Convection under rotation for Prandtl numbers near 1: Küppers-Lortz instability

Abstract

The K{umlt u}ppers-Lortz (KL) instability in Rayleigh-B{acute e}nard convection rotated about a vertical axis was studied experimentally using optical-shadowgraph imaging in the rotating frame for dimensionless rotation rates 6{lt}{Omega}{lt}20. Two cylindrical convection cells with radius-to-height ratios {Gamma}=40 and 23 were used. The cells contained CO{sub 2} at 33.1 bar and 16.6 bar with Prandtl numbers {sigma}=0.93 and {sigma}=0.83, respectively. Numerical solutions of the Boussinesq equations with parameter values corresponding to the experiments were obtained for comparison. For {Gamma}=40 and 8{lt}{Omega}{lt}10.5, the initial pattern above onset was time dependent. Its dynamics revealed a mixture of sidewall-nucleated domain-wall motion characteristic of the KL instability and of dislocation-defect motion. For {Omega}{gt}10.5, spontaneous formation of KL domain walls away from the sidewall was observed. For 8{lt}{Omega}{lt}12, there were differences between the two cells very close to onset, but for {epsilon}{approx_gt}0.02 the systems were qualitatively similar. For {Omega}{approx_gt}12 there was no qualitative difference in the behavior of the two cells at any {epsilon}. The average size of a domain containing rolls of approximately the same orientation decreased with increasing {Omega}, and the time dependence speeded up and became dominated by domain-wall propagation. The numerical solutions were qualitatively similar, although there was a tendency for themore » domains to be larger at the same {epsilon} and {Omega}. The replacement of domains of one orientation by those with another led to a rotation in Fourier space which was characterized by a rotation frequency {omega}{sub a} in the frame rotating at angular velocity {Omega}. Quantitative experimental measurements of {omega}{sub a}, of a correlation length {xi}, and of a domain-switching angle {Theta}{sub s} as functions of {epsilon}{equivalent_to}{Delta}T/{Delta}T{sub c}{minus}1 and {Omega} are presented. For 13{approx_lt}{Omega}{approx_lt}18, {Theta}{sub s} was independent of {Omega} and close to 58{degree}. We computed the angle of maximum growth rate {Theta}{sub KL} of KL perturbations, and found it to be 43{degree}, distinctly different from {Theta}{sub s}. The results for {omega}{sub a}({epsilon},{Omega}) over the range 13{approx_lt}{Omega}{approx_lt}20 can be collapsed onto a single curve {tilde {omega}}{sub a}({epsilon}){equivalent_to}{omega}{sub a}({epsilon},{Omega})/{omega}{sub r}({Omega}) by applying an {Omega}-dependent factor 1/{omega}{sub r}. Similar collapse can be accomplished for {tilde {xi}}({epsilon})={xi}({epsilon},{Omega})/{xi}{sub r}({Omega}). An analysis of {tilde {omega}}{sub a}({epsilon}) and {tilde {xi}}({epsilon}) in terms of various functional forms is presented. It is difficult to reconcile the {epsilon} dependence of {tilde {omega}}{sub a} and {tilde {xi}} at small {epsilon} with the theoretically expected proportionality to {epsilon} and {epsilon}{sup {minus}1/2}, respectively. {copyright} {ital 1998} {ital The American Physical Society}« less