## 3 Actuator Line Numerical Analysis
### Wind Turbine Rotor Numerical Analysis
Nevertheless, since \(\#strips\) adds no significant computational cost, the ratio \(\Delta r=1.5\Delta x\) will be used in the following WT aeroelastic simulations, in order to ensure a smooth representation of the radial distribution of aerodynamic forces. Figures 7 and 8 present a comparison between a uniform distribution of control points (\(\Delta r=1.5\Delta x\)) with a non-uniform one with the same number of points. Non-uniformity implies a coarse discretization close to the root that gradually gets finer as we approach the tip. As seen in Figures 7 and 8, where the normal and tangential force along the blade span are plotted for the two different discretizations, the two distributions almost coincide due to the large number of points used. Uniform distribution better resolves the loads at the radius of \(\approx\)30 m where a local peak appears in particular in the tangential force. No advantage is observed at the tip region when employing denser distribution of points. It is easy to understand that due to the large number of control points along the blade span, a non-uniform distribution adds no extra benefit. Consequently, constant strip length \(\Delta r\) is adopted for the sake of simplicity. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{e=2\Delta x}\) & \(\mathbf{e=0.47c}\) & \(\mathbf{e=max(2\Delta x,0.47c)}\) \\ \hline \(\Delta x=R/30\) & \((151\) _thousand cells_) & \(+31.4\%\) & \(+67.5\%\) & \(+9.2\%\) \\ \(\Delta x=R/60\) & \((283\) _thousand cells_) & \(+22.6\%\) & \(-37.6\%\) & \(+2.6\%\) \\ \(\Delta x=R/90\) & \((495\) _thousand cells_) & \(+15.1\%\) & \(-20.7\%\) & \(-0.8\%\) \\ \(\Delta x=R/120\) & \((884\) _thousand cells_) & \(+9.5\%\) & \(-10.1\%\) & \(-1.5\%\) \\ \(\Delta x=R/150\) & \((1.500\) _million cells_) & \(+4.5\%\) & \(-3.9\%\) & \(-0.9\%\) \\ \(\Delta x=R/180\) & \((2.290\) _million cells_) & \(7.921\) MW & \(9.202\) MW & \(9.362\) MW \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variation of power [MW] with increasing the structured region characteristic length \(\Delta x\) [m]. Reference power corresponds to minimum value \(\Delta x=R/180\). Figure 7: Normal force radial distribution. Comparison of uniform and tip dense distribution of strips. Another crucial aspect is the grid type employed. Fully unstructured, structured, and hybrid grids can be used. Structured grids are orthogonal but with high aspect ratio cells. On the other hand, unstructured grids result in a good aspect ratio, but orthogonality is lost. The projection method used in the AL approach can be sensitive to the type of grid. For this reason, mesh structure needs to be properly examined. In Figures 9-11, four different grid setups are visualized, whereas in Table 5 the total number of grid cells and the power outcome of each setup are listed. In Figure 9 a fully structured grid (_\"setup1\"_) is shown with 8.9 million cells. The area nearby the rotor is kept uniform and fine with a characteristic length of \(\Delta x=R/90\), followed by a gradual coarsening up to \(\Delta w=10\Delta x\) at the limits of the near wake region and a steeper one up to the outer boundaries of the mesh. Figure 10 illustrates two meshing strategies similar to the one used in the periodic, steady-state simulations. Particularly, in _\"setup2\"_ a fine and structured area is considered around the blade with length \(\Delta x=R/90\), while the rest of the domain is unstructured. Again the near wake region has a characteristic length of \(\Delta w=10\Delta x\) and the total amount of grid cells is 2.9 million. The only difference in _\"setup3\"_ is that the AL vicinity is unstructured, thus exploding the number of cells to 6.2 million. Finally, a hexahedral mesh (_\"setup4\"_) is shown in Figure 11 with a total number of 1.9 million cells. The characteristic lengths of \(\Delta x=R/90\) close to the rotor and \(\Delta w=10\Delta x\) in the near wake region have been maintained in this setup. Based on Table 5, all 4 mesh setups produce similar results.