Scaling relations in large-Prandtl-number natural thermal convection

TL;DR

This paper derives and confirms scaling laws for large-Prandtl-number natural thermal convection, showing a transition between boundary-layer and bulk-dominated regimes through numerical simulations.

Contribution

It provides a theoretical derivation of the large-Pr boundary-layer regime and validates it with numerical simulations, highlighting a transition between different dissipation regimes.

Findings

The large-Pr boundary-layer regime I$_ty^<$ has specific scaling laws for Nu and Re.

Numerical simulations confirm the regime I$_ty^<$ is similar to regime III$_ty$, bulk-dominated.

Transition to regime IV$_u$ occurs at higher Rayleigh numbers, with different dissipation mechanisms.

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.