The critical mass phenomenon in vortex-induced vibration at low $Re$

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.