Linear optimal control applied to instabilities in spatially developing boundary layers

Abstract

The work presented extends previous research on linear controllers in temporal channel flow to spatially evolving boundary layer flow. The flows studied are those on an infinite swept wedge described by the Falkner–Skan–Cooke (FSC) velocity profiles, including the special case of the flow over a flat plate. These velocity profiles are used as the base flow in the Orr–Sommerfeld–Squire equations to compute the optimal feedback control through blowing and suction at the wall utilizing linear optimal control theory. The control is applied to a parallel FSC flow with unstable perturbations. Through an eigenvalue analysis and direct numerical simulations (DNS), it is shown that instabilities are stabilized by the controller in the parallel case. The localization of the convolution kernels for control is also shown for the FSC profiles. Assuming that non-parallel effects are small a technique is developed to apply the same controllers to a DNS of a spatially evolving flow. The performance of these controllers is tested in a Blasius flow with both a Tollmien–Schlichting (TS) wave and an optimal spatial transiently growing perturbation. It is demonstrated that TS waves are stabilized and that transient growth is lowered by the controller. Then the control is also applied to a spatial FSC flow with unstable perturbations leading to saturated cross-flow vortices in the uncontrolled case. It is demonstrated that the linear controller successfully inhibits the growth of the cross-flow vortices to a saturated level and thereby delays the possibility of transition through secondary instabilities. It is also demonstrated that the controller works for relatively high levels of nonlinearity, and for stationary as well as time-varying perturbations.