Small-scale isotropy and ramp-cliff structures in scalar turbulence

TL;DR

This study investigates the impact of ramp-cliff structures on scalar turbulence, revealing how small-scale anisotropy diminishes at high Schmidt numbers through high-resolution simulations and a simple modeling approach.

Contribution

It introduces a simple model for ramp-cliff structures that explains the scalar derivative statistics and the restoration of isotropy at high Schmidt numbers.

Findings

Ramp-cliff structures cause odd-order moment anomalies in scalar turbulence.

Small-scale isotropy is restored at high Schmidt numbers, as shown by the model.

The model suggests a correction to the Batchelor length scale.

Abstract

Passive scalars advected by three-dimensional Navier-Stokes turbulence exhibit a fundamental anomaly in odd-order moments because of the characteristic ramp-cliff structures, violating small-scale isotropy. We use data from direct numerical simulations with grid resolution of up to $8192^3$ at high P\'eclet numbers to understand this anomaly as the scalar diffusivity, $D$, diminishes, or as the Schmidt number, $Sc = \nu/D$, increases; here $\nu$ is the kinematic viscosity of the fluid. The microscale Reynolds number varies from 140 to 650 and $Sc$ varies from 1 to 512. A simple model for the ramp-cliff structures is shown to characterize the scalar derivative statistics extremely well. It accurately captures how the small-scale isotropy is restored in the large-$Sc$ limit, and additionally suggests a slight correction to the Batchelor length scale as the relevant smallest scale in the scalar field.