## 3 Results and Discussion  
To illustrate the VIV characteristics of the oscillating cylinder at \(Re=4000\), we conducted a series of simulations for \(A/D=0.5\) to 1.5 and \(f_{e}\)/\(f_{s}=0.5\) to 1.35 in this paper, and the specific parameter settings are shown in Table 3. Additionally, the temporal evolutions of the lift and drag coefficients, and the frequency and mode of the vortex shedding is discussed in detail, respectively. Figure 5: Control function (\(Fr\)) distribution of the computational domain in \(10T\) at the amplitude-to-cylinder ratio, \(A/D=1,f_{e}\)/\(f_{s}=0.75\). (**a**) \(Fr\) contour and (**b**) \(Fr\) distribution with coordination of X direction. \(T=1\)/\(f_{e}\). Figure 6: The y+ distribution of the cylinder in \(10T\) at \(A/D=1,f_{e}\)/\(f_{s}=0.75\). Figure 7 shows the vorticities around and behind the oscillating cylinder at \(A/D=0.5\) and \(f_{c}/f_{s}=0.5\) to \(1.35\). Figure 8 shows the temporal evolutions of the lift coefficient (\(Cl\)) and drag coefficient (\(Cd\)) (left) and the power spectral density (PSD) of the lift coefficient of the oscillating cylinder (right) in the corresponding parameters. The PSD distribution of the lift coefficient of an oscillating cylinder is obtained by a fast Fourier transform (FFT). \[\int_{0}^{nT}Cl(t)e^{2\pi ft}dt=Cl(f), \tag{11}\]  
\[\text{PSD}=\lim_{T\rightarrow\infty}\frac{\left|Cl(f)\right|^{2}}{nT}. \tag{12}\]  
The fluid force acting on the cylinder includes two contributing parts based on pressure and shear stress. Additionally, the lift (coefficient) \(F_{l}\) (\(Cl\)) and drag (coefficient) \(F_{d}\) (\(Cd\)) are defined as, respectively:  
\[Cl=F_{l}/0.5\rho Du^{2}, \tag{13}\]  
\[Cd=F_{d}/0.5\rho Du^{2} \tag{14}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(A\)/\(D\) & \multicolumn{6}{c}{\(f_{c}/f_{s}\)} \\ \hline
1 & 0.5 & 0.75 & 0.95 & 1.05 & 1.2 & 1.35 \\ \hline
1.5 & 0.5 & 0.75 & 0.95 & 1.05 & 1.2 & 1.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Oscillating amplitudes and frequencies of present numerical simulation. Figure 7: Vorticities around and behind the oscillating cylinder in \(10T\) at \(A/D=0.5\) and different frequencies. (**a**) \(f_{c}/f_{s}=0.5\); (**b**) \(f_{c}/f_{s}=0.75\); (**c**) \(f_{c}/f_{s}=0.95\); (**d**) \(f_{c}/f_{s}=1.05\); (**e**) \(f_{c}/f_{s}=1.2\); and (**f**) \(f_{c}/f_{s}=1.35\). As per the Parseval’s identity, the lift coefficient can be expressed as follows by FFT:  
\[F_{l}=-\int_{0}^{2\pi}\biggl{(}p(t)cos\theta-\frac{2}{R_{e}}\omega_{z}(t)sin\theta \biggr{)}d\theta, \tag{15}\]  
\[F_{d}=-\int_{0}^{2\pi}\biggl{(}p(t)sin\theta+\frac{2}{R_{e}}\omega_{z}(t)cos \theta\biggr{)}d\theta. \tag{16}\]  
where \(p(t)\) and \(\omega_{z}(t)\) are obtained by a dimensionless transformation of the pressure and the axial vorticity on the surface of the oscillating cylinder, respectively. In Figure 7, the wake vortices are mainly distributed near the wake center line, showing no significant lateral deviation due to less excitation amplitude at \(A/D=0.5\). In addition, the number of the wake vortices at \(f_{e}/f_{s}=0.5\) obviously is greater than that at \(f_{e}/f_{s}=0.75\). This is caused by the fact that the vortex shedding is influenced by the excitation frequency \(f_{e}\) and, according to the PSD at \(A/D\)  
Figure 8: Temporal evolutions of the lift and drag coefficients (left) and the power spectral density (PSD) of the lift coefficient (right) of the oscillating cylinder at \(A/D=0.5\) and different frequencies. = 0.5 and _f_/_f_s = 0.5 in Figure 8, the Stroulhal frequency _f_s also impacts the vortex shedding. Both of the two frequencies combine to determine the vortex shedding in the wake, causing an obvious \"beating\" characteristic in the temporal evolutions of the lift and drag coefficients. Additionally, the vortex shedding mode also verifies that the frequency ratio _f_/_f_s = 0.5 is not within the range of the \"lock-in\" phenomenon at _Re_ = 4000.