2. Experimental set-up

2.1. Aerodynamic and acoustic measurements

A comprehensive measurement study on a small-scale rotor in edgewise flows was conducted in an open-circuit anechoic wind tunnel at the Hong Kong University of Science and Technology (HKUST), with an anechoic chamber that has dimensions of 3.3 m (L) $\times$ 3.1 m (W) $\times$ 2.0 m (H) (Yi et al. Reference Yi, Zhou, Fang, Guo, Zhong, Zhang, Huang, Zhou and Chen2021). The inflow turbulence intensity is lower than 0.27 % within the speed range of 16–30 m s⁻¹, and the cutoff frequency of the anechoic chamber is 200 Hz. A low-noise two-bladed rotor SP2 with a diameter of 0.217 m developed at HKUST was employed for measurements (Wu et al. Reference Wu, Chen, Jiang, Zhong and Zhang2023). Figure 1 shows the experimental set-up for rotor tests and acoustic measurements. A multiaxis load cell of ATI Mini 40 was attached to the rotor system to record blade aerodynamic loading. The sensing range is 120 N in the direction of rotor thrust, with the resolution of 1/50 N. For the torque, the sensing range is 2 Nm with a resolution of 1/4000 Nm. The sampling frequency for the aerodynamic force and torque is 20 kHz. The tilting angle was set to 0 $^\circ$ as a benchmark, where the rotor disc was parallel with the direction of the free stream velocity (see figure 1 b). In this case, BVI is most significant (Yang et al. Reference Yang, Liu, Li, Arcondoulis and Wang2020; Jamaluddin et al. Reference Jamaluddin, Celik, Baskaran, Rezgui and Azarpeyvand2024), which is beneficial for the fundamental investigation. The far-field noise was measured by a phased microphone array with 56 Brüel & Kjær type 4957 microphones placed 1.5 m away from the rotor. These microphones have a dynamic range of 32–134 dB, maintaining a flat frequency response between 50 Hz and 10 kHz. To ensure accuracy, each microphone was calibrated using a Brüel & Kjær 4231 piston phone calibrator at 94 dB kHz $^{-1}$ before the tests. All microphones were sampled simultaneously at 50 kHz for a total sampling period of 10 s. An NI PXIe-1082 chassis from National Instruments performed the data acquisition with four 24-bit PXI 4497 cards. The acoustic data were divided into 50 blocks for the averaged spectrum analysis by Welch’s method, resulting in a frequency resolution of 5 Hz.

Figure 1.

The experimental set-up of rotor noise measurements in the anechoic wind tunnel at HKUST: (a) photography and (b) schematic.

2.2. Flow field measurements

The PIV measurements were employed to visualize the flow fields surrounding the rotor blade during BVI, as shown in figure 2. The test section was seeded with di-ethyl-hexyl-sebacat particles with a mean diameter of 0.3 ${\unicode{x03BC}}$ m from Dantec Dynamics and illuminated by a Nd:YLF 527 nm dual-cavity laser. This dual pulse laser can yield up to 200 mJ per pulse. The laser sheet has a thickness of approximately 1.5 mm. Cartesian coordinates ( $x, y, z$ ) are used in this study, as shown in figure 2(b), with an origin fixed at the centre of the rotor disk, midway between the end planes of the rotor hub. The $x$ -axis is defined along the free stream direction, the $y$ -axis aligns with the direction of rotor’s rotation vector and $z$ -axis is perpendicular to the $x$ – $y$ plane, completing a right-handed system.

Figure 2.

The experimental set-up of flow field measurements in the anechoic wind tunnel at HKUST. (a) Photography and (b) schematic of the stereoscopic PIV measurements in the free stream direction. (c) Schematic of the two-dimensional (2-D) PIV measurements in the blade spanwise direction.

Stereoscopic PIV measurements were conducted in a plane parallel to the $x$ – $y$ plane, located near the rotor blade’s tip at $z=-0.8R$ , as illustrated in figure 2(b). Dimensions of field of view (FoV) are 85 mm $\times$ 41 mm, which corresponded to a domain of $0.75R\times 0.36R$ in the $x$ and $y$ directions. The set-up of FoV aimed to measure the detailed BVI process in the vicinity of the blade. Two SpeedSense M310 high-speed complementary metal-oxide-semiconductor (CMOS) cameras, each with a resolution of $1280\times 800$ pixels $^2$ and a frame rate of 1600 Hz, were used to obtain the velocity components in $x$ , $y$ and $z$ directions, denoted as $u$ , $v$ and $w$ , respectively. Two Nikon microlenses (AF-S VR Micro Nikkor 105 mm F/2.8G IF ED) were mounted with Scheimpflug adapters, each with a 105 mm focal length. A CALT PD32 encoder is mounted on the motor to generate a signal of one pulse per revolution for phase locking.

Additionally, 2-D PIV measurements were conducted in three planes that align with the spanwise direction perpendicular to the rotor disk at representative rotor phases, as illustrated in figure 2(c). The blade tip is located within these planes. An additional cylindrical coordinate ( $y, r, \theta$ ) is introduced to describe the flow fields along the blade spanwise direction, as depicted in figure 2(c). The $r$ -axis aligns with the rotor spanwise direction, and coordinate $\theta$ is defined as the rotor phase, representing the angle between the positive directions of $x$ and $r$ . The $y$ -axis aligns with the rotor’s rotational axis, consistent with the Cartesian coordinate system ( $x, y, z$ ), obeying the right-handed rule. Three measurement planes were selected at rotor phases of $\theta =60^\circ$ , $75^\circ$ and $105^\circ$ based on the locations of BVI occurring. A SpeedSense M310 camera equipped with the same Nikon microlens was used for imaging. Each measurement plane has a FoV of 52 mm $\times$ 78 mm, corresponding to a domain of $0.48R\times 0.72R$ in the $r$ and $y$ directions.

In this study, a total of 200 phase-locked image pairs were captured for both stereoscopic and 2-D PIV measurements to calculate the phase-averaged velocity field for each case. In addition, time-resolved flow fields of representative cases are continuously recorded with 4000 snapshots at 1600 Hz for time-domain analysis. Images are processed using Dantec DynamicStudio 7.6 software, and the final interrogation window of PIV images is $16\times 16$ pixels with an overlap factor at 75 %. This yields a pixel resolution of 0.30 mm pixel⁻¹ for the stereoscopic PIV measurements and 0.27 mm pixel⁻¹ for the 2-D PIV measurements, respectively. Note that illumination through the blade surface is affected by local refraction, resulting in a shadowed region on the opposite side where data are not available, as indicated by the covered shapes in all PIV results.

2.3. Range of test conditions

The entire test was performed at free stream conditions of $U_\infty$ from 0 to 20 m s⁻¹, and the rotor was operated at rotational speeds from 60 to 120 revolutions per second (r.p.s.). Therefore, the corresponding advance ratio, $\mu =U_{\infty }/\varOmega R$ , varied from 0 to 0.5, where $\varOmega$ is the rotational speed in radians and $R$ is the blade radius. The corresponding Reynolds number at 80 % of the blade radius, ${\textit{Re}}_{0.8R}$ , ranges from $3.9\times 10^4$ to $1.0\times 10^5$ at a measurement temperature of $25\,^\circ$ C. The operating conditions discussed in detail in this paper are listed in table 1 in terms of advance ratio ( $\mu$ ), rotor revolutions per second, maximum free stream relative tip velocity ( $U_{\mathit{tip, max}}$ ) and ${\textit{Re}}_{0.8R}$ . These cases correspond to the low advance ratio ( $\mu$ = 0), moderate advance ratio ( $\mu$ = 0.16) and high advance ratio ( $\mu$ = 0.33), respectively.

Table 1.

Detailed test conditions for select representative cases.

2.4. Methodology of transient beamforming

In this work, we used the wavelet-based beamforming method from Chen et al. (Reference Chen, Yang, Peng and Huang2021) to study the transient interaction noise between the rotor blade and vortices. By using wavelet transform, sound pressure obtained from the phased microphone array can be represented in the time–frequency domain as follows:

(2.1) \begin{align} P_{\psi }(\boldsymbol{r},\boldsymbol{\mathcal{X}},t,s)=\frac {1}{s}\int _{-\infty }^{+\infty }p(\boldsymbol{\mathcal{X}},\tau )\psi ^{*}\left (\frac {\tau -t}{s}\right )\textrm{d}\tau , \end{align}

where $\boldsymbol{\mathcal{X}}$ are the locations of microphones, $\boldsymbol{r}$ are the imaging points, $(\boldsymbol{\cdot })^*$ denotes complex conjugate, $\tau$ is translational value and $s$ is scale value of the Morse wavelet $\psi$ ( $s\gt 0$ ), which is related to the angular frequency $\omega$ as

(2.2) \begin{align} \omega (s)=2\pi \frac {f_{c}}{s}T_{s}, \end{align}

where $f_c$ is the peak frequency and $T_s$ is the sampling time interval (Lilly & Olhede Reference Lilly and Olhede2012). The array measurement forms the following vector from the relationship between the sound emission time $t_e$ and time $t$ ,

(2.3) \begin{align} Y(\boldsymbol{r},\omega ,t_e)=[P_{\psi }(\boldsymbol{r},\boldsymbol{\mathcal{X}}_{1},t_1,s)],\boldsymbol{\cdot}s ,P_{\psi }(\boldsymbol{r},\boldsymbol{\mathcal{X}}_{n},t_n,s)], \end{align}

where $t_i=t_e+ \lvert \boldsymbol{r}-\boldsymbol{\mathcal{X}}_{i} \rvert /c_0$ , $(\boldsymbol{\cdot })_i$ denotes the $i$ th sensor of the array and $c_0$ is the speed of sound. Similar to the conventional beamforming method, the associated array cross-spectral-matrix (CSM) would be

(2.4) \begin{align} {\textit{CSM}}(\boldsymbol{r},\omega ,t_e)=Y(\boldsymbol{r},\omega ,t_e)\boldsymbol{\cdot }Y^*(\boldsymbol{r},\omega ,t_e). \end{align}

Following Chen et al. (Reference Chen, Peng, Liem and Huang2020a ), the free space Green function is represented by $\boldsymbol{G}(\boldsymbol{r},\boldsymbol{\mathcal{X}})$ . Next, to achieve the weight vector, we define $\boldsymbol{G}=[G_1\boldsymbol{\cdot}s G_n]$ . Then, the weight vector of the wavelet-based beamforming can be constructed as

(2.5) \begin{align} \boldsymbol{W}(\boldsymbol{r},t_e)=\boldsymbol{G}(\boldsymbol{r},\boldsymbol{\mathcal{X}})/ \left \lVert \boldsymbol{G}(\boldsymbol{r},\boldsymbol{\mathcal{X}}) \right \rVert , \end{align}

where $\left \lVert \boldsymbol{\cdot }\right \rVert$ represents the $L_2$ -norm. By following the classical analytical form of conventional beamforming approaches, we proposed the wavelet-based beamforming as

(2.6) \begin{align} b(\boldsymbol{r},\omega ,t_e)=\boldsymbol{W}^*(\boldsymbol{r},t_e)\boldsymbol{\cdot }{\textit{CSM}}(\boldsymbol{r},\omega ,t_e)\boldsymbol{\cdot }\boldsymbol{W}(\boldsymbol{r},t_e), \end{align}

where $(\boldsymbol{\cdot })^*$ denotes the complex transpose of the matrix. In this work, we further apply the phase-averaged beamforming method, which would enable the calculations of the phase-averaged results. The phase-averaged beamforming can be conducted as

(2.7) \begin{align} \bar {b}\left(\boldsymbol{r},\omega ,\frac {2\pi t_e}{n_bT}\right)=\frac {1}{N}\sum _{j=0}^{N}b(\boldsymbol{r},\omega ,t_e+jT), \; 0\lt t_e\lt T, \end{align}

where $n_b$ is the number of the blades, $T$ is the rotating period and $N$ is the average number.

3. Aerodynamic performance

The averaged aerodynamic forces regarding thrust of $T$ and torque of $Q$ , were calculated for different rotational speeds from 60 to 120 r.p.s. and inflow speeds of $U_\infty$ from 0 to 20 m s⁻¹, based on the sampling data for 10 s. The thrust coefficient of $C_T$ , power coefficient of $C_{\!P}$ and rotor’s FM are defined as follows (Leishman Reference Leishman2006):

(3.1) \begin{align} C_T=\frac {T}{\rho A(\varOmega R)^2},\quad C_{\!P}=\frac {P}{\rho A(\varOmega R)^3},\quad \textrm{FM}=\frac {C_T^{3/2}}{\sqrt {2}C_{\!P}}, \end{align}

where $\rho$ is air density, $A=\pi R^2$ is area of the rotor disc and $P=\varOmega Q$ is power of the rotor. Figure 3 illustrates the aerodynamic performance in terms of thrust and FM of the rotor under varying advanced ratio ( $\mu$ ) for the representative rotational speeds of 60, 90 and 120 r.p.s. The uncertainty of the thrust measurement from the repeatability tests at the 95 % confidence interval is 0.12 N based on the $t$ -distribution (Student 1908). In general, figure 3(a) shows a significant increase in thrust with increasing rotational speed. The increase rates with $\mu$ for the cases at 60 and 90 r.p.s. are relatively low, while the case at 120 r.p.s. exhibits a rapid rise, with an amplitude of approximately 1.5 N from $\mu =0$ to 0.25. Additionally, a flat region exists at the advance ratio from 0.16 to 0.25 for the low revolutions per second cases, suggesting that thrust is insensitive to the free stream speed in this flow range. Figure 3(b) compares the FM values to demonstrate the aerodynamic efficiency under varying $\mu$ for the representative cases of 60, 90 and 120 r.p.s. The results exhibit evident fluctuations of FM values with an approximate average level of around 0.6, and no clear trend is observed.

Figure 3.

Aerodynamic performance for the cases of the free stream speeds from 0 to 20 m s⁻¹ at 60, 90 and 120 r.p.s. Here (a) thrust ( $T$ ), and (b) figure of merit (FM).

Figure 4.

Non-dimensional aerodynamic results under varying advance ratio of $\mu$ ranged from 0 to 0.5. Here (a) thrust coefficient of $C_T$ and (b) power coefficient of $C_{\!P}$ .

Figure 4 presents the non-dimensional aerodynamic results in terms of the thrust and power coefficients across varying $\mu$ for the representative cases of 60, 90 and 120 r.p.s., respectively. The result of $C_T$ shows a linear relationship of $C_T \sim 0.02\mu$ over a wide range of $\mu$ from 0 to 0.5, as indicated by the solid line. However, figure 4(a) shows deviations for $\mu \gt 0.35$ , with a maximum of 15 % relative to the fitted value at $\mu =0.37$ . Figure 4(b) presents a similar linear relationship between $C_{\!P}$ and $\mu$ of $C_{\!P} \sim 0.004\mu$ , as indicated by the solid line. Similarly, the deviation of $C_{\!P}$ becomes relatively large as $\mu$ above 0.35, with a maximum of 10 % relative to the fitted value at $\mu =0.37$ . Overall, the linear laws for $C_T$ and $C_{\!P}$ are well fitted for a wide range of advance ratios ( $0\lt \mu \lt 0.5$ ), which can provide useful information for the rotor design and applications. At high advance ratios, particularly in low-revolutions per second cases at high free stream speeds, deviations in aerodynamic behaviour may be caused by relatively high spanwise and reverse flows on the blade surface when the rotor operates under off-design conditions. However, these cases are not the focus of this study.

4. Classification of BVI regimes

The relative positions and vortex characteristics during when the vortex encounters the blade are important for the BVI process, leading to a significant impact on noise generation. In this work, the results of flow fields show that BVI characteristics vary significantly with increasing advance ratio, which can be categorized into three distinct flow regimes. This section discusses the comprehensive flow fields in different planes aligned with the streamwise and blade spanwise directions to quantify the detailed BVI process. First, the flow field results obtained from stereoscopic PIV measurements (see figure 2 b) are presented in terms of the velocity and vorticity fields. This information can describe the relative positions of the vortex centre and the blades during their interactions according to the principle of vector projection. Second, the properties of flow fields along the blade spanwise direction (see figure 2 c) are discussed, including the axial displacement of interaction, vortex strength and turbulence kinetic energy (TKE) distribution surrounding the tip region, for the same cases analysed in the stereoscopic PIV measurements.

Figure 5.

Representative phase-averaged velocity fields ( $w$ ) at 90 r.p.s. in the $x$ – $y$ plane for different advance ratios at three rotor phases of (i) $60^\circ$ , (ii) $90^\circ$ and (iii) $100^\circ$ : (a) $\mu =0$ , (b) $\mu =0.16$ and (c) $\mu =0.33$ . Dashed lines are the vortex-centre trajectories in the measurement plane. The free stream direction aligns with the $x$ axis.

Figure 6.

Representative phase-averaged velocity fields ( $v$ ) at 90 r.p.s. in the $x$ – $y$ plane for different advance ratios at three rotor phases of (i) $60^\circ$ , (ii) $90^\circ$ and (iii) $100^\circ$ : (a) $\mu =0$ , (b) $\mu =0.16$ and (c) $\mu =0.33$ . Grey dashed lines represent the vortex boundary trajectories in the measurement plane. Black dashed lines are the vortex-centre trajectories, which are the same as those in figure 5.

Figure 5 shows the $w$ fields of $U_\infty =0$ , 10 and 20 m s⁻¹ at 90 r.p.s., corresponding to advance ratios of $\mu =0$ , 0.16 and 0.33, respectively, in a plane parallel to the $x$ – $y$ plane at $z=-0.8R$ . In each case, the flow fields corresponding to three rotor phases at $60^\circ$ , $90^\circ$ and $100^\circ$ are used to explore the detailed interaction processes concerning the vortex trajectory and blade motion. The projection trajectory of vortex centres can be identified by the regions where $w=0$ , as indicated by the dashed lines in each phase. Here, positive $w$ represents an inward direction ( $+z$ ) towards the measurement plane, while negative $w$ indicates an outward direction ( $-z$ ) from this plane. At $\mu =0$ , as shown in figure 5(a), the vortex centre is located approximately 0.2 $R$ away from the blade, indicating that the vortex has insignificant impact on the blade surface and causing relatively weak interaction. When $\mu$ increases to 0.16, the vortex centre is still not yet cut by the blade, while the shear layer of the vortex (blue region) gradually interacts with the blade leading edge (LE), leading to an increase in interaction strength, as shown in figure 5(b). A change in flow direction compared with the case at $\mu =0$ can be observed at a phase of $60^\circ$ , indicating that the spanwise flow from the pressure side to the suction side is suppressed. Furthermore, as $\mu$ increases to a higher value ( $\mu =0.33$ ), figure 5(c) shows that BVI occurs earlier compared with that at $\mu =0.16$ , which is consistent with the results of noise source maps discussed in § 5, and the vortex–LE interaction remains the primary physical process. The velocity fields show that the blade directly passes through the vortex centre, resulting in a strong interaction.

Figure 6 shows the $v$ fields for the same cases presented in figure 5. The outer boundary of vortices during BVI can be identified by the regions where $v=0$ according to the projection principle, as indicated by the black dashed lines in each phase. At $\mu =0$ , the blade slightly interacts with the outer vortex boundary, resulting in an extremely weak interaction, as shown in figure 6(a). As $\mu$ increases to 0.16, the blade cuts through the vortex’s shear layer, significantly distorting the tip vortex, as shown in figure 6(b). As $\mu$ increases to 0.33, as shown in figure 6(c), the $v$ fields indicate that the upstream vortex aligns more parallel with the LE of the blade, forming a saddle point in the measurement plane (see figure 7 c). The induced velocity moves along the blade chordwise direction as the parallel interaction, potentially leading to streamwise fluctuations on the blade surface. This change in the cutting mechanism from low to high advance ratios may impact noise generation, which is discussed further in § 5.

Figure 7.

Representative phase-averaged vorticity component ( $\omega _z$ ) at 90 r.p.s. in the $x$ – $y$ plane for different advance ratios at three rotor phases of (i) $60^\circ$ , (ii) $90^\circ$ and (iii) $100^\circ$ : (a) $\mu =0$ , (b) $\mu =0.16$ and (c) $\mu =0.33$ . Red dashed lines represent the vortex centre trajectories projected onto the measurement plane.

To gain further insights into the vortex characteristics, the corresponding vorticity fields of $\omega _z$ are presented in figure 7. At $\mu =0$ (figure 7 a), the vortices are relatively invisible in the measurement plane because they are oriented approximately perpendicular to the LE of the blade. In this condition, small velocity fluctuations can be induced near the blade surface due to the relatively low vorticity, causing a vortex–boundary layer (BL) interaction. The tip vortex formation is not affected by the previous generated tip vortex, as shown in the phases of $60^\circ$ and $90^\circ$ . As $\mu$ increases to 0.16 (figure 7 b), the blade’s LE passes through the vortex shear layer, leading to a shear layer–LE interaction. Due to increased free stream speeds, the angle between the vortex tube’s axis and the measurement plane become smaller compared with the case at $\mu =0$ , based on projection principles. The vorticity contours exhibit opposite values on either side of the trajectory of the vortex centre, suggesting that the measurement plane is located within the vortex’s shear layer without intersecting the vortex core, resulting in a complex oblique interaction. At $\mu =0.33$ (figure 7 c), the blade cuts the vortex centre, causing a vortex centre–LE interaction. Only positive vorticity is present across three rotor phases from $60^\circ$ to $100^\circ$ . The upstream vortex region exhibits an elliptical shape rather than a circular one, due to the angle between the vortex tube’ axis and the measurement plane. These results illustrate that the interaction becomes more parallel to the LE of the blade, thereby causing a stronger BVI (Glegg Reference Glegg1991; Peng & Gregory Reference Peng and Gregory2015).

Figure 8.

Representative phase-averaged downwash velocity fields ( $u_d$ ) at 90 r.p.s. in the $r$ – $y$ plane for different advance ratios at three rotor phases of (i) $60^\circ$ , (ii) $75^\circ$ and (iii) $105^\circ$ : (a–c) $\mu =0$ , (e–g) $\mu =0.16$ and (h–j) $\mu =0.33$ . Here $r$ is the spanwise direction of the rotor at various phases.

The BVI process in a practical rotor is significantly complex due to its three-dimensional nature. To better understand this process, the flow field in the blade spanwise direction ( $r$ – $y$ plane) is also analysed to comprehensively reveal the interaction mechanism. The cases discussed in the following parts are the same as those from stereoscopic PIV measurements presented in figures 5, 6 and 7. Note that there is a slight difference in in the selection of rotor phases ( $\theta =60^\circ$ , $75^\circ$ and $105^\circ$ ) due to the challenges in PIV set-up when the rotor phase angle approaches $90^\circ$ .

Figure 9.

Representative phase-averaged vorticity component ( $\omega _\theta$ ) at 90 r.p.s. in the $r$ – $y$ plane for different advance ratios at three rotor phases of (i) $60^\circ$ , (ii) $75^\circ$ and (iii) $105^\circ$ : (a–c) $\mu =0$ , (e–g) $\mu =0.16$ and (h–j) $\mu =0.33$ . The negative value denotes the clockwise rotation of the vortex.

Figures 8 and 9 present the downwash velocity fields ( $u_d$ ) and corresponding vorticity fields ( $\omega _\theta$ ), respectively, to highlight the differences in BVI process at various $\mu$ . At $\mu =0$ , the downwash velocity contours (figure 8 a–c) show that the vortex-induced velocity has a negligible impact on the blade surface, resulting in weak interaction. Tip flow from the pressure side to suction side of the blade is observed near the blade tip and is unaffected by the downstream vortex. The vorticity results show that the vortex centre is located at approximately $y/R=0.02$ for each rotor phase (see figure 9 a–c), which is consistent with the results in figure 5(a). When $\mu$ increases to 0.16, the downwash velocity beneath the rotor disk initially increases, then decreases as $\theta$ , due to the relatively spanwise movement of the vortex towards the hub (see figure 8 e–g). This transient variation in velocity field alters the radial distribution of the effective angle of attack and relative inflow speeds during BVI, which in turn affects aerodynamic loading and the associated noise. As the vortex approaches the tip at $\theta =75^\circ$ and $105^\circ$ (see figures 8 f and 8 g), there is intensified convection from the pressure side to the suction side, leading to spatial velocity fluctuations in the mixing region near the tip. The vorticity results show that the axial displacement of the vortex to the rotor disk is gradually reduced as $\theta$ increases (see figure 9 e–g). The shear layers of the vortices begin to interact with the blade pressure side, and small-scale vortices can be observed at $\theta =75^\circ$ (see figure 9 f), resulting in moderate interaction. However, vortex centres remain unaffected, maintaining vorticity magnitudes similar to those at $\theta =60^\circ$ . At higher $\mu$ of 0.33, the vortex is closer to the blade surface, further enhancing the interaction strength (see figure 9 h–j). The tip flow is suppressed at $\theta =60^\circ$ , but is subsequently promoted as the vortex moves towards the hub at $\theta =75^\circ$ (see figure 8 i). At $\theta =105^\circ$ , the tip flow remains significant even when the vortex moves away from the tip region ( $r/R\lt 0.65$ , figure 8 j). Meanwhile, significant distortion of the vortex can be observed due to BVI at $\theta =75^\circ$ (figure 9 i), accompanied by a noticeable reduction in vorticity magnitude at the vortex centre compared with $\theta =60^\circ$ . There are small-scale vortices near the tip region generated by the BVI process (see figure 9 i,j). In addition, at low advance ratios ( $\mu =0$ and 0.16, figure 9 a–g), the vortices adjacent to the blade surface display a circular configuration, indicating that the vortex tube is perpendicular to the LE. In this case, the vortex-induced fluctuations are primarily oriented in the spanwise direction. In contrast, at a higher $\mu$ of 0.33 (figure 9 h–j), vortex shapes transition to elliptical due to twisting of the vortex tubes, resulting in parallel interactions.

Figure 10.

Vortex characteristics at varying advance ratios of $\mu$ during BVI: (a) axial displacement ( $\Delta y/R$ ) relative to the rotor disk, and (b) vorticity magnitudes ( $\omega _{\theta }$ ) of the vortex centres. The values are concluded in various rotor phases of $\theta =60^\circ$ , $75^\circ$ and $105^\circ$ .

Figure 11.

Representative TKE fields at 90 r.p.s. in the $r$ – $y$ plane for different advance ratios at three rotor phases of (i) $60^\circ$ , (ii) $75^\circ$ and (iii) $105^\circ$ : (a–c) $\mu =0$ , (e–g) $\mu =0.16$ and (h–j) $\mu =0.33$ .

Figure 10 summarizes the vortex characteristics during BVI regarding axial displacements ( $\Delta y/R$ ) relative to the rotor disk and the vorticity ( $\omega _{\theta }$ ) at the vortex centre, across a wide range of $\mu$ from 0 to 0.5. Note that different combinations of rotational and free stream speeds can result in multiple data points at the same advance ratio. Figure 10(a) illustrates that when $\mu \lt 0.23$ , $\Delta y/R$ rapidly decreases with increasing $\mu$ for each rotor phase. In contrast, the reduction rate of $\Delta y/R$ significantly slows when $\mu$ exceeds 0.23. This result indicates that the vortex moves rapidly towards the rotor disk as $\mu$ increases ( $\mu \lt 0.23$ ). However, when $\mu \gt 0.23$ , the axial displacement remains relatively constant due to the obstruction caused by the blade, resulting in a strong BVI. It is worthwhile to mention that the axial displacements remain relatively consistent in the different cases at the same $\mu$ , illustrated in the zoomed-in view in figure 10(a). This result implies that, under the tested conditions, the axial displacement is primarily dominated by the advance ratio, which reflects a combined effect of inflow speeds and rotational speeds. Figure 10(b) displays the vorticity magnitudes at the vortex centres for varying $\mu$ , generally exhibiting a ‘ladder-type’ structure. The magnitude of vorticity generally decreases as $\mu$ increases from 0 to 0.23, and then jumps suddenly at $\mu =0.23$ , after which it continues to decrease as $\mu$ increases further. The formation of the ‘ladder’ structure is due to the significant increase in tip vortex strength when both the rotational speed and inflow velocity are simultaneously increased, while maintaining the same $\mu$ . However, the trend does not hold when $\mu =0$ , as vorticity magnitudes consistently increase with rising revolutions per second

To quantify the unsteadiness surrounding the blade during BVI, the contours of TKE for representative cases are presented in figure 11. At $\mu =0$ (figure 11 a–c), the TKE reaches its maximum at the vortex centre, approximately 10 times higher than that in the wake region. Relatively small unsteady fluctuations are visible beneath the blade surface due to the vortex development. This behaviour changes as $\mu$ increases to 0.16, when the blade passes through the shear layer of the vortex. In this case, extensive unsteady regions appear in the vicinity of the blade tip at $\theta =75^\circ$ due to the interaction, with the same TKE level found at the vortex centre (see figure 11 f). The region with high TKE levels remains a circular shape, suggesting that the blade does not directly cut the vortex centre. However, at $\theta =105^\circ$ (see figure 11 g), the unsteadiness in the wake region diminishes, as the clean edgewise flow likely sweeps away turbulence generated during the interaction process. At $\mu =0.33$ , figure 11(i) shows that a distinct unsteady region can be observed immediately adjacent to the blade pressure side, centred at around ( $y/R, r/R)=(0.1, 0.75)$ . The maximum TKE value reaches around 580 $\textrm{m}^2\textrm{s}^{-2}$ , which is approximately three times the level at the vortex centre before BVI occurring at $\theta =60^\circ$ in figure 11(h). The unsteady flow region exhibits a significant distortion compared with that shown in figure 11(f). This variation may lead to a stronger BVI and associated noise.

Figure 12.

Positions of the blade and vortex during interactions at the rotor phase of $90^\circ$ in various regimes: (a) regime 1, vortex–BL interaction, $\mu \lt 0.1$ , (b) regime 2, shear layer–LE interaction, $0.1\lt \mu \lt 0.23$ and (c) regime 3, vortex centre–LE interaction (passing the vortex centre), $\mu \gt 0.23$ . (d) Summary of vortex centre trajectories in the different regimes against the advance ratio.

Figure 13.

Sketches of the interaction mechanism between the vortex and blades in various regimes. (a) Three-dimensional schematic (not to scale), (b) regime 1, (c) regime 2 and (d) regime 3. The left-hand side shows the view from the spanwise plane, while the right-hand side displays the view from the chordwise plane.

Based on abovementioned analysis, we can conclude that the BVI process is influenced by several factors, including the vortex’s relative position to the blade, the vortex attitude (the angle between the vortex tube’s axis and the rotor disk plane) and vorticity strength. The vortex strength can directly improve the noise level. However, the position and attitude of the vortex tube can deeply affect the noise generation mechanism during BVI, which is primarily dominated by $\mu$ . Therefore, the flow fields containing the trajectory of the vortex can be used to distinguish the key features of BVI across various flow regimes. Figure 12 concludes all trajectories of vortex centre (see figure 5) at various free stream speeds at 90 r.p.s., which can be categorized into three regimes based on their interaction characteristics. In regime 1 ( $\mu \lt 0.1$ , figure 12 a), the trajectories of the vortex centre are located away from the blade, indicating weak interaction. In regime 2 ( $0.1\lt \mu \lt 0.23$ , figure 12 b), the vortex centre is closer to the blade compared with that in regime 1, and the blade begin to interact with the shear layer of the vortex. In regime 3 ( $\mu \gt 0.23$ , figure 12 c), the blade directly passes through the vortex centre, leading to a significant interaction. Figure 12(d) summarizes the BVI process in three flow regimes with the varying advance ratios, including vortex–BL interaction in regime 1, shear layer–LE interaction in regime 2 and vortex centre–LE interaction in regime 3.

Figure 13 sketches BVI regimes and their associated effects on blade surfaces. The definition of the coordinate system used is shown in figure 13(a), where views from the $y$ – $r$ plane and the $x$ – $y$ plane are provided to illustrate the differences in the BVI process across the various flow regimes. In regime 1 (low advance ratio, figure 13 b), the upstream vortex is located below the blade pressure side during interactions with a relatively large separation distance. In this regime, vortex–BL interaction is inherently dominated by viscous effects, which depends on the ${\textit{Re}}$ (Taylor & Green Reference Taylor and Green1937; Peng & Gregory Reference Peng and Gregory2015). In this case, the upwash and downwash velocities induced by the vortex can affect the blade surface pressure distribution (Peng & Gregory Reference Peng and Gregory2015). This interaction likely induce instability in the blade BL and alter the rotor noise generation, particularly trailing edge (TE) noise. Therefore, in this viscous-type regime, the vortex strength, separation distance and ${\textit{Re}}$ can affect the noise generation during BVI process. The vortex tube is nearly perpendicular to the LE from the side view, which may induce spanwise fluctuations on the blade surface and result in weak interactions (Glegg Reference Glegg1991). In regime 2 (moderate advance ratio, figure 13 c), the interaction occurs at a smaller separation distance, leading to a higher pressure gradient near the LE being imposed onto the vortex, resulting in significant BVI noise. This process is dominated by the pressure gradient rather than viscous effects alone (Peng & Gregory Reference Peng and Gregory2015). Moreover, the velocity induced by the vortex aligns with the direction from the suction to the pressure side, which is opposite to the direction of the tip flow, potentially suppressing the tip vortex formation and tip noise. In regime 3 (high advance ratio, figure 13 d), the blade directly passes through the vortex centre due to a small separation distance, and the adverse pressure gradient near the surface decelerates the flow. Additionally, from the side view, the vortex tube gradually changes from perpendicular to parallel with the LE of blades during interactions due to the increasing free stream speed. The induced velocity on the blade surface aligns with the tip flow, possibly enhancing the tip vortex strength.

5. Acoustic characteristics

Generally, rotor noise consists of tonal and broadband components. The tonal part in the noise spectrum is generated at blade passing frequency (BPF) and its harmonics. Two main mechanisms contribute to the generation of this tonal component: loading noise and thickness noise. The former results from the aerodynamic forces exerted on the rotating blades relative to the surrounding air, while the latter derives from fluid displacement by the blade surfaces (Kurtz & Marte Reference Kurtz and Marte1970; Glegg & Devenport Reference Glegg and Devenport2017; Jamaluddin et al. Reference Jamaluddin, Celik, Baskaran, Rezgui and Azarpeyvand2024). In addition to these tonal noise sources, rotor noise also comprises the broadband component, which is stochastic in nature, is mainly generated by turbulent flow across the surface of rotor blades, significantly contributed by both BL noise and interaction noise (Brooks, Pope & Marcolini Reference Brooks, Pope and Marcolini1989). The vortex dynamics has impacts on both tonal and broadband components due to the inherent periodic and turbulent characteristic of the BVI process. In this section, we first provide an overview of the acoustic results, including spectral and beamforming analyses across various flow regimes. Then, the effects of BVI on the tonal and broadband components are separately discussed.

Figure 14.

Representative rotor noise spectra combined with motor and background noise: (a) $U_\infty =0$ m s⁻¹, 60 r.p.s., (b) $U_\infty =10$ m s⁻¹, 60 r.p.s., (c) $U_\infty =0$ m s⁻¹, 90 r.p.s., (d) $U_\infty =10$ m s⁻¹, 90 r.p.s., (e) $U_\infty =0$ m s⁻¹, 120 r.p.s. and (f) $U_\infty =10$ m s⁻¹, 120 r.p.s. The label of $m$ is non-dimensional number of BPF. The background noise includes the wind-off conditions in (i) and two wind-on conditions in (ii), with and without the rotor, for comparison.

Figure 14 presents the sound pressure level (SPL) spectra for representative cases at free stream speeds of 0 and 10 m s⁻¹, with rotational speeds of 60, 90 and 120 r.p.s., respectively. The results were obtained from a microphone positioned directly above and closest to the centre of rotation. The non-dimensional number of the BPF, $m$ , is defined as $m=f/\textrm{BPF}$ . The peak at $m=1$ corresponds to the fundamental BPF, and the peaks at $m=2$ and 3 represent the second and third harmonics of the BPF. Note that the spectra exhibit a high signal-to-noise ratio, as the motor noise is relatively lower than the rotor noise. Additionally, this motor can generate discrete tones near the 13 $\times$ r.p.s., 14 $\times$ r.p.s. and 42 $\times$ r.p.s. (see figure 14 c), which are likely caused by its inherent vibration (Huff & Henderson Reference Huff and Henderson2018; Wu et al. Reference Wu, Jiang, Zhou, Zhong, Zhang, Zhou and Chen2022). Therefore, in subsequent calculations, we have subtracted the discrete tones related to the motor noise. The SPL of tonal noise at the first BPF and the overall SPL of broadband noise (1–10 kHz) are used for uncertainty analysis. The corresponding uncertainties at the 95 % confidence interval are 2.1 and 0.52 dB based on the $t$ -distribution (Student 1908), respectively. Additionally, background noise is measured under the wind-off condition and two wind-on conditions, both with and without the rotor, to facilitate better comparison. The spectra show that subharmonic tonal components ( $m= 1.5, 2.5, \ldots$ ) are not evident, indicating the stability of the electric motor during the rotor operation since they can be regarded as the rotor shaft noise (Jamaluddin et al. Reference Jamaluddin, Celik, Baskaran, Rezgui and Azarpeyvand2024). Overall, the broadband noise level in edgewise flow cases is much higher than those in the hovering cases. Additionally, the spectrum shifts towards a more broadband characteristic, with several humps appearing near the tonal frequencies, particularly in the case at 120 r.p.s. (see figure 14 f). In the edgewise flow cases, particularly at low revolutions per second (see figure 14 b), several tones are superposed on a hump in the frequency range of 3–4 kHz. This portion of the noise is believed to be aerodynamic noise resulting from BVI, as BVI involves periodic and turbulent processes and occurs more frequently at higher advance ratios.

Figure 15.

Representative rotor noise spectra at various free stream speeds of 0, 10 and 20 m s⁻¹: (a) 60 r.p.s., (b) 90 r.p.s., (c) 100 r.p.s. and (d) 120 r.p.s. The corresponding $\mu$ is indicated in the legends.

Figure 15 shows the representative rotor noise spectra at free stream speeds of 0, 10 and 20 m s⁻¹ for rotational speeds ranged from 60 to 120 r.p.s. The corresponding advance ratios cover a practical range from 0 to 0.49. At $\mu =0$ (black line), the spectra are characterized by numerous harmonic peaks across a wide frequency range for all cases. These high-amplitude harmonic tones indicate the nonlinear effects of turbulence on the blade surface. In this regime, the vortex centre is located away from the blade, and small-scale turbulence from the boundary of vortices can be ingested by the blade (see figures 6 a and 9 a–c). These flow structures can cause instability in the BL of blades, distorting sound pressure signals within a small time scale compared with a rotational period (Li et al. Reference Li, Chen, Zhou, Huang and Zhang2024a ). These distortions result in the appearance of the high-order BPFs across the wide frequency range from 5 to 10 kHz. As the advance ratio increases to a moderate range (blue lines), the first BPF exhibits a significant increase, with an accompanying rise in broadband noise level due to the increased load. The harmonic peaks remain visible below 5 kHz, while the high-order BPFs become several humps above 5 kHz, indicating that interaction with large-scale turbulence (relatively low frequency) dominates the BVI process rather than ingesting small-scale turbulence (relatively high frequency). As the advance ratio increases to higher value (red lines), a wider frequency range of noise spectra exhibits a broadband nature, which overwhelms the high-order tones. The periodic noise associated with BVI is concentrated in the low-frequency range below 2 kHz. This is consistent with the flow physics, where the blade passes the vortex centre with significant velocity fluctuations over a large time scale, of the order of the rotational period (see figure 5 c). In this case, coherent turbulent structures can reach length scales comparable to the blade chord, impacting tonal noise generation.

Figure 16.

The SPL contour plots as a function of the advance ratio examined in this experiment. The rotational speed at (a) 60 r.p.s., (b) 90 r.p.s. and (c) 120 r.p.s. Specific spectral features are labelled in the callouts.

Figure 17.

The phase-averaged beamforming results of 6 kHz across various rotating angles in a period at $U_\infty =$ 0 m s⁻¹ and 90 r.p.s. Here (a) $\theta =0^\circ$ , (b) $\theta =30^\circ$ , (c) $\theta =60^\circ$ , (d) $\theta =90^\circ$ , (e) $\theta =120^\circ$ and (f) $\theta =150^\circ$ . The rotational direction is clockwise.

Figure 16 presents rotor noise spectra as a function of advance ratio in the form of SPL contours for 60, 90 and 120 r.p.s., respectively. In all cases, there is an evident increase of the first BPF with increasing advance ratio due to the enhanced loading on the blades, and the shaft tones are gradually overwhelmed by the broadband noise when $\mu \gt 0.2$ . Meanwhile, multiple high-amplitude tones significantly arise in the frequency range below 2 kHz from the moderate advance ratio ( $\mu \gt 0.1$ ) except in the case at 60 r.p.s. These tones are related to the periodical BVI process. In the cases with 90 and 120 r.p.s. (see figures 16 b and 16 c), the humps aligned with the BPF tones rise at the high advance ratio, forming a haystacking phenomenon (Murray et al. Reference Murray, Devenport, Alexander, Glegg and Wisda2018; Huang Reference Huang2023), and become more marked as the advance ratio further increases. Additionally, frequency shifting of the humps is observed near the high-order BPFs (i.e. $m=3$ , 5), indicating that the turbulence interaction with the blade alters the ideal periodicity of the rotational motion (Martinez Reference Martinez1996). It is worthwhile to mention that the harmonics of BPF ( $m\gt 2$ ), along with the broadband component, present an discontinuity feature at $\mu \approx 0.1$ for the cases with 90 r.p.s. (figure 16 b) and 120 r.p.s. (figure 16 c). However, it should be acknowledged that, at present, there is no clear physical explanation for the discontinuity observed in SPL spectra, which needs further investigation. The variation in spectral characteristics suggests that the underlying flow mechanism changes rapidly at different advance ratios, consistent with the flow mechanism variation across the three regimes. Based on the overview of the acoustic characteristics and flow field in various regimes, the BVI process impacts both the tonal noise, due to the periodicity of blade rotation, and the broadband noise, related to turbulence.

Figure 18.

The phase-averaged beamforming results of 6 kHz across various BVI-related rotating angles at $U_\infty =$ 10 m s⁻¹ and 90 r.p.s. Here (a) $\theta =45^\circ$ , (b) $\theta =60^\circ$ , (c) $\theta =75^\circ$ , (d) $\theta =90^\circ$ , (e) $\theta =105^\circ$ and (f) $\theta =130^\circ$ . The rotational direction is clockwise, and the free stream direction aligns with the $x$ axis.

As suggested by figure 14, broadband noise is significantly affected by the BVI process. To illustrate the broadband noise source location and strength, the wavelet-based beamforming method is employed (Chen et al. Reference Chen, Yang, Peng and Huang2021). A representative frequency of 6 kHz is selected as the broadband noise is dominant at this frequency, and a relatively high resolution of the acoustic imaging can be achieved (Chen et al. Reference Chen, Zhong and Huang2020b ). Figure 17 shows the phase-averaged acoustic image results at various rotor phases across a rotational period at $U_\infty =0$ m s⁻¹ and 90 r.p.s. ( $\mu =0$ ), corresponding to regime 1. The imaging plane is positioned at the rotor disk ( $y/R=0$ ), and the rotational direction is clockwise, as indicated in figure 17(a). The dashed lines represent tip locations, while the solid outlines of the rotor correspond to specific rotor phases ( $\theta$ ). The results are phase-averaged using 45 images for each rotor phase, which contain a total of 2 250 000 points in the sound signal. The uncertainty of SPL for the case of $\mu =0$ at the 95 % confidence interval is 1.19 % of the mean SPL at the blade tip for $\theta =0^\circ$ , as calculated using $t$ -distribution (Student 1908). The dynamic range of beamforming results is set to 8 dB as it can effectively illustrate the variation in BVI-related noise. The acoustic images show the noise sources span the entire radius of the blade, with respective maxima located at approximately 80 % of the blade radius. The noise source exhibits comparable strength on the two blades of the rotor for each rotor phase. At $\mu =0$ , the BVI noise is not dominated for the noise generation, which is consistent with the flow field results (see figure 5 a). However, there are deviations in the SPL strength across the various rotor phases, with a maximum variation of approximately 3 dB. This variation is likely caused by practical environmental uncertainties, such as the reflection and motor-induced vibration (Chen & Huang Reference Chen and Huang2018).

In contrast, figure 18 presents the acoustic image results at $U_{\infty }=10$ m s⁻¹ and 90 r.p.s. ( $\mu =0.16$ ), corresponding to regime 2. The direction of the free stream flow aligns with the $x$ axis, indicated by the arrows in figure 18(a). Note that some rotor phases, which have the noise levels far below those experience BVI process, are not included, as they do not provide useful information. Other set-ups are the same as those in figure 17. The beamforming results show that BVI occurs at approximately $\theta =45^\circ$ , and completes at approximately $\theta =130^\circ$ . The BVI-related noise exhibits obvious transient characteristics. For example, the noise intensity gradually increases as $\theta$ increases, reaching to the largest level at around $90^{\circ }$ , and rapidly decreases from around $105^{\circ }$ . The noise level reaches to a level higher than the hovering conditions around 5 dB when BVI occurs, indicating the BVI process is crucial for the rotor noise generation. The location of the maximum noise level gradually shifts towards the tip, moving from $\theta =45^\circ$ to $\theta =75^\circ$ . After reaching $\theta =75^\circ$ , it reverses direction, moving back towards the rotor hub. At $\theta =130^\circ$ , the maximum noise source shifts to the middle of the blade, indicating a migration of the BVI location.

Figure 19.

The phase-averaged beamforming results of 6 kHz across various BVI-related rotating angles in a period at $U_\infty =$ 20 m s⁻¹ and 90 r.p.s. Here (a) $\theta =15^\circ$ , (b) $\theta =30^\circ$ , (c) $\theta =45^\circ$ , (d) $\theta =60^\circ$ , (e) $\theta =75^\circ$ and (f) $\theta =90^\circ$ . The rotational direction is clockwise, and the free stream direction aligns with the $x$ axis.

Figure 19 presents the acoustic image results at $U_{\infty }=20$ m s⁻¹ and 90 r.p.s. ( $\mu =0.33$ ) corresponding to regime 3. Other set-ups are the same as those in figure 18. The rotor phase at which BVI begins to occur shifts earlier due to the increasing free stream speed. The rotor phases associated with BVI process range from approximately $15^\circ$ to $90^\circ$ . The results show that the strength of BVI-related noise rises to a maximum of 90 dB, indicating that the interaction becomes more significant due to the reduced axial distance between the blade surface and vortices. This finding is consistent with the flow field results (see figure 9). The beamforming results reveal that BVI-related noise is the primary source of rotor noise across moderate to high advance ratios, highlighting the significant impact of vortex dynamics during the BVI process on noise generation.

Figure 20.

Decomposition of a typical noise spectrum ( $U_\infty =10$ m s⁻¹ and 90 r.p.s.). The spectrum of narrowband, broadband and 1/3 octave band of the broadband component are indicated by black, red and blue lines, respectively.

Following an overview of the acoustic results across various flow regimes, we analyse the effects of BVI on both tonal and broadband noise components in the subsequent sections. This analysis aims to enhance our understanding of the underlying fluid mechanisms associated with each noise component. The initial step is to extract the tonal and broadband parts in the narrowband spectrum for further noise analysis, and the decomposition process of a representative noise spectrum based on a local regression method (Fattah et al. Reference Fattah, Chen, Wu, Wu and Zhang2019; Wu et al. Reference Wu, Chen, Jiang, Zhong and Zhang2023), presented in figure 20. The SPL of tones is extracted by using the peak value in the spectrum. Meanwhile, the 1/3 octave band spectrum of the broadband component is calculated, as indicated by the blue line in figure 20. The result clearly shows that the broadband component is successfully extracted from the spectrum.

5.1. Effects of vortex dynamics on tonal noise

Figure 21 presents the SPL values at BPF and its harmonics ( $m=$ 1–4) under various advance ratios for 60, 90 and 120 r.p.s. For the first BPF tone ( $m=1$ , figure 21 a), the SPL value rapidly increases with the advance ratio increasing as $\mu \lt 0.23$ , corresponding to regime 1 and regime 2. The increment of SPL values arrives at approximately 20 dB. Then, in regime 3 ( $\mu \gt 0.23$ ), the SPL values of first BPF demonstrate a slow growth and finally becomes insensitive to the advance ratio even as thrust continues to increase (see figure 3 a). Figure 21(b) shows that a large fluctuation of the second BPF with no clear trends based on the advance ratio. For higher-order BPFs ( $m=3$ and 4), there is a distinctive feature that the SPL values rapidly reduce in regime 1 ( $\mu \lt 0.1$ ), and then experience a continuous increase from regime 2 to regime 3, as shown in figure 21(c) and figure 21(d). The presence of an inflection point in SPL values between regime 1 and regime 2 indicates a change in the noise generation mechanism due to different interaction characteristics of the BVI process.

Figure 21.

The SPL of BPF and its harmonics for various advance ratios. (a) The first BPF ( $m=1$ ), (b) the second BPF ( $m=2$ ), (c) the third BPF ( $m=3$ ) and (d) the fourth BPF ( $m=4$ ).

Reynolds stress fields surrounding the blade are calculated to quantify the unsteady characteristic of flow conditions during BVI occurring, which is crucial for the generation of harmonic tones. Figure 22 shows the normal turbulence stresses of $u'^{2}$ , $v'^{2}$ and $w'^{2}$ in the $x$ , $y$ and $z$ directions for representative cases at 90 r.p.s. and a rotor phase of $60^\circ$ in the corresponding flow regimes. In general, the values of $u'^{2}$ , $v'^{2}$ and $w'^{2}$ reach respective maxima near the vortex centres in the measurement plane and increase with increasing free stream speeds. At $\mu =0$ (regime 1), the strength of turbulence stress is relatively weak compared with the cases in regime 2 and regime 3. In this case, a large distribution of unsteadiness can be observed in the vicinity of the rotor disk plane (see figure 22 a), suggesting that the blade likely ingests the small-scale turbulence. This turbulence ingestion can occur at all the rotor phases owing to symmetry of flow fields, likely causing sustained unsteadiness on the blade surface, which contributes to high-amplitude harmonic tones at high-frequencies (see figure 15 b). As $\mu$ increases, the ingested small-scale turbulence can be convected downstream by the clean inflow. In this case, the sustained interaction with small-scale turbulence is gradually replaced by the BVI process. However, the BVI strength is relatively weak due to the low advance ratio, leading to reduced amplitudes of the harmonic BPFs ( $m=3$ and 4, figures 21 c and 21 d) in regime 1. At a moderate advance ratio ( $\mu =0.16$ , regime 2), the blade begins to cut through the vortex’s shear layer, causing a relatively strong interaction, resulting in increasing amplitudes of the harmonic BPFs, as shown in figures 21(c) and 21(d). At $\mu =0.33$ (figure 22 c), the blade is closer to the core region of unsteady flows between $x/R=0.35$ and 0.55, where the maximum value of normal turbulence stresses can reach 50 times those in the undisturbed free stream. This significant interaction likely causes a strong BVI coupled with both periodic and turbulent processes, leading to increasing amplitudes of tones in the midfrequency range from 1 to 2 kHz, and particularly in broadband noise from 2 to 10 kHz (see figure 14 f).

Figure 22.

Normal turbulence stress fields of $u'^{2}$ , $v'^{2}$ and $w'^{2}$ in the $x$ , $y$ and $z$ directions in the measurement plane for representative cases at a rotor phase of $60^\circ$ and 90 r.p.s., corresponding to three flow regimes: (a) $U_\infty =0$ m s⁻¹, $\mu =0$ , (b) $U_\infty =10$ m s⁻¹, $\mu =0.16$ and (c) $U_\infty =20$ m s⁻¹, $\mu =0.33$ .

Figure 23.

The comparison of normal turbulence stresses at four specific streamwise locations of $x/R=-0.2$ , 0, 0.2 and 0.4 along the $y$ direction, extracted from the contours in figure 22: (a) $u'^{2}$ , (b) $v'^{2}$ and (c) $w'^{2}$ . Other set-ups are the same as those in figure 22.

Figure 23 presents line plots of normal turbulence stresses ( $u'^{2}$ , $v'^{2}$ and $w'^{2}$ ) at four specific streamwise locations of $x/R=-0.2$ , 0, 0.2 and 0.4 along the $y$ direction, corresponding to three flow regimes in figure 22. At $x/R=-0.2$ , the levels of normal turbulence stresses at $U_\infty =0$ m s⁻¹ are much higher than these at $U_\infty =10$ and 20 m s⁻¹, indicating the disturbed flow conditions at the upstream region for a low advance ratio. At $x/R=0$ , the turbulence stress values at $U_\infty =0$ m s⁻¹ are comparable to those at the upstream location $x/R=-0.2$ . The stress values at $U_\infty =10$ m s⁻¹ begin to increase, whereas those at $U_\infty =20$ m s⁻¹ remain low. At $x/R=0.2$ , all stress values are generally at the same level, with the values at $U_\infty =10$ and 20 m s⁻¹ slightly higher than those at $U_\infty =0$ in the vicinity of the blade, from $y/R=0$ to $y/R=0.1$ . At $x/R=0.4$ , there is a distinctive peak at $y/R=0.1$ for the case with $U_\infty =20$ m s⁻¹, which could lead to significant fluctuations on the blade surface during BVI.

Additionally, the wavenumber–frequency decomposition of the velocity fluctuations near the blade is employed to characterize specific waves associated with the BVI process and to determine their propagation direction and phase velocity. The primary objective is to decompose the signals into both spatial and temporal frequencies using a 2-D Fourier transform, mapping energy in a wavenumber–frequency diagram (Choi & Moin Reference Choi and Moin1990; Gloerfelt & Berland Reference Gloerfelt and Berland2013; Pröbsting et al. Reference Pröbsting, Serpieri and Scarano2014). The detailed calculation follows the methodology outlined by de Kat & Ganapathisubramani (Reference de Kat and Ganapathisubramani2015), which is well-established for studies on turbulence. Firstly, a space–time signal of the downwash velocity fluctuation near the blade, $v'(t, x; y_0)$ , is obtained along a line at $y_0/R=0.08$ , parallel to the free stream direction (see figure 22). This signal is derived from time-resolved PIV results with 4000 snapshots at a sampling rate at 1600 Hz. Then, a windowed and truncated 2-D Fourier transform of the velocity fluctuations, denoted as, $\skew 2\hat {v}(f, \xi _{x}; y)$ , is calculated by

(5.1) \begin{align} \skew 2\hat {v}(f, \xi _{x}; y_0)=\frac {1}{\sqrt {XT\int _{0}^{X}\int _{0}^{T}w(x,t)^{2}\textrm{d}t \textrm{d}x}}\int _{0}^{X}\int _{0}^{T}w(x,t)v'(t, x; y_0)\textrm{e}^{-\textrm{i}2\pi (ft+\xi _{x}x)}\textrm{d}t\textrm{d}x, \end{align}

where $f$ represents the frequency, $\xi _{x}$ is the wavenumber in the $x$ direction, defined as $\xi _{x}=1/\lambda _{x}$ ( $\lambda _{x}$ is the corresponding wavelength). Here $X$ denotes the domain length, $T$ is the sample time interval length and $w$ is the weighting function (Hamming window). Then, the autospectrum of $\skew 2\hat {v}(f, \xi _{x}; y_0)$ can be calculated by

(5.2) \begin{align} {\varPhi }_{vv}(f, \xi _{x}; y_0)=\skew 2\hat {v}^{*}(f, \xi _{x}; y_0)\skew 2\hat {v}(f, \xi _{x}; y_0), \end{align}

where $\skew 2\hat {v}^{*}$ is the complex conjugate of $\skew 2\hat {v}$ . Then, $L_{v}(f, \xi _{x}; y_0)$ is given by

(5.3) \begin{align} {L_{v}(f, \xi _{x}; y_0)}=10\log _{10}({\varPhi }_{vv}(f, \xi _{x}; y_0)). \end{align}

Figure 24.

Wavenumber–frequency spectra of downwash velocity fluctuations, $v'(t, x; y_0)$ , along a line parallel to the free stream direction at $y/R=0.08$ for representative cases at 90 r.p.s. Here (a) $U_\infty =0$ m s⁻¹ in regime 1, (b) $U_\infty =10$ m s⁻¹ in regime 2 and (c) $U_\infty =20$ m s⁻¹ in regime 3. Also indicated are the convection velocity (solid lines), dominant tone (first BPF) and the corresponding wavelength (dashed lines).

Figure 24 shows the corresponding wavenumber–frequency map of downwash velocity fluctuations, $v'(t, x; y_0)$ , for representative cases with advance ratios of 0, 0.16 and 0.33 at 90 r.p.s. At a low advance ratio ( $\mu =0$ , figure 24 a), the wavenumber–frequency contour of $L_{v}(f, \xi _{x}; y_0)$ shows that the maximum value is located at a frequency of 180 Hz, which is the same frequency of first BPF (see figure 14 c). Here, the wave velocity or phase velocity ( $u_{p}$ ) of the flow structures associated with first BPF can be calculated by (de Kat & Ganapathisubramani Reference de Kat and Ganapathisubramani2015)

(5.4) \begin{align} u_{p}=f\lambda _{x}=\frac {f}{\xi _{x}}. \end{align}

Therefore, the wave velocity can be calculated by examining the slope of the solid lines in figure 24. These solid lines are defined as extending from the origin to the point of maximum value on the wavenumber–frequency contour at the first BPF. The wavelength can be determined by examining the vertical coordinate corresponding to the peak value ( $\lambda _{x}=1/\xi _{x}$ ). It is worthwhile to mention that energy content at positive wavenumbers indicates downstream propagating waves, while negative wavenumbers represent upstream propagation (Pröbsting et al. Reference Pröbsting, Serpieri and Scarano2014). The wavelength relating this wave velocity to the tonal frequency indicates the length scale of the associated flow structures. In regime 1 ( $\mu =0$ ), figure 24(a) shows that the corresponding wavelength $\lambda _x$ is 9.6 mm with an upstream propagating direction at a convection speed of 2.2 m s⁻¹. Note that the upstream propagating direction is the opposite to the free stream direction and aligned with the $-x$ axis. This result indicates the movement of the rotating blade dominates the propagation direction of tone-related flow structures at a low advance ratio. Additionally, this wavelength is relatively small, consistent with the condition of blade–BL interaction in regime 1, where small-scale turbulence near the vortex boundary affecting the noise generation. In regime 2 ( $\mu =0.16$ , figure 24 b), the wavenumber–frequency contour shows the wavelength of tone-related vortices increases to 23.8 mm because shear layer of vortices dominates the interaction process. In this case, the propagation direction remains upstream as the free stream speed is relatively low. In regime 3 ( $\mu =0.33$ , figure 24 c), the wavelength continues to increase as the vortex centre moving closer to the blade surfaces. The propagation speed of $u_p$ is 5.1 m s⁻¹ with a downstream direction, indicating that inflow dominates motion of the tone-related flow structures. This change may provide insights into noise reduction for eVTOL aircraft at various flow regimes, considering the propagation direction of tone-related flow structures.

5.2. Effects of vortex dynamics on broadband noise

The 1/3 octave band spectra of representative cases across a wide range of $\mu$ with rotational speeds from 90 to 120 r.p.s. and $U_\infty$ from 0 to 20 m s⁻¹, corresponding to three flow regimes, are shown in figure 25. In general, the broadband noise level gradually increases with revolutions per second and $U_\infty$ . However, the spectral features in different flow regimes show distinct incremental behaviours in terms of frequency range and the central frequency. The power law of $M_{\mathit{tip}}^5$ is commonly used to scale the noise amplitude for different broadband noise sources based on the study from Brooks et al. (Reference Brooks, Pope and Marcolini1989), where $M_{\mathit{tip}}$ is the tip Mach number. However, to account for the effects of $\mu$ on the BVI process, this study uses the non-dimensional parameters of $M_{\mathit{tip}}$ and $\mu$ for improved scaling. It should be acknowledged that the current scaling law remains valid for a single rotor under edgewise flow conditions at the low Reynolds number range ( $3.9\times 10^4\lt Re\lt 1.0\times 10^5$ ). Nevertheless, the available experimental data set and well-fitted results provide insights into the effects of vortex dynamics on rotor broadband noise generation associated with BVI.

Figure 25.

The 1/3 octave band spectra comparison: unscaled results in (a) regime 1, (b) regime 2 and (c) regime 3; scaled results considering $M_{\mathit{tip}}$ and $\mu$ in (d) regime 1, (e) regime 2 and (f) regime 3. The black lines in scaled results are obtained by averaging the scaled data. A grey dashed line is used to separate the frequency ranges in regime 2.

In regime 1 (figure 25 a), the spectral shapes at different advance ratios are highly similar with the same increase rate for low and high frequency parts, as indicated by the red arrows. This phenomenon illustrates that the BVI is insignificant, and small-scale turbulence affects the noise at a broad frequency range. The similar spectral behaviours at different rotational speeds indicate the potential of performing non-dimensional analysis on the broadband spectra to obtain a universal spectral shape function for rotor blades under a certain operation range in regime 1. Here, the scaling law can be written as

(5.5) \begin{align} \mathcal{SPL}_{1/3}=\textrm{SPL}_{1/3}-10\log _{10}\left(K_1\mu ^2M_{\textit{tip}}^5\right)\!, \end{align}

where $\mathcal{SPL}_{1/3}$ is the scaled 1/3 octave band SPL, and $K_1=1/(\mu _{\mathit{ref}}^{2}M_{\textit{tip,ref}}^{5})=7.8\times 10^5$ is an empirical constant determined at the reference condition of 90 r.p.s. and $U_{\infty }=5$ m s⁻¹. Notably, in practice, noise scaling requires various physical information of BL parameters, which might vary spanwisely. However, the spanwise correlation of the blade BL is difficult to measure and model. Therefore, in this work, an empirical parameter of $K_1$ is used to scale the noise amplitude. The scaling law can collapse data over the frequency range of 0.1–10 kHz for the current test configuration, with a maximum deviation of less than 2 dB, as shown in figure 25(d).

In regime 2 (figure 25 b), a different spectra behaviour exists with a significant spectral shape change. The increasing rate of noise level for the low-frequency range ( $f\lt 2$ kHz) is much higher than that in the high-frequency range ( $f\gt 2$ kHz), as indicated by the red arrows. The increasing rate at higher frequency range ( $f\gt 2$ kHz) is comparable to that observed in regime 1. These spectra features indicate that the interaction between blade and the vortex’s shear layer primarily affects low-frequency broadband noise. The non-comparable spectral behaviours at different advance ratios are difficult to model with a single scaling law. Thus, two separate scaling laws according to the frequency range are used to perform non-dimensional analysis on the broadband spectra. Here, the scaling law in regime 2 can be written as

(5.6) \begin{align} \mathcal{SPL}_{1/3}=\textrm{SPL}_{1/3}-10\log _{10}\left(K_2\mu ^2M_{\textit{tip}}^5\right)+\Delta \mathcal{K}, \end{align}

where $K_2$ is an empirical parameter determined at the reference condition of 90 r.p.s. and $U_\infty =10$ m s⁻¹ in regime 2, and is given by

(5.7) \begin{align} K_2=\left \{ \!\!\begin{array}{c} 1/\left(\mu _{\mathit{ref}}^{3.5}M_{\textit{tip,ref}}^{5}\right)=4.3\times 10^6, \qquad (f\lt 2 \textrm{kHz}) \\[7pt] 1/\left(\mu _{\mathit{ref}}^{2}M_{\textit{tip,ref}}^{5}\right)=2.4\times 10^5. \,\qquad (f\gt 2 \textrm{kHz}) \end{array} \right . \end{align}

Here $\Delta \mathcal{K}$ is a constant to adjust the SPL levels between two frequency ranges, which is given by

(5.8) \begin{align} \Delta \mathcal{K}=\left \{ \!\!\begin{array}{cc} 0,\quad \qquad (f\lt 2 \textrm{kHz}) & \\ 1.53. \qquad (f\gt 2 \textrm{kHz}) & \end{array} \right . \end{align}

The scaling result presents a great agreement over the frequency range of 0.1–10 kHz, as illustrated in figure 25(e).

In regime 3 (figure 25 c), the increase in broadband noise with the advance ratio is primarily observed in the high-frequency range ( $f\gt 1$ kHz), while the variation in the low-frequency range below 1 kHz is not evident. It is worthwhile to mention that we employed a single scaling law for engineering simplicity and convenience as the intervals between the lines at various $\mu$ are not substantial. The spectra feature suggests that the BVI-related noise moves to the high frequency components rather than the low frequency in regime 2. The significant interaction between the blade and vortex centre induces high-frequency disturbances on the blade surface. Besides, the more parallel interaction induces chordwise fluctuations, which could result in a stronger turbulent BL trailing-edge noise source. The scaling law for regime 3 can be fitted as

(5.9) \begin{align} \mathcal{SPL}_{1/3}= \textrm{SPL}_{1/3}-10\log _{10}\left(K_3\mu ^{3}M_{\textit{tip}}^5\right)\!, \end{align}

where $K_3=1/(\mu _{\mathit{ref}}^{3}M_{\textit{tip,ref}}^{5})=1.5\times 10^5$ is an empirical constant determined at the reference condition of 90 r.p.s. and $U_{\infty }=20$ m s⁻¹ in regime 3. Notably, the scaling parameter of $\mu$ is 3, which lies between the two values of the scaling parameter of $\mu$ in regime 2, indicating a comprehensive effect of the BVI process. The scaling results are shown in figure 25(f), which can also collapse data over the wide frequency range of 0.1–10 kHz.