Summary
Invariant solutions to the Navier-Stokes equations, known as Exact Coherent Structures (ECSs), are known to represent important physical mechanisms of turbulent flows. Recently a variational optimisation approach based on gradient descent was proposed as a robust numerical method for finding these solutions. However, its practical implementation has proven challenging due to the high dimensionality of the problem and the slow convergence rate. Moreover, the method is not applicable to domains containing walls. In this work the variational optimisation methodology is extended to find time periodic solutions in domains that contain walls using a Galerkin projection approach. The basis used to construct the subspace onto which the projection is performed is derived using Resolvent analysis, which has been shown to provide an efficient representation of the invariant solutions for some wall-bounded flows. By truncating the modes used a significant dimensionality reduction is achieved while retaining much of the important dynamical information. This effectively acts as a preconditioner and thereby increases the convergence rate of the optimiser. Additionally, the effect on the convergence rate when replacing the gradient descent algorithm with quasi-Newton algorithms for the optimisation is analysed. The Rotating Plane Couette Flow (RPCF) is chosen to demonstrate the modified optimisation methodology.
