PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Acoustic source detection inside a pipe using vibro-acoustic
beamforming: assessment of the array gain from virtual
experiments
Souha Kassab1,2, Sanae Serbout1,2, Frédéric Michel2, Laurent Maxit1
1
2

Laboratoire Vibrations-Acoustique (LVA), INSA Lyon, Villeurbanne, France

DTN/STCP/LISM, Commissariat of Atomic Energy and Alternative energies, Cadarache, France

ABSTRACTIn the R&D framework on sodium water heat exchangers, a monitoring technique based on vibration
measurements is developed for detecting a leak of water into the sodium. However, leak-induced vibrations
could be smeared in the ambient vibrations. In order to increase the signal-to-noise ratio (SNR), theconventionnel and MaxSNR beamforming treatments have been considered. In order to study their efficiency
for the present application, one has developed a laboratory mock-up composed by a pipe coupled to a
hydraulic circuit through two flanges. The source to detect is a sound emitter introduced into the pipe whereas
the background vibration noise is controlled by changing the speed of the flow. In parallel, numerical vibroacoustic tools were developed for predicting the vibratory response of the pipe excited by a monopole source
or a turbulent boundary layer. These models allow us to realize virtual experiments mimicking the behaviorof our laboratory test case. The virtual signals induced by the monopole source to be detected and by the
turbulent flow are used for assessing the performances of the vibro-acoustic beamforming treatments. Aftera presentation of the vibro-acoustic models, we illustrate the array gain estimated using the virtual
experiments for the two beamforming treatments.
Keywords: beamforming, detection, pipe, vibro-acoustic modeling

1. INTRODUCTION
This paper describes the study of a non-intrusive vibro-acoustic beamforming technique aimed at
the detection of sodium-water reactions in the steam generator of a liquid sodium fast reactor (SFR).
Vibro-acoustic beamforming previously developed within a PhD thesis by J. Moriot is reconsidered[1]. Beamforming over an array of sensors is of main interest due to its ability to increase the signal
to-noise ratio (SNR) of the chemical reaction signals, generally masked by the high power plantbackground noise. Thus, we can provide a quantitative estimation of the SNR increase at the
beamforming output relative to the SNR on the reference sensor using the “effective gain”. From thedetection on a threshold criterion at the output of beamforming (instead of the reference sensor), thisgain allows the improvement of the detection rate while limiting the sensibility to false alarms.
Moriot’s thesis showed promising results. However, the latter were obtained by on academic
numerical test models (plate or infinite shell) or from experimental data at some harmonic frequencies[2]. New investigations have been carried out in the framework of the Kassab thesis in order to assess
more accurately the interest of the vibro-acoustic beamforming for detecting an acoustic source inside
a pipe. Numerical models of the pipe test section illustrated in the figure 1 have been developed. In
these models, the coupling between the pipe and the flanges is described as well as the coupling
between the pipe and the inner fluid (water). These models allows us to simulate the vibrations on thepipe excited, either by a monopole source or by a turbulent flow. These simulations allows us
generating virtually the array signals due to the source to be detected (i.e. a monopole) and due to thebackground noise (i.e. turbulent flow). From these virtual signals, one can test different types of
beamforming treatments and different physical/array parameters. In the section 2, one recal ls theprinciple of two types of beamforming treatments whereas the theoretical principles of the vibro-

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acoustic simulations are presented in section 3. An illustration of the treatments of the virtual
experiment data is proposed in section 4.

Figure 1. Schematic representation of the configuration considered for assessing the performanceof the vibro-acoustic beamforming

2. CLASSICAL AND MAXSNR BEAMFOMRING
In this section, one recalls the principle of two beamforming treatments that will be evaluated on
the pipe test section through virtual experiments.
Let us denote by Ȟ the cross-spectral matrix of the signals received by the array of sensors and by‫ ݑ ܨ‬the steering vector (i.e. spatial filtering vector) when the treatment focus at the position u of thedetection space. The beamforming output, ‫ ݑ ݕ‬is given by:‫ݕ‬௨ ൌ ‫ܨ‬௨‫ כ‬Ȟ‫ܨ‬௨ Ǥ(1)Assuming that the source to be detected and the background noise are independent, we candecompose the matrix Ȟ as such:Ȟ ൌ Ȟ ௦ ൅ Ȟ௡  ǡ(2)where Ȟ ௦ is the cross-spectral matrix of the signals induced by the source alone and Ȟ ௡  is the
cross-spectral matrix of the signals induced by the noise alone.
When the background noise is spatially homogeneous and incoherent (i.e. Ȟ ܾ ൌ ߪܾ ‫ ܫ‬with theidentity matrix and ߪ ܾ, the power spectrum density of the background noise), it can be shown that thearray gain is maximum when the steering vector is given by:‫ܨ‬௨௖௟௔௦௦ ൌ‫ܪ‬௨ǡԡ‫ܪ‬௨ ԡ(3)

where ԡ ԡ represents the Euclidean norm and  is the vector containing the transfer functions
between the (assumed) position u of the source and the antenna’s accelerometers (i.e. sensors).
This beamforming technique relies on a prior knowledge of the source (through the transfer
functions) as well as on the assumption that the noise would appear spatially incoherent. In thefollowing, one refers as the classical beamforming for this treatment.
The classical beamforming is widely used in the world for various applications. However, when
the background noise is partially correlated (spatially), its performances can be strongly deteriorated
compared to those obtained when the noise is perfectly incoherent.
To overcome this drawback, different variants of beamforming based on prior knowledge on thenoise have been developed [3]. In particular, the so-called MaxSNR beamforming seeks to maximize
the signal-to-noise ratio at the beamforming output, knowing the cross-spectral matrix of the noise.

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The steering vector is then defined as follows:
௢௣௧

‫ܨ‬௨

‫ܨ‬௨‫ כ‬Ȟ௨௦ ‫ܨ‬௨ ሺ߱ሻൌ  ቈ ‫ כ‬௡቉ǡ‫ܨ‬௨ Ȟ ‫ܨ‬௨ ሺ߱ሻிೠ(4)where Ȟ௨௦ is the cross-spectral matrix of the signals induced by the source located at the positionu in the detection space.
From bilinear algebra considerations, it can be shown that the solution of Eq. ( 4) corresponds toone eigenvector associated with the greatest eigenvalue of the matrix ሺȞ ௡ ሻିଵ‫ܪ‬௨ ‫ܪ‬௨‫ כ‬:ሺȞ ௡ ሻିଵ ‫ܪ‬௨ ‫ܪ‬௨‫ܨ כ‬௨௢௣௧ ൌ ɉ௠௔௫ ‫ܨ‬௨௢௣௧ Ǥ

(5)This treatment can be relevant for the source detection in the pipe with a threshold criteria as it
permits to increase the SNR at the output of beamforming compared to the one of a single sensor.
Nevertheless, this technique requires solving a generalized eigenvalues problem which can induce
numerical instabilities. It also requires knowledge of the cross-spectral matrix of the noise. In practice,
this can be estimated from in-situ measurements with the sensor array when the source is (supposed)
off. An average over a set of measured signals samples can be performed regularly to take into accountthe variation of the cross-spectral noise matrix in the system as a function of time.

3. VIRTUAL EXPERIMENTS
In this section, one presents briefly the principle of the numerical simulations for generating thesensor signals induced by the monopole source to be detected and by the turbulent flow constituting
the background noise.
3.1

Vibratory response of the shell excited by a monopole source

3.1.1 Circumferential admittance approach
One considers the system shown in Figure 1-a composed of an infinite cylindrical shell filled witha heavy fluid (water) and coupled to two ring stiffeners. In this section, one supposes that the system
is excited either by a monopole inside the fluid or by a radial point force applied on the shell. The
excitation is supposed harmonic with a time dependency in ݁ ௝ఠ௧ where ߱ represents the angularfrequency. The dynamic behavior of the thin cylindrical shell will be described by the Flûgge model
whereas the acoustic behavior of the fluid will be represented by the Helmholtz model. Only the
coupling along the radial direction between the shell and the stiffener will be considered for these firstdevelopments. The ring stiffener will be represented by a ring model (i.e. rod of circular curvature).
A cylindrical coordinate system (‫ݔ‬, r, ߠ) is considered. The two ring stiffeners are located at the axial
positions, ‫ݔ‬ଵ  and ‫ݔ‬ଶ .
The global system is partitioned in two subsystems as shown in Figure 2. The fluid filled cylindricalshell constitutes one subsystem whereas the ring stiffeners constitutes the second subsystem.

(a)

(b)

Figure 2. Partitioning of the vibro-acoustic problem: (a), fluid filled cylindrical shell; (b), ring stiffeners.
One will use the circumferential admittance approach (CAA) [4] for assembling the two
subsystems. One reminds here the principle of the approach (for more details, see [4]).

6263• First, as the considered system is axisymmetric (excepted for the excitation), the different spatial
fields (displacement, acoustic pressure, force) noted ݂ can be represented by a Fourier series
decomposition about the circumference:݂ሺߠሻ ൌ ෍ ݂ሚሺ݊ሻ݁ ௝௡ఏ Ǥ

(6)

௡‫א‬Ժ

• Second, for each circumferential order n (omitted of the notation in the following), one defines
the circumferential admittances ܻ෨௜௝ఈ (m2 /N) of the shell (α=1) and of the ring stiffeners (α=2) between
the junction i and j by:෩ഀௐܻ෨௜௝ఈ ൌ ೔ǡೕ
(7)ഀ ǡ ݅ ‫ א‬ሼͳǡʹሽǡ ݆ ‫ א‬ሼͳǡʹሽ,ி෨೔where:ఈ෩௜ǡ௝െ ܹ(m) is the circumferential radial displacement at the junction j (of the shell for ߙ ൌ ͳ orof the stiffeners for ߙ ൌ ʹ ) for a circumferential radial force applied at junction i;െ ‫ܨ‬෨௜ఈ (N/m) is the circumferential radial force applied at junction i on the shell by the stiffenersfor ߙ ൌ ͳ or the circumferential radial force applied on the stiffeners by the shell for ߙ ൌ ʹǤMoreover, for each subsystem α, one defines the circumferential free displacement at the junction෩ഥ
i, ܹ௜ఈ as the circumferential radial displacement induced by the external excitation at the junction iwhen all the junctions are let free

• Finally, using the superposition principle for passive linear systems and considering the radialdisplacement continuity and the force equilibrium at the junctions, one finally obtains the matrix
system [4]:
෩ଵ െܹ
෩ ଶ,
(8)
ഥ
ഥ
൫ܻ෨ଵ ൅ ܻ෨ ଶ ൯‫ܨ‬෨ ௖ ൌ ܹఈ෩ఈഥܻ෨ ఈ ܻ෨ଵଶ
‫ܨ‬෨ଵ ഥ෩ ఈ ൌ ቈܹଵ ቉.
where ܻ෨ ఈ ൌ ቈ ଵଵ቉ ǡ ‫ܨ‬෨ ௖ ൌ ቈ ଵଵ ቉ ǡ ܹఈఈ෩ఈഥܻ෨ଶଵ ܻ෨ଶଶ
‫ܨ‬෨ଶܹଶ
Resolving this equation system, one deduces the coupling forces exerted by the stiffeners on theshell. In a last step, one can introduce these forces in the model of the infinite fluid filled shell inorder to estimate the vibratory field of the shell when it is coupled to the ring stiffeners. In the෩ ଶ ൌ Ͳ in Eq.
ഥfollowing, as no external excitation is contained in subsystem #2, one considers that ܹ௜(4).
3.1.2 Spectral approach for estimating the shell circumferential admittance and freedisplacement
The problem describing the vibro-acoustic behaviour of the fluid loaded shell excited by aharmonic radial force is resolved using a spectral approach. The 2D Fourier transform (i.e. space
Fourier transform about x-axis + Fourier serie decomposition about the circumference) is considered:ାஶ ଶగͳ݂ሺ‫ݔ‬ǡ ߠሻ ՜ ݂ሚሺ݇௫ ǡ ݊ǡ ሻ ൌන න ݂ሺ‫ݔ‬ǡ ߠሻ݁ ି௝௞ೣ ௫ ݁ ି௝௡ఏ ݀ߠ݀‫ݔ‬
ʹߨ

(9)

ିஶ ଴

where ݂ is a generic function representing either the shell displacements or the acoustic pressure.
Applying this transform to the Flügge, Helmholtz, and Euler equations describing the behaviourof the fluid filled shell, one obtains finally [4]
෨ ෨
෨ మ
(10)
෩ ൌ ߛ‫ܨ‬෨ ௓ೠೠ ௓ೡೡି൫௓ೠೡ ൯ ,
ܹ
௪

෩
෩
୼

෩ is the determinant of the spectral
where ܼ෨఍క are the components of the spectral Flügge matrix, ȟFlügge matrix taking into account of the spectral fluid loading impedance and ߛ is a constantdepending on the shell characteristics.
In order to estimate the circumferential admittances of the fluid filled cylindrica l shell, one
considers the shell excited by a radial circumferential force at x=0 of unit magnitude for eachcircumferential order, n. As the considered system is invariant in translation about the axial direction,

6264

the admittance between junctions i and j can be deduced from෩ ൫௫೔ ି௫ೕ ǡ௡൯ௐܻ෨௜௝ଵ ൌ
,

(11)
෩
‫ܨ‬
෩
where ܹ is the radial displacement of the shell excited by a unitary radial circumferential force‫ܨ‬෨  at x=0 (‫ܨ‬෨ =1 N/m).
The spectral radial force for the unitary circumferential radial force applied at x=0 is ‫ܨ‬෨௪ ൌ ͳ N/m.
෩ can then be calculated analytically using Eq. (6). The
The spectral radial displacement ܹ෩circumferential displacement ܹ can be obtained from an inverse Fourier transform about x.

3.1.3 Model of the ring stiffenerଶଶൌ ܻ෨ଶଵ
ൌ Ͳ and as they are identical,
As the two stiffeners are not physically coupled, one has ܻ෨ଵଶଶଶܻ෨ଵଵ ൌ ܻ෨ଶଶ ൌ ܻ෨௥ . Each ring is represented by a rod of circular curvature and of rectangular cross -section.
݈௥ ǡ ݄௥ ǡ ߩ௥ ǡ ܿ௥ represent the width, height of the cross section, density of the material and celerity ofthe extensional waves. Neglecting inertia and rotary effect, one can determinate its circumferential
admittance, ܻ෨௥ (m/N) as given in reference [5].
3.2 Vibratory response of the shell excited by the turbulent flow
Now let us considered the fluid filled shell excited by a turbulent flow. The flow speed is supposed
to be low compared to the acoustic speed and the turbulent boundary layer (TBL) excited the wall of
the shell is supposed homogeneous, stationary and fully developed. The effect of the convection on
the acoustic behaviour is neglected. The Cross Spectrum Density (CSD) of the Wall Pressure Field(WPF) induced by the TBL, ߶௣௣ will be estimated from the TBL parameters and a mixed GoodyChase 1987 model [6,7]. The CSD of the radial shell displacement between the points of coordinatesோሺ‫ݔ‬ǡ ߠሻ and ሺ ‫ ݔ‬ᇱ ǡ ߠ ᇱ ሻ, ܵ௪௪can be written (see [8-12]):ோ ሺ‫ݔ‬ǡܵ௪௪ߠǡ ‫ ݔ‬ᇱ ǡ ߠ ᇱ ǡ ߱ሻ ൌା҄ ෩ ఠ ‫כ‬
෩ఠ
ା҄
ᇱ ᇱ
ோ
ʹߨ σ௡ୀି҄ ‫ܪ  ҄ି׬‬௪ ሺ‫ݔ‬ǡ ߠǡ ݇௫ ǡ ݊ሻ ߶௣௣ ሺ݇௫ ǡ ݊ǡ ߱ሻ ‫ܪ‬௪  ሺ‫ ݔ‬ǡ ߠ ǡ ݇௫ ǡ ݊ሻ݀݇௫ Ǥ(12)෩ ఠ are expressed (using the reciprocity principle as described inwhere the sensitivity functions, ‫ܪ‬௪[8, 10]),෩ ఠ ሺ‫ݔ‬ǡ ߠǡ ݇ ǡ ݊ሻ ൌ ଵ ‫׬‬ାஶ ‫׬‬ଶగ ‫ ܪ‬൫‫ݔ‬෤ǡ ߠ෨ ǡ ‫ ݔ‬ǡ ߠǡ ߱൯ି௝௞ೣ ୶෤ ݁ ି௝௡ఏ෩ ݀෤ ݀ߠ෨.
‫ܪ‬
(13)
௪
௫
௪
ଶగ

ିஶ

଴

෩  of
In Eq. (13), ‫ܪ‬௪ ൫‫ݔ‬෤ǡ ߠ෨ǡ ‫ ݔ‬ǡ ߠǡ ߱൯ corresponds to the radial displacement of the shell at point ෨coordinates ൫‫ݔ‬෤ǡ ߠ൯ when the shell is excited by a unitary radial point force at point  of coordinatesሺ‫ ݔ‬ǡ ߠሻ. Considering the definition of the Fourier transform given by Eq. (9), its results that the෩௪ఠ given by Eq. (13) correspond to the Fourier transform of the shell radial
sensitivity functions ‫ܪ‬displacement field (expressed in ሺ ݇௫ ǡ ݊ሻ space) when the shell is excited by a unitary radial point
force at point . For the fluid filled shell coupled to the two ring stiffeners, these quantities can becalculated using the CAA approach described in section 2. For calculating the sensitivity functions at
a given point , it is necessary to consider a unitary radial point force at this point, to calculate thecoupling forces by resolving Eq. (8), and to evaluate the spectral radial displacement of the shell
directly from Eq. (10).
The CSD of the radial displacement excited by the TBL can then be estimated using Eq. (12), the෩ ఠ estimated with the CAA approach and the CSD of the WPF, ߶ ோ given thesensitivity functions ‫ܪ‬௪௣௣Goody-Chase model ߶௣௣.

6265

4. ASSESSMENT OF THE BEAMFORMING TREATMENTS USING VIRTUAL
EXPERIMENT
Only an illustration of the beamforming assessment with the virtual experiment data obtained withthe models developed in section 3 is presented. More results will be presented during the oralpresentation and will be published in an international journal.
The test section is composed by a cylindrical shell made of stainless steel and filled with water.
The length, the diameter and the thickness are respectively, 3.1 m, 219 mm and 8 mm. The vibrations
are measured by a linear array of 24 virtual accelerometers fixed on the pipe. The spacing between
the sensors is ߂‫ ݔ‬ൌ Ͷ  and the first accelerometers is positioned at ͳͷ cm from the upstream
flange of the flow.
Figure 3 shows the acceleration levels at the sensor position for two positions of the source. The
patterns appearing on this figure are due to the circumferential and pseudo-axial modes of the pipe.
Figure 4a illustrates the acceleration levels at the sensor position for a turbulent flow whereas
figure 4b presents the coherence between sensor #12 and the other sensors for the same turbulent flow.
One can observe the signals due to the background noise may be significantly correlated at certain
frequencies.
The data related to the two previous figures have been filtered by the two beamforming treatments
described in section 2. One illustrates one result of these treatments by showing the array gain for awide band analysis. This gain expresses the increase of SNR by the beamforming compared to the
SNR on the reference sensor. This result shows that the interest of the classical beamforming is weakfor this case, with a negative gain at some frequencies and a gain always lesser than 4 dB (due to thestrong coherence of the background noise) whereas a significant gain, over 18 dB in mean, is obtainedwith the MaxSNR beamforming, whatever the frequency band.

(a)(b)
Figure 3. Acceleration levels (ref. dB, 1 μm.s -2.Hz -0.5) measured by the sensor array for two
radial positions of the acoustic source: (a), r s=-0.03 m; (b), r s=0.09 m. x s=0.6 m, θ s=0°. Results of
CAA calculation.

6266

(a)
(b)
Figure 4. (a), Acceleration levels (ref. dB, 1 μm.s-2 .Hz -0.5 ) measured by the sensor array when
the pipe is excited by the turbulent flow. (b), Normalized cross spectrum density function of theshell displacement between sensor #12 and the sensor #i‫ͳۤ א‬ǡʹͷ‫ۥ‬. Flow rate: 140 l/s.

Figure 5. Array gain versus frequency.
Wide band analysis of 500 Hz bandwidth

5. CONCLUSIONSThe present study was achieved in the framework of a novel non-intrusive vibro-acoustic technique
for the detection of a sodium-water reaction in a sodium-cooled nuclear reactor steam generator. In
order to study the efficiency of different beamforming treatments for increasing the SNR, a laboratory
mock-up composed by a pipe coupled to a hydraulic circuit through two flanges has been developed.
In parallel of the experimental investigations, the numerical simulation abilities are explored forassessing the performances of these treatments for different configurations. In the present paper, one
has described the numerical vibro-acoustic tools that we have developed for predicting the vibratory
response of the pipe excited by a monopole source or a turbulent boundary layer. These models a llow
us to realize virtual experiments mimicking the behavior of our laboratory test case. The virtual signals
induced by the monopole source to be detected and by the turbulent flow are used to test the vibroacoustic beamforming treatments. Different configurations of pipe, of array, of source and of flow canbe easily considered with these virtual experiments. Obtained results show that the interest of theclassical beamforming is weak for this configuration, whereas a significant gain is obtained with the

6267

MaxSNR beamforming.

ACKNOWLEDGEMENTS
This work was carried out in the framework of the LabEx CeLyA ("Centre Lyonnais d'Acoustique",ANR-10-LABX-60) and the Sodium Technology project of GEN4 program (R4G/TECNA) of the
Nuclear Energy Division of CEA (CEA/DEN).

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