# Optimizing Airfoil Aerodynamic Characteristics by Using Proposed CSA-KJ Method
## 3 Example Analysis and Simulation
### CFD Simulations  
The density of the grid significantly affects the precision of the calculation. In order to obtain accurate results, the absolute relative error of the lift force of the CSA-KJ4412 airfoil is taken as the standard for selecting the optimal grid number. In this study, grid independence verification is also carried out.  
Figure 1: Flow chart of CSA-KJ airfoil optimization method.  
Figure 2: Airfoil shape before and after treatment.  
The accuracy of the CFD model is verified by calculating the lift and drag of the airfoil. The calculation formulas of the lift and drag are as follows  
\[F_{L}=1/2C_{l}\rho U^{2}c, \tag{9}\]  
\[F_{D}=1/2C_{d}\rho U^{2}c, \tag{10}\]  
where \(C_{l}\) is the lift coefficient, \(C_{d}\) is the drag coefficient, and \(\rho\) is the fluid density.  
In grid independence verification and subsequent simulation, determine SA as the turbulence model and set the boundary condition of the semicircular grid as the velocity inlet, the upper and lower boundaries of the rectangle as the wall, and the right boundary condition as free outflow. Select the air as fluid, and set its temperature as 298.15 K, its viscosity as \(1.79\times 10^{-5}\) kg/(m\(\cdot\)s), and its density as \(1.225\) kg/m\({}^{3}\). The iterative residual threshold is given as \(1\times 10^{-3}\), and the maximum number of iterations is \(1\times 10^{3}\). The incompressible URANS equation is solved by the computational fluid dynamics software Fluent 19.2. The SIMPLEC algorithm is applied to the coupling solution of pressure and velocity. The second-order upwind scheme is set as for the convection term, the second-order central difference scheme is selected as the diffusion term, and the central difference scheme is applied to the pressure term and viscous flux. The verification results of lift force under different grid numbers are listed in Table 2.  
It can be observed from Table 2 that the lift force difference between Grid 3 and Grid 4 is less than 1% when the number of grids is increased by about 30-50%. Due to the smaller grid number, Grid 4 is used to calculate the airfoil aerodynamic characteristics.  
The airfoil chord length \(c\) and the radius \(R\) of the semicircular flow field are separately set to 1 m and 12.5\(c\), the grid area of the flow field on the right is \(2R\times 14\)\(c\), and the number of grids on the airfoil surface is selected as 140. The grid height of the first layer is \(1\times 10^{-5}\), \(y^{+}\) value is approximately equal to 1. The flow field on the right is divided into upper and lower parts and symmetrical to the \(x\)-axis, the upper and lower ends are both divided into 24 segments, and the height is separated into 48 segments, so the total number of grids in the calculation domain is 9024. The generated mesh is presented in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Cells** & **Grid Growth/\%** &
\begin{tabular}{c} **Lift Force** \\ **F\({}_{\text{L}}\)/N** \\ \end{tabular} & **Absolute Relative Error\%** \\ \hline Grid 1 & 3072 & - & 6578.3 & - \\ Grid 2 & 4176 & 35.94 & 6253.6 & 4.94 \\ Grid 3 & 6384 & 52.87 & 6327.1 & 1.18 \\ Grid 4 & 9024 & 41.35 & 6367.9 & 0.64 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Absolute relative error of lift force.  
Figure 3: Generated mesh of CSA-KJ4412 airfoil.