[American Institute of Aeronautics and Astronautics 14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference) - Vancouver, British Columbia, Canada ()] 14th AIAA/CEAS

American Institute of Aeronautics and Astronautics

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Influence of Source Characteristics on the Liner Performance in a Circular Duct

Teresa Bravo∗ and Cedric Maury† Département de Génie Mécanique – Secteur Acoustique,

Université de Technologie de Compiègne, BP20529, 60205 Compiègne, France

The prediction and the attenuation of the tonal and broadband components of fan noise emissions is a research subject of considerable importance within the aeronautical sector. It requires a proper selection of the absorbing liner parameters for particular design duct conditions. In this work, a parametric study is carried out on the influence of the source characteristics on the sound attenuation in a uniform lined circular duct using the multi-modal propagation method. Different types of source models are considered under either tonal or broadband excitation conditions and in the latter case, for varying degree of correlations between the strengths of the distributed elementary sources. It has been found that the detailed nature of the excitation source significantly influences the optimal liner parameters, especially the liner depth, thus motivating the need for reliable equivalent source models under both tonal and broadband excitations. Such models are difficult to obtain since they require delicate blade surface pressure measurements. Alternatively, in-duct model-based sound source reconstruction techniques are being developed. Although they are intrinsically sensitive to the presence of noise in the measured data, especially for low modal density, and resolution limited when they cannot be performed from near-field measurements, such equivalent models would be extremely valuable to provide input information about the relative phasing between the multiple incident modes, especially when dealing with complex broadband noise source generation mechanisms.

Nomenclature A = duct cross-section area

mnA = upstream modal amplitudes for the increasing z values

mnB = downstream modal amplitudes for the decreasing z values

0c = sound speed C = modal cross-coupling matrix E = expectation operator

zF0 = axial dipole source strength h = liner depth H = complex acoustic transfer matrix in the SDM J = mean square error function

mJ = Bessel function of the first kind of order m

0k , ±mnk = acoustic wavenumber and axial wavenumber

l = index for the sensor location 1l , 2l = axial positions of the liner terminations

∗ Post-doctoral Researcher, Département de Génie Mécanique – Secteur Acoustique, Laboratoire Roberval, Université de Technologie de Compiègne, BP20529, F-60205 Compiègne, France, [email protected]. † Associate Professor, Département de Génie Mécanique, [email protected].

14th AIAA/CEAS Aeroacoustics Conference (29th AIAA Aeroacoustics Conference)5 - 7 May 2008, Vancouver, British Columbia Canada

AIAA 2008-2976

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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rL = radial correlation length M = number of error sensors

zM = axial Mach number m , n = azimuthal and radial order of the duct spinning modes Ω = rotor shaft speed p = acoustic pressure

mp , dp = acoustic pressure due to monopole- and dipole-type sources ±

mnP = pressure amplitude of the mnth mode spinning towards the upstream and downstream directions p , p = modeled and measured pressure vector P = vector of the modal pressure amplitudes

0Q = monopole source strength q , mnq = amplitude and modal amplitude of the source strength

optq , opt~q = optimal source strength vector (for the ESM and the SDM)

R = liner resistance dR = duct radius

zr ,,θ = cylindrical coordinates

0S = point-power spectrum

XXS = cross-spectral density matrix associated to the variable X

cU , ∞U = convection and free-stream velocity

zv = axial component of the acoustic velocity

mzv , , dzv , = axial component of the acoustic velocity due to monopole- and dipole-type sources ±

mnV = velocity amplitude of the mnth mode spinning towards the upstream and downstream directions V = vector of the modal velocity amplitudes Z = complex acoustic transfer matrix in the ESM ωZ = liner impedance

rα = empirical coefficient for the radial correlation length

mnκ = cross sectional wavenumber

mnΛ = modal normalization factors ω = angular frequency

0ρ = fluid density ±mnξ = specific modal mobility ratio

mnψ = rigid duct normal modes

I. Introduction IRCRAFT noise constitutes an important annoyance for airport communities. Even with the reduction that has been achieved with the development of quieter engines and the improvement of landing procedures, aircraft

noise continues to be of major concern as the traffic density increases. The development of modern turbofan engines has led to higher bypass ratio with reduced exhaust jet velocity and an increase of the contribution of fan noise sources to the overall radiated acoustic power, particularly at take-off and approach conditions.1 Existing noise reduction technologies or those currently being developed need a complete characterization of the noise source generation and propagation mechanisms that would help to develop strategies to control the radiated sound field and achieve compliance with the noise emission requirements. However, a direct characterization of such noise sources in real situations is difficult to achieve as it requires advanced measurement techniques. Although directivity measurements from static ground testing have shown some promise to reliably detect the radiated mode amplitudes, they can hardly provide a detailed description of the excitation mechanisms at source level. 2, 3

An alternative methodology for noise source identification consists of assuming a discretized model of the original source distribution and a suitable propagation path between the modeled sources and the sensor positions. A

A


3

reconstructed solution for the strength of the original sources can be derived from the analysis of the measured data at discrete points over an observation surface and the inversion of the acoustic transfer matrix.4 The main drawbacks of this method is the ill-conditioning of the transfer function matrix that usually occurs in real situations,5 and the assumption that has to be made for the choice and/or the location of the reconstructed source model. The lack of a priori information on the real source distribution can lead to large inaccuracies in the reconstructed solution.6

However, it has been shown that the use of model-based inverse methods such as the source spectral decomposition analysis with an appropriate choice of the regularization procedure or their combination with high-resolution beamforming techniques allows to obtain prior information on the source distribution and thus provide useful reconstruction results.7 The reconstructed noise sources then constitute a model of primary sources that can be used as an input to assess the efficiency of several noise control strategies. In particular, the optimization of absorbing treatments for noise reduction at the turbofan inlet constitutes an important subject of research and a great amount of effort is devoted to the proper selection of liners parameters for particular design duct conditions. Early efforts were concentrated on uniform liners, and recent studies have focused on axially segmented and circumferentially spliced liners, for which the specific acoustic impedance presents abrupt variations along the length and the circumference of the duct, respectively.8, 9

As for the process of liner optimization, a number of design parameters can be varied. A widely used representation involves a two-dimensional plot of iso-attenuation contours as a function of the porous facing-sheet acoustic resistance and the liner depth, as other properties are mostly depending on manufacturing restrictions. If one assumes a locally reacting model for the acoustic liner specific impedance, the optimization is then carried out in a straightforward manner when plotting the sound power Transmission Loss (TL) as a function of the liner design parameters. However, an important point concerns the determination of the optimal liner properties which are dependent on a priori information about the source modal (or spatial) characteristics. In the current design approaches, assumptions often have to be made concerning the distribution of the squared modal amplitudes to be introduced in the propagation model.

Previous authors have already noted the influence of the source characteristics on the attenuation obtained in a lined duct and have performed comparisons when using different assumptions on the excitation term. Optimization procedures were initially based on assuming an incident plane wave with uniform acoustic pressure distribution, without considering the propagation of higher-order, transversal modes.10 Other models are based on a superposition of hard-walled duct modes with different hypothesis concerning the weighting amplitudes that describes the interaction between the source and the in-duct pressure field. Snow11 studied the influence of a planar incident wave in comparison with a number of incident radial pressure distributions either with an amplitude scaling on the square of the local velocity or with constant amplitude and a phase varying linearly along the duct radius. More recently, Joseph et at.12 have investigated the relationship between the mean-square pressure and the in-duct sound power under the assumptions of equal energy per mode and equal energy density per mode in comparison with those obtained from different arbitrary distribution of elementary sources. Zlavog and Eversman13 have studied the effects of statistical random modal power and phase distribution as inputs to a finite element program. They have performed an analysis providing the source statistical distribution corresponding to the realized attenuations.

The general conclusion that can be obtained from the published works is that the characterization of the noise source plays a critical role in the liner optimization and the attenuation results obtained. However, this information is not known in detail and the experimental determination of the modal amplitudes can be difficult to obtain due to the large number of modes involved in the practical configurations analyzed. This lack of fundamental information and the requirement to improve optimization procedures has led to the alternative described in this work to obtain an equivalent model of source distribution that could be used to optimize the liner properties.

The paper is organized as follows: the formulation of inverse techniques for source strength reconstruction is outlined and an analytical model that describes the transfer path between the excitation and the in-duct measurement positions is formulated in terms of the rigid duct normal modes. Numerical simulations illustrate the capabilities and the physical limitations of inverse techniques to characterize the unknown noise sources. The information extracted is then used as an input for the optimization of the physical properties of a uniform liner. The main results obtained are summarized in the conclusions.

II. Inverse methods for sound source reconstruction The basis of the Equivalent Source Method (ESM) for source strength reconstruction is briefly outlined in this section. Further details can be found in the references.4-5 This approach belongs to the category of model-based inverse methods, that builds on the knowledge of a model of the transmission path between a source and a set of measurement positions. An acquisition of the acoustic pressure or velocity field generated by the real source is made


4

at the sensor positions. Assuming a set of reconstruction sources, their strength is sought to provide a field that best matches the one due to the real source. The reconstructed source strengths are optimised using the mean square difference between the measured pressure field, p , and the one generated with the optimised reconstructed sources, p , at each measurement location l , such that

∑=

−=M

lll ppJ

1

2ˆ . (1)

When there are more measurements points than assumed sources, then Eq. (1) has a unique global minimum which corresponds to the optimal source strength vector which gives the best match to the measured data. It reads4

pZpZZZq ˆˆ][ †H1Hopt == − , (2)

where †Z is the pseudo-inverse of the transfer matrix Z between the assumed reconstruction source model and the measurement positions. The combination of the ESM with proper regularization techniques can lead to useful results in terms of the accuracy and the resolution reconstructed source strengths. However, the question on the a priori knowledge of the source location remains the main drawback of the method, as the performance results significantly degrade when errors in the spatial distribution of the assumed reconstructed sources are introduced.

We note that, for the particular configuration under study, i.e. the reconstruction of sources inside ducted geometries, the authors have proposed an alternative methodology7 that decomposes the equivalent ducted source strengths distribution as a Fourier-Bessel expansion, in the form

( ) ( )∑=mn

mnmn rqrq ','',' θψθ , (3)

where ''nmq are the unknown amplitudes for the decomposition and ( )θψ ,rmn are the rigid duct normal modes.14 To find the optimal solution to the inverse problem, we calculate the transfer matrix between the source distribution Fourier-Bessel coefficients and the pressure field at each measurement positions, H . The solution of the inverse reconstruction problem is thus expressed as

pHq ˆ~ †opt = , (4)

where opt~q is the unknown vector of the excitation Fourier-Bessel decomposition from which the source strength

spatial distribution can be recovered using Eq. (3). Although the performance results of the Spectral Decomposition Method (SDM) are not as accurate as those obtained with the ESM (with correct assumed source location), it does not need any prior information concerning the original source distribution. The two methods could thus be used sequentially, with the SDM that would first determine a proper distribution of the equivalent sources and then provide an input to the ESM that would allow to refine the estimated source strength amplitudes. Since we are primarily concerned with the identification of the source characteristics for liner optimization, we will only present reconstruction results obtained when using the ESM.

III. Modal-based description for the lined duct Although finite-element methods provide an accurate prediction of the in-duct acoustic field for complex geometries and/or impedance conditions, analytical modeling, based on an expansion of the acoustic field as a series of rigid duct normal modes, has been used as it allows a deeper understanding of the physical phenomena. This method also presents clear advantages from the point of view of the computational cost. A schematic representation of the system is presented in Fig. 1. It consists of an infinite duct with a circular cross section of radius dR , excited by a source distribution situated at 0zz = . The duct has a lined section situated between the positions 1lz = and 2lz = , as indicated in Fig. 1. It is assumed that a uniform flow, with axial Mach number zM , is established along the z-


5

direction. For the case we are interested in, a turbofan engine inlet, the Mach number is negative since the flow is oriented towards the negative z-direction.

In the next subsections, we will first present expressions for the calculation of the sound field due to a set of elementary sources inside a rigid duct of infinite length with a uniform flow. We will then extend the model to account for the effect of a uniform liner designed to reduce the in-duct sound pressure levels.

A. Acoustic pressure field in a rigid duct with a uniform axial flow The calculation of the pressure and the axial velocity inside an infinite rigid duct with a uniform axial flow has

been derived following a classical formulation for a number of elementary source models.14-15 Assuming a harmonic excitation of the form )( tie ω , the in-duct sound pressure and velocity fields are respectively expressed as a sum of circumferential and radial modes traveling and/or decaying in the positive and negative z-directions,

( ) ( ) ( ) ( ) PΨT,,),,( =+=−+ −−−+∑∑ zik

mnmnzik

m nmnmn

mnmn erzPerzPzrp θψθψθ , (5)

( ) ( ) ( ) ( ) VΨT,,),,( =+=−+ −−−+∑∑ zik

mnmnzik

m nmnmnz

mnmn erzVerzVzrv θψθψθ , (6)

where ( )zPmn± and ( )zVmn

± are the upstream and downstream modal amplitudes for the pressure and the velocity (as

components of the modal vectors P and V respectively, with 210321 ,,;,,; BBBAAAPPP == −+ in relation to Fig. 1), that can be deduced from the characteristics of the excitation source. The ( )θψ ,rmn are the rigid duct normal modes and the mnk are the axial wavenumbers in the presence of an axial flow which read above cut-on

2

22200

1)1(

z

zmnz

MMkkM

kmn −

−−±−=± κ

, (7)

where 0k is the acoustic wavenumber, the mnκ are radial wavenumbers, each of them corresponds to the nth zero of the normal derivative of the mth Bessel function evaluated at the duct wall.

As the goal of this work is to show the influence of the source characteristics on the optimal selection of the liner parameters, we will provide expressions for the pressure and velocity fields generated by monopole-like and dipole-

Figure 1. Representation of the lined duct with corresponding modal amplitudes


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like source distributions which respectively correspond to the thickness noise and to the loading noise induced by the interaction between an incoming flow and the rotor blades. The general expression for the in-duct acoustic pressure radiated by a monopole source of volume velocity 0Q located at ( )000 ,, zr θ can be expressed as

( ) ( ) ( ) ( )

( )⎪⎩

⎪⎨⎧

≤

≥−−

=−−

−−

−+ −

+

∑∑0

02

00*

00

forfor

))(1(,,,,

0

0

zzezze

kkMrr

AQzrp

zzik

zzik

m n mnmnz

mnmnm

mn

mnθψθψωρθ , (7)

where 0ρ is the air density and A is the duct cross section. The corresponding expression for the axial velocity can be deduced from the linearized momentum equation as

( ) ( ) ( ) ( )

( )⎪⎩

⎪⎨⎧

≤

≥−−

=−−−

−−+

−+ −

+

∑∑0

02

00*

00,

forfor

))(1(,,,,

0

0

zzezze

kkMrr

AkQzrv

zzikmn

zzikmn

m n mnmnz

mnmnmz

mn

mn

ξξθψθψθ , (8)

where ±mnξ is the specific modal mobility ratio defined as

±

±±

−=

mnz

mnmn kMk

k

0ξ . (9)

The expression for the in-duct pressure field due to a dipole can be deduced considering the superposition of the field generated by two closely-spaced monopoles with opposite strengths. It is expressed as

( ) ( ) ( ) ( )

( )⎪⎩

⎪⎨⎧

≤

≥−−

=−−−

−−+

−+ −

+

∑∑0

02

00*

00

forfor

))(1(,,,,

0

0

zzekzzek

kkMrr

AFzrp

zzikmn

zzikmn

m n mnmnz

mnmnzd

mn

mnθψθψρθ , (10)

where dp is the pressure field due a point force, zF0 , per unit mass density. The corresponding axial velocity along the z-axis is given by

( ) ( ) ( ) ( )

( )⎪⎩

⎪⎨⎧

≤

≥−−

=−−−−

−−++

−+ −

+

∑∑0

02

00*

0

0,

forfor

))(1(,,,,

0

0

zzekzzek

kkMrr

AcFzrv

zzikmnmn

zzikmnmn

m n mnmnz

mnmnzdz

mn

mn

ξξθψθψ

θ , (11)

The expressions (7-11) ensure appropriate continuity conditions on either side of the source plane, namely for monopole-like sources

[ ] 0=−= −++− mnmnmn PPP , (12)

[ ] ( ) −+−+

+− −= mnmn

mnmnz

mnmn kk

kMr

AQV ξξθψ 2

02

00*

0

)1(, , (13)

where [ ]+−mnP and [ ]+−mnV denotes the jump in the modal pressure and velocity amplitude values from += 0zz to −= 0zz . The corresponding expressions for dipole-like sources are given by

[ ] ( ))1(

,2

00*

00

z

mnzmn M

rAFP

−=+

−θψρ , (14)


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[ ] ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡+

−= −

+++

− mnmn

mnz

mnzmn k

kMr

AcFV ξξθψ 0

200

*

0

0 1)1(

, . (15)

These equations will be used in the next section to impose suitable continuity conditions across the source position.

B. Pressure field for a lined duct with a mean axial flow along the axis direction The system to be analyzed consists of a rigid duct covered with an absorbent material situated at a distance

1lz = from the source section. The solution of the equations for the sound propagation in a non-uniform, longitudinal liner duct is based on the Multi-Modal Propagation Method (MMPM) derived by Pagneux et al. to model sound propagation in varying cross-section ducts with rigid boundary conditions and to the case of waveguides fitted with regions of non-uniform liners.16 This mode-matching method has been suitably modified to deal with a uniform flow along the duct axis.17

In this approach, the pressure and axial velocity field inside each duct section are expressed in terms of a weighted sum of the normal modes of the rigid walled duct (Eqs. 5-6). To obtain a governing equation for the field modal amplitudes in each duct section, it is necessary to combine the linearised equations of motion (the Euler equation, the mass conservation and the state equation) with the corresponding rigid or impedance boundary conditions, so that one can calculate the evolution of the projection coefficients along the duct axis. Following this approach, we arrive to a second order differential equation in the pressure field modal amplitudes as

0'22'')1( 000

20

000

2

0

002 =⎥⎦

⎤⎢⎣

⎡+−−⎥

⎦

⎤⎢⎣

⎡+−⎥

⎦

⎤⎢⎣

⎡+− PCILPCIPCI

ωωω

ρρρZ

cikkZ

cMMikMZkciM z

zzz , (16)

where ωZ is the liner acoustic specific impedance modeled as locally reacting, so that )(cot 0hkiRZ −=ω , with R the porous facing-sheet acoustic resistance and h the liner depth. In Eq. (16), I is the identity matrix, L is a diagonal matrix of the radial wavenumbers mnκ , and C is the modal cross-coupling matrix, which in the case of a uniform absorbing material, can be defined as

( ) ( )'

''

''''',

2mm

nmmn

nmmmnmnmmn

RJRJ δκκπΛΛ

=C , (17)

where mnΛ is the normalisation factor and ( )rJ mnm κ is the Bessel function of the first kind of order m . With an appropriate vector transformation, Eq. (16) can be reduced to a generalised eigenvalue problem, that provides a vector of eigenvalues, izK , , where Ni 2,1K= , and a matrix of eigenvectors, X , of dimensions

)22( NN × . From the eigenvalues associated with the modal pressure amplitudes, half of them will correspond to

plane waves propagating towards the increasing z -direction, +X , so that 0)Im( , <+izK , and the other half will

correspond to planes waves propagating towards the decreasing z -direction, −X , so that 0)Im( , >−izK . Applying

the appropriate conditions (continuity of pressure and axial velocity at the junctions together with suitable pressure and axial velocity jump or continuity conditions across the source section), one obtains a global solution for the field modal amplitudes at any position inside the duct. To illustrate the results calculated with the MMPM, numerical simulations have been performed to calculate the sound pressure field at a particular frequency of analysis for both tonal and broadband excitation conditions.

The physical parameters concerning the model have been taken representative of a real engine configuration. Hallez and Burdisso have presented an experimental study made on the TFE731-60 engine with the purpose of fan noise control using Herschel-Quincke resonators.15 The same parameters have been taken into account in this work for the optimisation of the liner attenuation. An inlet duct of diameter 0.787 m and axial length 0.51 m has been considered with a lined section of length 0.255 m located 0.127 m apart from the rotor section. One assumes a uniform inlet flow speed with axial Mach number equal to -0.12. Two different noise source configurations have been considered, a uniform distribution of monopoles and a set of axial dipoles over the duct diameter on the rotor


8

cross-section. The results obtained are represented in Fig. 2 at the engine BPF frequency of 2250 Hz, under tonal excitation conditions.

From the analysis of this figure, it can be seen that large differences can be observed in the acoustic pressure

field distribution when considering either a monopole-like or a dipole-like configuration for the excitation, both in the hard wall or in the lined duct case. The resulting pressure field due to a monopole excitation clearly shows a distinctive modal pattern (left) as the excitation frequency is just above the cut-on frequency of the (2,4) duct mode, whereas this wavenumber filtering behavior does not appear when considering a dipole excitation (right). Indeed, this is related in the hard-walled case to the pressure modal amplitudes which roughly depend on the inverse of the modal axial wavenumber under monopole source excitation whereas one observes a smoother distribution of the modal pressure amplitudes for a dipole excitation, with therefore a greater number of low-order modes contributing to the in-duct pressure field and which are less efficiently attenuated when traveling through the lined section. In the no-flow case, a dipole excitation would exactly create an equal modal pressure amplitude distribution.

The simulation results confirm, as it has already pointed out, that the coupling between the different combination of sources and the generated sound pressure field inside the duct is strongly dependent on the nature of the excitation. For further comparisons between the results published in the literature and those provided by the MMPM, the normalised squared modal amplitudes have been calculated as a function of the modal index for different model of excitations. Joseph et al.12 have presented two alternative source models: the assumption that all cut-on modes carry equal power, i.e. Equipartition of Energy per Mode (EEpM), and that all cut-on modes produce Equal Energy Density per Mode (EEDpM). They have also provided approximate analytical expressions for the modal amplitudes excited in a semi-infinite duct by an incoherent source distribution uniformly distributed over the duct cross section. These expressions can be particularised when assuming uncorrelated volume velocity sources and uncorrelated dipoles sources.

The results obtained with the MMPM when using 19 uniformly distributed incoherent monopoles and dipoles are compared with those provided in the analytical approximations in Fig. 3, when assuming a uniform flow of Mach number 12.0−=zM . As it can be appreciated, the MMPM is able to provide results very close to those obtained with the analytical approximations for both the monopole (black) and the axial dipole excitations (red). As already observed,12 it can be seen that the normalised squared modal amplitudes in the case of a uniform flow for either a monopole-type or a dipole-type excitation are quite different from those provided under the assumption of EEpM or EEDpM, especially when increasing the modal index. Although it is not shown here, simulations have also been carried out when no axial mean flow is assumed.

Figure 2. Sound pressure level predicted from an assumed model of 19 monopoles (left) and 19 dipoles (right) uniformly distributed over the duct diameter when comparing the rigid (top) and the lined (bottom) configurations (dB rel. Papref

5102 −⋅= )


9

The results show that with zero flow speed, the squared modal amplitudes for a uniform distribution of incoherent dipoles collapse on the squared modal amplitudes assuming EEDpM. However, there is not a clear correspondence in the flow case between the equal energy source types and the assumption of uniformly distributed uncorrelated sources. Therefore, the reconstruction of the strength of such source distributions from in-duct pressure-based measurements appears as a useful strategy to provide input source characteristics for liner optimization.

IV. Inverse problem for noise source reconstruction

In this section, a simulation study is presented for the analysis of the source strength reconstruction in lined duct geometries. The set of equations presented in Section II has been implemented for the calculation of the optimal source strength amplitudes. The primary excitation is supposed to be a number of point monopoles or axial dipoles uniformly distributed at 0zz = along a diameter of the duct cross section. A total number of 19 volume velocity sources or 19 axial dipoles oriented along the z -axis have been assumed for the reconstruction process. A radial distribution has also been considered for the simulated noisy pressure measurements, with an array of 32 microphones distributed over a duct diameter located at various axial separation distances with respect to the primary excitation we aim at recovering.

Fig. 4 shows the estimated source strengths in the tonal case assuming either a set of monopoles or axial dipoles as a function of their radial positions along the duct diameter, comparing the cases when measurements are performed in a fully rigid or a partially lined cylindrical duct. The axial distance between the sensor array and the source array has been fixed to 0.25 m so that the pressure measurements are acquired within the treated section in the lined case, and the frequency of analysis is taken as 2250 Hz ( 16≈dkR ). It corresponds to an absorption coefficient of 0.82 for the liner treatment which has been chosen with a resistance of 2.41 rayls and a depth of 3.1 cm. These parameters provide the liner with a maximum attenuation at 2760 Hz. For this particular configuration, the original transfer function matrix Z between the assumed reconstructed sources and the microphone positions is ill-conditioned and direct inversion leads to an unbounded solution.

Figure 3. The squared incident modal amplitudes normalized on the plane wave quantity with 19 uniformly distributed incoherent monopoles (black circles: bold, analytical approximation; thin, MMPM), 19 uniformly distributed incoherent dipoles (red diamonds: bold, analytical approximation; thin, MMPM), Equal Energy per Mode (green stars) and Equal Energy Density per Mode (blue plus).

Figure 4. The estimated source strengths as function of their radial positions along the duct diameter when assuming 19 monopoles (top) and 19 dipoles (bottom) for a rigid (blue) or a lined (red) cylindrical duct at a frequency of 2250 Hz. The assumed primary source strengths are shown by the dashed line.


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In order to obtain reliable results, it has been necessary to use a regularization technique such as the Tikhonov approach in combination with the L-curve method18 for the proper selection of the regularization parameter, when a normally distributed random noise with a SNR of 40 dB has been added to the original input data. We note that regularization is required up to a critical frequency (about 3200 Hz in our case) above which the number of propagating modes becomes higher than the number of singular values of the acoustic transfer matrix Z , i.e. the number of assumed sources, and for which direct pseudo-inversion of Z would then lead to meaningful results. From Fig. 4, it can be seen that the inverse model-based method is clearly able to provide an accurate estimation of the original source strength distribution for both the rigid and lined cases.

It has been shown previously19 that the main parameters affecting the performance results for this study are the separation distance between the source and the measurement array, the frequency range considered and the level of noise perturbation present in the measurements. Fig. 5 presents the normalized mean squared error in the source for the physical configuration when the axial separation distance between the source array and the measurement array is increased while Tikhonov regularization in combination with the L-curve method has been used for a proper selection of the regularization parameter for each separation distance. As it can be seen, the mean squared error in the regularized solution stays between -30 dB and -10dB, which guaranties accurate reconstruction results.

Also, the error globally increases with the source-sensor separation distance although in a non regular way. At low frequency (typically below 600 Hz), i.e. out of the frequency range of the liner maximum efficiency, it has been observed that the accuracy in the reconstruction results are almost the same in the rigid and in the lined configurations, as expected. However, at frequencies for which the liner has a large efficiency, monopole-type sources are usually reconstructed with a greater accuracy in the lined case with respect to the rigid case whereas the difference is not so well marked for the identification of dipole-type sources, as illustrated in Fig. 5. This difference is more pronounced, i.e. over a large range of separation distances, when one attempts to reconstruct monopole-type sources at higher frequencies but still within the liner maximum efficiency bandwidth. This is due to the fact that monopole-type sources efficiently excite a fewer number of just cut-on modes than dipole-type sources, as shown in Fig. 3. Hence, the attenuation per mode is greater in the former case with respect to the latter, so that the in-duct field reactivity (and the transfer matrix condition number) is lowered for monopole-type sources in a lined duct.

The results obtained with the inverse source strength reconstruction method can reduce the uncertainty existing in the input data to be introduced for the correct optimization of the physical liner properties. It can also provide further understanding about the physical phenomena involved in the duct excitation.

V. Optimization of the liner properties This section discusses the optimization of the physical properties of acoustic treatments that are used inside the

nacelle to reduce the turbofan engine noise. The parameter under consideration is the acoustic impedance of the absorbing material. This quantity is comprised of a real part, the resistance, and an imaginary part, the reactance, that has been expressed as a function of the liner cavity depth which corresponds to the thickness of the honeycomb sandwiched between a porous facing-sheet (with resistance R) and a rigid back plate. For the optimization, the attenuation values as a function of the real and imaginary part of the impedance are calculated using the MMPM propagation model detailed in Section III. However, in this procedure, an assumption has to be made about the source characteristics, i.e. about their location and also their degree of correlation in the broadband case.

Figure 5. The normalized mean square error for the source strengths as function of its radial distance along the duct diameter when assuming 19 monopoles (top) and 19 dipoles (bottom) for a rigid (blue) or a lined (red) cylindrical duct at a frequency of 2250 Hz.


11

Indeed, in the optimization process, we have considered different types of correlation structures between the broadband sources. To the best of authors’ knowledge, this analysis has not been considered before as the results presented up to now have been usually carried out assuming completely uncorrelated sources or EEpM/EEDpM modal distributions. In the first subsection we will present optimization results in the case of uncorrelated sources, and the results obtained will be confronted in the second subsection to the cases of a set of totally coherent sources and a set of partially correlated sources.

A. Set of incoherent uniformly distributed sources The sound power transmission loss is calculated considering the ratio between the incident power before the

liner and the transmitted power after the liner carried by all the cut-on modes, as

( )

( ) ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +++⎥⎦

⎤⎢⎣⎡

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +++⎥⎦

⎤⎢⎣⎡

=

∑

∑

++

++

mnmnzmnz

mnmnzmnz

MMA

MMATL

mn

mn

2223

2221

10

1Re)1(~E

1Re)1(~Elog10

ξξ

ξξ, (18)

where ⎥⎦⎤

⎢⎣⎡ 21~E

mnA and ⎥⎦

⎤⎢⎣⎡ 23~E

mnA correspond

respectively to the calculated auto-power spectra for the incident and transmitted pressure modal amplitudes, with E the expectation operator. The auto-power spectra correspond to the diagonal elements of the Cross-Spectral Density (CSD) matrices,

11 AAS and 33 AAS for the incident and

transmitted pressure modal amplitudes, respectively. Using Eq. (18), the sound power transmission loss has been calculated when considering four different assumptions on the source characteristics. The results shown in Fig. 6 as a function of frequency, have been obtained when considering the same sources than those used for Fig.3, namely a set of 19 uniformly distributed incoherent monopoles, 19 uniformly distributed incoherent axial dipoles, together with EEpM and EEDpM excitations. One generally observes that the transmission loss increases with frequency up to about 1 kHz and then stays at an upper level over the frequency bandwidth of the maximum efficiency of the liner, i.e. between 1 kHz and 4 kHz. Below 1 kHz, maxima in the transmission loss are mostly due to the fact that a small fraction of the incident modal amplitude is transmitted, most of it being reflected (without azimuthal conversion because of the axi-symmetric lined section) at the rigid/lined wall discontinuity. Moreover, these low order modes which preferentially convey the acoustic energy towards the duct axis are inefficiently dissipated through the liner material. Above about 1 kHz, the liner appears to be more efficient to attenuate in-duct pressure fields excited by uncorrelated monopole-type sources with respect to those due to EEpM, incoherent dipole-type and EEDpM source models, by decreasing order of magnitude for the overall transmission loss. This scales on the relative distribution of the incident squared modal amplitudes between each type of assumed source model, in accordance with the results shown in Fig. 3, thus providing some guidelines on the dependency of the liner attenuation properties with the respect to the nature of the sources.

Indeed, a narrowband modal filtering, as the one induced by a monopole-type excitation, provides an incident field dominated by a few number of modes just above cut-on. Such modes typically convey acoustic energy all the more close to the wall as one deal with high-order modes which also present a low azimuthal phase speed so that

Figure 6. Sound power Transmission Loss predicted from an assumed model of 19 uniformly distributed incoherent monopoles (blue), 19 uniformly distributed incoherent dipoles (red), EEpM (green) and EEDpM (black).


12

these modes are more easily dissipated by the liner. In the opposite case, an EEDpM excitation is an all-pass modal filter. In this case, the liner will be efficient to absorb the high-order modes, but it will hardly dissipate the energy of the low-order modes which equally contribute to the in-duct pressure field. Therefore, it appears that the narrower is the modal filtering induced by the sources over the liner absorption bandwidth, the larger is the transmission loss that can be achieved. In particular, it is found that, in the chosen configuration, an EEDpM source model tends to underestimate the transmission loss over the frequency range corresponding to the liner maximum efficiency whereas an EEpM source model provides transmission loss values in closer agreement with those obtained assuming incoherent monopole or dipole-type sources. These trends are still valid if one considers a larger or a fewer number of elementary sources.

The optimization results obtained for the transmission loss at a given frequency and as a function of the liner parameters are represented in Fig. 7 using the MMPM. It can be seen that the optimal liner depth is significantly influenced by the monopole-like or the dipole-like nature of the excitation, with differences up to 25 mm. However, the optimal liner resistance appears to be more robust to changes in the nature of the excitation. In accordance with the above comments, it is noted that the TL is larger when assuming incoherent monopole-type source distributions with respect to dipole-type sources for a wide range of the liner parameters.

B. Set of partially coherent uniformly distributed sources The optimization procedure is carried out when considering the CSD matrix between the source strength

amplitudes assuming a frequency-dependent exponential decay of the correlation function along the radial sources distributions, i.e. partially correlated sources distributed along a blade span. We have studied the case of fully correlated broadband sources, with a CSD matrix full of ones, and the case of partially frequency-dependent correlated sources, characterized by a CSD matrix taken from the model proposed by Blake20 to describe the statistics of Turbulent Boundary Layer (TBL) wall-pressure fluctuations, as

)(0, e),()( ωω r

ji

L

rr

ijiqq rSrrS

−−

= , (19)

where ),(0 ωirS is the point-power spectrum of the TBL and )(ωrL is the boundary layer radial correlation length given by ( )ωαω rcr UL =)( . Although this model assumes fully correlated wall-pressure fluctuations at low frequencies (typically below 100 Hz in our case), a more refined model at low frequencies should account for an upper bound for the correlation length depending on the turbulent boundary layer thickness (as in the Efimtsov model). rα is an empirical constant that has been taken as 0.7 for the simulations and cU is the convection velocity

Figure 7. Predicted sound power transmission loss (in dB) at the blade passing frequency in the (R, h) plane when R varies between 0.01 and 3 and h varies between 1 and 100 mm: 19 uncorrelated monopoles (left figure) and 19 uncorrelated axial dipoles (right figure) uniformly distributed over the duct diameter.


13

proportional to the relative free-stream velocity which reads ( )202

0 crMcU z Ω+≈∞ with Ω the angular shaft speed.

Figure 8 shows how the TL varies as a function of frequency for either monopole- or dipole-type type sources under different degrees of correlation between the sources. As already observed in the extreme cases of coherent or incoherent sources, the TL is weakly influenced by the duct cut-on frequencies under dipole-like excitations whatever the degree of correlation between the sources whereas this is a strong sensitivity of the TL at the duct cut-on frequencies under monopole-like excitations. Moreover, the number of cut-on modes highly coupled to the sources distribution increases as the degree of correlation between the sources decreases. Indeed, in the case of an axisymmetric distribution of fully correlated sources, only the duct modes with even azimuthal order well couple into the excitation and this is clearly seen on Fig. 8 when comparing the TL due to monopole sources either partially or fully correlated. Hence, for a given type of excitation, this spatial filtering induced by a fully correlated excitation tends to lower the values of the TL with respect to a partially correlated excitation. In our case, this occurs typically above 500 Hz for which the radial correlation length falls below about 20% of the duct radius. We should note that the TL induced by partially correlated sources coincides at low frequencies (typically below 200 Hz) with the TL due to fully correlated sources and at very high frequencies (above 4 kHz) with the TL due to incoherent sources.

Fig. 9 shows the TL optimization results assuming a distribution of 19 partially correlated monopole or dipole-

type sources at the BPF frequency, i.e. for a correlation length of about 5% the duct radius. As in the incoherent case, it is the optimal liner depth that is influenced by the different nature of the sources, with up to 20 mm predicted differences. In particular, the optimal liner depth assuming axial dipole excitations is lowered with respect to monopole-type excitations. Moreover, when comparing with Fig. 7, it appears that the liner optimal resistance that

Figure 8. Sound power Transmission Loss predicted from an assumed model of 19 partially (dashed) or fully (solid) correlated monopoles (black) and axial dipoles (red) uniformly distributed over the duct diameter

Figure 9. Predicted sound power transmission loss (in dB) at blade passing frequency in the (R, h) plane when R varies between 0.01 and 3 and h varies between 1 and 100 mm: 19 partially correlated monopoles (left figure) and axial dipoles (right figure) uniformly distributed over the duct diameter.


14

maximizes the TL is sensitive to the degree of correlation between the sources and is lowered when one increases the amount of correlation between the sources.

VI. Conclusions The methodologies used for the optimal design of liners require assumptions, either on the excitation source

characteristics, or on the duct squared modal amplitude distributions, that are included as an input in the propagation models to calculate the liner transmission loss for a range of parameters. In this work, a parametric study has been achieved to further understand how the liner efficiency depends on the ducted source characteristics. The MMPM method has been extended to calculate the pressure field in a lined duct for a given spatial distribution of elementary sources. It accounts in the flow case for the discontinuity conditions satisfied by the pressure and velocity modal amplitudes across the source section which comprises monopole- or dipole-like excitations. It has been found that, at a given frequency, the modal filtering effect significantly increases when one considers, by increasing order of magnitude, an EEDpM source model, a set of dipoles, an EEpM distribution and a set of monopoles. Hence, it appears that the reconstruction of elementary source strengths distributions from in-duct pressure-based measurements would be a useful strategy to provide input source characteristics for liner optimization. A simulation study has been undertaken to show the limitation performances of a MMPM-based inverse method, eventually regularized, to identify assumed source strengths distributions from noisy pressure data acquired in the lined duct. The presence of a lined section usually improves the accuracy of the reconstruction with respect to the rigid case, especially for monopole-type excitations. It has been observed that, the accuracy degrades when increasing the source-sensor separation distance (but non uniformly, usually at a reduced rate in the lined section), when increasing the level of noise perturbations in the measurements and in the low frequency range, especially below a critical frequency at which the number of propagating modes becomes lower than the number of assumed sources, and for which regularization is required. A simulation study has shown that the transmission loss scales on the modal filtering properties of the assumed source model. It appears that the higher is the modal filtering induced by the sources over the liner absorption bandwidth, the larger is the transmission loss that can be achieved. Furthermore, for a given type of excitation, the spatial filtering induced by a fully correlated excitation tends to lower the values of the TL with respect to a partially correlated excitation. For both incoherent and partially correlated elementary sources, it has been observed that the optimal liner depth may be greatly influenced by the nature of the sources (monopoles or dipoles) whereas the optimal liner resistance appears to be more robust to changes in the nature of the excitation.

A sensitivity analysis of the TL function with respect to the liner parameters (including the liner length) would be useful to further identify the underlying mechanisms that would explain the dependence of the optimal parameters on the source characteristics. This work would also benefit from experimental validations.

Acknowledgments This research has been supported by the European Commission under the Sixth Framework program, contract

INDUCT-MEIF-CT-2006-022579.

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