A new segmentation approach for sound propagation in non-uniform lined ducts with mean flow

Journal of Sound and Vibration 330 (2011) 2369–2387

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A new segmentation approach for sound propagation in non-uniform lined ducts with mean flow Xiaoyu Wang, Xiaofeng Sun n Fluid and Acoustic Engineering Laboratory, School of Jet Propulsion, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

a r t i c l e in f o

abstract

Article history: Received 15 January 2009 Received in revised form 25 September 2010 Accepted 20 November 2010 Handling Editor: Y. Auregan Available online 21 December 2010

An analytical model is presented to study sound attenuation in a non-uniform lined duct with mean flow in this paper. This work is an extension of segmentation approach with emphasis on the physical and mathematical descriptions of locally or non-locally reacting liners in flow ducts. Various numerical results show that the model not only gives the same prediction as the existing models for both continuously and abruptly varying crosssectional area ducts without flow, but also provides a convenient tool for the optimization of sound attenuation in a non-uniform flow duct at the preliminary design stage of acoustic treatments. As one of the objectives of the present model, attention is also paid to how to improve the sound attenuation in a varying flow environment by the change of the wall impedance using bias flow. In particular, a better physical understanding to the interaction mechanism of different liner combinations may be expected with the help of the proposed model. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction Prediction and control of sound propagation in ducts with a varying cross-sectional area has long been an important topic in acoustics. There are many situations occurring in practice where sound propagates in such non-uniform ducts. In particular many applications, including the acoustic design of mufflers, nozzles and aeroengine nacelle, are closely associated with this problem. Thus, there has been a considerable amount of research on the sound propagation of non-uniform ducts, and the relating work can be categorized as analytical and numerical methods. Segmentation approach is one of the analytical methods which attracted some attention previously due to its low computational cost [1,2]. Especially for continuous crosssectional area duct, it was verified that this method could be extended to include the effect of higher-order modes and flow on sound propagation [3,4]. But there is no relevant work for non-uniform lined ducts in the same way. However, many other methods have been developed to handle these difficult problems, such as the perturbation methods [5,6], the finite element method [7,8] and CAA method [9–11]. Recently, a Multi-Modal Propagation Method (MMPM) [12] was proposed to study non-uniform lined ducts in the absence of mean flow, including the non-locally reacting liner with porous materials [39]. This method is an extension of the multimodal formulation proposed by Pagneux et al. [13], which can be used to handle both circumferentially and axially non-uniform liner without requiring the calculation of radial complex eigenvalues in a lined duct [38]. However, to authors’ knowledge, there is little work to study the combination effect of both local and non-local reaction liners in non-uniform ducts. On the other hand, it is noted that for a non-uniform duct, the most important problem is to study how to design the acoustic treatments in order to realize the maximum sound attenuation under various restrictions. Especially in recent ten years, some research in connection with the concept of advanced liner or

n

Corresponding author. Tel.: + 86 10 82317408; fax: + 86 10 82317408. E-mail addresses: [email protected] (X. Wang), [email protected] (X. Sun).

0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2010.11.022

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Nomenclature A area An,Bn,Cn,Dn amplitude of acoustic pressure aaim defined by Eq. (29) c speed of sound in air ccBbm defined by Eq. (27) csin defined by Eq. (17) dh perforated hole diameter f frequency G+ Green function in the cavity G Green function in the duct k0 wavenumber in air kmn radial wavenumber k^ 0 complex wavenumber in the absorbing material l liner length m circumferential mode Ma mean flow Mach number Mb bias flow Mach number n radial mode ~ p,p,pu pressure, mean pressure, acoustic perturba~ ¼ pueiot ) tion pressure (pu pþu acoustic pressure in the cavity

pu peu psu R Rmn r, j, z sci ssbm A Tmn TL tw ~ v, U,vu Vn Z Z0 Z^ 0

o s F, f

r,r,r~u

acoustic pressure in the duct standing wave (peu ¼ pBu þ pCu ) disturbance wave or scattering wave flow resistively of the absorbing material reflection coefficient cylindrical coordinate defined by Eq. (18) defined by Eq. (16) transmission coefficient transmission loss thickness of the perforated liner velocity, mean velocity, acoustic perturbation velocity acoustic particle velocity in the perforated hole non-dimensional characteristic impedance characteristic impedance in air complex characteristic impedance of absorbing material circle frequency open-area ratio of perforated eigenfunction for duct density, mean density, acoustic perturbation density (r~u ¼ rueiot )

hybrid acoustic treatment has received increasing attention [14–16]. At present, although there are different strategies to realize the goal [15], it is generally recognized that the advanced liner at least contains two main features. First, such acoustic liner has good performance as a passive means for noise suppression; secondly, it has adaptive functions so that to make itself effectively work in varying acoustic fields. The control of wall acoustic impedance using bias flow through a perforated plate is one of the techniques used to actively control sound propagation. Dean and Tester [17] made a pioneer experimental investigation for the change of the impedance using bias flow as early as 1975. In the past, attention was focused on how to obtain more accurate the impedance model [18–21] and how to control the impedance under relatively simple conditions [22]. But in recent years, a lot of research results have shown that the non-locally reacting liner may have a comparatively broad frequency range corresponding to a better sound absorption property in a duct [23–25]. Some investigators [26–29] also realized that the combination of a non-locally reacting liner and bias flow was one of the technical methods worthy of further analysis and discussion for the hybrid control of sound propagation in ducts. However, these studies were concentrated on uniform ducts [23–29], it is hard to find the relating investigations for non-uniform ducts. The objective of the present study is to develop a new segmentation approach in order to include the effect of lined ducts on the sound propagation and attenuation. At the same time, in order to make full use of the features of both locally and non-locally reacting liners, emphasis will be put on how the different combinations of locally and non-locally reacting liners take effect on the control of acoustic attenuation in flow ducts. To meet this goal, we will establish the model in the following three steps. First, an acoustic field solution for a subsection or an element is constructed after dividing a continuously varying sectional duct into finite subsections, while the effect of acoustic liners is replaced by unknown monopole sources. Secondly, in term of the conservation law the interface matching conditions is determined with emphasis on the differences with uniform ducts. In this way, all unknowns will be given on the interface plane in the form of matrix expression. Finally, different combination models are established by applying the interface matching conditions so that to study the optimizing strategies for the sound attenuation in a non-uniform duct. On the basis of the proposed model, various numerical results show that our work not only gives the same prediction as the existing models for both continuously and abruptly varying cross-sectional area ducts without flow, but also provides a convenient tool for the optimization of sound attenuation in a non-uniform flow duct at the preliminary stage of acoustic treatments. In particular, with the help of the model, it is expected to have a better physical understanding to the interaction mechanism of different liner combinations. 2. Mathematical model 2.1. Basic solution for a subsection with a rigid wall As we mentioned in the introduction, the objective of the model is to predict the behavior of sound attenuation in a nonuniform duct which is lined with an impedance wall. For convenience of discussion, we will first investigate the sound propagation in a hard duct with flow.

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As Fig. 1(a) shows, we considered a slowly varying shape of duct which can be approximated by a series of subsections whose sides are parallel to the axis of the duct. Therefore, a physical model should be established to investigate the behavior of sound as it propagates in an abrupt changing duct, as shown in Fig. 1(b). Besides, we assume that the non-uniform duct contains one dimensional mean flow. So, the application of mass and momentum conservations results in the following linearized equations: ! ! U U r1 v1u þ p1u 21 A1 ¼ r2 v2u þ p2u 22 A2 , (1) c1 c2 p1u þ p1u

U12 U2 þ 2r1 U1 v1u ¼ p2u þ p2u 22 þ 2r2 U2 v2u , 2 c1 c2

(2)

~ and pu ~ are the fluctuating part, which vary harmonically in time where r, U and p are the stationary part of duct flow, and r~ u, vu iot ~ with angular frequency o (i.e., pu ¼ pue ). In addition, c is sound speed and A stands for the cross-section area. Subscripts 1, 2 refer to the expansion and contraction sections, respectively. As Fig. 2 shows, for Eq. (1), assuming A1 oA2, we have ðr1 v1u þ r1u U1 ÞA1 ¼ ðr2 v2u þ r2u U2 ÞðA1 þ DAÞ:

(3)

The axial fluctuating velocity in Section 2 is zero at the discontinuity DA, i.e., v1u ¼ 0,

(4)

hence, it can be further shown that

r1 v1u þ p1u

1

U1 U2 ¼ r2 v2u þ p2u 2 : c12 c2

(5)

2 1

2

nd nd − 1

L

pi′

nd+1 nd

pB′

j

pD′ pC′ pA′ 2 1 1

L

j

nd − 1 nd

nd nd+1

2 Fig. 1. (a and b) Schematic of varying cross-sectional duct.

1

2 ΔA

A1

1, U1, p1

2, U2, p2

1′, v1′, p1′

2′, v2′, p2′

A2

ΔA

Fig. 2. Configuration for the interface match.

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Especially, it is noted that when there is no mean flow in the duct, for a non-uniform duct, Eqs. (2), (5), and (4) become, respectively, p1u ¼ p2u ,

(6a)

v1u ¼ v2u ,

(6b)

v2u ¼ 0 ðfor DAÞ

(6c)

p1u ¼ p2u ,

(7a)

v1u ¼ v2u ,

(7b)

and for a uniform duct

Eqs. (6a, b, c) and (7a, b) represent the matching conditions for non-uniform and uniform ducts without mean flow. After giving the matching conditions on the interface, we are able to discuss how to construct an acoustic field solution in the duct. For this purpose, we take a contraction duct as an example because a single discontinuity only includes two forms, i.e., expansion and contraction sections. As shown in Fig. 3, the cylindrical coordinates r, j and z are used to describe the sound pressure p0 . For subsection 0, there are incident wave p0i and reflected wave p0A from subsection 1, while the acoustic field in subsection 1 can be expressed as a standing wave pBu þ pCu . The p0 D in subsection 2 stands for a transmission wave from subsection 1. So, the problem has now become how to find the acoustic field solution in subsection 1 with the known parameter p0i. For this specific objective, we can first describe the acoustic field in the subsections as X 0 piu ¼ pin fm ðk0mn rÞeigmn z , (8a) n

pAu ¼

X 0þ An fm ðk0mn rÞeigmn z ,

(8b)

n

pBu ¼

X 1 Bn fm ðk1mn rÞeigmn z ,

(8c)

n

X 1þ Cn fm ðk1mn rÞeigmn ðzlÞ ,

pCu ¼

(8d)

n

X 2 Dn fm ðk2mn rÞeigmn ðzlÞ :

pDu ¼

(8e)

n

Consider an axial momentum equation for the subsection 0, 1 or 2   @ @ @pu þU vu ¼  : r @t @z @z

(9)

If we take the reflecting wave as an example, substituting Eq. (8b) into Eq. (9) gives vAu ¼ 

1 X

r0

n

0þ g0þ mn An fm ðk0mn rÞeigmn z : o þ U0 g0þ mn

Applying the momentum equation for the interface b  b yields ( X X An fm ðk0mn rÞð1 þ Ma20 Þ  2U0 An fm ðk0mn rÞ n

n

b

(10)

g0þ mn

)

o þ U0 g0þ mn

c

r z

r0

pi′

Ma

pA′ 0

pB′

r1

b

pC′

1

pD′

c

l

Fig. 3. Subsection of a transfer element.

r2 2

X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387



( X X Bn fm ðk1mn rÞð1 þ Ma21 Þ  2U1 Bn fm ðk1mn rÞ n

n

n

n

n

)

g1 mn o þ U1 g1 mn

( X X  Cn fm ðk1mn rÞð1 þ Ma21 Þ  2U1 Cn fm ðk1mn rÞ X  ¼  pin fm ðk0mn rÞð1 þ Ma20 Þ  2U0 fm ðk0mn rÞ

2373

g1þ mn

)

o þ U1 g1þ mn

g0 mn

1þ

eigmn l



(11a)

o þ U0 g0 mn

and using the continuity equation for the interface results in ( ) ( ) X X U0 X g0þ U1 X g1 0 0 1 1 mn mn An fm ðkmn rÞ 2  An fm ðkmn rÞ Bn fm ðkmn rÞ 2  Bn fm ðkmn rÞ  o þ U1 g1 c0 c1 o þ U0 g0þ mn mn n n n n ( ) ( ) 1þ X X X 1þ U g U g0 1 0 igmn l 0 0 mn mn  Cn fm ðk1mn rÞ 2  Cn fm ðk1mn rÞ ¼  p f ðk rÞ  f ðk rÞ e : in m mn m mn o þ U0 g0 c1 c02 o þ U1 g1þ mn mn n n n (11b) In addition, from the rigid wall boundary condition at the sidewall, r1 or or0, the relating relation can be derived in the form of X n

X pin g0 An g0þ mn mn fm ðk0mn rÞ þ fm ðk0mn rÞ ¼ 0: 0þ 0 o þ U0 gmn n o þ U0 gmn

(11c)

Multiplying Eqs. (11a) and (11b) by fm ðk0mi rÞr and integrating it with respect to r from 0 to r1, and also for (11c) in the same way and integrating it with respect to r from r1 to r0, then substituting Eq. (11c) into Eqs. (11a) and (11b) yields Z r1 X g0þ mi An ð1 þ Ma20 Þ fm ðk0mn rÞfm ðk0mi rÞr dr  2U0 Ai G0mi o þ U0 g0þ 0 n mi  Z r1 X  g1 mn  Bn ð1 þ Ma21 Þ  2U1 f ðk1 rÞf ðk0 rÞr dr 1 o þ U1 gmn 0 m mn m mi n   Z r1 X 1þ g1þ mn f ðk1 rÞf ðk0 rÞr dr  Cn eigmn l ð1 þ Ma21 Þ  2U1 1þ o þ U1 gmn 0 m mn m mi n Z r1 X g0 mi pin ð1 þ Ma20 Þ fm ðk0mn rÞfm ðk0mi rÞr dr þ 2U0 pii G0mi ¼  o þ U0 g0 0 n mi ði ¼ 1,2,. . .,NÞ, (12a) X U0 Z r1 g0þ mi An 2 fm ðk0mn rÞfm ðk0mi rÞr dr  Ai G0mi c0 0 o þ U0 g0þ n mi ( )Z r1 X U1 g1 mn  Bn 2  fm ðk1mn rÞfm ðk0mi rÞr dr 1 o þ U g c 1 mn 0 n 1 ( )Z r1 X 1þ U1 g1þ mn fm ðk1mn rÞfm ðk0mi rÞr dr  Cn eigmn l 2  1þ c1 o þ U1 gmn 0 n X U0 Z r1 g0 mi ¼  pin 2 fm ðk0mn rÞfm ðk0mi rÞr dr þ pii G0mi c o þ U0 g0 0 n 0 mi ði ¼ 1,2,. . .,NÞ: Similarly, for the interface c–c, the matching equations can be derived as Z r2 X 1 Bn ð1 þ Ma21 Þeigmn l fm ðk1mn rÞfm ðk1mi rÞr dr  2U1 Bi 0

n

Z X Cn ð1 þ Ma21 Þ n

r2 0

fm ðk1mn rÞfm ðk1mi rÞr dr  2U1 Ci

X   Dn ð1 þ Ma22 Þ  2U2 n

g2 mn o þ U2 g2 mn

Z

r2 0

g1 mi o þ U1 g1 mi

g1þ mi o þ U1 g1þ mi

(12b)

1

G1mi eigmi l

G1mi

fm ðk2mn rÞfm ðk1mi rÞr dr ¼ 0

ði ¼ 1,2,. . .,NÞ, X U1 1 Z r2 g1 1 mi Bn 2 eigmn l fm ðk1mn rÞfm ðk1mi rÞr dr  Bi G1 eigmi l 1 mi c o þ U g 0 1 mi n 1 X U1 Z r2 g1þ 1 1 mi Cn 2 fm ðkmn rÞfm ðkmi rÞr dr  Ci G1mi c1 0 o þ U1 g1þ n mi

(13a)

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( )Z r2 X U2 g2 mn  Dn 2  fm ðk2mn rÞfm ðk1mi rÞr dr ¼ 0 2 o þ U g c2 2 mn 0 n ði ¼ 1,2,. . .,NÞ:

(13b)

In terms of the above mathematical treatment, it is seen that each of the equations such as Eqs. (12a), (12b), (13a) and (13b) will produce N equations by using the orthogonal conditions ði ¼ 1,2,. . .,NÞ. Therefore, we have 4N linear algebraic equations available and also the 4N unknown coefficients (i.e., An, Bn, Cn and Dn). Obviously, the close solution for the configuration shown in Fig. 3 can be obtained by introducing the following matrix equation: 2 bm 3 2 3 2 bm 3 ssbm ssbm 0 ssA An pi B C 6 bf 7 7 bf bf 6 ss 7 6 Bn 7 6 bf ss ss 0 6 B 7 6 pi 7 6 A 76 C 7: (14a) 7¼6 6 cm cm cm 7 6 ssC ssD 7 4 Cn 5 4 0 7 ssB 6 0 5 4 5 Dn 0 sscf sscf sscf 0 B D C Furthermore, if there are more sections as shown in Fig. 1(dash line), we can apply the matching conditions on each interface in the non-uniform duct, a close solution can be still expressed as 9 8 bm ssbm ssbm ... ss 0 1 0 bm 1 > > B1 C1 > > > > A An pi > > > > bf bf bf > > B C B bf C ss ss ss . . . > > 1 1 1 > > A B C B B C B C > > > > B n1 C B pi C > > > > sscm ... ... sscm BC C B C > > B1 C1 > > > > B n C B 0 C > > > > B C B C = < sscf sscf ... ... B ^ C B ^ C B1 C1 B C¼B C : (14b) bm bm B C B C . . . . . . ssBnd ssC nd > > > > B ^ C B ^ C > > > > B C B C n > > d > > B Bn C B ^ C > > . . . . . . ssbf ssbm > > B n C B C C nd Bnd > > > > BC d C B C > cm bm cm > > > n A @ @ A ^ . . . ss ss ss > > n n D > > > B d C d > > 2ðnd þ 1ÞN Dn > > cf cf > 0 bm ; : 2ðnd þ1ÞN . . . ssBnd ssC nd ssD  2ðnd þ 1ÞN In Eq. (14b), the inlet and outlet of the non-uniform duct are treated as an anechoic end in order to make the direct comparison with the literatures [6,31,33,34]. If considering a finite length duct with the acoustic reflection at the both side of the duct, there are some methods which can be used to include the effect of acoustic reflection on the sound propagation inside the duct. The one is to use the mode reflection coefficients which were derived in Refs. [40,41]. The second is to apply the finite element method or boundary element method to establish the radiating field. For a uniform duct, such example can be found in Ref. [29], which shows how to combine the mode-matching method with the boundary integral method to include the effect of acoustic reflection. In addition, it is observed that for each subsection, a transfer matrix can be defined as 2 bm 3 ssB ssbm C 6 bf 7 6 ss 7 ssbf 6 B C 7 , (15) 6 cm cm 7 ssC 7 6 ssB 4 5 sscf sscf B C 4N2N

which is only determined by the acoustic characteristics of the subsection itself, so we can take it as an independent element. In addition, ss in the matrix (15) denotes coefficient matrices, for example 2 3 2 3 cs11 ... cs1N 0 ... 0 sc1 6 7 6 6 7 6 0 & & ^ 7 7 6 7 6 7 7 6 ^ i bm 7, ^ 7þ6 csn ^ sc ssA ¼ 6 (16) i 7 6 7 6 6 7 6 7 4 & & 0 5 4 5 ... csN csN 0 ... 0 scN N 1 where Z r1 fm ðk0mn rÞfm ðk0mi rÞr dr csin ¼ ð1 þ Ma20 Þ 8 0    2 r2 > > u ðk0mi r1 Þ2 þ 1  ðk0mr Þ2 ½fm ðk0mi r1 Þ2 , n ¼ i > ð1 þ Ma20 Þ 21 ½fm < mi 1 ¼ > r1 k0mn fum ðk0mn r1 Þfm ðk0mi r1 Þr1 k0mi fum ðk0mi r1 Þfm ðk0mn r1 Þ > 2 > ð1 þ Ma Þ , : 2 0 2 0 0

ðkmi Þ ðkmn Þ

(17)

nai

X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387

2375

c

b

l

pi′ rd0

pA′

pB′

pC′

rh0

rd1

pD′

rh1

rd2 rh2

b

c

Fig. 4. Approximation to a contraction duct.

and sci ¼ 2U0

g0þ mi o þ U0 g0þ mi

where

G0mi ¼

Z

r0 0

fm ðk0mn rÞfm ðk0mi rÞr dr ¼

and the integral expressions are given as Z r1 0

G0mi

8  > < r02 1  2

> :

fm ðk1mn rÞfm ðk0mi rÞr dr ¼

m2

(18)

 ½fm ðk0mi r0 Þ2

n¼i

0

nai

ðk0mi r0 Þ2

r1 k0mi fm u ðk0mi r1 Þfm ðk1mn r1 Þ ðk1mn Þ2  ðk0mi Þ2

:

(19)

(20)

Finally, it can be noted that when the incident mode is plane wave (i.e. m= 0, n = 0) the integral expressions can be simplified to a special form [3]. Besides, as Fig. 4 shows, for an annular duct, the sidewall conditions include two velocity equations, for example, at b b interface X

An g0þ pin g0 mn mn fm ðk0mn rÞ þ fm ðk0mn rÞ ¼ 0 for rd1 o r ord0 , o þ U0 g0 o þ U0 g0þ mn mn

(21)

X Cn g1þ eigmn l Bn g1 mn mn fm ðk1mn rÞ þ fm ðk1mn rÞ ¼ 0 for rh1 o r orh0 : 1þ o þ U1 g1 mn n o þ U1 gmn

(22)

n

X n

1þ

We can use the same way as Eqs. (11c) to orthogonalize the equations. 2.2. Basic solution for a subsection with an impedance wall According to the above discussion, the solution for each subsection is only related to the interface parameters. However, for an impedance wall or soft-walls, we need to include the influence of soft-wall on the interface parameters, i.e., Bn and Cn in the solution. A typical method is to calculate the radial eigenvalues with the soft-wall condition just as Zorumski suggested previously in his mode-matching approach [30]. However, the calculation of complex eigenvalues is generally troublesome and even awkward task. An alternative way is to use the singularity method to consider the influence of soft walls on the sound field, which was first proposed by Namba and Fukushige [31]. Now, we shall apply the same principle to construct the solution in a subsection with soft walls. However, we will focus on the mathematical description to a non-locally reacting liner, which is more complex than the case analyzed by Namba and Fukushige [31]. As shown in Fig. 5, we consider a non-locally reacting liner. Assuming the pressure in the cavity is p0+ and in the duct is p0 0 which is decomposed into two components: the standing wave peu ¼ pBu þ pCu and the disturbance wave ps. The standing wave can be solved by satisfying rigid wall condition. The disturbance wave is due to the existence of the perforation which can be considered as distributed monopole sources. In terms of the definition of the acoustic impedance condition, the relevant impedance equation can be expressed as pþu  peu  psu ¼ rcVn Z,

(23)

where Z is non-dimensional characteristic impedance of the perforated liner, and the normal acoustic particle velocity Vn is continued across a perforation of small thickness. For a local reaction liner the pressure in the cavity p0+ will vanish. For solving the equation, we need to find the relation between p0+ , p0s and Vn. As mentioned above, the soft wall can be regarded as a monopole source, so we can obtain the relevant expression in terms of aeroacoustic equation [32] Z 1 1 7 X ½o þ U gm,n  ig 7 ðzzuÞ r X Fm,n ðx,yÞ ! psu ¼ V Fm,n ðxu,yuÞ e m,n dsð r uÞ, (24a) 2 m ¼ 1 n ¼ 1 Gm,n k n,m sðtÞ

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X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387

b

c

l

r z

p+′ pi′ pA′

pB′

n

ps′ p−′ = ps′ + pB′ + pC′ pC′ pD′ c

b

Fig. 5. Subsection with soft walls.

XXX Fm,n ðx,yÞcosðqpz=lÞ Z ! qpzu dSð ru Þ: pþu ¼ iro Vn Fm,n ðxu,yuÞcos l Gm,n,q ðk20  k2m,n,q Þ sðtÞ m n q

(24b)

There is the particle displacement continuous condition between V and Vn   U @ Vn : V ¼ 1þ io @zu

(25)

0

The key step lies in how to express the scattering wave ps as the explicit function of the standing waves p0B and p0C. Ref. [29] should be consulted for more details about the mathematical description of this transfer element. It can be proved that 8 9 7 N 1 = rX Fm,n ðx,yÞeigm,n z < X Bm Cm psu ¼ ½B Q þ C m Qn 7  , (26) :m ¼ 1 m n 7 ; 2n¼1 Gm,n B

C

m can also be find in Ref. [29]. where Qn m7 and Qn 7 Therefore, the scattering wave p0s can be expressed as the functions of standing wave coefficients Bm and Cm. Since the singularity method [31] will not affect the hard wall condition mathematically, we can still use the orthogonality of the eigenfunctions on the matching interface. To illustrate how to construct the solution in a subsection with soft walls, a circular duct with varying cross-section is considered here. For this situation, the eigenfunction in Eq. (26) will be replaced by Fmn ðxu,yuÞ ¼ fmn ðkmn rÞeimj . Substituting Eq. (26) and Eqs. (8a)–(8e) into Eqs. (2), (4) and (5) and orthogonalizing these equations, we obtain the coefficients of Bn and Cn in the form of   Z r1 N r X fm ðk1mn rÞ Bm g1þ mn ccBbm : 1 Qnþ ð1 þ Ma21 Þ  2U1 fm ðk1mn rÞfm ðk0mi rÞr dr, (27a) 2 n ¼ 1 Gm,n o þ U1 g1þ 0 mn

N r1 X fm ðk1mn rÞ

ccCbm :

2

n¼1

ccBbf :

ccCbf :

ccBcm :

ccCcm :

Gm,n

N r1 X fm ðk1mn rÞ

2

n¼1

n¼1

2

n¼1

Gm,n

N r1 X fm ðk1mn rÞ

2

ccBcf

ccCcf

:

:

n¼1

Gm,n

N r1 X fm ðk1mn rÞ

N r1 X fm ðk1mn rÞ

2

 C Qnþm ð1 þ Ma21 Þ  2U1

Gm,n

Gm,n B

2

n¼1

Gm,n

N r1 X fm ðk1mn rÞ

2

n¼1

Gm,n

g

r1 0

fm ðk1mn rÞfm ðk0mi rÞr dr,

(27b)

( )Z r1 U1 g1þ mn  fm ðk1mn rÞfm ðk0mi rÞr dr, 2 1þ c1 o þ U1 gmn 0

(27c)

C

( )Z r1 U1 g1þ mn  fm ðk1mn rÞfm ðk0mi rÞr dr, 2 1þ c1 o þ U1 gmn 0

(27d)

Qnþm (

1

( 1

Qnm eigmn l ð1 þ Ma21 Þ

N r1 X fm ðk1mn rÞ

oþ

U1 1þ mn

B

Qnþm

Qnm eigmn l ð1 þ Ma21 Þ

C

Z

g1þ mn

Z

r2

0

Z 0

r2

fm ðk1mn rÞfm ðk1mi rÞr dr 

) 2U1 g1 1 mi G mi , o þ U1 g1 mi

fm ðk1mn rÞfm ðk1mi rÞr dr 

2U1 g1 mi G1mi o þ U1 g1 mi

(27e)

) (27f)

1 B Qnm eigmn l

( Z ) g1 U1 r2 1 1 1 mi fm ðkmn rÞfm ðkmi rÞr dr  Gmi , c12 0 o þ U1 g1 mi

(27g)

1 C Qnm eigmn l

( Z ) g1 U1 r2 1 1 1 mi fm ðkmn rÞfm ðkmi rÞr dr  Gmi , c12 0 o þ U1 g1 mi

(27h)

X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387

where each cc stands for the coefficient matrices, for example 2 aa1 ... 6 1 6 & 6 6 aaim ccBbm ¼ 6 ^ 6 6 4 ... aaN 1

aa1N

3

7 7 7 ^ 7 7 7 7 5 N aaN

&

2377

(28)

and aaim ¼

N r1 X fm ðk1mn rÞ

2

n¼1

Gm,n

 B Qnþm ð1 þ Ma21 Þ 

2U1 g1þ mn o þ U1 g1þ mn

Z

r1 0

fm ðk1mn rÞfm ðk0mi rÞr dr,

(29)

So, Eq. (14a) is changed into 2

ssbm A

6 bf 6 ss 6 A 6 6 0 4 0

bm ssbm B þ ccB

bm ssbm C þ ccC

ssbf B sscm B sscf B

ssbf C sscm C sscf C

þ þ þ

ccBbf ccBcm ccBcf

Especially, it is noted that Eq. (15) has become 2 bm ssB þ ccBbm 6 bf 6 ss þ ccbf B 6 B 6 cm 6 ssB þ ccBcm 4 cf sscf B þ ccB

þ þ þ

ccCbf ccCcm ccCcf

3

2 3 2 bm 3 An pi 7 7 6 Bn 7 6 bf 0 7 6 7 6 pi 7 76 7: ¼ 6 7 7 7 sscm 4 Cn 5 6 D 7 4 0 5 5 cf D n ssD 0 0

bm ssbm C þ ccC

(30)

3

7 bf 7 ssbf C þ ccC 7 cm 7 sscm C þ ccC 7 5 cf ssC þ ccCcf

(31)

4N2N

As we know, cc in the matrices represents the effect of acoustic treatments in a subsection, and the matrix including the effect of a lined subsection can still be considered as an independent element since all unknowns remain on the interface. By applying the interface matching condition for different element or subsection, we can obtain the final system of linear algebraic equations like Eqs. (14) and (30) for the acoustic solution. It should be noticed that we only discuss the contraction duct here, and the method in this paper can also be applied to the expansion duct. In contrast to the mode-matching approach [30], the above model may have the following three main features. One is to give up the calculation of complex eigenvalues by applying the singularity method to simulate the effect of acoustic liners. The second is to be capable of including the effect of both locally and non-locally reacting liners, and even sound source on the acoustic attenuation. The third is to have the function of considering the influence of both soft walls and varying crosssectional ducts on the sound propagation and attenuation. In order to describe the characteristics of ‘‘transfer element’’ consisting of Eqs. (30) and (31), and also for a comparative analysis, we still name our proposed method as ‘‘transfer element method’’ (shorten as TEM) as done in Ref. [29]. On the other hand, if we make a comparison with the existing work [29] for a uniform duct, the most obvious differences consist in the introduction of the additional boundary conditions like Eqs. (11c), (21) and (22). This results in more complicated matching conditions and completely different matrix coefficients, as shown in Eq. (30).

3. Results and discussion 3.1. Numerical results for a rigid duct To validate the model in this paper, we will make some comparison with the existing work [33]. If we define the refection and transmission coefficients as Rmn ¼ 9pA,mn u =pi,mn u 9, Tmn ¼ 9pD,mn u =pi,mn u 9, Fig. 6 shows the results for the reflection and transmission coefficients in the expansion and contraction ducts without mean flow. In spite of the different method, Fig. 6 almost gives the same results as Ref. [33]. This also means the present model can give a relatively accurate prediction to the non-uniform duct with a rigid wall.

3.2. Effect of soft walls To illustrate the influence of soft walls on the sound propagation in the duct, we first consider an abrupt varying duct as shown in Fig. 7(a), and then make the comparison with the example provided by Selamet et al. [34]. In fact, Fig. 7(b) shows a reasonable agreement with the calculations based on boundary element method [34].

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1.0

present Ref.33

0.8 r1

pi′ pA′

Rmn Tmn

r2 pD′

0.6

T21

T11

0.4

l

R11

R21

0.2 0.0 6

5

8

7

10

9

f (kHz)

present Ref.33

1.6 pi′ pD′

pA′ l

T21

1.2 r2

Rmn Tmn

r1

T11

0.8 R21

R11

0.4 0.0 6

7

8 f (kHz)

9

10

11

Fig. 6. Transmission and reflection coefficients for a rigid varying cross-sectional duct ((a) r1 = 0.0208, r2 = 0.02645, l =0.0207; (b) r1 =0.0208, r2 = 0.01785, l = 0.0207).

3.3. Flow boundary condition in non-uniform ducts We now assess the validity of the method in the higher-order mode incidence and slowly varying cross-sectional duct. For this respect, the only example available in literatures was provided by Rienstra and Eversman [6]. In this example, ‘‘soft-wall’’ is locally reacting liners, so in Eq. (23), p0+ = 0. The outer wall (0 rzr1.86393) is soft-wall with impedance Z= 2  i and the inner wall is hard. Besides, because the duct includes flow and slowly varying cross-section, the impedance boundary condition should be given in terms of what Myers [36] and Everman [37] suggested previously. Therefore, the impedance equation shown in Eq. (23) is an approximate relation under the condition of non-uniform mean flow. In order to consider the effect of mean flow variation on the impedance equation, we have attempted the following three kinds of expressions (i.e., TEMa, TEMb and TEMc) to calculate the radial perturbation component. But we have to assume that all perturbation solutions for a subsection still exist in this course. For TEMa, we have the same expression as Eq. (25), that is   Uz @ V ¼ 1þ Vn (32a) io @zu and for TEMb   Uz @ @Uz  sin2 y Vn V ¼ 1þ io @zu @zu

(32b)

   Uz @ @Uz @Ur @Ur  sin2 y þ sin y cos y þ cos2 y Vn , V ¼ 1þ io @zu @zu @zu @r

(32c)

and for TEMc

where Uz in Eqs. (32a) and (32b) is calculated based on one dimensional duct flow, y is the axial inclination angle of a nonuniform duct, while Uz and Ur in Eq. (32c) includes the radial component of main stream. The relevant axial and radial flow Mach number are depicted in Fig. 8, which are obtained by the calculation of the potential equation using the finite element method. The above three equations will reduce to the same form without flow.

X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387

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fibrous material r3

r2

pi′ pA′

r1

′ pD

r1

fibrous material l 35 Ref.34(Fig.5) r2 = 3.45 Ref.34(Fig.5) r2 = 4.45 present r2 = 3.45 present r2 = 4.45

Transmission Loss (dB)

30 25 20 15 10 5 0 0

500

1000 1500 2000 2500 3000 3500 Frequency (Hz)

Fig. 7. Effect of radius on the transmission loss, (a) configuration for an expansion chamber with porous materials, (b) r1 = 2.45 cm, r3 = 8.22 cm, l = 25.72 cm, dh = 0.249 cm, tw = 0.09 cm, s =8%, R = 4896 Rayls/m.

0.4

Mz1 (TEMa,TEMb) Mz2 (TEMc) Mr (TEMc)

0.2

M

0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0.0

0.5

1.0 Z

1.5

Fig. 8. Flow Mach number in the varying cross-sectional duct.

With the description to the example, the predicted sound attenuation is given in Table 1, in which m is circumferential mode and k0 is non-dimensional angular frequency. The case ‘a’ is without flow and ‘b’ denotes that the Mach number is equal to  0.5 at the source plane, while the variation of the flow Mach number is shown in Fig. 8. In addition, the present solution has set up the source to be represented in terms of input modal amplitudes for eigenmodes for a duct with hard walls, which is the same as the FEM in Ref. [6]. In the given example shown in Table 1, we just used the first-order radial mode as the input parameter. As shown in Table 1, if the radial flow and axial inclination angle of a duct are included in the impedance equation, the results for TEMb and TEMc tend to approach the prediction finished by the finite element method [6] and perturbation method [6].

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Table 1 Acoustic attenuation comparison of various methods.

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b

m

k0

Ma

FEM

MS

TEMa

TEMb

TEMc

Cut-on mode

Cut-off ratio

10 10 10 10 20 20 30 30 40 40

16 16 50 50 50 50 50 50 50 50

0  0.5 0  0.5 0  0.5 0  0.5 0  0.5

51.6 27.2 4.7 1.5 12.5 3.9 29.0 9.7 196 28.4

51.6 27.1 3.5 0.9 12.3 3.3 28.7 8.9 210 28.6

50.4 30.3 5.9 2.5 13.3 4.9 28.7 11.2 182.8 31.4

50.4 30.1 5.9 2.5 13.3 4.9 28.4 11.2 182.8 31.4

50.4 28.1 5.9 2.7 13.3 4.7 28.4 10.7 182.8 29.0

1 2 9 11 8 9 4 6 2 3

1.36 1.25 3.38 3.97 1.91 2.26 1.34 1.59 1.03 1.23

35 34 33

1b 5b

TL (dB)

32 31 30 29 28 27 26 25

0

50

100 150 200 Number of section

250

300

Fig. 9. Variation of sound attenuation with the number of section; 1b, m =10, k0 =16, Ma =  0.5, 5b, m= 40, k0 = 50, Ma =  0.5

3.4. Convergence of numerical prediction for different subsection number It is indeed very difficult to directly prove whether or not the segmentation approach is reasonable under flow condition. However, as an alternative way, we still chose the example in Ref. [6] to check ‘‘one dimension flow+ all sound modes assumption’’. After all, Rienstra and Eversman gave the detailed parameter setup and relevant results from both finite element method and perturbation method for a very typical non-uniform duct in Ref. [6]. It is shown in Fig. 9 that when there is higher-order mode as an incident wave with flow, with the increase of the subsection number, the calculating results for TEMa will approach an interval of convergence, for 1b case listed in the Table 1, the value is about 28.5 dB, for 5b case, the value is around 29.4 dB. Up to 300 subsection number, we have not found there is a value which is completely outside the interval. For the two cases, Table 1 shows the comparison with the finite element method (27.2 dB, 28.6 dB) and perturbation method (27.1 dB, 28.4 dB). As for Fig. 10, we can seen that with the increase of the subsection number, the variation of sound attenuation for the cases, 2b, 3b and 4b listed in Table 1 almost converge on the same values, which still have some discrepancies with the results in Ref. [6]. For convenience, we have also listed the result in Table 2 for the several typical subsection number of the varying sectional duct. It is found that the prediction values of sound attenuation for TEMa have no obvious variation from the subsection number 20–50. The relative error is even below 1% except for the case 5a in Table 2. The later is related to the effect of the cutoff condition. In fact, it is extremely difficult to predict the attenuation well under this condition almost for all methods. In a word, the results in Table 2 have shown that the present segmentation approach has an acceptable numerical convergence. 3.5. Application of the present model for the design of various acoustic liners As discussed previously, with the different combination of each transfer element for subsection, the present model can effectively include the effect of both locally and non-locally reacting liners on the sound propagation. This also means that we may have more choices for the optimum design of acoustic liners.

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12 10 2b 3b 4b

TL (dB)

8 6 4 2 0

0

20

40

60

80 100 120 140 160 180 200 220 Number of section

Fig. 10. Variation of sound attenuation with the number of section; 2b, m =10, k0 = 50, Ma =  0.5, 3b, m= 20, k0 =50, Ma =  0.5, 4b, m= 30, k0 = 50, Ma =  0.5

Table 2 Comparison of prediction accuracy for different subsection number.

1a 1b 2a 2b 3a 3b 4a 4b 5a 5b

m

k0

Ma

20

30

40

50

10 10 10 10 20 20 30 30 40 40

16 16 50 50 50 50 50 50 50 50

0  0.5 0  0.5 0  0.5 0  0.5 0  0.5

50.4 30.3 5.9 2.5 13.3 4.9 28.7 11.2 182.8 31.4

50.2 29.8 5.9 2.6 13.3 4.9 28.3 11.1 180.0 30.8

49.9 29.7 6.1 2.6 13.4 4.9 28.1 11.1 178.4 30.6

49.7 29.7 6.0 2.7 13.4 4.9 28.0 11.2 177.2 30.5

Fig. 11 gives three kinds of different liners, which are locally reacting liner, expansion chamber and non-locally reacting liner, respectively. In Fig. 12, r0 = 0.1 m, this means that first-order cut-off frequency is 2073 Hz. For the locally reacting liner A, rd = 0.125 m, l = 0.1 m, dh = 2.5 mm, tw = 2 mm and s =2%, the impedance model Z= (r, x) is [25,35] pffiffiffiffiffiffiffiffiffiffi   9Ma 9 tw 8vo 1þ r¼ , (33a) þ 0:3 sc dh s pffiffiffiffiffiffiffiffiffiffi





oðtw þ 0:85dh Þ oðrd  r0 Þ 8vo tw , þ  cot x¼ c sc sc dh

(33b)

where v= 1.5  10  5. In addition, in this impedance model, the second term of the acoustic resistance r is zero without mean flow, while for a non-locally reacting liner, there is no last term in Eq. (33b). For the expansion chamber B, rd = 0.18 m, l= 0.3 m; and for the non-locally reacting liner C, rd = 0.125 m, l = 0.3 m, dh = 1.83 mm, tw = 0.87 mm and s = 2%. The transmission loss is given by TL ¼  20 log10 9pHu =piu 9

(34)

and the acoustic absorption coefficient is defined as AC ¼

9pAu 9þ9pHu 9 9piu9

(35)

Fig. 12 shows the transmission loss corresponding to different frequency values. In addition, Fig. 12(a) indicates that locally reacting liner has great sound attenuation near the resonance peak but a very narrow frequency range for effective sound absorption. Fig. 12(b) reveals that the varying cross-section duct has the transmission loss consisting of the arch curves with different amplitudes. In this situation, the sound wave reflection actually plays dominant role in the computational

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l

r0

rd

p′i

pH′

pA′ perforated plate

l

r0

rd

pi′

pH′

pA′

l

r0

pi′

rd pH′

pA′ perforated plate

Fig. 11. Schematic of three kinds of liners: (a) local reaction liner, (b) expansion chamber and (c) non-local reaction liner.

value of transmission loss, while real sound absorption is always zero. From Fig. 12(c), it is found that non-locally liner has multi-resonance peaks, which naturally results in a broad frequency range for sound attenuation. However, the transmission loss corresponding to the resonance peak is not so big as the locally reacting liner. Especially, the performance for the lower frequency range is relatively poor for such a parameter setup. So, it may be expected that with different combinations of three kinds of liners, we may obtained a better liner design. Fig. 13 is a schematic diagram for our computation. Fig. 14 shows that for different combinations, we can indeed obtain a relatively optimum sound attenuation corresponding to a broad frequency range. Especially, Fig. 15(a) shows the two kinds of combinations have almost the same transmission loss but Fig. 15(b) indicates that they have different sound absorption coefficients. In some case, we may also hope to control the unnecessary reflection; therefore, an acoustic engineer needs to carefully choose different design combination in order to absorb more sound energy inside the liner for different purpose. In the followings, we will give a more complicated example in order to show how to control the sound propagation in a varying cross-sectional duct with flow as shown in Fig. 16. The last two columns of Table 1 give the cut-on modes and cut-off ratio with the sound source position corresponding to a rigid wall. As for the data shown in Table 1, it is noted that the more cut-on modes the duct possesses, the less sound attenuation the liner can realize. In fact, since different cut-on mode corresponds to different incidence to the liner, this means a fixed acoustic impedance design will not effectively meet the sound attenuation requirements for multi-propagating modes. Therefore, we will use different combinations of locally and non-locally reacting liners to improve the sound attenuation performance shown in Table 1.

0.8

25

0.6

20 15

0.4

10

0.2

5 0 500

0

1.0

TL AC

35 Transmission Loss (dB)

0.0 2000

1000 1500 Frequency (Hz)

0.8

30 25

0.6

20 15

0.4

10

0.2

5 0 500

0

Transmission Loss (dB)

0.0 2000

1500 1000 Frequency(Hz)

1.0

TL AC

35

0.8

30 25

0.6

20 15

0.4

10

0.2

5 0 500

0

Absorption Coefficient

30

Absorption Coefficient

1.0

TL AC

35

2383

Absorption Coefficient

Transmission Loss (dB)

X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387

0.0 2000

1000 1500 Frequency (Hz)

Fig. 12. Transmission loss of three kinds of liners: (a) local reaction liner (r0 = 0.1 m, rd =0.125 m, l = 0.1 m, dh =2.5 mm, tw = 2 mm, s =2 %), (b) expansion chamber (r0 = 0.1 m, rd = 0.18 m, l = 0.3 m) and (c) non-local reaction liner (r0 = 0.1 m, rd = 0.125 m, l = 0.3 m, dh = 1.83 mm, tw =0.87 mm, s = 2 %).

A

B

C

pi′

pH′

pA′

Fig. 13. Schematic of combination liners.

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40 A+B+C B+A+C A+C+B

Transmission Loss (dB)

35 30 25 20 15 10 5 0 0

500

1000 Frequency (Hz)

1500

2000

Fig. 14. Effect of different combinations on transmission loss.

40 A+B+C C+B+A

Transmission Loss (dB)

35 30 25 20 15 10 5 0 0

500

1000 Frequency (Hz)

1500

2000

Absorption Coefficient

1.0 A+B+C C+B+A

0.8 0.6 0.4 0.2 0.0 0

500

1000 Frequency (Hz)

1500

2000

Fig. 15. (a and b) Optimum sound attenuation with respect to a broad frequency range.

As shown in Fig. 16, we assume that the non-locally reacting liner of length l1 = 0.485 m has constant inner radius, i.e., R1 = 0.995 m, and the other variation of R1 and R2 along the axial direction is the same as introduced in Ref. [6]. In addition, l1 + l2 = 1.86393 m, Rd = 1.2 m. The perforated plate has its parameters: tw = 0.001 m, dh = 0.001 m and s = 0.02. In this case, for

X. Wang, X. Sun / Journal of Sound and Vibration 330 (2011) 2369–2387

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non-locally reacting liners, the impedance model [18] Z =r + ix is r¼

CD2 ¼

Mb

sCD2

,

x¼

4ksg ktw þ , s pdh

1 2

ðk1 þ k2 =l þ k3 =l Þ

,

l¼

(36)

Mb , Ma

(37)

where k1 = 2.10, k2 =  0.332, k3 = 0.0556 (for outflow) and the x was given by Howe [18] without the effect of main stream, while the r was derived by Sun and Jing [20] with the effect of main stream. For the locally reacting liner, the impedance model is the same as shown in Eq. (33). Table 3 gives some numerical results and its comparison with the previous predictions

l1

Source plain

r

l2

Mb

Rd

R1 Ma R2

z

Fig. 16. Schematic of combination of locally and non-locally reacting liners with flow.

Table 3 Comparison of local reaction liner (TEMa) and combination liners. k0

Ma (z =0)

TEMa

Combination liners

10 10 20 30 40

16 50 50 50 50

 0.5  0.5  0.5  0.5  0.5

30.0 2.5 4.9 11.2 31.4

22.58 5.29 12.74 16.85 27.35

25

20

TL

1b 2b 3b 4b 5b

m

15

10

1b 3b 4b

5 0.00

0.05

0.10 Mb

0.15

Fig. 17. Effect of bias flow on the transmission loss.

0.20

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(TEMa). Besides, in this calculation, the bias flow Mach number is assumed that Mb = 0.03, which is introduced to go through the orifices shown in Fig. 16. It is known that the bias flow can greatly enhance the resistance, so the relevant sound attenuation corresponding to larger cut-off ratio such as case 2b, 3b and 4b can obviously increase. However, for the attenuation related to smaller cut-off ratio such as 1b and 5b may slightly decrease. Besides, Fig. 17 shows that variation of sound attenuation with the change of bias flow. This means that with the bias flow control, the frequency range for the sound attenuation becomes broader, which is very useful for practical application. 4. Conclusions This work is an extension and new application of transfer element method (TEM) suggested in Ref. [29]. However it is noted that although we still use the same way as Ref. [29], more concrete difficulties need to be overcome for non-uniform lined ducts, such as the determination of various interface matching relations for a non-uniform duct, the establishment of ‘‘a transfer element’’ of non-locally reacting liner. In fact, almost all formulation and matching relations are completely different from the work in Ref. [29] for uniform ducts. Based on constructing the solution of a subsection for both locally and nonlocally reacting liners, this paper presents an analytical model to study sound attenuation in a non-uniform duct. This model contains more functions compared with the existing mode-matching approach and boundary element method, especially the capability of handling different types of liner combination. Various numerical results show that the present model not only gives the same prediction as the existing models for both continuously and abruptly varying cross-sectional area ducts without flow, but also make a relatively reasonable prediction to non-uniform duct with flow. As one of the objectives of the present model, attention is also paid to how to improve the sound attenuation in a varying flow environment by the change of the wall impedance using bias flow. In fact, the example from a nacelle configuration may show some valuable information. However, for a complicated duct flow like aeroengine nacelle, the model cannot include enough flow details, so it is difficult to obtain high accuracy prediction for this situation. Anyway, it is expected that the model may provide a useful tool for the optimization of sound attenuation in a non-uniform duct at the preliminary design stage of acoustic treatments.

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