Optimal initial condition of passive tracers for their maximal mixing in finite time

TL;DR

This paper identifies the optimal initial tracer pattern for maximal mixing in finite time using the singular vectors of a truncated Perron-Frobenius operator, applicable to complex fluid flows.

Contribution

It introduces a novel method to determine optimal initial conditions for passive tracers based on the singular vectors of a truncated Perron-Frobenius operator, applicable to real-world flow data.

Findings

Optimal initial conditions align with the right singular vector of the truncated PF operator.

Method demonstrated on sine flow and 2D turbulence simulations.

Framework requires only particle positions at initial and final times.

Abstract

The efficiency of a fluid mixing device is often limited by fundamental laws and/or design constraints, such that a perfectly homogeneous mixture cannot be obtained in finite time. Here, we address the natural corollary question: Given the best available mixer, what is the optimal initial tracer pattern that leads to the most homogeneous mixture after a prescribed finite time? For ideal passive tracers, we show that this optimal initial condition coincides with the right singular vector (corresponding to the smallest singular value) of a suitably truncated Perron-Frobenius (PF) operator. The truncation of the PF operator is made under the assumption that there is a small length-scale threshold $\ell_\nu$ under which the tracer blobs are considered, for all practical purposes, completely mixed. We demonstrate our results on two examples: a prototypical model known as the sine flow and a direct numerical simulation of two-dimensional turbulence. Evaluating the optimal initial condition through this framework only requires the position of a dense grid of fluid particles at the final instance and their preimages at the initial instance of the prescribed time interval. As such, our framework can be readily applied to flows where such data is available through numerical simulations or experimental measurements.