Mode competition in galloping of a square cylinder at low Reynolds number

Abstract

Galloping is a type of fluid-elastic instability phenomenon characterized by large-amplitude low-frequency oscillations of the structure. The aim of the present study is to reveal the underlying mechanisms of galloping of a square cylinder at low Reynolds numbers ($Re$) via linear stability analysis (LSA) and direct numerical simulations. The LSA model is constructed by coupling a reduced-order fluid model with the structure motion equation. The relevant unstable modes are first yielded by LSA, and then the development and evolution of these modes are investigated using direct numerical simulations. It is found that, for certain combinations of $Re$ and mass ratio ($m^{\ast }$), the structure mode (SM) becomes unstable beyond a critical reduced velocity $U_{c}^{\ast }$ due to the fluid–structure coupling effect. The galloping oscillation frequency matches exactly the eigenfrequency of the SM, suggesting that the instability of the SM is the primary cause of galloping phenomenon. Nevertheless, the $U_{c}^{\ast }$ predicted by LSA is significantly lower than the galloping onset $U_{g}^{\ast }$ obtained from numerical simulations. Further analysis indicates that the discrepancy is caused by the nonlinear competition between the leading fluid mode (FM) and the SM. In the pre-galloping region $U_{c}^{\ast }<U^{\ast }<U_{g}^{\ast }$, the FM quickly reaches the nonlinear saturation state and then inhibits the development of the SM, thus postponing the occurrence of galloping. When $U^{\ast }>U_{g}^{\ast }$, mode competition is weakened because of the large difference in mode frequencies, and thereby no mode lock-in can happen. Consequently, galloping occurs, with the responses determined by the joint action of SM and FM. The unstable SM leads to the low-frequency large-amplitude vibration of the cylinder, while the unstable FM results in the high-frequency vortex shedding in the wake. The dynamic mode decomposition (DMD) technique is successfully applied to extract the coherent flow structures corresponding to SM and FM, which we refer to as the galloping mode and the von Kármán mode, respectively. In addition, we show that, due to the mode competition mechanism, the galloping-type oscillation completely disappears below a critical mass ratio. From these results, we conclude that transverse galloping of a square cylinder at low $Re$ is essentially a kind of single-degree-of-freedom (SDOF) flutter, superimposed by a forced vibration induced by the natural vortex shedding. Mode competition between SM and FM in the nonlinear stage can put off the onset of galloping, and can completely suppress the galloping phenomenon at relatively low $Re$ and low $m^{\ast }$ conditions.