## 3 Results and Discussion
### Extraction and Analysis of Streak Structures
Therefore, to better describe the performance of near-wall streak structures in the flow field, a correlation function is used to calculate their spanwise length scale. The spatial two-point correlation coefficient of streamwise fluctuating velocity has been widely proven in previous studies to indicate the existence of streak structures and to determine the size of the streak spacing [51; 52; 53], and its expression is:  
\[R_{u^{\prime}u^{\prime}}(x_{0},y_{0},z_{0},\Delta x,\Delta z)=\frac{\frac{u^{ \prime}_{A}(x_{0},y_{0},z_{0})u^{\prime}_{B}(x_{0}+\Delta x,y_{0},z_{0}+ \Delta z)}{\sigma_{A}\sigma_{B}}}{\sigma_{A}\sigma_{B}} \tag{5}\]  
Here, \(\sigma_{A}\) and \(\sigma_{B}\) represent the standard deviations of \(u^{\prime}(t)\) at spatial points A and B, respectively, \(\Delta x\) is the streamwise separation between the two points, and \(\Delta z\) is the spanwise separation between the two points. To reduce the interference of small-scale structures in the flow field, low-order modes containing 50% of the energy were selected to reconstruct the flow field, followed by the calculation of the correlation function of the flow field. Figure 9 shows the two-point correlation coefficient graph of the streamwise fluctuating velocity; in this graph, the characteristic spanwise scale \(\overline{\lambda^{+}}\) can be defined as the gap between two negative valleys, that is, the distance between the two lines in the figure. It can be seen from the graph that the two-point correlation coefficient can fully display the structural morphology of the  
Figure 8: The first five orders POD mode cloud image of different wall streamwise fluctuation velocity at \(y^{+}=8.8\) in two-phase flow field: (**a**) smooth flat wall; (**b**) streamwise riblet wall; (**c**) 30\({}^{\circ}\) divergent riblet wall; (**d**) 30\({}^{\circ}\) convergent riblet wall. flow field, which is distributed in a strip-like pattern and stretched along the streamwise direction. By comparing the results longitudinally, it can be observed that the two-point correlation coefficients for different walls in the two flow fields all show the same trend of change. From the smooth flat to the 30\({}^{\circ}\) divergent riblet walls, the distribution of the correlation coefficient \(R_{u^{\prime}u^{\prime}}\) continuously expands in the spanwise direction and is also elongated in the streamwise direction, representing an increase in the range of low-speed fluids. This trend is consistent with the previously mentioned trend in the streak spacing \(\overline{\lambda^{+}}\) increasing with the wall-normal height \(y^{+}\), indicating that the motion of the streaks in the spanwise direction is suppressed. In contrast, the distribution graph of the 30\({}^{\circ}\) convergent riblet shows a completely opposite change, indicating that it promotes the motion of the streaks and enhances the spanwise mixing of turbulent structures in the near-wall region. By comparing the left and right sides of Figure 9, it can be seen that the addition of particles can effectively suppress the motion of the streak structures and enhance the stability of the flow field. The degree of mixing of high- and low-speed fluids is reduced, which means fewer vortex structures appear and the transfer of momentum and kinetic energy is decreased. To more intuitively compare the spanwise spacing of streaks under each type of wall, further processing of the correlation function of streamwise fluctuating velocity in space is carried out. By performing an autocorrelation operation on \(u^{\prime}\) only in the spanwise direction, the distribution of \(R_{u^{\prime}u^{\prime}}\) along \(\Delta z^{+}\) can be obtained, and its expression is:  
\[R_{u^{\prime}u^{\prime}}(\Delta z)=\frac{\left\langle u^{\prime}(z)u^{\prime}( z+\Delta z)\right\rangle}{\left\langle u^{\prime}(z)^{2}\right\rangle} \tag{6}\]  
Figure 9: Two-point correlation coefficient of streamwise fluctuation velocity in space \(R_{u^{\prime}u^{\prime}}\) (\(x_{0}\), \(z_{0}\)): (**a**) clean water flow field; (**b**) two-phase flow field.