13 research outputs found
Lagrangian heat transport in turbulent three-dimensional convection
Spatial regions that do not mix effectively with their surroundings and thus
contribute less to the heat transport in fully turbulent three-dimensional
Rayleigh-B\'{e}nard flows are identified by Lagrangian trajectories that stay
together for a longer time. These trajectories probe Lagrangian coherent sets
(CS) which we investigate here in direct numerical simulations in convection
cells with square cross section of aspect ratio , Rayleigh number
, and Prandtl numbers and . The analysis is
based on Lagrangian tracer particles which are advected in the
time-dependent flow. Clusters of trajectories are identified by a graph
Laplacian with a diffusion kernel, which quantifies the connectivity of
trajectory segments, and a subsequent sparse eigenbasis approximation (SEBA)
for cluster detection. The combination of graph Laplacian and SEBA leads to a
significantly improved cluster identification that is compared with the
large-scale patterns in the Eulerian frame of reference. We show that the
detected CS contribute by a third less to the global turbulent heat transport
for all investigated compared to the trajectories in the spatial
complement. This is realized by monitoring Nusselt numbers along the tracer
trajectory ensembles, a dimensionless local measure of heat transfer.Comment: 8 pages, 5 figure
Identification of individual coherent sets associated with flow trajectories using coherent structure coloring
We present a method for identifying the coherent structures associated with individual Lagrangian flow trajectories even where only sparse particle trajectory data are available. The method, based on techniques in spectral graph theory, uses the Coherent Structure Coloring vector and associated eigenvectors to analyze the distance in higher-dimensional eigenspace between a selected reference trajectory and other tracer trajectories in the flow. By analyzing this distance metric in a hierarchical clustering, the coherent structure of which the reference particle is a member can be identified. This algorithm is proven successful in identifying coherent structures of varying complexities in canonical unsteady flows. Additionally, the method is able to assess the relative coherence of the associated structure in comparison to the surrounding flow. Although the method is demonstrated here in the context of fluid flow kinematics, the generality of the approach allows for its potential application to other unsupervised clustering problems in dynamical systems such as neuronal activity, gene expression, or social networks.
In recent years, there has been a proliferation of techniques that aim to characterize fluid flow kinematics on the basis of Lagrangian trajectories of collections of tracer particles. Most of these techniques depend on the presence of tracer particles that are initially closely spaced, in order to compute local gradients of their trajectories. In many applications, the requirement of close tracer spacing cannot be satisfied, especially when the tracers are naturally occurring and their distribution is dictated by the underlying flow. Moreover, current methods often focus on determination of the boundaries of coherent sets, whereas in practice it is often valuable to identify the complete set of trajectories that are coherent with an individual trajectory of interest. We extend the concept of Coherent Structure Coloring, an approach based on spectral graph theory, to achieve identification of the coherent set associated with individual Lagrangian trajectories. The method does not require a priori determination of the number of coherent structures in the flow, nor does it require heuristics regarding the eigenvalue spectrum corresponding to the generalized eigenvalue problem. Importantly, although the method is demonstrated here in the context of fluid flow kinematics, the generality of the approach allows for its potential application to other unsupervised clustering problems in dynamical systems such as neuronal activity, gene expression, or social networks
Identification of individual coherent sets associated with flow trajectories using coherent structure coloring
We present a method for identifying the coherent structures associated with individual Lagrangian flow trajectories even where only sparse particle trajectory data are available. The method, based on techniques in spectral graph theory, uses the Coherent Structure Coloring vector and associated eigenvectors to analyze the distance in higher-dimensional eigenspace between a selected reference trajectory and other tracer trajectories in the flow. By analyzing this distance metric in a hierarchical clustering, the coherent structure of which the reference particle is a member can be identified. This algorithm is proven successful in identifying coherent structures of varying complexities in canonical unsteady flows. Additionally, the method is able to assess the relative coherence of the associated structure in comparison to the surrounding flow. Although the method is demonstrated here in the context of fluid flow kinematics, the generality of the approach allows for its potential application to other unsupervised clustering problems in dynamical systems such as neuronal activity, gene expression, or social networks.
In recent years, there has been a proliferation of techniques that aim to characterize fluid flow kinematics on the basis of Lagrangian trajectories of collections of tracer particles. Most of these techniques depend on the presence of tracer particles that are initially closely spaced, in order to compute local gradients of their trajectories. In many applications, the requirement of close tracer spacing cannot be satisfied, especially when the tracers are naturally occurring and their distribution is dictated by the underlying flow. Moreover, current methods often focus on determination of the boundaries of coherent sets, whereas in practice it is often valuable to identify the complete set of trajectories that are coherent with an individual trajectory of interest. We extend the concept of Coherent Structure Coloring, an approach based on spectral graph theory, to achieve identification of the coherent set associated with individual Lagrangian trajectories. The method does not require a priori determination of the number of coherent structures in the flow, nor does it require heuristics regarding the eigenvalue spectrum corresponding to the generalized eigenvalue problem. Importantly, although the method is demonstrated here in the context of fluid flow kinematics, the generality of the approach allows for its potential application to other unsupervised clustering problems in dynamical systems such as neuronal activity, gene expression, or social networks
Network Measures of Mixing
Transport and mixing processes in fluid flows can be studied directly from
Lagrangian trajectory data, such as obtained from particle tracking
experiments. Recent work in this context highlights the application of
graph-based approaches, where trajectories serve as nodes and some similarity
or distance measure between them is employed to build a (possibly weighted)
network, which is then analyzed using spectral methods. Here, we consider the
simplest case of an unweighted, undirected network and analytically relate
local network measures such as node degree or clustering coefficient to flow
structures. In particular, we use these local measures to divide the family of
trajectories into groups of similar dynamical behavior via manifold learning
methods
Fast and robust computation of coherent Lagrangian vortices on very large two-dimensional domains
We describe a new method for computing coherent Lagrangian vortices in
two-dimensional flows according to any of the following approaches: black-hole
vortices [Haller & Beron-Vera, 2013], objective Eulerian Coherent Structures
(OECSs) [Serra & Haller, 2016], material barriers to diffusive transport
[Haller et al., 2018, Haller et al., 2019], and constrained diffusion barriers
[Haller et al., 2019]. The method builds on ideas developed previously in
[Karrasch et al., 2015], but our implementation alleviates a number of
shortcomings and allows for the fully automated detection of such vortices on
unprecedentedly challenging real-world flow problems, for which specific human
interference is absolutely infeasible. Challenges include very large domains
and/or parameter spaces. We demonstrate the efficacy of our method in dealing
with such challenges on two test cases: first, a parameter study of a turbulent
flow, and second, computing material barriers to diffusive transport in the
global ocean.Comment: 25 pages, 10 figures (partially of very low quality due to size
constraint by arxiv.org), postprin
From large deviations to semidistances of transport and mixing: coherence analysis for finite Lagrangian data
One way to analyze complicated non-autonomous flows is through trying to
understand their transport behavior. In a quantitative, set-oriented approach
to transport and mixing, finite time coherent sets play an important role.
These are time-parametrized families of sets with unlikely transport to and
from their surroundings under small or vanishing random perturbations of the
dynamics. Here we propose, as a measure of transport and mixing for purely
advective (i.e., deterministic) flows, (semi)distances that arise under
vanishing perturbations in the sense of large deviations. Analogously, for
given finite Lagrangian trajectory data we derive a discrete-time and space
semidistance that comes from the "best" approximation of the randomly perturbed
process conditioned on this limited information of the deterministic flow. It
can be computed as shortest path in a graph with time-dependent weights.
Furthermore, we argue that coherent sets are regions of maximal farness in
terms of transport and mixing, hence they occur as extremal regions on a
spanning structure of the state space under this semidistance---in fact, under
any distance measure arising from the physical notion of transport. Based on
this notion we develop a tool to analyze the state space (or the finite
trajectory data at hand) and identify coherent regions. We validate our
approach on idealized prototypical examples and well-studied standard cases.Comment: J Nonlinear Sci, 201