## 2 Materials and Methods
### Numerical Analysis Methodology  
Three major numerical analysis techniques, namely, direct numerical simulation (DNS) [33], large eddy simulation (LES) [34], and RANS simulation, can be used to analyze turbulent flow fields. To analyze the general vortex behavior precisely in a turbulent flow field containing vortices of various scales, the calculation grid size must be smaller than the minimum space scale of the vortex structure and the time interval should be less than the minimum time scale of the vortex variation. LES uses the spatially averaged Navier-Stokes equations, which directly calculate vortices greater than the grid scale and indirectly refer to the subgrid-scale model for vortices smaller than the grid scale. DNS and LES require extensive resources for their calculations, making them unfeasible in practical engineering applications. In comparison, RANS uses the time-based ensemble-averaged Navier-Stokes equations and models all of the effects caused by turbulence. Although RANS yields a lower resolution analysis than DNS or LES, it is widely used in engineering applications because it does not require high-resolution calculation grids. RANS uses the following equations:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}\rho U_ {j}\big{)}\ =0, \tag{1}\]  
\[\frac{\partial(\rho U_{i})}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho U_{i}U_{j}\big{)}\ =-\frac{\partial\rho}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\big{(} \tau_{ij}-\rho\overline{u_{i}u_{j}}\big{)}\ +\ S_{M} \tag{2}\]  
Turbulence models that are widely used for RANS numerical analysis in engineering include \(k\)-\(\varepsilon\), \(k\)-\(\omega\), and SST. The \(k\)-\(\varepsilon\) turbulence model [35] precisely analyzes the turbulence behavior in free-stream regions with small pressure gradients; however, it inaccurately estimates the boundary layer separation in viscous sub-layer regions. Although the \(k\)-\(\omega\) turbulence model, which was developed by [PERSON] [36], accurately analyzes the separation caused by adverse pressure gradients, it is sensitive to the free-stream region. [PERSON] [37] combined the advantages of the \(k\)-\(\varepsilon\) and \(k\)-\(\omega\) models and proposed the SST turbulence model. Simulation of the wind turbine flow generally involves the use of CFD to analyze the performance of the cross-sectional airfoil of the blade, and the numerical analysis and experimental results are often in agreement [38; 39]. The complete formulation of the SST model is given by:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho U_{i}k)}{\partial x_{i }}=\widetilde{P}_{k}-\beta^{*}\rho k\omega+\frac{\partial}{\partial x_{i}} \bigg{[}(\mu+\sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{i}}\bigg{]} \tag{3}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho U_{i}\omega)}{ \partial x_{i}}=\alpha\rho S^{2}-\beta\rho\omega^{2}+\frac{\partial}{\partial x _{i}}\bigg{[}(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{i }}\bigg{]}\ +\ 2(1-F_{1})\rho\sigma_{w2}\frac{1}{\omega}\frac{\partial k}{\partial x_{i}} \frac{\partial\omega}{\partial x_{i}} \tag{4}\]  
The blending function \(F_{1}\) is defined by:  
\[F_{1}=tanh\Bigg{\{}\Bigg{\{}\min\bigg{[}max\Bigg{(}\frac{\sqrt{k}}{\beta^{*} \omega y},\frac{500\
u}{y^{2}\omega}\bigg{)},\frac{4\rho\sigma_{\omega^{2}}k} {CD_{k\omega}y^{2}}\bigg{]}\Bigg{\}}^{4}\Bigg{\}} \tag{5}\]  
where \(CD_{k\omega}=max\Big{(}2\rho\sigma_{\omega^{2}}\frac{1}{\omega}\frac{\partial k }{\partial x_{i}}\frac{\partial\omega}{\partial x_{i}},\ 10^{-10}\Big{)}\) and \(y\) is the distance to the nearest wall.