Scalar decay in a three-dimensional chaotic flow

TL;DR

This study uses high-resolution simulations to analyze passive scalar decay in a 3D chaotic flow, confirming theoretical predictions and identifying two decay regimes with different controlling factors.

Contribution

It extends 2D mixing theories to 3D flows and compares scalar decay in flows with different expanding directions, validating theoretical models.

Findings

Two decay regimes identified: local and global control.

Variance decay rates match asymptotic predictions.

Scalar decay rates are the same in flows with one or two expanding directions.

Abstract

The decay of a passive scalar in a three-dimensional chaotic flow is studied using high-resolution numerical simulations. The (volume-preserving) flow considered is a three-dimensional extension of the randomised alternating sine flow employed extensively in studies of mixing in two dimensions. It is used to show that theoretical predictions for two-dimensional flows with small diffusivity carry over to three dimensions even though the stretching properties differ significantly. The variance decay rate, scalar field structure, and time evolution of statistical moments confirm that there are two distinct regimes of scalar decay: a locally controlled regime, which applies when the domain size is comparable to the characteristic lengthscale of the velocity field, and a globally controlled regime, which when applies when the domain is larger. Asymptotic predictions for the variance decay rate in both regimes show excellent agreement with the numerical results. Consideration of both the forward flow and its time reverse makes it possible to compare the scalar evolution in flows with one or two expanding directions; simulations confirm the theoretical prediction that the decay rate of the scalar is the same in both flows, despite the very different scalar field structures.