## 3 Results and Discussion
### Extraction and Analysis of Streak Structures
The presence of particles intensifies the fragmentation of large-scale structures, leading to a decrease in the energy content ratio of the main structure in the entire flow field, which in turn reduces the difference in the riblet walls' influence. Figure 7 shows the cloud images of low-order modes in the clean water flow field. The lower the order, the higher the energy content ratio. In the first-order mode diagrams of the four types of walls, it is clear that there are high and low speed streak structures that are alternately distributed along the spanwise direction near the wall. Their streamwise length scales basically cover the entire field of view, indicating that the high and low speed streaks are the dominant structures at this normal height. As the order increases, the scale of the streak structures gradually decreases, and more small-scale structures appear. Among them, in the fifth-order modes of the flow fields of the streamwise riblet and 30\({}^{\circ}\) divergent riblet walls, the structural scale is significantly reduced, and their streamwise scale no longer covers the entire field of view and is more narrow in the spanwise direction, especially for the 30\({}^{\circ}\) divergent riblet wall, which shows a trend of reduction from the third-order mode. In contrast, in the fifth-order modes of the smooth wall and 30\({}^{\circ}\) convergent riblet wall, the streamwise scale has not yet decreased significantly. Compared with the others, the streak structures in the 30\({}^{\circ}\) convergent riblet wall are clearer and appear more in the form of high-speed streaks. The above situation indicates that the drag-reduction riblet  
Figure 6: Energy distribution of POD modes: (**a**) clean water flow field; (**b**) two-phase flow field. walls can lead to a certain degree of reduction in the spatial scale of the streak structures in the streamwise and normal directions. Figure 8 presents the low-order modal cloud images of streamwise fluctuating velocity for the four types of walls under the liquid-solid two-phase flow field. Compared to the clean water flow field, after the addition of particles, the spacing of the streak structures near the wall region for all four types of walls has increased significantly. Under the same dimensionless spanwise spacing, the number of high and low speed streaks has decreased. The turbulent structure scale of the streamwise riblets and 30\({}^{\circ}\) divergent riblets begins to decrease noticeably in the third-order mode, and the structure no longer exhibits a clear streak-like pattern in the fifth-order mode. The streamwise scale of the fifth-order mode structure for the smooth wall has also decreased, indicating that the energy content ratio of the streak structures has decreased, and the energy content ratio of small-scale structures has increased. From the conclusions drawn earlier, it can be seen that this is due to the particles breaking up the large-scale structures into smaller-scale structures. Due to the integrity of the self-sustaining mechanism, the streak structures are more broken by particles, which will inevitably affect their transformation into streamwise vortices. Whether the transformation process from streamwise vortices to streak structures is affected still requires further research. Figure 7: POD mode cloud image of different wall streamwise fluctuation velocity at \(y^{+}\) = 8.8 in clear water flow field for the first five orders: (**a**) smooth flat wall; (**b**) streamwise riblet wall; (**c**) 30\({}^{\circ}\) divergent riblet wall; (**d**) 30\({}^{\circ}\) convergent riblet wall. The motion of high- and low-speed streaks in the spanwise direction accelerates the mixing of high- and low-speed fluids, making the flow in the near-wall region more chaotic. The spanwise spacing between streaks can reflect the spanwise motion of the streaks well; the narrower the spanwise spacing, the more intense the spanwise mixing of high- and low-speed fluids, and the stronger the momentum transfer.