Mean flow structure in thermal convection in a cylindrical cell of aspect ratio one half

Abstract

In this paper we propose a simple model that, by comparing different time scales, allows a prediction for the mean flow structure and its dynamics in confined thermal convection in a cylindrical cell of aspect ratio (diameter over cell height) $\Gamma\,{=}\,1/2$. It is shown that the break-up of the mean elongated recirculation into two counter-rotating unity-aspect-ratio rolls, sometimes referred to as flow bimodality, occurs only in a narrow range of Rayleigh numbers whose extrema depend on the Prandtl number. The predictions of the present model are consistent with the published literature, according to which the dual mean flow structure has been observed in numerical simulations at $\hbox{\it Pr}\,{=}\,0.7$ and experiments in gaseous helium ($\hbox{\it Pr}\,{\approx}\,0.7$) but never in water at ‘ambient’ temperature ($\hbox{\it Pr} \,{\approx}\,5$) and only once in water at $T\,{=}\,80\,^\circ$C ($\hbox{\it Pr}\,{=}\,2$). Another prediction of the model is that the thermal properties of the sidewall affect the mean flow unsteadiness and, sometimes, prevent transitions via a subtle anchoring mechanism that has been identified and verified by ad hoc numerical simulations.