A review of aerodynamic noise

Bibliotheek TU Delft

Faculteit derl.lx;hIvaart.911lil1i1i1î1!Wi'iit1IèdîilIlf Kluyverweg 1

2629 HS DeKt A REVIEW OF AERODY AMIC NOISE

BY

L. R. FOWELL AND

G. K. KORBACHER

- _ -- .J UL Y; 1955

A REVIEW OF AERODYNAMIC NOrsE

BY

L.R. FOWELL AND

G. K. KORBA CHER

..

ACKNOWLEDGEMENTS

The authors wish to express their gratitude to

Dr. G. N. Patter-aon , Director of the Institute of Aer-ophyslcs, for his initiation of and int e r e s t in this work.

Thanks are expressed to Prof. B. Etkin and

Dr. H.S. Ribner for their careful review of the manusc r ipt . The financial assistance received from the Defenc e Research Board of Canada is gratefully acknowledged.

.

...

( i )

SUMMARY

A detailed review is presented of the theoretical and exper imenta l advances that have bee n made in th e stu dy of

aerodynamic noise and in devising me a ns for its suppression . Since many wor ke r s in the fields of aero dy na mic s and aerophys ic s .m a y be unfam ili ar with acoustic pr-i nc iples, the ne c e s s ary

backgrou nd of laws and ideas from the field of acoustics is include d. The theories for nois e caused by sub son i c dis tu r bances, which ma y include tur-bu lené e fields in overchok ed jets , (Lrg h thiIl, Pr oudm a n ) and for those noise sourees pecu lia r to overc h oked je t s (Ligh th i. Il,

Powe ll, Rîbne r] are cons ider-e d, Ex p e r iment a l resu lts are quoted comple t e with num e r ou s gr-aphs, notes on cor r ela ti on of da ta for mode l and engi n e jets and a comparison with theory . The re s u lt s of attem pt s at noise su pp r e ssio n are disc u ssed, noting bo t h

unt ried an d extensive ly tested sugge s tions , A list of refe ren c es is included .

Th o s e phases of aerodynamic no i se re s e a r c h con c e rni ng whic h there is disa g re e m e nt or in which th e re is confusio n becaus e of insuffici ent theo r eti c a l and experimenta l work, or whi ch appear to have been neglecte d , are th u s br- ought to attenti on.

It is ho p e d that th is review will prove usefu l in the initi a tion of research programs in the field of aero dynamic noise and its co n tr o l.

( ii )

II INTRODUCTION

III THE NOISE PROBLEM

V " SOME BASIC ACOUSTIC CONCEPTS

IV THE CONCEPT OF "AERODYNAMIC NOISEIl

Page 1 5 5 6 7 7 7 8 9 11 11 12 12 13 15 16 ,,~ 17 ~. \<'18 20 22 Intro d uctory N ot e

Derivat ion of Equivalent So u r e e Dis tribu tio n-.

Deduction of Quad r up o le Field Den s.ity Fi e ld

Dimens iona l Ana ly s i s

Frame of Refere nc e in Tr a n sla tion 6. 1. 3 6.1.4 6. 1.5 6. 1. 6 6.1. 1 6.1. 2

6.3 Ligh thill's Extens ion of Subsonic Noise Theory CONTE NTS

6.1 .. Ligh t h ill's Theo r y for Subso ni c Jet Noise

6.2 Pr ou dm an' s Theory for Isot r opie Tu r b u lenc e 5.1 Sound Intensity an d Sound Pr e s s u r e

5. 2 Sound Level an d Sound Loudne s s 5.3 Sound Leve l an d the Hum a n Ear 5.4 Sou nd Ener-gyA tt enua tio n

5.5 Some Useful Shor tc ut s GENERAL THEORY VI I NOTA T ION

...

6. 3 . 1 6.3.2 6. 3. 3 6.3.4

..

6.3.5

Turbulence Eddi e s Con ve c ted at 22 Non-Negligib le Mach Numbers

The Impor t anc e of Sh e a r in 23 Aerodynamic Noise

Note on Eddy Size 24

Aerodynamic Noise in th e Pr esenc e of 25

\

Large Shea r

Aerodynamic NoiseA ri s i n g from a Diffe r enc e 26 of the Mean Sour c e Ve lo city of the Turbulent Region from tha t in th e Surrounding Medium

( Hi ) _Page

6.4 Some Physical Implications of the Theory 28

6.4. 1 The Equivalent Stress Tensor, Tij ₂₈

6.4.2 Directionality ₂₉

6.4.3 Dimensional Reasoning ₃₀

6.4.4 Mach Number, Reynolds Number, and Strouhal L _{·3 1.'} ... Number

6.5~ Theory for the Choked and Overchoked Jet 32

VII EXPERIMENTAL INVESTIGATION OF JET NOISE 34

7. 1 '{he Structure of Jets 35

70'•.

1..

L Subsonic Jets 35

7.1.2 , Choked and Overchoked Jets 35

7.2 Location of Aerodynamic Noise Sources 36

7.2. 1 ,:S ubs oni c Jets 36

7.2.2 Choked and Overchoked Jets 37

';,.'

.

.

• .l, _,.

7.3 Noise Fields and Directionality 38

7. 3.1 Subsonic Jets 39 "

7.3.2 Choked and Overchoked Jets 40

7.4';1' Factors Governing Jet Noise 40

7.4. 1 Turbulence and Shear 40

7.4.2 Jet Velocity and Noise 42

7.4.3 Jet Diameter 43

7.4.4 Mach Number, Reynolds Number, and Strouhal .1.. ..44.. .'

7.4.5 Frequency Number 44

7.4.6 Jet Flow Density 47

7.4.7 CeU Length 47

7.4.8 Acoustic Power Coefficient 48

7.4.9 Re-heated Jets (After-burning) 49

VIIIr. MODEL AND FULL SCALE CORRELATION 50

8. 1 Noise Frequency Spectrum 51

8.2 Jet Noise Intensity 52

8.3 Jet Noise Directionality 52

8.4 Jet Flow Temperature 52

8.5 Jet Flow Velocity 53

8.6 Initial Jet Flow Turbulence 54

8.7 Combustion Effects 54

( iv )

Page

IX A THEORY -TEST RESULT COMPARISON 55

9.1 Jet Flow Velocity 55

9.2 Eddy Convection Mach Number 56

9.3 Jet Diameter 57

...

9.4 Noise Field and Directionality 57

9.5 Near Field - Far Field 59

9.6 Jet Temperature 59

9.7 Noise Frequency 59

9.8 Jet Flow Density 60

X NOISE SUPPRESION DEVICES 60

10. 1 Mechantsm of Nofse.Reductiön.. 60

10.1.1 Subsonic Jets 60

10.1.2 Choked and Overchoked Jets 61

10.2 Methods of Noise Reduction 61

10.2.. 1 Lower Jet Flow Velocity 62

10.2.2 Teeth (Fingers) or Notches 62

10.2.3 Gauze Cylinders 64

10.2 .4 Radial Swirl Vanes 64

10.2.5 By-Pass Air and Diffusers 65

10.2.6 Noise Suppresion Devices under 66

Preliminary Test

10.2.7 Final Remarks 67

XI ASSOCIATED PROBLEMS 68

11.1 Various Propulsion Jets 68

11. 2 Non-Aerodynamic Noise Sour-ees 69

11.3 B oundary Layer arideWake Noise 69

11.4 G round Running of Jet Engine s 71

11. 5 Aerodynamic Noise at Altitudes 73

11.6 Jet Noise as a Vibration Source 74

11. 7 Edge Tones 75

XII POSSIBLE RESEARCH PROJECTS 75

."

REFERENCES FIGURES

I

I

"p .

( 1 )

NüTATlüN

Intensity of sound (Eq. 1)

Perturbation sound pressure relative to atmospheric

pressure (Po.) IJ.

9

y

(1,0

T

f

•

I

_o

Is

HP

db

'Po

'F

'

I-V'

"

t

tL

x.~,y~

p

P~j

TLJ

Instantaneous partiele velocity Density at a point x at time t

The angle between the instantaneous partiele velocity and a specified direction (Eq. la)

Velocity of sound in a uniform medium (atmosphere) The period or a time much longer than a period A typical frequency

Intensity of sound at the threshold of audibility

(= 10-16 watts per s q, crn , )

Intensity of a pure tone Horsepower

Decibels

Sound pressure (rr;;~ value) at the threshold of

audibility (= 2 x 1Oëly_nes, : per sq. cm , )

Cartesian components of external force Cartesian velocity components

Time

Dynamic viscosity

...

Cartesian components of vectors x and y Statie pressure

Compressive stress tensor

( 2 )

À. Wave length of sound

Q

Mass per unit volume per unit time

Arbitr.a~S~allincrementin the space co-ordinate

U

A velocity typical of the flow ). A length typical of the flow

X Magnitude of the vector -:

1:(2')

The component of mean shear in the direction of shear

<-.~

at the origin (Eq. 43)

't Kinematic viscosity of atmosphere

Fe

Reynolds number

'

..

M Mach number ,.

St Strouhal number (Sectjon 6.1.5)

ÓLj-

Unit diagonal tensor

e

Angle of sound direction with jet axis (direction of flow motion)

cL

Jet pipe or jet diameter at flow exit ~o A tmosphere density

0'0_{) The density appropriate to the mean temperature at the fluctuating} fluid

\-+ - . \

.r:

The magnitude. X -

Y

•

of the distance from source to observer

Me Mach number of quadrupole convection and of reference frame trans la tion

S :-.. Cel! length or distance between two successive shock waves

'rld. Number of cycles of the stream disturbance created in a given time interval

'n Number of wave lengths between the orifice and the sound source

A

Distance between the effective sound source and the orifice

•

( 3 )

N Integer

q,

Rate of arnplffi.c ati.on of the flow disturbances

Yb

The factor in the gain criterion associated with the actual

sound production by the disturbances interacting with

the cellular shock wave pattern (Eq. Bl)

YLt

C

cc

~ -+

'Y\,d.

, y

PQ,

Pc'

"P'

p..

0

f*

L

S

The factor in the gain criterion associated with transmission

of sound to the orifice and its directionality (Eq. Bl)

A constant

Sign of proportionality Sign of approximation

The factor in the gain criterion associated with the

initiation of the flow disturbances and the pressure ratio across the jet boundary at the orifice (Eq, Bl)

Atmospheric (s t a ti c ) pressure

Critical stagnation pressure in jet flow reservoir

(c o rre s ponding to sonic exit ve locity)

Stagnation pr'e ssu re in jet flow reservoir Static pressure in jet flow at nozz le exit

Instantaneous local power output per unit mass for isotropic turbulence

Double correlation function Length scale of turbulence

Space correlation of the second time derivatives of

.... ...>

a t points y and y

e~. Angle of maxim urn intensity peak with jet axis

( 4 )

~

F requency of the intensity peaks

Stop

Peak Str-ouhal number

Efficiency of noise production (Eq. 58)

d.. Expansion angle of overchoked jets (see Fig. 5)

1J

Ang1e of spread of a jet measured from the jet axis (see SecUon 7. 1.1)

'DH'

r.

Stagnation pressure in jet flow at nozz ie exit

p*

Jet pressure ratio

=

("?: -

"Pa,) . "?a,-'

Subscripts

i. j, k, equal to 1. 2, 3

Superscripts - average va lues

( 5 )

II INTRODUCTION

The high noise levels characteristic of powerful aircraft engines and propell e r-s are a source of annoyance to all in their

vicinity an d may be the cause of actual injury to those who must work in close proximity to aircraft. The problem of aeroplane noise has become progressively more important with the introduction of pis ton engines of ever increasing power with propellers achieving

su pe rs on i c tip speeds.

A new urgency was attached to noise suppression with the

abr u pt inc rea s e in power'provided by jet engines , especially when

such engine s wer.e considered as the propulsive units for civil

air craft. Civil airports, for practical use, must be close to centres of population; and as a resu lt are usually surrounded by or close

to ex pa n din g su b u r bs of met ropo litan areas.

.Each new je t engine features another increase in thrust and

in nois e, with the result that extensive research is now under way, pa r tic ula rly in the U.

x

l

and the United Sta tes , with the aim of noise redu cti on, Significant advances have been made.

Th is review result s from a study of the published research in the subj e ct to date. and was uildertaken to provide a background for

a re s earch program at th e Inst itute of Aerophysics. It is essentially

a lite r a tUre surv e y . and is not meant to be a critical review of

resea r ch in this fie ld.

III TI!E~OIS~PROBLElVÏ

Thos e most; affected by noise from jet engines are:··

(a) th e turbojet manufacturers (testing engines)

(b) military users (g r o un d crews. carrier crews, and flying personne 1). and

(c) civil aviation (ground crews, flying personnel, pas s enge r s, and the public near airports). In cases (a) and (b) the problem is to avoid possible physical darna ge , espec ially to the ea rs , besides the impairment of communication by speech. In case (c) it is more the annoyance caused by th e noise of normal airport activities. In all cases there

( 6 )

(a) the reduction of the overall noise level (in decibels), and (b) the reduction of the noise nuisance.

The noise nuisance means the offensiveness of the no is e , i.e. the adverse psychological effects of the various noises on those subjected to them. These effects should be studied in a joint effort w ith

psychologists. Especially noteworthy are the ëffects of ultrasonic noise and that of a very low frequency of about ten cycles per second known as "belly sha king", which is most unpleasant. The noise of frequencies within the range of human speech has to be reduced in all cases to decrease their highly annoying disturbance of conversation.

The noise nuisance is difficult to define since:

(a) it is different with different people and the circumstances in which it is heard may not be comparable,

(b) noise coming from all around the observer is more annoying than an even louder noise from an easily localized s our c e,

(c) noise of a distinct frequency is more annoying than noise

of a more indiscriminate nature. and

(d) the preparedness for noise has an important bearing on

its nuisance value (e.g. the sonic bang gives na warning ).

There is still another and most serious aspect of jet noise, i.e. its effe c t on adjacent aeroplane structures. The pressure fluctuations

r adiatin g from the je t flow. especially from an o ve r-choke d je t. cause

vib ra tions and possible fatigue failures when the je t passes in close

proximity to the aeroplane structure. Research work in th i s field must

als o be given high priority.

rv

THE CONCEPT OF "AERODYNAMIC NOrSE"

"Aerodynamic noise" is noise generated as a by-product of an

unsteady (turbulent ) airflow, or noise produced without participation of

solid boundaries. lts frequency spectrum reaches from the sub-audib le to th e u ltr-as onic, encompassing a wide range of frequencies of comparable

in t ensity. Therefor-e, it is called "noise " as opposed to "sound". which

is usually cons idered to be composed of a few discrete frequencies .

Aerodynamic noise is produced by air jets. boundary laye r-s,

vortices and wakes. edge tones and related phenomena. Propeller nois e does not belang in this category as it is noise created by solid bodies in m otion,

t

( 7 )

As stated above., aerodynamic noise compniaes

ultrasonic noise as well. lt cannot be "hear-d", but its effect on

human beings or animals can be detrimental, and requires inv e stigati on by medical research workers .

V SOME BASIC ACOUSTIC CONCEPTS 5. 1 Sound Intensity and Sound Pressure

Sound intensity

I

in a specified direction at a point, is the average rate of sound energy transmitted in the specified direction thr ou gh a unit area normal to this direction at the point. In general

T

I -.

T-'

J

"P

u,

~:r' dt

(La )

o

whe re

T

is the period or a time much longer than a period,

"P

is the instantaneous sound perturbation pressure at the point , Ik is

the instantaneous particle velocity , and

y

is the angle between the instanta n e ou s particle velocity and the specified direction. In a gas of density ~o ' for a plane or spherical wave having a velocity of

propagation 0.₀ ' the average sound intensity over a cycle at a

point,'X. , is :

rex)

=

r:a~

,J,.~o 0-0

From Eq. (1) it follows that

Iex)

~

"P1x)

(1)

and it should be noted that the root mean square (r-ms) value of the perturbation pressure is referred to in acoustics as "sound pressure".

5.2 Sound Level and Sound Loudness

It is standard practice in acoustic measurements to relate the sound intensity

I

and the sound pres'sure"'P to reference values

I

_o and

r;

which correspond approximately to th e intensity and rms pressure fluctuation of the faintest audible sound at a frequency of 1000 cps. The nqise level (a.lso sound or intens ity level ) is measured in decibels (db).

Intensity level (db) Pressure level (db)

( 8 )

These two levels are equal for pressure and particle velocity in phase

as in the case of a plane wave. This phase r'e lat Ionahi p, of course, does

not always exist. Numerically

-1&

10 watts per sq. cm.

-ft.

- do·10 dynes per sq. cm.

The noise loudness (also sound loudness) is a subjective concept. and is

measured in terms of the response of the "average" human ear. The

loudnes s , measured in phons, of a tone of frequency

+

equals the

intensity (I~) in d.b of a tone of frequency 1000 cps which sounds equivalently loud to the human ear (see Fig. 1).

loudness leve1(phons)

=

lO..to.o

I~

~IO T

- 0

From Fig. 1 it can be seen that a reading in decibels must be numerically

greater than a reading in phons at frequencies below and above

approximately 600 and 6000 cps respectively. The phon scale gives the

loudness in close agreement with the characteristics of the human ear. Normally, noise levels are quoted in decibels relative to "Po

or

I

_o . Both values correspond only approximately to the threshold of

hearing (a rather difficult value to define, as it can only be an ave rage value),

and 2 x 10-4 dynes per square cm. is about 0.2 decibels less than 10-16

watts per square cm.

5.3 Sound L-evel: and t'he Human Ear

The values which represent the threshold of hearing have been disc us s ed in Section 5.2. The human ear is capable of tolerating without

di scomfo rt an intensity up to about 10l~ times the lowest audible intensity or

I::>" -lb

10 2~ 10 . 10 -. I~O

db

' 0 IO-'b

Measurements of's ound waves show that the maximum pressure variations

in the loudest sounds which our ear can tolerate are of the order of 280 dynes

per square cm. above and below the atmospheric pressure of about lOb dynes

per square cm.

The following noise table indicates decibel va lues which we

Description of Noise Rus tle of leaves

Aver a ge whisper

Liv ing room, quiet office Typica 1office

Ordinary conversation He a vy tr a ffic

Pa s s i ng tr a i n Pneumatic drill

A Derw ent jet engine on ground 1/2 mile a way

Ri v e t e r

Corne t tak e-off (4 ghost engines)

( 9 ) Noi.se Level (db) 10 20 40 60 65 80 and over 90 90-100 80-90 85 95 110 120 140 160 Noise Effect on Humans None lnterference with speech The highest noise level in which one can work for normal periods Discomfort Actual pain Permanent damage to .th e ear

...

Tab le-I - Noi se Levels (db) Due to Various Sources

The average power developed as sound waves by a person speaking in a normal conversational tone is about 10-5 watts. Thus, one mi 11ion per sons would only develop a power of 10 watts if speaking at the same time. On the other hand, imagine an auditorium in the shape of a sphere which is 25 meters in radius, in which the sound 'i n t en sity at the surface is to be 10-5 wat ts per cm~. Here the

acous tic powe r output of a ~speaker at the centre of the au d ito riu m would have to be 250 watts •

5.4 Sound Energy.Attenuation

A modern jet engine produces noise ener-gy at a rate of the order of 100 HP, which if radiated uniformly in all directions is sti ll pa inful at

a

distance of about 100 feet, and in spite of attenuation with di s tance would still interfere with conversation at a distance of 3000 ft.

( 10 )

With actual jet engines , the noise is not radiated uniformly

but has peaks in certain directions, hence the effect can be even worse

in thes e directions.

The following measurements may illustrate the effect of

di stance on jet no is e , The distance is given ahead of various aeroplanes

on a five degree take -off path, at which their noise levels still interfere

with speech. According to Table I, those noise levels are at about 80

or above,

A Dakota would produce 80 db up to 2.5 miles ahead,

AStratocruiser would produce 80 db up to 4 miles ahea d, A 6 engine jet airliner with reheat would produce 80 db

up to 12 miles ahea d,

A 6 engine airliner .w ith supersonic propellers would

produce 80 db up to 16 miles ahead,

Th e atm os phe r e àffects the transmission of sound energy by

(a) absorption (due to an acoustic resistance

=

f'

0.,,0)'

(b) dissipation caused by tur-bu lenc e,

(c) refraction caused by wirid,

(d) temperature and humi di ty,

(e) density gr-adients ,

(f) trans lationa l effects of wind. and

(g) re fle ction s from clouds or the stratosphere.

The atmospheric attenuation of sound energy by absorption - that is the

convers ion of so u n d en e r gy (orderly mass motion) into heat (random

mole cu la r motio n ) - amount tor

1 db per 1000 ft. in the frequency range of 37.5 to 75 cps .

~30 db per 1000 ft. in the frequency range of 4800 to 9600 cps,

Noise with a "flat" spectrum (called "white" nois e, the sound intensity of

whic h is th e sa m e in each consecutive octave) at 100 ft. distance from the

sound source is no longer flat at 1000 fe e t, where the high frequencies are

much reduced in intensity . The rate of absorption of sound energy by gases

at low pressures is known to increase with the square of the frequency,

if f

<

50. 000 cps , When the sound vibrations are of a frequency at which

mol ecu les can absorb energy i.nte r-na lly, we get "superclassical"

( u )

va lues which may be up to several hundred times higher than those

pr edic ted by th e f~ law,

As, an illustration of th e effect of wind on sound en e rgy attenu a ti o n, a very light breeze of only 10 knots would cause a

reduc ti on of 4 db per 1000 feet in the frequency range of 37.5 to 75 cps, and of '~ 18 db per 1000 feet in the frequency range of 4800 to 9600 cps , in the downstream direction. In the upstream directio n

the lo u dn es s level drops by !:::!20 phons compared with the

downstr e am va lues ,

5.5 Some Useful Shortcuts

4»p

,

Below are quoted some re lationships, which ar e us e fu l to remem be r .

(a) An increase in loudness level by 9 phons make s a sound appea r twice as loud to our ear.

(b) White noise with a sound in t e ns ity of 100 db in ea c h

con s e c u ti ve octa ve th r o ugho u t th e audible range would yie ld a total

noise level of 109 db or a loudness level of 125 phons.

(c) If the distance between a noise source and an

observer is doubled, the noise intensity, due to the inverse squa r e

la w is reduc ed to a quarter or the noise leve 1 is reduced by 6 db.

(d) If the int e n sity at any point is doub le d, the noise leve l is increased by 3 db. If it is te n fo l d or hundredfold of the

origina l inten sity , the noise le vel is increased by 10 or 20 db

re spe ctively.

(e) The displacement amplitude in the loudes t

-?l

tolerab le sounds of a frequency f = 1000 cps, is only about 10 cm,,

and in the fa intes t sounds about 10-b cm. (for yellow light À

=

6 x 10- 5 cm,; .a n d the diameter of a molecule d,

=

lO-Scm. for

comparison).

VI GENERAL THEORY

r ,~.

Lighthill's paper "On Sound Generated Ae r o dyna mically"

(Re L 2 an d 3) provides the general ·:th e o r y of the production of aero dy nam i c noise by subsonic jets. Since its publication (Re L 2) in 195 2, this paper has become a standard reference for sound or

noise crea ted without the aid of vibrating or moving solids, as

( 12 )

I

Pr-oudman (ReL 4) has applied Lighthill's theory to the case

of isotropic homogeneous (uniform throughout) turbulence.

No theory for choked and overchoked jet noise comparable to.

that of Lighthill for subsonic jets is at present available. Lighthill

contributed a paper (ReL 5) in which he shows how turbulence creates sound

when it traverses a shock wave. Ribner (Refs. 6 and 78) worked on the

sa m e problem. Lighthill's theory is used by Powell (Ref. 7) as a basis to

explain the production of noise in overchoked jets.

6.1 Lighthill's Theory for Subsonic Jet Noise

6~ _l~ ~. Irttroductory N ote

\

The momentum equation for a continuous medium subject to --:

external forces F~ per unit volume is in Euler's form

--(>

v:

~Vi,

+

J'

t

'é)X' d (4)

"

Here

ptj

is the compressive stress tensor representing the force in the xL

direction acting on a portion of flui d, pe r unit surface area with

inwar d normal in the

Xi

direction.

For air, assuming the stress components are independent of the rate of change of density (a Stokesian gas),

(5 )

Inserting in the left side of Eq. .(4) the equation of continuity

- 0

(6)

multiplied by v~ , one obtains

o

'C)

v~

+

V

~

+ V

Iêl

(fJ

v.)

+

(QVj)dV~

+

It)

(p"\

~L'

(7)

J

'è>t

~

ê>t

i. 0>)(.\\ I ) 0

'é>x'

~)(. l.d)

( 13 )

or

(8)

which is the Reynolds form of the momentum equation for a medium

subject to external forces Tt .

From Eq, (8) one obtains the appropriate equation for

soun d propagation in a field subject to external forces, by noting that

fo r the case of infinitesimal disturbances the viscous effects and the

se c ond order terms in the velocities wil l be negligible. Eq. (8)

the n becomes where

~

'Vx~

;t(~"L)

+

=~~

Ië;)~ 'è>x~ = ~.I.

has been replaced by

(9)

In case of no external forces or

rio -

0 , Eq. (8)

becomes

(10)

whi ch represents the exact equation of momentum in an arbitrary

continu ou s medium under no external forces. Similarly Eq. (9)

bec ome s

=

o

(11 )

whic h is an approximate equation of momentum governing the

propaga tion of sound in a uniform medium, without sources of matter

or exte r na l forces.

6.1.2 Derivation of Equivalent Source Distribution

A fluctuating fluid flow of limited extent (e. g. an air jet)

is'considered embedded in an infinite volume (e. g. the atmosphere).

The exact momentum equation for such an arbitrary

( 14 )

section given by

-

0

(10)

which may be rewritten as

(12) with =

+

(13 ) ~ --~

..

'ê>x'

l.~

The tensor

Tt~ m~y

be neglected outside the fluctuating flow

(jet flow) where the velocities Vt and v~ are the infinitesimal loca l

velocities of sound motten, and both viscous stresses and heat conduction effects are negligible (see Eq.s. (5) and (13}.). Thus outside the jet flow the density satisfies the usual equations of sound propagation in a

uniform medium at rest, free from external stresses and without sources

of matter, as given by Eq. (6) and Eq. (11).

Eq. (l~) has the form of Eq. (9), which governs the propagation of sound in

a continuous medium at rest due to external for-ces. rio . Interpreting

Eq. (12) in this manner one considers the tensor

Tti

as an externa11y

applied fluctuating stress and the applied for-ce. Fi, , per unit volume

equals its inwards flux

Using Eq. (6) for substitution in Eq. (9) results in the relation

One now notes that these equations governing the medium in the case of

applied forces bear, a certain resemblance to those describing the

fundamental case of a medium free from external stresses but cont ...'ng

in some limited space a continuous distribution of fluctuating sources of

additional matter suc~ that a mass

Q(X,

t)

per unit volume per unit time is

introduced at

"t

at time

t

.

The equation of continuity in this case is

( 15 )

which taken together with the momentum equation (11 ) for infinitesimal disturbances yields

/C)Q

=

_C>t

'I>

The density field for such a source distribution in an

unbounded medium is given by the volume integral over all s pace,

o(t

t)-

D

=

'

J~ Q(~

t -

\1-"Yt)

rly, rJ.y:..d."3

(17)

y I )0

4-'1I'o..~ ~t

'I)

0.₀

l~-'f'

the '·'retarded potential" of electromagnetic theory. The e s s eritia l

factor governing sound production is seen (from Eq. (17)) to be the t:·

time rate change of the rate of mas s introduction é>%t:: and for the

present purpose it is denoted as the "source strength" per unit volume.

Comparing the continuity -momenturn relationships of

Eq. (14) and (16) for the cases of applied forces and a source

distribution respectively one sees that

uQ

corresponds to _ ~Fi.

'Ot 0)(L

such that the fluctuating force field is equivalent in its effect on density

to a source distribution whose strength per unit volume ~~ . equals

the inwards force flux -

~:~

Hence in the case of applied stresses,

Tt.. '

in which the equivalebt force field "Ft is given by _ 'C>Ti.i

o 'C>x. •

~

the effect on density is seen to be the same as for a source distribution of strength per unit volume

6.1.3 Deduction of the Quadrupole Fie ld

In the estimation of the acoustic power output of the above equivalent source field, one must take due note of the fact that the

equivalent source s tr ength is a space derivative and not a simple source. The effect of the double differentiation in space is shown as follows.

In thé equivalent source distribution of strength _

~'F~

ê>)(t

the term -

~:'.

is equivalent in the limit

(ê ....

0)

to the source

- I ( ) _I

distributions é. 'F, X(1)X:l.)X~ and -é.

r;

(X,:+E,X~,X~) ,such that any

value

e-'F.(x., ,

X;t)X~) occurs with positive sign at (XI) x~) ~:!l')

( 16,)

dipole of strength

r,

with axis in the positive x, direction. For the

_.-general vector"F~ it is seen that a choice of coordinates such that the

XI axis is in the direction of "FL ' yields by the above argument the fact

th a t the corresponding di po l e is of strength equal to the magnitude of FL

and has its axis in the direction of the vector

rio .

Hence the force fie ld

Fi. per unit volume emits sound as a volume distribution of dipoles with the vector strength per unit v olume,

rio

In the present case, however,"Fi. is itself the space derivative

._ ê)Ti.~ , thus the term _ 'dTLI for example being equivalent in the

ê>xi

'

ê>x,

limit

Ce

-+0) to the dipole fields

ê.-T~\(x.,»)(~)x.?»

and

-ê-\Ti.,(~,+é)X.:L)'X.~)

such that e.-ITi.I

(x.,

X:l) X. ) occurs with positive sign at

(x,)

x.:L) X!l)

and negative sign at )

(~,

- E)

x~

)

)(3\.

This pair of dipoles in the limit yields a quadrupole with strength equal to the magnitude of the vector Tt.!

and ax es in the directions of

11.,

and x, . Again generalizing the

argument as done for

rL

one sees that an applied fluctuating stress field

Ti.j per unit volume emits sound as a volume distribution of quadrupoles of s1:rength per unit volume TLi . Since there is no indication that Tt.d

is in turn a space derivative it is concluded that the sound field may be treated as the effect of this quadrupole distribution.

6.1.4. Density Field

In Lighthill's consideration of the density field the equivalent so u r c e distribution _

dF~

replaces

(~\

I

~

-11-

1

'd

Xi. ot)

the integrand for the simple source (see E q. (17) by

in

such that the R. H.S. of Eq. (17) becomes

- I

Replacing

rio

by _

d"TI.~

and repeating the ar-gument yields

OYi

~(t)t)- ~o

==

1

~~

fTv(y\t

+-

,t

-~1:\

d.'f,c1.

y

. d.'j)

4'1

0.,;

OXt

oXi

~

0.0 )

\"t

-'1\

One obtains the simpler form

'

.

( 17 )

for p o l nt a a t a distance by retaining .o n ly the d i f f e r ' e n t i a t i o n s of

TLJ

a n d ' n egl e c tt n g those terms involving d i f f er e n t i at i o n of \~-

"0/\-\

which vary as the inverse second and third powers of the distance

from the fluctuating flow field. Eq. (19) is considered to be valid .f or X at a distance from the flow field large compared with À~'1t'

where À is a typical wave length since fluctuations of a function are less than those of its time der.ivat ive by a factor of the 'o r -de r

.l.JüÄ '.(a

~luctuating

quantity

<t •

.'B

e - ..

J:'Itft

having the time. derivative -I.~~

19.

)

.

A further simplification can be introduced for distance s., large relative to the dimensions of the fluctuating flow (the "far " field) such tha t

ex

i.. - Yi.. ) in Eq. (19) may be replaced by x.i.

provided the origin is taken within the flow. Thus without neglecting

1-' 1-'

any te r ms of order X Eq. (19) becomes

I

Equations (l ~ and (20), depending principally on the

se c on d time derivative of the equivalent applied stress

Tti

are the bas ic result of Lighthill's paper. Note that Eq. (20) is not

app licab l e to the so- ca l le d " nea r" field consisting of the space

extèrnal to the fluctuating jet flow at a distance not large relative to its dimensions. The more exact forms of Eq. (19) or- even Eq. (18)

are'needed in this case.

6.1.5 Dimensional A nalys is

I

The dependence of the density variations in Eq. (19) on

~Q )o...o~

'Jó

,a typical velocity and length of the flow (U and

1. )

is

mfer r ed for geometrica lly similar mechanisms by noting that Ti.{ is proportional to~oU~ (see Eq. (13 )). In general, a typical flow f'r-equency

-t

'

varies as

U/;,

(from the relative constancy of the

St rouhal number St

=

~

of the order of 1). He nc e , the fluctuations

. U

in

d~i..i

are proportional to

(fU'U):(:l..) .

Density variations

'at:L

(~

-)0)

at a given directed distance )( are then proportional (see Eq. (19)) to

I X.

( 18 ) The intensity of sound is

(22)

where the term in square brackets is the mean square fluc tuat lón of ~ at th e point considere-d. From Eq , (19) one sees that (~_fa) is a time derivative and hence has a zero mean value in time such that the mean density is ~o and the mean square fluctuation is the mean value at the point of

(

f. -

fo

)

squared. Dimensionally the intensity is thus pro-portiona 1 to

U

il -5

(J..

\~

=

~o

0..0

x )

(23)

The total acoustic power output obtained by integrating the intensity over the su r f a c e of a sphere of radius large compared with the dimensions of the flow field is then proportional to

(24)

Th e proportionalities referred to above are not exact and hence

ex pe c ta ti ons should he conservative regarding these dimensional _, fo r m u la e.

A

·

6.1.6 Frame of Reference in Trans lation

One considers a coordinate system in translation with uniform

velo city

a..)1

_{c (Me}

<I)

and coinciding with the previously considered fixed

axes at time

t

.

The axes then move a distance ~

pt

-Y\

while so u n d travels fr om

y

to ~ such that

y

in the argument of TL}

is replace d by

li

=

y

+ ~

pt-

y

I

(see Fig. 2). The density field

of Eq . (18 ) then becomes

(t t) -

= I

';;)~fT;·~t-'X-YL\

J

l1 l

ci

lJ:t ç1 YQ (25)

~

, .

~

o

4-J:.Ov;

ex

_L

8xJ

l.l"l)

0..0

Jl'x-YI-Mc.(t-Y)

At points at a distance from the je.t flow large compared

~ith

;1C:

this equ a ti on sim p lifi es , by applying the differentiation to

Tt

t

only,

( 19 )

As previously stated (see Eq. (22)) the intensity field is given by

~ f~' times th e mean sq ua r e at a point of

f

-~o and hence in this

case taking the mean sq ua r e of Eq. (26)

I(x)~

I fJ(XL-Yi.)(xJ-Ji)(X-k-y....

)(x.t-y~.)

16Jt~~oa,05

{lx-YI-Me.(X-Y)Y

{lx-il-~.(x-Z)}3

(2. '1)

?l

T~j(Yi,t

_Ix-YI)

~;l..

TkJ.

(~\

t _ \

~

-

Y\)d.~lcl'1

..

d.TJ.~d,)1 Q.:~~d)o

~.. a L ~ (Yt:l.. \: Ovo

In integrating thi s mean square over a large sphere to obtain power output, th e y~

.

and XL may be neglected relative to the xL

and the integral

~

mus t be eva lua t e d. With the Xl axis chosen in the direction of

M(!

(28) vantsh e s by symmetry unless the sufftxe s i, j, k, 1 are equal in

pairs. The power output for the case

Me -

0: (coordinate system at

rest) is th e n multiplied by

b)

1+

5M~

(I -

M~)lt-

or

9

I

+

10

M2

+

5

Mc4

(1-

M~)s 't

ac c o r di n g as netther, one or both the pairs of suffixes agree with the

direction of ~ . These case s correspond to quadrupoles with

neithe r, one or both the i r axes in the direction of motion.

At distances large compared with the maxim ûm.ee.pat- àtion

of poin ts

y

an d

X

in th e flow for which there is a significant

cova rianc e, both (XL-Yt) and (XL -

'Xi.)

in Eq. (27 ) may be replaced

by

r-X - y\

cos

e

or

\x -

y

1 sin Gdepe nding on whether

i.

is

1

'o r not (ass u m i n g one of the lateral quadrupole axes is

,...-+

coincide n t with the di r ec tion of Me ).

In th e spe cia l cas e with source velocity .

CLc·

Me in the di r e c tion

x

,

and cos

e

= )(.1-Y. assumed to be constant for all

Jt

positions of the so u r c es in the fluctuating flow relative to an observer

at

X

,

the intensity may be written in the form

( 20 )

where KI, K:l,. ~ K~ are functions of M(!

(se e ReL 8).

and are independent of

e

The effect of trans lation on frequency is to cause the band of

fre quencies considered in

TLrCYr,

t)

to be responsible for radiation to a

distant point X of a band of frequencies obtained by multiplying those of the

original band by the factor

The fre quency is increasedfor so un d emitted forwards and decreased for

bac kw a r ds emi ssion . In the. spe céal case considered above for th e

deriva tion of Eq . (3 0), the so u n d'i nte ns ity

I-i-(t')

measured in a sm a ll

ban d ar ound th e frequency

r

by an observer at ~ , allowing for Doppler

effe c t is .

wh ere . Klj., _{Ks , K}El

(see ReL 8) .

are functions of M(4. and

r ,

independent of

e

6.2 Proudman' s Theory of Isotropic Turbulence

Proudman, considering an isotropic region of turbulence with

stationary boundaries embedded in an infinite expanse of compressible fluid,

derived the rate of conversion of the kinetic energy of turbulence into

ac o ustic energy . Assuming th e turbulent medium to be incompressible

reduce s the number of parameters on which mean values are dependent to

thr e e (by Heisenberg' s hypothesis ) . Only eddies not dissipating energy

by vi scosity are considered to contribute to noise generation at large Reynolds

num b ers., their contribution being independent of Reynolds numbe r , These

considerations, dimenaional ana lys is and Lighthill's result of Eq. (24) yield

(32)

t'

for the acoustic power output per unit ma s s, 6) Here

f3

is a constant

to be determined and

e.

,

the mean rate of dissipation of energy per unit

mas s is given by

( 21 )

where V~ is the mean square of one of the three velocity

components and the Mach number is defined as

*"

7,;).

M

_

I

V,-Q.o

It is further assumed that..the mean temperature of the

tu r b u le n c e approximates that of the surrounding fluid, that the

turbulence is a diabaticç.that the Reynolds number i~ large, the Mach

number sma ll, and that the space correlation,

S,

of the; secend

timederivatives of

[v/- v;

1

at points

y

and

-y'

(where

v~

is the velocity component in the

X

direction) is zero for points

further se pa r a te d than some distance

.J, ,

and the time correlation

for these quantities is unity at a point for a .time intervalless than

~o

(h e n c e the smallest wavelength of significant acoustic energy

must be much larger than the largest eddies gene rating noise.)

Lighthill's equation for the density (Eq. (20 )), under the

assumptions there noted, reduces in this case to

,

~o

4l:

0,_o4

si n c e

TLk

is simplified to

fo

vLVi by t he assumption of the present

article . The equation for the intensity (Eq. (~2)) then becomes

or

leX';

t)

.

1

6

Jo"

a.;

x..

JU

5

a(l-y,)d.(

y;-

~d.(y;_y~dy,:y

..

cl.

Y,

(34l

l -Cl..o

being independent of the direction of

X

for this case of isotropy

su ch that the intensity fie ld is non- direc tional like t hat for a simple .

sou r e e .

The local instantaneous power output per unit mass is then

( 22 )

an d (3 is dete r min d by evalua tion of this integral.

The corr elation

S

being of fourth orde r in th e velocity and

its first tw o time der-Iva tives, the assumption is made that thes e three

quantities at two poi n ts in space have a norma l jo i nt proba bility

dis tribu tion allowing ex pression of

S

in terms of second order corr ela tions . Th e s e in turn, by th e as sum pti on of incom p re ss i b i lity,

are ele me n ts of so le noida l tensors (te nsors for which'the div ergence is

zero) and may be exp resee d in terms of thr e e double correlation

functions. Th e terms involvi n g tw o of thes e in the expres sion for ~ may be ex pressed in terms of the th i r d ('-{* ) by use of the dynamica1

equations and norma li ty hypoth es is such tha t 6) is a funct i on invo1ving only VI:J. ,

1-'*

and its 'der i vativ e s with respec t to

\1'-

y\

and

t .

~ is then given b:r a sum of integra 1s inv ol ving .

{*

an d its deriva tives

with respect to

x ..

I

y

'

-

~l·C'

(

w

h

er

e

L-.

(V~)~E.-I

is th e 1en gt h sc ale

of th e tu r b u1e nce) in w ich the terms depending on th e ins tan taneous forms

of the correlation functi ons occ ur sepa ra tely from thos e depe nding on th e i r

time rat es of c an ge, allowing separate calc ula tion of the con tributions of

insta n ta n ous and decar terms . Using Hei senbe rg's form of the

c or-r- elation,function. r1 ~(X) (wh ich has stron g expe rimenta1su ppo r t for large Re ynolds numbers ) ~ was found to be 37. 7. Considering th e _1t)l.:t

greatest re asona ble va r-la tion in th e sha pe of

~*()()

by using

t*CX) -

e

"'1j:"""

yielded (3 = 13. 2. T e contr-ibution of the decay terms in ach case

was roug 1y 10/0.

In h case of ste ady anisotropic turbu1en ce fie 1ds, maintained

by an ex te r na1 e ier-gy sou r ce ( ê. becoming ow th e rate of en rgy su pp1y)

any var-ia tton in pow e r output is ex pe c ted to be expla ine d by the form of the

corr e1ation functions, since decay alone is not an im po rta nt fea t u r e in the

isotr opic ca s e. It has been show n, howe ver, th a ~ is r-eIa ti vely

insensitive to c an ges in the cor rela tion function and he nce for mos t ty pe s

of tu rbu1e nce it is expe cted that ~ wiU be in th e range of 10 to 100. 6.3 Ligh t h ill's Exte nsion of Su b s oni c Nois e Theory

6. 3.1 Tu rbu1ence Ed dies Conv ect ed at Non -Ne g ligib1e Mach Number In Ref. 3 Lighthi ll considers an ex tension of th e theory of

aero dyn am ic nois e to th e case of non-neglrgtble fluctua t ion Ma c h number obtain ing th e expres sion

( 23 }

for the intensity at

X

per unit volume of turbulence at the origin. This

exp r-ession is de r ive d from th e genera1 intensity equation of ReL 2

(see Secti on 6. 1) gi ven as

I(x)~ ( 37)

by lettin g

y

= 0, dropping the in t egration with respect to

y

and

negle c ting differenc es betw e e n

(

t -

y)

and

ex

-z)

on the assumption

that ~ is far from the origi n compared with an average eddy size.

The average edd y volum e has a diame ter

.t

such that the correlation

in Eq . (37) is neg ligib le for points sepa r a t e d by a distance exceeding

J...

The differen ce in the retarded time s of the two values of ~~ 1:'

~t~

l.k

is made negligib le by ba sing the analys i s in a frame of reference,

moving at the loca l eddy convec tion velocity

~o ~

, which is such

that th e eddies alter slowe s t when vi ewed from this frame. The time

scale of th e tu rbu len t fluc tuations is then of the order of an eddy size

di vtded by a ty p ica l departure of the velocity from its mean. This ratio

is in general lar ge re la tive to

1.

0.;'

since the Mach number of the

velo city flu c tuations is alm ost always smal l. The difference between

the re tar-de d.time s (bei ng less th a n l

a.,:'

)

is then ne g ltgfb leac om pa red with th e times signif ica nt in the turb ulent fluctuations. The results of

Re L 2 conce rning a moving fr a m e of reference are then applied,

mu ltiply ing the int ensity field of an element ciy,.ciY:l.:

d.y3.

by the factor

(38)

~

While the eddy con vection velocity Q.,o·

M<!,

need not be

.uniforrn as in ReL 2. but varies ove r the field. the result may still

be us ed provided ~ shows sm all variati on in a distance of order

1. ,

sincetwo points radi a te so und independeritly if separated by a greater

dista nce .

6.3.2 The_Importance of th e Shear in Aerodynamic Noise

The amplifying effect of large mean shear is seen by

consi dering the time rat e of ch ange of momentum flux (t h e chief factor

( 24 )

(using the equations of mome ntum and continuity to express time

de r-ivattveaas space de r-ivative s}. The last term being a pure space

derivative represents an octupole field and hence may be.d r o p pe d as a

negligibly weak radiation source. The viscous contribution to the

stresses mayalso be ignored for Reynolds n umbers of the order found in fully developed tur-bulence , yielding

i.e. the product of the pressure

p

(measured from atmospheric as

zero) and the rate of strain (e;. 'be ing the rate of strain tensor).

.

"i

6.3 .3 Note on Eddy Size

An intermediate size of eddy is shown (Ref. 3) to be of

mos t importance in s ound generation by replacing

~Tt~

,

in

Prolidman's theory (ReL 4) for is otr-opic turbulence .t by

p-ei.}

The

power output per unit volume is then

6)* -

,

(~( av,\

~('Ov,'\

di

(41)

-

)(

~o

(LoS

J

ê>t

P

'ax,7

x•

0

'C>t

P

1Yx,-)~

..

~

b'"

-I

-s

fS..)~

=

~8 ~o\"lj

L

CL

_o

ec -,

~8 E·~

•

~o

CL_o5

as sh own by Eq. (32). :wh e r e

L=

(v,:1.)~.

€-l and

e

is the mean rate

of en e r gy dissipation per.unit mass. Proudman evaluated the integral

( 42)

The order of magnitude of the fluctuations in velocity gradient cont r ibuting

mos t to pe~k is inferred from the square root of the ratio of Eq. (41) to

( 42 ) to be

8 (

VI·)~

.C'

.

Hence the effective ve10city gradients in

sound production are roughly 8 times those typical of the largest energy

( 25 )

6.3 .4 Aerodynamic No is e in th e Presence of Large Shear

In tur b u le nc e with large mean shear a single term (e.g.

pel~

)

predom inates, benc e th~ qua ? ru p o le s a.re aligned to ra~ia te

maximum so un d at 450 _{to the direc ti.on of motton. The sound 15}

amp lifi e d if th e mean shear

e

l~

ex c e e ds 30

lV;:)~

·t:'

(using the

facts that the effe c ti ve fluc t uating velocity gradients in isotropic

tu rb ule n c e ar e

~ 8(V;:~L

-'

an d the sound radiated by one latera.l

quadr upole orientation is 1/15 that by all orientations in equal strength ).

Th e inte n sity fi e ld per unit vo lu m e of turbulence at the

or igin is

·

L(X)

~

_·

_v

(43)

obtaine d Irom Eq . (36) by assu m i n g th at

~

T

_LL

~t Cl

is pr edomina n tly

In

e

oe:

p

~v,

,

that the dir ection of eddy convection is parallel

r,~

/":\

O'x~

to the only significant me a n ve locity

v,

and that only the compo ne n ts

e

e

of

e

(x.)

need be cons idered (as the covaria nc e is

IJ.. ' .;tI ~.e.

si gn ifica nt only when

Z.

is close to the origin at whic h time the

me a n shears sho u l d be sim ila rly oriented at

X.

an d the or-Ig in),

"'t

Cz.)

~ e l:!.

ex)

is th e com p one nt of mean she a r in the di r- e c t ion

of shear at the origine

V

is the avera ge ed dy vo lume, de fined as

v

=

J

1;(

1')

"t(o)

~

p(

o)t).

~ p(i)t)

(~ p(o)t))~

(44 )

The power output per unit volume under these conditions

i

( 45)

...

( 26 )

The to t a l power output of a th i n shear layer is

Assuming the covariance varies much more slowly than

'"t(1)

or

'"t(Z)

as

"lX

.

cross the layer, integration yields the

power output per unit area of turbulent shear layer.

(47)

where

Ó

V is the velocity change across the shear layer and

(48)

is the average eddy area at a point

y

with <iÀJ.an element of area of

the shear layer in

X

space. . .

The r-es ult of Eq. (47) applies also to the annular shear layer

of a jet if area in Eq. (48) is taken to mean projected area on the ta ng e n t

plane at

y .

In addition, the combination of lateral quadrupoles in the

( XI , xJ.. ) and ( XI , x:!> )planes yields a directional distribution

proportional to

(49)

The frequencies in shear flow turbulence are expected to be of the order

(

;}..Jt

) - '

times the mean shear because of the term

v.(d.v,

~ in the

J.

d.

X

.

l

acce leration

~

~~'

) . ;).

6.3.5 Aerod:inamic Noise ArisingFrom a Difference of the

Mean Sonic Velocity in the Turbulent Region From That in the Surrounding Medium.

~ 27 )

general of small im p or an ce m generating aerodynamic noise through

th ei ... direct sonie fie ld, by con sidering th e main te r m in TL~ oth er

t a momentum flux. This is the souree field due to that part 0 the

turbulent pressure f uc ua ion s whi ch is not balanced by aceompanying lo cal density fluctuation s in the free ac ous tic vib rations of the

atm os phe r-e , having a sour e e st r e n gth pe r uni t volume of

{50)

where

a:.

is the mean ve lo c ity of sound at th e po int and pressure

fluctuationa are consider ed to he ne arly adiabatic . Viseau s

contr-ibut tons to PL~ are ne glected .

The powe r output of the ab ov e sour e e field pe r unit

volume of tu r b u l e n c e at the origin is shown by the pr evious arguments

to

be

orni ting quadrupo le convecti n.

The ampli ication du e to pres sure fluctuations can only

be Im pcr-ta t if thi s ter m is at lea st com par able to the power output

due to momenturn flux.

For is otr opi c tur b u le n c e with (i >'» ~ {a s in a

heated jet) Eq. (51) dïffer s from Eq . (4 2) by a singl e time

differenHation. The same integral with na tim e diffe ren t iations is obtained from Bat che lor 's results {Ref. 24) as

J.fo~(V(-t

I:

showing

the domina nt frequency far pressure fluctuations to be

-)

J:

,

of the order of (VI~ J.. •

L

-

and Eq. (51) to be of the order of

(--,.)4 _,

?o

VI .

L

4

x.

.

0.

0S

Hence this term is negligible relative to th e .mom e n t u m flux cont rib u -i n (s ee Eq.

B

JIJ ) .

This result is considere d applicabl e to the case

o the jet .

In heavi.ly shear e d turbu len c e (ecg, the maxmg region of a

jet) the power output of ~omentum flux (w hich in general takes the

fer-m 0 Eq. {51) with ~ re p l aced by Olr·eLj

~~ ~

and hence exceeds Eq, (51) if

(.;).~y

'

.

ei.i excee ds th e dom inant frequencies of pr essur e fluctuation) is.redu c e d by 4/1 5 by

consideration of a si ngle lateral quadrupole fïeld in ste a d of

, (28) and (5 1). The relative importance of

to

is greatest in the high Ire quençy bands where a possible amplification of 10~(I + I~) c : 6db is thus an upper limit.

In heavy gases having

a

<, Q,o • greater amplification at high

frequency in shea re d ' turbulence is expected; the theoretical ratio of

the intensity field to the above case with

ex..» a

_o being 18 to 1 for Freon as an example.

6.4 Some P.hy.siê'al Implications oLthe Theory

The following sections may help one to a better understanding of Lighthill's theor-y, and aid in its comparison later with experimental results•

6.4. 1 The Equivalent Stress Tensor, TL}__. __

In the region outside the fluctuating jet flow (the atmosphere)

TLJ

may be neglected as explained in Section 6.1~. In the region of the

fluctuating flow itself, the assumption that the viscosity term in P~j (see Eq. (5)) may be neglected permits Ti.} to re duce to

V

vi,vi • H, in addition, the flow system is adiabatic (as it may be assumed, e. g. in a cold jet) the further si.mplified form

(52)

may be us e d, The error involved in this form is of the order of M*;l. since the relative changes in density are of the order of

M*.1.

and the ratios of fluctuations in pressure to those in density depart from

a~ by a proportional error of the order of M*;). The error is sm a ll,th e r e fo r e , fo r low fluctuation Mach numbers.

In a "c o ld" jet flow- one expects only the first term (~vi.v~) in

ll'

representing momentum flux to be important (ReL 3). The sounJ field will therefore, be large when there are large variations in

vI.. and Vj as, e. g. : in regions of high velocity shear and high turbulence. f,viY,Y can fluc tuate ~ost wide ly when the fluctuations of ?vi. are amplified by a heavy mean value of

vf

In other words, turbulence of given intensity can generate more sound in the presence of a large mean shear.

( 29 )

In a jet flow of non-uniform temperature or fluid composition (e. g. a heated jet) the speed of sound varies widely in the turbulent

flow from that in the surrounding atmosphere, and the pressure terms

in Ti..j (see Eq. (13)), PLi - o..~~dLt ' may become important. In

this case, the pressure fluctuations in the turbulence are only partly balanced by

0,:

times the density fluctuations and the remainder contributes to Ttj

Lighthill assumes this contribution to Ti..~ is still small compared with that resulting from momentum flux ~vi.vJ

except in the case of sheared turbulence in heavy gases (Section 6.3.5).

6. 4. 2 Dire ctiona lity

Lighthill's theory considered in previous sections deals with the genera 1 quadrupole fie ld in the case of a frame of reference at res t, and extends it to the cas e of a frame of reference in

tra ns lat ion, He shows that an analysis of the equivalent stress

tensor Ti..~ into a pressure and a single pure shearing stress effects an analysis of the quadrupole into three equal mutually orthogonal

longitudinal quadrupoles, each of strength

T

=-TLLI .3

(53)

and one lateral quadrupole. The sum of these three longitudinal quadrupoles is equivalent to a simple source of strength

(54)

at least as far as their sound radiation field outside the stress field is concerned.

(a) Stationary Field

The sound radiation fields of a stationary lateral

quadrupole and a simple source are shown in Fig. 3. It shows that the lateral quadrupole radiation field has intensity maxima at angles of

e

=

45 and 1350 to the jet axis and shows no radiation of sound ener-gy at

e

= .9 00• The simple source radiation is uniform in all

direc tions.

(b) Moving Field

( 30 )

rest and in motion are given by EqS.,(19) and (26) respectively for the same

sim p lify i n g assumptions. A cornpar ison of these two equations shows _ ,_

that Eq. (26) is nothing else than Eq. (19) multiplied bya factor

Cl

-Me~er! due to the motion of the quadrupole field.

Applyipg this "factor"to the radiation field equations of the stationary lateral quadrupole and the simple source, we get the

changes in the radiation fields due to trans lation as shown in Fig. 3 for a Mach number of M_è= 0.9. Lt-clearly indicates the heavy increase in emission of sound energy forwards, which is much greater than the reduction in rearwards emission. Fig. 4 shows the rapid increase of the sound intensity maxima in the downstream direction and their shifting towards smaller angles

e

with increasing Mach number of translation. One also notes a distinct decrease in upstream noise levels.

One of the weaknesses of Lighthill's theory is that it does not allow for refraction of noise. The turbulence of a jet may cause severe refraction of noise passing through it.

6.4. 3 Dimensional Reasoning

When the typical flow velocity

U

and length

1

are chosen to be the jet flow exit velocity

U

and the nozzle diameter

ei

,

the total power output (Eq.( 24))is seen to be roughly proportional to

8 -5'

:1-~o

U

0.,0

cl

(55)

Thus theory predicts that the sound intensity increases roughly as the

ei gh t h power of the "jet exit velocity

U

,

and the square of the jet diameter.

The sound pressure

"P

being proportional to the density fluctuations is roughly proportiona 1 (see Eq . (21)) to

U

4 and cl, .

:t It may be noted here that in th e dimensional expr e s s ion

~o

U

for Ti.j the

Vo

is a reference density appropriate to th e

fluctuating flow field and hence in the case of heated jets or jets of a gas

different from the surrounding medium (atm os phere) is not.; the "atmosphe rIc"

density

f.o

exis;.ting outside the jet. In such cases the re lations hip is

clarified by using ~~

UJ..

for T~} such that the sound pressure ~ is seen to be proportional to the mean density in the jet

( 29 )

In a jet flow of non-uniform temperature or fluid composition

(e. g. a heated jet) the speed of sound varies widely in the turbulent

flow from that in the surrounding atmosphere, and the pressure terms

in T~j (see Eq. (13)). Pi.j - o.,~~J'~t

'

may become important. In

this case, the pressure fluctuations in the turbulence are only partly

balanced by a..~ times the density fluctuations and the remainder

contributes to Tt,

Lighthill assumes this contribution to T~~ is still

small compared with that resulting from momentum flux ~ y~vJ

except in the case of sheared turbulence in heavy gases (Section 6.3 .5).

6.4.2 Directionality

Lighthill's theory considered in previous sections deals with the genera 1 quadrupole fie ld in the case of a frame of reference

at rest, and extends 'i t to the case of a frame of reference in

trans lation. He shows that an analysis of the equivalent stress

tensor T~! into a pressure and a single pure shearing stress effects

an analysis of the quadrupole into three equal mutually orthogonal longitudinal quadrupoles, each of strength

T

=-TLLI

,3

(53)

and one lateral quadrupole. The sum of these three longitudinal

quadrupoles is equivalent to a simple source of strength

I

(I

-a~LL)

=

3"

a;

''dt~

(54)

at least as far as their sound radiation field outside the stress field is concerned.

(a) Stationary Field

The sound radiation fields of a stationary lateral

quadrupole and a simple source are shown in Fig. 3. It shows that the

lateral quadrupolè radiation field has intensity maxima at angles of

e

::

45 and 1350 _{to the jet axis and shows no radiation of sound}

energy at

e ::

.900• The simple source radiation is uniform in all

di r e c tions.

(b) Moving Field

( 30 )

rest and in motion are given by Eqs,(19) and (26) respectively for the same

sim p lify i n g assumptions. A c om par-Ison of these two equations shows .. ~

th a t Eq. (2 6) is nothing else than Eq: (19) multiplied by a factor('-Me~ey!

due to the motion of the quadrupole field.

Applyipg this"f a c t or "to the radiation field equations of the

stationary lateral quadrupole and the simple source, we get the

changes in the radiation fields due to trans lation as shown in Fig. 3 for

a Mach number of M_t

=

0.9. It-clear-ly indicates the heavy increase in

emission of sound energy forwards, which is much greater than the

reduction in rearwards emission. Fig. 4 shows the rapid increase of

the sound intensity maxima in the downstream direction and their

shifting towards smaller angles

e

with increasing Mach number of

translation. One also notes a distinct decrease in upstream noise levels.

One of the weaknesses of Lighthill's theory is that it

does not allow for refraction of noise. The turbulence of a jet may

cause severe refraction of noise passing through it.

6.4. 3 Dimensional Reasoning

When the typical flow velocity

U

and length

1.

to be the jet flow exit velocity

U

and the nozzle diameter

d

power output (Eq.( 24)) is seen to be roughly proportional to

8 -s a,

~o

U

<Lo

d

are chosen

, the total

(55)

Thus theory predicts that the sound intensity increases roughly as the

eighth power of the jet exit velocity

U ,

and the square of the jet diameter.

The sound pressure .'? being proportional to the density

fluctuations is roughly proportional (see Eq. (21)) to

U

4 and cl, .

It may be noted here that in the dimensional expression

~o U~

for TV the

Vo

is a reference density appropriate to the

f'l.uctuat ing

flo~

field and hence in the case of heated jets or jets of a gas

different from the surrounding medium (atm os phe-re) is note the "atmos pheric "

density

f.o

exisJ:ing outside the jet. In such cases the re lat ions hip is

c larified by using ~~ U~ for

T\.k

such that the sound pressure

'"P

is

seen to be proportional to the mean density in the jet

( 31 )

and the intensity is proportional to the mean density square d

- I 1;).

U

8 - 5 ...J~ -~

D ' O ' , a, ' U,, ' )(

) 0 ) 0 0 (57)

Finally, using dimensional reasoning an efficienc y of nois e

production

'1.N

can be defined as the ratio of the total ac oustic power output (~o-,I ~~~ U~ Q,~~ cL~) to the supply of je t energy

(~~

,

cL~ U~)

or

(58)

This relationship indicates that turbulence at low Mach numb e rs is a

quite exceptionally inefficient producer of sound (for M -

~.:

J... ,

'1

N ~ 0.3%; for M = " ytN':::.:! 0.01~o). The factor K is ca lled the

"acoustic power coefficient" and defined as

K

= acoustic power (measured)

c

,\j 8, a,-5,

d:1.

) 0 0

(59a)

when the density of the jet flow is equal to that of the atmospher- e,

fo

.

If the jet flow density~" differs from ~o we get

K

=

acoustic power (measured)

~o-I. ~~~,

Us,

0.;5.

a-

.

6.4 .4 Mach Number, Reynolds Number, and

stroUhal Number i

(59b)

.The density variations in the sound radia tion field (at

distances from the turbulent flow large compared with

>-Ax

)

are

proportiona 1 to M 4 (Eq, (21)). Density changes in the flow itself

are of the order of ~..M;l. . This difference indicates that sound

radiation is a Mach number effect, because of its origin fr-om-a

( 32 )

".

An increase in ~e is expected to increase K (ReL 2) ~

si n ce the energy of the turbulence is borneprincipally by frequencies

su c h that the Stroupà.l~. number

(St

-1-

:

cl ·

U-I)

is less than one~ and these frequencies grow gradually with"Re . Counter to this , the corresponding eddy sizes (and hence range of

I

y -

X \

for which the covariance in Eq. (37) is not negligible) are smaller.

From previous experience a constant value of unity may be considered for

St

such that the predominant flow frequency would satisfy the relationship ~ C! U.d,-I . The frequency has a lower limit

of ~ o.a

%

(im posed by the scale of the jet system ) while its high values are damped by viscous actlori. Hence the Strouhal number

ha s a corresponding lower limit and grows slowly with increasing

Reynolds numbe r , The Strouhal number would prove a useful parameter in an analysis of the dependence of

K

on frequency at various "Re

va lue s . The variations in sh a p e of the-frequency spe.ctra might give

.In r or- ma tion as to which aspects of the turbulent flow contribute most to the noise .

6.5 Theory for the Choked and Overchoked Jet

So fa r there does riot.exist a quantitative theory for the choked and overchoked jet comparable to that of Lighthill fot' the

subsonic jet flow. All knowledge available is based on empirical

results.

In the choked jet

flo~ w~

have again a fie ld 'of moving

turbulence (eddies ) as in the subsonic jet. In a ddifi on, a standing shock wave pa ttern is formed which grows in strength the more the jet flow becomes overchoked. Experimental evidence shows a marked change in the nature of th e noise produced when the pressure ratio at the nozzle

ex it is increased beyond the c ritica l (L.e, that at which the jet exit ve lociîy? ifi rst becomes sonic or the jet becomes "choke d"). The conclusion was drawn that an eddy-shock wave interaction may be respons ible for th i s change in the nature of jet noi se ,

Lighthil l (ReL 5) has put the general case of the noise produce d by the interaction of tu r bu le nc e with shock waves on a firm

mathematical basis. He derived formulae for the sound energy

scatter-ed when a so un d wave passes through a turbulent fluid flow or when a unit of turbulence (eddy) pas ses through a shock wave. Ribner

(R e f s. 6 and 78) examined the same problem theoretically. Powell (R e L 7)

used the theorY:of noise gene ra ted by eddyeshock wave interaction to explain the sound production in choked jets as follows:

If a moving eddy crosses a standing shock wave sound is generated according to theory. This will happen at each shock of the