3.1. Wind turbine wake

Here, we discuss the characteristics of the wake that develops from the turbine downstream to $8D$ . To highlight the coherent structures composing the turbulent wake before their instability and loss of synchronisation with the wind turbine rotation, we compute phase-averaged quantities. The phase averaging operation over a cylindrical frame of reference $(r, \theta , x)$ for a generic quantity $\varPsi$ is defined as follows:

(3.5) \begin{equation} \hat {\varPsi }(r,\theta ,x) = \frac {1}{N} \sum _{i=1}^{N} \varPsi (r, \theta +\varOmega t_i, x, t_i). \end{equation}

In figure 9, we show the phase-averaged axial velocity, vorticity magnitude and turbulent kinetic energy along the entire wind turbine wake in the meridian plane. In addition, in figure 10, we show the snapshot of the three-dimensional wake structures individuated through the isosurface of the non-dimensional second invariant of the velocity gradient tensor $Q$ :

(3.6) \begin{equation} \frac {{\textit{QD}}^2}{U_\infty ^2} = \frac {D^2}{2U_\infty ^2}\big(\|W\|^2-\| S\|^2\big), \end{equation}

where $S$ and $W$ are the symmetric and deviatoric parts of the velocity gradient tensor, respectively.

Figure 9. Phase-averaged quantities in the meridian plane passing through the wind turbine axis: (a) axial velocity field normalised with $U_\infty$ ; (b) vorticity magnitude normalised with $U_\infty /R$ ; (c) turbulent kinetic energy (TKE) normalised with $U_\infty ^2$ .

Figure 10. Snapshot of instantaneous isosurface ${\textit{QD}}^2/U_ \infty ^2=12.66$ .

The wind turbine placed in a uniform laminar flow develops a complex wake composed of three main regions, as shown in figures 9 and 10. A momentum deficit, more evident in the region between the tip and the hub, together with wake expansion (figure 9 a) are associated with the extraction of kinetic energy by the wind turbine. Behind the hub, a small recirculation zone is present beyond which the velocity in the axial direction increases rapidly to approximately $U_\infty$ , in a region extending up to $5D{-}6D$ . Up to roughly $1.0D$ , the wake exhibits robust coherence, with structures synchronised with the rotation of the wind turbine, characterising the near wake. Beyond that, a decrease in the magnitude of vorticity is observed, which corresponds to an increase of the TKE at the outer radii. This phenomenon can be interpreted as the loss of coherence and synchronisation of the tip vortex system due to instability mechanisms, leading to the formation of turbulent eddies defining the intermediate wake up to approximately $7D$ . Finally, a significant amount of TKE is spread throughout the remaining part of the wake, reducing the momentum deficit which identifies the transition to the far wake. Figure 10 also shows that in the very far wake, few large-scale structures populate the turbulent field, especially at the centre of the wake, according to the theory of decaying isotropic turbulence. Hereafter, we separately discuss the three different regions of the wake.

3.1.1. The near wake

The near wake extends roughly up to $1.0D$ , with the presence of organised structures, namely the sheet vortical structures released by the trailing edge of the blades, the root vortices and the helicoidal tip vortex. The Q-criterion (figure 10) shows these structures, with the maximum value of vorticity observed in the tip vortex core (figure 9 b). As expected, in the near wake, the tip vortex is stable and coherent exhibiting the typical helical path. The turbulent boundary layer that develops over the blades generates a thin trailing edge vortex sheet, detectable downstream of the wind turbine in figures 9(b) and 10 as a thin sheet of intense vorticity. The root vortices of a wind turbine are composed of several coherent structures released at the root of the blades, interacting with the nacelle boundary layer, as discussed by Sherry et al. (Reference Sherry, Nemes, Lo Jacono, Blackburn and Sheridan2013a ,Reference Sherry, Sheridan and Jacono b ). This is evident in our simulation in figures 9(b) and 10, where it is possible to detect the interaction between the root vortex structures at approximately $0.2R$ and the vorticity downstream the nacelle; this involved vorticity system is more susceptible to destabilisation and viscous dissipation compared with that of a marine propeller (see Kumar & Mahesh Reference Kumar and Mahesh2017 and Posa et al. Reference Posa, Broglia, Felli, Falchi and Balaras2019 for a discussion).

There is a clear difference between the isolated marine propeller and the wind turbine. In the former case, a strong hub vortex is present (see, among others, Felli et al. Reference Felli, Camussi and Di Felice2011; Kumar & Mahesh Reference Kumar and Mahesh2017; Posa et al. Reference Posa, Broglia and Felli2022b ), which, moving downstream, eventually may start to oscillate and to interact with the tip vortex structures, generating a complex vortex interaction and breakdown (see, for example, figures 22 and 23 of Felli et al. Reference Felli, Camussi and Di Felice2011). In contrast, in the case of the wind turbine rotor, the hub vortex is insignificant (see figure 9), and the mechanism of interaction between the hub vortex and the tip vortex systems is clearly absent. This is probably due to significant geometry differences, among them: the ratio between the hub radius and the wind turbine radius is much smaller than in the case of a marine propeller; the wind turbine blade is very elongated with a very large aspect ratio, which is not the case for the marine propeller; the ratio between the expanded area and the disc area is typically larger than 1 for the marine propeller and much smaller than 1 for the three-blade wind turbine; finally, the wind turbine is less bulky than a marine propeller, especially in the root/hub region, promoting a central region in the wake where the mean velocity is almost undisturbed (see figure 9 a).

In general, in a wind turbine wake, the thin trailing edge vortex sheet and the root vortices dissipate rapidly and are weaker than the system of tip vortices (Lignarolo et al. Reference Lignarolo, Ragni, Scarano, Ferreira and van Bussel2015; Posa et al. Reference Posa, Broglia and Balaras2021). Figure 11 shows the contour of the phase-averaged axial velocity field, vorticity magnitude and turbulent kinetic energy over cross-sectional planes at increasing distance from the wind turbine, in the very near wake ( $x / D = 0.05,0.2,0.4,0.8$ ). The tip vortex system, the coherent filaments composing the root vortices and the blade wakes are distinguishable. Near the wind turbine, the axial velocity peaks in the region of the blade tip; a high level of vorticity is present along the blades with peaks in the region of the tip due to the presence of the tip vortex. The turbulent kinetic energy is large downstream of the trailing edge of the blades due to the presence of the blade trailing edge vortex sheet and in correspondence with the tip vortex. As the distance increases downstream, the streamwise velocity maintains large values in correspondence with the tips of the blades, it keeps increasing around the hub, and the defect velocity appears more homogeneous within an annular region between the hub and the tip of the blades. The blade trailing edge vortex dissipates rapidly with a corresponding decrease of turbulent kinetic energy, whereas the tip vortex dissipates TKE slower. It should be noted that a decreased level of vorticity magnitude does not correspond to a strong distortion of the vorticity field and to an increased level of TKE. In fact, at $x=0.6D$ , the blade wakes and the coherent helicoidal structures composing the root vortices are weak but still distinguishable, suggesting their rapid dissipation. The tip vortex exhibits gradual diffusion and loss of vorticity intensity with decreasing values of turbulent kinetic energy and remains stable up to $1.0D$ as depicted in figures 9 and 10. This provides further confirmation that the near wake of the wind turbine is dominated by the tip vortex system, as previously discussed in the literature by, among others, Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Ferreira and van Bussel2015) and Posa et al. (Reference Posa, Broglia and Balaras2021).

Figure 11. Phase-averaged axial velocity (left column), vorticity magnitude (central column) and turbulent kinetic energy (right column) normalised using $U_{\infty }$ and $R$ . From the top to the bottom, each row of panels corresponds at a certain distance downstream of the wind turbine: $x / D = 0.05$ , $0.2$ , $0.4$ , $0.8$ .

3.1.2. The intermediate wake

Beyond $1.0D$ , the tip vortex starts losing its own coherence and synchronisation with the wind turbine rotation (figures 9 and 10), due to instabilities as already discussed by Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Ferreira and van Bussel2015) and Posa et al. (Reference Posa, Broglia and Balaras2021). Specifically, Posa et al. (Reference Posa, Broglia and Balaras2021) showed that the deviation from the helicoidal and stable configuration starts with the short-wave instability with wavelength deviations of the order of the vortex core diameter; after that, the long-wave instability increases the wavelength deviations up to few vortex core diameters and promotes mutual inductance between two successive tip vortices. Our results are consistent with the literature (see figure 12).

Figure 12. Snapshots of isosurfaces $QD^2/U_\infty ^2=25.00$ computed at four time instants: $T/4$ , $T/2$ , $3T/4$ , $T$ , and phase-averaged turbulent kinetic energy normalised using $U_{\infty } ^2$ on the meridian plane.

Figure 13. Instantaneous vorticity magnitude normalised using $U_{\infty }$ and $R$ ; $x / D = (a)\, 1.0$ ; (b) $2.0$ ; (c) $4.0$ ; (d) $6.0$ .

As the helicoidal path of the tip vortex starts to exhibit short- and long-wave instabilities, an increase of turbulent kinetic energy is detected; the trajectory of the vortical structures starts to deviate from the helicoidal path, promoting the mutual inductance mode and leap-frogging mechanism. Figure 12 clearly shows the exchange of the relative position of successive tip vortices and vortex grouping mechanisms, as described by Felli et al. (Reference Felli, Camussi and Di Felice2011) for a ship propeller, and finally the three-dimensional breakdown of the helical vortices. Leap-frogging produces the breakdown of large vortical structures into smaller eddies, thus increasing turbulent mixing, resulting in the spread of turbulent kinetic energy over a large flow region and loss of synchronism with the wind turbine revolution. This is also highlighted in figure 13, which shows the instantaneous vorticity field over four significant cross-sectional planes. The exchange of the position between two successive tip vortices is evidenced in figure 13(b), leading to a reduction of vorticity magnitude, a significant distortion of the vorticity field and generation of a wide range of turbulent eddies populating the intermediate-to-far wake (figure 13 c,d). Tip vortex destabilisation is also observable looking at phase-averaged quantities in cross-flow planes at increasing distances from the turbine (figure 14); in particular, figure 14 shows a reduction in vorticity and an increase in turbulent kinetic energy moving downstream in the wake.

In addition, the shear layer present in the tip vortex region promotes turbulent transport of axial momentum (left column of figure 14) and TKE (right column of figure 14) from the tip vortex region along the radial direction.

Figure 14. Phase-averaged axial velocity (left column), vorticity magnitude (central column) and turbulent kinetic energy (right column) normalised using $U_{\infty }$ and $R$ . From the top to the bottom, a row corresponds to a certain distance downstream of the wind turbine, $x / D = 1.0, 2.0, 4.0, 6.0$ .

3.1.3. The far wake

In the far wake, the phase-averaged axial velocity and TKE tend to be smooth and homogeneous over the disk. The velocity fluctuations generated by the tip vortex destabilisation and shear layer in the intermediate part of the wake are transported across the entire cross-section in the far wake. In other words, turbulence progressively propagates from the tip region to the interior (figure 9 c) with significant homogenisation over the disc of diameter roughly equal to $D$ . As for the other regions of the wake, here, we show the phase-averaged streamwise velocity, the vorticity magnitude and the turbulent kinetic energy (figure 15) at increasing distances from the turbine ( $x / D = 7.0, 8.0$ ).

The phase-averaged velocity field in the axial direction becomes increasingly more homogeneous, due to the turbulent mixing and momentum wake recovery (left column of figure 15). The low value of vorticity magnitude, compared with the other regions of the wake, shows loss of coherence in the tip vortices and the absence of large, energetic vortical structures synchronised with the wind turbine’s rotation (central column of figure 15).

Finally, TKE appears distributed across the section with a substantially smaller magnitude than in the other parts of the wake (right column of figure 15). This indicates the presence of a population of smaller-scale turbulent eddies in the far wake, covering the entire disk as depicted in figure 10.

Figure 15. Phase-averaged axial velocity (left column), vorticity magnitude (central column) and turbulent kinetic energy (right column) normalised using $U_{\infty }$ and $R$ . From the top to the bottom, a row is relative to a certain distance downstream of the wind turbine: $x / D = 7.0, 7.5$ .

4.1. Noise sources

As discussed in § 2.2, the FWH equation (2.3) contains seven terms that represent the acoustic sources associated with different fluid-dynamic processes. The thickness noise source is related to the rotational velocity ( $\omega \boldsymbol{r}$ ), which is constant in time. However, a time dependency is introduced because of the periodic variation of the relative position between the microphone and the solid surfaces, leading to a tonal noise at the blade passing frequency ( $f_{bp}$ ). As for the thickness term, the loading term peaks at the blade passing frequency, although its own magnitude also depends on the pressure fluctuations and their time derivative over the surfaces. These fluctuations contribute to the high-frequency noise, as discussed later in the paper. To highlight the noise source originated by the pressure field over the blades, we analyse the pressure fluctuation coefficient ( $C_{\!p_{\textit{rms}}} = p_{\textit{rms}} / 0.5\rho U_{\infty }^2$ ) acting on the blade’s surface (figure 16). We observe that the downwind side of the blade’s surface is affected by pressure fluctuations more than the upwind blade’s surface; these fluctuations are associated with the transition from laminar to turbulent boundary layer and flow separation. Significant pressure fluctuations are present in the trailing edge and moving towards the tip of the blade; this may be ascribed to the increased value of the cord-based Reynolds number. At the tip of the blade (figure 16 c), the release of the tip vortex promotes pressure fluctuations which are weaker compared with the corresponding trailing edge fluctuations (compare figures 16 a and 16 c). As discussed by Wagner et al. (Reference Wagner, Bareiss and Guidati1996), the trailing edge noise source is among the major noise sources in wind turbine aeroacoustics. In particular, the trailing-edge noise source dominates the high-frequency spectrum, perceived as a swishing sound.

Figure 16. Phase-averaged pressure fluctuation coefficient on the blade’s surface: downwind side (a); upwind side (b); tip of the blade (c).

Figure 17. Phase-averaged r.m.s. of the Lighthill source term made non-dimensional with $\varOmega$ : (a) meridian plane; transversal planes located at $x / D=$ (b) 0.5; (c) 1.0; (d) 2.0; (e) 3.0; (f) 5.0; (g) 7.5.

The analysis of the kinematic Lighthill source term in the fluid region,

(4.1) \begin{equation} S = \frac {\partial ^2 u_i u_{\!j}} {\partial x_i \partial x_{\!j}}, \end{equation}

allows us to quantify the noise source associated with the wake. In figure 17, we show the phase-averaged root mean square (r.m.s.) of the Lighthill source term ( $S_{\textit{rms}}$ ) along the whole wind turbine wake, in the meridian plane and at cross-sectional planes at increasing distance from the turbine. This quantity gives information on the noise sources mainly at the blade passing frequency. Higher values of $S_{\textit{rms}}$ are detected in the near wake corresponding respectively to: the thin trailing edge vortex sheet released by each blade; the coherent structures composing the root vortices; the system of tip vortices syncronised with the blade passing frequency (figure 17 a,b). Similar results were obtained by Posa et al. (Reference Posa, Broglia and Felli2022b ) for the case of marine propellers. The $S_{\textit{rms}}$ rapidly decreases in the blade wake (figures 17 b and 17 c), consistently with a strong decay of vorticity and TKE in the same region (figure 11). In the hub and tip regions, we observe large values of $S_{\textit{rms}}$ which decrease rapidly moving downward (figures 17 a and 17 c– e). To be noted that, beyond $1D$ , noise is generated mainly by the tip vortex whose dynamics is described in the previous section. Finally, $S_{\textit{rms}}$ decreases in intensity further downstream over the whole cross-sectional area (figures 17 a and 17 e–g), indicating that the near-to-intermediate wake contributes more to the production of noise. In agreement with the results of Posa et al. (Reference Posa, Broglia and Felli2022b ) for ship propellers, large values of $S_{\textit{rms}}$ are found in regions of large coherence of the fluid flow, such as the tip and root vortices. However, in contrast to the case of marine propellers, the region of large fluctuations persists at the outer radii more than in the inner region, suggesting that the acoustic signature of the wind turbine is dominated to a large extent by tip vortices and their instability rather than by the hub vortex (which is insignificant as already discuss in the previous section) and the root vortices as the distance downstream increases. This is due to the geometrical differences between marine propellers and wind turbines, as we already pointed out in § 3.1.1.

4.2. Acoustic spectra

This section presents the analysis of the wind turbine’s acoustic signature. We show the streamwise and radial evolution of the sound pressure level (SPL) expressed in dB. The SPL is defined as

(4.2) \begin{equation} \textit{SPL} = 20 \log _{10} \left (\frac {\textit{fft}(\hat p)}{ \hat p_0}\right )\!, \end{equation}

where $\textit{fft}$ is the Fast-Fourier transform of $\hat p$ and $\hat p_0 = 20 \,\unicode{x03BC} {\mathrm{Pa}}$ is the reference sound pressure level in air. As discussed in § 2.2, the integrals that compose the FWH equation must be evaluated at the emission time $\tau$ , which may introduce a significant increase in computational efforts. Cianferra et al. (Reference Cianferra, Ianniello and Armenio2019a ) introduced the maximum frequency parameter (MFP),

(4.3) \begin{equation} \textit{MFP} = \frac {1}{\varDelta _{\textit{del}} f_{\textit{max}} } = 1, \end{equation}

to evaluate the maximum frequency $f_{\textit{max}}$ beyond which neglecting the time delays may have an impact on $\hat p$ . The maximum difference in time delay $\varDelta _{\textit{del}}$ is defined as

(4.4) \begin{equation} \varDelta _{\textit{del}} =\frac { \max \limits _{\boldsymbol{y}\in S,V}(\boldsymbol{y}(\tau ) - \boldsymbol{x}(t)) - \min \limits _{\boldsymbol{y}\in S,V}(\boldsymbol{y}(\tau ) - \boldsymbol{x}(t))}{c_0}, \end{equation}

where $\boldsymbol{x}(t)$ and $\boldsymbol{y}(\tau )$ are the microphone position at the reception time $t$ and the position of the acoustic source at the emission time $\tau$ , respectively. The criterion $\textit{MFP}\gt 1$ gives the range of frequencies $f\lt f_{\textit{max}}$ , where the computation of the time delays is not required. The MFP criterion provides a quantification of the general definition of a compact source, identifying an exact frequency threshold within which computation of time delays can be neglected. Note that, in the case of sources fixed in space, $\boldsymbol{y}(\tau )$ is constant and coincides with grid nodes, making the computation of $\varDelta _{\textit{del}}$ straightforward. However, for bodies in motion, $\boldsymbol{y}(\tau )$ is unknown and can be computed by solving an implicit equation

(4.5) \begin{equation} t - \tau - \frac {\|\boldsymbol{x}(t) - \boldsymbol{y}(\tau )\|}{c_0} = 0. \end{equation}

This equation is also adopted to compute the retarded surface defined by Ianniello, Muscari & Di Mascio (Reference Ianniello, Muscari and Di Mascio2013) and based on the retarded time approach of Brentner & Farassat (Reference Brentner and Farassat2003). We solve it numerically using the bisection method to obtain the emission times $\tau$ , the related positions $\boldsymbol{y}(\tau )$ and finally $\varDelta _{\textit{del}}$ . The values of $f_{\textit{max}}$ are then computed for each microphone for the surface elements (table 3) and for the volume elements (table 4). Note that a rough estimation of the parameter $\varDelta _{\textit{del}}$ is given by $D/c_0 = 5.4 \times 10^{-4} \ {\mathrm{s}}$ for the linear part and $7D/c_0 = 0.0038 \ {\mathrm{s}}$ for the nonlinear part, for the less favourable condition (corresponding to the microphone closest to the turbine); this thumb rule estimation is consistent with the results obtained with the use of (4.4) and (4.5).

Table 3. $\varDelta _{\textit{del}}$ and non-dimensional maximum frequency $f_{\textit{max}}/\!f_T$ , where $f_T$ is the revolution frequency, that can be resolved correctly neglecting the time delays for the linear terms of the FWH equation at different microphones.

Table 4. $\varDelta _{\textit{del}}$ and non-dimensional maximum frequency $f_{\textit{max}}/\!f_T$ , where $f_T$ is the revolution frequency, that can be resolved correctly neglecting the time delays for the nonlinear terms of the FWH equation at different microphones.

The minimum value of $f_{\textit{max}}$ that can be solved without introducing significant errors is obtained for the microphone placed in the very near field at $(0,0,D)$ ; its value is $30 f_T$ for the linear terms and $3.85 f_T$ for the nonlinear ones, where $f_T$ is the revolution frequency. Then, the value of $f_{\textit{max}}$ increases with distance from the source. Note that the compressibility delay is more significant for the quadrupole terms than for the linear contribution, as expected. Here, we do not consider the compressibility delay in the computation of the acoustic pressure, since it is computationally expensive, in particular, for the quadrupole terms of the FWH equation. However, in our specific case, we show that the nonlinear terms are significant in the lower frequency range only ( $f \lt 4.0 f/f_T$ ), which makes our assumption acceptable.

We first calculate separately the linear and nonlinear contributions to the SPL of acoustic pressure $\hat {p}$ (figure 18) at microphones placed at the radial distance $r / D = 1.0$ at four streamwise positions, that is, $x / D = 0.0, 0.5, 1.0, 2.0$ (figure 6).

Figure 18. Linear and nonlinear noise contributions and total noise in terms of SPL of acoustic pressure at microphones placed at $r / D = 1.0$ along different axial positions $x / D=$ (a) $0.0$ ; (b) $0.5$ ; (c) $1.0$ ; (d) $2.0$ . The shaded grey area is the range of frequencies in which the compressibility delay may play a role in the evaluation of the nonlinear terms.

As expected, the linear part dominates the acoustic signature of the wind turbine, especially in the intermediate- to high-frequency range $f \geqslant 3 f_{T}$ . However, the lowest frequency part of the spectra is dominated by the nonlinear source terms. A similar outcome was found by several authors, among the others, Cianferra & Armenio (Reference Cianferra and Armenio2021), Cianferra et al. (Reference Cianferra, Petronio and Armenio2019b ), Posa et al. (Reference Posa, Broglia and Felli2022b ), Turner & Kim (Reference Turner and Kim2022), Zamponi et al. (Reference Zamponi, Avallone, Ragni, Schram and Van Der Zwaag2024), showing the predominance of the nonlinear terms in the near-to-intermediate field. The linear contribution (figure 18 a) exhibits a fundamental peak at the blade passing frequency and is broadband in the high frequency range; the former aspect is associated with rigid rotation, while the latter is associated with pressure fluctuations over the blades (see figure 7). Moving downstream (figure 18 b–d), the peak decreases in amplitude practically to disappear at the farthest microphone, resulting in a broadband profile. Regarding the nonlinear contribution to the noise, figure 18(a,b) show several peaks at the blade passing frequency and its own multiples, in the low range of frequency; they are related to the large and organised inertial vortical structures, such as the tip vortices, trailing edge vortex sheet and root vortices. In particular, the near-wake coherent structures of the flow, associated with high vorticity and high levels of the Lighthill noise source term (figures 9 b and 17) are synchronised with the turbine rotation and thus produce tonal noise. Moving downstream, the nonlinear tonal peaks tend to be smoothed out and spread, leading to a low-frequency broadband profile. The latter is associated with the vortex instability mechanism and loss of synchronisation, occurring in the intermediate wake (see figure 9 b–c). In other words, the loss of synchronisation, at the expense of the growth of disorganised turbulence structures, results in a transition from a tonal spectrum to a purely broadband one, with an increase in amplitude at low frequencies. Similar behaviour was highlighted by Posa et al. (Reference Posa, Broglia and Felli2022b ) for the case of a marine propeller. The authors suggested that the main mechanisms that come into play are tip-vortex instability and successive grouping in the intermediate wake (see Felli et al. Reference Felli, Camussi and Di Felice2011).

The problem of the time delays in the range of frequencies $3.85 \lt f/f_T\lt 30$ is related to the evaluation of the nonlinear terms; however, the results of figure 18 clearly show that neglecting the time delays does not introduce significant errors in the evaluation of the overall noise. In fact, for $ f/f_T\gt 3.85$ , the nonlinear contribution to the noise is in general negligible compared with the linear one, and the noise spectrum practically coincides with that of the linear part. An exception is present in figure 18(c,d) in which an increased level of the nonlinear noise is observable in the range of high frequencies ( $ f/f_T\gt 20$ ). Although in this range of frequencies the time delays may play a role, a similar behaviour was found by Wolf, Azevedo & Lele (Reference Wolf, Azevedo and Lele2012) and Turner & Kim (Reference Turner and Kim2022), when analysing the quadrupole part of the noise in the wake of an aerofoil and it was attributed to the Kelvin–Helmholtz instability developing in the separated shear layer.

The use of the FWH equation allows us to discern among the single source contributions to the overall noise spectrum. Figure 19 shows the contribution of each term of the FWH equation at the four microphones, the same as in the previous figure. The thickness term exhibits the main tonal peaks at the blade passing frequency and a few super-harmonics (figure 19 a). The main peak at $f/f_T=3$ remains well visible moving downstream in the wake, although the signal decays rapidly over the entire frequency range. The loading terms (figure 19 b,c) exhibit a tonal behaviour at the blade passing frequency; super-harmonics are also present, as well as a broadband contribution in the high-frequency range. The amplitude of the acoustic pressure fluctuations exhibits non-monotonic decay; specifically, it increases moving from $x / D = 0.0$ to $x / D = 0.5$ and decreases afterward; the directivity of the loading term explains this characteristic and will be discussed in § 4.3. The three nonlinear terms (figure 19 d,e,f) exhibit a similar behaviour, with the third term (namely the seventh one of the FWH equation, panel f) being predominant over the others. This is due to the position of the microphones. Indeed, the FWH terms have different directivity, magnitude and decay rates. In the very near field (for small values of $r$ , see (2.3)), the third nonlinear term may play a major role. As the distance $r$ increases, the contribution of the second and first terms becomes more significant. This will be further discussed in § 4.3. All nonlinear terms (figure 19 d,e,f) exhibit a tonal behaviour for the microphones placed in the region of the near wake, while a broadband character is detected for the microphones placed in the region of the intermediate wake, as already discussed previously.

Figure 19. SPL of the six terms composing the FWH equation at microphones placed at the radial coordinate $r / D = 1.0$ along different axial positions: $x / D = 0.0, 0.5, 1.0, 2.0$ . (a) First and second linear term; (b) third linear term; (c) fourth linear term; (d) first nonlinear term; (e) second nonlinear term; (f) third nonlinear term. The shaded area of grey is the range of frequencies in which the compressibility delay may play a role in the evaluation of the nonlinear terms.

The direct formulation of the FWH equation allows us to assess the contributions of different regions of the domain to the overall acoustic spectrum. With this purpose, we perform two sets of computations, considering different shapes of subdomains to detect which fluid-dynamic structures contribute the most to the noise: the first set considers annular subdomains comprehensive of the surface of the rotor; the second set considers different cylindrical regions of the wake, roughly corresponding to the near wake and two regions of the intermediate wake.

For the first set of computations, the linear and nonlinear parts are evaluated in the inner region ( $0.0 \lt r / D \lt 0.2$ ), in the middle region ( $0.2 \lt r / D \lt 0.45$ ) and in the outer region ( $0.45 \lt r / D \lt 0.9$ ). The three integration volumes are concentrical cylinders that extend from $x / D = -0.5$ to $x / D = 7.0$ . In figure 20, we show the linear contribution to the noise, considering the inner, middle and outer regions of the turbine surface. At the nearest microphone (figure 20 a), peaks at blade passage frequency $f/f_{T}=3$ and its first super-harmonic $f/f_{T}=6$ are visible, mainly generated by the thickness term (see figure 19 a; here, the outer and middle sections contribute equally). In the high-frequency range, pressure fluctuations in the middle region of the blades (see figure 16), feeding the loading term, are found to contribute the most.

Figure 20. Linear contribution of the inner ( $0\lt r / D\lt 0.2$ ), middle ( $0.2\lt r / D\lt 0.45$ ) and outer ( $0.45\lt r / D\lt 0.9$ ) surfaces in terms of SPL at microphones placed at $r / D = 1.0$ along different axial positions $x / D=$ (a) $0.0$ ; (b) $0.5$ ; (c) $1.0$ ; (d) $2.0$ .

Figure 21. Nonlinear contribution of the inner ( $0\lt r / D\lt 0.2$ ), middle ( $0.2\lt r / D\lt 0.45$ ) and outer ( $0.45\lt r / D\lt 0.9$ ) volume in terms SPL at microphones placed at $r / D = 1.0$ across different axial positions $x / D=$ : (a) $0.0$ ; (b) $0.5$ ; (c) $1.0$ ; (d) $2.0$ .

At downstream locations (figure 20 b–d), we observe that the middle section of the blades still dominates the linear noise, being mainly generated by the loading term (figure 19 b,c). This is consistent with the analysis of pressure fluctuations over the blades (figure 16). The analysis suggests that trailing edge pressure fluctuations are the main source of noise associated with the wind turbine surface.

In figure 21, we report the nonlinear term, considering the inner, middle and outer regions of the turbine wake. In the region of the near wake (figure 21 a,b), the outer and the middle regions contribute equally to the nonlinear noise at the blade passing frequency peak and at its super-harmonics. This originates from intense sources, namely the tip vortex system and the trailing edge vortex sheets (see figure 17 a–c). Moving downstream, the outer volume governs the noise generated in the low-frequency range (figure 21), and we observe the progressive disappearance of the peaks at blade passing frequency and its super-harmonics. The analysis of the fluid-dynamic field (figure 14) has shown that in the intermediate wake, TKE is primarily produced in the outer section due to tip vortex instability and grouping. This is the primary reason for the broadband character of the nonlinear noise. The inner volume contribution to the overall noise is negligible at all locations, highlighting that the root vortices dynamics is, by far, less important than the tip and blade vortices. For the sake of completeness, we computed the acoustic signal at additional microphones positioned at $x / D = 3, 4, 5, 6, 7$ at the same radial distance $r / D = 1$ (not shown). The spectral component of the linear term remains similar to that depicted in figure 20(d), with amplitude decaying with the distance from the turbine. The behaviour of the nonlinear term with increasing distance from the turbine, still in the region of the intermediate-to-far wake, remains consistent with that shown in figure 21(d) and the sound level remains relatively high. Overall, the turbulent fluctuations occurring in the outer region are found to be the primary source for all the microphones analysed in this section. A detailed analysis of the decay of linear and nonlinear terms is in § 4.3.

For the second set of computations, we evaluate the nonlinear terms of the noise associated with different regions of the wake, namely the near wake ( $-0.5 \lt x / D \lt 1.0$ ), the first part of the intermediate wake (FPIW) ( $1.0 \lt x / D \lt 5.0$ ) and the downstream part of the intermediate wake (DPIW) ( $5.0 \lt x / D \lt 7$ ). The microphones are placed at $r / D = 1.0$ along different axial positions $x / D=0.5$ , $3.0$ , $5.0$ , $7.0$ , thus spanning over the three regions herein considered.

As the microphone moves along the streamwise direction, the dominant acoustic contribution clearly comes from the region directly beneath it; this trend is evident across all panels. Specifically, close to the rotor (figure 22 a), the near wake is dominant, exhibiting the peak at the blade passing frequency and super-harmonics.

As the distance from the source increases, the intermediate wake contributes the most (figure 22 b–d). The relative importance of the two sub-regions of the intermediate wake is inverted moving further downstream, with the DPIW dominant in the farther microphone. This highlights the necessity to consider the whole wake in the computation of the noise in the near field.

In figure 23, we report the SPL computed at microphones placed in the turbine plane $x / D = 0.0$ , at three radial distances, $r / D = 2.0, 4.0, 8.0$ . Note that the spectra at $r / D =1.0$ are already shown in figure 18(a).

For the linear part, the behaviour with increasing distance is similar to what was already observed for the microphones positioned along the x-axis. Specifically, as the radial distance increases, the tonal peaks gradually vanish, along with the broadband part of the spectrum in the high-frequency range.

For the nonlinear part, in contrast to what was observed for the microphones placed along the wake, the broadband noise at low frequencies associated with intermediate-to-far wake turbulence is not detected. Instead, the signal decays uniformly across the whole frequency range.

Figure 22. Nonlinear contributions to the noise associated with different regions of the wake, at four microphones placed along the wake: near wake ( $-0.5\lt x / D\lt 1.0$ ), blue lines; first part of the intermediate wake (FPIW) ( $1.0\lt x / D\lt 5.0$ ), yellow lines; downstream part of the intermediate wake (DPIW) ( $5.0\lt x / D\lt 7.0$ ), violet lines. The total nonlinear term is shown with a red line. The microphones are placed at $r / D = 1.0$ along different axial positions $x / D=$ (a) $0.5$ ; (b) $3.0$ ; (c) $5.0$ ; (d) $7.0$ .

Figure 23. Linear and nonlinear noise contributions in terms of SPL of acoustic pressure at microphones placed at $x / D = 0.0$ along different radial positions $r / D=$ (a) 2.0; (b) 4.0; (c) 8.0.

4.3. Directivity and decay rate

In the previous section, we discussed spectra at specific points along the streamwise direction out of the wake and along the radial direction. The analysis along two directions is not exhaustive since the acoustic signal typically exhibits directivity depending on the source characteristics (monopole, dipole, quadrupole). In addition, the decay rate of each FWH term (2.3) plays a role in the propagation of the signal in the near-to-far field.

Here, we analyse the directivity, which gives the preferential direction of noise propagation in the meridian plane $z=0$ . We consider $20$ microphones placed in concentric circles around the source, at different radii. To choose the radii, we perform a rough estimation of the acoustic near-to-far field transition, in accordance with the literature. The choice depends on a reference wavelength, for example, $\lambda = c_0 /3f_T$ , which is the wavelength associated with the blade passage frequency. The condition $K_{\lambda } = d/\lambda \lt 1$ , where $d$ is the distance from the source, may be used to delimit the acoustic near field from the far field. Based on this discussion, we consider the radii $2D$ , $10D$ , and $100D$ , which respectively give $K_{\lambda } \sim 0.22, 1.1, 10.8$ . We consider separately the linear and nonlinear contributions, and the overall noise, in figure 24.

Figure 24. Directivity of the root mean square of the acoustic pressure fluctuations $C_{\hat p_{\textit{rms}}}$ at microphones placed in concentric circles on the $x{-}y$ plane, at: (a–c) $2D$ ; (d –f) $10D$ ; (g–i) $100D$ . (a,d,g) Linear contribution; (b,e,h) nonlinear contribution; (c,f,i) overall contribution. The arrow indicates the streamwise direction. Note that at the radial distance of 2D the microphones between $30^{\circ}$ – $330^{\circ}$ are not shown since they lie inside the FWH integration volume.

The linear part exhibits the typical dipole shape with lobes aligned with the flow direction and with decreasing amplitude with increasing distance from the source (figure 24 a,d,g). The dipole-type pattern aligns with expectations and is consistent with findings from other rotor studies (see, for example, Wang et al. (Reference Wang, Wang and Wang2021), in spite of differences in test case and operating conditions). The directivity of the linear term, which exhibits a dipole-like shape, explains the non-monotonic decay of the acoustic intensity moving downstream, as already observed in figure 19(b,c) about the spectral analysis.

The nonlinear part exhibits a quadrupole shape (figure 24 b,e,h). In the near field, the observed asymmetry is due to the wake, the lobe in the streamwise direction being much more pronounced than the others. With increasing distance from the source, the magnitude of the signal decays rapidly and a more symmetric quadrupole shape tends to recover.

The overall noise level in the near-field ( $K_{\lambda } = 0.21$ ), figure 24(c), exhibits a non-symmetric dipole shape, given by the summation of the two contributions. This highlights the importance of the nonlinear terms in the direction of the wake, near the source. To be noted is that the directivity is expected to be symmetric with respect to the $0^\circ {-}180^\circ$ axis. The slight asymmetry has to be ascribed to imperfect statistical convergence.

The directivity in the rotor plane is not reported because the axial symmetry of the wind turbine motion results in a homogeneous noise propagation over the circumferential directions. In this case, the decay rate along the radial direction is more suited to evaluate the relative importance of the linear and nonlinear noise source contributions. For this purpose, we calculate the root mean square of the acoustic pressure coefficient ( $C_{\hat p_{\textit{rms}}} = \hat p_{\textit{rms}} /0.5 \rho U_{\infty } ^2$ ) considering three revolution periods, up to a distance of $1000D (K_{\lambda } \sim 108)$ . In figure 25, for the sake of clarity, we show the directivity up to $100D\, (K_{\lambda } \sim 10.8)$ .

Figure 25. Decay of the root mean square of the acoustic pressure coefficient: (a) radial decay; (b) decay along the streamwise direction out of the wake.

In figure 25(a), we report the decay rate of the linear and nonlinear terms together with that of the overall noise along the radial direction in the turbine plane.

The linear and nonlinear terms decay as $r^{-1}$ in the far field, with the linear term dominating over the other; this finding is consistent with the literature, where the linear term is recognised as the principal contributor to the overall noise in the far field. In the near field, linear and nonlinear contributions decay much faster than $r^{-1}$ ; specifically, the nonlinear terms decay roughly as $r^{-4}$ ; the term-by-term analysis (not reported) shows that the third nonlinear term contributes the most, in accordance with results shown in figure 19 (compare panels d,e,f).

The distance at which the decay rate changes to $r^{-1}$ , in this case, around $z=4D$ , may depend on various features, such as the rotational Mach number and the number of blades. In the near-field, the linear term decays as $r^{-4}$ .

This pronounced decay would seem anomalous, given (2.3), which indicates $r^{-2}$ in the near-field and $r^{-1}$ in the far-field. It can be explained by the destructive interference associated with the rotational motion of the noise sources, as discussed in detail by Ianniello (Reference Ianniello2016). In that paper, the author found that the kinematic auto-interaction of the noise sources produces destructive interference, leading to a decay rate higher than the expected $r^{-1}$ law. In particular, the author found that considering $N$ rotating sources, the decay rate of the linear term approaches the $r^{-(N+1)}$ law in the rotor plane in the near field; our results are in accordance with that of the rotpole model. This applies to the linear terms, due to geometrical reasons. Interestingly, our results are consistent with those of Posa et al. (Reference Posa, Broglia and Felli2022a ) for a system composed of a seven-blade marine propeller placed upstream of a hydrofoil and Posa et al. (Reference Posa, Broglia, Shi and Felli2025) for conventional and tubercled five-blade marine propellers. Indeed, along the radial direction, the authors observed decay trends and relative contribution of the different FWH terms with increasing distance similar to those herein reported. However, such comparisons should be approached with caution due to the influence of several key parameters, such as the rotational Mach number, the rotation frequency and the number of blades.

Figure 25(b) reports the decay rates along a line parallel to the wind turbine axis, at a distance of $r / D=1$ , out of the wake. The linear term decays as $r^{-1}$ as the distance increases. Up to $9D$ , the nonlinear term dominates; this is expected since in the region up to $8D$ , the selected microphones are placed close to the turbulent wake, leading to a non-monotonic behaviour peaking at $4.5 D$ . Beyond that, the nonlinear part drops very rapidly, decaying asymptotically as $r^{-1}$ .

To summarise, beyond $10D$ , the decay rate of the overall noise fits the $r^{-1}$ law, due to the linear part, indicating that the latter becomes dominant in the far field.

When comparing the decay rates (panel a versus panel b of figure 25), mutual interaction between rotating noise sources is evident in the turbine disk since it gives rise to a destructive effect (higher decay rate), while the mutual interaction is not clearly identified out of the rotor plane, along the streamwise direction. The destructive effect is described using the rotpole model by Ianniello (Reference Ianniello2016), where the author showed decay rates steeper than $r^{-1}$ along axial and radial directions, by analysing a multi-rotpole configuration. In contrast, in the present study, we found that the decay along the wake axis adheres consistently to the $r^{-1}$ law. One possible explanation for this discrepancy, as previously asserted by Ianniello (Reference Ianniello2016), is that the rotpole model relies on a constant pressure point-like source, while in the present case, realistic unsteady pressure loads on the blade surface are present. The broadband pressure distribution over the wind turbine surface tends to break up the synchronisation of the signals, being more similar to a set of time-dependent distributed sources along the surface. This is particularly true when investigating turbulent flow-induced pressure disturbances, such as the turbulent boundary layer at the trailing edge of a blade. Conversely, the concept of multiple rotpoles is fully representative of the phenomenon when the observer is placed on the turbine disk and produces accurately the mechanisms of kinematic auto-interaction. This can be attributed to the nearly constant pressure distribution in the tip region resulting from the reduced instability induced by the tip vortex.

To show the importance of unsteady pressure fluctuations on the decay rate of the linear terms, we extend the rotpole model considering a stochastic pressure signal (stochastic rotpole). In particular, we consider a tri-rotpole model with the rotating point-sources characterised by pressure $\tilde {p}$ being a stochastic quantity; we use the same geometrical and rotational properties of the model-scale wind turbine herein considered (radius $R$ , rotational velocity $\varOmega$ , speed of sound $c_0$ ) to have the same rotational Mach number. We consider the classical formulation of the FWH equation for the loading component in terms of multiple rotating noise sources:

(4.6) \begin{equation} 4\pi\! \hat p(\boldsymbol{x},t) = \frac {1}{c_0} \frac {\partial }{\partial {t}} \sum _{l=1} ^{N} \frac {\tilde {p}_l \boldsymbol{n}_{l} \boldsymbol{\cdot }\hat {\boldsymbol{r}}_{l}}{r_{l} |1-M_{r_l} |} + \sum _{l=1} ^N \frac {\tilde p_l {\boldsymbol{n}}_{l} \boldsymbol{\cdot }\hat {\boldsymbol{r}}_{l}}{{r_{l}}^2 |1 - M_{r_l} |} ,\end{equation}

where $\tilde {p}_l = p_s + p_{rand}$ is the stochastic (normal distribution) pressure fluctuation, in which $p_s = 1 \,{\mathrm{Pa}}$ and $p_{rand}$ is a random distribution with zero mean value and assigned standard deviation, $\boldsymbol{n}_l$ is the normal vector associated with the $l$ th rotating source, $\hat {\boldsymbol{r}_{l}}$ is the versor connecting the source, $r_l$ is the source-observer distance, $M_{r_l} = \boldsymbol{v}_l\boldsymbol{\cdot }\boldsymbol{r}_l/c_0$ is the rotational Mach number, where $\boldsymbol{v}_l$ is the velocity vector of the rotating source (for more details, see Ianniello Reference Ianniello2016) and $N$ is the number of rotating sources, which is equal to 3 in this case. For clarity, we consider the normal versors $ \boldsymbol{n}_1$ , $\boldsymbol{n}_2$ and $\boldsymbol{n}_3$ , the first pointing in the radial direction, the second is tangent to the circumferential trajectory of the rotating sources and the last is normal to the rotor plane (for more details, see figure 12 of Ianniello Reference Ianniello2016). Three case are here considered, namely $p_{\textit{rms}}/\!p_s=0$ , $p_{\textit{rms}}/\!p_s=0.01$ , $p_{\textit{rms}}/\!p_s=0.1$ . The first one coincides with the original rotopole model.

Figure 26. Acoustic pressure coefficient decay rate in the radial direction, considering the tri-rotpole model: $\tilde {p}_{\textit{rms}} /p_s = 0.0$ yellow solid line; $\tilde {p}_{\textit{rms}} /p_s = 0.01$ red-dot line; $\tilde {p}_{\textit{rms}} /p_s= 0.1$ blue-diamond marker line. Contribution associated to: (a) the normal $\boldsymbol{n}_1$ l; (b) the normal $\boldsymbol{n}_2$ . Note that the contribution associated to the normal $\boldsymbol{n}_3$ is zero.

Figure 27. Acoustic pressure coefficient decay rate in the streamwise direction, considering the tri-rotpole model: $\tilde {p}_{\textit{rms}} /p_s = 0.0$ , yellow solid line; $\tilde {p}_{\textit{rms}} /p_s = 0.01$ , red-dot line; $\tilde {p}_{\textit{rms}} /p_s= 0.1$ , blue-diamond marker line. Contribution associated to: (a) the normal $\boldsymbol{n}_1$ ; (b) the normal $\boldsymbol{n}_2$ ; (c) the normal $\boldsymbol{n}_3$ .

In figures 26 and 27, we show the acoustic pressure coefficient decay rate along the radial and streamwise direction, respectively. The microphones are placed along the same lines as shown in the previous figure 6. The original tri-rotpole model, in the near field, exhibits the $r^{-4}$ and $r^{-7}$ decay rate in the radial and streamwise direction, respectively; moving towards the far field, an asymptotic decay is evident (compare our figures 26 and 27 with figures $27\;\mathrm{and}\;28$ of Ianniello Reference Ianniello2016). However, we observe that pressure fluctuations induce a faster transition from the near-field high-order decay rate to the asymptotic ones. Specifically, the increase in the level of fluctuations anticipates this transition. As also argued by Ianniello (Reference Ianniello2016), this indicates that the presence of pressure fluctuations, always present in a turbulent flow field, reduces or even eliminates the destructive kinematic auto-interaction present in the case of rotating constant-pressure signals. This stochastic rotpole model highlights the role of fluid dynamic pressure fluctuations in noise propagation, providing additional conceptual support to our results.

To conclude, it is evident that in the rotor plane, where the tip of the blade contributes the most to loading noise generation, pressure fluctuations given by the released tip vortex are weak (as depicted in figure 16 c), thereby promoting a higher decay rate due to the phenomena of kinematic auto-interaction. However, along the wake of the rotor, the downwind face of the blade where the fluid flow exhibits separation and turbulent transition induces high turbulent pressure fluctuations, so that the phenomenon of kinematic auto-interaction is practically absent and the decay is linear.