Control of transient growth in plane Poiseuille flow

The feedback control of perturbations of plane Poiseuille flow is considered. Due to the non-normal nature of the dynamics of the Poiseuille flow, small perturbations experience transient growth before decaying to zero in the spectrally stable regime. For high enough amplitudes, the path to zero is blocked by exciting nonlinear effects, a process that leads to turbulence. The linearised Navier–Stokes equations that describe the perturbations are transformed to a finite-dimensional state-space representation by the Galerkin method. Vortex injection through the boundaries is devised as an actuation mode. The latter is implemented by the L1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_1$$\end{document} peak-to-peak control scheme through an appropriate set of LMIs that allow the optimised trade-off between transient energy reduction and control effort. The control scheme is first applied to the linearised equations and is consequently integrated into numerical simulations of the full nonlinear three-dimensional Navier–Stokes equations. Significant reduction in the transient energy growth is achieved, and suppression of the transition to turbulence in the full numerical simulations is demonstrated for perturbations of finite initial amplitudes.