Finite-time Lyapunov Exponent Method Research Articles

Dynamics of multi-scale vortical structures behind a barchan dune

In this study, multi-scale vortical structures and vortex dynamics around a fixed-bed barchan dune have been experimentally investigated based on the particle image velocimetry technique, wavelet transform, and the finite-time Lyapunov exponent method. It was found that the dynamic characteristics of a dune wake are predominated by large- and intermediate-scale coherent structures. Quadrant analysis of the Reynolds-stress distribution for the corresponding wavelet components revealed that ejection and sweep events are the main contributors to the whole field, while outward and inward interaction events just dominate the region near the dune crest. In addition, the process of ejection and sweeping motions associated with the turbulent bursting sequence can also be captured by applying proper orthogonal decomposition analysis of the decomposed velocity field for the different wavelet components. Finally, a continuous development process of the different wavelet scale structures in the shear layer was visualized in the Lagrangian framework. The small-scale waves grow exponentially and gradually develop into larger-scale vortices when convected downstream until the reattachment point, and larger-scale vortices break into the smaller ones.

The effect of surface roughness on the Lagrangian coherent structures in turbulent Rayleigh–Bénard convection

We perform direct numerical simulations of turbulent Rayleigh–Bénard (RB) convection in a closed square cell with roughness plates at Rayleigh number fixed at Ra=108 and the Prandtl number fixed at Pr = 1. To gain insight into the effect of surface roughness on material transport in turbulent Rayleigh–Bénard convection, the Lagrangian coherent structures (LCSs) are extracted using the finite-time Lyapunov exponent method in the cases of different roughness heights. First, we find that lobe structures are widely present in RB convection and we elucidate how they play a part in transporting heat from coner-flow rolls to large-scale circulation. Then, we quantify the heat flux along the LCSs, which contributes to 80% of the total flux. This implies that the LCSs play an important role in heat transport regardless of the roughness height. Furthermore, two different mechanisms of heat transport in RB convection induced by roughness heights are explained in the Lagrangian perspective: the decrease in Nu number in the cases of h<hc is caused by the LCSs between the roughness elements which hinders the exchange of material between the fluid in the cavity and the bulk region; whereas, the increase in Nu number in the case of h>hc is produced by the enhanced mixing events of the convection that enhance the contribution of heat transport in the bulk region.

Applying finite-time lyapunov exponent to study the tidal dispersion on oil spill trajectory in Burrard Inlet

This study used a combination of Finite-Time Lyapunov Exponent (FTLE) values, residual currents, and tidal excursion lengths to systematically investigate the effects of tidal dispersion on oil spill trajectories in Burrard Inlet, BC, Canada, which is a tidally dominated estuary. The FTLE analysis results showed that tidal type and tidal phase significantly influenced the FTLE fields because the flow structure and the location of saddle points varied as a function of the tidal type and tidal phase. Some transport barriers formed in the Inner Harbour, which blocked the water exchange between the western and eastern parts of the inlet. Moreover, tidal mixing in the wider regions of Burrard Inlet (i.e., the western Outer Harbour) was relatively weak than in the narrower areas (i.e., First Narrows and Second Narrows). The observations from the FTLE analysis agreed well with the residual currents and tidal excursion results. The drifter trajectories were consistent with the Lagrangian coherent structure lines extracted from the FTLE analysis. To verify the tidal dispersion impact on an oil spill trajectory in the inlet, the FTLE fields were compared with a real oil spill that occurred in Burrard Inlet in 2015 (the M/V Marathassa oil spill). The FTLE fields reasonably explained the spilled oil’s trajectories from the real event. In addition, a set of stochastic oil spill models were run in this study and found that the FTLE analysis was a reliable tool for oil spill tracking. Overall, the FTLE method would be a valuable addition to practical oil spill response planning.

Time-Resolved PIV Measurements and Turbulence Characteristics of Flow Inside an Open-Cell Metal Foam.

Open-cell metal foams are porous medium for thermo-fluidic systems. However, their complex geometry makes it difficult to perform time-resolved (TR) measurements inside them. In this study, a TR particle image velocimetry (PIV) method is introduced for use inside open-cell metal foam structures. Stereolithography 3D printing methods and conventional post-processing methods cannot be applied to metal foam structures; therefore, PolyJet 3D printing and post-processing methods were employed to fabricate a transparent metal foam replica. The key to obtaining acceptable transparency in this method is the complete removal of the support material from the printing surfaces. The flow characteristics inside a 10-pore-per-inch (PPI) metal foam were analyzed in which porosity is 0.92 while laminar flow condition is applied to inlet. The flow inside the foam replica is randomly divided and combined by the interconnected pore network. Robust crosswise motion occurs inside foam with approximately 23% bulk speed. Strong influence on transverse motion by metal foam is evident. In addition, span-wise vorticity evolution is similar to the integral time length scale of the stream-wise center plane. The span-wise vorticity fluctuation through the foam arrangement is presented. It is believed that this turbulent characteristic is caused by the interaction of jets that have different flow directions inside the metal foam structure. The finite-time Lyapunov exponent method is employed to visualize the vortex ridges. Fluctuating attracting and repelling material lines are expected to enhance the heat and mass transfer. The results presented in this study could be useful for understanding the flow characteristics inside metal foams.

Fluctuations of Lyapunov exponents in homogeneous and isotropic turbulence

In the context of the analysis of the chaotic properties of homogeneous and isotropic turbulence, direct numerical simulations are used to study the fluctuations of the finite time Lyapunov exponent (FTLE) and its relation to Reynolds number, lattice size and the choice of the steptime used to compute the Lyapunov exponents. The results show that using the FTLE method produces Lyapunov exponents that are remarkably stable under the variation of the steptime and lattice size. Furthermore, it reaches such stability faster than other characteristic quantities such as energy and dissipation rate. These results remain even if the steptime is made arbitrarily small. A discrepancy is also resolved between previous measurements of the dependence on the Reynolds number of the Lyapunov exponent. The signal produced by different variables in the steady state is analyzed and the self decorrelation time is used to determine the run time needed in the simulations to obtain proper statistics for each variable. Finally, a brief analysis on MHD flows is also presented, which shows that the Lyapunov exponent is still a robust measure in the simulations, although the Lyapunov exponent scaling with Reynolds number is significantly different from that of magnetically neutral hydrodynamic fluids.

Time Recurrence Analysis of a Near Singular Billiard

Billiards exhibit rich dynamical behavior, typical of Hamiltonian systems. In the present study, we investigate the classical dynamics of particles in the eccentric annular billiard, which has a mixed phase space, in the limit that the scatterer is point-like. We call this configuration the near singular, in which a single initial condition (IC) densely fills the phase space with straight lines. To characterize the orbits, two techniques were applied: (i) Finite-time Lyapunov exponent (FTLE) and (ii) time recurrence. The largest Lyapunov exponent λ was calculated using the FTLE method, which for conservative systems, λ > 0 indicates chaotic behavior and λ = 0 indicates regularity. The recurrence of orbits in the phase space was investigated through recurrence plots. Chaotic orbits show many different return times and, according to Slater’s theorem, quasi-periodic orbits have at most three different return times, the bigger one being the sum of the other two. We show that during the transition to the near singular limit, a typical orbit in the billiard exhibits a sharp drop in the value of λ, suggesting some change in the dynamical behavior of the system. Many different recurrence times are observed in the near singular limit, also indicating that the orbit is chaotic. The patterns in the recurrence plot reveal that this chaotic orbit is composed of quasi-periodic segments. We also conclude that reducing the magnitude of the nonlinear part of the system did not prevent chaotic behavior.

Chaotic mixing analysis of a novel single-screw extruder with a perturbation baffle by the finite-time Lyapunov exponent method

Abstract The single-screw extruder with a perturbation baffle is a novel piece of equipment for polymer processing, in which the polymer melts undergo complex chaotic mixing. In this paper, from a new Lagrangian perspective, the fluid transporting mechanism in chaotic flow of the unwound screw channel was analyzed based on the finite element method. Firstly, two-dimensional velocity distributions in the unwound screw channel were calculated based on the mesh superposition technique. Fluid particle evolution processes in the extruder were tracked based on the fourth-order Runge-Kutta scheme. The numerical method used in this paper was validated by grid independence and experiments obtained from literature. Moreover, the finite-time Lyapunov exponent (FTLE) and Poincaré sections were adopted to discuss the chaotic mixing in the novel single-screw extruder. The effects of baffle width and height on the manifold structures in the flow dynamic system were analyzed. The results show that the homoclinic point of the manifold structure can give rise to chaotic mixing in the single-screw extruder. The height of the baffle is an important parameter to control the chaotic strength. In a way, increasing the height of the baffle can enlarge the kink scale and increase the stretching and folding actions, which results in the decrease of regular regions and an increase of the mixing efficiency in the single-screw extruder.

Flowfield and Force Evolution for a Symmetric Hovering Flat-Plate Wing

The unsteady flow around a hovering flat-plate wing has been investigated experimentally using particle image velocimetry and direct force measurements. The measurements are conducted on a wing that rotates symmetrically about the stroke reversal at a reduced frequency of and Reynolds number of . The Lagrangian finite time Lyapunov exponent method is used to analyze the unsteady flowfields by identifying dynamically relevant flow features such as the primary leading-edge vortex, secondary vortices, and topological saddles as well as their evolution within a flapping cycle. The flow evolution is divided into four stages that are characterized by the leading-edge vortex: 1) emergence, 2) growth, 3) liftoff, and 4) breakdown and decay. The saddle-point trajectory helps in identifying the leading-edge vortex liftoff, which occurs at the maximum stroke velocity. The flowfields are correlated with the aerodynamic forces, revealing that the maximum lift and drag are observed just before leading-edge vortex liftoff. The end of wing rotation in the beginning of the stroke stimulates a change in the direction of the leading-edge vortex growth, and the start of rotation at the end of the stroke triggers the breakdown of the leading-edge vortex.

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.

Separated Flow Response to Single Pulse Actuation

The response of a separated flow over a two-dimensional wing to short-duration disturbances from a Lorentz force leading-edge actuator is presented. At a chord Reynolds number of and an angle of attack of , the flow is initially laminar but undergoes transition within the separated shear layer.The transient flow structures and lift force measurements were obtained with varying actuator pulse duration, pulse amplitude, and direction of actuation. The peak amplitude of the lift is shown to depend on the pulse duration when the pulse duration is less than 0.5 convective times, after which it saturates. Saturation of the lift amplitude also occurs when the effective actuator pulse amplitude exceeds . Detailed flow structures that develop in the separated shear layer were identified using the finite-time Lyapunov exponent method. The direction of the actuator pulse has a significant influence on the initial development of the shear layer, but the larger-scale envelope of the separated flow has essentially the same response, irrespective of the direction of actuation. A reconstruction of the velocity field using only three proper orthogonal decomposition modes is sufficient to reproduce the dominant features of the flow response to a single pulse. The second proper orthogonal decomposition mode correlates with the transient lift signal.

Experimental investigation of the wall shear stress in a circular impinging jet

The influence of the large-scale vortical structures on the wall shear stress in a circular impinging jet is investigated experimentally for a Reynolds number of 1260. Time-resolved particle image velocimetry and polarographic measurements are performed simultaneously. It is found that the instantaneous wall shear stress is strongly dependent on the vortex dynamics, particularly for different parts of the transverse vortex. The influence of the vortex ring, the secondary and tertiary vortices on the ejection/sweep process near the wall is the main mechanism involved in the wall shear stress variation. In the region of the boundary layer separation, the wall shear stress amplitude increases just upstream of the separation and dramatically decreases in the recirculation zone downstream from the separation. The interaction between primary and secondary structures and their pairing process with the tertiary structure affects the sweep/ejection process near the wall and subsequently the wall shear stress variation. A comparison between the Finite Time Lyapunov Exponent (FTLE) method and the phase average technique is performed. It is shown that both methods describe the flow dynamics in the impinging region of the vortex ring. However, the FTLE method is more suitable for describing the unsteady separation of the boundary layer.

Lagrangian coherent structures in the California Current System – sensitivities and limitations

We report on numerical experiments to test the sensitivity of Lagrangian coherent structures (LCSs), found by identifying ridges of the finite-time Lyapunov exponent (FTLE), to errors in two systems representing the California Current System (CCS). First, we consider a synthetic mesoscale eddy field generated from Fourier filtering satellite altimetry observations of the CCS. Second, we consider the full observational satellite altimetry field in the same region. LCS are found to be relatively insensitive to both sparse spatial and temporal resolution and to the velocity field interpolation method. Strongly attracting and repelling LCS are robust to perturbations of the velocity field of over 20% of the maximum regional velocity. Contours of the Okubo–Weiss (OW) parameter are found to be consistent with LCS in large mature eddies in the unperturbed systems. The OW parameter is unable to identify eddies at the uncertainty level expected for altimetry observations of the CCS. At this expected error level, the FTLE method is reliable for locating boundaries of large eddies and strong jets. Small LCS features such as lobes are not well resolved even at low error levels, suggesting that reliable determination of lobe dynamics from altimetry will be problematic.

Lagrangian Coherent Structures with Guaranteed Material Separation

AbstractGiven an unsteady flow field, one common way to compute Lagrangian Coherent Structures (LCS) is to extract extremal structures of the Finite Time Lyapunov Exponent (FTLE). Experience has shown that the resulting structures are often close to material structures (i.e., material lines or material surfaces). Moreover, it has been proven that for an integration time converging to infinity, they converge to exact material structures. However, due to the finite integration time in FTLE, they are generally not exact material structures.In this paper we introduce a modification of the FTLE method which is guaranteed to produce separating material structures as features of a scalar field. We achieve this by incorporating the complete available integration time both in forward and backward direction, and by choosing an appropriate definition for separating structures. We apply our method to two test data sets and show the differences to classical FTLE.

Accurate extraction of Lagrangian coherent structures over finite domains with application to flight data analysis over Hong Kong International Airport

Locating Lagrangian coherent structures (LCS) for dynamical systems defined on a spatially limited domain present a challenge because trajectory integration must be stopped at the boundary for lack of further velocity data. This effectively turns the domain boundary into an attractor, introduces edge effects resulting in spurious ridges in the associated finite-time Lyapunov exponent (FTLE) field, and causes some of the real ridges of the FTLE field to be suppressed by strong spurious ridges. To address these issues, we develop a finite-domain FTLE method that renders LCS with an accuracy and fidelity that is suitable for automated feature detection. We show the application of this technique to the analysis of velocity data from aircraft landing at the Hong Kong International Airport.

Identifying invariant manifold using phase space warping and stochastic interrogation

Advances in the generalization of invariant manifolds to finite time, experimental (or observational) flows have stimulated many recent developments in the approximation of invariant manifolds and Lagrangian coherent structures. This paper explores the identification of invariant manifold like structures in experimental settings, where knowledge of a flow field is absent, but phase space trajectories can be experimentally measured. Several existing methods for the approximation of these structures modified for application when only unstructured trajectory data is available. We find the recently proposed method, based on the concept of phase space warping, to outperform other methods as data becomes limited and show it to extend the finite-time Lyapunov exponent method. This finding is based on a comparison of methods for various data quantities and in the presence of both measurement and dynamic noise.