Onset of Rayleigh-Bénard convection in cylindrical containers

Abstract

We determined the critical Rayleigh numbers Ra{c} for the onset of convection in cylindrical containers with aspect ratios 1 approximately <Gamma [ triple bond] D/L approximately <9 ( D is the diameter and L the height) and the patterns that form just above Ra{c}, both from experiment and by direct numerical simulation (DNS). Results for Ra{c} agree well with the linear stability analysis by Buell and Catton for containers with finite sidewall conductivity. For Gamma<or=1.58+/-0.10 , we found that the patterns correspond to an azimuthal Fourier mode with mode number m=1 , corresponding to a single convection roll. For 1.58 approximately < Gamma approximately <3.26+/-0.02 , the pattern was a concentric roll, corresponding to m=0 . For 3.26<or=Gamma approximately >4, an m=1 mode was found again, but near Gamma=4 either m=1 or m=2 was observed in different runs. These results are consistent with the marginal stability curves calculated by Buell and Catton in the sense that the mode that is the first as a function of Ra to acquire a positive growth rate is the one that is observed. For Gamma approximately >4, the theoretical marginal curves for the four lowest modes lie very close together. There we found patterns near onset that corresponded to various modes, including m=2 and 4. At relatively large Gamma approximately > 6, we observed parallel straight rolls quite close to onset. Our patterns agree with several DNS investigations by others, but at some Gamma values differ from those observed experimentally by Stork and Müller. Some results for the pattern evolution with increasing Ra are reported as well.