Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows

Blazevski, Daniel; Haller, George

doi:10.1016/j.physd.2014.01.007

Cited by 70 publications

(91 citation statements)

References 33 publications

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“…While in some cases particular structures may continue to have influence after the interval of study, 58 more results on the longevity of structures outside the given time interval would be significant. And while there are now more results for threedimensional analytic flows, 11 there is a need to exploit GPUs and other numerical techniques 6 to aid in efficient calculation and visualization of this highly parallelizable problem.…”

Section: Discussionmentioning

confidence: 99%

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Lagrangian based methods for coherent structure detection

Allshouse

Peacock

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Section: Discussionmentioning

confidence: 99%

“…11 In addition to this transition to 3D calculations, a tool set to calculate many of these material lines has been developed for Matlab.…”

Section: A Finite-time Lyapunov Exponentsmentioning

confidence: 99%

Lagrangian based methods for coherent structure detection

Allshouse

Peacock

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…The importance of the large class of hyperbolic trajectories outlined above is underscored by their effect on the global behavior. For example, it is well known that stable and unstable manifolds form important flow separators in fluid flows, thereby providing a skeleton which distinguishes between regions of anomalous motion [4,[6][7][8][9][10]. Hyperbolic trajectories are entities to which these stable and unstable manifolds are attached, and hence their motion with time governs the time variation of these flow separators.…”

Section: Introductionmentioning

confidence: 99%

Accurate control of hyperbolic trajectories in any dimension

Balasuriya

Padberg‐Gehle

2014

Phys. Rev. E

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The unsteady (nonautonomous) analog of a hyperbolic fixed point is a hyperbolic trajectory, whose importance is underscored by its attached stable and unstable manifolds, which have relevance in fluid flow barriers, chaotic basin boundaries, and the long-term behavior of the system. We develop a method for obtaining the unsteady control velocity which forces a hyperbolic trajectory to follow a user-prescribed variation with time. Our method is applicable in any dimension, and accuracy to any order is achievable. We demonstrate and validate our method by (1) controlling the fixed point at the origin of the Lorenz system, for example, obtaining a user-defined nonautonomous attractor, and (2) the saddle points in a droplet flow, using localized control which generates global transport.

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“…The operator P uses spatial fields of these eigenvalues and eigenvectors to identify curves (in 2D [69,68,70]) or surfaces (in 3D [71,72]) of extremal attraction or repulsion in the time-slice t, with respect to the time evolution from t to τ . The variational methods also exist for elliptic and parabolic structures [9], and the excellent review article [9] is recommended for readers requiring additional information on how to compute variational LCSs.…”

Section: Lcs Diagnostic Methodsmentioning

confidence: 99%

Generalized Lagrangian coherent structures

Balasuriya

Ouellette

Rypina

2018

Physica D: Nonlinear Phenomena

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The notion of a Lagrangian Coherent Structure (LCS) is by now well established as a way to capture transient coherent transport dynamics in unsteady and aperiodic fluid flows that are known over finite time. We show that the concept of an LCS can be generalized to capture coherence in other quantities of interest that are transported by, but not fully locked to, the fluid. Such quantities include those with dynamic, biological, chemical, or thermodynamic relevance, such as temperature, pollutant concentration, vorticity, kinetic energy, plankton density, and so on. We provide a conceptual framework for identifying the Generalized Lagrangian Coherent Structures (GLCSs) associated with such evolving quantities. We show how LCSs can be seen as a special case within this framework, and provide an overarching discussion of various methods for identifying LCSs. The utility of this more general viewpoint is highlighted through a variety of examples. We also show that although LCSs approximate GLCSs in certain limiting situations under restrictive assumptions on how the velocity field affects the additional quantities of interest, LCSs are not in general sufficient to describe their coherent transport.

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Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows

Cited by 70 publications

References 33 publications

Lagrangian based methods for coherent structure detection

Lagrangian based methods for coherent structure detection

Accurate control of hyperbolic trajectories in any dimension

Generalized Lagrangian coherent structures

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