Geometry of the turbulence in wall-bounded shear flows: periodic orbits

The dynamical theory of moderate Reynolds number turbulence triangulates the infinite-dimensional Navier-Stokes state space by sets of exact solutions (equilibria, relative equilibria, periodic orbits, etc), which form a rigid backbone that enables us to describe and predict the sinuous motions of a turbulent fluid. We report on the determination of a set of unstable periodic orbits from close recurrences of the turbulent flow. A few equilibria that closely resemble frequently observed but unstable coherent structures are used for constructing a low-dimensional state-space projection from the extremely high-dimensional data sets. The turbulent flow can then be visualized as a sequence of close passages to unstable periodic orbits, i.e the time-recurrent dynamical coherent structures typical of a turbulent flow.