Modelling Jet Noise Installation Effects Associated with Close-Coupled, Wing-Mounted, Ultra High Bypass Ratio Engines

UNIVERSITY OF SOUTHAMPTON

Modelling Jet Noise Installation Eﬀects Associated with Close-coupled, Wing-mounted, Ultra High Bypass Ratio Engines

by Juan Vera

A thesis submitted in partial fulﬁllment for the degree of Doctor of Philosophy

in the Faculty of engineering and the environment Institute of Sound and Vibration Research

May 2018

Declaration of Authorship

I, Juan Vera, declare that the thesis entitled Modelling Jet Noise Installation Eﬀects Associated with Close-coupled, Wing-mounted, Ultra High Bypass Ratio Engines and the work presented in the thesis are both my own, and have been generated by me as the result of my own original research. I conﬁrm that:

• this work was done wholly or mainly while in candidature for a research degree at this University;

• where any part of this thesis has previously been submitted for a degree or any other qualiﬁcation at this University or any other institution, this has been clearly stated;

• where I have consulted the published work of others, this is always clearly at- tributed;

• where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work;

• I have acknowledged all main sources of help;

• where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself;

• none of this work has been published before submission

Signed: ......

Date: ...... • • • •. • • • • • • • • • • • • • • • • • • • • • •

III ”We live on a placid island of ignorance in the midst of black seas of inﬁnity, and it was not meant that we should voyage far.” H. P. Lovecraft UNIVERSITY OF SOUTHAMPTON

ABSTRACT

FACULTY OF ENGINEERING AND THE ENVIRONMENT INSTITUTE OF SOUND AND VIBRATION RESEARCH

Doctor of Philosophy

Modelling Jet Noise Installation Eﬀects Associated with Close-coupled, Wing-mounted, Ultra High Bypass Ratio Engines

by Juan Vera

The research has studied the jet noise installation effects associated with close-coupled, wing-mounted, ultra high bypass ratio engines, which main source is produced by the scattering of the hydrodynamic field passing the wing trailing edge. The objectives of the thesis were to understand the noise generation associated with this mechanism and to produce an analytical methodology able to predict the Sound Pressure Level that is radiated to the far-field.

The hydrodynamic pressure field of a jet is studied from the theoretical and experimental point of view. Whether Lighthill’s acoustic analogy is extended into the near-field by removing the far-field assumptions, three different terms can be found. These terms (near-field term, mid-field term and far-field term) were named regarding the geometric region in which each of them is dominant. When compared against experimental data, the three terms are found and the scaling laws, obtained from the theory, are proof.

Wiener-Hopf technique is then used to obtain an expression for the scattered pressure ﬁeld and the result is compared with the classic solution from Amiet. The knowledge learnt is used to create a prediction methodology for a static single-stream jet interacting with a parallel ﬂat plate. Acknowledgements

This research was sponsored by the Rolls-Royce University Technology Centre in Gas Turbine Noise of the University of Southampton, where the research has been car- ried out.

I am specially thankful to Dr. Michael Kingan, his support and supervision was essential in the development of this thesis. Thanks to my academic supervisor Prof. Rod Self, I am grateful for his guidance. Also, I would like to thank Dr. Aldo Rona and Prof. Phillip Joseph for the recommendations given during the examination of the thesis.

My thanks also go to Adolfo Serrano who awakened my interest in aeroacoustic and to Dr. Paul Strange who introduced me to jet noise. Thank you both.

Dr. Jack Lawrence is responsible for the Doak laboratory data used in this thesis. I would like to acknowledge him for the help analysing it. This data was part of the test campaign that was conducted during the SYMPHONY project, funded by the UK Technology Strategy Board and Rolls-Royce Plc.

Large eddy simulations f rom University of Cambridge was used i n this Thesis. I am grateful to Prof. Paul G. Tucker, Dr. I ﬃ Z. Naqavi, and Dr. Zhong-Nan Wang f or sharing the data.

I would like to thank my parents for raising me the way they did and to my brother for being a role model to me; to my friends at ISVR for living with me this experience; to the people of Leyenda.net for helping me keep my sanity intact and last, but no means least, to my partner, Susana, for believing in my more than I did, thank you with all my heart.

VI Contents

Declaration of AuthorshipIV

Abstract V

AcknowledgementsVI

ContentsXI

List of FiguresXV

List of Tables XVII

NomenclatureXX

1 Introduction3 1.1 Thesis outline...... 4

2 Background and literature review7 2.1 Chapter overview...... 7 2.2 Noise pollution...... 8 2.2.1 Effects of noise pollution...... 9 2.2.2 Aircraft noise pollution...... 10 2.3 Civil aviation legislation...... 11 2.4 Aircraft noise sources...... 17 2.4.1 Airframe noise...... 17 2.4.2 Engine noise...... 19 2.4.3 Jet-Wing installation effects...... 20 2.4.4 Computation of jet flow field using CFD...... 23 2.5 Jet mixing noise theory...... 24 2.5.1 Jet flow field...... 24 2.5.2 Jet acoustics...... 25 2.6 Trailing edge noise...... 28 2.6.1 Curle’s theory for solid bodies...... 28 2.6.2 Extension for moving sources...... 30 2.6.3 Noise produced by turbulent flow past a trailing edge...... 31 2.6.4 Amiet’s model for noise due to turbulent flow past a trailing edge. 34

3 Study of the jet near-ﬁeld 39 3.1 Chapter overview...... 39

VII VIII CONTENTS

3.2 Near-field extension of Lighthill’s acoustic analogy...... 41 3.2.1 Cross-power spectral density...... 43 3.2.2 Scaling of the jet near-field...... 46 3.3 Near-field experimental data...... 48 3.3.1 Near-field experimental setup...... 48 3.3.2 Radial Scaling...... 53 3.3.3 Velocity Scaling...... 60 3.4 Jet near-field prediction...... 63 3.4.1 A model for the fourth order correlation tensor...... 63 3.4.2 A prediction model based on the extended Lighthill’s theory... 63 3.4.3 Comparison with experimental data...... 66

4 Theoretical model for the scattered pressure field 71 4.1 Chapter overview...... 71 4.2 Problem description...... 72 4.3 Scattering process...... 72 4.3.1 Incident pressure field...... 72 4.3.2 Scattered pressure field...... 74 4.3.2.1 Solution for P s using the Wiener-Hopf technique..... 75 4.3.2.2 Solution for P s on the flat plate...... 80 4.3.2.3 Solution for P s far away from the flat plate...... 83 4.4 Far-field radiation...... 86 4.4.1 Method 1:...... 87 4.4.2 Method 2:...... 88 4.5 Far-field directivity...... 91 4.6 Acoustic spectrum...... 92 4.7 Comparison with Amiet’s trailing edge model...... 93

5 A prediction methodology for the jet-surface interaction 95 5.1 Chapter overview...... 95 5.2 Problem specification and assumptions...... 97 5.3 Hydrodynamic field model...... 98 5.3.1 Numerical LES data...... 99 5.3.2 Jet source model...... 99 5.4 Near-field propagation...... 103 5.4.1 Benchmark of the propagation of a monopole source...... 107 5.4.2 Near-field propagation of LES data...... 108 5.5 Trailing edge pressure field...... 109 5.5.1 Small-scale experimental data...... 110 5.5.2 Trailing edge pressure Vs. Doak Laboratory data...... 111 5.6 Jet-surface interaction prediction...... 112 5.6.1 Small-scale experimental setup...... 113 5.6.2 Far-field prediction...... 114 ...... 116 5.7 Jet-surface interaction methodology and its use in more realistic cases.. 116

6 Summary and further work 121 CONTENTS IX

6.1 Conclusions...... 121 6.2 Further work...... 123

A Schwarzchild’s method 125

B Remarks on the eigenvalues of the associated Legendre function with application to conical problems 127 B.1 Numerical results...... 127 B.2 Summary...... 132

C Jet-surface interaction prediction against Doak laboratory data. 133

D Jet-surface interaction methodology and its use in more realistic cases145 D.1 Effect of the bypass ratio to the JSI...... 145 D.2 Effect of the flight stream in the JSI...... 147

List of Figures

1.1 A350-XWB in an approach trajectory at the Airbus site in Toulouse, 2013.3

2.1 Number of people exposed to noise in Europe Lden > 55 dB...... 9 2.2 Pyramid of noise effects...... 10 2.3 Evolution of the air traffic in UK from 1990 to 2015...... 11 2.4 History of increased ICAO noise stringency...... 12 2.5 Operation points for noise standards...... 14 2.6 EPNL levels for Chapter 2 and revised version of Chapter 2...... 15 2.7 Different climb profiles to satisfy one engine failure criterion...... 15 2.8 EPNL levels for Chapter 3...... 15 2.9 Number of aircraft in different Chapters operating over time...... 16 2.10 The progression of ICAO Noise Standard...... 18 2.11 Example of share between engine and airframe noise for a generic civil aircraft...... 18 2.12 Comparison of noise sources in a turbojet, LBPR turbofan and HBPR turbofan...... 19 2.13 A HBPR turbofan engine installed under a wing...... 20 2.14 Generic 1/3rd octave band breakdown of jet installation effects H/D = 0.67, L/D = 10, M = 0.75 [data from Doak Laboratory, ISVR]...... 21 2.15 Schematic of convecting and scattered pressure fields in the hydrodynamic near field...... 22 2.16 Schematic of a simple jet flow showing the classical three jet regions... 25 2.17 Cartisian coordinate system. Origin in nozzle exhaust exit center...... 26 2.18 Trailing edge coordinates and relation between them and nozzle exit coordinates...... 31

3.1 Cartisian coordinate system. Origin in nozzle exhaust exit center...... 41 3.2 DOAK laboratory. Near-field array parallel to the jet plume...... 49 3.3 DOAK laboratory. Near-field array schematic...... 49 3.4 SPL decay with the radial distance for a single-stream jet at M = 0.3... 50 3.5 SPL change with the jet velocity for a single-stream jet at rs/D = 13.09.. 51 3.6 SPL change with the jet velocity for a single-stream jet at rs/D = 0.45.. 52 3.7 Radial separation vs. Sound Pressure Level for an isothermal single- stream jet...... 54 3.8 Slopes for the radial separation decay in the Sound Pressure Level for an isothermal single-stream jet...... 55 3.9 Far-field corrected Sound Pressure Level for an isothermal single-stream jet...... 56

XI XII LIST OF FIGURES

3.10 Far-field corrected Sound Pressure Level for an isothermal single-stream jet...... 57 3.11 Mid-field corrected Sound Pressure Level for an isothermal single-stream jet...... 58 3.12 Near-field corrected Sound Pressure Level for an isothermal single-stream jet...... 59 3.13 Sound Pressure Level scaled by exit Mach number for an isothermal single- stream jet at rs/D = 13.00...... 61 3.14 Sound Pressure Level scaled by exit Mach number for an isothermal single- stream jet at rs/D = 0.45...... 62 3.15 Variation of the near-field, mid-field and far-field terms with the distance for l1 = D = 0.0381m, uj = 150m/s, τs = l1/uj ...... 67 3.16 Extended Lighthill’s theory predictions showing the contribution of near- field, mid-field and far-field terms...... 68 3.17 Doak Lab. experimental (coloured solid line) data compared with prediction using extended Lighthill’s theory at 4 different axial locations L/D = 3.15, 4.20, 5.25 & 6.30...... 69

4.1 Decomposition of the pressure field into incident pressure from the jet and scattered pressure due to the jet-surface interaction...... 73 4.2 Branch cuts defining κ3 in the complex κ1-plane...... 76 4.3 Real and imaginary components of κ3, κ3+ and κ3− in the complex κ1- plane for K = 1...... 77 4.4 Integration path in the complex κ1 (a) and t (b) planes (for K = 1). Black dotted lines represent the branch cuts associated with the definition of κ3...... 85 4.5 Closed contour of integration in the complex κ1 (a) and t (b) planes (for K = 1). Black dotted lines represent the brach cuts associated with the definition of κ3...... 86 4.6 Polar directivity for a semi-infinite chord...... 92

5.1 Schematic of the jet-surface interaction problem...... 97 5.2 Control surface and coordinate system used in the near/field model.... 99 5.3 probe distributions of near-field LES hydrodynamic array [data from Cambridge]...... 100 5.4 Source model parameters obtained from LES data using least-square method. Isothermal jet with Mj = 0.875...... 102 5.5 Comparison of LES data (symbols) and Gaussian model (line) for Auto- Spectra density of the hydrodynamic field for different Strouhal numbers and different azimuthal modes...... 103 5.6 Comparison of LES data (left) and Gaussian model (right) for cross spectral density at different Strouhal numbers; (a) m=0 & (b) m=1..... 104 5.7 Comparison of LES data (left) and Gaussian model (right) for cross spectral density at different Strouhal numbers; (c) m=2 & (d) m=3..... 104 5.8 Comparison of LES data (left) and Gaussian model (right) for cross spectral density at different Strouhal numbers; (a) m=0 & (b) m=1..... 105 5.9 Comparison of LES data (left) and Gaussian model (right) for cross spectral density at different Strouhal numbers; (c) m=2 & (d) m=3..... 105 LIST OF FIGURES XIII

5.10 Comparison of LES data (left) and Gaussian model (right) for cross spectral density at different Strouhal numbers; (a) m=0 & (b) m=1..... 106 5.11 Comparison of LES data (left) and Gaussian model (right) for cross spectral density at different Strouhal numbers; (c) m=2 & (d) m=3..... 106 5.12 Monopole source located inside the conical surface...... 108 5.13 Near-field projection of the pressure amplitude from a monopole source vs. exact solution using different number of Legendre degrees...... 108 5.14 Comparison of LES data (symbols) and projected (dashed line) power spectral density at different axial locations for 5 azimuthal modes..... 109 5.15 (a) General near-field set-up for horizontal flat plate installed jet configuration; (b) T-array of near-field surface pressure transducers [1]..... 111 5.16 Comparison of DOAK Lab. data (continuous line) and projected (dashed line) power spectral density at different span locations on the plate the trailing edge...... 113 5.17 Comparison of DOAK Lab. data (continuous line) and projected (dashed line) power spectral density at different axial locations on the plate centreline...... 113 5.18 Polar directivity patterns for an isothermal single stream jet, M = 0.3.. 115 5.19 Polar directivity patterns for an isothermal single stream jet, M = 0.5.. 115 5.20 Polar directivity patterns for an isothermal single stream jet, M = 0.75. 115 5.21 Polar directivity patterns for an isothermal single stream jet, M = 0.9.. 115 5.22 JSI prediction for an isothermal single stream jet and a flat plate H/D = 1.25, L/D = 2, Mj = 0.3...... 117 5.23 JSI prediction for an isothermal single stream jet and a flat plate H/D = 1.5, L/D = 3, Mj = 0.3...... 118 5.24 JSI prediction for an isothermal single stream jet and a flat plate H/D = 2, L/D = 4, Mj = 0.3...... 119 5.25 JSI prediction for an isothermal single stream jet and a flat plate H/D = 4, L/D = 7, Mj = 0.3...... 120

m B.1 Roots of the associated Legendre function Pµ (cos θc) = 0 with respect ◦ to µ, θc = 175 ...... 129 m B.2 Roots of the associated Legendre function Pµ (cos θc) = 0 with respect ◦ to µ, θc = 150 ...... 129 m B.3 Roots of the associated Legendre function Pµ (cos θc) = 0 with respect ◦ to µ, θc = 135 ...... 130 m B.4 Roots of the associated Legendre function Pµ (cos θc) = 0 with respect ◦ to µ, θc = 120 ...... 130 m B.5 Roots of the associated Legendre function Pµ (cos θc) = 0 with respect ◦ to µ, θc = 95 ...... 131 m B.6 Roots of the associated Legendre function Pµ (cos θc) = 0 at different cone angles for the first to orders (m = 0, 1)...... 131 m B.7 Roots of the associated Legendre function Pµ (cos θc) = 0 at different cone angles for the first to orders (m = 2, 3)...... 132

C.1 JSI prediction for an isothermal single stream jet and a ﬂat plate H/D = 1.25, L/D = 2...... 135 C.2 JSI prediction for an isothermal single stream jet and a ﬂat plate H/D = 1.5, L/D = 2...... 136 XIV LIST OF FIGURES

C.3 JSI prediction for an isothermal single stream jet and a flat plate H/D = 1.25, L/D = 3...... 137 C.4 JSI prediction for an isothermal single stream jet and a flat plate H/D = 1.5, L/D = 3...... 138 C.5 JSI prediction for an isothermal single stream jet and a flat plate H/D = 1.25, L/D = 4...... 139 C.6 JSI prediction for an isothermal single stream jet and a flat plate H/D = 1.5, L/D = 4...... 140 C.7 JSI prediction for an isothermal single stream jet and a flat plate H/D = 2, L/D = 4...... 141 C.8 JSI prediction for an isothermal single stream jet and a flat plate H/D = 4, L/D = 4...... 142 C.9 JSI prediction for an isothermal single stream jet and a flat plate H/D = 2, L/D = 7...... 143 C.10 JSI prediction for an isothermal single stream jet and a flat plate H/D = 4, L/D = 7...... 144

D.1 JSI prediction for an UHBPR engine versus CEPRA 19 experimental data.147 D.2 Empirical mexp scaling of the in-flight LES near-field data onto the static DOAK laboratory data...... 150 D.3 In-flight JSI prediction for an UHBPR engine versus CEPRA 19 experimental data...... 150 List of Tables

2.1 Noise Exposure Limits with the Criterion Level 85 dB(A) for a 3 dB(A) exchange rate...... 12

C.1 Index of ﬁgures with JSI prediction for diﬀerent plate locations...... 133

Nomenclature

Bn(.) Bessel function of the ﬁrst kind c wing chord length [m] c0 ambient speed of the sound [m/s]

D nozzle inner exit diameter [m] d total plate span [m]

E error function

E∗ complex error function f(.) generic function

2 Fi force per unit area in the xi direction exerted by solid boundaries on ﬂuid [N/m ]

G Green’s function

H vertical distance from jet centreline to aerofoil trailing edge

1 Hn(.) Hankel function of the ﬁrst kind

K 2D wavenumber in the xt1 − xt2 plane k acoustic wavenumber ki wavenumber in xi direction

L axial distance from nozzle exhaust exit to aerofoil trailing edge l2 spanwise correlation length

M jet Mach number m azimuthal mode

Mc convective Mach number

Mf ﬂight stream Mach number

XVII XVIII Nomenclature n unitary normal vector outward to the ﬂuid p pressure [P a]

n Pµ (.) Legendre function rx − y distance between observer and source locations [m] rs distance, orthogonal to the jet axis, measured from the jet shear layer [m] rt distance from trailing edge centre to observer location [m]

S surface of a solid boundary

Spp power spectral density

Rijlm fourth-order source correlation tensor

St Strouhal number t time [s]

Tij Lighthill’s stress tensor

U jet mean Velocity [m/s]

Uc convective velocity [m/s]

Uf ﬂight stream velocity [m/s] uj fully expanded jet veloctiy [m/s]

V reference volume v ﬂuid particle velocity (v1, v2, v3) [m/s] xt Observer location, origin at trailing edge centre (xt1, xt2, xt3)(rt, θt, φt) [m] x Observer location, origin at nozzle exhaust exit centre (x1, x2, x3)[m] yt Source location, origin at trailing edge centre (yt1, yt2, yt3)[m] y Source location, origin at nozzle exhaust exit centre (y1, y2, y3)[m]

Greek letters

δ(.) Dirac’s delta

δij Kronecker delta

η y − y0 separation vector between two source points

θ polar angle measured form the nozzle exhaust exit centre Nomenclature XIX

θt polar angle measured form the trailing edge centre

Π0 streamwise-integrated wavenumber spectral density of wall-pressure ﬂuctuations

ρ ﬂuid density [kg/m3]

σij compressive stress tensor

τ delayed time [s]

Φ wall pressure power spectral density

φ azimuthal angle measured form the nozzle exhaust exit centre

φt azimuthal angle measured form the trailing edge centre

ω angular frequency [rad/s]

Symbols

(.)∗ complex conjugate

(.)0 value in steady background ﬂow

(.)n value in the normal direction

()¨ second time derivative

()˙ ﬁrst time derivative

() time average value

Abbreviations

ACARE Advisory Council for Aeronautics Research in Europe

CAEP Committee on Aviation Environmental Protection

CAN Committee on Aircraft Noise

EEA European Environment Agency

EPNdB Eﬀective Perceived Noise decibel

EPNL Eﬀective Perceived Noise Level

FAA Federal Aviation Administration

FAR Federal Aviation Regulation

HBPR High Bypass Ratio

ICAO International Civil Aviation Organization XX Nomenclature

IFX jet-wing Installation Eﬀects

IMP Impingement noise

ISVR Institute of Sound & Vibration Research

JSI Jet-Surface Interaction

JSV Journal of Sound and Vibration

JWR Jet-wing reﬂection

LBPR Low Bypass Ratio

LES Large Eddy Simulation

MTOM Maximum Take-Oﬀ Mass

MTOW Maximum Take-Oﬀ Weigth

PSD Power Spectral Density

RANS Reynolds Averaged Navier Stokes

SPL Sound Pressure Level

TWT Trapped Wave Tones

UHBPR Ultra High Bypass Ratio

Physical constants

Acoustic reference pressure pref =20 µ Pa

Kinematic viscosity of dry air (20◦) ν = 1.4 × 10−6m2/s

Speciﬁc gas constant for dry air Rc = 287.1J/(kgK) To Susana. Thank you for keeping me aﬂoat all these years. This thesis could not have been possible without your truly support.

Chapter 1

Introduction

Noise pollution is a major problem in modern society. It aﬀects both physical and psychological health and can lead to hearing loss (temporary or permanent), sleep disturbance, behaviour disorders and other detrimental eﬀects [2]. With the number of aircraft operations increasing, aircraft noise is a major contributor to these problems, especially for population living near airports. For a modern turbofan engine, the noise is primarily aerodynamic and is mainly generated by the airframe and the fan, jet and turbine.

The ﬁrst commercial aircraft powered by a turbojet, the de Havilland Comet, started operating in 1952. Since then, aircraft engines have evolved signiﬁcantly. From single stream to double-stream with low bypass ratio engines and from these to the high bypass ratio (HBPR) turbofans.

Figure 1.1: A350-XWB in an approach trajectory at the Airbus site in Toulouse, 2013 [photo of Vincent Privat].

3 4 Chapter 1 Introduction

The utilization of HBPR and the coming ultra high bypass ratio (UHBPR) engines has resulted in a reduction of jet mixing noise. However, due to the large fan diameter, engines are increasingly placed closer to the wing, as it can be seen in Figure 1.1 for an Airbus A350 powered by Rolls-Royce Trent XWB engines. This configuration reduces the distance between the wing and the jet and augmenting the interaction between them, which translates to a rise in the noise level. This rise is known as jet-wing installation effects (IFX) and is produced by both the modification of the jet mixing source and the addition of new sources. The main two new mechanisms are: 1) the reflection of jet mixing noise on the aerofoil (wing and flap) surface; and 2) the jet-surface interaction noise generated when the pressure field (acoustic and hydrodynamic), produced by the jet, is scattered by the aerofoil trailing-edge.

The core of this thesis relates to the study of jet-surface interaction noise, SPLjsi, and the development of a prediction methodology for it. A study of the generation and radiation of the jet-surface interaction has been conducted in this thesis an can be found in Chapters3 and4. The near hydrodynamic field is investigated and characterized in Chapter3, while the scattering process and the far-field radiation is explained in Chapter4. In addition, a prediction methodology, able to obtain accurate predictions for the jet-surface interaction noise, has been developed in this thesis and can be found in Chapter5, where an analytical model for the interaction noise between a circular isothermal single-stream jet an a parallel flat plate is presented.

1.1 Thesis outline

The thesis is laid out as follows:

Chapter2 presents a literature review of the jet noise problem. Topics covered include an overview of noise pollution, noise associated with aviation and more speciﬁcally jet- mixing noise and jet-wing installation eﬀects.

Chapter3 explores the near-field of a jet from the theoretical and experimental points of view. Ligthill’s acoustic analogy is extended to the near-field in order to calculate both the acoustic and hydrodynamic pressure fields. Experimental data is used to characterize the jet pressure field in the near-field. The first novel aspect of work is presented here - a revised definition of the near-field of an isolated jet with the three terms found in the near-field region being characterized. In addition, a semi-empirical model to calculate the hydrodynamic near-field of a jet is presented. Chapter 1 Introduction 5

In Chapter4, a mathematical model is presented for predicting the radiated pressure produced by the interaction of the hydrodynamic field of a jet with a flat plate. The problem is formulated by assuming that the hydrodynamic field of the jet may be approximated by a single, subsonically convecting, harmonic gust. The Wiener-Hopf technique is used to obtain a theoretical expression for the scattered pressure field both on the plate surface and in the acoustic far-field. The second novel aspect of work is presented here - It is proven that the result from Wiener-Hopf technique for the scattered pressure is equivalent to the result from the boundary layer noise model of Amiet [3] but without the spurious term Amiet’s formulation has.

The third novel aspect is presented in Chapter5 - a novel theoretical prediction model for the jet-surface interaction noise. The model combines the theories of Chapter3 and4. With this methodology the far-field power spectral density of the noise produced by the interaction between an isothermal single-stream jet and a flat plate can be calculated. In this chapter, the methodology is benchmarked against different model scale experimental databases as well as against numerical large eddy simulation data.

Chapter6 presents the conclusions and recommendations for future work.

Chapter 2

Background and literature review

Contents 2.1 Chapter overview...... 7 2.2 Noise pollution...... 8 2.2.1 Effects of noise pollution...... 9 2.2.2 Aircraft noise pollution...... 10 2.3 Civil aviation legislation...... 11 2.4 Aircraft noise sources...... 17 2.4.1 Airframe noise...... 17 2.4.2 Engine noise...... 19 2.4.3 Jet-Wing installation effects...... 20 2.4.4 Computation of jet flow field using CFD...... 23 2.5 Jet mixing noise theory...... 24 2.5.1 Jet flow field...... 24 2.5.2 Jet acoustics...... 25 2.6 Trailing edge noise...... 28 2.6.1 Curle’s theory for solid bodies...... 28 2.6.2 Extension for moving sources...... 30 2.6.3 Noise produced by turbulent flow past a trailing edge..... 31 2.6.4 Amiet’s model for noise due to turbulent flow past a trailing edge 34

2.1 Chapter overview

Aircraft and airport noise has been considered as a problem since 1952, when the ﬁrst jet powered commercial aircraft ﬂew. That year, Sir James Lighthill published On Sound Generated Aerodynamically. I. General Theory [4], his seminal publication

7 8 Chapter 2 Background and literature review about aeroacoustics, establishing the foundations of the aeroacoustic ﬁeld as a branch of acoustics concerned with noise generated by aerodynamic sources.

Due to the high exhaust velocities expelled by turbojets, jet noise was the dominant noise source in these engines. Jet noise, as defined here, refers to the noise produced by the turbulent jet downstream of the engine exhaust nozzle and the interaction in this turbulent flow with the airframe. Over the years, the desire for more efficient engines led to the double flow turbojets or turbofans. Since then, with the increase of bypass ratio, jet noise has been reduced, while the fan noise has increased. However, jet noise has remained one of the dominant sources, in particular during take-off [5].

This chapter serves as an introduction to aircraft noise. Noise pollution and its effects are explained first in section 2.2. In section 2.3, an introduction to the civil aviation regulations is given and a noise source breakdown are presented. In the source breakdown, the noise sources associated with aircraft noise are briefly explained and compared at different aircraft operation points.

In section 2.4 the aircraft noise sources are reviewed indicating the diﬀerent sources contributions with emphasis in jet-wing installation eﬀects. Ligthill’s theory is then explained in section 2.5 and extended into Curle’s analogy [6] in section 2.6 in order to take into account the interaction with solid boundaries.

2.2 Noise pollution

By environmental noise or noise pollution, it is meant the disturbing or excessive noise that may alter the normal ambient condition in an speciﬁc area. Noise cannot be accumulated, transferred or stored as other pollutions do. However, it can damage the quality of life if is not mitigated adequately. Noise pollution is produced by unwanted sound that can produce harmful psycho-physiological eﬀects. The main causes of noise pollution are the ones related to human activity, such as transportation systems: motor vehicles, aircraft, and trains; construction activities or industrial noise.

Noise is a bigger problem in urban environments. This pollution aﬀects a wide variety of spheres in human society. According to WHO [2], the excessive exposure to noise pollution can damage people’s physical and mental health, decreasing their quality of life, altering calm and serenity. WHO considers that 50 dB is the desirable upper limit for acoustic comfort, considering that, when this limit is exceeded, the sound is annoying and may be harmful. Chapter 2 Background and literature review 9

Figure 2.1: Number of people exposed to noise in Europe Lden > 55 dB, [7]

The European Environment Agency (EEA) estimates in the 2014 report Noise in

Europe [7] that more than 120 million people in the EU are exposed to Lden levels (annual average day, evening and night period of exposure) from road traﬃc that are above 55 dB, see Figure 2.1. Night-time road traﬃc is another major source of noise exposure, with over 83 million Europeans being exposed to harmful Lnight levels above 50 dB. In addition, many people are also exposed to rail (nearly 8 million people exposed above 55 dB Lden), aircraft (almost 3 million people exposed above 55 dB Lden) and

industrial noise (300,000 people exposed above 55 dB Lden), particularly in towns and cities. More general impacts of exposure to harmful levels of environmental noise include annoyance, stress reactions, sleep disturbance and an increase in the risk of hypertension and cardiovascular disease, which can lead to premature death. While aircraft noise does not affect a wide geographical area, its documented harmful effects extend beyond health impacts on nearby populations to also impairing the ability of younger generations to concentrate in schools that are affected by aircraft flight paths [2].

2.2.1 Eﬀects of noise pollution.

Noise acts through the hearing organ upon the central and autonomic nervous systems. When the stimulus exceeds certain limits, hearing losses and pathological eﬀects in both systems, immediate and delayed, are produced. At lower levels, noise produces malaise and diﬃculties in attention, communication, concentration, rest and sleep. 10 Chapter 2 Background and literature review

Figure 2.2: Pyramid of noise eﬀects [8].

The reiteration of noise stimuli may cause nervous and chronic stress. This in turn, leads to psycho-physic disorders, cardiovascular diseases and immune system distur- bances.

Finally, other effects should not be overlooked. This include: decrease in work and academic performance, antisocial behaviour, traffic and occupational accidents, aban- donment of affected urban centres or the decrease of property value.

2.2.2 Aircraft noise pollution

Air traffic as a cause of pollution factors is a relative novel phenomenon. This is because of the extraordinary increase in this method of travel, both, nationally and internationally. This increase can be seen in Figure 2.3, where the rise of passengers and commercial flights in the UK for the last 25 years is shown [9]. In the figure, it can be seen how both the number of passengers and the number of commercial flights have constantly increased since 1990, except for a gap of drop over 3 years, between 2007 and 2010, due to the international financial crisis.

The pollution associated with aircraft noise is concentrated in the areas around airports. These areas are small compared to the ones aﬀected by road or rail noise. How- ever, the practice of surrounding airports with residential neighbourhoods and the increase in number of operations has led to an increase in the population aﬀected by aircraft noise even though modern aircraft are quieter (individual aircraft have become some 75 % less noisy over the last 30 years). Chapter 2 Background and literature review 11

Figure 2.3: Evolution of the air traﬃc in UK from 1990 to 2015. Data from Civil UK Aviation Authority [9]

Acoustic signals by aircraft are sporadic events that have a consistent and stable behaviour emerging from the background noise. They are perceived by someone on the ground as a gradual increase in loudness to reach a maximun, following by a gradual decrease, until towards it is hidden by the background noise [10].

The human ear is frequency selective; between 500 Hz and 6 kHz the human ear is more sensitive compared to sound at lower and higher frequencies. For this reason, frequency weighted correlations are used in order to correlate objective sound measure- ments with the subjective human response. The most common frequency weighting is the A-weighting, that covers the full human audio range, from 20 Hz to 20 kHz. The concept of noise exposure is the A-weighting Sound Pressure, squared and integrated over a stated period of time or event. If the time is fixed as an 8-hour working day, the noise exposure can be defined as noise dose. It is common to define noise dose as a percentage. Dose% varies according to National Standards. In the UK, the 100% level is 85 dB(A) using the 3 dB(A) Exchange Rate. The 3 dB(A) exchange rate means that for each increment of 3 dB(A) the maximum permitted duration is halved, see Table 2.1. For example, the maximum permitted duration for a 100 dB(A) noise exposure is 15 minutes. With the 5 dB(A) exchange rate, it is one hour.

2.3 Civil aviation legislation

Aircraft noise regulations require all aircraft operating to comply with noise standards and recommended practices introduced under the Convention on Civil Aviation. 12 Chapter 2 Background and literature review

Noise Exposure Limits; Criterion Level = 85 dB(A) 3 dB(A) Exchange Rate Maximum Permitted Allowable Level dB(A) Daily Duration (hours) 85 8 88 4 91 2 94 1 97 0.5 100 0.25

Table 2.1: Noise Exposure Limits with the Criterion Level 85 dB(A) for a 3 dB(A) exchange rate.

Figure 2.4: History of Increased ICAO Noise Stringency [11]

These standards are set out in the International Civil Aviation Organization’s (ICAO) document Annex 16, Environmental Protection - Volume I. These standards and recommended practices for aircraft noise were first adopted by the Council on 2 April 1971 and became applicable in 1973, setting noise limits as a direct function of Maximum Take-off Weight (MTOW) or Maximum Take-off Mass (MTOM). The limitations have become more restricted through the years in the so called Chapters or Stages, as can be seen in Figure 2.4 that shows the progressive restriction in aircraft noise emissions over time across various weight categories.

The entry into force of each new chapter implies two main changes in the noise legislation: more stringent limits for noise certiﬁcation and a deadline for aircraft that were in service before the change of chapter and that do not meet the new limits. Chapter 2 Background and literature review 13

The chronological evolution of the main noise aviation legislation is now summarised.

Chapter 1

In 1966, discussions on the control of aircraft noise and the requirement for certiﬁcation commenced. European countries and the American Federal Aviation Administration (FAA) began a series of meetings and agreements in order to create an international plan. The Committee on Aircraft Noise (CAN), now called CAEP (Committee on Aviation Environmental Protection), was created in 1969 as part of the ICAO.

The fleet of aircraft manufactured before 1969 is known as Chapter 1 and they did not require to fulfil any regulation regarding noise. A deadline for the end of service of this fleet was not imposed in 1969. However, nowadays, most aircraft from this chapter have stop flying.

Chapter 2

The ﬁrst aircraft noise regulations appear in 1971 in the USA as the Federal Aviation Regulations (FAR) part 36: Noise standards and later in the same year in Annex 16 to the Convention on International Civil Aviation (the “Chicago Convention”). The regulation had a retroactive eﬀect to the date Chapter 2 started, January 1969.

The regulation was limited to civil aeroplanes, imposing an EPNL limit as a direct function of MTOM for three operation points around the airport. The Eﬀective Per- ceived Noise Level (EPNL), measured in Eﬀective Perceived Noise decibels (EPNdB), is a metric of the relative loudness of an individual aircraft pass-by event and represents the integrated sum of loudness over the period within which the aircraft noise is within 10 dB of the maximum noise. These operation points or events, that are schematically presented in Figure 2.5, are still in use nowadays. These are:

• Sideline: The point on a line parallel to and 450 m from the runway centre line, where the noise level is a maximum during take-oﬀ because the engines are at maximum power in order to obtain the required thrust for take-oﬀ.

• Cutback: After reaching the critical height, the engines power is cut, reducing thrust and the ascending angle. The emitted noise at Cutback is typically lower than in Sideline. However, the aﬀected area and the residence time are bigger.

• Approach: Low angle path used in approach for landing. Engines are at minimum power but, due to the small approach angle the emission time increases, increasing the EPNL. 14 Chapter 2 Background and literature review

Figure 2.5: Operation points for noise standards.

Together with the operation points, a maximum of 108 EPNdB, that could not be exceeded in the surrounding of airports was established. This level meant a reduction between 6dB and 10dB for the noisiest aircraft to deliver a reduction between a 25-50% of the noise perceived by the airport neighbouring population.

In 1975, noise standards were revised and more stringent levels were introduced. The comparison between the original Chapter 2 levels and the revised levels can be seen in Figure 2.6. The noise level was lightly reduced at approach, with 1 EPNdB reduction only for small aeroplanes. Noise limits were reduced more for take-off, with a stringent 5 EPNdB lower that Chapter 2 for small aeroplanes and 2 EPNdB for large ones. Ad- ditionally, a new criteria was implemented for sideline, applying a different noise limit depending on the number of engines that the aeroplane has, being more stringent the less engines it has. The reason for this criterion is that the climb angle is typically associated with the number of engines, for less engines the climb angle is steeper and, consequently, the residence time is shorter. This reflects the safety requirement that an aircraft has to be able to safetly take-off with the failure of one engine. The rest of the engines are able to provide the required thurst for the operation. This requirement means that a twin-engined aircraft has 100% more thrust than the minimum it requires under normal circumstances, a three-engined aircraft 50% more thrust and a four-engined aircraft has 33% more thrust.

Chapter 3:

Together with the improvement in thrust, the introduction into service of higher bypass ratio jet engines resulted in reductions in jet noise and, consequently, engine noise reduction. This allowed for the ICAO noise standard to become more restrictive in 1977. The EPNL limits of Chapter 3 can be seen in Figure 2.8. Chapter 2 Background and literature review 15

Figure 2.6: EPNL levels for Chapter 2 and revised version of Chapter 2.

Figure 2.7: Diﬀerent climb proﬁles to satisfy one engine failure criterion.

Figure 2.8: EPNL levels for Chapter 3. 16 Chapter 2 Background and literature review

Figure 2.9: Number of aircraft in diﬀerent Chapters operating in the US, EU and elsewhere over time [12].

However, aircraft belonging to Chapter 2 were kept in service and it was not until 1998 when Europe imposed 2002 as the date when Chapter 2 aircraft had to be adapted to fulﬁl Chapter 3 requirements by either equipping them with new quieter engines or by retiring them from service. The number of aircraft belonging to each chapter over the years can be seen in Figure 2.9 for US and EU ﬂeet.

Chapter 4:

After the Montreal convention in 1999, a more stringent regulation was agreed. It was signed in 2001, with its standards applied in 2006. The new limits were 10 EPNdB cumulative lower than imposed in Chapter 3 and were more stringent by at least 2 EPNdB for each operation point. Cumulative levels are defined as the arithmetic sum of the certification levels at each of the three certification points [13]. Chapter 2 Background and literature review 17

The same year, ACARE (Advisory Council for Aeronautics Research in Europe) was launched as a joint eﬀort of over 40 member organisations and associations, including representation from the European Member States, the European Commission and stakeholders: manufacturing industry, airlines, airports, service providers, regulators, research establishments and academia. ACARE’s main focus was to outline the strategy that Europe should take to meet society’s needs for aviation as a public mode of transport as well as noise and emissions reduction requirements in a sustainable way.

Chapter 14:

In February 2013 the CAEP/9 meeting recommended an amendment to Annex 16, Volume I involving noise limit reduction of 7 EPNdB (cumulative) relative to the current Chapter 4 levels. In 2014, the ICAO Council adopted the new Chapter 14 noise standards for jet and propeller-driven aeroplane that are the current ICAO noise limit standard for subsonic jets and propeller-driven aircraft noise. This new standards are applicable to new aircraft types submitted for certiﬁcation on or after 31 December 2017, and on or after 31 December 2020 for aircraft less than 55 tonnes in mass.

ICAO expects that, as a result of this new Chapter 14 for Noise Standards, more than one million people could be removed from “Day Night average sound Level (DNL) of 55 dB aﬀected areas” between 2020 and 2036.

The evolution of the noise levels through the diﬀerent Chapters can be seen in Figure 2.10 that shows the total cumulative EPNL allowed for each Chapter as the sum of the allowed EPNL for the three operation points shown in Figure 2.5.

2.4 Aircraft noise sources

Aircraft noise sources can be classiﬁed by their origin as: mechanic, combustion and aerodynamic. However, aircraft noise contribution produced by mechanical sources has a lower level than the sources with a combustion or aerodynamic origin, the last ones being the most important. Additionally, the aerodynamic noise can be decomposed in two, noise produced by the airframe and noise produced by the engines.

2.4.1 Airframe noise

Main noise sources in an aeroplane, if the engine is excluded, are the parts that creates turbulent ﬂow and the interaction of this turbulence with airframe edges. This is the 18 Chapter 2 Background and literature review

Figure 2.10: The progression of ICAO Noise Standard [14].

Figure 2.11: Example of share between engine and airframe noise for a generic civil aircraft [16] reason why the fuselage and the wing, despite being the biggest part of the airframe, are the ones that contribute least to the airframe noise. Three main components can be identiﬁed [15,16]: (1) wings including tail surfaces and fuselage, (2) high-lift devices including trailing-edge ﬂaps, leading-edge slats and brackets and (3) undercarriage including main and nose wheels, axles, oleo legs and struts, fairings, brake cables and wheel wells and doors.

The noise produced from airframe components is most evident during approach, when engine power is lowest and so it is the engine noise contribution, as it can be seen in Figure 2.11. At this operation condition the airframe noise is maximum because the landing gear is extended and the high-lift surfaces are deployed. Chapter 2 Background and literature review 19

Figure 2.12: Comparison of noise sources in a turbojet, LBPR turbofan and HBPR turbofan.

2.4.2 Engine noise

The main source for a subsonic turbojet is the turbulence mixing noise produced by the exhaust jet. Downstream of the engine and wings, the jet generates strong turbulence as it mixes with the atmosphere. The intensity of the jet noise scales as the eight 8 power of the exhaust velocity (∼ Uj ) as theorised by Lighthill [4] and demonstrated by experiment by Lush [17].

The addition of a propeller or a fan to the engine produces a big reduction in the jet mixing noise. This is because the jet core speed is reduced in order to propel a secondary air stream which typically has a lower downstream velocity but a bigger mass ﬂow. In addition, the compressor noise is greatly attenuated due to the presence of an upstream fan. However, the total noise reduction is not as big because the fan or propeller becomes an additional source, emitting both upstream and downstream of the jet. This can be observed in the schematic Figure 2.12, which shows the evolution of the diﬀerent noise engine sources importance from the old turbojet to the low bypass ratio turbofan and to the actual high bypass ratio (HBPR) turbofan engines.

Turbine, compressor and fan noise sources are classiﬁed in turbomachinery noise. The generation and propagation of this kind of noise is given by the interaction between the rows of blades and vanes. The most noteworthy types of noise associated with turbomachinary are:

• Tonal noise: the main noise mechanism for turbines and compressors is the interaction between the turbulent wakes of rotors with the stator vanes, and vice- versa, that produces harmonic perturbations or pure tones. The frequencies of these tones depend on the ratio between the number of blades in the two rows and the rotor shaft frequency. 20 Chapter 2 Background and literature review

Figure 2.13: A HBPR turbofan engine installed under a wing [photo courtesy of B. J. Venkatesh.]

• Broadband noise: it is a no harmonic noise contribution that aﬀects a wide range of frequencies. The main mechanism is the known as trailing edge noise produced by the scattering along the blades trailing edge of the turbulent boundary layer.

Combustion noise is produced by two diﬀerent mechanisms: (1) direct combustion noise, generated by acoustic waves propagating to the outlet, and (2) indirect combustion noise, caused by the acceleration of entropy waves (or hot spots) and of vorticity waves through turbine blades [18].

2.4.3 Jet-Wing installation eﬀects

Jet-wing installation effects include sources and mechanisms that increase the noise level of an isolated jet when an engine is installed beneath the aircraft wing, as shown in Figure 2.13. This type of noise combines physical processes in engine noise and in airframe noise because the augmentation is produced by the interaction between the turbulent jet and the airframe, specifically with the wing and the flaps. The importance of jet-wing installation has increased in the last 20 years, due to the development of high bypass ratio engines (HBPR), which are now used in commercial aviation. The increase of the bypass ratio, with larger engine diameters, has led to the engines being placed closer to the wings, reducing the distance between the wing and the jet. Thus, the interaction between jet and wing has increased, resulting in an increase in its noise contribution.

The pressure field generated by the turbulent shear layer of a jet has been widely studied during the last 60 years. It can be decomposed into propagating (acoustic field) and non-propagating (hydrodynamic field) parts, as shown by Howes 1960 [19], Morgan 1961 [20], Ribner 1962 [21] or Ffowcs-Williams 1969 [22] among others. The acoustic Chapter 2 Background and literature review 21

Figure 2.14: Generic 1/3rd octave band breakdown of jet installation effects H/D = 0.67, L/D = 10, M = 0.75 [data from Doak Laboratory, ISVR] pressure field propagates to the far-field at the speed of the sound and decays linearly with distance. The hydrodynamic field propagates at the convective velocity of the jet and has a much larger amplitude than the acoustic field, but decays much rapidly with the distance than the acoustic field and is therefore not perceived in the far-field. The interaction of these two pressure fields with a solid surface (i.e. the aircraft wing and flaps) is the origin of jet-wing installation effects (IFX).

Figure 2.14 shows an example of this phenomenon from measured data at the Anechoic Doak Laboratory, at the Institute of Sound & Vibration (ISVR), within the University of Southampton, UK. The dotted-line represents the 1-metre lossless sound pressure level (SPL) measured at a polar angle of 90◦ produced by an isothermal single stream jet. The jet is expelled from a convergent nozzle with an exhaust diameter, D, of 38.1mm and a nozzle exit Mach number, M, of 0.75. The dashed and continuous lines represent the measurement when a semi-infinite flat plate is located parallel to the jet, behind the jet (unshielded) or between the jet and the microphone (shielded). Apart from the reflection, which happens at medium and high frequencies and is easily seen, there are two main source mechanisms that generate an increase of low frequency broadband noise: (1) jet- surface interaction (JSI) that is labelled in Figure 2.14, and (2) jet impingement noise (IMP), which was not present in the experiment that generated Figure 2.14. However, as both sources share similar spectral characteristics, it is complicated to separate from one another [23]. In addition, a third mechanism that generates low frequency tonal noise can be found, although that was absent in the experiment of Figure 2.14 which is (3) trapped waves tones (TWT). 22 Chapter 2 Background and literature review

Figure 2.15: Schematic of convecting and scattered pressure ﬁelds in the hydrodynamic near ﬁeld.

Jet-wing reflection (JWR) refers to the reflection of the jet mixing noise under the wing. This is a well understood effect that produces up to 3dB of noise due to the incoherent addition of the reflected noise to the direct noise level. Ffowcs Williams and Hall [24] calculate the reflection on a plate of the noise generated by an eddy, using method of the images. For jets, a 3D ray-theory prediction methodology was developed by Moore and Mead [25, 26]. On it, the aerofoil geometry is represented by a number of flat plates and the jet is decomposed in point sources distributed in the jet plume. This theory was lately extended by Mc Laughlin [27, 28] by creating a semi-empirical prediction tool for reflection and shielding configurations that includes an empirical hot-jet blockage model.

Jet-surface interaction (JSI) effects have been studied in references [24, 29, 30], sug- gesting that the pressure field, due to the jet shear layer turbulence, is scattered at the trailing edge. As specified above, the hydrodynamic pressure field decays evanescently with the distance. However, in the presence of a solid surface (i.e. the wing), it is acoustically scattered. A schematic of the process is shown in Figure 2.15. The process, then, is one of an evanescent pressure field produced by the jet turbulence, being scattered by a trailing edge, where becomes a truly propagated acoustic pressure field.

Impingement noise (IMP) originates where the wing or flap surfaces are scrubbed by the jet flow due to a close-coupled configuration [31]. In this mechanism, the hydrodynamic near-field is diffracted by the trailing edge, generating an augmentation in the broadband noise.

Trapped waves tones (TWT) are strong tonal noise that is observed in both near and far-field when a sharp edge is introduced in the near-field of a transonic jet 0.8 ≤ M ≤ 1. Neuwerth [32] first observed jet-edge tonal noise in 1974, however it was associated with the impingement process. In 2016, Jordan et at [33], inspired by Lawrence & Self Chapter 2 Background and literature review 23 measurement [31], related the tonal noise to acoustic waves that are trapped in the potential core of subsonic jets and that travel upstream [34, 35].

Over the years, jet-wing installation eﬀects have been studied extensively [23, 36–45] and advancement in computing have enabled numerical simulation studies [46–49], in an attempt to obtain a deeper understanding of these mechanisms. Progress has been obtained on deﬁning the physical nature of the sources involved and in determining the main parameters that govern the jet-wing installation noise generation.

2.4.4 Computation of jet ﬂow ﬁeld using CFD

Computational methods of calculating flow generated sound can be classified into direct computations and indirect or hybrid computations. A direct computation involves solving for the noise sources along with unsteady fluid dynamic source field represented by the compressible flow equations. Methods like the Direct Numerical Simulation (DNS) computate of the acoustic near field by solving the full compressible Navier-Stokes equations. Large Eddy Simulations (LES) resolve only the large scales by applying spatial filters and models the effects of the smaller scales on the resolved ones.

On the other hand, modelling jets using Reynolds Averaged Navier Stokes (RANS) methods has become attractive due to its simplicity and reduced computational costs. This method provides the time-averaged jet flow field. The effect of turbulence on the mean flow is modelled by using a turbulence closure model such as the k − or the k − ω models. Some information about the turbulent unsteady motion in the mixing regions of the jet can be obtained from the computed turbulence parameters such as from the specific turbulent kinetic energy, k, and from the specific turbulence dissipation rate, , in the case of the k− model. The main advantage of RANS CFD methods are that they are computationally cheaper in comparison with DNS and LES, and that they are capable of modelling the flow field from complex nozzle geometries. RANS based solutions are coupled with aeroacoustic models to predict jet noise. These methods use mean turbulent flow information to statistically represent the sound source, to predict the radiated acoustic far-field. Ribner’s work [50] on separating the two-point space-time correlation function forms the basis of many statistical jet noise models. A combination of such a model with a CFD mean-flow field has been used by Mani et al. [51] , Bechara [52], Bailly [53,54] and Khavaran [55–57]. Some of these works assumed isotropic turbulence where the use of a single characteristic length-scale and time-scale is justified to describe the spatial and temporal decay rate of the velocity correlation, respectively. Some [57,58] have investigated the effect of anisotropic turbulence on the far-field noise. 24 Chapter 2 Background and literature review

In jet engine exhaust noise predictions, installation effects can include those due to the pylon-jet interaction, wing downwash, and flap-jet interaction. Hunter and Thomas used the Jet3D code developed at NASA Langley to compute complex three-dimensional flow fields for isolated and installed jets [59]. Dezitter et al. used RANS CFD to characterise the installation effects for a typical high bypass ratio engine using advanced numerical and measurement techniques [60]. In this paper, an assessment of the capability of the CFD solvers to predict the aerodynamic flow for different nozzles installed under the wing was presented.

2.5 Jet mixing noise theory

In this section, a review on isolated jet aeroacoustics is presented. This is done because the jet is an integral part of the installation eﬀects problem. A circular isothermal isolated single-stream jet discharged in a quiescent ambient medium is considerer. Firstly, in section 2.5.1, the physics of the jet is presented. Then, in section 2.5.2, Lighthill’s Acoutic Analogy is reported.

2.5.1 Jet ﬂow ﬁeld

In the first third of the twentieth century, jet flows were experimentally characterized and their properties were studied and documented. Examples of these studies are in Forthmann [61] or Abramovich [62]. A comprehensive review of the many jet flows that have been studied experimentally over the years can be found in “The Theory of Turbulent Jets”, written by Abramovich in 1963 [63], which last edition was revised in 2003.

The ﬂow of a circular isolated jet is axisymmetric, i.e it has symmetry around the longitudinal axis and there is no dominant azimuthal directivity. The jet is statistically stationary, i.e all statistical properties are constant in time. Three regions can be identiﬁed longitudinally [63, 64], sketched in Figure 2.16.

1. Initial mixing region: Just downstream of the nozzle exhaust exit, the jet exhibits a potential core, which is a conical region of quasi-laminar parallel flow with constant velocity. For a single air jet, the potential core axial length is four or five nozzle diameters. The laminar flow mixes with the ambient in a sheared flow region known as the turbulent boundary layer or shear layer. This region grows with axial distance downstream of the nozzle as more ambient fluid is entrained by and mixes with the jet. Chapter 2 Background and literature review 25

Figure 2.16: Schematic of a simple jet ﬂow showing the classical three jet regions

2. Transition region: Downstream of the potential core, the shear layer continues to spread into the transitional region. The spreading angle here is slightly bigger than in the initial region.

3. Fully developed region: downstream of the end of the transition region at around eight diameters from the nozzle exit a fully developed turbulence region forms.

2.5.2 Jet acoustics

In 1954, Lighthill [65] published “On sound generated aerodynamically II. Turbulence as a source of sound” as an extension of the general theory presented in 1952 [4] to predict the broadband noise that is generated by the fluctuations in a jet shear layer during the mixing process. Lighthill rearranged the Navier-Stokes equations into an inhomogeneous wave equation for unbounded flows, by which the nonstationary fluctuations due to the turbulent flow can be represented by a distribution of quadrupole sources that occupy the same volume. Lighthill showed that the inhomogeneous wave equation can be written as 2 2 ∂ ρ 2 2 ∂ 2 − c0∇ ρ = Tij, (2.1) ∂t ∂yi∂yj where 2 Tij = ρvivj − σij + (p − c0ρ)δij, (2.2) is Lighthill’s turbulence stress tensor, which represents a quadrupole source. This term includes the Reynolds stresses, the viscous stresses and a term that measures the depar- ture of the flow from small-amplitude isentropic motion.

Equation (2.1) describes how the internal fluctuating stresses of a fluid flow, on the right hand side of the equation, generate sound that propagates as a wave at the speed of sound c0 in a quiescent medium. The most general solution for the inhomogeneous 26 Chapter 2 Background and literature review

Figure 2.17: Cartisian coordinate system. Origin in nozzle exhaust exit center. wave equation, given by Stratton 1941 [66] is

1 Z 1 ∂2 ρ (x, t) − ρ0 = 2 Tij (y, τ) dy+ 4πc0 V r ∂yi∂yj τ=t− r c0 1 Z 1 ∂ρ 1 ∂r 1 ∂r ∂ρ + + 2 ρ + dS (y) , (2.3) 4π S r ∂n r ∂n rc0 ∂n ∂t τ=t− r c0 where n is the unit vector normal to the surface S (y) and outward the ﬂuid, x =

(x1, x2, x3) represents the observer location, y = (y1, y2, y3) is the source locations and r = |x − y| is the separation distance from source to observer, as sketched in Figure 2 2.17. Note that the quantities ∂ Tij , ρ, ∂ρ and ∂ρ are evaluated at the retarded time ∂yi∂yj ∂n ∂t τ = t − r/c0. The ﬁrst integral of Equation (2.3) is taken over the total volume external to solid boundaries, V , while the second integral is taken over the surface S of the solid boundaries. The surface integral will be zero in a free ﬁeld, i.e. in the absence of solid boundaries only the volume integral remains,

∞ 1 ZZZ 1 ∂2 ρ (x, t) − ρ0 = 2 Tij (y, τ) dy, (2.4) 4πc0 r ∂yi∂yj τ=t− r −∞ c0 that is Lighthill’s general solution for the free field. Equation (2.4) can be simplified. Making use of the chain rule of partial differentiation, it can be expanded to

 r 2 T y, t − 1 ∂ Z ij co ρ (x, t) − ρ0 = 2  dy 4πc0 ∂xi∂xj V r

r r  T y, t − 2 T y, t − ∂ Z ∂ ij co Z ∂ ij co +2 dy + dy , (2.5) ∂xi V ∂yj r V ∂yi∂yj r in which can be observed that the second and third integrands are the divergence of a vector. The Gauss divergence theorem states that the volume integral of a vector Chapter 2 Background and literature review 27 divergence can be written as a surface integral of the vector. Z Z ∇ · Ady = A · dS, (2.6) V S where S is a surface that surround the volume V . Applying the Gauss divergence theorem to Equation (2.5) gives

 r 2 T y, t − 1 ∂ Z ij co ρ (x, t) − ρ0 = 2  dy 4πc0 ∂xi∂xj V r  ∂T y, t − r ∂ Z r dS (y) Z ij co dS (y) + njTij y, t − + ni  (2.7) ∂xi S co r S ∂yj r

For an unbounded ﬂow, the surface S can be chosen as an sphere of radius R → ∞.

Assuming that Tij is smooth and decays faster than 1/y for large y, the two surface integrals will then vanish, i.e. the second and third integrals of Equation (2.7) can be neglected and thus,

∞ r 2 T y, t − 1 ∂ ZZZ ij co ρ (x, t) − ρ0 = 2 dy . (2.8) 4πc0 ∂xi∂xj r −∞

Equation (2.8) is an exact solution of the inhomogeneous wave equation in the free ﬁeld.

For an observer located many wavelengths away from the source region, which is always true for the far-ﬁeld, the dependence with y of r is small and the following assumption can be used ∂ 1 = O x−2 , (2.9) ∂xi r which leads to the next equivalence

r 2 Tij y, t − 2 ∂ co xixj ∂ r −2 = 3 2 2 Tij y, t − + O x , (2.10) ∂xi∂xj r x c0 ∂t co where the following equivalence, only valid at the far-ﬁeld, has been used

∂ r x ∂ r f t − = − i f t − . (2.11) ∂xi c0 c0r ∂t c0 28 Chapter 2 Background and literature review

Equation (2.8) then reduces to

∞ ZZZ 2 1 xixj ∂ r ρ (x, t) − ρ0 ∼ 2 3 2 2 Tij y, t − dy. (2.12) 4πc∞ x c0 ∂t co −∞

This expression is known as Lighthill’s analogy for an acoustically compact source. It demonstrates the quadrupole structure of the sound source and allows one to calculate the density ﬂuctuations in the far-ﬁeld once the source term is known.

However, in the near-field, because the distance to the observer is of the same order as the wavelength, Equation (2.12) is not valid. Later, in Chapter3, Lighthill’s analogy will be extended by means of evaluating Equation (2.8) without the far-field assumption (i.e. retaining all the radiating terms) in order to make it valid in the near-field region. This will be important in order to calculate the jet surface interaction noise because it is the near-field pressure that interacts with the aerofoil and it is scattered by the trailing edge, see section 2.4.3. Two additional terms, that decay faster and vanish in the far-field, are present. These terms provide a similar or bigger noise contribution in the near-field.

2.6 Trailing edge noise

In this section, Lighthill’s theory is extended to predict the noise associated with the presence of a solid body. This is the basis of the classical trailing edge noise theory developed by Amiet which will help the reader to understand the mechanisms involved in jet-surface interaction noise.

2.6.1 Curle’s theory for solid bodies

A year after Lighthill published his analogy, in 1955, Curle extended it by including a formal solution to take hard surfaces into account [6]. Because of the solid boundary, the free ﬁeld simpliﬁcations made by Lighthill do not apply. Neither of the surface integrals in Equation (2.3) can be neglected nor the two surface integrals in Equation (2.7) vanish. The solution of the inhomogeneous wave equations then yields

2 Z Z 1 ∂ Tij 1 ∂ dS (y) ρ (x, t) − ρ0 = 2 dy + 2 njTij + 4πc0 ∂xi∂xj V r 4πc0 ∂xi S r Z Z 1 ∂Tij dS (y) 1 1 ∂ρ 1 ∂r 1 ∂r ∂ρ + 2 ni + + 2 ρ + dS (y) . (2.13) 4πc0 S ∂yj r 4π S r ∂n r ∂n rc0 ∂n ∂t Chapter 2 Background and literature review 29

2 Note that the quantities Tij , ∂Tij , ∂ Tij , ρ, ∂ρ and ∂ρ are taken at retarded times r ∂yi ∂yi∂yj ∂n ∂t τ = t − r/c0 but it has been omitted in the equation for brevity. The ﬁrst integral in the right hand side of the Equation (2.13) is taken over the total volume external to the solid boundaries. The other terms are integrals over the surface S of the solid boundaries. The volume integral can be evaluated in the same way as was done above for Lighthill’s analogy. The surface integral, however, requires a more cautious approach that can be found in the literature. Curle transforms the last surface integral into a more convenient form

Z 1 ∂ρ 1 ∂r 1 ∂r ∂ρ + 2 ρ + dS (y) S r ∂n r ∂n rc0 ∂n ∂t Z ∂ dS (y) Z 1 ∂r 1 ∂r ∂ρ = ni (ρδij) − ni 2 ρ + dS (y) S ∂yj r S r ∂xi c0r ∂xi ∂t Z Z ni ∂ ∂ nj = (ρδij) dS (y) + (ρδij) dS (y) , (2.14) S r ∂yj ∂xi S r where the last integral is obtained by applying

∂ 1 r 1 ∂r r 1 ∂r ∂ r f t − = − 2 f t − − f t − . (2.15) ∂xi r c0 r ∂xi c0 c0r ∂xi ∂t c0

Substituting Equation (2.14) and (2.2) into (2.13) then leads to

2 Z 1 ∂ Tij ρ (x, t) − ρ0 = 2 dy+ 4πc0 ∂xi∂xj V r Z 1 ∂ nj + 2 (ρvivj − σij + pδij) dS (y) + 4πc0 ∂xi S r Z 1 ni ∂ + 2 (ρvivj − σij + pδij) dS (y) . (2.16) 4πc0 S r ∂yj

The partial derivative ∂ in the third integral on the right hand side of Equation ∂yj (2.16) can be written as a time derivative

∂ ∂ (ρvivj − σij + pδij) = − (ρvi) . (2.17) ∂yj ∂t

If the solid boundary is ﬁxed and the ﬂow velocity vi is zero on its surface S, then Equation (2.16) reduces to

r r 2 Z T y, t − Z F y, t − 1 ∂ ij c0 1 ∂ i c0 ρ (x, t) − ρ0 = 2 dy − 2 dS (y) , 4πc0 ∂xi∂xj V r 4πc0 ∂xi S r (2.18) where

Fi = −ni (pδij − σij) , (2.19) 30 Chapter 2 Background and literature review

is the force per unit area in the xi direction that the solid boundaries exert on the ﬂuid. Equation 2.18 is known as Curle’s analogy. In it, the surface integral represents the contribution of solid boundaries to Lighthill’s source term. It can be interpreted as the sound generated in a quiescent medium by an equivalent dipole distribution of strength

Fi.

The sound ﬁeld, for a quiescent medium, can be expressed then as the summation of:

(1) a volume distribution of quadrupoles Tij in the region external to a solid body, and

(2) A surface distribution of dipoles Fi on the surface of this body.

2.6.2 Extension for moving sources

By retaining all the terms from Equation (2.16), Ffowcs Williams and Hall extended Curle’s work to include moving sources [24].

r 2 Z T y, t − 1 ∂ ij c0 ρ (x, t) − ρ0 = 2 dy 4πc0 ∂xi∂xj V r Z F y, t − r 1 ∂ i c0 − 2 dS (y) 4πc0 ∂xi S r Z ρv y, t − r 1 ∂ n c0 + 2 dS (y) . (2.20) 4πc0 S ∂t r

In a similar way as in Equation (2.1), the noise sources are on the right hand side of Equation (2.20). The volume integral on the right hand side is non-zero outside of the solid boundary and the surface integrals are non-zero on the surface of the solid boundary. Note that the second surface integral is the one that was neglected in Equation (2.16) when the solid boundary was ﬁxed. It is easy to understand that this additional term represents the density ﬂuctuations due to the body movement:

• Tij - Flow source, Lighthill’s stress tensor which represents a quadrupole source distribution outside of the body volume.

• Fi - Loading source, a dipole or loading source which represents the noise generated by a normal force distribution exerted by the ﬂuid on the body surface.

• ρvn - Thickness source, a monopole source, located on the body surface, which models the noise generated by the displacement of ﬂuid as the body passes. Chapter 2 Background and literature review 31

Figure 2.18: Trailing edge coordinates and relation between them and nozzle exit coordinates

The monopole source can be neglected if the mean ﬂow is assumed to be steady and the body is at rest. This is the case for jet-wing installation eﬀects in which the aerofoil is located above the jet. The interaction noise is generated by the loading source that represents a dipole distribution along the trailing edge.

2.6.3 Noise produced by turbulent ﬂow past a trailing edge

Jet-Surface Interaction noise produced by a single-stream jet interacting with a parallel ﬂat plate in a quiescent ambient medium can be calculated by evaluating the surface integral on Curle’s analogy (equation (2.18)),

Z F y, t − r 1 ∂ i c0 ρJSI (x, t) − ρ0 = − 2 dS (y) . (2.21) 4πc0 ∂xi S r

A sample computation is performed in an idealised geometry. The plate has a chord c, a span length d and is assumed to have zero thickness. For simplicity, the coordinate system origin of x and y is located on the plate. The origin of this coordinate system

is at the trailing edge centre, the xt1 coordinate is parallel to the jet axis, pointing

downstream, and the xt3 coordinate is perpendicular to the aerofoil, pointing at the

ground, towards the jet. The the xt2 direction is parallel to the plate trailing edge,

satisfying the right hand rule, as shown in Figure 2.18. xt = (xt1, xt2, xt3) and yt =

(yt1, yt2, yt3) are, respectively, the observer and source locations, both measured from

the origin at the trailing-edge centre. yt is then contained on the plate surface and

consequently yt3 = 0. The relation between this coordinate system and the nozzle exit coordinate system is

xt1 = x1 − L

xt2 = x2 (2.22)

xt3 = x3 + H, 32 Chapter 2 Background and literature review where L is the axial distance from the nozzle exhaust exit to aerofoil trailing edge and H is the vertical distance from jet centreline to aerofoil trailing edge.

Figure 2.18 shows, as well, a spherical coordinate system, xt = (rt, θt, φt), centred on the trailing edge centre that will be used later in Chapter4

xt1 = rt sin φt cos θt = R cos θt

xt2 = rt cos φt (2.23) xt3 = rt sin φt sin θt = R sin θt,

R = rtsinφt.

The force exerted on the ﬂuid by the aerofoil Fi is generated by the lift ﬂuctuations caused by the pressure disturbance on the two sides of the plate due to the scattered pressure along the trailing edge. The force per unit area is then Fi = F3 = ∆p, the pressure jump between the two sides of the plate which is normal to the plate surface as pressure forces are. Equation (2.21) then yields

Z 0 Z d/2 ∆p y , t − r 1 ∂ t c0 ρJSI (x, t) − ρ0 = − 2 dyt2dyt1. (2.24) 4πc0 ∂xt3 −c −d/2 r

For any isentropic flow in which the pressure and density fluctuations are small compared with their mean values (which happens for subsonic flows), the pressure and density fluctuations can be related by

2 p = c0 (ρ − ρ0) , (2.25) where p represents the pressure ﬂuctuations. Equation (2.24) can be written for the pressure as Z 0 Z d/2 ∆p y , t − r 1 ∂ t c0 p (xt, t) = − dyt2dyt1. (2.26) 4π ∂xt3 −c −d/2 r

By means of the same knowledge of partial derivation that was used in Lighthill’s theory, equations (2.9-2.12), Equation (2.26) reduces in the far-ﬁeld to

Z 0 Z d/2 xt3 ∂ rt p (xt, t) = − 2 ∆p yt, t − dyt2dyt1. (2.27) 4πc0rt −c −d/2 ∂t c0 Chapter 2 Background and literature review 33

Before continuing, we will deﬁne the convention for the Fourier transforms in time and space that will be used in this thesis

∞ Z f (t) = F (ω) exp {iωt} dω, −∞ ∞ (2.28) Z g (xi) = G (ki) exp {−ikixi} dki, −∞

where ω is the angular frequency and ki is the wavenumber associated to the coordinate

xi. The inverse Fourier transform are

∞ 1 Z F (ω) = f (t) exp {−iωt} dt. 2π −∞ ∞ (2.29) 1 Z G (k ) = g (x ) exp {ik x } dx . i 2π i i i i −∞

For a harmonic pressure ﬂuctuation of the form p (t) = P (ω) expiωt, the far-ﬁeld pressure produced by the dipole distribution along the aerofoil can be written as

Z 0 Z d/2 iωxt3 rt P (xt, ω) = − 2 ∆P (yt, ω) exp iω dyt2dyt1. (2.30) 4πc0rt −c −d/2 c0

The ratio xt3 can be written as rt

xt3 = sin (θt) sin (φt) (2.31) rt if the spherical coordinate system deﬁned in Equation (2.23) is used for xt, Equation (2.30) yields

Z 0 Z d/2 iω rt P (xt, ω) = − sin (θt) sin (φt) exp iω ∆P (yt, ω) dyt2dyt1, (2.32) 4πc0rt c0 −c −d/2 that represents the pressure fluctuations at a far-field point xt and at a given frequency ω caused by the pressure jump on the plate. These fluctuations travel as acoustic waves, i.e they decay as 1/rt, and present a dipole directivity sin (θt) sin (φt) that presents a π maximum when travel normal to the plate, θt = φt = , and are zero when travel n o 2 parallel to it, θ = 0 or φ = 0. Finally, exp iω rt represents the change in phase t t c0 between the pressure jump on the plate and the pressure fluctuation at xt.

The pressure jump, ∆P can be calculated using the Wiener-Hopf technique as it will be shown in Chapter4 or using Schwarzschild’s solution, see AppendixA, as Amiet did 34 Chapter 2 Background and literature review in 1975 [3] for the noise generated by a frozen turbulent gust convecting past a trailing edge. In Chapter4, the power spectral density of the far-ﬁeld sound obtained from Wiener-Hopf technique will be compared with Amiet’s solution. For this reason, it is worthy to introduce Amiet’s model for noise due to a turbulent gust past a trailing edge. A more detailed explanation of it can be found easily in the literature, for example, in the original Amiet’s publications [3, 67, 68], in Roger and Moreau [69] or in Lawrence thesis [70].

2.6.4 Amiet’s model for noise due to turbulent ﬂow past a trailing edge

Amiet developed a theoretical model [3] in 1975 to predict noise generated by a frozen turbulent gust convecting past a trailing edge. The model takes into consideration the wing chord, span size and ﬂow compressibility eﬀects. The incident wall pressure is split by Fourier analysis into 2D wavenumbers in the streamwise and spanwise directions, k1 and k2. For each frequency, each wavenumber pair represents an oblique gust convecting past the trailing edge. These incident pressure gusts can be generalized as

i P0 = exp {−ik1xt1 − ik2xt2} . (2.33)

With Equation (2.30) as a starting point, and making use of the Schwarzchild solution that can be found in AppendixA, the far-field acoustic pressure at xt and ω corresponding to the scattering of an incident pressure gust of wavenumbers k1 and k2 is given by f iωxt3dc xt2 d p (xt, ω) = − 2 sinc k2 − k exp {−ikrt} I(k1, k2) , (2.34) 4πc0rt rt 2 where k = ω is the acoustic wavenumber, sinc (x) = sin(x) is the cardinal sine function c0 x and I (k1, k2) is the factor relating the incident wall pressure to the far-field pressure fluctuations when the aerofoil is artificially extended to infinity upstream

n o 2i exp i k − k xt1 c 1 rt I(k1, k2) = − k − k xt1 c 1 rt ( s k1 + K xt1 ∗ xt1 × (1 + i) exp −i k1 − k c E K + k c k xt1 + K r r rt t t ) ∗ − (1 + i) E [(k1 + K) c] + 1 , (2.35)

p 2 2 ∗ where K = k − k2 is the wavenumber in the xt1-xt3 plane and E [x] is the complex function derived from Fresnel integrals [71], C2 and S2, for which the following convention Chapter 2 Background and literature review 35 will be used in this thesis

Z xt it ∗ e E (xt) = √ dt = C2 (xt) − iS2 (xt) . (2.36) 0 2πt

As stated above, Equation (2.35) is valid for a semi-inﬁnity aerofoil. In their publication in the Journal of Sound and Vibration (JSV), Roger and Moreau extended Amiet’s model to account back-scattering eﬀects due to the leading edge [69, 72] by adding an

correction term to I (k1, k2). For simplicity, this correction term is not included in the calculation of this section.

If Equation (2.35) is written in terms of the error function as (using Abramowitz & Stegun [73] Equations 7.1.10, 7.3.7, 7.3.8 and 7.3.22) and introduced in Equation (2.34), it yields

(s √ r f kxt3 K + k1 xt1 √ p (xt, ω) = 2 K+x erf i K + k c 2παr k t1 rt t rt ) h√ p √ i xt2 d + 1 − erf i K + k1 c exp {iαc} − 1 exp {−ikrt} d sinc k2 − k , rt 2 (2.37) where α = k − k xt1 . The term -1 at the end of the curly brackets can be considered 1 rt to balance the contribution of the incident ﬁeld to the sound radiation and it must be discarded as suggested by Amiet [3, 68] and Roger and Moreau [69]. Equation (2.37) will be used in Chapter4 to compare Amiet’s trailing edge solution with the solution obtained with the Wiener-Hopf technique.

Equation (2.34) only holds for a unit gust with wavenumbers k1 and k2 at a given angular frequency ω. The power spectral density of the far-field sound at the same frequency results from an integration over all gusts with 2D wavenumbers that contribute to this frequency. The turbulent intensity at the trailing edge is small and satisfies Taylor’s hypothesis, as detailed by Amiet in his JSV paper [67]. Therefore, it can be assumed that the incident pressure field is frozen when convected past the trailing edge at velocity a U in the x -direction, i.e k = ω . The acoustic far-field pressure is then c t1 1 Uc given by

f iωxt3dc p (xt, ω) = − 2 exp {−ikrt} 2πc0rt Z ∞ ω ω xt2 d × P0 , k2 I , k2 sinc k2 − k dk2, (2.38) −∞ Uc Uc rt 2 where P0 is the amplitude of the gust with (k1, k2) wavenumber. 36 Chapter 2 Background and literature review

As turbulence is a random quantity it is necessary to work with statistical quantities such as the power spectral density of the far-ﬁeld pressure, Spp (xt, ω). This can be obtained by multiplying Equation (2.38) by its complex conjugate to give

π h f f∗ i Spp (xt, ω) = lim E p (xt, ω) p (xt, ω) , (2.39) T →∞ T where E [...] denotes the expected value. The only statistical quantity of p (xt, ω) is P0. Thus if Equation (2.38) is introduced in Equation (2.39) it can be written as,

2 Z ∞ 2 ωxt3dc ω ω 2 xt2 d Spp (xt, ω) = 2 Π0 , k2 I , k2 sinc k2 − k dk2, 4πc0rt −∞ Uc Uc rt 2 (2.40) where Π0 is the wavenumber spectral density of the incident gust amplitudes P0. It represents for a given spanwise wavenumber, the energy of the incident wall pressure ﬂuctuations at frequency ω.

A final simplification can be made by assuming that the characteristic scales of the near pressure field close to the trailing edge are small when compared to the chord span. The sinc term can be written as, 2 xt2 d 2π xt2 2π sinc k2 − k ' δ k2 − k = δ (k2 − k cos φt) (2.41) rt 2 d rt d which means that only one oblique gust is selected for each angle of radiation off the mid-span plane. Equation (2.40) then yields,

2 2 ωxt3c ω xt2 ω xt2 Spp (xt, ω) = 2 2πd I , k Π0 , k . (2.42) 4πc0rt Uc rt Uc rt

According to Corcos’ model [74] as is explained by Roger and Moreau [69], the energy of the incident wall perssure ﬂuctuations Π0 at a given frequency ω can be calculated as, 1 Π (k , k ) = Φ(ω) l (k , k ) , (2.43) 0 1 2 π 2 1 2 where Φ (ω) is the wall pressure power spectral density (assumed to be statistically homogeneous close to the trailing edge). l2 is the spanwise correlation length, which is related with the spanwise coherence, and is found as

k1 l (k , k ) = bc , (2.44) 2 1 2 2 k2 + k1 2 bc with bc = 2.1 as a constant value, as suggested by Amiet [3]. Chapter 2 Background and literature review 37

Equation (2.42) can then ﬁnally be written as,

2 2 ωxt3c ω xt2 Spp (xt, ω) = 2 πl2d I , k Φ(ω) . (2.45) c0rt Uc rt that allows to calculate the PSD of the far-field sound produced by an uniform flow that is convected at a subsonic Mach number along and aerofoil and past its trailing edge, which is orthogonal to the flow. It is important to notice that the flow was assumed to have frozen turbulence. It has been demonstrated that this assumption is not valid for a non-uniform flow like the one presented in an spreading jet that will produce spanwise-varying flow conditions along the aerofoil [1, 75]. Later in Section 4.6, this problem will be tackled, taking into account the spanwise-varying conditions by means of a Wigner-Ville spectral decomposition distribution.

Chapter 3

Study of the jet near-ﬁeld

Contents 3.1 Chapter overview...... 39 3.2 Near-field extension of Lighthill’s acoustic analogy..... 41 3.2.1 Cross-power spectral density...... 43 3.2.2 Scaling of the jet near-field...... 46 3.3 Near-field experimental data...... 48 3.3.1 Near-field experimental setup...... 48 3.3.2 Radial Scaling...... 53 3.3.3 Velocity Scaling...... 60 3.4 Jet near-field prediction...... 63 3.4.1 A model for the fourth order correlation tensor...... 63 3.4.2 A prediction model based on the extended Lighthill’s theory. 63 3.4.3 Comparison with experimental data...... 66

3.1 Chapter overview

Jet-surface interaction is generated by the scattering and diffraction of the hydrodynamic pressure field generated by the jet passing by a wing or flap trailing edge. A good understanding of the generation and propagation of the hydrodynamic field is an important requirement in order to understand the interaction process and create an accurate prediction tool.

In the past, the majority of jet noise studies have focused on the propagating acoustic field, relegating the non-propagating terms out of the spotlight. However, in the presence of a solid boundary (i.e. the aircraft wing and flaps) the non-propagating pressure field is

39 40 Chapter 3 Study of the jet near-field scattered by the trailing edge where it becomes a truly propagating field. Therefore, the necessity of understanding and modelling the near-field has become a pressing problem in recent years, due to the advent of high-bypass ratio (HBPR) engines and the potential to move towards ultra high-bypass ratio (UHBPR) engines. This is because, as engines grow larger, they have the jet flow closer to the wing.

The pressure field that surrounds an unbounded jet has been extensively studied during the last 60 years. It is worth mentioning that Morgan, in 1961 [20], and Miller, in 2015 [76], both analysed the near pressure field and showed that it can be decomposed into three terms. Miller called these terms: near-field, mid-field and far-field to relate the region in which each one is dominating.This nomenclature is used throughout this chapter.

Morgan pointed out that the near-field term represents the linear hydrodynamic field or pseudo-sound [20]. This is a non-propagating low frequency noise weakly influenced by compressibility effects, as shown by Howes 1960 [19], Ribner 1962 [21] and Ffowcs Williams 1969 [22] among others. Additionally, Howes and Ffowcs-Williams argued that the hydrodynamic pressure can be obtained by solving the incompressible part of the mass and momentum conservation equations. The intensity of the hydrodynamic field then decays with r−6, where r is the distance to the source. This term is characterized by a phase velocity that is a fraction of the jet velocity, whereas the acoustics propagates at the speed of the sound.

The mid-field term, as described by Morgan [20], dominates the area outside the hydrodynamic pressure field but is close enough to be within the region where the radiated pressure wave can be assumed to be planar and in phase with the particle velocity. For a given frequency, the sound perceived in this region does not have a well defined directivity because it arrives from many different sources which are spaced along the jet, i.e. is a non-compact source region.

Finally, the far-field term represents the acoustic field that is associated with compressible pressure fluctuations. The intensity decays with r−2 and has a propagation velocity close to the speed of sound.

It is well known that Lighthill’s Acoustic Analogy [65] predicts the far-field noise produced by a turbulent flow. As pointed out in section 2.5, to develop his analogy, Lighthill assumed that the observer is located many wavelengths away from the source region. In 2013, Miller [76] showed that a similar expression can be obtained for regions in which these two lengths (the wavelength and the distance between source and observer) Chapter 3 Study of the jet near-field 41

Figure 3.1: Cartisian coordinate system. Origin in nozzle exhaust exit center. are equivalent (i.e. in the near-field). The near-field extension of Lighthill’s analogy consists of three terms, the original one and two new terms that are related to the three regions identified by Morgan and Miller.

This chapter is organized as follows. In section 3.2, Lighthill’s Acoustic Analogy is extended to obtain the pressure in the near-field. In section 3.3, the near-field of a jet is characterized in magnitude, frequency content, radial decay and velocity dependence. This section includes detailed descriptions of the experiment, used to identify each of the three terms and their scaling laws, and discussion of the trends identified by correlation and spectral analyses. Section 3.4 shows how the near pressure field can be predicted using the extended Lighthill’s theory with a similar approach to the one that is used for the far-field.

3.2 Near-ﬁeld extension of Lighthill’s acoustic analogy

In section 2.5, it was shown how Lighthill obtained an expression, Equation (2.12), for the far-field pressure, assuming that the observer is located many wavelengths away from the source. In the near-field, because the distance to the observer is of the order of a wavelength, Equation (2.12) is no longer valid. In this section, Lighthill’s Acoustic Analogy will be extended by means of evaluating Equation (2.8) without the far-field assumption (i.e. retaining all the radiating terms) in order to extend its validity to the near-field region. Equation (2.8) is included here for simplicity.

∞ r 2 T y, t − 1 ∂ ZZZ ij co ρ (x, t) − ρ0 = 2 dy, (2.8 revisited) 4πc0 ∂xi∂xj r −∞ where x is the observer location, y is the source location and r = |x − y| is the distance between the observer and the source as illustrated in Figure 3.1. 42 Chapter 3 Study of the jet near-ﬁeld

If the ﬁrst divergence of the Reynolds stress tensor ∂ T is expanded, Equation (2.8) ∂xi ij yields Z Z 1 ∂ (xj − yj) (xj − yj) ∂Tij ρ (x, t) − ρ0 = − 2 3 Tijdy + 2 dy (3.1) 4πc0 ∂xi V |x − y| V c0 |x − y| ∂t where the following equivalence has been used

∂ r x − y ∂ r f t − = − i i f t − . (3.2) ∂xi c0 rc0 ∂t c0

The second divergence can also be expanded. The ﬁrst term on the right hand side of Equation (3.1) yields

Z Z ∂ (xj − yj) (xi − yi)(xj − yj) 3 Tijdy = −3 5 Tijdy ∂xi V |x − y| V |x − y| Z Z (xi − yi)(xj − yj) ∂Tij δij − 4 dy + 3 Tijdy, (3.3) V c0 |x − y| ∂t V |x − y| an the second term

Z Z ∂ (xj − yj) ∂Tij (xi − yi)(xj − yj) ∂Tij 2 dy = −2 4 dy ∂xi V c0 |x − y| ∂t V c0 |x − y| ∂t Z (x − y )(x − y ) ∂2T Z δ ∂T − i i j j ij dy + ij ij dy, (3.4) 2 3 ∂t2 2 ∂t V c0 |x − y| V c0 |x − y|

2 ∂Tij ∂ Tij Arranging together the terms that are multiplied by Tij, ∂t and ∂t2 the density ﬂuctuation can be written as

 Z 2 1 (xi − yi)(xj − yj)  1 ∂ Tij ρ (x, t) − ρ0 =  2 2 2 |x − y| ∂t2 4πc0 V c0 |x − y|  | {z } A 3 δij ∂Tij + 2 − c0 |x − y| (xi − yi)(xj − yj) ∂t | {z } B  3 δ c2  + − ij 0 T  dy. (3.5) 2 ij |x − y| (xi − yi)(xj − yj) |x − y|  | {z } C

Assuming that the process is isentropic by Equation (2.25), the pressure ﬂuctuations at a location x and a time t can then be written as

1 Z (x − y )(x − y ) p (x, t) = i i j j AT¨ + BT˙ + CT dy, (3.6) 4π 2 2 ij ij ij V c0 |x − y| Chapter 3 Study of the jet near-ﬁeld 43 where the dot and double dot represent the ﬁrst and second time derivatives.

Equation (3.6) is valid for observer locations in both the near-field and the far-field and not just the far-field as is Equation (2.8). For far-field locations, i.e. x much larger than y, Equation (3.6) reduces to

1 Z (x − y )(x − y ) p (x, t) = i i j j AT¨ + o 1/x2 dy, (3.7) 4π 2 2 ij V c0 |x − y|

which is equivalent to Lighthill’s analogy of Equation (2.12) if |x| |y|.

It is interesting to note that the term multilpied by C represents the pressure perturbations that are not influenced by compressibility effects. In other words, the pressure fluctuations described by

1 Z (x − y )(x − y ) p (x, t) = i i j j CT dy (3.8) 4π 2 2 ij V c0 |x − y| are, as pointed by Howes [19], the solution of the incompressible part of the inhomogeneous wave equation, Equation (2.1), namely

2 2 −1 ∂ ∇ ρ = 2 Tij, (3.9) c0 ∂yi∂yj

and represents the hydrodynamic ﬂuctuations, as it was pointed in references [19,21,22].

3.2.1 Cross-power spectral density

Turbulence is a deterministic process but only statistical quantities describing the flow can normally be predicted. This is obviously a function of the accuracy of CFD. In principle, DNS gives a fully deterministic solution. The most commonly used statistic is a two-point cross-correlation of Equation (3.6) is obtained. The two-point cross-correlation of the pressure can be obtained with the time average of the fluctuating pressure at x and t, p (x, t), multiplied by the pressure fluctuations, p (x0, t0), at a different location x0 and time t0 = t + τ. The cross-correlation, then, yields

1 Z Z 0 0 ¨ ˙ 0 ¨0 0 ˙ 0 0 0 p (x, t) p (x , t ) = 2 ATij + BTij + CTij A Tlm + B Tlm + C Tlm × 16π V V (x − y )(x − y )(x0 − y0)(x0 − y0 ) × i i j j l l m m dydy0, (3.10) 4 2 0 0 2 c0 |x − y| |x − y |

where the overline denotes the time average, deﬁned as

1 Z T f (t) f 0 (t + τ) = lim f (t) f 0 (t + τ) dt. (3.11) T →∞ 2T −T 44 Chapter 3 Study of the jet near-ﬁeld

The term between braces in Equation (3.10) can be evaluated, noting that the only time-dependent term is the Reynolds stress tensor. This yields

1 c 3 δ ¨ 0 ˙ 0 0 ˙ 0 0 0 ¨ ¨0 0 lm ¨ ˙ 0 A2 = ATij A Tlm + B Tlm + C Tlm = 0 TijTlm + 02 − 0 0 TijTlm+ rr r r rlrm 2 c0 3 δlm ¨ 0 + 0 02 − 0 0 TijTlm (3.12) rr r rlrm

3 δ hc ˙ 0 ˙ 0 0 ˙ 0 0 0 ij 0 ˙ ¨0 B2 = BTij A Tlm + B Tlm + C Tlm = 2 − 0 TijTlm+ r rirj r 3 2 3 δlm ˙ ˙ 0 c0 3 δlm ˙ 0 +c0 02 − 0 0 TijTlm + 0 02 − 0 0 TijTlm (3.13) r rlrm r r rlrm

3 δ c2 0 ˙ 0 0 ˙ 0 0 0 ij 0 ¨0 C2 = CTij A Tlm + B Tlm + C Tlm = 2 − 0 TijTlm+ r rirj rr 3 4 c0 3 δlm ˙ 0 c0 3 δlm 0 + 02 − 0 0 TijTlm + 0 02 − 0 0 TijTlm (3.14) r r rlrm rr r rlrm where r = |x − y| and ri = xi − yi have been used. Equation (3.10) can then be written as Z Z 0 0 0 0 1 rirjrlrm 0 p (x, t) p (x , t ) = 2 4 2 02 A2 + B2 + C2 dydy . (3.15) 16π V V c0r r

¨ ˙ 0 ˙ 0 ¨ 0 ˙ ˙ 0 Note that TijTlm = 0, TijTlm = 0 and that TijTlm = −TijTlm, which can be demonstrated integrating by parts as follow

Z T ¨ 0 1 ¨ 0 0 TijTlm = lim Tij (y, t) Tlm y , t + τ dt = T →∞ 2T −T T Z T 1 h ˙ 0 0 i 1 ˙ ˙ 0 0 lim Tij (y, t) Tlm y , t + τ − lim Tij (y, t) Tlm y , t + τ dt = T →∞ 2T −T T →∞ 2T −T Z T 1 ˙ ˙ 0 0 ˙ ˙ 0 − lim Tij (y, t) Tlm y , t + τ dt = −TijTlm, (3.16) T →∞ 2T −T and

Z T ˙ 0 1 ˙ 0 0 TijTlm = lim Tij (y, t) Tlm y , t + τ dt = T →∞ 2T −T Z T 1 0 0 T 1 ˙ 0 0 lim Tij (y, t) Tlm y , t + τ −T − lim Tij (y, t) Tlm y , t + τ dt = T →∞ 2T T →∞ 2T −T Z T 1 ˙ 0 0 ˙ 0 lim Tij (y, t) Tlm y , t + τ dt = −TijTlm = 0. (3.17) T →∞ 2T −T Chapter 3 Study of the jet near-ﬁeld 45

A2, B2, C2 can be then simpliﬁed to

2 1 ¨ ¨0 3 δlm c0 ˙ ˙ 0 A2 = 0 TijTlm − 02 − 0 0 0 TijTlm, (3.18) rr r rlrm rr 2 3 δij 3 δlm ˙ ˙ 0 B2 = c0 2 − 02 − 0 0 TijTlm, (3.19) r rirj r rlrm 2 4 c0 3 δij ˙ ˙ 0 c0 3 δij 3 δlm 0 C2 = − 0 2 − TijTlm + 0 2 − 02 − 0 0 TijTlm. (3.20) rr r rirj rr r rirj r rlrm

¨ ¨0 Note that second derivatives of the stress tensor, TijTlm are multiplied by terms that decay with the second power of the observer distance, ≈ r−2. Also, the first derivatives, ˙ ˙ 0 −4 TijTlm are multiplied by terms that decay with the forth power of the distance, ≈ r , 0 −6 and TijTlm by terms that decays with the sixth power of the observer distance, ≈ r . If we group them, we can call them the far-field term, Fijlm, mid-field term, Mijlm, and near-field term, Nijlm. Later in this chapter it will be shown that the names relate them with the region in which they make their main contribution. Equation (3.15) can be then written as

1 Z Z 0 0 0 ˙ ˙ 0 ¨ ¨0 0 p (x, t) p (x , t ) = 2 NijlmTijTlm + MijlmTijTlm + FijlmTijTlm dydy . 16π V V (3.21) where

0 0 2 02 2 02 2 rirjrlrm r r r r 1 Nijlm = 5 05 9 − 3 δij − 3 0 0 δlm + 0 0 δijδlm ∼ 3 , (3.22) r r rirj rlrm rirjrlrm r

0 0 2 02 0 2 rirjrlrm r + r rr − 3r Mijlm = 4 04 2 9 − 3 0 + δij+ r r c0 rr rirj rr0 − 3r02 r2r02 1 2 + 0 0 δlm + 0 0 δijδlm ∼ 2 , (3.23) rlrm rirjrlrm r

0 0 2 rirjrlrm 1 Fijlm = 3 03 4 ∼ . (3.24) r r co r

Assuming that the pressure is statistically stationary in time, the cross-power spectral density (cross-PSD) of the pressure is deﬁned as

Z +∞ 0 0 1 0 r − r Spp x, x , ω = p (x, t) p (x , t + τ) exp −iω τ + dτ, (3.25) 2π −∞ c0 46 Chapter 3 Study of the jet near-ﬁeld which is obtained by taking the time Fourier transform of Equation (3.21). Substituting Equations (3.21, 3.22, 3.23& 3.24) in Equation (3.25), the cross-PSD is ﬁnally reached

Z Z Z +∞ 2 4 0 1 ∂ ∂ Spp x, x , ω = 3 Nijlm + Mijlm 2 + Fijlm 4 × 32π V V −∞ ∂τ ∂τ 0 0 i r − r TijTlm (y, η, τ) exp −iω τ + dτdηdy, (3.26) c0 and can be written as

Z Z Z +∞ 2 4 0 1 ∂ ∂ Spp x, x , ω = 3 Nijlm + Mijlm 2 + Fijlm 4 × 32π V V −∞ ∂τ ∂τ r − r0 Rijlm (y, η, τ)] exp −iω τ + dτdηdy, (3.27) c0

0 where Rijlm (y, η, τ) ≡ Tij (y, t) Tlm (y + η, t + τ) is the fourth-order space-time source correlation tensor deﬁned by Goldstein [77] and η = y0 − y is the separation vector between two source points. This equation allows us to calculate the cross-PSD between two locations if the fourth order correlation tensor is known within the jet plume. The double volume integral can be interpreted as an integral over the volume of an eddy R dη, where the correlation tensor is non-zero and a second integral over the volume of the entire jet R dy.

3.2.2 Scaling of the jet near-ﬁeld

In this section the cross-PSD of the near-field pressure of a jet will be characterized. This will help to understand and model the jet near-field appropriately. A similar scaling approach will be used as per Lighthill’s far-field analysis. Note that Spp (Equation (3.27)) is the sum of three functions with different decay behaviours (1/r6, 1/r4 and 1/r2), which can be written as:

1 Z Z Z +∞ r − r0 SNF = 3 NijlmRijlm (y, η, τ) exp −iω τ + dτdηdy, 32π V V −∞ c0 (3.28) 1 Z Z Z +∞ ∂2 r − r0 SMF = 3 Mijlm 2 Rijlm (y, η, τ) exp −iω τ + dτdηdy, 32π V V −∞ ∂τ c0 (3.29) 1 Z Z Z +∞ ∂4 r − r0 SFF = 3 Fijlm 4 Rijlm (y, η, τ) exp −iω τ + dτdηdy, 32π V V −∞ ∂τ c0 (3.30) 1 1 1 where Nijlm ∼ 6 , Nijlm ∼ 4 2 and Fijlm ∼ 2 4 . For an isothermal jet, that is the case r r c0 r c0 of study, Lighthill’s stress tensor can be approximated by Reynolds stress

Tij ∼ ρ0uiuj, (3.31) Chapter 3 Study of the jet near-ﬁeld 47

2 and it is easy to note then that Lighthill’s stress tensor then scales as Tij ∼ U . The 0 4 fourth order space-time correlation tensor then scales as Rijlm (y, η, τ) ≡ TijTlm ∼ U . The volume integrals are proportional to D3 and the integral over time is proportional to the time scale τ0, which is proportional to D/U. Thus, the following scaling laws, for each of the terms are obtained

U 3D3 D4 U 3D7 S ∼ = , (3.32) NF r2 r4 r6

U 3D3 U 2 D2 U 5D5 SMF ∼ 2 2 2 = 2 4 , (3.33) r c0 r c0r U 3D3 U 4 U 7D3 SFF ∼ 2 4 = 4 2 . (3.34) r c0 c0r

The above equations show that each term scales differently with velocity as well as 7 with distance. The far-field cross-PSD term, SFF , scales with U , a factor which is expected for the PSD of the acoustic field, related to the famous Lighthill’s U 8 law for 5 acoustic power. The mid-field cross-PSD term SMF , Equation (3.33), scales with U 3 and the near-field cross-PSD term SNF , Equation (3.32), scales with U . 48 Chapter 3 Study of the jet near-field

3.3 Near-ﬁeld experimental data

In section 3.2, Lightill’s Acoustic Analogy was extended into the near-field and the near pressure field of a jet was found to consist as the sum of three terms: a near-field, a mid-field and a far-field term. In this section, experimental data will be examined in order to confirm the existence of these three terms. The scaling with the observer location, r, and the jet velocity, U will be analysed and the frequency content associated with each term will be identified.

3.3.1 Near-ﬁeld experimental setup

This section examines the data from an experimental campaign conducted on a small model-scale isolated single-stream jet. It was conducted by Lawrence in the DOAK laboratory at the ISVR [1]. The jet is unheated and it is exhausted from a conical nozzle with a 38.1 mm inner exit diameter, D.

The data was collected with the traversable microphone array pictured in Figure 3.2. The array consisted of twelve microphones and was angled parallel to the nominal edge ◦ of the jet shear layer, at θj = 7 to the jet axis. The distance, orthogonal to the jet axis, between each microphone and the shear layer, rs, is kept constant, see Figure 3.3. The array was traversed perpendicular to the edge of the jet shear layer between

0.45 ≤ rs ≤ 13.09. The data was recorded at 22 radial locations and for four diﬀerent exit acoustic Mach numbers (M = 0.30, M = 0.50, M = 0.75 and M = 0.90).

In Figure 3.4, data from the traversable array can be seen for six different axial locations. It is easy to observe two separate areas in the figures: (1) A low frequency hump that becomes dominant for the closest locations to the jet and almost disappears at the farthest locations. (2) A region that decays much slower with the distance and becomes dominant in the farthest locations. These two regions can be identifieed as the linear hydrodynamic field which is just perceived in the near-field and the acoustic field that propagates to the far-field. It will be shown that these two regions scale as the near-field and far-field terms of Equation (3.27) do.

The change with the velocity can be observed in Figures 3.5 and 3.6. In both figures, the SPL is shown for four Mach number (M = 0.3, M = 0.5, M = 0.75 & M = 0.9) at six different axial location (L/D = 1.05, L/D = 2.10, L/D = 3.15, L/D = 4.20, L/D = 5.25 & L/D = 6.30). The data shown in Figure 3.5 corresponds to the farthest radial location of the traverse rs/D = 13.09 in which, as stated above, the contribution of the hydrodynamic term is minimum. Data in Figure 3.6, instead, corresponds to the closest radial location rs/D = 0.45 where the hydrodynamic field is the dominant term. Chapter 3 Study of the jet near-field 49

Figure 3.2: DOAK laboratory. Near-ﬁeld array parallel to the jet plume

Figure 3.3: DOAK laboratory. Near-ﬁeld array schematic 50 Chapter 3 Study of the jet near-ﬁeld

Figure 3.4: Radial decay of the Sound Pressure Level for an isothermal single- stream jet. Data from the near-ﬁeld array at DOAK laboratory, D = 0.0381m, M = 0.30, ∆f = 10Hz. The data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for 22 radial positions, from rs/D = 0.45 to rs/D = 13.00. Chapter 3 Study of the jet near-ﬁeld 51

Figure 3.5: Change with the jet velocity of the Sound Pressure Level for an isothermal single-stream jet. Data from the near-ﬁeld array at DOAK laboratory, D = 0.0381m, ∆f = 10Hz at the farthest location rs/D = 13.00. The data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for 4 jet Mach numbers M = 0.30, 0.50, 0.75 & 0.90. 52 Chapter 3 Study of the jet near-ﬁeld

Figure 3.6: Change with the jet velocity of the Sound Pressure Level for an isothermal single-stream jet. Data from the near-ﬁeld array at DOAK laboratory, D = 0.0381m, ∆f = 10Hz at the closest location rs/D = 0.45. D data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for 4 jet Mach numbers M = 0.30, 0.50, 0.75 & 0.90. Chapter 3 Study of the jet near-ﬁeld 53

It should be noted that the SPL drop at low Strouhal numbers (St < 0.1) for the data at M = 0.3 that can be observed in Figures 3.4 and 3.5 at the axial location x/D = 5.25 and radial location rs/D = 13.00 is due to an acquisition issue and it is not representative of the SPL at those frequencies.

3.3.2 Radial Scaling

It has been stated in section 3.2.1 that the three terms (Near-field, mid-field & far- field) scale different with the source to observer distance r. They scale as r−2, r−4 and r−6, respectively. These three scaling factors will be applied to the experimental data in order to identify the regions in which each term dominates.

Figure 3.7 shows the radial decay of the SPL at six axial locations (L/D =1.05, 2.10, 3.15, 4.20, 5.25 & 6.30) for diﬀerent Strouhal numbers. The radial decay is consistent along the axial locations of the jet as it can be observed in Figure 3.7. In Figure 3.8, two slopes can be observed. At low frequencies and at locations close to the jet, the slope of −60·log10 (H) dominates, where H is the radial distance from jet centreline, see Figure 3.3. The maximum H at which this slope is valid decreases as St increases. On

the other hand, at high frequencies the slope is −20·log10 (H) and the minimum H of validity for this slope increases as St decreases. Is it easy now to relate these slope with the radial scaling of the near-ﬁeld term, 1/r6, and far-ﬁeld term, 1/r2, respectively (as stated in Equations (3.32)&(3.34)). One would expect then an intermediate region

region associated with the mid-ﬁeld term that scales as −40·log10 (H). This region of H

and St would be between the −60·log10 (H) and the −20·log10 (H) extremes.

The far-field term, Equation (3.24), was identified with the acoustic field and decays with r−2. Figure 3.9 shows the SPL for the 6 previously defined axial locations. The

SPL has been scaled by adding 20·log10 (H). As expected, the mid and high frequencies,

identified with the acoustic field, present the best data collapse from the 20·log10 (H) scaling. The data collapse better at the farthest locations, in which the far-field term is dominant and the contribution of the near-field and mid-field terms is minimal. This

can be seen in Figure 3.10, that shows the best collapse with the 20·log10 (H) scaling

for rs/D ≥ 3.

The mid-ﬁeld term, Equation (3.23), decays with r−4. A region can be found 1 <

rs/D < 3 in which the SPL scales with 40·log10 (1/H). Figure 3.11 shows the data collapse with this scaling for the locations within the range. 54 Chapter 3 Study of the jet near-ﬁeld

Figure 3.7: Radial separation Vs. Sound Pressure Level for an isothermal single- stream jet. Data from the near-ﬁeld array at DOAK laboratory, D = 0.0381m, M = 0.30, ∆f = 10Hz. The data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for 14 Strouhal numbers, from St = 0.10 to St = 10.0. Chapter 3 Study of the jet near-ﬁeld 55

Figure 3.8: Slopes for the radial separation decay in the Sound Pressure Level for an isothermal single-stream jet. Data from the near-field array at DOAK laboratory, D = 0.0381m, M = 0.30, L/D = 3.15. Black dashed lines show the three different slopes associated with the near-field (−60·log10 (H)), mid-field (−40·log10 (H)) and far-field (−20·log10 (H)).

A similar approach can be followed for the near-field term, that represents the linear hydrodynamic field. This is present as the low frequency hump found in the near-field. According to Equation (3.22) in section 3.2.1, it decays as r−6. Figure 3.12 shows this

scaling for the closest radial locations rs/D ≤ 1. The SPL has been corrected with

60·log10 (H). Note that the data collapse observed at this location is not as good as the one observed for the farther locations with the far-field and mid-field terms, Figures 3.10 and 3.11. This is because, as the microphone is moved away into the far-field region, the contribution of the near-field and mid-field terms can be neglected. In the mid-field region, only the near-field term can be neglected. For the near-field region, neither the far-field term nor the mid-field term can be neglected. 56 Chapter 3 Study of the jet near-field

Figure 3.9: Far-field corrected Sound Pressure Level for an isothermal single- stream jet. Data from the near-field array at DOAK laboratory, D = 0.0381m, M = 0.30, ∆f = 10Hz. The data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for 22 radial positions, from rs/D = 0.45 to rs/D = 13.9. The SPL is corrected with 20·log10 (H). Chapter 3 Study of the jet near-field 57

Figure 3.10: Far-field corrected Sound Pressure Level for an isothermal single- stream jet. Data from the near-field array at DOAK laboratory, D = 0.0381m, M = 0.30, ∆f = 10Hz. The data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for the farthest radial positions, from rs/D = 3.02 to rs/D = 13.9. The SPL is corrected with 20·log10 (H). 58 Chapter 3 Study of the jet near-field

Figure 3.11: Mid-field corrected Sound Pressure Level for an isothermal single- stream jet. Data from the near-field array at DOAK laboratory, D = 0.0381m, M = 0.30, ∆f = 10Hz. The data is presented at six axial locations (L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3). For each of these axial locations the SPL is shown for the radial positions from rs/D = 1.25 to rs/D = 2.01. The SPL is corrected with 40·log10 (H). Chapter 3 Study of the jet near-field 59

Figure 3.12: Near-field corrected Sound Pressure Level for an isothermal single- stream jet. Data from the near-field array at DOAK laboratory, D = 0.0381m, M = 0.30, ∆f = 10Hz. The data is presented at six axial locations L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3. For each of these axial locations the SPL is shown for the closest radial positions, from rs/D = 0.45 to rs/D = 1.00. The SPL is corrected with 60·log10 (H). 60 Chapter 3 Study of the jet near-field

3.3.3 Velocity Scaling

The same approach followed in section 3.3.2 with the radial distance is followed in this section with the velocity. Section 3.2.2 showed that the three components of the cross-PSD (near-field, mid-field and far-field) scale differently with velocity. They scale as U 3, U 5 and U 3, respectively. These three factors will be applied to the experimental data in order to validate them and to identify the regions in which each term dominates.

The cross-PSD of the far-ﬁeld or acoustic term scales as U 7, as stated in Equation

(3.34) in section 3.2.2. Figure 3.13 shows the SPL scaled by with −70·log10 (M). A very good data collapse can be observed at medium and high frequencies above St = 1.0. For low frequencies, the agreement increases with the Mach number.

The near-field or hydrodynamic term, Equation (3.32), scales as U 3. Figure 3.14 shows the SPL scaled by −30·log10 (M). A good data collapse is observed for frequencies below the hydrodynamic peak for all cases but M = 0.3, where the the curve only matches up to St = 0.1. The data spread at medium and high frequencies is produced, as it was stated in section 3.3.2, by the contribution of the mid-field and far-field terms, that cannot be neglected at these locations.

According to Equation (3.33), the mid-field term scales as U 5. It is then expected that a region of the cross-PSD scales with −50·log10 (M), in the same way that it has been done with the far-field and near-field terms. However, in the used dataset, it is not possible to find this region. This is probably produced because it is not possible to find a region where the mid-field term dominates. What can be observed, instead, is probably a mix of the three terms. Chapter 3 Study of the jet near-field 61

Figure 3.13: Sound Pressure Level scaled by exit Mach number for an isothermal single-stream jet at the farthest radial location rs/D = 13.00 of the near-ﬁeld array at DOAK laboratory, D = 0.0381m, ∆f = 10Hz. The data is presented at six axial locations (L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3). For each of these axial locations the SPL is shown for 4 jet Mach numbers (M = 0.30, 0.50, 0.75 & 0.90). The SPL is scaled by 70·log(M). 62 Chapter 3 Study of the jet near-ﬁeld

Figure 3.14: Sound Pressure Level scaled by exit Mach number for an isothermal single-stream jet at the closest radial location rs/D = 0.45 of the near-ﬁeld array at DOAK laboratory, D = 0.0381m, ∆f = 10Hz. The data is presented at six axial locations (L/D = 1.05, 2.1, 3.15, 4.2, 5.25 & 6.3). For each of these axial locations the SPL is shown for 4 jet Mach numbers (M = 0.30, 0.50, 0.75 & 0.90). The SPL is scaled by 30·log(M). Chapter 3 Study of the jet near-ﬁeld 63

3.4 Jet near-ﬁeld prediction

With an appropriate model for the fourth order correlation tensor, Rijlm, Lighthill’s Acoustic Analogy allows to obtain the far-field mixing noise produced by a jet from a RANS CFD solution, [78]. This section, shows that the same approach followed for the far-field predictions can be used, together with the extended Lighthill’s theory, to predict the jet near-field pressure. The intention of this section is not to present an optimum prediction for the near-field but to show that, using a model for Rijlm like the one used for the far-field predictions, the near-field pressure can be calculated.

3.4.1 A model for the fourth order correlation tensor

The model proposed by Tam [79] for Rijlm (y, η, τ) is adapted, as this can be integrated analytically and subsequently produce a simpliﬁed model:

( 2 |η1| η1 − ujτ Rijlm (y, η, τ) = Aijlm (y) exp − − − ujτs (y) l1 (y) ) η 2 η 2 − 2 − 3 , (3.35) l2 (y) l3 (y) where k (y) A (y) = C (ρk (y))2 ; τ (y) = C ; ijlm ijlm s τ 3 k 2 (y) l (y) l (y) = C ; l (y) = l (y) = 1 . (3.36) 1 l 2 3 3 where k is the speciﬁc turbulent kinetic energy and is the speciﬁc dissipation of the turbulent kinetic energy. These two values can be obtained from RANS solutions.

3.4.2 A prediction model based on the extended Lighthill’s theory

In section 3.2, a new expression for the pressure field of a jet, valid in the near-field, was developed by removing all far-field assumptions and re-deriving Lighthill’s analogy. The exact solution of the cross-PSD of a jet, represented by Equation (3.27), is reproduced here for convenience:

Z Z Z +∞ 2 4 0 1 ∂ ∂ Spp x, x , ω = 3 Nijlm + Mijlm 2 + Fijlm 4 × 32π V V −∞ ∂τ ∂τ r − r0 ×Rijlm (y, η, τ)] exp −iω τ + dτdηdy. (3.27 revisited) c0

The model for the fourth order correlation tensor presented in section 3.4.1 is now used. If Equation (3.35) is substituted in Equation (3.27), it is easy to note that the 64 Chapter 3 Study of the jet near-ﬁeld derivatives and integral over time can be easily evaluated. The integral over τ yields

Z +∞ " 2 4 ( 2)# ∂ ∂ η1 − ujτ Nijlm + Mijlm 2 + Fijlm 4 exp − × −∞ ∂τ ∂τ l1 × exp {−iωτ} dτ, (3.37) in which the time derivatives only aﬀect the ﬁrst exponential, that for convenience of notation, is referred to as D0

( 2) η1 − ujτ D0 = exp − . (3.38) l1

The second and fourth time derivatives of D0 are

2 2 2 ! ( 2) ∂ uj η1 − ujτ η1 − ujτ D2 = 2 D0 = 2 2 − 1 exp − , (3.39) ∂τ l1 l1 l1 and

3 4 4 ! ∂ uj η1 − ujτ η1 − ujτ D4 = 3 D0 = 4 4 − 12 + 3 × ∂τ l1 l1 l1 ( ) η − u τ 2 × exp − 1 j . (3.40) l1

Substituting equations (3.38, 3.39& 3.40) in Equation (3.37), the integral over τ can be written as the sum of three integrals

Z +∞ (NijlmD0 + MijlmD2 + FijlmD4) exp {−iωτ} dτ = −∞

= NijlmI0 + MijlmI2 + FijlmI4, (3.41) where I0, I2 and I4 are the following integrals

Z +∞ I0 = D0 exp {−iωτ} dτ, (3.42) −∞ Z +∞ I2 = D2 exp {−iωτ} dτ, (3.43) −∞ Z +∞ I4 = D4 exp {−iωτ} dτ, (3.44) −∞ that can be evaluated as follows

2 2 √ l1 η1 l1ω I0 = π exp −iω − , (3.45) uj uj 4uj Chapter 3 Study of the jet near-ﬁeld 65

2 I2 = ω I0, (3.46)

4 I4 = ω I0, (3.47)

Equation (3.27) then yields

√ Z Z 0 π l1 2 4 Spp x, x , ω = 3 Nijlm + ω Mijlm + ω Fijlm Aijlm× 32π V V uj ( 0 2 2 2 2) η1 r − r l1ω |η1| η2 η3 × exp −iω + − 2 − − − dηdy. (3.48) uj c0 4uj ujτs l2 l3

Since eﬀects in the axial direction are dominant, the dependence on η2 and η3 for

Nijlm, Mijlm and Fijlm are minimal and their integrals result in

Z +∞ ( 2) ηi √ exp − dηi = πli; for i = 2, 3, (3.49) −∞ li

Equation (3.48) then yields

√ Z Z +∞ 0 π l1l2l3 2 4 Spp x, x , ω = 2 Nijlm + ω Mijlm + ω Fijlm Aijlm× 32π V −∞ uj 0 2 2 η1 r − r l1ω |η1| × exp −iω + − − dη1dy. (3.50) uj c0 4uj ujτs

An approximation for the cross-PSD of Equation (3.50) can be obtained by assuminge

that the integral for η1 can be evaluated in the same way that the ones for η2 adn η3,

i.e. the dependence on η1 for Nijlm, Mijlm and Fijlm is minimal and therefore the term 2 4 Nijlm + ω Mijlm + ω Fijlm can be moved out of the integral. The integral on η1 then yields, Z +∞ |η1| η1 exp − − iω dη1, (3.51) −∞ ujτs uj that can be evaluated by splitting the integral into two intervals on which it doesn’t change sign, from −∞ to 0 and from 0 to +∞.

Z +∞ Z 0 |η1| η1 1 iω exp − − iω dη1 = exp − η1 dη1+ −∞ ujτs uj −∞ ujτs uj Z +∞ 1 iω + exp − + η1 dη1. (3.52) −0 ujτs uj

The ﬁrst part of the integral is

Z 0 1 iω 1 exp − η1 dη1 = 1 iω , (3.53) −∞ ujτs uj − uj τs uj 66 Chapter 3 Study of the jet near-ﬁeld and the second

Z +∞ 1 iω 1 exp − + η1 dη1 = 1 iω . (3.54) 0 ujτs uj + uj τs uj

The integral on η1 (Equation (3.51)) then yields

Z +∞ |η1| η1 2ujτs exp − − iω dη1 = 2 2 (3.55) −∞ ujτs uj 1 + ω τs and ﬁnally the cross-power spectral density of Equation (3.50) can be written as

√ Z 0 π τsl1Aijlm 2 4 Spp x, x , ω = 2 2 2 Nijlm + ω Mijlm + ω Fijlm × 16π 9 V 1 + ω τs ( 0 2 2 ) r − r l1ω × exp −iω − 2 dy. (3.56) c0 4uj

Equation (3.56) gives Spp as the integral over the jet plume of the sum of three modiﬁed Gaussian functions of ω (angular frequency) that represent the three diﬀerent decaying with the distance ratios (1/r6, 1/r4 and 1/r2) of the spectral density:

( 2 ) Nijlm l1 2 GNF (y, ω) = 2 2 exp − 2 ω , (3.57) 1 + τs ω 4uj

2 ( 2 ) ω Mijlm l1 2 GMF (y, ω) = 2 2 exp − 2 ω , (3.58) 1 + τs ω 4uj

4 ( 2 ) ω Fijlm l1 2 GFF (y, ω) = 2 2 exp − 2 ω , (3.59) 1 + τs ω 4uj

1 1 1 where Nijlm ∼ 6 , Nijlm ∼ 4 2 and Fijlm ∼ 2 4 , as stated in section 3.2.2. The three r r c0 r c0 Gaussian functions, GNF , GMF and GFF , are then named by the region (near-field, mid-field and far-field) in which their contribution is dominant. An example of these three terms is shown in Figure 3.15, for an arbitrary value of the parameters τs and l1.

3.4.3 Comparison with experimental data

In this section, Equation (3.56) is solved numerically. The mean ﬂow and the values of k and are obtained directly from RANS calculations provided by University of

Cambridge [80]. Once k and are sourced, the time scale τs, the length scale ls and the amplitude factor Aijlm are determined by Equation (3.36). A simple isotropic model 2 has been chosen for Rijlm (y, η, τ) and Aijlm = C (ρk (y)) . C, Cτ and Cl have to be chosen to obtain the best fit to the data, and have been kept constant throughout the different predictions shown in this section. Chapter 3 Study of the jet near-field 67

Figure 3.15: Variation of the near-field, mid-field and far-field terms with the distance for l1 = D = 0.0381m, uj = 150m/s, τs = l1/uj

An example prediction is shown in Figure 3.16. For the same axial location L/D = 5.25, three radial locations have been chosen, one representative of each geometric region: near-field (H/D = 1.59), mid-field (H/D = 4.12) and far-field (H/D = 14.05). The black solid lines represents the total prediction which consists of the sum of the three terms (NF , MF and FF terms). It can be noted that each term dominates in its corresponding geometric and frequency regions, as was stated in section 3.3. 68 Chapter 3 Study of the jet near-field

Figure 3.16: Extended Lighthill’s theory predictions showing the contribution of near- field, mid-field and far-field terms at 3 radial locations H/D = 1.59, 4.12 & 14.05 for an isothermal single-stream jet. Data from the near-field array at DOAK laboratory, D = 0.0381m, M = 0.30, L/D = 5.25, ∆f = 10Hz is shown in coloured solid line. The total predictions are shown in black solid line and consist of the sum of the three terms: near-field term in the blue dashed line, mid-field term in the red dashed line and far-field term in the green dashed line. Chapter 3 Study of the jet near-field 69

Figure 3.17: Doak Lab. experimental (coloured solid line) data compared with prediction using extended Lighthill’s theory at 4 diﬀerent axial locations L/D = 3.15, 4.20, 5.25 & 6.30

Figure 3.17 shows the prediction at four axial locations L/D = 3.15, 4.20, 5.25 & 6.30. For each of these axial location three radial position has been chosen: 1) near-field location (blue), mid-field location (red) and far-field location (yellow). The agreement between prediction and the experimental data decreases approaching the nozzle exhaust exit.

Chapter 4

Theoretical model for the scattered pressure ﬁeld

Contents 4.1 Chapter overview...... 71 4.2 Problem description...... 72 4.3 Scattering process...... 72 4.3.1 Incident pressure field...... 72 4.3.2 Scattered pressure field...... 74 4.4 Far-field radiation...... 86 4.4.1 Method 1:...... 87 4.4.2 Method 2:...... 88 4.5 Far-field directivity...... 91 4.6 Acoustic spectrum...... 92 4.7 Comparison with Amiet’s trailing edge model...... 93

4.1 Chapter overview

This chapter presents a model for the scattering process between a flat plate and the near pressure field of an isothermal jet. The plate is located parallel to the jet axis and close enough to perceive the incident hydrodynamic field radiated by the jet, that is assumed as a single, subsonically convecting, harmonic gust. A theoretical expression for the scattered pressure field is derived in terms of the near-field pressure on the plate using the Wiener-Hopf method. The scattered field is then propagated to the far-field calculating the acoustic power spectral density (PSD). The solution is equivalent to the boundary layer noise model of Amiet [3].

71 72 Chapter 4 Theoretical model for the scattered pressure ﬁeld

Note that, in aeroacoustic, the Wiener-Hopf technique is used for a two-dimensional gust passing by a trailing edge or for a three-dimensional gust with stationary statistics along the span-wise direction. The jet-surface interaction is a three-dimensional problem but the hydrodynamic ﬁeld cannot be considered homogeneous in the span-wise direction. In order to make use of Wiener-Hopf technique, a Wigner-Ville distribution on the span-wise direction is used.

4.2 Problem description

In the development of the model several assumptions are made. The aerofoil is assumed to be a flat plate with zero thickness and zero angle of attack. The plate has a chord c, a span length d and it sees the turbulence of the jet as an incident pressure field that can be assumed as a single, subsonically convecting, harmonic gust. Even if the aerofoil has a finite chord, it will be assumed to be semi-infinite with a trailing edge but without a leading edge. Later, this assumption will be relaxed and finite chord effects will be analysed at the end of the chapter.

The Cartesian coordinate system xt = (xt1, xt2, xt3), deﬁned in Section 2.6.3 by Equa- tion (2.23) and sketched in Figure 2.18 has been chosen for the development of the model.

The coordinate system origin is at the trailing edge centre, the xt1 coordinate is parallel to the jet axis, pointing downstream, and the xt3 coordinate is perpendicular to the aerofoil, pointing at the ground, towards the jet. Figure 2.18 shows, as well, a spherical coordinate system, xt = (rt, θt, φt), centred on the trailing edge that will be used later.

4.3 Scattering process.

The acoustic response of the aerofoil can be expressed as a dipole distribution at the trailing edge with an amplitude equal to the surface pressure force. For the jet problem, the force is the one generated by the pressure jump between the two sides of the aerofoil. The pressure at each side of the aerofoil can be written as the sum of two fields, p = pi + ps, as is shown in Figure 4.1. pi represents the incident pressure field that would be there if the jet were isolated, i.e. the pressure field defined in Chapter 3 by equation (3.6), while ps represents the interaction of this incident field with the aerofoil, i.e. the scattered pressure field.

4.3.1 Incident pressure ﬁeld

The incident pressure field pi must satisfy the homogeneous wave equation in the region exterior to the jet where it is assumed that the fluid is at rest and there are no Chapter 4 Theoretical model for the scattered pressure field 73

Figure 4.1: Decomposition of the pressure ﬁeld into incident pressure from the jet and scattered pressure due to the jet-surface interaction. acoustic sources

2 2 2 2 ∂ ∂ ∂ 1 ∂ i 2 + 2 + 2 − 2 2 p (xt1, xt2, xt3, t) = 0. (4.1) ∂xt1 ∂xt2 ∂xt3 c0 ∂t

the Fourier transform in time and at the span-wise coordinate xt2 of the incident pressure is deﬁned as

∞ T 1 Z Z p˜i (x , k , x , ω) = pi (x , x , x , t) exp {−iωt + ik x } dtdx , (4.2) t1 2 t3 2π t1 t2 t3 2 t2 t2 −∞ −T

where T → ∞ is assumed such that an inverse transform can be defined. As stated in Chapter2, section 2.6.4, the turbulent intensity at the trailing edge is small and satisfies Taylor’s hypothesis, as detailed by Amiet in his JSV paper [67]. Therefore, it is assumed that the incident pressure field is frozen when convected past the trailing edge at velocity a U in the x -direction, i.e k = ω . The incident pressure in the frequency domain c t1 1 Uc can be written as

∞ Z i i pˆ (xt1, xt2, xt3, ω) = P (xt3, k2, ω) exp {−ik1xt1 − ik2xt2} dk2, (4.3) −∞

i i where k1 = ω/Uc is the convective wavenumber and P (xt3, k2, ω) is equal top ˜ (xt1, k2, xt3, ω),

Equation (4.2), at xt1 = 0. Applying the Fourier transform of Equation (4.2) to the wave equation, Equation (4.1), it yields

2 ∂ 2 i 2 − k3 P (xt3, k2, ω) = 0, (4.4) ∂xt3

where q 2 2 2 ω k3 = k − k1 + k2, k = = k1Mc. (4.5) c0 74 Chapter 4 Theoretical model for the scattered pressure ﬁeld

√ Note that indicates the principal square root (branch cut along negative real axis).

The ﬁeld decays towards the aerofoil, that is located at xt3 = 0 (i.e. the jet occupies a region xt3 > 0). The solution to this equation for xt3 below the jet is thus

i i P (xt3, k2, ω) = P0 (k2, ω) exp {k3xt3} . (4.6)

i i where P0 is the value of P at the plate (xt3 = 0).

4.3.2 Scattered pressure ﬁeld

The scattered pressure ﬁeld, ps, is now calculated using the Wiener-Hopf technique i s as a function of the incident pressure at the plate, P0. p must satisfy the homogeneous wave equation

2 2 2 2 ∂ ∂ ∂ 1 ∂ s 2 + 2 + 2 − 2 2 p (xt1, xt2, xt3, t) = 0, (4.7) ∂xt1 ∂xt2 ∂xt3 ∂c0 ∂t and the following boundary conditions

∂ps ∂pi = − , xt1 < 0, xt3 = 0; (4.8) ∂xt3 ∂xt3 s p = 0, xt1 > 0, xt3 = 0. (4.9)

The first boundary condition refers to the hard wall condition on the aerofoil surface. Since the aerofoil is assumed to be perfectly rigid, the normal derivative of the total fluctuation field p on the aerofoil surface must be zero. The second boundary condition is a consequence of the dipole behaviour of the scattered field, being then an odd function s in xt3. Then, to be a continuous function outside the plate (xt1 > 0), p has to be zero at xt3 = 0. If the Fourier transform in t and xt2 are taken, equation 4.7 yields,

2 2 ∂ ∂ 2 s 2 + 2 + K P (xt1, k2, xt3, ω) = 0, (4.10) ∂xt1 ∂xt3 where 2 2 2 K = k − k2 (4.11)

2 is the wavenumber in the xt1 − xt3 plane. Note that for cases where K is negative, the Fourier components of the scattered ﬁeld will not radiate. Therefore only consider cases 2 where K > 0 are considered. The transformed boundary conditions are for a given k2 and ω

s ∂P (xt1, xt3) i = −k3P0 exp {−ik1xt1} , xt1 < 0, xt3 = 0; (4.12) ∂xt3 s P (xt1, xt3) = 0, xt1 > 0, xt3 = 0; (4.13) Chapter 4 Theoretical model for the scattered pressure ﬁeld 75

s s where the dependence of P and P˜ on k2 and ω is implied.

4.3.2.1 Solution for P s using the Wiener-Hopf technique

Sections 4.3.1 and 4.3.2 define a problem with separable geometry and different boundary conditions on different regions of the boundary (i.e. mixed boundary conditions). Therefore, the Wiener-Hopf technique [81] can be used to calculate P s, as it is explained by Vera [82].

The Fourier transform in xt1 and its inverse are deﬁned as follows

∞ 1 Z P˜s (κ , k , x , ω) = P s (x , k , x , ω) exp {iκ x } dx , (4.14) 1 2 t3 2π t1 2 t3 1 t1 t1 −∞

∞ Z s s P (xt1, k2, xt3, ω) = P˜ (κ1, k2, xt3, ω) exp {−iκ1xt1} dκ1, (4.15) −∞

Applying this Fourier transform to the governing Equation (4.10) yields

2 ∂ 2 ˜s 2 + κ3 P (κ1, k2, xt3, ω) = 0, (4.16) ∂xt3

where q 2 2 κ3 = K − κ1. (4.17)

Only cases in which K2 > 0 are considered because they will be the ones that will

radiate, as it was explained above. κ3 has two branch points at κ1 = ±K. The root to take when we move away from the branch points has to be chosen. A single branch cut

joining the two roots would be possible. However, in order to ensures that κ3 is analytic

in the strip between − < Im (κ1) < , two separate branch cuts are chosen from each

zero of κ3 to inﬁnity in each half of the complex κ1-plane., so that these branches do not cut the real axis. These branch cuts are shown in Figure 4.2.

The solution of equation 4.16 is thus

s P˜ (κ1, k2, xt3 ≥ 0, ω) = A (κ1, k2, ω) exp {−iκ3xt3} , (4.18) s P˜ (κ1, k2, xt3 ≤ 0, ω) = −A (κ1, k2, ω) exp {iκ3xt3} . 76 Chapter 4 Theoretical model for the scattered pressure ﬁeld

Figure 4.2: Branch cuts deﬁning κ3 in the complex κ1-plane.

This solution satisﬁes the governing ordinary diﬀerential equation and the constrains of the boundary conditions applied on xt3 = 0. However, in order to use Wiener-Hopf technique, the terms of equation 4.18 have to be rearranged so that one side is analytic in the upper half of the complex κ1-plane, while the other side is analytic in the lower half. Thus, k3 is factorised as κ3 = κ3+κ3− where

1 κ3+ = (K − κ1) 2 , 1 (4.19) κ3− = (K + κ1) 2 .

Note that the subscript indicates in which half of the complex κ1-plane the function is analytic, i.e. κ3+ is analytic in the upper half of the complex κ1-plane, while κ3− is analytic in the lower half of the complex κ1-plane. Figure 4.3 shows the real and imaginary components of κ3, κ3+ and κ3− in the complex κ1-plane for → 0 and the branch cuts have been aligned parallel to the imaginary κ1-axis. The branch cuts are deﬁned as follows

Im (K − κ ) 3π π κ = p|K − κ | exp {iφ /2} , tan φ = 1 , − < φ ≤ ; 3+ 1 + + Re (K − κ ) 2 + 2 1 (4.20) p Im (K + κ1) 3π π κ3− = |K + κ1| exp {iφ−/2} , tan φ− = , − < φ− ≤ . Re (K + κ1) 2 2

Note that the branch cut that satisﬁes that lim κ3 = −i |κ1| along the real axis has κ1→±∞ been selected.

Because of the mixed boundary condition, it is convenient to rewrite the Fourier transform, Equation (4.14), as the sum of two half-transforms

˜s ˜s ˜s P (κ1, k2, xt3, ω) = P− (κ1, k2, xt3, ω) + P+ (κ1, k2, xt3, ω) , (4.21) Chapter 4 Theoretical model for the scattered pressure ﬁeld 77

Figure 4.3: Real and imaginary components of κ3, κ3+ and κ3− in the complex κ1-plane for K = 1.

where 78 Chapter 4 Theoretical model for the scattered pressure ﬁeld

0 1 Z P˜s (κ , k , x , ω) = P s (x , k , x , ω) exp {iκ x } dx , (4.22) − 1 2 t3 2π 1 2 t3 1 t1 t1 −∞ ∞ 1 Z P˜s (κ , k , x , ω) = P s (x , k , x , ω) exp {iκ x } dx . (4.23) + 1 2 t3 2π t1 2 t3 1 t1 t1 0

Note that each integrand contains a factor, exp {−Im (κ1) xt1}, so that for each integral there are values of Im (κ1) which will cause the integrand to grow exponentially. This is s only allowable provided P shrinks at a greater rate as xt1 → ±∞, implying a maximum ˜s ˜s value of Im (κ1) for which P− can exist and a minimum value of Im (κ1) for which P+ can exist. Whether , the imaginary part of K, is then chosen to meet both requirements, ˜s ˜s Im (κ1) < therefore P− can exist and Im (κ1) > − so that P+ can exist. There is thus ˜s ˜s a strip between − < Im (κ1) < in which both P+ and P− are analytic. Any equation ˜s ˜s containing both P+ and P− must lie in this strip to be valid.

Taking the half-transforms of the boundary conditions yields

˜s P+ (κ1, k2, 0, ω) = 0, (4.24)

0 i 0 −k P (k , ω) Z P˜s (κ , k , 0, ω) = 3 0 2 exp {i (κ − k ) x } dx , (4.25) − 1 2 2π 1 1 t1 t1 −∞ where the prime denotes the derivative with respect to xt3 and k1, k2 and k3 are the wavenumbers of the incident gust. Provided Im (κ1 − k1) ≤ 0, i.e. Im (k1) is always greater than or equal to Im (κ1), the integral in Equation (4.25) converges to give

i ˜s0 ik3P0 (k2, ω) P− (κ1, k2, 0, ω) = (4.26) 2π (κ1 − k1)

s s Note that P is odd in xt3 and therefore P˜ is odd in xt3. The solution on the upper surface of the aerofoil can then be derived and used to recover the other half-ﬁeld. Equation (4.18) and the deﬁnition of the half-Fourier transforms give the following two equations ˜s P− (κ1, k2, 0, ω) = A (κ1, k2, ω) , (4.27) i ik3P0 (k2, ω) ˜s0 + P+ (κ1, k2, 0, ω) = −iκ3A (κ1, k2, ω) . (4.28) 2π (κ1 − k1)

Eliminating A and dividing by κ3+ yields

i ˜s0 ik3P P (κ1, 0) 1 0 + 2 ˜s 1 + 1 = −i (K + κ1) P− (κ1, 0) . (4.29) 2π (κ1 − k1)(K − κ1) 2 (K − κ1) 2 Chapter 4 Theoretical model for the scattered pressure ﬁeld 79

i ˜s0 ˜s0 where the dependence of P0, P+ and P− on k2 and ω is implied. The ﬁrst term on the left-hand side can be decomposed as follows:

i ik3P0 1 = 2π (κ1 − k1)(K − κ1) 2  

i   i ik3P0  1 1  ik3P0  1 − 1  + 1 . (4.30) 2π (κ − k ) 1 1  (K − κ1) 2 (K − k1) 2  2π (κ1 − k1) (K − k1) 2 | {z }   pole at k1 | {z } | {z } branch cut from K pole at k1 | {z } | {z } analytic above K analytic below K

Note that the ﬁrst term on the right-hand side of this expression is analytic in the

upper half of the complex κ1-plane and the second term is analytic in the whole plane but at the pole κ1 = k1. Thus, by sustituting Equation (4.30) in Equation (4.29) the following identity is obtained

i ik3P 1 0 2 ˜s 1 + i (K + κ1) P− (κ1, 0) = 2π (κ1 − k1)(K − k1) 2 0 " # P˜s (κ , 0) ik P i 1 1 − + 1 − 3 0 − (4.31) 1 2π (κ − k ) 1 1 (K − κ1) 2 1 1 (K − κ1) 2 (K − k1) 2

The left hand side is analytic in the lower half of the complex κ1-plane whereas the right hand side of the equation is analytic in the upper half, and both are analytic in

the strip between − < Im (κ1) < . Then, there exists an unique function, J (κ1), that is analytic over both domains

i ik3P 1 0 2 ˜s J (κ1) = 1 + i (K + κ1) P− (κ1, 0) (4.32) 2π (κ1 − k1)(K − k1) 2

0 " # P˜s (κ , 0) ik P i 1 1 J (κ ) = − + 1 − 3 0 − (4.33) 1 1 2π (κ − k ) 1 1 (K − κ1) 2 1 1 (K − κ1) 2 (K − k1) 2

Following Crighton [29], let J (κ1) = 0. Therefore

i ˜s −k3P0 P− (κ1, 0) = 1 1 , (4.34) 2π (κ1 − k1)(K − k1) 2 (K + κ1) 2

˜s and therefore, making use of the fact, from equation (4.27), that P− (κ1, 0) = A, equation (4.18) then yields

i ˜s −k3P0 P (κ1, xt3 ≥ 0) = 1 1 exp {−iκ3xt3} . (4.35) 2π (κ1 − k1)(K − k1) 2 (K + κ1) 2 80 Chapter 4 Theoretical model for the scattered pressure ﬁeld

Finally the scattered pressure ﬁeld can be calculated if the inverse Fourier transform in xt1 is taken

∞ i Z s −k3P0 (k2, ω) exp {−iκ1xt1 − iκ3xt3} P (xt1, k2, xt3 ≥ 0, ω) = 1 1 dκ1 . (4.36) 2π (K − k ) 2 (κ − k )(K + κ ) 2 1 −∞ 1 1 1

The integral in the right hand side of equation (4.36) does not have an analytical solution. However it can be evaluated in two diﬀerent regions: (1) On the surface of the plate (xt3 = 0) and (2) far away of the plate (R → ∞).

4.3.2.2 Solution for P s on the ﬂat plate

The integral in equation (4.36) can be evaluated, for points on the ﬂat-plate (where xt1 < 0, xt3 = 0), by closing the contour in the upper half-plane and wrapping the contour around the branch cut. It can be written as

s P (xt1 < 0, k2, xt3 = 0, ω) = C1 + C2, (4.37) where C1 is a contribution from the residue at κ1 = k1 and C2 is the contribution from the integration along the branch cut. C1 then yields

i −k3P0 (k2, ω) exp {−ik1xt1} i C1 = −2πi 1 1 = P0 (k2, ω) exp {−ik1xt1} , (4.38) 2π (K − k1) 2 (K + k1) 2 which has been simpliﬁed by introducing the value of k3 from equation (4.5). The integral along the branch-cut C2 yields

 −K+δ i Z −k3P0 (k2, ω) exp {−iκ1xt1} C2 = 1 lim  1 dκ1 δ→0 2π (K − k1) 2 (κ1 − k1)(K + κ1) 2 −K+δ+i∞ −K−δ+i∞  Z exp {−iκ1xt1} + 1 dκ1 . (4.39) (κ1 − k1)(K + κ1) 2 −K−δ

Reversing the direction of integration on the ﬁrst integral and noting that

1 1 √ lim (δ + ix) 2 = − lim (δ + ix) 2 = ix, (4.40) δ→0+ δ→0− gives −K+i∞ i Z k3P0 (k2, ω) exp {−iκ1xt1} C2 = 1 1 dκ1. (4.41) π (K − k1) 2 (κ1 − k1)(K + κ1) 2 −K Chapter 4 Theoretical model for the scattered pressure ﬁeld 81

Making the substitution κ1 = −K + is (s = −i(κ1 + K)), where (K + k1) is assumed

to be real and positive. C2 yields

∞ i Z k3P0 (k2, ω) exp {iKxt1} exp {xt1s} C2 = √ 1 √ ds. (4.42) π i (K − k ) 2 (s + i (K + k1)) s 1 0

The integral in equation (4.42) can be solved by making the substitution u = (−xt1)s,

note that xt1 < 0. The integral I yields

∞ ∞ Z Z exp {xt1s} √ exp {−u} I = √ ds. = −xt1 √ du. (4.43) s (s + i (K + k1)) u (u + i (K + k1)(−xt1)) 0 0

Using the following identity, see [83]:

∞ exp {−u} Z = exp {− [u + i (K + k1)(−xt1)] s + i (K + k1)(−xt1)} ds, u + i (K + k1)(−xt1) 1 (4.44) and changing the order of integration, Equation 4.43 yields

∞ ∞ √ Z Z exp {−us} I = −x exp {i (K + k )(−x )} exp {−i (K + k )(−x ) s} √ duds. t1 1 t1 1 t1 u 1 0 (4.45)

The integral over u is analytically determined as

∞ √ Z exp {−us} π √ du = √ . (4.46) u s 0

Equation 4.45 then yields

∞ √ √ Z exp {−i (K + k )(−x ) s} I = π −x exp {i (K + k )(−x )} √ 1 t1 ds, (4.47) t1 1 t1 s 1

that can be solved provided the substitution v = (K + k1)(−xt1) s is used

∞ Z √ exp {i (K + k1)(−xt1)} exp {−iv} I = π p √ dv, (4.48) (K + k1) v (K+k1)(−xt1) 82 Chapter 4 Theoretical model for the scattered pressure ﬁeld that can be written as   ∞ (K+k1)(−xt1) √ exp {i (K + k )(−x )} Z exp {−iv} Z exp {−iv} 1 t1  √ √  I = π p  dv − dv , (4.49) (K + k1) v v 0 0

∗ that is the sum of two Fresnel integrals, that are deﬁned by function E2 (z), equation 8.251 in reference [83], as follows

z 1 Z exp {−iu} E∗ (z) = √ √ du, (4.50) 2 2π u 0

Noting that 1 − i lim E∗ (z) = , (4.51) z→∞ 2 2 equation (4.48) can then be rewritten as

√ exp {i (K + k1)(−xt1)} 1 − i ∗ I = π 2 p − E2 ((K + k1)(−xt1)) . (4.52) (K + k1) 2

Introducing the expression of Equation (4.52) in Equation (4.42), the integral along the brach cut, C2, yields

i ∗ C2 = P0 (k2, ω) exp {−ik1xt1} [(1 + i) E2 ((K + k1)(−xt1)) − 1] , (4.53)

Therefore we have

s + i ∗ P xt1 < 0, k2, xt3 = 0 , ω = P0 (k2, ω) exp {−ik1xt1} (1 + i) E2 ((K + k1)(−xt1)) , (4.54) that can be rewritten in terms of the error function as (using Abramowitz & Stegun [73] Equations 7.1.10, 7.3.7, 7.3.8 and 7.3.22) √ s + i p √ P xt1 < 0, xt3 = 0 = P0 exp {−ik1xt1} erf i K + k1 −xt1 . (4.55) √ in which the positive branch of i has been used. Note that the same expression for the scattered pressure on the aerofoil can be obtained using Schwarzchild’s solution, which can be found in the AppendixA.

As it was stated above, the total pressure is the sum of the incident and scattered ﬁelds. The pressure on each side of the plate is then √ √ √ P (x < 0, x = 0+) = P i exp {−ik x } 1 + erf i K + k −x , t1 t3 0 1 t1 √ √ 1 t1 (4.56) − i √ P (xt1 < 0, xt3 = 0 ) = P0 exp {−ik1xt1} 1 − erf i K + k1 −xt1 . Chapter 4 Theoretical model for the scattered pressure ﬁeld 83

Well upstream of the trailing edge, xt1 → −∞ the total pressure field amplitude is dou- + i ble that of the incident pressure on the upper surface P (xt1 → −∞, xt3 = 0 ) → 2P0, and zero change in pressure on the lower surface, i.e the scattered field cancels the con- − tribution from the incident pressure, P (xt1 → −∞, xt3 = 0 ) → 0. Thus well upstream of the trailing edge, the pressure on the surface of the flat-plate is approximately equal to that which would be produced by an infinite plane surface. The change in pressure across the flat-plate is given by the difference in pressure between the two sides

+ − ∆P (xt1 < 0, k2, ω) = P xt1 < 0, k2, xt3 = 0 , ω − P xt1 < 0, k2, xt3 = 0 , ω , (4.57) and as P is an odd function, ∆P can be written as

s + ∆P (xt1 < 0, k2, ω) = 2P xt1 < 0, k2, xt3 = 0 , ω , (4.58) and then √ i p √ ∆P (xt1 < 0, k2, ω) = 2P0 (k2, ω) exp {−ik1xt1} erf i K + k1 −xt1 . (4.59)

4.3.2.3 Solution for P s far away from the ﬂat plate

According to Equation (4.36), the scattered pressure ﬁeld for xt3 > 0 is

∞ i Z s −k3P0 (k2, ω) exp {−iκ1xt1 − iκ3xt3} P (xt1, k2, xt3 ≥ 0, ω) = 1 1 dκ1. 2π (K − k ) 2 (κ − k )(K + κ ) 2 1 −∞ 1 1 1 (4.36 revisited)

In this section, this integral will be evaluated for an observer location well away from the trailing edge using contour integration in the complex plane. To evaluate the integral, a spherical coordinate system has been used, see Figure 2.18

xt1 = rt sin φt cos θt = R cos θt

xt2 = rt cos φt (4.60) xt3 = rt sin φt sin θt = R sin θt,

R = rt sin φt. 84 Chapter 4 Theoretical model for the scattered pressure ﬁeld

In addition, the following variable substitution for the wavenumbers has been done

K = k cos φk,

k1 = K cos θk = k cos φk cos θk,

k2 = k sin φk, (4.61) k3 = K sin θk = k cos φk cos θk,

κ1 = K cos t,

κ3 = K sin t, where k = ω is the acoustic wavenumber, k = ω = k is the convective wavenumber, c0 1 Uc Mc p 2 2 p 2 2 p 2 2 k3 = K − k1 (Equation (4.5)), K = k − k2 (Equation (4.11)) and κ3 = K − κ1 (Equation (4.17)). Note that, since is smaller than k1 = ω/Uc, θk is purely imaginary and it is chosen to be negative Im (θk) < 0. Along the real κ1 axis, κ3 can be factorised as

1 √ κ3− = (K + κ1) 2 = 2K cos (t/2) , 1 √ (4.62) κ3+ = (K − κ1) 2 = 2K sin (t/2) .

Equation 4.36 can be then written as

i Z s −k3P0 (k2, ω) exp {−iKR cos (θt − t)} P (xt1, k2, xt3 ≥ 0, ω) = √ 1 sin (t/2) dt, (cos t − cos θ ) π 2K (K − k1) 2 k C (4.63) where C is the integration path in the complex t plane that can be shown in Figure 4.4 in which the path in the complex κ1-plane is as well shown. The path of the branch cuts associated with κ3 is deﬁned so that it do not cut the real axis and it has been deformed so that each branch cut is no longer vertical. Figure 4.4 shows branch cuts from κ1 = −1 to −∞ by the green line in Figure 4.4 and κ1 = 1 to ∞ by the red line in Figure 4.4. In order to determine the path of integration in the complex t-plane the contour passes along the lower side of the branch cut for κ1 < −1 and along the upper side of the branch cut for κ1 > 1. The contour integral in the complex t-plane may be evaluated making use of the method of residues whether the contour is closed. Note that the integrand in equation 4.63 will become exponentially small for

Im (t) → −∞, θt − π < Re (t) < θt; (4.64) Im (t) → +∞, θt < Re (t) < θt + π;

The contour can be closed using the blue and magenta curves shown in Figure 4.5.

The blue portions of the integration path run from t = −i∞ to t = θt − δ − i∞ and from t = θt + δ + i∞ to t = π + i∞ where 0 < δ < π is a parameter which ensures the Chapter 4 Theoretical model for the scattered pressure ﬁeld 85

Im( 1)

-5 5 Im(t) 0.5 1 1.5 2 2.5 3 Re(t) Re( 1)

(a) (b)

Figure 4.4: Integration path in the complex κ1 (a) and t (b) planes (for K = 1). Black dotted lines represent the branch cuts associated with the deﬁnition of κ3. value of the integrand along these curves is exponentially small and therefore that there is no contribution to the integral from these curves. The two ends of the blue paths are connected with a path of steepest descent through the stationary point at t = θt. This path is denoted by D and is shown in magenta in Figure 4.5 (b). This closed integration path is also shown in the complex κ1-plane in Figure 4.5 (a). Note that the curve avoids the branch cuts, or rather the branch cuts have been deformed to avoid the path of integration, and winds once around the pole at κ1 = k1 in an anticlockwise direction. The residue theorem thus gives

s P (xt1, k2, xt3 ≥ 0, ω) = C1 + C2, (4.65)

where C1 is the residue at the pole κ1 = k1 and C2 is the contour integral, which only contributor, as stated above, is the integral along the path D. The residue C1 yields

−ik P i (k , ω) n 1 o 3 0 2 2 2 2 C1 = 1 exp −ik1xt1 − i K − k1 xt3 2 2 2 K + k1 i = −iP0 (k2, ω) exp {−iKR cos (θt − θk)} , (4.66) 86 Chapter 4 Theoretical model for the scattered pressure ﬁeld

Im( 1)

-5 5 Im(t) 0.5 1 1.5 2 2.5 3 Re(t) Re( 1)

(a) (b)

Figure 4.5: Closed contour of integration in the complex κ1 (a) and t (b) planes (for K = 1). Black dotted lines represent the brach cuts associated with the deﬁnition of κ3.

and the contour integral C2 yields

i Z k3P0 (k2, ω) exp {−iKr cos (θ − t)} C2 = √ 1 sin (t/2) dt. (4.67) (cos t − cos θ ) π 2K (K − k1) 2 k D

Since θk is negative imaginary, as stated above, C1 does not radiate to the acoustic far-ﬁeld and therefore for R → ∞ this term can be neglected. The integration along D can be evaluated using the method of steepest descent to give

i s k3P0 (k2, ω) sin (θt/2) n π o P (xt1, k2, xt3 ≥ 0, ω) ≈ √ 1 exp −iKR + i . (4.68) 4 πR (K − k1) 2 (K cos θt − k1) which is valid for KR → ∞

4.4 Far-ﬁeld radiation.

The far-ﬁeld pressure can be calculated from any of the two expressions obtained in section 4.3; direclty from P s away from the plate, Equation (4.68) or by means of a Chapter 4 Theoretical model for the scattered pressure ﬁeld 87

Green’s function with Equation 4.59, the pressure jump between the two sides of the plate. As shown in sections 4.4.1 and 4.4.2, both results are equivalent and they are comparable to the boundary layer noise model of Amiet [3].

4.4.1 Method 1:

Making use of the expression for the scattered pressure ﬁeld obtained in equation 4.68, the total pressure above the half-plane is given by performing an inverse Fourier transform in the span-wise wavenumber k2

∞ Z i π s k3P0 (k2, ω) sin (θt/2) exp −ik2xt2 − iKR + i 4 p˜ (xt1, xt2, xt3 ≥ 0, ω) ≈ √ 1 dk2. πR (K − k ) 2 (K cos θ − k ) −∞ 1 t 1 (4.69)

Whether the spherical coordinates deﬁned in Equation (4.60) and the wavenumber substitutions deﬁned in Equation (4.61) are used,p ˜s can then be written as follow

Z i s k3P0 (k2, ω) sin (θt/2) n π o p˜ ≈ √ 1 exp −ikrt cos (φt − φk) + i k sin φkdφk 4 πR (K − k1) 2 (K cos θt − k1) D (4.70) where D is the contour, shown in Figure 4.5 and used in Equation (4.65). This integral may be evaluated for krt → ∞ using the method of steepest descents (noting there is a stationary point at φk = φt at which point k2 = k cos φt and K = k sin φt). The scattered pressure ﬁeld for xt3 ≥ 0 then yields √ √ θt s 2 sin 2 Mc sin φt p exp {−ikrt} i p˜ (xt, ω) ≈ Mc sin φt + 1 P0 (k cos φt, ω) , (4.71) 2π 1 − Mc sin φt cos θt rt where the equivalence k1 = k/Mc has been used. Equation (4.71) can be rearrange as

s exp {−ikrt} i p˜ (xt, ω) ≈ D (θt, φt,Mc) P0 (k cos φt, ω) , (4.72) rt where D (θt, φt,Mc) represents the far-field directivity pattern of the scattered pressure field for observer locations far away from the aerofoil. √ √ θt 2 sin 2 Mc sin φt p D (θt, φt,Mc) = Mc sin φt + 1. (4.73) 2π 1 − Mc sin φt cos θt 88 Chapter 4 Theoretical model for the scattered pressure field

4.4.2 Method 2:

Assuming that the pressure jump on the infinite half-plane can be used as an approximation for the pressure jump on a finite panel. The amplitude of the pressure jump across xt3 = 0 at a particular frequency ω is defined by

∞ Z ∆˜p (xt1, xt2, ω) = ∆P (xt1, k2, ω) exp {−ik2xt2} dk2, xt1 < 0; (4.74) −∞

∆˜p (xt1, xt2) = 0, xt1 > 0. (4.75)

The (frequency domain) radiated acoustic pressure, pf or far-ﬁeld pressure, must satisfy the following inhomogeneous Helmholtz equation:

2 2 f 0 ∇ + k p˜ (xt, ω) = ∆˜p (xt1, xt2) δ (xt3) , (4.76) where δ represents the Dirac’s delta and the prime denotes the derivative with respect to xt3. According to Goldstein [77], Equation (4.76) has the solution

∞ 0 Z Z f ∂G p˜ (xt, ω) = − ∆˜p (yt1, yt2) dyt1dyt2, (4.77) ∂y3 −∞ −∞ where G is a Green’s function which satisﬁes the inhomogeneous wave equation

2 2 ∇ + k G (xt|yt) = −δ (xt1 − yt1) δ (xt2 − yt2) δ (xt3 − y3) . (4.78)

In the far-ﬁeld, rt → ∞, the Green’s function can be written as n o xt1 xt2 exp −ikrt + ikyt1 + ikyt2 ∂G (xt|yt) rt rt ≈ ikxt3 2 , (4.79) ∂y3 4πrt and the pressure then yields

∞ 0 Z Z f ikxt3 xt1 xt2 p˜ (xt, ω) ≈ − 2 exp −ikrt + ikyt1 + ikyt2 ∆˜p (yt1, yt2) dyt1dyt2. 4πrt rt rt −∞ −∞ (4.80) Chapter 4 Theoretical model for the scattered pressure ﬁeld 89

p 2 2 2 where rt = xt1 + xt2 + xt3. By substituting the Making use of Equation (4.75), Equa- tion (4.80) yields

∞ 0 Z Z f ikxt3 xt1 xt2 p˜ (xt, ω) ≈ − 2 exp −ikrt + ikyt1 + ikyt2 4πrt rt rt −∞ −∞ ∞ Z × ∆P (yt1, k2, ω) exp {−ik2yt2} dk2dyt1dyt2, (4.81) −∞

and rearranging its terms

∞ 0 Z Z f ikxt3 xt1 p˜ (xt, ω) ≈ − 2 exp {−ikrt} ∆P (yt1, k2, ω) exp ik yt1 dyt1 × 4πrt rt −∞ −∞ | {z } I1 ∞ Z xt2 exp i k − k2 yt2 dyt2 dk2. (4.82) rt −∞ | {z } I2

The integral over y , I , is evaluated as the Dirac’s delta δ k xt2 − k and then the t2 2 rt 2 f integral over k2 can be evaluated.p ˜ then yields

0 Z f ikxt3 xt2 xt1 p˜ ≈ − 2 exp {−ikrt} ∆P yt1, k , ω exp ik yt1 dyt1 . (4.83) 4πrt rt rt −∞ | {z } I1

Note that ∆P (yt1) is a function of yt1, k2 and ω. Since k2 has a ﬁxed value due to the Dirac’s delta, k = k xt2 = k cos φ , it can be said that only the acoustic wavefronts that 2 rt 2 t are ortogonal to the line joining the centre of the aerofoil trailing edge and the observer can be heard in the far-ﬁeld (geometric and acoustic).

The integral over yt1, I1, is evaluated by splitting the integral into two parts giving

0 Z xt2 xt1 I1 = ∆P yt1, k , ω exp ik yt1 dyt1 = rt rt −∞ −c 0 Z Z xt1 xt1 ∆P (yt1) exp ik yt1 dyt1 + ∆P (yt1) exp ik yt1 dyt1. (4.84) rt rt −∞ −c 90 Chapter 4 Theoretical model for the scattered pressure ﬁeld

It is assumed that for the purposes of evaluating the ﬁrst integral in the right hand side of Equation (4.84), c is suﬃciently far upstream of the trailing edge such that it can i be assumed that ∆P (yt1) ≈ 2P0 exp {−ik1yt1}. Substituting and evaluating the integral gives

−c Z xt2 xt1 i xt2 exp {iαc} ∆P yt1, k , ω exp ik yt1 dyt1 = −2P0 k , ω , (4.85) rt rt rt iα −∞ where α = k − k xt1 6= 0. 1 rt

The second integral in the right hand side of Equation (4.84) can be evaluated by s substituting equation (4.59) for ∆P (yt1) and integrating by parts. It yields

0 Z xt2 xt1 i xt2 exp {iαc} ∆P yt1, k , ω exp ik yt1 dyt1 = −2P0 k , ω rt rt rt iα −c   √ r K + k1 √ xt1 √ √ p √ × q exp {−iαc} erf i K + k c − erf i K + k1 c  . (4.86) K + k xt1 rt rt

Adding the results of the two integrals, I1 yields

 √ √ r i xt2 exp {iαc} K + k1 xt1 √ I1 = −2P0 k , ω q exp {−iαc} erf i K + k c rt iα K + k xt1 rt rt √ p √ i −erf i K + k1 c + 1 . (4.87)

f By substituting the value of I1 in Equation (4.83),p ˜ yields

 √ √ r f kxt3  K + k1 xt1 √ p˜ (xt, ω) ≈ 2 q erf i K + k c 2παrt  K + k xt1 rt rt √ ) h p √ i i xt2 − 1 − erf i K + k1 c exp {iαc} exp {−ikrt} P0 k , ω , (4.88) rt

The first integral on the right hand side of I1, Equation (4.84) is the radiation produced by the reflection of the evanescent incident wave by a plane surface. If the surface were infinitely large, c → ∞, then there would be no radiation. However, this integral produces a radiating term because of the truncation as c is finite, the term −exp (iαc) within the brackets of Equation (4.88). Provided that c is large enough such that the Chapter 4 Theoretical model for the scattered pressure field 91

pressure at yt1 = −c is adequately modelled by Equation (4.88), this radiating term will be cancelled by a corresponding term from the second integral on the right hand side of √ √ √ Equation (4.84), the −erf i K + k1 c that tends to −1 as c → ∞.

Equation (4.88) can be simplified if the spherical coordinate system defined in Equa- tion (4.60) is used, see Figure 2.18. The far-field pressure can be then written as

f exp {−ikrt} i xt2 p˜ (xt, ω) ≈ −D (θt, φt,Mc, kc) P0 k , ω , (4.89) rt rt where D (θt, φt,Mc, kc) represents the far-ﬁeld directivity pattern of the scattered pressure ﬁeld

"s 1 Mc sin φt sin θt 1 + Mc sin φt p D = erf i sin φt (1 + cos θt) kc 2π 1 − Mc sin φt cos θt Mc sin φt (1 + cos θt) !!# 1 + M sin φ r 1 + M sin φ + exp i c t kc 1 − erf i c t kc , (4.90) Mc Mc with kc = ωc being the non-dimensional acoustic wavenumber normalized with the chord c0 length. In the limit kc → ∞, the directivity loses its frequency dependency and yields √ √ θt 2 sin 2 Mc sin φt p D (θt, φt,Mc) = Mc sin φt + 1, (4.91) 2π 1 − Mc sin φt cos θt that is the same expression as the one obtained with Method 1 in section 4.4.1, Equation (4.73). It is clear then, that the far-ﬁeld pressure obtained with Method 1, Equation (4.71), is equivalent to the one obtained with Method 2, Equation (4.88). The diﬀerence between both yields in that Equation (4.73) is only valid for observer locations far away from the plate while Equation (4.88) is valid at any observer location.

4.5 Far-ﬁeld directivity

By analysing Equation (4.91), it can be seen that D, and consequentlyp ˜f , will be

0 for φt = 0 and φt = π and maximum for φt = π/2, i.e beneath the trailing edge centre. Regarding the polar directivity, for an observer beneath the aerofoil, φt = π, the directivity term yields √ θt p 2 sin 2 Mc (1 + Mc) D (θt, π, Mc) = . (4.92) 2π 1 − Mc cos θt

An example of the polar directivity pattern obtained with Equation (4.92) is shown in Figure 4.6 for diﬀerent convective Mach numbers. It is easy to see that it can be split in two distinguished terms: 92 Chapter 4 Theoretical model for the scattered pressure ﬁeld

90 120 60

150 30

M =0.90 c M =0.75 180 c 0 M =0.50 c 0.5 M =0.30 c 1 1.5 210 330 2 2.5 240 300 270

Figure 4.6: Polar directivity for a semi-inﬁnite chord

θt 1. sin 2 which represents the expected cardioid directivity of the jet-wing interaction p 2. Mc (1 + Mc)/ (1 − Mc cos θt) which represents the stretching of the directivity pattern due to the jet convection.

4.6 Acoustic spectrum

In the far-field, the microphone at location xt, will measure the time-history of the auto- f correlation of the far-field pressure p (xt, t). The auto-correlation can be decomposed in frequency and expressed as a Power Spectral Density (PSD) if the Fourier transform in time is taken. In addition, the Power Spectral Density of the far-field pressure,

Spp (xt, ω), can be obtained analytically if equation (4.89) is multiplied by its complex conjugate, i.e. π h f f∗ i Sff (xt, ω) = lim E p˜ (xt, ω)p ˜ (xt, ω) , (4.93) T →∞ T giving 2 D (θt, φt,Mc, kc) Sff (xt, ω) = 2 Φii (k2, ω) , (4.94) rt where π i i∗ Φii (k2, ω) = lim E P0 (k2, ω) P0 (k2, ω) (4.95) T →∞ T is the auto-power spectral density of the incident pressure at the plate, xt3 = 0. Note that k2 has a fixed value for each observer location k2 = k cos φt. Since the turbulent Chapter 4 Theoretical model for the scattered pressure field 93 pressure is not statistically stationary in the span-wise direction, Equation (4.95) has no obvious solution. However, a Wigner-Ville spectral decomposition distribution can be used in order to estimate its spectrum. This will be used in the next chapter, section 5.5, where a prediction methodology for the jet-surface interaction will be presented.

4.7 Comparison with Amiet’s trailing edge model

In this section, the solution obtained with the Wiener-Hopf technique is compared with Amiet’s trailing edge solution. For this purpose, Amiet’s solution for the far-ﬁeld acoustic pressure at xt and ω corresponding to the scattering of an incident pressure gust of wavenumbers k1 and k2, Equation (2.37), and the Wiener-Hopf solution for the far-ﬁeld acoustic pressure, Equation (4.88), are reproduced below

(s √ r f kxt3 K + k1 xt1 √ p (xt, ω) = 2 K+x erf i K + k c 2παr k t1 rt t rt ) h√ p √ i xt2 d + 1 − erf i K + k1 c exp {iαc} − 1 exp {−ikrt} d sinc k2 − k , rt 2 (2.37 Revisited)

 √ √ r f kxt3  K + k1 xt1 √ p˜ (xt, ω) ≈ 2 q erf i K + k c 2παrt  K + k xt1 rt rt √ ) h p √ i i xt2 − 1 − erf i K + k1 c exp {iαc} exp {−ikrt} P0 k , ω , rt (4.88 Revisited)

Two diﬀerences can be observed between Equation (2.37) and Equation (4.88)

1. The term -1 at the end of the curly brackets in Equation (2.37) is not present in Equation (4.88). According to Amiet, this term balances the contribution of the incident ﬁeld to the sound radiation. However, it must be discarded as suggested by Amiet [3, 68] and Roger and Moreau [69].

i 2. The incident pressure at the trailing edge is P (k2, ω) for Equation (4.88) while 0 h i the gust contribution for Equation (2.37) is d sinc k − k xt2 d . Only the 2 rt 2 i acoustic wavefronts of P0 (k2, ω) that are ortogonal to the line joining the centre of the aerofoil with the observer (k = k xt2 ) contribute to the noise. In 2 rt Amiet’s solution, Equation (2.37), the contribution of the unit gust is deﬁned 94 Chapter 4 Theoretical model for the scattered pressure ﬁeld

h i by sinc k − k xt2 d . The sinc functions acts as a ﬁlter and, as it happens with 2 rt 2 P i (k , ω) in Equation (4.88), only one oblique gust (k = k xt2 ) is selected for each 0 2 2 rt angle of radiation oﬀ the mid-span plane. In Amiet’s solution the turbulence is assumed to be statistically stationary in the span-wise direction. For this reason, the span length of the aerofoil, d multiplies the contribution of the unit gust.

It can be said, then, that Equation (4.88) is essentially Amiet’s trailing edge noise equation, Equation (2.37), for an incident pressure at the trailing edge P i k xt2 , ω 0 rt and without the term that balances the contribution of the incident ﬁeld to the sound radiation. Chapter 5

A prediction methodology for the jet-surface interaction

Contents 5.1 Chapter overview...... 95 5.2 Problem specification and assumptions...... 97 5.3 Hydrodynamic field model...... 98 5.3.1 Numerical LES data...... 99 5.3.2 Jet source model...... 99 5.4 Near-field propagation...... 103 5.4.1 Benchmark of the propagation of a monopole source...... 107 5.4.2 Near-field propagation of LES data...... 108 5.5 Trailing edge pressure field...... 109 5.5.1 Small-scale experimental data...... 110 5.5.2 Trailing edge pressure Vs. Doak Laboratory data...... 111 5.6 Jet-surface interaction prediction...... 112 5.6.1 Small-scale experimental setup...... 113 5.6.2 Far-field prediction...... 114 5.7 Jet-surface interaction methodology and its use in more realistic cases...... 116

5.1 Chapter overview

Chapter2 showed that jet-surface interaction noise is originated when the evanescent near pressure field produced by the jet turbulence is scattered by a trailing edge, where it becomes a truly propagated acoustic pressure field. This noise source was studied in details in Chapters3 and4. In Chapter3, the near-field of a jet was studied and 95 96 Chapter 5 A prediction methodology for the jet-surface interaction

Lighthill’s Acoustic Analogy was extended into this region, providing the scaling laws that explain how the pressure field of a jet behaves in the near-field. In Chapter4, the Wiener-Hopf technique was used to obtain a theoretical expression for the scattered pressure field produced by an incident pressure field passing by the trailing edge of a flat plate. These two theories are the foundations of the prediction methodology that is presented in this chapter.

This new methodology is designed to predict the far-field noise produced by the interaction of an isothermal single stream jet with a finite flat plate, parallel to the jet centre line, in a quiescent ambient medium. The near pressure field of an isolated, non-heated, single stream jet is calculated on a conical surface surrounding the jet. This pressure is then propagated with a Green’s function to the aerofoil surface, where it is scaled appropriately with knowledge obtained from the extended Lightill’s theory. The wall pressure is then scattered by the trailing edge by means of the Wiener-Hopf technique.

In section 5.2, the problem speciﬁcation and the required assumption for the prediction model are presented.

In section 5.3, a semi-empirical model for the jet hydrodynamic ﬁeld is presented. The model consists on a source distribution along a conical surface surrounding the jet. The source is presented for each azimuthal mode as a modiﬁed Gaussian function.

In section 5.4 a Green’s function is used to radiate the source distribution through the near-ﬁeld. The Green’s function is benchmarked for the propagation of a monopole source and later for the near-pressure ﬁeld of a single-stream jet using numeric LES data.

In section 5.5, the near-field model is used to predict the cross power spectral density of the incident pressure along the trailing edge of a flat plate where it is scaled with learned knowledge from the near-field extension of Lighthill’s theory from Chapter3. The section also shows the agreement between the calculated cross-PSD and experimental data from pressure kulites on the plate surface.

In section 5.6 a Wigner-Ville distribution is used to calculate the incident wall pressure as a function of the span-wavenumber. This wall pressure is the input for the scattering model. Far-field results are finally shown and compared against experimental data. Together with the far-field results, a directivity study, for polar and azimuthal directivities, is included in this section. Chapter 5 A prediction methodology for the jet-surface interaction 97

Figure 5.1: Schematic of the jet-surface interaction problem.

5.2 Problem speciﬁcation and assumptions

The problem considers an isothermal single-stream jet interacting with a parallel ﬂat plate in a quiescent ambient medium. The prediction methodology can be decomposed in the following steps and a schematic of this process can be seen in Figure 5.1.:

1. A source model. The cross-PSD of the hydrodynamic ﬁeld is obtained on a control surface

2. A Green’s function is used to propagate the hydrodynamic ﬁeld from the control surface to the plate surface.

3. The wall pressure at the trailing edge of the plate is obtained with a Wigner-Ville distribution.

4. The far-ﬁeld PSD is obtained by scattering the wall pressure at the trailing edge using the Wiener-Hopf technique.

The jet is assumed to be unaffected by the presence of the flat plate, i.e it is not deflected and it keeps its axisymmetry. The hydrodynamic pressure field of the jet is modelled on a control surface. This control surface is defined as a cone that surrounds the jet and it is close enough to be dominated by the hydrodynamic field. 98 Chapter 5 A prediction methodology for the jet-surface interaction

The ﬂat plate has to be parallel to the jet centreline, i.e. with zero angle of attack. The plate has a chord and a span length of C and d respectively. The plate is assumed to have zero thickness. Local to the trailing edge, the wall pressure is assumed to have stationary turbulence statistics in the streamwise direction.

The main limitation of the model lays in that the trailing edge of the plate cannot be inside of the conical surface.

5.3 Hydrodynamic ﬁeld model

In Chapter3, it was shown that, by modelling the fourth order correlation tensor

Rijlm (y, η, τ), the near-ﬁeld hydrodynamic pressure can be calculated using extended Lighthill’s theory with a similar approach than that used to obtain the acoustic far- ﬁeld. However, the lack of an optimum model for the fourth order correlation tensor

Rijlm (y, η, τ) in the near-field has lead the author to look into a different way to calculate the hydrodynamic field analytically. In this section, a model for the hydrodynamic field is presented. The model was first shown and benchmarked against LES data in a paper, “Hydrodynamic pressure field propagation model for the prediction of the far-field sound produced by jet-wing interaction” [84], based in a Reba and Colonius publication [85]. In the model, the hydrodynamic pressure field of a isothermal single-stream jet is calculated using a Gaussian correlation function as a two-point space-time correlation on a control surface surrounding the jet plume.

The control surface is chosen as a conical surface surrounding the jet. It must be close enough to the turbulent flow in order to be dominated by the hydrodynamic field, but with enough separation to keep the linear behaviour of the pressure field. A spherical coordinate system, xc = (rc, θc, φc), is used, see Figure 5.2, with the origin on the virtual vertex of the conical surface and the polar angle measured from the upstream angle. The cone surface is defined by a constant polar angle θc = θ0 and the nozzle exit plane is located at rc = r0. The relation between this new coordinate system and the nozzle exit coordinate system is

x1 = rc cos (π − θc) − X0 (5.1)

x2 = rc sin (θc) cos φc (5.2)

x3 = rc sin (θc) sin φc. (5.3) Chapter 5 A prediction methodology for the jet-surface interaction 99

Figure 5.2: Control surface and coordinate system used in the near/ﬁeld model

5.3.1 Numerical LES data

Numerical data from a large eddy simulation (LES), computed by the University of

Cambridge [80], for a cold single-stream round jet (Tj/T0 = 1) with an acoustic Mach number M = 0.875, has been used to benchmark the Source model. The jet has a nozzle exit diameter D of 10.16 mm corresponding to a jet Reynolds (Re) number of 2×105. As explained by Wang in ref [80], a grid with 34 million points was used within a HYDRA code with a time step of 5 × 10−7s and a valid frequency range for ﬁeld sound from St = 0.068 to St = 1.2.

The LES data is deﬁned on two conical arrays with a half angle α = 9.98◦. Both arrays have 41 rings with an axial separation between rings of 0.25D, extending from the nozzle exit to 10D downstream. Each ring has 16 probes with an equal spacing of 22.5◦ azimuthally. Figure 5.3 shows the probe distribution of the inner conical array. The outer conical array is concentric to the inner one but one diameter, D, away from it.

5.3.2 Jet source model

In 1979, Glegg demonstrated that the acoustic pressure in the far-ﬁeld for an isolated jet can be obtained by modelling the jet as a source distribution along the jet centreline [86]. The Source distribution could be written as an exponential function along the centreline: y n−1 n x o q (x , ω) = 1 exp −2πσ 1 , (5.4) 1 λ λ

where λ = c0/ω is the acoustic wavelength and n and σ are the two parameters that deﬁne the distribution. The cross-spectral density of the pressure between two points 100 Chapter 5 A prediction methodology for the jet-surface interaction

-3 9.98°

-2

-1 1.5D 5.02D /D

3 0 x

1 Nozzle exit

2 -5

0 3 -2 0 2 4 6 8 5 10 12 x /D x /D 2 1

Figure 5.3: probe distributions of near-ﬁeld LES hydrodynamic array [data from Cambridge]. along the jet centreline is given by

( 2) 1 x − x0 0 0 2 1 1 R x1, x1, ω = q (x1, ω) q x1, ω exp − exp {−2iκ} , (5.5) Lcor

0 0 0 where x1 and x1 are the axial coordinates of two points x = (x1, 0, 0), x = (x1, 0, 0) on the jet centreline. κ and Lcor are the axial spatially varying wave-number and correlation length scale, respectively.

A similar approach can be used to model the near hydrodynamic pressure field. How- ever, while in the far-field, where the observer distance is much bigger than distances inside of the jet, an axisymmetric jet can be seen as a line source along the jet centreline; in the near-field the distances inside of the jet are equivalent to the distance to the observer and the azimuthal distribution must be considered and cannot be omitted. For this reason, instead of defining the source distribution along the centreline, this is defined on a conical surface that surrounds the jet. For each azimuthal mode, m, the cross-spectral density of the pressure between two points on the conical surface, θ = θ0, can be defined by the equation:

( 2) 1 rc1 − rc2 Rm (rc1, rc2, ω) = (Fm (rc1, ω) Fm (rc2, ω)) 2 exp − exp {−2iκ} , (5.6) Lcor where rc1 and rc2 are points on the control surface, κ is the axial wave-number, Lcor is the correlation length scale and Fm (rc) is the auto-spectral density at rc for the azimuthal mode m. The ﬁrst exponential of Equation (5.6) represents the decaying of Chapter 5 A prediction methodology for the jet-surface interaction 101 the cross-PSD with the separation between the two points and the second exponential represents the phase diﬀerence between the two points. Lcor can be modelled with a simple linear model of the form [85]

r + r L = B c1 c2 + C. (5.7) cor 2

The auto-spectral density, Fm (r, ω), along the conical surface, θ = θ0, can be modelled using the source distribution deﬁned by Glegg. For each azimuthal mode, we deﬁne

Fm (r, ω) as:

r − r n−1 r − r F (r , ω) = A c 0 exp −2πσ c 0 , (5.8) m c λ λ

that corresponds to the Equation (5.4) multiplied by an amplitude factor A. Note that

rc − r0 is the distance along a generatrix of the conical surface, from the nozzle exhaust plane, as it can be seen in Figure 5.2

X (X + x ) x r = 0 , r = 0 1 , r − r = 1 . (5.9) 0 cos α c cos α c 0 cos α

The three parameters required for Fm (r, ω), n, σ and A, are frequency dependent. Using a non-linear least-square method, the three parameters involved in the model can be obtained from the LES data. An example of these parameters is shown in Figure 5.4. The three parameters are shown for the axisymmetric mode m = 0 and the first three asymmetric or helical modes (m = 1, m = 2 and m = 3) as function of the Strouhal number. Two differentiated frequency region can be observed in Figure 5.4.A low frequency region (St < 0.5) that corresponds with the hydrodynamic field as it was shown in Chapter3 and a second region for higher frequencies ( St > 0.5) that represents the acoustic pressure field.

These parameters can be used together with Equation (5.8) in order to evaluate the LES data at the inner conical array. At each axial location, discrete Fourier transforms are applied in time and in the azimuthal direction. Figure 5.5 shows a comparison of this model against LES data for the axisymmetric mode and the ﬁrst three helical modes for a range of frequencies up to St = 1. A good agreement can be observed for most of the frequencies except at St = 0.2, where the model works well only up to 4 diameters downstream.

The cross spectral density on the conical surface is then obtained with Equation (5.6) as shown in Figures 5.6 to 5.11, where the cross-spectral density along the cone surface for the ﬁrst four azimuthal modes is compared between LES data and the model. The 102 Chapter 5 A prediction methodology for the jet-surface interaction

LES parameter n 8 m=0 6 m=1 m=2

n 4 m=3 2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 St=f*D/U j LES parameter 0.8 m=0 0.6 m=1 m=2 0.4 m=3 0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 St=f*D/U j LES parameter A 10000 m=0 m=1 100 m=2

A m=3 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 St=f*D/U j

Figure 5.4: Source model parameters obtained from LES data using least-square method. Isothermal jet with Mj = 0.875

ﬁrst two ﬁgures, Figure 5.6 and Figure 5.7, show the absolute value of the CPSD. Figures 5.8 and 5.9 show the real value of the CPSD. Finally, Figures 5.10 and 5.11 show the imaginary value of the CPSD. Both the amplitude and the phase are well caught for the whole range of frequencies and modes. The spectral shape, however, presents a good agreement for low frequencies (i.e. St < 0.5). The agreement deteriorates as the Strouhal increases for St > 0.5.

The hydrodynamic field arises from unsteady turbulence within the jet, analogous to the generation of the acoustic field. The source model of Glegg [86] was developed for the acoustic field. However in view of the results shown in this section. it is possible to assume that the hydrodynamic and acoustic fields have similar sources although they differ in their spatial decay rates. This is not a surprise, since both arise from the Lighthill stress tensor as it was shown in Chapter3. Chapter 5 A prediction methodology for the jet-surface interaction 103

Auto-Spectral Density, m=0 Auto-Spectral Density, m=1 150 150

130 130

110 110

PSD [dB] 90 PSD [dB] 90

70 70

50 LES data, St=0.2050 0 2 4 6 8 10LES data, St=0.400 2 4 6 8 10 Radial distance (r-r )/D LES data, St=1.00 Radial distance (r-r )/D 0 LES data, St=2.00 0 Model, St=0.20 Auto-Spectral Density, m=2 Model, St=0.40 Auto-Spectral Density, m=3 150 Model, St=1.00150 Model, St=2.00

130 130

110 110

PSD [dB] 90 PSD [dB] 90

70 70

50 50 0 2 4 6 8 10 0 2 4 6 8 10

Radial distance (r-r0)/D Radial distance (r-r0)/D

Figure 5.5: Comparison of LES data (symbols) and Gaussian model (line) for Auto- Spectra density of the hydrodynamic field for different Strouhal numbers and different azimuthal modes.

5.4 Near-ﬁeld propagation

Once the hydrodynamic pressure ﬁeld is known on a control surface, it can be propagated through the near-ﬁeld using a Green’s function. The cross-power spectral density 0 0 0 0 of the incident pressure between two points rc = (rc, θc, φc) and rc = (rc, θc, φc) in the spherical coordinates centred in the cone vertex is

ZZ 0 2 2 X Spp rc, rc, ω = 4π sin (θ0) exp {im (φc1 − φc2)} Rm (rc1, rc2, ω) m ∂ 0 0 ∂ ∗ × gm rc2, θ0; rc, θc gm (rc1, θ0; rc, θc) drc1drc2, (5.10) ∂θc ∂θc 104 Chapter 5 A prediction methodology for the jet-surface interaction

2 2 P a P a P a2 P a2 Hz Hz Hz Hz