Applied Acoustics 90 (2015) 54–63

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Applied Acoustics

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A wave expansion method for acoustic propagation in lined flow ducts

http://dx.doi.org/10.1016/j.apacoust.2014.10.0150003-682X/� 2014 Elsevier Ltd. All rights reserved.

E-mail address: ciaran@kth.se

Ciarán J. O’ReillyKTH Royal Institute of Technology, Department of Aeronautical and Vehicle Engineering, Teknikringen 8, SE-100 44 Stockholm, Sweden

a r t i c l e i n f o

Article history:Received 14 February 2014Received in revised form 8 September 2014Accepted 18 September 2014

Keywords:Acoustic linerFrequency-domain propagationWave expansion methodFlow impedance boundary condition

a b s t r a c t

Acoustic liners are used extensively in engineering applications, particularly in aero-engines and automo-tive exhaust systems. In this paper, a flow impedance boundary conditions is introduced into the waveexpansion method with the aim of providing an efficient methodology for computing the acousticpropagation through a lined duct with flow. For a potential flow, the boundary layer and the lined wallare included in the discretisation scheme by the Myers flow impedance boundary condition. The acousticpropagation through a flow impedance tube is computed in order to validate the implementation of theimpedance boundary condition in this scheme. The results show that this computationally lightmethodology provides generally good agreement with the experimental data.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Acoustic liners are used to absorb sound in many engineeringapplications, but particularly in aero-engines [1,2]. As aero-engineshave become progressively larger in diameter, with theintroduction of higher bypass-ratio flows, fan noise has become aprominent noise source. The accompanying reduction in jet noisehas also unmasked combustion and turbine noise as significantnoise sources. This has resulted in their already extensive, and everincreasing, use in aero-engines to treat fan, combustion andturbine noise.

Liners are typically composed of a single layer honeycombstructure with a perforated facing sheet and a rigid backing sheet,although more complex novel configurations are under investiga-tion [3]. Quite often, it is the effect of the liner on sound propagatingpast them which is of primary interest, and liners are included inpropagation predictions through an impedance model, rather thanincluding individual cells, the perforations in the facing sheet, etc.Liners are therefore characterised by a complex impedance value,which is dependent on the grazing flow velocity and acoustic pres-sure spectrum at the liner surface. This impedance value is usuallyinversely determined from experimental data using an eductiontechnique with numerical or analytical propagation methods [4–7]. For typical liner applications (i.e. with a honeycomb structure),the liner may reasonably be assumed to be locally reacting.

In this paper, a flow impedance boundary condition is imple-mented in a numerical propagation scheme known as the waveexpansion method (WEM) [8,9] with the intention of investigating

acoustic propagation in lined ducts. The implementation of a flowimpedance boundary condition in the WEM has not beenexamined in the past, but is of interest due to the inherent effi-ciency of the scheme. The WEM uses fundamental solutions ofthe wave operator and so accurate solutions to linearised propaga-tion equations may be obtained with only two-to-three points perwavelength [10], which is significantly less than the six-to-tenpoints per wavelength of comparable traditional finite differenceand finite element methods. The method is robust to meshingand could be implemented in a meshless manner (only points, noelements). These properties make it well suited for use in the opti-misation of realistic complex lined engine nacelles.

This paper is structured as follows. The WEM discretisationscheme is presented in Section 2. The propagation equation andthe flow impedance boundary condition are implemented into theWEM in Section 3. The methodology is then used to compute thepropagation through a flow impedance tube and to educe imped-ance values in Section 4. The results are compared against the datafrom the grazing incidence tube (GIT) at the NASA Langley ResearchCenter, published by Jones et al. [11] in Section 5. These data are themost extensive available for such a configuration and have beenwidely used to benchmark the use of flow impedance boundary con-ditions in numerical schemes [12–14]. These results are discussedfurther along with the improved efficiency of the methodology inSection 6, before conclusions are presented in Section 7.

2. Wave expansion discretisation scheme

The wave expansion method (WEM) was first proposed byCaruthers et al.[8], and was more recently used by Ruiz and Rice

C.J. O’Reilly / Applied Acoustics 90 (2015) 54–63 55

[10] and Barrera Rolla and Rice [15] to investigate sound propaga-tion in quiescent media. The discretisation scheme may be used forsolving linearised time-harmonic propagation equations. The WEMis a physically-based numerical scheme, in that it uses fundamen-tal solutions of the wave operator. As such, it has a very low disper-sion error compared to those associated with other numericalschemes [8]. As a result, it is a highly efficient numerical procedure,requiring only two-to-three points per wavelength to obtain accu-rate solutions. This discretisation is perhaps optimal, as it is validdown to the Nyquist limit of two points per wavelength. The valueof the unknown at each discrete point in the domain is related tothe values at a selected set of neighbouring points by plane-wavefunctions. The WEM represents a local interpolation formula toobtain the field values in the domain.

The solution to a linear propagation equation of the form L/ ¼ 0at each point in a computational lattice may be approximated bythe superposition of the field generated by N hypothetical planewaves of strength cn and with unit propagation in direction vn.The solution at a discrete point x0 at the centre of a small compu-tational stencil, as depicted in Fig. 1, is

/0 ¼XN

n¼1

cn exp �iqn vn:x0ð Þ½ � ¼ hc; ð1Þ

where h is a 1� N vector containing fundamental plane-wavesolution hn ¼ exp �iqn vn:x0ð Þ½ �; c is a N � 1 vector containing cn,and qn is the wave number. Similarly, the solution at neighbouringpoints at xm in the computational stencil (see Fig. 1 again), wherem ¼ 1;2; . . . ;M and M is the number of neighbours, is given as

/m ¼XN

n¼1

cn exp �iqn vn:xmð Þ½ � ¼ Hc; ð2Þ

where H is a M � N matrix containing Hm;n ¼ exp �iqn vn:xmð Þ½ �.Within the computational stencil, the wave strengths c must

simultaneously satisfy the central and neighbouring points so inthis way the central point is linked to its neighbours. This assumesthat within the stencil it is possible to approximate any mean flowas locally uniform.

Eq. (2) is therefore used as a constraint on Eq. (1) by solving Eq.(2) and substituting into Eq. (1). By choosing the number of wavesto be greater than the number of neighbours, N > M, Eq. (2) is

Fig. 1. Schematic of the wave expansion discretisation stencil with a central node,its neighbours and hypothetical plane waves.

under-constrained and so there exists an infinite set of solutionsfor c. However, if Hþ is the Moore–Penrose pseudo-inverse of H then

c ¼ Hþ/m; ð3Þ

is the minimum-norm out of all solutions. Upon substituting backinto Eq. (1) this leads to

/0 � hHþ/m ¼ 0; ð4Þ

which may be written in vector form as

1 � hHþ� � /0

/m

� �¼ 0: ð5Þ

The central point is thus linked to its neighbours in an optimal orleast-squares way.

For the total computational lattice made up of overlapping sten-cils, a linear system of equations may be constructed, using Eq. (5)to determine each row. This takes the form

K/ ¼ f; ð6Þ

where K is the overall sparse unsymmetric stiffness matrixndof � ndof in dimension, where ndof is the number of degrees offreedom in the overall problem. The vector f is been added to theright-hand-side and may be used to add point sources or imposeboundary values. This system of equations may be solved for /, avector containing the solution at each point in the overall computa-tional lattice.

It is assumed in this present formulation that the values of thecoefficients are constant within a computational stencil so thatthere are only small variations across overlapping stencils. The gridmust therefore be sufficiently fine to resolve flow variations. How-ever, this is also true of other numerical schemes. Alternatively, ahigher-order stencil could be employed if there are large variationsin the background flow as discussed by Caruthers et al. [8].

From experience it has been found that an approximatelyuniform distribution of plane-wave vectors in all directions givesthe most accurate results. In 2D this can be achieved easily, whilstin 3D a Buckminster Fuller geodesic dome gives an effective distri-bution of 60 uniformly spaced vectors.

Additionally, out-going and in-coming waves may both beincluded for qþ and q�, although one set is sufficient. This isachieved by augmenting the number of columns in h and H suchthat h becomes fhqþ hq�g and similarly for H. For example, if adistribution of 20 vectors is used then N ¼ 40.

It is worth noting that although the computational stencildepicted in Fig. 1 shows a central point and its eight neighbourson a regular lattice, there is nothing in the present formulationto restrict the method to this type of grid. Indeed all that isrequired for the WEM are the grid points and information aboutneighbouring points. In practice, it is often easiest to use the meshelement connectivity generated by (regular or irregular) meshingroutines to determine the point neighbours. There may, however,be circumstances when it is more efficient or reliable to determinethe neighbours directly from a list of points (or use a combinationof surface meshes with distributed points within the domain). Inthis regard the WEM may be regarded mesh-less or loosely meshdependent.

3. A WEM-based methodology for computing the propagationin a lined flow duct

In order to compute the linear acoustic propagation in a linedduct, it is necessary to decide how the lined wall or impedanceboundary is to be introduced into the computation and whatassumption can be made about the background flow. In theabsence of a mean flow, the impedance relates the Fourier

1 An over-line is used throughout to denote mean flow, as opposed to fluctuating,uantities. Bold characters denote vector or matrix quantities.2 u:r u:r/ð Þ ¼ u2r2/þ u:r/ruþ u: u� r�r/ð Þð Þ þ u: r/� r� uð Þð Þ but for

coustic propagation in a potential flow r�r/ ¼ 0 and r� u ¼ 0, so the last tworms disappear.

56 C.J. O’Reilly / Applied Acoustics 90 (2015) 54–63

transformed acoustic pressure and normal acoustic particle veloc-ity at the surface. This is also the case when mean flow is presentwhere the velocity goes to zero at the wall (no-slip condition).However, in this case, the effect of the viscous boundary layer overthe liner must be included in the propagation. This may be done bysolving the linearised Navier–Stokes equations. The linearisedNavier–Stokes equations may also support vorticity and entropywaves but these are not of primary interest here. A pragmaticalternative frequently employed has been to include the boundarylayer in the impedance model and to assume that the flow ispotential [2]. This is computational much lighter and is attractiveparticularly for inverse searches of liner impedance values fromexperimental data or large-scale design optimisations.

As the viscous boundary layers of high-Reynolds-number flowsare typically much thinner than a characteristic wavelength,Ingard [16] and later Myers [17] derived an impedance wallmodel for sound at a lined wall in a mean flow with slip at thewall. This is the limit of a vanishing boundary layer and assumesthat the acoustic pressure and particle velocity are continuousacross the thin boundary layer. This boundary condition is knownas the Myers boundary condition (or alternatively as the Ingard-Myers boundary condition). As the boundary layer has beeneffectively removed from the propagation, the mean flow maybe assumed to be inviscid and potential. The perturbed field canbe written in terms of an acoustic velocity potential and thepropagation equations may be reduced down to a convectedHelmholtz equation.

The Myers boundary condition has been widely used to achievenumerical predictions of lined duct propagation [2,18], which areconsidered to be sufficiently accurate for engineering applications.However, there have been a number of studies in recent years,which have identified problems with Myers condition resultingfrom the modelling assumptions. There are doubts as to whetherthe Myers model deals correctly with hydrodynamic modes inthe boundary layer [19–21]. Since the boundary condition is thelimit of an infinitely thin boundary layer, or the boundary layerbeing collapsed into a vortex sheet, when applied with slippingmean flow it may exhibit an instability comparable to theKelvin–Helmholtz free-shear-layer instability. Furthermore, themodel has been shown to be mathematically ill-posed [22]. Thesetwo issues are problematic for time-domain simulations, whereerrors generate perturbations at every frequency, and may resultin difficulties with numerical convergence. For frequency-domaincomputations, however, they are of minor practical importance[23]. It has also been shown that the Myers model fails to resultin a single impedance value for wave propagation downstreamand upstream in a lined duct when the acoustic boundary layeris not much smaller that the mean flow boundary layer thickness,i.e. at low frequencies [24]. This finding does not negate the use ofthe model altogether but rather it limits its applicability. Cautionshould be exercised when applying the Myers boundary conditionto cases which are not consistent with those used to evaluate theimpedance.

In this paper, the acoustic propagation in a potential flow issolved with Myers boundary condition implemented for the linedwall impedance condition. This choice could be changed later.The questions at hand here are how to implement this type ofboundary condition in the WEM and how does this computation-ally light approach compare with available experimental dataand alternative computationally more demanding approaches?

3.1. Acoustic propagation in inhomogeneous irrotational steady flows

Wave propagation in an inhomogeneous irrotational homentro-pic steady flow may be described in terms of an acoustic velocitypotential, /, and is governed by [25]

1qr: qr/ð Þ � ixþ u:rð Þ 1

c 2 ixþ u:rð Þ/� �

¼ 0; ð7Þ

where u;q; c, are the local mean velocity vector, density, and speedof sound respectively.1 x is the angular frequency and i �

ffiffiffiffiffiffiffi�1p

.r isa vector differential operator, which depends on the chosen dimen-sions and coordinate system. The acoustic pressure and velocity arerelated to the potential by p ¼ �ixq/� qu:r/ and u ¼ r/.

Eq. (7) may be expanded into the form

r2/� M:rð Þ2/� B:r/þ K2/ ¼ 0; ð8Þ

where

B ¼ 2ikMþ u:r 1c 2

� �uþ 1

c 2 u:ru� 1qrq; ð9Þ

K2 ¼ k2 � ixu:r 1c 2 ; ð10Þ

and where M ¼ u=c is the local Mach vector and k ¼ x=c.At a discrete point x0 in the small sub-domain covering any

stencil of the entire computational lattice the coefficients of Eq.(8) may be considered to be constant, and a fundamental plane-wave solution may be determined, via solving for the roots of itscharacteristic equation, as

/0 ¼ exp �iq� v:x0ð Þ½ �; ð11Þ

where /0 ¼ / x0ð Þ;v is an arbitrary unit vector in the direction of theplane-wave propagation, q� is the wave number for out-going andin-coming waves [8]

q� ¼iB0:v �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4K2

0 1� M0:vð Þ2

� B0:vð Þ2r

2 1� M0:vð Þ2 ; ð12Þ

and B0 ¼ B x0ð Þ;K20 ¼ K2 x0ð Þ and M0 ¼M x0ð Þ are constant coeffi-

cients. These waves may be used in the WEM discretisationdescribed in the previous section.

3.2. Implementation of a flow impedance boundary condition in theWEM

In the absence of flow the complex impedance on the boundaryis simply Z ¼ p=n:u, where p and u are the acoustic pressure andvelocity. Z � ZR þ iZX , where ZR and ZX are the resistance and reac-tance of the liner. For a steady mean flow that slips at the bound-ary, this boundary condition must be modified [16,17]. If the fluidvelocity at the boundary is uþ u expðixtÞ, then a flow impedanceboundary condition may be given by

n:u� pZ� u:r p

ixZ

þ p

ixZn: n:ruð Þ ¼ 0: ð13Þ

This is known as the Myers boundary condition [17].In order to impose this boundary condition in the discretisation

scheme outlined above, it must firstly be expressed in terms of theacoustic velocity potential. The acoustic velocity and pressure arerelated to the potential by u ¼ r/ and p ¼ �ixq/� qu:r/.Inserting these relations into the Myers equation and expandingusing the vector dot product identity2 leads to

n:r/þ ixqZ

/þ 2qZ

u:r/þ qixZ

u2r2/þ qixZ

u:r/r:u

� qZ

/n: n:ruð Þ � qixZ

u:r/n: n:ruð Þ ¼ 0: ð14Þ

q

ate

Table 1Normalised exit impedances from Jones et al. [11].

f (Hz) Mx = 0.0 Mx = 0.335

ZR=ðq1c1Þ ZX=ðq1c1Þ ZR=ðq1c1Þ ZX=ðq1c1Þ

500 0.99 0.06 0.85 �0.131000 1.02 0.06 1.09 0.281500 1.03 0.08 0.86 �0.172000 0.98 0.13 1.04 0.242500 1.02 0.11 1.00 �0.203000 1.00 0.13 0.95 0.16

Table 2Normalised GIT liner impedances at Mx ¼ 0:0.

f (Hz) Jones et al. educed WEM educed

ZR=ðq1c1Þ ZX=ðq1c1Þ ZR=ðq1c1Þ ZX=ðq1c1Þ

500 0.51 �1.68 0.57 �1.611000 0.46 0.00 0.47 0.001500 1.02 1.30 1.08 1.302000 4.05 0.62 4.68 0.772500 1.54 �1.60 1.55 �1.633000 0.70 �0.29 0.72 �0.27

Table 3Normalised GIT liner impedances at Mx ¼ 0:335.

f (Hz) Jones et al. educed WEM educed

ZR=ðq1c1Þ ZX=ðq1c1Þ ZR=ðq1c1Þ ZX=ðq1c1Þ

500 0.61 �0.59 0.43 �0.721000 0.17 0.14 0.43 0.071500 1.18 1.27 1.15 1.432000 4.40 �1.64 5.08 �2.132500 0.93 �1.43 1.00 �1.393000 0.73 �0.24 0.66 �0.25

C.J. O’Reilly / Applied Acoustics 90 (2015) 54–63 57

As mentioned previously, Eq. (2) is used as a constraint on Eq.(1) in the discretisation scheme. Therefore, this boundary conditionmay be imposed by augmenting Eq. (2) at each boundary pointwith an additional constraint row [10]. As the potential at eachpoint is the superposition of plane-wave functions, its derivativesin Eq. (14) may be evaluated explicitly in the WEM so that H is aug-mented with a row containing

�iqnn:vn þixq

Z

�� 2iq

Zqnu:vn �

qixZ

q2n u2 v2

n�qxZ

qnu:vnr:u

�qZ

n: n:ruð Þ þ qxZ

qnu:vn n: n:ruð Þ�

hn ð15Þ

in each column for each vector direction n. H becomes a ðM þ 1Þ � Nmatrix denoted as Haug .

Taking the pseudo-inverse of Haug gives

c ¼ Hþaug

/m

0

� �; ð16Þ

where Hþaug is a N � ðM þ 1Þ matrix. As before, this is then insertedinto Eq. (1). However, Eq. (4) is now partitioned on boundary points,and so becomes

/0 � hHþaug;L /m � hHþaug;R 0 ¼ 0; ð17Þ

where Hþaug;L is the first M columns of Hþaug and Hþaug;R is theremaining column. As the right-hand-side of Eq. (14) is zero, the lastterm here does not contribute. However, it is worth noting thatother boundary conditions with non-zero right-hand-sides may beimposed in this way with last term being entered in the forcing vec-tor f. The first two terms are used to assemble the overall stiffnessmatrix exactly as for non-boundary points. This implementationmay be used to impose both soft- and hard-wall boundary condi-tions, with the impedance Z set to infinity in the latter case. It is alsoused to implement the exit impedance at the end of the duct.

4. Validation of the propagation methodology

The application of the WEM with a flow impedance boundarycondition to acoustic liner problems is validated by comparingresults obtained using this methodology with experimental datafrom the grazing incidence tube (GIT) at the NASA LangleyResearch Center. This data, presented by Jones et al. in Ref. [11],was collected specifically for the purpose of validating acousticpropagation codes and is the most comprehensive available. Assuch, it has been widely used to benchmark the use of flow imped-ance boundary conditions in numerical schemes [12,26,14,13].

4.1. Validation cases

The GIT test section, illustrated in Fig. 2, was 0.8128 m long andhad a square cross-section with an internal width and height of0.0508 m. It contained an axially centred liner with a length of0.4064 m, which formed the centre of the upper wall. The remain-ing walls were rigid. Acoustic waves in the plane-wave range weregenerated on the upstream side and a near anechoic termination isdownstream. The tube exit impedances from Ref. [11] are given inTable 1. Two flow conditions are considered here – firstly, with nomean flow (Mx ¼ 0:0) and, secondly, with a mean flow of average

Fig. 2. Schematic of the GIT test section from Ref. [11]. Acoustic mea

axial Mach number Mx ¼ 0:335. Acoustic pressure measurementswere made at thirty-one points along the lower surface of theGIT (opposite the liner).

4.2. Liner impedance eduction procedure

Normalised impedance values for the liner have been educedfrom the pressure data by Jones et al. [11] using a 2D finite elementmethod (FEM)-based procedure. In addition to using these Joneset al. educed liner impedance values, impedance values have alsobeen educed here to best-fit the Jones et al. complex pressure datausing the WEM-based methodology. This is achieved, in the sameway as used by Jones et al. for consistency, by iteratively minimis-ing the objective function

F Zð Þ ¼XNp

i¼1

pWEM xið Þ � pData xið Þð Þ p�WEM xið Þ � p�Data xið Þ� �

; ð18Þ

where p is the complex acoustic pressure along the lower surface ofthe tube, Np is the total number of sampled points and ⁄ denotes thecomplex conjugate. These WEM educed values are presented along-side the Jones et al. educed values in Tables 2 and 3 for the two flowcases, and are discussed further in the following sections.

surements were made on the lower surface, opposite the liner.

58 C.J. O’Reilly / Applied Acoustics 90 (2015) 54–63

4.3. Scattering matrix components

The transmission and reflection coefficients are also computedas a further test of the agreement between the experimental dataand the WEM predictions for the upstream source. The acousticpressure values on either side of the liner are decomposed inupstream and downstream propagating plane waves

p ¼ pþ exp �ikþxð Þ þ p� exp ik�xð Þ; ð19Þ

where k� ¼ k=ð1�MxÞ and Mx is the mean axial Mach number. Fora number of sample points 1;2; . . . Ns with Ns P 2 a system can beformed [27]

exp �ikþx1ð Þ exp ik�x1ð Þexp �ikþx2ð Þ exp ik�x2ð Þ

..

. ...

exp �ikþxNsð Þ exp ik�xNsð Þ

266664

377775

pþp�

�¼

p1

p2

..

.

pNs

266664

377775; ð20Þ

which upon solving via a pseudo-inverse gives the values of pþ andp�. The transmission and reflection coefficients are definedrespectively as

T ¼ pdþ

puþ;R ¼ pu

�puþ; ð21Þ

where the superscript u denotes the value on the upstream side ofthe liner and d the downstream side of the liner.

5. GIT validation results

The propagation of acoustic plane waves through the GIT hasbeen computed using the WEM with the newly implementedMyers boundary condition. The computed results presented havebeen made on a uniform lattice of 65� 9 points. Grid convergenceof the solution was checked with finer grids with no appreciabledifference observed. The number of nodal neighbours, M, is takento be eight for internal grid points, i.e. those immediatelysurrounding the point in question as illustrated in Fig. 1. Tenuniformly spaced wave vectors were used for both qþ and q� soN ¼ 20.

5.1. No-flow case

In the absence of a mean flow, Eq. (7) effectively reduces to theHelmholtz equation and the impedance condition to n:u ¼ p=Z.Fig. 3 shows a contour plot of the solution at 3000 Hz. The effec-tiveness of the liner in reducing the amplitude of the acoustic waveis clearly visible. The impedance boundary condition described inSection 3.2 has been used in this solution to impose the hard-wall,soft-wall, and exit impedance boundary conditions. At the inlet ofthe tube (the left-hand boundary) a Dirichlet condition is used tointroduce plane waves. Dirichlet type boundary conditions may

Fig. 3. Normalised contours of the acoustic pressure at 3000 Hz and with Mx ¼ 0:0mean flow computed with the WEM. The boundary conditions are illustrated withplane-wave input (cyan), hard-wall impedance (black), soft-wall impedanceZ ¼ 0:72� 0:27i (grey) and exit impedance Z ¼ 1þ 0:13i (green). (For interpreta-tion of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

be implemented in the WEM by simply constraining the appropri-ate entries in the overall stiffness matrix K [10], i.e. setting theappropriate row to zeros with a one on the diagonal and enteringthe desired solution values into f.

The SPL along the wall opposite the liner is shown in Fig. 4. Itcan be seen that there is very good agreement across all frequen-cies, with only small disagreements at 500 Hz and 2000 Hz, wherethe WEM educed impedance values provide slightly better agree-ment. This is to be expected, of course, as these values have beeneduced using the WEM to best-fit the data. The phase across theGIT is plotted in Fig. 5. The agreement here between the computa-tional values and the data is excellent throughout.

Importantly, there is little difference in the impedance valuesobtained using the WEM- and FEM-based eduction procedures.Fig. 6 shows a plot of the impedance values from Table 2. Thisshows that there is a robustness to these impedance values, in thatthey are variations of the same globally converged value ratherthan quite different values that are perhaps locally optimal. Thereis a consistency in the characterisation of the liner and so in theperformance of the two methods. Fig. 7 shows the transmissionand reflection coefficients, and once again the agreement is good.

5.2. Grazing-flow case

The WEM methodology has also been used to compute thepropagation of acoustic plane waves through the GIT configurationwhen a grazing flow of Mx ¼ 0:335 is present. This uniform-Mach-number flow is the average of the measured profile acrossthe tube. For this flow, the mean flow velocity divergence is zero,so the impedance condition in Eqs. (14) and (15) is simplified asthe last three terms are equivalent to zero. Also, the other gradientterms in the propagation operator (in B and K) are all zero so thewave number q ¼ �k=ð1�M0:vÞ.

A contour plot of the solution at 3000 Hz with the uniform flowis shown in Fig. 8. The boundary condition types used are the sameas in the no-flow case. The longer wavelength in this case due themean flow is visible. The SPL along the wall opposite the liner isshown in Fig. 9. The SPL reduction exiting the liner is capturedquite well at all frequencies, with the exception of at 1000 Hz.The behaviour of the SPL does not show as good an agreementacross the liner as in the no-flow case3 and there are some smalldeviations between the two, particularly at lower frequencies. Thiswill be discussed in more detailed in the next section.

A comparison of the phase values is presented in Fig. 10. Hereagain the agreement between the experimental data and thenumerical values is excellent, with the exception of at 1000 Hz.Again, the WEM educed impedance values provide an improvedfit. The educed impedance values are illustrated in Fig. 11. Thesame consistency noted for the no-flow case is evident here andthe linear behaviour of this particular liner is of note.

6. Discussion

In the previous section, it has been demonstrated the WEM pro-vides a reasonably robust method for computing the propagationin a flow impedance tube. When used to educe the impedance ofa liner, the values obtained are consistent with those found usinga FEM to solve the propagation equation. Despite the limitationsof the Myers boundary condition, the methodology has capturedthe reduction in SPL across the liner using impedance values whichare consistent with those educed using more computationallydemanding procedures in Refs. [11,26,14].

3 Note that the y-axis range used is different for the different frequency plots soariations are more obvious on plots with a narrower SPL range.

v

Fig. 4. Sound pressure levels (SPL) on the surface opposite the liner for Mx ¼ 0:0 with experimental results from Ref. [11] (þ), and WEM computed levels using WEM educedimpedance values (—–) and impedance values from Ref. [11] (---).

Fig. 5. Acoustic pressure phase on the surface opposite the liner for Mx ¼ 0:0 with experimental results from Ref. [11] (þ), and WEM computed levels using WEM educedimpedance values (—–) and impedance values from Ref. [11] (---).

C.J. O’Reilly / Applied Acoustics 90 (2015) 54–63 59

In order to further justify the use of the present methodology,two issues need to be discussed in greater detail – the small depar-tures between the experimental and numerical results that are vis-ible in the grazing flow validation case in the previous section, andthe efficiency of the methodology compared to alternativemethods.

6.1. Differences between the experimental and numerical values

The agreement between the experimental and numerical valuesin the grazing flow validation case is poorest at 1000 Hz. This is vis-ible in both the SPL and phase plots in Figs. 9 and 10. A relativelysmall difference in the impedance value results in quite a large

Fig. 6. Educed liner impedance values with Mx ¼ 0:0 using the WEM-based procedure (—–) and the FEM-based procedure from Ref. [11] (-- -).

Fig. 7. Transmission and reflection coefficients with Mx ¼ 0:0 from the WEM (—–) and from the data from Ref. [11] (þ).

Fig. 8. Normalised contours of the acoustic pressure at 3000 Hz and with a uniformMx ¼ 0:335 mean flow computed with the WEM. The boundary conditions areillustrated with plane-wave input (cyan), hard-wall impedance (black), soft-wallimpedance Z ¼ 0:66� 0:25i (grey) and exit impedance Z ¼ 0:95þ 0:16i (green).(For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Fig. 9. Sound pressure levels (SPL) on the surface opposite the liner for Mx ¼ 0:335 wieduced impedance values (—–) and impedance values from Ref. [11] (-- -).

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change in the acoustic field. Even with the improvement of theWEM educed impedance value, there remains a difference of about6 dB in the exit SPL compared to the experimental data – muchlarger than at any other frequency. The disagreement in the phaseis also the largest at this frequency. This disagreement may beattributed to the presence of a hydrodynamic instability in theexperimental test at this frequency. This instability has beenreported previously in Refs. [26,14]. As it is not included in thepresent methodology, the disagreement with the data here is tobe expected. Interestingly, it can be noted that the WEM educedimpedance value is still comparable to a value which has beenshown to give good agreement in propagation methods that

th experimental results from Ref. [11] (þ), and WEM computed levels using WEM

Fig. 10. Acoustic pressure phase on the surface opposite the liner for Mx ¼ 0:335 with experimental results from Ref. [11] (þ), and WEM computed levels using WEM educedimpedance values (—–) and impedance values from Ref. [11] (---).

Fig. 11. Educed liner impedance values with Mx ¼ 0:335 using the WEM-based procedure (—–) and the FEM-based procedure from Ref. [11] (---).

C.J. O’Reilly / Applied Acoustics 90 (2015) 54–63 61

include this instability – Burak et al. for example used an imped-ance value of Z ¼ 0:48þ 0:03i (compared to the WEM educed valueof Z ¼ 0:43þ 0:07i) in a LES simulation and achieved excellentagreement with the same data. This strengthens the argument thatthe present methodology is robust, at least for impedanceeduction.

At 500 Hz, the numerical values underestimate the strength ofthe standing-wave-like formation across the liner that is visiblein the experimental data. Note, the peaks and troughs are presentin the numerical values at the same axial locations that appear inthe experiments, but the range of variation is much lower, and sothe lines appear to be almost smooth on the axis presented here.Note that this agreement does not occur if the impedance condi-tion is taken to be the no flow impedance boundary conditionn:u� p=Z ¼ 0. The phase values at 500 Hz do not fully capturethe shape of the experimental curves either although the differenceis not so obvious. The same trend is also visible at 1500 Hz but to amuch lesser extent. Both the length of the domain and the compu-tational mesh have been investigated and excluded as possibleexplanations for this behaviour. Also, the exit impedance has beeninvestigated, but this was seen to effect mostly the SPL down-stream of the liner, and not across the liner itself. This same trendis also visible in the results presented in Ref. [26] for example

(which also uses the Myers boundary condition), where it wasnot specifically discussed.

Some insight may be gained from looking at the transmissionand reflection coefficients shown in Fig. 12 (the WEM resultsshown here where computed with the WEM educed impedancevalues). Although the transmission coefficient displays the sameagreement as has been discussed for the total pressure values,the reflection coefficient shows considerable divergence particu-larly at lower frequencies. The higher reflection coefficient in theexperimental data explains the standing wave across the linernoted in Fig. 9. Three influences may account for this difference.Firstly, that a hydrodynamic instability is influencing the result.As already discussed, this is most likely the case at 1000 Hz. How-ever, at other frequencies, this seems unlikely as the results pre-sented in the literature do not indicate that this is the case.Secondly, it may be that the implementation of the numericalboundary condition is done in a way that causes less reflection atthe end of the liner than in the experimental setup. This explana-tion must be counterbalanced with the fact that the no-flow casedoes capture a similar formation. Thirdly, and perhaps most likely,this disagreement at lower frequencies is consistent with the typeof failure of the Myers boundary condition discussed in Ref. [24],where the model impedance values were found to be dependent

Fig. 12. Transmission and reflection coefficients with Mx ¼ 0:335 from the WEM (—–) and from the data from Ref. [11] (þ).

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on the direction of propagation. Therefore, waves reflected backupstream see a different impedance.

6.2. WEM efficiency

The advantage of using the present methodology is that itrequires less computational resources to achieve these positiveresults, compared to alternative numerical schemes and/or flowimpedance assumptions, which makes it interesting for manypractical applications. To demonstrate this, the accuracy of theWEM is compared to a finite element method (FEM) solution (com-puted using the commercial solver COMSOL Multiphysics) withlinear shape functions. Using the same domain as the GIT, this timewith hard walls, and varying the frequency, the error relative to theexact reference solution was determined and is shown in Fig. 13.Included in this plot is the WEM error for different numbers ofwave vectors, i.e. varying N in the discetisation. As can be seen,the relative error is considerable lower for the WEM for a givennumber of points per wavelength. The WEM requires roughly halfthe number of points per wavelength for the same level of accu-racy. This translates into a significant saving in computationalresources as the size of the overall problem to be solved – the sizeof the overall stiff matrix ndof to be inverted – will be 1=2D the size,where D is the number of spatial dimensions.

As this discretisation method works by finding an optimal solu-tion to the local wave vector amplitudes via a Moore–Penrosepseudo-inverse (computed here via a singular value decomposi-tion), it is important that the size of the matrix H is kept to aminimum so as to limit the assembly requirements for the overallstiffness matrix. For the present 2D computation, the number ofneighbours on the regular grid is M ¼ 8, so the method requiresthat N > 8 wave vectors be used for internal points and N > 9 forboundary points, which have an additional row. Fig. 13 shows thatthe accuracy of the solution with N ¼ 10 used throughout thedomain is almost identical to that with a larger number of wave

100 101 10210−8

10−6

10−4

10−2

100

102

Fig. 13. Relative error for propagation in the GIT domain with hard walls using FEM(—–) and WEM with N ¼ 10 (� � �), N ¼ 20 (---) and N ¼ 40 (- �-) wave vectors.

vectors used. The size of the pseudo-inverse should therefore notbe adversely large.

7. Conclusions

In this paper, a methodology is presented which couples theWEM with the Myers flow impedance boundary condition in orderto examine acoustic propagation through a lined duct with flow.The Myers boundary condition has been implemented for the firsttime in the WEM by explicitly evaluating the derivatives of thebase plane-wave functions and augmenting the local stiffness val-ues accordingly. The computed reduction in SPL across the NASAGIT configuration agrees well with the measured data in terms ofboth SPL and phase across the liner. Deviations in this agreementhave been accounted for. The method has been used to educeimpedance values, which are consistent with those determinedusing alternative, often more demanding, approaches. Analysis ofthe reflection coefficient shows that the Myers boundary conditionfails to fully model the wave behaviour across the liner at low fre-quencies. Aside from this, the discretisation scheme itself has beenshown to be considerably more efficient than a comparable FEMdiscretisation. The WEM with the Myers boundary condition offersa highly efficient numerical methodology for computing acousticpropagation in lined ducts with a grazing flow.

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