  We investigate a theoretical framework for modeling fluid turbulence based on
the formalism of exact coherent structures (ECSs). Although highly promising,
existing evidence for the role of ECSs in turbulent flows is largely
circumstantial and comes primarily from idealized numerical simulations. In
particular, it remains unclear whether three-dimensional turbulent flows in
experiment shadow any ECSs. In order to conclusively answer this question, a
hierarchy of ECSs should be computed on a domain and with boundary conditions
exactly matching experiment. The present study makes the first step in this
direction by investigating a small-aspect-ratio Taylor-Couette flow with
naturally periodic boundary condition in the azimuthal direction. We describe
the structure of the chaotic set underlying turbulent flow driven by
counter-rotating cylinders and present direct numerical evidence for shadowing
of a collection of unstable relative periodic orbits and a traveling wave,
setting the stage for further experimental tests of the framework.
