Steady three-dimensional convection at high Prandtl numbers

Abstract

Three-dimensional solutions are computed describing convection in a layer of a Boussinesq fluid of infinite Prandtl number. Rigid boundaries of constant temperature are assumed. As many as four physically different solutions are found for a given rectangular horizontal periodicity interval. These are two solutions describing bimodal convection, and two ‘square-pattern’ solutions which correspond to two orthogonally superimposed convection rolls of nearly equal amplitude. The Galerkin method used in obtaining the steady solutions can also be employed for the investigation of their stability. The stability of the bimodal solutions agrees with the experimental determination of the stability region by Whithead & Chan (1976). The square-pattern solution is unstable in the investigated parameter range, even though it exhibits the highest Nusselt number.