## 3 Results and Discussion  
### Statistical Characteristics  
Research indicates that the downward or upward shift in the mean velocity profile in the logarithmic region corresponds to an increase or decrease in drag [39]. For the smooth flat plate model, the friction velocity can be obtained using the Clauser method [40], and the logarithmic law expression for the mean velocity of a smooth wall is given by Equation (1):  
\[u^{+}=\tfrac{1}{\kappa}lny^{+}+B \tag{1}\]  
where \(y^{+}=yu_{\tau}/\
u,u^{+}=u/u_{\tau,}u_{\tau}\) is the wall friction velocity, \(\kappa\) is the Karman constant, \(B\) is the logarithmic region velocity intercept. The logarithmic law is fitted using the Newton-Raphson method and the steepest descent method to obtain the average wall friction velocity of the turbulent boundary layer [41]. However, for riblet walls, because their theoretical zero point is below the rib crest plane, it causes a shift in the intercept B of the logarithmic region of the velocity profile [42; 43], rendering the traditional Clauser  
Figure 3: Schematic diagram of the experimental facility: (**a**) \(x\)-\(y\) plane; (**b**) \(x\)-\(z\) plane. method ineffective. Therefore, a modified Clauser method is employed [44], and the logarithmic law region of the riblet wall boundary layer satisfies Equation (2):  
\[u^{+}=\frac{1}{\kappa}lny^{+}+B+F \tag{2}\]  
where \(F\) is the intercept offset caused by the riblets. Taking F as a given discrete point varying from \(-10\) to \(10\) with an interval of \(0.1\), the error Err generated by fitting the wall friction velocity for each F value was calculated [45], defining Equation (3):  
\[Err=\sqrt{\sum_{i=NS}^{NE}(u_{e}^{+}(i)-\big{(}2.5\text{log}\big{(}y_{e}^{+}(i) \big{)}+B+F\big{)}\big{)}^{2}/\big{(}NE-NS+1\big{)}} \tag{3}\]  
where \(y_{e}^{+}\) and \(u_{e}^{+}\) represent the dimensionless normal coordinates and streamwise average velocity of the experimental values, respectively. \(NS\) and \(NE\) are the starting and ending points for the logarithmic region experimental data values, respectively. Based on the minimum experimental deviation, the corresponding \(F\) value and wall friction velocity can be obtained. Figure 4 shows the dimensionless velocity profiles of the wall under different conditions. It can be observed from the figure that, in both flow fields, the velocity profiles of the streamwise riblet and divergent riblet walls tend to shift upwards, with a slight decrease in boundary layer thickness, while the convergent riblet wall exhibits a significant downward shift, resulting in an increased boundary layer thickness. This indicates that the first two types of walls possess certain drag reduction characteristics, with the divergent riblet wall being more effective, while the convergent riblet wall leads to an increase in wall resistance. The addition of particles causes the velocity profiles of all three types of walls to shift upwards, but the increased boundary layer thickness can result in a section of the wake region having lower velocities than in the clear water flow field. However, the overall drag reduction characteristics are still satisfied. The specific flow parameters are shown in Table 1.  
Figure 4: Mean streamwise velocity profile. Table 1 presents the flow parameters for each wall surface in both flow fields, with the inner scale Reynolds number defined as \(Re_{\tau}=u_{\tau}\delta/\
u\), where \(\delta\) is the nominal boundary layer thickness. Combined with the wall friction shear stress \(\tau_{w}=\rho u_{2}\), and the average free stream velocity \(U_{\infty}\), the wall friction coefficient can be obtained as \(C_{f}=2\tau_{w}/\rho U_{\infty}^{2}\), and the drag reduction rate is given by \(DR=(C_{f_{8}}-C_{f\beta})/C_{f_{8}}\times 100\%\), where \(C_{f_{8}}\) and \(C_{f\beta}\) are the friction coefficients for the smooth and riblet walls, respectively.