## 3 Methodology
### Computational Model  
As additional validation of the resulted airfoil, a computational study conceded out by solving a compressible form of Reynolds Averaged Navier-Stokes equation (RANS) and continuity equation. The mass and momentum conservation equations are expressed as follows [31; 32; 33]:  
\[\frac{\partial}{\partial x_{i}}(\rho u_{i})=0 \tag{14}\]  
\[\frac{\partial\big{(}\rho u_{i}u_{j}\big{)}}{\partial x_{j}}=\frac{\partial p}{ \partial x_{i}}+\frac{\partial}{\partial x_{i}}\Bigg{(}\mu\Bigg{(}\frac{ \partial u_{i}}{\partial x_{i}}+\frac{\partial u_{j}}{\partial x_{i}}-\frac{2} {3}\delta_{ij}\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}\Bigg{)}+\frac{ \partial}{\partial x_{j}}\big{(}-\rho\overline{i_{i}\overline{u_{j}}}\big{)}+ \rho\ f_{b} \tag{15}\]  
Figure 5: Optimization flow chart. where \(\rho=\) density, \(p=\) pressure, \(\mu=\) dynamic viscosity and \(f_{b}\) denotes body forces. \(u_{i}\) and \(u_{j}\) denotes velocity vectors for \(i,\ j=1,2,3\). Another unknown term in Equation (15) are Reynold stresses, i.e., \(\left(\frac{\partial}{\partial x_{j}}\left(-\rho\overline{u_{i}}\overline{u_{ j}}\right)\right)\) where \(\hat{u_{i}}\) and \(\hat{u_{i}}\) are known as fluctuating parts of the corresponding velocity vectors \(u_{i}\) and \(u_{j}\), respectively [34; 35]. As per the requirement of solution due to unknown Reynolds stress term, RANS turbulence model is mandatory to close the equation but the RANS equation is not closed. The wall function is also used by some people to model flow in the RANS turbulence model within the boundary layer but these types of turbulence models are not appropriate for accurate prediction of the boundary layer, flow recirculation, pressure drops, and flow separations [29]. There are some very expensive computationally techniques, for instance, Direct Numerical Simulation (DNS), Large Eddy simulations (LES), and Detached Eddy Simulation (DES). Therefore SST turbulence model is used for the numerical study to obtain the advantages of accurate aerodynamic data and computational economy [30]. The shear stress transport turbulence model is a mixture of Wilcox \(k-\omega\) and \(k-\epsilon\) turbulence models. The \(k-\epsilon\) turbulence model is used to solve the flow in full turbulence region to get benefit from its economy, robustness, and free flow independence, while Wilcox \(k-\omega\) is used to solve the flow near the wall to accurately predict the boundary layer [36; 37]. The two equations of turbulence energy (\(k\)) and turbulence dissipation rate (\(\omega\)) in the SST turbulence model are as follows in Equations (15) and (16):  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(u_{i}\rho k )=\frac{\partial}{\partial x_{i}}\left(\mu_{k}\frac{\partial}{\partial x_{i}} k\right)+\widetilde{P}_{k}-\beta^{*}\rho\omega k \tag{16}\]  
\[\frac{\partial}{\partial t}(\rho\omega)+\frac{\partial}{\partial x_{i}}(u_{i} \rho\omega)=\frac{\partial}{\partial x_{i}}\left(\mu_{\omega}\frac{\partial} {\partial x_{i}}\omega\right)+\alpha\ \rho S^{2}-\beta\rho\omega^{2}+2\rho(1-F_{1})\sigma_{\omega 2} \frac{1}{\omega}\frac{\partial}{\partial x_{i}}k\ \frac{\partial}{\partial x_{i}}\omega \tag{17}\]  
where, \(F_{1}=\) blending function shown in Equation (17) and \(F_{1}\) value is zero away from the wall and enables \(k-\epsilon\) model. The value within the boundary layer is approximately 1 which allows \(k-\omega\) model.