Flow in a rectangular enclosure with a square vertical cross-section normal to the longitudinal coordinate direction and having a strip on the lower horizontal surface which is heated to a uniform high temperature has been numerically studied. Two wall thermal boundary conditions have been considered. In one, the longitudinal vertical side walls are cooled to a uniform low temperature and the horizontal top surface is adiabatic while in the other the longitudinal vertical side walls and the horizontal top surface are cooled to a uniform low temperature. In both cases, the square vertical end 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. The unsteady, three-dimensional governing equations, 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 Rayleigh Number, *Ra*, the Prandtl number, *Pr*, the dimensionless longitudinal length of the enclosure relative to the size of the square cross-section, *Ay *, the dimensionless width of the heated strip on the lower surface relative to the size of the square cross-section, *wH *, and the thermal boundary condition on the upper surface. Results have only been obtained for a Prandtl number of 0.7 and only results for *wH * = 1/3 will be presented. Results have been obtained for values of *Ay * between 0.5 and 2 for Rayleigh numbers up to 5×105 . In all cases, three-dimensional unsteady flow has been found to exist at the higher Rayleigh numbers. The conditions under which this unsteady flow develops and the effect of *Ay * on the variation of the mean Nusselt number with Rayleigh number and the effect of the wall surface boundary condition on these results has been investigated.