Quasi-two-dimensional Kolmogorov flow: Bifurcations and exact coherent structures

Abstract

Fluid turbulence is nearly ubiquitous in natural and human-made systems. Despite systematic research for over hundred years, scientists are still searching for efficient ways to forecast and control the evolution of turbulent flows. The research presented in this thesis tests and extends recent ideas aimed at developing a simplified description of turbulent evolution, with the final goal being its efficient forecasting and control. The underlying methodology includes identifying ``Exact Coherent Structures'' (ECS), which are unstable, nonchaotic solutions of the Navier-Stokes equation that describes the evolution of fluid flows. While ECS are observed only fleetingly in turbulence, they display relatively simple spatiotemporal features. Hence, being more tractable to analysis than turbulence, ECS may serve as simple building blocks in developing a simplified description of turbulent evolution. The present study explores the role of ECS in turbulence generated in a shallow electrolyte layer which is driven using a horizontal electromagnetic force with a sinusoidal spatial profile. The flow in the experiment, often termed quasi-two-dimensional (Q2D) Kolmogorov Flow, is nearly horizontal. The Q2D flow is described theoretically using a strictly 2D model, which is validated by showing quantitative agreement between its numerical simulation and the experiment in the comparison of pre-turbulent flow states and the transitions between them. Analyzing the dynamics in the weakly turbulent regime, it is identified that dramatic slowing-down in the evolution of the flow is related to turbulent trajectories in the state space visiting the neighborhoods of unstable equilibrium solutions, a class of ECS. The dynamical role of ECS is validated by showing that turbulent trajectories in the neighborhood of an unstable equiliurbium depart following its unstable manifold. Hence, turbulent evolution in the neighborhood of the equilibrium can be forecast by constructing its unstable manifold, which is demonstrated using both the experiment and simulations. This study offers unambiguous experimental evidence for the dynamical role of ECS in turbulence, as well as the first ECS based forecasting of turbulent evolution.