On Turbulent Swirling Jets:
Vortex Breakdown, Coherent Structures,
and their Control
von
Diplom-Ingenieur
Kilian Oberleithner
aus Innsbruck
Von der Fakult¨at V – Verkehrs- und Maschinensysteme
der Technischen Universit¨at Berlin
zur Erlangung des akademischen Grades
Doktor der Ingenieurwissenschaften
– Dr.-Ing. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. sc. techn. J¨orn Sesterhenn
Gutachter: Prof. Dr.-Ing. Christian Oliver Paschereit
Gutachter: Prof. Dr. Israel J. Wygnanski
Tag der wissenschaftlichen Aussprache: 18. Juni 2012
Berlin 2012
D 83
Danksagung
Diese Dissertation entstand w¨ahrend meiner T¨atigkeit als Wissenschaftlicher Mitarbeiter
am Institut f¨ur Str¨omungsmechanik und Technische Akustik des Fachgebiets Experimentelle
Str¨omungsmechanik der TU Berlin. An dieser Stelle m¨ochte ich allen denen danken, die
durch ihre Unterst¨utzung zum Gelingen dieser Arbeit beigetragen haben.
Herzlich danken m¨ochte ich meinem Doktorvater Christian Oliver Paschereit, der mir
durch eine ausgewogene Mischung aus Freiraum und F¨uhrung optimale Arbeitsbedingungen
schuf. Das mir entgegengebrachte Vertrauen erm¨oglichte mir eine freie, durch Neugierde
und pers¨onliche Neigung gelenkte Entwicklung.
Mein besonderer Dank richtet sich an meinen Mentor Israel Wygnanski, der alle meine
wissenschaftlichen T¨atigkeiten w¨ahrend der letzten sechs Jahre verfolgte. Seine kompro-
misslose Durchdringung fachlicher Probleme und sein unbedingter Wille, die Physik hinter
den Dingen zu verstehen, haben mich von Anfang an stark gepr¨agt. Die immerw¨ahrende
pers¨onliche Hingabe, die er mir und meiner Arbeit entgegenbrachte war f¨ur mich sehr
wichtig. Ich danke ihm f¨ur die ¨
Ubernahme der Gutachtert¨atigkeiten im Rahmen des Pro-
motionsverfahrens und daf¨ur, dass er die M¨uhen auf sich genommen hat, f¨ur die Verteidi-
gung dieser Arbeit nach Berlin zu reisen. Außerdem m¨ochte ich Navid Nayeri danken. F¨ur
große wie kleine Probleme war er f¨ur mich meist der erste Ansprechpartner. Ob bei der
Betreuung von Diplomanden, der Planung von Versuchskampagnen oder gr¨oßeren Investi-
tionen in Laborger¨at, man konnte sich immer auf seinen einf¨uhlsamen und pragmatischen
Ratschlag verlassen.
Bernd Noack m¨ochte ich hier ausdr¨ucklich f¨ur die gute Zusammenarbeit danken. Sein
verl¨asslicher F¨uhrungsstil, seine exakte und schnelle Beurteilung des wissenschaftlichen
Potentials der erbrachten Resultate und der F¨ahigkeit diese Resultate zu b¨undeln und
zu pr¨asentieren waren essentiell f¨ur unsere gemeinsamen Ver¨offentlichungen.
Des Weiteren bedanke ich mich herzlich f¨ur die gute Zusammenarbeit bei Christoph
Petz, Roman Seele und Stephan Kallweit, ohne deren Betr¨age diese Arbeit so nicht h¨atte
verfasst werden k¨onnen.
Außerdem will ich den Studenten danken, die f¨ur mich als studentische Hilfskr¨afte gear-
beitet haben oder deren Diplom-/Master-/Bachelorarbeit ich betreut habe. Meinem ersten
Diplomanden, Martin L¨uck, sei gedankt f¨ur seine strukturierte Arbeitsweise und die wun-
derbaren PIV-Daten, die er aufgenommen hat. Phoebe Kuhn sei ausdr¨ucklich gedankt f¨ur
die unglaubliche Geduld w¨ahrend der Hitzdraht-Messungen und der sauberen Auswertung,
auf die sie wirklich sehr stolz sein kann. Moritz Sieber danke ich sehr f¨ur die mehrj¨ahrige Un-
terst¨utzung als studentische Hilfskraft, die gute Zusammenarbeit w¨ahrend der Anfertigung
unserer gemeinsamen Ver¨offentlichungen und der selbstst¨andigen Arbeitsweise w¨ahrend
seiner Diplomarbeit. Genauso m¨ochte ich Lothar Rukes danken, f¨ur die Weiterentwicklung
des Stabilit¨atsl¨osers. Es hat mich besonders gefreut, dass aus der von mir begonnenen Ar-
beit an Drallstrahlen nun zwei Folgeprojekte entstehen zu denen Moritz Sieber und Lothar
Rukes promovieren. Meinen beiden letzten Master-Absolventen Amir-Reza Afkhami und
Schaham Schoar sei gedankt f¨ur die professionellen Umbauten des Drallkanals.
Ausdr¨ucklicher Dank geb¨uhrt auch den Mitarbeitern der Metallwerkstatt, unserem Elek-
troingenieur Heiko Stolpe, unserer Systemadministratorin Angela P¨atzold sowie Kristin
iii
Halboth, Sandy Meinecke und Lilli Lindemann aus dem Sekretariat. Die Letztere konnten
mir, durch leichtfertiges Negieren gewisser b¨urokratischer Regeln in Bedr¨angnis gekommen,
stets mit geschickten Man¨overn aus der Patsche helfen.
Außerdem m¨ochte ich hier Stefan Vey danken, mit dem ich mir ¨uber Jahre das B¨uro
geteilt habe. Trotz eines sehr unterschiedlichen Empfindens der Raumtemperatur und der
daraus folgenden Differenzen im Heizbed¨urfnis, habe ich diese Zeit mit ihm sehr genossen.
Ebenso m¨ochte ich Jonas Moeck danken, mit dem ich mir im letzten Jahr das B¨uro teilte.
Sein analytischer Verstand, seine wissenschaftliche Neugierde und seine Hilfsbereitschaft
machen ihn zu einem sehr wertvollen Kollegen.
Danken m¨ochte ich auch Sebastian G¨oke f¨ur die das t¨agliche Stelldichein am Kicker-
tisch, auf das ich mich jeden Tag aufs Neue gefreut habe. Zuletzt m¨ochte ich noch den
Mitarbeitern danken, die diese Arbeit Korrektur gelesen haben.
Finanzielle Unterst¨utzung erhielt ich in dieser Zeit durch ein Auslandsstipendium der
Gottlieb Daimler- und Karl Benz-Stiftung, durch ein NaF¨oG-Stipendium des Landes Berlin,
durch eine Anschubfinanzierung der Technischen Universit¨at Berlin und durch die Deutsche
Forschungsgemeinschaft im Rahmen der Projekte PA-PA920/10-1 und PA-PA920/10-2.
Den Geldgebern sei f¨ur ihre Unterst¨utzung gedankt.
Berlin, im Juni 2012
Zusammenfassung
Diese Arbeit umfasst die experimentelle und theoretische Untersuchung der Entstehung
großskaliger koh¨
arenter Strukturen in turbulenten Drallstrahlen. Die untersuchte Str¨
o-
mungskonfiguration zeichnet sich durch eine hohe Komplexit¨
at aufgrund gleichzeitig auf-
tretender axialer und azimutaler Scherschichten aus. Dar¨
uber hinaus tritt ab einer gewissen
Drallintensit¨
at ein Ph¨
anomen auf, was zu einer abrupten ¨
Anderung der gesamten Str¨
omung
f¨
uhrt. Durch das sogenannte Aufplatzen des Wirbelkerns (vortex breakdown) entsteht eine
R¨
uckstr¨
omblase auf der Strahlachse wodurch sich eine innere und eine ¨
außere Scherschicht
manifestiert. Die Analyse und Kontrolle dieser komplexen Str¨
omungskonfiguration stellt all-
gemein eine große Herausforderung f¨
ur die Grundlagenforschung dar und ist dar¨
uber hinaus
von großem Nutzen f¨
ur die Optimierung drallstabilisierter Verbrennung in Gasturbinen.
Die Arbeit ist im Wesentlichen in vier Untersuchungen unterteilt. Im ersten Teil wer-
den die verschiedenen Str¨
omungszust¨
ande bei ansteigender Drall charakterisiert. Dieser
rein experimentelle Teil gibt einen ¨
Uberblick ¨
uber die wesentlichen Str¨
omungsph¨
anomene,
mit dem Augenmerk auf das Aufplatzen des Wirbels und das damit verbundene Ein-
setzen einer globalen Instabilit¨
at. Die Experimente zeigen, dass eine globale Oszillation
dann einsetzt, wenn die R¨
uckstromblase eine kritische Gr¨
oße erreicht hat. Das diesem
Vorgang vorangehende Einsetzen des Wirbelaufplatzens wird anhand stehender Wellen
im Wirbelkern erkl¨
art. Die zweite Untersuchung geht detailliert auf die Charakteristik
r¨
aumlich anwachsender koh¨
arenter Strukturen im global stabilen Drallstrahl ein. Die hy-
drodynamischen Instabilit¨
aten, welche die Entstehung dieser Strukturen antreiben, werden
mittels r¨
aumlicher Stabilit¨
atsanalyse systematisch untersucht. Es wird gezeigt, dass trotz
des Einsetzens einer Zentrifugalinstabilit¨
at mit ansteigendem Drall, die Kelvin–Helmholtz-
Instabilit¨
at dominiert. Jedoch beeinflusst der Drall die Phasengeschwindigkeit der helikalen
Instabilit¨
atsmoden und erh¨
oht dabei die Dispersivit¨
at der Scherschicht. Außerdem sorgt zu-
nehmender Drall f¨
ur eine Destabilisierung stehender Moden, was dazu f¨
uhren kann, dass
die mittlere Str¨
omung ihre Axialsymmetrie verliert. Diese Ergebnisse werden durch Experi-
mente des harmonisch und des gepulst angeregten Strahls best¨
atigt. Im dritten Teil dieser
Arbeit werden die koh¨
arenten Strukturen der globalen Mode untersucht. Basierend auf
der proper orthogonal decomposition, wird die phasengemittelte Geschwindigkeit bez¨
uglich
der globalen Oszillationsfrequenz aus zeitlich unkorrelierten Daten extrahiert. Die resultie-
renden koh¨
arenten Strukturen werden mit der globalen Eigenmode verglichen, die mittels
r¨
aumlicher Stabilit¨
atsanalyse berechnet werden kann. Die globale Mode zeichnet sich durch
einen pr¨
azedierenden Wirbelkern (precessing vortex core) stromauf der R¨
uckstromblase aus,
der als der Taktgeber (’wavemaker’) interpretiert wird. Von dort wird die Oszillation auf
die gesamte Str¨
omung aufgepr¨
agt, was zu einem synchronen Anwachsen helikaler Struk-
turen in der ¨
außeren Scherschicht f¨
uhrt. Die vierte Untersuchung geht auf die Kontrolle
der globalen Stabilit¨
atsmode mittels sinusf¨
ormiger Anregung ein. Aus den Ergebnissen der
ersten drei Untersuchungen l¨
asst sich eine Kontrolltechnik ableiten, die bei geringer An-
regungsamplitude gr¨
oßtm¨
ogliche Wirkung erzielt. Die Anregung konvektiv instabiler Mo-
den am D¨
usenaustritt f¨
uhrt dabei zu einer ¨
Anderung des mittleren Str¨
omungsfeldes und
zu einer Unterdr¨
uckung der R¨
uckkopplung in Strahlinneren, was zu einer D¨
ampfung des
pr¨
azedierenden Wirbelkerns f¨
uhrt.
v
Abstract
This thesis provides an experimental and theoretical investigation of turbulent swirling jets
with the emphasis on the formation of large-scale coherent flow structures. The involved
mechanisms are highly three-dimensional due to the coexistence of an axial and an az-
imuthal shear layer. Moreover, swirling jets are prone to a unique flow phenomenon that
causes an abrupt change of the entire flow. The so-called vortex breakdown is manifested
in the appearance of an internal recirculation bubble, which creates an inner and an outer
shear layer. The complexity of this flow configuration poses a great challenge to fundamen-
tal research and deals as a benchmark for recent theoretical concepts. Furthermore, the
present work is of great importance for the gas turbine industry, where swirling flows are
frequently applied to improve combustion processes.
The work consists of four major investigations. First, the dominant flow dynamics
are characterized at different swirl intensities via time-resolved stereo particle image ve-
locitmetry. This experimental work provides an overview of the main flow features with
particular focus on the formation of vortex breakdown and the onset of global instability.
A self-excited single-helical mode is found to arise from the axisymmetric breakdown state
when the recirculation bubble reaches a sufficient streamwise extent. The preceding onset of
breakdown is explained by the criticality of the rotating base flow. The second investigation
consists of a detailed examination of the coherent flow structures that amplify in the swirled
shear layer. The impact of swirl on the driving instabilities is addressed theoretically by
means of spatial stability analysis based on the mean turbulent flow. The most dominant in-
stability is of Kelvin–Helmholtz type, similar to the non-swirling jet. However, swirl effects
the phase velocity of the helical instability waves and renders the shear layer as strongly
dispersive. Moreover, the azimuthal shear destabilizes steady modes that gain significant
amplitudes, which may lead to a breaking of the mean flow symmetry. The theoretical
predictions are confirmed by hot-wire measurements of the pulsed and the single-mode ac-
tuated swirling jet. The third investigation focuses on the coherent structures associated
with the swirling jet’s global mode that arises from the axisymmetric vortex breakdown
state. A method based on the proper orthogonal decomposition is developed that allows
to reconstruct the phase-averaged velocity field of the dominant coherent structures from
uncorrelated flow snapshots. The obtained coherent structures are compared to the global
eigenmode derived theoretically from a local spatial stability analysis. The global mode is
characterized by a precessing vortex core located upstream of the breakdown bubble. It is
interpreted as the global wavemaker that imposes its frequency onto the highly receptive
outer shear layer, causing large-scale helical flow structures to evolve in the periphery of
the jet. The last investigation focuses on the control of the swirling jet undergoing vortex
breakdown. By combining the findings of the preceding studies, it is possible to derive a
control scheme that enables to dampen the natural global mode at low amplitude forcing.
The natural flow is shown to be globally unstable to a single-helical mode resulting in the
precession of the vortex core. Convectively unstable modes forced at the nozzle lip lead to
an enhanced growth of the outer shear layer. The resulting change of the mean flow leads
to a global stabilization of the single-helical mode and to the suppression of the internal
feedback that causes the precession of the vortex core.
vii
Contents
Zusammenfassung v
Abstract vii
List of Figures xiii
Nomenclature xvii
1 Introduction 1
1.1 Overview of the Thesis .............................. 4
2 Theoretical Concepts 9
2.1 Triple Decomposition as the Basis for Coherent Structure Extraction . . . . 9
2.2 Coordinate Systems ............................... 10
2.3 Normal Mode Ansatz and Sign Convention .................. 10
2.4 Linear Stability Analysis ............................. 12
2.4.1 Eigenvalue Problem of the Parallel Flow ................ 12
2.4.2 Temporal, Spatial, and Spatio-temporal Analysis ........... 13
2.4.3 Numerical Method ............................ 15
2.4.4 Modeling Small-scale Turbulence .................... 18
2.5 Proper Orthogonal Decomposition ....................... 19
3 Experimental Arrangements and Measurement Techniques 21
3.1 The Swirling Jet Water Facility ......................... 21
3.2 The Swirling Jet Air Facility .......................... 21
3.3 Stereo-PIV Measurements ............................ 24
3.3.1 Measurements in Water ......................... 24
3.3.2 Measurements in Air ........................... 24
3.4 Hot-wire Measurements in the Moderately Swirling Jet ............ 25
3.4.1 Coherent Velocity Extraction of the Excited Flow . . . . . . . . . . 26
3.4.2 Ensemble Average of the Pulsed Experiments ............. 26
4 The Onset of Vortex Breakdown and Global Instability 29
4.1 Literature Review ................................ 29
4.2 Objective and Approach ............................. 31
4.3 The Mean Flow Configuration .......................... 32
4.4 Momentum Balance and Swirl Number Discussion .............. 34
4.5 Onset of Vortex Breakdown ........................... 35
ix
4.6 Vortex Core Criticality .............................. 43
4.7 Coherent Structures ............................... 46
4.7.1 Local Examination of Azimuthal Waves ................ 46
4.7.2 The Onset of the Global Mode ..................... 49
4.8 Summary and Discussion ............................ 54
5 Instabilities in the Moderately Swirling Jet 59
5.1 Background and Scope .............................. 60
5.2 The Mean Flow Configuration .......................... 61
5.2.1 Analytic Representation of the Mean Flow .............. 62
5.2.2 The Velocity Distribution at the Nozzle ................ 62
5.2.3 Quantification of Swirl at the Nozzle Exit ............... 63
5.2.4 Streamwise Distribution of Mean Velocities and Turbulent Stresses . 64
5.2.5 Parametrization of the Divergent Mean Flow ............. 65
5.2.6 Streamwise Distribution of the Eddy Viscosity ............ 68
5.3 Stability at the Nozzle Exit ........................... 68
5.3.1 Shear Instability ............................. 69
5.3.2 Centrifugal Instability .......................... 75
5.4 Streamwise Evolution of Instability ....................... 80
5.4.1 Shear Instability ............................. 80
5.4.2 Centrifugal Instability .......................... 85
5.5 Single-mode Actuation .............................. 85
5.5.1 Streamwise Growth of the Co-rotating Single-helical Mode . . . . . . 87
5.5.2 Streamwise Phase Velocity in the Swirling Jet ............. 90
5.6 Impulse Response ................................. 92
5.6.1 Trajectory of the Wave Packet Envelope ................ 93
5.6.2 Modal Decomposition of the Wave Packet ............... 96
5.6.3 Morphology of the Wave Packet .................... 98
5.7 Summary and Discussion ............................ 100
5.7.1 The Purpose of the Present Investigation ............... 100
5.7.2 The Main Observations ......................... 101
5.7.3 Final Remarks .............................. 103
6 Coherent Structures in the Strongly Swirling Jet 105
6.1 Motivation, and Approach ............................ 105
6.2 A Brief Survey on Empirical Mode Construction ............... 106
6.3 Description of the Flow Configuration ..................... 107
6.3.1 Characteristic Numbers ......................... 107
6.4 Mean Flow Properties .............................. 107
6.4.1 Analytic Representation of the Mean Flow .............. 109
6.4.2 Self-excited Oscillations ......................... 111
6.5 Empirical Construction of the Global Mode .................. 113
6.5.1 Spatial and Temporal POD Modes ................... 113
6.5.2 Linking the POD Modes to the Coherent Velocity . . . . . . . . . . 117
6.5.3 Construction of Three-dimensional Coherent Structures . . . . . . . 119
6.6 Most Unstable Spatial Modes .......................... 119
6.7 Three-dimensional Shape of the Global Mode ................. 123
6.8 Summary and Discussion ............................ 124
7 Open-loop Control of the Self-excited Swirling Jet 127
7.1 Objectives and Approach ............................ 127
7.2 Description of the Unforced Flow ........................ 129
7.2.1 Characteristic Numbers ......................... 129
7.2.2 Flow Features ............................... 130
7.3 Open-loop Control ................................ 132
7.3.1 Amplitude Variation ........................... 133
7.3.2 Frequency Variation ........................... 136
7.4 Impact of Forcing on the Flow Characteristics ................. 137
7.5 Impact of Forcing on Global Instability .................... 141
7.6 Summary and Discussion ............................ 142
8 Concluding Remarks 145
A Flow Properties of the Swirling Water Jet 147
A.1 Mean Flow Properties .............................. 147
A.1.1 Velocity Profiles ............................. 147
A.1.2 Characteristic Velocity and Length Scales ............... 149
A.2 Turbulent Fluctuations .............................. 150
A.2.1 Turbulent Normal Stress ......................... 150
A.2.2 Turbulent Shear Stress .......................... 152
B Supplemental Information of the Swirling Air Jet Facility 155
B.1 The Swirling Jet Facility ............................. 155
B.2 The Actuation Device .............................. 156
B.3 Actuator Calibration ............................... 162
C Visualization of the Global Mode 163
Bibliography 165
List of Figures
1.1 Smoke visualization of the laminar jet at increasing swirl ........... 2
1.2 Smoke visualization of the laminar and turbulent vortex breakdown . . . . 4
2.1 Coordinate systems ................................ 10
2.2 Normal mode sign convention .......................... 11
2.3 Comparison of the stability analysis with previous work ........... 17
3.1 Sketch of the swirling jet water facility ..................... 22
3.2 Sketch of the swirling jet air facility ...................... 23
3.3 Hot-wire traversing system ........................... 25
3.4 Hot-wire signal of a traveling wave packet ................... 27
4.1 Streamlines of the jet at increasing swirl .................... 33
4.2 Streamwise distribution of axial momentum flux with swirl ......... 36
4.3 Development of the swirl number Swith rotation rate Ω of the honeycomb 37
4.4 Decay of axial velocity at a local minimum with increasing swirl . . . . . . 37
4.5 Mean location of the breakdown bubble with increasing swirl . . . . . . . . 38
4.6 Azimuthal vorticity contours at increasing swirl ................ 39
4.7 Statistic evaluation of onset of vortex breakdown ............... 42
4.8 Criticality of the vortex core at increasing swirl ................ 45
4.9 PSD of azimuthal modes derived at x/D = 1.1................ 47
4.10 Total energy distributed among azimuthal modes at x/D = 1.1. . . . . . . . 48
4.11 Turbulent kinetic energy captured in the POD modes (S= 1.37) . . . . . . 50
4.12 Comparison of POD and Fourier analysis ................... 52
4.13 Phase-locked vorticity distribution of the global mode ............ 53
4.14 Limit-cycle amplitude versus swirl number ................... 54
4.15 Conceptual drawing summarizing the experimental results. ......... 56
5.1 Mean velocity profiles at the nozzle for different swirl intensities . . . . . . 63
5.2 Mean flow of the non-swirling jet S1...................... 65
5.3 Mean flow of the swirling jet S3......................... 66
5.4 Streamwise distribution of mean flow parameters ............... 67
5.5 Streamwise distribution of eddy viscosity and turbulent Reynolds number . 68
5.6 Growth rate of the shear instability at the nozzle exit ............ 70
5.7 αr,αi, and cph of the shear instability at the nozzle exit ........... 72
5.8 Amplitude distribution of the shear instability at the nozzle exit . . . . . . 74
5.9 Axial wavenumber and frequency selection at the nozzle exit . . . . . . . . 75
5.10 Radial stratification of angular momentum ................... 76
5.11 Growth rate of the centrifugal instability at the nozzle exit ......... 77
xiii
5.12 αr,αi, and cph of the centrifugal instability at the nozzle exit . . . . . . . . 78
5.13 Amplitude distribution of the centrifugal instability at the nozzle exit . . . 79
5.14 Streamwise evolution of spatial growth rate of the shear mode in the non-
swirling jet .................................... 81
5.15 Streamwise development of the maximum growth rate of the shear mode in
the non-swirling jet ................................ 82
5.16 Streamwise evolution of spatial growth rate in the swirling jet . . . . . . . . 84
5.17 Streamwise evolution of spatial growth rate of the centrifugal instability in
the swirling jet .................................. 86
5.18 Radial amplitude distribution of the m= 1 mode ............... 88
5.19 Streamwise amplitude distribution of actuated co-winding m= 1 mode for
various swirl configurations ........................... 88
5.20 Streamwise development of spatial growth rate of actuated co-winding m= 1
mode ........................................ 89
5.21 Phase distribution of mode m= 1 for the swirling jet ............. 91
5.22 Streamwise evolution of the phase delay with swirl .............. 91
5.23 Streamwise evolution of the phase velocity with swirl ............. 92
5.24 Trajectory of the wave packet envelope for the swirling and non-swirling jet 95
5.25 Modal amplitude distribution derived from linear stability analysis . . . . . 97
5.26 Modal amplitude distribution derived from hot-wire measurements . . . . . 99
5.27 3D-wireframe visualization of the traveling wave packet . . . . . . . . . . . 100
6.1 Profiles of the mean axial and plane-normal velocity ............. 108
6.2 Velocity profiles of the approximated mean flow ................ 110
6.3 Peak in the PSD from the global mode ..................... 111
6.4 Hopf bifurcation diagram ............................ 112
6.5 Lock-in regime .................................. 113
6.6 POD spectrum .................................. 114
6.7 First five POD modes of the crossflow plane of measurement . . . . . . . . 114
6.8 Phase portrait of the POD for the crossflow plane .............. 115
6.9 First four POD modes of the streamwise measurement plane . . . . . . . . 116
6.10 Phase portrait of the POD for the streamwise plane ............. 116
6.11 Comparison of streamwise and crosswise coherent data ............ 118
6.12 Schematic diagram of the 3D flow construction ................ 120
6.13 Spatial amplification rate of the mode m= 1 ................. 121
6.14 Streamwise evolution of the spatial amplification rate and axial wavelength 122
6.15 Coherent velocity of the most amplified normal mode ............. 123
6.16 Three-dimensional flow field visualizing the global mode . . . . . . . . . . . 125
7.1 Swirl number versus axial distance ....................... 130
7.2 Natural flow at S= 1 and ReD= 20000 .................... 131
7.3 Cross-section of the jet undergoing vortex breakdown ............. 134
7.4 Linear growth and nonlinear saturation of the forced mode . . . . . . . . . 135
7.5 Streamwise distribution of kinetic energy of the forced and natural mode . . 135
7.6 Spatial linear stability analysis for m= 2 ................... 136
7.7 Streamwise development of coherent energy at varying forcing frequency . . 137
7.8 Contours of coherent energy for the natural and actuated flow . . . . . . . . 138
7.9 Impact of forcing on the momentum thickness of the outer shear layer . . . 139
7.10 Impact of forcing on location and size of the recirculation bubble . . . . . . 140
7.11 Spatio-temporal analysis of the natural flow .................. 143
A.1 Mean profiles of the swirling jet ......................... 148
A.2 Characteristic velocity and length scales versus axial distance. . . . . . . . . 149
A.3 Profiles of turbulent normal stresses ...................... 151
A.4 Profiles of turbulent shear stresses. ....................... 153
B.1 Sketch of the swirling jet facility (first state) .................. 156
B.2 Photographs of the swirling jet facility (second state) ............. 157
B.3 Photographs of the actuation device ...................... 158
B.4 Technical drawing of the complete excitation device ............. 158
B.5 Technical drawing of the nozzle plate with details on the wave guide geometries.159
B.6 Technical drawing of the nozzle front plate with details. . . . . . . . . . . . 160
B.7 Technical drawing of the nozzle adapter. .................... 161
B.8 Technical drawing of the nozzle ring. ...................... 161
B.9 Speaker calibration at no-flow condition .................... 162
C.1 3D-Visualization of the global mode in swirling jets .............. 163
Nomenclature
(·)′stochastic part of the Reynolds decomposition
(·)ccoherent part of the triple decomposition
(·)pphase-averaged part
(·)sstochastic part of the triple decomposition
(·)x,y,... vector components
α=αr+iαicomplex axial wavenumber
αmax
imaximum spatial growth rate
αmax
rreal axial wavenumber at maximum growth rate
δxmomentum thickness
˙
Qxaxial flux of axial momentum
˙
Qθaxial flux of tangential momentum
γintermittency factor
ˆ
(·) complex Fourier coefficient
λiith POD eigenvalue
ReDglobal Reynolds number
Retturbulent Reynolds number
RMS root mean square
νkinematic viscosity
νteddy viscosity
Ω rotation rate of the honeycomb
ω=ωr+iωicomplex angular frequency
Ωerotation rate of the wave packet envelope
ω0=ω0,r +iω0,i complex absolute angular frequency
ωmax
rreal angular frequency at maximum growth rate
ωs=ωs,r +iωs,i complex angular frequency at the saddle point
Ωcl rotation rate on the jet centerline
ωθ,cl azimuthal vorticity near the jet centerline
φcBenjamin’s test function
Π all control parameters of the mean flow
ϕmazimuthal mode phase angle
Φiith spatial POD mode
Vmean velocity vector
vtime-dependent velocity vector
x=x, r, θ cylindrical coordinates
x=x, y, z Cartesian coordinates
ωvorticity vector
xvii
e
Ktot total coherent energy
A∗
spk =Aspk/Aspk,0normalized speaker input voltage
Aeamplitude of wave packet envelope
aiith temporal POD mode
Ammode amplitude integrated across the shear layer
Ae
max maximum amplitude of wave packet envelope
Asat global mode limit-cycle amplitude
Aspk,0critical speaker input voltage of mode m= 2
Aspk speaker input voltage
Bshape parameter of reversed flow
b1,2fit parameters of axial velocity profile
b3,4fit parameters of azimuthal velocity profile
Dnozzle diameter
ffrequency
fsPIV acquisition sampling frequency
fact single mode actuation frequency
H, F, G, P complex disturbance Eigenfunctions
Kturbulent kinetic energy (TKE)
mazimuthal mode number
Nnumber of PIV snapshots
Nx, Nθshape parameter of axial and azimuthal shear layers
N1,2shape parameters of axial inner and outer shear layers
N3,4shape parameters of azimuthal inner and outer shear layers
PRF probability of reversed flow
PVB probability of vortex breakdown
R.05 radial location where Vx= 0.05Vcl
R.5radial location where Vx= 0.5Vcl
R.95 radial location where Vx= 0.95Vcl
Rcore radial position where ωx= 0
Rcrit Benjamin’s critical radius
Rmax radial location of maximum axial velocity
Rθradial location of maximum mean azimuthal velocity
Rvvortex core radius
Sintegral swirl number
Siith swirl configuration
SVB minimum integral swirl number for vortex breakdown
Scrit critical integral swirl number of Hopf bifurcation
Sloc local swirl number
St =fD/V Strouhal number
Tperiod time
ttime
Tens duration of measurement ensemble
Vaxial bulk velocity at the nozzle exit
Vcl axial velocity on the jet centerline
Vmax maximum axial velocity
Xs=Xs,r +iXs,i complex axial coordinate at the saddle point
Znumber of Chebyshev points
PDF probability density function
PSD power spectral density
Chapter 1
Introduction
Swirling jets represent a flow configuration that is both easy to generate in an experiment
and astonishingly rich in physical problems. It often serves as a benchmark for new theo-
retical concepts in the field of unbounded shear flows. Moreover, swirling jets are of great
importance for the combustion industry, due to their ability to stabilize flames and enhance
turbulent mixing. The scientific work presented here focuses on the fundamental aspects
of swirling jets with the prospect of gaining deeper insights into the dynamics of turbu-
lent, three-dimensional shear flows and to derive analysis and control strategies that are
applicable to industrial flows.
The swirling jet considered in this work emanates from a round nozzle into an uncon-
fined domain of quiescent fluid. The investigations described in this thesis target the flow
dynamics in the jet nearfield that are dominated by the following two flow phenomena. (i)
Streamwise growing waves emerge in the thin shear layer between the jet and the steady
surrounding fluid. These waves cause the shear layer to roll-up to large-scale eddies that en-
train quiescent fluid from the surrounding and pump it towards the jet center. This causes
the jet diameter to increase continuously in downstream direction. (ii) Vortex breakdown
occurs when a critical swirl level is exceeded and the jet switches to a completely different
flow state. This flow phenomenon occurs abruptly and is characterized by the appearance
of flow reversal on the jet axis. This leads to the formation of a recirculation bubble located
on the jet center that transforms the flow into that of an annular jet.
The smoke visualizations of a laminar jet depicted in figure 1.1 illustrate these flow
phenomena and may serve as an intuitive introduction to the dominant features of swirling
jets. The photographs represent snapshots of jets emanating from a circular nozzle that is
located near the left image margin. The plane along the jet axis is made visible by mixing
smoke particles to the jet, which are illuminated by a vertical laser sheet coming from down-
stream. The different gray levels along the jet cross-section are due to an inhomogeneous
smoke distribution in the jet. The steady surrounding fluid was not filled with smoke and
appears white.
Figure 1.1a represents a non-swirling jet. The shear layer between the jet and the steady
fluid rolls up and forms into ring-like axisymmetric vortices that propagate in the down-
stream direction at approximately half the jet velocity. These orderly eddy-like structures
are caused by the well-known Kelvin–Helmholtz instability. With increasing distance from
the nozzle these vortices merge, causing the shear layer to spread radially until it reaches
1
2Chapter 1 Introduction
(a) (b)
(c) (d)
(e) (f)
Figure 1.1: Experimental smoke visualization of the plane aligned to the jet axis of
laminar non-swirling (a) and swirling jets (b-f). The swirl intensity is successively increased
from image a to f.
Chapter 1 Introduction 3
the jet axis. By introducing swirl to the jet (figures 1.1b-c), the roll-up of the shear layer
occurs closer to the nozzle and the eddies break down to irregular structures at an earlier
state. At sufficiently strong swirl, the axisymmetry of the eddy-like structures is lost. Flow
structures on the upper boundary of the jet appear at different streamwise locations than
on the bottom boundary, indicating the existence of inclined, spiral-shaped vortices that
replace the axisymmetric, ring-like vortices of the non-swirling jet (compare figure 1.1a
with figure 1.1c). Hence, swirl effects the development of the shear layers and the dynam-
ics of the eddies that form therein. The understanding of these phenomena is of crucial
importance to predict the mixing characteristic of swirling jets. A detailed experimental
and theoretical investigation on this topic is given in chapter 5of this thesis.
Vortex breakdown occurs when the swirl intensity exceeds a certain threshold. Figure
1.1d displays a snapshot of a swirling jet undergoing vortex breakdown, visible by a small
mushroom-shaped white structure in the center of the right half of the image. This structure
indicates fluid that is sucked from downstream to the jet center, creating reversed flow on
the jet axis for a short instant of time. At that swirl intensity, vortex breakdown occurs
intermittently and at a varying streamwise position. Furthermore, the image shows that
the entire jet column is deformed to a wave-like shape, indicating traveling waves in the jet
column. The onset of vortex breakdown and its connection to the formation of standing
waves in the vortex core is discussed in detail in chapter 4.
By further increasing swirl, vortex breakdown becomes more orderly and a recirculation
bubble stabilizes near the nozzle. For instance figure 1.1e depicts a steady bubble located in
the center of the image. Its upstream end is bounded by a dark trident-shaped flow pattern
that rolls up into two inwardly-directed opposing eddies. In fact, these internal eddies are
synchronized to the large rollers that are visible at the periphery of the jet (e.g., see bottom
half of figure 1.1e). Hence, swirling jets undergoing vortex breakdown exhibit large-scale
flow structures that reside in the center of the jet and in the outer shear layers, thereby,
dominating the entire nearfield dynamics. These structures are analyzed in chapter 6, while
their active control is discussed in chapter 7.
The smoke visualizations depicted in figure 1.1 have been conducted for laminar jets at
a very low Reynolds number (inertia to viscosity ratio). The photographs clearly depict
how flow structures grow in the downstream direction, saturate, and break down into small-
scale structures, thereby, marching along the classical road from a orderly laminar flow to
chaos and turbulence. The growth of the (laminar) flow structures near the nozzle are
determined by the linearized Navier–Stokes equations providing the theoretical framework
of linear stability analysis.
Figure 1.2 displays snapshots of a laminar and a turbulent swirling jet undergoing vor-
tex breakdown. Both flows correspond to the same flow states, although the turbulent flow
refers to a Reynolds number that is an order of magnitude higher than for the laminar flow.
By carefully comparing these two images, one may find similarities, such as the recirculation
zone in the center of the image or the periodic waves in the periphery of the jet. For the
turbulent flow, these large-scale flow features are covered with small-scale turbulent ‘noise’
due to the high Reynolds number that causes the laminar-to–turbulent transition to occur
already upstream of the nozzle exit. The existence of so-called coherent structures in tur-
bulent flows and their similarity to the orderly laminar structures is of immense importance
for the theoretical approach presented in this thesis. The linear stability analysis, a well
4Chapter 1 Introduction
(a) (b)
Figure 1.2: Experimental smoke visualization of the plane aligned to the jet axis of a
laminar (a) and a turbulent (b) swirling jet undergoing vortex breakdown.
established theoretical framework for laminar flows, is applied to turbulent flows, yielding
an effective analytical tool.
The present work focuses exclusively on fully turbulent flows. Their stochastic nature
requires statistical methods in order to separate the random turbulent noise from the co-
herent structures. In chapter 6of this thesis, a method is derived to reconstruct the three-
dimensional shape of such flow structures from an uncorrelated sequence of flow snapshots.
The experimental description of the coherent structures is always compared to the results
from linear stability analysis of the turbulent mean flow. The experimental and theoretical
methods used throughout this work turn out to complement one another allowing for a
comprehensive analysis of the flow dynamics in turbulent swirling jets.
1.1 Overview of the Thesis
A detailed description of the underlying theoretical concepts and definitions is given in
chapter 2. Therein, the triple decomposition as the corner stone of coherent structure
extraction and the decomposition of the coherent velocity in normal modes is introduced
first, followed by an outline of the theoretical concept of the linear stability analysis and
its numerical implementation. The chapter ends with a brief introduction to the proper
orthogonal decomposition (POD) used in this work.
Chapter 3contains a detailed description of the two experimental setups used in this
work. Experiments were conducted at the swirling jet water facility at the University of
Arizona and the swirling jet air facility at the TU Berlin. The chapter provides details of
the conducted experiments and the applied measurement techniques.
The main part of this thesis is organized in four chapters, which correspond to four
independent investigations (chapters 4–7). Each chapter begins with a short abstract,
followed by a brief introductory section, a comprehensive result section, and a summary
and discussion of the major results.
In the following, the major concepts and outcome of the four investigations are summa-
rized and the connections between them are pointed out.
Chapter 1 Introduction 5
•Chapter 4:This investigation focuses on the formation of vortex breakdown at
increasing swirl and the onset of global instability. The swirling jet water facility
described in section 3.1 is used and the flow field at increasing swirl is mapped via
time-resolved PIV. A consistent swirl number definition is proposed in section 4.4
that is based on the full equation of motion. The commonly used swirl number based
on boundary layer approximations results in inaccurate values within the region of
vortex breakdown. The formation of vortex breakdown in the turbulent jet is ob-
served by increasing the swirl number incrementally at very small steps.
Breakdown occurs first intermittently thereby heavily oscillating in streamwise di-
rection (section 4.5). Thereby, the location of vortex breakdown coincides with the
super-to-subcritical transition of the mean flow in excellent agreement with the invis-
cid theory of Benjamin (section 4.6). By further increasing the swirl, vortex break-
down stabilizes and the axial extent of the recirculation region increases. At a critical
swirl number, the swirling jet undergoes a supercritical Hopf bifurcation to a global
mode. The global oscillations is manifested in a precession of the vortex core and the
roll-up of the outer shear layer to helix-like eddies (section 4.7.2). The experiments
provide clear evidence that vortex breakdown occurs at a lower swirl number than
the onset of global instability.
•Chapter 5:This chapter deals with the swirling jet below the onset of vortex break-
down. As shown in chapter 4, this flow configuration is globally stable. The shear
layers act as amplifiers to upstream perturbations. The investigation discussed in
chapter 4already reveals a breaking of the rotational symmetry of the mode selection
with increasing swirl (section 4.7.1). The impact of swirl on the convectively unsta-
ble modes is investigated in more detail in chapter 5. A spatial stability analysis is
conducted employing the mean turbulent flow of the swirling jet below the threshold
of vortex breakdown. Details to the applied stability analysis are given in section
2.4, whereas the mean flow configuration is described in section 5.2.1. Results are
compared to the instabilities in a non-swirling jet.
At the nozzle, the swirling jet is unstable to a centrifugal and a Kelvin–Helmholtz
instability (section 5.3). Due to the rotational motion, the symmetry breaks and
the phase velocity of the helical instabilities depends on the frequency and the az-
imuthal mode number. It is concluded that this affects the ability of the swirling jets
to promote subharmonic resonance and intermodal interactions of helical instability
waves, which is a prominent feature of non-swirling jets. With increasing streamwise
distance, only co-winding shear modes remain unstable. The swirling jet inherits a
double-helical co-winding mode as the preferred mode in the nearfield that decelerates
and becomes nearly steady at the end of the potential core (section 5.4).
The theoretical results are validated by experimental investigations. A wave packet is
generated at the nozzle lip and its streamwise development is mapped via ensemble-
averaged hot-wire measurements (section 5.6). The theoretically derived mode se-
lection is validated by decomposing the wave packet into Fourier modes. Details
on the experimental procedure are given in 3.4. The existence of steady modes is
clearly confirmed by the pulsed experiments (section 5.6). The impact of swirl on the
streamwise phase velocity is validated by forcing the flow at a single mode. Phase-
averaged hot-wire measurements clearly support the theoretical findings (section 5.5).
6Chapter 1 Introduction
Furthermore, this investigation shows that the streamwise growth of instabilities in a
swirling jet is much lower than in a non-swirling jet due to the rapid increase of the
thickness of the axial shear layer. The swirl component had only a marginal effect on
the growth rates for the considered flow configurations.
•Chapter 6:As shown in chapter 4, the axisymmetric vortex breakdown state bifur-
cates to a global mode when a certain swirl number is exceeded. Chapter 6focuses
on the description of the coherent structures associated with the global mode at its
limit-cycle oscillations. PIV measurements are conducted at a swirl number beyond
the threshold of the Hopf bifurcation. Details to the considered flow configuration
are given in section 6.3. The phase-averaged velocity field of the global mode is de-
rived from uncorrelated PIV snapshots employing a POD-based phase reconstruction
procedure. Although the method has already been used in chapter 4to obtain the
critical swirl number of the Hopf bifurcation (section 4.7), its application is described
in section 6.5 in a more general scope.
The dynamics associated with the global mode of the swirling jet is characterized by
a precession of the vortex core upstream of the breakdown bubble and the roll-up
of large-scale coherent structures in the outer shear layer (section 6.5). The three-
dimensional shape of this global mode is reconstructed from the two-dimensional PIV
snapshots (section 6.5.3) and compared to the global mode derived from a local spa-
tial stability analysis (section 6.7). The stability eigenmodes and the phase-averaged
velocities indicate that the coherent kinetic energy of the global mode is concentrated
in the periphery of the jet where coherent structures grow rapidly with downstream
distance.
•Chapter 7:This chapter deals with the open-loop control of the global mode. A
swirling jet is generated using the same swirling air facility as in chapter 5and 6.
The swirl is adjusted to a level where vortex breakdown occurs with the jet–to–wake
transition to be located downstream of the nozzle. Details to the flow configuration
are given in section 7.2. A spatio-temporal stability analysis employing the natural
mean flow reveals that the single-helical mode is the only mode that exhibits a pocket
of absolutely unstable flow. The analysis further reveals that the wavemaker of this
mode is located upstream of the breakdown bubble, which coincides with the location
where the precessing vortex core is most energetic. Forcing of the flow at the azimuthal
mode number of the global mode results in a global lock-in (section 6.4.2). Forcing the
flow at a different azimuthal wavenumber leads to the amplification of convectively
unstable modes in the outer shear layer (section 7.4). These excited modes saturate
nonlinearly, which results in a significant mean flow correction. This has a stabilizing
effect on the natural global mode, revealing an effective suppression of the precession
of the vortex core (section 7.5).
While the investigation of the moderately swirling jet of chapter 5is rather self-contained,
the results from the chapters 4,6, and 7can be condensed to the following statements.
Spiral-shaped vortex breakdown is a consequence of a globally unstable axisymmetric vor-
tex breakdown. The associated large-scale coherent structures are most energetic in the
periphery of the jet due to the thin outer axial shear layer that enables strong streamwise
amplification of upstream perturbations. The pacemaker of these perturbations is located
Chapter 1 Introduction 7
upstream the breakdown bubble and is characterized by a precessing vortex core that rep-
resents the wavemaker of the global mode. This wavemaker can be controlled indirectly by
forcing the outer shear layer at a different mode, thereby utilizing the amplifier dynamics
in the outer axial shear layer.
Chapter 2
Theoretical Concepts
2.1 Triple Decomposition as the Basis for Coherent Struc-
ture Extraction
Large-scale organized structures in turbulent flows were investigated for more than 40
years. For a comprehensive summary of earlier work, the reader is referred to Laufer
(1975), Roshko (1977), Cantwell (1981) and Ho & Huerre (1984). A variety of definitions
and techniques have been developed to reveal these so-called coherent structures. These
include statistical approaches, pattern recognition methods, stability theory, conditional
sampling and averaging and topological methods from dynamical system theory.
Some of these methods are based on a triple decomposition introduced by Hussain &
Reynolds (1970). Accordingly, the time and space dependent flow v(x, t) is decomposed into
a time-averaged part V(x), a coherent part vc(x, t), and a randomly fluctuating (stochastic)
part vs(x, t), yielding
v(x, t) = V(x) + vc(x, t) + vs(x, t),(2.1)
The coherent part is assumed to be periodic in time and space and may be decomposed into
normal modes. The triple decomposition method represents a refinement of the classical
Reynolds decomposition
v(x, t) = V(x) + v′(x, t),(2.2)
with v′(x, t) simply representing the fluctuating part of the velocity.
The triple decomposition has become a conventional tool in active flow control experi-
ments that distill coherent disturbances by means of phase-locked averaging. Its application
to experiments is easy when the coherent structure is tagged by external excitation where a
simple synchronization of the data acquisition with the forcing signals is required. Without
such external phase trigger, POD-based techniques can provide another means for phase
identification (Depardon et al. 2007). The triple decomposition method has been implicitly
used in a number of theoretical articles where the stability analysis was applied for turbu-
lent flows (e.g., see Crighton & Gaster 1976;Gaster et al. 1985). Liu (1989) has developed
a local turbulence model for many shear flows utilizing the triple decomposition.
9
10 Chapter 2 Theoretical Concepts
51
154
one of eight
speaker
x
z
yr
θ
one of eight
speaker
Figure 2.1: Coordinate systems used throughout this work displayed with the nozzle.
2.2 Coordinate Systems
The orientation of the two coordinate systems used in the present work is shown in figure
2.1. Cylindrical coordinates are used to describe the flow quantities in the crossflow plane
whereas Cartesian coordinates are used for data shown in the streamwise plane. The
two coordinate systems are necessary, as the latter does not cause a singularity along the
jet axis. On the other hand, in the crossflow plane, cylindrical velocity components are
necessary to correctly describe the flow quantities of the axisymmetric flow. Furthermore,
the normal mode decomposition, necessary for the linear stability analysis, is strictly based
on an axisymmetric flow given in cylindrical coordinates. In order to compare theoretical
results with experimental data we will switch between the two coordinate systems.
2.3 Normal Mode Ansatz and Sign Convention
In order to avoid confusion during the discussion of the spatio-temporal characteristics of the
instabilities, it is important to clarify the sign conventions used in this work. The coherent
velocity of a disturbance traveling in an axisymmetric shear layer can be decomposed in
normal modes in the following form (e.g., see Gallaire & Chomaz 2003):
vc(x, r, θ, t) = ˆ
v(r)ei(αx+mθ−ωt)+ˆ
v∗(r)e−i(α∗x+mθ−ω∗t)(2.3)
where αis the complex axial wavenumber, ωis the complex frequency, mis the real az-
imuthal wavenumber, and the asterisk ∗denotes the complex conjugate. Hence, instability
modes are equally represented by
(α, m, ω) or (−α∗,−m, −ω∗).(2.4)
Chapter 2 Theoretical Concepts 11
mωr
mαr
0
co-rotating
counter-winding
counter-rotating
counter-winding
counter-rotating
co-winding
co-rotating
co-winding
streamwise modes
(αr= 0)
steady modes
(ωr= 0)
Figure 2.2: Schematic drawing of the six non-axisymmetric variants of mode alignments
in swirling jets; m,ωr, and αrmay take both signs and are equally expressed by (αr, m, ωr)
or (−αr,−m, −ωr).
Without loss of generality, we consider only mean flows with Vθ≥0. Cases with negative Vθ
can be deduced by the following symmetry (Gallaire & Chomaz 2003;Olendraru & Sellier
2002)
(Vθ, m)→(−Vθ,−m).(2.5)
In experimental studies, it is common to let ωrand αronly take positive values (e.g., see
Liang & Maxworthy 2005;Panda & McLaughlin 1994). The phase function (αrx+mθ−ωrt)
then implies that at a fixed axial location xmodes with m > 0 rotate in time in the direction
of the basic flow and are called co-rotating, while modes with m < 0 are called counter-
rotating. The phase function further implies that at fixed time tand increasing x, modes
with m > 0 have a line with constant phase that winds in opposite direction to the basic flow
rotation and are called counter-winding and modes with negative mare called co-winding.
Although the restriction to positive frequencies and wavenumbers appears more intu-
itive, it constrains the investigation of certain spatio-temporal mode configurations that
may occur in reality. It is therefore more general to let αrtake both signs. In that case,
co-winding modes correspond to αrm < 0 and counter-winding modes to αrm > 0. Hence,
a co-rotating co-winding mode corresponds to positive mand ωrand negative αr. These
modes exist especially in swirling jets and they can only be expressed by negative wavenum-
bers. In fact, for the presentation of certain diagrams, it is convenient to let also ωrtake
both signs with negative values being derived from equation (2.4). This implies that co-
rotating modes refer to ωrm > 0 and counter-rotating modes to ωrm < 0. In figure 2.2 the
six non-axisymmetric mode variants are summarized in a schematic drawing. It may serve
as a visual aid that comes in handy during the discussion of the stability analysis presented
in chapter 5.
12 Chapter 2 Theoretical Concepts
2.4 Linear Stability Analysis
Computing spatially growing disturbances in shear layers by means of linear stability anal-
ysis has a long history. Michalke (1965) calculated the spatial stability characteristics for
the hyperbolic-tangent velocity profile according to inviscid theory. Spatial growth rate and
amplitude distribution agreed well with measurements conducted by Freymuth (1966), but
they failed in some detail when the flow was divergent. As refinement, several attempts have
been made to account for non-parallel effects (Cohen et al. 1994;Crighton & Gaster 1976;
Gaster 1974;Gaster et al. 1985;Plaschko 1979). Gaster et al. (1985) applied the inviscid
linear stability analysis to the periodically forced turbulent and slightly divergent mixing
layer. The computed normalized phase and velocity amplitudes agreed with experimental
data but the amplification rates in the direction of streaming were strongly overpredicted.
The robustness of the stability analysis was demonstrated by Weisbrot & Wygnanski (1988),
whose computed eigenmodes correctly predicted the measured phase and amplitude distri-
butions of the excited waves, although the latter were forced at high amplitudes clearly
exceeding the linear regime. It is important to note that for high Reynolds numbers, the
stability analysis is based on the time-averaged turbulent flow which is not a stationary
solution of the Navier–Stokes equation. It is argued that this infringement of the linear
stability theory is possible ’knowing that the random changes in the mean velocity occur on
a time scale that is short in comparison with the period associated with the large coherent
structures’ (Weisbrot & Wygnanski 1988).
For the linear stability analysis conducted in this work quasi-parallel flow is assumed.
Hence, the streamwise development of a traveling wave is derived by successively solving
the local parallel flow problem. Therefore, the Orr–Sommerfeld eigenvalue problem must
be solved for fictitious parallel flows at each streamwise location. The global shape of an
instability mode is then constructed from the local solutions by applying a multiple-scale
approximation (Ho & Huerre 1984;Huerre & Monkewitz 1990).
2.4.1 Eigenvalue Problem of the Parallel Flow
To analyze the linear stability at a given axial location of the mean flow, velocity and
pressure disturbances (vc
x, vc
r, vc
θ, pc) are superposed onto the corresponding mean velocity
profile. These perturbations are periodic in time and are decomposed in normal modes,
yielding
(vc
x, vc
r, vc
θ, pc) = ℜn[H, iF, G, P] ei(αx+mθ−ωt)o,(2.6)
where H,F,Gand Pare the complex eigenfunctions. Upon substituting the modal decom-
position (2.6) into the Navier–Stokes equations, linearized about the mean flow V(x, r, θ),
we obtain the linear system of ordinary differential equations for continuity
F′+F
r+mG
r+αH = 0,(2.7a)
Chapter 2 Theoretical Concepts 13
for the x-momentum
H′′
Re +H′
rRe +iω −imVθ
r−iαVx−m2r2ReH
−iV ′
xF−iαP =α2H
Re (2.7b)
for the r-momentum
iF′′
Re +iF′
rRe −ω−mVθ
r−αVx+i(m2+ 1)
r2Re F
−2im
r2Re −Vθ
rG−P′=iα2F
Re (2.7c)
and for the θ-momentum
G′′
Re +G′
rRe +iω −imVθ
r−iαVx−m2+ 1
r2Re G
−iV ′
θ+2m
r2Re +iVθ
rF−imP
r=α2G
Re (2.7d)
where the primes denote d/dr. For the free jet, the boundary conditions in the farfield are
(Khorrami et al. 1989)
F(∞) = G(∞) = H(∞) = P(∞) = 0 (2.8)
and in the limit along the centerline (r= 0) impose
F(0) = G(0) = H(0) = P(0) = 0 if |m|>1 (2.9a)
H(0) = P(0) = 0
F(0) + mG(0) = 0
2F′(0) + mG′(0) = 0
if |m|= 1 (2.9b)
F(0) = G(0) = 0
H(0) and P(0) finite )if m= 0 (2.9c)
For a given mean velocity profile, the system of equations (2.7–2.9) describes an eigenvalue
problem. A non-zero solution of (F, G, H, P) exists if and only if the complex pair (α, ω)
satisfies the dispersion relation D(α, ω, m, Γ,Re) = 0. The symbol Γ represents all control
parameters describing the mean velocity profiles.
2.4.2 Temporal, Spatial, and Spatio-temporal Analysis
The above introduced eigenvalue problem can be solved for a complex ωand a given real
α, for a complex αand a given real ω, or for a complex αand a complex ω.
The first approach is called temporal analysis, yielding temporally growing (ωi>0) or
decaying (ωi<0) modes. It is most appropriate for the analysis of bounded flows that
have no free stream velocity such as the Taylor–Couette flow.
14 Chapter 2 Theoretical Concepts
The second approach is called spatial analysis, yielding spatially growing (−αi>0)
or decaying (−αi<0) modes. It is applicable to open shear flows that are convectively
unstable (noise amplifiers, see Michalke 1965). In weakly non-parallel flows, the spatial
analysis describes the streamwise growth and decay of flow perturbations initiated at a
certain axial location, e.g. x= 0. Within the framework of the multiple-scale analysis, the
global disturbance velocity field is given by
vc(x, t) = ℜA0(x)[H(x, r), iF(x, r), G(x, r)] exp iZx
0
α(ξ)dξ +mθ −ωrt.(2.10)
It is assumed that the (small) length scale of the instability is separated from the (large)
length scale that characterizes the streamwise non-uniformity of the mean flow (Huerre &
Monkewitz 1990). This implies a slow streamwise variation of the term A0[H, iF, G] with
xand a fast streamwise variation of the wavenumber αwith ξ.
Crighton & Gaster (1976) have developed a first order correction for weakly non-parallel
flows that enables to derive a slowly varying amplitude scaling A0(x) from a ordinary differ-
ential equation employing the eigenfunction and its adjoint. Accounting for the streamwise
and radial varying term A0[H, iF, G] implies that the growth rate and phase velocity of a
traveling wave depends on the streamwise and radial coordinate and on the velocity com-
ponent. Hence, αiand cph =ωr/αrmust be considered as rough estimates of these two
quantities. This is confirmed by experimental observations of streamwise traveling waves
in the mixing layer Gaster et al. (1985).
However, for the sake of simplicity, we omit the weakly non-parallel correction and
assume A0to be uniform throughout this work. This quasi-parallel approach is justified
by the following reasons. The swirling jet flows considered here are highly non-parallel
due to strongly enhanced jet spreading and vortex breakdown. Under these conditions,
the local analysis exceeds its strict limits of validity and so does the weakly non-parallel
correction scheme. The resulting inaccuracies in the prediction of the fast variable αis
much more crucial than the inaccuracies in the prediction of the slowly varying A0. The
present argumentation is in line with a previous study on the instabilities in the wake of a
cylinder conducted by Juniper et al. (2011).
Within the framework of quasi-parallel stability analysis, a disturbance at frequency ωr
and azimuthal wavenumber mthat travels downstream is amplified in flow regions where
−αi>0 and damped in regions where −αi<0. Assuming A0= 1, the amplitude distri-
bution of the axial velocity component of the traveling wave corresponds to the modulus
of the eigenfunction H(x, r) computed at each streamwise location and weighted by the
amplitude ratio g(x) = exp[−Rx
0αi(ξ)dξ], yielding
|ˆvx(x, r)|=H(x, r)exp −Zx
0
αi(ξ)dξ,(2.11)
where the eigenfunction Hdepends parametrically on x. In order to assemble the eigen-
functions computed for different x, a consistent normalization is required, which intro-
duces an ambiguity to the present analysis. Here, a uniform total kinetic energy K=
R∞
0|[H, iF, G]|2rdr is used. Note that the normalization with other quantities such as the
Euclidean norm or the radially integrated amplitude did not noticeable effect the results.
Chapter 2 Theoretical Concepts 15
The streamwise phase velocity for the parallel flow is given by cph =ωr/αr. This corre-
sponds approximately to the phase velocity of the disturbance traveling in the center of the
shear layer (Gaster et al. 1985). Nonetheless, an exact prediction of the phase velocity is
impossible within the quasi-parallel approach.
The third approach, where the eigenvalue problem is solved for complex αand complex
ωis called spatio-temporal analysis. It is applicable to flows that undergo self-excited oscil-
lations at a discrete tone, as for instance wakes (Monkewitz 1988;Provansal et al. 1987), hot
jets (Monkewitz et al. 1990), or cold swirling jets undergoing vortex breakdown (Gallaire
et al. 2006;Liang & Maxworthy 2005). The intrinsic oscillations of these so-called globally
unstable modes are strongly connected to a spatial domain where the flow is absolutely
unstable (Chomaz et al. 1988;Chomaz et al. 1991). In that region, disturbances are pro-
moted that grow in time in upstream and downstream direction, ultimately contaminating
the entire flow (flow oscillators). In contrast, in convectively unstable regions, modes grow
solely in downstream direction and are swept away from their source (flow amplifiers). In
order to distinguish between absolute and convective instability, the absolute frequency ω0
must be derived (Briggs, R. 1964;Huerre & Monkewitz 1990). This implies tracking for
saddle points in the complex α-plane by minimizing dω/dα. The complex frequency at
the saddle point ω0=ω0,r +iω0,i is then associated with the absolute frequency and the
streamwise distribution of ω0(x) can be considered as a global dispersion relation. A key
feature of globally unstable flows is the existence of a wavemaker from which the global
oscillations arise. In case of a linear global mode, this wavemaker is located at a saddle
point in the complex x-plane (Chomaz et al. 1991). The complex frequency of the global
mode ωsis selected at the wavemaker location, yielding
ωs=ω0(Xs) with dω0
dXs
(Xs) = 0.(2.12)
Hence, if the imaginary part of the global frequency ωsis larger than zero, the flow is
considered as linearly globally unstable. In case of a nonlinear global mode, the wavemaker
is located at the convective absolute transition point (Pier & Huerre 2001).
2.4.3 Numerical Method
Khorrami et al. (1989) demonstrated that the eigenvalue problem can be efficiently solved
by using a Chebyshev spectral collocation method. Following this study, the system of
ordinary differential equations (2.7) is solved numerically by discretizing the three velocity
components and the three momentum equations at the Chebyshev collocation points. The
continuity equation is enforced at the mid grid points. This approach has been successively
applied by Khorrami (1991) to the temporal problem and recently to the spatial problem
by Parras & Fernandez-Feria (2007). For a detailed description of the numerical procedure,
the reader is referred to Khorrami et al. (1989), and thus, only a brief summary is given
here.
The boundary conditions (2.8) are enforced at a large but finite radius rmax ≫1 in
consistency with the work of Olendraru & Sellier (2002) and Parras & Fernandez-Feria
(2007). A coordinate transformation is necessary to map the Chebyshev collocation points,
in the interval −1≤ξ≤1, onto the physical domain of the problem, in the range 0 ≤
16 Chapter 2 Theoretical Concepts
r≤rmax. Here, the two-parameter transformation proposed by Malik et al. (1985) is used,
which reads r
rc
=1 + ξ
1−ξ+ 2rc/rmax
.(2.13)
Since the Chebyshev collocation points are known to be distributed in the vicinity of r= 0
and rmax, the parameter rcis necessary to redistribute the collocation points. It allows half
of the points to be distributed in the region 0 ≤r≤rc. Finally, the eigenvalue problem
(2.7) for the case of spatial stability (given real ω, complex eigenvalue α) is linearized
by introducing a generalized eigenvector X= [F, G, H, αF, αG, αH, P]T. Discretizing the
system of ordinary differential equations (2.7) in terms of the variable ξand enforcing the
boundary conditions (2.8–2.9), we may write the generalized eigenvalue problem as
DX =αEX .(2.14)
Taking Zas the number of Chebyshev points, both Dand Eare square matrices with
dimensions of 7Z. Note that the last 14 rows of matrix Dcontain the boundary conditions.
The eigenvalue problem (2.14) is solved using a standard EIG routine embedded in the
software environment MATLABTM. Spurious eigenmodes, caused by the discretization,
are discarded by two independent criteria: first, all eigenmodes are discarded that do not
diminish at r→ ∞, that is to say that only those eigenvalues are considered that satisfying
Z/10
X
i=1 |F(ri)|2
Z
X
i=1 |F(ri)|2
< ǫ1,(2.15)
with ribeing the radial points and ǫ1a given tolerance. Second, spurious eigenvalues are
filtered out by comparing the computed spectra SZand SZ′for Z′> Z. The location of
the spurious modes in the complex α-plane is very sensitive to the number of Chebyshev
points Z, in contrast to the few physical eigenvalues of the problem. Thus, the eigenvalues
αare considered as spurious if min|α−α′|> ǫ2.
The accuracy of the calculations is first checked by comparing the computed eigenvalues
with those calculated by Khorrami et al. (1989) and Parras & Fernandez-Feria (2007). The
computed eigenvalues agree for all shown digits which is not surprising as these authors use
exactly the same numerical method. A comparison of the computations with the results
presented by Gallaire & Chomaz (2003) is more challenging as their results are retrieved
by direct numerical simulations of the linear impulse response. Unfortunately, Gallaire &
Chomaz do not explicitly present computed eigenvalues of their spatio-temporal analysis
but only display the complex frequency ωof the temporal problem. Hence, for the sake of
comparison, the temporal modes are computed using the same base flow as used by Gallaire
& Chomaz. Figure 2.3 clearly shows that for m= 1 both numerical methods arrive at
the same solutions. The correctness of the computed eigenvalues makes provides enough
confidence to apply the computations to the present flow configuration. In comparison to
the base flow used by Gallaire & Chomaz and Parras & Fernandez-Feria, our mean flow is
more complex as it consists of two axial and two azimuthal shear layers. Thus, the problem
is more demanding and the number of collocation points has to be increased. To satisfy the
Chapter 2 Theoretical Concepts 17
0 2 4
0.1
0.2
0.3
i
α
+
_1
0
+
_2
+
_3
Figure 2.3: Temporal stability analysis of a non-swirling jet. The growth rate ωiis
plotted over the axial wavenumber α. The corresponding azimuthal wavenumbers mare
shown close to the corresponding curves. The solid lines represent computations reprinted
from Gallaire & Chomaz (2003). Symbols represent the computations at m±1 using a
Chebyshev collocation method (Z= 180, a= 3, rmax = 100, ǫ1= 10−11,ǫ2= 10−3).
filter criterion (2.15) and to successfully discard the spurious eigenvalues, the number of
Chebyshev points is increased to Z= 300. A convergence study optimizes the parameters
aand rmax of the transformation (2.13) yielding a= 3 and rmax = 100 for all computations
presented here. It was observed that the calculated eigenvalues are relatively insensitive to
the radial distribution of the collocation points.
A further challenging aspect of the computation of spatial instability is the sorting of the
eigenvalues and eigenfunctions with respect to the corresponding modes. Spatial branches
have to be identified and tracked in the complex wavenumber plane while a parameter of
the dispersion relation is changed. A well-designed sorting routine is of great importance
to accurately follow the spatial branches into the region of negative growth αi>0. In this
region, many modes coexist and several spatial branches intersect. A routine is developed
which sorts the eigenvalues and eigenfunctions of the dispersion relations Dand D′by
incorporating two criteria: first, eigenvalues are sorted by minimizing the distance |α−α′|;
second, the normalized eigenvectors Xare sorted by minimizing |1−hXX′i|. The efficiency
of the sorting routine is validated by visually checking the spatial branches.
For the study of the moderately swirling jet (chapter 5), the numerical scheme is fur-
ther simplified by implementing the MATLABTMroutine EIGS which solves an eigenvalue
problem and returns only one eigenvalue that is closest to a given estimate. The eigenvalue
problem is first solved for one combination of the parameters (ωr, m, Γ) by using the stan-
dard MATLABTMroutine EIG, which returns numerous eigenvalues including the spurious
ones. The physically meaningful solutions are then sorted out, a posteriori, by using two
independent criteria mentioned earlier. Starting from this solution, the eigenvalue problem
is solved for a new set of parameter (ωr, m, Γ,Ret) using the EIGS routine. The required
estimate is thereby derived from cubic extrapolation of the already obtained solutions while
going in small increments through the m-ωr-Γ-parameter space. Particular attention must
be payed when several modes exist for one parameter setting since the EIGS routine may
18 Chapter 2 Theoretical Concepts
switch from one mode to the other. All results are cross-checked by using the standard
EIG routine.
For the spatio-temporal analysis, the absolute frequency ω0is tracked while traversing
through the flow field in streamwsie direction, using a method introduced by Rees (2009). It
is based on a truncated Taylor series expansion that is expected to converge in the vicinity
of a saddle point. The required starting values of ω0are derived iteratively from cubic
extrapolation. The routine is started at an axial location where ω0is derived visually from
global pictures computed at two closely spaced axial locations (see Suslov (2006) for details
on global pictures and required eigenvalue sorting routines). Details of the procedure are
given in in the work of Rukes (2010). The location of the wavemaker Xsand the associated
global complex frequency ωsare derived from equation (2.12) by analytic continuation of
ω0(X) into the complex Xplane. Therefore, ω0(x) is fitted to a Pad´e polynomial.
2.4.4 Modeling Small-scale Turbulence
To apply linear stability analysis to a turbulent mean flow, it is assumed that the turbulent
fluctuations are at a much smaller time and length scale than the coherent structures that
arise due to hydrodynamic instability. An interaction between these coherent structures,
the mean flow, and turbulent fluctuations is neglected. Notwithstanding, it is assumed that
fine-scale turbulence provides additional mixing and behaves like an added eddy viscosity.
The instabilities in free shear flows are typically inviscid, and hence, the additional eddy
viscosity due to fine-scale turbulence has primarily a stabilizing effect (e.g., see Liu 1971;
Marasli et al. 1991). This fictitious viscosity νtthat adds to the kinematic viscosity ν
is integrated into the stability analysis by writing the dispersion Dwith the turbulent
Reynolds number
Ret=V δx
ν+νt
.(2.16)
The eddy viscosity is derived from the well-known Boussinesq’s approximation, yielding
−v′
xv′
r=νt∂Vx
∂r +∂Vr
∂x .(2.17)
As stated by Townsend (1956), this simple eddy viscosity model is only valid within the
turbulent flow and must therefore be weighted by an intermittency factor γthat is derived
from the PIV snapshots using the following approach: First, the instantaneous azimuthal
vorticity is calculated from each PIV snapshot. Second, the noise from the obtained vortic-
ity field is removed by setting values below 5 % to zero. The resulting vorticity distributions
provide a reasonably well description of the instantanious boundary between rotational and
irrotational flow. Third, the number of events of irrotational flow Npot are derived for each
measurement location. The intermittency function is then defined as γ=Npot/N, where
Nis the total number of PIV snapshots. The eddy viscosity weighted by the intermittency
function is assumed to be constant in radial direction but may vary in axial direction.
Chapter 2 Theoretical Concepts 19
2.5 Proper Orthogonal Decomposition
This section provides a brief introduction to the principal concept of the proper orthog-
onal decomposition (POD). In this work, POD is used to extract the dominant coherent
structures that are naturally prevalent in the globally unstable swirling jet.
We consider the fluctuation snapshots of a velocity field
v′(x, tk) = v(x, tk)−V(x),
where xis a point in a spatial domain Ω ⊂R3, and tk,k= 1,...,N, are the sampling
instants. The goal is to find a least-order expansion of the snapshots
v′(x, tk) =
I
X
p=1
ap(tk)Φp(x),+vres(x, tk),(2.18)
which minimizes the residual vres in a sense specified below. The snapshots are considered
as elements of the Hilbert space of square integrable vector fields L2(Ω). This Hilbert space
is equipped with the inner product in Ω between two vector fields vand wdefined by
(v,w)Ω:= ZΩ
v·wdx,(2.19)
and the related norm kvkΩreads
kvkΩ:= q(v,v)Ω.(2.20)
The velocity fields are provided by PIV snapshots taken at Nuncorrelated points at the
times tk,k= 1,...,N. In addition to the inner product in space, we define the ensemble
average of a quantity ζas
ζ:= 1
N
N
X
k=1
ζ(tk).(2.21)
The quantity ζmay be a scalar, a vector or any other tensor. The norm and ensemble
average allows one to formulate an optimal property of the Galerkin expansion (2.18). We
require that the spatial modes are chosen such that the time-averaged L2error is minimal
for the number of modes I= 1,...,N:
χ2(Φ1,...,ΦI) := kvresk2= min.(2.22)
Note that the minimized residual vres ≡0 for I=N. This optimality property is fulfilled
by the snapshot POD modes introduced by Sirovich (1987). The corresponding algorithm
is based on the N×Nautocorrelation matrix R= (Rkl) defined by
Rkl := 1
Nv′(x, tk),v′(x, tl)Ω(2.23)
20 Chapter 2 Theoretical Concepts
quantifying the relation between the snapshots. The correlation matrix is symmetric and
positive semi-definite, i.e. the eigenvalue problem
Rap=λpap(2.24)
has real and non-negative eigenvalues λp≥0. Without loss of generality, we assume the
eigenvalues to be sorted by magnitude:
λ1≥λ2≥...≥λN= 0.
Note that λN= 0 since Nlinearly independent snapshots cannot span the whole N-
dimensional space. Two points, for instance, define a one-dimensional line. The corre-
sponding eigenvectors ap:= [ap(t1), ..., ap(tN)]t, called temporal modes, are orthogonal by
construction. For reasons of convenience, we require
apaq=1
N
N
X
k=1
ap(tk)aq(tk) = λpδpq.(2.25)
Now, the spatial POD modes can be calculated as a linear combination of the fluctuation
snapshots
Φp(x) = 1
Nλp
N
X
k=1
ap(tk)v′(x, tk).(2.26)
These spatial POD modes are orthonormal by construction:
(Φp,Φq)Ω=δpq.(2.27)
The eigenvalues λprepresent twice the amount of the fluctuating kinetic energy contained
in each POD mode, Kp:= (v′,Φp)2
Ω/2 = λp/2. The total fluctuation energy is defined as
the sum of the modal contributions owing to orthonormality (2.27):
K:= 1
2kv′k2
Ω=
N
X
p=1
Kp=1
2
N
X
p=1
λp.(2.28)
Kis generally referred to as turbulent kinetic energy (TKE). Snapshot POD extracts the
most energetic structures representing them as linear combinations of the snapshots and
imposes orthogonality in spatial and temporal modes. Snapshot POD is the time-discrete
variant of a general continuous formulation (Holmes et al. 1998). In turbulent flows, the
large-scale structures usually contain a major portion of the TKE, so the POD modes with
high energy content can hence be expected to span the basis for the dominant coherent
structures.
Chapter 3
Experimental Arrangements and
Measurement Techniques
3.1 The Swirling Jet Water Facility
The water facility belongs to the laboratory of Prof. I. Wygnanski at the University of
Arizona. A schematic view of the experimental apparatus is shown in figure 3.1. The
facility consists of a horizontal swirling jet discharging into a large transparent water tank
that has a 1000 mm by 1000 mm cross section and measures 1300 mm in length. A 762 mm
long plexiglass cylinder with an inner diameter of 254 mm serves as a settling chamber and
houses the apparatus generating the swirl. Similar to the method used by Billant et al.
(1998), the swirl is generated by passing the water through a rotating cylinder, placed in
the interior of the settling chamber. The jet axial velocity is generated by a pump and
the flow is set into a state of solid body rotation by inserting a 559 mm long honeycomb
into the inner rotating cylinder. The swirling flow is then guided through a converging
nozzle attached to the outer cylinder and mounted onto the tank. A serrated ring glued
to the interior surface of the contraction trips the flow, thus preventing transitional effects
associated with a change of Reynolds number. The nozzle diameter is D= 52.4 mm. Effects
of confinement are minimized by the relatively large size of the tank and the large diameter
of the scooping nozzle. Consequently, the measured recirculation currents are found to
be negligible. The water surface is covered by a floating foam in order to minimize the
vertical temperature gradients resulting from evaporation. Additional information about
this facility are given in the work of Richard (2003), Oberleithner (2006), and Seele (2008).
3.2 The Swirling Jet Air Facility
The air facility belongs to the laboratory of Prof. C. O. Paschereit at the TU Berlin. A
turbulent swirling jet is generated using an apparatus that resembles the one built by
Chigier & Chervinsky (1965). The schematic arrangement of the facility is shown in figure
3.2. The primary axial stream of air passes through a deep honeycomb prior to entering a
swirler through which a secondary air stream is introduced through four tangential slots,
ech 80 mm long. The flow is then guided through a 600 mm long tube, before entering
21
22 Chapter 3 Experimental Arrangements and Measurement Techniques
water tank
1300
flowmeter
pump
dc motor
floating top
axisym. contraction
nozzle
800
254
559
52.4
1000
Figure 3.1: Sketch of the swirling jet water facility (all lengths are expressed in mm)
the contraction forming the nozzle. A perforated plate is mounted in the tube to minimize
possible inhomogeneities resulting from the tangential inlets in the swirl chamber. The
swirl levels generated by the facility depend on the ratio of mass flows coming through
the two inlets: a non-swirling jet is generated when no air enters tangentially through
the swirler and the maximum swirl level is attained when the axial inflow is zero. Two
frequency-controlled blowers provide the necessary airflow. The volume flow of each blower
was measured using calibrated orifices connected to BARATRON gauges. The blowers are
feed-back controlled to provide a constant volume flux. The nozzle diameter is D= 51 mm.
Particular attention was paid to the design of the excitation device located at the nozzle
lip where the thin shear layer between the jet and the quiescent surrounding fluid is unstable
to all azimuthal modes (Cohen & Wygnanski 1987;Gallaire et al. 2004). Acoustic excitation
is applied using an array of eight loudspeakers equally spaced along the azimuth (figure 3.2).
Such an array provides radial fluctuations that trigger the inviscid shear layer instabilities.
An acoustic wave-guide from each actuator terminates in a rectangular duct leading to a
narrow slot that does not interfere with the jet flow when the speakers are inactive. The
loudspeakers are driven by a set of digital–to–analog converters under program control. The
actuators are adjusted to equal amplitudes under no–flow conditions using a microphone
located at the centerline in the exit plane of the nozzle. The azimuthal disturbances can
be controlled by varying the phase difference between the actuators. With an array of
eight actuators the highest azimuthal mode numbers that can be excited are m=±4.
A similar excitation device was successfully used by Long & Petersen (1992) to study
instabilities in non-swirling jets and by Panda & McLaughlin (1994) for swirling jets. For
the measurements of the impulse response, a pulse is generated by using only one of the
eight loudspeakers. A delta-pulse in time and space is generated by running the speaker
with a saw-tooth signal with a sharp rising edge and a linearly decaying falling edge. The
peak velocities of the zero-mass-flux jets created by the actuators are measured at the slots
using a hot-wire probe. This characteristic velocity scales linearly with the speaker input
Chapter 3 Experimental Arrangements and Measurement Techniques 23
600
51
154
one of eight
speaker
perforated
plate
swirler
swirler
axial inelt
honeycomb
one of four
swirler inlets
swirler
inlet
tangential
inlets
x
z
yr
θ
Figure 3.2: Sketch of the swirling jet air facility (all lengths are expressed in mm)
24 Chapter 3 Experimental Arrangements and Measurement Techniques
voltage Aspk. Details about the design of the actuators and their calibration are given in
appendix B.
3.3 Stereo-PIV Measurements
Stereoscopic particle image velocitmetry (Stereo-PIV) was used to measure the flow field.
It consists of velocity measurements of particles going through a laser sheet generated by a
double-pulsed Nd:Yag laser at 532 nm. Two CCD cameras were positioned at a 45◦angle
in order to measure all three velocity components in a 2D plane. The cameras observed
the light sheet in a Scheimpflug configuration, each inclined at 45◦to the light sheet plane.
For calibration, the cameras were focused on a target that was placed in the measurement
domain and aligned with the laser light-sheet. Datum marks were automatically mapped
on the picture of the target by the PIV evaluation software to define the physical length
scale of the images. A multigrid evaluation strategy was used (64x64, 32x32 interrogation
size at 50 % overlap) including window deformation, Whittaker peak fitting, and B-Spline
reconstruction. Errors due to misalignment of the laser sheet were minimized by the cor-
rected mapping functions. Therefore, the datum marks for the initial calibration were
back-projected onto the actual light sheet plane via linear triangulation using the pinhole
model.
3.3.1 Measurements in Water
The system consisted of two X-STREAM VISION CCD cameras with a resolution of
1024 ×1020 pixels, a 40 mJ flash lamp pumped Nd:YAG laser, an articulated arm, stan-
dard light sheet forming optics, a synchronization and timing unit to control the laser and
camera timing, and a commercial PIV evaluation software (INTELLIGENT LASER AP-
PLICATIONS GmbH). To minimize optical distortion and to avoid total reflexion at the
air–water interface, the water tank was equipped with water-filled prisms on both sides.
The light sheet entered from the bottom into the tank.
The pump operated continuously to mix the flow and to provide a constant temperature
in the entire tank. 20 minutes before acquiring data, the tank was stirred manually to
obtain homogeneous seeding. Data were taken in the x-r-plane (streamwise plane) for 0.3<
x/D < 3 and r/D ≤1.1 and in the r-θ-plane (crosswise plane) at x/D = 1.1 for r/D ≤1.1.
Each ensemble of PIV snapshots consists of 400 events captured at approximately 3 Hz.
The oscillation frequencies of the dominant coherent structures in water at the considered
Reynolds number were orders of magnitude below the acquisition frequency and, hence,
the measurement can be considered as time-resolved.
3.3.2 Measurements in Air
The first measurement campaign at the air facility was conducted with a double-pulsed
Nd:Yag laser at 532 nm and 25 mJ in 5 ns burst. Two CCD cameras with a resolution
of 1.3 million pixels were used. The cameras and the laser were mounted onto a single
traversing system. Data were taken in the crossflow plane as well as in the axial plane.
Each ensemble of PIV snapshots consists of 800 events captured at approximately 3 Hz.
The results of that measurement campaign are presented in chapter 6.
Chapter 3 Experimental Arrangements and Measurement Techniques 25
Figure 3.3: Hot-wire traversing system
During the PhD work, a new PIV system was purchased. It consists of a double pulsed
Nd:Yag laser at 532 nm and 160 mJ and two CCD cameras with a resolution of 2048×2048
pixels. The PIV data presented in the chapters 5and 7were acquired using the new system.
Both cameras were positioned at a 45◦angle in back-scattering mode in order to measure
all three velocity components in the x-r-plane. Each ensemble of PIV snapshots consists of
900 events captured at approximately 6 Hz. The camera view angle allows to obtain data
even inside the nozzle which is necessary to have reliable data at x/D = 0.
3.4 Hot-wire Measurements in the Moderately Swirling Jet
In order to capture the downstream traveling instability waves in th moderately swirling
jet, time-resolved volumetric measurements were conducted using eight hot-wire probes
simultaneously (chapter 5). A unique traversing mechanism was used that allows to si-
multaneously move all hot-wires in radial and axial direction (figure 3.3). The hot-wire
anemometers, which were built locally, were used in conjunction with a A.A.LAB SYS-
TEM LTD anemometer system. The hot-wires, made of tungsten, were 5 pm in diameter.
They were kept at a constant overheat ratio of 1.6 and had a maximum frequency re-
sponse of 50 kHz. An analog low-pass filter with a cut-off frequency of 6.2 kHz was used
to condition the measured signals prior to the AD-converter. All channels were sampled
simultaneously at 20 kHz, giving a maximum frequency response (Nyquist frequency) of
10 kHz.
26 Chapter 3 Experimental Arrangements and Measurement Techniques
In order to retain the phase information of the forced experiments, the excitation signal
was recorded together with the signal from the hot-wire anemometer. In case of sinusoidal
forcing, the typical length of a velocity record used for averaging was equivalent to 1500
periods of the excitation frequency. In case of pulsed experiments, the record length is
equivalent to 600 pulses which are generated at 10 Hz.
The probes were distributed circumferentially around the jet center, with the wires being
parallel to the tangential velocity. This enabled to measuring the axial velocity with a high
spatial accuracy in radial direction. The hot-wires were calibrated in the exit plane of the jet
at no-swirl conditions against a standard Pitot tube at seven different velocities. Particular
attention was payed to the radial adjustment of the hot-wire probes. This was done in
no-swirl conditions by placing each of the eight hot-wires at the center of the shear layer,
where the mean velocity is reduced to one-half of the centerline value. After completing the
fine alignment of the probes, all wires were traversed simultaneously in the radial and axial
direction. Details to the experimental procedure are given in the work of Kuhn (2010).
3.4.1 Coherent Velocity Extraction of the Excited Flow
The time resolved axial velocity component vx(x, t) of the jet being excited at sinusoidal
perturbations derived from hot-wire measurements, is decomposed into the three parts of
the triple decomposition, yielding
vx(x, t) = Vx(x) + vc
x(x, t) + vs
x(x, t).(3.1)
The coherent velocity which is periodic in tand θis decomposed into a Fourier series with
coefficients
ˆvx(x, r, m, n) = 1
2πT Z2π
0ZT
0
vc
x(x, r, θ, t)ei(mθ−2πnt/T)dφdt. (3.2)
The coefficients ˆvxare complex, and mand ncorrespond to the azimuthal wavenumber
and the time harmonic, respectively. Throughout this investigation, we neglect higher
harmonics of the excited waves and set n= 1. The amplitude of the forced mode at a given
axial location xis then derived by integrating the radial amplitude distribution across the
axial shear layer (Delbende et al. 1998), yielding
Am(x) = ZR.95
R.05 |ˆvx|2rdr1/2
.(3.3)
The radial phase distribution of the forced mode is
ϕm(x, r) = arg(ˆvx).(3.4)
3.4.2 Ensemble Average of the Pulsed Experiments
The pulsed experiments were conducted in order to measure the impulse response in the
shear layer of the moderately swirling jet (chapter 5). The pulse was generated by a single
speaker at the nozzle exit. Its downstream development was captured via hot-wire measure-
ments. In order reduce noise caused by turbulent fluctuations, the pulsed experiments were
repeated 600 times and were ensemble-averaged. A detailed description of this method is
Chapter 3 Experimental Arrangements and Measurement Techniques 27
vc
xin m/s
T
0 0.25 0.5 0.75 1
-2
-1
0
1
2
Figure 3.4: Hot-wire signal of a traveling wave packet in the swirling jet (x/D = 0.27;
r/D = 0.47; θ=π/4)
given in Gaster & Grant (1975). The method was further finalized to reduce the ambiguity
in the mean quantities due to jitter in the arrival time and location. These irregularities,
which are induced by random turbulent motion of the base flow, may blur the actual shape
of the wave packet. Therefore, the signal of each individual recorded wave packet, which
is a function of θand t, was cross-correlating with the ensemble-averaged signal. The cor-
relation peak provides an estimate of the t- and θ-displacement of each pulse in respect
to the mean values. The displacement was then compensated for each pulse and the en-
semble average was redone. This procedure was repeated iteratively until convergence was
achieved. For a detailed description of this method, the reader is referred to Zhou et al.
(1996) and references therein. An example of the resulting phase-averaged signal of a wave
packet passing a hot-wire is shown figure 3.4.
Chapter 4
The Onset of Vortex Breakdown
and Global Instability
The investigation described in this section is aimed to provide quantitative
insights into the onset of vortex breakdown and the bifurcation to a global spiral
mode. Several flow features that accompany vortex breakdown are combined
to one consistent picture, starting from a weakly swirling jet and ending with
a strongly swirling jet. The different flow states that evolve at incrementally
increasing swirl are characterized by means of time-resolved stereo PIV mea-
surements in conjunction with post-processing tools, including Fourier analysis
and proper orthogonal decomposition. The experiment presented in this section
is properly scaled by the swirl number based on the axial momentum flux when
omitting the commonly used boundary layer approximations. Vortex breakdown
occurs first intermittently, accompanied by strong axial oscillations. By further
increasing the swirl, vortex breakdown stabilizes and a region of reversed flow
appears in the mean flow. This region grows linearly with increasing swirl un-
til the flow undergoes a supercritical Hopf bifurcation to a global single-helical
mode and vortex breakdown becomes spiral shaped. The appearance of an inter-
nal stagnation point is accompanied by a supercritical–to–subcritical transition
of the inflow profiles, in accordance to Benjamin’s inviscid theory (Benjamin
1962). This critical swirl number is found to be smaller than the one for the
supercritical Hopf bifurcation. The observed mean flow sequence compares well
with the transient formation of spiral vortex breakdown in laminar swirling jets
as reported by Bruecker & Althaus (1995), Liang & Maxworthy (2005), and
Ruith et al. (2003).
4.1 Literature Review
Free and confined strongly swirling jets are of great interest due to their unique feature,
commonly known as vortex breakdown. This phenomenon occurs when the ratio of the
azimuthal to axial momentum exceeds a certain threshold, while both quantities have to be
of the same order of magnitude. Breakdown in swirling jets is characterized by a transition
of a jet-like axial velocity profile to a wake-like profile with a local minimum on the axis.
29
30 Chapter 4 The Onset of Vortex Breakdown and Global Instability
This leads to a stagnation point to be followed by a highly turbulent region of reverse flow
farther downstream. It can play a crucial role – from desired to detrimental – in a variety of
technical applications. For example, vortex breakdown stabilizes the flame of a gas turbine
combustor and enhances mixing, thus leading to a reduction of NOxemissions (Huang &
Yang 2009). On the other hand, bursting of leading-edge vortices adversely affects the lift
distribution on delta wings resulting in poor flight performance. Understanding the cause
of the vortex breakdown is therefore of great importance in order to develop appropriate
control strategies. Furthermore, the transition of the flow from jet-like to wake-like that
generates coexisting inner and outer shear layers and the concomitant axial and azimuthal
shear makes this flow complex and highly three-dimensional, and thus, poses a formidable
challenge to fundamental studies. A detailed introduction to the fundamental physics of
vortex breakdown is given, for example, by Hall (1972), Leibovich (1978,1984), Escudier
& Keller (1985), and Lucca-Negro & O’Doherty (2001).
Several types of vortex breakdown have experimentally been observed. Lambourne &
Bryer (1962) were the first to describe the axisymmetric and spiral type of vortex break-
down. Swirling jet experiments in pipes conducted by Sarpkaya (1971) and Faler & Lei-
bovich (1978) identified three different types of vortex breakdown, namely the single-helical,
the double-helical, and the bubble-shaped vortex breakdown. Billant et al. (1998), inves-
tigating a swirling jet at a low Reynolds number, observed an additional conical-shaped
breakdown type.
Several theories have been proposed to explain the vortex breakdown phenomenon. They
can be roughly divided into three categories: The first associates vortex breakdown with
a critical state related to wave phenomena. The basic ideas were developed by Benjamin
(1962) for a steady inviscid axisymmetric vortex. The abruptness of vortex breakdown is
described by a downstream transition from a supercritical to a subcritical flow, analog to a
hydraulic jump. The supercritical flow supports only downstream traveling waves, whereas
the subcritical flow supports both downstream and upstream traveling waves. The second
concept considers vortex breakdown that is analogous to a boundary layer separation (Hall
1972) and the third idea suggests that vortex breakdown is a consequence of hydrodynamic
instability (Lessen et al. 1974;Ludwieg 1961).
However, experiments conducted by Escudier & Keller (1985) and Liang & Maxworthy
(2005) and direct numerical simulations conducted by Ruith et al. (2003) indicate a clear
separation of flow criticality and flow instability. It was proposed that the criticality of
the flow determines the basic, wake-like character of the flow and that instability waves
are a superimposed fine detail. Recent quantitative investigations could significantly con-
tribute to the understanding of the dynamics accompanying the onset of vortex breakdown.
Time-resolved measurements conducted by Liang & Maxworthy (2005) indicate that a re-
circulation bubble with nearly axisymmetric shape accompanies the first appearance of a
stagnation point. It was further noticed that in the wake of this dividing streamline a
single-helical vortex arises near the jet center that amplifies until it imposes its frequency
onto the entire nearfield. The authors suggest this to be a self-excited/globally unstable
mode, supposably arising from a region of local absolute instability in the lee (downstream)
of vortex breakdown. Forced experiments using vortex generators mounted on a rotating
nozzle support the absolute/convective nature of the dominating instabilities. Gallaire et al.
(2006) performed a linear stability analysis based on numerical simulations of a swirling
Chapter 4 The Onset of Vortex Breakdown and Global Instability 31
jet at Re = 200 that were conducted by Ruith et al. (2003). They found a convective to
absolute instability transition in the lee of the recirculation-bubble with a single-helical
mode being most unstable. Thus, it is likely that the precessing of the vortex core and the
appearance of strong oscillations that have been observed in experiments and simulations
(Duwig & Fuchs 2007;Liang & Maxworthy 2005;Martinelli et al. 2007;Ruith et al. 2003)
can be attributed to a self-excited global mode initiated by flow instabilities in the region
of vortex breakdown.
4.2 Objective and Approach
The investigation presented in this chapter provides an overview of the different flow states
at increasing swirl that ultimately leads to the formation of a spiral vortex breakdown.
The experiments were conducted with the swirling jet water facility of the University of
Arizona. Details to the experimental setup and procedure are given in section 3.1.
This study is closely related to the experimental work of Liang & Maxworthy (2005) who
study a laminar swirling jet with particular interest on the onset of global instability. Due to
the absence of a contraction, their facility generates a swirling jet that differs considerably
from the one presented here. A more similar – but also laminar – flow configuration was
investigated experimentally by Billant et al. (1998) and Loiseleux & Chomaz (2003) and
theoretically by Gallaire & Chomaz (2003). All authors provide valuable insight into the
instabilities dominating laminar swirling jets with particular interest in the formation of
a global mode. The present experiments supplement their work by investigating vortex
breakdown in a turbulent base flow at precisely controlled conditions.
The flow is investigated at a swirl intensity that is successively increased at very small
increments. At each swirl configuration, PIV measurements are conducted to reveal the
onset of vortex breakdown, the bifurcation to a globally unstable flow, and the formation of
a spiral vortex breakdown. Since the flow is turbulent, a quantitative description of the fluc-
tuating flow field is only possible by employing statistical means as time- or phase-averaging
and proper orthogonal decomposition (POD). The PIV measurements were partially car-
ried out by Roman Seele during his Master thesis work that he conducted at the University
of Arizona (Seele 2008). His contribution to this work shall be greatly acknowledged here.
The description of the investigation is organized as follows. First, an overview of the flow
configurations considered throughout this chapter is given in section 4.3. A discussion of
the used swirl number definition is presented in section 4.4. The onset of vortex breakdown
and its associated flow field is quantitatively described in section 4.5 and its relation to the
flow criticality is discussed in section 4.6. In section 4.7 the dominant coherent structures
in the shear layers are quantified for various swirl numbers. Thereby, particular attention
is paid to swirl numbers beyond the onset of breakdown where self-excited oscillations
dominate the flow dynamics. In section 4.8 the main observations are summarized and
conclusions are drawn. The reader is also referred to the appendices A.1 and A.2, which
provide additional information about the mean and fluctuating flow field.
32 Chapter 4 The Onset of Vortex Breakdown and Global Instability
4.3 The Mean Flow Configuration
All PIV measurements presented throughout this chapter are conducted at a Reynolds
number of
ReD=V D
ν= 3300,(4.1)
based on the nozzle diameter and the bulk velocity. The latter is defined as the time-
averaged axial velocity in the potential core of the non-swirling jet. It is derived from PIV
measurements with a non-rotating honeycomb.
To give an overview of the investigated flow configurations, streamlines are computed
from the axial and transverse mean velocity components in the x-r-plane and shown in
figure 4.1. The displayed swirl configurations cover the different flow regimes that are ob-
served while successively increasing the rotation rate of the honeycomb. The corresponding
swirl numbers are given in the caption of figure 4.1 and are defined in the next section.
At a rotation rate of Ω = 1.5 min−1, the streamlines in the core of the jet are approx-
imately parallel (r/D < 0.5 at x/D < 3). Entrainment of the steady ambient fluid is
indicated by the nearly vertical streamlines outside the jet (e.g. r/D > 0.7 at x/D = 0.5).
At Ω = 4.0 min−1, the streamlines indicate a strong widening of the jet downstream of
x/D = 0.7. It will be shown later that this flow configuration exhibits reversed flow on
the jet axis at some instances in time that is not detectable from the time-averaged flow.
Therefore, it is still considered as a pre-breakdown state. At Ω = 4.2 min−1, two stagnation
points are visible with a region of reversed flow in between them. Its approximate axial
extent is 1.3< x/D < 1.6 with its maximum diameter being 0.3D. This region appears
to be similar to the recirculation bubble that can often be observed in swirling pipe flows
(Escudier & Keller 1985;Faler & Leibovich 1978;Leibovich 1978). However, in the present
case the streamlines do not converge towards the jet axis and contours of the azimuthal
vorticity do not suggest a closed recirculation bubble (shown later).
To emphasize this difference, the phrase ’region’ instead of ’bubble’ is used throughout
this investigation. A further increase of the revolution speed to Ω = 4.4 min−1causes
a drastic change of location and size of the recirculation region. It moves upstream and
grows in size, having an axial extent of 0.7< x/D < 1.6 and a maximum diameter of 0.7D.
When the amount of swirl is increased to Ω = 5.4 min−1, the recirculation region moves
closer to the nozzle, but its size remains approximately the same. At the highest rotation
rate, the recirculation region has moved close to the nozzle and its size has decreased. This
behavior is attributed to the influence of the nozzle walls on the development of the jet
near the orifice. For very high swirl, the recirculation region is literally ”swallowed” by the
nozzle. The streamlines indicate that this also reduces the spreading rate of the jet. These
confinement effects can already be observed at Ω = 5.4 min−1but in a less pronounced
way.
One may, therefore, qualitatively divide the presented data into three regimes, the first
describing the pre-breakdown state with Ω <4.2 min−1, the second characterizing the
breakdown state with 4.2 min−1≤Ω<5.4 min−1, and the third describing the confined
breakdown state being influenced by the nozzle orifice with Ω ≥5.4 min−1.
Chapter 4 The Onset of Vortex Breakdown and Global Instability 33
Ω=1.5 min−1
r/D
Ω=4.0 min−1
r/D
Ω=4.2 min−1
r/D
Ω=4.4 min−1
r/D
Ω=5.4 min−1
r/D
Ω=7.0 min−1
r/D
x/D
0.5 1 1.5 2 2.5 3
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
Figure 4.1: Streamlines in the x-r-plane for different rotation rates of the hon-
eycomb Ω. From top to bottom, the corresponding swirl numbers are S=
(0.38,1.01,1.07,1.12,1.37,1.78). Black stars mark stagnation points on the jet axis.
34 Chapter 4 The Onset of Vortex Breakdown and Global Instability
4.4 Momentum Balance and Swirl Number Discussion
In this section an attempt is made to define a swirl number that consistently scales the flow
for all configurations considered. Several swirl number definitions exist in the literature,
each being applicable to a certain flow regime but none being universal. Their definitions
are typically based on either velocity or momentum ratios. The former are conveniently
derived from single point measurements, but they strongly depend on the axial location
at which they are derived. It seems more appropriate to use the swirl number definition
that is based on the ratio of axial flux of azimuthal momentum ˙
Gθto axial flux of axial
momentum ˙
Gx, yielding
S=˙
Gθ
D/2˙
Gx
,(4.2)
as this quantity is conserved in axial direction. The calculation of the axial momentum
flux based on the full equations of motion demands very accurate data at decent spatial
resolution. Simplifications that include boundary layer approximations and the negligence
of the turbulent stresses are typically introduced. These stipulations are valid for non-
swirling and moderately swirling jets and for the farfield of strongly swirling jets (Chigier &
Chervinsky 1965,1967;Pratte & Keffer 1972;Semaan et al. 2009). However, the nearfield of
strongly swirling jets is highly divergent and the validity of boundary layer approximations
is questionable. The swirl number is therefore suggested to be based on the full equations
of motion. According to Rajaratnam (1976), the equations for the axial flux of axial and
angular momentum of an inviscid, incompressible and axisymmetric swirling jet can then
be written as
˙
Gθ= 2πρ Z∞
0
VxVθ
|{z}
I
+v′
xv′
θ
|{z}
II
r2dr (4.3)
˙
Gx= 2πρ Z∞
0
V2
x−V2
θ
2
|{z }
III
+v′2
x−v′2
r+v′2
θ
2
|{z }
IV
r+
Vx
∂Vr
∂x +Vr
∂Vr
∂r
|{z }
V
+∂
∂xv′
xv′
r
|{z }
VI
r2
2
dr.
(4.4)
The expressions are grouped to six terms marked by Roman numerals. (Chigier & Beer
1964), who introduced the most simplified swirl number definition, assume that v′2
x≈v′2
r≈
v′2
θ, resulting in term IV to vanish. They further neglect the terms containing the radial
velocity component (term V) and turbulent shear stresses (terms II and VI) and arrive at
the most commonly used definition for the axial flux of momenta
˙
Gθ= 2πρ Z∞
0
VxVθr2dr. (4.5)
˙
Gx= 2πρ Z∞
0V2
x−V2
θ
2rdr. (4.6)
The validity of these stipulations is investigated in detail for Ω = 4.4 min−1(figure 4.2).
The streamwise distribution of ˙
Gxand ˙
Gθare derived from equations (4.3-4.4) for each
term marked by a Roman numeral (figure 4.2a-b). The comparison reveals that the terms
II, IV, and V contribute significantly to the axial momentum flux within the region of
Chapter 4 The Onset of Vortex Breakdown and Global Instability 35
vortex breakdown and should therefore not be neglected. In particular ˙
Gxbased on term
V reaches considerable high magnitude, which is attributed to the considerably high radial
component due to the jet divergence (see also appendix A.1). The turbulent shear stress
v′
xv′
θ, represented by term II, increases around the upstream end of the recirculation zone
and approximately balances the decaying term I upstream of x/D = 1. However, the tur-
bulent shear stress v′
xv′
θand corresponding term VI is nearly zero within the measurement
domain. Hence, all terms except for VI contribute to the momentum balance and need to be
considered within the region of vortex breakdown. The swirl number based on these terms
(I, II, III, IV, and V) remains approximately constant within 0.3< x/D = 1.1 (figure 4.2c).
Further downstream, the determination of the swirl number becomes erroneous because
the jet exceeds the measurement domain. The simplified swirl number based on the terms
I and III varies significantly in streamwise direction within 0.3< x/D = 1.1, revealing the
invalidity of the boundary approximations. It is likely that the two swirl numbers are equal
at the nozzle exit where the flow is parallel and the boundary approximations hold (confer
with Toh et al. (2010)).
Hence, for the present investigation the swirl number is used that is based on the terms
I, II, III, IV, and V averaged within 0.3< x/D < 1.1 as the characteristic swirl number.
From here on, this quantity will be referred to as S. Figure 4.3 displays how Sis related
to the rotation rate of the honeycomb, revealing an approximate linear increase of Sfor
Ω≤4.8 min−1. For the confined breakdown state (Ω ≥5.4 min−1) the values decrease and
increase again. This somewhat implausible behavior for the two highest swirl configurations
is caused by the confinement of the flow. The calculations of the axial momentum is
erroneous as pressure forces induced by the nozzle walls are not incorporated. The swirl
number that represents the present experiment best is, therefore, derived from a linear fit
of Sneglecting these two highest values (see dashed line in figure 4.3). The fitted values
are expressed by S= Ω0.25 min and will be used throughout this investigation.
To conclude, one may state that the calculation of the swirl number based on momentum
conservation is very delicate within the region of vortex breakdown. The terms containing
the radial velocity component and its gradients as well as the turbulent normal and shear
stresses, particularly v′
xv′
θ, need to be taken into account. The conservation of axial momen-
tum flux within the entire measurement domain can be seen as a challenging benchmark to
validate the consistency and quality of the acquired data. However, it should be noted that
the invalidity of the boundary approximations for the strongly swirling jet could probably
overcome by using spherical coordinates.
4.5 Onset of Vortex Breakdown
This section contains a quantitative investigation of the mean and instantaneous flow field
at swirl numbers around the onset of vortex breakdown. Thereby, vortex breakdown is
strictly associated with a stagnation point on the jet axis. Thus, breakdown can be tagged
from the time-averaged or from the instantaneous flow field. The latter is evaluated by
statistical methods and its results are linked to the observed mean flow pattern.
Considering the mean flow at swirl numbers below breakdown, flow stagnation is ob-
served to be foreshadowed by the formation of a local minimum of the axial velocity on the
jet axis. Hence, vortex breakdown (in the mean flow) occurs when the minimum local axial
36 Chapter 4 The Onset of Vortex Breakdown and Global Instability
˙
Gx/V 2/(D/2)2
III
IV
V
VI
(b)
˙
Gθ/V 2/(D/2)3
I
II
(a)
S
I, III
I, II, III, IV, V
(c)
r/D
x/D
(d)
0.5 1 1.5 2
0
0.5
0.5
1
1.5
-0.1
0
0.1
0.2
0
0.1
0.2
0.3
Figure 4.2: Streamwise distribution of axial flux of azimuthal (a) and axial (b) momentum
and the swirl number (c). The Roman letters refer to the terms of equations (4.3-4.4) that
are used to calculate the respective quantities. The streamlines visualize the corresponding
flow configuration (Ω = 4.4 min−1,S= 1.12).
Chapter 4 The Onset of Vortex Breakdown and Global Instability 37
Ω in min−1
S
breakdown regime
0 1 2 3 4 5 6 7
0
0.5
1
1.5
Figure 4.3: Development of the swirl number Swith rotation rate Ω of the honeycomb;
black stars refer to measurements averaged within 0.3≤x/D ≤1.1. Dashed line represents
linear fit expressed by S= Ω0.25min
S
(Vcl/V )min
breakdown regime
ւSVB = 1.06
0.8 1 1.2 1.4 1.6 1.8
-0.2
-0.1
0
0.1
0.2
Figure 4.4: Decay of axial velocity at a local minimum on the jet axis with increasing
swirl; for S≥SVB a stagnation point exists on the jet axis.
velocity is equal to or smaller than zero. This quantity is plotted versus the swirl number in
figure 4.4. The root of the curve defines the lowest swirl number SVB = 1.06 at which vor-
tex breakdown occurs. It separates the pre-breakdown regime from the breakdown regime.
For S < 0.96 no minimum is found as the axial velocity decreases monotonically in axial
direction. For swirl numbers near SVB, the minimum decreases approximately linearly with
increasing swirl number (0.99 ≥S≥1.16). For higher swirl, the minimum levels off at
approximately (Vcl/V )min =−0.1 and gently increases for S≥1.37.
The axial location of that minimum is drawn in figure 4.5 (black stars). For swirl
numbers below SVB the local minimum figure at x/D ≈2.24. Exceeding SVB causes
the minimum to jump abruptly upstream to x/D = 1.42. In the breakdown regime, the
minimum successively propagates upstream with increasing swirl indicating an upstream
movement of the recirculation region that is slowed down at the proximity to the nozzle
exit. The location of the upstream stagnation point follows the same trend (white circles
38 Chapter 4 The Onset of Vortex Breakdown and Global Instability
x/D
S
Ucl = 0
ωθ,cl = 0
(Ucl)min
↓
breakdown regime
recirculation region
0.8 1 1.2 1.4 1.6 1.8
0.5
1
1.5
2
2.5
Figure 4.5: Axial locations of the local minimum of axial velocity, the stagnation points,
and the root of azimuthal vorticity along the jet centerline with increasing swirl.
in figure 4.5). Its distance from the nozzle exit decreases even more rapidly with increasing
swirl for SVB < S ≤1.11 than the local minimum does. The downstream stagnation
point only moves marginally in that range of swirl, which results in a rapid growth of the
recirculation region with increasing swirl. It reaches its maximum axial extent at S≈1.11.
For higher swirl, it is shifted further upstream and shrinks for S≥1.37 due to confinement
effects.
Another characteristic of vortex breakdown is the existence of a change of sign in the
azimuthal vorticity ωθnear the jet axis (e.g., see Akilli et al. 2003;Brown & Lopez 1990;
Gallaire et al. 2004). In principal, this change of sign is caused by a transition of the
axial velocity profile from jet-like to wake-like. Certainly, this criterion is weaker than the
criterion of a stagnation point, but it is nevertheless a useful indicator for the precursor to
vortex breakdown. Contours of the axial and azimuthal vorticity component are displayed
in figure 4.6 for increased swirl numbers. For moderate swirl, azimuthal vorticity is con-
centrated in the outer part of the jet. For S≥0.86 a pocket of negative azimuthal vorticity
appears in the jet core that increases in size and intensity while moving upstream with
increased swirl. Its leading edge comes closer to the nozzle exit and presumably reaches it
at S > 1.12. The present experiment does not provide information about the downstream
end of the negative ωθpocket. It may end due to turbulent diffusion, although the ωθ= 0
curve seems to be open toward the downstream end of the measurement domain.
The point where the azimuthal vorticity changes its sign near the jet axis is included in
figure 4.5 and is marked by white diamonds. It is detectable for swirl numbers S≥0.86,
which is far below the onset of vortex breakdown. Except for a small plateau near S= 1,
the point of vorticity transition moves continuously upstream with increasing swirl. Around
SVB a linear relation of this movement is evident. For values beyond S= 1.09 the point
of ωθreversal has moved out of the measurement domain and possibly into the nozzle
(indicated by the black arrow).
Chapter 4 The Onset of Vortex Breakdown and Global Instability 39
r/D
S= 0.38
r/D
S= 0.86
r/D
S= 0.94
r/D
S= 1.04
r/D
S= 1.07
r/D
S= 1.12
azimuthal vorticity ωθin s−1
x/D
-2 -1 0 1 2
0.5 1 1.5 2 2.5 3
0
0.25
0.5
0.75
0
0.25
0.5
0.75
0
0.25
0.5
0.75
0
0.25
0.5
0.75
0
0.25
0.5
0.75
0
0.25
0.5
0.75
Figure 4.6: Azimuthal vorticity contours at increasing swirl; the black thick line marks
ωθ= 0
40 Chapter 4 The Onset of Vortex Breakdown and Global Instability
The smooth upstream progression of the position where the azimuthal vorticity com-
ponent changes its sign indicates that there must be a mechanism for vortex breakdown
that develops progressively with increasing swirl at swirl numbers that are considerably
lower than SVB. On the other hand, the rapid decrease in the distance where the axial
velocity component has its minimum and the rapid decrease in the value of this minimum
magnitude when SVB is exceeded suggests an abrupt appearance of vortex breakdown.
In order to clarify this contradiction, the onset of vortex breakdown is determined from
the instantaneous velocity field. The analysis of the time-resolved velocity field demands
for additional statistical quantities that are defined as following: The probability Pof a
random continuous variable v(t) to take a value in the interval [v1, v2] is commonly defined
as
P[v1≤v≤v2] = Zv2
v1
PDF[v]dv,
with PDF[v] being the probability density function of v(t). The PDF is non-negative
everywhere, and the probability of v(t) to take an arbitrary value can be lower than one,
yielding
1≥Z∞
−∞
PDF[v]dv.
By considering the instantaneous axial velocity component vx(x, t, S), which depends para-
metrically on S, as the random variable, the corresponding probability density function
PDF[vx](x, S) becomes parametrically dependent on the position vector xand the swirl
number. However, for axisymmetric jets, vortex breakdown occurs (in the mean) centered
to the jet axis and hence, it seems sufficient to consider the instantaneous axial velocity on
the jet centerline vcl(x, t, S), which makes PDF[vcl](x, S) parametrically dependent on the
streamwise coordinate xand the swirl number S.
The results of the statistical analysis of the emergence of vortex breakdown are sum-
marized in figure 4.7. The first diagram (figure 4.7a) shows the probability of reversed
flow
PRF(x, S) = Z0
−∞
PDF[vcl](x, S)dvcl.(4.7)
This quantity provides information about the size and location of an intermittently appear-
ing recirculation region that might not be detectable from the mean flow. The diagram
on the left hand side of figure 4.7b shows contours of PDF[xst](S), the probability density
function of the streamwise location of the most upstream stagnation point xst(t, S). The
associated zero-crossing of vcl is thereby estimated from each PIV snapshot by cubic inter-
polation. The diagram reveals the probability of vortex breakdown to occur at a specific
axial location for a given swirl number. The probability of vortex breakdown to occur at
any point inside the measurement domain is obtained from the integral
PVB(S) = Z3.5D
0.3D
PDF[xst](S)dx. (4.8)
It is displayed on the right hand side of figure 4.7b. PVB ranges from 0 to 1, representing 0 to
400 snapshots that exhibit flow stagnation, respectively. The gray area in the background of
the diagrams shown in figure 4.7b indicate the swirl number regime where an internal stag-
nation point exists in the mean flow. Figure 4.7c displays the power spectral density PSD of
Chapter 4 The Onset of Vortex Breakdown and Global Instability 41
the upstream stagnation point as a function of the dimensionless frequency St =fD/V for
three selected swirl numbers. This quantity, defined as PSD(f) = |FFT[xst(t)/D]|2, pro-
vides information about the streamwise oscillation frequency and amplitude of the vortex
breakdown location.
Figure 4.7a shows that for S= 0.38 the instantaneous flow exhibits no reversed flow.
For S= 0.86 a few events of back-flow are detectable, with a maximum probability of 0.023
at x/D ≈2.3. Intensifying the swirl to S= 1.01 increases this probability to 0.15 with the
maximum still located at x/D ≈2.3. In addition, a second maximum appears, which is
located at x/D = 1.68. For the same swirl number, the contours of PDF[xst] exhibit two
peaks located at x/D ≈1.5 and x/D ≈2.1 (figure 4.7b). This is a strong indication for an
oscillating vortex breakdown location. A further increase of swirl to S= 1.04 increases the
upstream maximum of PRF and displaces it in upstream direction. The upstream maximum
of PDF[xst] is correspondingly displacement upstream. This indicates that the stagnation
point oscillates between a fixed downstream location of x/D ≈2.1 and an upstream location
which moves upstream with increasing swirl. When SVB is exceeded, the upstream peak
of PDF[xst] increases rapidly while the downstream peak disappears. For S > 1.1, the
streamwise fluctuations of the stagnation point are strongly reduced, indicated by a narrow
PDF[xst]. Moreover, the probability of vortex breakdown PVB asymptotes one, which
corresponds to the permanent occurrence of vortex breakdown. The unsteadiness of vortex
breakdown in the pre-breakdown flow regime is confirmed by the PSD shown in figure 4.7c.
The strongest oscillations of the stagnation point are detected at S= 1.1, where PDF[xst]
is very broad and exhibits two distinct but nearly equally valued maxima. The dominant
frequency is St ≈0.08, which is significantly lower than the frequencies associated with
hydrodynamic instability waves (typically St >0.3 (Loiseleux & Chomaz 2003)). This is
in qualitative agreement with experimental observations on vortex breakdown over a delta
wing conducted by Gursul & Yang (1995).
Combining the statistical observations (figure 4.7) with the time-averaged flow features
(figures 4.4-4.6), one may draw the following conclusions: For S= 0.86 vortex breakdown
occurs very rarely but at approximately the same axial location, thus resulting in a local
axial velocity deficit on the jet axis and the associated negative ωθ. For higher swirl
numbers, which are still below SVB, the occurrence of vortex breakdown becomes more
frequent and it is accompanied by strong oscillations of the stagnation point. Thereby, it
oscillates between the point where breakdown initially emerged (x/D ≈2.1) and an axial
location that moves upstream with increasing swirl. This explains the continuous growth
of the pocket of concentrated azimuthal vorticity and its upstream movement in the pre-
breakdown regime (figures 4.5 and 4.6) and also clarifies the discontinuous movement of
the local minimum of axial velocity (figure 4.5). This minimum occurs first at x/D ≈2.24
which corresponds to the intermittent emergence of reversed flow. The vortex breakdown
stabilizes once SVB is exceeded, as indicated by the decaying oscillations in the PSD, and the
minimum is discontinuously displaced to x/D ≈1.42. Furthermore, SVB coincides with the
swirl number beyond which vortex breakdown occurs permanently (PVB ≈1). Below that
value, a stagnation point appears intermittently within the measurement domain. Thus
vortex breakdown, like transition to turbulence, occurs intermittently over a certain range
of swirl numbers.
42 Chapter 4 The Onset of Vortex Breakdown and Global Instability
x/D
PRF
S= 0.38
S= 0.86
S= 1.01
S= 1.04
S= 1.07
S= 1.12
S= 1.37
0.5 1 1.5 2 2.5 3 3.5
0
0.2
0.4
0.6
0.8
1
(a) probability of reversed flow versus axial distance
pre-breakdown
breakdown regime
S
x/D PVB
0 0.5 10.5 1 1.5 2 2.5
0.9
1
1.1
1.2
(b) left:probability density function of xst (contour lines: 0.1 to 2 at 0.1 increments);
right probability of vortex breakdown
St
PSD
S= 1.01
S= 1.07
S= 1.12
00.1 0.2 0.3 0.4
0
2
4
6×10−3
(c) power spectral density of xst
Figure 4.7: Statistic evaluation of onset of vortex breakdown. All quantities are derived
on the jet axis.
Chapter 4 The Onset of Vortex Breakdown and Global Instability 43
These findings are qualitatively consistent with previous observations. Billant et al.
(1998) and Liang & Maxworthy (2005) report the occurrence of an unstable recirculation
region at x/D ≈4, which grows and propagates upstream with increasing swirl. Their
experiments were conducted at a much lower initial turbulence level (the flow in the nozzle
was not tripped) and may, therefore, be contaminated by transition to turbulence. Liang
& Maxworthy observed high amplitude oscillations of vortex breakdown and concluded
that time-averaged flow measurements are not representative in this specific flow regime.
However, a comparison of the mean flow field in the pre-breakdown regime as displayed in
figure 4.1 with the time dependent measurements conducted by Bruecker & Althaus (1995)
shows clear similarities. Hence, the time average of the intermittent flow field reveals the
same flow features that can be observed during the temporal formation of vortex breakdown.
One may pose the question of whether the time-averaged flow at swirl numbers close to
SVB approximates the time sequence of the laminar flow undergoing a controlled vortex
breakdown as observed by Bruecker & Althaus (1995).
4.6 Vortex Core Criticality
An attempt is made to relate the onset of vortex breakdown, observed in the present
experiment, to the inviscid model developed by Benjamin (1962). His derivations are
based on the assumption that the sudden occurrence of vortex breakdown is similar to
the appearance of a hydraulic jump. The existence of a critical state is suggested that
separates a subcritical from a supercritical flow state. A supercritical flow supports only
downstream traveling waves while a subcritical flow supports upstream and downstream
traveling waves. Experimental and numerical investigations confirm that vortex breakdown
represents a transition from supercritical to subcritical flow (e.g. Escudier & Keller (1985);
Ruith et al. (2003); Sarpkaya (1971)).
Following the derivations of Benjamin (1962) and using notations similar to Ruith et al.
(2003) the criticality of the base flow can be predicted by the criticality condition
d2φc
dr2−1
r
dφc
dr+1
r3V2
x
d(rVθ)2
dr−r
Vx
d
dr1
r
dVx
drφc= 0 (4.9)
that delivers the test function φcexcept for an arbitrary constant multiplier. The boundary
conditions are φc(r= 0) = 0 and dφc(r= 0)/dr= 1. For pipe flows, Benjamin shows
that a necessary and sufficient condition for a subcritical state is that φchas to vanish
at least once within the pipe radius. Following Mager (1972) and Ruith et al. (2003)
who apply Benjamin’s analysis to radially unbounded flows, φchas to diminish within the
characteristic core radius for subcritical flow. Ruith et al. emphasize the importance of φc
within the core radius in determining the criticality of the flow, arguing that the ability
of rotating flows to support traveling waves depends on the magnitude of axial vorticity
(Escudier & Keller 1985;Ruith et al. 2003).
The present flow is not parallel, particularly in the vicinity of vortex breakdown. None-
theless, Benjamin’s analysis is applied, which is strictly speaking only valid for columnar
flows. Ruith et al. demonstrate that the analysis may successfully predict the location
of vortex breakdown of a laminar flow retrieved from direct numerical simulation. Their
approach is extended by investigating the applicability of the analysis to time-averaged
44 Chapter 4 The Onset of Vortex Breakdown and Global Instability
turbulent flow. Hence, the criticality condition (4.9) is applied to the measured velocity
profiles at different axial locations. The flow at a certain axial location is considered
subcritical if the test function φcdiminishes within the core radius.
The critical radius rcrit at which φc= 0 is shown on the left-hand side of figure 4.8 as a
black solid line together with contours of measured axial vorticity ωx. The radial extent of
the vortex core rcore is defined as the contour line on which ωx= 0 and is marked as a red
thick line. The right-hand column shows the projected streamlines together with contours
of ωθ.
For S= 0.38 the flow does not exhibit deceleration and the profiles remain supercritical
everywhere. Upon increasing the swirl to S= 1.01, vortex breakdown occurs intermittently
and negative azimuthal vorticity is generated in the jet core. The critical radius does not
exceed the core radius, and thus the profiles are supercritical everywhere. At S= 1.04, the
largest swirl number available that is still lower than SVB, the profiles become subcritical
at 1.3< x/D < 2.2. This region agrees well with the area where vortex breakdown appears
intermittently (figure 4.7a-b). It is interesting that the locations of the two local minima
of rcrit agree with the maxima of PDF[xst] (figure 4.7b). In spite of the somewhat loose
definition of rcore the latter agreement provides an explanation for the oscillating vortex
breakdown in the pre-breakdown range: For S= 1.04 there are two separated subcritical
regions causing vortex breakdown to oscillate between them.
For swirl numbers higher than SVB = 1.06, vortex breakdown stabilizes and two stagna-
tion points exist on the jet axis (see black stars in figure 4.8). The flow becomes subcritical
upstream of the first stagnation point and returns to supercriticality downstream of the
second stagnation point. Hence, the region of reversed flow is entirely trapped within the
subcritical region.
According to the columnar assumption of Benjamin’s model, the criticality of the inflow
profile should suffice for predicting vortex breakdown. Due to measurement limitations,
the criticality of the inflow profile can only be assessed by extrapolating rcrit and rcore
to x/D = 0. It turns out that the inflow profiles undergo a supercritical–to–subcritical
transition when SVB is exceeded, yielding a surprisingly good agreement with Benjamin’s
inviscid theory. This differs from the findings of Ruith et al. (2003), who successfully
predict breakdown from the local criticality, but fail to predict breakdown solely from the
inflow profiles. They attribute the discrepancy to the influence of viscosity due to very low
Reynolds numbers. The high initial turbulence level in the present experiment eliminates
the significance of viscous effects (see appendix A.2 for a detailed discussion of the turbulent
quantities).
Concluding, the recirculation region that appears for S > SVB is explained by the criti-
cality character of the inflow profiles and hence this swirl number can be attributed to the
critical swirl number for vortex breakdown. Furthermore, the location of vortex breakdown
and its intermittent occurrence for S < SVB is well predicted by the local criticality char-
acter of the time-averaged velocity profiles. The good applicability of Benjamin’s model
suggests that the mechanism that drives breakdown is inviscid for the present experiment.
Chapter 4 The Onset of Vortex Breakdown and Global Instability 45
r/D
S= 0.38
r/D
S= 1.01
r/D
S= 1.04
r/D
S= 1.07
r/D
S= 1.09
r/D
S= 1.12
axial vorticity ωxin s−1azimuthal vorticity ωθin s−1
x/D x/D
-2 -1 0 1 2-6 -4 -2 0 2 4 6
0.5 1 1.5 2 2.5 30.5 1 1.5 2 2.5 3
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
0
0.5
Figure 4.8: Left-hand column: Criticality of the vortex core according to Benjamin
(1962); flow is subcritical when test function φcof equation (4.9) vanishes inside the vortex
core, i.e. rcrit (black solid line) is smaller than core radius rcore (red line). Inflow profiles
undergo supercritical–to–subcritical transition for S > 1.04 in correspondence with the
appearance of an internal recirculation region (black stars mark stagnation points). The
local criticality characteristic successfully predicts intermittent breakdown (S= 1.04) and
stationary breakdown (S≥1.07).
46 Chapter 4 The Onset of Vortex Breakdown and Global Instability
4.7 Coherent Structures
Numerous theoretical and experimental investigations have shown the importance of helical
instabilities in swirling jets (e.g., see Gallaire & Chomaz 2003;Liang & Maxworthy 2004;
Loiseleux & Chomaz 2003;Panda & McLaughlin 1994;Ruith et al. 2003). The break-
ing of rotational symmetry and the corresponding appearance of spiral-shaped vortices are
attributed to instabilities generated by axial and azimuthal shear. Experimental investi-
gations of laminar swirling jets reveal that the azimuthal mode number and axial phase
velocity of the dominant instabilities strongly depend on the amount of swirl and the rela-
tive location of the axial and azimuthal shear layers (Liang & Maxworthy 2004;Loiseleux
& Chomaz 2003). Despite the different experimental setups and consequently different
base flow configurations, there is a consensus when considering laminar swirling jets: At
weak swirl, instabilities with azimuthal wavenumbers m= 0 and m=±1 are equally
dominant in agreement with non-swirling jets. At moderate swirl, non-axisymmetric in-
stabilities become dominate while their overall amplification is reduced. At strong swirl
beyond breakdown, single-helical large-scale oscillations are observed that are caused by a
supercritical Hopf bifurcation to a self-excited global mode (Gallaire et al. 2006;Liang &
Maxworthy 2005;Ruith et al. 2003). For an introduction to the concept of global stability,
the reader is referred to the review articles of Huerre & Monkewitz (1990) and Chomaz
(2005).
In this section, the azimuthal modes that are detected from crosswise measurements are
briefly described, followed by a detailed discussion of the self-excited oscillations that arise
at very strong swirl. Furthermore, links are established between the vortex breakdown
unsteadiness and the appearance of the global mode.
4.7.1 Local Examination of Azimuthal Waves
In order to obtain information about azimuthal waves, PIV-measurements are conducted
along the r-θ-plane at the axial location x/D = 1.1. The sequences of PIV snapshots
are Fourier decomposed in time and in space providing the spectral distribution of spatial
modes. Thus, the time resolved three-component PIV snapshots, depicted as v(x, r, θ, t),
are decomposed into complex Fourier coefficients ˆ
vyielding
ˆ
v(x, r, m, f) = 1
2πT ZT
0Z2π
0
v(x, r, θ, t)ei(mθ−2πft)dθdt. (4.10)
Note that m > 0 modes rotate with the mean flow and are referred to as co-rotating while
m < 0 modes are counter-rotating. The PIV snapshots are recorded for N= 400 samples at
a rate of 2.25 Hz yielding a maximum resolvable frequency of 1.125 Hz. In order to reduce
noise, the spectra are averaged with the drawback of doubling the maximum resolvable
frequency to ∆f= 2fs/N = 0.0056 Hz. The power spectral density contained in a single
azimuthal mode is defined as the square of the modulus, yielding PSD = |ˆ
v|2. Figure 4.9
displays the PSD of −3≤m≤3 modes. The spectra are taken at the radial position of the
highest energy content. It is interesting to note that these locations coincide approximately
with the center of the shear layer of the jet-like axial velocity profile (figure 4.9a-b) and with
the center of the inner shear layer of the wake-like axial velocity profile (figure 4.9c-d). The
overall picture is presented in figure 4.10. It shows the distribution of the total energy e
Ktot
Chapter 4 The Onset of Vortex Breakdown and Global Instability 47
St
PSD
m=-3
m=-2
m=-1
m=0
m=1
m=2
m=3
0 0.2 0.4 0.6 0.8 1
0
0.5
1
1.5
2
2.5 ×10−3
(a) S= 0 r/D = 0.36
St
PSD
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8×10−4
(b) S= 0.38 r/D = 0.4
St
PSD
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2 ×10−3
(c) S= 1.07 r/D = 0.3
St
PSD
00.2 0.4 0.6 0.8 1
0
0.5
1
1.5 ×10−3
(d) S= 1.37 r/D = 0.52
Figure 4.9: PSD of azimuthal modes derived at x/D = 1.1 at the radial location of
highest energy contents
among the azimuthal modes ranging from m=−5 to m= 5. The total energy corresponds
to the accumulated energy of all considered frequencies and radial positions.
In absence of swirl, mode m= 0 is clearly dominant for 0.6<St <1 and all other modes
are weak (figures 4.9a and 4.10). By increasing the swirl to S= 0.38, the energy of mode
m= 0 is strongly reduced. Figure 4.9b shows that the counter-rotating modes (m < 0),
marked by open symbols, become more energetic than the axisymmetric and co-rotating
ones and they peak at lower oscillation frequencies (0.1<St <0.4). These results can be
compared to the linear stability analysis conducted by Gallaire & Chomaz (2003). Their
analysis is based on the velocity profiles measured by Billant et al. (1998), which agree
reasonably well with the present flow (see appendix A.1 for detailed comparison). They
predict mode m= 0 and m=±1 to be most unstable for S= 0 with a somewhat higher
amplification rate of mode m=±1. However, it is usually m= 0 which dominates in
laboratory jets because random disturbances generated upstream of the settling chamber
become axisymmetric at the nozzle exit, due to the high area ratio of contractions used in
48 Chapter 4 The Onset of Vortex Breakdown and Global Instability
m
e
Ktot/DV 2
S= 0.00
S= 0.38
S= 0.76
S= 1.07
S= 1.37
-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.05
0.1
0.15
0.2
0.25
Figure 4.10: Total energy distributed among azimuthal modes at x/D = 1.1.
those jet facilities (Cohen & Wygnanski 1987;Long & Petersen 1992). For moderate swirl,
Gallaire & Chomaz (2003) predict mode m=−1 to be most unstable, followed by mode
m=−2 and m=−3. This agrees well with the present results, revealing mode m=−1
to be dominant, followed by m= 0, m=−2, and m=−3 for S= 0.38 (figure 4.10). The
relative significance of the axisymmetric mode is likely to be caused by the random noise
created by the facility upstream of the contraction, in the same way as for the non-swirling
jet. The modal distribution reported by Liang & Maxworthy (2005) differs from the present
results as they found only azimuthal waves that co-rotate with the base flow (m > 0). Due
to the absence of a contraction, the mean flow in their experiment differs substantially
from the one investigated by Billant et al. (1998); Gallaire & Chomaz (2003); Loiseleux &
Chomaz (2003) and the time-averaged flow presented here. The reduction of the dominant
frequency with increasing swirl (figure 4.9a-b) disagrees with the observations of Loiseleux
& Chomaz (2003), who found the local Strouhal number to decrease only slightly with
increasing swirl. Presumably, the high initial turbulence level in the present experiment
enhances the shear layer growth, thus reducing the dominant frequencies more rapidly with
increasing distance from the nozzle. A detailed investigation of the instability at moderate
swirl is given in chapter 5.
At the swirl number S= 1.07, breakdown occurs and a recirculation region is generated.
Spectra taken at the center of the inner shear layer show considerably higher energy for
m= 1 and m= 0 mode at St ≈0.2 (figure 4.9c). This is representative for the flow regime
associated with unsteady vortex breakdown. The axial fluctuation of the recirculation
region generates m= 0 fluctuations while a meandering of the recirculation region in the
direction of the basic flow rotation causes m= 1 fluctuations. This modal distribution is
typical for the inner jet region only. In the outer shear layer, m= 1 and m= 2 modes
dominate as indicated by the total energy distribution (figure 4.10). At very strong swirl,
Chapter 4 The Onset of Vortex Breakdown and Global Instability 49
where vortex breakdown has stabilized, a clear peak of mode m= 1 arises that indicates
the existence of a dominant coherent structure (figure 4.9d). Its total energy exceeds mode
m= 0 for the non-swirling jets (figure 4.10).
4.7.2 The Onset of the Global Mode
The emergence of large-scale oscillations for swirl numbers exceeding breakdown has been
observed by many researchers. It may be characterized by a precessing vortex core (PVC)
that is a well known flow feature arising in swirl-stabilized combustors (Syred 2006). Recent
theoretical and experimental investigations on laminar swirling jets show that this preces-
sion can be interpreted as a globally unstable single-helical mode (Gallaire et al. 2006;Liang
& Maxworthy 2005;Ruith et al. 2003).
The introduction to the main idea of this concept shall be recapitulated here: According
to local parallel linear stability theory, the velocity profile at a specific axial location is said
to be absolutely unstable, if a localized disturbance spreads in upstream and downstream
direction, ultimately contaminating the entire parallel flow (Huerre & Monkewitz 1990).
If, by contrast, disturbances are swept away from the source, the velocity profile is said
to be convectively unstable. Examples for convectively unstable flows are non-swirling jets
(Cohen & Wygnanski 1987) and mixing layers (Gaster et al. 1985) and examples for abso-
lutely unstable flows are the cylinder wake (Yang & Zebib 1989) and hot jets (Monkewitz
1988). The two latter examples are subjected to intrinsic large-scale (global) oscillations
that arise at a critical control parameter. These flow configurations are globally unstable
and are typically treated by employing a global stability analysis with a two-dimensional
perturbation ansatz (e.g., see Sipp & Lebedev 2007). Weakly nonlinear theory allows to
connect the local and global stability. Accordingly, only shear flows are globally unstable
that exhibit a sufficiently large region of absolute instability (Chomaz 1992;Chomaz et al.
1988;Chomaz et al. 1991). The large-scale oscillations that are distinct for global modes
are synchronized to one frequency by a so-called wavemaker that is located near the region
of absolute instability. For swirling jets, this region is usually associated with reversed flow
on the jet axis.
The transient experiments conducted by Liang & Maxworthy (2005) show that in con-
junction with vortex breakdown, a global mode arises from the jet center that grows rapidly
in time, reaches the outer shear layer, and ultimately imposes its frequency onto the entire
flow. Hence, it is likely that the global mode m= 1 observed in experiments originates
from a pocket of absolutely unstable flow that exists near the jet axis due to vortex break-
down. This scenario is supported by the stability analysis of a simulated laminar swirling
jet (Gallaire et al. 2006) and by the spatio-temporal stability analysis presented in chapter
7. Once the global mode has saturated at its limit-cycle oscillation it can be characterized
by a convectively unstable mode m= 1 that amplifies in the outer shear layer at a fre-
quency that is imposed by the wavemaker located on the jet centerline. This flow state is
also often referred to non-axisymmetric or spiral vortex breakdown (e.g., see Billant et al.
1998;Bruecker & Althaus 1995;Liang & Maxworthy 2005;Ruith et al. 2003).
A characteristic of global modes is a single (global) oscillation frequency within a spatial
domain (Monkewitz et al. 1990;Provansal et al. 1987), and indeed, the high amplitude
50 Chapter 4 The Onset of Vortex Breakdown and Global Instability
POD mode
TKE in %
streamwise
crosswise
110 100 400
10−1
100
101
Figure 4.11: Turbulent kinetic energy captured in the POD modes (S= 1.37)
oscillation frequency shown in figure 4.9 is detected at several positions in the flow (not
shown).
In case of a dominant global frequency, the phase-averaged flow field can be accurately
derived by means of proper orthogonal decomposition (POD). The reader is referred to
chapter 6.5 for a detailed description of the phase-averaging procedure. A brief introduction
to the POD applied in the present investigation is given section 2.5. For a comprehensive
survey on POD, the reader is referred to (Berkooz et al. 1993;Holmes et al. 1998;Lumley
1967). For the phase-averaging procedure, the PIV snapshots are decomposed into spatial
POD modes Φi(x) and temporal POD modes ai(t) with i= 1,2...N. The eigenvalues λi
represent twice the turbulent kinetic energy contained in the corresponding POD mode.
The energy distribution of the POD modes of the highly swirling jet at limit cycle os-
cillations is shown in figure 4.11. The hollow circles correspond to POD modes derived
from streamwise measurements while the star symbols correspond to modes derived from
crosswise measurements at x/D = 1.2. The two leading POD modes contain approximately
equal energy and span the convecting vortex pattern that corresponds to the global oscil-
latory motion. The energy content of the streamwise modes is somewhat smaller since the
streamwise domain also covers regions where the oscillations are weak (see section 6.5.1 for
details).
To validate whether the first two POD modes represent the global mode, one may com-
pare the corresponding temporal POD modes a1and a2with the Fourier coefficients derived
from equation (4.10). The normalized PSD of mode m= 1 is displayed in figure 4.12a and
compared to the normalized PSD of a1and a2of the crosswise and streamwise POD modes
respectively (figure. 4.12b-c). Evidently, the first two POD modes correspond to the same
Chapter 4 The Onset of Vortex Breakdown and Global Instability 51
oscillating structure and they accurately represent the dynamics of mode m= 1. The
spectra of the first two streamwise temporal POD modes do not perfectly agree with each
other and they also show a broader peak compared to the spectrum of mode m= 1. This
is again attributed to the fact that the streamwise domain contains regions of weak os-
cillations subjected to more jitter and noise. Nonetheless, the characteristic frequency is
well captured, which implies that the periodically fluctuating global mode is represented
by the first two POD modes in both measurement planes. It is a straight forward proce-
dure to assign a phase angle ϕjof mode m= 1 to each PIV snapshot indexed by jusing
ˇajeiϕj=a1(tj) + ia2(tj). Hence, it is possible to define a flow phase via the POD and
consequently obtain phase-averaged quantities of the dominant oscillatory mode.
The structure of the global mode along the streamwise and crosswise plane of the mea-
surement is shown in figure 4.13 for S= 1.37. The first row corresponds to the mean
flow, the second row to the coherent flow, and the third row shows the sum of both. The
projected streamlines emphasize the eddy-like structures. The coherent structures associ-
ated with the global mode are best visualized by the coherent component (figure 4.13c-d).
Traveling waves are created in the outer shear layer that lead to a roll-up of a helical vortex
that is wrapped around the recirculation region. Vorticity on the jet axis upstream of the
recirculation region indicates the precession of the vortex core. The shape of the global
mode agrees well with the one found at higher Reynolds numbers at a similar configuration
(see chapter 6and 7). The structure on the jet centerline is identified as the wavemaker
that excites instabilities in the convectively unstable outer shear layer. Adding the time-
averaged flow to the coherent flow yields the phase-averaged flow field shown here at an
arbitrary phase (figure 4.13e-f). The streamwise plane indicates the waviness of the syn-
chronized inner and outer shear layer. The crosswise plane shows an eccentric vortex – the
precessing vortex core (PVC).
Generally, the amplitude of a global mode is governed by the forced Landau equation
(Huerre & Monkewitz 1990;Landau & Lifshitz 1987). Near critical conditions and in the
absence of forcing, the limit cycle amplitudes should increase proportionally to the deviation
from a control parameter, yielding
Asat ∝pS−Scrit ,(4.11)
where Scrit is the critical control parameter for a constant Reynolds number and S≥Scrit.
In the present experiment, Asat is derived from the PSD of the leading two temporal POD
modes a1and a2(figure 4.12b-c). This is preferable to the PSD of the Fourier modes
(figure 4.12a) since the POD modes capture the oscillation amplitude of the entire spatial
domain yielding a more representative global value. Both, the saturation amplitude derived
from streamwise or crosswise data are found to be proportional to √S−Scrit (figure 4.14).
The linear dependence suggests that the oscillation is of the supercritical Hopf bifurcation
type. As mentioned earlier, the amplitudes differ between streamwise and crosswise POD
modes due to the different measurement domains. Note that the streamwise POD modes
also capture the weak oscillations for S= 1.12 that are not detectable by the crosswise
measurement located at x/D = 1.1. A linear fit of the streamwise and crosswise saturation
amplitudes yields a critical swirl number Scrit = 1.1, which is clearly higher than the critical
swirl number SVB = 1.06 for vortex breakdown. This disagrees with the observations of
52 Chapter 4 The Onset of Vortex Breakdown and Global Instability
St
PSD/PSDmax
(a) FFT (r-θ-plane)
(b) POD (r-θ-plane)
(c) POD (x-r-plane)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Figure 4.12: (a) PSD of mode m= 1 derived via spatial Fourier analysis of the r-θ-plane
at x/D = 1.1 according to equation (4.10). (b) PSD of temporal POD mode a1(symbol)
and a2(line) derived from the r-θ-plane at x/D = 1.1. (c) PSD of temporal POD mode
a1(symbol) and a2(line) derived from x-r-plane.
Chapter 4 The Onset of Vortex Breakdown and Global Instability 53
(b)
(d)
z/D
(f)
y/D
(a)
y/D
(c)
y/D
x/D
(e)
-0.5 0 0.50.5 1 1.5 2 2.5
-1
0
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Figure 4.13: Portrait of co-rotating counter-winding global instability mode m= 1;
projected streamlines and through-plane vorticity component ωzin the streamwise and
crosswise (x/D = 1.1) measurement plane; (a-b) time-averaged flow; (c-d) coherent flow
(times 5 for equal color scale); (e-f) time-averaged plus coherent flow; coherent flow is
displayed at an arbitrary phase; (S= 1.37)
54 Chapter 4 The Onset of Vortex Breakdown and Global Instability
S
A2
sat
crosswise
streamwise
0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8×10−3
Figure 4.14: Square of limit cycle amplitudes of global oscillation derived from temporal
POD modes of the crosswise (white circles) and streamwise (black stars) plane of measure-
ment. Linear extrapolation predicts Scrit = 1.1 to be the critical swirl number where the
flow undergoes a supercritical Hopf bifurcation
Liang & Maxworthy (2005), who suggest that both swirl numbers coincide. In fact, they
found Scrit to be below SVB but argued that due to noise, the former is underestimated.
The present findings support the sequence leading to the global mode m= 1 that is
described at the beginning of this section: Axisymmetric vortex breakdown is established
and persists permanently at swirl numbers below a critical bifurcation parameter Scrit.
Exceeding that, the region of reversed flow that generally promotes absolute instability,
reaches a sufficient size to set up a feedback mechanism that leads to a global mode and
vortex breakdown becomes non-axisymmetric. It is interesting to note that the critical
swirl number Scrit coincides with the swirl number for which the linear growth of the
recirculation region stagnates (see figure 4.5). Any further increase in swirl only leads to
an upstream movement of that region. Furthermore, the critical swirl number coincides
with the swirl number at which the axial fluctuations of the stagnation point reach their
minimum value (figure 4.7b). Apparently, the onset of the self-excited oscillations dampen
the low frequency breakdown oscillations and interact with the mean flow in such a way
that any further growth of the recirculation zone is inhibited.
4.8 Summary and Discussion
This chapter describes the experimental investigation of the nearfield of an unconfined
swirling jet by means of time-resolved stereoscopic PIV. Experiments were conducted in
Chapter 4 The Onset of Vortex Breakdown and Global Instability 55
water at Re = 3300. The flow is fully turbulent at the nozzle exit due to a serrated tip-ring
mounted upstream of the nozzle. The amount of swirl was adjusted carefully near the
onset of vortex breakdown in order to map the different flow states ultimately leading to its
non-axisymmetric state. The investigation provides a quantitative characterization of the
mean and the instantaneous flow field with an emphasis on the vortex breakdown dynamics
and the associated coherent structures.
The attempt is made to scale the experiments conducted for a wide range of swirl
levels by using one universal swirl number definition. It is found that the commonly used
swirl number – the axial flux ratio of axial to azimuthal momentum – is inaccurate for
strong swirl due to the invalidity of the underlying boundary layer approximations. A
detailed investigation of the individual terms of the equations of motion show that the
turbulent stresses and radial velocity component contribute significantly to the momentum
conservation in regions of strong jet divergence. Taking this into account, the swirl number
is conserved in the axial direction and increases linearly with the rotation rate of the swirl-
generating honeycomb. However, for very strong swirl, vortex breakdown moves into the
nozzle and the presented swirl number definition is still inaccurate, presumable due to the
negligence of pressure forces induced by the nozzle walls. For these cases the swirl number
is derived from a linear extrapolation.
The various flow states observed for increasing swirl intensity and the corresponding
dominant vortical structures are characterized by three different methods. First, the oc-
currence of vortex breakdown, associated with a stagnation point on the jet axis, is tagged
from the mean flow and from each individual PIV snapshot. Second, Benjamin’s (Ben-
jamin 1962) criticality concept is applied to the mean flow and the axial positions of
supercritical–subcritical transition are derived. Third, azimuthal instability modes are de-
tected via Fourier decomposition of PIV measurements taken along the crosswise plane. In
case of large-scale intrinsic oscillations, PIV snapshots, taken along the streamwise plane,
are phase-averaged a posteriori by employing POD.
Figure 4.15 provides an schematic overview of the major findings. Starting with the
non-swirling jet, the axisymmetric mode m= 0 is found to be dominant near the nozzle.
By increasing the swirl, the symmetry breaks and counter-rotating modes with m < 0 are
destabilized. For 0.86 < S < 1.06 vortex breakdown appears intermittently and oscillates
in axial direction at low frequencies (St <0.1), but it remains undetectable from the mean
flow. At S= 1.04 two nearly separated subcritical regions exist that match the axial lo-
cations of these oscillations. Thus, the unsteady vortex breakdown is well predicted from
the local criticality of the mean flow that does not exhibit vortex breakdown at that swirl
intensity. This supplements the findings of Ruith et al. (2003), who successfully apply the
local criticality concept to localize steady vortex breakdown in a laminar jet. By further
increasing swirl, the occurrence of vortex breakdown increases progressively until it ap-
pears in the mean flow at SVB = 1.06. At that swirl number, the inflow profiles transition
from supercritical to subcritical, in agreement with the theoretical prediction of Benjamin
(1962). The good applicability of Benjamin’s model suggests that the mechanism that
drives breakdown is inviscid for the present experiment. For swirl numbers above SVB,
a recirculation region appears that grows continuously with increasing swirl until its size
stagnates at S≈1.11. At that flow configuration, the low frequency axial oscillations of
56 Chapter 4 The Onset of Vortex Breakdown and Global Instability
0 0.38 0.86 1.04 1.06 1.1 1.11 S
never permanent
intermittent
V
B
o
s
c
i
l
l
a
t
i
o
n
globally stable
l
i
m
i
t
c
y
c
l
e
SVB Scrit
maximum size of
recirculation region
subcritical core
at 1.3< x/D < 2.2
subcritical core
at x/D > 0
supercritical
Hopf bifurcation
0−2, −1 0, 1 1
swirl number
VB occurrence
VB dynamics
global stability
dominant
modes m
Figure 4.15: Conceptual drawing summarizing the experimental results.
vortex breakdown have completely decayed whereas strong helical waves at higher frequen-
cies (St >0.44) are found. Experiments indicate that the flow undergoes a supercritical
Hopf bifurcation to a global mode m= 1 at Scrit = 1.1. The selection of mode m= 1 is
in agreement with previous numerical, theoretical, and experimental investigations and is
well documented in recent literature (Gallaire et al. 2006;Liang & Maxworthy 2005;Ruith
et al. 2003). However, the observation that SVB < Scrit is new and stands in contrast to the
experiments of Liang & Maxworthy (2005) who found SVB > Scrit. They report that their
Scrit is likely to be underestimated due to unwanted external forcing from their facility.
In conclusion, the following scenario can be derived from the presented experiment: The
turbulent swirling jet undergoes intermittent and heavily oscillating vortex breakdown at
swirl numbers far below SVB = 1.06 that is initiated by two small and spatially separated
regions of subcritical flow. By increasing the swirl to SVB, the inflow profiles undergo a
supercritical–to–subcritical transition and axisymmetric vortex breakdown occurs perma-
nently. A further increase in swirl causes the recirculation zone to grow linearly until the
promoted pocket of absolutely unstable flow is sufficiently large for the mode m= 1 to
become globally unstable. At Scrit = 1.1, a feedback mechanism sets in that stimulates
the vortex core to precess in the direction of the mean flow rotation and vortex breakdown
becomes non-axisymmetric. A further increase in swirl enhances the global oscillations
Chapter 4 The Onset of Vortex Breakdown and Global Instability 57
and associated nonlinear mean flow interactions which inhibits a further increase of the
recirculation zone.
The presently observed time-averaged turbulent flow states that describe the formation
of vortex breakdown with increasing swirl, compare well with the transient flow states of a
laminar jet undergoing vortex breakdown that were observed by Bruecker & Althaus (1995)
. Hence, the present investigation provides representable and repeatable measurements of
the stepwise formation of spiral vortex breakdown in a turbulent base flow.
Chapter 5
Instabilities in the Moderately
Swirling Jet
This chapter focuses on the streamwise growth of coherent structures in jets
at swirl intensities below the onset of vortex breakdown. All flow configurations
considered are globally stable. The main question addressed in this investigation
is, how swirl effects the evolution of convectively unstable modes. The problem
is approached theoretically by conducting a quasi-parallel spatial linear stability
analysis employing the mean flow. The impact of small-scale turbulence on
the stability is incorporated using an eddy viscosity model derived from the
measured Reynolds stresses. The major theoretical results presented throughout
this part are compared to hot-wire measurements of the single-mode actuated
and pulsed actuated flow, showing good agreement. The swirling jet at the
nozzle is receptive to a shear instability and a centrifugal instability mode.
The previous is primarily generated by the radial stratification of axial velocity
(axial shear) and is much stronger than the latter that is generated by the
radial stratification of azimuthal velocity (azimuthal shear). With increasing
downstream distance, the centrifugal instability stabilizes much more rapidly
than the shear instability. Swirl affects the streamwise phase velocity of helical
(oblique) traveling waves, which is explained by simple kinematic considerations.
The non-zero azimuthal group velocity is thereby the key driver. The swirling
jet selects a co-winding m= 2 mode as the preferred mode in the nearfield and
a co-winding m= 1 mode further downstream. The swirling jet is unstable
to streamwise modes with zero axial wavenumber and steady modes with zero
frequency. The latter may reach significant high amplitudes, which explains the
breaking of rotation symmetry in the mean flow that is often observed in swirling
jet experiments. Comments to swirl-enhanced jet spreading are given at the end
of this part and connections to the presently found stability characteristic are
made.
59
60 Chapter 5 Instabilities in the Moderately Swirling Jet
5.1 Background and Scope
The investigation described in this chapter focuses on axisymmetric unconfined turbulent
jets at swirl intensities below the onset of vortex breakdown, which are referred to as mod-
erately swirling jets. Due to the ability of swirl to enhance turbulent production and shear
layer spreading, these flows are commonly used in industrial applications where efficient
turbulent mixing is required. The great importance of swirling flows for the combustion
industry is reflected by the large number of related publications (see review articles of Gut-
mark et al. 1995;Huang & Yang 2009;Knowles & Saddington 2006;McManus K. et al.
1993;Syred 2006, and references therein).
During the last two decades, various ideas have come up to explain the enhanced mixing
in swirling flows. There is a common consensus that the vortical structures in swirled shear
layers, which differ to those found in non-swirling jets, must be the driving force for the
enhanced turbulent production and entrainment rate. However, the characteristics and
the sources of the flow structures that reside in swirled shear layers are still unclear. It is
known that with the addition of swirl, the flow promotes shear instabilities and centrifugal
instabilities, but their ability to enhance jet spreading is still a controversial issue. Linear
stability analysis based on swirling jet models reveal that centrifugal instabilities become
successively destabilized with increasing swirl (Gallaire & Chomaz 2003;Loiseleux et al.
2000;Lu & Lele 1999;Martin & Meiburg 1996,1998;M¨uller & Kleiser 2008). Several
investigators suggest these instabilities, which promote disturbances at smaller scales than
shear instabilities, to be the cause for the enhanced jet spreading (Cutler et al. 1995;Mehta
et al. 1991;Panda & McLaughlin 1994;Wu et al. 2006). This stands in contrast to the
experimental observation of Naughton et al. (1997), who found swirl-enhanced mixing for
a centrifugally stable profile, which is confirmed by the numerical studies of Hu et al.
(2001a,b). These authors conclude that centrifugal instability is not a necessary condition
for enhanced jet spreading. In fact, their direct numerical simulations indicate that the
growth of swirled shear layers is augmented by a nonlinear interaction of primary vortex
ring and the columnar vortex that leads to a rapid breakdown of large flow structures into
smaller scales.
The three-dimensional nature of the vortical structures in swirling jets was demonstrated
by several experimental investigations. Panda & McLaughlin (1994) successfully excited
axisymmetric and non-axisymmetric modes in the shear layer of a turbulent swirling jet,
revealing a lower receptivity for axisymmetric modes in comparison to the non-swirling
jet. Billant et al. (1998) and Loiseleux & Chomaz (2003) investigated the dynamics in the
nearfield of a natural laminar jet for various swirl intensities. They found large-scale co-
winding double- and triple-helical structures in the pre-breakdown state that were perfectly
steady for the experiments conducted by Billant et al. (1998), whereas Loiseleux & Chomaz
(2003) observed them to rotate at very low frequencies in direction of the base flow rotation.
A spatio-temporal stability analysis conducted by Gallaire & Chomaz (2003) for a mean
flow measured by Billant et al. (1998) at approximately half a diameter downstream of
the nozzle lip revealed an absolute–convective instability transition of the double-helical
mode when exceeding a certain amount of swirl. The authors suggest this mode to be a
self-excited globally unstable mode with its wavemaker located at the nozzle lip.
Chapter 5 Instabilities in the Moderately Swirling Jet 61
The temporal analysis of Gallaire & Chomaz (2003) confirm that swirl destabilizes modes
of high negative azimuthal wavenumbers due to the centrifugal instability in good agreement
with the analysis of Leibovich & Stewartson (1983). However, the mode selection predicted
by their temporal analysis disagrees with the experimental observations of Billant et al.
(1998) and Loiseleux et al. (2000), which they explain with the above mentioned onset of
global instability. The authors admit that their arguments remain vague as the analysis is
based on a single measurement location, which does not reflect the streamwise varying base
flow and associated stability. The mode selection in the swirling jet experiments conducted
by Liang & Maxworthy (2005) differs significantly from the previously reported one. For
various swirl intensities, all detected helical modes are co-rotating counter-winding and are
aligned with the local helical vortex lines of the mean flow. The investigators argue that
the relative location and thickness of the axial and azimuthal shear layers have a great
influence on the streamwise evolution of the dominant modes. Hence, a comparison of
their results to stability analysis based on simplified velocity models, as attempted by the
authors, remains only qualitative.
In consideration of the complexity of swirling jet instabilities and their sensitivity to base
flow variations, a linear stability analysis is conducted employing the entire nearfield of an
unconfined spatially evolving swirling jet derived from time-averaged flow measurements.
The theoretical results are compared to measurements of the externally excited flow. The
author is aware that there exists no universal swirling jet configuration and that the one
investigated presently represents only one of several possibilities. Nonetheless, a detailed
description of the stability of this particular flow in conjunction with a quantitative exper-
imental validation will enhance the fundamental understanding of the dynamics in swirling
jets, in order to properly interpret other experimental findings and to develop flow control
methods for mixing enhancement.
Experiments are conducted at the swirling air jet facility at the TU Berlin. The experi-
mental setup is described in section 3.2 and details of the conducted hot-wire measurements
and data treatment are given in section 3.4. The investigation is organized as follows. The
flow configurations discussed throughout this chapter are described in section 5.2, ranging
from zero swirl to strong swirl below onset of vortex breakdown. A detailed investigation
of the linear stability of the swirling jet at the nozzle is presented in section 5.3, followed
by the examination of the streamwise evolution of instability in section 5.4. The results of
the experimental investigation of the single-mode and pulsed actuated flow are described
in section 5.5 and 5.6, respectively. The major findings are summarized and discussed in
section 5.7.
5.2 The Mean Flow Configuration
A swirling air jet was generated using the facility described in section 3.2. The mean flow in
the x-r-plane was derived from the Stereo-PIV measurements for different swirl intensities
at ReD=DV/ν = 20000.
62 Chapter 5 Instabilities in the Moderately Swirling Jet
5.2.1 Analytic Representation of the Mean Flow
The experimental data is approximated by profiles given by the following analytic expres-
sions:
Vx=Vcl
1 + (exp [(r/R.5)2log(2)] −1)Nx,(5.1a)
Vr= 0,(5.1b)
Vθ= Ωclrexp −(r/Rθ)Nθ/Nθ.(5.1c)
A slightly modified version of the expression for the axial velocity component Vxwas
introduced by Monkewitz & Sohn (1988). Later, Gallaire & Chomaz (2003) used this
profile to approximate the measurements conducted by Billant et al. (1998). Vcl represents
the centerline axial velocity and R.5the jet radius, defined as the radial distance where
Vx= 0.5Vcl. The dimensionless parameter Nxis related to the radial gradient of the axial
velocity profile and is inversely proportional to the axial shear layer thickness. The model
for the azimuthal velocity component Vθwas first introduced by Carton & McWilliams
(1989) and more recently used by Gallaire & Chomaz (2003). Ωcl represents the rotation
rate on the jet axis and Rθthe radial location of the maximum azimuthal velocity. The
dimensionless parameter Nθis inversely proportional to the azimuthal shear layer thickness.
The parameters Nxand Nθare determined from a least squares fit of the expressions (5.1a)
and (5.1c) to the axial and azimuthal mean velocity, respectively.
The contraction upstream of the nozzle generates an overshoot of the axial velocity
profile in agreement with previous studies (Billant et al. 1998;Panda & McLaughlin 1994;
Semaan et al. 2009). To account for this hump in the axial velocity profile, Gallaire &
Chomaz (2003) introduced an additional term to (5.1a), yielding the more complex expres-
sion
Vx=Vcl
1 + (exp [(r/R.5)2log(2)] −1)Nx+Vos exp[−(r/rc)2],(5.2)
where Vos represent the strength of the overshoot and rcits radial extent. According to
Gallaire & Chomaz (2003), the axial overshoot has minor effect on the instability of the
base flow. Preliminary computations conducted for both models confirm the insignificance
of the overshoot for the present flow. Hence, the stability analysis presented throughout
this study is based on the mean flow approximated by the simpler and far more generic
model (5.1a).
5.2.2 The Velocity Distribution at the Nozzle
The back-scattering configuration of the PIV measurements enables to access the flow
quantities at the nozzle exit. The corresponding axial and azimuthal mean velocity profiles
are shown in figure 5.1. Measurements are conducted at four different swirl configurations
ranging from zero swirl, labeled by S1, to strong swirl, labeled by S4. The solid lines
correspond to the simple model (5.1) and the dashed lines correspond to model (5.2) that
compensates for the axial overshoot. The model parameters are given in table 5.1. Despite
the formation of an overshoot, the swirl does not significantly alter the axial velocity profile,
which is in agreement with measurements of laminar swirling jets (Billant et al. 1998). The
axial shear layer thickness increases slightly with increasing swirl, indicated by a decrease of
Chapter 5 Instabilities in the Moderately Swirling Jet 63
y/D
Vx/V
S4
S3
S2
S1
y/D
Vθ/V
S4
S3
S2
S1
S4
S3
S2
S1
S4
S3
S2
S1
S4
S3
S2
S1
S4
S3
S2
S1
S4
S3
S2
S1
S4
S3
S2
S1
-1 -0.5 0 0.5 1-1 -0.5 0 0.5 1 -1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
1
1.5
2
Figure 5.1: Mean velocity profiles at the nozzle for different swirl intensities. The solid
lines correspond to the simple model (5.1) and the dashed lines correspond to model (5.2)
that compensates for the axial overshoot. The model parameters are given in table 5.1.
S1S2S3S4
Vcl/V 1 1 1 1
ΩclR.5/Vcl 0 0.54 1.54 2.23
R.5/D 0.47 0.47 0.47 0.47
Rθ/D −0.36 0.32 0.29
Nx17.78 16.74 14.61 12.75
Nθ−10.37 6.14 4.45
rc−0.42 0.42 0.42
Vos/V −0.15 0.52 0.78
Table 5.1: Parameters of the mean flow approximation (5.1) determined by a least order
fit to the measured velocity profiles at at x/D = 0.
Nx. The azimuthal velocity profile is certainly effected by an increase in swirl. The rotation
rate near the jet axis, expressed as the dimensionless ratio ΩclR.5/Vcl, is gained up to 2.23
with increasing swirl. Moreover, the radial position of the maximum azimuthal velocity Rθ
moves towards the jet center and the azimuthal shear layer thickens considerable, indicated
by an decrease of Nθ. The latter is not observed in the laminar swirling jet experiments
conducted by Billant et al. (1998); Loiseleux & Chomaz (2003), where Nθand Rθremain
constant with increasing swirl.
5.2.3 Quantification of Swirl at the Nozzle Exit
Several swirl number definitions exist in the literature, each being applicable to a certain
flow regime, but none being universal. For the sake of comparison, the most common defi-
nitions of swirl numbers are derived at the nozzle exit and listed in table 5.2 together with
64 Chapter 5 Instabilities in the Moderately Swirling Jet
definition references S1S2S3S4
2R∞
0VxVθr2dr
DR∞
0(V2
x−V2
θ/2)rdr
Chigier & Chervinsky
(1965); Lu & Lele (1999);
M¨uller & Kleiser (2008);
Panda & McLaughlin (1994)
0 0.19 0.46 0.6
√R∞
0V2
θ/rdr
Vcl Billant et al. (1998)0 0.28 0.55 0.62
2Vθ,max
Vx,max Escudier & Keller (1985)0 0.66 1.17 1.25
2Vθ(r=D/4)
Vcl
Billant et al. (1998);
Loiseleux & Chomaz (2003)0 0.5 1.04 1.19
ΩclD/2
VLiang & Maxworthy (2005)0 0.58 1.64 2.38
ΩclRθ
V−1Billant et al. (1998); Spall
et al. (1987)∞2.41 0.96 0.72
arctan Vθ
Vxmax
Sarpkaya (1971)0 20.52 40.64 46.01
Table 5.2: Values for different swirl number definitions at x= 0
the corresponding references. Throughout this investigation, the four flow configurations
considered will be simply referred to as S1,S2,S3, and S4. A comparison with the listed
literature reveals that the highest swirl number S4is still below the critical level where
vortex breakdown occurs.
5.2.4 Streamwise Distribution of Mean Velocities and Turbulent Stresses
The figures 5.2 and 5.3 depict the mean flow distribution along the streamwise plane of
measurement derived from PIV measurements for the non-swirling and swirling jet, respec-
tively, together with the velocity profiles approximated by the simplified model (5.1). The
agreement between the fit and the actually measured values is excellent for the non-swirling
jet. For swirl configuration S3, the analytic approximation of the axial velocity component
is also good, despite a slight underestimated shear layer thickness near the nozzle exit. The
swirl profiles are represented reasonably well by the model, however, upstream of x/D = 1
the inflection point inside the vortex core region is not reproduced.
The contour surfaces shown in the figures 5.2 and 5.3 indicate regions of high turbulent
production. they represent the normalized turbulent shear stress component v′
xv′
rfor the
non-swirling jet, and v′
xv′
rand v′
rv′
θfor the swirling jet. Without swirl, v′
rv′
θis orders of mag-
nitude smaller than for the swirling case and is therefore covered in noise (not shown). The
contours reveal the radial maximum is located approximately at the highest radial gradient
of Vθ, indicating that v′
rv′
θis primarily generated by the azimuthal shear. The turbulent
shear stress component v′
xv′
ris also enhanced by the addition of swirl, which presumably
results in an enhanced downstream growth of the axial shear layer and a shortening of
the potential core region. Furthermore, the turbulent shear stresses confirm that the axial
overshoot near the nozzle, which creates an additional inflection point in the axial velocity
Chapter 5 Instabilities in the Moderately Swirling Jet 65
r/D
x/D, V/U ·0.25
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.005
0.01
0.015
0.02
0.025
-1
-0.5
0
0.5
1
Figure 5.2: Non-swirling jet configuration S1: Profiles of the axial velocity are shown
together with contours of the turbulent shear stress v′
xv′
r/V 2. Black dots refer to measure-
ments and black solid lines refer to the analytic approximation (5.1a).
profile, does not contribute significantly to the turbulent production, and hence, instabil-
ities growing in the inner shear layer seem to be negligible. Similarly, the aforementioned
inflection point of the azimuthal profile near the jet center upstream of x/D = 1 does not
generate turbulent shear stress and it is not expected to promote dominant instabilities.
The poor approximation of this feature is therefore considered to be insignificant. Note that
PIV measurements are only conducted for the unforced flow where the coherent velocity
cannot be extracted. Hence, the turbulent shear stresses discussed here are derived from
the fluctuating velocity field that comprises stochastic and coherent motion.
5.2.5 Parametrization of the Divergent Mean Flow
The evolution of the mean flow with downstream distance is characterized by the streamwise
distribution of the model parameters. The parameters that scale the axial velocity profiles
are shown in figures 5.4a, 5.4c, and 5.4e and those corresponding to the azimuthal velocity
profile are shown in figures 5.4b, 5.4d, and 5.4f. The figures 5.4g display the streamwise
development of the momentum thickness δx. It is used as the characteristic length scale
throughout this chapter and represents a measure for the axial shear layer thickness, defined
as
δx=ZR.95
R.05
Vx
Vcl 1−Vx
Vcl dr. (5.3)
Figure 5.4a indicates the axial extent of the potential core, the region where Vcl is
equal to the bulk velocity V. For the non-swirling jet (S1), the potential core exceeds the
measurement domain. By increasing the swirl, the potential core region is significantly
shortened due to the increased growth of axial shear layer thickness. For S3,Vcl declines
for x/D > 2, whereas Loiseleux & Chomaz (2003) measures a potential core length of
66 Chapter 5 Instabilities in the Moderately Swirling Jet
r/D
Vx/V
r/D
x/D, V/U ·0.25
Vθ/V
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.005
0.01
0.015
0.02
0.025
0
0.005
0.01
0.015
0.02
0.025
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Figure 5.3: Swirling jet configuration S3: Profiles of the axial velocity are shown. Black
dots refer to measurements and black solid lines refer to the analytic approximation (5.1).
Symbols refer to experimental data. Contours represent turbulent shear stresses v′
xv′
r/V 2
(top) and v′
rv′
θ/V 2(bottom)
x/D ≈3.8 at comparable swirl intensity. Hence, the presently found streamwise extent of
the potential core is much smaller than for laminar swirling jets.
It is interesting to note that the swirl parameter (inverse Rossby number) shown in
figure 5.4b remains perfectly constant within the potential core, despite the downstream
decay of the maximum swirl velocity Vθ,max. It is the only local swirl number of the
definitions listed in table 5.2 that remains constant within the potential core. All other
numbers show a dependence on the axial distance already close to the nozzle exit, which
makes a quantitative comparison to the investigations of Billant et al. (1998); Gallaire
& Chomaz (2003); Loiseleux et al. (1998) very difficult. It is further worth noting that
downstream of the potential core, the Rossby number decays at the same rate independent
of the swirl intensity and it ultimately collapses to one curve that asymptotes zero.
The quantities shown in figure 5.4c, 5.4e, and 5.4f indicate the enhanced jet spread-
ing with increasing swirl. The non-monotonic development of Rθis somewhat unexpected
Chapter 5 Instabilities in the Moderately Swirling Jet 67
Vcl/V
(a)
ΩclRmax/V
(b)
R.5/D
(c)
Rmax/D
(d)
Nx
(e)
Nθ
(f)
x/D
δx/D
(g)
S1
S2
S3
S4
x/D
Vθ,max/V
(h)
0 1 2 3 40 1 2 3 40
0.5
1
0
0.1
0.2
0.3
0
5
10
0
10
20
0.2
0.4
0.6
0.8
0.4
0.6
0.8
1
0
0.5
1
1.5
0.6
0.8
1
Figure 5.4: (a-f) Streamwise development of mean flow parameters of (5.1); Streamwise
development of axial momentum thickness (g) and maximum swirl velocity (h).
68 Chapter 5 Instabilities in the Moderately Swirling Jet
x/D
νt×10−3in m2/s
x/D
Ret
S1
S2
S3
S4
(a) (b)
0 1 2 3 40 1 2 3 40
20
40
60
80
0
2
4
6
8
Figure 5.5: Spatial development of eddy viscosity (a) and turbulent Reynolds number
(b) for the swirl configurations considered.
(figure 5.4d). Within the potential core, the downstream growth of the azimuthal shear
layer leads to a displacement of the maximum of the swirl component to the jet center,
while the constant swirl parameter (figure 5.4b) implies that Ωcl must increase correspond-
ingly. Downstream of the potential core, the maximum of Vθmoves outwards and the
azimuthal shear layer thickness decreases, indicated by an increasing shape parameter Nθ.
The maximum azimuthal velocity Vθ,max decelerates drastically within the potential core
and collapses for all swirl configurations at x/D > 3.5 in a similar manner as the inverse
Rossby number. Apparently, the growth of the azimuthal shear layer is coupled to the
growth of axial shear layer in the potential core region, which leads to a constriction of the
vortex core. Further downstream, the axial shear layer merges on the jet axis, the vortex
core spreads radially, and the inverse Rossby number decays.
5.2.6 Streamwise Distribution of the Eddy Viscosity
Figure 5.5a displays the streamwise distribution of the weighted eddy viscosity νtaveraged
in radial direction. A detailed description of how this quantity is derived from the PIV data
is given in section 2.4.4. For all swirl configurations, the turbulent viscosity is nearly equal at
the nozzle exit and grows rapidly with downstream distance. The growth is enhanced with
increasing swirl in consistency with the enhanced shear layer growth discussed in section
5.2.5. Without swirl, νtasymptotes to a constant level of approximately 0.002 at x/D > 3
(Figure 5.5b). The turbulent Reynolds number decays rapidly upstream of x/D = 1 and
then asymptotes at similar constant values within the range 10 <Ret<20. The growth of
eddy viscosity appears to be balanced by the growth of momentum thickness.
5.3 Stability at the Nozzle Exit
From a preliminary spatio-temporal analysis of the considered configurations, the existence
of absolute instability for |m| ≤ 3 can be excluded. This implies the flow is globally stable
Chapter 5 Instabilities in the Moderately Swirling Jet 69
and the signaling problem is valid. Hence, the spatially traveling disturbance waves are best
modeled by the spatial stability analysis (Huerre & Monkewitz 1990). The Orr–Sommerfeld
type eigenvalue problem is, therefore, solved for complex eigenvalues αand given real ω.
Within the quasi-parallel approximation, −αirefers to the streamwise (spatial) growth rate
and αrto the streamwise wavenumber. The flow is considered as spatially unstable when
a disturbance grows with x, i.e. when αiis negative. The mean flow is made dimensionless
by using δxas the characteristic length scale and Vas the characteristic velocity scale. The
eigenvalues αand the frequencies ωthat are presented throughout this investigation are
always normalized with respect to δxand V, which vary significantly in axial direction.
A linear stability analysis is applied to the mean flow at the nozzle exit (x/D = 0) for
different swirl intensities. As described in section 5.2.2, the mean flow at the nozzle exit
is characterized by a top-hat axial velocity profile that is nearly independent of the swirl.
The azimuthal velocity profile of the swirling jet is characterized by a linear region near the
jet center (solid body rotation) and an outward decaying region (azimuthal shear layer).
Both depend strongly on the amount of swirl. Hence, the stability of the flow at the nozzle
exit is solely altered by the varying swirl component. For the following discussion, the
reader is encouraged to use the schematic representation displayed in figure 2.2 in order to
distinguish between the different mode alignments.
5.3.1 Shear Instability
Figure 5.6 provides an overview of the influence of swirl on the dominant instability. This
rather unusual plotting style appears several times throughout this chapter and is therefore
described in more detail at this point. The filled contours represent the dimensionless spa-
tial growth rate −αiδxcomputed for various frequencies ωrand azimuthal wavenumbers
m. All contour plots shown within one figure have always the same contour levels. Re-
gions of negative growth rate (−αiδx<0) are blanked. The labeled contour lines refer to
the dimensionless axial wavenumbers αrδxand are also only displayed for unstable modes
(−αiδx>0). For better visibility, the contour lines of αrδxare black, while the lines sep-
arating the filled contours of −αiδxare light-gray. For most cases, we allow the frequency
and axial wavenumber to have positive and negative values. This implies redundant infor-
mation for αiδxcontours shown in the ωr-m-plane, as modes at negative frequencies have
the same growth rate and streamwise wavenumber with the latter having an opposite sign
(see transformation (2.4)). Recall that the label S1refers to the non-swirling jet, while S2,
S3and S4refer to swirling jets at increasing swirl intensity. A detailed description of the
considered mean flow configurations are given in chapter 5.2, while the corresponding swirl
numbers are listed in table 5.2.
The stability analysis applied at the nozzle reveals two unstable eigenvalues, each cor-
responding to a different instability mode. In this section we focus on the mode with
higher amplification rate, while the additional, less unstable, mode is described in the sub-
sequent section. At the nozzle exit, the non-swirling jet (figure 5.6a) is most unstable to
axisymmetric disturbances in agreement with previous investigations (e.g., see Cohen &
Wygnanski 1987;Crighton & Gaster 1976;Gallaire & Chomaz 2003). In the absence of
swirl, co-rotating and counter-rotating modes are equally unstable. The growth rate de-
cays continuously with increasing azimuthal wavenumber, yielding a cutoff at |m|= 18.
The streamwise wavenumber of maximum amplification is approximately αrδx= 0.2 and
70 Chapter 5 Instabilities in the Moderately Swirling Jet
S1
-0.3
-0.2
-0.1
0.1
0.2
0.3
ωrδx/V
S2
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
S3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
ωrδx/V
m
S4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
m
-30 -20 -10 0 10 20 30-30 -20 -10 0 10 20 30
0
0.02
0.06
0.08
0.1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Figure 5.6: Instability of the shear mode at the nozzle exit (x/D = 0) for the non-
swirling jet (S1) and the swirling jets (S2-S4). Filled contours refer to the spatial growth
rate −αiδx. Dark-gray labeled contour lines refer to the streamwise wavelength −αrδx.
Stable modes −αiδx<0 are blanked. Quantities are derived theoretically for varying
azimuthal wavenumbers mand frequencies ωr.
remains constant for varying m. The overall maximum amplification and corresponding
frequency agrees well with the values reported by Cohen & Wygnanski (1987). This brings
credibility to the measured mean flow on which the analysis is based.
By imposing swirl onto the flow (confer with S2-S4in figure 5.6), the symmetry breaks.
Counter-rotating modes at low frequencies are destabilized, whereas co-rotating modes at
high frequencies become less unstable. By increasing swirl, the filled contours cross the
ωr= 0 line, indicating the steady modes are unstable, in contrast to the non-swirling jet.
However, the maximum growth rates and associated streamwise wavenumbers are nearly
unaffected by the swirl component. This indicates that this instability is driven by the axial
shear layer similar to the Kelvin–Helmholtz instability of the non-swirling baseline case.
Throughout the remaining part of this thesis, we will, therefore, refer to this instability as
Chapter 5 Instabilities in the Moderately Swirling Jet 71
the shear instability mode in differentiation to the centrifugal instability mode that is also
found in the present flow configuration.
The contours in figure 5.6 reveal that the growth rates of the shear mode is not sig-
nificantly affected by the swirl and solely depends on the axial velocity profile. However,
there is yet a considerable influence of the swirl component on the selected frequencies and
azimuthal wavenumbers that are most amplified. This is discussed in detail by consid-
ering profiles of the streamwise growth rate αi, the streamwise wavenumber αr, and the
streamwise phase velocity cph =ωr/αrfor the non-swirling (S1) and strongly swirling jet
(S3) presented in figure 5.7. To avoid redundancy, mtakes only positive values, with α
of negative mand positive ωrbeing transformed to positive mand negative ωr. Hence,
the four variants of possible mode alignments refer conveniently to the four quadrants of
the αr-ωr-diagram (see labels in figure 5.7a). Note that figure 5.7a and c depict only those
quantities that refer to unstable modes with −αr>0.
For the non-swirling baseline case (figure 5.7a, left), unstable modes are in the first
and third quadrant, depicting this flow to be unstable to co-rotating counter-winding and
counter-rotating co-winding modes. The wavenumbers increase approximately linearly with
ωrand do not change considerably with m. Moreover, the non-swirling jet is stable to
streamwise modes (αr= 0) and steady modes (ωr= 0).
For the swirling jet configuration S3, the streamwise wavenumbers and growth rates of
the axisymmetric modes (m= 0) are very similar to the non-swirling jet. These modes
are unaffected by the addition of swirl, as they are purely driven by the axial shear layer.
The non-axisymmetric modes (m6= 0) are affected by the azimuthal shear. With swirl, the
streamwise wavenumber depends on the azimuthal mode number, yielding decreasing αrfor
increasing m(figure 5.7a, right). This relation implies that, within a certain frequency and
wavenumber band, unstable non-axisymmetric modes exist in the fourth quadrant, which
renders swirling jets to be receptive to steady modes, to co-rotating co-winding modes, and
to streamwise modes. The non-winding (streamwise) modes are purely driven by azimuthal
shear in contrast to the axisymmetric mode that is purely driven by the axial shear. The
spatial growth rate displayed in figure 5.7b for the S1and S3configuration replicates the
results shown in figure 5.6a and c. These diagrams emphasize that the swirling and non-
swirling jet is unstable to the same finite frequency band. Hence, the axial shear layer acts
as a band-pass filter with the band being determined by the shape of the axial velocity
profile. The selected wavenumbers, however, are altered by the addition of swirl (figure
5.7a).
The streamwise phase velocity cph for the non-swirling jet shows typical features of the
Kelvin–Helmholtz instability (figure 5.7c, left). For frequencies near neutral amplification,
cph/V asymptotes 0.55, which implies that waves propagate in downstream direction at
approximately half the bulk velocity. Moreover, in the absence of swirl, waves are weakly
dispersive (∂cph/∂ωr≈0) for a wide range of frequencies. According to Cohen & Wygnan-
ski (1987), non-dispersive waves in shear layers that travel downstream with equal phase
velocity interact nonlinearly, provided that specific resonance conditions are satisfied. The
weak dispersiveness of non-swirling jets allows for nonlinear interactions between a down-
stream traveling wave at neutral amplification and its subharmonic (Paschereit et al. 1995).
This is associated with the process of vortex merging that is the key driver for the stream-
wise growth of the shear layer in the near-field of axisymmetric non-swirling jets (Gutmark
72 Chapter 5 Instabilities in the Moderately Swirling Jet
−αiδx
(b)
αrδx
counter-rotating
counter-winding
co-rotating
counter-winding
counter-rotating
co-winding
co-rotating
co-winding
co-rotating
co-winding
(a)
S1
cph/V
m= 0
m= 5
m= 10
m= 15
ωrδx/V
(c)
counter-rotating
counter-winding
co-rotating
counter-winding
counter-rotating
co-winding
co-rotating
co-winding
S3
ωrδx/V
-0.2 -0.1 0 0.1 0.2-0.2 -0.1 0 0.1 0.2
-0.5
0
0.5
1
1.5
0
0.02
0.04
0.06
0.08
0.1
0.12
-0.4
-0.2
0
0.2
0.4
Figure 5.7: The streamwise wavenumber αr(a), the spatial growth rate −αi(b), and
the axial phase velocity cph =ωr/αr(c) versus the frequency ωrof the shear mode; only
unstable modes are shown. Quantities are derived from linear stability analysis of the
mean flow at (x/D = 0) for the non-swirling jet (S1) and the swirling jet (S3).
Chapter 5 Instabilities in the Moderately Swirling Jet 73
et al. 1995;Ho & Gutmark 1987). Moreover, the constancy of cph for varying menables
nonlinear interactions between waves at different azimuthal mode numbers, which may lead
to a drastic distortion of the mean flow (Long & Petersen 1992).
These mechanisms are not as rigorously applied to swirling jets. As shown on the
right side of figure 5.7c, only the modes with m < 2 may resonate with their subharmonics.
For higher azimuthal modes, the phase velocity at the neutral frequency differs significantly
from that at the subharmonic. Moreover, the dependency of cph on minhibits an interaction
between two azimuthal modes at the same frequency. A subharmonic resonance between
two different azimuthal modes is principally possible. For instance, the m= 1 mode at
ωr= 0.2 could resonate with the subharmonic ωr= 0.1 of the m= 0 mode. However, at
these frequencies, the phase velocity of the two modes depend differently on ωrδx/V and,
thus, on x. This implies that the resonance conditions for two modes are only fulfilled at a
single streamwise location and it is, thus, very unlikely to find a mode pair that interacts
within a sufficiently large streamwise domain. Nonetheless, the axisymmetric mode should
be the best candidate for a subharmonic resonance as its streamwise phase velocity is not
effected by the swirl.
The dependence of cph and αron mis directly related to the swirl component. In a
swirling jet, a disturbance initiated at the nozzle lip is convected in axial and azimuthal
direction. Hence, the group velocity of a traveling wave has a non-zero axial and azimuthal
component. This allows to define a group rotation rate Ωgr = (∂ωr/∂m)αr=const.in addition
to the commonly known axial group velocity cgr = (∂ωr/∂αr)m=const.. Ωgr represents the
slope of the αrδxcontour lines shown in figure 5.6. Accordingly, it increases successively
with increasing swirl for all unstable modes. A positive group rotation rate implies that the
streamwise wavenumber must decrease with increasing azimuthal wavenumber at constant
ωr. Moreover, the non-zero Ωgr is related to the appearance of unstable streamwise modes
(αr= 0) for the swirling jet. For these modes, the phase rotation rate Ωph =ωr/m is
equal to the group rotation rate Ωgr. For the non-swirling jet, a disturbance travels only
in axial direction (Ωgr ≈0) and the group velocity is solely given by cgr that is, assuming
non-dispersiveness, equal to cph. The non-zero Ωgr for swirling jets is supported by the
experimental investigation of the impulse response that will be discussed in section 5.6.
The radial amplitude distribution of the most unstable modes, given by their eigenfunc-
tions, are shown in figure 5.8 for the non-swirling (S1) and swirling jet (S3). The eigenvector
X= (H, iF, G, P) is normalized with respect to the Euclidean norm kXk=p(X, X). The
radial coordinate is normalized using the jet half width R.5and the momentum thickness
δx. Hence, the zero coordinate corresponds to the center of the axial shear layer. The
eigenfunctions are not significantly altered by the swirl. All velocity components peak ap-
proximately at the center of the axial shear layer (r=R.5), indicating the importance of
the axial shear layer for this type of instability. It is interesting to note that the azimuthal
component of the axisymmetric mode m= 0 is not zero for the swirling flow in contrast to
the non-swirling flow.
Summarizing this section, the most unstable modes at the nozzle exit are driven by a
shear instability, with the overall maximum spatial growth rate determined by the shape
of the axial velocity profile. The selection of the most amplified axial wavenumbers αmax
r
depends only weakly on Sand m, as shown in figure 5.6 and, more explicitly, on the left
side of figure 5.9. However, the corresponding frequencies ωmax
rdepend strongly on Sand
74 Chapter 5 Instabilities in the Moderately Swirling Jet
S3
(r−R.5)/δx
|vc
x|
(a)
S1
m=−15
m=−10
m=−5
m= 0
m= 5
|vc
r|
(b)
|vc
θ|
(c)
(r−R.5)/δx
|pc|
(d)
-8 -6 -4 -2 0 2 4 6 8-8 -6 -4 -2 0 2 4 6 8
0
0.02
0.04
0.06
0.08
0.1
0
0.05
0.1
0.15
0.2
0
0.05
0.1
0.15
0
0.1
0.2
0.3
0.4
Figure 5.8: Amplitude distribution of the shear mode at the nozzle exit (x/D = 0) for
the non-swirling jet S1(left column) and the swirling jet S3(right column) calculated for
frequencies at maximum spatial amplification ωmax
r. Rows refer to the axial (a), radial (b),
and azimuthal (c) velocity component and the coherent pressure (d).
Chapter 5 Instabilities in the Moderately Swirling Jet 75
m
αmax
rδx
S1
S2
S3
S4
m
ωmax
rδx/V
-10 -5 0 5 10-10 -5 0 5 10 0
0.05
0.1
0.15
0.2
0
0.1
0.2
Figure 5.9: Axial wavenumber αmax
r(left) and frequency ωmax
r(right) at maximum
growth rate −αifor different azimuthal wavenumbers and swirl intensities at the nozzle
exit (x/D = 0)
mdue to the non-zero azimuthal group rotation rate Ωgr caused by the swirling motion
of the mean flow. This dependency is a kinematic effect, as stated by Martin & Meiburg
(1994), and can be illustrated by considering inclined waves on a infinite long cylinder
as a model for instability waves traveling along a swirling jet. Consider first waves fixed
to the rotating cylinder. For a given wavenumber (here αmax
r), the axial phase velocity
depends on the rotation rate of the cylinder (here Ωgr or S) and the inclination of the
waves (m/αmax
r), yielding cph ∝Ωgrm/αmax
r. Now, consider a non-rotating cylinder that
moves in axial direction at constant velocity (here V/2). The phase velocity is then given
by cph =V/2, independent of the inclination. Superimposing these two cases results in
the relation cph ∝V/2 + Ωgrm/αmax
rand with cph =ωr/αmax
rwe get the proportionality
ωr∝V/2αmax
r+ Ωgrm, which describes the influence of swirl on ωrand mof the most
unstable wavenumber αmax
r.
5.3.2 Centrifugal Instability
The radial stratification of angular momentum can lead to centrifugal instability in swirling
jets. Rayleigh’s well known inviscid criterion provides a necessary and sufficient condition
for axisymmetric disturbances to be centrifugally unstable. For steady ambient fluid and
the circulation Γ being positive, the stratification of angular momentum is unstable if Γ
decays monotonically in radial direction, yielding
dΓ/dr < 0.
In the present investigation, the azimuthal shear layer is indeed centrifugally unstable as
shown in figure 5.10. The flow at the nozzle exit should, ideally, reveal a dimensionless
circulation that is parabolic in radial direction with a sharp peak at r=D/2 with value
unity that drops thereafter to zero (solid body rotation). In practice, however, a boundary
layer generated by the nozzle wall causes a smooth drop of the azimuthal velocity component
in radial direction. The radial extend of this initial azimuthal shear layer, which is Rmax <
76 Chapter 5 Instabilities in the Moderately Swirling Jet
r/Rmax
Γ/2πΩcl/R2
max
S2
S3
S4
S2
S3
S4
S2
S3
S4
S2
S3
S4
0 0.5 1 1.5 2
0
0.5
1
Figure 5.10: Radial stratification of angular momentum Γ
r < D/2, depends on the swirl intensity. Hence, Γ is nearly parabolic for r < Rmax and
drops off smoothly in outward direction for r > Rmax.
In contrast to the shear mode, a centrifugal mode is expected to primarily depend on
the azimuthal profile, and indeed, a detailed inspection of the dispersion relation reveals a
second unstable mode that exists only for the swirling jets. The corresponding contours of
−αiδxin the ωr-m-plane are shown in figure 5.11 for different swirl intensities. Its overall
maximum growth rate increases with increasing swirl, but it gains no more than a third
of the maximum growth rate of the shear mode for the swirl intensities considered. For
the weakly swirling jet S2, centrifugal modes with m=−7 and ωrδx/V ≈0.3 are most
unstable. For stronger swirl (S3and S4), modes with m=−2 and ωrδx/V ≈0.5 are most
amplified. The cutoff azimuthal wavenumber for S3is m= 28 for negative ωrand m= 4
for positive ωr, revealing a strong tendency of this instability to counter-rotating modes.
The streamwise wavenumber αr, indicated as black contour lines in figure 5.17, is much
larger in regions of maximum amplification in comparison to the shear instability. The
centrifugally unstable azimuthal shear layer selects short axial and azimuthal wavelengths
at high frequencies, a typical characteristic of centrifugal instability.
Figure 5.12 displays profiles of streamwise growth rate, axial wavenumber, and axial
phase velocity of the centrifugal instability modes. The magnitude of αrincreases linearly
with increasing frequency, yielding an axial group velocity cgr ≈1. The axial wavenum-
ber decays with increasing msimilar to the shear instability. However, neither co-rotating
co-winding nor counter-rotating counter-winding modes are unstable. Moreover, this type
of instability does not promote steady or streamwise modes. The axisymmetric mode
(m= 0) is unique, as its axial phase velocity is equal unity for all unstable frequencies
(figure 5.12c), revealing that the centrifugal instability promotes non-dispersive axisym-
metric waves that travel in downstream direction with the same speed as the bulk velocity.
All non-axisymmetric modes asymptote cph =Vfor higher ωr, but they remain dispersive
within their unstable frequency band.
As mentioned earlier, the shear modes of the non-swirling jet are non-dispersive for
higher frequencies with cgr ≈cph ≈0.5V(figure 5.7c) meaning that the individual waves
and the entire wave packet travel in axial direction with approximately half the jet velocity.
This is consistent with the amplitudes of the shear modes that peak at R.5(see figure 5.8),
Chapter 5 Instabilities in the Moderately Swirling Jet 77
S1
ωrδx/V
S2
-0.6
-0.5
-0.4
-0.3
0.2
0.3
0.4
0.5
0.6
S3
-1.2
-1
-0.8
-0.6
-0.4
0.2
0.4
0.6
0.8
1
1.2
ωrδx/V
m
S4
-1
-0.8
-0.6
-0.4
0.4
0.6
0.8
1
1.2
m
-30 -20 -10 0 10 20-30 -20 -10 0 10 20
0
0.005
0.01
0.015
0.02
0.025
0.03
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 5.11: Instability of centrifugal mode at x/D = 0 for the non-swirling jet (S1) and
the swirling jets (S2-S4). Same plot style as in figure 5.6.
the radial location where Vx= 0.5V. In contrast, the amplitudes of the centrifugal modes do
not peak at R.5as they are generated in the azimuthal shear layer with its center located at
the vortex core radius Rv. This length scale is derived from the mean flow model parameters
using Rv=Rmax(1/Nθ)−1/Nθ, with Nθbeing the shape parameter of the azimuthal shear
layer (see section 5.2 for details). It corresponds to the center of the azimuthal shear
layer (Gallaire & Chomaz 2003). Eigenfunctions of the centrifugal instability reveal that
the magnitude of vc
r,vc
x, and vc
θof the most energetic modes (3 < m < 6) are indeed
located around Rv(figure 5.13). The modes at higher negative azimuthal wavenumbers
(m < −10) are less unstable and their peaks are shifted towards the jet centerline. The
vortex core radius Rvis located closer to the jet axis than the center of the axial shear layer
(Rv/R.5= 0.91). Hence, the centrifugal modes are located inside the potential core and
are convected downstream with the bulk velocity V. This explains the axial group velocity
of cgr ≈V.
78 Chapter 5 Instabilities in the Moderately Swirling Jet
−αiδx
(b)
αrδx
counter-rotating
counter-winding
co-rotating
counter-winding
counter-rotating
co-winding
co-rotating
co-winding
co-rotating
co-winding
(a)
cph/V
ωrδx/V
m= 0
m= 3
m= 8
m= 15
m= 20
(c)
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.5
1
1.5
0
0.01
0.02
-1.5
-1
-0.5
0
0.5
1
1.5
Figure 5.12: The streamwise wavenumber αr(a), the spatial growth rate −αi(b), and
the axial phase velocity cph =ωr/αr(c) versus the frequency ωrof the centrifugal mode;
only unstable modes are shown. Quantities are derived from linear stability analysis of the
mean flow at (x/D = 0) for the swirling jet (S3).
Chapter 5 Instabilities in the Moderately Swirling Jet 79
|vc
x|
Rv/R.5= 0.91
(a)
|vc
r|
m=−20
m=−15
m=−8
m=−3
m= 0
(b)
|vc
θ|
(c)
(r−Rv)/δx
|pc|
(d)
-10 -5 0 5 10
0
0.002
0.004
0.006
0.008
0.01
0
0.05
0.1
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
0
0.05
Figure 5.13: Amplitude distribution of the centrifugal mode at the nozzle exit (x/D = 0)
for the swirling jet S3calculated for frequencies at maximum spatial amplification ωmax
r.
Rows refer to the axial (a), radial (b), and azimuthal (c) velocity component and the
coherent pressure (d).
80 Chapter 5 Instabilities in the Moderately Swirling Jet
In conclusion, a second, less unstable, mode is found that is of centrifugal type. It
promotes primarily counter-rotating co-winding modes at short axial and azimuthal wave-
lengths at much higher frequencies than the shear instability. The overall maximum spatial
growth rate increases with increasing swirl. Waves are most energetic in the azimuthal
shear layer that is mainly located inside the potential core region. This causes the modes
to travel downstream with the bulk velocity.
5.4 Streamwise Evolution of Instability
The stability analysis of the mean velocity profiles at the nozzle exit reveal the coexistence
of a shear instability and a centrifugal instability, with the previous being more unstable.
Both instabilities promote modes at a wide range of m,ωr, and αr. As a next step, the
downstream development of these instabilities is investigated. Due to the divergence of
the jet, the mean flow and corresponding stability characteristics change significantly in
streamwise direction, which implies that certain modes stabilize with downstream distance
and die away, while other modes are continuously amplified. To obtain the overall growth of
the instability modes, it is necessary to perform the stability analysis at various streamwise
locations following the quasi-parallel approach introduced in section 2.4. It is understood
that the quasi-parallel assumption is responsible for an increase in amplification rates of the
linear modes and may not accurately predict second order, nonlinear interactions among
modes that may resonate with one another to dominate the large structure observed. The
complex analysis of such interactions is mostly beyond the scope of this thesis. In line with
the previous section, we will first consider the shear instability followed by a discussion of
the centrifugal instability.
5.4.1 Shear Instability
The Kelvin–Helmholtz instability in axisymmetric non-swirling jets have been investigated
by numerous researchers. It will be considered as a benchmark to validate the present
numerical results and the mean flow on which the analysis is based on. At the nozzle exit,
where the shear layer is thin, the non-swirling jet is known to be unstable to various axial
and azimuthal wavenumbers and frequencies (e.g., see Cohen & Wygnanski 1987). With
increasing distance from the nozzle, the number of unstable azimuthal modes decreases
successively. At the end of the potential core, only the bending modes with m=±1
remain unstable.
The present investigation shows very similar results. Figure 5.14 displays the contours
of spatial growth rate in the ωr-x-plane for 0 ≤m≤3. Recall that in the absence of
swirl, co-rotating counter-winding modes have equal growth rates and wavenumbers as the
counter-rotating co-winding modes and hence, we can restrict their presentation, as for
instance in figure 5.14, to positive mand ωrwithout loss of generality. The axisymmetric
mode, which is most unstable at the nozzle exit, becomes stable at x/D ≈3. All non-
axisymmetric modes except for m=±1 stabilize in downstream direction with their neutral
point located closer to the nozzle for higher |m|. The frequency and wavelength of the most
unstable mode remains approximately constant with x, indicating the appropriate choice
of the velocity and length scale.
Chapter 5 Instabilities in the Moderately Swirling Jet 81
m= 0
0.1
0.2
0.3
0.4
ωrδx/V
m= 1
0.1 0.1
0.2 0.2
0.3
0.4
ωrδx/V
m= 2
0.1
0.2
0.3
0.4
ωrδx/V
m= 3
0.1
0.2
0.3
0.4
ωrδx/V
x/D
0 1 2 3 4
0
0.02
0.04
0.06
0.08
0.1
0
0.1
0.2
0.3
0
0.1
0.2
0.3
0
0.1
0.2
0.3
0
0.1
0.2
0.3
Figure 5.14: Streamwise evolution of spatial growth rate of the shear mode for the non-
swirling jet S1for the azimuthal wavenumbers m= (0,1,2,3) (from top to bottom). Same
plot style as in figure 5.6.
82 Chapter 5 Instabilities in the Moderately Swirling Jet
m= 0
m= 1
m= 2
m= 3
−αmax
iδx
(a)
ωmax
rδx/V
x/D
(b)
0 1 2 3 4
0.06
0.08
0.1
0.12
0.14
0.16
0
0.02
0.04
0.06
0.08
0.1
0.12
Figure 5.15: Streamwise development of the maximum growth rate −αmax
iδx(a) and
corresponding frequency ωmax
r(b) of the shear modes with m= (0,1,2,3) for the non-
swirling jet S1.
The maximum growth rates αmax
iand the corresponding frequencies ωmax
rare shown
explicitly in figure 5.15. It can be directly compared to the results from the inviscid analysis
conducted by Cohen & Wygnanski (1987). Close to the nozzle exit, the agreement is
good. However, the viscous analysis presented here predicts a faster downstream decay of
−αmax
i. In the present analysis, the axisymmetric modes (m= 0) become neutrally stable
at x/D ≈3.3, while the inviscid analysis of Cohen & Wygnanski (1987) predicts x/D ≈4.5.
Perhaps more importantly, the maximum growth rate of the bending modes with m=±1
presented here asymptotes to 0.22, which is less than half of the value derived by Cohen &
Wygnanski (1987).
The discrepancy between the viscous and the inviscid analysis is due to the eddy viscos-
ity model used in the present investigation. Near the nozzle exit, the turbulent viscosity
νtis very small and it has only a marginal effect on the growth of instability. With larger
distance from the nozzle, the turbulent fluctuations increase, resulting in a rapid increase
of νt. This fictitious viscosity drastically dampens the growth rate of the instability modes.
Consequently, the inviscid analysis overestimates the growth rate that is observed in ex-
periments, as shown by Cohen & Wygnanski (1987), while the present approach yields a
theoretical prediction that agrees favorably with the experimental results (see section 5.5).
The streamwise distribution of αiof the swirling jet configurations S2and S3is shown
in figure 5.16 for 0 ≤m≤3. By introducing swirl to the flow, the symmetry breaks and
Chapter 5 Instabilities in the Moderately Swirling Jet 83
co-rotating modes undergo different amplification cycles than counter-rotating modes and
hence, contours of −αiδxmust be derived for positive and negative frequencies, separately.
During the ensuing discussion of the divergent swirling, it is more meaningful to classify
the instability modes by their sense of winding than by their direction of rotation. In this
spirit, the contour line αr= 0 in figure 5.16 that refers to streamwise modes is emphasized
as it separates the regime of co-winding modes from the regime of counter-winding modes.
It is located in a ‘valley’ between regimes of high amplification and thus, the streamwise
modes are always less unstable than the most unstable winding modes. This contour line
must not necessarily coincide with ωr= 0, as it is the case for the non-swirling jet (confer
with figure 5.14). As can be seen, for instance, in figure 5.16c, co-rotating streamwise
modes, depicted by the αr= 0 contour line, are unstable in the swirling jet S2and S3. The
rotation rate of these modes is related to the base flow rotation and thus, their frequencies
must be larger than zero and increase with swirl.
The analysis of the streamwise evolving flow reveals a different mode selection than one
would expect from the analysis of the mean flow at the nozzle exit. The streamwise decay of
the growth rates of the co-winding modes, corresponding to regions of αr<0 in the figures
5.16b-d, is reduced with increasing swirl. In contrast, the streamwise decay of the growth
rates of the counter-winding modes (αr>0 in figures 5.16b-d) and the axisymmetric modes
(figure 5.16a) is enhanced with increasing swirl although these modes are most unstable at
the nozzle exit (confer with figure 5.6). Hence, the modes that are most amplified at the
nozzle ext, which are presently the counter-winding modes, are not necessarily the modes
which undergo the strongest overall amplification.
For the swirling jets S2and S3, the streamwise wavenumbers of the most amplified modes
are approximately constant in axial direction at αrδx≈0.2, independently of the azimuthal
wave number (figure 5.16). This agrees with the non-swirling jet (figure 5.14). However,
the most amplified frequencies are not constant in axial direction unlike the non-swirling
jet. For the modes with m > 1 (figure 5.16c-d), the frequency at maximum amplification
increases in downstream direction. This results in a destabilization of co-winding steady
modes (ωr= 0, αr<0) at streamwise wavenumbers that are similar to the most amplified
ones in the non-swirling jet. At sufficient strong swirl (right column in figure 5.16), the
αr=−0.2 contour line, depicting roughly to the most amplified modes, asymptotes the
ωr= 0 line. This implies that, with increasing downstream distance from the nozzle, the
rotation rate of the most amplified co-winding counter-rotating modes decrease continuously
until they stand still (ωr= 0).
The ability to promote steady modes at considerable high amplification rates is a unique
feature of swirling jets. The tendency of the co-winding modes to lower rotation rates
(frequencies) appears plausible when considering the proportionality ωr∝V/2αmax
r+Ωgrm,
derived in section 5.3.1. It implies that with increasing swirl (increasing Ωgr), the frequency
ωrof the most unstable counter-rotating co-winding mode (m > 0, ωr<0, αmax
r<0) must
increase in order to maintain constant wavenumber. In the same train of thought, steady
modes become most unstable when V/2αmax
r=−Ωgrm, which seems to be fulfilled at
certain axial locations for m > 1.
Furthermore, figure 5.16 reveals that all modes become successively more stable with
increasing downstream distance from the nozzle, except for the co-winding modes with
84 Chapter 5 Instabilities in the Moderately Swirling Jet
m= 3
-0.5
-0.4
-0.3
-0.2
0
x/D
m= 2
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
m= 1
-0.4
-0.3
-0.2
-0.1
00
0.1
0.2
0.3
m= 0
-0.3
-0.2
0
0.1
0.2
0.3
S3
m= 3
-0.3
-0.2
-0.1
0
(d)
ωrδx/V
x/D
m= 2
-0.3
-0.2
-0.1
0
0.1
0.2
(c)
ωrδx/V
m= 1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
(b)
ωrδx/V
m= 0
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
(a)
ωrδx/V
S2
0 1 2 3 40 1 2 3 4
0
0.02
0.04
0.06
0.08
0.1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Figure 5.16: Streamwise evolution of spatial growth rate −αiof the shear mode at
azimuthal wavenumbers m= (0,1,2,3) for the swirling jets S2(left column) and S3(right
column). Same plot style as in figure 5.6.
Chapter 5 Instabilities in the Moderately Swirling Jet 85
m= 1. Computations based on a larger domain could reveal that these modes remain the
only unstable ones within 0 < x/D < 8 (not shown). Interestingly, their spatial growth
rate asymptote to the same value for all swirling jets considered, which is twice as high as
for the non-swirling jet. It appears that the swirl-induced destabilization of the co-winding
modes and the stabilization of counter-winding modes is maintained for a wide streamwise
distance, although the swirl component decays very rapidly in downstream direction (see
figure 5.4). The collapse of the growth rates downstream the potential core is presumably
linked to the formation of an universal swirl velocity profile, as indicated by the collapse of
the Rosby number, Nθ, and Vθ,max (see figure 5.4).
Hence, the swirling jet selects a counter-rotating co-winding m= 1 mode to be the
remaining unstable mode at sufficient downstream distance. This mode is expected to
dominate the farfield dynamics. The nearfield is dominated by various co-winding modes
with the azimuthal wave number m= 2 to be most amplified. The frequencies of the most
amplified modes decay in downstream direction due to the rotational motion of the mean
flow, which results in a significant destabilization of steady modes.
5.4.2 Centrifugal Instability
The downstream development of the centrifugal instability is shown in figure 5.17 for S3.
The analysis at the nozzle exit has revealed that the centrifugal instability promotes modes
at high negative and low positive azimuthal wave numbers (figure 5.11). In order to track
their downstream development, the analysis is conducted at various streamwise locations
for positive ωrand azimuthal mode numbers m= (−10,−5,−2,−1,0,1,2). The results
are shown in figure 5.17. Accordingly, the modes driven by the centrifugal instability decay
with downstream direction at a much faster rate than the shear mode, and they all become
neutrally stable upstream of x/D = 1. The rapid downstream decay of the growth rate
is presumably caused by the decreasing radial gradient of angular momentum Γ due to
the continuous decay of the maximum azimuthal velocity Vθ,max that goes in hand with a
thickening of the azimuthal shear layer (see model parameters in figure 5.4 in section 5.2).
Hence, the centrifugal instabilities are restricted to a small spatial region near the nozzle lip
and, therefore, are not expected to have significant influence on the streamwise evolution
of large-scale flow structures.
5.5 Single-mode Actuation
In a first experiment, the jet at different swirl intensities was forced at the azimuthal
mode m= 1. The disturbance field of the excited wave was measured with hot-wires and
compared to the theoretical predictions discussed in the previous sections. Good agreement
will bring credibility to the stipulations that are made for the presented stability analysis.
Theses assumptions are summarized here again:
•The instabilities of the mean turbulent flow represent the coherent structures.
•Small-scale turbulence is well modeled by the eddy viscosity.
•The axial overshoot is negligible.
•The quasi-parallel approximation is sufficiently accurate.
86 Chapter 5 Instabilities in the Moderately Swirling Jet
m=−10
1
ωrδx/V
m=−5
1
ωrδx/V
m=−2
0.4
0.6
0.8
1
ωrδx/V
m=−1
0.4
0.6
1
ωrδx/V
m= 0
1
ωrδx/V
m= 1
1
ωrδx/V
m= 2
1
ωrδx/V
x/D
0 0.5 1 1.5 2
0
0.01
0.02
0.03
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
Figure 5.17: Streamwise evolution of spatial growth rate −αiof the centrifugal mode
at azimuthal wavenumbers m= (−10,−5,−2,−1,012) for the swirling jet S3. Same plot
style as in figure 5.6.
Chapter 5 Instabilities in the Moderately Swirling Jet 87
In the second experiment, the swirling jet configuration S3was forced at different az-
imuthal wavenumbers −2≤m≤2 at a frequency of 100 Hz. This experiment is aimed
to validate the dependency of the phase velocity on the azimuthal mode number for the
swirling jet.
5.5.1 Streamwise Growth of the Co-rotating Single-helical Mode
Hot-wire measurements were conducted at several axial and radial positions within the
range 0 < x/D < 1.2 and 0 < r/D < 0.8. The single-helical m= 1 mode was actuated at
the frequencies 150 Hz, 100 Hz, and 80 Hz for the swirl numbers S1,S3, and S4, respectively.
This corresponds to dimensionless frequencies ranging from ωrδx/V ≈0.07 at the nozzle
lip to ωrδx/V ≈0.4 at x/D = 1.2. The frequencies were selected as such that the actuated
modes go through their entire amplification cycle within the measurement domain. The
frequencies do not correspond to the maximum overall amplification.
The choice of the actuation amplitude is not trivial as it has to be large enough to
overcome the random turbulent noise in the shear layers. On the other hand, it must
be sufficiently low to provide linear amplification. Moreover, forcing the flow at too high
amplitudes would alter the mean flow substantially and thereby falsify the theoretical pre-
dictions that are based on the non-forced flow. The most appropriate amplitude for each
flow configuration was derived through a preliminary parameter study. Therefore, the flow
was forced at various amplitudes and a trade-off between good signal-to-noise level and a
clear linear response was found.
As outlined in section 3.4.1, the flow excited at a single mode is decomposed into a
mean, a coherent, and a turbulent part. The coherent part is derived from phase-averaged
hot-wire measurements and is Fourier decomposed in time and azimuthal direction. The
obtained amplitude distribution of the axial component |ˆvx|can be compared to the sta-
bility eigenfunctions. Figures 5.18 shows the amplitude of the co-rotating counter-winding
traveling m= 1 mode at selected axial locations. The black lines refer to the theoretical
predictions derived from spatial linear stability analysis. Symbols are derived from phase-
locked measurements. The amplitudes are normalized with respect to the area below the
graph. Good agreement between theory and experiments are found in the unstable region
of the forced instability, which is approximately upstream of x/D = 0.75.
The radially integrated amplitude measure Am(x) is derived using equation (3.3). Fig-
ure 5.19 shows the streamwise amplitude distribution of the modes actuated at m= 1. The
amplitudes are shown in a logarithmic scale and are normalized by their initial value at
x/D = 0. Hence, the graphs represent an amplitude ratio of the radially integrated axial
velocity component. For all cases, the excited waves saturate at similar axial locations. The
waves in the non-swirling jet undergo an overall amplification that is an order of magnitude
higher than for the swirling jet.
The spatial growth rate of the axial velocity component can be derived from the slope
of the amplitude ratio displayed in figure 5.19. The black stars in figure 5.20 show its
streamwise development for the swirling jet (left) and non-swirling jet (right) normalized
by the local momentum thickness δx. For the non-swirling jet, neutral amplification of the
excited mode is reached at approximately x/D = 0.75. For the swirling jet, the excited
mode is neutrally stable at approximately x/D = 0.5.
88 Chapter 5 Instabilities in the Moderately Swirling Jet
r/D
normalized amplitude
x/D = 0.25
0 0.5 1
r/D
x/D = 0.5
0 0.5 1
r/D
x/D = 0.75
0 0.5 1
r/D
x/D = 1
0 0.5 1
Figure 5.18: Radial amplitude distribution of the axial velocity component of the m= 1
mode for the swirling jet configuration S3; markers correspond to measurements and lines
represent viscous linear theory
x/D
Am/Am(x= 0)
S1
S3
S4
0 0.5 1
100
101
Figure 5.19: Streamwise amplitude distribution of actuated co-winding m= 1 mode for
various swirl configurations. Actuation frequencies are 150 Hz, 100 Hz, and 80 Hz for the
swirl numbers S1,S3, and S4, respectively, to obtain neutral amplification at similar axial
locations.
Chapter 5 Instabilities in the Moderately Swirling Jet 89
x/D
d(ln(Am))
dx δx
S1
← −αiδx
x/D
S3
← −αiδx
0 0.25 0.5 0.75 10 0.25 0.5 0.75 1
-0.1
-0.05
0
0.05
0.1
Figure 5.20: Streamwise development of spatial growth rate of actuated co-winding
m= 1 mode for the non-swirling jet S1(left) and the swirling jet S3(right); the black
stars refer to measurements; the black thick line refers to the theoretical prediction using
equation (5.5) that incorporates the shear layer spreading; the black dots represent the
theoretical predicted eigenvalue αiδxthat does not corroborate shear layer spreading.
The growth rate of the axial velocity component can also be derived from the stabil-
ity analysis. As outlined in section 2.4, the amplitude distribution of the axial velocity
component of the downstream traveling wave is approximated by
|ˆvx(x, r)|=H(x, r)exp −Zx
0
αi(ξ)dξ,(5.4)
where His the eigenfunction of the axial velocity component, which depends parametrically
on x. In line with the experimental approach, the integrated amplitude of the axial velocity
component is then derived from
Am(x) = ZR.95
R.05 |ˆvx|2rdr1/2
.(5.5)
The actual growth rate of the theoretically derived Am(x) is shown as a black thick line in
figure 5.20. The streamwise development of the growth rate −αiδxthat corresponds to a
parallel flow is displayed in the same figure.
At the nozzle lip, the growth rate of Amis well represented by −αiδx. Further down-
stream, the growth rate of the parallel flow overestimates the actually growth rate of Am.
This is more pronounced for the swirling jet. The non-uniformity of the flow implies differ-
ent growth rates for different velocity components, and hence, −αiδxdoes not represent the
growth rate of the axial velocity component. However, the growth rates of the axial veloc-
ity component derived from the equations (5.4-5.5) agree well with the experimental data.
This indicates that the negligence of the amplitude scaling A0of the weakly non-parallel
correction does not significantly effect the accuracy of the predicted amplitudes.
One may conclude that the local quasi-parallel analysis approximates the streamwise
growth of a single-mode excited wave at a sufficient measure of accuracy. The growth rate
−αiδxof the parallel flow provides an approximate measure of the growth of instability, but
90 Chapter 5 Instabilities in the Moderately Swirling Jet
for an accurate comparison with experiments the streamwise varying eigenfunction must
be considered.
The theoretical results presented in figure 5.20 correspond to the shear instability. The
growth rates of the centrifugal instability do not match the experimental results at all (not
shown), which indicates that centrifugally unstable modes are not excited by the applied
actuation.
5.5.2 Streamwise Phase Velocity in the Swirling Jet
One important finding of the stability analysis of the swirling jet is the dependence of
the axial phase velocity and wavelength on the azimuthal wavenumber. This is validated
experimentally by actuating the swirling jet S3at various azimuthal wavenumbers. The
corresponding phase distribution ϕm(x, r) is derived from the phase-locked hot-wire mea-
surements using equation 3.4. The phase velocity for a given mis derived by
cph(x, r) = 2πfact
∂ϕm(x, r)/∂x.(5.6)
It is a function of the axial and radial coordinate. Figure 5.21 shows the phase distribution
of the co-rotating counter-winding traveling waves m= 1 at selected axial locations. The
agreement between measurment and theoretical prediction is reasonably well in the region
of amplification x/D < 0.7.
The phase cannot be integrated across the shear layer, as it is done for the amplitude
distribution. Hence, the streamwise development of the phase velocity of an excited wave
must be considered for each radial location separately. However, it is here assumed, that the
phase velocity in the center of the shear layer r=R.5represents approximately the average
phase velocity of the entire forced coherent structure. This approximation is supported by
measurements in the forced mixing layer (Gaster et al. 1985).
Figure 5.22 shows the measured phase delay of modes actuated at −2≤m≤2 derived
at the center of the axial shear layer R.5for the swirling jet S3. All modes are excited
at the same frequency. Accordingly, for lower mthe phase increases more rapidly with
downstream distance. At x/D = 0.7, the m=−2 mode has gone through more than one
period while the m= 2 mode has gone only through somewhat more than half a period.
The phase velocity of the excited modes is derived from the slope of the phase delay
shown in figure 5.22. Results are displayed in figure 5.23. The maximum cph is found at
the nozzle for all modes considered. cph decreases rapidly with downstream distance to a
local minimum that is followed by a slight increase in the region of neutral amplification
x/D ≈0.7. As already indicated by the phase delay, modes at lower mhave lower axial
phase velocities than modes with higher m. This confirms the trend predicted by the
stability analysis. As discussed in section 5.3, the mean flow rotation results in a dependency
of the streamwise phase velocity on the azimuthal wavenumber. The analysis confirms that
co-rotating waves at higher mtravel at a higher streamwise phase velocity.
A quantitative comparison between the measured phase velocity distribution cph(x, r)
with the results from the stability analysis is impossible within the present approach. The
phase of the eigenfunctions computed at different streamwise locations cannot be aligned
with streamwise distance in a consistent way. A quantitative comparison of the phase
Chapter 5 Instabilities in the Moderately Swirling Jet 91
phase
r/D
x/D = 0.25
0.25
0.5
phase
x/D = 0.5
phase
x/D = 0.75
phase
x/D = 1
Figure 5.21: Phase distribution of mode m= 1 for the swirling jet at S=0.4, markers
correspond to measurements, lines represent viscous linear theory
x/D
ϕm(x, R.5)−ϕm(0, R.5)
m=−2
m=−1
m= 0
m= 1
m= 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0
π
2π
Figure 5.22: Streamwise evolution of the phase delay ϕm(x, R.5)−ϕm(0, R.5) for modes
excited at various azimuthal wavenumbers mfor the swirling jet S3. Values are taken in
the center of the axial shear layer R.5.
92 Chapter 5 Instabilities in the Moderately Swirling Jet
x/D
cph(x, R.5)/V
m=−2
m=−1
m= 0
m= 1
m= 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.5
1
1.5
2
Figure 5.23: Streamwise evolution of the phase velocity cph for modes excited at various
azimuthal wavenumbers for the swirling jet S3. Values are taken in the center of the axial
shear layer R.5.
velocity, would require to solve a second differential equation describing the streamwise
development of the eigenfunctions. Nonetheless, the experiments confirm the theoretical
prediction, revealing that, for swirling jets, the axial phase velocity depends on the az-
imuthal wave number m.
5.6 Impulse Response
In the second experimental approach the response of the shear layer to a single pulse
is investigated experimentally. A time- and space-discrete disturbance is created at the
nozzle lip using one of the eight loudspeakers that are mounted circumferentially around
the nozzle lip. The downstream development of the excited wave packet is captured via
ensemble-averaged hot-wire measurements. Details of the data acquisition and treatment
are given in section 3.4.2. In contrast to the single-mode actuation discussed in the previous
section, the delta pulse excites an infinite number of modes, which are either amplified or
damped, depending on the stability of the mean flow. At sufficient distance from the nozzle,
all stable modes should have died out and the wave packet is expected to consist solely of
unstable modes. The modal content of the wave packet is compared to the results from
the stability analysis for various streamwise locations. The aim of this experiment is to
validate the mode selection predicted from the local stability analysis.
The downstream evolution of a traveling wave packet is modeled accurately by linear
stability analysis provided that the unstable modes do not interact with each other. In
order to force all modes at equal amplitude, the initial disturbance should ideally be a delta
function in time and space. However, in the present experiment, the pulse is generated by
one loudspeaker driven at a saw-tooth signal, which generates a pulsed jet with an azimuthal
extent of ∆θ=π/4 and an axial extent of x/D = 0.02. This differs significantly from a
delta function in space. Forcing artifacts are introduced that influence the development
of the wave packet between the nozzle lip and the next downstream measurement position
Chapter 5 Instabilities in the Moderately Swirling Jet 93
(x/D = 0.25). Further downstream, the influence on the particular forcing characteristic
becomes insignificant and the wave packet evolution is in line with the linear prediction, as
will be shown later.
In the following, the development of the wave packet envelope will be described first,
revealing the shape, the location and the velocity of the wave packet. Thereafter, the modal
content of the wave packet is discussed and compared to the spatial stability analysis.
Results are shown for the non-swirling jet S1and the swirling jet S3.
5.6.1 Trajectory of the Wave Packet Envelope
In general, the Fourier coefficients of a signal vjsampled at j= 1...N time points is given
by
ˆvn=1
N/2
N
X
j=0
v(tj) exp −iπnj
N/2(5.7)
and the envelope of the signal is given by
ve(tj) =
N/2
X
n=0
ˆvnexp iπnj
N/2.(5.8)
The measured signal of the wave packet is represented by the coherent axial velocity com-
ponent vc
x(x, t). The experimental arrangement allows for a very good temporal reso-
lution, a reasonably good spatial resolution in radial direction, and, due to the specific
ensemble averaging procedure, a good spatial resolution in azimuthal direction. The res-
olution in axial direction is poor, as data are acquired only for the streamwise locations
x/D = (0.02,0.25,0.6,1,1.5,2,2.5,3). Hence, the signal of the wave packet is Fourier de-
composed in r-, θ-, and t-direction for each streamwise position, separately, and the corre-
sponding envelope ve
x(x, t) is derived from the inverse 3-dimensional Fourier transformation,
in accordance with equation (5.8). In consistency with the single-mode investigation, the
radial dependence of the envelope is omitted by integrating the envelope across the axial
shear layer, yielding the following expression for the envelope amplitude
Ae(x, θ, t) = ZR.95
R.05
(ve
x)2rdr1/2
.(5.9)
Figure 5.24 shows the trajectory of the wave packet envelope in a 3D-plot for the non-
swirling jet S1and the swirling jet S3. 2D-contours are displayed for each streamwise
measurement location, showing the envelope amplitude distribution along the θ-t-plane.
The time is made dimensionless using the half bulk velocity V/2, which is approximately
the streamwise convection velocity of the disturbance. At each axial location, Aeis normal-
ized by its maximum value Ae
max within the corresponding θ-t-plane. This normalization
facilitates to compare the wave packet envelope between different crosswise measurement
planes. Contour surfaces with Ae/Ae
max <0.5 are blanked and the lowest contour line
Ae/Ae
max = 0.5 is selected arbitrarily as the characteristic outer bound of the wave packet.
The corresponding coordinates of the leading and trailing edge of the wave packet are pro-
jected on the axis planes (see dotted lines at x/D = 1.5). They are marked by white-filled
black circles in the x-t-axis-plane and by gray-filled black circles in the x-θ-axis-plane. The
94 Chapter 5 Instabilities in the Moderately Swirling Jet
center of the wave packet is associated with Ae/Ae
max = 1. Its coordinate is marked on the
axis planes by big black dots. The streamwise development of Ae
max are shown in the small
images placed near the bottom-right corner of the figures 5.24a and b.
To get familiar with this rather complex plotting style the non-swirling jet is discussed
first (figure 5.24a). The pulse is initiated at the nozzle exit at t= 0 and −π/8≤θ≤π/8.
This creates a wave packet that peaks at θ= 0, with an azimuthal extend of ∆θπ/4
(see marker in the θ-x-axis-plane at x/D = 0.02). The leading edge arrives at the first
measurement position (x/D = 0.02) only shortly after t= 0 followed by the maximum and
the trailing edge (see marker in the x-t-axis-plane at x/D = 0.02). While traveling to the
next downstream measurement point (x/D = 0.25) the wave packet spreads significantly in
azimuthal direction and in time and Ae
max decays slightly. The significant deformation of the
wave packet near the nozzle is presumable caused by the imperfect forcing. However, with
further downstream distance, amplification sets in and the wave packet maximum grows
continuously up to a downstream distance of x/D = 2.5 (see small image in figure 5.24a).
Within this region of amplification, the envelope maintains its shape remarkably well despite
the streamwise variation of the mean flow. This confirms the weak dispersiveness of the
shear layer of the non-swirling jet as predicted from the stability analysis. The wave packet
maximum is found to propagate in axial direction at a velocity of approximately 0.7V
near the nozzle (0 < x/D < 0.6) and at 0.54Vfor x/D > 0.6. The lower value, which
corresponds to the amplifying region, compares well with the phase-velocity derived from
the linear theory of 0.5V(confer with figure 5.7c).
Figure 5.24b shows the trajectory of the wave packet for the swirling jet S3. The
envelope at x/D = 0.02 is very similar to the non-swirling jet, showing equal shape and
maximum amplitude Ae
max. The abrupt spreading of the pulse upstream of x/D < 0.25 is
also observed for the swirling jet, although the widening of the envelope in time direction
is less pronounced compared to the non-swirling jet. The streamwise evolution of Ae
max
does not indicate a region near the nozzle lip where the wave packet decays. In fact, the
wave packet is amplified already at the nozzle exit and gains amplitude up to a streamwise
distance of x/D = 2.5. For the swirling jet, the irregularities introduced by the imperfect
forcing seem less significant than for the non-swirling jet and linear amplification set in at
a shorter distance to the nozzle lip. It is further interesting to note that the wave packet in
the swirling jet gains higher amplitudes than in the non-swirling jet although the maximum
growth rates of the modes are higher for the non-swirling jet.
The wave packet in the swirled shear layer propagates in axial direction and in the
direction of the base flow rotation, as indicated by the coordinates of Ae
max projected
on the θ-x-axis plane (figure 5.24b). The azimuthal propagation velocity, expressed as
an azimuthal rotation rate Ωe, is 2πΩeD/V ≈1.1 within the region 0 < x/D = 1.5,
which coincides roughly with the potential core region and the region of constant Rossby
number (see figure 5.4a-b). With further downstream distance, Ωedecays rapidly, yielding
2πΩeD/V ≈0.1 at x/D = 3. It is interesting to note that the theoretically derived
azimuthal phase velocity ωr/m of the streamwise modes (αr= 0) with m= 2 and m= 3
agree quite well with the rotation rate of the wave packet envelope. This is consistent
with the kinematic relation ωr∝V/2αmax
r+ Ωgrmderived in section 5.3, if one interprets
Ωeas the group rotation rate Ωgr. The streamwise propagation velocity of the envelope
is very similar to the non-swirling jet, ranging from 0.7Vnear the nozzle to 0.5Vfurther
Chapter 5 Instabilities in the Moderately Swirling Jet 95
tV/(2D)
x/D
θ/π 1
0.5
0
-0.5
-1
0.5
0.6
0.7
0.8
0.9
1
0
1
2
3
4
5
0.020.25
0.6
1
1.5
2
2.5
3
0 1 2 3
0
.05
.1
Ae
max/(DV )
x/D
Ae/Ae
max
(a) non-swirling jet S1
tV/(2D)
x/D
θ/π 1
0.5
0
-0.5
-1
0.5
0.6
0.7
0.8
0.9
1
0
1
2
3
4
5
0.020.25
0.6
1
1.5
2
2.5
3
0 1 2 3
0
.05
.1
Ae
max/(DV )
x/D
Ae/Ae
max
(b) swirling jet S3
Figure 5.24: Trajectory of the wave packet envelope for the swirling and non-swirling jet.
The pulse is initiated at θ= 0, t= 0 and x/D = 0. Contours show the envelope amplitude
distribution Aeat each r-θ-plane of measurement normalized by the corresponding maxi-
mum Ae
max. Big black dots refer to the maximum of the envelope projected onto the θ-x-
and t-x-plane. White-filled black circles refer to the leading and trailing edge of the wave
packet in time direction. Gray-filled black circles refer to the leading and trailing edge of
the wave packet in θdirection.
96 Chapter 5 Instabilities in the Moderately Swirling Jet
downstream (x/D > 0.6). This seems plausible, as the stability analysis predicts modes
with low azimuthal wavenumbers, which are dominant further downstream, to have a similar
axial phase velocity as for the non-swirling jet.
However, a prediction of the wave packet propagation speed based on the phase velocities
of the individual modes remains cumbersome for the swirling jet due to the dispersiveness of
the shear layers. The latter is indicated by a strong deformation of the wave packet envelope
during its downstream propagation. In the region of strong amplitude gain (0.6≤x/D ≤
2.5), the envelope looses its symmetry and its maximum is shifted closer to the leading
azimuthal bound, revealing a steep front and a smooth tail of the disturbance envelope
in θ-direction. It is assumed that this pattern results from the superposition of modes
with different azimuthal wavenumbers traveling at different streamwise phase velocities, as
predicted by the linear theory. Contrarily, the wave packet is not deformed significantly
in t-direction while traveling downstream. The most unstable modes have small azimuthal
wavenumbers (m < 3) and small inclination angles αr/m and thus, different phase velocities
result in strong amplitude variations in θ-direction but only weak variations in t-direction.
5.6.2 Modal Decomposition of the Wave Packet
Within the framework of linear stability analysis, the wave packet created by a short pulse
consists initially of an infinite number of modes and frequencies that are either linearly
stable or unstable. While the wave packet travels downstream, stable modes decay and un-
stable modes grow at a rate that is given by the local dispersion relation D(α, ωr, m, Γ,Ret).
Hence, in a non-parallel flow, the dominant modes in the traveling wave packet successively
replace each other in accordance to the stability of the downstream varying base flow.
The theoretical investigation in section 5.4 reveals that for the non-swirling jet, the
axisymmetric modes are most unstable at the nozzle lip and are replaced by the bending
modes with m=±1 for x/D > 2, which remain the only unstable modes downstream
of x/D = 3.3. For the swirling jet, the mode selection is more complex as positive and
negative modes have different growth rates. Close to the nozzle, the axisymmetric and
co-winding modes are more unstable than the counter-winding modes, but they stabilize
more rapidly with downstream distance and the counter-winding modes become dominant
with m= 2 to be most unstable within a large streamwise region. The counter-winding
mode with m= 1 remains unstable throughout the entire measurement domain.
Consistent with the single-mode analysis, the amplitude distribution of a instability
mode excited at x= 0 is derived from equation 2.11 and the radially integrated amplitude
is given by the expression
Am(x, ωr) = ZR.95
R.05 |ˆvx|2rdr1/2
.(5.10)
Contrarily to the single-mode amplitude distribution (5.5), the modal amplitude distribu-
tion of the wave packet Am(x, ωr) depends now parametrically on mand is a function of
xand ωr. It is computed for the entire measurement domain for modes with |m| ≤ 5 and
ωr≤0.3. Results are displayed as contours in the m-ωr-plane at the axial locations where
the hot-wire measurements are conducted (figure 5.25).
Chapter 5 Instabilities in the Moderately Swirling Jet 97
ωrδx/Vωrδx/Vωrδx/V
ωrδx/Vωrδx/V
mm
x/D = 1
x/D = 1.5
x/D = 2
x/D = 2.5
x/D = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
Figure 5.25: Modal amplitude distribution derived from linear stability analysis; left
column refers to the non-swirling jet S1and right column to the swirling jet S3, contours
refer to amplitude Anormalized by its overall maximum.
98 Chapter 5 Instabilities in the Moderately Swirling Jet
The theoretical predictions are compared to the modal content of the measured wave
packet. Therefore, the measured coherent axial velocity of the wave packet vc
x(x, r, θ, t) is
transformed into Fourier space, yielding the coefficients ˆvc
x(x, r, m, ωr). The modal ampli-
tude distribution is then derived from equation (5.10). Results are shown in figure 5.26 for
the swirling and non-swirling jet. It can be directly compared to the theoretical prediction
shown in figure 5.25.
For the non-swirling jet, the agreement between measurements and theoretical predic-
tion is reasonably well downstream of x/D = 1. The measurements confirm that the wave
packet in the non-swirling shear layer is first dominated by axisymmetric waves at fre-
quencies around ωrδx/V = 0.1 (see second frame in left column). Further downstream,
the bending m=±1 modes become most dominant. The normalized frequency of highest
amplitude remains constant with downstream distance, confirming that the most amplified
wavelengths in the packet scales with the local length scale δx. The mismatch between
theoretical prediction and experiment at x/D ≤1 is probably attributed to the imperfect
forcing and to spatially decaying modes at low frequencies that are excited at the nozzle.
For the swirling jet, the mode selection is in line with the theoretical prediction. Note
that the sense of winding of the helical waves cannot be derived from the measurements
and the modes can only be classified in co-rotating and counter-rotating modes, which
corresponds in the figures 5.25 and 5.26 to m > 0 and m < 0, respectively. This implies
a special caution in interpreting these figures, since for the swirling jet, modes with the
same sense of winding may have negative or positive m. This is the case for the co-winding
double-helical mode that is most amplified within the measurement domain. Its maximum
amplitude is located at a frequency of ωrδx/V ≈0.06 at m=−2, yielding that the most
amplified mode is the co-winding counter-rotating double helical mode. In agreement with
the theoretical prediction, the dimensionless frequency of the most unstable mode is reduced
with the addition of swirl. The experiments further show that this co-winding double-helical
mode can be counter-rotating (ωr>0, m=−2), steady (ωr= 0 m=±2), or co-rotating
(ωr>0, m = 2). Hence, the existence of steady modes at considerable high amplitude are
confirmed by the present experiments. Moreover, the measurements conducted downstream
of x/D = 2 indicate the amplification of the single-helical counter-rotating mode in good
agreement with the theoretical prediction.
5.6.3 Morphology of the Wave Packet
The shape of the wave packet for given xand ris displayed in figure 5.27 for the non-
swirling and swirling jet. The visualizations are based on hot-wire measurements conducted
in the center of the axial shear layer at the downstream end of the measurement domain.
The shear layer of the non-swirling jet is dominated by equally unstable co-rotating and
counter-rotating single-helical modes that create a wave packet with a v-shaped front that
propagates downstream with its open end pointed in upstream direction. The morphology
of the disturbance traveling in the swirling jet can be described as a wave packet with
two wave crests that move in negative θ-direction with increasing t. This corresponds to
the dominant counter-rotating double-helical mode. One wave crest is somewhat higher
than the other indicating the appearance of the single-helical mode. The passage time of
a wave crest is much longer for S3than for S1which corresponds to lower frequencies and,
assuming similar convection velocities, to larger streamwise wavelengths.
Chapter 5 Instabilities in the Moderately Swirling Jet 99
ωrδx/V
ωrδx/Vωrδx/Vωrδx/Vωrδx/V
m m
x/D = 1
x/D = 1.5
x/D = 2
x/D = 2.5
x/D = 3
-5 -4 -3 -2 -1 0 1 2 3 4 5-5 -4 -3 -2 -1 0 1 2 3 4 5
0
0.5
1
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
0
0.1
0.2
Figure 5.26: Modal amplitude distribution derived from hot-wire measurements; left
column refers to the non-swirling jet S1and right column to the swirling jet S3, contours
refer to amplitude Anormalized by its overall maximum.
100 Chapter 5 Instabilities in the Moderately Swirling Jet
(a) non-swirling jet S1
(b) swirling jet S3
Figure 5.27: 3D-wireframe visualization of the traveling wave packet based on vc
xmea-
sured in the center of the shear layer r=R.5at x/D = 3. Dashed contour lines refer to
negative vc
x.
5.7 Summary and Discussion
5.7.1 The Purpose of the Present Investigation
The nearfield of a turbulent axisymmetric unconfined swirling and non-swirling jet is inves-
tigated for a Reynolds number of ReD= 20000. The swirl intensity is below that for the
onset of vortex breakdown. This work focuses on the formation of vortical structures in
the streamwise growing shear layer between the swirling jet and the quiescent surrounding
fluid. The spatially growing disturbance waves therein are modeled theoretically by means
of quasi-parallel linear stability analysis based on the measured mean flow and experimen-
tally by acoustically forced experiments.
Within the last two decades, the stability of swirling jets has been investigated exten-
sively. However, most studies were restricted to parallel flows using generic model-based
velocity profiles that do not reflect the experimental findings. Although these model-based
studies are useful to study the effects of certain parameters on the stability, their relevance
Chapter 5 Instabilities in the Moderately Swirling Jet 101
to real flows is often difficult to assess. In the present work we depart from model-based
studies and apply stability analysis to the measured swirling jet flow. This allows for a
quantitative comparison of the empirically derived vortical structures with the theoreti-
cal results. The theoretical approach implicitly accounts for nonlinear effects induced by
Reynolds stresses that alter the mean flow and, more importantly, for the strong divergence
of the swirling jet. The major question addressed in this investigation is: What are the
dominant coherent structures that evolve in turbulent swirling jets and what instability
mechanism is responsible for their spatial growth.
5.7.2 The Main Observations
Stability analysis is carried out at the nozzle lip, where the swirling jet differs only by its
swirl component from the non-swirling jet. The swirling jet is found to promote a shear
instability and a less unstable centrifugal instability, represented by two solutions of the
dispersion relation. The characteristics of the these disturbance modes are similar to those
found in swirled shear layers (Cooper & Peake 2002;Lu & Lele 1999;M¨uller & Kleiser
2008). However, in the present flow, only the growth rates of the centrifugal modes in-
crease with increasing swirl while the growth rates of the shear modes remain constant.
Note that it seems to be not always possible to strictly distinguish between centrifugal and
shear instabilities. Gallaire & Chomaz (2003) applied linear stability analysis to a jet with
a thicker axial shear layer than the present one, revealing only one perturbation mode that
is a clear Kelvin–Helmholtz instability for no-swirl, while it comprises characteristics of a
centrifugal instability for strong swirl. Nonetheless, for the present study, the two types
of instabilities occur always as two individual wavenumber branches and are considered as
two modes driven by two different mechanisms. This allows the instabilities to be phe-
nomenologically separated into centrifugal modes that exist only for the swirling jet and
shear modes that are primarily driven by the strong shear of the axial velocity profile. The
centrifugal instability promotes chiefly co-winding modes at high frequency and wavenum-
bers, whereas the shear layer instability promotes co-winding and counter-winding modes
at the same frequency and wavenumber band as the Kelvin–Helmholtz instability of the
non-swirling jet.
The centrifugal instability is weak even near the nozzle where it is somewhat relevant.
All centrifugal modes are stable within a distance of half a nozzle diameter due to the
rapid growth of the azimuthal shear layer. The rapid downstream decay of the centrifugal
instability confirms the weakly non-parallel stability analysis of Cooper & Peake (2002),
who investigated the stability of a swirled shear layer that spreads radially due to viscosity.
However, they found the shear mode to be significantly amplified by the addition of swirl,
which contradicts the present findings. In sharp contrast, the growth rate of the shear
mode is presently found to scale inversely with the axial shear layer thickness and, thus, it
decreases with increasing swirl due to enhanced shear layer spreading. These correlations
are confirmed by the present experiments and support the qualitative considerations of
Panda & McLaughlin (1994).
Moreover, the swirl component is found to alter the frequencies selected by the most
unstable non-axisymmetric modes. As noticed by Martin & Meiburg (1994), this is a pure
kinematic effect of the mean flow rotation. As explained in more detail in the present
work, the rotational motion implies a non-zero azimuthal group velocity that results in the
102 Chapter 5 Instabilities in the Moderately Swirling Jet
dependence of streamwise wavelength and phase velocity on the azimuthal wavenumber.
In other words, the inclination of the helical waves for a given frequency are altered with
increasing swirl due to an azimuthal propagation velocity of disturbances in the swirled
shear layer. This further implies that the shear layers of the swirling jets are dispersive.
The wave crests of modes at different frequencies or azimuthal wavenumbers travel at
a different streamwise velocities, which is not the case for the non-swirling jets. This
inhibits the subharmonic resonance to occur in contrast to the non-swirling jets (Paschereit
et al. 1995), which explains the absence of vortex pairing in the swirled shear layer as
observed by Panda & McLaughlin (1994). Moreover, the dispersiveness hinders controlled
intermodal interactions, which may occur at certain conditions in non-swirling jets (Long
& Petersen 1992). These findings are consistent with the presented experimental results.
Phase-locked measurements of the single-mode actuated flow clearly reveal the dependence
of the streamwise phase velocity on the azimuthal wavenumber, whereas ensemble-averaged
measurements of the pulsed flow confirm the enhanced dispersiveness of the shear layers
with increasing swirl.
In the spirit of non-swirling jet studies, the question is targeted, whether or not a swirling
jet selects a preferred mode. As reexamined in this work, the non-swirling jet is unstable
to a large wavenumber and frequency band at the nozzle exit with the axisymmetric mode
being most unstable. However, with increasing downstream distance all modes successively
stabilize except for the m=±1 modes that remain unstable. They are usually considered
as the preferred modes (e.g., see Cohen & Wygnanski 1987;Petersen & Samet 1988) of
non-swirling jets. Consistently, the present experiments reveal that a pulse initiated at the
nozzle lip creates a wave packet that consists solely of the preferred modes at sufficient
downstream distance from the nozzle.
The mode selection in swirling jets is more complex as symmetry breaks and co-rotating
modes undergo different amplification than counter-rotating modes. Experimental and nu-
merical results consistently show that the co-winding double-helical mode gains maximum
amplitude in the potential core region and is, therefore, considered as the preferred mode of
the swirling jet nearfield. However, downstream of the potential core, this mode stabilizes
and the co-winding single-helical mode remains the only unstable mode and is expected to
dominate the farfield dynamics. The present mode selection is in line with model-based
studies that indicate that swirl tends to destabilize co-winding modes (αrm < 0) and to
stabilize counter-winding modes (αrm > 0). Note that the dominance of the axisymmetric
mode observed in swirling and non-swirling jet experiments (e.g., see Ho & Gutmark 1987;
Liang & Maxworthy 2005;Loiseleux & Chomaz 2003) is usually attributed to an accidental
axisymmetric forcing caused by the facility upstream of the contraction and must not be
mistaken with the preferred mode (Cohen & Wygnanski 1987).
Moreover, swirl is found to destabilize two rather exotic types of shear modes, namely the
streamwise modes with αr= 0 and the steady modes with ω= 0. The first is purely driven
by azimuthal shear revealing relatively small grow rates due to the thick azimuthal shear.
The steady modes, however, are driven by the axial shear and azimuthal shear, which allows
for significant spatial growth. In fact, the modes m=−2 and m=−3 reveal significant
amplification rates at frequencies around zero. The measurements of the traveling wave
consistently confirm the existence of steady and weakly rotating double-helical modes.
Chapter 5 Instabilities in the Moderately Swirling Jet 103
Steady and nearly steady co-winding spiral modes with m= 2 or m= 3 have been
observed in experimental arrangements similar to the present one (Billant et al. 1998;
Loiseleux & Chomaz 2003;Oberleithner et al. 2007). Gallaire & Chomaz (2003) assign
the weakly rotating m= 2 mode observed by Loiseleux & Chomaz (2003) to a self-excited
mode that arises from a convective/absolute transition point near the nozzle exit. In
other words, they associate the preferred mode with a globally unstable mode with its
wavemaker located near the nozzle exit. Their argumentation remains vague as their spatio-
temporal analysis is based on one axial measurement location neglecting the divergence of
the mean flow. Recall that a spatio-temporal analysis preliminarily applied to the present
flow configuration could not reveal any region of absolutely unstable flow. It is, therefore,
suggested that the appearance of double or triple-helical structures, as reported by (Billant
et al. 1998), Loiseleux & Chomaz (2003), and Oberleithner et al. (2007) correspond to
convectively unstable steady or nearly steady modes. Their spatial phase may be tagged
by a small irregularity of the experimental arrangement, which leads to a breaking of
rotational symmetry of the mean flow. Moreover, for very high swirl below breakdown,
Loiseleux & Chomaz (2003) observe a counter-rotating m= 1 mode at high rotation rates
(frequencies). This mode presumably corresponds to the presently found preferred mode
m= 1 of the farfield that also counter-rotates at much higher frequencies than mode m= 2.
5.7.3 Final Remarks
The present work describes the formation of three-dimensional disturbance modes in tur-
bulent swirling and non-swirling jets. The major theoretical findings derived from linear
stability analysis are supported by an experimental investigation. At the nozzle exit, the
swirling jet differs only by its swirl component from the non-swirling jet, revealing a weak
influence of the swirl component on the stability. The different initial swirl strengths,
however, cause a significantly different downstream development of the mean flow which
effects the selected modes and their growth rates. Therefore, it appears that the stability
analysis of a spatially evolving flow derived from measurements leads to significantly dif-
ferent results than studies based on fictitious velocity models. The present linear stability
analysis is based on the mean flow, which intrinsically corroborates nonlinear interactions
between the spatially growing waves and the mean flow. The results can be especially used
to predict the dynamics, shape, and receptivity of large-scale flow structures that reside in
the mean flow, which is of immense importance for effective flow control applications. The
present theoretical approach is incapable of explaining with sufficient certainty the cause
for the swirl-enhanced jet spreading that leads to the presently observed reduction of the
shear layer receptivity. However, the present results may support or cancel out one or the
other of the recent arguments. The present investigation confirms that vortex merging,
the nonlinear mechanism that leads to a successive shear layer spreading in the potential
core region of a non-swirling jet (Gutmark et al. 1995;Ho & Gutmark 1987), is inactive
in swirling jets due to the swirl-enhanced dispersiveness of the shear layer. Moreover, the
idea of a significant swirl-induced destabilization of centrifugal and/or shear modes that
would intensify the growth of coherent structures and enhance the entrainment rates (e.g.,
see Cooper & Peake 2002;Cutler et al. 1995;Lu & Lele 1999;Mehta et al. 1991;Panda
& McLaughlin 1994;Wu et al. 1992) is not supported by the present work. The strong
increase of jet spreading is unlikely to be attributed to the rather small change in the base
104 Chapter 5 Instabilities in the Moderately Swirling Jet
flow stability found at the nozzle exit. It is more likely that complex nonlinear effects are
the driving force. Recent investigations indicate that these nonlinearities might be trig-
gered by an interaction of the centrifugal and the shear modes (Martin & Meiburg 1994,
1996) or by the coexistence of axial and azimuthal shear which causes a modification of the
involved turbulent structures (Hu et al. 2001a,b;Martin & Meiburg 1998;Naughton et al.
1997;¨
Orl¨u & Alfredsson 2008).
Chapter 6
Coherent Structures in the
Strongly Swirling Jet
In this chapter the large-scale coherent structures are investigated that arise
in a turbulent swirling jet undergoing vortex breakdown. Experiments suggest
the existence of a self-excited global mode having a single dominant frequency.
This oscillatory mode is characterized by a co-rotating counter-winding helical
structure that is located at the periphery of the recirculation zone. It arises
from a precessing vortex core that is located further upstream. The result-
ing time-periodic 3D velocity field is derived theoretically by means of spatial
stability analysis employing the mean flow data from experiments. The 3D os-
cillatory flow is constructed from uncorrelated 2D snapshots of PIV data, using
the proper orthogonal decomposition (POD), a phase-averaging technique and
an azimuthal symmetry associated with helical structures. Stability-derived
modes and empirically derived modes correspond remarkably well, yielding pro-
totypical coherent structures that dominate the investigated flow region. The
proposed method of constructing 3D time-periodic velocity fields from uncorre-
lated 2D data is explained in detail. It is applicable to a large class of turbulent
shear flows. The employed phase-averaging technique is also used in chapters 4
and 7of this thesis.
6.1 Motivation, and Approach
The present investigation deals with coherent structures of a strongly swirling jet under-
going vortex breakdown. As shown in chapter 4, this flow configuration is subjected to
self-excited flow oscillations. The corresponding 3D time-periodic coherent structures are
predicted by linear stability theory and are constructed from 2D PIV data via a proposed
identification method.
In principle, self-excited oscillations are known to arise from a region of absolutely
unstable flow. They can be described by an unstable global mode (Chomaz 2005) or by
local spatio-temporal stability analysis with complex frequency and wavenumber (Huerre &
Monkewitz 1990;Monkewitz et al. 1993;Pier & Huerre 2001). However, for the underlying
flow configuration a simple local spatial stability analysis is employed to approximate the
105
106 Chapter 6 Coherent Structures in the Strongly Swirling Jet
velocity of the global mode. This simplification serves the main purpose of this study,
which is to enhance the understanding of turbulent coherent structures in highly turbulent
swirling flows. It is in line with similar studies (Juniper et al. 2011).
As described in chapter 4, strong oscillations at the global frequency are found upstream
of vortex breakdown, revealing a precession of the vortex core that acts as the global
wavemaker. In the outer shear layer, downstream traveling instabilities are detected that
are internally forced by the wavemaker and are synchronized to its frequency. These waves
serve as amplifiers to external forcing, which suggests that the signaling problem is valid
for the outer flow region. Assuming that the outer shear layer responds to internal forcing
in the same way as to external forcing, the large-scale fluctuations downstream of the
wavemaker may be approximated by convectively unstable modes that oscillate at the
global frequency. Hence, the spatial analysis presented here is conducted with an unknown
complex streamwise wavenumber and the known real global frequency.
This approach can also be justified within the generalized mean field model proposed
by Noack et al. (2003). Accordingly, the mean flow at limit cycle oscillation is globally
marginally stable. This was validated for the cylinder wake by Barkley (2006). This
implies that the absolute frequency from the local analysis should be real at the wavemaker
location. Hence, it is justified to model the coherent structures of the global mode from a
spatial analysis with real frequency and unknown complex streamwise wavenumber.
For this study, the swirling jet air facility at the TU Berlin is used. The experimental
setup is described in in section 3.2 and details on the PIV measurements are given in section
3.3.
The outline of this chapter is as follows. A brief introduction to empirical mode con-
struction is given next. Then, the main features that characterize the strongly swirling jet
undergoing vortex breakdown are summarized in section 6.3. This flow is dominated by
strong oscillations resulting from a self-excited global mode. A POD-based method to ex-
tract the three-dimensional coherent velocity of this mode from uncorrelated PIV snapshots
is explained in section 6.5. In section 6.6, the global mode shape is derived from a local
spatial stability analysis. The results of both methods are compared in section 6.7. and a
three-dimensional reconstruction of the global mode is presented. It is based on stability
analysis and on PIV data providing a portrait of the dominant coherent structures. The
main observations are summarized in section 6.8.
6.2 A Brief Survey on Empirical Mode Construction
Large-scale coherent structures in many turbulent shear flows are visually similar to pre-
dominant instability modes persisting over a wide range of Reynolds numbers (e.g., see
Van Dyke 1975). This similarity applies to flows whose mean velocity profiles are invis-
cidly unstable and whose shape of these profiles does not materially change during the
transition from laminar to turbulent flow. This observation suggests that stability consid-
erations can be applied to the mean turbulent flow field, although there is no theoretical
basis for this step. However, weakly nonlinear stability approaches explicitly assume that
instability modes and most energetic (POD) modes are the same, at least near the onset
of a supercritical Hopf bifurcation (Noack et al. 2003;Stuart 1958).
Chapter 6 Coherent Structures in the Strongly Swirling Jet 107
Stability theory approximates the flow as a given mean flow and a superposition of
space- and time-dependent modes. In similar spirit, turbulent coherent structures can
be conceptualized as an expansion of modes. A corresponding least-order representation
of a flow snapshot ensemble is obtained by the POD. This approach minimizes a time-
averaged residual of the POD expansion for a given number of modes, and it is equipped
with further useful analytical properties (Holmes et al. 1998;Lumley 1967). Historically,
Lumley (1967) introduced POD as a least-biased definition of coherent structures following
up on the analytical approach by Townsend (1956) and the well-known Karhunen–Lo´eve
decomposition from the 1940s.
Meanwhile, many other empirical expansions of flow snapshots have been proposed serv-
ing dynamical systems or control theory goals. For instance, the dynamic mode decompo-
sition (DMD) extracts modes from snapshots that are more related to stability eigenmodes
(Rowley et al. 2009;Schmid 2010). Furthermore, the balanced POD serves as economic
expansion for linear input-output relationships (Rowley 2005). The present study is re-
stricted to the classical POD since it targets an optimal kinematical representation of the
flow.
6.3 Description of the Flow Configuration
This investigation focuses on the nearfield of a turbulent jet at a very high rate of swirl.
The basic features of this flow will be described in the following section in order to explain
the motivation for investigating the evolution of the coherent structures.
6.3.1 Characteristic Numbers
The Reynolds number and the swirl number are the two independent dimensionless numbers
that characterize the global behavior of the flow. The previous is based on the nozzle
diameter Dand on the axial plug flow velocity V, which is derived from the mean mass
flow rate. Throughout this chapter, the Reynolds number is set to ReD= 20000. The swirl
number Sis the commonly used parameter that quantifies the amount of swirl (Chigier
& Chervinsky 1965;Panda & McLaughlin 1994). It is defined as the ratio between the
axial flux of angular momentum ˙
Gθand the axial flux of axial momentum ˙
Gx. According
to the conservation of momentum, the swirl number is conserved in the axial direction
(Rajaratnam 1976). It is set to S= 1.22 throughout this chapter.
6.4 Mean Flow Properties
Figure 6.1 illustrates the streamwise distribution of the time averaged flow. Due to the
occurrence of vortex breakdown, the maximum axial velocity is displaced from the jet
center. The axial velocity profiles have a local velocity minimum in the inner region of
the jet. Thus, a wake with a region of reversed flow on and near the jet axis is resembled
with a recirculation bubble that is bound by upstream and downstream stagnation points.
This reversed flow region is similar to one created by an obstacle placed on the jet center.
Hence, the flow emanating the nozzle is a swirling ring-jet with inner and an outer axial
and azimuthal shear layers.
108 Chapter 6 Coherent Structures in the Strongly Swirling Jet
y/Dy/D
x/D, V/V ·0.4
Vx
V
Vz
V
0.2 0.6 1 1.4 1.8 2.2 2.6 3
-0.5
0
0.5
-0.5
0
0.5
Figure 6.1: Profiles of the mean axial and plane-normal velocity at various axial loca-
tions; velocities are normalized by bulk-velocity V; Streamlines indicate the location of the
recirculation bubble (ReD= 20000; S= 1.22);
Chapter 6 Coherent Structures in the Strongly Swirling Jet 109
The streamlines shown in figure 6.1 illustrate how the flow is guided around the recircu-
lation zone causing a rapid increase of the jet diameter. Downstream the recirculation zone,
at approximately x/D > 1.4, the inner shear layers begin to merge and the axial velocity
on the jet center increases gradually with increasing downstream distance. The azimuthal
velocity profiles may be divided into a vortex core, the region between the jet center and the
maximum azimuthal velocity and into the outer azimuthal shear layer located between the
maximum azimuthal velocity and the quiescent surrounding fluid. The axial velocity profile
has two inflection points and thus, in terms of inviscid hydrodynamic stability they possess
as many plane instability modes. Since the flow is axisymmetric these could combine with
azimuthal modes. The convex streamlines over the frontal part of the recirculation bubble
coupled with the decelerating outer flow provide the necessary conditions for centrifugal
instability, as do the concave streamlines in the lee of the bubble coupling with the inner
shear layer.
6.4.1 Analytic Representation of the Mean Flow
The incompressible mean flow of the unconfined swirling jet is expressed by an axial velocity
Vxand a circumferential velocity Vθ. The radial velocity is neglected. The characteristic
velocity scale Vmax =Vx,max is defined as the maximum axial velocity at a certain axial
location, and the characteristic length scale Rmax =rx,max is represented by the radial dis-
tance of the maximum velocity. The ‘Monkewitz profile’ approximates the axial component
if we normalize all velocities by Vmax and all lengths by Rmax. In the current study, we
employ a modification of this profile introduced by Michalke (1999):
Vx= 4BF1[1 −BF2],(6.1a)
where F1and F2are given by Monkewitz & Sohn (1988) as
Fj=h1 + (er2bj−1)Nji−1, j = 1,2.(6.1b)
The quantity Bdepends on the axial velocity on the jet centerline Vcl
B= 0.5h1 + (1 −Vcl)1/2i(6.1c)
N1,2≥1 are the shape parameters that control the thickness of the jet shear layers. In
contrast to Michalke (1999), two parameters N1and N2are used in order to approximate the
inner and outer shear layer, respectively. The normalized swirl component is represented by
the same equations with the simplification that B= 1. Therefore, the local swirl parameter
Sloc =Vθ,max/Vx,max is introduced yielding:
Vθ= 4F3[1 −F4]Sloc ,(6.2)
Note that all flow parameters and fitting parameters of equations (6.1–6.2) vary in the
axial direction due to the non-parallelism of the mean flow. Their quantities are displayed
in table 6.2 for profiles taken at x/D = (0.25,0.5,1,1.5). The fitted velocity profiles are
displayed in figure 6.2 together with the measured mean axial and azimuthal velocities for
distances in the range 0.5≤x/D ≤3. Both velocity components are well represented
110 Chapter 6 Coherent Structures in the Strongly Swirling Jet
r/Rmax
Vx
Vmax
r/Rmax
x/D,V/Vmax ·0.5
Vz
Vmax
0.5 1 1.5 2 2.5 3
2
1
0
1
2
2
1
0
1
2
Figure 6.2: Velocity profiles of the approximated mean flow field using the shape param-
eters listed in table 6.2. Symbols refer to experimental data.
x/D Vmax/V Sloc Rmax/D B
0.25 0.87 0.91 0.45 1.01
0.50 0.76 0.90 0.50 1.03
1.00 0.63 0.81 0.56 1.04
1.50 0.51 0.71 0.58 1.00
Table 6.1: Flow parameters of mean flow approximation (6.1-6.2)
x/D N1N2N3N4b1b2b3b4
0.25 4.23 1.10 4.18 0.73 0.51 0.38 0.51 0.27
0.50 3.14 1.35 2.23 1.09 0.56 0.52 0.73 0.66
1.00 2.92 1.06 1.87 0.55 0.53 0.46 0.63 0.35
1.50 2.38 0.87 1.17 0.56 0.46 0.35 0.71 0.57
Table 6.2: Fitting parameters of mean flow approximation (6.1-6.2)
by the suggested approximation. Note that this rather complex mean flow approximation
is necessary to accurately represent the two axial and azimuthal shear layers. Simpler
models as introduced by Michalke (1999) and Gallaire et al. (2004) did not approximate
the underlying mean flow well enough for an accurate linear stability analysis.
Chapter 6 Coherent Structures in the Strongly Swirling Jet 111
St = 0.49
PSD
St
0.2 0.51 1.5
100
101
102
103
104
Figure 6.3: Power spectral density of hot-wire-anemometer voltage fluctuations for the
unforced swirling jet at (x/D, y/D) = (0.57,0.38). The same dominant frequency is mea-
sured in the inner and outer axial shear layers.
6.4.2 Self-excited Oscillations
Former experimental investigations by Liang & Maxworthy (2005) and numerical simula-
tions of Ruith et al. (2003) revealed that the onset of vortex breakdown is accompanied
by energetic large-scale fluctuations. In the present investigation, these strong oscillations
had a distinct frequency (figure 6.3). By traversing a hot-wire probe in radial and axial
directions, a constant dominant frequency is observed throughout the region of interest,
with highest amplitudes occurring in the inner and outer axial shear layers. As will be
shown later, the spectral peak at St = 0.49 corresponds to the precession of the vortex core
or, in terms of hydrodynamic stability, to the appearance of a strong helical instability with
azimuthal wavenumber m= 1. This relatively sharp peak is attributed to a self-excited
global mode with its origin being located on the jet centerline in the region of reversed flow
(Gallaire et al. 2006;Liang & Maxworthy 2005;Ruith et al. 2003).
There are several experimental techniques to confirm that the flow has transitioned to
a global mode via a supercritical Hopf bifurcation. According to Huerre & Monkewitz
(1990), near critical conditions, the amplitude of the global mode Ais governed by the
forced Landau equation (Landau & Lifshitz 1987):
dA/dt =c1A−c2A3+g(6.3)
where c1is the temporal amplification rate during the time of exponential growth and gis
proportional to the external forcing amplitude. In the absence of forcing, the limit cycle
amplitude should increase proportionally to the deviation from a control parameter,
Asat ∝pS−Scrit (6.4)
where Scrit is the critical control parameter for a constant Reynolds number and S≥Scrit.
The amplitude of the global mode is measured with a calibrated hot-wire placed in the
112 Chapter 6 Coherent Structures in the Strongly Swirling Jet
S
A2
sat
Scrit = 0.88
0.9 1 1.1 1.2 1.3 1.4
0
0.1
0.2
0.3
0.4
0.5
Figure 6.4: The squared saturation amplitude A2
sat of the dominant mode as a function of
the increasing control parameter S. The open circles represent measurements; the straight
line represents a least-squares fit to this data. This linear dependence is characteristic for a
supercritical Hopf bifurcation. The zero marks the critical control parameter Scrit = 0.88.
Asat is measured with a single hot-wire anemometer placed at (x/D, y/D) = (0.57,0.38)
in the center of the inner axial shear layer (ReD= 20000).
center of the inner axial shear layer at (x/D, y/D) = (0.57,0.38). At this radial location
the oscillations reach their maximum amplitude. A single-wire probe is used with the
wire aligned parallel to the azimuthal velocity. The signal is Fourier transformed and the
amplitude at the dominant frequency is derived. Figure 6.4 shows the growth of the global
amplitude with increasing swirl while keeping the Reynolds number constant. Evidently,
the saturation amplitude is proportional to √S−Scrit. The linear dependence suggests
that the oscillation is of the supercritical Hopf bifurcation type. The critical swirl number
is found to be Scrit = 0.88.
A second experimental technique that may confirm the existence of a self-excited global
mode is to investigate the lock-in characteristic. According to Provansal et al. (1987),
Sreenivasan et al. (1989) and Juniper et al. (2009), the critical forcing amplitude at which
the frequency of the natural mode Stnat locks onto the forcing frequency Stfshould linearly
depend on |Stnat −Stf|. The lock-in region is defined as the forcing amplitude at which
the spectral peak of the natural mode disappears and the spectrum peaks at the forcing
frequency. Measurements conducted with a single hot-wire, placed on the center of the inner
axial shear layer at (x/D, y/D) = (0.57,0.38), revealed that the critical lock-in amplitude is
proportional to |Stnat −Stf|(figure 6.5). This provided additional evidence for the existence
of a supercritical Hopf bifurcation needed to establish a global mode.
Concluding this chapter, the base flow under investigation is a swirling ring-jet whose
conical boundaries originate at the orifice due to a recirculation zone located on the jet
axis. The axial and azimuthal shear layers coexist in the outer region of the jet and in
the jet core. The swirl number considered presently is above the critical value at which
a supercritical Hopf bifurcation takes place. Thus, the strong coherent fluctuations that
are dominating the entire flow field near the nozzle are attributed to the existence of a
self-excited global mode.
Chapter 6 Coherent Structures in the Strongly Swirling Jet 113
St
Speaker input in V
Lock-in
0.4 0.5 0.6
0
0.1
0.2
Figure 6.5: Critical loudspeaker input voltage at which the global mode locks onto the
forcing frequency. The open circles mark measurement values and the solid lines represent
fits to these data. The linear dependence of the threshold amplitude on |Stnat −Stf|is
another indicator of a supercritical Hopf bifurcation to a global mode. The flow is forced
at the orifice at m= 1 which is the azimuthal wavenumber of the natural mode. The
natural frequency is Stnat = 0.49 (ReD= 20000; S= 1.22)
6.5 Empirical Construction of the Global Mode
In this section, the path is outlined leading from the POD of 2D PIV data to the con-
struction of full 3D time-dependent velocity field. This method is applicable whenever the
coherent structure is solely characterized by two POD modes that span the traveling wave
pattern. This section was written in collaboration with Moritz Sieber, who worked as a
research assistant for the author of this thesis. His great work is acknowledged here.
6.5.1 Spatial and Temporal POD Modes
The snapshot POD, as described above, is applied to the data taken in the crossflow and
streamwise planes of measurement. Both sets of measurement consist of 800 snapshots. The
observation domains have a spatial extent of −1.1< y/D < 1.1 and −1.1< z/D < 1.1 at
x/D = 0.57 for the crossflow plane and 0.25 < x/D < 3 and −1.1< y/D < 1.1 at z/D = 0
for the streamwise plane. The eigenvalue spectrum of POD modes for both measurement
planes is shown in figure 6.6.
For both cases the POD shows that the first two eigenvalues contain substantial amount
of energy. In the crossflow plane the first two modes contain already 30 % of TKE while
in the streamwise plane these modes contain more than 14 %. In both cases, the energy
contained in the two leading modes is nearly equal suggesting that the two modes span a
traveling wave.
In the crossflow plane of measurement, the first and second spatial POD modes resemble
one another as do the fourth and fifth modes (figure 6.7). The first pair of modes describes
an azimuthal wave represented by two modes having a π/2 phase shift. The second pair
having twice the azimuthal wavenumber has a π/4 shift between them with respect to the
114 Chapter 6 Coherent Structures in the Strongly Swirling Jet
p
λp
crosswise
streamwise
100101102
10−1
100
101
Figure 6.6: POD spectrum of the velocity modes for the crossflow and streamwise plane
of measurement. TKE is expressed in per cent of the sum K=PN
p=1 Kp.
1
15.0% TKE
2
14.9% TKE
3
3.1% TKE
4
2.9% TKE
5
2.6% TKE
1
Figure 6.7: First 5 POD modes of the crossflow plane of measure-
ment; radial velocity component is shown with contour lines vr/max(vr) =
(−0.8,−0.6,−0.4,−0.2,0.2,0.4,0.6,0.8). The POD mode-number pis written in
the top left corner and the percentage of TKE at the bottom. The dashed circle indicates
the nozzle diameter.
dominant harmonics. The first pair represents a traveling azimuthal wave with wavenumber
m= 1 and the second pair indicates a traveling azimuthal wave m= 2.
To elucidate the temporal behavior of the identified structures, the phase portraits of
the corresponding temporal amplitudes apare investigated. Considering the phase portrait
of a1and a2(figure 6.8), it is clearly seen that the modes describe a oscillating process.
In addition, the comparison of a1and a4reveals that the second mode-pair is the second
harmonics with respect to the first one, as indicated by the eight-like form of the Lissajous
figure. Both mode pairs describe rotating structures that rotate with the same revolution
time with azimuthal wavenumber m= 1 for the first and second modes and with m= 2
for the fourth and fifth modes. The third mode describes an axisymmetric fluctuation of
the flow, which is not correlated with the identified harmonic structures (figure 6.8). This
mode is related to the axial fluctuation of the location of vortex breakdown.
Figure 6.9 displays the POD modes of the streamwise cross-section. Again, the first
two modes have the same azimuthal wavenumber and frequency. These similar modes are
axially shifted by a quarter wavelength. They describe coherent structures that are first
Chapter 6 Coherent Structures in the Strongly Swirling Jet 115
a1/hrmi
a2/hrmia3/hrmia4/hrmi
-1 0 1-1 0 1-1 0 1
-1
0
1
Figure 6.8: Phase portrait of the POD modal amplitudes apfor the crossflow plane of
measurement. The dots represent the experimental data. The solid line is a smoothed fit
and similar to Lissajous figures. A=pa2
1+a2
2is the mean amplitude of the first two
modes (see (6.5)).
growing and than decaying in the streamwise direction. The temporal representation of the
first two streamwise modes corroborates an oscillating process (figure 6.10). The first two
modes in both crossflow and streamwise planes of measurement describe a harmonically
fluctuating structure that the spectral analysis (figure 6.3) picked up as containing one
fundamental frequency. Consequently one may assume that both modes are tied to the
same oscillatory structure. Linking the information from both measurements suggests that
the dominant structure is a helical instability mode with azimuthal wavenumber m= 1
winding around the recirculation bubble. In this context, the structure at the jet center
(i.e. r/D = 0) near the nozzle exit (x/D < 0.5), that is visible in the two streamwise
modes, is interpreted as being the wavemaker for the global mode.
Note that the relative levels of the first mode-pair with respect to its TKE content
are unequal for crosswise and streamwise observation planes. This is due to the dominant
structure governing only half of the analyzed streamwise measurement domain. Hence, the
energy content with respect to the entire streamwise plane is approximately half as high in
comparison to the crosswise plane. Using an appropriate sub-domain for streamwise POD
analysis can decrease this difference.
The third and forth streamwise modes (figure 6.9) are coupled and represent the mean-
dering of the recirculation bubble. The phase portrait reveals no relation to the dominant
structure. Hence, the meandering is affected by other processes.
In conclusion, the periodically fluctuating global mode is represented by the first two
POD modes in both measurement planes. This harmonic process is indicated by the phase
portraits. It is possible to extract the phase information of the dominant coherent structure
by identifying the corresponding POD modes. It is then a straight forward procedure to
use the temporal amplitudes of these POD modes to obtain the phase angle ϕkfor each
snapshot k, yielding
ˇakeiϕk=a1(tk) + ia2(tk) (6.5)
This phase angle corresponds to the phase position of a snapshot with respect to the
dominant structure, in the manner that the optimal amount of kinetic energy of each
snapshot is represented by these modes. Hence, it is possible to define a flow phase via the
POD. In the following this definition of phase is used to extract the coherent structures.
116 Chapter 6 Coherent Structures in the Strongly Swirling Jet
r/D
1
8.1% TKE
2
6.1% TKE
x/D
r/D
3
2.0% TKE
x/D
4
1.6% TKE
0.5 1 1.5 2 2.5 30.5 1 1.5 2 2.5 3
0.5
0
0.5
0.5
0
0.5
Figure 6.9: First 4 POD modes of the streamwise measurement plane;
transversal velocity component is shown with contour lines vy/max(vy) =
(−0.8,−0.6,−0.4,−0.2,0.2,0.4,0.6,0.8). The mode number is written in the top right
corner and the percentage of TKE in the bottom right corner. The vertical dashed line
indicates the position of crossflow plane of measurement.
a1/hrmi
a2/hrmia3/hrmi
-1 0 1
-1 0 1
-1
0
1
Figure 6.10: Same as figure 6.8, but with apof the streamwise measurement plane.
Chapter 6 Coherent Structures in the Strongly Swirling Jet 117
6.5.2 Linking the POD Modes to the Coherent Velocity
First, we define the coherent velocity vc.Holmes et al. (1998) recommend to identify and
exploit symmetries in experimental data. This additional filter reduces the complexity of
the POD and yields a better understanding of the underlying process. For an axisymmetric
swirling jet, the azimuthal direction θcan be regarded as a homogeneous direction, as it
is also assumed for linear stability analysis. This direction can be represented through
Fourier modes with respect to the azimuthal wavenumber m= 1. The triple decomposition
assumes the coherent component as being a phase-dependent average of a harmonic signal
with temporal period T. Hence, the time can also be regarded as a homogeneous direction
and can be decomposed by a Fourier representation according to the phase information ϕk
of the POD (6.5). With these simplifications, the coherent velocity reads
vc(x, r, θ, t) = ℜnˆ
vmn(x, r)ei(mθ−nωrt)o(6.6)
where mindicates the azimuthal wavenumber, nis a multiple of the fundamental frequency
ωr= 2πf and ˆ
vmn(x, r) is a complex valued vector field containing the radial and axial
dependence of the coherent component. The Fourier modes ˆ
vmn are obtained through a
Fourier transform of the fluctuating part of the velocity v′
ˆ
vmn(x, r) = 1
2πT ZT
0Zπ
−π
v′(x, r, θ, t)e−i(mθ−nωrt)dθdt(6.7)
As described in the previous section, the POD relates each snapshot to a phase angle of
the dominant fluctuations. Thus, the Fourier modes ˆ
vmn are obtained for a discrete time
(according to ϕk) and continuous space Fourier transform:
ˆ
vmn(x, r) = 1
2πN
N
X
k=1 Zπ
−π
v′(x, r, θ, tk)e−i(mθ−nϕk)dθ(6.8)
This is valid only if the phase angles ϕkare equally distributed in [0,2π]. The phase angles
obtained from the POD fulfill this condition, so these angles correspond to a oscillation
with uniform frequency ϕk=ωrtk.
The assumed rotational symmetry is examined for the crossflow measurement plane of
the vector field. If we omit the assumption of homogeneity in θof equation (6.6), then the
complex coherent component is given by
ˆ
vn(x, r, θ) = 1
N
N
X
k=1
v′(x, r, θ, tk)einϕk(6.9)
This equals the definition of POD modes (2.26) provided the first two POD modes are
considered in a complex representation with ˇakeiϕk=a1(tk) + ia2(tk), except that the
amplitude ˇakof the temporal modes is neglected. We assume the amplitude variations to
be caused by turbulent noise, which is indicated by the phase portrait in figure 6.11 (see
also Depardon et al. (2007)). In consequence, the spatial POD modes are similar to the
coherent component, in detail Φ1+iΦ2≈ˆ
v1(with ˆ
vnas in (6.9)).
118 Chapter 6 Coherent Structures in the Strongly Swirling Jet
Figure 6.11: Coherent radial velocity vc
rderived from crossflow measurement via vc=
ℜ{ˆ
v1(r, θ)}(left) is compared with the one derived from streamwise measurement via
vc=ℜˆ
v11(r)eiθ(right). The flow is visualized by the contour lines vc
r/max(vc
r) =
(−0.8,−0.6,−0.4,−0.2,0.2,0.4,0.6,0.8). The dashed circle indicates the nozzle diameter.
In figure 6.11, the coherent component vcin the crosswise measurement plane is com-
pared between the simple phase-average derived from crosswise data vc=ℜ{ˆ
v1(r, θ)}
and the phase-average constructed from streamwise data assuming azimuthal symmetry
vc=ℜˆ
v11(r)eiθ. The contour plots of vcare very similar proving that the assumed
symmetry is legitimate. The discrepancy between the two plots is attributed to an insuffi-
cient number of snapshots for more precise averaging.
It should be noted, that the triple decomposition, as outlined in this section, is not
limited to the use of POD. The phase information of oscillatory fluctuations is often inferred
in three different ways. First, if the flow is externally actuated, the phase information can
be directly derived from the actuation signal. Second, it is possible to obtain the required
phase information from a time resolved point measurement (e.g. hot-wire), using a bandpass
filter and a Hilbert transformation. Third, it is provided by a statistical approach such as
the POD described above. In the present investigation, the POD is chosen because it has
some advantages with respect to the other techniques which will be shortly depicted.
Most flow oscillations do not occur at a prescribed frequency, as there is always jitter.
Filtering with respect to a fixed frequency will ignore large portions of the flow affected
by frequency modulation, reappearing in vs. With a locally adjusted frequency, we reduce
the amount of TKE captured by vs, while vclumps a narrow frequency band into a single
frequency. In the case of a phase average, triggered by external forcing, no phase jitter is
incorporated at all. If a time-resolved sensor is used, the amount of phase jitter accounted
for depends on the bandwidth of the bandpass filter used. When using the POD phase,
obtained according to (6.5), no fixed frequency has to be assumed, and the calculated phase
is the optimal one in terms of energy representation. The optimal phase is related to the
optimality of spatial POD modes, where these modes are understood as a prior guess for
coherent structures providing the phase through projection on the snapshots. Furthermore,
in contrast to a time-resolved point measurement, the POD approach predicts the phase
angle from spatial modes and not from one single point in the flow yielding a more accurate
prediction of the global phase.
Chapter 6 Coherent Structures in the Strongly Swirling Jet 119
6.5.3 Construction of Three-dimensional Coherent Structures
It is now straightforward to construct 3D-velocity data from the two measurement planes
utilizing the identified symmetries of the coherent velocity vcdiscussed in the previous
section. The approach for this construction is graphically outlined in figure 6.12, illustrating
the main steps in this section.
(i) Identification of a fundamental frequency by time-resolved point measurement.
(ii) POD analysis of the PIV data in the two measurement planes, which identifies dom-
inant structures and provides the related phase information.
(iii) Calculation of coherent structures with phase information and identification of az-
imuthal symmetries through a Fourier transform.
(iv) Construction of 3D data of the coherent structure using the identified symmetries to
extrapolate data from 2D measurement plane.
In the final step, the streamwise measurement plane (r–x-plane) gives the axial and radial
dependence of the coherent complex amplitude ˆ
vmn(x, r) (6.7). As shown in the previous
section, the fundamental frequency is related to the azimuthal wavenumber m= 1 and first
harmonic n= 1. Combining this information in (6.6), a spatio–temporal representation of
the coherent velocity is given by
vc(x, r, θ, t) = ℜnˆ
v11(x, r)ei(θ−ωrt)o(6.10)
6.6 Most Unstable Spatial Modes
Prior the discussion of the results obtained from the linear stability analysis, the simplifying
assumptions are recalled here. Due to the occurrence of vortex breakdown, the underlying
time-averaged flow exhibits strong gradients in the axial direction. Hence, the employed
parallel-flow assumption for the normal mode decomposition (2.6) is violated due to these
gradients. Various approaches have been developed to overcome this shortcoming of the
linear theory, generally restricted to small deviations. Intriguingly, wavelength and ra-
dial amplitude distribution are often found to be reasonably good approximated by the
eigensolution of the Orr–Sommerfeld equation solely based on a locally parallel flow.
In the scheme of linear stability analysis, the spatial amplification rate αiis computed
for the azimuthal wavenumber m= 1 in order to derive the coherent velocity of the single-
helical instability mode that is observed in the experiment. The complex wavenumber is
calculated for real frequencies in the range 0 ≤ωrRmax/Vmax ≤6 at four different axial
locations. The spatial branches are displayed in figure 6.13. Note that the frequency ωris
normalized by the length scale Rmax and velocity scale Vmax (see section 4.1.1) whereas the
Strouhal number St associated with the global oscillation frequency is based on the bulk
velocity Vand nozzle diameter D. The dimensionless frequency expressed by ωrRmax/Vmax
increases in the downstream direction as the downstream increase of Rmax is more rapid
than the decay of Vmax (see the open circles in figure 6.13). At the axial locations where
the flow is unstable to the global frequency only one unstable spatial branch is found and
the spatial amplification rate can easily be tracked in the downstream direction. Waves
120 Chapter 6 Coherent Structures in the Strongly Swirling Jet
Construction of 3D velocity Data
0D / hotwire2D / POD
3D
streamwise data crosswise data
PIV/POD phase information PIV/POD phase information
Hotwire/FFT dominant frequency
phase-average coherent structures phase-average coherent structures
spatial rotation + symmetry
3D data azimuthal symmetry
2D /phase-average
Figure 6.12: Schematic diagram of the 3D flow construction. The illustrated steps are
as follows: (a) identification of the fundamental frequency; (b) POD analysis of the PIV
data yielding the flow phase; (c) Identification of coherent structures and symmetries; (d)
Construction of 3D data from the 2D measurements.
Chapter 6 Coherent Structures in the Strongly Swirling Jet 121
−αiRmax
x/D = 0.25 x/D = 0.5
ωrRmax/Vmax
−αiRmax
x/D = 1
ωrRmax/Vmax
x/D = 1.5
01 2 3 4 50 1 2 3 4 5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Figure 6.13: Spatial amplification rate of the mode m= 1 versus dimensionless real
frequency at different cross-sections. The open circle marks the measured frequency of the
mode m= 1 indicating the growth (x/D ≤1) and decay (x/D = 1.5) of most unstable
mode.
forced at the global frequency grow rapidly near the nozzle exit where the shear layer is
thin relative to the nozzle radius. As the shear layer spreads in the radial direction with
increasing downstream distance, the spatial amplification rate at St = 0.49 decreases. The
corresponding spatial branch downstream of the point of neutral amplification is stable for
all frequencies.
The evolution of the spatial growth rate αiwith downstream distance is investigated in
detail by computing the eigenmodes for various streamwise locations at the global frequency
of St = 0.49 (figure 6.14). Accordingly, the spatial growth rate decreases continuously
with downstream distance reaching natural amplification at x/D = 1.28. Downstream of
the neutral point, the decaying rate increases in the axial direction up to x/D = 2. In
the decaying region several modes coexist and the spatial branch that corresponds to the
most unstable mode is tracked from the unstable regime where only one mode exists by
minimizing the distance between the eigenfunctions and eigenvalues (see the last paragraph
of section 2.4.3 for further details). Note that in the bottom right frame of figure 6.13, only
the spatial branch that corresponds to the most unstable mode is shown. The computed
122 Chapter 6 Coherent Structures in the Strongly Swirling Jet
x/D
−αiRmax
0 1 2 3
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x/D
1
αrD
01 2 3
0.6
0.8
1
1.2
1.4
1.6
Figure 6.14: Streamwise evolution of the spatial amplification rate (left) and axial wave-
length (right) of the most unstable mode m= 1 based on local analysis.
axial wavelength of the most amplified mode is displayed as a function of x/D in figure
6.14. The rapid decrease in wavelength in the axial direction is caused by non-parallel
effects. The convectively unstable modes are amplified at constant frequency while they
travel downstream. Due to the pronounced jet widening, their convection velocity, that
is related to U, is rapidly decaying in the axial direction, thus causing the wavelength to
decrease.
The accuracy of the stability analysis is validated by comparing the computed eigenfunc-
tions with the actually measured phase-averaged velocities described in section 6.5. The
eigenfunction of the most amplified mode is computed at x/D = 0.57, the axial location
where crossflow measurements are conducted. Comparing the empirical mode, displayed
in figure 6.11, to the stability mode, displayed in figure 6.15 (b) shows obvious similari-
ties. Both modes agree well in the outer region of the jet where they are most energetic
(|y/D|>0.5). In the inner region the structures seem to be out of phase and the coherent
velocity near the jet core that is evident from the measurements is not modeled by the
eigenmodes. As stated previously, the present stability analysis is only valid for the outer
convective unstable region of the jet. In particular, near the jet core close to the nozzle
exit where coherent velocity indicates the location of the wavemaker, the present analysis
should produce wrong estimations. Further downstream of the wavemaker, the agreement
of the eigenmodes to the phase-averaged measurements improves even for the inner region
of the jet. This is shown by constructing the streamwise shape of the global mode from the
locally computed eigenfunctions and eigenvalues. The overall growth of a disturbance can
be calculated by integrating the complex αalong x, yielding
[vc
x, vc
r, vc
θ, pc] = ℜ
[H, iF, G, P] e
i(
x
R
x0
αdx+mθ−ωrt)
(6.11)
where αis complex and x0is the location of the first measured profile with x0/D = 0.2. In
order to consistently ensemble the local eigenmodes to a global solution, the eigenvector X
is normalized using the Euclidean norm kXk=p(X, X), and the phase angle is equalized
Chapter 6 Coherent Structures in the Strongly Swirling Jet 123
x/D
y/D
0.5 1 1.5 2 2.5 3
-0.5
0
0.5
(a) Transversal coherent velocity vc
y
y/D
-0.5 0 0.5
(b) Radial coherent velocity vc
rat
x/D = 0.57
Figure 6.15: Coherent velocity of the most amplified normal mode. Eigenfunctions
are shown at an arbitrary phase-angle. The contour lines represent vc
r/max(vc
r) =
(−0.8,−0.6,−0.4,−0.2,0.2,0.6,0.4,0.8). The eigenvalue problem (2.14) is solved for the
azimuthal wavenumber m= 1 and global frequency St = 0.49. The dashed circle in (b)
indicates the nozzle diameter which approximately separates the inner and the outer shear
layer.
at the characteristic jet radius Rmax. The resulting coherent velocity distribution along the
stream-wise plane is shown in figure 6.15 (a).
Except for the jet core region upstream x/D = 0.7 where the wavemaker is located,
the computations agree well with the empirical POD modes shown in figure 6.9. The axial
wavelength and the radial amplitude distribution are well estimated. Even the overall
amplification is well approximated with the maximum coherent amplitude at x/D ≈1.25
and the decaying region further downstream.
6.7 Three-dimensional Shape of the Global Mode
Finally, the empirically extracted mode and the mode derived from the stability model,
are constructed in the entire three-dimensional domain. By adding the mean flow to the
coherent velocities, the global mode renders a physically plausible shape as can be seen
in figure 6.16. The blue helix represents an iso-surface having constant phase-averaged
azimuthal vorticity. For the upper figure the phase-average is derived applying the empirical
approach based on the POD as outlined in section 6.5, whereas for the lower figure the
phase-averaged velocity is derived from the linear stability analysis based on the analytically
approximated mean flow as outlined in 6.6. Both methods reveal the same structure. It
represents the m= 1 mode that rotates in the clockwise direction with the base flow at
a rotation rate corresponding to St = 0.49. The yellow streamlines and the gray LIC-
surface (linear integral convolution Cabral & Leedom (1993); Stalling & Hege (1995)) are
computed from the actually measured time-averaged flow. The streamlines are intended to
visualize the direction of rotation of the mean flow and also indicate the mean boundary
of the recirculation bubble. Note that the streamlines are orientated perpendicularly to
124 Chapter 6 Coherent Structures in the Strongly Swirling Jet
the roll-axis of the helical mode. Accordingly, the helical structure is co-rotating counter-
winding. Similar results have been found in experiments by Liang & Maxworthy (2005)
and in DNS computations conducted by Ruith et al. (2003). The LIC surface visualizes the
flow structures inside the recirculation bubble showing a stationary ring-like vortex. This
structure must be carefully interpreted because the time-averaged flow field of the inner
region of the recirculation bubble differs strongly from the phase-averaged or instantaneous
flow. It is observed that the entire recirculation bubble meanders around the jet center in
phase with the outer coherent structure. As mentioned above, the frequency of the global
mode is dictated by the wavemaker located at the jet center. This precessing of the vortex
core is not visible in figure 6.16 since the level of the vorticity iso-contour is selected to most
properly visualize the structures in the outer region of the jet and is therefore too high for
visualizing the structures located in the interior of the jet. The 3D flow visualizations have
been prepared with Amira software by Christoph Petz (Zuse-Institute, Berlin).
6.8 Summary and Discussion
The coherent structures of strongly swirled jets undergoing vortex breakdown are investi-
gated. The nature of these structures is phenomenologically observed, whereupon power
spectra measured by a hot-wire indicate that this flow regime exhibits pronounced harmonic
oscillations. The origin of these oscillations is studied by two independent experimental
techniques. First, the lock-in behavior of the forced flow is investigated and it reveals a lin-
ear relationship between the forcing frequency and the critical lock-in amplitude. Second,
the limit-cycle amplitude of the dominant mode is observed to be proportional to the square
root of the deviation from the swirl magnitude considered as a control parameter. Both
observations corroborate quantitative relationships of a self-excited oscillatory global mode
originating from a supercritical Hopf bifurcation. In other words, the results are consistent
with mean-field theory. The phase-averaged velocities show the existence of a co-rotating
counter-winding helical mode. Most energy of its intensity is located in the outer region
of the jet. A similar structure was observed experimentally by Liang & Maxworthy (2005)
and in numerically simulations by Ruith et al. (2003). Theoretical considerations of Gal-
laire et al. (2006) suggest that the self-excited global mode originates from a local region
of absolute instability located in the jet center. These results are confirmed by the present
observation where a strong precessing of the vortex core sets the pace of the amplified
instabilities in the outer part of the jet.
The observed oscillatory helical structures are then closely examined. Starting with an
experiment using 2D uncorrelated PIV snapshots in a streamwise and a cross-flow plane,
the 3D time-dependent coherent structures are extracted by kinematic and dynamic consid-
erations, exploiting the observed dominant periodicity of the flow. The kinematic velocity
field reconstruction starts from the uncorrelated 2D streamwise velocity fields determined
by PIV. The pronounced oscillatory nature of the fluctuations is evidenced by two leading
POD modes spanning a corresponding convecting vortex pattern. These two POD modes
allow one to attribute a phase to each snapshot taken. A continuous time dependence
is imposed by assuming a single oscillation frequency, consistent with the experimental
measurements. This assumption allows for restoration of time dependence provided that
small amplitude variations of the turbulent flow such as the less-energetic higher harmonics
Chapter 6 Coherent Structures in the Strongly Swirling Jet 125
(a) Empirically extracted coherent structure using POD
(b) Theoretically extracted coherent structure using linear stability analysis
Figure 6.16: Three-dimensional flow field visualizing the global mode m= 1 that is
dominating the nearfield of a strongly swirled jet (ReD= 20000; S= 1.22). The blue iso-
contour represents the constant azimuthal vorticity of phase-averaged velocities indicating
the streamwise growth of a single-helical instability in the outer region of the jet. Stream-
lines and LIC surface are based on the time-averaged flow, visualizing the recirculation
bubble and the sense of rotation of the swirling jet.
126 Chapter 6 Coherent Structures in the Strongly Swirling Jet
and stochastic small-scale fluctuations are negligible. A 3D flow pattern is reconstructed
by exploiting an azimuthal symmetry of the observed helical coherent structures. The
resulting 3D flow pattern corroborates PIV observations in crossflow planes. Thus, the
spatio-temporal evolution of the full 3D helical structure is obtained from 2D uncorrelated
data sets.
The velocity field of the global mode is also derived by means of a linear stability analysis
employing the measured mean flow. A spatial approach is justified by previous experiments,
showing that the wavemaker generating the global oscillations is located inside the jet
central core upstream of the recirculation bubble imposing its frequency on the outer shear
layer where instabilities are convective. Hence, for a purely spatial analysis, the precessing
vortex core upstream of the vortex breakdown location is considered as a ’natural oscillator’
and the convectively unstable surrounding flow field is modeled as being externally forced.
The spatial approach is corroborated a posteriori by the good agreement of the stability
eigenmodes, amplification rates, and wavelengths with the corresponding quantities of the
measured phase-averaged flow, particularly in the periphery of the recirculation bubble.
The theoretical prediction is less accurate in the interior of the jet near the wavemaker
location due to the convective type of the analysis. The good agreement in the convective
unstable region also gives credibility to the above-mentioned empirical reconstruction of
the 3D time-periodic flow, using the same velocity field ansatz.
Finally, a three-dimensional portrait of the global mode is constructed from experimental
data and from the theoretical model. Both approaches represent a co-rotating, counter-
winding single-helical coherent structure that is wrapped around the recirculation zone – in
a remarkable agreement. The vortex axis is perpendicular to the mean flow direction which
is characteristic for the Kelvin–Helmholtz type of instability. The good agreement between
instability and POD-based eigenmodes is neither self-evident nor completely unusual in free
shear flows. This study reveals that the highly swirled jet has similar dynamics as, e.g. the
wake flow with an absolutely unstable clock-work of vortex shedding in the recirculation
zone and convectively moving structures in the far wake. Moreover, the study provides
simple and effective kinematic and dynamic tools that complete the coherent-structure
extraction from 2D PIV data. The proposed approach is expected to be applicable to a
large class of other shear flows.
Chapter 7
Open-loop Control of the
Self-excited Swirling Jet
In this investigation, harmonic actuation is applied at the nozzle lip with
the aim to control the self-excited oscillations associated with the spiral vor-
tex breakdown. A spatio-temporal analysis of the mean natural flow associates
these flow oscillations to a global instability mode. The wavemaker of this mode
is located upstream of a large pocket of absolutely unstable flow that is located
in the vortex breakdown bubble. The limit-cycle oscillations are associated with
a co-rotating counter-winding single-helical coherent structure that originates
from a precessing vortex core that is most energetic in the wavemaker region.
The self-excited oscillations can be synchronized to an external signal by forcing
at the same azimuthal mode (global lock-in). By actuating the flow at a differ-
ent azimuthal mode, it is possible to excited convectively unstable modes in the
outer shear layer that gain large amplitudes with downstream distance. The
resulting coherent structures enhance the streamwise growth of the outer shear
layer and cause a downstream displacement and reduction of the vortex break-
down bubble. This leads to a reduction of the internal feedback mechanism
and the natural global mode is suppressed. In consistency, the spatio-temporal
stability analysis applied to the forced flow reveals that the pocket of absolutely
unstable flow diminishes with increasing forcing amplitude.
7.1 Objectives and Approach
The investigation described in this chapter deals with open-loop control of a turbulent
swirling jets undergoing vortex breakdown. This flow phenomenon is known to occur when
the ratio of axial flux of azimuthal to axial flux of axial momentum exceeds a certain
threshold. It is characterized by the appearance of a stagnation point on the jet centerline
and the creation of an internal recirculation zone. A comprehensive investigation of the
formation of vortex breakdown and the onset of global instability is given in chapter 4.
Swirling jets undergoing vortex breakdown are commonly used to improve the efficiency
of swirl-stabilized combustors. In these devices, the internal recirculation zone acts as a
flame holder, and the enhanced turbulent mixing leads to a reduction in NOxemissions.
127
128 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
As a drawback, swirl-stabilized (in particular lean premixed) combustors are susceptible to
self-excited combustion oscillations (see review article of Huang & Yang 2009)). The driving
mechanism of these thermo-acoustic instabilities is the dynamic coupling of heat release rate
and acoustic pressure oscillations. This feedback cycle can be interrupted by controlling
the flow oscillations caused by large-scale flow structures. Their occurrence in swirling jets
has been extensively studied within the last decade. The most prominent structure that is
consistently found in experimental and numerical investigations is a self-excited large-scale
oscillation that is associated with an unstable global mode (Gallaire et al. 2006;Liang &
Maxworthy 2005;Ruith et al. 2003). It is characterized by a co-rotating, counter-winding
instability wave with an azimuthal wavenumber m= 1 that originates from a pocket of
absolutely unstable flow (see section 2.4 for reviews on local/global instability concepts).
This scenario is confirmed by the spatio-temporal stability analysis of Gallaire et al. (2006)
based on the laminar swirling jet that was simulated numerically by Ruith et al. (2003). In
chapter 6of this thesis, the shape of this global mode is predicted by computing the most
convectively unstable mode based on the measured turbulent mean flow. Its agreement
with phase-averaged measurements is found to be reasonably good within the shear layers
surrounding the region of reversed flow.
The applicability of the convective analysis is of great importance for efficient flow
control. It implies the validity of the signaling problem, meaning that the outer shear layer
acts as an amplifier to upstream perturbations. To recall, the convectively unstable nature
of axisymmetric (Cohen & Wygnanski 1987;Crighton & Gaster 1976) and plane shear
layers (Oster & Wygnanski 1982) have enabled efficient jet-noise and separation control by
means of periodic excitation (Greenblatt & Wygnanski 2000).
In the present investigation, the feasibility of controlling this global mode with azimuthal
wavenumber m= 1 that dominates the strongly swirling jet is investigated experimentally.
Periodic excitation is applied at the nozzle lip where the outer shear layer is most receptive
to external forcing (Cohen & Wygnanski 1987;Long & Petersen 1992). The flow response
is assessed using stereo-PIV. A spatio-temporal stability analysis is applied to the natural
and actuated mean flow states. This allows a theoretical study of the impact of forcing on
the global stability of the flow.
Experiments are conducted using the swirling jet air facility at the TU Berlin (see
section 3.2). Details to the PIV measurements are given in section 3.3. The outline of this
chapter is as follows. The base flow configuration considered is described in section 7.2. The
results of the open-loop control experiments are described in section 7.3, containing mode,
amplitude, and frequency variations. Moreover, the impact of the forcing on the mean flow
and its spatio-temporal stability is discussed. The main observations are summarized in
section 7.6.
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 129
7.2 Description of the Unforced Flow
7.2.1 Characteristic Numbers
The global parameters that characterize the forced swirling jets are the swirl number S,
Reynolds number ReD, and the Strouhal number St, defined as
ReD=DV
ν, S =˙
Gθ
(D/2) ˙
Gx
, and St = fD
V.
The Reynolds number ReDis based on the nozzle diameter Dand on the bulk velocity
V, which is derived from the mean mass flow rate. It is set to ReD= 20000 throughout
this investigation, yielding V= 5.8 m/s. The Strouhal number St characterizes the forcing
frequency fand it is based on the nozzle diameter Dand the bulk velocity V. The swirl
number Sis defined as the ratio between the axial flux of angular momentum ˙
Gθand the
axial flux of axial momentum ˙
Gx. Since the axial momentum flux must be conserved in
axial direction, the swirl number Smust remain constant with axial distance. However, as
demonstrated in chapter 4, the commonly used expressions for the axial flux of momenta
that are
˙
Gθ= 2πρ
∞
Z0
VxVθr2dr and ˙
Gx= 2πρ
∞
Z0V2
x−V2
θ
2rdr (7.1)
are inaccurate in the region of vortex breakdown. The underlying boundary layer approx-
imations are invalid in the vicinity of vortex breakdown due to the strong jet divergence.
Hence, it is necessary to omit these simplifications and consider additional terms of the
equations of motion (Rajaratnam 1976), yielding the more complex expressions for the
axial flux of azimuthal momentum
˙
Gθ= 2πρ
∞
Z0
(VxVθ+v′
xv′
θ)r2dr (7.2)
and for the axial flux of axial momentum
˙
Gx= 2πρ
∞
Z0 V2
x−V2
θ
2+(v′
x)2−(v′
θ)2+ (v′
r)2
2!r+Vx
∂Vr
∂x +Vr
∂Vr
∂r r2
2dr. (7.3)
Figure 7.1 depicts the swirl number based on the simplified expressions (7.1) together with
the swirl number based on the expressions (7.2-7.3), both derived from the PIV measure-
ments of the present flow configuration. It is clearly shown that upstream of x/D = 2, the
appearance of vortex breakdown and the associated jet divergence falsifies the swirl number
based on the simplified equations. Downstream of the recirculation bubble (x/D > 2), both
quantities merge to an approximate value of S= 1, which remains constant throughout the
remaining measurement domain.
130 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
Swirl numbers
x/D
S
S(simplified)
0 1 2 3 4
0.5
1
1.5
Figure 7.1: Swirl number versus axial distance. In the region of vortex breakdown
(x/D < 2) the simplified swirl number based on (7.1) differs strongly from the one based
on (7.2-7.3).
7.2.2 Flow Features
The swirling jet configuration considered is dominated by a large-scale oscillatory mode.
A detailed discussion of the corresponding coherent flow structures is given in chapter 6.
This mode is characterized by a helical instability wave with azimuthal wavenumber m= 1
rotating in the same direction as the base flow but winding in opposite direction. It is
shown in chapter 6of this thesis that this mode arises from a supercritical Hopf bifurcation
with a critical swirl number of S= 0.88. Hence, the presently considered swirl number,
S= 1, is supercritical and the global mode oscillates at its limit-cycle.
Figure 7.2 highlights the main features of the base flow configuration. The swirling
jet at S= 1 undergoes vortex breakdown and flow stagnation occurs on the jet axis at
x/D ≈0.51 (see streamlines in figure 7.2a). A large recirculation bubble is created near
the orifice and the jet emanating from the nozzle is distorted to an annular swirling jet. At
the nozzle exit, the axial velocity profile is nearly top-hat shaped with strong axial shear
between the jet and the quiescent surrounding fluid (see Vx-profiles in figure 7.2b). At
x/D ≈0.3 an inner shear layer appears and the axial velocity profiles become wake-like
near the jet center.
The thick black lines in figures 7.2b and c mark the center of the axial shear layers,
which coincide with the locations of maximum axial shear ∂Vx/∂r. Note that the axial
shear at the jet center near the nozzle exit is caused by an overshoot of the axial velocity
profile (confer with most upstream velocity profile in figure 7.2b). This velocity hump is
generated by the swirling jet passing the contraction upstream of the orifice (Batchelor
1967;Billant et al. 1998;Panda 1990).
The global mode is driven by a precessing vortex core (PVC) which is located upstream
of the recirculation bubble. The outer shear layer, which is thin near the nozzle exit and
highly receptive to external perturbations, amplifies the dynamics of the PVC emanating
from the jet core. This leads to synchronized helical coherent structures that grow rapidly
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 131
Figure 7.2: Natural flow at S= 1 and ReD= 20000: (a) 2D-streamlines and contours
of coherent energy; (b) thin black lines represent the normalized axial velocity profiles
Vx/V , thick black lines indicate the center of axial shear layer where Vx/V = 0.5, contours
represent the axial shear intensity;(c) contours (video in online version) showing contours
of coherent vorticity Ωc
z, black thick lines represent the center of axial shear layer.
y/D
(a) |vc|2
V2
y/D
(b) ∂Vx/∂r
D/V
y/D
x/D
(c) Ωc
z
Ωc
z,max
01 2 3
-1
0
1
0
2
4
6
0
0.02
0.04
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
132 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
with downstream distance, ultimately dominating the entire nearfield dynamics. A 3D-
visualization of this global mode can be found in Petz et al. (2011). It is reprinted in
appendix C. The kinetic energy distribution of this mode is depicted in figure 7.2a. It
is derived from PIV measurements that are phase-averaged with respect to the global
oscillation. At the nozzle exit, energy is concentrated around the jet axis depicting the
PVC. With larger downstream distance, coherent energy increases in the outer and in the
inner shear layer, indicating the synchronized oscillations. At x/D ≈1, where the bubble
has its maximum diameter, the oscillations in the inner and outer shear layer merge and
the energy peaks at the maximum of the axial velocity profile. Downstream of this axial
position, the distinction between inner and outer shear layer becomes meaningless as the
disturbance waves do not travel independently at each side of the annular jet. In fact, the
annular jet itself oscillates at the global frequency. Contours of phase-locked through-plane
vorticity of the m= 1 mode, depicted at an arbitrary phase angle in figure 7.2c, reveal
consistent wave-like structures that arise from the jet core and move outwards around the
recirculation bubble where they merge with the structures in the outer shear layer. Note
that the through-plane vorticity Ωzof the mode m= 1 is axisymmetric by definition
and shows no discontinuity on the jet axis, while the azimuthal vorticity component that
is skew-symmetric for m= 1 would yield a singularity on the jet axis and is, therefore,
inappropriate to visualize the flow structures in the streamwise plane. Hence, the strong
eddy centered at x/D = 1.5 and y/D = 0.7 corresponds to a spiral-shaped vortex that is
connected to the oppositely signed eddies centered at x/D = 1 and x/D = 2 at y/D =−0.7.
7.3 Open-loop Control
The following section describes how the swirling jet at S= 1 responds to sinusoidal ex-
citation applied at x= 0. The experimental facility allows to force axisymmetric and
azimuthal modes. To demonstrate the effect of forcing, data taken along the crosswise
plane of measurement are considered first. The first row in figure 7.3 shows contours of the
mean natural flow and the mean forced flow. The second and third row in figure 7.3 show
contours of the phase-locked natural and forced flow at an arbitrary phase angle. For the
unforced flow the phase-average is derived with respect to the naturally prevailing global
mode oscillations.
The contours shown in the first row of figure 7.3 reveal that the mean flow is nearly ax-
isymmetric for forced and natural conditions. No deformation of the mean flow is observed
for any forcing case that would indicate the interaction of two modes.
The phase-locked flow forced at m= 1 differs only marginally from the natural flow,
showing a precessing vortex core near the jet center. By forcing m= 1, a lock-in to the
natural helical instability is achieved. Details on the lock–in characteristic of mode m= 1
are given in section 6.4.2, revealing typical oscillator dynamics.
By forcing the flow at higher modes, helical waves amplify in the outer shear layer and
double- and triple-spiral vortices are generated. The phase-locked jet core is deformed into
an elliptical (m= 2) or triangular (m= 3) shape. By forcing m6= 1, instabilities are
amplified that evolve primarily in the outer shear layer.
In fact, preliminary studies have shown that an actuation of mode m= 0, m= 2, m= 3,
or m= 4 have a similar affect on the vortex breakdown and on the associated global mode
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 133
m= 1 (L¨uck 2009). However, mode m= 2 undergoes the highest overall amplification and
is, thus, considered as most efficient for active flow control. Hence, the following discussion
focuses on the flow forced at m= 2. The forcing frequency of St = 0.44 used for the mode
and amplitude variation studies. The frequency dependency is discussed in detail in section
7.3.2.
7.3.1 Amplitude Variation
The nature of the excited instability is of great importance for flow control applications.
The question is whether the forced mode m= 2 is convectively or absolutely unstable.
According to linear stability theory, shear flows that are convectively unstable everywhere,
act as a linear amplifier to infinitesimal disturbances. This stands in contrast to absolutely
unstable flows that behave as oscillators (Huerre & Monkewitz 1990).
Figure 7.4a shows the amplitude of mode m= 2, integrated across the shear layers,
at x/D = 0.6 versus speaker input voltage Aspk. At this axial location the modes grow
exponentially in downstream direction. The linear relation between forcing amplitude and
mode amplitude proves the validity of the signaling problem for mode m= 2. Thus, the
downstream exponential growth of mode m= 2 can be predicted by spatial linear stability
analysis. This important finding enables to apply flow control at small forcing amplitudes
in order to effectively control the flow further downstream.
Figure 7.4b shows the square of the saturation amplitude of the forced mode m= 2
versus speaker amplitude. Accordingly, the forced instabilities saturate at amplitudes that
correlate linearly with the square root of the forcing amplitude, yielding
Z|vc|2rdr1/2
∝pAspk −Aspk,0,
with Aspk,0= 20.7 mV being the critical forcing amplitude above which m= 2 is amplified.
The growth or decay rate of the forced instabilities are determined by the local stability
of the mean flow. Without a significant change of the mean flow, the overall maximum
amplitude of the mode forced at the same frequency should scale linearly with the input
amplitude. Hence, the presently observed nonlinear dependence of the overall maximum
amplitude on the forcing amplitude, as shown in figure 7.4b, indicates a significant manip-
ulation of the mean flow.
Throughout the remaining chapter, the input amplitude voltage of the speaker is nor-
malized by the critical forcing amplitude, yielding the dimensionless amplitude
A∗
spk =Aspk
Aspk,0
.(7.4)
Thus, the input voltage Aspk = [25,50,75,100,125] mV corresponds to the dimensionless
amplitude A∗
spk ≈[1.2,2.4,3.6,4.8,6].
Figure 7.5 shows the streamwise energy distribution of mode m= 1 and mode m= 2,
for the latter being forced at various amplitudes. Recall that the amplitude of mode m= 1
can only be extracted from the PIV snapshots when its energy is high enough to perform a
reliable phase average based on POD. For A∗
spk = 0, no forcing is applied and mode m= 1
134 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
natural flow (mean) forced m= 2 (mean)
natural flow (phase-locked) forced m= 1 (phase-locked)
forced m= 2 (phase-locked) forced m= 3 (phase-locked)
Figure 7.3: Cross-section of the jet undergoing vortex breakdown in the center of the
recirculation zone (x/D = 0.6). Contours refer to the mean (first row) and phase-locked
(second and third row) axial velocity, depicted at an arbitrary phase angle. 2D-streamlines
are computed from vp
yand vp
z. The dashed white circle indicates the diameter of the nozzle
exit. All modes rotate clockwise in the same direction as the base flow and are forced at
the same frequency (St = 0.44).
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 135
Aspk in mV
R|vc|2rdr1/2/D/V x/D=0.6
(a)
20.7 mV
Aspk in mV
R|vc|2rdr/D2/V 2sat
(b)
0 25 50 75 100 125
0 25 50 75 100 125 0
0.01
0.02
0.03
0
0.05
0.1
0.15
0.2
Figure 7.4: (a) Amplitude of mode m= 2 versus forcing amplitude. Linear fit (black line)
confirms the convective nature of the excited mode. (b) Square of saturation amplitude of
mode m= 2 versus forcing amplitude.
x/D
R|vc|2rdr/D2/V 2
m= 1 m= 2
x/D
A∗
spk = 0
A∗
spk = 1.2
A∗
spk = 2.4
A∗
spk = 3.6
A∗
spk = 4.8
A∗
spk = 6
01 2 30 1 2 3
0
0.01
0.02
Figure 7.5: Streamwise distribution of kinetic energy of the natural mode m= 1 (left
frame) and the forced mode m= 2 (right frame) forced at 40 Hz corresponding to St = 0.44
is dominant. Its energy saturates at x/D ≈1, which coincides approximately with the
maximum diameter of the recirculation bubble. Forcing mode m= 2 at A∗
spk = 1.2 does
not influence mode m= 1 and there is hardly any energy for m= 2 detectable. However,
for A∗
spk >1.2, mode m= 2 undergoes significant streamwise amplification upstream of
x/D ≈1 and mode m= 1 gets successively damped. For A∗
spk >3.6 mode m= 1 is too
small to be accurately identified. Note that for A∗
spk = 3.6, mode m= 1 peaks at x/D ≈2,
indicating that the forcing dampens and displaces the global mode peak amplitude.
The mean flow correction that ultimately leads to a dampening of the global mode
is characterized in section 7.4. It is interesting to note that the modes saturate at the
136 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
x/D
St
0
St = 0.22
St = 0.44
St = 0.66
St = 0.88
x/D
iRx
0αidx
0 0.5 1 1.5 2
0 1 2 0
0.5
1
1.5
0
1
2
3
4
0
0.5
1
Figure 7.6: Results from spatial linear stability analysis employing the natural mean
flow. Left: Contours of spatial amplification rate −αiDof mode m= 2; dashed lines
indicate the forcing frequencies of the experiments; right: overall amplification of forced
mode m= 2.
same axial location regardless of the actuation amplitude. At this streamwise location, the
inner and outer shear layers merge and one may speculate whether the nonlinear mean
flow correction at this streamwise location interrupts the resonance principle of the global
mode.
7.3.2 Frequency Variation
Within the framework of spatial linear stability analysis, modes forced at different fre-
quencies undergo different amplification cycles. Figure 7.6a depicts contours of the spatial
amplification rate αiof mode m= 2 derived from a spatial stability analysis of the natural
mean flow. Figure 7.6 (b) displays the integral −Rx
0αidx, which refers approximately to
the overall amplification of mode m= 2 forced at a constant frequency St. As shown in
chapter 5, this quantity overestimates the actually measured growth rate due to the non-
uniformity of the flow. For the four frequencies considered, the computations predict the
highest overall amplification for St = 0.22, followed by St = 0.44,0.66, and 0.88. This
disagrees with the energy distribution shown in figure 7.7. Experiments assign mode m= 2
forced at St = 0.44 to reach the highest overall amplification, causing the strongest sup-
pression of mode m= 1. Forcing at St = 0.88 does not show any significant impact on the
global mode, at all. The stability analysis based on the natural flow fails to predict the
growth rates due to the significant mean flow change caused by the nonlinear interaction
of the forced instability. Nevertheless, qualitatively speaking one may derive the following
conclusions: Forcing at St = 0.44 is most effective in dampening the global mode because
it goes through its complete amplification cycle before it reaches x/D = 1, the streamwise
location where m= 1 is naturally most energetic. This is not the case for St = 0.22. In
contrast, instabilities forced at St = 0.88 grow rapidly, saturate, and decay before they
interact with the global mode. This scenario is supported by the fundamental derivations
of Pier (2003). He demonstrates that flows with absolutely unstable regions are controlled
by upstream harmonic forcing only when the forced mode reaches nonlinear saturation at
sufficiently high amplitude upstream of the absolutely unstable region.
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 137
x/D
R|vc|2rdr/D2/V 2
m= 1 m= 2
x/D
natural
St = 0.22
St = 0.44
St = 0.66
St = 0.88
0 1 2 30 1 2 3
0
0.01
0.02
Figure 7.7: Streamwise development of coherent energy of the natural mode m= 1 (left
frame) and the mode m= 2 forced at A∗
spk = 4.8 (right frame)
7.4 Impact of Forcing on the Flow Characteristics
As mentioned above, the actuated instabilities saturate nonlinearly at x/D = 1, thereby
significantly manipulating the mean flow and its stability. The streamlines shown in figure
7.8 depict how the forcing alters the mean flow. The colored contours refer to the kinetic
energy of the natural mode (shown for y > 0) and the kinetic energy of the actuated mode
(shown for y < 0). The streamlines indicate how the recirculation bubble is diminished and
displaced downstream with increasing forcing amplitude. The coherent energy distribution
reveals a successive dampening of mode m= 1 with increasing forcing amplitude. Mode
m= 2 forced at sufficiently high amplitude grows initially in the outer shear layer in contrast
to the natural mode that arises from the PVC located at the jet center. The maximum
coherent energy of the actuated mode is located at x/D ≈1 which agrees quite well with the
theoretically predicted overall amplitude (confer with figure 7.6). In that region coherent
energy is transfered to the inner shear layer, the origin of the global mode, presumably
leading to the nonlinear saturation depicted in figure 7.4b. Hence, the forced mode m= 2
originates in the outer shear layer and spreads to the inner one, whereas for the natural
flow, the mode m= 1 originates from the jet center and spreads to the outer shear layer. It
should be mentioned here that the actuated system as depicted in figure 7.8 is described by
two oscillators (mode m= 1 and m= 2) with their amplitudes being nonlinearly coupled.
A dynamic model based on such system has been proposed by Luchtenburg et al. (2009)
and has been successfully applied to the present flow configuration. Details can be found
in the work of Sieber et al. (2011) and Sieber (2012).
The instabilities excited by the external actuation may reach higher overall amplitudes
than the waves forced by the PVC. This causes higher entrainment rates of the jet near the
nozzle resulting in an enhanced streamwise growth of the outer shear layer. The thickness
of the outer shear layer is derived from the PIV measurements by computing its momentum
138 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
y/Dy/Dy/D
y/D
x/D
A∗
spk = 4.8|vc|2
V2
natural
A∗
spk = 2.4
A∗
spk = 3.6
0 0.5 1 1.5 2 2.5 3
0.02
0.04
0.06
0.08
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
natural mode
m= 1
actuated mode
m= 2
Figure 7.8: Contours of coherent energy for the natural and actuated flow and 2D-
streamlines depicting the mean flow change due to forcing of mode m= 2 at St = 0.44;
the contours shown for positive yrefer to the natural mode and the contours shown for
negative yrefer to the actuated mode.
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 139
x/D
δx/D
A∗
spk = 0
A∗
spk = 1.2
A∗
spk = 2.4
A∗
spk = 3.6
A∗
spk = 4.8
A∗
spk = 6
0 0.5 1 1.5
0
0.02
0.04
0.06
0.08
0.1
Figure 7.9: Streamwise distribution of the momentum thickness of the outer shear layer
for the swirling jet forced at m= 2 and St = 0.44 at various amplitudes.
thickness
δx=Z∞
Rmax
Vx
Vx,max 1−Vx
Vx,max dr, (7.5)
where Rmax refers to the radial location of the maximum axial velocity Vx,max. Figure
7.9 depicts the streamwise development of δxfor increasing forcing amplitudes. At the
nozzle, δxis unaffected by the actuation, which implies that the imposed amplitudes are
sufficiently small at the nozzle exit not to alter the mean flow. With increasing downstream
distance, the growth of the outer shear layer is enhanced significantly for the jet forced at
A∗
spk ≥3.6 compared to the natural flow. Surprisingly, for A∗
spk ≤2.4, a forcing amplitude
at which the global mode is already somewhat damped, the outer shear layer thickness
remains equal to the natural case. Hence, a thickening of the outer shear layer is not the
cause for the global mode dampening. It is solely a consequence of the enhanced overall
amplitude of the coherent flow structures caused by sufficiently strong forcing. For the flow
actuated at A∗
spk ≥3.6, δxremains constant within the region 0.5< x/D < 1, indicating the
region where the actuated waves interact with the inner shear layer. Further downstream,
the forced waves cover the inner and the outer shear layers and definition (7.5) becomes
meaningless.
Finally, the impact of the open-loop forcing on the location and the strength of the
vortex breakdown is considered. In vortex breakdown studies it is a challenging task to
define a global variable that characterizes the state of vortex breakdown. An intuitive
quantity might be the size and the location of the recirculation bubble. The white-filled
black circles in figure 7.10 depict the mean streamwise location of the upstream and down-
stream stagnation points for the natural and the forced flow. The error bars represent the
corresponding RMS values of the instantaneous streamwise locations. Hence, a long error
bar refers to a stagnation point that oscillates heavily in streamwise direction. For the
natural flow, the upstream stagnation point is located at x/D ≈0.5 with RMS value of
0.15. The corresponding downstream stagnation point is located at x/D ≈1.5 and shows
much higher fluctuations. The vortex breakdown bubble for the natural flow has a nearly
140 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
x/D
A∗
spk
stagnation points on the jet axis
recirculation region
0 1.2 2.4 3.6 4.8 6
0
0.5
1
1.5
2
Figure 7.10: Impact of forcing mode m= 2 on location and size of the recirculation
bubble. White-filled black circles refer to the mean streamwise locations of the upstream
and the downstream stagnation point. Error bars indicate the RMS of the corresponding
instantaneous streamwise locations.
steady upstream stagnation point, while its ’tail’ oscillates heavily in streamwise direction.
This is consistent with the observations reported in chapter 4.
Forcing mode m= 2 at moderate amplitudes shifts the upstream end of the recirculation
bubble downstream and enhances its streamwise fluctuation. It appears that a dampening
of the precessing vortex core destabilizes the vortex breakdown bubble, in agreement with
the experiments discussed in chapter 4, revealing decreasing vortex breakdown oscillations
with the onset of the global mode m= 1. Moreover, it is found that the downstream
displacement of vortex breakdown correlates linearly with pAspk −Aspk,0, revealing the
same nonlinearity as the saturation amplitude of the forced mode (confer with figure 7.4).
This indicates a link between the downstream displacement of vortex breakdown and asso-
ciated mean flow change and the interaction between the inner and the outer shear layer
associated with the nonlinear saturation.
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 141
7.5 Impact of Forcing on Global Instability
A major interest in applied fluid mechanics is the prediction of the impact of external
forcing on the mean flow and its dynamics. Forced instabilities that grow in shear flows can
reach sufficiently high amplitudes to trigger nonlinearities that alter the mean flow and, in
turn, its instability. The analysis of a flow dominated by large-scale oscillations, caused by
either high amplitude forcing (convectively unstable) or self-excitation (absolutely/globally
unstable), is most accurate when based on the mean flow that incorporates nonlinear effects.
In contrast, an analysis of the non-forced, non-oscillating flow yields inaccurate results, as
the mean flow distortion, which goes in hand with the instability, cannot be assessed from
linear stability theory. Modeling the link between the coherent velocity component of a
strong instability and the Reynolds stresses that alter the mean flow is still an unsolved
problem of turbulence closure (Reau & Tumin 2002). The analysis based on the non-forced,
non-oscillating flow (base flow) is capable to predict the linear growth of instabilities forced
at low and moderate amplitudes. For heavily oscillating flows, as considered here, the
analysis must be based on the actual mean flow and is used as an analytic tool rather than
a predictive tool.
The main theoretical approach is in line with the spatial stability analysis conducted
in the investigations described in chapter 5and 6. However, in the present investigation,
a spatio-temporal analysis is performed that involves the tracking of saddle points in the
complex wavenumber plane for complex frequencies. This procedure is numerically more
demanding than the spatial analysis conducted in the previous parts. The theoretical
background and numerical implementation is briefly described section 2.4.
In this section, results are presented for the spatio-temporal analysis of the natural and
the forced swirling jet. As mentioned in section 2.4, the analysis is capable of detecting
regions of absolute instability, which correspond to a positive absolute temporal growth
rate ω0,i. Several questions can be answered by this theoretical approach:
•Is the flow globally unstable to the m= 1 mode?
•Is the frequency of the precessing vortex core in agreement with the theoretically
derived global frequency?
•Where is the region of absolute instability located?
•Where is the wavemaker located?
•Is the flow absolutely unstable to other modes than m= 1?
•How does the forcing of mode m= 2 alter the global stability?
These questions can be answered by considering the theoretical results depicted in fig-
ure 7.11. The black thick line represents the streamwise distribution of the absolute tem-
poral growth rate ω0,i of mode m= 1 for the baseline configuration (figure 7.11d). There
is a region of absolutely unstable flow located inside the recirculation bubble. The global
frequency ωsand the wavemaker location Xsare derived from the streamwise distribu-
tion of ω0by employing the criterion (2.12). The wavemaker is located at Xs≈0.4D,
indicated in figure 7.11a–d by the dashed vertical line. The global frequency is found to
be ωs= 52 Hz in very good agreement with the measured PVC oscillation frequency of
51.5 Hz. Moreover, the spatio-temporal analysis conducted for modes other than m= 1
142 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
could not reveal any absolutely unstable flow – neither for the natural nor for the forced
flow. Hence, mode m= 1 remains the only absolutely unstable mode for all considered
flow configurations. The present analysis confirms the experimental findings described in
the chapters 4and 6, and it is line with the spatio-temporal analysis conducted by Gallaire
et al. (2006) employing the base flow of a laminar swirling jet simulated by Ruith et al.
(2003).
Winding up, the swirling jet undergoing vortex breakdown becomes globally unstable
to mode m= 1. The limit cycle oscillation of this mode is characterized by a precessing
vortex core and the helical roll-up of the outer shear layer. The present analysis predicts
the wavemaker to be located upstream of the recirculation bubble, contrasting the results
of Gallaire et al. (2006), who found the wavemaker to be located inside the bubble. The
dashed vertical line in figure 7.11a–d depicts that for the present flow, the source of the
global mode is located exactly where the center of the inner shear layer (thick black solid
line) intersects the jet axis. This axial position coincides with the maximum coherent
energy along the jet axis (7.11a), corresponding to the location of strongest dynamics of
the precessing vortex core.
Figure 7.11d further depicts the absolute growth rate of mode m= 1 for the jet forced
at m= 2. As mentioned in earlier, the analysis is not capable to predict the complicated
nonlinearities that lead to the drastic mean field changes discussed in the previous section.
However, a spatio-temporal analysis of the forced flow may shed some light on the involved
mechanisms a posteriori. Forcing at A∗
spk = 1.2 does not suffice to excite instabilities, and
the mean flow and corresponding ω0,i remain unchanged (not shown). Forcing the flow at
A∗
spk = 2.4 reduces slightly the spatial extent of the region of reversed flow (figure 7.8
and 7.10). This results in a quite significant change of ω0,i, leading to a reduction of the
region of absolutely unstable flow. At a forcing amplitude of A∗
spk = 3.6, the mean flow
is significantly altered and the flow becomes convectively unstable in the bubble, which
implies that the flow must be globally stable. Hence, one may interpret the dampening
of the global mode as a consequence of the mean flow change due to the excitation of
convectively unstable modes. This scenario is in good agreement with the generalized
mean-field model (Luchtenburg et al. 2009;Noack et al. 2003).
7.6 Summary and Discussion
Open-loop control of a turbulent swirling jet undergoing vortex breakdown was investigated.
Experiments were conducted at S= 1 and ReD= 20000. At this swirl number, the jet
has undergone a supercritical Hopf bifurcation to a global mode m= 1. The aim of this
investigation was to control the limit cycle oscillations of this global mode by means of low
amplitude sinusoidal forcing at the nozzle lip.
The phase-averaged velocities of the natural flow reveal that the global oscillations
originate from a precessing vortex core that is located upstream of the internal recirculation
zone. Perturbations emanating from the vortex core propagate to the outer shear layer
where large-scale helical coherent structures evolve. A spatio-temporal analysis based on the
natural mean flow reveals that mode m= 1 is the only globally unstable mode with its global
frequency to be equal to the measured frequency of the precessing vortex core. The analysis
Chapter 7 Open-loop Control of the Self-excited Swirling Jet 143
y/D
(a) |vc|2
V2
y/D
(b) ∂Vx/∂r
D/V
y/D
(c) Ωc
z/Ωc
z,max
ω0,iD/V
x/D
(d)
A∗
spk = 0
A∗
spk = 1.2
A∗
spk = 2.4
A∗
spk = 3.6
00.5 1 1.5 2 2.5 3
-1
0
1
0
2
4
6
0
0.02
0.04
-2
-1
0
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
Xs
Figure 7.11: Natural flow at S= 1 and ReD= 20000 : (a) 2D-streamlines together with
contours of coherent energy; (b) black thin lines refer to normalized axial velocity profiles
Vx/V , black thick lines refer to center of axial shear layer Vx/V = 0.5, contours refer to
axial shear intensity;(c) contours of coherent vorticity Ωc
zdepicted at an arbitrary phase
angle, black thick lines refer to center of axial shear layer Vx/V = 0.5;(d) spatio-temporal
analysis: absolute temporal growth rate ω0,i versus xfor baseline and forced configurations,
regions with ω0>0 refer to absolute unstable flow.
144 Chapter 7 Open-loop Control of the Self-excited Swirling Jet
predicts the wavemaker of this mode to be located upstream of the vortex breakdown bubble
at the streamwise location of strongest vortex core precession (x/D ≈0.4).
A manipulation of the mean flow at the wavemaker location should be most efficient
to control the global mode. However, in the present experiment, external actuation was
applied at the nozzle lip, which can only indirectly influence the wavemaker dynamics.
Forcing mode m= 1 caused the global mode to lock-in to the actuation (confer with chapter
6). Forcing mode m= 2 excited a double-helical instability wave in the jet periphery
that grew exponentially in downstream direction. Its amplitude scales linearly with the
actuation amplitude, revealing that mode m= 2 is convectively unstable. Hence, the outer
shear layer acts as a linear amplifier for upstream perturbations – a necessity for efficient
active flow control.
The actuated instabilities saturated nonlinearly at x/D > 0.7. In this saturation region,
coherent energy is transferred from the outer to the inner shear layer, causing a drastic
change of the mean flow in the jet interior that results in a significant suppression of the
precessing vortex core.
The impact of the forcing on the mean flow was validated by tracking the location of the
recirculation bubble for different forcing amplitudes. With increasing amplitude, the size
of the recirculation zone is successively decreased and the bubble is displayed downstream.
This drastically changes the global stability of the entire flow. A spatio-temporal analysis
of the forced flow reveals that the flow becomes globally stable at sufficiently strong forcing
amplitudes.
Hence, the receptivity of the outer shear layer allows for stabilization of the global-
mode associated precession of the vortex core using low actuation amplitudes. The excited
convectively unstable modes amplify to sufficient amplitudes to interact nonlinearly with
the mean flow, thereby reducing the absolute growth rate of the m= 1 mode. In other
words, the actuation at the jet periphery is capable to shift the critical swirl number of the
Hopf bifurcation to values beyond S= 1.
Moreover, a frequency dependence of the forced mode m= 2 and its effectiveness in sup-
pressing the global mode is observed. Waves that reach their maximum amplitude (neutral
instability) in the region of inner and outer shear layer interaction are most effective in sup-
pressing the global mode. This scenario is supported by recent theoretical considerations
of Pier (2009).
Chapter 8
Concluding Remarks
The swirling jet configuration combines a variety of physical mechanisms whose complex
interactions are very challenging. Despite the scientific challenge, the understanding and
control of this flow is particularly useful for improving combustion processes (see review ar-
ticles by Candel (2002); Huang & Yang (2009); Lieuwen & Yang (2005)). The present work
is intended to shed some light on the natural and externally controlled coherent structures
that evolve in swirling jets below and above the onset of vortex breakdown. Through-
out this work, results are collected by means of state-of-the-art experimental methods in
conjunction with linear stability analysis.
The theoretical concepts used in this work are rather classic. However, their applica-
tion is usually restricted to purely theoretical investigations that deal flows obtained from
direct numerical simulations at very low Reynolds numbers or from simplified models. In
the present work, the analysis is applied to actually measured turbulent flows and the re-
sults are quantitatively compared to the experiments. The good agreement between the
theoretical and experimental results presented in this thesis confirm that linear stability
analysis is more than a tool to predict the laminar–to–turbulent transition. When applied
to a turbulent mean flow, this method serves as a theoretical framework to predict the
dominant flow dynamics, thereby, intrinsically capturing the involved nonlinearities. This
allows to reconstruct reduced order models that are of great importance for active control
of turbulent flows.
Previous investigations on swirling jets suffer from the fact that there is no universal
swirl configuration. The stability of a swirling base flow depends significantly on the consid-
ered velocity profiles, and hence, each proposed swirling jet configuration exhibits different
stability mechanisms. This brings up a zoo of different instability modes that are proposed
to be crucial in swirling jets. The investigation of the stability of moderately swirling jets
described in chapter 6is, therefore, chosen to be strictly in line with an experimental in-
vestigation, thereby, accepting a loss in generality. As a consequence, the derived stability
characteristics quantitatively agree with the corresponding experimental results, but they
can only be qualitatively related to other swirling jet experiments. Nonetheless, the ex-
haustive examination of the theoretical results brings up several general statements that
should be valid for various types of swirling jet configurations. These are, for instance, the
relevance of an azimuthal group velocity that results in a manipulation of the streamwise
phase velocity, or the swirl-induced destabilization of steady and streamwise modes. It
further clarifies recent theoretical arguments regarding the swirl-enhanced jet spreading.
145
146 Chapter 8 Concluding Remarks
The major part of this thesis deals with swirling jets undergoing vortex breakdown.
State-of-the-art measurement techniques are employed to extract the dynamics that dom-
inate this flow configuration. Theoretical concepts are applied in order to predict the
onset of vortex breakdown and the formation of a global mode. The detailed investiga-
tion of the formation of vortex breakdown (chapter 4) reveals that vortex core criticality
and shear layer instability are two independent mechanisms. With increasing swirl, the jet
first undergoes vortex breakdown and then becomes globally unstable. The corresponding
global mode, discussed in detail in chapter 6, is characterized by a single-helical co-rotating
counter-winding mode, a robust feature that was observed in numerous previous studies of
laminar swirling jets (e.g., see Gallaire et al. 2006;Liang & Maxworthy 2005;Ruith et al.
2003). The precessing vortex core, which is often observed in swirl-stabilized combustor
flows, reveals the same characteristic as the presently observed global mode. The compar-
ison of the present results with previous investigations suggest that the flow states that
occur during the formation of the spiral-shaped vortex breakdown are universal and do not
depend on the initial conditions.
Furthermore, this work reveals control methods that enable this global mode to be
locked-in or dampened, thereby, utilizing the receptivity of the outer shear layer. The
actuation was applied at the nozzle lip where the outer shear layer is most receptive,
ensuring the lowest input cost for the strongest streamwise amplification. However, as
described in chapter 7, the source of the natural global mode is located at the jet center
and can, therefore, only be indirectly affected by the imposed forcing. A future project will
focus on active and passive control methods that act directly on the source of the global
instability.
It is surprising to note that the precession of the vortex core that is often observed
in combustor flows has so far not been directly related to global stability concepts. This
is partly attributed to the poor interconnection between applied research and theoretical
work. In the field of combustion, there is still some confusion about the connection between
shear layer instabilities and the vortex core dynamics. The previous is assumed to be
responsible for the roll-up of the outer shear layer to large-scale flow structures that may
significantly effect the flame dynamics. The present work shows that the precessing vortex
core corresponds to the wavemaker of the global mode, and it is located upstream of the
breakdown bubble. It acts as an internal clockwork, perturbing the convectively unstable
outer shear layer. The receptivity of that outer shear layer is related to its radial thickness
and so is the streamwise growth rate of the large-scale flow structures. The precession of
the vortex core at the jet center and the streamwise growing waves in the jet periphery are
synchronized to the same global frequency and both correspond to the same global mode.
Appendix A
Flow Properties of the Swirling
Water Jet
A.1 Mean Flow Properties
A.1.1 Velocity Profiles
Dimensionless axial, radial, and azimuthal velocity components are shown in the first three
columns of figure A.1 for swirl numbers S= 0.38,1.01 below SVB and the swirl numbers
S= 1.07,1.12,1.37 above SVB. The axial distance between two profiles corresponds to a
magnitude of one. Note that the radial velocity Vris scaled four times larger. Azimuthal
vorticity contours ωθtogether with projected streamlines are shown in the right column.
The black dots correspond to measurements conducted by Billant et al. (1998) at similar
S. The maxima of the profiles agree well, whereas the present data shows thicker shear
layers indicating enhanced turbulent diffusion.
Vx-component: For moderate swirl (S= 0.38, figure A.1a), the axial velocity profile is
jet-like everywhere. The overshoot near the jet axis at x/D = 0.5 is created by the rotating
flow passing through the contraction (Batchelor 1967). This is consistent with comparable
investigations (Billant et al. 1998;Panda & McLaughlin 1994;Shiri et al. 2008;Toh et al.
2010). For higher swirl (S= 1.01, figure A.1b), the profiles render a downstream transition
from jet-like to wake-like. Upon further increasing swirl, this velocity deficit is enhanced,
leading to the appearance of an internal recirculation zone (S= 1.07) that moves upstream
and grows in axial and radial distance, rendering the profiles wake-like within the entire
measurement domain (S= 1.12,1.37). The profiles describe an annular jet with an inner
and outer shear layer that strongly spreads in radial direction.
Vr-component: For moderate swirl (S= 0.38), negative values for x/D > 0.5 indicate
entrainment of steady surrounding fluid. For higher swirl (S= 1.01), the radial profiles
become positive, indicating enhanced jet spreading. For swirl beyond breakdown, Vris
considerably higher upstream of the first stagnation point, where the streamlines indicate
highly divergent flow. E.g. for S= 1.12 at x/D = 0.5, Vrpeaks at 20 % of the bulk
velocity, forming strong gradients in radial direction. Negative radial velocity for r/D < 1
is found downstream of the recirculation region, indicating the closure of the jet.
147
148 Appendix A Flow Properties of the Swirling Water Jet
x/D
x/Dx/Dx/Dx/D
Vx/V
(a)
4Vr/V Vθ/V ωθin s−1
(b)
(c)
(d)
(e)
(·)
Vb=1
r/D r/D r/D r/D
-2 0 2
0 0.5 10 0.5 10 0.5 10 0.5 1
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
Figure A.1: From left to right: mean axial, radial, azimuthal velocity profiles,
and azimuthal vorticity contours with projected streamlines; rows (a-e) represent S=
(0.38,1.01,1.07,1.12,1.37), respectively; black dots represent data recasted from (Billant
et al. 1998).
Appendix A Flow Properties of the Swirling Water Jet 149
Vcl/VVmax/V
Vmin
r,max/V
S= 0.00
S= 0.38
S= 1.01
Vθ,max/V
S
S= 1.07
S= 1.12
S= 1.37
x/D
0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
1.2
-0.1
0
0.1
0.2
0.3
0
0.5
1
1.5
0
0.5
1
1.5
Rmax/DR.5/D
Rθ/DRcore
x/D
0.5 1 1.5 2 2.5 3
0.4
0.6
0.8
1
0.2
0.4
0.6
0.8
1
0.4
0.6
0.8
1
1.2
0
0.2
0.4
0.6
0.8
1
Figure A.2: Characteristic velocity and length scales versus axial distance.
Vθ-component: Below vortex breakdown (S= 0.38), the swirl component can be
characterized by an inner region where it increases approximately linear with r(solid body
rotation) and an outer region where it decays smoothly in outward direction. The decaying
region forms an azimuthal shear layer that spreads radially with increasing downstream
distance. In the case of vortex breakdown (e.g. S= 1.12), the azimuthal component in
the recirculation region is very low (e.g. x/D = 1). There, the profiles render a concave
curvature, revealing a second deflection point in the jet core.
A.1.2 Characteristic Velocity and Length Scales
The spatial development of the mean flow is summarized in figure A.2 for the same swirl
numbers as presented in figure A.1. The right column shows the characteristic velocity
scales Vcl,Vmax,Vmin
r,max, and Vθ,max versus axial distance and the left column shows the
quantities Rmax,R.5, and Rθversus axial distance.
150 Appendix A Flow Properties of the Swirling Water Jet
Vx-component: For moderate swirl (S= 0.38), the velocity on the centerline decays
for x/D ≈1.2, depicting the axial extent of the potential core. The growth rate of the half-
width of the jet R.5is comparable to the non-swirling jet. With higher swirl (S= 1.01),
the decay of Vcl is strongly enhanced and a local minimum appears at x/D ≈2.25. The
jet half-width increases rapidly with xat an approximately constant growth rate. The
radial location of Vmax is displaced from the jet axis at x/D ≈1.1, indicating the point of
jet-wake-transition. Further downstream, Rmax saturates at Rmax/D ≈0.6. Vcl indicates
reversed flow when the swirl is increased to S= 1.07. The rapid growth of R.5is initiated
further upstream but the growth rate remains equal to S= 1.01. The location, where Rmax
is displaced from the jet axis, is shifted upstream. This trend is continued for S= 1.12. At
S= 1.37 the recirculation region has reached the vicinity of the nozzle. The streamwise
growth rate of the jet diameter is reduced and develops nonlinearly with x.Rmax decays
for x/D > 1.5 indicating the jet-closure downstream of the recirculation region. Vcl,Vmax,
and Rmax converge to the same values at the downstream end of the measurement domain
for S≥1.01, indicating that a change in swirl drastically alters the nearfield but does not
substantially influence the flow downstream of x/D > 2.
Vr-component: Vmin
r,max corresponds to either the maximum or the minimum of Vr,
depending on which quantity has a higher magnitude. The radial component is negative
in regions of strong entrainment and positive in regions of strong jet spreading. Symbols
on different sides of the abscissa correspond to different characteristic points of the ve-
locity profiles and are, therefore, not connected. For moderate swirl (S= 0.38), strong
entrainment is indicated by negative radial velocity for x/D < 2.25. With 3 % of the bulk
velocity, it is approximately four times higher than for non-swirling jets. For higher swirl
(S= 1.01), the entrainment near the nozzle is enhanced. Downstream of x/D = 0.76,
jet divergence becomes more pronounced and the outwards directed flow reaches 8 % of
the bulk velocity at x/D = 1.2. Upon increasing the swirl to S= 1.07, this maximum is
shifted upstream and increases to 28 % at x/D = 0.6. Downstream of x/D = 1.25, Vmin
r,max
is negative, indicating the inward directed flow downstream of the recirculation region.
Vθ-component: For moderate swirl (S= 0.38), the azimuthal component decays down-
stream of x/D = 1. This correlates to the end of the potential core and is presumably
attributed to turbulent diffusion. Note that the jet spreading indicated by Rθand R.5
is minor. For higher swirl, the maximum of the swirl component is shifted outwards at
the jet-wake transition point. The conservation of tangential momentum requires a rapid
downstream decay of Vθ,max. In consistency with Rmax, this point is shifted upstream with
a further increase in swirl.
A.2 Turbulent Fluctuations
A.2.1 Turbulent Normal Stress
Dimensionless turbulent normal stresses are shown in the first three columns of figure A.3
for swirl numbers S= 0.38,1.01 below SVB and the swirl numbers S= 1.07,1.12,1.37 above
SVB. The axial distance between two profiles corresponds to a magnitude of 0.25. Contours
of the total turbulent kinetic energy together with projected streamlines are shown in the
right column.
Appendix A Flow Properties of the Swirling Water Jet 151
x/Dx/D
x/Dx/Dx/D
qv′2
x/V
(a)qv′2
r/V qv′2
θ/V v′2
x+v′2
r+v′2
θ
3V2
(b)
(c)
(d)
(e)
(·)
Vb=.25
r/D r/D r/D r/D
-0.2 0 0.2
00.5 10 0.5 1
0 0.5 10 0.5 1
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
Figure A.3: From left to right: axial, radial, azimuthal profiles of turbulent normal
stress, and contours of total turbulent kinetic energy with projected streamlines; rows
(a-e) represent S= (0.38,1.01,1.07,1.12,1.37), respectively.
152 Appendix A Flow Properties of the Swirling Water Jet
For all configurations, the axial and azimuthal fluctuations peak at similar magnitude
and location. The radial component is always somewhat weaker. Near the nozzle exit
upstream of the recirculation region, qv′2
xand qv′2
θpeak on the jet axis and in the center
of the outer shear layer. The previous indicates strong oscillation of vortex breakdown and
associated meandering of the jet core (e.g. S= 1.07, x/D = 0.5) while the latter indicates
eddies that grow in downstream direction due to the unstable axial and azimuthal shear
layer. Approaching the region of vortex breakdown from upstream, the inner peak moves
outwards and merges with the peak of the outer shear layer, yielding very weak fluctuations
in the recirculation region.
A.2.2 Turbulent Shear Stress
Dimensionless turbulent shear stresses are shown in the first three columns of figure A.4 for
swirl numbers S= 0.38,1.01 below SVB and the swirl numbers S= 1.07,1.12,1.37 above
SVB. The axial distance between two profiles corresponds to a magnitude of 0.05. Contours
of the total turbulent shear stresses together with projected streamlines are shown in the
right column.
For moderate swirl (S= 0.34), high levels of u′w′are generated due to the coexistence
of an axial and an azimuthal shear layer. The peak is located on the inner boundary of the
outer shear layer. With incipient breakdown (S≥1.01), a second peak of v′
xv′
θappears in
the inner shear layer upstream of the recirculation region. It appears that v′
xand v′
θare
strongly correlated in the region of vortex core precessing. In the recirculation region, the
turbulent shear stresses diminish completely.
Appendix A Flow Properties of the Swirling Water Jet 153
x/Dx/Dx/Dx/Dx/D
v′
xv′
r
V2
(a)
v′
xv′
θ
V2
v′
rv′
θ
V2
v′
xv′
r+v′
xv′
θ+v′
rv′
θ
3V2
(b)
(c)
(d)
(e)
(·)
V2b=.05
r/D r/D r/D r/D
-.02 .02
00.5 10 0.5 1
0 0.5 10 0.5 1
1
2
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
0.5
1
1.5
2
Figure A.4: Profiles of turbulent shear stresses; rows (a-e) represent S=
(0.38,1.01,1.07,1.12,1.37), respectively.
Appendix B
Supplemental Information of the
Swirling Air Jet Facility
B.1 The Swirling Jet Facility
The flow enters the facility through two blowers that permit the independent setting of
the Reynolds number and the swirl number. The volume flow was derived by measuring
the pressure drop across a calibrated orifice. In order to provide constant volume flow
for long time measurements, the volume flow was controlled in a closed-loop manner by
using a real-time data acquisition setup that allows to re-adjust the blowers automatically
and monitor the actual mass flow. A LabView program was written by Moritz Sieber that
allows for zeroing the Baratrons connected to the measuring orifices and adjusting the swirl
or Reynolds number independently.
After passing a honeycomb, the axial flow concentrically enters the swirl generator.
The azimuthal flow enters the swirl chamber through a divider and four symmetrically
arranged flexible tubes. Therein, the flow is guided through the slots between four curved
and azimuthally displaced guiding plates to the inside of the swirl generator and is merged
with the axial flow. In figure B.2 the swirl generator is hidden behind a curved perforated
plate which improves the homogeneity of the azimuthal flow. The swirled flow leaves the
swirl generator through a 800 mm long duct, before it is converged in a nozzle. The section
from the axial blower to the nozzle inlet has a diameter of 154 mm. The diameter of the
nozzle outlet is D= 51 mm.
The first design depicted in figure B.1 was further improved for the following reason.
Measurements of mean velocity profiles of the jet using Laser Doppler Velocimetry revealed
an asymmetry of both the non-swirling and swirling jet. Thus, several adjustments on the
swirling jet facility have been made. Figure B.1 shows the first state of the experimental
setup, and figure B.2 shows the second state. The search for the cause of the asymmetry
involved the rotation of the nozzle, the upstream duct, the swirl generator, and the whole
swirl chamber. Comparing the asymmetric velocity profiles of each setup, all changes had
an influence to a greater or lesser extent but none was solely responsible for the asymmetry.
In order to make the azimuthal flow into the swirl generator symmetrical, a divider, four
flexible tubes, additional wooden plates in the swirl chamber, and a curved perforated plate
surrounding the swirl generator have been added to the setup. Moreover, the four guiding
155
156 Appendix B Supplemental Information of the Swirling Air Jet Facility
O51
550
1
24°
honey comb
axial inlet
tangential
inlet
swirl generator
grid
excitation device
contraction
actuators
slot
contraction wall
slot
nozzle lip
35011566751125191,8
251,586
3,5
O154
Figure B.1: Sketch of the swirling jet facility (first state)
plates of the swirl generator were glued in place more accurately. The rebuild of the swirl
chamber significantly improved the symmetry of the jet. Another perforated plate was
added right downstream of the swirl generator. At last, a somewhat wavy grid upstream
of the nozzle was removed, which also made an improvement.
B.2 The Actuation Device
The facility allows for zero mass flow excitation at the nozzle lip. Commercial hi-fi speakers
are used as actuators that are mounted to a metal plate that houses the wave guides
that terminate at slots that are 1 mm thick. The following pages show photographs and
technical drawings of the excitation device. It was made from aluminum plates using a
CNC-controlled milling machine.
Appendix B Supplemental Information of the Swirling Air Jet Facility 157
measuring nozzle
6
divider
?
blowers
?measuring orifices for
axial and azimuthal
flow
J
J^
divider
*
flexible tubes -
location of honeycomb
?
curved perforated plate
?
swirl chamber
?
contractionAAA
AU
actuators :
excitation device
:
Figure B.2: Photographs of the swirling jet facility (second state)
158 Appendix B Supplemental Information of the Swirling Air Jet Facility
Figure B.3: Photographs of the actuation device with unmounted front plate giving
optical excess to the speakers and the wave guides.
Figure B.4: Technical drawing of the complete excitation device
Appendix B Supplemental Information of the Swirling Air Jet Facility 159
Figure B.5: Technical drawing of the nozzle plate with details on the wave guide geome-
tries.
160 Appendix B Supplemental Information of the Swirling Air Jet Facility
Figure B.6: Technical drawing of the nozzle front plate with details.
Appendix B Supplemental Information of the Swirling Air Jet Facility 161
Figure B.7: Technical drawing of the nozzle adapter.
Figure B.8: Technical drawing of the nozzle ring.
162 Appendix B Supplemental Information of the Swirling Air Jet Facility
input voltage in mV peak slot velocity in m/s peak slot mass flow in %
25 1.42 0.23
50 2.79 0.46
75 4.16 0.68
100 5.53 0.9
125 6.9 1.13
Table B.1: Speaker input voltage used for the forced experiments and corresponding peak
velocity and slot mass flow relative to the mass flow of the main jet.
(a) Hot-wire measurements
speaker input voltage in mV
peak velocity in m/s
0 50 100 150 200
0
2
4
6
8
10
(b) Speaker input voltage versus peak slot velocity
for 50 Hz actuation frequency. Symbols refer to hot-
wire measurements and black line refers to a linear
fit.
Figure B.9: Speaker calibration at no-flow condition
B.3 Actuator Calibration
The peak velocity of the zero mass-flow jet generated by a single actuator was measured
at no-flow conditions using a hot-wire probe that was placed at the slot exit (see figure
B.9a). This was done for several excitation amplitudes. The frequency that was used for
this study was 50 Hz. Figure B.9b shows the linear relation between the speaker input
voltage and the peak velocity at the slot exit. Table B.1 lists the values corresponding to
the actuation amplitude used in this thesis. The peak slot mass flow is calculated using
the area of one slot. Moreover, all eight speakers were adjusted to the same sound pressure
level by using a microphone place at the nozzle exit at r= 0.
Appendix C
Visualization of the Global Mode
Figure C.1: Visualizations are based on the three-dimensional phase-averaged velocity
field (Re = 20000) that is constructed from uncorrelated 2D PIV snapshots. Thereby
the focus is placed on three flow features: the internal recirculation zone that is charac-
teristic for vortex breakdown (semi-transparent gray pathline-surface in the center); the
meandering vortex core that acts as the pacemaker for the global oscillations (central
streak-lines and bluish streak-surface); helical waves in the outer shear layer that amplify
near the nozzle and roll up to spiral vortices (semi-transparent greenish streak-surface).
The visualization was prepared by Christoph Petz using the AIMIRA software environ-
ment developed at the Zuse Institute, Berlin. The picture won the gallery of fluid motion
award 2011 and is published in Physics of Fluids (Petz et al. 2011)
163
Bibliography
Akilli, H., Sahin, B., & Rockwell, D. (2003). “Control of vortex breakdown by a
coaxial wire.” Physics of Fluids,15:123–133.
Barkley, D. (2006). “Linear analysis of the cylinder wake mean flow.” EPL (Europhysics
Letters),75:750–756.
Batchelor, G. K. (1967). An Introduction to Fluid Dynamics. Cambridge University
Press.
Benjamin, T. B. (1962). “Theory of the vortex breakdown phenomenon.” Journal of
Fluid Mechanics,14:593–629.
Berkooz, G., Holmes, P., & Lumley, J. L. (1993). “The proper orthogonal decompo-
sition in the analysis of turbulent flows.” Annual Review of Fluid Mechanics,25:539–575.
Billant, P., Chomaz, J.-M., & Huerre, P. (1998). “Experimental study of vortex
breakdown in swirling jets.” Journal of Fluid Mechanics,376:183–219.
Briggs, R., J. (1964). Electron-Stream Interaction with Plasmas. MIT Press, Cambridge,
Massachusetts.
Brown, G. L. & Lopez, J. M. (1990). “Axisymmetric vortex breakdown. II - Physical
mechanisms.” Journal of Fluid Mechanics,221:553–576.
Bruecker, C. & Althaus, W. (1995). “Study of vortex breakdown by particle tracking
velocimetry (PTV) Part 3: Time-dependent structure and development of breakdown-
modes.” Experiments in Fluids,18:174–186.
Cabral, B. & Leedom, L. C. (1993). “Imaging vector fields using line integral con-
volution.” In: Proceedings of the 20th Annual Conference on Computer Graphics and
Interactive Techniques, SIGGRAPH 1993, pp. 263–270. ACM, New York, NY, USA.
Candel, S. (2002). “Combustion dynamics and control: Progress and challenges.” Pro-
ceedings of the Combustion Institute,29(1):1–28.
Cantwell, B. J. (1981). “Organized motion in turbulent flow.” Annual Review of Fluid
Mechanics,13:457–515.
Carton, X. & McWilliams, J. (1989). “Barotropic and baroclinic instabilties of ax-
isymmetric vortices in a quasi-geostrophic model.” In: J. Nihoul & B. Jamart (eds.),
Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence, p. 225¨ı¿1
2244. Else-
vier.
165
166 BIBLIOGRAPHY
Chigier, N. & Chervinsky, A. (1965). “Experimental and theoretical study of turbulent
swirling jets issuing from a round orifice.” Israel Journal of Technology,4:44–54.
Chigier, N. & Chervinsky, A. (1967). “Experimental investigation of swirling vortex
motion in jets.” Journal of Applied Mechanics,34:443–451.
Chigier, N. A. & Beer, J. M. (1964). “Velocity and static-pressure distribution in
swirling air jets issuing from annular. and divergent nozzles.” ASME Journal of Basic
Engineering,86:788–796.
Chomaz, J. M. (1992). “Absolute and convective instabilities in nonlinear systems.”
Physical Review Letters,69(13):1931–1934.
Chomaz, J.-M. (2005). “Gobal Instabilities in Spatially Developing Flows: Non-Normality
and Nonlinearity.” Annual Review of Fluid Mechanics,37:357–392.
Chomaz, J. M., Huerre, P., & Redekopp, L. G. (1988). “Bifurcations to local and
global modes in spatially developing flows.” Physical Review Letters,60:25–28.
Chomaz, J.-M., Huerre, P., & Redekopp, L. G. (1991). “A frequency selection
criterion in spatially developing flows.” Studies in Applied Mathematics,85:119.
Cohen, J. & Wygnanski, I. (1987). “The evolution of instabilities in the axisymmetric
jet. part 1. the linear growth of disturbances near the nozzle.” Journal of Fluid Mechanics,
176:191–219.
Cohen, J. & Wygnanski, I. (1987). “The evolution of instabilities in the axisymmetric
jet. Part 2. The flow resulting from the interaction between two waves.” Journal of Fluid
Mechanics,176:221–235.
Cohen, J., Marasli, B., & Levinski, V. (1994). “Interaction between the mean flow and
coherent structures in turbulent mixing layers.” Journal of Fluid Mechanics,260:81–94.
Cooper, A. J. & Peake, N. (2002). “The stability of a slowly diverging swirling jet.”
Journal of Fluid Mechanics,473:389–411.
Crighton, D. G. & Gaster, M. (1976). “Stability of slowly diverging jet flow.” Journal
of Fluid Mechanics,77:397–413.
Cutler, A. D., Kraus, D. K., & Levey, B. S. (1995). “Near-field flow of supersonic
swirling jets.” AIAA Journal,33:876–881.
Delbende, I., Chomaz, J.-M., & Huerre, P. (1998). “Absolute/convective instabilities
in the Batchelor vortex: a numerical study of the linear impulse response.” Journal of
Fluid Mechanics,355:229–254.
Depardon, S., Lasserre, J. J., Brizzi, L. E., & Bor´
ee, J. (2007). “Automated
topology classification method for instantaneous velocity fields.” Experiments in Fluids,
42:697–710.
Duwig, C. & Fuchs, L. (2007). “Large eddy simulation of vortex breakdown/flame
interaction.” Physics of Fluids,19:5103.
BIBLIOGRAPHY 167
Escudier, M. P. & Keller, J. J. (1985). “Recirculation in swirling flow - A manifesta-
tion of vortex breakdown.” AIAA Journal,23:111–116.
Faler, J. H. & Leibovich, S. (1978). “An experimental map of the internal structure of
a vortex breakdown.” Journal of Fluid Mechanics,86:313–335.
Freymuth, P. (1966). “On transition in a separated laminar boundary layer.” Journal of
Fluid Mechanics,25:683–704.
Gallaire, F. & Chomaz, J.-M. (2003). “Mode selection in swirling jet experiments: a
linear stability analysis.” Journal of Fluid Mechanics,494:223–253.
Gallaire, F., Chomaz, J.-M., & Huerre, P. (2004). “Closed-loop control of vortex
breakdown: a model study.” Journal of Fluid Mechanics,511:67–93.
Gallaire, F., Rott, S., & Chomaz, J.-M. (2004). “Experimental study of a free and
forced swirling jet.” Physics of Fluids,16:2907–2917.
Gallaire, F., Ruith, M., Meiburg, E., Chomaz, J.-M., & Huerre, P. (2006).
“Spiral vortex breakdown as a global mode.” Journal of Fluid Mechanics,549:71–80.
Gaster, M. (1974). “On the effects of boundary-layer growth on flow stability.” Journal
of Fluid Mechanics,66:465–480.
Gaster, M. & Grant, I. (1975). “An experimental investigation of the formation and
development of a wave packet in a laminar boundary layer.” Royal Society of London
Proceedings Series A,347:253–269.
Gaster, M., Kit, E., & Wygnanski, I. (1985). “Large-scale structures in a forced
turbulent mixing layer.” Journal of Fluid Mechanics,150:23–39.
Greenblatt, D. & Wygnanski, I. J. (2000). “The control of flow separation by periodic
excitation.” Progress in Aerospace Sciences,36(7):487 – 545.
Gursul, I. & Yang, H. (1995). “On fluctuations of vortex breakdown location.” Physics
of Fluids,7:229–231.
Gutmark, E. J., Schadow, K. C., & Yu, K. H. (1995). “Mixing enhancement in
supersonic free shear flows.” Annual Review of Fluid Mechanics,27:375–417.
Hall, M. G. (1972). “Vortex Breakdown.” Annual Review of Fluid Mechanics,4:195–218.
Ho, C.-M. & Gutmark, E. (1987). “Vortex induction and mass entrainment in a small-
aspect-ratio elliptic jet.” Journal of Fluid Mechanics,179:383–405.
Ho, C.-M. & Huerre, P. (1984). “Perturbed free shear layers.” Annual Review of Fluid
Mechanics,16:365–424.
Holmes, P., Lumley, J. L., & Berkooz, G. (1998). Turbulence, Coherent Structures,
Dynamical Systems and Symmetry. Cambridge University Press.
Hu, G.-H., Sun, D.-J., & Yin, X.-Y. (2001a). “A numerical study of dynamics of a
temporally evolving swirling jet.” Physics of Fluids,13:951–965.
168 BIBLIOGRAPHY
Hu, G.-H., Sun, D.-J., Yin, X.-Y., & B., T. (2001b). “Studies on stability and dynamics
of a swirling jet.” Acta Mechanica Sinica,17:237–244. 10.1007/BF02486879.
Huang, Y. & Yang, V. (2009). “Dynamics and stability of lean-premixed swirl-stabilized
combustion.” Progress in Energy and Combustion Science,35(4):293 – 364.
Huerre, P. & Monkewitz, P. A. (1990). “Local and global instabilities in spatially
developing flows.” Annual Review of Fluid Mechanics,22:473–537.
Hussain, A. K. M. F. & Reynolds, W. C. (1970). “The mechanics of an organized
wave in turbulent shear flow.” Journal of Fluid Mechanics,41:241–258.
Juniper, M. P., Li, L. K., & Nichols, J. W. (2009). “Forcing of self-excited round jet
diffusion flames.” Proceedings of the Combustion Institute,32(1):1191 – 1198.
Juniper, M. P., Tammisola, O., & Lundell, F. (2011). “The local and global stability
of confined planar wakes at intermediate Reynolds number.” Journal of Fluid Mechanics,
686:218–238.
Khorrami, M. R. (1991). “On the viscous modes of instability of a trailing line vortex.”
Journal of Fluid Mechanics,225:197–212.
Khorrami, M. R., Malik, M. R., & Ash, R. L. (1989). “Application of spectral collo-
cation techniques to the stability of swirling flows.” Journal of Computational Physicsl,
81(1):206–229.
Knowles, K. & Saddington, A. J. (2006). “A review of jet mixing enhancement for
aircraft propulsion applications.” Proceedings of the Institution of Mechanical Engineers
Part G Journal of Aerospace Engineering,220(2):103–127.
Kuhn, P. (2010). Untersuchung der r¨aumlichen und zeitlichen Anfachung von akustisch
angeregten Instabilit¨atsmoden in Drallstrahlen. Bachelorarbeit, TU-Berlin. Bachelorar-
beit 897.
Lambourne, N. & Bryer, D. (1962). “The bursting of leading-edge vortices: some
observations and discussion of the phenomenon.” Tech. rep., Ministry of Aviation.
Landau, L. & Lifshitz, E. (1987). Fluid Mechanics, Vol. 6.
Laufer, J. (1975). “New trends in experimental turbulence research.” Annual Review of
Fluid Mechanics,7:307–326.
Leibovich, S. (1978). “The structure of vortex breakdown.” Annual Review of Fluid
Mechanics,10:221–246.
Leibovich, S. (1984). “Vortex stability and breakdown - Survey and extension.” AIAA
Journal,22:1192–1206.
Leibovich, S. & Stewartson, K. (1983). “A sufficient condition for the instability of
columnar vortices.” Journal of Fluid Mechanics,126:335–356.
Lessen, M., Singh, P. J., & Paillet, F. (1974). “The stability of a trailing line vortex.
Part 1. Inviscid theory.” Journal of Fluid Mechanics,63:753–763.
BIBLIOGRAPHY 169
Liang, H. & Maxworthy, T. (2004). “Vortex Breakdown and Mode Selection of a
Swirling Jet in Stationary or Rotating Surroundings.” APS Meeting Abstracts, pp. 6–+.
Liang, H. & Maxworthy, T. (2005). “An experimental investigation of swirling jets.”
Journal of Fluid Mechanics,525:115–159.
Lieuwen, T. C. & Yang, V. (eds.) (2005). Combustion Instabilities in Gas Turbine
Engines, vol. 210 of Progress in Astronautics and Aeronautics. AIAA, Inc.
Liu, J. T. C. (1971). “Nonlinear Development of an Instability Wave in a Turbulent
Wake.” Physics of Fluids,14:2251–2257.
Liu, J. T. C. (1989). “Coherent structures in transitional and turbulent free shear flows.”
Annual Review of Fluid Mechanics,21:285–315.
Loiseleux, T. & Chomaz, J.-M. (2003). “Breaking of rotational symmetry in a swirling
jet experiment.” Physics of Fluids,15:511–523.
Loiseleux, T., Chomaz, J. M., & Huerre, P. (1998). “The effect of swirl on jets
and wakes: Linear instability of the Rankine vortex with axial flow.” Physics of Fluids,
10:1120–1134.
Loiseleux, T., Delbende, I., & Huerre, P. (2000). “Absolute and convective insta-
bilities of a swirling jet/wake shear layer.” Physics of Fluids,12:375–380.
Long, T. A. & Petersen, R. A. (1992). “Controlled interactions in a forced axisymmet-
ric jet. Part 1. The distortion of the mean flow.” Journal of Fluid Mechanics,235:37–55.
Lu, G. & Lele, S. K. (1999). “Inviscid instability of compressible swirling mixing layers.”
Physics of Fluids,11:450–461.
Lucca-Negro, O. & O’Doherty, T. (2001). “Vortex breakdown: a review.” Progress
in Energy and Combustion Science,27(4):431 – 481.
Luchtenburg, D. M., G¨
unther, B., Noack, B. R., King, R., & Tadmor, G. (2009).
“A generalized mean-field model of the natural and high-frequency actuated flow around
a high-lift configuration.” Journal of Fluid Mechanics,623:283–+.
L¨
uck, M. (2009). Control of Absolute Instability in Highly Swirled Jets using Azimuthal
Forcing. Diplomarbeit, TU-Berlin. Diplomarbeit 849.
Ludwieg, H. (1961). “Erg¨anzung zu der arbeit: Stabilit¨at der str¨omung in einem zylin-
drischen ringraum.” Z. Flugwiss,9:359.
Lumley, J. L. (1967). Atmospheric Turbulence and Radio Wave Propagation. Elsevier
Publishing Co., Nauka, Moscow.
Mager, A. (1972). “Dissipation and breakdown of a wing-tip vortex.” Journal of Fluid
Mechanics,55:609–628.
Malik, M. R., Zang, T. A., & Hussaini, M. Y. (1985). “A spectral collocation method
for the Navier-Stokes equations.” Journal of Computational Physics,61:64–88.
170 BIBLIOGRAPHY
Marasli, B., Champagne, F. H., & Wygnanski, I. J. (1991). “On linear evolution of
unstable disturbances in a plane turbulent wake.” Physics of Fluids,3:665–674.
Martin, J. E. & Meiburg, E. (1994). “On the stability of the swirling jet shear layer.”
Physics of Fluids,6:424–426.
Martin, J. E. & Meiburg, E. (1996). “Nonlinear axisymmetric and three-dimensional
vorticity dynamics in a swirling jet model.” Physics of Fluids,8(7):1917–1928.
Martin, J. E. & Meiburg, E. (1998). “The growth and nonlinear evolution of helical
perturbations in a swirling jet model.” European Journal of Mechanics B Fluids,17:639–
651.
Martinelli, F., Olivani, A., & Coghe, A. (2007). “Experimental analysis of the
precessing vortex core in a free swirling jet.” Experiments in Fluids,42:841–841.
McManus K., R., Poinsot, T., & Candel S., M. (1993). “A review of active control
of combustion instabilities.” Progress in Energy and Combustion Science,19:1–29.
Mehta, R. D., Wood, D. H., & Clausen, P. D. (1991). “Some effects of swirl on
turbulent mixing layer development.” Physics of Fluids,3:2716–2724.
Michalke, A. (1965). “On spatially growing disturbances in an inviscid shear layer.”
Journal of Fluid Mechanics,23:521–544.
Michalke, A. (1999). “Absolute inviscid instability of a ring jet with back-flow and swirl.”
European Journal of Mechanics,18:3–12.
Monkewitz, P. A. (1988). “The absolute and convective nature of instability in two-
dimensional wakes at low Reynolds numbers.” Physics of Fluids,31:999–1006.
Monkewitz, P. A. & Sohn, K. D. (1988). “Absolute instability in hot jets.” AIAA
Journal,26:911–916.
Monkewitz, P. A., Bechert, D. W., Barsikow, B., & Lehmann, B. (1990). “Self-
excited oscillations and mixing in a heated round jet.” Journal of Fluid Mechanics,
213:611–639.
Monkewitz, P. A., Huerre, P., & Chomaz, J.-M. (1993). “Global linear stability
analysis of weakly non-parallel shear flows.” Journal of Fluid Mechanics,251:1–20.
M¨
uller, S. B. & Kleiser, L. (2008). “Viscous and inviscid spatial stability analysis of
compressible swirling mixing layers.” Physics of Fluids,20(11):114103–+.
Naughton, J. W., Cattafesta, III, L. N., & Settles, G. S. (1997). “An experimental
study of compressible turbulent mixing enhancement in swirling jets.” Journal of Fluid
Mechanics,330:271–305.
Noack, B. R., Afanasiev, K., Morzy´
nski, M., Tadmor, G., & Thiele, F. (2003).
“A hierarchy of low-dimensional models for the transient and post-transient cylinder
wake.” Journal of Fluid Mechanics,497:335–363.
BIBLIOGRAPHY 171
Oberleithner, K. (2006). Vortex Breakdown in a Swirling Jet with Axial Forcing. Mas-
ter’s thesis, University of Arizona, Technical University Berlin.
Oberleithner, K., Paschereit, C. O., & Wygnanski, I. (2007). “Vortex break-
down in a swirling jet with axial forcing.” In: 18eme Congres Francais de Mechanique,
colloquium: Stratified and rotating flows, 5010.
Olendraru, C. & Sellier, A. (2002). “Viscous effects in the absolute convective insta-
bility of the Batchelor vortex.” Journal of Fluid Mechanics,459:371–396.
¨
Orl¨
u, R. & Alfredsson, P. (2008). “An experimental study of the near-field mix-
ing characteristics of a swirling jet.” Flow, Turbulence and Combustion,80:323–350.
10.1007/s10494-007-9126-y.
Oster, D. & Wygnanski, I. (1982). “The forced mixing layer between parallel streams.”
Journal of Fluid Mechanics,123:91–130.
Panda, J. (1990). Experiments on the instabilities in swirling and non-swirling free jets.
Ph.D. thesis, Pensylvania State University, USA.
Panda, J. & McLaughlin, D. K. (1994). “Experiments on the instabilities of a swirling
jet.” Physics of Fluids,6:263–276.
Parras, L. & Fernandez-Feria, R. (2007). “Spatial stability and the onset of absolute
instability of Batchelor’s vortex for high swirl numbers.” Journal of Fluid Mechanics,
583:27–+.
Paschereit, C. O., Wygnanski, I., & Fiedler, H. E. (1995). “Experimental investi-
gation of subharmonic resonance in an axisymmetric jet.” Journal of Fluid Mechanics,
283:365–407.
Petersen, R. A. & Samet, M. M. (1988). “On the preferred mode of jet instability.”
Journal of Fluid Mechanics,194:153–173.
Petz, C., Hege, H.-C., Oberleithner, K., Sieber, M., Nayeri, C. N., Paschereit,
C. O., Wygnanski, I., & Noack, B. R. (2011). “Global modes in a swirling jet
undergoing vortex breakdown.” Physics of Fluids,23(9):091102.
Pier, B. (2003). “Open-loop control of absolutely unstable domains.” Proceedings of the
Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences,
459(2033):1105–1115.
Pier, B. (2009). “Signalling problem in absolutely unstable systems.” Theoretical and
Computational Fluid Dynamics, pp. 57–+.
Pier, B. & Huerre, P. (2001). “Nonlinear self-sustained structures and fronts in spatially
developing wake flows.” Journal of Fluid Mechanics,435:145–174.
Plaschko, P. (1979). “Helical instabilities of slowly divergent jets.” Journal of Fluid
Mechanics,92:209–215.
172 BIBLIOGRAPHY
Pratte, B. D. & Keffer, J. F. (1972). “The swirling turbulent jet.” ASME Journal of
Basic Engineering,93:739.
Provansal, M., Mathis, C., & Boyer, L. (1987). “Benard-von Karman instability:
transient and forced regimes.” Journal of Fluid Mechanics,182:1–22.
Rajaratnam, N. (1976). Turbulent jets (Vol. 5 of Developments in water science). Elsevier
Scientific Publishing Co.
Reau, N. & Tumin, A. (2002). “Harmonic Perturbations in Turbulent Wakes.” AIAA
Journal,40:526–530.
Rees, S. R. (2009). Hydrodynamic instability of confined jets & wakes and implications
for gas turbine fuel injectors. Ph.D. thesis, University of Cambridge.
Richard, E. (2003). “Swirling jet project.” Tech. rep., University of Arizona.
Roshko, A. (1977). “Errata: Structure of Turbulent Shear Flows: A New Look.” AIAA
Journal,15:768–768.
Rowley, C. (2005). “Model reduction for fluids using balanced proper orthogonal decom-
position.” Intnational Journal of Bifurcation and Chaos,15(3):997–1013.
Rowley, C. W., Mezi´
c, I., Bagheri, S., Schlatter, P., & Henningson, D. S.
(2009). “Spectral analysis of nonlinear flows.” Journal of Fluid Mechanics,641:115–+.
Ruith, M. R., Chen, P., Meiburg, E., & Maxworthy, T. (2003). “Three-dimensional
vortex breakdown in swirling jets and wakes: direct numerical simulation.” Journal of
Fluid Mechanics,486:331–378.
Rukes, L. (2010). Convective/absolute instability transition in swirling jets: linear theory
and its application to experimental data. Diplomarbeit, TU-Berlin. Diplomarbeit 905.
Sarpkaya, T. (1971). “On stationary and travelling vortex breakdowns.” Journal of Fluid
Mechanics,45:545–559.
Schmid, P. J. (2010). “Dynamic mode decomposition of numerical and experimental
data.” Journal of Fluid Mechanics,656:5–28.
Seele, R. (2008). ON THE MEAN PROPERTIES OF THE SWIRLED FLOW. Master’s
thesis, University of Arizona.
Semaan, R., Naughton, J. W., & Ewing, D. (2009). “Approach toward similar be-
havior of a swirling jet flow.” In: 47th AIAA Aerospace Sciences Meeting.
Shiri, A., George, W. K., & Naughton, J. W. (2008). “Experimental Study of the
Far Field of Incompressible Swirling Jets.” AIAA Journal,46:2002–2009.
Sieber, M. (2012). Closed-loop control of turbulent swirling jets undergoing vortex break-
down: Heuristic design and empirical validation of a reduced order model. Diplomarbeit,
TU-Berlin.
BIBLIOGRAPHY 173
Sieber, M., Oberleithner, K., Nayeri, C. N., & Paschereit, C. O. (2011). “Model
design and calibration for closed-loop control of swirling jet instabilities.” In: Colloquium
FLUID DYNAMICS. Institute of Thermomechanics AS CR, v.v.i., Prague,.
Sipp, D. & Lebedev, A. (2007). “Global stability of base and mean flows: a general
approach and its applications to cylinder and open cavity flows.” Journal of Fluid Me-
chanics,593:333–358.
Sirovich, L. (1987). “Turbulence and the dynamics of coherent structures. part i: Coher-
ent structures.” Quarterly of Applied Mathematics,XLV:561–571.
Spall, R. E., Gatski, T. B., & Grosch, C. E. (1987). “A criterion for vortex break-
down.” Physics of Fluids,30:3434–3440.
Sreenivasan, K. R., Raghu, S., & Kyle, D. (1989). “Absolute instability in variable
density round jets.” Experiments in Fluids,7:309–317.
Stalling, D. & Hege, H.-C. (1995). “Fast and resolution independent line integral
convolution.” In: Proceedings of the 22nd annual conference on Computer graphics and
interactive techniques, SIGGRAPH 1995, pp. 249–256. ACM, New York, NY, USA.
Stuart, J. (1958). “On the non-linear mechanics of hydrodynamic stability.” Journal of
Fluid Mechanics,4:1–21.
Suslov, S. A. (2006). “Numerical aspects of searching convective/absolute instability
transition.” Journal of Computational Physics,212:188–217.
Syred, N. (2006). “A review of oscillation mechanisms and the role of the precessing
vortex core (pvc) in swirl combustion systems.” Progress in Energy and Combustion
Science,32(2):93 – 161.
Toh, I. K., Honnery, D., & Soria, J. (2010). “Axial plus tangential entry swirling jet.”
Experiments in Fluids,48:309–325.
Townsend, A. (1956). The Structure of Turbulent Shear Flow. Cambride University Press.
Van Dyke, M. (1975). Perturbation Methods in Fluid Mechanics. The Parabolic Press,
Stanford, annotated edn.
Weisbrot, I. & Wygnanski, I. (1988). “On coherent structures in a highly excited
mixing layer.” Journal of Fluid Mechanics,195:137.
Wu, C., Farokhi, S., & Taghavi, R. (1992). “Spatial instability of a swirling jet -
Theory and experiment.” AIAA Journal,30:1545–1552.
Wu, J.-Z., Ma, H.-Y., & Zhou, M.-D. (2006). Vorticity and vortex dynamics. Springer.
Yang, X. & Zebib, A. (1989). “Absolute and convective instability of a cylinder wake.”
Physics of Fluids,1:689–696.
Zhou, M. D., Heine, C., & Wygnanski, I. (1996). “The effects of excitation on the
coherent and random motion in a plane wall jet.” Journal of Fluid Mechanics,310:1–37.
