A comprehensive numerical investigation on the natural convection in a hydrodynamically anisotropic porous enclosure is presented. The flow is due to nonuniformly heated bottom wall and maintenance of constant temperature at cold vertical walls along with adiabatic top wall. Brinkman-extended non-Darcy model, including material derivative, is considered. The principal direction of the permeability tensor has been taken oblique to the gravity vector. The spectral element method has been adopted to solve numerically the governing conservative equations of mass, momentum, and energy by using a stream-function vorticity formulation. Special attention is given to understand the effect of anisotropic parameters on the heat transfer rate as well as flow configurations. The numerical experiments show that in the case of isotropic porous enclosure, the maximum rates of bottom as well as side heat transfers ($Nub$ and $Nus$) take place at the aspect ratio, $A$, of the enclosure equal to 1, which is, in general, not true in the case of anisotropic porous enclosures. The flow in the enclosure is governed by two different types of convective cells: rotating (i) clockwise and (ii) anticlockwise. Based on the value of media permeability as well as orientation angle, in the anisotropic case, one of the cells will dominate the other. In contrast to isotropic porous media, enhancement of flow convection in the anisotropic porous enclosure does not mean increasing the side heat transfer rate always. Furthermore, the results show that anisotropy causes significant changes in the bottom as well as side average Nusselt numbers. In particular, the present analysis shows that permeability orientation angle has a significant effect on the flow dynamics and temperature profile and consequently on the heat transfer rates.

1.
Holm
,
N. G.
,
Cairns-Smith
,
A. G.
,
Daniel
,
R. M.
,
Ferris
,
J. P.
,
Hennet
,
R. J. C.
,
Shock
,
E. L.
,
Simoneit
,
B. R. T.
, and
Yanagawa
,
H.
, 1992, “
Future Research
,”
Marine Hydrothermal Systems and the Origin of Life
,
N. G.
Holm
, ed.,
,
Dordrecht
.
2.
Kaviany
,
M.
, 1995,
Principles of Heat Transfer in Porous Media
,
Springer
,
New York
.
3.
Sahimi
,
M.
, 1995,
Flow and Transport in Porous Media and Fractured Rock
,
VCH
,
Weinheim
.
4.
Helmig
,
R.
, 1997,
Multiphase Flow and Transport Processes in the Subsurfaces
,
Springer-Verlag
,
Berlin
.
5.
Koponen
,
A.
,
Kandhai
,
D.
,
Hellen
,
E.
,
Alava
,
M.
,
Hoekstra
,
A.
,
Kataja
,
M.
,
Niskanen
,
K.
,
Sloot
,
P.
, and
Timonen
,
J.
, 1998, “
Permeability of Three Dimensional Random Fiber Webs
,”
Phys. Rev. Lett.
0031-9007,
80
, pp.
716
719
.
6.
King
,
P. R.
,
Buldyrev
,
S. V.
,
Dokholyan
,
N. V.
,
Havlin
,
S.
,
Lee
,
Y.
,
Paul
,
G.
, and
Stanley
,
H. E.
, 1999, “
Applications of Statistical Physics to the Oil Industry: Predicting Oil Recovery Using Percolation Theory
,”
Physica A
0378-4371,
274
, pp.
60
66
.
7.
Dando
,
P. R.
,
Stuben
,
D.
, and
Varnavas
,
S. P.
, 1999, “
Hydrothermalism in the Mediterranean Sea
,”
Prog. Oceanogr.
0079-6611,
44
, pp.
333
367
.
8.
Nield
,
D. A.
, and
Bejan
,
A.
, 2006,
Convection in Porous Media
,
Springer
,
New York
.
9.
Poulikakos
,
D.
, 1985, “
A Departure From Darcy Model in Boundary Layer Natural Convection in a Vertical Porous Layer With Uniform Heat Flux from the Side
,”
J. Heat Transfer
0022-1481,
107
, pp.
716
720
.
10.
Nithiarasu
,
P.
,
Seetharamu
,
K. N.
, and
Sundararajan
,
T.
, 1997, “
Natural Convective Heat Transfer in the Fluid Saturated Variable Porosity Medium
,”
Int. J. Heat Mass Transfer
0017-9310,
40
, pp.
3955
3967
.
11.
Baytas
,
A. C.
, and
Pop
,
I.
, 2002, “
Free Convection in a Square Porous Cavity Using a Thermal Non-Equilibrium Model
,”
Int. J. Therm. Sci.
1290-0729,
41
, pp.
861
870
.
12.
Walker
,
K. L.
, and
Homsy
,
G. M.
, 1978, “
Convection in Porous Cavity
,”
J. Fluid Mech.
0022-1120,
87
, pp.
449
474
.
13.
Saeid
,
N. F.
, and
Pop
,
I.
, 2004, “
Transient Free Convection in a Square Porous Cavity Filled With Porous Medium
,”
Int. J. Heat Mass Transfer
,
47
, pp.
1917
1924
. 0017-9310
14.
Ni
,
J.
, and
Beckermann
,
C.
, 1991, “
Natural Convection in a Vertical Enclosure Filled With Anisotropic Porous Media
,”
ASME J. Heat Transfer
0022-1481,
113
, pp.
1033
1037
.
15.
Degan
,
G.
,
Vasseur
,
P.
, and
Bilgen
,
E.
, 1995, “
Convective Heat Transfer in a Vertical Anisotropic Porous Layer
,”
Int. J. Heat Mass Transfer
0017-9310,
38
, pp.
1975
1987
.
16.
Degan
,
G.
, and
Vasseur
,
P.
, 1996, “
Natural Convection in a Vertical Slot Filled With an Anisotropic Porous Medium With Oblique Principal Axes
,”
Numer. Heat Transfer, Part A
,
30
, pp.
397
412
. 1040-7782
17.
Dhanasekaran
,
M. R.
,
Das
,
S. K.
, and
Venkateshan
,
S. P.
, 2002, “
Natural Convection in a Cylindrical Enclosure Filled With Heat Generating Anisotropic Porous Medium
,”
ASME J. Heat Transfer
0022-1481,
124
, pp.
203
207
.
18.
Bera
,
P.
,
Eswaran
,
V.
, and
Singh
,
P.
, 1998, “
Numerical Study of Heat and Mass Transfer in an Anisotropic Porous Enclosure Due to Constant Heating and Cooling
,”
Numer. Heat Transfer, Part A
,
34
, pp.
887
905
. 1040-7782
19.
Bera
,
P.
, and
Khalili
,
A.
, 2002, “
Double-Diffusive Natural Convection in an Anisotropic Porous Cavity With Opposing Buoyancy Forces: Multi-Solutions and Oscillations
,”
Int. J. Heat Mass Transfer
,
45
, pp.
3205
3222
. 0017-9310
20.
Lage
,
J. I.
, and
Bejan
,
A.
, 1993, “
The Resonance of Natural Convection in an Enclosure Heated Periodically From the Side
,”
Int. J. Heat Mass Transfer
0017-9310,
36
, pp.
2027
2038
.
21.
Sarris
,
I. E.
,
Lekakis
,
I.
, and
Vlachos
,
N. S.
, 2002, “
Natural Convection in a 2D Enclosure With Sinusoidal Upper Wall Temperature
,”
Numer. Heat Transfer, Part A
1040-7782,
42
, pp.
513
520
.
22.
Bilgen
,
E.
, and
Yedder
,
R. B.
, 2007, “
Natural Convection in Enclosure With Heating and Cooling by Sinusoidal Temperature Profiles on One Side
,”
Int. J. Heat Mass Transfer
0017-9310,
50
, pp.
139
150
.
23.
Saeid
,
N. F.
, 2005, “
Natural Convection in Porous Cavity With Sinusoidal Bottom Wall Temperature Variation
,”
Int. Commun. Heat Mass Transfer
0735-1933,
32
, pp.
454
463
.
24.
Basak
,
T.
,
Roy
,
S.
,
Poul
,
T.
, and
Pop
,
I.
, 2006, “
Natural Convection in a Square Cavity Filled With a Porous Medium: Effect of Various Thermal Boundary Conditions
,”
Int. J. Heat Mass Transfer
0017-9310,
49
, pp.
1430
1441
.
25.
Zahmatkesh
,
I.
, 2008, “
On the Importance of Thermal Boundary Conditions in Heat Transfer and Entropy Generation for Natural Convection Inside a Porous Enclosure
,”
Int. J. Therm. Sci.
,
47
, pp.
339
346
. 1290-0729
26.
Patera
,
A. T.
, 1984, “
A Spectral Element Method for Fluid Dynamics: Laminar Flow in a Channel Expansion
,”
J. Comput. Phys.
0021-9991,
54
, pp.
468
488
.
27.
Canuto
,
C.
,
Hussaine
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T. A.
, 1986,
Spectral Method in Fluid Dynamics
,
Springer
,
New York
.
28.
Neale
,
G.
, 1977, “
Degrees of Anisotropy for Fluid Flow and Diffusion (Electrical Conduction) Through Anisotropic Porous Media
,”
AIChE J.
0001-1541,
23
, pp.
56
62
.
29.
Wooding
,
R. A.
, 1976, “
Large-Scale Geothermal Field Parameters and Convection Theory
,”
Second Workshop Geothermal Reservoir Engineering
, December 1–3,
Stanford University, Stanford
, pp. 339–345.
30.
Tyvand
,
P. A.
, and
Storesletten
,
L.
, 1991, “
Onset of Convection in an Anisotropic Porous Medium With Oblique Principal Axes
,”
J. Fluid Mech.
0022-1120,
226
, pp.
371
382
.
31.
Storesletten
,
L.
, 1998, “
Effects of Anisotropy on Convective Flow Through Porous Media
,”
Transport Phenomena in Porous Media
,
D. B.
Ingham
and
I.
Pop
, eds.,
Pergamon
,
Oxford
, pp.
261
283
.
32.
Tyvand
,
P. A.
, 1980, “
Thermohaline Instability in Anisotropic Porous Media
,”
Water Resour. Res.
,
16
, pp.
325
330
. 0043-1397
33.
Lauriat
,
A.
, and
,
V.
, 1989, “
Non-Darcian Effects on Natural Convection in a Vertical Porous Enclosure
,”
Int. J. Heat Mass Transfer
0017-9310,
32
, pp.
2135
2148
.
34.
Bera
,
P.
, and
Khalili
,
A.
, 2007, “
Stability of Buoyancy Opposed Mixed Convection in a Vertical Channel and Its Dependency on Permeability
,”