Rayleigh and Prandtl number scaling in the bulk of Rayleigh-Benard turbulence

Abstract

The Rayleigh (Ra) and Prandtl (Pr) number scaling of the Nusselt number Nu, the Reynolds number Re, the temperature fluctuations, and the kinetic and thermal dissipation rates is studied for (numerical) homogeneous Rayleigh-Benard turbulence, i.e., Rayleigh-Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient. This system serves as model system for the bulk of Rayleigh-Benard flow and therefore as model for the so called ``ultimate regime of thermal convection''. With respect to the Ra dependence of Nu and Re we confirm our earlier results \cite{loh03} which are consistent with the Kraichnan theory \cite{kra62} and the Grossmann-Lohse (GL) theory \cite{gro00,gro01,gro02,gro04}, which both predict $Nu\sim R{a}^{1/2}$ and $Re\sim R{a}^{1/2}$. However the Pr dependence within these two theories is different. Here we show that the numerical data are consistent with the GL theory $Nu\sim P{r}^{1/2}$, $Re\sim P{r}^{-1/2}$. For the thermal and kinetic dissipation rates we find $\eps_θ/(κΔ^{2}L^{-2}) \sim (Re Pr)^{0.87}$ and $\eps_u/(ν^3 L^{-4}) \sim Re^{2.77}$, also both consistent with the GL theory, whereas the temperature fluctuations do not depend on Ra and Pr. Finally, the dynamics of the heat transport is studied and put into the context of a recent theoretical finding by Doering et al. \cite{doe05}.

arXiv comment8 pages, 9 figures

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