# 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  
This section presents the lift and drag coefficient evolutions of the square cylinder during the entire numerical simulation process, starting from the flow around the fixed square cylinder (simplified as 'FR') up to the MSBC stage. The time histories of the aerodynamic coefficients are compared before and after the rotation of the small circular cylinders. Figures 18 and 19 present the time histories of the lift and drag coefficients of the square cylinder for inward rotation and different \(k\) values when the wind angle is \(0^{\lx@math@degree}\) and \(15^{\lx@math@degree}\), respectively. At these two wind angles, the MSBC plays an important role in suppressing the oscillating wake. In Figure 18a,b, each time history during the interval of \(0\leq t<100\) s indicates the lift and drag coefficients of the 'FR' stage. After the small circular cylinders have started to rotate at \(t\geq 100\) s, the mean drag coefficient \(\overline{C}_{d}\) decreases from 1.015 to 0.9, and the fluctuation of the lift coefficient decreases at \(k=1\), as well, as shown in Figure 18a; however, the vortex street in the wake of the square cylinder has not been completely eliminated in this case. With the increase in the rotating speed of the small circular cylinders, the input of momentum to the flow field gradually increases, as well. When \(k=2\), the fluctuation of the lift coefficient seems to have been eliminated, and \(\overline{C}_{d}\) further decreases to 0.8. In Figure 18c,d, the MSBC was initialized from the fully developed flow at \(t\geq 150\) s. When \(k=3\), the time history of the lift coefficient has the form of an almost straight line, without fluctuations, which indicates that the MSBC method completely eliminates the oscillating wake of the square cylinder. The \(\overline{C}_{d}\) further decreases to 0.71, and the fluctuation of the drag coefficient decreases accordingly. When \(k=4\), \(\overline{C}_{d}\) continues to decrease to 0.65, which indicates that the drag coefficient decreases as \(k\) increases; however, the fluctuation of the lift and drag coefficient is larger than that for \(k=3\). This suggests that the case of \(k=2\) can be regarded as the optimal choice of the MSBC method, in terms of the control effect and the external energy consumed. When \(t\geq 200\) s, the square cylinder activates the MSBC from the fully developed flow when the wind angle is \(15^{\lx@math@degree}\), as shown in Figure 19. With the increase in \(k\) from 1 to 4, the fluctuation value \(C_{l}t\) of the lift coefficient decreases significantly, whereas \(\overline{C}_{l}\) increases from 0.29 to 1.54. The drag coefficient \(\overline{C}_{d}\) decreases to 0.87 at \(k=4\), which corresponds to a 54.2% decrease compared to of its value when \(k=0\). It may be observed that the fluctuation of the lift and drag coefficients decreases with the increase in the input of momentum, whereas the mean of the lift coefficient increases, and the mean of the drag coefficient decreases. The MSBC plays an important role in the decrease in the mean of the drag force and in the increment of the mean lift force. Figure 19: Lift and drag coefficient time histories of the square cylinder with different velocity ratios at \(\theta=15\lx@math@degree\) (inward rotation), **(a)**\(k=1\), **(b)**\(k=2\), **(c)**\(k=3\), **(d)**\(k=4\). Figure 20 shows the comparative analysis of time histories of the lift and drag coefficients for inward rotation and \(\theta=0^{\lx@math@degree}\) when MSBC becomes activated at different times. The simulation is divided into two cases. In the first case, the small circular cylinders are not rotating, and the square cylinder is at the 'FR' stage at \(0\leq t<100\) s; when the alternating vortex shedding was steady; the MSBC became activated at \(t=100\) s with \(k=1.5\).