Title: 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 $Nu\sim
Pr^0\,Ra^{1/3}$ and $Re\sim Pr^{-1}\,Ra^{2/3}$. 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.