In this paper the effect of a transverse magnetic field on buoyancy-driven convection in a shallow rectangular cavity is numerically investigated (horizontal Bridgman configuration). The enclosure is insulated on the top and bottom walls while it is heated from one side and cooled from the other. Both cases of a cavity with all rigid boundaries and a cavity with a free upper surface are considered. The study covers the range of the Rayleigh number, Ra, from 102 to 105, the Hartmann number, Ha, from 0 to 102, the Prandtl number, Pr, from 0.005 to 1 and aspect ratio of the cavity, A, from 1 to 6. Comparison is made with an existing analytical solution (Garandet et al., 1992), based on a parallel flow approximation, and its range of validity is delineated. Results are presented for the velocity and temperature profiles and heat transfer in terms of Ha number. At high Hartmann numbers, both analytical and numerical analyses reveal that the velocity gradient in the core is constant outside the two Hartmann layers at the vicinity of the walls normal to the magnetic field.

Issue Section:
Natural and Mixed Convection
Keywords:
Enclosure Flows, Materials Processing and Manufacturing Processes, Natural Convection
Topics:
Flow (Dynamics), Heat transfer, Magnetic fields, Natural convection, Cavities, Approximation, Buoyancy, Convection, Manufacturing, Materials processing, Numerical analysis, Prandtl number, Rayleigh number, Temperature profiles
1.
Birikh
R. V.
,
1966
, “
Thermocapillary Convection in a Horizontal Layer of Liquid
,”
J. Appl. Mech. Tech. Phys.
, Vol.
3
, pp:
69
–
72
.
2.
Chandrasekhar, S., 1961, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, United Kingdom.
3.
Cormack
D. E.
,
Leal
L. G.
, and
Imberger
J.
,
1974
a, “
Natural Convection in a Shallow Cavity With Differentially Heated End Walls. Part I, Asymptotic Theory
,”
J. Fluid Mech.
, Vol.
65
, pp.
209
–
229
.
4.
Cormark
D. E.
,
Leal
L. G.
, and
Seinfeld
J. H.
,
1974
b, “
Natural Convection in a Shallow Cavity With Differentially Heated End Walls, Part II: Numerical Solutions
,”
J. Fluid Mech.
, Vol.
65
, pp.
231
–
246
.
5.
De Vahl Davis
G.
, and
Jones
I. P.
,
1983
, “
Natural Convection in a Square Cavity: A Comparison Exercise
,”
Int. J. Numerical Methods in Fluids
, Vol.
3
, pp.
227
–
248
.
6.
Garandet
J. P.
,
Alboussiere
T.
, and
Moreau
R.
,
1992
, “
Buoyancy Driven Convection in a Rectangular Enclosure With a Transverse Magnetic Field
,”
Int. J. Heat Mass Transfer
, Vol.
35
, pp.
741
–
748
.
7.
Hart
J. E.
,
1972
, “
Stability of Thin Non-rotating Hadley Circulations
,”
J. Atmospheric Sci.
, Vol.
29
, pp.
687
–
697
.
8.
Hunt
R.
, and
Wilks
G.
,
1981
, “
Low Prandtl Number Magnetohydrodynamic Natural Convection in a Strong Cross Field
,”
Numerical Heat Transfer
, Vol.
4
, pp.
303
–
316
.
9.
Moreau, R., 1990, Magnetohydrodynamics. Kluwer Academic Publishers, The Netherlands.
10.
Ozoe
H.
, and
Maruo
E.
,
1987
, “
Magnetic and Gravitational Natural Convection of Melted Silicon—Two-Dimensional Numerical Computations for the Rate of Heat Transfer
,”
J.S.M.E
, Vol.
30
, pp.
774
–
784
.
11.
Ozoe
H.
, and
Okada
K.
,
1989
, “
The Effect of the Direction of the External Magnetic Field on the Three-Dimensional Natural Convection in a Cubical Enclosure
,”
Int. J. Heat Mass Transfer
, Vol.
32
, pp.
1939
–
1954
.
12.
Peaceman
D. W.
, and
Rachford
H. H.
,
1955
, “
The Numerical Solution of Parabolic and Elliptic Differential Equations
,”
J. Soc. Indust. Appl. Mathematics
, Vol.
3
, pp.
28
–
41
.
13.
Roache, P. J., 1985, Computational Fluid Mechanics, Hermosa, Alberquerque, NM.
14.
Roux
B.
,
Ben Hadid
H.
, and
Laure
P.
,
1989
, “
Hydrodynamical Regimes in Metallic Melts Subject to a Horizontal Temperature Gradient
,”
Eur. J. Mech.
, Vol.
5
, pp.
375
–
396
.
15.
Shiralkar
G. S.
, and
Tien
C. L.
,
1981
, “
A Numerical Study of Laminar Natural Convection in Shallow Cavities
,”
ASME JOURNAL OF HEAT TRANSFER
, Vol.
103
, pp.
226
–
231
.
16.
Tabeling
P.
,
1982
, “
Convection Flow Patterns in Rectangular Boxes of Finite Extent Under an External Magnetic Field
,”
J. de Physique
, Vol.
43
, pp.
1295
–
1303
.
17.
Toh, K., and Ozoe, H., 1992, “Three-Dimensional Czochralski Flow of Liquid Metal in a Lateral Magnetic Field,” Proc. of the First Intern. Conf. on Transport Phenomena in Processing, S. I. Guceri, ed., pp. 14–24.
18.
Utech
H. P.
, and
Flemmings
M. C.
,
1966
, “
Elimination of Solute Banding in Indium Atimonide Crystals by Growth in a Magnetic Field
,”
J. Appl. Phys.
, Vol.
37
, pp.
2021
–
2024
.
19.
Vasseur
P.
,
Hasnaoui
M.
,
Bilgen
E.
, and
Robillard
L.
,
1995
, “
Natural Convection in an Inclined Fluid Layer With a Transverse Magnetic Field: Analogy With a Porous Medium
,”
ASME JOURNAL OF HEAT TRANSFER
, Vol.
117
, pp.
121
–
129
.
20.
Venkatachalappa
M.
, and
Subbaraya
C. K.
,
1993
, “
Natural Convection in a Rectangular Enclosure in the Presence of a Magnetic Field With Uniform Heat Flux From the Side Walls
,”
Acta Mechanica
, Vol.
96
, pp.
13
–
26
.
21.
Vives
C.
, and
Perry
C.
,
1987
, “
Effects of Magnetically Damped Convection During the Controlled Solidification of Metals and Alloys
,”
Int. J. Heat Mass Transfer
, Vol.
30
, pp.
479
–
496
.
22.
Wilks
G.
,
1976
, “
Magnetohydrodynamic Free Convection About a Semi-infinite Vertical Plate in a Strong Cross Field
,”
J. Appl. Math. Phys.
, Vol.
27
, pp.
621
–
631
.
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