## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
For the description of single WT wakes, they considered expressions derived from the Betz-Lanchester theory, which link the thrust and power coefficient of the WT to the flow speed deficit in the wake. A cylindrical control volume with constant cross-sectional area equal to the wake area, with horizontal axis parallel to the mean wind velocity vector, and with no flow across its surface was used to perform a momentum balance between the free-stream flow and the WT wake, considering a simplified integral version of the momentum equation, to determine the momentum deficit in the wake. The simplified momentum equation disregarded pressure gradients (e.g., the control volume extends upwind, \(0.5D<x<D\), and downwind, \(2D<x<3D\), far enough for the pressure to become equal to the free-stream pressure), the gravity force, viscous forces (e.g., shear forces in the cylinder surface were neglected), rotational effects, and tower effects. The free-stream wind speed was considered constant, homogeneous, stationary, and incompressible. The wake was considered axisymmetric and its transversal profile constant (_i.e._, a top-hat profile was considered). The model defines the wake wind speed (\(U_{\mathrm{W}}\)) as shown in Equation (33):  
\[\frac{U_{\mathrm{W}}(x)}{U_{\infty}}=\frac{1}{2}\pm\frac{1}{2}\sqrt{1-2\,\frac {A}{A_{\mathrm{W}}(x)}\,c_{T}} \tag{33}\]where \(A\) [m\({}^{2}\)] is the swept area of the WT rotor, \(A_{\rm W}(x)\) is the area of the wake, which is assumed to be a function of the downstream distance (\(x\) [m]), and \"+\" in Equation (33) applies when \(0\leq c_{T}\leq 0.75\) and \"-\" when \(c_{T}>0.75\). The wake area after it initially expands (assumed to happen at the rotor position, so \(A_{\rm W}(x=0)=A_{\rm W}^{I}\)) is given by \(A_{\rm W}^{I}=\beta A\), where \(\beta=(1+\sqrt{1-c_{T}})/(2\sqrt{1-c_{T}})\). The evolution of the wake diameter downstream the WT was defined as shown in Equation (  
where \(s_{f}\) is the dimensionless distance between neighboring rows, \(x_{0}\) and \(h_{0}\) are the axial distance and wake height when the neighboring wakes start to merge. The third scenario is when the wind farm is infinitely large and the wake-affected flow is in balance with the atmospheric boundary layer. The interested reader is referred to [283,298] for a complete description of the model, which links the surface wind speed to the geostrophic wind speed by defining two flow layers divided at the WT hub height and by introducing the geostrophic drag law. The most widely used EWM is the Jensen-Katic model [21,23,282], as shown in \[\hat{d}=d_{0}\left(1+7.12\left(\dfrac{\sigma_{\theta}x}{b_{\text{W}}}\right)^ {2}\right)^{-\frac{1}{2}} \tag{25}\]0 The Jensen-Katic wake model defines the velocity deficit as shown in Equation (39):  
\[\hat{d}=d_{0}\left(1+7.12\left(\dfrac{\sigma_{\theta}x}{b_{\text{W}}}\right)^ {2}\right)^{-\frac{1}{2}} \tag{25}\]1  
where \(I_{a}\) [%] is the ambient turbulence intensity, representing the effect of turbulent forces on the WT wake recovery and wake expansion, \(c_{T}\) is the thrust coefficient, and \(D\) [m] is the WT rotor diameter. \(I_{a}\) is sometimes written as \(I_{a}=2\alpha\), where \(\alpha\) denotes the momentum entrainment constant (or wake decay coefficient) modulating both the wind speed recovery and the wake expansion, as first proposed in [23,298] and as shown in Equation (40):  
\[\hat{d}=d_{0}\left(1+7.12\left(\dfrac{\sigma_{\theta}x}{b_{\text{W}}}\right)^ {2}\right)^{-\frac{1}{2}} \tag{25}\]2  
where z is the height above ground [m], and z\({}_{0}\) is the roughness length [m], representing the height above ground level at which a zero wind speed is found. Equation (40) implies that the wake recovery and expansion is lower with increasing height, since lower ambient turbulence intensities are found far from the surface.