## 3 Results and Discussion
### Analysis of the CFD Calculation Results
When \(\lambda/H=7\), at \(x>20H\), the streamlines began to bend due to the counter-pressure gradient effect, and the flow became unsmooth. When \(\lambda/H=9\), at \(x>20H\), the streamlines were twisted, which indicated that the VGs were not able to adequately suppress the flow separation. In Figure 11b, it can be observed that with the increase of the VG spacing, the distance between the two vortices gradually increased. Downstream of \(x/H=15\), when \(\lambda/H=7\) and \(\lambda/H=9\), the shape of the vortices was irregular with no obvious vortex center, while the area surrounding the vortices increased. When \(\lambda/H=3\) and \(\lambda/H=5\), the two vortices were closer to each other, and the shape of the concentrated vortices was still intact. Since the concentrated  
Figure 10: \(Q\) isosurfaces (\(Q=1\times 10^{5}\)) colored according to velocity with VGs: (**a**) computational domain and (**b**) inlet section. vortices rotated upward at the inner side of the VGs and were subjected to upward forces, the highest vortex core height from the wall was observed at \(\lambda/H=3\). When the distance between the vortices was small enough, the upward rotation force made the vortices lift off the wall. This implies that in order to restrain the flow separation using VGs, the vortices need to be closer to the wall to give a stronger energy exchange effect in the near-wall area, and the more beneficial the flow control. When \(\lambda/H=3\), the spacing was somewhat small, which was not conducive to flow control. Figure 12 shows the relationship between the vortex core radius and vortex core spacing with flow distance, where \(r\) represents the vortex radius and \(\Delta z\) represents the distance between the two vortex cores. With the increase of the VG spacing, the \(\Delta z\) increased gradually. The vortex core radius represents the radial range of action. Theoretically, when VGs are used to suppress flow separation, the optimal distance is when the vortex radius \(r\) is equal to the distance between the two vortex cores \(\Delta z/2\) (half the distance). However, the radius of the vortices increased along the flow direction, and the vortices slightly deviated toward the flow direction. Therefore, the radius of the vortices and spacing between vortices cannot be equal everywhere along the flow direction. When the distance between vortices was \(\Delta z/2<r\), VGs hindered the development of vortices. When \(\Delta z/2>r\), a region where the vortices could not act was developed, which led to the decrease of the added value of fluid kinetic  
Figure 11: (**a**) Vorticity streamlines and contours in the flow domain and (**b**) vorticity contours at different cross-sections. energy in the boundary layer. By comparing the four different VG spacing results, it can be seen that, except at \(x/H=1\), the distance between the vortices \(\Delta z\) was smaller than the vortex radius \(r\) at the other locations when \(\lambda/H=3\). This shows that this VG spacing hindered the development of vortices at most positions along the flow direction, and that the VG spacing was too small to facilitate flow control. When \(\lambda/H=7\) and \(9\), \(\Delta z>r\) for each direction, which reduced the effective range of the vortices and was not conducive to flow control. When \(\lambda/H=5\), it was found that \(\Delta z>r\) before \(x/H=10\), while after \(x/H=10\), \(\Delta z<r\). Consequently, a VG spacing of \(\lambda/H=5\) was more suitable for flow control. Second-order spatial correlation is a statistical method commonly used to study turbulence. Two-point spatial correlation can reflect the spatial relationship of one or more pulsations in the turbulent flow field. This method can be used to analyze the correlation of the structural characteristics of the downstream VG vortices.