# Investigation on Numerical Simulation of VIV of Deep-Sea Flexible Risers
## 4 Results and Discussion
### Riser Vibration Response Frequency Analysis  
In Figure 17, the oscillation frequencies of the IL and CF directions of the riser at nine different positions along the riser axis are presented. The dominating frequency was consistent with the main mode obtained in Sections 4.4 and 4.5. The ratio of the dominating frequency of the IL vibration to the dominating frequency of the CF vibration was approximately 2.  
Figure 16: Time-history nephogram of riser vibration response 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.  
In Figure 17a, the dominating frequency of the riser in the IL direction was 1.8456 Hz, which is close to the second-order natural frequency of 1.7136 Hz shown in Table 5. The vibration presented as the second-order main control mode with weak third-order and fourth-order frequency components. The dominating frequency of the riser in the CF direction was 0.9408 Hz, which is close to the first-order natural frequency of 0.7837 Hz, and the vibration presented as the first-order main control mode.  
Figure 17: _Cont._  
Figure 18 shows the variation in the dominating frequency of the riser in both the IL and CF directions as the slenderness ratio varied. With an increase in the slenderness ratio, the broadband characteristic of frequencies in both IL and CF directions became more prominent, leading to an aggravated degree of fatigue damage of the riser.  
Figure 17: Frequency spectra of IL and CF vibration responses at different axial positions 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.  
Comparative analysis of Figure 17c-f reveals that with an increase in flow velocity, the vibration frequency increased continuously, resulting in more obvious multi-frequency and wide-frequency vibration phenomena. Moreover, it can be observed from the frequency spectra in the CF direction that there were secondary frequencies which were twice or even three times the dominating frequency, representing the coupling effect of vibrations in both the IL and CF directions.  
Furthermore, comparative analysis of Figure 17a-c shows that as the slenderness ratio increased, the amplitude of lower-order frequencies below the dominating frequency in the CF vibration response spectra increased continuously, which had a significant effect on the fatigue damage of the riser. This was due to the transverse average fluctuation that occurred when the transverse vibration equilibrium position of the riser deviated from y = 0, as can be vaguely seen from the transverse envelope diagram of the riser.  
To conclude, Figures 17 and 19 provide valuable insights into the frequency analysis of the riser vibration response in both the IL and CF directions, demonstrating the impact of the slenderness ratio and flow velocity on the vibration frequency and characteristics of the riser.  
Figure 19: Dominating frequencies of IL and CF directions at different flow rates (L/D = 700).  
Figure 18: Dominating frequencies of IL and CF directions at different slenderness ratios (U = 0.2 m/s).