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

TL;DR

This paper addresses the challenge of quantifying mixing in chaotic advection at low Reynolds numbers using Lagrangian particle tracking. It introduces an adaptive grid strategy aligned with representative elementary volume ideas to compute the dilution index, an entropy-based measure of mixing, while preserving its monotonic growth in time. The authors apply the method to two chaotic injection-extraction systems, the pulsed source-sink and the rotated potential mixing flow, showing that diffusion is crucial for filling KAM islands and that mixing enhancement is not guaranteed by chaos alone. The work provides a practical approach to evaluating mixing potential in porous-media and microfluidic contexts and highlights the importance of diffusion and island topology for design decisions in groundwater remediation and related applications.

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.

Figures (18)

• Figure 1: Two-dimensional plane with one sink and one source at $\pm\left(10\right)$. All particles with a distance to the sink smaller than $\Lambda$ are extracted during the next stroke of the sink. Within the following stroke of the source, all of them are reinjected and occupy the circle around the source of radius $\Lambda$.
• Figure 2: (a) Configuration of sources and sinks that motivates the periodic-like boundary condition which was applied for the PSS system. (b) and (c) show two examples of particle paths that start at the white filled circle and that cross the domain quasi-periodic boundary at the top and at the left side. The blue filled circles indicate the sources, red circles indicate the sinks.
• Figure 3: Poincaré sections of the pulsed source-sink system for $\Lambda^2 \in \{0.1, 0.2, 0.3, 0.4, 0.5\}$. The blue filled circle indicates the location of the source while the red circle indicates the sink. White areas not occupied by particles are the KAM islands.
• Figure 4: Poincaré sections of the RPM flow for $\Theta \in \left\{0, \pi/6, \pi/3, \pi/2, 2\pi/3, 5\pi/6\right\}$ and $\tau \in \{0.2, 0.5\}$. White areas not occupied by particles are the KAM islands.
• Figure 5: Grid with $h = 0.15$ for approximating the dilution index of the RPM flow.