## 3 The Results of the Plates with Riblets and Grooves  
The total frictional resistance to the plate can be obtained by integrating the wall shear stress:  
\[D = \int\limits_{A}\tau\mathrm{d}A\;=\;\overline{\tau}l \tag{19}\]  
where \(A\) is the area of the flat plate. Since it is a two-dimensional flat plate and the span direction is infinite, then \(A=1\), and \(\overline{\tau}\) represents the average shear stress. \(C_{f}\) is the coefficient of frictional resistance and \(C_{p}\) is the coefficient of pressure resistance. \[c_{f} = \frac{f}{\frac{1}{2}\rho V_{\infty}{}^{2}l} \tag{20}\] \[c_{p} = \frac{p}{\frac{1}{2}\rho V_{\infty}{}^{2}l} \tag{21}\]  
\(f\), \(p\), \(\rho\), \(V_{\infty}\), and \(l\) are friction resistance, pressure resistance, density, flow velocity, and length of the plate, respectively. It can be seen from Figure 3 that the friction resistance coefficient of the plates with grooves and riblets are lower than that of the smooth flat plate. The friction resistance coefficients of triangular protrusion and trapezoidal concave groove are the lowest, and the curves of both are almost consistent. The ability of triangular concave grooves to reduce frictional resistance is highest at 0.4 Ma, and the friction resistance coefficient is lower than that of trapezoidal protrusions and is very close to circular grooves. The effect of circular concave grooves on reducing frictional resistance is better than that of the triangular concave grooves and the trapezoidal protrusions. When the speed is higher than 0.7 Ma, the drag reduction effect of circular concave grooves is very close to that of triangular protrusions and trapezoidal grooves. Figure 3: Friction resistance coefficient of different riblets and grooves varies with Mach number. \(\eta\) is the drag reduction rate, which is defined as follows:  
\[\eta\ =\ \frac{\left(C_{f}\right)_{smooth}-\left(C_{f}\right)_{groove}}{\left(C_ {f}\right)_{smooth}} \tag{22}\]  
From the point of view of the frictional resistance coefficient and drag reduction rate of different riblets at different Mach number, as shown in Table 3, in general, the triangular concave grooves have the worst drag reduction effect, only higher at 0.4 Ma, reaching 57.37%, and drag reduction rates at other speeds are below 30%, indicating that the triangular concave grooves have a good drag reduction effect at about 0.4 Ma. The drag reduction rates of trapezoidal concave grooves and triangular protrusion riblets can reach more than 80% when the Mach number is above 0.6 Ma, and below 0.6 Ma, the drag reduction rate can also reach more than 70%. The two types of riblets have the best drag reduction effect. When the Mach number of the circular concave grooves is lower than 0.7 Ma, the drag reduction rate is about 65%. When the speed is higher than 0.7 Ma, the drag reduction rate can reach more than 75%. The drag reduction rate of trapezoidal protrusion riblets is between 45% and 49%, and the drag reduction rate will drop to 35.19% when Mach number is 0.6 Ma. Compared to longitudinal riblets, the experiments by [PERSON] [1] report a drag reduction of 2% and the DNS(Direct Numerical Simulation) by [PERSON] et al. [12] give drag reduction of 5%. The effect of lateral riblets on the reduction of friction resistance is very significant. The lateral grooves and riblets structure can greatly reduce the frictional resistance, but it also causes additional pressure resistance, as shown in Table 4. It can be found that under the same geometry and speed, the pressure resistance coefficient is one order of magnitude higher than the friction resistance coefficient, and pressure resistance accounts for a considerable proportion of total resistance, which is the sum of pressure resistance and friction resistance.