Robust and efficient identification of optimal mixing perturbations using proxy multiscale measures

TL;DR

This paper introduces a robust, computationally efficient method to identify optimal initial perturbations for scalar mixing in fluid flows using proxy multiscale measures called mix-norms, applicable at high Péclet numbers.

Contribution

It demonstrates that minimizing mix-norms over short time horizons effectively finds initial conditions that enhance mixing, with the development of coherent vortical structures depending on Péclet number and time horizon.

Findings

Minimizing mix-norms over short horizons efficiently identifies effective mixing perturbations.

Optimal perturbations induce coherent vortical flow structures that enhance mixing.

The method is robust and applicable at high Péclet numbers.

Abstract

Understanding and optimizing passive scalar mixing in a diffusive fluid flow at finite P\'eclet number $Pe=U h/\kappa$ (where $U$ and $h$ are characteristic velocity and length scales, and $\kappa$ is the molecular diffusivisity of the scalar) is a fundamental problem of interest in many environmental and industrial flows. Particularly when $Pe \gg 1$, identifying initial perturbations of given energy which optimally and thoroughly mix fluids of initally different properties can be computationally challenging. To address this challenge, we consider the identification of initial perturbations in an idealized two-dimensional flow on a torus that extremize various measures over finite time horizons. We identify such `optimal' initial perturbations using the `direct-adjoint looping' (DAL) method, thus requiring the evolving flow to satisfy the governing equations and boundary conditions at all points in space and time. We demonstrate that minimising multiscale measures commonly known as `mix-norms' over short time horizons is a computationally efficient and robust way to identify initial perturbations that thoroughly mix layered scalar distributions over relatively long time horizons, provided the magnitude of the mix-norm's index is not too large. Minimisation of such mix-norms triggers the development of coherent vortical flow structures which effectively mix, with the particular properties of these flow structures depending on $Pe$ and also the time horizon of interest.