Searching for invariant solutions to wall-bounded flows using resolvent-based optimisation

We present a robust optimisation framework for computing invariant solutions of wall-bounded flows by recasting the Navier–Stokes equations as a variational problem as established in Ashtari & Schneider ( J. Fluid Mech ., vol. 977, 2023, A7). The approach minimises the residual of the governing equations over a finite-time horizon, seeking periodic or equilibrium solutions. A novel contribution is made by including a Galerkin projection onto a basis of divergence-free modes that satisfy the no-slip boundary conditions. This projection not only makes the variational framework applicable to wall-bounded flows but it also yields a low-order representation of the dynamics. The basis is derived from resolvent analysis, which provides an orthonormal set. We demonstrate the method on a streamwise invariant formulation of rotating plane Couette flow, obtaining exact equilibrium and periodic solutions consistent with direct numerical simulations. The conditioning of the optimisation problem is analysed in detail, showing that convergence rates depend on the stability properties of the targeted solutions. Finally, we highlight a direct link between the conditioning of the optimisation and the structure of the resolvent operator, suggesting a unifying perspective on both the efficiency of the optimisation and the dynamical significance of resolvent modes.