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New techniques promise better forecasting of where damaging contaminants in the ocean or atmosphere will end up.

We show how the location of Lagrangian coherent structures, such as unstable manifolds of Lagrangian separation points, can be controlled via feedback control in two-dimensional shear flows. Such control can be used, for instance, to guide fuel transport into designated regions of the flame in a combustor. Motivated by this example, we consider an unsteady vortex model for flow past a bluff body, and create unstable manifolds in this model at prescribed locations by applying control along the boundary. We find that oscillating the newly created unstable manifolds in 1:1 resonance with the von Karman vortex shedding frequency enhances mixing in the wake significantly.

This paper examines whether hyperbolic Lagrangian structures—such as stable and unstable manifolds—found in model velocity data represent reliable predictions for mixing in the true fluid velocity field. The error between the model and the true velocity field may result from velocity interpolation, extrapolation, measurement imprecisions, or any other deterministic source. We find that even large velocity errors lead to reliable predictions on Lagrangian coherent structures, as long as the errors remain small in a special time-weighted norm. More specifically, we show how model predictions from the Okubo–Weiss criterion or from finite-time Lyapunov exponents can be validated. We also estimate how close the true Lagrangian coherent structures are to those predicted by models.

We derive analytic criteria for the existence of hyperbolic (attracting or repelling), elliptic, and parabolic material lines in two-dimensional turbulence. The criteria use a frame-independent Eulerian partition of the physical space that is based on the sign definiteness of the strain acceleration tensor over directions of zero strain. For Navier–Stokes flows, our hyperbolicity criterion can be reformulated in terms of strain, vorticity, pressure, viscous and body forces. The special material lines we identify allow us to locate different kinds of material structures that enhance or suppress finite-time turbulent mixing: stretching and folding lines, Lagrangian vortex cores, and shear jets. We illustrate the use of our criteria on simulations of two-dimensional barotropic turbulence.

Lapeyre, Hua, and Legras have recently suggested that the detection of finite-time invariant manifolds in two-dimensional fluid flows, as described by Haller and Haller and Yuan, can be substantially improved. In particular, they suggested (a) a change of coordinates to strain basis before the application of Theorem 1 of Haller and (b) the use of a nondimensionalized time computed from Theorem 1. Here we discuss why these proposed steps will not result in a significant overall improvement. We verify our arguments in a more detailed computation of the example analyzed in Lapeyre, Hau, and Legras (the Kida ellipse), as well as in a two-dimensional barotropic turbulence simulation. While in both of these examples the techniques suggested by Lapeyre, Hau, and Legras reveal additional thin regions of hyperbolicity near vortex cores, they also lead to an overall loss of detail in the global computation of finite-time invariant manifolds. (c) 2001 American Institute of Physics.

The relationship between statistical and geometric properties of particle motion in aperiodic, two-dimensional flows is examined. Finite-time-invariant manifolds associated with transient hyperbolic trajectories are shown to divide the flow into distinct regions with similar statistical behavior. In particular, numerical simulations of simple, eddy-resolving barotropic flows indicate that there exists a close correlation between such geometric structures and patchiness plots that describe the distribution of Lagrangian average velocity over initial conditions. For barotropic turbulence, we find that Eulerian velocity correlation time scales are significantly longer than their Lagrangian counterparts indicating the existence of well-defined Lagrangian structures. Identification of such structures shows a similar, close relationship between the invariant manifold geometry and patchiness calculations at intermediate time scales, where anomalous dispersion rates are found.

We propose here the use of the variational level set methodology to capture Lagrangian vortex boundaries in 2D unsteady velocity fields. This method reformulates earlier approaches that seek material vortex boundaries as extremum solutions of variational problems. We demonstrate the performance of this technique for two different variational formulations built upon different notions of coherence. The first formulation uses an energy functional that penalizes the deviation of a closed material line from piecewise uniform stretching [Haller and Beron-Vera, J. Fluid Mech. 731, R4 (2013)]. The second energy function is derived for a graph-based approach to vortex boundary detection [Hadjighasem et al., Phys. Rev. E 93, 063107 (2016)]. Our level-set formulation captures an a priori unknown number of vortices simultaneously at relatively low computational cost. We illustrate the approach by identifying vortices from different coherence principles in several examples.

Lagrangian coherent structures (LCSs) are material surfaces that shape 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, $\xi_{2}(x_{0})$, of the Cauchy-Green strain tensor. This $\xi_{2}$-system allows for the detection of LCSs in any unsteady flow by classic methods, such as Poincar\'e maps, developed for autonomous dynamical systems. As examples, we consider both steady and time-aperiodic flows, and use their dual $\xi_{2}$-system to uncover both hyperbolic and elliptic LCSs from a single computation.

We introduce an approach to identify elliptic transport barriers in three-dimensional, time-aperiodic flows. Obtained as Lagrangian Coherent Structures (LCSs), the barriers are tubular non-filamenting surfaces that form and bound coherent material vortices. This extends a previous theory of elliptic LCSs as uniformly stretching material surfaces from two-dimensional to three-dimensional flows. Specifically, we obtain explicit expressions for the normals of pointwise (near-) uniformly stretching material surfaces over a finite time interval. We use this approach to visualize elliptic LCSs in steady and time-aperiodic ABC-type flows.

Recent developments in dynamical systems theory have revealed long-lived and coherent Lagrangian (i.e., material) eddies in incompressible, satellite-derived surface ocean velocity fields. Paradoxically, observed drifting buoys and floating matter tend to create dissipative-looking patterns near oceanic eddies, which appear to be inconsistent with the conservative fluid particle patterns created by coherent Lagrangian eddies. Here we show that inclusion of inertial effects (i.e., those produced by the buoyancy and size finiteness of an object) in a rotating two-dimensional incompressible flow context resolves this paradox. Specifically, we obtain that anticyclonic coherent Lagrangian eddies attract (repel) negatively (positively) buoyant finite-size particles, while cyclonic coherent Lagrangian eddies attract (repel) positively (negatively) buoyant finite-size particles. We show how these results explain dissipative-looking satellite-tracked surface drifter and subsurface float trajectories, as well as satellite-derived \emph{Sargassum} distributions., Submitted to \emph{Chaos} Focus Issue on Objective detection of Lagrangian Coherent Structures. Revised 23-Feb-15

Undulatory swimmers flex their bodies to displace water, and in turn, the flow feeds back into the dynamics of the swimmer. At moderate Reynolds number, the resulting flow structures are characterized by unsteady separation and alternating vortices in the wake. We use the flow field from simulations of a two-dimensional, incompressible viscous flow of an undulatory, self-propelled swimmer and detect the coherent Lagrangian vortices in the wake to dissect the driving momentum transfer mechanisms. The detected material vortex boundary encloses a Lagrangian control volume that serves to track back the vortex fluid and record its circulation and momentum history. We consider two swimming modes: the C-start escape and steady anguilliform swimming. The backward advection of the coherent Lagrangian vortices elucidates the geometry of the vorticity field and allows for monitoring the gain and decay of circulation and momentum transfer in the flow field. For steady swimming, momentum oscillations of the fish can largely be attributed to the momentum exchange with the vortex fluid. For the C-start, an additionally defined jet fluid region turns out to balance the high momentum change of the fish during the rapid start.

Using the recently developed variational theory of hyperbolic Lagrangian coherent structures (LCSs), we introduce a computational approach that renders attracting and repelling LCSs as smooth, parametrized curves in two-dimensional flows. The curves are obtained as trajectories of an autonomous ordinary differential equation for the tensor lines of the Cauchy-Green strain tensor. This approach eliminates false positives and negatives in LCS detection by separating true exponential stretching from shear in a frame-independent fashion. Having an explicitly parametrized form for hyperbolic LCSs also allows for their further in-depth analysis and accurate advection as material lines. We illustrate these results on a kinematic model flow and on a direct numerical simulation of two-dimensional turbulence.

We study the validity of various models for the dynamics of finite-sized particles in fluids by means of a direct comparison between theory and experimentally measured trajectories and velocities of large numbers of particles in chaotic two-dimensional flow. Our analysis indicates that finite-sized particles follow the predicted particle dynamics given by the Maxey-Riley equation, except for random correlated fluctuations that are not captured by deterministic terms in the equations of motion, such as the Basset-Boussinesq term or the lift force. We describe the fluctuations via spectral methods and we propose three different Lagrangian stochastic models to account for them. These Lagrangian models are stochastic generalizations of the Maxey-Riley equation with coefficients calibrated to the experimental data. We find that one of them is capable of describing the observed fluctuations fairly well, while it also predicts a drag coefficient in near agreement with the theoretical Stokes drag.

Lagrangian coherent structures (LCSs) are time-evolving surfaces that act as organizers of trajectory patterns in dynamical systems. The detection of such surfaces from observed or simulated finite-time velocity data sets is important in a number of applications, including atmospheric and oceanic tracer spread analysis. One of the most efficient diagnostic tools for LCSs has been the largest finite-time Lyapunov exponent (FTLE) field, whose ridges appear to mark repelling LCSs. In this note we show that from the same numerical run that generates the forward FTLE field, one can also extract attracting LCSs as troughs of the minimum forward FTLE field graphed over evolving trajectory positions.

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.

Finite-size particle motion in fluids obeys the Maxey–Riley equations, which become singular in the limit of infinitesimally small particle size. Because of this singularity, finding the source of a dispersed set of small particles is a numerically ill-posed problem that leads to exponential blowup. Here we use recent results on the existence of a slow manifold in the Maxey–Riley equations to overcome this difficulty in source inversion. Specifically, we locate the source of particles by projecting their dispersed positions on a time-varying slow manifold, and by advecting them on the manifold in backward time. We use this technique to locate the source of a hypothetical anthrax release in an unsteady three-dimensional atmospheric wind field in an urban street canyon.

We develop a nonlinear theory for separation and attachment on no-slip boundaries of three-dimensional unsteady flows that have a steady mean component. In such flows, separation and attachment surfaces turn out to originate from fixed lines on the boundary, even though the surfaces themselves deform in time. The exact separation geometry is not captured by instantaneous Eulerian fields associated with the velocity field, but can be determined from a weighted average of the wall-shear and wall-density fields. To illustrate our results, we locate separation surfaces and attachment surfaces in an unsteady model flow and in direct numerical simulations of a time-periodic lid-driven cavity.

We derive analytic criteria for the location and angle of unsteady particle separation and reattachment in two-dimensional flows with free-slip boundary conditions. Our wall-based criteria show that, in general, fluid breakaway from the boundary takes place at locations different from either instantaneous or averaged stagnation points. Indeed, for time-varying flows, separation does not occur along a free streamline or along an average free streamline. We apply the formula to transport in randomized Rayleigh–Benard convection cells, as well as to boundary current separation and reattachment in high-frequency radar data collected in Monterey Bay, California.

The asymptotic dynamics of finite-size particles is governed by a slow manifold that is globally attracting for sufficiently small Stokes numbers. For neutrally buoyant particles (suspensions), the slow dynamics coincide with that of infinitesimally small particles, therefore the suspension dynamics should synchronize with Lagrangian particle motions. Paradoxically, recent studies observe a scattering of suspension dynamics along Lagrangian particle motions. Here we resolve this paradox by proving that despite its global attractivity, the slow manifold has domains that repel nearby passing trajectories. We derive an explicit analytic expression for these unstable domains; we also obtain a necessary condition for the global attractivity of the slow manifold. We illustrate our results on neutrally buoyant particle motion in a two-dimensional model of vortex shedding behind a cylinder in crossflow and on the three-dimensional steady Arnold–Beltrami–Childress flow.

We develop an analytic approach to two-dimensional flow separation control by feedback. With two wall-based actuators enclosing an array of distributed wall-shear sensors, we control the wall-shear evolution equation through its boundary values at the actuators. Using this approach, we induce separation at prescribed locations in steady and unsteady channel flows, and reduce the recirculation length behind a backward-facing step to a prescribed value.