Localised estimation and control of linear instabilities in two-dimensional wall-bounded shear flows

Abstract

A new framework is presented for estimation and control of instabilities in wall-bounded shear flows described by the linearised Navier–Stokes equations. The control design considers the use of localised actuators/sensors to account for convective instabilities in an ${\mathcal{H}}_{2}$ optimal control framework. External sources of disturbances are assumed to enter the control domain through the inflow. A new inflow disturbance model is proposed for external excitation of the perturbation modes that contribute to transition. This model allows efficient estimation of the flow perturbations within the localised control region of a conceptually unbounded domain. The state-space discretisation of the infinite-dimensional system is explicitly obtained, which allows application of linear control theoretic tools. A reduced-order model is subsequently derived using exact balanced truncation that captures the input/output behaviour and the dominant perturbation dynamics. This model is used to design an ${\mathcal{H}}_{2}$ optimal controller to suppress the instability growth. The two-dimensional non-periodic channel flow is considered as an application case. Disturbances are generated upstream of the control domain and the resulting flow perturbations are estimated/controlled using point wall shear measurements and localised unsteady blowing and suction at the wall. The controller is able to cancel the perturbations and is robust to both unmodelled disturbances and sensor inaccuracies. For single-frequency and multiple-frequency disturbances with low sensor noise a nearly full cancellation is achieved. For stochastic forced disturbances and high sensor noise an energy reduction in perturbation wall shear stress of 96 % is shown.