Chaotic mixing in a bounded three-dimensional flow

Abstract

Even though the first theoretical example of chaotic advection was a three-dimensional flow (Hénon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincaré section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.

Nonlinear vortex development in rotating flows

“…Although over small exposure times the deviation from being a perfect Lagrangian tracer of the underlying flow is not significant, over the larger time scales required to capture weak vortex motion, the presence of non-axisymmetric particle paths is clearly visible. There are obvious similarities between this image and those obtained by (for example) Fountain et al (2000), showing chaotic advection in similar flows that are forced to have a significant non-axisymmetry by a sloped rotating wall (visualized by dye).…”

Three-dimensional Lagrangian transport phenomena in unsteady laminar flows driven by a rotating sphere

“…as closed-form analytical perturbation satisfying conditions (14). This in essence artificial perturbation may at first glance seem of limited physical relevance.…”

“…This enables topological analyses based on 3D coherent structures formed by Lagrangian fluid trajectories and has found application in a great variety of 3D transport problems. [10][11][12][13][14][15][16][17][18][19][20] Such structures geometrically determine the transport properties and insight into their formation, characteristics and response to parametric variations is key to better understanding -and, ultimately, systematic manipulation -of 3D transport. The traditional objective is accomplishment of efficient mixing, which here…”

Optimal control of mixing in Stokes fluid flows

“…In this paper, we assume velocity fields of the form α 1 (t)u 1 (x) + α 2 (t)u 2 (x). In the three-dimensional case, it is possible to have chaotic advection even with a steady velocity field (Bajer & Moffatt 1990;Stone, Nadim & Strogatz 1991, Fountain et al 2000. However, in the presence of symmetries (Mezić & Wiggins 1994;Grigoriev 2005) it may be necessary to have two or three independent velocity fields with complex time-dependence to cause efficient mixing.…”