# Aerodynamic Analyses of Airfoils Using Machine Learning as an Alternative to RANS Simulation
## 2 Background Theory  
### Numerical Simulations  
The purpose of performing numerical simulations was to obtain the values of the coefficients of drag and lift that were produced by different airfoils under diverse flow conditions, which could then be utilized as the required dataset for training and testing the artificial neural network that was based on the regression learning model. The training dataset for the current study was obtained by performing CFD simulations on four different NACA series airfoils, including NACA 0012, NACA 2415, NACA 23024 and NACA 24112. The first two airfoils belong to the NACA four-digit family and the last two airfoils belong to the NACA five-digit family. The NACA 0012 airfoil is a symmetric airfoil and is one of the most frequently used and widely researched airfoils due to its lift to drag ratio [22]. Similarly, the NACA 2415 is also one of the most widely used airfoils in the unsymmetrical airfoil category. It also has applications in wind turbines for alternate energy resources [23]. The NACA 23024 and NACA 24112 are also unsymmetrical airfoils that have various applications in aeronautics and wind turbines [24]. Numerical simulations were performed on each of the selected airfoils using 10 different Reynolds numbers, in the range of 0.5 to 5 million, at a fixed Mach number of 0.5. This range of Reynolds numbers lies in the moderate regime on the low-to-high Reynolds number scale and represents the onset of turbulence to the fully turbulent boundary layer over the airfoil [25]. The Reynolds numbers that were used in our simulations were \(0.5\times 10^{6}\), \(1.0\times 10^{6}\), \(1.5\times 10^{6}\), \(2.0\times 10^{6}\), \(2.5\times 10^{6}\), \(3.0\times 10^{6}\), \(3.5\times 10^{6}\), \(4.0\times 10^{6}\), \(4.5\times 10^{6}\) and \(5.0\times 10^{6}\). Additionally, each airfoil with each Reynolds number was numerically simulated at different angles of attacks from \(0^{\circ}\) to \(20^{\circ}\), with increment of \(2^{\circ}\) between each case. In total, CFD simulations of 440 cases were performed under different flow conditions. A summary of these cases is presented in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Airfoil** & **R No. ** & & & & & & & & & & \\ \hline \multirow{8}{*}{\(\mathrm{C_{D}}\)} & 0.5 \(\times\) 10\({}^{6}\) & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\  & 1.0 \(\times\) 10\({}^{6}\) & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\  & 1.5 \(\times\) 10\({}^{6}\) & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\  & 2.0 \(\times\) 1 For the numerical simulations, the commercially available CFD software package Ansys Fluent 16.0 was used. The RANS-based one-equation turbulence model Spalart-Allmaras, which is available in Fluent, was selected for the simulations. This turbulence model was specially developed for aerospace applications [26] and is commonly used in wall-bounded flow simulations. It has been proven to produce correct estimations for cases where boundary layers are subjected to adverse pressure gradients [27]. Elaborated mathematical details regarding the governing equations of the Spalart-Allmaras model, as implemented in the Fluent software package, can be found in the Fluent user guide [28]. The most common C-type flow domain was chosen for the simulations using a distance that was equivalent to 15 chord lengths between all sides of the airfoil and the domain boundaries in order to ensure the smooth dissipation of the flow effects and avoid backflow at the exit. The flow domain was resolved using unstructured grids for the computations. Inflation layers were introduced around the airfoils to achieve y+ values of less than 1 in each case that was simulated. The boundary condition at the airfoil was selected as \"wall\", whereas the \"velocity inlet\" and \"pressure outlet\" boundary conditions were selected at the flow domain inlet and exit, respectively. The airfoils are shown in Figure 1 with the meshing that was created around them, along with a zoomed-in view of the inflation layers.