Theoretical results on wall-normal heat flux and wall shear stress in turbulent vertical convection between two differentially heated plates

Turbulent vertical convection in a fluid between two differentially heated vertical plates is ubiquitous in nature and in engineering applications. It is also a common model system for studying buoyancy-driven turbulent flows. We report our theoretical study of turbulent vertical convection between two infinite vertical plates with constant temperature difference. The mean vertical velocity and temperature are governed by the mean momentum balance equation and the mean thermal energy balance equation in terms of the wall-normal turbulent heat flux and the Reynolds shear stress. By solving the mean momentum balance equation, we show that mean wall shear stress at the plate is equal to the buoyancy force per unit area in the velocity boundary layer and, as a result, obtain a relation between the shear Reynolds number Re_τ and the Nusselt number Nu in terms of the Rayleigh (Ra) and Prandtl numbers (Pr). By analysing the mean thermal energy balance equation, we estimate a relation between Nu, Pr and Re_τ. These two results lead to scaling laws for Nu and Re_τ in terms of Ra and Pr.