# Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder
## 3 Results and Discussion  
### Analysis of Influence of \(D/L\) and \(h/D\) Parameters  
The diameter (\(D\)) of the small rotating circular cylinders and the spacing (\(h\)) between the rotating circular cylinders and the controlled square cylinder are important parameters that affect the control effect on the wake of the square cylinder. When wind angle \(\theta=0^{\circ}\), velocity ratio \(k=2\), and the rotation direction of the circular cylinders is fixed to 'inward' rotation, as shown in Figure 3a, the influences of \(D/L\) and \(h/D\) on the statistical values and frequency characteristics of the aerodynamic coefficients of the square cylinder are mainly analyzed. The two dimensionless parameters \(E_{C_{I}}=(C_{I}-C_{I,c}t)/C_{I}t\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Investigation** & \(\mathbf{-}\) & \(\mathbf{C_{d}}\) & \(\mathbf{C_{d}^{{}^{\prime}}}\) & \(\mathbf{C_{I}^{{}^{\prime}}}\) & \(\mathbf{S_{t}}\) \\ \hline [PERSON] [41], Experimental & 1.45 & - & - & 0.14–0.148 \\ [PERSON] et al. [39], Numerical, 2D & 1.462 & - & 0.377 & 0.15 \\ [PERSON] et al. [40], Numerical, 2D & 1.45 & - & 0.372 & 0.15 \\ [PERSON] and [PERSON] [42], Numerical, 2D & - & - & - & 0.148 \\ [PERSON] and [PERSON] [43], Numerical, 2D & 1.488 & 0.027 & 0.332 & 0.153 \\ Present, Numerical, 2D (\(N_{c}=70\)) & 1.465 & 0.020 & 0.373 & 0.148 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the force coefficients statistics and Strouhal number of flow around a single square cylinder at \(\mathrm{Re}=200\). Figure 4: Comparisons of results of flow around a standard square cylinder and a single square cylinder, (**a**) Time histories of lift and drag coefficients, (**b**) Spectral analysis of \(C_{l}\). Figure 5: Vortex shedding in the wake of a standard square cylinder and a single square cylinder, (**a**) Standard square cylinder, (**b**) Single square cylinder. \(E_{C_{d}}=(\overline{C}_{d}-\overline{C}_{d,c})/\overline{C}_{d}\) are defined, which represent the relative variation in the RMS value of the lift coefficient and the mean value of the drag coefficient, respectively. These two parameters are used to evaluate the control effectiveness of the MSBC on the wake of flow around the square cylinder and determine the optimal parameters of the \(D/L\) and \(h/D\). When \(E_{C_{l}}\) and \(E_{C_{d}}\) are close to 1.0, the suppression effect on the wake is optimal. In contrast, when \(E_{C_{l}}\) and \(E_{C_{d}}\) are close to 0, no control exists. Figure 6 shows the aerodynamic statistical parameters and Strouhal number (\(S_{t}\) and \(S_{t,c}\)) of flow around the uncontrolled (\(k=0\)) and controlled (\(k=2\)) square cylinder versus \(D/L=0.05\sim 0.4\) when \(h/D=1/4\), together with the control effectiveness. The specific values corresponding to Figure 6 are shown in Table 3, where the aerodynamic statistical parameters include the RMS value of the lift coefficient (\(C_{l}\)/ and \(C_{l,c}\)), the mean value of the drag coefficient (\(\overline{C}_{d}\) and \(\overline{C}_{d,c}\)), and the RMS value of the drag coefficient (\(C_{d}\)/ and \(C_{d,c}\)). For different \(D/L\) cases, the centers of the rotating circular cylinders can ensure that the side extended lines of the square cylinder are tangent to the circumference of the rotating circular cylinders. For the uncontrolled standard square cylinder, \(C_{l}\)/ is slowly decreased by 8%, \(\overline{C}_{d}\) and \(C_{d}\)/ are decreased by 29% and 50%, respectively, while \(S_{t}\) is kept unchanged at 0.15 with the increase of \(D/L\) from 0.05 to 0.4. When there are two nonrotating cylinders at the corners of the square cylinder, it is equivalent to the rounded treatment of the leeward sharp corners of the square cylinder. This has less influence on the fluctuation degree of the lift coefficient in the cross-flow direction and vortex shedding frequency but can reduce mean value and fluctuation of the drag coefficient in the along-wind direction.