Linear and nonlinear optimal control in spatial boundary layer

Authors: Chevalier, M.C., Högberg, M., Berggren, M, Henningson, D.S.H.
Document Type: Conference
Pubstate: Accepted
Journal: AIAA 2002-2755
Year: 2002


Instabilities in a spatially-developing three-dimensional boundary layers are controlled through blowing and suction at the wall. The performance of the control is tested in direct numerical simulations (DNS) of the incompressible Navier-Stokes equations for Tollmien-Schlichting (TS) waves, optimal transiently growing streaks in a Blasius boundary layer, and cross flow vortices in a Falkner-Skan-Cooke (FSC) flow. Two control strategies are compared. First, a quasi-Newton optimization algorithm is applied to solve an off-line optimal control problem. A solver for the adjoint equations has been implemented in the spectral DNS code used. This method adapts naturally, without modification, to nonlinearities such as a strongly varying mean flow. However, it is computationally expensive and storage demanding, needing numerous solves of the Navier-Stokes and associated adjoint equations. Second, feed-back optimal control is applied, using a strategy designed to operate locally on a spatially-developing flow. The feed-back operator is constructed from the Orr-Sommerfeld-Squire equations. Assuming the flow to be locally parallel makes it feasible to solve the associated Riccati equations for each wave number pair in the stream- and spanwise directions. The feed-back is applied to a DNS of the flows mentioned above. This method is much less computationally costly than the first nonlinear method. The method performs surprisingly well, in spite of the limitations with respect to being able to account for strong nonlinear effects. It is demonstrated that TS waves are stabilized and that transient growth is considerably lowered by the controller. Moreover, the controller successfully inhibits growth of steady cross flow vortices in the FSC flow.