## 4 Results and Discussions  
### VIV of a Cylinder Close to a Plane Wall  
With \(Re=150\), \(\zeta=0\), and \(m^{*}=\frac{8}{\pi}\), the effects of the gap ratio \(e/D\) on structural responses and vortex shading patterns, including their interactions, are investigated by setting \(e/D=0.1\), 1.0, and 2.0. To compare with the cases in unbounded flows, the reduced velocity is also varied between 2 and 9. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(Re\) & \(m^{*}\) & \(U_{r}\) & \(\text{Max}\,A_{ave}\) \\ \hline Current simulation & 150 & \(8/\pi\) & 4 & 0.556\(D\) \\ [PERSON] and [PERSON] [36] & 150 & \(8/\pi\) & 4 & 0.56\(D\) \\ [PERSON] et al. [34] & 150 & \(8/\pi\) & 4 & 0.57\(D\) \\ [PERSON] et al. [35] & 150 & 2 & 4 & 0.57\(D\) \\ [PERSON] & 300–\(10^{6}\) & - & - & \(\sim\)1.0D \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the current results with other numerical and experimental results. The variation of the non-dimensional amplitude \(A_{ave}\) and the predominant frequency of cylinder vibration \(f_{d}/f_{n}\) with \(U_{r}\) is presented in Figure 7, together with the results of the cylinder VIV in unbounded flows. In Figure 7a, it can be seen that the response amplitude \(A_{ave}\) in the case of \(e/D=0.1\) is severely affected by the plane wall and it increases with \(U_{r}\). The cylinder-wall impact occurs at \(U_{r}=7\sim 9\), which is marked in the figure. In the cases of \(e/D=1\) and \(2\), the general trends of \(A_{ave}\) are similar to that in the unbounded flow and the response curves tend to have \"initial\", \"upper\", and \"lower\" branches. This is apparently due to the weaker effect of the wall. However, the maximum amplitudes of \(A_{ave}\approx 0.59\) for \(e/D=1\) and \(2\) are predicted at \(U_{r}=5\); they are larger than that for the unbounded case predicted at \(U_{r}=4\). When \(U_{r}\leq 5\), \(A_{ave}\) for \(e/D=1\) and \(2\) are very close and the deviations become obvious when \(f_{d}/f_{n}>1\). It is interesting to find that when \(U_{r}>5\), the trend of \(A_{ave}\) for \(e/D=1\) is much more similar to that in the unbounded case, even though the plane wall is farther away in the case of \(e/D=2\). A similar phenomenon is found for \(f_{d}/f_{n}\) (Figure 7b) as the lock-in region for \(e/D=1\) and the unbounded case is \(4\leq U_{r}\leq 7\) while that for \(e/D=2\) is broadened, which results in a large amplitude at \(U_{r}=8\). [PERSON] [38] reported that the size of the lock-in region increases with a decreasing gap ratio. This difference between the present study and the previous work of [PERSON] would originate from the different Reynolds numbers or wall kinematics, which play important roles in determining the vibration amplitude [39]. In order to understand how the hydrodynamic forcing causes the above amplitude response, Figure 8 presents two different statistics of the lift coefficients versus \(U_{r}\), including the root-mean-square value of the lift coefficient \(C_{L,\,rms}\) and the time-averaged value of the lift coefficient \(C_{L,\,mean}\). To conveniently find the relationship between the amplitude response and the hydrodynamics, \(A_{ave}\) is also plotted in the figures. It can be seen that the peaks of \(C_{L,\,rms}\) for all three \(e/D\) differ from the points where the maximal values of \(A_{ave}\) occur. Impact with the wall in the case of \(e/D\) causes the trend of \(C_{L,\,rms}\) to be changed. So, it can be guessed that \(C_{L,\,rms}\) must approach the maximal value where the impact has the potential to occur. The curves for \(C_{L,\,\,mean}\) show that the small-gap-ratio case acquires a higher \(C_{L,\,mean}\) than the large-gap-ratio one in the entire range of computed \(U_{r}\). For \(e/D=0.1\), \(C_{L,\,mean}\) decreases with the increasing \(U_{r}\), almost in the entire computed range, except for the reduced velocities where impact occurs. The above regularities found for \(C_{L,\,mean}\) are consistent with those by [PERSON] [38].