Unstable Equilibria and Invariant Manifolds in Quasi-Two-Dimensional Kolmogorov-like Flow

Recent studies suggest that unstable, non-chaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role of unstable equilibrium solutions and their invariant manifolds in a weakly turbulent, electromagnetically driven, shallow fluid layer. Identifying instants when turbulent evolution slows down, we compute 31 unstable equilibria of a realistic two-dimensional model of the flow. We establish the dynamical relevance of these unstable equilibria by showing that they are closely visited by the turbulent flow. We also establish the dynamical relevance of unstable manifolds by verifying that they are shadowed by turbulent trajectories departing from the neighborhoods of unstable equilibria over large distances in state space

Flow of cerebrospinal fluid is driven by arterial pulsations and is reduced in hypertension

Arterial pulsations are thought to drive CSF flow through perivascular spaces (PVSs), but this has never been quantitatively shown. Using particle tracking to quantify CSF flow velocities in PVSs of live mice, the authors show that flow speeds match the instantaneous speeds of the pulsing artery walls that form the inner boundaries of the PVSs

The Search for Exact Coherent Structures in a Quasi-Two-Dimensional Flow

Jeffrey Tithof is a Graduate Research Assistance in the School of Physics at Georgia Tech.Presented on October 17, 2014 at 1:00 p.m. in the Jesse W. Mason Building, room 3133.Runtime: 49:35 minutesRecent theoretical advances suggest that turbulence can be characterized using unstable solutions of the Navier-Stokes equations having regular temporal behavior, called Exact Coherent Structures (ECS). Due to their experimental accessibility and theoretical tractability, two-dimensional flows provide an ideal setting for the exploration of turbulence from a dynamical systems perspective. In our talk, we present a combined numerical and experimental study of electromagnetically driven flows in a shallow layer of electrolyte. On the numerical front we present our research concerning the search for ECS in a two-dimensional Kolmogorov-like flow. We discuss the change in the dynamics of the flow as the Reynolds number is varied. For a weakly turbulent flow, we show that the turbulent trajectory explores a region of state space which contains a number of ECS. We then discuss the occurrence of states similar to these numerically computed ECS in an experimental quasi-two-dimensional Kolmogorov-like flow

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

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.Ph.D