Novel methods of dimensionality reduction applied to a two-dimensional fluid flow

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Tithof, Jeffrey
Schatz, Michael F.
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Fluid turbulence is a ubiquitous phenomenon that has been called "the greatest unsolved problem in classical physics." Despite the fact that fluid flows are governed by the deterministic Navier-Stokes equation, turbulence is notoriously difficult to predict. This difficulty largely arises because turbulence is chaotic (i.e., it exhibits extreme sensitivity to initial conditions) and has a very large number of degrees of freedom because of its continuous spatial dependence. However, a growing body of research suggests that turbulent dynamics are effectively low-dimensional, but it is not yet known how to optimally perform dimensionality reduction to capture the dynamically-relevant dimensions. In this dissertation, two dimensionality reduction methods are explored in the context of a quasi-two-dimensional (Q2D) fluid flow. This Q2D flow can be treated as effectively 2D, making the experimental and numerical aspects of the study more tractable than that of a fully three-dimensional flow. The first method involves the calculation of exact, unstable solutions of the Navier-Stokes equation, often called "exact coherent structures" (ECS). ECS exist in the same parameter regime as turbulence and play an important role in guiding the dynamics. In this work, experimental evidence for the existence and dynamical relevance of ECS is provided, as well as the first experimental demonstration of how ECS can be used to forecast weak turbulence. The second method, known as "persistent homology," provides a powerful mathematical formalism in which well-defined geometric features of a flow field are encoded in a so-called "persistence diagram." The results presented herein demonstrate how persistence diagrams can be used to characterize individual flow fields, make pairwise comparisons, and identify periodic dynamics. The substantial progress presented in this dissertation suggests that Q2D flows provide an excellent platform for testing new approaches to understanding turbulence.
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