# Study on Actuator Line Modeling of Two NREL 5-MW Wind Turbine Wakes
## 4 A Single NREL 5-MW Turbine Case
### Validation of Numerical Model  
In order to quantify the influence of the mesh quality, a series of computations with variable grid numbers are carried out for \(U_{0}\) = 11.4 m/s, which are 375,432, 744,792, 2,903,144, and 5,958,336, respectively. As there are no experimental data for a full scale NREL 5-WM wind turbine, it adopts the results of FAST as the benchmark data, which can be referred to in [21]. The simulation time is set 30 s. The power and thrust coefficients are introduced and defined as below:  
\[C_{p}=\frac{P}{0.5\rho U_{0}^{3}\pi R^{2}} \tag{27}\]  
\[C_{T}=\frac{T}{0.5\rho U_{0}^{2}\pi R^{2}} \tag{28}\]  
where \(P\) is the mechanical power; \(T\) is the thrust on the blades; and \(R\) is the radius of the rotor blade and \(R\) = 63 m, \(\rho\) = 1.225 kg/m\({}^{3}\). According to reference [21], \(C_{p}\) = 0.4418, \(C_{t}\) = 0.8099 when \(U_{0}\) = 11.4 m/s. The errors of \(C_{p}\) and \(C_{t}\) of different cell numbers are shown in Table 6. The errors of the power and thrust are compared in Figure 9, which shows that when the numbers of the grid increase, the differences in the power and thrust begin to stabilize. Considering the computational efficiency, the total number of 2,903,144 cells should be high enough to obtain accurate results, considering the errors of \(C_{p}\) and \(C_{t}\) which are 4.858% and 5.981%, respectively. Figure 8: Computational domain mesh of A with refinement. The blue mesh is the fluid domain; the red mesh is the part of wind turbine. Figure 7: The schematic setup of computational domain: (**a**) Top view; (**b**) Side view. To examine the accuracy of different wind speeds, the thrust and power are computed and compared with the FAST code [21], which are shown in Figure 10. The results agree with FAST excellently for the wind speeds up to 11.4 m/s. The quantitative values of interest and errors are shown in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Cell** & \(C_{p}\) & **Error (\%)** & \(C_{t}\) & **Error (\%)** \\ \hline
375,432 & 0.4678 & 5.864 & 0.8665 & 6.987 \\
744,792 & 0.4642 & 5.047 & 0.8599 & 6.170 \\
2,903,144 & 0.4634 & 4.858 & 0.8584 & 5.981 \\
5,958,336 & 0.4627 & 4.715 & 0.8572 & 5.838 \\ \hline \end{tabular}
\end{table} 
Table 6: The mesh independence test. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Wind** & **Thrust of** & **Power of** & **Thrust of** & **Power of** & **Thrust** & **Power** \\
**Speed (m/s)** & **TurbinesFoam (kN)** & **TurbinesFoam (MW)** & **FAST (kN)** & **FAST (MW)** & **Errors (\%)** & **Errors (\%)** \\ \hline
5.0 & 223.578 & 0.4400 & 264.9066 & 0.4236 & 15.60 & 3.88 \\
6.0 & 288.945 & 0.8265 & 331.8101 & 0.7652 & 12.92 & 8.01 \\
7.0 & 357.735 & 1.3088 & 397.2068 & 1.2240 & 9.94 & 6.92 \\
8.0 & 433.465 & 1.9026 & 467.6329 & 1.8251 & 7.31 & 4.25 \\
9.0 & 548.601 & 2.7090 & 577.9532 & 2.5955 & 5.08 & 4.37 \\
10.0 & 677.285 & 3.7160 & 691.1475 & 3.5587 & 2.01 & 4.42 \\
11.4 & 722.471 & 5.0363 & 803.9102 & 5.000 & 3.91 & 0.73 \\ \hline \end{tabular}
\end{table} 
Table 7: Comparison of the computed power and thrust between FAST and turbinesFoam at different wind speeds. Figure 10: Comparison of the computed power and thrust between FAST and turbinesFoam at different wind speed: (**a**) Thrust, (**b**) Power. Figure 9: The mesh independence test: (**a**) Error distributions of \(C_{p}\); (**b**) Error distributions of \(C_{t}\). As for the high wind speed, the local Reynolds number of the blades is calculated at the rated wind speed, as defined below:  
\[Re=\frac{U_{ blade}c}{\upsilon} \tag{29}\]  
where \(c\) is the chord length; \(\upsilon\) is the air kinematic viscosity; \(U_{ blade}\) is the local velocity, and can be calculated by \(U_{ blade}=\sqrt{U_{axial}^{2}+\left(\omega r\right)^{2}}\), where \(U_{axial}\) is the axial velocity; and \(\omega\) is the rotating angular velocity.