Vortex-induced vibrations of three staggered circular cylinders are investigated via two-dimensional finite element computations. All the cylinders are of equal diameter (D) and are mounted on elastic supports in both streamwise (x−) and transverse (y−) directions. The two downstream cylinders are placed symmetrically on either side of the upstream body at a streamwise gap of 5D, with the vertical distance between them being 3D. Flow simulations are carried out for Reynolds numbers (Re) in the range of Re = 60-160. Reduced mass (m*) of 10 is considered and the damping is set to zero value. The present investigations show that the upstream cylinder exhibits initial and lower synchronization response modes like an isolated cylinder does at low Re. Whereas for both the downstream cylinders, the upper lock-in branch also appears. The initial and the upper modes are characterized by periodic oscillations, while the lower lock-in branch is associated with nonperiodic vibrations. The 2S mode of vortex shedding is observed in the near wake of all the cylinders for all Re, except for the upper branch corresponding to the downstream bodies. In the upper branch, both the downstream cylinders shed the primary vortices of the P+S mode. For the upstream cylinder, the phase between lift and the transverse displacement exhibits a 180° jump at certain Re in the lower branch. On the other hand, the downstream bodies undergo transverse oscillations in phase with lift in all lock-in modes, while the phase jumps by 180° as the oscillation response reaches the desynchronization regime.
The oscillation responses and the associated wake structures of three rotating tandem cylinders, mounted on elastic supports in both the streamwise (x-) and transverse (y-) directions, are numerically studied. Finite element computations are carried out at Reynolds number Re = 150, where the flow is two-dimensional (2-D), and at Re = 2000 (3-D flow), varying the reduced velocity (U*) in the range U* = 2–14. Each cylinder is placed at a distance of 5 diameters (5D) from its immediate neighbor. In the two-dimensional flow, the upstream and the second downstream cylinders are rotated at the rotation rate (α) of 1, while for the first downstream cylinder, α = 0, 0.5, and 1 are employed. In the 3-D computations, α = 1 is considered for the three bodies. It is observed that at Re = 150, all the cylinders exhibit three distinct lock-in modes, namely, modes I, II, and III. Depending on the rotation configuration, 2S, P + S, and 2P shedding patterns appear in the wake of upstream cylinder, while the downstream cylinders shed 2S, P, P + S, 2P, and T + P modes of primary vortices. In the 3-D flow (at Re = 2000), the upstream cylinder exhibits a bell-shaped profile in the variation of amplitude response as a function of U*. Oscillation responses of the two downstream bodies appear in three distinct regions. During high amplitude oscillations, the upstream cylinder sheds the 2P mode of primary vortices, while in the near wakes of the two downstream bodies, small and incoherent primary vortical structures form due to the presence of three-dimensional instability. Low and high amplitude responses are associated with weak and strong 3-D flow instabilities, respectively.
Galloping cross-flow vibration responses of three in-line identical square cylinders are numerically studied for the mass ratio m*=2, streamwise gaps Lx=3B and 5B, reduced velocity U*=3−50, and Reynolds numbers Re = 150 in two dimensions (2-D) and 2000, where the flow is three-dimensional (3-D). Here, B is the side of the cylinder. An isolated cylinder does not gallop since the mass ratio m*=2 is less than the critical value in the Re = 150 flow, whereas for the three in-line bodies, galloping instability is triggered at the upstream cylinder due to the interference effect caused by the presence of downstream bodies. The interaction with the wake of galloping upstream cylinder promotes galloping instability for the two downstream cylinders almost immediately at Re = 150. In the three-dimensional wake at Re = 2000, downstream cylinders interact with less coherent Karman vortices shed by the galloping upstream cylinder, compared to the 2-D case. This phenomenon leads to the delayed on-set of galloping response for the first downstream cylinder, while the second one never gallops.
Oscillation responses and wake modes of three staggered rotating cylinders, free to move in streamwise and transverse directions, are numerically studied in two- (2-D) and three-dimensional (3-D) flows. 2-D computations are carried out for Reynolds number Re = 60–150, employing the following rotation rates (α), respectively, for the upstream, upper, and lower downstream cylinders: 1, 1, 0; 1, 1, 1; 1, 1, −1. Here, the clockwise rotation is positive. 3-D simulations are performed at Re = 2000 and reduced velocity, U* = 2–11, with the three cylinders being rotated at α = 1. Bell-shaped amplitude profiles are observed for all the rotating cylinders, indicating that the bodies undergo vortex-induced vibrations. In 2-D flow, the considered Re regime can be categorized into three distinct regions, based on the oscillation and frequency responses. Cylinders exhibit negligible amplitudes in the first region, whereas the second region is characterized by high amplitude lock-in oscillations for all three cylinders. In the third region, the downstream cylinders exhibit lock-in response in certain rotation configurations. The oscillation responses and wake modes appear sensitive to the direction of rotation of the lower downstream cylinder for the streamwise and transverse gaps of 5 diameters (5D) and 3D, respectively, between the cylinders. Depending on the rotation configuration, 2S, P, and P + S modes of primary shedding are observed. In 3-D flow also, the cylinders exhibit bell-shaped amplitude profiles, contrary to the galloping response noticed for isolated rotating cylinders in few previous studies. Higher and lower amplitude oscillations induce stronger and weaker 3-D flow instabilities, respectively, in the wake region.
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