# Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder
## 3 Results and Discussion
### Detailed Results and Analysis of \(D/L=0.3\) and \(h/D=1/4\)
#### 3.2.1 Aerodynamic Statistics and Frequency Characteristics
When \(\theta=15^{\circ}\), the values of \(C_{d}\prime\) for \(k>0\) can decrease by \(25\%\) compared to the value of \(C_{d}\prime\) for \(k=0\). Figure 8: Aerodynamic coefficient statistics of the square cylinder for the wind angles (\(\theta\)) and the velocity ratios (\(k\)) under inward rotation, (**a**) \(\overline{C}_{l}\), (**b**) \(C_{l}\prime\), (**c**) \(\overline{C}_{d}\), (**d**) \(C_{d}\prime\). Through the above analysis, it may be observed that the control method may help to improve the wake stability of a square cylinder when \(\theta\leq 60^{\lx@math@degree}\), and it is unfavorable to the stability of the wake for the remaining wind angles. The control effect on the oscillating wake of the square cylinder is optimal when \(\theta=0^{\lx@math@degree}\). When \(\theta=15^{\lx@math@degree}\), this particular control method resulted in a great decrease in \(\overline{C}_{d}\) and a great increase in \(\overline{C}_{l}\) of the square cylinder. From this viewpoint, the optimum effect of 'drag reduction and lift increment' is achieved at this particular wind angle. In Figure 9, the variation of the statistical parameters of the aerodynamic forces with wind angles is similar for \(k=2\) and \(k=4\); the difference lies only in terms of the individual angle. When \(k=0\), the aerodynamic parameters of the square cylinder (outward rotation) are consistent with the results in Figure 8. When \(\theta=0^{\lx@math@degree}\), the values of \(\overline{C}_{l}\) for different \(k\) values are near 0; however, the value of \(C_{l}\prime\) increases from 0.32 at \(k=0\) to 0.97 at \(k=4\). In a similar manner, the mean and fluctuating values of the drag coefficient have increased significantly, thus suggesting that the momentum generated by the rotating cylinder leads to a more intense vortex shedding in the wake of the square cylinder. When \(\theta=30^{\lx@math@degree}\) and \(\theta=60^{\lx@math@degree}\), \(\overline{C}_{l}\) is not equal to 0, and \(C_{l}\prime\), \(\overline{C}_{d}\), and \(C_{d}\prime\) present growth trends as the \(k\) value incrementally increases; this illustrates that the wake vortex pattern has changed, and the MSBC presents an effect of 'lift and drag increment'. When \(\theta=90^{\lx@math@degree}\) and \(k=4\), \(C_{l}\prime\) and \(C_{d}\prime\) increase slightly compared with their values when \(k=0\). \(\overline{C}_{d}\) remains unchanged, and \(\overline{C}_{l}\) slightly decreases. This indicates that the momentum injection does not improve the flow structure. When \(\theta=120^{\lx@math@degree}\) and \(\theta=150^{\lx@math@degree}\), \(\overline{C}_{l}\) increases significantly with the increase in \(k\); however, \(C_{l}\prime\), \(\overline{C}_{d}\), and \(C_{d}\prime\) present a significantly decreasing trend. This suggests that the oscillating wake of the square cylinder has been suppressed and that the MSBC presents an effect of 'drag reduction and lift increment'. When \(\theta=120^{\lx@math@degree}\) and \(k=4\), \(\overline{C}_{d}\) decreases by 90% compared with the value of \(\overline{C}_{d}\) when \(k=0\), and \(C_{d}\prime\) decreases by 60%; therefore, the 'drag reduction' effect is obvious. In addition, the value of \(\overline{C}_{l}\) becomes six times higher than the value of \(\overline{C}_{l}\) when \(k=0\), whereas \(C_{l}\prime\) decreases by 75%; thus, the 'lift increment' effect is obvious. When \(\theta=180^{\lx@math@degree}\), \(\overline{C}_{d}\) increases slightly with the increase in \(k\); the values of the remaining parameters do not present significant differences with their respective values when \(k=0\). The results show that the MSBC method has no obvious effect on the oscillating wake for this particular wind direction. Figure 9: Cont. As shown in Figure 10, when \(k>0\), the aerodynamic statistics of the square cylinder present a similar change for different wind angles. With the increase in \(k\) from 0 to 4, the momentum of the rotation that the circular cylinders transfer to the flow field gradually increases.