Patterns in cylindrical Rayleigh-Benard convection

We simulate the Boussinesq equations for Rayleigh-B\'{e}nard convection in a cylindrical container of aspect ratio {\it radius}$/${\it height}$=2$ and either perfectly insulating or perfectly conducting sidewalls. We investigate the phenomenon of coexisting stable states for a fluid with Prandtl number $6.7$. Varying initial conditions, we obtain various convective patterns for the same Rayleigh number. The results for perfectly insulating sidewalls are in good agreement with experiment of Hof {\it et~al}. We follow the stationary solutions using a steady-state solver and obtain a bifurcation diagram covering the range of Rayleigh numbers from convection onset up to $Ra=30\,000$.