Robust Gradient-Based Solver for Invariant Solutions to the Navier-Stokes Equations using Resolvent Analysis

Low-dimensional chaotic dynamics can be modelled in terms of periodic solutions of the governing strange attractor. Analogously, exact nonlinear solutions to the Navier-Stokes equations, called Exact Coherent Structures (ECSs), are expected to serve a similar purpose for turbulence. To find these solutions, specialised numerical techniques have been devised, which largely suffer from difficulties resulting from the high dimensionality and sensitivity of the chaotic dynamics. In addition, they are rarely applied to domains that contain walls, limiting them to flows of mostly theoretical interest. In this work, the variational optimisation methodology of Schneider (2022) is extended to find periodic flows in domains that include no-slip boundary conditions. The method relies on a Galerkin projection of the “optimisation dynamics”, defined by a modal basis obtained from Resolvent Analysis (RA). RA is a technique that takes a base/mean flow and the Navier-Stokes equations as input and provides a set of response modes ranked by their receptivity to harmonic forcing of the Navier-Stokes equations. These Resolvent response modes have been shown to provide an efficient basis of ECSs for certain wall-bounded flows. This methodology is demonstrated on the Rotating Plane Couette Flow, a limiting case of the general Taylor-Couette flow, focusing on the convergence rates and accuracy of the solutions obtained.