Quadrupole correlations governing the pattern of jet noise

QUADRUPOLE CORRELATIONS GOVERNING THE

.

LtEGTUIGBOUWKU O~ by

G

.

IOT iK:

H.

s.

Ribner

NOVEMBER 1967 UTIAS REPORT NO. 128

•

QUADRUPOLE CORRELATIONS GOVERNING THE PATTERN OF JET NOISE

Manuscript received 28 Ju1y 1967

NOVEMBER 1967

by

H. S. Ribner

UTIAS REPORT NO. 128 AFOSR 67-2159

ACKNOWLEDGEMENT

This research was sponsored by the Air Force Office of Scientific Research Office of Aerospace Research, United States Air Force, under "AFOSR Grant Nr. AF-AFOSR

672-67.

The research was monitored under the technical supervision of Major Donald L. Calvert, AFOSR.

SUMMARY

The effects of convection and refraction dominate the heart-shaped pattern of jet noise. These can be corrected out to yield the "basic directivity" of the eddy noise generators. The observed quasi-ellipsoidal pattern was pre-dicted by Ribner (1963, 1964) in a variant of the Lighthill theory postulating isotropic turbulence superposed on a mean shear flow. This had the feature of dealing with the joint effects of the quadrupoles without displaying them indi-vidually. The present paper reformulates the theory so as to calculate the relative contributions of the different quadrupole self- and cross-correlations to the sound emitted in a given direction.

Of the thirty-six possible quadrupole correlations only nine yield distinct non-vanishing contributions to the axisymmetric noise pattern of a round jet. The correlations contribute either cos 4e, cos2 esin4Bor sin4e directional patterns, where e is the angle with the jet axis. A separation into parts called "self-noise" (from turbulence alone) and "shear-noise" (jointly from turbulence and mean flow) may be made.

The nine self-noise patterns combine as A cos 4e(1) +

A

cos28sin2e(~ +

~ + ~ +~) +

A

sin4e(~~

+

~~

+

3~

+

3~)

= A(cos 2e + sin2ef= A: this is uniform 'in all directions as it must be, arising fr om isotropic turbulence. The two nonvanishing shear-noise correlation patterns combine as B cos 48(1) +

B cos28sin2e(~) = B(cos4e + cos 4e)/2.

']he overall "basic" pattern - self-noise plus shear-noise - thus has the form A + B (cos 48 + cos4e)/2. The dimensional constants A and B are of comparable magnitude: the pattern in any plane through the jet axis resembles an ellipse of modest eccentricity.

Frequency spectra are also discussed, following the earlier work. Finally the predictions·are shown to be compatible with recent experimental results.

1. 2.

3·

4.

5.

6.

7.

8.

9·

10. TABLE OF CONTENTS NOTATION INTRODUCTION GOVERNING EQUATIONS

INFERENCES FROM AXISYMMETRY BASIC QUADRUPOLE CORRELATIONS SHEAR NOISE

SELF NOISE

BASIC BROAD BAND NOISE PATTERN; ASSOCIATED QUADRUPOLE CORRELATIONS

EFFECTS OF CONVECTION AND REFRACTION SPECTRA AND NARROW BAND NOISE PATTERNS COMPARISON WITH EXPERIMENT

v 1 2 3 4

6

7 9 11 12

14

APPENDIX A: Proudman Formulation: Correction of Ribner

(1963,1964)

17

APPENDIX B: Nonradiating and Radiating. Quadrupole Correlations 19

A B B c b(Cf') C c o dir (ijkt) f' M c P p(z,e) p(z,

e

,f) NOTATION self-noise 'constant' ( A

=

shear-noise 'constant' A modified by convection (A

=

C-

5

A) c B modif'ied by convection (B

=

C-5A) c

spectral density of A (~ al(f)df'

=

A) spectral density of B (~bl(f)df'

=

B)

spectral density of' overall self' noise (a(Cf) =

j

al (Cf)d3,l) jet

spectral density of overall shear noise (b(Cf')

=

~bl(Cf)d3,l)

1 convection-Doppler factor (C

=

[(l-M cos e~+

a

2M 2] 2 )

c c

ambient sound speed (local deviations from Co within jet are neglected af ter Eq. (2) )

directional f'actor (eqns. (la) and

(12))

f'requency, cps. or Hz.

correlation integral (eqn. (la))

a scale of turbulence (L3 = correlatiQmv.à)lum~)

convection Mach number (M

=

U jc )

c c 0

total acoustic power (p

=

27T

j7T

sine de

r

d3,l p(,l, e)

7T 7T 0

~et

=

J

'

dep

1

sin9de ( d3X p(,l,e,ep), etc.)

o

~et

acoustic power radiated in direction(e, ep)f'rom unit volume at ,l, per steradian

axisymmetric part of' sound power radiated in direct ion e (any ep) f'rom unit volume at ,l, per steradian

spectral density of' p(,l,e)(jOOp(,l,e,f)df

=

p(y,e)) o

P (y,e,f) c - . P (e,f) c T .. lJ R .. lJ r t

u

5. 1 5 .. lJ

e

p cr T

spectral densi~~ of p(~,e) with allowance for source convection (p (be,f) :::: C p(~,e,Cf))

c

spectral density of directional sound power, per steradian, emitted by jet as a whole (p (e,f)::::

J

pc(~,e,f)d3~)

c jet

instantaneous 'Reynolds stress' (T .. ::::pv.v. + T .. + (1T-C 2p)5 .. ;

approximated herein as p v.

v.)

lJ 1 J lJ 0 lJ

o 1 J

velocity correlation (R .. (rjT) :::: u.u.))

lJ - 1 J

source point separation

(!: ::::

~'

-

i')

time

local mean velocity (xl-direction) effective source convection velocity component of local turbulence velocity ~

component of resultant velocity ::!.. (v. :::: u·

1 1 + U5 i ) component of resultant

observer)

velocity ::!.. in direction x (pointing

component of observation point vector ~ ( I~I :::: x) component of source point vector ~

vector to centroid of source points (~ :::: ~' + _{2 )} ~" pure number (eqn.

(40));

taken as

0.55

special symbol (5i :::: {;

~/î})

special symbol (5 .. :::: 5. 5

J.)

lJ 1

polar angle in spherical polar coordinates x,e,cp (Fig. 1) azimuth angle in spherical polar coordinates x,e,cp (Fig. 1) local pressure in flow

local density in flow

local mean density (assumed uniform)

pure number (eqn.

(18));

estimated as

0.45

toward

time delay between emissions of two-source points (T

=

t' - til); value for simultaneous reception at x, t given by eqn.

(15).

T • • l.J w ( }cpave

J=

Subscripts i,j,k,.t se sh

compressive viscous stress tensor: force in xi direction on unit surface area with inward nor mal in x j direction.

radian frequency (w

=

2nf)

characteristic radian frequency in th~ local turbulence, defined -wflTI

by velocity correlation of form e

time averages are denoted by overbars, e.g., in forming correlations average over azmuith angle cp

symbol separating two expressions that make equal contributions to the jet noise integral; it does not imply direct equality.

tensor subscripts with values 1, 2,or 3

self-noise shear-noise

1. INTRODUCTION

Experiments on the refraction of sound by gas jets (Atvars et al

1965,

Grande

1966)

have given strong and unambiguous support for a very simple model of the directional pattern of jet noise. In brief, the directivity is dominated by the competing effects of convection and refraction. Convection wants to beam the sound waves downstream into a broad fan enveloping the jet, whereas refraction wants to bend the waves out of the jet, weakening the core. The result is a heart shaped pattern.

The effects of refraction and convection can be corrected out of measured jet noise patterns to yield the "basic directivity" of the eddy noise generators. Some recent results for this (Grande

1966)

and for corresponding "basic spectra" (MacGregor, unpublished work) bear a close relationship to recent theoretical results.

The theoretical model (Ribner

1963,

and in amplified ferm,

1964)

is formulated from Lighthill's (195~,

1954)

basic equations, but the develop-ment and some of the assumptions are different. One of the results is a pre-diction of a "basic directivity" pattern that is quasi-circular in each fre-quency band, which is similar to the measured patterns.

Another result is the decomposition of the "basic spectrum" into two primary spectra, one for "self-noise" from the turbulence alone and the other from "shear-noise" arising jointly from the turbulence and the mean shear flow. The proportions of the two spectra vary markedly with direction. Taken in com-bination (with some assistance from the refraction effect) they explain the ob-served deeper pitch of noise radiated at small angles with the jet as compared with 900 . This overrides the opposite effects of Doppler shift.

The Ribner model,alt~ough incorporating Lighthill's quadrupoles P ViVj, does not display them individually. Instead only their joint effects are dealt with. This is accomplished by use of the Proudman

(1952)

formalism wherein P vx2 _{governs the combined effect for the x - direction of emission. The r~­}

sulting simplicity effects a great reduction in the volume and labor of analytical work.

The price, however, is a loss of detail and perhaps even a loss of credibility. The thirty-six possible self- and cross- correlations of the P viv j do not appear separately and their role is bypassed.

The aim of the present paper is to supply this missing detail by areversion from the Proudman form to the basic Lighthill formalism. The

physical model, as in the earlier work, postulates isotropic turbulence super-imposed on a specified mean shear flow. The space-time velocity correlation functions are assumed. All the quadrupole correlations that contribute to the axisymmetric jet noise are evaluated. Their respective directional patterns are combined to produce the joint p~ttern. The results are further broken down into frequency spectra.

The final results for the directional pattern and spectra merely confirm the results - af ter a minor correction - of Ribner

(f963, 1964).

What is new is the display of the relative contributions of various self- and cross-correlations to the sound emitted in a given direction. Also new is the

2. GOVERNING EQUATIONS

Lighthill

(1952, 1954)

has shown that the sound pressure radiated to a point ~ in the far field by a localized unsteady or turbulent flow is given by

p(~, t)

where T .. is a quadrupole strength density,

l.J

T. .

=

P v. v . + T. . + (7T - Cl 215 ) 5 .. ,

l.J l. J l.J 0 l.J

(1)

(2)

that is normally dominated by the unsteady momentum flux p v· v ., e. g. in a turbu-lent jet at ambient temperature. Here Tij is the viscous c6m~ressive stress , tensor, 7T is the local pressure, p the density, _{c q} the ambient speed of sound, vi the velocity, and 5ij

=

0 or 1 according as i f j or i

=

j; the symbol [ ] designates retarded time, i, j

=

1, 2 or

3,

and repeated indices are summed over. The origin of coordinates is taken within the flow.

On retaining only PViVj in (2) the sound power x2 ~/poëo radiated in direction 8,~ iu polar coordinates (per unit solid angle) may be written x. x. x k Xn P(8,~)~ l. J

4

16~p

c

5

x o 0

11

02(p1_{VI VI} k ~

where the first term under the overbar is evaluated at ~I, tI and the second term at

i ' ,

til. The generation times tI, til are suitabily retarded relative to the reception time t, which is averaged over at fixed t. Alternatively an ensemble average may be used.

The product average or quadrupole correlation shown with an over-bar can be expressed (e.g. Ribner

1962)

as a function of the midpoint ~ and the separation in time and space (Fig. 1)

~ = ~ (~I +

i') ;

.!:

=

~I -

i'

T = tI - til

(4)

If the observer di stance x is large compared with the flow dimensions

C T ~.!: •

EX

o

(Meecham and Ford,

1958).

A convenient transformation of

(3)

is then

where p( 8 ,CP

,x..)

= P X.X ,xkxO l . J ~XI n

J

0 ~4

16~

c

5

_x

4

00

OT4

o V V v' v' i j k ~

Here P, p' have been approximated by the constant ambient value pand the

04/0T4

operation is to be applied before insertion of relation (g) for

T

divided by x,

' ... "' .. -.t_

.,.;;....,

The summations xi vi' Xj v. etc. implicit in

(7j

are each merely the component of v in the direction of x.J Thus equation (7) may be reexpressed in the very neat form

P(8,cp, X)

=

po(16

~ co5)-1~ O~:

00

due to Proudman (1952).

V 2 V '2

X X (8)

The quantity P(8,cp, x..) is the acoustic power radiated in direct ion 8,cp (per unit solid angle) from a unit volume element at

x..;

it takes the form (7)

when based on the Lighthill quadrupole formulation, and the form (8) when based on the Proudman formulation. The Proudman form is by far the simpler: the single correlation V_X'2 V_X'2 replaces some thirty-six quadrupole correlations vi Vj v

_k

vt~. T~is simplification was exploited by Ribner (1963, 1964): see Sectl.on

7

and Appendix A here in.

In what follows the quadrupole formulation is reduced to nine basic terms that are evaluated explicitly. The equivalence with the corrected results of the Proudman formulation (Appendix A herein) is demonstrated.

3 • INFERENCES FROM AXISYMMETRY

The sound power emission from a round jet, being axisymmetric, possesses no cp-dependence. There is accordingly no change in (6) on taking the cp-average,

Thus although the emission p(X' 8,cp) from an individual volume element of the jet is not in general axisymmetric, only the cp-average - or axisymmetric part as it were - contributes to the overall axisymmetric emission. Physically, the various volume elements of the jet will mutually cancel - on a time-average basis - all devïations from their respective cp-average sound power emissïons.

The required cp-average of (7) reads

where

(10)

27r

dir(ijk.!)

=

(21Tr1

i

_{(Xi x j}

"k

xi

X4 ) dqJ

Since

Xl = X cos 9, x

2 = X sin 9 cos cp , X 3

=

X sin 9 sin cp ( 11)

it is found that dir(ijkt) is nonzero only when ijkt are equal in pairs. The nonzero directional factors are

dir(1212) : (1/2)cos2esin2e = dir(1313 -. (12)

dir (1111) - cos40e

1

dir(2222)

=

(3/8)sin4e

=

dir (3333) _ dir(2323) = (1/8)sin49

together with those obtained by permutations ofl the indices, which do not alter the value.

It is remarkable that equation (7) can predict a negative power emission in certain directions for a single quadrupole correlation, when ijkt are not equal in pairs. This is not a spurious effect. Instead it points up the fact that the cross-correiations arise from the average of the square of a sum quadrupole terms: a single cross-correlation by itself is physically

inadmiss:ib1é'~. Thus negative contributions fr om one cross-correlation are

com-pensated by positive contributions from autocorrelations and other cross-correlations.

The expanded form of (10) reads

+

where

~

implies the proportionality factor _{po(16n2 co}5)-1 on the right hand side. In this expression we have ~sed relations like

I

1212dir(1212) = I2112 dir(2112)

=

I1221 dir(1221)

=

I2121 dir(2121)

(14)

because neither the directiona1 factor nor the value of the integral I .. kt is alter:ed by permutation of pairs of indices at ~'(ij) or

i'

(kt); that1ïs, each of the four is an equivalent noise generator, and their sum may be replaced by I 1212 dir(1212) with a weight factor of four. In addition we note that

vi v· v

_k

_{vl and vk}

Vt

vi v~ . possessidentical integrals over r - space if,

anti~ipating

a later step, *e set T 0 in the integrand of

I.~kt.

Thus, for

example ;tJ

(15)

so that I l122 replaces the sum with a weight factor of two. For the complete array of weight factors (designated "permissible permutations of ijkt") see equat'ion (17) below.

Part of the formalism leading to equation (13) and (13) itself are similar to steps in the work of Kotake and Okazaki (1964), which came to the present author's attention af ter developing the present independent approach. However, the differences - particularly in the later steps - exceed the simUlarities, and the final results are very divergent.

4. BASIC Q,UADRUPOLE CORRELATIONS

Equation (13)completes the reduction of the directional acoustic power emission from unit volume to an expression involving nine basic correlation integrals. The next step is to set forth expressions for the correlation func-tions involved. Theconstituent velocities may be written, e.g.,

v. = U B. + U.

1. 1. 1. (16)

where U is the local mean velocity which is directed along the Yïaxis and u. is the contribution of the turbulence, assumed locally homogeneousand isotr~pic in our model; the special symbol Bi

= 1 if i

=

1 and is otherwise zero. Thus the cases where i, j, k or

t

= 1 lead to a multiplicLty of terms which are dealt

with in Appendix B. A number of these cont ribute nothing to the noise: either they are constant and differentiate out, or they possess a zero integral over r because of the postulated isotropy. ~he surviving terms contributing to the

lijkt integrals (10) appearing in (13) are

Shei.à.:rt N6i~ë SeLF N6isè Permissible Permutations of ijkt vl V_{l vi vi}

1

=

4 uU ' ului + u 2U '2 _{1 1} 1 v_l V_{2 vi v}

2]

=

UU' u₂u₂ + U_l U₂

_UJ..

u' ₂ 4 v l v3 v' 1 v

3

j

=

UU' u3u

3

+ ul u3 u' 1 u' 3 4 v l vl v' 2 v

2

1

=

~2u22 2 v l vl v' 3 v

3

1

=

u 2U '2 1 3 2 (17) v_{2 v2} v' v' ₂

₌

u 2U '2 1 2 2 2 v 3 v3 v' 3 v' 3 u 2u '2 3 3 1 v₂ v 3 v' 2 v' ₃ u2 u₃ u2 u' ₃ 4 v 2 v 2 v

3

v:3 _{u2 2U3} 2 2

where the notationj

=

signifies (following Ffowcs Williams and Maidanik 1965) that the right and left hand sides make equal contributions to the noise integral: terms making no contribution have been discarded from the right hand side. The meaning of the designations shear noise and self noise will now be discussed.

5. SHEA1:\ NOISE

Consid er the terms like

uur

ului in (17). For isotropic ~Urbu­

lence it is known (e.g. Batchelor 1953) that the volume integrals of ului, etc., over r - space must vanish. If, then, the mean-flow factor

uur

were constant with

E

the cited terms would contribute exactly nothing to the noise integrals. The contribution will be nonzero only when the mean flow is nonuniform - i.e., possesses shear. The noise associated with such source terms is thus called shear noise.

The terms like u_l?ui2 , ul U

3 ui u~, etc., in (17) contain turbu-lent velocity components only and are independent of the mean flow. The noise

associated with such source terms may be called self-noise. (Lilley 1958 intro-duced the phrases shear-amplified noise and self noise; the formalism was quite different so that the correspondence is rather loose).

We shall evaluate in this section the shear-noise contribution to the directional sound power

P(X,8)

from a representative unit volume in a jet. For the shear noise the location

X

of this volume element is restricted to lie in the annular turbulent mixing region of the jet, at the radius of maximum shear (cf. Fig. 1). This region is known (e.g. Ribner 1958, Dyer 1959) to make the major contribution to the total jet noise. Following Ribner (1963,1964) the mean flow correlation at X is taken as

(18) in the present model of jet flow; the parameter ~ was estimated to be of the order 0.45.

Equation (10) for P(8,y) refers to a stationary reference frame. It will be more convenient, however, to employ a frame moving with the local convection speed Uc

=

_{Mcc o , in which the correlations take their simplest form.}

The Lighthill transformation (1952) allows the d4/dT4 operation in (10) to be -5 carried out in the moving frame and applies a multiplicative factor (1-Mccos8) (as corrected by Ffowcs Williams 1960) and an exaggerated time delay. We shall reduce this factor to unity by allowing Uc to approach zero so that (10) is formally unaltered, being changed only in interpretation. The effects of con-vection at finite Mc will be approximated later. With this low-speed stipulation it can be argued that the ratio (eddy size)/(wave length of sound) is small com-pared with unity. This implies that the time delay is negligible throughout the volume - approximately the correlation volume - that makes the major contri-bution to the integral, so we may set T

=

0 therein.

We postulate that the two-point velocity correlations in (17) are factorable into a space factor and a time factor:

-~f

IT I

=

Rok(r,T)

=

Rok(r) e

1. - 1.

-in the specified reference frame mov-ing with the loc al convection speed U. The time factor approximates exper.imental data (Davies , Fisher & Barratt 1963J

except for the cusp at T

=

O.

The space factor in (19) is taken to be appro~riate to homogeneous isotropic turbulence. Thus Rik(E) must have the general form (e.g. Batchelor 1953)

(20)

where f is some universal function of r; in our model of turbulence this is taken

to be 2

_TIT2/L . r2 _r 2 + r 2 + r 2

f::;e

, =

1 2 3 (21)

which has been used by Lilley (1958).

Equations (18) to (21) assemble all the data needed for evalua-tion of the shear noise integrals. Therequired integrals take the form

(11111) sh = 4 w_f

4Joo

uur

(r)

R₁₁ (E) d3E (22) and two similar integrals (11212) h and (1l~13) h by virtue of (10), (17) and

(19), together with the low-speedsapproximationsT ~ O. The integration gives

( ) _{11111 sh = 2} W 4 3 2 - 2 ( )-3/2 4 )

f L U u1 rr l+rr = (11313 sh (23)

and (1

1212)sh vanishes. These results may be inserted into (13) interpreted as

P(~,8) = P(~,8)sh + P(~,8)se

Shear Noise Self Noise

(24)

With all the (Iij~t) _{equal to zero except (I1ll1 )sh and (11313)Sh the} shear-noise part of (13) t~es the form

where

= B(cos48 + ~ cos28sin28) = B ~ (cos48+ cos28) B P OWf4 L 3_U2 _{~ rr} 87fc 5 (1+rr)3!2 o

j

(25) (26)

is constant with respect to 8; its value may vary with source position ~ in the jet.

6 . SELF NOISE

We proceed to the evaluation of that part of P(~,8) arising from turbulence alone - free of cross-coupling with the mean flow - and labeled self noise in (24). The contribution of self noise to lijkt' by (10) and (17), is

under our assumptions. There are nine of these: ijkt = 1111, 2222, 3333; 1122, 2233, 3311; 1212, 1313, 2323 • . Members in each set of three are equal by isotropy.

We further assume nor mal joint probability of u. and ~, from

which it follows that 1

+ u; u~ • . u . u..'

... XI J ~.K

=

R. . ( 0) R,. n ( 0) + R. kR . n + R. oR • k

1J r.JJ 1 JXI 1X1 J

(28)

(see e.g. Batchelor 1953). Thus for example

u1u1uiui = (?)2 + 1 2 R11 2 u

2u

a

u

2

u

3

=

0 + R22R33 + R23 2 since R .. (0)

=

u

2

1J 1 for i

=

j and = 0 for i

f

j

- wflT

I

By virtue of the time factor e in (19) and the results for Rij(O) the integral (27) reduces to, for T = 0,

(I. .,.0) = (2W

f)4(R.kR. o + R. oR·k)d3r

1JlVI se

J

1 JXI 1X1 J

-The integrals come out to be (1 1111) (12222) ( ) _ ( )~ -3/2 (~)2 3 = 13333 - _2wf 2 _u1 L se se se (1 1122) _se . (12233) _se

(I

1133 se

)

=

l

8 (2w f

)42-3/2(~2)2L3

(1₁₂₁₂) = (1 1313) = (12323)

=

ïb

7 (2w f )42-3/2 (ül2)2L3 1 se se se

of which the first was eva1uated in.cR.ibner (1964).

insertion

With p(l,e) split into shear noise ~~~2self. noise by (24),

into (13) yie1ds, on absorbing (2wf)4 2 / (u12)2L3 into a factor A,

p(I,8)se :

A[COS48 + 2cos2esin28

(~

+

!ö

+

Îb

+

Ïb)

where

+ sin48

(~~

+

~~

+

~2

+

~2

)

=

A(cos28+ sin2_e)2

= A A

=

../2Po W f 4 (ü()2L3 4n-2 c o5 (32) \ _ I (33)

and like B is constant with respect to

e,

but may vary with source position

l:

in the jet.

7. BASIC BROAD BAND N01SE PATTERN; ASSOCIATED QUADRUPOLE CORRELATIONS

Upon adding the contributions of self-noise (~~) and shear-noise (25} there results

p(l:, e)

= A

SELF

+ B(cos4

e

+ cos2

e)/2

SHEAR

(34)

for the total noise power from emitted unit volume at

l:.

A and B are of com-parable order of magnitude (Ribner

1964).

This is termed a "basic" pattern because the nprmally important effects of eddy convection and refraction of the sound by the mean flow are not allowed for: these are dealt with in a later section.

The two parts of the directional pattern

(34)

are shown in Fig. 2.

One part is a non-directional contribution from the self-noise (turbulence alone); the other is a dipole-like contribution from the shear-noise (turbulence acting on mean flow). The combined pattern for A

=

B is a quasi-ellipsoid with the

~ong axis in the direction of the jet axis.

Precisely the same pattern

(34)

is obtained but far more simply -by use of the Proudman formulation (8) in place of the basic Lighthill formulation

(7).

The procedure is indicated in Ribner

(1963, 1964)

as amended by Appendix B herefn. The earlier analysis led to

p(v,e,~)

=

A + B cos4

e

JI... cp =0

which is the basic directional pattern in a certain plane ~

=

0: this was mis-takenly taken to be the ~-average pattern

p(l:,e).

The inappropriate form (35)

has figured in some comparisons (e.g. Grande

1966)

but fortunately the differ-enee is not large.

Let us now consider the basis for the sound pattern

(34).

1t is clear that the isotropic directivity

(32),

p( v, e) • = A

JI... se . constant,

of the self noise is a necessary consequence of the isotropy of the turbulence. But it wasfar from clear a priori how the self- and cross-correlation of the quadrupole strength densities - the source term in the noise integrals - would be proportioned to bring this ab out • Let us examine this point.

The (I, 'kP,) comprising the quantitative part of

(32)

are de-fined in (10) as integ~alsSinvolving self- and cross-correlations of quadrupole strength density (Tij)se

=

_{Po(ViVj)se. Tll , T}

_{22 ,}

T₃₃ are longitudinal quadru-poles, and the others are lateral quadrupoles.

It will be convenient to rewrite (10) - specialized to the self-noise part, but omitting the ( )se designation to simplify the notation - 'as

P 21

J

êl

4

T T '

d3~

-- (2w f

)4

/;T;"JoTk'

> .

L 3 o ijk.€ =

êh

4 ij :K.e " ... (36)

on noting th at

êl4/êlT4

merely applies a multiplicative factor. Here

<

TijTk,e

>

is an effective spatial mean value of the correlation such that an

arbi trary "eddy volume" L3 characterized by this value inside and zero outside will radiate equivalently to the actual flow. Accordingly, in terms of two-point correlations 11111 '"

<

Tll Til>

=

11122 '" <Tll T22

>

etc. self-correlation~ longitudinal quadrupole cross-correlation, longitudinal quadrupoles self-correlation, lateral quadrupole

Thus the relative contributions of the different quadrupole correlations to the directional factors in the self-noise (32) may be discerned, with the aid of

(13) and (36):

cos4_e

<T ll Til

>

"

'" 1

cos2esin2e: _<T12Ti2

>

_{+ (T13Ti3>} + <T T' _{11"22 +}

Î

<TT' _{11 33}

>

-

_"'"8

7

-tl

_8+8*8 1 ,1

sin4e:

<'r

_22T22

>

_{+ (T}

33T33) +

<

TT' ) 23;;3 +

('r

22 33 T'

>

'" 32 +32 12]}2 + 32+ 32 7 1 That is, for the cos4e term only the self-correlation of the longitudinal quadru-pole T 1 contributes. The cos2esin2e term depends on the self-correlations of

two lateral quadrupoles T12 and T13 and two cross-correlatioris (TllT22

>,(

TllT₃₃

>

of longitudinal quadrupoles, in the proportions 7: 7: 1: 1. Finally,

the sin4e term depends on the self-correlations of two longitudinal quadrupoles T22 and T3"1 and one later al quadrupole T_2V and one cross correlation

<

_T22T33

>

of longituainal .quadrupoles, in the propörtions 12: 12: 7: 1. Note that the nine correlations cited here represent through the equivalences (14) and (15) a much larger number of corr~lations, which are not distinct.

The right-hand sides of (37) add up to 1, 2, and 1, respectively, giving the pattern cos4e + 2 cos2esin2e + sin4e which equals unity. Thus the inference that the self-noise must be independent of e is confirmed by the detailed examination.

Turning now from the self noise to the shear noise (25), we may make a similar interpretation. The effective mean quadrupole correlations con-tributing to the directional factors are

cos4e

<T ll Till

<

P02UU' ului)

sh (38) cos2esin2e 013Ti3> = (P0 2_{UU' u} 3u

3 )

sh (37)

and their contributions are in the ratio 1: 1/2 respectively. A third quadrupole correlation

<

T12Ti2

>

has a zero integral (or mean). That is, for the cos4e term only the self-correlation of the longitudinal quadrupole PoUul contributes. For the cod2 esiife only the self -correlation of the lateral quadrupole p_oUu'1

contributes, The form of these shear-noise quadrupbles deserves special note: they _~nvolve a cross-coupling between the mean flow U and the turbulence ui' They correspond physically to a transport of mean flow momentum by the turbu-lent fluctuations.

The reasons for the basic broadband noise pattern intensity ~ A

SELF

+ B(cos4_{e + cos}2_e)/2,

SHEAR

may now be summed up. (By "basic" we mean convection and refr.action effects are excluded, as mentioned earlier). The non-directional self-noise Aresults from the joint contribution of nine quadrupole correlations having cos4_{e, cos}2_esin2_S,

or sin4e directionality.., The proportions are such that they combine to the nondirectional pattern. The directional shear-noise B (cos4e + cos2e)/2 results from two quadrupole correlations having cos~e and cos2esin2e respective

directionalities.

Thus in the present model - taking due account of the shear in the mean flow - the noise pattern is contributed to by a number of quadrupoles, both longitudinal and lateral. The average spatial "phasing" am ong them is automa-tically taken into account in the magnitudes of the cross-correlations. This leads to a definite resultant directionality, equation (34), for the noise from unit volume in the mixing region of a jet.

These results contrast with the Lighthill (1954) model which suggests that the single T12 lateral quadrupole dominates in the mixing region. In his view" ... the most important term in the rate of change of momentum flux [êl/êlt (Tij)] is the product of the pressure and rate of strain " . The higher frequency sound from the heavily sheared mixing region close to the orifice of a jet is found to be of this [single lateral quadrupole]character.tI

The axisymmetric part of the associated directional pattern has a cos2esin2S directionality. This looks like a four-leaf clover and bears little resemblance to the quasi-el1ipsoidal pattern deduced herein (Fig. 2).

For the most part we must rely on experimental evidence to choose between the two models. Af ter suitable preliminaries, Sections 8 and 9, com-parisons with experiment are given in Section 10.

8. EFFECTS OF CONVECTION AND REFRACTION

The analysis so far has been postulated on a negligibly small flow Mach number so that the effects of source convection do nqt appear. The result

obtained for the basic broadband directional pattern radiated from unit volume at

may be generalized to allow for an eddy convection speed Uc the mixing region) in the form

where

pc(l,e) = C-5 [A convection

factor

+ B (cos4

e

+ cos 2e)/2] Basic Pattern

1 1

C

=

[(1 - Mccose)2 + w~2L2/~02]2

=

'O(1-Mccose)2 + a2Mc2]2

(40)

is a refinement of Lighthill's well-known factor (1 - Mc cos e) (Ffowcs Williams 1963),(Ribner 1962). The basis for this generalization is developed in Ribner (1964) in connection with the pattern (35). (The later developments in Section 9 mayalso be traced to this same reference.) Here wf and L are a characteristic frequency and scale of the turbulence (not necessarily those in the earlier

analysis), and the equality of the two forms of (40) defines the nondimensional parameter á. The empirical value

a

=

0.55 gives good agreement with experiment for turbojets (see Comparison with Experiment).

These broadband equations and the narrow band equations to follow omit the powerful effect of refraction of the sound by the jet mean velocity field; t his dominates for

lel

< 300 for ·all but the lowest frequencies (cf. Fig.

3). The refraction effect has been evaluated quantitatively by means of

experi-ments with a pure-tone point source placed in an air jet (Atvars, Schubert, Grande and Ribner 1965, Grande 1966). The result may be expressed as a

REFRACTION FACTOR =

(~ntens~ty

at

eOe)

(41)

~ntens~ty at 90 f

for frequency f which multiplies the narrow-band version of (39) (see later).

Thus experimental narrow band sound measurements should in effect be multiplied by the inverse of (41) to correct out the refraction effect before comparison with theory.

The effects of convection and refraction are illustrated in ~ig.

4; this shows the sequence as the basic pattern is modified first by convection and next by refraction to produce the final directional pattern of jet noise. The figure refers to the sound power radiated from unit volume in the mixing region, but the spatial pattern may be taken as typical for the jet as a whol~.

9. SPECTRA AND NARROW BAND NOISE PATTERNS

The constants A and B in the basic broad band spectrum (34) radiated from unit volume may be decomposed into their spectral components (Ribner 1963, 1964)

Self Noise Shear Noise

The spectrum a1(f) is predicted on his model to peak an octave higher than the spectrum b1(f), because the turbulence velocities appear quadratically in the self-noise terms as against linearly in the shear-noise terms: the frequ~ncy

These spectralare generalized to allow for eddy convection by writing

where the reception frequency f now incorporates an "effective11

Doppler shift C-l relative to the source frequency Cf. This effective Doppler shift is taken as the shift of the spectrum peak, which is less than the true Doppler shift

(1 - Mc cos e)-l of the constituent lines because of a spectrum distortion. The overall convective a~plifiaation C-5 contained in Ac_md Bc can be seen to consist of an amplification C- plus a "Doppler" shift C of spectral elements. These points are elaborated in Ribner (1964).

If these narrow sharply peaked spectra are summed for all the radiating volume elements of a jet, the overall spectrum is obtained.

Symbolically,

j

a(Cf)

=

.

.

tal (Cf) d3~

Je

b(Cf)

=

J

_{b l (Cf)}

d3~

jet

The narrow elementary spectra al(Cf), bl(Cf) summed over in

and a(Cf), b(Cf) peak at progressively lower frequencies as the source distance from the nozzle increases. Thus a(Cf) and b(Cf) cover a broad frequency band.

volume

To summarize, the basic broad band directional pattern fr om unit

p(~,e)

=

A

self

+ B(cos4e + cos2e)/2) Shear

(34)

is effectively the integral over frequency of the narrow-band pattern (or spectral density for given e)

(42)

self shear

Allowance for eddy convection at Mach number Mc generalizes this to

pc(~,e,f)

= C-4 [al(Cf) + bl(Cf) (cos4e + cos2e)J2] (43)

self shear

Integration over source-position ~ yields the narrow-band pattern (for spectral density for given e) emitted by the jet 'as a whole as

p (e,f)

c

=

C-4

[a(Cf) + b(Cf)(cos4

e

+ cos2 e)/2]

self shear

(44)

The general forms of the component spectrum functions a(Cf) and b(Cf) have been discussed by Ribner (1964) based on tlle work of Powell (1958) ("f2 and f-2 laws").

A simplifying assumption in the foregoing must be pointed out. In the step from (43) to (44) - the integration over the jet - it is implicit that the self-noise and shear-noise directivities derived for the mixing region apply without change in the developed jet. In addition, the convection factor C

is to be taken constant and associated with the constant convection Mach number Mc ~ (~)U./c along the annular central surface of the mixing region: the decay

of Mc in ihe developed jet is neglected. These assumptions are made for expediency and justified in part on the ground that the bulk of the noise originates from the mixing and transition regions. The remainder of the noise - essentially the low frequency part of the spectrum - originates from the developed jet and for this the assumptions are clearly faulty.

We content ourselves herein with attributing the directivity in the general form (44) to theory, with the spectral forms of a(Cf) and b(Cf) being only loosely specified. A closer specification cannot be made with confidence in view of the oversimplifications in the theory, e.g. those of the last para-graph together with the assumption of isotropy in the turbulence. The integrals of a(Cf) and b(Cf) over frequency are predicted to be approximately equal 80 tha~A ~ B in (34). Further, the peak of the a(Cf) spectrum (self-noise) should be substantially above the peak of the b(Cf) spectrum (shear noise). The rela-tive shift according to the notions of the first paragraphs of Section 9, should be a fair portion of an octave if al(Cf) and bl(Cf) were in the same proportion for all parts of the jet. There are various uncertainties here so that the theory must remain only qualitative as to this spectrum shift.

The present viewpoint is se en to represent a relaxation of the over-restrictive mathematiball .. ind>del given in Ribner (1964). The major features are retained, but some flexibility is allowed as to the details.

10. COMPARIS ON WITH EXPERIMENT

The broad band pattern (34), as modified to allow for the source convection, has been given as (39)

pc(~,e) =

C-5 [A + B (cos4e + cos2e)/2 ] convection basic pattern

factor

This refers to the sound power radiated from unit volume in the mixing region, but the spatial pattern may be taken as typical for the jet as a whole. This pattern with A

=

Band

a

(in C, Eqn. (39) ) taken as 0.55 for best fit is com-pared with measurements for several turbojets in Fig. 5, adapted from Pietra-santa (1956) and Ribner (1963, 1964). The agreement is quite good over the wide range from e

=

400 to 1800 , covering a 200-fold variation (23 db) in intensity. The failure below 400 is due to refraction (Fig. 3).

The powerful convection factor C- 5 together with the refraction jointly dominate the pattern at these turbojet flow speeds. The basic directional pattern 1 + (cos4e + cos2e)/2 accounts for only 3 db of the 23 db variation.

Thus although the final pattern is markedly directional the basic pattern (which omi ts convection and :.'refraction effects ) is quasi-circular.

The narrow band spectrum (45) reads

P (e,f)

=

c-

4 [a(Cf) + b(Cf) (cos4_{e + cos}2_e)/2]

c

This corresponds to the passage of the broad band radiation in direction e through a filter of unit band width centered at frequency f. The possibility of testing

an equation of this kind (based on

(35)

in place of

(34))

motivated the work of MacGregor (not yet published); he has been making comparisons of the theory with

spectral measurements on a

3/4

inch air jet in the Institute for Aerospace Studies anechoic chamber.

I am indebted to MacGregor for Figures 6 and 7 here in. is a two-component spectrum obtained by fitting the theoretical model experimental data measured at 9

= 45

0 and

90

0_{• We identify the a(Cf)} the self noise aild',lthe b(Cf) peak with shear noise. The shift of the peak well above the shear noise peak is in general agreement with the argument. The amount of shift is, however, much less than the factor suggested in Ribner

(1963, 1964).

Figure 6

(45)

to his peak with self noise theoretical of two

The factor of two is clearly peculiar to thelc~ice of the mathe-matical model. If the assumed correlation time factor e-wf T (cf. first para-graph of Sec. 9) were replaced by e_wf2T2, for example, the factor 2 would be replaced by ~ in better agreement with figure

5.

Figure 7 shows the narrow band "basic" directional pattern at f =

1500

,

Hz. The dashed curve is ~btained from

(44)

wi th the empirical a( Cf)

and b(Cf) of Figure

6,

and with C- omitted. The solid curve is direct experi-mental data af ter cerrection for refraction (Grande

1966)

and for convection by multiplication by C. It is clear th at

(44)

has been used essentially as an interpolation - .. extrapolation scheme, the two curves having been matched at

45

0 and

90

0• One can conclude, however, that the a(Cf) + b(Cf) (cos4e + cos2e)/2

form is adequate to predict the general shape of the basic directional pattern. This pattern like that inferred for Figure

5,

is seen to differ little from a circle or mild ellipse: there is no tendency here toward a four-leaf clover (cosFesin2_{e) pattern as suggested by Lighthill}

₍₁₉₅₄₎

_{for the high frequency}

noise. Similar quasi-circular patterns were found experimentally (with use of less accurate reduction methods) by Grande

(1966).

The theoretical predictions for spectra and dirèctivity mayalso be compared with recent results obtained from hot-wire measurements in a jet by

W.T. Chu

(1

9

66).

In a very exacting investigation Chu measured in detail the space-time correlations of turbulent velocity figuring in the Lighthill integral in the Proudman form

(8).

He was able to evaluate a development of this inte-gral and also its Fourier transform to obtain the sound pressure

auto-correlation and spectrum. This is a major advance; however, the results can still be regarded as only semi-quantitative, the uncertainties arising mainly in the evaluation of a fourth derivative of an experimental curve.

Chu's results are a prediction of the sound radiated from unit volume - and in three specific directions 00,

45

0, 600 - based on measurements

of turbulence within the unit volume. The spectrum he obtained - broken into its two constituents of self noise and shear noise - is shown for the 9 = 00 case in Fig. ~. The shape s are qui te similar to those predicted by the theore-tic al model (Ribner

1964,

Fig.

16)

fo~ al(f) and bl(f). In this case the sepa-ration of the peaks is even greater than the one octave expected from the theory.

The corresponding broad band pattern obtained by Chu is shown in Fig.

9.

This is compared with the pattern

(39)

predicted herein as being emitted from unit volume, with A set equal to B. The agreeme;ntt of the directionali ty is qui te good.

Since Chu used a very low speed jet (142 ft/sec) the convection factor C-5 is near unity. Thus the bracketed "basic" directional pattern accounts for most (3db) of the small variation between 0° and 90°. Here again the quasi-circular nature of this basic pattern is evident.

APPENDIX A

Proudman Formulation: Correction of Ribner (1963, 1964)

The directional pattern (35) is the radiation pattern fr om a

volume element of a jet evaluated in the plane ~ = 0 containing the element and,

the jet axis. This was obtained in Ribner (1963, 1964) on the basis of the very

simple Proudman formulation

(8);

the result was confused with the ~-average. In

what follows the procedure is generalized to obtain the radiation pattern in a

plane through the jet axis making some arbitrary angle ~ with the cited plane.

Then this pattern is averaged over ~ to obtain the axisymmetric pattern (34).

Paralleling the procedures herein, but bypassing many of them,

Ribner (1964) shows that the integral in (8) is proportional to

2 W_f

1

Rn2(!:) d3r + -è:- wf4cos2e

J

DU' (!:) _Rl1(!:) d3r (Al)

self-noise 00 shear noise

where ul and the l-axis has been chosen parallel to the vector ~ drawn 5rom the

origin to the observation point. The proportionality factor is p _{, 0} (n2c ₀

)-1,

which corrects amisprint.

It will be necessary to re-express (Al) relative to the present

reference,.flI1arne ~in,:whioh;,the l-axis is parallel to the mean jet flow U. Because

of the assumed isotropy of ~, Rlf in 'the first integral can be retained without

change of form, sin:~ its volume integral will be unaltered. The first

inte-gral is then just 2 times the first of equations (31):

self -noise term (A2)

This is a constant and is unaltered upon averaging over ~.

For the second integral in (Al) the transformation is taken in two steps. First a rotation is made through an angle e in the plane containing

~ and

Q

to bring the l-axis into alignment with

Q.

Call this the ri' r

_2,

r

₃

frame. Next a rotation is made about the l-axis such that the new r

₂

-axis

makes an angle ~ with the ~,

Q

plane. Call this the rï, r

_2,

r

₃

frame. The

connection between coordinates in this and the initial r

l , r2, r3 frame is r l = rï cose + r

2

sine r₂ =-rï sine + r

2

cose r"sin~ 2 cos~ - rIt 3 sin~ - rIt 3 + rIt 3 sine simp cose sin~ cos~

These are to be inserted in Rll of (20) and (21)

. Rll (~) = u₁2 [1 - (rrr2/L2) + (77T₁2/L2) ]

(A3)

(A4)

In doing this we drop the " .so that the final r

l , r2, r frame is identified

with the r

1TT 2 rrr 2 rrr 2

1 _{s 2} 2 _{s 2} _3_ _{s 2 (A5)}

----='"2 _L = 1 _L2

=

2 _L2

=

₃ is

Rll (:!:) =~ ₁ [l-s2 + s 2cos 2e ₁ + s 2sin2ecos2@ ₂ + s ₃ 2sin2esin2~

_s2_S2_S2

+ cross-product terms] e 1 2 3 (A6) Upon combining (A6) and (18) the second integral of (Al) may now be evaluated. The cross-product terms, being odd, will integrate to zero. The integration yields

w 4U2~ cr L3

f 1

shear noise term

=

~---~~~~-8(l+a-)3/2

This reduces to equation (9.14) of Ribner (1964) in the plane ~ 0, as it should.

Now tion of the volume and the ~-average

if we average overall azimuth angles ~ describing the orienta-element around the jet mixing region, sin2~ averages to 1/2, is, af ter reduction,

shear noise term

=

w 4U2

U'"""2

cr L 3 f 1

(A8)

The sum of equations (A2) and (A8) are equivalent to equation (34) of the main text.

APPENDIX B

Nonradiating and Radiating Quadrupole Correlations The general quadrupole correlation

V V v , v'

i j k .t

with the special notation

etc. reads in expanded form

5_1. . U U u , u' i j k .t

{~

+ U( 5 . u. 11.' u' 1. J ! C t + 5

_it

_uJ.u.' !C (Bl)

upon noting th at ui

=

_{u j}

=

uk U]

=

0 by definition. The terms in U2 , U'2and U2U,2 are constant wi th

-f:~re

not sources of sound; they will be eliminated by the

d,-dT

Á _{operation of (10).}

In order to deal with correlatibns~ like U u.u~ul we first postula(oo.; that UjUkUl is factorable into a function of T h1.me delay) and

function of E (cf. (l9)). Then setting

T

=

0 af ter applying

d

4

/dT

4 as speci-fied in the text leaves (10) as merely a spatial integral. Next we reverse the coordinate transformation

(4)

reverting to the original coordinates X' and

i'

appearing in (3). Thus d3X' replaces d3X in (6) and d3i' replaces d3E in (7)

and (10 )for these correlations. The value of P~

e

,cp) will be unal tered thereby. Since U refers to point X' it is invariant as X' is held fixed while i'is varied in performing

u' _k u'

.t

For the assumed homogeneous isotropic turbulence this reduces to the form

and the volume integral of the triple correlation vanishes (Batchelor 1953). A1together, then, those corre1ations (designated by the symboli~)

contributing to the noise integra1 are contained in

viv jVk vi

J

=

Ui ujuk ui

(B3) Of these the correlations contributing to the net radiation are those with indices as selected in (12) or (13). Insertion of these values of ijkt gives the nine basic se1f- and cross-correlations of equation

(7).

Permissib1e num-bers of permutations of ijkt are tabulated, to be used as weight factors.

REFERENCES

Atvars, J., SChubert, L.K., Grande, E. & Ribner. 1965. "Refraction of Sound by'Jet Flow or Jet Temperature" , University of Toro:qto Inst. Aerospace Studies, UTIAS TN 109; NASA Contractor Rep. NASA CR-494 (1966).

Batchelor, G.K. 1953. "The Theory of Homogeneous Turbulence" Cambridge Univ. Press. Chu, W.T. 1966. "Turbulence Measurements Relevant to Jet Noise". U. of Toronto,

Institute for Aerospace Studies, UTIAS Report No. 119.

Davies , P. O.A.L., Fishe-r, M.J., and Barratt, M.J., 1963. "The Characteristics of the turbulence in the mixing region 0f a round jet". J. Fluid Mech. 15, 337-367. Dyer, I. 1959. "Distribution 0f Sound Sources in a Jet Stream". J. Acoust. Soc. Amer.

31, 1016-1021.

Ffowcs Williams, J.E. 1960. "Some thoughts on the effects of aircraft motion and eddy convection on the n0ise from air jets". Univ. Southampton, Aero. Astro. Rep. no. 155.

, Ffowcs Wil1iams, J.E. 1963, "The Noise from Turbulence Convected at High Speed", Phil. Trans. Roy. Soc. (London) Series A, 255, 469-503.

Ffowcs Williams, J.E. and Maidanik, G., 1965. "The Mach Wave Field Radiated by Supersonic Turbulent Shear Flows", J. Fluid Mech. 21, 641-657.

Grande, E. 1966, "Refraction of Sound by Jet Flow and Jet Temperature 11", Univ. Toronto Inst. Aerospace Studies, UTIAS TN 110, NASA Contractor Rep. NASA CR-840 (1967). Kotake, S.G, Okazaki, T. 1964. "Jet Noise", Bull. Japan Soc. Mech. Engrs.

1,

No. 25,

153-163.

Lighthil1 , M.J. 1952. "On Sound Generated'Aerodynamical1y: 1. General Theory", Proc. Roy. Soc. A211, 564-587.

Lighthil1, M.J. 1954 "On Sound Generated Aerodynamically: Ir. Turbulence as a Source of Sound", Proc. Roy. Soc. A222, 1-32.

Lilley, G.M. 1958. "On the Noise from Air Jets" Aeronaut. Research Council, ARC 20, 376-N40-FM2724.

Meecham, W.C. & Ford G. W. 1958. "Acoustic Radiation from Isotropie Turbulence"

J. Acoust. Soc. Amer. 30, No. 4, 318-322.

Pietrasanta, A.C. 1956. "Noü;e Measurements around some Jet Aircraft" , J. Acoust. Soc. Amer. 28~,,434-442.

Powell , Alan 1958. "Similarity Considerations of Noise Production from Turbulent Jets, Both Statie and Moving", Douglas Aircraft Co., Rep. SM-23246; abridged in J. Acoust. Soc. Amer. 31,812-813 (1959).

Proudman, 1. 1952. "The Generation of Noise by Isotropie Turbulence", Proc. Roy Soc. A214, 119-132.

Ribner, H.S. 1958. "On the Strength Distribution of Noise Sources Along a Jet". U. of Toronto, Inst. of Aerospace Studies, UTIA Rep. 51 (AFOSR TN 58-359, AD 154264); abridged in J. Acoust. Soc. Amer. 31, 245-246.

Ribner, H.S.

1962.

"Aerodynamie Sound from Fluid Dilatations - A Theory of Sound from Jets and Other Flows", U. of Toronto, Institute for Aerospaee Studies,

UTIA Rep.

86

(AFOSR TN

3430).

Ribner, H.S.

1963.

"On Spectra and Directivity of Jet Noise", J. Acotist. Soc.

Am.

35,

no.

4, 614-616.

Ribner, H.S.

1964

tlThe Generation of Sound by Turbulent Jets" in Advances in

Applied Meehanics, Vol. VIII, pp

103-182,

Academie Press, New York and

X

2

OBSERVER

~,

_ _

+-~

_ _

~.OBSERVER

.J \

X,

X

₃

~'Y2

ol

8

_ -i

~:;~ADRUPOLE

STRENGTH PVj

,

vJ

---

_- - -_--... J

X

_{I' 1}

Y

QUADRUPOLE STRENGTH

pV ' Vi

FIG. 1

_______

K

1

GEOMETRY FOR QUADRUPOLE CORRELATIONS IN THE JET NOISE INTEGRAL

.

r

3db

+

SELF NOISE

A

SHEAR NOISE

B (

cos

4

8

+

cos

2

8)

2

FIG. 2 BASIC PATTERN OF JET NOISE (before convection and refraction). THE SELF-NOISE IS A SUPERPOSITION OF NINE 2-LOHE AND 4-LOBE QUADRUPOLE PATTERNS;

BASIC PATTERN

/

/

/ ' / / /

.,.,

/

f= 1500 Hz

/

I

/

I.

I

I

I

" , . . .

-"

/ /

/

/ '

"

/ '

--/

/

/

/

I

I

I

- - - PURE TONE

--FILTERED JET NOISE

FIG. 3 MATCH OF NOISE PATTERNS TO INDICATE

REFRACTION-DOMINATED ZONE OF FILTERED JET NOISE (M = 0.9).

BASIC

(FIG.2)

FIG. 4

•

CONVECTION

(C-

5 FACTOR)

,~

.

REFRACTION

JET NOISE AS A BASIC PATTERN NOT VERY DIRECTIONAL -WHICH IS POWERFULLY MODIFIED BY CONVECTION OF THE EDDY SOURCES AND REFRACTION OF THE SOUND WAVES BY THE MEAN FLOW.

40~1

---~

Ol

"0

..

~ +-'Cf)

c

Q)

30

20

c I O

·'

Q)

>

+-e

- . - Me= 0.76 (J33-A-10

engine)

- - Me

= 0.78 (J34-WE-34 engine)

~

Pietrasanta

---- Me= 0.84(J48-P-8 engine)

_._-- Me=0.82

[(I-Mecos8)2+0.3M~J-5/2

X

(I

+

cos

4

8

+cos

2

8)

2

Q) . """"",- I

I

I

I

I

I

c::

Ol

I '

~

I~

•

s~.-1200

·

14001600

.

.

!~--...

._---.

~

...

_~

--

_{--~--. ...}

_,,-

...

'.

-

...

-10

0°

40°

60°

·

-... _{~'-=4_} - . &:.

--"'-'"

----20~'

---~

~

"0

rc

~

::110 6

"'"

0:: : l 5

en

en

"'"

0::>

G.t-4

en

enZ

en"'"

"'" 0 3 ~ z~ O~ -0::

ent-Zu

""'"", ~A.

-en

0 0.1 FIG. 6 ~SHEAR NOISE ~SELF NOISE 0.2 0.3 . 0.4 0.5 0.6 0.7

STROUHAL NO. Cf D/U

TWO- COMPONENT SPECTRUM OBTAINED BY FITTING

THEORETICAL EQUATION (45) TO EXPERIMENT AT '9

=

45°

AND 90° DOMINANCE OF THE SHEAR NOISE BY LOW

FREQUENCIES AND THE SELF- NOISE BY HIGHER FRE-QUENCIES IS SHOWN.

-90°'--1--""""---+---\ _{----40dB- 90°}

I

MACH NO. = 0.5 - - - <%>(8) C" (e) (EXPERIMENT) Cf

=

1500 Cpl FIG. 7

S4f-

=

0.17

"BASIC" DIRECTIONAL PATTERN OF JET NOISE IN NARROW FREQUENCY BAND AT 1500 cps. COMPARISON OF EXPERI-MENT WITH THEORY (curves of Fig. 6) ADJUSTED FOR TWO POINT FIT.

-

Q)

o

U Cf) ~

o

Q) c: Cf) c: Q)

-

c: Q) >

-

o

Q) ~

-E

::l ~

-12 10 8 6 0 0 I

~

3 Cl.

en

2 0 FIG. 8 0 I ·1 ·2 O'S I

- ,

"

·2 ·4 1 I ·3 ·4 1 I (0)

Fourier Cosine Transform of ASech(b,)Cos(c"t") Numerical FCT of 4th Derivative Curve up to third zero crossing

Hflt!

Uncertainty band

·5 _\

\

,

45° and 60° cases will have approximately the same shapes and peaks except for the relative intensities

Mc'" 0·08 ·8 ,·9

'--/ ,_-1'5 2 I I ~ 2·5

O· CASE. SHEAR NOISE

ft

D/Uj

1·1 ft = f ( ,- Mccos 8) kHz \ \ ·6 ·8 1·0 1'2 _"- 1·4 ... 1·6 1·8

--,_

....

' 2 3 4 5 I I I I

(b) O· CASE, SELF NOISE ftD/Uj

FIG. 9

SELF NOISE

DOMINATES _{• EXPERIMENTAL POINTS}

I

I

I

/

,

\

AFTER REFRACTION

-A\

(QUALITATIVE)

[1+(cos 2

e

+cos4

e

)/2]

Ic

5

( Ribner )

BROAD BAND NOISE EMITTED FROM UNIT VOLUME OF A JET IN THREE DIRECTIONS, AS COMPUTED BY CHU (1966) FROM HOT-WIRE MEASUREMENTS OF TURBULENCE CORRELATIONS.

•

UNCLASSIFIED

Securi ty Classi fication

DOCUMENT CONTROL DATA· R&D

(Secut/ty cle •• U/catlon ol tltle. body ol .b.tt.ct .nd Ind.x/n, anno te ti on muet be .nt.t.d wtt.n the ovet." ,epott /. c/ ••• llled) 1. ORIGINATIN G ACTlvl"!"Y (Corpot.t • • uthot) 2 •. REPORT .ECURI TV e LA.SIFICA TION

Institute for Aerospace Studies,

University of Toronto,

'1', •• C) Onb'l.ri 1"\ (""'n~ilA.

J. REPORT TlTLE

UNCLASSIFIED

Zb. GROUP

QUADRUPOLE CORRELATIONS GOVERNING THE PATTERN OF JET NOISE •. DIESCRIPTIVIE NOTES (Type ol .. port end Inclu./ . . . t.e)

Scientific Interim

5. AUTHOR(S) (L.et ... IIt.t name. 'nltl.I)

Rerbert S. Ribner

11· REPORT DATE November, 1967

8 •. CONTRACT OR ORANT NO.

AF-AFOSR 672-67

b. PRO.JECT NO. 9781-02

c. 61445014

d. 681307

10. AVA IL ABILITY!LIMITATION NOTICIES

7.· TOTAL NO. OF PAGE.

29 _!

7b. NO. OF

"E".

21

• •• ORIGINATO"'. REPORT NU .... ERfS)

UTIAS Report No. 128

AFOSR 67-2159

1. Distribution of this document is unlimited.

11. SUPPL EMENTARY NOTIES TECR., OTHER

13. ABSTRACT

12. SPONSORING MILITARY ACTIVITV (SREM) Air Force Office of Scientific Reseàrch 1400 Wilson Blvd.,

Arlington, Virginia 2220~

The effects of convection and refraction dominate the heart-shaped pattern of jet noise. These can be corrected out to yield the "basic directivity" of the eddy noise generators. The observed quasi-ellipsoidal pattern was predicted by Ribner (1963, 1964) in a variant of the Lighthill theory, postulating isotropk turbulence superposed on a mean shear flow. This had the feature of dealing with the joint effects of the quadrupoles without displaying them individually. The present paper reformulates the theory so as to calculate the relative contribu-tions of the different quadrupole self- and cross-correlacontribu-tions to the sound emitted in a given direction. Spectra are also discussed, following the earlier work. Finally, the predictions are shown to be compatible with recent experi-mental results. Of the 36 possible quadrupole correlations only 9 yield distinct nonzero contributions to the axisymmetric noise pattern of a round jet. The individual directional patterns have either 2 or

4

lobes, but they combine to yield a quasi-ellipsoidal overall pattern ("basic" directivity before convection or refraction are allowed for). This is compounded of partial patterns called

'self-noise' and 'shear-noise'.

DD

FORM