# Wind Turbine Enhancement via Active Flow Control Implementation
## 2 Numerical Method
### Non-Dimensional Parameters  
The Reynolds number is defined as the ratio of inertial to viscous forces. Mathematically, it is expressed as follows:  
\[Re=\frac{\rho u_{rel}C}{\mu} \tag{17}\]  
where \(u_{rel}\) is the relative velocity, \(C\) the chord length, and \(\mu\) the fluid absolute viscosity.  
The most common aerodynamic coefficients are the lift \(C_{l}\) and drag \(C_{d}\) ones, which are mathematically expressed as follows:  
\[C_{l}=\frac{L}{\frac{1}{2}\rho u_{rel}^{2}C}\hskip 56.905512 ptC_{d}=\frac{D}{ \frac{1}{2}\rho u_{rel}^{2}C} \tag{18}\]  
L and D represent the lift and drag forces, respectively.  
The aerodynamic efficiency is the ratio between the lift and drag coefficients:  
\[\eta=\frac{C_{l}}{C_{d}} \tag{19}\]Pressure Coefficient  
It characterizes the relative pressure field (\(p-p_{\infty}\)) along the body.  
It is mathematically expressed as  
\[C_{p}=\frac{p-p_{\infty}}{\frac{1}{2}\rho u_{rel}^{2}} \tag{20}\]  
where (\((1/2)\rho u_{rel}^{2}\)) is defined as the dynamic pressure.  
The friction coefficient, also known as the skin friction coefficient or tangential friction coefficient, evaluates the frictional force per unit area \(\tau_{w}\) acting on a surface in contact with the fluid.  
\[C_{f}=\frac{\tau_{w}}{\frac{1}{2}\rho u_{rel}^{2}} \tag{21}\]  
Momentum Coefficient  
It represents the momentum associated with the jet divided by the incoming fluid momentum. According to reference [34], \(C_{\mu}\) is the primary AFC parameter. Mathematically it is expressed as follows:  
\[C_{\mu}=\frac{h\big{(}\rho u_{max}^{2}\big{)}\sin\theta_{\rm{pet}}}{C\big{(} \rho u_{rel}^{2}\big{)}} \tag{22}\]  
where (\(h\)) is the jet width, (\(u_{max}\)) is the maximum jet velocity, and (\(\theta_{\rm{jet}}\)) is the jet inclination angle with respect to the airfoil surface.  
Forcing Frequency  
The forcing frequency, given as a non-dimensional number, is defined as follows:  
\[F^{+}=\frac{f}{f_{0}} \tag{23}\]  
where \(f\) is an arbitrary frequency and \(f_{0}\) is the vortex shedding frequency (the frequency at which vortices are shed from the airfoil).  
Courant-Friedrichs-Levy Number  
The Courant number, often denoted as CFL, quantifies how the numerical scheme/setting correctly captures the propagation of the information in the flow due to its velocity. Mathematically, it is defined as the fluid velocity measured at any given mesh cell multiplied by the simulation time step and divided by the mesh cell length scale (\(\Delta x\)).  
\[CFL=\frac{u\Delta t}{\Delta x} \tag{24}\]  
For time-dependent simulations, the Courant number is an important parameter in order to ensure the stability of the system, and it is recommended to keep it below 1.  
The dimensionless wall distance \(y^{+}\) represents the distance from the wall to the first mesh cell, \(y\), normalized by a characteristic velocity \(u_{\tau}\) and the relative viscosity of the fluid \(\
u\). Mathematically, it is expressed as follows:  
\[y^{+}=\frac{u_{\tau}y}{\
u} \tag{25}\]  
where \(u_{\tau}\) is the friction velocity, defined as  
\[u_{\tau}=\sqrt{\frac{\tau_{w}}{\rho}} \tag{26}\]