537935

8/18/2019 537935

1/11

Research Article3D Acoustic Modelling of Dissipative Silencers withNonhomogeneous Properties and Mean Flow

E. M. Sánchez-Orgaz, F. D. Denia, J. Martínez-Casas, and L. Baeza

Centro de Investigacíon de ecnolog ́ıa de Vehı́culos, Universitat Polit ̀ecnica de Val ̀encia, Camino de Vera s/n, Valencia, Spain

Correspondence should be addressed to F. D. Denia; [email protected]

Received April ; Accepted June ; Published July Academic Editor: Chao-Nan Wang

Copyright © E. M. Śanchez-Orgaz et al. Tis is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A nite element approach is proposed or the acoustic analysis o automotive silencers including a per orated duct with uni ormaxial mean ow and an outer chamber with heterogeneous absorbent material. Tis material can be characterized by means o its equivalent acoustic properties, considered coordinate-dependent via the introduction o a heterogeneous bulk density, and thecorresponding material air ow resistivity variations. An approach has been implemented to solve the pressure wave equation or anonmoving heterogeneous medium, associated with the problem o sound propagation in the outer chamber. On the other hand,the governing equation in the central duct has been solved in terms o the acoustic velocity potential considering the presence o a moving medium. Te coupling between both regions and the corresponding acoustic elds has been carried out by means o

a per orated duct and its acoustic impedance, adapted here to include absorbent material heterogeneities and mean ow effectssimultaneously. It has been ound that bulk density heterogeneities have a considerable in uence on the silencer transmission loss.

1. Introduction

Te acoustic behaviour o dissipative silencers strongly depends on the propertieso the absorbent material. In many modelling applications, it is necessary to predict accurately the acoustic per ormance in a wide requency range. Froma practical and computational point o view, it is easier orthe silencer designer to consider homogeneous materials.Nevertheless, this assumption is ofen ar away rom real-ity, where material properties can present relevant spatial variations. Tere ore, it can be important to take materialheterogeneities into account when modelling bulk reacting

brous materials as these variations are expected to signi -cantly affect the acoustic per ormance o the silencer [– ].Material heterogeneities can be caused by an uneven llingo the chamber, or example, when the bre is rolled aroundthe central duct or it is pushed in the chamber. Selamet et al.[ ] studied a dissipative silencer containing two concentricannular layers o absorbent material with different air ow resistivities in the absence o mean ow. In this work, a Danalytical approach was used to compute the wavenumbersand transversal pressure modes in the central airway and the

chamber. Finally, the transmission loss was obtained throughthe application o the mode matching technique consideringthe continuity conditions o the acoustic pressure and axial velocity at the geometrical discontinuities.A good agreementwas ound between the results derived rom this methodand FE calculations. In a later work o the same authors [ ],the acoustic effect o voids inside the silencer, modelled by meanso axially staggering lled/empty segments in theouterchamber, was studied considering a similar approach as theprevious re erence. In this case, the method provided goodcorrelation with both experimental measurements and FEcalculations. Antebas et al. [ , ] presented a pressure-basedFE approach to compute the transmission loss o per orateddissipative silencers including a continuously varying bulk density distribution. In these investigations, a linear unctionwas proposed to model the axialvariation o the bulk density,leading to heterogeneous material properties such as the ow resistivity, equivalent complex density, and speed o sound.Some numerical issues were ound at very low requencies inthe presence o a moving propagation medium [ ].

On the other side, anisotropy is likely to appear orsilencers manu actured in such a way that the bres are

Hindawi Publishing CorporationAdvances in Mechanical EngineeringVolume 2014, Article ID 537935, 10 pageshttp://dx.doi.org/10.1155/2014/537935

http://dx.doi.org/10.1155/2014/537935http://dx.doi.org/10.1155/2014/537935

8/18/2019 537935

2/11

Advances in Mechanical Engineering

aligned in a speci c direction or when a strongly directionalmean ow exists within the absorbent material. Peat andRathi [ ] presented a FE approach to model the acousticbehaviour o dissipative silencers with anisotropic and het-erogeneous properties caused by an induced ow, even i the material is initially isotropic and homogeneous. Despite

being usual to place a per orated sur ace to protect the breand reduce the static pressure losses, a per orated duct wasnot considered in this study.

Heterogeneities can be also caused by the soot particlescontained in the exhaust gases rom the engine [ ]. Froma modelling point o view, this can lead to a variablematerial resistivity and there ore to a coordinate-dependentequivalent density and speed o sound [ , , ]. Te presenceo a per orated screen [ , ] has an impact on the silencerper ormance, and the brous backing material has a largeeffect on the acoustic impedance o the per orations [ , ].Tere ore, material heterogeneities are expected to produceper orated duct impedance nonhomogeneities. Sullivan andCrocker studied the acoustic behaviour o a per oratedsur ace in the absence o absorbent material [ ]. Kirby andCummings [ ] presentedempirical ormulae orthe acousticimpedance o a per orated sur ace with absorbent materiallocated closely to one side o the plate in the presence o mean ow [ ]. More relevant in ormation describing thein uence o the brousmaterial on the acoustic per ormanceo a per orated sur ace can be ound in the work o Lee et al.[ ].

Te aim o the current work is to assess the in u-ence o a continuously varying bulk density distribution onthe acoustic per ormance o a dissipative silencer in thepresence o mean ow. An approach based on the nite

element method is presented, combining a velocity potential-based ormulation in the central pipe with a pressure-basedwave equation in the outer dissipative chamber [ ]. Tishybrid approach overcomes some numerical issues [ ] o the pressure ormulation at very low requencies. Due to thelow Mach numbers usually ound in exhaust systems [ ],

ow noise [ ] is not taken into account, while mean ow convective effects on the sound propagation and per oratedduct impedance are retained.

2. Governing Equations

. . Derivation of the Finite Element Equations. Figure shows an outline o the per orated dissipative silencer understudy, consisting o a central duct (Ω) carrying mean ow withaxialvelocity m andan outerchamber ( Ω) containingnonhomogeneousabsorbentmaterial.Te boundary sur aceso Ω and Ω are Γ and Γ, respectively, while Γ and Γdenote the inlet and outlet sections and Γ represents theper orated duct sur ace. From an acoustic point o view,the central airway can be characterized by means o the airdensity and the speed o sound , and the per oratedscreen can be modelled considering its acoustic impedancẽ . Due to an uneven distribution o the bulk density ( , , ) = (x ), the equivalent acoustic properties o theabsorbent material are also heterogeneous [ ]; that is, the

equivalent complex density ( , , ) = (x ) and speedo sound ( , , ) = (x ) are modelled as coordinate-dependent unctions (see Section . ), leading to a spatially varying acoustic impedance ̃ (x ) ( urther details will beprovided in Section . ).

In the central airway

Ω carrying mean ow, a wave

equation in terms o acoustic velocity potential is used [ ],

ΔΦ − 12 2Φ = 0, ( )where Δ is the Laplacian operator and Φ is the acoustic velocity potential, its gradient being the acoustic velocity U ;that is,

U = = ∇Φ. ( )In addition, is the total time derivative, given by [ ]

= +U m ∇ ( )with U m = m m m = m 0 0. Te acousticpressure is related to the velocity potential by means o the

ollowing expression [ ]:

= −Φ. ( )Assumingharmonic behaviourwith angular requency [ ]and combining ( ) and ( ) yield

1− 2m 2 2Φ2 + 2Φ2 + 2Φ2

− 2m

2 Φ + 2

2 Φ = 0,( )

being the imaginary unit.In the outer chamber Ω, the equivalent acoustic prop-erties are coordinate-dependent, and there ore a suitable

version o the wave equation is required to account orthe heterogeneous properties o the absorbent material. Tegoverning wave equation can be written in terms o acousticpressure [ , , , ] as ollows:

∇ 1∇ + 22 = 0, ( )where both the equivalent complex density

and speed o

sound [ , ] appear in the governing equation.Te nite element approach is now applied to ( )and( ).

First, the air subdomain Ω is considered, which yields [ ]∑=1 ∫Ω∇NM∇N Ω+ 2 m 2 ∫Ω N N Ω

− 22 ∫Ω N N Ω ̃Φ= ∑=1 ∫Γ N n M∇ΦΓ ,

( )

8/18/2019 537935

3/11


Heterogeneousabsorbentmaterial

Perforatedduct

Uniformmean owAir

x

y

zΩa

Ωm

Γi

Γa Γp

Γo

Γm

ca

a

cm

m

b

R

M =Umf

ca

̃Zp (x)

(x)

(x) (x)

(x)

F : Dissipative silencer with uni orm mean ow and heterogeneous material properties.

where N contains the shape unctions and ̃Φ the unknownnodal potentials, is the number o elements o the airsubdomain Ω, n is the outward unit normal vector to theboundary sur aceΓ, and nally M is a matrix de ned as

M = 1− 2m

2

0 00 1 00 0 1. ( )

I a similar procedure is applied to ( ) associated with theabsorbent material subdomain Ω, the ollowing expressionis obtained:

∑=1 ∫Ω 1∇N∇N Ω−2∫Ω 12 N N Ω P

= ∑=1∫Γ 1N Γ,

( )

where P represents the unknown nodal pressures and isthe number o elements belonging to the absorbent materialsubdomain Ω.Equations ( ) and ( ) are now combined to obtain the

nal system o equations. First, to achieve a more compactormulation, ( ) can be written as

K +C −2M ̃Φ = F , ( )where the corresponding matrices and load vector are

K = ∑=1 ∫Ω∇NM∇N Ω , ( )M = ∑=1 12 ∫Ω N N Ω , ( )C = ∑=1 2m 2 ∫Ω N N Ω , ( )

F = ∑=1 ∫Γ N n M∇ΦΓ

= ∑=1 ∫Γ ∩Γ / N 1− 2m 2

Φ Γ+∫Γ ∩Γ N Φ Γ .

( )Te integral over Γ that appears in ( ) is taken only overthe inlet/outlet sections Γ/ = Γ ∪ Γ and the per oratedduct Γ, as the rigid wall condition is assumed or theremaining sur aces. For the transmission loss ( L) computa-tions presented in Section , an acoustic velocity potentialboundary condition is considered at the inlet, while ananechoic termination is supposed at the outlet section. Teseconditions are suitable or attenuation calculations, since Lis de ned as the difference between the power incident onthe muffler and that transmitted downstream into a tail pipeterminating anechoically [ ].

Equation ( ) is expressed as

K −2M P = F ( )and the FE matrices and load vector corresponding to theabsorbent material subdomain are

K = ∑=1∫Ω 1∇N∇N Ω, ( )M = ∑=1∫Ω 12 N N Ω, ( )F

=

∑=1∫Γ ∩Γ1N

Γ. ( )

In this case, the integral over Γ in ( ) is taken on theper orated sur ace Γ, as the chamber walls are consideredrigid.o combine ( ) and ( ), the acoustic coupling between

the subdomains de ning the silencer ( Ω and Ω) is donethrough the per orated duct. Te integrals carried out overthe per orated boundary Γ in ( ) and ( ) are computedtaking into account the de nition o the acoustic impedance[ ],

̃ = − = − Φ −, ( )

8/18/2019 537935

4/11


where is the normal acoustic velocity. Note that in thecentral duct the acoustic velocity potential Φ is the eld variable explicitly used or the FE computations, related tothe acoustic pressure in the air through ( ). Accordingto the literature [ , , ], the acoustic impedance dependson several parameters such as requency, thickness, hole

diameter, and porosity. In addition, a dependence exists ontheabsorbent materialbacking theper orationsandthe meanow. Further details will be provided in Section . .

Assuming continuity o the acoustic velocity through theper orations [ ], the a orementioned load vector F in ( )evaluated over Γ is given by

F = ∑=1 ∫Γ ∩Γ N Φ Γ= ∑=1 ∫Γ ∩Γ N −̃ Γ= ∑=1 ∫Γ ∩Γ N − Φ − m Φ/̃

−̃ Γ .

( )

Te previous equation in compact orm is

F = −K ̃Φ −K P −C ̃Φ , ( )where the matrices have been de ned as ollows:

K =

∑=1 m ∫Γ ∩Γ Ñ N Γ ,K = ∑=1 ∫Γ ∩Γ N Ñ Γ ,C = ∑=1 ∫Γ ∩Γ N Ñ Γ .

( )

Finally, the load vector F o ( ), related to the brousabsorbent material, is given by

F =

∑=1∫Γ ∩Γ N Γ= ∑=1∫Γ ∩Γ

N −̃ Γ

= ∑=1∫Γ ∩Γ N2Φ − m Φ/̃

−̃ Γ,

( )

the compact notation being

F = −C P −C ̃Φ +2M ̃Φ , ( )where the matrices are

C =

∑=1∫Γ ∩ΓN N

̃ Γ,C =

∑=1 m ∫Γ ∩Γ ÑN Γ ,

M = ∑=1 ∫Γ ∩Γ N Ñ Γ .

( )

By combining ( ), ( ), ( ), and ( ) the nal system o equations is obtained:

K

+K K

0 K + [C

+C 0

C C ]−2 M 0M M ̃ΦP = F 0 ,

( )

where F corresponds to the part o load vector F in( ) calculated over the inlet/outlet sections Γ ∪ Γ with theboundary conditions indicated previously. Te next sectionsprovide details or the computation o the heterogeneousequivalent density and speeds o sound and as well asthe nonuni orm acoustic impedance o the per orated duct.

. . Acoustic Model of the Material Heterogeneity due to Den-sity Variations. An absorbent material can be characterizedby equivalent acoustic properties, such as the wavenumber = / and the characteristic impedance = ;both o them are complex and requency-dependent values[ , ]. Tese properties are usually uni orm in the litera-ture due to the assumed homogeneity o the steady air ow resistivity [ – ]. Tis act means that also the bulk density o the absorbent material has to be constant, sinceits relationship with the material resistivity is given by the

ollowing expression [ ]:

= 1 2 , ( )where the coefficients 1 and 2 can be obtained rom acurve tting process ollowing experimental data. E glassbre is considered in the calculations, whose coefficients are

1 = 5.774 and 2 = 1.792, thus providing a resistivity = 30716rayl/m or = 120kg/m 3 [ ].Te bulk density is usually considered constant in thebib-liography[ , , – ]. Heterogeneitiescan appear, however,due to the silencer manu acturing process, leading to spatial variations o the bulk density and material resistivity [ – ].Tese are included in the current investigation by assumingthat the extension o ( ) produces good predictions; that is,

(x ) = 1 (x )2. ( )

8/18/2019 537935

5/11


Te numerical test cases considered in the calculationshereafer are axisymmetric or computation purposes (seeFigure ). Te in uence o the material heterogeneity on theacoustic per ormance is assessed as ollows. Te density isapproximated by a bilinear unction in the context o thecurrent work, the dependence on the axial and the radial

coordinates being ( , ) = 0 + 1 + 2 + 3 . Accordingto ( ), the resistivity also depends on the coordinatesand there ore the usual expressions o the characteristicimpedance and the wavenumber [ , ] have been modi edhere to include the presence o heterogeneities, giving

(x ) = 1+0.095 (x )−0.669

− 0.169 (x )−0.571 ,

(x

) = 1+0.201 (x )−0.583

− 0.220 (x )−0.585 ,

( )

where = is thecharacteristic impedanceo theairandis the requency.. . Perforated Screen: Extension of the Acoustic Impedance

Model. Kirby and Cummings [ ] studied the acousticimpedance o a per orated pipe in the presence o anabsorbent material and a relevant dependence was ound.

Tis was taken into account by including the material prop-erties in the acoustic model associated with the per orations.Te bre considered in the current work is heterogeneous,so its acoustic properties vary rom one point to another.Tere ore, these spatial variations will modi y the behaviouro theper oratedscreen andhavetobe included in itsacousticmodel to guarantee accurate predictions. In addition, theimpedance o the per orations also depends on the mean

ow, as shown in the works [ , , ]. Here, an extensiono a orementioned results is implemented, the per oratedduct acoustic impedance being modi ed to include theheterogeneities produced by the bulk density variations [ , ]together with the mean ow effects. Te ollowing expressionis considered [ , ]:

̃ () = ()+ 0.425ℎ ()/ −1() ,( )

where it is worth noting (according to Figure ) that thecentral per orated passage is parallel to the -axis and it hasuni orm cross-section, being the only relevant coordinatewhen computing the acoustic impedance o the sur ace. Inaddition, = / is the wavenumber, ℎ the hole diameter, the porosity, and ( ) a actor that considers the acousticinteraction between the ori ces o the per orated duct [ ].

: Lee and Ih model or the acoustic impedance computation[ ].

(real part) (imaginary part) crit0 = 3.94⋅10−4 0 = −6.00⋅10−3 1 = 4121 = 7.84⋅10−3 1 = 194 2 = 1042

= 14.9 2

= 432 3

= 2743 = 296 3 = −1.72 —4 = −127 4 = −6.62⋅10−3 —Tis actor is obtained here as the average value o the Ingardand Fok corrections:

() = 1−1.055√ +0.17√ 3 +0.035√ 5. ( )Finally, ( ) is the nondimensionalized per orated ductimpedance considering the in uence o the mean ow with-out absorbent material, which can be expressed as ollows[ ]:

() = ()= + . ( )Te real and the imaginary parts o the previous expressionare de ned as

= 0 1+1 −crit 1+2 1+3 ℎ 1+4 , = 0 1+1 ℎ 1+2 1+3 1+4 .

( )

In ( ), = m / is the mean ow Mach number, is theper orated duct thickness, and crit is given by crit = 1 1+21+3 ℎ . ( )

Te coefficients o expressions ( )-( ) or the computationo , , and crit were obtained by Lee and Ih [ ] by theuse o experimental data and a curve tting procedure. Teassociated values are shown in able .

3. Results

Te outline o the axisymmetric per orated dissipativesilencer under study is shown in Figure . Te dimensionsde ning the geometry are 1 = 0.0268m, 2 = 0.05m, = 0.3m, and = = 0.1m (both long enough to guar-antee plane wave propagation in the inlet/outlet sections).Te mesh considered to compute the silencer transmissionloss consists o eight-noded quadrilateral elements whoseapproximate size is . m to obtain an accurate solution. Terelevant characteristics o the per orated sur ace are = 0,( %), = 0.001m, and ℎ = 0.0035m. E glass is consideredin all the calculations.

As shown in the gure, the bulk density o the absorbentmaterial is variable, being expressed by a bilinear unction

8/18/2019 537935

6/11


r

xR1

Li Lm Lo

R2

M

b(x,r) = c0 + c1x + c2r + c3xrb r1

b r2 b r4

b r3

F : Axisymmetric per orated dissipative silencer understudy.

( , ) that depends on the axial and the radial coordinates.For a particular numerical test case, the density distributionis de ned rom the “corner” density values , = 1,2,3,4,shown in Figure . In all the computations hereafer the bulk density law has been de ned in SI units (kg/m 3).

. . Validation. o validate the current approach, severalcomputations have been carried out or different bulk den-sity distributions in the absence o mean ow. Silencer

transmission loss has been calculated or three uni ormdensity con gurations, whose values are = 100kg/m 3,kg/m3, and kg/m3, respectively, and or an axially varying distribution ( ) = 366.67 − 666.67 (de ned by

1 = 2 = 300kg/m 3 and 3 = 4 = 100kg/m 3)with an average value o kg/m3. Each analysis has beencomputed twice or comparison purposes, by using thehybrid ormulation (velocity potential/pressure) presentedhere and also a pressure approach, similar to that describedin [ ] in the absence o ow.

Figure shows that the two sets o predictions (hybridormulation and pressure approach) agree very well in the

whole requency region or the our density distributions.

Increasing valueso the bulk density produce higher materialresistivities and deliver lower soundattenuation. Te resistiv-ity o E glass is relatively high and a blocking effect appears,reducing the penetration and dissipation o acoustic energy in the absorbent material. As expected, the transmission lossobtained with the variable density distribution lies betweenthose obtained with uni orm density computations corre-sponding to = 300kg/m 3 and = 100kg/m 3, since thelatter are the upper and lower bounds o the variable density

unction, respectively. In addition, the comparison betweenthe con gurations with = 200kg/m 3 and ( ) = 366.67−666.67 (both having the same average value) shows thattransmission loss discrepancies can be signi cant i materialheterogeneity is not considered. Tis is especially true in themid- requency range,where an examination o the L curvesprovides differences over dB (more than %). Hence, toavoid accuracy to be sacri ced, it is reasonable to employ theactual density distribution i possible rather than an average value.

. . Effectof Axially Varying Density on the ransmissionLoss.Several computations have been carried out in this section tostudy the in uence o axial density variations on the silencerper ormance. All the bulk density distributions used in thecalculations are linear. Te particular unctions are given by

1

( ) = 216.67 − 66.67,

2

( ) = 283.33 − 333.33, and

0 400 800 1200 1600 2000 2400 2800 32000

10

20

30

40

50

60

70

Frequency (Hz)

T L

( d B

)

F : L (dB) o dissipative silencer with different bulk density distributions: blue solid line, uni orm,

= 100kg/m 3, hybrid; blue

circles, same, pressure; red solid line, uni orm, = 200kg/m3,

hybrid; red circles, same, pressure; black solid line, uni orm, =300kg/m 3, hybrid; black circles, same, pressure; magenta solidline, ( ) = 366.67−666.67kg/m 3, hybrid; magentacircles, same,pressure.

0 400 800 1200 1600 2000 2400 2800 32000

10

20

30

40

Frequency (Hz)

T L (

d B

)

F : L (dB) o dissipative silencer with different axially varying bulk density distributions (with the same average density o kg/m3): red solid line, uni orm, = 200kg/m3; blue solidline, variable, 1( ); magenta solid line, variable,

2

( ); black solidline, variable, 3( ).3( ) = 366.67 − 666.67 (all expressed in SI units, kg/m 3).Note that the same average density is considered in the three

con gurations, given by = 200kg/m 3, which is takenhere as a re erence. A value = 0.2 has been assignedor the mean ow Mach number in all the nite elementcomputations.

In Figure , a comparison between transmission loss pre-dictions or the three bulk density distributions is made. Anadditionalcomputationwithuni orm density

= 200kg/m 3

8/18/2019 537935

7/11


: Density values or thede nitiono ( , ) = 0+1 +2 +3 ( 1 = 3 = 0).

1, 3 2, 4 0 2(kg/m ) (kg/m )Con g. I . .Con g. I . .Con g. I − . .Con g. D . − .Con g. D . − .Con g. D . − .is included. As can be observed in the gure, it is clear thatincreasing axial variations o the density distribution leads tohigher attenuation mainly in the mid- requency range, whileonly a slight acoustic impact is ound at low requencies,where density variations do not seem to have a relevantin uence. A transition is ound in the L curves, since these

tend to intersect at high requencies (approximately Hzin Figure ), where the previous trend seems to change; thatis, lower attenuation is achieved as the density distributionhas larger axial variation.

. . Effect of a Radially Varying Density on the Acoustic Per- formance. o assess the acoustic in uence o radial bulk density variations on the silencer attenuation, two types o

unctions have been considered: (a) increasing density in thedirection o increasing and (b) decreasing bulk density asthe radial coordinate increases. For both types, three differ-ent con gurations have been evaluated. Te corresponding values

,

= 1,... ,4, or the computation o the density

distribution are listed in able . Common with the previoussection, all the con gurations yield an average bulk density = 200kg/m 3.Te results associated with the con gurations listed in

able are depicted in Figure . A computationwith uni ormdensity = 200kg/m 3 is also included or comparisonpurposes. Again, a value = 0.2 has been assigned orthe mean ow Mach number in all the transmission losscalculations.

It is evident rom Figure that the acoustic impact o radial variations is stronger than this associated with caseswith axially varying density (see Figure ). Te uni ormdensity distribution sets a borderline between con gurationsI and D, showing that it is necessary to consider the radialdensity variations in the nite element computations topredict accurately the acoustic behaviour. While modellingthe absorbent material as a uni orm propagation mediumallows computations to be made with some computationaladvantage, it yields a lack o accuracy in the transmission losspredictions. Ascanbe observed, thediscrepancies with radial variations are higher than those presented in the previoussection or axially varying bulk density distributions.

wo different trends can be distinguished, dependingon the type o bulk density distribution. Con guration Iprovides higher attenuations than D. In addition, the ormerexhibits higher transmission loss or larger density variations

0 400 800 1200 1600 2000 2400 2800 32000

10

20

30

40

Frequency (Hz)

T L

( d B )

F : L (dB) o dissipative silencer with different radially varying bulk density distributions (with the same average density o kg/m3): red solid line, uni orm, = 200kg/m

3; blue dashedline, con g. I ; black dashed line, con g. I ; magenta dashed line,con g. I ; blue solid line, con g. D ; black solid line, con g. D ;magenta solid line, con g. D .

(except at very low requencies), while the latter delivershigher attenuation curves as the radial variation decreases. Areasonable physical explanation o this behaviour is relatedto the act that in con guration I the density increases withthe radius, and hence the material resistivity is lower nearthe per orated sur ace. Tis allows a gradual absorption o the sound energy as the acoustic wave penetrates in the

outer chamber, thus providing more attenuation than incon guration D, where the high material resistivity near theper oratedsur ace makes thepenetration o theacoustic wavewithin the chamber more difficult. Tis effect is relatively strong in Figure due to the high resistivity presented by E glass. For example, a bulk density = 100kg/m 3 yieldsa resistivity = 22155rayl/m, while or a bulk density = 300kg/m 3 the resistivity is = 158664rayl/m. Tismeans that increasing three times thematerialdensity yieldsaresistivity increase o approximatelyseventimes. In principle,

or less resistive materials, lower discrepancies are expected.Finally, regarding the low requency range, transmission lossdifferences are not signi cant in Figure below Hz.

. . In uence of a Density Distribution with Axial and Radial Variations. In this section, general density variations includ-ing axial and radial coordinate dependence are studied. Tedetails are provided in able , where two different con g-urations are described: ( ) Te density increases rom 1 to

2 while an axial reduction takes place rom the inlet to theoutlet section (con guration I ); ( ) Te density decreases inboth the axial and radial directions (con guration D ).

For comparison purposes, con gurations I and D(considering only radial variation), as well as the re erencegeometry with uni orm bulk density, are also included in theanalysis. In all the cases the value o the average density is

8/18/2019 537935

8/11


: Density values or the de nition o ( , ) = 0 +1 +2 +3 ( 3 = 0). 1 (kg/m ) 2 (kg/m ) 3 (kg/m ) 4 (kg/m ) 0 1 2

Con g. I . − . .Con g. D . − . − .

0 400 800 1200 1600 2000 2400 2800 32000

10

20

30

40

Frequency (Hz)

T L

( d B

)

F : L (dB) o dissipative silencer with different radially andaxially varying bulk density distributions (with the same averagedensity o kg/m3): red solid line, uni orm, = 200kg/m3; bluedashed line, con g. I ; black dashed line, con g. I ; blue solid line,con g. D ; black solid line, con g. D .

kg/m3 and the mean ow Mach number is given by =0.2

. Te corresponding transmission loss curves are shownin Figure , where it can be observed that the results areconsistent with the previous FE computations.Con gurationI provideshigher attenuation than the uni orm density case,similar to the radial examples o Figure , and it also deliverslarger L than con guration I since the latter does notinclude axial density variation (see Figure , where axially varying distributions are shown to increase the attenua-tion). In addition, con guration D provides lower soundattenuation compared to the uni orm density case, whichis consistent with the previous results o radially decreasingdensity distributions (see Figure ), but it yields larger Lthan con guration D since the latter does not include anaxially varying distribution. o conclude, it is worth empha-sizing that radial density variations have more in uence onthe attenuation than the axial ones. Tere ore, when both areconsidered simultaneously with a similar magnitude in theFE model, the prediction o the silencer behaviour is mainly dictated by the radial density distribution.

Finally, the in uence o the mean ow Mach number isconsidered next or the general density distributions I andD . Figure shows the transmission loss results obtained

rom the computations or = 0, = 0.1, and = 0.2. In general, the attenuation is reduced as themean ow increases, as expected. Tis is consistent withearlier studies with homogeneous material [ ]. A changein trend is observed, however, or con guration I at high

0 400 800 1200 1600 2000 2400 2800 32000

10

20

30

40

50

Frequency (Hz)

T L

( d B

)

F : L (dB) o dissipative silencer with radially and axially varying bulk density distributions (with the same average density o

kg/m3) and different mean ow Mach numbers: blue solid line,con g. I , = 0; blue dashedline, con g. I , = 0.1; blue dashed-dotted line, con g. I , = 0.2; red solid line, con g. D , = 0;reddashed line, con g. D , = 0.1; red dashed-dotted line, con g.D , = 0.2.

requencies, with increasing Mach numbers yielding largersound attenuation. When the mean ow increases, the crosspoint between the curves is shifed beyond the upper boundo the requency range under analysis. For con guration Dthe previous comments are also valid, but the cross pointseems to be beyond the requency limits o the gure or allthe curves. In this case, the in uence o the mean ow on theacoustic per ormance is lower than in con guration I .

4. Concluding Remarks

A hybrid nite element approach combining an acoustic velocity potential ormulation in the central airway with apressure-based wave equation in the outer chamber has beenpresented to study the acoustic per ormance o per orateddissipative silencers with heterogeneous properties in thepresence o mean ow. Tis represents an improvementover previously published models concerning automotivesilencers. Te material heterogeneities have been introducedby means o a nonuni orm bulk density, thus providing aspatial-varying material air ow resistivity. Tis nally leadsto coordinate-dependent equivalent acoustic properties suchas the density and speed o sound, which are properly incorporated in the FE ormulation developed in the currentwork. Te coupling between the central per orated ductand the outer chamber has been carried out through a

8/18/2019 537935

9/11


nonuni orm acoustic impedance accounting or the materialheterogeneity and the presence o mean ow.

In order to validate the approach presented here, sev-eral computations have been carried out in the absence o

ow to compare the corresponding predictions with resultscalculated with a pressure-based wave equation, showing a

good agreement. Te in uence o a number o bulk density distributions on the silencer per ormance has been thenanalysed in the presence o mean ow. It has been shown thatthe actual density distribution plays an important role in theFE acoustic predictions. Tus, it seems reasonable to employ it i possible,rather than anaveragevalue,to avoidaccuracytobe sacri ced. Te acoustic impact o radial density variationshas been shown to be more relevant than the axial ones. Inaddition, as radial density increases higher transmission lossis obtained,while lowerattenuation is achieved ordecreasingbulk density in the radial direction. Finally, the mean ow has been ound to reduce the silencer attenuation or thecon gurations under analysis.

Conflict of Interests

Te authors declare that there is no con ict o interestsregarding the publication o this paper.

Acknowledgments

Tis work was supportedby Ministerio de Econom ı́a y Com-petitividad (Projects DPI - and RA - -C - -R), Conselleria d’Educacío, Cultura i Esport (ProjectPrometeo/ / ), and Programa de Apoyo a la Investi-gación y Desarrollo (PAID- - and Project SP ) o

Universitat Polit ècnica de Val ència.

References

[ ] A. Selamet, M. B. Xu, I. J. Lee, and N. . Huff, “Dissipativeexpansion chambers with two concentric layers o brousmaterials,” International Journal of Vehicle Noise and Vibration , vol. , pp. – , .

[ ] A. Selamet, M.B. Xu, I.J. Lee, and N. .Huff,“Effectso voids onthe acoustics o per orated dissipative silencers,” International Journal of Vehicle Noise and Vibration, vol. , no. , pp. – ,

.[ ] A. G. Antebas, F. D. Denia, E. M. Śanchez-Orgaz, andF. J. Fuen-

mayor, “Numerical transmissionlosscalculations or per orated

dissipative mufflers containing heterogeneous material andmean ow ,” in Proceedings of the Joint Meeting th Congr èsFrançais d’Acoustique and IOA Annual Meeting (Acoustics ’ ),Nantes, France, April .

[ ] A. G. Antebas, F. D. Denia, A. M. Pedrosa, and F. J. Fuenmayor,“A nite element approach or the acoustic modeling o per o-rated dissipative mufflers with non-homogeneous properties,” Mathematical and Computer Modelling , vol. , no. - , pp.

– , .[ ] K. S. Peat and K. L. Rathi, “A nite element analysis o the

convected acoustic wave motion in dissipative silencers,” Jour-nal of Sound and Vibration , vol. , no. , pp. – , .

[ ] S. Allam and M. Åbom, “Sound propagation in an array o narrow porous channels with application to diesel particulate

lters,” Journalof Soundand Vibration ,vol. ,no. – ,pp. –, .

[ ] J. F. Allard and N. Atalla, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, John Wiley & Sons, Chichester, UK, .

[ ] G. Montenegro, A. della orre, A. Onorati, and R. Fairbrother,

“A nonlinear Quasi- D approach or the modeling o mu -ers with per orated elements and sound-absorbing material,” Advances in Acoustics and Vibration, vol. , Article ID

, pages, .[ ] J. W. Sullivan and M. J. Crocker, “Analysis o concentric-

tube resonators having unpartitioned cavities,” Journal of the Acoustical Society of America, vol. , no. , pp. – , .

[ ] R. Kirby and A. Cummings, “Te impedance o per oratedplates subjected to grazing gas ow and backed by porousmedia,” Journal of Sound and Vibration , vol. , no. , pp. –

, .[ ] I. Lee, A. Selamet, and N. . Huff, “Acoustic impedance o

per orations in contact with brous material,” Journal of the Acoustical Society of America, vol. , no. , pp. – ,

.[ ] E. M. S´anchez-Orgaz, F. D. Denia, J. Mart́ınez-Casas, and F.

J. Fuenmayor, “FE computation o sound attenuation in dissi-pative silencers with temperature gradients and non-uni ormmean ow,” in Proceedings of the nd International Congressand Exposition on Noise Control Engineering (IN ER-NOISE’ ), Innsbruck, Austria, September .

[ ] M. L. Munjal, Acoustics of Ducts and Mufflers, John Wiley & Sons, New York, NY, USA, .

[ ] Q. Si, S. Yuan, J. Yuan, and Y. Liang, “Investigation on ow induced noise due to back ow in low speci c speed centri ugalpumps,” Advances in Mechanical Engineering , vol. , ArticleID , pages, .

[ ] A. D. Pierce, “Wave equation or sound in uids with unsteady inhomogeneous ow,” Journal of the Acoustical Society of Amer-ica, vol. , no. , pp. – , .

[ ] M. E. Delany and E. N. Bazley, “Acoustical properties o brousabsorbentmaterials,” AppliedAcoustics, vol. ,no. , pp. – ,

.[ ] O. C. Zienkiewicz, R. L. aylor, andJ. Z. Zhu,Te Finite Element

Method: Its Basis and Fundamentals, Elsevier, .[ ] I. J.Lee and A. Selamet, “Measurement o acousticimpedance o

per orations in contact with absorbing material in the presenceo mean ow,” Noise Control Engineering Journal , vol. , no. ,pp. – , .

[ ] R. KirbyandF. D.Denia, “Analytic mode matching or a circulardissipative silencer containing mean ow and a per orated

pipe,” Journal of the Acoustical Society of America, vol. , no., pp. – , .[ ] A. Selamet, M. B. Xu, I.-J. Lee, and N. . Huff, “Analyti-

cal approach or sound attenuation in per orated dissipativesilencers,” Journal of the Acoustical Society of America, vol. ,no. , pp. – , .

[ ] F. D.Denia, A. Selamet, F. J. Fuenmayor, andR. Kirby, “Acousticattenuationper ormance o per orated dissipative mufflers withempty inlet/outlet extensions,” Journal of Sound and Vibration, vol. , no. - , pp. – , .

[ ] F. D. Denia, A. G. Antebas, A. Selamet, and A. M. Pedrosa,“Acoustic characteristics o circular dissipative reversing cham-ber mufflers,” Noise Control Engineering Journal , vol. , no. ,pp. – , .

8/18/2019 537935

10/11


[ ] R. Kirby and A. Cummings, “Prediction o the bulk acousticproperties o brous materials at low requencies,” Applied Acoustics, vol. , no. , pp. – , .

[ ] A. Selamet, I. J. Lee, and N. . Huff, “Acoustic attenuation o hybrid silencers,” Journal of Sound and Vibration, vol. , no.

, pp. – , .

[ ] F. Payri, A. Broatch, J. M. Salavert, and D. Moreno, “Acousticresponse o brous absorbent materials to impulsive transientexcitations,” Journal of Sound and Vibration, vol. , no. , pp.

– , .[ ] S. H. Lee and J. G. Ih, “Empirical model o the acoustic impe-

dance o a circular ori ce in grazing mean ow,” Journal of the Acoustical Society of America, vol. , no. , pp. – , .

[ ] J. L. Bento, Acoustic characteristics of perforate liners in expan-sion chambers [Ph.D. thesis], University o Southampton, .

8/18/2019 537935

11/11

Submit your manuscripts athttp://www.hindawi.com

537935

Documents