## 2 CFD Simulations
### Grid Strategy  
#### 2.1.1 NACA0012 Models and External Flow Field of the Computational Domain  
The following are three general ways to establish the NACA0012 airfoil model:  
* Airfoil tools, which refers to the UIUC Airfoil coordinates database and provides 132 data points. It should be noted that the ends of the established NACA0012 trailing edge based on Airfoil tools are not closed, and we need to connect the ends manually to form the blunt trailing edge. If the ends of the trailing edge are not connected, after being imported into meshing tools, a blunt elliptic shape will be formed at the trailing edge of the generated two-dimensional airfoil, failing mesh generation
* NACA 4-digit airfoil generator, which provides a maximum of 200 data points. It gives a close trailing edge option to form the sharp trailing edge.
* The definition formula, which is Equation (1), where \(x\) is the \(x\)-axis location and \(y\) is the \(y\)-axis location. \[y=\pm 0.594689181\times[0.298222773\times x^{1/2}-0.127125232\times x-0.3579079 06\times x^{2}+\\ 0.291984971\times x^{3}-0.105174606\times x^{4}]\] (1)  
Considering there are two sources of data points, to further analyze the influences of different data points source and data points' numbers on the simulation accuracy, six NACA0012 models are built, which include one blunt trailing edge shape model and five sharp trailing edge shape models. For the blunt trailing edge shape model, we apply the Airfoil tools to establish. For the five sharp trailing edge models:  
* The NACA4-digital generator.
* The definition formula applies 132 data points provided by Airfoil tools.
* The definition formula applies 200 data points provided by the NACA4 generator.
* The definition formula applies 264 data points. We double the 132 points provided by Airfoil tools to 264 points by adding the average value between two points.
* The definition formula applies 400 data points by doubling 200 points from NACA4.  
Figure 1a depicts the above six NACA0012 models and there are apparent location differences among models, as shown in Figure 1b. The Airfoil tools and the NACA 4-digit airfoil generator directly provide the data points for modeling, which do not give the related formulas. We then adopt curve fitting to produce the related formulas based on the existing data points. For the Airfoil tools, the fitting formula is  
\[y=\pm\Big{(}a_{1}x^{8}+a_{2}x^{7}+a_{3}x^{6}+a_{4}x^{5}+a_{5}x^{4}+a_{6}x^{3}+ a_{7}x^{2}+a_{8}x+a_{9}\Big{)} \tag{2}\]  
where \(a_{1}=-32.25\), \(a_{2}=139.4\), \(a_{3}=-249.5\), \(a_{4}=239\), \(a_{5}=-132.5\), \(a_{6}=43.15\), \(a_{7}=-8.247\), \(a_{8}\) = 0.9256, \(a_{9}=0.005819\) and \(R^{2}=0.9971\).  
For the NACA 4-digit airfoil generator, the fitting formula is  
\[y=\pm\Big{(}a_{1}x^{8}+a_{2}x^{7}+a_{3}x^{6}+a_{4}x^{5}+a_{5}x^{4}+a_{6}x^{3}+ a_{7}x^{2}+a_{8}x+a_{9}\Big{)} \tag{3}\]  
Figure 1: (a) Six NACA0012 airfoil models; (b) Close view of different models.  
where \(a_{1}=-31.18\), \(a_{2}=134.8\), \(a_{3}=-241.4\), \(a_{4}=231.4\), \(a_{5}=-128.5\), \(a_{6}=41.93\), \(a_{7}=-8.052\), \(a_{8}=0.9116\), \(a_{9}=0.006113\) and \(R^{2}=0.9975\). Therefore, the differences among different NACA0012 models shown in Figure 1 could be represented through the subtraction between the related formulas.