*3.2. The E*ff*ect of Gaussian Width*

In this section, the simulation results using different value are compared with the experimental results to study the influence of the Gaussian width. Figure 8 shows the comparison of the torque result between the experiment and simulations. According to the discussion above, the value is related to the chord length (chord length of blade tip when using standard regularization kernel) to make this study more referential. The lift and drag coefficient data obtained by Sarlak [30] are used in this study and the data were gathered at Re = 100,000, which is a little higher than the Reynolds number of this study. It should be noticed that the cross-section changes from a circle to airfoil S826 in the transition section of the model turbine blade and there are no aerodynamic data for this section and thus the aerodynamic performance of the transition section is neglected in all simulations. It is believed that the difference between the simulation result and the experimental result when rotating speed is 350 and 400 RPM is because of this neglecting. However, when the rotating speed increases, the contribution of the transition section to the aerodynamic load of the whole rotor can be neglected because it is close to the hub and its velocity is low. Nevertheless, the comparison illustrates that the prediction of torque does not converge when the value grows too much.

Figures 10 and 11 show the velocity and attack angle of each blade element when the rotating speed is 400 RPM or 550 RPM. It shows that the tangential velocity of different cases is almost the same when value varies and is mainly determined by the rotating speed of the rotor. However, the normal velocity is strongly affected by and the normal velocity increase with the value of . This has a significant influence on the attack angle of each blade element as shown in Figure 11 and therefore has a significant influence on the lift and drag force of each blade element. It should be noticed that the influence of the value on the torque result is not linear. According to the Reynolds number of this study, airfoil S826 gives the best aerodynamic performance when the attack angle is about 8 degree. Therefore, although the simulation with = 0.83ctip gives a lower prediction of the normal velocity and the attack angle of each blade element, it gives a higher prediction of the torque, which shows a different trend compared with the results at other rotating speeds.

**Figure 10.** The Vn and Vt of each blade element along the blade when rotating speed is (**a**) 400 RPM, (**b**) 550 RPM.

**Figure 11.** The attack angle of each blade element along the blade when rotating speed is (**a**) 400 RPM, (**b**) 550 RPM.

#### *3.3. The E*ff*ect of the Chord Length Gaussian Width*

Figure 12 shows the comparison for torque between experiment and simulations using the anisotropic regularization kernel. The best result using the standard regularization kernel is also added to the comparison. This figure also shows that the prediction of torque will not converge with the increment of the c value. The empirical value of is not suitable for the anisotropic regularization kernel and <sup>c</sup> = 1.2c shows the best prediction of the torque. However, the torque result is less sensitive to the c value compared with the value when using standard regularization kernel. The simulation using anisotropic regularization kernel with <sup>c</sup> = 1.0c also gives a reasonable enough prediction of torque.

**Figure 12.** The comparison of torque between experiment, simulation using standard regularization kernel with = 0.83ctip, and simulations using anisotropic regularization kernel with different <sup>c</sup> values.

The velocity and attack angle results when using anisotropic regularization kernel are different from the standard one. Figure 13 shows the comparison for the velocity between the result using standard regularization kernel with = 0.83ctip and results using anisotropic regularization kernel with different <sup>c</sup> values. It is clear that the tangential velocity of each blade element is less affected by the regularization kernel. However, the normal velocity is strongly affected by the regularization kernel which has been discussed. Compared with the result using anisotropic regularization kernel, the normal velocity is underestimated near the blade root and overestimated near the blade tip when using constant value. This matches with the previous discussion because the chord length of the blade root is larger than the tip one and a constant value will mispredict the affect region of a blade element. However, the attack angle of each blade element is not experimentally measured in this study due to the equipment limitation. Further study is needed to make a quantitative conclusion.

**Figure 13.** The Vn and Vt of each blade element along the blade when rotating speed is (**a**) 400 RPM, (**b**) 550 RPM.

According to the results above, although the optimal values are different, the effect of in standard kernel and the effect of *<sup>c</sup>* in anisotropic is similar. Both and *<sup>c</sup>* have little influence on the simulation result of tangential velocity, but will significantly affect the result of *Vn* and therefore significantly affect the rotor torque result. Figure 14 illustrates the effect region of different value. With a larger value, a blade element has a larger effect region in the actuator line method. However, the total value of the force is the same due to the Gaussian function, which means a regularization kernel has a flatter strength distribution when value is larger. Therefore, the regularization kernel with larger Gaussian width will affect a larger region of the flow and will lead to a higher velocity. This will cause an incorrect prediction of the attack angle of the blade element and has a strong effect on the prediction of lift and drag force.

It should be noticed that the optimal value in this study is different with the study of Martínez-Tossas [13] and Churchfield [12]. Actually, they do not agree with each other. Martínez-Tossas recommended = 0.14c ∼ 0.24c, Churchfield studied NREL 5MW wind turbine with <sup>c</sup> = 0.4c and studied NREL Phase VI wind turbine with <sup>c</sup> = 0.85c (here, c is the chord length). The main difference between these studies and this paper is the scale of the wind turbine. It could be inferred that the optimal parameters are related to the Reynolds number and this still needs further study.

**Figure 14.** An illustration of the effective region of different value. The red one represents a regularization kernel with larger Gaussian width.

## *3.4. The E*ff*ect of the Thickness Gaussian Width*

Table 2 shows the torque result when using different values of the thickness parameter t. Since the thickness of the airfoil is much smaller than the chord length and usually smaller than the limit of > 2Δgrid, the t value is usually limited by an absolute value which is related to the grid size. Two different <sup>t</sup> values are compared here. The series results for <sup>t</sup> = 20 mm are calculated on the normal mesh and this value is equal to twice of the grid size. The series results for <sup>t</sup> = 9 mm are calculated on the special mesh and this value is equal to 1.5 times of the grid size in the main flow direction. Table 2 shows that the thickness parameter has little influence on the torque results. Figure 15 shows the velocity component of each blade element and there is only a small difference for Vn which appears at the blade tip. It can be concluded that the thickness parameter <sup>t</sup> has a little influence on the torque prediction of ALM.


**Table 2.** Torque result when using different <sup>t</sup> value (Unit: Nm).

**Figure 15.** The Vn and Vt of each blade element along the blade when rotating speed is 500 RPM calculated by different <sup>t</sup> value.
