Energy-conserving model of hexagonal pattern in Rayleigh-Bénard convection

Abstract

We have constructed an energy-conserving sixteen mode dynamical system to model hexagonal pattern in Rayleigh-Bénard convection of Boussinesq fluids with symmetric stress-free thermally conducting boundaries. The model shows stable roll pattern at the onset of convection. Hexagon is found to appear in the system via sausage and (or) stationary rhombus patterns. Both up and down hexagons arise periodically or chaotically with roll, sausage and rhombus patterns. Hexagonal patterns exist for all values of the Prandtl number, 1 ≤ Pr ≤ 5 explored here. However the pattern is more prominent for small Pr and k < kc , where k denotes the wave number. The plot of Nusselt number matches with previous theoretical result. In dissipationless limit, the total energy and the unavailable energy are constants though the kinetic energy, the potential energy and the available energy vary with time. The derived model does not diverge for large values of Rayleigh number Ra.