NASA CR 175067

R83AEB566

July, 1983

o

NASA

National Aeronautics and

Space Administration

Turbofan Aft Duct

Suppressor Study

A.A. Syed

R.E. Motsinger

G.H. Fiske

General Electric Company

by

M.C. Joshi

R.E. Kraft

Douglas Aircraft Company

(NASA-CR-175067) TURBOFAN AFT DUCT

SUPPRESSOR STUDY _General Electric Co.)

202 p Avail: NTIS HC AIO/MF A01 CSCL 21E

G3/O?

N87-29539

Unclas

0100_91

!

il

prepared for

National Aeronautics and Space Administration

Lewis Research Center

21000 Brookpark Road

Cleveland, Ohio 44135

Contract NAS3-22766
NASA CR 175067

R83AEB566

July, 1983

N/ A

National Aeronautics and

Space Administration

Turbofan Aft Duct

Suppressor Study

A.A. Syed

R.E. Motsinger

G.H. Fiske

General Electric Company

by

M.C. Joshi

R.E. Kraft

Douglas Aircraft Company

prepared for

National Aeronautics and Space Administration

Lewis Research Center

21000 Brookpark Road

Cleveland, Ohio 44135

Contract NAS3-22766
1. Re;)ort No.

NASA CR175067

4. Title and 5uiotitle

AFT DUCT SUPPRESSOR STUDY

2. Government Accmion No. 3. Recipient'$ Cgtalog No.

5. Report Date

7. Author(s}

A.A. Syed, R.E. Hotsinser, G.H. Fiske, M.C. Joshi, R.E. Kraft

9. Pmrfmming Organiuti_NomeandAddrlm

General Eleetrlc Company

_rcreft Engine Business Group

1 Neumann Way

Cincinnati, Ohio 45215

12. Soons_'ingAgat_:vNememdAddr,_

NASA Levis Research Center

21000 BrookparkRoad

Cleveland, Ohio 44135

6. Performing Organization Code

8. P_rformingOrganization Report No.

R83AEB566

10. Work Unit No,

11. Contract or Gnmt No.

_L_$3-22766

13. TyI_ of Report and Period Coverecl

Contract Report

14. Slxmsoring Agac_'y Code

15, Sup_ernentMVNotH

Program Manager: B. Clark

NASA Lewis Research Center

21000 Brookpark Road

Cleveland, Ohio 44135

16. Abstra_

Suppressions due to acoustic treatment in the annular exhaust duct of a model fan have been

theoretically predicted and compared with measured suppressions. The predictions are based on

the modal analysis of sound proDagatlon in a straigh_ annular flow duct with segmented treatment.

Modal distributions of the fan nolae source (fan-stator interaction only) were measured using induct modal probes. The flow profiles were also measured in the vicinity of the modal probes. The

acoustic impedance of the slngle-degree-of-freedom treatment was measured in the presence of

grazing flow. The measured values of mode distribution of the fan noise source, the flow velocity

profile and the acoustic impedance of the treatment in the duct were used as input to the prediction

program. The predicted suppressions, under the assumption of uniform flow in the. duct, compared

well with the suppressions measured in the duct for all test conditions.

The interaction modes generated by the rotor-stator interaction spanned a cut-off ratio range

from nearly 1 to 7.

The modal probe dare and a users' guide to the computer programs ere available as separate

publications.

17. Key Worm (Suggast_lby Author(s))

Fan exhaust duct noise suppression

Acoustic propagation In segmented annular ducts

Acoustic spinning mode measurement

In-situ acoustic impedance measurement

18. Dis_ibutionS_tement

Unclassified - Unllmlted

19, Secwity Oauif.(ofthismPort}

Unclassified

20. Sacurity _auif.(ofthi$ _ga)

Unclassified

21. No. of Pages

188

"For sale by the National Technical Inf(xmation Service, SpHnefleld, Virginia 22161

22. I_ice"

NASA-C-168 (Rev. 10-75)
TABLE OF CONTENTS

Section

1.0

2.0

3.0

4.0

List of Fisures

List of Tables

List of Symbols and Definitions

SUMMARY

INTRODUCTION

OBJECTIVES

THEORETICAL CONSIDERATIONS

4.1 Nave-Field Theory

4.1.1 Sound Propagation In Annular Ducts With Flow

4.1.2 Modal Analysis for Segmented Duct

4.1.3 Nozzle Reflection Coefficient

4.2 Mathematical Details of Modal Decomposition

4.3 The In-Situ Impedance Measurement Technique

5.0 " EXPERIMENTAL PROGRAM AND RESULTS

5.1 Test Apparatus

5.1.1 Test Facility

5.1.2 Fan Vehicle

5.1.3 Aerodynamic Instrumentation

5.1.4 Acoustic "Mode" Probes

5.2 Vehicle Aero-Acoustic Results

5.2.1 Selection of Fan Speed Points

5.2.2 Fan Performance

5.2.3 Test Section Aerodynamic Data

5.2.4 Acoustic Data

5.3 Acoustic Treatment Impedance

5.3.1

5.3.2

5.3.3

Treatment Design

In-Situ Impedance Measurements

Plunker Measurements on the Treatment Sections

of the Aft Duct

Pa_e

iii

ix

xi

1

3

9

II

11

11

19

29

31

37

41

42

42

48

50

50

59

59

60

63

67

88

92

95

109

£
TABLE OF CONTENTS (Concluded)

Section

6.0 THEORY-EXPERIMENT COMPARISON

6.1 Input to Prediction Prosram

6.2 In-Duct Suppressions

6.3 Effects of Sheared Flow

6.4 Sensitivity of Suppression to Treatment Impedance

6.5 Far Field Suppression

6.6 Nozzle Reflection Characteristics

6.7 Mode Shapes and Equivalent _mpedance

6.8 Effect of Aft-Duct Treatment on Fan Noise Generation

7.0 CONCLUDING REMARKS

REFERENCES

APPENDIX A - SCATTER IN NODAL COEFFICIENT FILES

APPENDIX B - AN ASSESSMENT OF ERRORS IN THE IN-SITU MEASUREMENTS

OF THE ACOUSTIC IMPEDANCE

APPENDIX C - EIGENVALUE EQUATIONS

ease

121

121

122

136

138

139

145

146

156

163

165

169

171

179

ii
LIST OF FIGURES

Figure No.

4.1

4.2

5.4

5.7

5.8

5.9

5.10

5. lla

5. llb

5.12

5.13

Title

The Annular Duct Geometry

Plane and Section Designation Convention for Two Adjacent

Duct Treatment Se_uents

Stacked System Matrix Equation for Three-Segment Duct

Schematic of the General Electric-Schenectady Anechoic

Chamber, Fan Exhaust Test Configuration

Installed Test Vehicle in the Exhaust Mode

Schematic Diagram of the Test Vehicle Showing Rotor 55 as

Operated in the Exhaust Mode

Kiel Probe Data at the Inlet to the Fan. Circumferential

Variation of the Mean Axial Flow Mach Number and Pressure

Ratio (Pt/Po) at Three Radial Locations

Rotor 55 Cross Section

Schematic Diagram of the Fan Exhaust Duct Showing Instrumentation Locations for Aerodynamic Measurements

Schematic Diagram Showing the Radial Locations of the Mode

Probes and the Axial Separations Between the Acoustic

Pressure Transducers on Each Axial Probe

Photograph Showing Two Axial Probes Mounted on a Radial

Strut

Photograph of a Complete Set of "Mode Probes" Showing the

Circumferential Traversing Actuator and Axial Probes at

Four Radial Locations Mounted on Two Diametrically Opposed

Radial Struts

A Schematic Diagram of the Data Acquisition and Analysis

System Used in the Modal Measurements

operating Map for the Rotor 55

Data Points Determined From the Upstream and Downstream

Locations for This Program at 7600 Physical RPM

Radial Profiles of the Mean Axial Flow Mach Number Measured

at the Same Circumferential Location; 7600 r_m

Circumferential Variations of the Mean Axial Flow Mach

Number at 67.3 mm (2.65 Inch) From the Outer Wall; 7600 r_m

20

21

28

44

45

46

47

49

51

54

55

56

58

61

62

64

65

iii
LIST OF FIGURES (Continued)

5.15

5.16

5.17

5.18

5.19a

5.19b

5.20a

5.20b

5.21a

5.21b

5.22a

5.22b

5.23a

Title

Circumferential Variations of the Mean Axial Flow Mach

Number at 101.6 nan (4.0 inch) From the Outer Wall; 7600 r?m

Circumferential Variations of the Mean Axial Flow Mach

Number at 63.5 _ra (2.5 Inch) From the Outer Wall; 7600 rpm

Radial Profiles of the Mean Axial Flow Mach Number

Measured in the Upstream Location at 7600 and 6700 rpm

Values

Radial Profiles of the Mean Axial Flow Mach Number

Measured in the Downstream Location at 7600 and 6700 rpm

Values

A Typical Comparison of Radial Profiles of Mean Axial

Flow Mach Number Obtained by a Boundary Layer Probe and

a Traversing Pitot-Tube

Modal Coefficient Amplitudes Measured at the Upstream

and the Downstream Locations in the Hardwall Configuration; 7600 rpm, 1900 Hz

Phase Values Corresponding to the Amplitude of Modal

Coefficients in Figure 5.19a; 7600 rpm, 1900 Hz

Modal Coefficient Amplitudes at the Upstream and Downstream Locations in the Treated Configuration; 7600 rpm,

1900 Hz

Phase Values Corresponding to the Amplitudes of Modal

Coefficients in Figure 5.20a; 7600 r_m, 1900 Hz

Modal Coefficient Amplitudes at the Upstream and the

Downstream Locations in the Hardwall Configuration;

6000 rpm, 1500 Hz

Phase Values Corresponding to the Amplitude of Modal

Coefficients in Figure 5.21a; 6000 rpm, 1500 Hz

Modal Coefficient Amplitudes Measured at the Upstream

and the Downstream Locations in the Treated Configuration; 6000 rpm, 1500 Hz

Phase Values Corresponding to the Amplitudes of the

Modal Coefficients in Figure 5.22a; 6000 rpm, 1500 Hz

Modal Coefficient Amplitudes Measured in the Upstream

and the Downstream Locations in the Hardwall Configuration; 4000 rpm, I000 Hz

_v

66

68

69

70

71

74

75

76

77

78

79

80

81

82
LIST OF FIGURES (Continued)

Figure No.

5.23b

5.24a

5.24b

5.25

5.26

5.27

5.28

5.29a

5.29b

5.30

5.31

5.32

5.33

5.34a,

b,c,d,e,

f

Title

Phase Values Corresponding to the Modal Coefficients in

Figure 5.23a; 4000 rpm, 1000 Hz

Modal Coefficient Amplitudes Measured in the Upstream and

the Downstream Locations in the Treated Configuration;

4000 rpm, 1000 Hz

Phase Values Corresponding to the Nodal Coefficients

in Figure 5.24a; 4000 r_m, I000 Hz

Nodal Coefficient Amplitudes of Forward (Downstream)

Propagating Nodes Measured at the Upstream Location;

7600 _pm, 1900 Hz

Plots of the Far Field (Narrow Band) Sound Pressure

Level (SPL dB) A_ainst Microphone Angular Position

Relative to the Nozzle Axis for Hardwall and Treated

Configurations at 7600 _'pm (1900 Hz)

Plots of the Far Field (Narrow Band) SPL Against

Microphone Angular Position Relative to Nozzle Axis for

Hardwall and Treated Configurations at 6000 Hz (1500 Hz)

Plots of the Far Field (Narrow Band) SPL Against MicrophoneAnsular Position Relative to Nozzle Axis for the

Hardwall and Treated Configurations at 4000 rpm

(I000 Hz)

Schematic Diagram of the Acoustic Treatment Used in

the Treated Sesment of the Aft Duct

Schematic Diagram Showing the Acoustic Treatment in the

Straight Annular Part of the Fan Exhaust Duct

Schematic Diagram of the apparatus Used for In-Situ Impedance Measurements in the Grazing Flow Duct

Schematic Diagram of the Data Acquisition and Analysis

Hardware Used in the In-Situ Impedance Measurements

Photograph of the Inner Treatment Section of the

Acoustically Treated Segment of the Aft Duct

Experimental Arrangement Used in the In-Situ Impedance

Measurements on the Outer and the Inner Treatment

Sections of the Aft Duct

In-Situ Impedance Measurements at Six Different Circumferential Locations on the Outer Treatment Section of the

Aft Duct

83

84

85

87

89

90

91

93

94

96

97

99

I00

101-103

v
LIST OF FIGURES (Continued)

5.36

5.38

5.39a

5.39b

5.40

5.41a

5.43

6.1

6.2

6.3

Title

Auto Correlation Spectra of the Acoustic Signals in the

In-Situ Measurement of Cavity A of Figure 5.34

The Coherence of Two Signals of Figure 5.35 and the

Phase Difference (CA - _B) as Functions of

Frequency

In-Situ Measurements at Three Different Circumferential

Locations on the Inner Treatment Section

Increase in Acoustic Resistance With Sound Pressure Level

(SPL, dB) of the Discrete Acoustic Signal; No-Flow

Increase in Acoustic Resistance With the Mean Flow Mach

Number in the Grazing Flow Duct

Increase in the Acoustic Reactance With the Mean Flow

Mach Humber in the Grazing Flow Duct

Auto Correlation Spectra of the Face Sheet Signal A

and Backwall Signal B During an In-Situ Impedance

Measurement at M - 0.4

Schematic Diagram of the Acoustic Plunker Placed Against

the Surface of a Flat Treatment Panel

Schematic Diagram of a Mormal Impedance Tube

Impedance Survey of the Outer-Treated Section of the

Aft Duct Based on 60 Different Plunker Measurements Over

Its Surface

Impedance Survey of the Inner Treated Section of the

Aft Duct Based on 30 Different Measurements Over Its

Surface

Measured and Predicted Complex Mode Amplitudes at the

Downstream Plane - Forward Propagating Modes - 1000 Hz,

m - -i, M - 0.21, Uniform Flow, Soft Wall

Measured and Predicted Complex Mode Amplitudes at the

Downstream Plane - Forward Propagating Modes - 1500 Mz,

m = -1, M ffi.32, Uniform Flow, Soft Wall

Measured and Predicted Complex Mode Amplitudes at the

Downstream Plane - Forward Propagating Modes - 1900 Hz,

m = -1, M = 0.4, Uniform Flow, Soft Wall

104

104

105-106

108

110

IIi

112

114

114

117

118

124

125

126

vi
LIST OF FIGURES (Continued)

6.5

6.6

6.7

6.8

6.9

6.10

6 .II

6.12

6.13

6.14

6.15

6.16

Title

Measured and Predicted Complex Mode Amplitudes at the

Downstream Plane - Forward Propagating Modes - 1900 Hz,

m = 7, M = 0.4, Uniform Flow, Soft Wall

Measured and Predicted Complex Mode Amplitudes at the

Upstream Plane - Backward Propasatin8 Modes - i000 Hz,

m = -1, M = .21, UniformFlow, Soft Wall

Measured and Predicted Complex Mode Amplitudes at the

Upstream Plane - Backward Propagating Modes - 1500 Hz,

m - -1, M - .32, UniformFlow, Soft Wall

Measured and Predicted Complex Mode Amplitudes at the

Upstream Plane - Backward Propagating Modes - 1900 Hz,

m = -I, M = .4, Uniform Flow, Soft Wall

Measured and Predicted Complex Mode Amplitudes at the

Upstream Plane - Backward Propagating Modes, 1900 Hz,

m = 7, M = 0.4, Uniform Flow, Soft Wall

Predicted Mode Distributions at Segment End Planes -

1000 Hz, m = -I, M = 0.21

Mode Distribution at Segment End Planes - 1900 Hz,

m = 7, M = 0.4

Sensitivity of Suppression to Treatment Resistance and

Reactance - I000 Hz; m = -I, n = 0,I,2,3; M = .21 Sheared

Flow

Sensitivity of Suppression to Treatment Impedance -

1500 Hz; m = -I; n = 0,I,2,3; M = .32; Sheared Flow

Sensitivity of Suppression to Treatment Resistance

and Reactance - 1900 Hz; m - -I, n = 0,1,2,3; M = .4;

Sheared Flow

Comparison of the Uniform Flow Eigenfunction With the

Actual Acoustic Mode Shape in Hardwall Duct - 1900 Hz,

m = 7, n = O, Mo = .37, _* = 0.02, Forward Propagation

Comparison of the Uniform Flow Eigenfunction With the

Actual Acoustic Mode Shape in Hardwa11 Duct - f = 1900 Hz,

m = 7, n = O, Mo = .37, 6* = .02, Backward Propagation

Comparison of the Uniform Flow Eigenfunction With the

Actual Acoustic Mode Shape in the Treated Duct - 1900 Hz,

m = 7, n = O, M0 = .37, _* = .02, _ = (.51 - .55i),

Forward Propagation

127

129

130

131

132

134

135

141

142

143

148

150

151

vii
LIST OF F_GURES (Concluded)

Fisure No.

6.17

6.18

6.19

6.20

6.21

6.22

6.23

A1

B1

B2

Title

Comparison of the Uniform Flow Eigenfunction With the

Actual Acoustic Mode Shape in the Treated Duct - 1900 Hz,

m = 7, n = 0, M o = .37, 6" = 0.02, C = (.51 - .55i),

Backward Propagation

Comparison of the Uniform Flow Eigenfunction Based on the

Equivalent _mpedance With Actual Mode Shape in the Treated

Duct - f = 1900 Hz, m = 7, n = 0, M = .37, 6" = .02,

NEX = 7, C = (.51 - .55i), Forward Propagation

Comparison of the UniformFlow Eigenfunction Based on the

Equivalent Impedance With the Actual Mode Shape in the

Treated Duct - 1900 Hz, m - 7, n = 0, M = .37, 6* = .02,

NEX = 7, C = (.51 - .55i), Backward Propagation

Complex Mode Amplitudes and Phase Values at Upstream

Plane - I000 Hz, m = -I, M = .21, Uniform Flow, Forward

Propagation

Complex Mode Amplitudes and Phase Values at the Upstream

Plane - 1500 Hz, m = -I, M = .32, Uniform Flow, Forward

Propagation

Complex Mode Amplitudes and Phase Values at the Upstream

Plane - 1900 Hz, m = -I, M = _.4, Uniform Flow, Forward

Propagation

Complex Mode Amplitude and Phase Values at the Upstream

Plane - 1900 Hz, m = 7, M = 0.4, Uniform Flow, Forward

Propagation

Scatter in the Modal Data DUe to Probe Elimination at the

Downstream Measurement Location

Typical Calibration of the Two Acoustic Transducers

Used in the In-Situ Measurements

Autospectra of the Signals Used in the Calibration of

Figure B1

152

153

154

157

158

159

160

170

175

177

viii
LIST OF TABLES

Table No.

5.1

5.2

5.3

5.4

5.7

6.1

6.2

6.3

6.4

6.5

B1

B2

Title

Far Field Microphone Distances and Angles

Rotor 55 Design Characteristics

Acoustic Modes Generated by Rotor 55 With an Eight Vane

Stator With Associated Cut-Off Frequencies and Ratios

Comparison of Computed Propagating Energy Flux, Upstream

and Downstream, for Hardwall Test Section

Index of Figures Showing Modal Amplitudes and Phases

Sound Power Levels From Far Field Data (Power Spectrum

Bandwidth = 4 Hz)

Average Normal Impedance of the Acoustic Treatment Hardware (OASPL = 140 dB)

Comparison of Measured Suppressions with Theoretical

Predictions

Comparison of Measured and Predicted In-Duct Suppressions

Based on Sheared Flow Mode Coefficients

Sensitivity of Predicted Suppressions to Treatment

Impedance

Comparison of In-Duct Suppressions with PWL Suppressions

From Far-Field Data.

Nozzle Reflection Coefficients (From Hardwall and

Treatment Case Data)

In-Situ Measurements - Estimated Errors in the Measured

Impedance Due to Errors in the Thickness (d) and

Temperature (T)

Expected Errors in the Measured Impedance Due to Errors

in the Measured Transfer Function (HAB) and Phase

Difference (_AB)

43

48

60

72

73

88

118

123

137

140

144

147

173

178

ix
LIST OF SYMBOLS AND DEFINITIONS

Except when otherwise defined in the text, the following nomenclature is used

in this report.

Symbols

A

c

d

E

f

H

H

xy

i

[I]

I

z

k

k

r

k e

K

L

m

M

n

N

P

Pr

r

R

T

v

X

z

(_)

complex mode amplitude

speed of sound

core thickness of the acoustic treatment

acoustic energy

frequency

duct height

transfer function of two acoustic signals at locations x and y

unit matrix

acoustic intensity

free space wave number

radial wave number

circumferential wave number

axial wave number (propagation constant)

seament len&th

spinning mode order

mean flow Nachnumber

radial mode order

number of segments

acoustic pressure

radial eigenfunction

radial coordinate

resistance of treatment

temperature

axial acoustic velocity

reactance of treatment

axial coordinate

admittance interface reflection matrix

sas

xi

P'REC;F..Dll_I PAGE _LA_K NOT F_i,V_--'O
L_IST OF SYMBOLS AND DEFINITIONS (Continued)

Symbols (Continued)

[S] stacked system matrix

[T] admittance interface transmission matrix

Iv]

{Q}

{A}

MEX

e

6

n

K

Po c

Po

uniform section transmission matrix

acoustic source vector

acoustic mode vector

Index n in the boundary layer velocity profile. Thus, NEX = 7 is

for the I/7th power boundary lay velocity profile.

angular location in the cylindrical polar coordinate system

angular frequency 2v f/c

duct wall acoustic admittance

complex dimensionless eigenvalue kr.r 2, where r2 is the

outer wall radius

Dirac delta

duct wall acoustic impedance

Hf/c=H/k = non-dimensional frequency

K/k = non-dimensional axial wave number

wavelength

characteristic impedance of medium

density of air in the duct

Subscripts/Superscripts

(J), (x)

.i-l, J,

k, k+l

m

n

S

T

Z

+

eq

duct segment designation

duct plane desisnation

spinning mode order

radial mode order

"source" plane

"termination" plane

axial direction

forward propagation

backward propagation

complex conjugate

equivalent

xii
I. 0 SUMMARY

Sound suppressions due to acoustically absorptive treatment in the

annular exhaust duct of a model fan have been theoretically predicted and

compared with measurements.

The predictions were based on the modal analysis of sound propagation

in a straight annular fan exhaust duct with axially segmented treatment.

Measured values of the mode distribution of the fan noise source (rotor-stator

interaction source only) and of the acoustic impedance of the treated segment

in the duct were used as input to the prediction program. The predicted

suppressions obtained with the assumption of uniform flow (no radial or

circumferential shear in the flow) compared well with the measured in-duct

suppressions for all test conditions.

Measurements of the acoustic modes were made on a fan test model

installed in an anechoic chamber at General Electric Company's Corporate

Research and Development Center in Schenectady, New York. The fan exhaust

duct was annular in cross section with a hub-to-tip ratio of 0.5. It

consisted of three axial segments. Acoustic mode probes were located in the

£icst and the third segmentswhich were hard walled. Initial tests were

conducted with a hard wall second segment. Later measurements were made with

acoustically absorptive inner and outer walls in the second sesment. The

suppression of sound due to the treated segment was determined from these data

by computing the difference between the acoustic energy flux at the upstream

and the downstream measurement locations. The quality of the measured modal

data was very good. The acoustic field in the duct at the frequency of

interest was observed to be stationary over the periods of the modal

measurements. This ensured accurate measurement of the modal amplitude and

phase data. Fan speeds for the tests were chosen so that the modes generated

by the fan-stator interaction spanned a cut-off ratio range from just over one

to seven.
The acoustically absorbins wall se_ents consisted o_ sinsle-degreeof-freedom ($DOF) treatment with linear characteristics. The measurements of

the normal acoustic impedance of the treatment were carried out in General

Electric Company's Acoustic Laboratory at Evendale, Ohio. These included

measurements under no-flow conditions on the treatment hardware and under

srazing flow conditions on a sample of the treatment.

The modal probe data and a users' suide to the computer prosrams are

available as separate publications (see list of regerences).

2
2.0 TFrRODUCTION

Acoustically absorptive treatment in aircraft engine ducts is an

essential part of the overall aircraft noise reduction effort. With the

increased emphasis on enersy efficiency, it is necessary to maximize the

effectiveness of the treatment. Duct treatments must be designed to suppress

specific noise sources in short ducts. By maximizing the treatment

effectiveness, a reduction in the treatment lenKth required to achieve a given

overall ensine noise level can be realized. This reduction translates into

reduced fan reverser length required and, consequently, results in lower

installed engine weight and higher perfo_aance consistent with the desired

soal of enersy efficient noise reduction. In order to achieve this soal,

analytical treatment desisn must be made an integral part of the advanced

energy efficient noise reduction technology.

Recognizing the importance of spinning modes in the design of the

acoustic treatment, several studies have been conducted to investigate the

propagation of acoustic modes in axi-sy_netric ducts. These studies, however,

are limited to semi-infinite ducts with unifot_nwall impedance. Zorumski

(Reference 1) developed an analysis to account for the axial impedance

changes. This analysis also included both the upstream and the downstream

travelling modes. Kraft, et. al., (Reference 2) measured the acoustic modal

distribution in a laboratory model inlet duct and designed and tested a

treatment to verify Zorumski's analysis.

This report contains the results of a study on the propagation of

acoustic modes in an annular exhaust duct. It involved the development of a

set of computer programs based on the modal analysis (Reference 3) of sound

propagation to predict in-duct suppression due to axially segmented acoustic

treatment. It also involved an experimental program designed to dete_uine

whether the analysis capability was sufficient to describe the physics in

turbofan aft duct suppressors. Although simplified, the experiment was

designed to contain the first-order physical effects which were thought to

gover_ the far field radiation attenuation due to an exhaust duct suppressor.

3
As the experiment was intended to verify the sound propagation theory in a

realistic turbofan exhaust duct, it contained the following features:

1. A realistic fan stage (NASA Lewis model fan--rotor 55) capable of

producing well-defined spinning modes.

2. An exhaust nozzle with a realistic contour and contraction ratio.

3. A straight annular section exhaust duct to model the simplified

sound propagation analysis.

4. A uniform, linear treatment design to minimize impedance

sensitivity to the operating conditions of the text vehicle.

Over the three fan speeds tested, spinning modes were generated with

cut-off ratios ranking between just over 1 to 7. The acoustic measurements

involved the measurement of (a) modal coefficients upstream and downstream of

a treated segment in the exhaust duct, (b) the aft radiated acoustic far

field, and (c) the acoustic impedance of the treatment used in the aft duct.

The test vehicle was mounted in an anechoic chamber in order to measure

the far field radiated noise under free field conditions.

The measurement of the acoustic impedance involved the use of the insitu method (References 4, 5, 6) and of the Acoustic Plunker (Reference 7), a

non-destructive portable transducer developed by the General Electric Company

to measure the normal acoustic impedance of finished treatment panels. A

total of 90 measurements on both the inner and the outer treatment sections

were made with the Plunker to establish that the normal acoustic impedance of

the treatment was uniform over the area of these sections and that both the

inner and the outer surfaces had equal impedance within prescribed

manufacturing tolerances.

The theory of sound propagation in a uniform duct with fluid flow and

its extension to ducts with axial impedance segmentation is described in

Section 4.1. Section 4.2 deals with the theoretical aspects of the

computation of the modal coefficients from the mode probe data and the

4
theoretically determined mode eigenfunctions and axial wave numbers in the

hardwall sesment of the duct. The problem has been considered for the uniform

flow and the radially sheared flow cases. It involves the solution of a set

of linear simultaneous equations for the complex modal coefficients. In the

case of uniform flow the eigenfunction of a downstream propagating (m,n) mode

is identical to that of the upstream propagating mode of the same order. This

allows decoupling of the problamwhichmakes the solution simpler in that

smaller size matrix equations have to be solved. In the case of radially

sheared flow, the eigenfunctions of the upstream and the downstream

propagating modes of the same (m,n) order are not identical which means that a

much bigger size matrix equation has to be handled. As the sensitivity of

matrix solutions to small input errors increases with the size of the matrix,

redundant measurements were included in the analysis giving a "least squares

fit" type of solution from the available data.

The theory of the in situ impedance measurement technique is described

in Section 4.3. This method assumes that the treatment is point reacting, and

there is no transmission of sound through the walls of the adjacent cavities.

The sound field inside the cavity is assumed to consist of plane waves only.

The method employs the complex ratio of two acoustic originals--one at the

surface of the treatment and the other at a known location in the cavity

(usually at the hard back wall). For this reason it has often been referred

to as the '2-microphone method.' An error analysis of the technique is

included in Appendix g.

The description of the experimental apparatus and a selection of

representative test data are included in Section 5. The test vehicle was

mounted in the Aeroacoustic Anechoic Facility in the General Electric

Corporate Research and Development Center, Schenectady, New York. The airflow

and noise source consisted of the 15 bladed NASA Lewis 0.504 m diameter fan

designated as Rotor 55. From the original eleven outlet guide vane set, eight

vanes were used at a spacing of 0.5 rotor tip chord lensth from the rotor to

generate a rotor stator interaction tone at levels appreciably higher then

noise produced by other mechanisms. To prevent rotor turbulence noise

5
generation, a turbulence control structure was incorporated in the inlet duct

of the fan. The annular flow path of the fan exhaust duct consisted of two

hardwall sections where in-duct acoustic probes were located, a treated

section and a nozzle termination which was representative of typical engine

nozzles. The acoustic measurement in the duct involved circumferential

traversin_ of an array of pressure transducers through 360 de_rees. Signals

were recorded at 18 degree intervals and then analyzed. In order to ensure

stationarity of the acoustic field over the time of recordins the sisnals from

the two arrays (of twelve tranducers each) used in the measurement, the fan

speed was maintained at a constant value durin_ a test.

Aerodynamic measurements were made to determine the velocity profile at

each of the in-duct probe locations. This information was necessary for the

calculation of the axial wave numbers and radial mode shapes (eigenfunctions). Instrumentation included: three total pressure and total

temperature rakes, each with five radial stations. In addition, two pitot

tubes which could traverse radially and circumferentially and sixteen static

pressure taps mounted in both inner and outer walls were employed. All

aerodynamic probes and rakes were removed durins acoustic tests.

The effects of grazins flow and of sound intensity on the impedance of

the treatment were investigated using laboratory samples constructed from the

same materials used in the construction of the annular duct hardware. Steady

flow resistance measurements were made first after removing the flexcore (hut

leavin_ the bonding agent intact). The sample was then instrumented for insitu impedance measurements in the Grazin_ Flow Duct usins a thick walled

cylindrical cavity to ensure local reaction aspect of the experiment.

The theory-experiment check is fully discussed in Section 6, and the

major conclusions drawn from this work are listed in Section 7. The exhaust

duct suppression prediction program (Reference 8) based on the modal analysis

was used to predict the suppression due to the treatment in the exhaust duct

at three fan speeds. The duct was modelled as a three segment straight

annular duct with the treated seEment placed between the two hardwall
seaments. Mode coefficients based on the assumption of uniform flow through

the duct were first obtained from the in-duct measurements. The mode

coefficients of the forward (downstream) traveling modes at the upstream

measurement plane (source plane) and of the backward (upstream) traveling

modes at the downstream measurement plane (termination plane) were used to

specify the source and the termination matrices. W£th these parameters and

the knowledge of the impedance of the treated segment, the duct geometry, the

flow conditions and the spinning mode order number, the program (Reference 8)

calculates the eigenvalues, the axial propagation constants (both forward and

backward), the uniform section transmission matrices for each segment and the

reflection and transmission matrices of the segment inter_aces. The program

then sets up the stacked system matrix equation and solves it to obtain the

forward and backward complex mode coefficients in each segment and the modal

energy flux at each plane. The net energy flux at each plane and the overall

sound power level (PWL) suppression are then calculated. Suppressions

predicted in this manner agreed well with the suppressions obtained from the

in-duct measurements.

The sensitivity of the predicted suppression to the treatment impedance

values was also examined by varying both the resistance and the reactance of

the input to the prediction program. The predicted suppression was found to

be more sensitive to variations in reactance, particularly at the maximum fan

speed tested (1900 Hz) when a variation of 0.1 _c in reactance resulted in

as much as 4 dB change in suppression.

Predictions were also made using the sheared flow mode coefficients.

These did not compare as well with the measured values as those obtained with

the uniform flow mode coefficients. This is considered to be due to a

sign£ficant energy mismatch that occurs at the segment interfaces when using

the sheared flow option in the prediction routines. This problem has not been

resolved.

Based on the theory experiment check presented in Section 6, it can be

concluded that the modal analysis for sound propagation in segmented ducts can
be used to predict the in-duct suppression due to acoustically absorptive

treatment in the exhaust duct of a turbofan. The modal distribution of the

acoustic source and the reflection characteristics of the duct termination are

required for the prediction. They may be obtained from measurements or from

analytical methods.

The acoustic data from the mode probes in terms of the linear amplitude

and phase measured at defined (z, r, e) locations in the hardwall segments

of the aft duct are included in the data report (Reference 9).
3.0 OBJECTXVES

The objectives of the work reported in this report are:

1. Development of the analytical tools required for rapid and cost

effective evaluation of acoustic treatment designs for fan exhaust

ducts using modal propagation analysis.

2. Provide experimental substantiation of the above analytical tools

in a realistic turbofan exhaust duct employing a model fan stage

and a realistic exhaust nozzle.

3. Study the effect of the selective reflection by the nozzle on the

far field suppression.
4.0 THEORETICAL CONSIDERATIONS

In this section of the report, the theoretical model of the acoustic

wave propasation in an acoustically lined cylindrical duct of uniform cross

section carrying radially sheared flow and its extension to axial segmentation

of the treatment is described. This theory is the basis of the computer

prosrams described in Reference 8. In addition, the theoretical bases of the

computations used in the determination of the modal coefficients from the mode

probe data obtained in the hardwall sesments of the fan exhaust duct and of

the in-situ impedance measurement technique are described.

4.1 WAVE-FIELD THEORY

The prediction of suppression of sound due to treatment in the exhaust

duct is based on a theoretical analysis of sound propagation in axially

sesmented annular ducts (Reference 3). Propasation in the duct is considered

in terms of the treatment and the presence of both hardwall and treated

sesments in the duct are incorporated in the analysis by considering the duct

to be axially sesmented and by allowin8 propagation of modes in both forward

and backward directions. Reflection and redistribution of acoustic energy at

segment interfaces and at the exhaust nozzle are also considered. The theory

of sound propagation in a uniform duct with flow and its extension to ducts

with axial impedance segmentation are described in the following paragraphs.

4.1.1 SOUND PROPAGATION IN ANNULAR DUCTS WITH FLOW

The propasation of sound in a duct carrying uniform flow is governed by

the converted wave equation

(4.1)

where V is the mean flow velocity in the axial direction, V 2 is the

z

Laplaclan operator and p is the acoustic pressure. To solve this equation by

_r_C,E_ING PAGE _' "_'_'_"

11
the separation of variables, assume that the acoustic pressure can be

represented in cylindrical coordinates (Figure 4.1) as

p(r,e,z,t) = Pr(r)p_(e) eiKz-iwt C4.2)

This reduces the wave equation to two ordinary differential equations, namely

C4.3)

and

d2pr 1 dPr ke = 0

dr2 r dr :-2 Pr

(4.4)

with

K2(I - M 2) + 2MkK + (k_-k 2) = 0

(4.5)

k is the wave number in free space and kr,k 8, "and K are the wave numbers

in the radial, circumferential, and axial directions respectively. M is the

mean flow Mach number (equal to V /c).

z

Equation (4.3) has. a solution of the form

Pe(e) = ce eime (4.6)

where values of k e are restricted to integers since the coordinate e is

periodic with period 2_ and the pressure must be sinsle valued. Physically

the integral values of k 8, to be denoted by m, represent circumferential

mode orders. If the duct contains radial splitters that are lined, k e may

be complex.

12
Equation (4.4) with k%fm is the standard Bessel equation and has

the solution

PrCr) = Jm Ck r) + C ¥ (k r) (4.7) r m m r

where J and Y are Bessel functions (of order m) of the first and second

m m

kind respectively. The acoustic admittance boundary condition that equation

(4.7) must satisfy at the duct walls can be expressed in terms of the

continuity of particle displacement as

dr lrfrl )2rrfrl

(4.8)

and

dPr -ik_2 - k M Pr

r=r2 = 1 r=r2

(4.9)

where B1 and B 2 are the specific acoustic admittance of the inner wall at

radius rl, and the outer wall at radius r2. Substitution of equation

(4.7) into equations (4.8) and (4.9) yields a set of two simultaneous

transcendental complex equations which must be solved for k and C . For

r m

a given circumferential mode order m, these equations have a sequence of roots

(krr 2) which represent the eigenvalues for a sequence of radial modes.

The functions pr(r) obtained from equation (4.7) represent the corresponding

eigenfunctions. Setting k r r 2 = T, rl/r 2 = P and eliminating Cm

from the simultaneous equations obtained by the substitution of equation (4.7)

into equations (4.8) and (4.9), we get a single equation for the eigenvalue of

the form F (T) = 0 (see Appendix C).

The eigenvalues can be obtained by solving this equation using the

second order Newton-Raphson iteration formula

13
7i+ 1- 7i - 2F(Yi)F' (_i) (4.10)

2

2[F'(7i)] - FCYi)F"(_i)

where T i is the initial (guessed) value of the root and 7i+ 1 is the

iterated value of the root. The primes denote differentiation with respect to

the argument. This formula is repeatedly applied until successive iterations

give roots (eigenvalues) whose absolute values differ by less than 10 -5.

The accurate and reliable determination of these eigenvalues is a

critical part of the calculation of sound propagation in du_ts. The ability

of the solution procedure to converKe to the correct sequence of eigenvalues

depends critically upon the initial value of each root at the start of the

iteration process. Xn order to assure convergence, the iteration is performed

in several steps as indicated below.

Rectangular duct hardwall eigenvalues are used as starting values and

the radius ratio is slowly decreased from unity to the annular duct radius

ratio in order to obtain annular duct hardwall eigenvalues. The latter are

used as starting values to obtain the eigenvalues for the case of hard outer

•wall and inner wall of admittance _I" This is done by slowly incrementing

the inner wall admittance magnitude from zero to I_iI and iterating along

the line of constant phase of _I" Using the new eigenvalues as the

starting values and slowly _ncrementing the outer wall admittance magnitude

from zero to I_21 (along the line of constant phase of _2 ) the

eigenvalue for the annular duct with inner wall admittance equal to _I and

outer wall admittance equal to _2 are determined.

When the inner and outer wall admittances are equal, the two step

process described above to obtain softwa11 eigenvalues from hardwall

eigenvalues reduces to a single step process. In this case the inner and

outer wall admittance magnitudes are incremented simultaneously and equally in

small steps from 0 to l_wl where _w is the admittance of both walls.

14
The same iteration procedure is used while the flow Mach number is

increased in small steps from zero to M in order to obtain the lined annular

duct eigenvalues in the presence of uniform flow.

This careful step-by-step iteration process provides a reasonably

reliable eigenvalue routine. However, the method is not totally infallible

and occasionally cases of missed modes or modes found more than once are

encountered.

An alternate method to determine the eigenvalues is used for cases for

which the iteration process is not successful. In this method the eigenvalue

equation is set up as a differential equation of the form (see Appendix C for

details of the eigenvalue equation)

(4.11)

where x is an independent parameter in the equation. To obtain the

eigenvalues corresponding to wall admittance _w (for both inner and outer

walls), equation (4.11) is integrated, using Runge-Kutta method, from x = 0.

The step size (&x) in the integration process is kept small to obtain good

accuracy. Furthermore, in order to minimize the error accumulated in the

integration process, a second order gewton-Raphson iteration of the eigenvalue

is performed at each step. Use of this eigenvalue solution procedure in

combination with the original iterative procedure improves the reliability of

the eigenvalue solution especially in identifying a mode that could have been

missed by the iterative procedure.

After the soft-wall eigenvalues have been determined, the axial wave

number K can be computed from

k

1 - M 2

(4.12)

15
In the case of a hard wall duct (81 = 8 2 = 0), when the expression under

the radical becomes negative, K becomes complex and causes the mode under

consideration to decay exponentially. Under such conditions the mode cannot

transport any acoustic energy. The cut-on frequency f* for a mode can thus be

defined by setting

k* = krl- _

and obtaining f. ffi cT _ (4.13)

2_r 2

The cut-off ratio _, which is the ratio of the modal frequency to the cut-on

frequency of the mode, is given by

(4.14)

where the frequency parameter . A mode is considered cut off for

_<1 but will be propagating when _>1. The definition for the cut-off

ratio as given in equation (4.14) is valid for a rectangular duct of height

r 2 if T is replaced by the-rectangular duct eigenvalu e kyr 2, and for a

cylindrical duct of radius r2 with 7 equal to k r r2.

The effects of sheared flow on the wave propagation are evaluated by

assuming that the boundary layers at the duct walls are sufficiently thin so

that the predominant part of the acoustic energy flux takes place in the

uniform flow region. In this case the solution to the differential equation

for acoustic pressure in the presence of shear layers is approximated by

choosing the eisenfunctions to be of the same form as in the uniform flow case

(i.e., equation (4.7), with eigenvalues modified by the effects of the

boundary layers. To obtain the eigenvalues in the presence of wall boundary

layers, the equation governing the radial variation of acoustic pressure, i.e.,

d2pr ldPr

--+--+

dr2 r dr I 2_ dM dPr + < kr2- m2 1 Pr = o

(4.15)

16
(where _ = K/k = non-dimensional propasation constant) is rewritten as two

simultaneous first order differential equations, namely

dM (k2r

2

- _)Pr

r

and dPr = q (4.16)

dr

and solved numerically by a combination of Runge-Kutta integration and

Newton-Raphson iteration as outlined below.

The no-flow softwall annular duct eigenvalue (7 = k r r2) and the

corresponding propagation constant . = K/k, as calculated from equation

(4.12), are used as the initial estimates. At the inner wall Pc is assigned

a value equal to the uniform flow eigenfunction value and the inner wall

boundary condition equation (4.8) is used to calculate q = dpr/dr. These

values of Pr and q provide the initial values to integrate equations (4.16)

by Runge-Kutta method across the annulus to obtain Pr and q at the outer

wall. The values of Pr and q at the outer wall are used to calculate the

admittance

z

-q

ik (l-Mx) 2p

r

r=r

2

(4.17)

If this value differs from the specified admittance _2' then the

process must be repeated with a new value of y until the condition

_=_2 is satisfied. This is done by second order Newton-Raphson

iteration. By setting the difference B-8 2 equal to F (T) a revised

eigenvalue is defined as

m

2F(y)F'(y)

2(F'(T)) 2 - F(y)F"(_)

(4.18)

17
where F' (T) = dFldy and F" (y) = d2Fld_ 2 are obtained by

performing Runge-Kutta integration of equations 44.16) with y ± ¢ and

± ic as initial estimates for the eigenvalue. The iteration process

is repeated until an eigenvalue that satisfies the outer wall admittance

boundary condition is obtained. This procedure can be used with any arbitrary

boundary layer flow profile by appropriately specifying the Math number

gradients dMldr. The linear or the one-seventh power law boundary layer

profiles are used for most calculations.

In order to obtain good accuracy of the integration process, a variable

step size is used. Logarithmically spaced steps for the power law profile

provide smaller steps in the region of high velocity gradient.

Having determined the eigenvalues and eigenfunctions, the general

solution for acoustic pressure is constructed in terms of a series

representation utilizing the eigenfunctions, i.e.,

p(r,e,z) =_n_IAmn Peru(e)

Prn(kr,nmr) eiKnmzl

(4.19)

where the summation is done over all spinning mode orders m and radial mode

orders n. As discussed before, the radial variation can be represented by the

eigenfunction form of equation (4.7). k and K are the eigenvalue

r,mn

(divided by r 2) and the axial propagation constant respectively for the

(m,n) mode. A is designated the mode coefficient and is normally

nM

determined by using the orthogonality property of the eigenfunctions.

Unfortunately, the eigenfunctions in the case of wave propagation in ducts

with flow are not orthogonal in the usual $turm-Liouville sense. A

generalized definition of orthogonality developed by Kraft and Wells

(Reference 10) can, however, be used to evaluate the mode coefficients.

18
4.1.2 MODAL ANALYSIS FOR SEGMENTED DUCT

The solution for wave propagation in a duct with multiple treatment

elements utilizes a transfer matrix principle (References 1 and 2) to connect

the solution at one end of the duct with the other. The duct is assumed to be

composed of axially uniform sections which adjoin at planes where a

discontinuity in wall admittance ocurs (Figure 4.1). Modal reflections and

redistributions at admittance discontinuity planes and the consequent

existence of forward and backward traveling waves in the uniform sections are

taken into account in the solution.

Based on the modal expansion of equation (4.19) the acoustic pressure

at an arbitrary axial position z in section J (see Figure 4.2) in the presence

of forward and backward traveling waves is given by

.÷(j)

p(J) (r,e,z) - Z Z A÷(J-1)ei_mn (z-zJ-1) (k +(J) r)

m n =n Pe (e)Pr r

m n mn

-iK -(J) (zj ) 1

+A-(j)_ mn -z

.m e PemCe)Prn (kr-(J)r)mn (4.20)

When the duct and liner geometries are axisymmetric, the spinning modes are

not coupled. The analysis can then be simplified by restricting attention to

one spinning mode order (m), i.e.,

r

p(J)(r,e,z) = pem(e)nE _ _+(j-1) x

L

iK+(J)(z-z_ .)

n (k +(J) r)

• d-_ Pr r

n n

-iK-(J)Cz'-z) 1

+ A -(j) • n J Pr (k; (J) r)

n n n (4.21)

19
Q_E

C_

C_

_u

l=. C

i2O
l.IJ

D

Z

m

l.IJ

Z

ul

0.

II

I-.-

m

=_

Im i_m_mmlm_

.m

Z

_o

l="

I-Ll

I

m

k.,

am

Ll-ul

Z

N"N

-- II

N

N

_ _

°_

N

E

e_

E

C

O

eeW

e,,

21
where the spinning order subscript m has been dropped from A, K and k for

r

brevity. In equations (4.20) and (4.21) the lower case superscripts are used

to indicate values at duct admittance discontinuity planes while the upper

case superscripts indicate values in uniform duct sections. The plus (+) and

minus (-) superscripts indicate forward and backward propagation

respectively. A completely analogous equation holds in section K and other

sections of the duct.

In any uniform section of the duct, since each mode propagates

independently, the mode coefficients at the end planes of the section are

related by the axial propagation constant and the section length. Thus, for

the radial mode n,

and

+(J)

iK Lj

a +(J) = a +(j-l) e n

n n

-iK -(J) Lj n A -(j-1) . A-CJ) •

n n 1

(4.22)

By representing the mode coefficients for the radial modes as elements of a

column matrix, the above equations can be written in the following matrix form

{ A+(._)}

for forward propagation, and

(4.23a)

I

(4.23b)

22
for backward propagation. Here [ U ] is the uniform section transmission

matrix, with elements

iK+CJ)n (zj-zj-1)

U+---'+---I)(_)(j= 6 e

sn sn

-iK: (J) (zj-zj_I)

u-(v-1)'----i(_) ffi6 e

sn sn

(4.24)

Across an axial admittance discontinuity, such as that from plane j to

plane k, solutions with different eigenfunction bases are related by requiring

the acoustic pressure and the axial component of acoustic velocity to remain

continuous across the discontinuity. The pressure continuity condition

p(r, z.) - p(r, z k) (4.25) 3

is written, using the modal expansion of pressure, in the matrix form

IpT

I 1 kl

(4.26)

where the superscript T represents the transpose of a column matrix. The

axial acoustic velocity is expressed in terms of the acoustic pressure (using

the momentum equation) as

K

n

Vzn(r) = PoC(k_ H K) Prn(r) (4.27)

so that the velocity continuity equation

Vz(r,z j) = Vz(r,z k) (4.28)

23
can be written as

p+(J)}

+

= lp+(K) } T

+ { p-(K) } T

where [ _z ] is the axial admittance matrix with elements

(4.29)

K+(J)

_+(J) (m,n) = _nm n

z k - M K +(J)

n

(4.30)

After some lengthy matrix algebra equations (4.26) and (4.29) can be

solved to obtain

A-(k) I

(4.31)

and

(4.32)

where [T] and [R] represent the transmission and reflection matrices for the

admittance discontinuity plane under consideration. The determination of IT]

and JR] matrices requires several matrix algebraic operations as shown in the

following equations

24
with

-I

[T+(K),+(j)] = [HI(K),+(K)] [HI(K),+(J) ]

and

[HI(K),+(J) ] = [GI][G +(K)'+(J) ] - [G +(K),+(J)

-1

[G 1] = [G+(x)'-(J)][ _z (J)][G +(x)'-(J)]

/' (,:",) G+(KI'+(J) (n,m) r r X= Prn n

r I

l[ _(J)]

(4.33)

where Pr is the eisenfunction represented in equation (4.7). Similar

equations hold for other transmission and reflection matrix elements. Since

the eigenfunctions are exprJssed in terms of the Bessel functions of the first

and second kind, calculation of the reflection and transmission matrix

elements requires the evaluation of the integrals of the products of the

Bessel functions.

Equations (4.23a) and (4.23b), involvins the uniform section matrix

[U], alons with equations (4.31) and (4.32), involving the transmission matrix

[T] and the reflection matrix [R], when written for each uniform section and

discontinuity in the duct, are sufficient to relate the forward and backward

wave solutions at one end of the duct with those at the other. Use of this

method to calculate the propagation of waves in a laboratory exhaust duct of

finite length would require specification of the axial admittances at the end

planes to establish the transmission and reflection matrices. (The axial

admittance at the duct termination depends on the method of flanging the duct

and the location of the boundaries in the free field. It is further

complicated by the nonuniform exhaust flow field. At the source location, the

axial admittance must be related to the internal impedance of the source.) An

25
alternate approach is to choose the two end planes at arbitrary stations in

hardwall sections of the duct upstream and downstream of treated sections and

specify the overall modal participation (i.e., the sum of the fouward and

backward traveling waves) at these planes. The modal participation can be

obtained from in-duct modal measurements.

The equation which relates the forward and backward modal vectors in

the source plane is

and at the termination plane is

where

(4.34)

(4.35)

[RS] = reflection matrix at source plane

[RT] = reflection matrix at termination plane

If] = unit matrix

and [Qs } and [QT ] are the generalized source and termination

vectors. If, as suggested above, the source and termination planes are chosen

as planes in hardwall sections of the duct then

[Rs] =-Ill = [_] (4.36)

and the source and termination vectors become the sum of the forward and

backward waves which can be obtained from measurement for input to the

program. Equations (4.34) and (4.35) along with equations (4.23a), (4.23b)

for each uniform admittance section of the duct and equations (4.31) and

(4.32) for each axial admittance discontinuity form a completely determined

system. This system of equations is written as a stacked system matrix

equation

26
ts] {M = [Q} (4.37)

where [S] is the stacked system matrix and [Q] is the generalized source

vector. Equation (4.37) is solved by a double-back substitution routine to

obtain

{M = [s]-X[Q} (4.38)

The stacked system matrix is shown in Figure 4.3 for a three segment duct.

Equations (4.34) and (4.35), along with (4.36), cannot be used for the

source and termination planes when the duct walls at these planes are lined or

when the presence of boundary layer is to be considered. Under these

conditions, the eigenvalues of the forward and backward modes are different

and the forward and backward mode coefficients cannot, therefore, be added

simply to obtain the complex pressure field. A separation of forward and

backward traveling modes at the source and termination planes can, however, be

obtained experimentally. This will define the mode content incident from the

source, i.e., the forward traveling mode coefficients at Plane 1,

{Q;}, and the mode content reflected from the nozzle; i.e., the

backward traveling mode coefficients at Plane 6, [QT]. The stacked

system matrix equation shown in Figure 4.3 can then be modified by setting

[Rs] - 0 - [_] and substitutins{Q_}_d {Q_}for

{Qs } and {QT }, respectively. Thus, by using the measured forward

traveling mode distribution at Plane I; i.e., {Q_} and the backward

traveling mode distribution on Plane 6; i.e., [QT}, the stacked

system matrix (Equation 4.37) can be solved for the forward and backward mode

coefficients at all planes.

Once the modal solution vector {A} is obtained, the acoustic energy

(E) at the end planes of all segments is obtained from the following

z

expression of the axial acoustic intensity due to Cantrell and Hart

(Reference 11)

27
:C) ®,@ ®',@

I

, I

* §i , §2 '

! I !

I I I

§3

@ I.._{____ Planes

I

I

!

I

I

m

"14 0 0 0

-p_._][qo o

o-[_.,.,][.,.] o

m

l

o oi._] o o

o 0

0' 0

o-_-_][-_.]o o

o o .R_]E._ o

o o o -_"']['_,1

0 0

0 0 0 0

0 0 ,{R+3"3] 0

0 0 0

0 0 0

0 0 0

0

0 0

0 0

0 0 0

0 -JR+5''] 0

0 0 0

o o o o o o [-_.]@-_- _1o o o o

o {_-_._] o o o o o [-q._-,-,]o o o

o o o .o o o o o [._.]._-,-_] o o

o o o.k_._]o o o o o C'q-_'_'_] o

o o o o o o o o o o [._,]_-,-,1

o o o o o_] o o o o o [.q

A+I Q5

A+2 0

A+3 0

A+'t* 0

A+5 0

A+6 0

A"I 0

A.2 0

A"3 0

A"4 0 i

A"$ 0

A'6 QT

b • •

Figure 4.3. Stacked System Matrix Equation for Three-Segment Duct

28
I z (r,z) = (l+M 2) Re [pv:] + _ pp + po c Mv v po c z z

(4.39)

and

E z (z) - 2_ /2

r I

Iz(r,z) rdr (4.40)

The energies at the first and the last planes in the segmented duct can be

used to calculate the total attenuation due to the treatment in the duct.

4.1.3 NOZZLE REFLECTION COEFFICIENT

If the te_nination plane of the fan duct analytical model is

represented by a plane in the hard wall section of the duct right at the

entrance to the nozzle, the stacked matrix IS] of the stacked system equation

(4.37) can be completely determined only if the nozzle reflection matrix

[RT] is specified. The matrix [RT] relates the backward propagatin_ wave

just upstream of the nozzle (and due to reflection from the nozzle) to the

forward propagating wave at the same location by the relation.

{x-(T)} = [_(T),+(T)] {,+(T)} (4.41)

No theory adequate for the purposes of this study is currently available for

the prediction of the nozzle reflection coefficients for multimodal

propagation. [RT] can, however, be determined experimentally.

The experimental determination of the nozzle reflection matrix [R T]

can be complicated due to the fact that the axisymmetric nozzle contraction

causes a scattering of acoustic modes among radial modes of the same spinning

mode order, i.e., a mode incident onto the nozzle can be reflected as a

combination of several modes. The difficulty in determining the elements of

29
the reflection matrix in the presence of cross-mode scattering amonz multiple

modes lies in the determination of the off-diagonal terms. Separating forward

and backward modes upstream of the nozzle gives a system of n equations (for n

modes) in the form of equation 4.41 in which the reflection coefficient matrix

2

[R T] is unknown; i.e., there are n equations in n unknowns. One

hypothetical method of determining the nozzle reflection matrix would be to

generate a radial mode in isolation which, upon incidence to the nozzle, is

scattered as a distribution of radial modes. The reflection coefficients for

the incident mode, i.e., Rnl, Rn2, Rn3 • • • etc., can then be

determined from the forward-backward modal separation of the complex pressure

profile measured just upstream of the nozzle. Each mode, in turn, would have

to be generated individually, zt is, however, impossible to generate an

isolated radial mode in a fan-duct system.

An alternate approach is to use the measured complex pressure profile

at a station just upstream of the nozzle for several different distributions

of the incident modes and separate the forward and backward mode

coefficients. The measurements made for each distribution of incident modes

will produce an independent equation, provided that these incident

distributions are not linearly related. If a total of n modes are

participating in the propagation and scattering phenomena, n different

distributions of incident modes must be generated. This can be achieved by

introducing axisymmetric phasing effects in the duct system, i.e., varying

treatment length, using liners of unequal admittance on opposite duct walls,

using mode-scattering obstacles, Helmholtz cavities, etc., to alter the source

radial mode distribution. For each distribution of incident modes, n

equations represented by a matrix equation like equation (4.41) is obtained,

so that for n different distributions of incident modes n x n equations will

be obtained which can be solved for the n x n elements of the reflection

matrix [RT]. To solve these equations, it is required that the incident

modal distributions are not equal or linearly related, to avoid numerical

problems caused by determinants with two proportional rows or columns.

30
When the nozzle reflection matrix is determined as suggested above,

Equation (4.41) can be written as

[RT(T)'+(T)]{A+(T)] - [A -(T)] = 0 (4.42)

Equation 4.42 can then be used to replace Equation 4.35 so that the stacked

system matrix equation in Figure 4.3 will have (QT}=O. Then, with

[Rs] set to zero, the matrix equation can be solved upon specification of

the forward traveling source vector {Q:}. As discussed before, the

source vector is specified by the measured separation of forward and backward

modes just downstream of the fan plane.

It should again be mentioned that in the case of an annular exhaust

duct-nozzle system where spinning modes propagate, scattering of modes by the

nozzle can be assumed to occur among radial modes only if the nozzle retains

the axisynmetry of the duct.

4.2 MATHEMATICAL DETAILS OF MODAL DECOMPOSITION

Starting with the equation that describes the pressure field in a duct

with flow,

+ + z

p(r. o, z, -- CA..pr

mn

m=-M n=O

-ik- z

+ AmnP _ (r) • mn] ei(me-_t) (4.43)

nm

consider measurement of the acoustic pressure amplitude and phase (relative to

a reference signal e i(_ - _c)). The measurement is made at a number of

circumferential positions, ej = (j-/M), j = O, 1, 2 . .., (2 M - 1),

where M = M + 1. Applying the discrete Fourier transform [Reference 4] to the

31
measurements, we obtain the order m coefficient of the measured

circumferential pressure distribution (for fixed values of r and z)

2_-Z

_m(r, z) ,, !_ _ _'(r, el, z_ _i(Zcm.n'/_) 2i

.,']-0

(4.44)

m = O0 + I, ..., t M

ApplyinK the discrete Fourier transform to equation (4.43) and comparing with

equation (4.44), we obtain

N i (kLz)

[AmnPr (r) e

nnl n=O

i (-k_n z)

+ A:nP; (r) • ] = Pm(r,z) (4.45)

The radial and axial mode separation will be considered next. When the radial

+

mode shapes, Pr (r) and Pr- (r) are equal, then the radial mode separation can

mn mn

be obtained by measuring Pm(r, z) at (N + 1) values of r, for a fixed z. The

results are then used to form the followinK matrix equation

K = ! (4.46a)

where

A g

m m

Pr (rl) Prml(rl ) "'" PrmN(rl )

mo

Pr (r2) " " "

mo

Pr (rN+l) " " PrmN(rN+l )

mo

_ m

(4.46b)

32
X

ik + z -ik- z

A + e mo + A- e mo

mo mo

A +

ml e + Aml e

(4.46c)

and

S

PmCrl, z)

Pm(r2, z)

Pm(rn+l, z)

(4.46d)

This matrix equation is then solved for the unknown vector, x. Note that the

value of x is a function of the axial coordinate, z.

To separate the order (m, n) forward and backward propagating modes,

the circumferential and radial mode separation described above is performed at

two axial locations, z l and z 2. Denoting the components of

correspondins to the (m, n) mode by X (z 1) and Xmn(Z2) , the following

matrix can be formed for the desired unknowns A+ and Amn in

Z = _ (4.47a)

33
l zleIk:n zl1

- (4.47b)

B z2 -ik- z 2 -- i n mn

e e

and

7_ = (4.47c)

b m

D

This matrix equation can be solved explicitly, with the results

(4.47d)

and

A = (4.48a)

mn i(k+mn + km)Az

1 - •

IX (z 2) - x (z z) • ]

e

A_ = i(k + k- )&z (4.48b)

mn + mn

1 - •

wheue

&z = z2 - z I (4.48c)

If we examine the uncertainties in the measurement of A+ and A_, we n_n

see that it is sensitive to the factor [I - cos(k+mn + k_)Az] in the denominator. In order to avoid this problem, two thinss can be done. First, careful

34
spacing can be chosen such that [(k_ + kL)_z] does not equal an integral

multiple of 2_. Secondly, redundant measurements can be introduced, which

will significantly improve the modal decomposition accuracy. The procedure for

using redundant data to effect a least squares matrix solution will now be explained. Consider the general matrix equation

R ._ " __ (4.49)

Here D is a square matrix of order q whose elements may be complex. _ is

the vector containing q complex unknowns, and _ is the (complex) input data

vector of order q. For matrices D, whose determinant is non zero, there is a

unique solution for _. However, when _ is determined from experimental

measurements, small measurement errors may lead to significant inaccuracies

in _.

To increase the accuracy in the calculation of _, additional

independent measurements may be added to the system. Assume that s additional

independent measurements are added to the system. The size of the matrix D is

then.(q + s) by q, and _ is of. order (q + s), while _ remains of order q.

The system is now overdetermined, and no exact solution for _ is possible.

The approach used is then to define the remainder _, Where

-r ,, D_._ - a_ (4.50)

We now search for the value of _ which minimizes the magnitude of _.

Noting that _ is complex, we define the magnitude squared of _ by T,

where

= _i_*i = Dij D*ik _i_*k

- "i Dik_*k - Dij _j "*i + ..a*. 1 l

(4.51)

where the * denotes the complex conjugate. We now minimize _ by setting

aT/_ i and 87/8_* i equal to zero. The resulting equations can be

written in matrix form as

35
D _T D _ = D_*T __ (4.52)

where T indicates the transpose. Equation (4.52) is the 8eneralization of the

least-squares fit for matrix equations, and can be applied to either equations

(4.46a) or (4.47a).

For ducts with sheared flow, Prm(r) + and pL (r) are not

identical. Under these conditions, the radial and axial mode separation

cannot be decoupled in the manner discussed above, but must be carried out

simultaneously. With only two axial locations, this would lead to inverting a

matrix of the order 2 (N + 1) as opposed to order (N + 1) when decouplins was

possible (N - q + s). As the sensitivity of matrix solutions to small input

errors increases with an increase of matrix size, redundant measurements

should be used in this case also. The resultin8 matrix equation is

c_*z c_gffic_*z c_ (4.53a)

where

C m

I

m

_:_z_ -_:_z_

+( Ikm+°Zl - -Ik:°Zl + )e - )e

Pr rl)e Pr (rl)e Prnd(rl "'" PrJrl

mo mo

_:o_ - -_:oZ_ -_:_

Pr + (rl)e Pr (rl)e " " Pr_rl )e

mo mo

+ _CoZ_ _ -_:oZ_ -_L_

Pr (rl)e Pr (rl)e " " Pr_rl )e

mo mO

+ ik:oZl

Pr (r2)e

mo

÷ i_oZ3 _ ik_z3

Pr (rN+l)e ..... PrJrN+l )e

mo

m

(4.53b)

36
and

°1

_o

_s

(4.53b)

(4.53c)

Pm (rl' zl)

Pm (r_, z2)

Pm (r1' z3)

Pm (r2' zl). (4.53d)

As before, the * denotes the complex conjugate and T implies the transpose of

the matrix.

4.3 THE IN-SITU IMPEDANCE MEASUREMENT TECHNIQUE

The In-Situ method (References 4, 5, 6) has been used for the

measurement of the Normal Acoustic Impedance of locally reacting treatment

panels. Dean 4 and Kooi and Satin 5 used this technique on single degree of

6

freedom (SDOF) panels while Zandbergen measured the impedance of 2 degrees

12

of freedom (2DOF) samples using this method. Zandbergen, et. al., also

measured the impedance of the inlet acoustic treatment of a Fokker F28

aircraft power plant during flight.

The theory of this method is illustrated here for an SDOF liner. The

sketch below shows a single cavity with an acoustically hard backwall at x = o

and a porous face sheet at x - d. The walls of the cavity are assumed to be

37
rigid so that no transmission of sound between cavities is allowed. The width

of the cavity is sufficiently small to allow only plane wave propagation

between the backwall and the face sheet.

X

A

nm ml ml m m mm

d

B

The acoustic pressure at any position x in the cavity and frequency f

is given by

p(x,f) - poei(wt + _B ) Cos (kx) (4.54)

where k is t_e acoustic wave number _/c, c is the speed of sound in the air

filling the cavity and the space outside it, w = 2vf and i = _-1.

The acoustic particle velocity u in the x direction is related to the

acoustic pressure by the linearized momentum equation:

u(x,f) - -I a

at p ax p(x,f) (4.55)

where p is the density of the air.

From equations (4.54) and 4.55)

i(_t + _B)

u(x,f) = -i Sin(kx)

pc Poe

= -i Sin(kx)

pc PB (f)

(4.56)

38
where pB(f) = po e , the acoustic pressure at the back wall.

It is assumed that the acoustic particle velocity is continuous across

the thin porous face sheet. That is

u (f) = uA(f) = -i sin(kd) pB(f) pc

44.57)

Thus the normal impedance at the face sheet is given by

PA(f) PA(f)

CA(f) - --7 - -i cosec (kd) pB(f ) 44.58)

-uA(f)pc

where pA(f) is the acoustic signal at the surface of the face sheet just

outside the cavity•

As illustrated by Equation (4•58), the In-Situ method requires the

measurement of the complex ratio of two acoustic siEnals. For this reason, it

is often referred to as the Two Microphone Method.

In the presence of srazing flow, the face sheet transducer signal

pA(f) may contain flow noise which may be regarded as random in nature and

uncorrelated with the acoustic signal. In order to remove this contamination

of the face sheet sisnal, Equation (4.58) is modified to

where

_A(f) - -i cosec(kd) • HAB(f) 44.59)

H--_= PA(f)PB*(f) / PB(f)PB*(f), the time averased complex

transfer function of the two signals at locations A and B.

_9
5.0 EXPERIMENTAL PROGRAM AND RESULTS

This section describes the critical experiment conducted to obtain the

data necessary to test the validity of the theory described in Section 4.1.

This experiment consisted of measurement of the necessary information required

to calculate the suppression achieved by an acoustically treated section in

the exhaust duct of a fan vehicle.

The vehicle was desisned for a stron8 blade-vane interaction tone,

making use of Rotor 55 with only 8 outlet guide vanes closely spaced against

the 15 bladed rotor. An inlet turbulence control structure was incorporated

into the anechoic-chamber test facility so as to reduce the production of

other acoustic modes.

The treatment section was designed to be as linear as possible in order

to establish the value of the wall admittance withminimumuncertainty.

Impedance measurements were made on the actual treatment panels used in the

test vehicle. Laboratory measurements were made to determine quantitatively

the effects of flow velocity and sound pressure level upon the acoustic

impedance.

Acoustic "Mode" probes were incorporated upstream and downstream of the

- treated section (of" the fan exhaust duct) to establish the acoustic modes

propagatins in both directions at two axial cross-sectional planes, one

upstream and the other downstream of the treated section. These probes were

desisned to measure the modes expected for the vane-blade interaction as well

as all other cut-onmedes at the tone frequency of the rotor blades.

Aerodynamic measurements were made at the same axial planes (by the

same probe actuators) to establish the mean flow profiles, both radially and

circumferentially, and to determine the boundary layer profiles incident upon

and leavins the treated sections.

The data from the acoustic probes were used in conjunction with the

aerodynamic measurements to establish the composition of the amplitudes and

I_C;..,GRiG PAGE BL._( _OT FiL_:_O

41
relative phases of the acoustic modes for use in the analytical prediction of

the acoustic suppression of the treated section, as discussed in the section,

"Theory-Experiment Comparison".

5.1 TEST APPARATUS

5.1.1 TEST FACILITY

The vehicle was tested in an anechoic-chamber 10.668 m wide by 7.62 m

long by 3.048 m high, measured from the tips of the foam wedges. The wedges,

.71 m deep polyurethane foam, provide less than ±1 dB standing wave ratio

down to 200 Hz. Far-field noise measurements were made by an array of twelve,

6.35mm-diameter, far-field microphones (B&K 4135) located on a 5.182 m radius

arc. The microphones were arranged at 10" intervals from 20" to 110" relative

to the fan exhaust. Table 5.1 summarizes the positions relative to the

nozzle. Calibration was by piston phone, B&K Model 4220, prior to each

far-fleld measurement.

The far field microphone levels were recorded on a Sangamo Sabre IV

tape recorder at 152 cm per second (60 ips) tape speed. Power spectra were

generated by processing the microphone signals through an HP 5451C Fourier

analyzer system. Power spectrum levels at blade passing frequency were

integrated for overall acoustic power levels. Fan operation was stabilized to

steady state before initiating any data recording.

The fan was driven in the exhaust mode as shown in Figure 5.1. Figure

5.2 shows Rotor 55 as it was installed in the chamber. The air is inducted

through a vertical stack and turned through 90" as illustrated in Figure 5.3;

an inlet turbulence control device was included.

The effectiveness of the 90 ° turn at the base of the vertical inlet

stack and the TCS structure was determined. Kiel probe data are shown in

Figure 5.4. Losses across the TCS and the flow contouring were found to be

small, and varied by less than 0.5% at any location except at the top of the

42
Table 5.1. Far Field Microphone Distances and An$1es

Relative to Fan

Distance

Meters

5.18

5.18

5.18

5.18

5.18

5.18

5.18

5.18

5.18

5.18

Angle

De_rees

20

30

4O

50

60

70

80

90

I00

110

Relative to Nozzle

Distance

Meters

3.54

3.73

3.97

4.25

4.56

4.87

5.18

5.49

5.77

6.04

Angle

Desrees

36

44

57

69

80

90

I00

109

118

126

43
0

o

t_

x

,2

E

,,c

u

0

I

I1;

0

E

.M,
ORIGI?¢AL PAGE I8

OF P()O_{ QUALiTy

O

am

_a

O

O

tO

O

45
See Insert Above

._TURBULENCE CONTROLSTRUCTURE

1_,=._-. ANECHOICCHAMBER WALL

NOZZLE

_%COUSTiC

ROTOR55 TREATMENT

OGV

Figure 5.3. Schematic Diagram of the Test Vehicle Showing Rotor 55 as

Operated in the Exhaust Mode

46
O_

O'J

! I

0

I!

4

t

O'J

0

0d/_d

_ .,-._

•,._ "1_ ._

rr

I--I "_" CD

! !

U'I

_J

0

0"_

O 0

o

0

ueqwnN

47

i

CM

0

a .I,..I

o

.2

m

@

0

0

0

o

I_ Itl

@.1

,-I _J

III I._

0

_"

o

0

III ,._

o,-'_
duct towards the outer wall. Nach numbers were calculated by using static

pressures that corresponded to the weighted average of the two closest static

wall taps. The flow was uniform at the measurement locations. The only point

of significant variation is at the top of the duct towards the outer wall

where the flow is about 3% above the mean.

5.1.2 FAN VEHICLE

The vehicle used in this program was the NASA Lewis Research Center fan

Rotor 55 modified for this program to a close-spaced rotor-OGV configuration

as shown in Figure 5.5. Rather than the original eleven outlet guide vane set

spaced at 1.5 chord length eight vanes were used at a spacing of 0.5 rotor tip

chord length. The annular exhaust duct consisted of two hardwall sections

where in-duct acoustic probes were located, a treated section, and a nozzle

termination. Test variations included a hardwall section in place of the

treated section.

Pertinent design characteristics of the vehicle are summarized in

Table 5.2.

Table 5.2. Rotor 55 Design Characteristics

Fan Diameter ....................... 50.8 cm (20 in.)

Radius Ratio ....................... 0.46

Number of Fan Blades ............... 15

Inlet Guide Vanes .................. None

Number of Stators .................. 8

DesiKn Tip Speed (I00_) Nl_O) .... 213 m/sec (700 fps)

Design Fan Speed (lO0_N/_e) ..... 8021 r_m

Stage Pressure Ratio ............... 1.16

Weight Flow ........................ 27.0 kg/sec (59.5 lbm/sec)

48
5. I. 3 AERODYNAMIC INSTRUMENTATION

Velocity profile measurements were made both upstream and downstream of

the treatment section by means of two traversing pitot tubes (radial and

circumferential traverse), three total-pressure (and total-temperature) pitot

tubes each with five radial locations, and sixteen wall static taps. Boundary

layer profiles were determined at both locations, up and downstream, by means

of two radially traversing boundary layer probes in conjunction with the

sixteen wall static taps.

All aerodynamic instrumentation information concerning type and

location is sunmarlzed in Figure 5.6.

Pressure data were individually sampled through two 48 channel

scanivalve systems. Temperature data were sampled though an HP scanner and

digital voltmeter. The data reduction program included corrections for

compressible flow, taking into account specific heat variation with

temperature and humidity.

5.1.4 ACOUSTIC "MODE" PROBES

In general, to determine modal coefficients the in-duct sound field

needs to be determined. The easiest quantity to measure in a duct is the

acoustic pressure, which for rotor/stator interaction can be written as

P(r, O, z, t)

+ + iKmnz

- (ampma(r) •

m_-M n_O

-iK- z

mn

+ *-__p--Or) e I ei(mO-et) (5.1)

_CEDING PAGE BL.Af'_t(HOT F_t.L_O

5O
,.,I

°t,,l

_ _, c_ _1 .,._

rJ3

C_

C_

c;3

v

Q.

c_U _J_ r"

C/300 CO

0

cJ 0O

4.1

_o

_0

.,-I o

c;3u

_o

co

rz_

EM

0_

u

_E

. 51
This expression is a solution of the convected wave equation using separation

of variables Here the complex constants A+ and A_ are the modal

•

coefficients of the forward and backward propasating (m, n) modes• Pmn(r)

and K are the radial mode shape functions and axial wave numbers, with the

m/t

superscripts + and - corresponding to forward and backward propagation,

respectively. The other variables are r, e, z, t, and w, which refer to

radial location, ansular location, axial location, time, and blade passing

ansular frequency, respectively.

Determination of the complex modal coefficients (amplitude and phase)

requires that the complex pressure (amplitude and phase) be measured. In the

case where primarily coherent noise is present, as was the case here, the

signal processing technique of phase averagins can be used [Reference 13].

This sisnal processins technique selects from a siven input signal only that

portion which is coherent with a specified reference signal, and the random or

noncoherent portion of the input sisnal is isnored (average approaches zero).

A phase lock amplifier, which uses this technique, can be used to measure

in-duct acoustic pressure amplitude and phase.

In the case of uniform flow in either cylindrical or annular ducts with

hard walls, the radial mode shapes are Bessel functions of the radial order n,

the circumferential mode order m, and of argument (krr). Also the forward

and backward propasatin8 values p+(r) and p-(r) are equal. When radially

sheared flow is present, then the radial mode shape is also a function of the

velocity profile, and forward and backward values p+(r) and p-(r) are not

equal.

When pressure, radial mode shapes, axial wave numbers, probe position,

and blade passins fequency are all known, the modal coefficients are

determined by the discrete Fourier transform and matrix manipulation. The

details of this modal decomposition are discussed in Section 4.2. An

important limitation of this process is that the use of only two axial

measurement positions can lead to large measurement inaccuracies [References

14, 15]. As is shown in Section 4.2, a third axial measurement position

sisnificantly improves the modal decomposition accuracy.

52
Probe Description

Two in-duct mode probe arrays were designed and fabricated so that all

cut-on circumferential and radial modes including forward and backward

propagating modes, could be determined. A sketch of the probe design and

geometry is given in Fisure 5.7. Each array consisted of four radially

located probes, each probe included three axial Kulite pressure transducers,

in order to provide a redundant measurement at each radial location. Two

probes mounted to a strut are shown in Figure 5.8. Each probe had three slots

containing recessed Kulite transducers (Model LQ-080-5). Each transducer is

connected by five wires (0.13 nun diameter) routed through the probe shell and

srooves in the support struts. Both the probe and strut were made as small as

possible based on transducer size and on the acoustic considerations discussed

below. The Kulite model used was chosen not only because of its small size

(1.29 n_n on each side) but also because of its linearity, stability, and

relatively high sensitivity. Each array was mounted on a circumferential

traversing actuator, as can be seen in Fisure 5.9.

Disturbance of the sound field due to the presence of the strut and

probe was of primary concern. Probe scattering can alter the transducer

readings significantly when an acoustic wave is traveling in a direction

perpendicular to the probe axis. This scattering problem has been previously

considered [Reference 16]. Also, scattering effects from the strut arise both

from the blockins of the acoustic motion by the body surface and from the mean

flow nonuniformity due to the strut thickness. This problem was treated using

the Taylor transformation [Reference 17], which reduces the low Nach number

problem to an equivalent stationary medium problem. Based on an analysis

usin8 that approach, the probe diameter was selected to be 4.76 nua and the

strut thickness 6.35 nee. With these values the error in phase measurement was

expected to be less than 4.5* and in masnitude, less than 1.0%.

Vortex shedding off the probe support strut was considered, and a

thickness of 6.35 nuwas selected to assure that the frequency was at least

twice the blade-passing-frequency.

53
X

m I

1-4 = ('_

G) 0

(_ _, I

(..* cn :<

q ,p4

,_=,o

_Z

r'_

U

I.o

.=-

II

r_

r_

,.,1"

II

I.o

(.<

I

-,1"

._= cO

u

0

0

X

m

0

0

4-1 4.1

0

_0._

k_

r_
O!_tG_NAL PAGE 1_

D_. POOR QUALITy

Figure 5.8. Photograph Showing Two Axial Probes Mounted on a Radial Strut

55
Figure 5.9. Photograph of a Complete Set of "Mode Probes" Showing the Circumferential

Traversing Actuator and Axial Probes at Four Radial Locations Mounted on

Two Diametrically Opposed Radial Struts

56
The probe and supporting strut were designed to withstand the stress at

velocities in excess of 0.4 Math number and to withstand regular handling

during calibration of the probes. In order to meet these requirements the

machine screws and the probe shell were made of tempered 403 SS with a yield

strength of 150,000 psi. The natural frequencies of the probe and strut were

determined to be well out of range of blade passing frequency.

A schematic of the measurement system is given in Figure 5.10. Data

acquisition was controlled by an HP 1000 computer that sampled pressure and

temperature data, and stored in-duct acoustic phase and magnitude

measurements. Two microprocessors assisted in this task under direction of

the HP 1000. One microprocessor controlled the actuator and directed a

multi-channel scanning amplifier (GR 1566) to sample each mode probe

transducer signal through a phase-lock amplifier (Brookdeal Model Ortholoc-SC

9505). The phase-lock amplifier passes only that portion of the signal which

is coherent with the blade passing frequency. The blade passing frequency was

measured by optically sensing the passage of each blade tip to trigger a

square wave generator. Both the phase and the magnitude of the coherent

portion of the mode probe signal were determined in this manner. The second

microprocessor controlled the operation of the phase-lock amplifier and

transferred phase quadrant information to the HP 1000. The HP 1000 also

applied calibration corrections. The acoustic signals were recorded on a

Sangamo IV tape recorder.

The mode probes were rotated in 18" increments around the entire

circumference of the duct, so that data were measured at 20 circumferential

positions for each test point. After a test point was completely surveyed,

the data files of phase and magnitude measurements were supplied to the

computer program which then calculated the individual modal coefficients,

using the method for modal decomposition described in Section 4.

The probes were calibrated for both magnitude and phase. The

calibration included all wiring and support equipment up to the phase lock

amplifier. The probes were calibrated by mounting each probe in an impedance

57
DC _,Iotor

Opticat

Tektronix AM 502 Amps

IIII[IIIII

Kulite Power Suppli.es

IIIlll llll

C,enerat Radio Scanner 1566

I

I

II

I

| I

m

!

, I

II

I,

I

Brookdeal SC 9505

Phase Lock Amp.

I T,k.Po501I i_"- si_a_/

i Pulse _ ._ITI KD245

__ Generator J lFotoni.c Sen. J

Sagamo

Sabre IV

Tape

Recorder

Encc!er

._[Lcroprocessor For

Phase Lock

Analog to DiEital

Converter HP 2240A

I

I

I

I

I

I

i

!

I

i

|

Microprocessor For

Actuator

HP 10O0-F/

T._..."= Storace

I

!

_i"--- Scan_.valve / (

HP Scanner

m

Figure 5.10. A Schematic Diagram of the Data Acquisition and Analysis

Modal Measurements

System Used in the

58
tube in the same axial plane as a previously calibrated 6.35 nun (0.25 in.)

Bruel and Kjaer model 4135 microphone and then using a speaker at the other

end of the tube to generate a plane wave at blade passing frequency. Because

only a plane wave was 5enerated, the calibration was valid for both level and

phase. With the speaker set at a fixed level, the masnitude of each

transducer as read with the phase lock amplifier was matched by use of

variable sain amplifiers for each transducer. Any difference in phase between

each transducer and the microphone was recorded and included as a correction

in the data reduction program.

The rotary actuator was used to locate the modal probes

circumferentially. The actuator consists of a stationary section and a

rotatins section. The stationary section, mounted on the adjacent ducting,

provides the support for the motor drive, optical locatins devices, and

transducer wire suides. The rotatins section, actuated by a motor driven

chain, contains the probes. As can be seen in Figure 5.9, the transducer

wires feed over a rotatins disk and wrap around the rotatins section of the

actuator as it moves. A microprocessor for each actuator controls the motor

to drive the rotatins section to a predetermined location. The location is

determined by a rotary optical encoder (Teledyne Gurley Model 8625). Although

the microprocessor controls the actuator, a visual check and manual adjustment

of the ansular location of the Node Probe was carried out.

5.2 VEHICLE AERO-&COUSTIC RESULTS

This section presents the results of the vehicle testing, including:

aerodynamic parameters, measured modal amplitudes, and farfield acoustic

data. Results from measurements of the treatment impedance, both statically

and with grazing sound and flow, are presented in the following section (5.3).

5.2.1 SELECTION OF FAN SPEED POINTS

The fan was operated at three speed points in order to vary the cut-off

ratio of the blade-vane-interaction tones. It was varied from a minimum value

59
near unity (near cut-off) to a relatively high maximum value, over seven, so

as to have a range of values between these two extremes that would provide

experimental data for a severe test of the validity of the analytic method

presented in Section 4. The interaction-tone modes were established by the 8

outlet guide vanes chosen for the 15 bladed rotor:

m = 15 +- 8k; k = 0, I, 2 ....

The cut-off ratios (defined as the frequency divided by the hardwall cut-off

frequency of the mode) for the propagating interaction modes at the speed

settinss selected are sunm_rized in Table 5.3.

Table 5.3. Acoustic Modes Generated by Rotor 55 With an Eight Vane

Stator With Associated Cut-Off Frequencies and Ratios

Cut- Hardwall Cut-Off Ratios at:

Off 1900 Hz 1500 Hz

Mode Freq. (7600 rpm) (6000 rpm)

I000 Hz

(4000 rpm)

-I, 0 270 7.04 5.56 3.70

-I, I 1305 1.46 1.15 --

7, 0 1695 1.12 ....

5.2.2 FAN PERFORMANCE

The operatins map for Rotor 55 is shown in Figure 5.11a. The data in

this plot shows the repeatability of the performance when three different vane

desiKns were used under the QCSEE Program [Reference 18]. In the current

prosrem, the aerodynamic data incorporated for use in determinin8 the flow

field into and from the test section was used to obtain an approximate

confirmation that the flow from the fan stase was representative of a typical

fan vehicle. The test section aerodynamic data are presented and discussed in

the next section (5.2.3). Data points determined from the upstream and the

downstream locations are shown in Figure 5.11b, for a fan speed of 7600 rpm

60
T_/_I_ CoT_e_ _nss_ g_e_S

0

0

0

0

u_

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m

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' 62
(95%). As noted on the figure, the two locations gave different values; this

was because of the radial velocity profiles used (see Figure 5.12) which made

the result dependent upon the particular circumferential location of the

instrumentation. These data were concluded to be sufficient for the purposes

needed, that is: the modified fan assembly was sufficiently close to design

values of pressure-ratio and air-flow-rate to be a representative turbofan

noise and airflow source.

As an interesting sidelight, these same type of data in the original

fan buildup assembly, revealed serious deficiencies that were traced to

inadvertant misalignment of the outlet-guide-vane angles. This condition was

detected and corrected, thereby resulting in the successful measurement of the

acoustic modal amplitudes and phases as described in the theory-experiment

comparison (Section 6).

5.2.3 TEST SECTION AERODYNAMIC DATA

Typical radial velocity pcofiles at the upstream and downstream

locations are shown in Figure 5.12; these profiles were detecmined by the

traversing boundary-layer rakes which were restricted to a single

circumferential position. There was sufficient difference in the mean values

of velocities to cause concern about the validity of the data.

Therefore, circumferential traverses were then also made with the

results shown in Figures 5.13 and 5.14, at radial insertion depths of .067

and .102 m, respectively. These data reveal periodic velocity defects around

the circumference. At the upstream station the peaks and valleys show a

circumferential offset between the two radii. The defects also shift in the

circumferential direction as they move downstream.

Swirling wakes off the outlet guide vanes were the cause. The

mixing-out of these defects resulted in the more uniform circumferential

profile at the downstream location. The particular circumferential location

chosen for the radial profile measurement was responsible for the discrepancy

63
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initially of concern in Figure 5.12. Finer resolution of the two

circumferential traverses is shown in Figure 5.15 which reveals that there

were no smaller scale defects.

The circumferential velocity non-uniformity is of concern because the

theory of Section 4 only accounts for radial velocity gradients. Averaged

Math numbers were used for calculation of axial wave numbers and radial mode

shapes.

Most aerodynamic measurements were made at 7600 rpm, although enough at

6700 rpm and 2800 rpmweremade to assure that the fan performance matched

previous tests (Figure $.11b). Figures 5.16 and 5.17 show the similarity of

the profiles both upstream and downstream at 7600 rpm and 6700 rpm.

During acoustic runs, all probes were removed from the airstream. Wall

static taps and recessed boundary-layer probes were monitored to assure that

the flow field was the same during the acoustic measurements as with the more

thorough aerodynamic surveys. Figure 5.18 shows a typical comparison between

the results from the boundary layer probe and earlier results with a

traversing pitot tube.

5.2.4 ACOUSTIC DATA

This section presents the results of the acoustic modal measurements in

the test section, at both upstream and downstream locations; measurements were

made at each of the three fan speeds for the test section with the treatment

in place and with the treatment replaced by a hardwall (zero admittance).

Modal Measurement Results

For the hardwall case, the results from the two separate mode probes

(upstream and downstream location) provided a quantitative measure of the

effect of the non-uniformity of the circumferential flow profile (e.g., per

Figures 5.13 and 5.14) and of the nozzle reflections upon the resolution of

67
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71
the modal coefficients. A typical result of this check is summarized in Table

5.4 in terms of the relative forward propagating energy (dB). The table shows

two cases, one considering only cut-on modes in the modal coefficient

determination and the other considering, in addition, cut-off modes up through

m ffi9 circumferential modes (maximum obtainable from 20 circumferential

measurement locations) and n = 3 radial modes (maximum obtainable from four

radial measurement locations).

Table 5.4, Comparison of Computed Propagating Energy Flux,

Upstream and Downstream, for Hardwall Test Section

Blade

Fan Passing Mode

Speed Freq. Index Cut-off

{RPM} {Hz) {Mtn) Ratio

Relative Enermy Flux* (dB)

Allowing Allowing

Cut-on Cut-on and

Only Cut-off

7600 1900 -I,0 7.04 1.2 2.1

-I,I 1.46 0.8 0.I

7,0 1.12 0.1 0.5

6000 1500 -I,0 5.56 0.4 0.2

-I,I 1.15 0.7 0.3

4000 1000 -I,0 3.70 2.5 3.4

_Relative energy flux is the difference between energy flux measured

upstream and that measured downstream.

The agreement, in general, is very good at the two higher fan speeds,

and within about 3 dB at the lowest speed; the agreement is best for the modes

nearest cut-off. (Further discussion of the effects of flow on acoustic

pressure profiles ratio modal components is given in the theory/experiment

comparison, Section 6).

The results of the modal decomposition (using maximum values of m and

n of 9 and 3, respectively, and eigenvalues for uniform flow with thin

boundary layer) are presented in Figures 5.19 through 5.24. The cases are

listed in Table 5.5.

72
Table 5.5. Index of Figures Showing Modal Amplitudes and Phases

Figure Case Modal

Number Freq. (Hz) Test Section Definition

5.19a 1900 Hardwall Amplitude

5.19b 1900 Hardwall Phase

5.20a 1900 Treated Amplitude

5.20b 1900 Treated Phase

5.21a 1500 Hardwall Amplitude

5.21b 1500 Hardwall Phase

5.22a 1500 Treated Amplitude

5.22b 1500 Treated Phase

5.23a 1000 Hardwall Amplitude

5.23b 1000 Hardwall Phase

5.24a i000 Treated Amplitude

5.24b i000 Treated Phase

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e_

E

co

(D

_Cl

E

Z

"0

O

W5

m

a.;

f.;

O

_J

Q

,,w

o

a.;

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e_

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N

0

_JO

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mO

.i

85
The fisures show the modes expected from rotor/stator interaction,

althoush other modes are present also. For the 1500 Hz case only the (-1, 0)

and (-1, 1) should be present. Figure 5.21a shows these modes were indeed

measured but there was also a strong (-5, 0) mode. This would be possible if

the inlet flow were not uniform, and may in fact be the result of interaction

between the one high velocity zone near the top of the duct (Fisure 5.4) and

the fan. The fact that this mode is so strong is due to the fact it is just

cut-on. Also because it is just cut-on it exhibits larse reflections, which

may explain its drop in level from upstream to downstream. In other cases the

expected rotor/stator modes were stronger. For example, the forward

propasatins (7, 0) mode was about 7 dB stronser than any other mode in the

1900 Hz hardwall case.

Fisure 5.22a (1500 Hz) is a typical case with treatment. As expected,

the upstream forward propagatins coefficients are about the same as in the

hardwall case, but the downstream coefficients are lower, due to the

treatment. Fisures 5.20 and 5.24 show similar results for the 1900 Hz and

1000 Hz cases. As is evident, the treatment proved to be optimally designed

for 1900 Hz. As expected, in the 1900 Hz case the (7, 0) mode which is just

cut-on is the dominant mode, althoush in the 1000 Hz case the (-1, 1) mode

which also is just cut-on is weak and the (1, 0) mode dominates.

Treatment was expected to have no effect on the upstream forward

propagatins modal coefficients. Figure 5.25 shows the effects of treatment

and temperature on these coefficients. Run A66 was a 7600 rpm hardwall case.

Run A67 was the same as A66 except for the fact the upstream and downstream

modal instrumentation was switched. When run A67 was carried out the outside

temperature was lower and because of this temperature chanse the (7, 0) mode

is less cut-on for A66 and therefore stronser, while the other modes are in

relatively sood asreement. Run a71 was the 7600 rpm treatment run with

temperatures similar to A67 and it shows sood asreement with the hardwall

cases except rheA66 (7, O) mode as would be expected. It should be noted

that the A67 downstream coefficients did not make sense and as far as can be

determined the only possible cause was calibration drift. Run A67 was the

only run where the probes were not calibrated the day of the run and the

downstream probes had a different set of calibration trim potentiometers.

This could explain why the upstream probes still save sood results.

86
Q

0

C_J

G

E

J.I

E_

._ M 0

! I I

CD CO r_

e-e-e-

_rrrr

!

C;

_- [_---_._

r_.

I

E

0

I

L_

r",.

i _ 1 | T "T 1

L_ 0 _ 0

,x'-t

(Jeeu_.I) epn_Tu6eN

Q O

E

C_

E

Z

C_

O

0

0

M

_0

M 0

0

0 ,_

t_

e_

G

0"_

_ m

¢M

_g
In general, the differences between carrying out the modal

decomposition using sheared flow values of Pr,mn and Knm or uniform values

were small, although the 1000 Hz case did show a difference of 1.5 dB in the

(-1, 0) mode. The same could be said whether the modal decomposition was

solved for cut-on modes only or for all modes.

Far-Field Measurement Results

Table 5.6 shows the sound power levels for both hardwall and treatment

cases along with the resultant suppression at each running speed. The

integration for acoustic power level was done using the nozzle exhaust as the

acoustic source and the revised distances and angles from Table 5.1.

Table 5.6. Sound Power Levels From Far Field Data

(Power Spectrum Bandwidth = 4 Hz)

Blade

Passing

Freq. PWL (dB) PWL (dB) Treatment

(Hz) Hardwall Treatment Suppression

I000 120.0 118.8 1.2

1500 129.2 125.0 4.2

1900 126.8 116.6 10.2

Figures 5.26 through 5.28 show the directivity patterns of the narrow

band SPLs. The lack of suppression in the 30 ° microphone position could well

be the result of the directivity shifting, which would be expected. In

general, the SPL at blade passing frequency was 5 to I0 dB higher than its

harmonics for 7600 rpm and 6000 rpm, while for 4000 rpm the difference was at

least 15 dB.

5.3 ACOUSTIC TREATMEFE IMPEDANCE

In the theory experiment check of Section 6, an accurately determined

value of the acoustic impedance of the treatment sections was required.

88
J

s

o

s

o

I

l

i I

l °

Q

I

I

l

I

o

-rI

eE

C..

[]

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!

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0

e,,,,

o

0 _ m N

0 oO

_ _r_

0 ,_ m m

c-" _ 0 =

t-_ ram. o

C_ _ _

(=) _: = =

0

_g

=B

(_P) "13A3-_ 3_(ISS3_d (INnOS

8_
i I

o

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s

o

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t

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(BP) [eAa-I

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i l

go

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R3E

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t_ {._

-r" _--

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m

i

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JaMOEI punos

E3 L_

C.;

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Q)

Q;

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°,-I

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0

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o

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o_1

c_

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< =

0

0

U

_ _ .

_ N

- O

¢1

,.2

¢N

u_

; 90
0

CSP) [aAa7

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91
Although the treatment was designed to be linear, measurements were carried

out to establish the effects of sound intensity variation and grazing flow.

In addition a comprehensive survey of the impedance was made over the surface

of both the inner and the outer treatment sections to ensure that (a) the

acoustic impedance of the two surfaces was equal and (b) the treatment

impedance was uniform over the entire surface. The details of the design of

the treatment, the measurement techniques and apparatus used and impedance

data obtained are presented below.

5.3.1 TREATMENT DESIGN

The acoustic treatment was designed to be locally reacting and to have

equal impedance on the inner and the outer walls of the treated section of the

aft duct. A single degree of freedom (SDOF) design was chosen consisting of a

25.4 nan thick Flex Core bonded to a 1.143 mm porous face sheet and the wooden

back wall of the treatment sections. The Flex Core used is made from 0.063 mm

thick aluminum sheet. The geometry of the cells is illustrated in Figure

5.29a.

The face sheet consisted of a 200 x 400 wire mesh bonded onto a 30%

porosity aluminum perforate (.83 mmhole size) sheet. The purpose of the wire

mesh was to make the treatment linear; that is, the effects of flow and sound

intensity on the impedance of the treatment were expected to be small as a

consequence of the wire mesh.

From previous experience of the blockase due to bonding the core to

face sheets, the resistance of the treatment was expected to be approximately

1.0 pc (= 41.6 cgsRayls). The resistance actually maasured on the

finished treatment sections and on the laboratory samples was approximately

0.5 pc.

The geometry of the treated segment of the aft duct is schematically

shown in Figure 5.29b.

92
200 x 400 Wiremesh

on Perforate Face Sheet

-V--!--]

Hardwood Casing

Fl__ex Core

J25.4 ram.

J_l

Cross Section of Flexcore

Figure 5.29a. Schematic Diagram of the Acoustic Treatment Used in the Treated Segment

of the Aft Duct

93
1 I + + ''+ lli: 'If

/

25.4 _ deep SDOF with 200 x 400

Wiremesh on 30% Porosity Perforate

as the Resistive Face Sheet

I I......... '-,'--'_ Mr,-"I

t

.@,,

1 I.... :.t,'.... _,u..'U_L.LI I i

Flow

li'7'''++---''" _''"TTTITiTII I

I _ 266.7 mm _I

v I

I---

r

Figure 5.29b. Schematic Diagram Showing the Acoustic Treatment

in the Straight Annular Part of the Fan Exhaust Duct

, 94
5.3.2 IN-SITU IMPEDANCE MEASUREMENTS

• Measurement Apparatus

The effects of grazin8 flow and sound intensity were studied in the

Grazing Flow Duct usin$ a sample of the face sheet of the treatment. The

sample was prepared using the same materials and procedures as used in the

manufacture of the treatment sections for the aft duct. The Flex Core was

removed while ensurin$ that the blockaKe of the face sheet due to the bonding

resin was preserved on the sample. A steady (dc) flow measurement on the

sample was made prior to its preparation for In-Situ measurements.

Figure 5.30 shows the test sample. The cavity consists of a thick

brass cylinder with inside diameter of 17.78 mm. The depth of the cavity

could be varied between 25.4 Ha and 45.72 m. Endevco pressure transducers

(Model 8514-10) were used to measure the acoustic signals at the face sheet

and the back wall. These transducers are of 1.587 nm diameter. The range of

these is 0-68950 pascals (0-10 psi) with a sensitivity of .004569 ± .0013

mY/pascal (31.5 ± 9.0 mY/psi). The size and the sensitivity of these

transducers made them suitable for the In-Situ measurements.

Figure 5.31 shows the data acquisition and analysis hardware used in

the measurements. The signals from the transducers were amplified by

Tektronix amplifiers (Model AM 502). These are DC-coupled differential

amplifiers with good coemon-mode rejection capabilities (rejection ratio of 50

dB DC to 50 kHz) and high gain for low voltage measurements. The amplified

signals were sampled and analyzed by the Time Data Analysis system based on a

12 bi t , 2 channel A to D converters and a DEC PDP 1135 minicomputer. The A to

D converter of the Time Data Analysis system has amplifiers and antialiasing

filters for each data channel which can be selected through software. This

ensures that spectral information is uncontaminated by higher frequency signal

and that the best use of the dynamic ranse of the A to D converter is made.

95
FACE SHEET

SEAL_

/,,,

-sJ

m

FLOW DUCT

WALL

TRANSDUCERS

Figure 5.30. Schematic Diagram of the Apparatus Used for In-Situ Impedance Measurements

in the Grazing Flow Duct

96
l.

,_J u'J

t

ORIGINAE PAGE IS

OF POOR QUALIT_ _

m

°_

_._ 0

m

o31

!

QJ

rJ3

o3

=

0

o

cn (_ Qj

• _, _.,_ _

4.)

D

u_

.-_ rO

-,4

I_ r._ -,'_

t"

E

I,=,I

!

xc

C

,w

E

,2

I1¢

m

: 97
In-Situ impedance measurements were also made on the treatment sections

prior to their installation in the vehicle. The outer treatment section was

made up from six circumferential segments and the inner section was made up

from three segments (see Figure 5.32). A single In-Situ measurement was made

in each segment. The purpose of these measurements was to establish an

averaged impedance value (without grazing flow) for the treatment to be tested

in the aft duct. The experimental arrangement used in these measurements is

shown in Figure 5.33. Broadband noise was used for these measurements. The

measurements on the inner section were done using a sampling rate of 12,800 Hz

and a bandwidth of 12.5 Hz while those on the outer section were carried out

with a sampling rate of 25,600 Hz and band width of 50 Hz.

• Impedance Data From the Treated Sections of the Aft Duct - No Flow

Resistances and reactances from the measurements on the outer treatment

section are presented in Figure 5.34. The narrowband spectra of the signals

measured at the face sheet and the back wall of the cavity at location A are

shown in Figure 5.35.

The coherence of the two signals and the phase difference between them

is shown in Figure 5.36.

The quality of the data in Figures 5.35 and 5.36 is typical of these

measurements.

With the exception of the impedance measured at cavity B, these

measurements look good. They show that the resistance varies by a small

amount between these cavities. For example at 1900 Hz, the resistance

measured at cavity A is 0.48 pc while that measured at cavity B is .67 pc.

As the inner treatment section is made up of three circumferential

seFjnents, only three In-Situ measurements were made on it. The data from

these measurements is shown in Figures 5.37. The reactances do not vary from

98
ORIGI'NAI_ PAGE _

OF POOR QUALITY

Figure 5.32. Photograph of the Inner Treatment Section of the Acoustically Treated

Segment of the Aft Duct

99
SOUND

SO_ FACE-SHE_

,._TRANSDUCER

Ill.ill/t

BACK WALL TRANSDUCER

(l.59mm DIAMETER)

Figure 5.33. Experimental Arrangement Used in the In-Situ Impedance Measurements

on the Outer and the Inner Treatment Sections of the Aft Duct

I00
s-%

i

m

>,

.v.,I

_nu

(_d) a_ue:lsT s_t

O4

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&J

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=:

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102
4.1

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q

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tt_

cm

or*

c,4

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=

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c

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=

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C

_g

_ xa

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u'3

103
_ouo_qoD

1

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(susTps_)

_ouoa_ggTa _ssq_

,,.1"

u

=

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=

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°_.,I

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= 0

,1_ I1)

=

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w

_.,_

_.,_

u_

e_

., 104
Q.

¢ 2

w

w

0

2

1

"_ 0

cO -i

-2

-3

(a) At Cavity G

2

/-

3

Frequency (kHz)

4 5

A

Q.

@

m 1

m

a

0

2

i -i0

_-2

d

(b) At Cavity H

!

f

1 2 3 4 5

Frequency (kHz)

Figure 5.37. In-Situ Measurements at Three Different Circumferential Locations

on the Inner Treatment Section

, 105
G

I

w

w

(c) At Cavity J

i

| ! I

2 3

Frequency (kHz)_

4 5

_" 0

G -I

G

-3

-4

2 3 4 5

Frequency (kHz)

Figure 5.37. In-Situ Measurements at Three Different Circumferential Locations

on the Inner Treatment Section (continued)

. 106
point to point, but the resistance at cavity c is appreciably higher than at

cavities a or b. This observation raised the question as to how

representative the measurements at a, b and c were of the impedance of the

panels on which they were located. There was a need to make several

additional measurements to get a good statistical measure of the averaged

impedance value of the treatment and the standard deviation from it. To do

this by the In-Situ method would be tedious and time consuming. Consequently

it was decided to carry out a comprehensive survey of the treatment impedance

using the Acoustic Plunker (See 5.3.3). It should be noted however that at

any point the acoustic resistance measured by the In-Situ method is virtually

constant between 1000 Hz and 2000 Hz, the range of interest in this study.

• Impedance Measurements in the Grazin_ Flow Duct (GFD)

These measurements were carried out on a laboratory sample of the

treatment. First the effects of sound intensity were measured without grazing

flow. Then the effects o_ grazins flow on the impedance were measured. It

was assumed that the correlations obtained from these measurements would

accurately represent the grazin_ flow and sound intensity effects occurrins on

the treatment in the aft duct during medal measurements.

• The E£_ects og Sound Intensity

Figure 5.38 shows the increase in acoustic resistance relative to the

value measured at 125 dB plotted against the sound pressure level at the three

frequencies of interest. At 1000 Hz, the effect of increasing SPL on the

acoustic resistance was much lower than that measured at 1500 Hz and 1900 Hz.

Negligible effects og sound intensity changes were observed on the reactance

of the sample.

The above data was obtained with discrete frequency sound field in the

duct.

107
o_

0.01

0 1_000 Hz

A 1500 Hz

n 1900 Hz

125 130 135 140 145

SOUND PRESSURE LEVEL (dB)

Figure 5.38. Increase in Acoustic Resistance With Sound Pressure Level (SPL, dB) of the

Discrete Acoustic Signal; No-Flow

, 108
• The Effects of Grazini_ Flow

A cavity depth of 4.572 cm instead of 2.54 cm was chosen for these

measurements because an error analysis (see Appendix B) had shown that the

magnitude of the error in the measured resistance would depend on the

magnitude of impedance at the frequency of interest. By increasing the cavity

depth, the magnitude of the reactance of the sample was reduced significantly

between 1000 Hz and 2000 Hz to yield more accurate measurements of the

acoustic resistance in this frequency range. It was assumed that the

reactance of the cavity did not influence the effects of grazing flow on the

lump impedance of the porous face sheet.

The effects of grazins flow on the resistance of the sample are shown

in Fisure 5.39a. As expected, the increase in the resistance with increasing

mean flow Mach number is roushly the same at the three frequencies of

interest. The data correspond to roushly 140 dB (OASPL) at the surface of the

sample.

Figure 5.39b shows the corresponding plots of reactance against the

mean flow Mach number.

Figure 5.40 shows typical narrowband spectra of the signals sensed by

the transducers at the face sheet and the back wall of the cavity at Mach 0.4.

The effect of sound intensity on the impedance in the presence of

grazing flow was found to be much less than that measured under no flow

conditions. In view of this the sound intensity effects under test conditions

in the aft duct may be neglected.

5.3.3 PLUNKER MEASUREMENTS ON THE TREATMENT SECTIONS OF THE AFT DUCT

As a considerable amount of variation in the impedance of the treatment

sections was revealed by the In-Situ measurements (as illustrated in Section

5.3.2). A much larger number of impedance measurements on these sections was

made using the Acoustic Plunker which is described below.

109
.2

0

0 1000 Hz

A 1500 Hz

- Q 1900 Hz

X'

0 .2 .4

MEAN FLOW MACH NUMBER, M

,,_- (_-_.o), sn.- 14o dS

Figure 5.39a. Increase in Acoustic Resistance With the Mean Flow Mach Number in the

Grazing Flow Duct

II0
u

.2

"I

0

0

• 1000 Hz

& 3.500 Hz

X 1900 Hz

X

i I I

.1 .2 .3 .4

MEAN FLOW MACH NUMBER, M

AxM- (xM - XM.o), sPL- 1,,o dS

Figure 5.39b. Increase in the Acoustic Reactance With the Mean Flow Mach Number in the

Grazing Flow Duct

• i!I
l&O

,.a

< I00 Z

b-#

80

.a

<

z

i_0

120

ioo .

80

• v-,,_

60

0 1 2 3

FREQUENCY (kHz)

Figure 5.40. Auto Correlation Spectra of the Face Sheet Signal A

and Backwall Signal B During an In-Situ Impedance

Measurement at M = 0.4

_12
The Acoustic Plunker [Reference 7] is a transducer developed by the

General Electric Company for the measurement of the normal acoustic impedance

of finished treatment panels. It is a portable and nondestructive device. It

has to be calibrated to measure the normal impedance of treatment panels of a

particular design. It has sufficient sensitivity to measure small variations

of the normal impedance on a given panel at a given OASPL as well as to sense

impedance changes with sound intensity.

The Plunker is schematically illustrated in Figure 5.41a. It consists

of a tube of circular cross section. The walls of the tube are acoustically

hard and smooth. A sound source is located at one end of the tube. The other

end is placed against the surface of the treatment panel such that the tube is

at right angles to the surface. The rubber flange is designed to minimize

sound leakage. The diameter of the Plunker tube is such that over the

frequency range of interest, only plane waves can propagate in it. Two

transducers located at distances xI and x2 from the treatment surface

sense the stationary sound field in the Plunker tube. The signals from the

Plunker transducers are analyzed to yield the time averaged values of the

complex.transfer function:

H21(f) = P2(f) Pl(f) / Pl(f) Pl(f) (5.2)

The apparent impedance measured by the Plunker using the above transfer

function is given by:

{Sin(kx 2) - H21(f)Sin(kxl) }

Cp(f) - -i {Cos(kx2 ) _ .21(f)Cos(_) } C5.3)

If a sample of the treatment is cut out of the panel at the location where the

plunker measurement _p(f) is made and its normal impedance C(f) is

measured in the Impedance Tube schematically illustrated in Figure 5.41b, then

the calibration of the Plunker is computed from the above data as follows:

C5.4)

_PCf) = Cp(f)

113
Flexible Flange

Surface of Treatment Panel

Figure 5.41a. Schematic Diagram of the Acoustic Plunker

Placed Against the Surface of a Flat Treatment Panel

F Acoustic Treatment Sample

- .. / -. -f_\_._--._- -..-I_ 2-"_-_ ""-'-_-

7/_-_#4 x_ x2 x3 (Tran_ducer

_j! ; 1 _ { Locations) Sound Source o

' _..-o- X

Figure 5.41b. Schematic Diagram of a Normal Impedance Tube

.. 114
The calibration of the Plunker as defined above for a given design of

treatment is needed because the apparent impedance Cp(f) measured by the

Plunker is different from the Impedance Tube data for the followin8 reasons:

lo The absorber volume behind the panel surface in the Plunker

measurement is significantly larger than in the equivalent

Impedance Tube measurement.

e Some sound leakage may occur due to imperfect seal between the

Plunker and the panel surface due to the curvature of the treatment

surface. The effect of this leakage can be calibrated out only if

the Plunker axis is always perfectly normal to the surface and that

the radius of curvature of the surface is constant everywhere.

3. There may be transmission of sound in the core of the panel at

risht ansles to the sound field in the tube due to

(a) non risid cavity vails

(b) drainase holes between cavities

(c) imperfect bonding of core to the face sheet and back wall.

Once the calibration _p(f) for a particular panel design is

obtained in the manne_ described above, the normal impedance of any other

panel of that desisn can be measured by simply measurins Cp(f) with the

Plunker and then multiplyin_ it with _p(f).

As it was not desirable to cut out a sample from the treatment sections

for the aft duct, the Plunker calibration was carried out using an In-Situ

measurement of impedance. It was established that this measurement was

similar to an Impedance Tube measurement on a laboratory sample of the

treatment. The inside diameter of the Plunker tube used in these measurements

was 17.78 mm. This size was re_arded sufficiently large to sive a measurement

of impedance on a statistically significant area of the panel. It was also

sufficiently small compared to the radius of curvature of the treatment

115
surfaces which could be regarded essentially flat over the area of the tube.

It also ensured minimum leakage of sound between the Plunker and the face

sheet due to imperfect seal caused by the curvature of the treatment surface.

Separate calibrations of the Plunker were required for the outer and the inner

sections due to significant differences in the curvatures of their surfaces

which affected the extent of leakage between the plunker and the treatment

surface.

On each of the six segments of the outer and the three segments of the

inner treatment sections, at least ten impedance measurements were obtained

with the Plunker. During each measurement the plunker was simply placed on

the location. Apart from its own weight, no additional pressure was applied.

Care was taken that the axis of the plunker was normal to the surface of the

treatment. Because of the curvature of the treatment surface the leakage of

sound was a problem. A very small tilting of the axis of the Plunker from the

normal caused significant leakage. This problem was particularly severe in

the case of the inner treatment section. This leakage of sound affected the

accuracy of the impedance data at lower frequencies (below 1500 Hz).

Figure 5.42 shows the results of the impedance survey of the outer

treatment section. It is based on sixty different measurements. The standard

deviation of the impedance data is quite small at 1500 Hz and 1900 Hz but at

1000 Hz the deviation in the dat_ is much larger due to reasons mentioned

above.

Figure 5.43 shows the results of the Plunker measurement on the inner

section. As expected, the standard deviation of the resistance values

measured on the inner section is larger than that on the outer section.

Moreover the resistance values below 1500 Hz are lower than expected due to

sound leakage.

Plunker measurements also confirmed that the In-Situ measurement on

cavity J of the inner treatment section is representative of the impedance of

116
t:

I=

t

-0.5

_e

i .........

Ol = "'= ' "'' -'''- '''''''''''''----"--'-- 3 "-- -'--"-'2 4

Fzeque=cy, k_z

0.5

...... ____ i=a_dar.= Dev£ac:Lon ........

0

-1.0_-

-1.._

-2.( 1 , , ; , 2, .... 3!

Frequency, Jd'lz

i i i i

Figure 5.42. Impedance Survey of the Outer Treated Section of the Aft Duct

Based on 60 Different Plunker Measurements Over Its Surface

117
TABLE 5.7

Average Normal Impedance of the Acoustic

Treatment Hardware

(OASPL = 140 dB)

Frequency Resistance Reactance

Hz (oc) (oc)

i000 .51 -1.82

1500 .51 -1.0

1900 .51 - .55

1.4

i 0._

1.0

0.5

0

i -0.$

-2.0

-I.$

-2.0

........... ...................

, I , ,

/

, , _ ' , , ,

2

Fv_lmcy,

i

/ •

PlrlmT. _u_

Figure 5.43. Impedance Survey of the Inner Treated Section of the Aft Duct

Based on 30 Different Measurements Over Its Surface

118
the entire panel and is midway between the In-Situ measurement results on

cavities H and J. The Plunker results show lower frequency dependence than

the In-Situ data.

In the light of the Plunker measurements the following conclusions may

be drawn.

1. The averaged impedance of the outer treatment section is

approximately equal to that of the inner section.

2. The treatment impedance is uniform over the entire surface within

the limits of manufacturing tolerances.

It was decided that the averaged acoustic impedance values obtained by

the Plunker at the frequencies of interest should be used in the

theory-experiment check. These values are presented in Table 5.7. The

In-Situ measurements had shown that the resistance of the treatment was

essentially constant between 1000 Hz and 2000 Hz. In view of this, the

resistance at 1000 Hz was assumed to be the same as that at 1500 Hz. It

should also be noted that the Plunker measurements were obtained with an OASPL

of 140 dB (approximately) at the treatment surface. As the in-duct sound

levels during the tests at Schenectady were significantly lower, appropriate

corrections to the resistance values would be required.- In addition

corrections should be made for the effects of grazing flow. The following

approach is suggested: Using the sound intensity correlation of Figure 5.38

and the average impedance at 140 dB from Table 5.7, the no-flow resistance at

130 dB is approximatey 0.44 pc. To this should be added the grazing flow

effect from Figure 5.39a. The reactance values of Table 5.7 should only be

corrected for flow effects using Figure 5.39b.

119
6.0 THEORY-EXPERIMENT cOMPARISON

The annular duct suppression prediction program based on the modal

analysis described in Section 4.1 was used to predict the suppression due to

the treatment in the Rotor-55 exhaust duct for all three test RPMs. The

theoretical predictions are compared with the measurements in this section.

The physical phenomenon of propagation in segmented ducts is discussed with

reference to this theory-experiment comparison.

6.1 INPUT TO PREDICTION PROGRAM

The exhaust duct was modeled as a three segment straight annular duct

with the treated sesment placed between two hardwall se_ents. The lengths of

the hardwall segments were taken to be the distances of the mode measurement

planes from the treated sesment. The effective length of the treated segment

was found to be 0.23 meters (L/H = 1.8) after accounting for the hole blockage

at segment ends and treatment panel interfaces.

The principal input parameters for the program are the duct geometry,

the mean flow conditions, the acoustic frequency, the inner and outer wall

impedances (or admittances) for all segments, the spinning mode order, the

radial mode distribution of the acoustic source and the reflection

characteristics of the duct termination. The mode coefficients of the forward

traveling modes measured at the upstream plane (or the "source" plane) and the

mode coefficients of the backward traveling modes measured at the downstream

plane (or the "termination" plane) were used to specify {Qs } and

{QT } respectively. As mentioned in Section 4.1.2 this required that

[R S] = 0 = [_]. For the blade passage frequencies corresponding to the

rotor RPMs in the test, a maximum of two radial modes (i.e., n=0 and n=l) were

expected to be cut-on in the hardwall segment of the duct. While utilizing

the prediction program, however, a minimum of four radial modes (i.e., n=O, 1,

2, 3) were considered to participate in the transmission and redistribution of

acoustic energy in the duct. The acoustic impedance of the treatment in the

duct was initially assumed to be equal to the average normal impedance

measured in the absence of flow using the acoustic Plunker (Section 5.3.3).

121
With these input parameters the prosram calculates the eigenvalues, the

axial propagation constants (for both forward and backward propagating modes),

the uniform section transmission matrices for each duct segment, and the

reflection and transmission matrices of the segment interfaces. The program

then sets up the stacked system matrix equation and solves it to obtain the

forward and backward complex mode coefficients and modal energy fluxes at each

plane. The net energy flux at each plane and the overall PWL suppression are

then calculated.

6.2 IN-DUCT SUPPRESSIONS

Mode coefficients obtained from the in-duct measurements and based on

the assumption of uniform mean flow in the duct were first used in the

predictions.

Table 6.1 lists the measured and predicted in-duct suppressions for the

fan-stator interaction modes at the three blade passage frequencies

corresponding to the test conditions. These suppression values represent the

difference in the forward energy fluxes between the downstream and upstream

measurement planes. The effect of reflections from the nozzle is taken into

account by specifying the backward traveling mode coefficients at the

downstream plane. The predicted suppressions can be seen to be in good

agreement with the measured suppressions for all three frequencies and for

both spinning mode orders. The largest difference between the predicted and

measured suppressions is 2.7 dB which is not large compared to the measured

suppression of 16.8 dB at that condition. Notice that six radial modes were

used in the prediction for m = 7 case (for reasons discussed later in this

section) while only 4 radial modes were used for m = 0-1.

The predicted and measured mode coefficient distributions of forward

propagating acoustic energy at the downstream plane are compared in Figures

6.1 through 6.4 for the four cases listed in Table 6.1. The magnitude as well

as the phase of the measured mode coefficients can be seen to be in good

122
Table 6.1. Comparison of Measured Suppressions With Theoretical Predictions

Uniform Flow

NO. OF

FREQUENCY SPINNING RADIAL TREATMENT

(Hz) MODE (m) mODES IMPEDANCE

IN-DUCT SUPPRESSION, &dBf

MEASURED PREDICTED

1,000 -1 4 0.51-1.82i 2.18 1.40

1,500 -I 4 0.51-1.00i 3.72 4.29

1,900 -1 4 0.51-0.55i 16.83 14.10

1,900 7 6 0.51-0.55i 22.587 24..32

123
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, 127
asreement with the predicted values for the cut-on modes. For m = -1 at

t000 Hz and for m = 7 at 1900 Hz the higher order radial modes (n > 0) are

theoretically cut-off in the hardwall duct. This causes the predicted

amplitudes of these modes at the downstream location to be several dB lower

than the predicted amplitude of the first radial mode. The measurements also

show the amplitudes of the higher order modes to be at least 10 dB lower than

the amplitude of the cut-on mode. The difference in the measured and

predicted values of the mode coefficients for cut-off modes will therefore

have only a small effect on the total suppression in the duct.

FiKures 6.5 throush 6.8 compare the predicted values of the mode

coefficients for the backward travelins modes at the upstream plane with the

correspondins measured values. The comparison for the backward travelin_

modes is not as good as that for the forward travelins modes. For m = -1 at

1900 Hz (Figure 6.7), the predicted mode amplitude for the second radial mode

(n=l) is much hisher than the amplitude for the first radial mode (n=O). The

measurements, however, show the n=0 mode to be slightly hisher than the n=l

mode.

In utilizing the suppression prediction prosram, it is essential to

check that the total acoustic energy is conserved across segment interfaces.

The modal analysis requires the acoustic pressure and velocity at all radial

locations to be continuous across segment interfaces. However, the pressures

(and consequently the velocities) on the two sides of the interface are

represented as expansions in sets of eisenfunctions which can be completely

different on either side. If the set of eigenfunctions used is not

mathematically complete, then the expansions, the pressure matchinK, and

consequently the acoustic enersy matchinK, can be in error. For the first

three conditions listed in Table 6.1 the predicted acoustic energy across the

segment interfaces in the rotor 55 duct was conserved within 0.1 dB. For the

m = 7 mode at 1900 Hz, however, an energy mismatch of 2.5 dB was observed at

the downstream sesment interface. This mismatch was reduced to 1.2 dB and 1.0

dB by including 6 and 8 radial modes respectively in the analysis. Assessment

of the effects of including radial modes of order higher than this was

prevented by limitations in the size of the computer prosram.

128
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Figure 6.9 shows the predicted mode coefficients (for both forwacd and

backward propagating modes) at the end planes of the hardwall and treated

segments for m = -1 modes at 1000 Hz. At this frequency only the lowest order

radial mode (n=0) is cut-on in the hardwall duct. Consequently the forward

propagating higher order modes (n > 0) at the "source" plane i.e., plane 1,

are suppressed to very low values at plane 2 just upstream of the treated

segment. Downstream of the hardwall/treatment interface i.e., at plane 3, the

mode coefficient distributions shown are required to match the energy at plane

2. Notice that a combination of forward modes n = 0, 1, 2 and backward modes

n = 0, 1 at plane 3 in the treated segment is able to match the energy carried

by the combination of forward mode n • 0 and backward modes n = 0 and n = 2 at

plane 2 in the hardwall segment. Thus a redistribution of mode coefficients

takes place at the segment interface. This is due to the fact that the radial

profiles of acoustic pressure and velocity at the interface are expressed as a

sunuuation of hardwall eigenfunctions at plane 2 and as a summation of softwall

eigenfunctions at plane 3. A similar mode coefficient redistribution takes

place at the downstream treatment/hardwall interface.

For m = 7 at 1900 Hz most of the incident energy is contained in only

one radial mode (n = 0) that is close to cut-off (see Figure 6.10). This mode

has a cut-off ratio equal to 1.1. Upon incidence to the upstream

hardwall/treatment interface this mode is reflected into several higher order

backward traveling modes. The backward modes at plane 2 in the hardwall can

be identified as due to reflection of forward modes at the same plane and not

as redistribution of backward modes at plane 3 in the treated segment because

the latter have near zero amplitudes as shown. Thus the acoustic energy

carried by the combination of the n • 0 forward mode and the backward modes of

several higher orders in the hardwall segment must be matched by distribution

of energy mostly in the forward modes in the treated segment. The backward

modes at plane 3 are small in amplitude because of the high suppression in the

treatment. At the downstream tceatment/hardwall interface the reflection of

the n = 0 forward mode in the treated segment into several higher order

backward tcaveling modes can be observed.

133
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135
It is felt that this combination of high reflection and suppression of

the near cut-off mode causes the problem of energy mismatch at segment

interfaces. The mismatch is reduced by including higher order radial modes in

the analysis. Since these modes are "cut-off" they are highly suppressed even

in the hardwall sesments of the duct. Their contribution to the total

acoustic energy at the "source" and "termination" planes is therefore

negligible and the in-duct suppression based on the forward energies at these

planes does not change much.

6.3 EFFECTS OF SHEARED FLOW

As mentioned in Section 5.2.3, the mean flow in the Rotor 55 exhaust

duct was far from uniform. In addition to the radial variation of velocity

due to the thick boundary layers at the duct walls, a periodic circumferential

variation of the mean velocity due to the persistent stator wakes was also

observed. It is not possible to account for the circumferential variations in

the prediction of suppression without considerably modifyin8 the analysis. An

attempt was, however, made to account for the thick boundary layers by using

the analysis described earlier for thin boundary layers. Sheared flow

eigenvalues were determined by using the measured velocity profile in the

integration of the governin8 differential equation across the duct annulus.

The eigenvalues and the acoustic pressure mode shapes obtained during this

integration process were used to obtain the sheared flow mode coefficients

from the mode probe measurements. These mode coefficients were used in the

suppression prediction program to obtain estimates of the suppressions due to

the treatment in the duct.

Table 6.2 compares the measured in-duct suppressions with the

suppressions predicted usin8 the sheared flow mode coefficients for the same

conditions as in Table 6.1. For the m = -1 mode at 1500 and 1900 Hz the two

suppression values can be seen to be in good agreement. Furthermore, the

measured values differ from the corresponding values in Table 6.1 by at most

3.2 dB. For the spinning mode m = 7 at 1900 Hz the in-duct measured

suppression based on the sheared flow mode coefficients is very similar to the

measured suppression based on the uniform flow mode coefficients. The

predicted and the measured suppressions based on the sheared flow mode

136
Table 6.2. Comparison of Measured and Predicted In-Duct Suppressions

Based on Sheared Flow Mode Coefficients

FREQUENCY SPI_ING

(Mz) MODE

1,000 -I

1,500 -1

1,900 -1

1,900 7

NO. OF

RADIAL TREATMENT

MODES IMPEDANCE

4 0.51-1.82i

4 0.51-1.00i

4 0.51-0.55i

6 0.51-0.55i

IN-DUCT SUPPRESSION, &dBf

MEASURED PREDICTED

5.36 1.60

5.32 5.28

18.96 17.10

22.29 28.08

137
coefficients for this case, however, do not compare as well as those based on

the uniform flow mode coefficients. This is due to the large mismatch of

energy in the prediction routine at segment interfaces for the sheared flow

case. In fact even for the cases where the measurement and prediction agree

(i.e., m = -1 mode at 1500 Hz and 1900 Hz), the energy conservation at the

segment interfaces is off by up to 0.8 dB. It is believed that this is due to

the assumption in the analysis that the complex acoustic pressure profile in

the duct in the presence of non-uniform flow can be expanded in terms of the

uniform flow eigenfunctions. This assumption is valid only for thin boundary

layers o.A _.. Introduce • °--o_ I_.. err_,e ;, highly sheared flows _Jrthe_emore

the expression for acoustic intensity (equation 4.39) is valid only for

uniform flow.

In an attempt to improve the energy conservation at segment interfaces

in the presence of sheared flow, the program was modified so that the acoustic

pressure mode shapes generated numerlcally during the determination of sheared

flow eigenvalues are used in the modal analysis instead of the uniform flow

eigenfunctions. This modification did not introduce a consistent improvement

in the energy conservation at sesment interfaces. It was, therefore, not

pursued further.

6.4 SENSITIVITY OF SUPPRESSION TO TREATMENT IMPEDANCE

All the predicted suppressions presented so far are based on the

average normal acoustic impedance of the treatment panels measured at 140 dB

in the absence of flow (see Section 5.3.3). Treatment impedance is known to

change due to the presence of flow (and boundary layer) over the treatment and

also with changes in sound pressure level in the duct (see References 19 and

4). Results of the experimental attempts to assess the flow and SPL effect on

the acoustic impedance of the treatment used in Rotor-55 exhaust duct are

described in Section 5.3.2. The average SPL in the Rotor-55 exhaust duct was

in the range 125-135 dB. This will cause the resistance of the treatment to

be lowered relative to the value at 140 dB. However, the grazing flow over

the treatment will cause the resistance to increase. The treatment reactance

138
was observed to become less negative due to the presence of flow. I_ was

estimated that the treatment resistance and reactance under the test

conditions will be within 0.1 pc of the average normal impedance values.

Table 6.3 (for uniform flow condition assumptions) shows that by varying the

treatment impedance within this limit the predicted in-duct suppression can be

brought into excellent agreement with the measured suppression.

The sensitivity of the predicted suppression to the treatment impedance

was established by changing the resistance and reactance over a range of

values close to the averaged measured value. The results for m = -I mode in

the presence of sheared flow are plotted in Figures 6.11, 6.12 and 6.13 for f

= 1000, 1500 and 1900 Hz respectively. At 1000 Hz the suppression is almost

equally sensitive to changes in resistance and reactance, varying by about 1

dB over 0.4 pc. At 1900 Hz the predicted suppression is extremely sensitive

to the reactance and even a change of 0.I pc in reactance can change the

suppression by 4 dB. The sensitivity to resistance is not that high, but

greater than the sensitivity at 1000 and 1500 Hz. In Figure 6.13 there is

also an indication of a local peak in the suppression curve at R = 0.45 pc.

6.5 FAR FIELD SUPPRESSION

The acoustic field radiated from the exhaust duct was measured in the

far field at all three rotor speeds for the two cases: a) all hardwall

segments in the duct and b) one treatment segment in the duct. By integrating

the SPL directivities (at the blade passage frequency) for the treated duct

and the hardwall duct and taking the difference we can obtain a power level

(PWL) suppression due to the treatment. Table 6.4 lists the power level

suppressions and the unifo_u-flow in-duct suppressions for all three

frequencies. At 1000 Hz and 1500 Hz, m = -1 is the only rotor-stator

interaction mode generated in the duct. At these frequencies significant

energy is carried by the (-1,0) mode. This mode is well cut-on at both

frequencies and will have a peak lobe of radiation in a direction close to the

duct axis. The jet flow exhausting from the duct termination (i.e., nozzle),

however, will introduce refraction effects and cause the peak lobe to move

139
Table 6.3. Sensitivity of Predicted Suppressions To Treatment Impedance

Uniform Flow

NO. OF IN-DUCT SUPPRESSION, AdBf

FREQUENCY SPINNING RADIAL TREATMENT

(Hz) MODE (m) MODES IMPEDANCE MEASURED PREDICTED

1,000 -1 4

1,500 -1 4

1,900 -I 4

0.51-1.82i 2.18 1.40

0.51-1.65i 2.18 1.77

0.60-1.82i 2.18 1.60

0.51-I.00i 3.72 4.29

0.40-I.00i 3.72 3.73

0.51-0.55i 16.83 14.10

0.51-0.45i 16.83 16.71

0.60-0.55i 16.83 14.54

140
I

__. -_

O

Q

I

U

=

U

U

4.l

E

0

U

=:C

0

U). ..

0 =

•,-I I

•,_ II

(n

rz.,

141
I

"!. ?

xW

I0I0 I

AA

II II X

Z

I,IJ -- u

wa._

n_ a.

A

n_ Z

I-- i

t_

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n_

II

e_

,.-¢I

II

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_J OJ M .,_

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_L_3 0

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. 142
A II

_. Q. m

II II "D

I- =x_

x

-

_" ! ! ! ! ! !

0

z

/.0

0

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¢d

E

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0

0 C

I>

_ E

=

143
Table 6.4. Comparison of In-Duct Suppressions With PWL

Suppressions From Far-Field Data

UniformFlow

FREQUENCY SPINNING IN-DUCT

(Hz) MODE PREDICTED

SUPPRESSION I

IN-DUCT

MEASURED

&dB

&PWL FROM

FAR-FIELD DATA

1,000 -1 1.40 2.18 1.2

1,500 -1 4.29 3.72 4.2

1,900 -1 14.10 16.83

1,900 7 24.32 22.58 10.3

144
away from the duct axis. Far-field measurements in the sector covering 20"

throush 110" should be able to capture the peak lobe of radiation and PWL

suppressions based on far-field radiation pattern should be representative of

the in-duct suppression due to the treatment (assuming the reflection at the

nozzle to be of the same order for both hardwall and treatment cases). In

Table 6.4, the PWL suppression (based on far-field data) can be seen to be of

the same order as the in-duct suppression at 1000 and 1500 Hz.

At 1900 Hz, the interaction modes m = 0-I and m = 7 coexist, with the

latter carrying a greater part of the energy. A good percent of this energy

is, however, reflected back into the duct at the nozzle termination.

Measurements in the downstream hardwall section showed the backward energy in

the mr7 modes to be only 4 dB below the forward energy. For the m = -1 modes

this difference was found to be nearly 10 dB. The (7,0) mode has a cut-off

ratio near 1.1 and a peak lobe of radiation almost normal to the duct axis.

The refractive effects of the jet will move the peak lobe to even higher

angles. This can cause the far-field measurement field (20" thru 110") to

miss part of the radiated %nersy. The m = -1 mode will, as in the case of

lower frequencies, radiate at a smaller angle to the duct axis. The

multi-modal _ suppression (from far-field data) is 10.3 dB compared to the

measured in-duct suppression values of 16.8 and 22.5 dB for the m = -1 and m =

7 modes respectively. Further examination of the causes for the variations in

these numbers is required.

6.6 NOZZLE REFLECTION CHARACTERISTICS

In Section 4.1.3 a method for measuring the reflection matrix of the

nozzle [_] was outlined. It was pointed out that all the elements of the

reflection matrix (includin8 the off-diagonal terms representin8 the mode

scatterins) can he determined by measuring the forward and backward mode

distributions just upstream of the nozzle for different distributions of the

incident modes. For the rotor-55 exhaust duct, in-duct measurements were made

with and without the treatment segments in the ducts. The mode distribution

incident to the nozzle for the all hardwall segment configuration is different

145
from the mode distribution for the one treated segment configuration. Mode

coefficients measured in these two cases can be used to solve equation 4.41

and determine four elements (namely R00, R01 R10 and Rll) for the

nozzle reflection matrix. The first subscript of R represents the radial

mode order of the reflected mode and the second subscript refers to the

radial mode order of the incident mode.

Table 6.5 lists the nozzle reflection matrix elements based on the

measured mode coefficients in the rotor-55 exhaust duct. No clear trend is

evident from these numbers. For m - 7 modes at 1900 Hz, the reflection matrix

element R01 can be seen to be very high. This points to the strong n = 0

backward mode near the nozzle for these conditions (Figure 6.10).

Theoretical prediction of the suppression due to treatment in the

exhaust duct required the inclusion of h£sher order radial modes (at least

four) in the analysis. This resulted in a nozzle reflection matrix with at

least 16 elements. The larger matrix cannot be calculated from the available

measurements. The nozzle reflection effects were therefore taken into account

by setting [R T] - 0 and specifying the backward mode coefficient vector for

the "termination" vector __{QT] .

6.7 NODE SHAPES AND EQUIVALENT IMPEDANCE

The predicted in-duct suppressions using the measured mode coefficients

based on sheared flow eisenvalues and eisenfunctions were compared with the

measured suppressions in Section 6.3. The calculations on which these

predictions were based showed discontinuity of total acoustic energy across

sezment interfaces. It is believed that this is due to the assumption in the

analysis that the complex acoustic pressure profile in the duct in presence of

non-uniform flow can be expanded in terms of the uniform flow eigenfunctions.

Figure 6.14 shows a comparison of the acoustic pressure profile in the

presence of sheared flow in the hardwa11 segment of the duct with the uniform

flow eigenfunction shape based on the sheared flow eigenvalue. This and the

146
Table 6.5. Nozzle Reflection Coefficients (From Hardwall and

Treatment Case Data)

Freq.

_Hz)

1000

1500

1900

1900

Spinning

Mode

Order

-I

-1

-I

7

RT(0,O)

-0.635+0.583i

-0.900-0.131i

0.544+0.491i

0.344+0.282i

Reflection Matrix Elements

RT(0,1)

0.958-0.111i

0.544+0.525i

-0.394+0.308i

-1.040-8.863i

RT(1,0)

-0.021+0.049i

-0.278+0.469i

-0.628-0.151i

-0.008-0.002i

RT(I,_)

-0.723+0.535i

0.135+0.319i

0.185+0.003i

0.379+0.599i

147
Figure 6.14.

.40| - f - 1900 HZ /

m-7

. - 0 (FOP,NARD)

c_0.370.02

.35 HAROk_L

.30

.25

.2O

.15

.10

.0E

Flow Eigenfunction

/ Q Acoustic Pressure Profile

/ From Numerical Integration

"Actual"

.00 I ! , ! I

.50 .60 .70 .80 .90 1.0

r/r 2

Comparison of the Uniform Flow Eigenfunction With the Actual

Acoustic Mode Shape in Hardwall Duct - 1900 Hz, m - 7, n - O,

Mo - .37, 6* - 0.02, Forward Propagation

148
following five figures (i.e., Figures 6.14 through 6.19) are for the m = 7

modes at 1900 Hz. The flow through the duct is assumed to have boundary

layers with 2 percent thickness and one-seventh power law velocity profile.

The uniform flow eigenfunction for the hardwall duct can be seen to be in

excellent agreement with the "actual" pressure profile (obtained by numerical

integration).

For the lowest order mode (n=O) propasatins backward in the hardwall

segment (see Figure 6.15), the uniform flow eigenfunction accurately

represents the complex pressure profile in a region of the duct annulus away

from the walls. Near the inner wall it overpredicts the actual acoustic

pressure while at the outer wall it underpredicts. The differences are,

however, small.

In contrast to the hardwall segment cases, the uniform flow

eigenfunctions in a treated segment are considerably different from the

"actual" acoustic pressure profiles for both forward and backward propagating

modes (see Figures 6.16 and 6.17). For the forward mode there is good

similarity between the two pressure profiles in the outer half portion of the

duct annulus. But near the inner wall even the shapes of the two pressure

profiles are different.

These figures show that the uniform flow eigenfunctions can be used to

represent the acoustic pressure profile in the hardwall segment of a duct in

the presence of thin boundary layers at the walls. In treated segments with

sheared flow, however, the uniform flow eigenfunction representation of the

acoustic pressure profile is not accurate.

In an attempt to improve the representation of acoustic pressure

profile (in ducts carrying sheared flow) by uniform flow eigenfunctions the

concept of treating the boundary layer as equlvalent wall impedance was

studied. As discussed in Section 4.1.1 the radial dependence of acoustic

pressure (pr) in an annular duct carrying uniform flow is

Pr (r) = Jm(krr) + CmmY (krr) (6 .I)

149
.41

.35

.3O

f = 1900 HZ

m=7

n - 0 (BAC ARD)

o_0.370.02

HAP,D_.L

°

Figure 6.15.

.o!

.0(

.50

---- Uniform Flow Eigenfunction

_) Acoustic Pressure Profile

From Numerical Integration

"Actual"

I 1 l t I

.60 .70 .80 .90 1.0

rlr2

Comparison of the Uniform Flow Elgenfunctlon With the Actual

Acoustic Mode Shape in Hardwall Duct - f - 1900 Hz, m - F, n -

O, Mo - .37, 6* - .02, Backward Propagation

150
1.0

.9O

.80

f • 1900 HZ

m-7

n " 0 (FORWARD)

t4- 0.37

0.02 0.51 - 0.55t

(D

.---- Uniform Flow Eigenfunction

O Acoustic Pressure Profile

From Numerical Integration

"Actual"

Figure 6.16.

.00

.50

(D

•60 .70 .80 .90 1.0

rlr 2

Comparison of the Uniform Flow Elgenfunctlon Wlth the Actual

Acoustic Mode Shape in the Treated Duct - 1900 Hz, m = 7, n - O,

Mo = .37, 6* - .02, _ - (.51 - .551), Forward Propagation

151
1.0 _

.9O

.8O

aA

.IU

.6O

.5O

f - 1900 Hz

m-7

a - 0 (SAC_,RO)

i'! " 0.37

002.51 - 0.55t

0

0 0

0

.==_ Uniform Flow Eigenfunction

(_ Acoustic Pressure Profile

From Numerical Integration

"Actual"

.00 , , , t ,

.50 .60 o70 .80 .90 1.0

Figure 6.17.

r/r 2

d

Comparison of the Uniform Flow Elgenfunction Wlth the Actual

Acoustic Mode Shape in the Treated Duct - 1900 Hz, m - 7, n - 0,

Mo - .37, 6* - 0.02, ¢ - (.51 - .55i), Backward Propagation

152
.90

.80

.7O

.60

.50

Pr I

.40

.3O

.20

.I0

.00

Figure 6.18.

f • 1900 HZ

-- Eigenfunction Based on

Equivalent Impedance

® Acoustic Pressure From

Numerical Integration

"Actual"

I I I I I

.50 .60 .70 .80 .90 1.0

r/r 2

Comparison of the Uniform Flow Eigenfunction Based on the

Equivalent Impedance With Actual Mode Shape in the Treated Duct

- f - 1900 Hz, m - 7, n - O, M = .37, 6* = .02, NEX = 7, _

(.51 - .55i), Forward Propagation

153
.80

f • 1900 HZ

m • 7

14" 0.37

&_- o.o2

NEX 7 .

"_n i 0.51 0.551

o (aACm,UU_,D)

.70

.6O

.5O

.4O

.2O

O

/

° /

I

I

/

I

I

/

/

G

/

/

.00

.50

G

J

J

J

,%.

%

%

%

Q

\

%

---- Eigenfunction Based on

Equivalent Impedance

® Acoustic Pressure From

Numerical Integration

"Actual"

\

%

! a m I I

.60 .70 .80 .90 1.0

r/r 2

Figure 6.19. Comparison of the Uniform Flow Eigenfunction Based on the

Equivalent Impedance With the Actual Mode Shape in the Treated

Duct - 1900 Hz, m - 7, n - O, M - .37, 6* - .02, NEX = 7,

- (.51 - .55i), Backward Propagation

154
and satisfies the admittance boundary conditions at the walls

aP__E_r

ar

fir I

= ik_ (1-gM)2pr

r=r I

(6.2)

and

ar

r=r 2

= ik_ (1-.M)2pr

r=r 2

(6.3)

These equations can be combined (by eliminatinE C ) and rearranged in the

m

form.

2

.,t,..(13w.i) + A2(13V..H)+ a3 :. 0 (6.4)

where H is the duct heiEht and A 1, A2 and A 3 are functions of the

eiEenvalue, axial propasation constant, duct Eeometry and flow Mach number.

The eigenvalues calculated for the boundary layer cases can be used to

evaluate the coefficient A1, A2 and A3 and equation (6.4) can then be

solved to obtain the equivalent uniform flow admittance, i.e.,

-a2 +_

B = (6.5)

eq 2_kH

This value of the equivalent uniform flow admittance can then be used with the

uniform flow eiEenvalues correspondinE to it (or equivalently with the sheared

flow eiEenvalues correspondinE to the actual wall admittance) and

C s

m

(m - iBeqkr2A) Jm(krr2) - (krr2)Jm+l(krr2)

(m - i_eqkr2A) Ym(krr2) - (krr2)Ym+l(krr2)

(6.6)

2

with A = (1-xM)

in equation (6.1) to obtain the radial acoustic pressure profile.

155
Figure 6.18 and 6.19 show comparisons of the actual mode shapes with

the uniform flow eigenfunctions based on the calculated equivalent impedance.

For the forward mode in the treated segment the uniform flow eigenfunction

based on the equivalent impedance is in good agreement with the acoustic

pressure profile. This is in contrast to the comparison of Figure 6.16. For

the backward mode, however, the use of equivalent impedance does not improve

the representation of the acoustic pressure profile by the uniform flow

eigenfunction. Further investigation is needed to seek ways of improvin8 the

eigenfunction representation.

6.8 EFFECT OF AFT-DUCT TREATMENT ON FAN NOISE GENERATION

Recent analysis [Reference 20] on the generation of fan tones has

suggested that the fan noise characteristics (both mode distribution and

acoustic power) could change when the wall impedance of any section of the

duct (at or away from the fan plane) is changed. In the present study the

source characteristics were determined from measurements in the duct. For the

prediction of the in-duct suppression the source characteristics obtained from

measurements with the treatment sections in place were used. The effects of

the duct treatment on the fan noise characteristics are thus taken into

account in the predicted suppressions.

By comparing the mode coefficients at the upstream measurement plane

for the cases of the hardwall duct and the duct with the treatment sections,

we should be able to assess the effect of the duct treatment on the fan noise

generation. The measured mode distributions of the fan noise in the hardwall

duct and in the duct with a treatment section (downstream of the fan plane)

are shown in Figures 6.20 thrugh 6.23 for the four test conditions of Table

6.1. The mode coefficients of the lowest order radial mode (n=O) for the

m = -1 modes (Figures 6.20, 6.21 and 6.22) in the case of the hardwall duct

differ by at most 2 dB from the mode coefficients in the presence of the

treatment in the duct. This suggests that the treatment in the exhaust duct

of the fan does not have a significant effect on the fan noise generation.

For the (7,0) mode (Figure 6.23) the difference in mode coefficients is of the

156
÷

Amn

(riB)

30-

20

10

m

\\

\\

i\\

n

f : I000 Hz.

m = -I

Mo-- 0.21

UNIFORM FLOW

Duct with one treatment segment

C= 0.51 - 1.82i

._Hardwal] Duct

4oo'

3oo'

Amn

PHASE

zoo'

1od

I

I

\\I

\\1

0 2

RADIAL MODE ORDER

1

Figure 6.20. Complex Mode Amplitudes and Phase Values at Upstream Plane -

i000 Hz, m - -i, M - .21, Uniform Flow, Forward Propagation

].57
÷

Amn

(dB)

40

30

20

10-

0

i \

, _,,,_

_-,,_

I N "_M

"'4

I "'N

%. "4

% "4,

%'4J

\\

',.%.

-I!

I

!

k\'_

k\N

L\M

f - 1500 Hz.

m ---I

Mo-- 0.32

UNIFORM FLOW

Duct with one treatment seoment

C = 0.51 - 1.00i

_Hardwall Duct

400"

300 °

Amn

PHASE

200"

100"

0 ° n

,,\

,.\

,.\

0 1 2

RADIAL HODE ORDER

3

Figure 6.21. Complex Mode Amplitudes and Phase Values at the Upstream Plane -

1500 Hz, m - -i, M - .32, Uniform Flow, Forward Propagation

158
30

20-

+

Amn

(dB)

I0

0

,

T_

I

% %

f = 1900 Hz.

m = -I

M== 0.40

UNIFORM FLOW

Duct with one

treatment segment

C = 0.51 - 0.55i

_Hardwa]l Duct

3oo -

200"-

Amn

PHASE o

100 -

o' n

Figure 6.22. Complex Mode Amplitudes and Phase Values at the Upstream Plane -

1900 Hz, m - -1, M - 0.4, Uniform Flow, Forward Propagation

159
40

I

(dB)!

I

20

1C-

\

\

\

\

!

\\1

\-,q

.--- \",1

\",1

\"J F__ _'_1

, \\' n

f = 1900 Hz.

m=7

Mo= 0.40

UNIFORM FLOW

Duct with one

treatment segment

C " 0.51 - 0.55i

_Hardwall Duct

400'

300"-

Amn

PHASE'

200%

t

ffl

1 2

RADIAL MODE ORDER

n

Figure 6.23. Complex Mode Amplitude and Phase Values at the Upstream Plane -

1900 Hz, m = 7, M - 0.4, Uniform Flow, Forward Propagation

160
order of $ dB. The differences in mode coefficients for the (7,1) and 7,2)

modes are also of the same order thus indicating a change in the ambient or

fan operating conditions.

It should be mentioned that the data for the hardwall duct and the duct

with the treatment section were obtained on different days and the operating

conditions were not identical (except for the fan rpm). The comparisons of

the mode coefficients in Figures 6.20 through 6.23 are thus appropriate for

qualitative analysis only. Notice that the treatment in the duct was located

a few duct heights downstream of the fan OGV plane and therefore may not have

a significant effect on the fan noise generation.

161
7.0 CONCLUDING REMARKS

Based on the theory-data comparisons presented above it can be

concluded that the modal analysis for sound propagation in sesmented annular

ducts can be used to predict the in-duct suppression due to treatment in the

exhaust duct of a turbomachine. The modal distribution of the acoustic source

and the reflection characteristics of the duct termination are required for

the prediction. These can be obtained from measurements (at least in a

laboratory duct) or from analytical methods.

The predicted values of the in-duct suppressions based on measured

source characteristics and treatment acoustic impedance were found to be in

sood agreement with the measured suppressions. This is true in spite of the

presence of thick boundary layers and circumferential flow variation in the

test duct, two effects not fully accounted for in the theory. The presence of

thick boundary layers did, however, cause numerical problems in the

calculations manifested as enerKy mismatches at duct segment interfaces. This

problem will require further investisation. The predicted suppression was

found to be quite sensitive to treatment impedance indicating the importance

of accurately determining this parameter.

The analysis should include at least two radial modes above those

theoretically "cut-on" in the hardwall seKment of the duct. This permits a

valid expansion of the acoustic pressure in terms of the mode eigenfunctions

and maintains conservation of acoustic enerKy across segment interfaces. More

modes may be required when a significant amount of energy is carried by a mode

that is close to cut-off.

163
REFER_CES

1. Zorumski, W., "Acoustic Theory of Axisymmetric Multisectional Ducts,"

NASA TR R-419, May 1974.

. Kraft, R.E., "Theory and Measurement of Acoustic Wave Propagation in

Multi-Segmented Rectangular Flow Ducts," Ph.D Dissertation, University of

Cincinnati, June 1976.

. Joshi, M.C., Kraft, R.E., and Son, S.Y., "Analysis of Sound Propagation

in Annular Ducts With Segmented Treatment and Sheared Flow," AIAA Paper

80-0123, January 1982.

4,

,

Dean, D.P., "An In-Situ Method of Wall Impedance Measurement in Flow

Ducts, Journal of Sound and Vibration, Vol. 34(1), pp. 97-130, 1974.

Kooi, J.W. and Satin, S.L., "An Experimental Study of the Acoustic

Impedance of Helmholtz Resonator Arrays Under a Turbulent Boundary

Layer," AIAAPaper 81-1998, October 1981.

o Zandbergen, T., "On the Practical Use of Three-Microphone Technique for

In-Situ Acoustic Impedance Measurements on Double Layer Flow Duct

Liners", AIAAPaper 81-2000, October 1981.

7. Syed, Asif A, "Acoustic Impedance Measurement, Patent Application, No.

499, 641 filed on May 31, 1983.

8. Joshi, M.C. and Kraft, R.E., "Turbofan Aft Duct Suppressor Study -

Program Listing and User's Guide."

o Fiske, G.H., Syed, A.A., Motsinger, R.E., Joshi, M.C., Kraft, R.E.,

"Contractors Data Report of Mode Probe Signal Data', NASA Contract No.

NAS3-22766.

165
10. Kraft, R.E., and Wells, W.R., "Adjointness Properties of Differential

Systems with Eigenvalue Dependent Boundary Conditions With Application to

Flow Duct Acoustics," J. Acoustical Society of America, Vol. 61, No. 4,

April 1977.

11.

12.

13.

14.

Cantrell, R.H. and Mart, R.W., "Interaction Between Sound and Flow in

Acoustic Cavities: Mass, Momentum and Enersy Considerations," J.

Acoustical Society of America, Vol. 36, No. 4, 1964, pp. 694-706.

ZandberKen, T., Losan, J.N., Zeemans, H.J., and Sarin, S.L., "In-Flight

Acoustic Measurements in the EnKine Intake of a Yokker F28 Aircraft",

AIAA Paper No. 83-0677, April 1983.

Moore, C.J., "In-Duct Investigation of Subsonic Fan 'Rotor Alone' Noise,"

Journal of Acoustical Society of America, Vol. 51, No. 5 (Part 1), 1972.

Kline, S.J. and McClintock, "The Description of Uncertainties in Sinsle

Sample Experiments," Mechanical EnaineerinR, January, 1958.

15. Moore, C.J., "'Measurements of Radial and Circumferential Modes in Annular

and Circular Fan Ducts," Journal of Sound and Vibration, Vol. 62(2), pp.

235-256, 1979.

16. Morse, M.M. and Ingard, K.U., Theoretical Acoustics, McGraw-Hill, New

York, 1968.

17.

18.

19.

Taylor, K., "A Transformation of the Acoustic Equation With Implications

for Wind-Tunnel and Low-Speed Flight Tests", Proceedings of the Royal

Society, 363, 271-281, 1978.

Tyler, 3.H. and Sofrin, T.G., "Axial Flow Compressor Noise Studies," SA__EE

Trans., 70, pp. 309-332, 1962.

Hersh, A.S. and Walker, B., "The Acoustic Behavior of Helmholtz

Resonators Exposed to High Speed Grazing Flows," AIAA Paper 76-563, July

1976.

166
20. Nunsur, P. and Gliebe, P.R., "Acoustic Analysis of Fan Tone Generation

and Propagation", AIAA Paper 83-0769, AIAA 8th Aeroacoustics Conference,

Atlanta, CA, April 1983.

21. Fiske, G.H., Syed, A.A., Notsinger, R.E., Joshi, N.C. and Kraft, R.E.,

"Measurement of Acoustic Modes and Wall Impedance in a Turbofan Exhaust

Duct", AIAA Paper No. 83-0733, April 1983.

22. Joshi, N.C., Kraft, R.E., Fiske, G.H., Syed, A.A. and Notsinger, R.E.,

"Sound Propagation in Segmented Exhaust Ducts - Theoretical Prediction

and Comparison with Measurementss", AIAA Paper No. 83-0734, April 1983.

167
APPENDIX A

SCATTER IN MODAL COEFFICIENT FILES

Extensive work was done to determine the reliability of a particular

probe through the elimination of that probe in the modal decomposition scheme

and its subsequent effect on the results. This could be done by eliminating

one or two probes from the input data to the modal decomposition prosram and

solving for less radial mode orders. This was done for all probes and then

the modal coefficient files were examined for scatter. To determine the

desree of scatter, all modal coefficient files that were generated without a

particular probe were combined and each mode had its average value and

standard deviation calculated. The reasoning was that if a probe was not

functioning properly its elimination from the data would result in reduced

scatter (lower standard deviation) assuming all the other probes were

functionin8 well. In the case where a probe was calibrated to read high as

was the case in the 7600 rpmhardwall run, this proved to be an excellent mode

of detectin_ a bad probe. However, in the case where a probe was slishtly

miscalibrated or reading low, this method was not as effective. For example,

in the case where a broken diaphragm gave no reading, as was the case in all

downstream treatment runs, the scatter was less by leaving out this probe but

only slightly less. Fisure A1 shows a comparison of scatter versus probe

elimination for these two cases. Since the degree of confidence in using this

method was low for low reading probes, it was not utilized in these cases.

In summary, modal decomposition was carried out using all functionin_

probes, except in the case of the 7600 rpm hardwall run. In that case, the

cause of miscallbration was known before the scatter techniques were applied,

and the scatter techniques confirmed that fact. In the downstream treatment

cases the probe with the defective transducer was eliminated from modal

decomposition. That is, the scatter criteria was not used in the data

reduction process.

169
m,-i

E

(_

rOE

"0 r'O

rid __

"r I--

O0

O0

COO

I

I

0

f.D

o

anleA

o

ueeM/'^eO "_S

P

J

iI

I

t

/

P

o

0

C_

::n::

E

0

z

o

r_

E

L_

..G

0

L

_d

0

I=

0

t_

,m

E

m-i

0

0

0 ,_

E

u

170
APPENDIXB

AN ASSESSMENT OF ERRORS IN THE IN-SITU MEASUREMENTS

OF THE ACOUSTIC IMPEDANCE

Honeycomb I

Core

_Acoustic Transducers

PA (f) Signal at A

PB (f) Signal at B

The sketch above shows two transducers sensing signals at the surface

of the porous face sheet (A) and at the backwall of the cavity (B).

The Acoustic Zmpedance of the locally reactins SDOF panel in terms of

the two signals, its thickness d and the frequency f is given by

_(f) = -i HAB exp (i _aBCf)} cosec(kd) (B1)

where k = 2vf/c, c is the speed of sound and

HAB ex'p {i _ABCf)} = PACf) PB(f) / PBCf) PB(f) CB2)

The transfer function HABCf) and the phase difference _ABCf) are

obtained from the analysis of the two signals; cosec(dk) is computed from

specified values of the panel thickness d, and temperature T in the cavity.

First consider the error in the measured impedance due to errors in the values

171
of d and T. It can be shown that an error in the impedance value 641(£)

due to an error 6(kd) is given by

6el(f) = (_R 1 + i_X 1)

= -(R+iX) cotCkd) . 6(kd) (B3)

where

6(kd) ffi0.000313 f _- T_-- (B4)

T is in °K and d is in millimeters.

It is clear from (B3) that the error 6el(f) is zero when cot(kd) is

zero, that is, close to the tuning frequency of the SDOF panel. Sample

calculations of the expected errors in the measured impedance of an SDOF panel

due to errors in the specified thickness and temperature are presented in

Table B1. The errors in the resistance and reactance values due to errors in

the temperature and thickness ere quite large near the antiresonance frequency

given by c/2d. Away from this frequency, these errors are quite small and may

be ignored. In practice the cavity thickness d may be measured with much

greater accuracy than specified in Table B1. The accuracy of temperature may

be improved by accurate calibration of the thermocouples used in the

laboratory.

The other sources of error in the measured impedance are errors

_HAB(f) and 50AB(f) in the measurement of the transfer function and

the phase difference of the signals. These errors may be either of the bias

type or of random type as discussed in Reference 22. These types of errors

may be minimized by increasing the number of averages in the data analysis as

well as by reducing the bandwidth. However significant errors in the measured

values of HAB(f) and _AB(f) may exist due to the following:

I. Transducers not flush with the surfaces at A and B.

2. Leakages from the cavity other than through the porous sheet.

172
Table BI. In-Sltu Measurements

e

e

Estimated Errors in the Measured Impedances Due to

Errors in the Thickness (d) and Temperature (T)

R = 0.4 pc, d - 46.48mm, T - 294"K, 6d = 0.762 mm,

5T = I.I°K

fJ

Hz

250.0

500.0

750.0

I000.0

1250.0

1500.0

1750.0

2000.0

2250.0

2500.0

2750.0

3000.0

3250.0

3500.0

37_0.0

4000.0

4250.0

4500.0

4750.0

5000.0

5250.0

5500.0

5750.0

6000.0

6250.0

R, X:

6R1 d ,

6R1 T,

Xj

Oc

=4. 604

-2.125

-1.217

-0. 698

-0. 330

-0.031

O. 236

0.495

O. 766

1.073

I .459

2.019

3.050

6.273

-27.150

-3. 213

-1. 233

-0.425

0.058

0.411

0.702

0.967

1. 226

1.501

1.818

Nominal

6R1 d , 6R1 T,

pc oc

O.006 O.001

0.006 0.001

O. 006 0.001

0.005 0.001

O.004 0.000

O.003 0.000

O.001 O.000

-0.001 -0.000

0.004 -0.001

-0.008 -0.001

-0.015 -0.002

-0.024 -0. 003

-0.044 -0.005

-0.109 -0.013

0.580 0.067

O.088 O. 010

O. 648 O.005

0.031 0.004

0.022 0.002

O.014 O.002

0.008 0.001

O.002 0.000

-0.005 -0.001

-O.013 -0.001

-0. 023 -0. 003

6Xld ,

DC

6Xld:

6X1T:

-0.074

-0.033

-0.017

-0.009

-0.003

-0.000

0.001

-0.002

-0.008

-0.023

-0.053

-0.'123

-0.337

-1.716

_A

-- aT l 381

-0.706

-0.147

-0.033

0.003

0.015

0.014

0.004

-0.016

-0.049

-0.104

6Xl T,

Oc

-0.009

-0.004

-0.002

-0.001

-0.000

-0.000

0.000

-0.000

-0.001

-0.003

-0.006

-0.014

-0.039

-0.197

-0.081

-0.017

-0.004

0.000

0.002

0.002

0.000

-0.002

-0.006

-0.012

values of resistance and reactance

Errors in resistance/reactance due to error

6d (nun)

Error in resistance/reactance due to error

6T (" K)

173
3. Inaccuracies in the calibration of transducers.

4. Inaccuracies due to insufficient signal-to-noise ratio.

Under conditions of srazing flow, the measurement of impedance is found

to be extremely sensitive to the position of transducer A relative to the

surface of the porous face sheet. Experience showed that for accurate

measurement the face sheet transducer must be absolutely flush with it.

Leakages were eliminated by constructing a sinsle cavity with hard walls.

Inaccuracies in the calibrations of the transducers and insufficient

sisnal-to-noise ratio are the most sisnificant sources of error in the

measured impedance. The calibrations of the transducers, the associated

signal conditionins equipment and the A to D converter of the data analysis

system are required because of sisnificant differences in their response

characteristics. The method used for the calibration of the transducers, the

sisnal conditionins amplifiers and the analos-to-disital converters consisted

of placins them at the end of a tube flush with a hard wall termination and

excitins a broadband acoustic sisnal at the other end. Under these

conditions, both the transducers were subjected to plane wave signals at all

frequencies up to the cut-on frequency of the (1,0) mode. The transfer

function and the phase difference measured under these conditions represented

the calibration to be used when measuring impedance. Clearly, the accuracy of

this calibration would depend on the signal-to-noise ratio during its

measurement. Figure B1 shows the calibration of the transducer system used in

the In-$itu measurements in the Grazing Flow Duct. The reliable part of this

data lies roughly between 600 Hz and 2800 Hz. At frequencies outside this

range, signal-to-noise ratio is a problem. The difficulty lies in poor

coupling between the speaker and the tube at these frequencies and a

relatively high level of electronic-system noise. A way to eliminate this

problem would be to use discrete frequency signals for calibration.

From error analysis, it can be shown that the error 6C 2 in the

measured impedance due to errors 6HAB and 6_AB is given by

174
G

uo++ou_ _,+sum_

q_ I

I

In

P)

Idm

i

n.l

I=I

G_

I

(eoo-,lop) _ o:mo-,o_:_"l:¢ oslq,:T

q)

.+,,4

I

i,-i

_J

,,.c:

I.,+

13.1

l,-i

E-,

<,J

,I.i

0

Q

t_

,==t

175
6_ 2 = 6R 2 + i _X 2

(B5)

where R and X are the acoustic resistance and reactance values respectively of

the panel being tested. Table B2 shows sample calculations of the expected

errors in the measured impedance due to given errors in the measured phase

difference and the transfer function of the signals at A and B. A small error

in phase has little effect on the measured reactance over the entire frequency

ranse but gives rise to significant errors in the measured resistance at

frequencies Where the reactance is large. A small error in measured transfer

function gives rise to a small error in both the measured resistance and

reactance values.

Over the frequency ranse of interest in this study, the errors in the

measured impedance due to errors in the measured values of HAB(f) and

_AB(f) were considered to be Insisnlficant.

176
(

kO

V Teu2_s _ ]euSTg

M

0

0

I,,

@

_4

177
Table B2. Expected Errors in Measured Impedance Due to Errors

in the Measured Transfer Function (HAB) and Phase Difference (_AB)

• R=0.40c

• _#AB = 0.025 radians (1.43")

_AS(f)

• 6HAB/HAB = 0.05

error in phase difference

f_

Hz

250.0

500.0

750.0

1000.0

1250.0

1500.0

1750.0

2000.0

2250.0

2500.0

27_0.0

3000.0

3250.0

3500.0

3750.0

4000.0

4250.0

4500.0

4750.0

5000.0

5250.0

5500.0

5750.0

6000.0

6250.0

Xj

oc

-4.604 .0

-2.125 .0

-1.217 .0

.0. 698 .0

• 0.330 .0

.0.031 .0

0.236 0

0.495

0. 766

1.073

1.459

2.019

3.050

6. 273

-27. 150

-3.213

-1.233

.0.425

0.058

0.411

0.702

0.967

1.226

• 1.501

1.818

6R2t, 6X20: Errors due to 64AB

6R2H, 6X2H: Errors due to 6HAB/HAB

.115

.053

.030

.017

.008

.001

.006

0.012

0.019

0.027

0.036

0.050

0.076

0.157

.0.679

.0.080

.0.031

.0.011

0.001

0.010

0.018

0.024

0.031

0.038

0.045

5X2 H,

QC

O. 230

O. 106

0.061

0.035

0.016

0. 002

0.012

0.025

0. 038

0.054

0.073

O.i01

0.153

O.314

-1.357

-0.161

-0. 062

.0.021

0.003

0.021

0.035

0. 048

0.061

0.075

0.091

178
APPENDIX C

EIGENVALUE EQUATIONS

The complete form of the eisenvalue equation (referred to in Section

4.1.1) for sound propasation in a lined duct carryin8 uniform flow is

F(7) = I"2_-- (1 - _.)2 + iBkHA m(l_ C) 2 + (BlcH)2A 2 1T1

- T (I - _) m(l - _) + iBkHA T3

- 7 2 (I - C) 2 T 4 = 0

where B = acoustic admittance of the duct wails

= krr 2

C =rllr 2

H = r 2 - r 1

T1 = Om (C7) YmC_) - Jm C7) ¥m (CY)

T2 = Jm COT) ¥m+l (7) - Jm+l (Y) Ym (C7)

T3 = Jm+l CrY) Ym (7) - Jm (Y) Ym+l (_y)

T4 = Jm+l(Y) Ym+l(_Y ) - Jm+l(Cy) Ym+l(7 )

A = (I - _ M) 2

I

72(1 C) 2

-M +-_1 - (1 - M 2)

K " (kH)2

1 - M 2

(c.1)

CC.2)

CC.3)

CC.4)

(C.5)

rI and r2 are the inner and outer wall radii

179
and Jm( ) and Ym( ) are Bessel £unctions o£ the first and

second kinds of order m.

As mentioned in Section 4.1.1, the eiaenvalues can be determined by a

Newton-Raphson iteration of Equation (C.1).

alternately, Equation (C.1) can be recast as the differential equation

.d,, _1" l_._-C__ _ _ l

-_C]. - C')il_'IA(¢ 2 - T3) ] _dx

I_'lere

(C.6)

÷ 2 (ilkH) 2 A(& + E A)t

Y

)

÷ T2 l (1 - C) 2 (C"f 2 - _'-) + ilSlc1.1(1 - C) ["1'A .nu_(lC- _') ]- (BkH)2A2 }

+ T3 zl (1 - _')2('I'2 - xll2) - iBlcll(]. - C') I'I' }& ÷ !_.(1 - C')1 - C'(BkH)2A 2

+ T4 (1 _')2m(_' 1) ÷ i81r_(1. _)(C - _') (C.7)

and T1, T2, T3, and T4 are represented in Equations (0.4).

Eisenvalues can be obtained by integratinK Equation (0.6). Any variation

o£ admittance (B) with respect to the independent variable x can be prescribed. For a linear variation B m xB where B is the wall admittance,

W W

Equation (C.6) and (C.7) reduced to

180
(C.8)

and

T- _(1 + ¢2)

+ 2(Bw_)2A(A + m_ A) x2 f

/

t '

m Id{(l- ¢) + T2 (1 - _)2(Cy2"- _) + iB w

+ T3 I (1 - C)2(y 2 - m2) - iBwkH(1 - ¢)

+ T4 t (1 - _)

(

I

2 m(y - 1) + Lswk_u_(1- ¢)(_ - _)x } (c.9)

To find the so£twall eisenvalues, Equation (C.8) must be integrated from

x _ 0 to x = 1. The initial values for y (at x - O) are the hardwall

eisenvalues.

181
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188