# The Influence of Tilt Angle on the Aerodynamic Performance of a Wind Turbine
## 2 Numerical Modeling
### Turbulence Model  
The \(SST\)\(k-\omega\) turbulence model can consider the complex flow of the adverse pressure gradient near the wall region and the flow in the free shear region. Thus, the \(SST\)\(k-\omega\) turbulence model is suitable for simulating the rotational motion of the blade [10]. In addition, this turbulence model can accurately capture wind turbine wake [11; 12].  
In the Reynolds-averaged N-S equations, \(\tau_{ij}\ =\ -\rho\overline{u_{i}^{\prime}u_{j}^{\prime}}\) refers to the Reynolds stress tensor. Reynolds stress tensor and mean strain rate tensor (\(S_{ij}\)) are related by the Boussinesq eddy viscosity assumption:  
\[\tau_{ij}\ =\ 2\
u_{t}S_{ij}-\frac{2}{3}\rho k\delta_{ij} \tag{1}\]  
where \(\
u_{t}\) refers to the eddy viscosity, \(\rho\) refers to the density, \(k\) refers to the turbulence kinetic energy and \(\delta_{ij}\) refers to the Kronecker delta function.  
To provide closure equations, in the \(SST\)\(k-\omega\) turbulence model, the turbulent kinetic energy (\(k\)) and specific dissipation of turbulent kinetic energy (\(\omega\)) also need governing transport equations, which are given as follows:\[\frac{D\rho k}{Dt}\ =\ \tau_{ij}\frac{\partial u_{i}}{\partial x_{j}}-\beta^{*} \rho\omega k+\frac{\partial}{\partial x_{j}}\bigg{[}(\mu+\sigma_{k}\mu_{t})\frac {\partial k}{\partial x_{j}}\bigg{]} \tag{2}\]  
\[\frac{D\rho\omega}{Dt}\ =\ \frac{\gamma}{\
u_{t}}\tau_{ij}\frac{\partial u_{i}}{ \partial x_{j}}-\beta\rho\omega^{2}+\frac{\partial}{\partial x_{j}}\bigg{[}( \mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{j}}\bigg{]}+2(1- F_{1})\rho\sigma_{\omega 2}\frac{1}{\omega}\frac{\partial k}{\partial x_{i}}\frac{ \partial\omega}{\partial x_{j}} \tag{3}\]  
In the formulas above, the model coefficients are defined as follows:  
\[\beta^{*}\ =\ F_{1}\beta_{1}{}^{*}+(1-F_{1})\beta_{2}{}^{*} \tag{4}\]  
\[\beta\ =\ F_{1}\beta_{1}+(1-F_{1})\beta_{2} \tag{5}\]  
\[\gamma\ =\ F_{1}\gamma_{1}+(1-F_{1})\gamma_{2} \tag{6}\]  
\[\sigma_{k}\ =\ F_{1}\sigma_{k1}+(1-F_{1})\sigma_{k2} \tag{7}\]  
\[\sigma_{\omega}\ =\ F_{1}\sigma_{\omega 1}+(1-F_{1})\sigma_{\omega 2} \tag{8}\]  
The blending function \(F_{1}\) is defined as follows:  
\[F_{1}\ =\ \tanh\left\{\left\{\min\left[\max\left(\frac{\sqrt{k}}{\beta^{*} \omega y},\frac{500\upsilon_{\infty}}{y^{2}\omega}\right)\frac{4\rho\sigma_{ \omega 2}k}{CD_{\omega}y^{2}}\right]\right\}^{4}\right\} \tag{9}\]  
where \(CD_{\omega}\) refers to the cross-diffusion term, \(y\) refers to the distance to the nearest wall and \(\upsilon\) refers to the kinematic viscosity. \(F_{1}\) is equal to zero in the region away from the wall (\(k-\varepsilon\) turbulence model) and one in the region near the wall (\(k-\omega\) turbulence model).  
The eddy viscosity is  
\[\upsilon_{t}\ =\ \frac{a_{1}k}{\max(a_{1}\omega,\Omega F_{2})} \tag{10}\]  
where \(\Omega\) is the absolute value of the vorticity and \(F_{2}\) is the second blending function, defined as  
\[F_{2}\ =\ \tanh\left[\max\left(\frac{2\sqrt{k}}{\beta^{*}\omega y},\frac{500 \upsilon}{y^{2}\omega}\right)^{2}\right] \tag{11}\]  
A more detailed description of the \(SST\ k-\omega\) turbulence model is provided in [10]. In this study, the parameters for the \(SST\ k-\omega\) turbulence model are as follows:  
\[\begin{array}{