# Investigation on Numerical Simulation of VIV of Deep-Sea Flexible Risers
## 3 Quasi-3D Numerical Calculation Model
### Quasi-3D Numerical Model Validation  
To verify the accuracy of the quasi-3D numerical model, we selected a working condition from a top tension riser experiment [35] conducted by the Norwegian Institute of Marine Technology. Table 2 shows the properties of the riser model.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Boundary** & **Setting** \\ \hline Inlet boundary & Velocity-inlet, \(u=U_{0}\), \(v=0\) \\ Outlet boundary & Pressure-outlet, \(\partial u/\partial x=0\), \(\partial v/\partial x=0\), static pressure = 0 \\ Upper and lower boundary & Symmetry, \(\partial u/\partial x=0\), \(\partial v/\partial x=0\) \\ Cylinder boundary & No-Slip Wall, \(u=v=0\) \\ Interface of foreground grid & Overset Interface \\ and background grid & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Boundary settings.  
Figure 5: Computational domain.  
Figure 6: Background grid diagram.  
To accurately simulate the vibration pattern of the riser, it was ensured that at least three slices and seven finite element nodes were available for each mode. The axial direction of the riser was uniformly divided into 1-69 finite element nodes from bottom to top, with 17 two-dimensional flow field slices arranged. The 9 th slice was located at the 35 th finite element node, which was in the center of the riser. Figure 9 shows the schematic diagram of slice division.  
Numerical simulation of the vortex-induced vibration of a riser was performed. Figure 10 shows the comparison of the envelopes of the inline displacement between the present quasi-3D numerical simulation and the experiment. Figure 11 shows the comparison of the envelopes of the crossflow displacement between the present quasi-3D numerical simulation and the experiment.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Values** & **Units** \\ \hline Riser outer diameter (D) & 0.02 & m \\ Riser wall thickness (tw) & 0.0045 & m \\ Length of the riser (L) & 9.63 & m \\ Top tension (T) & 817 & N \\ Young’s modulus (E) & 1.025 \(\times\) 1011 & N/m\({}^{2}\) \\ Mass ratio (m \({}^{*}\)) & 2.23 & \\ First-order eigenfrequency (f1) & 1.79 & Hz \\ Second-order eigenfrequency (f2) & 3.67 & Hz \\ \hline \hline \end{tabular} \({}^{*}\): Dimensionless.  
\end{table}
Table 2: Properties of the riser model.  
Figure 8: Grid division diagram of single 2D flow field slice.  
Figure 7: Foreground grid diagram.  
It can be seen that the main control modes of inline and crossflow displacements obtained from the quasi-3D numerical simulation results were second-order and first-order, respectively, which is consistent with the experimental results. Although the amplitude ranges were slightly different, the envelope profile was essentially the same.  
As shown in Figure 12, by comparing the displacement-time history curves in the crossflow direction of the riser at \(\mathrm{z}/\mathrm{L}=0.22\), it is easy to find that the amplitude range of the present quasi-3D numerical simulation is in good agreement with the original experimental results. The original experimental period was about 0.60 s, and the present CFD period was about 0.55 s, which is slightly shorter than that of the original experimental data, but the error was controlled within 8.3%. The frequency calculated from the periodic value obtained from the numerical calculation results in this paper was 1.786 Hz, which is almost the same as the first-order eigenfrequency (1.79 Hz).  
Figure 10: Comparison of the riser response envelopes in IL direction. (**a**) Simulation results. (**b**) Experimental results [27].  
Figure 9: Slice position sketch.  
According to the above analysis, the quasi-three-dimensional numerical results in this paper are in good agreement with the experimental data, indicating that the quasi-three-dimensional numerical model in this paper can accurately predict the dynamics of vortex-induced vibration.