) of infinite extent and limited to low or moderate Darcy Numbers and effective Brinkman viscosity equal to fluid viscosity[l]. Vafai and Kim [2] have studied convection in a-vertical channel for the case of fully developed flow by considering either constant heatflux or temperatures at the boundaries. Their analysis was limited to the case of effective Brinkman viscosity equal to the fluid viscosity. Most of the practical problems cited above, particularly the industrial problems involve sparsely packed porous cavities with high permeability having effective Brikman viscosity much larger than the fluid viscosity. In spite of its importance much work has not been done on natural convection in cavities involving boundary and inertia effects with. effective viscosity different from fluid viscosity. In this paper we consider a rectangular porous cavity with side walls maintained at uniform but different temperatures with top and bottom walls closed and insulated. 2. Mathematical Formulation We consider a two dimensional rectangular porous cavity of height H and width L filled with homogeneous, isotropic, sparsely packed porous material of high permeability K. The top and bottom walls are closed and insulated and side walls are maintained at constant but different
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temperatures. Based on the Boussinesq approximation, the governing equations using the vorticity - stream function formulation in nondimensional form are
s aT + U aT + V aT = 1- V2 T i7t ax ay Pr 1-
(1)
E2 [ i7t
+ U
ax
+ V
E; =
ay]
Gr aT + A
2 ay
( ax2
ay2 J
- Da-lt,
(2) (3)
2 V'=-t"
where
u = a'l' v=_a' ay , ax '
v _, v 't=~, tof, u= -, -aiL a/L-.2 co \jI 8 - 80 a2 t, = , ' = - , T =. ., 80 = aiL + 8c V2 = ~+ ~ T:1 8h -, 2 ax ar alL2 ex. 8h - 8c
(4)
r - av - au x - x
"'-ax BY' -L'
y - .r
-L'
The dimensionlessparametersare: Pr = ~ ,aD = K 2' ex. L
A = J ,r G = gP(8h-2'8c)L3 ~ v
=H . L
Initially the fluid is quiescent and the temperature is uniform throughout. The initial and boundary conditions in the non-dimensional form are: 't = 0 : U = v.. = ' = 0, T = t, = 0, 0 ~ X ~ A , 0 ~ y ~ 1 't>o: '=~=O T=-lO at y= O
ay" .
a' ' = - = 0 T = + 1.0 ay , a' aT '=-=O -=0 ax ' ax
3. Numerical Method
at at
Y = 1
x= 0 andX=A
The modified two step ADI technique is employed to advance the fields of temperature and vorticity at the interior grid points across a time step n l:1't to the new level ( n + 1 ) l:1'to In order to improve the stability of the numerical scheme, the nonlinear convective terms are evaluated using second order upwind difference method, the diffusion terms are approximated using central difference and the time derivative is by forward difference. The method of SLOR is then employed to solve the elliptic stream function equation for new stream function field. Finally the velocities are computed from the stream function solutions using equation (4). The whole process described above is repeated for each time steps until steady state ~lution is obtained. Prior to the calculations, as a partial verification of the computational procedure,. the results are compared with the solutions given by Wilkes and Churchill [3] in the absence of
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porousmediumand are found to be in good agreement. 4. Results and Discussion
Nonlinear natural convection in a rectangular porous cavity is investigated numerically for the values of Gr = 50000, Pr = 7, A = 0.1,1 & 3, S = 1 and porosity E = 0.9 and the results are shown in Figs. 1 to 4. Figures 1a and 1b show that effect of Darcy number on streamlines for the different values of viscosity ratio. We found that for Da = 0.01 the flow structure is symmetric to the axes (see Fig. la). But for Da = 0.001 the streamlines distorted slightly and oriented along the principal diagonal of the cavity and the central stream lines are almost in elliptic shape. (see Fig. Ib). From these figures it is also evident that the effect of viscosity ratio on the streamline pattern is to change the position of maximun value of stream function. Figures 2a and 2b illustrate the isotherm contours. From Fig. 2a, we find that the temperature stratification is dominant near the top and bottom of the cold and hot walls respectively. As Da decreases the temperature stratification also decreases (see Fig. 2b). This is because when Da decreases the Darcy resistance is dominant and the convection reduces. It is observed that as the viscosity ratio increases, the shift of isotherms towards the hot wall is less pronounced and also their effect is to reduce the thermal boundary layer growth. To know the influence of aspect ratio, the numerical results for A = 3 are reported in Figs. (3a) and (3b) in terms of streamlines and temperature contours. We note that increase in the value of A distorts the pattern of streamlines considerably compared to those for smaller values of A. In the case of isotherms, there exists large temperature gradients which indicates a boundary layer structure in the regions adjacent to the thermally active ( hot and cold) walls. The rate of heat transfer across the cavity is obtained by evaluating the average Nusselt number NU at the hot wall and the variation of mean Nusselt number with dimensionless time is depicted in Figs. 4a and 4b. From this it is evident that NU decreases with the increase in viscosity ratio and decrease in Da. The effect of A on the heat transfer rate is noticeable for Da = 0.01 but not so significantfor Da = 0.001 (see Figs. 4a and 4b). We also found that the steady state can be obtained faster with the decrease in the value of Da. Acknowledgement The authors are very thankful to Prof. N. Rudraiah, INSA Senior Scientist, for his valuable discussions. This work was supported by UGC under DSA and COSIST Programmes. References
[1] Hong,J.T., Tien, C.L, and Kaviany,M, Int. J. Heat Mass Transfer,28, 11,2149 (1985).
563
{:J/J
[2] Kim, SJ and Vafai, K, Int. 1. Heat Mass Transfer, 32, 665 (1989). [3] Wilkes, J.O and Churchill, S.W, AI.Ch.E. J, 12,161 (1966). Nomenclature A aspect ratio Gr Grashof number H height of the cavity L width of the cavity Pr Prandtl number S ratio of specific heat t time T dimensionless temperature u vertical velocity U dimensionless vertical velocity v horizontal velocity V dimensionless horizontal velocity x vertical co-ordinate X dimensionless vertical co-ordinate y horizontal co-ordinate Y dimensionless horizontal co-ordinate
a 8 80 v
\If
thermal diffusivity coefficientof thermal expansion temperature referencetemperature kinematicviscosity
stream function dimensionless stream function dimensionless time vorticity dimensionless vorticity viscosity ratio porosity
'Ii
't
ro A E
0 0
y 0.5 0,
y 0.5 0
y 0.5
--
,0.-0.1 -1 -3
X 0.5
lHt:
~
'/--l,
.--~--
0 :-::\\
y 0.5
.~
"~ Fig. 1 STREAMLINES (e) D.
y 0.5 0
\', \'\... --:.:
I(bJ
O .
\
C:\:'
':
-,0..0.1 --1
-3 x
:\'
'
:I~ :
::
,': '../:1
. 0.01 & (b) D. = 0.001
0 y 0.5
3.1 , ~=-~,-.:.-~
~'J
~bI
:..i
Fig. 3 STREAMLINES & ISOTHERMS De = 0.01. A = : 10 10
- - -1 ----- - 3
X 0.5
-,0.-0.1
"'.",
Nu
"'.a.on'
Nu 5
-"".<
,,~
Fig. 2 ISOTHERMS (e) D. = 0.01 & D. = 0.001 '(b, 0.1 0.2 0.3 0.4 0.5
i 2l 0 0.2
0.4 0.5 0.8
,.,
b,
Fig.4 VARIATION FMEANNUSSELT UMBER O N WITHDIMENSIONLESS 1 TIME
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