2.1. Beamforming in acoustic imaging

In this work, both the classical beamforming and the novel wavelet-based beamforming methods are used to produce aeroacoustic source maps of the propeller. The basic idea of the beamforming technique is introduced here briefly for those unfamiliar with this imaging method. For a more detailed introduction to beamforming for aeroacoustic applications, interested readers are referred to Huang et al. (Reference Huang, Bai, Vinogradov and Peers2012) for classical beamforming and Chen et al. (Reference Chen, Peng, Liem and Huang2020a) for wavelet-based beamforming.

In the classical frequency-domain beamforming, the acoustic pressure signal $p(\,\boldsymbol {y},t)$ is first sampled from each sensor of the array, where $\boldsymbol {y}$ gives the associated spatial coordinates. Then the frequency-domain signal $P(\,\boldsymbol {y},f)$ is obtained by applying a Fourier transform. Next, the data from multiple sensors are collected to construct the so-called cross-spectral matrix (CSM):

(2.1)\begin{equation} \boldsymbol{\mathsf{A}} = \boldsymbol{Y}(\,\boldsymbol {y},f) \boldsymbol{\cdot} \boldsymbol{Y}^*(\,\boldsymbol {y},f), \end{equation}

where $\boldsymbol {Y}(\,\boldsymbol {y},f)=[P(\,\boldsymbol {y}_1,f), \ldots, P(\,\boldsymbol {y}_n,f)]$ is the vector form of the array measurements, $\boldsymbol {y}_n$ gives the coordinates of the $n$th array sensor, and $({\cdot })^*$ denotes the complex conjugate. Diagonal removal of the CSM is applied to mitigate the interference and background noise. Finally, the imaging output is computed by

(2.2)\begin{equation} b(\boldsymbol{x},f) = \boldsymbol{w}^*(\boldsymbol{x}) \boldsymbol{\cdot} \boldsymbol{\mathsf{A}} \boldsymbol{\cdot} \boldsymbol{w}(\boldsymbol{x}), \end{equation}

where $\boldsymbol {x}$ is the location of the imaging point, $\boldsymbol {w}(\boldsymbol {x}) = \boldsymbol {G}(\boldsymbol {x},\boldsymbol {y}) / \Vert \boldsymbol {G}(\boldsymbol {x},\boldsymbol {y}) \Vert$ is the beamforming weight vector, $\Vert {\cdot } \Vert$ denotes the L2-norm, and $\boldsymbol {G}(\boldsymbol {x},\boldsymbol {y}) = [G(\boldsymbol {x},\boldsymbol {y}_1), \ldots, G(\boldsymbol {x},\boldsymbol {y}_n)]$ is the free-space Green's function. Thus the source distribution and the associated source amplitude at any frequency are obtained by calculating iteratively $b(\boldsymbol {x},f)$ over a given plane.

For wavelet-based beamforming, the procedure is similar but slightly modified to include the temporal dimension. The time–frequency domain data $P(\,\boldsymbol {y},t,s)$ are obtained from the sensor signals by the continuous wavelet transform

(2.3)\begin{equation} P(\,\boldsymbol {y},t,s) = \frac{1}{s} \int_{-\infty}^{\infty} \psi^*\left(\frac{\tau-t}{s}\right) p(\,\boldsymbol {y},\tau) \,{\rm d}\tau, \end{equation}

where $\psi$ is the Morse wavelet, and $s$ is the scale parameter. The CSM is then constructed with a form similar to (2.1):

(2.4)\begin{equation} \boldsymbol{\mathsf{A}} = \boldsymbol{Y}(\,\boldsymbol {y},t_e,s) \boldsymbol{\cdot} \boldsymbol{Y}^*(\,\boldsymbol {y},t_e,s), \end{equation}

where $\boldsymbol {Y}(\,\boldsymbol {y},t_e,s)=[P(\,\boldsymbol {y}_1,t_1,s), \ldots, P(\,\boldsymbol {y}_n,t_n,s)]$, and $t_n$ is the corresponding receiving time of the noise emitted at time $t_e$. Finally, the imaging output is computed by

(2.5)\begin{equation} b(\boldsymbol{x},s,t_e) = \boldsymbol{w}^*(\boldsymbol{x},t_e) \boldsymbol{\cdot}\boldsymbol{\mathsf{A}} \boldsymbol{\cdot} \boldsymbol{w}(\boldsymbol{x},t_e), \end{equation}

where $\boldsymbol {w}(\boldsymbol {x},t_e) = \boldsymbol {G}(\boldsymbol {x},\boldsymbol {y},t_e) / \Vert \boldsymbol {G}(\boldsymbol {x},\boldsymbol {y},t_e) \Vert$, and $\boldsymbol {G}(\boldsymbol {x},\boldsymbol {y},t_e) = [G(\boldsymbol {x},\boldsymbol {y}_1,t_e), \ldots, G(\boldsymbol {x},\boldsymbol {y}_n, t_e)]$ is the free-space Green's function for moving sources (Chen et al. Reference Chen, Peng, Liem and Huang2020a). In this way, the source maps at different time $t_e$ are obtained in the time–frequency domain.

2.2. Near-field aeroacoustic source analysis

One of the main drawbacks of the numerical beamforming approach is that the resolution of the sources is frequency-dependent and can result in inadequate resolutions at lower frequencies. More of this limitation of the beamforming approach will be demonstrated and discussed in § 4. In this subsection, we propose a high-resolution near-field aeroacoustic source analysis approach that reconstructs directly the aeroacoustic sources on a rotating surface from the computational fluid dynamic solutions. The derivation starts from the classical FW–H equation, which, however, can provide only the far-field noise signals. By introducing the assumptions of low Mach number flow, constant rotating motion and far-field receivers, the FW–H equation is so simplified that it leads to the definition of the so-called near-field source terms, which will be used in the following near-field aeroacoustic source analysis in the rest of this paper and constitutes one of the main contributions of this work.

In the FW–H equation, the source terms are separated into various parts that are defined in the volume or on the surface. Let $g(\boldsymbol {x},t)=0$ denote the control surface that moves at a velocity $\boldsymbol {U}(\boldsymbol {x},t)$, and let $\boldsymbol {v}(\boldsymbol {x},t)$ denote the velocity of the fluids. Then the FW–H equation takes the classical form

(2.6)\begin{align} \left(\frac{1}{c_0^2}\,\frac{\partial^2}{\partial t^2} - \nabla^2 \right) \left\{ H(g)\,c_0^2 (\rho-\rho_0) \right\} &= \frac{\partial^2 \{H(g)\,T_{ij}\}}{\partial x_i\,\partial x_j} - \frac{\partial}{\partial x_i} \left\{ \left[ \rho v_i (v_j - U_j ) + P^\prime_{ij} \right]\delta(g) \right\} \nonumber\\ &\quad + \frac{\partial}{\partial t} \left\{ \left[ \rho v_j + \rho (v_j - U_j) \right] \delta(g) \right\}, \end{align}

where $T_{ij} = \rho v_i v_j + [ (p-p_0) - c_0^2 (\rho -\rho _0) ]\, \delta _{ij} - \sigma _{ij}$ is the Lighthill stress tensor, $P^\prime _{ij} = (p-p_0) \delta _{ij} - \sigma _{ij}$ is the compressive stress tensor, and $\sigma _{ij}$ is the viscous stress tensor. Here, $p_0$ and $\rho _0$ denote the mean values of pressure $p$ and density $\rho$, respectively, $c_0$ is the speed of sound, $\boldsymbol {v}$ is the velocity of the fluids, $H(g)$ is the Heaviside function, and $\delta (g)$ is the Dirac delta function.

The three terms on the right-hand side of (2.6) are the well-known aeroacoustic sources. The generic functions $H(g)$ and $\delta (g)$ inside those terms can be incorporated seamlessly into far-field integral solutions but cannot be calculated directly at the near field due to their mathematical features (defined only in far-field integrals). Therefore, we need to introduce further simplifications to bypass those generalized functions in the calculations of the proposed near-field aeroacoustic source model.

To achieve the simplified source terms, we assume that the derivation is for low Mach number flow and the associated control surfaces rotating around a fixed axis at a constant speed, which is often the case for aerodynamic rotors. As the flow Mach number is low and the focus of this study is at low to mid frequencies, where the wavelengths of the sound waves from the rotor are much greater than the characteristic dimension of the rotor, the propeller can be regarded as acoustically compact, and consequently, the Lighthill stress tensor can be neglected (Turner & Kim Reference Turner and Kim2022). In addition, the term $v_j-U_j$ in (2.6) is zero when the control surface – which is the blade surface in this work – is impermeable and no-slip. Hence, the derivation starts from the simplified FW–H equation

(2.7)\begin{equation} \left( \frac{1}{c_0^2}\,\frac{\partial^2}{\partial t^2} - \nabla^2 \right) \left\{ H(g)\,p_a^\prime \right\}= \frac{\partial}{\partial t} \left\{ \rho U_n\,\delta(g) \right\} - \frac{\partial}{\partial x_i} \left\{ p n_i\,\delta(g) \right\}, \end{equation}

where $p_a^\prime = c_0^2 (\rho -\rho _0)$ denotes the acoustic pressure exterior of the control surface, and $\delta (g)$ limits the rest terms to be evaluated on the control surface. Here, $\boldsymbol {n}$ is the surface normal, and $U_n = \boldsymbol {U} \boldsymbol {\cdot } \boldsymbol {n}$ denotes the projection of the surface local velocity in the direction of $\boldsymbol {n}$. The first term on the right-hand side of (2.7) represents the thickness noise source, and the second term denotes the loading noise source.

To solve the FW–H equation, integral relations have been provided to calculate noise from moving sources. With the help of the free-space Green's function, the solution (Farassat's formulation 1A) to (2.7) is obtained by (Farassat & Succi Reference Farassat and Succi1980)

(2.8)\begin{equation} p_a^\prime(\boldsymbol{x}, t) = p^\prime_T(\boldsymbol{x}, t) + p^\prime_L(\boldsymbol{x}, t), \end{equation}

in which the thickness and loading components are

(2.9)\begin{equation} 4{\rm \pi}\,p^\prime_T(\boldsymbol{x}, t) = \rho_0 \int_{f=0} \left[ \frac{\dot{U}_n}{r(1-M_r)^2} + \frac{ U_n \hat{r}_i \dot{M}_i}{r(1-M_r)^3} + \frac{c_0 U_n (M_r - M^2)}{r^2(1-M_r)^3} \right]_{{ret}} \mathrm{d}S \end{equation}

and

(2.10)\begin{align} 4{\rm \pi}\,p^\prime_L(\boldsymbol{x}, t) &= \frac{1}{c_0} \int_{f=0} \left[ \frac{\dot{p} \cos\theta}{r(1-M_r)^2} + \frac{\hat{r}_i \dot{M}_i p \cos\theta}{r(1-M_r)^3} \right]_{{ret}} \mathrm{d}S \nonumber\\ &\quad + \int_{f=0} \left[ \frac{p(\cos\theta - M_i n_i)}{r^2(1-M_r)^2} + \frac{(M_r-M^2)p\cos\theta}{r^2(1-M_r)^3} \right]_{{ret}} \mathrm{d}S, \end{align}

where $\boldsymbol {M}=\boldsymbol {U}/c_0$ is the local Mach number, $\boldsymbol {r}=|\boldsymbol {x}-\boldsymbol {y}|$ is the distance vector from the source to the receiver, and $\hat {\boldsymbol {r}} = \boldsymbol {r}/r$, $\cos \theta = \boldsymbol {n} \boldsymbol {\cdot } \hat {\boldsymbol {r}}$ and $M_r=\boldsymbol {M} \boldsymbol {\cdot } \hat {\boldsymbol {r}}$. Here, $[ {\cdot } ]_{{ret}}$ means that a function is evaluated at the retarded time, and $\dot {({\cdot })}=\partial /\partial \tau$ denotes the time derivative of any quantity. The solution is based on a simplified substitution from the spatial derivative to the time derivative, and adopts the following relationship between the receiver time $t$ and the source time $\tau$:

(2.11a,b)\begin{equation} \frac{\partial}{\partial x_i} \approx{-}\frac{\hat{r}_i}{c_0}\,\frac{\partial}{\partial t},\quad \frac{\partial}{\partial t} [ {\cdot} ]_{{ret}} = \left[ \frac{1}{1-M_r}\, \frac{\partial}{\partial \tau} \right]_{{ret}}. \end{equation}

In (2.9) and (2.10), the terms that decay in the order of $1/r^2$ are ignored by assuming that the receivers are at the far field with $r \gg 1$. Furthermore, with the solid blade surfaces of the propeller used as the control surfaces, and rotating around a fixed axis $x$ with a constant speed, the noise at far-field receivers can be simplified as

(2.12)\begin{equation} 4{\rm \pi}\,p^\prime_T(\boldsymbol{x}, t) = \frac{\rho_0}{c_0} \int_{f=0} \left[\frac{U_n a_r}{r(1-M_r)^3} \right]_{{ret}} \mathrm{d}S \end{equation}

and

(2.13)\begin{equation} 4{\rm \pi}\,p^\prime_L(\boldsymbol{x}, t) = \frac{1}{c_0} \int_{f=0} \left[ \frac{\dot{p} \cos\theta}{r(1-M_r)^2} + \frac{a_r p \cos\theta}{c_0 r(1-M_r)^3} \right]_{{ret}} \mathrm{d}S, \end{equation}

where $a_r=\boldsymbol {a} \boldsymbol {\cdot } \hat {r}$, and $\boldsymbol {a}$ denotes the local acceleration on the surface.

The aeroacoustic sources are then defined from the right-hand-sides of (2.12) and (2.13). Taking the first right-hand-side term in (2.13) as an example, the term inside the retarded time square bracket can be divided into three parts:

(2.14) \begin{equation} \underbrace{\frac{\partial p}{\partial \tau}\,\boldsymbol{n}}_{{\rm source}}\boldsymbol{\cdot} \underbrace{\frac{\boldsymbol{\hat{r}}}{r}}_{{\rm receiver}}\!\! \underbrace{\frac{1}{(1-M_r)^2}}_{{\rm Doppler}}, \end{equation}

where $1/r$ denotes the order of the far-field emission, $\hat {r}$ is the direction from local sources to the far-field receivers, and $1/(1-M_r)^2$ are the Doppler effects caused by the relative motion between local sources and stationary receivers, and the remaining part to be the near-field aeroacoustic source.

Under similar decompositions, the following three near-field aeroacoustic sources are defined:

(2.15a)\begin{gather} \boldsymbol{S}_{T} = \rho_0 U_n \boldsymbol{a}, \end{gather}

(2.15b)\begin{gather}\boldsymbol{S}_{L1} = \dot{p} \boldsymbol{n}, \end{gather}

(2.15c)\begin{gather}\boldsymbol{S}_{L2} = (a_r p/c_0) \boldsymbol{n}. \end{gather}

The above model is a straight extension from the classical FW–H acoustic analogy but enables the examination of near-field sources based directly on computational fluid dynamics solutions. Physically, the direction of the thickness noise source $\boldsymbol {S}_{T}$ is determined by the local acceleration, and the loading noise source $\boldsymbol {S}_{L1}$ has the same direction as the local surface normal direction. It is expected that the thickness noise would be louder on the propeller disc plane, and the loading noise radiating from $\boldsymbol {S}_{L1}$ would be louder for far-field receivers in the downstream and upstream directions. The direction of another loading noise source $\boldsymbol {S}_{L2}$ is slightly more complicated, due to the difficulty to be fully decoupled from the effect of far-field receivers. More specifically, although its direction is defined to be the same as surface normal in (2.15c), the term $a_r$ suggests that the directivity is somewhat between $\boldsymbol {S}_{T}$ and $\boldsymbol {S}_{L1}$. More discussions are to be given in § 5.1.

Amongst the above three noise sources, we found that $\boldsymbol {S}_{L1}$ possesses mathematical beauty mostly, which is also dominant compared to $\boldsymbol {S}_{L2}$ for the current problem of interest. When considering a constantly rotating surface, the thickness noise source depends on only the surface itself (i.e. the revolving speed and geometry), while the loading noise source terms depend on both the surface and the flow that induces pressure perturbations. For simplicity, the amplitudes of the above sources are denoted by $S_{T}=\rho _0 U_n a$, $S_{L1}=\dot {p}$ and $S_{L2}=ap/c_0$, respectively. The effect of the receiver's direction on the acceleration part of $S_{L2}$ is removed for later convenience, as always $a_r \leq a$. Moreover, the relative ratio between the two loading noise source terms is well defined. By applying the Fourier transform to $S_{L1}$ and $S_{L2}$, the two terms become $S_{L1}(f) = 2{\rm \pi} f\,p(f)$ and $S_{L2}(f) = a\,p(f)/c_0$, which lead to $S_{L1}(f)/S_{L2}(f) = 2{\rm \pi} f c_0 / a = f c_0 / 2{\rm \pi} N^2 r_s$, where $N$ is the revolving speed of the propeller, and $r_s$ is the distance to the revolving axis. It is clear that the ratio between $S_{L1}$ and $S_{L2}$ increases along with the frequency. In the applications of rotors, the most concerned noise is often around the BPF, which is the revolving speed times the number of blades. Setting $f=N$ for brevity, we have $S_{L1}(f)/S_{L2}(f) \sim c_0 / 2{\rm \pi} N r_s = 1/M$. Hence for low Mach number flows, the amplitude of $S_{L1}$ is much larger than that of $S_{L2}$ at those frequencies with $f \geq N$.

3.1. Numerical set-up

The numerical set-up of a two-bladed propeller ingesting an upstream aerofoil wake is shown in figures 2(a) and 2(b). In the present set-up, the $x$-axis is along the freestream direction and is the revolving axis of the propeller, the $y$-axis is along the spanwise direction of the aerofoil, the $z$-axis is perpendicular to the aerofoil, and the coordinate origin is at the centre of the propeller. The revolving phase $\varphi$ is defined such that at $\varphi =0^\circ$ the propeller is parallel to the aerofoil, at $\varphi =90^\circ$ they are perpendicular, and $\varphi$ increases with the rotation direction.

Figure 2. (a,b) Schematics of the computational domains and the boundary conditions (not to scale), where the dotted frame and circle denote the boundary of the rotating domain around the propeller. Line 1 in (b) is $0.1c$ upstream from the propeller disc plane. (c) The coupling interfaces of computational domains. (d) The artificial microphone arrays used in numerical beamforming.

The propeller has diameter $D=240$ mm, and the upstream NACA 0020 aerofoil has chord length $c=100$ mm. The distance from the trailing edge of the aerofoil to the propeller disc plane ($x=0$ plane) is $0.25c$, which ensures well-built interactions between the wake turbulence and the propeller. Two distinct revolving speeds $N$ of the propeller are investigated, i.e. 3000 (with medium thrust) and 4500 (with high thrust) revolutions per minute (rpm), and the freestream speed $U_0$ is 5 m s$^{-1}$, corresponding to advance ratios $J=U_0/ND$ of 0.83 and 0.56, respectively. The Reynolds number based on the freestream velocity and aerofoil chord is $Re_c=U_0 c/\nu =3.4\times 10^4$, and the Reynolds number based on the propeller diameter is $Re_D=U_0 D/\nu =8.1\times 10^4$, where $\nu$ is the kinematic viscosity of the fluid.

The computational domains consist of an outer static domain and an inner rotating domain. Figures 2(a) and 2(b) show the sizes of the domains and the employed boundary conditions. The outer static domain is a rectangular box of length $7.5D$ in the streamwise direction and length $5D$ in the two lateral directions. The inner rotating domain is a cylinder with diameter $1.5D$ and height $0.15D$. The velocity inlet boundary condition is imposed $2.6D$ upstream of the propeller, and the pressure outlet boundary condition is employed $5D$ downstream of the propeller. The other four side boundaries are all $2.5D$ away from the propeller's centre, where the symmetry boundary conditions are applied in the spanwise direction of the aerofoil, and the far-field boundary conditions are applied to the other two lateral boundaries. The no-slip boundary conditions are prescribed on the surfaces of the propeller and the aerofoil.

The computational mesh is fully structured for both the static domain and the rotating domain, and the mesh interface between the two domains is shown in figure 2(c). For the 3000 rpm case, there are 35 million cells in the rotating domain, and 61 million cells in total. For the 4500 rpm case, there are 39 million cells in the rotating domain, and 66 million cells in total. The wall-adjacent cells are refined to ensure $y^+<1$ for the first layer and maintain ${\rm \Delta} y^+ \sim 1$ for $y^+<25$ for each case to resolve the boundary layer accurately.

Wall-resolved large eddy simulations are performed to simulate the propeller wake-ingestion turbulence flow. As the Mach number is less than 0.3 in this work, the spatially filtered incompressible Navier–Stokes equations are solved:

(3.1)\begin{equation} \left.\begin{gathered} \frac{\partial \bar{v}_i}{\partial x_i} = 0, \\ \frac{\partial \bar{v}_i}{\partial t} + \frac{\partial \bar{v}_i \bar{v}_j}{\partial x_j} ={-} \frac{1}{\rho}\,\frac{\partial \bar{p}}{\partial x_i} + \nu\, \frac{\partial^2 \bar{v}_i}{\partial x_j\,\partial x_j} - \frac{\partial \tau_{ij}}{\partial x_j}, \end{gathered}\right\} \end{equation}

where an overbar $\overline {({\cdot })}$ denotes the spatial filter, and $\tau _{ij}=\overline {v_i v_j} -\bar {v}_i \bar {v}_j$ is the subgrid-scale stress tensor. The needed subgrid-scale viscosity $\nu _{{sgs}}$ is modelled by the Smagorinsky–Lilly model (Smagorinsky Reference Smagorinsky1963)

(3.2)\begin{equation} \nu_{{sgs}} = (C_s \varDelta)^2\,|\bar{S}_{ij}|, \end{equation}

where $C_s=0.1$ is the Smagorinsky constant suggested by van Balen, Uijttewaal & Blanckaert (Reference Van Balen, Uijttewaal and Blanckaert2009) and Yao et al. (Reference Yao, Cao, Wu, Yu and Huang2020), $\varDelta$ is the filter width, and $\bar {S}_{ij}$ is the filtered strain rate tensor. A standard van Driest damping function (van Driest Reference Van Driest1956) is applied to prescribe $\nu _{{sgs}} \rightarrow 0$ near the solid walls.

Figure 2(d) depicts the two multi-armed spiral arrays for the later use of numerical beamforming. Array I is on the side of the propeller, and array II locates downstream of the propeller. Each of the artificial microphone arrays has a distance 0.55 m ($2.3D$) from the array's centre to the propeller's centre, and consists of 99 virtual microphones to acquire the acoustic pressure.

The acquisition of flow field and acoustic data is performed for 10 revolutions of the propeller after the first 20 revolutions to ensure fully developing of the turbulent flow. The time step corresponds to a $0.09^\circ$ rotation of the propeller for the 3000 rpm case, and $0.135^\circ$ for the 4500 rpm case. At both the microphone arrays and far-field receivers, the radiated noise is computed by solving the aforementioned FW–H equation in (2.8)–(2.10).

3.2. Validation

To ensure the quality of numerical results, a mesh convergence study has been conducted to check the validity of the current computational mesh by comparing with coarse and fine meshes. The coarse mesh has 36 million cells in total, and 21 million in the rotating domain, and the fine mesh has 108 million cells in total, and 59 million cells in the rotating domain.

Figure 3(a) compares the phase-averaged axial velocity on Line 1, which is shown in figure 2(b), across different grid refinements. Unless specifically mentioned, the phase average in this paper is performed by averaging over the values at fixed revolving phases. The medium and fine meshes give similar distributions of the axial velocity, while the result of the coarse mesh shows velocity mismatch near $z=0$. Figure 3(b) compares the phase-averaged axial force of the propeller. The result of the medium mesh is close to that of the fine mesh, and the coarse mesh underestimates the force by 10 %. Given that the differences in the results between the medium grid refinement of 61 million and the fine grid refinement of 108 million are less than 2 % for the majority of the phase angles, we chose the medium grid refinement in the subsequent numerical studies. Moreover, the simulations are validated by comparing the numerical results with the measured data from previous PIV experiments conducted by Chen et al. (Reference Chen, Peng, Liem and Huang2020a). Figure 4 compares the axial velocity distribution on Line 1. Both the results are phase-averaged at three revolving phases, $-9^\circ$, $0^\circ$ and $9^\circ$, which correspond to the propeller's three representative stages of entrance into, cutting through and departure from the wake region. The results in figure 4 show good agreement between the experiments and the present numerical results. The velocity profiles are well captured, and the amplitude discrepancy is within 10 % for most of the points on the lines.

Figure 3. (a) The mean axial velocity on Line 1, and (b) the axial force of the propeller, predicted with different grid refinements. The revolving speed is 3000 rpm.

Figure 4. Measured and simulated mean velocity on Line 1 (see figure 2b) at the revolving phases (a) $\varphi =-9^\circ$, (b) $\varphi =0^\circ$, and (c) $\varphi =9^\circ$. The revolving speed is at 4500 rpm. The measured flow field data are from Chen et al. (Reference Chen, Peng, Liem and Huang2020a).

Figure 5 compares the computed and measured far-field sound pressure level (SPL) values, both of which are averaged over 56 microphones to reduce the potential interference. The background noise of the wind tunnel is subtracted from the measured results by subtracting the corresponding acoustic energy at each frequency. Strong peaks are found at the BPF and its harmonics, which is a distinctive feature of this wake-ingestion set-up. The peaks at half-BPF and its harmonics in the measurements are caused by the asymmetry blade–rotor interaction and the imperfect blade geometry (Wu et al. Reference Wu, Jiang, Ma, Chen and Huang2022), which is not modelled in the current numerical simulations and therefore results in discrepancies at the half-BPF and its harmonics in figure 5.

Figure 5. Measured and simulated sound pressure levels at the revolving speeds (a) 3000 rpm and (b) 4500 rpm. The measured sound data are from Chen et al. (Reference Chen, Peng, Liem and Huang2020a).

The comparison with measured results shows good agreement around the BPF harmonics. The difference between simulated noise and the measurements at the first two harmonics of the BPF is less than 3 dB. At the next few BPF harmonics, the differences are within 5 dB. Due to the upstream turbulent flow in the wind tunnel, the measured broadband noise is louder than the simulation at low to mid frequencies. In summary, the current simulation set-ups and numerical schemes are validated by the measurements; the calculated flow field results will be used below for the near-field noise source analysis.

3.3. Flow characteristics

To provide a more comprehensive perspective on the wake-ingestion problem, the flow characteristics are first introduced. Figure 6(a) shows the instantaneous flow structures at 3000 rpm by plotting isosurfaces of the $Q$-criterion (Haller Reference Haller2005) that are coloured by the normalized vorticity. Four kinds of structures are denoted: ${\unicode{x2460}}$ denotes the tip vortices that are generated by the pressure difference between the pressure side and suction side of the propeller blades; ${\unicode{x2461}}$ denotes the trailing edge shedding vortices of the propeller blades; ${\unicode{x2462}}$ denotes the root vortices; and ${\unicode{x2463}}$ denotes the shedding vortex from the upstream aerofoil.

Figure 6. (a) Flow structures at 3000 rpm shown by isosurfaces of $Q(R/U_0)^2=10$ and coloured by $\omega _z R/U_0$. Here, ${\unicode{x2460}}$ indicates the blade tip vortex, ${\unicode{x2461}}$ the blade trailing edge vortex, ${\unicode{x2462}}$ the rotor root vortex, and ${\unicode{x2463}}$ the aerofoil shedding vortex. (b) Contours of instantaneous radial vorticity $\omega _r R/U_0$ at 3000 rpm on cylindrical surfaces $r/R=0.25$ (inner) and $r/R=0.75$ (outer) at phase angle $\varphi =0^\circ$.

Figure 6(b) shows the instantaneous radial vorticity of the flow field at 3000 rpm on the $r/R=0.25$ and $r/R=0.75$ cylindrical surfaces at $\varphi =0^\circ$, where $r$ denotes the local radius to the revolving axis. In addition to the vorticity of the aerofoil wake, the trailing edge vortex shedding of the blade is shown clearly on the surface at $r/R=0.75$. However, due to the smaller radius and lower relative velocity, the trailing edge vortex shedding is not observed on the surface at $r/R=0.25$.

The phase-averaged axial velocities at the blade leading edge plane ($x/R=-0.05$), rotor midplane ($x/R=0$) and blade trailing edge plane ($x/R=0.05$) are given in figure 7. It is clear that the aerofoil wake resulted in the low-velocity band in the middle of the plane, while the rotation of the blade accelerates the flow and leaves the high axial velocity region behind the blade. At the rotor mid-plane and blade trailing edge plane, a low-velocity band above the blade tip can be seen, which is the result of the blade tip vortex. At the blade trailing edge plane, the centre low-velocity region is due to the blade hub, and the thin low-velocity band lagging behind the blade is due to the trailing edge vortex shedding.

Figure 7. Phase-averaged axial velocity on different cross-sections: (a) $x/R=-0.05$, (b) $x/R=0$, and (c) $x/R=0.05$. The propeller rotates in the clockwise direction.

The wake–rotor interaction process can be demonstrated by the correlation of pressure fluctuations, which is analysed on the suction surface for three sample lines from leading edge to trailing edge at 10 %, 50 % and 90 % local chord, as shown in figure 8(a). The space–time correlation coefficients of pressure fluctuations $C_{p^\prime p^\prime }(0.6R,{\rm \Delta} r,{\rm \Delta} t)$ for the base points at $r/R=0.6$ to other points on the same sample lines are plotted in figures 8(b)–8(d), and the correlation coefficient is defined as (Wang et al. Reference Wang, Wang and Wang2021)

(3.3)\begin{equation} C_{p^\prime p^\prime}(r,{\rm \Delta} r,{\rm \Delta} t) = \frac{\langle p^\prime(r,t)\,p^\prime(r,r+{\rm \Delta} r,t+{\rm \Delta} t) \rangle}{\sqrt{\langle p^\prime(r,t) \rangle^2}\, \sqrt{\langle p^\prime(r,r+{\rm \Delta} r,t+{\rm \Delta} t) \rangle^2}}, \end{equation}

where $\langle {\cdot } \rangle$ denotes time averaging.

Figure 8. (a) Sample lines on the suction side of the blade surface. Solid line is 10 % local chord; dashed line is 50 % chord; dash-dotted line is 90 % chord. The solid circles denote the $r/R=0.6$ points on each line. (b–d) Space–time correlation coefficients of the pressure fluctuations $C_{p^\prime p^\prime }(0.6R,{\rm \Delta} r,{\rm \Delta} t)$ on (b) 10 %, (c) 50 % and (d) 90 % chord sample lines.

It is observed that the correlation coefficients on all three sample lines show a periodic pattern, and the high values align with the aerofoil wake-cutting timing. The bias of the high correlation coefficient to the left near ${\rm \Delta} r/R=-0.3$ is due to the fact that the blade at $r/R=0.3$ enters the wake of the aerofoil earlier than the rest of the blade due to the bumped geometry. Figures 8(b)–8(d) also show that the correlation is stronger for the lines closer to the leading edge, which indicates that the wake-cutting effect and the generated fluctuations decay from leading edge (LE) to trailing edge (TE).

Figures 9(a) and 9(b) show the blade-to-blade space–time correlation coefficient $C_{p^\prime p^\prime }(0.6R,{\rm \Delta} r,{\rm \Delta} t)$, where the base point and the correlated points are both at the 10 % chord sample line but on two different blades. Due to the symmetrical set-up of the wake-ingesting configuration, the blade-to-blade correlations are very similar to the correlations of the same blade. Moreover, the correlation coefficient of the 4500 rpm case is higher than that of the 3000 rpm case, which was also observed in the work of Wang et al. (Reference Wang, Wang and Wang2021).

Figure 9. (a,b) Blade-to-blade correlation coefficient of the pressure fluctuations $C_{p^\prime p^\prime }(0.6R,{\rm \Delta} r,{\rm \Delta} t)$ between two different blades for (a) 3000 rpm and (b) 4500 rpm on the 10 % chord sample line. (c,d) Space–time correlation coefficient of the time derivative of the pressure $C_{\dot {p} \dot {p}}(0.6R,{\rm \Delta} r,{\rm \Delta} t)$ for (c) 3000 rpm and (d) 4500 rpm on the 10 % chord sample line.

Figures 9(c) and 9(d) plot the correlation coefficient of the time derivative of the pressure $C_{\dot {p} \dot {p}}(0.6R,{\rm \Delta} r,{\rm \Delta} t)$ defined in the same way as in (3.3) for 3000 and 4500 rpm cases at the 10 % chord sample line. Similar to the correlation coefficient of the pressure fluctuations, the $C_{\dot {p} \dot {p}}(0.6R,{\rm \Delta} r,{\rm \Delta} t)$ level changes periodically with the rotation, and the high value corresponds to the wake-cutting process. The correlation of the time derivative of the pressure is weaker than that of pressure fluctuations because the time derivative puts more weight on high-frequency components, while the correlated sources at leading edge are mainly at lower frequencies.

5.1. Contribution of noise sources

Figure 12(a) shows the directivity of overall sound pressure level (OASPL) at far-field receivers $400D$ away from the propeller centre on the $y=0$ plane, which are far enough away to neglect the near-field effects. The directivity angle $0^\circ$ denotes the downstream direction with the receiver at $(400D, 0,0)$, and the directivity angle increases anticlockwise. The OASPL of the noise at the receiver is calculated by

(5.1)\begin{equation} \text{OASPL} = 10 \log_{10}\int \frac{\phi_{pp}(f)}{p^2_{{ref}}}\,{\rm d} f, \end{equation}

where $\phi _{pp}(f)$ is the power spectral density, and the integration is performed in the frequency range from $25\,\textrm {Hz}$ to $8000\,\textrm {Hz}$. The directivity results show that the radiated noise is strongest in the upstream and downstream directions, and is relatively weak on the propeller disc plane, which is also observed in other propeller studies with the ingestion of an upstream turbulent wake (Wang et al. Reference Wang, Wang and Wang2021). Figure 12(a) shows that dipole-type directivity patterns at 3000 and 4500 rpm are almost the same, and the OASPL increases with the revolving speed.

Figure 12. Directivity of the noise at far-field receivers on the $y=0$ plane. (a) OASPL of the total noise. (b) Contribution of each noise source component at the BPF with 4500 rpm. (c) OASPL of the noise source components at 3000 rpm. (d) OASPL of the noise source components at 4500 rpm.

Figures 12(b)–12(d) compare the contribution of each noise source term to the far-field noise, where the thickness source term $\boldsymbol {S}_{T}$ and the two loading source terms $\boldsymbol {S}_{L1}$ and $\boldsymbol {S}_{L2}$ are defined in (2.15). The directivity of noise contributions from the three components are evaluated at far-field receivers that are the same as those used in figure 12(a). Figure 12(b) shows the contributions of each noise component at BPF for the 4500 rpm case. At the BPF, the thickness noise from $\boldsymbol {S}_{T}$ is dominant in the $90^\circ$ (above the propeller) and $270^\circ$ (under the propeller) directions, and has a shape of ‘$\infty$’. The noise from $\boldsymbol {S}_{L1}$ is loudest in the $0^\circ$ (upstream) and $180^\circ$ (downstream) directions, and is dominant in most directions. The noise from $\boldsymbol {S}_{L2}$ reaches its peak around $60^\circ$ and $300^\circ$, and decreases greatly in other directions. By comparison, $\boldsymbol {S}_{L2}$ is much weaker than the other two source terms. The results agree well with the theoretical discussions in § 2.2. The only exception is that the noise from $\boldsymbol {S}_{L2}$ is more than expected in the directions from $60^\circ$ to $90^\circ$ and from $270^\circ$ to $300^\circ$, where it is comparable or even louder than the noise from $\boldsymbol {S}_{L1}$. The reason for this is that although stronger sound sources could produce a louder noise, they can also possibly cancel each other in certain directions, making the resulting noise less loud than expected at those angles.

Figures 12(c) and 12(d) demonstrate the OASPL of each noise component for the 3000 and 4500 rpm cases, respectively. The OASPL results include higher-frequency noise, and $\boldsymbol {S}_{L1}$ generates larger overall noise than the other two source terms. At 3000 rpm, the noise contributions from $\boldsymbol {S}_{T}$ and $\boldsymbol {S}_{L2}$ are too small to be shown. As the thickness noise is related directly to the revolving speed, $\boldsymbol {S}_{T}$ at 4500 rpm is around 15 dB larger than at 3000 rpm. And the noise from $\boldsymbol {S}_{L2}$ at 4500 rpm is around 18 dB larger than at 3000 rpm.

Figure 13 demonstrates a snapshot at phase $\varphi =0^\circ$ for the contributions of each surface computational element on the blade suction side to the noise at downstream receiver $(400D,0,0)$ and above the propeller receiver $(0, 400D,0)$. For each surface element, the contributions are calculated by the quantities inside the integrals in (2.12) and (2.13), which are labelled as thickness, loading 1 and loading 2, corresponding to the aeroacoustic sources $\boldsymbol {S}_T$, $\boldsymbol {S}_{L1}$ and $\boldsymbol {S}_{L2}$, respectively. The contributions are divided by the element area $\mathrm {d}S$ and evaluated at the receiver time, so that the noise signal of the receiver can be calculated by summing the product of the contribution and area of each surface element. Negative and positive contributions are represented by the blue and red colours, respectively. The coexistence of negative and positive contributions would lead to the aforementioned cancellation at the far-field receivers in figure 12.

Figure 13. A snapshot of the contribution to far-field noise at two receivers by the surface elements on the suction side of the propeller blades. From top to bottom, rows are the elementary contributions from $\boldsymbol {S}_T$, $\boldsymbol {S}_{L1}$, $\boldsymbol {S}_{L2}$ and the total acoustic pressure, respectively. The left side is for the downstream receiver at $(400D,0,0)$, and the right side is for the receiver above the propeller at $(0,400D,0)$.

It is clear that the contributions from the source terms to different receivers are different. For the downstream receiver, the contributions from the thickness noise and $\boldsymbol {S}_{L2}$ are nearly invisible, and the total contribution is almost the same as that from $\boldsymbol {S}_{L1}$. This means that the noise in the downstream direction is dominated by the $\boldsymbol {S}_{L1}$ source, which agrees with the directivity patterns in figure 12. For the receivers above the propeller, the contribution from thickness noise $\boldsymbol {S}_{T}$ and loading noise $\boldsymbol {S}_{L2}$ becomes more distinctive, although the total contribution still remains largely dependent on $\boldsymbol {S}_{L1}$. The results of surface element contributions support the observation that $\boldsymbol {S}_{L1}$ is the major acoustic source for the current physical problem studied, and justify the choice of $\boldsymbol {S}_{L1}$ as the focus of discussion in most of the following subsections.

5.2. On-surface source distributions

Figure 14 shows the surface distribution of the r.m.s. value of the normalized loading noise sources, i.e. $(S_{L1})_{{rms}}/(\rho U_0^2 N)$ and $(S_{L2})_{{rms}}/(\rho U_0^2)$. The time average is implemented for 10 revolutions, and the results show the average of the sources at all phases. The abbreviations ‘LE’ and ‘TE’ in figure 14 stand for leading edge and trailing edge, respectively. For the distribution of $(S_{L1})_{{rms}}$, strong sources are found in three regions that are classified and noted by the markers in figure 14, where: ${\unicode{x2460}}$ denotes the sources at the leading edge that appear on both sides of the blade, which is due to the interaction between the propeller blades and the ingesting wake; ${\unicode{x2461}}$ denotes the sources at the trailing edge of the blade, which is also observed on both sides of the blades; and ${\unicode{x2462}}$ denotes the sources that are found only in the middle section of the pressure side of the blade. The other surface area covered in blue colour indicates weaker sources.

Figure 14. The amplitude of the loading noise sources (a,b) $(S_{L1})_{{rms}}/(\rho U_0^2 N)$ and (c,d) $(S_{L2})_{{rms}}/(\rho U_0^2)$ on the propeller surface. The revolving speed is (a,c) 3000 rpm and (b,d) 4500 rpm. The abbreviations ‘LE’ and ‘TE’ stand for leading edge and trailing edge, respectively.

The distributions are generally similar for both revolving speeds. At the higher rpm speed, the leading edge and trailing edge sources are stronger, manifesting a positive relation between on-surface aeroacoustic sources and the far-field noise emission. However, the pressure side mid-span sources are stronger at 3000 rpm. This is because the relative angle of attack (AoA) is larger at lower revolving speeds leading to stronger separation when the incoming mean flow is fixed at $U_0$.

In figures 14(c) and 14(d), the secondary loading source $(S_{L2})_{{rms}}$ presents distributions similar to those of $(S_{L1})_{{rms}}$. A distinctive distribution of sources can also be found at the leading edge, trailing edge and mid-span of the blade pressure side. It should be noted that the value of $S_{L2}$ is lower compared to $S_{L1}$ and plotted with different scales. At the trailing edge, the source of $(S_{L2})_{{rms}}$ is absent at 3000 rpm on the suction side.

In § 2, it has been discussed that the loading sources are affected by the background flow. This is particularly true for the current turbulent ingestion propeller problem, which contains the direct interaction between the upstream shedding vortex and the downstream propeller blades. Pressure fluctuations are produced near the leading edge as the blade cuts through the shedding vortex of the aerofoil generating the leading edge sources. Figure 15 demonstrates the interaction procedure at several different revolving phases on the cross-section plane at the radial height $0.5R$, where $R$ is the radius of the blade. The normalized vorticity $\omega _y R/U_0$ perpendicular to the plane is plotted to reflect the flow interaction procedure. At $\varphi =-9^\circ$, the propeller blade is about to enter the wake region, and the leading edge of the blade has just cut into the shedding vortex of the aerofoil. At $\varphi =0^\circ$, the blade cuts and separates the shedding vortex into two parts, resulting in the discontinuity of the wake of the aerofoil. At $\varphi =9^\circ$, the blade is leaving the wake region, and at the downstream to the blade, the shedding vortex of the aerofoil is slightly brought upwards due to the shear force from the blade. The shedding vortex of the blade itself is also left behind and would further interact with the wake of the upstream aerofoil.

Figure 15. Evolution of the vortices on the cross-section plane at the radial height $0.5R$ at phases (a) $\varphi =-9^\circ$, (b) $\varphi =0^\circ$, and (c) $\varphi =9^\circ$. The vorticity is normalized to $\omega _y R/U_0$. The current revolving speed is 3000 rpm.

Apart from the interaction, the rotating motion has also led to flow mechanisms that generate noise sources. Figure 16 exhibits snapshots of the static pressure coefficients $C_p$ at phase angle $\varphi =0$ on four section planes with radial heights $0.25R$, $0.5R$, $0.75R$ and $R$. Near the blade leading edge, the stagnation points have moved slightly rearward on the suction side because of the negative relative AoA. More specifically, the relative AoA is defined by $\mathrm {AoA}_{{rel}}=\theta _r-\mathrm {atan}(U_0/2{\rm \pi} N r_s)$, where $\theta _r$ is the pitch angle between the chord line and the $z$-axis, and $r_s$ is the radial height of the section plane. The corresponding $\mathrm {AoA}_{{rel}}$ values at $0.25R$, $0.5R$ and $0.75R$ are $-8^\circ$, $-2^\circ$ and $-1^\circ$, respectively.

Figure 16. Pressure coefficients on several section planes at phase $\varphi =0^\circ$. The section plane has a distance to the revolving axis (a) $0.25R$, (b) $0.5R$, (c) $0.75R$, and (d) $R$. The current revolving speed is 3000 rpm.

On the section plane of $r_s=0.25R$, the negative $\mathrm {AoA}_{{rel}}$ leads to the adverse pressure gradient close to the pressure side of the blade. At increased distances to the rotation axis $0.5R$ and $0.75R$, $\mathrm {AoA}_{{rel}}$ increases and the flow on the pressure side shows less adverse pressure gradient, and the suction side would become more normal in terms of the pressure gradient. At $r_s=R$, which is over the blade tip, the pressure difference between the pressure and suction sides would lead to the generation of a tip vortex.

Figure 17 shows the phase-averaged limiting streamlines on the propeller blade over the distribution of phase-averaged static pressure coefficients. The phase average is implemented by averaging over the results at the same revolving phase $\varphi$ for more than 10 revolutions to ensure statistical confidence. Only the result at 3000 rpm is presented for brevity, since the higher revolving speed case has a similar pressure distribution and streamlines. The pressure difference between the suction and pressure sides is larger at the outer half of the blade, suggesting that the lift is mostly generated therein. The negative pressure gradient at the leading edge and trailing edge on the pressure side is observed.

Figure 17. Phase-averaged limiting streamlines and pressure coefficients on the propeller blade at phase $\varphi =0^\circ$. (a,b) Phase-averaged static pressure distributions overlapped with limiting streamlines. (c,d) Close views of limiting streamlines on the pressure side of the blade. The current revolving speed is 3000 rpm.

The limiting streamline is obtained from the wall shear stress that is computed from the relative velocity. Generally, the flow on the suction side of the blade is smooth, except for the gathering of streamlines near the trailing edge that drives the fluid towards the blade tip. The reversed streamlines from the trailing edge to the gathering line indicate flow separation near the trailing edge, which is a potential cause of noise sources. On the pressure side, however, the streamlines reflect complicated flow patterns in the boundary layer. The aggregated streamlines near the leading edge reveal the movement from the root half of the blade to different destinations, including the leading edge, the trailing edge of the tip half, and other places of the blade. Moreover, on the pressure side, the streamlines from the root to the middle part indicate the existence of complicated patterns that are further illustrated in the enlarged views in figures 17(c) and 17(d), where the nodal and saddle points can be identified by following the definitions from Tobak & Peake (Reference Tobak and Peake1982). Flow separations and interactions occur at these spots, which lead to the aeroacoustic sources.

5.3. Sources at different phases

Due to the unsteady flow interaction between the spatially non-uniform wake and the propeller, the distribution of on-surface aeroacoustic sources varies along with the revolving phases. Figure 18 presents an overview of the relationship between on-surface sources and the flow phenomenon for a single blade at various revolving phases. The shedding vortices separated from the aerofoil and the propeller blade on two section planes at $0.3R$ and $0.6R$ radial heights are shown by rainbow-scale colours. Only the 4500 rpm case is presented for brevity; the results at 3000 rpm are similar in distribution.

Figure 18. Variations of on-surface aeroacoustic sources as a single blade of the propeller rotates and interacts with the ingesting wake flow. The red–blue colour on the surface of the blade denotes the amplitude of the loading noise source $\boldsymbol {S_{L1}}$. The rainbow scale in the field denotes the normalized vorticity $\omega _y R/U_0$. The surface sources are phase-averaged, superimposed for the single blade at 12 representative rotating angles from $0^\circ$ to $360^\circ$, while the vorticity plot is a snapshot when $\varphi =0^\circ$. The revolving speed of the propeller is 4500 rpm.

At phases $0^\circ$ and $180^\circ$, where the blade is cutting through the wake of the aerofoil, figure 18 shows that strong aeroacoustic sources are generated at the leading edge of the blade. At other revolving phases, the leading edge noise sources are less prominent, which reflects the direct relationship between the leading edge sources and the interaction between the ingesting wake and the blade. On the contrary, noise sources at the trailing edge persist at a certain strength at each revolving phase, suggesting that they are less correlated to the wake-ingestion.

To have a more detailed view of the interaction process, figure 19 shows the aeroacoustic sources on the blade suction side over various revolving phases by smaller revolving steps. During the interaction procedure, the leading edge sources grow stronger from $-15^\circ$ to $0^\circ$; at $-15^\circ$, the leading edge sources barely exist at the root half blade, which has already entered into the wake region, whereas the tip half blade is still outside the wake region; when the blade moves further into the wake region, the aeroacoustic sources grow stronger and finally reach the peak at $0^\circ$. Then the sources become gradually weaker; at $5^\circ$, the leading edge source is still evident, while at $10^\circ$ and later revolving phases, the aeroacoustic sources are almost absent at the leading edge. In contrast, the trailing edge aeroacoustic sources are less affected by the interaction, and their distribution over the blade remains nearly the same over the whole interaction procedure.

Figure 19. Amplitude of the aeroacoustic source $S_{L1}$ on the blade suction side at various revolving phases from $-15^\circ$ to $15^\circ$. The revolving speed is 4500 rpm.

The aeroacoustic sources associated with the wake impingement are revealed in figure 19. In particular, when the revolving phase is between $5^\circ$ and $10^\circ$, a vertical strip that represents strong sources appears at the middle of the blade surface, which is the result of the wake impingement. Additionally, the proposed approach helps to show the location of the impingement that moves from the leading edge to the trailing edge as the blade rotates through the wake region.

Figure 20 shows the on-surface aeroacoustic sources on the pressure side of the blade. The variation of the leading and trailing edge sources along with the revolving phase is similar to the results in figure 19. The mid-span sources, which are observed only on the pressure side, preserve a similar distribution at different phases. This is consistent with the observation that the mid-span sources are created mainly by the complicated local flows rather than the wake interaction process as discussed in § 5.2. Moreover, since the direct wake impingement occurs only on the upstream-facing suction side, the effect of impingement is not observed on the pressure side.

Figure 20. Amplitude of the aeroacoustic source $S_{L1}$ on the blade pressure side. Other set-ups are the same as those in figure 19.

5.4. Sources at different frequencies

The frequency properties of the aeroacoustic sources are discussed in this subsection. Figure 21 compares the amplitude of the aeroacoustic source at several representative points on the blade surface to illustrate the frequency-domain properties. As shown in figure 21(a), the points are on the section plane at radius $0.6R$, of which P1–P4 are on the suction side, and P5–P8 are on the pressure side. Figure 21(b) illustrates further the locations of the points on the blade section; in particular, P1 and P8 are at the leading edge, and P4 and P5 are at the trailing edge.

Figure 21. Spectra of $S_{L1}$ at various surface points. (a,b) Locations of the points of interest. (c,d) Spectra at 3000 rpm. (e,f) Spectra at 4500 rpm.

Figures 21(c)–21(f) show that the distinctive peaks of the spectral results of $S_{L1}(f)$ are at the BPF and its harmonics. At the leading edge points (P1 and P8) those peaks have the highest values, and become gradually lower at points further downstream, which indicates the weakening of the wake interaction effects moving from leading edge to trailing edge on the propeller surfaces. Moreover, at the trailing edge points (P4 and P5), steep humps appear at high frequencies, which indicates the rise of a high-frequency noise source.

The spectra present certain differences in the shape and amplitude at the two revolving speeds. At 4500 rpm, the peak values at the BPF and its harmonics are higher, which manifests a stronger interaction at higher speeds. While the broadband values do not show much difference, the hump appears at a higher frequency in the 4500 rpm case. The peak value of the hump is also much higher, which would result in a louder noise emitting from the trailing edge.

Figure 22 shows the distribution of surface sources at different frequencies. The frequency-domain source $S_{L1}(\boldsymbol {x},f)$ is calculated by directly taking a Fourier transform to the time-domain source amplitude $S_{L1}(\boldsymbol {x},t)$. Although the frequency of the source does not coincide exactly with the frequency of the far-field noise due to the Doppler effect, the frequency-domain results can still be used to isolate surface sources of different frequencies.

Figure 22. Distribution of normalized sources $S_{L1}$ at various frequencies. The frequencies BPF, 20 BPF and 40 BPF are investigated. The revolving speed is (a,c,e) 3000 rpm, and (b,d,f) 4500 rpm.

Three representative frequencies are compared, i.e. BPF, 20 BPF and 40 BPF in each case. Unlike the beamforming results shown in figures 10 and 11, the on-surface aeroacoustic sources have a resolution as high as the computational mesh, and the distribution of sources at each frequency is shown clearly. At the BPF, strong sources appear at the leading edge on both the pressure and suction sides, and also in the middle of the blade's pressure side. At the mid frequency 20 BPF, the most significant sources are still at the leading edge, while the mid-span sources are weaker. At the higher frequency, significant sources appear only at the trailing edge with radius from $0.5R$ to $0.8R$, indicating a shift from the leading edge noise to the trailing edge noise. The locations of the high-frequency source coincide with the high-frequency imaging results in figures 11(a) and 11(b) at both revolving speeds.

The source distributions are similar under the two revolving speeds. The mid-span sources on the pressure side at 3000 rpm cover a larger area due to the complex flows and separation caused by the large negative $\mathrm {AoA}_{{rel}}$. With regard to source strength, the leading edge noise sources at BPF and 20 BPF are much stronger at 4500 rpm, which agrees with the far-field noise and beamforming results.

5.5. Comparison of source analysis methods

In the above study, the propeller wake-ingesting noise has been analysed comprehensively using the near-field aeroacoustic source approach. A comparison with other source identification methods, such as surface pressure fluctuations visualization and numerical beamforming, would show their strengths and limitations.

As reviewed in § 1, the surface pressure fluctuation $p^\prime =p-p_0$ has been used in many previous works to represent the aeroacoustic sources, where $p_0$ is the time-averaged pressure. A Fourier transform of the time derivative of the pressure gives $\mathcal {F}(\dot {p}) = 2{\rm \pi} \textrm {i}f\,\mathcal {F}(p)$. As a result, in the proposed source model $\boldsymbol {S_{L1}}$, its high-frequency components are less likely to be overwhelmed by low-frequency components. Hence, as shown below, fine-scale structures usually remain in the near-field distribution of aeroacoustic sources when the proposed source model approach is adopted.

Figures 23(a) and 23(b) compare the surface distributions of the r.m.s. of the proposed source $S_{L1}$ and pressure fluctuations $p^\prime$. The distributions at the leading edge are similar, but the noise source at the trailing edge in the $p^\prime$ plot is not as evident as that in $S_{L1}$ plot. This is due to the fact that trailing edge noises are mostly high-frequency sources, as has been discussed in § 5 and figure 22. Also, the sources in the middle of the blade pressure side show differences.

Figure 23. Amplitude of (a) loading noise sources $(S_{L1})_{{rms}}$ and (b) pressure fluctuations $p^\prime _{{rms}}$ on the propeller surface. (c,d) Spectra at P1 and P4, respectively. The revolving speed is 4500 rpm.

Figures 23(c) and 23(d) compare the spectra of $S_{L1}$ and $p^\prime$ at two points on the suction side of the blade, i.e. P1 and P4 in figure 21. At the leading edge point (P1), the amplitudes of $S_{L1}$ and $p^\prime$ at the BPF and its first few harmonics are close, while the broadband component and harmonic components of $S_{L1}$ at higher frequencies show higher amplitudes than those of $p^\prime$. At the trailing edge point (P4), the high-frequency hump for $S_{L1}$ is augmented to even exceed low-frequency peaks, indicating the importance of trailing edge noise sources at this point. It is worthwhile to mention that when visualizing frequency-domain sources like that in figure 22, $p^\prime$ would produce the same distribution as that of $S_{L1}$ because the relative strengths of sources are the same.

We wish to mention that the results of the numerical beamforming in § 4 are consistent with that of the near-field aeroacoustic source analysis. First, at low to mid frequencies, source images are greater at $\varphi =0^\circ$ when the blades are cutting through the wake. Second, at high frequencies, the source images are more evident at the outer half of the blade tip, which is aligned with the trailing edge $\boldsymbol {S_{L1}}$ distribution. Finally, the sources are stronger at higher revolving speeds.

Compared with the proposed near-field aeroacoustic source approach, several drawbacks of the numerical beamforming are recognized in this work. The first is the aforementioned resolution limitation. Second, beamforming needs an a priori hypothesis of point sources, which are usually equivalent monopole (Sijtsma Reference Sijtsma2006) or dipole (Chen et al. Reference Chen, Jiang and He2022) sources, while the practical aeroacoustic sources could be of multiple types simultaneously. Third, when applied to numerical data, beamforming still makes use of the noise signals from finite microphone arrays, as was used originally in the experiments. The array signals are discrete, and the loss of information is unavoidable. Finally, the flow field solution acquired in the simulations can be used by the near-field aeroacoustic source method directly, while the numerical beamforming needs additional calculations of solving the FW–H equation to generate the noise signals at the sensor arrays. Such a feedforward procedure is straightforward but time-consuming, and time can be saved if the proposed approach is adopted.