Coherent structures in incompressible fluid flows

Abstract

The work is broadly related to the transition to turbulence in pipe at intermediate Reynolds numbers and includes a discussion of two classes of structures observed during the transition to turbulence: numerically-extracted solutions of the Navier-Stokes equations (the ``exact'') and localized/patterned turbulent spots that are not themselves solutions of the Navier-Stokes equation but are nonetheless pervasive during the transition (the ``inexact''). High-dimensional descriptions of turbulence is predicted by periodic orbit theory (POT) which expects to describe turbulence exactly, as opposed to approximately. The search for relative periodic orbits and traveling waves in pipe---the constituents solutions to periodic orbit theory---are discussed. The successful search for relative periodic orbits at transitional Reynolds numbers gave a catalogue of invariant solutions; many of these solutions were continued in parameter space to find that solutions coexist with the transitional regime. In addition to collecting solutions to eventually test periodic orbit theory, an investigation into the Barkley's pipe model---a system that successfully models the transition to turbulence in pipe---was undertaken.