# Wind Turbine Enhancement via Active Flow Control Implementation
## 2 Numerical Method
### Governing Equations and Turbulence Model
Some RANS turbulence models are based on solving the Navier-Stokes equations by incorporating the concept of turbulence viscosity (\(\mu_{t}\)), which can be mathematically implemented into the momentum equations described above using the subsequent definition (called the Boussinesq hypothesis):  
\[\mu_{t}=\frac{-\rho\overline{u_{x}^{\prime}u_{y}^{\prime}}}{2\frac{\partial \bar{u}_{x}}{\partial x}}=\frac{-\rho\overline{u_{y}^{\prime}u_{y}^{\prime}}}{2 \frac{\partial\bar{u}_{y}}{\partial y}}=\frac{-\rho\overline{u_{x}^{\prime}u_ {y}^{\prime}}}{\frac{\partial\bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{ y}}{\partial x}} \tag{7}\]  
The X and Y momentum terms of the NS equations can therefore be expressed as follows:  
\[\frac{\partial\bar{u}_{x}}{\partial t}+\bar{u}_{x}\frac{\partial\bar{u}_{x}}{ \partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{x}}{\partial y}=g_{x}-\frac{1}{ \rho}\frac{\partial p}{\partial x}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}2(\mu+\mu_{t})\frac{\partial\bar{u}_{x}}{\partial x}\bigg{)}+\frac{1} {\rho}\frac{\partial}{\partial y}\bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial \bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{y}}{\partial x}\bigg{)}\bigg{)} \tag{8}\]  
\[\frac{\partial\bar{u}_{y}}{\partial t}+\bar{u}_{x}\frac{\partial\bar{u}_{y}}{ \partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{y}}{\partial y}=g_{y}-\frac{1}{ \rho}\frac{\partial p}{\partial y}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}(\mu+\mu_{t})\bigg{(}\frac{\partial\bar{u}_{x}}{\partial y}+\frac{ \partial\bar{u}_{y}}{\partial x}\bigg{)}\bigg{)}+\frac{1}{\rho}\frac{\partial}{ \partial y}\bigg{(}2(\mu+\mu_{t})\frac{\partial u_{y}}{\partial y}\bigg{)} \tag{9}\]  
We have chosen to use the \(k\)-\(\omega\) SST turbulence model, which relies on using the \(k\)-\(\omega\) model near the wall, the \(k\)-\(\epsilon\) model far away from the object, and a blending function between these two. Mathematically, \(\mu_{t}\) is calculated as follows:  
\[\mu_{t}=\frac{\rho k}{\omega}\quad\longrightarrow\quad \begin{cases}\rho:\text{ density}\\ k:\text{turbulent kinetic energy}\\ \omega:\text{turbulent kinetic energy specific dissipation rate}\end{cases}  \tag{10}\]As stated, this model is a two-transport equation model in order to solve \(k\) and \(\omega\). The transport equations used to define each parameter are as follows:  
\[\frac{\mathrm{D}(\rho k)}{\mathrm{D}t}=\tau_{ij}\frac{\partial u_{i}}{\partial x _{j}}-\beta^{*}\rho k\omega+\frac{\partial}{\partial x_{j}}\left[(\mu+\sigma_{k }\mu_{t})\frac{\partial k}{\partial x_{j}}\right] \tag{11}\]  
\[\frac{\mathrm{D}(\rho\omega)}{\mathrm{D}t}=\frac{\gamma}{\
u_{t}}\ \tau_{ij}\frac{ \partial u_{i}}{\partial x_{j}}-\beta\rho\omega^{2}+\frac{\partial}{\partial x _{j}}\left[(\mu+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{j}} \right]+2\rho(1-F_{1})\sigma_{\omega}2\frac{1}{\omega}\frac{\partial k}{ \partial x_{j}}\frac{\partial\omega}{\partial x_{j}}, \tag{12}\]  
where the turbulent stress tensor \(\tau_{ij}\) is defined as  
\[\tau_{ij}=\mu_{t}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}}-\frac{2}{3}\frac{\partial u_{k}}{\partial x_{k}}\delta_ {ij}\right)-\frac{2}{3}\rho k\delta_{ij} \tag{13}\]  
The SST model combines the constants of the \(k\)-\(\omega\) and \(k\)-\(\epsilon\) turbulence models, defined as \(\phi_{1}\) and \(\phi_{2}\), respectively, using a blending function \(F_{1}\), which depends on the distance of the first cell to the wall.