## 4 Results and Discussion
### Flow Induced Vibration of Two Staggered Cylinders
#### 4.2.3 Wake Vortex Structure  
The amplitude and frequency of the two cylinders can be determined by the fluid force on the two cylinders, which is closely related to the vortex shedding behind the bluff body. At the same time, the vortex generation and shedding process can be reversed by the vibration characteristics of two cylinders. The amplitude frequency response of FIV of two cylinders is closely related to its wake structure. When the vibration branch changes with the velocity of the incoming flow, the wake structure will also undergo modal transformation, among which are the typical wake vortex modes S, P (Pair), P+S (A pair of vortices add a single vortex), and so on. This section mainly selects four different working conditions to analyse the calculation results. The wake vortex structure of two cylinders with roughness strips at different reduced velocities at \(T/D=0\) is shown in Figure 12. As can be seen from Figure 12a, in the case with small reduced velocity (\(U^{*}=2\)), the two cylinders hardly vibrated, and the upstream cylinders could not form vortices. The downstream mode of the vortices was S. With the increase of the reduced velocity, the upstream cylinder could form a vortex, but it could not fall off. The vortices which could not fall off from the upstream cylinder were then attached to the surface of the downstream cylinder, which affected the downstream cylinder and in turn promoted its own vibration. When the reduced velocity increased to \(U^{*}=6\), the gap flow occurred between the two cylinders, which means that the interaction between the two cylinders was strengthened. After that, the amplitude of the downstream cylinder was larger than that of the upstream cylinder. As shown in Figure 12b, the vortex of shedding in P mode from the upstream cylinder was attached to the downstream cylinder surface again Meanwhile, enveloping  
Figure 11: The frequency responses of two cylinders under different spacing ratios. Figure 10: The amplitude responses of two cylinders under different spacing ratios. the S mode vortex formed by the downstream cylinder and behind the downstream cylinder, the P+S mode was shedding. When the reduced velocity continued to increase, the falling vortices from the rear of the upstream cylinder were still attached to the surface of the downstream cylinder and fell off with the downstream cylinder vortices. The falling mode was P and S, as shown in Figure 12c. When the reduced velocity increased to \(U^{*}=11\), the vortices formed by the upstream cylinder were difficult to fall off. The unsealed vortices acted on the surface of the downstream cylinder to promote the vibration of the downstream cylinder. The backward vortices of the downstream cylinder were similar to the P mode, as shown in Figure 12d. After \(U^{*}>11\), the shape and size of vortices were quite different. The downstream cylinder generally shed off in the P and S mode, as shown in Figure 12e. Figure 13 shows the wake vortex structure of the two cylinders with roughness strips at different reduced velocities at \(T=0.4D\). From Figure 13a, it can be seen that the upstream and downstream cylinders hardly vibrated at a low reduced velocity, but there was a tendency of gap flow between the two cylinders. As shown in Figure 13b, the upstream cylinder could form completely vortices in the P mode and was hardly affected by the downstream cylinder. The vortex formed independently by the downstream cylinder merged with the vortex shedding of the upstream cylinder, and it still shed in the P mode at the rear. At \(U^{*}=9\), the upstream and downstream cylinders fell off in the P mode, but the two cylinders did not merge near the wake. The P mode is shown behind the downstream cylinder in Figure 13c. It is worth noting that, in the case of \(U^{*}=11\), the downstream cylinder vortex shedding was more complicated than that under other conditions as shown in Figure 13d.