Vortex-induced vibration of four cylinders in an in-line square configuration

This paper presents a numerical study of vortex-induced vibration of four rigidly connected and four separately mounted circular cylinders in an inline square configuration at a Reynolds number of 150, a low mass ratio of 2.5, and a range of spacing ratio L from 1.5 to 4, where the spacing ratio is defined as the centre-to-centre distance of two adjacent cylinders normalized by the cylinder diameter. For the rigidly connected cylinder array, the maximum and minimum response amplitudes occur at L = 1.5 and L = 2.0, respectively, for the range of spacing ratio covered in this study and the maximum response amplitude at L = 1.5 is accompanied by a wide lock-in range. The large response amplitude at a small spacing ratio L = 1.5 is because the cylinder array responds to the flow as a single cylinder with an overall size that is much larger than the diameter of the single cylinder while the small response amplitude observed at L = 2.0 is attributed to the strong interaction of the vortices through the gap between the top and bottom rows of the cylinder and also in the wake of the cylinder array. For spacing ratio L ≥ 2.5, the lock-in regime of four rigidly connected cylinders is similar to that of a single cylinder and the response amplitudes in the lock-in regime are slightly higher than that of a single cylinder. The energy transfer analysis between fluid flow and individual cylinders in the array shows that the hydrodynamic forces on individual cylinders either excite or damp the vibration, depending on the reduced velocity. An interesting flow feature observed at L = 2, 2.5, and 3 is the biased vortex street in the wake of four rigidly connected cylinders. The biased vortex street leads to a shift of the mean position of the cylinder array with the largest mean position shift being observed at L = 3. Four response modes are identified for four separately mounted cylinders. These are the in-phase mode, the anti-phase mode, the correlated out-of-phase mode, and the uncorrelated mode. It is found that the response mode chosen by the cylinders is dependent not only on the spacing ratio but also on the initial condition of the flow. The response amplitude under the in-phase mode is generally higher than that under the anti-phase mode at identical spacing ratios. This is attributed to the interaction of vortices in the wake of the cylinders.