## 3 Results and Discussion
### Extraction and Analysis of Burst Events
All four types of walls show a clear stratification phenomenon in both flow fields, with the number of ejection events ranked from small to large as follows: 30\({}^{\circ}\) divergent riblets, streamwise riblets, smooth flat plate, and 30\({}^{\circ}\) convergent riblets. This also clearly explains that the inhibitory and promotional effects of riblet walls on burst events are not only targeted at their scales, but also significantly reduce and increase their number. Table 2 summarizes the average values of ejection events under eight conditions, confirming not only the existence of the above phenomena but also finding that the addition of particles reduces the frequency of ejection events under all wall surfaces, rather than breaking down burst events into more small-scale structures. Figure 12: Normal fluctuating velocity cloud image of ejection–sweep events under different wall surfaces at \(y^{+}=30.5\): (**a**) clear water field ejection event; (**b**) liquid–solid two-phase flow field ejection event; (**c**) clear water field sweep event; (**d**) liquid–solid two-phase flow field sweep event. Since the aforementioned texts mention that burst events are an important reason for high Reynolds shear stress, analyzing only the structure and quantity of burst events is not sufficient to demonstrate their contribution to Reynolds shear stress. Figure 14 shows the distribution of Reynolds shear stress caused by burst events at two normal heights for each wall, which can intuitively show the specific impact of the change in yaw angle and the addition of particles on the Reynolds shear stress produced by burst events. It can be seen from the figure that at both normal heights, the drag-reduction riblet walls can effectively reduce the Reynolds shear stress produced by ejection events, while the drag-increasing riblet walls enhance the contribution of ejection events to Reynolds shear stress. By comparing the two normal heights, it can be seen that, unlike the intensity of the burst events themselves, the contribution of ejection events to Reynolds shear stress at a height of \(y^{+}=30.5\) does not show a significant decrease. On the contrary, the core high Reynolds shear stress area has expanded. This indicates that even at the \(y^{+}=90\) normal height, where burst events are weaker, their contribution to Reynolds shear stress is also very high. Therefore, the contribution of burst events to frictional resistance at both normal heights is very important. In the two-phase flow field, due to the addition of the particulate phase, the high Reynolds shear stress area caused by ejection events is fragmented, the area is reduced, and the intensity is decreased, indicating that the particulate phase effectively reduces the Reynolds shear stress produced by ejection events, thereby reducing the contribution to frictional resistance. \begin{table}
\begin{tabular}{c c c c c} \hline
**Flow Field Type** & **Smooth** & **Riblet-0\({}^{\circ}\)** & **DL-30\({}^{\circ}\)** & **CL-30\({}^{\circ}\)** \\ \hline clean water flow field & 15,659 & 13,415 & 10,590 & 17,073 \\ two-phase flow field & 15,043 & 12,769 & 10,383 & 16,126 \\ \hline \end{tabular}
\end{table} 
Table 2: Average number of eject events on different wall surfaces. Figure 13: Distributions of the number of eject events along normal-wall positions: (**a**) clean water flow field; (**b**) liquid–solid two-phase flow field.