Nonlinear equilibration of two-dimensional optimal perturbations in viscous shear flow

Abstract

Two-dimensional perturbations configured for maximum energy growth in laminar viscous shear flow are shown to develop into quasisteady finite amplitude structures, provided that the initial perturbation has sufficient energy and a nearby nonlinear mode exists. For Poiseuille flow, which supports finite amplitude equilibria for Reynolds numbers above ∼2900, an optimal perturbation with initial energy density equal to or greater than 0.1% of the mean flow energy density closely approaches the quasiequilibrium state within 10 advective time units. For Couette flow, which has no finite amplitude solution, the optimal perturbations decay rapidly after reaching maximum amplitude unless the configuration is sufficiently close to a linear mode with slow exponential decay rate. While the quasiequilibrium structure for Poiseuille flow is locally inflectional, it supports only weak instabilities with scales larger than the local region.