Quasisteady theory for the hydrodynamic forces on a circular cylinder undergoing vortex-induced vibration

Abstract

Vortex-induced vibrations of a rigid circular cylinder were studied by constructing a theory based on a wake oscillator model under quasisteady approximations, thereby evaluating vortex-induced hydrodynamic forces acting on the cylinder. A lock-in limit line representing the boundary for the occurrence of frequency lock-in was also theoretically derived. Hydrodynamic forces in forced oscillation problems estimated by the theory were compared with measured ones. Although some discrepancies were found, particularly in cases with high-frequency oscillations, good agreement was achieved in most cases. Accordingly, we conclude that the present theory captures well real phenomena in the wake downstream of a cylinder subjected to a flow.

Appendix

According to Kutta-Joukowsiki’s theorem, the lift force acting on a flat rigid bar subject to a potential flow is written as

$$ \begin{aligned} L &= \rho V( - \Upgamma ) \\ &= \rho V(2\pi lV\sin \alpha ), \\ \end{aligned} $$

(26)

where the circulation Γ is determined by the additional condition that the velocity of the potential flow must be finite at the trailing edge of the bar (Kutta’s condition). Under the assumption that the length between an application point of the lift force and the center of rotation is $\frac{d}{2} + l$ (Fig. 2), the restoring lift moment acting on the rotation of the wake oscillator can be written as

$$ \begin{aligned} M &= \pi \rho lV^{2} \left( {\frac{d}{2} + l} \right)\sin 2\alpha \\ & \cong 2\pi \rho lV^{2} \left( {\frac{d}{2} + l} \right)\alpha \equiv k\alpha . \\ \end{aligned} $$

(27)

Namely, the coefficient of the restoring moment k (Eq. 2) is derived under the assumption of potential flow, quasisteady state, and small angle of attack.