# Wind Turbine Enhancement via Active Flow Control Implementation
## 2 Numerical Method  
### Governing Equations and Turbulence Model  
The Navier-Stokes (NS) equations under incompressible flow conditions take the following form:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+\frac{\partial u_{i}u_{j}}{\partial x_{j}}=- \frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\
u\frac{\partial^{2}u_{i}}{ \partial x_{j}\partial x_{j}} \tag{2}\]  
As Reynolds number increases, the mesh and the associated time step NS equations require drastic decreases in order to reach the Kolmogorov length and time scales, but reaching such scales involves extremely large computational times; such simulations are called Direct Numerical Simulations (DNS). Due to its drawbacks, nowadays DNS is just used for research purposes and for relatively small Reynolds numbers. For highly turbulent flow and to obtain reasonably accurate results, turbulence models are still needed. The use of Large Eddy simulation (LES) as a turbulence model is a very good option in 3D flows, yet it still requires very fine meshes and large computational times; therefore, LES simulations require supercomputers to shorten the computational time. As a result, simulations at large Reynolds numbers still need to be performed using Reynolds Averaged Navier Stokes (RANS) or URANS unsteady-RANS turbulence models. Their precision is not as accurate as LES or DNS, but the computational time needed shortens drastically. Under incompressible flow conditions, the only variables associated with the NS equations are the pressure and the three velocity components, which are generically called \(\phi\). In order to be able to apply URANS models, each variable from the NS equations needs to be substituted by its average \(\tilde{\phi}\) and a fluctuation \(\phi^{\prime}\) term. \[\phi=\tilde{\phi}+\phi^{\prime} \tag{3}\]  
After the substitution, the resulting NS equations (considering just two dimensions) take the following form:  
\[\frac{\partial\bar{u}_{x}}{\partial x}+\frac{\partial\bar{u}_{y}}{\partial y}=0 \tag{4}\]  
\[\frac{\partial\bar{u}_{x}}{\partial t}+\bar{u}_{x}\frac{\partial\bar{u}_{x}}{ \partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{x}}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+g_{x}+\frac{1}{\rho}\frac{\partial}{\partial x} \bigg{(}2\mu\frac{\partial\bar{u}_{x}}{\partial x}-\rho(\bar{u}_{x}^{\prime} )^{2}\bigg{)}+\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}\mu\bigg{(} \frac{\partial\bar{u}_{x}}{\partial y}+\frac{\partial\bar{u}_{y}}{\partial x }\bigg{)}-\rho\overline{u_{x}^{\prime}u_{y}^{\prime}}\bigg{)} \tag{5}\]  
\[\frac{\partial\bar{u}_{y}}{\partial t}+\bar{u}_{x}\frac{\partial\bar{u}_{y}}{ \partial x}+\bar{u}_{y}\frac{\partial\bar{u}_{y}}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial y}+g_{y}+\frac{1}{\rho}\frac{\partial}{\partial x }\bigg{(}\mu\bigg{(}\frac{\partial\bar{u}_{y}}{\partial x}+\frac{\partial\bar {u}_{x}}{\partial y}\bigg{)}-\rho\overline{u_{y}^{\prime}u_{x}^{\prime}} \bigg{)}+\frac{1}{\rho}\frac{\partial}{\partial y}\bigg{(}2\mu\frac{\partial \bar{u}_{y}}{\partial y}-\rho(\bar{u}_{y}^{\prime})^{2}\bigg{)} \tag{6}\]  
It is now relevant to consider the concept of the apparent Reynolds stress tensor (\(\tau_{app}\)), which comprises a symmetric matrix encompassing the time-averaged fluctuations in the \(x\), \(y\), and \(z\) directions. It is important to observe that in the case of two-dimensional flows, this matrix explicitly consists of four terms, whereas in three dimensions, it expands to a \(3\times 3\) symmetric matrix.