Relative periodic orbits form the backbone of turbulent pipe flow

Abstract

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 also be 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, 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.

26 pages, 11 figures, 1 table

References in corpus (9)

• Asymmetric, helical and mirror-symmetric travelling waves in pipe flow
• Recurrence of Travelling Waves in Transitional Pipe Flow
• Heteroclinic connections in plane Couette flow
• Statistical analysis of coherent structures in transitional pipe flow
• Highly-symmetric travelling waves in pipe flow
• Reduction of SO(2) symmetry for spatially extended dynamical systems
• Heteroclinic path to spatially localized chaos in pipe flow
• Coherent structures in localised and global pipe turbulence
• Simple invariant solutions embedded in 2D Kolmogorov turbulence

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• Inferring symbolic dynamics of chaotic flows from persistence
• Deep Learning of Conjugate Mappings
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• Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow
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