Multi-modes approach to modelling of vortex-induced vibration

In this work the fluid–structure interactions are considered by investigating a straight but slender pipe interacting with uniform water flow. Two configurations are studied, namely vertically and horizontally positioned pipes, which are modelled as an Euler–Bernoulli beam with flexural stiffness. Both pretension and length-wise mass distribution are considered. The structure is assumed to be moving only in the direction normal to flow (cross-flow motion) hence its in-line motion is neglected. The external fluid force acting on the structure is the result of the action of sectional vortex-induced drag and lift forces. Only mean drag force is considered, with time varying lift force modelled using a non-linear oscillator equation of the Van der Pol type. The obtained coupled system of non-linear partial differential equations is simplified employing Galerkin-type discretisation. The resulting ordinary differential equations are solved numerically providing multi-mode approximations of cross-flow displacement and non-dimensional lift coefficient. The comparison between the responses of vertical and horizontal structures shows that, as expected, due to a balancing between pretension and weight, in general a higher amplitude of vibration is observed for the vertical configuration than in the same location along the pipe for the horizontal configuration in the lower part of the structure. However, lower amplitudes are obtained in the upper part of the pipe. The horizontal configuration solutions are identical in symmetrical locations along the pipe due to constant pretension. The influence of the wake equation coefficients and the fluid force coefficients on the response amplitudes has been also considered together with the length of the pipe and pretension level, and the appropriate response curves are included. Finally, for the higher mode approximations it has been shown that the vibrations level at lower frequencies is predicted reasonably well by retaining only a small subset of modes.