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Steady and oscillatory convection in rigid vertical cylinders heated from below studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. Both adiabatic and ideal conducting sidewalls are considered. The effect of the geometry of the container on the onset of convective instability and the structure and symmetry of the flow are analysed and compared with the results of linear stability theories. The nonlinear evolution and stability of convective flows at Rayleigh numbers beyond the critical number for the onset of convective motion are investigated for Prandtl numbers of 0.02 to 6.7. The limits of stable axisymmetric solutions are an important finding of this study. The onset and the frequency of oscillatory instability are calculated for the small Prandtl number 0.02 and compared with experimental data. Calculated stream patterns and velocity profiles illustrate the three-dimensional structure of steady convection and the time-dependent behaviour of oscillatory flows.

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