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

1998; American Physical Society; Volume: 58; Issue: 5 Linguagem: Inglês

10.1103/physreve.58.5821

1538-4519

Yuchou Hu, W. Pesch, Guenter Ahlers, Robert E. Ecke,

Theoretical and Computational Physics

The K\"uppers-Lortz (KL) instability in Rayleigh-B\'enard convection rotated about a vertical axis was studied experimentally using optical-shadowgraph imaging in the rotating frame for dimensionless rotation rates $6<\ensuremath{\Omega}<20.$ Two cylindrical convection cells with radius-to-height ratios $\ensuremath{\Gamma}=40$ and 23 were used. The cells contained ${\mathrm{CO}}_{2}$ at 33.1 bar and 16.6 bar with Prandtl numbers $\ensuremath{\sigma}=0.93$ and $\ensuremath{\sigma}=0.83,$ respectively. Numerical solutions of the Boussinesq equations with parameter values corresponding to the experiments were obtained for comparison. For $\ensuremath{\Gamma}=40$ and $8<\ensuremath{\Omega}<10.5,$ 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 $\ensuremath{\Omega}>10.5,$ spontaneous formation of KL domain walls away from the sidewall was observed. For $8<\ensuremath{\Omega}<12,$ there were differences between the two cells very close to onset, but for $\ensuremath{\epsilon}\ensuremath{\gtrsim}0.02$ the systems were qualitatively similar. For $\ensuremath{\Omega}\ensuremath{\gtrsim}12$ there was no qualitative difference in the behavior of the two cells at any \ensuremath{\epsilon}. The average size of a domain containing rolls of approximately the same orientation decreased with increasing \ensuremath{\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 the domains to be larger at the same \ensuremath{\epsilon} and \ensuremath{\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 ${\ensuremath{\omega}}_{a}$ in the frame rotating at angular velocity \ensuremath{\Omega}. Quantitative experimental measurements of ${\ensuremath{\omega}}_{a},$ of a correlation length \ensuremath{\xi}, and of a domain-switching angle ${\ensuremath{\Theta}}_{s}$ as functions of $\ensuremath{\epsilon}\ensuremath{\equiv}\ensuremath{\Delta}T/\ensuremath{\Delta}{T}_{c}\ensuremath{-}1$ and \ensuremath{\Omega} are presented. For $13\ensuremath{\lesssim}\ensuremath{\Omega}\ensuremath{\lesssim}18,$ ${\ensuremath{\Theta}}_{s}$ was independent of \ensuremath{\Omega} and close to $58\ifmmode^\circ\else\textdegree\fi{}.$ We computed the angle of maximum growth rate ${\ensuremath{\Theta}}_{\mathrm{KL}}$ of KL perturbations, and found it to be $43\ifmmode^\circ\else\textdegree\fi{},$ distinctly different from ${\ensuremath{\Theta}}_{s}.$ The results for ${\ensuremath{\omega}}_{a}(\ensuremath{\epsilon},\ensuremath{\Omega})$ over the range $13\ensuremath{\lesssim}\ensuremath{\Omega}\ensuremath{\lesssim}20$ can be collapsed onto a single curve ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{a}(\ensuremath{\epsilon})\ensuremath{\equiv}{\ensuremath{\omega}}_{a}(\ensuremath{\epsilon},\ensuremath{\Omega})/{\ensuremath{\omega}}_{r}(\ensuremath{\Omega})$ by applying an \ensuremath{\Omega}-dependent factor $1/{\ensuremath{\omega}}_{r}.$ Similar collapse can be accomplished for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\xi}}(\ensuremath{\epsilon})=\ensuremath{\xi}(\ensuremath{\epsilon},\ensuremath{\Omega})/{\ensuremath{\xi}}_{r}(\ensuremath{\Omega}).$ An analysis of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{a}(\ensuremath{\epsilon})$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\xi}}(\ensuremath{\epsilon})$ in terms of various functional forms is presented. It is difficult to reconcile the \ensuremath{\epsilon} dependence of ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\omega}}}_{a}$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\xi}}$ at small \ensuremath{\epsilon} with the theoretically expected proportionality to \ensuremath{\epsilon} and ${\ensuremath{\epsilon}}^{\ensuremath{-}1/2},$ respectively.