A spectral force representation and its physical implication for vortex shedding past a stationary or an oscillating circular cylinder at low Reynolds number

Vortex shedding is an ubiquitous phenomenon behind a bluff body (such as circular cylinder) and becomes more complicated when the body is also in oscillation. It is apparent that periodic behavior must be accompanied by the time-varying force, such as lift and drag (coefficients) with known distinguished cases (say, at Re=200) of low-frequency modulation (LFM), sub-harmonic synchronization (SHS), and normal harmonic synchronization (NHS). In a classical analysis, the force spectrum is often analyzed by the Fourier transform or some more recent methods, and typically, a quite complex frequency spectrum is obtained owing to the inherent nonlinearity in the flow system. In the present study, we extend the principal frequency analysis [Lu et al., “An EMD-based principal frequency analysis with applications to nonlinear mechanics,” Mech. Syst. Signal Process. 150, 107300 (2021)] to the principal spectrum analysis (PSA) with both its amplitude and phase in a composite functional form and provide a spectral representation (SR) of the force coefficients only in terms of the characteristic frequencies. In particular, we consider the unsteady laminar flow past a stationary circular cylinder or an oscillating circular cylinder (with frequency f0), while the resulting vortex shedding frequency is denoted by fVS. The spectral representation via the proposed PSA can reveal nonlinear interactions of the two characteristic frequencies (f0 and fVS) in influencing the force coefficients and distinguish direct and interactive modes in which f0 and fVS interact with each other. As a matter of fact, the successively shed vortices are not identical in the strength (amplitude) nor in the phase function. The spectral representation further enables us to identify complicated vorticity activity near around the bluff body: the periodicity of the strength of the shed vortices and the phase shift in the successive vortex shedding—all at the integer multiples of the greatest common-divisor (gcd) of the (two) characteristic frequencies. The gcd frequency of ⟨f0, fVS⟩ is identified as the genuine (slow, long-term) frequency of the entire vortex shedding process in contrast to the (fast, short-term) vortex shedding frequency. It turns out in this scheme of classification by the PSA-SR that all the distinguished types of the above-mentioned LFM, SHS, and NHS can be considered to be gcd-frequency synchronization.