Aeroacoustics of a forward-flight propeller ingesting small- and large-scale turbulent wakes | Journal of Fluid Mechanics

3.1. Aerodynamic characteristics of turbulence ingestion

In order to understand the physical mechanisms of turbulence ingestion noise when a propeller is subjected to turbulent inflows of distinct characteristics, i.e. ingestion of SST and LST, a comprehensive examination of the near-field flow must be conducted and subsequently, correlating the flow features to the far-field acoustics. In this section, the flow and turbulence characteristics will first be presented.

Figure 7.

Flow field visualisation of the (a–c) normalised mean root-mean-square (r.m.s.) of axial velocity component,

$\bar {u}_{\mathit{rms}}/U_\infty$

, and (d–f) normalised mean spanwise vorticity,

$\bar {\omega }_zR/U_\infty$

, for the isolated, SST ingestion and LST ingestion cases.

3.1.1. Overview of the propeller flow field

To begin, an overview of the flow development for all three configurations is presented. Figure 7 shows the r.m.s. of axial velocity normalised by the free stream velocity,

$u_{\mathit{rms}}/U_\infty$

, for the isolated, SST ingestion and LST ingestion cases, respectively. In the isolated configuration, shown in figure 7(a), velocity fluctuations are minimal, confined primarily to the immediate blade-tip region and the downstream of the tip region, as a result of the tip vortices developing and being convected downstream. The wake structure is essentially symmetrical. The SST ingestion case, displayed in figure 7(b), introduces heightened axial velocity fluctuations directly upstream of the propeller, induced by small-scale vortices shed from the upstream cylinder. These fluctuations are limited to the region above the axis of rotation, resulting in localised blade–vortex interactions. The propeller wake region becomes asymmetric, with the top side (i.e. where the cylinder is located) having notably higher

$u_{\mathit{rms}}/U_\infty$

magnitudes. The tip–vortex trajectory remains largely unaltered, although a visible increase in velocity fluctuation strength can be seen downstream of the propeller blades. Conversely, the LST ingestion case, shown in figure 7(c), exhibits substantial axial velocity fluctuations in the wake of the cylinder, which is extended into the propeller wake, covering a wide spatial extent. The magnitude of the fluctuations reaching the propeller plane is lower than that in the SST ingestion case, as the cylinder wake decays over a larger distance in LST ingestion; however, the much greater ‘extent’ of turbulence significantly enhances its interaction with the propeller. Here, the propeller wake region becomes highly fluctuating, with complete loss of a clearly defined tip–vortex trajectory above the rotation axis (i.e. top ‘half-plane’). Unlike the isolated and SST ingestion cases, this large-scale interaction also impacts the flow development at the bottom side, with elevated velocity fluctuations surrounding the blades and in the wake, when compared with the isolated and SST ingestion cases. In addition, despite the fact that it retains the development of the tip vortices, the trajectory is noticeably more widespread with corresponding elevated velocity fluctuations. To complement the r.m.s. contours, figure 7(d –f) shows the normalised mean spanwise vorticity,

$\bar {\omega }_z R/U_\infty$

, for the three configurations. The isolated case in figure 7(d) shows the vorticity with symmetric development of the propeller wake, with the tip vortices being well defined in terms of magnitude and shape. For the SST ingestion case in figure 7(e), the additional vorticity introduced by the cylinder wake is largely restricted to the upper-half of the propeller plane. The propeller tip–vortex structures remain relatively compact, with signs of lower magnitude and mixing in the upper part. The lower vortex trajectory and magnitude remain unchanged. In contrast, the LST ingestion case shown in figure 7(f) exhibits a much broader vorticity variation, with the wake encompassing the majority of the propeller plane. As a result, the interaction is no longer confined to the upper half-plane, but spreads across a substantial portion of the propeller disk, including the lower side. The resultant tip vortices show a significant decrease in magnitude for the whole wake, as well as a loss in structure for the positive y section of the propeller wake.

Figure 8.

Instantaneous radial vorticity,

$\omega _rR/U_{\infty }$

, on two cylindrical surfaces: (a,c,e)

$r/R=0.7$

, (b,d,f)

$r/R=0.99$

) for (a,b) the isolated propeller, (c,d) SST ingestion and (e,f) LST ingestion cases. Upstream cylinders are omitted for clarity.

Figure 8. Long description

Panel A: A 3D rendering showing radial vorticity on a cylindrical surface for an isolated propeller. The propeller is depicted in the center with a color gradient indicating vorticity values. Panel B: A 3D rendering showing radial vorticity on a cylindrical surface for an isolated propeller from a different angle. The propeller is depicted in the center with a color gradient indicating vorticity values. Panel C: A 3D rendering showing radial vorticity on a cylindrical surface for the SST ingestion case. The propeller is depicted in the center with a color gradient indicating vorticity values. Panel D: A 3D rendering showing radial vorticity on a cylindrical surface for the SST ingestion case from a different angle. The propeller is depicted in the center with a color gradient indicating vorticity values. Panel E: A 3D rendering showing radial vorticity on a cylindrical surface for the LST ingestion case. The propeller is depicted in the center with a color gradient indicating vorticity values. Panel F: A 3D rendering showing radial vorticity on a cylindrical surface for the LST ingestion case from a different angle. The propeller is depicted in the center with a color gradient indicating vorticity values. The color gradient ranges from blue to red, representing different vorticity values.

Figure 8 illustrates the instantaneous radial vorticity (

$\omega _r R/U_{\infty }$

) on cylindrical surfaces with radii of

$r/R=0.7$

(figure 8
a,c,e) and

$r/R=0.99$

(figure 8
b,d,f), for the isolated propeller (figure 8
a,b), SST ingestion (figure 8
c,d), and LST ingestion (figure 8
e,f) cases. The radius

$r/R=0.7$

is selected as the location where the phase-averaged r.m.s. of blade surface pressure fluctuation mapsindicate the strongest turbulence–blade interaction (see § 3.1.4 on the blade-level analysis), while

$r/R=0.99$

allows the analysis of the effect of ingestion on the tip–vortex organisation for the instantaneous field. In the isolated case, coherent tip vortices dominate the vorticity field, clearly defined as organised helical structures being convected downstream from the blade, suggesting that these tip vortices are periodic with reference to the blade pass frequency, typical of propeller wake flow fields. For the SST ingestion case shown in figures 8(c) and 8(d), small vortical structures between the helical tip–vortex trajectory are visible, which interact directly with the blades and slightly alter the otherwise coherent wake. This interaction is only visible in the positive y-direction, which corresponds to the location of the cylinder. Larger fluctuations can be observed at

$r/R=0.99$

(see figure 8
d), where the cylinder wake can be discerned to disrupt the coherent organisation of the tip vortices, resulting in higher levels of vorticity, i.e. spanwise vorticity as seen in figure 7(e). Nonetheless, the tip vortices remain recognisable despite the heightened level of turbulence, indicating limited distortion by the small-scale turbulent structures. In the negative y-direction, the vortices appear less disturbed than for the direction exposed to the cylinder wake, although appearing weaker than in the isolated case. This reduction can be attributed to the global effect of the ingested wake on the propeller loading and the overall near-wake development. The radial vorticity for the LST ingestion case, as displayed in figures 8(e) and 8(f), is most disorganised, with a large area of the ‘surface’ having elevated vorticity. This level of enhanced interaction between the incoming turbulence and the propeller wake disrupts the coherent formation of the tip vortices as observed in the isolated and SST ingestion cases. At the inner radius (

$r/R=0.7$

), large-scale turbulent eddies are evident, promoting spatial spread of turbulent structures, manifested as regions with greater vorticity. Moreover, the vorticity field is perturbed both at the top and bottom sides of the propeller, suggesting that the propeller–turbulence interaction can encompass the entire diameter of the propeller (this is verified by observing a series of consecutive instantaneous vorticity fields over several propeller revolutions). At the outer radius (

$r/R=0.99$

), the tip vortices become heavily distorted, and their clear helical shape is replaced by broad patches of disorganised and intense vorticity at the top side. This is consistent with the wider interaction footprint of the LST ingestion case and explains why stronger velocity fluctuations are observed despite the weaker apparent amplitude in the immediate wake as noted in figures 7(c) and 7(f).

The direct flow field results indicate that the interactions between the propeller and the two cylinder wakes are distinct for the current configuration. In the SST ingestion case, the wake reaches the propeller as a relatively compact disturbance, confined largely to the upper side of the disk, and the resulting perturbation remains correspondingly localised. In contrast, although the local fluctuation amplitude in the immediate upstream wake of the LST ingestion case is lower than in the SST ingestion case because of the greater decay distance, the incoming vortical structures are distributed over a much larger spatial extent when they reach the propeller plane. The sectional vorticity fields show that these structures perturb the upper and lower half-plane of the propeller through the wider redistribution of the wake-induced vorticity. This elucidates why the LST ingestion case produces more elevated velocity fluctuations over a larger portion of the disk, despite appearing weaker than the SST ingestion case in the immediate wake core. The present comparison, therefore, highlights that the propeller response is governed not only by the local amplitude of the incoming wake but by the combined effect of wake scale and spatial footprint at the interaction plane. This interaction is likely to promote significant unsteady loading on the blade, which can lead to an increase in both tonal and broadband components of the propeller noise, which will be further illustrated in the discussion below.

Figure 9.

Profiles of (a,b) normalised mean axial velocity

$u/U_{\infty }$

; (c,d) normalised mean upwash velocity

$v/U_{\infty }$

; (e,f) r.m.s. of axial velocity

$u’/U_{\infty }$

; (g,h) r.m.s. of upwash velocity

$v’/U_{\infty }$

in the cylinder wake at (a,c,e,g)

$0.5R$

and (b,d,f,g)

$0.1R$

upstream of the propeller.

3.1.2. Propeller–cylinder wake characterisation

The introduction of a propeller in the wake of the turbulent von Kármán street is representative of a realistic eVTOL design and operation, given the vastly different configurations and inflow conditions these vehicles will experience. It is therefore essential to characterise the effect the cylinders have on the propeller and vice versa. Figure 9 presents the profiles of mean and r.m.s. velocity at two axial stations upstream of the propeller:

$x/R=0.5$

(figure 9
a,c,e,g) and

$x/R=0.1$

(figure 9
b,d,f,h). The normalised mean axial and upwash velocities are plotted in figure 9(a–d), whilst the r.m.s. of the axial and upwash velocity fluctuations are plotted in figure 9
(e–h). For the isolated case,

$u/U_\infty$

is nearly uniform and close to unity, with a symmetric velocity increase due to flow acceleration by the propeller visible at the second station. Upstream turbulence modifies these mean profiles in a way consistent with the characteristics of the cylinder wake. At

$x/R=0.5$

, the LST ingestion case exhibits a broader axial deficit

$u/U_\infty \lt 1$

centred about the midspan, together with a weak upwash profile

$v/U_\infty$

, indicating the presence of cross-stream momentum associated with large-scale wake structures. The SST ingestion case exhibits a stronger axial distortion with a more localised impact location, with positive and negative values of

$v/U_\infty$

, consistent with well-defined ‘gust’ eddies. At

$x/R=0.1$

, the suction effect of the propeller increases

$u/U_\infty$

near the hub and flattens the mean velocity profile, but a residual skewness remains for the LST ingestion case, while SST is nearly recovered to the profile shape of the isolated case, with only a small velocity deficit at

$y/R=0.5$

. The

$v/U_\infty$

profile shows minor differences between the cases, with only the LST ingestion case exhibiting a lower upwash velocity in the top-half of the propeller disk.

The results from the velocity r.m.s. show a consistent trend with those observed in the mean profile. In the isolated case,

$u_{\textit{rms}}/U_\infty$

and

$v_{\textit{rms}}/U_\infty$

do not show any fluctuation upstream of the propeller as expected; however, at

$x/R=0.1$

the shape closely follows the velocity profiles. With SST ingestion,

$u_{\textit{rms}}/U_\infty$

and

$v_{\textit{rms}}/U_\infty$

develop distinct peaks that are confined to a narrow band about the wake centre line. These peaks diminish close to the propeller at

$x/R=0.1$

, caused by the increase in local velocity due to the propeller, largely alleviating the extent of velocity deficit of the cylinder wake. In the LST ingestion case, the fluctuation profiles are of similar magnitude but broader in

$y/R$

. It is observed that

$u_{\textit{rms}}/U_\infty$

increases across the top side of the disk, whilst

$v_{\textit{rms}}/U_\infty$

appears to have a larger effect on

$y/R$

, past the hub location, which implies strong cross-stream unsteadiness carried into the propeller.

Figure 10.

Normalised phase averaged turbulent kinetic energy,

$\mathrm{TKE}/U_{\infty }^2$

, along the streamwise coordinate

$x/R$

for the isolated (a,b), SST (c,d) and LST (e,f) ingestion cases along the top shear layer of the cylinder,

$y/R = 0.5+(d_{\mathit{SST}}, d_{\mathit{LST}})$

, and along the centre line of the cylinder,

$x/R=0.5$

. Dashed lines mark the periodic shedding.

Figure 10. Long description

Panel A: A heat map showing turbulent kinetic energy distribution along the top shear layer of the cylinder for the isolated case. The x-axis represents the streamwise coordinate x/R, and the y-axis represents the angle ψ. The color scale ranges from blue to red, indicating increasing turbulent kinetic energy. Panel B: A heat map showing turbulent kinetic energy distribution along the center line of the cylinder for the isolated case. The x-axis represents the streamwise coordinate x/R, and the y-axis represents the angle ψ. The color scale ranges from blue to red, indicating increasing turbulent kinetic energy. Panel C: A heat map showing turbulent kinetic energy distribution along the top shear layer of the cylinder for the SST ingestion case. The x-axis represents the streamwise coordinate x/R, and the y-axis represents the angle ψ. The color scale ranges from blue to red, indicating increasing turbulent kinetic energy. Dashed lines mark the periodic shedding. Panel D: A heat map showing turbulent kinetic energy distribution along the center line of the cylinder for the SST ingestion case. The x-axis represents the streamwise coordinate x/R, and the y-axis represents the angle ψ. The color scale ranges from blue to red, indicating increasing turbulent kinetic energy. Dashed lines mark the periodic shedding. Panel E: A heat map showing turbulent kinetic energy distribution along the top shear layer of the cylinder for the LST ingestion case. The x-axis represents the streamwise coordinate x/R, and the y-axis represents the angle ψ. The color scale ranges from blue to red, indicating increasing turbulent kinetic energy. Panel F: A heat map showing turbulent kinetic energy distribution along the center line of the cylinder for the LST ingestion case. The x-axis represents the streamwise coordinate x/R, and the y-axis represents the angle ψ. The color scale ranges from blue to red, indicating increasing turbulent kinetic energy.

Phase-averaging is then performed with the time-series velocity data mapped to the phase angle of the propeller along the propeller azimuth,

$\psi$

, with

$\psi = 0^\circ$

corresponding to the blades being horizontal, as previously in figure 1(b). Figure 10 shows the normalised phase-averaged turbulent kinetic energy,

$\mathrm{TKE}/U_{\infty }^2$

, along the streamwise coordinate

$x/R$

for the isolated (figure 10
a,b), SST (figure 10
c,d) and LST (figure 10
e,f) ingestion cases, respectively. The results are presented at two radial locations: figure 10
(a,c,e) is along the top shear layer of the cylinder,

$y/R = 0.5+(d_{\mathit{SST}}, d_{\mathit{LST}})$

, respectively, whilst figure 10(b,d,f) is along the centre line of the cylinder,

$y/R = 0.5$

. Regardless of the vertical location (

$y/R$

), the isolated case exhibits a steady, low value of turbulent kinetic energy for essentially all phase angles and distances upstream of the propeller plane except the two hot spots at

$\psi =90^\circ \text{ and } 270^\circ$

, which coincide with the blade passing across the probe lines. Figures 10(c) and 10(d) show that for the SST ingestion case, the same blade-pass features appear close to the propeller plane, and an additional pattern emerges from

$x/R \approx 0.5$

upstream. These bands of elevated turbulent kinetic energy slope with

$\psi$

in a manner consistent with the vortex-shedding frequency of the small cylinder,

$f_{0,\textit{ SST}}\approx {0.5}\,\mathrm{BPF}$

, confirming that the propeller is ingesting a coherent von Kármán vortex street, and more importantly, the turbulent wake is phase-locked to the propeller. Along the wake centre line, these ‘bands’ are broader, indicating greater turbulence intensity at the vortex cores. In contrast, along the shear layer, the energy is confined to the previously delineated markings. Figures 10(e) and 10(f) show a high turbulent kinetic energy region developing at

$x/R\approx 2.5$

, similar to the SST case; however, it is nearly ‘homogeneously’ increased across phase angle,

$\psi$

, and gradually and monotonically decays monotonically with streamwise distance,

$x/R$

, albeit on average, remains significantly higher than those of the isolated and SST ingestion cases. This is a consequence of the lower vortex-shedding frequency

$f_{0,\textit{ LST}}\approx {0.1}\,\mathrm{BPF}$

of the large cylinder. Since the propeller completes several revolutions during a single shedding cycle, the phase-averaging captures the ‘spread’ development due to the lack of synchronisation. The absence of blade-pass hot spots along the top shear layer is due to the vertical offset of the probes, which coincide with

$y/R\approx 0.9$

, where the effect of the blade pass is significantly reduced. The observed behaviour depends on the coupling between the cylinder shedding frequency and the BPF, which is controlled by the propeller rotational speed, the cylinder diameter and the cylinder Reynolds number. Varying the propeller revolutions per minute at fixed cylinder geometry or varying the cylinder diameter and Reynolds number at fixed revolutions per minute would modify the ratio between the shedding frequency and the BPF, and hence the degree of phase-locking, the compactness of the side peaks and the likelihood of haystacking. These parameters are not varied independently in the present work and are identified as important directions for future investigation.

Figure 11 shows the power spectral density of

$\phi _{uu}$

,

$\phi _{vv}$

and

$\phi _{ww}$

versus the streamwise coordinate

$x/R$

for the SST (figure 11
a–c) and LST (figure 11
d–f) ingestion cases along the cylinder wake centre line,

$y/R = 0.5$

. For the SST ingestion case, each velocity component exhibits ‘tone-like’ narrow peaks at the BPF and its integer harmonics, with its principal maximum occurring at the closest measurement station to the propeller plane (

$x/R=0.1$

). The tone remains discernible upstream to

$x/R\approx 0.5$

, beyond which it decays as the wake region is characterised by recirculation. Along the centre line, the cylinder vortex shedding at

$f_{0,\textit{ SST}}$

is not observed and is overwhelmed by the stronger blade-pass effect. Higher harmonics (

$f/\text{BPF}=2,3,\ldots$

) are present, but weaken rapidly with both frequency and distance. The cross-stream spectra

$\phi _{vv}$

exceed

$\phi _{uu}$

, reflecting the dominance of transverse fluctuations in the near wake and revealing a high level of turbulence, and potentially anisotropy in the flow. For the LST ingestion case shown in figure 11(d –f), a broad spectral hump at

$f/\text{BPF}=0.1$

can be observed, which corresponds to the shedding frequency of the large cylinder. This low-frequency hump is most prominent in the

$\phi _{vv}$

component and has the highest magnitude downstream of the cylinder, where the wake has fully developed. In contrast with the SST case, the ‘tone-like’ narrow peaks at the BPF and its harmonics are not quite visible. Moreover, the three velocity components exhibit comparable levels of fluctuation energy, signalling the transition towards isotropic turbulence in the wake of the larger cylinder. Both the phase-averaged velocity and the power spectral density of the velocity fluctuations upstream of the cylinder reveal that there exists a notable flow ‘phase-locking’ effect between the wake of the small cylinder and the propeller, which is mostly absent for the large cylinder. As a result, the hydrodynamic and acoustic behaviour of the propeller–turbulence interaction is likely to reflect this effect and exhibit distinct characteristics.

Figure 12.

Comparison of normalised length scales

$L_{u,\,x}(x)$

,

$L_{v,\,x}(x)$

and

$L_{w,\,x}(x)$

between the isolated cylinder cases and (a–c) the SST ingestion and (d–f) the LST ingestion cases extracted at three spanwise locations,

$z/R=0,\, 0.5,\, 1$

.

Figure 12. Long description

The image contains six line graphs comparing normalized length scales for isolated cylinder cases and ingestion cases at different spanwise locations. Panel A: A line graph shows the normalized length scale Lu,x/R against x/R for the isolated cylinder and SST ingestion cases. The x-axis ranges from 0.15 to 0.60, and the y-axis ranges from 0 to 0.15. The legend indicates different spanwise locations with solid, dashed, and dotted lines. Panel B: A line graph shows the normalized length scale Lv,x/R against x/R for the isolated cylinder and SST ingestion cases. The x-axis ranges from 0.15 to 0.60, and the y-axis ranges from 0 to 0.15. The legend indicates different spanwise locations with solid, dashed, and dotted lines. Panel C: A line graph shows the normalized length scale Lw,x/R against x/R for the isolated cylinder and SST ingestion cases. The x-axis ranges from 0.15 to 0.60, and the y-axis ranges from 0 to 0.15. The legend indicates different spanwise locations with solid, dashed, and dotted lines. Panel D: A line graph shows the normalized length scale Lu,x/R against x/R for the isolated cylinder and LST ingestion cases. The x-axis ranges from 0.50 to 2.75, and the y-axis ranges from 0 to 0.15. The legend indicates different spanwise locations with solid, dashed, and dotted lines. Panel E: A line graph shows the normalized length scale Lv,x/R against x/R for the isolated cylinder and LST ingestion cases. The x-axis ranges from 0.50 to 2.75, and the y-axis ranges from 0 to 0.15. The legend indicates different spanwise locations with solid, dashed, and dotted lines. Panel F: A line graph shows the normalized length scale Lw,x/R against x/R for the isolated cylinder and LST ingestion cases. The x-axis ranges from 0.50 to 2.75, and the y-axis ranges from 0 to 0.15. The legend indicates different spanwise locations with solid, dashed, and dotted lines.

3.1.3. Length scale and convection of the inflow turbulence

To estimate the size of the turbulent eddies ingested by the propeller and determine the average distance over which velocity fluctuations are correlated, the integral length scales for each component of velocity evaluated along the streamwise direction,

$L_{i,\,x}=(L_{u,\,x},\,L_{v,\,x},\,L_{w,\,x})$

, are calculated based on the following equation (Nicolaides, Honnery & Soria Reference Nicolaides, Honnery and Soria2004):

(3.1)

\begin{align} L_{i,\,x}=\overline {U}\int _{0}^{\tau _{0,\,i}}\rho _{ii}(\tau )\,\mathrm{d}\tau ,\qquad i\in \{u,v,w\}, \end{align}

where

$\overline {U}$

is the local mean velocity magnitude,

$\rho _{ii}(\tau )$

is the normalised autocorrelation of the corresponding fluctuating velocity component and

$\tau _{0,\,i}$

denotes the first zero-crossing of the autocorrelation. This provides a measure of the average streamwise distance over which each velocity component remains correlated. Figure 12 traces the streamwise evolution of the length scales

$L_{u,\,x}$

,

$L_{v,\,x}$

, and

$L_{w,\,x}$

normalised by the propeller radius

$R$

for three spanwise locations (

$z/R=0,\,0.5,\,1$

) of both the SST (figure 12
a–c) and LST (figure 12
d–f) ingestion cases upstream of the plane of rotation. The results for the isolated cylinder cases are overlaid on the installed cases for comparison, extracted as a mean over five spanwise locations in the wake of the cylinders. For the SST case (figure 12
a,c,e), the length scales of all three velocity components remain small and exhibit an overall increasing trend as the turbulent eddies are convected downstream, with values

$L_{i,\,x}\lt 0.02R$

in the wake. A small rise in

$L_{u,\,x}$

to

$\approx 0.05R$

between

$x/R=0.3$

and the propeller plane is observed towards the blade tip (

$z/R=1$

), where the blades largely distort the flow. The transverse and spanwise length scales,

$L_{v,\,x}$

and

$L_{w,\,x}$

, exhibit only minor variations along the span. Comparison with the isolated-cylinder case shows that the SST integral length scales remain similar in magnitude, indicating that the propeller does not substantially alter the characteristic correlation scale of the compact structures before ingestion. Rather, the primary effect appears to be local distortion of small-scale eddies, largely distorted by the blade tip, which produce predominantly tonal noise through periodic tip–vortex interactions (Gonzalez-Martino et al. Reference Gonzalez-Martino, Romani, Wang and Casalino2018). This observation aligns closely with the far-field spectra presented in figure 6(b), where a number of distinct tones in the midfrequency range were observed for the SST ingestion case. In fact, the sizes of the turbulent eddies are generally less than 3 % of the blade radius, and their interaction with the propeller is primarily coupled through ‘locked’ temporal coherence rather than size, as seen in figures 10 and 11. In contrast, the LST ingestion case (figure 12
b,d,f) contains much larger structures as well as a wider spread of length scales between the different spanwise locations. Here

$L_{u,\,x}$

shows a constant growth up to very close to the propeller plane, reaching

$0.12R$

. In this case, comparison with the isolated cylinder shows a more pronounced deviation, indicating that the presence of the propeller alters the correlation scale of the incoming large structures more substantially than in the SST case. This suggests that the propeller actively stretches the incoming coherent structures as they approach the blades, as it has been observed in figure 8. Interestingly, The spanwise variations of

$L_{u,\,x}$

reveal an interesting development: the trend is opposite to that of the SST case (e.g. compare figures 12
a and 12
d), which the length scale is the smallest at the blade tip location (

$z/R=1$

), suggesting that interacting with the blade tip results in a breakdown of the turbulent eddies, rather than a growth of the length scale possibly through vortex-stretching. The same trend is observed in

$L_{v,\,x}$

and

$L_{w,\,x}$

, though the length scales are slightly smaller and diverge less along the spanwise direction. For the LST ingestion, the sizes of the eddies are generally less than 12 % of the blade radius at the ingestion plane, approximately four times that of the SST ingestion case. The presence of large turbulent structures, the spatial extent and nature of their interaction (e.g. stretching and breakdown at different locations of the blade) could yield significant aerodynamic loading fluctuations on the blades of more broadband nature, consistent with the broadband increase as well as the presence of clear haystacking hump observed in far-field spectra as shown in figure 6(c). These results are in agreement with previous work by Molinaro et al. (Reference Molinaro, Balantrapu, Hickling, Alexander, Devenport and Glegg2017) and Kankanwadi & Buxton (Reference Kankanwadi and Buxton2023), and highlight a fundamental shift in the turbulence–blade interaction mechanism governed by the spatial scales of the incoming turbulence.

To relate the measured turbulence length scales of the inflow to the unsteady loading of the propeller, the local convection velocity at the ingestion plane,

$U_{c}$

, was estimated using the probes in the wake of the cylinder aligned with the free stream. For each pair of consecutive probe locations, denoted by subscripts

$j$

and

$j+1$

, the time-series data were bandpass filtered around the cylinder shedding frequency, and the time lag,

$\tau ^*_{j,\,j+1}$

, was obtained from a band-limited generalised cross-correlation with phase transform weighting (Romano Reference Romano1995; Wallace Reference Wallace2014). The weighting used normalises the cross-spectrum by its magnitude, emphasising phase agreement rather than signal amplitude. This results in the delay estimate being more robust to differences in fluctuation level between probes and to broadband contamination unrelated to the dominant convecting wake structures. Since the computation is restricted to a narrow band around the cylinder shedding frequency, the resulting delay is associated primarily with the convection of the shedding-related wake structures. The baseline estimate is then given by

(3.2)

\begin{align} U_{c,\, j} \;=\; \frac {\Delta x_j}{\tau ^*_{j,\,j+1}}. \end{align}

The convection velocity at the ingestion plane is then calculated as

$U_{c} = \mathrm{median}\{U_{c,\, j}\}$

, giving a value which is robust to spurious delays and heavy-tailed scatter (Knapp & Carter Reference Knapp and Carter1976). When applied to the SST and LST ingestion cases, this procedure yields

$U_c/U_\infty$

values of

$1.15$

and

$1.04$

, respectively, consistent with axial acceleration into the propeller disk. For the SST ingestion case, taking a maximum of

$L_{u,\,x}/R = 0.05$

, the time taken for the eddy to be convected past the blade is approximately

${0.5}\,\mathrm{ms}$

, which is notably shorter than the time for the two blades to consecutively cut through a given azimuthal location, approximately

${3.7}\,\mathrm{ms}$

.

A further flow analysis follows the classical gust-response approach (Amiet Reference Amiet1975) through evaluating the convective Strouhal number,

$St_c$

, which is given by

(3.3)

\begin{align} St_c=\frac {U_c/L_{u,\,x}}{\mathrm{BPF}}, \end{align}

where

$U_c$

is the convective velocity and

$L_{u,\,x}$

is the streamwise turbulent length scale at the ingestion plane obtained from (3.1). The convective Strouhal number,

$St_c$

, compares the convective rate of the turbulence,

$U_c$

, with the blade-passing rate,

$L_{u,\,x}$

. Large

$St_c$

indicates that the turbulence seen by a blade section varies rapidly within a blade-pass interval, favouring ‘tone-like’ response, whilst a small value of

$St_c$

indicates slowly varying eddies over a blade passage and is associated with tone broadening (Homicz & George Reference Homicz and George1974). In the present work, the SST ingestion case yields

$St_c\simeq 7.9$

(taking

$L_{u,\,x} = 0.05R$

), consistent with interactions with relatively small eddies; the LST ingestion case gives

$St_c\simeq 1.64$

(taking

$L_{u,\,x} = 0.15R$

), consistent with the blade response to slowly varying eddies.

Following this analysis, it can be concluded that for the SST ingestion case, the majority of the incoming turbulent eddies pass through the propeller plane without interacting multiple times with the blades, resulting clearly in a ‘phase-locked’ behaviour, modulating the periodic motion of blade passes. In contrast, in the LST ingestion case, the much larger turbulent eddies can potentially interact multiple times with the blades, giving rise to more complex and stochastic processes, centred around the periodic blade motion.

3.1.4. Blade-level analysis – unsteady sectional forces

To shed further light on the turbulence interaction and understand the mechanism for noise generation, it is essential to conduct a blade-level analysis. This includes the unsteady lift experienced by the blades as well as identifying the noise sources from blade-level information, such as pressure fluctuations and blade-to-blade correlation.

Figure 13 shows the phase-dependent thrust coefficient,

$C_T(\psi )$

, calculated from a single propeller blade for the isolated, SST and LST ingestion cases, extracted over all 40 revolutions. Plotting the time-varying

$C_T$

as a function of phase angle,

$\psi$

, allows for the visualisation of the turbulence interaction and the extent to which it affects the thrust. For the isolated case shown in figure 13(a), the thrust coefficient appears to be steady with a level around 0.045 for all phase angles. A similar level is observed in the SST ingestion case in figure 13(b), with narrow fluctuations observed as the blade passes through the turbulent wake region. This effect can be observed in the shaded region between the angles of

$\psi = {15}^\circ$

and

$\psi = {175}^\circ$

, with a trough at

$\psi = {90}^\circ$

. This aligns with the confined region of interaction observed in figure 8. The thrust coefficient measured for these angles is within

$\pm 10\,\%$

of the average

$C_T$

. However, the LST ingestion case shown in figure 13(c) reveals much stronger variations with reference to blade phase, exhibiting large fluctuations of

$\pm 30\,\%$

of the mean

$C_T$

, and thus

$C_T$

becomes highly phase-dependent. The shaded region is extended to encompass the angles between

$\psi = {300}^\circ$

and

$\psi = {240}^\circ$

as shown in the figure. This region reflects the constant interaction between the blade and the large-scale turbulent structures present in the inflow, as identified in figures 8 and 7. Unlike the SST ingestion case, the LST ingestion case produces thrust perturbations that are distributed over a larger portion of the blade’s rotation. The unsteady loading variations of 10 % and 30 % are approximately proportional to the differences in the upwash velocity r.m.s. and turbulent length scales between the two cases at the propeller planes, providing a direct link between the nature of the incoming turbulence, the unsteady aerodynamic loading on the blades, and later the spectral characteristics of the radiated noise.

Figure 13.

Phase-dependent thrust coefficient

$C_T(\psi )$

of a singular blade for (a) isolated, (b) SST ingestion and (c) LST ingestion cases. The shaded area signifies the turbulence interaction region.

Figure 14 presents the power spectral density of the total propeller thrust for the isolated, SST ingestion and LST ingestion cases with marked

$f=\text{BPF}- nf_{0}$

(dashed line) and

$f= \text{BPF}+ nf_{0}$

(solid line) for

$n=1,\,2,\,3$

. In the isolated propeller configuration, the spectrum exhibits no discernible tone at the BPF, with levels at

$f/\text{BPF}=1$

lying below the neighbouring harmonics. This is due to the ‘total’ thrust being fundamentally steady with respect to the blade-passing frequency. The SST ingestion case shows a similar lack of BPF energy, despite the presence of inflow turbulence and imbalance in the blade loading. However, the length scales were observed to remain below 5 % of the blade radius (see figure 12), resulting in the induced unsteady loading remaining largely blade-local between opposed azimuthal positions. In addition, clear tones appearing at

$f\approx \text{BPF}+ nf_{0,\textit{ SST}}$

can be observed as marked in the figure, with subsequent harmonics at multiples of these frequencies (

$m\mathrm{BPF}\pm nf_0$

). The combination of

$m$

and

$n$

creates several overlapping tones, resulting in the comb-like structure in the midfrequency range observed in the SST ingestion case. Since

$f_{0,\textit{ SST}}/\mathrm{BPF}\approx 0.5$

, these modulation tones fall at integer and half-integer multiples of the BPF, which results in the appearance of multipeak tones at 2, 2.5, 3, 3.5 and higher frequencies. In contrast, the LST ingestion spectrum contains side peaks at

$f\approx \text{BPF}\pm nf_{0,\textit{ LST}}$

, which fall within the broad hump centred on the BPF, consistent with the haystacking-like broadening observed in the far-field spectra. It can be noted that each of the marked

$\mathrm{BPF}\pm nf_{0,\textit{ LST}}$

frequencies is coincident with a narrow tone within the hump. These arise when LST wake flow structures, with a large integral length scale, impose simultaneous loading variations across the two blades. As the blades cut through spatially and temporally correlated structures, it allows energy at these specific frequencies to show in the spectrum. Further harmonics are displayed as humps around the multiples of the BPF. This correlated interaction between blades and large turbulent structures is a known mechanism for enhancing broadband noise generation (Brooks et al. Reference Brooks, Pope and Marcolini1989; Jamaluddin et al. Reference Jamaluddin, Celik, Baskaran, Rezgui and Azarpeyvand2023; Raposo & Azarpeyvand Reference Raposo and Azarpeyvand2024), as observed beyond

$f/\text{BPF}\gt 1.5$

.

Figure 15 maps the phase-averaged r.m.s. of blade surface pressure fluctuation,

$p_{\mathit{rms}}(\psi ,\,r/R)$

, normalised by the tip dynamic pressure

$p_{\mathit{rms}}^*=p_{\mathit{rms}}/(\rho U_{\mathit{tip}}^2)$

for the isolated, SST and LST ingestion cases. As there is no disturbance introduced for the isolated case, the

$p_{\mathit{rms}}^*$

contour map is essentially identical at each azimuthal position, with narrow footprints appearing in the midspan of the blade. With SST ingestion, shown in figure 15(b), the result shows a significant difference in

$p_{\mathit{rms}}$

, with the leading edge (LE) of the blade experiencing larger pressure fluctuations as it moves from

$\psi =30^\circ$

to

$\psi =150^\circ$

. The radial location of the effect of turbulence ingestion migrates, producing alternating hot spots that are not locked to a single sector and result in a larger area of the blade having high fluctuations. The azimuthal effect is, however, restricted to the region directly in the wake of the cylinder. For LST ingestion, shown in figure 15(c), the azimuthal angles affected increase as also observed previously in figure 13. The fluctuations appear to involve most of the blade area, with the most significant fluctuations located towards the LE, TE and the blade tip, while the root region remains relatively undisturbed. Zooming into the

$p_{\mathit{rms}}^*$

at

$\psi =10^\circ$

and

$180^\circ$

, which represent one revolution of the ‘two-bladed’ propeller, both the isolated and SST ingestion cases show similar behaviour with no significant fluctuations at the LE. On the contrary, the LST ingestion case exhibits an intense level of fluctuations at the LE with comparable patterns, suggesting that the interaction between the two ‘consecutive’ blades is likely to remain correlated, reaffirming the fact that the turbulent structures in the LST ingestion case can potentially interact multiple times with blades as it is convected downstream.

To quantify the footprint of the turbulence–blade interaction, a turbulence-interaction area

$A_{TI}$

fraction is defined as

(3.4)

\begin{align} A_{TI}=\frac {1}{A_b}\int ^{2\pi }_0\int ^R_0H(p^*_{\mathit{rms}}-p^*)r\,\mathrm{d}r\,\mathrm{d}\psi , \end{align}

where

$A_b$

is the blade disk area,

$H(\boldsymbol{\cdot })$

is the Heaviside function and

$p^*$

is a reference threshold chosen as

$0.9p_{\mathit{rms}}^*$

. Consistent with the results displayed in figure 15, the results show

$A_{TI,\mathit{\,LST}}\,(0.6)\gt A_{TI,\mathit{\,SST}}\,(0.4)\gg A_{TI,\mathit{\,isol}}\,(0.05)$

, with the largest increase from SST ingestion to LST ingestion arising from the large wake size as shown in figure 7(c).

Figure 15.

Phase-averaged r.m.s. of pressure fluctuation

$p_{\mathit{rms}}^*=p_{\mathit{rms}}/(\rho U_{\mathit{tip}}^2)$

for (a) the isolated, (b) SST ingestion and (c) LST ingestion cases. The shaded area represents the cylinder with respect to the propeller disk.

Taken together, the

$C_T(\psi )$

,

$\phi _{TT}(f)$

,

$C_T(r,\,\psi )$

and the

$p^*_{\mathit{rms}}(\psi ,\, R)$

results provide a direct link between inflow turbulence structure and unsteady blade loading. The SST ingestion case highlights a regime where discrete inflow disturbances lead to phase-dependent load fluctuations that may contribute to tonal noise. In contrast, the LST ingestion case illustrates a regime dominated by random inflow turbulence, which disrupts coherent loading patterns and is likely to lead to an increase in broadband noise emission. These distinctions align with theoretical models of propeller inflow-noise generation (Gutin Reference Gutin1948; Amiet Reference Amiet1975) and reinforce the need for turbulence characterisation when predicting propeller noise in complex inflow environments.

3.2. Acoustic characteristics of turbulence ingestion

Numerical simulations allow for a more comprehensive study, and performing noise source identification and decomposition will shed light on the turbulence interaction noise generation mechanisms. The far-field analysis is carried out using the acoustic modules in PowerFLOW, based on the impermeable surface formulation of the FW-H acoustic analogy Ffowcs Williams & Hawkings (Reference Ffowcs Williams and Hawkings1969) and a forward-time solution of Farassat’s Formulation 1A (Farassat & Succi Reference Farassat and Succi1982; Casalino Reference Casalino2003). The solid surfaces included for the FW-H include both the cylinder and the propeller (hub included).

3.2.1. Far-field noise directivity and power spectrum density

The power spectral density of the pressure signals at each microphone was computed with a frequency resolution of

$\Delta f={8}\,\mathrm{Hz}$

using Welch’s estimate with a 50 % overlap. The sound pressure level (SPL) is then calculated using

(3.5)

\begin{align} \text{SPL}(f,\,\theta ) = 10\,\text{log}_{10}\left (\frac {\phi _{p}(f)\Delta f}{p^2_{\textit{ref}}}\right ), \end{align}

where

$\phi _{p}(f)$

is the power spectral density based on the unsteady pressure fluctuations

$p'(t)$

(where

$p'(t) = p(t) – p_{\textit{atm}}$

) and

$p_{\textit{ref}}={20}\,{\unicode{x03BC}}\mathrm{ Pa}$

is the reference pressure. The OASPL at different angles

$\theta$

can be evaluated by integrating over the range of frequency of interest,

(3.6)

\begin{align} \text{OASPL}(\theta ) = 10\log _{10}\left (\frac {\displaystyle \int {\phi _{pp}(f)\,\mathrm{d}f}}{p^2_{\textit{ref}}}\right ), \end{align}

which is calculated over the range of

$160{-}{10\,000}\,\mathrm{Hz}$

. This range is chosen to eliminate the contribution from the lift harmonic of the cylinder to the OASPL results. The directivity of the first and second BPF is calculated by taking the value of the SPL spectra at

$f/\text{BPF}=1,2$

.

Figure 16 compares far-field acoustic power spectral density at

$\theta =90^\circ$

and

$180^\circ$

, respectively (recall figure 1 for the

$\theta$

angle definition). Figure 16(a) shows the loading-dominated radiation as the observer location is aligned with the plane of rotation. The isolated propeller exhibits a sharp tonal peak at the BPF and its harmonic, as expected for steady-loading dominant acoustic spectra (i.e.

$m=1$

, where

$m = 1,\, 2,\, \ldots$

represents integer multiples of BPF) (Scharpf & Mueller Reference Scharpf and Mueller1995). For the SST ingestion case, a noticeable number of discrete tonal peaks appear in the midfrequency range. Their spacing is set by the cylinder shedding frequency

$f_0$

, resulting in a comb structure with tones at

$m\mathrm{BPF}\pm nf_0$

(where

$n=1,\,2,\,\ldots$

denotes the integer multiples of fundamental vortex shedding frequency of a given cylinder), consistent with amplitude modulation characteristics observed in the literature (Marte & Kurtz Reference Marte and Kurtz1970; Robison & Peake Reference Robison and Peake2014; Yao et al. Reference Yao, Huang, Davidson, Niu and Chen2022; Dróżdż et al. Reference Dróżdż, Niegodajew, Romańczyk and Elsner2023). This results in a spectral signature that is dominated by coherent cylinder shedding at

$f_0$

, producing rotation-locked modulation of the blade-pass harmonics and tones. The broadband level increases only modestly relative to the isolated case since the stochastic part of the ingested turbulence is weaker than the coherent shedding component at the interaction region. In contrast, the LST ingestion case displays broad humps centred around the BPF and its harmonics rather than distinct ‘pairs of side-peaks’. The energy spreads around the BPF and its harmonics, showing signs of haystacking. This broadening is consistent with a slowly varying envelope imposed by large structures (see figure 26
b). The polar angle for figure 16(b) was chosen to minimise the steady loading contribution and allow a better focus on the interaction noise. The spectrum presented for the isolated case does not display any tones at the BPF or its harmonics. For the SST ingestion case, a series of tonal peaks remains visible at

$m\,\mathrm{BPF}\pm n f_0$

with no distinct tone at the BPF. The BPF and its harmonics show reduced contributions at this microphone position, confirming that the tones are a product of the blade–turbulence interaction rather than a direct loading contribution. For ease of discussion, these

$m\mathrm{BPF}\pm nf_0$

tonal peaks are referred to as sidebands. For the LST ingestion case, the BPF and its harmonics are surrounded by ‘broadband-like’ humps, reaffirming the presence of haystacking. At

$f/\mathrm{BPF}\gtrsim 10$

, all three cases converge towards similar decay rates, indicating that ingestion primarily reorganises energy around the BPF and harmonics, while at high frequencies, the TE noise becomes essentially dominant.

Figure 16.

Comparison of SPL spectra between isolated, SST ingestion and LST ingestion cases for the polar microphones corresponding to (a)

$\theta =90^\circ$

and (b)

$\theta =180^\circ$

.

Figure 17 shows the spectral density spectra of each of the observer locations probed as a function of the polar position

$\theta$

versus normalised frequency

$f/\text{BPF}$

for the isolated, SST and LST ingestion cases. In all cases, a narrow ridge at

$ f/\text{BPF}=1$

exhibits two pronounced ‘hotspots’ centred at

$\theta = 90^\circ\text{ and }{270}^\circ$

with relative minima upstream and downstream, consistent with compact-dipole radiation due to disk-wide periodic loading. The isolated case displays weak higher harmonics with a similar directivity pattern, while the rest of the broadband levels appear to remain between

$40\text{ and }{50}\,\textrm{dB}$

. For the SST ingestion case, displayed in figure 17(b), an ordered set of additional narrow bands appears at

$f/\text{BPF}=m\pm 0.5$

, which are produced by modulation at the cylinder shedding frequency

$f_{0,\textit{ SST}}\approx 0.5\text{ BPF}$

. The visible peak at

$f/\text{BPF}=0.5$

is characteristic of the cylinder vortex shedding lift harmonic and mirrors the propeller directivity (Jacob et al. Reference Jacob, Boudet, Casalino and Michard2005; Giret et al. Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015). Nonlinear harmonic loading excites higher-order harmonics to appear in the spectra, with increased spanwise phase variation, resulting in less compact source radiation. These tones do not exhibit the same angular pattern as the tonal peaks at

$f/\text{BPF}= 0.5$

and 1, indicating that the harmonics and the extra tones produced by turbulence ingestion have different directivities. The partial spanwise and interblade decorrelation render these tones to have a more uniform directivity. The broadband content, however, remains similar to the isolated case. The LST ingestion case shows a mechanism that redistributes energy from discrete tones into a ‘broadband-like’ spread around the BPF and its harmonics, as shown in figure 17(c). These broadband components are azimuthally diffuse and do not display the same directivity as the BPF, and are attributed to the diagonal track taken by the blades as they cut through the convecting turbulence structures, as recognised in the literature by Martinez (Reference Martinez1996) and Murray et al. (Reference Murray, Devenport, Alexander, Glegg and Wisda2018). Moreover, their amplitudes are more elevated, with values between

$50\text{ and } {60}\,\mathrm{dB}$

, and hence are manifested as broadening of the tones. Additional sidebands can be observed mainly around the BPF, suggesting that acoustic modulation due to the cylinder shedding frequency occurs to some extent at this fundamental blade-passing frequency. Overall, the SST ingestion case retains the classical loading-dipole character at the first BPF, with minima along the propeller axis at

$\theta = 0^\circ\text{ and } {180}^\circ$

due to the localised low-order loading fluctuations (see figure 13
b). In contrast, the LST ingestion case is subject to a broader inflow distortion across the disk, producing an azimuthal asymmetry in the blade loading that alters the classic dipole pattern at the first BPF. Turbulence ingestion additionally results in spectral broadening around

$m\mathrm{BPF}\pm nf_{0,\textit{ LST}}$

, which accentuates the region around the first BPF. As a result, a finite first BPF component becomes visible for the majority of the directivity angles.

Figure 17.

Spectral density contour plots as a function of far-field microphone position

$\theta$

at all angles examined in this study.

Figure 18 shows the directivities of the propeller noise for the three cases in terms of the OASPL in decibels and the directivity of the first and second BPFs, denoted by

$\mathrm{SPL}_{m}$

, (

$m=1,\,2$

). The OASPL curves are essentially axisymmetric about the horizontal plane for all cases. However, the ingestion cases show an increase in noise levels compared with the isolated propeller for both the upstream and downstream microphone locations. The LST case exhibits the largest increase, with a gain between

$5\text{ and } {10}\,\mathrm{dB}$

, while the SST case shows an increase below

${5}\,\mathrm{dB}$

. This indicates that the principal OASPL increase is broadband in nature, consistent with the additional harmonics in figure 17(b) and the haystacking effect observed in figure 17(c). At the first BPF (

$\mathrm{SPL}_{m=1}$

), the directivity patterns are almost identical for the isolated and the SST ingestion cases. These display a typical dipole-like pattern with two broad side-lobes having minima at

$\theta = 0^\circ\ \text{and}\ {180}^\circ$

, which suggests that the radiation is governed by the disk-wide periodic loading and is only weakly sensitive to the incoming SST. This is expected as the asymmetry in the loading remains small (Gutin Reference Gutin1948). The LST ingestion case exhibits an increase in levels on both the upstream and downstream angles, indicating that the large-scale inflow non-uniformity introduces an azimuthal contribution to the low-order BPF tone that ‘contaminates’ the reference dipole. Similar behaviour has been observed for propellers with an angle of attack, where the blade loading acquires an azimuthal dependence resulting in a tilt in directivity (Goyal et al. Reference Goyal, Sinnige, Ferreira and Avallone2025). Figure 18(c) shows that the second BPF tone (

$\mathrm{SPL}_{m=2}$

) has a greater sensitivity to turbulence for both ingestion cases. The SST ingestion case shows a consistent rise towards the upstream and downstream angles, reminiscent of the first BPF directivity of the LST ingestion case. This is expected as higher-order BPFs receive increasing contribution from the unsteady loading components, and thus the directivity of the SST ingestion case resembles more that of the LST ingestion case (Moreau et al. Reference Moreau, Mendonca, Qazi, Prosser and Laurence2005). Moreover, the LST ingestion case shows a more uniform increase in second BPF amplitude for all polar angles. As will be shown later, the second BPFs of both the SST and LST ingestion cases are further subjected to amplitude modulations, and hence, take a more ‘omnidirectional’ pattern.

Figure 18.

Directivities of propeller noise in terms of (a) OASPL (

${160}\,\mathrm{Hz}$

–

${10}\,\mathrm{kHz}$

), (b) first BPF (

$\text{SPL}_{m=1}$

) and (c) second BPF (

$\text{SPL}_{m=2}$

) for isolated, SST ingestion and LST ingestion cases.

Figure 18. Long description

Panel A: A polar plot displays the Overall Sound Pressure Level (OASPL) in decibels (dB) for isolated, SST ingestion, and LST ingestion cases. The radial axis represents the sound pressure level in dB, ranging from 45 to 90 dB. The angular axis represents the angle theta (θ) in degrees, ranging from 0 to 360 degrees. The black line represents the isolated case, the red line represents the SST ingestion case, and the blue line represents the LST ingestion case. Panel B: A polar plot displays the Sound Pressure Level (SPL) at the first Blade Passing Frequency (BPF) in decibels (dB) for isolated, SST ingestion, and LST ingestion cases. The radial axis represents the sound pressure level in dB, ranging from 45 to 75 dB. The angular axis represents the angle theta (θ) in degrees, ranging from 0 to 360 degrees. The black line represents the isolated case, the red line represents the SST ingestion case, and the blue line represents the LST ingestion case. Panel C: A polar plot displays the Sound Pressure Level (SPL) at the second Blade Passing Frequency (BPF) in decibels (dB) for isolated, SST ingestion, and LST ingestion cases. The radial axis represents the sound pressure level in dB, ranging from 45 to 75 dB. The angular axis represents the angle theta (θ) in degrees, ranging from 0 to 360 degrees. The black line represents the isolated case, the red line represents the SST ingestion case, and the blue line represents the LST ingestion case.

Figure 19 shows the near-field dilatation field,

$\partial p/\partial t$

, for the (figure 19
a) isolated, (figure 19
b) SST ingestion and (figure 19
c) LST ingestion cases, bad passed at the first BPF. In all three configurations, as observed in figure 18(b), the first-BPF radiation is characterised by two dominant lobes located above and below the rotor plane, consistent with a loading-dominated dipole-type source distribution. For the isolated case, these lobes are compact and symmetric, which agrees with the directivity pattern discussed previously. The SST ingestion case retains the same global structure, although local perturbations appear in the vicinity of the wake-interaction region. In contrast, the LST ingestion case shows a more distorted dilatation field, indicating that the larger turbulent structures modify the spatial distribution of the loading fluctuations and the tip–vortex shedding of the propeller blades, while preserving the general dipole-like organisation of the radiated field.

Figure 19.

Dilatation field,

$\partial p/\partial t$

, for the (a) isolated, (b) SST ingestion and (c) LST ingestion cases.

3.2.2. Noise impact assessment

In the present work, a total of 256 far-field observer locations (16 parallels and 16 meridians) are arranged in a spherical pattern aligned with the rotational axis. For clarity, the lower hemisphere of microphone locations is presented in figure 20 and used in figure 21.

Figure 20.

Lower hemisphere of the spherical far-field microphone array used for the noise-impact assessment and patchwise sound power level (PWL) calculation. The reference microphone used for the coherent output power (COP) analysis is marked by

$\boldsymbol{\times }$

(propeller not to scale).

Figure 21 illustrates the change in OASPL,

$\Delta \mathrm{OASPL}=\mathrm{OASPL}_{\mathit{ingestion}}-\mathrm{OASPL}_{\mathit{isolated}}$

, on a hemispherical surface placed at a radius of

${1.5}\,\mathrm{m}$

beneath the assembly, shown from a meridional cut and a three-dimensional view for the SST case (figure 21
a,b) and the LST case (figure 21
c,d). In both configurations, the increase concentrates along the rotation axis both upstream and downstream, while a band of near-zero change appears aligned with the plane of rotation. This result is expected since there is tonal and broadband unsteady loading noise generation due to ingestion. The SST ingestion produces a compact, axis-centred hotspot with

$\Delta \mathrm{OASPL}\approx 2$

–

${4}\,\mathrm{dB}$

over a relatively narrow angular sector. In contrast, the LST ingestion yields a broader region with noise level increases of

$\Delta \mathrm{OASPL}\approx 4$

–

${6}\,\mathrm{dB}$

that spreads omnidirectionally. This pattern is consistent with the source changes observed earlier. As previously seen in figure 16, SST ingestion results in multiple narrow tones that affect the midfrequency region of the spectra, whereas LST ingestion generates broadband humps near the BPF and its harmonics (i.e. over low frequencies to midfrequencies that more significantly raise the OASPL). The weak response in the propeller plane reflects partial cancellation of blade-section dipoles across azimuth and the fact that ingestion primarily amplifies the axial (thrust-aligned) component of the unsteady loading rather than the in-plane component. These results are crucial to establishing flight paths and regions of high noise for pedestrian zone planning.

Figure 21.

Two- and three-dimensional views of the change in OASPL (

$\Delta$

OASPL) over the noise hemisphere placed underneath the propeller–cylinder assembly for the (a,b) SST ingestion and (c,d) LST ingestion cases.

Figure 21. Long description

Panel A: A heat map showing the change in OASPL for the SST ingestion case. The heat map is a two-dimensional view with the x and y axes representing spatial coordinates. The color scale ranges from blue to red, indicating the magnitude of the change in OASPL, with blue representing lower values and red representing higher values. The heat map shows a gradient from blue to red, indicating varying levels of OASPL change across the hemisphere. Panel B: A three-dimensional view of the heat map for the SST ingestion case. The x, y, and z axes represent spatial coordinates. The color scale is the same as in Panel A, with a gradient from blue to red. The three-dimensional view provides a more detailed perspective of the OASPL change distribution. Panel C: A heat map showing the change in OASPL for the LST ingestion case. The heat map is a two-dimensional view with the x and y axes representing spatial coordinates. The color scale ranges from blue to red, indicating the magnitude of the change in OASPL, with blue representing lower values and red representing higher values. The heat map shows a gradient from blue to red, indicating varying levels of OASPL change across the hemisphere. Panel D: A three-dimensional view of the heat map for the LST ingestion case. The x, y, and z axes represent spatial coordinates. The color scale is the same as in Panel C, with a gradient from blue to red. The three-dimensional view provides a more detailed perspective of the OASPL change distribution.

3.2.3. Noise source decomposition and identification

Noise source identification is based on the time-domain FW-H analogy Ffowcs Williams & Hawkings (Reference Ffowcs Williams and Hawkings1969), and is performed using the Opty

$\partial$

B toolkit embedded in the MAAS workflow. The surface distribution of SPL over representative frequency bands is first shown in figure 22 to support the blade partition used in the subsequent analysis. The three representative frequency bands include the first BPF, a broadband band centred on

${2000}\,\mathrm{Hz}$

and a higher-frequency band between 3000 to 10 000 Hz. At the BPF, most of the blade surface displays elevated SPL levels; however, the LE and tip regions show a significantly higher magnitude. In the broadband, the elevated levels remain distributed over much of the blade surface, but with a clearer concentration over the aft section of the midchord region. The high-frequency band contains the prescribed range, showing the transition of the high-SPL region from the midchord towards the trailing edge. These distributions indicate that different parts of the blade dominate different spectral regions. The blade surface, therefore, is partitioned into several physically meaningful regions, namely the midchord blade region (blade), LE, tip, TE and root, together with the full propeller surface, in order to isolate blade areas with distinct surface-pressure-fluctuation behaviour for the subsequent patchwise PWL analysis. For each of these predefined surface patches, Opty

$\partial$

B evaluates the corresponding acoustic contribution to the far-field observers distributed over a spherical surface centred on the propeller.

Figure 22.

Sound pressure level of surface-pressure fluctuations for three frequencies of interest: (a) the BPF, (b) broadband, (c) high-frequency band showing the surface partitioning for contribution analysis.

Figure 23.

The PWL calculated using the microphone sphere for (a) all the propeller surfaces, and (b–f) blade, LE, tip, TE and root, respectively) contributions from each surface as outlined in (g).

Figure 23. Long description

The image contains six line graphs labeled Panel A to Panel F, each depicting the power spectral density (PWL in dB) against the frequency ratio (f/BPF). Panel A shows data for the propeller, Panel B for the blade, Panel C for the leading edge, Panel D for the tip, Panel E for the trailing edge, and Panel F for the root. Each graph includes three lines representing different conditions: Isolated (black), SST ingestion (red), and LST ingestion (blue). The x-axis represents the frequency ratio (f/BPF) ranging from 0.5 to 50, and the y-axis represents the power spectral density (PWL in dB) ranging from 25 to 75. The graphs illustrate the acoustic power spectral density for different propeller surfaces under various conditions. The isolated propeller exhibits sharp tonal peaks at the BPF and its harmonics. The SST ingestion case shows discrete tonal peaks in the midfrequency range with a comb structure. The LST ingestion case displays broad humps centered around the BPF and its harmonics, indicating haystacking. The right side of the image includes a labeled diagram of a propeller blade, indicating the locations of the tip, leading edge, trailing edge, and root.

Following the definition adopted by Yunus et al. (Reference Yunus, Casalino, Romani and Snellen2025), the sound power level represents the acoustic energy generated by the propeller independently of observer distance and directivity, and is obtained by integrating the far-field acoustic power spectral density over the surrounding observer sphere. Accordingly, the total propeller sound power level may be written as

(3.7)

\begin{align} \mathrm{PWL}_{\mathit{PROP}}(f)= \int _{0}^{\pi }\int _{0}^{2\pi } \frac {R_s^2 \sin \theta }{2\rho _0 c_0} \, \mathrm{PSD}(f,\eta ,\theta )\, \mathrm{d}\eta \,\mathrm{d}\theta , \end{align}

where

$R_s$

is the radius of the observer sphere,

$\theta$

and

$\eta$

are the polar and azimuthal angles, respectively,

$\rho _0$

is the ambient density,

$c_0$

is the speed of sound and

$\mathrm{PSD}(f,\eta ,\theta )$

is the far-field acoustic power spectral density. In the present study, the PWL is evaluated not only for the complete propeller surface but also for the predefined blade patches, allowing the acoustic contribution of each blade region to be assessed over the frequency range of interest. By assessing the time-domain unsteady pressure fluctuations, it can attribute the noise contribution emitted towards the far-field observers to the designated surfaces on the blade. A Fourier transform of the resulting unsteady pressure fluctuations allows for visualisation of the contribution to far-field noise (in decibels) of the different regions on the blade at different integrated frequency bands (Casalino et al. Reference Casalino, Grande, Romani, Ragni and Avallone2021). Since the blade patches have different surface areas, a direct comparison of their sound power levels would bias the interpretation towards the largest regions. To remove this geometric effect, the PWL is normalised by the ratio of the patch area (

$A_{\mathit{patch}}$

) to the total propeller area (

$A_{\mathit{PROP}}$

) such that

(3.8)

\begin{align} \mathrm{PWL} = \mathrm{PWL}_{\mathit{patch}}(f) – 10\log _{10}(A_{\mathit{patch}}/A_{\mathit{PROP}}). \end{align}

This quantity therefore represents a per-unit-area measure of the acoustic contribution of each blade region, allowing a direct comparison of the acoustic efficiency of the different surface patches. The results are shown in figure 23 for the total propeller surface (figure 23
a) and then decomposed into five blade patches: midchord blade region (figure 23
b), LE (figure 23
c), tip (figure 23
d), TE (figure 23
e) and root (figure 23
f). The total propeller PWL in figure 23(a) confirms the trends seen in the far-field spectra. The SST ingestion case adds narrow BPF- and harmonics-centred sidebands, whereas the LST ingestion results in broader humps around the BPF and its harmonics, as well as a broadband rise, with a level increase of approximately 8–

${10}\,\mathrm{dB}$

relative to the isolated case at low frequencies to midfrequencies. It should be noted that in essentially all patches, these same features emerge. Above

$f/\text{BPF}=10$

, the spectra largely collapse, indicating that ingestion primarily affects the low frequencies to midfrequencies. It should be noted that the tone at

$f= f_{0,\,\textit{SST}}$

is absent in the SST ingestion case, as it corresponds to the lift harmonic of the cylinder. The surface breakdown helps identify where the ingestion signature is generated. The blade patch shown in figure 23(b) exhibits a substantial ingestion sensitivity in the low to midfrequency range. Compared with figure 23(a), this patch significantly contributes to the total increase in PWL around the harmonics. The LE displayed in figure 23(c) is the most direct patch affected by the incoming wake, and contributes primarily to the noise increase due to turbulence ingestion. For both SST and LST ingestion, the spectra are significantly higher than in the isolated case across the entire frequency range. The tip contribution shown in figure 23(d) also exhibits a pronounced increase similar to the one observed in the blade and LE patches previously displayed in figure 23(b), however, with a reduced amplitude of increase and a more extended frequency range. Together with the TE contribution shown in figure 23(e), they contribute primarily to the noise at high frequencies (e.g.

$ f\geqslant {8000}\,\mathrm{Hz}$

), corroborating the fact that TE noise is likely to become dominant at higher frequencies. Lastly, the root patch displayed in figure 23(f) is consistently the weakest contributor, except for a few sidebands as a result of the acoustic modulation. This reflects both the lower local radiation efficiency near the root and the limited exposure of the root region in the SST ingestion case.

Following the blade-level PWL contribution analysis, the COP per unit surface for the three cases is computed as a means to identify far-field noise source locations. Unlike a conventional coherence function, the COP retains amplitude information and therefore reflects both correlation with the observer signal and the associated acoustic weighted contribution, making it useful as a source-localisation metric. In the present work, the COP is evaluated for a single prescribed far-field observer location outlined in figure 20 by

$\boldsymbol{\times }$

, to identify the blade regions contributing most coherently to the selected frequency band. A prescribed bandwidth

$\Delta f_{\textit{COP}}$

is used for the source analysis, and the Fourier transform of the total far-field noise signal

$\hat {p}_t(f)$

and of each

$n_s$

surface element contribution

$\hat {p}_{n_s}(f)$

to the far-field noise signal are computed, giving the coherent acoustic pressure spectral density as

(3.9)

\begin{align} \hat {P}_{n_s}(f)=\frac {\hat {p}_{n_s}(f)\hat {p}_t^{*}(f)}{\Delta f_{\textit{COP}}|\hat {p}_t(f)|}, \end{align}

where ‘

$^*$

’ denotes the complex conjugate and the Welch-averaged

$\hat {P}_{n_s}(f)$

of this complex quantity is computed for the full transient simulation of 40 revolutions. This is a novel FW-H-based noise source identification technique previously introduced by Casalino et al. (Reference Casalino, Romani, Pii and Colombo2023). Since the formulation is based on the complex far-field acoustic contribution of each surface element relative to the total observer pressure, the COP should be interpreted as a coherence-based acoustic source-localisation metric over the prescribed frequency interval. It is computed directly from the corresponding FW-H-based acoustic contribution of each surface element to the selected observer location. In the present study, the selected observer is the upstream microphone at

$\theta = 180^\circ$

, where the steady-loading contribution is minimal, and the turbulence-interaction noise is most clearly exposed. Due to the low tip Mach number in this study, quadrupole contributions are considered negligible, and therefore the impermeable formulation of the FW-H is utilised.

Figure 24.

The COP contribution per unit of area (dB) at the upstream microphone for the pressure side at the first (a–c) and second (d–f) BPF and three broadband frequencies (g–o) for the isolated propeller (a,d,g,j,m), SST ingestion (b,e,h,k,n), LST ingestion (c,f,i,l,o).

Figure 25.

The COP contribution per unit of area (dB) at the upstream microphone for the pressure side for

$\text{BPF}\pm f_{0,\textit{ LST}}$

.

Figure 24 shows the noise source maps in terms of COP contribution per unit area for the pressure side of the blade. All three configurations are analysed and the frequencies are divided into five distinct and acoustically important intervals, covering the first and second BPF (figure 24
a–f) as well as two midfrequencies (

$1000 \text{ and }{2000}\,\mathrm{Hz}$

) (figure 24
g–l) and a high-frequency (

${8000}\,\mathrm{Hz}$

) (figure 24
m–o). The COP has been computed using the surface data extracted from a singular blade and projected towards the upstream observer location of

$\theta = {180}^\circ$

. This observer location was selected given the mean-loading BPF contribution is weakest in the isolated case, allowing the ingestion-induced acoustic response to be examined with reduced steady-loading contamination, as shown in figure 17. The minima observed in figure 18(b) are aligned with the rotational axis, resulting in an equivalence between

$\theta = {0}^\circ$

and

$\theta = {180}^\circ$

. In the ingestion cases, however, the cylinder placement introduces a fore–aft asymmetry, and therefore the observer locations at

$\theta =0^\circ$

and

$180^\circ$

can no longer be assumed as equivalent. At the first BPF (figure 24
a–c), the isolated case exhibits a compact source at the midspan along the TE of the blade, consistent with a coherent disk loading pattern. With the SST ingestion, this footprint migrates to the LE and extends to

$r/R\geqslant 0.6$

, caused by interaction with the wake vortices. The LST ingestion produces the largest change, with the BPF source extending over most of the outer two-thirds of the span and showing distinct hotspots at the LE and TE. The contribution appears mostly around the LE area, as that is the region of the blade which first encounters the large-scale turbulent structures. The second BPF (figure 24
d–f) displays changes in the noise-contributing regions. The isolated case shows a region of dominant noise source encompassing most of the outboard section of the TE. However, a significant change can be observed for SST ingestion, where the TE becomes a more significant contributor compared with the LE, with a larger region of the blade displaying higher COP levels. In contrast, for LST ingestion, there is a reduction in the overall noise-contributing area as the hotspots previously observed shift closer together towards the tip region of the blade. The midfrequencies at

${1000}\,\mathrm{Hz}$

and

${2000}\,\mathrm{Hz}$

(figure 24
g–l) show that turbulence ingestion increases COP levels over the outer span not only at the LE but also across midchord towards the TE. This is likely a result of the pressure fluctuations originating at both LE and TE in response to the turbulence. For the SST ingestion case, the contribution becomes diffuse spanwise and covers a larger planform area, whereas for the LST ingestion case, it retains a degree of organisation and shows a clear shift from LE- to TE-dominated noise sources. These trends are consistent with the patchwise PWL decomposition shown in figure 23.

The LE dominates the low frequencies to midfrequencies, whilst the tip and TE become more influential at higher frequencies. At

${8000}\,\mathrm{Hz}$

the COP localises sharply at the TE and tip for all configurations. The isolated and LST ingestion cases display modest differences, matching the expected rise of TE scattering of the turbulent boundary layer at high frequencies, whilst the SST ingestion case retains comparatively higher COP levels over the rest of the blade surface. In order to assess the modulation behaviour of the LST ingestion case previously observed in figure 17, the sidebands centred around

$\text{BPF}\pm f_{0,\,\textit{LST}}$

are shown in figure 25. The COP contour maps retain the same pattern but lower overall levels than the BPF map in figure 24(c), which implies amplitude modulation of the existing loading source by a slowly varying inflow. The modulation redistributes energy around the BPF without creating a distinct source family since there are no new contribution regions in the COP maps.

Figure 26.

Time history of cylinder lift coefficient for the (a) SST ingestion, and (b) LST ingestion cases showing the modulation envelope. Shedding amplitude is noted.

3.2.4. Acoustic haystacking and modulation mechanisms

Figure 26 shows the time histories of the cylinder lift coefficient for the two ingestion cases, plotted against the propeller revolution. In the SST ingestion case, shown in figure 26(a), the lift coefficient exhibits a nearly periodic waveform with varying amplitude whose envelope is traced by the dashed curves Gonzalez-Martino et al. (Reference Gonzalez-Martino, Romani, Wang and Casalino2018). This behaviour reflects persistent vortex shedding over many propeller revolutions, which is convected towards the disk at frequency

$f_{0,\,\mathit{SST}}$

. The spanwise modulation of the shedding is a known feature of cylinder wakes (Casalino Reference Casalino2003). Because this modulation rate is in the same order of magnitude as the BPF, it results in phase-locked sidebands at

$m\mathrm{BPF}\pm n f_0$

in the spectra as observed previously in figures 14 and 16 (Schlinker & Amiet Reference Schlinker and Amiet1981; Paterson & Amiet Reference Paterson and Amiet1982). The LST ingestion case displayed in figure 26(b) shows a much slower, nearly sinusoidal change in the lift coefficient with only very minor low-frequency modulation traced by the envelope. The dominant structures are not phase-locked to the propeller, resulting in a blade–turbulence interaction that varies across revolutions and along the blade span. With large-scale ingestion, the BPF and its harmonics are expected to display low-frequency humps, i.e. haystacking around the BPF tone and its harmonics (McAlpine et al. Reference McAlpine, Powles and Tester2009; Huang Reference Huang2023; Raposo & Azarpeyvand Reference Raposo and Azarpeyvand2024). The lift coefficient behaviour supports that if amplitude modulation occurs in the turbulence ingestion cases, it is likely to be notably stronger in the SST ingestion case, which is likely to promote the ‘carrier-wave-led’ variations.

Since radiated acoustic fields depend not only on the source location and strength but also on the phase relationship between blades, it is essential to evaluate space–time correlations and modulation of the blade sectional thrust coefficient. Blade-to-blade correlations are relevant to determining the presence of the haystacking phenomenon observed for the turbulence ingestion case (Wang et al. Reference Wang, Wang and Wang2021; Zhou et al. Reference Zhou, Wang and Wang2024). The normalised blade-to-blade cross-correlation of sectional thrust coefficient along the LE of the blades is computed using

(3.10)

\begin{align} C_{12}(\Delta t;r)=\frac {\langle C’_1(r,t)\,C’_2(r,t+\Delta t)\rangle } {\sqrt {\langle C_1^{^{\prime }2}(r,t)\rangle \,\langle C_2^{^{\prime }2}(r,t)\rangle }}, \end{align}

with

$C’_1(r,t)$

and

$C’_2(r,t)$

being the sectional thrust coefficient fluctuations taken for segments along the span

$r$

on blade 1 and blade 2, respectively. The contour maps are plotted in figure 27 as a function of the correlation time-lag normalised by the BPF, i.e.

$\Delta t\,{\mathrm{BPF}}$

. Expectedly, the isolated case in figure 27(a) shows no clear correlation between the blades, with levels remaining close to zero for all time-lags, consistent with the absence of any strong inflow imprint on consecutive blades. The SST ingestion case in figure 27(b) exhibits correlation peaks predominantly at odd integer lags, most notably at

$\Delta t\mathrm{BPF} \approx \pm 1$

and weaker

$\pm 3$

, with the strongest levels concentrated over the inboard portion of the blade. This behaviour arises from the spatial phase of the ingested turbulence with shedding frequency

$f_{0,\mathit{SST}}/\mathrm{BPF} \approx 0.5$

. Since the blades are separated by

$\psi =180^\circ$

, the blade response samples the turbulent wake with an approximate half-period shift, which produces an alternating blade-to-blade similarity pattern at successive integer lags. The correlation is strongest for

$r/R \lesssim 0.5$

, where the ingestion imprint is larger, and the local convective velocity is lower, thereby reducing the extent of dephasing between the two blades. As

$r/R$

increases towards the tip, the correlated region becomes thinner and weaker, indicating that the blade response becomes less repeatable across consecutive blades. This pattern in correlation is consistent with the multiple tones observed earlier at the distinct frequencies in the spectrum for the SST case.

The most distinct difference between the SST and LST ingestion cases, as shown in figure 27(c), is the strong and broad positive correlation centred around

$\Delta \,t{\mathrm{BPF}}\approx \pm 1$

across a large fraction of the blade span. This indicates that consecutive blades respond to the same incoming turbulent structures at the blade-passing delay, which is consistent with the haystacking behaviour observed in the spectra. This is also confirmed by the large spread in frequency bins (

$\Delta \,t{\mathrm{BPF}}\approx \pm 0.2$

around the BPF). In comparison with the SST case, the LST ingestion case exhibits a more spatially extended and more intense blade-to-blade similarity, particularly over the midspan and inboard regions. At larger integer lags, alternating regions of correlation and anticorrelation appear towards the root and tip, reflecting the progressive loss of phase coherence as the turbulent structures are convected and deform between successive blade encounters. Whilst one radial region of the blade appears correlated, the opposite radial region is anticorrelated. This can partly be attributed to the phase shift of the vortex shedding being ingested by the different parts of the blades. Overall, the correlation maps confirm that the SST case is characterised by a more compact and phase-structured blade response, whereas the LST ingestion case exhibits repeated blade interaction with coherent large-scale structures that underpin the haystacking mechanism.

Figure 27.

Space–time correlation coefficient,

$C_{12}(\Delta t \mathrm{BPF},\, r/R)$

, of blade-to-blade sectional thrust coefficient for the (a) isolated, (b) SST ingestion and (c) LST ingestion cases.

Figure 27. Long description

Panel A: A heat map showing the spacetime correlation coefficient of blade-to-blade sectional thrust coefficient for the isolated case. The x-axis represents the delta t BPF ranging from -4 to 4, and the y-axis represents the r/R ratio ranging from 0 to 0.75. The color scale on the right indicates the correlation coefficient values, with blue representing lower values and red representing higher values. The heat map shows a relatively uniform distribution with minor variations. Panel B: A heat map showing the spacetime correlation coefficient of blade-to-blade sectional thrust coefficient for the SST ingestion case. The x-axis represents the delta t BPF ranging from -4 to 4, and the y-axis represents the r/R ratio ranging from 0 to 0.75. The color scale on the right indicates the correlation coefficient values, with blue representing lower values and red representing higher values. The heat map shows distinct patterns with higher values concentrated around specific regions. Panel C: A heat map showing the spacetime correlation coefficient of blade-to-blade sectional thrust coefficient for the LST ingestion case. The x-axis represents the delta t BPF ranging from -4 to 4, and the y-axis represents the r/R ratio ranging from 0 to 0.75. The color scale on the right indicates the correlation coefficient values, with blue representing lower values and red representing higher values. The heat map shows distinct patterns with higher values concentrated around specific regions.

To assess how turbulence ingestion modulates the radiated spectrum, the MID is evaluated as a cyclostationary metric. For a selected carrier-frequency range

$[f_1,\,f_2]$

, the integrated MID (IMD) is

(3.11)

\begin{align} \mathrm{IMD}^{f_2}_{f_1}(\alpha ,\,\Delta f) = \int _{f_1}^{f_2} \mathrm{MID}_{\Delta f}(f,\,\alpha )\,\text{d}f, \end{align}

where MID is the spectral correlation statistical value calculated in the carrier frequency range from

$f_1$

to

$f_2$

, and

$\alpha$

is the cyclic frequency. For brevity, the derivation of the spectral correlation density as input to the MID is not included in this work but was presented by Urbanek et al. (Reference Urbanek, Antoni and Barszcz2012). The

$\mathrm{IMD}$

is normalised by its casewise maximum

$\mathrm{IMD}^*$

, and plotted in figure 28. Peaks in

$\mathrm{IMD}(\alpha )$

, therefore, indicate periodicity in the second-order statistics at cyclic frequency

$\alpha$

. Integrating MID over all carriers emphasises periodic modulations of the turbulence ingestion case, e.g. pairs of ingestion-induced sidebands at

$f=m\mathrm{BPF}\pm nf_0$

contribute at

$\alpha /\mathrm{BPF} = 1,\,2,\,3,\,\ldots$

. The isolated propeller produces values close to zero and no peaks across

$\alpha$

, consistent with an unmodulated signal. Under SST ingestion,

$\mathrm{IMD}(\alpha )$

exhibits sharp peaks at

$\alpha /\mathrm{BPF} = 1,\,2,\,3$

, indicating rotation-locked modulation of the blade-pass harmonics. This is consistent with periodic vortex shedding from the upstream cylinder imposing a strong envelope on the tones (Keefe Reference Keefe1962). The absence of a visible

$\alpha /\mathrm{BPF} = 4$

peak is consistent with the rapid decay of higher harmonics and the reduced number of harmonic pairs inside the integration band. The LST ingestion case exhibits only low-level, broadband humps centred near

$m\mathrm{BPF}$

generated when large, slowly convecting structures spread the modulation over a wider band (Martinez Reference Martinez1997; Yangzhou et al. Reference Yangzhou, Wu, Ma, Huang, Yangzhou, Wu, Ma and Huang2023). A small feature at

$\alpha = 2f_{0,\,\mathit{LST}}$

is noted, representing a moderate correlation between the two sidebands.

Figure 28.

Normalised MID integrated over carrier frequencies for the isolated, SST ingestion and LST ingestion cases.

Following from the

$\mathrm{IMD}(\alpha )$

in figure 28, the distinction between amplitude-modulated sidebands and turbulence-induced haystacking can be further drawn using several indicators and spectral organisation. In particular, the SST case is characterised by compact modulation and limited blade-to-blade correlation at the blade-passing delay, whereas the LST case exhibits stronger repeated blade interaction with coherent inflow structures, leading to broader humps around the BPF harmonics. The first acoustic metric quantifies the sharpness of the sidebands at

$\mathrm{BPF}\pm f_0$

using the

${3}\,\mathrm{dB}$

bandwidths

$B_{-{3}\,\mathrm{dB}}$

and

$B_{+{3}\,\mathrm{dB}}$

. The averaged width is computed as

$B_{{\pm 3}\,\mathrm{dB}}=(1/2)\! (B_{-{3}\,\mathrm{dB}}+B_{+{3}\,\mathrm{dB}} )$

and the sideband compactness index is defined as

$S_{sb}={f_0}/{B_{{\pm 3}\,\mathrm{dB}}}$

. A large value of

$S_{sb}$

indicates narrow sidebands, while haystacking produces broader humps, and is indicated by a low

$S_{sb}$

value (Yangzhou et al. Reference Yangzhou, Wu, Ma, Huang, Yangzhou, Wu, Ma and Huang2023). The second acoustic metric compares the sidebands (

$f_{\pm sb}=\mathrm{BPF}\pm f_0$

) and tone power using the same symmetric integration over half-bandwidth

$W$

(full bandwidth

$B=2W$

),

(3.12)

\begin{align} R_{\mathit{sp}}(W)= \frac {\displaystyle \int _{f_{\textit{-}sb}-W}^{f_{\textit{-}sb}+W}\!\phi _{pp}(f)\,\mathrm{d}f +\int _{f_{+sb}-W}^{f_{+sb}+W}\!\phi _{pp}(f)\,\mathrm{d}f} {\displaystyle \int _{\mathit{BPF}-W}^{\mathit{BPF}+W}\!\phi _{pp}(f)\,\mathrm{d}f}, \end{align}

where

$\phi _{pp}(f)$

is the far-field pressure spectral density at the

$\theta =90^\circ$

observer. For compact peaks,

$R_{\mathit{sp}}(W)$

saturates quickly as

$W$

is increased, since most of the sideband energy is confined to the tone. Conversely, a smooth trend indicates that additional bandwidth continues to capture distributed broadband energy and is therefore characteristic of a hump of a broadband nature (Beckenbauer, Stemplinger & Selter Reference Beckenbauer, Stemplinger and Selter1996). In the SST ingestion case the sideband sharpness has a value of

$S_{sb}=8.6$

, analogous of narrow peaks, and

$R_{\mathit{sp}}(W)$

changes by

$\approx 5\,\%$

between

$W=40$

and

$200\,\mathrm{Hz}$

, whereas in the LST ingestion case, the sidebands are broad with an

$S_{sb}\approx 2.1$

and

$R_{\mathit{sp}}(W)$

decreases over the same range by

$\approx 40\,\%$

.

The cyclostationary metric probes coherence at the modulation scale directly. The cyclic coherence

$\gamma _c(f,\,\alpha )$

, evaluated at the cyclic frequency

$\alpha$

, is summarised by a carrier-centred band average with frequency resolution

$\Delta f={4}\,\mathrm{Hz}$

and given by

(3.13)

\begin{align} \langle \gamma _c\rangle _f(\alpha )=\frac {1}{\Delta f}\int _{\mathit{BPF}-\Delta f/2}^{\mathit{BPF}+\Delta f/2}\gamma _c(f,\,\alpha )\,\mathrm{d}f, \end{align}

so that a clear peak near

$\alpha \simeq f_0$

indicates coherent amplitude modulation (Grizewski et al. Reference Grizewski, Behn, Funke and Siller2021). The SST ingestion case shows a pronounced peak of

$\langle \gamma _c\rangle _f=0.45$

at

$\alpha \simeq f_{0,\,\mathit{SST}}$

, consistent with the compact sidebands, the results shown in figure 28 and the large

$St_c$

noted above. The LST ingestion case shows a broad maximum of

$\langle \gamma _c\rangle _f=0.40$

at

$\alpha \simeq f_{0,\,\mathit{LST}}$

, consistent with broad sidebands and the lower

$St_c$

. Taken together, these metrics convey a consistent picture that aligns with previous results and reinforces that the SST ingestion case aligns with compact amplitude modulation, while the LST ingestion case exhibits haystacking, agreeing with the

$\mathrm{IMD}(\alpha )$

and with the spectra shown earlier. These flow and acoustic indicators have provided further evidence that the far-field acoustic characteristics of the propeller in response to turbulence inflows with different turbulence intensity, length scales and the extent of interaction are governed by distinct mechanisms with the SST ingestion case having significant acoustic modulations at the BPF and higher harmonics and the LST ingestion case experiencing more broadband increase with haystacking and a coexisting, yet weaker modulation at the BPF.

The present study generates inflow by two physical upstream cylinders of different diameters, representative of eVTOL configurations; the resulting wakes differ not only in characteristic scale but also in spatial extent, coherence and intensity at the propeller plane. In addition, other key parameters, such as the wake offset relative to the propeller axis, the cylinder Reynolds number and the ratio between the wake-shedding and propeller revolutions per minute, are not varied independently in this study. These quantities are all expected to influence the resulting blade loading and radiated sound through changes in the interaction area, persistence of blade-to-blade correlation and the extent to which the spectral response remains harmonic-dominated versus forming haystacking-like humps. Moreover, a continuous transition is expected between these two responses as the length scale and coherence are varied, such that intermediate inflow conditions would be anticipated to exhibit a combination of compact sideband modulation and haystacking spreading (see Appendix B for the noise spectra from ingesting the turbulent wakes of an intermediate-sized cylinder). These results can inform predictive noise models as they further the understanding of the underlying mechanisms of noise generation for turbulence ingestion configurations of forward flight propellers.