## 5 Conclusions  
In this paper, the effects of superimposed current, different velocity ratios and different oscillation frequencies on vortex-induced vibration characteristics of tubes are studied, and the main conclusions are as follows:  
1. With the increase of the velocity ratio, the non-dimensional displacement of the vertical pipe in the cross-flow direction and forward-flow direction increases continuously and decreases with the increase of the oscillation frequency. The difference is that the variation amplitude is small in the cross-flow direction but large in the forward-flow direction.
2. With an increasing velocity ratio, a higher order pattern appears in the vertical cross-flow vibration envelope. As the oscillation frequency increases, the vertical vibration envelope always consists of first-order and second-order formations, but the vertical vibration envelope changes from \"symmetrical up and down\" to \"wide up and narrow down\", and the process of vertical mode switching can be observed.
3. With the increase of the velocity ratio, the main control frequencies of the cross-flow direction and forward-flow direction of the riser increase. However, the change of the oscillation frequency of the superimposed flow does not cause a change of the vibration frequency of the riser.
4. There are obvious processes of \"establishment\", \"locking\" and \"decay\" in the transverse-flow of the riser to dimensionless displacement. During the process from 0.50 to 0.62, the proportion of the \"locking\" phase gradually increases. The effect of the superimposed flow oscillation frequency on the proportion of the \"locked\" stage is not obvious in which the proportion of the \"locked\" stage of the riser reaches 50% when f is 0.25f1.
5. Due to the cyclic change of superposition flow velocity, the dimensionless displacement in both the transverse-flow direction and forward-flow direction of the riser has amplitude modulation, and the amplitude modulation period is close to the oscillation period of superposition flow. The lift coefficient and drag coefficient are also modulated. Furthermore, the modulation period is close to the oscillation period of superimposed flow.
6. In the section of the riser structure, the three-dimensional characteristics of the riser structure are considered by dividing three-dimensional finite elements; however, in the fluid domain, this paper adopts the slicing method and replaces the three-dimensional flow field with multiple two-dimensional flow field slices without considering the influence of the three-dimensional flow field characteristics.
7. When studying the influence of the velocity ratio on the vortex-induced vibration characteristics of the opposite tube, this paper only considers the condition that the velocity ratio is greater than or equal to 0.50. Corresponding to the condition where the velocity of the uniform flow is greater than or equal to the maximum velocity of the relative oscillating flow (the condition where the velocity ratio is less than 0.50 is not studied), the influence of different velocity ratios on the vortex-induced vibration characteristics of the tube has not been perfected.  
Conceptualization, S.S.; methodology, F.L. and S.S.; software, F.L. and W.Z.; investigation, J.Z.; resources, S.S.; data curation, F.L. and W.Z.; writing--original draft preparation, F.L. and W.Z.; writing--review and editing, S.S. and F.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the NSFC and Shandong Province (U2106223); the Major Science and Technology Innovation Project in Shandong Province (2021 CXGC010707); and the Natural Science Foundation of Shandong Province (ZR2022 ME092).  
Not applicable.  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
Data Availability Statement: Not applicable.