Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow

Abstract

Subcritical transition in plane Poiseuilleflow is investigated by means of aLagrange-multiplier direct-adjoint optimization procedure with the aim offinding localized three-dimensional perturbations optimally growing in a giventime interval(target time). Space localization of these optimal perturbations(OPs)is achieved by choosing as objective function either a p-norm(withp1)of the perturbation energy density in a linear framework; or theclassical(1-norm)perturbation energy, including nonlinear effects. This workaims at analyzing the structure of linear and nonlinear localized OPs forPoiseuilleflow, and comparing their transition thresholds and scenarios. Thenonlinear optimization approach provides three types of solutions: a weaklynonlinear, a hairpin-like and a highly nonlinear optimal perturbation,depending on the value of the initial energy and the target time. The formershows localization only in the wall-normal direction, whereas the latterappears much more localized and breaks the spanwise symmetry found atlower target times. Both solutions show spanwise inclined vortices and largevalues of the streamwise component of velocity already at the initial time. Onthe other hand, p-norm optimal perturbations, although being strongly loca-lized in space, keep a shape similar to linear 1-norm optimal perturbations,showing streamwise-aligned vortices characterized by low values of thestreamwise velocity component. When used for initializing direct numericalsimulations, in most of the cases nonlinear OPs provide the most efficientroute to transition in terms of time to transition and initial energy, even whenthey are less localized in space than the p-norm OP. The p-norm OP follows a transition path similar to the oblique transition scenario, with slightly oscil-lating streaks which saturate and eventually experience secondary instability.On the other hand, the nonlinear OP rapidly forms large-amplitude bentstreaks and skips the phases of streak saturation, providing a contemporarygrowth of all of the velocity components due to strong nonlinear coupling.