The main theme of this thesis is application of recently developed tools in dynamical systems theory in the study of incompressible fluid flows. These applications fall into two general categories: the first one is study of the flow evolution as an infinite-dimensional dynamical system and it is related to classical topics like flow stability and transition. The second area is the study of flow kinematics where tools of dynamical systems are used to study the trajectory of particles immersed within the flow, and includes topics like mixing enhancement, and prediction of pollution movement in the ocean or atmosphere.
In studying flow dynamics, we utilize the Koopman operator framework for data-driven study of dynamical systems (introduced in chapter 1). The increasing popularity of this framework is due to a versatile combination of rigorous theory and data analysis algorithms which allows extraction of dynamic information from almost any type of data from a dynamical system. In chapter 2, we use the spectral properties of the Koopman operator, computed from data, to interpret the flow dynamics: we use the Koopman spectra to determine the attractor geometry, the Koopman eigenfunctions to map the state space linear coordinates, and the Koopman modes to characterize the unsteady motion in the flow domain.
We also discuss the numerical computation of the Koopman spectral properties. As of now, this computation is mostly done through a class of numerical algorithms known as Dynamic Mode Decomposition (DMD). In chapter 3, we prove the convergence of a class of DMD algorithms, called Hankel-DMD, for systems with ergodic attractors. Our proofs are based on the fact that projections in the space of functions can be approximated via vector projections in DMD by the virtue of Birkhoff's ergodic theorem. This new result also provides insight on dynamics of chaotic systems with continuous spectrum and computation of Koopman eigenvalues for dissipative systems. We compare the performance of Hankel-DMD to the signal processing techniques used for fluid flows in chapter 2.
One of the important questions in the study of flow kinematics is how to characterize the mixing in flows with aperiodic time dependence. This question has given rise to a variety of methodologies that strive to describe the mixing in a given aperiodic flow by detecting the coherent structures or other objects of special interest. In chapter 4, we study this problem from a different perspective, namely, we
consider how the mixing portrait is changed while the temporal regime of a bounded flow -
the lid-driven cavity flow - changes from steady to aperiodic. We use the Koopman spectral properties of the flow (studied in chapter 2) and the so-called hypergraphs to isolate and characterize the effect of different elements from the flow dynamics
on the mixing. For example, we will see how the interaction between vorticity distribution in the mean flow and non-zero Koopman frequencies determines the regions of slow mixing.
In chapter 5, we report on application of hypergraphs combined with high-frequency radar data in the study of surface mixing in Santa Barbara channel (i.e the patch of Pacific Ocean between Santa Barbara coastline and Channel Islands) with special focus on prediction of oil slick movements in the aftermath of 2015 Refugio oil spill.