A Numerical Model of Convection Driven by a Surface Stress and Non-Uniform Horizontal Heating

Abstract

This paper examines the two-dimensional convective motion of a nonrotating incompressible Boussinesq fluid heated non-uniformly from below. The fluid container is rectangular; the side and top boundaries are insulating and rigid. A linear temperature field is maintained along the bottom boundary. Using the DuFort-Frankel scheme for diffusion and the Arakawa scheme for advection, the governing vorticity and temperature equations are integrated numerically for two cases, the first having a stress-free bottom boundary and the second having a constant stress along the bottom boundary. In the first case, a single convective cell develops; an intense buoyant jet of fluid rises from the warmer section of the bottom while there is a more uniform sinking motion over the cooler section of the bottom. The cell asymmetry, the circulation, and the convective heat transfer increase markedly with increasing Rayleigh number (based here on fluid properties, cell height, and the horizontal temperature difference along the bottom). The flow is insensitive to changes in the Prandtl number σ, provided σ>1. A review of previous literature on this convective problem is given. In the second case, a uniform stress is applied along the heated bottom in opposition to the thermal driving. The magnitude of the stress is varied while the horizontal Rayleigh and Prandtl numbers are held constant. For weak stresses, the flow is that of the first case. A two-cell circulation pattern containing both thermal and stress-driven sections emerges for a moderate stress, while for large stresses the predominantly thermally driven cell disappears.