  The turbulent flow around a squared-section cylinder at $Re=22000$ exhibits
both a low-frequency Vortex-Shedding (VS), a common feature of all bluff-body
flows, and high-frequency Kelvin-Helmholtz (KH) structures evolving on the
shear layers formed at the top and bottom side of the cylinder. The VS motion
triggers a periodic movement of these shear layers, inducing oscillations of
the strength of their velocity gradients. The high-frequency KH structures
respond to these low-frequency oscillations of the base-flow gradients in a
quasi-steady manner. In this paper we will propose a general framework to
capture and reconstruct stochastic high-frequency fluctuations (the KH
structures) evolving on top of a low-frequency deterministic motion (the
periodic VS). Based on a generic triple decomposition, that allows separation
of the stochastic component from the deterministic one, we propose a
Phase-Conditioned Localized Spectral Proper Orthogonal Decomposition
(PCL-SPOD), which isolates high-frequency spatio-temporal structures within the
stochastic component at different phases of the VS motion, with a short-time
Fourier transform. Finally, we compare these structures to those obtained with
a Phase-Conditioned Localized Resolvent formalism (PCL-Resolvent), which
consists in a Resolvent analysis around instantaneous snapshots of the periodic
VS.
