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Budapest University of Technology and Economics

Department of Hydrodynamic Systems

Flow and acoustics of the edge tone configuration

Written by: Istvan Vaik

June 8, 2013

Supervised by: Gyorgy Paal, Ph.D.

Ph.D. dissertation


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Abstract

In this Ph.D. dissertation the well-known low Mach number edge tone configuration is investigated. As

described in Chapter1 the edge tone has been the subject of research for more than one hundred years,

still the phenomena are not fully understood.

In Chapter2 the flow of the edge tone is investigated by experimental and numerical means. Beside

several minor phenomena (e.g. the frequency drop of the first stage in the multi stage coexistence mode)

the result of a detailed parametric study to explore the Reynolds number and dimensionless nozzle-to-

wedge distance dependence of the Strouhal number in the case of top hat and parabolic edge tones is

presented. Also the hysteresis, the mode switching and the jumps between the stages of the edge tone

are studied. Moreover, it is also shown that the phase of the jet disturbance between the nozzle and the

wedge does not vary linearly with the distance from the nozzle, thus the convection velocity of the jet

disturbance is not constant, as it is usually assumed in the theoretical models.

The edge tone is a planar flow that under certain circumstances generates an audible tonal sound

with three-dimensional dipole characteristics. To numerically calculate its sound production, a CFDsimulation has to be coupled to an acoustic simulation. Until now, in spite of the planar nature of the

flow the calculation of the 3D acoustic field of a planar flow was only achievable by carrying out a 3D

CFD simulation that is highly inefficient. In Chapter 3 a new method is proposed with which a 2D CFD

simulation can be coupled to a 3D acoustical simulation. Then the presented newly developed method is

applied to investigate the acoustic attributes of the edge tone.

At last, Chapter 4 opens a window to further research possibilities. A well known real-world

appearance of the edge tone, the flue organ pipe is investigated. Since for the subjective perception

of the sound the attack transient is crucial, that is mostly determined by the specialised edge tone

configuration formed by the jet blowing from the foot of the organ pipe and the upper lip (which acts

as a wedge in the system) only the foot model is investigated (i.e. without the resonator). It is shown

that the strongest and most stable edge tone oscillation occurs (and thus the strongest and most stable

sound production can be achieved) if the upper lip is placed exactly in the centreline of the jet. The main

outcome of this chapter is that the flow in the foot and mouth of an organ pipe can be reliably simulated

with a commercial CFD code which is of great value in organ pipe research.

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Kivonat

Ebben a Ph.D. disszertacioban a jol ismert alacsony Mach szamu elhang konfiguraciot vizsgalom. Amint

azt az 1. fejezetben bemutatom, az elhangot mar tobb mint szaz eve vizsgaljak, megis jelensegeit meg

mindig nem teljesen ertettuk meg.

A2. fejezeteben az elhang aramlasi jelensegeit vizsgalom kserleti es numerikus eszkozokkel. Szamos

kisebb jelenseg (mint peldaul az elso modus frekvenciajanak lecsokkenese kevert modusok eseten) mellett

bemutatom az elhang Strouhal szamanak Reynolds szamtol es dimenziotlan fuvoka-ek tavolsagtol valo

fuggeset feltaro reszletes parametertanulmany eredmenyet. Az elhang modusaiban bekovetkezo hisztere-

zist, modusugrast es modus kapcsolasokat is tanulmanyozom. Tovabba megmutatom, hogy a szabadsugar

zavarasanak fazisa nemlinearis modon valtozik a fuvoka es az ek kozott, tehat a zavaras terjedesi sebessege

nem allando, mint ahogy azt altalaban feltetelezik a jelenseget lero elmeleti modellek.

Az elhang egy skaramlas ami bizonyos korulmenyek kozott haromdimenzios dipolus karakterisztikaju

hallhato hangot hoz letre. Az elhang altal letrehozott hang numerikus kiszamtasahoz egy CFD szimulaciot

kell egy akusztikai szimulacioval osszekapcsolni. Eddig ehhez annak ellenere kellett 3D CFD szimulaciotvegezni, hogy az aramlas skaramlas jellegu, ami nagy mertekben megnoveli a szamtas eroforrasigenyet.

A3.fejezetben egy uj modszert javaslok, amivel 2D CFD szimulaciot lehet 3D akusztikai szimulacioval

kapcsolni. Majd a bemutatott modszerrel vizsgalom az elhang akusztikai tula jdonsagait.

Vegezetul a 4. fejezet uj lehetosegekre mutat ra a tovabbi kutatasok teren. Egyik jol ismert valos

eletbeli megvalosulasa az elhang jelensegnek az orgonaspban kialakulo aramlas. Mivel a szubjektv

hangerzekeles szempontjabol a kezdeti tranziens nagy jelentoseggel br, amit leginkabb az orgonasp

lababol kilepo szabadsugar es a felso ajak altal letrehozott specialis elhang konfiguracio hataroz meg,

ezert csak az orgonasp labanak modelljet vizsgalom (azaz rezonator nelkul, csak az orgonasp labanak

es a felso ajaknak kornyezetet). Megmutatom, hogy a legerosebb es legstabilabb elhang konfiguraciot, es

ezaltal a legerosebb es legstabilabb hangkepzest akkor erheto el, ha a felso ajkat pontosan a szabadsugar

kozepvonalaba helyezzuk. A fejezet legfobb, az orgonakutatas szamara nagy jelentosegu eredmenye, hogy

megmutatja, hogy kereskedelmi CFD szoftverrel megbzhatoan szimulalhato az orgonasp labanak es

szajreszenek kornyeken kialakulo aramlas.

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Acknowledgements

I would like to express my deep and sincere gratitude to my supervisor, Gorgy Paal. His help, stimulating

suggestions and encouragement were invaluable for me during my research.

I am also indebted to Gisbert Stoyan, the supervisor of my MSc Thesis at the E otvos Lorand

University. He was not only supervising my MSc thesis, but he was also the one who arose my interest

in numerical methods. I am really grateful for the consultations with him during my Ph.D. research.

The acoustic part of my dissertation would not have been possible without the acoustic solver (part

of CFS++) developed by Manfred Kaltenbacher and his group. Special thanks must go to Simon Trieben-

bacher who always had enough patience to solve the emerging problems I encountered during the coupled

simulations. I also would like to express my thanks to the staff at Erlangen (Stefan Becker, Irfan Ali and

Max Escobar) for their cooperative work.

This dissertation would not have been possible without my family. My wife, Sarolt and our sons, Vince

and Lorinc gave me encouragement day-by-day, and always cheered me up when I was a bit annoyed. I

also received indescribably much help from my parents who heartened me to start and finish the Ph.D.study.

Last but not least I have to thank God for giving me always what I need, would it be strength,

courage, patience or wisdom. AMDG

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Nomenclature

Lists of abbreviations and symbols (with their SI units) used throughout the dissertation are collected

in the following pages. Nomenclatures that were only used during the linear algebraic deduction for the

necessary time step (pages2023) are collected in a separate group.

Abbreviations

2D two-dimensional

3D three-dimensional

avr average (mean) value

CAA Computational AeroAcoustic

CFD Computational Fluid Dynamics

cmv cubic mean value

DES Detached Eddy Simulation

DNS Direct Numerical Solution

est.rel.err. estimated relative error

FFT Fast Fourier Transformation

LES Large Eddy Simulation

PML Perfectly Matched Layer

qmv quadratic mean value

rms root mean square

SAS Scale Adaptive Simulation

Nomenclature used only in pages2023

j see eq. (2.12)

j

see page21

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A see eq.(2.12)

Bj see eq.(2.12)

ej accumulated global error of the Second Order Backward Euler method after thejth time

step

f(t, y (t)) right-hand side of the ordinary differential equation used in the time step study (eq.2.4)

gj local error of the Second Order Backward Euler method in thejth time step

I identity matrix

Lf Lipschitz constant off(t, y (t))

N index of the last time step

S the matrix with which S AS1is the Jordan normal form ofA

v amplitude of the velocity oscillation

vj see eq.(2.12)

y (t) solution of the initial value problem of the ordinary differential equation (2.4)

y0 initial value condition of equation (2.4)

yj numerical approximation ofy (tj)

Symbols

/2 number of element layers in the source region of the acoustic mesh (half space only because

of the symmetry condition) -

width of the jet m

vc width of the jet in the vena contracta point (in Section4.2) m

phase lag in the feedback loop (eq. (1.17)) -

scaling factor during the 2D CFD 3D CAA simulations, when the sources are extruded,= wacou/wCFD -

c scaling factor during the 2D CFD 3D CAA simulations, when the sources are concentrated

on the symmetry plane, c = W/wCFD -

acoustical wavelength m

wavelength of jet disturbance m

dynamic viscosity kg/ms

kinematic viscosity m2/s

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angular frequency rad/s

angle between the centreline of the jet and the y axis (in Section4.2) rad

Phase delay relative to the reference point rad

/2 -

density of the fluid kg/m3

density perturbation kg/m3

0 density of the fluid at a reference state kg/m3

time step s

ij elements of the stress tensor Pa

angle between the position vector of the observation point and they axis rad

A area of the cross section of the nozzle m2

a0 speed of sound m/s

b half-thickness of the jet m

c coefficient ofS t (Re) in eq. (2.43) -

c1, c2, c3 coefficients ofSt (Re, h/) in eq. (1.16) or (2.48) -

d coefficient ofS t (h/) in eq. (2.45) -

ef error of the frequency measurement Hz

e error of the jet width measurment m

e m error of the mass flow rate measurement kg/s

eA error of the nozzle cross section measurment m2

eh/ error of the dimensionless nozzle-to-wedge distance measurement -

eh error of the nozzle-to-wedge distance measurement m

eRe error of the Reynolds number measurement -

eSt error of the Strouhal number measurement -

F force acting on the wedge N

F in Lighthills analogy (eq. (3.2)), external force density N/m3

f frequency of oscillation Hz

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f Frequency resolution of a spectrum Hz

Fy amplitude of the force acting on the first 7.2 mm of the wedge NFx the x component of the force acting on the upper lip N

G strength of dipole sound source NG(t) instantaneous strength of a dipole sound source (eq. (3.8)) N

h nozzle-to-wedge distance m

h the length of an element between the nozzle and the wedge m

h/ Dimensionless nozzle-to-wedge distance -

H1, H2, V sizes of the CFD domain of the 2D edge tone simulation (Figure2.1) m

k exponent of h or h/ in the h dependence of f (eq. (1.15)) or h/ dependence of St

(eq. (1.16)) -

L cut-up length (organ pipe foot model) m

l number of layers in that the acoustical sources are extruded -

m order of accuracy -

m mass flow rate kg/s

min mass flow rate that is injected into the system through the nozzle kg/s

Mz torque Nm

Ma Mach number,Ma= ua0 -

n ordinal number of the stage -

nv number of vortices passing at a fixed spatial point next to the wedge -

p pressure Pa

p amplitude of the acoustic pressure fluctuation Pap pressure perturbation Pa

p0 pressure of the fluid at a reference state Pa

Q mass source kg/m3s

q refinement ration between meshes during the mesh study (Secton2.1.1) -

R correlation coefficient b

r distance from the dipole sound source m

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r distance from the tip of the wedge m

R2 coefficient of determination -

Re Reynolds number,Re = u -

S dimensionless oscillation frequency as defined by Crighton (eq.(1.8)), S= bu -

Sel surface of an element in the mesh m2

St Strouhal number,S t= fu -

St coefficient ofS t (h/) in eq. (2.45) -

St coefficient ofS t (Re) in eq. (2.43) -

T in eq. (2.34), temperature C

T period time s

t time s

T size of the window during the sliding window Fourier transformation s

t time step of the signal s

tj time value of thejth time step, tj =j s

To oscillating part of the simulation s

TS duration of simulation in simulated time s

ts elapsed time during the mass flow rate sensor calibration s

Tt transient part of the simulation before the quasi-steady oscillation sets in s

tv elapsed time during the frequency measurement with the vortex counting method s

tw elapsed time between two subsequent photographs s

Tij elements of Lighthills stress tensor Pa

u mean exit velocity of the jet m/s

uc centreline velocity of the jet m/s

uconv convection velocity of jet disturbance m/s

uvc jet velocity in the vena contracta point (in Section4.2) m/s

v velocity vector,v = [v1; v2; v3] [m/s; m/s; m/s]

v1

, v2

, v3

velocity component in thex, y and z direction, respectively m/s

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Vel volume of an element in the mesh m3

W height of the flow m

wacou height of an element in the source region of the acoustic mesh m

wCFD thickness of the element layer in the 2D CFD simulation m

x the distance that the disturbance travels between two subsequent photographs

(in Chapter2) m

x upper lip offset measured from they axis (in Chapter4) m

xF mean value ofxF m

x,y,z Cartesian coordinate directions

x0 the x position of the jet centreling at y = 0 (in Section 4.2) m

xF instantaneous point of force action m

xr the x position where the jet velocity is r times of the maximum (in Section 4.2) m

SPL Sound Pressure Level, SPL = 20 lgp/20 Pa dB

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Contents

Abstract i

Kivonat iii

Acknowledgements v

Nomenclature vii

1 Introduction 1

1.1 What is the edge tone?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Numerical simulations and other research . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Frequency and phase characteristics of the edge tone . . . . . . . . . . . . . . . . . . . . . 10

1.4 Why the edge tone? Aims of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 The flow of the edge tone 15

2.1 The CFD setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Two-dimensional simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Three-dimensional simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 The experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Experimental system and instrumentation . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Derived quantities and their error estimation . . . . . . . . . . . . . . . . . . . . . 28

2.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.1 Varying the Reynolds number at a fixed geometric configuration . . . . . . . . . . 35

2.3.2 Varying the nozzle-to-wedge distance. . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.3 St (Re; h/) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.3.4 Stage jumps and mode switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3.5 Miscellaneous CFD results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.4.1 Theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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2.4.2 Tezisek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3 Acoustics of the edge tone 67

3.1 Hybrid Computational AeroAcoustic methods . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.1.1 Lighthills analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2 Basic ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.1 Computational algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Definition of test setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.3.1 Computational domains for the acoustic simulation. . . . . . . . . . . . . . . . . . 73

3.3.2 2D CFD with 3D acoustics - the treatment of sources . . . . . . . . . . . . . . . . 75

3.3.3 3D CFD with 3D acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4.1 2D-2D: acoustic mesh study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.4.2 3D-3D computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4.3 2D-3D: source extrusion with different numbers of layers. . . . . . . . . . . . . . . 77

3.4.4 2D-3D: comparison with theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4.5 2D-3D: different Reynolds numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.4.6 The computational costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.5.1 Theses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5.2 Tezisek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4 CFD simulations on an organ pipe foot model 874.1 Geometrical configuration, boundary conditions, mesh and solver settings . . . . . . . . . 87

4.2 Free jet simulation: velocity profiles and jet centreline . . . . . . . . . . . . . . . . . . . . 91

4.3 Edge tone simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.4.1 Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.2 Tezis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Bibliography 101

Own publications 105

Appendix 107

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Chapter 1

Introduction1.1 What is the edge tone?

The edge tone is one of the simplest aero-acoustic flow configurations. It consists of a planar free jet that

impinges on a wedge-shaped object (traditionally called the edge). The main parameters of an edge tone

configuration are the mean exit velocity of the jet (u), the width of the jet () and the nozzle-to-wedge

distance (h) (Figure1.1). Secondary parameters may also influence the flow, such as the velocity profile

of the jet (top hat and parabolic profiles are the most common ones), the offset of the wedge from the jet

center line, the shape of the nozzle or the angle of the wedge. Despite its geometric simplicity the edge

tone displays a remarkably complex behaviour. Under certain circumstances a self-sustained oscillation

evolves with a stable oscillation frequency. The oscillating jet creates an oscillating force on the wedge,

that generates a dipole sound source, that under certain circumstances creates an audible tone.

Figure 1.1: Snapshot from a CFD simulation of a first stage edge tone flow with the main parameters of

the edge tone configuration ( width of the slit on the nozzle; h nozzle-to-wedge distance; u mean

exit velocity of the jet)

The oscillating jet can take different shapes, these are called the stages of the edge tone. Their ordinal

number corresponds roughly to the number of half waves between the nozzle and the wedge. Figure 1.1

shows a snapshot from the result of a computational fluid dynamics (CFD) simulation of a first stage

edge tone flow. The flow is visualised by virtual smoke introduced in the central part of the nozzle.

Figure 1.2(a) shows what happens with the oscillation frequency when the velocity is varied in a

fixed geometrical configuration (at constant and h values). At low velocities the wedge cuts the jet in

half and a steady flow is formed. Increasing the velocity above a certain threshold velocity that value

is of course dependent on the geometric configuration the first stage of the edge tone sets in (position

A). Further increasing the jet velocity the second stage comes into being with a sudden jump in the

frequency to a higher value (positionB). Usually the first stage still coexists with the new, second stage,

thus a multi-stage operation mode can be observed, but the second stage can be present purely as well.

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CHAPTER 1. INTRODUCTION

(a) mean exit velocity of the jet (b) nozzle-to-wedge distance

Figure 1.2: Characteristics of frequency variation as a function of the

At higher velocities the third stage of the edge tone is formed (position C) again with a sudden jump

in the frequency to a higher value either with some of the lower stages coexisting, or purely. Further

increasing the velocity, even higher stages may evolve, with a jump in the frequency to a higher value at

the onset of each new stage. Similar behaviour can be observed when the velocity of the jet is decreased.

The frequency of the oscillation decreases, and at a point the highest stage disappears (at positions C

andB the third and the second stage disappears, respectively) with a sudden drop in the frequency and

at last at low jet speeds the first stage of the edge tone disappears and a steady flow is formed (position

A). It can be that the velocity value at the point where a stage disappears during the decrease of the

jet velocity (position C, B or A) may differ from the velocity value at the point where this stage first

appeared when the velocity was increased (position C, B orA), so hysteresis may occur.On the other hand, when the nozzle-to-wedge distance is varied while the velocity of the jet is kept

constant (Figure1.2(b)), the following can be observed. At low distances no oscillation occurs. Increasing

the distance, at a certain lower limiting value the first stage of the edge tone forms (position A). Further

increasing the distance, the frequency of the oscillation decreases. At a point the second stage sets in

with a sudden jump in the frequency to a higher value (position B). Further increasing the distance, the

frequency again decreases until the next stage forms with the sudden jump in the frequency again to a

higher value (position C). Now, if the nozzle-to-wedge-distance is decreased from a higher value, then

the frequency of oscillation increases until a certain position where the jet jumps back to a lower stage

where the frequency is also lower (position C and B) and at last the oscillation disappears completely

(position A). Just as in the case when the velocity is varied hysteresis may occur, it can be that the

jump between the stages forth and back are at different positions (A =A orB=B or C=C).

The following dimensionless numbers will be used throughout my dissertation:

- Reynolds numberbased on the mean exit velocity of the jet and the width of the jet will be used

as the dimensionless jet velocity:Re = u- Strouhal number based on the frequency of oscillation (f), the width of the jet and the mean

exit velocity of the jet will be used as the dimensionless oscillation frequency: S t= fu- h/ will be used as the dimensionless nozzle-to-wedge distance

Because of scaling laws, two edge tone configurations with different jet velocities and geometric sizes,

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1.2. LITERATURE OVERVIEW

but at the same Reynolds number and h/ dimensionless nozzle-to-wedge distances produce oscillations

with different frequencies but with the same Strouhal numbers, therefore when comparing results from

different sources (theoretical and/or experimental and/or numerical) comparison of the Strouhal numbersat same Reynolds numbers and at same h/values will be carried out.

1.2 Literature overview

Brown[1, 2] gives a detailed overview on the early research on the edge tone phenomenon from the first

80 years after it was first noted by Sondhaus in 1854. Several researchers tried to explain theoretically

the mechanism of the edge tone production, others made extensive experimental investigations in the

field. Without being exhaustive I shall give a short introduction to the most important studies. At first,

I shall review the theories (in chronological order), then the experimental and numerical studies. There

are a couple of other experimental works carried out to investigate some specific aspects of the edge tone.

Some of them will be cited in the corresponding chapters.

1.2.1 Theories

Curle [3]published his purely hydrodynamic, vortex theory explaining the edge tone in 1953. He claims

that vortices of opposite circulation are produced at the nozzle (embryo vortex) and at the tip of the

wedge (secondary vortex) at the same time. The formation of the secondary vortex takes place when

the transverse velocity at the tip of the wedge is maximal, that occurs halfway between the alternate

vortices below and above the wedge. Thus the relationship between the nozzle-to-wedge distance and

the wavelength is: h = n+ 14, where n denotes the ordinal number of the stage and the distance

between two consecutive vortices on the same side of the jet. Independently from this result he deduced a

semi-empirical formula for the velocity with which the vortices moves ( uconv) in the case when h/ >10:

uconvu

= 1

2

1

30

, (1.1)

thus the frequency of oscillation of the nth stage is:

f= 1

2u

n+ 1

4

h 1

30

(1.2)

He also emphasises that if Savics [4] result of uconv

u = 1.024 is used instead of his semi-empiricalformula (1.1), then the frequency of oscillation becomes:

f= u

12

h32

c, (1.3)

wherec= 1.43, 3.46, 6.00 and 8.98 for the first four stages.

He suggests that n has a value such that = hn+ 14

is near the wavelength for which an edgeless jet

is most sensitive, and after stage jumps gets closer to this value.

In 1954 Nyborg published his dynamic theory explaining the edge tone phenomena[5]. His theory assumes

that due to a kind of sources of hydrodynamic origin at the wedge transverse forces act on each particle

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of the jet as it travels toward the wedge. He dealt with only the centreline of the jet, and supposed that

the vertical acceleration acting on any particle travelling towards the wedge depends only on two factors:

its actual horizontal distance from the wedge and the actual position of the jet displacement at the wedge.With his theory, Nyborg was able to describe the shape of the centreline of the jet. Although he was not

able to determine the frequency of oscillation, he was able to determine the ratio of the frequencies of

the different stages (1 : 2.44 : 3.86 : 5.29 for the first four stages). He also indicated that as h increases

higher modes becomes possible. He found the lower limit of the nth stage to be h/ > 2n, however the

theory is not capable to predict the position where the jet jumps from one stage to another. Because

of his dynamic theory only deals with the centreline of the jet it fails to predict that the frequency of

oscillation depends on the width of the jet.

Powell first published his feedback loop theory in 1953 and then later, in 1961 he gave a detailed discussion

of it [6, 7]. He suggests that an infinitesimal excitation at the nozzle exit grows along the jet via an

instability displacement wave. This distortion generates an oscillating force on the wedge that creates

a dipole sound source, which then closes the loop by exciting the jet at the nozzle exit. Despite of

the feedback loop being based on the dipole sound source, he stated that the feedback loop is purely

hydrodynamic, the sound radiation itself does not play an essential role in the mechanism. From this

feedback loop a phase criterion can be deduced inducing the oscillation frequency of the nth stage to be:

f=

n+

1

4

uconv

h , (1.4)

where uconv is the convection velocity of the disturbances. He emphasises that the frequencies of the

stages are not to bear ratios of 5

/4: 9

/4: 13

/4, because sinuosities of different wavelengths have differentconvection velocities.

For the stages jumps he gave the following explanation: For any given uconv andh values a certain f

frequency can be calculated for each of the stages from equation (1.4). For low nozzle-to-wedge distances

these frequencies will be well above the region where the edge-less jet is sensitive to acoustic excitations

even for the first stage, thus the edge tone phenomenon will not occur. As the nozzle-to-wedge distance is

increased to a point the frequency of the first stage reaches the region of sensitivity, and the first stage of

the edge tone sets in. Further increasing the nozzle-to-wedge distance the oscillation frequency decreases

and at a point it reaches the lower boundary of the sensitivity region and the first stage disappears. At

this point the frequency of the second stage is already in the sensitivity region, thus by this point the

edge jumps to the second stage. Similar explanation can be given for the onset of the stages in the case

of a fixed geometric configuration with varying jet velocity.

Powell notes that it can happen that the new stage is superimposed on the old stage, and the two

stages coexist. He also notes that it is more likely than not that the jumps between the stages will be

hysteretic.

He experimentally showed the dipole characteristics of the edge tone sound field, and that the am-

plitude of the acoustic pressure is proportional to the third power of the jet velocity.

Holger et al. developed a vortex street theory in 1977 [8]. Their assumptions were that the wavelength

of the jet disturbance, the width of the vortex street and the propagation velocity of the vortices are

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constant. While Curle based his vortex theory on the formation of secondary vortices at the edge, their

analysis does not depend on secondary vortices, and used an entirely different formula for describing the

oscillation frequency. Contrary to Powell, they assumed that the vortex street is fully formed by the timeit interacts with the edge. They found the frequency of oscillation to be:

f= 0.925

h

12 u (n+n)

32

h , (1.5)

where= 0.4, 0.35 and 0.5 for the first, the second and the third stages, respectively.

In 1980 Holger et al. [9] extended their theory and gave an approximation on the vertical force acting

on the wedge. From this, they were able to calculate the acoustic pressure at an arbitrary point in the

far field with Lighthills equation. They found that the integration length on the wedge should be chosen

as 2, and in this case the calculated force is

F 1.08W u2, (1.6)

where is the density of the fluid and Wis the height of the flow. From this, the amplitude of the acoustic

pressure at a distance ofr in the direction of maximum radiation is

|pa| = f F2ra0

0.5

h

32

(n+n)32

u3W

ra0, (1.7)

wherea0is the speed of sound. They also noticed that the vortex pair nearest to the tip of the wedge gives

the most significant part of the force, and the instantaneous force has its maximum when the distance

between the tip of the wedge and the first vortex downstream of it is 0.1.

In 1992 Crighton [10] created a linear analytical model to predict the frequency characteristics of the

edge tone oscillation. He dealt with a top hat jet impinging on a plate placed parallel in the center of the

jet. He assumed inviscid flow with vortex-sheet shear layers, and solved the problem asymptotically by

Wiener-Hopf methods. He found that the dimensionless oscillation frequency S= bu, where = 2f

is the angular frequency and b = 2 is the jet half-thickness is

S=

b

h

32

4

n 3

8

32

, (1.8)

and the h/ ratio is n 38. He found that his Strouhal number (defined as at the beginning of theintroduction) is much larger than the values reported by Holger et al. For the relative convection velocity

of the disturbance he used the uconvu 2S13 formula, while Holger et al. used uconvu 0.645S

13 . Without

essentially finding the cause of this large difference, he concludes that his formula would give a better

prediction if uconvu 2S13

1+43S13

would be used, but (I cite,[10] p. 386) all such expressions would lead to the

same behaviour, namely preservation of essentially the form of equation (1.8), but with (4)32 replaced

by a smaller coefficient.

In 1996 and 1998 Kwon [11,12] presented a theoretical model in which the jet-edge interaction was

modelled by an array of dipoles on the edge. By assuming the jet to be sinusoidally oscillating and

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the convection velocity of the disturbances to be constant, his model can estimate the surface pressure

distribution on the wedge from that an array of acoustic dipoles on the wedge can be deduced. He found

that the peak value of the spatial pressure distribution on the wedge can be found approximately quarterwavelength downstream from the tip of the wedge. He found that the phase criterion is: h +

h

=n 14

,

where is the wavelength of the upstream propagating disturbance (the acoustic field of the dipole

sources). Thus, he claims that the point of the wedge surface where the pressure has its maximum

(quarter wavelength downstream from the tip) is the position of the effective acoustical source. He also

found that the convection velocity of the disturbance on the jet is approximately 60 % of the mean exit

velocity of the jet, and thus the Strouhal number of the oscillation can be approximated as:

St =h

n 14

1.667 +Ma, (1.9)

whereMais the Mach number of the mean jet velocity (Ma= u/a0).

1.2.2 Experiments

In 1937 Brown [1,2]investigated an edge tone setup of a = 1 mm wide, top hat jet with a wedge with

an angle of 20 experimentally. He found that the whole edge tone phenomenon occurs at frequencies for

that the edgeless jet is sensitive to sound and the frequency of the stages depends on the exit velocity of

the jet and the nozzle-to-wedge distance through the following formula:

f= 0.466 j (u 40)

1

h 0.07

, (1.10)

where u and h are measured in

cm

s and in cm, respectively, and j = 1, 2.3, 3.8 and 5.4 for the first,the second, the third and the fourth stage, respectively. He claimed that the deviation between his

measurement and his formula for jet velocities u = 120 2000 cms (that is in nondimensional valuesRe = 75 1300) and frequencies f = 20 5000 Hz was maximum 6%. He found the limits of h to be0.31 cm and 6 cm, so the nondimensional nozzle to wedge distance was between 3.1 and 60.

He found that for higher stages the first stage could also be coexisting, and in this case the frequency

of the first stage is about 7 % lower than the frequencies predicted by his formula. As the formula can

have as much as 6 % deviation from the measured values he concluded that this drop in the frequency

practically can be neglected.

In the case of higher stages he measured the wavelength of the jet disturbance as the distance between

two successive vortices on the same side of the stream (from the photographs he took of the visualised

flow), while for Stage I. he assumed that = h. With this and the measured oscillation frequency he

calculated the convection velocity of the vortices as uconv =f that resulted in values of about 40 % ofthe jet exit velocity ( uconvu 0.4).

He also investigated how sound production effects the edge tone, and concluded that in some cases

acoustical excitation can control the stages of the edge tone.

In 1942 Jones[13]investigated an edge tone configuration with a 0.8 mm wide top-hat jet, at velocities up

to 50 ms , and with nozzle-to-wedge distances between 5 and 25 mm. In his experiments the wedge angle

was 25. He reported two types of the edge tone: In the first type that occurs at lower jet velocities

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he found three stages, between which jumps occurs in the frequency of oscillation. In the second type of

the edge tone that occurs at higher jet velocities (above 37 ms ) the jet is probably turbulent and

no jumps occurs if the parameters are varied, but the frequency changes continuously. He found that thefrequencies of the three stages of the first type edge tone oscillation and also of that of the second type

can be described as:

f=j uhk

, (1.11)

whereu is measured in cms

, hin mm. The values ofj andk for the three stages of the first type and for

the second type are: j = 3.9, 11.8, 24 and 6.8;k= 1, 1.14, 1.22 and 1.43, respectively.

In 1952 Nyborg et al. [14]made an extensive experimental research in mapping the stage boundaries in

the h qplane (where q is the volumetric flow rate of the air, thus in a given geometric configuration

proportional to the velocity of the jet) of small edge tones ( = 0.25 1.02 mm) with high frequencyoscillations (fup to 200 kHz). They used parabolic jets with different widths and several (in some cases

asymmetric) wedges that sometimes were placed with a transversal offset from the center of the jet.

They compared their measured frequencies to a somewhat simplified form of Browns semi-empirical

formula (equation (1.10)), namely:

f= 0.466 j u/h, (1.12)and found that, the measured values agree well with the simplified formula at f 2.5 kHz but are a bitlower than the formula below 2.5 kHz and are a bit higher than the formula above 2.5 kHz.

They found that the regions of the stages in the h q plane can overlap, indicating that hysteresismay occur when changing the mean exit velocity of the jet or the nozzle-to-wedge distance, and the

overlapping regions are independent of the wedge angle if it is less than 40.

They also made measurements on the directivity of the sound emitted by the edge tone, and found

that atf


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approximatelyRe = 100 900. In their experiments they used two different nozzles (with = 0.5 mmand 1 mm width) and nozzle-to-wedge distances between 2.2 mm and 8.7 mm. They concentrated on the

first stage oscillation only, and found that the oscillation frequency in this stage is proportional to themaximum jet velocity and inversely proportional to the nozzle-to-wedge distance:

f =cdumax

h , (1.14)

where as they claim cddepends on the width of the nozzle. From the experiments they concluded that

for the = 0.5 mm wide nozzlec0.5 = 0.339 0.02 and for the = 1 mm wide nozzlec1 = 0.344 0.02,which in my opinion differs within the uncertainty of the values. However, from the CFD simulations

they obtained somewhat (about 13 15 %) lower values:c0.5= 0.29 0.04 and c1= 0.3 0.03.

1.2.3 Numerical simulations and other researchNumerical simulations have been carried out on the edge tone already in the last century. Although they

were able to catch certain typical characteristics of the edge tone flow, but they were still in early states,

and no detailed parameter study was carried out.

Ohring[18] in 1986 carried out CFD simulation on the edge tone. He used a finite difference method

for solving the vorticity/stream-function formulation of the Navier-Stokes equations. He managed to

reproduce the basic features of the edge tone (oscillating flow, two stages) and certain results reported by

Lucas and Rockwell who did experimental research with a underwater parabolic edge tone configuration

in 1984[19]. He carried out simulations at three different jet velocities (at dimensionless velocity values

ofRe= 250, 450 and 650) and found different edge tone stages when Re = 450 was computed from theresult of the Re = 250 simulation with increasing the velocity or when Re = 450 was reached from the

result of the Re = 650 simulation with decreasing the velocity.

In 1994 Dougherty et al. [20]submitted a report to NASA about the numerical simulations of the edge

tone. They managed to reproduce Browns experimental data with a finite volume method based Navier-

Stokes solver (USA - Unified Solutions Algorithm) having first order time and third order spatial accuracy.

They dealt with an edge tone configuration which geometric sizes matched Browns experimental setup.

They had a fixed Reynolds number (Re = 1083), and made simulations at different nozzle-to-wedge

distances in the range of h/ = 3 16. In this nozzle-to-wedge range they managed to produce thefirst four stages of the edge tone. Although the duration of their simulations (in virtual time) was only

sufficient to obtain four or five periods of the lowest stage, they found it to be adequate for FFT analysis,

and the resulting frequencies of the stages agreed well with Browns experimental results.

In the last decade a couple of research were made about the edge tone that instead of carrying out a full

parameter study was investigating only certain aspects of the phenomenon. In the following I will give a

short overview on these new ideas and trends of the edge tone research in the new millennium.

In 2001 Lin and Rockwell [21] carried out experiments on high speed, turbulent underwater edge

tones. They had a Reynolds number approximately 5500 and varying nozzle to wedge distances up to

h/= 7.5. They found two clearly identifiable modes of instability in their experiments. The first one is

a large scale, global mode, whose frequency corresponds well to the frequency of the first stage measured

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by previous researchers for low Reynolds number, laminar edge tones. The other one is a small scale, local

mode, in that vortical structures develop at the nozzle. The frequency of this mode is much higher

than the large scale, global mode, and the phase shift between the development of the vortices on thetwo sides of the jet has no particular numerical value.

In 2003 Fujisawa and Takizawa[22]made a study on how the edge tone could be weakened by feedback

control. In their experiments they used a low speed underwater edge tone configuration ( Re = 200,

h/= 5) that resulted in an oscillation at a frequency of 0.26 Hz. They mounted two control nozzles at

the two sides of the jet next to the nozzle. An image processing system (processing pictures taken of the

visualised flow at 30 Hz) allowed them to precisely control the secondary nozzles to weaken the evolved

edge tone oscillation.

In 2004 Segoufin et al [23] made an experimental study on how the geometry of the nozzle affects

the evolved flow. They tested four nozzle types: a nozzle creating a top hat jet (top hat nozzle); a nozzle

creating a parabolic jet (parabolic nozzle); a nozzle creating a parabolic jet with chamfered jet exit

(chamfered nozzle); and a nozzle created a parabolic jet with rounded jet exit (rounded nozzle). They

found that the rounded nozzle does not produce the edge tone phenomenon. Comparing the oscillation

frequencies with the top hat and the parabolic nozzle they found that with the same maximum velocity

values the oscillation frequency is about 50 % higher in the top hat case and also the oscillation sets in

at a lower velocity (by about a factor of 1.5).

In 2005 Devillers and Coutier-Delgosha[24]published their results on the investigation of the influence

of the gas nature on the edge tone. They investigated both experimentally and numerically parabolic

edge tone configurations with three different nozzle-to-wedge distances h/= 58.3, several jet velocities

from Re= 50 in some cases up to 1000 with three different gases (air, CO 2, Neon), all of them injectedinto air. Beside their extensive experimental and numerical work they also discussed a linear analysis of

the instability. From their investigations they concluded, that the density has a significant effect on the

growth of the vortices on both sides of the edge. When the density of the gas used for the jet is higher

than the density of the environmental medium (CO2 air, CO2/air = 1.52), then the frequency of

oscillation is higher (in this case, by a ratio ofStCO2/Stair = 1.1), and when the density of the gas of

the jet is lower (Neon air, Neon/air = 0.64), then the frequency is lower (in their case by a ratio of

StNeon/Stair = 0.9).

Tsuchida et al. [25, 26]carried out incompressible three-dimensional (3D) CFD simulations on the

edge tone as a test for their numerical code. They only made simulations at a couple of Reynolds numbers

(Re 195, 325, 350 and 455) at a fixed geometry (h/= 6). They found that the Strouhal numbers oftheir simulations is consistent with Browns experimental results.

Nonomura et al. [27, 28] investigated the effect of the Mach number on the edge tone phenomenon

numerically. In their simulations they used jets with Reynolds numbers equal to 208 , 416 or 624 at Mach

numbers equal to 0.087, 0.174, 0.261, 0.348 or 0.435 on a top hat edge tone configuration with h/= 5.

They found that when increasing the Mach number at a fixed Reynolds number that was realised by

changing the size of the model the Strouhal number decreases, that agrees well with Powells feedback

loop theory for high speed edge tones [6]. They also investigated the phase lag in the feedback loop (

in equation (1.17)), and found that its almost constant ( between0.17 and0.21) in their Reynolds

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number and Mach number regions. They found that the relative disturbance propagation velocity is

approximately 0.5 0.6.

In the last few years, a couple of authors made research in the field of the compressible CFD simulations

on the edge tone. This field is a step forward to the direct noise simulation of the edge tone, but holds

several challenges. For example, the typical boundary conditions of CFD simulations tends to generate

spurious reflections of compressibility waves that totally disrupt the acoustical results.

In 2005 Kang and Kim [29] employed a finite difference-based lattice Boltzmann method for the

direct numerical simulation of the sound of a two-dimensional (2D) parabolic edge tone configuration

withh/= 6 for a couple of different jet velocities. They have found that the oscillation frequencies agrees

acceptably with the ones from the experiments of Bamberger et al. [17]. They succeeded in capturing

very small pressure fluctuations resulting from the edge tone oscillation.

In 2010 Gao and Li [30] carried out compressible large eddy simulations (LES) on a high speed

edge tone configuration with Mach numbers of 0.18 and 0.23. They encountered reflections from the

boundaries. They claim that these are because that their computational domain is not large enough,

and suggest that for further investigation a perfectly matched layer (PML) boundary condition should

be adopted in their solver. Nonetheless they drew the following conclusions from their simulations. They

found five frequency peaks in the pressure spectrum, out of which only the third and fifth are harmonically

related. Therefore they claimed that they had found four modes of the edge tone of which the second

one is dominant. Except for the frequencies of the third and fifth peaks they are close to the frequencies

measured experimentally by Krothapalli and Horne [31] in a similar setup. Although the wavelength

of the sound at these frequencies is much higher than the dimensions of their CFD domain they alsoinvestigated the directivity of the stages at a distance of approximately 0.592 measured from the tip of

the wedge, where 2 is the acoustic wavelength of their second mode. They found that the first mode

is omni-directional while the second one radiates more to the downstream direction. The directivity

patterns they had computed are clearly not like that of a dipole that can be because of the reflections

they encountered from the boundaries.

Takahashi et al. [32] published their results on the 2D and 3D compressible LES simulations with

OpenFOAM. In the 2D case they made a parameter study changing the mean exit velocity of the jet

with Reynolds numbers between 320 and 1940 on a fixed geometry with h/= 5 and made a single 3D

simulation at Re = 650 with the same nozzle-to-wedge distance. From the 2D CFD simulations they

obtained oscillation frequencies about 10% higher than those of Brown, but the 3D simulation agreed

well. They also investigated the sound sources computed from the simulated flow.

1.3 Frequency and phase characteristics of the edge tone

As has been shown, the literature is consequent in the proposition that the oscillation frequency is

roughly proportional to the mean exit velocity of the jet and inversely proportional to the kth power of

the nozzle-to-wedge distance:

f

u

hk. (1.15)

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1.3. FREQUENCY AND PHASE CHARACTERISTICS OF THE EDGE TONE

Sometimes an additive constant in one or both of the relationships is also present (such as f u+cuhk+ch

).

About the value of the exponent k there has been a long debate. In the early phase of the research

ratherk = 1 was favoured (Brown and other researchers before him [1], Curle [3]) later it became generallyaccepted thatk = 3/2(Curle using Savics results [3], Holger et al. [8], Crighton[10]). In 1942 Jones [13]

found a variety of exponents, all between 1 and 3/2, depending on the stage number. Recent research

(Bamberger et al.[17]) and also the results of my experimental and numerical studies indicate thatk = 1

is more correct.

In order to ensure comparability, the discussed frequency formulae were transformed to Strouhal

numbers and doing so it turned out that all of them can be described in the following form:

St

Re,

h

c1 c2Re

1(h/)k

c3

(1.16)

Table 1.1 shows the value of the coefficients for the first three stages, while Figures 1.3 and 1.4 show

the Strouhal number of the first stage plotted as a function of the Reynolds number at h/= 10 and

as a function ofh/at Re= 200, respectively. To avoid the overloading of the figures with curves only

three theoretical (Curles two formulae[3] and Holgers [8]formula) and three experimental (Browns [1],

Jones[13] and Brackenridges [16] formulae) results are plotted. The experiments of Brown and Jones

fit acceptably well (for h/= 10 above Re 150), the experiments of Brackenridge has a mentionablebut still not too high deviance from their results (above Re = 200). The formulae from the theoreti-

cal considerations tend to over-estimate the results of the measurements. It can be seen that for low

Reynolds numbers and/or low nozzle-to-wedge distances the curves separate and the difference can easily

be more than 100 %. For higher Reynolds numbers or nozzle-to-wedge distances the differences between

the formulae are somewhat more bounded, but still can reach 25 %.

All of the above mentioned theories can be summarised as the disturbances on the jet that born somewhere

near the nozzle have to travel to the wedge, where they somehow interact with it. As a result of their

interaction a signal is sent to the place where disturbances of the jet are born.

As Powell suggests: the oscillating jet creates an oscillating force on the wedge, that creates a dipole

sound source. The generated sound then excites the jet at low Mach numbers with no time delay and

a new disturbance is born that grows as it travels downstream to the wedge.

The phase relation of this loop can be summarised in the following equation:

h= (n+) (1.17)

where is the wavelength of the disturbance, n is a whole number corresponding to the stage number,

and is a small number indicating that the effective resonance length of the edge tone system somewhat

differs from h.

There is no agreement in the literature about the value of, it may also depend on the details of the

configuration and on the stage number. The most often occurring value is 0.25 (Curle [3], Powell [7]),

Holger et al. [8] found values between 0.35 and 0.5 depending on the stage, but negative values are also

suggested0.2 (Nonomura et al.[27, 28]),0.25 (Kwon [11,12]) or3/8(Crighton [10]). One reason forthe uncertainty in the dependence of the frequency of oscillation on h is the uncertainty of. The exact

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Table 1.1: Parameters of the S t (Re, h/) relationships (equation (1.16)) by different authors

Stage Author c1 c2 c3 k

Stage I

Brown[1] 0.4659 12.06 0.007 1

Jones[13] 0.39 0 0 1

Curle [3] 0.625 0 0.0267 1

Curle-Savic [3] 1.43 0 0 3/2Brackenridge [16] 0.6298 38.4 0.0235 1

Holger[8] 1.532 0 0 3/2

Crighton [10] 2.477 0 0 3/2

Kwon[12] (with Ma 0) 0.45 0 0 1Bamberger et al. [17] (exp.) 0.513 0 0 1

Bamberger et al. [17](CFD) 0.443 0 0 1

Stage II

Brown 1.072 27.74 0.007 1

Jones 1.217 0 0 1.14

Curle 1.125 0 0.0148 1

Curle-Savic 3.46 0 0 3/2

Brackenridge 1.512 92.2 0.0235 1

Holger 3.332 0 0 3/2

Crighton 10.385 0 0 3/2

Kwon (with Ma 0) 1.05 0 0 1

Stage III

Brown 1.77 45.83 0.007 1

Jones 2.52 0 0 1.22

Curle 1.625 0 0.0103 1

Curle-Savic 6 0 0 3/2

Holger 6.057 0 0 3/2

Crighton 21.32 0 0 3/2

Brackenridge 2.645 161.3 0.0235 1

Kwon (with Ma 0) 1.65 0 0 1

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1.3. FREQUENCY AND PHASE CHARACTERISTICS OF THE EDGE TONE

Figure 1.3: Dependence of the Strouhal number of the first stage edge tone oscillation on the Reynolds

number ath/= 10 in various results reported in the literature [1, 3, 8, 13, 16]

Figure 1.4: Dependence of the Strouhal number of the first stage edge tone oscillation on the dimensionless

nozzle-to-wedge distance atRe= 200 in various results reported in the literature[1,3, 8, 13, 16]

positions where the dipole source is located (i.e. at the tip of the wedge or at a certain distance away

downstream from the tip) and where the sound generated by the acoustic dipole source excites the jet

(directly at the nozzle, or somewhat further downstream) are still not explored.

Also the theories presented usually assumes that the wavelength and convection velocity of the distur-

bance do not change between the nozzle and the wedge, and thus the phase of the disturbance decreases

linearly in proportion to the distance, but this was found to not to be true (Stegen and Karamcheti [ 33],

Section2.3.5).

Neglecting these minor problems, assuming that the disturbance has to travel the nozzlewedge

distance (Stage I with = 0, thus= h) with the mean speed of disturbance propagation that is about

40 % of the mean exit velocity of jet (Section2.3.5), the period of one feedback loop is about T h0.4u

and so the frequency of the first stage oscillation would be about f 0.4uh , which is very close to theabove formula of Jones for the first stage. For the higher stages this heuristic model does not work.

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1.4 Why the edge tone? Aims of the work

The edge tone is a very interesting aero-acoustic flow phenomenon. On the flow side, it is a planarflow (as long as the depth (W) of the jet is much larger than its width()) while on the acoustic side,

under certain circumstances it generates an audible, tonal sound that has a true three-dimensional dipole

directivity at least in the far field and as long as the depth of the jet is not comparable to the acoustic

wavelength (). Usually these conditions ( W, andW ) are fulfilled. Therefore the edge tone is aperfect subject of testing a newly developed method of coupling a 2D flow simulation with a 3D acoustic

simulation that will be demonstrated in Chapter3. At the same time, in spite of its geometric simplicity,

the edge tone produces flow phenomena that are interesting in themselves (Chapter 2) and not yet fully

understood. Despite its intensive research in the previous more than one hundred years the literature is

still not concordant even about its most basic attribute, its frequency characteristics.

The aims of this dissertation can be divided to three groups, that also makes the structure of the disser-

tation:

1. The aim of Chapter 2 is to investigate the flow of the edge tone phenomena with experimental

and numerical tools, verify one of the formulae for the Reynolds number and dimensionless nozzle-

to-wedge distance dependence of the Strouhal number published already and to point out a weak

point, an incorrect assumption of all the models that were published, namely that the convection

velocity of the disturbance on the jet is not constant.

2. The aim of Chapter3 is to create a method of coupling a 2D flow simulation with a 3D acoustic

simulation, and to investigate the acoustic attributes of the edge tone with this newly developedmethod.

3. At last, the aim of Chapter4is to investigate a real-world appearance of the edge tone phenomenon,

namely the flow inside the foot model of an organ pipe.

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Chapter 2

The flow of the edge tone

The flow field of the edge tone was investigated both by numerical and experimental methods. This

chapter will describe the Computational Fluid Dynamic (CFD) and the experimental setups and discuss

the results obtained by the two methods.

2.1 The CFD setup

ANSYS-CFX (Releases from CFX-5.7.1 to ANSYS-CFX v14; product of ANSYS Inc. Southpointe 275

Technology Drive Canonsburg, PA 15317, [34]) was used to simulate the flow. This solver is based on a

finite volume scheme, and uses an iterative method to solve the Navier-Stokes equation system. In our

case the iteration targets are the conservation of mass and momentum.

2.1.1 Two-dimensional simulations

The flow was assumed to be two-dimensional (2D). This assumption was justified: all the experimental

studies used a high aspect ratio nozzle and edge and no three-dimensional effects have been found. A

three-dimensional (3D) simulation was also performed, and in the central region the flow proved to be

almost perfectly two-dimensional and similar to the results of the 2D simulations (Section 2.3).

Although the software is only capable of calculating flows in 3D domains discretised with 3D elements,

it is still possible to calculate planar flows. For this the domain of the planar flow and the mesh discretising

it have to be extruded in the third direction with only one layer of elements. The height of this layer

can be chosen arbitrarily. As long as the aspect ratio of the elements is moderate, the magnitude of the

extrusion does not affect the result of the simulation. With symmetry boundary conditions prescribed on

the bottom and top surfaces of the extruded domain, the simulation leads to a planar flow.

Geometry, boundary conditions and solver settings

The geometry and the mesh for the CFD simulations were prepared using ANSYS ICEM CFD. The

geometric parameters were the following: the width of the slit on the nozzle () was 1 mm, the nozzle-

to-wedge distance (h) was varied between 3 and 15 mm and the angle of the wedge was 30. This edge

tone configuration was placed inside a rectangular domain as shown in Figure 2.1(the figure is not to

scale). It is advantageous for the flow development if the back boundary is placed somewhat behind

the nozzle exit (H1 = 12.5 mm) otherwise non-physical vortices might appear in the flow. The other

boundaries were placed far enough from the region interest not to have any disturbing effect on the flow

(H2= V= 75 mm). The height of the domain in thez direction was 1 mm.

On the two x-yplanes bordering the slice symmetry boundary conditions were prescribed. At the

solid walls of the wedge no slip wall boundary conditions whereas at the outer wall of the nozzle free

15


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CHAPTER 2. THE FLOW OF THE EDGE TONE

h

H1 H2

V

V

xy30u wedgenozzle

Figure 2.1: Sketch of the CFD domain (not to scale)

slip wall conditions were given. At the back wall a small inflow was prescribed with an inlet velocity

of about 1% of the exit velocity. This was done in order to stabilize the flow, while not influencing the

parameters studied. The spectral peaks appear at the same frequencies but they get sharper. Without

this, experience shows that there is an increased risk that non-physical vortices appear and remain in the

domain. At the nozzle exit inflow boundary condition was prescribed with uniform or parabolic velocity

distribution. The mean exit velocity of the jet was determined from the required Reynolds number. All

other boundaries (dashed lines) were set to opening boundary condition, i.e. prescribed static pressure

without prescribed flow direction.

Air at 25 C (= 1.185 kgm3

, = 1.831 105 kgms

) was used as fluid. Because of the moderate Reynolds

number and low Mach number regions (Re= 60 2000,Ma = 0.003 0.09) the flow was assumed to belaminar thus no turbulence model was used and incompressible. Second order accurate spatial (High

resolution scheme) and temporal (Second order backward Euler scheme) discretisations were used.

It has been tested to what extent the initial condition influences the result. Simulations with initiallyquiescent fluid and initially steady state flow have been performed. It turned out that the initial condition

has no influence on the final character of the flow. No special measures had to be taken to initiate the

oscillation; the oscillation set in spontaneously after a short transient period.

The target root mean square (rms) residuum of the iteration was set to 105. Some other values

(104 and 106) were also tested. It was found that the permissive target is not sufficient and the stricter

one is not necessary.

Mesh study

In order to increase the simulation accuracy a block-structured hexagonal mesh was used. First of all a

mesh convergence study has been performed on one configuration (h= 10mm and Re = 200). Table2.1

shows the main parameters of the three meshes that were tested: the number of elements (NoE), the

number of elements along the width of the nozzle exit (Nozzle res.) and the number of elements along

the nozzle exit - wedge tip distance (Nozzle-Wedge res.). Although the element sizes in the nozzle-wedge

region in both direction changed always by a factor of 2, the number of elements in the entire mesh

changed less than by a factor of 4 as the refinement in the outer regions was a bit smaller. The two most

important attributes of the evolved flow the frequency of oscillation (f) and the rms of the force acting

on the wedge (Frms) are also presented in the table. Only the y component of the force was investigated

as that is the direction of maximum radiation of the acoustic dipole.

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2.1. THE CFD SETUP

Table 2.1: Mesh parameters (NoE- Number of Elements;Nozzle res.- number of elements along the width

of the nozzle exit; Nozzle-Wedge res. - number of elements along the nozzle exit - wedge tip distance; f

- frequency of oscillation; Frms - rms of the force acting on the wedge; est.rel.err. - estimated relativeerror)

Mesh no. NoE Nozzle res. Nozzle-Wedge res. f[Hz] Frms[mN] est.rel.err.

1 13 024 10 16 104 0.0255 4.0 %

2 36 300 20 32 112 0.0263 1.0 %

3 81 920 40 64 114 0.0265 0.25 %

With the generalised Richardson extrapolation [35] one can estimate the exact value ofFrms from

extrapolating the values ofFrms gathered from simulations with three different meshes. The refinementratio between the 1st and the 2nd and the 2nd and the 3rd meshes is nowq= 2. Them order of accuracy

can be estimated as:

m= ln[(Frms,1 Frms,2) / (Frms,2 Frms,3)]

ln (q) 2 (2.1)

From that, the estimation of the exact value follows as:

Frms,ex = Frms,3+Frms,3 Frms,2

qm 1 0.0266 mN (2.2)

The last column of Table 2.1shows the relative error ofFrms comparing to the estimated exact value.

Figure2.2shows the monotonic convergence of the values ofFrmsto theFrms,exestimated exact value (red

line) in the limit of h 0, where h is the length of an element between the nozzle and the wedge.The dashed curve in the figure is the curve fitted to the points during the process of the Richardson

extrapolation

Frms[mN] 0.0266 0.00272(h[mm])2

.

Figure2.3 shows snapshots of the velocity field as vector plots with the three different meshes. It can

0 0.1 0.2 0.3 0.4 0.5 0.60.025

0.0255

0.026

0.0265

0.027

h [mm]

Frm

s[mN]

Figure 2.2: The y component of the rms of the force (Frms) acting on the wedge versus the

length of an element between the nozzle and the wedge (h); red curve: the estimated exact

value (Frms,ex); dashed curve: the curve fitted to the points during the Richardson extrapolation

Frms[mN] 0.0266 0.00272(h[mm])2

17


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(a) Result with the 1st mesh

(b) Result with the 2nd mesh

(c) Result with the 3rd mesh

Figure 2.3: Velocity vector plots in case of the three different meshes; scale 3:1

18


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2.1. THE CFD SETUP

be concluded from the mesh study that the coarser mesh was not satisfactory because certain important

flow structures were not well resolved (such as the vortex at the wedge wall near the tip, denoted by red

rectangle in Figure2.3(a)) and also the oscillation frequency was nearly 10% lower than that on the finestmesh. The results with the second and the third meshes differ negligibly, while the run time increased

dramatically for the third one.

Therefore the medium mesh was chosen as the best affordable. In this mesh the nozzle width was

resolved by 20 and the nozzle exit-wedge tip distance by 32 uniformly spaced elements resulting in element

sizes of about 0.3 mm x 0.05 mm (Figure2.4(a)). When the distanceh was varied, the number of elements

in this region was varied proportionally. At the wall of the wedge the flow domain was resolved by roughly

0.05 mm thick elements. Near the tip these elements had the same length as the elements between the

nozzle and the wedge and started to grow only after 50 elements (Figure 2.4(b)). In the outer regions of

the CFD domain the largest dimension of the elements was about 2.5mm.

(a) Between the nozzle and the wedge (h= 10 mm case; scale 10:1)

(b) At the wedge wall near the tip (scale 10:1)

Figure 2.4: Snapshots of the CFD mesh

Time step study

Great care was also taken to determine the optimum temporal resolution. The optimum time step was

determined for the same reference case as in the mesh study. Simulations with time steps of = 0.05,

0.1, 0.2 and 0.4 ms were carried out to determine the optimum time step. Table2.2 shows the oscillation

frequency and the rms of the force acting on the wedge in the case of the four simulations. Again only

the y component of the force was dealt with.

Using the generalised Richardson extrapolation this time for the time discretisation (with = 0.05, 0.1

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Table 2.2: Results of the time step study ( - time step; f- frequency of oscillation; Frms - rms of the

force acting on the wedge; est.rel.err. - estimated relative error; CPU time- CPU time required for the

simulation using one core of an Intel Core i7-3770 CPU running at 3.4 GHz)

# [ms] f[Hz] Frms[mN] est.rel.err. CPU time

1 0.05 114 0.0259 0.8 % 11 h 57 min

2 0.1 112 0.0260 1.3 % 8 h 59 min

3 0.2 112 0.0263 2.3 % 6h 5 min

4 0.4 110 0.0268 4.3 % 4 h 30 min

and 0.2 ms) the exact value ofFrms can be estimated as:

Frms,ex 0.0257 mN (2.3)The fifth column of Table 2.2shows the relative error ofFrms comparing to the estimated exact value.

The CPU time required for the simulation grew by 48% from the 3rd to the 2nd simulation, and by

another 33% to the 1st simulation. The estimated error of Frms and the error of oscillation frequency

were both around 2% in the = 0.2 ms case, therefore it was decided that = 0.2 ms is accepted as a

golden mean between computation resources and accuracy. The frequency of the oscillation here is 112 Hz,

which means about 45 time steps per cycle.

It is well known that the main source of the error in the numerical solution of a time-dependent partial

differential equation is the accumulated error of the discretised time derivatives over many time steps.Thus after the time step offering the best compromise was found, an analytical criterion to keep the error

of the temporal discretisation constant was derived as follows.

ANSYS CFX uses the Second Order Backward Euler time discretisation scheme. Let us analyse

this scheme via the numerical solution of an ordinary differential equation (thereby assuming that the

error from the spatial derivatives remains constant):

y

t =f(t, y (t)) , y (0) =y0 (2.4)

wheref(t, y (t)) is Lipschitz continuous with a Lipschitz constantLf

i.e. |f(t,y1)f(t,y2)||y1y2| Lf

.

Let the time step, resulting a time discretisation oftj =j be small enough to satisfy the following

criterion:3

2 Lf 1

c (2.5)

wherec >0 is a constant.

The Second Order Backward Euler differentiation scheme takes the following form:

y

t (tj) 1

3

2y (tj) 2y (tj1) +1

2y (tj2)

f(tj , y (tj)) (2.6)

With approximating the exact valuesy (tj) by the numerical valuesyj the algebraic equation to be solved

is the following:1

3

2

yj

2yj1+

1

2

yj2 =f(tj , yj) (2.7)20


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2.1. THE CFD SETUP

whereyj1 andyj2 are known from the previous time steps and yj is to be determined.

The local error (gj) of this discretisation is the difference of the two sides of (2.6):

gj :=g (tj , ) = 1

32

y (tj) 2y (tj1) + 12

y (tj2) f(tj , y (tj))

= 1

3

2y (tj) 2

y (tj) y

t(tj) +

2

2

2y

t2 (tj)

3

6

3y

t3 (tj) + O

4

+1

2

y (tj) 2y

t (tj) +

42

2

2y

t2 (tj) 8

3

6

3y

t3 (tj) + O

4 f(tj , y (tj))

= 1

y

t(tj)

3

3

3y

t3 (tj) + O

4 f(tj , y (tj)) = 2

3

3y

t3 (tj) +O

3

(2.8)

subtracting (2.7) from (2.8) with the notation ofej =y (tj) yj we obtain:

132 ej 2ej1+ 12 ej2 =f(tj , y (tj)) f(tj , yj) +gj (2.9)Introducing the j =

f(tj ,y(tj))f(tj ,yj)y(tj)yj

notation equation (2.9) transforms into

ej

3

2 j

= 2ej1 1

2ej2+ gj (2.10)

As 132j

= 23

+ 2j

3( 32j)it yields to:

ej =4

3ej1 1

3ej2+

32 j

4

3jej1 1

3jej2+gj

(2.11)

Introducing the following notations:

j =

ej1

ej

; A=

0 1

13

43

; Bj =

0 0

13

j1

32j

43

j1

32j

; vj =

0

132j

gj

(2.12)

(2.11) in a matrix notation becomes:

j =Aj1+ Bjj1+ vj (2.13)

LetSbe a matrix with which S AS1 is the Jordan normal form ofA:

S= 12 32 12

12 , SAS 1 = 1 00 1

3 SAS1 = 1 (2.14)

whereP := maxi

j |pij |.After multiplying both sides of (2.13) from the left side by S, with S1S= Iwe get:

Sj =SAS1Sj1+ SBjS

1Sj1+ Svj (2.15)

with the normjS := Sj and the usual inequalities for norms:

jS

SAS1

j1S+ S Bj

S1

j1S+S v (2.16)

j

S

j1

S+ 2

Bj

4

j1

S+2

v

(2.17)

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Using the Lipschitz property off:

B

j=

|

j | 132 j 13+43 Lf 132 j 53 53 Lf 132 Lf (2.18)vj =

132 j

gj

132 Lf

|gj| (2.19)

(2.17) takes the following form:

jS j1S+ 85

3Lf

132 Lf

j1S+ 2 1

32 Lf

|gj | (2.20)

Since 32 Lf 1c ,

j

S

1 +40

3

Lfc j1S+ 2 c |gj |

1 +40

3 Lfc

j0S+ 2 c

jk=1

|gk|

1 +40

3 Lfc

jk

1 +40

3 Lfc

j 0S+ 2 c

jk=1

|gk|

e 403 jLfc

0S+ 2 cj

k=1

|gk|

(2.21)

The-norm ofj can be limited from below and above with the S-norm:

jS= Sj S j = 2 j, and (2.22)j =

S1Sj S1 Sj = 4 jS (2.23)thus (2.21)can be written as:

j 4 jS 4e403 jLfc

0S+ 2 c

jk=1

|gk|

4e 403 jLfc

2 0+ 2 cj

k=1

|gk|

(2.24)

Wherej = max(|ej1| ; |ej |), thus|ej | j. Let us assume that the initial values are correctand the initial error0 is zero, so:

|ej | 8e 403 jLfc c jk=1

|gk| (2.25)From (2.8):

gj = 2

3

3y

t3 (tj) + O

3 |gj | 2

3 max

3yt3+ O 3 (2.26)

Thus at TS=N ,i.e. at the end of the simulation the global error will be:

|eN| 8e 403 TSLfc cN

2

3 max

3yt3+O 3 = 8e 403 TSLfcTSc 23 max

3yt3+ O 3 (2.27)

|eN| const eTSTS2 max

3y

t3 + O

3

(2.28)

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2.1. THE CFD SETUP

whereconst is a constant, independently of the time step.

If we assume that the flow velocity at a fixed spatial point is a harmonic function of time, v sin(t),

the maximum of the absolute value of the third derivative can be approximated with v3

. The globalerror of the simulation after the last time step can be estimated as (with f=/(2)):

|eN| const eTT 2vf3 +O

3

(2.29)

whereT is the duration of the simulation,is the time step, v is the amplitude of the velocity oscillation

and f is the expected oscillation frequency.

For our reference case, Re = 200 a certain optimum time step was determined. Our task is to keep the

same error at other Reynolds numbers.

The frequency of oscillation as will be shown later is nearly proportional to the mean exit velocity

(u) of the jet in each stage. The duration of the simulations (in simulated time, TS) can be divided into

the transient part before the quasi-steady oscillation sets in (Tt) and the oscillation part (To):T =Tt+To.

This transient part was omitted from the signal when performing FFT. To was mainly determined by

the required frequency resolution of the spectra (f = 1/To). Of course, with decreasing time step, it

becomes increasingly difficult to get the same absolute frequency resolution in a reasonable time. At

higher Reynolds numbers it was decided that the duration of the simulation should ensure a frequency

resolution of 1 2% of the expected maximum frequency. Fortunately it turned out that the durationof the transient part decreased linearly with increasing frequency. That means that the total duration

of the simulation (TS) could be reduced inversely proportionally with frequency (f) thus also with the

mean exit velocity of the jet (u) while keeping a constant relative frequency resolution. The amplitude(v) of the velocity oscillation is also proportional to the mean velocity u. Finally, the exponential factor

can be ignored since then we are on the conservative side. This factor decreases anyway with increasing

velocity and tends to 1.

Putting all the information together, it can be concluded that, in order to keep the absolute error

constant, the time step has to be decreased more quickly than inversely proportional to the mean velocity;

it has to be proportional to u3/2. Thus the number of time steps per cycle increases from about 34 at

Re = 150 to 272 at Re = 1800. Below Re = 150 smaller time steps were used in order to have the

period resolved fine enough. Although the simulations were carried out keeping this condition, it has to

be mentioned that if the relative error is to kept constant it is sufficient if the time step is proportional

to u1 resulting in a constant period resolution.

When a simulation is started the expected oscillation frequency and thus the required time step can

be estimated by linear extrapolation. This frequency and time-step estimation was always a posteriori

verified. When a higher stage appears with a sudden frequency rise then the simulation had to be repeated

with the time step adjusted accordingly.

The frequency of oscillation was determined by means of FFT from pressure and velocity histories

in several points of the flow field or from the history of the force acting on the wedge. No significant

differences were found in the frequencies whether the force, velocity or pressure histories were used and

whether this or that point was used in case of the velocity or pressure signals. Further post processing of

the CFD simulations are described in Section 2.3.5.

23


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2.1.2 Three-dimensional simulation

A 3D CFD simulation was also carried out at a Reynolds number of 225 to verify the planar nature of the

flow. Re = 225 was choose as that was the highest Reynolds number where a pure first stage oscillation

was found among the 2D simulations. The width of the nozzle was the same as in the 2D simulations

( = 1 mm) and the nozzle-to-wedge distance was h = 10 mm. The nozzle and the wedge had different

heights: 25 mm and 70 mm, respectively. This was done to allow the jet to spread in the z direction and

to have the full effect on the wedge while minimize the end effects. This edge tone setup was placed in

a 90 mm x 151 mm x 70 mm rectangular domain. Again a block-structured hexahedral mesh was used

(Figure 2.5). In the central region (12.5 mm < z < +12.5 mm, wherez = 0 is the middle plane) theelements had a height of 1 mm. From the end of the nozzle in the following 7.5 mm the height of the

elements grew up to 3 mm from where the elements had a height of 3 mm. The mesh in any cross sectionperpendicular to the z direction had the same resolution in the nozzle-wedge region as the mesh used

in the 2D simulations and was coarser at the boundaries. The mesh contained about 685 000 elements.

Comparison between the results of the 2D and the 3D simulations will be given later in Section 2.3.

Figure 2.5: Mesh of the 3D CFD simulation

24


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2.2. THE EXPERIMENTAL SETUP

2.2 The experimental setup

2.2.1 Experimental system and instrumentation

The rig to produce the edge tone flow is depicted in Figure 2.6(a). Shop air with a pressure reduced

to 0.5 bar by a pressure reducing valve was led by 3/4 reinforced flexible plastic tubes to a cylindrical

pressure reservoir with a volume of 57 l.

A mass flow rate sensor (Sensortechnics, Honeywell AWM700, working on a heated element principle with

a voltage output) was built into the line between the pressure reducing valve and the reservoir tank to

determine the mean velocity of the jet. There was a long copper pipe section before the sensor to ensure

undisturbed inflow. The sensor was placed between two throttle valves to keep the pressure at the sensor

constant as the calibration of the sensor showed that the sensor might be sensitive to the pressure inside.

This was examined, and it was found out that this is only crucial if the pressure inside the sensor exceeds

10 kPa. To be on the safe side the calibrations and the measurements were all done at an inside pressure

of 0.4 kPa.

The control mass flow rate for the calibration was generated by a sinking bell vessel. A bell vessel

was plunged into an oil bath upside down. The air beneath the vessel was lead out through a pipe

onto which the mass flow rate sensor was mounted. With two valves at both sides of the mass flow rate

sensor it was possible to keep the pressure inside the sensor at a constant 0.4 kPa value. The sinking

height (z = z1 z2, where z1 and z2 are the two levels read at the beginning and at the end of the

measurement both with 0.5 mm uncertainty) was measured during a certain time (ts, measured by astopwatch having precision of 0.01 s). Although the value of ts always exceeded 60 s (thus the error in

the time measurement is negligible) still at extremely low mass flow rates the value of z was only

10 15 mm, thus its uncertainty was to around 5 7 %. Fortunately these mass flow rates were below thevalues used during the experiments on the edge tone. In the domain of interest the value of the error was

less than 3 %. From these values, the mass

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