Efficient Mode Matching Based on Closed-FormIntegrals of Pridmore-Brown Modes

M. Oppeneer∗

National Aerospace Laboratory (NLR), 8316 PR Marknesse, The Netherlands

S. W. Rienstra†

Eindhoven University of Technology, 5612 AP Eindhoven, The Netherlands

and

P. Sijtsma‡

National Aerospace Laboratory (NLR), 8316 PR Marknesse, The Netherlands

DOI: 10.2514/1.J054167

A new mode-matching method is developed for acoustic modes in axially sectioned lined ducts with parallel, but

otherwise arbitrarymean flow. Themethod is evaluated in detail for a circularly symmetric configuration (where the

mean flow depends only on the radial coordinate), with the so-called Pridmore-Brown modes, satisfying the radial

Pridmore-Brown equation. Classically, the matching of the modal representations at the interfaces between the

sections is basedon continuity of pressure andvelocity in combinationwithprojection to a suitable but unrelated set of

test functions by means of a standard integral inner product. The alternative proposed here uses essentially the same

Pridmore-Brown modes, but instead of the standard inner product it applies a particular bilinear form that can be

evaluated in closed form. Apart from numerical efficiency, this approach features also a higher accuracy because it

avoids the inherently inaccurate numerical quadrature of oscillating functions. The results are comparedwith results

obtained by an implementation of the classical approach, and the agreement is excellent, with higher accuracy and

with greater computational efficiency.

Nomenclature

A�, B� = mode-matching matrices (matching pressure)A�νμ, B

�νμ = mode-matching inner products

(matching pressure)A, Γ = integration area with boundarya�l;μ, b

�l;μ = modal amplitudes

a�l , b�l = modal amplitudes vectors

C�, D� = mode-matching matrices (matching axialvelocity)

c = sound speedD0 = convective derivative, �∂∕∂t� � u0�∂∕∂x�D, �D = energy dissipation term, time averagedd = diameter ductE = energy densityF, ~F, F = vector of modal shape functions

U; etc.; ~U; etc.; U; etc.hl = xl − xl−1, segment thicknessI, �I = energy flux vector, time averagedJm = Bessel function of order mk, ~k = axial modal wave number

(unspecified boundary condition)k�l;μ = axial modal wave number of order

μ in segment lM = Mach numberm = circumferential mode numbern = outward normal vector of the wall

(i.e., pointing into the wall)

P = modal pressure, p1 � P�y; z�eikx-iωtP�l;μ = modal shape functionp = pressureR = gas constantSl, �Sl, S = segment scattering matrix, cumulative

scattering matrix, interface scattering matrixs = entropyT = temperatureU, V,W, P = modal velocities and pressure,

�U�r�; V�r�;W�r�; P�r��e−iωt�ikx�imθ~U, ~V, ~W, ~P = idem (modal velocities and pressure),

corresponding with ~kU, V, W, P = associate modal velocities and pressure of

inhomogeneous equations; depending on ku0 = mean flow velocity in axial (x) directionv = velocity vector�X�l �μμ = diagonal matrix element eik

�l;μhl

�X−l�1�μμ = diagonal matrix element e−ik

−l�1;μhl�1

x, r, θ = cylindrical coordinatesx, y, z = Cartesian coordinatesx = position vectorxl = right end of segment lZ, ~Z = wall impedanceα, ~α = auxiliary constantsμ, ν = radial mode numberρ = densityϕ, ~ϕ, ϕ, Φ, f = auxiliary functionsΨν = test functions for mode

matchingω = radial frequencyω = vorticityΩ, ~Ω = ω − ku0, ω − ~ku0⟪ ·; · ⟫ = inner product-like bilinear map over y, z areah·; ·i = inner product-like bilinear map over r range�·; ·� = standard inner product over r range

Subscripts

0 = mean flow variable1 = perturbation variable

Presented as Paper 2013-2172 at the 19th AIAA/CEAS AeroacousticsConference, Berlin, Germany, 27–29 May 2013; received 5 February 2015;revision received 29 May 2015; accepted for publication 15 June 2015;published online 2 November 2015. Copyright © 2015 by the authors.Published by the American Institute of Aeronautics and Astronautics, Inc.,with permission. Copies of this paper may be made for personal or internaluse, on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-385X/15 and $10.00 in correspondence with the CCC.

*Ph.D. Student; currently VORtech BV, Delft.†Associate Professor. Senior Member AIAA (Corresponding Author).‡Senior Scientist; currently PSA3 Advanced AeroAcoustics, Wezep.

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I. Introduction

T HE subject of this paper is motivated by the problem of soundpropagation through the auxiliary power unit (APU) exhaust

duct. This is typically a straight, circular duct surrounded bymultipleannular segments, which constitute an axially varying, locallyreacting, or bulk absorbing liner (Fig. 1). The duct carries amean flowof a (radially) highly nonuniform velocity profile, and a hightemperature. Because of insertion of cooling air upstream, thetemperature profile is also (radially) highly nonuniform.Because the full problem, ofmean flow and temperature varying in

all directions, is too formidable to allow any other approach than afully numerical simulation, we restrict our study to the lined part ofthe exhaust duct, where a parallel flow model is reasonable [1]. Asuitable approach to compute the sound propagation for such ageometry is mode matching, which consists of expressing theacoustic field in each segment as a series of modes, and thendetermining the modal amplitudes by matching the fields at eachinterface between two segments by suitable continuity conditions [2].The system of mode-matching equations results from taking inner

products of the duct eigenfunctions with certain test functions. Foruniform flow and temperature, the duct eigenfunctions are Besselfunctions. Because closed-form integrals are known for products ofBessel functions [3], the choice of a suitable set of Bessel functions astest functions allows us to determine the inner products analyticallyexactly. For nonuniform flow and temperature, however, the ducteigenfunctions (modal solutions of the Pridmore-Brown equation)are not Bessel functions anymore, and no exact integrals of productsof modes are known. For the classical mode-matching (CMM)approach a set of Bessel functions may still be used as test functions,although the inner products cannot (to our knowledge) be computedin closed form, and numerical quadrature is needed.In this paper we present a new approach that uses the same

Pridmore-Brown modes as test functions, but instead of the standardinner product we use an associated bilinear form§ that resembles an“inner product.”We will show that in this way the integrals we needare available in closed form.The use of this new inner product for mode matching may require

to compute an extra set of Pridmore-Brown modes to be used as testfunctions, or the solution of an inhomogeneous Pridmore-Brownequation. Any of these solutions have to be computed only once,regardless of the number of segments, whereas for the classicalapproach we need to compute all inner products at each interface.Furthermore, in some occasions the off-diagonal inner products arezero, which simplifies the calculations even more.Because the computational work is comparable to the numerical

quadrature for a single interface while the inherent numericalintegration errors are avoided, we conclude that our new approach isboth more accurate and cheaper than the CMM methods.We note that the issue of possible instabilities due to the interaction

of shear layer and impedancewall (as, e.g., discussed in [4,5])will notbe addressed here. Ill-posedness, associated with a vanishingboundary layer, only occurs in time domain calculations, while thedetection of possible unstablemodes requires a causality analysis thatwe have not undertaken in the present context.

II. Problem Formulation

A. General Parallel Flow

We start with linearized Euler equations for time-harmonic modalsolutions in general parallel mean flow. Only the mean pressure isassumed constant. If the mean flow is uniform in x direction and onlyvaries in (y, z), the coefficients of the equations do not depend on xand solutions proportional to eikx (modes) are possible. The resultingequation can be recognized as a preform of the Pridmore-Brownequation.Although the final application will be for a circular symmetric

mean flow in a circular duct, we derive these general equations

because later wewill derive and use somegeneral results for solutionsin the form of analytically exact integrals.Assume an inviscid, non-heat-conducting ideal gas with the

uniform mean pressure p0, parallel mean flow velocity v0 in xdirection with density ρ0, and sound speed c0 varying in (y, z) only:

v0 � u0�y; z�ex; ρ0 � ρ0�y; z�;c0 � c0�y; z�; p0 � ρ0RT0 � γ−1ρ0c

20 � constant (1)

It follows from the linearized Euler equations that perturbations(subscript 1) of a parallel mean flow (subscript 0) are governed by

D0ρ1 � v1 · ∇ρ0 � ρ0∇ · v1 � 0 (2a)

ρ0�D0v1 � v1 · ∇v0� � ∇p1 � 0 (2b)

c20�D0ρ1 � v1 · ∇ρ0� � D0p1 (2c)

where the linearized convective derivative is denoted byD0 � �∂∕∂t� � u0�∂∕∂x�. The convective derivative of thedivergence of momentum Eq. (2b) becomes with mass Eq. (2a) andagain momentum Eq. (2b)

c20D30ρ1 � 2c20

∂∂x�∇u0 · ∇p1� −D0�c20∇2p1� � 0 (3)

Since c20�D0v1 · ∇ρ0� � ∇c20 · ∇p1, we have with Eq. (4) finally(cf. [6])

D30p1 � 2c20

∂∂x�∇u0 · ∇p1� −D0∇ · �c20∇p1� � 0 (4)

If we look for solutions of the formp1�x; y; z; t� � P�y; z�eikx−iωt,we obtain a preform of the Pridmore-Brown equation [7,8]:

iΩ3P� 2ikc20�∇u0 · ∇P� � iΩ�−k2c20P� ∇ · �c20∇P�� � 0 (5)

where

Ω � ω − ku0

By noting that −k∇u0 � ∇Ω, this equation can be furthersimplified to

∇ ·

�c20Ω2

∇P���1 −

k2c20Ω2

�P � 0 (6)

If u0 ≡ 0, this reduces to

∇ · �c20∇P� � �ω2 − k2c20�P � 0 (7)

which is just equivalent to the Helmholtz equation if c0 is constant.

B. Circular Symmetric Mean Flow in Circular Duct

The final application will be for a circular symmetric mean flow ina circular duct with radius d, which models an APU exhaust duct(Fig. 1). Therefore, we will develop the equations in cylindricalcoordinates and consider Fourier modes in the circumferentialcoordinate.

§Mathematically, it is not exactly an inner product. Although imprecise, wewill refer to it here as inner product because of the role it plays in the mode-matching procedure.

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In a radially symmetric mean flow u0�r�, ρ0�r�, c0�r� withperturbations v1 � u1ex � v1er �w1eθ, ρ1, p1 of the form

�u1; v1; w1; ρ1; p1� � �U;V;W; R;P�e−iωt�ikx�imθ (8)

we have the equations

−iΩR� ρ0

�ikU� V 0 � 1

rV � im

rW

�� ρ 00V � 0 (9a)

ρ0�−iΩU� u 00V� � ikP � 0 (9b)

−iΩρ0V � P 0 � 0 (9c)

−iΩρ0W �im

rP � 0 (9d)

c20�−iΩR� ρ 00V� � iΩP � 0 (9e)

where a prime denotes a derivative to r. R can be eliminated byreplacing Eqs. (9a) and (9e) by

ρ0c20

�ikU� V 0 � 1

rV � im

rW

�− iΩP � 0 (9f)

This system can be reduced to one equation for P, known as thePridmore-Brown equation:

Ω2

rc20

�rc20P

0

Ω2

���Ω2

c20− k2 −

m2

r2

�P � 0 (10)

which is to be compared with Eq. (6). Another form is

P 0 0 ��1

r� T

00

T0

� 2ku 00Ω

�P 0 �

�Ω2

c20− k2 −

m2

r2

�P � 0 (11)

C. Boundary Conditions

Physically, the relevant boundary conditions may involve locallyreacting liners as well as bulk absorbers. For duct modes, however,the relevant boundary conditions are equivalent in form [9,10], andwe will use an impedance condition only.

Because we want to include cases with slipping flow (like, e.g.,uniform flow), we use the Ingard–Myers boundary condition for aboundary layer with vanishing thickness (modeled as a vortex sheet)along a straight wall [11]. This is for time harmonic perturbations

−iω�v1 · n� � �−iω� v0 · ∇�p1

Z(12)

For a circularly cylindrical duct, after elimination of the radialvelocity, Eq. (12) becomes

P 0 −iρ0Ω2

ωZP � 0 at r � d (13)

III. Exact Integrals of Pridmore-Brown Eigenfunctions

A. Exact Integrals of Solutions of the Helmholtz Equation

Modematching is a particularly successful method for acoustic wavepropagation in segmented ducts of circular or rectangular cross sectionwith a uniform medium. The necessary integrals of the modaleigenfunctions (Bessel functions or (co)sine functions) appear to beavailable in closed form, which greatly simplifies the numericalevaluation. These closed-form integrals are a manifestation of a moregeneral propertyof solutionsof theHelmholtz equation. Suppose thatwehave, for parameters α and ~α, the solutions ϕ and ~ϕ in a regionA ∈ R2

with boundary Γ (as yet, boundary conditions do not play a role) of

∇2ϕ� α2ϕ � 0 (14a)

∇2 ~ϕ� ~α2 ~ϕ � 0 (14b)

Whenwe subtractϕ times the second equation from ~ϕ times the first andintegrate overA, we obtain

�α2 − ~α2�ZZ

Aϕ ~ϕ dS �

ZZA�ϕ∇2 ~ϕ − ~ϕ∇2ϕ�dS

�ZZ

A∇ · �ϕ∇ ~ϕ − ~ϕ∇ϕ�dS (15)

After using the divergence theorem, this inner product of ϕ and ~ϕ isgiven by an integral along the boundary

ZZAϕ ~ϕ dS � 1

α2 − ~α2

ZΓ�ϕ∇ ~ϕ · n − ~ϕ∇ϕ · n�dl (16)

Ifα � ~α, this result cannot beused.Suppose thatwe replaceEq. (14b)by the far more general

∇2ϕ� α2ϕ � f (17)

where f is an arbitrary (integrable) function.When we again cross-wisemultiply and subtract as before, we find

ZZAϕf dS �

ZZA�ϕ∇2ϕ − ϕ∇2ϕ�dS

�ZΓ�ϕ∇ϕ · n − ϕ∇ϕ · n�dl (18)

This result was derived without specifying any boundary conditionson ϕ, and so they can be chosen arbitrarily as long as there exists asolution ϕ. This is guaranteed¶ if (the boundary conditions on ϕ are suchthat) α is not an eigenvalue of the homogeneous version of Eq. (17).

hard wall resistive sheet

liner cavity

cool air inlet

exhaust

temperatureprofile

mean flow velocityprofile

Fig. 1 APU geometry. The present problem refers to the section withsegmented linear and parallel flow.

¶This result is related to the Fredholm alternative for linear operators [12].Assume that Aϕ � B∇ϕ · n on Γ and suppose that there exists a non-zero solution w of ∇2w� α2w � 0 with the same boundary conditions.Then we obtain the, for arbitrary f, contradiction

RRA wf dS �

∫ Γ�w∇ϕ · n − ϕ∇w · n�dl � 0.

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The advantage of this result is its generality.We can substitute for fany function we need, for example, f � ϕ, solution of Eq. (14a),leading to

∇2ϕ� α2ϕ � ϕ (19a)

ZZAϕ2 dS �

ZΓ�ϕ∇ϕ · n − ϕ∇ϕ · n�dl (19b)

The disadvantage is, of course, that it requires the solution of theadditional inhomogeneous Eq. (17). So in practice we will use thisresult only if Eq. (16) breaks down.In the specific case of a circular disk of radius 1 and circularly

symmetric solutions ϕ � Jm�αr�eimθ and ~ϕ � Jm� ~αr�e−imθ (wechoose opposite signs ofmθ for nontrivial results later), substituted inEq. (16), we obtain the well-known [3] relation for Bessel functions:

Z1

0

Jm�αr�Jm� ~αr�r dr �1

α2 − ~α2� ~αJm�α�J 0m� ~α� − αJ 0m�α�Jm� ~α�

(20)

For the case when α � ~α one approach is to take the limit and usel’Hôpital’s rule, with result

Z1

0

Jm�αr�2r dr �1

2

�1 −

m2

α2

�Jm�α�2 �

1

2J 0m�α�2 (21)

Amore generic approach is the one discussed above. Suppose thatwe have ϕ�r; θ� � Φ�r�eimθ, regular in r � 0, where Φ is a solutionof the inhomogeneous Bessel equation

1

r�rΦ 0� 0 �

�α2 −

m2

r2

�Φ � Jm�αr� (22)

(e.g., Φ�r� � −rJ 0m�αr�∕2α), then we have the equivalent result

Z1

0

Jm�αr�2r dr � Jm�α�Φ 0�1� − αJ 0m�α�Φ�1� (23)

Again, the boundary conditions on ϕ can be selected

arbitrarily, except for the restriction that AΦ�1� � BΦ 0�1� ≠ 0if AJm�α� � αBJ 0m�α� � 0.The abovemanipulations (14–18) can be repeated for Eq. (7) (with

u0 ≡ 0) to obtain weighted inner product integrals of the type

ZZAc20P ~P dS

but for the general case (6) this is not possible because Ω � Ω�k�.Indeed, no closed-form expressions can be found for the standardinner products with eigenfunctions of the Pridmore-Brown equation,which are required to set up the mode-matching equations, and so itseems that we have to resort to numerical quadrature to compute theintegrals. With increasing radial order, the eigenfunctions becomemore and more oscillatory, resulting in increasingly more difficultnumerical computations.All this is not the case if we change the standard inner product

integrals into a dedicated integral, associated to the prevailingequations, which we call (as it is not a real inner product) a bilinearform.In the following wewill construct two bilinear forms consisting of

products of eigenfunctions. One for the general case of parallel meanflow (applicable, e.g., in distortion mode problems [13]), much in thesame fashion as discussed above, and one for the particular case ofcircular ducts with radially symmetric mean flow, the so-calledPridmore-Brown modes. These results may be used to compute thecoefficients ofmode-matching equations in closed form.Anumerical

implementation will be given, as well as some numerical examplescomparing the classic approach and the present new one.This result was inspired by [14], in which a related integral was

used to obtain a solvability condition for a multiple scales solutionof the disturbance field for a slowly varying duct with meanswirling flow.

B. Exact Integrals of Parallel-Flow Modal Eigenfunctions

Analogous to Eq. (16) we want to construct an integral involvingproducts of Pridmore-Brown eigenfunctions and use the divergencetheorem to evaluate its value through the eigenfunction values on theboundary. Suppose that we have parallel mean flow in x direction andmodes of the form

�ρ1; p1; v1� � �R�y; z�; P�y; z�;U�y; z�ex � V�y; z�ey � W�y; z�ez�eikx−iωt (24)

U, V, and W are the velocity components in the x, y, and zdirections of a Cartesian coordinate system. For modal solutions ofthis form that are governed by Eq. (2), we have

−iΩP� iρ0c20kU� ρ0c20�Vy �Wz� � 0 (25a)

−iρ0ΩU� ρ0�u0yV � u0zW� � ikP � 0 (25b)

−iρ0ΩV � Py � 0 (25c)

−iρ0ΩW � Pz � 0 (25d)

where Ω � ω − ku0, and R follows directly from the otheramplitudes, for example, with (2c). (Note that the system reduces toEq. (6) if U, V, and W are eliminated.) Together with suitableboundary conditions, this is an eigenvalue problem with eigenvaluek, but this will not be used here; k will be considered as a givenconstant.When the individual equations in Eq. (25) are multiplied by

suitable combinations** of other solutions of the same equations (say~P, ~U, ~V, and ~W) with constant ~k and corresponding auxiliary function~Ω � ω − ~ku0, and subsequently added together, we obtain

�−iΩP� iρ0c20kU� ρ0c20Vy � ρ0c

20Wz�

~P

ρ0c20

� �−iΩρ0U� ρ0u0yV � ρ0u0zW � ikP�~k ~P

ρ0 ~Ω

− �−iΩρ0V � Py� ~V − �−iΩρ0W � Pz� ~W � 0 (26)

After reordering and splitting off a cross-wise divergence, this isequivalent to

**This choice is clearly not self-evident, and not the result of randomtrying. It was found by first taking the products of the governing equationswith arbitrary functions, and then imposing the required conditions on thesefunctions. The resulting equations appeared to be equivalent to our originalequations for P, U, V, andW.

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−i�

Ωρ0c

20

−k ~k

ρ0 ~Ω

�~PP − i

Ω ~k − ~Ωk~Ω

~PU� iρ0Ω� ~VV � ~WW�

− V ~Py −W ~Pz � � ~Vy � ~Wz�P�~k

~Ω�u0y ~V � u0z ~W�P

� ~Ω�~PV − ~VP

~Ω

�y

� ~Ω�~PW − ~WP

~Ω

�z

� 0 (27)

After using the defining equations in Eq. (25) this becomes

− i�

Ωρ0c

20

−k ~k

ρ0 ~Ω

�~PP − i

Ω ~k − ~Ωk~Ω

~PU� iρ0Ω� ~VV � ~WW�

− i ~Ωρ0� ~VV � ~WW� � i�

~Ωρ0c

20

−~k2

ρ0 ~Ω

�~PP

� ~Ω�~PV − ~VP

~Ω

�y

� ~Ω�~PW − ~WP

~Ω

�z

� 0 (28)

Recombining and dividing by ~Ω yields

−i�k − ~k� 1~Ω

��u0ρ0c

20

�~k

ρ0 ~Ω

�~PP� ω

~Ω~PU − ρ0u0� ~VV � ~WW�

�

� ∂∂y

�~PV − ~VP

~Ω

�� ∂

∂z

�~PW − ~WP

~Ω

�(29)

In case we want to write the left-hand side in terms of P only, wecan use the defining equations for U, V, and W. When we integrateEq. (29) over a cross section A with boundary Γ and use thedivergence theorem, we obtain an integral over A of parallel-flowshape functions, in particular (with suitable boundary conditions andeigenvalues k) parallel-flow eigenfunctions, expressed as an integralalong boundary Γ. We introduce the vector of shape functions F ��P;U; V;W� and denote this integral by††

⟪F; ~F⟫�Z Z

A

1

~Ω

��u0ρ0c

20

�~k

ρ0 ~Ω

�~PP�ω

~Ω~PU−ρ0u0� ~VV� ~WW�

�dS

� i

k− ~k

ZΓ

~P�Vny�Wnz�−� ~Vny� ~Wnz�P~Ω

dl (30)

whereny andnz denote the y and z components of the outward normalvector on Γ and k ≠ ~k.If u0 ≡ 0, this reduces to a regular integral inner product (with a

weight function ∝ ρ−10 ∝ c20):

⟪F; ~F⟫ � k�~k

ω2

ZZA

~PP

ρ0dS

� 1

�k − ~k�ω2

ZΓ

~P�Pyny � Pznz� − � ~Pyny � ~Pznz�Pρ0

dl

(31)

a result, very similar to Eq. (16), and which could have been obtaineddirectly from Eq. (7).For a slipping mean flow along an impedance wall at Γ,

we apply Ingard–Myers conditions Vny �Wnz � ΩP∕ωZ and~Vny � ~Wnz � ~Ω ~P ∕ω ~Z and obtain

⟪F; ~F⟫ � i

k − ~k

ZΓ

~PP

~Ωω

�ΩZ−

~Ω~Z

�dl (32)

The integrals vanish for hard walls, when Z � ~Z �∞. For ano-slip mean flow with u0 � 0 along Γ and impedance boundaryconditions P � Z�Vny �Wnz� and ~P � ~Z� ~Vny � ~Wnz�, weobtain

⟪F; ~F⟫ � i

k − ~k

ZΓ

~PP

ω

�1

Z−1

~Z

�dl (33)

Interestingly, the integral vanishes for different modes thatcorrespond with the same Z.Although this surface integral resembles a nonstandard inner

product between vectorsF and ~F, it is not an inner product because itlacks positive definiteness for ⟪F;F⟫, mainly because of the PUterm. Therefore, we refer to it as a bilinear map, although,occasionally, because it plays the same role of an inner product as inthe CMM procedure, it may be referred to as an inner product.The above result is evidently not valid for k � ~k. In practice, when

we deal with modal eigenfunctions, all satisfying the same boundarycondition, the limit of k � ~k goes together with F � ~F and we willconsider that situation here.We start with the following associated inhomogeneous system of

Eq. (25) with solution �P; U; V; W�, with the same k as in the originalsystem, and a solution vector �P;U; V;W� satisfying Eq. (25)

−iΩP� iρ0c20kU� ρ0c20�Vy � Wz� � i�u0P� ρ0c

20U� (34a)

−iΩρ0U� ρ0u0yV � ρ0u0zW � ikP � i�ρ0u0U� P� (34b)

−iΩρ0V � Py � iρ0u0V (34c)

−iΩρ0W � Pz � iρ0u0W (34d)

with boundary conditions such that k is not an eigenvalue ofthe left-hand side, in order to guarantee the existence of a solution�P; U; V; W�.We multiply left- and right-hand sides withP∕ρ0c20, kP∕ρ0Ω,−V,

and−W, respectively; add; and do exactly the samemanipulations asbefore.We find that no factor k − ~k appears, and obtain the final result

⟪F;F⟫�Z Z

A

1

Ω

��u0ρ0c

20

� k

ρ0Ω

�P2�ω

ΩUP−ρ0u0�V2�W2�

�dS

�iZΓ

P�Vny�Wnz�−�Vny�Wnz�PΩ

dl (35)

If u0 ≡ 0, we obtain

⟪F;F⟫ � 2k

ω2

ZZA

P2

ρ0dS

� 1

ω2

ZΓ

P�Pyny � Pznz� − �Pyny � Pznz�Pρ0

dl (36)

C. Exact Integrals of Radial Pridmore-Brown Modes

A special application of the above results will be for a circularlysymmetric mean flow u0�r�, c0�r�, ρ0�r� in a circular duct of radius d

††If a symmetric form is preferred, we can replace ⟪F; ~F⟫ by�1∕2�⟪F; ~F⟫� �1∕2�⟪ ~F;F⟫ and the RHS correspondingly.

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and cross section A (an annular duct would require only littlechanges) with polar coordinate system (x, r, θ) and v1 �uex � ver �weθ. In this case the solution can be written as a sumover circumferential Fourier modes, and we can assume modalshape solutions of the formF�r�eimθ��P�r�;U�r�;V�r�;W�r��eimθ,satisfying

−iΩP� iρ0c20kU� ρ0c20

�V 0 � 1

rV � im

rW

�� 0 (37a)

−iρ0ΩU� ρ0u00V � ikP � 0 (37b)

−iρ0ΩV � P 0 � 0 (37c)

−iρ0ΩW �im

rP � 0 (37d)

whereΩ � ω − ku0 and the 0 denotes an r derivative. As before, wewill assume k to be just a constant, but with suitable boundaryconditions this system is an eigenvalue problem with eigenvalue k.Because of the symmetry, it is no restriction to assume another

solution of Eq. (37) with ~k ≠ k, of the form ~F�r�e−imθ �� ~P�r�; ~U�r�; ~V�r�;− ~W�r��e−imθ, such that the surface integral inEq. (30) (divided by 2π) simplifies to

hF; ~Fi �Zd

0

r

~Ω

��u0ρ0c

20

�~k

ρ0 ~Ω

�P ~P�ω

~ΩU ~P− ρ0u0�V ~V�W ~W�

�dr

� id

k− ~k

�~PV − ~VP

~Ω

�r�d

(38)

where we assumed that the solutions are regular in r � 0.If u0 ≡ 0, we obtain

hF; ~Fi � k�~k

ω2

Zd

0

r

ρ0P ~P dr � d

�k − ~k�ω

�~PP 0 − ~P 0P

ρ0

�r�d

(39)

With slipping flow and impedance walls along r � d, we applyIngard–Myers boundary conditions V � ΩP∕ωZ and Ω � ω −ku0�d� for both solutions, and obtain

hF; ~Fi � id ~PP

�k − ~k� ~Ωω

�ΩZ−

~Ω~Z

�����r�d

(40)

which vanishes ifZ � ~Z � ∞. With no-slip flow and u0�d� � 0 thissimplifies to

hF; ~Fi � id ~PP

�k − ~k�ω

�1

Z−1

~Z

�����r�d

If k and ~k are different eigenvalues from the same impedancecondition, so ~Z � Z, then all integrals vanish in this case.To find the degenerate case of ~k � k and ~F � F, we consider with

constant k the solution.Fe−imθ � �P;U; V;−W�e−imθ [equally given by Eq. (37)] and the

associated solution.Fe−imθ � �P; U; V;−W�e−imθ of the inhomogeneous system

(with the same k)

−iΩP� iρ0c20kU� ρ0c20

�V 0 � 1

rV � im

rW

�� i�u0P� ρ0c

20U�

(41a)

−iΩρ0U� ρ0u00V � ikP � i�ρ0u0U� P� (41b)

−iΩρ0V � P 0 � iρ0u0V (41c)

−iΩρ0W �im

rP � iρ0u0W (41d)

In actual practice, the system (41) will be reduced to the followinginhomogeneous Pridmore-Brown equation in P, which may besolved by almost the same routine as is used for the Pridmore-Brownequation itself

Ω2

rc20

�rc20Ω2

P 0� 0��Ω2

c20− k2 −

m2

r2

�P � 2

ωu 00Ω2

P 0 − 2

�u0Ωc20� k

�P

(42)

The surface integral (divided by 2π) of Eq. (35) now simplifies to

hF;Fi �Zd

0

r

Ω

��u0ρ0c

20

� k

ρ0Ω

�P2 � ω

ΩUP − ρ0u0�V2 �W2�

�dr

� id�PV − VP

Ω

�r�d

(43)

where we assumed that the solutions are regular in r � 0. As before,it should be noted that the inhomogeneous Eq. (42) has no solutions ifthe problem for P is an eigenvalue problem with homogeneousboundary conditions, and the same conditions are applied to P.If u0 ≡ 0, we obtain

hF;Fi � 2k

ω2

Zd

0

r

ρ0P2 dr � d

ω2

�PP 0 − P 0P

ρ0

�r�d

(44)

IV. Mode Matching

A. Construction of Matrix Equations (Classical Mode Matching)

We consider a duct that is divided inN axial segments, and assumethat the wall properties are constant within each segment.We assume that the perturbation field for each segment can be

expressed as a summation of eigenmodes of the Pridmore-Brownequation, as discussed previously. This is not obvious and in factgenerally not true. Our representationwithmodes proportional to eikx

can be justified [5] by considering it as a Fourier transform in x. Theobtained solution in k can be transformed back to x domain by inverseFourier transformation. The residues of the poles in the complex kplane then become the modes that were anticipated. In shear flow,however, there are more singularities in k than just the modal poles.There is also a branch cut, also known as “the continuous spectrum”

of values of k with Ω � ω − ku0 � 0 at (so-called) critical layers,which cannot be evaluated asmodes. This contribution, in the presentmodel, is ignored as it can be shown, as has been reported in [5], thatin general it is small.To compute the field inside the entire duct we set up a system of

equations that relates the modal amplitudes in adjacent segments byapplying suitable continuity conditions at the interface betweenthem. This is themode-matchingmethod. Subsequently, we computethe amplitudes in all segments with the aid of the numerically stableS-matrix formalism, as described in the next section.In this section we describe the CMMapproach based on continuity

of pressure and axial velocity. The total field for a given circumfer-ential wavenumberm in each segment is a superposition of all right-and left-running modes:

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pl�x; r� �X∞μ�1�a�l;μP�l;μ�r�e

−ik�l;μ�x−xl−1� � a−l;μP−

l;μ�r�e−ik−

l;μ�x−xl��;

xl−1⩽x⩽xl (45)

In a numerical implementation this infinite series has to betruncated; the finite number of modes μl to represent the field of thelth segment depends in general on the type of liner. At the interface atx � xl (see Fig. 2) we have for the pressure in segment l

pl�r� �Xμlμ�1�b�l;μP�l;μ�r� � a−l;μP−

l;μ�r�� (46)

Consider the hard-wall uniform flow eigenfunctions byΨν�r� � Jm�ανr�, where αν are the hard-wall radial wavenumbers,which satisfyΨ 0ν�d� � 0. These functions form a complete L2 basis;are at least locally, for high orders, similar in behavior as thePridmore-Brown modes; and are therefore suitable to serve as testfunctions when we set up the matrix system for the modal vectors.Imposing continuity of pressure (approximated due to the

truncation) at an interface at xl and subsequent projection onto the setof test functions Ψν, ν � 1; : : : ; νmax, yields

Xμlμ�1

b�l;μ�P�l;μ;Ψν� � a−l;μ�P−l;μ;Ψν�

�Xμl�1μ�1

a�l�1;μ�P�l�1;μ;Ψν� � b−l�1;μ�P−l�1;μ;Ψν� (47)

where we use the standard inner product

�f; g� �Zd

0

f�r�g�r�r dr

The set of equations for the continuity of axial velocity is foundanalogously by using the relation

U � k

ρ0ΩP −

u 00ρ0Ω2

P 0 (48)

which follows from Eqs. (9b) and (9c). All matching conditionstogether yield the following system of equations:

�A� A−

C� C−

��b�la−l

���B� B−

D� D−

��a�l�1b−l�1

�(49)

The matrix entries are inner products of Pridmore-Browneigenfunctions and Bessel functions; for thematrices that correspondto the pressure equations, we have

A�νμ � �P�l;μ;Ψν� �Zd

0

P�l;μ�r�Ψν�r�r dr;

B�νμ � �P�l�1;μ;Ψν� �Zd

0

P�l�1;μ�r�Ψν�r�r dr (50)

The matrix entries of C� and D� corresponding to the axialvelocity equations are computed analogously.

We consider the amplitudes of the waves propagating toward theinterface as the unknowns. Rearranging leads to

μl�1 μlνmax

νmax

�B� −A−

D� −C−

� �a�l�1a−l

��

μl�1 μlνmax

νmax

�A� −B−

C� −D−

� �b�lb−l�1

�

(51)

where the dimensions of the submatrices have been included. In thispaperwe restrict ourselves to locally reacting liners, and sowe chooseμl � μmax, and choose νmax � μmax. Thus, we have 2μmax equationsand μl � μl�1 � 2μmax unknowns.We note in passing that this approach can be extended to bulk

absorbing liners, in which case the number of modes μl > μmax (i.e.,the number of degrees of freedom) depends on the depth of the liner,and consequently the system of equations needs to be extended withextra conditions [1,9].

B. Scattering Matrix Formalism

A naive coupling of the duct sections via the transmission andreflection matrices is possible, but this process is unstable in case of alarge numbers of sections due to the exponentially decaying andincreasing cutoff modes that are involved. An alternative approachwould be an iterative one, where the propagation of a wave is onlyconsidered in the direction in which it decays. With this approach,the amplitudes are updated as more and more reflections andtransmissions at different interfaces are taken into account at eachnew iteration, until the change in the amplitudes is below a certainthreshold. However, this procedure may not converge for geometrieswith a large number of segments. We therefore follow the so-called“scattering matrix formalism” (see, e.g., [15]), which has noconvergence issues and is numerically stable.We want to express the modal amplitude vectors of the outgoing

waves in terms of the amplitude vectors of the incident waves. Thus,we write

�a�l�1a−l

���B� −A−

D� −C−

�−1�A� −B−

C� −D−

��b�lb−l�1

�

�μl μl�1

μl�1μl

�S11 S12

S21 S22

� �b�lb−l�1

�(52)

where we introduced the interface scattering matrix S, including thesizes of the submatrices. Next, we want to combine the effect ofscattering at the interface and propagation through the segment.Therefore, we introduce the segment scattering matrix Sl as

follows:

�a�l�1a−l

���S11l S12lS21l S22l

��X�l 0

0 X−l�1

��a�la−l�1

�

��S11l S12lS21l S22l

��a�la−l�1

�(53)

where the propagation is accounted for by the diagonal matrices

a+l

a−l

b+l

a−l+ 1b−

l+ 1

a+l+ 1

a−l− 1

a+l− 1

a−l+ 2

a+l+ 2

xl xl+ 1xl− 1

Fig. 2 Mode matching at several interfaces.

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�X�l �μμ � eik�l;μhl ; �X−

l�1�μμ � e−ik−

l�1;μhl�1 ; hl � xl − xl−1(54)

Note that the exponentials are always decaying.The effect of all layers up to layer l can be combined in the

cumulative scattering matrix �Sl (see also Fig. 3):�a�l�1a−1

����S11l �S12l�S21l �S22l

��a�1a−l�1

�(55)

The cumulative scattering matrix of a certain set of segments canbe computed by using the Redheffer star product, which is defined as

�A11 A12

A21 A22

��B11 B12

B21 B22

���

B11�I −A12B21�−1A11 B12 �B11A12�I − B21A12�−1B22

A21 �A22B21�I −A12B21�−1A11 A22�I −B21A12�−1B22

�(56)

By using this definition, �Sl can be computed as

�Sl � �Sl−1 Sl � S1 · · · Sl (57)

The Redheffer star product can be constructed as follows. Wewould like to construct the cumulative scatteringmatrix �Sl in order tocompute the effect of segment l. Let us assume that we can describethe effect of all segments up to segment l − 1 via

�a�la−1

����S11l−1 �S12l−1�S21l−1 �S22l−1

��a�1a−l

�(58)

Furthermore, we have at interface l

�a�l�1a−l

���S11l S12lS21l S22l

��a�la−l�1

�(59)

Substitute the second row of Eq. (59) into the first row of Eq. (58)to obtain

a�l � �S11l−1a�1 � �S12l−1a

−l � �S11l−1a

�1 � �S12l−1�S21l a�l � S22l a−l�1�

(60)

Collecting the terms gives

�I − �S12l−1S21l �a�l � �S11l−1a

�1 � �S12l−1S

22l a

−l�1 (61)

and so

a�l � �I − �S12l−1S21l �−1 �S11l−1a�1 � �I − �S12l−1S

21l �−1 �S12l−1S22l a−l�1 (62)

Substituting this into the first row of Eq. (59) yields

a�l�1 � S11l �I − �S12l−1S21l �−1 �S11l−1

z�����������������}|�����������������{a�1

�S11l

� fS11l �I − �S12l−1S21l �−1 �S12l−1S22l � S12l g|�����������������������������{z�����������������������������}a−l�1

�S12l

(63)

Analogously, we can substitute the first row of Eq. (58) into thesecond row of Eq. (59) to obtain

�I − S21l �S12l−1�a−l � S21l �S11l−1a�1 � S22l a−l�1 (64)

from which follows

a−l � �I − S21l �S12l−1�−1S21l �S11l−1a�1 � �I − S21l �S12l−1�−1S22l a−l�1 (65)

Substituting this into the second row of Eq. (58) yields

a−1 � f �S21l−1 � �S22l−1�I − S21l �S12l−1�−1S21l �S11l−1gz�������������������������������}|�������������������������������{�S1l

a�1

� �S22l−1�I − S21l �S12l−1�−1S22l|�����������������{z�����������������}�S12l

a−l�1 (66)

These equations can still be used when the number of modes isdifferent for each segment (i.e., when the four blocks of the scatteringmatrices are not square). To see this, consider the total number ofamplitudes for the three segments numbered 1, l, and l� 1, which is2μ1 � 2μl � 2μl�1. Of these amplitudes the μ1 � μl�1 amplitudes ofthe incident waves in segments 1 and l� 1 are known. Hence, theother amplitudes can be determined by using the μ1 � 2μl � μl�1equations of Eqs. (58) and (59). Also note that the dimensions of theterms between square brackets in Eqs. (63) and (66) are μl × μl, andso only the solution of square systems is required.Consequently, if all of the segment scattering matrices and the

incoming amplitudes a�1 and a−N of the outer segments are known,then the outgoing amplitudes and hence the total field in the outer

Fig. 3 Schematic representation of the S-matrix algorithm.

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segments can be computed. To compute the field inside the entireduct, the amplitudes in intermediate segments are required as well.We compute these by using Eq. (60) and the second row of Eq. (59).Note that we did not invert the propagationmatricesXl, which wouldhave caused growing exponentials (which might provoke numericalproblems).

C. Matching Conditions Based on the Bilinear Map

In this section we set up a system of equations that has the samestructure as Eq. (49) for the CMM approach.Let us define the vector f, whose components are the acoustic

pressure and the velocity components:

f l�x; r� � �pl�x; r�; ul�x; r�; vl�x; r�; wl�x; r�� (67)

The total field for a given circumferential wavenumber m in eachsegment is a superposition of all modes:

f l�x; r� �X∞μ�1�a�l;μF�l;μ�r�e

ik�l;μ�x−xl−1� � a−l;μF−

l;μ�r�eik−l;μ�x−xl��;

xl−1⩽x⩽xl (68)

where F again denotes the vector of perturbation amplitudes. At theinterface at x � xl we have

f l�r� �Xμlμ�1�b�l;μF�l;μ�r� � a−l;μF−

l;μ�r�� (69)

f l�1�r� �Xμl�1μ�1�a�l�1;μF�l�1;μ�r� � b−l�1;μF−

l�1;μ�r� (70)

Inside the duct we impose continuity ofp�x; r�,u�x; r�, v�x; r� andw�x; r� by applying the bilinear form to f l � f l�1 with the solutionof the associated problem Ψν, which results in

Xμlμ�1

b�l;μhF�l;μ;Ψνi � a−l;μhF−l;μ;Ψνi

�Xμl�1μ�1

a�l�1;μhF�l�1;μ;Ψνi � b−l�1;μhF−l�1;μ;Ψνi (71)

for ν � −νmax; : : : ;−1; 1; : : : ; νmax. When we split the range of νinto left (ν < 0) and right (ν > 0) running parts, we again obtain inmatrix format

�A� A−

C� C−

��b�la−l

���B� B−

D� D−

��a�l�1b−l�1

�(72)

To prevent unnecessary computation of an extra set ofeigenfunctions, we can choose as test functionΨν the eigensolutionsof, say, segment l. In that case the matrix entries can be computed as

fA;Cg�νμ � hFlμ;Flνi (73a)

fB;Dg�νμ � hFl�1;μ;Flνi (73b)

where

fA; Bg�: μ > 0; ν > 0; fC;Dg�: μ > 0; ν < 0;

fA; Bg−: μ < 0; ν > 0; fC;Dg−: μ < 0; ν < 0

The values of the bilinear forms in Eq. (73) can be computed asfollows. Suppose that we know the set of eigensolutions for twosegments l and n, with possibly different liner properties. WhenZl ≠ Zn then the sets of axial wavenumbers have in general (exceptfor a rare coincidence) no values in common, and so it holds thatklμ ≠ knν. Consequently, we can use Eq. (40) to compute

hFlμ;Fnνi �idPlμPnν

Ωnνω�klμ − knν�

�ΩlμZlμ

−ΩnνZnν

�����r�d; for Zl ≠ Zn

(74)

For the case when Zl � Zn ≔ Z we can have μ ≠ ν, which meansthat klμ ≠ knν, and so we can compute

hFlμ;Fnνi � −idPlμPnνu0ΩnνωZ

����r�d; for Zl � Zn; μ ≠ ν

(75)

which is identically zero for nonslipping flow (u0�d� � 0) or a hardwall (Z → ∞). When μ � ν we have klμ � knν ≔ kμ andPlμ � Pnν ≔ Pμ. We require the solution Pμ of the inhomogeneousPridmore-Brown equation (42) to compute

hFlμ;Fnνi �d

ρ0Ω2μ

�PμP

0μ −

u0ΩμP 0μPμ − P

0μPμ

�r�d;

for Zl � Zn; μ � ν (76)

Note that this implies that A� and C− are diagonal matrices, andA− and C� are zero matrices for nonslipping flow or hard wall.

V. Numerical Results

To compare the results of the classical (CMM) and the bilinear-map-based (BLM) mode-matching approaches, we consider thetest cases that are listed in Table 1. Configurations I, II, and III have a1-m-long duct with radius 0.15 m, split into two segments atx � 0.5 m. The left segment has a hard wall, and the right-hand sidesegment has a locally reacting impedance wall. The incident fieldconsists of one right-runningmode, either μ � 1 or 2 for the pertinentconfigurations.‡‡ For the BLM-based results, the modes of the left(hard-wall) segment are used as test functions. The configurations I,II, and III are included to compare the two mode-matching methodsin isolation. Configuration IV is a duct withN � 20 segments, and isincluded to illustrate the total effects ofmodematching and scatteringmatrices for a fully configured multisectioned duct with lining ofvarying impedance.

Table 1 Test configurations

Configuration I II III IV

Helmholtz & azi. ω � 13.86, m � 5 ω � 8.86, m � 5 ω � 15, m � 5 ω � 15, m � 5Temperature T � 1 T � 1 T � 2 log�2��1 − r2

2� T � 1

Mean flow M � 0.5�1 − r2� M � 0.3 × 43�1 − r2

2� M � 0.3 tanh�10�1 − r�� M � 0.3 tanh�10�1 − r��

Soft-wall impedance Z � 1 − 1i Z � 1� 1i Z � 1 − 1i Z � 1 − 3i; : : : ; 1� 3i, N � 20Incident rad. mode nr. μ � 1 μ � 1 μ � 2 μ � 1

μmax � 50 for all configurations.

‡‡For nonuniform flow and/or temperature, the numbering of the modes isnot unambiguous;we number themaccording their similarity in eigenfunctionshape and axial wavenumber value to the uniform flow modes.

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The results in this section are made dimensionless by scaling onthe duct radius d, a reference density ρ∞, and a reference temperatureT∞. This implies that velocities are scaled on reference soundspeed c∞ �

��������������γRT∞p

and time on d∕c∞. The nondimensional meanflow axial velocity is denoted by M (the Mach number); thatis, u0 � c∞M.Configuration II has a slipping flow; hence, it is necessary to use

the Ingard–Myers boundary condition here, and this is in contrast toconfigurations I and III, which have nonslipping flow. Themean flowprofile for configuration II has the equivalent mass flow as a uniform

flow with Mach number 0.3. The nonuniform temperature profileof configuration III has the equivalent mass flow of a constant meantemperature of T � 1. The flow profile of configuration III ismore representative of a uniform flow with a thin boundary layer. Inconfiguration IV the impedances of the 20 segments have animaginary part that varies linearly between −3 and 3. For allconfigurations we use μmax � 50 modes to represent the field.To compute the solutions of the boundary value problems, we use a

path-following approach as described in [16]. We start the processfrom the solutions for a uniform flow and constant temperature.Subsequently, the liner and flow properties are gradually changed tothe target configuration.Finding the uniform-flow constant-temperature solutions amounts

to finding the complex-valued roots of an analytic function. Tocompute these roots we employ the method of Delves and Lyness[17]. This method first constructs a polynomial that has the sameroots as the analytic function of interest by using contour integration(i.e., numerical quadrature). Subsequently, the roots of the poly-nomial can be computed in a standard manner. The advantage of thismethod lies in the fact that it is guaranteed that all roots inside a givenarea are found (apart from finite numerical accuracy issues), whereasa Newtonmethod converges to all of the roots only if it is started fromsufficiently close initial guesses. The location of these initial guessesfor the Newton method is not always self-evident, as, for example, incase of surfacewaves [18]. Figure 4 shows an example of the paths ofthe axial wavenumbers for configuration I.It is, moreover, important to note that it is not necessary to use path

following for the inhomogeneous problem (42), because the axialwave number kμ is given and not part of the solution, and theinhomogeneous problem (with inhomogeneous boundary conditionat r � d) has a unique solution. Consequently, the inhomogeneousproblem may be extra, and it is much cheaper to solve than theoriginal eigenvalue problem.

−50 −40 −30 −20 −10 0 10 20 30 40

−30

−20

−10

0

10

20

30

Re (k)

Im (

k)

Fig. 4 Axial wavenumber paths for configuration I, moving from blue(hard) to red (soft). First 10 modes in both directions. × indicates awavenumber of the soft-walled section.

x (m)

r (m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

−1

−0.5

0

0.5

1

Fig. 5 Real part of pressure field for configuration IV.

x (m)

r (m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

−1

−0.5

0

0.5

1

a) Classical mode-matching

x (m)

r (m

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

−1

−0.5

0

0.5

1

b) Bilinear map-based mode-matching

Fig. 6 Real part of pressure field for configuration I.

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To illustrate that a high number of segments pose no problem formode matching with the scattering matrix formalism, we includedFig. 5, which depicts the pressure field for a configuration withN � 20 segments. From our experience, an iterative procedure oftendoes not converge for more than 10 segments. The artifacts near

r � 0 are because the field there is almost zero, very close to the levelcurve at zero.Figure 6 shows the acoustic pressure field for both the classical

(CMM) and the bilinear form (BLM)-based mode-matchingapproaches for configuration I. Figures 7–9 compare the pressure,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

x (m)

Re(

P)

(dim

less

)

Re(P) (BLM), r=0.035mRe(P) (CMM), r=0.035mRe(P) (BLM), r=0.075mRe(P) (CMM), r=0.075mRe(P) (BLM), r=0.15mRe(P) (CMM), r=0.15m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x (m)

Re(

U)

(dim

less

)

Re(U) (BLM), r=0.035mRe(U) (CMM), r=0.035mRe(U) (BLM), r=0.075mRe(U) (CMM), r=0.075mRe(U) (BLM), r=0.15mRe(U) (CMM), r=0.15m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

x (m)

Re(

V)

(dim

less

)

Re(V) (BLM), r=0.035mRe(V) (CMM), r=0.035mRe(V) (BLM), r=0.075mRe(V) (CMM), r=0.075mRe(V) (BLM), r=0.15mRe(V) (CMM), r=0.15m

Fig. 7 Comparison of CMM and BLM mode-matching approaches for configuration I. Pressure, axial, and radial velocity at radial locations:r � f0.035;0.075;0.15g m.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

x (m)

Re(

P)

(dim

less

)

Re(P) (BLM), r=0.035mRe(P) (CMM), r=0.035mRe(P) (BLM), r=0.075mRe(P) (CMM), r=0.075mRe(P) (BLM), r=0.15mRe(P) (CMM), r=0.15m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x (m)

Re(

U)

(dim

less

)

Re(U) (BLM), r=0.035mRe(U) (CMM), r=0.035mRe(U) (BLM), r=0.075mRe(U) (CMM), r=0.075mRe(U) (BLM), r=0.15mRe(U) (CMM), r=0.15m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x (m)

Re(

V)

(dim

less

)

Re(V) (BLM), r=0.035mRe(V) (CMM), r=0.035mRe(V) (BLM), r=0.075mRe(V) (CMM), r=0.075mRe(V) (BLM), r=0.15mRe(V) (CMM), r=0.15m

Fig. 8 Comparison of CMM and BLM mode-matching approaches for configuration II. Pressure, axial, and radial velocity at radial locations:r � f0.035;0.075;0.15g m.

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axial, and radial velocity for both mode-matching methods at severalradial locations. As can be seen, the results of the two approaches arein very good agreement for configurations I–III.The validity of the numerical results can be further assessed by

checking whether they satisfy the balance of energy. For that purposewe use the exact Myers’s energy corollary (see Appendix A).Figure 10 shows, for configuration I, the sum of the acoustic fluxesthrough the wall, the inlet, and the outlet plane, and the volumeintegral over the source term.We use a relative numerical accuracy of10−6 for the boundary value problem solver, and use Simpson’s rulefor the numerical quadrature on a grid of 151 by 1001 grid points.Thus, this sum (which is normalized on the flux through the inletplane), which ideally should be zero, is not expected to be bigger than

10−6. Figure 10 shows that the energy balance is satisfied very wellfor both methods and better as the number of modes μmax increases,which is to be expected. Most interesting, however, is that thebilinear-form-basedmode-matchingmethod performs better than theclassical one.Incidentally, the energy integral being so accurately satisfied is not

straightforward, as we represent the field by a sum of discrete modesonly [5]. Therefore, it confirms the assumption of a negligiblecontinuous spectrum contribution.To verify the regular behavior of the solution at the hard–soft

singularities along thewall (where a so-called edge condition [19,20]has to be satisfied, which is that any volume surrounding the edgecarries finite energy),we check the uniform convergence of themodalseries via the convergence rate of the found amplitudes a�l;μ, heregenerically denoted by aμ.If we assume that aμ � O�μq� for μ → ∞ such that

log jaμj � q log μ�O�1�, then for qμ defined by

qμ �log jaμjlog μ

(77)

qμ is expected to approach q with qμ � q�O�1∕ log μ�.Anticipating a local behavior of the modal functions that is

asymptotically similar to a Fourier series, a convergence rate q < −1will be sufficient for uniform convergence. For the configurationsconsidered, with q ≃ −2, we see that this is indeed the case, inparticular for both mode-matching methods in the same way(see Fig. 11).Moreover, there is another interesting observation possible from

these plots. The behavior of qμ from the classical matching method isnot as smooth as from the BLM matching as μ becomes larger.Apparently, the amplitudes from the classical method are moreinaccurate for large μ. This is an interesting confirmation of the factthat the inner products based on exact relations are not prone to theoscillation quadrature errors for large μ.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5

−1

−0.5

0

0.5

1

1.5

x (m)R

e(P

) (d

imle

ss)

Re(P) (BLM), r=0.035mRe(P) (CMM), r=0.035mRe(P) (BLM), r=0.075mRe(P) (CMM), r=0.075mRe(P) (BLM), r=0.15mRe(P) (CMM), r=0.15m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x (m)

Re(

U)

(dim

less

)

Re(U) (BLM), r=0.035mRe(U) (CMM), r=0.035mRe(U) (BLM), r=0.075mRe(U) (CMM), r=0.075mRe(U) (BLM), r=0.15mRe(U) (CMM), r=0.15m

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x (m)

Re(

V)

(dim

less

)

Re(V) (BLM), r=0.035mRe(V) (CMM), r=0.035mRe(V) (BLM), r=0.075mRe(V) (CMM), r=0.075mRe(V) (BLM), r=0.15mRe(V) (CMM), r=0.15m

Fig. 9 Comparison of CMM and BLM mode-matching approaches for configuration III. Pressure, axial, and radial velocity at radial locations:r � f0.035;0.075;0.15g m.

5 10 15 20 25 30 35 40 45 50−4.5

−4

−3.5

−3

−2.5

−2

−1.5

mumax

log1

0 of

nor

mal

ized

ene

rgy

bala

nce

test1 (BLM)test1 (CMM)

Fig. 10 Comparison ofCMMandBLMmode-matching approaches forconfiguration I. Energy balance (normalized on flux through inlet plane)versus μmax.

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VI. Conclusions

Modematching is a particularly successfulmethod for problems oftime harmonic wave propagation in ducts of circular or rectangularcross section, with a uniform medium. The simple geometry and thesimple medium properties result in just Bessel functions, or cosineand sine functions, respectively, for the modal shape functions. Forthese functions the standard L2 integral inner products can beexpressed in closed form, and the elements of the mode-matchingmatrices can be determined analytically exactly, leaving the mode-matching problem as a relatively easy linear algebra problem that canbe solved fast and accurately.All these advantages disappear for ducts with a (cross-wise)

nonuniform medium. Modes, that is, solutions of the form∼f�y; z�eikx, still exist, but their mode shape is not simple anymore,and certainly integrals of their products are not available in closedform. As a result, the method of mode matching will require a largeamount of numerical quadrature, on top of the numerical exercisenecessary to determine the mode shape functions. Because themodalshape functions will show increasingly oscillatory behavior withincreasing modal order, the integrals for the highest orders willinherently be prone to numerical error and difficult, or at leastexpensive, to determine.The method presented in this paper avoids these problems by

constructing an alternative “inner product,” such that, when appliedto modal shape functions, they are explicitly available in closed form(for the radially symmetric problem) or simplify to a line integral

along the boundary (in the general 2-D problem). Although theproposed “inner product” is not a real inner product, which iswhy it iscalled here a bilinear form, the behavior in the mode-matchingmethod is entirely the same as for the inner product in the classicalapproach. By their construction, the present integrals can beconsidered as the natural generalizations of the Bessel functionproduct integrals.To make a start with establishing a firm mathematical basis for the

method, the new method was compared with an implementation ofthe classical method with hard-wall radial modes (Bessel functions)as test functions, known to form a complete basis. The results are infull agreement with each other, while the anticipated efficiency andaccuracy are indeed realized.Although the elegance is very appealing, a lot of research is to be

done, both numerically and functional analytically.

Acknowledgments

We gratefully acknowledge the fruitful discussions with R. M. M.Mattheij of the Mathematics Department at Eindhoven University ofTechnology, and particularly appreciate his insights and guidance inmatters of numerical analysis.

Appendix: Myers’s Energy Corollary

When the equations for conservation of mass and momentum andthe general energy conservation law for fluid motion are expanded to

0 10 20 30 40 50−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

µ

qµ

BLM

CMM

0 10 20 30 40 50−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

µ

qµ

BLM

CMM

0 10 20 30 40 50−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−1

µ

qµ

BLM

CMM

Fig. 11 Convergence rate of amplitudes for classical and BLM mode matching, for configurations I, II, and III.

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quadratic order, this second-order energy termmay be reduced to thefollowing conservation law for perturbation energy densityE, energyflux I, and dissipation D:

∂E∂t� ∇ · I � −D (A1)

where

E � p21

2ρ0c20

� 1

2ρ0jv1j2 � ρ1v0 · v1 �

ρ0T0s21

2Cp(A2a)

I � �ρ0v1 � ρ1v0��p1

ρ0� v0 · v1

�� ρ0v0T1s1 (A2b)

D � −ρ0v0 · �ω1 × v1� − ρ1v1 · �ω0 × v0�� s1�ρ0v1 � ρ1v0� · ∇T0 − s1ρ0v0 · ∇T1 (A2c)

as shown byMyers [21]. The vorticity here is denoted byω � ∇ × v.Note that these equations only contain zeroth- and first-order terms.Consequently, this conservation law is exactly valid for small lineardisturbances (p1, v1, etc.) of a mean flow (p0, v0, etc.) that can havenonuniform velocity and temperature. Taking the time average ofEq. (A1) for time harmonic perturbations yields

∇ · �I � − �D (A3)

By taking the volume integral of Eq. (A3) over a volume V withboundary ∂V and applying the divergence theorem, we find

Z∂V

�I · ndA�ZV

�D dV � 0 (A4)

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[2] Astley, R., Hii, V., and Gabard, G., “A Computational Mode-MatchingAproach for Propagation in Three-Dimensional Ducts with Flow,” 12thAIAA/CEAS Aeroacoustics Conference, AIAA Paper 2006-2528,May 2006.

[3] Watson,G.N.,ATreatise on the Theory of Bessel Functions, CambridgeUniv. Press, London, 1966, p. 134.

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[9] Sijtsma, P., and van der Wal, H., “Modelling a Spiralling Type of Non-Locally Reacting Liner,” 9th AIAA/CEAS Aeroacoustics Conference

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[11] Myers, M. K., “On the Acoustic Boundary Condition in the Presence ofFlow,” Journal of Sound and Vibration, Vol. 71, No. 3, 1980, pp. 429–434.doi: 10.1016/0022-460X(80)90424-1

[12] Kreyszig, E., Introductory Functional Analysis with Applications,Wiley Classics Library ed., Wiley, New York, 1989, pp. 451–458.

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[16] Oppeneer, M., Lazeroms, W. M., Rienstra, S. W., Sijtsma, P., andMattheij, R. M., “Acoustic Modes in a Duct with Slowly VaryingImpedance andNon-UniformMeanFlow andTemperature,”17thAIAA/CEAS Aeroacoustics Conference, AIAA Paper 2011-2871, June 2011.

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L. TichyAssociate Editor

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