rodriguez verdugo (2012 phd) - experimental investigation of flow past open and partially covered...

DOTTORATO DI RICERCA IN INGEGNERIA MECCANICA E

INDUSTRIALE

CICLO XXIV

Experimental investigation of flow past open and partially covered

cylindrical cavities

Francisco Rodriguez Verdugo

Tutor: Prof. Roberto CamussiCoordinatore: Prof. Edoardo Bemporad

Thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Rome, Italy, March 2012

Abstract

Flows past wall-mounted cavities are found in a wide variety of applications, including side-branches, organ

pipes, automobile sunroofs, inter-car gaps in trains and aircraft bays. Under certain conditions, flow excited

cavities can generate large pressure fluctuations, undesirable noise and significant structural loads. To

date, most of the studies have been focused on rectangular cavities while little attention has been given to

cylindrical cavities despite their widespread use.

Two different types of cylindrical cavities were experimentally investigated in low speed wind tunnels: an

open mouth cavity and a deep cavity with a small rectangular opening. The measurements included hot wire

anemometry, particle image velocimetry (PIV) and unsteady surface pressure measurements. Additionally,

numerical analysis of the test section/cavity systems were carried out with the finite element program

COMSOL Multiphysics and with a wave expansion method (WEM) code developed by the Trinity College

Dublin.

Important flow features are described by evaluating the pressure measurements conducted in several

positions over the walls of an open mouth cavity, the PIV measurements performed over horizontal planes

inside the cavity and the hot-wire measurements on the shear layer and on the wake of the cavity.

Pressure Fourier spectra evidence the presence of the first three shear layer hydrodynamic modes at

frequencies well predicted by classical formulation for rectangular cavities (Rossiter). When the cavity is

open, the acoustic modes of the test section are found to be excited by the flow but when the cavity is

partially covered, the shear layer hydrodynamic modes are more likely to lock on the natural frequencies of

the cavity. The position of the opening has an influence on the lock-on acoustic modes.

The acoustic energy generated by the shear layer is calculated by applying the vortex sound theory of

Howe: the flow velocity and the vorticity are extracted from the PIV data and the acoustic particle velocity

field from the WEM calculation. The acoustic sources are localised in space and quantified over an acoustic

period providing insight into the sound production of flow-excited partially covered cylindrical cavities.

i

Acknowledgements

Vorrei ringraziare innanzitutto il prof. Roberto Camussi, per avermi accolto nel suo gruppo di ricerca.

Questi quattro anni passati a Roma Tre, prima per la tesi magistrale e dopo per il dottorato, sono stati

molto formativi, non solo dal punto di vista professionale ma anche dal punto di vista personale.

Un particolare ringraziamento va ai miei cari colleghi della sezione di termofluidodinamica e aerodinamica

per i loro aiuti, i loro consigli, e per tutti i momenti passati insieme all’interno del dipartimento e al di fuori.

In ordine alfabetico ringrazio: Giovanni Aloisio, Alessandro Di Marco, Dajana Giulieti, Emanuele Giulietti,

Daniele Grassucci, Silvano Grizzi, Riccardo Moscatelli, Tiziano Pagliaroli, Alessandra Parlato e Francesco

Tomassi.

Ringrazio a tutti gli stagisti e i tesisti che mi hanno circondato durante la mia attivita di ricerca. In

particolare ringrazio in ordine cronologico a Chistophe Perge, Stefano Valerio, Alessandro Guerriero, Federico

Gargano, Gabriele Baiocco e Ludovica Pentene per il loro supporto tecnico in galleria del vento e ad Andrea

Serrani per le sue simulazioni acustiche.

Non mancherei di ringraziare i dottorandi del GRACO per le pause pranzo, per i loro aiuti con LATEX,

per le cene, per i loro consigli durante la mia ricerca di lavoro, per i TrovaRoma, per l’Internazionale, per le

giornate in montagna, per il calcetto e tant’altro. Loro sono: Alessandro Anobile, Paolo Gradassi, Eugenio

Lombardi, Simone Menicucci, Emanuele Piccione.

This research project has been supported by a Marie Curie Early Stage Research Training Fellowship of

the European Community’s Sixth Framework Programme under Contract number MEST CT 2005 020301.

This financial support was greatly appreciated. I thank Prof. Aldo Rona for leading the AeroTraNet project,

for bringing us a nice cavity model back in 2007 and also for the useful acoustic measurement equipment that

he lent us. I would like to express my thanks to all the AeroTraNet fellows for the interesting discussions

that we had during the meetings and the conferences. Many thanks to Marco Grottadaurea for the fructose

collaboration.

Parmi les participants de l’AeroTraNet, je tiens egalement a citer Antoine Guitton et Julien Grilliat avec

qui j’ai partage de magnifiques moments a Roma Tre. Je souhaite leur exprimer mes sinceres remerciements

pour leurs precieux conseils et l’appui qu’ils ont pu me fournir.

I would like to thank my second advisor, Dr. Gareth Bennett for giving me the opportunity to spend

thirteen month in the Trinity College. Dublin was, without doubts, a wonderful experience. Moreover I

iii

would not have been able to complete this work without his guidance and motivation. Many thanks to

Dr. David Stephens who designed the Helmholtz resonator experimental rig that I used. His contribution to

this work has been invaluable.

I also thank Dr. Craig Meskell and Prof. John Fitzpatrick for the interesting discussions about the physical

phenomena studied. Special thanks go to Shane Finnegan whose experimental methodology has stimulated

much of my research. I also want also to acknowledge my Fluids Lab’s colleagues Miguel Pedroche, Ian

Davis and John Mahon. Furthermore I want to give a special thanks to Eoin King, Donal Lynch, Dorota

Skupinska, Emer Walsh and Rory Stoney with who I spent many late evenings in the ‘dungeon’. I will not

forget to thank my lunch-break mates from the Mechanical and Manufacturing Engineering Department:

Paul Ervine, Paul Harris, Karl Brown, Peadar Golden, Emma Brazel, Stuart Murphy, Kevin Kerrigan and

Robert Smyth.

Many thanks to Prof. Marc Jacob for reading this thesis. I have greatly appreciated his meticulous

corrections and suggestions.

Desde luego, mi mas amplio y sincero agradecimiento a mi familia, cuyo apoyo incondicional hizo posible

la culminacion mi formacion universitaria en Europa.

Ringrazio Sandro e Ivana che mi hanno generosamente accolto nella loro famiglia facendomi sentire a

casa.

Vorrei infine ringraziare Ambra, per la sua pazienza e il suo amore e per essermi stata vicina in tutti

questi anni.

Grazie a tutti di cuore.

Table of contents

Abstract i

Acknowledgements iii

Table of contents vii

Introduction 1

1 Background 3

1.1 Open mouth cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Rectangular cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Cylindrical cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Partially covered cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Helmholtz resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.2 Shear layer over the mouth of a Helmholtz resonator . . . . . . . . . . . . . . . . . . . 15

1.3 Noise source characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

I Cylindrical cavity with open mouth 19

2 Experimental set-up 21

2.1 Wind tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Test model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.1 Pitot-static tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Hot wire anemometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.3 Microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.4 Particle image velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.5 Data acquisition card and processing of the pressure signals . . . . . . . . . . . . . . . 26

2.4 Test conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

v

TABLE OF CONTENTS

2.4.1 Flow conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.2 Incoming boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.3 Background pressure fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Measurement matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Acoustic mode calculation 31

3.1 The acoustic modes of an open-closed cylindrical cavity . . . . . . . . . . . . . . . . . . . . . 31

3.2 The acoustic modes in a wind tunnel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 The computational model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 The geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.1 Acoustic modes without the cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.5.2 Acoustic modes with the cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Experimental results 43

4.1 Overall aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.1 Shear layer topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1.2 Wake topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Description of some flow features inside the cavity . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Unsteady response to a grazing flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Pressure response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.2 Nondimensionalization process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.3 Velocity response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Spectral decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.1 Spectral decomposition and analysis on the symmetry plane . . . . . . . . . . . . . . . 52

4.4.2 Analysis on the cavity walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Conclusion 59

II Cylindrical cavity with partially closed mouth 61

6 Experimental set-up 63

6.1 Overview of the experimental rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Design of the experimental rig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.3 Opening details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.4 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4.1 Pitot-static tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.4.2 Microphones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

vi

TABLE OF CONTENTS

6.4.3 Hot wire anemometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4.4 Particle image velocimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.4.5 Data acquisition card . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.5 Phase-averaging technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.6 Boundary layer characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7 Acoustic mode calculation 73

7.1 Analytical solution of the Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7.2 Wave Expansion Method (WEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2.1 Overview of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.3 Helmholtz resonance frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.4 Response of the resonator to an external excitation . . . . . . . . . . . . . . . . . . . . . . . . 80

7.4.1 Acoustic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.4.2 Boundary layer excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8 Experimental results 83

8.1 Response of the resonator to a grazing flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.1.1 Baseline opening: case L45EU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

8.1.2 Strength of lock-on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.1.3 Influence of the location of the opening . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.2 Shear layer dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.2.1 First shear layer mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.2.2 Second shear layer mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.3 Acoustic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.3.1 Computation of the acoustic particle velocity . . . . . . . . . . . . . . . . . . . . . . . 89

8.3.2 Time-averaged acoustic power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

9 Conclusion 99

10 Summary 101

A Acoustic power 103

References 105

List of publications 113

vii

TABLE OF CONTENTS

viii

Introduction

Motivation for the research

Flows past wall-mounted cavities are found in a wide variety of applications, including side-branches, organ

pipes, automobile sunroofs, inter-car gaps in trains and aircraft bays. Under certain conditions, flow excited

cavities can generate large pressure fluctuations, undesirable noise and significant structural loads.

To date, most of the studies have been focused on rectangular cavities while little attention has been

given to cylindrical cavities despite their widespread use. In the aerospace sector, cylindrical cavities are

present as pressure relief valve of the fuel vents (figure 1), circular anti-icing vent holes or cylindrical landing

gear wheel wells, just to cite some examples.

The purpose of this thesis was to extend the existing knowledge in this area through an experimental

investigation. Two different cases were studied: an open mouth cavity and a partially covered cavity. The

first cavity has an aspect ratio of 1.357 which was dictated by the need of reproducing typical geometries

present on commercial aircraft. The second cavity (mouth partially covered) was designed in such a way

that different acoustic resonances can be excited.

Frameworks

Part of the PhD work presented here was performed in the framework of the AeroTraNet project, an

Early Stage research Training (EST) program funded by Marie-Curie Actions. The AeroTraNet project

was launched in 2006 in order to study the unsteady flow in selected airframe cavities of a wide-body civil

transport aircraft. This four-year European initiative brought together the University of Leicester, the Uni-

versita degli Studi Roma Tre, the Politecnico di Torino and the Institut de Mecanique des Fluides de Toulouse

around a common research topic. This successful program brought to an extensive list of publications which

can be found in the AeroTraNet official webpage: http://aerotranet.imft.fr.

The Universita degli Studi Roma Tre encourages PhD students to spend a period of time in a foreign

research center. Part of the experimental results was therefore obtained during a thirteen months stay

at the Trinity College of Dublin (TCD). The School of Engineering at Trinity was founded in 1841 and

is one of the oldest Engineering Schools in the English-speaking world. The Department of Mechanical

1

Introduction

Figure 1: Photograph of a commercial aeroplane. Detail of circular pressure relief valve on the lower surface of the

wing.

and Manufacturing Engineering has been conducting research for many years on modelling and analysis of

flow/structure interactions and vibro-acoustic problems. The results achieved by the author was an invited

research student at TCD were presented in four different international conferences (refer to list of publication

on page 113).

Outline of the thesis

This thesis is organized as follows: the first chapter introduces the state of the art; the two main parts,

each of which contains four chapters, treat the open mouth case and the partially closed mouth case; the

final chapter summarises the main results. Hereafter the organization of the two main parts of the thesis is

described: chapters 2 and 6 detail the experimental facilities and the measurement techniques; in chapters 3

and 7 the acoustic modes are calculated analytically and numerically; the experimental results are given in

chapters 4 and 8; chapters 5 and 9 draw the conclusions of each part.

2

Chapter 1

Background

This chapter is not intended to be an exhaustive literature survey on cavity flow studies: special emphasis

is given to research on deep cylindrical cavities with open or partially covered mouths in low Mach number

flows. For general surveys on cavity flow, see for instance Rockwell & Naudascher (1978, 1979). More

recently, Cattafesta et al. (2003) and Rowley & Williams (2006) have reviewed the studies on cavity flow

with a particular focus on the control of flow-induced resonance.

1.1 Open mouth cavities

1.1.1 Rectangular cavities

Early studies and oscillations classification

Anatol Roshko and Krishnamurty Karamcheti, both from California Institute of Technology, performed the

first mayor studies on rectangular cavities in the 1950’s. Roshko (1955) reported the pressure distribution

on the walls of cavities tested in a low speed wind tunnel. Karamcheti (1955, 1956) studied the acoustic

radiation of a rectangular cavity in a transonic wind tunnel by means of schlieren observation, interferometry

and hot-wire anemometry. These pioneer researchers are still largely cited nowadays and have inspired many

studies on flow-excited cavities over the years.

Rockwell & Naudascher (1978) divided the shear layer driven cavity oscillations into three categories:

fluid-elastic, fluid-dynamic, and fluid-resonant. The primary condition for fluid-elastic interactions to exist

is the structural elasticity of the cavity walls. Fluid-dynamic oscillations are triggered by the interaction

between the upstream edge of the cavity and the pressure wave generated on the cavity’s downstream edge

by the impingement of the shear layer. A fluid-resonant interaction may occur if the acoustic wavelength is

of the same order of magnitude than the dimensions of the cavity. The last two cavity oscillation categories

are further analyzed in the next two sections.

3

1.Background

Fluid-dynamic oscillations

The fluid-dynamic oscillations, also called self-sustained cavity oscillations or shear layer hydrodynamic

modes are the cavity oscillations in an acoustic-free system. Figure 1.1 shows the essential features involved

in the feedback mechanism which can be summarized in the following way: creation of an instability at the

upstream edge which is amplified as it is convected downstream until it interacts with the downstream edge

of the cavity; the impingement generates an acoustic perturbation which propagates upstream and triggers

the generation of another instability at the upstream edge of the cavity.

Many studies have been conducted in order to predict of the frequencies of the shear layer modes over

a rectangular cavity. Rossiter (1964) proposed a semi-empirical formula based on the vortex shedding

phenomenon:

St =f LcharU

' n− αM + 1

κ

(1.1)

where n is the order of the shear layer mode, Lchar is the characteristic length of the shear layer,

α describes the phase delay between the hydrodynamic forcing and the acoustic feedback and κ is ratio

between the convection velocity in the shear layer and the free stream velocity. This formula correctly fitted

Rossiter (1964)’s experimental data which were obtained with pressure transducers and flow visualisation

(shadowgraph) for Mach numbers between 0.3 and 1.2.

Even if many different values for the convection speed coefficient κ can be found in the literature, the

original value proposed by Rossiter (κ = 0.57) is often used. Chatellier et al. (2004) and El Hassan et al.

(2007) argued that there is no need to consider a phase delay when the convection speed is much lower than

the speed of sound and therefore α = 0. Usually the selected characteristic length Lchar is the length of the

cavity in the streamwise direction.

Over the years, equation 1.1 has been subjected to small changes introduced after analytical developments:

see for instance Bilanin & Covert (1973), Heller & Bliss (1975), Block (1976) and Howe (1997).

Flow-acoustic coupling

The shear layer hydrodynamic modes can excite different types of acoustic modes. Plumblee et al. (1962)

showed that for shallow rectangular cavities the predominant excited acoustic mode is the lengthwise mode

whereas for cavities of aspect ratio higher than unity the depth acoustic mode is the one excited. Similar

flow-acoustic coupling mechanisms exist in the case of coaxial side branches (Arthus & Ziada (2009); Dequand

et al. (2003); Oshkai & Yan (2008); Ziada & Shine (1999)) or in the case of axisymmetric internal cavities

mounted on pipes (Aly & Ziada (2010)).

Ziada et al. (2003) noticed that when a shallow rectangular cavity is mounted in a closed test section

wind tunnel a flow-acoustic coupling can occur between the shear layer modes and the acoustic modes of

the cavity-tunnel. Alvarez & Kerschen (2005) analytically evaluated the influence of the confinement on the

acoustic resonances of a two-dimensional cavity. In Kerschen & Cain (2008) a good agreement was found

4

1.Background

Figure 1.1: Schematic diagram of cavity flow.

between the predicted and the experimental trapped modes. In order to avoid the confinement effects, Yang

et al. (2009) studied a deep rectangular cavity mounted at the outlet of a duct.

1.1.2 Cylindrical cavities

Mean flow

As opposed to rectangular cavities for which the flow can be considered two-dimensional when the spanwise

length is large compared to the streamwise length, the flow in a cylindrical cavity is always fully three-

dimensional. The organization of the mean flow depends exclusively on the aspect ratio H/D of the cavity,

where H and D denote respectively the depth and the diameter. Gaudet & Winter (1973), Hiwada et al.

(1983) and Dybenko & Savory (2008) showed that the flow is asymmetric with respect to the central stream-

wise plane for aspect ratios between 0.2 and 0.8. Through long pressure measurements, Hiwada et al. (1983)

identified two different dynamics: the flow can either flap for H/D = 0.2 − 0.4 or switch orientation for

H/D = 0.4− 0.7. For aspect ratios such as H/D < 0.2 or 0.8 < H/D, the flow inside the cavity was found

to be stable and symmetric.

The parametric study of Gaudet & Winter (1973) includes a set of oil-flow visualisations at the walls of

9 different cavities (H/D between 0.04 and 1.34). The employed experimental technique revealed impor-

tant information about the flow near the walls. Gaudet & Winter (1973) presented such information with

streamline pattern drawing. However, as they mentioned, “a certain amount of imagination has been used

in drawing the streamline patterns”. An example of these results is shown in figure 1.3: a strong asymmetric

case (H/D = 0.47) as well as a symmetric one (H/D = 1.07) are presented.

Haigermoser et al. (2009) investigated a cylindrical cavity of aspect ratio H/D = 0.5 with stereo and

tomographic particle image velocimetry achieving the first detailed description of the three-dimensional

macroscopic organisation of the flow inside a cylindrical cavity. Chicheportiche & Gloerfelt (2010), Desvi-

5

1.Background

Figure 1.2: Schematic of a cylindrical cavity

gne et al. (2010), Marsden et al. (2010) and Mincu et al. (2009), within a Fondation de Recherche pour

l’Aeronautique & l’Espace (FRAE) project called AEROCAV, studied numerically a cylindrical cavity (H

= D = 10 cm). Their results were compared to experiments performed in two different wind tunnels: the

Ecole Centrale de Lyon’s anechoic wind tunnel and the ONERA’s F2 closed section wind tunnel (experi-

mental details can be found in Marsden et al. (2008) and Mincu et al. (2009)). Even if the main objective of

the AEROCAV project was the investigation of the flow unsteadiness and the noise generated by cylindrical

cavities, some important results on the mean flow were also obtained. Some results are reported in figure 1.4.

Grottadaurea (2009) simulated the flow over a deep cylindrical cavity (H/D = 1.4) with a Detached Eddy

Simulation (DES). One of the important results of this study, from the aerodynamic point of view, is the

analysis of the wake behind the cavity. Specific features are described as for example the counter-rotating

convective eddies generated by the cavity downstream edge. Figure 1.5 gives an example of the results

obtained.

Shear layer hydrodynamic mode

An important question for the aeroacoustic viewpoint is if Rossiter’s formula can be used to estimate the

frequencies of the shear layer modes on cylindrical cavities. Bruggeman et al. (1991) assumed that circular

side branches can be treated as rectangular side branches for the prediction of the shear layer modes as long

as an effective length Weff is used:

Weff =πD

4(1.2)

This effective length represents the streamwise dimension of a rectangular opening with the same surface

area and the same spanwise dimension (D) as the original circular opening. If the upstream edge of the side

branch is not sharp, an additional term should be added to Weff (see Bruggeman et al. (1991) or Tonon

et al. (2011)).

Czech et al. (2006) recently studied an array of circular vent holes in a wind tunnel. In order to accom-

modate Rossiter’s formula, another expression for the effective length, based on an equivalent streamwise

6

1.Background

(a) (b)

(c) (d)

(e)

Figure 1.3: Oil-film flow visualisation and streamline patterns of a cylindrical cavity of aspect ratio H/D = 0.47 (a,

b) and H/D = 1.07 (c, d), Gaudet & Winter (1973). Legend (e).

7

1.Background

(a) (b) (c)

Figure 1.4: Numerical results: (a) Non-dimensionalized mean streamwise U velocity profiles, in black lines, and vector

plot of in-plane velocity (U,W ) as grey arrows in the middle spanwise plane of the cavity; (b) Non-dimensionalized

mean cross-stream V velocity profiles and vector of in-plane velocity (V,W ) in the middle streamwise plane of the

cavity. Free-stream flow velocities of 70 m/s, Marsden et al. (2010). (c) Flow dynamics using Q criterion applied to

the mean flow, iso-surfaces of Q = 0.25(U∞/D)2, Mincu (2010)

Figure 1.5: Grottadaurea numerical simulation of a H/D = 1.375 cylindrical cavity. ReD = 34800 and M∞ = 0.235.

(a) View of the cavity with streamlines over streamwise planes. The mean velocity contours are also plotted over to

the walls. A color-bar range from blue to red is used (0.1 < u/u∞ < 0.5). Image taken from Rodrıguez Verdugo

et al. (2010)

8

1.Background

length for a square vent hole of the same area, has been introduced:

Leff =

√πD

2(1.3)

Figure 1.6 shows the equivalent rectangular openings according to Bruggeman et al. (1991) and Czech

et al. (2006).

Figure 1.6: Schematic representing the original circular opening and the equivalent rectangular opening according

to Bruggeman et al. (1991) and Czech et al. (2006).

Mery et al. (2009) found a better agreement with Block (1976) ’s formula in the prediction of the shear

layer modes over a cylindrical cavity:

St =n

1kR

+M(1 + 0.514L/D )

(1.4)

where kR is the real part of the wave number of the disturbance travelling downstream. This formula

includes the effect of the bottom-reflected acoustic wave originated at the downstream edge.

Acoustic modes of an open mouth cylindrical cavity

When studying the acoustic resonances of a cavity without mean flow, it is fundamental to distinguish the

open and the closed mouth cases: the boundary conditions influence indeed drastically the acoustic modes.

The presence of a mean flow, especially a shear layer over the mouth of the cavity, is believed to change

the characteristics (shape and frequency) of the modes. In Rona (2007), an extensive discussion about

which boundary condition should be imposed at the opening of a cavity when solving the acoustic eigenvalue

problem for low Mach number flows is given. Rona (2007) chose a simple acoustic reflecting boundary

condition for the cavity open end while admitting that this represents a “strong hypothesis” in his model.

In the next section, important experimental results will show that the acoustic modes of an open mouth

cylindrical cavity excited by a low Mach number flow are better estimated when a no reflection condition

is imposed at the open end. Therefore, in the following the acoustic modes of a cylindrical cavity with an

open end condition are described.

9

1.Background

The first longitudinal mode of a deep cylindrical cavity, sometimes referred as the depth acoustic mode or

as the quarter-wavelength, is the lowest tone which can resonate in the cavity. Euler and Lagrange assumed

the pressure at the mouth of an open pipe to be zero and therefore the wave-length of the first longitudinal

mode to be exactly 4H. Rayleigh (1894) introduced a correction for the open end of a tube with an infinite

flange. The general expression for the frequency of the first longitudinal mode of a H-long open-closed pipe

is:

fc =c

4(H + αR)(1.5)

where c is the speed of sound and α is the end correction factor and R = D/2 is the radius of the pipe.

The original value calculated by Rayleigh (1894) for an infinite flange tube is α = 0.8242. Nomura et al.

(1960) studied the sound radiation from a flanged circular pipe with Weber-Schafheitlin type integrals and

Jacobi’s polynomials. They reported how the end correction varies with the frequency of the incident plane

wave and found limka→0

α = 0.8217 which is close to the coefficient calculated by Rayleigh. Unfortunately,

Nomura et al. (1960) did not propose a simple expression fitting their results for α. Norris & Sheng (1989)

studied the same problem with Green’s functions and proposed a rational function to approximate their

numerical solution:

α ≈ 0.82159− 0.49(kR)2

1− 0.46(kR)3(1.6)

where k is the wavenumber of the incident acoustic wave. According to Norris & Sheng (1989), their own

results graphically match the Nomura et al. (1960) ones. Another function that approximates correctly the

numerical results of Nomura et al. (1960) and that removes the singularity of equation 1.6 is (Dalmont et al.

(2001)):

α ≈ 0.8216

[1 +

(0.77kR)2

1 + 0.77kR

]−1(1.7)

Figure 1.7: Correction coefficient α as a function of kR predicted by equation 1.7.

10

1.Background

The frequencies of higher order longitudinal modes can be predicted by the generalization of equation 1.5.

fc =c(2q + 1)

4(H + αR)(1.8)

where q ∈ N and 2q+1 is the order of the mode. This expression guarantees a pressure node at a distance

of αR from the opening.

The longitudinal modes are not the only resonant acoustic modes of a cylindrical cavity: there are also

the azimuthal and the radial modes and all the combinations of these three types of modes. Marsden et al.

(2008), while testing a cylindrical cavity in an anechoic wind tunnel, found a peak on the pressure spectra

whose frequency was much higher than the frequencies of the quarter wave length: “The peak at 2160 Hz

[...] is more strongly visible on the cavity floor, and it exhibits a strong symmetry with respect to the flow

direction, almost disappearing for φ = ±π. This peak is not a multiple of one of the two main frequencies

present, nor of the cavity resonant frequency. Its origin is at present not clearly identified.”. Mincu et al.

(2009) performed a numerical simulation of a cylindrical cavity with exactly the same dimensions. They

demonstrated that the 2160 Hz peak reported by Marsden et al. (2008) corresponds in fact to the first

azimuthal mode. Furthermore, they described four other modes present in the pressure spectra. The shape

of the modes are given in figure 1.8.

Figure 1.8: Shape of the acoustic modes (pressure amplitude) inside a cylindrical cavity at the half depth plane

given by Large Eddy Simulation (Mincu (2010)). For the coordinate system see figure 1.4.

The acoustic modes can be calculated analytically from the Helmholtz equation which is an eigenvalue

problem: the eigenvectors represent the pressure distribution of the acoustic modes of the cavity and the

eigenvalues represent the square of the natural frequencies. In section 3.1, the acoustic modes of an open

mouth cylindrical cavity are calculated analytically whereas in section 7.1 the closed case is treated.

11

1.Background

Acoustic-hydrodynamic coupling

Similarly to rectangular cavities, the hydrodynamic modes of a shear layer spanning a cylindrical cavity can

be amplified when their frequencies match the acoustic modes of the cavity. Parthasarathy et al. (1985)

performed a series of experiments on deep cylindrical cavities at low Mach numbers (M = 0.12 - 0.24). They

reported a strong whistle when the quarter wave length matches the first shear layer mode. Marsden et al.

(2008) found a strong coupling mechanism between the first two shear layer modes and the quarter wave

length of a cylindrical cavity tested in an anechoic wind tunnel. Dybenko & Savory (2008) did not find a

fluid acoustic coupling for any of their cylindrical cavities (H/D = 0.20, 0.47 and 0.70) tested at 27 m/s

because, as it is mentioned in the paper, the first shear layer mode was expected at 145.5 Hz and the quarter

wave length at 1156 Hz for the deepest cavity studied.

1.2 Partially covered cavities

Cavities whose openings are partially covered, commonly called Helmholtz resonators, are found not only in

the transportation industry (window buffeting in cars/trains, aircraft landing gear wheel well) but also in

musical instruments (jug bands) or in duct applications (side branches).

When these systems are exposed to a grazing flow, periodic pressure fluctuations can be generated inside

the cavity. However, as noticed by Rayleigh (1894) in §310, Panton & Miller (1975a) and De Metz et al.

(1977), two different oscillatory mechanisms can be excited: either periodic compressions of the volume of

fluid inside the cavity, commonly called the Helmholtz resonance, or the excitation of a standing acoustic wave

in the resonator. Both mechanisms fall into the Rockwell & Naudascher (1978) category of fluid-resonant

oscillations.

1.2.1 Helmholtz resonance

The ‘gravest mode of vibration’ of a cavity communicating with the exterior space though an orifice is the

Helmholtz resonance. This acoustic mode can be excited either by acoustic pressure fluctuations or by fluid-

dynamic pressure fluctuations. The response of a Helmholtz resonator acoustically excited is presented first,

followed by the examination of the flow excited case.

The classical Helmholtz theory

The easiest way to model the Helmholtz resonance is with the mechanical analogy of a spring-mass system.

The air in the opening acts as lumped mass and the air volume inside the cavity acts as spring. If an external

perturbation (incident acoustic waves for example) displace the slug of air in the neck, the volume inside

the cavity undergoes an adiabatic transformation (compression or dilatation). By applying Newton’s second

law, one can find the second order differential equation governing the displacement of the slug of air in the

neck. The natural frequency of vibration of the system, Helmholtz resonance, takes the flowing expression:

12

1.Background

(a) (b)

Figure 1.9: Two different types of Helmholtz resonator: (a) thin opening cavity and (b) long neck resonator

fHR =c

2π

√S

V leq(1.9)

where c is the sound speed, V the volume of the resonator, S is the cross-section area of the orifice, leq

is the equivalent neck length. For the extended development, refer any textbook of acoustics, for example

Rienstra & Hirschberg (2011). This generic formula can be applied to different types of Helmholtz resonators

by correctly estimating the equivalent neck length which is the height of the slug of air. Figure 1.9 gives a

schematic drawing of two types of cavities: the slug of air is represented with a dashed line.

Conceptually, leq is the actual length of neck plus an interior and an exterior end correction (∆lint and

∆lext):

leq = l + ∆lint + ∆lext (1.10)

Different expressions for the end corrections have been developed depending on the shape of the aperture,

on the flange and on the dimensions of cavity. For circular openings, the Rayleigh end correction formerly

introduced in equation 1.5 is often used for the exterior part (∆lext = 0.8242a). In order to apply the

Rayleigh end correction for non-circular openings, an effective radius aeff = 1.06S3/4U−1/2, where U is the

perimeter of the opening, has proven to give reasonable agreement with experiments (Crighton et al. (1994)).

Ingard (1953) proposed different expressions for the interior end correction, based on the shape of both the

cavity and the opening and on the position of the opening. Interior and exterior end corrections are often

grouped in a single parameter 2∆l.

Influence of the geometry

Some researchers have shown that the classical Helmholtz theory presented so far does not correctly predict

the frequency of the Helmholtz resonance because it does not take into account the geometrical characteristics

13

1.Background

of the cavity. Selamet et al. (1995), for example, investigated the effect on the resonance frequency of changing

the aspect ratio of a cylindrical cavity while maintaining its volume. Their computational and experimental

results showed that the resonance frequency depends on the cavity’s aspect ratio: the deeper the cavity is,

the lower the frequency is.

Panton & Miller (1975b) proposed a transcendental equation for the resonant wavenumbers k of a cylin-

drical Helmholtz resonator based on one-dimensional wave propagation inside the cavity:

leq A

H SkH = cot(kH) (1.11)

where A is the cavity cross section area. In Panton & Miller (1975b)’s development, the slug of air in

the neck was treated as a lumped mass. A more sophisticated analysis was performed by Tang & Sirignano

(1973) who assumed one-dimensional wave propagation not only inside the cavity but also on its neck. The

first-order approximation gives:

tan(kl) tan(kH) = S/A (1.12)

This expression yields to equation 1.11 when the cavity has a short neck (kl � 1) because in this

case tan(kl) ≈ kl. Furthermore, when the wavelength λ is much larger than the dimensions of the cavity

(kH � 1), equation 1.11 reduces to the classical formula for the Helmholtz resonance (equation 1.9) because

cot(kH) ≈ 1/kH. It is interesting to note that Selamet et al. (1995), through a different conceptual develop-

ment, found equation 1.12 as well. The trend of the Helmholtz resonance frequency observed experimentally

is correctly predicted by equation 1.12 even if some quantitative discrepancies have been noticed for small

aspect ratios H/D (Selamet et al. (1995)). Tang & Sirignano (1973) recommended the addition of the end

corrections to the length of the orifice and the inclusion of the flow contraction effects to the model for small

aspect ratio cavities in order to improve the prediction of the resonant modes.

The model of Panton & Miller (1975b) and Tang & Sirignano (1973) does not only predict the frequency

of the Helmholtz resonance but also the longitudinal modes of the cavity. In 1.10 a graphical resolution of

equations 1.11 and 1.12 is given.

Panton & Miller (1975b) also proposed an explicit expression for the Helmholtz resonator that improves

the classical Helmholtz resonance formula (equation 1.9) by retaining two terms in equation 1.11:

fHR =c

2π

√S

V leq +H2S/3(1.13)

This formula has been used by different authors to predict the Helmholtz resonance (Panton (1990) or

Mechel (2002)).

14

1.Background

(a) (b)

Figure 1.10: Graphical resolution of two different transcendental equations for the resonant wavenumbers. (a) Panton

& Miller (1975b) : blue line cot(kH) and red lineleq A

H SkH. (b) Tang & Sirignano (1973): black line cot(lkH/H)S/A

and magenta line tan(kH)

Flow effect on the resonance frequency

Several researchers have reported an increase of the Helmholtz resonance frequency when the cavity is excited

by a flow. Anderson (1977), while testing side branch Helmholtz resonators of different sizes in a circular

duct with fully developed turbulent flow, reported an increase of the Helmholtz resonance frequency for

flow velocities higher than 30 m/s. Their 53.1 mm long cavity underwent a 100 Hz shift of the resonance

frequency from 250 Hz at 0 m/s to 350 Hz at a flow velocity of 80.7 m/s. For flow velocities lower than 30

m/s the resonant frequency did not change. Phillips (1968) reported an increase of fHR between 20 m/s and

80 m/s for a partially covered cavity tested in a wind tunnel. The experiments of Panton & Miller (1975a)

on the fuselage of a glider with a free-stream speed of 30 m/s also indicated an increase in the frequency of

the fundamental acoustic mode of different Helmholtz resonators. Their hot wire measurements assessed a

fully developed boundary layer in the orifice region.

Zoccola (2000), reported in his PhD thesis a decrease of the resonance’s frequency of flow excited

Helmholtz resonators. He tested three different cavities and measured the frequency fHR for a purely acous-

tic excitation (0 m/s) and for a 6.9 m/s flow: 346 Hz (333 Hz), 636 (554 Hz) and 436 Hz (416 Hz). Zoccola

(2000)’s experiments were done at a flow speed inferior than the velocity range analysed by Anderson (1977),

Phillips (1968) and Panton & Miller (1975a). Unfortunately Zoccola (2000) did not give the characteristics

of his boundary layer.

1.2.2 Shear layer over the mouth of a Helmholtz resonator

Of particular interest is the understanding of the shear layer organization when the cavity is subject to a fluid-

resonant oscillations. Elder (1978) experimentally investigated a deep cylindrical cavity with a rectangular

opening. He characterized the shear layer with a hot wire probe sampled simultaneously with a microphone

15

1.Background

at the bottom wall the cavity: after phase-averaging the velocity, shear layer profiles at different phases

during an acoustic cycle were presented. Elder (1978) also introduced the concept of the “interface wave”

surface which is the locus of the inflection points of the shear layer profiles along the opening. He proposed

a model to predict interface wave surface based on an initial displacement wave generated by the traverse

acoustic particle motion. Good agreement was found between the experiments and the proposed model.

Nelson et al. (1981) used two-component Laser Doppler Anemometry (LDA) and a flow visualisation

technique (stroboscopic light) to characterize the shear layer spanning the rectangular opening of a cuboidal

Helmholtz resonator. The free stream velocity chosen for the experiments was 22 m/s which corresponds to

the first shear layer mode strongly coupled with the Helmholtz resonance of the cavity. At the upstream lip

they observed the shedding of a single vortex per acoustic cycle.

Ma et al. (2009) studied the shear layer over a rectangular Helmholtz resonator with PIV. They analysed

the same resonant case as Nelson et al. (1981): the first shear layer mode coupled with the Helmholtz

resonance. However Ma et al. (2009) reported phase-averaged contours of the spanwise vorticity for three

different velocities corresponding to one strongly resonant case and two weakly resonant cases. They found

that the shear layer has a sheet-like character in the region closer to upstream edge and tends to roll up into

a single discrete vortex in the downstream portion of the opening (figure 1.11).

Figure 1.11: Contours of phase averaged vorticity for three different flow speeds. In all the cases the predominant

acoustic mode is the Helmholtz resonance and the amplified shear layer instability is the first hydrodynamic mode

(Ma et al. (2009)).

16

1.Background

1.3 Noise source characterisation

Aeroacoustic source characterization has been facilitated by the development of full field velocity mea-

surement techniques, as indicated by the number of publications in the last decade: Geveci et al. (2003),

Haigermoser (2009), Velikorodny et al. (2010) and Finnegan et al. (2010) just to cite a few. Morris (2010)

has recently provided a review on the use of PIV to examine how shear layer instabilities and turbulence lead

to radiated sound. The common strategy is to apply an acoustic analogy to the experimental data, either

the Curle (1955) acoustic analogy or the vortex sound theory introduced by Powell (1964) and extended

by Howe (1975), in order to identify the spatial distribution of the sound sources. Both analogies were

applied by Koschatzky et al. (2010) to estimate the cavity sound emission: the overall sound pressure level

was correctly predicted by the two methods even if the vortex sound theory appears to predict better the

amplitude of the tonal component.

Howe (1975, 1980) estimated that with the low Mach number, inviscid, constant entropy approximation,

the total flow velocity can be decomposed into an incompressible vorticity-bearing velocity and an irrotational

acoustic velocity. The generation of acoustic power by the vortical field can be calculated by the following

formula:

Π = −ρ0∫V

(~w × ~v) · ~uacoust dV (1.14)

where ρ0 is the fluid density, ~v and ~w are the fluid velocity and vorticity, and ~uacoust is the acoustic

particle velocity. The sign of Π determines if vorticity acts as a source or sink for the acoustics. For the

details on the derivation of equation 1.14 see Appendix A.

Interesting results have been found by applying equation 1.14 to strongly resonant cases: Velikorodny

et al. (2010), for example, estimated the distribution of the acoustic sources and sinks on a duct with coaxial

side branches (figure 1.12).

17

1.Background

Figure 1.12: Patterns of time-averaged acoustic power on a duct with coaxial side branches corresponding to the

second hydrodynamic oscillation mode (Velikorodny et al. (2010)).

18

Part I

Cylindrical cavity with open mouth

19

Chapter 2

Experimental set-up

2.1 Wind tunnel

The experimental investigation was conducted in a closed circuit low speed wind tunnel designed by the

Mechanical and Industrial Engineering Department (DIMI) of Roma Tre University. The facility is located

in the Italian National Agency for New Technology Energy and Environment (ENEA) research center of

Casaccia, 28 km from Rome. The closed test section is 2.49 m long (Lx) and has a 0.89 × 1.16 m2 cross

section (Ly×Lz). Other important dimensions of the wind tunnel are given in figure 2.1. The fan is able to

generate a flow ranging from 0 to 90 m/s in the centreline of the test section with a relative turbulence level

of 0.1% at a velocity of 40 m/s. Further details about the wind tunnel properties can be found in Camussi

et al. (2006a).

2.2 Test model

The test model was designed within the framework of the AeroTraNet project and manufactured by the Uni-

versity of Leicester. A Perspex cylindrical pipe with an interior diameter (D) of 210 mm was mounted flush

to the bottom wall of the test section, 1780 mm downstream the end of the convergent section (figure 2.2).

A flat Perspex disk sealed the cylinder from underneath, creating a 285 mm deep (H) cavity (figure 2.3).

These dimensions lead to a cavity aspect ratio of 1.357 . According to the above mentioned literature studies

(section 1.1.2), the selected aspect ratio should guarantee the symmetry of the flow with respect to the

central streamwise wall-normal plane.

21

2.Experimental set-up

Figure 2.1: Schematic drawing of the ENEA Casaccia wind tunnel (top view).

(a) (b)

Figure 2.2: Schematic drawing of the test section with the cylindrical cavity studied. Dimensions of the test section

are: Lx = 2.49 m, Ly = 0.89 m, Lz = 1.16 m. An azimuthal angle ϕ has been introduced to follow the pressure at

the walls of the cavity.

22


Figure 2.3: Photography of the cavity before the preliminary characterization campaign. During the experiments,

one microphone was mounted in one of the orifices. See figure 2.5 for further details on the set-up of the microphones.

2.3 Instrumentation

2.3.1 Pitot-static tube

The flow velocity in the test section was monitored with a Pitot tube connected to a Kavlico pressure

transducer model P592. The probe was introduced in the wind tunnel through the “breather”, a small slot

around the perimeter at the downstream end of the test section. The purpose of this vent is to keep the test

section close to atmospheric pressure and to isolate it from vibration that could otherwise be transmitted

through the diffuser. The calibration of the pressure transducer was done with a pressure pump and a U-tube

manometer.

2.3.2 Hot wire anemometry

Velocity in the shear layer and in the wake of the cavity was measured with a 55P11 single component Dantec

probe connected to a constant temperature hot-wire anemometer (A.A. Lab System AN-1003). To reach the

desired positions on a given yz-plane, the probe was mounted on a two axis traverse system equipped with

stepping motors (Rexroth Compact Module CKK). The calibration of the probe was done with the Pitot

tube.

Because the hot wire anemometry is an intrusive technique, a verification that the presence of the hot-

wire probe does not affect the flow significantly was performed. A microphone in the downstream cavity

23


wall was used as controller. A hot-wire probe was introduced in the test section and placed at 16 different

locations above the cavity. Figure 2.4 presents two pressure spectra corresponding to two different cases:

the hot-wire probe in the shear layer (y/H = 0) and outside the shear layer, far above the cavity (y/H =

0.35). Significant differences are not observed, leading to the conclusion that the presence of the hot-wire

probe in the shear layer does not strongly influence the flow.

Figure 2.4: Auto-spectra of the pressure signal with a hot-wire probe in the shear layer (black line) and outside

the shear layer (grey line). The hot-wire probe was located in the center plane at 0.05D of the downstream edge.

U∞ = 40 m/s.

2.3.3 Microphones

Two 1/4-inch Bruel&Kjær free-field microphones (type: 4939 and 4135) were used to measure the fluctuating

pressure at the cavity walls. The microphones were connected to pre-amplifiers B&K 2670 and to a signal

conditioner Bruel&Kjær NEXUS 2692. Even if several holes for the microphones are available on the walls

of the cavity, a toothed graduated ring mechanism (precision of 1◦) was designed in order to reach any

azimuthal angle by rotating the cavity (figure 2.3).

The microphones were connected to the interior of the cavity through 1 mm diameter pinholes. The

geometry of the pinholes is the same as that adopted in Camussi et al. (2006a,b, 2008). By using pinholes

the spatial averaging effects are minimized. The main drawback of this layout is the presence of a non

negligible volume between the microphone diaphragm and the pinhole. This volume can act as a Helmholtz

acoustic resonator. In the present case the corresponding cut-off frequency has been accurately calculated

to 3347 Hz with equation 1.9. The expression leq = l + 1.64a was taken for the equivalent neck length

24


(a) (b)

Figure 2.5: Detail on the microphone set-up. The volume of air between the microphone diaphragm and the interior

surface of the cavity represents a Helmholtz resonator which dimensions are: length and cross surface area of the

pinhole l = 1.3 mm and S = π0.52 mm2 respectively and volume of the resonator V = 99 mm3.

(Blake (1986)), where a represents the pinhole’s radius, which is a simplification of a double Rayleigh end

correction. The maximum frequency of interest (shear layer modes) is expected to be around 300 Hz for the

velocity range explored, thus the Helmholtz resonator should not affect the frequency range of interest.

2.3.4 Particle image velocimetry

To investigate the flow inside the cavity, Particle Image Velocimetry (PIV) was used. The planes of interest

were illuminated by a BigSky Twins Ultra/CFR 200 Nd:YAG laser with the maximum energy of 50 mJ

per pulse. The images were taken with a PCO Pixelfly VGA (1280 × 1024 pixels) camera equipped with a

Docter Optics Tevidon 1.8/16 lens. Synchronization between the laser and the camera was achieved with a

BNC digital pulse/delay generator 575 controlled with an in-house Labview program.

Velocity fields over horizontal streamwise planes (Oxz) were explored by introducing the laser light sheet

horizontally through the cavity sidewall and placing the camera under the cavity. The measurements were

performed at different depths by moving the laser along the y-axis. A picture and a schematic of the

experiment are given in figure 2.6.

The PIV processing was done with PIVDEF, a software developed by the Istituto Nazionale per Studi ed

Esperienze di Architettura Navale (INSEAN) also known as the “Italian Ship Model Basin”. This software

is based on a standard PIV cross-correlation algorithm (Cotroni et al. (2000)), a recursive window offset

(Westerweel et al. (1997)) and a multiplication between adjacent correlation tables (Hart (1998)). A window

deformation techniques similar to the one developed by Lecordier et al. (1999) is also used. Further details

on the processing algorithm are given in Di Florio et al. (2002).

For each plane of interest and for each flow velocity investigated, 600 couples of images were recorded.

The output velocity fields obtained with PIVDEF were exported to Matlab for further processing.

25


(a) (b)

Figure 2.6: (a) Photograph taken from the exterior of the test setion showing, from left to right, the laser

beam, a cylindrical and a spherical lens, the generated laser sheet, the cavity and the camera underneaths.

(b) Schematic of the experimental rig showing the position of the PIV apparatus.

2.3.5 Data acquisition card and processing of the pressure signals

Signals from all the instruments were acquired using a National Instrument SCXI-1600 Data Acquisition

Module. The signal from the hot wire was sampled at a frequency of 40 kHz, acquired for 5 seconds and

low-pass filtered at a cut-off frequency of 10 kHz to avoid aliasing. The pressure signals were acquired for 4

seconds using a sample rate of 25 kHz and were band-pass filtered (between 20 Hz and 10 kHz) by the signal

conditioner.

The pressure signals were processed in Matlab using a Fast Fourier Transform (FFT) technique. The

number of points per segment was 25000 and therefore the frequency resolution was 1 Hz. The 4 data sets

were then averaged.

2.4 Test conditions

2.4.1 Flow conditions

The wind tunnel velocity was set manually by rotating a knob which controls the speed of the fan: the RPM

can be then read on a digital display located in the control room. Before every measurement, enough time is

waited in order to obtain stable flow conditions. Upstream of the wind tunnel’s contraction, the temperature

was kept at 21◦C ±1◦C by a heat exchanger. For the aeroacoustic characterization of the cavity, the flow

speed was increased from 5 to 55 m/s (M = 0.015 - 0.161) by steps of 1 m/s.

26


2.4.2 Incoming boundary layer

The boundary layer over the test section floor was characterized by hot-wire anemometry. The properties of

the incoming boundary layer at 46 mm from the upstream corner of the cavity are summarized in table 2.1 for

three velocities (20, 30 and 40 m/s). The length scale of the boundary layer (momentum thickness) represents

less than 2 % of the diameter of the cavity thus this parameter should not influence the aerodynamics and

acoustics of the cavity. Velocity profiles are given in figure 2.7 jointly with the 1/7 power profiles. The

velocity measurements match the theoretical turbulent boundary layer trend.

U∞ (m/s) δ (mm) δ∗ (mm) θ (mm)

19.9 35.2 4.9 3.8

30.4 34.4 4.7 3.6

40.2 31.7 4.3 3.3

Table 2.1: Boundary layer properties for 3 free stream velocities (20, 30 and 40 m/s). Boundary layer thickness δ (99 %),

displacement thickness δ∗ and momentum thickness θ.

Figure 2.7: Velocity profiles of the boundary layer measured at 0.22D from the upstream corner of the cavity at 3

different flow velocities: U∞ = 20, 30 and 40 m/s.

2.4.3 Background pressure fluctuations

The background pressure fluctuations in the test section were explored in order to check if their intensity is

lower than the flow-excited cavity unsteady pressure level. Camussi et al. (2000) have already clarified most

27


of the background unsteady pressure features of the ENEA Casaccia wind tunnel. Therefore the results from

this previous study were used to identify some of the frequency peaks in the pressure spectra.

The measurements were done by means of an in-flow microphone fitted with a nose cone and located 150

mm (y) above the cavity. The mouth of the cavity was covered with a wooden plate. The pressure spectra

obtained for flow velocities between 5 and 55 m/s are given in figure 2.8 with the theoretical blade passing

frequency and its harmonics superimposed. The pure tone at 300 Hz, present for all the flow velocities, has

been ascribed by Camussi et al. (2000) to the constant speed fan installed for cooling the wind tunnel blower.

Figure 2.8: Background pressure fluctuations in the test section without the cavity. The white dots represents the

blade passing frequency and its first two harmonics.

2.5 Measurement matrix

The fluctuating pressure was measured at the side and bottom walls of the cavity. A total of 325 different

positions were surveyed over 6 different depths (y), 4 different radii (r) and 36 different azimuthal angles (ϕ).

Figure 2.9 shows the positions adopted by the microphone. In this schematic drawing, the lateral wall has

been unrolled and the circular bottom reproduced underneath. With this plane representation, the reader

can have a panoramic view of the cavity walls. To achieve this measurement matrix, a single microphone

was moved from one pinhole to another and the cavity rotated by steps of 10 degrees. Results can be found

in section 4.4.

The aerodynamic campaign consisted in a three-dimensional grid of 2520 points distributed over 8 dif-

ferent streamwise (x), 7 different vertical (y) and 45 different spanwise (z) positions. A schematic drawing

indicating the hot wire probe positions adopted for the velocity measurements is given in figure 2.10: a top

28


and a lateral view of the experimental rig are represented. In section 4.1, the results from this campaign are

discussed.

For the PIV measurements, three different depths inside the cavity were explored: y/H = -0.25, -0.50

and -0.75. The obtained mean velocity fields are reported in section 4.2.

Figure 2.9: Schematic drawing of the wall pressure measurements grid, see figure 2.2 for the reference frame adopted.

The lateral wall has been unrolled and the circular bottom reproduced underneath.

29


(a)

(b)

Figure 2.10: Schematic drawing of the velocity measurements grid, see figure 2.2 for the reference frame adopted.

Positions taken by the hot wire probe for the aerodynamic campaign: top view (a) and lateral view (b)

30

Chapter 3

Acoustic mode calculation

3.1 The acoustic modes of an open-closed cylindrical cavity

The aim of this section is to find an analytical expression for the natural acoustic modes of a cylindrical cavity

with an open mouth. The starting point is the three-dimensional wave equation that is the second-order

linear partial differential equation governing the acoustic pressure p:

∇2p =1

c2∂2

∂t2p (3.1)

where c is the speed of sound. The pressure can be decomposed into a sum of orthogonal Fourier

components pheiωht which individually satisfy the wave equation at every moment t. Therefore equation 3.1

becomes the Helmholtz equation:

∇2ph + k2h ph = 0 (3.2)

where kh = ωh/c the wave number of the corresponding Fourier component.

In cylindrical coordinates (r, θ and y, see figure 3.1) the Helmholtz equation becomes:

∂2ph∂r2

+1

r

∂ph∂r

+1

r2∂2ph∂θ2

+∂2ph∂y2

+ k2h ph = 0 (3.3)

This equation can be solved analytically by the separation of variables technique (details are not given

here for sake of brevity and because it can be found in many acoustic textbooks). The general solution of

equation 3.3 takes the following form:

ph(r, θ, y) = Jm(krr) eimθ (C1e

ikyy + C2e−ikyy) (3.4)

where Jm is the Bessel function of the first kind of order m, C1 and C2 two constants.

31

3.Acoustic mode calculation

Figure 3.1: Definition of the cylindrical coordinates for the analytical resolution of the Helmholtz equation.

In order to find the acoustic eigenfrequencies of the cavity, the boundary conditions have to be specified:

the wall-normal derivative of p has to vanish at the walls (bottom and side) and an assumption has to be

made for the mouth of the cavity (y = H). As mentioned in section 1.1.2, physically the pressure node is not

exactly at the opening of the cavity. A simple way to proceed is to take an end correction: the additional

boundary condition adopted here is that the pressure has to vanish at y = H + αR. The general expression

for an eigenfrequency is:

fm,n,q =ω

2π=

c

2π

((j′m,nR

)2

+

((2q′ + 1)π

2(H + αR

)2)1/2

(3.5)

where j′m,n is (n + 1)th positive zero of J ′m and m, n and q are the azimuthal, radial and longitudinal

order of the mode and q = 2q′ + 1 where (m,n, q′) ∈ N3. It is interesting to notice that if m = n = 0,

equation 3.5 reduces to equation 1.8.

With the dimensions of the present cavity (H and D) and the temperature and the ambient pressure

during the tests (for the calculation of c), the frequencies of the acoustic modes of the cavity can be calculated.

The end correction coefficient α that depends on ky can be evaluated with equation 1.7 (figure 3.2). The

predicted frequencies are given in table 3.1 and the shapes of the first eight modes are reported in figure 3.3.

Mode λ/4 3λ/4 AZ1(λ/4) AZ1(3λ/4) 5λ/4 AZ1(5λ/4) AZ1R1(λ/4) AZ1R1(3λ/4)

(m,n, q) (0,0,1) (0,0,3) (1,0,1) (1,0,3) (0,0,5) (1,0,5) (1,1,1) (1,1,3)

Frequency 236 760 994 1246 1333 1663 2792 2904

Table 3.1: Frequencies, in hertz, of the acoustic modes of the cavity. Mode order (m,n, q): azimuthal, radial, longitudinal.

32


Figure 3.2: Correction coefficient predicted by equation 1.7. The red cross represent the quarter wavelength (α =

0.7536), the black dot the mode 3λ/4 (α = 0.5153) and the green star the mode 5λ/4 (α = 0.3562).

3.2 The acoustic modes in a wind tunnel

When performing aeroacoustic experiments in a closed test section, it is important to study the acoustic

behaviour of the facility in addition to its aerodynamic performance. If the walls of the test section are not

acoustically treated, the pressure measurements may be affected by the acoustic resonant modes of the wind

tunnel. Furthermore the presence of a model in the test section can modify the acoustic resonances of the

wind tunnel and, more important, excite strongly localized resonant modes with zero radiation loss called

trapped modes.

In order to predict the frequency and the shape of the resonant acoustic modes of an experimental rig,

finite element analysis has become popular in the past decade: Aly & Ziada (2010) used the commercial

software ABAQUS to study the trapped modes of a ducted axisymmetric internal cavity, Finnegan et al.

(2010) calculated the acoustic field around a bluff body with ANSYS and Oshkai et al. (2008) predicted

the acoustic modes of a coaxial side branch resonator with COMSOL. Some advantages of the use of a

commercial software for the calculation of the acoustic modes are the promptness and the reliability of the

results when the boundary conditions are properly set.

3.3 The computational model

The acoustic modes of the test rig were studied using the commercial software COMSOL Multiphysics

4.0. This package includes a finite element solver able to evaluate partial differential equations of different

disciplines. Even if several physical models can be solved jointly, a no-flow case was studied. The default

eigenfrequency formulation available within this software was used for the calculation:

∇(−1

ρ∇p)

+λ2p

ρc2= 0 (3.6)

33


(a) (0,0,1) (b) (0,0,3)

(c) (1,0,1) (d) (1,0,3)

(e) (0,0,5) (f) (1,0,5)

(g) (1,1,1) (h) (1,1,3)

Figure 3.3: Shape of the first height acoustic modes of a cylindrical cavity with a closed bottom wall and an open

top surface. The infinite flange has not been represented. The color bar gives the normalized real pressure values.

See table 3.1 for the frequency of each mode (m,n, q). 34


(a) (b)

Figure 3.4: Overview (a) and detail (b) of the mesh. The spatial domain chosen for the calculation includes the

contraction and the test section of the wind tunnel.

where ρ is the air density, c the speed of sound, p the pressure and λ an eigenvalue. From this equation, it

is clear that the fluid is modeled as a lossless medium. Once the equation has been solved, the eigenfrequency

can be evaluated through:

λ = i2πf (3.7)

3.4 The geometry

A preliminary investigation brought into evidence that the simulation of the test section without the con-

vergent nozzle does not predict correctly the frequencies measured experimentally. Thereby a simplified

contraction was included into the calculation leading to a five-meter-long geometry.

As already mentioned in the previous chapter, the frequency range of interest is [0 - 300 Hz]. In order to

resolve correctly the highest frequency of this range, every mesh element does not have to exceed 0.2 meters.

This is dictated by the five-mesh-elements-per-wavelength rule. The wind tunnel section was meshed with

approximately 53 000 tetrahedrons and the cavity with 6 000 elements. Figure 3.4 gives an idea of the

meshed geometry.

All the walls (contraction, test section and cavity) were modelled with rigid boundaries (called ‘sound-

hard’ in the COMSOL Multiphysics) meaning that the normal derivative of the pressure is set to zero. At

the end of the contraction and at the end of the test section ‘soft boundary’ conditions were imposed: the

acoustic pressure vanishes.

35


Without cavity With cavity

Frequency Mode number Mode number Frequency Difference

Hz Hz %

42.64 1 1 42.63 0.02

66.72 2 2 66.67 0.07

99.95 3 3 99.72 0.23

102.42 4 4 102.42 0.00

118.07 5 5 118.07 0.00

130.86 6 6 130.62 0.18

136.11 7 7 136.11 0.00

152.05 8 8 152.06 -0.01

154.98 9 9 154.81 0.11

160.25 10 10 160.28 -0.02

165.67 11 11 165.47 0.12

167.84 12 12 167.84 0.00

181.13 13 13 181.19 -0.03

183.76 14 14 183.76 0.00

186.95 15 15 186.87 0.04

191.16 16 16 191.16 0.00

196.16 17 17 196.16 0.00

198.32 18 19 198.66 -0.17

199.83 19 18 197.78 1.03

204.28 20 20 204.3 -0.01

212.2 21 21 211.96 0.11

215.95 22 22 212.78 1.47

217.59 23 23 217.61 -0.01

223.86 24 24 223.86 0.00

225.32 25 26 225.32 0.00

226.96 26 25 224.6 1.04

232.04 27 27 232.06 -0.01

232.52 28 28 232.53 0.00

241.29 29 29 236.77 1.87

244.71 30 30 244.72 0.00

do not exist 31 246.76

249.47 31 32 249.47 0.00

250.08 32 33 250.13 -0.02

251.21 33 34 251.29 -0.03

253.97 34 35 256.64 -1.05

258.69 35 36 258.69 0.00

260.52 36 37 260.52 0.00

262.13 37 38 263.08 -0.36

Table 3.2: Frequency of the acoustic modes of the wind-tunnel with (right) and without (left) cavity.

36


3.5 Results

3.5.1 Acoustic modes without the cavity

In the range of frequency [0 - 263 Hz], 37 different eigenmodes were found and their frequencies, sorted from

smallest to largest, are reported in table 3.2. Six modes were identified with a zero isobar central horizontal

plane but without a zero isobar central vertical plane. The frequencies of these modes are highlighted in

the table 3.2 and their pressure distributions over the walls of the test section are plotted in figure 3.5. The

shapes of six modes not having the characteristics previously described are given in figure 3.6.

3.5.2 Acoustic modes with the cavity

When the cavity is considered, 38 different eigenmodes are found in the same frequency range. The eigenfre-

quencies are reported in table 3.2 jointly with the corresponding eigenfrequencies without the cavity. Two

minor changes can be noticed: small frequency shifts and two order permutations (modes 18-19 and 25-26).

Furthermore, the addition of the cavity into the geometry brings one mayor change: another eigenfrequency

appears. The discussion of this extra mode will be detailed later.

As in the previous case, the modes can be divided in two categories according to the pressure pattern on

the horizontal and vertical planes of the test section. The second group (31 eigenmodes) is weekly influenced

by the introduction of the cavity: the biggest frequency shift is 0.23 %. When comparing figure 3.6 to

figure 3.8 it can be seen that the shapes of the modes have not changed significantly. Another observation

is that the pressure inside the cavity is constant and equal to zero. It is important to point out that most

of the modes of this group have a central vertical plane where the pressure is equal to zero (pressure node).

The shape of these modes would has probably changed if the cavity had been added in a plane other than

the central one.

Figures 3.7 gives the shapes of the 6 acoustic modes for which the central horizontal plane (0xz) was

a pressure node and the central vertical plane was not a zero isobar before the introduction of the cavity.

The frequency shifts are reported in table 3.2: the maximum difference is of 1.87 % which is a much higher

percentage than for the former category. It can be seen, in figures 3.7, that the pressure amplitude in the

cavity is not constant. All the modes of this class have a pressure pattern similar to the cavity quarter-wave

length: the bottom of the cavity is a pressure anti-node and the fluctuations at the orifice are minimal. This

common shape can be illustrated by the mode 35 (figure 3.10). At the cavity orifice, the pressure pattern is

not uniform as it should be for an unconfined case: this mode, as the 6 others, are indeed a combination of

a cross-ducted mode with the quarter-wave length of the cavity.

The only eigenfrequency added by the cavity is 246.76 Hz (pressure reported in figure 3.9). This value

is close to 326 Hz, the frequency of the quarter-wave length of the cavity analytically calculated (table 3.1).

This result is relevant for the present analysis since the signature of this mode will be evidenced by the wall

pressure measurements presented in section 4.4 of the next chapter.

37


(a) Mode 19 (b) Mode 22

(c) Mode 26 (d) Mode 29

(e) Mode 34 (f) Mode 37

Figure 3.5: Pressure distribution (Pa) on the walls for six different eigenmodes (without cavity).

38





Figure 3.6: Pressure distribution (Pa) on the walls for six different eigenmodes (without cavity).

39





Figure 3.7: Pressure distribution (Pa) on the walls for six different eigenmodes (with cavity).

40





Figure 3.8: Pressure distribution (Pa) on the walls for six different eigenmodes (with cavity).

41


Figure 3.9: Extra mode: acoustic mode 31

Figure 3.10: Isosurface plot of the acoustic pressure distribution (Pa) inside the cavity at 256Hz. This mode has a

quarter-wave length shape

42

Chapter 4

Experimental results

4.1 Overall aerodynamics

A single component hot-wire probe was used to measure the velocity above and downstream of the cavity

in order to obtain a general understanding of the shear layer and of the wake topology. The incoming free

stream velocity was fixed to U∞ = 40 m/s. The flow velocity at a given position (x, y, z) in the test section

is ~U = ~Ux+ ~Uy+ ~Uz. As the wire was orientated along the z direction, the velocity measured by the hot-wire

is Uhw = ||~Ux + ~Uy||. Profiles of the non-dimensional mean velocity (Uhw/U∞) and of the non-dimensional

standard deviation of the measured velocity (σhw/U∞) are given in figure 4.1 and figure 4.2 respectively.

According to the literature, because H/D > 0.8 both quantities are expected to be symmetric with respect to

the central streamwise plane (Oxy) within the range of experimental uncertainties. The shear layer topology

is first presented followed by the wake description.

4.1.1 Shear layer topology

Three different streamwise planes are reported in figure 4.1: x/D = 0, 0.25 and 0.50. For each one of these

planes, different heights were explored. In the region close to the vertical symmetry plane (z/D = 0) a mean

velocity defect is clearly observed as an effect of the shear layer generated over the cavity opening. Because

the geometry is cylindrical, the greatest shear layer effects are observed in the central region. The turbulent

velocity fluctuations (σhw) confirm this trend since an increase of the turbulent kinetic energy is observed in

the region close to the plane z/D = 0. When moving downstream, the shear layer grows: the mean velocity

decreases and the velocity fluctuations increase. This effect is shown both for the cases at x/D = 0.25 and

0.5.

It is interesting to note that when a given longitudinal position x/D is considered, the perturbation

induced by the presence of the cavity is restricted to the wall proximity region. Indeed, in the region close

to the downstream side of the cavity and for around y/D = 1/10, Uhw and σhw profiles are almost flat and

43

4.Experimental results

the signature of the cavity shear layer is no longer observed.

Closer to the wall, further interesting features are detected as an effect of the interaction between the

shear layer and the downstream edge of the cavity. In the plane x/D = 0.5, wiggles at about z/D = −0.25

and z/D = 0.25 are present in the y/D = 0.015 and y/D = 0.025 profiles. These localized bumps are present

both in the mean velocity and in the root-mean-square velocity fluctuations profiles at the same positions.

According to surface oil-film pattern visualisations presented by Gaudet & Winter (1973), these bumps can

be ascribed to the effect of the reattachment of the shear layer in the downstream cavity wall region. The

shear layer impacting the wall indeed leads to a localized variation of the mean velocity magnitude and,

consequently, to a relevant increase of the velocity fluctuations.

4.1.2 Wake topology

The presence of wiggles is also detected in the velocity and turbulence profiles in the wake region (figure 4.2).

According to the physical features described above, the signature of the shear layer is still present in the

vicinity of the symmetry plane even though for x/D > 0.65 its effect declines rapidly. Similar results have

been obtained in a rectangular deep cavity by Ukeiley & Murray (2005) as an effect of the reattachment of

the shear layer on the downstream lip. A decrease of the turbulent velocity fluctuations is indeed observed in

the wake near the wall, while for larger distances from the wall, the turbulence level increases as an effect of

the wake development. The amplitude of the velocity defect and the corresponding increase of the turbulence

level gradually disappear when moving downstream from the cavity, thus the effect of wiggles on the velocity

profiles becomes more evident in the far region. With respect to the velocity bumps observed in the shear

layer region above the cavity, the wiggles are now observed at spanwise positions of about z/D = −0.5 and

z/D = 0.5, and for larger distances from the wall, up to y/D ≈ 0.1 for x/D = 2.5. These results seem to

suggest that the trend observed in the wake region is due to physical mechanisms different from the one

observed in the region above the cavity. With the support of the Gaudet & Winter (1973) visualisations,

an interpretation can be addressed: the observed wiggles can be ascribed to the perturbation caused by

two vortical structures convected from the cavity downstream lip, having streamwise vorticity and detaching

symmetrically with respect to the vertical plane at z = 0, from the cavity wall. The signature of the two

vortices is indeed not present in the region above the cavity thus suggesting that they are formed downstream

of the cavity opening. With the current measurements it is impossible to draw definite conclusions about

the topology of the tip vortices. However, preliminary results from a numerical simulation of a cylindrical

cavity (Grottadaurea & Rona (2008)), confirm the existence of longitudinal tip vortices.

4.2 Description of some flow features inside the cavity

Before discussing the flow inside a cylindrical cavity, it is important to describe first the well documented

flow inside a rectangular cavity. For a deep rectangular cavity, the flow detaches at the upstream edge

44


(a) (b)

(c) (d)

(e) (f)

Figure 4.1: Spanwise profiles of non-dimensional mean velocity Uhw/U∞ (left) and of the non-dimensional root-

mean-square velocity fluctuations σhw/U∞ (right) in the shear layer at three different streamwise positions (x/D =

0, 0.25 and 0.5) and for different heights (legend in the left column).

45


(a) (b)

(c) (d)

(e) (f)

Figure 4.2: Spanwise profiles of non-dimensional mean velocity Uhw/U∞ (left) and of the non-dimensional root-

mean-square velocity fluctuations σhw/U∞ (right) in the wake at three different streamwise positions (x/D = 0.64,

1.5 and 2.5) and for different heights (legend in the left column).

46


creating a shear layer over the mouth of the cavity. This shear layer hits the downstream-wall and one part

of the flow turns downwards into the cavity. This ‘turned down jet’ behaves like a decelerating stagnation

flow while approaching the cavity the bottom. Afterwards the flow mainly recirculates inside the enclosure.

This description is also valid in the symmetric plane (z/D = 0) of a deep cylindrical cavity (Mincu et al.

(2009) and Grottadaurea & Rona (2008)). However, because for circular based cavity the flow is strongly

3-dimensional, the knowledge of the flow in a single plane gives only a local understanding and not a global

description of the flow.

Results from PIV measurements taken at three different horizontal planes are analysed in this section.

At each plane, 600 couples of images were captured and a mean velocity field was calculated (figure 4.3).

The velocity field in an upper plane inside the cavity (y/H = −0.25) is presented in figure 4.3(a). Near

the downstream-wall, the turned down jet gives rise to two vertical counter-rotating vortices (x = 0.07 m,

figure 4.3(a)). The main reasons for the generation of these vortices are the curvature of the wall and the

opposite spanwise velocity components at each side of the cavity. The negative streamwise component of

the velocity indicates that the jet detaches from the vertical wall as it goes down. By analyzing the plane

at half depth (y/H = −0.5, figure 4.3(b)), one can notice that the jet has spread and that the vortices have

moved apart from each other.

As expected for a deeper plane, at y/H = −0.75 the fluid was found to flow in an opposite direction

than for an shallower plane: on the downstream half part of the cavity (x/D > 0), the velocity vectors are

orientated upstream (figure 4.3(c)). For the half plane x/D < 0 however, velocity field is governed by the

‘sink’ point located at x = −0.04 m over the x-axis. These 2D PIV measurements do not give information

about the vertical velocity component (Uy). However it is easy to imagine that the ‘sink’ point corresponds

to the center of the turn up jet over the upstream wall.

4.3 Unsteady response to a grazing flow

4.3.1 Pressure response

In order to characterize the cavity pressure response to a turbulent boundary layer, different flow velocities

were investigated. The pressure spectra at three different flow velocities are reported in figure 4.4: a case

with no relevant resonance (30 m/s), a case exhibiting two spectral peaks of almost the same intensity (41

m/s) and a strong resonating case (53 m/s). When the velocity increases, the pressure fluctuations at the

walls also increase by several orders of magnitude, thus justifying the use of a logarithmic scale. As already

mentioned in chapter 2, the blower’s cooling system generates a tone at 300 Hz that can be noticed in all

the spectra. For comparison, the spectra of the background pressure fluctuations in the test section are also

reported in figure 4.4 (see section 2.4.3 for experimental details).

In order to have a global view of the pressure response for velocities ranging from 4 to 56 m/s, a total of 53

spectra are presented together in figure 4.5. Three different spectral peaks can be seen, evolving in frequency

47


(a)

(b)

(c)

Figure 4.3: Mean velocity field for 3 different planes: (a) y/H = −0.25; (b) y/H = −0.50; (c) y/H = −0.75

for a free stream velocity in the test section of 10 m/s.

48


Figure 4.4: Pressure fluctuations measured with a microphone mounted on the downstream wall of the cavity

(ϕ = 180◦, y/H = -0.1875) and with a in-flow microphone (cavity’s mouth covered) for different velocities: U∞ =

30, 41 and 53 m/s.

and in amplitude with the wind tunnel velocity: they are the signature of the shear layer hydrodynamic

modes. The agreement with the Rossiter (1964) prediction is very good. For the present data, the values

showing the best agreement with the Rossiter semi-empirical equation are α = 0 and κ = 0.53 as can be

seen in figure 4.5. It should be noted that the characteristic length chosen was the diameter of the cavity

because it fits better the data compared the other expressions as those used by Bruggeman et al. (1991) and

Czech et al. (2006).

The shear layer modes follow a linear trend for velocities up to 30 m/s. For higher flow speeds, a stepwise

evolution characterizes the progression of the modes: the second and third hydrodynamic modes start to

increase by steps after reaching 197 Hz (figure 4.5). This phenomenon can be ascribed to the fact that the

cavity model is installed in a closed test section, an environment propitious to the establishment of a flow-

acoustic coupling. The shear layer instabilities are responsible for the initial excitation of the acoustic modes

of the wind-tunnel/cavity system. The acoustic eigenmodes of the system dictate the lock-on frequencies. As

can be seen in figure 4.5, lock-on is generated only at 6 frequencies. When comparing to the finite element

study (chapter 3), these lock-on frequencies correspond to acoustic modes that have a quarter-wave length

shape. A stepwise evolution of the shear layer modes was also observed by Czech et al. (2006) for an array of

cylindrical cavities in a closed section. As a matter of fact, Marsden et al. (2008) performed measurements

in a cavity installed in an open test section and reported a linear evolution of the shear layer modes.

The coupling between a hydrodynamic mode and an acoustic mode not only generates lock-on but also

49


(a)

(b)

Figure 4.5: Three-dimensional and plan view of the pressure amplitude on a logarithmic scale (dB) as a function of

the flow speed and the frequency. The microphone was located at: (ϕ = 90◦, y/H = -0.25). Solid lines represent the

Rossiter (1964) prediction of the shear layer modes with [α, κ] =[0 , 0.53 ]. The frequencies of the acoustic modes

calculated with the FEM simulation are also plotted (dotted and dashed lines and diamonds). The dash lines are the

eigenvalues having a quarter-wave length shape pressure distribution inside the cavity. The depth mode (diamonds)

has a frequency of 246 Hz.

50


amplification of the pressure oscillations. At 53 m/s, for example, the shear layer mode SL2 is amplified

because coupled with an acoustic mode of the wind-tunnel/cavity: the peak in the pressure spectrum raises

40 dB over the background noise levels (figure 4.5). This should be compared to a non-resonating state, at

30 m/s for example, where the peaks corresponding to hydrodynamic modes are only 10 dB higher than the

overall sound pressure level.

4.3.2 Nondimensionalization process

The pressure spectra are now presented in non-dimensional form in figure 4.6. The Strouhal number based on

the cavity diameter is the selected dimensionless frequency. It has been shown previously that the pressure

fluctuations increase with the flow velocity: in order to take into account the hydrodynamic effect on the wall

measurement, the pressure is normalized with respect to the dynamic pressure 0.5ρU2∞. The dimensionless

pressure spectra are presented on a logarithmic scale. For Strouhal numbers up to 0.8, a satisfactory collapse

of the spectra is observed. The signature of the first hydrodynamic shear mode is a bump in the spectra at

St = 0.57. In contrast, the signature of the second hydrodynamic mode (St = 1.00) is either a bump (U∞

= 15, 22 and 32 m/s) or a sharp peak (U∞ = 42 and 52 m/s) depending whether or not is it coupled with

an acoustic modes. For Strouhal numbers higher than 1.1, a collapse of the spectra is no longer observed.

Figure 4.6: Pressure spectra for different velocities: U∞ = 15, 22, 32, 42 and 52 m/s. The frequency and the pressure

intensity have been normalized.

For the sake of completeness, the pressure response for velocities between 4 and 56 m/s, is reported in

the form of plane contour plot with non dimensional frequencies in figure 4.7. The frequencies of the three

Rossiter modes and the 246 Hz mode predicted with the FE simulation are also plotted. This quarter wave

51


length-like mode has the propriety to be the only trapped mode generated by the cavity itself.

Figure 4.7: Plan view of dimensionless pressure amplitude on a logarithmic scale as a function of flow speed and the

Strouhal number based on the diameter of the cavity. Lines represent the Rossiter (1964) prediction and diamonds

the 246 Hz mode found with the FEM calculation.

4.3.3 Velocity response

A similar study was conducted from the velocity measurements in the shear layer. The velocity spectra

at different flow speed are reported in figure 4.8. The plane contour plot representation is used and the

velocity amplitudes are normalized by the flow speed. This figure is similar to the pressure response given

in figure 4.5: the three first shear layer hydrodynamic modes are identifiable as well as their lock-on with

the test section/cavity acoustic modes.

4.4 Spectral decomposition

4.4.1 Spectral decomposition and analysis on the symmetry plane

In order to extend the spectral pressure study, several measurement positions have been analysed. Figure 4.9

shows the curvilinear coordinate s adopted in order to locate the 19 pinholes on the vertical central plane of

the cavity. The pressure spectra at three different points (s = 0.93, 0.5 and 0.07) are presented in figure 4.10

for a flow speed of 40 m/s. The signature of the three shear layer modes can be identified (peaks at f

= 100, 195, 256 Hz). When considering the spectra quantitatively, pressure amplitudes are higher on the

downstream wall (s = 0.93) than on the upstream wall (s = 0.07) for all the frequencies except for f = 256

52


Figure 4.8: Plane view of the dimensionless velocity amplitude on a logarithmic scale as a function of flow speed

and frequency. The hot-wire was located in the shear layer region near the downstream edge (x/D = 0.45, y/H =

0.14, z/D = 0).

Hz. For this specific frequency, the pressure fluctuations on the downstream wall are as important as the

pressure fluctuations on the upstream wall. When the microphone is located at the bottom of the cavity (s

= 0.5), the spectrum is entirely dominated by the 256 Hz peak.

An original signal post-processing has been applied to the pressure data in order to explain the variations

of spectral content from one position to another. The idea is to evaluate the importance of some frequencies

to the overall fluctuating pressure level which is defined as:

OAFPL = 10 log

+∞∫f=0

Wp(f) df

P 2ref

(4.1)

where Wp is the power spectral density of the wall pressure and Pref is the reference sound pressure in

air (20 µ Pa) . The fluctuating pressure level of the frequency f can be estimated by:

FPLf = 10 log

f+10Hz∫

f ′=f−10Hz

Wp(f′) df ′

P 2ref

(4.2)

By this procedure, four different quantities have been calculated: the FPL30Hz from the low frequencies

and FPL100Hz, FPL195Hz and FPL256Hz from the shear layer hydrodynamic modes (SL1, SL2 and SL3).

53


Figure 4.9: Schematic drawing of the cavity’s middle plane with the curvilinear coordinate s adopted. The positions

of the pressure transducers are represented with red circles. The recirculation pattern present inside a deep cavity is

also shown: the shear layer impacts on the downstream wall (s ≈ 1), then a near-wall flow moves towards the bottom

and impacts the bottom wall of the cavity (s ≈ 0.6).

Figure 4.10: PSD measured at three different positions (s = 0.07, 0.50 and 0.93) along the cavity wall for a flow

velocity of 40 m/s. The first three hydrodynamic modes are annotated.

54


(a) (b)

Figure 4.11: (a) Power spectral density of the pressure fluctuations at the bottom wall (s = 0.5). The four stacked

areas represent the integrals in FPL30Hz, FPL100Hz, FPL195Hz and FPL256Hz. (b) Fluctuating pressure levels of some

frequencies (LF, SL1, SL2 and SL3) along the curvilinear coordinate s for 40 m/s.

The selected frequencies are shown in figure 4.11(a) and the related fluctuating pressure levels are given in

figure 4.11(b). When considering the low frequency fluctuations, one can notice that FPL30Hz evolves as

the flow progresses inside the cavity: it is maximal where the shear layer hits the downstream wall (near

s = 1) and decreases inside the cavity as a consequent loss of energy of the internal recirculating flow. The

detachment of the turned down jet from the downstream wall and the second impingement on the bottom

of the cavity (at s = 0.6) can be detected with FPL30Hz. Such a flow organization has been observed,

for example, in the Detached Eddy Simulation (DES) of Grottadaurea & Rona (2008), where streamlines

indicate a detachment of the turned down jet and an impact on the bottom wall. The mean pressure

measurements of Hering et al. (2006) for a circular cavity (H/D = 0.7, U∞ = 27 m/s), also indicate that the

maximum pressure region on the bottom is not exactly in the corner but more towards the bottom center

leading to a recirculation region in the downstream corner of the cavity. The FPL100Hz trend is similar

to the FPL30Hz. The explanation for that is given by the dimensionless pressure study (figure 4.6): for

low frequencies (St < 0.7), the pressure spectra collapse because there is no strong coupling between the

fluid-dynamics and the acoustics inside the cavity.

A very different evolution is found for FPL256Hz: its maximum is reached on the cavity bottom and its

minimum at the cavity mouth. This trend is related to an acoustic mode having a pressure anti-node at the

cavity bottom and a node at its opening. The velocity presented (40 m/s) corresponds indeed to a case in

which the third shear layer mode excites a trapped mode (f = 256 Hz) which shape is similar to the quarter

wavelength mode of the cavity.

55


4.4.2 Analysis on the cavity walls

The fluctuating pressure levels are further analyzed by considering their azimuthal variations. The quantities

OAFPL, FPL256Hz and FPL30Hz over the side wall and the bottom of the cavity are given in figure 4.12.

When comparing the spatial distribution of OAFPL and FPL256Hz, it appears that the acoustic effects

related to the 256 Hz mode dominate the wall pressure fluctuating almost everywhere inside the cavity. The

pressure level of the low frequencies gives an indication of the flow proprieties inside the cavity. In order

to facilitate the description of the flow, the side wall was divided in five different regions based on FPL30Hz

amplitude and on Gaudet & Winter (1973) observations. Arrows representing the flow direction were also

included (figure 4.12(c)). High values of FPL30Hz on the downstream wall (A) and on the bottom wall

correspond for example to impingement areas. The interface between a region where the air flows towards

the cavity bottom (B), and a region where it flows towards the top of the cavity (D) is characterized by

medium FPL30Hz values (C).

56


(a) (b)

(c)

Figure 4.12: Fluctuating pressure levels over the internal walls of the cavity (side and the bottom) for a flow speed

of 40 m/s. (a) overall fluctuating pressure levels OAFPL, (b) fluctuating pressure level of the third shear layer mode

FPL256Hz, and (c) fluctuating pressure level of the low frequencies FPL30Hz. The arrows represent the flow velocity

direction in the different regions (A, B, C, D) of the side wall.

57


58

Chapter 5

Conclusion

An open mouth cylindrical cavity was characterized experimentally by means of velocity and wall-pressure

measurements.

Velocity and turbulence profiles revealed that the flow is symmetric as expected. Through the analysis

of the cavity’s wake, some three-dimensional effects were identified: the signature of a vortex pair detaching

from the cavity downstream lip and the shear layer reattachment on the downstream wall.

Concerning the flow inside the cavity, PIV measurements gave important insight into the three-dimensional

characteristics of the mean flow. The turned down jet was found to generate two vertical symmetric vortices

near the downstream wall.

The spectral analysis of both the wall-pressure and velocity signals revealed the presence of three shear

layer hydrodynamic modes. The frequencies of these modes are correctly predicted by the classical empirical

formulation for rectangular cavities. A flow-acoustic coupling has been found to occur for some specific

flow velocities. A finite element simulation of the experimental rig showed that the lock-on frequencies

correspond to some acoustic modes of the test section/cavity. These acoustic resonances have a common

pressure distribution inside the cavity: a quarter-wave length shape.

Through a processing of the wall pressure fluctuations, which calculates the contribution of some specific

frequencies to the overall fluctuating pressure level, some important flow features were clarified. After the

impingement of the shear layer on the downstream wall, a fraction of the fluid enters into the cavity creating

a turn down jet along the downstream sidewall. Before it impinges on the bottom, the wall-jet moves away

from the sidewall.

59

5.Conclusion

60

Part II

Cylindrical cavity with partially

closed mouth

61

Chapter 6

Experimental set-up

6.1 Overview of the experimental rig

Experiments were performed in the Fluids Laboratory of the Mechanical and Manufacturing Engineering

Department at the Trinity College in Dublin. The rig consists of a 493 mm (H) deep cylindrical cavity with

a 238 mm internal diameter (D) mounted on the lateral wall of a 335 mm long test section with a 125 mm

× 125 mm cross section (W × W ). A section view of the experiment is given in figure 6.1. A rectangular

orifice connects the interior of the cavity to the test section of the elliptical bell-mouth inlet draw-down wind

tunnel. More details about the opening are given in section 6.3. The flow was generated by a centrifugal

blower driven by a motor. A picture of the Perspex cavity is given in figure 6.2. Additionally, a schematic

of the wind tunnel (figure 6.3) gives an overview of the experimental rig.

6.2 Design of the experimental rig

As the available facility for the testing was a small low speed wind tunnel, a preliminary analysis was required

in order to optimally design the experimental rig. One of the goals of this research is to demonstrate that

acoustic resonant modes other than longitudinal modes can be excited in deep cylindrical cavities. Therefore

the cavity had to be designed in such a way that the frequencies of the desired acoustic modes match the

frequencies of one of the shear layer hydrodynamic modes within the velocity range of the wind tunnel.

The geometrical parameters that could be chosen were the length of the opening L and the dimensions

of the cavity (diameter and depth). The constraints were mainly the dimensions of the test section (335 mm

× 125 mm × 125 mm) and the flow speed range (0 - 53 m/s). The shear layer hydrodynamic modes were

estimated with the Rossiter (1964)’s formula (equation 1.1) and only the first three orders (n = 1, 2 and 3)

were taken into account. The frequencies of the acoustic modes were computed analytically: see section 7.1

for more details.

63


Figure 6.1: Schematic of the wind tunnel test section and the cavity. Section taken in the central plane. Only one

microphone is represented for simplicity. Not to scale.

Figure 6.2: Photograph of the experimental rig. One can see the digital camera for the PIV measurements and two

rings of eight microphones each.

64


Figure 6.3: Schematic of the wind tunnel, adapted from Finnegan (2011).

Figure 6.4: Design of the experimental rig. The depth of the cavity was set to H = 492.5 mm. Coloured curves

represent the first three shear layer hydrodynamic modes (red I, blue II and green III): the thickness of the curves

is proportional to the opening length L: 10 mm (thin), 40 mm (normal) and 70 mm (thick). The horizontal lines

represent the acoustic modes of the cavity (H1, AZ1 and AZ1H1): the thickness of the lines is proportional to the

diameter of the cavity D: 150 mm (thin), 200 mm (normal) and 250 mm (thick).

65


According to Rossiter (1964)’s formula, the frequencies of the hydrodynamic modes do not depend on

D nor on H; they depend only on the length of the opening (L). Secondly, the frequencies of the acoustic

modes do not depend on L; they depend only on the dimensions of the cavity: the bigger the cavity is,

the lower the modal frequencies are. These two considerations are the main concepts for the design of the

experimental rig.

Figure 6.4 shows some of the results of the parametric study performed during the design stage. For

simplicity, H was set to 492.5 mm and only L and D were allowed to variate. Three different opening lengths

(10, 40 and 70 mm) and three different cavity diameters (150, 200 and 250 mm) were analysed. In order to

keep the figure as simple as possible, the second and the third longitudinal modes were voluntarily omitted.

The lines representing the acoustic modes have been represented with a thickness which is proportional to

D. The thickness of the hydrodynamic mode curves is proportional to L.

The first longitudinal mode H1 depends only on the cavity depth which is constant in the example given

in figure 6.4: the three curves are indeed one over the other at 348 Hz.

If the rig had not been correctly designed, for example D = 150 mm and L = 70 mm, the frequencies of

the shear layer modes would not have been high enough (SL3 at 904 Hz for U∞ = 53.5 m/s) to excite the

first azimuthal mode (1340 Hz).

6.3 Opening details

In order to describe the effect of the opening length and location, a parametric study was performed.

Because of the small dimensions of the test section, no technical solution was found to gradually change

both the position and the streamwise length of the opening. Six different covering plates each one with

one opening were instead manufactured. By rotating a plate of 180deg, the position of the opening can be

changed. Therefore, with the six different plates, ten different cases can be studied. Figure 6.5 and table 6.1

represents and summarizes all the possible openings.

The covering plates were manufactured from a 7.75 mm thick Perspex panel. A single rectangular orifice

with sharp (45◦) chamfered leading and trailing edges was machined into each plate. All the openings have

a spanwise length of 40 mm and were centred in the spanwise direction. Their position is given by ∆, the

distance between the trailing edge of the opening and the downstream edge of the cavity (figure 6.5).

L (mm) 40 45

Case L40EU L40HU L40CC L40HD L40ED L45EU L45HU L45CC L45HD L45ED

∆ (mm) 189 159 99 39 9 184 157 97 37 9

Table 6.1: Cavity opening streamwise length and position.

66


Figure 6.5: Schematic of the opening

(a) (b)

Figure 6.6: Sketch showing the different rectangular openings. (a) L = 40 mm and (b) L = 45 mm. Cases: EU,

HU, CC, HD and ED. Refer to table 6.1 for the values of ∆.

67


6.4 Instrumentation

6.4.1 Pitot-static tube

The flow speed was measured with a Pitot-static tube located at the end of the test section and connected

to a micromanometer Furness Controls FCO 510. The probe was introduced in the wind tunnel through

a small hole on the upper wall of the diffuser section. During the PIV measurements, the Pitot tube was

removed from the test section in order to avoid flow disturbances. The micromanometer internally calculates

the velocity; a verification of the calculated mean velocity was done with a pressure transducer connected to

a liquid column manometer. The mean velocity given by the FCO 510 was found in good agreement with

the liquid column manometer measurements.

6.4.2 Microphones

Two 7 mm diameter G.R.A.S microphones (model 40PR) and fourteen Sennheiser KE4 electret microphones

were used to measure the sidewall pressure fluctuations. The GRAS microphones were connected to an

amplifier PCB Piezotronics 482A16. The Sennheiser microphones have a 20 - 20 000 Hz range and integrated

pre-amplifiers (Jordan et al. (2002)). They were calibrated before the experimental campaign in an impedance

tube with a white noise produced by a B&K Noise Generator (type 1405) up to 20 kHz. The impedance tube

has a radius of 25 mm: its cut-off frequency is therefore around 4 kHz. All the microphones were calibrated

with respect to one of the two G.R.A.S microphones. The transfer function is plotted in figure 6.7.

Figure 6.7: Transfer functions. Calibration of the 14 Sennheiser microphones.

68


6.4.3 Hot wire anemometry

A Dantec CTA Module 90C10 with a Dantec 50P11 single component hot-wire (HW) probe was used for

the boundary layer characterization. The velocity measurements are automatically compensated for flow

temperature changes through a thermocouple based system.

The HW probe was directly calibrated inside the test section and the procedure is described hereafter.

The probe mounted on its holder was introduced in the test section at the position desired for the BL

characterization. A Pitot tube was also introduced in the wind tunnel and placed near the HW probe. The

location of the Pitot tube in the same plane that the HW probe was not always possible because of the finite

number of holes to introduce the Pitot tube. After the calibration, the Pitot tube was removed from the

wind tunnel and the BL characterization was done straight after. The main advantage to this procedure is

that the calibration and the measurements are done with the same conditions (temperature, pressure, HW

orientation).

6.4.4 Particle image velocimetry

The flow in the orifice region was explored with a low speed LaVision PIV system. The seeding particles of

Di-Ethyl-Hexyl-Sebacat (DEHS), which had a typical particle size of 1 µm, were produced by an LaVision

aerosol generator. A double pulsed Nd:YAG laser was used to illuminate the plane of interest. Images were

taken with a digital Flow Master CCD camera equipped with a 1279× 1023 pixel CCD sensor and a 28 mm

focal length lens. The images were processed using Davis 7.2 software. The computed velocity fields were

then exported into Matlab for further post processing. The PIVMat Toolbox (www.fast.u-psud.fr/pivmat)

developed by Frederic Moisy from the Universite Paris-Sud was used for the production of the vector field

figures.

6.4.5 Data acquisition card

The output signals from the instruments were acquired using a National Instrument PXI-4472B Data Ac-

quisition Card. The three signals were sampled at a frequency of Fs = 40 kHz for 8 seconds. The pressure

signals were processed in Matlab using a Fast Fourier Transform (FFT) technique. The number of points

per segment was 8192, therefore the frequency resolution was 2.44 Hz. The 19 data sets were then averaged.

6.5 Phase-averaging technique

The highest recording rate of the camera is 4.03 Hz. This value is far below the range of frequency of the

shear layer modes [0 - 1500 Hz] predicted in Section 6.2. In order to provide an average view of the velocity

fluctuations, a phase-averaging technique was used. This method is applicable only when the pressure signal

has a single dominant peak in the spectrum. To identify the phase of the PIV images, the wall pressure

69


Figure 6.8: Schematic the PIV configuration

fluctuations and the laser trigger signal were recorded simultaneously. The Φ = 0◦ condition was chosen

at an arbitrary location during the pressure evolution (one quarter of period previous to the position of

the local maxima). The phase in degrees is defined as Φ = 360◦ τ/T (figure 6.9(a)) where τ is the relative

time at which the velocity is measured and T is the local period of the pressure fluctuations. In practice,

the signal from the microphone was filtered to reduce temporal disturbances. For every velocity studied,

1000 PIV images were acquired and divided in 8 different bins according on their phases. Each bin contains

between 100 and 150 PIV realizations as reported in figure 6.9(b). All the velocity fields of each bin are then

averaged in order to obtain a phase-averaged velocity field.

6.6 Boundary layer characterization

The knowledge of the boundary layer’s nature is of special interest when dealing with wall bounded flows.

For the boundary layer characterization, the panel with a rectangular orifice was replaced by a panel without

orifice. The boundary layer was measured in a position corresponding to the upstream edge of the openings

L40EU and L45EU. Two different free stream velocities were investigated: 7.2 m/s and 48.3 m/s. Velocity

profiles are reported in figure 6.10. At low velocities, the boundary layer was found to be laminar whereas

for high flow speed it becomes turbulent. Table 6.2 gives the characteristics of the boundary layer.

70


(a) (b)

Figure 6.9: (a) Temporal evolutions of the pressure signal and the laser trigger electric signal. The relative time

of the PIV realization τ and the local period of the pressure fluctuations T . (b) Histogram of 1000 PIV realizations

distributed into height different phase bins

Figure 6.10: Boundary layer profiles for two different flow velocities: 7.2 and 48.3 m/s. The Blasius (plain line) and

1/7 power profiles (dashed line) are also represented.

71


U∞ (m/s) δ (mm) δ∗ (mm) θ (mm)

7.2 3.98 1.34 0.50

48.3 3.6 0.51 0.40

Table 6.2: Boundary layer properties for 2 free stream velocities: 7.2 and 48.3 m/s. Boundary layer thickness δ (99 %),

displacement thickness δ∗ and momentum thickness θ.

72

Chapter 7

Acoustic mode calculation

7.1 Analytical solution of the Helmholtz equation

In chapter 3 the acoustic modes of an open-closed cylindrical cavity were estimated by analytically solving

the Helmholtz equation (eq. 3.3). In the case of a closed-closed cavity, the boundary conditions are simpler:

the wall-normal derivative of p has to vanishes at all the walls. Therefore the general expression for the

mode frequency is:

fm,n,q =ω

2π=

c

2π

((j′m,nR

)2

+(qπH

)2)1/2

(7.1)

where j′m,n is (n + 1)th positive zero of J ′m and m, n and q are the azimuthal, radial and longitudinal

order of the mode ((m,n, q) ∈ N3). The frequencies and the shapes of the first seven modes, calculated with

equation 7.1 and 3.4, are reported in table 7.1 and figure 7.1 respectively.

Mode H1 H2 AZ1 AZ1H1 H3 AZ1H2 AZ1H3

(m,n,q) (0,0,1) (0,0,2) (1,0,0) (1,0,1) (0,0,3) (1,0,2) (1,0,3)

Analytical (without opening) 350 699 846 915 1049 1097 1347

WEM 365 706 842 930 1054 1111 1354

Speaker 375 712 852 930 1054 1110 1350

Flow 377 706 851 927 1050 1102 1345

Panton & Miller (1975b) 364 699 − − 1049 − −Tang & Sirignano (1973) 364 706 − − 1052 − −

Table 7.1: Frequencies, in hertz, of the acoustic modes of a cylindrical cavity (D = 238 mm and H = 493 mm) at 22◦ C.

Mode order (m,n, q): azimuthal, radial, longitudinal.

73


(a) H1 (b) H2

(c) AZ1 (d) AZ1H1

(e) H3 (f) AZ1H2

(g) AZ1H3

Figure 7.1: Shape of the first seven acoustic modes of a cylindrical cavity with a closed bottom wall and a closed

top surface. The color bar gives the normalized real pressure values.

74


7.2 Wave Expansion Method (WEM)

In the previous section, the rectangular opening was neglected intentionally in order to calculate the modes

analytically. However, the presence of the opening modifies the shape and the frequency of the modes. In

order to characterize the acoustic behaviour of our experimental rig (cavity and test section), a numerical

simulation was performed. A highly efficient finite difference method originally introduced by Caruthers

et al. (1996) was used for the analysis. The approach uses wave functions which are exact solutions of the

governing differential equation. The wave expansion method (WEM) code used for this study was developed

by Ruiz & Rice (2002) and has been examined by Bennett et al. (2009) for its applicability in ducts.

7.2.1 Overview of the method

The framework of this method is the Helmholtz equation:

∇2p+ k2p = 0 (7.2)

The aim of the WEM is to solve this equation numerically for a finite number of point in a domain. The

pressure p at a given position ~x0 can be approximated locally by a combination of m plane waves

p( ~x0) =

m∑j=1

γje−i k ~dj · ~x0 (7.3)

where k is the wavenumber, ~dj is the unit propagation direction vector of the jth plane wave with complex

amplitude γj .

By using the matrix form, equation 7.3 can be re-written as

p0 = h0 γ (7.4)

where h0 is a row vector (1 × m) of the plane wave functions evaluated at ~x0 and γ is a column vector

(m × 1) of the wave strengths. The same approximation can be applied to other positions leading to

p = H γ (7.5)

where p is an (n × 1) vector of the pressures at each position i and Hij = e−i k~dj · ~xi

By combining equations 7.4 and 7.5,

p0 = h0 H+ p (7.6)

where H+ is a pseudo-inverse of matrix H. Equation 7.6 represents a so-called finite element template,

i.e. a parametrized algebraic forms that reduces to specific finite elements by setting numerical values to the

free parameters (Felippa & Onate (2007)). The raw vector (1 × n) formed by the product −h0 H+ is called

local stiffness κ0.

75


Figure 7.2: Computational domain meshed for the WEM calculation. Detail of the mesh in the opening area (section

taken in the middle plane).

Once the templates are formed (after the inclusion of the boundary conditions, Ruiz & Rice (2002)), an

overall sparse equation system may be assembled with each template independently contributing a row:

K p = f (7.7)

where K is the overall stiffness matrix and f represents a source vector.

7.2.2 Implementation

A three-dimensional unstructured mesh encompassing the wind tunnel test section and the cavity was gen-

erated with the commercial software Gambit, resulting in approximately 240 000 tetrahedral elements (fig-

ure 7.2). As the largest cell in the mesh measures 0.0185 m, the highest frequency of interest (1838 Hz) is

therefore resolved with at a minimum of 10 grid points per wavelength. In the opening region the mesh was

refined in order to have at least the same resolution as the PIV measurements.

The system was excited by a monopole source located in the center of the opening. A preliminary

parametric study showed that the location of the monopole source does not have any influence on the shape

and the frequency of the modes. The inlet and the outlet of the test section were modelled with radiation

boundary conditions.

The response of the system as a function of frequency was determined by running the code in a loop over

1000 different frequencies in the range [46 Hz - 1838 Hz].

76


7.2.3 Results

In the studied frequency range, 17 different acoustic modes were found. The shape of the first seven is

reported in figure 7.3. Contrarily to the open mouth case (chapter 3), here the test section does not have

an influence on the acoustic resonances of the cavity. The WEM study reveals indeed that there are no test

section/cavity modes excited by the acoustic source. The shapes of modes calculated by the WEM are very

similar to the shapes of the modes inside a completely closed cylinder (figure 7.1). In order to appreciate the

differences between these two cases, the real part of the complex pressure is analysed in the middle plane of

the cavity (figure 7.4). Unsurprisingly, the opening does not only modify the pressure in the orifice vicinity

as shown by the first longitudinal mode but in some cases the pressure pattern changes everywhere inside

the cavity: the third longitudinal mode clearly illustrates this shape distortion.

The frequencies of the WEM modes is given in table 7.1: small differences can be noticed when they

are compared with the analytically case. These differences are generated by the distortion of the pressure

pattern. The pressure vanishes in the test section which induces a pressure adaptation in proximity to the

orifice. The effects of the opening are similar to a reduction of the cavity depth. Consequently the frequencies

of the modes increase. An illustration of this is given by the mode AZ1H1 (figure 7.4(b)): there is a clear

virtual reduction of the cavity depth compared to the closed case.

7.3 Helmholtz resonance frequency

The frequency of the Helmholtz resonance was calculated with the classical Helmholtz theory and with

the Panton & Miller (1975b) and Tang & Sirignano (1973) transcendental equations which also predict the

frequencies of the longitudinal modes of the cavity. Table 7.1 summarizes the results for the longitudinal

modes and table 7.2 the frequency of the Helmholtz resonance. The Rayleigh end correction with an effective

radius was used to estimate the effective depth of the opening (neck length) for all the calculations:

leq = l + 2∆l = l + 2× 0.8242× 1.06S3/4U−1/2 (7.8)

Method Equation Frequency (Hz)

Prediction: Classical Helmholtz theory Eq. 1.9 73.2

Prediction: Improved Helmholtz theory Eq. 1.13 68.4

Prediction: Panton & Miller (1975b) Eq. 1.11 68.3

Prediction: Tang & Sirignano (1973) Eq. 1.12 68.3

Experimental: Speaker 84.2

Table 7.2: Frequencies, in hertz, of the Helmholtz resonance of the cylindrical cavity (D = 238.5 mm and H = 492.5 mm)

with a rectangular opening (L = 45 mm) at 22◦ C. Effective length leq = 45.9 mm.

77


(a) H1 (b) H2

(c) AZ1 (d) AZ1H1

(e) H3 (f) AZ1H2

(g) AZ1H3

Figure 7.3: Shape of the first seven acoustic modes. The color bar gives the normalized real pressure values.

78


(a) (b) (c)

(d) (e) (f)

Figure 7.4: Normalized real pressure on the middle plane of the cavity. WEM results (first row) and analytical

solutions (second row). Three acoustic resonance are given: (a and d) first longitudinal mode H1; (b and e) first

azimuthal-longitudinal mode AZ1H1; (c and f) third longitudinal mode H3. The position of the microphone for the

PIV campaign (section 8.2) is indicated by a black square.

79


7.4 Response of the resonator to an external excitation

7.4.1 Acoustic excitation

The cavity response to an acoustic excitation was experimentally investigated in order to validate the numer-

ical acoustic simulation. For this purpose, a small loudspeaker radiating broadband noise was used. After

trying different locations (inlet and outlet of the wind-tunnel), the lateral wall of the test section opposite

to the cavity was removed and replaced by the loudspeaker. A disadvantage of such a configuration is that

the influence of the closed test section on the acoustics could not be studied. However, as shown previously

by the numerical simulation, the test section/cavity mode should not be excited.

The transfer function between the speaker’s input signal and the pressure measured with a GRAS mi-

crophone on the walls of the cavity was calculated and its magnitude is plotted in figure 7.5. The WEM

transfer function between the monopole source and a location corresponding to the experimental microphone

position is also given. There is a good qualitative agreement between the experimental and the numerical

data. Table 7.1 gives the frequencies of the peaks in the experimental transfer function.

Figure 7.5: Response of the cavity to different excitations: monopole acoustic source (WEM, red), broadband noise

(experimental, black) and flow (experimental, blue).

7.4.2 Boundary layer excitation

The resonator was exposed to a flow at room temperature through the wind tunnel. The unsteady pressure

inside the cavity was measured with a flush mounted microphone. Figure 7.5 shows the pressure autospectrum

at a flow velocity of 21 m/s. The only conclusion given here is that the grazing flow excites all the acoustic

80


modes of the cavity. A deepest analysis is made is the next chapter.

81


82

Chapter 8

Experimental results

8.1 Response of the resonator to a grazing flow

8.1.1 Baseline opening: case L45EU

An automated velocity sweep of the tunnel was performed by controlling the centrifugal blower motor speed

with a LabView program. Results are presented in figure 8.1 together with of the shear layer modes calculated

according to the Rossiter (1964)’s formula (equation 1.1). Values of α = 0 and κ = 0.46 have been used here

as a best fit to the measured data.

In the flow speed range [4 - 9 m/s], the first shear layer mode is found to excite the Helmholtz resonance.

The pressure fluctuations over the walls of the cavity are reported in figure 8.2(a) for U∞ = 7.4 m/s. The

frequency of the dominant peak of the spectra is 67 Hz, which, if the frequency resolution of the sprectrum

(2.44 Hz) is considered, is in good agreement with the frequency found with equation 1.12. The spectral

analysis reveals that the pressure signal contains, apart from the harmonics of the Helmholtz resonance, the

acoustic modes of the cavity (H1, AZ1, ...).

For velocities between 28 m/s and 48 m/s, the first longitudinal mode H1 is excited by the shear layer

mode I. The pressure fluctuations on the sidewall of the cavity are strong for the whole flow speed range.

The case U∞ = 46.3 m/s is reported in figure 8.2(b). The oscillations are ten times higher than in the former

case (figure 8.2(a)). The pressure signal contains also the harmonics of the H1 mode: the sharp peak at 751

Hz, between H2 and AZ1, corresponds to the first harmonic of H1. The intensity of the second harmonic is

surprisingly as high as the first harmonic: this is mainly due to the fact that the frequency of the second

harmonic is very close to the AZ1H2 mode.

When the flow velocity reaches 48 m/s, the first hydrodynamic mode (SL1) is overtaken by SL2. This

switch occurs because the shear layer instability II gets closer to the frequency of a different acoustic reso-

nance: in this case, the combination of the first azimuthal and the first longitudinal mode (AZ1H1). It is

interesting to notice that two eigenmodes are skipped (H2 and AZ1). Figure 8.2(c) gives the pressure both

83


Figure 8.1: Acoustic response inside the cavity as a function of tunnel flow-speed. Superimposed on the plot are the

theoretical shear layer modes (SL), the WEM acoustic modes (H1, AZ1...) and the Helmholtz resonance (HR). The

first longitudinal cavity/test section mode (HW1) calculated analytically is also reported. Frequencies are given in

non-dimensional form (Helmholtz number = 2πRf/c).

in the time and in the frequency domain for a flow velocity of 48.4 Hz. Even if AZ1 is just 11 Hz higher

than the theoretical frequency of the shear layer instability II, the AZ1H1 mode, that is 89 Hz higher, is

the one coupled with the shear layer mode II. The vortex sound theory developed by Howe (1975) is a good

start in order to explain the predilection for certain specific eigenmodes for a flow-acoustic coupling. In the

Howe’s acoustic analogy, the Coriolis density forces ρ0 ~w × ~u are identified as the principal source of sound.

The acoustic power generated by the vortical field Π can be calculated by equation 1.14 which states that

the Π is proportional to the triple ~uacoust · (~w× ~u). From this formula it is clear that if the acoustic particle

velocity at the opening has the same orientation as the velocity or the vorticity, there is no acoustic power

generated. Let us now take the example of the first azimuthal mode (AZ1). It has been shown numerically,

in section 7.2.3, that this mode is symmetrical about plane Oyz (it could has been symmetrical about any

other vertical plane; however the opening location imposes the symmetry plane) and the 16 simultaneous

wall pressure measurements have confirm this (results not given here). The mode AZ1 does not radiate

because the acoustic particle velocity has the same orientation as the mean flow. This is a very simplistic

explanation especially because it assumes the directions of ~uacoust, ~w and ~u to be known a priori.

The hydrodynamic mode II remains locked on AZ1H1 mode until 51 m/s. When the flow speed reaches

51.5 m/s there is again a switch of predominant acoustic resonance. Figure 8.2(d) shows that the third

longitudinal mode dominates the spectrum. It has been found that for the highest blower rotational speed

(corresponding to U∞ = 52 m/s), this particular flow-acoustic coupling persisted. It is expected that for

84


(a) (b)

(c) (d)

Figure 8.2: Wall pressure recorded with a microphone (top) and the corresponding spectrum (bottom). The flow

velocity was fixed to 7.4 m/s (a), 46.3 m/s (b), 48.4 m/s (c) and 51.5 ms/s (d). The frequencies of the acoustic modes

numerically predicted (table 7.1) are also reported (H1, AZ1, ...). The frequency of the Helmholtz resonance (68.3

Hz). The first three hydrodynamic shear layer modes (I, II and III) calculated with the Rossiter formula (equation 1.1,

[α, κ] = [0, 0.42]) are plotted in red.

85


(a) (b)

Figure 8.3: Pressure amplitudes at 73 Hz (a) and 376 Hz (b) as a function of the wind tunnel flow speed. Opening

L45ED.

higher flow speeds the first shear layer will lock-on the second longitudinal eigenfrequency of the cavity.

8.1.2 Strength of lock-on

The receptivity of the cavity to shear layer excitation was quantified using a “strength of lock-on” parameter,

as suggested by Mendelson (2003) and described by Yang et al. (2009). The parameter chosen was the

amplification of the cavity pressure level above the background noise level, as defined by a linear scaling in

log-log space. For each frequency, a linear fit was made to the spectral density in decibels. The magnitude

of the pressure above this linear fit was designated as the “strength of lock-on” (SoL). At the flow speed

U∞, the SoL of the frequency f can be calculated through:

SoL(M∞, f) = pdB(M∞, f)− [pdB(1, f) + 20n log(M∞)] (8.1)

where pdB denotes the pressure amplitude (in dB) function of the frequency f and the Mach number of

the flow (M∞ = U∞/c) and n the exponential at which the broadband turbulent noise grows. Two examples

of the linear fitting are given in figure 8.3. The same procedure was applied to all the frequencies of the

spectra and the results are given in figure 8.4 in the form of a colour map.

8.1.3 Influence of the location of the opening

The effect of the opening location was quantified using the SoL parameter previously introduced. Three

different positions were explored: ∆ = 99, 39 and 9 mm (L40CC, L40HD and L40ED respectively). The

results are summarized in figure 8.5, where the contour lines encircle values of SoL higher than 13 dB. The

86


(a) (b)

Figure 8.4: Acoustic response (a) and the corresponding strength of lock-on (b) inside the cavity as a function of

tunnel flow-speed. The opening L40CC.

threshold criterion is an useful and straightforward technique to identify the resonance conditions in a flow

excited cavity.

There are some interesting differences between the three opening analysed. It is clear that the resonance

lock-on for H1 is much stronger when the opening is in the center of the cavity (L40CC): the first shear

layer hydrodynamic mode remains locked-on H1 for a wider range of velocities than for the other two orifice

positions. Especially noteworthy is the cut-on of the azimuthal mode AZ1H1 at velocities above 45 m/s for

L40ED and L40HD, which does not occur for L40CC. When the opening is off-center, the shear layer pressure

fluctuations tend to excite AZ1H1 because they are closer to an acoustic anti-node. On the contrary, the

central location is a pressure node for this acoustic mode as seen in chapter 7. The third longitudinal mode

is excited by the shear layer mode II from 50.3 m/s to 52.3 m/s for L40HD while this resonance occurs only

for the highest tested flow speed for L40ED.

Chanaud showed, first numerically and then experimentally (Chanaud (1994, 1997)), that a displacement

of the orifice away from the center results in decrease of the resonant frequency . In the present experiments,

a change of frequency is not observed. However, from figure 8.5, the Helmholtz resonance is more likely to

be excited when the opening is centred or half centred.

For low velocities (U∞ < 15 m/s), the amplitudes of the second and third shear layer modes are higher

when the opening is at ∆ = 39 mm. This observation could not be explained by the author and its

interpretation is a start of future studies.

87


8.2 Shear layer dynamics

Velocity measurements in the orifice region allow greater insight into the fluid dynamics of the shear layer

to be obtained. The four different velocities explored (7.4, 46.3, 48.4 and 51.5 m/s) have already been

presented in section 8.1.1. They correspond to: the first shear layer mode locked on the Helmholtz resonance

(figure 8.6), the first shear layer mode locked on the first longitudinal mode (figure 8.7), the second shear

layer mode locked on AZ1H1 (figure 8.8) and the second shear layer mode locked on H3 (figure 8.9) for the

baseline opening (L45EU). The phase-averaged velocity and vorticity fields are reported. Height different

phases were calculated during the processing but 45◦, 135◦, 225◦ and 315◦ were omitted for brevity’s sake.

8.2.1 First shear layer mode

Helmholtz resonance

At a velocity of 7.4 m/s, the first shear layer hydrodynamic mode locks on the Helmholtz resonance (fig-

ure 8.2(a)). The phase average velocity and vorticity fields are reported in figure 8.6. In the upstream

portion of the opening (−1 < x/L < 0), the shear layer appears to flap whereas in the downstream portion

(0 < x/L < 1) it rolls up into a single vortex. As the vortex is convected downstream along the cavity

opening, it grows and when it reaches the downstream edge, it splits in two parts: one part is captured by

the cavity while the other part escapes from it. The splitting mechanism is not very clear mainly because

the area under the edge is not correctly illuminated by the laser.

Acoustic mode H1

The phase averaged velocity and vorticity fields are given in figure 8.7 for U∞ = 46.3 m/s. During an acoustic

cycle, a single vortex is generated as in the previous case. In fact, the 7.4 m/s and the 46.3 m/s cases are

very similar even if the nature of the resonant acoustic mechanism is different: periodic compression of the

fluid inside the cavity in the former case as opposed to a standing acoustic wave in the second case. For

both velocities however the predominant instability in the shear layer is the first hydrodynamic mode.

8.2.2 Second shear layer mode

Acoustic mode AZ1H1

The phase averaged velocity and vorticity fields are given in figure 8.8. The dynamics is less clear than

for the two velocities formerly presented where the shear layer was rolling up into one single vortex per

acoustic cycle. Here instead, it seems to roll-up into two smaller vortices. The flapping noticed for the first

hydrodynamic mode is not discernible here. Furthermore, the air does not appear to enter into the cavity,

not even near the downstream edge.

88


Acoustic mode H3

In figure 8.9 the phase averaged velocity and vorticity fields are reported. During an acoustic cycle, two

vortices are generated in the shear layer. From the velocity fields, the rolling-up appears to take place in

the first half of the orifice, sooner than when the shear layer mode I is dominant. Again, the flow does not

seem to enter into the cavity at any phase of the cycle. The main difference between AZ1H1 and H3 is the

phase of the velocity fields: the φ = 180◦ for a flow speed of 48.4 m/s corresponds to the φ = 0◦ at 51.5

m/s. This change of phase is due to the position of the microphone (figure 7.4): the acoustic oscillations at

the opening are in phase with the fluctuations at the microphone location for the mode AZ1H1 and out of

phase for the mode H3.

8.3 Acoustic power

In this section the Howe (1975) vortex sound theory is applied to the opening region for the baseline orifice

(L45EU). The generation of acoustic power by the vortical field is calculated through the Howe’s integral

(equation 1.14). The velocity and the vorticity fields were extracted from the PIV data and the particle

velocity numerically from the WEM simulation. An important assumption in the development followed is

that the hydrodynamic and to acoustic fields can be computed independently. This assumption is motivated

by the fact that the orifice, i.e the region where the acoustics is generated, is compact compared to the

acoustic wavelengths.

8.3.1 Computation of the acoustic particle velocity

As seen in section 7.2, the acoustic simulation gives a complex pressure field whose amplitude depends on

the intensity of the monopole source. In order to scale the pressure, the experimental data acquired with a

flush-mounted microphone was used. It is assumed that the pressure inside the cavity has a simple harmonic

behaviour. This assumption is good for strongly resonant states. Therefore, the pressure Pacoust at any

point inside the cavity (x, y, z) and at any moment of time (t) can be expressed as:

Pacoust(x, y, z, t) = cst PWEM (x, y, z, facoust) sin(2πfacoust t) (8.2)

where, PWEM is the pressure from the numerical simulation, facoust is the frequency of the dominant

acoustic mode during the experimental testing and cst a constant necessary to match the experimental

pressure measurements. The calculation of the particle velocity field was done by integrating the linearized

Euler momentum equation:

ρ0∂~uacoust∂t

+∇Pacoust = 0 (8.3)

89


The integration is straightforward because the field is time harmonic. As an example, pressure and

velocity fields in the orifice region are given in figure 8.10 for the modes H1, AZ1H1 and H3. The area of

interest for this study is delimited by a rectangle. Even if inside the cavity the acoustic modes have very

different shapes (figure 7.4), locally, around the opening the acoustic velocity fields are very similar for the

three cases (figure 8.10).

8.3.2 Time-averaged acoustic power

Three free stream velocities were chosen for the study: 46.3, 48.4 and 51.5 m/s. These velocities correspond

respectively to the first shear layer mode (SL1) locked-on with the first longitudinal mode (H1), the second

shear layer mode (SL2) exciting the first azimuthal-longitudinal mode (AZ1H1) and the second shear layer

mode (SL2) amplified by the third longitudinal mode H3. The shape of these three acoustic modes is

displayed in figure 7.4 and 8.10. The instantaneous acoustic power was found for 8 different phases by

computing the integrand of equation 1.14. This intermediate result, even if essential for understanding of

the sound production, is not presented here for sake of brevity.

The acoustic power generated by the vortices in the orifice region over an entire acoustic period can be

obtained by averaging the computed instantaneous acoustic powers:

< Π >=< −ρ0∫V

~uacoust · (~w × ~u) dV > (8.4)

where <> denotes the time averaging over one period of oscillation.

The net acoustic energy E = <Π> /f produced by a free stream during an acoustic cycle of 46.3 m/s

(SL1-H1) is given in figure 8.11(a). The spatial distribution of the acoustic energy is characterized by a

distinct source-sink pair. This corresponds to the fact that when the first shear layer mode predominates, a

single large-scale vortex is generated during an acoustic period.

The two cases for which the second shear layer mode is dominant are presented in figure 8.11(b) and

8.11(c). It is interesting to observe the degree of similarity between these two patterns: two source-sink

pairs are found above the opening. Again, the shear layer rolling up into large-scale vortices generates this

pattern: the second shear layer mode produces two vortices during an acoustic cycle. Oshkai et al. (2008)

(alternatively Velikorodny et al. (2010)) also reported two source-sink pairs when calculating the acoustic

power for coaxial side branches for a dominant second shear layer mode (figure 1.12).

90


(a) L40CC (b) L40HD

(c) L40ED (d)

(e)

Figure 8.5: Contour of strength of lock-on higher than 13 dB. Three different orifice position are given: L40CC,

L40HD and L40ED.

91


Figure 8.6: Phase-averaged velocity (right) and vorticity (left) fields at the opening of the cavity. Four phases are

given: 0◦, 90◦, 180◦ and 270◦. Flow speed 7.4 m/s. Predominant shear layer mode: SL1. Predominant acoustic

mode: Helmholtz resonance.

92




mode: the first longitudinal resonance (H1).

93




mode: AZ1H1.

94




mode: the third longitudinal resonance (H3).

95


(a) (b)

(c) (d)

(e) (f)

Figure 8.10: Acoustic pressure field Pacoust (left), acoustic velocity field ~uacoust in green and streamlines in blue

(right) at the orifice region calculated with the WEM simulation and scaled by the experimental data. From top to

bottom: H1, AZ1H1 and H3. Pressure fields are given at φ = 90◦ whereas velocity fields at φ = 0◦.

96


(a)

(b)

(c)

Figure 8.11: Distribution of the net acoustic energy generated per acoustic cycle on the orifice. (a): 46.3 m/s,

SL1-H1. (b): 48.4 m/s, SL2-AZ1H1. (c): 51.5 m/s, SL2-H3.

97


98

Chapter 9

Conclusion

A cylindrical Helmholtz resonator was experimentally studied by wall pressure measurements and 2D PIV.

The designed experiment allows different resonant modes to be excited depending on both flow speed and

orifice location.

At low flow speeds, the Helmholtz resonance was found to be excited by the first shear layer hydrodynamic

mode at the frequency predicted by an improved Helmholtz resonance formulation.

For higher velocities, lock-on between the first two shear layer hydrodynamic modes and different eigen-

modes of the cavity was observed. The location of the cavity’s opening was found to be a major factor in

determining which acoustic mode is excited. Specifically, a combination mode (azimuthal and longitudinal)

was found to generate lock-on only when the opening was located off-center. This was ascribed to the fact

that the main axis of the cavity is a pressure node for the first azimuthal mode.

The dynamics of the shear layer were then explored through PIV measurements for four flow speeds

corresponding to strongly resonant cases. Phase averaged PIV allowed the coherent structures present in

the shear layer to be examined and the interaction between the cavity resonances and the shear layer to be

analysed. The first hydrodynamic mode is characterized by a flapping movement in the upstream portion

of the resonator’s opening and by the rolling up of the shear layer in a single vortex in the downstream

portion of the orifice. The number of vortices generated per acoustic cycle increases to two when the

second hydrodynamic instability becomes dominant. This observation is consistent with previous studies.

The experimental results show that the order of the dominant acoustic resonant mode does not affect the

organization of the shear layer. A possible explanation is that in the opening region, the acoustic velocity

fields are similar.

The vortex sound theory of Howe was applied in order to characterize the energy transfer mechanisms

on the orifice region. The velocity and the vorticity fields were extracted from the PIV data and the particle

velocity was calculated using a Wave Expansion Method (WEM) simulation. Three different flow conditions

generating acoustic resonance were analysed. For each case, the acoustic sources were localized revealing

99

9.Conclusion

that the spacial organization of the sound production depends exclusively on the predominant shear layer

hydrodynamic mode.

100

Chapter 10

Summary

The main objective of this research was to increase the understanding of mechanisms governing flow-induced

resonance in cylindrical cavities. The dissertation was divided into two main parts corresponding to the

two cases studied: an open mouth cylindrical cavity and a cylindrical Helmholtz resonator with a rectan-

gular orifice. Specific conclusions have been provided in chapters 5 and 9 and a summary of the major

accomplishments is given hereafter.

By using different measurement techniques (hot-wire anemometry, Particle Image Velocimetry and wall

mounted microphones), the shear layer hydrodynamics modes were identified and characterised. Extensive

data were obtained over a broad subsonic speed range for the two cavities studied. In both cases, the

classical formulation for the prediction of the frequency of the sheartones proved to be in agreement with the

experimental data. The constant accounting for the phase delay in the Rossiter (1964)’s feedback mechanism

was found to be negligible for the low Mach numbers tested. The description of the shear layer over the

Helmholtz resonator was achieved by phase averaging the PIV data. The first shear layer mode was found

to correspond to one single vortex whereas the second mode to two vortices as described by previous studies.

The first hydrodynamic mode is also characterized by a flapping movement of the shear layer especially on

the upstream portion of the opening. An important result is that the dominant acoustic mode does not have

an influence on the shear layer morphology.

The acoustic resonances of the two test rigs were investigated numerically and good agreement was found

with pressure measurements. When the cavity opening is partially covered, the shear layer hydrodynamic

modes tend to couple with specific eigenmodes of the cavity. In particular, lock-on was detected at three

different acoustic modes using a “strength of lock-on” parameter. The position of the orifice has an influence

on the flow-acoustic coupling: a combination of the first longitudinal and the first azimuthal modes is strongly

excited when the opening is off-center of the cavity axis.

The effect of the confinement has a substantial influence on the acoustics of the flow excited open mouth

cavity: lock-on between the shear layer modes and the acoustic resonances of the test section was reported.

101

10.Summary

The common characteristic of the lock-on modes is the quarter wave length shape inside the cavity.

A semi-empirical approach was used to estimate the acoustic power generated by a flow-excited Helmholtz

resonator. The vortex sound theory of Howe (1975) was used to quantify the energy transfer between the

acoustic field and the turbulent flow. Velocity field measurements in conjunction with a numerical simulation

of the acoustic field were used for the calculation of the physical quantities in the Howe’s integral. The spatial

distribution of the acoustic power generated over the orifice was found to depend only on the predominant

shear layer hydrodynamic mode: the first sheartone produces one acoustic source-sink pair whereas two

source-sink pairs were identified when the second sheartone is dominant. Results show this approach to be

promising for the understanding of flow-induced acoustic resonance.

The presented experimental work also led to the description of some flow features of an open mouth

cylindrical cavity past by a grazing flow. These results were obtained using hot-wire anemometry, 2-D PIV

and wall pressure measurements. The three-dimensionality and the symmetry of the flow, expected for the

aspect ration studied, was confirmed. The velocity measurements, obtained from hot-wire anemometry using

a single wire probe on the shear layer and the wake of a relatively deep open mouth cylindrical cavity (H/D

= 1.357), are the first of this kind in the literature because they concern planes normal to the streamwise

direction. They provide a robust database for CFD validation. Other original results that can be used for

comparison, are the velocity fields at different horizontal planes inside the cavity (PIV) and the fluctuating

pressure levels at the walls of the cavity.

102

Appendix A

Acoustic power

By following the vortex sound theory of Howe (1975, 1980), an expression for the acoustic power generated

is derived bellow.

The total fluid velocity ~u can be decomposed in an irrotational part∇ϕ and a rotational part ~urot = ∇× ~ψ:

~u = ∇ϕ+∇× ~ψ (A.1)

A homentropic flow satisfies Crocco’s form of the Euler momentum equation:

∂~u

∂t+∇B = −~ω × ~u (A.2)

where B is the total enthalpy and where the vorticity ~ω is defined by ~ω = ∇× ~u. Note that the friction

and heat transfer are neglected.

By using the decomposition of ~u, equation A.2 becomes:

∂~urot∂t

+∂∇ϕ∂t

+∇B = −~ω × ~urot − ~ω ×∇ϕ (A.3)

By taking the scalar product of ~urot and by simplifying:

1

2

∂

∂tu2rot + ~urot · ∇

(∂ϕ

∂t+B

)= −~urot · (~ω ×∇ϕ) (A.4)

This equation can be simplifies further by recalling that ∇ · (~urot) = 0 :

1

2

∂

∂tu2rot +∇ ·

(~urot

(∂ϕ

∂t+B

))= −~urot · (~ω ×∇ϕ) (A.5)

By integrating over a volume:

1

2

∂

∂t

∫V

u2rotdV +

∫V

∇ ·(~urot

(∂ϕ

∂t+B

))dV = −

∫V

~urot · (~ω ×∇ϕ)dV (A.6)

103

A.Acoustic power

The divergence theorem states that:

∫V

∇ ·(~urot

(∂ϕ

∂t+B

))dV =

∫S

(~urot · ~n)

(∂ϕ

∂t+B

)dS (A.7)

The volume V chosen for the integration is a large sphere whose radius can be set arbitrarily. For a large

sphere’s radius, the normal component of ~urot vanishes on the surface S and therefore:

1

2

∂

∂t

∫V

u2rotdV = −∫V

~urot · (~ω ×∇ϕ)dV (A.8)

Furthermore, by assuming small acoustic perturbation, the unsteady irrotational part of the velocity field

can be defined as the acoustical velocity: ~uacoust = ∇ϕ.

1

2

∂

∂t

∫V

u2rotdV = −∫V

~urot · (~ω × ~uacoust)dV (A.9)

By multiplying this equation by ρ0 and by rearranging its right hand side:

ρ01

2

∂

∂t

∫V

u2rotdV = ρ0

∫V

~uacoust · (~ω × ~urot)dV (A.10)

The left hand side of the equation gives the rate at which the the vorticity-bearing part of the velocity field

generates kinetic energy. Right hand side represents the work performed by the lift ρ0~ω×~urot experienced by

vortex elements in the velocity field of the sound. Equation A.10 quantifies Πabs, the rate at which acoustic

energy is absorbed by the vortical field. Therefore the acoustic power generated by the flow is given by:

Π = −ρ0∫V

~uacoust · (~w × ~urot) dV (A.11)

104

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112

List of publications

Journal Publications

1. Rodriguez Verdugo F., Guitton A., Camussi R., Experimental investigation of a cylindrical cavity in a

low Mach number flow, Journal of Fluids and Structures 28, pp. 1-19, January 2012.

Conference Proceedings

1. Rodriguez Verdugo F., Camussi R., Bennett G.J., Aeroacoustic source characterization technique ap-

plied to a cylindrical Helmholtz resonator, International Conference on Sound and Vibration, Rio de

Janeiro, 10-14 July 2011.

2. Rodriguez Verdugo F., Bennett G.J., Stephens D.B., Dynamics of the shear layer in the orifice of a

cylindrical Helmholtz resonator using PIV, XVIII A.I.VE.LA. National Meeting, Rome, Italy, 15-16

December 2010.

3. Bennett G. J., Rodriguez Verdugo F., Stephens D. B., Shear layer dynamics of a cylindrical cavity

for different acoustic resonance modes, 15th Int. Symp. Appl. Laser Techn. Fluid Mech., Lisbon,

Portugal, 05 - 08 July 2010.

4. Stephens D. B., Rodriguez Verdugo F., Bennett G. J., Shear layer driven acoustic modes in a cylindrical

cavity, 16th AIAA/CEAS Aeroacoustics Conference, Stockholm, Sweden, 07 - 09 June 2010.

5. Rodriguez Verdugo F., Guitton A., Camussi R., Di Marco A., Grottadaurea M., Investigation of the

flow and the acoustics generated by a cylindrical cavity, 16th AIAA/CEAS Aeroacoustics Conference,

Stockholm, Sweden, 07 - 09 June 2010.

6. Verdugo F.R., Camussi R., Guitton A., Experimental characterisation of a cylindrical cavity in a low

Mach number flow, XX AIDAA Congress, Milano, Italy, July 2009.

7. Rodriguez Verdugo F., Guitton A., Di Marco A., Camussi R., Aeroacoustic characterization of a

cylindrical cavity, XVII A.I.VE.LA. National Meeting, Ancona, Italy, 26 - 27 November 2009.

113

List of publications

8. Rodriguez Verdugo F., Guitton A., Camussi R. Grottadaurea M., Experimental investigation of a

cylindrical cavity, 15th AIAA/CEAS Aeroacoustics Conference, Miami, USA, 11 - 13 May 2009.

114

rodriguez verdugo (2012 phd) - experimental investigation of flow past open and partially covered...

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