Heat transport in high-Rayleigh-number convection

The heat flux (Nusselt number) as a function of Rayleigh number is deduced from the presence of a mean flow and the nesting of the thermal boundary layer within the viscous one.

Abstract

The heat flux (Nusselt number) as a function of Rayleigh number, ${\mathit{N}}_{\mathrm{Nu}}$\ensuremath{\approxeq}0.3${\mathit{N}}_{\mathrm{Ra}}^{2/7}$, is deduced from the presence of a mean flow and the nesting of the thermal boundary layer within the viscous one. The numerical coefficients are obtained from those known empirically for turbulent boundary layers. The consistency of our assumptions as a function of Prandtl number limits this regime to (${10}^{7}$--${10}^{8}$)${\mathit{N}}_{\mathrm{Pr}}^{5/3}$\ensuremath{\lesssim}${\mathit{N}}_{\mathrm{Ra}}$\ensuremath{\lesssim} (${10}^{13}$--${10}^{15}$)${\mathit{N}}_{\mathrm{Pr}}^{4}$. The Bolgiano-Obukhov ${\mathit{k}}^{\mathrm{\ensuremath{-}}7/5}$ spectrum for the temperature fluctuations is inconsistent with a simple scaling treatment of the equations.