  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.
