## 4 Results and Discussions
### VIV of a Cylinder Close to a Plane Wall
However, for a larger \(e/D\) (\(e/D=1\) or \(2\)), the overall trend of \(C_{L,\,mean}\) is that it increases in the range of \(2\leq U_{r}\leq 4\) and decreases beyond \(U_{r}=4\). The vibration frequency is also a major concern with regard to structure fatigue and noise generation. So, it is worth it to explore the relation between the frequency of hydrodynamic force and cylinder vibration. Figure 9 shows the variation in the predominant frequency of cylinder vibration \(f_{d}\) and hydrodynamic force \(f_{cl}\) with \(U_{r}\) for three gap ratios, in comparison with the straight line representing the Strouhal frequency of an isolated stationary circular cylinder at \(Re=\)150, \(f_{0}=0.182\). For \(e/D=2\), \(f_{d}=f_{cl}\) in the range of \(2\leq U_{r}\leq 7\). However, \(f_{cl}\) deviates from \(f_{d}\) at \(U_{r}=8\) and 9. A similar deviation can be seen in the work of [PERSON] [38] for the large-gap-ratio case. In addition, it is predicted that \(f_{cl}>f_{0}\) in the low-\(U_{r}\) part and \(f_{cl}<f_{0}\) in the high-\(U_{r}\) part. For \(e/D=1\), \(f_{cl}\) exceeds \(f_{0}\) in the whole range of \(U_{r}\) and \(f_{cl}\) deviates from \(f_{d}\) at \(U_{r}=6\) and 7, i.e., the high-\(U_{r}\) part in the lock-in region. For \(e/D=0.1\), \(f_{cl}\) and \(f_{d}\) are the same as before \(U_{r}=7\) while \(f_{cl}\) is considerably larger than \(f_{0}\) and \(f_{d}\) at \(U_{r}=8\) and 9, where \(f_{d}\) approaches \(f_{n}\). The case in [40] with \(m^{*}=1.89\) and \(e/D=0.2\), which is analogous to the present case with \(m^{*}=8/\pi\) and \(e/D=0.1\), also shows that \(f_{cl}\) is considerably larger than \(f_{0}\) in the lock-in region. This may be due to the impact between the cylinder and the plane wall. The minimal and maximal gaps, \(e_{min}\) and \(e_{max}\), encountered through the course of cylinder vibration for three \(e/D\) are presented in Figure 10. It is observed that the vibrating  
Figure 8: Variation of \(C_{L}\), \(r_{ms}\) and \(C_{L}\), mean with \(U_{r}\) for three gaps together with a variation of \(A_{\text{arc}}\). (a) \(e/D=0.1\); (b) \(e/D=1\); (c) \(e/D=2\). cylinder can move to positions very close to the plane wall in nearly the whole range of \(U_{r}\) and \(e_{max}\) increases with the increasing \(U_{r}\) of the current considered range in the case of \(e/D=0.1\). Two conclusions can be obtained from the figure. One is that \(e_{min}\) and \(e_{max}\) increase with an increasing gap ratio. This is easy to explain as the effects of the wall on the cylinder are decreased with larger gap ratios. The second is that a larger \(e_{max}\) always accompanies a smaller \(e_{min}\) in the lock-in region for all the three cases with a different \(e/D\). For quite a long time, the vortex structures in the wake behind a bluff body have attracted the attention of researchers. Deep insight into the physics involved would be very useful for engineering applications. So, the vortex shedding patterns of the near-wall cylinder at different gap ratios are presented to illustrate the influence of the wall proximity under various reduced velocities. Figure 11 shows instantaneous vorticity contours \(\omega=\left(\frac{\partial\varphi}{\partial x}-\frac{\partial u}{\partial y }\right)/(U_{\infty}/D)\) for e/D = 0.1 at \(U_{r}=2-9\) when the cylinder is at the highest (left) and lowest (right) positions during the vibration cycle. Overall, the vortices shed from the gap-side surface of the cylinder are suppressed due to the interaction with negative vortices generated on the plane wall. With increasing \(U_{r}\), the induced separation zone of the plane boundary layer is getting larger. At \(U_{r}=2\), the vibration of the negative vortices shed from the freestream-side surface is not obvious due to the extremely small \(A_{ave}\). Even though the minimum gap \(e_{min}\) and the cylinder amplitude \(A_{ave}\) are very close at \(U_{r}=3\) and 4, the vortex shedding patterns are different in that there exists periodic shedding of the positive vortices behind the cylinder at \(U_{r}=4\).