An experimental study on the aeroacoustics of wall-bounded flows: Sound emission from a wall-mounted cavity, coupling of time-resolved PIV and acoustic analogies

An experimental study on the aeroacoustics of

wall-bounded flows

Sound emission from a wall-mounted cavity, coupling of time-resolved PIV and acoustic analogies

An experimental study on the aeroacoustics of

wall-bounded flows

Sound emission from a wall-mounted cavity, coupling of time-resolved PIV and acoustic analogies

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op donderdag 22 december 2011 om 10.00 uur

door

Valentina KOSCHATZKY

Ingegnere aerospaziale, Politecnico di Milano, Italia. Geboren te Milaan, Italië.

Prof. dr. ir. J. Westerweel

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof. dr. ir. B. J. Boersma, Technische Universiteit Delft, promotor Prof. dr. ir. J. Westerweel, Technische Universiteit Delft, promotor Prof. dr. ir. W. Schröder, RWTH Aachen University

Prof. dr. ir. C. Schram, Vrije Universiteit Brussel Prof. dr. ir. A. Hirschberg, Universiteit Twente

Prof. dr. ir. A. Gisolf, Technische Universiteit Delft Prof. dr. D.J.E.M. Roekaerts, Technische Universiteit Delft

Dit werk maakt deel uit van het onderzoekprogramma van de Stichting voor Fundamenteel Onderzoek der Materie (FOM), die deel uit maakt van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO). Copyright c 2011 by V. Koschatzky

All rights reserved. ISBN 978-94-6108-252-7 Printed by Gildeprint, Enschede

Contents

Summary vii

Samenvatting xi

1 Introduction 1

1.1 Background and motivation . . . 1

1.2 Objectives of the research . . . 3

1.3 Outline of the thesis . . . 4

2 Theoretical Background 5 2.1 What is aeroacoustics . . . 5

2.1.1 Human perception of sound . . . 5

2.1.2 Flow disturbances and sound . . . 6

2.2 Historical overview . . . 7

2.3 Cavity flow . . . 10

2.3.1 Cavity geometries . . . 10

2.3.2 Cavity modes . . . 10

2.3.3 Feedback mechanisms, resonance and sound emission . . . 14

2.4 Particle Image Velocimetry . . . 16

2.4.1 Bias towards integer pixel displacements . . . 18

2.4.2 Illumination and reflections . . . 19

2.4.3 Large dynamic range . . . 20

2.5 Fluid mechanics governing equations . . . 21

2.6 Aeroacoustics governing equations . . . 23

2.6.1 Green’s function . . . 24

2.6.2 Acoustic analogies . . . 29

3 High speed PIV applied to aerodynamic noise investigation 41 3.1 Introduction . . . 41

3.2 Acoustic pressure prediction from PIV data . . . 42

3.2.1 On the viscous terms . . . 44

3.3 Experimental setup . . . 45 v

3.3.1 PIV setup . . . 47

3.3.2 Wall pressure and sound measurements . . . 49

3.4 Flow characterization . . . 50

3.5 Pressure from PIV data . . . 54

3.6 Acoustic emission with Curle’s analogy . . . 58

3.7 span-wise coherence of the cavity flow . . . 59

3.8 Conclusions . . . 64

4 Two different approaches 69 4.1 Introduction . . . 69

4.2 Methodology . . . 73

4.2.1 PIV and Microphones . . . 73

4.2.2 Acoustic analogies . . . 75

4.3 Results . . . 82

4.3.1 PIV results . . . 82

4.3.2 Results obtained with the acoustic analogies . . . 85

4.4 Discussion and Conclusions . . . 88

5 Tomographic PIV 91 5.1 Introduction . . . 91

5.2 Experimental setup . . . 92

5.3 Results . . . 94

5.3.1 PIV results . . . 95

5.3.2 Source terms comparison in two and three dimensions . . . 98

5.3.3 Sound emission . . . 104

5.4 Conclusions . . . 106

6 Conclusions and outlook 109 6.1 Overall conclusions on this thesis . . . 109

6.2 Outlook . . . 111

Bibliography 113

Acknowledgements 119

List of publications 121

Summary

An experimental study on the aeroacoustics of wall-bounded flows - V. Koschatzky

This thesis deals with the problem of noise. Sound is a constant presence in our lives. Most of the times it is something wanted and it serves a purpose, such as communication through speech or entertainment by listening to music. On the other hand, quite often sound is an annoying and unwanted by-product of some other activity necessary to us. This is what we usually refer to as noise. Noise does not simply create an unpleasant en-vironment but can severely affect human health and have a serious impact on the natural environment as well. The noise pollution problem is addressed in many different ways: by clever urban planning, for instance, and by providing the heavily exposed buildings with acoustic barriers and insulation. Legislation also provides a limit on the maximum acceptable noise levels in many contexts such as working environments, discotheques and music headphones devices. An important effort in the fight against noise is obvi-ously placed at the source of the problem itself, trying to make our tools and devices quieter. In order to do so it is essential to understand, and eventually be able to model, the processes through which sound is produced and propagated.

The objective of this work is to develop and to employ experimental methods, based on the particle image velocimetry (PIV) measurement technique, for the estimate of the emitted sound from flow field measurements. In particular, the focus will be on the study of the noise that is generated by the unsteady loads on solid bodies immersed in the flow. The proposed technique, based on time-resolved PIV measurements, provides insight into the sound sources that is not possible to achieve with other established experimental methods. It also permits experimental investigation in those situations where other mea-surements would not be possible, for example, in the presence of a noisy environment or when a suitable anechoic tests facility is unavailable. In addition, it allows the estimation of the acoustic emission, coupled to the proper acoustic model, in the same fashion as done in hybrid computational approaches. This kind of experimental approach seems to be particularly suited for the study of aerodynamic sound from wall-bounded flows at low Mach number. The main limitation of a PIV-based method in general noise applica-tions is in fact the poor temporal resolution currently available. However, for low enough velocities this is no longer a concern since the resolution is good enough to perform time resolved measurements. Moreover, an experimental technique might be advantageous with respect to computational studies for low Mach number and high Reynolds number

flows with immersed solid bodies. The numerical simulation of the flow in the prox-imity of solid boundaries is in fact particularly complex. In this thesis, a rectangular wall-mounted shallow open cavity has been chosen as a test case, a series of experiments have been performed and different methods and solutions has been tested. In chapter 3 the main details of the experimental setup are given and Curle’s analogy is applied in its classical formulation. In this chapter we also give an estimate of the span-wise coher-ence of the flow, based on stereoscopic PIV measurements. Both the process of pressure calculation from PIV data and the application of Curle’s analogy are affected by several uncertainties, and simplifying hypotheses is necessary in the process. For this reason we also perform direct microphones measurements of the pressure fluctuations at the walls of the cavity and of the sound emitted. We can therefore compare those values with the estimated quantities from the PIV data in order to validate the results and to check the range of validity of the approach. We find that both the hydrodynamic pressure com-puted from the PIV data and the sound emission obtained applying Curle’s analogy have a frequency spectrum that is comparable to that of the direct microphones measurements. In section 3.7 we demonstrate that the larger flow structures, responsible for most of the sound, are rather two-dimensional and coherent in the span-wise direction. In chapter 4 we discuss the details of the implementation and solution of both Curle’s Analogy and the theory of vortex sound. In this chapter, in contrast with the implementation of the model performed in chapter 3, we take into account the presence of the non-compact wall in which the cavity is located by using the image principle in the derivation of the solution. The two methods are derived under the assumption of low Mach number and high Reynolds number and for a listener positioned in the far field. The two analogies perform quite well for the present test case and give very similar results, both in total intensity and in the spectral distribution of the emitted sound. In their application, how-ever, they each have different strengths and weaknesses. The two solutions are in fact derived through quite different pathways, and the mathematical schemes used to solve the equations are sensitive to different factors. The choice for either of the two methods needs therefore to be carefully done in relation to the specific application. The use of the image principle seems to be crucial to properly estimate the sound emission for compact geometries placed in large non-compact surfaces. In chapter 5 we investigate the effect of the three-dimensional velocity fluctuations on the final result. We discuss the results obtained by time-resolved thin tomographic PIV measurements. These measurements provided enough velocity vectors in the span-wise direction to allow for the calculation of the differential quantities in that direction and therefore for the computation of the full source terms of both Curle’s analogy and Theory of Vortex Sound. Details about the new experimental setup and measurement technique were given as well. Results showed that the flow is indeed rather two-dimensional and that there is little difference between the analogies source terms computed two or three-dimensionally. Compared to two-dimensional measurements, volumetric PIV measurements are more expensive, re-quire more complex setups and the obtained data rere-quires a much longer processing time. Moreover, the quality of the data is often lower than that of planar PIV. At the same time thin tomographic measurements do not seem to add relevant information to the compu-tation of the sound emission for our experimental case, that is representative of quasi

Summary ix

two-dimensional wall-bounded flows where the main source of sound is determined by large span-wise coherent flow structures.

Samenvatting

Een experimentele studie naar de aeroakoestiek van wandbegrensde stromingen -V. Koschatzky

Dit proefschrift behandeld het probleem van geluidshinder. Geluid is voortdurend aanwezig in ons leven. Meestal is geluid gewenst en heeft het een doel, zoals commu-nicatie door middel van spraak of tijdens het luisteren naar muziek. Aan de andere kant is geluid vaak vervelend en een ongewenst bijproduct van zaken die noodzakelijk zijn. Dit is waar we gewoonlijk naar refereren als geluidshinder. Geluidshinder veroorzaakt niet alleen een onaangename omgeving, maar kan ook ernstige schade toebrengen aan de gezondheid. Tevens kan geluidshinder een serieuze impact hebben op het milieu. Geluidsoverlast kan aangepakt worden op verschillende manieren, o.a. door slimme ste-denbouw en door het verstrekken van akoestische barri ´Rres en isolatie voor de zwaar blootgestelde gebouwen. Wetgeving voorziet een limiet aan de maximaal toegestane geluidswaarde in allerlei werkomstandigheden, discotheken en koptelefoons. Erg belan-grijk in de bestrijding van geluidshinder, is uiteraard bestrijding aan de geluidsbron zelf. Om dit te bewerkstelligen, is het essentieel om de manier waarop geluid zich voortplant te begrijpen, maar uiteindelijk ook om dit te modelleren.

Het doel van dit onderzoek is het ontwikkelen en gebruiken van experimentele meth-oden, gebaseerd op de meettechniek Particle Image Velocimetry (PIV), voor het verkri-jgen van een schatting van geproduceerd geluid aan de hand van metingen van het stro-mingsveld. In het bijzonder ligt de nadruk op de studie van geluid gegenereerd door een instationaire belasting afkomstig van een stroming over een vast lichaam. De voorgestelde techniek, welke gebaseerd is op tijdsopgeloste PIV-metingen, geeft inzicht in de gelu-idsbronnen. Het verkrijgen van dit inzicht is niet mogelijk met andere experimentele methoden. De techniek laat namelijk experimenteel onderzoek toe in omstandigheden, waar andere metingen niet mogelijk zijn. Bijvoorbeeld, in de aanwezigheid van een lawaaierige omgeving of wanneer een geschikte echo-vrije testopstelling niet beschik-baar is. Daarnaast is het mogelijk om een schatting te maken van de akoestische emissie gekoppeld met een geschikt akoestisch model. Dit wordt op eenzelfde manier gedaan als in computational hybride methodes. Deze experimentele aanpak lijkt in het bijzonder geschikt voor de studie van geluid afkomstig van een aerodynamische wand-begrensde stroming met een laag Mach-getal. De voornaamste limitatie van een PIV-methode voor algemene toepassingen, is de beperkte huidige tijdsresolutie. Voor snelheden laag

noeg is dit echter geen probleem, omdat de resolutie voldoende is om tijdsopgeloste metingen uit te voeren. Bovendien zou een experimentele meettechniek voordelig kun-nen zijn ten opzichte van numerieke studies voor stromingen over oppervlakken met een laag Mach getal en een hoog Reynolds getal. De numerieke simulatie van de stro-ming in de buurt van vaste wanden is in feite bijzonder complex. In dit proefschrift is een rechthoekige ondiepe open holte aan het oppervlak gekozen als testmodel. Verder zijn een aantal experimenten uitgevoerd, en zijn verschillende methoden en oplossin-gen getest. In hoofdstuk 3 worden de voornaamste details van de experimentele op-stelling omschreven en is Curle’s analogie toegepast in zijn klassieke formulatie. In dit hoofdstuk geven we ook een afschatting coherentie structuren in de dwarsrichting van de stroming, gebaseerd op stereoscopische PIV-metingen. Zowel het berekenen van de druk aan de hand van PIV-data, als de toepassing van Curle’s analogie zijn be´’invloed door een aantal omzekerheden. Daarom is een simplificatie van de hypothese noodza-kelijk. Om deze reden voeren we ook directe metingen uit van de drukfluctuaties aan de wanden van de holte en de geluidsemissie met behulp van een microfoon. Hierdoor is het mogelijk om deze gegevens te vergelijken met geschatte waarden verkregen uit de PIV-data. Het doel hiervan is het valideren van de resultaten en om het geldighei-dsbereik van de methode na te gaan. We vinden dat zowel de hydrodynamische druk, berekend aan de hand van PIV-data en de geluidsemissie verkregen door het toepassen van Curle’s analogie, een frequentiespectrum heeft dat overeenkomt met de microfoon-metingen. In paragraaf 3.7 tonen we aan dat de grotere structuren in de stroming, welke verantwoordelijk zijn voor het grootste deel van het geproduceerde geluid, min of meer twee-dimensionaal zijn en coherent in de dwarsrichting van de stroming. In hoofdstuk 4 staan de details van de implementatie en oplossing van zowel Curle’s analogie als de Theory of Vortex Sound ter discussie. In dit hoofdstuk wordt de aanwezigheid van een niet-compacte wand, waar de holte is gelegen, in acht genomen door het gebruik van het afbeeldingsprincipe in de afleiding van de oplossing. Dit is in tegenstelling tot hoofdstuk 3, waarin de implementatie van het model wordt beschreven. De twee methodes zijn afgeleid onder de veronderstelling van een laag Mach getal, een hoog Reynolds getal en een opnemer ver weg geplaatst. De twee analogieiën doen het goed in het huidige test geval, en geven gelijkwaardige resultaten in zowel de totale intensiteit en de spec-trale verdeling van het geluid. Beide analogieën hebben echter hun sterke en zwakke punten in deze toepassing. De twee oplossingen zijn eigenlijk afgeleid door behoorlijk verschillende benaderingen, en de gebruikte rekenkundige methoden ter oplossing van de vergelijkingen zijn gevoelig voor schillende factoren. De keuze voor een van de twee methodes moet daarom gedaan worden met het oogpunt op de specifieke applicatie. Het gebruik van het afbeeldingsprincipe lijkt cruciaal om adequaat de geluidsemissie voor compacte geometrieiën in grote niet-compacte oppervlakken af te schatten. In hoofdstuk 5 onderzoeken we het effect van drie-dimensionale snelheidsfluctuaties op het uiteindeli-jke resultaat, en bespreken we de resultaten verkregen met tijdsopgeloste tomografische PIV-metingen. Deze metingen geven voldoende snelheidsvectoren in de dwarsrichting van de stroming om berekening van differentiaal grootheden in deze richting mogelijk te maken. De berekening van de bron termen in de Curle’s analogie en de Theory of Vortex Sound is daarom mogelijk. Tevens zijn de details van de nieuwe meetopstelling

Samenvatting xiii

en meettechniek omschreven. De resultaten laten zien dat de stroming inderdaad min of meer twee-dimensionaal is. De brontermen in de analogieën verschillen weinig indien deze twee- of drie-dimensionaal bepaald worden. Vergeleken met de twee-dimensionale metingen, zijn volumetrische PIV-metingen duurder, vereisen een complexere opstelling en de data-analyse kost veel meer tijd. Verder is de kwaliteit van de data minder dan die van PIV-metingen in een vlak. Tegelijkertijd lijken dunne tomografische metingen geen relevante informatie toe te voegen aan de berekening van de geluidsemissie in de experimenten. De experimenten representeren een twee-dimensionale wand-begrensde stroming, waar de voornaamste geluidsbron is bepaald door grote coherente structuren in de dwarsrichting van de stroming.

Chapter 1

Introduction

Abstract

We give here a general introduction to the role of sound in our life and on the problem of unwanted sound, generally referred to as noise. We discuss the objective of the thesis, displaying the framework it falls within, specifying the particular problem under investi-gation and justifying the chosen research methodology. We finally provide an outline of the thesis, briefly describing the content of each of the following chapters.

1.1

Background and motivation

Sound is a constant presence in our life. Most of the times it is something wanted and it serves a purpose, such as communication through speech or entertainment by listening to music. On the other hand, quite often sound is an annoying and unwanted by-product of some other activity necessary to us. This is what we usually referred to as noise.

Noise does not simply determine an unpleasant environment, but it can severely affect human health with problems such as hearing impairment, hypertension and ischemic heart disease, annoyance, sleep disturbance, and decreased school performance. Other effects, such as changes in the immune system and birth defects, have been suggested but the evidence is still limited (Passchier-Vermeer and Passchier, 2000). Exposure to high noise levels can also create stress, increase workplace accident rates, and stimulate aggression and other anti-social behaviors (Kryter, 1994). Noise has a serious impact on nature as well. It can reduce the usable habitat of certain animal species. In the case of endangered species, this may be part of the path to extinction. It is often reported in the news of whales that beached themselves after being exposed to the loud sound of military sonar. Underwater noise also makes whales communicate louder and for a longer time. This would eventually cause the whole ecosystem to ‘speak’ louder (Fristrup et al., 2003). Noise pollution seems to severely affect the mating rituals of many species of birds as well. European robins living in urban environments, for example, are more likely to

sing at night when it is quieter and their message can propagate more clearly (Fuller et al., 2007). Zebra finches seem to become less faithful to their partners when exposed to traffic noise (Milius, 2007). . This behaviors could alter a population’s evolutionary trajectory and lead to profound genetic and evolutionary consequences. It is clear then, that limiting the impact of noise is of vital importance for our well-being and for the environment in general.

The noise pollution problem is addressed in many different ways: by clever urban development planning, for instance, and by providing the heavily exposed buildings with acoustic barriers and insulation. In many countries specific laws exist that regulate the hours during which it is allowed to perform noisy activities, such as house renovations, the disposal of glass in recycling containers, and even the usage of home appliances like washing machines or vacuum cleaners. Legislation also provides a limit on the maximum acceptable noise levels in many contexts like working environments, discotheques and music headphones devices. An important effort in the fight against noise is obviously placed at the source of the problem itself, trying to make our tools and devices quieter. In order to do so, it is essential to understand, and eventually be able to model, the processes through which sound is produced and propagated. Noise production from a mechanical system can be seen as a dissipation process through which the system looses a small portion of its energy. The acoustic dissipation though, is extremely small compared to the total energy of the system and even compared to the other forms of dissipation. The system can be therefore usually described ignoring its acoustic dissipation for most of the others purposes, such as efficiency or losses for example. A description of the system including the generated acoustic field usually requires a much higher level of accuracy, thus increasing the complexity of the study. This also stands for those system involving unsteady fluid motions. Unsteady flows, by themselves or interacting with surfaces and moving bodies, are the source of what is commonly known as aerodynamic sound (to be distinguished from sound due to vibrating structures). The investigation of the aerodynamic sound production mechanisms started fairly recently, in the 1950’s, with the work of Lighthill (1952), in relation with the problem of noise from the exhaust jet of aircrafts engines. At that time, in fact, jet engines were introduced on passenger airplanes, determining a serious problem of noise pollution in the environment close to airports. Nowadays, there is still a strong focus on aeronautical applications. Jet noise is now considerably reduced compared to those early times, but other sound sources in the meanwhile have become significant, such as the rotating blades of the engine fan. The development of helicopters and their extensive use in urban areas also raises concerns about noise from the rotor. There is therefore a strong research interest in developing tools for the study of noise produced by moving solid bodies and wall bounded flows in general. The knowledge acquired by investigating problems of aeronautical nature can also be fruitfully applied to less sophisticated technologies, such as cooling fans in cars or computers, air conditioning system, and even hair-dryers and vacuum cleaners. People ask for low noise levels not only if they live near to an airport, but in every aspect of their lives, even when they are vacuuming their floors! Ground transportation is also a field where there is a significant concern and research on noise of aerodynamic nature. In cars, but even more in high-speed trains, an important source of noise is the presence

1.2. Objectives of the research 3

of singularities such as cavities, steps and spoilers on large flat surfaces. More recently, the development and wide diffusion of wind turbines brought up the interest also on the problem of noise emission from these machines (Hubbard and Shepherd, 1991).

1.2

Objectives of the research

This study was carried out as part of a larger research project involving computational and experimental aspects. Previous PhD students (Moore, 2009; Slot et al., 2009) focused their attention on numerical and experimental studies of noise produced by turbulent disturbances and in particular on jet noise.

The objective of this work is to develop and to employ experimental methods, based on the particle image velocimetry (PIV) measurement technique, for the estimate of the emitted sound from flow field measurements. In particular, the focus will be on the study of the noise that generates because of the unsteady loads on solid bodies immersed in the flow. The production of sound by unsteady loads on solid bodies is extremely efficient even at very low subsonic Mach numbers, in contrast to noise produced by turbulent disturbances such as jet noise (Curle, 1955). Typical applications are therefore in ground transportation as well as underwater applications. The high efficiency at low speed is due to two different factors: the creation or augmentation of flow features that generates noise, such as the unsteady shear layer fluctuations on top of a cavity, the chosen test case for this thesis, and the imposition of boundary inhomogeneities that enhance the conversion of flow energy into acoustic energy, such as the aft wall of the cavity on which act the unsteady drag due to the shear layer fluctuations.

The approach that was investigated, based on time-resolved PIV measurements (Adrian and Westerweel, 2011), provided insight in the sound sources that is not possible to achieve with any other established experimental method. It also permitted the experimen-tal investigation in those situations where other measurements would not be possible, for example, in the presence of a noisy environment or when a suitable anechoic tests facility is unavailable. In addition, it allowed the estimation of the acoustic emission, coupled to the proper acoustic model, in the same fashion as done in hybrid computational ap-proaches. This kind of experimental approach seems to be particularly suited for the study of aerodynamic sound from wall-bounded flows at low Mach number. The main limitation of a PIV-based method is in fact the poor temporal resolution currently avail-able (in the order of 10kHz). However, for low enough velocities (up to about 20 m/s, in some circumstances) this is no longer a concern since the resolution is good enough to perform time resolved measurements. Moreover, an experimental technique might be advantageous with respect to computational studies for low Mach number and high Reynolds number flows with immersed solid bodies. The numerical simulation of the flow in the proximity of solid boundaries is in fact particularly complex. Firstly, because of the uncertainty in the definition of the in-flow and out-flow conditions, and secondly because of the computational cost that can become a limiting factor due to the high Reynolds numbers involved and the necessity to accurately resolve the velocity fluctua-tions near immersed solid bodies of complex geometries.

1.3

Outline of the thesis

In Chapter 2 of this thesis, we give an introduction of the topic followed by an historical overview of the research performed in the field. We give a description of the specific problem under investigation and of the measurement technique used in this thesis. We finally derive the analytical models that are used in the study, along with the assumptions made in their application.

Chapter 3 gives the main details of the experiment, and the results obtained applying Curle’s analogy in its classical formulation. In this chapter, we also give an estimate of the span-wise coherence of the flow, based on stereoscopic PIV measurements.

Chapter 4 we discusses details of the implementation and solution of both Curle’s Analogy and Theory of Vortex Sound. In this chapter, in contrast with what was done in chapter 3, we take into account the non-compact plane wall that includes the cavity. The methods are derived under the same assumptions, and they are compared pointing out the different advantages and disadvantages when applied to experimental PIV data.

Chapter 5 investigates the effect of the three-dimensional velocity fluctuations on the final result. We do so through high-speed tomographic PIV measurements. That allow the measurement of all three instantaneous velocity components, and their spatial derivatives in all three directions, i.e. the full deformation tensor. Details about the new experimental setup and measurement technique are given as well.

Chapter 6 summarizes the conclusions for the thesis as a whole. We also discuss some suggestions where such research could be carried in the future.

Chapter 2

Theoretical Background

Abstract

We introduce material relevant for the work presented in this thesis. Firstly, the basic theory of sound and aeroacoustics is discussed. Next, a historical overview of aeroa-coustic research is given, and the context of the present work is made clear. A detailed description of the problem under investigation is given, along with details of the problem geometry and the expected flow behaviors. Next, a brief description of the measurement technique, particle image velocimetry, is given together with a discussion on the main challenges related to the specific application. Finally, the governing equations of fluid mechanics are introduced followed by the derivation of the acoustic analogies used in this thesis and a discussion of the assumptions made in their application.

2.1

What is aeroacoustics

Acoustics is the science that deals with the study of the generation and propagation of sound waves in gases, liquids and solids. In fluids such as air and water, sound waves propagate as disturbances in the ambient pressure level. While this disturbance is usually small, it is still noticeable to the human ear. Aeroacoustics is a particular branch of acoustics that studies aerodynamic sound, i.e. the sound generated by unsteady fluid motion.

2.1.1

Human perception of sound

The smallest sound level that a person can observe is known as the threshold of hearing. This level is nine orders of magnitude smaller than the ambient pressure. The loudness of the sound is called the sound pressure level (SPL), and is measured on a logarithmic scale in decibels. This is because the perceive loudness of humans changes approximately

logarithmically with the root mean square of the pressure variation. The threshold of hearing is commonly set at:

pre f = 2 × 10−5Pa (2.1)

and is the reference value for the logarithmic scale. The definition of sound pressure level is then:

SPL= 20 log₁₀ prms

pre f



. (2.2)

where Prmsis the root mean square of the measured pressure. On this scale, an increase

of 20 dB corresponds roughly to a doubling in the perceived loudness of sound, although this is quite arbitrary since it differs from individual to individual and it varies with the frequency of the sound. In order to make this scale comprehensible, some typical sounds along with their sound pressure level in decibels are given in table 2.1.

Table 2.1: Noise associated with some common sound sources.

Source of sound Sound Pressure [Pa] SPL [dB] Jet engine at 30 m 6.32 × 102 ₁₅₀ Threshold of pain 6.32 × 101 ₁₃₀ Vuvuzela at 1 m 2× 101 ₁₂₀ Hearing damage 6.32 × 100 ₁₁₀ Jack hammer at 1 m 2× 100 ₁₀₀ Busy road at 10 m 2× 10−1 ₈₀ TV at 1 m 2× 10−2 60 Conversation at 1 m 2× 10−3 ₄₀ Breathing 6.35 × 10−5 10 Threshold of hearing 2× 10−5 ₀

2.1.2

Flow disturbances and sound

Sound can be generated by unsteady fluid motion in different ways. A typical classifi-cation of the sources of aerodynamic sound is done according to their multipole order. Sources associated with unsteady mass addition to the fluid radiate as monopoles. This is typically the case of a vibrating body. Sources related to unsteady forces acting on the fluid radiate as dipoles. An example is the unsteady drag acting on the back wall of the cavity object of this thesis or the nearly periodic lift variations caused by vortex shedding in bluff bodies wakes. Sound sources directly associated with unsteady disturbances in the flow radiate as quadrupoles. This is typically the case of highly turbulent flows.

The generating mechanism of the acoustic perturbation can therefore be very dif-ferent. A common aspect to all is that the acoustic perturbations are in general small

2.2. Historical overview 7

compared to other disturbances present in the flow. These include turbulent disturbances, related to pressure and velocity gradients and entropic disturbances, associated with tem-perature gradients. These flow disturbances convect with the flow and have a little effect away from the flow region. In contrast, acoustic disturbances can efficiently propagate over long distances and animals and human beings are extremely sensitive to such per-turbations. These aspects make the study of aerodynamically generated sound extremely important. At the same time, the weakness of the acoustic perturbation, compared to the other present disturbances in the region where it is generated, makes the problem highly complex to investigate.

2.2

Historical overview

The modern theory of aeroacoustics was pioneered by Lighthill (1952) who firstly put together the fields of acoustics and aerodynamics. Lighthill developed his theory in the attempt to understand the mechanisms of noise generation by jet engines of recently introduced passenger jet aircraft. He rearranged the fluid mechanics governing equations in a wave equation now known as Lighthill acoustic analogy. Lighthill’s acoustic analogy is valid only for the acoustics generated by isolated regions of free turbulence, such as found by a jet in an open location. Later, Curle (1955) extended Lighthill’s analogy to include the presence of solid bodies in the source region. Curle’s equation is usually referred to as Curle’s analogy. Curle’s analogy was generalized by Ffowcs and Hawkings (1969) to allow for the arbitrary motion of the solid bodies in the source region. Lighthill, Curle and Ffowcs Williams-Hawkings derived their analogies choosing the fluid density, and precisely the term

p− p0= c20(ρ − ρ0), (2.3)

as the acoustic variable where p0 pis the pressure, p0is an arbitrarily chosen reference

pressure, c0is the reference speed of sound andρ − ρ0the flow density fluctuation.

Alternative analogies were later derived, that choose different quantities as acoustic variables. Phillip’s (Phillips, 1960) and Lilley’s (Lilley, 1974) analogies were derived choosing the pressure-related field quantity

Π = 1 γln  p p0  (2.4) as the acoustic variable, whereγ = Cp/Cvis the adiabatic constant i.e. the ratio between

the specific heat for constant pressure and the specific heat for constant volume. Phillip and Lilley both wanted to take into account the convection and refraction of the sound by a non-stationary mean flow. To do so, they tried to separate such mean flow-acoustic field interaction effects from the sound generation process itself, by moving parts of the source term from the right hand side of the wave equation into the wave operator on the left hand side, and calculate them therefore as part of the solution.

Powell (1964) reformulated Lighthill’s analogy choosing the total enthalpy

B= Z _{d p} ρ + 1 2u 2 (2.5)

as the acoustic variable. His purpose was to emphasize the role of vorticity in the sound production mechanism by showing it explicitly in the source term of the obtained wave equation. Howe (2003) extended Powell’s analogy to account for solid bodies in the source region, analogously to what was done by Curle in respect to Lighthill’s analogy. Möhring (1978) derived a slightly different analogy that further applies for bodies ex-ecuting any combination of translational and rotational motion. As we can see, many analogies have been derived since the first formulation of Lighthill. However, the diffi-culties related to direct solving the fluid mechanics equations for the acoustic pressure do not simply disappear by rewriting the equation into the different form of a wave equation. In fact, the wave equations of the acoustic analogies cannot be solved exactly. The advan-tages of the formulation become only evident when simplifying assumptions are made. The neglected phenomena must be therefore of secondary importance and the dominant mechanisms must be preserved. It follows that certain analogies may be more suited for different applications. It is in this spirit that Goldstein (2003) recently proposed a model, that he called the generalized acoustic analogy, in which he attempts to combine the classical and the more recent sound emission prediction methods into a unified rational model.

All the aforementioned acoustic analogies were originally derived in the attempt of understanding the sound generation processes and of identifying the sources of sound. For more than twenty years from their first introduction, they were mostly used as a di-mensional tool, or in combination with simplified analytical model of specific flows. It is only recently, with the progress of computational fluid mechanics, that they also started to be widely used in combination with simulated flow fields in the so-called computa-tional aeroacoustic (CAA) hybrid methods for the prediction of the sound emission (see Wang et al. (2006) and Colonius and Lele (2004) for a complete review and analysis of the CAA different methods).

From an experimental point of view, the commonly limited available data did not allow for the use of acoustic analogies in the computation of the sound emission. In fact, experimental investigation commonly focus on the simple microphone measurement of the acoustic emission or on the localization of the sound sources by surface pressure sensors and microphone beam-forming (see Dougherty (2002) for an overview of the method, and Brooks and Humphreys (2006) for recent improvements). These measure-ments are able to locate the sound sources and to follow their temporal evolution but cannot provide any information about the nature of the sound source. In research envi-ronments, the interest is more focused on understanding the processes leading to sound generation. Therefore the microphone measurements in the propagation region are usu-ally performed in combination with flow measurements in the source region (by means of laser doppler anemometry, LDA, hot wire anemometry, HWA, and particle image ve-locimetry, PIV), for two points space-time correlation studies (See the work of Henning, (Henning et al., 2008) and (Henning et al., 2010) for some recent results), or for POD (proper orthogonal decomposition) analyses (See for example the recent studies of Jacob et al. (2005), Kang and Sung (2009) and Murray et al. (2009)). Those studies can provide useful information; their limitation lies in the need of anechoic facilities to be properly performed and on the choice of the flow quantity to be used for the correlation that is not

2.2. Historical overview 9

always well related with the actual acoustic source.

The recent developments of high-speed cameras and lasers have opened the way to use velocity data fields, obtained from time-resolved experimental PIV measurements, in the evaluation of the acoustic analogies sources. This new methodology provides insight in the sound sources that was previously not possible to achieve with the established experimental methods. The technique, in contrast to the experimental measurements de-scribed before, does not rely on actual acoustic measurements. It therefore permits the experimental investigation in those situations where reliable acoustic measurements are not feasible, such as a noisy test environment or when a suitable anechoic tests facility is unavailable. Moreover, since the method permits to identify the sound sources regard-less of the noisiness of the environment, it also makes it easier to identify the acoustic sources in specific components of complex system. Finally, it allows the estimation of the acoustic emission, coupled to the proper acoustic model, in the same fashion as done in hybrid computational approaches. Compared to computational studies, the estimates of the acoustic emission via acoustic analogies applied to PIV data have both advantages and disadvantages. The temporal and spatial resolutions achievable by PIV, as well as the size of the investigated domain, are currently still considerably modest in comparison to what can be achieved with CFD, which limits the range of applications. On the other hand, the results are computed from measured velocity fields rather than from simulated ones. This allows the investigation of those cases for which the numerical simulation of the flow field would be difficult, because of the uncertainty in the definition of the in-flow and out-flow conditions, or would be expensive, such as high Reynolds number flows and wall-bounded flows over complex geometries.

Only recently, researchers started to investigate the potential of the combination of PIV and acoustic analogies. Schram (2003), performed phase-locked PIV measure-ments of a subsonic jet of which the periodicity was acoustically triggered (pseudo-time-resolved measurements) and applied Powell’s analogy on the velocity data fields in order to estimate the acoustic emission due to the vortex rings pairing in the jet. Later on, Haigermoser (2009) presented time-resolved PIV measurements in combination with Curle’s analogy. Despite its innovative character, the work of Haigermoser should be considered as a preliminary study: the experiments were performed in a water flow facil-ity at a low free-stream velocfacil-ity (Re= 7.8 × 103_{); such conditions are not representative}

of those for significant acoustic noise generation. Furthermore, his study did not pro-vide any comparison of the estimated sound emission with direct acoustic measurements. With the work presented in this thesis, and other recent publications and reports (such as (Moore et al., 2011)), (Lorenzoni et al., 2010), (Violato et al., 2010), and (Moore et al., 2010)) a growing interest has been shown in the application of acoustic analogies to time-resolved PIV data demonstrating the potentiality of this new measurement technique.

These measurements seems to be particularly suited for the study of sound emission from wall-bounded flows. The main source of sound in often related to the unsteady loads on the body immersed in the flow. This kind of sound sources is extremely efficient even at low subsonic Mach numbers, unlike sound due to turbulence disturbances within the flow. The limited time resolution of the PIV systems currently available is therefore enough to provide time-resolved measurements in relevant situations, allowing at the

same time for a comparison with direct measured sound. Moreover, this experimental technique might be advantageous with respect to computational studies for low Mach number and high Reynolds number flows with immersed solid bodies. The numerical simulation of the flow in the proximity of solid boundaries is in fact particularly complex. Firstly, because of the difficulties in setting proper boundary conditions and secondly, because of the high numerical cost due to the numerous grid points necessary to properly resolve the flow close to the body.

2.3

Cavity flow

In this thesis, a rectangular cavity in a flat surface is used as test case for the study of the application of PIV to aeroacoustics. The flow that is generated over rectangular cavities can develop in rather different ways, depending on multiple factors such as the cavity geometry, the free stream velocity and the approaching boundary layer, (Ahuja and Mendoza, 1995).

2.3.1

Cavity geometries

From a geometrical point of view, rectangular cavities can be classified into different families: (I) deep cavities (figure 2.1(a)), for which the depth is larger than the length; (II) shallow cavities (figure 2.1(b)), for which the length is a few times larger than the depth and the shear layer passes over the cavity with a recirculation (and eventually secondary recirculation) developing inside the cavity; (III) closed cavities (figure 2.1(c)), for which the length is several times larger than the depth and the shear layer reattach at the bottom of the cavity; (IV) and Helmholtz resonators, (figure 2.1(d)) for which the opening of the cavity is narrower than the cross-sectional area of its volume. Cavities can be also classified in wide cavities, when the width is larger than the length, and narrow cavities in the opposite case.

2.3.2

Cavity modes

Different flow regimes arise in flows over rectangular open cavities. Gharib and Roshko (1987) showed that an important role in the selection of the “mode” in which a particular cavity will operate is played by the boundary layer momentum thickness. They observed the occurrence of three different regimes depending on the ratio of the length of the cav-ity and the boundary layer momentum thickness, L/θ. When this ratio is smaller than a certain value (L/θ < 70 ÷ 80, depending on other parameters) the shear layer that de-velops over the cavity does not organize itself into periodical structures (figure 2.2(a)). This is typically the case for thick turbulent boundary layers. At higher values of L/θ (> 80 ÷ 100) the shear layer that separates at the leading edge of the cavity develops in large-scale coherent span-wise vortices. Gharib and Roshko (1987) named this flow regime “shear layer mode” (figure 2.2(b)). At even higher values of L/θ (> 120) the “wake mode” occurs (figure 2.2(c)). In the “wake mode” the vortical structures become

2.3. Cavity flow 11

(a) Deep cavity (b) Shallow open cavity

(c) Shallow closed cavity (d) Helmholtz resonator

Figure 2.1: Cavity geometries

comparable in size to the cavity depth. The external flow alternatively enters into the cavity, reattaches to the bottom and then ejects out of it. The “wake mode” has been rarely observed in experiments, while it occurs often in numerical simulations where three dimensional perturbations of the incoming flow can be completely avoided and pe-riodic boundary conditions are a common practice in the transverse direction, (Suponit-sky et al., 2005) and (Larchevêque et al., 2007). For the cavity considered in this thesis,

(a) Broadband turbulence (b) Shear layer mode

(c) Wake mode Figure 2.2: Cavity modes

the incoming boundary layer is naturally laminar. The length of the flat plate portion before the cavity was designed in order to obtain a thin boundary layer just before the cavity and therefore to get a cavity operating in the shear layer mode. Some measure-ments were performed to check the effective presence of the desired mode and to see the

effect of a thicker and turbulent incoming boundary layer. The measurements were per-formed with the same set-up described in chapter 3. The boundary layer was measured at the leading edge of the cavity with a hot wire anemometer. We performed experiments with both laminar and turbulent boundary layer conditions at free-stream velocities of 10 m/s, 12 m/s and 15 m/s, giving a Reynolds number based on the cavity length L, between 20,000 and 30,000. Turbulence is induced in the naturally laminar boundary layer by means of a cylindrical tripping wire positioned upstream, (Preston, 1957). To character-ize the boundary layer approaching the cavity a hot wire was translated normal to the wall just before the cavity leading edge. Figure 2.3 shows the measured boundary layer profiles for the 15 m/s case together with the theoretical curves for boundary layers on a flat plate. Table 2.2 summarizes the results for all test cases. The values for the bound-ary layer momentum thickness, θ, were deduced from the boundary layer thickness, δ, assuming a ratioδ/θ = 8 for the laminar boundary layer and δ/θ = 10 for the turbulent one. The measurements show how, with a laminar boundary layer, the L/θ ratio is always above the threshold given by Gharib and Roshko (1987) for self-sustained oscillations, while this is not the case for a turbulent boundary layer. The wall pressure fluctuations

0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 u/U_∞ δ [mm] Power Law Blasius Turbulent Laminar

Figure 2.3: Boundary layer profiles at 15 m/s

were measured with microphones mounted flush with the cavity walls. The microphones were used here as high-sensitivity and high-frequency pressure transducers. The sound emission was also recorded in the far field by microphones positioned approximately one acoustic wavelength away from the cavity. In the experiments with the laminar bound-ary layer both the wall pressure fluctuations and the far field noise measurements show a strong periodicity and the tones occur at the same frequency spectrum (figure 2.4).

2.3. Cavity flow 13

Table 2.2: Boundary layer thickness,δ, and estimated momentum thickness, θ, relative to the cavity length, L, for different testing free-stream velocities, U∞, and approaching boundary layers.

U∞[m/s] Turbulent Laminar δ [mm] L/θ [-] δ [mm] L/θ [-] 10 4.74 63 2.60 92 12 4.30 70 2.30 104 15 3.90 77 2.14 112

With a turbulent boundary layer, the broadband spectrum is comparable to the laminar spectrum but lacking the tonal component. This can be explained by looking at the flow in the two different cases (figure 2.5). PIV measurements, performed together with the

102 103 104 0 10 20 30 40 50 60 70 80 90 100 110 Frequency [Hz] En e rg y sp e ct ra o f w a ll p re ssu re f lu ct u a ti o n s a n d n o ise [ d b ]

Figure 2.4: Power spectra at 15 m/s. Black: turbulent b.l.; Red: laminar b.l.; On top: hydrodynamic pressure at the cavity wall; On the bottom: far field sound.

.

microphone measurements, show two completely different behaviors of the shear layer crossing the cavity. In the laminar boundary layer case (figure 2.5(b)) the shear layer operates in the shear layer mode described earlier in this section. The impingement of the eddies on the aft wall of the cavity causes a periodic pressure fluctuation on the cavity walls and consequentially the generation of noise with a strong tonal character (see figure 2.4). In the turbulent boundary layer case (figure 2.5(a)), the thickness of the boundary layer approaching the cavity is larger and no periodic shedding occurs. As a result, the periodic pressure fluctuation on the wall does not occur, and tonal noise is not generated.

These experiments show the importance of the boundary layer thickness in the selection of the operating mode of the cavity and therefore on the generation of sound.

(a) Turbulent (b) Laminar

Figure 2.5: Out-of-plane vorticity contours for both turbulent and laminar boundary layers approaching the cavity, at a free-stream velocity of 15 m/s. The cavity is 30 mm long, 15 mm deep and 600 mm wide.

2.3.3

Feedback mechanisms, resonance and sound emission

In the shear layer mode, the vortical structures periodically impinge on the aft wall of the cavity, producing pressure fluctuations that can radiate as acoustic waves. At the same time this generates a self-sustaining oscillation mechanism that triggers and enhances the oscillation. This feedback mechanism is both acoustic and hydrodynamic in nature. The acoustic feedback mechanism is due to the acoustic waves generated by the interaction of vortices with the downstream edge, which propagates directly toward the upstream edge of the cavity. The hydrodynamic feedback mechanism consists of a Biot-Savart induction (which is incompressible) from the motion of the vortices in the shear layer to the separation point. Both feedback mechanisms coexist, but which one predominates depends on the relative dimensions of the cavity and the wavelength of the perturbation, and therefore on the flow speed and on the characteristic dimension of the cavity, i.e. the cavity length. At low subsonic Mach numbers, the length of the cavity is in general much smaller than the acoustic wavelength. Therefore, the acoustic perturbation is not able to directly influence the flow behavior in the cavity, and the self-sustaining oscillation mechanism can be considered as purely hydrodynamic. At high subsonic Mach numbers, it is no longer possible to distinguish the hydrodynamic interaction from the acoustic feedback. Rossiter (1964) proposed a semi-empirical formula to estimate the frequency of the shear layer instability:

St= f L

U0

= ₁m− γ

k+ Ma

. (2.6)

Here, St is the Strouhal number, f is the frequency of the oscillation, L the length of the cavity, U0the free-stream velocity, Ma the Mach number, k the ratio between the shear

layer convection velocity and the free-stream velocity, n= 1, 2, 3, ... the mode number, andγ is an empirical constant. The value of γ is usually set to 0.25, to account for the

2.3. Cavity flow 15

time lag between the passage of a vortex and the emission of a sound pulse at the down-stream corner of the cavity. Equation (2.6) was derived for flows in which the feedback mechanism is of acoustic nature and gives accurate predictions for flows with a high sub-sonic Mach number. However, it seems to fail at lower Mach numbers (Ma< 0.2), where the flow can be regarded as effectively incompressible and the feedback mechanism is of hydrodynamic nature ((Howe, 1997b) and (Suponitsky et al., 2005)). Studies at very low Mach number give results that significantly diverge from Rossiter’s prediction and show at the same time a broad scattering of the Strouhal number of the oscillation ( (Tam and Block, 1978), (Heller and Bliss, 1975) and (Suponitsky et al., 2005)). For incom-pressible flows (Ma<< 1), the frequency of the oscillation is governed by the convective velocity of the vortices in the shear layer. The convective velocity depends on the incom-ing boundary layer properties and especially on its momentum thickness. The Strouhal number is commonly defined using the free-stream velocity rather than the convective velocity, which may explain the scattering of the results. The vorticity in the shear layer, triggered by the hydrodynamic feedback mechanism described earlier, organizes itself in an integer number of discrete vortices along the cavity length. The oscillation frequency is then simply given by:

St= f L

Uc

= n, (2.7)

where Ucis the shear layer convective velocity. In other words, the hydrodynamic mode

nis defined by the number of discrete vortices along the cavity length and is proportional to the inverse of the boundary layer momentum thickness.

At low Mach numbers and in the case of a deep cavity, there may occur an indirect acoustic feedback, involving the resonance of the cavity. The resonance occurs when the frequency of the shear layer oscillation matches the frequency of a depth-wise acoustic mode of the cavity,

fD=

c(2n + 1)

4D , (2.8)

where c is the speed of sound, n= 0, 1, 2... the order of the mode, and D the depth of the cavity. For a Helmholtz resonator (figure 2.1(d)) the resonance frequency is given by

f = c

2π r

S

V d, (2.9)

where S is the area of the opening, V is the volume of the cavity and d is the length of the neck.

The cavity considered in this thesis is a shallow, wide and open cavity, with an aspect ratio between the length and depth of L/D = 2 and an aspect ratio between length and width of L/W = 0.05. The cavity operates in the “shear layer mode” for the range of free-stream velocities that is considered and with a naturally laminar incoming boundary layer. The feedback mechanism is of hydrodynamic nature, and the shedding frequency is given by equation (2.7). Hence, no acoustic resonance is expected to occur.

2.4

Particle Image Velocimetry

In this section a short introduction to the particle image velocimetry (PIV) measurement technique is given, followed by a discussion of the main challenges faced in performing such measurements for the purposes of this thesis.

Figure 2.6: A schematic of a typical PIV setup

PIV is an optical measurement technique for the measurement of the instantaneous velocity field in fluids ((Adrian, 1991), (Westerweel, 1993) and (Adrian and Westerweel, 2011)). A schematic of a typical PIV measurement is showed in figure 2.6. The fluid flow is seeded with tracer particles, which are assumed to faithfully follow the flow dy-namics. A thin slice of the flow is illuminated by a laser light sheet. To record the motion of the tracer particles, the laser emits two subsequent pulses. The light scattered from the particles at the time of the pulses is recorded onto two separate image frames by a digital camera. Between the recorded frames the particles have moved in the il-luminated plane by a certain displacement, which is measured by cross-correlation of the two image frames. The cross-correlation is performed considering the particles im-ages in sub-domains (called interrogation windows) of the recordings. One then obtains a single velocity vector per interrogation window. These velocity vectors represent the spatial average of the displacement of the particle in the interrogation window as well as the temporal average over the time separation between the two laser pulses. It is

there-2.4. Particle Image Velocimetry 17

fore important to use small interrogation windows, compatible with the seeding density limitations in the flow and the pixel resolution of the camera, and to keep a separation between pulses as short as possible but sufficiently long to detect the displacement at ev-ery time scale present in the flow. Using the single-camera approach depicted in Figure 2.6, only the components of the velocity in the plane of the light sheet can be measured. Techniques have been developed to reduce the size of the interrogation windows, (West-erweel et al., 1997) and (Scarano, 2002), to decrease the vector spacing by oversampling the frames by considering overlapping of subsequent interrogation windows by 50 and 75 per cent, and to use optimized time separations for different flow regions.

Stereoscopic PIV

Stereoscopic PIV (figure 2.7) is used in chapter 3 to measure the span-wise coherence of the flow. By stereoscopic PIV measurements, it is possible to reconstruct all three veloc-ity components in one illuminated planar view. This is possible by using two cameras looking at the same field of view from two different viewing directions. The reconstruc-tion of the three velocity components relies on the perspective of the displacement of the particles when observed from different directions.

Figure 2.7: Stereoscopic setup.

Tomographic PIV

Tomographic PIV (figure 2.8) is used in chapter 5 to evaluate the effect of ignoring the small three-dimensional fluctuations on the computation of the source terms in the acous-tic analogies. By means of tomographic PIV measurements, it is possible to reconstruct the three velocity components for three-dimensional positions within an illuminated vol-ume. This is possible by using at least three cameras (usually more for redundancy)

looking at the same volumetric field of view from different viewing directions. In tomo-graphic PIV the particle locations in the measurement volume are back-projected from the recorded particle images. Then a three-dimensional correlation in volumetric sub-domains is performed to determine the particle displacements.

Figure 2.8: Tomographic setup.

In the following sections, we discuss the main challenges faced in the PIV measure-ments described in this thesis.

2.4.1

Bias towards integer pixel displacements

In this thesis we used a high-speed PIV system. The cameras and laser allow a frame rate in the order of 10000 Hz (compared to a typical repetition rate of around 10 Hz per couple of frames of a standard PIV camera). This makes it possible to perform the time-resolved velocity measurements needed for the computation of the source terms of the acoustic analogies described in section 2.6. The main drawbacks of such systems, compared to a standard PIV system, are the lower sensitivity to light and the much larger pixel size of the cameras sensor. These factors, together with the small size and the limited scattering properties of the smoke particles used as tracers in this study and the limitations in the available laser power, result in rather small particle-image sizes with low exposure levels (i.e., gray level values) in our experiments. This leads to a bias error towards integer pixel displacements, known as pixel locking, which reduces the accuracy of the measurements. Pixel locking becomes apparent when the particle images have a diameter that is less than 1-2 pixels. A method that is often used to reduce such error is to slightly de-focus the images to obtain larger particle images with respect to the size of the pixel dimensions. For the results presented in this thesis this technique seems to actually lead to even larger errors, probably because of the lower signal to noise ratio for defocused images. The defocusing reduces the signal amplitude (i.e. gray value) of the particle images relative to the sensor dark-image noise level. Therefore, image defocussing was

2.4. Particle Image Velocimetry 19

not used. Figure 2.9 shows the typical image frames (after the normalization described in paragraph 2.4.2) when focusing on the illuminated plane and when slightly de-focusing. Figure 2.10 shows the PDF over 1500 PIV vector fields, of the sub-pixel displacement of the particles, reconstructed with a Gaussian fitting, (Ronneberger et al., 1998). The coefficients Cf and Cd are the ratio between the number of vectors with a fractional

displacement in pixel units of 0.5±0.05 pixels and the number of vectors with a fractional displacement in pixel units of 1±0.05 pixels. This indicates the level of pixel-locking on a scale from 0 to 1, where 0 indicates fully locked vectors and 1 that no pixel-locking is present.

(a) Focused frame (b) De-focused frame

Figure 2.9: Typical focused and de-focused image frames, after normalization

−0.50 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.003 0.006 0.009 0.012 0.015 Sub−pixel displacement [px] F ra ct io n o f to ta l n u mb e r o f ve ct o rs [−] C f=0.6

(a) Focused frame

−0.50 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.003 0.006 0.009 0.012 0.015 Sub−pixel displacement [px] F ra ct io n o f to ta l n u mb e r o f ve ct o rs [−] C d=0.4 (b) De-focused frame Figure 2.10: PDF of sub-pixel displacement for focused and slightly de-focused image frames

2.4.2

Illumination and reflections

Performing PIV near a reflective surface is a challenge. The image quality is easily de-teriorated by light reflections as the particle images can have very low exposure levels.. In order to minimize the surface reflections, the set-up used in this thesis was made in transparent Plexiglas. However, because of the sharp corners present in the geometry,

still significant reflections were present (figure 2.11(a)). The images were therefore nor-malized before being processed. In particular, the background intensity was removed by subtracting the minimum intensity value for each pixel averaged over all recorded frames. The intensity distribution over the whole frame was then normalized by dividing the intensity value by the mean intensity for each pixel. In this step the average value was chosen over the minimum value to avoid the eventuality of a “divide by 0” error. The minimum and the average value for every pixel were calculated for each series of images independently, over the whole series and separately for the first and the second frame. The dynamic range of the images was preserved through the two steps of subtraction and division by re-scaling the intensity range over 16 bits before each of the two steps. Figure 2.11 shows the improvement of the image quality after normalization and figure 2.12 shows a detail, close to the aft wall of the cavity, of the vector fields obtained by PIV on raw and normalized images. The positive effect of normalization is clear, especially for the vectors close to the wall and in the area above the upper aft cavity corner where a strong reflection is visible in the raw images. Red vectors represent vectors that have been detected as spurious by the PIV processing software.

(a) Raw image (b) Normalized image

Figure 2.11: Image normalization

2.4.3

Large dynamic range

The flow under investigation exhibits a large dynamic range in the displacement over a fixed time step. In particular, it is possible to identify different regions with specific characteristics: (I) the free-stream flow, characterized by an almost uniform and large dis-placement in the stream-wise direction; (II) the incoming boundary layer region, where strong gradients are present in the direction normal to the wall; (III) the shear layer, that shows displacements of the same order of magnitude in the stream-wise and normal di-rections; (IV) the region after the cavity near the wall, where the flow has strong gradients in the normal direction, similarly to the incoming boundary layer region, but without be-ing as uniform in the stream-wise direction; and (V) the region inside the cavity, where the displacement is rather small and of the same order of magnitude in every direction.

In chapter 3, the images were interrogated with different setting in each of the men-tioned regions (table 2.3 and figure 2.13), because of the variety of flow behaviors in the

2.5. Fluid mechanics governing equations 21

(a) Vector field from a raw images pair (b) Vector field from a normalized images pair

Figure 2.12: Improvement after normalization

different areas of the investigated flow region. This allowed a tailored and optimal result for each specific flow region minimizing at the same time the processing time.

In chapter 4, on the other hand, there was the need to carefully compute the vorticity field and its time derivative as part of the source term for the vortex sound equation (2.84). The small discontinuities in the velocity field, obtained by merging the separately interrogated regions, lead to local high intensities of the source term _∂t∂ (ω × u). For this reason a different approach was chosen, and the images were processed using the same setting everywhere. In this way, discontinuities at the borders of the different regions were avoided, but at the cost of a longer processing time.

2.5

Fluid mechanics governing equations

Let us consider the equations that describe the motion of a generic fluid. These govern-ing equations are derived by applygovern-ing the conservation of mass, momentum and energy across the boundaries of a fluid volume. The continuity equation, that expresses the mass conservation, states that:

∂ρ

∂t + ∇ · (ρu) = 0. (2.10)

The Navier-Stokes equations, i.e. the conservation of momentum, are: ρDui Dt = − ∂p ∂xi +∂σi j ∂xj + Fi. (2.11)

III II

I

IV

V

Figure 2.13: Velocity vector field obtained using different settings in different field zones.

Table 2.3: PIV interrogation settings as used in chapters 3 and 4. Everywhere was used a 50% overlap of the interrogation windows. ps: preshift, wd: window deformation

Zone Pass 1 Pass 2 Pass 3 Pass 4

Chapter 3

Free flow 32×32, ps 16×16 16×16 none

Incoming boundary layer 64×8 32×4 16×4 16×4

Shear layer 64×64, wd 32×32, wd 16×16, wd 16×16, wd After the cavity 64×32, wd 32×16, wd 16×16, wd 16×16, wd Inside the cavity 32×32, wd 16×16, wd 16×16, wd none

Chapter 4

Everywhere 64×64, wd 32×32, wd 16×16, wd 16×16, wd

For a Newtonian and Stokesian fluid, σi j= 2µ  ei j− 1 3ekkδi j  (2.12) where ei jis: ei j= 1 2  ∂ui ∂xj +∂uj ∂xi  . (2.13)

When the variation of viscosity, µ, is negligible, the Navier-Stokes equations take the form: ρDu Dt = −∇p + µ  ∇2_u+1 3∇ (∇ · u)  + F. (2.14)

2.6. Aeroacoustics governing equations 23

Figure 2.14: Source and listener representation.

In the previous set of equationsρ is the density of the fluid, p is the pressure, u is the velocity vector, µ is the dynamic viscosity of the fluid and F are generic external forces. In flows, like the one considered in this thesis, for which friction and thermal conduction effects are small and thus entropy increases negligibly slightly, we can treat the flow as isoentropic and the conservation of energy reduces to:

∇s ≃ 0. (2.15)

The motion of the flow can be therefore completely determined from a set of equations that includes the continuity equation (2.10), the Navier-Stokes equations (2.14), or the more general form equation (2.11), and the equation of state for an ideal gas:

p= ρRT. (2.16)

Equation (2.16) relates the thermodynamic quantities p, ρ and T . Here R is the gas constant. For an ideal gas the speed of sound c is c=√γRT where γ is the specific heat ratio. For an ideal gas the speed of sound is thus only a function of the temperature and of the composition of the gas and is independent of the density and the pressure. The viscosity of the flow varies with temperature. The ambient value µ0occurs when

the temperature is at its ambient value T0. This dependence is empirically given by the

Sutherland’s Law: µ µ0 = T T0 3₂ 1.4T0 T+ 0.4T0 . (2.17)

2.6

Aeroacoustics governing equations

We first introduce the notation we will use in the reminder of the discussion. We will talk about the source region that is the area where sound is produced. The position of the single source locations is identified by the vector y= (y1, y2, y3) in a Cartesian

direction and 3 the span-wise direction with respect to the mean flow. The origin of the coordinate system, O will usually lie within the source region. The position where we want to compute the emitted sound will be often referred as the listener position and is identified by the vector x= (x1, x2, x3).

2.6.1

Green’s function

We make use of the Green’s function construct to find the integral solution of the acoustic analogies we will derive in section 2.6.2. These are inhomogeneous wave equations and have the general form:

 1 c2 0 ∂2 ∂t2− ∇ 2  q= F(x,t). (2.18)

Free-space Green’s function

The Green’s function for the wave equation (2.18) is the solution G(x, y,t − τ) of the wave equation generated by an impulsive point sourceδ(x − y)δ(t − τ):

 1 c2₀ ∂2 ∂t2− ∇ 2  G_{= δ(x − y)δ(t − τ).} (2.19)

It can be easily verified, (see (Abramowitz and Stegun, 1965)), that the solution of equa-tion (2.19) is: G_{(x, y,t − τ) =} 1 4π|x − y|δ  t_{− τ −}|x − y| c0  . (2.20)

This represent an impulsive, spherical and symmetric wave expanding from the source at y at the speed of sound. The wave amplitude decreases inversely with the distance from the source,|x − y|. The solution in an unbounded fluid of the inhomogeneous wave equation (2.18) can be found by noticing that:

F(x,t) =

Z Z +∞ −∞

F_{(y, τ)δ(x − y)δ(t − τ)dy}3dτ (2.21) and therefore:

q(x,t) =

Z Z +∞ −∞

F_{(y, τ)G(x, y,t − τ)dy}3dτ = 1 4π Z +∞ −∞ Fy_{,t −}|x−y|_c 0  |x − y| dy 3_. (2.22) Equation (2.22) is called a retarded potential. It represents the variable q at the listener position x at the time t as a linear superposition of contribution from the sources located at position y that radiate at an earlier time t− |x − y|/c0, being|x − y|/c0the time that