*2.4. Gust Model*

Different gust shapes and intensities were imposed to impact one of the three blades. In order to investigate the worst-case scenario, the gusts were chosen to collide on the blade when the blade pointed upward (i.e., at a 90◦ azimuth angle); in this position, the blade was attacked by the highest wind speed, and therefore, the solicitations acting on it were also the highest. In this work, as in other works on wind gusts [29,44], the gusts were modelled as a local streamwise velocity increase to be superimposed on the local wind velocity field. Using *vg* to indicate the gust's additional velocity and *v* the local wind velocity, the velocity in "gusted" conditions *v* can be expressed as:

$$
v' = \upsilon + v\_{\mathcal{K}}.\tag{6}$$

Notice that, at each time instant, all these velocities are functions of the position in the numerical domain. In order to minimize the computational time and numerical dissipation of the gusts [29,44], the gusts were added in the proximity of the turbine when the blade to be hit was positioned at a 60◦

azimuth angle, namely 30◦ and 20 time steps in advance with respect to the impact of the gust on the blade. They affected only a cylindrical region of 25 m in diameter (i.e., half of the blade span) and 12 m in length, whose axis was aligned with the wind direction. The frontal tip of this cylindrical region was positioned at an appropriate axial coordinate (approximately 1 m downstream of the blade tip's axial coordinate) in order to ensure the gust hit the targeted blade. This means that only in this cylindrical region, the gust velocity used in Equation (6) could be written as:

$$
\upsilon\_{\mathcal{S}} = s(a) \cdot f(r) \cdot A\_{\mathcal{S}} \; \neq \; 0. \tag{7}
$$

In Equation (7), *s* and *f* are shape functions of the gust, depending respectively on the axial (*a*) and radial (*r*) coordinates inside the gusted cylindrical region. In particular, the function *s* sinusoidally ramped up from 0 to 1 in the first 3 m of the axial length of the cylinder, was kept constant and equal to 1 in the middle, and then sinusoidally ramped down from 1 to 0 in the last 3 m of the axial length. Figure 4 illustrates the initial position and shape of the imposed gusts, as well as the coordinates *a* and *r*.

**Figure 4.** (**left**) Side view with detail of the blade tip axial position relative to the gusted region and (**right**) front view of the initial position and shape of the imposed gusts (in red) with respect to the wind turbine.

For what concerns the gust amplitude *Ag*, probabilistic analyses of wind gusts have shown that a gust amplitude of 5 m/s has more than an 80% probability of daily occurrence over central Europe when the wind conditions are similar to the ones chosen in this work [45]. Similarly, gusts exceeding this speed are commonly reported in Germany [46]. Also, more intense gusts (18 to 25 m/s) are fairly common in Europe, especially in coastal areas [25,47]. For these reasons, two gust amplitudes *Ag* were used in this work, namely 5 and 10 m/s.

Lastly, two shape functions *f* were used in this work. In the first subsection, a novel gust shape function was proposed, imposing a local redistribution of the flow rate and no global change in it, providing a velocity deficit on its border and a velocity increase in its center. This gust shape was conceptually similar to the "extreme operating gust" from the 61400-IEC standards for wind turbines. Subsequently, in the second subsection, a consistently positive gust velocity was used, analogous to the "extreme coherent gust" from the 61400-IEC standards for wind turbines.
