2009a Matrix-free methods for the stability and control of boundary layers

This Lecture presents matrix-free methods for the stability analysis and control design of high-dimensional systems arising from the discretized linearized Navier–Stokes equations. The methods are applied to the two-dimensional spatially developingBlasius boundary layer. A critical step in the process of systematically investigating stability properties and designing feedback controllers is solving very large eigenvalue problems by storing only velocity fields at different times instead of large matrices. For stability analysis, in which the entire dynamics of perturbations in space and time is of interest, iterative and adjoint-based optimization techniques are employed to compute the global eigenmodes and the optimal initial conditions. The latter are the initial conditions yielding the largest-possible energy growth over a finite time interval. The leading global eigenmodes take the shape of Tollmien– Schlichting wave packets located far downstream in the streamwise direction, whereas the leading optimal disturbances are tilted structures located far upstream in the boundary layer. For control design, on the other hand, the input–output behavior of the system is of interest and the snapshot method is employed to compute balanced modes that correctly capture this behavior. The inputs represent external disturbances and wall actuation and the outputs represent sensors that extract wall shear stress. A low-dimensional model that captures the input–output behavior is constructed by projection onto balanced modes. The reduced-order model is then used to design a feedback control strategy such that the growth of disturbances is damped as they propagate downstream. I