AERODYNAMIC NOISE OF TURBOMACHINES

Wolfgang Neise and Ulf Michel

DLR Internal Report 22314-94/B5 (1994)

German Aerospace Center (DLR) Institute of Propulsion Technology, Department of Engine Acoustics

Previous affiliation: Institute of Fluid Mechanics, Department of Turbulence Research

Mller-Breslau-Str. 8 10623 Berlin,

Germany

Notes of a Short Course performed at

Pennsylvania State University, University Park, Pennsylvania, USA,

18-20 July 1994; United Technology Research Center, UTRC, East Hartford, Conn., USA,

28-30 November 1994; BMW Rolls Royce Aeroengines, Dahlewitz,

12-13 December 1994.

Contents

1 INTRODUCTION 1.1 GENERAL REMARKS

9 9

1.2 INDUSTRIAL, VENTILATING AND AIR CONDITIONING FANS. . . .. 11 1.2.1 FAN TYPES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11 1.2.2 BASIC FLUID MECHANICS OF FANS . . . . . . . . . . . . . . .. 14 1.2.3 AFFINITY LAWS AND NON-DIMENSIONAL FAN PERFOR-

MANCE PARAMETERS ............. . . . . . . . . . .. 17 1.2.4 TYPICAL FAN NOISE SPECTRA . . . . . . . . . . . . . . . . . .. 20

1.3 INDUSTRIAL COMPRESSORS ........................ 20 1.4 AIRCRAFT ENGINES. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21 1.5 PROPELLERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 24 1.6 HELICOPTERS ................................. 26 1.7 BIBLIOGRAPHY OF CHAPTER 1 ...................... 26

2 BASIC AEROACOUSTIC THEORY 31 2.1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31 2.2 LIGHTHILL EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32

2.2.1 DERIVATION OF LIGHTHILL EQUATION . . . . . . . . . . . .. 32 2.2.2 CONVECTIVE LIGHTHILL EQUATION . . . . . . . . . . . . . .. 34 2.2.3 ELEMENTARY SOUND SOURCES . . . . . . . . . . . . . . . . .. 35 2.2.4 SOURCES IN TERMS OF THE LIGHTHILL EQUATION ..... 37

2.3 SOLUTION OF THE LIGHTHILL EQUATION. . . . . . . . . . . . . . .. 37 2.3.1 KIRCHHOFF INTEGRAL. . . . . . . . . . . . . . . . . . . . . . .. 37 2.3.2 INTEGRAL OF CURLE. . . . . . . . . . . . . . . . . . . . . . . .. 38 2.3.3 FAR-FIELD SOLUTION. . . . . . . . . . . . . . . . . . . . . . . .. 39

2.4 TIME AVERAGED SOLUTIONS. . . . . . . . . . . . . . . . . . . . . .. 40 2.4.1 MEAN SQUARE VALUE. . . . . . . . . . . . . . . . . . . . . . .. 40 2.4.2 AUTO-CORRELATION FUNCTION ................. 41 2.4.3 POWER-SPECTRAL DENSITY ..... .

2.5 BIBLIOGRAPHY OF CHAPTER 2 ....... .

1

41 43

3 APPLICATION TO JET NOISE 45 3.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45 3.2 SOUND PRESSURE IN THE FAR FIELD. . . . . . . . . . . . . . . . .. 46 3.3 TIME AVERAGED SOLUTIONS. . . . . . . . . . . . . . . . . . . . . . .. 49

3.3.1 AUTOCORRELATION OF SOUND PRESSURE. . . . . . . . . . 50 3.3.2 POWER-SPECTRAL DENSITY . . . . . . . . . . . . . . . . . . . 51 3.3.3 DIRECTIVITY OF JET NOISE . . . . . . . . . . . . . . . . . 53

3.4 SCALING OF FLOW FIELD OF JET . . . . . . . . . . . . . . . . . . . . 56 3.5 SCALING OF SOUND PRESSURE ..................... 57 3.6 SUPERSONIC JETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.6.1 MACH WAVE RADIATION. . . . . . . . . . . . . . . . . . . . . . 59 3.6.2 BROADBAND SHOCK NOISE . . . . . . . . . . . . . . . . . . . . 60 3.6.3 SCREECH ............................... 63

3.7 RELATION BETWEEN FLYOVER AND WIND-TUNNEL CASES ... 63 3.8 PREDICTION OF JET NOISE . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8.1 STATIC JET NOISE. . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8.2 JET NOISE IN FLIGHT. . . . . . . . . . . . . . . . . . . . . . . . 64

3.9 BIBLIOGRAPHY OF CHAPTER 3 ..................... 64

4 INFLUENCE OF SOLID SURFACES 67 4.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 SOUND GENERATION BY SURFACES. . . . . . . . . . . . . . . . . . . 68

4.2.1 SPACE-FIXED COORDINATE SYSTEM . . . . . . . . . . . . . . 68 4.2.2 ROTOR-FIXED COORDINATE SYSTEM ............. 68

4.3 FAR-FIELD APPROXIMATION . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.1 ROTATING SURFACES. . . . . . . . . . . . . . . . . . . . . . . . 71 4.3.2 STATIONARY SURFACES . . . . . . . . . . . . . . . . . . . . . . 72

4.4 BOUNDARY LAYER NOISE . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.1 INFINITE FLAT PLATE . . . . . . . . . . . . . . . . . . . . . . . 73 4.4.2 PLATES WITH EDGES . . . . . . . . . . . . . . . . . . . . . . . . 75

4.5 BIBLIOGRAPHY OF CHAPTER 4 ..................... 77

5 AERODYNAMIC SOUND GENERATION MECHANISMS IN TURBO-MACHINES 79 5.1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 80 5.2 BLADE THICKNESS NOISE . . . . . . . . . . . . . . . . . . . . . . . . .. 81 5.3 TONAL NOISE DUE TO STEADY AND UNSTEADY AERODYNAMIC

FORCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81 5.3.1 ROTOR IN SPATIALLY UNIFORM STEADY FLOW FIELD

(STEADY BLADE FORCES) . . . . . . . . . . . . . . . . . . . . .. 81 5.3.2 ROTOR IN SPATIALLY NON-UNIFORM STEADY FLOW FIELD

(UNSTEADY BLADE FORCES) . . . . . . . . . . . . . . . . . . .. 82

2

5.3.3 NOISE GENERATION BY ROTOR/STATOR AND ROTOR/CASING INTERACTION ............ .

5.3.4 IMPULSIVE NOISE .......................... . 5.3.5 ROTATING STALL .......................... . 5.3.6 NON-UNIFORM ROTOR GEOMETRY ............... . 5.3.7 NARROW-BAND NOISE DUE TO A ROTOR OPERATING IN UN-

STEADY FLOW FIELD ....................... . 5.4 RANDOM NOISE DUE TO UNSTEADY AERODYNAMIC FORCES .. .

5.4.1 GENERAL REMARKS ........................ . 5.4.2 TURBULENT BOUNDARY LAYER NOISE ............ . 5.4.3 NOISE DUE TO INCIDENT TURBULENCE ............ . 5.4.4 VORTEX SHEDDING NOISE .................... . 5.4.5 FLOW SEPARATION NOISE ..................... . 5.4.6 TIP VORTEX NOISE ......................... .

5.5 QUADRUPOLE NOISE ............................ . 5.5.1 RANDOM NOISE ........................... .

85 86 86 87

89 91 91 92 92 93 94 96 98 98

5.5.2 DISCRETE NOISE. . . . . . . . . . . . . . . . . . . . . . . . . . .. 101 5.6 CONCLUSIONS ................................. 101 5.7 BIBLIOGRAPHY OF CHAPTER 5 ...................... 103

6 DUCT ACOUSTICS 6.1 INTRODUCTION 6.2 WAVE EQUATION FOR FLOW DUCTS WITH FLOW AND THERMAL

107 107

BOUNDARY LAYERS .............................. 108 6.3 SOUND PROPAGATION IN RIGID-WALLED RECTANGULAR DUCTS

WITH NO FlOW AND NO TEMPERATURE GRADIENTS ........ 110 6.3.1 GENERAL SOLUTION OF THE HOMOGENEOUS WAVE EQUA-

TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.3.2 BOUNDARY CONDITIONS AT THE RIGID DUCT WALLS . 112 6.3.3 BOUNDARY CONDITIONS AT THE DUCT TERMINATION 114 6.3.4 BOUNDARY CONDITIONS AT THE SOUND SOURCE ... 115

6.4 SOUND PROPAGATION IN HARD-WALLED CYLINDRICAL OR ANNU-LAR DUCTS IN THE ABSENCE OF TEMPERATURE GRADIENTS AND MEAN FLOW .................................. 116

6.5 SOUND PROPAGATION IN RECTANGULAR DUCTS WITH UNIFORM FLOW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.6 CONCLUSIONS ......................... . 6.7 BIBLIOGRAPHY OF CHAPTER 6 .............. .

3

. . . . . .. 122 123

7 GENERATION OF DUCT MODES BY TURBOMACHINES AND THEIR EXPERIMENTAL ANALYSIS 125 7.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2 MODES GENERATED BY A ROTOR ALONE. . . . . . . . . . . . . 126 7.3 DECAY OF NON-PROPAGATIONAL MODES. . . . . . . . . . . . . 127 7.4 MODES GENERATED BY ROTOR/STATOR INTERACTION ... 127 7.5 MODES GENERATED BY THE INTERACTION OF TWO COUNTER-

ROTATING ROTORS OF EQUAL BLADE NUMBER AND SPEED. . .. 131 7.6 ANALYSIS OF DUCT MODES TO DETERMINE THE DOMINANT

AEROACOUSTIC SOURCE MECHANISMS IN A PROPFAN MODEL .. 133 7.6.1 GENERAL REMARKS . . . . . . . . . . . . . . . . . . . . . . 133 7.6.2 TEST FACILITIES. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.6.3 ANALYSIS OF AZIMUTHAL AND RADIAL MODES. . . . . . .. 136 7.6.4 EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . . . .. 136 7.6.5 PREDICTION OF THE FAR-FIELD SOUND BASED ON NEAR-

FIELD DATA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.7 CONCLUSIONS ... . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.8 BIBLIOGRAPHY OF CHAPTER 7

8 OPEN ROTORS 8.1 INTRODUCTION

142

145 145

8.2 FREE FIELD RADIATION OF OPEN ROTORS . . . . . . . . . . . . . . . 146 8.3 DISCRETE TONES DUE TO ROTATING POINT FORCES . . . . . . . . 148 8.4 BIBLIOGRAPHY OF CHAPTER 8 ...................... 153

9 PROPELLERS 157 9.1 INTRODUCTION 9.2 EFFECT OF HELICAL BLADE-TIP MACH NUMBER .. . 9.3 EFFECTS OF INFLOW CONDITIONS ........... .

157 158 159

9.4 EFFECTS OF NONUNIFORM BLADE DISTRIBUTION . . . . . . . 161 9.5 EFFECTS OF COUNTER-ROTATION . . . . . . . . . . . . . . . . . 165 9.6 NOISE REDUCTION MEASURES. . . . . . . . . . . . . . . . . . . . 165 9.7 BIBLIOGRAPHY OF CHAPTER 9 ................... 167

10 HELICOPTERS AND WIND TURBINES 10.1 INTRODUCTION ....................... .

169 169

10.2 HELICOPTERS ................................. 170 10.2.1 DIFFERENCE BETWEEN HELICOPTER NOISE AND PRO-

PELLER NOISE . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 170 10.2.2 MAIN ROTOR NOISE. . . . . . . . . . . . . . . . . . . . . . . . . . 171 10.2.3 TAIL ROTOR NOISE . . . . . . . . . . . . . . . . . . . . . . . . .. 172

10.3 WIND TURBINES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 10.3.1 DIFFERENCE BETWEEN WIND TURBINE NOISE AND PRO-

PELLER NOISE . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 174

4

10.3.2 AERODYNAMICS OF HORIZONTAL AXIS WIND TURBINES .. 175 10.3.3 LOADING NOISE AND TRAILING EDGE NOISE. . . . . . . 178 10.3.4 NOISE REDUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 182

10.4 BIBLIOGRAPHY OF CHAPTER 10 . . . . . . . . . . . . . . . . . . . . .. 182

11 EFFECTS OF ACOUSTIC LOADING ON FAN NOISE 185 1l.1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 1l.2 PASSIVE ELECTRICAL ONE-PORTS AND TWO-PORTS . . . . . . 187 1l.3 PASSIVE ACOUSTIC ELEMENTS. . . . . . . . . . . . . . . . . . . . 189 1l.4 FANS MODELLED AS ACTIVE ACOUSTIC ELEMENTS ...... 190 1l.5 EXPERIMENTAL DETERMINATION OF THE PASSIVE TWO-PORT

PARAMETERS .................................. 191 11.6 MODELLING OF FAN NOISE BASED ON ACOUSTIC TWO-PORT DATA 194 11.7 CONCLUSIONS .............................. 196 1l.8 BIBLIOGRAPHY OF CHAPTER 11 . . . . . . . . . . . . . . . . . . . . .. 196

12 NOISE REDUCTION METHODS FOR AXIAL-FLOW MACHINES 199 12.1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 12.2 INCREASING THE DISTANCE BETWEEN ROTOR AND STATOR . . . 200 12.3 INTRODUCING A PHASE DISTRIBUTION INTO THE UNSTEADY

FORCES DUE TO ROTOR/STATOR INTERACTION . . . . . . . . 201 12.3.1 LEANED STATOR VANES . . . . . . . . . . . . . . . . . . . . 201 12.3.2 TILTED STATOR VANES .................... 201 12.3.3 IRREGULAR VANE SPACING. . . . . . . . . . . . . . . . . . 203 12.3.4 STEPPED STATOR VANES . . . . . . . . . . . . . . . . . . . 203

12.4 NOISE REDUCTION BY SUITABLE CHOICE OF BLADE AND VANE NUMBERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

12.5 NOISE REDUCTION BY IMPELLER DESIGN. . . . . . . . . . . . . . . . 206 12.5.1 IRREGULAR BLADE SPACING. . . . . . . . . . . . . . . . . 206 12.5.2 LEANED IMPELLER BLADES .................... 207 12.5.3 SWEPT IMPELLER BLADES . . . . . . . . . . . . . . . . . . . . . 208 12.5.4 INFLUENCE OF THE RADIAL BLADE LOADING DISTRIBUTION209 12.5.5 NOISE REDUCTION BY BLADE DESIGN . . . . . . . . . . . 211

12.6 REDUCTION OF TIP CLEARANCE NOISE . . . . . . . . . . . . . . . . . 213 12.7 CASING MODIFICATIONS. . . . . . . . . . . . . . . . . . . . . . . . 214 12.8 CONCLUSIONS .............................. 215 12.9 BIBLIOGRAPHY OF CHAPTER 12 . . . . . . . . . . . . . . . . . . . 216

5

13 NOISE REDUCTION METHODS FOR CENTRIFUGAL FANS 221 13.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 222 13.2 CASING MODIFICATIONS TO REDUCE THE STRENGTH OF THE IN-

TERACTION FORCES BETWEEN IMPELLER FLOW AND THE CUTOFF222 13.2.1 INCREASING THE CUTOFF CLEARANCE. . . . . . . . . . .. 222 13.2.2 RECTANGULAR FAN CASING . . . . . . . . . . . . . . . . . . . . 226 13.2.3 CIRCULAR FAN CASING .. . . . . . . . . . . . . . . . . . . . . . 226 13.2.4 ACOUSTICAL OPTIMIZATION OF A CENTRIFUGAL FAN CAS-

ING .................................... 227 13.2.5 RIB PLACED AT THE INNER CIRCUMFERENCE OF AN IM-

PELLER WITH FORWARD CURVED BLADES. . . . . . . . . . . . 230 13.3 INTRODUCING A PHASE SHIFT INTO THE INTERACTION FORCES

BETWEEN IMPELLER FLOW AND CASING . . . . . . . . . . . . . . . . 231 13.3.1 ANGLE OF INCLINATION BETWEEN IMPELLER BLADES AND

CUTOFF EDGE ............................. 231 13.3.2 STAGGERING THE BLADES OF DOUBLE INLET BLOWERS

AND DOUBLE ROW IMPELLERS ................. 234 13.3.3 IRREGULARLY SPACED IMPELLER BLADES . . . . . . . . .. 234 13.3.4 TRIANGULAR GUIDE BELT AROUND THE IMPELLER . . .. 235

13.4 IMPELLER MODIFICATIONS . . . . . . . . . . . . . . . . . . . . . . .. 235 13.4.1 TRANSITION MESHES AT THE INNER AND OUTER CIRCUM-

FERENCE OF THE BLADE ROW ................... 235 13.4.2 ANNULAR CLEARANCE BETWEEN FAN INLET NOZZLE AND

IMPELLER MOUTH. . . . . . . . . . . . . . . . . . . . . . . . . . . 237 13.4.3 INFLUENCE OF THE BLADE NUMBER. . . . . . . . . . . . . . . 238 13.4.4 ROTATING DIFFUSER. . . . . . . . . . . . . . . . . . . . . . . . . 238 13.4.5 AIRFOIL BLADES. . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 13.4.6 COMPARISON OF DIFFERENT IMPELLER DESIGNS ...... 240

13.5 LOW NOISE BLOWER DESIGN. . . . . . . . . . . . . . . . . . . . . . . . 241 13.6 ACOUSTICAL MEASURES. . . . . . . . . . . . . . . . . . . . . . . . . . . 241

13.6.1 MISMATCH BEWEEN THE ACOUSTIC IMPEDANCES OF FAN AND DUCT SYSTEM . . . . . . . . . . . . . . . . . . . . . . . . . . 241

13.6.2 ACOUSTICAL LINING OF THE FAN CASING . . . . . . . . . . . 242 13.6.3 MINIMIZING THE ACOUSTIC RADIATION EFFICIENCY OF A

FAN .................................... 242 13.6.4 RESONATORS MOUNTED IN THE CUTOFF. . . . . . . . . .. 243 13.6.5 ACTIVE SOURCES MOUNTED IN THE CUTOFF . . . . . . . . . 245

13.7 CONCLUSIONS ................................ 247 13.8 BIBLIOGRAPHY OF CHAPTER 13 . . . . . . . . . . . . . . . . . . . .. 247

6

14 INSTALLATION EFFECTS ON FAN NOISE 14.1 INTRODUCTION ....................... .

251 251

14.2 EFFECTS OF INFLOW CONDITIONS ON FAN NOISE .. . . . . . . . . 252 14.3 EFFECTS OF FAN OPERATION CONTROL ON FAN NOISE. . . . . . . 259 14.4 ACOUSTIC LOADING EFFECTS ON FAN NOISE . . . . . . . . . . .. 261 14.5 CONCLUSIONS ................................ 268 14.6 BIBLIOGRAPHY OF CHAPTER 14 . . . . . . . . . . . . . . . . . . . .. 269

15 SOUND POWER MEASUREMENT PROCEDURES FOR FANS 273 15.1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 273 15.2 REVERBERATION-ROOM METHOD. . . . . . . . . . . . . . . . . . . . . 274 15.3 FREE-FIELD METHOD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 15.4 IN-DUCT METHOD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 15.5 FAN NOISE TESTING ............................. 281 15.6 EXPERIMENTAL COMPARISON OF FAN NOISE MEASUREMENT

STANDARDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 15.7 CONCLUSIONS ................................. 288 15.8 BIBLIOGRAPHY OF CHAPTER 15 . . . . . . . . . . . . . . . . . . . . . . 290

16 ACOUSTIC SIMILARITY LAWS FOR FANS 293 16.1 INTRODUCTION .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 16.2 FORMULATION OF SIMILARITY RELATIONSHIPS GOVERNING FAN

NOISE ....................................... 294 16.2.1 SIMILARITY RELATIONSHIPS FOR THE AERODYNAMIC FAN

PERFORMANCE ........... . . . . . . . . . . . . . . . . . 294 16.2.2 SIMILARITY RELATIONSHIPS FOR FAN NOISE ......... 294

16.3 EXPERIMENTAL VERIFICATION OF THE ACOUSTIC SIMILARITY LAWS ....................................... 297 16.3.1 VARIATION OF THE REYNOLDS NUMBER VIA THE DENSITY

OF THE WORKING FLUID . . . . . . . . . . . . . . . . . . . . . . 297 16.3.2 VARIATION OF THE REYNOLDS NUMBER VIA THE FAN SIZE 298

16.4 SCALING FAN NOISE DATA FROM A MODEL TO A FULL SIZE FAN . 307 16.4.1 GENERAL REMARKS ......................... 307 16.4.2 IDENTICAL WORKING FLUID AND IMPELLER TIP SPEED IN

MODEL AND FULL SIZE FAN ..................... 307 16.4.3 SCALING OF FAN NOISE SPECTRA FOR ARBITRARY CONDI-

TIONS ................................... 308 16.4.4 SCALING THE FAN TOTAL SOUND POWER ........... 308

16.5 CONCLUSIONS ................................. 310 16.6 NOMENCLATURE .............. . 310 16.7 BIBLIOGRAPHY OF CHAPTER 16 ...................... 312

7

17 SOUND POWER ESTIMATION OF INDUSTRIAL AND VENTILA-TION FANS 315 17.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 315 17.2 HISTORICAL BACKGROUND. . . . . . . . . . . . . . . . . . . 316

17.2.1 MADISON'S FAN SOUND LAW . . . . . . . . . . . . . . 316 17.2.2 PREDICTION FORMULA AFTER GROFF, SCHREINER & BUL-

LOCK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 317 17.2.3 GRAHAM'S FAN NOISE ESTIMATION METHOD . . . . . . . . . 318 17.2.4 FAN SOUND POWER PREDICTION AFTER REGENSCHEIT . . 318

17.3 FAN SOUND POWER PREDICTION ACCORDING TO VDI 3731 BLATT 2318 17.3.1 LINEAR AND A-WEIGHTED SOUND POWER LEVEL IN THE

FAN OUTLET DUCT . . . . . . . . . . . . . . . . . . . . . . . .. 318 17.3.2 OUTLET DUCT FAN NOISE SPECTRA . . . . . . . . . . . . .. 320 17.3.3 INLET DUCT FAN SOUND POWER . . . . . . . . . . . . . . .. 321 17.3.4 FREE-FIELD SOUND POWER SPECTRA . . . . . . . . . . . .. 321

17.4 ASHRAE-METHOD OF FAN SOUND POWER PREDICTION. . . . .. 322 17.5 FAN SOUND POWER PREDICTION FOR AXIAL FANS AFTER WRIGHT322 17.6 CONCLUSIONS ..................... . . . . . 324 17.7 BIBLIOGRAPHY OF CHAPTER 17 . . . . . . . . . . . . . . . . . . . . . . 325

18 FAN SELECTION ON THE BASIS OF THE SPECIFIC SOUND POWER LEVEL 329 18.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 18.2 DEFINITION OF THE SPECIFIC SOUND POWER LEVEL . . . . . 330 18.3 EXPERIMENTAL DATA . . . . . . . . . . . . . . . . 331

18.3.1 AERODYNAMIC FAN PERFORMANCE . . . . . . . . . . . . 331 18.3.2 TOTAL SPECIFIC SOUND POWER LEVEL. . . . . . . . . . 331 18.3.3 A-WEIGHTED SPECIFIC SOUND POWER LEVEL ..... 333 18.3.4 NORMALIZED SPECIFIC SOUND POWER SPECTRA ..... 334 18.3.5 COMPARISON WITH OTHER STUDIES. . . . . . . . . . . . .. 336

18.4 CONCLUSIONS ................................ 337 18.5 BIBLIOGRAPHY OF CHAPTER 18 ......... . 338

19 BIBLIOGRAPHY 341

8

Chapter 1

INTRODUCTION

Contents of Chapter 1

1 INTRODUCTION 1.1 GENERAL REMARKS 1.2 INDUSTRIAL, VENTILATING AND AIR CONDITIONING FANS.

1.2.1 FAN TYPES ....................... .

9 9

11 11

1.2.2 BASIC FLUID MECHANICS OF FANS . . . . . . . . . 14 1.2.3 AFFINITY LAWS AND NON-DIMENSIONAL FAN PERFOR-

MANCE PARAMETERS ..... 17 1.2.4 TYPICAL FAN NOISE SPECTRA

1.3 INDUSTRIAL COMPRESSORS 1.4 AIRCRAFT ENGINES. 1.5 PROPELLERS . . . . . . . . . . 1.6 HELICOPTERS ........ . 1.7 BIBLIOGRAPHY OF CHAPTER 1

1.1 GENERAL REMARKS

20 20 21 24 26 26

In turbomachines the working fluid is compressed and moved by the dynamic action of the rotating blades of one or several impellers. The rotating impeller transfers mechanical energy from the fan shaft to the working fluid in the form of increased velocity and/or pressure. There is a wide variety of machines that fall under the above definition: fans, blowers, compressors, turbines, pumps, propellers, wind energy converters, helicopter rotors, etc. Only turbomachines handling gaseous media are treated in this course, which excludes all sorts of pumps and ship propulsors. Of the various kinds of compressors, only turbo-compressors are treated in this chapter, because both the mechanism of energy transfer from the machine to the working fluid and the mechanism of noise generation in reciprocating compressors, roots compressors or rotary compressors are entirely different from those in turbomachines.

The field of applications of fans and compressors handling gases or vapors is extremely wide, and in nearly all cases the noise emitted by these machines is of considerable annoyance.

9

The dominant part of the total noise emitted by these machines is the aerodynamically generated noise. Only this noise contributor is treated here, other noise sources like gear noise, bearing noise, vibrational noise and noise of drive motors are excluded.

Turbo-machines are usually categorized with respect to the mean flow direction of the working fluid. The main types are axial-flow machines, radial-flow (centrifugal) machines, mixed flow fans and tangential flow fans. Examples of axial flow machines are the propellers of airplanes and hovercrafts, helicopter rotors, the fans, turbines and most compressors of modern aircraft engines, and the fans installed in all kinds of ventilating and air conditioning systems and industrial plants. The dimensions of industrial axial flow fans range from a few centimeters in diameter, for example the cooling fans in electronic equipment, over the medium size fans used for ventilation purposes, over wind-tunnel fans with impeller diameters of the order of 10m, up to machines as big as several ten meters which operate in the cooling-towers of electric power plants. Rotor diameters of as much as 100 m are in use for wind energy converters.

The range of sizes of radial flow machines is not quite as big as that of axial flow machines. Small fan units are in household appliances such as vacuum-cleaners or electric heaters, medium sizes provide air-flow in ventilating and air-conditioning duct systems, and impellers up to several meters in diameter are employed in the mining industry, steel making plants and chemical industry. Radial compressors operate in small aircraft turbo engines, refrigerating equipment, compressed air installations, automotive vehicles (turbo chargers) and again in various industrial plants. Radial turbines are used for turbo chargers.

Although the basic mechanisms of aerodynamic sound generation are the same for axial and radial flow machines, by far more detailed knowledge has been established for the axial type, mainly because of its wide application in aviation and the demand for quieter aircrafts which has given the research work on turbomachinery noise problems a tremendous impact.

It is the aim of this course to provide some fundamental understanding of the basic aero-dynamic sound generation by rotating blades and to give some insight into the theoretical background relevant to turbomachinery noise rather than to present detailed solutions and results for particular applications. After the presentation of the basic acoustic theory de-veloped by Lighthill (1952) in chapter 2 and its application to jet noise in chapter 3 , the influence of solid surfaces on the sound generation is derived in chapter 4. This leads to a discussion of the sound generation mechanisms relevant to turbomachinery in chapter 5. An important aspect is the generation of acoustic modes by subsonic and supersonic rotors, by the interaction between rotor blades and stator vanes, and by the interaction between adjacent rotors. Sound propagation effects are described in the chapters on duct acoustics and free field radiation of open rotors. The next three chapters 8, 9 and10 are concerned with propeller, helicopter, and wind-turbine noise. Methods of noise control for axial and centrifugal flow machines are discussed on the basis of the sound generation mechanisms in chapters 12 and 13. The remainder of the course is devoted to the noise of industrial, ventilating and air conditioning fans covering effects of acoustic loading, installation effects on fan noise, fan sound power measurement procedures, acoustic similarity laws, fan sound power estimation, and fan selection with respect to noise.

In this first chapter, some general aerodynamic and acoustic characteristics of various turbomachines are discussed.

10

1.2 INDUSTRIAL, VENTILATING AND AIR CONDITIONING FANS

There is no definite distinction between compressors, blowers and fans. Broadly speaking, fans and blowers are medium-pressure or low-pressure compressors. A boundary between fans and compressors can be derived from the international standard on fan performance testing of industrial fans ISO 5801 (1994), where an upper limit for the fan work per unit mass of 25 kJ jkg is set, corresponding to an increase of fan pressure approximately equal to 30 kPa for a mean density of 1.2 kgjm3 . This is equivalent to a ratio of inlet to outlet pressure of 1.3, assuming standard atmospheric pressure of 100 kPa.

1.2.1 FAN TYPES Different fan applications require fans with different performance characteristics which are determined primarily by the design of the rotating impeller and the fan casing. The types of fans described here are normally used in air conditioning systems, in industrial ventilating systems and industrial process applications. Typical designs of axial fans, centrifugal fans, half-axial fans (mixed flow impeller in axial type casing), half radial fans (mixed-flow impeller in centrifugal type casing), and cross-flow fans are schematically shown in Figures 1.1 to 1.7.

Axial-flow fans are categorized into the three following major groups: propeller fans (no casing), axial fans with casing and no guide vanes (tubeaxial fans), and axial fans with outlet guide vanes (vaneaxial fans). Inlet guide vanes are rarely used nowadays for axial fans, primarily for noise reasons.

The designs of the two types of mixed flow fans (half-axial and half-radial fans) lie somewhere between those of axial and centrifugal fans, and so does their performance which is the technical reason for these constructions.

Centrifugal impellers are normally operated in volute casings, most often without outlet guide vanes with the casing spiral being the only means to guide the flow and to convert kinetic flow energy to potential energy (increased static pressure). Centrifugal fans are designed with single inlet or double inlet. Depending on the application, various impeller designs and blade shapes are employed. The best design from an aerodynamic point of view is that with backward curved airfoil blades with a nicely curved shroud. Radial or radial ending blades are used for pneumatic transport applications or when large centrifugal stresses are encountered. Centrifugal fans incorporating a drum type impeller with a large number of forward curved blades of short radial extent are often called scirocco blowers or simply blowers.

The cross-flow fan is a unique type of fan, operating in a fundamentally different way than axial or centrifugal flow machines. The impeller is of the drum type with forward curved blades which are usually simple circular arcs, similar to the centrifugal fan impeller with forward curved blades. Unlike the scirocco blower, however, the cross-flow impeller is closed at both ends, and the working fluid enters the impeller perpendicular to its axis of rotation over its full axial length. The fluid passes through the blade row in the radially inward direction, through the interior, then through the blade row a second time, this time in the radially outward direction, to discharge on the other side of the impeller. For this reason, the cross-flow fan is, in principle, a two-stage fan. The flow inside the impeller is characterized by the formation of an eccentric line vortex parallel to the rotor axis which rotates in the same direction as the impeller. The presence of this vortex is of utmost importance for the

11

0)

Support

b)

Motor support

c) Impeller Guide vanes Tail fairing

Figure 1.1: Axial-flow fans a) propeller fan; b) vaneless axial fan (tubeaxial fan); c) axial fan with outlet guide vanes (vaneaxial fan).

operation of the cross-flow fan. Other names of this fan type are tangential fan, line-flow fan, and transverse-flow fan.

Typical performance characteristics of axial and centrifugal fans, all of D = 600mrn impeller diameter, are shown in Figure 1.8. The impeller speed is n = 3000/min for the axial and the centrifugal fan with backward curved blades and n 600/min for the scirocco blower.

12

Impeller Outlet guide vanes Tail fairing I

Figure 1.2: Half-axial fan.

Half-radial fan

Figure 1.3: Half-radial fan.

The operating points of maximum efficiency are marked by circles. Generally speaking, axial-flow fans are characterized by large flow rates and moderate fan pressures, while the inverse applies to centrifugal fans. The examples shown in Figure 1.8 were chosen purposely to illustrate this general behavior, i.e., a high pressure centrifugal fan design and a low pressure axial fan. If the performance characteristics of a low or medium pressure centrifugal fan and a high pressure axial fan were compared, the difference would not be quite as drastic.

Because of their weak structural design, scirocco blowers and cross-flow fans are normally run at much lower tip speeds than the other fan types. Due to their blade shapes, however, they handle large flow rates against relatively high back pressures. On the other hand, while the peak efficiencies of well designed axial fans and centrifugal fans with backward curved blades are in the range 80 to 90%, the optimum efficiency of scirocco blowers lies between 60 and 70%; the values reported for cross-flow fans are scattered in the range 30 to 70%, obviously depending on both the fan design and the test method.

13

Single inlet Double inlet

~ Discharge nozzle --+-..::.--

Diffuser

Impeller

Volute

Figure 1.4: Centrifugal fan designs.

Shroud Impeller backplate

Figure 1.5: Centrifugal impeller designs.

1.2.2 BASIC FLUID MECHANICS OF FANS The transfer of mechanical energy from the impeller to the working fluid results in a change of the thermodynamic steady state values of the medium when flowing from the inlet (subscript i) to the discharge (subscript d). Assuming reversible processes and a compressible working fluid, the fan work per unit mass is governed by the following equation (see Baehr (1962)):

d

J dps 1 2 2 Yt = - + -(Cd - Ci) + g(Zd - Zi), . p 2

(1.1 ) ,

14

Sheet metal blades Airfoil blades

!'.

@!) a) b)

@@G c) d) e)

Figure 1.6: Blade shapes of centrifugal impellers; a) backward curved blades; b) backward inclined blades; c) radial ending blades; d) radial blades; e) forward curved blades.

rvlotor

--

'-

II I -------- -----f--

I I

~~~ .. < " Rear guide wall Velocity profile

at outlet

Figure 1. 7 : Cross-flow fan.

where Ps is the static pressure of the flow, p the medium density, z the geodetical height, 9 gravitational constant, and c the flow velocity. All flow quantities are averaged over the cross sections of the fan inlet and outlet, respectively. The actual fan work per unit mass is of course higher than Yt because of the effect of friction.

It is customary in fan engineering to treat the working fluid as an incompressible one because the ratio of inlet pressure to outlet pressure is limited to about 1.3. For the case

15

7000 Po

6000

Cl 5000

The impeller shaft power is transmitted in the form of torque Timp times angular shaft speed Wimp' The impeller torque is balanced by the torque exerted by the working fluid, which is caused by the deflection of the mean flow between intake (subscript 1) and outlet of the impeller (subscript 2). The condition of angular momentum conservation leads to the following expression for the impeller torque:

(1. 7) where Tl and T2 are the radii of the blade leading and trailing edges, respectively, and Cl u and C2u the circumferential components of the absolute velocities at Tl and T2.

In most practical cases the flow entering the impeller is without swirl (Cl u = 0); for this case the velocity triangles of axial and centrifugal blade rows are sketched in Figures 1.9 and 1.10.

Outlet guide vanes'

Impeller ~

u

Figure 1.9: Velocity triangles for the inlet and outlet of an axial fan for the case of swirl-free intake flow (adapted from Schreck (1961)).

1.2.3 AFFINITY LAWS AND NON-DIMENSIONAL FAN PERFORMANCE PARAMETERS

The affinity laws are based on the geometric similarity of the members of one fan family, i.e., fans of different size but identical dimensional design, and on the kinematic similarity of the interior flow fields at different impeller speeds and sizes. In this case, the following relationships apply:

(1.8)

(1.9) where D, n, and U are impeller diameter, speed and peripheral velocity (tip speed). Based on the affinity laws, the following non-dimensional fan performance parameters can be defined:

17

Forward curved Radial ending

w~ 1 \\

Figure 1.10: Velocity triangles for the inlet and outlet of a centrifugal fan for the case of swirl-free intake flow (adapted from Schreck (1961)).

'P 4Q fl ffi . 7r D2 U -ow coe Clent (1.10)

'ljJt 2!:lpt (1.11) ---uz total pressure coefficient po

'ljJs 2!:lps. ffi . --2 statIc pressure coe Clent poU

(1.12)

,\ 'P . 'ljJt . (1.13) -- power coeffiCIent 'lJt

Note that different definitions of the flow coefficient are sometimes used for centrifugal fans ('Pcentr Q/7rbDU; b impeller width) and cross-flow fans ('Pc!! = Q/bDU). Typical dimensionless fan performance curves are shown in Figure 1.11, where the flow coefficient is defined as in equation (1.10), except for the cross-flow fan which is defined as above. In this example the performances of axial and centrifugal fans differ not as much as in Figure 1.8.

The affinity laws do not provide an exact description of the behavior of fans of different size and speed, because of the influence of viscosity and compressibility on the aerodynamic performance. These effects are accounted for by two additional parameters,

UD Re = -- Reynolds number

1/

18

(1.14)

_ - 0- -

4~

2 c OJ 0- - --

u .~ 4_ 4-ill a u

ill '-::J (f) (f) OJ '-0..

~~-----.71------.~2------.~3------.~4------.~5------.~6--~ Flow coefficient

n 0 u z 1/min m m/s

+--+Axial fan, DGIf, airfoil blades 3000 0.45 70.7 24 +---+Axial fan, OGV, airfoil blades 3000 0.45 70.7 12 If---

The speed coefficient indicates how much faster a fan has to run than a fictitious reference fan characterized by cp 1 and 'IjJ = 1, and the diameter coefficient indicates how much bigger the impeller diameter is than that of the reference fan. Speed coefficient and diameter coefficient are usually defined only for the best efficiency points of each fan. Their usefulness in characterizing the performance of different fan designs is demonstrated best in the so-called Cordier diagram. Cordier (1953) showed that in a plot of specific diameter versus specific speed coefficient, the data for all fans of good aerodynamic design (efficiency) collapse in single curve with reasonable scatter, see Figure l.12. Here it is obvious that centrifugal fans are characterized by small values of the speed coefficient and large values of the diameter coefficient, which is equivalent to high pressure and low flow rates, as was stated before. Similarly for the other fan types. Also shown in Figure l.12 are the ranges of total fan pressure, flow coefficient and total fan efficiency, each at the operating point of maximum efficiency. This presentation is a valuable help for the initial selection of a fan type for a given pumping task. A further overview over the performance characteristics of the various fan designs in terms of their dimensionless performance parameters is given in Figure l.13.

1.2.4 TYPICAL FAN NOISE SPECTRA In Figures l.14 to l.19 are shown examples of the one-third octave band sound power spectra of most of the fan types discussed above. The spectra were measured at the respective best efficiency points and represent the sum of inlet and outlet sound power (total sound power level). A general observation is that the spectral distribution of the broadband noise of axial fans is characterized by a level maximum at medium frequencies, with a fall off towards higher and lower frequencies, while the random noise spectra of centrifugal fans of all kinds, of half-radial fans and cross-flow fans show a typical maximum at the very low frequency end with a continuous drop in level with increasing frequency. The blade passage frequency component is very distinctly visible in the spectra of axial fans and centrifugal fans with backward curved and radial blades. The spectrum of the scirocco blowers is mainly broadband in nature with a weak tone component. No published information is available for cross-flow fans, and one has to rely on the assumption that the noise spectrum of this fan type is similar to that of scirocco blowers, because of the similar blade shapes and numbers. As will be shown later, see chapter 16 on acoustic similarity laws, the general spectral shape of each spectrum is shifted with increasing fan speed (towards higher frequencies) and increasing fan size (towards lower frequencies).

1.3 INDUSTRIAL COMPRESSORS Axial and radial compressors are used in aircraft engines, compressed air installations, re-frigeration systems, industrial process applications, turbo-chargers, etc. The tip speed of compressors is typically much larger than that of fans because of the high pressure rise re-quirements. Typical designs of multistage axial and centrifugal compressors are shown in Figures l.20 and l.2l. The axial compressor has a set of inlet guide vanes and a series of stages consisting of impeller blades and stator vanes. Note that the last stage has a cen-trifugal impeller. Flow rate control is achieved by adjustable guide vanes in the first few stages. The centrifugal compressor has two intakes and two discharge ducts. On both sides, the working fluid leaving the first impeller passes a vaned outlet diffuser to decelerate the flow and thereby increase the pressure before it is guided radially inward to the intake of the second stage, etc.

Different designs of single stage centrifugal impellers are schematically shown in Figure

20

Centro backw. curved

High- Low-pressure pressure

Scirocco blowers (S8) Tubeaxial

4.0

2.0 0.4 tV t 0.2 1.0

0.1 0.5 0.05

ljl

0.25 0.01

0.9

Ilt 0.7

0.5 20

0 10 1.0

5 OjO 0.5

0.5 ...L-.jilillJlC:.....j.--l---+-+-+-+-+----t--t---t--"- 0.2 0.06 0.1 0.2 0.4 0.6 1.0 2.0 4.0

~~ "~ IW "~9, 9,"~'~5"O51 j 0, /0;:;;; 0.25

Figure 1.12: Cordier diagram (adapted from VDr 3731 Blatt 2 (1990)).

1.22. The two designs on the left have two-dimensional impeller blades (2D), i.e., curved in only one direction, whereas the blades of the others have a 3D-curvature. The two designs shown on the right are open impellers (with no shroud) which run at substantially higher speeds than the shrouded ones. The change of the characteristic performance parameters with the design are described relative to a reference impeller, which was chosen to be the second from the left.

1.4 AIRCRAFT ENGINES

The noise emission of industrial compressors is normally of not so much concern because of the heavy ducting connected to them. The situation is entirely different in the case of the aircraft engine compressors which radiate sound almost directly into the environment. On the other hand, while the aeroacoustics of axial fans and compressors have been investigated

21

q; I a

~ I I I I ;~ L+- 1,0 2 .. 4 2 .... ! 0,35 .. 0,6 1,14 .. 1,19 40 .. 95 \ I _:1 1 2 .. 3 2 .. 3 !0,438. ,502 1,10 .. 1,32 69 .. 93

-- - ----"----

~ 0,3 O,i5 0,225 0,68 1,7 107,5 ohj ,> ,- 0,6 0,12 0,657 1,065 101

-_._--------

I '8 J. 3~a_ 0,13 1,0 0,13 0,:J61 2,72 57,1 I _._- ---------.

otd 0,03 1,1 0,033 0,162 5,02 26,6 l-' I I ,

O,ISTaJ I 0,00185 ! 1,1 0,00203 0,04 24,4 6,3 -I' I

I I

--~----- O,1 ... 0J 0,05 .. 0,01 0,005 .. 0,02 1,6 .. 3,8 1,0 .. 1,78 250600

~ 0,3 0,5 0,15 0,924 1,535 146 e 0,3 0,7 0,21 0,715 1,62 113

Figure 1.13: Ranges of the dimensionless fan performance parameters for different fan types (after Eck (1972)).

thoroughly because of their aeronautical use, hardly any information is available on centrifu-gal compressor noise. Examples of early, contemporary, and future aircraft engine designs are sketched in Figures 1.23 to 1.26.

In Figure 1.27 is shown how the noise emission of aircraft engines has changed with the design concept. As a result of the increased bypass ratio (fan mass flow to core engine mass flow) the jet velocity has been lowered and with it the noise level as well. It should be pointed out that the increase in bypass ratio was not made for acoustic considerations but for reasons of fuel economy. While jet nOIse was the dominating contributor in the early

22

110

m 100 D

90

Axial fan, no guide vanes 80 0 = 600 mm,

Z=8 n = 2925/min Q = 5.1 m'/s

7031 63 125 250 500 li< 21< 41< 81< A lin f. rlz

Figure 1.14: Total sound power spectrum of an axial fan with no guide vanes.

120 Axial fan, outlet guide vanes 0= 500 mm

110 Z = 12 ..* n = 2843/min Q = 3.3 m'/s )\ \

100 \ m I 1H\ D ., \

I ... ",

"" I \

.i, "--' 90 )"

"" *-,*-~)I...;r"A-,( \ "-

....

"'-"'-80 "'-"

7031 63 125 250 500 1k 2k 4k 8k A Iln f, Hz

Figure 1.15: Total sound power spectrum of an axial fan with outlet guide vanes.

80

70

m D

60 -,

_J

50-

40

31

Half-radial fan 0= /,50 mm Z=8 n = 1000/min Q=1.1/'m'/s

t ! J

63 125 250 500 lk 21< ~I< 81< A Iln f, Hz

Figure 1.16: Total sound power spectrum of a half-radial fan.

engine designs, the noise emitted by the fan and compressor have become important with the reduction in jet speed.

23

100

90

co -C1

80 ",

-,

70

60

3:

Centrifugal fan, b.c. airfoil blades D=510mm Z = 12 n = 1900/min 0= 2.1 m'/s

I

63 125 250 500 1k 2k 4k 8k A 1 in f, Hz

Figure 1.17: Total sound power spectrum of a centrifugal fan with backward curved blades.

o:J "CO

100

90

80 Centrifugal fan, flat radial blades D = 510 mm

70 Z = 15 n = 1800/min O=1.24m'/s

6031 63 125 250 500 lk 2k 4k 8k A lin f, Hz

Figure 1.18: Total sound power spectrum of a centrifugal fan with radial blades.

100

90 Q J

.>$0- J

80 "'", 1>

'-~

63 125 250 500 1K 2k ilk 8k A lln f, i-lz

Figure 1.19: Total sound power spectrum of a centrifugal fan with forward curved blades (scirocco blower).

1.5 PROPELLERS Typical propeller noise spectra are depicted in Figure 1.28. The levels of the blade tone fundamental are almost the same for clean inflow and disturbed inflow, However, while the

24

Radial impeller Guide vane control

\,~.~ =..,~ , ,

I ~~v.T------~==========~

Discharge Intermediate Intake discharge

Figure 1.20: Multistage axial compressor.

Figure 1.21: Multistage radial compressor.

harmonics fall off sharply when the propeller operates under clean flow conditions, high har-monic tone levels are generated by disturbed flow conditions. Note that tone level differences of more than 30dB exist, and the raise in random noise level is of the order of 20dB. The influence of the helical tip flow Mach number on the noise emission is shown in Figure 1.29.

25

~ 4.00 -- Shrouded impellersi

Unshrouded impellers

~ 3.00 1-------+--------.----I-------I----t7;;;wZZZ~ ..c

2.00 f-----

O~ lOO --o o

0.16

0.12

9- 0.08

0.04 o

~ ~

/ ~ 7 ! / / ~ I !

I I

i

~~~8~ LJu'mlsJ------cM i ~~~ 10vuM+"nU"+"'7n,~ t:1

ZO(min) ZO(mox) 30(norm) 30(mox) s R

____ Reference des!.9D __

Figure 1.22: Impeller designs of radial compressors (hp=polytropic enthalpy coefficient; 'ljJp=hp/U 2 = polytropic pressure coefficient).

1.6 HELICOPTERS Helicopters are much needed aircrafts in civil as well as in military life, but they are notorious for their noise. A typical narrow band spectrum of helicopter noise showing the influences of the various noise sources involved is shown in Figure 1.30. Contributors to the overall noise are the main rotor, the tail rotor, the gear box, and the engine noise. The spectrum shown does not include the impulsive blade slap noise which occurs when the main rotor blade chops through the wake from the preceding blade.

1.7 BIBLIOGRAPHY OF CHAPTER 1 BAEHR, H. D., 1962. Thermodynamik. Springer Verlag, Berlin, Germany.

CORDIER, 0., 1953. A.hnlichkeitsbedingungen bei Ventilatoren. Brennstoff Warme Kraft 5, 337-340.

26

Applications

1950's design Comet, Caravelle early B707 & DCS Business jets (NB: Concorde uses purejet wilh atterburning)

1960's design B707, DC8 & VC10 B727, 737 DC9 BAC-111 Trident F28 Business jets

1960's designs (1,2 stage fan) 707 & DCS, military early 747 1970's and 1980's designs (single stage fan) B737, 747,757, 767 L 1011 OC10 A300 M080 737-300 BAe 146 A320 F100 Business jets

Air inlel Compressor

a ~~t

Contra-rotating swept propeller '::';~ _ '""

Free power turbine \

Gas generator

/ Reduclion gear and pilch change mechanism

Figure 1.25: Unducted counter-rotating propfan (from Smith (1989)).

-----

1-------\ 1m

Figure 1.26: Ducted counter-rotating propfan (MTU-CRISP; from Heinig, Kennepohl & Traub (1992)).

ECK, B., 1972. VentilatoT'en, 5th ed. Springer Verlag, Berlin, Germany.

HECKL, M. & MULLER, H. A., 1994. Taschenbuch deT' Techniscl{en Akustik. Springer Verlag, Berlin, Germany.

HEINIG, K., KENNEPOHL, F. & TRAUB, P., 1992. Acoustic design of a counter rotating shrouded propfan. Conference Report DGLRj AIAA 14th Aeroacoustics Conference,

28

o o o

00

" ) ~.or'------\---5' o~ 1';:

- %: Unsuppressed \): mUlti-stage

-10 Noise (EPNdB)

-15

-20

-25

-30

compressors

Aircraft noise data corrected to constant thrust

Modern suppressed

turbofilns

Iy. olse 'flo . Or Set b ;,.::-..-. __ _

Y let mIxing noise

-35~--------~----------~----------~--------~---o 5 10 20 40

Bypass ratio

Figure 1.27: Aircraft engine noise as a function of bypass ratio (from Smith (1989)).

ill > ill -I

E ::J I-

.....

U ill 0.

!J)

ill lf) '0 z

Figure 1.29: Influence of the helical blade tip Mach number on propeller noise (after Do-brzynski et al in Heckl & Muller (1994)).

CD u

30

20

10

o Main rotor rotational noise Transmission noise

Chapter 2

BASIC AEROACOUSTIC THEORY

Contents of Chapter 2

2 BASIC AEROACOUSTIC THEORY 31 2.1 INTRODUCTION .......... ... , .... 31 2.2 LIGHTHILL EQUATION ...... ........ 32

2.2.1 DERIVATION OF LIGHT HILL EQUATION 32 2.2.2 CONVECTIVE LIGHTHILL EQUATION .. 34 2.2.3 ELEMENTARY SOUND SOURCES ..... 35 2.2.4 SOURCES IN TERMS OF THE LIGHTHILL EQUATION 37

2.3 SOLUTION OF THE LIGHTHILL EQUATION. 37 2.3.1 KIRCHHOFF INTEGRAL. 37 2.3.2 INTEGRAL OF CURLE .. 38 2.3.3 FAR-FIELD SOLUTION .. 39

2.4 TIME AVERAGED SOLUTIONS. 40 2.4.1 MEAN SQUARE VALUE. 40 2.4.2 AUTO-CORRELATION FUNCTION 41 2.4.3 POWER-SPECTRAL DENSITY 41

2.5 BIBLIOGRAPHY OF CHAPTER 2 .... 43

2.1 INTRODUCTION Although the first papers on rotor noise were published at the beginning of this century and the first propeller noise studies appeared in the thirties (for a historical survey see the review papers by Morfey (1973b) and Cumpsty (1977)), most of the recent progress in understand-ing aerodynamic sound generation by turbulent flows and by rotating blades is based on the acoustic analogy developed by Lighthill (1952), Curle (1955), and Ffowcs Williams & Hawkings (1969a).

The basics of aeroacoustic sound generation were presented in many papers and short courses. The following presentation is based on the work of Fuchs & Michalke (1973), Goldstein (1974), Fuchs (1976), and Michalke & Michel (1979).

31

The Lighthill equation is derived in this chapter and a general solution for the sound radiation into an unbounded space is presented which includes the effects of stationary surfaces. These equations are applied to the problem of jet noise in the following chapter 3. The influence of stationary and moving surfaces is treated in chapter 4.

2.2 LIGHTHILL EQUATION 2.2.1 DERIVATION OF LIGHTHILL EQUATION The basic equations of aeroacoustics can be derived from the equations for the conservation of mass and momentum. The conservation of mass is given in differential form and cartesian space fixed coordinates Xi by

ap apci _ 0 at + aXi - , (2.1 )

where p is the fluid density and Ci the velocity vector. The corresponding equation for the conservation of momentum is

(2.2)

Here, p is the pressure and Tij is the viscous stress tensor in the fluid. Many aeroacoustic wave equations have been derived based on these two equations. The

first work was published by Lighthill (1952). He derived an inhomogeneous wave equation for the description of the aeroacoustic noise generation of a small region of turbulent flow which is embedded within a homogeneous fluid at rest. The procedure is as follows.

Take the momentum equation (2.2) and add Ci times the continuity equation (2.1). This yields

apc< apc

Similarly, one obtains a wave equation for the density by adding 82p18x; - a682 pi 8x; on both sides of equation (2.6) which yields

82 P 2 82 P 82 82 8t2 - a08x2 = 8 .a . (pCiCj - Tij) + -8 2(P - a~p). (2.8) , x, X J Xi

Both equations (2.7 and 2.8) are inhomogeneous wave equations. Both equations are exact for any value for ao because the conservation laws of mass and

momentum were only used for their derivation and ao was introduced as an arbitrary constant.

If we consider a restricted turbulent flow region surrounded by an inviscid fluid at rest and take ao to be equal to the speed of sound in the ambience, then all four terms on the right hand side of these equations vanish (or are at least quadratically small in the fluctuations) outside the turbulent flow region.

In regions where the right hand sides of the equations are zero, they describe the acoustic wave propagation in a homogeneous fluid at rest with a sound speed of ao.

The right hand sides are nonzero inside the flow region. All effects due to the presence of the turbulent flow are viewed as an equivalent acoustic source distribution.

Lighthill used equation (2.8) in his original work, which is often formulated as 82 P 2 82 P 82Tij 8t2 - ao 8x~ = 8x 8x . '

, 'J (2.9)

where Tij is Lighthill's stress tensor,

(2.10) ao denotes the speed of sound in the undisturbed fluid, p is the pressure in the flow and Xi is the space-fixed coordinate system in which the mean velocity is zero outside the confined turbulent flow region. Equation (2.9) is a wave equation that governs the propagation of sound emitted by the source distribution on the right hand side. For a calculation of the emitted sound it is necessary that this source distribution is known.

Lighthill derived equation (2.9) in terms of the density p. The pressure p is preferred in the following, because it is the quantity measured by a microphone. The corresponding equation for the pressure is

1 82p 82p 2"8t2 - -8 2 = q, ao Xi

where q expresses the acoustic source distribution. The source term q is given by

r:"]- ~ (p-~) 'J 8t2 a6'

(2.11)

(2.12)

The first term of equation (2.12) is identical to the first term of 82Tij I ox i 8x j on the right hand side of equation (2.9) when equation (2.10) is evaluated.

For q = 0, equation (2.11) describes the sound wave propagation in a uniform medium at rest. Within a turbulent flow region, q ::J O. Thus, the right hand side can be interpreted as a source term which is produced by the turbulent flow, but which is zero, or at least quadratically small in the fluctuations, outside the region occupied by the turbulent flow. The situation is described in Figure 2.1.

33

q=O

turbulent flow r,egion

Figure 2.1: Turbulent flow region as a source of sound surrounded by a fluid at rest. Ci(Xi, t) is the turbulent velocity vector.

.... ~(t)

U' I

Figure 2.2: Turbulent flow region as a source of sound surrounded by a fluid in uniform motion. Ui(Xi, t) = Ci(Xi, t) - Ui is the turbulent velocity vector relative to the ambient flow velocity, Ui .

2.2.2 CONVECTIVE LIGHTHILL EQUATION The situation of a medium at rest outside the source region is a rare condition in the case of aerodynamic noise generation. A more realistic assumption is that the fluid flows past a noise generating region. In such a situation, the source term q in equation (2.11) is not quadratically small outside the noise generating region. This case is better approximated by the assumption of a uniform flow speed outside the turbulent flow region as shown in Figure 2.2. The convective form of the Lighthill equation (2.11) can be derived for this situation as was first shown by Ribner (1959) for the study of flight effects on jet noise.

1 ([) [) ) 2 [)2p a6 fJt + Ui [)Xi P - [)x; = q. (2.13)

Xi denotes the coordinates in a coordinate system fixed relative to a point within the source region (e.g., the center of the nozzle in the case of jet noise or a position on the rotor axis of an axial fan), Ui is the velocity vector of the ambient flow relative to this coordinate system, and ao is the sound speed in the ambience.

The source term q for equation (2.13) is given by

[)2 ( [) [) ) 2 ( P ) q = [)Xi[)X j (pUiUj - Tij) - [)t + Ui [)Xi P - a6 ' (2.14)

34

where Ui = Ci-Ui is the local velocity vector relative to the velocity vector U i in the ambience. Equation (2.13) has the property that the left hand side describes the sound wave prop-

agation in a medium with uniform velocity Ui and sound speed ao. The right hand side is produced by the turbulent flow and is zero, or at least quadratically small in the fluctuations, outside the region occupied by the turbulent flow. It was shown, e.g., by Michalke & Michel (1979) with the assumption that the entropy is conserved along particle path lines (viscous stresses Tij = 0, no heat conduction) that equation (2.14) can be approximated by

(2.15)

where (2.16)

and qi = p/~ (Po) .

OXi P (2.17)

pi = P _ Po is the difference between the local pressure p and the ambient pressure po. Terms of order p'2/ (poa6) were neglected, which might not be permissible in expansion cells of supersonic jets or in rotor flows with higher pressure ratios.

The first term on the right hand side of equation (2.15) which is defined by % in equation (2.16) describes a quadrupole source distribution following the argumentation for the term Tij of Lighthill (1952).

The second term on the right hand side of equation (2.15) describes a dipole source distribution which was overlooked by Lighthill and is even rarely considered in more recent studies of aerodynamic noise generation. It can be concluded from equation (2.17) that this unsteady density source term qi will only be significant if the local density p differs from the ambient density po. This unsteady density source term was discussed by Ffowcs Williams (1969) but it was first shown by Morfey (1973a) that these terms are important for hot jets. Dipole sources are generally more efficient than quadrupole sources and it will be shown later in chapter 3, that this dipole source term is an important contribution to jet mixing noise if the density of the jet is different from the density of the ambient fluid.

2.2.3 ELEMENTARY SOUND SOURCES Elementary sources, e.g., monopoles, dipoles, and quadrupoles, play important roles in acous-tic theory. The sound field of a monopole source can be created by a pulsating sphere as shown in Figure 2.3. The directivity of a monopole is uniform in all directions. A good approximation of a monopole source is an oscillating diaphragm in an infinite baffle if the diameter of the diaphragm is small compared to the acoustic wave length.

The sound field of a dipole source can be created by two monopoles with opposite sign (the source rate of one monopole is absorbed by the sink rate of the second) whose distance is very small as shown in Figure 2.4. The dipole strength M = hQ is defined as the monopole strength Q times the distance h between the two monopoles. The distance must be very small compared to the acoustic wave length. The directivity of a dipole is proportional to cos (), where () = 0 is alined with the axis of the two monopoles (derivation in Fuchs & Michalke (1973)).

The sound field of a quadrupole source is equal to the sound field of two closely spaced dipoles of opposite strength or four closely spaced monopoles with opposite strength as shown

35

directivity

pulsating sphere

streamlines

diaphragm in infinite baffle

monopole

Figure 2.3: Sound field of a monopole source created by a pulsating sphere or an oscillating diaphragm in an infinite baffle.

directivity streamlines

oscillating sphere

dipole

Figure 2.4: Sound field of a dipole source created by two adjacent monopoles in antiphase or an oscillating sphere.

longitudinal quadrupole

lateral quadrupole

Figure 2.5: Sound field of quadrupole sources created by four adjacent monopoles located in a linear array (qii,i = 1,2,3) or in a square (qij,i -I- j).

in Figure 2.5. A longitudinal quadrupole (qii, i = 1,2,3) is formed when the two dipoles or the four monopoles are located on a line. A lateral quadrupole (qij, i -I- j) corresponds to two dipoles located side by side or four monopoles located in a square. The distance between the sources must be very small compared to the acoustic wave length. The directivity of a longitudinal monopole is proportional to cos2 0, where e = 0 is the axis of the two dipoles. The directivity of a lateral dipole is proportional to cos 0 sin e (derivation in Fuchs & Michalke

36

(1973) ).

2.2.4 SOURCES IN TERMS OF THE LIGHTHILL EQUATION

The basic assumption in Lighthill's analysis is that the source distributions on the right hand sides of equations (2.9), (2.11), or (2.13) are known. This is generally not the case, because a turbulent motion is the result of the same equations for the conservation of mass and momentum that were used for the derivation of the above mentioned wave equations. The importance of Lighthill's approach comes from its capability for deriving scaling laws when certain scaling assumptions are made for the turbulent source terms. This capability is based on the existence of closed form solutions for the Lighthill equation. The first, now famous scaling law was the result of Lighthill (1954) that the acoustic power of a free jet is proportional to the eighth power of the jet's speed. It will be shown in the next chapter, that this is only valid for a jet with constant density because the dipole source terms lead to a sixth power law which agrees with measured data from hot jets like those of jet engines.

The right hand side q of equation (2.13) on page 34 includes the influence of all deviations of the mean velocity from the velocity Ui in the ambience (e.g., due to the mean velocity profile in a jet) and of all deviations of the local speed of sound from the ambient value ao (e.g. in areas with different temperatures) on sound propagation. Through this mechanism, the effects of convection and refraction of sound waves in high speed jet flows are considered as source effects despite being the result of propagation effects. Sources have to be known a priori while propagation effects can act only on the generated sound waves.

To resolve this contradiction, other acoustic wave equations have been derived for mean velocity profiles which more resemble the actual mean velocity distributions in the flow. It was assumed that the right hand sides would better describe the "true" sources in the flow. One of these equations was derived by Lilley. He assumed that only terms that are nonlinear in the fluctuations are true source terms. Therefore, all terms that are linear in the fluctuations were moved to the left hand side of the equation. The final differential equation has the order three. Unfortunately, solutions for this partial differential equation can be obtained only numerically. An additional complication originates from the fact that the equation has non-trivial solutions for the homogeneous equation (when the source term on the right hand side of the equation, q _ 0). The great advantage of Lighthill's equation in comparison to these other aeroacoustic wave equations is that closed form solutions exist.

2.3 SOLUTION OF THE LIGHTHILL EQUATION 2.3.1 KIRCHHOFF INTEGRAL Supposed the source function q in equation (2.13) is known as function of time and space. Then, general solutions are available from classical theories. For the case of a source region surrounded by an infinite space which may contain solid surfaces, aI,ld for Ui = 0 (only to simplify the following equations for these notes, the external motion will we included in chapter 3) and for a uniform sound speed ao in the ambience, we obtain the Kirchhoff integral for the pressure perturbation p'(Xi' t) = p(Xi' t) - Po in a field point Xi

pi (Xi, t) 1 J1 {[ [J2%] [Oqi]}dV() 47r r OYiOYj + 0Yi Yi +

V

37

~ L/s1 source point

field point

turbulent region

Figure 2.6: Explanation of source point Yi, field point Xi and radiation distance r = hi.

1 J {1 [op] 1 or 1 or [op]} - - - +--[p]+-- - dS(Yi). 41r r an r2 an aor an at

s

(2.18)

The notation is explained in Figure 2.6. The volume integral can be limited to the volume Va that contains the turbulent flow. r = Iril = IYi - xii is the radiation distance between the source point Yi and the field point Xi. The brackets indicate that the enclosed terms have to be evaluated at the retarded time tr = t - r / ao which considers the time needed by the sound wave to propagate the distance r from the sorce position Yi to the observer position Xi with a sound speed ao. The integral (2.18) is valid in any position, even within the source region Va. Later, far field approximations will be presented that require that Xi is far outside Va.

It can be seen in equation (2.18), that the pressure perturbation pi in the field point Xi is made up by a volume integral involving the source terms qij and qi which are described by equations (2.16) and (2.17), and by a surface integral involving the pressure p. a/an is the gradient in the direction of the outward normal of a surface element. The volume integral was first studied with respect to aerodynamic noise generation by Lighthill (1952), the surface integral by Curle (1955).

2.3.2 INTEGRAL OF CURLE A more convenient form of the solution (2.18) can be derived according to Curle (1955).

(2.19)

It can be concluded from this equation that the sound field in the field point Xi is created by a volume distribution of quadrupoles qij, a volume distribution of dipoles qi, and a surface distribution of dipoles ii, where ii are the forces per unit area acting on the fluid at each surface element dSi . The brackets indicate again, that the source distributions have to be

38

evaluated at the appropriate retarded times. The integral (2.19) is valid in any observer position Xi.

It must be noted, here, that the solution for the sound radiation into a duct is different from this solution into free space. A solution for a circular duct was discussed by Goldstein (1974).

2.3.3 FAR-FIELD SOLUTION In almost all practical applications (e.g., the sound generation by jets, propellers, wind turbines, and helicopters) one is interested in the sound radiation at large distances from the source region. This allows two simplifications:

Field points Xi in the acoustic far field (far in comparison to the sound wave lengths): the space derivatives 0/ OXi and 02 / (OXiOXj) in equation (2.19) can be replaced by time derivatives .

Field points Xi in the geometric far field (far in comparison to the dimensions of the source region): the factor l/r in equation (2.19) can be considered identical for all source points Yi and can be determined with the distance r r between a reference position Yri within the source region and the field point Xi.

With the reference position arbitrarily defined as the origin of the coordinate system, Yri = 0, equation (2.19) simplifies to

(2.20)

with r = IXil. This equation can be written in a form that is physically easier to understand. The three

integrals shall be studied, separately:

(2.21 )

The first contribution is described by a volume integral over a quadrupole source distri-bution

I 1 J [02qq] Pq(Xi, t) = 47ra6r ot2

v

(2.22)

where the quadrupole source term is described by

(2.23)

This can be approximated for small pi to

(2.24)

39

Here, U r is the component of the velocity Ui in the direction to the observer point Xi. The integral (2.22) describes the sound due to those velocity fluctuations in the source region that are directed toward the observer. The brackets indicate evaluation at the retarded time,

(2.25)

The second term on the right hand side of equation (2.21) IS described by a volume integral over a dipole source distribution,

p~( X" t) = 47r~or I [~~d 1 dV(Yi), (2.26) where the dipole source term is defined by

, a (po) qd = P aYr P . (2.27)

a / aYr is the component of the gradient in the direction toward the observer. The integral (2.26) describes the sound due to the pressure fluctuations in a density gradient toward the observer.

The third term in equation (2.21) is described by a surface integral over a dipole source distri bu tion,

pj(x/, t) = 41r~or! [8f; 1 dS(Yi), (2.28) where ir is the component of the surface force per unit area acting on the fluid, ii, in the direction to the observer. The integral is the result of fluctuations of the surface force component directed toward the observer.

The volume integrals will be studied in more detail in the next chapter about jet noise. The influence of surfaces is then studied in chapter 4.

2.4 TIME AVERAGED SOLUTIONS Many aeroacoustical studies are content with a solution for the sound pressure in the time domain. However, in acoustics, one is generally not interested in time histories of pressure fluctuations but in the mean square of the pressure fluctuations and their frequency spectra.

2.4.1 MEAN SQUARE VALUE The mean square of the pressure fluctuations P'(Xi, t) is defined by

(2.29)

The mean square is independent of the integration boundaries t1 and t2 only if p'(Xi, t) is stationary random. This means that the integration volumes and integration surfaces in

40

equations (2.22), (2.26), and (2.28) must be stationary or at least periodic (for the cases of rotors) .

The sound pressure level is then defined by

(2.30)

where Pref = 2 . 10-5 Pa in air.

2.4.2 AUTO-CORRELATION FUNCTION The distribution of the mean-square sound pressure in the frequency domain is expressed by the power-spectral density of the pressure fluctuations which can be derived from the autocorrelation function Rpp( Xi, T) of the pressure fluctuations in the far-field point Xi

(2.31 )

With the far-field solution (2.21) we obtain

(2.32)

Rppqq(Xi' T) is the result of equation (2.31) when solution (2.22) is inserted in equation (2.31). Rppdd(Xi,T) and Rppff(Xi,T) are the corresponding results with solutions (2.26) and (2.28). It is assumed here for simplicity that the three source distributions a2qq/at2, aqd/at, and a iT / at are not correlated. Otherwise, six additional terms with the cross-correlations between the different source distributions would have to be considered, in addition.

2.4.3 POWER-SPECTRAL DENSITY The power spectral density Wpp ( Xi, 1) of the pressure fluctuations in the far field point Xi can be obtained by Fourier transforming equation (2.31). This yields

00

Wpp (Xi, 1) = j Rpp(Xi,T) exp(i27r}T) dT. (2.33) -00

Like the autocorrelation function, the power-spectral density function Wpp(.Ti, 1) consists of contributions from the three source distributions. Under the condition that these are uncorrelated, we obtain

(2.34)

The power spectral density Wppqq ( Xi, 1) of the pressure fluctuations in the far field point Xi due to the quadrupole source distribution is given by the double integral

Wppqq (Xi, 1) = (47r:a6)2j j WQQqq(Yi,1]i,1)exp(i~R)dV(1]i)dV(Yi)' v v

41

(2.35)

toward observer

Figure 2.7: Definition of separation vector 1]i and 1]r.

where WQQqq(Yi' 1]i, 1) is the cross-spectral density 00

WQQqq(Yi' 1]i, 1) = J --=-Q-q('---Yl-" t---,--)---=-Q----,q(,---Yl- +-1]i-, t-+-T-'-) exp(i21f IT) dT, (2.36) -00

of the source function (2.37)

between the two source positions Yi and Yi + 1]i. as a function of frequency 1. The separation vector 1]i between the two source points is illustrated in Figure 2.7.

The phase difference 'l/JR in equation (2.35) is determined by the time difference !:ltr between the retarded times in the two source positions Yi and Yi + 1]i,

(2.38)

where 1]r is the component of the separation vector 1]i in the direction () of the observer. The other two power-spectral densities in equation (2.34) are defined, correspondingly.

The volume dipole source distribution is defined by

Wppdd ( Xi, 1) = (41f:ao)2 J J WQQdd(Yi, 1]i, 1) exp( i'l/JR) dV( 1]i) dV(Yi) v v

with the source function

Q ( . t) = aqd(Yi, t) d Yn at' and the surface dipole distribution by

Wppf f( Xi, 1) (41f:ao)2 J J WQQf f(Yi, 1]i, f) exp( i'l/JR) dS( 1]i) dS(Yi) s s

with the source function Q ( . t) = alr(Yi, t) f Yl, at'

42

(2.39)

(2.40)

(2.41 )

(2.42)

The power-spectral densities Wppqq(Xi' j), Wppdd(Xi, j), and Wppff(Xi, j) include the ef-fects of interference between the sources. The importance of these interference effects will be shown in the next chapter.

2.5 BIBLIOGRAPHY OF CHAPTER 2 CUMPSTY, N. A., 1977. Review A critical review of turbomachinery noise. ASME-

Transactions) Journal of Fluids Engineering 99, 278~293. CURLE, N., 1955. The influence of solid boundaries upon aerodynamic sound. Proceedings

of the Royal Society (London) A 231, 505~514. FFOWCS WILLIAMS, J. E. & HAWKINGS, D. L., 1969. Sound generated by turbulence and

surfaces in arbitrary motion. Philosophical Transactions of the Royal Society (London) A 264, 321 ~342.

FFOWCS WILLIAMS, J. E., 1969. Hydrodynamic noise. Annual Review of Fluid Mechanics 1, 197~222.

FUCHS, H. V. & MICHALKE, A., 1973. Introduction to aerodynamic noise theory. Progress in Aerospace Science 14, 227~297.

FUCHS, H. V., 1976. Basic aerodynamic noise theory. AGARD-VKI Lecture Series 80 on "Aerodynamic Noise".

GOLDSTEIN, M. E., 1974. Unified approach to aerodynamic sound generation in the pres-ence of solid boundaries. Journal of the Acoustical Society of America 56, 497~509.

LIGHTHILL, M. J., 1952. On sound generated aerodynamically. I. General theory. Proc. Royal Soc. London A 211, 564~587.

LIGHTHILL, M. J., 1954. On sound generated aerodynamically. II. Turbulence as a source of sound. Proc. Royal Soc. London A 222, 1~32.

MICHALKE, A. & MICHEL, U., 1979. Prediction of jet-noise in flight from static tests. Journal of Sound and Vibration 67, 341~367.

MORFEY, C. L., 1973a. Amplification of aerodynamic noise by convected flow inhomo-geneities. Journal of Sound and Vibration 31, 391~397.

MORFEY, C. L., 1973b. Rotating blades and aerodynamic sound. Journal of Sound and Vibration 28, 578~617.

RIBNER, H. S., 1959. New theory of jet-noise generation, directionality, and spectra. The Journal of the Acoustical Society of America 31, 245~246.

43

44

Chapter 3

APPLICATION TO JET NOISE

Contents of Chapter 3

3 APPLICATION TO JET NOISE 45 3.1 INTRODUCTION ........ . . . . . . 45 3.2 SOUND PRESSURE IN THE FAR FIELD. 46 3.3 TIME AVERAGED SOLUTIONS ...... 49

3.3.1 AUTOCORRELATION OF SOUND PRESSURE. 50 3.3.2 POWER-SPECTRAL DENSITY 51 3.3.3 DIRECTIVITY OF JET NOISE 53

3.4 SCALING OF FLOW FIELD OF JET 56 3.5 SCALING OF SOUND PRESSURE 57 3.6 SUPERSONIC JETS .......... 59

3.6.1 MACH WAVE RADIATION .. 59 3.6.2 BROADBAND SHOCK NOISE. 60 3.6.3 SCREECH ............ 63

3.7 RELATION BETWEEN FLYOVER AND WIND-TUNNEL CASES 63 3.8 PREDICTION OF JET NOISE . 64

3.8.1 STATIC JET NOISE ..... 64 3.8.2 JET NOISE IN FLIGHT ... 64

3.9 BIBLIOGRAPHY OF CHAPTER 3 64

3.1 INTRODUCTION The sound generated by a free jet when it mixes with the ambience is generally not important for turbomachinery noise generation. However, this sound generation process was the first application of Lighthill's acoustic analogy and it is a remarkable example for the capabilities of analytical considerations concerning aerodynamic sound generation. In addition, it is an important contribution to the noise of aircraft jet engines.

The now famous scaling law for the sound power of a jet (though only valid for a jet with constant density) was derived from equation (2.9). The sound power was expected to be

45

proportional to the eighth power of the jet's speed (Lighthill (1954)). In addition, the theory explained why the sound pressure level in the far field of a jet is larger in the rear arc (in the direction of the jet's mean velocity vector) than in the forward arc. This was possible, despite of the fact that the source term q on the right hand side of the inhomogeneous wave equation (2.9) was not known. Lighthill only assumed that the sources are convected with the flow and made certain assumptions on how the sources scale on mean flow quantities.

However, other experimental findings could not be explained with Lighthill's results, e.g., the deviations from the eighth power law for low jet speeds and for hot jets, the additional noise generated by supersonic jets, and the unexpected high sound pressure levels of aircraft in flight. In addition, the frequencies found experimentally in the rear arc are not higher than in the forward arc as was expected from the assumption of moving sources and the frequencies in flight are not reduced as expected from the reduced relative jet speed.

It will be shown in this chapter, that all these experimental findings can be explained and described with the acoustic analogy provided the correct source model is chosen and the dipole source term is included.

3.2 SOUND PRESSURE IN THE FAR FIELD

We start with the convective form of the Lighthill equation (2.13) which is repeated here,

(3.1 )

and use definition (2.15) for the source function q,

(3.2)

in which the original Lighthill source function EPTij / oxJix j of quadrupole type is replaced by a quadrupole source function 02%/OXiOXj and a dipole source function Oqi/OXi. qij and qi are defined by equations (2.16) and (2.17), respectively. It was already concluded in the previous chapter that the dipole source term qi may be important when density gradients are present in the jet flow.

Equation (3.1) has the property that the left hand side describes the sound wave prop-agation in a medium with uniform velocity Ui and sound speed aQ. The right hand side is produced by the turbulent flow and is zero, or at least quadratically small in the fluctuations, outside the region occupied by the turbulent flow.

A solution of equation (3.1) for an unbounded field and for the static case Ui = {Uj, 0, o} = was already derived as equation (2.18). In jet noise, the surface integrals playa role for some installation effects, but this influence shall not be considered, here.

The influence of the external velocity vector Ui cannot be neglected when the important effects of flight speed on jet noise are to be studied. This situation is described in Figure 3.1 in a coordinate system fixed on the nozzle (wind tunnel coordinate system) in which the flow is stationary random, i.e., all time averaged flow quantities are independent of the start time of the averaging process.

The solution of equation (3.1) for an unbounded field that includes the effects of Ui

46

u I

y. I

Figure 3.1: Free jet in an external stream Ui = {Uj,O,O} described in a coordinate system Yi fixed on the nozzle of the jet.

{Uf , 0, O} was derived by Michalke & Michel (1979),

(3.3) --------~vr------~ ~----~vr-------

quadrupole sources dipole sources

This solution for the pressure fluctuations is valid in any position Xi inside or outside the integration volume V which must include the turbulent flow field of the jet. With qi 0 and flight speed Uf = 0, the solution corresponds to that discussed by Lighthill (1952) for the density fluctuations, qij being very similar to Lighthill's stress tensor Tij . The dipole term qi in the second integral was overlooked by Lighthill but is important for hot jets.

In comparison to equation (2.19), the uniform motion of the external fluid is now consid-ered while the surface integral is neglected. The external motion is considered by replacing the distance l' in the denominators in the two integrands by the term roD f. D f is the Doppler factor defined by

Df = 1- MafcosBo (3.4) with the flow (or flight) Mach number Ma f = Uf / ao. The emission distance 1'0 and the emission angle Bo are explained in Figure 3.2. The Figure shows a source point Yi within the jet flow field and a field (or observer) point Xi in the external stream. l' is the distance between these two points and B the corresponding geometrical angle relative to the upstream direction. 1'0 is the distance the wave has to propagate with the speed of sound relative to the moving medium to reach the observer. Maf 1'0 is the distance by which the wave is convected downstream by the external flow during the time of propagation, 1'0/ ao. Bo is the angle normal to the wave fronts of the propagated waves. It is therefore also called wave-normal angle (Morfey (1979)). Here, we use the term emission angle because it is the angle into which the sound is emitted from the source. The bracketed terms have to be evaluated at the retarded time ir which is defined by

(3.5) where t is the time of observation. The choice of emission coordinates (Xi) rather than observer coordinates (1', B) is causal for the compact form of equation (3.3). The observer

47

source point ---- ------

8 o r 8

Figure 3.2: Relation between source and observer position III a wind tunnel coordinate system.

coordinates can be transformed to emission coordinates by

(3.6)

and cos eo = cos e [Jl - Ma} sin2 e - Maj cos e] + Maj. (3.7)

The near field of jets is of interest when the acoustic loads on structures must be de-termined which is a problem for some high performance aircraft. Generally, one is more interested in the sound received in positions far away from the jet. The solution (3.3) can then be simplified .

Acoustic far field - emission distance "'0 far in comparison to the wave length A: the space derivatives O/OXi and fJ2/(OXiOXj) can be replaced by time derivatives, e.g., aoo / OXi = a/at.

Geometric far field - emission distance "'0 far in comparison to the dimensions Yi within the source volume: the Doppler factor D j can be considered to be identical for all source positions and one emission distance "'0 relative to a reference position can be applied to all source positions except for its influence on retarded times.

It was shown by Michalke & Michel (1979) that the sound pressure in the acoustic and geometric far field point Xi is given by the following integral over the source volume V:

where po) / - P P

(3.9)

48

is a quadrupole source term, and

I fJ (po) qd =p-- -fJYra P

(3.10)

is a dipole source term. The emission angle eo and the emission distance /0 as well as the source point Yi and

the field point Xi are now defined relative to a reference position within the source volume. Generally, the center of the nozzle exit plane is used as shown in Figure 3.3.

Source position 1 position 2

I Observer position

Figure 3.3: Definition of source and observer coordinates in the geometric far field.

Yra and UTa are the components of Yi and Ui, respectively, in the direction of Bo. fJ / fJYra is the gradient in that direction.

The retarded time for the terms in brackets in equation (3.8) is now defined by

iT = t - /0 + ~. (3.1l) ao aoDj

The last term considers the retarded time difference between the source position Yi and the reference position.

The far field solution (3.8) consists of two integrals. The first is Lighthill's quadrupole contribution, the second is a dipole contribution. Note, that the exponent of the Doppler factor D j is different for both integrals which means that flight speed has a larger effect on the quadrupole noise of a jet (which is the only sound contribution of a jet with constant density, e.g., many laboratory jets) than on dipole noise (which requires density gradients in the jet flow, e.g., like in an engine jet). This explains why different flight effects were found for pure jet engines which have a large dipole contribution and for high-bypass-ratio engines which have a relatively large quadrupole contribution.

3.3 TIME AVERAGED SOLUTIONS One is generally interested in time averaged solutions, e.g., the mean square or the power-spectral density of the sound pressure in the far field. One-third octave spectra or octave spectra can then be computed from the power-spectral density. The mean-square value is

49

defined by equation (2.29) from which the sound-pressure level can be derived with equation (2.30). The autocorrelation function is the basis for the determination of mean-square values as well as power-spectral densities.

3.3.1 AUTOCORRELATION OF SOUND PRESSURE The far-field solution of equation (3.8) is stationary random in the coordinate system fixed on the nozzle. The autocorrelation function Rpp (Xi, T) is then defined by

(3.12)

The mean square value of the sound pressure is given for T = 0,

(3.13)

If only the quadrupole term of solution (3.8) is inserted in equation (3.12), we obtain the autocorrelation due to quadrupole terms,

where the observer position Xi is expressed in terms of emission distance TO and emission angle eo. With Yli = Yi and Y2i = Yi + T/i, and after exchanging the sequence of integrations we obtain

R (x T) = 1 J J 1 Jt2 {[82qq(Yi' t)] [82qq(Yi + T/i, t + T)] } dt d . d . ppqq~, 16 2 2 4D6 t - t 8t2 8t2 T/~ Y~ 7r Toao j 2 1 t t V V t1 r r

~------------------v--------------------~ Rqq(Yi,7)i,r)

(3.14) The integrand of the double integral over the source volume is the cross-correlation

function Rqq (Yi, T/i, T) of the quadrupole source terms between the positions Yi and Yi + T/i The separation vector T/i between the two source points is illustrated in Figure 3.4.

Equation (3.14) can then be abbreviated as follows:

(3.15)

The double integral over the source volume will be discussed later. Here, we shall discuss the terms in front of the integral. The autocorrelation function of the sound pressure (as well as the mean square value) is inversely proportional to the square of the wave normal distance TO. This is expected because T5 p'2=T5Rpp ( Xi, 0) is independent of TO in the geometric far field (spherical sound propagation). Notable is the Doppler factor D j 1 Maj cos eo in the denominator which appears with a power of six. Since D j < 1 in the forward arc (toward the flight direction) and D j > 1 in the rear arc, flight speed causes a dramatic increase of the sound pressure in the forward arc and a corresponding decrease in the rear arc. This explains the strong forward arc amplification of jet noise observed in flight which was not understood for a long time.

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Source position 1 Source position 2

I

I Observer position

Figure 3.4: Definition of separation vector 1]i and 1]7'0'

A corresponding equation can be derived for the dipole-dipole-term:

(3.16)

where the cross-correlation function Rdd(Yi, 1]i, r) of the dipole-dipole terms is defined by

(3.17)

Please note that the power of the Doppler factor D j has decreased to four in equation (3.16) in comparison to six in equation (3.15). This means that the flight effect is larger for the quadrupole source terms.

The evaluation of equation (3.12) by use of equation (3.8) also results in terms containing the cross-correlation functions between quadrupole and dipole terms. These will be neglected here for simplicity which is equivalent to the assumption that quadrupole and dipole source terms are not correlated.

The auto-correlation function Rpp( Xi, r) is then given by the sum of only the two equations (3.15) and (3.16).

Rpp(Xi,r) = Rppqq(xi,r)+RpPdd(xi,r) (3.18) This equation demonstrates that the autocorrelation function (as well as the mean-square

value of the sound pressure, see equation (3.13)) in the far field of a jet consists of contri-butions from quadrupole sources and of dipole sources. The solution is exact, except for the neglect of the cross-correlation terms between quadrupole and dipole source terms and the assumption that viscous stresses and heat conduction do not playa significant role in aerodynamic sound generation (see page 35 in chapter 2).

3.3.2 POWER-SPECTRAL DENSITY The power-spectral density Wpp(Xi' f) of the pressure fluctuations in the far field point Xi can be obtained by Fourier transforming the autocorrelation function as defined by equation

51

(3.18). This yields (3.19)

The contribution of the quadrupole terms is given by the double integral

(3.20)

with the cross spectral density Wqq

(Xl

Wqq (Yi, 1]i, 1) = J --=Q-q (-:-Y-i , ---:-t )---:-Q-q ('--y l-' +-1]i-, t-+-T---:-) exp ( i 2 7f j T ) d T, (3.21 ) -(Xl

of the quadrupole source function

(3.22)

between the two source positions Yi and Yi + 1]i where qq is defined by equation (3.9). j is the frequency in the coordinate system fixed on the nozzle.

The phase difference 'l/;r in equation (3.20) is determined by the difference !:::"tr between the retarded times in the two source positions Yi and Yi + 1]i,

(3.23)

where 1]r