## 3 Results
### Integral Forces  
In the following, integral forces resulting from sectional forces from Section 3.2 are discussed. Time series of the aerodynamic thrust and torque are shown in Figure 12. As can be observed, the thrust as well as the torque evolution is very similar for BR and AL. With the definition of the correlation coefficient  
\[\rho=\frac{\langle\left(F_{1}(t)-F_{1}\right)\left(F_{2}(t)-\bar{F}_{2}\right) \rangle_{t}}{\sigma_{1}\sigma_{2}} \tag{4}\]  
where \(F_{1}\) and \(F_{2}\) are the considered forces and \(\sigma_{1}\) and \(\sigma_{2}\) the corresponding standard deviations, the correlation is calculated as 0.90 and 0.93 for thrust and torque respectively, even if the distribution of sectional forces and the wake dynamics are more different (see Figures 4-6). A comparison between BR and BEM reveals a much smaller correlation with values of 0.08 and 0.09 for thrust and torque. This supports the argument that the wind dynamics are handled in a very different way for BEM and CFD simulations for the current setup. In Table 2 the means, standard deviations and EFLs of torque and thrust for all three methods are illustrated with their corresponding errors. Those errors are obtained by separating the corresponding time series' into five distinct equally sized ensembles and calculating the ensemble standard deviation. For comparison reasons the ratios of the calculated quantities for AL and BR as well as BEM and BR are shown, because the BR case is assumed to be the highest fidelity approach. For aerodynamic thrust and torque AL and BEM show lower values than the BR case with a minimum ratio of 91.7%, which can also be estimated from the sectional forces in Figure 5.  
Analog to the relation between mean sectional forces and mean integral forces, the standard deviations of tangential and axial forces in Figure 6 correspond to a standard deviation in the thrust and torque. The standard deviation for the AL case is for both quantities approximately 10% smaller than for the BR simulation, because of the missing variance in the root and tip region. Especially for the torque the tip region plays a significant role. In contrast to that, for the BEM simulation an approximately 20 to 30% increase in the standard deviation is obtained which is also very clear from the much higher sectional standard deviation except at the root. A reason for the high standard \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & **BR** & **AL** & **BEM** & **AL/BR [\%]** & **BEM/BR [\%]** \\ \hline \multirow{3}{*}{Thrust [kN]} & mean & 761 \(\pm\) 7.3 & 698 \(\pm\) 4.0 & 729 \(\pm\) 9.6 & 91.7 & 95.8 \\  & \(\sigma\) & 37.9 \(\pm\) 4.3 & 33.9 \(\pm\) 2.6 & 49.7 \(\pm\) 6.5 & 89.4 & 131.1 \\  & EFL & 147.6 \(\pm\) 9.7 & 136.7 \(\pm\) 4.2 & 190.1 \(\pm\) 8.0 & 92.6 & 128.8 \\ \hline \multirow{3}{*}{Torque [MNm]} & mean & 4.43 \(\pm\) 0.10 & 4.32 \(\pm\) 0.08 & 4.35 \(\pm\) 0.13 & 97.5 & 98.2 \\  & \(\sigma\) & 0.495 \(\pm\) 0.044 & 0.452 \(\pm\) 0.038 & 0.599 \(\pm\) 0.061 & 91.3 & 121.0 \\ \cline{1-1}  & EFL & 1.90 \(\pm\) 0.15 & 1.90 \(\pm\) 0.13 & 2.29 \(\pm\) 0.12 & 99.6 & 116.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thrust and torque comparison by mean, standard deviation \(\sigma\) and EFL between BR, AL and BEM. Figure 12: Time series for (**a**) thrust and (**b**) torque for Blade Resolved (BR), Actuator Line (AL) and Blade Element Momentum (BEM) methods (from top to bottom). deviation could be the equilibrium wake model used here, where the wake dynamics are not captured in a proper way. The relation between EFL and the standard deviation of integral forces is shown in Figure 13 where we additionally added the diagonal line governed by \(EFL=4\sigma\) like for sectional forces in Figure 11. Sectional as well as integral forces seem to follow the same linear trend. Further analysis of the EFL by the underlying range histograms in Figure 14 shows that for the thrust and torque force all three methods have in general a similar trend.