## 2 Numerical Simulation Method of Riser Vortex Shock Vibration
### Vibration Displacement Analysis  
The dimensionless displacement time-history curve of the riser can directly mirror the characteristics of the motion trajectory of the superimposed flow neutral pipe. In superimposed flows with different flow rate ratios, the time-history changes of transverse-flow direction and forward-flow direction dimensionless displacement at the middle point of the riser (z/L = 1/2) are shown in Figure 11.  
With the velocity ratio of the superposition flow increasing, the amplitude of the displacement of the transverse-flow to dimensionless flow increases, but the amplitude of the increase is relatively small. When \(\alpha\) increases from 0.57 to 0.62, the amplitude of the dimensionless displacement increases nearly twice.  
By analyzing the characteristics of the forward-flow dimensionless displacement, it can be obviously observed that there is a delay in the peak value of the forward-flow dimensionless displacement of the riser relative to the maximum velocity, and the transverse-flow dimensionless displacement of the riser within one cycle is taken, as shown in Figure 12.  
Figure 11: Time\(-\)history variation plot of dimensionless displacement of riser under different flow rate ratios.  
Figure 10: Overlay of the flow velocity time-history change plot.  
By observing the above figure, it becomes clear that in a period of amplitude modulation change, there are three obvious development processes of \"establishment\", \"locking\" and \"decay\" in the transverse-flow of the riser to dimensionless displacement. When \(\alpha=0.50\), the transverse-flow of the riser to the dimensionless displacement \"establishment\" and \"decay\" stages occupies a large proportion, while the \"locking\" stage accounts for only about 25%. As \(\alpha=0.57\), the proportion of the \"lock\" phase has increased by about 35%. When \(\alpha=0.62\), the instantaneous velocity of the superposition flow increases further, and the \"locking\" phase of the transverse-flow to dimensionless displacement of the riser occupies more time, accounting for 43%. When \(\alpha=0.67\), the proportion of the \"locking\" phase decreases, accounting for about 36%. The proportion of the \"locking\" phase increases gradually when \(\alpha\) increases from 0.50 to 0.62. When \(\alpha\) increases up to 0.67, the proportion of the \"locking\" phase ceases increasing.  
We can understand that from Figure 13, under the action of superimposed flow with different flow ratios, the RMS value of transverse-flow to dimensionless displacement in the deceleration stage is greater than that in the acceleration stage. Moreover, there is a \"delay effect\", which is because the riser speed accelerates and decelerates periodically under the action of superimposed flow. When the velocity changes, the drain vortex and the fluid lift force do not change immediately, so there is a hysteresis between the vibration amplitude and the velocity change of the riser.  
Figure 12: Diagram of the development process of displacement in the amplitude modulation period of the riser.  
Figure 13: Root mean square value of dimensionless displacement in the cross-flow direction of the riser.