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A model of confined thermal convection driven by non-uniform heating from below

Published online by Cambridge University Press:  19 April 2006

Peter D. Killworth  and
Peter C. Manins
Peter D. Killworth
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, England
Peter C. Manins
Affiliation:
Division of Atmospheric Physics, C.S.I.R.O., P.O. Box 77, Mordialloc, Victoria 3195, Australia
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Abstract

A model is presented of the two-dimensional boundary-layer and interior flow in a rectangular box resulting from the application of a quadratic temperature variation on its lower surface. The other walls are insulating. It is shown that a similarity form exists for the narrow, thermocline-like layer near the lower surface, and that this can satisfy all known consistency conditions with the interior, together with either laminar or turbulent side-wall regions. The interior temperature and Nusselt number are shown to be insensitive to Prandtl number, and to be primarily functions of the horizontal Rayleigh number RaL.

Specifically, the interior temperature, relative to the coldest applied value, is 60% of the total applied temperature range. The Nusselt number is predicted to vary as $0.26 Ra^{\frac{1}{5}}_L$ for a box with unit aspect ratio. The dynamics of the side-wall region, and the details of the imposed temperature variation appear to be unimportant in determining the overall buoyancy exchange. The solutions are compared with numerical and observational results, and generally good agreement is found.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 3 , 12 June 1980 , pp. 587 - 607
Copyright
© 1980 Cambridge University Press

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