Lock-in in vortex-induced vibration

Abstract

The phenomenon of lock-in in vortex-induced vibration of a circular cylinder is investigated in the laminar flow regime ($20\leqslant Re\leqslant 100$). Direct time integration (DTI) and linear stability analysis (LSA) of the governing equations are carried out via a stabilized finite element method. Using the metrics that have been proposed in earlier studies, the lock-in regime is identified from the results of DTI. The LSA yields the eigenmodes of the coupled fluid–structure system, the associated frequencies ($F_{LSA}$) and the stability of the steady state. A linearly unstable system, in the absence of nonlinear effects, achieves large oscillation amplitude at sufficiently large times. However, the nonlinear terms saturate the response of the system to a limit cycle. For subcritical $Re$, the occurrence of lock-in coincides with the linear instability of the fluid–structure system. The critical $Re$ is the Reynolds number beyond which vortex shedding ensues for a stationary cylinder. For supercritical $Re$, even though the aeroelastic system is unstable for all reduced velocities ($U^{\ast }$) lock-in occurs only for a finite range of $U^{\ast }$. We present a method to estimate the time beyond which the nonlinear effects are expected to be significant. It is observed that much of the growth in the amplitude of cylinder oscillation takes place in the linear regime. The response of the cylinder at the end of the linear regime is found to depend on the energy ratio, $E_{r}$, of the unstable eigenmode. $E_{r}$ is defined as the fraction of the total energy of the eigenmode that is associated with the kinetic and potential energy of the structure. DTI initiated from eigenmodes that are linearly unstable and whose energy ratio is above a certain threshold value lead to lock-in. Interestingly, during lock-in, the oscillation frequency of the fluid–structure system drifts from $F_{LSA}$ towards a value that is closer to the natural frequency of the oscillator in vacuum ($F_{N}$). In the event of more than one eigenmode being linearly unstable, we investigate which one is responsible for lock-in. The concept of phase angle between the cylinder displacement and lift is extended for an eigenmode. The phase angle controls the direction of energy transfer between the fluid and the structure. For zero structural damping, if the phase angle of all unstable eigenmodes is less than 90°, the phase angle obtained via DTI evolves to a value that is close to 0°. If, on the other hand, the phase angle of any unstable eigenmode is more than 90°, it settles to 180°, approximately in the limit cycle. A new approach towards classification of modes is presented. The eigenvalues are tracked over a wide range of $U^{\ast }$ while keeping $Re$ and mass ratio ($m^{\ast }$) fixed. In general, for large values of $m^{\ast }$, the eigenmodes corresponding to the two leading eigenvalues exhibit a decoupled behaviour with respect to $U^{\ast }$. They are classified as the fluid and elastic modes. However, for relatively low $m^{\ast }$ such a classification is not possible. The two leading modes are coupled and are referred to as fluid–elastic modes. The regime of such occurrence is shown on the $Re{-}m^{\ast }$ parameter space.