Turbulent natural convection in a vertical two-dimensional square cavity, isothermally heated from below and cooled at the upper surface, is numerically analyzed using the finite volume method. The enclosure has a thin horizontal porous obstruction, made of a highly porous material and extremely permeable, located at the cavity midheight. Governing equations are written in terms of primitive variables and are recast into a general form. For empty cavities, no discrepancies result for the Nusselt number when laminar and turbulent model solutions are compared for Rayleigh numbers up to 107. Also, in general the porous obstruction decreases the heat transfer across the heated walls showing overall lower Nusselt numbers when compared with those without the porous obstruction. However, the presence of a porous plate in the cavity seems to force an earlier separation from laminar to turbulence model solutions due to higher generation rates of turbulent kinetic energy into the porous matrix.

1.
Jones
,
I. P.
, 1979,
A Comparison Problem for Numerical Methods in Fluid Dynamics: The Double-Glazing Problem, Numerical Methods in Thermal Problems
,
R. W.
Lewis
and
K.
Morgan
, eds.,
Pineridge Press
,
Swansea
, pp.
338
348
.
2.
de Vahl Davis
,
G.
, 1983, “
Natural Convection in a Square Cavity: A Benchmark Numerical Solution
,”
Int. J. Numer. Methods Fluids
0271-2091,
3
, pp.
249
264
.
3.
de Vahl Davis
,
G.
, and
Jones
,
I. P.
, 1983, “
Natural Convection in a Square Cavity—A Comparison Exercise
,”
Int. J. Numer. Methods Fluids
0271-2091,
3
, pp.
227
248
.
4.
Markatos
,
N. C.
, and
Pericleous
,
K. A.
, 1984, “
Laminar and Turbulent Natural Convection in an Enclosed Cavity
,”
Int. J. Heat Mass Transfer
0017-9310,
27
, pp.
755
772
.
5.
Henkes
,
R. A. W. M.
,
van der Vlugt
,
F. F.
, and
Hoogendoorn
,
C. J.
, 1991, “
Natural-Convection Flow in a Square Cavity Calculated With Low-Reynolds-Number Turbulence Models
,”
Int. J. Heat Mass Transfer
0017-9310,
34
(
2
), pp.
377
388
.
6.
Fusegi
,
T.
,
Hyun
,
J. M.
, and
Kuwahara
,
K.
, 1991, “
Three-Dimensional Simulations of Natural Convection in a Sidewall-Heated Cube
,”
Int. J. Numer. Methods Fluids
0271-2091,
3
, pp.
857
867
.
7.
Barakos
,
G.
,
Mitsoulis
,
E.
, and
Assimacopoulos
,
D.
, 1994, “
Natural Convection Flow in a Square Cavity Revised: Laminar and Turbulent Models With Wall Function
,”
Int. J. Numer. Methods Fluids
0271-2091,
18
, pp.
695
719
.
8.
Nield
,
D. A.
, and
Bejan
,
A.
, 1992,
Convection in Porous Media
,
Springer
,
New York
.
9.
Ingham
,
D. B.
, and
Pop
,
I.
, 1998,
Transport Phenomena in Porous Media
,
Elsevier
,
Amsterdam
.
10.
Walker
,
K. L.
, and
Homsy
,
G. M.
, 1978, “
Convection in Porous Cavity
,”
J. Fluid Mech.
0022-1120,
87
, pp.
449
474
.
11.
Bejan
,
A.
, 1979, “
On the Boundary Layer Regime in a Vertical Enclosure Filled With a Porous Medium
,”
Lett. Heat Mass Transfer
0094-4548,
6
, pp.
93
102
.
12.
Prasad
,
V.
, and
Kulacki
,
F. A.
, 1984, “
Convective Heat Transfer in a Rectangular Porous Cavity-Effect of Aspect Ratio on Flow Structure and Heat Transfer
,”
ASME J. Heat Transfer
0022-1481,
106
, pp.
158
165
.
13.
Beckermann
,
C.
,
Viskanta
,
R.
, and
Ramadhyani
,
S.
, 1986, “
A Numerical Study of Non-Darcian Natural Convection in a Vertical Enclosure Filled With a Porous Medium
,”
Numer. Heat Transfer
0149-5720,
10
, pp.
557
570
.
14.
Gross
,
R. J.
,
Bear
,
M. R.
, and
Hickox
,
C. E.
, 1986, “
The Application of Flux-Corrected Transport (Fct) to High Rayleigh Number Natural Convection in a Porous Medium
,”
Proc. 8th Int. Heat Transfer Conf.
,
San Francisco
,
CA
.
15.
Manole
,
D. M.
, and
Lage
,
J. L.
, 1992, “
Numerical Benchmark Results for Natural Convection in a Porous Medium Cavity, Heat and Mass Transfer in Porous Media
,” Asme Conference, Htd,
216
, pp.
55
60
.
16.
Baytas
,
A. C.
, and
Pop
,
I.
, 1999, “
Free Convection in Oblique Enclosures Filled With a Porous Medium
,”
Int. J. Heat Mass Transfer
0017-9310,
42
, pp.
1047
1057
.
17.
Hsu
,
C. T.
, and
Cheng
,
P.
, 1990, “
Thermal Dispersion in a Porous Medium
,”
Int. J. Heat Mass Transfer
0017-9310,
33
, pp.
1587
1597
.
18.
Bear
,
J.
, 1972,
Dynamics of Fluids in Porous Media
,
American Elsevier
,
New York
.
19.
Whitaker
,
S.
, 1966, “
Equations of Motion in Porous Media
,”
Chem. Eng. Sci.
0009-2509,
21
, pp.
291
300
.
20.
Whitaker
,
S.
, 1967, “
Diffusion and Dispersion in Porous Media
,”
Am. Inst. Chem. Eng. Symp. Ser.
0065-8812,
13
(
3
), pp.
420
427
.
21.
Masuoka
,
T.
, and
Takatsu
,
Y.
, 1996, “
Turbulence Model for Flow Through Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
39
(
13
), pp.
2803
2809
.
22.
Kuwahara
,
F.
,
Nakayama
,
A.
, and
Koyama
,
H.
, 1996, “
A Numerical Study of Thermal Dispersion in Porous Media
,”
ASME J. Heat Transfer
0022-1481,
118
, pp.
756
761
.
23.
Kuwahara
,
F.
, and
Nakayama
,
A.
, 1998, “
Numerical Modeling of Non-Darcy Convective Flow in a Porous Medium
,”
Heat Transfer 1998: Proceedings. 11th Int. Heat Transf. Conf., Kyongyu, Korea
,
Taylor & Francis
,
Washington, D. C.
, Vol.
4
, pp.
411
416
.
24.
Nakayama
,
A.
, and
Kuwahara
,
F.
, 1999, “
A Macroscopic Turbulence Model for Flow in a Porous Medium
,”
ASME J. Fluids Eng.
0098-2202,
121
, pp.
427
433
.
25.
Lee
,
K.
, and
Howell
,
J. R.
, 1987, “
Forced Convective and Radiative Transfer Within a Highly Porous Layer Exposed to a Turbulent External Flow Field
,”
Proceedings of the 1987 ASME-JSME Thermal Engineering Joint Conf., Honolulu, Hawaii
,
ASME
,
New York
, Vol.
2
, pp.
377
386
.
26.
Antohe
,
B. V.
, and
Lage
,
J. L.
, 1997, “
A General Two-Equation Macroscopic Turbulence Model for Incompressible Flow in Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
40
(
13
), pp.
3013
3024
.
27.
Getachewa
,
D.
,
Minkowycz
,
W. J.
, and
Lage
,
J. L.
, 2000, “
A Modified Form of the k-ε Model for Turbulent Flow of an Incompressible Fluid in Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
43
, pp.
2909
2915
.
28.
Pedras
,
M. H. J.
, and
de Lemos
,
M. J. S.
, 2000, “
On the Definition of Turbulent Kinetic Energy for Flow in Porous Media
,”
Int. Commun. Heat Mass Transfer
0735-1933,
27
(
2
), pp.
211
220
.
29.
Pedras
,
M. H. J.
, and
de Lemos
,
M. J. S.
, 2001, “
Macroscopic Turbulence Modeling for Incompressible Flow Through Undeformable Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
44
(
6
), pp.
1081
1093
.
30.
Pedras
,
M. H. J.
, and
de Lemos
,
M. J. S.
, 2001, “
Simulation of Turbulent Flow in Porous Media Using a Spatially Periodic Array and a Low-Re Two-Equation Closure
,”
Numer. Heat Transfer, Part A
1040-7782,
39
(
1
), pp.
35
59
.
31.
Pedras
,
M. H. J.
, and
de Lemos
,
M. J. S.
, 2001, “
On the Mathematical Description and Simulation of Turbulent Flow in a Porous Medium Formed by an Array of Elliptic Rods
,”
ASME J. Fluids Eng.
0098-2202,
123
(
4
), pp.
941
947
.
32.
Rocamora
,
F. D.
, Jr.
, and
de Lemos
,
M. J. S.
, 2000, “
Analisys of Convective Heat Transfer of Turbulent Flow in Saturated Porous Media
,”
Int. Commun. Heat Mass Transfer
0735-1933,
27
(
6
), pp.
825
834
.
33.
de Lemos
,
M. J. S.
, and
Rocamora
,
F. D.
, 2002, “
Turbulent Transport Modeling for Heated Flow in Rigid Porous Media
,”
Proceedings of the Twelfth International Heat Transfer Conference
,
Grenoble
,
France
, August 18–23, pp.
791
795
.
34.
de Lemos
,
M. J. S.
, and
Braga
,
E. J.
, 2003, “
Modeling of Turbulent Natural Convection in Saturated Rigid Porous Media
,”
Int. Commun. Heat Mass Transfer
0735-1933,
30
(
5
), pp.
615
624
.
35.
Braga
,
E. J.
, and
de Lemos
,
M. J. S.
, 2004, “
Turbulent Natural Convection in a Porous Square Cavity Computed With a Macroscopic k-ε Model
,”
Int. J. Heat Mass Transfer
0017-9310,
47
, pp.
5639
5650
.
36.
Saito
,
M.
, and
de Lemos
,
M. J. S.
, 2005, “
Interfacial Heat Transfer Coefficient for Non-Equilibrium Convective Transport in Porous Media
,”
Int. Commun. Heat Mass Transfer
0735-1933,
32
(
5
), pp.
667
677
.
37.
Saito
,
M.
, and
de Lemos
,
M. J. S.
, 2006, “
A Correlation for Interfacial Heat Transfer Coefficient for Turbulent Flow Over an Array of Square Rods
,”
ASME J. Heat Transfer
0022-1481,
128
(
5
), pp.
444
452
.
38.
de Lemos
,
M. J. S.
, and
Mesquita
,
M. S.
, 2003, “
Turbulent Mass Transport in Saturated Rigid Porous Media
,”
Int. Commun. Heat Mass Transfer
0735-1933,
30
(
1
), pp.
105
113
.
39.
de Lemos
,
M. J. S.
, and
Tofaneli
,
L. A.
, 2004, “
Modeling of Double-Diffusive Turbulent Natural Convection in Porous Media
,”
Int. J. Heat Mass Transfer
0017-9310,
47
(
19–20
), pp.
4221
4231
.
40.
Assato
,
M.
,
Pedras
,
M. H. J.
, and
de Lemos
,
M. J. S.
, 2005, “
Numerical Solution of Turbulent Channel Flow Past a Backward-Facing Step With a Porous Insert Using Linear and Nonlinear k-ε Models
,”
J. Porous Media
1091-028X,
8
(
1
), pp.
13
29
.
41.
Santos
,
N. B.
, and
de Lemos
,
M. J. S.
, 2006, “
Flow and Heat Transfer in a Parallel Plate Channel With Porous and Solid Baffles
,”
Numer. Heat Transfer, Part A
1040-7782,
49
(
5
), pp.
471
494
.
42.
Silva
,
R. A.
, and
de Lemos
,
M. J. S.
, 2003, “
Numerical Analysis of the Stress Jump Interface Condition for Laminar Flow Over a Porous Layer
,”
Numer. Heat Transfer, Part A
1040-7782,
43
(
6
), pp.
603
617
.
43.
Silva
,
R. A.
, and
de Lemos
,
M. J. S.
, 2003, “
Turbulent Flow in a Channel Occupied by a Porous Layer Considering the Stress Jump at the Interface
,”
Int. J. Heat Mass Transfer
0017-9310,
46
, pp.
5113
5121
.
44.
de Lemos
,
M. J. S.
, 2005, “
Turbulent Kinetic Energy Distribution Across the Interface Between a Porous Medium and a Clear Region
,”
Int. Commun. Heat Mass Transfer
0735-1933,
32
(
1–2
), pp.
107
115
.
45.
de Lemos
,
M. J. S.
, and
Silva
,
R. A.
, 2006, “
Turbulent Flow Over a Layer of a Highly Permeable Medium Simulated With a Diffusion-Jump Model for the Interface
,”
Int. J. Heat Mass Transfer
0017-9310,
49
(
3–4
), pp.
546
556
.
46.
de Lemos
,
M. J. S.
, and
Pedras
,
M. H. J.
, 2001, “
Recent Mathematical Models for Turbulent Flow for Saturated Rigid Porous Media
,”
ASME J. Fluids Eng.
0098-2202,
123
(
4
), pp.
935
940
.
47.
de Lemos
,
M. J. S.
, 2006,
Turbulence in Porous Media: Modeling and Applications
,
Elsevier
,
New York
.
48.
Kuwahara
,
F.
,
Kameyama
,
Y.
,
Yamashita
,
S.
, and
Nakayama
,
A.
, 1998, “
Numerical Modeling of Turbulent Flow in Porous Media Using a Spatially Periodic Array
,”
J. Porous Media
1091-028X,
1
(
1
), pp.
47
55
.
49.
Ergun
,
S.
, 1952, “
Fluid Flow Through Packed Columns
,”
Chem. Eng. Process.
0255-2701,
48
, pp.
89
94
.
50.
Ochoa-Tapia
,
J. A.
, and
Whitaker
,
S.
, 1995, “
Momentum Transfer at the Boundary Between a Porous Medium and a Homogeneous Fluid—I. Theoretical Development
,”
Int. J. Heat Mass Transfer
0017-9310,
38
, pp.
2635
2646
.
51.
Lee
,
K.
, and
Howell
,
J. R.
, 1987, “
Forced Convective and Radiative Transfer Within a Highly Porous Layer Exposed to a Turbulent External Flow Field
,”
Proceedings Of The 1987 ASME-JSME Thermal Engineering Joint Conf.
, Vol.
2
, pp.
377
386
.
52.
Khosla
,
P. K.
, and
Rubin
,
S. G.
, 1974, “
A Diagonally Dominant Second-Order Accurate Implicit Scheme
,”
Comput. Fluids
0045-7930,
2
, pp.
207
209
.
53.
Patankar
,
S. V.
, and
Spalding
,
D. B.
, 1972, “
A Calculation Procedure for Heat, Mass and Momentum Transfer in Three Dimensional Parabolic Flows
,”
Int. J. Heat Mass Transfer
0017-9310,
15
, pp.
1787
1806
.
54.
Stone
,
H. L.
, 1968, “
Iterative Solution of Implicit Approximations of Multi-Dimensional Partial Differential Equations
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429,
5
, pp.
530
558
.
55.
Ozoe
,
H.
,
Mouri
,
A.
,
Ohmuro
,
M.
,
Churchill
,
S. W.
, and
Lior
,
N.
, 1985, “
Numerical Calculations of Laminar and Turbulent Natural Convection in Water in Rectangular Channels Heated and Cooled Isothermally on the Opposing Vertical Walls
,”
Int. J. Heat Mass Transfer
0017-9310,
28
, pp.
125
138
.
You do not currently have access to this content.