Session D17: Nonlinear Dynamics II: Coherent Structures II

An accurate computation of the flow map gradient

The flow map gradient (tangent map) is often used in dynamical systems for computation of Lagrangian coherent structures. In more sophisticated methods it is important to recover the complete spectrum of this operator, as well as the eigenvectors. Traditional methods to compute the tangent map using finite differencing often fail in accurately computing these quantities. Due to nonlinear effects of the flow, perturbations of trajectories mapped forward by the tangent map may grow excessively and they collapse on the dominant eigenvector of the map. We describe alternative techniques to overcome these issues. Both continuous or discrete QR factorization and singular value decompositions are used to automatically carry out computation of Lyapunov exponents and directions. Results on sum of Lyapunov exponents for divergent free flow, as well as sensitivity to integration time are compared in contrast to previous methods.   

Variability of reaction in chaotic flows - an approach based on Lagrangian coherent structures

The study of reactive-diffusive systems in the presence of background flows is an important problem of biological, geophysical and engineering interest. The coupling between stirring and reaction brings new complexity, which may lead to strong variability of the outcome of reaction, as compared to homogeneous reaction processes. In this talk, I will discuss several examples of reaction processes, whose variability can be tied to Lagrangian coherent structures (LCS), the deterministic tool developed to address passive scalar transport in chaotic flows. We find that LCS plays different roles in different reaction processes, but the overall strategy to approach such problems has a unified theme.   

Effects on finite-time scalar statistics by partitioning metric

When partitioning a nonlinear aperiodic dynamic system into different regions identified by Lagrangian coherent structures (LCS) there are two approaches, geometric and probabilistic; each offering a handful of different metrics. We consider stochastic scalar dispersion associated with LCS and compare the statistics of the separate flow partitions as identified by several partitioning methods. The differences of the resident time curves between methods indicate the effectiveness of that partitioning method for objectively partitioning the flow into topologically distinct regions. In this talk we explore such correlation between methods and statistics, and effective mixing.   

Getting Things Sorted With Lagrangian Coherent Structures

The dispersion of a tracer in a fluid flow is influenced by the Lagrangian motion of fluid elements. Even in laminar regimes, the irregular chaotic behavior of a fluid flow can lead to effective stirring that rapidly redistributes a tracer throughout the domain. For flows with arbitrary time-dependence, the modern approach of Lagrangian Coherent Structures (LCSs) provide a method for identifying the key material lines that organize flow transport. When the advected tracer particles possess a finite size and nontrivial shape, however, their dynamics can differ markedly from passive tracers, thus affecting the dispersion phenomena. We present details of numerical simulations and laboratory experiments that investigate the behavior of finite size particles in 2-dimensional chaotic flows. We show that the shape and the size of the particles alter the underlying LCSs, facilitating segregation between tracers of different shape in the same flow field.   

Lagrangian Coherent Structures are templates for reaction initiation between initially distant scalars

Lagrangian Coherent Structures (LCS) are shown to be effective templates for the location of reactions between initially distant scalars in 2D flows. Computations of reactions and finite-time Lyapunov exponent (FTLE) fields demonstrate that reactions are initiated when the scalars come into contact on a common FTLE ridge at a time that depends upon the initial condition.~ To show robustness of the phenomenon, a hierarchical set of three numerical flows is used:~ the periodic wake downstream of a stationary cylinder, a chaotic double gyre flow, and a chaotic, aperiodic flow consisting of interacting Taylor vortices. Coalescence of highly concentrated filaments leads to transient reaction rates that are orders of magnitude greater than predicted by the well-mixed state. As a consequence, we show that chaotic flows, known for their ability to efficiently dilute scalars, also have the competing effect of organizing initially distant scalars along the LCS at timescales shorter than that required for dilution.   

Lagrangian coherent structures and the dynamics of inertial particles

In this work we investigate the dynamics of inertial particles using the finite-time Lyapunov exponent (FTLE). In particular, we analyze preferential concentration of particles with nonzero Stokes number, $St$, and varying density ratio, $R$, for the double gyre vector field. We find that heavy particles (aerosols) tend to accumulate strongly onto negative-time (attracting) FTLE ridges of the non-inertial fluid particles, while lighter particles (bubbles) tend to repel from these ridges and accumulate at vortex cores. The transition of the negative-time FTLE ridges from attractors to repellers, based on the value of $R$, partially explains the preferential concentration of inertial particles. We also investigate the inertial finite-time Lyapunov exponent (iFTLE) based on the trajectories of inertial particles. The iFTLE is used to quantify the effect of $St$ and $R$ on particle stirring, and we present preliminary results establishing a connection between iFTLE and the two-point dispersion. Finally, we analyze the low-pass filtering effect of Stokes number on particle trajectories.   

Search strategy in a complex and dynamic environment (the Indian Ocean case)

The disappearance of Malaysia Airlines Flight 370 (MH370) in the early morning hours of 8 March 2014 has exposed the disconcerting lack of efficient methods for identifying where to look and how to look for missing objects in a complex and dynamic environment. The search area for plane debris is a remote part of the Indian Ocean. Searches, of the lawnmower type, have been unsuccessful so far. Lagrangian kinematics of mesoscale features are visible in hypergraph maps of the Indian Ocean surface currents. Without a precise knowledge of the crash site, these maps give an estimate of the time evolution of any initial distribution of plane debris and permits the design of a search strategy. The Dynamic Spectral Multiscale Coverage search algorithm is modified to search a spatial distribution of targets that is evolving with time following the dynamic of ocean surface currents. Trajectories are generated for multiple search agents such that their spatial coverage converges to the target distribution. Central to this DSMC algorithm is a metric for the ergodicity.   

Untangling tracer trajectories and clarifying coherence in 2D flows using braid theory

Interpreting ocean surface transport is crucial to many areas of oceanography, ranging from marine ecology to pollution management. To better understand surface mixing, we investigate a braid theory method to detect transport barriers bounding coherent structures in two-dimensional fluid flows. Whereas most existing techniques rely on an extensive spatiotemporal knowledge of the flow field, we seek to identify these structures from sparse data sets involving trajectories of a few tracer particles or floats. We present the results of model and laboratory experimental studies to test the robustness and applicability of the braid theory method, and discuss the potential applicability to oceanic data sets.   

Locating coherent material vortices in three-dimensional unsteady flows

Recent work has shown that coherent material vortices in two-dimensional unsteady flows are bounded by closed stationary curves of the averaged material strain [1]. These material vortex boundaries are objective (frame-invariant) Lagrangian coherent structures (LCSs) of the elliptic type, which turn out to stretch uniformly under the flow. We extend this approach to three-dimensional unsteady flows to locate toroidal and cylindrical material vortex boundaries as two-dimensional, elliptic LCS surfaces. We provide a detailed numerical procedure building on the approach in [2] and discuss several examples.\\[4pt] [1] G. Haller and F.J. Beron-Vera, Coherent Lagrangian vortices: The black holes of turbulence. J. Fluid Mech. 731 (2013) R4 \newline [2] D. Blazevski and G.Haller, Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows, Physica D, 273--274 (2014), 46-62.   

Optimal transport of diffusive scalar from the boundary

Motivated by the problem of microfluidic heat transfer, we consider the optimal control of advection-diffusion in Stokes flows in two dimensional bounded domains. Our aim is to identify the incompressible velocity fields which result in most efficient transport of a diffusive scalar from boundary. We discretize the PDE using a spectral formulation, and derive the optimality conditions for the resulting system of ODEs. We assume that the optimal velocity field can be constructed by a linear combination of the available finite set of basis velocity fields. We compare the results obtained under constraints of fixed energy and fixed enstrophy. We also compare the numerical results with some theoretical predictions and bounds, and discuss the role of chaotic mixing in this process.