  We apply the phase-reduction analysis to examine synchronization properties
of periodic fluid flows. The dynamics of unsteady flows are described in terms
of the phase dynamics reducing the high-dimensional fluid flow to its single
scalar phase variable. We characterize the phase response to impulse
perturbations, which can in turn quantify the influence of periodic
perturbations on the unsteady flow. These insights from the phase-based
analysis uncover the condition for synchronization. In the present work, we
study as an example the influence of periodic external forcing on unsteady
cylinder wake. The condition for synchronization is identified and agrees
closely with results from direct numerical simulations. Moreover, the analysis
reveals the optimal forcing direction for synchronization. The phase-response
analysis holds potential to uncover lock-on characteristics for a range of
periodic flows.
