# Investigation on Numerical Simulation of VIV of Deep-Sea Flexible Risers
## 4 Results and Discussion
### Vibration Modal Analysis of Riser  
Figure 15 exhibits the instantaneous dimensionless displacement envelopes of the riser in the IL and CF directions, revealing that an increase in the slenderness ratio or flow velocity leads to increasingly obvious multi-modal vibration phenomenon.  
Figure 13: Displacement–time history curves of each node in the CF and IL directions.  
Figure 14: Trajectories of the mass centers at slices of the riser. (**a**) C1, U = 0.20 m/s, L/D = 400. (**b**) C2, U = 0.20 m/s, L/D = 550. (**c**) C3, U = 0.20 m/s, L/D = 700. (**d**) C4, U = 0.10 m/s, L/D = 700. (**e**) C5, U = 0.15 m/s, L/D = 700. (**f**) C6, U = 0.25 m/s, L/D = 700.  
Figure 15: _Cont._  
When comparing Figure 15a-c, it was observed that an increase in the slenderness ratio resulted in a higher number of vibration modes in both the IL and CF directions. The dominant mode for the IL direction was fourth-order, while for the CF direction, it was second-order with a weak third-order mode. Such findings can be explained by two reasons: first, the Reynolds number was constant in the subcritical stage, resulting in a stable vortex shedding frequency equal to the vibration frequency when the locking phenomenon occurred; second, an increase in the slenderness ratio decreased natural frequencies while increasing the order of the natural frequencies, which resulted in more vibration modes. Moreover, due to the reduction in the difference between the natural frequencies of each order, the vibration frequency was more likely to jump to adjacent orders, leading to a further aggravation of the multi-modal vibration phenomenon.  
When comparing Figure 15c-f, it was concluded that an increase in the flow velocity also resulted in an increase in the number of vibration modes in both the IL and CF directions. As the flow velocity increased, the vortex shedding frequency of the riser also increased. When the locking phenomenon occurred, the vibration frequency of each finite element node was equal to the vortex shedding frequency of the riser. Thereby, an increase in flow velocity did not affect the natural frequency of the riser but rather increased the  
Figure 15: The VIV displacement envelops of the riser in the IL and CF directions. (**a**) C1, U = 0.20 m/s, L/D = 400. (**b**) C2, U = 0.20 m/s, L/D = 550. (**c**) C3, U = 0.20 m/s, L/D = 700. (**d**) C4, U = 0.10 m/s, L/D = 700. (**e**) C5, U = 0.15 m/s, L/D = 700. (**f**) C6, U = 0.25 m/s, L/D = 700.  
Figure 16a shows that the response of the riser in the IL direction was a weak traveling wave, while in the CF direction, it was a standing wave. Figure 16b shows the traveling wave response in the IL direction and the combination of a traveling wave and standing wave in the CF direction. Figure 16c shows significant traveling wave responses in both the IL and CF directions. This phenomenon indicates that with an increase in the slenderness ratio, the standing wave response gradually disappears, and the traveling wave response gradually dominates.  
Figure 16d shows that the response of the riser in the IL direction was an unstable traveling wave, while in the CF direction, it was a standing wave. Figure 16e shows the traveling wave response in the IL direction and the combination of a traveling wave and standing wave in the CF direction. Figure 16c,f shows significant traveling wave responses in both the IL and CF directions. These phenomena lead to a conclusion consistent with the above: with an increase in the flow velocity, the traveling wave effect gradually dominates.