Stability enhancement by boundary control in 2-D channel flow

In this paper, we stabilize the parabolic equilibrium profile in a two-dimensional (2-D) channel flow using actuators and sensors only at the wall. The control of channel flow was previously considered by Speyer and coworkers, and Bewley and coworkers, who derived feedback laws based on linear optimal control, and implemented by wall-normal actuation. With an objective to achieve global Lyapunov stabilization, we arrive at a feedback law using tangential actuation (using teamed pairs of synthetic jets or rotating disks) and only local measurements of wall shear stress, allowing to embed the feedback in microelectromechanical systems (MEMS) hardware, without need for wiring. This feedback is shown to guarantee global stability in at least H(2) norm, which by Sobolev's embedding theorem implies continuity in space and time of both the flow field and the control (as well as their convergence to the desired steady state). The theoretical results are limited to low values of Reynolds number, however, we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher.