Pattern Formation and Wave-Number Selection by Rayleigh–Bénard Convection in a Cylindrical Container

We describe an apparatus and procedures for simultaneous heat transport measurements and computer enhanced shadowgraph flow-pattern imaging in a shallow horizontal layer of fluid heated from below. The heat transport measurements have a resolution of better than 0.1%, and the shadowgraph technique can detect the flow field for ≡ ( R – R c )/ R c as small as 10 -2 ( R c is the critical value of the Rayleigh number R for onset of convection). The apparatus and procedures were used to study pattern and wave-number evolution in a cylindrical layer of water with radius-to-height ratio L = 7.5 and Prandtl number σ = 6.1. We found that dynamic sidewall forcing during the early thermal transients after a change in the heat current from a subcritical to a supercritical value establishes a cylindrical flow pattern. Once created, this pattern is stable in our apparatus over the wide range 0.16 ≲ ≲ 8 even after the transients have decayed. With changing , adjustment in the wave number k takes place discontinuously by hysteretic changes at the cell center in the number of convection roll pairs. When is increased, the discontinuous changes at the cell center are towards smaller k and are preceded by a continuous loss of cylindrical symmetry (the middle roll pair moves off center). The selected wave numbers coincide neither with the zig-zag instability of the infinite system, as once suggested, nor with a linear extrapolation to = 0 (1) of the recent prediction to lowest order in of Manneville and Piquemal and of Cross. Comparison of the selected k with measurements by others reveals no dependence upon L and σ. For < 0.16, the cylindrical pattern is unstable and decays on a time scale much longer than a horizontal diffusion time to patterns of rolls which tend to be perpendicular to the sidewalls and which contain defects. Once formed, these latter patterns will persist at large values of . These patterns also undergo a wave-number adjustment process with hysteretic changes mediated mostly by focus singularities near the walls. In these cases, larger values of also tend to produce smaller values of k .