

Artikel
Matrixfree methods for the stability and control of boundary layers
AbstractThis paper presents matrixfree methods for the stability analysis and control design of highdimensional systems arising from the discretized linearized NavierStokes equations. The methods are applied to the twodimensional spatially developing Blasius boundarylayer. 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, where the entire dynamics of perturbations in space and time is of interest, iterative and adjointbased 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 TollmienSchlichting wavepackets located far downstream in 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 inputoutput behavior of the system is of interest and the snapshotmethod is employed to compute balanced modes that correctly capture this behavior. The inputs are external disturbances and wall actuation and the outputs are sensors that extract wall shear stress. A lowdimensional model that capture the inputoutput behavior is constructed by projection onto balanced modes. The reducedorder model is then used to design a feedback control strategy such that the growth of disturbances are damped as they propagate downstream.

