  We obtain an optimal actuation waveform for fast synchronization of periodic
airfoil wakes through the phase reduction approach. Using the phase reduction
approach for periodic wake flows, the spatial sensitivity fields with respect
to the phase of the vortex shedding are obtained. The phase sensitivity fields
can uncover the synchronization properties in the presence of periodic
actuation. This study seeks a periodic actuation waveform using phase-based
analysis to minimize the time for synchronization to modify the wake-shedding
frequency of NACA0012 airfoil wakes. This fast synchronization waveform is
obtained theoretically from the phase sensitivity function by casting an
optimization problem. The obtained optimal actuation waveform becomes
increasingly non-sinusoidal for higher angles of attack. Actuation based on the
obtained waveform achieves rapid synchronization within as low as two vortex
shedding cycles irrespective of the forcing frequency whereas traditional
sinusoidal actuation requires O(10) shedding cycles. Further, we analyze the
influence of actuation frequency on the vortex shedding and the aerodynamic
coefficients using force-element analysis. The present analysis provides an
efficient way to modify the vortex lock-on properties in a transient manner
with applications to fluid-structure interactions and unsteady flow control.
