## 2 Materials and Methods
### Computational Model and Sensitivity Study  
A grid sensitivity study is conducted by employing four meshes to analyze the flow past an oscillation cylinder at \(A/D=1\) and \(f_{c}/f_{s}=0.75\). The cylinder oscillates as a harmonic mode, expressed as: \(y(t)=\text{Asin}(2\pi\eta_{c}t)\). Computational domain is discretized by a boundary layer and an external unstructured grid, and the boundary layer oscillates as a rigid body along with the boundary of the cylinder. The external unstructured grid is updated by a spring-based smoothing and a remeshing method to avoid a large mesh deformation. The detailed nodes and the total number of elements are shown in Table 1. Table 2 shows the comparison between the Strouhal number (\(St=f_{s}D/u\)) obtained by the four meshes and those reported by previous works. The \(St\) numbers from the former are all around 0.22 and increase slightly with an increase in the total number of elements. Its growth rate is not more than 2.26%. In the case of Mesh-III and Mesh-IV, the \(St\) numbers exhibit a tiny change and tend to a stable state, which indicates that the calculated results are insensitive to a further increase in the number of elements. Additionally, the results also verify the conclusion that the VLES method has good predictability for a coarse grid, and high efficiency for dynamic boundary problems. According to the empirical formula provided by [PERSON] et al., the \(St\) number is 0.21 when \(Re=3900\)[21]. [PERSON] et al. employed the variational multiscale LES (VMS-LES) to simulate the 3D stationary cylinder at \(Re=3900\) and obtained a \(St\) number of 0.22 [22]. However, [PERSON] et al. and [PERSON] et al. found that the \(St\) number at \(Re=3900\) was within the range 0.2-0.22 [16; 23]. Additionally, [PERSON] et al. also studied the flow structures on the onset and development of the shear layer instability at \(Re=3900\) and 4000 and \(Re=10{,}000\). The results showed that the velocity fluctuations and the Reynolds stress from the particle image velocimetry (PIV) results at \(Re=4000\) agreed well with the simulation studies at \(Re=3900\) using the DNS method, illustrating that the fluid mechanics between \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Nodes (\(\mathbf{X}\times\mathbf{Y}\))** & **Nodes (Cylinder)** & **Total Elements** \\ \hline Mesh-I & \(51\times 34\) & 160 & 6626 \\ \hline Mesh-II & \(101\times 68\) & 198 & 17,894 \\ \hline Mesh-III & \(201\times 134\) & 240 & 60,418 \\ \hline Mesh-IV & \(301\times 201\) & 300 & 131,426 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Elements and nodes distribution of the grid convergence study. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Mesh-I** & **Mesh-II** & **Mesh-III** & **Mesh-IV** & **Ref. [21]** & **Ref. [22]** & **Experiments [16; 23]** \\ \hline
0.21995 & 0.21995 & 0.22493 & 0.22495 & 0.21 (\(Re=3900\)) & 0.22 (\(Re=3900\)) & 0.2-0.22 (\(Re=3900\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Strouhal (\(St\)) numbers calculated by the four grids as compared with the results from the literatures. the two had exactly similar characteristics. Thereby, the \(St\) number from experimental data at \(Re=3900\) is reasonably selected to validate the simulation results at \(Re=4000\) in this section. Combined with the above analysis, the \(St\) number of a 3D stationary cylinder at \(Re=4000\) should be 0.2 to 0.22 and tends toward 0.22. In this section, the \(St\) numbers from the sensitivity study are around 0.22, which also verify the validation of simulation results. The reason for the small difference in the \(St\) numbers could be caused by the adoption of a 2D model, thus, deviating from the reported results of the 3D model from Ref. [16; 21; 22; 23], because the 2D simulation does not capture the 3D effects. However, several 2D studies verified that the computed results are quite consistent with the experimental results, despite this limitation. [10; 11; 16; 23] Moreover, [PERSON] et al. [16] and [PERSON] et al.