On the interactions of sound waves and vortices César Legendre

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On the interactions of sound waves

and vortices

César Legendre

Academic advisor: Prof. Jean-Louis Migeot

1,2

Industry advisor: Dr. Gregory Lielens

1

Thesis for the Degree of Doctor of Engineering Science

Université Libre de Bruxelles

1

Free Field Technologies S.A. 2Civil Engineering Department

MSC software company, Université Libre de Bruxelles,

Mont-Saint-Guibert, Belgium Brussels, Belgium

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ii

Assessment committee:

Prof. L.M.B.C. Campos

Center for Aeronautical and Space Science and Technology (CCTAE), Instituto Superior Técnico (IST), Technical University of Lisbon, 1049-001 Lisboa codex, Portugal.

Prof. Jean-François Mercier

POEMS, CNRS, ENSTA ParisTech, INRIA, 828 boulevard des Maréchaux, Palaiseau 91762, France.

Prof. Jean-Pierre Coyette

École Polytechnique de Louvain, EPL - Place du Levant 1, 1348 Louvain-la-Neuve, Belgium.

Prof. Gérard Degrez

Université Libre de Bruxelles, Aero-Thermo-Mechanics Department, Avenue F.D. Roosevelt 50 B-1050 Brussels, Belgium.

Prof. Arnaud Deraemaeker

Université Libre de Bruxelles, Faculté de Sciences Appliquées, Franklin Roo-seveltlaan 50, 1050 Brussels, Belgium.

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On the interactions of sound waves and vortices

Short abstract:

The effects of vortices on the propagation of acoustic waves are numerous, from simple convection effects to instabilities in the acoustic phenomena, including absorption, reflection and refraction effects. This work focusses on the effects of mean flow vorticity on the acoustic propagation. First, a theoretical background is presented in chapters 2-5. This part contains: (i) the fluid dynamics and thermodynamics relations; (ii) theories of sound generation by turbulent flows; and (iii) operators taken from scientific literature to take into account the vorticity effects on acoustics. Later, a family of scalar operators based on total enthalpy terms are derived to handle mean vorticity effects of arbitrary flows in acoustics (chapter6). Furthermore, analytical solutions of Pridmore-Brown’s equation are featured considering exponential boundary layers whose profile depend on the acoustic parameters of the problem (chapter7). Finally, an extension of Pridmore-Brown’s equation is formulated for predicting the acoustic propagation over a locally-reacting liner in presence of a boundary layer of linear velocity profile superimposed to a constant cross flow (chapter8).

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I

Theoretical Framework

9

2 Fundamental equations for fluid dynamics 11 2.1 Introduction. . . 12

2.2 The Reynolds transport theorem . . . 12

2.3 Conservation of mass, momentum and energy . . . 13

2.4 Heat flux, stress and gravity force. . . 16

2.5 A brief note on thermodynamics . . . 17

2.6 Alternative forms of fluid dynamics equations . . . 19

2.7 Concluding remarks . . . 21

3 On vortex motion 23 3.1 Introduction. . . 24

3.2 Streamlines, pathlines and stream functions . . . 24

3.3 The vorticity equation . . . 25

3.4 Vortex representations . . . 26

3.5 Vortex solutions . . . 29

II

Acoustics in moving media

33

4 Aerodynamic Noise 35 4.1 Introduction. . . 36

4.1.1 Linearisation . . . 37

4.2 Acoustics solutions in free field space . . . 38

4.3 Lighthill’s analogy . . . 39

4.4 Solid boundary and flow effects . . . 42

4.5 A generalised acoustic field . . . 44

4.6 Theory of vortex sound . . . 47

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iv Contents

5 Effects of non-uniform mean flows on sound propagation 51

5.1 Introduction. . . 52

5.2 The linearised Navier-Stokes equations . . . 54

5.3 The linearised Euler equations . . . 55

5.4 Hydrodynamic instabilities in shear flows . . . 56

5.5 Lilley’s equation . . . 60

5.6 Goldstein’s equations. . . 62

5.7 Acoustic propagation in unidirectional shear flows . . . 65

5.7.1 On simplifications as impedance boundary conditions . . . 67

III

Theoretical and Numerical Contributions

71

6 Weakly-coupled acoustic-vortical wave equations 73 6.1 Introduction. . . 74

6.2 Möhring’s equation . . . 76

6.3 Linearisation of wave equation . . . 79

6.4 The acoustic-vortical conjecture. . . 81

6.5 Family of acoustic-vortical wave equations . . . 88

6.6 Acoustic propagation in a linear shear layer . . . 96

6.7 Acoustic propagation in a dipole vortex . . . 106

6.8 Acoustic propagation in a cylindrical jet . . . 125

6.9 Acoustic propagation in a realistic aircraft engine . . . 148

7 Acoustics involving exponential boundary layers 163 7.1 Introduction. . . 164

7.2 The wave equation . . . 165

7.3 Exponential boundary layer profile . . . 165

7.4 Solution of the wave equation . . . 167

7.4.1 Asymptotic solution I: infinitely thin boundary layers . . . 168

7.4.2 Asymptotic solution II: infinitely thick boundary layers . . . . 169

7.4.3 Asymptotic solution III: infinitely thick boundary layers . . . . 171

7.5 Boundary conditions . . . 171

7.6 Numerical results . . . 173

8 Effects of shear and cross flow on acoustic liners 181 8.1 Introduction. . . 182

8.2 Sound in superimposed shear and cross flows . . . 184

8.2.1 Linear unidirectional shear with uniform cross flows . . . 184

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Contents v

8.2.3 The absence of a critical layer . . . 188

8.3 Solution of the third-order wave equation . . . 190

8.3.1 Pressure perturbation inside the boundary layer . . . 190

8.3.2 Matching to the acoustic field in the free stream . . . 192

8.3.3 Wall impedance and scattering coefficients. . . 194

8.4 Impedance liner with bias flow . . . 196

8.4.1 Zones of silence and propagation sectors . . . 197

8.4.2 Baseline case and variations of parameters. . . 201

8.4.3 Pressure fields and scattering coefficients. . . 204

9 Conclusions and perspectives 215 9.1 Overview . . . 216

9.2 Results, perspectives and future ideas . . . 217

9.2.1 Comments on chapter 6 . . . 217

Appendix

227

A Solution of Pridmore-Brown’s equation 229 B Asymptotic solution of Pridmore-Brown’s equation 231 C Generic second-order scalar wave operator 233 C.1 Introduction. . . 233

C.2 The generic wave operator . . . 233

C.2.1 Case I: Helmholtz’s equation . . . 234

C.2.2 Case II: Convected wave equation . . . 234

C.2.3 Cases III: Acoustic-vortical wave equations . . . 235

C.3 Finite element method: weak formulation . . . 235

C.3.1 3D, 2D finite element formulation . . . 236

C.3.2 Axi-symmetric finite element formulation. . . 236

C.3.3 Addendum: calculus properties . . . 238

D Contributions to Linear Algebra 241 D.1 Introduction. . . 241

D.2 Inverse of a vector-composed pseudo-symmetric matrix . . . 241

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vi Contents

List of Figures 245

List of Tables 253

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Preface

T

his Ph.D. thesis entitled “On the interactions of sound waves and vortices” contains the results of the research undertaken at the product development group of Free Field Technologies S.A. (FFT) in collaboration with the civil engineering department of the Université Libre de Bruxelles. This research was realized within the project: ATCoMe (Advance Techniques in Computational Mechanics) under the European commission’s 7th_{framework programme for research (2007-2013).}

This challenge began three and half years ago when I decided to move to Brussels to start a new adventure. In April 2011, after discussions with my supervisors Jean-Louis Migeot, Jean-Pierre Coyette and Gregory Lielens, I joined the product development group in FFT. Since the beginning, this research was focussed on the vast subject of vorticity effects on acoustics. The original idea was simple and general, the study of acoustics in vortical flows but, when I started to progressively learn about the subject, this became more complex (and interesting!) than I had expected.

In June 2011 and June 2013, I participated in the congresses and meetings organized by the ATCoMe consortium; ADMOS 2011 (Paris-France) and ADMOS 2013 (Lisbon-Portugal). Other successive meetings within the frame of ATCoMe project were also held in Barcelona-Spain on October 2011 (midterm meeting), January 2012 (annual meeting), January 2013 (annual meeting) and October 2013 (closure meeting). Later, my supervisors and I participated in two congresses where I presented my work; EuroNoise 2012 (Prague, Czech Republic) and InterNoise 2012 (New York, USA). And finally, in March 2014 following an invitation from professor Jean-François Mercier, part of this work was presented in POEMS seminar held in ENTSA ParisTech.

In addition, an important contribution to this work was performed during my stay in the Instituto Superior Técnico of Lisbon (March and October 2013). During few months I had the opportunity to work with professor L.M.B.C. Campos in vorticity and thermal effects on acoustics. This was a fruitful experience.

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viii Contents

Actually, the operators presented in chapter6were made available in Actran software for industrial applications.

Now of course, trying to resume all the experiences of my Ph.D. in a few paragraphs is very difficult if not impossible. I had plenty of advice of what I should put in the preface in order to give the readers a taste of my experiences and therefore capture their attention. Some pieces of advice like, “you should make a very professional and sober preface”, or “Make a fun preface of lived jocular experiences” or one that I particularly liked, which was “why not starting the thesis like a typical magic realism novel, as for instance Cien años de soledad of García-Márquez?”. So, I thought of something along the lines of “Many years later, as he faced the firing squad, Colonel Aureliano Buendía was to remember that distant afternoon when his father took him to discover ice” but changing, “firing squad” by “thesis jury”, “Colonel Aureliano Buendía” by César and “ice” by car mechanics. Might be funny they said. Truly, I had wonderful experiences (and not that wonderful) during the conception and drafting of this Ph.D. thesis, such as my tremendous fears during my first congress camouflaged by a fake over-confident behaviour (which at the end worked well by the way). Or the original derivations and formulas made during sleepless nights only to realize the next morning that this “original derivation” was published many years before I was born (not that original hein!, that happens!). I believe it is these experiences that made possible this work ...

I hope you enjoy the lecture of this work as much I did ...

César Legendre

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acknowledgements

I

t would not have been possible to write this work without the help and support of the people around me, to only some of whom it is possible to give particular mention here.

This thesis could not have been materialised without the help, support and patience of my supervisors: Dr. Gregory Lielens, for his knowledge and dexterity in facing mathematical and programming problems which have been of great value for me; professor Jean-Louis Migeot for the pieces of advice and deep interest in acoustics; and professor Jean-Pierre Coyette for his way of thinking and the methodology used in all academic matters and of course for all the interesting mathematical discussions that we shared. Thank you for your confidence and trust on me.

I would like to acknowledge the financial, academic and technical support of the ATCoMe project, funded under the seventh framework program of the Euro-pean commission and the university of Brussels (ULB) for all academic facilities. Special thanks to professors Antonio Huerta, Sonia Fernández-Méndez and

Adeline de Montlaur for organizing with the highest enthusiasm the ATCoMe

project. Also, special thanks to Carlos Manuel Tiago Tavares Fernandes, José

Paulo Moitinho de Almeidaand Pham Tiencuong from the civil engineering

department of the University of Lisbon. In addition, I would like to thank the ESR of ATCoMe: Vladimir Ivannikov, Augusto Emmel Selke, Miquel Aguirre,

Sudharsana Raamunu Raman, Sudhakar Yogaraj, Aleksandar Angeloski, Omid Javadzadeh Moghtader and Clément Sambuc. Right now as I write

these lines, I remember the numberless times that we share a beer in Barcelona after the meetings, you are the best guys!.

I would also like to thank professor L.M.B.C. Campos for giving me the oppor-tunity to come to Portugal and to undertake a part of my research at the Instituto Superior Técnico of Lisbon. I am very grateful to you for all the discussions about my thesis and other work-related issues and for accepting being part of my jury thesis. I admire your enthusiasm for science and the time you were always happy to share with everyone. Thank you for your invaluable kindness professor.

Special thanks to my present and ex-colleagues with whom I have enjoyed the past three and a half years, starting from the PDG group: Joël Davis, Boris

De-sauvage, Benoît Van den Nieuwenhof, Yves Detandt, Benjamin DeBrye,

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x Contents

Jacqmot, Bernard Van Antwerpen, Ze Zhou (Jo), Lou Long, Dave Burd, Diego d’Udekem, Diego Copiello, Marie Cabrol, Alexis Talbot, Bastien Ganty, Gai Vo Thi, Prateek Mustafi, Morgan Jacq, Romain Baudson,

Steve Laldjee, Xavier Robin. And of course, the administration team:

Cé-cile Vandenplas, Marie-Laure Lefebvre, Julie Blaise, Valérie Larsille and Pino Mbutu.

I would also like to take this opportunity to sincerely acknowledge the jury mem-bers, professor Gérard Degrez, professor Jean-François Mercier and professor

Arnaud Deraemaeker for their careful reading, valuable comments and suggestions

on my thesis.

Many thanks to my close friend Carlos Chávez (a.k.a. Compa) for your uncon-ditional friendship along these years. I lost track of how many times you have read my thesis for english corrections. José Eduardo Mendoza (a.k.a. Meke) for the exciting discussions about quantitative finances around a beer. I must confess that sometimes I barely followed your explanations Meke. Marina Ramírez for her way of seeing life and passion to do everything. And my friends Dr. Daniel Palhazi and Dr. César Urbina, you are an example to be followed.

My time in Brussels was made enjoyable to a great extend due to the many friends and groups that I became a part of. I am grateful for the time spent with my colloqs (a.k.a. le Manoir): Chloé, Christel, Audrey (mou), Celestine, Patrick, Jony,

Vincent, Florence, Max, Guillaume, Camille, Damien, Simon-Pierre and Lio. And my dear friends: Pierre-Alexandre (Minh) and Greg Vettas (Steve Tragg), thank you guys!.

Last but not least, I would like to thank my family for all their love and encour-agement. My parents Constante and Clara who raised me with a love for science and supported me on all my pursuits. My brothers, sisters and in-laws in Venezuela:

Lucho, Nando, Mari, Poly, Leonel and Maricarmen. The presence of my little

brother Daniel not that far from Brussels, in Finland, you help me to feel closer to home. I am sure you will succeed in your Ph.D. brother. And most of all to my loving, supportive, encouraging, and patient girlfriend Olalla Lopéz Álvarez whose faithful support during final stage of this Ph.D. is so appreciated. Thank you!.

César Legendre

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Declaration Of Authorship

I, César Legendre, declare that this thesis and the work presented in it are my own and has been generated by me as the result of my own original research.

On the interactions of sound waves and vortices I confirm that:

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

(ii) 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;

(iii) where I have consulted the published work of others, this is always clearly attributed;

(iv) 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;

(v) I have acknowledged all main sources of help;

(vi) 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; (vii) either none of this work has been published before submission, or parts of this

work have been published in references.3,33,81,82

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Abbreviations

Actran TM Actran Turbo Machine

Actran DGM Actran Discontinuous Galerkin Method

APU Auxiliary Power Unit

ATCoMe Advance Techniques in Computational Mechanics

BEM Boundary Element Method

BPF Blade Passing Frequency

CAA Computational AeroAcoustics

CFD Computational Fluid Dynamics

DG Discontinuous Galerkin

DNS Direct Numerical Simulation

ESR Early Stage Researchers

EPNL Effective Perceived Noise Level

FWH Ffowcs-Williams and Hawking analogy

FE Finite Element

FFT Free Field Technologies

GPA Goldstein-Perez’s Asymptotic model

HVE High Vorticity Effects model

K-H Kelvin-Helmholtz

l.h.s Left-hand side

LEE Linearised Euler Equations

LES Large Eddy Simulation

LLE Linearised form of Lilley’s equation

LNSE Linearised Navier-Stokes Equations

MVE Medium Vorticity Effects model

NRBC Non-Reflecting Boundary Condition

PDG Product Development Group

POT Classic convected-type wave equation for irrotational mean flows

PML Perfectly Matched Layer

PSG Product Support Group

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RANS Reynolds Averaged Navier-Stokes

SPL Sound Pressure Level

sAbrinA Solver for Acoustic BRoadband INteraction with

Aerody-namics, developed by ONERA

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Nomenclature

Roman symbols

Symbol Units Description

(dx, dy, dz) − differentials in Cartesian coordinates (x, y, z) − Cartesian coordinates

(ωx, ωy, ωz) − vorticity components, Cartesian coordinates (u, v, w) m/s velocity components, Cartesian coordinates (ur, uθ) m/s velocity components, polar coordinates (xp, yp, zp) − pathlines components

(ex, ey, ez) − unitary vectors, Cartesian coordinates (er, ez, eθ) − unitary vectors, cylindrical coordinates (r, θ, z) − cylindrical coordinates

(r, θ, φ) − spherical coordinates A kg−1.m2_.s−1 _{acoustic admittance}

B J total enthalpy

cp J/(kg.K) heat capacity at constant pressure cv J/(kg.K) heat capacity at constant volume ¯

e J/kg internal energy per unit mass

f N external forces, chapter2

f − generic vector field, chapter4

f Hz frequency

F − generic source distribution, chapter4

Gm Nm2kg−2 gravitational constant

g m/s2 acceleration of gravity

h J enthalpy

H − arbitrary intensive property

I W/m2 energy flux

kc 1/m constant, Lamb-Chaplygin vortex.

k 1/m wavenumber in x−direction

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K − advection–diffusion tensor, generic scalar oper-ator

L m/s2 Lamb’s vector

lt m characteristic turbulent length M a, M∞, M0 − Mach number

m − azimuthal order

MT kg mass of the earth

n − normal vector

p Pa pressure

P − pressure perturbation spectrum

pm Pa mechanical pressure

Q − arbitrary sources

Q J heat energy

Q(z) − pressure perturbation spectrum, chapter 8

q J/m2 heat flux

qm(t) − compact sources

Re − Reynolds number

R − reflection coefficient

Rc m jet radius

R J.K−1mol gas constant

s J/K entropy

Sn − thermodynamic state

T K temperature

T − transmission coefficient

Tij − generic tensor function

t s time

T kg/(s2.m) Lighthill tensor U∞ m/s far field mean velocity

Ua m/s velocity, Hill’s vortex.

u? m/s vortical distortions, Perez’s equations Uc m/s jet central velocity

v m/s velocity

v1, v2 − convecting vectors, generic scalar operator

Vc m/s jet core velocity

Vd m/s jet velocity in the fully developed zone

W W acoustic power

Wt W power radiated by turbulence Z kg.m−2s acoustic impedance

Greek symbols

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δ m boundary/shear layer thickness

ηt m Kolmorogov scale

η(x, t) m geometric deformation of vortex sheet

εt W turbulent dissipation

ε − horizontal compactness

εw m wake thickness

εj m wake-jet transition thickness φ m2/s velocity potential

ϕ m2/s second potential

ϕ − vertical compactness, chapter8

ψ m2/s stream function

Γ − generic Surface

γ − specific heat ratio

K m2/s circulation κ 1/m wavenumber in y−direction ¯ κ − wavenumber ratio λ W/(m.K) thermal conductivity λa m wavelength

µ Pa.s dynamic viscosity

ν m2/s kinematic viscosity

Ω − generic volume

Ω − Helmholtz Number

ω s−1 vorticity

ω s−1 angular frequency

ω∗ s−1 Doppler shifted frequency

˜

Θ s−3 sources terms of the Lilley equation

ρ kg/m3 density

σ N/m2 stress tensor

ς1 − vector source, generic scalar operator

ς0 − scalar source, generic scalar operator

τ N/m2 stress vector

τν _N/m2 _{viscous tensor}

τ s arbitrary time

ϑ m3/kg specific volume

ξ? m/s vortical distortions, Goldstein’s equations

ζf − reduced vorticity

ζr − rotational ratio

Operators

∇. divergence

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∇2 _Laplacian

: tensor contraction

⊗ dyadic product

H

closed surface integral D

Dt total or material derivative ∂

∂x partial derivative with respect to x LHowe Howe’s operator

LDoak Doak’s operator

Lh Helmholtz’s operator La D’Alembert’s operator d differential

Mathematical symbols

Ai, Bi Airy functions I identity matrix γ, β Clebsch potentials G Green function

G vector Green function H Heaviside function

Jn Bessel function of nthorder and first kind Yn Bessel function of nthorder and second kind

Γ Gamma function

∇2 Laplacian

δ(x) Dirac’s delta

n normal vector

Acoustic and fluid dynamics notation

H scalar variable

H vector or tensor variable ˙

H time derivative ˜

H complete variable, from chapter4

H acoustic variable, from chapter4

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1 Introduction

El escritor escribe su libro para explicarse a sí mismo lo que no se puede explicar.

Gabriel García Márquez

1.1 Historical context

The early years of the theory of aerodynamic sound ...

T

he mechanisms of sound generation and propagation on turbulent flows have been studied over the last six decades. This study began in 1952 with the publication of the paper “On sound generated aerodynamically I ” by James Lighthill.84 This paper represented the birth of the research area “Aeroacoustics”. The motivation that led Lighthill to write this paper was the increasing noise emissions in airports, due to emerging jet aircraft technologies at that time. Consequently, Lighthill’s work was intended to establish a theory of sound produced by turbulence with special emphasis on jet noise. Now of course, the jet noise phenomena are inherently linked to the understanding of turbulence in jet flows which even nowadays has not been completely elucidated71

Undoubtedly, the research field of aeroacoustics began with the study of sound generation mechanisms by turbulence rather than propagation of sound in non-uniform flows. For this purpose, Lighthill defined the concept “Acoustic Analogy”. The idea of this concept is to reformulate fluid dynamics equations into a form of wave operator on the left hand-side. Instead, the remaining terms on the right hand-side of equation are the sources produced by turbulence. The turbulent sources in Lighthill’s derivation were expressed as spatial distributed quadrupoles. This gave an idea of how the turbulence is translated into noise. In addition, Lighthill’s acoustic analogy led to a noise scaling law. By means of dimensional analysis, Lighthill established that the acoustic power radiated by a jet should scale with the eighth power of the jet’s velocity. Although Lighthill’s acoustic analogy does not take into account flow convection effects, the results of the eighth power law were confirmed by the available experimental evidence at that time.

First convection effects ...

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1.1. Historical context 3

The role of vorticity ...

With the improvements of noise measurement techniques in the 1970s, the gener-ation mechanisms predicted by Lighthill and its extension with regards to uniform flow convection developed by Ffowcs-Williams were not sufficiently precise to coincide with the experimental evidence, according to the measurements on jet noise made by Lush.89 These measurements showed that, in the jet downstream, the peak frequency decreased and the convective amplification over-predicted the sound pressure levels. It was argued by several authors57,87,108 _{that these discrepancies were probably due} to the convection effects of non-uniform flows not included in the original derivations. Later, the effects of non-uniform flows on the sound propagation were taken into account in works such as Lilley’s,87 _{Goldstein’s}57_{(with vorticity) and Howe’s}66 (with-out vorticity). Therefore, not only the non-uniformity of the mean flow affects the acoustic propagation, but in particular the mean flow vorticity has an important role that has to be taken into account for noise prediction. In parallel, the role of vorticity was emphasized also in sound generation. The theory of vortex sound established by Powell110 _{with further works of Howe,}66 _Möhring98 _{and Müller}100_{pointed out the} fluctuations of vorticity as the ultimate turbulence source.

The analytical solutions ...

Following the line of reasoning of the importance of vorticity in acoustics, from the 1950s, an important part of the study of theoretical acoustics was focussed on the interaction of acoustic waves with simplified vortex patterns. First, the study of acous-tic waves interacting with two-dimensional37,69 _{and cylindrical vortex sheets}11,93,102 evidenced the close relation between the vorticity and acoustic instabilities. Sec-ondly, Pridmore-Brown109 _{derived an acoustic operator for unidirectional shear flows,} namely shear and boundary layers. Since then, numerous analytical solutions for this operator were proposed considering different flow configurations, for instance: (i) linear shear and boundary layers;24,70,73(ii) exponential boundary layers;23(iii)

hyperbolic tangent shear layers;25_{and (iv) parabolic profiles in cylindrical ducts.}31

Numerical methods ...

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Möhring’s,99 _{or convection effects of irrotational flows in the acoustic propagation, i.e.} Howe’s66 _{or Doak’s}43 _{operators. The commercial software Actran TM and Actran} Aeroacoustics developed by FFT offers an implementation for these purposes. In addition, the rotational flow effects on the acoustic propagation are predicted by solving the linearised Euler equations (LEE). Numerical methods as discontinuous Galerkin (DG) applied to the LEE are capable to handle efficient parallelism schemes and scalability for industrial problems. The commercial software Actran DGM uses DG methods for solving the LEE in time domain. Although the strengths of DG methods are predominantly present in time domain, frequency domain approaches have been implemented to avoid the problems associated with K–H instabilities.3 Finally, the non-linear Euler equations are also used for acoustic noise predictions. Non-linear terms present in the formulation control the occurrence of K-H instabilities leading to robust solutions. The sAbrinA code, developed by ONERA, solves the Euler equations with a sixth-order interpolation to assure spatial accuracy.115

1.2 Thesis outline

This thesis is conceived in the form of a self-contained book divided in three parts:

Part I: Theoretical Framework

In this part the theoretical framework of fluid dynamics is given. This part is divided in two short chapters as it follows. Chapter 2 “Fundamental equations for fluid dynamics” introduces the fundamental equations of fluid dynamics and basic principles of thermodynamics frequently used in acoustics. By manipulation of the equations presented in this chapter, alternative forms of the fluid dynamics equations are derived. These forms will be used in further chapters for the theoretical derivations of this work. Similarly, Chapter3“On vortex motions” introduces the concept of vorticity and the vorticity equation. In this chapter, vortex patterns, i.e. boundary layers, jet flows, wakes, etc, are explained. Finally, analytical solutions of the Navier-Stokes equations are featured.i

Part II: Acoustics in moving media

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1.2. Thesis outline 5

are featured emphasizing the role of the vorticity in sound generation. Moreover, arguments are given to sustain the use of the fluctuating total enthalpy as a generalized acoustic variable.

Chapter4“Effects of non-uniform mean flows on sound propagation” moves from the right hand-side to the left hand-side of the operator. In this chapter, arbitrary flows are taken into account in the wave propagation, namely mean flows with vorticity. The LEE, Goldstein’s equations among others are presented. Finally, a brief note on hydrodynamic instabilities is also addressed.

Part III: Theoretical and Numerical Contributions

This part reflects the theoretical and numerical contributions to the theory of noise propagation in vortical flows. The contributions are divided in three chapters with a complementary chapter of conclusions and perspectives:

Chapter6“Weakly-coupled acoustic-vortical wave equations” presents the deriva-tion of three scalar models for the acoustic propagaderiva-tion in vortical flows. These models, based on fluctuating enthalpy terms, explicitly contain in their coefficients the mean flow vorticity scaled by the angular frequency. The chapter begins with the linearisation of the fluid dynamics equations from chapter2. Then, four equations in terms of fluctuating enthalpy, vorticity and entropy are obtained. In order to condense such equations, the vorticity and entropy waves are neglected. The justification of this is made a posteriori using numerics. Finally, to prove the efficacy of the weakly-coupled acoustic-vortical wave equations, they are compared with both the LEE and the classic convected wave equation for irrotational flows in four cases of acoustic propagation through: (i) a shear layer with linear profile; (ii) a vortex dipole; (iii) a cylindrical jet; and (iv) a generic aircraft engine.

Chapter7“Acoustics involving exponential boundary layers” features an analytical solution to Pridmore-Brown’s equation. The solution considers an exponential bound-ary layer profile whose shape depends on the acoustic parameters of the problem. Furthermore, three asymptotic solutions are derived: (i) for infinitely thin bound-ary layers; and (ii)(iii) for infinitely thick boundbound-ary layers. Comparisons with FE computations are made to verify the quality of the analytical solutions.

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Table 1.1: Changes in priorities for civil aviation.138

1950-1970 1970-1990 1990-2020

Flight safety Flight safety Flight safety Speed Economic indices Environmental protection

(including noise)

Range Noise around airports Resources

Economic indices Regularity of operations Regularity of operations

Comfort Comfort Economic indices

Regularity of operations Speed Comfort

Noise around airports Range Speed and range

And finally, chapter9“Conclusions and perspectives” makes a compilation of the results achieved in this work. Additionally, all the theoretical contributions (original) to the theory of sound propagation in vortical media are highlighted. Finally, some recommendations are given to the reader and perspectives and future research ideas to continue dome parts of this work are provided.

1.3 Motivation and goals

The impact of the air transportation on modern society has been enormous. Since the advent of the first jet commercial aircraft, the De Havilland Comet in 1949,137_air transportation has connected nations and boosted world markets in many areas like tourism, business or freight services among others. New airports have been created to deal with the increasing number of travellers and cargo of commercial aviation in the last years. In particular, within Europe, the number of commercial flights will be up to 25 million in 2050, compared to 9.4 million in 2011.49 _{These airports are} now reaching their operating limits, resulting in environmental impacts upon local communities or air routes passing over the cities,83 _{i.e. noise and CO}

2emissions.

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1.3. Motivation and goals 7

Since the development of the first airliner, a significant progress in terms of noise reduction has been made. The noise levels produced by modern jet aircrafts are about 20 dB lower than those of the first generation. In addition, in order to deal with the problem of aircraft noise, this has been divided in several specific sources, see figure

1.1. First, the airframe noise is the result of the aerodynamic interactions between air and wings, flaps or landing gears; the aifoil-tip vortex is as well a consequence of airframe noise. In addition, the sonic boom is another form of airframe noise. Fan and compressor noise may fit inside the topic of aircraft noise. For instance, a fan or a compressor exhibit both: (i) tonal noise due to the interaction of the fan’s blade at the blade passing frequency and its harmonics; (ii) broadband noise due to fine-scale turbulence. In addition, turbine noise is similar to fan noise, but the tonal component will be dominant in this case. Furthermore, the jet noise covers several topics that have been studied separately in the scientific literature.122 _{For instance, one of the} most important sub-topics of jet noise is the jet mixing noise. This kind of noise is predominantly broadband due to the fluctuating shear stress in the mixing process behind the nozzle, i.e. shear layers and turbulence. In fact, it was the study of jet mixing noise that led Lighthill to pioneer the science of aeroacoustics. Finally, other sources of noise may be identified, for instance cabin noise produced by turbulence in the boundary layer on the outer surface of the cabin, combustion noise, installation effects, APU systems, among others.

Engine sources

Landing gear sources

Lift-devices

Propulsion/airframe interations

Figure 1.1: Aircraft noise sources.8

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aircraft noise, giving place to the science of Computational AeroAcoustics (CAA). The problem of numerical prediction of aircraft noise has been tackled in several fronts: (i) the development of theories of noise generation and propagation; (ii) the development of new techniques for solving partial differential equations in complex geometries; and (iii) indirectly, the progress in computer science, efficiency and speed-up in computations. Therefore, understanding aircraft noise from a theoretical point of view and/or by using numerical tools is of essential importance in the noise reduction of the next generation aircrafts.

The general goal of this thesis proposal is the study of the acoustic propagation in vortical flows from a theoretical and numerical point of view. Specifically, the sub-goals of this work are listed below:

(i) to extend the classic-convected wave equation (scalar operator) including vortical mean flow effects, this is achieved with the family of weakly-coupled acoustic-vortical wave equations featured in chapter6. Numerical evidences show a range of validity for the the family of weakly-coupled acoustic-vortical wave equations closely related to the mean flow vorticity and the frequency of the acoustic phenomena (reduced vorticity);

(ii) to contribute to the scientific literature with analytical solutions of acoustic propagation in boundary layers (chapter7);

(iii) to extend the limits of Pridmore-Brown’s equation in order to study the effects of bias and shear flows over locally-reacting liners (chapter8);

(iv) to make available the family of weakly-coupled acoustic-vortical wave equations in the commercial software Actran for industrial purposes (reduction of aircraft noise).

(v) to pave the way towards more complete acoustic analogies taking into account both propagation and generation effects. This is suggested in chapter6with the derivation of aerodynamic source terms for the family of weakly-coupled acoustic-vortical wave equations;

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2 Fundamental equations for

fluid dynamics

There is a conservation of matter and of energy, there may be a conserva-tion of life; or if not of life, of something which transcends life.

Sir Oliver Joseph Lodge

2.1 Introduction

T

he derivations in this section rely on the assumption that the matter is consid-ered as a continuum. For analysing the fluid motion, one might take two paths: (i) to describe the phenomenon as an individual volume moving through time and

space, i.e. Lagrangian frame of reference; or (ii) to work in a finite region where the fluid passes through it, and then, to perform a balance to determine the flow effects like, force or total energy exchange, i.e. Eulerian frame of reference. In this work the latter approach is considered. The Reynolds transport theorem is used in a control volume to find the rate of change of an arbitrary fluid property, e.g. density ρ, as a result the conservation laws of mass, momentum and energy are derived. Then, the Navier-Stokes equations are formulated from these conservation laws. Particular cases of fluid dynamics equations are presented, namely Euler equations, where the viscous effects are neglected. Finally, other forms of momentum and energy equations are derived using thermodynamic principles and vector identities.

2.2 The Reynolds transport theorem

The Reynolds transport theorem (2.1_{) refers to any extensive property H in a particular}

control volume Ω. It states that the material change of an extensive property H associated to continuum medium defined inside the control volume Ω is equal to the local rate of change of such a property plus the flux that enters or leaves the boundaries Γ : DH Dt = ∂ ∂t Z Ω ρH dΩ + Z Γ ρHv.n dΓ , (2.1)

where H is the associated intensive property (property H per unit mass), ρ is the fluid density, v is the fluid velocity and defining the material derivative (2.2) as follows:

D Dt∗ =

∂

∂t ∗ +v.∇∗, (2.2)

In addition, using the Reynolds transport theorem (2.1), consider again a control volume Ω fixed in space, bounded by a surface denoted by Γ , as represented in figure

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2.3. Conservation of mass, momentum and energy 13

by an external or internal source denoted as Q in the following form: ∂ ∂t Z Ω ρH dΩ = − Z Γ ρHv.n dΓ + Z Ω Q dΩ. (2.3)

Equation (2.3) shows that the rate of change of the property H defined inside the volume Ω depends on what is generated inside Ω by the sources Q and what is convected through the boundaries Γ by means of the fluid velocity, this relation (2.3) is known as law of conservation. Then, applying the divergence theorem to the surface integral in equation (2.3) results:

∂ ∂t Z Ω ρHdΩ + Z Ω ∇.(ρHv)dΩ − Z Ω QdΩ = 0, (2.4)

by combining all the terms of equation (2.4) in one single integral, it is proven that: ∂ρH

∂t + ∇.(ρHv) = Q, (2.5)

is the law of conservation (differential form) for an intensive property H.

Ω n

Q Γ

H v

Figure 2.1: Quantity conservation of a fixed volume Ω in the space.

2.3 Conservation of mass, momentum and energy

In the case of the conservation of mass, since the mass cannot be created or destroyed, there are no mass sources or sinksiand therefore, equation (2.1) may be used to write its conservation law:

∂ρ

∂t + ∇.(ρv) = 0, (2.6)

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commonly known as the continuity equation. There are many forms to write the continuity equation, for instance, expressing equation (2.6) in an expanded form:

∂ρ

∂t + v.∇ρ + ρ∇.v = 0, (2.7)

as a result, the continuity equation (2.6) is written in the following form: Dρ

Dt + ρ∇.v = 0. (2.8)

Moreover, the continuity equation may be represented in terms of the specific volume ϑ = ρ−1 as follows:

Dϑ

Dt − ϑ∇.v = 0. (2.9)

The momentum ρv can be considered as an intensive property. It is defined as the product of the fluid density and velocity, being a vector quantity. Without loss of generality, the conservation law (2.3) as a consequence of the Reynolds transport theorem (2.1) may be used to derive the conservation law for a vector property (similar to a scalar property), hence:

∂ ∂t Z Ω ρvdΩ + Z Γ ρvv.ndΓ = Z Ω ρf dΩ + Z Γ τ dΓ , (2.10) equation (2.10) is the integral form of the momentum equation. The source or sink terms of equation (2.10) are divided as volume forces per unit mass denoted by f , and forces per unit area over a surface Γ . Defining a stress vector τ to be applied over a surface Γ , a force acting on a unit of area dΓ may be represented as τ dΓ . The relation between the stress vector τ and the normal vector n to the surface Γ is expressed by means of the stress tensor σ, so τ = σ.n. Applying the divergence theorem on equation (2.10), the differential form of the momentum equation results:

∂ρv

∂t + ∇.(ρvv) = ∇.σ + ρf , (2.11)

expanding and then rearranging terms, equation (2.11) is now written as:

v ∂ρ ∂t + ∇.(ρv) + ρ ∂v ∂t + v.∇v = ∇.σ + ρf . (2.12)

By means of the continuity equation (2.6), the first term of equation (2.12) vanishes and the momentum equation takes its classic form:

ρ ∂v

∂t + v.∇v

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2.3. Conservation of mass, momentum and energy 15

One can define two kinds of energy within a fluid. First, there is an internal energy per unit mass, here denoted as ¯e, or rather per unit volume ρ¯e. Secondly, since the fluid is in motion, a kinetic energy per unit volume must be considered and it is defined as 1₂ρv2. Then, considering again a fluid contained in a volume Ω and bounded by a surface Γ , the law of conservation of energy guarantees that the rate of change of the total energy, i.e. the sum of internal and kinetic energies, plus the energy that leaves the boundary by convection effects is equal to the work done by the body/surface forces plus the heat flux that enters (or leaves) into Ω through the surface Γii_{. Using, therefore equation (}_2.3_{), the aforementioned statement is} mathematically written as:

∂ ∂t Z Ω ρ¯e +1 2ρv 2 dΩ + Z Γ ρ¯e +1 2ρv 2 vdΓ = Z Ω ρv.fdΩ + Z Γ v.τ dΓ − Z Γ q.dΓ (2.14) where the heat flux is denoted as q and the stresses are represented by τ = σ.n. Using the divergence theorem, equation (2.14) reduces to:

∂ ∂t Z Ω ρ¯e +1 2ρv 2 dΩ + Z Ω ∇. ρ¯e +1 2ρv 2 v dΩ = Z Ω ρv.fdΩ + Z Ω ∇. (v.σ) dΩ − Z Ω ∇.qdΩ, (2.15)

while the differential form of equation (2.15) is given by: ∂ ∂t ρ¯e +1 2ρv 2 + ∇. ρ¯e +1 2ρv 2 v = ρv.f + ∇. (v.σ) − ∇.q. (2.16)

Expanding terms of equation (2.16) and rearranging such that:

¯ e ∂ρ ∂t + ∇.(ρv) + ρ ∂¯e ∂t + v.∇¯e +1 2v 2 ∂ρ ∂t + ∇.(ρv) +ρv.Dv Dt = ρv.f + ∇. (v.σ) − ∇.q, (2.17)

it is noticed that the first and the third term of equation (2.17) vanish due to the definition of the continuity equation (2.6). Then, using the momentum equation (2.13) to simplify equation (2.17), the classic form of the energy equation can be written as:

ρ ∂¯e

∂t + v.∇¯e

= σ.∇v − ∇.q. (2.18)

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2.4 Heat flux, stress and gravity force

The conservation of momentum (2.13), energy (2.18) and mass (2.6) only depends on the nature of the fluid and external sources/forces. So far, the conservation laws represent an under-determined system of partial differential equations, for instance, the stress tensor σ and the heat flux q are commonly unknowns. Such unknowns may be represented as function of classical fluid dynamics variables, i.e. pressure, velocity and density and their derivatives, depending on parameters that must be determined experimentally. These relations are known as constitutive relations. In particular, the heat flux in the energy equation (2.18) may be described by means of Fourier’s Law, stating that the rate of change of heat flux passing through a surface is proportional to the temperature gradient and the thermal conductivity of the medium, here denoted as λ. Fourier’s law is written as:

q = −λ∇T. (2.19)

The thermodynamics relations usually involve the fluid pressure piii_{. In fact, the} fluid pressure p depends on the mechanical pressure pm expressed as the diagonal of the stress tensor, using index notation this is denoted as:

pm= − 1

3σii= − 1

3(σ11+ σ22+ σ33) . (2.20) On the other hand, for the most commons fluids, the stress tensor is linearly related to the velocity gradientsiv _{by a scalar, the dynamic viscosity µ. Therefore, the stress} tensor can be written as:

σ = −pmI + µ

∇v + ∇vT−2

3µ(∇.v)I, (2.21)

where I is the identity matrix. Usually the fluid pressure p is not equal to the mechanical pressure pm, but these may be related by means of the bulk viscosity µν with the expression p = pm+ µν∇.v. Consequently, the stress tensor is written as

σ = −pI + τν. Here τν is called viscous tensor and it has the following form:

τν = µ∇v + ∇vT + I µν−2 3µ ∇.v. (2.22)

Continuum mechanics studies the matter at a macroscopic scale, at this scale gravitational forces might be dominant when compared to electrical or magnetic

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2.5. A brief note on thermodynamics 17

forces. Without considering relativistic effects, Newton’s law of universal gravitation states that two bodies with mass attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. In the earth this force is expressed as GnmM_r2Tr, where

m is an arbitrary mass, MT is the mass of the earth, r is the distance between the center of masses and Gn is the gravitational constantv. In the earth GnMT/r2 is considered as constant known as the acceleration of gravity g ≈ 9.81 m/s2. The force per unit volume due to gravitational effects is expressed as ρg.

When the expression of the viscous tensor (2.22), the gravity forces and thermal conductivity (2.19) are combined with the momentum (2.13) and energy (2.18) equations, it results: ρ ∂v ∂t + v.∇v = −∇p + ∇.τν+ ρg, (2.23a) ρ ∂¯e ∂t + v.∇¯e = τν.∇v − p∇.v + ∇.(λ∇T). (2.23b)

Equations (2.23a,b) together with the continuity equation (2.6) are known as the Navier-Stokes equations. They form a closed system of non-linear partial differential equations which must be solved numerically. The so-called Euler equations is a particular case of the Navier-Stokes equations (2.23) when the viscous, thermal and gravity effects are neglected. The Euler equations are represented as follows:

ρ ∂v ∂t + v.∇v = −∇p, (2.24a) ρ ∂¯e ∂t + v.∇¯e = −p∇.v. (2.24b)

2.5 A brief note on thermodynamics

Let an infinitesimal fluid volume Ω in an initial state of equilibrium S1 (local

ther-modynamic equilibrium) that undergoes a process of change to another state of equilibrium S2 as depicted in figure2.2. A process is called “reversible” if such a

change takes place through a sequence of equilibrium states. Thus, considering a reversible process between the states S1 and S2, the second law of thermodynamics

v_G

nis approximatively 6.67 × 10−11Nm

2

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defines the entropy s as follows:

ds = dQ T , s = Z S2 S1 dQ T . (2.25)

Moreover, if the process of change from S1 to S2takes place without any heat flow,

i.e. dQ = 0, the process is called “adiabatic”. As a consequence of the second law of thermodynamics (2.25), a reversible adiabatic process is always isentropic, i.e. dQ = 0 implying that ds = 0. Furthermore, the first law of thermodynamics (2.27):

dQ = d¯e + dw, (2.26)

states that the heat flux dQ released (or absorbed) across the boundaries equals the increase (or decrease) of internal energy d¯e plus the work done dw. Since in this case the process of change is considered isentropic, the work of the system may be expressed as the deformation of the boundaries pdϑ. Therefore, the first law of thermodynamics may be rewritten as:

dQ = d¯e + pdϑ. (2.27)

Combining the first (2.27) and second (2.25) law of thermodynamics, one gets an important thermodynamic relation between entropy, energy and work:

Tds = d¯e + pdϑ. (2.28)

Albeit equation (2.28) was derived for a reversible process, one can extend its validity to an irreversible process, because it is a combination of the first and second law of thermodynamics. To be precise, if a process is irreversible, equation (2.28) is valid because the heat released is always equal to the increase of the internal energy plus the work done. Additionally, the local reversibility does not imply global reversibility. In other words, global reversibility is associated to global thermodynamic equilibrium where the intensive properties of the fluid are homogeneous in the whole system. In contrast, local reversibility is related to local thermodynamic equilibrium when those intensive properties may vary in time and space, nonetheless they vary slowly. Furthermore, the thermodynamic fluid properties like ¯e, p, ρ, T may be related between them by the equation of state, for instance ρ = ρ(p, s). Actually, for a system in thermodynamic equilibrium all the properties can be determined by means of the equation of state if at least two independent properties are known.

On the other hand, defining the enthalpy h as:

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2.6. Alternative forms of fluid dynamics equations 19 d Q Ω Ω S₁ S₂ ρ+dρ, T +dT ¯ e+d¯e, ϑ+dϑ, p+dp ρ, ϑ, ¯e, T, p

Figure 2.2: Fluid element in two states S1 and S2 of thermodynamic equilibrium, this change is caused by a heat flow dQ.

consisting in the sum of the internal energy ¯e and the product of the pressure p and the specific volume ϑ. The enthalpy is considered as a thermodynamic potential. Using equation (2.28) a relation between enthalpy h and entropy s is written as follows:

dh = Tds + ϑdp. (2.30)

Finally, the adiabatic speed of sound is the rate of propagation of a pressure pulse through a fluid, being a thermodynamic property of the fluid itself. The formal definition is c2_{= (∂p/∂ρ)}

s. To be precise, the square of the adiabatic speed of sound c is defined as the rate of change of the pressure p with respect to the density ρ at constant entropy s. Changes in density dρ and entropy ds produce a pressure change dp related to the speed of sound, such that:

dp = c2dρ + ∂p ∂s ρ ds or dρ = c−2dp + ∂ρ ∂s p ds. (2.31)

2.6 Alternative forms of fluid dynamics equations

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The curl of the fluid velocity v is known as the vorticity ω, that is ω = ∇ × v. It is convenient to express the momentum equation (2.23-a) in terms of vorticity using the following identity:

∇ 1

2v

2

= v.∇v + v × ω, (2.32)

the momentum equation (2.23-a), can now be written as: ∂v ∂t − v × ω = − ∇p ρ − ∇ 1 2v 2 +1 ρ∇.τ ν_{+ g.} _(2.33)

Considering equation (2.30), hence ∇h = T∇s + (1/ρ)∇p may be replaced in relation (2.33), rearranging such that:

ρ ∂v ∂t + ∇ h +1 2v 2 = ρv × ω + ρT∇s + ∇.τν+ ρg. (2.34)

Conveniently, one defines the total enthalpy as B = h +1₂v2_{. Therefore, equation}

(2.34) results in:

ρ ∂v ∂t + ∇B

= ρv × ω + ρT∇s + ∇.τν+ ρg, (2.35)

known as Crocco’s equation.10,35

Similarly, from the conservation of specific volume (2.9), ∇.v = ρDϑ/Dt is obtained and combining with the energy equation (2.23-b) results:

ρ D¯e Dt + p Dϑ Dt = τν.∇v + ∇.(λ∇T). (2.36)

Using the first and the second law of thermodynamics for a reversible process (2.28), then TDs/Dt = D¯e/Dt + pDϑ/Dt and equation (2.36) becomes:

ρTDs Dt = τ

ν_{.∇v + ∇.(λ∇T),} _(2.37)

as a matter of fact, when the thermal and viscous effects are neglected Ds/Dt = 0, since always T > 0. Using TDs/Dt = Dh/Dt − 1/ρDp/Dt in (2.37) and combining with the product of the momentum equation (2.23-a) and v, it gives:

ρD Dt h +1 2v 2 = ∂p ∂t + (∇.τ ν_{) .v + τ}ν_{.∇v + ∇.(λ∇T) + ρv.g,} _(2.38)

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2.7. Concluding remarks 21

equation (2.38), an alternative form of the energy equation is written as follows: ρDB

Dt = ∂p

∂t + ∇.(τ

ν_{.v) + ∇.(λ∇T) + ρv.g,} _(2.39)

relating the thermodynamic pressure p to the total enthalpy B.

2.7 Concluding remarks

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3 On vortex motion

What comfort can the vortices of Descartes give to a man who has whirl-winds in his bowels! .

Benjamin Franklin

3.1 Introduction

T

urbulence remains a challenging problem in theoretical physics.61,132 This phenomenon is characterized by a chaotic behaviour of the fluid properties. In turbulent flows, unsteady vortices appear in many scales and they interact with each other until the smallest scale where they are dissipated by viscous effects. This latter scale is commonly known as Kolmogorov’s scale.77 _{Fluid dynamics equations may be} expressed in terms of vorticity, being this representation, natural and transparent to understand physical mechanisms of energy generation or dissipation. In this chapter, the fluid dynamics equations and some formulas emphasizing the vorticity’s effects are presented. Later, several concepts concerning vortex motion, namely: (i) vortex lines; (ii) vortex sheets, among others are given. Finally analytical solutions of Navier-Stokes

equations considering vortical behaviour are described.

3.2 Streamlines, pathlines and stream functions

In fluid dynamics, the velocity field may be represented and visualized by means of streamlines tangent to the particle velocity or pathlines corresponding to the trajectory followed by a particle in the fluid. The streamlines have interesting properties. They are defined as a group of curves locally tangent to the velocity vector at a given instant. For instance, let an infinitesimal arc dl = dx ex+ dy ey+ dz ez tangent to the velocity vector v = u ex+ v ey+ w ez, the streamlines represent a family of solutions of: dx u = dy v = dz w. (3.1)

On the other hand, the concept of the pathlines relies upon the track followed by an individual fluid particle over a certain period of time. Mathematically, the components xp, yp, zp of a pathline are defined by the integral of the velocity components over time as follows: xp= Z udt, yp= Z vdt, zp= Z wdt. (3.2)

Finally, incompressible flows are characterized by a divergence-free velocity field, i.e. ∇.v = 0. Therefore, the components of the velocity vector can be expressed as spatial derivatives of a scalar function ψ commonly known as stream function. The stream function may be regard as the only non-zero component of a vector potential

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3.3. The vorticity equation 25

∇ × A.10 _{For instance, the velocity components (u, v) of a two-dimensional flow are}

expressed as follows:

u = ∂ψ

∂y, v = − ∂ψ

∂x, (3.3)

or expressed in polar coordinates (θ, r), the velocity components (ur, uθ) are given by: ur= 1 r ∂ψ ∂θ, uθ= − ∂ψ ∂r. (3.4)

3.3 The vorticity equation

The local motion of any fluid may be expressed as a combination of: (i) a rigid-body translation driven by the velocity v; (ii) an isotropic volume expansion proportional to

∇.v; (iii) a pure straining motion without volume change; and (iv) a local rigid-body

rotation described by an angular velocity. The vorticity is defined as twice this angular velocity of the fluid particle, i.e. ω = ∇ × v. The components of the vorticity vector in Cartesian coordinates are:

ωx= ∂w ∂y − ∂v ∂z, ωy= ∂u ∂z − ∂w ∂x, ωz = ∂v ∂x− ∂u ∂y, (3.5)

therefore, by definition, the divergence of the vorticity vector always vanishes ∇.ω = 0. Taking the curl of the momentum equation (2.35) results:

Dω Dt = ω.∇v − ω(∇.v) + ∇T × ∇s + 1 ρ∇ × (∇.τ ν_{) −} 1 ρ2∇ρ × (∇.τ ν_), _(3.6)

and it is known as the vorticity equation. It is worth noticing by equation (3.6) that the vorticity is closely related to viscous and thermal mechanisms. For the sake of simplicity, consider a two dimensional incompressible flow without thermal effects. So, the term ω(∇.v) vanishes due to the definition of the continuity equation (2.6) and

ω.∇v vanishes since the flow is bi-dimensional. Consequently, the vorticity equation

(3.6) results:

Dω Dt = ν∇

2

ω, (3.7)

where the ν = µ/ρ is the kinematic viscosity. The l.h.s. term of equation (3.7) is the common rate of change of the vorticity ω due to the fluid convection. The r.h.s. term of equation (3.7) represents the molecular diffusion by means of vorticity, similar to the diffusion of the velocity ν∇2v in an incompressible Stokesian fluid. The

term ν∇2ω plays a very important dissipative role. On the other hand, when the

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26 3. On vortex motion

instabilities may appear because, damping viscosity mechanisms do not exist.

3.4 Vortex representations

The complex behaviour of the fluid dynamics equations cannot be expressed in terms of analytical functionsi. Nonetheless, some idealized vortex shapes can be mathematically described. In fact, some of these are simplified solutions forms of the Navier-Stokes equations and they are presented in §3.5.

Vortex lines

Let us consider a line within the fluid which is locally tangent to the vorticity vector at any instant, given by:

dx ωx = dy ωy = dz ωw , (3.8)

this is the mathematical representation of a vortex line, similar to the definition of streamlines (3.1), see figure3.1-a. Since ∇.ω = 0, by means of the divergence theorem H ωdΓ = 0 is obtained. Therefore, a vortex line cannot begin or end at any point within the fluid, it should exist inside closed surfaces or cross the fluid beginning and ending on its boundaries.79

Vortex sheets

A vortex sheet is an idealized model of a thin layer of vorticity for an inviscid flow (figure 3.1-c). This is defined by an imaginary surface Γ across which the velocity changes instantaneously from v+ on upper domain to v− on the lower domain. In

other words, the normal components of the velocity are equal while the tangent components are discontinuous and described in the following form:

v = H(Γ )v++ H(−Γ )v−, (3.9)

where H is the Heaviside step function. The discontinuity of the velocity field implies singularities in the vorticity distributions, such singularities make the vortex sheets unstable even for small disturbances. These phenomena were first observed by Helmholtz62 and then by Kelvin72 and nowadays referred as Kelvin-Helmholtz instabilities.

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3.4. Vortex representations 27

Free shear layers

In viscous fluids, vortex sheets do not exist because either: (i) the velocity gradients are made by very thin but stable shear layers; or (ii) the vortex sheets are diffused and degenerated in free shear layers by means of viscous and turbulent effects. The free shear layers are characterized by a smooth velocity transition between two streams where v → v+∀ y → +∞ and v → v−∀ y → −∞ as depicted in figure 3.1-e. The

thickness of this transition will depend on the velocities of the upper and lower streams and the diffusion mechanisms like the kinematic viscosity ν. It is worth noticing that, free shear layers are important because it is there where the mixing process of the properties of the two streams occurs, commonly known as mixing layers.

Boundary layers

One of the most studied topics in fluid dynamics is the boundary layers. At high Reynoldsiinumbers the inertial forces are greater than the viscous forces. Nevertheless, the viscosity effects, namely stresses and forces due to the vorticity, viscosity diffusion among others, are significant and comparable to inertial effects, in layers next to the solid boundaries, called boundary layers. Therefore, a boundary layer may be defined as a thin layer where the vorticity varies rapidly due to a combination of viscous and convection effects, as depicted in figure3.1-b.

Prandtl107 was the first to suggest this idea. He simplified the equations of fluid dynamics dividing the flow in two regions: (i) inside the boundary layer where the viscous effects are important; and (ii) outside the boundary layer where the viscous effects can be neglected without important changes in the solution. A boundary layer is limited by two velocities, the velocity of the solid boundary and the free stream velocity where the solid is immersed in. Since the transition of these two velocities is asymptotic, the boundary layer thickness is defined as the location where the internal velocity reaches the 99% of the free stream velocity.

Boundary layers are very important in aerodynamics. Aircraft manufacturers pay attention to control the turbulent transition of the boundary layer to minimize the drag coefficient. Certainly, for fuselages and wing profiles it is preferred to have laminar instead of turbulent boundary layers. Moreover, in acoustics applications, the boundary layer plays an important role on the use and configuration of locally-reacting liners to absorb or to attenuate noise inside jet engines.23,33,55