Turbulent Thermal Convection at Arbitrary Prandtl Number

The mixing-length theory of turbulent thermal convection in a gravitationally unstable fluid is extended to yield the dependence of Nusselt number H/H0 on both Prandtl number σ and Rayleigh number Ra. The analysis assumes a layer of Boussinesq fluid contained between infinite, horizontal, perfectly conducting, rigid plates. Also obtained is the dependence of mean temperature deviation T̄(z), rms temperature fluctuation ψ̃(z), and rms velocity upon height z above the bottom plate. The theory gives H/H0 ∝ Ra1/3 (high σ), H/H0 ∝ (σ Ra)1/3 (low σ), and H/H0 ∼ 1 (very low σ). The boundaries of the several σ ranges are determined. At one intermediate Prandtl number only, the behavior of T̄(z) and ψ̃(z) reduces to that previously found by Priestly. At high σ, there is a range of z, outside the molecular conduction region, where T̄(z) ∝ z−1, ψ̃(z) ∝ z−1. The results at very low σ reduce to those of Ledoux, Schwarzschild, and Spiegel. The dynamics are found to be importantly modified at extremely large Ra because of the stirring action of small-scale turbulence generated in shear boundary layers attached to the eddies of largest scale. The consequent corrected asymptotic law of heat transport at fixed σ is H/H0 ∝ Ra/(In Ra)31/2.