State-space formulation and controller design for three-dimensional channel flows

Robust reduced-order feedback control of near wall turbulence of three dimensional Poiseuille flow in a periodic channel is investigated. Control of near-wall disturbances is achieved through wall-transpirations, and measurements of wall-shear stresses are fed back to the controller. Linear-Quadratic-Gaussian/loop-transfer-recovery synthesis and model reduction techniques are used to construct the robust feedback controllers from the linearized three dimensional Navier-Strokes equations. The quadratic cost function is composed of the total wall-shear stresses. Using Galerkin's method, the system, controller, and output are decomposed into a set of independent systems, controllers, and outputs for each pair of streamwise and spanwise wave numbers. This decomposition allows reduced-order controllers for each wave number pair to be implemented in parallel. It is assumed that the controllers and sensors will be distributed in fine enough resolution to provide the measurements and control needed. This preliminary study develops a controller for a single pair of wave numbers. The single wave number pair controller produces closed loop poles that show a more rapid decay of disturbances than the open loop system.