Optimized Waveforms for Feedback Control of Vortex Shedding

Optimal control theory is combined with the numerical simulation of an incompressible viscous flow to control vortex shedding in order to maximize lift. A two-dimensional flat plate model is considered at a high angle of attack and a Reynolds number of 300. Actuation is provided by unsteady mass injection near the trailing edge and is modeled by a compact body force. The adjoint of the linearized perturbed equations is solved backwards in time to obtain the gradient of the lift to changes in actuation (the jet velocity), and this information is used to iteratively improve the controls. The optimized control waveform is nearly periodic and locked to vortex shedding. We compare the results with sinusoidal open- and closed-loop control and observe that the optimized control is able to achieve higher lift than the sinusoidal forcing with more than 50% lower momentum coefficients. The optimized waveform is also implemented in a simple closed-loop controller where the control signal is shifted or deformed periodically to adjust to the (instantaneous) frequency of the lift fluctuations. The feedback utilizes a narrowband filter and an Extended Kalman Filter to robustly estimate the phase of vortex shedding and achieve phase-locked, high lift flow states.