## 3 Results and Discussion
### Influence of Different Propagation Directions
The time history curves of lift and drag coefficients for uncontrolled cylinder are taken for reference in the following parts to measure the influence of different control parameters on the control effect. Figure 6 shows the lift and drag coefficients characteristics with different traveling wave propagation directions, while the dashed curves inside the same figure demonstrate the results of the standard circular cylinder. As shown in Figure 6a, \(\overline{C}_{l}\) approaches to 0 for \"Downstream\" and \"Upstream\" propagation direction, while the \(\overline{C}_{l}\) increases dramatically to 3.917 and \(-\)4.057 for co-propagations. For the \"Downstream\" propagation, the \(C_{l}^{\prime}\) approaches to 0, which indicates that the cylinder oscillating wake is eliminated by TWW. For the \"Upstream\" propagation, \(C_{l}^{\prime}\) increases dramatically to 1.825. In addition, \(C_{l}^{\prime}\) for the co-propagations is close to that of the standard circular cylinder. These illustrate that the other three propagations are not able to suppress the cylinder wake vortex except \"Downstream\". Figure 6b shows that the fluctuating drag coefficients \(C_{d}^{\prime}\) vary between 0.172 and 0.763 in various propagation directions, and all of them are larger than that of the standard circular cylinder, which means that TWW leads to a larger amplitude fluctuating drag. For \"Downstream\" propagation, \(\overline{C}_{d}\) decreases dramatically to \(-\)0.152, meaning the cylinder is subjected to reverse thrust due to TWW. For \"Upstream\" propagation, \(\overline{C}_{d}\) increases dramatically to 2.809, i.e., the inline flow drag increases dramatically. The \(\overline{C}_{d}\) of other two propagation directions equal to 0.905 and 0.902, respectively, which are slightly lower than the standard circular cylinder. Compared with the standard circular cylinder results, \(C_{l}^{\prime}\) and  
Figure 5: Lift and drag coefficient time history curves under various propagation directions. (**a**) Downstream. (**b**) Upstream. (**c**) Corotating-Clockwise. (**d**) Corotating-Counterclockwise. \(\overline{C}_{d}\) of \"Downstream\" propagation decreases by 99.48% and 112%, respectively. It illustrates the cylinder wake oscillation in FR stage can be eliminated by the TWW propagating downstream. The vorticity contours surrounding TWW cylinder with \"Downstream\" propagation are shown in Figure 7. In the interval \(t<1.8\) s, the traveling wave was not activated, and the vortex sheds from both side of the cylinder alternately, as shown in Figure 7a. Figure 7b shows that the TWW activated shortly has little effects, and the separated free shear layer still rolls into vortices. Form Figure 7c-g, as time advanced, the control effect of TWW became more obvious, and the vortex street appeared in uncontrolled cylinder was eliminated. The shear layer detaches from the cylinder surface, and then enters the wake region without rolling into vortices due to the \"fluid roller bearing\" (FRB) effect of TWW, and the vorticity is distributed steadily and symmetrically in the cylinder wake region, which means that the TWW cut off the energy source for generating vortex. When \(t=2.97\) s, the TWW eliminates oscillating wakes of the cylinder successfully. In addition to the \"Downstream\" propagation, the rest three kinds of propagations are less effective in eliminating the cylinder oscillating wake, which can be seen from the vorticity contours obtained after activating TWW, as shown in Figures 8 and 9. In Figure 8, the vortices in the wake formed by FR are replaced by stronger vortices, so that the fluctuating lift coefficient and the mean drag coefficient increase dramatically. In Figure 9, an overall downward vortex street is formed in the cylinder wake region because of the \"Corotating-Clockwise\" traveling wave.