Correlation of noise and flow of a jet

August, 1971.

CORRELATION OF NOISE AND FLOW OF A JET

by

Hie K.

L~e

UTIAS Report No.168

AFOSR-TR-71-2572

CORRELATION OF NOISE AND FLOW OF A JET by

Hie K. Lee

Manuscript received ~u1y, 1971.

August, 1971. UTIAS Report NO.168

ol'

'11'

ACKNOWLEDGEMENT

The author wishes to express his sincere gratitude to his supervisor Dro H. S. Ribner, who initiated and guided the research with patience.

Thanks are due to Dr. T. E. Siddon for suggesting the basic motion of cross-correlating the turbulence signal with the acoustic signal, and to Dr. W. T. Chu and Dr. G. W. Johnston for many other helpful ideas and discussions. Thanks are due also to his fellow student

Mr.

C. S. Kim for the proof-reading.

At the last, but not the least, thanks go to his wife Youngoak who typed the manuscript and gave constant support throughout the work.

This research was sponsored by the Nat~onal Research Council of Canada under Grant No. A 2003 and by the Air Force Office of Scientific Research,

•

SUMMARY

. Two approaches have been tried to develop and apply a new metnod for experimentally estimating, at a far field position (r =

96

D, ~=

40

0) , the spectra and intensity of noise originating from unit volume of a ~ound jet, extending to

7

D downstream.

They were:

(1) through the cross-correlation between the broad-band jet turbulence signal (via hot-film) and the broad-band acoustic pressure signal (via microphone) induced by the turbulence, and

(2) through the cross-correlation between their filtered narrow-band counterparts.

. Af ter exhaustive exploration of the approach (1) it was finally abandoned because of severe experimental difficulties. Appro~ch (2) has been developed and exploited because it avoids these difficulties, although eventu~lly providing same total information.

The measurements were analyzed v.ia Proudman's form of Lighthill's integral further manipulated to apply to the approaches ~entioned above. The results showed qualitative agreement with some Of the earlier theoretical pre-dictions made by Ribner and by Powell.

\

I.

•

Ilo 111.

IV.

V. TABLE OF CONTENTS i Notation INTRODUCTION 1.1 Historical Background 1.2 Present Investigation THEORY 2.1 Governing Equations

2.2 Correlation Between Broad-Band Signals

2.3

Correlation Between Narrow .. Band Signals

2.4

Separation of Self-Noise and Shear-Noise EXPERIMENTAL APPARATUS AND INSTRUMENTATION

3.1

Anechoic Chamber

3.2

Model Air Jet System

3.3

Hot-Film Probe Traversing Gear

3.4

Hot-Film Anemometer System

3.5

Signal Squarer

3.6

Microphone System

3.7

Frequency Analyzers

3.8

Random Noise Voltmeters

3.9

Signal Correlator and Recorder EXPERIMENTAL PROCEDURES

4.1

Range of Measurements

4.2

Hot-Film Position and Orientation

4.3

Unwanted Vibrations

4.4

Correlations Between Broad-Band Signals

4.5

Correlations Between Narrow-Band Signals

4.6

Data Treatments

RESULTS AND DISCUSSIONS

5.1

Power Spectra and Intensity of Noise from a Unit Volume of the Jet

5.2

Power Spectra and Intensity of Noise from a Unit Length of the Jet

5.3

Overall Power Spectra REFERENCES APPENDICES FIGURES PAGE 1 1 2

3

3

5

7 10 12 12

13

13

13

14

14

14

15

15

16

16

1

6

17

17

18

19

20 20 21 22

24

..

NOTATION

Notatinn's newly introduced into Appendices are not included here. They are separately defined wherever necessary.

A B D e. l e 0 f [:., f f P G ,(x;f) pp -G. ,(x,y;f) pp -G (x y·f) se pp' -'-' hG ,(x,y;f) s pp -G 2 ' (x,y;f) _v x p -I(~) I(~,l)

Ise(~'Z)

Ish(~'Z)

j

proportionality constant defining the bandwidth of the narrow-band filter as a fraction of its centre frequency

(M/f)

local, instantaneous speed of sound at x and t ambient speed of sound

constant (

p~

) 41Ta

lxi

o -jet nozzle diameter input voltage

output voltage frequency

bandwidth of narrow-band filter

peak frequency of noise from a slice of jet

physically realizable, one-sided power spectral density function of p(~;t)

physically realizable, one-sided power spectral density function of p(~, Z; t) portion of portion of G ,(x,y;f) pp -G _pp ,(x,v;f) _.:I... due to self-noise due to shear-noise

physically realizable, one-sided cross-power spectral density function of v

x2 (Z;t) and p'(~,z;t) acoustic intensity at x

-acoustic intensity at x due to unit volume of turbulence

-at Z

0

3

I

(!)

(

₀₃

)

Z

portion of I(~,Z) due to self-noise portion of I(~,X) due to shear-noise imaginary uni t ( .J-l)

· M p(~;t) ,pI (~;t)

P

(x) 0 -R R I (X;T) pp -R I (X,Y;T) pp -R 2 I (X,Y;T) V x

P

-Mach number vector of turbulent jet flow (!Ia ) o acoustic pressure at ~ and t

ambient pressure at !

local, instantaneous pressure at x and t

(Pt(~;t)

=

po(~) t p(!;t)

spectral content of p(x;t) passing through a narrow-band filter of bandwidth 6f-centered at f

portion of p(~;t) contributed by unit volume of turbulence at I

electrical resistance

auto-correlation function of p(~;t) with variable time delay T

(p(~;t) pI (!; t + T )

auto-correlation function of p(~,l;t) with variable time delay T

cross-correlation function of v 2(y;t) and pl(X;t) with

x -

-variable time delay T

6R 2 I (x,v;T,f)cross-correlation function of 6v 2(y;t,f) and

~p'(_x;t,f)

v p _:l.. X - .

x with variable time delay T

(6v 2(y;t,f) 6p'(X;t + T,f ) x 6R~ 2 I

(!,I;T ,f)

2 x p cross-correlatioI). function of 6V x (.I;t,f) an,d ~pl (~;t,f) r S(X) S _{ppl _,_,} (x yof) S (x vof) se ppl

_':l..'

with variable time delay T using a pair of identical con-stant percentage bandwidth filters

(AfMv 2p' (x, y; T ,f) )

x

distance between the source point

.x..

and the field point x. (I~

- .II).

cross-sectional area of a slice of jet turbulence at X

power spectral density function of p(~,

I;

t) portion of Spp,(!,.I;f) due to self-noise

S (x yof)

sh pp' -'-' S 2. ,

(

x

,

y;

f)

V

x P

-portion of Spp,(~,Z;f) due to shear-noise

cross-power spectral density function of v 2(y;t) and

) x

-p(~,z;t

T computation period of the correlogram

t time

Q(z)

mean flow velocity vector at

Z

U_j absolute value ~f

Q

at the jet nozzle

U

X (Z) component of Q(Z) in ~-direction

~(Z;t) local, instantaneous turbulent velocity vector at zand t ux(Z;t) component of ~(Z;t) in ~-direction

V

volume of the jet flow

~(Z;t),VI(Z;t) local, instantaneous flow velocity vector at zand t

Vx(z;t),V~(z;t) component of ~(Z;t) or~'

(Z;t)

in x - direction

2 6v (y;t,f) x

-x,

Y,

Z

x x l., z 5

e

p (~;

t)

T 0 ' T 0

spectral content of v 2(y;t) passing through a narrow-band x

-filter of bandwidth 6f centered at f

right-handed system of Cartesian coordinates with its orlgln

at the center of the jet nozzle and X-axis along the jet

axis (see Fig. 1)

position vector of the field point

absolute value of ~ (I~

I)

position vector of the source point

Dirac delta function

angle between the jet nozzle radius and the Z-axis fluctuation in air density associated with acoustic pressure p(~;t)

ambient air density at x

local, instantaneous air density at x and t (Pt(~;t)

=

po(~) + p(~;t)

)

time or variable time delay

time delay, i.e., travelling time of sound from the source point to the field point

+

T T - T '

o

angle between the field point vector ~ and the X-axis difference in accumm~lated phase lags of two input trans-ducer systems at f

•

I . INTRODUCTION

Even though there have been a few excellent review papers on the subject

of "Jet Noise" (e.g. Ref's. 1 and 2 among others), it would be worthwhile to present

here a brief but up-to-date review of prior work by others in the field and find a

proper place for the present paper. 1.1 Historical Background

The modern theory of the aerodynamic noise was initiated by Lighthill (Refis. 3 and 4) wh en he described, mathematically, the far-field acoustic pressure generated by turbulence in terms of fluctuations of momentum flux (quadrupole source

theory). Later on, Ribner (Ref's. 5 and 6) developed an alternative but mathemati-cally equivalent approach by describing the generation process in terms of pressure fluctuations in the turbulence (simple source theory or dilatation theory).

As in other fields of fluid dynamics, similarity considerations have been

ingeniously employed from the existing knowledge of dynamic similarity in jet turbu-lence (e.g. Ref. 7) to provide many valuable insights in aerodynamic noise. First of all, Lighthill indicated that the total radiated acoustic power from a jet is proportional to the eighfupower of its exit velocity (U8 law). This law is now well

established with numerous experimental confirmations (e.g. Ref's. 8,

9

and 10 among others). Ribner (Ref. 11), followed closely by Powell (Ref. 12) and Lilley (Ref.

13)~ first showed that the acoustic power emitted by a slice of jet should be nearly

constant along the jet axis in the initial mixing region (Xo law), then further

downstream, fall off extremely fast in the fully developed jet (x-7 law). Although this analysis has enjoyed wide acceptance with some empirical evidence (Ref's. 14, 15 and 16), it is still a matter of interest and controversy (e.g. Ref. 17) waiting

for more conclusive experimental results. Powell went out one step further in his treatise to predict the frequency dependence of the spectral shape (f2 and f-2

laws), which turned out to be similar to the measured spectra, af ter assuming in

effect that a given slice of jet emits just a single characteristic frequency. First prompt theoretical application of Lighthill's theory came from Proudman (Ref. 18) for a relatively simple case of decaying homogeneous isotropic turbulence. A real jet is, however, a shear flow and differs markedly from this simple model, The presence of the mean jet flow in the jet,moreover, makes the

generation and propagation of noise within it much more complicated than within

homogeneous turbulence in three important ways, namely, the mean flow-turbulence interaction, convection and refraction.

In dealing with the effect of shear flow in jet turbulence, Lilley

(Ref.13) first introduced the concepts of "shear-noise" and "self-noise" to sepa-rate the lower frequency noise due to shear-turbulence interaction from the higher frequency noise due to turbulence-turbulence interaction. Ribner (Ref's. 1, and ~9) redefined them in a different fashion by dividing the instantaneous

velocity into its mean and fluctuating components. Using these new definitions,

he showed the differences in their directivities as well as peak frequencies; "self-noise" is non-directional and its peak occurs roughly one octave higher than

that of "self-noise" whose directionality resembles afigure "8" with the lobes

aligned with the flow. By adding them back, he successfully reconstructed a

"basic" directivity of jet noise compatible with experimental results obtained at UTIAS (Ref. 20.), and at the same time explained the so-called "reverse Doppler shift paradox!! (Ref. 2:,?1,22).

The convection effect of the mean flow on the jet noise directivity was considered already in Lighthill's original paper and refined by Ribner (Ref's.

5,

6

and

23)

and Ffowcs Williams (Ref's.

24

and

25).

Jones (Ref.

26)

later showed the different effects of convection on "self-" and "shear-noise", a matter of controversy.

The important effect of refraction on the jet noise directivity due to

the velocity gradient in the flow was indicated by Powell (Ref. 27) and independently by Ribner (Ref.

23).

Ribner and his students at this laboratory, by means of

exhaustive investigations analytically (Ref. 28) as well as experimentally (Ref. 29 among others), brought this speculation into an accepted fact and ended the long controversy regarding the "valley-of-quiet" in and near the jet axis (Ref. 30). Furthermore, the analysis of Ref. 28 revealed that the process involved is far

more complicated than Powell's original speculation.

Lately Pao and Lowson (Ref.

31)

opened a promising new analytical approach using the spectral analysis technique by expressing the source term in Lighthill's integral in terms of its wave number-frequency spectrum from the very beginning. They were able to regenerate several of Ribner's results in a simple manner and calculate overall acoustic power of a jet in acceptable agreement with experiment. Lowson has applied the theoretical results to turbulence data in a reasonably

successful attempt to predict jet noise properties (Ref.

57).

A somewhat different approach was initiated in Grande's paper (Ref. 29) and developed further by

MacGregor, Lam and Ribner (unpubLished) at this laboratory.

As usual, experimental investigations have progressed hand in hand with

theoretical developments in jet noise. Stimulated by Lighthill's theory a number of workers (e.g. Ref's.

8,

9,

32-35

among others) had carried out acoustic measure-ments of jets to test the empirical validity of his theory in a rather simple way.

Meanwhile, another group of investigators (Ref's.

36-42)

had concentrated their efforts to extend their knowledge on the structure of the jet turbulence by measuring the characteristics of its velocity and pressure fluctuations, hoping to exploit it for better understanding of acoustic characteristics of a jet flow. However, a major advance in this +ine of experiments came only lately when Chu at this laboratory (Ref.

42)

succeeded, to a certain extent, in estimating the

characteristics of the noise emitted from a unit volume of jet turbulence by analy-zing his detailed measurements of the space-time correlations of turbulent velocities via the Proudman form of Lighthill's integrale Unfortunately, the uncertainties arising mainly in the evaluation of a fourth derivative of an experimental curve prevented him from achieving a total success. Jones (Ref.

43)

followed to report a similar attempt from somewhat different set of measurements, i.e., measurements of fluctuating turbulent stresses in a jet flow, analyzed from a different analy-tical framework of his own.

1.2 Present Investigation

The initial objective of the present investigation was to estimate, at a far-field point, the intensity and spectrum of the noise originating from a unit volume of the jet. This data was then to be processed to yield the distri-bution of the noise source strength along the jet and the overall spectrum of the jet noise. Cross-correlation with variable time delay between the turbulence sig-nal of the jet flow and the acoustic sigsig-nal at the far-field point was selected as the means of experimental investigation. This approach (suggested by Dr. T. E. Siddon while at UTIAS) is more direct than Chu's, since it cross-correlates the cause-effect pair, whereas Chu processed the 'cause' signal alone.

Even though the correlation method has become fashionable in the study

of stochastic phenomena, it has not been used extensively, chiefly because of

serious limitations in capability of handling vast amounts of data. Fortunately,

recent developments in electronic data handiing technique made the method feasible

for the present investigation.

Two kinds of corre1ation have been tried, namely,

(1) the braad-band turbulenee signal (hot-film) with the broad-band

acoustic signal (microphone);

(2) the narrow-band filtered turbulence signal with the narrow-band

filtered acoustic signal.

Eventually, combination (2) won the preference and had been used throug

h-out the experiment , sinee combi nation

(1)

showed incompatibility with the present instrumentation.

Perhaps combination (2) can be said to be the most direct and powerful

approach available so far sinee it does not only relate directly the acoustic

measurement to the turbulence measurement but also incorporates aspectral

analysis.

The Proudman form of Lighthill's integral for aerodynamic noise has

been chosen and manipulated further to adapt it to this experimental approach. Ilo THEORY

2.1 Governing Equations

In a turbulent air jet flow, the Reynolds number is large enough to

neglect the viscous stresses compared with the Reynolds stresses. Molecular heat conduction is likewise negligible-if the jet is cold (Ref. 1). Thus the main acoustie features are exhibited when the fluid is'treated as inviscid and ise

n-tropic. For sueh a fluid, if there is no source or sink of mass as well as no

external force in the field, the conservation equations for mass and momentum

read and where èP t è ~t _01; + ~ _oX.

(

p

_t Vl·)

=

0 1

è

2 ~ (p t v. v.)

=

-a oX j 1 J (2.1) (2.2) 2

a èp~èxi has been written in place of the pressure gradient èPt/èxi

by virtue of the isentropy. Einstein's dummy index has been adopted.

Elimination of Ptv

2

o

P

ot

2 2

o

(p v.

v.)

o 1. J

dX. dX.

1. J (2.3)

approximately, on replacing the local speed of sound a by its space-time average a

o' and the local instantaneous density of air Pt in the RHS by the ambient air

density P , if the jet flow Mach number M is so lowthat the fluid can be regarded

o

as quasi-incompressible (Ref.

1).

Or, equation (2.3) can be rewritten in terms of pressure as

~

_{2 '::: P}

dx

.

0 1. 2

o

(v. v.) 1. J

dx. dx.

1. J (2.4)

Being a classical wave equation of the acoustic pressure with a source distribution, the strength of which per unit volume is

0

2 (v. v.) 1. J

Po

dX. d X.

1. J

equation (2.4) states in effect that an aerodynamic flow provides a forcing term for generating a sound field.

The solution of equation (2.4) in an unbounded medium is given by the Kirchhoff retarded potential solution (Ref. 44)

02[V.V.(y;t')] I_x-_y+ 3 1. J --.,-:~----

_{dy. dy}

_.

5( t' -t + )dt' dl., a 1. J 0 (2.5)

where 5(T) is the Dirac delta function and ~ and X are position vectors of the

field point and the source point (see Fig.

1).

Two applicationsof the divergence theorem yield (Ref. 45),

P(x·t)

_-'

';;

I~

-

l.

I

5(t'-t + a ) dt' d9_X o

(2.6)

P _o è\2 ₀

iJr

v. v _{1. J -}. (y;

t')

41T

è \ " : \

I~

-

l.

I

oX. oX. 1. J 00

if the surface integral of

v

.

v.

at infinity is taken to vanish.

1. J

In the "far field" where the distance 1 x 1 is much larger than atypical wavelength, the solution can be rewritten as (Ref.-5)

d

2 [v. v . (X; t ' ) ]

I x-xl

3

_ _ 1.~Ji:--_ _ _ 5( t' -t + -a ) dt' d X ,

dt

2 0

where

V

is the volume of the jet flow. Furthermore, when lxi is much larger than the dimensions of the flow as well, eq. (2.7) becomes (Ref.-5)

P x.x.

Ir

02[v.v.(y;t ' )] 3

"'P(x·t):: 0 ~ J ~ J - 5 (tl-t +

E-.. )

dt ' d Y

- ' 2

3

2 " a

-47rao I~

1

V ~oo ot 0

(2.8)

where r

=

I~

- xl

is the distance between the source point and the field point.

Equation may be rewritten in a very neat form,

47ra o 2 2

I r

0

v x

Cl;

tI) 2 2 r v ot Voo

due to Proudman (Ref. 18), since

5(t ' -t +

E-.. )

_a dt ' d31 o

xiXjV_{iV j}

=

1

~

1

2

Vx2 .

Note that x is a vector from the origin to the field point~mere p is measured,

and v is the component of instantaneous flow velocity parallel to x. i.e., pointtng toward the observer.

2.2 Correlation Between Broad-Band Signals

The auto-correlation function of the acoustic pressure p(~;t) with an

arbitrary time delay TI is defined by

Rpp I (~; TI ) - P (~; t) pI (~; t + TI)

~ JTp(~;t)

pl(~;t

+ TI) dt,.

lim (2.10 )

T-7oo _o

where overbar (---) denotes time average.

Anr~ if the turbulence is statistically stationary, it can be reduced

formally into

(2.11)

Using the description of p(_x;t) in terms of v 2(y;t - ria) in eq.

x - 0

(2.9), eq. (2.11) can be rewritten as 2 ,2 Po

11

o v (Y;T) T )dTd

3

y R (X'T I ) '::: X - _{P I} (~; _{TI) 5(T}

+

ppi _, ₂ ot2

,

47ra r o - _(2.12) 0 Voo r

I~-ll

where T = - - = is the time delay, which is the ttmvemIingetimanóffth:w

0 a a

0 0

the source point

_I

_{to the field} point ~.

sound from

resu1t of the time average depends on1y upon the re1ative time, and thus is inde-pendent of t. Hence, eq. (2.12) becomes (Ref's.

6,

25 and

46),

'" 47T P ao 2r

vfJ

c/

2 R (X'T') - [v (y,'T)p'(X,'T')] pp' -'

èh

2 x - -o V 00

Or, writing it in a convenient form for later use,

R _{ppr _,} (X'T')'" -

~

R- 2-'" (x.v_'T)5(T + T*) dT d

3

v , '" 2 v~~ -pI

_'Jl..'

Jl.. OT X where 2 2 R 2 .:::',(x, V;T)= V (y;t)p' (x,t+ T) = V (Y'O)p' (X'T) V _x P - Jl.. X - - x -' -' (2,13) (2.14) 2

=

V x (1..;O)p' (!,1..; T) (2.15) 2

is actua11y the cross-corre1ation of p at !, and V

x at 1.. with an arbitrary time de1ay T, and From eq. (2.14), where R _{pp _} ,(x,v; _Jl..

T')

T*

=

T'

+

T o (2.16) (~.17)

R

pp

'(!'

1..; T') is the auto-corre1ation function of pC!;

1..;

t), the acoustic

pressure at ! contributed by a unit volume of turbu1ence at 1...

Since the acoustic intensity at ! ' I(!), is defined as

=

R ,(x; 0)

pp

-from eq's. (2.14) and (2.16),

I(!) Z 1

r

J

47Ta 3r v o V 00

0

2 3 R 2

1)

/

(x, 1..; T)5(T-T ') dT d y, OT2 V_x P - 0 (2.18) (2.19)

and further in detail, the acoustic intensity at x due to a unit volume of the turbu1ence at 1.., I(~, 1..), is

I(~,l) -

d3I(~)

d31 ~ 1

r

0

2 R T) t (~,l; _T) 5(T-T ) dT

-47Ta 3r J dT2 v 2 pi _X 0 (2.20) 0 00 or I(~,,~:> 1

[

d 2

l

3r - R 2 'J' (x Y'T) 47T a dT 2 v x pi -' -' T T 0 0 (2.21)

Hence, ouce the R 2 ': (x, y; T) curve and the retarded time T

=

ria

v

x p - - 0 0

are known, nothing more than a methodical procedure for carrying out numerical

differentiations twice is needed to calculate the acoustic intensity of the noise at ~ originating from a unit volume of the turbulence at

I'

Comparison of the eq. (2.21) with Chuls working equation in his report

eq. (6.2) of Ref. 42 clearly shows that the number of error-introducing numerical

differentiations of an experimental curve is reduced to two instead of four.

Through this reduction, it would appear possible to enhance the accuracy of the

experimental results substantially; section 4,4, however, brings out a serious

additional source of error.

The error is inherent in the instrumentation for the broad band approach,

but may be avoided via a narrow band approach discussed in Sec. 2.3. The following

section - still dealing with the broad-band signals - lays the foundations.

2.2.2 Spectral Analysis

It is well known (e.g. Ref's. 47 or 48) that, for stationary random

processes, correlation functions and power spectral density functions are Fourier

transform pairs. That is, in the case of the auto-correlation function of

p(~,

I;

t)

,

-j27TfT' (2.22) e dT' , - 0 0 and

r

00 j27TfT I R. ,(x, y;T')

=

S . ,(x, y; f) e df, pp - - ~ '- 0 0 pp - -(2.23)

where Sppi (~'

K;

f) is the power spectral density function of p (~,

I;

t).

00 P -j2?TfT'

J~r

cl

s ,(x

r

f) " - 0

2

R 2

i(x'l;T).

5(T-T'-T )

-

₂ e pp' - ' ,

o

_{v p -} 0 or

s.

,(x,y; f) pp -47ra r 0 Po = 47ra 0

2

47ra r o 2 r T x -00'

Joo

-j2?Tf(T-T O)

0

2 R I e OT2 vx2p' -00

Jf we make the reasonable assumption that

(~,DT ) dT

dT.

R .,,' (_X,V,'T) =

0

R , , ( ) = 0

v 2 p' .:i...

dT

v 2 p'l ~,l; T at T

=

±

00 ,

x x

we can rewrite eq. (2.24) as

7rP_o f2. j2?TfT

1

00 -j2?TfT\ spp, (~,

l.,

f) _{- 2 -} e 0 R t)' (x

_l.;

_{T) e} v 2 p' - ' a r x 0 -00 or j2?TfT 7rP 2 0 S ,(x,

_l.;

f) _ 0 _ f _e 8 ), (~,

l;

f) pp - 2 _{v 2 p'} a r x 0 where ==

~

r

00 -j2?TfT 8

,

(~,

Y'

f) R ' (x

l;T)

e dT

v

_x 2p' _,

-'

v_x2p' -' -00 dT, dTdT' (2.24)

(2.25)

(2.26) (2.27) 2

is the cross-power spectral densi ty function of v x

(l.;

t) and p(~,

l.;

t).

Consequently, the Fourier transform of the cross-correlogram R 2'Ji

(X,.

,l;T),

weighted with V_x P -7rP o 2 j2?TfTO - 2 - f . e a r o

would provide the power spectral pressure p(~,

l;

t) whicn is the v61ume of turbulence at

l..

denis ty function S _{pp _} ,(x, v; .:i... f) of the acoustic .

acoustic pressure at ~, originated from a unit

As is well known, the physically realizable one-sided ~)power spectral density function of p(~,

l;

t) is

G

(x

Y' f)

pp' - ' - ' 28 pp -, (x, _y; f) "-

(2.28)

for 0

<

f

<

00 ; otherwise zero.

2.3 Correlation Between Narrow-Band Signals

An alternative, and perhaps more straightforward, approach to getting

S ',-i (x yo f), the cross-power spectral density function of v 2(y; t) and

V

2

1>

'

-~

-'

x

-X

p(~~ l;t)~ in eq. (2.26) is to obtain its spectral components experimentally

through cross-correlating the narrow-band velocity and pressure signals.

Since

J

(Xl ' j27TfT R ':) '/ (x Y'

T)

= S r I (x Y' f) . e ' v 2p' - ' -' v 2p' -' -' x x df , (2.29) -00

Llli 2-~ (x, y; T ~ f), the cross-correlation between the velocity and pressure

v

x p -

-signals filtered through an identical pair of narrow constant bandwidth filters

of 1L:,it gain centered at frequency f, can be written (Appendix A) as j27TfT

(2.30)

where 6f is the filter bandwidth.

to

Using this relation, eq. (2.26) can be reduced

S _{ppi _}(x _{, _,} Y' f) 7TP o 2 2a r o LRv 2p} 'J~,

x,;

T 0 ' f)

x

'

(2.31)

Furthermore, if a pair of identical constant percentage bandwidth filters,

which happen to be more popular, are used instead of constant bandwidth filters,

the bandwidth of each channel is proportional to its center frequency (i.e.,

6f

=

A.f)~ and~ in turn, the power spectral density of pressure p(~, X,; t) is

7TP o 2 2a Ar o . L LR' 2J' (x, y;T, f) , V x p' - - 0 (2.32)

where LR' denotes the correlation using constant percentage bandwidth filters.

Equation (2.32) clearly reveals marked advantages over the broad-band

approach, L e, a simple measurement oflR' 20 \(x, y; T , f), the height of the

v

x p - - 0

correlogram at the retarded time replaces a digital computer calculation: the

Fourier transform and frequency weighting in eq. (2.26), which includes a multib plication of a complex number ej 27TfTo, is replaced by the on-Hne spectral analys:i's.

In addition,the difficulty encountered in measuring the broad-band cross-correlation,

which has inherently a very small correlation coefficient ( ~ 0.02), is somewhat

relieved by the filtering procedure.

As is well known, the physically reàlizable, one-sided, power spectral

density function of p(~,

z;

t ) is G ppi (x _~ _Y' ,

f

)

=

7TP o 2 a Ar o f 6R' p' (x, y;T ,f) v 2p' - - 0 x

forO

<

f

<

00 ; otherwise zero.

And the aco~stic intensity at x due to unit volume of the turbulence at

1:.

is or, I (!, y)

~ a6~

0

J

00 G_pp'(!'

l;

f) df o 7T a

3

_Ar o .61\' . , (x

v,·

v 2-:p' - ' Jl... X T , o f) df (2.34)

It should be pointed out here that S ,(x, y; f) in eq. (2.32) is a

pp -

-positive quantity in spite of its negative sign: since R 2:;' (T) has the nature of v p'

a double derivative of the autocorrelation function of . x v 2(y, t) as shown x

-in Appendix B, R 2 ~(T )is already negative, and the two negative signs cancel. vx ·p 0

In addition, it might be interesting to note that it is possible to rebuild the broad-band correlogram R 2" (x, v; T) by superimposing each

narrow_-v p' - Jl...

X

band correlogram 6R 2- ' (x, y; T, f) (Appendix A). The relation between them is v p' -

-x

!:::,f (2.35)

where a pair of ideal constant bandwidth filters of unit ga in is assumed. 2.4 Separation of Self-Noise and Shear-Noise

By definition, the instantaneous resultant, mean, and turbulent velocities are related as

(2.36) And the x-component of this equation can be written as

v (y, t) == U (y) + u (y, t) •

x - x - x - (2.371.

Thus these are velocity components pointing toward the observer located at the end of the vector x. Therefore, the cross-correlation Rv 2:p', can be expressed in

x terms of two cross-correlations R 2 " and R ")', i.e.,

U_x p uxP'

,

2 2 - _up' 2 p' . R v 21" v p' x U x p' + 2 U x x + u x x or R

,

== 2 U R ., ... ' + _R .• ,1 v 21l' _\x x u 'p' u 2p' X x (2.38)

since pi

=

0 by definition.

Now~ on the ana10gy of eq. (B-12) of Appendix B, we can write

cl

]

(

)

~ B

r

[--2 R

V X I

2

_{(TI) T I}

₌

_T+ T -

eh

l . U_x ~ V But~ 2 - 2 + 2 UI R ₁₂ = u Vi = U Vi uv X x x x _x x x

-

_o

_{and u Ui} 2 or~ since u =

»

u Ui X Consequent1y, R r: I (T) - 2 U pi X Likewisej since or R X X X x u Vl2

""

2 UI X X X

L

[U

12

èh

l2 x U Ui + U Ui 2 x x x x R _{U Ui} X X

"2

_{u + 2 UI} -2-_{U Ui + U Ui} 2 2 J _{, .} X X xx xx from the facts that u 2 x 2 ;::> 2

-constant, and u Ul~> u Ui

X X X X

(2.40)

(2.41)

(2.42)

(2.43)

From eqls (2.41) and (2.42), it is easy to identify R Tl II and R 2·'" in eq (2.38)

up u p

as the "shear-noise" term and the "se1f-noise" term re:gpective1yx(fo11owing termino1ogy of Ribner)and it is evident that they are separab1e experimenta1~y.

In the case of the broad-band corre1ation, substituting eq. (2.38)

into eq. (2.20) yie1ds the "se1f-noise" and the " s hear-noise" portion of the acoustic intensity I (~, ~),

1

(2.44)

2

O

2 R

~

:(x,

y;T).5(T-T) dT.

OT

UxP -

-

0

(2.45)

I

00

And, using eg.

(2.38),

the "self-noise" and the "shear-noise". portion of the one-sided power spectrum can be separated as

G (x Y'f) se pp' -'-' 217P 2 j 27TfT

r

00 _ _ 0_ f e 0 R 2 ' I (x 2 . u · I - ' a r ~ -00 x p (2.46) -j27TfT 'i.; T) e dT o

for 0

<

f

<

00 , otherwise zero, and

G (x Y' sh pp' - ' -' 7TU p j 27TfT

r

00 - j 27TfT f) ':::: -

~

₂ f2 e 0 R _{u p' _,} " (x _'i.; ) T e a r ~ x-dT

(2.47)

o -00

for 0 < f < 00, otherwise zero.

Similarly, in the case of the narrow-band correlation, the separated intensities can be obtained from eg.

(2.34)

as

I (x, y) se - -and 47TU x 6R I 2 I (_x, _y; U x

p'

T , o f) df,

J

00 f. DB I I (x, v; T 0 ' f) df. u p' - ~ o x

The separated one-sided spectra came from eg.

(2.33)

as

G (x Y' f) ~ se pp' -' - ' -27TP o - 2 - f. a 'Ar o l:iR I n I (_x, _y; T , f) UXl:::p' 0

for 0

<

f < 00 otherwise zero, and

47TU P ---,~x---.,;..o f. 6R I I (_x, Y' T , f) 2 u p' - ' 0 a Ar x

G

(x V' f) ~ sh pp' - ' ~, -o

for 0 ~ f < 00; otherwise zero.

III. EXPERIMENTAL APPARATUS AND INSTRUMENTATION

3.1 Anechoic Chamber

(2.48)

(2.50)

All measurements had been made in an anechoic chamber (Fig. 2) located at the Institute for Aerospace Studies, University of Toronto. The chamber is 13 ft. long by

9

ft.

4

in. wide by

6

ft. 10 in. high, as measured between the

tips of the

8

in. deep Fiberglas wedges. A wire grid, spring tension mounted

4

ft. above the wedges, serves as the floor. A 1/16 in. layer of lead sheet in the

A 2 ft.

7

in. square, 20 ft. long curved duct lined with Fiberglas is provided

as an outlet for the jet flow.

The residual background noise level in the chamber is acceptably low

(~ 20 dB with weighing network "curve A"). For test signals above 300 Hz, (the

low frequency cut-off), deviations from the inverse square law for far-field

acoustic intensity are of the order of 1 dB to within about 1 foot of the wedge

tips. Further details of its free-field simulating properties are described in

Ref. 49 along with constructional details. 3.2 Model Air Jet System

The model air jet (Fig. 3) installed in the UTIAS anechoic chamber has a circular nozzle of

3/4

in. diameter and can be operated witnin the range of the

flow Mach number 0 to l.O. The nozzle contraction was designed to produce a

uniform velocity profile at the orifice. The contraction is preceded by a section

of

8

in. pipe about

4

ft. long, housing a heating section, a settling chamber,

and a silencer to eliminate the flow noise in the pipe.

The jet is located off to one side in the anechoic chamber in order to

maximize the radïus (7 ft. 5 in.) of the semi-circular microphone traverse.

The air flow is supplied by a continuously operating compressor and

controlled with a Fisher Governor type 99 Pressure Regulator located outside the

anechoic chamber.

Further details of the jet facility, and the velocity and temperature

profiles of the flow are described in Ref's.

4

9

and 50.

3.3 Hot-Film Probe Traversing Gear

The traversing mechanism for the hot-fiLm probe (Fig.

4)

is of a

design providing the free and independent motion of the probe along the three

Cartesian coordinates as well as a rotation around the probe axis. It covers

the full length of 12 in. along the jet downstream, + 3 in. across the jet stream

in horizontal plane, + 1.5 in. along the vertical line and unlimited rotation

around the probe axis: The repeatability of the hot-film position is within + 1/64 in. and + 2°.

-

-3

.4

Hot-Film Anemometer System

For measuring the jet turbulence velocities, the TSI (Thermo-Systems

Inc.) anemometer system has been used throughout the research work.

30401 Probes

Astrong mechanical structure was required to withstand a large number of runs without breakage. Therefore, quartz-coated hot-film type probes were

preferred to hot-wire types in spite of their lower sensitivity and inferior

frequency response.

Sommercially avaiiable Model 1210-10 probes had been used successfully

in trial runs; however, for the most part, custom desigped probes (Model 1210

AG-IO) had to be used to eliminate the probe vibration due to the interaction

models are given in Fig. 5. To take into account the change in the hot-film sen-sitivity due to contamination from the dirt in air stream, the change in supposedly constant mean flow velocity during the spectral analyses at a fixed sensor position has been monitored with a Fairchild Model 7050 Digital Voltmeter.

3 • 4 ~ Anemometer

A Model 1034A Constant Temperature Anemometer was used with the probes. It has a built-in linearizer circuit accurate within less than

2%

of full scale covering up to 1,000 ft/sec. The overheat ratio is 1.5, and the frequency res-ponse, when used with the hot-film probes, has been custom tailored to be flat up to 20 KHz. Test results by the square wave method (Ref. 51) has been affirmative. 3.5 &ignal Squarer

The turbulent velocity signal v must be squared before being fed into

the correlator to get the correlation fun~tion R 2!", (T). This operation was

V

x 1>

achieved through an analog squaring circuit (Ref's. 52 and 53) set up on a

Philbrick Model RP Operational Manifold with a Model PSQ-N Transconductor and two Model EP 85AU Operational Amplifiers. The schematic diagram is given in Fig. 6.

With this circuit, the output voltage e is given in terms of the input o

voltage e. as

l

e o =

where R is in ohms, and e. and e in volts. For present case, the value of R

l 0 2

has been taken as 100 KD, and consequently e

=

e. /2. o l

3.6 Microphone System

In measuring the far-field ihstantaneous acoustic pressure, a B

&

K (Bruel and Kjaer) Type 4131 1 in. Condenser Microphone has been incor,porated

with B

&

K Type 2612 Cathode Follower. Although i t has a somewhat inferior frequency response (Fig. 7) in comparison with other smaller B

&

K microphones

(e.g. 1/2 in., 1/4 in., or 1/8 in. ), it has been used to increase the sensi-tivity and the resolving power of the system, since the correlation is inherently very small.

The free-field characteristic of the system is essentially flat between 50 Hz and 16 KHz with a maximum deviation of + 1 dB at the resonant frequency of 7.5 KHz. The microphone was periodically calibrated using B

&

K Type 4220

Pistonphone. The sensitivity was found to remain constant (s

=

4.42 mV/~bar). The system was driven,by aB

&

K Type 2801 Microphone Power Supply and its output was "DIRECT" coupled to the filter system.

3.7 Frequency Analyzers

The B

&

K Type 2107 Frequency Analyzer, which is essentially a versatile filter - amplifier combination (broad-band as well as narrow-band) (Ref. 54) was placed immediately before the correlator as a means of processing the signal, one filter in each channel.

3.7.1 Filters

As a broad-band filter, the Weighting Network "Curve B" was used (Fig.

8). As a narrow-band filter, the Octave Selectivity "40 dB" under the Weighting

Network "Linear 20-40,000 els" was used. That is equivalent to a constant

per-centage bandwidth of

8.5

percent or 0.13 octave bandwidth (Fig.

9).

In order ~o synchronize the center frequencies of the two variabIe filters

accurately, a sine wave signal fr om a Hewlett-Packard 3310 A Function Generator

was fed into the Analyzers through a DPDT switch as a common frequency standard

before each correlation took place (Fig's. 10 and 11). In this way accurate

on-line frequency alignments were aChieved, eliminating the critical error due to

the slight deviations between the two nominally identical frequency scales.

3.7.2 Amplifiers

There are two amplifiers (input and output) in the FrequetIcy Analyzer.

The input amplifier has a crest factor of 4.5 and variabIe amplificatiön factors

of 1, 10, 100 and 1,000. The output amplifier has amplification factors of 1, 3, 10, 30 and 100. Due to its inferior characteristics, the output amplifier is

recommended to be used at the amplification factors of 1 and

3,

-

whenever

possible. By combining these with the amplification factors (1, 2 and 5) of the correlator input amplifier, it is possible to compose a sequence of amplification factors of 1, 1.5, 2,

3,

5,

6

and 10 and so on.

The Frequency Analyzer has a built-in standard square-wave generator

for reference purposes, and the amplifier sensitivity was calibrated everyday

accordingly.

3.8

Random Noise Voltmeters

Two B

&

K Type 2417 Random Noise Voltmeters were used in on-line rros

measurements performed at both inputs of the correlator for later normalization

operations. The time constant of the Random Noise Voltmeter is variable within

0.3 and 100 seconds (from

6

to 20 KHz) in 1, 3 sequence and very convenient for

measuring phenomena of various frequencies.

3.9

Signal Correlator and Recorder

In order to obtain real-time correlations with variable time delay,

a PAR (Princeton Applied Research) Model 101 Signal Correlator was used. utili-zing both analog and digital techniques, this instrument operates as a hybrid computer and is capable of computing a cross-correlation function of the two input signals (from DC to approx. 250

KHz)

with a computation period ranging from 0 to

6

T in integer multiples of the computation period T (Ref.

55).

As it is being computed, the correlation function is stored in the

100 channel analog memory; this is actually 100 RC integrating networks, the

time constant of which determines the averaging time of the computation and can

be changed by replacing the resistors. In this particular work, it was found

that the time constant of 20 seconds as supplied by the manufacturer wasrnot long enough to get a státionary correlogram, so it was subsequently increased

to 60 seconds. By doing this, the author was able to get a stationary

corre-logram of higher signal-to-noise ratio at the expense of longer measurement

Versatile readout circuitry allows the function to be display~d on an oscilloscope as i t is being computed, and to be non-destructively read out either af ter or dliring computation. Suitable recording instruments include both strip-chart and X-Y recorders, as well as oscilloscope-camera combinations. A Moseley Autograph Model 2 D-4 X-Y recorder was employed to record the outputs of the correlator on the paper.

At the same time, a Tektronix Type

555

Dual Beam Oscilloscope with a Keithley Instruments Model 102 B Decade Isolation Amplifier was used in moni-toring the correlatoroutputs as well as the signals at various stages in both channels to detect any abnormalities, such as probe vibration, anemometer oscillation, undue clipping or distortion, etc.

IV • EXPERlMENTÀL PROCEDURES

4

~l Range of Measurements

To determine the total acoustic power radiated from the jet, the

whole jet should be surveyed as the source region for each field point located

on a sphere of radius r where 21TT

»À.

By utilizing the symmetry properti.es of the flow and noise fields, this can be greatly reduced as the upper half of the jet for each field point on a quadrant in the horizontal plane.

In. an attempt to materially reduce this still enormous amount of

work, the upper half of the mixing region of the jet was replaced by

35

points

on the surface of the upper half of an imaginary cylinder where the turbulence intensities are greatest, as shown in Fig. 12. They were located in such a way that a group of

5

points were distributed at equi-distance apart around the circumference of the semi-circle on Y-Z plane at each X = nD, where n is

an integer extending from 1 to

7.

For the field points, only one point was selected at is outside the refractive "valley-of-quiet", with x

=

6.0 ft. present work something short of an exhaustive investigation of invites more extensive efforts by others to follow.

4.2

Hot-Film Position and Orientation

cp

=

400, which

This made the

i t s kind, and

To determine the position of the hot-film sensor accurately, five equi-distant points were marked around the upper half of the outer rim of the jet nozzle and were used as the reference points (Fig. 12).

As the first step in positioning the hot-film to a desired position (X,Y,Z), it was brought to its corresponding reference point (O,Y,Z) to

determine the Y and Z coordina tes • Af ter that, i t was moved downstream to a desired X position to complete the operation.

Due to the uncertainties introduced by the effects of refraction of sound in the jet flow, a geometric set-up of the hot-film alone did not

guarantee the correct orientation to give the V_{x of the mathematical} formula-tion. Dynamic readjustments were tried by finding the orientations which give the maximum cross-correlations. Test runs at various hot-film positions and different frequencies showed that the maximum correlations occurred very near the geometric set-ups within the accuracy of the rotating gear

«

+ 20) .

This was not surprising at all,considering the very weak effect of the mean jet flow on the

phase characteristics of a sound ray (Ref.

28).

Consequently, the orientation

of the hot-film was judged not to be seriously affected by the refraction and

was set up entirely by geometry before each run throughout the measurements.

4

0

3

Unwanted Vibrations

Unwanted vibrations in the hot-film signal originated from three distinctive sourceso

The first, at a relatively low frequency (~

4

KHz) accompanying an

overall increase of sound level, occurred when, usually far downstream (~

5

D),

a substantial part of the probe support of Model 1210-10 was immersed in the turbulence. (See Figo

5

for the names of the probe parts). The modified Model 1210 AG-IO, areplacement for Model 1210-10 with the probe length extended from 1/2 in. to 1-1/2 in., has shown no such vibration down to

7

D.

The second, observed occasionally at a relatively high frequency (~ 40 KHz), came from the vibration of the probe when the tip of the probe"was placed

upstream (In) near the jet n ozzle where strong vortices were known to existo In

this case~ the probe was displaced slightly outward of the jet to avoid the

strong vortices, with the expectation that the general calibration procedures

described at Section

4.7

would take care of it.

The last vibration (~

8.5

KHz) was caused by the sensor itself when

the hot sensor, through a long exposure, had collected and baked on very small

dirt=oil mixture droplets formed from the oil vapour mixed in the air stream

from the compressor. Since there was no known method to remove this kind of

contaminant from the sensor, the vibrating probe was simply rep1.aced by a new one.

Although irrelevant to the present work, it might be worthwhile to note that, when the probe was accidently placed in the laminar flow of the

potential core, astrong vibration (~

5

KHz) was observed.

In order to ensure no such vibrations were causing error,

auto-correlations of the broad-band signals from both the hot-film and microphone

were inspected for any trace of pure tone (discrete frequency) components prior to each measurementso

In the microphone signal, there has been no peculiar prominent dis-crete frequency component af ter eliminating the causes in the hot-film signal

mentioned above.

4.4

Correla.tions Between Broad-Band Signals

Theory, treated in Section

2.2,

indicated that, once the

cross-correlogram bet ween the broad-band turbulence and acoustic signals with variable

time delay was measured, the intensity of the portion of noise at the

micro-phone which originated from a unit volume of the turbulence around the

hot-film could be calculated from its double derivative at proper retarded time.

The spectrum of this noise could be obtained from the Fourier transform of the

cross-correlogramo As a check, the integra~ of the spectrum over frequency

should equal the noise intensity calculated from the double-time derivative.

However, as shown in Fig. 13, the quasi-odd function shape of the

sensitive to selection of the point of evaluation, determined by the actual delay, time. The problem would be trivial in the case of the even function shape

pre-dicted by simple analysis as shown in Appendix B. In fact, the nominal delay

time, figured from the speed of sound, would locate the point of evaluation close

to the point of inflection where the curvature passes through zero. Therefore,

the position of T sho~ld be determined within an accuracy of 2 x 10- 2 msec. for

any meaningful ev~luation of the curvature.

In an attempt to explain the discrepancy of the experimental

correlo-gram with the theoretical.one of Appendix Band to find a clue for the proper

retarded time, the possibility of correlogram deformation due to the convection was explored theoretically using a mathematical model (Appendix C) af ter

Ribner's (Ref. 6) fashion, only to learn that convection does not affect the

predicted symmetry of the correlogram around the proper retarded time T •

o

On the other hand, an attempt to determlne T experimentally by placing

a pure tone sound source at the close vicinity of the got-film sensor revealed that the apparent proper retarded time on the correlógram, which could be identi-fied as where the maximum correlation occlirred, is not unique but differs as much as 0.5 msec. for different frequencies.

Since it was well known that the speed of sound is unique at fixed temperature regardless of its frequency, it was understood that the variation was caused by the difference in the accummulated phase characteristics of the

two input systems.from the transducers up to the input stages of the correlator.

Simple consideration would show the relation

where

ln(f)

=

2.78

~(f)

f

~(f): difference between apparent and true delay time at

frequency f, in ~.

~(f): difference between accummulated phase lags, at fre-quency f, of two input transducer systems at the input stages of the correlator, in degrees.

(4.1)

Consequently, it is important to note that, in cross-correlation

measurements using two non-identical transducer systems, careful considerations

should be given to their phase characteristics. Unless they are matched closely

enough to make ~:(f) in eq.

(4.1)

negligibly smallover the frequency range of

interest (e.g. Ref.

56),

the resultant correlogram, and naturally the information

it carries, cannot be guaranteed from serious distortion.

Since adequate matching of the phase characteristics of both systems with each other was not practical, it was decided to explore the alternative

narrow-band correlation approach, where, if the bandwidth of the filter was

taken narrow enough, the variation of the delay time could be neglected within that filter frequency range.

4.5

Correlations Between Narrow-Band Signals

As mentioned earlier in Section 2.3, this narrow-band approach has

(1) A simple measurement of the height of the correlogram at its maximum would eliminate both the error of intro-ducing double differentiations of the experimental curve and the difficult task of locating the exact proper

re-tarded time.

(2) The input signals have been filtered to a common fre-quency band before being fed into the correlator. Therefore, the signal-to-noise ratio and, in turn, the

correlation coefficient has been greatly enhanced, leading to clearer correlograms available.

(3) The necessity for a Fourier transform operation of the correlogram by digital computer would be replaced by an on-line power spectral density analysis.

As shown in Fig. 14, the frequency range for meaningful measurements

in the present experiment lies between 300 Hz and 16 KHz. The center frequency of the narrow-band filter was placed at each 1/3 octave point between 640 Hz and 16 KHz, totalling 15 measurement points for each hot-film position.

The arrangement of the delay time ranges of the correlator were given

in Table I for various frequencies. In the last column, for example, the item 5-6 msec, means that the time delay period 0-5 msec. is deleted from the corr-logram, allowing an expanded scale between 5 and 6 msec. delay.

TABLE I. Freq. Range (KHz) 0.64

_-

1.0 1.25 - 1.6 2.0 - 4.0 5.0 - 16.0

LAYOUT OF CORRELATOR DELAY TIME ARRANGEMENT FOR NARROW-BAND CORRELATIONS

Precomp. Period Computation Period Range of

(msec. ) T(msec.) (mse

0 (= OT) 10

o

-

10 5 (= IT) 5 5 - 10 4 (= 2T) 2 4 - 6 5 (= 5T) 1 5 - 6 Dèlay Time c. )

A typical cross-correlation between narrow-band signals was given

in Fig. 15.

4.6 Data Treatment

Working on normalized (nondimensional) correlations instead of directly measured correlations has been a very useful practice. This manipu-lation makes it possible to eliminate errors introduced by drift or uncertainty

in the system. The same method was adopted for present investigation in the data smoothing operation as described below.

First of all, each cross-correlation was normalized with its rms inputs measured with Random Noise Voltmeters (see Section 3.8). The result

was plotted against the center frequency of the narrow-band filter with the hot-film position fixed, then smoothed out by observation. In parallel, the

same operation was applied for each rms input measurement. These smoothed rms

inputs were used to convert the normalized quantities back to di~sional quantities

later when the smoothed correlations were required. This dimensionalization is necessary because the rms inputs are different for different center frequencies as weIl as for different hot-film positions; thus, the normalized correlations would not be suitable in reconstructing the spectra of noise originating from the

hot-film position, or comparing the intensities of noise from different hot-film

positions to each other.

To begin with, each individual hot-film sensor has a somewhat different

sensitivity of its own. Furthermore, as mentioned briefly in Sections

3.4.1

and

4.3,

the hot-film would collect dirt-oil mixture in the air stream from the

compressor during measurements and this would inevitably lead to the slow decrease of its sensitivity.

The normalizing operation would take care of most of the problem of

drift; however, to reconve~t the data later to dimensional (non-normalized) form

proper calibrations should be taken to compensate the sensitivity variations and

inject consistency into the collection of measurements. The most conventional

way of doing this is by calibrating the sensi tivity of each hot-film sen sor before and af ter each measurement.

In the present case, a somewhat different calibration method was used to substitute for the above mentioned process. A single hot-film probe was used at

the same filter frequency (f =

4.0

KHz) to measure cross-correlations at all

35 measurement points with other experimental conditions kept unchanged. These measurements provided a kind of calibration so that a direct comparison between the data collected with probes of different sensitivities was made possible. With use of the calibration the difficult hot-film vibration problem encountered

at the region of strong vortices near the jet exit was overcome. Where astrong

hot-film vibration was observed, the measurement position was moved to a nearby point. The noise spectral shape at the nearby point was taken as that of the

origin~l measurement point, and its relative level was determined by the

cali-bration measurement. This method is valid when the spatial variation of the

noise spectra is not large, which was generally the case.

Finally, it should be noted that one-half the peak-to-peak value of the correlogram at its maximum was measured in practice, as an evaluation of the peak height called for in the theory; this was to avoid possible error coming from the zero-level drift of the correlator output.

V. RESULTS AND DISCUSSION

5.1

Power Spectra and Intensity of Noise from a Unit Volume of the Jet

Deleting the constant factors, eq's.

(2.33)

and

(2.34)

can be rewritten

as

G ,(x, y; f)oef. 6R' 2')' (_x, y; T , f)

pp - - v p' - 0

x

for 0

<

f

<

00 otherwise zero,

"2

P where

"2

p (~,

_xJ

0<: I (~, y)

oe

JOO

f. tR' _v

_2p/

(~, y. T -' 0' 0 x (~,

I)

is defined as

p2(~)

=

J

p2

(~,

.~)

dV (I) V f) df

Hence, the height of the narrow - bandcross - correlogram at its absolute extremum tRI -:iI?i(T, f), weighted with the filter center frequency f, would give

v~'p 0

x

the relative power spectral density of noise at the microphone position generated

from a unit volume of turbulence centered at the hot-film. This in turn, if integrated over the entireOfrequency spectrum, would yield the total

(broad-band) noise power emitted from the unit volume. However, the present investigation has been mainly aimed to determine the relative power spectra and intensity of

noise from a slice of unit ~ength of the jet. Therefore, undue side-stepping

was limited to merely presenting the 6R' 2l?:(x, y; T , f) versus frequency f for

v l' - - 0

x

each series of measurements in Fig's. 16-20. (The scales are in arbitrary units; cf~Appendix E for calibration of the units).

By inspection, no more can be said than Series 3 exhibits somewhat lower peak frequencies compared with the others, and counterparts in symmetry are rather similar to each other.

5.2 Power Spectra and Intensity of Noise from a Unit Length of the Jet

Instead of dealing with the acoustic characteristics of individual unit

volumes of the jet, their averages over the jet circumference have been taken

as the principal quantities. In obtaining the average the smoothed measurements

for Series 2, 3 and

4

were given a weight factor of 2 to allow for both top and

bottom of the jet by symmetry, and the measurements for Series 1 and 5 were given

a weight factor of 1. Shown in Fig. 2~ are these angular averages of 6R'v 2p:(T

O,f)

x

versus frequency f at each downstream position down to 7D. Their relative power spectra per unit volume, which are proportional to f. tR' 20; (T ,f), were

cal-v

p

0

culated from Fig. 21 and presented in Fig. 22. x

Extracted from Fig. 22, Fig. 23 shows the dependence of the peak fre-quency fo of the acoustic power spectra upon the downstream distance X. It shows

qualitat~ve agreement with Powell's prediction (Ref.12) of a zone of X-l depen-dence (mixing region) followed by a zone of X-2 dependepen-dence (developed jet). This breaks down near the jet exit where the flow Reynolds number Re

= uxjv

of the present experiment is smaller than the critical Reynolds number (~

4

x 105 ) for the self-preservation and the similarity consideration employed by Powell to be applicable (Ref's.

7,

11 and 13).

Since the volume element dV(X) of the noise-generating turbulence in a slice of the jet at X can be written as

dV(X) = S(X) dX (5.4) where S(X) is thecross-sectional area of the zone of turbulence in the slice, eq. (5.3) would beceme

"2

~ p (~, X) S(X)

"2

if P (~,

z)

were uniform within dV(X).

However, p2(x, y) presumably varies according to a bell-shaped curve across S(X) radially. -If-such curves are similar for different axial positions

X - a plausible assumption - then the two sides of (5.5) are proportional rat her

than equal. This proportionality is expected to apply separately in the mixing

region and in the developed jet where the profiles are different, with perhaps

slightly different constants. In what follows a uniform proportionality is assumed

for the entire region 0 ~X ~ 7D as an approximation.

Therefore, the relative acoustic power from a unit length of the jet

dp2(x)/dX was calculated by multiplying the relative ~(x, X) integrated from

-

-Ftig. 22 with the relative S(X) in Table 11, and shown in Fig. 24 versus the

downstream position X. S(X), which has a form of an annulus in the mixing region

or of a disc in the fUlly developed region, was calculated at each X in Appendix

D from a jet model based on the actual measurements.

TABLE Ir. S(X) X S(X)jS(X = lD) 1 1.000 2 2.034 3 3.103 4 · 4.205 5 5.342 6 6.788 7 7.856

Figure 24 supports the validity of Ribner's (Ref. 11) Xo law in the

IDlxlng region for the acoustic source strength distribution along the jet axis.

Again

tbè

slight discrepancy near the jet exit can be interpreted without

diffi-culty in the light of the same argument regarding the critical Reynolds number

for self-preservation.

The roll-off from 5D might be interpreted as the start of Ribner's

X-7 law even though it is felt a little too early and is not conclusive.

5.3 Overall Power Spectra

From,eq. (5.1), the overall power spectral density function at x from