## 2 Materials and Methods
### Governing Equations  
The solver used is Fluent, which is CFD software that utilizes a finite volume method to solve the complete Reynolds-averaged Navier-Stokes (RANS) equations [20]. The finite volume solver is used for numerical simulations to solve the three-dimensional Navier-Stokes (N-S) equations. N-S equations embody the fundamental concepts of mass and momentum preservation for a fluid in motion. The utilization of a turbulence model entails the substitution of the N-S equation with the RANS equation. The continuity equation and RANS equations for steady, incompressible, and turbulent flow can be expressed as follows [21]:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{i}}(\rho u_{i})=0, \tag{1}\]  
\[\frac{\partial}{\partial t}(\rho u_{i})+\frac{\partial}{\partial x_{i}}\Big{(} \rho u_{i}u_{j}\Big{)}=-\frac{\partial\rho}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\Bigg{[}n\Bigg{(}\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}-\frac{2}{3}\delta_{ij}\frac{\partial u_{i}}{ \partial x_{j}}\Bigg{)}\Bigg{]}+\frac{\partial}{\partial x_{j}}\Big{(}- \overline{\rho u_{i}u_{j}}\Big{)}, \tag{2}\]  
Here, \(\rho\) is the density of air, \(t\) is the time, \(\mu\) is dynamic viscosity, and \(u_{i}\) is the velocity components. The last term in the equation \((-\rho u_{i}u_{j}^{\prime})\) indicates Reynolds turbulence stress. There is no analytical way to determine these values because turbulence is chaotic. The turbulence stresses in the momentum equation are computed using turbulence models. To compute these values, turbulence models have been constructed [21].  
The most commonly used turbulence models in CFD analysis are the Spalart-Allmaras, \(k\)-\(\varepsilon\), and \(k\)-\(\omega\) turbulence models. The Spalart-Allmaras model is a one-equation turbulence model. Both k turbulence models are two-equation models. Although a one-equation solver is a low-cost model, two-equation models can meet the accuracy requirements of most flow problems. Also, the \(k\)-\(\omega\) turbulence model has two types, Standard and Shear Stress Transport (SST). The \(k\)-\(\omega\) SST turbulence model is applied in this work. The \(k\)-\(\omega\) SST two-equation model can simulate large separation flow by combining the \(k\)-\(\varepsilon\) and \(k\)-\(\omega\) two-equation models through a mixing function. This allows the model to fully utilize the advantages of the \(k\)-\(\varepsilon\) model for free flow and the \(k\)-\(\omega\) model restricted flow on the wall [11]. Developed by [PERSON] [22], the SST model makes use of the following equations for turbulent kinetic energy (\(k\)) and specific dissipation (\(\omega\)):  
\[\frac{\partial k}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}kU_{j} \big{)}=\tau_{ij}\frac{\partial U_{i}}{\partial x_{j}}-\beta_{k}\omega k+ \frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}v+\frac{v_{t}}{\sigma_{k}} \bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}, \tag{3}\]  
\[\frac{\partial\omega}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \omega U_{j}\big{)}=\frac{\gamma}{v_{t}}\tau_{ij}\frac{\partial U_{i}}{ \partial x_{j}}-\beta_{\omega}\omega^{2}+\frac{\partial}{\partial x_{j}} \Bigg{[}\bigg{(}v+\frac{v_{t}}{\sigma_{\omega}}\bigg{)}\frac{\partial\omega}{ \partial x_{j}}\Bigg{]}+2(1-F_{1})\sigma_{cd}\frac{1}{\omega}\frac{\partial k }{\partial x_{j}}\frac{\partial\omega}{\partial x_{j}}, \tag{4}\]  
where \(t_{ij}\) is the turbulence stress tensor and \(v_{t}\) is the turbulence viscosity (eddy viscosity). \(F\) is a function designed over the near-wall area, and \(B_{k}\), \(B_{\omega}\), \(\sigma_{k}\), \(\sigma_{\omega}\), and \(\sigma_{cd}\) are constants. The values of these constants are left as the default values of the solver (Fluent). [PERSON] has adjusted these values with reference to the work of [PERSON].