# Numerical validation of shear flow corrections for beamforming acoustic source localisation in open wind-tunnels

Thomas Padois, Christian Prax, Vincent Valeau

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Thomas Padois, Christian Prax, Vincent Valeau. Numerical validation of shear flow corrections for beamforming acoustic source localisation in open wind-tunnels. Applied Acoustics, 2013, 74 (4), pp.591-601. 10.1016/j.apacoust.2012.09.013. hal-04540351

HAL Id: hal-04540351

https://univ-poitiers.hal.science/hal-04540351

Submitted on 10 Apr 2024

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Distributed under a Creative Commons Attribution 4.0 International License
Numerical validation of shear flow corrections for
beamforming acoustic source localisation in open
wind-tunnels.

Thomas Padois<fnref n="1" /> Christian Prax<fnref n="2" /> and Vincent Valeau<fnref n="3" />

Institut Pprime - CNRS, Université de Poitiers, ENSMA

Département Fluides, Thermique, Combustion

ENSIP - Bâtiment B17 - Campus Sud - 6, rue Marcel Doré - BP 633

86022 Poitiers Cedex - France

Abstract

The beamforming technique has become a common method to localise aeroa-
coustic sources. However, in the case of anechoic wind tunnel experiments,
some shear flow corrections have to be taken into account in order to estimate
accurately the position and power of the source. This study deals with the
efficiency of such corrections. Several correction methods are presented. The
first one is based on the convected wave equation and predicts that, if no
flow model is used, the source is localised with an apparent source shift, that
is approximately equal to the product of the Mach number by the mean flow
thickness. Two other shear flow corrections, based on acoustic ray theory,
are considered. They allow the modeling of both the effective source-receiver
propagation time and pressure amplitude. The efficiency of such corrections

<footnote n="1">¹thomas.padois@univ-poitiers.fr</footnote>

<footnote n="2">²christian.prax@univ-poitiers.fr</footnote>

<footnote n="3">³vincent.valeau@univ-poitiers.fr</footnote>
Numerical validation of shear flow corrections for
beamforming acoustic source localisation in open
wind-tunnels.

Thomas Padois<fnref n="1" /> Christian Prax<fnref n="2" /> and Vincent Valeau<fnref n="3" />

Institut Pprime - CNRS, Université de Poitiers, ENSMA

Département Fluides, Thermique, Combustion

ENSIP - Bâtiment B17 - Campus Sud - 6, rue Marcel Doré - BP 633

86022 Poitiers Cedex - France

Abstract

The beamforming technique has become a common method to localise aeroa-
coustic sources. However, in the case of anechoic wind tunnel experiments,
some shear flow corrections have to be taken into account in order to estimate
accurately the position and power of the source. This study deals with the
efficiency of such corrections. Several correction methods are presented. The
first one is based on the convected wave equation and predicts that, if no
flow model is used, the source is localised with an apparent source shift, that
is approximately equal to the product of the Mach number by the mean flow
thickness. Two other shear flow corrections, based on acoustic ray theory,
are considered. They allow the modeling of both the effective source-receiver
propagation time and pressure amplitude. The efficiency of such corrections

<footnote n="1">¹thomas.padois@univ-poitiers.fr</footnote>

<footnote n="2">²christian.prax@univ-poitiers.fr</footnote>

<footnote n="3">³vincent.valeau@univ-poitiers.fr</footnote>
is assessed by using a numerical simulation of the acoustic pressure mea-
surement with a linear microphone array in an anechoic wind-tunnel. The
simulation, based on a program solving the linearised Euler Equations, is first
validated at a low Mach number (0.11) by a careful comparison, in terms of
source localisation, with experimental data for several array positions. The
efficiency of the shear flow corrections is then investigated numerically for
a set of flow parameters. The apparent source shift is shown to be inde-
pendent of the ratio of the shear flow thickness to the sound wavelength,
and the shear flow corrections perform well, at low Mach numbers for thin
and thick shear layers. Conversely, the mean flow thickness and the Mach
number are shown to be the two main parameters having an influence on the
apparent source shift. The estimate of this shift by the product of the Mach
number with the mean flow thickness is shown to be a good approximation,
and can be used for a posteriori corrections up to Mach number 0.4. For
higher subsonic Mach numbers, the ray-based shear flow correction models
associated to beamforming processing are shown to be robust enough to lo-
calise the source position in a flow, and to avoid significant source power
under-estimation.

***Keywords:*** beamforming, aeroacoustics, localisation, anechoic wind-tunnel,
shear flow correction
is assessed by using a numerical simulation of the acoustic pressure mea-
surement with a linear microphone array in an anechoic wind-tunnel. The
simulation, based on a program solving the linearised Euler Equations, is first
validated at a low Mach number (0.11) by a careful comparison, in terms of
source localisation, with experimental data for several array positions. The
efficiency of the shear flow corrections is then investigated numerically for
a set of flow parameters. The apparent source shift is shown to be inde-
pendent of the ratio of the shear flow thickness to the sound wavelength,
and the shear flow corrections perform well, at low Mach numbers for thin
and thick shear layers. Conversely, the mean flow thickness and the Mach
number are shown to be the two main parameters having an influence on the
apparent source shift. The estimate of this shift by the product of the Mach
number with the mean flow thickness is shown to be a good approximation,
and can be used for a posteriori corrections up to Mach number 0.4. For
higher subsonic Mach numbers, the ray-based shear flow correction models
associated to beamforming processing are shown to be robust enough to lo-
calise the source position in a flow, and to avoid significant source power
under-estimation.

***Keywords:*** beamforming, aeroacoustics, localisation, anechoic wind-tunnel,
shear flow correction
# 1. Introduction

Transport noise has attracted some considerable scientific attention over the past several decades. Due to some substantial improvement concerning wheel-track interaction noise and engine noise, some attention has been brought out more recently on the noise generated aerodynamically. In this context, researchers and engineers have focused their effort onto the experimental localisation of aeroacoustic noise. Among the available techniques, the use of a microphone array, associated with data processing algorithms such as beamforming, is certainly one of the most classical techniques for aeroacoustic applications [1]. The sound source localisation experiments can be carried out *in situ* (see eg., references [2], [3]...). An alternative approach is to perform an experiment in an anechoic wind-tunnel (see eg., reference [4, 5]). In this case, the microphone array is located outside the flow in the quiescent air while the sound source is located within the flow. This is why, in this type of wind-tunnel aeroacoustic measurements, the shear flow effects (i.e., convection and refraction) on the acoustic propagation of the sound waves from the source to the array have to be taken into account.

In the early 1970’s, Amiet has studied the sound propagation in the shear layer [6, 7]. The Amiet’s method considers the propagation of acoustic rays through a simplified flow assuming a zero-thickness shear layer and that the phenomena are independent to sound frequency. A set of equations is obtained accounting for the shear layer effects on the acoustic propagation. This set of equations can be solved by an iterative process [8] and the travel
# 1. Introduction

Transport noise has attracted some considerable scientific attention over the past several decades. Due to some substantial improvement concerning wheel-track interaction noise and engine noise, some attention has been brought out more recently on the noise generated aerodynamically. In this context, researchers and engineers have focused their effort onto the experimental localisation of aeroacoustic noise. Among the available techniques, the use of a microphone array, associated with data processing algorithms such as beamforming, is certainly one of the most classical techniques for aeroacoustic applications [1]. The sound source localisation experiments can be carried out *in situ* (see eg., references [2], [3]...). An alternative approach is to perform an experiment in an anechoic wind-tunnel (see eg., reference [4, 5]). In this case, the microphone array is located outside the flow in the quiescent air while the sound source is located within the flow. This is why, in this type of wind-tunnel aeroacoustic measurements, the shear flow effects (i.e., convection and refraction) on the acoustic propagation of the sound waves from the source to the array have to be taken into account.

In the early 1970’s, Amiet has studied the sound propagation in the shear layer [6, 7]. The Amiet’s method considers the propagation of acoustic rays through a simplified flow assuming a zero-thickness shear layer and that the phenomena are independent to sound frequency. A set of equations is obtained accounting for the shear layer effects on the acoustic propagation. This set of equations can be solved by an iterative process [8] and the travel
time from the source to the microphone can be computed. Since a decade
now, the Amiet's method (or other similar methods [9]) are conventionally
associated with the beamforming technique [10]. They are used as shear
layer correction models improving the sound source localisation, and are
powerful in the sense that they allow to predict the shear flow effects based
on little information about the flow, i.e., the mean flow velocity and thickness.
However few studies deal with a quantitative assessment of the efficiency of
such corrections.

Recently, the beamforming technique associated to the Amiet’s correc-
tions has been likened to an in-flow calibration proposed by Kroeber et
al. [11]. Both experimental methods showed that the methods accurately
localise the source position for low Mach number flows (less than 0.15). The
shear flow correction has also been studied by Bahr et al. [12] by using a
plasma point source in an anechoic wind-tunnel. Preliminary tests in an
empty test section at Mach number M=0.17 showed an apparent shift of
the source position in the downstream direction if no shear flow corrections
are made. The shear layer corrections are used for different flow configu-
rations (the sound source being located behind an airfoil trailing-edge or a
nose gear). In spite of the complexity of these flow configurations, the overall
trends given by the Amiet’s method are correct.

These two studies state that the Amiet’s method properly corrects the
results given by the beamforming technique. But for both studies, the Mach
number is low and no discussion about the parameters affecting the apparent
time from the source to the microphone can be computed. Since a decade
now, the Amiet's method (or other similar methods [9]) are conventionally
associated with the beamforming technique [10]. They are used as shear
layer correction models improving the sound source localisation, and are
powerful in the sense that they allow to predict the shear flow effects based
on little information about the flow, i.e., the mean flow velocity and thickness.
However few studies deal with a quantitative assessment of the efficiency of
such corrections.

Recently, the beamforming technique associated to the Amiet’s correc-
tions has been likened to an in-flow calibration proposed by Kroeber et
al. [11]. Both experimental methods showed that the methods accurately
localise the source position for low Mach number flows (less than 0.15). The
shear flow correction has also been studied by Bahr et al. [12] by using a
plasma point source in an anechoic wind-tunnel. Preliminary tests in an
empty test section at Mach number M=0.17 showed an apparent shift of
the source position in the downstream direction if no shear flow corrections
are made. The shear layer corrections are used for different flow configu-
rations (the sound source being located behind an airfoil trailing-edge or a
nose gear). In spite of the complexity of these flow configurations, the overall
trends given by the Amiet’s method are correct.

These two studies state that the Amiet’s method properly corrects the
results given by the beamforming technique. But for both studies, the Mach
number is low and no discussion about the parameters affecting the apparent
position shift is presented. In this paper, the position shift due to the shear
flow effect is investigated by using a numerical simulation of the beamforming
localisation. We consider the parameters affecting the source shift and the
reliability of the shear flow correction for different flow configurations in terms
of Mach number, shear layer thickness, mean flow thickness, etc...

Additionally, this paper considers the errors carried out by the beamform-
ing technique for estimating the acoustic source power. The model developed
by Schlinker and Amiet is then considered [8, 13], modeling the flow effects
on the amplitude of the wave measured by the microphones. The efficiency
of such a model to avoid source power estimation errors is then assessed
by using the same numerical methodology than for the study of localization
errors, and considering different Mach numbers.

It is worth noting that the diffusion of acoustic waves by turbulence is
not addressed by the correction models that are considered in this study.
Before reaching the microphones outside the flow, sound waves pass through
the shear layer where interactions with the turbulent structures result in
a redistribution of energy among different wave numbers, or in directional
broadening of the same wave number. Phase and amplitude fluctuations are
stronger when the frequency or the flow velocity increase. This phenomenon
has been the subject of experimental and numerical studies (e.g., [14] and
[15]). For the tone frequencies under test in this study (less than 10 kHz), the
rate of acoustic diffusion is supposed to be moderate enough to concentrate
the study on refraction and convection effects.
position shift is presented. In this paper, the position shift due to the shear
flow effect is investigated by using a numerical simulation of the beamforming
localisation. We consider the parameters affecting the source shift and the
reliability of the shear flow correction for different flow configurations in terms
of Mach number, shear layer thickness, mean flow thickness, etc...

Additionally, this paper considers the errors carried out by the beamform-
ing technique for estimating the acoustic source power. The model developed
by Schlinker and Amiet is then considered [8, 13], modeling the flow effects
on the amplitude of the wave measured by the microphones. The efficiency
of such a model to avoid source power estimation errors is then assessed
by using the same numerical methodology than for the study of localization
errors, and considering different Mach numbers.

It is worth noting that the diffusion of acoustic waves by turbulence is
not addressed by the correction models that are considered in this study.
Before reaching the microphones outside the flow, sound waves pass through
the shear layer where interactions with the turbulent structures result in
a redistribution of energy among different wave numbers, or in directional
broadening of the same wave number. Phase and amplitude fluctuations are
stronger when the frequency or the flow velocity increase. This phenomenon
has been the subject of experimental and numerical studies (e.g., [14] and
[15]). For the tone frequencies under test in this study (less than 10 kHz), the
rate of acoustic diffusion is supposed to be moderate enough to concentrate
the study on refraction and convection effects.
The remainder of the paper is then organized as follows. In section 2, the beamforming technique is briefly presented and several flow correction methods are presented: one is a simple analytical formulation of the source apparent shift, and the other ones are correction methods using acoustic rays, allowing the modeling of both the effective source-receiver propagation time and pressure amplitude. Then, an experimental set-up in an anechoic wind-tunnel is presented and the numerical code, based on the linearised Euler equations (LEE) used to simulate the acoustic propagation in the wind-tunnel shear flow, is described (section 3). The beamforming results obtained from experimental and numerical data are discussed and compared. Section 4 is devoted to numerical results. The flow parameters affecting the source shift and the source power estimation are investigated and the efficiency of the shear flow corrections is studied. Finally, some conclusions are given in section 5.

## 2. Beamforming algorithm and shear flow correction

### 2.1. *Principle of beamforming*

Beamforming is a technique used to process microphone array data in order to find the direction of incident acoustic waves [16]. The origin of a coherent sound propagating across the array is located by using a position-search for the best multichannel constructive interference, once the signals from each array element have been delayed for the theoretical source-to-receiver travel time. At short source-receiver distances, spherical waves reach the
The remainder of the paper is then organized as follows. In section 2, the beamforming technique is briefly presented and several flow correction methods are presented: one is a simple analytical formulation of the source apparent shift, and the other ones are correction methods using acoustic rays, allowing the modeling of both the effective source-receiver propagation time and pressure amplitude. Then, an experimental set-up in an anechoic wind-tunnel is presented and the numerical code, based on the linearised Euler equations (LEE) used to simulate the acoustic propagation in the wind-tunnel shear flow, is described (section 3). The beamforming results obtained from experimental and numerical data are discussed and compared. Section 4 is devoted to numerical results. The flow parameters affecting the source shift and the source power estimation are investigated and the efficiency of the shear flow corrections is studied. Finally, some conclusions are given in section 5.

## 2. Beamforming algorithm and shear flow correction

### 2.1. *Principle of beamforming*

Beamforming is a technique used to process microphone array data in order to find the direction of incident acoustic waves [16]. The origin of a coherent sound propagating across the array is located by using a position-search for the best multichannel constructive interference, once the signals from each array element have been delayed for the theoretical source-to-receiver travel time. At short source-receiver distances, spherical waves reach the
microphones. As the sound speed $c_0$ is constant, the travel time only depends on the source-microphone distance. Therefore, a phase-shift difference occurs between the microphones and this phase-shift depends on the source position. The basic idea of the Delay-And-Sum Beamforming (DASB) technique is then to compensate the time-delay between the signals received by the microphones in order to localise the source.

Therefore, we assume a virtual source at position $\mathbf{r}_F$. For all microphones, the travel time of the acoustic waves is calculated from this focus position. Then the microphone signals are delayed by this travel time by using a phase shift in the frequency domain and summed. The array output in the frequency domain $Y(\mathbf{r}_F, f)$ can be written under the following matrix formulation

$$Y(\mathbf{r}_F, f) = \mathbf{e}^* \mathbf{w} \mathbf{P}, \quad (1)$$

where $\mathbf{e}$ is the so called steering vector that steers the array in a direction corresponding to the assumed source position $\mathbf{r}_F$, $\mathbf{w}$ is a weighting function (usually used to compensate the geometrical decreasing of the acoustic waves) and $\mathbf{P}$ is the Fourier transform of the microphone signals (the $(\cdot)^*$ symbol represents the conjugate transposed matrix or vector). The array output power is given by

$$Z(\mathbf{r}_F, f) = \mathbf{e}^* \mathbf{w} \mathbf{R} \mathbf{w}^* \mathbf{e}, \quad (2)$$

where $\mathbf{R} = \mathbf{P}\mathbf{P}^*$ is the cross-spectral matrix. Eq. (2) is computed to localise the source position. In the following, a standard DASB result is presented.
microphones. As the sound speed $c_0$ is constant, the travel time only depends on the source-microphone distance. Therefore, a phase-shift difference occurs between the microphones and this phase-shift depends on the source position. The basic idea of the Delay-And-Sum Beamforming (DASB) technique is then to compensate the time-delay between the signals received by the microphones in order to localise the source.

Therefore, we assume a virtual source at position $\mathbf{r}_F$. For all microphones, the travel time of the acoustic waves is calculated from this focus position. Then the microphone signals are delayed by this travel time by using a phase shift in the frequency domain and summed. The array output in the frequency domain $Y(\mathbf{r}_F, f)$ can be written under the following matrix formulation

$$Y(\mathbf{r}_F, f) = \mathbf{e}^* \mathbf{w} \mathbf{P}, \qquad (1)$$

where $\mathbf{e}$ is the so called steering vector that steers the array in a direction corresponding to the assumed source position $\mathbf{r}_F$, $\mathbf{w}$ is a weighting function (usually used to compensate the geometrical decreasing of the acoustic waves) and $\mathbf{P}$ is the Fourier transform of the microphone signals (the $(.)^*$ symbol represents the conjugate transposed matrix or vector). The array output power is given by

$$Z(\mathbf{r}_F, f) = \mathbf{e}^* \mathbf{w} \mathbf{R} \mathbf{w}^* \mathbf{e}, \qquad (2)$$

where $\mathbf{R} = \mathbf{P}\mathbf{P}^*$ is the cross-spectral matrix. Eq. (2) is computed to localise the source position. In the following, a standard DASB result is presented.
We assume a monopolar sound source, located at a distance $H = 1$ m from the array center, radiating spherical waves at frequency $f = 10$ kHz. The acoustic pressure is measured by a 31 uniform linear microphone array. The microphone spacing is constant and equal to $d = 0.017$ m. In order to apply the DASB processing, we assume a focal point moving along an axis parallel to the array, and distant from 1 m to the array. The variable $x$ represents the supposed source position on this axis, and the position $x = 0$, is located in front of the array central microphone. For every position of the virtual source the array output power is calculated according to Eq. (2). The obtained result is displayed on Figure 1 and is called array pattern or beampattern. The array output power exhibits a main lobe and side lobes. The $x$-position of the main lobe gives the source position estimated with the DASB technique ($x = 0$ in the present case).

During an experiment in an anechoic wind-tunnel the acoustic waves propagate in a shear flow undergoing convection and refraction phenomena and substantial localisation errors can be made if the flow effects are not taken properly into account. Two kinds of flow corrections are presented in this paper. The first one is an *a posteriori* correction, based on a coarse modeling of the flow effect on the waves phase (section 2.2). The second type of correction takes into account the flow effect on propagation in the DASB processing by using a modified expression of the steering vector **e** (section 2.3.1). In the same manner, flow effects involve a modification of the observed pressure amplitude, and flow corrections should be used in order to
We assume a monopolar sound source, located at a distance $H = 1$ m from the array center, radiating spherical waves at frequency $f = 10$ kHz. The acoustic pressure is measured by a 31 uniform linear microphone array. The microphone spacing is constant and equal to $d = 0.017$ m. In order to apply the DASB processing, we assume a focal point moving along an axis parallel to the array, and distant from 1 m to the array. The variable $x$ represents the supposed source position on this axis, and the position $x = 0$, is located in front of the array central microphone. For every position of the virtual source the array output power is calculated according to Eq. (2). The obtained result is displayed on Figure 1 and is called array pattern or beampattern. The array output power exhibits a main lobe and side lobes. The $x$-position of the main lobe gives the source position estimated with the DASB technique ($x = 0$ in the present case).

During an experiment in an anechoic wind-tunnel the acoustic waves propagate in a shear flow undergoing convection and refraction phenomena and substantial localisation errors can be made if the flow effects are not taken properly into account. Two kinds of flow corrections are presented in this paper. The first one is an *a posteriori* correction, based on a coarse modeling of the flow effect on the waves phase (section 2.2). The second type of correction takes into account the flow effect on propagation in the DASB processing by using a modified expression of the steering vector **e** (section 2.3.1). In the same manner, flow effects involve a modification of the observed pressure amplitude, and flow corrections should be used in order to
estimate correctly the source power; such corrections are presented in section 2.3.2.

## 2.2. Model using the convected wave equation

Due to the convection of waves, the source-microphone travel times in a flow are modified and the phases of the microphones signals are different from what they would be in a medium at rest. Consequently, the source position estimated by the DASB technique is incorrect if a propagation model that does not take the flow into account is used, and this position is shifted into the downstream direction [12].

An expression of this apparent source shift can be obtained by solving the convected wave equation for pressure $p'$. The propagation of waves in a uniform flow at Mach number $M$ orientated according to $x$ can be written in a cartesian coordinate system $(x, y, z)$, with a monopolar source term in the right-hand side

$$ (1 - M^2) \frac{\partial^2 p'}{\partial x^2} + \frac{\partial^2 p'}{\partial y^2} + \frac{\partial^2 p'}{\partial z^2} - \frac{1}{c_0^2} \frac{\partial^2 p'}{\partial t^2} - 2 \frac{M}{c_0} \frac{\partial^2 p'}{\partial x \partial t} = - \frac{\partial q}{\partial t}. \quad (3) $$

One way to solve Eq. (3) is proposed by Wu et al. in reference [17] by using the velocity potential and a Prandtl-Glauert transformation. Haddad points out that the amplitude and the phase of the acoustic pressure are changed by the mean flow [18]. The phase term $\phi$ of the solution of Eq. (3) can be expressed as
estimate correctly the source power; such corrections are presented in section 2.3.2.

## 2.2. Model using the convected wave equation

Due to the convection of waves, the source-microphone travel times in a flow are modified and the phases of the microphones signals are different from what they would be in a medium at rest. Consequently, the source position estimated by the DASB technique is incorrect if a propagation model that does not take the flow into account is used, and this position is shifted into the downstream direction [12].

An expression of this apparent source shift can be obtained by solving the convected wave equation for pressure $p'$. The propagation of waves in a uniform flow at Mach number $M$ orientated according to $x$ can be written in a cartesian coordinate system $(x, y, z)$, with a monopolar source term in the right-hand side

$$ (1 - M^2) \frac{\partial^2 p'}{\partial x^2} + \frac{\partial^2 p'}{\partial y^2} + \frac{\partial^2 p'}{\partial z^2} - \frac{1}{c_0^2} \frac{\partial^2 p'}{\partial t^2} - 2 \frac{M}{c_0} \frac{\partial^2 p'}{\partial x \partial t} = - \frac{\partial q}{\partial t}. \quad (3) $$

One way to solve Eq. (3) is proposed by Wu et al. in reference [17] by using the velocity potential and a Prandtl-Glauert transformation. Haddad points out that the amplitude and the phase of the acoustic pressure are changed by the mean flow [18]. The phase term $\phi$ of the solution of Eq. (3) can be expressed as
$$ \phi = k_1 \sqrt{(x_1^2 + y^2 + z^2)} - k_1 M x_1, \qquad (4) $$

with $k_1 = k/\sqrt{1-M^2}$ and $x_1 = x/\sqrt{1-M^2}$, the wave number and the streamwise coordinate respectively, after applying the Prandtl-Glauert transformation. Eq. (4) can be rewritten as

$$ \left(x - \frac{\phi M}{k}\right)^2 + y^2 + z^2 = \frac{\phi^2}{k^2}. \qquad (5) $$

The equi-phase surfaces are then given by Eq. (5) that exhibits the equation of a sphere with center $(\frac{\phi M}{k}, 0, 0)$ [18]. So, the origin of the spherical equi-phase surfaces, which would be seen as the origin of the waves if they were propagating in free field, is shifted in the downstream direction. Now, by using Eq. (4) together with the expressions of $k_1$ and $x_1$, the sphere center $x_{sh} = \phi M/k$ reduces to the simple expression:

$$ x_{sh} = \sqrt{M^2(y^2 + z^2)}. \qquad (6) $$

The wave propagation in the plane $(x, y)$ (i.e., for $z=0$) is now considered. Eq. (6) then becomes $x_{sh} = My$. The interpretation of this last expression is that when the spherical wavefront reaches a point at height $y$, its center is located at $My$. Now, instead of considering a uniform flow, let us consider a flow that is bounded at $y = h$, so that the propagation medium is at rest for $y > h$. When the spherical wavefront reaches the shear layer at $y = h$
$$ \phi = k_1 \sqrt{(x_1^2 + y^2 + z^2)} - k_1 M x_1, \qquad (4) $$

with $k_1 = k/\sqrt{1-M^2}$ and $x_1 = x/\sqrt{1-M^2}$, the wave number and the streamwise coordinate respectively, after applying the Prandtl-Glauert transformation. Eq. (4) can be rewritten as

$$ \left(x - \frac{\phi M}{k}\right)^2 + y^2 + z^2 = \frac{\phi^2}{k^2}. \qquad (5) $$

The equi-phase surfaces are then given by Eq. (5) that exhibits the equation of a sphere with center $(\frac{\phi M}{k}, 0, 0)$ [18]. So, the origin of the spherical equi-phase surfaces, which would be seen as the origin of the waves if they were propagating in free field, is shifted in the downstream direction. Now, by using Eq. (4) together with the expressions of $k_1$ and $x_1$, the sphere center $x_{sh} = \phi M/k$ reduces to the simple expression:

$$ x_{sh} = \sqrt{M^2(y^2 + z^2)}. \qquad (6) $$

The wave propagation in the plane $(x, y)$ (i.e., for $z=0$) is now considered. Eq. (6) then becomes $x_{sh} = My$. The interpretation of this last expression is that when the spherical wavefront reaches a point at height $y$, its center is located at $My$. Now, instead of considering a uniform flow, let us consider a flow that is bounded at $y = h$, so that the propagation medium is at rest for $y > h$. When the spherical wavefront reaches the shear layer at $y = h$
(Fig. 2), the sphere center is located at:

$$x_{sh} = Mh. \qquad (7)$$

After passing through the shear layer, the spherical wavefront continues its propagation in the quiescent medium. If we neglect the refraction effect at the shear layer, we can then consider that the wavefront will propagate as if its origin was located at $x_{sh} = Mh$.

Let us consider now a microphone array measuring the sound field outside the flow, as in typical anechoic wind tunnel measurements. If the DASB processing is applied together with a quiescent flow propagation model, the source position estimated should then be located at a distance $Mh$ downstream the real source position. The relation $x_{sh} = Mh$ can then be considered as a straightforward relation to estimate the apparent source shift, and can be used to correct the indication of the source position *a posteriori*. In this case, $h$ is the distance between the source and the shear layer in the direction perpendicular to the flow; this quantity will be called mean flow thickness in the following.

## 2.3. Shear flow correction based on ray theory

### 2.3.1. Propagation time correction

As shown in the previous section, a wave-equation based approach can be used to assess the source shift. However, this approach can not be im-
(Fig. 2), the sphere center is located at:

$$x_{sh} = Mh. \qquad (7)$$

After passing through the shear layer, the spherical wavefront continues its propagation in the quiescent medium. If we neglect the refraction effect at the shear layer, we can then consider that the wavefront will propagate as if its origin was located at $x_{sh} = Mh$.

Let us consider now a microphone array measuring the sound field outside the flow, as in typical anechoic wind tunnel measurements. If the DASB processing is applied together with a quiescent flow propagation model, the source position estimated should then be located at a distance $Mh$ downstream the real source position. The relation $x_{sh} = Mh$ can then be considered as a straightforward relation to estimate the apparent source shift, and can be used to correct the indication of the source position *a posteriori*. In this case, $h$ is the distance between the source and the shear layer in the direction perpendicular to the flow; this quantity will be called mean flow thickness in the following.

## 2.3. Shear flow correction based on ray theory

### 2.3.1. Propagation time correction

As shown in the previous section, a wave-equation based approach can be used to assess the source shift. However, this approach can not be im-
plemented as a modification of the phase of each microphone in the steering
vector of the DASB data processing. In order to correct the DASB, several
other models have been suggested in the literature. Most of them are based
on the ray theory of sound, like the Amiet's theory [6–8, 10].

These models suppose an infinitely thin shear layer separating the flow
and the quiescent air. Amiet considers the path between the source, lo-
cated inside the flow, and a microphone, located outside the flow region
(Fig. 3). Three unknown angles corresponding to three directions are con-
sidered, as the ray direction in absence of flow (θ), the convected ray di-
rection (θc), the refracted ray direction (θ0), while the geometrical direction
source-microphone (θm) is known. Geometrical considerations and the Snell-
Descartes law give a set of three equations that can be solved by an iterative
process to yield the three unknowns angles. The effective traveling distance
r1 + r2 from the source to the microphone can then be calculated, leading
to a propagation time taking into account the shear flow effect [10]. How-
ever, Wooley et al. [19] have shown that, for some values of angle and Mach
number, limits of the model are reached in the downstream and upstream
directions.

In this study, the correction model used for calculating the effective prop-
agation time between the source and a given microphone is the one developed
by Koop et al. [9]. This ray-based model gives results that are almost identi-
cal to the Amiet's model, but does not suffer from the limit angles mentioned
above. The aim is to define a new travel time for the steering vector of Eq. (2),
plemented as a modification of the phase of each microphone in the steering
vector of the DASB data processing. In order to correct the DASB, several
other models have been suggested in the literature. Most of them are based
on the ray theory of sound, like the Amiet's theory [6–8, 10].

These models suppose an infinitely thin shear layer separating the flow
and the quiescent air. Amiet considers the path between the source, lo-
cated inside the flow, and a microphone, located outside the flow region
(Fig. 3). Three unknown angles corresponding to three directions are con-
sidered, as the ray direction in absence of flow (θ), the convected ray di-
rection (θc), the refracted ray direction (θ0), while the geometrical direction
source-microphone (θm) is known. Geometrical considerations and the Snell-
Descartes law give a set of three equations that can be solved by an iterative
process to yield the three unknowns angles. The effective traveling distance
r1 + r2 from the source to the microphone can then be calculated, leading
to a propagation time taking into account the shear flow effect [10]. How-
ever, Wooley et al. [19] have shown that, for some values of angle and Mach
number, limits of the model are reached in the downstream and upstream
directions.

In this study, the correction model used for calculating the effective prop-
agation time between the source and a given microphone is the one developed
by Koop et al. [9]. This ray-based model gives results that are almost identi-
cal to the Amiet's model, but does not suffer from the limit angles mentioned
above. The aim is to define a new travel time for the steering vector of Eq. (2),
where the convection and refraction effects are accounted for.

Fig. 4 illustrates the situation where an acoustic source at $(x_s, y_s)$ is radiating in an air-flow. The model assumes that the flow direction is $x$ and that the velocity profile is uniform between $y = 0$ and $y = y_0$. The shear layer is supposed infinitely thin and located at $y = y_0$. The microphone is located in the quiescent air at coordinates $(x_m, y_m)$. The acoustic source emits waves that propagate in the uniform flow. When these waves reach the shear layer, new waves are generated from these points according to Huygens principle. Then these new waves reach the microphone located in the quiescent air. The travel time from the source to the shear layer and from the shear layer to the microphone are denoted $t_f$ and $t_a$ (for time in flow and time in air) respectively, and given by [9]

$$t_f = \frac{-U_0(x_s - x_0)}{(c_0^2 - U_0^2)} \pm \sqrt{\frac{U_0^2(x_s - x_0)^2 - (c_0^2 - U_0^2)((y_s - y_0)^2 - (x_s - x_0)^2)}{(c_0^2 - U_0^2)^2}}, \quad (8)$$

$$t_a = \frac{1}{c_0} \sqrt{(x_m - x_0)^2 + (y_m - y_0)^2}, \quad (9)$$

where $U_0$ is the flow velocity. The minimisation with respect to the position at the shear layer $x_0$ of the sum $\tau = t_f + t_a$ gives the travel time from the source to the microphone. So the flow effects can be taken into account by introducing this new travel time in the steering vector of Eq. (2) for every microphone.
where the convection and refraction effects are accounted for.

Fig. 4 illustrates the situation where an acoustic source at $(x_s, y_s)$ is radiating in an air-flow. The model assumes that the flow direction is $x$ and that the velocity profile is uniform between $y = 0$ and $y = y_0$. The shear layer is supposed infinitely thin and located at $y = y_0$. The microphone is located in the quiescent air at coordinates $(x_m, y_m)$. The acoustic source emits waves that propagate in the uniform flow. When these waves reach the shear layer, new waves are generated from these points according to Huygens principle. Then these new waves reach the microphone located in the quiescent air. The travel time from the source to the shear layer and from the shear layer to the microphone are denoted $t_f$ and $t_a$ (for time in flow and time in air) respectively, and given by [9]

$$t_f = \frac{-U_0(x_s - x_0)}{(c_0^2 - U_0^2)} \pm \sqrt{\frac{U_0^2(x_s - x_0)^2 - (c_0^2 - U_0^2)((y_s - y_0)^2 - (x_s - x_0)^2)}{(c_0^2 - U_0^2)^2}}, \quad (8)$$

$$t_a = \frac{1}{c_0} \sqrt{(x_m - x_0)^2 + (y_m - y_0)^2}, \quad (9)$$

where $U_0$ is the flow velocity. The minimisation with respect to the position at the shear layer $x_0$ of the sum $\tau = t_f + t_a$ gives the travel time from the source to the microphone. So the flow effects can be taken into account by introducing this new travel time in the steering vector of Eq. (2) for every microphone.
### 2.3.2. Pressure amplitude correction

Based on the physical configuration of Fig. 3, Schlinker and Amiet developed a model for predicting the effect of shear layer on the amplitude of the wave received by the microphone [7, 8, 13]. The theory derives a model for the ray tube divergence behaviour above the shear layer. By combining the obtained model with an expression of the acoustic transmission coefficient of the shear layer, an expression of the ratio of the pressure amplitude $p_c$ that would be generated by the source without flow over the measured pressure amplitude $p_m$ is found to be:

$$ \frac{p_c}{p_m} = \frac{1}{2} \alpha_1 \sqrt{(\alpha_2 \alpha_3)(1 + M^2 \xi^2)} \left( \frac{\xi}{\sin(\theta_0)} + (1 + M \cos(\theta_0))^2 \right), \quad (10) $$

with

$$ \xi = \sqrt{(1 + M \cos(\theta_0))^2 - \cos^2(\theta_0)}. \quad (11) $$

In Eq. (10), $\alpha_1$, $\alpha_2$ and $\alpha_3$ are geometrical factors depending on the source-microphone distance $R_m$, the distance $h$ from the source to the shear layer, the angles $\theta_m$ and $\theta_0$, and $\xi$ (see Fig. 3). Following [13], they are given by

$$ \begin{align} \alpha_1 &= \frac{\sin(\theta_m)}{\sin(\theta_0)} \nonumber \\ \alpha_2 &= \frac{h}{R_m \sin(\theta_m)} \left[ \left( \frac{\sin(\theta_0)}{\xi} \right)^3 - 1 \right] + 1 \tag{12} \\ \alpha_3 &= \frac{h}{R_m \sin(\theta_m)} \left[ \left( \frac{\sin(\theta_0)}{\xi} \right) - 1 \right] + 1 \nonumber \end{align} $$
### 2.3.2. Pressure amplitude correction

Based on the physical configuration of Fig. 3, Schlinker and Amiet developed a model for predicting the effect of shear layer on the amplitude of the wave received by the microphone [7, 8, 13]. The theory derives a model for the ray tube divergence behaviour above the shear layer. By combining the obtained model with an expression of the acoustic transmission coefficient of the shear layer, an expression of the ratio of the pressure amplitude $p_c$ that would be generated by the source without flow over the measured pressure amplitude $p_m$ is found to be:

$$ \frac{p_c}{p_m} = \frac{1}{2} \alpha_1 \sqrt{(\alpha_2 \alpha_3)(1 + M^2 \xi^2)} \left( \frac{\xi}{\sin(\theta_0)} + (1 + M \cos(\theta_0))^2 \right), \quad (10) $$

with

$$ \xi = \sqrt{(1 + M \cos(\theta_0))^2 - \cos^2(\theta_0)}. \quad (11) $$

In Eq. (10), $\alpha_1$, $\alpha_2$ and $\alpha_3$ are geometrical factors depending on the source-microphone distance $R_m$, the distance $h$ from the source to the shear layer, the angles $\theta_m$ and $\theta_0$, and $\xi$ (see Fig. 3). Following [13], they are given by

$$ \begin{align} \alpha_1 &= \frac{\sin(\theta_m)}{\sin(\theta_0)} \nonumber \\ \alpha_2 &= \frac{h}{R_m \sin(\theta_m)} \left[ \left( \frac{\sin(\theta_0)}{\xi} \right)^3 - 1 \right] + 1 \tag{12} \\ \alpha_3 &= \frac{h}{R_m \sin(\theta_m)} \left[ \left( \frac{\sin(\theta_0)}{\xi} \right) - 1 \right] + 1 \nonumber \end{align} $$
Eq. (10) is valid in the case where the shear layer thickness is of the same order of magnitude than the sound wavelength. By using Eq. (10), a correction of the measured pressure amplitude can be used in order to assess the sound power of the acoustic source.

### 3. Localisation of an acoustic source in a shear flow based on experimental and numerical data

#### 3.1. *Experimental set-up*

An experimental device has been implemented so as to check the reliability of the beamforming technique to localise an acoustic source in a flow and to check the ability of the LEE solver to simulate the DASB source localisation. The experiments have been carried out in an anechoic Eiffel-type wind-tunnel. The dimensions of the anechoic room are (2.8×3.8×3) m³. The free field condition is obtained for acoustic frequencies higher than 250 Hz. The nozzle exit has a square cross-section of (0.46 × 0.46) m². The test section is open and its length is 1.7 m. The flow velocity is set to $U_0 = 37$ m/s which corresponds to a Mach number $M = 0.11$.

A diagram of the experimental set-up is displayed on Fig. 5. The exit of the nozzle is connected with a plywood plate to the collector. The acoustic source is a Visaton FR 13 WP loudspeaker flush-mounted in the plywood plate in order to generate a monopolar source. It is located at 0.60 m downstream the nozzle exit. The loudspeaker operating frequency range is [70–18000] Hz and the sensitivity is 85 dB at 1 m for a 1 W electrical input.
Eq. (10) is valid in the case where the shear layer thickness is of the same order of magnitude than the sound wavelength. By using Eq. (10), a correction of the measured pressure amplitude can be used in order to assess the sound power of the acoustic source.

### 3. Localisation of an acoustic source in a shear flow based on experimental and numerical data

#### 3.1. *Experimental set-up*

An experimental device has been implemented so as to check the reliability of the beamforming technique to localise an acoustic source in a flow and to check the ability of the LEE solver to simulate the DASB source localisation. The experiments have been carried out in an anechoic Eiffel-type wind-tunnel. The dimensions of the anechoic room are (2.8×3.8×3) m³. The free field condition is obtained for acoustic frequencies higher than 250 Hz. The nozzle exit has a square cross-section of (0.46 × 0.46) m². The test section is open and its length is 1.7 m. The flow velocity is set to $U_0 = 37$ m/s which corresponds to a Mach number $M = 0.11$.

A diagram of the experimental set-up is displayed on Fig. 5. The exit of the nozzle is connected with a plywood plate to the collector. The acoustic source is a Visaton FR 13 WP loudspeaker flush-mounted in the plywood plate in order to generate a monopolar source. It is located at 0.60 m downstream the nozzle exit. The loudspeaker operating frequency range is [70–18000] Hz and the sensitivity is 85 dB at 1 m for a 1 W electrical input.
In this work, the input signal is a pure-tone signal at frequency $f = 10$ kHz.

The acoustic pressure is recorded outside the flow by a uniform linear
microphone array. We use 31 Brüel&Kjaer phased microphones, model 4957.
The microphone spacing is fixed to $d = 0.017$ m, that corresponds to half-
a-wavelength at $f = 10$ kHz. The source-array distance is $H = 1.36$ m.
The array is fixed on a three-dimensional moving system in order to locate
carefully the array center with respect to the loudspeaker location. The array
is aligned in the streamwise direction and can be moved in the upstream
or downstream. In the following, $x$ and $y$ denote the streamwise and the
vertical directions, respectively. The acquisition system is a 32 synchronised
lines system ETEP. The time data are sampled at 100 kHz.

Hot-wire measurements are performed to characterise the flow field. Four-
teen vertical velocity profiles have been carried out along the streamwise di-
rection. Each vertical profile is composed of 81 measurement points over
0.8 m. This flow characterisation defines the mean flow field used in the
numerical solver (see section 3.2).

## 3.2. Numerical solver based on linearised Euler Equations (LEE)

Numerical simulation is now a powerful tool for noise predictions. Thereby,
numerical sound propagation has been widely studied in computational aeroa-
coustics [20, 21]. The ability of the LEE to reproduce accurately the be-
haviour of sound propagation through a shear mean flow is well established
and efficient solvers have been developed in numerous studies [22–24]. The
In this work, the input signal is a pure-tone signal at frequency $f = 10$ kHz.

The acoustic pressure is recorded outside the flow by a uniform linear
microphone array. We use 31 Brüel&Kjaer phased microphones, model 4957.
The microphone spacing is fixed to $d = 0.017$ m, that corresponds to half-
a-wavelength at $f = 10$ kHz. The source-array distance is $H = 1.36$ m.
The array is fixed on a three-dimensional moving system in order to locate
carefully the array center with respect to the loudspeaker location. The array
is aligned in the streamwise direction and can be moved in the upstream
or downstream. In the following, $x$ and $y$ denote the streamwise and the
vertical directions, respectively. The acquisition system is a 32 synchronised
lines system ETEP. The time data are sampled at 100 kHz.

Hot-wire measurements are performed to characterise the flow field. Four-
teen vertical velocity profiles have been carried out along the streamwise di-
rection. Each vertical profile is composed of 81 measurement points over
0.8 m. This flow characterisation defines the mean flow field used in the
numerical solver (see section 3.2).

## 3.2. Numerical solver based on linearised Euler Equations (LEE)

Numerical simulation is now a powerful tool for noise predictions. Thereby,
numerical sound propagation has been widely studied in computational aeroa-
coustics [20, 21]. The ability of the LEE to reproduce accurately the be-
haviour of sound propagation through a shear mean flow is well established
and efficient solvers have been developed in numerous studies [22–24]. The
benefits of the numerical simulation is the possibility of considering any mean velocity profiles in terms of shear layer thickness or Mach number.

The LEE are derived from the full Euler equations [24] (the equation for conservation of mass, momentum and energy). The density, the two velocity components (in the coordinate system of Fig. 5) and the pressure ($\rho, u, v, p$) are split into mean and fluctuating quantities, denoted respectively by the subscript (.)₀ and the symbol (.)'. Finally, the LEE can be written as a first order linear system in the bidimensional-case with a source term $\mathbf{S}$ in the right hand side as follows

$$ \frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{E}}{\partial x} + \frac{\partial \mathbf{F}}{\partial y} + \mathbf{H} = \mathbf{S}, \qquad (13) $$

where $\mathbf{U}$ is the unknown vector, $\mathbf{E}$ and $\mathbf{F}$ are the two-dimensional flux vectors and $\mathbf{H}$ a vector containing terms related to the gradients of the mean flow. The expressions of these vectors are

$$ \mathbf{U} = \begin{pmatrix} \rho' \\ \rho_0 u' \\ \rho_0 v' \\ p' \end{pmatrix}, \quad \mathbf{E} = \begin{pmatrix} \rho_0 u' + \rho' u_0 \\ \rho_0 u' u_0 + p' \\ \rho_0 u_0 v' \\ p' u_0 + \gamma \rho_0 u' \end{pmatrix}, \quad \mathbf{F} = \begin{pmatrix} \rho_0 v' + \rho' v_0 \\ \rho_0 u' v_0 \\ \rho_0 v_0 v' + p' \\ p' v_0 + \gamma \rho_0 v' \end{pmatrix}, $$
benefits of the numerical simulation is the possibility of considering any mean velocity profiles in terms of shear layer thickness or Mach number.

The LEE are derived from the full Euler equations [24] (the equation for conservation of mass, momentum and energy). The density, the two velocity components (in the coordinate system of Fig. 5) and the pressure ($\rho, u, v, p$) are split into mean and fluctuating quantities, denoted respectively by the subscript (.)₀ and the symbol (.)'. Finally, the LEE can be written as a first order linear system in the bidimensional-case with a source term $\mathbf{S}$ in the right hand side as follows

$$ \frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{E}}{\partial x} + \frac{\partial \mathbf{F}}{\partial y} + \mathbf{H} = \mathbf{S}, \qquad (13) $$

where $\mathbf{U}$ is the unknown vector, $\mathbf{E}$ and $\mathbf{F}$ are the two-dimensional flux vectors and $\mathbf{H}$ a vector containing terms related to the gradients of the mean flow. The expressions of these vectors are

$$ \mathbf{U} = \begin{pmatrix} \rho' \\ \rho_0 u' \\ \rho_0 v' \\ p' \end{pmatrix}, \quad \mathbf{E} = \begin{pmatrix} \rho_0 u' + \rho' u_0 \\ \rho_0 u' u_0 + p' \\ \rho_0 u_0 v' \\ p' u_0 + \gamma \rho_0 u' \end{pmatrix}, \quad \mathbf{F} = \begin{pmatrix} \rho_0 v' + \rho' v_0 \\ \rho_0 u' v_0 \\ \rho_0 v_0 v' + p' \\ p' v_0 + \gamma \rho_0 v' \end{pmatrix}, $$
$$ \mathbf{H} = \begin{pmatrix} 0 \\ (\rho_0 u' + \rho' u_0) \frac{\partial u_0}{\partial x} + (\rho_0 v' + \rho' v_0) \frac{\partial u_0}{\partial y} \\ (\rho_0 u' + \rho' u_0) \frac{\partial v_0}{\partial x} + (\rho_0 v' + \rho' v_0) \frac{\partial v_0}{\partial y} \\ (\gamma - 1) p' \left( \frac{\partial u_0}{\partial x} + \frac{\partial v_0}{\partial y} \right) - (\gamma - 1) \left( u' \frac{\partial p_0}{\partial x} + v' \frac{\partial p_0}{\partial y} \right) \end{pmatrix}, \quad (14) $$

where $\gamma$ is the ratio of specific heat at constant pressure and volume. In this work, the spatial discretization is based on a Dispersion Relation Preserving scheme (DRP) [21] which needs at least six points per wavelength to ensure an accurate solution. The time integration is performed with a 4th order Runge-Kutta scheme. The non-reflective boundary conditions proposed by Tam et al. in [21] are used to outgoing the disturbances at the limits of the calculation domain.

The mesh size $\Delta x$ and $\Delta y$ are chosen as $\Delta x = \Delta y = \lambda/n$ with $n$ the number of point per wavelength $\lambda$, here $n = 10$. In order to match the experimental data, the numerical dimensions are expressed in the physical domain. In order to share the geometric similarity and the dynamic similarity, the reference length in the computational domain is taken as a tenth of the wavelength at $f = 10$ kHz. The characteristic dimensions of the experimental study and the corresponding dimensions in the simulation are provided in Table 1.

The numerical domain is slightly oversized compared to the size of the experimental device in order to avoid boundary effects during the simulation. A regular mesh of 491 × 491 points allows to place the source and the array
$$ \mathbf{H} = \begin{pmatrix} 0 \\ (\rho_0 u' + \rho' u_0) \frac{\partial u_0}{\partial x} + (\rho_0 v' + \rho' v_0) \frac{\partial u_0}{\partial y} \\ (\rho_0 u' + \rho' u_0) \frac{\partial v_0}{\partial x} + (\rho_0 v' + \rho' v_0) \frac{\partial v_0}{\partial y} \\ (\gamma - 1) p' \left( \frac{\partial u_0}{\partial x} + \frac{\partial v_0}{\partial y} \right) - (\gamma - 1) \left( u' \frac{\partial p_0}{\partial x} + v' \frac{\partial p_0}{\partial y} \right) \end{pmatrix}, \quad (14) $$

where $\gamma$ is the ratio of specific heat at constant pressure and volume. In this work, the spatial discretization is based on a Dispersion Relation Preserving scheme (DRP) [21] which needs at least six points per wavelength to ensure an accurate solution. The time integration is performed with a 4th order Runge-Kutta scheme. The non-reflective boundary conditions proposed by Tam et al. in [21] are used to outgoing the disturbances at the limits of the calculation domain.

The mesh size $\Delta x$ and $\Delta y$ are chosen as $\Delta x = \Delta y = \lambda/n$ with $n$ the number of point per wavelength $\lambda$, here $n = 10$. In order to match the experimental data, the numerical dimensions are expressed in the physical domain. In order to share the geometric similarity and the dynamic similarity, the reference length in the computational domain is taken as a tenth of the wavelength at $f = 10$ kHz. The characteristic dimensions of the experimental study and the corresponding dimensions in the simulation are provided in Table 1.

The numerical domain is slightly oversized compared to the size of the experimental device in order to avoid boundary effects during the simulation. A regular mesh of 491 × 491 points allows to place the source and the array
<table><caption>Table 1: Experimental dimensions versus numerical dimensions.</caption><thead><tr><td></td><td>Experimental (cm)</td><td>Numerical (dimensionless)</td></tr></thead><tbody><tr><td>Wavelength λ</td><td>3.4</td><td>10</td></tr><tr><td>Microphone spacing (d = λ/2)</td><td>1.7</td><td>5</td></tr><tr><td>Source-array distance H</td><td>136</td><td>400</td></tr><tr><td>Flow thickness h</td><td>46</td><td>135</td></tr><tr><td>Array Aperture L</td><td>51</td><td>150</td></tr></tbody></table>

away from the boundary of the grid. The CFL number, $CFL = \frac{\Delta t}{\Delta x}c_0(1+M)$, is fixed to 1 in all the following simulations. The velocity of sound in air is also fixed to $c_0 = 1$. The time step is then deduced from the CFL in the following simulations and computation is performed over 1200 time steps.

The mean velocity profiles can be obtained by using experimental measurements or by using an analytical expression. Hot-wire measurements profiles are interpolated on the computational grid. The corresponding mean flow velocity field is displayed on Fig. 6. The origin of the coordinates system corresponds to the sound source location. The y-origin is moved away from the boundary of the computational domain to avoid disturbances. From $y/λ = 0$ up to $y/λ ≈ 13.5$ (ie., $y ≈ h = 0.46$ m), the velocity of the mean flow is constant ($≈ 40$ m/s i.e. $M = 0.11$). Then the thickening shear layer is clearly visible. Finally, there is no flow for $y/λ > 17.6$. Of course, before being introduced in the LEE calculation, this velocity field is sized according to the Mach number.

An analytical expression can be used to define the mean velocity profile that will make possible to control the flow parameters in section 4. Candel et
<table><caption>Table 1: Experimental dimensions versus numerical dimensions.</caption><thead><tr><td></td><td>Experimental (cm)</td><td>Numerical (dimensionless)</td></tr></thead><tbody><tr><td>Wavelength λ</td><td>3.4</td><td>10</td></tr><tr><td>Microphone spacing (d = λ/2)</td><td>1.7</td><td>5</td></tr><tr><td>Source-array distance H</td><td>136</td><td>400</td></tr><tr><td>Flow thickness h</td><td>46</td><td>135</td></tr><tr><td>Array Aperture L</td><td>51</td><td>150</td></tr></tbody></table>

away from the boundary of the grid. The CFL number, $CFL = \frac{\Delta t}{\Delta x}c_0(1+M)$, is fixed to 1 in all the following simulations. The velocity of sound in air is also fixed to $c_0 = 1$. The time step is then deduced from the CFL in the following simulations and computation is performed over 1200 time steps.

The mean velocity profiles can be obtained by using experimental measurements or by using an analytical expression. Hot-wire measurements profiles are interpolated on the computational grid. The corresponding mean flow velocity field is displayed on Fig. 6. The origin of the coordinates system corresponds to the sound source location. The y-origin is moved away from the boundary of the computational domain to avoid disturbances. From $y/λ = 0$ up to $y/λ ≈ 13.5$ (ie., $y ≈ h = 0.46$ m), the velocity of the mean flow is constant ($≈ 40$ m/s i.e. $M = 0.11$). Then the thickening shear layer is clearly visible. Finally, there is no flow for $y/λ > 17.6$. Of course, before being introduced in the LEE calculation, this velocity field is sized according to the Mach number.

An analytical expression can be used to define the mean velocity profile that will make possible to control the flow parameters in section 4. Candel et
al. [25] propose an expression of the shear layer which is written as

$$u_0(y) = \frac{U_0}{2} (1 - \tanh(\eta)), \qquad (15)$$

where $\eta = \frac{2(y - y_c)}{\delta}$ is the reduced coordinate, $y_c$ is the inflexion point and $\delta$ the shear layer thickness. This expression allows to investigate the influence of the shear layer thickness, the uniform flow thickness and the mean velocity. In the computational domain the boundary layer is not simulated, we assume that its effects on the sound source shift is negligible compared to the ones of the others parameters.

### 3.3. Validation of the simulation method

The first step of this study is to validate the numerical simulation of the sound source localisation process by using the DASB technique. In this purpose, the experimental and numerical results are compared.

In the experimental case, the sound source is located in a shear flow as described in section 3.1, and generates a monopolar harmonic sound field at frequency 10 kHz. The microphone array is located outside the flow. Different microphone array positions are studied. The microphone array is moved 0.3 m into the upstream and downstream directions by steps of 0.05 m. An extreme position at 0.4 m in the downstream direction is also studied. Beyond this position, the collector of the wind tunnel disturbs the acoustic measurement.

In the numerical case, the acoustic source is a harmonic monopole (see
al. [25] propose an expression of the shear layer which is written as

$$u_0(y) = \frac{U_0}{2} (1 - \tanh(\eta)), \qquad (15)$$

where $\eta = \frac{2(y - y_c)}{\delta}$ is the reduced coordinate, $y_c$ is the inflexion point and $\delta$ the shear layer thickness. This expression allows to investigate the influence of the shear layer thickness, the uniform flow thickness and the mean velocity. In the computational domain the boundary layer is not simulated, we assume that its effects on the sound source shift is negligible compared to the ones of the others parameters.

### 3.3. Validation of the simulation method

The first step of this study is to validate the numerical simulation of the sound source localisation process by using the DASB technique. In this purpose, the experimental and numerical results are compared.

In the experimental case, the sound source is located in a shear flow as described in section 3.1, and generates a monopolar harmonic sound field at frequency 10 kHz. The microphone array is located outside the flow. Different microphone array positions are studied. The microphone array is moved 0.3 m into the upstream and downstream directions by steps of 0.05 m. An extreme position at 0.4 m in the downstream direction is also studied. Beyond this position, the collector of the wind tunnel disturbs the acoustic measurement.

In the numerical case, the acoustic source is a harmonic monopole (see
reference [24] for the expression) located at the origin of the computational grid. The mean flow introduced in the LEE results from the interpolation of the measured velocity profiles on the computational grid described in Fig. 6. In the same way as the experimental measurements, the microphone array is moved into the upstream and downstream directions. The microphone array is moved 0.3 m in the upstream direction and 0.7 m in the downstream direction by steps of 0.1 m. Therefore, the numerical extreme position downstream is almost twice the experimental one, and the efficiency of the shear layer corrections will then be assessed in the case of high value of the array steering angle. This angle is approximately equal to 30° from broadside, that corresponds to a limit for the array scanning range, leading to a resolution worsening of about 50% [26].

For all cases, the acoustic signals are Fourier-transformed and the DASB technique is applied. The cross-spectral matrix $\mathbf{R}$ of Eq. (2) is calculated from the experimental microphone data by performing a statistical mean with 70 block of 4096 samples; conversely, no statistical mean is carried out for the calculation of $\mathbf{R}$ from the simulated data, as there is no noise corruption and the number of simulated time steps is only 1200. The scanning area is extended from $x = -0.5$ m to $x = 0.5$ m by steps of 0.005 m for the experimental data and from $x = -0.5$ m to $x = 0.8$ m by steps of 0.005 m for the numerical data. For all cases, the position of the peak value of the beampattern ($x_{DASB}$) is searched and plotted in relation to the array center positions ($x_{array}$).
reference [24] for the expression) located at the origin of the computational grid. The mean flow introduced in the LEE results from the interpolation of the measured velocity profiles on the computational grid described in Fig. 6. In the same way as the experimental measurements, the microphone array is moved into the upstream and downstream directions. The microphone array is moved 0.3 m in the upstream direction and 0.7 m in the downstream direction by steps of 0.1 m. Therefore, the numerical extreme position downstream is almost twice the experimental one, and the efficiency of the shear layer corrections will then be assessed in the case of high value of the array steering angle. This angle is approximately equal to 30° from broadside, that corresponds to a limit for the array scanning range, leading to a resolution worsening of about 50% [26].

For all cases, the acoustic signals are Fourier-transformed and the DASB technique is applied. The cross-spectral matrix $\mathbf{R}$ of Eq. (2) is calculated from the experimental microphone data by performing a statistical mean with 70 block of 4096 samples; conversely, no statistical mean is carried out for the calculation of $\mathbf{R}$ from the simulated data, as there is no noise corruption and the number of simulated time steps is only 1200. The scanning area is extended from $x = -0.5$ m to $x = 0.5$ m by steps of 0.005 m for the experimental data and from $x = -0.5$ m to $x = 0.8$ m by steps of 0.005 m for the numerical data. For all cases, the position of the peak value of the beampattern ($x_{DASB}$) is searched and plotted in relation to the array center positions ($x_{array}$).
The results are shown graphically in Fig. 7 concerning the source position estimation from both experimental and numerical data. When the array center is located at a given position $x_{array}$, the beampattern should exhibit a maximum at $x_{DASB} = -x_{array}$, if the estimation is correct. As a consequence, the real source position is depicted by the plain line of equation $x_{DASB} = -x_{array}$ on Fig. 7. However, it is noticed that the estimated source positions (experimental and numerical) depict a line that is parallel and above the plain line: this means that the apparent position of the source is shifted downstream the real position, and that this shift seems to be nearly independent from the array position: its averaged value is equal to $1.59\lambda$ (i.e, about 5.1 cm in the present case). It is then concluded that the beamforming algorithm leads to a systematical localisation error if the flow effects are not taken into account in the propagation model.

The apparent source shift as predicted by the coarse estimation of Eq. (7) is equal to $Mh \simeq 1.5\lambda$, which provides a reasonable estimation of the localisation error. It is then shown that in the present case study, the product of the Mach number by the mean flow thickness is a good estimation that provides a satisfactory correction of the apparent source shift.

The other way of improving the beamforming localisation method is to use the flow propagation model as presented in section 2.3.1 in the DASB algorithm. The beampatterns are then computed by using this model, and the graph of Fig 8 depicts the obtained results. All source position estimations are now located on the line of equation $x_{DASB} = -x_{array}$. It is then con-
The results are shown graphically in Fig. 7 concerning the source position estimation from both experimental and numerical data. When the array center is located at a given position $x_{array}$, the beampattern should exhibit a maximum at $x_{DASB} = -x_{array}$, if the estimation is correct. As a consequence, the real source position is depicted by the plain line of equation $x_{DASB} = -x_{array}$ on Fig. 7. However, it is noticed that the estimated source positions (experimental and numerical) depict a line that is parallel and above the plain line: this means that the apparent position of the source is shifted downstream the real position, and that this shift seems to be nearly independent from the array position: its averaged value is equal to $1.59\lambda$ (i.e, about 5.1 cm in the present case). It is then concluded that the beamforming algorithm leads to a systematical localisation error if the flow effects are not taken into account in the propagation model.

The apparent source shift as predicted by the coarse estimation of Eq. (7) is equal to $Mh \simeq 1.5\lambda$, which provides a reasonable estimation of the localisation error. It is then shown that in the present case study, the product of the Mach number by the mean flow thickness is a good estimation that provides a satisfactory correction of the apparent source shift.

The other way of improving the beamforming localisation method is to use the flow propagation model as presented in section 2.3.1 in the DASB algorithm. The beampatterns are then computed by using this model, and the graph of Fig 8 depicts the obtained results. All source position estimations are now located on the line of equation $x_{DASB} = -x_{array}$. It is then con-
cluded that the beamforming algorithm combined to a shear-flow correction based on a flow propagation model retrieves correctly the source position in the flow, even for array positions involving a high value of the array steering angle.

The results of Fig. 7 and Fig. 8 show that the simulation of the experiment by using the two-dimensional LEE-based solver reproduces perfectly the wind-tunnel experimental results. The solver can then confidently used in order to investigate the efficiency of the DASB technique associated to flow corrections, in the case of a wide range of the parameters describing the experimental configurations: Mach number, shear layer thickness, mean flow thickness (as controlled by Eq. (15)).

# 4. Numerical study of flow correction robustness

In this section, the efficiency of the DASB corrections for estimating the position and amplitude of the source in the presence of a shear flow are investigated by means of the numerical tool validated in the preceding section. Section 4.1 consider corrections for source position: they are based on the models presented in Sections 2.2 and 2.3.1. Then Section 4.2 considers the correction for the amplitude, based on the model presented in Section 2.3.2.

## 4.1. Investigation of the position correction

### 4.1.1. Influence of the Shear Layer Thickness (SLT)

In this section the influence of the SLT on the sound source localisation with the beamforming technique is studied. The SLT is controlled by the
cluded that the beamforming algorithm combined to a shear-flow correction
based on a flow propagation model retrieves correctly the source position in
the flow, even for array positions involving a high value of the array steering
angle.

The results of Fig. 7 and Fig. 8 show that the simulation of the exper-
iment by using the two-dimensional LEE-based solver reproduces perfectly
the wind-tunnel experimental results. The solver can then confidently used
in order to investigate the efficiency of the DASB technique associated to
flow corrections, in the case of a wide range of the parameters describing the
experimental configurations: Mach number, shear layer thickness, mean flow
thickness (as controlled by Eq. (15)).

4. Numerical study of flow correction robustness

In this section, the efficiency of the DASB corrections for estimating the
position and amplitude of the source in the presence of a shear flow are in-
vestigated by means of the numerical tool validated in the preceding section.
Section 4.1 consider corrections for source position: they are based on the
models presented in Sections 2.2 and 2.3.1. Then Section 4.2 considers the
correction for the amplitude, based on the model presented in Section 2.3.2.

4.1. *Investigation of the position correction*

4.1.1. *Influence of the Shear Layer Thickness (SLT)*

In this section the influence of the SLT on the sound source localisation
with the beamforming technique is studied. The SLT is controlled by the
parameter $\delta$ of Eq. (15). A thin and a thick shear layer are simulated and their thickness are $\delta = 0.5\lambda$ and $\delta = 8\lambda$, respectively. The SLT is constant in the streamwise direction in order to take into account only the thickness influence. The inflexion point position $y_c$ is similar for both simulations which involves that the mean flow thickness is the same in both cases. The boundary layer is replaced by a uniform flow. The Mach number is set to $M = 0.11$. The mean velocity profiles are displayed on Fig. 9.

The acoustic source parameters and positions are the same than in section 3.3. The estimated DASB positions in relation to the array center positions are plotted on Fig. 10. For the two SLT values, the estimated DASB positions are moved into the downstream direction and the source shift is the same. It is then concluded that the source shift does not depend on the shear layer thickness. The use of a flow propagation model that is not shear layer thickness dependent is then reasonable and its application retrieves the real source position (Fig. 8).

However, the above results show that, when using the flow propagation model described in section 2.3.1, the position of the theoretically infinitely thin shear layer must be located at the inflexion point of the real velocity profile. A practical consequence is that the shear layer thickening phenomenon, typical of open wind-tunnel mean flow profiles (see eg., the data of Fig. 6), does not need to be taken account in the shear layer model, provided that the position of the shear layer inflexion point is properly identified.
parameter $\delta$ of Eq. (15). A thin and a thick shear layer are simulated and their thickness are $\delta = 0.5\lambda$ and $\delta = 8\lambda$, respectively. The SLT is constant in the streamwise direction in order to take into account only the thickness influence. The inflexion point position $y_c$ is similar for both simulations which involves that the mean flow thickness is the same in both cases. The boundary layer is replaced by a uniform flow. The Mach number is set to $M = 0.11$. The mean velocity profiles are displayed on Fig. 9.

The acoustic source parameters and positions are the same than in section 3.3. The estimated DASB positions in relation to the array center positions are plotted on Fig. 10. For the two SLT values, the estimated DASB positions are moved into the downstream direction and the source shift is the same. It is then concluded that the source shift does not depend on the shear layer thickness. The use of a flow propagation model that is not shear layer thickness dependent is then reasonable and its application retrieves the real source position (Fig. 8).

However, the above results show that, when using the flow propagation model described in section 2.3.1, the position of the theoretically infinitely thin shear layer must be located at the inflexion point of the real velocity profile. A practical consequence is that the shear layer thickening phenomenon, typical of open wind-tunnel mean flow profiles (see eg., the data of Fig. 6), does not need to be taken account in the shear layer model, provided that the position of the shear layer inflexion point is properly identified.
### 4.1.2. Influence of the sound wavelength

The shear flow correction presented in section 2.3.1 is independent to the wavelength. In this section, we study the position shift given by the beamforming technique in relation to the wavelength. The new wavelength values are $λ_1 = 2λ$ and $λ_2 = 4λ$. The source position is not changed. As no flow parameters are changed, the mean flow field described in Fig. 6 is used.

The beamforming technique is applied for the different microphone array positions. The estimated DASB positions (without flow corrections) are plotted in relation to the array center on Fig. 11. For the two wavelengths, the apparent source positions are shifted into the downstream direction, but the position shift is the same for the two wavelengths. So, the wavelength is not a predominant parameter on the position shift and it can be neglected for the flow effect correction.

### 4.1.3. Influence of the mean flow thickness

The shear flow correction takes into account the mean flow thickness. The influence of this parameter is studied in this section. The shear layer thickness is set to $δ = 2λ$ in this simulation. The mean flow thickness values are $h = 6.76λ$ and $h = 27.06λ$, that correspond to half and twice the experimental mean flow thickness. The mean velocity fields are displayed on Fig. 12.

The source parameters and positions have been described in section 3.3. The beamforming technique is applied for the different microphone array po-
### 4.1.2. Influence of the sound wavelength

The shear flow correction presented in section 2.3.1 is independent to the wavelength. In this section, we study the position shift given by the beamforming technique in relation to the wavelength. The new wavelength values are $λ_1 = 2λ$ and $λ_2 = 4λ$. The source position is not changed. As no flow parameters are changed, the mean flow field described in Fig. 6 is used.

The beamforming technique is applied for the different microphone array positions. The estimated DASB positions (without flow corrections) are plotted in relation to the array center on Fig. 11. For the two wavelengths, the apparent source positions are shifted into the downstream direction, but the position shift is the same for the two wavelengths. So, the wavelength is not a predominant parameter on the position shift and it can be neglected for the flow effect correction.

### 4.1.3. Influence of the mean flow thickness

The shear flow correction takes into account the mean flow thickness. The influence of this parameter is studied in this section. The shear layer thickness is set to $δ = 2λ$ in this simulation. The mean flow thickness values are $h = 6.76λ$ and $h = 27.06λ$, that correspond to half and twice the experimental mean flow thickness. The mean velocity fields are displayed on Fig. 12.

The source parameters and positions have been described in section 3.3. The beamforming technique is applied for the different microphone array po-
sitions. The estimated DASB positions are plotted in relation to the array
center on Fig. 13. For the two mean flow thickness values, the source posi-
tions are shifted into the downstream direction, this shift being independent
from the array position. But the smallest mean flow thickness presents an
overall source shift equal to 0.91λ, whereas for the highest mean flow thick-
ness the overall source shift is 3.24λ. So, the mean flow thickness is the
most relevant parameter for the shear flow corrections. By applying the *a
posteriori* correction of section 2.2 (Eq. (7)), we find for the small and the
high mean flow thickness respectively, 0.74λ and 3.00λ that is in fairly good
agreement with the numerical results. So, this estimation is a powerful tool
to estimate the source shift, at least for low Mach numbers.

However, this method does not correct the DASB beampattern. Flow
effect corrections can be taken into account following the model described in
section 2.3.1. The results given by the beamforming technique associated to
shear flow corrections are presented on Fig. 14. For the two mean flow thick-
ness values and all source positions the beamforming technique estimates
very accurately the real source positions. It is concluded that the beamform-
ing technique associated to shear flow corrections is very robust in terms of
sound source localisation for a wide range of mean flow thickness values.

4.1.4. *Influence of the Mach number*

Finally the influence of the Mach number is investigated. In this case the
mean velocity field has a mean flow thickness *h* = 13.53λ and a shear layer
sitions. The estimated DASB positions are plotted in relation to the array
center on Fig. 13. For the two mean flow thickness values, the source posi-
tions are shifted into the downstream direction, this shift being independent
from the array position. But the smallest mean flow thickness presents an
overall source shift equal to 0.91λ, whereas for the highest mean flow thick-
ness the overall source shift is 3.24λ. So, the mean flow thickness is the
most relevant parameter for the shear flow corrections. By applying the *a
posteriori* correction of section 2.2 (Eq. (7)), we find for the small and the
high mean flow thickness respectively, 0.74λ and 3.00λ that is in fairly good
agreement with the numerical results. So, this estimation is a powerful tool
to estimate the source shift, at least for low Mach numbers.

However, this method does not correct the DASB beampattern. Flow
effect corrections can be taken into account following the model described in
section 2.3.1. The results given by the beamforming technique associated to
shear flow corrections are presented on Fig. 14. For the two mean flow thick-
ness values and all source positions the beamforming technique estimates
very accurately the real source positions. It is concluded that the beamform-
ing technique associated to shear flow corrections is very robust in terms of
sound source localisation for a wide range of mean flow thickness values.

4.1.4. *Influence of the Mach number*

Finally the influence of the Mach number is investigated. In this case the
mean velocity field has a mean flow thickness *h* = 13.53λ and a shear layer
thickness $\delta = 2\lambda$. The acoustic and geometrical parameters are unchanged (Table 1) and the source position is set to $(x, y) = (0,0)$. For all Mach number values, the estimated DASB position is calculated with and without the flow correction model of section 2.3.1. The *a posteriori* source shift estimate of Eq. (7) is also calculated. The results are plotted on Fig. 15. If the DASB estimation is correct, the values of $x_{DASB}$ should be equal to zero whatever the Mach number.

A linear increase of the apparent source shift (the solid line on Fig. 15) with Mach number is observed, as expected from Eq. (7). However, the results of Fig. 15 show that the rate of increase of the apparent source shift with Mach number is underestimated by Eq. (7). This *a posteriori* correction is found to provide a good correction up to Mach numbers of 0.4. Beyond this value, the error becomes significant. Conversely the flow propagation model associated to DASB technique leads to a good estimate of the source position whatever the Mach number.

Comparatively, the lower robustness of the *a posteriori* correction can find an explanation by considering the convection-based model (section 2.2) in terms of ray theory. Indeed, this model provides the spatial shift suffered by a ray launched in a direction normal to the flow direction, and subjected to deflection by convection during its path through the mean flow thickness. This spatial shift is considered at the ray exit of the flow region. However, the main drawback of this model is to ignore the position of the receiver. Conversely, the Huygens principle-based model associated to beamforming
thickness $\delta = 2\lambda$. The acoustic and geometrical parameters are unchanged (Table 1) and the source position is set to $(x, y) = (0,0)$. For all Mach number values, the estimated DASB position is calculated with and without the flow correction model of section 2.3.1. The *a posteriori* source shift estimate of Eq. (7) is also calculated. The results are plotted on Fig. 15. If the DASB estimation is correct, the values of $x_{DASB}$ should be equal to zero whatever the Mach number.

A linear increase of the apparent source shift (the solid line on Fig. 15) with Mach number is observed, as expected from Eq. (7). However, the results of Fig. 15 show that the rate of increase of the apparent source shift with Mach number is underestimated by Eq. (7). This *a posteriori* correction is found to provide a good correction up to Mach numbers of 0.4. Beyond this value, the error becomes significant. Conversely the flow propagation model associated to DASB technique leads to a good estimate of the source position whatever the Mach number.

Comparatively, the lower robustness of the *a posteriori* correction can find an explanation by considering the convection-based model (section 2.2) in terms of ray theory. Indeed, this model provides the spatial shift suffered by a ray launched in a direction normal to the flow direction, and subjected to deflection by convection during its path through the mean flow thickness. This spatial shift is considered at the ray exit of the flow region. However, the main drawback of this model is to ignore the position of the receiver. Conversely, the Huygens principle-based model associated to beamforming
is a more sophisticated implementation, but it is more efficient. The reason
is that it considers the relative positions of source and receivers. The time
required by a ray to reach the limit of the flow now depends on the receivers
positions and on the Mach number. Following the behavior of sound energy
propagating in two media with different speeds, it optimizes the path so as
to obtain a minimum source-receiver traveling time, which explains its better
performances when the Mach number increases.

4.2. Investigation of the amplitude correction

In Section 2.3.2, it was pointed out that the propagation of the waves
through the shear flow modifies their amplitude. The pressure amplitude
measured at a given microphone position is then theoretically governed by
Eq. (10), where $p_m$ is the measured amplitude at a given Mach number
and $p_c$ the amplitude that would be measured without flow. In this section,
some numerical simulations are carried out in order to estimate the validity
of this analytical modeling, and to evaluate the consequence of shear flow
effects on the estimation of the source power by using the DASB technique.
The experimental configuration considered for this calculation is the same
as in Section 4.1.4: the shear layer thickness of the mean velocity field is
$\delta = 2\lambda$, the acoustic and geometrical parameters are unchanged (Table 1)
and the source position is set to $(x, y) = (0, 0)$. The considered positions of
the microphones are located along the array (located at the dimensionless
distance $H=400$ from the source).
is a more sophisticated implementation, but it is more efficient. The reason
is that it considers the relative positions of source and receivers. The time
required by a ray to reach the limit of the flow now depends on the receivers
positions and on the Mach number. Following the behavior of sound energy
propagating in two media with different speeds, it optimizes the path so as
to obtain a minimum source-receiver traveling time, which explains its better
performances when the Mach number increases.

4.2. Investigation of the amplitude correction

In Section 2.3.2, it was pointed out that the propagation of the waves
through the shear flow modifies their amplitude. The pressure amplitude
measured at a given microphone position is then theoretically governed by
Eq. (10), where $p_m$ is the measured amplitude at a given Mach number
and $p_c$ the amplitude that would be measured without flow. In this section,
some numerical simulations are carried out in order to estimate the validity
of this analytical modeling, and to evaluate the consequence of shear flow
effects on the estimation of the source power by using the DASB technique.
The experimental configuration considered for this calculation is the same
as in Section 4.1.4: the shear layer thickness of the mean velocity field is
$\delta = 2\lambda$, the acoustic and geometrical parameters are unchanged (Table 1)
and the source position is set to $(x, y) = (0, 0)$. The considered positions of
the microphones are located along the array (located at the dimensionless
distance $H=400$ from the source).
The ratio $\frac{p_c}{p_m}$ has been calculated by using Eq. (10) for a wide range of Mach number values. The position of the microphone along the array is indicated by using the variable $\theta_m$ (defined in Fig. 3): when $\theta_m = 90^\circ$, the microphone is located at $x = 0$; a value of $\theta_m$ of less than $90^\circ$ indicates that the microphone is located upstream the source, while the microphone is located downstream the source for $\theta_m < 90^\circ$. The distribution of the theoretical value of $\frac{p_c}{p_m}$ as a function of $M$ and $\theta_m$ is displayed in Fig. 16a. For Mach numbers values lower than 0.3, the flow effects on the pressure amplitude appear as negligible, whereas they are increasingly significant for higher Mach numbers, resulting in an attenuation of the pressure amplitude measured by the microphone. This effect is more pronounced for microphones located upstream the source.

In order to validate this theoretical formulation, the $\frac{p_c}{p_m}$ has been calculated for the same configuration and Mach number values, by using the simulation method. The results are presented in Fig. 16b. The agreement between the analytical and numerical results is found to be very satisfactory. It is then concluded that the available model for pressure amplitude correction is very robust, over a wide range of subsonic Mach numbers.

The consequence of this flow effect on the DASB results must now be assessed; given the fact that flow effects involve some pressure attenuation, an under-estimation of the power of the source in the flow is expected if no amplitude correction is carried out. The simulation data were then used to collect the pressure virtually measured by the microphones, and to apply
The ratio $\frac{p_c}{p_m}$ has been calculated by using Eq. (10) for a wide range of Mach number values. The position of the microphone along the array is indicated by using the variable $\theta_m$ (defined in Fig. 3): when $\theta_m = 90^\circ$, the microphone is located at $x = 0$; a value of $\theta_m$ of less than $90^\circ$ indicates that the microphone is located upstream the source, while the microphone is located downstream the source for $\theta_m < 90^\circ$. The distribution of the theoretical value of $\frac{p_c}{p_m}$ as a function of $M$ and $\theta_m$ is displayed in Fig. 16a. For Mach numbers values lower than 0.3, the flow effects on the pressure amplitude appear as negligible, whereas they are increasingly significant for higher Mach numbers, resulting in an attenuation of the pressure amplitude measured by the microphone. This effect is more pronounced for microphones located upstream the source.

In order to validate this theoretical formulation, the $\frac{p_c}{p_m}$ has been calculated for the same configuration and Mach number values, by using the simulation method. The results are presented in Fig. 16b. The agreement between the analytical and numerical results is found to be very satisfactory. It is then concluded that the available model for pressure amplitude correction is very robust, over a wide range of subsonic Mach numbers.

The consequence of this flow effect on the DASB results must now be assessed; given the fact that flow effects involve some pressure attenuation, an under-estimation of the power of the source in the flow is expected if no amplitude correction is carried out. The simulation data were then used to collect the pressure virtually measured by the microphones, and to apply
the DASB technique. For this calculation, the flow effect on the source-
microphone propagation time was taken into account by using the flow prop-
agation model described in section 2.3.1. The DASB output of Eq. (2) was
then used in order to estimate the source power at the emission frequency.
This source power estimation was carried out in two situations: by using no
amplitude correction, and then by using the analytical amplitude correction
of Eq. (10) for each microphone.

The results of such an analysis are displayed in Fig. 17. The estimated
power of the source is normalized by its real value (ie., the one that would be
found in a quiescent medium at *M*=0). As expected, if no amplitude correc-
tion is carried out, the source power is under-estimated (for Mach numbers
higher than 0.2), and this under-estimation increases with increasing Mach
number (reaching -60% at Mach number 0.9). Meanwhile, when applying the
analytical amplitude correction, the source power is satisfactorily estimated
(with an error of less than 10%) up to Mach number values of 0.7. Beyond,
the under-estimation increases, reaching about -20%. It is then concluded
that an amplitude correction for the collected sound pressure is necessary and
efficient for a wide range of practical situations, in order to avoir significant
source power under-estimation.

**5. Conclusions**

The present study has concerned the sound source localisation in an ane-
choic wind-tunnel by using the beamforming technique. The beamforming
the DASB technique. For this calculation, the flow effect on the source-
microphone propagation time was taken into account by using the flow prop-
agation model described in section 2.3.1. The DASB output of Eq. (2) was
then used in order to estimate the source power at the emission frequency.
This source power estimation was carried out in two situations: by using no
amplitude correction, and then by using the analytical amplitude correction
of Eq. (10) for each microphone.

The results of such an analysis are displayed in Fig. 17. The estimated
power of the source is normalized by its real value (ie., the one that would be
found in a quiescent medium at *M*=0). As expected, if no amplitude correc-
tion is carried out, the source power is under-estimated (for Mach numbers
higher than 0.2), and this under-estimation increases with increasing Mach
number (reaching -60% at Mach number 0.9). Meanwhile, when applying the
analytical amplitude correction, the source power is satisfactorily estimated
(with an error of less than 10%) up to Mach number values of 0.7. Beyond,
the under-estimation increases, reaching about -20%. It is then concluded
that an amplitude correction for the collected sound pressure is necessary and
efficient for a wide range of practical situations, in order to avoir significant
source power under-estimation.

**5. Conclusions**

The present study has concerned the sound source localisation in an ane-
choic wind-tunnel by using the beamforming technique. The beamforming
data processing usually lies on a model of sound propagation in a medium at rest, but in the context of anechoic wind-tunnel applications, the microphone array is located outside the flow and the source in the flow, so that the sound waves undergo convection and refraction, due to mean and shear flow effects, respectively. Some ray-based models are classically used to include those effects in the beamforming processing, and the object of the present study has been to validate these models for a wide range of flow parameters, by using a numerical simulation of the microphone array measurement in an anechoic wind-tunnel.

After a presentation of the principle of beamforming processing, a simple model based on the convected wave equation is presented, showing that if the flow effects are not taken into account, the sound source is localized downstream its real position; this apparent source shift can be estimated as proportional to the Mach number and the mean flow thickness, which should then be the main parameters having an influence on the wave propagation. Other models based on ray-theory, currently used for applications of beamforming to aeroacoustics, are also presented, allowing to describe the effects of the shear flow on the amplitude and phase of the microphone signals.

The numerical simulation used in this study is based on a program solving the linearised Euler Equations. A two-dimensional version is used to simulate the wind-tunnel measurements. The results are compared with experimental results where a loudspeaker is flush-mounted in a plate bounding the flow, and emitting sound measured by a linear uniform microphone array located
data processing usually lies on a model of sound propagation in a medium at rest, but in the context of anechoic wind-tunnel applications, the microphone array is located outside the flow and the source in the flow, so that the sound waves undergo convection and refraction, due to mean and shear flow effects, respectively. Some ray-based models are classically used to include those effects in the beamforming processing, and the object of the present study has been to validate these models for a wide range of flow parameters, by using a numerical simulation of the microphone array measurement in an anechoic wind-tunnel.

After a presentation of the principle of beamforming processing, a simple model based on the convected wave equation is presented, showing that if the flow effects are not taken into account, the sound source is localized downstream its real position; this apparent source shift can be estimated as proportional to the Mach number and the mean flow thickness, which should then be the main parameters having an influence on the wave propagation. Other models based on ray-theory, currently used for applications of beamforming to aeroacoustics, are also presented, allowing to describe the effects of the shear flow on the amplitude and phase of the microphone signals.

The numerical simulation used in this study is based on a program solving the linearised Euler Equations. A two-dimensional version is used to simulate the wind-tunnel measurements. The results are compared with experimental results where a loudspeaker is flush-mounted in a plate bounding the flow, and emitting sound measured by a linear uniform microphone array located
in the quiescent air. The experimental and numerical data show a very good agreement, which validate the simulation procedure. The apparent source shift is well observed, and in the considered case at Mach number 0.11, the two correction methods are shown to perform very well.

The correction results are further investigated by using numerical sim-
ulations, for a wider range of flow configurations and array positions. The
flow correction models are independent of the shear layer thickness and the
wavelength. Some sensitivity of the flow effects to the ratio of the shear layer
thickness to the wavelength could be expected, so that two different shear
layer thicknesses and two wavelengths are simulated to compute the acoustic
propagation in a 0.11 Mach number flow. Then, the beamforming technique
is applied without shear flow correction. We show that the source shift is not
dependent of these two parameters, so that the shear flow corrections models
are able to retrieve the correct source position, provided that the inflection
point of the shear flow profile is properly identified.

Then, the influence of the mean flow thickness on beamforming results
without shear flow correction is studied. The apparent source shift appears
to be highly dependent to the shear flow thickness, and approximately pro-
portional to the product of the mean flow thickness by the Mach number.
The beamforming technique associated to the shear flow correction localises
accurately the source position for the range of mean flow thickness under
study.

Lastly, the influence of the Mach number is investigated. The beam-
in the quiescent air. The experimental and numerical data show a very good agreement, which validate the simulation procedure. The apparent source shift is well observed, and in the considered case at Mach number 0.11, the two correction methods are shown to perform very well.

The correction results are further investigated by using numerical sim-
ulations, for a wider range of flow configurations and array positions. The
flow correction models are independent of the shear layer thickness and the
wavelength. Some sensitivity of the flow effects to the ratio of the shear layer
thickness to the wavelength could be expected, so that two different shear
layer thicknesses and two wavelengths are simulated to compute the acoustic
propagation in a 0.11 Mach number flow. Then, the beamforming technique
is applied without shear flow correction. We show that the source shift is not
dependent of these two parameters, so that the shear flow corrections models
are able to retrieve the correct source position, provided that the inflection
point of the shear flow profile is properly identified.

Then, the influence of the mean flow thickness on beamforming results
without shear flow correction is studied. The apparent source shift appears
to be highly dependent to the shear flow thickness, and approximately pro-
portional to the product of the mean flow thickness by the Mach number.
The beamforming technique associated to the shear flow correction localises
accurately the source position for the range of mean flow thickness under
study.

Lastly, the influence of the Mach number is investigated. The beam-
forming technique associated to the shear flow correction permits to localise
the source position up to a Mach number of 0.9. It is then confirmed that
the beamforming method, associated to a ray-based shear flow correction,
is a very robust technique to localise the source position in a flow, even at
high subsonic Mach numbers. It is also confirmed that Mach number and
mean flow thickness are the two main parameters having an influence on the
beamforming localisation results, and that their product gives a fairly good
estimation of the apparent source shift, up to Mach number 0.4. Beyond this
Mach number value, this correction should not be applied. In addition, for
subsonic Mach numbers higher than 0.2, it is demonstrated that a correction
model for simulating the flow effect on the amplitude of the acoustic waves
during their propagation is necessary and efficient in order to avoid some
significant under-estimation of the sound source power.

Acknowledgments

The authors warmly acknowledged David Marx for his invaluable help con-
cerning the EEL numerical solver, and Pascal Biais, Janick Laumonier, Lau-
rent Philippon and Jean-Christophe Vergez for their technical support.

[1] Michel U. History of acoustic beamforming. 1st Berlin Beamforming Conference 2006.

[2] Poisson F, Valiere J-C and Herzog P. High speed sound sources local-
ization using bilinear time-frequency transformation. Applied Acoustics
1998;53(1-3):1-13.
forming technique associated to the shear flow correction permits to localise
the source position up to a Mach number of 0.9. It is then confirmed that
the beamforming method, associated to a ray-based shear flow correction,
is a very robust technique to localise the source position in a flow, even at
high subsonic Mach numbers. It is also confirmed that Mach number and
mean flow thickness are the two main parameters having an influence on the
beamforming localisation results, and that their product gives a fairly good
estimation of the apparent source shift, up to Mach number 0.4. Beyond this
Mach number value, this correction should not be applied. In addition, for
subsonic Mach numbers higher than 0.2, it is demonstrated that a correction
model for simulating the flow effect on the amplitude of the acoustic waves
during their propagation is necessary and efficient in order to avoid some
significant under-estimation of the sound source power.

Acknowledgments

The authors warmly acknowledged David Marx for his invaluable help con-
cerning the EEL numerical solver, and Pascal Biais, Janick Laumonier, Lau-
rent Philippon and Jean-Christophe Vergez for their technical support.

[1] Michel U. History of acoustic beamforming. 1st Berlin Beamforming Conference 2006.

[2] Poisson F, Valiere J-C and Herzog P. High speed sound sources local-
ization using bilinear time-frequency transformation. Applied Acoustics
1998;53(1-3):1-13.
[3] Guérin S, Michel U. Aero-engine noise investigated from flight tests. 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, Massachusetts (USA), May 2006.

[4] Padois T, Prax C, Valeau V, Marx D. Experimental localization of an acoustic sound source in a wind-tunnel flow by using a numerical time-reversal technique. J. Acoust. Soc. Am. 2012;132(4):2397-2407.

[5] Blacodon D. Spectral estimation method for noisy data using a noise reference. Applied Acoustics 2011;72(1):11-21.

[6] Amiet RK. Correction of open jet wind tunnel measurements for shear layer refraction. 2nd AIAA Aero-Acoustic conference, Hampton, VA, March 1975.

[7] Amiet RK. Refraction of sound by a shear layer. Journal of Sound and Vibration 1978;58(4):467-482.

[8] Humphreys Jr WM, Brooks TF, Hunter Jr WW, Meadows KR. Design and Use of Microphone Directional Arrays for Aeroacoustic Measurements. 36th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1998.

[9] Koop L, Ehrenfried K, Krober S. Investigation of the systematic phase mismatch in microphone-array analysis. 11th AIAA/CEAS Aeroacoustics Conference (26th AIAA Aeroacoustics Conference), Monterey (USA), May 2005.
[3] Guérin S, Michel U. Aero-engine noise investigated from flight tests. 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, Massachusetts (USA), May 2006.

[4] Padois T, Prax C, Valeau V, Marx D. Experimental localization of an acoustic sound source in a wind-tunnel flow by using a numerical time-reversal technique. J. Acoust. Soc. Am. 2012;132(4):2397-2407.

[5] Blacodon D. Spectral estimation method for noisy data using a noise reference. Applied Acoustics 2011;72(1):11-21.

[6] Amiet RK. Correction of open jet wind tunnel measurements for shear layer refraction. 2nd AIAA Aero-Acoustic conference, Hampton, VA, March 1975.

[7] Amiet RK. Refraction of sound by a shear layer. Journal of Sound and Vibration 1978;58(4):467-482.

[8] Humphreys Jr WM, Brooks TF, Hunter Jr WW, Meadows KR. Design and Use of Microphone Directional Arrays for Aeroacoustic Measurements. 36th Aerospace Sciences Meeting and Exhibit, Reno, NV, January 1998.

[9] Koop L, Ehrenfried K, Krober S. Investigation of the systematic phase mismatch in microphone-array analysis. 11th AIAA/CEAS Aeroacoustics Conference (26th AIAA Aeroacoustics Conference), Monterey (USA), May 2005.
[10] Mueller T. Aeroacoustic measurement. Springer 2002.

[11] Kroeber S, Ehrenfried K, Koop L, Lauterbach A. In-flow calibration approach for improving beamforming accuracy. 3rd Berlin Beamforming Conference 2010.

[12] Bahr C, Zawodny NS, Yardibi T, Liu F, Wetzel D, Bertolucci B, Cattafesta L. Shear layer correction validation using a non-intrusive acoustic point source. 16th AIAA/CEAS Aeroacoustics Conference, Stockholm, Sweden, June 2010.

[13] Schlinker RH, Amiet RK. Shear layer refraction and scattering of sound. 6th AIAA Aeroacoustics Conference, Hartford (USA), June 1980.

[14] A. Lauterbach, K. Ehrenfried and L. Koop, Array measurements in wind tunnels with open test sections, 2nd Berlin Beamforming Conference, 2008

[15] J-F. Pinton, G. Brillant, Sound and vorticity interactions: transmission and scattering, Theoretical Computational Fluid Dynamics, Vol. 18, N.6, 2005, 413-433.

[16] Johnson DH, Dudgeon DE. Array Signal Processing: Concepts and Techniques. Prentice Hall 1993.

[17] Wu TW, Lee L. A direct boundary integral formulation for acoustic radiation in a subsonic uniform flow. Journal of Sound and Vibration 1994;175(1):51-63.
[10] Mueller T. Aeroacoustic measurement. Springer 2002.

[11] Kroeber S, Ehrenfried K, Koop L, Lauterbach A. In-flow calibration approach for improving beamforming accuracy. 3rd Berlin Beamforming Conference 2010.

[12] Bahr C, Zawodny NS, Yardibi T, Liu F, Wetzel D, Bertolucci B, Cattafesta L. Shear layer correction validation using a non-intrusive acoustic point source. 16th AIAA/CEAS Aeroacoustics Conference, Stockholm, Sweden, June 2010.

[13] Schlinker RH, Amiet RK. Shear layer refraction and scattering of sound. 6th AIAA Aeroacoustics Conference, Hartford (USA), June 1980.

[14] A. Lauterbach, K. Ehrenfried and L. Koop, Array measurements in wind tunnels with open test sections, 2nd Berlin Beamforming Conference, 2008

[15] J-F. Pinton, G. Brillant, Sound and vorticity interactions: transmission and scattering, Theoretical Computational Fluid Dynamics, Vol. 18, N.6, 2005, 413-433.

[16] Johnson DH, Dudgeon DE. Array Signal Processing: Concepts and Techniques. Prentice Hall 1993.

[17] Wu TW, Lee L. A direct boundary integral formulation for acoustic radiation in a subsonic uniform flow. Journal of Sound and Vibration 1994;175(1):51-63.
[18] Haddad K. Contributions expérimentales et théoriques à l'étude des perturbations apportées aux images acoustiques obtenues par voies préformées en présence d'écoulement. PhD thesis, Université de Paris VI 1996.

[19] Wooley J, Smith C, Karamcheti K. Propagation of sound through a sheared flow. NASA CR-152299 1979.

[20] Colonius T, Lele SK. Computational aeroacoustics: progress on non-linear problems of sound generation. Progress in Aerospace Sciences 2004;40:345-416.

[21] Tam CKW, Webb JC. Dispersion relation preserving finite difference schemes for computational acoustics. Journal of Computational Physics 1993;107(2):262-281.

[22] Bogey C, Bailly C, Juvé D. Computation of flow noise using source terms in linearized Euler's equations. AIAA Journal 2002;40(2):235-243.

[23] Prax C, Golansky F, Nadal L. Control of the vorticity mode in the linearized Euler equations for hybrid aeroacoustic prediction. Journal of Computational Physics 2008;227:6044-6057.

[24] Bailly C, Juvé D. Numerical solution of acoustic propagation problems using linearized Euler equations. AIAA Journal 2000;38(1):22-29.

[25] Candel S, Guedel A, Julienne A. Radiation, refraction and scattering of
[18] Haddad K. Contributions expérimentales et théoriques à l'étude des perturbations apportées aux images acoustiques obtenues par voies préformées en présence d'écoulement. PhD thesis, Université de Paris VI 1996.

[19] Wooley J, Smith C, Karamcheti K. Propagation of sound through a sheared flow. NASA CR-152299 1979.

[20] Colonius T, Lele SK. Computational aeroacoustics: progress on non-linear problems of sound generation. Progress in Aerospace Sciences 2004;40:345-416.

[21] Tam CKW, Webb JC. Dispersion relation preserving finite difference schemes for computational acoustics. Journal of Computational Physics 1993;107(2):262-281.

[22] Bogey C, Bailly C, Juvé D. Computation of flow noise using source terms in linearized Euler's equations. AIAA Journal 2002;40(2):235-243.

[23] Prax C, Golansky F, Nadal L. Control of the vorticity mode in the linearized Euler equations for hybrid aeroacoustic prediction. Journal of Computational Physics 2008;227:6044-6057.

[24] Bailly C, Juvé D. Numerical solution of acoustic propagation problems using linearized Euler equations. AIAA Journal 2000;38(1):22-29.

[25] Candel S, Guedel A, Julienne A. Radiation, refraction and scattering of
acoustic waves in a free shear flow. 3rd AIAA Aero-Acoustic conference,
Palo alto, California (USA), July 1976.

[26] Cigada A, Ripamonti F, Vanali M. The Delay & Sum algorithm applied to microphone array measurements: Numerical analysis and experimental validation. Mechanical Systems and Signal Processing 2007;21:2645-2664.

List of Figures

1 Example of beampattern for a monopolar sound source ($f = 10$ kHz) in front of the array ($x=0$). 31 microphones are used and the microphone spacing is $d = 0.017$ m. The amplitude is normalised by the peak value of the beampattern. . . . . . . . . 42

2 Diagram illustrating the method for calculating the apparent source shift by using the DASB technique without a flow-propagation model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Diagram of the ray-based model for deriving a correction of the measured sound pressure [7] . . . . . . . . . . . . . . . . . . . 44

4 Diagram of the ray-based model for calculating the acoustic travel time in the zero-thickness shear layer [9]. . . . . . . . . . 45

5 Diagram of the experimental set-up in the anechoic wind-tunnel. 46

6 Mean flow field obtained from hot-wire measurements (color-bar in m.s⁻¹). The black line is the array position. . . . . . . . 47
13 Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two flow thickness values. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted with (●) symbols depict the real source positions, the estimated DASB positions with $h = 6.76λ$ and the estimated DASB positions with $h = 27.06λ$, respectively. . . . . . . . . . . 54

14 Estimated DASB positions versus array positions (with shear flow corrections) positions with flow ($M = 0.11$) for two mean flow thickness values. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted with (●) symbols depict the real source positions, the estimated DASB positions with $h = 6.76λ$ and the estimated DASB positions with $h = 27.06λ$, respectively. . . . . . . . . . . 55
<figure><img src="image_1.png" /><figcaption>Figure 1: Example of beampattern for a monopolar sound source ($f = 10$ kHz) in front of the array ($x=0$). 31 microphones are used and the microphone spacing is $d = 0.017$ m. The amplitude is normalised by the peak value of the beampattern.</figcaption></figure>
<figure><img src="image_2.png" /><figcaption>Figure 1: Example of beampattern for a monopolar sound source ($f = 10$ kHz) in front of the array ($x=0$). 31 microphones are used and the microphone spacing is $d = 0.017$ m. The amplitude is normalised by the peak value of the beampattern.</figcaption></figure>
<figure><img src="image_2.png" /><figcaption>Figure 2: Diagram illustrating the method for calculating the apparent source shift by using the DASB technique without a flow-propagation model.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Diagram illustrating the method for calculating the apparent source shift by using the DASB technique without a flow-propagation model.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 3: Diagram of the ray-based model for deriving a correction of the measured sound pressure [7]</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Diagram of the ray-based model for deriving a correction of the measured sound pressure [7]</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 4: Diagram of the ray-based model for calculating the acoustic travel time in the zero-thickness shear layer [9].</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4: Diagram of the ray-based model for calculating the acoustic travel time in the zero-thickness shear layer [9].</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 5: Diagram of the experimental set-up in the anechoic wind-tunnel.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5: Diagram of the experimental set-up in the anechoic wind-tunnel.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 6: Mean flow field obtained from hot-wire measurements (colorbar in m.s⁻¹). The black line is the array position.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6: Mean flow field obtained from hot-wire measurements (colorbar in m.s⁻¹). The black line is the array position.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 7: Estimated DASB positions versus array positions (with no shear layer correction). The harmonic monopolar source (10 kHz) is located in a shear flow ($M = 0.11$). The array is a 31 uniform linear microphone array located outside the flow at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the experimental estimated DASB positions and the numerical estimated DASB positions, respectively.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7: Estimated DASB positions versus array positions (with no shear layer correction). The harmonic monopolar source (10 kHz) is located in a shear flow ($M = 0.11$). The array is a 31 uniform linear microphone array located outside the flow at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the experimental estimated DASB positions and the numerical estimated DASB positions, respectively.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 8: Estimated DASB positions versus array positions (with shear layer correction). The harmonic monopolar source (10 kHz) is located in a shear flow ($M = 0.11$). The array is a 31 uniform linear microphone array located outside the flow at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the experimental estimated DASB positions and the numerical estimated DASB positions, respectively.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: Estimated DASB positions versus array positions (with shear layer correction). The harmonic monopolar source (10 kHz) is located in a shear flow ($M = 0.11$). The array is a 31 uniform linear microphone array located outside the flow at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the experimental estimated DASB positions and the numerical estimated DASB positions, respectively.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 9: Contour representation of the mean flow field for two different values of the shear layer thickness a) $\delta = 0.5\lambda$ and b) $\delta = 8\lambda$ (colorbar in m.s⁻¹). The black line is the array position.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9: Contour representation of the mean flow field for two different values of the shear layer thickness a) $\delta = 0.5\lambda$ and b) $\delta = 8\lambda$ (colorbar in m.s⁻¹). The black line is the array position.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 10: Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two different shear layer thicknesses. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the estimated DASB positions with $\delta = 0.5\lambda$ and the estimated DASB positions with $\delta = 8\lambda$, respectively.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 10: Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two different shear layer thicknesses. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the estimated DASB positions with $\delta = 0.5\lambda$ and the estimated DASB positions with $\delta = 8\lambda$, respectively.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 11: Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two different wavelengths. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the estimated DASB positions with $\lambda_1 = 2\lambda$ and the estimated DASB positions with $\lambda_2 = 4\lambda$, respectively.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 11: Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two different wavelengths. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the (●) symbols and the dashed line depict the real source positions, the estimated DASB positions with $\lambda_1 = 2\lambda$ and the estimated DASB positions with $\lambda_2 = 4\lambda$, respectively.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 12: Contour representation of the mean flow field for two different flow thickness values: a) $h = 6.76\lambda$ and b) $h = 27.06\lambda$ (colorbar in m.s⁻¹). The black line is the array position.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 12: Contour representation of the mean flow field for two different flow thickness values: a) $h = 6.76\lambda$ and b) $h = 27.06\lambda$ (colorbar in m.s⁻¹). The black line is the array position.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 13: Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two flow thickness values. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted with (●) symbols depict the real source positions, the estimated DASB positions with $h = 6.76λ$ and the estimated DASB positions with $h = 27.06λ$, respectively.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 13: Estimated DASB positions versus array positions (without shear flow corrections) positions with flow ($M = 0.11$) for two flow thickness values. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted with (●) symbols depict the real source positions, the estimated DASB positions with $h = 6.76λ$ and the estimated DASB positions with $h = 27.06λ$, respectively.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 14: Estimated DASB positions versus array positions (with shear flow corrections) positions with flow ($M = 0.11$) for two mean flow thickness values. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted with (●) symbols depict the real source positions, the estimated DASB positions with $h = 6.76λ$ and the estimated DASB positions with $h = 27.06λ$, respectively.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 14: Estimated DASB positions versus array positions (with shear flow corrections) positions with flow ($M = 0.11$) for two mean flow thickness values. The source is harmonic and monopolar. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted with (●) symbols depict the real source positions, the estimated DASB positions with $h = 6.76λ$ and the estimated DASB positions with $h = 27.06λ$, respectively.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 15: Estimated DASB positions versus Mach number. The source is in front of the array center. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted line with (●) symbols depict the estimated DASB positions without shear flow correction, with shear flow correction (section 2.3.1) and the *a posteriori* correction of Eq. (7), respectively.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 15: Estimated DASB positions versus Mach number. The source is in front of the array center. The array is a 31 uniform linear microphone array located at $H = 1.36$ m from the source and the microphone spacing is $d = 0.017$ m. The solid line, the dashed line and the dashed dotted line with (●) symbols depict the estimated DASB positions without shear flow correction, with shear flow correction (section 2.3.1) and the *a posteriori* correction of Eq. (7), respectively.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 16: Distribution of the $\frac{p_c}{p_m}$ ratio as a function of microphone position ($\theta_m$) and Mach number: a) analytical formulation of Eq. (10) and b) numerical calculation.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 16: Distribution of the $\frac{p_c}{p_m}$ ratio as a function of microphone position ($\theta_m$) and Mach number: a) analytical formulation of Eq. (10) and b) numerical calculation.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 17: Simulation of DASB results: estimated power of the acoustic source, normalized by the power obtained for $M = 0$, as a function of the Mach number. Solid line, no amplitude correction; dashed line, the amplitude correction of Eq. (10) is carried out prior to the application of the DASB technique.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 17: Simulation of DASB results: estimated power of the acoustic source, normalized by the power obtained for $M = 0$, as a function of the Mach number. Solid line, no amplitude correction; dashed line, the amplitude correction of Eq. (10) is carried out prior to the application of the DASB technique.</figcaption></figure>