Mode transformation and interaction in vortex-induced vibration of laminar flow past a circular cylinder

An investigation of the mode transformation and interaction underlying the behavior of vortex-induced vibration (VIV) of a flow past a circular cylinder elastically mounted on a linear spring is conducted using a high-fidelity full-order model (FOM) based on computational fluid dynamics (CFD), a reduced-order model (ROM), and a dynamic mode decomposition (DMD) of the velocity. A reduced-order model for the fluid dynamics is obtained using the eigensystem realization algorithm (ERA), which is subsequently coupled to a linear structural equation to provide a state space model for the coupled VIV system, in order to provide a simplified computationally inexpensive mathematical representation of the system. This methodology is used to study the dynamics of laminar flows past an elastically mounted circular cylinder with Reynolds number Re ranging from 20 to 180, inclusive. The results of the simulations conducted using FOM/CFD and ROM/ERA, in conjunction with the power spectral analysis and DMD, are used to identify the characteristic natural frequencies and the growth/decay of various modes (including the complex interactions between the myriad wake modes and the structural mode) of the VIV system as a function of the Reynolds number and the reduced natural frequency Fs (or, equivalently, the reduced velocity Ur). A detailed analysis of the distribution of the eigenvalues of the transfer (or, system) matrix of the reduced VIV system shows that the frequency range of the lock-in can be partitioned into resonance and flutter lock-in regimes. The resonance lock-in (lower branch of the VIV response) dominates the fluid-structure interaction. Furthermore, it is shown that when the structural natural frequency is close to one of the eigenfrequencies associated with the wake modes, resonance lock-in (rather than flutter lock-in) will be the primary mechanism governing the VIV response even though the real part of the eigenvalues associated with structural mode is positive. With increasing Reynolds number, the instability of each wake mode is enhanced resulting in a transformation of the wake modes interacting with the structural mode. It is suggested herein that the weakened interaction between the wake modes and the structural mode at Re = 180 (associated with the greater separation between the root loci of the modes) results in the premature termination of the resonance lock-in at Fs=0.155 with increasing Ur. The DMD and power spectral analysis of the time series of the transverse displacement and lift coefficient are used to support the results obtained from ROM/ERA and, more specifically, to provide a clear demonstration of the balanced interaction between the wake modes and the structural mode. This result is used to explain the beating phenomenon, which occurs in the initial branch and the significant lag time that arises between the initial branch and the occurrence of a fully developed response in the lower branch.