A scale analysis for natural convective flows over vertical surfaces

Abstract

A scale analysis is presented for natural convection from the face of a vertical plate. Three types of thermal boundary conditions are considered: (1) constant surface temperature; (2) constant surface heat flux, and (3) plate heated from its back surface. Unlike previous studies, the scale for the thickness of the hydrodynamic boundary layer is here obtained by strictly restricting the continuity and momentum equations to the region between the thermal and hydrodynamic boundary layers. This results in expressions for the local Nusselt number that are valid for all values of the Prandtl number and that approach the solutions found in the literature for the limiting cases of very high and very low Prandtl numbers. Furthermore, for all cases the order of magnitude of the local Nusselt number was found to be a function of the parameter grouping Bo/(1 + Pr), where Bo and Pr are the Boussinesq and Prandtl numbers, respectively, raised to a power that depends on the boundary condition. For condition (3), the local Nusselt number is an implicit function of the Biot number characterizing the convective heating on the backside of the plate. The order of magnitude of the local Nusselt number was therefore evaluated numerically for three values each of the Boussinesq, Prandtl, and Biot numbers.