Turbulent Prandtl Number & Convection

Rayleigh-Bénard convection: Ra = g·α·ΔT·H³/(ν·κ_T), Pr = ν/κ_T. The Nusselt number Nu = Q·H/(k·ΔT) measures heat transport enhancement. Ultimate regime theory (Kraichnan 1962; Grossmann-Lohse 2000): Nu ~ Ra^(1/2) asymptotically, but in practice Nu ~ Ra^(0.3) for Ra < 10¹². The turbulent Prandtl number Pr_t = ν_t/κ_t ≈ 0.7–0.9 in the bulk, rising near walls. At Pr>>1 (viscous oils): thin thermal boundary layers inside thick viscous ones. At Pr<<1 (liquid metals): opposite regime.