Effects of finiteness on the thermo-fluid-dynamics of natural convection above horizontal plates

A rigorous and systematic computational and theoretical study, the first of its kind, for the laminar natural convective flow above rectangular horizontal surfaces of various aspect ratios phi (from 1 to infinity) is presented. Two-dimensional computational fluid dynamic (CFD) simulations (for phi -> infinity) and three-dimensional CFD simulations (for 1 <= phi < infinity) are performed to establish and elucidate the role of finiteness of the horizontal planform on the thermo-fluid-dynamics of natural convection. Great care is taken here to ensure grid independence and domain independence of the presented solutions. The results of the CFD simulations are compared with experimental data and similarity theory to understand how the existing simplified results fit, in the appropriate limiting cases, with the complex three-dimensional solutions revealed here. The present computational study establishes the region of a high-aspect-ratio planform over which the results of the similarity theory are approximately valid, the extent of this region depending on the Grashof number. There is, however, a region near the edge of the plate and another region near the centre of the plate (where a plume forms) in which the similarity theory results do not apply. The sizes of these non-compliance zones decrease as the Grashof number is increased. The present study also shows that the similarity velocity profile is not strictly obtained at any location over the plate because of the entrainment effect of the central plume. The 3-D CFD simulations of the present paper are coordinated to clearly reveal the separate and combined effects of three important aspects of finiteness: the presence of leading edges, the presence of planform centre, and the presence of physical corners in the planform. It is realised that the finiteness due to the presence of physical corners in the planform arises only for a finite value of f in the case of 3-D CFD simulations (and not in 2-D CFD simulations or similarity theory). The presence of physical corners is related here to several significant aspects of the solution-the conversion of in-plane velocity to out-of-plane velocity near the diagonals, the star-like non-uniform distribution of surface heat flux on heated planforms, the three-dimensionality of the temperature field, and the complex spatial structure of the velocity iso-surfaces. A generic theoretical correlation for the Nusselt number is deduced for the averaged surface heat flux for various rectangular surfaces (1 <= phi < infinity) over a wide range of Grashof number. Innovative use of numerical visualization images is made to generate a comprehensive, quantitative understanding of the physical processes involved. Published by AIP Publishing.