Aspects of vortex-induced in-line vibration at low Reynolds numbers: Simulation and prediction by a reduced-order model

Abstract

The purpose of this study is to develop a reduced-order model (ROM) for the prediction of vortex-induced vibration of an elastically mounted circular cylinder constrained to oscillate purely in-line with a uniform free stream. The fluid force in the streamwise direction was modeled by an extended version of Morison’s equation that comprises an inviscid inertial component opposing the cylinder acceleration, a viscous drag component proportional to the square of the relative velocity of the free stream and the oscillating cylinder, and a wake drag component representing the excitation from vortex shedding. A van der Pol equation was employed to model the oscillating wake drag component. The coupling between the oscillating wake and the oscillating cylinder was accomplished by forcing the wake oscillator with a term proportional to either the acceleration or the velocity of the cylinder. The ROM parameters were tuned against data from numerical solutions of the full system of non-linear equations governing the fluid and cylinder motions for a cylinder with a mass ratio of 10 and a Reynolds number of 180. The results showed that acceleration forcing can predict well the amplitude response when the ROM is properly tuned, whereas velocity forcing leads to underestimation of the amplitude response. In addition, acceleration forcing can predict satisfactorily the magnitude of the total streamwise force and of the wake drag. However, the trend in the frequency response was not well predicted by acceleration forcing, which was predicted more satisfactorily by velocity forcing. Both acceleration and velocity forcing predicted satisfactorily the phase lag between the wake drag and the cylinder displacement. Predictions with the calibrated ROM at different mass ratios captured well the amplitude response as a function of the reduced velocity, including the narrowing of the range where relatively high amplitudes occur, as in the full numerical simulations. Linear stability analysis based on the ROM equations indicated that the interaction between the fluid and the vibrating cylinder has the characteristics of nonlinear resonance, where the selection of the common frequency of the system is sensitive to nonlinearities, which cannot be faithfully captured by the van der Pol wake oscillator.