Geometry of 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, ...) which form a rigid backbone which enables us to describe and predict the sinuous motions of a turbulent fluid. We report 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 to construct 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, time-recurrent dynamical coherent structures typical of the turbulent flow. PACS numbers: 05.45.-a, 47.10.ad, 47.10.Fg, 47.11.-j, 47.27.-i, 47.27.De, 47.27.ed, 47.27.ek, 47.27.N-, 47.27.nd Submitted to: Phys. Scr.