Wind and boundary layers in Rayleigh-Bénard convection. II: Boundary layer character and scaling

The scaling of the kinematic boundary layer thickness ${\ensuremath{\lambda}}_{u}$ and the friction factor ${C}_{f}$ at the top and bottom walls of Rayleigh-B\'enard convection is studied by direct numerical simulation (DNS). By a detailed analysis of the friction factor, a new parameterisation for ${C}_{f}$ and ${\ensuremath{\lambda}}_{u}$ is proposed. The simulations were made of an $L/H=4$ aspect-ratio domain with periodic lateral boundary conditions at $\text{Ra}={{10}^{5},{10}^{6},{10}^{7},{10}^{8}}$ and $\text{Pr}=1$. The continuous spectrum, as well as significant forcing due to Reynolds stresses, clearly indicates a turbulent character of the boundary layer, while viscous effects cannot be neglected, judging from the scaling of classical integral boundary layer parameters with Reynolds number. Using a conceptual wind model, we find that the friction factor ${C}_{f}$ should scale proportionally to the thermal boundary layer thickness as ${C}_{f}\ensuremath{\propto}{\ensuremath{\lambda}}_{\ensuremath{\Theta}}/H$, while the kinetic boundary layer thickness ${\ensuremath{\lambda}}_{u}$ scales inversely proportionally to the thermal boundary layer thickness and wind Reynolds number ${\ensuremath{\lambda}}_{u}/H\ensuremath{\propto}{({\ensuremath{\lambda}}_{\ensuremath{\Theta}}/H)}^{\ensuremath{-}1}{\text{Re}}^{\ensuremath{-}1}$. The predicted trends for ${C}_{f}$ and ${\ensuremath{\lambda}}_{u}$ are in agreement with DNS results.