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

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.

Automated summary

This article proposes a family of methods to converge unstable periodic orbits for the incompressible Navier-Stokes equations. The methods are based on variations of an integral objective functional and traditional gradient-based optimization strategies. The variational methods are applied to a specific case of two-dimensional Kolmogorov flow and compared with existing Newton iteration-based shooting methods. The methods are computationally slow but able to converge from inaccurate initial guesses.