## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
The wake is assumed to grow by turbulent entrainment at its edge, which introduces free stream momentum as well as mass into the wake, so these quantities are not conserved in the wake. Nonetheless, the momentum deficit in the wake, considering an idealistic constant pressure field, is conserved. The momentum deficit is defined by the integral shown in Equation (16) across the wake:  
\[D=\int\ \frac{U_{\text{W}}}{U_{\infty}}\Big{(}1-\frac{U_{\text{W}}}{U_{\infty}} \Big{)}\,r\,\text{d}r \tag{16}\]  
where the free-stream wind speed (\(U_{\infty}\)) was assumed constant and homogeneous. Equation (16), along with the wake growth (\(\text{d}R_{\text{W}}/\text{d}x\)), and the wake profile at a radius \(r\) [m], measured from the wake centerline and given by a function of the form \(F(r/R_{\text{W}})=U_{\text{W}}/U_{\infty}\) (where \(R_{\text{W}}\) is the effective outer radius of the wake), are the most important mechanisms modeling the wake. The momentum deficit of the wake can be computed from the initial state of the wake, which can be directly related to the thrust force, or the thrust coefficient (\(c_{T}\)), of the WT rotor and its output power. The thrust coefficient of the WT represents the normalized axial force produced by the wind on the WT rotor, or conversely, of the rotor on the wind. The thrust coefficient can be viewed as a basic parameter representing the aero-structural loading of the WT and is dependent on the same variables as the power coefficient (e.g., \(U\), \(\rho\), TI and \(\Delta\theta\)), as discussed in Section 3.1.1. Similarly to the power coefficient, the thrust coefficient is strongly influenced by the adopted control strategy and is commonly reported by WT manufactures as an empirical function of the wind speed at hub height, while assuming standardized inflow conditions and disregarding any other dependence on the previously mentioned atmospheric variables. [PERSON] assumed that turbulence near the edge of the wake controls the lateral growth. This turbulence is a functional combination of: (1) the ambient turbulence intensity; (2) the turbulence generated by the shear due to the wind speed gradients within the wake itself (or wake shear); and (3) the mechanically added turbulence generated by the viscous interactions of the free-stream flow and the WT rotor (_i.e._, the mechanically generated turbulence). These terms have a different relative importance in the different regions of the wake. The ambient turbulence (assumed isotropic and uniform in the free stream) controls the wake growth after the mechanically added turbulence and the turbulence generated by the wake shear have decayed. Thus in the near wake region, the three turbulent components drive the wake growth, and in the far region only the ambient component drives the wake growth. At the beginning of the near wake region, the radius of the wake is given by \(r_{0}=R\sqrt{(m_{\mathrm{W}}+1)/2}\) and at the end of the near wake region is given by \(r_{c}=r_{0}/(0.214+0.144m_{\mathrm{W}})^{1/2}\), where \(R\) [m] is the WT rotor radius and \(m_{\mathrm{W}}=U_{\infty}/U_{\mathrm{CW}}\) is the inverse of the normalized wake speed at the wake core (\(U_{\mathrm{CW}}\)). In the near wake region, the transversal wake profile has a potential core of uniform velocity (\(U_{\mathrm{CW}}\)) and radius \(r^{\prime}\). Outside the wake core (e.g., \(r^{\prime}<r<R_{\mathrm{W}}\)), the transversal wake profile is defined by a characteristic shear layer, which in turn defines the wake velocity deficit, as shown in Equation (17):  
\[U_{\mathrm{Def}}=\left(\frac{U_{\infty}(m_{\mathrm{W}}-1)}{m_{\mathrm{W}}} \right)\left(1-\left(1-\eta_{\mathrm{W}}^{1.5}\right)^{2}\right) \tag{17}\]  
where \(\eta_{\mathrm{W}}=(R_{\mathrm{W}}-r)/(R_{\mathrm{W}}-r^{\prime})\), so that \(R_{\mathrm{W}}-r^{\prime}\) is the width of the turbulent mixed zone in the transversal wake profile.