Lagrangian Transport and Chaotic Advection in Three-Dimensional Laminar Flows

Abstract

Transport and mixing of scalar quantities in fluid flows is ubiquitous in industry and Nature. Turbulent flows promote efficient transport and mixing by their inherent randomness. Laminar flows lack such a natural mixing mechanism and efficient transport is far more challenging. However, laminar flow is essential to many problems and insight into its transport characteristics of great importance. Laminar transport, arguably, is best described by the Lagrangian fluid motion ("advection") and the geometry, topology and coherence of fluid trajectories. Efficient laminar transport being equivalent to "chaotic advection" is a key finding of this approach. The Lagrangian framework enables systematic analysis and design of laminar flows. However, the gap between scientific insights into Lagrangian transport and technological applications is formidable primarily for two reasons. First, many studies concern two-dimensional (2D) flows yet the real world is three dimensional (3D). Second, Lagrangian transport is typically investigated for idealised flows yet practical relevance requires studies on realistic 3D flows. The present review aims to stimulate further development and utilisation of know-how on 3D Lagrangian transport and its dissemination to practice. To this end 3D practical flows are categorised into canonical problems. First, to expose the diversity of Lagrangian transport and create awareness of its broad relevance. Second, to enable knowledge transfer both within and between scientific disciplines. Third, to reconcile practical flows with fundamentals on Lagrangian transport and chaotic advection. This may be a first incentive to structurally integrate the "Lagrangian mindset" into the analysis and design of 3D practical flows.

Fouling Mitigation via Chaotic Advection in a Flat Membrane Module with a Patterned Surface

“…As far as the herringbone pattern in a duct flow is concerned, it has been demonstrated that two asymmetric counter-rotating flows are created by the herringbone-shaped grooves fabricated in a rectangular microchannel working in the creeping flow ( ) [ 22 , 37 , 38 , 39 , 40 , 41 , 42 ]. When the two flow portraits are projected onto a single plane normal to the ‒axis, the cross-sectional flow patterns from the two flows intersect each other, satisfying the necessary condition for chaotic advection in a three-dimensional geometrically periodic flow [ 20 , 21 , 43 ]. In the case of the flat membrane module ( Figure 1 ), unlike the two side walls of the microchannel, the two side surfaces of the computational domain (depicted in Figure 1 b) are not rigid walls but open boundaries, where a symmetric boundary condition is applied.…”

“…It is an interesting subject to relate the chaotic behavior of the feed fluid to the fouling mitigation for the flat membrane module. If a flow system is chaotic, fluid elements exhibit strong deformations due to repeated stretching and folding, resulting in an exponential growth of the distance between two neighboring fluid particles [ 19 , 20 , 21 ]. Once chaotic advection occurs in a filtration module working in a laminar flow, it can uniformly distribute foulants via chaotic mixing, thereby mitigating membrane fouling if properly combined with downwelling flows [ 16 ].…”

“…The present study employs a crossflow filtration module that consists of a flat membrane and a patterned non-permeable solid surface with herringbone-shaped grooves, which stirs the feed fluid via chaotic advection, thereby suppressing fouling and concentration polarization in the crossflow filtration module. For more details on chaotic advection, we refer to three review papers [ 19 , 20 , 21 ].…”

Passive tracer advection in the equatorial Pacific region: statistics, correlations, and a model of fractional Brownian motion

“…in the phase space related to chaotic advection (see e.g. Péntek et al, 1995;Cartwright et al, 1999;Tél et al, 2000;Speetjens et al, 2021). Recent important applications are connected to the exploration of Lagrangian coherent structures (Haller, 2015;Hadjighasem et al, 2017;Haller et al, 2018;Beron-Vera et al, 2018;Callies, 2021;Haller et al, 2021), mostly in the context of ocean flows.…”

Comment on os-2021-94

“…In most cases, it is possible to reconstruct invariant sets of fractal nature in the phase space related to chaotic advection (see e.g. Péntek et al, 1995;Cartwright et al, 1999;Tél et al, 2000;Speetjens et al, 2021). Recent important applications are connected to the exploration of Lagrangian coherent structures (Haller, 2015;Hadjighasem et al, 2017;Haller et al, 2018;Beron-Vera et al, 2018;Callies, 2021;Haller et al, 2021), mostly in the context of ocean flows.…”