  We present a dynamic decomposition analysis of the wake flow in
fluid-structure interaction (FSI) systems under both laminar and turbulent flow
conditions. Of particular interest is to provide the significance of
low-dimensional wake flow features and their interaction dynamics to sustain
the free vibration of a square cylinder at a relatively low mass ratio. To
obtain the high-dimensional data, we employ a body-conforming variational
fluid-structure interaction solver based on the recently developed partitioned
iterative scheme and the dynamic subgrid-scale turbulence model for a moderate
Reynolds number. The snapshot data from high-dimensional FSI simulations are
projected to a low-dimensional subspace using the proper orthogonal
decomposition (POD). We utilize each corresponding POD mode for detecting
features: the vortex street, the shear layer and the near-wake bubble. We find
that the vortex shedding modes contribute solely to the lift force, while the
near-wake and shear layer modes play a dominating role to the drag force. We
further examine the fundamental mechanism of this dynamical behavior and
propose a force decomposition technique via low-dimensional approximation. We
ascertain quantitatively that the shear layer feeds the vorticity flux to the
wake vortices and the near-wake bubble during the wake-body synchronization.
Based on the decomposition of wake dynamics, we suggest an interaction cycle
for the frequency lock-in during the wake-body interaction, which provides the
inter-relationship between the high amplitude motion and the dominating wake
features. Through our investigation of wake-body synchronization below critical
Re range, we discover that the bluff body can undergo a synchronized
high-amplitude vibration due to flexibility-induced unsteadiness. The
interaction cycle for the wake-synchronization is found to be valid for the
turbulent wake flow.
