A Numerical Investigation of Thermal Convection in a Heat-Generating Fluid Layer

Finite difference solutions of the equations governing thermal convection driven by uniform volumetric energy sources are presented for two-dimensional flows in a rectangular domain. The boundary conditions are a rigid, (i.e., zero slip), zero heat-flux lower surface, rigid adiabatic sides, and either a rigid or free (i.e., zero shear) isothermal upper surface. Computations are carried out for Prandtl numbers from 0.05 to 20 and Rayleigh numbers from 5 × 10⁴ to 5 × 10⁸. Nusselt numbers and average temperature profiles within the layer are in good agreement with experimental data for rigid-rigid boundaries. For rigid-free boundaries, Nusselt numbers are larger than in the former case. The structure of the flow and temperature fields in both cases is dominated by rolls, except at larger Rayleigh numbers where large-scale eddy transport occurs. Generally, low velocity upflows over broad regions of the layer are balanced by higher velocity downflows when the flow exhibits a cellular structure. The hydrodynamic constraint at the upper surface and the Prandtl number are found to influence only the detailed nature of flow and temperature fields. No truly steady velocity and temperature fields are found despite the fact that average Nusselt numbers reach steady values.