AbstractReview of existing methods of liner impedanceeduction was carried out and as most universalone the finite element method was selected . Thecomputational program of liner impedanceeduction by finite element method has beendeveloped. The verification of computing resultswas made on the basis of experimental dataobtained in duct with flow facility. Acceptableagreement of computed results withexperimental ones was noted. It was found thatthe objective function in search domain has onlyone minimum, which simplifies definingimpedance. It was established that reduction ofmicrophones number in the experiment slightlyeffects on calculating values of liner impedance .

1 IntroductionDevelopment of an effective noise reductionsystem is an important subject area in aircraftengine creation. The ICAO Assembly(September, 2010) approved the targetspecification of noise and emission reductionaccording to which, for example, in the mediumterm (by 2018) an achievable progress of noisereduction in the class of regional aircrafts is tobe 13.0 EPNdB, for short and medium -haulaircrafts is to be 21.0 EPNdB. The problem ofnoise reduction is being solved actively by thecountries of the world “aviation club”. Russiandesigners have faced the problem of effectivenoise reduction system development whilecreating the PD-14 engine for airliner of thenew generation MS-21. By now various wayspotentially providing necessary levels of noisereduction have been offered, but their level isnot sufficiently high. Research and developmentof new technologies depend on successful

solution of fundamental problems in the fieldsof aeroacoustics and gasdynamics directlyconnected with the turbulence phenomenon andimprovement of computational experiments ofnoise formation and propagation.

To solve these problems Perm NationalResearch Polytechnic University (PNRPU) andAviadvigatel OJSC (Perm, Russia) have createdthe Acoustic Research Center. Material andtechnical base of Acoustic Research Center is aduct with flow facility (Fig. 1) created inaccordance with the Central AerohydrodynamicInstitute (Zhukovsky, Russian Federation) andallowing to research acoustic properties ofvarious liners in the air flow.

Similar facilities already exist in suchdepartments and research centers as NASALangley Research Center (USA) [1], BoeingWichita Noise Lab (USA) [2], UnitedTechnologies Research Center (USA) [3],Department of Aeronautical and VehicleEngineering (Sweden) [4], Institute ofPropulsion Technology, Engine Acoustics(Germany), Aerodynamics and EnergeticsModeling Department (France) [5].

Primary tasks when creating the duct withflow facility were:

- review of existing methods of the linerimpedance eduction;

- selection of the most universal method;- computational program development on

the basis of the method chosen;- verification of computational results.Review of existing methods of the liner

impedance eduction on the basis of acousti cpressure measurements in the duct with flowfacility reveals that the mos t widely spreadmethods among them are those based on theanalytical solution of an initial mathematicalmodel. The analytical solutions contain values

THE PROCEDURE OF LINER IMPEDANCE EDUCTIONBY FINITE ELEMENT METHOD

R.V. Bulbovich, V.V. Pavlogradskiy, V.V. PalchikovskiyPerm National Research Polytechnic University , Russian Federation

Keywords: liner, impedance, duct with flow, finite element method

R.V. BULBOVICH, V.V. PAVLOGRADSKIY, V.V. PALCHIKOVSKIY

2

of acoustic pressure amplitudes of direct andreflected waves being defined commonly on thebasis of measurements by the 2- [6-8], 3- and 4-microphone [9] methods. These methods have anumber of weaknesses, some of them arementioned in [10]. Also a final form of theanalytical solution significantly depends on theduct geometry and microphone positions. Thu s,the field of application of the analyticalsolutions is quite limited.

Among numerical approaches the finiteelement method is commonly used [11 -12]. Ithas a number of advantages compared to theanalytical solutions:

- flexibility (in case of changing positionsand number of microphones the computationalprocedure does not change, the method suits forthe duct of any configuration);

- automatic account of high-order modes incalculation;

- account of transversal modes propagationbeyond the liner edges.

Thus, to solve the problem of the linerimpedance eduction in the duct with flowfacility the finite element approach has beenchosen.

2 Mathematical Model and SolutionThe mathematical model describing propagationof acoustic disturbances in the duct with gasflow is based on the convected Helmholtzequation. For the two-dimensional case it hasthe following form:

02)1( 202

2

2

220

pkxpikM

yp

xpM (1)

where p is acoustic pressure; ck / is wavenumber; c is sound velocity; is angularfrequency; M is average Mach number ofnonperturbed gas flow; i is imaginary unit; x ,y are longitudinal and transverse coordinates ofthe duct.

Boundary conditions to solve the equationare the following:- at the duct inlet e

1pp (2)

- at the duct outlet enpp (3)

- on the rigid wall 0

yp

(4)

- on the liner wall

2

22002

xp

ikZM

xp

ZM

Zikp

yp

(5)

Fig. 1. Duct with flow facility in the PNRPU Acoustic Research Center

3

THE PROCEDURE OF LINER IMPEDANCE EDUCTION BY FINITE ELEMENT METHOD

where ep1 , enp are acoustic pressure measured

by microphones at inlet and outlet of duct; Z isnondimensional impedance.

Application of Galerkin’s finite elementmethod to equation (1) gives

02][

)1(][

20

2

2

2

220

dVpkxpikMN

dVy

px

pMN

V

T

V

T

(6)

where ][N is shape function of finite element.To solve it the finite element is used with shapefunctions formed on the basis of Lagrangepolynomial; it ensures continuity of the requiredfunction in nodes after integration, but not itsderivatives. Therefore in equation (6) it isnecessary to reduce the second order ofderivatives to the first one. Applying Green’stheorem yields

0][

][

][][)2

01(

][][02

][][][][)2

01(

][][2

PL

dLyNTNyl

LdL

x

NTNxlM

AdA

x

NTNikM

AdA

y

N

y

TN

AdA

x

N

x

TNM

A

T dANNk

(7)

where xl , yl are directional cosines; A is areaof finite element. Application of the sameconversions to expression (5) yields

dLL x

N

x

TN

L ikZ

MdL

x

NTNZ

M

LdLNTN

Z

ik

LdL

y

NTN

][][20][

][02

][][][

][

(8)

In equation (7) contour integrals are 0 forall walls except the liner, where expression ( 8)is used. Application of equation (7) for all finiteelements and taking into account the boundaryconditions (2-4) results in the system of linearequations

}{}]{[ FPK (9)

where ][K is complex coefficient global matrix ;}{F is global vector formed by boundary

conditions; }{P is vector of acoustic pressuresin finite element nodes. Solution of system (9)provides all values of acoustic pressures in finiteelement nodes.

3 Liner Impedance Eduction ProcedureProcedure for determining the liner impedancecan be divided into two stages: initial estimatesearch and solution adjustment. The first stage(Fig. 2) consists of the following steps:

Fig. 2. Initialization for the method of alternatingvariables.

1) The searching range is assigned. For themost liners

,100 1010

Here , are real and image part ofimpedance.

2) All searching range is meshed.3) System (9) is solved for every center of

the cell accepted as the value of impedance iz .

4) The objective function is defined by theexpression

n

jjj pp

1

ec (10)

5) The value of impedance correspondingto the minimum of is accepted as an initialpoint for the method of alternating variables.

R.V. BULBOVICH, V.V. PAVLOGRADSKIY, V.V. PALCHIKOVSKIY

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Fig. 3.1 Objective function (M=0.000).

Fig. 3.2 Objective function (M=0.172).

Fig. 3.3 Objective function (M=0.255).

Fig. 3.4 Objective function (M=0.400).

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THE PROCEDURE OF LINER IMPEDANCE EDUCTION BY FINITE ELEMENT METHOD

Then the second stage comes. Impedancevalue z is defined more precisely by the fastvariant of the method of alternating variables.From the point found at the previous stagemovement starts along a real axis; the step ofdescending is equal to 0.1. The movementcontinues while the objective function isdecreasing. If the decrease stops, the movementstarts along an imaginary axis in the direction ofdecreasing of the objective function. If theobjective function does not decrease along bothaxes, the step of descending is divided into twoand again the movement starts along the realaxis. The search continues until the descendingstep is less than 0.001. Thus the impedancecorresponding to the objective functionminimum is accepted as the liner impedancerequired.

The objective function being obtainedunder expression (10) has the only minimum(Fig. 3.1-3.4). Thus it ensures reliableidentification of the impedance at any stepreduction coefficient and any initial point of thegiven search range. In the process the first stagecan be omitted, however an improper initialpoint increases considerably the time forcalculations by the method of alternatingvariables.

4 ResultsThe suggested procedure of the liner impedancecalculation shows acceptable agreement ofobtained acoustic pressures with experimentaldata of paper [12]. Some impedance valuesobtained by computation of the objectivefunction on 31 microphones and used forcomputation of acoustic fields are indicated inTable 1. The objective function valuescorresponding to the obtained impedance s areindicated in Table 2. The amplitudes ofcalculated acoustic pressures on the duct upperwall as well as experimental acoustic pressures,and also acoustic fields in the duct are shown inFig. 4-6.

The figures show that accuracy of linerimpedance eduction becomes worse at high

Mach numbers of flow. It is mainly caused bythe fact that when increasing the flow velocityits profile is distorted greatly .

Table 1. Impendence values (31 microphones)

1000 Hz 2000 Hz 3000 HzM

θ χ θ χ θ χ0.000 0.472 0.025 4.149 1.350 0.674 -0.3700.079 0.425 0.059 3.459 1.823 0.612 -0.3200.172 0.389 0.143 4.996 0.187 0.685 -0.2180.225 0.358 0.180 4.791 0.027 0.630 -0.2220.335 0.290 0.282 4.881 -1.399 0.598 -0.2060.400 0.240 0.362 2.939 -2.001 0.566 -0.192

Table 2. Objective function values, Pa(31 microphones)

M 1000 Hz 2000 Hz 3000 Hz0.000 13.224 49.733 50.3920.079 21.665 71.729 86.9160.172 58.791 82.160 117.5710.225 90.793 94.031 116.8580.335 120.292 110.294 92.7310.400 226.398 115.878 112.572

The analysis of impedance determinationaccuracy depending on the number andarrangement of microphones (Fig. 7) has beenconducted. Results of the liner impedancecalculations at different microphone number areshown in Fig. 8. Objective function values arepresented in tables 3.1-3.3. In the table cells for19, 11, 7 microphones the upper value is anobjective function obtained when searching theimpedance under corresponding microphonenumber. The lower value is an obje ctivefunction obtained on 31 microphones with theimpedance calculated previously oncorresponding number of microphones.

One can see that the impedance obtai ned atdifferent number of microphones gives closeobjective functions on 31 microphones. Thisindicates the possibility to use the finite elementapproach for simulation of optimal microphonequantity in terms of the equipment -cost-to-solution-quality principle.

R.V. BULBOVICH, V.V. PAVLOGRADSKIY, V.V. PALCHIKOVSKIY

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Fig. 4.1. Acoustic pressures on microphon es (M = 0.000, 31 microphones)1000 Hz, 2000 Hz, 3000 Hz.

Fig. 4.2. Acoustic pressures in the duct with flow(M = 0.000, 31 microphones).

7

THE PROCEDURE OF LINER IMPEDANCE EDUCTION BY FINITE ELEMENT METHOD

Fig. 5.1. Acoustic pressures on microphones (M = 0.225, 31 microphones)1000 Hz, 2000 Hz, 3000 Hz.

Fig. 5.2. Acoustic pressures in the duct with flow(M = 0.225, 31 microphones).

R.V. BULBOVICH, V.V. PAVLOGRADSKIY, V.V. PALCHIKOVSKIY

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Fig. 6.1. Acoustic pressures on microphones (M = 0.400, 31 microphones)1000 Hz, 2000 Hz, 3000 Hz.

Fig. 6.2. Acoustic pressures in the duct with flow(M = 0.400, 31 microphones).

9

THE PROCEDURE OF LINER IMPEDANCE EDUCTION BY FINITE ELEMENT METHOD

Fig. 7. Arrangement of microphones.

M=0.000

M=0.255

M=0.400

Fig. 8. Liner impedance at different number of microphones 31 microphones, 19 microphones, 11 microphones, 7 microphones.

R.V. BULBOVICH, V.V. PAVLOGRADSKIY, V.V. PALCHIKOVSKIY

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Table 3.1 Objective function (1000 Hz)

Microphone numberM

31 19 11 7

0.000 13.22 9.9513.31

6.0113.23

2.5116.39

0.079 21.67 16.4521.78

8.1521.93

4.0326.30

0.172 58.79 43.7259.62

20.4859.62

9.9765.21

0.225 90.79 67.3590.92

39.8891.12

12.7896.39

0.335 120.29 82.46120.29

38.57124.57

15.12141.28

0.400 226.40 157.02226.90

84.08233.38

25.25238.17

Table 3.2 Objective function (2000 Hz)

Microphone numberM

31 19 11 7

0.000 49.73 28.1651.24

14.3350.06

6.3349.76

0.079 71.73 42.8371.92

21.5572.76

8.2075.76

0.172 82.16 48.3283.23

26.5784.23

13.3885.42

0.225 94.03 53.4594.15

33.71101.71

13.79104.18

0.335 110.29 72.95114.47

41.49120.54

18.49116.17

0.400 115.88 79.50119.42

44.50129.76

23.22138.02

Table 3.3 Objective function (3000 Hz)

Microphone numberM

31 19 11 7

0.000 50.39 31.4650.41

17.8453.41

7.1155.49

0.079 86.92 56.1986.96

30.6388.51

11.3989.59

0.172 117.57 73.46118.02

44.57118.05

14.17117.70

0.225 116.86 73.28116.94

55.93116.91

11.74119.21

0.335 92.73 64.4093.04

41.5893.07

14.6393.37

0.400 112.57 76.13113.54

48.14120.52

15.35120.29

5 ConclusionThe research conducted has lead to thefollowing conclusions.

1) Review of existing scientificpublications on the problem of liner impedanceeduction in the duct with flow facility showedthat the most universal meth od is the finiteelement method (its realization does not dependon the number and arrangement ofmicrophones; it automatically takes into accountthe high-order modes and transverse modesbeyond the liner edges; it is suitable for a ductof any configuration).

2) Investigation of objective functionbehavior in impedance search revealed that theobjective function is unimodal. Thus, the searchprocedure can be started in any point of a setarea with any initial step and any step reductionfactor.

3) Comparison of the calculation resultsof acoustic pressures at microphones with theexperiment results revealed that increment inflow velocity decreases accuracy of impedances.It is mainly caused by the fact that whenincreasing the flow velocity its profile isdistorted greatly (especially in a narrow duct).Thus, for precise calculation of acousticpressures it is necessary to take into accountderivatives of steady flow parameters in themathematical model.

4) Calculations on the basis of differentnumber of microphones show that impedanceslightly changes at a certain decreases of themicrophone number. Thus, the given type ofcalculation can be used to determine optimalmicrophone number in terms of the equipment -cost-to-solution-quality relation.

AcknowledgmentThe authors would like to express deepappreciation to Jones M. G., Watson W. R.,Parrott T. L. for wide experimental material onacoustic researches presented in paper [12], thatwas used for verification of the liner impedancevalues computed by the develope d program.

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THE PROCEDURE OF LINER IMPEDANCE EDUCTION BY FINITE ELEMENT METHOD

References[1] Jones M. G., Watson W. R., Parrott T. L. Design and

evolution of modifications to the NASA Langleyflow impedance tube. AIAA Paper 2004-2837, 2004.

[2] Gallman J. M., Kunze R. K. Grazing flow acousticimpedance testing for the NASA AST Program.AIAA Paper 2002-2447, 2002.

[3] Simonich J. C., Morin B. L. Grazing flow impedancemeasurement facility. AIAA Paper 2006-2640, 2006.

[4] Elnady T., Boden H. On semi-empirical linerimpedance modeling with grazing flow. AIAA Paper2003-3304, 2003.

[5] Heuwinkel C., Fischer A., Roehle I., Enghardt L.,Bake F., Piot E., Micheli F. Characterization of aperforated liner by acoustic and opticalmeasurements. AIAA Paper 2010-3765, 2010.

[6] Elnady T., Boden H. An inverse analytical methodfor extracting liner impedance from pressuremeasurements. AIAA Paper 2004-2836, 2004.

[7] Elnady T., Musharrof M., Boden H., Elhadidi B.Validation of an inverse analytical technique to educeliner impedance with grazing flow. AIAA Paper2006-2639, 2006.

[8] Auregan Y., Leroux M., Pagneux V. Measurement ofliner impedance with flow by an inverse method.AIAA Paper 2004-2838, 2004.

[9] Sobolev A. F. On the technique of liner impedanceeduction in interferometer with flow . Third OpenRussian Conference on Aeroacoustics , Zvenigorod,Russian Federat ion, pp. 84-86, 2013.

[10] Jones M. G., Stiede P. E. Comparison of methods fordetermining specific acoustic impedance. Journal ofAcoustical Society of America , Vol. 101, No. 5, pp.2694-2704, 1997.

[11] Eversman W., Gallman J. M. Impedance Eductionwith an Extended Search Procedure. AIAA Paper2009-3235, 2009.

[12] Jones M. G., Watson W. R., Parrott T. L. BenchmarkData for Evaluation of Aeroacoustic PropagationCodes with Grazing Flow. AIAA Paper 2005-2853,2005.

Contact Author Email Addressrkt@pstu.ru, vvpal@perm.ru

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