Aspect Ratio Dependence of Heat Transfer in a Cylindrical Rayleigh-Bénard Cell

Abstract

While the heat transfer and the flow dynamics in a cylindrical Rayleigh-B\'enard (RB) cell are rather independent of the aspect ratio $\mathrm{\ensuremath{\Gamma}}$ (diameter/height) for large $\mathrm{\ensuremath{\Gamma}}$, a small-$\mathrm{\ensuremath{\Gamma}}$ cell considerably stabilizes the flow and thus affects the heat transfer. Here, we first theoretically and numerically show that the critical Rayleigh number for the onset of convection at given $\mathrm{\ensuremath{\Gamma}}$ follows ${\mathrm{Ra}}_{c,\mathrm{\ensuremath{\Gamma}}}\ensuremath{\sim}{\mathrm{Ra}}_{c,\ensuremath{\infty}}(1+C{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{-}2}{)}^{2}$, with $C\ensuremath{\lesssim}1.49$ for Oberbeck-Boussinesq (OB) conditions. We then show that, in a broad aspect ratio range $(1/32)\ensuremath{\le}\mathrm{\ensuremath{\Gamma}}\ensuremath{\le}32$, the rescaling $\mathrm{Ra}\ensuremath{\rightarrow}{\mathrm{Ra}}_{\ensuremath{\ell}}\ensuremath{\equiv}\mathrm{Ra}[{\mathrm{\ensuremath{\Gamma}}}^{2}/(C+{\mathrm{\ensuremath{\Gamma}}}^{2}){]}^{3/2}$ collapses various OB numerical and almost-OB experimental heat transport data $\mathrm{Nu}(\mathrm{Ra},\mathrm{\ensuremath{\Gamma}})$. Our findings predict the $\mathrm{\ensuremath{\Gamma}}$ dependence of the onset of the ultimate regime ${\mathrm{Ra}}_{u,\mathrm{\ensuremath{\Gamma}}}\ensuremath{\sim}[{\mathrm{\ensuremath{\Gamma}}}^{2}/(C+{\mathrm{\ensuremath{\Gamma}}}^{2}){]}^{\ensuremath{-}3/2}$ in the OB case. This prediction is consistent with almost-OB experimental results (which only exist for $\mathrm{\ensuremath{\Gamma}}=1$, $1/2$, and $1/3$) for the transition in OB RB convection and explains why, in small-$\mathrm{\ensuremath{\Gamma}}$ cells, much larger Ra (namely, by a factor ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{-}3}$) must be achieved to observe the ultimate regime.

Topics & Concepts