Turbulent Prandtl number from isotropically forced turbulence

TL;DR

This study uses simulations and experiments to measure turbulent diffusion coefficients and the Prandtl number in isotropic turbulence, finding that it approaches a constant at high Reynolds and Peclét numbers, independent of microscopic Prandtl number.

Contribution

The paper provides the first direct computation of turbulent Prandtl number in isotropic turbulence, challenging existing models that suggest its dependence on microscopic Prandtl number.

Findings

Turbulent Prandtl number approaches an asymptotic value at high Reynolds and Peclét numbers.

No significant dependence of ${ m Pr}_{ m t}$ on microscopic Prandtl number was observed.

Results contrast with $k- ext{epsilon}$ model predictions.

Abstract

Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ($\nu_{\rm t}$) and turbulent thermal diffusivity ($\chi_{\rm t}$) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, ${\rm Pr}_{\rm t} = \nu_{\rm t}/\chi_{\rm t}$, approach asymptotic values at sufficiently high Reynolds and Pecl\'et numbers. We also do not find a significant dependence of ${\rm Pr}_{\rm t}$ on the microscopic Prandtl number ${\rm Pr} = \nu/\chi$. These findings are in stark contrast to results from the $k-\epsilon$ model which suggests that ${\rm Pr}_{\rm t}$ increases monotonically with decreasing ${\rm Pr}$. The current results are relevant for the ongoing debate of, for example, the nature of the turbulent flows in the very low ${\rm Pr}$ regimes of stellar convection zones.