  We establish the theoretical framework for adjoint-based phase reduction
analysis for incompressible periodic flows. Through this adjoint-based method,
we obtain spatiotemporal phase sensitivity fields through a single pair of
forward and backward direct numerical simulations, as opposed to the
impulse-based method that requires a very large number of simulations.
Phase-based analysis involves perturbation analysis about a periodically
varying base state and hence is tailored for the analysis of periodic flows. We
formulate the phase description of periodic flows with respect to the potential
and vortical perturbations in the flow field. The current phase-reduction
analysis can also be implemented consistently in the immersed boundary
projection method, which facilitates the analysis over arbitrarily-shaped
bodies. We demonstrate the strength of the phase-based analysis for periodic
flows over circular cylinder and symmetric airfoils at high incidence angles.
The critical regions for phase modification in the cylinder flow are
investigated and the locations of flow separation are shown to be the most
sensitive regions. Further, the results reveal the influence of the angle of
attack and airfoil thickness on the phase-sensitivity distribution of flows
over various airfoils. The phase for such flows is defined based on the lift
coefficient, and hence is influenced by the vortical structures responsible for
lift production. The present framework sheds light on the connection between
phase-sensitivity and vortex formation dynamics.
