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Heat source-driven thermal convection at arbitrary Prandtl number

Published online by Cambridge University Press:  19 April 2006

F. B. Cheung
F. B. Cheung
Affiliation:
Reactor Analysis and Safety Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439
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Abstract

A theoretical investigation is made of turbulent thermal convection in a horizontally infinite layer of fluid confined between a rigid isothermal upper plate and a rigid adiabatic lower plate, driven by a temperature difference between the plates that is totally induced by volumetric heating of the layer. The dependence of upper surface Nusselt number, Nu, on both Prandtl number, Pr, and internal Rayleigh number, RaI, is obtained from considerations of the Boussinesq equations of motion. Also obtained is the dependence of various turbulence quantities upon distance from the upper plate. At a sufficiently high Rayleigh number, the present theory gives $Nu \sim Ra^{\frac{1}{4}}_I$ for large Pr and Nu ∼ Pr¼Ra¼I for small Pr. At lower Rayleigh numbers, however, the Nusselt number is found to vary according to $Nu \sim (a-b\,Ra^{-\frac{1}{12}}_I)^{-1} Ra^{\frac{1}{4}}_I$, where a and b are coefficients dependent upon Pr. The asymptotic $Ra^{\frac{1}{4}}_I$ law tends to support the boundary layer instability model of Howard (1966), although significant deviation from the model is predicted by the present theory over the range of Rayleigh numbers explored experimentally (Kulacki & Nagle 1975; Kulacki & Emara 1977). Based upon the results of this study the empirical power-law representation of Nu is critically examined and found to be adequate within finite ranges of RaI. Comparison of the present flow situation is made with the corresponding case of turbulent Bénard convection.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 97 , Issue 4 , 29 April 1980 , pp. 743 - 758
Copyright
© 1980 Cambridge University Press

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