Convection under rotation for Prandtl numbers near 1: Linear stability, wave-number selection, and pattern dynamics

Abstract

Rayleigh-B\'enard convection with rotation about a vertical axis was studied with the shadowgraph imaging method up to a dimensionless rotation rate \ensuremath{\Omega} of 22. Most of the results are for a cylindrical convection cell with a radius-to-height ratio \ensuremath{\Gamma}=40 that contained ${\mathrm{CO}}_{2}$ at 33.1 bars with a Prandtl number \ensuremath{\sigma}=0.93. Measurements of the critical Rayleigh number ${\mathrm{R}}_{\mathrm{c}}$ and wave number ${\mathrm{k}}_{\mathrm{c}}$ for 022 agree well with predictions based on linear stability analysis. Above onset and with rotation, the average wave number and details of the pattern dynamics were studied. For \ensuremath{\Omega}\ensuremath{\lesssim}5, the initial onset was to a pattern of straight or slightly curved rolls. For 0.1\ensuremath{\lesssim}\ensuremath{\epsilon}\ensuremath{\equiv}\ensuremath{\Delta}T/\ensuremath{\Delta}${\mathrm{T}}_{\mathrm{c}}$-1\ensuremath{\lesssim}0.5 but below the onset of spiral-defect chaos, rotation with \ensuremath{\Omega}\ensuremath{\lesssim}8 produced weak perturbations of nonrotating patterns. Typically, this gave an ``S-shaped'' distortion of the zero-rotation pattern of straight or somewhat curved rolls. Rotation had a stronger effect on the source and motions of dislocation defects. For \ensuremath{\Omega}g0 the defects were generated primarily at the wall, whereas for \ensuremath{\Omega}=0 they were nucleated in the bulk via the skewed-varicose instability. Rotation picked a preferred direction of motion for the defects once they formed. For \ensuremath{\epsilon}\ensuremath{\gtrsim}0.5, recognizable spiral-defect chaos and the oscillatory instability were observed for \ensuremath{\Omega}\ensuremath{\lesssim}12. For \ensuremath{\Omega}\ensuremath{\geqslant}8, domain growth and front propagation suggestive of the K\uppers-Lortz instability were observed from onset up to an \ensuremath{\epsilon} value that increased with \ensuremath{\Omega}. Increasing \ensuremath{\epsilon} at fixed \ensuremath{\Omega}\ensuremath{\lesssim}12 enhanced dislocation-defect dynamics over K\uppers-Lortz front propagation. Quantitative measurements of average pattern wave numbers, correlation lengths, and spatially averaged roll curvature as functions of \ensuremath{\epsilon} and \ensuremath{\Omega} are presented. At a fixed \ensuremath{\Omega}\ensuremath{\gtrsim}10, the average wave number had two distinct wave-number-selection regions with different slopes as a function of \ensuremath{\epsilon}, one above \ensuremath{\epsilon}\ensuremath{\approx}0.45 and the other near onset. The slope for \ensuremath{\epsilon} near onset reached a minimum at \ensuremath{\Omega}=12.1 and increased linearly for 1220.