Understanding the effect of Prandtl number on momentum and scalar mixing rates in neutral and stably stratified flows using gradient field dynamics

TL;DR

This study investigates how the Prandtl number influences momentum and scalar mixing rates in stratified flows, revealing mechanisms behind energy dissipation changes and proposing a new parameter to better estimate buoyancy effects at small scales.

Contribution

The paper provides a mechanistic explanation for Prandtl number effects on turbulence dissipation rates and introduces an alternative parameter for buoyancy impact estimation.

Findings

TPE-DR decreases and TKE-DR increases with higher Prandtl numbers.

Buoyancy effects intensify at smaller scales as Prandtl number increases.

A new parameter better estimates buoyancy effects at small scales.

Abstract

Recently, direct numerical simulations (DNS) of stably stratified turbulence have shown that as the Prandtl number ($Pr$) is increased from 1 to 7, the mean turbulent potential energy dissipation rate (TPE-DR) drops dramatically, while the mean turbulent kinetic energy dissipation rate (TKE-DR) increases significantly. Through an analysis of the equations governing the fluctuating velocity and density gradients we provide a mechanistic explanation for this surprising behavior and test the predictions using DNS. We show that the mean density gradient gives rise to a mechanism that opposes the production of fluctuating density gradients, and this is connected to the emergence of ramp-cliffs. The same term appears in the velocity gradient equation but with the opposite sign, and is the contribution from buoyancy. This term is ultimately the reason why the TPE-DR reduces while the TKE-DR increases with increasing $Pr$. Our analysis also predicts that the effects of buoyancy on the smallest scales of the flow become stronger as $Pr$ is increased, and this is confirmed by our DNS data. A consequence of this is that the standard buoyancy Reynolds number does not correctly estimate the impact of buoyancy at the smallest scales when $Pr$ deviates from 1, and we derive a suitable alternative parameter. Finally, an analysis of the filtered gradient equations reveals that the mean density gradient term changes sign at sufficiently large scales, such that buoyancy acts as a source for velocity gradients at small scales, but as a sink at large scales.