A numerical study of natural convection with surface radiation in an air filled square enclosure with a centrally heated bottom wall and cooled upper wall is presented. The vertical walls and the rest of the bottom wall are assumed to be insulated. The problem is studied for Rayleigh numbers Ra, ranging from $103$ to $4×106$ and surfaces emissivity $ε$, varying from 0 to 1. The governing equations, written in terms of stream function-vorticity formulation, are solved using a finite difference approach. It is found that, under these heating/cooling conditions, three different steady-state solutions are possible in the ranges of the parameters considered. Results are presented detailing the occurrence of each steady-state solution and the effect of Ra and $ε$ on its range of existence. It is found that the surface radiation alters significantly the existence ranges of the solutions. For each solution, convective and radiative contributions to the global heat transfer are also quantified for various Ra and $ε$. The influence of the heated surface dimension on the fluid flow and thermal patterns is also presented by comparing the present results against those obtained by the authors in an earlier study within a square cavity totally heated from below.

1.
Chong
,
L. C.
,
Yang
,
K. T.
, and
Lloyd
,
J. R.
, 1983, “
Radiation–Natural Convection Interactions in Two-Dimensional Complex Enclosures
,”
J. Heat Transfer
0022-1481,
105
(
1
), pp.
89
95
.
2.
Webb
,
B. W.
, and
Viskanta
,
R.
, 1987, “
Analysis of Radiation-Induced Natural Convection in a Rectangular Enclosure
,”
J. Thermophys. Heat Transfer
0887-8722,
1
(
2
), pp.
146
153
.
3.
Siegel
,
R.
, and
Howell
,
J. R.
, 1981,
Thermal Radiation Heat Transfer
, 2nd ed.,
McGraw-Hill
, New York.
4.
Modest
,
M. F.
, 2003,
, 2nd ed.,
, CA.
5.
Balaji
,
C.
, and
Venkateshan
,
S. P.
, 1993, “
Interaction of Surface Radiation with Free Convection in a Square Cavity
,”
Int. J. Heat Fluid Flow
0142-727X,
14
(
3
), pp.
260
267
.
6.
Balaji
,
C.
, and
Venkateshan
,
S. P.
, 1994, “
Correlations for Free Convection and Surface Radiation in a Square Cavity
,”
Int. J. Heat Fluid Flow
0142-727X,
15
(
3
), pp.
249
251
.
7.
Akiyama
,
M.
, and
Chong
,
Q. P.
, 1997, ‘
Numerical Analysis of Natural Convection with Surface Radiation in a Square Enclosure
,”
Numer. Heat Transfer, Part A
1040-7782,
31
, pp.
419
433
.
8.
Mezrhab
,
A.
, and
Bchir
,
L.
, 1999, “
Radiation-Natural Convection Interactions in Partitioned Cavities
,”
Int. J. Numer. Methods Heat Fluid Flow
0961-5539,
9
(
1
), pp.
186
203
.
9.
Mahapatra
,
S. K.
,
Sen
,
S.
, and
Sarkar
,
A.
, 1999, “
Interaction of Surface Radiation and Variable Property Natural Convection in a Differentially Heated Square Cavity—A Finite Element Analysis
,”
Int. J. Numer. Methods Heat Fluid Flow
0961-5539,
9
(
4
), pp.
423
443
.
10.
Han
,
C. Y.
, and
Baek
,
S. W.
, 2000, “
The Effect of Radiation on Natural Convection in a Rectangular Enclosure Divided by Two Partitions
,”
Numer. Heat Transfer, Part A
1040-7782,
37
, pp.
249
270
.
11.
Colomer
,
G.
,
Costa
,
M.
,
Consul
,
R.
, and
Oliva
,
A.
, 2004, “
Three Dimensional Numerical Simulation of Convection and Radiation in a Differentially Heated Cavity Using the Discrete Ordinates Method
,”
Int. J. Heat Mass Transfer
0017-9310,
47
, pp.
257
269
.
12.
Ramesh
,
N.
, and
Venkateshan
,
S. P.
, 1999, “
Effect of Surface Radiation on Natural Convection in a Square Enclosure
,”
J. Thermophys. Heat Transfer
0887-8722,
13
(
3
), pp.
299
301
.
13.
Ramesh
,
N.
,
Balaji
,
C.
, and
Venkateshan
,
S. P.
, 1999, “
Effect of Boundary Conditions on Natural Convection in an Enclosure
,”
Int. J. Transp. Phenom.
1028-6578,
1
, pp.
205
214
.
14.
Hasnaoui
,
M.
,
Bilgen
,
E.
, and
Vasseur
,
P.
, 1992, “
Natural Convection Heat Transfer in Rectangular Cavities Partially Heated from Below
,”
J. Thermophys. Heat Transfer
0887-8722,
6
(
2
), pp.
255
264
.
15.
Ridouane
,
E. H.
,
Hasnaoui
,
M.
,
Amahmid
,
A.
, and
Raji
,
A.
, 2004, “
Interaction Between Natural Convection and Radiation in a Square Cavity Heated from Below
,”
Numer. Heat Transfer, Part A
1040-7782,
45
, pp.
289
311
.
16.
Ridouane
,
E. H.
,
Hasnaoui
,
M.
, and
Campo
,
A.
, 2006, “
Effects of Surface Radiation on Natural Convection in a Rayleigh-Benard Square Enclosure: Steady and Unsteady Conditions
,”
Heat Mass Transfer
0947-7411,
42
, pp.
214
225
.
17.
Woods
,
L. C.
, 1954, “
A Note on Numerical Solution of Fourth Order Differential Equations
,”
Aeronaut. Q.
0001-9259,
5
, pp.
176
184
.
18.
Hottel
,
H. C.
, and
Sarofim
,
A. F.
, 1967,
,
McGraw-Hill
, New York.
19.
De Vahl Davis
,
G.
, 1983, “
Natural Convection of Air in a Square Cavity: A Benchmark Numerical Solution
,”
Int. J. Numer. Methods Fluids
0271-2091,
3
, pp.
249
264
.
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