Hyperbolic Lagrangian Coherent Structures Research Articles

Quantitative analysis of enhanced mixing and combustion by lobed mixer in a ramjet engine: Study using hyperbolic Lagrangian coherent structures

Mixing and combustion are the core processes in the operation of the ramjet engine, and the contact line length (2D) or contact area (3D) is an important bridge to analyze the link between mixing and combustion. In this paper, a new formula for calculating the contact area of two components in an arbitrary complex region is proposed, which can analyze the dynamic mixing process of the two components. It can be used to evaluate the mixing efficiency in the dynamic mixing process, the larger the contact area, the higher the mixing efficiency. This paper establishes a new method to evaluate the mixing in ramjet engines based on the Lagrangian coherence structures, and uses this method to study the feasibility of the application of the lobed mixer in ramjet engines. The study shows that the vortex induced by the lobed mixer can significantly increase the contact area between gas and air in the combustion chamber, effectively enhancing the mixing and combustion of gas and air therein. The average mixing efficiency during the operation increased by 11.046% and the combustion efficiency increased by 8.318%.

An improved numerical method for hyperbolic Lagrangian Coherent Structures using Differential Algebra

In dynamical systems, it is advantageous to identify regions of flow which can exhibit maximal influence on nearby behaviour. Hyperbolic Lagrangian Coherent Structures have been introduced to obtain two-dimensional surfaces which maximise repulsion or attraction in three-dimensional dynamical systems with arbitrary time-dependence. However, the numerical method to compute them requires obtaining derivatives associated with the system, often performed through the approximation of divided differences, which can lead to significant numerical error and numerical noise. In this paper, we introduce a novel method for the numerical calculation of hyperbolic Lagrangian Coherent Structures using Differential Algebra called DA-LCS. As a form of automatic forward differentiation, it allows direct computation of the Taylor expansion of the flow, its derivatives, and the eigenvectors of the associated strain tensor, with all derivatives obtained algebraically and to machine precision. It does so without a priori information about the system, such as variational equations or explicit derivatives. We demonstrate that this can provide significant improvements in the accuracy of the Lagrangian Coherent Structures identified compared to finite-differencing methods in a series of test cases drawn from the literature. We also show how DA-LCS uncovers additional dynamical behaviour in a real-world example drawn from astrodynamics.

Detecting Lagrangian coherent structures from sparse and noisy trajectory data

Many complex flows such as those arising from the collective motion of ocean plastics in geophysics or motile cells in biology are characterized by sparse and noisy trajectory datasets. We introduce techniques for identifying Lagrangian coherent structures (LCSs) of hyperbolic and elliptic nature in such datasets. Hyperbolic LCSs, which represent surfaces with maximal attraction or repulsion over a finite amount of time, are computed through a regularized least-squares approximation of the flow map gradient. Elliptic LCSs, which identify regions of coherent motion such as vortices and jets, are extracted using DBSCAN – a popular data clustering algorithm – combined with a systematic parameter selection strategy. We deploy these methods on various benchmark analytical flows and real-life experimental datasets ranging from oceanography to biology and show that they yield accurate results, despite sparse and noisy data. We also provide a lightweight computational implementation of these techniques as a user-friendly and straightforward Python code.

Lagrangian coherent structures in flow past a backward-facing step

This paper investigates flow past a backward-facing step (BFS) in a duct at Reynolds number $Re = 5080$ based on step height, mean inflow velocity and fluid kinematic viscosity. The flow configuration matches a combustion experiment conducted by Pitz and Daily in 1983. High-resolution velocity fields are obtained in OpenFOAM by direct numerical simulation, and the flow field analysed by Lagrangian approaches. Trajectories of fluid particles in areas of interest are obtained by high-order numerical integration, and used to compute finite-time Lyapunov exponents (FTLEs) and polar rotation angles. Lagrangian coherent structures (LCSs) are extracted using geodesic theory, including hyperbolic LCSs and elliptic LCSs. We use complementary qualitative and quantitative LCS analyses to uncover the underlying flow structures. Notably, we find that a flow pathway in which fluid particles rarely diverge from adjacent particles is opened and closed by FTLE ridges determined by the periodic shedding of vortices from the BFS. Two dominant vortices with significant Lagrangian coherence, generated respectively by the separated boundary layer and shear layer, are self-sustaining and of comparable strength. Hyperbolic repelling LCSs act as transport barriers between the pathway and cores of the coherent vortices, thus playing a major part in the fluid entrainment process. Interactions between these different geometric regions partitioned by LCSs lead to intrinsic complexity in the BFS flow.

Quantitative analysis of fluid transport in dynamic stall of a pitching airfoil using variational Lagrangian coherent structures and lobe dynamics

The evolution of flow structures during dynamic stall of a two-dimensional pitching National Advisory Committee for Aeronautics 0012 airfoil is studied using the variational Lagrangian coherent structures (LCSs), and the mass transport and vorticity transport are precisely analyzed using LCSs and lobe dynamics for further understanding the nature of flow phenomena in dynamic stall. First, the variational LCS algorithm is improved to be efficiently used in the accurate extraction of flow structures. Then, both the hyperbolic LCSs and elliptic LCSs are computed numerically in the whole process of dynamic stall to analyze the evolution of flow structures in detail. Further, a high-accuracy LCS-advection method is used in the advection of LCSs to quantitatively analyze the mass transport and vorticity transport in the evolution of LCSs utilizing lobe dynamics based on nonlinear dynamics. Finally, the evolution and motion of primary leading edge vortex (LEV) and trailing edge vortex (TEV) identified by elliptic LCSs are analyzed in depth. The results obtained can provide a deeper insight into the nature of flow phenomena in dynamic stall from the viewpoint of nonlinear dynamics. Specifically, the nature of evolution of primary LEV and the TEV and the reasons for the changes of lift coefficients are clarified from the viewpoint of fluid transport. To explain it briefly, the variational LCSs and lobe dynamics are powerful tools to quantitatively analyze the evolution of flow structures and fluid transport.

On the Lagrangian dynamics of saliva particles during normal mouth breathing

The flow and saliva particulate transport dynamics during normal human breathing through the mouth are simulated numerically using an Eulerian large-eddy simulation (LES) approach for the flow coupled with a Lagrangian approach for the transport of saliva particles. The coupled Eulerian–Lagrangian (EL) simulation results reveal new striking insights into the rich dynamics of the Lagrangian coherent structures (LCS) that arise from saliva particles during normal breathing. Specifically, they uncover a new time-periodic mechanism via which particles are introduced into the flow as individual breathing pulses and accumulate to form a slowly propagating vortex front that persists long distances away from the source. The simulated LCS reveal a wealth of recurrent material motion through which the biosols propagate forward while their cloud expands laterally with a slowly evolving vortex front. Also, the finite-time Lyapunov exponent (FTLE) field of human breathing was calculated using the flow map from the LES velocity field. The ridges of the calculated FTLE field revealed distinct hyperbolic LCS, which closely resemble trajectories of saliva particles obtained from the coupled EL simulation. Finally, simulation results for normal breathing with a non-medical face mask show that the mask can effectively disrupt the formation of coherent particle surfaces and, thus, effectively limit saliva particle propagation.

Quasi-objective coherent structure diagnostics from single trajectories

We derive measures of local material stretching and rotation that are computable from individual trajectories without reliance on other trajectories or on an underlying velocity field. Both measures are quasi-objective: they approximate objective (i.e., observer-independent) coherence diagnostics in frames satisfying a certain condition. This condition requires the trajectory accelerations to dominate the angular acceleration induced by the spatial mean vorticity. We illustrate on examples how quasi-objective coherence diagnostics highlight elliptic and hyperbolic Lagrangian coherent structures even from very sparse trajectory data.

A Modified Double Gyre with Ground Truth Hyperbolic Trajectories for Flow Visualization

AbstractThe model of a Double Gyre flow by Shadden et al. is a standard benchmark data set for the computation of hyperbolic Lagrangian Coherent Structures (LCS) in flow data. While structurally extremely simple, it generates hyperbolic LCS of arbitrary complexity. Unfortunately, the Double Gyre does not come with a well‐defined ground truth: the location of hyperbolic LCS boundaries can only be approximated by numerical methods that usually involve the gradient of the flow map. We present a new benchmark data set that is a small but carefully designed modification of the Double Gyre, which comes with ground truth closed‐form hyperbolic trajectories. This allows for computing hyperbolic LCS boundaries by a simple particle integration without the consideration of the flow map gradient. We use these hyperbolic LCS as a ground truth solution for testing an existing numerical approach for extracting hyperbolic trajectories. In addition, we are able to construct hyperbolic LCS curves that are significantly longer than in existing numerical methods.

Finite-time Lyapunov exponents in the instantaneous limit and material transport

Lagrangian techniques, such as the finite-time Lyapunov exponent (FTLE) and hyperbolic Lagrangian coherent structures (LCS), have become popular tools for analyzing unsteady fluid flows. These techniques identify regions where particles transported by a flow will converge to and diverge from over a finite-time interval, even in a divergence-free flow. Lagrangian analyses, however, are time consuming and computationally expensive, hence unsuitable for quickly assessing short-term material transport. A recently developed method called OECSs [Serra, M. and Haller, G., `Objective Eulerian Coherent Structures', Chaos 26(5), 2016] rigorously connected Eulerian quantities to short-term Lagrangian transport. This Eulerian method is faster and less expensive to compute than its Lagrangian counterparts, and needs only a single snapshot of a velocity field. Along the same line, here we define the instantaneous Lyapunov Exponent (iLE), the instantaneous counterpart of the FTLE, and connect the Taylor series expansion of the right Cauchy-Green deformation tensor to the infinitesimal integration time limit of the FTLE. We illustrate our results on geophysical fluid flows from numerical models as well as analytical flows, and demonstrate the efficacy of attracting and repelling instantaneous Lyapunov exponent structures in predicting short-term material transport.

Probability of Ship High-Runs from Phase-Space Data

A clustering scheme has been applied for capturing qualitatively different surge motion patterns in the phase space. The scheme enables the identification of "high-run" incidents as soon as such motions are triggered and while their phenomenology has not yet been well developed. A "high run" is a surf-riding-like behavior, appearing in irregular following seas. The concept of finite-time coherent sets is exploited for deriving estimates of the probability of high-runs. The method is verified by identifying independently the corresponding hyperbolic Lagrangian coherent structures; then, consistency is sought between the two approaches. An important feature of the method is that it does not rely on the use of some empirical criterion for the high-run threshold, such as one based on the exceedance of an arbitrary high-speed level. Despite its computational burden, the proposed scheme offers "objective" statistical information on a ship's high-run tendency that can be used for benchmarking simpler (approximative) probability calculation schemes. 1. Introduction Current efforts to assess a ship's tendency for abnormal behavior in extreme seas are still limited from our inadequate grasp of the full variety of nonlinear ship motion phenomena that could be realized in an irregular seaway. A classification of these motion patterns would provide a sound basis for developing probabilistic calculation methods of ship operability and safety in extreme seas. A few recent research efforts in our group have been related to this target. In one case, it was endeavored to distinguish ship high-runs from ordinary surging, by engaging the concept of instantaneous wave celerity (Spyrou et al. 2014). In another, the derivation of a practical metric for the probability of high-run was pursued (Belenky et al. 2016). Also, high-run and broaching-to statistics were produced through a direct approach based on assigning prescriptive exceedance thresholds (Spyrou et al. 2016b). Moreover, the theory of surf-riding was extended for bichromatic waves, revealing some rather unexpected types of motion (Spyrou et al. 2018). Even richer phenomena could be conjectured for a multifrequency environment.

Extracting Lagrangian coherent structures in the Kuroshio current system

In this paper, we propose to extract Lagrangian coherent structures (LCSs) based on MADT data to describe the structure of the Kuroshio Current System, and this method helps study the monthly variation in the intensity. By exploiting the improved variational principles, we extract LCSs through a geostrophic velocity vector field derived from the absolute dynamic topography. Specifically, we address hyperbolic LCSs and parabolic LCSs, with the former as the outer bounds of the jet flow and the latter as a more robust depiction of the jet axis. These LCSs delineate the jet with theoretical underpinnings, as opposed to common contours identified in an anecdotal manner. We employ the top 20% most repelling and attracting hyperbolic LCSs to approximate the outer bounds of the Kuroshio Current System, and these structures are apt to enclose the jet. Compared with the results of FTLEs and FSLEs, our results are more reliable. A comparison in the intensity based on the jet axis of months in 2012 demonstrates the effectiveness of our method in extracting LCSs. In situ data, such as drifter tracks and temperature-salinity profiles, are applied to validate the reliability of the axis and outer bounds extracted from the Kuroshio.

Trajectory-free approximation of phase space structures using the trajectory divergence rate

This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximating hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for 2-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions.

Probability of Ship High-Runs from Phase-Space Data

A clustering scheme has been applied for capturing qualitatively different surge motion patterns in the phase space. The scheme enables the identification of “high-run” incidents as soon as such motions are triggered and while their phenomenology has not yet been well developed. A “high run” is a surf-riding–like behavior, appearing in irregular following seas. The concept of finite-time coherent sets is exploited for deriving estimates of the probability of high-runs. The method is verified by identifying independently the corresponding hyperbolic Lagrangian coherent structures; then, consistency is sought between the two approaches. An important feature of the method is that it does not rely on the use of some empirical criterion for the high-run threshold, such as one based on the exceedance of an arbitrary high-speed level. Despite its computational burden, the proposed scheme offers “objective” statistical information on a ship’s high-run tendency that can be used for benchmarking simpler (approximative) probability calculation schemes.

Probability of Ship High-Runs from Phase-Space Data

A clustering scheme has been applied for capturing qualitatively different surge motion patterns in the phase space. The scheme enables the identification of “high-run” incidents as soon as such motions are triggered and while their phenomenology has not yet been well developed. A “high run” is a surf-riding–like behavior, appearing in irregular following seas. The concept of finite-time coherent sets is exploited for deriving estimates of the probability of high-runs. The method is verified by identifying independently the corresponding hyperbolic Lagrangian coherent structures; then, consistency is sought between the two approaches. An important feature of the method is that it does not rely on the use of some empirical criterion for the high-run threshold, such as one based on the exceedance of an arbitrary high-speed level. Despite its computational burden, the proposed scheme offers “objective” statistical information on a ship’s high-run tendency that can be used for benchmarking simpler (approximative) probability calculation schemes.

Accelerated Monte Carlo Rendering of Finite-Time Lyapunov Exponents.

Time-dependent fluid flows often contain numerous hyperbolic Lagrangian coherent structures, which act as transport barriers that guide the advection. The finite-time Lyapunov exponent is a commonly-used approximation to locate these repelling or attracting structures. Especially on large numerical simulations, the FTLE ridges can become arbitrarily sharp and very complex. Thus, the discrete sampling onto a grid for a subsequent direct volume rendering is likely to miss sharp ridges in the visualization. For this reason, an unbiased Monte Carlo-based rendering approach was recently proposed that treats the FTLE field as participating medium with single scattering. This method constructs a ground truth rendering without discretization, but it is prohibitively slow with render times in the order of days or weeks for a single image. In this paper, we accelerate the rendering process significantly, which allows us to compute video sequence of high-resolution FTLE animations in a much more reasonable time frame. For this, we follow two orthogonal approaches to improve on the rendering process: the volumetric light path integration in gradient domain and an acceleration of the transmittance estimation. We analyze the convergence and performance of the proposed method and demonstrate the approach by rendering complex FTLE fields in several 3D vector fields.

Chaotic Manifold Analysis of Four-Screw Extruders Based on Lagrangian Coherent Structures.

The four-screw extruder (FSE) is a novel equipment for polymer processing. In this paper, from a new viewpoint of Lagrangian coherent structures (LCS), two-dimensional fluid transport and chaotic mixing characteristics within three kinds of novel industrial FSEs are explored based on LCS to better understand the flow and mixing natures in the FSEs. Firstly, based on the finite-time invariant manifold theory, the finite-time Lyapunov exponent (FTLE) and LCS of FSEs are calculated by considering the different initial time. Hyperbolic LCSs from the FTLE maps are adopted to identify chaotic mixing manifolds in FSEs. Moreover, particle tracking and Poincaré sections are used to illustrate the different fluid motions in the above three isolated regions. Finally, the effects of relative rotating directions and layout of four screws on the chaotic manifolds in FESs are discussed in order to enhance local mixing performance. Furthermore, quantitative mixing measures, such as the segregation scale, logarithmic of stretching, and mean-time mixing efficiency are employed to compare the mixing efficiencies in three kinds of FSEs. The results show that the relative rotating directions and positions of four screws can change the chaotic manifolds and increase mixing performance in local poor mixing regions. FTLE and LCS analysis are helpful to better understand the chaotic mixing nature in the novel screw extruders.

The kinematic genesis of vortex formation due to finite rotation of a plate in still fluid

We present a combined experimental and numerical study of an idealized model of the propulsive stroke of the turning manoeuvre in fish. Specifically, we use the framework of Lagrangian coherent structures (LCSs) to describe the kinematics of the flow that results from a thin plate performing a large angle rotation about its tip in still fluid. Temporally and spatially well-resolved velocity fields are obtained using a two-dimensional, incompressible finite-volume solver, and are validated by comparisons with experimentally measured velocity fields and alternate numerical simulations. We then implement the recently proposed variational theory of LCSs to extract the hyperbolic and elliptic LCSs in the numerically generated velocity fields. Detailed LCS analysis is performed for a plate motion profile described by$\dot{\unicode[STIX]{x1D703}}(t)=\unicode[STIX]{x1D6FA}_{max}\sin ^{2}(\unicode[STIX]{x1D714}t)$during$0\leqslant t\leqslant t_{o}$and zero otherwise. The stopping time$t_{o}$is given by$t_{o}=\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}=10~\text{s}$, the value of$\unicode[STIX]{x1D6FA}_{max}$chosen to give a stopping angle of$\unicode[STIX]{x1D703}_{max}=90^{\circ }$, resulting in a Reynolds number$Re=c^{2}\unicode[STIX]{x1D6FA}_{max}/\unicode[STIX]{x1D708}=785.4$, where$c$is the plate chord length and$\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$the kinematic viscosity of water. The flow comprises a starting and a stopping vortex, resulting in a pair of oppositely signed vortices of unequal strengths that move away from the plate in a direction closely aligned with the final plate orientation at$t/t_{o}\approx 2$. The hyperbolic LCSs are shown to encompass the fluid material that is advected away from the plate for$t>t_{o}$, henceforth referred to as the advected bulk. The starting and stopping vortices, identified using elliptic LCSs and hence more objective than Eulerian vortex detection methods, constitute only around two thirds of the advected bulk area. The advected bulk is traced back to$t=0$to identify five distinct lobes of fluid that eventually form the advected bulk, and hence map the long-term fate of various regions in the fluid at$t=0$. The five different lobes of fluid are then shown to be delineated by repelling LCS boundaries at$t=0$. The linear momentum of the advected bulk region is shown to account for approximately half of the total impulse experienced by the plate in the direction of its final orientation, thus establishing its dynamical significance. We provide direct experimental evidence for the kinematic relevance of hyperbolic and elliptic LCSs using novel dye visualization experiments, and also show that attracting hyperbolic LCSs provide objective characterization of the spiral structures often observed in vortical flows. We conclude by showing that qualitatively similar LCSs persist for several other plate motion profiles and stopping angles as well.

Backward Finite-Time Lyapunov Exponents in Inertial Flows.

Inertial particles are finite-sized objects that are carried by fluid flows and in contrast to massless tracer particles they are subject to inertia effects. In unsteady flows, the dynamics of tracer particles have been extensively studied by the extraction of Lagrangian coherent structures (LCS), such as hyperbolic LCS as ridges of the Finite-Time Lyapunov Exponent (FTLE). The extension of the rich LCS framework to inertial particles is currently a hot topic in the CFD literature and is actively under research. Recently, backward FTLE on tracer particles has been shown to correlate with the preferential particle settling of small inertial particles. For larger particles, inertial trajectories may deviate strongly from (massless) tracer trajectories, and thus for a better agreement, backward FTLE should be computed on inertial trajectories directly. Inertial backward integration, however, has not been possible until the recent introduction of the influence curve concept, which - given an observation and an initial velocity - allows to recover all sources of inertial particles as tangent curves of a derived vector field. In this paper, we show that FTLE on the influence curve vector field is in agreement with preferential particle settling and more importantly it is not only valid for small (near-tracer) particles. We further generalize the influence curve concept to general equations of motion in unsteady spatio-velocity phase spaces, which enables backward integration with more general equations of motion. Applying the influence curve concept to tracer particles in the spatio-velocity domain emits streaklines in massless flows as tangent curves of the influence curve vector field. We demonstrate the correlation between inertial backward FTLE and the preferential particle settling in a number of unsteady vector fields.

An autonomous dynamical system captures all LCSs in three-dimensional unsteady flows.

Lagrangian coherent structures (LCSs) are material surfaces that shape the finite-time tracer patterns in flows with arbitrary time dependence. Depending on their deformation properties, elliptic and hyperbolic LCSs have been identified from different variational principles, solving different equations. Here we observe that, in three dimensions, initial positions of all variational LCSs are invariant manifolds of the same autonomous dynamical system, generated by the intermediate eigenvector field, ξ2(x0), of the Cauchy-Green strain tensor. This ξ2-system allows for the detection of LCSs in any unsteady flow by classical methods, such as Poincaré maps, developed for autonomous dynamical systems. As examples, we consider both steady and time-aperiodic flows, and use their dual ξ2-system to uncover both hyperbolic and elliptic LCSs from a single computation.

Coherent structures in phase space, governing the nonlinear surge motions of ships in steep waves

In steep multi-chromatic seas, ship surge dynamics can become intricate and the full variety of exhibited motions is unknown. This accrues, partly, from the nonlinear nature of surge motion; and partly because, for multi-frequency waves, the phase-space flow of the dynamical system becomes time-dependent. Accordingly, conventional concepts that were applied in the past for analyzing stationary phase-space flows are rendered incapable to support in-depth exploration of ship dynamics. Towards overcoming this limitation, use of the concept of hyperbolic Lagrangian Coherent Structures (LCSs) is proposed. These phase-space objects can be regarded as the “finite-time” generalizations of the stable and unstable manifolds of hyperbolic fixed points defined in “time-invariant” dynamical systems. They can be described as, locally, the strongest repelling or attracting material surfaces (curves in the case of 2-dimensional systems) advected with the phase flow. We have identified hyperbolic LCSs that are innate to the phase-flow associated with the surge motion of a ship in astern seas. To the global approach of LCS identification, a supplementary computational scheme is incorporated, aiming to track, in space-time, local “features” of the flow, connected with surf-riding. The emerging toolset can enhance current efforts towards a rigorous assessment of ship dynamic stability in following seas.