An adjoint-based approach for finding invariant solutions of Navier-Stokes equations

TL;DR

This paper introduces an adjoint-based method to efficiently find invariant solutions of the Navier-Stokes equations, successfully discovering new solutions in a chaotic flow and linking them to flow intermittencies.

Contribution

The paper presents a novel adjoint-based approach for locating invariant solutions of Navier-Stokes equations, enabling efficient convergence to equilibria and traveling waves in complex flows.

Findings

Achieved 100% convergence to invariant solutions in a chaotic flow

Discovered 21 new steady and traveling wave solutions at Re=40

Linked a specific equilibrium to flow intermittency phenomena

Abstract

We consider the incompressible Navier--Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and traveling wave solutions of the Navier--Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of 100% is observed, leading to the discovery of 21 new steady state and traveling wave solutions at Reynolds number Re=40. Some of the new invariant solutions have spatially localized structures that were previously believed to only exist on domains with large aspect ratios. We show that one of the newly found steady state solutions underpins the temporal intermittencies, i.e., high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.