Nonplanar vortex-induced vibrations of cantilevered pipes conveying fluid subjected to loose constraints

In this paper, a theoretical model for vortex-induced vibrations (VIVs) of a cantilevered pipe conveying fluid subjected to loose constraints is proposed and analyzed. The unsteady hydrodynamic forces associated with the wake dynamics are modeled by two distributed van der Pol wake oscillators. Taking into account the coupling between the structure and fluid, the full nonlinear partial differential equations of the flexible pipe are derived by using Hamilton's principle. The nonlinear partial differential equations for the pipe and wake are further discretized by the aid of the Galerkin's technique, resulting in a set of ordinary differential equations. These ordinary differential equations are further numerically solved by using a fourth-order Runge-Kutta integration algorithm. Via varying the internal and cross-flow velocities, the derived high-dimensional reduced-order model displays some interesting dynamical behaviors of the coupled system. Phase portraits, bifurcation diagrams, Argand diagram and oscillation shape diagrams are plotted, showing the existence of lock-in phenomenon and figure-of-eight trajectory. When the internal fluid velocity is small and below the critical value for flutter instability, the oscillation of the pipe is mainly induced by the cross flow. For internal flow velocity beyond the critical value for flutter instability, interestingly, the effect of low cross-flow velocity is not pronounced. For relatively high cross-flow velocity, both internal and external fluid flows can significantly contribute to the responses of the pipe. This work is expected to be useful for predicting the responses of marine oil-conveying pipes under the combined effects of internal and external fluid flows.