Optimising energy growth as a tool for finding exact coherent structures

Abstract

We discuss how searching for finite amplitude disturbances of a given energy which maximise their subsequent energy growth after a certain later time $T$ can be used to probe phase space around a reference state and ultimately to find other nearby solutions. The procedure relies on the fact that of all the initial disturbances on a constant-energy hypersphere, the optimisation procedure will naturally select the one which lies nearest to the stable manifold of a nearby solution in phase space if $T$ is large enough. Then, when in its subsequent evolution, the optimal disturbance transiently approaches the new solution, a flow state at this point can be used as an initial guess to converge the solution to machine precision. We illustrate this approach in plane Couette flow by: a) rediscovering the spanwise-localised `snake' solutions of Schneider et al. (2010b); b) probing phase space at very low Reynolds numbers ($< 127.7$) where the constant linear-shear solution is believed to be the global attractor; and finally c) examining how the edge between laminar and turbulent flow evolves when stable stratification kills the turbulent attractor. We also show that the steady snake solution smoothly delocalises as unstable stratification is gradually turned on until it connects (via an intermediary global 3D solution) to 2D Rayleigh-Benard roll solutions.