2017/05/31 by Nazmi Burak Budanur, Kimberly Y. Short, Kimberly Short +3
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Attractor #Chaotic #Classical mechanics #Computer science #Dynamical systems theory #Flow (mathematics) #Fluid Dynamics and Turbulent Flows #Invariant (physics) #Lorenz system #Mathematical analysis #Mathematical physics #Mathematics #Mechanics #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Periodic orbits #Physics #Quantum chaos and dynamical systems #Saddle #Statistical physics #Turbulence #physics.flu-dyn
paper · pdf · doi:10.1017/jfm.2017.699
published as J. Fluid Mech. 833, 274-301 (2017) · 26 pages, 11 figures, 1 table
arxiv created 2017/08/03 · openalex publication_date 2017/11/06 · arxiv updated 2019/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
The chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this is also the case for the infinite-dimensional dynamics of Navier–Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics – specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier–Stokes equations as they do in lower-dimensional systems – is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell at Re=2500 , and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.