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Optimal control of plane channel flow, aiming at reducing skin friction or increasing flow rate with a net power saving, is usually studied assuming a continuous distribution of zero-net-flux blowing-suction actuators at walls. This work investigates the macroscopic effects of the wall-distributed actuator pores on the con- troller synthesis by modeling solid walls as rigid layers of homogeneous porous materials. Porous regions are described by the Volume Averaged Navier-Stokes equations, while the coupling between flow in the channel and porous regions is modeled by the Ochoa-Tapia interface conditions.
A multidomain spectral discretization is specifically studied to numerically approximate the linearized dynamical system without the occurrence of any spurious eigenvalue which could spoil the dynamical analysis. An optimize-then-discretize approach is used to overcome the additional mathematical difficulties caused by the flow coupling through the interfaces.
In order to compute the Linear Quadratic Regulator (LQR) gains to be applied in the nonlinear Direct Numerical Simulation (DNS) of the coupled flow system, the minimum energy controller design technique introduced by Bewley, Pralits and Luchini has been extended to the continuous adjoint approach. The Adjoint of the Direct-Adjoint (ADA) technique has been used to evaluate the feedback gains for higher degrees of controller authority. Results show that the actuator modeling could affect the robustness properties of LQRs for plane channel flow when applied in a real control setup.