## 4 Results and Discussions
### VIV of a Cylinder Close to a Plane Wall
An obviously periodic shedding of the negative vortices from the freestream-side surface of the cylinder is shown at \(U_{r}=6\), 7, 8, and 9. Furthermore, it can be seen clearly from the lowest-position contours (right column in Figure 11) that the strength of the positive vortex immediately behind the cylinder is stronger with increasing \(U_{r}\) and even a negative vortex is induced to be shed from the plane wall at \(U_{r}=6\), 7, 8, and 9 (marked with circles in Figure 11 d-h). By comparing with the vorticity contours at the highest and lowest positions, one apparent finding is that the vortex formation length, which is defined as the streamwise distance from the cylinder center to the core of the fully formed vortex [41], at the highest position, is shorter than that at the lowest position. Figure 10: Variation of the minimal and maximal gaps for three \(e/D\). Figure 12 plots instantaneous vorticity contours for \(e/D=1\) and \(e/D=2\) when the cylinder is at the lowest position during the vibration cycle. The vortex shedding pattern for \(e/D=1\) is very different from that for \(e/D=0.1\). For \(e/D=1\) (left column in Figure 12), the negative vortex accompanied by a positive vortex can be seen at each \(U_{r}\), even though the strength of the positive one is much weaker. The strength of the interaction between the cylinder and the plane wall changes with the cylinder amplitude. An obvious separation of the plane boundary layer can be seen in the lock-in region when the amplitude value is relatively large. However, a slighter separation in the boundary layer is seen at the other reduced velocities. Figure 11: Instantaneous vorticity contours for \(e/D=0.1\). (**a**–**h**): \(U_{r}=2,3,4,5,6,7,8,9\). “vor” in the figure represents vorticity. Figure 12: Instantaneous vorticity contours for \(e/D=1\). **(left)** and \(e/D=2\). (right). (a–h): \(U_{r}=\)2, 3, 4, 5, 6, 7, 8, 9. “vor” in the figure represents vorticity. [PERSON] and [PERSON] [8] showed that the initial and upper branches are associated with 2S and 2P vortex modes in the unbounded case. As shown in Figure 12 (right column), for the case of \(e/D=2\), the positive vortices become stronger and the vortex patterns are arranged in 2S at \(U_{r}=2\) and \(U_{r}=3\), i.e., two opposite-sign vortices per shedding cycle. Due to the small \(A_{ave}\) and large \(e_{min}\), the shedding vortices are substantially symmetric to the cylinder centerline at \(U_{r}=2\) while asymmetry is caused by the plane wall at \(U_{r}=3\). At \(U_{r}=4\), the transverse spacing between the opposite-signed and negative-signed vortices becomes obviously larger, causing the shedding vortices in the wake near the cylinder to be arranged in 2P mode. But it tends to be in 2S mode away from the cylinder. Despite the amplitude, the minimal and maximal gaps at \(U_{r}=5\) are very close to that at \(U_{r}=4\), the vortex pattern seems to be in 2S at \(U_{r}=5\). At \(U_{r}=7\) and \(U_{r}=8\), the negative-signed vortices are weaker than the positive ones and the vortex shedding mode is very different from the traditional vortex street in that the transverse spacing is larger and the streamwise spacing is smaller, as marked in Figure 12f. At \(U_{r}=9\), the mode tends to be in 2S near the cylinder. However, the transverse spacing is larger and the streamwise spacing is smaller away from the cylinder. In Figure 13, the Fast Fourier Transform (FFT) of the lift coefficient \(C_{L}\) and displacement \(Y\) to the physical time at \(U_{r}=4\), \(U_{r}=5\), and \(U_{r}=9\) is given. It can be concluded from the FFT of \(C_{L}\) that the vortices shed in different modes near and away from the cylinder are always with different frequencies.