Relative periodic orbits form the backbone of turbulent pipe flow Journal Article

Author(s): Budanur, Nazmi B; Short, Kimberly Y; Farazmand, Mohammad; Willis, Ashley P; Cvitanović, Predrag
Article Title: Relative periodic orbits form the backbone of turbulent pipe flow
Affiliation IST Austria
Abstract: 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.
Keywords: Nonlinear dynamical systems; turbulent flows; turbulence modelling
Journal Title: Journal of Fluid Mechanics
Volume: 833
ISSN: 1469-7645
Publisher: Cambridge University Press  
Date Published: 2017-12-25
Start Page: 274
End Page: 301
DOI: 10.1017/jfm.2017.699
Notes: We are indebted to J. F. Gibson for many inspiring discussions. We are grateful to the Kavli Institute for Theoretical Physics, where the collaboration was supported in part by the National Science Foundation under grant no. NSF PHY11-25915, for hospitality. A.P.W. was supported by the EPSRC grant EP/K03636X/1. K.Y.S. was supported by the NSF Graduate Research Fellowship Grant NSF DGE-0707424, P.C. was partly supported by NSF Grant DMS-0807574, and thanks the family of G. Robinson Jr for support.
Open access: yes (repository)