A SYSTEMS-THEORY APPROACH TO THE FEEDBACK STABILIZATION OF INFINITESIMAL AND FINITE-AMPLITUDE DISTURBANCES IN PLANE POISEUILLE FLOW

A systems theory framework is presented for the linear stabilization o f two-dimensional laminar plane Poiseuille flow. The governing lineari zed Navier-Stokes equations are converted to control-theoretic models using a numerical discretization scheme. Fluid system poles, which are closely related to Orr-Sommerfeld eigenvalues, and fluid system zeros are computed using the control-theoretic models. It is shown that the location of system zeros, in addition to the well-studied system eige nvalues, are important in linear stability control. The location of sy stem zeros determines the effect of feedback control on both stable an d unstable eigenvalues. In addition, system zeros can be used to deter mine sensor locations that lead to simple feedback control schemes. Fe edback controllers are designed that make a new fluid-actuator-sensor- controller system linearly stable. Feedback control is shown to be rob ust to a wide range of Reynolds numbers. The systems theory concepts o f modal controllability and observability are used to show that feedba ck control can lead to short periods of high-amplitude transients that are unseen at the output. These transients may invalidate the linear model, stimulate nonlinear effects, and/or form a path of 'bypass' tra nsition in a controlled system. Numerical simulations are presented to validate the stabilization of both single-wavenumber and multiple-wav enumber instabilities. Finally, it is shown that a controller designed upon linear theory also has a strong stabilizing effect on two-dimens ional finite-amplitude disturbances. As a result, secondary instabilit ies due to infinitesimal three-dimensional disturbances in the presenc e of a finite-amplitude two-dimensional disturbance cease to exist.