Heat transport by turbulent rotating Rayleigh–Bénard convection and its dependence on the aspect ratio

Abstract

AbstractWe report on the influence of rotation about a vertical axis on heat transport by turbulent Rayleigh–Bénard convection in a cylindrical vessel with an aspect ratio$\Gamma \equiv D/ L= 0. 50$($D$is the diameter and$L$the height of the sample) and compare the results with those for larger$\Gamma $. The working fluid was water at${T}_{m} = 4{0\hspace{0.167em} }^{\ensuremath{\circ} } \mathrm{C} $where the Prandtl number$\mathit{Pr}$ is 4.38. For rotation rates$\Omega \lesssim 1~\mathrm{rad} ~{\mathrm{s} }^{\ensuremath{-} 1} $, corresponding to inverse Rossby numbers$1/ \mathit{Ro}$between zero and twenty, we measured the Nusselt number$\mathit{Nu}$for six Rayleigh numbers$\mathit{Ra}$in the range$2. 2\ensuremath{\times} 1{0}^{9} \lesssim \mathit{Ra}\lesssim 7. 2\ensuremath{\times} 1{0}^{10} $. For small rotation rates and at constant$\mathit{Ra}$, the reduced Nusselt number${\mathit{Nu}}_{red} \equiv \mathit{Nu}(1/ \mathit{Ro})/ \mathit{Nu}(0)$initially increased slightly with increasing$1/ \mathit{Ro}$, but at$1/ \mathit{Ro}= 1/ {\mathit{Ro}}_{0} \simeq 0. 5$it suddenly became constant or decreased slightly depending on$\mathit{Ra}$. At$1/ {\mathit{Ro}}_{c} \approx 0. 85$a second sharp transition occurred in${\mathit{Nu}}_{red} $to a state where${\mathit{Nu}}_{red} $increased with increasing$1/ \mathit{Ro}$. We know from direct numerical simulation that the transition at$1/ {\mathit{Ro}}_{c} $corresponds to the onset of Ekman vortex formation reported before for$\Gamma = 1$at$1/ {\mathit{Ro}}_{c} \simeq 0. 4$and for$\Gamma = 2$at$1/ {\mathit{Ro}}_{c} = 0. 18$(Weisset al.,Phys. Rev. Lett., vol. 105, 2010, 224501). The$\Gamma $-dependence of$1/ {\mathit{Ro}}_{c} $can be explained as a finite-size effect that can be described phenomenologically by a Ginzburg–Landau model; this model is discussed in detail in the present paper. We do not know the origin of the transition at$1/ {\mathit{Ro}}_{0} $. Above$1/ {\mathit{Ro}}_{c} $,${\mathit{Nu}}_{red} $increased with increasing$\Gamma $up to${\ensuremath{\sim} }1/ \mathit{Ro}= 3$. We discuss the$\Gamma $-dependence of${\mathit{Nu}}_{red} $in this range in terms of the average Ekman vortex density as predicted by the model. At even larger$1/ \mathit{Ro}\gtrsim 3$there is a decrease of${\mathit{Nu}}_{red} $that can be attributed to two possible effects. First, the Ekman pumping might become less efficient when the Ekman layer is significantly smaller than the thermal boundary layer, and second, for rather large$1/ \mathit{Ro}$, the Taylor–Proudman effect in combination with boundary conditions suppresses fluid flow in the vertical direction.