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

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We present a framework to calculate the scale-resolved turbulent Prandtl number ${\textit{Pr}}_t$ for the well-mixed and highly inertial bulk of a turbulent Rayleigh–Bénard mesoscale convection layer at a molecular Prandtl number of ${\textit{Pr}}=10^{-3}$ . 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 definitions of turbulent (or eddy) viscosity and diffusivity do 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 $\varGamma =25$ for Rayleigh numbers $10^5\leqslant Ra\leqslant 10^7$ . We find that the turbulent Prandtl number is effectively up to four orders of magnitude larger than the molecular one, ${\textit{Pr}}_t\sim 10$ . This holds particularly for the upper end of the inertial subrange, where the eddy diffusivity exceeds the molecular value, $\kappa _e(r)\gt \kappa$ . Highly inertial low-Prandtl-number convection becomes effectively a higher-Prandtl-number turbulent flow, when turbulent mixing processes on scales that reach into the inertial range are included. This might have some relevance for prominent low-Prandtl-number applications, such as solar convection.

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