Heat transfer resulting from the natural convection in a fluid layer contained in an infinite horizontal slot bounded by solid walls and subject to a spatially periodic heating at the lower wall has been investigated. The heating produces sinusoidal temperature variations along one horizontal direction characterized by the wave number α with the amplitude expressed in terms of a suitably defined Rayleigh number Rap. The maximum heat transfer takes place for the heating with the wave numbers α = 0(1) as this leads to the most intense convection. The intensity of convection decreases proportionally to α when α→0, resulting in the temperature field being dominated by periodic conduction with the average Nusselt number decreasing proportionally to α2. When α→∞, the convection is confined to a thin layer adjacent to the lower wall with its intensity decreasing proportionally to α−3. The temperature field above the convection layer looses dependence on the horizontal direction. The bulk of the fluid sees the thin convective layer as a “hot wall.” The heat transfer between the walls becomes dominated by conduction driven by a uniform vertical temperature gradient which decreases proportionally to the intensity of convection resulting in the average Nusselt number decreasing as α−3. It is shown that processes described above occur for Prandtl numbers 0.001 < Pr < 10 considered in this study.

References

References
1.
Bénard
,
H.
,
1900
, “
Les tourbillons cellulaires dans une nappe liquide
,”
Rev. Gen. Sci. Pures Appl.
,
11
, pp.
1261
1271
.
2.
Rayleigh
,
J. W. S.
,
1916
, “
On Convection Currents in a Horizontal Layer of Fluid, When the Higher Temperature is on the Under Side
,”
Philos. Mag.
,
32
, pp.
529
546
.10.1080/14786441608635602
3.
Sparrow
,
E. M.
, and
Charmchi
,
M.
,
1980
, “
Heat Transfer and Fluid Flow Characteristics of Spanwise-Periodic Corrugated Ducts
,”
Int. J. Heat Mass Transfer
,
23
, pp.
471
481
.10.1016/0017-9310(80)90089-7
4.
Asako
,
Y.
,
Nakamura
,
H.
, and
Faghri
,
M.
,
1988
, “
Heat Transfer and Pressure Drop Characteristics in a Corrugated Duct With Rounded Corners
,”
Int. J. Heat Mass Transfer
,
31
, pp.
1237
1245
.10.1016/0017-9310(88)90066-X
5.
Ligrani
,
P. M.
,
Oliveira
,
M. M.
, and
Blaskovich
,
T.
,
2003
, “
Comparison of Heat Transfer Augmentation Techniques
,”
AIAA J.
,
41
, pp.
337
362
.10.2514/2.1964
6.
Kelly
,
R. E.
, and
Pal
,
D.
,
1976
, “
Thermal Convection Induced Between Non-Uniformly Heated Horizontal Surfaces
,”
Proceedings of Heat Transfer and Fluid Mechanics Institute
,
Stanford University Press
, pp.
1
17
.
7.
Yoo
,
J. S.
and
Kim
,
M. U.
,
1991
, “
Two-Dimensional Convection in a Horizontal Fluid Layer With Spatially Periodic Boundary Temperatures
,”
Fluid Dyn. Res.
,
7
, pp.
181
200
.10.1016/0169-5983(91)90057-P
8.
Vozovoi
,
L. P.
, and
Nepomnyashchy
,
A. A.
,
1974
, “
Convection in a Horizontal Layer in the Presence of Spatial Modulation of Temperature at the Boundaries
,”
Gidrodynamika
,
7
, pp.
105
117
.
9.
Kelly
,
R. E.
, and
Pal
,
D.
,
1978
, “
Thermal Convection With Spatially Periodic Boundary Conditions: Resonant Wavelength Excitation
,”
J. Fluid Mech.
,
86
, pp.
433
456
.10.1017/S0022112078001226
10.
Pal
,
D.
, and
Kelly
,
R. E.
,
1978
, “
Thermal Convection With Spatially Periodic Non-Uniform Heating: Non-Resonant Wavelength Excitation
,”
Proceedings of 6th International Heat Transfer Conference
,
Toronto
, pp.
235
238
.
11.
Riahi
,
D. N.
,
1995
, “
Finite Amplitude Thermal Convection With Spatially Modulated Boundary Temperatures
,”
Proc. R. Soc. London, Ser. A
,
449
, pp.
459
478
.10.1098/rspa.1995.0053
12.
Schmitz
,
R.
, and
Zimmerman
,
W.
,
1996
, “
Spatially Periodic Modulated Rayleigh-Bénard Convection
,”
Phys. Rev. E
,
53
, pp.
5993
6011
.10.1103/PhysRevE.53.5993
13.
Rees
,
D. A. S.
,
1990
, “
The Effects of Long-Wavelength Thermal Modulations on the Onset of Convection in an Infinite Porous Layer Heated Form Below
,”
Q. J. Mech. Appl. Math.
,
43
, pp.
189
214
.10.1093/qjmam/43.2.189
14.
Canuto
,
C.
,
Hussaini
,
M. Y.
,
Quarteroni
,
A.
, and
Zang
,
T. A.
,
1996
,
Spectral Methods
,
Springer
,
New York
.
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