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Side branch resonators modelling with Green’sfunction methods

E. Perrey-Debain, R. Marechal, J.-M. Ville

Laboratoire Roberval UMR 6253, Equipe Acoustique,Universite de Technologie de Compiegne, France

July 09-10, 2012

MATHmONDES 2012, Reading, U.-K.

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Outline

Introduction

Impedance matrix and Green’s function formalism

Simplified models

Low-frequency applications : Helmholtz and Herschel-Quinckeresonators

Medium-frequency applications : HQ-liner system for inlet fannoise reduction

Conclusions

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Introduction

Side branch resonators are commonly used for engine exhaust noisecontrol : (i) low-frequency applications with a single plane wavemode (automotive) (ii) medium-frequency applications and highlymultimodal context (aeronautics).

Pinc

Pref Ptr

Figure: Side branch resonator (cavity Ω) with two openings Γ.

Numerical predictions : (i) 1D approximations (with lengthcorrections) ; (ii) FEM, BEM -> computationally demanding (thisincludes mesh preparation etc...) especially in a highly multimodalcontext, lack of physical interpretation.

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Impedance matrix

In the frequency domain, the acoustic pressure must obey theHelmholtz equation

∆p + k2p = 0, k = ω/c ,

and q = ∂np = 0 everywhere except on Γ. The Green’s function forthe rigid-wall cavity is given by the infinite series

GΩ(r, r′) =

∞∑

n=0

φn(r)φn(r′)

λn − λ

where λ = ω2. Eigenfunctions φn are properly normalized so thatapplication of the Green’s theorem in the cavity yields

p(r) =

∫

ΓGΩ(r, r

′) q(r′)dΓ(r′)

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Impedance matrix

We need to precompute a finite set of eigenfunctions and estimatethe truncation error...Consider the eigenvector Φn, the FE discretization of ntheigenmode φn :

A(λn)Φn = 0 where A(λ) = K − λM

After reduction to the interfacial nodes, we obtain the impedancematrix

Z(λ) = ITΓ A−1(λ) IΓ

with

A−1(λ) =∞∑

n=0

ΦnΦTn

λn − λ= ΦD(λ)ΦT

Finally,

p = Z(λ) F q = GΩ q

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Truncation

Keeping the first N eigenmodes gives

Z(λ) = (ΦD(λ) ΦT)|N + R(λ).

The correction term R remains weakly dependent on the frequency,so we can take the first order Taylor expansion

R(λ) ≈ R(λ) + (λ− λ)∂R

∂λ+ . . .

The residual matrices are computed via

R = ITΓ A−1 IΓ − (ΦD(λ) ΦT)|N∂λR = ITΓ A−1MA−1IΓ − (Φ ∂λD ΦT)|N

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Scattering matrix

The theory starts by introducing the hard-walled duct Green’sfunction

G (r, r′) =∞∑

l=0

ψl(x , y)ψ∗l (x

′, y ′)

2iβleiβl |z−z ′|

The transverse eigenmodes ψl are solution of the boundary valueproblem

(∂2xx + ∂2yy)ψl + k2ψl = β2l ψl

with ∂nψl = 0 on the boundary line ∂S . These modes arenormalized as

∫

S|ψl |2 dS = 1

in particular ψ0 = 1/√Ad where Ad is the cross section area of the

main duct. For circular ducts :

ψl = Nm,nJm(αm,nr)eimθ, βl =

√

k2 − α2m,n

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Scattering matrix

The finite element discretization of the integral equation

p(r) =

∫

ΓG (r, r′) q(r′)dΓ(r′) + pI (r)

gives (ri is the FE node location)

pi =N∑

j=1

Gijqj + pIi with Gij =

∫

ΓG (ri , r

′) φj (r′)dΓ(r′)

The acoustic velocity is deduced from

q = (GΩ − G)−1 pI

Note :i. The computation is not trivial as ∂z′G is discontinuous at z′ = zi .

ii. The matrix GΩ − G is of small size.

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Simplified models : one opening

Starting with one opening only, the impedance matrix relation canbe averaged to give

p = Z (λ)q where Z(λ) =1

N

N∑

i=1

N∑

j=1

(Z(λ) F)ij

By the same token,

p =1

Wq + pI , where W =

2ikAd

A

and A denotes the area of the interface. An incident plane wavepI = e

ikz produces a transmitted pressure field

p = T eikz with T = 1 +

1

WZ(λ) − 1

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Helmholtz resonators

Figure: Classical (left) ; with extended neck (right).

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Helmholtz resonator, classical

60 70 80 90 100 110 1200

10

20

30

40

50

60

TL modèle mixteTL simplifiéTL Ingard

Frequency (Hz)

TL(dB)

60 70 80 90 100 110 120−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Zero100 modes1 mode2 modes10 modes50 modes

ZFrequency (Hz)

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Helmholtz resonator, with extended neck

60 70 80 90 100 110 1200

5

10

15

20

25

30

35

40

45

50

TL modèle mixteTL simplifié

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Simplified models : two openings

We consider a symmetric resonator connected to the main duct viatwo openings located at z = z1 and z = z2.

(

p1p2

)

=

(

Z11 Z12

Z12 Z11

)(

q1q2

)

Moreover,(

p1p2

)

=1

W

(

1 eikL

eikL 1

)(

q1q2

)

+

(

pI1pI2

)

where L = |z2 − z1|. This gives

T = 1 +2WZ11 − 2WZ12 cos(kL) + (e2ikL − 1)

(WZ11 − 1)2 − (WZ12 − eikL)2

Thus, no acoustic energy is transmitted if

Z 212 − Z 2

11 − A sin(kL)

kAdZ12 = 0

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Herschel-Quincke resonator

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Herschel-Quincke resonator

0 100 200 300 400 500 600 700 8000

10

20

30

40

50

60

70

80

TL simplifiéTL tube droitTL modèle mixte

Frequency (Hz)

TL(dB)

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Fan noise

Perforate sheet

The fan noise is one of the dominant components at take-offand landing for aircraft with modern high bypass ratioturbofan engines : broadband noise + Blade PassingFrequency (BPF) tones

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Fan noise

Actual configuration...

Soufflante

Redresseurs

Collecteur

Tube HQ

Model...

A+

A−

B+

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Validation on a small size configuration

0

0.1

0.2

0.3

0.4−0.05 0 0.05 0.1

−0.1

−0.05

0

0.05

0.1

x (m) y (m)

z(m

)

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

30

35

40

Frequency (Hz)

TL(dB)

Matrix size CPU time (Matlab)

Our model 500 1 h 50 min

FEM 82 000 31 h 15 min

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Optimal configuration (36 HQ tubes)

0 5 10 15 20 25 30 35 40 4586

88

90

92

94

96

98

100

ka

SPL(dB)

0 5 10 15 20 25 30 35 40 450

2

4

6

8

10

12

ka

TL(dB)

Incident Liner-HQ system

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What does the liner do ?

0 5 10 15 20 25 30 35 40 45

0

2

4

6

8

10

12

14

16

18

Re(β0,n)

Im(β

0,n)

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

Re(β20,n)

Im(β

20,n)

Surface wave modes at 1530 Hz :

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Influence of the number of tubes on the first BPF

(iso-surface)

Number of tubes

Modes

Modal

pow

er(dB)

72675545403736300 (Liner)0 (Incident)

(7,1)(4,2)

(8,1)(2,3)

(-5,2)(-9,1)

(-3,3)

30

50

70

90

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Conclusions and prospects

The proposed Green’s function based method allows to reduce thecomputational effort as only the acoustic velocity at the interfaceneeds to be calculated.A very high number of propagative modes (few hundreds) can behandled easily on a single PC.Gives access to physical interpretation in the low-frequency regime.

In prospect : - could be used for designing taylor-made resonatorsusing optimization procedures. - viscosity effects should be included