# Three-dimensional microphone array for the reconstruction of compact dipole aeroacoustic sources with spatially varying orientation

Louise Chiocchetti, David Marx, Vincent Valeau, François Ollivier, Régis Marchiano

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## To cite this version:

Louise Chiocchetti, David Marx, Vincent Valeau, François Ollivier, Régis Marchiano. Three-dimensional microphone array for the reconstruction of compact dipole aeroacoustic sources with spatially varying orientation. Applied Acoustics, 2026, 246, pp.111243. 10.1016/j.apacoust.2026.111243. hal-05485483

HAL Id: hal-05485483

https://hal.science/hal-05485483v1

Submitted on 30 Jan 2026

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# Three-dimensional microphone array for the reconstruction of compact dipole aeroacoustic sources with spatially varying orientation

Louise Chiocchetti(1), David Marx(1), Vincent Valeau(1), François Ollivier(2), Régis Marchiano(2)

(1) Institut PPRIME, CNRS-Université de Poitiers-ENSMA, Poitiers, France

(2) Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, 4 place Jussieu, 75005 Paris, France

**Abstract** This work investigates, using a microphone array, the localization of aeroacoustic sources resulting from the interaction of a flow with rods. In similar investigations in the literature, arrays often have a planar geometry and the spatial region in which acoustic sources are searched is a plane parallel to that of the array. However, the sources are not always distributed in such a plane. Moreover, aeroacoustic sources resulting from flow-obstacle interaction are often dipoles, and for some complex geometries, the dipoles' orientation can vary in space, such as for curved obstacles or arrangements of rods with different orientations. In order to identify such dipoles, this work uses a three-dimensional array composed of four flat arrays, forming a tunnel of 1024 microphones around the open vein of an anechoic wind tunnel. Microphone signals are processed by an inverse beamforming technique to identify equivalent dipole sources producing the measured sound field at the array, using classical Tikhonov regularization. Taking advantage of the acoustic compactness of the cross-section of the rods located in the flow, the dipoles are sought along the axis of the rods, with a spacing of the order of the vortex shedding coherence length. The technique does not require any prior assumption on dipole orientation. Results from simulated or experimental data are presented to assess the effectiveness of the method, in the cases of a rectilinear rod and a bended rod forming a ring.

## 1. Introduction

The use of microphone arrays for aeroacoustic source identification in wind-tunnels has become widespread over the past two decades [1]. In many applications, the sound produced by the interaction of objects with turbulence is studied, and the spatial distribution and sound levels of the emitted sources are searched for. In such studies, the main acoustic emissions are due to the
pressure fluctuations produced at the surface of the obstacles in the flow. Such pressure fluctuations
produce acoustic dipoles located on the surface of the obstacles [2].

For identifying such sources of sound, classical imaging techniques are often efficient, performing
some mapping of the acoustic sources in an observation plane, based on a measurement from a
planar microphone array parallel to the latter. The data processing techniques are generally based
on the beamforming technique, associated with deconvolution techniques to improve the spatial
resolution, such as the CLEAN-SC [3] or the DAMAS technique [4]. However, using 2D arrays
associated with a planar focusing domain can be limited practically in aeroacoustic applications
when studying the sound produced by objects with three-dimensional (3D) geometries that are
not contained in a plane.

To increase the capacity of imaging methods to tackle 3D configurations, some volumetric array
processing techniques have been developed over the last years. In such studies, the potential sources
of sound are not located in a plane but in a volume. Such studies involve the use of an array made
of several planar arrays (generally perpendicular to one another), in order to obtain a satisfactory
resolution in several directions. Porteous et al. [5] developed several implementations in 3D of
the conventional beamforming technique, using two perpendicular arrays acquired simultaneously.
Another option is to combine asynchronous microphone array measurements to perform volumetric
source estimation [6], but the phase information between each measurement is lost in this case.
Zhou et al. [7] used a tunnel-shaped array made of three perpendicular planes of 256 microphones.
By applying the 3D conventional beamforming technique with a dipole source model, the authors
reported some successful identification of several sources of sound generated by a wall-mounted
airfoil. The application of a deconvolution technique, the 3D CLEAN-SC technique in this case, was
shown to be compulsory to make the acoustic maps usable. The limits of conventional beamforming
techniques come from the fact that the model is restricted to a given source model: monopoles, or
dipoles with a fixed orientation, the orientation being imposed by the physics of the problem (for
example, for an airfoil, the direction perpendicular to the chord).

Instead of using the conventional beamforming technique, Battista et al. [8] chose to model and
solve within an investigation volume an inverse problem, using a sparsity constraint and the IRLS
(Iteratively Re-weighted Least Squares) algorithm and two perpendicular arrays, but the study was
restricted to monopolar source models. The technique was shown to identify some aeroacoustic
sources on a scale model of an aircraft. To make the inverse problem more versatile, Suzuki [9]
developed a technique referred to as GIB (Generalized Inverse Beamforming), which solves the
inverse problem using a catalog of several source models (monopole and dipoles with arbitrary
orientation in a plane), potentially correlated. The method requires the eigenmode decomposition
of the cross-spectral matrix of the array data, and the resolution of the inverse problem for each
eigenmode. The GIB technique has proven its efficiency in two-dimensional applications [10, 11, 15]. Eventually, it is possible to go beyond the dipole approximation and include higher order radiation patterns in the GIB source dictionary, by using spherical harmonics for example [14]. Other methods (SODIX [12, 13]) have been developed to study the directive sound radiation from plane engines.

One of the main difficulties of volumetric identification of sound sources is the great number of sources to consider. A 3D grid needs to be defined with a sufficient resolution, linked to the explored wavelength, which can involve prohibitive computational cost. Gridless compressive beamforming methods have been proposed [16] to limit the number of sources, and recently, a gridless method [17] based on equivalent monopole point sources was employed to reconstruct the pressure level and directivity of the measured sound field.

The present paper adopts a different strategy in order to limit the number of potential sources to be identified in a 3D domain. Following the physical understanding of aeroacoustic sources produced by obstacles in flows, the potential sources are dipoles located on a grid meshing the surface of the obstacles, as Legg et al. [24]. The potential sources considered are dipoles with unknown orientations in a 3D cartesian reference frame, each dipole having its proper orientation. A 3D inverse problem is written following the GIB technique developed in two dimensions by Suzuki [9]. A major difference with the classical GIB method is that the number of dipole sources being by definition limited to the obstacle boundaries, no sparsity constraint is used for solving the inverse problem. The presented technique will be refered to as TGIB (Tikhonov Generalized Inverse Beamforming) in this paper.

The application experiments presented in this paper focus on the aeolian tones produced by cylinders with circular cross-sections in flows [18, 19]. These sources of sound manifest in various situations, including building facades, pantographs, boat shrouds, among others. More specifically, this study focuses on two setups: a straight cylinder, and a curved cylinder. The acoustic sources at the vortex shedding frequency are assumed to be dipole distributions positioned along the cylinder and oriented perpendicular to the latter [2]. Given the form of the structures, the induced sources will be dipoles with spatially varying orientations in three dimensions, and make an interesting benchmark case for the proposed identification method. The microphone array used for source identification is a tunnel made of four planar arrays (256 microphones each) enclosing the wind-tunnel test-section.

The structure of the paper is as follows: Section 2 presents the array processing methodology employed to reconstruct the acoustic dipole sources. Section 3 outlines the experimental setup (wind-tunnel, array and application cases). Section 4 presents simulations conducted with configurations reproducing those used in the experiments, and Section 5 details the source reconstruction
results obtained for the wind-tunnel experiments.

## 2. Inverse beamforming method

The objective is to estimate the sound source distribution due to rods placed in a flow using a three-dimensional array of microphones. The method employed here is inspired from Suzuki [9], who developed the GIB algorithm for investigating sources in a scanning plane. Each potential source is represented by a superposition of dipoles, oriented towards $x, y$ and $z$ axes. Monopole sources, which are originally included in the GIB, are not relevant in this physical application, and it was checked that the results are little affected by the addition of monopoles in the source catalog. The physics of the interaction of the flow with rods considered here allows to make some choices about the position and number of scanned sources. First, the section of the rods, specified later in Table 1, is acoustically compact at the shedding frequency. As a result, the scanned dipole source locations are searched along the axis of the considered rods. Second, the sources are related to vortex shedding, and this has a typical correlation length along the cylinder span, of the order of a few cylinder diameters [28, 29]. As a result, the scanned sources are typically set a few diameters apart. This tends to limit the number of scanned sources, compared to the situation where these would be searched in an entire cubic-like volume. It follows that, when solving the inverse problem, unlike in [9], we do not seek to increase the sparsity of the source by using the L1 norm for regularization, preferring a more direct L2 norm. However, to put things in perspective, we will present one example of applying the standard GIB method to our configuration. In the remaining of this section we present the details of the inverse problem.

### 2.1. Description of the method

The first step is to calculate the cross-spectral matrix (CSM) $C(\omega)$ between the microphone signals. We denote by $p(\omega)$ the vector of size $M \times 1$ having for components $p(\mathbf{r}_m, \omega)$, the Fourier transform of pressure at the $m^{th}$ microphone located at position $\mathbf{r}_m$, $m = 1...M$, where $M$ is the number of microphones. The cross spectral matrix, of size $M \times M$, is then $C = \overline{\mathbf{p}\mathbf{p}^\dagger}$ where $\dagger$ is the complex conjugate transpose, and $\cdot$ denotes the statistical average. Its components are:

$$C_{ml}(\omega) = \overline{p(\mathbf{r}_m, \omega)p^*(\mathbf{r}_l, \omega)}. \qquad (1)$$

where $*$ denotes the complex conjugate. The average is computed by using the Welch's method [30]: the microphone pressure signals are segmented into blocks of size $N_{fft}=1024$ with 50% overlap, and a Hanning window is used for each of the blocks. Therefore, $C(\omega)$ is computed for $N_{fft}$ different values of the angular frequency $\omega$, with a frequency resolution $\Delta\omega/(2\pi) = 48$ Hz. Below, the algorithm for source localization is computed one frequency at a time, and the frequency
dependence of **C** is not indicated. Since it is positive definite and Hermitian, **C** has the following eigendecomposition:

$$ \mathbf{C} = \mathbf{U} \mathbf{\Sigma} \mathbf{U}^{\dagger} = \sum_{i=1}^{M} \mathbf{u}_i \sigma_i \mathbf{u}_i^{\dagger}, \qquad (2) $$

where **U** = (u₁, ..., u_M) is a unitary matrix with orthonormal eigenvectors of **C** in its columns, and **Σ** is a diagonal matrix with the real and positive eigenvalues σᵢ on its diagonal. The uᵢ's are the POD (Proper Orthogonal Decomposition) modes of the CSM [31]. Equivalently, they are the SPOD (Spectral POD) modes of the pressure field measured at the antenna [32]. These are (uncorrelated) vectors of size M × 1 containing complex pressure fields maximally correlated with the different realizations of the pressure field at the microphones (at a given ω). The eigenvalue σᵢ measures the energy of the pressure field captured by the mode uᵢ. In the following, we will keep only the modes which contribute significantly to the energy to avoid irrelevant features of the pressure field, considered as noise. In addition, since we will solve one system for every mode (Eqs. (7a-7b) below), this avoids, in case the CSM eigenvalue spectrum has a long tail of small eigenvalues, solving a huge number of systems with minimal gain on the source amplitude. Below, we will consider the N_mode modes capturing 90% of the energy, that is, such that:

$$ \sum_{i=1}^{N_{\text{mode}}} \sigma_i > 0.9 \sum_{i=1}^{\infty} \sigma_i \qquad (3) $$

For the present configurations, N_mode will be at most 10. The 90% threshold is arbitrary and can be modified. We have verified that, in our configurations, the source amplitude is only slightly affected by this choice, compared to 99%.

The objective is now to find the acoustic source giving birth to the measured pressure field on the antenna. Following Suzuki [9], we search for a complex source vector sᵢ responsible for pᵢ = √σᵢuᵢ, where pᵢ is the pressure mode on the antenna incorporating its amplitude. This should satisfy, for every mode i = 1, ..., M :

$$ \mathbf{H}\mathbf{s}_i = \mathbf{p}_i, \qquad (4) $$

where sᵢ is a vector of size N × 1. The number of sources, N, included in this vector depends on the number of possible (or scanned) source locations and the number of source types at each location. The different types will be discussed below. The M × N matrix **H** represents the transfer function between the sources and the microphones, that is, its components Hmn are the Green's function between the n-th source, n = 1, ..., N, and the m-th microphone, m = 1, ..., M. In Eq. (4), the pᵢ's have been found by processing the microphone signals, **H** needs to be modelled, and the sᵢ's are the source vectors which need to be solved for.
The subsequent objective involves calculating the transfer matrix. Given our choice to consider only dipoles located on the bodies' centerline, **H** is modeled in this paper by free-field dipolar Green functions. Three dipoles per source point are required in three dimensions, oriented respectively along *x*, *y*, and *z*. The three corresponding transfer functions in the frequency domain are [33, 39]:

$$ 
\left\{
\begin{aligned}
H_{mn}^{dip,x}(\omega) &= \frac{jk}{4\pi|\mathbf{r}_{mn}|} \frac{\mathbf{r}_{mn} \cdot \mathbf{e}_x}{|\mathbf{r}_{mn}|} e^{-jk|\hat{\mathbf{r}}_{mn}|}, \\
H_{mn}^{dip,y}(\omega) &= \frac{jk}{4\pi|\mathbf{r}_{mn}|} \frac{\mathbf{r}_{mn} \cdot \mathbf{e}_y}{|\mathbf{r}_{mn}|} e^{-jk|\hat{\mathbf{r}}_{mn}|}, \\
H_{mn}^{dip,z}(\omega) &= \frac{jk}{4\pi|\mathbf{r}_{mn}|} \frac{\mathbf{r}_{mn} \cdot \mathbf{e}_z}{|\mathbf{r}_{mn}|} e^{-jk|\hat{\mathbf{r}}_{mn}|},
\end{aligned}
\right.
\qquad (5) $$

where $k = \omega/c_0$ is the wave number, $c_0$ the speed of sound, and $\mathbf{e}_x, \mathbf{e}_y, \mathbf{e}_z$ are the basis vectors associated with the Cartesian coordinates. $\mathbf{r}_{mn}$ is the vector from the $n^{th}$ scanning source point to the $m^{th}$ microphone, while $\hat{\mathbf{r}}_{mn}$ represents the same quantity but corrected by the Amiet method [23, 7]. This takes into account the geometry of the shear layer of the wind-tunnel jet, in order to calculate the time delay between the acoustic source and each microphone, modeling the effects of flow on sound propagation, ie., convection and refraction. A dipole which is not oriented towards one of the three directions *x*, *y* or *z* will be a linear combination of the predefined types of dipoles.

It remains to solve Eq. (4) to obtain the source vector $\mathbf{s}_i$. This vector has size $N = N_{type} \cdot N_{grid}$, where $N_{type} = 3$ is the number of possible types, here 3 dipoles, and $N_{grid}$ is the number of points in the source domain, that is, the number of source points distributed along the rod centerline.

The matrix **H** is rectangular in general, since the number of sources is never equal to the number of microphones, and is not directly invertible. In the overdetermined case, $M > N$, there is no solution but a least-square solution may be computed by using the left pseudo-inverse $\mathbf{H}_{\text{left}}^{-1} = (\mathbf{H}^{\dagger}\mathbf{H})^{-1}\mathbf{H}^{\dagger}$. In the underdetermined case, $M < N$, the solution is not unique and the minimum-norm solution may be computed using the right pseudo-inverse, $\mathbf{H}_{\text{right}}^{-1} = \mathbf{H}^{\dagger}(\mathbf{H}\mathbf{H}^{\dagger})^{-1}$. However, in both situations, the problem is ill-conditioned and the solution computed with the pseudo-inverse is dominated by errors, so that some regularization is needed [34]. In this work, we use the widespread Tikhonov regularisation method [35], which adds a constraint on the energy of the solution to the least square minimisation. The quantity to minimize is the cost function $J_2$:

$$ J_2(\mathbf{s}_i) \equiv \|\mathbf{H}\mathbf{s}_i - \mathbf{p}_i\|_2^2 + \lambda \|\mathbf{s}_i\|_2^2, \qquad (6) $$

where $\|\cdot\|_2$ is the L2 norm and $\lambda$ is a regularization parameter determined by finding the maximum curvature [36] of the L-curve [34]. An example of L-curve for one of the studied experimental configurations is provided in Appendix C, together with values of $\lambda$. The minimum of function
$J_2$ in Eq. (6) is then finally computed as:

$$ \text{Underdetermined: } \mathbf{s}_i \approx \mathbf{H}^{\dagger}(\mathbf{H}\mathbf{H}^{\dagger} + \lambda\mathbf{I})^{-1}\mathbf{p}_i, \qquad (7a) $$

$$ \text{Overdetermined: } \mathbf{s}_i \approx (\mathbf{H}^{\dagger}\mathbf{H} + \lambda\mathbf{I})^{-1}\mathbf{H}^{\dagger}\mathbf{p}_i. \qquad (7b) $$

For the configurations studied here, we will take about 60 scanned point sources for the inclined cylinder, and about 360 for the ring and its support bars. When all the 1024 microphones of the antenna are used, the system to solve is overdetermined.

## 2.2. Results layout

In this section we specify the quantities which will be used to characterize and plot the dipole sources, namely the dipole angles and amplitude.

The source vector $\mathbf{s}_i$ for the $i$-th mode computed from Eqs. (7a-7b) contains, for each scanning source point $n = 1, ..., N_{grid}$, three complex amplitudes denoted by $s_{i,n}^{dip,x}$, $s_{i,n}^{dip,y}$ and $s_{i,n}^{dip,z}$, which correspond to the dipoles along $x$, $y$, and $z$, respectively, and are associated with each of the Green's function in Eq. (5). If a dipole vector $\mathbf{D}_{i,n}$ is present at scanning point $n$ for mode $i$, then in ideal circumstances the found amplitudes would be its projections on the three axes:

$$ \left\{ \begin{aligned} s_{i,n}^{dip,x} &= \|\mathbf{D}_{i,n}\| e^{j\phi_{i,n}} \cos(\varphi_{i,n}), \\ s_{i,n}^{dip,y} &= \|\mathbf{D}_{i,n}\| e^{j\phi_{i,n}} \sin(\varphi_{i,n}) \cos(\theta_{i,n}), \\ s_{i,n}^{dip,z} &= \|\mathbf{D}_{i,n}\| e^{j\phi_{i,n}} \sin(\varphi_{i,n}) \sin(\theta_{i,n}). \end{aligned} \right. \qquad (8) $$

where $\|\mathbf{D}_{i,n}\|$ is the dipole modulus, $\theta_{i,n}$ and $\varphi_{i,n}$ its spherical angles, and $\phi_{i,n}$ is a phase angle shared by the three components. Spherical coordinates are indicated in Fig. 1, with azimuth $0 \le \theta \le 2\pi$ and elevation $0 \le \varphi \le \pi$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Schematic indicating the spherical coordinates of the problem. The blue vector shows the direction of the dipole, D. The origin O is at the location of the source.</figcaption></figure>

These quantities may be computed from $s_{i,n}^{dip,x}$, $s_{i,n}^{dip,y}$ and $s_{i,n}^{dip,z}$ by inverting Eq. (8). In practice, though, the three dipole components found by the inverse beamforming method are necessarily
approximate and do not represent the exact projection of a dipole vector on the three axes. As a result, the inversion of Eq. (8) typically returns slightly complex-valued modulus and angles. To avoid this problem, we obtain real modulus, angles and phase which fit the found amplitudes in the least square sense by minimizing the following objective function:

$$ \mathcal{F}(\|\mathbf{D}\|_{i,n}, \varphi_{i,n}, \theta_{i,n}, \phi_{i,n}) = \left|s_{i,n}^{\text{dip},x} - \|\mathbf{D}_{i,n}\| e^{j\phi_{i,n}} \cos(\varphi_{i,n})\right|^2 \\ + \left|s_{i,n}^{\text{dip},y} - \|\mathbf{D}_{i,n}\| e^{j\phi_{i,n}} \sin(\varphi_{i,n}) \cos(\theta_{i,n})\right|^2 \\ + \left|s_{i,n}^{\text{dip},z} - \|\mathbf{D}_{i,n}\| e^{j\phi_{i,n}} \sin(\varphi_{i,n}) \sin(\theta_{i,n})\right|^2 \quad (9) $$

In all the plots below, the dipole at scanning point n for a given mode i is symbolized by a segment with two arrows at its extremities, centered on the scanning point and oriented according to $\theta_{i,n}$ and $\varphi_{i,n}$. The size of the arrow is proportional to $\|\mathbf{D}\|_{i,n}$. The phase $\phi_{i,n}$ of the dipole is represented by a color scale.

To obtain a total source amplitude at scanning point n, one needs to sum over the modes. For a given realization $\xi$ of the sound and its associated source field, the total source is a weighted sum of the modes, given by:

$$ \mathbf{D}_n(\xi) = \sum_{i=1}^{N_{\text{mode}}} b_i(\xi) \mathbf{D}_{i,n} \quad (10) $$

where $b_i(\xi)$ are expansion coefficients. These are uncorrelated, in the sense that $\langle b_i(\xi)b_j(\xi) \rangle = \delta_{ij}$, where the average $\langle \cdot \cdot \rangle$ is performed over the realizations $\xi$. This follows from properties of the SPOD modes $\mathbf{p}_i$ (see Eq.(2.20) in [32]). Equation (10) implies that merely summing the complex modes ($\sum_{i=1}^{N_{\text{mode}}} \mathbf{s}_i$) is not meaningful. However, since the modes are uncorrelated it is allowed to compute the RMS value of the total dipole source at scanning point n by:

$$ \|\mathbf{D}_n\| = \sqrt{\sum_{i=1}^{N_{\text{mode}}} \|\mathbf{D}_{i,n}\|^2}, \quad (11) $$

The sum is computed over the $N_{\text{modes}}$ modes whose eigenvalue $\sigma_i$, cumulatively, account for 90% of the total energy of the sound field, as expressed in Eq. (3).

Let us stress here that $\|\mathbf{D}_n\|$ in Eq. (11) is a norm and does provide any orientation for the vector $\mathbf{D}_n$ from the the vectors $\mathbf{D}_{i,n}$. For this reason, in the applications considered, the direction will be assumed normal to the rod cross-sections and perpendicular to the flow. This is legitimate if each of the obtained modes $\mathbf{D}_{i,n}$ bears this orientation (see comment of Fig. 8(c)).

### 3. Experimental set up

The experiments were carried out in the anechoic wind tunnel BETI (Bruit Environnement Transport Ingérierie) of the PPRIME institute, shown in Fig. 2. The global set up is presented
in Fig. 3 and is similar to Zhou [7, 20], but with more microphones and a different collector. The coordinate system is indicated in the figure, its origin being located at the center of the square cross-section of the nozzle exit. The room volume is 90 m³ and its cutoff frequency is 200 Hz. The section of the square nozzle is (0.7 × 0.7)m², and the distance between the nozzle and the collector is 1220 mm. The collector is covered with foam to avoid reflections. The maximal flow speed is 60 m.s⁻¹.

<figure><img src="image_4.png" /><figcaption>Figure 2: Photography of the experimental set-up, in the wind tunnel BETI at Institut PPRIME. The perspective captured is from the collector, observing the nozzle. Positioned in front of the nozzle is a black-painted ring fixed to a square frame with rods.</figcaption></figure>

The microphone array is based on the MegaMicros system [21], developed by Sorbonne University. It is made of 4 planes of microphones, assembled to form around the flow a diverging conical tunnel with a growing square section so as to maintain the microphones out of the turbulent flow. Each planar array is made of 16 bars oriented approximately parallel to the flow, and each bar comprises 16 equi-spaced microphones, with a spacing of 72 mm. In total, one plane of 256 microphones is 1024 mm long in the streamwise direction *x*, and 1427 mm long in the spanwise direction (*y* or *z*), and has a trapezoidal shape (Fig. 3). The microphones are MEMS digital microphones, which are cheap and compact, allowing a big array to be built at a reasonable cost. The microphones have a variability of only 1.7 dB on 180° on the directivity, so they are considered omnidirectional. Moreover, the frequency responses of the microphones are consistent among the
<figure><img src="image_5.png" /><figcaption>Figure 3: Sketch of the experimental set-up, represented with a ring (black, configuration $C_2$) and a cylinder (red, configuration $C_1$) in the test section. The flow comes from the nozzle on the left of the picture. The colored points represent the microphones from the four arrays. The black dot on the red array plane shows the microphone used for plotting the power spectrum densities in this paper.</figcaption></figure>

different microphones: a common correction is applied to the measured signals through spectral equalization [25]. Each measurement lasts 15 s, and the sampling frequency is $F_e = 50$ kHz. In this work, all 1024 microphones in the array are used. One reason for this is that the signal-to-noise ratio (SNR) is proportional to the number of sensors, and a high SNR provides a larger dynamics of the solution, allowing to reveal low level sources. Also, another advantage is that densifying the microphone array increases the smallest wavelength of interest without diminishing the spatial resolution.

Different objects can be placed in the wind tunnel so that their aeroacoustic noise can be studied. A metallic square frame positioned at $x=0.5$ m serves to support these objects, it is indicated in Fig. 3. This frame spans a width of 1.15 m and is located out of the flow so that it does not generate aeroacoustic noise itself. The bars comprising this structure have a diameter of 4.5 cm. This structure is equipped with detachable fixing devices, which are customized to fit to various size and shapes of cylinders used in the study. Those fixing devices can be affixed either upstream or downstream of the square structure, facilitating the simultaneous placement of multiple cylinders with diverse configurations.

The objects considered in this work are cylinder-like. The nature of the flow around a cylinder
depends on the Reynolds number (Re) defined by

$$Re = U_{\infty}d/\nu, \qquad (12)$$

with $U_{\infty}$ the freestream flow speed, $d$ the diameter of the object, and $\nu$ the kinematic viscosity of air. For $Re$ larger than about 50, the flow becomes unsteady, which results in the periodic shedding of vortices alternately from each side of the cylinder. As the Reynolds number increases, this vortex street becomes more and more turbulent [22], but persists up to at least a Reynolds number of 200,000. Above a critical Reynolds of about 200,000 the boundary layer itself becomes turbulent and vortex shedding may be suppressed. The frequency of vortex shedding is characterized by the Strouhal number ($St$), defined as:

$$St = f_0d/U_{\infty}, \qquad (13)$$

where $f_0$ is the vortex shedding frequency. This number stays close to $St=0.2$ for a cylinder with a circular cross-section as long as $Re$ is larger than about 200. From an acoustic viewpoint, vortex shedding generates dipole tonal acoustic sources [18] distributed along the spanwise length of the cylinder, with frequency equal to the shedding frequency. These dipoles are oriented parallel to the lift force, namely orthogonally to both the flow direction and the local curvature of the structure [26, 27]. The resulting sound is known as aeolian tone. This paper focuses on two cylinder-like configurations, which are detailed in Tab. 1. The first one involves a rectilinear cylinder of circular cross section inclined at an angle $\beta$ relative to the y-axis. Although the dipoles are expected to align uniformly, this orientation does not coincide with any of the Cartesian coordinate axes or microphone array normals. The goal is to demonstrate how accurately the dipole orientation can be recovered. In the second scenario, a challenging ring configuration with circular cross-section is employed. The difficulty is that the orientation of the dipoles around the ring is now varying. Thus, this setup tests the algorithm's ability to estimate varying dipole orientations. The Reynolds number range specified in Tab. 1 is well within the subcritical regime ($Re < 200,000$): the boundary layer on the cylinder remains laminar, and the wake has a turbulent vortex street generating tones at $St \sim 0.2$.

## 4. Simulations

In this section, in order to validate the inverse method, the localization of known sources of sound is performed. These sources mimic tonal dipole sources that would be generated by compact objects placed in the wind tunnel flow, although the effect of flow on sound propagation is ignored. The simulated sources are distributed along the two geometries considered in Sec. 3 and Table 1: the rectilinear cylinder, and the curved cylinder forming a ring. The cross-section of these objects
<table><caption>Table 1: Configurations and velocities.</caption><thead><tr><th>Configuration</th><th>(C1)<br>Inclined cylinder</th><th>(C2)<br>Ring</th></tr></thead><tbody><tr><td>Sketch</td><td><img src="sketch_image.png" alt="Sketch of a cylinder with an inclined plane and a line segment from the top of the cylinder to the plane."></td><td><img src="ring_image.png" alt="Ring of a cylinder with an inclined plane and a line segment from the top of the cylinder to the plane."></td></tr><tr><td>Dimensions</td><td>d=6 mm<br>L=140 mm<br>β=33°</td><td>d=6 mm<br>d<sub>f</sub>=10 mm<br>R<sub>c</sub>=230 mm</td></tr><tr><td>Material</td><td>stainless steel</td><td>copper, painted black</td></tr><tr><td>U<sub>∞</sub></td><td>40 m/s</td><td>50 m/s</td></tr><tr><td>Re</td><td>15 385</td><td>19 231</td></tr><tr><td>Shedding frequency</td><td>1300 Hz</td><td>1600 Hz</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

is considered acoustically compact, so that they are modeled by point dipole sources distributed along the cylinder axis. The synthetic dipoles are locally perpendicular to the cylinder axis, and parallel to the lift force. This force is orthogonal to the flow direction [37], the latter being parallel to the x-axis. Thus, the dipole orientation, denoted by $\mathbf{e}_d$, is contained within the plane $(y, z)$, with a null x-component.

The microphone signals corresponding to the sources are created numerically in the time domain, by taking the convolution product between these sources and the Green function. The latter has been specified in Eq. (5) for a dipole. The temporal pressure field at the microphone at position $\mathbf{r}$ is thus calculated from [38]:

$$p(\mathbf{r}, t) = \sum_{q=1}^{N_q} a_q \frac{\sin\left[\omega\left(t - \frac{|\mathbf{r} - \mathbf{r}_q|}{c_0}\right)\right]}{4\pi|\mathbf{r} - \mathbf{r}_q|} k \frac{(\mathbf{r} - \mathbf{r}_q) \cdot \mathbf{e}_d}{|\mathbf{r} - \mathbf{r}_q|} + n_q(t), \quad (14)$$

where index $q$ in the sum goes through the point sources, with $N_q$ the number of imposed sources. $\omega$ is such that $St = 0.2$. $\mathbf{r}_q$ and $a_q$ are respectively the position and amplitude of the q-th source, $c_0$ is the speed of sound in air. All sources are in phase and in the following they have the same amplitude: $a_q = 1, \forall q$. An additive normally distributed random noise $n_q(t)$ can be added to each point source. When $n_q = 0$ ($\forall q$) the signal is the same at all the source points, these points being fully correlated. Setting $n_q \neq 0$ allows to simulate partially uncorrelated sources. This will be
done below in order to illustrate how noise influences the eigenvalues of the CSM. In this case, the
standard deviation of the additive noise is set to $\sigma = 40$.

In order to show the effect of the array geometry on source localization, in the reaming of
this paper, two different sets of microphones are considered: the first set (S1) consists of the entire
array of 1024 microphones; the second set (S2) includes only 256 microphones located on one planar
array (the red one in Fig. 3 is utilized, located in the region $y > 0$). In all cases, microphone signals
at these arrays are simulated over a duration of $t = 1$ s and sampled with the same sampling rate as
in the experiment (50 kHz). Based on the microphone signals, inverse beamforming is performed
to recover the source, that is, Eq. (4) is solved by computing Eqs. (7a-7b).

4.1. *Simulation of a inclined cylinder*

First, a synthetic source mimicking the configuration (C1) described in Table 1, and also shown
in Fig. 3, is considered. $N_q$ is selected so that the distance between dipoles is slightly smaller than
the spanwise coherence length, namely $l_c = 5d = 3$ cm [29]. A total of $N_q = 42$ equi-spaced point
dipole sources are simulated, arranged along a line tilted at an angle of 33° to the y-axis, operating
at a frequency of 1600 Hz (the spacing between the point sources is then about one eighth of the
wavelength). This means that the angles are set to $\theta = 33^\circ$, and $\phi = 90^\circ$ all along the distribution
(dipoles are in the yz-plane). These dipoles are positioned within the range $-0.35 \text{ m} \le y \le 0.35 \text{ m}$,
to represent the portion of the cylinder immersed in the wind-tunnel flow. The microphone pressure
signals due to these sources are computed using Eq. (14), and no additive noise is used for this
case ($n_q(t) = 0$). Hence, the point sources are fully correlated, and so is the pressure field at
the microphones. As a result the CSM has rank 1, with a single non-zero eigenvalue ($N_{\text{mode}}=1$),
associated with a single pressure mode $\mathbf{p}_1$, which provides a single source mode $\mathbf{s}_1$ upon inversion
of Eq. (4).

Concerning the scanning points, we have mentioned in the beginning of section 2 that they are
placed along the cylinder axis and are separated by a distance of the order of the vortex shedding
correlation length. Of course, in this simulated case, this fits perfectly the way the dipoles are
imposed. It is possible to use more or less scanned point sources than imposed ones, this is done
in Appendix A, but here we just use the same number ($N_{\text{grid}} = N_q$) and the same spacing. As
a result, the retrieved dipole amplitudes are directly comparable to the imposed amplitudes $a_q$.
The total number of sources is $N = N_{\text{grid}} \times N_{\text{type}} = 42 \times 3 = 126$ unknowns, which leads to an
overdetermined system whatever the considered antenna, solved with Eq. (7b).

The retrieved source mode is shown in Fig. 4(a) when the full microphone array (S1) is used,
and in Fig. 4(b) when only a single planar array (S2) is used. In these and equivalent figures in
the following, one arrow per detected point source is plotted and is aligned with the direction of
the dipole, while the norm of the arrow indicates the source amplitude. The color of the arrow
indicates the phase of the source, and the regularization parameter is set to $\lambda = 0.05$.

<figure><img src="image_8.png" /><figcaption>Figure 4: Source mode $s_1$ obtained by inverse beamforming when the simulated source consists of a line of correlated dipoles oriented at 33° relative to the y-axis. Results obtained for a microphone array made of: (a) 1024 microphones (four panels); (b) 256 microphones (single, red panel).</figcaption></figure>

The error made on the dipole source angle and amplitude obtained by computing the differ-
ence between the retrieved (see Eqs. (8-9)) and known values, are shown in Fig. 5(a) for the full
microphone array (S1), and in Fig. 5(b) for a single panel (S2).

The source retrieved with the 1024 microphones accurately captures the known source behavior:
within the range of $-0.3 \text{ m} \le y \le 0.3 \text{ m}$, the arrows exhibit similar amplitude and phase. For $y$ in
<figure><img src="image_9.png" /><figcaption>Figure 5: Black lines (left axis): error on the orientation of the dipoles (dashed line: $\Delta\theta$, solid line: $\Delta\varphi$). Grey lines (right axis): simulated (dotted line) and recovered (dashed dotted line) amplitudes. The result is plotted for fully correlated distribution of simulated dipoles along a line. Calculated with (a) the 1024 microphones; (b) only the red array.</figcaption></figure>

the range $0.3 \text{ m} \le |y| \le 0.5 \text{ m}$, the amplitude decreases, eventually approaching near-zero values (Fig. 4(a)). Moreover, the dipoles remain perpendicular to the cylinder: the error in orientation is inferior to $\Delta\theta < 5^\circ$ for $|y| < 0.4$ (Fig. 5(a)). When utilizing a single planar panel of 256 microphones (S2), the precise recovery of the line in 3D is compromised (Fig. 4(b)). The directivity of the point sources located near the array is discernible, but that of sources located further away from the array suffers degradation. The error of orientation is inferior to $\Delta\theta < 10^\circ$ between $0 \text{ m} < y < 0.3 \text{ m}$, but otherwise the dipoles are not recovered properly (neither in amplitude, nor in orientation, Fig. 5(b)). In Appendix A, it is shown that these conclusions are unchanged if the number of scanned source points differs from the number of imposed source points, as long as the lineic amplitude of sources is considered.

Hence, the reconstruction of the sound source is not possible with a single array, due to the
fact the spatial resolution of a planar array is poor in the direction perpendicular to it. However, using the complete 3D array results in an accurate reconstruction of the sources in amplitude and orientation.

In order to highlight the advantage of the present TGIB method, we apply the GIB as presented in [9] to the present configuration, by searching sources in a volume enclosing the cylinder. Results of source identification are shown in Appendix B, Fig. B.1, they show that dipoles' orientation is not as well recovered with GIB as with TGIB.

### 4.2. Simulation of a ring

A synthetic source mimicking the configuration (C2) described in Table 1, and also shown in Fig. 3, is now considered, and the same procedure as previously is employed. $N_q = 60$ equi-spaced dipoles emitting tonal noise at a frequency 1600 Hz and an amplitude of $a_q = 1$ are imposed along the ring centerline, with a spacing equal to about four times the cylinder diameter. Each dipole is orthogonal to the local curvature of the ring and to the flow, so that the azimuthal and elevation angles satisfy: $\varphi = 90^\circ$ (ie, the dipole component on $x$ is 0) and $0 \le \theta \le 2\pi$.

Again the scanning points are the same as the imposed source points for the ring part. Also, to replicate the experimental case, 60 scan points are placed at the location of holding bars. This number of grid points corresponds to $N = N_{grid} \times N_{type} = 120 \times 3 = 360$ unknowns, which leads to an overdetermined (resp., underdetermined) system for antenna S1 (resp., S2).

Inverse beamforming is performed at 1600 Hz. Again, a single source mode $s_1$ is found in this case. Figure 6 presents its source map obtained with the two different microphone arrays: Fig. 6(a) is obtained using all 1024 microphones (S1), while Fig. 6(b) depicts the results obtained with only 256 microphones (S2). Also, the errors in orientation and amplitude of the dipoles reconstructed on the ring are plotted in Fig. 7(a,b) for the two arrays. In both cases, the regularization parameter is set to $\lambda = 0.012$.

When employing all microphones (S1), the dipole sources are reconstructed satisfactorily along the ring, despite a decrease in amplitude at the junctions with each of the four support bars. The identified dipoles exhibit the anticipated correct directivity along the ring, being within the ring's plane and orthogonal to it (Fig. 6(a)). The phase, calculated for each $s^{dip}.n$, where $n$ is a unitary vector locally oriented normal to the ring, is constant, which is consistent with the imposed model. Figure 7(a) confirms that at the grid points where the simulated ring is not crossing the simulated holding bars, the errors on the dipole orientation angles are inferior to 6°, and the amplitude is close to 1.

For the array (S2), Fig. 6(b) shows that correct source reconstruction is limited to the areas corresponding to the right and the left portions of the ring (where the dipoles are oriented towards,
<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Figure 6: Source mode $s_1$ obtained by inverse beamforming performed at 1600 Hz for simulated sources made of correlated dipoles distributed along a ring (1600 Hz) and its holding bars (1025 Hz). Results obtained for a microphone array made of: (a) 1024 microphones (four panels); (b) 256 microphones (single, red panel).</figcaption></figure>

and close to, the panel of microphones). Moreover, the reconstructed sources demonstrate a bias in their directivities and amplitude (see Fig. 7(b)). Similarly to the inclined cylinder, the amplitude is better on the grid points that are the closest to the used array.

A second set of simulated sources is produced by setting $n_q(t) \neq 0$ in Eq. (14), meaning the simulated sources along the ring are now partially uncorrelated. The same is true for pressure at the microphones, and the CSM presents multiple nonzero eigenvalues $\sigma_i$, each associated with
<figure><img src="image_12.png" /><figcaption>Figure 7: Source reconstructed relative to the source imposed at 1600 Hz along the circumference of the ring. Black lines (left axis): error on the orientation of the dipoles (dashed line: $\Delta\theta$, solid line: $\Delta\varphi$). Grey lines (right axis): simulated (dotted line) and recovered (dashed dotted line) amplitudes. Plotted for fully correlated distribution of simulated dipoles around a circle. Calculated with (a) the 1024 microphones; (b) only the red array.</figcaption></figure>

an eigenmode $\mathbf{p}_i$, and subsequently to a source mode $\mathbf{s}_i$ upon inversion of Eq. (4). Figure 8 (a) shows a typical pressure spectrum at one of the microphones, showing the broadband noise added to the pure sine at 1600 Hz. Figure 8 (b) shows the eigenvalues of the CSM, normalized so that they represent the contribution of each mode to the total pressure energy, that is, $\sigma_i/(\sum_j \sigma_j)$ is represented for each value of the mode number $i$. The most energetic mode contributes to 27% of the energy, the second and third ones to between 12% to 14%, and the other mode contributions gradually decrease.

Figure 8 (d,e,f) show the first three source modes ($\mathbf{s}_1$, $\mathbf{s}_2$, $\mathbf{s}_3$): the retrieved dipoles are consistently orthogonal to the curvature of the ring, and as before the first mode captures much of the source known behavior, since it consists of regularly distributed dipoles that are mostly in phase, orthogonal to the ring, and approximately of equal amplitudes. However, the other modes have a
<figure><img src="image_13.png" /><figcaption>Figure 8: Simulation results for decorrelated dipoles. (a) Power spectral density measured at one microphone, (b) Percentage of energy contribution of each mode, (c) Total amplitude for the sum of the 12 most energetic modes (those contributing for 90% of the modes), (d,e,f) maps for the first three modes. These results are obtained with the 1024 microphones.</figcaption></figure>

non-negligible amplitudes with respect to the first one, as seen from the energetic contributions in Fig. 8 (b). It is observed that the modes are functions that are close to azimuthally sinusoidal and this can be explained as follows. These modes, verifying $Hs_i = p_i$, are the inverse by $H$ of the POD modes $p_i$ of the pressure CSM. While these are certainly not the POD modes of the source CSM, it has been verified that they are reasonably close to them. And it is known that the POD modes are Fourier modes in homogeneous directions [31]. Since the imposed source field is spatially homogeneous in the azimuthal direction (at least in between holding bars, which locally perturb homogeneity), it results that the $s_i$, being approximations of the POD modes of this source, are close to azimuthal Fourier modes. A more formal derivation is left for future research.

Consequently, numerous modes collectively contribute to the final source distribution. The total source amplitude, obtained by summation over the $N_{modes} = 12$ modes representing 90% of the sound field energy (see Eq. (11)), is given in Fig. 8(c). Regularization parameter is set to $\lambda=0.03$ in this case. Note that the total amplitude in Eq. (11) just consists of an amplitude (no direction, no phase). Since all the modes considered separately contain dipoles that have an
orientation close to the normal to the ring and flow direction, we have decided to represent the
total resulting amplitude with vectors pointing in this direction, in Fig. 8(c) and similar figures
in the following. The uniform distribution of amplitudes is in good agreement with the simulated
sources. In conclusion, the partial lack of correlation of the sources (here obtained by adding noise)
results in an increase of the number of significant modes. This will be observed for experiments
presented below.

**5. Experiments**

Experiments are performed in the anechoic wind-tunnel introduced in Section 3. The sound
pressure level is established across the 1024 microphones, and equals the diagonal elements of
matrix **C** with dimensions in Pa²/Hz. The corresponding sound level in decibels (dB) is computed
using the reference pressure P₀ = 2.10⁻⁵ Pa. The PSD is visualized for the microphone marked in
black within Fig. 3. The mean level indicated in the figures corresponds to the mean level averaged
over the 1024 microphones, at the peak frequency.

5.1. *Inclined cylinder*

First we analyse the configuration (C1) in Tab. 1. The cylinder is inclined at an angle of
33° with respect to the y-axis and extends beyond the limit of the array. However, the shear
layers of the wind-tunnel jet are localized at y =-0.35 m and y =0.35 m. Hence, the effective
length of the cylinder immersed in the flow and emitting sound is limited to a length of 0.7 m.
The Reynolds number, Re = 15,385, indicates that the wake behind the cylinder is turbulent.
However, the shedding of turbulent vortices remains quasi-two-dimensional within this regime,
so that an homogeneous distribution of dipole sources is expected along the cylinder span. In
Fig. 9, the PSD exhibits a notable peak at frequency f₀ = 1300 Hz, corresponding to vortex
shedding at a Strouhal number St = 0.195. The inverse problem is solved at this frequency,
and the regularization parameter is set to λ = 0.79. The Ngrid = 60 scanned source points are
located on the centerline of the cylinder, with a spacing of about 5 diameters. This corresponds
to N = Ngrid × Ntype = 60 × 3 = 180 unknowns.

Fig. 9(b) shows that the eigenvalues decrease rapidly, and the number of significant modes is $N_{\text{mode}} = 5$. Figures 9(d,e,f) illustrate the three most energetic modes. The first mode reveals a consistent source correlation spanning the entire cylinder, resembling the simulated correlated sources. The next two modes are Fourier-like, which probably means that the source is approximately homogeneous along the cylinder (see discussion of section 4.2). The maximal error in the dipole orientation relative to the expected one is about $\theta = 8^\circ$. A summation of the modes according to Eq. (11) is shown in Fig. 9(c), plotted in the same way as in Fig. 8(c). It confirms that
<figure><img src="image_14.png" /><figcaption>Figure 9: Experimental results for the 33° inclined bar, at the vortex shedding frequency: $f_0 = 1300$ Hz. (a) Power spectral density measured on one microphone, (b) Percentage of contribution of each modes, (c) Sum of the 5 most energetic modes, (d,e,f) maps obtained for the first three modes. These results are obtained with the 1024 microphones, at a velocity $U_\infty = 40$ m.s$^{-1}$</figcaption></figure>

there exists a uniform distribution of dipole amplitude along the entire span, which is physically relevant, and is well captured by the 3D tunnel antenna.

As for the simulated cylinder, to put things in perspective and see the advantage of using scanning source points based on physical considerations, we apply the GIB as presented in [9] to the present configuration, by searching sources in a volume enclosing the cylinder. The results are given in Appendix B, in Fig. B.1. TGIB estimates the sources with more accuracy than GIB for the present physical setup.

## 5.2. Ring

The experiment with the ring, case (C2) in Tab. 1, is now analysed. The sources are expected to be dipoles oriented orthogonal to the flow and to the local curvature of the ring, as in the simulation. The PSD in Fig. 10(a) exhibits three main peaks. The dominant peak at 1600 Hz, due to the lift dipole, corresponds to the shedding frequency of the ring. Its harmonic at 3200 Hz is associated with the drag dipole of the ring. The first peak at 1025 Hz corresponds to the shedding
frequency of the bars holding the ring. Scanning point sources are placed on both the main ring and the support bar centerlines, with a spacing of the order the shedding frequency correlation length, which is slightly different for the ring and supports. A total of $N_{grid} = 120$ source points are taken, 60 on the ring, and 15 on each of the four holding bars. All together, this represents $N = N_{grid} \times N_{type} = 120 \times 3 = 360$ unknowns. The inverse problem is solved for two different vortex shedding frequencies: at 1600 Hz in Fig. 10, and at 1025 Hz in Fig. 11.

At 1600 Hz, the regularization parameter is set to $\lambda = 2.01$. The eigenvalues shown in Fig. 10(b) do not decrease as rapidly as for the inclined cylinder, due to the increased complexity of the configuration (geometry, flow speed). It takes $N_{mode} = 9$ to account for 90% of the energy, with the first mode representing 23% of the total energy. Since this configuration is the most complex described in this article, it is chosen to report the L-curve, which is done in Appendix C. The predominant sources are localized on the ring. They are mainly orthogonal to the local curvature of the ring and to the flow (having a dipole component in the x-direction which is small). The total dipole amplitude, accounting for all the modes, is shown in Fig. 10(c), again plotted in the same way as in Fig. 8(c). It shows a rather even distribution of dipoles along the ring, slightly altered at the junctions with the four holding bars which probably disturb locally the vortex shedding. This even distribution is obtained thanks to the 3D configuration of the antenna and would not be observed for an antenna made of a single plane array, as already illustrated in Fig. 6 for the simulated case. The amplitude of the dipoles on the holding bars is not negligible at 1600 Hz, although their own vortex shedding frequency is at 1025 Hz.

At 1025 Hz, the shedding frequency of the holding bars, only $N_{mode} = 5$ account for 90% of the energy, with the first one contributing 40%. The regularization parameter is set to $\lambda = 2$. The sources sequentially appear along the four holding bars, with a non-negligible contribution from the ring. The sources on each of the bars are seemingly uncorrelated from those on the other bars, since they appear quite separately on each of the four dominant modes. However, the 4 modes, having different eigenvalues, do not correspond to the same energy level. Hence, the ideal picture in which each support bar acts independently of the others and contributes an equal amount of acoustic energy, is not completely obtained. Similarly to Fig. 9, the amplitudes of the dipoles on the holding bars are diminishing around the shear layers, namely where $(y, z) \notin [-0.35, 0.35]$ m.

## 6. Conclusion

In this work, we consider acoustic sources due to bodies placed in a flow, which principal noise source mechanism is vortex shedding and whose cross-section is acoustically compact, i.e., the emitted sound wavelength is much larger than the cross-section dimensions. The orientation of the
<figure><img src="image_15.png" /><figcaption>Figure 10: Experimental results for the ring, at the vortex shedding frequency: 1600 Hz. (a) Power spectral density measured on the microphone shown before, (b) Percentage of contribution of each modes, (c) Sum of the 9 most energetic modes, (d,e,f) maps obtained for the three first modes. These results are obtained with the 1024 microphones, at a velocity $U_∞ = 50$ m.s$^{-1}$.</figcaption></figure>

bodies center line is possibly varying, causing dipoles of varying orientation in the direction normal to the flow. The localization, strength, and orientation of these dipole sources are estimated from pressure data measured with the microphones of a three-dimensional array forming a tunnel in the flow direction and surrounding the bodies. An inverse problem similar to GIB as presented in [9] is solved by inverting the matrix of the transfer function from given point sources in the flow to the microphones of the antenna. To stick to the physics under consideration the source points are searched along the bodies' centerline, with a spacing that corresponds to the correlation length of vortex shedding. At each point, three dipole components are searched, respectively along the x, y, and z-axes. Hence, no prior knowledge of the dipole orientation is necessary. The number of source points is limited, so that unlike GIB, making the source sparser is not useful and a minimization in L2-norm, regularized with a Tikhonov method, is sufficient.

The method has been applied to a rectilinear rod of circular cross section not aligned with any of the panels of the antenna, and to a ring of circular cross section presenting an important
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>Figure 11: Experimental results for the ring, at the vortex shedding frequency of the holding bars: 1025 Hz. (a) Percentage of contribution of each modes, (b,c,e,f) maps obtained for the four first modes. Those results are obtained with the 1024 microphones, at a velocity $U_\infty = 50 \text{ m.s}^{-1}$.</figcaption></figure>

variety of dipole angles. These configurations have first been simulated to assess the error level of the method, with correlated or partially uncorrelated sources. The three-dimensional nature of the array is demonstrated to be essential for the reconstruction of dipole sources with varying orientation: the error of orientation is less than 10° with the full array, against more than 20° with only one plane array. The amplitude is strongly altered if only one plane array is used, whereas the three-dimensional antenna recovers the amplitudes correctly.

The efficiency of the method in real configurations has been verified by conducting experiments with geometries similar to the simulated ones. These were placed in a flow in the open test section of a wind tunnel. The method is shown to successfully reconstruct dipole sources for aeolian tones configurations, finding dipoles essentially orthogonal to the flow and the bodies' local orientation, with an even amplitude distribution. The same technique could be applied to assess the dipolar sources generated by a twisted airfoil or blade, producing a line of dipoles with varying orientations along the span, provided that the chord remains acoustically compact. At higher frequencies, however, the question of source distribution over the surface of the body arises, since a given cross-section can no longer be represented by a single point dipole. This issue will be addressed in future studies. Another future investigation will also involve including higher-order
propagation models in the source catalog, in order to tackle more complex aeroacoustic radiation patterns, due to non-acoustically compact bodies or turbulence.

## 7. Acknowledgments

The authors thank Agence Nationale de la Recherche (ANR, grant ANR-18-CE22-0005-01), and Région Nouvelle-Aquitaine for the financial support. The authors are grateful to Pascal Challande, Jacques Marchal, Hélène Moingeon, and Christian Ollivon at Institut Jean Le Rond d'Alembert for designing the microphone array, and Laurent Philippon and Pascal Biais at Institut PPRIME for their technical support.

## Appendix A. Effect on the number of points on the scanning grid

In the main text, we have compared the source amplitude retrieved by TGIB to the imposed one for simulated dipole sources placed along an inclined line (Fig. 5). However, the spacing between the scanning source points was the same as the spacing between the known source points. This appendix discusses the influence of the number of scanning points. Figure A.1 shows TGIB result for simulated dipoles on a line, either with a scanning grid of 30 points (Fig. A.1(a)) or with one of 90 points (Fig. A.1(b)). In this representation, the found dipoles are scaled by the maximum amplitude over the line, which means that the shape of the dipoles' envelope remains the same. To be more quantitative, the amplitude and errors associated with these simulations are shown in Fig. A.2. Here, the amplitudes of the simulated sources are equal to $a = 0.04$, which corresponds to $a/\delta_l = 1$, and those of the estimated sources are plotted as $||D||/\delta_l$, with $\delta_l$ being the unit of length between two points of the grid. This figure shows that the number of sources on the grid does not have any influence on the TGIB result, as the amplitude per unit of length remains the same. In fact, the amplitude received by the array is distributed among the sources in the grid: the more sources there are on the grid, the less significant the amplitude per source is.

## Appendix B. Comparison with GIB method (Generalized Inverse Beamforming)

In this appendix, and as a basis for comparison, we apply the GIB method to the inclined cylinder configurations of sections 4.1 and 5.1. The scanning grid consists of points located not on the axis of the cylinder, but rather in a parallelepiped of volume $0.2 \times 1 \times 0.7 \text{ m}^3$ and spaced by 3.5 cm on the x, y and z-axes. GIB is based on L1-norm minimization, and sources are made sparser during successive iterations of an iteratively reweighted least-square procedure, each iteration removing the weakest sources. Results of source identification for several iterations are shown in Fig. B.1. For simulated sources (the counterpart of which using TGIB is Fig. 4(a)),
<figure><img src="image_22.png" /><figcaption>Figure A.1: Source mode $s_1$ obtained by inverse beamforming when the simulated source consists of a line of correlated dipoles oriented at 33° relative to the y-axis. Results obtained for a grid made of (a) 30 points, (b) 90 points.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Figure A.2: Black lines (left axis): error on the orientation of the dipoles (dashed line: $\Delta\theta$, solid line: $\Delta\varphi$). Grey lines (right axis): simulated (dotted line) and recovered (dashed dotted line) amplitudes. The result is plotted for fully correlated distribution of simulated dipoles along a line. Results obtained for a grid made of (a) 30 scanning points, (b) 90 scanning points. The source amplitude $||D||$ is normalized by the spacing $\delta_l$ between scanning points.</figcaption></figure>

the result is plotted for (a) 20 iterations, (b) 40 iterations, and (c) 50 iterations. The sources on the cylinder are recovered in the interval $y \in [-0.35, 0.35]$ m. However their orientation is not
constant over the cylinder. The same conclusion holds for the experimental case, in Fig. B.1(d-f)
(the counterpart of which using TGIB is Fig. 9). We thus found TGIB to be more reliable in this
investigation.

<figure><img src="image_24.png" /><figcaption>Figure B.1: (a,b,c): Simulation results obtained with GIB, corresponding to the case shown in Fig. 4(a). These results are obtained with the 1024 microphones. (a) after 20 iterations, (b) after 40 iterations, (c) after 50 iterations. (d,e,f): Experimental results obtained with GIB for the 33° inclined bar, at the vortex shedding frequency (1300 Hz), corresponding to the case shown in Fig. 9. These results are obtained with the 1024 microphones, at a velocity $U_\infty = 40 \text{ m.s}^{-1}$. (d) after 20 iterations, (e) after 40 iterations, (f) after 50 iterations.</figcaption></figure>

Appendix C. L-curves and regularization parameters for the ring experiment

In this appendix, typical L-curves are provided for the experimental ring configuration. At
1600 Hz, 9 modes are needed to account for 90% of the energy, as illustrated in Fig. 10(b).
Therefore, 9 problems like Eq. (6) are solved for $i = 1, ..., 9$. Each of these requires a regularization
parameter $\lambda$, and the 9 corresponding L-curves, one per mode, are shown in Fig. C.1. The detected
corners are indicated and they determine the optimal values of $\lambda$ to be used. In the present case, the
values of $\lambda$ for the nine solves vary between 1.3 and 5. For comparison, among the 360 eigenvalues of $H^\dagger H$ sorted in descending order, the largest being approximately 184, an eigenvalue close to 5 is found at rank 50, one close to 1.3 at rank 60. Smaller eigenvalues with a larger rank are filtered out by regularization.

In order to estimate the sensitivity of TGIB to regularization, we compute the total dipole

<figure><img src="image_25.png" /><figcaption>Figure C.1: L-curves (one per mode) for TGIB performed for the experimental ring configuration at 1600 Hz. The numbers attached to the curves indicate the modes $i = 1, ..., 9$ they correspond to. The symbols indicate the detected corners (fixing the optimal regularization parameter).</figcaption></figure>

amplitude (given by Eq. (11)) for three different values of the regularization parameter: $\lambda_{opt}$, which is the value given by the corner point used in this work; an increased value $3\lambda_{opt}$; and a decreased value $0.33\lambda_{opt}$. The corresponding positions on the L-curve are indicated in Fig. C.2(a) for the first mode only, the situation being equivalent for the other modes. The total dipole amplitude found at the 120 scanned source positions (see Section 5.2) for each of these 3 values of $\lambda$ is given in Fig. C.2(b). For the largest sources at 1600 Hz, found on the ring, it is observed that changing the regularisation parameter by a factor of 3 changes the dipole amplitude by up to 15%.

## AUTHOR DECLARATIONS

**Conflicts of interest** The authors have no conflicts to disclose.
<figure><img src="image_26.png" /><figcaption>Figure C.2: (a) L-curve, shown for the 1st mode, with symbols indicating the optimal, decreased, or increased regularization parameters $\lambda$. (b) Retrieved total dipole amplitude for these three different values of $\lambda$.</figcaption></figure>

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

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