## 3 Results and Discussion  
### Influence of Different Propagation Directions  
The control effectiveness of 4 types of TWW propagation should be considered first. The four propagation directions include the \"Downstream\", the \"Upstream\", the \"Corotating-Clockwise\" and the \"Corotating-Counterclockwise\", as shown in Figure 4. During calculation process, the effects of the TWW propagation directions are considered under fixed velocity ratio \(c/U_{\infty}~{}=~{}2.0\), wave number \(N~{}=~{}4\), and maximum wave amplitude \(\tilde{A}/D~{}=~{}0.02\). This numerical simulation begins with the flow around a fixed cylinder (FR). The TWW is activated to control the flow field after the flow field is brought into stability with alternating shedding vortices. The aerodynamic coefficients and vortex shedding patterns before and after activating the TWW could serve as a good measurement for the control effect of TWW control method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\Delta t\) (s) & \(N_{c}\) & \(N_{mesh}\) & \(\overline{C}_{d}\) & \(C^{{}^{\prime}}_{d}\) & \(C^{{}^{\prime}}_{I}\) & \(S_{t}\) \\ \hline \(1.0\times 10^{-3}\) & 250 & 111,600 & 1.1433 & 0.0591 & 0.7928 & 0.226 \\ \(5.0\times 10^{-4}\) & 250 & 111,600 & 1.2378 & 0.0745 & 0.9003 & 0.237 \\ \(2.5\times 10^{-4}\) & 250 & 111,600 & 1.2674 & 0.0812 & 0.9211 & 0.243 \\ \(1.0\times 10^{-4}\) & 250 & 111,600 & 1.2819 & 0.0853 & 0.9351 & 0.243 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparisons of the global parameters for the time step independence study. Figure 3: Comparisons of aerodynamic coefficient and the Strouhal number with previous results. (a) \(\overline{C}_{d}\) versus Re. (b) \(S_{t}\) versus Re. The time histories of lift and drag coefficient for the TWW cylinder with the four different propagation directions are shown in Figure 5. When \(t<1.8\) s, TWW was not activated, the result of FR was obtained. When \(t\geq 1.8\) s, TWW was activated, the suppression effect of TWW on the vortex shedding and wake under the four different propagation directions can be obtained. Figure 5 shows that the lift coefficient fluctuation reduces dramatically only when traveling wave propagates in the downstream direction. It indicates that the TWW propagating downstream has the effect of eliminating Vortex Street. In addition, the mean drag coefficient decreases at this propagation direction, while the fluctuating drag coefficient increases dramatically, as shown in Figure 5a. Figure 5b shows that the fluctuating lift coefficient and mean drag coefficient increase significantly when traveling wave propagates in the upstream direction. It illustrates that the traveling wave propagating upstream enhances the wake behind the cylinder. The drag coefficient fluctuation increases, and the mean lift coefficient varies significantly when traveling wave co-propagates along the cylinder rear surface. When traveling wave Co-rotates clockwise, the mean lift coefficient increases significantly to a positive value, as shown in Figure 5c. When traveling wave Co-rotates counterclockwise, the mean lift coefficient decreases significantly to a negative value, as shown in Figure 5d. It indicates that there is a negative pressure zone on the upper or lower surface with these two propagation directions, and the oscillating wake of traveling wave cylinder cannot be eliminated. Figure 4: Four types of propagation direction of TWW. (**a**) Downstream. (**b**) Upstream. (**c**) Corotating-Clockwise. (**d**) Corotating-Counterclockwise. The mean value reflects the variant characteristics within a period, and the fluctuation value (root-mean-square, RMS) reflects the discrete degree of the variant. Before activating the TWW, the mean lift coefficient \(\overline{C}_{l}\) approximates 0 and \(C_{l}^{\prime}\) is 0.921, the mean drag coefficient \(\overline{C}_{d}\) is 1.267 and \(C_{d}^{\prime}\) is 0.081.