Optimal control of Tollmien–Schlichting waves in a developing boundary layer

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Abstract
Optimal control theory is used to determine the wall transpiration (unsteady blowing/suction) with zero net mass flux capable of attenuating Tollmien–Schlichting waves in a spatially developing boundary layer. The flow state is determined from the parabolized stability equations, in a linear setting. An appropriate cost functional is introduced and minimized iteratively by the numerical solution of the equations for the state and the dual state, coupled via transfer and optimality conditions. Central to the control is the determination of the wall Green’s function expressing the receptivity of the flow to wall inhomogeneities. The optimal wall velocity is obtained in few iterations and a reduction of several orders of magnitude in output disturbance energy is demonstrated, as compared to the uncontrolled case, for control laws operating both over the whole wall length and over a finite strip. Finally, white noise disturbances are applied to the optimal wall velocities already determined, to assess the influence of an imperfectly operating controller on the final result, and to decide on the practical feasibility of the approach.