4.1. Space–time POD of the pressure fluctuation

The pressure fluctuations, denoted as $p'$ , sampled by a linear array of microphones, are plotted in a dimensionless space–time domain. Time and space are normalised using the blade passing frequency $f^*$ and the corresponding wavelength $\lambda ^*=c/f^*$ , where $c$ is the speed of sound at temperature $293.15\,\textrm K$ . The pressure fluctuations generated by an isolated rotor are shown in figure 5 over a dimensionless time interval equivalent to four times the period of a blade passage. The figure reveals a distinct pattern consisting of four alternating curved black-and-white streaks, corresponding to expansion fronts followed by compression waves. In the spatial sub-domains $x/\lambda ^*\lt 0$ and $x/\lambda ^*\gt 1$ , a reduction in wavefront intensity is observed due to the directivity of tonal noise and the increasing distance between the microphones and the source. The wavefronts are perfectly equispaced in time, an effect associated with the stability of the active control system regulating the rotational speed. Additionally, the field appears to be highly noisy, with irregular horizontal streaks that characterise the stochastic component of the sound.

Figure 5.

Pressure fluctuations generated by the isolated rotor with the wind tunnel turned off ( $U_\infty =0\, \textrm{m}\, \textrm{s}^{-1}$ ). Dimensionless representation of the pressure fluctuations in space and time, as recorded by microphone array 4. The red-and-white circles indicate the positions of the microphones along the $x$ -axis.

By applying a windowing process to the acquired time series, 1020 fields were generated, from which the correlation tensor $\boldsymbol{\mathcal{C}}(x,x',t,t')$ was constructed. The size of the correlation tensor $\boldsymbol{\mathcal{C}}$ is $1020 \times 1020$ , and its rank is 1020. This indicates that the tensor is of full rank, meaning that all its modes contribute to the data representation. The eigenvalues, modes and coefficients were then computed.

Figure 6.

(a) Spectrum of eigenvalues for an isolated rotor obtained by analysing microphone array 4. (b) The SPML on modal order $n$ . The dashed blue line represents $\text{OASPL}-20\ \textrm{dB}$ , while the blue dash-dotted line represents the value assumed by the OASPL.

In figure 6(a), the obtained eigenvalue spectrum is shown on a logarithmic scale. The first six dominant modes have relative energy greater than $1\, \%$ . These eigenvalues appear in pairs with similar values, suggesting a coupling between modes. In addition, figure 6(b) provides the same information as in figure 6(a) in terms of identifying coupled and dominant modes. However, figure 6(b) also provides additional information that is particularly valuable in acoustics and aeroacoustics, namely the amount of noise associated with each mode, measured in decibels (dB). This information in dB complements the relative energy information, offering a more comprehensive view of the modal dynamics and their acoustic implications. In figure 6(b), a blue band represents the new criterion for selecting coherent modes. It is noteworthy that all the coherent modes selected, following a conventional criterion, fall within this coherence band. The comparison between $\text{SPML}_i$ and the conventional SPSL is reported in table 2 . It is observed that the SPSL consistently appears higher than $\text{SMPL}_i$ . This discrepancy is explained by the fact that the SPSL is computed by using microphone 33, located closer to the noise source and on the rotor plane, where pressure fluctuations are higher. In contrast, $\text{SPML}_i$ represents the measurement of the entire microphone array.

Table 2.

Comparison between the values assumed by $\text{SPML}_i$ and SPLS for the first six modes.

Figure 7 shows the first six space-time modes that turn out to be very similar pairwise. Degenerate modes occur when two or more eigenvalues coincide. The most prevalent type of degeneracy is termed double modes, as in this case, where only two eigenvalues are identical. According to Hasan & Sanghi (Reference Hasan and Sanghi2007), the translational symmetry in both space and time of these degenerate modes confirms the presence of travelling waves. Moreover, all three pairs exhibit directivity effects.

Figure 7.

First six modes for an isolated propeller calculated using time history acquired by linear array of microphones 4.

Figure 8 shows the auto-power spectrum of the modes at a specific position $x/\lambda \approx 0.5$ (i.e. at microphone 33). The results indicate that the modes are perfectly harmonic over time. In particular, the modes are characterised by regular oscillations at distinct frequencies: (i) modes 1–2, $f/f^* = 2$ ; (ii) modes 3–4, $f/f^* = 1$ ; (iii) modes 5–6, $f/f^* = 3$ . The POD has effectively selected the Fourier basis as its kernel. As noted by Lumley (Reference Lumley1981) and Tinney & Jordan (Reference Tinney and Jordan2008), POD simplifies to harmonic orthogonal decomposition when the analysed field is homogeneous, periodic, or stationary in a given direction. This latest result confirms that the inequality $\text{SPML} \lt \text{SPSL}$ is related to directivity. Indeed, the signal energy outside the rotor plane reduces SPML, which accounts for this effect. In contrast, SPSL is computed only on the rotor plane and remains higher.

Figure 8.

Power spectrum of the first three modes calculated at specific position $x/\lambda \approx 0.5$ (i.e. for microphone 33) and reported in dimensionless form.

Figure 9.

Phase portraits of the first six POD coefficients.

Lissajous patterns are observed in the phase portraits of the POD coefficients $a_i$ and $a_j$ , as shown in figure 9. The shapes of the Lissajous curves are determined by the amplitude, frequency and phase of the two series of values. Observing them allows for the assessment of the frequency ratio between $a_i$ and $a_j$ (Oberleithner et al. Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011). Some useful information for interpreting the phase portraits is provided in Appendix A. In figure 9(a,b,c), circular patterns are observed, indicating a phase difference $\pi /2$ between the coefficients, and the same frequency. Conversely, in figure 9(d), an eight-like pattern is observed, corresponding to frequency ratio $f_i/f_j = 2$ and phase difference $\pi /2$ . Furthermore, in figure 9(e), an $\alpha$ -shaped pattern is observed, indicating frequency ratio $f_i/f_j = 2/3$ and phase difference $\pi /2$ . Finally, in figure 9(f), the curve exhibits a candy-like shape, which corresponds to frequency ratio $f_i/f_j = 1/3$ and phase difference $\pi /2$ . This result aligns with the nature of the phenomenon, as pure travelling waves will have orthogonal components of the same magnitude (Feeny Reference Feeny2008). The presence of these Lissajous curves indicates that the first six modes are regular waves propagating over time.

Figure 10.

Representation of the input–output of the separation technique: (a,d) the raw data, (b,e) the tonal component, and (c,f) the broadband component, for (a,b,c) the space–time fields, and (d,e,f) a single time series extracted at $x/\lambda ^* = 0.45$ , located in the rotor plane.

4.1.1. Pressure fluctuation separation

By applying the data decomposition technique to the raw data $p'(x/\lambda ^*,tf^*)$ shown in figure 10(a), the pressure fields were separated into the tonal component $p_{c}(x/\lambda ^*,tf^*)$ , depicted in figure 10(b), and the broadband component $p'_{s}(x/\lambda ^*,tf^*)$ , shown in figure 10(c). Figure 10(d,e,f) show the results of the decomposition for a single time series extracted at $x/\lambda ^* = 0.45$ , located in the rotor plane. The signal in figure 10(e) is much smoother, and perfectly in phase with the raw signal.

Furthermore, the separation technique was computed with $T_h=20\,\text{dB}$ and interrogation window $w=4/f^*$ . These values were selected by performing a sensitivity analysis. More specifically, figure 11 displays how the coherent component varies with the threshold. In particular, for threshold values $T_h\lt 17\,\text{dB}$ , there is a loss of signal energy, whereas for $T_h\gt 23\,\text{dB}$ , the stochastic component of the signal becomes noticeable. Therefore, threshold $T_h=20\,\text{dB}$ was selected, representing an intermediate value within the $[17,23]\ \text{dB}$ interval and corresponding to $\lambda _i/\sum \lambda _i=0.01$ .

Figure 11.

Pressure fluctuation associated with the coherent component calculated for different values of $T_h$ .

According to Pierce (Reference Pierce1981) and Mancinelli et al. (Reference Mancinelli, Pagliaroli, Di Marco, Camussi and Castelain2017), the SPSL of the data is computed as

(4.1) \begin{equation} \textit{SPSL}=10\log _{10}\frac {\textit{PSD}\,\Delta f_{0}}{p_{0}^2}, \end{equation}

where PSD stands for power spectral density, and $\Delta f_{0}=5\,\textrm{Hz}$ and $p_{0}=20\, \mu \rm{Pa}$ are, respectively, the frequency resolution and the reference pressure.

In figure 12, the spectrum of the raw signal is compared with those of the coherent and incoherent components. Furthermore, the frequencies are represented in a dimensionless form by normalising them with the blade passage frequency ( $f^*$ ). In this figure, the coherent component, $p'_c$ , exhibits an almost flat trend over the range $f/f^*\in [0.1,\, 1)$ . Furthermore, the first three harmonics of this component match those of the raw signal. Conversely, beyond the third harmonic, the spectrum gradually decays. Moreover, minor peaks present between the first three harmonics in the raw signal’s spectrum are cancelled by the separation technique. This characteristic is particularly intriguing, as it significantly distinguishes the technique from a standard band-pass filter. Notably, the spectrum of the coherent component exhibits its highest peak at the second harmonic rather than the first. This behaviour is attributed to the installation effect, which amplifies the second harmonic. Such a phenomenon aligns with findings reported in the literature (Whelchel, Alexander & Intaratep Reference Whelchel, Alexander and Intaratep2020; Zarri et al. Reference Zarri, Dell’Erba, Munters and Schram2022).

Commonly, the installation effect refers to the influence of surrounding structures (e.g. wings, fuselage, or ground surfaces) on the generation and propagation of aerodynamic noise. These structures can reflect, scatter or amplify acoustic waves, causing constructive or destructive interference that modifies the perceived noise intensity. In this study, the term installation effect specifically denotes pylon–rotor interaction noise.

Figure 12.

The SPSL calculated at $x/\lambda ^*=0.45$ for the raw data, tonal and broadband components.

In figure 13, the probability density function (PDF) of the incoherent signal follows a Gaussian distribution up to $3\sigma$ , suggesting that this component can be considered stochastic (Mancinelli et al. Reference Mancinelli, Pagliaroli, Di Marco, Camussi and Castelain2017). However, beyond $3\sigma$ , a departure from the standard Gaussian distribution is evident, with the PDF tails exceeding those of the reference Gaussian distribution. This discrepancy in the tails is commonly attributed to intermittent pressure events (Mancinelli et al. Reference Mancinelli, Pagliaroli, Di Marco, Camussi and Castelain2017; Pagliaroli et al. Reference Pagliaroli, Mancinelli, Troiani, Iemma and Camussi2018; Candeloro, Ragni & Pagliaroli Reference Candeloro, Ragni and Pagliaroli2024).

By contrast, the PDF of the coherent component exhibits a quadri-modal shape with four peaks of similar probability. The outer peaks, located at $p'/\langle p'^2 \rangle ^{1/2} \approx \pm 2$ , represent the main signature generated by the blade passage. The inner peaks are linked to additional fluctuations caused by installation effects, which increase the second harmonic in the spectrum. Each blade passage produces a pair of strong positive and negative peaks, followed by a weaker secondary pair. In conclusion, the Gaussian shape of the PDFs for the incoherent component and the quadri-modal shape for the coherent component indicate that the proposed decomposition technique is effective in separating the $p_c'$ and $p_s'$ components.

Figure 13.

The PDFs of the tonal and stochastic components of the pressure fluctuation.

4.2. Interference function: one-dimensional formulation

The overall tonal aeroacoustic field produced by two rotors is obtained by summing their individual, non-interacting sound fields. According to Tam, Salikuddin & Hanson (Reference Tam, Salikuddin and Hanson1988), rotor noise fields with the same number of blades and angular frequency either cancel or reinforce each other. This effect, known as acoustic interference, depends on the relative phase at the observation point, and determines whether the resulting interaction leads to constructive or destructive interference. A mathematical model for predicting acoustic interference between rotors is highly valuable and can be utilised to minimise tonal noise. For engineering applications, such a model can support the design of quieter distributed electric propulsion systems or even be integrated into active noise control systems.

An analytical model, hereafter referred to as the interference function, is introduced in this paper to predict the acoustic interference between propellers. For modelling purposes, the rotors are assumed to have the same number of blades, rotate with the same angular frequency, and maintain a constant phase shift. The mathematical formulation is carried out in a coordinate reference frame located at the midpoint between the propellers, as illustrated in figure 14(a).

Figure 14.

(a) Sketch of the two rotors labelled for identification, with corresponding angular positions and a half-distance between them. It includes reference microphones and a reference frame. (b) Pressure fluctuations generated by the passage of the propellers, specifically sampled by microphone 6. (c) Similarly, pressure fluctuations sampled by microphone 33. Time delays are depicted, taking into account both distance and phase delays, with their respective signs. The time location $t_0$ serves as a reference point used to define all delays.

Moreover, the mathematical formulation of the interference function is based on three characteristic time delays: (i) distance half time delay $\tau _d$ ; (ii) phase time delay $\tau _\psi$ ; (iii) coherent noise (or tonal) half time scale $\tau _{c}$ .

These time delays depend on the separation distance between the rotors, the rotational regime and the observer’s position.

More specifically, the half time delay due to the separation distance between rotors can be defined as

(4.2) \begin{equation} \tau _d=\frac {d}{2c}, \end{equation}

while the time delay resulting from the phase shift between rotors is investigated by Dobrzynski (Reference Dobrzynski1993) and Tinney & Valdez (Reference Tinney and Valdez2020), and can be defined as

(4.3) \begin{equation} \tau _\psi =\frac {\psi B}{2\pi f^*}, \end{equation}

where $B$ is the number of blades. Finally, the half time scale of the signature associated with the blade passage is defined as

(4.4) \begin{equation} \tau _c:\,\rho _{\textit{pp}}(\tau _c)\leqslant \rho _{\textit{pp}}(\tau )\quad \forall \tau \in [0, +\infty ), \end{equation}

where $\rho _{\textit{pp}}(\tau _c)$ represents the absolute minimum of the autocorrelation coefficient of $p'(t)$ .

Although the model can be extended easily to predict the phenomenon at a generic point in space, the following formulation is limited to a sub-domain along the $z$ -axis defined as

(4.5) \begin{equation} \mathcal{Z} = \left\{(y,z) \in \mathbb{R}^2 \mid y=0 \text{ and } z \in \left(-\infty , -\frac {d+D}{2}\right] \cup \left[\frac {d+D}{2}, +\infty \right)\right\}, \end{equation}

where the tonal component reaches maximum intensity.

In the interference function formulation, time delays must be considered with the appropriate signs. To this aim, figure 14 depicts two scenarios, illustrating the effects of time delays at microphones 6 and 33, both positioned on the rotor plane. Regarding microphone 6, as shown in figure 14(b), an increase in separation distance or phase shift leads to a greater delay between the signatures produced by the corresponding blades of the two rotors. In contrast, for microphone 33, an increase in phase shift leads to a reduction in the delay between the red and blue signals, as illustrated in figure 14(c). The function $\text{sgn}(z)$ takes into account the sign of $\tau _\psi$ , and it is defined as

(4.6) \begin{equation} \text{sgn}(z) = \left \{ \begin{array}{rl} 1, & \text{if } z \gt 0, \\ 0, & \text{if } z = 0, \\ -1, & \text{if } z \lt 0, \end{array} \right . \end{equation}

with $n\in \mathbb{N}$ .

The mathematical model is based on the fundamental principle that constructive interference occurs when the sum of the delays is zero or an integer multiple of the blade pass period, i.e.

(4.7) \begin{equation} \tau =2\tau _d+\text{sgn}(z)\,\tau _\psi =\frac {n}{f^*}. \end{equation}

By substituting the expressions for $\tau _d$ and $\tau _\psi$ into (4.7), we obtain

(4.8) \begin{equation} \frac {d}{c}+\text{sgn}(z)\,\frac {\psi B}{2\pi f^*}=\frac {n}{f^*}. \end{equation}

In practical applications, only two cases are relevant: $n=0$ and $n=1$ . For $n=0$ , the pressure fluctuations generated by the corresponding blades of the two rotors are perfectly synchronised. For $n=1$ , the fluctuation generated by the $i$ th blade of one rotor is synchronous with the pressure fluctuation generated by the $(i-1)$ th blade of the other rotor.

Consequently, starting from (4.8), it is possible to define the interference function as

(4.9) \begin{equation} \mathcal{I}_n(d,\psi ,f^*,B)= \left\{ \begin{array}{c@{\quad}l} \text{sgn}(z)\,\dfrac {d2\pi f^*}{c\psi B}, & \text{if } n=0,\\[8pt] \dfrac {df^*}{c}+\text{sgn}(z)\,\dfrac {\psi B}{2\pi }, & \text{if } n=1. \end{array} \right. \end{equation}

The interference function $\mathcal{I}_n$ depends on the rotational regime, number of blades, phase shift and separation distance between the rotors. When $\mathcal{I}_n$ assumes the unitary value, constructive interference occurs.

Two scenarios are described below to provide an example of the mathematical tool’s application. With reference to the scenario depicted in figure 14(c), for $n=0$ and $z\lt 0$ , the condition for constructive interference is given by

(4.10) \begin{equation} \mathcal{I}_0(d,\psi ,f^*,B)=\frac {d2\pi f^*}{c\psi B}=1. \end{equation}

Physically, this means that the signature generated by blade 2 of rotor 2 overlaps in time and space with that generated by the corresponding blade of rotor 1. The second scenario proposed corresponds to constructive interference for $n=1$ and $z\gt 0$ (see figure 14 b). In this case, a wavefront emitted by blade 2 of rotor 2 coincides with that generated by the preceding blade of rotor 1 (blade 1). The condition governing this scenario can be expressed as

(4.11) \begin{equation} \mathcal{I}_1(d,\psi ,f^*,B)=\frac {df^*}{c}+\frac {\psi B}{2\pi }=1. \end{equation}

Starting from the acoustic synchronisation conditions of the two rotors, (4.7), it is possible to define the phase opposition condition using $\tau _c$ :

(4.12) \begin{equation} \tau =2\tau _d+\text{sgn}(z)\,\tau _\psi =\frac {n}{f^*}\pm \tau _c. \end{equation}

Thus destructive interference occurs when

(4.13) \begin{equation} \mathcal{I}_1(d,\psi ,f^*,B)=1\pm \tau _c f^* \end{equation}

or

(4.14) \begin{equation} \mathcal{I}_0(d,\psi ,f^*,B)=-\text{sgn}(z)\pm \frac {\tau _c 2\pi f^*}{\psi B}. \end{equation}

In figure 15(a,b), the black dashed quarter-circle of radius 1 indicates the constructive interference condition, and the black solid lines denote destructive conditions. The blue, red and green curves depict interference functions at three distances for $n=0$ and $n=1$ . In figure 15(a), $\mathcal{I}_1$ is computed for the three distances $d/D = 1.02$ , $1.10$ and $1.20$ , which were also tested experimentally. The interference functions intersect only two interference conditions: one corresponding to destructive interference, and the other to constructive interference. Moreover, the phase shifts at which these intersections occur depend on the distance between the rotors. In figure 15(b), $\mathcal{I}_0$ is displayed for the same three distances. Three intersections with the black curves are observed, indicating that by varying the phase between $0^\circ$ and $90^\circ$ , two conditions of destructive interference and one condition of constructive interference can be achieved.

Figure 15.

Plot of the destructive interference conditions (black solid lines), constructive interference condition (black dashed line), and interference functions for $d/D=1.02$ (blue solid line), $d/D=1.10$ (red dash-dotted line), and $d/D=1.20$ (green dashed line). Here, $\mathcal{I}_n$ is computed for (a) $n=1$ , $z\gt 0$ , and (b) $n=0$ , $z\lt 0$ .

4.3. Comparison between experimental results and interference function

The pressure fluctuation time histories from microphones 6 and 33 were decomposed into tonal and broadband components, as described in § 2.2, to focus the analysis solely on the coherent signal. This decomposition is applied specifically because the interference phenomenon pertains exclusively to the tonal component. Subsequently, the $\text{OASPL}_c$ , the SPSL for the first and second harmonics, and the autocorrelation at $\tau =1/2f^*$ were computed and are presented in figures 16 and 17. These figures depict the relationship between the previously listed acoustic quantities, the phase difference, and the distance ratio $d/D$ . They also include dashed black circles, representing the isolated rotor values, as a reference.

Figure 16.

The $\text{OASPL}_c$ , SPSL and autocorrelation coefficient as functions of phase shift calculated from the time histories sampled by microphone 6, for (a,b,c,d) $d/D=1.02$ , (e,f,g,h) $d/D=1.10$ , and (i,j,k,l) $d/D=1.20$ .

In figure 16(a,e,i), the $\text{OASPL}_c$ is shown as a function of $\psi$ , calculated for three different distances. The first observation is that the $\text{OASPL}_c$ associated with the tandem rotor is generally higher than that of the isolated rotor. For $d/D = 1.02$ , the maximum $\text{OASPL}_c$ occurs at $\psi _M = 90^\circ$ (constructive interference), while two minima are visible at $\psi _m=0^\circ$ and $\psi _m=60^\circ$ , corresponding to destructive interference conditions. As the rotor distance increases from $d/D = 1.02$ to $d/D = 1.20$ , the maximum shifts to $\psi _M = 75^\circ$ . More generally, the entire curve shifts by $\Delta \psi = -15^\circ$ as the distance between the rotors increases (see figure 16 a,i).

Figure 17.

The OASPL, SPSL and autocorrelation coefficient as functions of phase shift, calculated from the time histories sampled by microphone 33, for (a,b,c,d) $d/D=1.02$ , (e,f,g,h) $d/D=1.10$ , and (i,j,k,l) $d/D=1.20$ .

This effect influences both the first and second harmonics, as is evident from the comparison between figures 16(b,f,j) and 16(c,g,k). While the impact of distance and phase is noticeable for both harmonics, it is particularly pronounced for the second harmonic. The presence of the second rotor significantly alters the dynamic system, with the signal’s energy and its dependence on phase and distance being more pronounced in the second harmonic. The signal closely resembles that emitted by a single rotor with six blades. Therefore, to better investigate this phenomenon, it is useful to define a new frequency as an alternative to the conventional blade pass frequency $\hat {f}=f^*N$ , where $N$ is the number of rotor.

Finally, in figure 16(d,h,l), the correlation coefficient at $\tau =1/2f^*$ shows the same trend as $\text{SPSL}_2$ , as expected. Specifically, for the phase angles where destructive interference occurs, the autocorrelation coefficient tends towards −1. Since this parameter is constantly bounded between 1 and −1, with −1 representing the condition of maximum noise mitigation, it could be used as the objective function in an active noise-cancellation control system.

Figure 17 displays a single constructive interference at $\psi _M=45^\circ$ , and two destructive interferences at $\psi _m=15^\circ, 75^\circ$ . The effect of varying the rotor’s separation distance is not visible, which is related to the time delay signs depicted in figure 14. This is a particularly interesting finding, as it indicates a lack of symmetry in the directivity of tandem rotor noise, in contrast to the behaviour observed with an isolated rotor. At this position, stronger effects of destructive interference are observed, reducing the sound level below that of the isolated rotor. This outcome highlights the potential of noise cancellation techniques for multi-rotor noise mitigation by appropriately selecting the phase shift and separation distance between propulsion systems.

Figure 18 compares the phase delays at which interference occurs, as detected experimentally, with those predicted using the interference function proposed in § 4.2. For the experimental data, a vertical error bar, corresponding to the discretisation step of the phase shift, represents the uncertainty in identifying the interference condition. Additionally, for destructive interference only, a horizontal error bar of $\varepsilon _\psi = 240f^*/f_s$ has been introduced to account for the uncertainty in determining the time scale of the signature associated with the blade passage.

Figure 18.

Comparison between experimentally measured phase delays corresponding to the occurrence of the interference phenomenon and those predicted by the interference function.

Although the experimental phase discretisation is relatively coarse, resulting in a significant margin of uncertainty in identifying interference conditions, the comparison between the data points and the bisector of the first and third quadrants demonstrates a satisfactory agreement between the experimental results and the formulated mathematical model.

4.4. Noise intermittency

The phenomenon of intermittency and the chaotic behaviour of pressure fluctuations has been systematically addressed within the context of thermoacoustic instability (Kabiraj & Sujith Reference Kabiraj and Sujith2012; Kashinath, Li & Juniper Reference Kashinath, Li and Juniper2018; Guan, Gupta & Li Reference Guan, Gupta and Li2020) through the use of time–frequency representations and recurrence quantification analysis. However, there are no studies that thoroughly investigate the phenomenon of rotor noise intermittency. Intermittency is one of the common routes to chaos and occurs when a dynamical system switches between irregularly spaced alternating intervals of chaotic/burst and steady/periodic states (Klimaszewska & Żebrowski Reference Klimaszewska and Żebrowski2009; Pagliaroli & Troiani Reference Pagliaroli and Troiani2020). In fluid dynamics, the term intermittency was introduced to describe signals that alternate between flat and bursting behaviour, interpreted, respectively, as laminar and turbulent states (Platt, Spiegel & Tresser Reference Platt, Spiegel and Tresser1993).

The current literature provides several criteria for the classification of intermittency. One of the most common classification systems is introduced by Bergé et al. (Reference Bergé, Dubois, Mannevillel and Pomeau1980) and Pomeau & Manneville (Reference Pomeau and Manneville1980), who categorise intermittency into three classes associated with a different bifurcation scenario: (i) type I, intermittency related to a saddle-node bifurcation; (ii) type II, intermittency due to a Hopf bifurcation, identified by the appearance of a quasi-periodic state; (iii) type III, intermittency related to a reverse period doubling bifurcation.

The following analysis focuses on four specific operating conditions, using microphone 33 ( $z\lt 0$ ) and rotor distance ratio $d/D=1.20$ :

1. (i) $\psi _m=15^\circ$ , destructive interference;

2. (ii) $\psi _i=60^\circ$ , transition between constructive and destructive interference;

3. (iii) $\psi _M=45^\circ$ , constructive interference.

In these three conditions, the nature of the signal changes completely; in particular, for one specific condition, the signal becomes strongly time-dependent. The time–frequency analysis of the noise enables a qualitative investigation of the time-dependent nature of the pressure fluctuations that is crucial for identifying transient or intermittent phenomena in propeller aeroacoustics.

This study employs the continuous wavelet transform to analyse localised time–frequency fluctuations in microphone signals. Similar to the Fourier transform, the energy density can be derived from wavelet coefficients as shown in Tinney & Sirohi (Reference Tinney and Sirohi2018):

(4.15) \begin{equation} E(s,\tau )=\frac {\vert w(s,\tau )\vert ^2}{\Delta s}, \end{equation}

where $w(s,\tau )$ are the wavelet coefficients. Finally, sound pressure wavelet level (SPWL) can be calculated as

(4.16) \begin{equation} \text{SPWL}(f,\tau )=20\log _{10}\frac {\vert w(f,\tau )\vert }{p_0}. \end{equation}

In particular, the present analysis was performed by selecting a complex Morlet wavelet, which is a solution of the wave equation (Visser Reference Visser2003) and fits the nature of the problem. For interested readers, further details on the wavelet transform can be found in Pagliaroli et al. (Reference Pagliaroli, Pagliaro, Patane, Tatí and Peng2020).

In figure 19 , the space–time pressure fluctuation field, the tonal component of the time series, and its wavelet transform in the normalised time–frequency domain are computed and presented for the three different conditions.

Figure 19.

(a,d,g) Pressure fluctuations represented in the space–time domain. (b,e,h) Time histories sampled by microphone 33. (c,f,i) The SPWL in the time–frequency domain. (a,b,c) Destructive interference ( $\psi _m=15^\circ$ ). (d,e,f) The transition condition between constructive and destructive interference ( $\psi _i=60^\circ$ ). (g,h,i) Constructive interference ( $\psi _M=45^\circ$ ).

In figure 19(b), only one dominant oscillation per blade passage period is observed, with a significantly smaller amplitude compared to the other cases (see figure 19 e,h). Destructive interference occurs, leading to substantial noise cancellation. For this condition, figure 19(c) shows that the signal energy is primarily concentrated in the first harmonic, as expected, although its intensity remains low. The second harmonic is completely cancelled. This reduction in energy at the second harmonic, associated with destructive interference, was recently observed and explained by Del Duchetto et al. (Reference Del Duchetto, Pagliaroli, Candeloro, Rossignol and Yin2025).

In figure 19(d–f), an intermediate case between constructive and destructive interference is observed. The energy associated with the second harmonic becomes intermittent, exhibiting amplitude modulation and forming irregular wave packets. This phenomenon has also been reported by Tinney & Sirohi (Reference Tinney and Sirohi2018), Zhong et al. (Reference Zhong, Zhou, Chen, Jiang, Wu and Zhang2023) and Celik et al. (Reference Celik, Jamaluddin, Baskaran, Meloni, Rezgui and Azarpeyvand2024). In this circumstance, the phenomenon becomes chaotic, and the observed intermittency in the second harmonic is of type II. In figure 19(i), the intermittency of the second harmonic fully transitions into periodic dominant radiation.

Zhong et al. (Reference Zhong, Zhou, Chen, Jiang, Wu and Zhang2023) associate the intermittency of sound with fluctuations in rotational speed. The results indicate that intermittency is linked to proximity to constructive interference conditions, where small variations produce significant effects. This local emergence or disappearance of a periodic oscillation is characteristic of subcritical Hopf bifurcations. The influence of proximity to constructive interference has not been emphasised previously, yet we believe that it is crucial for understanding this recently studied phenomenon. A deviation from destructive interference conditions leads to modulation of the tonal component, which, in turn, enhances the perception of annoyance (Celik et al. Reference Celik, Jamaluddin, Baskaran, Meloni, Rezgui and Azarpeyvand2024).