Basis for finding exact coherent states

M. Arslan Ahmed and Ati S. Sharma
Phys. Rev. E 101, 012213 – Published 24 January 2020

Abstract

One of the outstanding problems in the dynamical systems approach to turbulence is to find a sufficient number of invariant solutions to characterize the underlying dynamics of turbulence [Annu. Rev. Fluid Mech. 44, 203 (2012)]. As a practical matter, the solutions can be difficult to find. To improve this situation, we show how to find periodic orbits and equilibria in plane Couette flow by projecting pseudorecurrent segments of turbulent trajectories onto the left-singular vectors of the Navier-Stokes equations linearized about the relevant mean flow (resolvent modes). The projections are, subsequently, used to initiate Newton-Krylov-hookstep searches, and new (relative) periodic orbits and equilibria are discovered. We call the process project-then-search and validate the process by first applying it to previously known fixed point and periodic solutions. Along the way, we find new branches of equilibria, which include bifurcations from previously known branches, and new periodic orbits that closely shadow turbulent trajectories in state space.

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  • Received 12 July 2019
  • Revised 2 January 2020

DOI:https://doi.org/10.1103/PhysRevE.101.012213

©2020 American Physical Society

Physics Subject Headings (PhySH)

Nonlinear DynamicsFluid Dynamics

Authors & Affiliations

M. Arslan Ahmed*

  • Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA

Ati S. Sharma

  • Department of Aerospace Engineering, University of Southampton, Southampton SO17 1BJ, United Kingdom and Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106, USA

  • *arslan@caltech.edu
  • a.sharma@soton.ac.uk

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Vol. 101, Iss. 1 — January 2020

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