Constructing invariant solutions of wall-bounded shear flows by a Jacobian-free adjoint-based method*

The dynamics of fluid turbulence is underpinned by invariant solutions embedded in the state space of the governing equations. Finding an invariant solution of a certain type can be viewed as an optimization problem over space-time fields with prescribed temporal behavior: minimizing a cost function that penalizes the deviation of space-time fields from being a solution to the governing equations. We propose a Jacobian-free algorithm based on an adjoint-based minimization technique for constructing invariant solutions of wall-bounded shear flows. We demonstrate the feasibility of the algorithm by applying it to plane Couette and plane Poiseuille flows. Unlike the state-of-the-art Newton-based alternatives, this approach is robust to inaccurate initial guesses, and is not affected by the exponential separation of trajectories. We also propose a data-driven procedure for accelerating the convergence of the adjoint-descent algorithm.