Computation Of Lagrangian Coherent Structures Research Articles

Computation of Lagrangian coherent structures from experimental fluid trajectory measurements in a mechanically agitated vessel

• Lagrangian coherent structures are computed from 3D particle tracking measurements. • LCSs characterise the hidden complex topology of turbulent flow in a stirred vessel. • Results show that LCSs are related to mixer configuration and macroscale mixing. In mechanically agitated vessels, bulk flow circulation which plays a leading role in macroscale mixing is controlled by hidden Lagrangian coherent structures (LCSs). We use a numerical finite-time Lyapunov exponent (FTLE) approach, for the first time, to resolve such LCSs. Experimental 3D Lagrangian trajectories obtained from a unique positron emission particle tracking (PEPT) technique are used to drive the FTLE model. By computing forward and backward FTLE fields and extracting repelling and attracting FTLE ridges in various azimuthal planes of the flow, a highly complex flow topology is unravelled which varies significantly with azimuthal position. We demonstrate how LCSs organise and quantify the chaotic behaviour of fluid particle paths that underpin mixing through the exchange of fluid between zones of different kinematics. This new Lagrangian approach driven by unique PEPT data is able to unfold some of the complexities of turbulent flow that are beyond the capability of traditional methods.

A more efficient anisotropic mesh adaptation for the computation of Lagrangian coherent structures

The computation of Lagrangian coherent structures is more and more used in fluid mechanics to determine subtle fluid flow structures. We present in this paper a new adaptive method for the efficient computation of Finite Time Lyapunov Exponent (FTLE) from which the coherent Lagrangian structures can be obtained. This new adaptive method considerably reduces the computational burden without any loss of accuracy on the FTLE field.

Topology of Blood Transport in the Human Left Ventricle by Novel Processing of Doppler Echocardiography

Novel processing of Doppler-echocardiography data was used to study blood transport in the left ventricle (LV) of six patients with dilated cardiomyopathy and six healthy volunteers. Bi-directional velocity field maps in the apical long axis of the LV were reconstructed from color-Doppler echocardiography. Resulting velocity field data were used to perform trajectory-based computation of Lagrangian coherent structures (LCS). LCS were shown to reveal the boundaries of blood injected and ejected from the heart over multiple beats. This enabled qualitative and quantitative assessments of blood transport patterns and residence times in the LV. Quantitative assessments of stasis in the LV are reported, as well as characterization of LV vortex formations from E-wave and A-wave filling.

Integrated computation of finite-time Lyapunov exponent fields during direct numerical simulation of unsteady flows

The computation of Lagrangian coherent structures typically involves post-processing of experimentally or numerically obtained fluid velocity fields to obtain the largest finite-time Lyapunov exponent (FTLE) field. However, this procedure can be tedious for large-scale complex flows of general interest. In this work, an alternative approach involving computation of the FTLE on-the-fly during direct numerical simulation of the full three dimensional Navier-Stokes equations is developed. The implementation relies on Lagrangian particle tracking to compose forward time flow maps, and an Eulerian treatment of the backward time flow map [S. Leung, J. Comput. Phys. 230, 3500-3524 (2011)] coupled with a semi-Lagrangian advection scheme. The flow maps are accurately constructed from a sequence of smaller sub-steps stored on disk [S. Brunton and C. Rowley, Chaos 20, 017503 (2010)], resulting in low CPU and memory requirements to compute evolving FTLE fields. Several examples are presented to demonstrate the capability and parallel scalability of the approach for a variety of two and three dimensional flows.

Geometry of the ergodic quotient reveals coherent structures in flows

Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, and (b) to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold–Beltrami–Childress flow, which was the test-bed for alternative approaches: the Ulam’s approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincaré map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill’s vortex flow, discovering unknown coherent structures in its state space. Finally, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.

A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures

A ridge tracking algorithm for the computation and extraction of Lagrangian coherent structures (LCS) is developed. This algorithm takes advantage of the spatial coherence of LCS by tracking the ridges which form LCS to avoid unnecessary computations away from the ridges. We also make use of the temporal coherence of LCS by approximating the time dependent motion of the LCS with passive tracer particles. To justify this approximation, we provide an estimate of the difference between the motion of the LCS and that of tracer particles which begin on the LCS. In addition to the speedup in computational time, the ridge tracking algorithm uses less memory and results in smaller output files than the standard LCS algorithm. Finally, we apply our ridge tracking algorithm to two test cases, an analytically defined double gyre as well as the more complicated example of the numerical simulation of a swimming jellyfish. In our test cases, we find up to a 35 times speedup when compared with the standard LCS algorithm.

Computational analysis of an aortic valve jet with Lagrangian coherent structures

Important progress has been achieved in recent years in simulating the fluid-structure interaction around cardiac valves. An important step in making these computational tools useful to clinical practice is the development of postprocessing techniques to extract clinically relevant information from these simulations. This work focuses on flow through the aortic valve and illustrates how the computation of Lagrangian coherent structures can be used to improve insight into the transport mechanics of the flow downstream of the valve, toward the goal of aiding clinical decision making and the understanding of pathophysiology.

Lagrangian coherent structures in the planar elliptic restricted three-body problem

This study investigates Lagrangian coherent structures (LCS) in the planar elliptic restricted three-body problem (ER3BP), a generalization of the circular restricted three-body problem (CR3BP) that asks for the motion of a test particle in the presence of two elliptically orbiting point masses. Previous studies demonstrate that an understanding of transport phenomena in the CR3BP, an autonomous dynamical system (when viewed in a rotating frame), can be obtained through analysis of the stable and unstable manifolds of certain periodic solutions to the CR3BP equations of motion. These invariant manifolds form cylindrical tubes within surfaces of constant energy that act as separatrices between orbits with qualitatively different behaviors. The computation of LCS, a technique typically applied to fluid flows to identify transport barriers in the domains of time-dependent velocity fields, provides a convenient means of determining the time-dependent analogues of these invariant manifolds for the ER3BP, whose equations of motion contain an explicit dependency on the independent variable. As a direct application, this study uncovers the contribution of the planet Mercury to the Interplanetary Transport Network, a network of tubes through the solar system that can be exploited for the construction of low-fuel spacecraft mission trajectories.

Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows

This paper develops the theory and computation of Lagrangian Coherent Structures (LCS), which are defined as ridges of Finite-Time Lyapunov Exponent (FTLE) fields. These ridges can be seen as finite-time mixing templates. Such a framework is common in dynamical systems theory for autonomous and time-periodic systems, in which examples of LCS are stable and unstable manifolds of fixed points and periodic orbits. The concepts defined in this paper remain applicable to flows with arbitrary time dependence and, in particular, to flows that are only defined (computed or measured) over a finite interval of time. Previous work has demonstrated the usefulness of FTLE fields and the associated LCSs for revealing the Lagrangian behavior of systems with general time dependence. However, ridges of the FTLE field need not be exactly advected with the flow. The main result of this paper is an estimate for the flux across an LCS, which shows that the flux is small, and in most cases negligible, for well-defined LCSs or those that rotate at a speed comparable to the local Eulerian velocity field, and are computed from FTLE fields with a sufficiently long integration time. Under these hypotheses, the structures represent nearly invariant manifolds even in systems with arbitrary time dependence. The results are illustrated on three examples. The first is a simplified dynamical model of a double-gyre flow. The second is surface current data collected by high-frequency radar stations along the coast of Florida and the third is unsteady separation over an airfoil. In all cases, the existence of LCSs governs the transport and it is verified numerically that the flux of particles through these distinguished lines is indeed negligible.

Tricubic interpolation in three dimensions

The purpose of this paper is to give a local tricubic interpolation scheme in three dimensions that is both C^1 and isotropic. The algorithm is based on a specific 64 × 64 matrix that gives the relationship between the derivatives at the corners of the elements and the coefficients of the tricubic interpolant for this element. In contrast with global interpolation where the interpolated function usually depends on the whole data set, our tricubic local interpolation only uses data in a neighbourhood of an element. We show that the resulting interpolated function and its three first derivatives are continuous if one uses cubic interpolants. The implementation of the interpolator can be downloaded as a static and dynamic library for most platforms. The major difference between this work and current local interpolation schemes is that we do not separate the problem into three one-dimensional problems. This allows for a much easier and accurate computation of higher derivatives of the extrapolated field. Applications to the computation of Lagrangian coherent structures in ocean data are briefly discussed.