# Aerodynamic Optimization and Characterization of a Ducted Tail for a Box-Launched Aircraft
## 3 Parametric Modeling and Numerical Simulation Methods
### Numerical Simulation and Mesh-Independent Analysis
#### 3.2.2 Mesh-Independent Analysis  
In order to verify that the meshing and the numerical simulation methods in this paper are independent of each other, 2 models were chosen for validation, the clean NACA0012 wing model and the ducted tail model. Figure 2a shows the mesh model of a NACA0012 wing, which adopts Poly-Hexcore unstructured hybrid mesh, and the thickness of the first layer of mesh on the surface of the wall is \(5\times 10^{-6}\) m, which satisfies the requirements of viscous computation. The local mesh encryption is carried out on the leading and trailing edges of the NACA0012 wing model; the computational domain is square, and the dimensions of the computational domain are 10 times of the length of the feature. The computational region boundary condition is pressure far field, and the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Parameters** & **Value Range (mm)** \\ \hline
1 & H1 & 50–140 \\
2 & H2 & 50–140 \\
3 & L & 1200 \\
4 & L1 & 350 \\
5 & L2 & 250 \\
6 & L3 & 400–600 \\
7 & L4 & 0–363 \\
8 & \(\theta\) & 0\({}^{\circ}\)–20\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of relative position parameters. wall is bounded by no-slip mesh. Figure 2b shows the mesh model of the ducted tail, using Poly-Hexcore unstructured hybrid mesh. The thickness of the first layer of mesh on the surface of the tail is \(3\times 10^{-6}\) m, which meets viscous calculation requirements. The ducted tail model locally encloses the mesh in the coupling region between the rear of the fuselage and the tail, the computational domain is square, and the dimensions of the computational domain are 10 times the length of the features. The computational region boundary condition is pressure far field, the wall is bounded by no-slip mesh, and in order to simulate the effect of the engine jet on the ducted tail, the fuselage tail is set as the velocity inlet boundary condition. Due to the strong coupling relationship between the ducted tail and the fuselage, the efficiency and accuracy of numerical computation need to be considered comprehensively when performing the meshing. Therefore, the NACA0012 model selected for the tail, and the model before optimization of the ducted tail and fuselage assembly, are used to establish computational mesh models with different sparsity degrees, and to carry out the validation of mesh-independence and the assessment of numerical simulation accuracy. The basic information of the mesh is shown in Table 2, and the main differences are reflected in the first boundary layer meshes height, refined area meshes scale, and surface meshes scale settings. As shown in Figure 3a, the NACA0012 airfoil is firstly selected for CFD calculation. The lift coefficient (CL) is 1.0034 when the meshes number is 210,000, the altitude is H = 0 m, the reference chord length is \(c_{ref}=0.1\) m, the Reynolds number is \(\text{Re}=10^{6}\), and the angle of attack is \(10^{\circ}\), and tends to be stable with the increase of the meshes number. In order to further verify the accuracy of the CFD calculation, as in Figure 3b, the CL value is simulated and calculated when the attack angle is \(-4^{\circ}\)-\(16^{\circ}\), and compared with the wind tunnel test data [37]. In Figure 3b, the CL numerical simulation results match well with the wind tunnel data, the overall trend of the two groups of data is consistent, and the maximum error between numerical simulation and wind tunnel test data is \(\leq\)15%, so the numerical simulation accuracy meets the calculation requirements. On the premise that the CFD calculation method of the NACA0012 airfoil meets the mesh-independence and calculation accuracy, the independence of the mesh of the ducted tail and fuselage combination base configuration is further verified. Typical working conditions are selected for checking, specifically altitude H = 0 m, incoming flow velocity 0.35 Ma, reference area S = 0.01 m\({}^{2}\), reference chord length \(c_{ref}=0.1\) m, reference spread length 0.1 m, and angle of attack \(\alpha=0^{\circ}\).