Revealing the state space of turbulent pipe flow by symmetry reduction

Symmetry reduction by the method of slices is applied to pipe flow in order to obtain a quotient of the streamwise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their infinite-dimensional symmetry-reduced state space onto suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows. Visualizations of the flow within the slice and its linearization at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic saddle, which capture turbulent dynamics at transitional Reynolds numbers.