A Numerical Study of the Development of Unsteady Three-Dimensional Natural Convective Flow in a Horizontal Enclosure With a Uniform Heat Flux on the Lower Surface

Abstract

Abstract Flow in a rectangular enclosure with a square horizontal cross-section and with a uniform heat flux applied across the lower horizontal surface and with the upper horizontal surface cooled to a uniform low temperature has been considered. The vertical side-walls of the enclosure are adiabatic. It has been assumed that the flow is laminar and that the fluid properties are constant except for the density change with temperature which gives rise to the buoyancy forces. In this situation, there is no flow in the enclosure at low Rayleigh numbers, steady flow in the enclosure at intermediate Rayleigh numbers and an unsteady flow in the enclosure at high Rayleigh numbers. The conditions under which unsteady flow develops have been numerically investigated in the present study. The unsteady form of the governing equations, written in terms of the vector potential and vorticity vector functions and expressed in dimensionless form, have been solved using a finite-difference procedure. The solution was started with no flow in the enclosure. The solution, in general, has the following parameters: the heat flux Rayleigh number Raq, the Prandtl number and the aspect ratio A of the enclosure, i.e. the ratio size of the height of the enclosure to the size of the square cross-sectional shape of the enclosure. Results have only been obtained for a Prandtl number of 0.7. Results for values of A between 1.25 and 3 for various values of Rayleigh number (up to 80000) have been obtained. The results have been used to determine the effect of A on the value of Rayleigh number above which there is unsteady fluid motion in the enclosure.