Turbulent Thermal Convection at Arbitrary Prandtl Number

TLDR

The analysis assumes a layer of Boussinesq fluid between infinite, horizontal, perfectly conducting, rigid plates. The mixing‑length theory of turbulent thermal convection in a gravitationally unstable fluid is extended to determine how the Nusselt number depends on both Prandtl number σ and Rayleigh number Ra. The extended theory predicts how the Nusselt number, mean temperature deviation, rms temperature fluctuation, and rms velocity vary with σ, Ra, and height, and it delineates the boundaries of the σ‑range regimes. The theory yields H/H₀ ∝ Ra¹ᐟ³ for high σ, ∝ (σ Ra)¹ᐟ³ for low σ, ∼1 for very low σ, reproduces Priestley’s intermediate‑σ behavior, shows T̄∝z⁻¹ and ψ̃∝z⁻¹ outside the conduction layer at high σ, reduces to Ledoux–Schwarzschild–Spiegel results at very low σ, and predicts a corrected asymptotic law H/H₀ ∝ [Ra/(ln Ra)³]¹ᐟ² at extremely large Ra.

Abstract

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)3]1/2.