## 3 Relevant Research in Wind Farm Design and Optimization
### Sub-Modeling
#### 3.2.1 Wind Turbine Wakes and Turbulence Modeling  
Wake and turbulence modeling [57; 58; 59; 281] are two of the most challenging research topics in the field of fluid mechanics and are considered unsolved problems in classic physics. However, a significant amount of contributions have been performed in the field during the last four decades resulting in two different types of WT wake modeling approaches. The first modeling approach is based on Engineering Wake Models (EWMs) [18; 19; 21; 23; 282; 283; 284], which, in the present context, are analytical functions derived from empirical regressions, correlations, or simplified Newtonian-based laws of mass flow and momentum conservation, which describe in a simplified way the WT energy conversion and the WT wake evolution processes (e.g., the evolution of the streamwise wind speed at any radial position measured from the WT hub height) while making certain assumptions regarding the near and far wake expansion, the wake speed recovery, and the wake merging process, which in turn are a function of the WT aerodynamics and operation characteristics and the local atmospheric condition. The second modeling approach consist of advanced wake models that are based on the Reynolds-Averaged Navier-Stokes (RANS) equations or Large Eddy Simulations (LES), both complemented with turbulence models and subgrid-scale models respectively, and are combined with the concepts of actuator line and actuator disk approaches for the WT rotor modeling [285; 286; 57; 287; 288; 25]. The second approach (based on either the RANS or the LES approaches) has shown improved accuracy when describing the evolution and the side effects of WT wakes (in both steady and unsteady conditions) but requires a significant amount of computational resources, thus is commonly not adopted by wind farm designers nor by most of the authors of the reviewed works. The first approach (based on EWMs), although simple, physically limited and computationally inexpensive, has proven to describe the wind speed deficit in the far wake zone with reasonable accuracy [60; 65; 66; 67; 68]. However, EWMstypically overpredict wake effects [58, 60, 66, 281, 289]. Moreover, some commercial software packages (e.g., WindFarmer, WindFarm, WindPro, OpenWind) complement EWMs with Engineering Turbulence Models (ETMs) [288, 290, 291, 292, 293, 294, 295], which describe the net TI evolution within the WT wake. Nonetheless, none of the reviewed works considered the inclusion of ETMs, since TI effects on the energy conversion process are often neglected. The basic mathematical structure of the EWMs is shown in Equation (15):  
\[U_{\text{W}}(x,r)=U_{\infty}\big{(}1-U_{\text{Def}}(x,r)\big{)} \tag{15}\]  
where \(U_{\text{W}}(x,r)\) is the horizontal wind speed in the wake at some axial (\(x\) [m]) and radial (\(r\) [m]) position downwind from the WT hub, and \(U_{\text{Def}}(x,r)\) is the velocity deficit, which represents the fraction of the free-stream wind speed that was affected during the energy conversion process. The wind speed in the wake exhibits a natural recovery due to the turbulent and viscous interactions of the wake with the unaffected neighboring free stream. The wake recovery is commonly modeled through the use of wake decay coefficients, which are assumed to be exclusively dependent on the ambient turbulence intensity levels [19, 23, 282, 283]. The differences between common EWMs are essentially the way they approximate \(U_{\text{Def}}\). The first wake model, categorized as a Kinematic wake model [59], was proposed by [PERSON] in 1977 [19, 296, 297], and later was updated by him in 1982 [24]. He assumed that the wake of a single WT can be decomposed into two different main regions: the near and the far wake regions. For each region, self-similar wind speed profiles were assumed. Between these two regions is a transition zone, in which the flow profile changes from one self-similar profile to another.