Numerical simulation and analysis of mixing enhancement due to chaotic advection using an adaptive approach for approximating the dilution index

TL;DR

This paper introduces an adaptive grid approach to accurately quantify mixing enhancement in chaotic advection systems using the dilution index, addressing limitations of particle-tracking methods and emphasizing the role of diffusion.

Contribution

A novel adaptive grid method for the dilution index that improves the quantification of mixing in chaotic systems, considering the effects of diffusion and numerical accuracy.

Findings

The adaptive approach effectively captures mixing enhancement in chaotic systems.

Diffusion plays a crucial role in filling KAM islands and achieving complete mixing.

Proper grid size selection is vital for accurate dilution index approximation.

Abstract

Lagrangian particle-tracking methods are particularly suitable to study solute transport in velocity fields displaying chaotic advection. They can accurately resolve stretching and folding processes, the increase in the solute-solvent interface available for diffusion as well as Kolmogorov-Arnold-Moser (KAM) islands, non-mixing regions that limit the chaotic area in the domain and, thereby, the mixing enhancement. However, they also display limitations due to the finite number of discrete particles, particularly if we are interested in the quantification of mixing processes, which require an accurate description of the particle density or concentration gradients. In this work, we use the dilution index to quantify the temporal increase in mixing of a solute within its solvent. We introduce a new approach to select a suitable grid size for the approximation of the density function, motivated by the theory of representative elementary volumes. It preserves the central feature of the dilution index, which is monotonically increasing in time, highlighting the importance of a suitable choice for the grid size in the dilution index approximation. We use this approach to demonstrate the mixing enhancement for two chaotic injection-extraction systems that exhibit chaotic structures: a source-sink dipole and a rotated potential mixing. Using our new approach, we assess the choice of design parameters of the injection-extraction systems to effectively engineer chaotic mixing. We demonstrate the important role of diffusion in filling the KAM islands and reaching complete mixing and, consequently, the importance of avoiding numerical diffusion, which often affects Eulerian methods applied on the advection-diffusion equation.