Exact coherent structures and dynamical connections in a quasi 2D Kolmogorov like flow

Abstract

Turbulence in fluid flows is ubiquitous. Scientists and engineers have been trying to understand it for centuries, yet it remains a rather mysterious phenomenon. This thesis explores a modern approach for describing turbulence using unstable nonchaotic solutions of the Navier-Stokes equation, called exact coherent structures (ECS). It has been conjectured that a hierarchy of ECS forms the skeleton of fluid turbulence: each ECS guides the dynamics of the flow in its vicinity, with apparent randomness arising from the flow moving from the neighborhood of one ECS to the neighborhood of another. Some of the most unexpected discoveries in developing and testing this conjecture have been made in a system -- Kolmogorov-like flow in a thin fluid layer of electrolyte driven by Lorentz force suspended above a thin lubricating layer of a fluid dielectric. Due to its effective two-dimensionality, this flow offers an unprecedented level of access both experimentally and numerically, which would be too difficult in most three-dimensional flows. We have developed, and implemented numerically, an improved ``weakly-compressible'' model of the flow which retains the simplicity of the ``incompressible'' two-dimensional model introduced previously, but also accounts for the thickness variation of the two fluid layers. The improved model has been shown to offer a more accurate description of the transition to turbulence in this system, compared with its incompressible predecessor. There are almost no previous studies that explain how and why the turbulent flow moves from the neighborhood of one ECS to the neighborhood of another. One possibility is that the flow follows another type of unstable solutions -- heteroclinic connections between ECS. However, no reliable methods for computing them for such complicated systems as a fluid flow have been developed previously. We have developed and tested several robust and efficient numerical algorithms for computing both ECS and dynamical connections in high-dimensional dynamical systems. These algorithms should facilitate the next step in the development of the geometric, deterministic description of fluid turbulence.