Feedback control of three-dimensional optimal disturbances using reduced-order models

Abstract

The attenuation of three-dimensional wavepackets of streaks and Tollmien–Schlichting (TS) waves in a transitional boundary layer using feedback control is investigated numerically. Arrays of localized sensors and actuators (about 10–20) with compact spatial support are distributed near the rigid wall equidistantly along the spanwise direction and connected to a low-dimensional (r = 60) linear quadratic Gaussian controller. The control objective is to minimize the disturbance energy in a domain spanned by a number of proper orthogonal decomposition modes. The feedback controller is based on a reduced-order model of the linearized Navier–Stokes equations including the inputs and outputs, computed using a snapshot-based balanced truncation method. To account for the different temporal and spatial behaviour of the two main instabilities of boundary-layer flows, we design two controllers. We demonstrate that the two controllers reduce the energy growth of both TS wavepackets and streak packets substantially and efficiently, using relatively few sensors and actuators. The robustness of the controller is investigated by varying the number of actuators and sensors, the Reynolds number and the pressure gradient. This work constitutes the first experimentally feasible simulation-based control design using localized sensing and acting devices in conjunction with linear control theory in a three-dimensional setting.