*3.1. Zero Net Flow Rate Gust*

In this section, the effect of a gust corresponding to a local redistribution of the available flow rate is investigated. The parameter *r* indicates the non-dimensional radial coordinate in the gusted cylinder, ranging from 0 (gust center) to 1 (gust border). Imposing C0 and C1 continuity at the border of the cylindrical region (*f*(1) = 0 and *f* (1) = 0), maximum gust velocity at the center (*f*(0) = 1 and *f* (0) = 0), and zero net flow rate ( <sup>2</sup><sup>π</sup> 0 1 <sup>0</sup> *f*(*r*) *r dr d*ϑ = 0), the following polynomial expression is obtained for the radial shape function:

$$f(r) = (1 - 12r^2 + 20r^3 - 9r^4). \tag{10}$$

The obtained function, compared to the "extreme operating gust" from the 61400-IEC standards (used as a function of space, as is done in De Nayer et al. [44]), produced a higher velocity increase, and thus more severe wind conditions, as depicted by Figure 6.

Notice that since this gust shape does not modify the net mass flow rate in the affected area but only redistributes it, it is considered appropriate to be used on inlet boundaries in combination with incompressible solvers, without the necessity to correct the mass flow rate to preserve its steady value [44].

Depending on the radial position where the gust hits the blade, a different effect was found regarding its axial deformation. In order to assess where to hit the blade to obtain the highest effect, several simulations were carried out, positioning the center of the gust at a distance h (Figure 4) equal to 35 m, 40 m, and 45 m starting from the axis. Figure 7 summarizes the results of these simulations.

**Figure 6.** Comparison between the zero net flow rate gust shape function and the "extreme operating gust" from the 61400-IEC standards for wind turbines.

**Figure 7.** Effect of the radial position of the gust center on the tip axial displacement: (**left**) gust with 5 m/s amplitude and (**right**) 10 m/s amplitude.

Hitting the blade further from its axis of rotation (and thus from its constrained end) increased the lever of the increased axial force. At the same time, due to the tapering of the blade (Figure 5), a smaller area was affected by the pressure increase and thus a smaller axial force increase was obtained. As shown in Figure 7, hitting the blade at 40 m (80%) of its span led to the highest axial deformation, irrespective of the chosen amplitude. For this reason, all the gusts analyzed in the remainder of this work were positioned at a 40 m height from the axis of rotation of the turbine. It was also noticed that, despite the blade always being hit by the gust at a 90◦ azimuth angle, a higher delay in its peak deflection was obtained when the gust was imposed further from its tip as a result of the higher portion of blade being displaced, and thus, of the higher inertia.

The axial force over the span of the blade was highly influenced by the wind gust. However, the differences on the lower 60% of the blade span (i.e., strips #1 to #6) were negligible, being smaller than 0.77% for *Ag* = 5 m/s and smaller than 1.64% when *Ag* = 10 m/s. Figure 8 shows the axial force evolution over each strip of the outboard 40% span of the blade, as well as the total axial force.

**Figure 8.** Zero net flow rate gusts: (**top**) total aerodynamic axial force over the blade hit by the gust as a function of its azimuth angle and (**bottom**) axial force contribution of the 4 most outboard strips.

The effect of the ABL is clearly seen in this figure, as well as in the ungusted condition, resulting in a higher loading when the blade pointed upward (90◦ azimuth angle) and lower loadings when it pointed downward (270◦ azimuth angle). The strips most sensibly influenced by the gust were #8 and #9 since the center of the gust was positioned exactly between these sections. These strips also provided an important contribution to the total bending moment, as they were located far from the axis of rotation. For both the amplitudes tested, the axial force over each strip resembled the distribution of velocity imposed for the gust, having a positive peak surrounded by two drops. However, it can be noted that the second drop in axial force was larger than the first one, especially for the case with *Ag* = 10 m/s and on strips #8 and #9. This was due to the occurrence of flow separation in both cases, as illustrated in Figure 9. In this figure, the regions affected by separation were identified by marking the portions of the blade suction side where the tangential component of the wall shear stress was oriented according to the direction of rotation.

**Figure 9.** Zero net flow rate gusts separation region (in red) over the blade suction side at 90◦ azimuth angle: (**left**) ungusted case, (**center**) case with *Ag* = 5 m/s, and (**right**) case with *Ag* = 10 m/s.

On the root of the blade, a separation region was observed also when no gust was considered. In this region, the blade shape underwent a transition from a cylindrical root to an aerodynamically shaped body. For what concerns the outboard part of the blade, in the case with *Ag* = 5 m/s, separation occurred only on a limited portion of the blade span and only on a restricted portion of the local chord length. On the other hand, when the gust amplitude was increased to 10 m/s, the separation area grew, expanding both in the spanwise and chordwise directions. This separation region was not reported during the first dip in the axial force. As soon as separation occurred, a sudden drop in the axial force was found. This was most intense on strip #9. As a result, the highest axial force was never reached at a 90◦ azimuth angle (Figure 8), but always a few degrees earlier. Furthermore, the second drop became longer in time and more intense in the axial force deficit with respect to the first one. This phenomenon strongly influenced the axial tip displacement, as shown in Figure 10. No sensible difference was observed in the tangential displacement because this was strictly related to gravity, as recognized in Santo et al. [6].

In this figure, the impact of the structural inertia was evident. The highest tip displacement was reached with a delay with respect to the axial force. Furthermore, the tip axial displacement immediately started to decrease when the outer border of the gust (where the velocity was decreased) impacted on the blade. Then, when the positive core of the gust hit the blade surface, the displacement started increasing again, and in the case with *Ag* = 5 m/s, a higher tip displacement was achieved with respect to the ungusted configuration. In the case with *Ag* = 10 m/s, the higher inertia of the blade due to its initial faster forward movement prevented the blade from reaching high peaks in its tip displacement, even if the axial force was increased, resulting in a maximum displacement lower than the case with *Ag* = 5 m/s. In addition, when the flow separated (Figure 9), the tip displacement rapidly decreased, preventing the blade from reaching high displacements. This phenomenon was much more

intense in the case with *Ag* = 10 m/s since the area affected by the separation was larger (Figure 9), and consequently, the drop in the axial force was also more intense (Figure 8). When the blade moved out of the gust region, the tip showed an oscillatory motion, whose amplitude gradually decreased until the blade reached the (ungusted) regime condition again.

**Figure 10.** Zero net flow rate gusts: tip axial displacement as a function of its azimuth angle in all analyzed cases.

A slightly different behavior was reported for the torque provided by the blade, as shown in Figure 11. Similarly to the axial force distribution, the torque differences over strips #1 to #6 did not exceed 1.43% for the *Ag* = 5 m/s case and 3.22% for the *Ag* = 10 m/s case.

**Figure 11.** *Cont*.

**Figure 11.** Zero net flow rate gusts: (**top**) total aerodynamic torque over the blade hit by the gust as a function of its azimuth angle and (**bottom**) torque contribution of the four most outboard strips.

Similarly to what was observed for the axial force, the occurrence of separation was reflected in the fact that the highest peak in the torque was never reached at a 90◦ azimuth angle but always slightly in advance on strips #8 and #9. Differently, the difference in the two drops in torque was much smaller than what was observed for the axial force. This was due to the different effect of separation over the axial and tangential forces. When the flow angle increased, the tangential component of the lift and drag forces also increased, translating into an increase for the tangential force and a decrease for the axial force. At the same time, when separation occurred, the magnitude of lift and drag respectively decreased and increased, leading to a reduction for both the axial and tangential components (assuming that, as is typical, the lift-to-drag ratio was high enough). The two effects compensated for the tangential force and summed up for the axial force, leading to the observed difference in their dynamics. It is also remarked that the tip velocity induced by its transient deformation also sensibly affected the flow angle and the magnitude of the incoming relative velocity, as already observed in Santo et al. [6]. This was reflected in the regions around the 110◦ azimuth angle, where, on the most outboard strips, a higher torque contribution was found with respect to the ungusted condition as a consequence of the fast forward movement of the blade tip, which increased the incoming flow angle. Lastly, the total torque provided by the machine is provided in Figure 12.

**Figure 12.** Zero net flow rate gusts: total torque provided by the machine as a function of the azimuth angle of the blade hit by the gust.

The peaks induced by the gust were comparable to the peaks induced by the tower-dam effect. It is also noted that a small effect was observed when the following blade went through the gusted

region, i.e., in the azimuthal range around 210◦. At this moment, the gust intensity in this region had lowered but had not disappeared.

### *3.2. The 1* + *Cos Gust*

In this section, a different gust shape function will be reported on, while the size and position of the gust is the same. The shape function reported on in this paragraph is given in Equation (11), where *r* indicates the non-dimensional radial coordinate in the gusted cylinder:

$$f(r) = \frac{1}{2}(1 + \cos(\pi r)).\tag{11}$$

This gust shape corresponds to the "extreme coherent gust" from the 61400-IEC, as used in similar works [26,29,44]. The present gust shape, when plugged into Equation (7), corresponds to a velocity increase in the whole region affected by the gust, contrary to the gust shape tested in the previous section. As already done for the zero net flow rate gust, two gust amplitudes will be used, namely 5 and 10 m/s.

The total aerodynamic axial force and the contribution of the most outboard strips are illustrated in Figure 13.

**Figure 13.** The 1 + cos gusts: (**top**) total aerodynamic axial force over the blade hit by the gust as a function of its azimuth angle and (**bottom**) axial force contribution of the four most outboard strips.

Similarly to what was observed for the zero net flow rate gust, the maximum value in the axial force was always reached in advance of the 90◦ azimuth angle. Furthermore, a drop in the axial force was observed on each analyzed strip but not due to the gust shape function, indicating, also in this case, the occurrence of a flow separation, as depicted in Figure 14.

**Figure 14.** The 1 + cos gusts separation region (in red) over the blade suction side at a 90◦ azimuth angle: (**left**) ungusted case, (**center**) case with *Ag* = 5 m/s and (**right**) case with *Ag* = 10 m/s.

Strips #8 and #9 were the most affected by the imposed gusts. Flow separation induced the drops visible in Figure 13. In general, the area affected by the flow separation was broader compared to Figure 9, showing an expansion in both spanwise and chordwise directions. This was due to the more severe gust conditions imposed by the 1 + cos gust. As a result, the drop in axial force was more intense. In particular, when the gust amplitude was increased from 5 to 10 m/s, not only a larger separation region was obtained, but also a lower drop in the axial force. Importantly, the azimuthal range in which the axial force dropped below the ungusted case was wider (especially on strips #8, #9, and #10 in Figure 13) when the gust amplitude was doubled.

The tip axial displacement was also heavily affected by the flow separation, as shown in Figure 15.

It is remarked that, in the case with *Ag* = 5 m/s, the highest tip axial displacement was obtained. Furthermore, in both cases, separation lowered the tip axial displacement in comparison with the ungusted configuration, acting as a protection mechanism against extreme deflection. This was clearly visible in the case with *Ag* = 10 m/s, where separation prevented the blade from reaching high deflections and resulted in a maximum deflection lower than the case with *Ag* = 5 m/s. In the latter case, the effect of the separation on the tip axial displacement was less evident and was restricted only to a marginal influence after a 120◦ azimuth angle. When the blade surpassed the region affected by the gust, its tip continued oscillating around the ungusted deflection, until it went back into regime conditions by the end of the analyzed full revolution.

**Figure 15.** The 1 + cos gusts: tip axial displacement as a function of its azimuth angle in all analyzed cases. Lastly, the torque provided by the blade hit by the gust is given in Figure 16.

**Figure 16.** The 1 + cos gusts: (**top**) total aerodynamic torque over the blade hit by the gust as a function of its azimuth angle and (**bottom**) torque contribution of the four most outboard strips.

Similarly to what was explained in the previous section, the effect of separation on the torque was much less intense than on the axial force, resulting in very small drops below the ungusted condition. However, a sudden change in the trend was visible, especially on strips #8 and #9, around an 87◦ azimuth angle, due to the separation itself, which prevented the maximum torque being reached at a 90◦ azimuth angle. In the region downstream of the gust (from 110◦ azimuth angle onwards), the differences observed in the torque, especially in the case with the highest gust amplitude, were attributed to the fast tip movement (Figure 15), which sensibly acted on the incoming flow angle and relative velocity magnitude.
