## 4 Conclusions  
In the present research, the CFD numerical simulation was employed to investigate the oscillating wake of flow around a fixed cylinder with the traveling wave wall at a high Reynolds number. The influence factors of TWW flow control method were investigated, i.e., the TWW propagation direction, wave amplitude, wave number and wave velocity. The main conclusions are as follows.  
The TWW with \"Downstream\" propagation direction can successfully eliminate the alternating shedding vortex behind the cylinder under the condition of \(c\,/\,U_{\infty}\ =\ 2.0\), and \(C_{l}^{\prime}\) of TWW cylinder decreases by \(99.48\%\) from that of standard circular cylinder. The TWW cylinder in \"Upstream\" propagation direction enhances the cylinder wake. Neither \"Corotating-Clockwise\" nor \"Corotating-Counterclockwise\" could eliminate the cylinder oscillating wake.  
Wave amplitude should be taken as \(0.02D\) to obtain the optimal control effect. At the wave amplitude ratio \(\hat{A}\,/\,D\) of \(0.02\), \(C_{l}^{\prime}\) decreases dramatically to \(0.005\). When \(\hat{A}\,/\,D\) increases from \(0.02\) to \(0.05\), \(C_{l}^{\prime}\) increases slightly, while \(C_{l}^{\prime}\) dropped by \(98.7\%\) from the result of standard circular cylinder at \(\hat{A}\,/\,D\ =\ 0.05\). With the increase in \(\hat{A}\,/\,D\), \(\overline{C}_{d}\) decreases from \(-0.010\) to \(-0.456\), and \(C_{d}^{\prime}\) increases to almost \(4.7\) times the result of standard circular cylinder.  
Wave number \(N\) has relatively slighter effect on lift and drag coefficients. The \(\overline{C}_{l}\), \(C_{l}^{\prime}\) and \(C_{d}^{\prime}\) show little changes with the increase of wave number \(N\). \(\overline{C}_{d}\) is less than \(0\), and the  
Figure 15: Characteristic values of lift and drag coefficient under different wave velocities. (**a**) \(C_{l}^{\prime}\) & \(\overline{C}_{l}\). (**b**) \(C_{d}^{\prime}\) & \(\overline{C}_{d}\).  
absolute value of \(\overline{C}_{d}\) increases as wave number \(N\) increases, which indicates that the reverse thrust applied to the cylinder increases gradually with the increase in wave number \(N\).  
The fluctuating lift coefficient \(C_{I}^{\prime}\) reaches the minimum at \(c/U_{\infty}~{}=~{}1.5\), which drops by 99.63% from the result of standard cylinder. When \(c/U_{\infty}=2.0~{}\sim 5.0\), \(C_{I}^{\prime}\) takes a slow increasing tendency and increases to 3.91% of the result of standard circular cylinder. With the increase of \(c/U_{\infty}\), \(\overline{C}_{d}\) decreases gradually and finally approaches to zero at \(c/U_{\infty}~{}=~{}1.5\). However, \(C_{d}^{\prime}\) increases gradually with increasing \(c/U_{\infty}\) and finally reaches to about 12.8 times the result of standard circular cylinder at \(c/U_{\infty}~{}=~{}5.0\). Considering the control effect and energy input under the Reynolds number in this study, \(c/U_{\infty}~{}=~{}1.5\) is the optimal wave velocity for TWW flow control in this paper.  
Conceptualization, FX.; investigation, FX. and X.L.; writing--original draft preparation, FX.; writing--review and editing, X.L. and W.B.; visualization, W.B. 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 (2019A1515012205), the fundamental research funds of Shenzhen Science and Technology plan (JCYJ20190806144009332, JCYJ20180306172123896), and the Stability Support Program for colleges and universities in Shenzhen (GXWD20201230155427003-20200823134428001). Not applicable. Not applicable. Not applicable. Not applicable. Data Availability Statement: Data are contained within this article. The authors declare no conflict of interest.