# Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder
## 2 Numerical Calculation Model
### Validity Investigation  
The square cylinder with \(\theta=0^{\circ}\) and \(k=0\) is defined as the'standard square cylinder', whereas the square cylinder without rotating cylinder is referred to as a'single square cylinder'. First, a mesh independence study was conducted in order to quantify the effect of mesh density on computational results. Simulations of flow around the single square cylinder were then performed at \(\text{Re}=200\) using four different mesh densities as listed in Table 1, where \(N_{c}\), \(N_{mesh}\), and \(N_{nodes}\) represent, respectively, the node number on each side length of the square cylinder, the total number of grids, and nodes in the computational domain. The minimum grid size of each side of the single square cylinder \(l_{min}=L/N_{c}\)  
Figure 3: Wind angle, rotation direction and pressure monitoring point layout (\(h/D=1/4\), \(D/L=0.3\)), (**a**) Inward rotation, (**b**) Outward rotation, (**c**) Clockwise co-rotation. and the grids grow into the computational domain with each side of the square cylinder as source and the growth rate of 1.04, and the maximum grid size \(l_{max}=0.4L\). The comparison shown in Table 1 includes the fluctuation of lift coefficient (\(C_{l}\prime\)), the mean and fluctuation of drag coefficient (\(\overline{C}_{d}\), \(C_{d}\prime\)), and the Strouhal number (\(S_{t}\)). As shown in Table 1, as the mesh density increases, the \(C_{d}\prime\) and \(S_{t}\) of different mesh schemes do not change significantly, and both \(\overline{C}_{d}\) and \(C_{l}\prime\) show a decreasing trend. The differences in highlighted aerodynamic coefficients \(\overline{C}_{d}\) and \(C_{l}\prime\) between the normal (Scheme 3) and dense (Scheme 4) meshes are 0.6 % and 4.2 %, and that between the coarse (Scheme 2) and dense (Scheme 4) meshes are 2.3 % and 12.8 %. Therefore, the normal mesh of 'Scheme 3' is appropriate in the following simulations of this study. Then, the mesh independent results of flow around the single square cylinder are compared with the existing research results to verify the accuracy of numerical simulation, and the results from the flow around the standard square cylinder are compared with those from the single square cylinder. The minimum grid size, growth ratio, and maximum grid size of each side of the standard square cylinder and the surfaces of the two rotating small cylinders are kept consistent with the parameters of the 'Scheme 3' meshing scheme of the single square cylinder. Figure 4 shows the time histories of the lift and drag coefficients of the single and standard square cylinders. The dimensionless lift coefficient by a spectral analysis is converted to the Strouhal number (\(S_{t}\)). The dimensionless time is \(t^{*}=tU_{\infty}/L\). The alternating shedding vortex wake is the main cause of the periodic variation in the lift and drag coefficients, as shown in Figure 5. The figure shows the vorticity contours in the \(z\) direction (perpendicular to the \(x\)-o-\(y\) plane). The red color in the figure represents the positive vorticity of counterclockwise rotation, and the blue color represents the negative vorticity of clockwise rotation. The comparison between the results from the present numerical simulation and the experimental and numerical results from other previous work is presented in Table 2. The statistical results of aerodynamic parameters in the Table 2 are analyzed by using the time histories of lift and drag coefficients in the time period of \(t^{*}=80\sim 150\), shown in Figure 4a. In the subsequent calculation cases, all pressure and aerodynamic coefficient time histories are calculated to the amplitude stable state, and the results of at least five vortex shedding characteristic periods are taken for statistical analysis. The simulation results of the aerodynamic forces and the Strouhal number are consistent with the previous results, and it indicates that the present grid accuracy, time step, and numerical settings can accurately capture the characteristics of the flow around a square cylinder.