## 4 Results and Discussions
### VIV of Two Inline Cylinders Close to a Plane Wall  
In this subsection, the VIV of two inline cylinders close to a plane wall is investigated. The parameters, like \(Re\), \(m^{*}\), \(\zeta\), \(U_{r}\), and \(e/D\), are all set to be the same as those in Section 4.1 for comparison. The center-to-center spacing of the two cylinders \(L\) is \(1.5D\), which is categorized into the reattachment region by [PERSON] [19]. Figures 14 and 15 show the variation of the non-dimensional amplitudes \(A_{ave}\) and the predominant frequency of the cylinder vibration \(f_{d}/f_{n}\) with \(U_{r}\) for different \(e/D\) and the two inline cylinders are denoted by 'front' and'rear', respectively. For \(e/D=0.1\), the \(A_{ave}\) for the two cylinders is almost zero and \(f_{d}/f_{n}\) increases linearly when \(2\leq U_{r}\leq 4\). For \(5\leq U_{r}\leq 9\), the \(A_{ave}\) of the two cylinders increases  
of the two cylinders are excited, especially for the rear cylinder. The amplitude of the rear cylinder is larger than that of the front one when \(f_{d}/f_{n}\) approaches unity, i.e., in the lock-in region. For the front cylinder, \(A_{ave}\) for \(e/D=1\) is smaller than that for \(e/D=2\) when \(f_{d}<f_{n}\) while \(A_{ave}\) for \(e/D=1\) becomes larger than that for \(e/D=2\) when \(f_{d}>f_{n}\). For the rear cylinder, \(A_{ave}\) for \(e/D=1\) is always smaller than that for \(e/D=2\). From current computations, it can be guessed that the amplitude for the rear cylinder increases with \(e/D\). Compared to the case of an isolated cylinder near a plane wall, the vibration amplitudes are larger and the lock-in region is obviously broadened. Figure 16 shows the vorticity contours in the wake at various \(U_{r}\) values for \(e/D=0.1\) when the displacement of the left cylinder center or right cylinder center nearly reaches a local minimum during the vibration cycle, corresponding to the left or right column in the figure, respectively. For \(2\leq U_{r}\leq 4\) when the amplitudes of the cylinders are almost zero, the shear layer of the front cylinder encloses the two cylinders and exhibits no fluctuations in the wake, leading to a stable state. The gap-side shear layer is suppressed by the near-wall effects, so only a single and elongated freestream-side vortex exists in the wake. At \(U_{r}=5\), the cylinders vibrate slightly and the elongated freestream-side shear layer is broken and the boundary layer of the plane wall is induced to be separated. As \(U_{r}\) increases and the cylinders vibrate obviously, periodically shedding negative and positive vortices can be seen in the wake, even though the intensities of the positive ones are lower than the negative ones due to the suppression of the plane boundary layer (Figure 16e,f). Figure 17 shows the vorticity contours for \(e/D=1\) when the displacement of the left cylinder center or right cylinder center nearly reaches a local minimum during the vibration cycle, corresponding to the left or right column in the figure, respectively. It can be seen that the vortices freely shed from both sides of the rear cylinder. Due to the suppression of the plane boundary layer, the gap-side vortices quickly dampen out in the wake. When the amplitude of the rear cylinder becomes large, the vortex shedding pattern tends to be 2P (Figure 17c-h) near the rear cylinder. From the right column, the shear layer of the front cylinder encircles the rear cylinder at \(U_{r}=2-4\) (marked by the circle in Figure 17a-c) while the gap-side shear layer impinges on the rear cylinder and splits into two parts at \(U_{r}=5-9\) (marked by the circle in Figure 17d-h). This is due to the small gap between the rear cylinder and the plane wall in the lock-in region, which leads to strong interaction between the gap-side shear layer and the plane boundary layer. Figure 14: Variation of \(A_{ave}\) with \(U_{r}\) for three gap ratios \(e/D\). Figure 16: Instantaneous vorticity contours for \(e/D=0.1\). (**a–d**): \(U_{r}=2,3,4,5\); (**e**) \(U_{r}=\)7; (**f**) \(U_{r}=\)9.