The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

The finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here, we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second-order velocity structure function.

A Lagrangian Approach Towards Quantitative Analysis Of Flow-mediated Infection Transmission In Indoor Spaces With Application To SARS-COV-2

The ongoing SARS-CoV-2 (Covid-19) pandemic has ushered an unforeseen level of global health and economic burden. As a respiratory infection, Covid-19 is known to have a dominant airborne transmission modality, wherein fluid flow plays a central role. Quantification of complex non-intuitive dynamics and transport of pathogen laden respiratory particles in indoor flows has been of specific interest. Here we present a Lagrangian computational approach towards quantification of human-to-human exposure quantifiers, and identification of pathways by which flow organizes transmission. We develop a Lagrangian viral exposure index in a parametric form, accounting for key parameters such as building and layout, ventilation, occupancy, biological variables. We also employ a Lagrangian computation of the Finite Time Lyapunov Exponent field to identify hidden patterns of transport. A systematic parametric study comprising a set of 120 simulations, yielding a total of 1,320 different exposure index computations are presented. Results from these simulations enable: (a) understanding the otherwise hidden ways in which air flow organizes the long-range transport of such particles; and (b) translating the micro-particle transport data into a quantifier for understanding infection exposure risks.

The Finite Size Lyapunov Exponent and the Finite Amplitude Growth Rate

The Finite size Lyapunov exponent (FSLE) has been used extensively since the late 1990’s to diagnose turbulent regimes from Lagrangian experiments and to detect Lagrangian coherent structures in geophysical flows and two-dimensional turbulence. Historically, the FSLE was defined in terms of its computational method rather than via a mathematical formulation, and the behavior of the FSLE in the turbulent inertial ranges is based primarily on scaling arguments. Here we propose an exact definition of the FSLE based on conditional averaging of the finite amplitude growth rate (FAGR) of the particle pair separation. With this new definition, we show that the FSLE is a close proxy for the inverse structural time, a concept introduced a decade before the FSLE. The (in)dependence of the FSLE on initial conditions is also discussed, as well as the links between the FAGR and other relevant Lagrangian metrics, such as the finite time Lyapunov exponent and the second order velocity structure function.

Go with the FLOW: visualizing spatiotemporal dynamics in optical widefield calcium imaging

Widefield calcium imaging has recently emerged as a powerful experimental technique to record coordinated large-scale brain activity. These measurements present a unique opportunity to characterize spatiotemporally coherent structures that underlie neural activity across many regions of the brain. In this work, we leverage analytic techniques from fluid dynamics to develop a visualization framework that highlights features of flow across the cortex, mapping wavefronts that may be correlated with behavioural events. First, we transform the time series of widefield calcium images into time-varying vector fields using optic flow. Next, we extract concise diagrams summarizing the dynamics, which we refer to as FLOW (flow lines in optical widefield imaging) portraits . These FLOW portraits provide an intuitive map of dynamic calcium activity, including regions of initiation and termination, as well as the direction and extent of activity spread. To extract these structures, we use the finite-time Lyapunov exponent technique developed to analyse time-varying manifolds in unsteady fluids. Importantly, our approach captures coherent structures that are poorly represented by traditional modal decomposition techniques. We demonstrate the application of FLOW portraits on three simple synthetic datasets and two widefield calcium imaging datasets, including cortical waves in the developing mouse and spontaneous cortical activity in an adult mouse.

Particle position prediction based on Lagrangian coherency for flow over a cylinder in 4D-PTV

Recent developments in time-resolved Particle Tracking Velocimetry (4D-PTV) consistently improved tracking accuracy and robustness. We propose a novel technique named ”Lagrangian coherent predictor” to estimate particle positions within the 4D-PTV algorithm. We add spatial and temporal coherency information of neighbour particles to predict a single trajectory using Lagrangian Coherent Structures (LCS). We found that even a weak signal from coherent neighbour motions improves particle prediction accuracy in complex flow regions. We applied Finite Time Lyapunov Exponent (FTLE) to quantify local boundaries (i.e. ridges) of coherent motions. Synthetic analysis of the wake behind a smooth cylinder at Reynolds number equal to 3900 showed enhanced estimation compared with the recent predictor functions employed in 4D-PTV. Results of the experimental study of the same flow configuration are reported. We compared predicted positions with the optimised final positions of Shake The Box (STB). It was found that the Lagrangian coherent predictor succeeded in estimating particle positions with minimum deviation to the optimised positions.

Adjustable window for 2D PIV estimation based on local Lagrangian coherency

We present a novel approach to adjust shapes of the interrogation windows (IW) in Particle Image Velocimetry (PIV) measurements as a function of temporal and spatial local coherent motions. Lagrangian Coherent Structures (LCS) has been widely utilized to determine local flow boundaries. We propose using Finite-Time Lyapunov Exponent (FTLE) to quantify LCS separatrix boundaries (i.e. ridges) and adjust the interrogation window. We integrated the proposed method with a local optical flow PIV algorithm. The evaluation was performed using synthetic particle images of 2D homogeneous isotropic turbulence obtained from Direct Numerical Simulation (DNS). The results showed significant improvements in regions with complex flow behaviours, particularly shear, vortex and hyperbolic motions. We studied improvements of the velocity estimation in a real experiment of the wake flow behind a cylinder at Reynolds number equal to 3900. It was found that optical flow featured by coherency based interrogation window (coherent optical flow) reveals detailed vector field estimations in regions with complex behaviours inside the wake flow.

Sensitive dependence of trajectories on tracer seeding positions – coherent structures in German Bight backward drift simulations

Backward drift simulations can aid the interpretation of in situ monitoring data. In some cases, however, trajectories are very sensitive to even small changes in the tracer release position. A corresponding spread of backward simulations implies attraction in the forward passage of time and, hence, uncertainty about the probed water body's origin. This study examines surface drift simulations in the German Bight (North Sea). Lines across which drift behaviour changes non-smoothly are obtained as ridges in the fields of the finite-time Lyapunov exponent (FTLE), a parameter used in dynamical systems theory to identify Lagrangian coherent structures (LCSs). Results closely resemble those obtained considering two-particle relative dispersion. It is argued that simulated FTLE fields might be used in support of the interpretation of monitoring data, indicating when simulations of backward trajectories are unreliable because of their high sensitivity to tracer seeding positions.

The validation of new phase-dependent gait stability measures: a modelling approach

Identification of individuals at risk of falling is important when designing fall prevention methods. Current measures that estimate gait stability and robustness appear limited in predicting falls in older adults. Inspired by recent findings on changes in phase-dependent local stability within a gait cycle, we devised several phase-dependent stability measures and tested for their usefulness to predict gait robustness in compass walker models. These measures are closely related to the often-employed maximum finite-time Lyapunov exponent and maximum Floquet multiplier that both assess a system's response to infinitesimal perturbations. As such, they entail linearizing the system, but this is realized in a rotating hypersurface orthogonal to the period-one solution followed by estimating the trajectory-normal divergence rate of the swing phases and the foot strikes. We correlated the measures with gait robustness, i.e. the largest perturbation a walker can handle, in two compass walker models with either point or circular feet to estimate their prediction accuracy. To also test for the dependence of the measures under state space transform, we represented the point feet walker in both Euler–Lagrange and Hamiltonian canonical form. Our simulations revealed that for most of the measures their correlation with gait robustness differs between models and between different state space forms. In particular, the latter may jeopardize many stability measures' predictive capacity for gait robustness. The only exception that consistently displayed strong correlations is the divergence of foot strike. Our results admit challenges of using phase-dependent stability measures as objective means to estimate the risk of falling.