## 6 Results of Wake Simulations
### Model Assessment  
In this section, we adopt the root-mean-square error (RSME) to analyze the accuracy of the CFD predictions using various turbulence models [58]:  
\[\text{RSME}=\sqrt{\frac{\sum_{1}^{n}(y_{\text{CFD}}-y_{\text{EXP}})^{2}}{n}} \tag{47}\]  
Figure 17: Lateral added turbulence in wakes of Danwin 180 kW wind turbines at hub height under the stable condition; \(\sigma_{u}\) and \(\sigma_{u0}\) represent the standard derivations of wind speed in wakes and in the atmosphere, respectively. All turbulence models are combined with AD based on thrust coefficient.  
where \(n\) is the number of measurement points in the evaluation, \(y_{\text{EXP}}\) is the experimental data, and \(y_{\text{CFD}}\) is the simulated values from CFD. Note that measurement points out of the wake range are not applied to calculate the RSME, and the wake width is set \(2D\) due to wake simulations.  
The RMSE analysis of the wake deficit is presented in Figures 18 and 19. Over a range of 0.04 to 0.2, the lower the RSME value, the better the behavior of the numerical model. According to Figure 18, it can be depicted that the proposed model, either based on BEM in Case 1 or based on the thrust coefficient in Case 2, has the best performance with the lowest RMSE values, except for 4-5\(D\) in Case 1. The AM, El-Askary, and Laan models have similar performance in the wake prediction, but the El-Askary model has lower RMSE values among the various turbulence models based on MOST. As compared to AM and Laan model, the El-Askary model could reduce the RSME of the lateral wake deficit by 0.03 in Case 1 and 0.01 in Case 2. However, this improvement is smaller than that of the proposed model: introducing the new similarity functions into the Laan model could reduce the RSME of the lateral wake deficit by 0.05 averagely (Figure 18). The LES approach presented in [PERSON] et al. [1] predicts a decreased RSME of the lateral wake deficit with the distance downstream of the rotor. Note that all test models, except for the LES approach, have poor performance in predicting wake structure at \(5D\) downstream of the rotor. This could be a result of the downslope terrain or uncertainties due to LiDAR measurements [1].  
Using BEM to distribute force through the rotor significantly improves near wake prediction at 1-3\(D\) downstream of the rotor in Case 1: the RSME of vertical wake deficit predicted by the proposed model using BEM approach is lower, 0.02-0.05, than the proposed model using the \(C_{T}\) approach. However, it should be noted that the proposed model using \(C_{T}\) approach has a similar performance in the wake prediction with that using BEM approach at 4\(D\)-5\(D\) downstream of the rotor. This suggests that the distribution of the momentum source through the rotor only affects near wake structures up to 3\(D\) and that one can expect similar accuracy of wake prediction for far wake regardless of which method to distribute the force through the rotor.  
The RMSE analysis of the added turbulence is presented in Figure 20. The RMSE of the added turbulence is observed to decrease with the distance downstream the rotor. Turbulence intensities are significantly underpredicted at 4.2\(D\), where the RMSE of the added turbulence is \(\sim\)4. More than 4.2\(D\) the RMSE of the added turbulence predicted by the proposed model is lower by 30% than those of models based on MOST. This indicates that introducing the new similarity functions into the Laan model can eliminate the side effects of the breakdown of MOST on wake modeling.  
Figure 18: RSME of the wake deficit at hub height under the stable condition: “-CT” represents models using AD based on the thrust coefficient.