# Aerodynamic Characteristics of Supersonic Rocket-Sled Involving Waverider Geometry
## 3 Results and Discussion  
### Validation and Computational Reviews  
To verify the correctness of the chosen calculation method, we repeated the supersonic test results of [50] for an axisymmetric slender body by the above numerical method. The governing Equations (1)-(3) are integrated numerically based on the finite volume method solver. The second-order upwind difference scheme is used to discretize the convection term in the model. The Roe-FDS flux scheme is used to reduce the dissipation in the computation, which further improves the accuracy of supersonic simulation. According to experiments performed concerning supersonic axisymmetric flow [50], the free-stream Mach number \(M_{\infty}\) and Reynolds number \(Re\) based on the cylinder diameter (\(d\) = 37 feet) are chosen as 2.5 and 4.2 \(\times\) 10\({}^{6}\), respectively. We output the pressure coefficient at sections 3.5 \(d\) and 5.5 \(d\) from the nose. The pressure coefficient \(C_{P}\) can be defined as:  
\[C_{P}=\frac{p-p_{\infty}}{0.5p_{\infty}U_{\infty}^{2}}. \tag{16}\]  
Figure 2 shows the calculation results and experimental results [50], in the form of plots of \(C_{P}\) versus \(\theta\) (the angle between the radial direction and vertical direction), for the isolated body at two longitudinal locations. The pressure coefficients of the two sections are close to the experimental values. These results show that the chosen numerical method can be used to simulate the supersonic flow of a similar slender body.  
The axisymmetric slender shape is the most common configuration in existing rocket-sled test systems. Figure 3a shows the shape parameters of the axisymmetric slender rocket-sled (ASRS). For comparison, Figure 3b shows the reversed waverider rocket-sled (RWRRS) integrated configuration based on waverider theory [51]. The design of the waverider is carried out using the following methods: numerical simulation of conical flow field, upper surface design, obtaining leading edge parameters, streamline tracing and lower surface design. After obtaining the shape parameters of the waverider, an integrated design of the RWRRS is obtained by combining the lower surface of the body with a smooth transition connection. To make sure the contrast is reasonable, both structures have the same radius \(R\) in the lower half. To ensure the stability of the rocket-sled movement, it is necessary to set the computational domain to be sufficiently long. Therefore, we set the length of the computational domain to above 24 m, as shown in Figure 4. The rocket-sled is evaluated using a ground test system, so the operating environment can be determined according to commonly used atmospheric environmental parameters. The local Mach number of the rocket-sled \(M_{r}\) is chosen as 2.0, and the ambient pressure \(p_{\infty}\) is 101.325 kPa.  
Figure 2: Circumferential \(C_{P}\) distribution on axisymmetric slender body: (**a**) \(x/d\) = 3.5, (**b**) \(x/d\) = 5.5.  
With respect to computing parameters, a no-slip wall condition is used in the ground and rocket-sleds. The far-field condition is adopted at the rest of the boundaries to avoid the influence of the boundary on the calculation results. The no-slip wall condition is described as Equation (17) and the far-field condition is described as Equation (18). As shown in Figure 5, we use a structured grid to discretize the computational domains. The nearest grid height normal to the wall is equal to 2 \(\times\) 10\({}^{-5}\) m and the total quantity of grid elements is about 8 million. The computational time step is 3 \(\times\) 10\({}^{-6}\) s, which is small enough to obtain the unsteady phenomena of the rocket-sled flow field.  
\[u=v=w=0. \tag{17}\]  
\[\rho=\rho_{\infty},p=p_{\infty},u=0,v=0,w=0. \tag{18}\]