Reynolds-number measurements for low-Prandtl-number turbulent convection of large-aspect-ratio samples

Abstract

AbstractWe present experimental results for the Reynolds number ${\mathit{Re}}_{U} $ based on the horizontal mean-flow velocity $U$ and for ${\mathit{Re}}_{V} $ based on the root-mean-square horizontal fluctuation velocity $V$ for turbulent Rayleigh–Bénard convection in a cylindrical sample of aspect ratio $\Gamma = 10. 9$ over the Prandtl number range $0. 18\leq \mathit{Pr}\leq 0. 88$. The results were derived from space–time cross-correlation functions of shadowgraph images, using the elliptic approximation of He & Zhang (Phys. Rev. E, vol. 73, 2006, 055303). The data cover the Rayleigh number range from $3\times 1{0}^{5} $ to $2\times 1{0}^{7} $. We find that ${\mathit{Re}}_{U} $ is nearly two orders of magnitude smaller than the values given by the Grossmann–Lohse (GL) model (Grossmann & Lohse, Phys. Rev. E, vol. 66, 2002, 016305) for $\Gamma = 1. 00$ and attribute this difference to averaging caused by lateral random diffusion of the large-scale circulation cells in large-$\Gamma $ samples. For the fluctuations we found ${\mathit{Re}}_{V} = {\tilde {R} }_{0} {\mathit{Pr}}^{\alpha } {\mathit{Ra}}^{\eta } $, with ${\tilde {R} }_{0} = 0. 31$, $\alpha = - 0. 53\pm 0. 11$ and $\eta = 0. 45\pm 0. 03$. That result agrees well with the GL model. The close agreement of the coefficient ${\tilde {R} }_{0} $ must be regarded as a coincidence because the GL model was for $\Gamma = 1. 00$ and for a mean-flow velocity $U$.