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Throughflow effects in the Rayleigh-Bénard convective instability problem

Published online by Cambridge University Press:  21 April 2006

D. A. Nield
D. A. Nield
Affiliation:
Department of Theoretical and Applied Mechanics, University of Auckland, Auckland, New Zealand
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Abstract

The effect of vertical throughflow on the onset of convection in a fluid layer, between permeable horizontal boundaries, when heated uniformly from below, is re-examined analytically. It is shown that when the Péclet number Q is large in magnitude, the critical Rayleigh number Rc is proportional to Qn, where n = 0, 1, 2, 3 or 4, with a coefficient depending on the Prandtl number P, according to the types of boundaries. When the upper and lower boundaries are of different types, the effect of a small amount of throughflow in one direction is to decrease Rc. This is so when the throughflow is away from the more restrictive boundary. Contributions arise from the curvature of the basic temperature profile, and from the vertical transport of perturbation velocity and perturbation temperature. The decrease in Rc is small if P ∼ 1 but can be of significant size if P [Lt ] 1 or P [Gt ] 1.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 185 , December 1987 , pp. 353 - 360
Copyright
© 1987 Cambridge University Press

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References

Gershuni, G. Z. & Zhukhovitskii, E. M. 1976 Convective Stability of Incompressible Fluids. Israel Program for Scientific Translations.
Homsy, G. M. & Sherwood, A. E. 1976 Convective instabilities in porous media with throughflow. AIChE J. 22, 168–174.Google Scholar
Jones, M. C. & Persichetti, J. M. 1986 Convective instability in packed beds with throughflow. AIChE J. 32, 1555–1557.Google Scholar
Krishnamurti, R. 1975 On cellular cloud patterns. Part 1: mathematical model. J. Atmos. Sci. 32, 1353–1363.Google Scholar
Nield, D. A. 1987 Convective instability in porous media with throughflow. AIChE J. 33, 1222–1224.Google Scholar
Platten, J. K. & Legros, J. C. 1984 Convection in Liquids. Springer.
Shvartsblat, D. L. 1968 The spectrum of perturbations and convective instability of a plane horizontal fluid layer with permeable boundaries. J. Appl. Mech. (PMM) 32, 266–271. (Translation from Prikl. Mat. Mekh.).Google Scholar
Shvartsblat, D. L. 1969 Steady convective motions in a plane horizontal fluid layer with permeable boundaries. Fluid Dyn. 4, 54–59. (Translation from Izv. Akad. Nauk SSSR, Fiz. Zhidkosti i Gaza.)Google Scholar
Shvartsblat, D. L. 1971 Chislennoe issledovanie statsionarnogo konvektivnogo dvizheniya v ploskom gorizontal'nom sloe zhidkosti. Uchen. Zap. Perm Univ. No 248, Gidrodinamika, No 3: 97.
Somerville, R. C. J. & Gal-Chen, T. 1979 Numerical simulation of convection with mean vertical motion. J. Atmos. Sci. 36, 805–815.Google Scholar
Sutton, F. M. 1970 Onset of convection in a porous channel with net through Phys. Fluids 13, 1931–1934.Google Scholar
Wooding, R. A. 1960 Rayleigh instability of a thermal boundary layer in flow through a porous medium. J. Fluid Mech. 9, 183–192.Google Scholar