Finite-time Lyapunov Exponent Fields Research Articles

Lagrangian Coherent Structures and the Dispersion of Green Algal Bloom in the Yellow and East China Sea

ABSTRACT Son, Y.B.; Kim, Y.H.; Choi, B.-J., and Park, Y.-G., 2016. Lagrangian Coherence Structures and the dispersion of green algal bloom in the Yellow and East China Sea. In: Vila-Concejo, A.; Bruce, E.; Kennedy, D.M., and McCarroll, R.J. (eds.), Proceedings of the 14th International Coastal Symposium (Sydney, Australia). Journal of Coastal Research, Special Issue, No. 75, pp. 1237 - `1241. Coconut Creek (Florida), ISSN 0749-0208. During the summer 2011, the Geostationary Ocean Colour Imager (GOCI) detected patches of the recurrent massive floating green algae in the central part of the Yellow Sea (YS) and East China Sea (ECS). This study utilizes the Lagrangian Coherent Structures (LCSs) analysis to demonstrate the horizontal dispersion pattern of algal bloom patches. This approach is based on the assumption that unstable manifolds such as ridges in the finite-time Lyapunov exponent (FTLE) fields coincide with material transport barriers. To calculate the FTLE fields, flow fields derived from the Regio...

Reynolds number effects on mixing due to topological chaos.

Topological chaos has emerged as a powerful tool to investigate fluid mixing. While this theory can guarantee a lower bound on the stretching rate of certain material lines, it does not indicate what fraction of the fluid actually participates in this minimally mandated mixing. Indeed, the area in which effective mixing takes place depends on physical parameters such as the Reynolds number. To help clarify this dependency, we numerically simulate the effects of a batch stirring device on a 2D incompressible Newtonian fluid in the laminar regime. In particular, we calculate the finite time Lyapunov exponent (FTLE) field for three different stirring protocols, one topologically complex (pseudo-Anosov) and two simple (finite-order), over a range of viscosities. After extracting appropriate measures indicative of both the amount of mixing and the area of effective mixing from the FTLE field, we see a clearly defined Reynolds number range in which the relative efficacy of the pseudo-Anosov protocol over the finite-order protocols justifies the application of topological chaos. More unexpectedly, we see that while the measures of effective mixing area increase with increasing Reynolds number for the finite-order protocols, they actually exhibit non-monotonic behavior for the pseudo-Anosov protocol.

Investigation of a wake formation for flow over a cylinder using Lagrangian coherent structures

In this study, flow over a cylinder was investigated to understand the physics of fluid motion in the wake region. Eulerian velocity field attained from computational fluid dynamics model while Lagrangian information obtained from finite time Lyapunov exponent fields illustrated how a vortex formation takes place once it is detached from the cylinder.

Lagrangian coherent structure identification using a Voronoi tessellation-based networking algorithm

The quantification of Lagrangian coherent structures (LCS) has been investigated using an algorithm based on the tesselation of unstructured data points. The applicability of the algorithm in resolving an LCS was tested using a synthetically generated unsteady double-gyre flow and experimentally in a nominally two-dimensional free shear flow. The effects of two parameters on LCS identification were studied: the threshold track length used to quantify the LCS and resulting effective seeding density upon applying the threshold. At lower threshold track lengths, increases in the threshold track length resulted in finite-time Lyapunov exponent (FTLE) field convergence towards the expected LCS ridge of the double-gyre flow field at several effective seeding densities. However, at higher track lengths, further increases to the threshold track length failed to improve convergence at low effective seeding densities. The FTLE of the experimental data set was well-resolved using moderate threshold track lengths that achieved field convergence but maintained a sufficiently high seeding density. In contrast, the use of lower or higher track lengths produced an FTLE field characterized by an incoherent LCS ridge. From the analytical and experimental results, recommendations are made for future experiments for identifying LCS directly from unstructured data.

Lagrangian-based investigation of the transient flow structures around a pitching hydrofoil

The objective of this paper is to address the transient flow structures around a pitching hydrofoil by combining physical and numerical studies. In order to predict the dynamic behavior of the flow structure effectively, the Lagrangian coherent structures (LCS) defined by the ridges of the finite-time Lyapunov exponent (FTLE) are utilized under the framework of Navier–Stokes flow computations. In the numerical simulations, the $$k\hbox {-}\omega $$ shear stress transport (SST) turbulence model, coupled with a two-equation $$\gamma {-Re}_\theta $$ transition model, is used for the turbulence closure. Results are presented for a NACA66 hydrofoil undergoing slowly and rapidly pitching motions from $$0^{\circ }$$ to $$15^{\circ }$$ then back to $$0^{\circ }$$ at a moderate Reynolds number $$Re=7.5\times 10^{5}$$ . The results reveal that the transient flow structures can be observed by the LCS method. For the slowly pitching case, it consists of five stages: quasi-steady and laminar, transition from laminar to turbulent, vortex development, large-scale vortex shedding, and reverting to laminar. The observation of LCS and Lagrangian particle tracers elucidates that the trailing edge vortex is nearly attached and stable during the vortex development stage and the interaction between the leading and trailing edge vortex caused by the adverse pressure gradient forces the vortexes to shed downstream during the large-scale vortex shedding stage, which corresponds to obvious fluctuations of the hydrodynamic response. For the rapidly pitching case, the inflection is hardly to be observed and the stall is delayed. The vortex formation, interaction, and shedding occurred once instead of being repeated three times, which is responsible for just one fluctuation in the hydrodynamic characteristics. The numerical results also show that the FTLE field has the potential to identify the transient flows, and the LCS can represent the divergence extent of infinite neighboring particles and capture the interface of the vortex region. In this paper, the transient flow structures around a pitching hydrofoil are studied with the FTLE and the LCS. The observation of LCS and Lagrangian particle tracers elucidates the vortex development and interactions. The numerical results also show that the FTLE field has the potential to identify the transient flows, and the ridges of FTLE, LCS, can represent the divergence extent of infinite neighboring particles and capture the interface of the vortex region.

Strong transient effects of the flow around a harmonically plunging NACA0012 airfoil at low Reynolds numbers

The flow pattern around a NACA0012 airfoil undergoing harmonic plunging motion corresponding to the deflected wake phenomenon reported by Jones and Platzer (Exp Fluids 46:799–810, 2009) is investigated in detail using direct numerical simulations. An arbitrary Lagrangian–Eulerian formulation based on an unstructured side-centered finite volume method is utilized in order to solve the incompressible unsteady Navier–Stokes equations. The Reynolds number is chosen to be 252, and the reduced frequency of plunging motion (k = 2π fc/U ∞) and the plunge amplitude non-dimensionalized with respect to chord are set to be 12.3 and 0.12, respectively, as in the experimental study of Jones and Platzer (2009). The present numerical simulations reveal a highly persistent transient effect, and it takes two orders of magnitude larger duration than the heave period to reach the time-periodic state. In addition, the three-dimensional simulation reveals that the flow field is three-dimensional for the parameters used herein. The calculation reproduces the deflected wake and shows a good agreement with the experimental wake pattern. The instantaneous vorticity contours, finite-time Lyapunov exponent fields and particle traces are presented along with the aerodynamic parameters including the lift and thrust coefficients.

Refining finite-time Lyapunov exponent ridges and the challenges of classifying them.

While more rigorous and sophisticated methods for identifying Lagrangian based coherent structures exist, the finite-time Lyapunov exponent (FTLE) field remains a straightforward and popular method for gaining some insight into transport by complex, time-dependent two-dimensional flows. In light of its enduring appeal, and in support of good practice, we begin by investigating the effects of discretization and noise on two numerical approaches for calculating the FTLE field. A practical method to extract and refine FTLE ridges in two-dimensional flows, which builds on previous methods, is then presented. Seeking to better ascertain the role of a FTLE ridge in flow transport, we adapt an existing classification scheme and provide a thorough treatment of the challenges of classifying the types of deformation represented by a FTLE ridge. As a practical demonstration, the methods are applied to an ocean surface velocity field data set generated by a numerical model.

Attracting Lagrangian coherent structures on Riemannian manifolds

It is a wide-spread convention to identify repelling Lagrangian Coherent Structures (LCSs) with ridges of the forward finite-time Lyapunov exponent (FTLE) field and to identify attracting LCSs with ridges of the backward FTLE. However, we show that, in two-dimensional incompressible flows, also attracting LCSs appear as ridges of the forward FTLE field. This raises the issue of the characterization of attracting LCSs using a forward finite-time Lyapunov analysis. To this end, we extend recent results regarding the relationship between forward and backward maximal and minimal FTLEs, to both the whole finite-time Lyapunov spectrum and to stretch directions. This is accomplished by considering the singular value decomposition (SVD) of the linearized flow map. By virtue of geometrical insights from the SVD, we provide characterizations of attracting LCSs in forward time for two geometric approaches to hyperbolic LCSs. We apply these results to the attracting FTLE ridge of the incompressible saddle flow.

Detection of magnetic barriers in a chaotic domain: first application of finite time Lyapunov exponent method to a magnetic confinement configuration

Magnetic field lines embedded in a plasma confinement system are often characterized by a chaotic motion. This weakens the confinement properties of any magnetic configuration. However, even in case of chaotic domains, magnetic barriers can emerge and limit the field line motion itself. In the context of the numerical simulation of a Reversed-Field Pinch configuration a new magnetic topology analysis, borrowed from previous fluid dynamic studies, is discussed. This methodology relies on the behavior of the Finite Time Lyapunov Exponent (FTLE) associated with the magnetic field. By referring to a previous work in which the magnetic field is given in terms of analytical function (Borgogno et al 2011 Phys. Plasmas 18 102307) the FTLE field shows the presence of ridges, special gradient lines normal to the direction of minimum curvature, forming magnetic barriers. These ridges can be recognized as Lagrangian Coherent Structures (LCSs) for the system, actually opposing the penetration of magnetic field lines across them. In this article a more general numerical scheme for the detection of the LCSs has been adopted that allows analysis of realistic cases in which the magnetic fields are numerically known on a discrete mesh. After a validation test performed on the analytical case, a first application to a numerical magnetohydrodynamics simulation of the RFP, characterized by a broad chaotic region, has been performed. A strong magnetic barrier has been observed that effectively limits the field lines motion inside the chaotic sea.

Experimental studies of coherent structures in an advection-reaction-diffusion system.

We present experimental studies of reaction front propagation in a single vortex flow with an imposed external wind. The fronts are produced by the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. The flow is generated using an electromagnetic forcing technique: an almost-radial electrical current interacts with a magnetic field from a magnet below the fluid layer to produce the vortex. The magnet is mounted on crossed translation stages allowing for movement of the vortex through the flow. Reaction fronts triggered in or in front of the moving vortex form persistent structures that are seen experimentally for time-independent (constant motion), time-periodic, and time-aperiodic flows. These results are examined with the use of burning invariant manifolds that act as one-way barriers to front motion in the flows. We also explore the usefulness of finite-time Lyapunov exponent fields as an instrument for analyzing front propagation behavior in a fluid flow.

DG-FTLE: Lagrangian coherent structures with high-order discontinuous-Galerkin methods

We present an algorithm for the computation of finite-time Lyapunov exponent (FTLE) fields using discontinuous-Galerkin (dG) methods in two dimensions. The algorithm is designed to compute FTLE fields simultaneously with the time integration of dG-based flow solvers of conservation laws. Fluid tracers are initialized at Gauss–Lobatto quadrature nodes within an element. The deformation gradient tensor, defined by the deformation of the Lagrangian flow map in finite time, is determined per element with high-order dG operators. Multiple flow maps are constructed from a particle trace that is released at a single initial time by mapping and interpolating the flow map formed by the locations of the fluid tracers after finite time integration to a unit square master element and to the quadrature nodes within the element, respectively. The interpolated flow maps are used to compute forward-time and backward-time FTLE fields at several times using dG operators. For a large finite integration time, the interpolation is increasingly poorly conditioned because of the excessive subdomain deformation. The conditioning can be used in addition to the FTLE to quantify the deformation of the flow field and identify subdomains with material lines that define Lagrangian coherent structures. The algorithm is tested on three benchmarks: an analytical spatially periodic gyre flow, a vortex advected by a uniform inviscid flow, and the viscous flow around a square cylinder. In these cases, the algorithm is shown to have spectral convergence.

Temperature and velocity measurements of a rising thermal plume

The three-dimensional velocity and temperature fields surrounding an isolated thermal plume in a fluid with temperature-dependent viscosity are measured using Particle-Image Velocimetry and thermochromatic liquid crystals, respectively. The experimental conditions are relevant to a plume rising through the mantle. It is shown that while the velocity and the isotherm surrounding the plume can be used to visualize the plume, they do not reveal the finer details of its structure. However, by computing the Finite-Time Lyapunov Exponent fields from the velocity measurements, the material lines of the flow can be found, which clearly identify the shape of the plume head and characterize the behavior of the flow along the plume stem. It is shown that the vast majority of the material in the plume head has undergone significant stretching and originates from a wide region very low in the fluid domain, which is proposed as a contributing factor to the small-scale isotopic variability observed in ocean-island basalt regions. Lastly, the Finite-Time Lyapunov Exponent fields are used to calculate the steady state rise velocity of the thermal plume, which is found to scale linearly with the Rayleigh number, in contrast to some previous work. The possible cause and the significance of these conflicting results are discussed, and it is suggested that the scaling relationship may be affected by the temperature-dependence of the fluid viscosity in the current work.

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.

3D Finite Time Lyapunov Exponents in a left ventricle laboratory model

Finite Time Lyapunov Exponents (FTLEs) are a powerful means to infer characteristic features of the flow that cannot be revealed by other Eulerian criteria. Recently FTLEs are becoming popular also in the medical context, for instance in the analysis of vascular flow measured by means of Magnetic Resonance Imaging. However, many of the FTLE experimental works are based only on two-dimensional velocity fields, moreover those computed on in-vivo data cannot be obtained under controlled and repeatable conditions. Here we present the 3D FTLE evolution inside a Left Ventricle (LV) laboratory model mimicking physiological human conditions. The investigation of FTLE fields highlights distinctive features of the cardiac flow and gives an insight on the physiological development of the Lagrangian Coherent Structures (LCS) that optimize the LV refill.

Experimental determination of three-dimensional finite-time Lyapunov exponents in multi-component flows

We present an experimental approach for esti- mating finite-time Lyapunov exponent fields (FTLEs) in three-dimensional multi-component or multi-phase flows. From time-resolved sequences of particle images, we directly compute the flow map and coherent structures, while avoiding and outperforming the computationally costly numerical integration. Performing this operation independently on each flow component enables the deter- mination of three-dimensional Lagrangian coherent struc- tures (LCSs) without any bias from the other components. The locations of concurrent LCSs for different flow ele- ments (e.g., passive tracers, inertial particles, bubbles, or active particles) can provide new insight into the inter- penetrating FTLE structure in complex flows.

Local-heterogeneous responses and transient dynamics of cage breaking and formation in colloidal fluids.

Quantifying the interactions in dense colloidal fluids requires a properly designed order parameter. We present a modified bond-orientational order parameter, ψ̄6, to avoid problems of the original definition of bond-orientational order parameter. The original bond-orientational order parameter can change discontinuously in time but our modified order parameter is free from the discontinuity and, thus, it is a suitable measure to quantify the dynamics of the bond-orientational ordering of the local surroundings. Here we analyze ψ̄6 in a dense driven monodisperse quasi-two-dimensional colloidal fluids where a single particle is optically trapped at the center. The perturbation by the trapped and driven particle alters the structure and dynamics of the neighboring particles. This perturbation disturbs the flow and causes spatial and temporal distortion of the bond-orientational configuration surrounding each particle. We investigate spatio-temporal behavior of ψ̄6 by a Wavelet transform that provides a time-frequency representation of the time series of ψ̄6. It is found that particles that have high power in frequencies corresponding to the inverse of the timescale of perturbation undergo distortions of their packing configurations that result in cage breaking and formation dynamics. To gain insight into the dynamic structure of cage breaking and formation of bond-orientational ordering, we compare the cage breaking and formation dynamics with the underlying dynamical structure identified by Lagrangian Coherent Structures (LCSs) estimated from the finite-time Lyapunov exponent (FTLE) field. The LCSs are moving separatrices that effectively divide the flow into distinct regions with different dynamical behavior. It is shown that the spatial distribution of the FTLE field and the power of particles in the wavelet transform have positive correlation, implying that LCSs provide a dynamic structure that dominates the dynamics of cage breaking and formation of the colloidal fluids.

On the use of the finite-time Lyapunov exponent to reveal complex flow physics in the wake of a mechanical valve

The finite-time Lyapunov exponent (FTLE) is a popular tool to extract characteristic features of flows that cannot be revealed by other criteria. However, even if the computational cost of computing particle trajectories in space and time has been reduced and optimized in a considerable number of works, the main challenge probably consists in increasing the spatial resolution of Lagrangian coherent structures locally, i.e., where the FTLE field reaches its maximum values. On the other hand, most of experimental data are obtained in planes so that the FTLE field is computed without the out-of-plane particle movements. To investigate which physics the FTLE can capture in flows that are highly three dimensional, the criterion is computed from high-speed stereoscopic particle image velocimetry measurements of the pulsatile flow that develops behind a bi-leaflet mechanical heart valve. The similitude is based on the Womersley number, and experiments are performed for a lower Reynolds number than the physiologic value to obtain sufficiently resolved data in space and time. It is found that the vortex shedding is well captured and that its development can be decomposed into four successive phases. The longest phase occurs near the peak flow rate and exhibits a break of symmetry similar to the one appearing in the wakes of two side-by-side cylinders in the regime when the separation between the cylinders is of the order of their diameter. Specifically, a vortex street with alternating vortex sheddings is observed in a narrow wake behind one of the leaflets, whereas single large vortices develop inside a wide wake downstream of the other leaflet. It appears that these patterns are difficult or even impossible to discern with classical Eulerian vortex identification techniques. The Strouhal numbers of vortex-shedding frequencies, obtained from continuous wavelet transforms and based on the apparent height of the leaflets, are also close to those found in the flow behind two cylinders. By invoking the Taylor hypothesis, an approximate three-dimensional reconstruction of the flow can be obtained and a three-dimensional FTLE field is deduced, which provides a very detailed view of the vortex structures that form and develop in the wakes of the leaflets.

Lagrangian methods for water transport processes in a long-narrow bay- Xiangshan Bay, China

A better understanding of water transport processes is highly desirable for the exploitation of the ocean resources and the protection of the ocean ecological system. In this paper, the Lagrangian methods are used to study the water transport processes in Xiangshan Bay in China, a typical semi-closed and narrow-shaped bay with complex coastline and topography. A high-resolution 3-D hydrodynamic model is developed and verified, and the results from the model agree well with the field data. Based on the hydrodynamic model, the Lagrangian residual current is computed by using the particle tracking method. A concept based on the dynamical systems theory, the Lagrangian coherent structures (LCSs), is introduced to uncover the underlying structures which act as the transport barriers in the flow. The finite-time Lyapunov exponent (FTLE) fields are computed from the hydrodynamic model results to extract the LCSs. The results indicate that the LCSs act as the internal structures of the Lagrangian residual current and the Lagrangian residual current displays the residual current speed and direction of different water regimes separated by the LCSs. The water masses with different transport characteristics can be identified and their exchange ability with other water masses can be estimated by combining the Lagrangian particle tracking with the LCSs methods. The comprehensive applications of these Lagrangian methods reveal the underlying structures and the inhomogeneous characteristics of the water transport in Xiangshan Bay.

Lagrangian coherent structures analysis of gas-liquid flow in a bubble column

The objective of this paper is to apply a new identifying method to investigating the gas-liquid two-phase flow behaviors in a bubble column with air injected into water. In the numerical simulations, the standard k-ɛ turbulence model is employed to describe the turbulence phenomenon occurring in the continuous fluid. The Finite-Time Lyapunov Exponent (FTLE) and Lagrangian Coherent Structures (LCS) are applied to analyze the vortex structures in multiphase flow. Reasonable agreements are obtained between the numerical and experimental data. The numerical results show that the evolution of gas-liquid in the column includes initial and periodical developing stages. During the initial stage, the bubble hose is forming and extending along the vertical direction with the vortex structures formed symmetrically. During the periodical developing stage, the bubble hose starts to oscillate periodically, and the vortexes move along the bubble hose to the bottom of column alternately. Compared to the Euler-system-based identification criterion of a vortex, the FTLE field presents the boundary of a vortex without any threshold defined and the LCS represents the divergence extent of infinite neighboring particles. During the initial stage, the interfaces between the forward and backward flows are highlighted by the LCS. As for the periodical developing stage, the LCS curls near the vortex centers, providing a method of analyzing a flow field from a dynamical system perspective.

Applications of lagrangian coherent structures to expression of invariant manifolds in astrodynamics

This paper investigates the relationship between invariant manifold and Lagrangian coherent structure (LCS) in dynamical systems. LCS is defined as the ridge of finite-time Lyapunov exponent (FTLE) field, and is proving to be excellent platform for studies of stable and unstable manifold in flows with arbitrary time dependence. In this study, the LCS tool is applied to autonomous systems, simple pendulum and planar circular restricted three-body problem (PCR3BP), and also non-autonomous ones, double-gyre flow and bicircular problem (BCP). A comparison between LCS and invariant manifold is presented.