## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
However, although this alternative procedure eliminates the previously mentioned issue, it is not clear if it is valid throughout the wind field, where surface effects might not be important. Finally, the tower wake was introduced by treating the tower as a drag producing cylinder, which can be regarded as an approximately two-dimensional problem and has appropriate two-dimensional growth rates (_i.e._, the wake of the tower expands only in the plane parallel to the ground). Another interesting wake model, categorized as a parabolic field model [59], was developed by [PERSON] [26] in 1988. The model considers that the wake is axisymmetric, fully turbulent, having nil circumferential velocities, and stationary with time. In addition, the pressure gradients in the co-flowing fluid outside the wake were assumed negligible. The model is based on the Reynolds-Averaged Navier-Stokes equations, which are complemented with an eddy viscosity model as turbulence closure. The eddy viscosity is predicted by a simple analytical form of the [PERSON]'s free shear layer model and includes contributions by the ambient turbulence. The set of equations were simplified by adopting the thin shear layer approximation and by disregarding the viscous terms. The complete Ainslie far wake model is shown in Equations (21) and (22):  
\[\overline{u_{x}}\frac{\partial\overline{u_{x}}}{\partial x}+\overline{u_{r}} \frac{\partial\overline{u_{x}}}{\partial r}=-\frac{1}{r}\!\left(\frac{\partial (r\overline{u_{x}^{\prime}u_{r}^{\prime}})}{\partial r}\right) \tag{21}\]  
\[-\overline{u_{x}^{\prime}u_{r}^{\prime}}=\varepsilon\,\frac{\partial \overline{u_{x}}}{\partial r} \varepsilon=l_{\mathrm{W}}(x)U_{\mathrm{W}}^{S}(x)+\varepsilon_{a} \tag{22}\]  
where \(\overline{u_{x}}\) [m/s] is the axial mean velocity, \(\overline{u_{r}}\) [m/s] is the radial mean velocity, \(x\) [m/s] is the axial distance coordinate (downwind from the WT position), \(r\) [m] is the radial distance coordinate (measured from the wake centerline), \(\overline{u_{x}^{\prime}u_{r}^{\prime}}\) [m\({}^{2}\)/s\({}^{2}\)] is the Reynolds stress cross-correlation, \(\varepsilon\) [m\({}^{2}\)/s] is the eddy viscosity, \(l_{\text{W}}\) and \(U_{\text{W}}^{S}\) are suitable length and velocity scales describing the wake shear layer, and \(\varepsilon_{a}\) [m\({}^{2}\)/s] is the ambient turbulence contribution to the eddy viscosity. The length and velocity scales were taken to be proportional to the wake width \(b_{\text{W}}\) [m] and the velocity difference \(U_{\infty}-U_{\text{W}}^{C}\) [m/s] across the wake shear layer (where \(U_{\text{W}}^{C}\) is the centerline wake velocity), respectively. \(\varepsilon_{a}\) is given by the atmospheric eddy diffusivity of momentum \(\varepsilon_{a}=k_{\text{M}}=ku^{*}z/\phi_{\text{m}}(z/L)\), where \(k\) is the Von Karman constant (\(k=0.4\)), \(u^{*}\) [m/s] is the friction velocity, \(z\) [m] is the height above the ground, \(L\) [m] is the Monin-Obukhov length, and \(\phi_{\text{m}}(z/L)\) is a similarity function reflecting the influence of the thermal stability condition on the turbulent mixing process (e.g., \(\phi_{\text{m}}=1\) for neutral conditions). The value of the friction velocity depends on the atmospheric surface shear layer, which can be approximated with the use of the logarithmic laws. For neutral conditions, the logarithmic law states that the axial wind speed (\(\overline{u_{x}}\) [m/s]) at any height within the atmospheric surface layer (e.g., up to 100 m) is governed by the roughness length (\(z_{0}\) [m]) and the friction velocity as given by \(\overline{u_{x}}=(u^{*}/k)\ln(z/z_{0})\). A modification is required in order to use the eddy viscosity equation in the near wake region (which length was taken to be approximately five rotor diameters), since a lack of equilibrium between the mean velocity field and the turbulence field was observed. A filter function \(F_{A}\), shown in Equation (23), was proposed to modulate the build-up of turbulence in the shear layer of the wake:  
\[F_{A}=