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Solution of the non-linear equations of cellular convection and heat transport

Published online by Cambridge University Press:  28 March 2006

H. L. Kuo
H. L. Kuo
Affiliation:
Massachusetts Institute of Technology
The research reported in this paper has been sponsored in part by the Geophysics Research Directorate of the Air Force Cambridge Research Laboratories, Air Research and Development Command, under Contract No. AF 19(604)-6108.
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Abstract

By expanding the dependent variables in series of orthogonal functions on the one hand, and expanding the coefficients of these functions in power series of a parameter η on the other hand, a solution has been obtained for the system of non-linear equations of cellular convection. The expansion parameter η is chosen in such a way as to make it remain less than 1 for all finite values of the Rayleigh number. The solution so obtained is found to be valid for a large range of the imposed temperature difference, and converges rapidly. This solution provides a quantitative theory for the convective heat transport as a function of the temperature difference in the range of laminar flow.

The solution also reveals that when the actual Rayleigh number is greater than twice the critical Rayleigh number, a layer of isothermal (adiabatic lapserate in a gas medium) mean temperature develops in the middle of the fluid layer. The thickness of this layer increases as the actual Rayleigh number increases, and at the same time the temperature gradient increases in the boundary layer so that an increase in the heat transport is accomplished.

The solution reveals further that the large temperture gradients are concentrated in the region where the cold descending current approaches the lower boundary and where the warm ascending current approaches the upper boundary. It is also shown that these ascending and descending currents spread out in mushroom-like patterns, a feature characteristic of the convection of isolated hot bubbles, but one which never has been considered as the form for finite cellular convection. Recent optical observations indicate that this is the most common form of the temperature field.

The heat transport given by this solution fits a power law of exponent 1.24, which is very close to the observed power law of exponent 1.25 for laminar flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 10 , Issue 4 , June 1961 , pp. 611 - 634
Copyright
© 1961 Cambridge University Press

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References

Globe, S. & Dropkin, D. 1959 Natural convection heat transfer in liquids confined by two horizontal plates and heated from below. J. Heat Transfer, 81, 24–8.Google Scholar
Jakob, M. 1949 Heat Transfer, vol. 1. New York: John Wiley.
Kuo, H. L. & Platzman, G. W. 1960 A normal mode nonlinear solution of the Rayleigh convection problem-to be published in Beit. Phys. frei Atmos.
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225–60.Google Scholar
Nakagawa, Y. 1960 Heat transport by convection. Phys. Fluids, 3, 82–6.Google Scholar
Pellew, Anne & Southwell, R. V. 1940 On maintained convective motion in a fluid heated from below. Proc. Roy. Soc. A, 176, 312–43.Google Scholar
Rayleigh, Lord 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529–46.Google Scholar
Silveston, P. L. 1958 Warmedurchgang in waagerechten Flussigkeitsschichten. Forsch. Ing. 24, 29–32 and 59–69.Google Scholar