Heat source-driven thermal convection at arbitrary Prandtl number

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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 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.