A Quantitative Evaluation of Lagrangian Coherent Structure Detection Methods Based on Computational and Experimental Limitations

Abstract

Lagrangian coherent structures are used in uid mechanics and the analysis of dynamic systems to visualise the most in uential ow structures present within a velocity system over a nite period of time. Over the last two decades, a wide variety of methods have been conceptualised for the numerical detection of various forms of these structures within di erent ows. These include continuous curves of maximal particle repulsion which act as ow barriers, two dimensional objects such as jets or eddies formed from more robust ow behaviour, or larger partitions which remain separated from the rest of the domain over an entire ow interval. While some studies which focus on comparing the basic functionality of groups of these methods have been undertaken, the impact of certain computational factors such as the uncertainty of velocity data or the available resolution of said data on the resultant structures generated from these methods has seldom been investigated. In this Thesis, we address both of these issues by performing a systematic analysis of eight of these Lagrangian coherent structure detection methods using a variety of velocity systems including analytically de ned ows (such as the Double Gyre, a non-autonomous Stuart vortex system and the Bickley jet), computational uid dynamics velocity data (corresponding to ows which each contain two layers of Kelvin-Helmholtz instability) and an oceanographic velocity data set representing the Gulf Stream. The methods we consider here are the nite time Lyapunov exponent (a measure of the exponential stretching rate of ow trajectories), variational Lagrangian coherent structures (geodesic solutions of variational problems related to ow stretching), Lagrangian averaged vorticity deviation (an objective measure of the vorticity of a ow trajectory against that of the entire domain), stochastic sensitivity (the expected uncertainty of a Lagrangian ow trajectory), the transfer operator (a probabilistic method which seeks density distributions that remain coherent), the dynamic Laplace operator (an extension of the transfer operator method which explicitly includes di usivity), fuzzy c-means clustering (grouping together collections of ow trajectories based on their consistent proximity) and coherent structure colouring (identifying coherent ow objects from how similarly groups of ow trajectories evolve as a ow advances). We compare the types of Lagrangian coherent structure each method is able to produce, and test how these methods react to the addition of stochastic noise to the velocity data which represents a ow. From our results, methods which detect two-dimensional coherent ow structures rather than the boundaries which separate them, such as coherent structure colouring, Lagrangian averaged vorticity deviation, stochastic sensitivity, the transfer operator and dynamic Laplace operator; are less sensitive to velocity uncertainty and give a more thorough picture of the most in uential ow behaviour observable. We also perform a detailed analysis on the impact of spatial resolution in comparison to the size of coherent structures for each of the methods, both qualitatively by visually comparing the coherent structures produced and quantitatively using the absolute errors of various LCS quantities against a \reference case" produced from the best velocity data resolution available.