## 3 Methodology
### Computational Model  
The Reynolds-averaged Navier-Stokes (RANS) and continuity equations are used to compute the compressible flow field in the steady state. The equation for mass and the momentum conservation equations in each direction with no source term can be written as shown in Equations (22) and (23). In addition, to solve the Reynolds stress term, a shear stress transport (SST) turbulence model is used, due to its ability to accurately predict aerodynamic features such as the flow boundary layer, pressure drop, flow separation, and recirculation:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{22}\]  
Figure 5: Optimization flow chart elaborating on the MATLAB code for airfoil shape optimization. \[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial\big{(}\rho u_{i}u_{j}\big{)} }{\partial x_{j}}=\frac{\partial p}{\partial x_{i}}+\frac{\partial}{\partial x _{i}}\left(\mu\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{j}}{\partial x_{i}} \right)\right)+\frac{\partial}{\partial x_{j}}\left(-\rho\overline{\dot{u}_{i} \dot{u}_{j}}\right)+\rho\,f_{b} \tag{23}\]  
where \(\rho\) is the density, \(p\) is the pressure, \(\mu\) is the dynamic viscosity, and \(f_{b}\) is the body forces, while \(u_{i}\) and \(u_{j}\) are called the velocity vectors for the related values of \(i,\ j\ =\ 1,2,3\). Due to the unknown Reynolds stress term, the RANS turbulence model equation must be closed, but the RANS equation is not closed [32]. For the better prediction of aerodynamic data, pressure drop flow recirculation, precise boundary layer calculation, and flow separation are essential. Some people use the wall function of the flow model within the boundary layer in the RANS turbulence model, but this type of turbulent model is not suitable to provide an accurate prediction of the boundary layer, flow maintenance, pressure drop, and flow separation. Direct numerical simulations (DNS), large eddy simulations (LESs), and detached eddy simulations (DESs) [33; 34] are some very expensive computational techniques; therefore, the SST turbulence model was adopted for the numerical study to achieve aerodynamic data with precise prediction, along with the advantages of the computational economy [27]. The shear stress transport turbulence model is an amalgam of the Wilcox \(k-\omega\) and \(k-\epsilon\) turbulence models. The \(k-\epsilon\) turbulence model is used to solve the flow in a fully turbulent region to benefit from its economy, robustness, and free stream independence, while the Wilcox \(k-\omega\) turbulence model is employed to solve the flow near the walls for precise prediction of the boundary layer [35]. The two equations of the SST turbulence model for the turbulent kinetic energy (\(k\)) and the turbulent dissipation rate (\(\omega\)) are given as follows in Equations (24) and (25), respectively:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(u_{i}\rho k) =\frac{\partial}{\partial x_{i}}\left(\mu_{k}\frac{\partial}{ \partial x_{i}}k\right)+\widetilde{P}_{k}-\beta^{*  
where \(\mu_{t}\) is the turbulent viscosity, given as follows in Equation (30):  
\[\mu_{t}=\frac{a_{1}k}{\max(a_{1}\omega,SF_{2})} \tag{30}\]  
where \(F_{2}\) is a second blending factor, given as follows:  
\[F_{2}=\ \left[\left[\max\left(2\frac{\sqrt{k}}{\beta^{*}\omega y^{\prime}},\ \frac{500\
u}{y^{2}\omega}\right)\right]^{2}\right] \tag{31}\]  
Furthermore, the details and values of the constants for the SST turbulence model can be found in [35]. ### Computational Model  
In this paper, aerodynamic shape optimization of an NREL S-809 airfoil has been presented by using a genetic algorithm for the NREL Phase II, Phase III, and Phase VI HAWT blade designs. These blades were composed of an NREL S809 airfoil from root to tip. Therefore, for better performance, the blade design of the S809 airfoil was optimized using different boundary conditions.