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Problem Model Linear case Non Linear case Results Applications

A systematic impedance model fornon-linear Helmholtz resonator liner

Deepesh Kumar Singh

Mentor: Sjoerd Rienstra

CASA DAY

April 17, 2013



Outline

1 Problem Definition

2 The Model

3 The Linear case

4 The Non Linear case

5 Results

6 Applications



Acoustic Liner



Helmholtz resonator and its ImpedanceMass-spring damper system for attenuating noiseAir in the hole ≡ massAir in the cavity ≡ springViscosity and vortex shading ≡ dampingThe resonator is characterizedby its impedance Z = Z(ω)

Spatially averaged impedancerelates the pressure andvelocity

Z(ω) :=p̂

v̂· n̂Ideally, Z is a wall property and independent of the acoustic fieldIn reality, Z is amplitude dependent for high but relevant amplitudes.

A good precision in the value of impedance improve the fidelityof the liner optimization by e.g. CAA Solvers



Aim and Motivation

Aim

Systematic solution of non-linear Helmholtz resonator equationfrom first principles to obtain analytic expressions for impedanceclose to resonance with non-linear effects.

Motivation

The usual non-linear corrections for a Helmholtz type impedanceare not based on a systematic asymptotic solution of thepertaining equations. All of them are either ad-hoc or by fullyCFD simulation [1] [2] [3].

example

Re(Z) = A + BV



Outline


2 The Model

3 The Linear case


5 Results

6 Applications



Model

SbV p′in

u′in ' 0

u′n

`

p′exSn

Sb � Sn - p′ex and u′in are uniformk`� 1 - cavity neck is acoustically compactLine Integral of momentum equation along center streamlineyields:

ρ0ddt

∫ exin

v′·ds + 12ρ0(u′2ex− u′2in) + (p′ex− p′in) =∫ ex

inµ∇2v′·ds.



Model

With minor friction effects, the velocity integral scales on typicallength and typical velocity∫ ex

inv′·ds = (`+ 2δ)u′n

The stress integral depends on Reynolds number, wall heatexchange, turbulence, separation from sharp edges. We takethese effects together in a small factor R.∫ ex

inµ∇2v′·ds = −Ru′n

Using the inflow and outflow conditionsInflow: u′ex = 0, u

′in = u

′n

Outflow: u′in = 0, u′ex = −u′n



ModelMass conservation of volume V

Vdρ′indt

= −ρu′nSn ≈ −ρ0u′nSn

Isentropic relation p′in = c20ρ′in.

`Vc20Sn

d2p′indt2

+V2

2ρ0c40S2n

dp′indt

∣∣∣∣dp′indt∣∣∣∣+ RVρ0c20Sn dp

′in

dt+ p′in = p

′ex

Resonance frequency ω0 = (c20Sn`V )

1/2

Non-dimensionalize the variablesp′ex = 2�

2ρ0c20(`Sn/V)F p′in = 2�ρ0c

20(`Sn/V)y

τ = ω0t R = �ρ0c0(`Sn/V)1/2rd2ydτ 2

+ εdydτ

∣∣∣∣ dydτ∣∣∣∣+ εr dydτ + y = εF0 cos(Ωτ) Ω = ωω0 = 1 + ε∆

Most notorious term harmonic forcing



Outline


2 The Model

3 The Linear case


5 Results

6 Applications



Linear case - away from resonance

The governing equation:

d2ydτ 2

+εdydτ

∣∣∣∣ dydτ∣∣∣∣+εr dydτ+y = εF0 cos(Ωτ) Ω = ωω0 = 1+ε∆

Since 1− Ω2 = O(1), y = O(ε) and εy′|y′| = O(ε3)

y(τ) = εF0(1− Ω2) cos(Ωτ) + εrΩ sin(Ωτ)

(1− Ω2)2 + ε2r2Ω2+ . . .

A =εF0√

(1− Ω2)2 + ε2r2Ω2, tan θ =

εrΩ(1− Ω2)2

Invalid close to the resonance, i.e. 1− Ω2 = O(ε) so A = O(1)



Outline


2 The Model

3 The Linear case


5 Results

6 Applications



Non Linear case

Ωτ = τ̃ + θ(ε)absorb Ω into τ̃ to avoid secular terms cos(Ωτ)phase shift θ(ε) to fix the location of the sign change of y′

Ω2d2ydτ̃ 2

+ εΩ2dydτ̃

∣∣∣∣ dydτ̃∣∣∣∣+ εΩr dydτ̃ + y = εF0 cos(τ̃ + θ)

Substitute the assumed Poincaré expansion

y(τ̃ ; ε) = y0(τ̃)+εy1(τ̃)+ε2y2(τ̃)+. . . , and θ(ε) = θ0+εθ1+. . .

and expand

(1+2�∆+�2∆2)(y′′0 +�y′′1 +�

2y′′2 )+(�+2�2∆+�3∆2)(y′0+�y

′1+�

2y′2)∣∣∣(y′0 + �y′1 + �2y′2)∣∣∣

+(�r+�2r∆)(y′0+�y′1+�

2y′2)+(y′0+�y

′1+�

2y′2) = �F0 cos(τ̃+θ0)−�2θ1F0 sin(τ̃+θ0)



Amplitude calculation

d2y0dτ̃ 2 + y0 = 0, y

′0(Nπ) = 0 with y0(τ̃) = A0 cos(τ̃)

Determine the integration constants A0 and θ0 from next order ε1,d2y1dτ̃ 2

+ y1 = F0 cos(τ̃ + θ0)− 2∆d2y0dτ̃ 2−

dy0dτ̃

∣∣∣∣dy0dτ̃∣∣∣∣− r dy0dτ̃

= F0 cos(τ̃ + θ0) + 2∆A0 cos(τ̃) + A0|A0| sin(τ̃) |sin(τ̃)|+ rA0 sin(τ̃)

Only stationary state→ no sine - cosine excitation

F0 cos θ0 = −2∆A0, F0 sin θ0 =(

83π|A0|+ r

)A0

The next order y1

y1(t) = A1 cos τ̃ +1

4πA0|A0|

[− 89 sin τ̃ +

∞∑n=1

sin(2n + 1)τ̃

n(n + 1)(n2 − 14 )(n +32 )

]

A1 and θ1 determined in a similar way as with y0.



Amplitude calculationThe full solution is

y(τ ; θ) = A0 cos(τ−θ)+�

A1 cos(τ − θ) + 14π

A0|A0|

− 89 sin(τ − θ) + ∞∑n=1

sin(2n + 1)(τ − θ)n(n + 1)(n2 − 14 )(n +

32 )

(1)

∆ = O(1ε) A0 and θ0 → linear caseAmplitude and Phase (ε0)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-4 -2 0 2 4

Am

plitu

de

∆

A0

0

0.5

1

1.5

2

2.5

3

3.5

-4 -2 0 2 4Ph

ase

(rad

ians

)

∆

θ0

Solution of amplitude (A0, A1) and phase (θ0 , θ1) as a function of ∆,for r = 1 and F0 = 1Max amplitude at linear resonance



Comparison with the numerical solution (R Kiteration)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

400 402 404 406 408 410 412

y(τ)

arbitrary τ

y0y0 + �y1

numerical solution

Comparison of the amplitude �0 and �1 order with full numerical solution(∆ = 1,� = 0.1)



Impedance Calculation

In order to comply with the literature, we define Impedance asthe ratio of the Fourier transforms of pressure and velocity at theforcing frequency

Z(η) = −∫∞−∞ p

′exe−iηtdt∫∞

−∞ u′exe−iηtdt

= − p̂′ex(η)

û′ex(η). (2)

Remember

p′ex = 2�2ρ0c20(`Sn/V)F

Vc20

2ερ0c20`Sn/V

Ω

dy(t)dt

= −ρu′nSn and u′ex = u′nSnSb



Outline


2 The Model

3 The Linear case


5 Results

6 Applications



Impedance results

0

20

40

60

80

100

120

140

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

Re(

Z)

Ω

100dB110dB120dB130dB

-250

-200

-150

-100

-50

0

50

100

150

200

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

Im(Z

)

Ω

100dB110dB120dB130dB

Impedance (Re(Z) and Im(Z) of Helmholtz resonator Vs Ω at differentdriving amplitudes (i.e. ε). Sn/Sb = 0.05, r = 0.2, ω0 = 9086.9 rad/s, L =0.035 m, ` = 0.002 m.



Comparison with the existing measurements

0

0.2

0.4

0.6

0.8

1

1000 1500 2000 2500 3000 3500

Re(

Z)

f

125dB130dB135dB140dB145dB150dB

Comparison of the resistance of the cavity Re(Z) with the measurements in[3]. 0.2 6 ε 6 0.55



Comparison of the Impedance obtained from y0 andy0 + εy1

0

20

40

60

80

100

120

140

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

Re(

Z)

Ω

ε0 orderε1 order

-250

-200

-150

-100

-50

0

50

100

150

200

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

Im(Z

)Ω

ε0 orderε1 order

Comparison of the Impedence of the cavity obtained from the �0 and �1asymptotic analysis for 100dB, 110dB, 120dB and 130dB. 0.1 6 � 6 0.3



Outline


2 The Model

3 The Linear case


5 Results

6 Applications



N wave forcing

Model N wave ∑∞n=1

sin(nt)n

sin(t)

N-wave modes/shocks produce by fan blade tip at full enginepower take off

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 5 10 15 20

ampl

itude

y

time t

N wave

Shocks from the blade tip at 95% casing height

nwave.wavMedia File (audio/wav)

sine.wavMedia File (audio/wav)



Impedance for N wave excitation

(1 + 2ε∆)d2ydτ̃ 2

+ εdydτ̃

∣∣∣∣ dydτ̃∣∣∣∣+ εr dydτ̃ + y = εF0

∞∑n=1

sin n(τ̃ + θ)n

(3)

0

200

400

600

800

1000

1200

1400

1600

1800

1 2 3 4 5 6 7

Rea

l(Z

)

n

Real(Z)

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

1 2 3 4 5 6 7

Im(Z

)

n

Im(Z)

Impedance at observed frequency η = nω and N wave excitation Ω = 1.



References

A.W. Guess.Calculation of perforated plate liner parameters from specified acoustic resistance and reactance.Journal of Sound and Vibration, 40(1):119 – 137, 1975.

T.H. Melling.The acoustic impendance of perforates at medium and high sound pressure levels.Journal of Sound and Vibration, 29(1):1 – 65, 1973.

H.H. Hubbard and Acoustical Society of America.Aeroacoustics of Flight Vehicles: Noise sources.Aeroacoustics of Flight Vehicles: Theory and Practice. Volume 1 Noise Sources; Volume 2 Noise Control. Publishedfor the Acoustical Society of America through the American Institute of Physics, 1991.

K.U. Ingard and Massachusetts Institute of Technology. Industrial Liaison Office.On the Theory and Design of Acoustic Resonators.1953.

R.M.M. Mattheij, S.W. Rienstra, and J.H.M.T. Boonkkamp.Partial Differential Equations: Modeling, Analysis, Computation.Society for Industrial and Applied Mathematics, 1987.

C. Bréard, A. Sayma, M. Imregun, A.G. Wilson, and B.J. Tester.A cfd-based non-linear model for the prediction of tone noise in lined ducts.In 7th AIAA/CEAS Aeroacoustics Conference, 2001.AIAA-2001-2181.

Problem DefinitionThe ModelThe Linear caseThe Non Linear caseResultsApplications

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