Finite-time Lyapunov Exponent Fields Research Articles

Hyperbolicity in temperature and flow fields during the formation of a Loop Current ring

Abstract. Loop Current rings (LCRs) are among the largest mesoscale eddies in the world ocean. They arise when bulges formed by the Loop Current in the Gulf of Mexico close off. The LCR formation process may take several weeks, and there may be several separations and reattachments before final separation occurs. It is well established that this period is characterized by a persistent saddle point in the sea surface height field, as seen in both model and satellite data. We present here a detailed study of this saddle region during the formation of Eddy Franklin in 2010, over multiple days and at several depths. Using a data-assimilating Gulf of Mexico implementation of the HYbrid Coordinate Ocean Model (HYCOM), we compare the vertical structure of the currents and temperature fields on 5 and 10 June 2010. Finite-time Lyapunov exponents (FTLE) are computed from the surface down to 200 m to estimate the location of relevant transport barriers. Several new features of the saddle region associated with LCR formation are revealed: the ridges in the FTLE fields are shown to be excellent surrogates for the manifolds delineating the material flow structures with only slight degradation at depth. The intersection of the ridges representing stable and unstable manifolds drops nearly vertically through the water column at both times; remarkably, the material boundary shapes are maintained even as they are advected. Moreover, velocity stagnation points and saddle points in the temperature field are consistently found near the intersections at all depths, and their geographic positions are also nearly constant with depth.

Effect of exercise on patient specific abdominal aortic aneurysm flow topology and mixing

Computational fluid dynamics modeling was used to investigate changes in blood transport topology between rest and exercise conditions in five patient-specific abdominal aortic aneurysm models. MRI was used to provide the vascular anatomy and necessary boundary conditions for simulating blood velocity and pressure fields inside each model. Finite-time Lyapunov exponent fields and associated Lagrangian coherent structures were computed from blood velocity data and were used to compare features of the transport topology between rest and exercise both mechanistically and qualitatively. A mix-norm and mix-variance measure based on fresh blood distribution throughout the aneurysm over time were implemented to quantitatively compare mixing between rest and exercise. Exercise conditions resulted in higher and more uniform mixing and reduced the overall residence time in all aneurysms. Separated regions of recirculating flow were commonly observed in rest, and these regions were either reduced or removed by attached and unidirectional flow during exercise, or replaced with regional chaotic and transiently turbulent mixing, or persisted and even extended during exercise. The main factor that dictated the change in flow topology from rest to exercise was the behavior of the jet of blood penetrating into the aneurysm during systole.

Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: An application and error analysis

The language of Lagrangian coherent structures (LCSs) provides a new means for studying transport and mixing of passive particles advected by an atmospheric flow field. Recent observations suggest that LCSs govern the large-scale atmospheric motion of airborne microorganisms, paving the way for more efficient models and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans. In addition, having reliable predictions of the timing of hyperbolic LCSs may contribute to improved aerobiological sampling of microorganisms with unmanned aerial vehicles and LCS-based early warning systems. Chaotic atmospheric dynamics lead to unavoidable forecasting errors in the wind velocity field, which compounds errors in LCS forecasting. In this study, we reveal the cumulative effects of errors of (short-term) wind field forecasts on the finite-time Lyapunov exponent (FTLE) fields and the associated LCSs when realistic forecast plans impose certain limits on the forecasting parameters. Objectives of this paper are to (a) quantify the accuracy of prediction of FTLE–LCS features and (b) determine the sensitivity of such predictions to forecasting parameters. Results indicate that forecasts of attracting LCSs exhibit less divergence from the archive-based LCSs than the repelling features. This result is important since attracting LCSs are the backbone of long-lived features in moving fluids. We also show under what circumstances one can trust the forecast results if one merely wants to know if an LCS passed over a region and does not need to precisely know the passage time.

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.

A Time-Dependent Vector Field Topology Based on Streak Surfaces

It was shown recently how the 2D vector field topology concept, directly applicable to stationary vector fields only, can be generalized to time-dependent vector fields by replacing the role of stream lines by streak lines. The present paper extends this concept to 3D vector fields. In traditional 3D vector field topology separatrices can be obtained by integrating stream lines from 0D seeds corresponding to critical points. We show that in our new concept, in contrast, 1D seeding constructs are required for computing streak-based separatrices. In analogy to the 2D generalization we show that invariant manifolds can be obtained by seeding streak surfaces along distinguished path surfaces emanating from intersection curves between codimension-1 ridges in the forward and reverse finite-time Lyapunov exponent (FTLE) fields. These path surfaces represent a time-dependent generalization of critical points and convey further structure in time-dependent topology of vector fields. Compared to the traditional approach based on FTLE ridges, the resulting streak manifolds ease the analysis of Lagrangian coherent structures (LCS) with respect to visual quality and computational cost, especially when time series of LCS are computed. We exemplify validity and utility of the new approach using both synthetic examples and computational fluid dynamics results.

Formation process of the vortex ring generated by an impulsively started circular disc

AbstractWe present an experimental study on the axisymmetric vortex ring generated by a thin circular disc. The velocity and vorticity fields are measured by digital particle image velocimetry (DPIV). The finite-time Lyapunov exponent fields and the Lagrangian coherent structures (LCSs) of the vortex flow are computed in order to analyse the transport of the fluid during its formation and identify the boundary of the vortex ring. The volume, circulation and energy of the vortex ring are calculated. It is found that the formation of the vortex ring basically includes three phases: a rapid growth phase, a stable growth phase and a non-axisymmetric phase. In the rapid growth phase (dimensionless time $0\lt {T}_{n} \lt 0. 2$) during which Taylor’s inviscid estimation is valid, the circulation of the vortex ring grows and the translational velocity of the vortex ring decreases. In the stable growth phase ($0. 2\lt {T}_{n} \lt 4$), the growth rate of the circulation decreases gradually. In the non-axisymmetric phase (${T}_{n} \gt 4$), the ring loses axisymmetry due to instability. Compared with the vortex ring generated by the laminar flow from an orifice, the one generated by a circular disc always moves with the disc, and the entrained fluid decreases and the saturated circulation increases. The temporal impulse exerted by the moving disc on the fluid is estimated by DPIV measurements and is calculated using the direct momentum conservation method. The momentum of the control volume enclosing the LCS is found to occupy 64–68 % of the entire impulse exerted by the disc on the fluid.

Lagrangian analysis of hemodynamics data from FSI simulation.

We present the computation of lagrangian-based flow characterization measures for time-dependent, deformable-wall, finite-element blood flow simulations. Applicability of the algorithm is demonstrated in a fluid-structure interaction simulation of blood flow through a total cavopulmonary connection (Fontan procedure), and results are compared with a rigid-vessel simulation. Specifically, we report on several important lagrangian-based measures including flow distributions, finite-time Lyapunov exponent fields, particle residence time, and exposure time calculations. Overall, strong similarity in lagrangian measures of the flow between deformable and rigid-vessel models was observed.

Comment on “Second derivative ridges are straight lines and the implications for computing Lagrangian Coherent Structures, Physica D 2012.05.006”

The finite-time Lyapunov exponent (FTLE) has become a standard tool for analyzing unsteady flow phenomena, partly since its ridges can be interpreted as Lagrangian Coherent Structures (LCS). While there are several definitions for ridges, a particular one called second derivative ridges has been introduced in the context of LCS, but subsequently received criticism from several researchers for being over-constrained. Among the critics are Norgard and Bremer (2012) [3], who suggest furthermore that the widely used definition of height ridges was a part of the definition of second derivative ridges, and that topological separatrices were ill-suited for describing ridges. We show that (a) the definitions of height ridges and second derivative ridges are not directly related, and (b) there is an interdisciplinary consensus throughout the literature that topological separatrices describe ridges. Furthermore, we provide pointers to practically feasible and numerically stable ridge extraction schemes for FTLE fields.

Flow and mixing performance in helical ribbon mixers

This paper explores the effect of impeller design on the chaotic mixing in a helical ribbon mixer. Numerical simulations using the smoothed particle hydrodynamics (SPH) method are used to solve for the fluid flow around the moving impellor. The mixing flow is visualised by extracting the hyperbolic Lagrangian coherent structures (LCS) of the flow from the finite-time Lyapunov exponent (FTLE) field, which provides qualitative information about the topology of the mixing structures and highlights the major barriers to transport. This is combined with a mixing measure that calculates the degree of mixing between different regions. A single helical ribbon mixer (SHR) was found to create an axially symmetric circulation cell that moves fluid down the outside of the tank and upward near the centre. Smaller circulation cells near the inside edge of the ribbon superimpose a chaotic mixing flow over this primary flow, which stretches and folds fluid volumes around the ribbon according to the shape of the LCSs. The horizontal struts supporting the ribbon generate strong circumferential mixing, but this is only within a narrow horizontal plane with thickness comparable with the strut diameter. The addition of an extra ribbon, 180 degrees out of phase (DHR) doubles the number of smaller circulation cells while decreasing their efficiency in mixing the fluid between the inner and outer regions of the tank. Overall the mixing rate is improved over the other ribbon geometries. The addition of a central screw (CSR) creates a zone of low mixing immediately surrounding the screw. The mixing processes surrounding the helical ribbon are qualitatively identical to the SHR, but the overall mixing rate is slightly increased.

Characterization of the transport topology in patient-specific abdominal aortic aneurysm models

Abdominal aortic aneurysm (AAA) is characterized by disturbed blood flow patterns that are hypothesized to contribute to disease progression. The transport topology in six patient-specific abdominal aortic aneurysms was studied. Velocity data were obtained by image-based computational fluid dynamics modeling, with magnetic resonance imaging providing the necessary simulation parameters. Finite-time Lyapunov exponent (FTLE) fields were computed from the velocity data, and used to identify Lagrangian coherent structures (LCS). The combination of FTLE fields and LCS was used to characterize topological flow features such as separation zones, vortex transport, mixing regions, and flow impingement. These measures offer a novel perspective into AAA flow. It was observed that all aneurysms exhibited coherent vortex formation at the proximal segment of the aneurysm. The evolution of the systolic vortex strongly influences the flow topology in the aneurysm. It was difficult to predict the vortex dynamics from the aneurysm morphology, motivating the application of image-based flow modeling.

Interactive Computation and Rendering of Finite-Time Lyapunov Exponent Fields

In this paper, we present a novel technique that allows for the coupled computation and visualization of salient flow structures at interactive frame rates. Our approach is built upon a hierarchical representation of the Finite-time Lyapunov Exponent (FTLE) field, which is adaptively sampled and rendered to meet the need of the current visual setting. The performance of our method allows the user to explore large and complex data sets across scales and to inspect their features at arbitrary resolution. The paper discusses an efficient implementation of this strategy on graphics hardware and provides results for an analytical flow and several CFD simulation data sets.

Anisotropic mesh adaptation on Lagrangian Coherent Structures

The finite-time Lyapunov exponent (FTLE) is extensively used as a criterion to reveal fluid flow structures, including unsteady separation/attachment surfaces and vortices, in laminar and turbulent flows. However, for large and complex problems, flow structure identification demands computational methodologies that are more accurate and effective. With this objective in mind, we propose a new set of ordinary differential equations to compute the flow map, along with its first (gradient) and second order (Hessian) spatial derivatives. We show empirically that the gradient of the flow map computed in this way improves the pointwise accuracy of the FTLE field. Furthermore, the Hessian allows for simple interpolation error estimation of the flow map, and the construction of a continuous optimal and multiscale Lp metric. The Lagrangian particles, or nodes, are then iteratively adapted on the flow structures revealed by this metric. Typically, the L1 norm provides meshes best suited to capturing small scale structures, while the L∞ norm provides meshes optimized to capture large scale structures. This means that the mesh density near large scale structures will be greater with the L∞ norm than with the L1 norm for the same mesh complexity, which is why we chose this technique for this paper. We use it to optimize the mesh in the vicinity of LCS. It is found that Lagrangian Coherent Structures are best revealed with the minimum number of vertices with the L∞ metric.

Second derivative ridges are straight lines and the implications for computing Lagrangian Coherent Structures

Lagrangian Coherent Structures (LCS) have become a cornerstone of the analysis of unsteady fluid flow. Intuitively, LCS are material boundaries that locally maximize attraction, repulsion, or shearing. Based on a number of examples and numerical experiments it has been suggested that LCS are indicated as ridges of the Finite-Time Lyapunov Exponent (FTLE) field. This commonly accepted intuition has led to a school of thought that defines LCS as so called second derivative ridges in the FTLE field. This viewpoint has been supported by a proof that such ridges indeed admit (almost) no cross flow. Recently analytic counter examples have been discovered that demonstrate that FTLE ridges in general may produce both false positive and false negative LCS classifications. Furthermore, without additional restrictions, second derivative ridges in particular may admit a large amount of flux. Here we provide additional evidence that second derivative ridges are ill suited to define LCS (or ridges in general) by showing that for any smooth scalar field second derivative ridges are necessarily straight lines.

Zonal Jets as Meridional Transport Barriers in the Subtropical and Polar Lower Stratosphere

Abstract Applications of recent results from dynamical systems theory to the study of transport and mixing in incompressible two-dimensional flows lead to the expectation that, independent of the background potential vorticity (PV) distribution, weakly perturbed zonal jets are associated with barriers that inhibit meridional transport. Here the authors provide evidence in support of this expectation based on the analysis of isentropic winds in the lower stratosphere as produced by the Canadian Middle Atmosphere Model (CMAM), a comprehensive general circulation model. Specifically, barriers to meridional transport are found to be associated with the (eastward) austral polar night jet, for which the meridional gradient of background PV is large, and also for the (westward) boreal summer subtropical jet, for which the background PV gradient is quite small. The identification of the meridional transport barriers is based on the computation of finite-time Lyapunov exponents (FTLEs), which characterize the amount of stretching about fluid particle trajectories. Being composed of regular fluid particle trajectories lying on invariant tori, the meridional transport barriers are identified with topologically circular, local minimizing curves or trenches of the backward-plus-forward FTLE field. Results from explicit passive tracer advection experiments and flux computations are also presented, which are consistent with results inferred using the FTLE diagnostic.

The role of jets as transport barriers in the Earth's stratosphere

Theoretical results relating to Kolmogorov-Arnold-Moser (KAM) theory, particularly those dealing with the stability of degenerate one-degree-of-freedom Hamiltonian systems under a time-quasiperiodic perturbation, have led to the expectation that, independent of the background Potential Vorticity (PV) distribution, associated with zonal jets should be barriers which inhibit meridional transport. In a recently submitted paper (Beron-Vera et al., 2011) the aforementioned expectation is confirmed based on the analysis of isentropic winds in the lower stratosphere as produced by a comprehensive general circulation model. Concretely, barriers for meridional transport are found to be associated with the (eastward) austral polar night jet, for which the meridional gradient of background PV is very steep, and also for the (westward) boreal summer subtropical jet, for which the background PV is rather uniform. The identification of the meridional transport barriers is based on the computation of Finite-Time Lyapunov Exponents (FTLE), which measures the amount of stretching about fluid particle trajectories. The meridional transport barriers, which are formed by fluid particle trajectories lying on invariant tori, are identified with topologically circular locally minimizing curves or trenches of the backward-plus-forward FTLE field. Results from explicit passive tracer advection experiments and flux computations are also presented, which confirm those inferred using the FTLE diagnostic. In this paper I present results based on the analysis of analytically prescribed stratospheric model winds, which complement the results presented in Beron-Vera et al. (2011) and further allow one to better assess the scope of the FTLE diagnostic.

GPU and APU computations of Finite Time Lyapunov Exponent fields

We present GPU and APU accelerated computations of Finite-Time Lyapunov Exponent (FTLE) fields. The calculation of FTLEs is a computationally intensive process, as in order to obtain the sharp ridges associated with the Lagrangian Coherent Structures an extensive resampling of the flow field is required. The computational performance of this resampling is limited by the memory bandwidth of the underlying computer architecture. The present technique harnesses data-parallel execution of many-core architectures and relies on fast and accurate evaluations of moment conserving functions for the mesh to particle interpolations. We demonstrate how the computation of FTLEs can be efficiently performed on a GPU and on an APU through OpenCL and we report over one order of magnitude improvements over multi-threaded executions in FTLE computations of bluff body flows.

Lagrangian transport properties of pulmonary interfacial flows

Disease states characterized by airway fluid occlusion and pulmonary surfactant insufficiency, such as respiratory distress syndrome, have a high mortality rate. Understanding the mechanics of airway reopening, particularly involving surfactant transport, may provide an avenue to increase patient survival via optimized mechanical ventilation waveforms. We model the occluded airway as a liquid-filled rigid tube with the fluid phase displaced by a finger of air that propagates with both mean and sinusoidal velocity components. Finite-time Lyapunov exponent (FTLE) fields are employed to analyse the convective transport characteristics, taking note of Lagrangian coherent structures (LCSs) and their effects on transport. The Lagrangian perspective of these techniques reveals flow characteristics that are not readily apparent by observing Eulerian measures. These analysis techniques are applied to surfactant-free velocity fields determined computationally, with the boundary element method, and measured experimentally with micro particle image velocimetry (μ-PIV). We find that the LCS divides the fluid into two regimes, one advected upstream (into the thin residual film) and the other downstream ahead of the advancing bubble. At higher oscillatory frequencies particles originating immediately inside the LCS experience long residence times at the air-liquid interface, which may be conducive to surfactant transport. At high frequencies a well-mixed attractor region is identified; this volume of fluid cyclically travels along the interface and into the bulk fluid. The Lagrangian analysis is applied to velocity data measured with 0.01 mg ml(-1) of the clinical pulmonary surfactant Infasurf in the bulk fluid, demonstrating flow field modifications with respect to the surfactant-free system that were not visible in the Eulerian frame.

Barriers in the transition to global chaos in collisionless magnetic reconnection. I. Ridges of the finite time Lyapunov exponent field

The transitional phase from local to global chaos in the magnetic field of a reconnecting current layer is investigated. Regions where the magnetic field is stochastic exist next to regions where the field is more regular. In regions between stochastic layers and between a stochastic layer and an island structure, the field of the finite time Lyapunov exponent (FTLE) shows a structure with ridges. These ridges, which are special gradient lines that are transverse to the direction of minimum curvature of this field, are approximate Lagrangian coherent structures (LCS) that act as barriers for the transport of field lines.

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

Through a combined computational–experimental study of flow in a slowly rotating quasi-two-dimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed “streamline jumping.” This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., “strange” eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincaré section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.

Lagrangian coherent structures are associated with fluctuations in airborne microbial populations

Many microorganisms are advected in the lower atmosphere from one habitat to another with scales of motion being hundreds to thousands of kilometers. The concentration of these microbes in the lower atmosphere at a single geographic location can show rapid temporal changes. We used autonomous unmanned aerial vehicles equipped with microbe-sampling devices to collect fungi in the genus Fusarium 100 m above ground level at a single sampling location in Blacksburg, Virginia, USA. Some Fusarium species are important plant and animal pathogens, others saprophytes, and still others are producers of dangerous toxins. We correlated punctuated changes in the concentration of Fusarium to the movement of atmospheric transport barriers identified as finite-time Lyapunov exponent-based Lagrangian coherent structures (LCSs). An analysis of the finite-time Lyapunov exponent field for periods surrounding 73 individual flight collections of Fusarium showed a relationship between punctuated changes in concentrations of Fusarium and the passage times of LCSs, particularly repelling LCSs. This work has implications for understanding the atmospheric transport of invasive microbial species into previously unexposed regions and may contribute to information systems for pest management and disease control in the future.