Noise Produced by the Interaction of Acoustic Waves and Entropy Waves with High-Speed Nozzle ...

CALIFORNIA INSTITUTE OF TECHNOLOGY DANIEL AND FLORENCE GUGGENHEIM JET PROPULSION CENTER

NOISE PRODUCED BY THE INTERACTION OF ACOUSTIC WAVES AND ENTROPY WAVES WITH HIGH-SPEED NOZZLE FLOWS by Mark S. Bohn

May 1976

SUPPORTED THROUGH

GRANT NUMBER DOT-OS - 40057

U . S. DEPARTMENT OF TRANSPORTATION

0 FF I CE 0 F N 0 I S E ABATE M EN T

CALIFORNIA INSTITUTE OF TECHNOLOGY Division of Engineering and Applied Science Daniel and Florence Guggenheim Jet Propulsion Center Karman Laboratory of Fluid Mechanics and Jet Propulsion

NOISE PRODUCED BY THE INTERACTION OF ACOUSTIC WAVES AND ENTROPY WAVES WITH HIGH-SPEED NOZZLE FLOWS

MarkS. Bohn

May 1976

Approved: Frank E. Marble Principal Investigator

Performed with the Sup port of the U. S. Department of Transportation Office of Noise Abatement

Grant Numbe r DOT -OS -40057

-IABSTRACT Some aspects of the noise generated internally by a turbojet engine are considered analytically and experimentally.

The emphasis

is placed on the interaction of pres sure fluctuations and entropy fluetuations, produced by the combustion process in the engine, with gradients in the mean flow through the turbine blades or the exhaust nozzle.

The results are directly applicable to the problem of excess

noise in aircraft powerplants and suggest that the phenomenon described is the dominant mechanism. The one-dimensional interaction of pressure fluctuations and entropy fluctuations with a subsonic nozzle is solved analytically.

The

acoustic waves produced by each of three independent disturbances are investigated.

These disturbances, which interact with the nozzle

to augment the acoustic radiation, are (i) pres sure waves incident from upstream,

(ii) pressure waves incident from downstream, and

(iii) entropy waves convected with the stream.

It is found that results

for a lar ge number of physically interesting nozzles may be presented in a concise manner. Some of the second-order effects which result from the area variations in a nozzle are investigated analytically.

The interaction

of an entropy wave with a small area variation is investigated and the two-dimensional duct modes, which propagate away from the nozzle, are calculated. An experiment is described in which one-dimensional acoustic waves and entropy waves are made to interact with a subsonic nozzle.

-2-

The response of the nozzle to these disturbances is measured and compared with the response as calculated by the analytical model. The interaction of two-dimensional entropy waves with a subsonic nozzle and with a supersonic nozzle is investigated experimentally.

The results are explained in terms of an analysis of the acous-

tic waves and entropy waves produced by a region of arbitrary heat addition in a duct with flow.

-3T ABLE OF CONTENTS Page

Chapter

I.

II.

III.

Abstract

1

Table of Contents

3

Notation

5

INTRODUCTION

7

References

11

THE INTERACTION OF ACOUSTIC WAVES AND ENTROPY WAVES WITH A SUBSONIC NOZZLE THE ONE-DIMENSIONAL MODEL

12

2. 1 Introduction

12

2. 2

Development of the Analytical Model

14

2. 3

Numerical Solution

19

2. 4

High-Frequency Asymptotic Solution and Normalization

22

2. 5 Numerical Results

33

2. 6 Examples of the One-Dimensional Model

37

Figures

44

References SECOND-ORDER DUCT ACOUSTICS

67 69

3. 1 Introduction

69

3. 2

The Expansion to Second Order

71

3. 3

First-Order Solutions

73

3. 4 Second-Order Solutions

77

3. 5

90

Calculation of the Duct Modes

3. 6 Response to a General Two-Dimensional Entropy Wave 3. 7

IV.

Response to High-Frequency Disturbances

99 103

3. 8 Calculations and Discussion

106

Figure

109

References

110

EXPERIMENTS CONCERNING THE RESPONSE OF A SUBSONIC NOZZLE TO ONE-DIMENSIONAL PRESSURE AND ENTROPY DISTURBANCES

111

4. 1 Introduction

111

4. 2

112

Description of the Experiment

-4-

Chapter

V.

4. 3

Data Acquisition and Processing

119

4. 4

Results and Discussion

134

4. 5

Conclusion

143

Figures

145

References

163

EXPERIMENTS CONCERNING THE RESPONSE OF NOZZLE FLOWS TO TWO-DIMENSIONAL DISTURBANCES

164

5. 1 Introduction

164

5. 2 Experimental Apparatus -- The Two -Dimensional Pulse Heater

165

5. 3 Fluctuating Heat Addition in a Two-Dimensional Duct

167

5. 4

173

Results of the Experiment and Discussion

Figures

178

APPENDIX A.

First-Order Steady Solution

183

APPENDIX B .

Calc ulation of the Green's Function for the Second-Order Inhomogeneous Solution

18 9

APPENDIX C.

Forcing Function

194

APPENDIX D.

Second-Order Homogeneous Solution

196

APPENDIX E.

Integrals Represented as I

199

APPENDIX F. Figures

, , Etc. a-1/m Description of Electrical Circuits

203 210

-5-

NOTATION dimensionless axial position in duct

X

nozzle length inlet Mach number exit Mach number velocity perturbation normalized by local mean velocity pressure perturbation normalized by local mean pressure X'{

entropy perturbation normalized by C local mean velocity normalized by a reduced frequency

= Wt/ a>:<

p

>!<

for Sections 2. 2, 2. 3

= wt/{u2- u1) elsewhere

dimensionless wave numbers in upstream duct dimensionless wave numbers in downstream duct acoustic wave upstream of nozzle propagating downstream acoustic wave upstream of nozzle propagating upstream acoustic wave downstream of nozzle propagating downstream acoustic wave downstream of nozzle propagating upstream

p+ /P+ 2

R

T

R T R

a

p

p 1-

1

/P:

m

p 1- /P2-

m

P2 P2

e e

+/

P

-

+/a

2

p 1-/0 value of

.c. 3

at nozzle inlet

-6independent variable used in high-frequency analysis (and the value at inlet and exit) denoting axial position

p

r;

u

L

2 1

as used in high-frequency analysis as used in high-frequency analysis

i!3 zoe' z le' ~q,±

functions of M , M 2 1

T

indicates T

R

indicates R

p p

or T or R

m m

Subscripts e

indicates entropy disturbance

p

indicates P

m

indicates P - disturbance 2

co

high-frequency value

0

low-frequency (quasi-steady) value

+ disturbance l

-7-

I.

INTRODUCTION

One aspect of the aircraft engine noise problem which has received relatively little attention in comparison with turbulent acoustic sources, is the production of acoustic disturbances by the longitudinal variation in temperature of the gas passing through the nozzle.

That

these non-uniform temperature regions -- or entropy spots --interact with the nozzle geometry to modify the flow is evident from the fact that the mass flow through a choked nozzle varies inversely as the square root of the stagnation temperature and hence a fluctuation in stream temperature leads to a corresponding fluctuation in nozzle mass efflux.

Two aspects of this phenomenon suggest its possible relative

importance.

First, the fluctuating mass flow behaves as a monopole

(or at worst a dipole) singularity and is consequently a more efficient acoustic radiator than a quadrupole.

Second, the non-uniform temper-

ature fluctuations, which are the origin of the disturbances, are not necessarily small.

Fluctuations in absolute temperature of 20 per cent

are usual from even a very good prilnary burner of a gas turbine, and those from an afterburner may be considerably larger. It is interesting that this type of gasdynamic problem first arose

in studies of the interior ballistics of rocket motors.

Pressure pulsa-

tion in the chambers of liquid monopropellant rocket motors produce non-uniform temperature gas masses which, because they affect the mass flow through the . nozzle, react, in turn, to change the chamber pressure and thus influence combustion stability.

The first complete

analysis of the resulting acoustic response of the rocket chamber was carried out by Tsien

1

for a nozzle of finite length and particular

-8geometry.

These results were gener alized by Crocco

quency range and were utilized by Marble

3

2

for the full fre-

for studying the stability

problem of hi-propellant rockets where the fluctuations in mixture ratio can produce quite significant variations in local temperatures of the combustion products. All of this work was restricted to the upstream side of the nozzle, however, and it was not until work was performed under the present grant that the acoustic field radiated from the downstream section of the nozzle was studied.

The work of Candel

4

and Marble

5

treated the

source-like character of both the compact and extended nozzles under choked conditions.

The subsequent extensive experimental examination

7 6 8 by Zukoski , Auerbach , and Zukoski and Auerbach into the effects of entropy waves convected through a choked nozzle gave results that were both qualitatively and quantitatively in agreement with the analysis and demonstrated the limitations of the compact element analysis.

The

effects of finite length, or non-vanishing reduced frequency, were shown by Marble 9 to be associated with the behavior of the upstreamfacing and downstream-facing waves in the divergent, supersonic part of the nozzle.

Because both of these waves are transported downstream

by the supersonic flow, the phase between them is altered during passage through the nozzle and causes the observed changes in the pressure fluctuation at the nozzle exit. At this point, two important questions remained.

First, both

the analysis and experiments dealt with plane longitudinal entropy waves, while the condition in a real nozzle was certainly non-uniform over the nozzle cross-section.

Under such circumstances, it is not clear how,

-9or in what approximation, the analysis could be interpreted. Second, and equally important, was the question of the unchoked nozzle, the importance of which concerned the applications of theory to turbinegenerated noise, Cumpsty and Marble tuations originate in the main burner.

10

, where the temperature fluc-

5 9 The compact analysis '

showed that the general level of pressure pulsations was lower than that generated in a choked nozzle by the same temperature fluctuations.

For this reason, it appeared possible that the effect of finite

nozzle length might be relatively quite important, even at modest values of the reduced frequency. The present work aims to investigate these two issues, both experimentally and analytically.

The analytical study of the interac-

tion of one -dimensional pressure and entropy waves with a onedimensional subsonic flow with strong mean gradients is described in Chapter II. Because the discharge flow from the nozzle is subsonic, pressure waves may impinge upon the nozzle from downstream as well as upstream and these two interactions, in addition to the convected en tropy wave, complicate the pres entation of results.

It will be found,

however, that using a sort of similarity argument, the results may be presented in a fairly compact form. In Chapter IV, an experimental program is described which was carried out to examine the response of a subsonic nozzle to these impinging waves.

The results, although restricted in their range of

reduced frequencies, confirm the assumptions of one -dimensional flow that were employed in the analysis.

The analysis is then ex-

-10-

tended so that the results for a wide range of parameters may be presented concisely.

In Chapter III an analytical investigation is present-

ed examining some of the two -dimensional effects neglected in the one -dimensional analysis of Chapter II. One of the simplifications which made the one -dimensional analysis tractable was to neglect the waves transverse to the flow direction.

This assumption simplifies enormously the acoustic modes of

the nozzle and is quite reasonable so long as the nozzle's transverse dimension is small in comparison with its length.

To investigate the

complete two-dimensional problem in more generality, a secondorder perturbation analysis is carried out in which the contraction of the nozzle and the strengths of the waves are both assumed small. The other essential assumption in the one -dimensional analysis is that the entropy waves are planar, with their propagation vectors pointing along the nozzle axis.

While this is a more reas enable as-

sumption for the turbine nozzle than for the engine discharge, the question of complex structure to the entropy waves is one that must require attention.

Chapter V presents experimental results obtained

with non-planar entropy disturbances transported through the nozzle.

-11REFERENCES FOR CHAPTER I l.

Tsien, H. S. "The Transfer Function of Rocket Nozzles," J. Amer. Rocket Soc., V. 22, 3, (1952 ), 139-143.

2.

Crocco, L. "Supercritical Gaseous Discharge with High Frequency Oscillations," Aerotechnica, Roma, Vol. 33 (1953), 46.

3.

Marble, Frank E. "Servostabilization of Low-Frequency Oscillations in Liquid Propellant Rockets, " J. Appl. Math. and Physics (ZAMP), VI, Issue 1 (1955), pp . 1-35.

4.

Candel, S. M. "Analytical Studies of Some Acoustic Problems of Jet Engines, 11 Ph. D. Thesis, California Institute of Technology, Pasadena, California (1971).

5.

Marble, Frank E . 11 Acoustic Disturbances from Gas Nonuniformities Convected Through a Nozzle, 11 Proceedinfs, First Interagency Symposium on University Research inransportation Noise, Stanford University, California (March 28-30, 1973 ), pp. 547-561.

6.

Zukoski, Edward E. 11 Acoustic Disturbances Produced by Gas Nonuniformities Convecting through a Supersonic Nozzle, 11 Proceedings, Second Interagency Symposiwn on University Research in Transportation Noise, North Carolina State University (June 5-7, 1974), pp. 902-915 .

7.

Auerbach, Jerome M. 11 Experimental Studies of the Noise Produced in a Supersonic Nozzle by Upstream Acoustic and Thermal Disturbances, 11 Ph. D. Thesis, California Institute of Technology, Pasadena, California (1975).

8.

Zukoski, E. E. and Auerbach, J. M. 1 1Experiments Concerning the Response of Supersonic Nozzles to Fluctuating Inlet Conditions, 11 J. Eng. for Power, Vol. 98A, .!. (Jan. 1976), pp. 60-64.

9.

Marble, Frank E. 11 Response of a Nozzle to an Entropy Disturbance -Example of Thermodynamically Unsteady Aerodynamics, 11 Unsteady Aerodynamics, Symposiwn Proceedings edited by R. B. Kinney, University of Arizona (March 18-20, 1975), pp. 699-711.

10.

Cwnpsty, N. A. and Marble, F. E. 11 The Generation of Noise by the Fluctuations in Gas Temperature into a Turbine, 11 University of Cambridge, Dept. of Engineering, CUED/ A TURBO/TR 57 (1974).

-12-

II.

THE INTERACTION OF ACOUSTIC WAVES AND

ENTROPY WAVES WITH A SUBSONIC NOZZLE THE ONE-DIMENSIONAL MODEL 2. 1 Introduction In this chapter we investigate the effect of acousti c waves and entropy waves propagating through a nozzle wi th a subsonic mean flow.

The problem of pressure disturbances in ducts with mean flow

and area change has been studied by many, ref. 1-5 for example, but the effects cause d by entropy disturbances have not been as widely studied. Candel

6 solved the problem of acoustic and entropy waves con-

vected into a choked nozzle. nally developed by Tsien

7

He used a formulation which was origi-

to study the oscillations in a rocket engine.

This formulation is the basis of the model developed in this chapter. Auerbach

8

and Zukoski 9 showed the validity of the Candel model ex-

perimentally.

The (choked) mean flow in a rectangular (cross-section)

blowdown tunnel was perturbed with entropy waves .

The entropy

waves were created by electrically pulsing a resistance heater lo c ated upstream of the nozzle, and then, using a periodi c mass bleed system (also upstream of the nozzle), the pressure wave component of the disturbance was cancelled.

The production of acoustic waves by the en-

tropy disturbance was then verified by the dete ction of pressure disturbances throughout the nozzle. The solution fo r low-frequency disturban ces was investigated by Marble

10

For disturban ces with wavelengths w hich are l ong com-

pared to the nozzle length, the resultin g solution w ill give disturb-

-13ances with constant phase throughout the nozzle.

This solution is

called the compact or quasi-steady solution, and may be solved by considering only matching conditions at the nozzle inlet and exit.

The

details of the mean flow in the nozzle may be neglected. Cumpsty and Marble

11

have investigated the interaction of

pressure and entropy disturbances with one or more turbine blade rows.

Large deflections and accelerations in the mean flow were

considered; however, the disturbances were assumed to be quasisteady so that precise details of the mean flow in the blade passages could be neglected. Our aim here is to examine these effects of mean flow variations that occur 1n the flow through such blade passages and, equivalently, exhaust nozzles.

We consider only one-dimensional, sub-

sonic flow with small disturbances.

In the choked nozzle, the throat

essentially decouples the supersonic portion from the rest of the nozzle.

Two independent solutions result.

The first solution repre-

sents the effects of an entropy wave convected into the nozzle, when no acoustic wave is incident upon the nozzle entrance .

The second

solution represents the results of an acoustic wave incident upon the nozzle entrance when no entropy wave convects into the nozzle. In the subsonic nozzle, every portion of the nozzle can communicate with every other portion.

The result is that we must admit

a third independent solution which represents the effects of an acoustic wave propagating upstream and impinging upon the nozzle exit. In the following sections we develop the equations which will serve as the analytical model.

Next, we discuss a method of nu-

-14-

merical solution.

The emphasis here will be on choosing the appro-

priate boundary conditions to give the three independent solutions. The solution for high-frequency disturbances is then discussed.

The

results of this solution are then used to normalize some numerical calculati ons so that a concise presentation of the results may be made.

Finally, the results are discussed and several examples of

the use of these results are presented.

2. 2

(See Appendix G for notation.)

Development of the Analytical Model We are given a duct of constant cross -sectional area with a

mean flow of Mach number M

1

.

The cross -sectional area then

changes in such a way that after an axial distance .f, the Mach number is M

2

.

The flow then continues through a constant cross -section-

al area duct.

If we let the cross-sectional area (of the axial region

in which the mean flow is changing) be call ed A(x), we have the following diagram describing the duct.

At) M=M,

-

M=M2.

--x

I

Diagram for the Analytical Model We assume that the gas flowing in the duct is ideal and in viscid, and that the mean flow is isentropic and wholly subsonic.

We will neglect

-15-

two-dimensional effects, and simply use the area variations to give mean flow variations.

These assumptions allow us to describe the

flow with the equations of momentum, continuity, entropy conservation, and the equation of state as follows:

du + u.. du.. c)t dx

+ ds

dt

+

+-' oP p ax

1.. df.pu.A) A dx

u.. ds

ox

0

(2. l)

0

(2. 2)

0

(2. 3)

(2. 4)

We will linearize these equations by assuming that a solution exists which is the sum of a known function of axial position only, plus a small periodic function which also varies with axial position.

For

example, the velocity will be expressed as

where W

is the radial frequency.

The primed quantity is, in gener-

al, complex, but we let

lu'l J -2{l-;i'') [

J

I

'(-/

3 (t-1)

J

2 ·· 7 ~+/ + 8 (~+oz+·

(2.27)

-27To find

A , 8

we use the same procedure as we used in

the numerical solution.

For the plus solution we specify that

P{J;_)+M,U(Ji.)= Z P/ 7- 0 P(:Je)-Mz.U(J~)

=o

Inserting (2. 23) into these relations we can solve for terms of

C., (ae)

short:

l.oe

ro (J~)

and

=

In sol~ing for A

A , B

which we will call ~re

in

and :C..oe for

l.o (Je) 8

we retain only terms to

CJ{-1.) .

For the minus solution we specify

P(Ji)+M,U(J;_) = o

P(Je)-Mz.U(Je) ~ 2P; e'. Cz-(X~-Xi) "I= The calculation of calculate

Tp 1 T rn J

A ,B

R p 1 R..,

0

for both solutions now allows us to

Since this is the high-frequency solu-

tion, we use the subscript:

T,(l) Tma> Rp

• • + • + •+ + ~

OS~

AI

4>

AI

2

• +

•

f3

+ + +

+ +

=

5

6

wt/ (u2 -u 1 )

7

Ml

M2

~

.2

.8

X

.3

.8

J!!.

.4

.8

•

.5

.8

+

.6

.8

+

+

lJ

3

• I

•

4>

Q)

•

•

4>

-._}

~

•

~

0 lJ)

s::

(]

~

@>

AI

(no

E-t

~

X

t

Q)

~

AI

(!1.

(O'b

~

Phase of Transmitted Wave (Entropy Solution) Ve rsus Reduced Frequency

8

9

10

U1

00

Snl

+

CJ)~

* X

rot-

-+

A

(!)

L:~

* A

Fig. 2-16

+ +

*

+

X A

~~

Normalized Magnitude of Reflected Waves (Entr opd Solution) Versus Reduce Frequency

+

*

+

(!)

*

Lf)

X

0 I!)

~

---~

A

I

Ml

M2

(!)

.2

.8

X

.3

.8

A

.4

.8

*

.5

.8

.6

.8

::1' I!)

I

*

-

+

(!)

0"?

A

X

0JI-

*

(!)

X (!)

(!)

-cr

A

X

-to

* (!)

(!)

~

X

A

ji

I

I

I

I

I

.L

1

2

3

4

5

6

0

0

13

U1

...0

=

w 0 (~2 -ul)

*

*

+

+

+

+

A

(!)

A

*

I

I

I

~

7

8

9

10

X

*

X

(!)

&

X

Sr

&

CY!r

~

X

X

~

(!)

(!)

~t-

&

~

• X

~ Fig. 2-17

*

No rmali zed Magnitud e of Transmitted Waves (Pl us or Min us Solution) Versus Redu ced Frequen cy

*

&

X (!)

l':t-

*

X (!)

(.Q

.--,

*

X

...... I

&

&

Ln

I

(!)

E-1

81E-10

-

1-1

0' 0

::::1'

X

&

*

,-----, ...... I

-

M2

(!)

.2

.7

X

.3

.7

A

.4

.7

*

.5

.7

(!)

(Y)

X (\J

(!)

&

)(

oW 0

1

*

&

*

X ....... 1--

*

&

E-1 IE-! 0 1-1

Ml

* 2

3

lJ 13

=

5

6

wt; cu-2 - li)

7

8

9

10

Fig. 2 -1 8 Error in Using High Frequency Solution for Phase of Transmitted Waves (Plus or Minus Solution) Versus Reduced Frequency

ro ['-

en..

+ o ptz> = oX

d~

d!.J

( 3. 11)

0

(3. 12)

0

~ dt + u 0 X

_!_(Q_ (f

I

0o (11

+-U-

~

3. 3

d><

xp21_ J...(Po')z)_(Q_ d_ \(!I2'_J.(Pr'')2) 2 (J t + u d X AP 2( _p p

p

~ + v(•> ~

dlj

(

)p a

-76Then +(X)

has first, second, and third derivatives which are con-

tinuous. T h e boundary condition on V

(t)

(velocity is tangent to the

wall) gives

dlf>_(o oi.J.

Lj.=b

udf

~=o

dx

and therefore

ofP

_

I

d~j- (Lo) - (2 rr)~

~~(S,b)

f co olf - Q)

-ix~

O'cf (x,o)e

dx

(3. 25)

= 0

The solution for

~(~ J lcJ)

takes the form

(3.26)

where

a(i)

potential

l.f

and

b(f)

will be determined upon applying (3. 25).

is then found by the Fourier inversion of

The

p

a. (3. 62)

We are now in a position to perform the integration (3. 52) using (3. 54), (3. 55), and (3. 62).

I

X-'f

for X.

We substitute

in (3. 54) and (3. 55).

r

for

X

in (3. 62) and

The integration (3. 52) may be

written

c..

(zrr)'lz

p~ (x,'J) = LrcrJ Hfx- rJd r

If we are interested in

X< -a.

L a.

+ G(f)J(>a.

)wehaver>x

(3.63)

(orr0

, see (3. 54)).

that the radiation condition is satisfied. ing upstream for

X>) a

We a1s o must insure

We want no waves propagat-

and no waves propagatin g downstream for

-89-

Xi>

-f

(or for

r ) refers

to the appropriate form of

x- r

(or X-~0

).

In this man-

ner we may find

Since this is the homogeneous solution, we define

(3.65)

(3. 66)

which will give the waves propagating at large distances from the contraction and correspond to (3. 42) and (3. 43) for the inhomogeneous solutions.

-903. 5

Calculation of the Duct Modes The acoustic waves which propagate in a (constant-area) chan-

nel must satisfy the homogeneous wave equation: 2

d d ) z(cf {(dt +Udx - C(Jxz lf (X, I.J 1 f:)

where

Jz ) }

+ d~2

-

Cf - o

(3.67)

is the velocity potential as so cia ted with the wave.

The solutions to (3. 67) will be of the form (3. 68) which are waves with constant phase lines with normal at angle V X direction.

to

Substitution of (3. 68) into (3. 67) gives the dispersion

relation

fo_ = w/c

(3. 69)

Mcosv~J

Since the vertical velocity must vanish at the channel top and bottom, say ~ =-0,

2b

we have

o'f 0 iJ ~J-=0

o a l.&.

0

w

a

:::>

t-

_J

a..

~

c

-.!:)

I

'lit"

D

D ~

.

~----------~---------,-----------r----------~---------,-----------r----------,

C:O.oo

10.00

20.00

30.00

-40.00

50.00

tiO.OO

TINE NS. Fig. 4-5.

Ensemble averaged waveform for typical pressure fluctuation recorded from position x = 11. 511 (See Fig. 4 - l ). Fundamental frequency is 250 Hz.

10.00

ANFLITUOE SPECTRUN EXPERINENT 136 CHANNEL 8

c

.

~

G:)

.... lt)

c

c

wt!l~ cr: .....

I

_J

.......

a >tn

\.J1

0 I

.

c c

0

C

•

.tyv......-•

9J.Oil

~'y ·~ • ~m' I

81l.Oil

160-0il

I

2~0.00

FREQUENCY

320.00

I

~00.00

I

~ao.oo

•10 1

Fig. 4-6 The spectrum of the waveform given in Fig. 4-5. The peak (at 250 Hz) represents a pressure fluctuation of 123 db.

I

561l.Oil

EXPERIMENT

D

L36

CHANNEL 9

.

CD

.,c. D

D

wco

(.!).; a:

t-

I

_.J

......

c

......

l]1

I

>a ~

0

•

CD ~

• ~-------------r------------~-------------,~----------~~------------r-------------~------------,

9J .oa

LD.BD

20.BD

3o.oo

-to.ao

sa.oo

i)Q.(JQ

;a.oo

liNE MS. Fig. 4-7 Ensemble averaged waveform for typical pressure fluctuation recorded from position x = 13. 5" (See Fig. 4-l). Fundamental frequency is 250Hz.

AMPLITUDE SFECTRUN EXPERIMENT 136 CHANNEL 9

.... 0

c

co

.

e:J

c

tn LLJC (!)' q:e:J

.......

_,

I

CJ

>l\.1

. c e:J

0

c

•

I

...... U1 N

-vv,,,. ·· ··

c:n.oo

I

I

1

SD.OD

., '

I

'Vt~

lfiD.OD

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2iO.DD

FREQUENCY Fig. 4- 8

I

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The spectrum of the waveform given in Fig. 4-7. represents a pressure fluctuation of 115 db.

The peak (at 250 Hz)

'

CHANNEL 2

EXPERIMENT t36 IC

.,.

tf)

tf)

.

OJ

Cl

0

W':

I

......

c.!)IC

\}1

a: .,_

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.J Dtn >OJ

.

IC

' IC

.J

tf)

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'e.DD

lD.DD

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2 6-DD

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11ME MS.

Fig. 4-9 Ensemble averaged waveform for typi cal temperature fluctuation signal. Fundamental frequency is 250 Hz .

16.60 1

0

ANfLITUDE SfECTRUN EXfERINENT 136 CHANNEL 2

.

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Fig. 4-10 The spectrum of the waveform given in Fig. 4-9. The peak (at 250 Hz) represents a temperature fluctuation of . 40°C.

I

560.00

BETA

= .85

~

,_. 1.11 1.11 I

?>

Fig. 4-11

+

-

Phase plane representation of P 1 {denoted 11 P 11 ) and P~ (denoted 11 M") acoustic waves from several pressure measurements m the upstream duct. The respective vector averages are denoted 11 +11 and 11 - 11 • Phase angles are measured (positive) counter-clockwise from right horizontal axis. Magnitude scale is arbitrary, but consistent for the same type of wave.

(>.

~' ~

£''1 d 2'1 d _ _ _ _ _ _ ____:=~:::!!~¥~~~ ..... \]1

0' I

BETA= .85

lv1 t, .3 INDEX I

2

P{

3 4

POSITION

6. 75" 9.25" 11.25

11

13.25"

Fig. 4-12 Phase plane representation of and Pz as calculated from several positions in downstream duct. Notation and convention follows Fig. 4-11. Indices near end of vector can be used with the table to determine location of 2 points in the duct at which the pressure measurements were made which determined that vector.

BETA = .85

+

f2~

·~Re 4

P2T"J '

·10

/

/

-- - --~

P.+ 2•

-- - -- -

~ "'*""P--Rm 2

_..,.p2

I0-

4

/ /

/ /

...... 1..11

/

-.J

/

~:

/ Jl.

p+ I

P+R I

p

Fig. 4-13a + Phase plane representation of three (P 1 R • oR • P~- Tm) components making up comput~d Pvecto't P 1 ; . The measured values, P/ and P 1- are shown dashed and were taken from Fig. 4-11.

Fig. 4-13b + Phase plane representation of three (P1 Tp• o T e• P 2- _fro) components making up computed vector P 2 The measured values and P 2are shown dashed and were ta.ken from Fig. 4-12 (see vectors labeled"+" and 11 - " ) .

*.

Pt

0 0 0

.

I p' (x) I

(X) r -

_P

"::I' I

0 0 0

----

....... II

a:

(.!)

0

••

1-

w

X

1l.jtCL

BETA= 0.85

0

.

vs. X

a:

1.1....

0 0 0

w _J a:

.

LJ

(/')

=r

•

81.

~

0

.I

\

......

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C"\1

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SCRLE FRCT~fl = 10

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-1 .200

-0 .BOO

-O.ij00

X INCHES

0.0 FR~M

O.ijOO

O.BOO

1

l .200

ENTRRNCE

Fig. 4-14 Magnitude of pressure perturbation field in the tunnel as calculated by equation 4. 34. Measured data points are indicated with the symbol. (The reduced frequency BETA:: wi,/a* )

1.600

0 0 0

.

PHASE { :•

C\1 .----

-w (f)

0

w 0 a: 0 l? . w __.

0 ..__

I 0..

"'0

..-

vs. X

BETR= 0.85

II

a: 0

1-

u

a:

w (f) )

we may re-write (5. l 0) in the form

(5. 2 5)

Hence the entropy wave retains its shape as it convects towards the nozzle. We summarize the above results for the experimental conditions of interest : (i)

The entropy wave produced by the pulse heater retains its shape as it convects towards the nozzle.

This will, in

gene ral, be two-dimensional. (ii)

The plane waves produced by the pulse heater scale like (t-c 0

I -~t

.

we close the contour in the upper half

may be shown to be QL

exp[-i 1TJrcx-a)-i(I-M 2) '12 ( b-tJ))]

smh [(I-M~"b1T-f]

21 'f=rr.f/~

e tp[i rr§. ((x-a) -i (!-M )(b-CJ)) J 2

smh{0-M'hbrr-t'] and from

plane

The contour will be indented above the poles on the

real axis and in this way the contribution from the pole at

and from

f

"$= 0 2

-rri (a ) ct-M'J~b rr./ The residue of the poles

j={l-~;fzb

will be

f= -rr-f/a.

-188The principal value may be calculated by summing over n, multiplying by 2fr( and subtracting the contributions from the poles on the real axis. The calculation of Ift , Ill-~

then gives I.t (X/J)

.

Eq. (A8)

will then give

U (IJ =

Urrt. 4o..2(!-M1)'/2

+Ja [ cos(rrtf)cosh([{l-Mr)'lz(b-LJ)) + co~rr!)c.osr(~