# 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 \(h/D=1/4\)
#### 3.2.3 Aerodynamic Time History Analysis
In the second case, the MSBC becomes activated for \(k=1.5\) to inject the momentum into the flow field from \(t=0\) s. Although the activation time of the MSBC is different, the two cases have the same control effect on the lift and drag forces of the square cylinder. Only a small phase difference can be found in the comparison of the lift coefficient. Figures 21 and 22 show the lift and drag coefficient evolutions of the square cylinder for outward rotation and different \(k\) values when the wind angle is \(30^{\lx@math@degree}\) and \(120^{\lx@math@degree}\), respectively. In Figure 21, it may be observed that the fluctuation of the lift coefficient increases with the increase in \(k\) when \(\theta=30^{\lx@math@degree}\) and \(t\geq 200\) s; the mean and the fluctuation of the drag coefficient increases with the increase in \(k\), as well. This indicates that the input of momentum that has been generated by the MSBC method enhances the vortex shedding energy in the wake of the square cylinder. Figure 22 shows that the MSBC has an effect of 'drag reduction and lift increment' at \(\theta=120^{\lx@math@degree}\). The mean and the fluctuation of the drag coefficient decrease with the increase in the value of \(k\) after the small circular cylinders start to rotate. Simultaneously, the mean of the lift coefficient increases with the increase in \(k\), and the fluctuation of the lift coefficient decreases with the increase in \(k\), although it still exists when \(k=4\). This shows that the wake vortex shedding of the square cylinder has not been completely eliminated; however, the eddy energy has decreased significantly. Similar conclusions can be drawn from the peak value of the amplitude spectra in Figure 12c. Figure 20: Comparisons of lift and drag coefficient time histories of the square cylinder while starting MSBC at different time under \(\theta=0^{\lx@math@degree}\) and inward rotation, (**a**) \(C_{l}\), (**b**) \(C_{d}\). Figures 23 and 24 show the time histories of the lift and drag coefficients of the square cylinder with clockwise co-rotation and different \(k\) values when the wind angle is \(60^{\lx@math@degree}\) and \(90^{\lx@math@degree}\), respectively. As shown in Figure 23, the mean and the fluctuation of the drag coefficient significantly decrease; the mean of the lift coefficient increases, and the fluctuation level is still high when \(\theta=60^{\lx@math@degree}\) and \(k=2\). This indicates that the wake vortex shedding of the square cylinder is still in existence. With the increase in the momentum input, the mean of the drag coefficient continues to decrease, and the mean of the lift coefficient increases when \(k=4\); however, the fluctuation has been significantly reduced. This suggests that the wake vortex street of the square cylinder was suppressed; however, it was not entirely eliminated. As shown in Figure 24, as \(k\) further increases, the mean and the fluctuation of the drag coefficient of the square cylinder present a small change; however, the mean of the lift coefficient increases gradually, and the fluctuation of the lift coefficient significantly decreases when \(\theta=90^{\lx@math@degree}\). This suggests that the oscillating wake of the square cylinder is effectively suppressed. High-frequency parts of the lift and drag coefficients are  
Figure 21: Lift and drag coefficient time histories of the square cylinder with different velocity ratios at \(\theta=30^{\lx@math@degree}\) (outward rotation), (**a**) \(k=2\), (**b**) \(k=4\). Figure 22: Lift and drag coefficient time histories of the square cylinder with different velocity ratios at \(\theta=120^{\lx@math@degree}\) (outward rotation), (**a**) \(k=2\), (**b**) \(k=4\). irrelevant to the vortex frequency, which corresponds to the rotation frequency of the small circular cylinders.