# 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 \(\theta=0^{\lx@math@degree}\), \(\overline{C}_{l}\) increases from 0 to 1.6, and \(C_{l}t\) increases approximately by 100%; \(\overline{C}_{d}\) remains almost unchanged, whereas \(C_{d}t\) slightly increases. When \(\theta=30^{\lx@math@degree}\) and \(\theta=60^{\lx@math@degree}\), \(\overline{C}_{l}\) at \(k=4\) increases to 2.1 and 1.1, respectively, whereas \(C_{l}t\) decreases by 36% and 90%, respectively. This reveals that the oscillating wake of the square cylinder can be suppressed to different levels for these particular two wind angles. Although the fluctuation of the lift coefficient has been eliminated, the mean of the lift coefficient has increased. Simultaneously, the value of \(\overline{C}_{d}\) decreased by 54% and 58% for \(\theta=30^{\lx@math@degree}\) and \(\theta=60^{\lx@math@degree}\), respectively. The values of \(C_{d}t\) at \(\theta=30^{\lx@math@degree}\) slightly decrease with the increase in \(k\), whereas \(C_{d}t\) at \(\theta=60^{\lx@math@degree}\) decreases by 87%. When \(\theta=90^{\lx@math@degree}\), \(C_{l}t\) at \(k=2\) and \(k=4\) decreases to 0; this reveals that the alternate shedding vortices in the wake of the square cylinder have been eliminated. When \(\theta=120^{\lx@math@degree}\) and \(\theta=150^{\lx@math@degree}\), \(\overline{C}_{l}\) still presents a growth trend, whereas the remaining statistical parameters present lower values than their corresponding values when \(k=0\); therefore, the input of momentum presents an effect of 'drag reduction and lift increment'. When \(\theta=180^{\lx@math@degree}\), \(\overline{C}_{l}\) presents a negative increment, whereas the remaining statistics present higher values than their corresponding values when \(k=0\). This demonstrates that the input of momentum causes the wake vortex shedding to become enhanced in this particular wind direction. Figure 10: _Cont._  
Figure 9: Aerodynamic coefficient statistics of the square cylinder for the wind angles (\(\theta\)) and the velocity ratios (\(k\)) under outward rotation, (**a**) \(\overline{C}_{l}\), (**b**) \(C_{l}t\), (**c**) \(\overline{C}_{d}\), (**d**) \(C_{d}t\). Figures 11-13 show the dimensionless amplitude spectra of the lift coefficients of the square cylinder for different velocity ratios and different wind angles for the cases of inward rotation, outward rotation, and clockwise co-rotation, respectively. The two horizontal axes represent the wind angle and the Strouhal number, respectively; the vertical axis represents the amplitude of the fast Fourier transform (FFT). The control effect of the MSBC method on the oscillating wake of the square cylinder can be observed through the maximum amplitude for different \(k\) values; simultaneously, the high harmonics with small amplitudes caused by the input of momentum can be observed, as well. When the circular cylinders start to rotate, the amplitude of certain spectral analysis curves decreases, or the peak disappears completely. This shows that the alternate shedding vortex street of the square cylinder wake is suppressed or eliminated under this wind direction. Figure 11: _Cont._  
Figure 10: Aerodynamic coefficient statistics of the square cylinder for the wind angles (\(\theta\)) and the velocity ratios (\(k\)) under co-rotation, (**a**) \(\overline{C}_{l}\), (**b**) \(C_{l}\prime\), (**c**) \(\overline{C}_{d}\), (**d**) \(C_{d}\prime\). Figure 11: Amplitude spectra of Strouhal number under different wind angles with the change of velocity ratios (inward rotation), (**a**) \(k=0\), (**b**) \(k=2\), (**c**) \(k=4\). Figure 12: Amplitude spectra of Strouhal number under different wind angles with the change of velocity ratios (outward rotation), (**a**) \(k=0\), (**b**) \(k=2\), (**c**) \(k=4\). Figure 13: Amplitude spectra of Strouhal number under different wind angles with the change of velocity ratios (co-rotation), (**a**) \(k=0\), (**b**) \(k=2\), (**c**) \(k=4\).