Variational methods for finding periodic orbits in the incompressible Navier–Stokes equations

J.P. Parker (Lead / Corresponding author), T.M. Schneider

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)
16 Downloads (Pure)


Unstable periodic orbits are believed to underpin the dynamics of turbulence, but by their nature are hard to find computationally. We present a family of methods to converge such unstable periodic orbits for the incompressible Navier–Stokes equations, based on variations of an integral objective functional, and using traditional gradient-based optimisation strategies. Different approaches for handling the incompressibility condition are considered. The variational methods are applied to the specific case of periodic, two-dimensional Kolmogorov flow and compared against existing Newton iteration-based shooting methods. While computationally slow, our methods converge from very inaccurate initial guesses.

Original languageEnglish
Pages (from-to)A17-1 - A17-14
Number of pages14
JournalJournal of Fluid Mechanics
Early online date26 Apr 2022
Publication statusPublished - 25 Jun 2022


  • turbulent flows
  • nonlinear dynamical systems


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