## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling
The shear layer expands outward (_i.e._, \(r>R_{\mathrm{W}}\)) and inward (_i.e._, \(r<r^{\prime}\)) from the edge of the wake as it develops in the streamwise direction (\(x\) [m]). It is considered that the near wake region terminates when the outer shear layer meet the centerline of the wake, so the center velocity deficit starts to decrease with increasing distance from the WT. The downstream extent (\(x_{c}\) [m]) of the near wake can be computed using Equation (18):  
\[x_{c}=r_{0}\left(\left(\frac{\mathrm{d}R_{\mathrm{W}}}{\mathrm{d}x}\right)_{a }^{2}+\left(\frac{\mathrm{d}R_{\mathrm{W}}}{\mathrm{d}x}\right)_{m}^{2}+ \left(\frac{\mathrm{d}R_{\mathrm{W}}}{\mathrm{d}x}\right)_{\lambda_{\mathrm{W }}}^{2}\right)^{-\frac{1}{2}} \tag{18}\]  
where \((\mathrm{d}R_{\mathrm{W}}/\mathrm{d}x)_{a}=I_{a}/0.36\), \((\mathrm{d}R_{\mathrm{W}}/\mathrm{d}x)_{m}=0.27(0.214+0.144m_{\mathrm{W}})^{ 1/2}(m_{\mathrm{W}}-1)/(m_{\mathrm{W}}+1)\), and \((\mathrm{d}R_{\mathrm{W}}/\mathrm{d}x)_{\lambda_{\mathrm{W}}}=(c_{T}\,\sigma _{\mathrm{L}}\,m_{\mathrm{W}}\,\lambda_{\mathrm{W}}^{3}/(6m_{\mathrm{W}}+6))^ {1/2}\) are the wake growth contributions due to ambient turbulence, due to the turbulence generated by the wake shear, and due to the mechanically added turbulence from the WT rotor. Moreover, \(\sigma_{\mathrm{L}}\) is the rotor solidity, \(I_{a}\) [%] is the ambient turbulence intensity, \(c_{T}\) is the thrust (or drag) coefficient of the blade profile, and \(\lambda_{\mathrm{W}}\) is the tip speed ratio. The wake growth rate is assumed linear and given by \(\mathrm{d}R_{\mathrm{W}}/\mathrm{d}x=(r_{c}-r_{0})/x_{c}\). In the transition region, the velocity deficit is given by a linear combination of profiles at the extremes of the transition region. At the beginning of the transition region, the radius of the wake is \(r_{c}\) and at the end of the transition region is \(r_{F}=r_{0}/(0.134+0.124m_{\mathrm{W}})^{1/2}\). The extent of the transition region can be calculated using Equation (19):  
\[x_{F}=\frac{x_{c}(0.214+0.144)^{\frac{1}{2}}\left(1-(0.134+0.124m_{\mathrm{W }})^{\frac{1}{2}}\right)}{\left((0.134+0.12 The wake growth rate at the transition region is the same as in the near wake region, given by \(\mathrm{d}R_{\mathrm{W}}/\mathrm{d}x=(r_{c}-r_{0})/x_{c}\).  
In the far wake, the velocity deficit is given by Equation (20):  
\[U_{\mathrm{Def}}=U_{\mathrm{Def}}^{\mathrm{CM}}\!\left(\frac{-0.258m_{\mathrm{W} }+\left((0.258m_{\mathrm{W}})^{2}+0.536(1-m_{\mathrm{W}})\left(\frac{r_{0}}{R_{ \mathrm{W}}}\right)^{2}\right)^{\frac{1}{2}}}{0.268(1-m_{\mathrm{W}})}\right) \!\left(1-\left(\frac{r}{R_{\mathrm{W}}}\right)^{1.5}\right)^{2} \tag{20}\]  
where \(U_{\mathrm{Def}}^{\mathrm{CM}}\) is the velocity deficit at the wake core. The wake growth rate in the far region is governed only by the ambient turbulence intensity, resulting in \(\mathrm{d}R_{\mathrm{W}}/\mathrm{d}x=I_{a}/0.36\). Thus the radius of the far wake at any point downstream (\(x\geq x_{F}\)) can be calculated as \(R_{\mathrm{W}}=r_{F}+(I_{a}/0.36)(x-x_{F})\). The ground has the effect of reducing vertical downward wake growth, both by suppression of the turbulence near the ground and by the direct presence of the ground. The effects of the ground were modeled by classical reflection techniques (or imaging techniques) [281], in which an imaginary WT is added below the ground (in a symmetrical fashion), so the wakes of both WTs expand beyond the ground and then their velocity deficits are added, thus the momentum deficit of the original wake is conserved. However, as pointed out by [PERSON] _et al._[281], this approach is not completely valid as the wake velocity deficit is not actually conserved because of the friction (viscous) effects at the surface. To overcome this issue, [PERSON] _et al._[281] considered an antisymmetric WT wake, so that velocity deficits are subtracted and result in a zero perturbation at the ground.