Scaling relations in large-Prandtl-number natural thermal convection

Abstract

In this study we follow Grossmann and Lohse, Phys. Rev. Lett. 86 (2001), who derived various scalings regimes for the dependence of the Nusselt number $Nu$ and the Reynolds number $Re$ on the Rayleigh number $Ra$ and the Prandtl number $Pr$. We focus on theoretical arguments as well as on numerical simulations for the case of large-$Pr$ natural thermal convection. Based on an analysis of self-similarity of the boundary layer equations, we derive that in this case the limiting large-$Pr$ boundary-layer dominated regime is I${}_{\infty }^{<}$, introduced and defined in [1], with the scaling relations $Nu\sim P{r}^{0}R{a}^{1/3}$ and $Re\sim P{r}^{-1}R{a}^{2/3}$. Our direct numerical simulations for $Ra$ from $10^4$ to $10^9$ and $Pr$ from 0.1 to 200 show that the regime I${}_{\infty }^{<}$ is almost indistinguishable from the regime III${}_{\infty }$, where the kinetic dissipation is bulk-dominated. With increasing $Ra$, the scaling relations undergo a transition to those in IV${}_u$ of reference [1], where the thermal dissipation is determined by its bulk contribution.

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• Heat and momentum transport in turbulent horizontal convection at low Prandtl numbers