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

J.P. Parker; T.M. Schneider

doi:10.1017/jfm.2022.299

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

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

Mathematics

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

189 Downloads (Pure)

Abstract

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 language	English
Pages (from-to)	A17-1 - A17-14
Number of pages	14
Journal	Journal of Fluid Mechanics
Volume	941
Early online date	26 Apr 2022
DOIs	https://doi.org/10.1017/jfm.2022.299
Publication status	Published - 25 Jun 2022

Keywords

turbulent flows
nonlinear dynamical systems

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10.1017/jfm.2022.299Licence: CC BY

Final published versionFinal published version, 860 KBLicence: CC BY

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title = "Variational methods for finding periodic orbits in the incompressible Navier{\textendash}Stokes equations",

abstract = "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{\textendash}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. ",

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AU - Parker, J.P.

AU - Schneider, T.M.

PY - 2022/6/25

Y1 - 2022/6/25

N2 - 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.

AB - 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.

KW - turbulent flows

KW - nonlinear dynamical systems

UR - https://www.scopus.com/pages/publications/85129716087

U2 - 10.1017/jfm.2022.299

DO - 10.1017/jfm.2022.299

M3 - Article

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SP - A17-1 - A17-14

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -

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

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