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

TL;DR

This paper introduces variational methods to compute unstable periodic orbits in the incompressible Navier-Stokes equations, demonstrating their ability to converge from poor initial guesses in turbulent flow simulations.

Contribution

It develops a family of variational approaches for finding unstable periodic orbits in fluid dynamics, offering an alternative to traditional Newton-based methods.

Findings

Methods successfully converge from inaccurate initial guesses.

Approaches are applied to 2D Kolmogorov flow.

Compared with existing shooting methods, showing different convergence properties.

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.