How to compute periodic orbits and equilibria of Navier-Stokes without suffering from chaos

Unstable non-chaotic invariant solutions of the Navier-Stokes equations capture transitional turbulent flow dynamics, yet numerically identifying equilibria and periodic orbits using common shooting methods has remained challenging. We thus propose a class of alternative computational methods for identifying the non-chaotic building blocks of fluid turbulence. These are variational methods unaffected by exponential error amplification associated with time-marching a chaotic system. Technically, we use adjoints to solve an optimization problem whose global minima represent invariant solutions. For incompressible 3D shear flows we treat the nonlocal pressure constraint within the adjoint formulation via an adaptation of the Kleiser-Schumann influence matrix method. We compute multiple equilibria and periodic orbits of different canonical shear flows, highlighting the robustness of the methodology.