## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
\begin{cases}0.65+\left(\dfrac{x-4.5}{23.32}\right)^{\frac{1}{3}}&x<5.5\\ 1&x>5.5\end{cases}  \tag{23}\]  
The resultant eddy viscosity closure, incorporating the previous approximation, results in Equation (24):  
\[\varepsilon=F[k_{1}b_{\text{W}}(U_{\infty}-U_{\text{W}}^{C})+k_{\text{M}}] \tag{24}\]  
where \(k_{1}\) is a constant, which is a property of the shear layer and largely independent of the ambient turbulence and the WT characteristics. [PERSON] conducted wind tunnel experiments for different inflow conditions at low ambient turbulence intensities and found that the value \(k_{1}=0.015\) was adequate. The solution of the [PERSON] model should be calculated considering downstream distances of more than two rotor diameters, where pressure gradients no longer dominate the flow. The wake transversal profile was defined using a Gaussian profile of the form \(1-\overline{u_{x}}/U_{\infty}=D_{\text{M}}\text{exp}(-3.56(r/b_{\text{W}})^{2 })\), where \(D_{\text{M}}=c_{T}-0.05-(16c_{T}-0.5)I_{a}/1000\), \(I_{a}\) [%] is the ambient turbulence intensity, \(c_{T}\) is the thrust coefficient of the WT, and \(b_{\text{W}}=(3.56c_{T}/(8D_{\text{M}}(1-0.5D_{\text{M}})))^{1/2}\). To complement the wake model, [PERSON] provided a simple model for the treatment of the wake meandering [59,281], a phenomenon in which the wake meanders relative to an observer fixed on the ground due to fluctuations in wind direction. The wake meandering results in a reduced centerline velocity deficit measured by the fixed observer when compared with the centerline velocity deficit of a stationary case, because the measurement point sweeps across a region of the wake profile during the averaging period of the measurement. When meandering occurs (e.g., in neutral and unstable atmospheres), the deficit measured by a fixed observed can be determined using Equation (25):  
\[\hat{d}=d_{0}\left(1+7.12\left(\dfrac{\sigma_{\theta}x}{b_{\text{W}}}\right)^ {2}\right)^{-\frac{1}{2}} \tag{25}\]where \(d_{0}\) is the uncorrected centerline velocity deficit and \(\sigma_{\theta}\) is the standard deviation of wind direction fluctuations, measured over the same averaging time as the wake measurements (but not including very short time timescale fluctuations). Finally, the Ainslie model do not account for ground effects, tower effects, and variations in turbulent properties across the wake in its basic formulation. Another relevant wake modeling approach was proposed by [PERSON], who derived two similar wake models in 1988 [25], which were updated by him later in 2009 [287]. The models are based on the turbulent boundary equations (a simplified version of the Reynolds-Averaged Navier-Stokes equations), from which closed-from analytical solutions were obtained for the wake width as well as for the mean wind speed profile within the WT wake. Similarity assumptions and the [PERSON]'s mixing length theory for the description of the turbulent stresses were considered. Both models have different levels of sophistication; the first model only takes into account the dominant terms of the turbulent boundary equations, whereas the second model takes into account the full system of equations. The free-stream airflow was considered turbulent, homogeneous, incompressible and stationary. [PERSON] simplified the original set of parabolic differential equations, expressed in cylindrical coordinates, by assuming an axisymmetric wake, by neglecting some terms (e.g., the pressure terms and distance terms of the form \(x^{-7/3}\)), by conducting analyses of order of magnitude (e.g., the thin shear approximation), and by using similarity assumptions for the description of the wake profile evolution, resulting in the first order model.