## 3 Results and Discussion
### Time History of Displacements for All Three Risers  
The time history of displacements in in-line flow direction (X) and cross flow direction (Y) for all risers in different cases are plotted in Figures 6 and 7, respectively. In order to show the comparison more clearly, the displacements in both directions were normalized by the O.D. of the risers. For all the risers, three typical locations, i.e., \(-6\) m, \(-12.5\) m and \(-19\) m, were selected. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**Property**} & \multicolumn{3}{c}{**Riser 1**} & \multicolumn{3}{c}{**Riser 2**} & \multicolumn{3}{c}{**Riser 3**} \\ \cline{2-10}  & **Case 1** & **Case 4** & **Case 7** & **Case 2** & **Case 5** & **Case 8** & **Case 3** & **Case 6** & **Case 9** \\ \hline natural frequency \(\mathrm{f_{n}/(Hz)}\) & & 2.01 & & & 2.45 & & & 1.89 & \\ vortex shedding frequency & 0.24 & 0.85 & 1.56 & 0.22 & 0.78 & 1.42 & 0.23 & 0.82 & 1.51 \\ reduced velocity U\({}_{\rm r}\) & 0.60 & 2.02 & 3.53 & 0.45 & 1.51 & 2.64 & 0.61 & 2.06 & 3.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Natural frequencies, vortex shedding frequencies and reduced velocities of the risers. Figure 6: _Cont._  
From Figure 6, it is clear that the in-line flow direction (X) displacements maintained at a relatively stable value and vibrate slightly after the initial growth for all three risers and at all locations. More specifically, for velocities equal to 0.36 m/s, the maximum displacements in flow direction/outside diameter (Ax/D) of risers 1, 2 and 3 were around 0.0017, 0.003 and 0.0065 at \(-6\) m, 0.002, 0.0036 and 0.008 at \(-12.5\) m and 0.0009, 0.0016 and 0.0035 at \(-19\) m, respectively. For velocities equal to 1.22 m/s, the maximum displacements in flow direction/outside diameter (Ax/D) of risers 1, 2 and 3 were 0.02, 0.035 and 0.075 at \(-6\) m, 0.024, 0.042 and 0.095 at \(-12.5\) m and 0.011, 0.019 and 0.042 at \(-19\) m, respectively. For velocities equal to 2.13 m/s, the maximum displacements in flow direction/outside diameter (Ax/D) of risers 1, 2 and 3 were 0.06, 0.10 and 0.23 at \(-6\) m, 0.075, 0.128 and 0.274 at \(-12.5\) m and 0.033, 0.057 and 0.12 at \(-19\) m, respectively. To summarize: (1) In the cases with the same current velocity, the displacement of riser 3 was the largest, followed by riser 2, and riser 1  
Figure 6: Time history of displacements in in-line flow direction (X) at three typical locations. was the smallest for all locations of the three risers. (2) With the increase of current velocity, the Ax/D rose for all three risers. Take riser 3 at 12.5 m for example, the Ax/D(s) was about 11 times (1.22 m/s) and 32 times (2.13 m/s) of that at 0.36 m/s. (3) The end support conditions affected the displacement significantly, i.e., simple support lead to larger displacement. Using riser 3 at 2.13 m/s as an example, the Ax/D(s) was about 0.23 at \(-\)6 m (6 m to the simple support end) and 0.12 at \(-\)19 m (6 m to the fixed support end). Figure 7: _Cont._  
From Figure 7, it is clear that the cross-flow direction (Y) displacements of all risers vibrated obviously. The vibrations for the cases with 0.36 m/s were comparatively more stable than those for the cases with 1.22 m/s and 2.13 m/s. This was because for cases 1-3 (0.36 m/s), the Re (s) was 101,887, 111,736 and 105,623 and therefore the vortex streets for these cases were fully turbulent [50]. For cases 4-9, the Re (s) was between 300,000 and 3,500,000 and therefore the laminar boundary layer underwent turbulent transition and the wake was narrower and disorganized [50]. More specifically, for velocities equal to 0.36 m/s, the maximum range of displacements in cross-flow direction/outside diameter (Ay/D) of riser 1, 2 and 3 were around 0.0004, 0.001 and 0.002 at \(-\)6 m, 0.0005, 0.0018 and 0.003 at \(-\)12.5 m and 0.0002, 0.0006 and 0.0013 at \(-\)19 m, respectively.