Drag reduction in transitional linearized channel flow using distributed control

Abstract

This work focuses on feedback control of incompressible transitional Newtonian channel flow described by the twodimensional linearized Navier-Stokes equations. The control objective is to use distributed feedback to achieve stabilization of the parabolic velocity profile, for values of the Reynolds number for which this profile is unstable, and therefore to reduce the frictional drag exerted on the lower channel wall compared to the open-loop values. The control system uses measurements of shear stresses on the lower channel wall and the control actuation is assumed to be in the form of electromagnetic Lorentz forces applied to the flow near the bottom wall. Galerkin's method is initially used to derive a high-order discretization of the linearized flow field that captures the flow instability and accounts for the effect of control actuation on all the modes. Then, a low-order approximation of the linearized flow field is derived and used for the synthesis of a linear output feedback controller that enforces stability in the high-order closed-loop system. The controller is applied to a simulated transitional linearized channel flow and is shown to stabilize the flow field at the parabolic profile and significantly reduce the drag on the lower channel wall.