Integral properties of turbulent natural convection over a vertical flat plate

Abstract

This paper investigates global integrals of the volumetric, momentum and temperature fluxes for turbulent natural convection over a vertical plate (TNCVP). The global integral of the mean continuity equation gives a relation for the mean wall-normal velocity outside the boundary layer as W∞ = − d(∫0∞Udz)/dx where U is the mean velocity in the streamwise x direction and z is the wall-normal direction. The global integral of the mean momentum equation gives a relation for the specific wall momentum flux as uτ2 = gβ∫0∞(Θ∞ − Θ)dz − d(∫0∞U2dz)/dx where uτ is the friction velocity, g is the gravitational acceleration and β is the thermal expansion coefficient. Θ = Tw − T is the difference between the wall temperature Tw and the fluid temperature, and Θ∞ = Tw − T∞ is the difference between the wall temperature and the ambient fluid temperature. Finally, the global integral of the mean heat equation gives a relation for the wall temperature flux as uτθτ = − W∞Θ∞ − d(∫0∞UΘdz)/dx where θτ is the frictional temperature. Based on the global integrals of the momentum and heat fluxes, we propose an indirect determination of uτ and θτ. Given the extreme challenge in direct measurement of uτ and θτ in wall-bounded turbulent flows, the indirect determination proposed here offers a powerful tool to subsequent experimental studies of TNCVP.