Nonlinear spacing and frequency effects of an oscillating cylinder in the wake of a stationary cylinder

Nonlinear responses to a transversely oscillating cylinder in the wake of a stationary upstream cylinder are studied theoretically by using an immersed-boundary method at Re=100. Response states are investigated in the three flow regimes for a tandem-cylinder system: the vortex suppression regime, the critical spacing regime, and the vortex formation regime. When the downstream cylinder is forced to oscillate at a fixed frequency and amplitude, the response state of flow around the two cylinders varies with different spacing between the two cylinders, while in the same flow regime, the response state can change with the oscillating frequency and amplitude of the downstream cylinder. Based on velocity phase portraits, each of the nonlinear response states can be categorized into one of the three states in the order of increasing chaotic levels: lock-in, transitional, or quasiperiodic. These states can also be correlated with velocity spectral behaviors. The discussions are conducted using near-wake velocity phase portraits, spectral analyses, and related vorticity fields. A general trend in the bifurcation diagrams of frequency spacing shows the smaller the spacing, frequency, or amplitude, the less chaotic the response state of the system and more likely the downstream and upstream wakes are in the same response state. The system is not locked-in in any case when the spacing between the cylinders is larger than the critical spacing. The near-wake velocity spectral behaviors correspond to the nonlinear response states, with narrow-banded peaks shown at the oscillation frequency and its harmonics in the lock-in cases. High frequency harmonic peaks, caused by interactions between the upstream wake and the downstream oscillating cylinder, are reduced in the near-wake velocity spectra of the upstream cylinder when the spacing increases.