Scaling in large Prandtl number turbulent thermal convection

TL;DR

This paper develops a quasi-linear theory to analyze how heat transfer scales in turbulent thermal convection at large Prandtl numbers, revealing two regimes with different dependencies on Reynolds and Rayleigh numbers, and proposing a universal scaling function.

Contribution

It introduces a unified scaling theory for turbulent thermal convection at large Prandtl numbers, identifying two regimes and a bimodal universal function for heat transfer.

Findings

Two distinct scaling regimes depending on Reynolds number.

A universal bimodal scaling function for $Nu(Ra,Pr)$.

Explanation of experimental discrepancies between different fluids.

Abstract

We study the scaling properties of heat transfer $Nu$ in turbulent thermal convection at large Prandtl number $Pr$ using a quasi-linear theory. We show that two regimes arise, depending on the Reynolds number $Re$. At low Reynolds number, $Nu Pr^{-1/2}$ and $Re$ are a function of $Ra Pr^{-3/2}$. At large Reynolds number $Nu Pr^{1/3}$ and $Re Pr$ are function only of $Ra Pr^{2/3}$ (within logarithmic corrections). In practice, since $Nu$ is always close to $Ra^{1/3}$, this corresponds to a much weaker dependence of the heat transfer in the Prandtl number at low Reynolds number than at large Reynolds number. This difference may solve an existing controversy between measurements in SF6 (large $Re$) and in alcohol/water (lower $Re$). We link these regimes with a possible global bifurcation in the turbulent mean flow. We further show how a scaling theory could be used to describe these two regimes through a single universal function. This function presents a bimodal character for intermediate range of Reynolds number. We explain this bimodality in term of two dissipation regimes, one in which fluctuation dominate, and one in which mean flow dominates. Altogether, our results provide a six parameters fit of the curve $Nu(Ra,Pr)$ which may be used to describe all measurements at $Pr\ge 0.7$.