An investigation on three-dimensional (3-D) vortex-induced vibration (VIV) of a supported pipe conveying fluid with internal velocities in the subcritical regime is presented in this study. A set of nonlinear 3-D equations modeling both the structural motion and the vortex-induced forces are discretized via the Galerkin method. Then the transformed partial differential equations are solved by a fourth order Runge-Kutta scheme. Based on the numerical results, the cross-flow (CF) and in-line (IL) amplitudes, the jumping phenomenon, the domain of lock-in and the trajectories of the pipe are discussed. Moreover, bifurcation diagrams for vibration amplitudes of the pipe with varying internal fluid velocities are constructed. The results indicate that the VIV dynamical behaviors are more complicated while both CF and IL motions are coupled.
As a well-known phenomenon in ocean engineering, vortex-induced vibration (VIV) has been studied for decades because of its powerful effects on structures. It is commonly encountered for a variety of structures exposed in air and water, such as marine cables, pipelines and risers connected to offshore platforms. When the vortex shedding frequency is equal to the natural frequency of the structure, the structural vibration amplitude drastically increases, reaching the maximum at ‘lock-in’ (Blevins, 1990). Since VIV can lead to fatigue damage of structures and oil production facilities, it is of great significance to study the behavior and mechanism of it for ocean engineering. Plenty of fundamental researches on VIV have been performed, many of which are discussed in the comprehensive reviews of Sarpkaya (2004) and Williamson and Govardhan (2008).
For decades, continuous efforts have been made to the investigations on mechanism and dynamics of VIV for both rigid and flexible cylinders based on experimental and numerical methods. Different vortex shedding modes and branches of response were found by Khalak and Williamson (1997) and Williamson and Roshko (1988) by increasing external fluid velocity. It is reported by Khalak and Williamson (1999) that the peak amplitude of vibration is dependent on the mass ratio of the pipes. As observed by Feng (1968), three response branches, i.e. the initial branch, the upper branch and the lower branch, appear for pipes with low mass ratios, while only two response branches, i.e. the initial branch and the lower branch, exist for pipes with high mass ratios. Afterwards, the structure and the wake oscillators are coupled to simulate the features of VIV by many scholars (Balasubramanian and Skop, 1996; Facchinetti, de Langre and Biolley, 2002b; Kim and Perkins, 2002). Meanwhile, a forcing term proportional to the acceleration of the structure in the van der Pol equation was found best appropriate to represent the effect of the structural motion on the wake (Facchinetti, de Langre and Biolley, 2004).