A nonlinear model for a hanging tubular cantilever simultaneously subjected to internal and confined external axial flows

Abstract

A full nonlinear model is developed for the dynamics of a hanging tubular cantilever that is simultaneously subjected to internal and external axial flows. These two flows are dependent on each other and are in opposite directions. Also, the external flow is confined over the whole length of the cantilever. A nonlinear equation of motion for the cantilever is obtained via Hamilton's principle to third-order accuracy. The virtual work due to the fluid-related forces acting on the cantilever is derived, to the same accuracy, considering the non-conservative forces associated with the internal flow as well as the inviscid, viscous and hydrostatic forces related to the external one. In addition, a vortex-lift mechanism due to the external flow is considered, and the associated steady hydrodynamic forces are derived and added to the expression of the virtual work. The equation of motion is then discretized and solved using the pseudo-arclength continuation method and a direct time-integration technique. The dynamical behaviour obtained is compared to the one predicted by another linear theoretical model, from the literature, for the same system parameters. The two models are in good qualitative and quantitative agreement with each other in terms of the type of instability, namely flutter, that occurs with increasing flow velocity and its onset. However, the proposed model can also predict quantitative facets of the dynamical behaviour beyond the onset of instability, such as limit-cycle amplitude and frequency. Moreover, the influences of various system parameters are investigated theoretically, namely the degree of confinement of the external flow, gravity, mass ratio, drag coefficient, and the thickness of the tubular cantilever.