Small-Scale Isotropy and Ramp-Cliff Structures in Scalar Turbulence

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 Péclet 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 developed and shown to characterize the scalar derivative statistics very well. It accurately captures how the small-scale isotropy is restored in the large-$Sc$ limit, and additionally suggests a possible correction to the Batchelor length scale as the relevant smallest scale in the scalar field.