# Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method
## 4 Numerical Simulation and Results
### Vortex-Induced Vibration Response of Deep-Sea Mining Pipe
#### 4.1.1 Analysis of Vibration Track Characteristics of Deep-Sea Mining Pipe  
Figure 7 shows the time-history curve of cross-flow direction and parallel-flow direction displacement response of 24 nodes along the length direction of mining pipe from top to bottom when \(\mathbf{u}_{\mathrm{r}}=20\). It can be found that the down-flow displacement at each node vibrates slightly relative to the equilibrium position of each node. However, unlike the phenomenon that the closer the riser is to the middle of the riser, the greater the downstream displacement, because the mining pipe has the characteristics of free movement and no constraint at the bottom, its maximum displacement is located at the bottom, about 3.198 d. In the context of the cross-flow dynamics of the mining pipe, the displacement exhibits a reciprocating motion at \(\mathrm{y}/\mathrm{d}=0\), although the amplitude of the vibration remains inconsistent, reminiscent of the dynamic behavior observed in conventional risers. The nature of the vortex-induced vibrations, whether transverse or longitudinal, manifests greater complexity. The movement pattern of the mining pipe more closely resembles the irregular paths observed during \"beating\" phenomena in vortex-induced vibrations of a two-dimensional cylinder, devoid of the predictable sinusoidal or co-sinusoidal trajectories. Studies indicate that the amplitude of vibrations in the direction perpendicular to the flow is significantly larger than those aligned with the flow. Correspondingly, the frequency of vibrations in the direction of the flow is higher than in the perpendicular direction, revealing a binary relationship between these variables.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{u}\) (\(\mathbf{m}/\mathbf{s}\)) & \(\mathbf{u}_{\mathrm{r}}\) & \(\mathbf{\hat{y}}_{\mathrm{RMS}}\) / \(\mathbf{d}\) (Primary Experiment) & \(\mathbf{\hat{y}}_{\mathrm{RMS}}\) / \(\mathbf{d}\) (Numerical Simulation of Paper) \\ \hline
0.10 & 5.04 & 0.26 & 0.28 \\
0.15 & 7.55 & 0.28 & 0.30 \\
0.20 & 10.07 & 0.27 & 0.29 \\
0.25 & 12.59 & 0.23 & 0.24 \\
0.30 & 15.11 & 0.46 & 0.49 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of numerical simulation and experimental results.  
Figure 7: _Cont_.  
It can be found that the movement of the upper end of the mining pipe is relatively chaotic, and its shape is like a \"cashew nut\"; this is due to the locking of the downstream vibrations occurring, which affects the cross-flow vibration frequency and leads to chaotic vibration trajectories. From slice 12 to slice 10, the outline of the motion track starts to become confused gradually when slice 9 begins, but when it is near the bottom, it can be observed that the moving track of the mining pipe has an obvious \"8\" shape stage, which is caused by the binding force generated by the bottom of the mining pipe under the effect of the middle bin at the bottom, resulting in a change in the phase of the cross-flow and down-flow displacements, also making the bottom movement mixed with stability in the chaos. The randomness and complexity of mining-pipe movement are more obvious.