Resolvent-based estimation and control of airfoil wakes

We employ resolvent-based estimation and control tools to predict and mitigate velocity fluctuations in the wake of laminar and turbulent NACA 0012 airfoils. For the laminar airfoil ($Re=5000$), we apply random upstream perturbations to disrupt the periodic limit cycle caused by vortex shedding, inducing chaotic fluctuations. For the turbulent airfoil ($Re=23,000$), we consider natural turbulence on the mid-span plane and its spanwise-averaged flows. Estimation and control kernels are derived using both operator-based and data-driven approaches. The operator-based approach offers low computational costs without requiring \textit{a priori} model reduction and incorporates colored-in-time statistics of the nonlinear terms from the Navier-Stokes equations, which are interpreted as a forcing on the linear dynamics, to account for complex nonlinear dynamics. The data-driven approach, which avoids constructing linearized Navier-Stokes operators, naturally includes these colored statistics of the nonlinear terms. The Wiener-Hopf formalism is used to ensure optimal causality in the estimator and controller, enhancing real-time estimation and control performance. Specifically, we achieve real-time estimation of velocity fluctuations across the entire wake region of the laminar and turbulent airfoils and closed-loop control reducing chaotic fluctuations induced by unsteady vortex shedding and external disturbances for the laminar airfoil. Furthermore, we explore effective sensor placement for the resolvent-based estimator using \textit{a priori} estimation errors.