# Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder
## 4 Conclusions  
In this study, the computational fluid dynamics (CFD) numerical simulation was employed for the study of the MSBC effect on the oscillating wake of a square cylinder. We conducted a numerical simulation of the entire process, from the flow around a fixed square cylinder to the oscillating wake controlled via the MSBC, and the conclusions obtained from the study are as follows:  
For the inward rotation mode and wind angles of \(\theta<60^{\lx@math@degree}\), the MSBC method plays an important role in suppressing the oscillating wake of the square cylinder; the effectiveness of the control in suppressing the wake was more obvious when the wind angle was smaller. In the other wind direction, the MSBC method was detrimental to the wake stability of the square cylinder. The fluctuation value of the lift coefficient decreased by \(98\%\), and the mean value of the drag coefficient decreased by \(36\%\) when \(\theta=0^{\lx@math@degree}\) and \(k=4\). In this case, the elimination of the oscillation wake of the square cylinder produced the best effect. In terms of control effectiveness and the external energy consumed, the velocity ratio \(k=2\) can be regarded as the optimal choice of the MSBC control method at \(\text{Re}=200\).  
For the outward rotation mode and when \(\theta=120^{\lx@math@degree}\) and \(150^{\lx@math@degree}\), the effect of suppressing oscillating wake of the square cylinder continuously improved with the increase in \(k\). In  
Figure 28: Vorticity contours in the wake of the square cylinder under co-rotation mode, (**a**) \(\theta=60^{\lx@math@degree}\), (**b**) \(\theta=90^{\lx@math@degree}\).  
Figure 27: Vorticity contours in the wake of the square cylinder under outward rotation mode, (**a**) \(\theta=30^{\lx@math@degree}\), (**b**) \(\theta=120^{\lx@math@degree}\).  
other wind directions, the momentum input that was generated by the rotating cylinders plays an important role in enhancing the wake of the square cylinder.  
For the clockwise co-rotation and when \(\theta=0^{\lx@math@degree}\) and \(180^{\lx@math@degree}\), the momentum input of the rotating circular cylinders was unfavorable to the wake stability of the square cylinder. In other wind directions, the oscillating wake of the square cylinder was suppressed to different levels. Among them, the effect of eliminating the oscillation wake of the square cylinder was most obvious for wind angles of \(60^{\lx@math@degree}\) and \(90^{\lx@math@degree}\).  
The high suction region on the surface of the rotating circular cylinders can prevent streamline bending, delay the separation of the shear layer, and form a narrow wake region, which transforms the unsteady flow in the wake of the square cylinder into an approximately steady flow. This leads to the decrease in the mean of the drag coefficient and the fluctuation of the lift coefficient; thus, suppressing the flow-induced vibration of the square cylinder can be achieved.  
Conceptualization, T.S. and F.X.; methodology, T.S. and F.X.; software, X.L.; validation, T.S. and F.X.; formal analysis, F.X.; investigation, T.S. and X.L.; resources, F.X.; data curation, X.L.; writing--original draft preparation, T.S.; writing--review and editing, F.X.; visualization, X.L.; supervision, F.X.; project administration, F.X.; funding acquisition, F.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Sciences Foundation of China (NSFC) (52078175, 51778199 and U1709207), the Natural Science Foundation of Guangdong Province (2019A15 15012205), the fundamental research funds of Shenzhen Science and Technology plan (JCYJ2019080614 4009332, JCYJ20180306172123896), and the Stability Support Program for colleges and universities in Shenzhen (GXWD20201230155427003-20200823134428001).  
Data Availability Statement: Data are contained within this article.  
The authors declare no conflict of interest.