# Feedback and adjoint based control of boundary layer flows

Sammanfattning: Linear and nonlinear optimal control have been investigated in transitional channel and boundary layer .ows. The flow phenomena that we study are governed by the incompressible Navier–Stokes equations and the main aim with the control is to prevent transition from laminar to turbulent flows. A linear model-based feedback control approach, that minimizes an objective function which measures the perturbation energy, can be formulated where the Orr– Sommerfeld/Squire equations model the flow dynamics. A limitation with the formulation is that it requires complete state information. However, the control problem can be combined with a state estimator to relax this requirement. The estimator requires only wall measurements to reconstruct the flow in an optimal manner. Physically relevant stochastic models are suggested for the estimation problem which turns out to be crucial for fast convergence. Based on these models the estimator is shown to work for both in.nitesimal as well as finite amplitude perturbations in direct numerical simulations of a channel flow at Recl = 3000. A stochastic model for external disturbances is also constructed based on statistical data from a turbulent channel flow at ReT = 100. The model is successfully applied to estimate a turbulent channel flow at the same Reynolds number. The combined control and estimation problem, also known as a compensator, is applied to spatially developing boundary layers. The compensator is shown to successfully reduce the perturbation energy for Tollmien–Schlichting waves and optimal perturbations in the Blasius boundary layer. In a Falkner– Skan–Cooke boundary layer the perturbation energy of traveling and stationary cross-flow disturbances are also reduced. A nonlinear control approach using the Navier–Stokes equations and the associated adjoint equations are derived and implemented in the context of direct numerical simulations of spatially-developing three-dimensional boundary layer .ows and the gradient computation is veri.ed with .nite-di.erences. The nonlinear optimal control is shown to be more e.cient in reducing the disturbance energy than feedback control when nonlinear interactions are becoming signi.cant in the boundary layer. For weaker disturbances the two methods are almost indistinguishable.

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