## 2 Methodology
### Initial and Boundary Conditions of the Model
The transport equation for the k-\(\omega\) SST turbulence model is:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{j}k)}{\partial x_{ j}}=\tau_{ij}\frac{\partial u_{j}}{\partial x_{j}}-\beta^{*}k\omega+\frac{ \partial}{\partial t}\Bigg{[}\rho(v+\sigma_{k}v\tau)\frac{\partial k}{\partial x _{j}}\Bigg{]} \tag{7}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial\big{(}\rho u_{j} \omega\big{)}}{\partial t}=\frac{\gamma}{\
u_{T}}\tau_{ij}\frac{\partial u_{j }}{\partial x_{j}}-\beta\omega^{2}+\frac{\partial}{\partial x_{j}}\Bigg{[} \rho(v+\sigma_{\omega}v\tau)\frac{\partial\omega}{\partial x_{j}}\Bigg{]}+2(1 -F_{1})\frac{\rho\sigma_{\omega 2}}{\omega}\frac{\partial k}{\partial x_{j}}\frac{ \partial\omega}{\partial x_{j}} \tag{8}\]  
where \(k\)--turbulent kinetic energy, \(\omega\)--specific frequency of turbulent energy dissipation, \(u_{j}\)--component of the velocity vector along the axis \(x_{j}\), \(x_{j}\)--spatial coordinate, \(\tau_{ij}\)--Reynolds stress, \(\
u\)--molecular viscosity, \(v_{T}\)--turbulent viscosity, \(\sigma_{k}\), \(\sigma_{\omega}\), \(\sigma_{\omega 2}\)- empirical constants of the SST model, \(\beta^{*}\), \(\beta\), \(\gamma\)--empirical coefficients (model constants), \(F_{1}\)--the blending function used in the SST model to combine the k\(-\omega\) and k\(-\epsilon\) models. In this paper, the ANSYS Fluent CFD package is used to simulate the flow around the VAWT. ANSYS Fluent automatically calculates F\({}_{1}\) and switches the model between modes, ensuring accuracy at walls and stability in free flow. It also uses the default coefficients above. ANSYS Fluent software simulates the fluid flow around the vertical axis of a four-bladed wind turbine with a NACA 0021 airfoil. In the 2D flow analysis, the computational domain is divided into two domains: fixed and rotating [34]. A 2D model VAWT was considered within a rectangular area of size 20D \(\times\) 60D, as shown in Figure 2:0 The parameters of the calculation domain are shown in Figure 2:1 In this configuration, the ratio of rotor height to the total height of the computational domain is less than 5%, which falls within the acceptable blockage ratio limits where the blockage effect on the flow is considered negligible. Thus, the influence of the domain boundaries on the simulation results is minimized, as confirmed by the stability of the computations and the achievement of convergence by the 20 th rotation. The dimensions of the computational domain of the simulation are based on the work of [PERSON] et al. [22], which are sufficient to model the flow around the VAWT, and ensure that the space around the turbines in the leeward region is sufficient to characterize the flow behavior in the VAWT system; such a 2D model has been widely used in the literature to model the performance of VAWT instead of a complex 3D model without losing generality in the accuracy of numerical simulation. Model boundary conditions at the inlet, the flow velocity, turbulent kinetic energy, and specific dissipation frequency are specified. At the outlet (pressure outlet), zero static pressure is specified and gradients from the internal cells for k and \(\omega\) are used. On the blade surface (wall), the no-slip condition (U = 0) is applied, with k = 0 and \(\omega\) calculated automatically by wall functions. The stationary and rotating domains are connected via a sliding interface, ensuring conservation of both mass and momentum. Each case was simulated for at least 20 revolutions, and the average data value of the last rotation cycle \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Meaning** \\ \hline Rotor diameter & D \\ \hline Rotating area diameter & 1.5D \\ \hline Distances from the rotor center to the velocity inlet & 20D \\ \hline Distances from the rotor center to the pressure outlet & 40D \\ \hline Distances from the rotor center to the symmetry walls & 10D \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of computational domain. Figure 3: Computational domain of the CFD model.