Mean velocity and temperature profiles in turbulent Rayleigh–Bénard convection at low Prandtl numbers

We report a direct numerical simulation (DNS) study of the mean velocity and temperature profiles in turbulent Rayleigh–Bénard convection (RBC) at low Prandtl numbers ($Pr$). The numerical study is conducted in a vertical thin disk with $Pr$ varied in the range 0.17≤ Pr≤ 4.4 and the Rayleigh number ($Ra$) varied in the range 5× 10⁸≤ Ra ≤ 1× 10¹⁰. By varying $Pr$ from 4.4 to 0.17, we find a sharp change of flow patterns for the large-scale circulation (LSC) from a rigid-body rotation to a near-wall turbulent jet. We numerically examine the mean velocity equation in the bulk region and find that the mean horizontal velocity profile $u(z)$ can be determined by a balance equation between the mean convection and turbulent diffusion with a constant turbulent viscosity ν ₜ. This balance equation admits a self-similarity jet solution, which fits the DNS data well. In the boundary-layer region, we find that both the mean temperature profile $T(z)$ and $u(z)$ can be determined by a balance equation between the molecular diffusion and turbulent diffusion. Within the viscous boundary layer, both $u(z)$ and $T(z)$ can be solved analytically and the analytical results agree well with the DNS data. Our careful characterisation of the mean velocity and temperature profiles in low-$Pr$ RBC provides a further understanding of the intricate interplay between the LSC, plume emission and boundary-layer dynamics, and pinpoints the physical mechanism for the emergence of a pronounced LSC in low-$Pr$ RBC.