The critical mass phenomenon in vortex-induced vibration at low

Abstract

Vortex-induced vibration of a circular cylinder with low mass ratio ($0.05\leqslant m^{\ast }\leqslant 10.0$) is investigated, via a stabilized space–time finite element formulation, in the laminar flow regime where$m^{\ast }$is defined as the ratio of the mass of the oscillating structure to the mass of the fluid displaced by it. Computations are carried out over a wide range of reduced speed,$U^{\ast }$, which is defined as$U/f_{N}D$, where$U$is the free-stream speed,$f_{N}$the natural frequency of the spring mass system in vacuum and$D$the diameter of the cylinder. In particular, the situation where the lock-in regime extends up to infinite reduced speed is explored. Studies at large$Re$, in the past, have shown that the normalized amplitude of cylinder oscillation at infinite reduced speed,$A_{\infty }^{\ast }$, exhibits a sharp increase when$m^{\ast }$is reduced below the critical mass ratio ($m_{crit}^{\ast }$). This jump signifies a shift from desynchronized response to lock-in state. In this work it is shown that in the laminar regime, a jump in$A_{\infty }^{\ast }$occurs only beyond a certain$Re$($=Re_{j}\sim 108$). For$Re<Re_{j}$, the response increases smoothly with decrease in$m^{\ast }$with no discernible jump. In this situation, therefore, the identification of$m_{crit}^{\ast }$based on jump in response at$U^{\ast }=\infty$is not possible. The difference in the$A^{\ast }-m^{\ast }$variation on the two sides of$Re=Re_{j}$, is attributed to the difference in the transition between the lower branch of cylinder response and desynchronization regime. This transition is brought out more clearly by plotting$A^{\ast }$with$f_{v_{o}}/f$, where$f_{v_{o}}$is the vortex shedding frequency for the flow past a stationary cylinder and$f$is the cylinder vibration frequency. In the$A^{\ast }-f_{v_{o}}/f$plane, the response data as well as other quantities related to free vibrations, for different$m^{\ast }$, collapse on a curve. Unlike at high$Re$, the collapsed curves show a dependence on$Re$in the laminar regime. The transition between the lock-in and desynchronized state, as seen from the collapsed curves, is qualitatively different for$Re$on either side of$Re_{j}$. The collapsed curves, at a certain$Re$, are utilized to estimate$A^{\ast }$for the limiting case of$(U^{\ast },m^{\ast })=(\infty ,0)$. Interestingly, unlike at large$Re$, this limit value is found to be lower than the peak amplitude of cylinder vibration at a given$Re$. Hysteresis in the cylinder response, near the higher-$U^{\ast }$end of the lock-in regime, is explored. It is observed that the range of$U^{\ast }$with hysteretic response increases with decrease in$m^{\ast }$. Interestingly, for a certain range of$m^{\ast }$, the response is hysteretic from a finite$U^{\ast }$up to$U^{\ast }=\infty$. We refer to this phenomenon as hysteresis forever. It occurs because of the existence of multiple response states of the system at$U^{\ast }=\infty$, for a certain range of$m^{\ast }$. The study brings out the significant differences in the response of the fluid–structure system associated with the critical mass phenomenon between the low- and high-$Re$regime.