## 3 Convergence Study and Calculation Validation  
A mesh dependency study of a single cylinder is performed at \(Re=3\times 10^{6}\). Three types of meshes (coarse, medium, and fine meshes, respectively) are utilized. The ratio of cell length between coarse mesh and medium mesh, medium mesh and fine mesh is \(\sqrt{2}\) in three space directions, which was recommended by [PERSON] [38]. The boundary layer mesh remains the same. Representative dynamic force coefficients are calculated to evaluate the mesh convergence. The drag and lift coefficients \(C_{d}\) and \(C_{l}\) are defined as  
\[C_{d}=\frac{F_{d}}{\frac{1}{2}\rho U_{0}^{2}DH} \tag{15}\]  
\[C_{l}=\frac{F_{l}}{\frac{1}{2}\rho U_{0}^{2}DH} \tag{16}\]  
and the Strouhal number \(S_{t}\) is defined as  
\[S_{t}=\frac{f_{s}D}{U_{0}} \tag{17}\]  
where \(F_{d}\) and \(F_{l}\) are the fluid forces on the cylinder in the \(x\)-direction and \(y\)-direction, respectively; \(U_{0}\) is the velocity of the fluid, \(H\) is the length of the cylinder in \(z\)-direction, \(f_{s}\) is the frequency of the vortex shedding, which is obtained from the power spectrum density (PSD) of the lift force of the cylinder. The pressure coefficient (\(C_{p}\)) is defined as  
Figure 3: The grid resolution near the cylinder wall. Figure 2: The computational mesh for \(L/D=2.5\): (**a**) mesh refinement using multiple sub-domains; (**b**) mesh near the cylinders surface. \[C_{p}=\frac{P-P_{0}}{\frac{1}{2}\rho U_{0}^{2}} \tag{18}\]  
where \(P_{0}\) represents the reference pressure. \(C_{pm}\) is the minimum value of \(C_{p}\), and \(C_{pb}\) is the base pressure coefficient of the cylinder. The overline bar means the time average. The fluctuating lift coefficient \(C_{l}^{\prime}\) is the root-mean-square (RMS) value of \(C_{l}\). For statistical accuracy, at least 25 vortex shedding cycles are sampled, which means the flow passes through the computational domain more than 3 times in the statistical process. As shown in Table 1, the results from the coarse mesh differ from those of the medium mesh and fine mesh. \(S_{t}\) of the three sets of meshes are the same. As can be seen, the medium mesh and fine mesh yield insignificant changes of \(\bar{C}_{d}\), \(-\overline{C}_{pm}\) except for \(C_{l}\) and \(-\overline{C}_{pb}\), but the difference is in an acceptable range. Thus, the medium mesh is considered to be suitable for the following calculations. In order to validate the present numerical model, Table 2 summarizes the results of the present simulation with previous experimental and numerical calculations of other scholars for a single cylinder at \(Re=3\times 10^{6}\). Regarding the super-critical regime, the outcomes of the earlier experiments showed significant scattering. This may be primarily caused by a variety of experimental challenges, including surface roughness, turbulent intensity of free flow, cylinder end conditions, wind tunnel blockage ratio, and cylinder aspect ratio, which make it challenging to precisely measure flow at high Reynolds numbers [39]. Considering the aforementioned factors, the present simulation results are acceptable. In addition, Figure 4 compares the \(\bar{C}_{p}\) distribution on a single cylinder with published numerical studies ([PERSON] et al. [44] at \(Re=3\times 10^{6}\) and [PERSON] et al. [45] at \(Re=1\times 10^{6}\)) and experimental studies ([PERSON] [47] at \(Re=1.2\times 10^{6}\) and [PERSON] [48] at \(Re=6.7\times 10^{5}\)). As shown in Figure 4, the \(\bar{C}_{p}\) distribution on a single cylinder agrees well with previous results. The skin friction coefficient, \(C_{f}=\tau/(0.5\rho U_{0}^{2})\), where \(\tau\) is the tangential wall shear stress, is presented in Figure 5. The separation point location \(\theta=116^{\circ}\)