## 3 Validation of the Proposed Numerical Methods
### Preliminary Tests  
The flow around a cylinder problem was chosen for calculation at \(Re=3900\), with a cylinder diameter \(D=0.01\) m. Figure 6 illustrates the computational domain and grid layout for the flow around the cylinder. The cylinder was located \(10D\) away from the fluid inlet surface, \(30D\) away from the pressure outlet surface, and \(10D\) away from both the left and right sides. The height of the cylinder corresponded to the height of the computational domain. The grid size on the surface of the cylinder was \(0.1\) mm, with local refinement in the vicinity of the cylinder using a size of \(0.4\) mm. This mesh was further refined in the \(5D\) domain downstream of the cylinder, with a refinement size of \(0.8\) mm. The initial grid consisted of \(0.8\) million cells, with adaptive refinement applied to the \(5D\) downstream range of the cylinder. The computational results of the typical parameters for the flow around a cylinder as well as the experimental results [29] are presented in Table 3. The typical factors include the Strouhal number (\(St\)), separation angle (\(\psi\)), and minimum flow velocity along the centerline of the wake (\(U_{min}\)). It can be seen that the calculated values of the typical parameters for the flow around the cylinder using the SST-DDES model show small deviations from the experimental results, with errors of \(1.43\%\), \(1.39\%\), and \(8.24\%\) for \(St\), \(\psi\), and \(U_{min}\), respectively. Figure 7 presents the lift and spectrum plots as well as a schematic representation of the separation angle, where \(Sp\) signifies the separation point. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(St\) & \(\psi\) & \(U_{min}/U_{0}\) \\ \hline Experiment & \(0.208\pm 0.002\) & \(86.0^{\circ}\pm 2\) & \(-0.340\) \\ SST-DDES & \(0.213\) & \(87.2\) & \(-0.312\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Typical parameter results for the flow around a cylinder (\(Re=3900\)). Figure 6: Computational domain and the considered mesh for the flow around a cylinder. (**a**) domain; (**b**) domain mesh. Figure 7: Typical parameter plotted results. (**a**) Lift and spectrum; (**b**) Separation angle. Figure 8 displays the \(C_{p}\) curve for the average pressure coefficient on the cylinder surface, with a general trend that matches the experimental results. The calculated values almost agree with the experimental results, with an error of only 2.24% observed for the point with the minimum \(C_{p}\) value of \(-1.133\), compared to the experimental value of \(-1.159\). A further analysis of the wake velocity field was performed by comparing and analyzing the obtained results for three cross-sections at \(x\)/\(D\) = 1.06, 1.54, and 2.02 in the wake of the cylinder. Figure 9 demonstrates a comparison between the average velocity profiles in these three cross-section positions and the experimental results. Figure 9 illustrates the average velocity profiles of the \(x\)-direction at three positions. The average velocity distributions for all three cross-sections exhibit axially symmetric curves. The velocity distribution at section \(x\)/\(D\) = 1.06 follows a U-shaped profile about the \(y\)/\(D\) axis, while the other two sections exhibit a V-shape, which is in good agreement with the experimental results. Figure 9 displays the average velocity profiles of the \(y\)-direction at three cross-section positions. The plotted results reveal that the average velocity distributions for all three sections demonstrate centrally symmetric curves. Furthermore, there are differences in the results of section \(x\)/\(D\) = 1.54, but overall, the pertinent results are almost consistent with the experimental results. Figure 8: Average pressure coefficient on the cylinder surface. Figure 9: Average velocity profiles at three sectional positions. (**a**) Average velocity profiles of the \(x\)-direction; (**b**) average velocity profiles of the \(y\)-direction.