 
Summary: Onset of RayleighBénard convection in cylindrical containers
François Hébert,1
Ryan Hufschmid,1
Janet Scheel,2
and Guenter Ahlers1
1
Department of Physics, University of California, Santa Barbara, California 93106, USA
2
Department of Physics, Occidental College, 1600 Campus Road, M21, Los Angeles, California 90041, USA
Received 9 January 2010; published 28 April 2010
We determined the critical Rayleigh numbers Rac for the onset of convection in cylindrical containers with
aspect ratios 1 D/L 9 D is the diameter and L the height and the patterns that form just above Rac,
both from experiment and by direct numerical simulation DNS . Results for Rac agree well with the linear
stability analysis by Buell and Catton for containers with finite sidewall conductivity. For 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 3.26 0.02, the pattern was a concentric roll, corresponding to m=0. For
3.26 4, an m=1 mode was found again, but near =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
4, the theoretical marginal curves for the four lowest modes lie very close together. There we found
