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Azimuthal diffusion of the large-scale-circulation plane, and absence of significant non-Boussinesq effects, in turbulent convection near the ultimate-state transition
Published online by Cambridge University Press: 17 February 2016
Abstract
We present measurements of the orientation ${\it\theta}_{0}$ and temperature amplitude
${\it\delta}$ of the large-scale circulation in a cylindrical sample of turbulent Rayleigh–Bénard convection (RBC) with aspect ratio
${\it\Gamma}\equiv D/L=1.00$ (
$D$ and
$L$ are the diameter and height respectively) and for the Prandtl number
$Pr\simeq 0.8$. The results for
${\it\theta}_{0}$ revealed a preferred orientation with up-flow in the west, consistent with a broken azimuthal invariance due to the Earth’s Coriolis force (see Brown & Ahlers (Phys. Fluids, vol. 18, 2006, 125108)). They yielded the azimuthal diffusivity
$D_{{\it\theta}}$ and a corresponding Reynolds number
$Re_{{\it\theta}}$ for Rayleigh numbers over the range
$2\times 10^{12}\lesssim Ra\lesssim 1.5\times 10^{14}$. In the classical state (
$Ra\lesssim 2\times 10^{13}$) the results were consistent with the measurements by Brown & Ahlers (J. Fluid Mech., vol. 568, 2006, pp. 351–386) for
$Ra\lesssim 10^{11}$ and
$Pr=4.38$, which gave
$Re_{{\it\theta}}\propto Ra^{0.28}$, and with the Prandtl-number dependence
$Re_{{\it\theta}}\propto Pr^{-1.2}$ as found previously also for the velocity-fluctuation Reynolds number
$Re_{V}$ (He et al., New J. Phys., vol. 17, 2015, 063028). At larger
$Ra$ the data for
$Re_{{\it\theta}}(Ra)$ revealed a transition to a new state, known as the ‘ultimate’ state, which was first seen in the Nusselt number
$Nu(Ra)$ and in
$Re_{V}(Ra)$ at
$Ra_{1}^{\ast }\simeq 2\times 10^{13}$ and
$Ra_{2}^{\ast }\simeq 8\times 10^{13}$. In the ultimate state we found
$Re_{{\it\theta}}\propto Ra^{0.40\pm 0.03}$. Recently, Skrbek & Urban (J. Fluid Mech., vol. 785, 2015, pp. 270–282) claimed that non-Oberbeck–Boussinesq effects on the Nusselt and Reynolds numbers of turbulent RBC may have been interpreted erroneously as a transition to a new state. We demonstrate that their reasoning is incorrect and that the transition observed in the Göttingen experiments and discussed in the present paper is indeed to a new state of RBC referred to as ‘ultimate’.
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References
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