Lagrangian based methods for coherent structure detection

Abstract

There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. The transport of material by advection in fluid systems is a process of vital and ubiquitous importance, underlying scenarios as diverse as pollutant distribution and searchand-rescue operations in the ocean to blood flow in the human body. The rapidly advancing field of Lagrangian based methods for studying flow transport has demonstrated an effective ability to find robust and significant transport features that underly the organization of flow transport in complex, unsteady flow fields. In this review, we present an overview of four of the leading Lagrangian approaches, each with their own strengths and challenges. Details of each method are presented along with an example application to the same model system, the double-gyre flow. Furthermore, we highlight a number of exciting applications and future directions to bring Lagrangian based analysis closer to implementation in real-time, real-world decision making strategies.

Trajectory‐based computational study of coherent behavior in flows

“…The mathematical definition and numerical study of coherent flow structures has been an area of intense research over the last two decades, see [1] for a discussion and comparison of different methods. Most of the established techniques require high resolution trajectory data from (1), that is, from a dense grid of initial conditions. This can be prohibitively expensive in complex systems, such as turbulent flows.…”

Generalized Lagrangian coherent structures

“…4 are the computed diagnostic fields obtained from the flow from time 1 to 0, and the right panel shows extracted LCSs based on some simple choices of procedures P. We do not use the most sophisticated methods available for such extraction, because (i) we are only interested in a qualitative comparison, and (ii) more refined extraction methods will result in approximately the same entities. We moreover emphasize that our intention is not to compare or critically evaluate different LCS methods (readers are referred to papers such as [110,143] for this), but rather to illustrate that results arising from different LCS methods are not necessarily the same, and do not necessarily have a connection to GLCS sets chosen according to some specification.…”

“…We specify an unsteady velocity field v that has seen extensive use as a testbed for LCS analysis [142,143,109,144,1,145,38,146,102,77,147,78,79,148,76,88]: the double gyre flow as introduced by Shadden et al [75]. The velocity v = (v 1 , v 2 ) takes the form where the function h is defined by…”

Network-based study of Lagrangian transport and mixing

“…All these methods can deal with sparse and incomplete trajectory data and do respect the dynamics of the entire trajectories, not just the end points. While c-means clustering as used by Froyland and Padberg-Gehle (2015) is computationally inexpensive and works well in example systems (see also Allshouse and Peacock, 2015), spectral clustering approaches as in Hadjighasem et al (2016), Banisch andKoltai (2017), andSchlueter-Kuck andDabiri (2017) appear to be more robust, but require considerable computational effort.…”

“…The mathematical definition and numerical study of coherent flow structures has received considerable scientific interest for the last 2 decades. The proposed methods roughly fall into two different classes, geometric and probabilistic approaches; see Allshouse and Peacock (2015) for a discussion and comparison of different methods. Geometric concepts aim at defining the boundaries between coherent sets, i.e., codimension-1 material surfaces in the flow that can be characterized by variational criteria (see Haller, 2015, for a recent review).…”