Comments on the numerical investigation of Rayleigh and Marangoni convection in a vertical circular cylinder

Abstract

Convection in a cylindrical container was simulated with a three-dimensional, time-dependent code. For the case of purely Rayleigh convection, a completely rigid cylinder with adiabatic vertical walls and conducting horizontal walls was considered. The calculations showed that the individual velocity components could be in a transient state, while the total heat transfer was steady. This occurred in cases where the maximum azimuthal component of velocity was very small in magnitude, in comparison to the other two components, and this component decreased in time. On the other hand, the total kinetic energy along with the heat transfer reached a steady value. Consequently the present results have been shown to be at variance with the calculations of Neumann [J. Fluid Mech. 214, 559 (1990)]. Marangoni convection was modeled with a free flat surface on the upper side, assuming the superimposed second layer to be passive. The numerically obtained critical Marangoni numbers and flow patterns were compared favorably to earlier results from linearized stability. In addition, flow structural changes for supercritical Marangoni numbers were illustrated. Interestingly, axisymmetric disturbances led to nonsymmetric bifurcation diagrams, but three-dimensional disturbance calculations led to symmetry in the bifurcation plots. Another very interesting result was the observed transition from three-dimensional to two-dimensional patterns as the Marangoni number was increased. Large computational requirements precluded a detailed parametric study. The special case of Prandtl number equal to 6.7 (corresponding to water), and in the case of Marangoni convection, a surface Biot number of unity was assumed.