# 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 \(h/D=1/4\)
#### 3.2.4 Wake Vortex Structure  
The momentum injection of the MSBC method affects the vortex characteristics of the flow field, and the different vortex structures and shedding modes are directly related to the aerodynamic forces of the square cylinder. In order to demonstrate the control effect of the MSBC method on the flow field around the square cylinder, in this study, we present the vortex wake of the square cylinder with representative wind angles for different rotation modes. Figure 25 shows the vorticity contours of the flow field around the square cylinder at different \(k\) values for inward rotation and wind angles of \(0^{\circ}\) and \(15^{\circ}\). In Figure 25a, when \(k=0\), the shear layers separate from the leading edge corner of the square cylinder and enter into the wake region; then, they immediately curl and develop into an alternating shedding vortex street, which causes the fluctuation of the lift and drag coefficients. When \(k=1\), the rotating cylinder begins to transfer momentum into the flow field; however, the  
Figure 24: Lift and drag coefficient time histories of the square cylinder with different velocity ratios at \(\theta=90^{\circ}\) (co-rotation), (**a**) \(k=2\), (**b**) \(k=4\). Figure 23: Lift and drag coefficient time histories of the square cylinder with different velocity ratios at \(\theta=60^{\circ}\) (co-rotation), (**a**) \(k=2\), (**b**) \(k=4\). wake vortex street still exists. Two quasi-stable small vortex structures are formed near the two small circular cylinders at the leeward side of the square cylinder, and they lead to the shear layer separating from the leading edge corner, which rolls up to a slightly farther downstream location. This reduces the lift coefficient by 80%. As the \(k\) value increases, the separated shear layer rolls up farther downstream, and the vorticity intensity and scale slightly decrease. Two small symmetric stable vortices on the leeward side of the square cylinder have gradually stretched, thus preventing the interaction between the upper and lower shear layers. When \(k=4\), the large-scale alternate shedding vortex street in the square cylinder wake has been completely eliminated by the small-scale vortices induced by the two rotating cylinders, and the shear layer has not rolled up in the computational domain again. Obviously, for this rotation direction and wind angle, the momentum injected into the flow field by the small cylinders delays the separation of the shear layer and narrows the wake region. It finally eliminates the wake vortex street and forms a symmetrical stable vortex structure in the wake of a steady flow. Figure 25b shows that the vortex near the square cylinder wake is stable and develops into the alternate shedding vortex slightly farther downstream with the increase in \(k\) when \(\theta=15^{\circ}\). The fluctuation of the lift coefficient is significantly reduced, as well; however, the vortex street has not been completely eliminated at \(k=4\). The vortex street presents an inclined downward direction, which is the main cause of the increase in the mean lift coefficient. This is consistent with the results shown in Figure 8a. For the case of inward rotation mode and \(\theta=0^{\circ}\), the velocity contours and streamlines in the near wake of the square cylinder with different velocity ratios are shown in Figure 26, which can clearly show the close-up view around the region of rotating cylinders. In order to compare the results of different cases more clearly, the upper limit value of the velocity in the legend of each subgraph is consistent, which is the maximum speed in the flow field when \(k=0\). In Figure 26a, a large-scale vortex is generated in the upper region of the near wake immediately adjacent to the leeward side of the square cylinder, and the lift coefficient of  
Figure 25: Vorticity contours in the wake of the square cylinder under inward rotation mode, \(\left(\mathbf{a}\right)\theta=0^{\circ}\), \(\left(\mathbf{b}\right)\theta=15^{\circ}\).