# Study on Actuator Line Modeling of Two NREL 5-MW Wind Turbine Wakes
## 3 Uncertainty Analysis of a Typical Airfoil Flow
### Uncertainty Analysis
\begin{array}{*{20}c}{{\phi}_{0}}\\ \alpha\end{array} }\right]=\left[{ \begin{array}{*{20}c}\sum\limits_{i=1}^{n_{ g}}{{w}_{i}}{{\phi}_{i}}\\ \sum\limits_{i=1}^{n_{g}}{{w}_{i}}{{\phi}_{i}}h_{i}\end{array} }\right] \tag{17}\]  
\[\left[{ \begin{array}{*{20}c}1&\sum\limits_{i=1}^{n_{g}}{{w}_{i}}h_{i}^{2}\\ \sum\limits_{i=1}^{n_{g}}{{w}_{i}}h_{i}^{2}&\sum\limits_{i=1}^{n_{g}}{{w}_{i}}h _{i}^{3}\end{array} }\right]\left[{ \begin{array}{*{20}c}{{\phi}_{0}}\\ \alpha_{1}\\ \alpha_{2}\end{array} }\right]=\left[{ \begin{array}{*{20}c}\sum\limits_{i=1} ^{n_{g}}{{w}_{i}}{{\phi}_{i}}\\ \sum\limits_{i=1}^{n_{g}}{{w}_{i}}{{\phi}_{i}}h_{i}^{2}\end{array} }\right] \tag{18}\]  
\[\left[{ \begin{array}{*{20}c}1&\sum\limits_{i=1}^{n_{g}}{{w}_{i}}h_{i}&\sum \limits_{i=1}^{n_{g}}{{w}_{i}}h_{i}^{2}\\ \par{{n}_{g}}&{{w}_{i}}h_{i}^{2}&\sum\limits_{i=1}^{n_{g}}{{w}_{i}}h_{i}^{3} \end{array} }\right]\left[{ \begin{array}{*{20}c}{{\phi}_{0}}\\ \alpha_{1}\\ \alpha_{2}\end{array} }\right]=\left[{ \begin{array}{*{20}c}{{\sum\limits_{i=1 }^{n_{g}}{{w}_{i}}{{\phi}_{i}}}}\\ \sum\limits_{i=1}^{n_{g}}{{w}_{i}}{{\phi}_{i}}h_{i}^{3}\end{array} }\right] \tag{19}\]  
The standard deviations are given by Equations (20)-(22), respectively. \[\sigma_{1}=\sqrt{\frac{\sum\limits_{i=1}^{n_{g}}{{{n}}{{w}_{i}}{{\left({{\phi} _{i}}-\left({{\phi}_{0}}+\alpha{{h}_{i}^{2}}\right)}\right)}^{2}}}{{{n}_{g}}-2}} \tag{20}\]  
\[\sigma_{2}=\sqrt{\frac{\sum\limits_{i=1}^{n_{g}}{{{n}}{{w}_{i}}{{\left({{\phi} _{i}}-\left({{\phi}_{0}}+\alpha{{h}_{i}^{2}}\right)}\right)}^{2}}}{{{n}_{g}}-2}} \tag{21}\]  
\[\sigma_{12}=\sqrt{\frac{\sum\limits_{i=1}^{n_{g}}{{{n}}{{w}_{i}}{{\left({{\phi} _{i}}-\left({{\phi}_{0}}+\alpha{{h}_{i}}{{h}_{i}}+\alpha{{h}_{i}^{2}}\right)} \right)}^{2}}}{{{n}_{g}}-3}} \tag{22}\]  
(2) Determine the following data range parameter to assess the quality of the fit used to obtain the error estimate \(\epsilon_{\phi}\)  
\[\Delta_{\phi}=\frac{{{\left({{\phi}_{i}}\right)}_{\max}}-{{\left({{\phi}_{i}} \right)}_{\min}}}{{{n}_{g}}-1} \tag{23}\]  
(3) Determine the safety factor  
If \(0.5\leq p<2.1\) and \(\sigma<\Delta_{\phi}\), \(F_{\rm s}=1.25\); otherwise, \(F_{\rm s}=3\). (4) Obtain the uncertainty For \(\sigma<\Delta_{\phi}\):  
\[U_{\phi}(\phi_{i})=F_{S}\varepsilon_{\phi}(\phi_{i})+\sigma+\left|\phi_{i}-\phi _{fit}\right| \tag{24}\]  
For \(\sigma\geq\Delta_{\phi}\):  
\[U_{\phi}(\phi_{i})=3\frac{\sigma}{\Delta_{\phi}}\Big{(}\varepsilon_{\phi}( \phi_{i})+\sigma+\left|\phi_{i}-\phi_{fit}\right|\Big{)} \tag{25}\]  
The lift (\(C_{L}\)) and drag (\(C_{d}\)) coefficients with different attack angles are shown in Table 3.  
By using the LSR uncertainty estimator which is presented in the previous section, the convergence of lift and drag coefficients with the grid refinement are shown in Figures 4 and 5, respectively. The lift coefficient is nearly zero when the angle of attack is zero, so these lift coefficients are not analyzed. The error bars for the mesh ratio equal to 1.0, 2.0, and 4.0 are also included in those figures. Then, Table 4 displays the uncertainty of the lift and drag coefficient with the grid refinement for different angle of attacks. It is clear that the error and uncertainty are reduced with the mesh refinement. Finally, we can bound the exact value (\(\phi_{exact}\)) and the interval is:  
\[\phi_{i}-U_{\phi_{i}}\leq\phi_{exact}\leq\phi_{i}+U_{\phi_{i}} \tag{26}\]  
Table 5 shows the interval of the exact lift and drag coefficient with the grid refinement for different angle of attacks. Figure 6 shows the uncertainty of the drag and lift coefficient with the grid refinement for different attack angles. Considering the experiment results of NACA 0012 by [PERSON] [29] and [PERSON] [30], the results are well bounded by the OpenFOAM uncertainty interval.