Scale-resolved turbulent Prandtl number for Rayleigh-B\'{e}nard convection at $\boldsymbol{Pr =10^{-3}}$

TL;DR

This paper introduces a new framework to calculate the scale-resolved turbulent Prandtl number in Rayleigh-Bénard convection at low Prandtl number, revealing it can be up to 4 orders of magnitude larger than the molecular value, especially at larger scales.

Contribution

It develops a scale-dependent turbulence model for Prandtl number in low-Prandtl-number convection without relying on traditional gradient-based closures.

Findings

Turbulent Prandtl number can reach up to 10^4 times the molecular value.

Eddy diffusivity exceeds molecular diffusivity at larger scales.

Low-Prandtl-number convection behaves like high-Prandtl-number flow at inertial scales.

Abstract

We present a framework to calculate the scale-resolved turbulent Prandtl number $Pr_t$ for the well-mixed and highly inertial bulk of a turbulent Rayleigh-B\'{e}nard mesoscale convection layer at a molecular Prandtl number $Pr=10^{-3}$. It builds on Kolmogorov's refined similarity hypothesis of homogeneous isotropic fluid and passive scalar turbulence, based on log-normally distributed amplitudes of kinetic energy and scalar dissipation rates that are coarse-grained over variable scales $r$ in the inertial subrange. Our definition of turbulent (or eddy) viscosity and diffusivity does not rely on mean gradient-based Boussinesq closures of Reynolds stresses and convective heat fluxes. Such gradients are practically absent or indefinite in the bulk. The present study is based on direct numerical simulation of plane-layer convection at an aspect ratio of $\Gamma=25$ for Rayleigh numbers $10^5\leq Ra \leq 10^7$. We find that the turbulent Prandtl number is effectively up to 4 orders of magnitude larger than the molecular one, $\Pr_t\sim 10$. This holds particularly for the upper end of the inertial subrange, where the eddy diffusivity exceeds the molecular value, $\kappa_e(r)>\kappa$. Highly inertial low-Prandtl-number convection behaves effectively as a high-Prandtl number flow, which also supports previous models for the prominent application case of solar convection.