Scale-resolved turbulent Prandtl number for Rayleigh-Bénard convection at $\mathbf{Pr}=10^{-3}$

Abstract

We present a framework to calculate the scale-resolved turbulent Prandtl number $P{r}_t$ for the well-mixed and highly inertial bulk of a turbulent Rayleigh-Bénard 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 $10^5\le Ra\le 10^7$. We find that the turbulent Prandtl number is effectively up to 4 orders of magnitude larger than the molecular one, ${\mathrm{Pr}}_t\sim 10$. 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.

18 pages, 8 figures

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Cited by in corpus (1)

• Radial differential rotation leading to dipole collapse in pre-main-sequence stars