Heat flux in turbulent Rayleigh-Benard convection: Predictions derived from a boundary layer theory

Using a closed set of boundary layer equations [E. S. C. Ching et al., Phys. Rev. Research 1, 033037 (2019)] for turbulent Rayleigh-Benard convection, we derive analytical results for the dependence of the heat flux, measured by the Nusselt number (Nu), on the Reynolds (Re) and Prandtl (Pr) numbers and two parameters that measure fluctuations in the regime where the horizontal pressure gradient is negligible. This regime is expected to be reached at sufficiently high Rayleigh numbers for a fluid of any given Prandtl number. In the high-Pr limit, Nu = F-1(k(1))(RePr1/3)-Pr-1/2 and, in the low-Pr limit, Nu tends to pi(-1/2RePr1/2)-Pr-1/2, where F-1(k(1)) has a weak dependence on the parameter k(1) in the eddy viscosity that measures velocity fluctuations. These theoretical results further reveal a close resemblance of the scaling dependencies of heat flux in steady forced convection and turbulent RayleighBenard convection and this finding solves a puzzle in our present understanding of heat transfer in turbulent Rayleigh-Benard convection.