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Journal of Sound and Vibration

Journal of Sound and Vibration 333 (2014) 4458–4472

http://d0022-46

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

Side-branch resonators modelling with Green'sfunction methods

E. Perrey-Debain n, R. Maréchal, J.M. VilleLaboratoire Roberval UMR 7337, Université de Technologie de Compiègne, 60205 Compiègne, BP 20529, France

a r t i c l e i n f o

Article history:Received 7 June 2013Received in revised form23 April 2014Accepted 25 April 2014

Handling Editor: Y. Auregan

through such a system, the method is known to be extremely demanding, both in terms of

Available online 22 May 2014

x.doi.org/10.1016/j.jsv.2014.04.0600X/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ 33 44 23 46 41.ail address: emmanuel.perrey-debain@utc.fr

a b s t r a c t

This paper deals with strategies for computing efficiently the propagation of sound wavesin ducts containing passive components. In many cases of practical interest, thesecomponents are acoustic cavities which are connected to the duct. Though standardFinite Element software could be used for the numerical prediction of sound transmission

data preparation and computation, especially in the mid-frequency range. To alleviatethis, a numerical technique that exploits the benefit of the FEM and the BEM approach hasbeen devised. First, a set of eigenmodes is computed in the cavity to produce a numericalimpedance matrix connecting the pressure and the acoustic velocity on the duct wallinterface. Then an integral representation for the acoustic pressure in the main duct isused. By choosing an appropriate Green's function for the duct, the integration procedureis limited to the duct–cavity interface only. This allows an accurate computation of thescattering matrix of such an acoustic system with a numerical complexity that growsvery mildly with the frequency. Typical applications involving Helmholtz and Herschel–Quincke resonators are presented.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In a large number of sound generating devices, sound waves of large amplitude are being set up inside a tube and someof the acoustic energy propagate in the duct before being radiated into the open. The practical applications of such systemsrange from noise transmission in vehicle exhaust systems, through ventilation and air conditioning ducts, to soundpropagation in the ducted regions of turbofan aircraft engines.

The acoustic energy flow reduction passive techniques for the duct noise problem can be divided into two categories:(i) dissipative techniques specifically aim to absorb the sound field as it propagates down the duct and (ii) reactivetechniques specifically aim to alter the duct impedance by reflecting back most of the incident acoustic wave [1–3]. Typicaldevices of category (ii) mostly encountered in industrial systems are expansion chambers [4] and side-branch resonatorssuch as the quarter wave tube, the Helmholtz resonator [5] and the Herschel–Quincke tube (HQ tube) [6,7]. Side-branchresonators are added components that can be modeled as acoustic cavity resonators connected to the main duct via one ormore openings, this is illustrated schematically in Fig. 1.

(E. Perrey-Debain).

Fig. 1. Illustration of reactive systems in duct acoustics showing two Helmholtz resonators and a side-branch duct modified with an expansion chamber.The domain Ω refers to the acoustic cavity and Γ denotes the interface with the main duct. Given an incident acoustic pressure pI, the side-branch resonatorwill generate a scattered pressure pS.

E. Perrey-Debain et al. / Journal of Sound and Vibration 333 (2014) 4458–4472 4459

For specific geometries, the low frequency acoustic properties of such reactive systems can be obtained by means ofclassical one-dimensional acoustic theories, see for instance [8,6], or by using more advanced modal expansion techniques[9–11]. Though sophisticated, these semi-analytical models are restricted to the analysis of a specific class of cavitygeometry, generally circular, and are valid under the assumption that only the plane wave mode is allowed to propagate inthe main duct. There are however many configurations of practical interest where the cavity can have a more complicatedshape [11] and this especially concerns adaptive and tunable resonators [12–14], or for which the frequency range of interestis not necessarily low, i.e. well above the first transverse mode cut-off frequency in the main duct. The use of the HQ tubeconcept, for instance, has been investigated and tested for fan noise reduction in a highly multimodal context [15,16].

The assessment of the efficiency of such systems requires a precise knowledge of the acoustic field. Though standarddiscretization techniques [17] such as the very popular Finite Element Method could, in principle, be used for this purpose, afull FE model would be extremely demanding as the number of variables is expected to grow like f3 (f is the frequency). Tomake the matter worse, the FE method is known to suffer from pollution errors which can be avoided at a price of a veryhigh discretization level especially in the medium and high frequency ranges [18]. This can have a negative impact whensome efficient optimizations (geometry and positions of these added components for instance) are needed. An alternativeoption is to use the Boundary Element Method (BEM) [19] (see applications in [5,11] for instance). Again, boundary elementformulations typically give rise to fully populated matrices and this limits its use even for medium-sized problems. The BEMalso suffers from numerical difficulties when applied to interior problems as shown in a recent paper [20].

In this work, we present a new numerical procedure that judiciously exploits the benefit of the FEM and the BEMapproaches. The proposed approach can be regarded as an extension of a previous work concerning the acousticpropagation in branched tubes [21] to arbitrarily-shaped cavity resonators. The idea relies mainly on the conceptof impedance matrix that connects the pressure to the acoustic normal velocity on the duct-cavity interface. Thisconsiderably reduces the number of variables as only the interface needs to be discretized. The technique based on theuse of integral equations for the acoustic pressure can be viewed as a Boundary Element Method dedicated to duct acousticswhereby the traditional free space Green's function has been replaced by an appropriate Green's function satisfying the rigidboundary conditions at the duct walls. The theory is presented in Sections 3 and 4. The last section shows practicalapplications involving Helmholtz resonators and HQ tubes showing comparisons with numerical results previouslypublished in [6,11]. The benefit of the present approach is shown in terms of data reduction, when compared to standardFE models.

2. Impedance matrix

Consider a three dimensional acoustic cavity Ω with rigid walls. Let us introduce the interface Γ, the boundary surfacecorresponding to the different openings of the cavity (see Fig. 1). For the moment, the shape(s) of Γ is (are) rather arbitraryand its exact definition will be clarified later. In the frequency domain, the acoustic pressure must obey the Helmholtzequation

Δpþk2p¼ 0; (1)

and ∂np¼ 0 everywhere except on Γ. Here ω, c and k¼ ω=c denote respectively the angular frequency, the speed of soundand the wavenumber. The aim of this section is to determine the linear relationship connecting the pressure p to its normalderivative q¼ ∂np on Γ (n is the unit vector normal to the surface). Standard results show that there exists a complete set oforthogonal eigenfunction Φn belonging to the spectrum of the Laplacian operator [22]:

ΔΦn ¼ �k2nΦn; (2)

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subject to the boundary conditions: ∂nΦn ¼ 0 everywhere on the boundary. Any given pressure field in Ω can be written as aweighted sum of the eigenfunctions. In particular, Green's function for the cavity, is given by the infinite series

GΩ r; r0ð Þ ¼ ∑1

n ¼ 0

ΦnðrÞΦnðr0Þω2n�ω2 ; (3)

where for the sake of clarity, we put ωn ¼ knc. Eigenfunctions are properly normalized so that application of Green's theoremin the cavity yields

pðrÞ ¼ZΓGΩðr; r0Þqðr0Þ dΓðr0Þ: (4)

Note that, as opposed to classical boundary integral formulations encountered in BEM, Eq. (4) is not only valid for any pointr inside the cavity but also on the boundary Γ. Discretization of Eq. (4), using a standard collocation method for instance,gives rise to an impedance matrix connecting the pressure to its normal derivative (or equivalently, the acoustic velocity). Inmany instances, the dimensions of the openings are small compared to the wavelength and it may be assumed that thesetwo quantities are almost constant over each opening so that Eq. (4) can be averaged to produce a single impedance matrixcoefficient [23]. The use of Eq. (4) requires the precise knowledge of the eigenpair ðΦn; knÞ. Apart from separable geometriesfor which eigenfunctions are available analytically, the use of computational methods become unavoidable for arbitrarilyshaped cavities. In this regard, the FEM is probably the simplest and most reliable numerical tool for eigencomputation. Theassociated weak formulation of (1) readsZ

Ω∇p � ∇δp dΩ�k2

ZΩpδp dΩ¼

ZΓqδp dΓ; (5)

where δp is a properly chosen weighting function. We consider a finite element mesh made of linear elements (tetrahedralelement, T4) and we denote by ϕi the associated piecewise linear shape function. Application of the standard procedure tothe weak form (5) leads to the following algebraic problem:

AðλÞp¼ Fq where AðλÞ ¼K�λM; (6)

where matrices involved are obtained from

ðKÞij ¼ZΩ∇ϕi � ∇ϕj dΩ ðMÞij ¼

1c2

ZΩϕiϕj dΩ; (7)

and we put λ¼ ω2 for convenience. Here p contains the value of the pressure at all nodes of the FE mesh and q contains thenodal value of the normal derivative of the pressure on Γ. To ease the demonstration, we proceed to an appropriateelements reordering so that the rectangular matrix F looks like

F¼ IΓ ~F with ð ~FÞij ¼ZΓ

~ϕi~ϕj dΓ; (8)

where IΓ denotes the identity matrix for nodes located on Γ. Here, the tilde symbol signifies that we consider the restrictionof the three dimensional shape function ϕi to the interface. Thus, ~ϕi ¼ ϕijΓ stands for the piecewise linear interpolatingfunction associated with classical two dimensional triangular elements. We call qi ¼ qðriÞ the value of q at the J surface nodesri so we can write

q¼ ∑J

i ¼ 1

~ϕiqi: (9)

Now, let us introduce the eigenvector Φn the FE discretization of nth eigenmode Φn of the rigid cavity (we put q¼ 0)satisfying

AðλnÞΦn ¼ 0: (10)

The inversion of (6) can be performed formally by invoking the orthogonality properties of the eigenvectors to give

A�1ðλÞ ¼ΦDðλÞΦT: (11)

The matrixΦ¼ ðΦ1;Φ2⋯Þ contains in its columns the nodal value of the eigenmodeΦn and D stands for the diagonal matrixwith its diagonal entries: Dnn ¼ ðλn�λÞ�1. After reduction to the interfacial nodes, we obtain the impedance matrix

ZðλÞ ¼ ITΓA�1ðλÞIΓ : (12)

Finally, we have

~p ¼ ZðλÞ ~Fq: (13)

where the vector ~p contains the nodal values of the pressure on the interface, i.e. ~pi ¼ pðriÞ. Eq. (13) can be interpreted as thediscrete form of the integral formulation equation (4). The interest for the modal decomposition (11) becomes clear whenthe frequency of interest is taken well below the highest modal ‘resonant’ frequency ωN (i.e. λ5λN). In this frequency range,

E. Perrey-Debain et al. / Journal of Sound and Vibration 333 (2014) 4458–4472 4461

the truncation of the modal series keeping the first N eigenmodes gives

ZðλÞ ¼ ~ΦNDNðλÞ ~ΦTNþRðλÞ: (14)

Here again, the tilde symbol means that we only retain the nodal values of the eigenmodes on the interface Γ. The Ntheigenvalue must be chosen high enough to ensure that the correction term R remains weakly dependent on the frequency,so we can take the first-order Taylor expansion

R λð Þ � R λ� �þ λ�λ

� �∂R∂λ

λ� �þ⋯; (15)

where λ is a reference value. Once the matrix R¼ RðλÞ and its first derivative have been stored, the computation of theimpedance matrix (14) becomes a very fast and simple procedure. The residual matrix is computed simply using

R¼ ITΓA�1IΓ� ~ΦNDN ~ΦT

N ; (16)

and the first-order derivative can be computed from

∂R∂λ

¼ ITΓA�1MA�1IΓ� ~ΦN

∂DN

∂λ~ΦTN : (17)

Now exploiting the symmetry of A, we see that ITΓA�1 ¼ ðA�1IΓÞT. Thus the important point to make is that the full

inversion of A is not needed here. Only the first columns of A�1 corresponding to the interface nodes are active and thesecan be efficiently computed by solving successively Avi ¼ ei with appropriate sparse solvers (ei is the column vector withzero elements with the unity on the ith line). To avoid any round-off error during the inversion process, it is judicious tochoose λ away from a resonant value. Here we take simply λ ¼ λ2=2, recalling that the first eigenvalue λ1 ¼ 0 corresponds tothe constant pressure mode in the cavity.

3. Scattering matrix of the reactive system

For the sake of generality, we consider a hard-wall duct of arbitrary cross section S. We note (x,y) the transversecoordinates and z the longitudinal coordinate. The theory starts by introducing the hard-walled duct Green's functionsatisfying the usual modal radiation condition on both ends of the main duct, i.e.

G r; r0ð Þ ¼ ∑1

l ¼ 1

ψ lðx; yÞψn

l ðx0; y0Þ2iβl

eiβljz� z0 j: (18)

Here r ¼ ðx; y; zÞ and r0 ¼ ðx0; y0; z0Þ are two points in the duct. The βl's are the axial wavenumbers and transverse eigenmodesψl are solution of the boundary value problem

ð∂2xxþ∂2yyÞψ lþk2ψ l ¼ β2l ψ l (19)

with ∂nψ l ¼ 0 on the boundary line ∂S. These modes are normalized asZSjψ lj2 dS¼ 1; (20)

in particular the plane wave mode is simply given by ψ0 ¼ 1=ffiffiffiffiffiffiAd

pwhere Ad is the cross section area of the main duct. Once

the cavity resonator has been connected to the main duct, it becomes clear that the shape of the opening Γ follows that ofthe duct wall. Now, given the incident acoustic wave pI , the side branch resonator will generate a scattered field pS which isexplicitly given via the integral representation

pSðrÞ ¼ZΓGðr; r0Þqðr0Þ dΓðr0Þ (21)

and the total pressure is recovered as p¼ pIþpS. The finite element discretization of the integral in (21) taking r¼ ri yields

~pi ¼ ∑J

j ¼ 1GijqjþpIi ; (22)

where Green's matrix coefficients are calculated from

ðGÞij ¼ Gij ¼ZΓGðri; r0Þ ~ϕjðr0Þ dΓðr0Þ: (23)

This expression also arises in collocation BEM [19]. With classical Green's functions, G behaves like jr�r0j�1 as r-r0 andappropriate numerical integration procedures must be used. In the present context, the modal expansion (18) does notexhibit a singular behavior. However, a close inspection reveals that the first derivative in z is discontinuous at z0.Consequently, care must be followed as to ensure that the plane of discontinuity z¼ z0 is taken into account in order toavoid poor convergence and significant integration errors. In practice, the interface Γ is discretized with flat triangularlinear elements. For a given collocation point ri ¼ ðxi; yi; ziÞ, elements crossing the plane z¼ zi must be decomposed into

Fig. 2. Subdivision of a flat triangular element into three sub-elements: element I is located in the region z4zi whereas elements II and III are located inthe region zozi.

E. Perrey-Debain et al. / Journal of Sound and Vibration 333 (2014) 4458–44724462

three sub-elements of triangular shape as indicated in Fig. 2. Integration is then performed over each sub-element usingstandard Gaussian quadrature rules.

Combining (13) with (21) gives

q¼ ðZðλÞ ~F�GÞ�1pI ¼QpI (24)

and the pressure is recovered via (21). In this regard, the scattering properties of the reactive system are convenientlymeasured in terms of cut-on modes. Thus the incident pressure is split into right-going (þ) and left-going (�) componentsas

pIðrÞ ¼ ∑L

l ¼ 1ðAþ

l Ψ þl ðrÞþA�

l Ψ �l ðrÞÞ; (25)

where the summation is limited to cut-on modes, i.e. βl is purely real and positive. Here, we introduced the functionΨ 7

l ðrÞ ¼ ψ lðx; yÞe7 iβlz for conciseness. By calling ðΨ7 Þil ¼Ψ 7l ðriÞ, Eq. (25) takes the form

pI ¼ΨþAþ þΨ�A� : (26)

The scattered field in both regions, that is on the left (�) or the right (þ) of the resonator, is

pS;7 ðrÞ ¼ ∑L

l ¼ 0B7l Ψ 7

l ðrÞ: (27)

where scattered coefficients are given by

B7l ¼ 1

2iβl

ZΓΨ 8

l r0ð Þq r0ð Þ dΓ r0ð Þ: (28)

Using (9), this has the alternative form

B7l ¼ ∑

J

i ¼ 1X7li qi where X7

li ¼ ðX7 Þli ¼1

2iβl

ZΓΨ 8

l r0ð Þ ~ϕi r0ð Þ dΓ r0ð Þ: (29)

Finally, combining Eqs. (24), (25) and (29) yields

B7 ¼X7Q ðΨþAþ þΨ�A� Þ: (30)

The scattering matrix is formed by identifying the incoming wave coefficients A7 and the outgoing waves C7 ¼A7 þB7 .Finally

Cþ

C�

!¼ IþXþQΨþ XþQΨ�

X�QΨþ IþX�QΨ�

!Aþ

A�

!: (31)

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4. Simplified models

For typical low-frequency applications, the model can be simplified by assuming that the size of the opening(s) is (are)small enough so that the pressure and the velocity are almost constant over each interface, i.e. we put ~p � p and q� q, herethe bar symbol signifies that we take the mean value over the opening. Further simplifications can be made by consideringthat the frequency of interest is well below the cut-off frequency of the first transverse mode so we can limit Green'sfunction modal series to the plane wave mode component only. We shall now develop the models for two scenarios ofpractical interest.

4.1. Resonator with one opening

Starting with one opening only, the matrix relation (13) can be averaged to give

p ¼ Z λð Þq where Z λð Þ ¼ 1J∑J

i ¼ 1∑J

j ¼ 1ðZðλÞ ~FÞij: (32)

By the same token, the relation (22) yields a single equation for the mean quantities as

p ¼ 1W

qþpI where W ¼ 2ikAd

A(33)

and A denotes the area of the interface. These two equations allow us to obtain q as a function of the incident field. Withoutloss of generality we take simply pI ¼ eikz and assume that the opening is located around the origin z¼0. This yields pI ¼ 1and we can calculate the transmitted pressure field p¼ Teikz where

T ¼ 1þ 1WZ ðλÞ�1

: (34)

Thus, no acoustic energy is transmitted at the resonance frequency λR ¼ ω2R satisfying Z ðλRÞ ¼ 0 (it is reminded that Z is a

real-valued function).

4.2. Resonator with two openings

We consider a resonator connected to the main duct via two openings located at z¼ z1 and z¼ z2. By calling p1 and p2the associated mean pressure (similarly we define q1 and q2), we have (we omit the λ-dependence for clarity)

p1

p2

!¼ Z11 Z12

Z21 Z22

!q1

q2

!: (35)

We shall simplify the analysis further by considering symmetric resonators so that Z12 ¼ Z21, Z11 ¼ Z22 and A¼ A1 ¼ A2.From (22), we obtain

p1

p2

!¼ 1W

1 eikL

eikL 1

!q1

q2

!þ

pI1

pI2

!; (36)

where L¼ jz2�z1j. This finally gives the transmitted field p¼ Teikz where

T ¼ 1þ2WZ11�2WZ12 cos ðkLÞþðe2ikL�1ÞðWZ11�1Þ2�ðWZ12�eikLÞ2

: (37)

Thus, no acoustic energy is transmitted if

Z212�Z

211�

A sin ðkLÞkAd

Z12 ¼ 0: (38)

5. Numerical examples

We shall demonstrate the efficacy of our numerical model for two typical resonators, the Helmholtz resonator and theHerschel–Quincke tubes. In all our applications, the main duct has a circular cross-section with radius a. Transversenormalized eigenmodes can be found in any standard textbooks. They are given explicitly in polar coordinatesðx; yÞ ¼ rð cos θ; sin θÞ by

ψ l x; yð Þ ¼ Jmðαmnr=aÞeimθffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπa2 1�m2

α2mn

� �J2m αmnð Þ

s ; (39)

E. Perrey-Debain et al. / Journal of Sound and Vibration 333 (2014) 4458–44724464

where Jm stands for the Bessel function of the first kind of order m and αmn is the nth roots of Jm. The correspondencel2ðm;nÞ is given by ordering the roots αmn in ascending order. The corresponding axial wavenumbers are given by

βl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�ðαmn=aÞ2

q. The integration of these modes over the flat triangular elements of the interface Γ (see (29)) is carried

out by assuming that the radius remains constant r¼a, which is not a stringent condition for sufficiently refined meshes.Given incident acoustic modes with complex amplitude Aþ

l (we take A�l ¼ 0), the efficiency of the reactive system is

conveniently measured in term of the Transmission Loss (TL) defined as

TL¼ �10 log10∑L

l ¼ 1βljCþl j2

∑Ll ¼ 1βljA

þl j2

!; (40)

where coefficients Cþl are calculated from the scattering matrix (31).

5.1. Helmholtz resonator

This first numerical example concerns a Helmholtz resonator of cylindrical shape whose dimensions are taken from [9]and are reminded here for the sake of completeness (see Table 1, 3rd line in [9]): the resonator volume dimensions arelv¼24.42 cm (height), dv¼15.319 cm (diameter) and the dimensions of the neck are lc¼8.5 cm (height), dc¼4.044 cm(diameter). The diameter of the main duct is 2a¼4.859 cm. A 3D view of the FE mesh of the resonator is illustrated in Fig. 3(a). This mesh corresponds to approximately 2500 FE nodes and there are about J � 100 nodes on the saddle-shapedinterface between the neck and the main duct. The geometry was designed using the Open Source parametric 3D CADmodeler: FreeCAD [24] and the meshing is performed with the 3D finite element mesh generator: Gmsh [25]. Thecalculation of the FE matrices from (7) as well as the first N eigenmodes was implemented on Matlab. Once parameterized,the whole process is a quick operation (few minutes at most on a personal computer). Fig. 4 shows the evolution of theaverage calculated impedance Z with respect to the frequency using N¼1,10 and 100 cavity modes. This result is instructiveas it gives the number of eigenmodes needed to ensure convergence of the truncated series (14) (and in fact convergencewas reached with about 50 modes in this frequency range). The corresponding TL for the plane wave mode (using N¼100modes in the series) is reported in Fig. 5. It is striking to see how good results obtained with the simplified model are. Thismodel somehow takes into account, at least partially, the geometry and therefore the scattering mechanisms that take place

Table 1Cut-off frequencies for a rigid duct of radius a¼7.5 cm.

Mode number m n Cut-off freq. (Hz)

l¼1 0 1 0l¼2 �1 1 1338l¼3 1 1 1338l¼4 �2 1 2219l¼5 2 1 2219l¼6 0 2 2784

Fig. 3. Helmholtz resonator (a) conventional, (b) with extended neck.

Fig. 4. Average impedance with respect to the frequency using respectively N¼1 (dashed line), N¼10 (dot-dashed line) and N¼100 cavity modes(solid line).

Fig. 5. Transmission Loss for the Helmholtz resonator (conventional). Legend: full model (solid line); simplified model (dot-dashed line); 1D model(dashed line).

E. Perrey-Debain et al. / Journal of Sound and Vibration 333 (2014) 4458–4472 4465

at the main duct junction. For the sake of comparison, the 1D estimation from [9]:

TL¼ 10 log10 1þ d2c8a2

tan ðkl0cÞþðdv=dcÞ2 tan ðklvÞ1�ðdv=dcÞ2 tan ðkl0cÞ tan ðklvÞ

!224

35; (41)

is also shown. Here l0c stands for the modified length of the neck l0c ¼ lcþδ where the correction length must account for thefluid motion on both sides of the neck. Following Ingard's analysis [3], an approximated expression for the added length δthat accounts for the neck-volume interface can be found in [9]:

δ� 0:85dc2

1�1:25dcdv

� �: (42)

The scattering effects at the main duct junction are not included in (42) which explains the slight but noticeable deviationfrom the full model. In order to take advantage of the flexibility of the FEM, an extended neck is now included in theprevious resonator as shown in Fig. 3(b). Its presence allows us to shift the resonance frequency without changing thevolume. The length of the neck is half the height of the cylindrical cavity (12.21 cm). The mesh contains approximately 7000FE nodes and N¼100 eigenmodes are used in the truncated series (14). The corresponding TL for the plane wave mode isreported in Fig. 6. The simplified model yields accurate results for a negligible computational cost as it takes less than a

Fig. 6. Transmission Loss for the Helmholtz resonator (with extended neck). Legend: full model (solid line); simplified model with no sheet (dot-dashedline); simplified model with a perforate sheet (dashed line).

E. Perrey-Debain et al. / Journal of Sound and Vibration 333 (2014) 4458–44724466

minute for about a hundred of calculations. The presence of a perforate sheet located on the interface can be taken intoaccount in a straightforward manner by simply adding the perforate sheet normalized impedance, ζ, to the purely reactiveimpedance of the cavity. For the simplified model, the transmitted coefficient takes the form

T ¼ 1þ 1WðZ� iζ=kÞ�1

: (43)

The corresponding TL is also shown, here we took the perforate sheet impedance model from [26]. The dimensions of theplate are t¼0.8 mm (thickness), d¼1.5 mm (hole diameter) and s¼10 percent (porosity).

5.2. Herschel–Quincke tubes

5.2.1. 1 tube in the low frequency regimeThis second numerical example concerns a HQ tube having the form of a side-branch duct of circular cross-section

connected to the main duct. For a single HQ tube, it is interesting to observe that if the tube can be modelized as if it was astraight pipe of length l then one can calculate that Z11 ¼ �cotðklÞ=k and Z12 ¼ �1=ðk sin ðklÞÞ. The resonance relation (38)becomes

sin ðklÞsin ðkLÞ ¼ � A

Ad: (44)

The resonance frequency satisfying (44) is precisely the one given by Selamet et al. (see Ref. [6, Eq. (12)]) under thecondition that the area A in (44) must now be regarded as the cross-sectional area of the tube and not the one of the saddle-shaped area which can be substantially higher. The precision of the 1D formula also depends upon length corrections inorder to estimate the effective lengths, this is discussed in [6]. To illustrate this, we consider a HQ tube very similar to [6]having a minimum centerline distance between the junctions of l¼70.766 cm with a cross-sectional area πd2c=4(dc¼4.674 cm is the inside diameter of the tube). The average distance between the two openings along the main duct isL¼44.524 cm and the duct radius is a¼2.4295 cm. The corresponding TL for the plane wave mode is reported in Fig. 7. Thetwo results calculated with the full and the simplified model (Eq. (37)) are in excellent agreement. These are compared withthe 1D formula by replacing Z11 ¼ �cotðklÞ=k and Z12 ¼ �1=ðk sin ðklÞÞ in (37) and by using the effective area of the tube:πd2c=4. Following on from the previous discussion, we can anticipate that the knowledge of the modified length l0 ¼ lþδ inthe 1D formula would have led to much better results.

5.2.2. Herschel–Quincke tubes arrayThis example illustrates the acoustical effects of the HQ tubes concept in a multimodal context. It consists of an array of

HQ tubes aligned in a circumferential array around the main duct. Initially, the idea originated from [27] in order to reducenoise from real turbofan engines and in particular the tonal noise at discrete frequencies produced by the fan (this will bethe subject of the next section). Here, we shall restrict ourselves to much smaller acoustic systems in order to compare ourresults with standard FEM. The HQ system comprises 3 identical tubes equally spaced around the main duct with innerradius a¼7.5 cm. The numerical prediction of the acoustic pressure field using classical Finite Element Analysis requires asufficiently refined mesh to ensure accurate results. An example of the corresponding mesh comprising approximately

Fig. 7. Transmission Loss for the HQ tube. Legend: full model (solid line); simplified model (dot-dashed line); 1D model (dashed line).

Fig. 8. FEM mesh for the 3 HQ tubes system.

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83 000 nodes is shown in Fig. 8. The radiation conditions on both sides of the duct is implemented using the DtN map [28].The correspondence l2ðm;nÞ as well as the cut-off frequency for the cut-on modes are reminded in Table 1. TheTransmission Loss for an incident plane wave is shown in Fig. 10 where three resonance peaks are clearly identified. The FEMmesh of the corresponding HQ tube is shown in Fig. 9(a). Note that Green's function method only requires about 80 nodes ateach interface making a total of 480 degrees of freedom which is considerably smaller than the FEM. Furthermore, usingsymmetry arguments and rotations, the computation of Green's matrix coefficients can be accelerated by observing thatGreen's matrix for the combination of 3 tubes can be calculated from Green's matrix of a single HQ tube. Following the sameprinciple, it can be shown that the modes excited by an array of NT identical tubes equally spaced around the duct satisfy therelation [15]

mS ¼mI7qNT ; (45)

where mS is the circumferential mode order excited by the HQ system, mI is the circumferential mode order of the incidentfield and q is an integer number. If we apply this selection rule to the present case (NT¼3), we find that an incident planewave cannot be converted into other propagative modes in the frequency range f o3000 Hz. This explains why 3D results inFig. 10 exhibit a one-dimensional character. The results of Fig. 11 show the effect of the shape of the HQ tube on theresonance frequency of the first peak. In order to make a fair comparison, we ensured that the centerline length is identicalfor both tubes. Effects are clearly non-negligible on the HQ resonator especially when the resonator is efficient only in anarrow frequency band. Now, let the first transverse mode (ðm;nÞ ¼ ð�1;1Þ for instance) be the incident wave. The selectionrule (45) indicates that the mode ðm;nÞ ¼ ð1;2Þ should be excited above its cut-off frequency (2219 Hz). This is shown in

Fig. 10. Transmission Loss versus the frequency. Legend: Green's function method (solid line), FEM results (dashed line).

Fig. 11. Transmission Loss versus the frequency. Legend: sharp bend tube (solid line), circular (dashed line).

Fig. 9. Shape of the HQ tube (a) sharp bend, (b) circular.

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Fig. 12 where the corresponding modal coefficients of the transmitted field (in absolute value) are plotted with respect tothe frequency. We can identify two specific frequencies for which the incident mode is completely (or nearly) attenuated.Thus this HQ system can serve as a mode converter as well as a mode filter.

Fig. 12. Coefficients of the transmitted field (abs. value) for an incident mode amplitude Aþ2 ¼ 1. Legend: Cþ

2 (solid line), Cþ5 (dashed line).

Fig. 13. Illustration of the HQ system with 36 tubes.

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5.2.3. Application of the HQ system for inlet fan noise reductionThe concept of Herschel–Quincke (HQ) waveguide resonators has been the subject of intensive research for fan noise

control in aircraft turbofan engines. The approach has been numerically and experimentally tested in combination withtypical acoustic liners for both inlet and aft fan noise reduction (see for instance [15,16,29] and references therein). Inparticular, results have demonstrated the potential of the HQ tubes for the Blade Passing Frequency (BPF) tone attenuationwhereas the liner is effective to reduce broadband noise. In this section, our aim is to show the benefit of the proposedmethod in the early stage of the design process in order to find an optimized HQ system configuration.

The HQ system consists of a regular array of HQ tubes installed around the circumference of the inlet duct which acts as acircular acoustic waveguide as shown in Fig. 13. The dimensions are taken from a typical turbofan engine: the radius isa¼0.98 m and the length of the duct is 60 cm. The incident sound field generated by the fan is decomposed in terms ofmodes with different amplitudes:

Aþ ¼Aþrand:þAþ

BPF: (46)

The broadband noise is modeled as a set of incident modes Aþrand: with a random phase carrying equal power and the total

Sound Power Level (SPL) is fixed at 90 dB. At the BPF discrete tones 650 Hz, 1300 Hz, 1950 Hz, etc. which correspond to anengine speed of 4875 tr/min with B¼8 rotating blades, the additional acoustic power is carried by azimuthal modes Aþ

BPFaccording to the well known theory of Tyler and Sofrin [30]:

mBPF ¼ q0B7qV ; (47)

where q0 and q are integers and V is the number of vanes (V¼17 here). For instance, the 2 cut-on modes generated by (47) atthe BPF1 (with q0 ¼ 1) are (�9,1) and (8,1). By calling T¼ IþXþQΨþ the transmission matrix (see Eq. (31)), the transmittedpower is calculated via [31]

⟨Wt⟩¼1

2ρωðAþ

rand:Þ† diag T†DβT� �

Aþrand:þðAþ

BPFÞ†T†DβTAþBPF

h i; (48)

Table 2Convergence of the Transmission Loss with the number of terms in the modal series (18) at the BPF1 (f¼650 Hz). The total incident power is 100 dB.

Criterion C Number of modes ⟨Wt ⟩ (dB) TL (dB) ðWi� ⟨Wt ⟩� ⟨Wr ⟩Þ=Wi (%)

10 3477 93.3 6.7 0.0004715 7780 94.9 5.1 0.0002217.5 10 563 95.3 4.7 0.0001220 13 777 95.5 4.5 0.00009

Fig. 14. Acoustic performances of the HQ system with respect to the Blade Passing Frequency. Results have been calculated using C¼10 (cross), C¼15(circle), C¼17.5 (dots) and C¼20 (solid line). The total incident power is 100 dB (dot-dashed line).

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where ⟨ ⟩ refers to the ensemble average over realizations and Dβ is a diagonal matrix containing the modal wavenumbers.The second term in (48) is only included at the BPFs. On the basis of physical arguments, the length of the HQ-tube, or to bemore precise the centerline length, should be initially chosen as to ‘resonate’ at the BPF1, that is near the first resonancefrequency of the tube assuming pressure release boundary conditions [16]. Assuming that the tube is straight, this gives0.2633 m. For our calculation, we retained a smaller value of 0.2565 m. The HQ-tube is meshed using about 24 500 lineartetrahedral elements (around FE 4400 nodes) and the impedance matrix is recovered using 50 eigenmodes in (14). There are70 FE nodes on each side of the tube, thus the size of the impedance matrix is 140�140. The computation of the scatteredcoefficients only requires the inversion of a full matrix in (24) whose size is 5040 and this corresponds to the number of FEnodes on the interface of a single HQ tube multiplied by 36. Table 2 shows the convergence value of TL with respect to thenumber of terms in Green's function modal decomposition (18). The left column refers to a criterion C which is used toselect the total number of modes in the modal series and all modes (m,n) satisfying the inequality

αmnrCκa (49)

are included in the summation. For instance C¼1 means that only propagative modes are retained. Results show that C¼20guarantees converged results and this criterion has been verified in all results presented in this paper. The fact that thenumber of modes satisfying (49) grows roughly as the square of Cκa implies that, for relatively large κa (which is around 12at the BPF1), the criterion becomes penalizing in terms of computational time as each mode must be integrated numericallyover all the elements of the interface. This stems from the slow convergence of the series (18) especially as r0-r, this isdiscussed in [32]. There is obviously room for improvements for increasing the rate of convergence of the modal series. Thiscould be done, for instance, in the spirit of Refs. [32,33] for the bidimensional case and we leave this point for furtherstudies. From the last column, the energy conservation

⟨Wt⟩þ⟨Wr⟩¼Wi (50)

is satisfied and does not vary much with the number of modes and this indicates that, although the acoustic energy isconserved, results are not necessarily accurate.

Now the question may arise as to whether this HQ system is optimal? One way to answer this is to run many numericaltests with HQ-tubes of different lengths or shapes. An easier option is to vary the Blade Passing Frequency around the targetfrequency of 650 Hz (this corresponds to changing slightly the operating conditions). The acoustic performances of the HQsystem are shown in Fig. 14. The convergence of the numerical model is also shown. We may notice that the lack of accuracy(i.e. due to an insufficient number of modes in the series (18)) is reflected by a shift in frequency of the performance curve.

Fig. 15. Incident and transmitted modal Sound Power Level (in dB) at 627 Hz (left) and 650 Hz (right). Modes generated by the Tyler and Sofrin theory are(�9,1) and (8,1). Legend: incident power (back row) and transmitted power (front row). Modes are identified in Table 3.

Table 3The correspondence l2ðm;nÞ is given by ordering the roots αmn in ascending order. Mode numbers 29 and 35 (in bold) correspond to Tyler–Sofrin modes.

l (m, n) l (m, n) l (m, n)

1 (0,1) 13,14 (85;1) 26,27 (84;2)2,3 (81;1) 15,16 (82;2) 28,29 (88;1)4,5 (82;1) 17 (0,3) 30,31 (82;3)6 (0,2) 18,19 (86;1) 32 (0,4)7,8 (83;1) 20,21 (83;3) 33,34 (85;2)9,10 (84;1) 22,23 (81;3) 35,36 (89;1)11,12 (81;2) 24,25 (87;1) 37,38 (83;3)

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To the authors’ experience, this behavior can also be observed when using the FEM and the accurate simulation of resonanceeffects at high frequency is a challenging task. Converged results obtained with C¼20 tell us that the HQ system is in factoptimal for a BPF of 627 Hz giving rise to a 12 dB attenuation. A more detailed analysis of the transmitted modal content canbe seen in Fig. 15. This allows us to identify a specific family of modes which are affected by the HQ system. Note that due tosome modal conversion effects, some transmitted modes are slightly amplified. The interest for HQ resonators in order toeliminate tonal noise in a highly multimodal context is clearly demonstrated here. Of course, the HQ system could beredesigned in order to resonate in the close vicinity of the target frequency of 650 Hz, by simply changing the size of thetube accordingly.

6. Conclusion and prospects

In this paper, a new numerical procedure that judiciously exploits the benefit of the FEM and the BEM approach for theanalysis of the sound transmission through rigid ducts with side-branch resonators has been presented. Based on theconcept of impedance matrices connecting the pressure to the acoustic normal velocity on the interface between theresonator and the main duct, the technique allows a drastic reduction of the number of variables needed for the accuratemodeling of the acoustic system. Furthermore, it is not limited to any particular resonator geometry and can be used in ahighly multimodal context. In the low frequency range (i.e. by this we mean when the size of the resonator–duct interface ismuch smaller than the wavelength), the numerical problem can be simplified further by averaging the pressure and thevelocity over each interface. It is shown that this simplified model yields very accurate results and outperforms classical 1D

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models. Work is on going by the authors to apply the technique to the analysis of HQ tube arrays for fan noise reductionproblems when high-order modes can propagate in the main duct. One direction of particular interest to us is to explore therelation between the side-branch resonator impedance and its shape. It is hoped that this could have an impact in designoptimization of adaptive and tunable resonators. As in most of the intended applications shown in this work, there would bemean flow past the resonators, especially in the aeronautic industry, the method presented could be extended in order toaccount for the presence of flow in the main duct. Within the usual assumption that the flow is uniform over the cross-section, such modifications are feasible by using the generalized Green's function [34,35] and this could be done in the spiritof [36] for instance.

Acknowledgments

The authors would like to acknowledge the financial support of the CNRS and SNECMA, as well as insightful discussionswith SNECMA.

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