Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution

Abstract

Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-B\'enard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as $0.588Pr^{-1/2}$ for $Pr\ll Pr^*$ and as $0.982 Pr^{-1/3}$ for $Pr^*\ll\Pr$, with $Pr^*= 0.046$. Comparison of the Prandtl--Blasius velocity boundary layer thickness with that evaluated in the direct numerical simulations by Stevens, Verzicco, and Lohse (J. Fluid Mech. 643, 495 (2010)) gives very good agreement. Based on the Prandtl--Blasius type considerations, we derive a lower-bound estimate for the minimum number of the computational mesh nodes, required to conduct accurate numerical simulations of moderately high (boundary layer dominated) turbulent Rayleigh-B\'enard convection, in the thermal and kinetic boundary layers close to bottom and top plates. It is shown that the number of required nodes within each boundary layer depends on Nu and Pr and grows with the Rayleigh number Ra not slower than \sim\Ra^{0.15}. This estimate agrees excellently with empirical results, which were based on the convergence of the Nusselt number in numerical simulations