## 3 Actuator Line Numerical Analysis
### Wind Turbine Rotor Numerical Analysis  
The AL numerical parameters regarding the WT simulations have been tested on an axial case at a 11-m/s uniform wind speed. The blades are evenly distributed and considered to be rigid. Structural deflections alongside prebend, cone, and tilt angle of the blades are neglected, so that the configuration is as simple as possible for this parametric study. As a result, we end up with a purely axisymmetric case. In order to reduce computational cost, only one blade has been resolved in the rotating frame, with the use of periodic boundaries and in steady-state condition. The domain is a cylindrical section of 20 rotor diameters (\(20D\)) length (see Figure 3) and \(10D\) radius (see Figure 4). Structured and uniform meshing under maximum grid resolution is applied in a thin region surrounding the actuator line. The structured area extends adequately around the actuator line, so that maximum accuracy is acquired in resolving the Gaussian projection of loads (see Figure 5). The characteristic length of this region is denoted by \(\Delta x\). Moreover,  
Figure 2: Local bases vectors of interconnection points. an unstructured and slightly coarser region is considered in order to take into account the near wake impact on the aerodynamic forces. This area extends up to \(1D\) upstream, \(3D\) downstream, and \(1D\) radially from the actuator line, so that the wake expansion is regarded properly. The characteristic length of this region is denoted by \(\Delta w\). The first priority is to determine \(\Delta x\). The results from the analysis are presented in Figure 6 and Table 1. Each column lists the extracted power in MW, with respect to different values of \(\Delta x\) and for various strategies to define the Gaussian kernel value \(\epsilon\). The standard approach in defining \(\epsilon\) based solely on \(\Delta x\)[3] fails to provide a grid independent solution. Since the blade geometry is not resolved, flow does not perceive the presence of a solid body. In order to perform a more accurate simulation that takes into account the blade  
Figure 4: Periodic grid. Axial view. Figure 3: Periodic grid. Lateral view. geometry, the Gaussian distribution of loads is made using varying kernel \(\epsilon\). The value of \(\epsilon\) is independent of the changes in the grid size \(\Delta x\) and is proportional to the chord \(c\) of each blade section [13]. The proportionality factor 0.47 is chosen so that the Gaussian projection of the aerodynamic forces extends only up to \(1c\) around the emission points. The projection width is \(3\sigma=\frac{3}{\sqrt{2}}\epsilon\). If the projection width is decided to extend up to \(1c\) around the emission points, then \(c=\frac{3}{\sqrt{2}}\epsilon\) or \(\epsilon\approx 0.47c\). This approach can resolve in detail the blade geometry and thus allow very fine grid resolution in the AL vicinity. However, the reduction of the chord close to the tip results in very small values of \(\epsilon\), even less than \(\Delta x\), leading to numerical singularities. Therefore, a lower bound is set for \(\epsilon\), \(\epsilon_{min}=2\Delta x\), as proposed by [3]. Based on the results in Table 1, \(\Delta x=R/90\) seems to be a good trade-off between accuracy and computational cost. Figure 5: Periodic grid. Gaussian projection of aerodynamic forces. Figure 6: Grid dependency analysis. Power [MW] vs grid characteristic length \(\Delta x\) [m]. The outcome of the investigation of the effect of the number of strips (\(\#strips\)) (AL control points are located at the center of each strip) and their distribution pattern in the power output is shown in Table 2 and Figures 7 and 8. As a first step, the AL control points are uniformly distributed along the blade, thus leading to a constant strip length (\(\Delta r\)). The parameter \(\#strips\) is not independently studied, except through the correlation of \(\Delta r\) with \(\Delta x\). In Table 2, it is clear that \(\Delta r\leq 3\Delta x\) is a reasonable rule of thumb to define \(\Delta r\).