Convection near threshold for Prandtl numbers near 1

Abstract

Rayleigh-B\'enard convection in two cylindrical geometries with different sidewalls and with radius-to-height ratios \ensuremath{\Gamma}=41 and 43 was studied with shadowgraph imaging and heat-transport measurements. The working fluid was ${\mathrm{CO}}_{2}$ at 25.3 bar with a Prandtl number \ensuremath{\sigma}=0.93. For one of the cells (\ensuremath{\Gamma}=43), axisymmetric convection rolls were stable above onset up to \ensuremath{\epsilon}==\ensuremath{\Delta}T/\ensuremath{\Delta}${\mathit{T}}_{\mathit{c}}$-1=0.19, where \ensuremath{\Delta}${\mathit{T}}_{\mathit{c}}$ is the critical temperature difference. The amplitude of the center of the concentric patterns, the umbilicus, much larger near onset than that of the outlying region, grew as ${\mathrm{\ensuremath{\epsilon}}}_{1}^{\mathrm{\ensuremath{\beta}}}$, with ${\mathrm{\ensuremath{\beta}}}_{1}$=0.27\ifmmode\pm\else\textpm\fi{}0.04. Outside the umbilicus, the amplitude grew as ${\mathrm{\ensuremath{\epsilon}}}_{2}^{\mathrm{\ensuremath{\beta}}}$, with ${\mathrm{\ensuremath{\beta}}}_{2}$=0.48\ifmmode\pm\else\textpm\fi{}0.04. For \ensuremath{\Gamma}=41, axisymmetric convection rolls, coexisting with an annular cross-roll state confined near the sidewall, were the preferred patterns above onset and remained stable up to an \ensuremath{\epsilon} of 0.08. The umbilicus of the concentric convection pattern shifted slightly off center as \ensuremath{\epsilon} was increased, but the shift never exceeded 50% of the cell depth. For \ensuremath{\epsilon}g0.10, the patterns were unstable and evolved over many horizontal thermal diffusion times to straight-roll patterns with defects and grain boundaries. The process involved the umbilicus moving toward the sidewall, accompanied by radially traveling waves emitted by the umbilicus. While the umbilicus emitted, the crossrolls adjacent to the sidewall traveled parallel to the wall, and rolls were created and destroyed at a source and sink, respectively, presumably owing to large-scale flows generated by the off-centered position of the umbilicus.