Variational identification of minimal seeds to trigger transition in plane Couette flow

Abstract

A variational formulation incorporating the full Navier-Stokes equations is used to identify initial perturbations with finite kinetic energy E_{0} which generate the largest gain in perturbation kinetic energy (across all possible time intervals) for plane Couette flow. Two different representative flow geometries are chosen corresponding to those used previously by Butler & Farrell (1992) and Monokrousos et al. (2011). In the former (smaller geometry) case as E_{0} increases from 0, we find an optimal which is a smooth nonlinear continuation of the well-known linear result at $E_{0} = 0$. At $E_{0} = E_{c}$, however, completely unrelated states are uncovered which trigger turbulence and our algorithm consequently fails to converge. As $E_{0} \rightarrow E^{+}_{c}, we find good evidence that the turbulence triggering initial conditions approach a 'minimal seed' which corresponds to the state of lowest energy on the laminar-turbulent basin boundary or 'edge'. This situation is repeated in the Monokrousos et al. (2011) (larger) geometry albeit with one notable new feature - the appearance of a nonlinear optimal (as found recently in pipe flow by Pringle & Kerswell (2010) and boundary layer flow by Cherubini et al. (2010)) at finite $E_{0} < E_{c}$ which has a very different structure to the linear optimal. Again the minimal seed at $E_{0} = E_{c}$ does not resemble the linear or now the nonlinear optimal. Our results support the first of two conjectures recently posed by Pringle et al. (2011) but contradict the second. Importantly, their prediction that the form of the functional optimised is not important for identifying E_{c} providing heightened values are produced by turbulent flows is confirmed: we find the what looks to be the same E_{c} and minimal seed using energy gain as opposed to total dissipation in the Monokrousos et al. (2011) geometry.