## 2 Basic Equations and Parameter Settings
### Method and Calculation Model  
For the numerical simulation of the plate with riblets or grooves, as shown in Figures 1 and 2, five models of triangular concave grooves, trapezoidal concave grooves, circular concave grooves, and triangular protrusions and trapezoidal protrusions are set. The calculation domain can be divided into front and back smooth areas, and middle riblets area. The smooth area is 1 mm wide in flow direction and the riblets and grooves area is 8 mm wide. Triangular concave grooves and triangular protrusions consist of a regular triangle with a side length of 0.1 mm, trapezoidal concave grooves and trapezoidal protrusions consist of a trapezoid with a height of 0.2 mm, a base length of 0.4 mm and a base angle of 60\({}^{\circ}\), and the circular concave grooves consist of a semicircle with a diameter of 0.1 mm. For the grid generation of all riblets plates, a structural grid is used [22]. For this flow problem, the shape is relatively simple, and a structural grid can improve the efficiency and accuracy of the calculation. Considering the accuracy and efficiency of the calculation, the mesh in the smooth area of the plate is sparse, and the mesh in the riblets area needs to be dense. The influence of the number of grids on the calculation results must be considered. In this paper, the circular concave plate is tested to verify grid independence. Calculate the pressure of the first arc  
Figure 1: Calculation model. Figure 2: Different riblets and grooves plate model (not to scale). (**a**) Triangular concave grooves, (**b**) triangular protrusion riblets, (**c**) trapezoidal concave grooves, (**d**) trapezoidal protrusion riblets, and (**e**) circular concave grooves. starting position at the number of grid elements of 410,000, 660,000, 920,000, 1.19 million, and 1.43 million grid elements at a flow velocity of 0.5 Ma, as shown in Table 1.  
As can be seen from Table 1, in the flat plate with arc grooves, the pressure has no significant change when the number of grids is above 1.19 million. Therefore, in this paper, the number of grids set in the flat plate with triangular concave grooves, triangular protrusions, trapezoidal concave grooves, triangular protrusions, and arc concave grooves is more than 1.4 million, to ensure that the calculation results of flat plates are not affected by the number of grids. Y+ is the dimensionless distance from the first grid to the wall, it should be higher than the viscous sublayer. For the k-\(\varepsilon\) model, the value of Y+ is generally 30-100 and relatively good results can be obtained when Y+ = 30 is chosen [23]. Then, the distance of the first grid is calculated as 0.009827 mm. The cell growth rate is 1.5. In this paper, the computational fluid dynamics software Fluent 15.0 is used for numerical calculation, which is based on the finite volume method. In standard atmospheric conditions, the incoming flow is perpendicular to the direction in which the riblets extend. The incoming velocity ranges from 0.3 Ma to 0.8 Ma with an interval of 0.1. The specific parameters are shown in Table 2, where Ma, Re, T, and L are Mach number, Reynolds number, temperature and characteristic length, respectively. The solution algorithm uses a wide range of SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithms. In the flow simulation problem of riblets, it is necessary to choose a standardized \(k\)-\(\xi\) model to simulate the boundary layer of flat plate with riblets. Only given the definitive boundary conditions and initial conditions, the solution of the flow field exists and is unique. Considering the reflection at the boundary of the calculation domain, pressure far-field conditions are used at the inlet and the upper boundary. The pressure far-field conditions are applicable to the free-flow boundary conditions for compressible flows, and it needs to give the incoming pressure and incoming Mach number. The outlet boundary is a pressure outlet boundary, and the pressure of the incoming flow is used.