Onset of Rayleigh-Bénard convection for intermediate aspect ratio cylindrical containers

Abstract

The convection patterns that occur at and slightly above the onset of convection in cylindrical containers were determined as a function of aspect ratio, using simulations of Rayleigh-Bénard convection and linear stability analysis. The study focused primarily on aspect ratios 6≤Γ≤20, where Γ = diameter/depth, with conducting or insulating, and no-slip boundary conditions and Prandtl numbers Pr = 0.7 and 28.9. Simulations demonstrate azimuthally pure Fourier mode patterns at onset consistent with what is expected from bifurcation theory, with an m = 1 mode, for even values of Γ, and a concentric roll pattern, or m = 0 mode, for odd values of Γ. For Rayleigh numbers slightly higher than onset other pure or mixed mode patterns were found and then for even higher Rayleigh numbers, straight parallel rolls were found. A linear stability analysis was used to determine the critical Rayleigh number, Rac, and flow pattern for a large range of aspect ratios and was found to agree with the simulation results.