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LEABHARLANN CHOLAISTE NA TRIONOIDE, BAILE ATHA CLIATH TRINITY COLLEGE LIBRARY DUBLIN

OUscoil Atha Cliath The University of Dublin

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N

o i s e

S

o u r c e

I

d e n t i f i c a t i o n

F

o r

D

u c t e d

F

a n s

G

a r e t h

J . B

e n n e t t

Dej)artinent of Mechanical

h

Manufacturing Engineering

Parsons Building

Trinity College

Dublin 2

Ireland

A pril 2006

A thesis submitted to the University of Dublin in partial

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' TRINITY C O L L E G ^ 1 G J A N 2 0 M

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D eclaration

I declare th a t I am the author of this thesis and th a t all work described herein is niy own, unless otherwise referenced. Furtherm ore, this work has not been subm itted in whole or part, to any other university or college for any degree or qualification.

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A b stract

U n d e rsta n d in g com bustion noise source m echanism s, designing efficient acoustic liners an d optim ising active co n tro l algorithm s for noise red u ctio n requires th e identification of th e frequency an d m odal co n ten t o f th e com bustion noise co n trib u tio n . C oherence-based noise source identification techniques have been developed which can be used to identify th e c o n trib u tio n of com bustion noise to near and far-ficld acoustic m easurem ents of aero-engines. A num ber of existing identification techniques from th e lite ra tu re , as well as som e new' techniques, are im plem ented and evaluated u n d er controlled ex p erim en tal conditions. A series of tests are co nducted to exam ine th e efficacy of each of th e p rocedures for specific applications. An experim ental rig was designed an d b u ilt to gain a fu n d am en tal j)hysic:al u n d e rsta n d in g of th e convection of com bustion noise th ro u g h th e tu rb in e of an aero-engine. T h e identification techniques are applied to this rig, w ith th e objective of se p a ra tin g th e pressure field into its c o n stitu e n t jm rts.

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A ck n ow led gm en ts

I would like to express iny heartfelt gratitude to Professor John fltz p a tric k for his help and guid­ ance over the last four years, and for encouraging me to re tu rn to Trinity College to work on the SILENCE(R) project. I sincerely wish th a t our collaboration in projects professional, as well social, continue into the future.

The aduiiiiistrative and technical staff are a support w ithout which no thesis could be completed. Joan Gillen. Tom Haveron, Gabrielle, J J and Sean have been instrum ental in the com pletion of this work. I would like to niention, in particular, Alan Reid, the Chief Experim ental Officer, for his help in designing the experim ental rig.

This Ph.D . was ])artly supported by the SILENCE(R) project under EU contract no. G4RD- CT-2()()l-005()0. W hilst working on this contract I have come into contact with engineers from all over Europe and have had an oi)portunity to visit their w’ork places and hometowns. I have been impressed with their friendliness, engineering achievements, the beauty of their countries, bu t most of all, with their food! In particular. I would like to acknowledge A lastair Moore from Rolls Royce Pic. and Alain D ravet from SNECMA Moteurs.

My workm ates in the lab. have made my workplace both ])leasurable and inspirational. My post­ graduate and post-doctorate colleagues from the Sound and V ibration Laboratory: Orla, Damien, Ludo, Franck. Bjorn, P etr, David, Stephen, John. Eoin and M eaghan are exemplary engineers, pos­ sessing a th irst for knowledge and an enthusiasm to contribute, th a t has led to an environm ent of

learning true to the ideal of any University.

To my sister Edel. and my m other, Amy, I w'ould like to express my deepest gratitude. For their support in all m atters through the years, and endless love, I could never thank them enough. Happiness, health and long life to us all. My new and extended family. Dr. Sean D evitt and his wife

Ann, Seamus, Daniela, M arie and Jim, have all helped and supported me in this work. I count myself tru ly fortunate to have been welcomed into their lives and hearts. As for my two darling little nieces Sofia and Rebecca, they pu t things in perspective. I ’ll give them each a copy of my thesis when they tu rn eighteen!

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C ontents

N o m e n c la t u r e v iii

1 I n t r o d u c t io n 1

1.1 B a c k g r o u n d ... 1

1.2 O bjectives of I ' h e s i s ... 2

1.3 Thesis O utline ... 4

2 P r e s s u r e F ie ld In A C y lin d r ic a l D u c t 5 2.1 D uct A coustic M o d e s ... 5

2.2 P r o p a g a tio n ... 9

2.3 R a d i a t i o n ... 13

2.4 P lan e Wave D uct A c o u s t i c s ... 15

2.4.1 P lan e Wave D e c o m p o s itio n ... 16

2.4.2 Reflection C o e ffic ie n t... 17

2.4.3 A coustic I m p e d a n c e ... 19

2.4.4 Transm ission L o s s ... 20

2.5 C ore Noise G eneration M e c h a n is m s ... 21

2.5.1 R o to r O n l y ... 21

2.5.2 R o to r-S ta to r I n t e r a c t i o n ... 23

2.5.3 C o m b u s tio n ... 24

2.5.4 Inflow D isto rtio n -R o to r I n t e r a c t i o n ... 26

2.6 C o n trib u tio n of T h e s i s ... 26

3 E x p e r im e n ta l R ig 29 3.1 I n tr o d u c tio n ... 29

3.2 Anechoic T e r m i n a t i o n ... 29

3.2.1 H orn T h e o r y ... 30

3.3 Vaiieaxial F a n ... 38

3.4 In stru m e n ta tio n an d A n a l y s i s ... 39

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I

C O N TE N TS

CONTENTS

3.4.1 Signal P ro c e ssin g ... 40

4 P r e lim in a r y E x p e r im e n t a l R e s u lt s 41 4.1 Introduction and O b je c tiv e s ... 41

4.2 Microphone C a l i b r a t i o n ... 42

4.3 Closed Ended D u c t ... 43

4.3.1 Theory ... 43

4.3.2 Results ... 46

4.4 Open Ended Duct ... 48

4.4.1 Theory ... 48

4.4.2 Results ... 50

4.5 Fan Installed (Not R otating) ... 52

4.6 Fan Installed ( R o t a ti n g ) ... 54

5 S o u r c e I d e n tif ic a tio n 71 5.1 In tro d u c tio n ... 71

5.2 Identification T echniques... 71

5.2.1 Coherent O u tp u t S p e c t r u m ... 72

5.2.2 Signal E n h a n c e m e n t... 73

5.2.3 Conditional Spectral A n a ly s is ... 74

5.2.4 System of Non-Linear E c ju a tio tis ... 74

5.2.5 The Two Test Procedure ... 75

5.3 Experim ental Setu]) ... 75

5.4 Exi)erimental R e s u lts ... 76

5.5 Evaluation of Techniques ... 77

5.6 Linear Noise Source S e p a r a tio n ... 78

5.6.1 Conditional Spectral Analysis Applied to Experim ental R i g ... 78

5.6.2 Conditioned Frequency Response Functions Across The F a n ... 84

6 N o n -L in e a r A n a ly s is 97 6.1 In tro d u c tio n ... 97

6.2 Non-Linear Sinmlations ... 99

6.3 Tim e Domain A v e ra g in g ... 105

6.4 Investigation above plane wave r e g i o n ...106

6.4.1 Tyler and Sofrin A n a ly s i s ... 108

7 A c o u s t ic M o d a l D e c o m p o s it io n In a D u c t 113 7.1 In tro d u c tio n ... 113

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CONTENTS

CONTENTS

7.3 Experim ental R e s u lts ... 121

7.3.1 Test S e t-u p s ... 121

7.3.2 Decomposition R e s u lts ...123

7.4 D isc u ssio n ...127

8 C o n c lu s io n s 1 3 7 8.1 Future Work ... 138

A F an S p e c ific a t io n s 145 B S ig n a l P r o c e s s in g 1 4 7 B .l Ergodic Random D a t a ...147

B.2 Mean V a l u e ...148

B.3 V a ria n c e ... 148

B.4 Mean Square V^alue... 148

B.5 Fourier Transform ... 149

B.6 Power Spectrum and Power Spectral D e n s i t y ... 150

B.7 The PSD from Digital D ata A n a ly s is ...151

B.8 M atlab and the P S D ... 152

B.9 The dB S ])ectru m ...152

B.IO M atlab Code to Calculate the P S D ...153

B .ll The Cross Spectral Density F u n c tio n ...154

B.12 The Frequency Respf)nse F u n c t i o n ...154

B .l3 The O rdinary Coherence F u n c t i o n ...155

B.14 Phase Angle and Convection S p e e d ...156

C M ic r o p h o n e C a lib r a tio n 1 5 7

D F E A A n a ly s is o f D u c t 159

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N om enclature

r, e . X P o la r co -ordinates of duct p { r , 0 , x , t ) A coustic pressure in duct

V { x , y , z ) M ean flow velocity vector: iu{x, y, z) + j v { x , y, z) + kw{ x , y, z u C o n sta n t m ean flow velocity iti x direction

P a rticle velocity vector

k W avenuniber also integer index

k x Axial w avenum ber

k r R ad ial w avenum ber

771 A zim uthal or circum ferential m ode index.

n R adial m ode index.

Bessel function o f th e first kind of order m

w D erivative of Bessel fm iction of th e first kind of o rd er m ^ m , n ( r , 0) E igenfunction of (m , m ode

A

■^7Ti,n A m p litu d e of m ode i^ in .n C ut-off ra tio for (rn, m ode

c^ ' T j i . n N orm alisatio n c o n sta n t of (m , m ode

“ m . n W A m p litu d e of incident rad ial m ode ^ m . n (■^) A m p litu d e of reflected (m , rad ial m ode ^ V , 7 n , 7 ( T ransverse eigenvalue of th e (m , m ode

a R adius of d u ct

A A dded length correction

a L ength end correction

S C ross-sectional a re a

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C O N T E N T S C O N T E N T S

M Mach Number

V(-) G radient operator: ^^§7 ^ ^ § 7 OGV O utlet Guide Vane

IGV Inlet Guide Vane FEA Finite Element Analy.-.is

I Incident Reflected ^ U pstream •)rf Downstream ■)ui U pstream Incident •)* Downstream Incident ■)t T ransm itted

■)+ Incident •)“ Reflected

R Reflection Coeflici('nt

Ro Reflection Coeflicic'nt at x 0

Po Density of air a t afmosptieric conditions c Speed of sound

Za Acoustic impedance

Zc C haracteristic acoustic impedance ui C ircular frequency

TL Transmission Loss W Sound Power

A Cross-sectional area of duct

Gxx Averaged single-sided power spectral density of signal x(t) A / Frequency resolution bandwidth

PSD Power Spectral Density

li Harmonic index and horn shape factor B R otor blade mnnber

V S tato r Vane number 17 R otational speed [radians/s]

RPM R otational Speed [Revolutions Per Minute] RPS R otational Speed [Revolutions Per Second] U Spatially dependent volume flow’rate.

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C hapter 1

Introd u ction

1.1

B ackground

Growth ill air traffic and the desire to locate airports closer to built up areas has resulted in aircraft

pollution, by way of enii.ssions and noise, becoming an increasing environmental problem. The Euro­

pean Research Commission has responded to this pollution issue, through the Advisory Council for

Aeronautics Research in Europe (ACARE), by including the “Challenge to the Environment" as one

of the five objectives in its Strategic Research Agenda (SRA). The environmental goals of the SRA's

20 year agenda, “Vision 2020’’. are very challenging:

• Reduce the perceived noise by half

• Reduce the emitted

CO

2

by 50 %

• Reduce the emitted

NOx

by 80%

In the Fifth Framework Programme, two large-scale projects in the field of the environment:

EEFAE and SILENCE(R) have been funded. Whilst the EEFAE project, “Efficient and Environ­

mentally Friendly Aircraft Engine” , is principally concerned with emissions reduction, SILENCE(R),

(Significantly Lower Community Exposure to Aircraft. Noise) is concerned mostly with environmental

noise. It is within the SILENCE(R) project th at aspects of this research are based.

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1.2. Ob j e c t i v e s o f Th e s i s In t r o d u c t i o n

T h e aero-engines of civil aircraft are th e dom inant noise sources for m ost fliglit conditions although airfram e noise is a significant c o n trib u to r at landing. The two largest sources of aero-engine noise, th e fan and th e je t, have been significantly reduced as a consequence of years o f research. W ith tlieir red u ctio n , a th resh o ld is being reached, which will form th e new noise floor an d lim it the benefits to be g ained by reducing tliese d om inant com])onents, unless th e noise sources w hich set this th resh o ld are in tu r n reduced. O f these, com bustion, or core noise, is c u rren tly a focus of research activity. A t relativ ely low je t velocities, such as would occur a t engine idle, d u rin g taxiing, an d a t a p p ro ach an d cruise conditions, core noise is considered a significant co n trib u to r to th e overall sound level. T he tre n d of core noise a tta in in g increasing a tten tio n wall continue w ith th e in co rp o ratio n of low N O x co m b u sto rs an d increasingly higlier by-pass ratios into m odern engine designs. A b e tte r u n d e rsta n d in g of com bustion noise generation, propagation and ra d ia tio n w'ill aid in th e design of noise suppression devices to alleviate com m unity noise problem s.

1.2

O b je ctiv e s o f T h esis

U n d e rstan d in g com bustion noise source m echanism s, designing efficient acoustic liners an d optim ising active control alg o rith m s for noise red u ctio n requires th e identification of th e frequency and modal co n ten t of th e com bustion noise co n trib u tio n . An acoustic m easurem ent of a system of in terest will m ost often be th e su n m iatio n of a num ljer of sep arate acotistic sources along w ith som e extraneous noise. F igure 1.1 illu stra te s th e principle noise sources in an aero-engine. For th e case w here it is not possible to rem ove individual sources w ith o u t effecting the behaviour of th e o th e rs, th e challenge is to decom pose th e m easu rem en t signal into its co n stitu en t ])arts. For acoustic sources th a t are consid­ ered to be s ta tio n a ry ran d o m processes w ith zero m ean and w here system s are c o n sta n t-p a ra m ete r linear system s, figure 1.2, a m u ltip le -in p u t/sin g le-o iitp u t m odel, can be used to rep resen t th e sys­ tem . T he e x tra n e o u s noise term , n{t). accom m odates all deviations from th e m odel, such as acoustic sources g re a te r th a n M w hich are unaccounted for. n o n -statio n ary effects, acquisition, in stru m en t and m ath e m a tic a l noise along w ith u n ste a d y press\ne fluctuations local to th e sensor, such as flow or h y d ro d y n am ic noise. As th e acoustic noise sources in an aero-engine overlap in th e frequency dom ain, w ith vary in g am p litu d es, as show n in figure 1.3. it can be difficult to q u an tify th e individual co n trib u tio n s. O ne of th e p rin cip al objectives of this thesis is to develop alg o rith m s to accurately identify th e com bustion noise co n trib \itio n to a far-field acoustic m easurem ent.

C oherence-based identification techniques can be used to identify th e co n trib u tio n of com bus­ tion noise to n e a r and far field acoustic m easurem ents of aero-engines. T hese techniques condition from ex te rn a l m easu rem en ts ad d itio n a l core noise sources, as well as o th er noise sources generated ex terio r to th e engine. T h e u n d erly in g assum ption w ith these identification techniques is th a t th e p ro p a g a tio n \c o n v e ctio n p a th , from com bustion can to m easurem ent point, is a linear one. R esults from the E u ro p e a n c o n tra c t Resound, showed a drop in coherence betw een com bustion noise

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In t r o d u c t i o n 1.2. O b j e c t i v e s o f T h e s i s

t onii)rcNw»r Kail iiiU't noKo

C o i ii h u s t i o ii n o is e K a n i‘\ h i i u v t n o K e

F k j L ' RE 1.1: Principal aero-engine acoustic noise sources. Length of vector representative of average power level of individual source w ith its angle indicating the directivity of maxiniuni lobe.

nil) >■2

F i g u r e 1.2: M ultiple Source Acoustic Mea­ surem ent

SPL (dB)

Jet Com bustion

Fan

Turbine

FREQUENCY (Hz)

F i g u r e 1.3: Q ualitative spectra indicating frecjuency content of individual aero-engine noise sources.

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1.3. T h e s i s O u t l i n e In t r o d u c t i o n

Com bustor + Turbine + Extr<mcous ^ C om bustor -f Extnuicous

4- N o n -U n ea r Interactions

( (Miipresvor F a n InUi n u i\c

C o m h i i s t i o n n o i s e K in exhiiUNt noKe

F i g u r e 1.4: Schem atic of aero-engine d epicting m odification of com bustion noise due to convection th ro u g h tu rb in e.

1.3

T h esis O u tlin e

C h a p t e r 2 sum m arises th e th eo ry of acoustic m ode p ro p ag atio n in a d u c t an(i some core noise source g en eratio n m echanism s. C h a p t e r 3 d etails th e design of a n experim ental rig built to facilitate the realisatio n of th e thesis objectives. T h e p relim in ary experim ents of c h a p t e r 4 serve to v alid ate th e te s t eq u ip m en t an d processing techniques as well as to characterise; this principal te s t rig u nder different te s t conditions. C h a p t e r 5 m odifies and evahiates existing acoustic source identification techniques w ith ex p erim en tal d a ta from a se p a ra te te st rig designed for th e purpose. T hese techniques are th e n applied to ex p erim en tal d a ta from tlie principal rig in o rd er to sep arate fan an d flow noise from a flu ctu atin g pressu re signal convected th ro u g h th e fan. T he non-linear interaction, betw een th e flu ctu atin g pressure signal an d th e fan, is exam ined in c h a p t e r 6. T h e m odal content o f th e n B P F ’s an d th e in te ra c tio n tones are d eterm in ed w ith an acoustic m odal decom positiopn technique described in c h a p t e r 7. T h e final c h ap ter sum m arises th e findings of th is thesis and identifies a num ber of possible fu tu re d irections for th is research.

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C h a p ter 2

P ressu re F ield In A C ylindrical

D u ct

T he u su al ap p ro ach in th e analysis of d u c t acoustics is to ap p ro x im ate th e d u c t by an infinite cylinder and to solve th e differential equations by se p a ra tio n of variables. T h is leads to an eigenvalue problem , the so lu tio n of w hich give th e d u c t p ro p ag atio n m odes. E ach m ode rep resen ts a different way in wdiich sound m ay travel dow n th e duct. A com plete description of th e sound field in th e d u c t consists of know ing th e com plex am p litu d e of each m ode.

2.1

D u c t A c o u stic M od es

For acoustic pro])agation in an infinite h ard w alled cylindrical d u ct w ith sujjerim posed co n sta n t m ean flov velocity 1^. th e pressure, p = p { r , 0 , x , t ) , in cylindrical co o rd in ates as defined by figure 2.1, is found as a solution of th e hom ogeneous convective wave equation,

Df 2 dx'^ r d r \ d r ) dd'^

where th e su b sta n tiv e derivative is defined to be

D ^

Df d t d x

This solution is found as a com bination of th e c h aracteristic functions of eq u atio n (2.1) each of which satisfy s])ecific b o u n d a ry conditions. T h e sohition to (2.1) w ith m ean flow can be shown to be equal to th a t of th e no-flow condition w ith a m odification m ade to th e axial w avenum ber. To sim plify the deiivation, th e no-flow case will be presented here w ith th e m ean flow form of th e axial w avenum ber introduced in eciuations (2.21) and (2.22).

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2.1. D u c t A c o u s t i c M o d e s Pr e s s u r k Fi e l d In A Cy l i n d r i c a l Du c t

z

F i g u r e 2.1: P o lar co o rd in ate system for a cylindrical d u ct, { r , 0, x)

th e ra d ial com ponent of th e ])ressure g ra d ie n t vanishes a t th e b o u n d a ry walls: | ^ = 0 a t r = a:

a t b o th ends of an arc 2n r long, located anyw here in th e a z im u th a l direction, th e pressure and

pressu re g rad ien t m ust be th e sam e;

• a t a reference plane, x = 0, norm al to th e cylinder axis, th e pressure is s]5ecified as an a rb itra ry

function of r, satisfying th e first tw'o conditions and a periodic function of tim e;

T h e c h a ra c teristic functions are o b tain ed as solutions to th e o rd in a ry differential eq u atio n s into which

th e wave eq u atio n sep a ra te s on su b stitu tio n of th e form p = R { r ) B { 0 ) X { x ) T ( t ) . A ssm ning th e tim e

d ep en d en t term to be harm onic, th e sp atially d ep en d en t eq u atio n s can be shown to be equal to,

^2 Y

— = - k l X (2.2)

Z X ^

^ (2.,3)

r ^ ^ + r ^ + { { k ^ - k i y - m ^ ) R = Q (2.4)

dr-^ dr

w here eq u a tio n (2.4) is a Bessel eq u atio n for R( r) . T h e c h a ra c teristic functions, in com plex form ,

aj)plying th e b o u n d a ry conditions are

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Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t 2.1. D u c t A c o u s t i c M o d e s

X = (2.5)

© = (2.6)

R = J m i K r ) (2.7)

T = (2.8)

w ith

= k l + k l (2.9)

As th e ra d ial velocity a t th e wall (/' = a) m u st be equal to zero, k,- takes only such values as to satisfy th e eq u atio n

j ; „ ( f c , a ) = 0 (2.10)

w here J„,(-) is a Bessel fim ction of th e first kind of o rd er m , J'„ (-) is its derivative, an d is related to th e Bessel function by

4 , ( • ) = - ■ / « ( • ) m = 0 (2.11)

Jf , V J w - l ( ' ) ~ J m - l ( ' ) I „ .r,

,/„,(•) = --- rn = l , 2 , . . . (2.12)

Figure 2.2 shows jilots of th e Bessel function an d of its derivative for a num ber of m orders. A ccording to ecjuation (2.10), kr can be calcu lated from th e ro o ts (or zero crossings) of th is derivative. For a given Bessel function derivative of order m , an infinite n u m b er of ro o ts exist and from th is it will be shown th a t for each circum ferential m ode of o rd er rn th e re are an infinite num ber of asso ciated radial mo(ies.

D enoting th e value of A> corresponding to th e ro o t of th is eq u atio n as /c(,. th e general solution to eq u atio n (2.1), w ith or w ith o u t m ean flow, can b e expressed as a linear com bination of eigenfunctions.

/j(r, 0, X, t) = R e ^ ^ / l „ , , „ ( x ) ' I ' „ , , „ ( 7 - , (2.13)

_ m = O n = l

w here th e eigenfunctions, 0), of am p litu d e A„,,^n{x) will d epend uniquely on th e cross-sectional shape of th e d u ct. For th e case of a h ard walled cylindrical d u c t, th e eigenfunction is

' ^ r n A r , 0 ) = Jrn{kr,rn,nr)e^”^'^ (2.14)

[image:23.529.22.515.23.481.2]
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2.1. D u c t A c o u s t i c M o d e s Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t

0.5

-0 5

25

0 5 10 15 20

z

0.5

0.5

■f ’..I / -1 . .... 4 .... ...y . -. 4--- ,.. ..

0 5 10 15 20 25

z

Fi g u r e 2.2: T he first five o rd ers of th e Bessel function of th e first kind an d its derivative.

p e rp en d icu lar to th e x-axis. an d w hich m ay p ro p a g a te as a travelling wave u p stre a m or dow nstream in th e d u c t in accordance w ith th e x d ep e n d e n t am p litu d e. T h e eigenvalues o f this eq u atio n provide frequencies above which g en erated m odes p ro p a g a te u n a tte n u a te d b u t below w'hich excited modes ex p o n en tially decay. Several m odes m ay coexist in th e d u c t a t a frequency of ex citatio n , so long as th is frequency is above th e ir ind iv id u al cut-off frequencies. T he pressu re in th e d u c t is assum ed to flu ctu ate harm onically as can be seen from th e exponential tim e term , kr.,n.n is’ th e transverse eigenvalue of th e m ode a n d is also called th e tran sv erse, com bined rad ial-circu m feren tial or sim ply th e ra d ia l w avenum ber.

In o rd er to rem ain co n sisten t w ith th e n o ta tio n for re c ta n g u la r d u cts, w here rn an d n rep resen t the num ber of nodes in th e tra n sv erse plane, th e n index for a circular d u c t is used n o t in fact to indicate th e ro o t of eq u atio n (2.10) b u t ra th e r th e num ber of azim u th al (or circum ferential) pressure nodes in th e tran sv erse plane. T h is re su lts in th e p lane w'ave m ode being d en o ted as ( m , n ) = (0,0)

as discussed in M unjal [48] an d E riksson [19]. T able 2.1 presents som e of th e zero crossings for the first six circum ferential m ode orders. T h e eigenvalue or radial w avenum ber. kr,m,n, for a circular cross-section is d ep en d en t on th e ra d iu s of th e d u ct, w here these values have been calcu lated using th e fzero function in M a tla b ’s O p tim iz a tio n toolbox.

G iven th e values in tab le 2.1, th e fun ctio n Jm{kr.m.ur) is p lo tte d to illu stra te som e ra d ia l m ode shapes. A value of a= 0 .0 2 5 m is used, as th is is the rad iu s of th e rig to be used in th e ex p erim en ts as

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Pr e s s u r p; Fi e l d In A Cy l i n d r i c a l Du c t 2.2. P r o p a g a t i o n

ni, n 0 1 2 3 4 5

0 0 3.83 7.02 10.17 13.32 16.47

1 1.84 5.33 8.54 11.71 14.86 18.02

2 3.05 6.71 9.97 13.17 16.35 19.51

3 4.20 8.02 11.35 14.59 17.79 20.97

4 5.32 9.28 12.68 15.96 19.20 22.40

5 6.42 10.52 13.99 17.31 20.58 23.80

Ta b l e 2.1:

the Bessel function.

Full two-dimensional mode shapes, including the azinm thal variation, are plotted for some exam­

ples in figures 2.4, 2.5, 2.6 and 2.7. According to equation (2.14), it can be seen th a t the normal

modes are siimsoids in the circum ferential direction and Bessel functions in the radial direction where

ni specifies the circum ferential mode number and n indicates the associated radial mode number. The ({),()) mode indicates the plane wave mode, (1,0) the first circumferential (or azim uthal) mode and

(0.1) the first radial mode. Physically, the circumferential mode number indicates the periodicity of

the circumferential acoustic mode j^attern. For example, a mode of rn = 4 is periodic over | of the

duct. Alternatively, it may l)e observed th a t a mode of m = 4 has four nodal lines passing through the centre of the circular section. Although the radial modes are not })eriodic along the radius of the

duct, the radial mode num ber indicates the number of circinnferential node lines to be found in the

acoustic ])attern.

Ecjuation (2.13) may now be re-w ritten upon substittition of ecjuation (2.14) as

p(r, 0, X , t) = R e

+ 00 -foo

E E

^ 7 n , n r,rn,7 m = 0 71= 0

+ O C + C O

(2.15)

p{r,e,X,U>) = ^ Y^A,n^„(x,U))Jrn{kr.m,nr)e^’^^^ (2-16) m = 0 n = 0

in the frequency domain, where p{r,9, x.ui), the complex pressure is a solution of the Helmholtz ecjuation. and where A,nji{x,u>) is a complex modal m agnitude. It should be noted here th a t the sum over n is from zero to infinity.

2.2

P r o p a g a tio n

Another conmion way of expressing equation (2.16), as used by E nghardt et al [18] and Tapken et al [62]), and one where the functional form of /4„,,„,(j:,a;) is expressed explicitly, is where the comjjlex

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2.2. P r o p a g a t i o n P r e s s u r e F i e l d In A C y l i n d r i c a l D u c t

+ CX; ;3(r, 6»,.x,ai) = ^

m = 0

0 .5

-0.5

0 0.005 0.01 0.015 0.02 0.025

0.5

-0.5

0 0 .0 0 5 0.01 0.015 0.02 0.0 2 5

Radius [mj Radius (m]

0.5 0 .5

-0.5

-0.5

0 0 .0 0 5 0.01 0 .0 1 5 0 .0 2 0 .0 2 5 0 0 .0 0 5 0.01 0.015 0 .02 0.025

Radius [m]__________________________ Radius (m)

■‘2^2.2'^ I - ' 3 ( ^ 3 / * - • 's f V s / l

-F i g u r e 2.3: T h e first four rad ial luode sh ap es for Bessel functions of order 0 ^ 3. J m { k r , m . p lo tte d for a = 0.025m . T h e centerline of the d u c t is a t r = 0.

Mod« (0.0)

Mo4«(1.0j

F i g u r e 2.4: A coustic M ode Shapes; (0,0) a n d (1,0)

(2 .1 7 )

.nr) is

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Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t 2.2. P r o p a g a t i o n

I f

•0 02

-0 03

D.02S

0.02

0 015

0.01

0 005

•0 015

-0 02

-0 02S

-0.03 -0 02 -0.01 0 0 01 0.02 0.03

F i g u r e 2.5: Acoustic Mode Shapes; (2 ,0 ) and (0,1)

•0 01

-002

-0 03

i

•0 04 -0 03 -0 02 -0 01 0 0 01 0 02 0 03 0 04

0 03

0 0 2

-0 01

•0.02

-0 03

-0 04 -0 03 ^0 02 •OOl 0 0 01 0 02 0 03 0 04

F i g u r e 2.6: Acoustic Mode Shapes; (3 ,0 ) and (4,0)

where the aniphtude of the circumferential mode of order rn separates out into a series of radial modes

as

-f-cx;

A„,{x,r,u}) = ^ j ^ a + Jm{krjn.nr) (2.18)

r i = 0

Ecjuation (2.16) can thus also be w ritten as

f c x : -+-OC

m = 0 7 1 = 0

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2.2. P r o p a g a t i o n Pr e s s u r f; Fi k l d In A Cy l i n d r i c a l Du c t

F i g u r e 2.7: Acoustic Mode Shapes; (1,1)

where

(2.2 0)

E quation (2.18) inchides incident and refl('ct('d travelhng acoustic waves witii „(w) being the

m agnitude of the forward propagating mode and ^,(w) the m agnitude of the rearw ard propagating

mode. T he axial propagation characteristic of the (m, n) mode is defined by the axial wavenumber.

kx.m.n com puted from

—__

'^ x .T n .,n

p2

'0^

'1

-3 t

T, i T) .n,

- M

l l - 1 + M

(2.2 1)

(2.2 2)

where /3 = \ / l — A/^ and M is the Mach number of the mean flow inside the duct.

Focusing on one of the exponential term s containing the axial W'aveninnber for the incident wave,

e.g. it can be deduced th a t, if the waveimniber is real, th en the term will be oscillatory

and will propagate. If the term is imaginary, then the wave will exponentially decay. By defining a

cut-off ratio for a particular {m, n) mode as

0 m , nk

0 k r ,r .

(2.23)

it is evident th a t (3m,n < 1 results in a decaying mode whereas > 1 will result in a propagating

mode. T he transitional point between these two conditions, where Pm,n = 1, is used to define the

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Prf:s s u r e Fi e l d In A Cy l i n d r i c a l Du c t 2.3. R a d i a t i o n

cut-off frequency for a particular (m, n) mode, — = 1 can be re-w ritten as

- t - o f f = ^ x / l - M 2 (2 .24)

In snininary, if a mode is excited a t a frequency which is any frequency above its cut-off frequency

then the mode is said to l^e cut-on, and will thus propagate down a duct. For frequencies below, the

mode is cut off and will exponentially decay, the ra te of decay increasing as the cut-off ratio decreases.

For zero m ean flow, the axial wavenumber is more simply related to the free stream waveimmber,

k = ^ , and the transverse wavenurnber by

k x , m . n = „ (2.25)

Etjuation (2.24) dem onstrates how the cut-off frequency is lowered with increasing flow speed as

Morfey [46] observed experimentally.

.Although the transverse wavenumber is fixed for a particular mode due to the boundary conditions,

as the frequency lo may vary, so then may the axial wavenumber. The axial propagation velocity (or

phase velocity) for a p articular mode is given by

U)

^x.7ii,n T (2.26)

,n

and its wavelength is simply

27T

(2.27)

However, ju st above cut-on. the wavelength is appreciably greater than would be the case in free-held

and the phase velocity in the j:-direction is correspondingly high, as required to satisfy the relation

f = j = c o n s t . As the frequency increases above cut-off, X x , m . n and C x.m .n rapidly approach

their f'ree-space values. This behaviour m ay be seen in figure (2.8), where the expressions may be

derived from (2.25) as

A x , m . n = ( -

V/"

- f h n . n ^ ^ ^

771.71 f ^ X ,n i ,7 l (2.29)

with, for this example, k r ,,n ,n = where A Q O O H z is approxim ately the ])lane wave cut-off

frequency for the duct diam eter to be used in the experim ents described in this thesis in chapter 3.

2.3

R a d ia tio n

If a mode has been generated\excited at a frequency above its cut-on frequency, it will propagate

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2.3.Ra d i a t i o n Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t

0.6 0.4 0.2

4000 5000 6000 7000 8000 9000 10000

Frequency [Hz]

1000

800 600 400 200

7000 8000 9000

4000 5000 6000 10000

Frequency [Hz]

F i g u r e 2 .8 : V a ria tio n o f a x ia l w a v e le n g th a n d p h a s e v e lo c ity w ith fre q u e n c y c o m p a re d w ith t h a t in free-field. f c u t - o n = i k H z for th is e x a m p le .

o f th e d u c t. If th e d u c t is o p e n , th e d u c t face will be s u b je c t to a f lu c tu a tin g p re s s u re field w hich m a y r a d ia te to th e far field. S om e o f th e e a r lie s t w o rk c a r rie d o u t o n s o u n d r a d ia tio n fro m c y lin d ric a l d u c ts w as by L ev in e a n d S ch w in g er [43] u sin g th e W ien e r-H o ])f te c h n iq u e . T h e s u b je c t o f la te r w ork b y H om icz a n d L o rd i [28], L a n s in g [42], a n d R ice [51] e f al w as th e d e te r m in a tio n o f th e d ire c tio n s of p e a k r a d ia tio n for d u c t a c o u s tic m o d e s. F or a p a r tic u la r m o d e (m , n ), th e an g le for th e p e a k r a d ia te d lo b e c a n b e e x p re sse d as

^osieZll) =

V i

-1- ^

(2.30) 1 M 2 ( l

-, n

w h e re e q u a tio n (2.23) defin es th e c u t-o ff r a tio P m . n

-F ro m e q u a tio n (2.3 0 ), tw o m a in o b s e rv a tio n s m a y b e m a d e . T h e first is t h a t , a s th e M a ch n u m b e r in c re a se s, th e a n g le t h a t th e m a in lobe m a k e s w ith th e d u c t ax is d e c re a se s. T h e se c o n d is t h a t for a p a r tic u la r flow velo city , th e m o d e s t h a t a re w ell ab o v e th e ir c u t-o ff fre q u e n c y r a d ia te to w a r d s th e axis o f th e d u c t, w h e re a s, a s th e fre q u e n c y d e c re a s e s to w a rd s th e c u t-o ff freqiiency, th e m o d e s r a d ia te a t w id e r far field an g les. E a c h m o d e w ill p ro d u c e a r a d ia tio n p a t te r n , th e so u n d p re s s u r e level d ir e c tiv ity b e in g a s u m m a tio n o f th e in d iv id u a l p a tte r n s .

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P r e s s u r e F i e l d In A C y l i n d r i c a l D u c t 2.4. P l a n e W a v e D u c t A c o u s t i c s

2.4

P la n e W ave D u c t A c o u stic s

A duct scenario of great practical im portance is one where the frequencies considered he below the

first higher order mode cut-off frequency which, using equation (2.24), is given by

f c u t - o f f = 2 T i a

where a is the radius of the duct, M = 0 and the value 1.84 comes from table 2.1.

For the plane wave mode, or (m, n) = (0,0), equation (2.19) reduces to

p = Pie + pre^^'-'-^ (2.32)

as both J„,{krjn.nT) and reduce to unity and the sum m ations over rn and n disappear. The

complex am plitudes of the forward and backward travelling plane waves are defined here to be pi and

Pr- Equation (2.32) is a plane wave solution to the Helmholtz equation. Equation (2.32) could have

been derived directly from analysis of the one dimensional wave equation

d^p

2

- c ^ ^ = 0 (2.33)

ox

where it can be shown th a t any function of the form

p{x, t) = (j{ct ± x) (2.34)

is a solution. These can be shown to be waves travelling steadily along with speed c parallel to the

X axis, m aintaining a constant ])rofile, and, since there is no dependence on y or z, the wavefronts

are all planes parallel to the yz plane. These wave fronts are known as plane waves. For the x-axis

taken to be positive to the right, tlie plus sign in equation (2.34) nmst correspond to waves travelling

towards the left and the negative sign to waves travelling towards the right. Figure 2.9 (a), taken

from Hall [24], shows a rightw ard travelling sinusoidal wave.

Since equation (2.33) is a linear differential equation, any sum of two solutions is also a solution.

Thus in general the duct could sujiport waves g{ct + x) and g{ct — x) at the same time. An im portant

physical reason for such a situation could be the presence of waves reflected from the end of a duct.

For a duct of finite length L, describing the pressure distribution for the whole duct in term s of

travelling waves remains possible bu t becomes unwieldy because of repeated reflections between the

two ends. It is therefore more convenient to study standing waves. The example of two sinusoidal

waves w ith equal am plitudes moving in oj)i)osite directions

p{x, t) = A sin k{ct + x) — /4 sin k{ct — x)

can be used to illustrate the case. This expression has two im portant properties illustrated in figure

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2.4. P l a n e W a v e D u c t A c o u s t i c s Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t

(a)

(b)

F i g u r e 2.9: Sinusoidal travelling wave (a) contrasted w ith standing wave (b).

are some values of x where p remains zero at all times; such points are called nodes of the standing

w'ave. Secondly, there are some values of t for which p is zero sim ultaneously at all locations.

Consider now the duct in figure 2.10 where acoustic waves em anating from the speaker travel to

the right in the direction of the positive x-axis. These waves are know'u as incident waves and in

complex notation are given by The left running waves reflected from an obstruction are

denoted by . The different wave nvnnbers result from the Doppler shift caused by the

mean flow velocity u. if there is one. W ith u > 0 in the positiv'e x direction.

k oj k

c + » 1 + A/ ^2.35)

u k

’■ " ^ = i ^ i

which are equivalent to ec}uations (2.21) and (2.22) for plane waves.

^ . the particle velocity, comes from the m om entum equation (Euler equation);

P o - ^ = - V p (2.36)

and is given by

dp _ I

^

jw p o d x p o c

2 .4 .1

P la n e W ave D e c o m p o s itio n

As dem onstrated above, the pressure in a duct may consist of an incident plus a reflected wave

(depending on the end condition). W hilst it is the sum m ation of these two th a t a single microphone

located in the duct would measure, with inform ation from a second microphone, located at a different

axial location, the i)ressnre field may be decomposed into the incident and reflected components. As

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P r e s s u r e F i e l d I n A CY LtNDR iC AL D u c t 2.4. P l a n e W a v e D u c t A c o u s t i c s

the waves are j)laiiar, the microphones can be located anywhere on a plane of location x. Therefore, it

is preferable to tnonnt the microphones flush with the inside surface of such a duct to reduce scattering

from the microphones themselves, W ith regard to figure 2.10, the complex pressures measured by

microphones 1 and 2 are

(2.38)

Algebraic m anipulation of these equations gives the complex am plitudes of the incident and reflected

travelling waves

Pi e — P 2 e

- j [ k i + k r ) x i _ g - j ( f c i + f c r ) X 2 (2.39)

and

P j g j f c i S l _ p.^gjkiX2

(2.10)

g i ( k i + k r ) x i _ g j ( k i + k r ) X 2

It is im portant to note th a t the position of the reference plane w'ill determ ine the signs of and x2.

I

p. Pi x+

H_D___________

I End Condition

1 -X2

-XI

Tube Length

Reference Plane

Fi g u r e 2.10: Schematic of Test Rig

2 .4.2

R e fle ctio n Coefficient

Acoustic behaviour a t a term ination is often descriljed in term s of the reflection coefficient /?, defined

as the ratio of the reflected wave i)ressure to th a t of the incident wave

R = \R\e^^

where |/?[ and 0 are, respectively, the m agnitude and phase of the reflection coefficient. Now, with

reference to equation (2,32), if we divide the second term on the rh.s by we get

R = (2.41)

Pi

According to equation (2.41), we see th at for this case of plane wave propagation and neglecting losses

a t the duct wall, the m agnitude of the reflection coefficient is equal at all planes along the duct, A

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2.4. P l a n e W a v e D u c t A c o u s t i c s Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t

For X = 0, th e reflection coefficient R is therefore sim ply

Ro = - (2.42)

P t

C learly th e reflection coefficient m ay be calculated d irectly from eq u atio n 2.41 using eq u atio n s 2.39

and 2.40 w ith som e averaging to im prove th e signal to noise ratio . However, th e re exist num erous

p u b licatio n s describing a v ariety of techniques for th e ro b u st calculation of th e reflection coefficient.

O ne of th e m ore com prehensive p ap ers on two m icrophone ran d o m e x citatio n m eth o d s is th a t of

C hung an d B laser [12], w hich has been sta n d ard ised in th e S ta n d a rd , BS EN ISO 10534-2:2001

[2]. However th e following form ulation, which can be show n to be equivalent to th a t of C hung and

B laser [12], is q u ite an elegant one, w hich em ploys th e frequency response function betw een th e

two m icrophones, an average of w hich can usually be d eterm in ed reliably. U sing eq u atio n (2.32), a

frequency response function betw een th e two m icrophones a t + x l and + x 2 can be deflned as,

pi(w)

'

By dividing above an d below by pi and dropping th e u> n o ta tio n , th e freciuency response function

m ay be expressed in term s of th e reflection coefficient,

H^2 = — 1 - (2.44)

which can be re-arran g ed to present th e reflection coefficient in te rm s of th e frequency response

function, i.e.

p - j k i X - 2

_

ft,, = — --- (2.45)

D r o p O u t F r e q u e n c y

S eybert and Ross [55] and C hung and B laser [13] have b o th exam ined how th e reflection coefficient

carm ot be d eterm in ed a t d iscrete frequency points for which the m icrophone spacing is an integer

m ultiple of th e h a lf w avelength of sound, i.e. located a t nodes, or w henever

s = n ( ^ ) , n = 1,2, 3 . . .

o r (2.46)

s tl[ ^ ), Ti 1 , 2 , 3 . . .

As these frequency p o in ts can be p red icted , th e discontinuities can be ignored. A lternatively, to avoid

these p o in ts up to a frequency f c r i t , th e m icrophone spacing, s. can be chosen such th a t

5 < or fc r it < (2.47)

w here fc r it could be chosen to equal f c u t o f f , equation (2.31), for exam ple.

(35)

Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t 2.4. P l a n e W ^ v e D u c t A c o u s t i c s

A nother approach is to supplem ent the lost inform ation w ith th a t gained from a second test using

different m icrophone spacings. These two tests might be am algam ated into one by performing a least

scjuares solution using more than two microphones, such as used, for example, by Seung-Ho Jang and

.leong-Guon Ih [31], Chu [10] and Fujimori and M iura [22]. A schematic of the approach is shown in

figure 2.11.

Reference Plane

1

Pi P2 P3 Pn X-h

End C ondition

, -Xn ,

^__________-X i__________________ __

L

_____

_

_

F i g u r e 2.11: Schematic for least squares technique

2 .4 .3

A c o u s tic Im p e d a n c e

The acoustic impedance is given by

Za = ^ (2.48)

u

the dependency on freciuency understood. As impedance has units of pressure per unit velocity,

impedance can be interpreted as how nmch pressure is recjuired to cause a medium to move at 1

m eter ]>er second. Using equations (2.32), (2.37) and (2.48) with x = 0 the impedance can be

expressed as a function of the reflection coefficient by

Za = P()C ^ ^ (2.49)

In the case of an infinitely long duct of uniform cross section, waves would travel away from the input

and never retu rn from the other end. From equations (2.41) and (2.49) while letting Pr = 0, this

would result in an im pedance of

2c = P o c (2.50)

for a travelling wave which can be seen to be real and to not depend on frecjuency. The term , Zc,

is known as the characteristic impedance, which is a property of the medium alone. In addition,

by dividing equation (2.49) by the characteristic impedance, Zc, a normalised impedance can be

expressed

[image:35.529.33.518.65.518.2]
(36)

2.4. P l a n e W a v e D u c t A c o u s t i c s Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t

(2.51)

w ith

^ 7 1

Pac

T h e reflection coefficient is re la te d to th is norm alised im]5pdance by

" ' I

tt

2 .4 .4

T ra n sm issio n L oss

T ransm ission Loss (TL) is th e acoustical pow er level difference betw een th e incident an d tra n s m itte d

waves th ro u g h an elem ent of in terest assum ing an anechoic term in a tio n , i.e.

T L = 10 lo g io ||:J (2.53)

w here IT; is th e incident som id pow er an d U'( is th e tra n s m itte d sound pow er. T h e sound pow er for

each wave can be expressed in te rm s of th e incident and tra n s m itte d rm s pressu re am p litu d es by

'2

n ; = (2.54)

pc

and

UV = (2.55)

pc

w here th e rm s p ressu re am p litu d e, as a fun ctio n of freciuency, can be expressed as

Pi = \ l Oii * A f

(2.56)

Pt = y <^nt * A /

w here Gjj an d G tt are th e u p stre a m incident an d d ow nstream tra n s m itte d single-sided averaged

pow er sp e c tra l densities respectively. A f is th e freqtiency resolution bandw 'idth. an d Ad are th e

u p stre a m a n d d o w n stream cross-sectional areas. For an anechoic te rm in a tio n , G u if’ equal to th e

P S D of any m icrophone read in g d o w n stream w hich is equal to th e d o w n stream incident P S D ,

Gdi,di-It is im p o rta n t to note, however, th a t th e te rm in a tio n m ust be anechoic for th e T L to be calcu lated

in th is way. A com m on m istake is to ap p ly decom position d o w n stream in th e absence of an anechoic

te rm in a tio n an d to assum e G tt is equal to G,u.di for this condition. T h is is in co rrect as Gdi.di, w hen

th e re is no anechoic term in a tio n , is th e sum of Gtt plus th e successive d o w n stream reflections from

th e elem ent of th e dow n streaiu reflected wave

(37)

Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t 2.5. C o r e N o i s e G e n e r a t i o n M e c h a n i s m s

2.5

C ore N o ise G en er a tio n M ech an ism s

T h e sound field ra d ia te d from a d u c te d vane-axial fan, such as th e low pressu re tu rb in e of a n aero­

engine or gas tu rb in e , consists of a n u m b er of tones on a b ro ad b a n d noise floor. Ig n o rin g effects

which occur to th e ra d ia te d field o u tsid e th e engine, such as th e “H aystacking” or sp e c tra l broadening

of th e to n es d ue to sc a tte rin g in th e tu rb u le n t sh ear layer, th e ra d ia te d sound field is m ade up of

a su p e rp o sitio n of noise sources g en e ra te d from w ith in th e duct. T h e noise g en eratin g m echanism s

are im m erous. B ro a d -b a n d noise can be g en e ra te d from tu rb u len ce, tu rb u le n t b o u n d a ry layers in­

te ra c tin g w ith ro to r b lades as discussed in Jo sep h an d P a rry [34], convected b road I)and core noise

o rig in atin g from th e co m b u sto r, v o rtex shedding an d tra n sie n t aero d y n am ic loading v ariatio n s due to

tu rb u le n c e a n d blade v ib ra tio n . T h e to n al co n ten t can b e m ade up of convected d u c t to n es u p stream

of th e tu rb in e , th erm o -aco u stic in stab ilities in th e com bustor, th e spinning p o te n tia l field of th e ro­

to rs alone (m easu rab le only in close p roxim ity to th e ro to rs w hen the ro to r tip speed is subsonic),

"buzz-saw ” o r co m b in atio n to n e noise w hen ro to r tip speeds are supersonic, ro to r-s ta to r/ro to r-O G V

in teractio n noise and sum a n d difference to n es due to in teractio n s betw een different ro to r stages. In

a d d itio n to these, noise, w h e th e r n arro w b an d , b ro a d b a n d or to n al in n a tu re , m ay be g en erated by

th e in teractio n betw een a ro to r and a flow d isto rtio n .

2 .5 .1

R o t o r O n ly

A fu n d am en tal noise g e n e ra tin g elem ent in a tu rb in e o r fan is th e ro to r. C onsidering initially plug

flow w ith no flow d isto rtio n s, th e noise d ue to ste a d y aerodynam ic blade loading is th e source of

g re a te st im p o rtan ce. C onsider a fan w ith B blades each equally spaced ^ a p a rt. F ig u re (2.12),

taken from T yler an d Sofrin [63], is a developed view show ing typical ])ressure co n to u rs aro u n d a

ro to r blade assem l)ly a t an a rb itra ry rad iu s, th e d etails of which will vary depending on blade pitch

angle, cam b er, etc.

As th is ste a d y p ressu re field or p o te n tia l field is d ue to th e ro to r, it m ust spin w ith th e ro to r

angular velocity of $2 = 2ttN , w here N = w ith resp ect to a fixed co o rd in ate system . T he

jjressure, th erefo re dej^ends on a specific com bination of angle an d tim e co o rd in ates, viz. {9 — ilt).

T his form p{9, t) = p(9 — ilt) is a travelling wave an d is sim ilar to th e travelling wave solution to th e

l-D wave ecjuation, (^ — f). T h e pressu re m easured a t a fixed rad iu s an d axial position will therefore

be periodic w ith an g u la r p e rio d of ^ rad ian s. Unless th e pressure flu ctu atio n from one p erio d to th e

n ext is a p u re sinusoid, a F ourier d ecom position of th is periodic signal would reveal a fu n d am en tal

frequency equal to Bi l along w ith harm o n ics l i BQ of varying relative m agnitudes, w here h is th e

harm onic index. T h e F ourier series m ay be w ritte n as

OC

p{0, t) = ( I k coH[hB{6 - n t ) + 0/,] (2.57)

h=0

(38)

2.5. Co r e No i s e Ge n e r a t i o n Me c h a n i s m s Pr e s s u r e Fi e l d In A Cy l i n d r i c a l Du c t

(c)

PRESSURE WAVE FORM 0.1 CHORD LENGTH AHEAD OF THE BLADES

HARMONICS

PROBE MIKE

' / / / /

ANGULAR SPAONG- 7

ANGULAR COORIMNATE 9

( 0 ) ( b )

F i g u r e 2.12: Pressure field around a rotor-blade assembly.

F^(e) = a, cos (Be

( 8 LOBES)

\1L

P i t )

FUNDAMENTAL

p ( t ) 1 a , c o s ( S n t

( t « B N C P S )

f^(S ) • 0 , COS (2

P ,( f ) = 0 , c o s ( 2 0 f i t - ^ )

<28 LOBESl riLT€(»

La

f 'i B N

___ f \ ( t )

( f = 2 0 N C PS)

HARMONIC

| } 8 LO eESI FILTER

-1 A

— P , ( l ) l*38N

C O M P O N E N T S O F P A T T E R N A T F I X E D IN S T A N T

2 "“ HARMONIC

C O M P O N E N T S O F

P , ( t ) « 0 , c o s ( 3 8 Q t - ^

( f - S B N CP S )

P (t) . P , (t) * P , (t) « P , (t) *

H A R M O N IC C O M P O N E N T S P R E S S U R E D IS T R IB U T IO N AT F IX E D L O C A T IO N

F i g u r e 2.13: Spatial and tem poral pressure variation for a four bladed rotor.

By setting t or 6 equal to zero, spatial or tem poral representations of the pressure field may

Figure

Figure 2.2 shows jilots of the Bessel function and of its derivative for a number of m  orders
figure 2.11.
Figure 4.7 shows how standing waves might fit in a cylindrical duct of a single closed end
Figure 4.9 displays the magnitude and phase of this expression, in blue, plotted versus the exper­
+7

References

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