The compressible starting jet:
fl uid mec hanics and acoustics
v orgelegt v on
Dipl.-Ing.
Juan José P eña F ernández
geb. in Madrid
v on der F akultät V – V erk ehrs- und Masc hinensysteme
der T ec hnisc hen Univ ersität Berlin
zur Erlangung des ak ademisc hen Grades
Doktor der Ingenieurwissensc haften
-Dr.-Ing.-
genehmigte Dissertation
Promotionaussc h uss:
V orsitzender: Prof. Dr. Julius Reiß
Gutac h ter: Prof. Dr. sc. tec hn. habil. Jörn Sesterhenn
Gutac h ter: Prof. Dr. Christophe Bogey
T ag der wissensc haftlic hen Aussprac he: 21. Septem b er 2017
Berlin, 2018

T o my family, friends and those who help e d me on my way.
I I I

Zusammenfassung
F reistrahlen, auc h "Jets" genann t, k ommen häufig so w ohl in indus-
triellen An w endungen als auc h in der Natur v or, so zum Beispiel b ei der
T reibstoffeinspritzung, Strahltrieb w erk en, V ulk anausbrüchen, Airbags, ab er
auc h W asserstrahlsc hneidemasc hinen, Rak etenruc ksäc k e, Impfpistolen, k os-
misc he Jets, usw.
K on tin uierlic he F reistrahlen w erden seit ungefähr einem Jahrh undert,
so w ohl im ink ompressiblen als auc h im k ompressiblen Bereic h un tersuc h t.
Der F okus liegt dab ei hauptsäc hlic h auf der Besc hreibung der Bew egung des
Fluides b ezüglic h der F reistrahlac hsen so wie dem Einfluss der wic h tigsten
P arameter.
Der startende F reistrahl wurde bisher n ur im ink ompressiblen Bereich
un tersuc h t und der F okus lag auf der Besc hreibung der En t wic klung des
Ringwirb els. Der k ompressible, startende F reistrahl k ann als ein neues
Thema b etrac h tet w erden. In dieser Dissertation wird dieser F all un ter-
suc h t, verw endet wird dabei ein n umerisc her Ansatz, der auf finiten Dif-
ferenzen basiert und diese in einem parallelen n umerisc hen Co de auf einem
der größten Rec hnenzen tren Europas implemen tiert.
Zusätzlic h w erden Exp erimen te so w ohl im Lab or als auc h auf V ulk anen
durc hgeführt um die Akustik der auftretenden F reistrahlen zu un tersuc hen.
Ein theoretisc her Ansatz wird v erfolgt um die Wirkung des Hauptparame-
ters eines solc hen F reistrahles abzusc hätzen.
Der Hauptb eitrag dieser Dissertation ist die V erbindung zwisc hen der
Fluiddynamik und der Akustik des k ompressiblen startenden F reistrahls.
Durc h eine umfangreic he Analyse so w ohl der Ström ungsmec hanik als auc h
der Akustik, hab en wir zwei Art und W eisen gefunden, wie der ausgebildete
Jet mit dem Ringwirb el, w elc her b eim Start en tsteh t, in teragiert, w o durch
zw ei der lautesten Lärmquellen der startende F reistrahl en tstehen:
( i ) die Stoß–Scherschicht In teraktion ist der Mec hanism us w o durc h
der "Broadband sho c k noise" en tsteh t und
( ii ) die Stoß–Scherschicht–R ingwirb el In teraktion wurde erstmals
gefunden; ob w ohl sie n ur eine einzige Druc kw elle generiert, hat sie eine
höhere Amplitude als die des "Broadband sho c k noise".
Während der Detailstudie üb er die Grenze zwisc hen des Ringwirb els
und des trailing Jets hab en wir gefunden, dass die An w endung einer Wirb el-
stärk esc h w elle v on ω p o /ω v ortex = 0 . 1 eine geeignetere Definition des Pinc h-
off ist als die v orherigen Metho den aus der Literatur, w elc he hauptsäc hlic h
auf optisc he Ausw ertungen basiert sind.
Wir k onn ten b estätigen, dass die En t wic klung der P osition des Ring-
wirb els in der axialen und radialen Ric h tung x VR /D ∼ ( t ∗ ) 1 / 2 und R VR /D ∼
( t ∗ ) 1 / 3 üb er einen substan tiellen Zeitb ereic h folgt.
Wir k onn ten auc h b estätigen, dass die K ompressibilität die V erbreitung
V

des Ringwirb els v erzögert. Wir hab en zusätzlic h gefunden, dass für ein
T urbulenzgrad v on 10 %, die Stoßzellen-Struktur no c h erhalten ist.
Mit umfassende Ken tnissen üb er so w ohl die Ström ungsmec hanik als
auc h die Akustik des startenden F reistrahls, als auc h der V erbindung zwis-
c hen b eiden, sind wir in der Lage, die Hauptparameter durc h akustisc he
und optisc he Messungen abzusc hätzen. Das ist besonders wich tig für An-
w endungen, w o k ein direkter Zugang zum Strahl möglic h ist (zum Beispiel
im F all v on V ulk anausbrüchen), ab er auc h für die En t wic klung neuer vulk a-
nisc he Üb erw ac h ungsmetho den.
VI

Abstract
Jets are in v olv ed in n umerous industrial applications and in nature: fuel
injection, jet engines, v olcanic jets, air-bag devices, but also w ater jet cut-
ters, jet pac ks (ro c k et b elts), jet injectors for v accination, astroph ysical jets,
etc. Con tin uous jets ha v e b een already studied in the incompressible and
compressible case for almost a cen tury , fo cused mainly on the c haracteriza-
tion of the fluid motion at the jet cen terline and the effect of the go v erning
parameters. The starting jet has b een studied only in the incompressible
case, fo cused mainly on the radial and axial c haracterization of the v ortex
ring. The compressible starting jet can b e considered a new topic.
The compressible starting jet has b een studied in this thesis mainly with
a n umerical approac h using finite differences with a massiv ely parallel co de
running in one of the biggest sup ercomputing cen tres in Europ e. Nev erthe-
less, exp erimen ts in b oth lab oratory and activ e v olcano es ha v e b een also
carried out mainly to study the acoustics emanated. A small, but not in-
significan t theoretical w ork w as p erformed in order to estimate some flo w
prop erties when c hanging the go v erning parameters.
The main con tribution of this thesis is the link b et w een the fluid me-
c hanics and the acoustics in the compressible starting jet. With an exten-
siv e analysis of b oth the fluid flo w and the acoustics, w e found t w o w a ys in
whic h the trailing jet and the head v ortex ring in teract, pro ducing tw o of the
loudest noise sources of the compressible starting jet: ( i ) the sho ck–she ar
layer in teraction is the basic noise generation mec hanism of the broadband
sho c k noise and ( ii ) the sho ck–she ar layer–vortex ring in teraction w as
found for the first time and generates a single acoustic w a v e, but with a
higher amplitude than those from the broadband sho c k w a v e. Studying in
detail the in terface trailing jet – v ortex ring w e found that using a pinch-
off vorticity thr eshold ω p o /ω vortex = 0 . 1 is a more appropriate definition
of the pinc h-off (separation b et w een the v ortex ring and the trailing jet)
than the previous metho ds used in the literature, based mainly on optical
in terpretations.
W e ha v e also confirmed that the ev olution of the axial p osition of the
v ortex ring follo ws x VR /D ∼ ( t ∗ ) 1 / 2 and the radius of the v ortex ring
ev olv es as R VR /D ∼ ( t ∗ ) 1 / 3 during a significan t p erio d of time. Compar-
ing a lo w subsonic v ortex ring with a high subsonic and a sup ersonic one,
w e confirmed that compressibilit y reduces significan tly the propagation v e-
lo cit y of the v ortex ring. W e analysed the effect of turbulence in tensit y
at the inlet as w ell as three differen t nozzle geometries and w e found that
ev en with p erturbations in tensities 10 % of the mean flo w, the sho c k-cell
structure still remains presen t.
With a broad kno wledge of b oth fluid mec hanics and the acoustics of
the compressible starting jet as w ell as the link b et w een b oth of them w e
VI I

predicted some of the go v erning parameters from acoustic and/or optical
measuremen ts. This is particularly imp ortan t for those disciplines where
the direct access to the jet is not p ossible (suc h as v olcanic jets) or for the
dev elopmen t of new monitoring tec hniques.
VI I I

A c kno wledgemen t
Firstly , I would lik e to express my sincere gratitude to m y advisor Prof. Jörn
Sesterhenn for the supp ort during m y Ph.D. study and related researc h. I
am v ery grateful for the p ossibilit y to use the equipmen t of the departmen t
and for giving me the to ols to b ecome a b etter researc her and a stronger
p erson.
Besides m y advisor, I w ould lik e to thank the rest of m y thesis commit-
tee: Prof. Dr. Bogey and Prof. Julius Reiß, for accepting the in vitation
to b ecome part of m y thesis defense tribunal. Y our commen ts help ed to
impro v e the qualit y of this thesis indeed.
My sincere thanks also go es to Thomas Engels, Mathias Lemk e and
Lewin Stein for man y fruitful and funn y discussions and for their con tin-
uous help. Without their precious supp ort it w ould not b e p ossible to
conduct this researc h. I thank m y friends and colleagues who supp orted
me and motiv ated me to w ards m y goal. I will alw a ys k eep in mind the
mem b ers of the "Computational Fluid Mec hanics" Departmen t that I met
during these y ears: Stefano Alois, Sergio Bengo ec hea Lozano, Elias Büch-
ner, Gabriele Camerlengo, Martin F rank e, Marian F uc hs, Sonja Hoßbac h,
Sophie Knec h tel, Steffen Nitsch, Lars Oergel, Mario Srok a, Christian W est-
phal, W olfgang Wiese and Rob ert Wilk e.
Also I thank m y advisor during m y in ternship in spring 2013 at the
Hong K ong P olytec hnical Univ ersit y , Asso c. Prof. Dr. Randolph Leung,
for the supp ort and the fertile con v ersations. Also m y appreciation to m y
friends and colleagues during m y asian in ternship: Garret Lam, Ka Him
Seid, Harris F an, Horus Chan and Ian W ong.
During the exp editions to the v olcano es and the differen t meetings ab out
v olcano es I met man y p eople who help ed me to understand b etter ho w
a v olcano w orks, but also another completely differen t w a y of doing re-
searc h. Thank y ou Ulli Küpp ers, V aleria Cigala, Jacop o T adeucci, Pierre-
Y v es T ournigand, T ullio Ricci and Andrea Cannata.
Last but not the least, m y sp ecial thanks go es to m y paren ts Juan José
and Ana and m y brother Alb erto for b eing the b est family in the w orld,
b eing alw a ys there and encouraging me during the difficult times, and m y
girlfriend Sabrina for sho wing me endless supp ort and lo v e; without y ou all
this thesis w ould not ha v e b een p ossible. Thank y ou also, Martin, Birgit
and Kathi for y our w elcoming arms.
IX

X

Con ten ts
Zusammenfassung V
Abstract VI I
A c kno wledgemen t IX
Nomenclature XXI I I
1 In tro duction 1
1.1 Description of the flo w . . . . . . . . . . . . . . . . . . . . . 5
1.2 Set-up description . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Effects of the go v erning parameters . . . . . . . . . . . . . . 9
1.3.1 Reynolds n um b er ( R e D ) ................ 1 0
1.3.2 Dimensionless mass supply ( L/D ) .......... 1 2
1.3.3 Pressure ratio ( p 0 r /p ∞ ) ................ 1 5
1.3.4 T emp erature ratio ( T 0 r /T ∞ ) ............. 1 7
I Sim ulations of a compressible starting jet 21
2 Numerical bac kground 23
2.1 Na vier-Stok es equations . . . . . . . . . . . . . . . . . . . . 23
2.2 Grid considerations . . . . . . . . . . . . . . . . . . . . . . . 24
2 . 3 P a r a l l e l i z a t i o n .......................... 2 9
2.4 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 31
2.6 P orous media / v olume p enalisation metho d . . . . . . . . . 33
2 . 7 S p o n g e r e g i o n .......................... 3 4
2 . 8 S i m u l a t i o n s s e t u p ........................ 3 5
XI

CONTENTS
I I Exp erimen ts of a compressible starting jet 41
3 Exp erimen tal setup 43
3.1 Sc hlieren photography . . . . . . . . . . . . . . . . . . . . . 45
3.2 A coustic measurements in the lab oratory . . . . . . . . . . 48
3.3 A coustic measurements at real v olcano es . . . . . . . . . . . 49
3 . 3 . 1 M o u n t E t n a ....................... 4 9
3 . 3 . 2 S t r o m b o l i ........................ 5 0
I I I Results of jet sim ulations and exp erimen ts 53
4 Characterisation of the compressible starting jet 55
4 . 1 C o m p r e s s i o n w a v e ....................... 5 5
4.2 V ortex ring dynamics . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 V ortex ring radius and core radius . . . . . . . . . . 56
4.2.2 Axial lo cation and propagation velocity . . . . . . . 59
4.2.3 Effects of compressibilit y . . . . . . . . . . . . . . . 65
4.3 T railing jet formation and dynamics . . . . . . . . . . . . . 67
5 Effects of the inflo w condition 71
5 . 1 L a m i n a r i n fl o w ......................... 7 1
5.2 T ripp ed b oundary lay er syn thetic turbulence . . . . . . . . 72
5 . 3 N o z z l e fl o w ........................... 7 2
5.4 Effects on the flo w field . . . . . . . . . . . . . . . . . . . . 73
5.5 Effects on the radiated acoustics . . . . . . . . . . . . . . . 76
5.6 Effect of the nozzle geometry on the starting jet . . . . . . . 79
6 Pinc h-off definition 83
6.1 Pinc h-off v orticit y threshold ( ω po /ω vortex ) .......... 8 3
6.2 Circulation divergence la w . . . . . . . . . . . . . . . . . . . 85
6.3 Kelvin-Benjamin v ariational principle . . . . . . . . . . . . 91
7 In teraction b et w een the v ortex ring and the trailing jet 93
7.1 Sho c k–shear lay er–vortex in teraction . . . . . . . . . . . . . 93
7.2 Sho c k–shear lay er interaction . . . . . . . . . . . . . . . . . 99
7.3 Effect of sho c k-w a v es dynamics on BBSN . . . . . . . . . . 100
8 A coustics of the starting jet 103
8.1 Compression wa ve . . . . . . . . . . . . . . . . . . . . . . . 104
8 . 2 V o r t e x r i n g ........................... 1 0 5
8.3 T urbulen t mixing noise . . . . . . . . . . . . . . . . . . . . . 106
8.4 Broadband sho ck noise . . . . . . . . . . . . . . . . . . . . . 107
XI I

8 . 5 S c r e e c h ............................. 1 0 9
8.6 Directivit y of the noise radiated . . . . . . . . . . . . . . . . 109
8.7 F requency-time domain (wa velet) analysis . . . . . . . . . . 113
9 Go v erning parameters prediction from acoustic and optical
measuremen ts 123
9 . 1 R e y n o l d s n u m b e r........................ 1 2 3
9 . 2 P r e s s u r e r a t i o .......................... 1 2 5
9.3 Dimensionless mass supply . . . . . . . . . . . . . . . . . . 128
9.4 T emp erature ratio . . . . . . . . . . . . . . . . . . . . . . . 128
10 V olcanic jets 131
10.1 Crater lo cation from acoustic measuremen ts . . . . . . . . . 132
10.2 F requency-time domain (wa v elet and
S T F T ) a n a l y s i s ......................... 1 3 3
11 Conclusions and p ersp ectiv es 137
App endices 141
A Gas dynamics of nozzle flo ws 143
B F ully expanded conditions 149
C Quasi-steady reserv oir motion 151
C . 1 S u b s o n i c n o z z l e......................... 1 5 1
C . 2 C h o k e d n o z z l e .......................... 1 5 3
C.3 Effect of the main parameters . . . . . . . . . . . . . . . . . 159
D Cylindrical reserv oir analogy 165
E Instrumen tation 169
E.1 A coustic measurements . . . . . . . . . . . . . . . . . . . . 169
E.2 Sc hlieren comp onents . . . . . . . . . . . . . . . . . . . . . 170
Publications related to this w ork 170
Bibliograph y 175
XI I I

CONTENTS
XIV

List of T ables
2.1 Grid spacing considerations for the three-dimensional sim u-
l a t i o n s .............................. 2 6
2.2 Grid spacing considerations for the t w o-dimensional sim ula-
t i o n s . ............................... 2 9
2.3 P arameter summary for cases in v olv ed in the trailing jet
a n a l y s i s .............................. 3 3
2.4 Ov erview of the differen t set-ups for the analyses in this study . 36
2.5 Setup for the analysis of the v ortex ring dynamics. . . . . . 37
2.6 Setup for the analysis of the existence of the trailing jet. . . 37
2.7 Setup for the analysis of the trailing jet for an infinite reserv oir. 37
2.8 Setup for the analysis of the inflo w conditions. . . . . . . . 38
2.9 Setup for the analysis of the geometry , non-dimensional mass
supply and pressure ratio. . . . . . . . . . . . . . . . . . . . 38
2.10 Setup for the analysis of the gov erning parameters. . . . . . 39
2.11 Setup for the analysis of the Reynolds n umber. . . . . . . . 39
4.1 V ortex ring non-dimensional radius at the end of the forma-
t i o n s t a g e ............................ 5 7
4.2 Co efficien ts of the prop ortionalit y la w. . . . . . . . . . . . . 60
4.3 Setup for the analysis of the v ortex ring. . . . . . . . . . . . 66
5.1 Nozzle design parameters for the cases 11 .a − 13 .h . . . . . . 80
6.1 Prediction of the pinc h-off dimensionless time using the Kelvin-
Benjamin v ariational principle. . . . . . . . . . . . . . . . . 91
A.1 P arameters of the nozzle geometry used for the example. . . 145
C.1 Parameters of the example in the subsonic nozzle flo w. . . . 153
C.2 Parameters of the example in the sup ersonic nozzle flo w. . . 155
C.3 Set of parameters chosen as base flo w for this analysis. . . . 159
XV

LIST OF T ABLES
XVI

List of Figures
1.1 Stages of the starting jet. . . . . . . . . . . . . . . . . . . . 5
1.2 Flo w configuration sho wing the dev elopmen t of the Kelvin-
H e l m h o l t z v o r t i c e s . ....................... 6
1.3 Con tour plot of a laminar and a turbulen t v ortex ring. . . . 7
1.4 Three-dimensional kinetic energy sp ectra . . . . . . . . . . 8
1.5 Sk etc h of the system set-up. . . . . . . . . . . . . . . . . . . 9
1.6 V orticit y isosurface of jets with Re D = 5 000 and 10 000 in
the contin uous stage. . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Isosurface of the Q-criterion of jets with Re D = 5 000 and
15 000 in the starting stage. . . . . . . . . . . . . . . . . . . 12
1.8 Starting jet fluid flo w for differen t parameter set-ups to sho w
differen t configurations where compressibilit y and turbulence
p l a y a m a i n r o l e . ........................ 1 3
1.9 Fluid flo w of a compressible turbulen t v ortex ring without
t r a i l i n g j e t . ........................... 1 4
1.10 Side effect of L/D on the jet Mac h num b er. . . . . . . . . . 15
1.11 Effect of pressure ratio on the temp oral evolution of the jet
Mac h num b er for a div ergen t nozzle . . . . . . . . . . . . . 16
2.1 Grid stretching and spacing. . . . . . . . . . . . . . . . . . . 25
2.2 Computational grid for the finest sim ulation. . . . . . . . . 26
2.3 T urbulen t kinetic energy sp ectra for Re D = 5 000 , 10 000
and the highest resolution. . . . . . . . . . . . . . . . . . . . 27
2.4 T urbulen t kinetic energy sp ectra for Re D = 5 000 , 10 000
and different resolutions. . . . . . . . . . . . . . . . . . . . . 28
2.5 Strong and w eak scaling plots. . . . . . . . . . . . . . . . . 30
2.6 Nozzle exit conditions. . . . . . . . . . . . . . . . . . . . . . 32
2.7 Discretisation of the nozzle exit for differen t resolutions. . . 33
2.8 Sp onge damping function. Linear and logarithmic represen-
t a t i o n . .............................. 3 5
XVI I

LIST OF FIGURES
3.1 General set-up in the anechoic c ham b er. . . . . . . . . . . . 44
3.2 Sk etc h of a classical Z-type sc hlieren set-up. . . . . . . . . . 45
3.3 General set-up for Sc hlieren photograph y . Detail view of the
r a z o r b l a d e . ........................... 4 6
3.4 Detail view of the ligh t source. Detail view of the razor blade
a t t h e c a m e r a . ......................... 4 7
3.5 Long exp osure and high-sp eed Sc hlieren examples. . . . . . 47
3.6 Disp osition of the microphones in the acoustic measuremen ts. 48
3.7 A ctiv e craters at Moun t Etna. . . . . . . . . . . . . . . . . . 49
3.8 Deplo ymen t of the microphones at Moun t Etna. . . . . . . . 50
3.9 General view of the craters at Moun t Etna. Detail view of a
laminar vortex ring ejected b y Moun t Etna. . . . . . . . . . 50
3.10 Crater configuration at Strom b oli on the 31 th May 2016 . . . 51
3.11 Microphones measuring the acoustic from Stromboli. . . . . 51
4.1 Characteristic diagram of the classical sho c k tub e problem.
In v olv ed Mac h n um b ers in the region after the compression
w a v e for differen t reserv oir to am bien t pressure ratios. . . . 56
4.2 Geometrical parameters of the vortex ring . . . . . . . . . . 57
4.3 Ev olution of the v ortex ring non-dimensional radius. . . . . 58
4.4 R V R /D -mo del trend lines and comparison with the literature. 59
4.5 Dynamics of the v ortex ring core. . . . . . . . . . . . . . . . 60
4.6 Ev olution of the non-dimensional x -location of the vortex ring. 61
4.7 Ev olution of the axial lo cation of the v ortex ring with the
square ro ot of the non-dimensional time. . . . . . . . . . . . 62
4.8 V ortex ring propagation v elo cit y o v er the non-dimensional
t i m e ............................... 6 3
4.9 V ortex ring propagation v elo cit y for ev ery case. . . . . . . . 64
4.10 Ev olution of the fully expanded Mac h num b er and the cir-
culation Mach n um b er. . . . . . . . . . . . . . . . . . . . . . 65
4.11 Ev olution of the axial p osition, the radius and the core radius
o f t h e v o r t e x r i n g . ....................... 6 6
4.12 V orticity con tour when the v ortex ring is at x/D ∼ 11 . . . 67
4.13 Characteristic diagram sho wing the dynamics of the sho ck
w a v e s . .............................. 6 9
5.1 V elo cit y p erturbations imp osed at the inflo w for the TBL case. 72
5.2 Compressible Blasius inlet profile and sk etc h of the straigh t
nozzle used in case 8 ...................... 7 2
5.3 Instan taneous Mac h n um b er con tours for the analysis of the
i n fl o w c o n d i t i o n s . ........................ 7 4
5.4 Instan taneous en trop y con tours for the analysis of the inflo w
c o n d i t i o n s . ............................ 7 5
XVI I I

5.5 Distribution of a v erage Mac h n um b er and r.m.s. en trop y
p erturbations along the jet axis . . . . . . . . . . . . . . . . 76
5.6 Sound pressure lev el sp ectra at 90 ◦ for the three inlet condi-
tions in the analysis . . . . . . . . . . . . . . . . . . . . . . 77
5.7 W a v elet co efficien t con tour at 90 ◦ . .............. 7 8
5.8 Sk etc h of the nozzles used in cases 11 − 13 . ......... 8 0
5.9 T emp oral ev olution of the exit (left) and fully expanded
(righ t) Mach n um b er. . . . . . . . . . . . . . . . . . . . . . 81
6.1 Sk etc h of the pinc h-off v orticit y threshold metho d. . . . . . 84
6.2 Time ev olution of the separation v orticit y and the pinc h-off
v orticit y threshold. . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Sk etc h of the differen t metho ds used to compute the circu-
l a t i o n . .............................. 8 6
6.4 Ev olution of the circulation con tained in the fluid flo w for
t h e s t a r t i n g j e t . ......................... 8 7
6.5 En trainmen t affects the circulation during the v ortex ring
g r o w t h .............................. 8 8
6.6 When en trainmen t disapp ears, the circulation reco v ers the
n o r m a l b e h a v i o u r . ....................... 8 9
6.7 3D effects to tak e in to accoun t when computing the circulation 89
6.8 Pinc h-off prediction la w. . . . . . . . . . . . . . . . . . . . . 90
6.9 Application of the Kelvin-Benjamin v ariational principle to
predict the v ortex ring pinch-off. . . . . . . . . . . . . . . . 92
7.1 Stages of the sho c k cell structure during its in teraction with
t h e v o r t e x r i n g . ......................... 9 4
7.2 Optical comparison of the fluid flo w with the literature. . . 95
7.3 A coustic radiated b y the in teraction and measured b y the
microphone at r /D = 5 .................... 9 7
7.4 The noise pro duced b y the in teraction b et w een the v ortex
ring and the trailing jet is at least as loud as the loudest
noise source in the con tin uous jet in terms of sound pressure
l e v e l . ............................... 9 8
7.5 The in teraction w a v e propagates with the sp eed of sound. . 99
7.6 The acoustic w a v es radiated when a v ortex passes b y a sho c k
w a v e ( B B S N ) . .......................... 1 0 0
7.7 Characteristic diagrams sho wing the dev elopmen t of the first
acoustic w a v e, vortex ring and shock w a v es for differen t inlet
c o n d i t i o n s . ............................ 1 0 2
8.1 T ypical sound pressure lev el sp ectra for a con tin uous sup er-
s o n i c j e t . ............................. 1 0 3
XIX

LIST OF FIGURES
8.2 Pressure profile through the first w a v e for differen t inlet con-
ditions in original and similarity axes. . . . . . . . . . . . . 104
8.3 Sound pressure lev el sp ectrum in a mo ving reference frame
sho wing the directivit y of the v ortex ring noise. . . . . . . . 105
8.4 Large scale turbulence noise w a v es visible in a sc hlieren pic-
ture of an under-expanded jet. . . . . . . . . . . . . . . . . 107
8.5 Noise generation mec hanism of broadband sho c k noise. Nu-
merical Sc hlieren con tour with M = 1 iso-line. . . . . . . . . 108
8.6 Directivities of the differen t comp onen ts of the jet noise. . . 110
8.7 Sound pressure lev el sp ectra radiated do wnstream and in the
transv erse direction. . . . . . . . . . . . . . . . . . . . . . . 111
8.8 Sound pressure lev el sp ectra of subsonic and sup ersonic jets. 112
8.9 Time series and sound pressure lev el sp ectrum of case 7 .a . . 113
8.10 W av elet coefficient con tour of case 7 .a ............. 1 1 4
8.11 Time series and sound pressure level spectrum of case 5 . . . 115
8.12 W av elet coefficient con tour of case 5 . ............. 1 1 6
8.13 Time series and sound pressure level spectrum of case 4 . . . 117
8.14 W av elet coefficient con tour of case 4 . ............. 1 1 8
8.15 Time series and sound pressure level spectrum of case 3 . . . 119
8.16 W av elet coefficient con tour of case 3 . ............. 1 2 0
8.17 Time series and sound pressure level spectrum of case 1 .a. . 121
8.18 W av elet coefficient con tour of case 1 .a. ............ 1 2 2
9.1 V ariation with the Reynolds n um b er of the p eak Strouhal
n um b er for the fine-scale turbulen t mixing noise. . . . . . . 124
9.2 V ariation of the BBSN p eak Helmholtz n um b er with the
p r e s s u r e r a t i o . .......................... 1 2 7
9.3 Sc hlieren picture of an under-expanded jet to predict the
fully expanded Mac h n um b er. . . . . . . . . . . . . . . . . . 127
10.1 Crater lo cation results of the eruptions measured during the
moun t Etna campaign in July 2014 .............. 1 3 4
10.2 Short time F ourier coefficient con tour of the acoustics at
S t r o m b o l i ............................ 1 3 5
10.3 W av elet coefficient con tour of the acoustics recorded at Strom-
b o l i . ............................... 1 3 6
A.1 Area - Mac h n um b er function for γ = 1 . 4 . .......... 1 4 4
A.2 Geometry , pressure ratio and Mach n um ber along the Lav al
nozzle axis for the example case. . . . . . . . . . . . . . . . 146
B.1 F ully expanded conditions in a jet. . . . . . . . . . . . . . . 150
XX

C.1 Dimensionless pressure ev olution in the reserv oir. . . . . . . 153
C.2 Exit and fully expanded Mach n um ber evolution in the sub-
s o n i c n o z z l e . ........................... 1 5 4
C.3 Dimensionless pressure evolution in the reserv oir for the su-
p ersonic nozzle flo w. . . . . . . . . . . . . . . . . . . . . . . 155
C.4 Exit and fully expanded Mac h n um b er ev olution in the c hok ed
n o z z l e ............................... 1 5 6
C.5 Prandtl-Meyer expansion angle ev olution in the c hok ed
n o z z l e ............................... 1 5 7
C.6 Evolution of the thermodynamic state in the reserv oir during
the discharge of the c hok ed nozzle. . . . . . . . . . . . . . . 157
C.7 F ully expanded and exit densit y and temp erature ev olution
of the c hok ed nozzle. . . . . . . . . . . . . . . . . . . . . . . 158
C.8 Evolution of the sho c k-cell spacing and fully expanded di-
ameter in the c hok ed nozzle. . . . . . . . . . . . . . . . . . . 158
C.9 Evolution of the reserv oir pressure and exit and fully ex-
panded Mac h n um b er for the baseline case for the analysis
of the main parameters. . . . . . . . . . . . . . . . . . . . . 160
C.10 F ully expanded Mach n um b er con tour plot o v er the pressure
ratio and the non-dimensional time. . . . . . . . . . . . . . 161
C.11 F ully expanded Mac h n um b er ev olution with the non-dimensional
time as a function of the non-dimensional mass supply . . . . 162
C.12 F ully expanded Mac h n um b er ev olution with the non-dimensional
time when c hanging the reserv oir to am bien t temp erature ratio. 163
C.13 F ully expanded Mach n umber evolution with the normalised
non-dimensional time t ∗ / ( T r /T ∞ ) when changing the reser-
v oir to am bien t temp erature ratio. . . . . . . . . . . . . . . 164
E.1 F requency resp onse of the PCB microphone with the grid
cap at 0 degrees incid ence. . . . . . . . . . . . . . . . . . . . 169
E.2 F requency resp onse of the G.R.A.S. microphone with the
grid cap at 0 degrees incidence. . . . . . . . . . . . . . . . . 170
XXI

LIST OF FIGURES
XXI I

Nomenclature
Abbreviations
BBSN Broadband sho c k noise
DSLR Digital single-lens reflex
FSS Fine scale similarit y (sp ectrum), see T am et al. [1996]
LSS Large scale similarit y (sp ectrum), see T am et al. [1996]
LRZ Leibniz-Rec henzen trum (Leibniz Sup ercomputing Cen tre)
MPI Message Passing In terface
NPR Nozzle pressure ratio. NPR = p 0 r /p e
r.m.s. ro ot mean square
STFT Short Time F ourier T ransform
SPL Sound pressure lev el
TBL T ripp ed b oundary la y er
TMN T urbulen t mixing noise
Greek Sym b ols
θ As suggested b y T am [1995], the jet angle from the upstream direc-
tion
σ Damping function of the sp onge region to treat n umerically the
outflo w condition
ρ Densit y (kg/m 3 )
µ Dynamic viscosit y . F or air at normal conditions µ = 1 . 7 × 10 − 5
kg/(m s)
XXI I I

LIST OF FIGURES
λ Heat conductivit y (m 2 /s)
γ Isen tropic exp onen t: γ = C p /C v
ν Kinematic viscosit y (m 2 /s)
η K olmogoro v length scale. η =  ν 3 /ε  1 / 4
ω v ortex Maxim um v orticit y in the v ortex ring
K P ermeabilit y of a p orous medium (m 2 )
ω p o Pinc h-off v orticit y threshold
φ P orosit y of a p orous medium φ = V f /V
τ ij Shear stress tensor
ε T urbulen t kinetic energy dissipation rate. ε = ν h ∂ u 0
i
∂ x j
∂ u 0
i
∂ x j i
δ V ortex ring core radius
ω V orticit y v ector ω = ∇ × u
Mathematical op erators
h · i θ A v erage in the azim uthal direction
c
( · ) V ariable in F ourier Space
Roman Sym b ols
x VR Axial p osition of the v ortex ring
U Characteristic v elo cit y
A ∗ Critical nozzle cross section area
R Gas constan t. F or air at normal conditions R = 287 . 058 J kg − 1 K − 1
d grid Grid c haracteristic length. d grid = ( dx min + dy min + dz min ) / 3 for
the three-dimensional case and d grid = ( dx min + dy min ) / 2 for the
t w o dimensional case.
H Helmholtz n um b er H = f D /c ∞
V 0 Injected fluid v olume
K n Kn udsen n um b er
XXIV

L Length of the h yp othetical cylindrical reserv oir with diameter D
that con tains the same mass as the real reserv oir
M Γ Mac h n um b er based on the v ortex ring circulation
m Mass con tained in the reserv oir
( p r /p ∞ ) blast Necessary pressure ratio to generate a blast w a v e. See Ishii
et al. [1999]
t ∗ Non-dimensional time t ∗ = t/ ( D /u j )
A Nozzle cross section area
D Nozzle diameter
P r Prandtl num ber
R VR Radial p osition of the v ortex ring
Re Reynolds n um b er
L s Sho c k cell length
c Sp eed of sound (m/s)
S t Strouhal n um b er S t = f D /u j
S ij Symmetric part of the shear stress tensor. S ij = 1
2  ∂ u i
∂ x j + ∂ u j
∂ x i 
T T emp erature (K)
u V elo cit y (m/s)
( L/D ) lim F ormation numb er , the limiting v alue that defines the existence
of a trailing jet, see Gharib et al. [1998]
Subscripts
( · ) j F ully expanded conditions. See app endix B
( · ) e Nozzle exit conditions
( · ) r Reserv oir conditions
( · ) ∞ Un b ounded c ham b er conditions
XXV

LIST OF FIGURES
XXVI

Chapter 1
In tro duction
Con tin uously blo wing jets ha v e b een rep orted in an extensiv e literature
b oth in the incompressible (e.g. [Ryhming, 1973]) and in the compressible
case (e.g. [T am, 1995]). This w ork fo cuses on the starting phase of a
compressible jet, this is, the formation of the jet flo w from a quiescen t
condition if the jet is ejected from an infinite reserv oir or the generation of
a v ortex ring with no jet at all in the extreme case of a v ery small reserv oir
compared to the nozzle diameter. A full range of starting and deca ying jets
can b e generated for differen t v alues of the go v erning parameters b et w een
these t w o extremes.
The starting jet has b een already rep orted in the literature: the in-
compressible case has initiated a considerable literature, mainly pla ying a
secondary role in the study of v ortex ring dynamics, which are t ypically gen-
erated b y a piston-driv en device in incompressible flo ws; the compressible
case has b een recen tly presen ted in [ J.J. P eña F ernández and Sesterhenn,
2017a]. In this work, w e examine the compressible starting and deca ying
jet in detail and discuss the effects of the go v erning parameters in the jet
flo w and the radiated acoustics.
T urner [1962] dev elop ed the first mo del of the starting jet comp osed b y
a spherical v ortex follo w ed b y a steady jet exp erimen ting with a buo y an t
plume and found that the tip of the spherical v ortex mo v es sligh tly faster
than half of the maxim um v elo cit y in the jet at the same lo cation for a
later time. In a similar manner, rep orting for incompressible flows experi-
men tally , Witze [1980, 1983] prop osed a theoretical mo del that assumes the
transien t jet as a spherical v ortex in teracting with a steady jet and demon-
strated that the ratio of the nozzle diameter ( D ) to the c haracteristic jet
v elo cit y ( U ) defines a time constan t τ = D /U that uniquely c haracterises
the b eha viour of impulsiv ely started jets leading to the dimensionless time
t ∗ = t/ ( D /U ) . Recen tly , Gao and Y u [2015] review ed the basic ideas on
1

Chapter 1. In tro duction
the incompressible starting jet and stated that the t w o broad directions of
the curren t researc h are ( i ) the vortex ring pinc h-off and ( ii ) en trainmen t
enhancemen t in pulsed-jet propulsion systems.
Starting jets ha v e b een t ypically used in most of the exp erimen tal w ork
to pro duce v ortex rings, b eing only the latter the fo cus of the researc h.
Based on exp erimen tal observ ations, Maxw orth y [1972] describ ed the flow
field, the vortex ring’s v elo city and the gro wth rate for stable rings. The
reviews of Shariff and Leonard [1992] and Lim and Nic k els [1995] pro vide
a detailed bac kground of v ortex ring prop erties and their dynamics. In the
literature, most of the exp erimen tal studies generate the v ortex rings b y
ejecting an incompressible fluid mo ving a piston inside a cylindrical tub e of
diameter D for a length L in to an op en c ham b er, giving rise to the definition
of the parameter ( L/D ). Ho w ev er, L/D can b e regarded with a more gen-
eral view through the con tin uit y equation as the non-dimensional mass sup-
ply: the mass con tained in a cylinder of diameter D and length L ( m cyl , L ),
compared to the one con tained in a cylinder with L = D ( m cyl , L=D ):
m cyl , L
m cyl , L=D
= π ( D
2 ) 2 Lρ
π ( D
2 ) 2 D ρ = L
D , (1.1)
whic h is more con v enien t in compressible flo ws, b ecause it is the av ailable
budget to generate the flo w and it directly relates to the basic equation
(con tin uit y), rather than relating to geometrical factors.
Concerning the v ortex ring dynamics, Didden [1979] prop osed that the
axial and radial p ositions of the v ortex ring ( x VR and r VR , resp ectiv ely)
v ary in the v ery early stage of the pro cess ( 0 . 1 < t ∗ < 1 ) as x VR /D ∼ ( t ∗ ) 3 / 2
and R VR /D ∼ ( t ∗ ) 2 / 3 . In a later stage, for t ∗ > 1 , the v ariation of the axial
and radial p ositions of the v ortex ring w ere found to b e x VR /D ∼ ( t ∗ ) 1 / 2
Witze [1980] and R VR /D ∼ ( t ∗ ) 1 / 3 Kelvin [1867], resp ectiv ely . W e presen t
in c hapter 4 the c haracterisation of the compressible starting jet. Gharib
et al. [1998] found a limiting v alue for the non-dimensional mass supply
( L/D ) lim of appro ximately four that univ ersally defines the existence of a
trailing jet after the v ortex ring. This limiting v alue was called ’formation
n um b er’. These results were reproduced by Rosenfeld et al. [1998] and
man y others but it w as rep orted that this limiting v alue is not univ ersal
and it ma y v ary within the range 1 < ( L/D ) lim < 5 dep ending on th e
spatial and temp oral distribution of the inlet condition or the Reynolds
n um b er, lik e Zhao et al. [2000], Gao and Y u [2010] or Rosenfeld et al. [1998].
This problem is discussed for the compressible starting jet in section 4.3.
Hermanson et al. [2000] defined the limiting parameter as ( L/D ) 1 / 3
lim in order
to relate it to the c haracteristic length of a cylinder defined b y the authors as
the cubic ro ot of the injected v olume ( V 1 / 3
0 ), resulting ( L/D ) 1 / 3
lim ∼ V 1 / 3
0 /D .
The n umerical studies related to this w ork ha v e b een t ypically fo cused
2

on the v ortex ring, b ecause of the relativ ely lo w requiremen ts of computa-
tional resources. The most closely related n umerical studies in the literature
are presen ted here.
James and Madnia [1996] found that during the formation of the v ortex
ring, the momen tum and the total circulation are appro ximately the same
generating the v ortex ring through an orifice in a w all or through a nozzle,
where no w all is around the lip of the nozzle. W e address the effect of the
inflo w conditions in c hapter 5.
In v estigating the trailing jet instabilities n umerically , Zhao et al. [2000]
found that the Kelvin-Helmholtz instabilities from the shear la y er driv e the
pinch-off pro cess, defined as the ph ysical separation of the v ortex ring from
the trailing jet. Sev eral attempts ha v e b een made to mo del the pinch-off of
the incompressible starting jet in the last decade (e.g. [Gao and Y u, 2010]).
An appropriate definition of the pinc h-off pro cess and a discussion of the
pinc h-off in the compressible starting jet are giv en in c hapter 6. Shortly
b efore the pinc h-off tak es place, the v ortex ring in teracts with the trailing
jet in the compressible case, leading to one of the loudest noise sources of
the compressible starting jet. This in teraction is examined in c hapter 7.
Zaitsev et al. [2001] compared the noise measuremen ts of a turbulen t
v ortex ring with theory and confirmed that the turbulen t v ortex ring noise
can b e represen ted as the sum of three quadrup oles. Ran and Colonius
[2009] prop osed three stages for the sound generation of a turbulen t v ortex
ring: the instabilit y w a v es whic h generate relativ ely w eak sound, the v ortex
breakdo wn (connected with the maxim um sound pressure lev el) and the
turbulen t deca y , whic h leads to a deca y in the sound pressure lev el of 30 dB.
There exists a v ery extensiv e literature of the jet flo w. In terested readers
are directed to the review b y Ball et al. [2012], whic h pro vides a v ery clear
and detailed o v erview of the differen t con tributions. W e giv e here a brief
o v erview of the milestones in the jet flo w researc h.
T ollmien [1926], one of the first works in turbulen t jets, considered an
incompressible jet flo w through Prandtl’s mixing length theory . The first
exp erimen tal studies of a round jet w ere carried out b y Ruden [1933] and
Kuethe [1935], who rep orted the similarit y of v elo cit y profiles w as reac hed
b efore 10 D . The jet similarity and the r egion in whic h this tak es place is
still under debate. The first direct numerical sim ulation of a sup ersonic jet
w as rep orted b y F reund et al. [2000], who considered a jet with a Reynolds
n um b er Re D = 2000 based on the nozzle diameter and on the fully ex-
panded Mac h n um b er M j = 1 . 92 at adapted conditions, fo cusing mainly
on the confirmation of the direct n umerical sim ulation and the acoustic ra-
diated b y the sup ersonic jet. As shown, the main fo cus un til no w has b een
to c haracterise the fluid flo w through statistical analyses to obtain univ ersal
la ws that define the b eha viour of the jet flo w.
3

Chapter 1. In tro duction
The noise generated b y the jet flo w has b een in tensiv ely studied since
the decade of 1950 , starting with Po well [1953], who considered the noise
radiated b y a c hok ed nozzle flo w. The jet noise, esp ecially from sup ersonic
con tin uous jets, is a v ery useful reference for the acoustics of the compress-
ible starting jet, whic h can b e considered a new topic. A short o v erview of
the closest jet noise references is presen ted here fo cusing on sup ersonic jets.
The reviews of Seiner and Y u [1984] and T am [1995] fo cus on the su-
p ersonic jet noise, pro viding a detailed discussion ab out the three differen t
noise comp onen ts and their generation. The turbulent mixing noise is pro-
duced b y the turbulen t scales of the shear la y er and therefore it is radiated
in b oth subsonic and sup ersonic cases. The br o adb and sho ck noise is gen-
erated b y the in teraction b et w een the sho c k w a v es and the v ortices of the
shear la y er where the sho c k w a v es are reflected. The scr e e ch tones are gen-
erated b y a feedbac k lo op in whic h the acoustic w a v es tra v el bac kw ards (in
the subsonic surroundings of the jet) to the lip of the nozzle triggering new
disturbances that are again con v ected do wnstream un til the p oin t in whic h
these acoustic w a v es w ere generated, where the shear la y er v ortices in teract
with the sho c k w a v es, closing in this wa y the lo op with a sp ecific frequency .
The acoustics of the starting jet are presen ted in c hapter 8.
As t ypical in jet noise, the angle θ that characterises the directivit y is
measured from the jet axis in the upstream direction, T am [1995]. W e use
here the same con v en tion.
As prop osed b y T am et al. [1996], the TMN can b e describ ed b y t w o
similarit y sp ectra pro duced b y t w o noise sources: ( i ) the lar ge-sc ale simi-
larity (LSS) sp ectrum pro duced b y the large and coheren t structures and
( ii ) the fine-sc ale similarity (FSS) sp ectrum pro duced b y the fine turbulen t
scales. Some time later, T am et al. [2008] confirmed that only t w o differen t
sources of jet TMN exist: the fine-scale turbulence and the large turbulent
structures of the jet flo w. Comparing the noise of a p erfectly expanded
with an under-expanded jet from a con v ergen t-div ergen t nozzle, Kim et al.
[1994] pro vided evidence that the presence of a sho c k cell structure do es
not mo dify the TMN.
Linking the go v erning parameter of the compressible starting jet (sec-
tion 1.3) to the acoustics generated (c hapter 8) w e can predict the go v erning
parameters from acoustic measuremen ts. W e handle this problem in c hap-
ter 9. This is esp ecially useful when no direct access to the jet flo w is
p ossible, as in v olcanic flo ws. This is exp ected to b e relev ant for v olcanic
monitoring purp oses. W e included as w ell in this w ork a c hapter dedicated
to v olcanic jets, c hapter 10.
4

1.1 Descript ion of the flo w
( a ) Pressure w a v e ( b ) V ortex ring ( c ) T railing jet ( d ) Decay stage
Figure 1.1: Stages of th e starting jet. N umerical sc hlieren ( |∇ ρ | ) is sho wn. The
case 7 .c is represen ted in ( a ) for t ∗ =0 . 8 ,( b ) t ∗ =3 . 2 and ( c ) t ∗ =5 . 9 and case
6 is represen ted in ( d ) for t ∗ = 72 .
During the impulsive disc harge of the reserv oir, just aft er the release
of the pressure through the nozzle, a compression w a v e is formed in the
vicinit y of the nozzle exit. This compress ion w a v e is generated b y the
impulsiv e start of the jet. The compression w a v e has a half-spherical form
and tra v els in to the un b ounded c ham b er with the sp eed of sound if it is a
pressure w a v e, as in figu re 1.1 a , or faster if it is a blast w a v e. The basic idea
of a blast w a v e is a viole n t propagating disturbance, pro duce d historically
b y an explosion, that co nsist of an abrupt rise in pressure fol lo w ed b y a
drop to or b elo w atmospheric pressure. The in terested reader is directed to
T a ylor [1950] f or more information.
During the propagatio n of the compression w a v e in the first few diam -
eters, a v ortex ring is a lw a ys generated due to the large veloc it y gradien ts,
figure 1.1 b . The vortex ring gro ws un til a critical size is reached ( 1 . 08 D
prop osed b y Didden [1 979]) and then it starts propagatin g in the axial di-
rection. Shortly after th e b eginning of the v ortex ring prop agation, the
v ortex ring separates from th e trailing jet (if existing) and propa gates fur-
ther, detac hed from the rest of the flo w. The pinc h-off pro cess is driv en b y
the Kelvin-Helmholt z instabilities from the shear lay er in the trailing jet,
Zhao et al. [2000]. When the successive Kelvin-Helmholtz v ortices in the
shear la y er are generate d, they start rotating and drawing in v orticit y from
upstream and downstream of the v ortex in the shear lay e r and this leads to
a v orticit y distributi on along the shear la y er with concentrated regions of
large v orticity (the v ortices) separated b y regions of r elativ ely lo w v orticit y
(the spaces b et w een t w o consecutiv e v ortices). These spaces b et w een v or-
tices are the p otential loc ations where the pinc h-off w ould take place, see
5

Chapter 1. In tro duction
figure 1.2. The first Kelvin-Helmholtz v ortex after the v ortex ring is the
first whic h is generated and therefore the one whic h will create the first a
region of v ery lo w v orticit y around it (if it is not ingested b y the head v ortex
ring), leading to the pinc h-off when the lo w v orticit y reac hes the pinc h-off
criterion. The dynamics of the shear la y er v ortices pla y a crucial role in the
pinc h-off pro cess. A detailed analysis of the pinc h-off definition is presen ted
in section 6.1. In order to predict the pinc h-off, a relationship b et w een the
normalised circulation injected un til the pinc h-off and the non-dimensional
time is pro vided in section 6.2.
Figure 1.2: Flow configuration for t ∗ = 5 . 5 sho wing the dev elopmen t of the
Kelvin-Helmholtz v ortices in the case 7 .c . The v orticit y magnitude is sho wn in a
blac k and white colour-scale.
Figure 1.3 sho ws a snapshot of the Mac h n um b er flo w field of a laminar
and a turbulen t v ortex ring. A v ery w ell organised flo w with one main big
structure describ es the laminar case in figure 1.3 a . As opp osed to that,
the turbulen t v ortex ring in figure 1.3 b includes a wide range of scales
with differen t sizes, but still with the global structure of a v ortex ring.
T o v erify that the instabilities presen t in the turbulen t v ortex ring are
true turbulence w e sho w in figure 1.4 the three-dimensional energy sp ectra
calculated using the v elo cit y fluctuations with resp ect to the azim uthal
a v erage: u 0 = u − h u θ i , where u is the instantaneous v elo cit y and h u θ i is
the mean v elo cit y a v eraged in the azim uthal direction. The K olmogoro v
length scale w as calculated as η =  ν 3 /ε  1 / 4 , where ε = ν h ∂ u 0
i
∂ x j
∂ u 0
i
∂ x j i and ν is
the kinematic viscosit y . The same pro cedure was performed for b oth cases.
6

F or the laminar case, we used the K olmogorov length scale of the turbulen t
case b ecause w e fo cus on the shap e of the curv es rather than their p osition
in the diagram. The dashed line corresp onds to the laminar case, sho wing
one big structure with the most of the energy and a rapid deca y for small
scales. The solid line, corresp onding to the turbulen t case, shows a big
structure with a lot of energy , but for smaller scales, w e can see a deca y of
the energy v ery close to the − 5 / 3 exp onent of the K olmogoro v la w. This
confirms that inside the v ortex ring a wide range of scales with differen t
sizes dissipate the kinetic energy in the turbulen t w a y: this is a turbulen t
v ortex ring.
( a ) ( b )
Figure 1.3: Mac h n umber magnitude contour plot of ( a ) a laminar (M in the
range [0 , 0 . 16] ) corresp onding to case 1 .a and ( b ) a turbulen t v ortex ring (M in
the range [0 , 1 . 85] ) corresp onding to case 7 .a .
In the case of the existence of a trailing jet, the jet in teracts with the
v ortex ring already for a Reynolds n um b er of 5 000 . This in teraction, here
called sho ck-she ar layer-vortex ring inter action , is explained in detail in
section 7, leading to one of the loudest noise sources of the starting jet and
comparable to those of the con tin uous jet.
The trailing jet can b e either subsonic during the whole pro cess or su-
p ersonic during the impulsiv e p erio d and subsonic during the final deca y ,
(see figure 1.1 d ), but it cannot b e subsonic during the first stage, follo w ed
b y a sup ersonic stage and a final subsonic deca y b ecause w e fo cus on the im-
pulsiv ely starting jet: the acceleration time in terv al to ha v e sup ersonic flo w
at the nozzle exit should b e shorter than the time in whic h the v ortex ring
is completely generated, so in the onset of the trailing jet there are already
sup ersonic conditions. Since the pressure at the inlet c hanges con tin uously
with time, the jet will b e adapted for a sp ecific time (when the pressure ratio
is the one the nozzle w as designed for), b eing imp erfectly expanded for the
rest, whic h means that the sup ersonic asso ciated phenomena such as shock
w a v es and expansion fans will b e presen t for most of the sup ersonic stage.
7

Chapter 1. In tro duction
10 −2 10 −1 10 0
10 −4
10 −3
10 −2
10 −1
10 0
10 1
10 2
10 3

( κ η ) − 5 / 3
κ η
E / ( η u 2
η )

Figure 1.4: Three-dimensional kinetic energy sp ectra. Re D = 5000 . The solid
blac k line corresp onds to a turbulen t v ortex ring. Case 7 .a . The dashed blac k
line corresp onds to a laminar v ortex ring. Case 1 .a . The vertical red dashed line
indicates the K olmogoro v scale and the grey solid line the − 5 / 3 reference deca y .
The com bination of sho c k w a v es and expansion w a v es is usually called in
the literature ’sho c k cell structure’, and its prop erties dep end mainly on
the w orking fluid via the isen tropic exp onen t γ , the geometry of the nozzle,
via the exit to critical area ratio A e / A ∗ and the op eration conditions via
the reserv oir to am bien t pressure ratio p 0 r /p ∞ . The relationship b et w een
the op erating conditions and the geometrical prop erties of the sho c k cell
structure of the trailing jet can b e used to c haracterise the jet flo w using the
correlations of the literature when the go v erning parameters are unkno wn.
Also due to the ev olution of the inlet condition during the deca y phase,
the sho c k w a v es presen t in the trailing jet translate with a sp ecific v elo cit y ,
whic h can b e comparable to the sp eed of sound, esp ecially during the decay
stage, ha ving a strong effect on the acoustic radiated. More details ab out
this phenomenon are giv en in section 7.3.
The noise radiation directionalit y of the con tin uous jet is v ery w ell
kno wn in the comm unit y . In section 8.6 w e study the angular dep endence
of the sound pressure lev el for the starting jet and compare it with the
con tin uous one.
The ph ysical pro cess finishes when the reserv oir reac hes the pressure
of the un b ounded c ham b er, assuming that the effects of buo y ancy or mass
8

diffusion due to temp erature differences b et w een the reserv oir and the un-
b ounded c ham b er can b e neglected.
1.2 Set-up description
The basic configuration considered here for a compressible starting jet
is a pressurised reserv oir separated from an un b ounded c ham b er b y a con-
v ergen t nozzle ( A e /A ∗ =1 , where A e is the nozzle exit cros s section area
and A ∗ is the critical cross s ection area) through whic h the fluid is d is-
c harged, see figure 1.5. The conditio ns in the reserv oir are denoted with
a subscript ’r’ ( p r ) and the t otal conditions at the reservoir are indicated
with a subscript ’0r’ ( p 0 r ), while the conditions in the un b ounded c ham b er
are denoted with a subscript ’ ∞ ’( p ∞ ).
Figure 1.5: System set-up. The reserv oir (left) is pressurised initially at p 0 r
and the fluid is injected thro ugh a con v ergen t nozzle into the un b ounded c ham b er
(righ t), at a pressure p ∞ . The pressure at the ce n tre of the nozzle exit is p e .
W e restrict our ana lysis to the neutral temp erature case, this is, T 0 r =
T ∞ , where T ∞ is the te mp erature in the un b ounded c ham b er. The inlet
is c hosen to b e adiabat ic and therefore an isen tropic flow tak es place from
the reserv oir up to the l ip of the nozzle.
1.3 Effects of the go v erning parameters
The b eha viour of the starting jet is go v erned b y four main dimensionless
parameters: the Rey nolds n um b er ( Re D ), affec ting mainly the size of the
Kelvin-Helmholtz in stabilities relativ e to the nozzle d iameter and therefore
the turbulence; the non-dimensi onal mass supp ly ( L/D ) giv en b y the
length ( L ) of a hypoth etical cylindrical reservoir with a constan t diameter
( D ) (the same diameter as the nozzle) and the same v olume than the real
reserv oir, influen cing mainly the existence of the trailing je t, the reserv oir
9

Chapter 1. In tro duction
to un b ounded c ham b er pressure ratio ( p 0 r /p ∞ ), ha ving an effect on
the compressibilit y and the reserv oir to un b ounded c ham b er temp er-
ature ratio T 0 r /T ∞ .
In this section, w e address the questions of what c hanges tak e place
in the compression w a v e, v ortex ring or trailing jet when c hanging the
go v erning parameters and what are the limiting v alues of the go v erning
parameters that define a sp ecific b eha viour.
1.3.1 Reynolds n um b er ( Re D )
The main effect of the Reynolds n um b er in the starting jet is the size
of the Kelvin-Helmholtz instabilities in the shear la y er of the trailing jet
compared to the nozzle diameter assuming a constan t shear la y er thic kness
of 0 . 1 D . A larger Reynolds n um b er leads to a smaller size of instabilities,
and vice-v ersa. This has sev eral consequences.
W e observ ed that a smaller size of the Kelvin-Helmholtz instabilities
leads to an earlier pinc h-off. The smaller v ortices are earlier formed in the
jet and they start earlier to absorb v orticit y upstream and do wnstream of
the v ortex whic h leads to a situation in whic h the v orticit y threshold defined
for the pinc h-off is reac hed earlier. Therefore, there is a critical Reynolds
n um b er ( Re D , interaction ) that makes the first Kelvin-Helmholtz v ortex to be
already formed when it passes through the first sho c k w a v e, see figures 1.1 c
and 1.2. The sho ck-she ar layer-vortex in teraction tak es place only if the
Reynolds n um b er is larger than this critical v alue. The Reynolds num b e rs
in this study ( Re D = 5 000 and 10 000 ) exceed this critical v alue, but the
Reynolds n um b er in Zhao et al. [2000], ( R e D = 3 800 ) is clearly under
the critical one. The critical Reynolds n um b er R e D, interaction for the sho ck-
she ar layer-vortex in teraction is an op en question and future w ork should
answ er this question. F or more details see section 7.
Ricou and Spalding [1961] suggested that the en trainmen t co efficien t
do es not v ary in turbulen t jets for Reynolds n um b er larger than 10 000
and this v alue migh t b e the starting of the Reynolds indep endence regime
for turbulen t jets. Based on the relative magnitude of dimensional spatial
scales, Dimotakis [2000] rep orted that the fully-dev elop ed turbulen t flo w
requires an outer-scale Reynolds n um b er of Re D & 10 000 − 20 000 . With
these argumen ts, and with the limitations of computational capacit y , w e
c hose a Reynolds n um b er of 10 000 for the starting jet without deca y (case
7 .c in table 2.7) as the largest practical Reynolds n um b er to b e fully re-
solv ed. A Reynolds n um b er of 15 000 w as simulated (case 7 .d in table 2.7)
with the same resolution as c hosen for Re D = 10 000 to explore the limits
of the spatial resolution. As a direct comparison w e sho w in figure 1.6 the
v orticit y isosurface of t w o jets with R e D = 5 000 and 10 000 in the con-
tin uous stage, where it can b e seen, that the smallest turbulent scales in
the case with Re D = 10 000 are smaller than the ones for Re D = 5 000 ,
10

( a )
( b )
Figure 1.6: V orticit y isosurface. ( a ) R e D = 5 000 , ω /ω max = 0 . 1 . Case 7 .b ( b )
Re D = 10 000 , ω /ω max = 0 . 1 . Case 7 .c .
while the largest turbulen t scales remain the same size. In figure 1.7 w e
sho w a comparison b et w een t w o jets in their starting stage at R e D = 5 000
and 15 000 , with the same conclusion, but where the inner structure of the
v ortex ring can b e seen.
Examining the needed conditions to generate an incompressible laminar
v ortex ring, a laminar v ortex ring that transitions to turbulence and an
initially turbulen t v ortex ring, Glezer [1988] generated a map, sp ecifying
that for Reynolds n um b ers (defined as Γ /ν ) larger than 2 . 5 × 10 4 an initially
turbulen t v ortex ring w as generated. F or laminar-generated v ortex rings,
an azim uthal instabilit y (similar in app earance to the Kelvin-Helmholtz
instabilit y of a free shear la y er) amplifies and mak es the v ortex ring to
breakdo wn to turbulence. They also rep orted that for larger v elo cities at
the nozzle exit, the axial distance to the turbulen t transition of the v ortex
ring w as shorter. This interaction is hereafter called she ar layer-vortex
in teraction. W e observ ed that compressibilit y accelerates the onset of this
in teraction. This in teraction is examined in detail in c hapter 7.
11

Chapter 1. In tro duction
( a ) ( b )
Figure 1.7: Isosurface Q-criterion. ( a ) Re D = 5 000 , Q = 10 9 . Case 7 .b ( b )
Re D = 15 000 , Q = 10 9 . Case 7 .d .
1.3.2 Dimensionless mass supply ( L/D )
Often can b e seen L/D as the ’strok e ratio’ in exp erimen tal w orks of the
literature when dealing with an incompressible starting jet. T ypically the
starting jet of an incompressible fluid w as driv en b y a piston in a cylinder
of diameter D o v er a strok e of length L , giving rise to the definition of
L/D as one of the go v erning parameters. In compressible flo ws, we sho wed
already in the in tro duction (equation 1.1) ho w L/D can b e seen in a more
general w a y as the non-dimensional mass supply . The main effect of the
dimensionless mass supply is on the existence of the trailing jet.
The limiting v alue of the non-dimensional mass supply ( L/D ) lim dic-
tates the existence of a trailing jet after the v ortex ring in the starting jet.
The more mass is injected b y the nozzle, the more v orticit y is pro vided to
the v ortex ring. If the v ortex ring tak es all the v orticit y ejected b y the
nozzle w e call it not satur ate d , b ecause it could tak e some more v orticit y .
If the v ortex ring do es not tak e all the v orticit y ejected b y the nozzle, a
trailing jet is generated with the rest of the v orticit y that the v ortex ring is
not able to tak e. In this case, the vortex ring is satur ate d . A limit has b een
already b een set for the v alue of L/D to saturate the vortex ring Gharib
et al. [1998], ( L/D ) lim ≃ 3 . 6 − 4 . 5 , but it has b een stated that this limit
is not univ ersal and it dep ends on the temp oral and spatial inlet condition
distributions, as w ell as the Reynolds n um b er, Zhao et al. [2000]; Rosenfeld
et al. [1998]; Gao and Y u [2010]. More details are giv en in 6.2. The limiting
12

v alue ( L/D ) lim for compressible flo ws is an op en question that should b e
addressed b y future w ork, but from observ ations in this study the v alue
app ears to b e v ery close to the incompressible one.
( a ) ( b )
( c ) ( d )
Figure 1.8: Starting jet fluid flo w with differen t sets of parameters of sev eral
t w o-dimensional sim ulations to exemplify the differences not only b et ween the
incompressible and the compressible cases, but also among the compressible cases
for differen t regimes. ( a ) Incompressible, case 15 . t ∗ = 6 . 41 . M j = 0 . 1 . |∇ ρ | ∈
[0 , 0 . 025] . ( b ) Sup ersonic laminar, case 20 . t ∗ = 13 . 37 . M j = 1 . 27 . |∇ ρ | ∈ [0 , 2] .
( c ) Sup ersonic turbulen t, case 18 . t ∗ = 12 . 72 . M j = 1 . 32 . |∇ ρ | ∈ [0 , 3] . ( d )
Sup ersonic with blast w a v e, case 22 . t ∗ = 6 . 91 . M j = 1 . 70 . |∇ ρ | ∈ [0 , 4] . The
axial p osition of the v ortex ring is x V R /D = 2 for all cases.
The stabilit y of the shear la y er migh t also b e influenced b y this param-
eter. On the one hand, the jet in figure 1.8 c sho ws an unstable shear la y er.
This jet comes from an infinite reserv oir ( L/D → ∞ ), blo wing con tin uously
and the com bination of the go v erning parameters lead to an unstable shear
la y er and a turbulen t jet. On the other hand, the jet in figure 1.8 b sho ws a
stable shear la y er. This jet comes from a finite reservoir ( L/D = 8 ) and is
already in the deca y stage, so it do es not blo w with the maxim um v elo cit y
and the sho c k w a v es are tra v elling to w ards the nozzle exit. Ho w ev er, the
13

Chapter 1. In tro duction
leading v ortex ring is propagated in the axial direction. This leads to a
stable shear la y er and a laminar jet.
Figures 1.8 a and d are included to sho w ho w similar are the effects
of the pressure ratio, Reynolds num ber and dimensionless mass supply .
Comparing optically the figures 1.8 a and c , w e w ould exp ect a c hange in the
Reynolds n um b er b et w een b oth cases, although the only differen t parameter
is the pressure ratio p 0 r /p ∞ = 1 . 007 in a and p 0 r /p ∞ = 2 . 84 in figure 1.8 c ;
the rest of the parameters ha v e the same v alue: L/D → ∞ , Re D = 5000
and T 0 r /T ∞ = 1 . While figure 1.8 a shows a laminar and incompressible
starting jet, figure 1.8 c shows a supersonic and turbulent starting jet.
Figure 1.9: Numerical sc hlieren for the case p 0 r /p ∞ = 50 , L/D = 0 . 1659 ,
Re = 5000 . |∇ ρ | ∈ [0 , 4] . t ∗ = 38 . 0378 . M j = 1 . 6979 . A very w eak trailing jet
is presen t although the jet w as generated with a v ery large pressure ratio, due to
the lo w v alue of the non-dimensional mass supply ( L/D = 0 . 1659 ).
Figure 1.8 d sho ws a starting jet generated with a pressure ratio of
p 0 r /p ∞ = 50 , a Reynolds n um b er of R e D = 5 000 (same as the other sub-
figures) and with a relativ ely small non-dimensional mass supply ( L/D =
0 . 1345 ). Comparing this set of parameters with those of figure 1.8 b , w e
agree on exp ecting a larger Mac h n um b er, b ecause of a stronger expansion,
but due to the small v alue of L/D , one could exp ect a fully laminar v ortex
ring, but we can iden tify the turbulent c haracter of the v ortex ring. Fig-
ure 1.9 sho ws, for a later stage, the turbulent and compressible v ortex ring
14

generated in the same case as figure 1.8 d .
The starting jet with a finite L/D will not reach the same maxim um
fully expanded Mac h n um b er exp ected for the con tin uous jet ( L/D → ∞ )
at the same pressure ratio ( p 0 r /p ∞ ). The maxim um fully expanded Mac h
n um b er decreases with lo w er L/D v alues. After the release of the pressure,
the expansion w a v e that tra v els inside the nozzle is reflected at the b ottom
of the reserv oir for an earlier time, so the expansion tak es place during a
shorter p erio d.
0 20 40 60 80
0
0.2
0.4
0.6
0.8
1

t ∗
M e

0 20 40 60 80
0
0.2
0.4
0.6
0.8
1

t ∗
M j

( a ) ( b )
Figure 1.10: Effect of the non-dimensional mass supply on the temp oral ev o-
lution of the exit ( a ) and fully expanded ( b ) Mac h n um b er. Straigh t nozzle
( A e / A ∗ = 1 ), p 0 r /p ∞ = 4 . Dashed: L/D = 2 (case 11 .c ), solid: L/D = 8 (case
11 .d ).
Figure 1.10 represen ts the effect of the non-dimensional mass supply
L/D in the exit and fully expanded Mac h n um b er. With a straight nozzle
( A e / A ∗ = 1 ) and a pressure ratio of p 0 r /p ∞ = 4 we see ho w a bigger
reserv oir (solid line, L/D = 8 ) leads to a faster flow during a longer time
than the case with a smaller reserv oir (dashed line, L/D = 2 ).
1.3.3 Pressure ratio ( p 0 r /p ∞ )
The steady fully expanded Mac h n um b er ( M j ) of a con tin uous jet can
b e determined just b y p 0 r /p ∞ , assuming γ = 1 . 4 for diatomic gases, by
using the one-dimensional isen tropic theory , see equation (1.2). The larger
is p 0 r /p ∞ the stronger is the expansion through the nozzle and the larger is
the fully expanded Mac h n um b er. In the compressible starting jet, p 0 r /p ∞
pla ys a crucial role determining the existence of sho c k w a v es, expansion
fans and their geometrical distribution. This parameter do es not app ear in
the incompressible case.
15

Chapter 1. In tro duction
M j = v
u
u
t 2
γ − 1  p 0 r
p ∞  γ − 1
γ
− 1 ! (1.2)
The minim um reserv oir to un b ounded c ham b er pressure ratio needed to
c ho c k a one-dimensional isen tropic con v ergen t nozzle is giv en b y  p 0 r
p ∞  ∗
=
 γ +1
2  γ
γ − 1 (e.g. Bec k er [1968]). The pressure ratio m ust ha v e at least this
v alue to ac hiev e sup ersonic flo w. In this case, due to the time dep endency of
the nozzle exit pressure ratio and the fixed geometry , the flow is imperfectly
expanded. This leads to the existence of the sup ersonic phenomena and the
corresp onding acoustic comp onen ts (mainly broadband sho c k noise).
Ishii et al. [1999] found b y applying the Rankine-Hugoniot relations
across a mo ving sho c k that the required pressure ratio to generate a blast
w a v e in a sho c k tub e is ( p 0 r /p ∞ ) blast = 41 . 2 . An example of a set-up with
a blast w a v e is sho wn in figure 1.8 d . In this study , the t ypical v alues for
p 0 r /p ∞ are ab out 3 . 6 and therefore only pressure w a v es are formed.
Didden [1979] prop osed a critical size of an incompressible v ortex ring
( 2 R V R /D = 1 . 08 ) at whic h it stops gro wing and starts propagating in the
axial direction. F or large v alues of p 0 r /p ∞ , a Prandtl-Mey er expansion
w as observ ed, leading to a widening of the flow after the nozzle exit and
therefore leading to larger v ortex rings b efore starting to propagate, see
figure 1.8 d . F or v alues of p 0 r /p ∞ that lead to a sup ersonic flow, the v ortex
ring has a sho c k w a v e in its fron t, so that the pressure of the sup ersonic
flo w inside the v ortex ring is matc hed to the pressure ahead of the v ortex
ring. This is presen ted in figures 1.8 b , 1.8 c and 1.8 d .
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3

t ∗
M e

0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3

t ∗
M j

( a ) ( b )
Figure 1.11: T emp oral ev olution of the exit (left) and fully expanded (righ t)
Mac h n um b er. Effect of the pressure ratio. 30 ◦ , L/D = 2 . Dotted: p 0 r /p ∞ = 3
(case 13 .a ), dash-dotted: p 0 r /p ∞ = 4 (case 13 .c ), dashed: p 0 r /p ∞ = 50 (case
13 .e ), solid: p 0 r /p ∞ = 80 (case 13 .g ).
16

Figure 1.11 sho ws the effect of the pressure ratio p 0 r /p ∞ on the exit
and fully expanded Mac h n um b er. With a div ergent nozzle c haracterised
b y A e / A ∗ = 4 and non-dimensional mass supply L/D = 2 ; the larger the
pressure ratio the larger the exit Mac h n um b er. F or the largest pressure
ratio, p 0 r /p ∞ = 80 , the exit Mach n um ber remains in the sup ersonic regime
for a longer time than the case with p 0 r /p ∞ = 50 . The tw o smaller pressure
ratios ( p 0 r /p ∞ = 3 , 4 ) sho w a subsonic Mac h n um b er at all times b ecause
their pressure ratio leads to a sho c k w a v e in the div ergen t part of the nozzle
and they ha v e a subsonic flo w at the nozzle exit. Concerning the fully
expanded Mac h n um b er, the b ehaviour is v ery similar with the exception
that the fully expanded Mac h n um b er do es not remain constan t while the
nozzle is c hok ed, while the exit Mac h n um b er do es.
1.3.4 T emp erature ratio ( T 0 r /T ∞ )
In this study , w e fo cused on the compressible starting jet where T 0 r
T ∞ = 1 ,
so this analysis is purely based on applying the principles of the isen tropic
one-dimensional theory of gas dynamics and results from the literature.
As a general trend, an increase of the jet temp erature ( T 0 r ) leads to an
increase in the fully expanded temp erature ( T j ), since their ratio dep ends
only on the pressure ratio p 0 r /p ∞ . Keeping a fixed pressure ratio, the fully
expanded Mac h n um b er ( M j ) remains constant (see equation (1.2) in the
previous p oin t). The com bination of b oth conditions leads to:
T 0 r
T j
= 1 + γ − 1
2 M 2
j =  p 0 r
p ∞  γ − 1
γ
. (1.3)
An increase of the fully expanded temp erature causes an increase in the
fully expanded sp eed of sound ( c 2
j = γ RT j ). A straigh tforw ard consequence
of this is an increasing fully expanded v elo cit y ( u j = M j c j ).
A similar analysis can b e done for the fully expanded densit y . The
ratio ρ 0 r /ρ j remains constan t for a fixed pressure ratio, but an increase
in the reserv oir temp erature leads to a decrease of the reserv oir densit y
( ρ 0 r = p 0 r / ( RT 0 r ) ).
No w, in order to ev aluate the c hange in momen tum, w e ha v e to ev aluate
the c hanges in the fully expanded v elo cit y and densit y . While the v elo cit y
increases b y a factor of the square ro ot of the c hanges in the temp erature
ratio, the densit y decreases b y the same factor than the c hanges in the
temp erature ratio, and due to this, the c hange in the momen tum decreases
b y a factor of the square ro ot of the c hanges of the temp erature ratio, see
17

Chapter 1. In tro duction
equation (1.4c).
u j, hot
u j, neutral
=
M j, hot q γ RT 0 r , hot T j, hot
T 0 r , hot
M j, neutral q γ RT 0 r, neutral T j , neutral
T 0 r , neutral
= s T 0 r , hot
T 0 r , neutral
(1.4a)
ρ j, hot
ρ j, neutral
= p 0 r , hot
RT 0 r, hot
RT 0 r, neutral
p 0 r , neutral
= T 0 r , neutral
T 0 r , hot
(1.4b)
ρ j, hot u j, hot
ρ j, neutral u j, neutral
= s T 0 r , neutral
T 0 r , hot
(1.4c)
A ccording to Cigala et al. [2017], the effect of the jet temp erature ratio is
p ositiv ely correlated with the maxim um particle ejection v elo cit y . This ex-
p erimen tal observ ation agrees with the previous analytical argumen tation.
The same authors confirm that the v elo cit y deca y rate is not correlated to
the temp erature ratio.
18

Ob jectiv es of this thesis
As seen in the in tro duction, the con tin uously blo wn jet has b een in ten-
siv ely studied in the literature b oth in the incompressible and the compress-
ible case. The starting-deca ying jet has b een also extensiv ely in v estigated,
but only in the incompressible case. The assumption of incompressibility
excludes some imp ortan t features of the starting jet that are crucial for
some applications, lik e pulse-jets, fuel injection in engines, air-bag devices
and v olcano es.
The ob jectiv es of this thesis are first of all to iden tify the go v erning
parameters of the compressible starting jet and to analyse the effect of them
on the fluid flo w and on the radiated acoustics. Second, the characterisation
of the differen t elemen ts of the jet when c hanging the go v erning parameters.
Third, the effects of the inflo w conditions b y means of turbulence lev el and
in terms of the nozzle geometry . F ourth, a full description of the acoustics
radiated b y the starting-deca ying jet separated b y its noise sources. Last,
but not least, the prediction of the go v erning parameters of a compressible
starting jet from acoustic and/or optical measuremen ts.
With this set of ob jectiv es, a very goo d understanding of such a complex
problem should b e ac hiev ed.
Ov erview of this thesis
This thesis is divided in to three main parts: The first part deals with
the n umerical framew ork used to p erform the n umerical w ork of this thesis.
The description of the n umerical mo del, initial and b oundary conditions
as w ell as the grid considerations are describ ed in this part. The second
part describ es the exp erimen tal set-up used in this thesis as w ell as the
equipmen t and the configuration used. The third part is the b o dy of this
thesis, w e presen t here most of the outcome of this thesis, togeth er with
the discussion of the results. W e tried to separate the parts concerning the
fluid flo w and the acoustics, with an o v erlap there where it w as necessary .
Chapter 4 deals with the c haracterisation of the differen t elemen ts of the
compressible starting jet, comparing where necessary to the incompressible
19

Chapter 1. In tro duction
case. Chapter 5 handles the effect of the inflo w conditions in terms of the
turbulence lev el. Chapter 6 defines the pinc h-off in a more general and
con v enien t w a y for compressible and turbulen t flo ws. Chapter 7 presen ts
ho w the trailing jet in teracts with the v ortex ring in the sup ersonic case,
giving rise to one of the dominan t acoustic sources of the sup ersonic start-
ing jet. In this direction, c hapter 8 in tro duces the acoustic sources of the
starting jet. Ha ving describ ed the effect of the go v erning parameters and
the acoustics of the starting jet, c hapter 9 gives an insigh t in the prediction
of the go v erning parameters from acoustic measuremen ts. W e included in
this c hapter also some relation with optical measuremen ts when appropri-
ate. As the biggest and most impressive of the applications in the nature
related to the compressible starting jet, c hapter 10 has b een dedicated to
v olcanic jets.
20

P art I
Sim ulations of a
compressible starting jet
21

Chapter 2
Numerical bac kground
2.1 Na vier-Stok es equations
An unsteady turbulen t flo w is generally describ ed b y the Na vier -
Stokes equations. They describ e the conserv ation of mass, momentum
and energy . These equations are highly non-linear and coupled with eac h
other, but it is p ossible to de-couple them b y applying a linearisation of
small p erturbations around a base-flo w, obtaining three main mo des: ( i )
v orticit y , ( ii ) entrop y and ( iii ) sound-w a v e. T o p erform this linearisation,
the equations are written in pressure ( p ), v elo cit y ( u i ) and en trop y ( s )
form ulation and they read:
∂ p
∂ t + u i
∂ p
∂ x i
= − γ p ∂ u i
∂ x i
+ p
C v  ∂ s
∂ t + u i
∂ s
∂ x i  (2.1a)
∂ u i
∂ t + u j
∂ u i
∂ x j
= − 1
ρ
∂ p
∂ x i
+ 1
ρ
∂ τ ij
∂ x j
(2.1b)
∂ s
∂ t + u i
∂ s
∂ x i
= 1
ρT  − ∂
∂ x i  − λ ∂ T
∂ x i  + Φ  (2.1c)
where
τ ij := 2 µ  S ij − 1
3 S k k δ ij  (2.2)
S ij := 1
2  ∂ u i
∂ x j
+ ∂ u j
∂ x i  (2.3)
Φ := τ ij S ij . (2.4)
In this set of equations, δ ij denotes the Kronecker delta.
23

Chapter 2. Numerical bac kground
T o close this set of equations, the classical assumption of an ideal gas
w as made:
p = ρRT (2.5)
with the gas constan t R = C p − C v and the isentropic exponent γ =
C p /C v = 1 . 4 . C p is the heat capacit y at constan t pressure and C v is
the heat capacit y at constan t v olume.
The viscosit y is mo delled using Sutherland ’s la w:
µ
µ 0 r
= T 0 r + T S
T + T S  T
T 0 r  3 / 2
, (2.6)
where T S = 110 . 4 K is the Sutherland’s temp erature. T 0 r denotes the total
temp erature and µ 0 r the corresp onding viscosit y .
W e mo delled the heat conductivit y ( λ ) follo wing F ourier’s la w:
λ = µ C p
Pr , (2.7)
with Pr = 0 . 71 as Prandtl n um b er.
The three-dimensional compressible Na vier-Stok es equations ((2.1) a − c )
are solv ed in c haracteristic form as in [Sesterhenn, 2000]. This has t w o main
adv an tages: ( i ) the implementation of boundary conditions for aeroacous-
tic applications is straigh tforw ard (for example a non-reflecting b oundary
condition is implemen ted b y setting the incoming w a v e to zero) and ( ii )
the suitabilit y of up wind sc hemes, where the up- and down tra v elling w a v es
are discretised with the up- and do wn wind sc hemes, resp ectiv ely .
The h yp erb olic part of the Na vier-Stok es equations is discretised with a
fifth-order up wind sc heme follo wing [A dams and Shariff, 1996]. The remain-
ing terms of the equations (heat flux and friction terms) are of parab olic
nature and are discretised with a sixth-order cen tral sc heme with sp ectral-
lik e resolution follo wing [Lele, 1992].
F or the time in tegration, a classical lo w-storage Runge-Kutta metho d
of fourth-order is used. All sim ulations ha v e an explicit CFL n um b er in
the range [0 . 3 − 0 . 7] , leading to a time step size of dt = 1 . 545 · 10 − 5 for the
largest resolution (2048 × 1024 × 1024) , dt = 1 . 768 · 10 − 5 for the medium
resolution (1024 × 512 × 512) and dt = 2 . 788 · 10 − 5 for the lo w est resolution
(512 × 256 × 256) .
2.2 Grid considerations
Since w e mainly fo cus on turbulence and acoustics, the strongest re-
quiremen t for the computational grid in our direct n umerical sim ulations
(DNS) is to resolv e the smallest turbulen t scales presen t in the flo w field.
As a consequence, the smallest resolv ed acoustic w a v elengths are of the
24

Sim ulations of a compressible starting jet
same size than the K olmogoro v length. The Cartesian geometry of length
25 D × 15 D × 15 D w as c hosen to b e discretised with 2048 × 1024 × 1024
grid-p oin ts for the cases 7 .b (with R e D = 10 000 ) and 7 .c (with Re D = 5000
highly resolv ed). The resolution 512 × 256 × 256 w as c hosen for the rest.
While the grid is equidistan t in the axial direction, b oth transv erse direc-
tions ha v e b een discretised using a grid stretc hing, as prop osed b y Anderson
et al. [1984] and sho wn in figure 2.1:
y ( η ) = y 0 + η c  sinh( τ y ( η − B ))
sinh( τ y B ) + 1  L y , (2.8)
where L y represen ts the length of the domain in y direction (the same
applies for z direction) and y 0 is the offset to the centre of the domain
in the corresp onding direction, to w ork with η c and τ y in non-dimensional
units from 0 to 1 . η c is the cen tre of the refinemen t in non-dimensional
units (w e used the v alue η c = 0 . 5 ) and τ y is the amount of refinemen t (we
used the v alue τ y = 4 ), so that the ratio biggest to smallest elemen t are in
the ratio τ y . The parameter B is defined as:
B = 1
2 τ y
ln  1+( e τ y − 1) η c
1+( e − τ y − 1) η c  . (2.9)
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1

n / N
y / L y

0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2

y / L y
∆ y / d h

( a ) ( b )
Figure 2.1: Grid stretc hing used in a typical sim ulation of this thesis. ( a ) Grid
distribution presen ting the ph ysical space v ersus the computational one. ( b ) Grid
cell spacing normalised to the equiv alen t equidistan t cell spacing ( dh = L y / N ).
V alues b elow one sho w the refined area.
A sc hematic represen tation of the finest grid is sho wn in figure 2.2.
The resolution c hosen for all sim ulations is sufficien t to resolv e the K ol-
mogoro v scale, see figure 2.3. This is supp orted and extended in tables 2.1
and 2.2 and figure 2.4. The turbulent spectra of figures 2.3 and 2.4 were
calculated using the v elo cit y fluctuations with resp ect to the azim uthal a v-
erage: u 0 = u − h u i θ , where u is the instan taneous v elo cit y and h u i θ is the
25

Chapter 2. Numerical bac kground
Figure 2.2: Computational grid for the finest simulation (case 7 .c ). Only every
50 th p oin t is sho wn.
T able 2.1: Grid spacing considerations for the three-dimensional cases in this
study . The c haracteristic length of the smallest elemen t in the domain is defined
as d grid = ( dx min + dy min + dz min ) / 3 . η min is the minim um K olmogoro v length
scale found in the flo w field. ν ∞ = ν ( T ∞ ) is the kinematic viscosity at the
temp erature of the un b ounded c ham b er in m 2 /s . Here λ is, exceptionally , the
free mean path of the fluid.
Case Re D n 1 × n 2 × n 3 D
d grid
η min
d grid
10 λ
d grid
1 − 7 .a 5 000 512 × 256 × 256 25 . 84 0 . 91 0 . 11
7 .b. 10 000 2048 × 1024 × 1024 103 . 09 1 . 93 0 . 23
7 .c. 5 000 2048 × 1024 × 1024 103 . 09 2 . 99 0 . 23
7 .d. 15 000 2048 × 1024 × 1024 103 . 09 0 . 87 0 . 23
7 .e. 5 000 512 × 256 × 256 25 . 84 0 . 91 0 . 11
8 − 10 5 000 640 × 256 × 256 25 . 84 0 . 91 0 . 11
11 − 13 3 000 1024 × 512 × 512 51 . 68 1 . 365 0 . 017
mean v elo cit y a v eraged in the azim uthal direction. The minim um v alue
of the K olmogoro v length scale η min compared to the c haracteristic length
26

Sim ulations of a compressible starting jet
10 −2 10 −1 10 0
10 −2
10 −1
10 0
10 1
10 2

( κ η ) − 5 / 3
k η
E / ( η u 2
η )

Figure 2.3: Non-dimensional turbulent kinetic energy spectra for the cases 7 .b
(solid red) with a Re D = 10 000 and 7 .c (dashed red) with a Re D = 5 000 . The
grey dot-dashed line represen ts the w a v enum b er asso ciated to the Kolmogoro v
length scale. As a reference, the − 5 / 3 deca y line has b een plotted in grey .
of the smallest elemen t in the domain ( d grid = ( dx min + dy min + dz min ) / 3 )
is v ery close to one in the w orst case ( 7 .d ). The K olmogoro v length scale
(calculated as η =  ν 3 /ε  1 / 4 , where ε = ν h ∂ u 0
i
∂ x j
∂ u 0
i
∂ x j i ) is time dep enden t
and decreases during the starting phase, while it increases during the de-
ca y phase. Comparing b oth sp ectra corresp onding to the large resolution
( 2048 × 1024 × 1024 ), see figure 2.3, we find a go o d agreemen t in the dis-
sipation range as w ell as in the inertial sub-range. The sp ectra of case 7 .b
corresp onding to R e D = 10 000 has a longer inertial sub-range and there is
more energy asso ciated with big scales, as exp ected. Only a few p oin ts are
in the energy con taining scales and since the data are not time a v eraged,
but for a single realisation, the agreemen t app ears to b e satisfactory . All
sp ectra presen t a small p eak for high w a v e-n um b ers, whic h corresp onds to
the error asso ciated to the in terp olation to an equidistan t grid to p erform
the F ourier transformation. In the energy sp ectrum normalised with the
nozzle diameter D (figure 2.4), the maxim um turbulen t kinetic energy can
b e seen for all three cases at scales of the same order of magnitude as the
nozzle diameter represen ted b y the grey dot-dashed line, where the simu-
lation with the largest Reynolds n um b er has a longer inertial sub-range,
27

Chapter 2. Numerical bac kground
10 0 10 1
10 −4
10 −3
10 −2
10 −1
10 0

( κ η ) − 5 / 3
k D
E / ( K D )

Figure 2.4: Non-dimensional tu rbulen t kinetic energy sp ectra for the cases 7 .a
(blac k) with a Re D = 5 000 , 7 .b (solid red) with a Re D = 10 000 and 7 .c (dashed
red) with a Re D = 5 000 . The grey dot-dashed line represen ts the w a v en um b er
asso ciated to the nozzle diameter. As a reference, the − 5 / 3 deca y line has b een
plotted in grey .
follo w ed b y the dissipation range.
The width of an in viscid sho c k w a v e is appro ximately one order of mag-
nitude larger than the mean free path, Salas and Iollo [1996]. Using non-
dimensional analysis, the Kn udsen n um b er can b e expressed in the form
K n = λ/D = M j p γ π / 2 /R e and since the c haracteristic length in this
study (the nozzle diameter, D ) is c hosen to b e unit y , the mean free path
( λ ) –and therefore the thic kness of the sho c k w a v es ( 10 λ )– is m uc h smaller
than the grid elemen ts ( 10 λ/d grid in tables 2.1 and 2.2). W e used the sho c k
capturing filter from Bogey et al. [2009].
The cases with a Re D = 5000 ha v e a ν ∞ = 0 . 04 m 2 /s , while the cases
with Re D = 10 000 ha v e a ν ∞ = 0 . 02 m 2 /s .
28

Sim ulations of a compressible starting jet
T able 2.2: Grid spacing considerations for the t wo-dimensional cases in this
study . The c haracteristic length of the smallest elemen t in the domain is defined
as d grid = ( dx min + dy min ) / 2 . η min is the minim um K olmogoro v length scale.
ν ∞ = ν ( T ∞ ) is the kinematic viscosit y at the temp erature of the un b ounded
c ham b er in m 2 /s . Here is λ exceptionally the free mean path of the fluid. An
estimation of the sho c k w a v e thickness (as proposed by Salas and Iollo [1996])
compared to the smallest elemen t in the domain is denoted b y 10 λ/d grid .
Case Re D n 1 × n 2 D /d grid η min /d grid 10 λ/d grid
14 8 000 3072 × 1536 409 . 47 5 . 95 0 . 49
15 5 000 1536 × 768 204 . 67 4 . 67 0 . 06
16 5 000 3072 × 1536 409 . 47 10 . 5 1 . 47
17 10 000 3072 × 1536 409 . 47 5 . 17 0 . 80
18 5 000 1536 × 768 204 . 67 1 . 73 0 . 80
19 5 000 3072 × 1536 409 . 47 3 . 97 1 . 85
20 5 000 3072 × 1536 409 . 47 2 . 70 1 . 85
21 5 000 3072 × 1536 409 . 47 2 . 34 2 . 93
22 5 000 3072 × 1536 409 . 47 2 . 49 3 . 89
23 2 000 4608 × 2304 307 . 13 6 . 82 1 . 42
24 5 000 4608 × 2304 307 . 13 6 . 28 0 . 568
25 10 000 4608 × 2304 307 . 13 6 . 15 0 . 284
2.3 P arallelization
Considering a grid lik e the one in tro duced in the previous section, with
more than t w o billion ( 2 × 10 9 ) grid p oints, running appro ximately 10 5 time
steps to analyse statistically the flo w, the need of millions of CPU hours
is ob vious. A massiv e parallelization is necessary for the feasibilit y of this
researc h pro ject.
With this approac h, ev ery CPU solv es the Na vier-Stok es equations in a
part of the domain. The cores need to comm unicate with eac h other since
their solution dep ends on the neigh b ours one. When computing deriv a-
tiv es, the finite difference stencil ne eds information ab out the solution in
the neigh b our p oin ts. In this study , the comm unication is based on a h y-
brid approac h using the Message P assing In terface (MPI) and Op enMP
metho ds. While Op enMP manages the comm unication of the cores within
a pro cessor to a v oid b ottlenec ks, the MPI manages the comm unication
in ter-pro cessor.
All sim ulations w ere p erformed in the Leibniz Rec hnenzen trum (LRZ),
equipp ed with Xe on E5-2680 8C processors at 2 . 70 GHz and it is curren tly
in p osition 36 of the most p o w erful sup ercomputing cen tres in the world 1 .
1 A ccording to T op500 List in No v em b er 2016. h ttp://www.top500.org.
29

Chapter 2. Numerical bac kground
16 64 256 1024 4096
16
64
256
1024
4096

N um b er o f pr o c e sso rs
Sp e e dup

16 64 256 1024 4096
0.5
0.6
0.7
0.8
0.9
1

N um b e r o f pro c e sso rs
E ffi c i e nc y

( a ) ( b )
Figure 2.5: ( a ) Scaling plot k eeping the grid constan t and increasing the par-
alellisation (strong scaling). A small case ( 128 × 256 × 32 ) normalised to 32 cores
is represen ted b y ( • ). A large case ( 1024 × 512 × 512 ) normalised to 512 cores
corresp onds to ( N ). The ideal speed-up is represented b y the dashed line. ( b )
Scaling plot k eeping the grid p er core constan t and increasing the n umber of
blo c ks (w eak scaling). A constan t blo c k of ( 16 × 16 × 16 ) grid p oin ts p er core w as
analysed.
T o the b est kno wledge of the author, the biggest simulations in this study
are the largest sim ulations of a jet un til no w.
With a lo w loading of the cores as ( • ) in figure 2.5a, the comm unica-
tion time b ecomes dominan t and a larger n um b er of cores do es not imply a
sp eedup. The last p oin t of this curv e corresp onds to a paralellisation with
6 grid p oin ts p er core in ev ery direction, and the communication time dom-
inates the computing time. F or a high loading of the cores, corresp onding
to ( N ) in the same figure, starting with the first p oin t, it corresp onds to
80 grid p oin ts p er core and direction, and the computing time dominates
the comm unication time. Therefore, adding more cores implies an ideal
sp eedup, as long as this condition holds.
W e found an optimal load of the cores at 29 grid p oin ts p er core and
direction and b ecause of this, w e used t ypically in this study , for the lo w-
resolution ones w e used 1024 cores (whic h means 32 grid p oin ts p er core and
direction) and for the high-resolution sim ulations 8192 cores (whic h means
64 grid p oin ts p er core and direction). F or the high-resolution sim ulations,
w e used 64 grid p oin ts p er core and direction to fit the sim ulations within
a single ’island’ in the sup ercomputing cen tre, a unit of the sup ercomput-
ing cen tre where the CPUs are connected via a high-sp eed non-blo c king
net w ork (Infiniband).
2.4 Initial conditions
Concerning the initial conditions, quiescen t flo w is imp osed o v er the
whole computational domain in order to study an impulsiv ely starting jet,
30

Sim ulations of a compressible starting jet
setting the pressure to the am bien t pressure and the en trop y to the am bien t
en trop y .
2.5 Boundary conditions
The truncation of the computational domain includes additional un-
w an ted information in the solution, and to minimise the effect of this addi-
tional un w an ted information, w e imp ose a kno wn solution at the b oundaries
(b oundary conditions), so that the solution inside the computational do-
main reac hes the kno wn solution at the b oundary of the computational
domain and its surroundings.
In the four lateral faces of the computational domain, non-reflecting
b oundary conditions are set in order to b e able to study the acoustic phe-
nomena that are presen t in the starting jet.
A t the outlet, a sp onge region is set in order to a v oid the reflection of
the w a v es that are tra v elling to w ards the end of the computational domain.
Since the flo w inside the nozzle is not computed in most of the cases,
the v ariation of the inlet condition with the non-dimensional time ( t ∗ =
t/ ( D j /U j ) , where D j and U j are the fully expanded diameter and v elo cit y ,
resp ectiv ely) is mo delled as
p e ( t ∗ )
p ∞     y,z =0
= 1 + p 0 r /p ∞
NPR exp  − t ∗
C  tanh( K t ∗ ) , (2.10)
where NPR is the nozzle pressure ratio, defined as the ratio of reserv oir to-
tal pressure to nozzle exit pressure p 0 r /p e . The one-dimensional isen tropic
theory predicts an exp onen tial deca y of the pressure in the dep osit when
disc harging. W e link ed the initial state of quiescence with the exp onen tial
deca y through a h yp erb olic tangen t. This is similar to the exp erimen tal
w ork of Cimarelli et al. [2014], where the pressure time distribution mea-
sured at the nozzle exit matc hes qualitativ ely with the mo del used in this
study . If the nozzle is not c hok ed ( M e < 1 ), p e = p ∞ b ecause the nozzle
is alw a ys adapted in the subsonic regime. Ho w ev er, if a con v ergen t noz-
zle is c hok ed ( M e = 1 ), NPR ∗ = (( γ + 1) / 2) ( γ / ( γ − 1)) [Bec k er, 1968], and
p e 6 = p ∞ , b ecause the flow in a con v ergen t nozzle in supersonic regime is
nev er adapted. Equation (2.10) is plotted in figure 2.6 a . The constant K
mo difies the h yp erb olic tangen t term, whic h is resp onsible for the impulsiv e
stage, corresp onding to the stage ( i ). The larger K , the more impulsive is
the first stage. W e used K = 60 as a trade-off b et w een the stabilit y of the
co de and high enough gradien ts in the first stage to giv e the jet the im-
pulsiv e c haracter. The same w a y , the constant C mo difies the exp onen tial
term, which is responsible for the decay , corresp onding to the stage ( ii ).
The larger C , the faster is the decay , whic h also implies a smaller v olume
31

Chapter 2. Numerical bac kground
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1

t ∗
( p e /p ∞ − 1 ) / ( p r /p ∞
N P R − 1 )

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1

( D/ 2 − y ) / D
( p e /p ∞
− 1 ) / ( p r /p ∞
N P R
− 1 )

( a ) Pressure temp oral distribution at
the nozzle exit cen terline. Lines as in
table 2.3.
( b ) Pressure distribution at the nozzle
exit plane for the case 7 .a for t ∗ = 50 .
Figure 2.6: Nozzle exit conditions for the starting jet.
of the reserv oir.
The spatial pressure distribution imp osed at the inlet ( p e ( y , z ) /p ∞ ),
sho wn in figure 2.6 b , w as mo delled by ,
p e ( y , z )
p ∞     t 0
= 1 +  p 0 r /p ∞
NPR − 1 
( 1
2 − 1
2 " tanh p y 2 + z 2 − 1
2 ( D − δ ω 0 )
l mass !#) ,
(2.11)
with a shear la y er thic kness of δ ω 0 = 0 . 1 D . The most common spatial
distribution of the inlet condition when n umerically sim ulating a jet is a
h yp erb olic tangen t, due to the go o d control o v er the gradien ts of the dis-
tribution to generate unstable Kelvin-Helmholtz v ortices. W e used this
distribution as w ell, adding the unsteady part to tak e in to accoun t the
time ev olution of the inlet condition. This, together with a relativ ely large
Reynolds n um b er led to a fully turbulen t shear la y er shortly after the noz-
zle exit. The parameter l mass = δ ω 0 / (2 π ) con trols the p osition of the shear
la y er in the nozzle, imp osing the b eginning of the shear lay er at the w all
and not cen tred in to the w all M ( r /D = 0 . 5) = 0 .
Figure 2.7 sho ws the discretisation of the inlet condition at the nozzle
exit for sim ulations with differen t resolutions. The colour scale represen ts
the relativ e v elo cit y to the corresp onding in the cen terline, giving an idea
ab out the t w o-dimensional profile. By using a resolution of 512 × 256 × 256 ,
w e discretised the nozzle exit diameter with 25 grid p oin ts, while using a
resolution of 1024 × 512 × 512 , the nozzle diameter was discretised with
51 grid p oin ts. In the largest resolution, 2048 × 1024 × 1024 , the nozzle
32

Sim ulations of a compressible starting jet
T able 2.3: Short summary of the cases 1 .a, 2 − 6 , 7 .a − e inv olv ed in the
trailing jet analysis to sho w the parameters used for the b oundary condition.
p 0 r /p ∞ = 3 . 60 . The sup ersonic non-dimensional time interv al is represen ted with
t ∗
M j > 1 .
Case C L/D max( M j ) t ∗
M j > 1 legend
1 .a 1 0 . 45 0 . 40 -
2 2 1 . 17 0 . 55 -
3 5 3 . 80 0 . 81 -
4 10 8 . 60 1 . 01 4 . 66 − 6 . 92
5 15 13 . 55 1 . 11 3 . 75 − 14 . 16
6 30 28 . 59 1 . 25 3 . 60 − 32 . 75
7 .a − e 10 7 ∞ 1 . 49 3 . 83 −
( a ) 512 × 256 × 256 ( b ) 1024 × 512 × 512 ( c ) 2048 × 1024 × 1024
Figure 2.7: Discretisation of the inlet condition at the nozzle exit for different
resolutions. The colour-scale indicates the relativ e v elo cit y to the corresp onding
in the cen terline of the jet ( u/u c ).
diameter w as discretised with 103 grid p oints.
2.6 P orous m edia / v olume p enalisation metho d
In order to create a prop er b oundary condition, p orous media ha v e b een
commonly used in the literature using the v olume p enalisation metho d. In
this section, we briefly describe the metho d used. P orous media w as used in
this thesis for the generation of a nozzle geometry in case 8 and the analysis
of the straigh t and div ergen t nozzles in cases 11 .a − 13 .h .
T w o parameters are needed to c haracterise a p orous medium: the p oros-
it y φ and the p ermeabilit y K . While φ describ es the v olume ratio of v oid
space V f to the v olume of the whole p orous material ( φ = V f /V ), K stands
for the p ermeabilit y of the material.
A p orosit y equal to one represen ts a v oid space and a p orosit y of zero
a solid b o dy . The p ermeabilit y of the material K and is a symmetric and
p ositiv e definite tensor and the en tries in this tensor can reac h v alues of zero
33

Chapter 2. Numerical bac kground
for a material whic h is not p ermeable (solid) and infinit y for a material with
no influence on the fluid.
T o include the effect of a p orous medium in the simulation, the equations
of con tin uit y , momen tum and energy ha v e to b e mo dified. The main idea is
based on a relation rep orted b y Darcy [1856] that relates the flo w v elo cities
and the pressure gradien t with the p ermeabilit y of the p orous medium in
a linear w a y . This relation, called Darcy’s la w, has b een used in a similar
sim ulation in Sc h ulze [2011] and reads:
v = − K
µ ∇ p,
with the so called Darcy v elo cit y v = φ u . This additional volume force
is added to the momen tum equation and acts as a sink term damping all
v elo cities in the p orous medium. Moreo v er, the c hanges in densit y ha v e to
b e implemen ted as w ell:
φ = V f
V = ρ
ρ f
.
The mo dified con tin uit y equation can b e expressed as:
∂ φρ
∂ t + ∂ φρu i
∂ x i
= 0 , (2.12)
where the densit y of the fluid ρ f = φρ has b een introduced.
Based on the same argumen tation line, the mo dified momen tum equa-
tion is:
∂ u i
∂ t + u j
∂ u i
∂ x j
= − 1
φρ
∂ p
∂ x j
+ 1
φρ
∂ τ ij
∂ x j − µ
ρ  K − 1  ij u j , (2.13)
and the mo dified energy equation is:
C v
D T
D t + D
D t  1
2 u i u i  = 1
φρ
∂
∂ x j  − pu j + u i τ ij +  λ ∂ T
∂ x j  . (2.14)
The n umerical implemen tation of the equations and the v alidation of the
metho d are out of the scop e of this thesis and can b e follo w ed in [Sch ulze,
2011].
2.7 Sp onge region
When a v ortex pair reac hes the outflo w b oundary of the computational
domain it generates ’reflected w a v es’ when they are truncated b y the b ound-
ary . Therefore w e n umerically treat the la y er close to the outflo w b oundary
b y adding a damping term σ ( x ) :
∂ q
∂ t = NS( q ) − τ sp σ ( x )( q − q 0 ) , (2.15)
34

Sim ulations of a compressible starting jet
where q represen ts the flo w state, NS ( q ) is the Na vier-Stok es op erator, σ ( x )
is the damping function, q 0 represen ts the reference flo w state and τ sp is
the amplitude or strength of the sp onge la y er. This damping term mak es
the sp onge region not ph ysical but allo ws us to k eep ph ysical results in the
rest of the domain. The choice of σ ( x ) , q 0 and τ sp has to b e done carefully
to a v oid n umerical instabilities when damping to o m uc h and acoustic re-
flections when damping to o less. In the compressible starting jet, w e c hose
the reference flo w state q 0 as quiescence.
The damping function σ ( x ) is required to b e smo oth to dissipate the
energy slo wly without reflect an y information bac kw ard to the rest of the
domain. The sp onge function implemen ted in this study can b e expressed
as:
σ ( x ) = 1
2  1 + erf  2
L X p
( x − C X p )  , (2.16)
where L X p = 5 D is the length of the sp onge region in x direction starting
from C X p = 25 D to ha v e enough p oin ts to damp the solution close to the
outlet. This function is sho wn in figure 2.8.
0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5

x/ D
σ ( x )

0 5 10 15 20 25
10 −10
10 −8
10 −6
10 −4
10 −2
10 0

x /D
σ ( x )

( a ) ( b )
Figure 2.8: Sp onge damping function σ ( x ) . ( a ) Linear represen tation. ( b )
Logarithmic represen tation. The b eginning of the sp onge la y er ( • ) w as set to
x/D = 20 .
2.8 Sim ulations se tup
T able 2.4 sho ws a general o v erview of the differen t set-ups considered in
this study . The vortex ring dynamics w ere analysed with the simulations
1 .a − c . Sim ulations 2 − 6 sim ulated starting and deca ying jets with differen t
reserv oir sizes in order to analyse ho w this affects the trailing jet. The
trailing jet for the starting jet without deca y w as analysed b y the cases
7 .a − e . Until this point, w e used the basic configuration of a starting
35

Chapter 2. Numerical bac kground
T able 2.4: General set-up of the sim ulations in this study . Sim ulations from 1
to 13 are 3 D and from 14 to 26 are 2 D .
Case starting deca ying p orous inflo w Analysis
jet jet media condition
1 .a − c X X — laminar V ortex Ring
2 − 6 X X — laminar T railing jet
7 .a − e X — — laminar T railing jet
8 — — straigh t nozzle Inflo w
nozzle conditions
Inflo w
9 — — — TBL conditions
10 — — — laminar Inflo w
conditions
straigh t
11 .a − h X X nozzle nozzle p 0 r /p ∞ , L/D
12 .a − h X X div ergen t nozzle p 0 r /p ∞ , L/D
nozzle
div ergen t
13 .a − h X X nozzle nozzle p 0 r /p ∞ , L/D
14 − 26 X X — laminar Main
parameters
jet as describ ed in the in tro duction with differen t v alues for the go v erning
parameters. Cases 8 − 10 fo cused on the effects of the inflo w condition
b y c hanging the spatial distribution of the inlet condition. The analysis
of the effect of the nozzle geometry on the fluid flo w and acoustics w as
done based on cases 11 .a − 13 .h ; here differen t geometries w ere analysed.
All cases en umerated un til here w ere sim ulated in three dimensions. Cases
14 − 26 w ere sim ulated in t w o dimensions to scan differen t regions of the
parameter space with sp ecial imp ortance.
Cases 1 .a − c fo cused on the v ortex ring dynamics and for this purp ose
w e sim ulated lo w incompressible, high compressible and sup ersonic starting
and deca ying jets with v ery short reserv oirs, generating a single v ortex ring
without trailing jet. The v alues used for the parameters can b e seen in table
2.5. W e sho w in section 4.2.3 ho w the compressibilit y affects the v ortex ring
dynamics.
Cases 2 − 6 together with 1 .a and 7 .a , which are the natural con tin uation
of the parameter v alues, fo cused on the trailing jet formation and dynamics
for differen t reserv oir sizes. The parameters used are summarised in table
2.6. Case 7 .a was in tro duced in this analysis to compare as w ell with the
36

Sim ulations of a compressible starting jet
T able 2.5: Go v erning parameters of the analysis of the v ortex ring dynamics.
The temp erature ratio T 0 r /T ∞ = 1 for all cases. All cases are generated with
a con v ergen t nozzle: A e / A ∗ = 1 . The Reynolds num ber based on the nozzle
diameter w as set to R e D = 5 000 for these cases.
Case L/D p 0 r /p ∞
1 .a 0 . 45 3 . 6
1 .b 0 . 0436 20 . 63
1 .c 0 . 091 96 . 36
T able 2.6: Gov erning parameters in the analysis of the existence of the trailing
jet. The temp erature ratio T 0 r /T ∞ = 1 for these cases. These cases are generated
with a con v ergen t nozzle: A e / A ∗ = 1 . The Reynolds num ber is set to b e Re D =
5 000 for these cases.
Case L/D Re D p 0 r /p ∞
1 .a 0 . 45 5 000 3 . 6
2 1 . 17 5 000 3 . 6
3 3 . 80 5 000 3 . 6
4 8 . 6 5 000 3 . 6
5 13 . 55 5 000 3 . 6
6 28 . 59 5 000 3 . 6
7 .a ∞ 5 000 3 . 6
starting jet without deca y .
The trailing jet w as analysed for the starting jet without deca y for the
cases 7 .a − e with the parameters sho wn in table 2.7. The starting jet
without deca y (from an infinite reserv oir) allo ws to study the prop erties of
the trailing jet assuming a statistically steady system.
T able 2.7: Go v erning parameters analysis trailing jet for an infinite reservoir.
T 0 r /T ∞ = 1 , A e / A ∗ = 1 , L/D → ∞ .
Case Re D p 0 r /p ∞ Resolution
7 .a 5 000 3 . 6 Lo w
7 .b 5 000 3 . 6 High
7 .c 10 000 3 . 6 High
7 .d 15 000 3 . 6 High
7 .e 5 000 4 . 35 Low
In order to analyse the effect of the inflo w conditions in the jet flo w
and its acoustics, we sim ulated three differen t cases 8 , 9 and 10 with the
parameters summarised in table 2.8. Here w e did not tak e in to account the
37

Chapter 2. Numerical bac kground
effect of the geometry , but the effect of the turbulence lev el at the inlet.
T able 2.8: Go v erning parameters analysis inflow conditions. The temp erature
ratio T 0 r /T ∞ = 1 for all cases. A e / A ∗ = 1 for all cases. L/D → ∞ , R e D = 5000 .
Case p 0 r /p ∞ inlet
8 3 . 786 nozzle
9 4 . 35 TBL
10 3 . 956 laminar
W e to ok in to accoun t the effects of the geometry in cases 11 .a − h ,
12 .a − h and 13 .a − h b y sim ulating differen t nozzle geometries, c haracterised
b y A e / A ∗ . T able 2.9 sho ws the parameters used here.
T able 2.9: Gov erning parameters in the analysis of the geometry , non-
dimensional mass supply and pressure ratio. The temp erature ratio T 0 r /T ∞ = 1
for all cases. R e D = 3 000 .
Case L/D p 0 r /p ∞ A e / A ∗
11 .a 2 3 1
11 .b 8 3 1
11 .c 2 4 1
11 .d 8 4 1
11 .e 2 50 1
11 .f 8 50 1
11 .g 2 80 1
11 .h 8 80 1
12 .a 2 3 2 . 358
12 .b 8 3 2 . 358
12 .c 2 4 2 . 358
12 .d 8 4 2 . 358
12 .e 2 50 2 . 358
12 .f 8 50 2 . 358
12 .g 2 80 2 . 358
12 .h 8 80 2 . 358
13 .a 2 3 4
13 .b 8 3 4
13 .c 2 4 4
13 .d 8 4 4
13 .e 2 50 4
13 .f 8 50 4
13 .g 2 80 4
13 .h 8 80 4
38

Sim ulations of a compressible starting jet
In order to scan imp ortan t regions of the parameter space, w e p erformed
as w ell t w o-dimensional sim ulations of the cases 14 − 22 , whose parameters
are presen ted in table 2.10.
T able 2.10: Gov erning parameters in the analysis of the go v erning parameters.
The temp erature ratio T 0 r /T ∞ = 1 for all cases. All cases are generated with a
con v ergen t nozzle: A e / A ∗ = 1 .
Case L/D Re D p 0 r /p ∞
14 20 8 000 1 . 3251
15 ∞ 5 000 1 . 007
16 100 5 000 2 . 46
17 100 10 000 2 . 84
18 ∞ 5 000 2 . 84
19 2 5 000 3 . 786
20 8 5 000 3 . 786
21 2 5 000 15
22 0 . 1345 5 000 50
An imp ortan t part of the parameter space for n umerical sim ulations
is the Reynolds n um b er, so w e segregated this cases from the rest of the
t w o-dimensional sim ulations for a Reynolds n um b er analysis. T able 2.11
indicates the v alues used for the go v erning parameters in these sim ulations.
T able 2.11: Go v erning parameters in the analysis of the Reynolds n um b er.
T 0 r /T ∞ = 1 , A e / A ∗ = 1 . p 0 r /p ∞ = 2 . 46 , L/D → ∞ .
Case Re D
23 2 000
24 5 000
25 10 000
39

Chapter 2. Numerical bac kground
40

P art I I
Exp erimen ts of a
compressible starting jet
41

Chapter 3
Exp erimen tal setup
T w o differen t families of exp erimen ts ha v e b een p erformed in this work:
( i ) natural exp erimen ts in whic h w e measured directly on a v olcano and
( ii ) analogic exp erimen ts in the lab oratory repro ducing partially the same
ph ysical pro cess of the natural case but with simplified b oundary condi-
tions in order to study the effect of a sp ecific parameter. In the course of
gas dynamics offered at the departmen t prof. Sesterhenn, together with
the tec hnical staff (Christian W estphal) and the researc h assistan t (m y-
self ) do once a y ear an exp erimen t of the jet flo w for academic purp oses;
some of the results obtained there are also presen t in this w ork. F or the
sak e of simplicit y , we will start describing the analogic experiments in the
lab oratory .
W e p erformed the measuremen ts in the anec hoic c ham b er facilit y of the
Berlin Institute of T ec hnology (Departmen t of T echnical A coustics). Figure
3.1 sho ws the general set-up of the anec hoic c ham b er. It has a free volume
of 1070 m 3 and a w alk able area of 126 m 2 . The lo w er frequency limit is 63
Hz.
In the anec hoic c ham b er facilit y , we performed acoustic measurements
of con tin uous and starting jets as w ell as sc hlieren photograph y . They are
describ ed in more detail in the follo wing sections.
43

Chapter 3. Exp erimen tal setup
Figure 3.1: General set-up in the anechoic c ham b er.
44

Exp erimen ts of a compressible starting jet
3.1 Sc hlieren photograph y
Sc hlieren photograph y is based on the c hanges of the refractiv e index of
fluids when c hanging their thermo dynamical state. W e use the dep endence
of the refractiv e index with the densit y (see the Snell’s la w in equation
3.1) b y illuminating the ob ject under study with p erfect parallel ligh t; this
will b e refracted in a differen t w a y b y the regions with differen t densities.
F o cusing this ligh t through a lens in to a camera w e can record the densit y
gradien ts of the fluid flo w.
sin θ 1
sin θ 2
= n 2
n 1
(3.1)
Figure 3.2: Sketc h of a classical Z-t yp e sc hlieren set-up.
Figure 3.2 sho ws a sk etc h of a classical Z-t yp e sc hlieren set-up that w e
used for our sc hlieren exp erimen ts (figure 3.3 a sho ws the set-up used). In
order to obtain p erfect parallel ligh t w e ha v e a p oin t ligh t source (see figure
3.4 a ) lo cated exactly at the fo cal p oin t of a parab olic mirror; when the
div erging ligh t ra ys from the source are reflected in the parab olic mirror,
they are reflected as p erfect parallel ligh t ra ys. These parallel ligh t ra ys
illuminate the starting jet. Here several things happen: some of the ligh t
ra ys are completely blo c k ed b ecause they illuminate the b o dy of the noz-
zle; some of the ra ys are undistorted b ecause they do not pass through the
starting jet, but b eside it; and some of the rays pass through the starting jet
and are refracted with differen t angles due to the spatial distribution of the
refractiv e indices in the flo w field. In order to record the information, w e
collect the ligh t ra ys through a second parab olic mirror fo cusing the ligh t
in to a camera (see figures 3.3 b and 3.4 b ). This second parab olic mirror will
fo cus all p erfect parallel ligh t ra ys in a single p oin t, but the deflected ra ys
45

Chapter 3. Exp erimen tal setup
will b e fo cused in a differen t p oin t out of the camera sensor. This generates
a picture at the camera where shado ws are sho wn for the regions where the
ligh t w as refracted. T ypically this has b een called shadowgraph photogra-
ph y . In order to blo c k the ligh t ra ys not fo cused b y the parab olic mirror in
the fo cal p oin t, w e placed a razor blade at the fo cal p oin t. This razor blade
mak es the difference b et w een shado wgraph and sc hlieren photograph y .
( a ) ( b )
Figure 3.3: ( a ) General set-up for Sc hlieren photograph y . On the righ t side
at the fron t, w e can see the light source emitting ligh t to the bac k, where the
first parab olic mirror is lo cated. In the first parab olic mirror, the ligh t rays are
parallelised and sen t through the jet (in the cen tre of the image) un til the second
parab olic mirror (on the fron t left side of the image). There, the light is focused
in to the camera (in the rear righ t side of the image) and just b efore arriving at
the camera a part of the ligh t is blo c k ed with a razor blade. ( b ) Razor blade at
the fo cal p oin t of the second parab olic mirror and the camera. Detail view.
T w o differen t cameras w ere used in this w ork: a digital single-lens reflex
(DSLR) Nik on D7100 with a video frame rate of 30 frames p er second and a
minim um exp osure time of 125 µs and an ultrahigh-sp eed camera Phantom
v2512, with a video frame rate of 10 6 frames p er second and a minim um
exp osure time of 265 ns . While the DSLR w as used to w ork with long
exp osures and record the mean v alue of the flo w field, the ultrahigh-sp eed
camera w as used to trac k the sup ersonic structures in the starting jet.
T ogether with the cameras, w e used mainly t w o differen t lenses: ( i ) a
Nik on zo om lense 18-105mm f/3.5-5.6 VR and ( ii ) a tele-zo om Nik on AF-S
200-500mm f/5.6E ED VR. W e used the zo om lense to visualise a region of
ab out 5 − 7 jet diameters, while the tele-zo om w as used to fo cus in a v ery
small region of ab out 1 − 2 jet diameters. Figure 3.5 sho ws an example of
b oth kinds of sc hlieren pictures.
46

Exp erimen ts of a compressible starting jet
( a ) ( b )
Figure 3.4: ( a ) T wo razor blades at the ligh t source to generate a slit source.
( b ) Razor blade at the fron t of the camera lense.
( a ) ( b )
Figure 3.5: Schlieren examples. ( a ) Long exp osure sc hlieren photograph y . The
com bination of DSLR with the zo om lense w ere used. ( b ) High-sp eed sc hlieren
photograph y . The high sp eed camera with the tele-zo om w ere used.
47

Chapter 3. Exp erimen tal setup
3.2 A coustic m easuremen ts in the lab oratory
W e p erformed acoustic measuremen ts as w ell in the anec hoic c ham b er of
the Berlin Institute of T ec hnology . With up to 16 microphones in different
configurations, we measured the jet noise in the near and the far field. Both
subsonic and sup ersonic jets w ere tak en in to consideration.
Figure 3.6 sho ws t w o differen t microphone arrangemen ts in the jet near
field.
( a ) ( b )
Figure 3.6: ( a ) Disp osition of the microphones to measure the acoustics of the
jet and its dep endence with the jet angle. ( b ) Differen t set-up of the microphones
fo cusing on the sup ersonic comp onen ts of the jet noise.
The half-inc h microphones used are man ufactured b y PCB with a fre-
quency range of 3 . 75 − 20 000 Hz, the sp ecifications of these microphones
are included in section E.1. W e used also an infrasound microphone man-
ufactured b y G.R.A.S. with a frequency range of 0 . 5 − 20 000 Hz, the
sp ecifications of these microphones are included in section E.1. They w ere
con trolled b y a data acquisition system man ufactured b y Oros.
The air ejected from the nozzle w as stored at 8 bars in a 3 m 3 dep osit
outside the anec hoic c ham b er. A pressure regulator led us to p erform ex-
p erimen ts with in termediate pressures. The nozzle, made of aluminium,
has an inner exit diameter of 10 mm . This c hoice w as not optimal since
the asso ciated frequency of the sup ersonic jet noise comp onen ts w ere par-
tially outside the frequency range of the microphones, but for some set-ups,
w e w ere able to measure also the sup ersonic comp onen ts.
48

Exp erimen ts of a compressible starting jet
3.3 A coustic measuremen ts at real v olcano es
In order to study as w ell a natural case, w e did sev eral exp editions
to activ e v olcano es to analyse the noise radiated during an eruption as
a natural example of a compressible starting jet. I w as in v olv ed in t w o
campaigns: Moun t Etna in July 2014 and Stromboli in May 2016 .
3.3.1 Moun t Etna
A collab oration with the Europ ean pro ject ’Mediterranean Sup ersite
V olcano es’ (MED-SUV) ga v e us the p ossibility to attend a campaign in
July 2014 . There, prof. Sesterhenn and myself measured the acoustics of
an eruptiv e fissure at the North-East Crater for sev eral hours during three
consecutiv e da ys. In total, w e measured more than 11 000 even ts. Figure
3.7 sho ws the t w o craters of the eruptiv e fissure of the North-East Crater
during the 15 − 16 / 07 / 2014 exp edition and figure 3.8 sho ws the deplo ymen t
of the microphones on that exp edition.
Figure 3.7: Craters of the eruptiv e fissure of the North-East Crater of moun t
Etna during the 15 − 16 / 07 / 2014 exp edition.
During this campaign, w e deplo y ed three microphones at different cir-
cumferen tial angles. W e fixed aluminium rods on the ground and the mi-
crophones at the top of the ro ds. The microphones were connected to the
data acquisition b o xes with BNC cables.
The microphones w ere lo cated co v ering a circumferen tial angle of ab out
120 ◦ around the craters of the eruptiv e fissure, although not at the same
altitude, whic h w as dep enden t on the orograph y of the v olcano.
In order to sho w that ev en at v ery large Reynolds n um b ers is p ossible
to observ e a laminar v ortex ring w e sho w in figure 3.9 a v ortex ring ejected
b y an eruption at Moun t Etna.
49

Chapter 3. Exp erimen tal setup
Figure 3.8: Deplo ymen t of the microphones around the craters at the field
during the Moun t Etna exp edition.
( a ) ( b )
Figure 3.9: ( a ) Image tak en at the eruptiv e fissure in the North-East crater at
Moun t Etna in 2015 . The t w o craters are visible in the lo w er part of the image
and ejected v ortex ring in the upp er part of the image. ( b ) Detail view of the
v ortex ring ejected b y the same eruption.
3.3.2 Strom b oli
A collab oration with the Europ ean pro ject ’V olcanic ash: FiEld, ex-
p eRimenT al and n umerIcal in v estiGations of prOcesses during its lifecycle’
(VER TIGO) ga v e us the p ossibilit y to assist to the field campaign of Ma y
2016 in Strom b oli. Figure 3.10 sho ws a general o v erview of the crater con-
figuration at Strom b oli on the 31 th May 2016 .
A group of up to four p eople (prof. Sesterhenn, Stefano Alois, Steffen
Nitsc h and m yself ) w ere 9 times in the summit of the volcano to perform
50

Exp erimen ts of a compressible starting jet
Figure 3.10: Crater configuration at Stromboli on the 31 th May 2016 .
acoustic measuremen ts of the eruptions. Sev eral h undreds of ev en ts w ere
recorded during this campaign. Similar to the campaign at Moun t Etna, we
fixed aluminium ro ds on the ground and one microphone p er ro d, see figure
3.11. During this campaign, we had up to four completely independent
stations, p erfectly sync hronised.
Dep ending on the da y , the w eather conditions and the plan of the rest of
the campaign, w e used only some of the stations, all lo cated at the summit
of the v olcano, or w e used all stations, lo cating them as w ell far a w a y from
the v olcano, close to the sea lev el.
( a ) ( b )
Figure 3.11: Measuremen ts at Strom b oli. In b oth figures the alumin um ro d
(fixed to the ground) can b e seen. A ttac hed to it are the microphones, whic h are
connected to the computer through a BNC cable.
The stations w ere basically comp osed of a microphone, a data acquisi-
tion system man ufactured b y Data T ranslation, a NUC computer and an
51

Chapter 3. Exp erimen tal setup
external p o w er bank. While the data acquisition system, the NUC and the
p o w er bank w ere pac k ed in a P eli case, the microphone was deplo y ed in the
field connected to the P eli case through a BNC cable.
The sync hronisation of the indep enden t stations to ok place in a v ery
simple, but efficien t w a y . When the differen t stations in v olv ed w ere ab out
to b e placed far a w a y from eac h other, w e started the measuremen ts at the
p oin t when the stations w ere still together. W e connected the microphones
for a short p erio d of time and w e used an athletics starting flap to generate
a sharp noise when holding all microphones at the same distance. The
pressure w a v e should arriv e at all microphones with a dela y smaller than
the sampling rate of the microphones, leading to enough accuracy in our
measuremen ts.
52

P art I I I
Results of jet sim ulations
and exp erimen ts
53

Chapter 4
Characterisation of the
compressible starting jet
In this c hapter, w e describ e ho w is the ev olution of the dynamics of the
differen t features in the compressible starting jet. W e proceed chronolog-
ically starting with the compression w a v e, follo wing with the dynamics of
the v ortex ring and ho w the compressibilit y affects its dynamics and closing
the c hapter with the formation of the trailing jet and its dynamics.
4.1 Compression w a v e
When impulsiv ely releasing the pressure of the reserv oir, a compression
w a v e is formed at the nozzle exit. It trav els into the un bounded cham ber
with a half-spherical shap e. F or lo w pressure ratios, this compression w a v e
is a pressure w a v e, that propagates at M = 1 . F or larger pressure ratios, a
blast w a v e is created, whic h propagates at M > 1 . Ishii et al. [1999] found
b y applying the Rankine-Hugoniot relations across a mo ving sho c k that
the required pressure ratio for a blast w a v e is p 0 r /p ∞ = 41 . 2 . This is shown
in figure 4.1 b . The blue line represen ts the Mac h n um b er of an isen tropic
expansion from the reserv oir to the region 2 represen ted in figure 4.1 a . F or
a pressure ratio p 0 r /p ∞ = 4 . 1 the Mac h n um b er in the region 2 b ecomes
M 2 = 1 . This condition remains for pressure ratios up to 41 . 2 , represen ted
b y the red line in figure 4.1 b . F rom p 0 r /p ∞ = 41 . 2 on, the Mach n umber
in region 2 b ecomes M 2 > 1 and the w a v es mo v es with sup ersonic sp eed.
4.2 V ortex ring dynamics
The c haracterisation of the v ortex ring dynamics can b e determined b y
the v ortex ring radius and core radius and the axial p osition and v elo cit y .
55

Chapter 4. Characterisation of the compressible starting jet
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
1.4

p 0 r /p ∞
M

( a ) ( b )
Figure 4.1: ( a ) Characteristic diagram of the classical sho ck tube problem.
Region 1 is the am bien t or the lo w-pressure region; 2 is the region b etw een the
first compression w a v e (it migh t b e a pressure w av e or a blast wa v e dep ending
on the pressure ratio) and the con tact surface (the region where the fluid coming
from the reserv oir and the corresp onding to the am bien t are in contact assuming
no mass diffusivit y); 3 denotes the region b et w een the contact surface and the
expansion fan and 4 denotes the reservoir region. ( b ) In v olv ed Mac h num bers in
the region 2 of the sho c k tub e problem. The real path is represen ted in solid lines;
the dashed lines are for reference. The blue lines (dashed and solid) represen ts
the Mac h n um b er of an isen tropic expansion from the reservoir un til the region
2 (represen ted in a ). The blac k lines (dashed and solid) represen ts the Mac h
n um b er b ehind a sho c k w a v e through the Rankine-Hugoniot equations until the
region 2 (represen ted in a ). The real ev olution is the com bination of the differen t
solid lines, this is, a subsonic flo w after the compression w a v e from p 0 r /p ∞ = 1 to
4 . 1 , sonic flow from p 0 r /p ∞ = 4 . 1 to 41 . 2 and sup ersonic flo w from p 0 r /p ∞ = 41 . 2
on.
W e discuss this features in the follo wing sections.
T o p erform this analysis w e used the results of the simulations 1 .a ,
2 − 6 and 7 .a , where the only free parameter w as L/D , see table 2.6 in the
in tro duction.
4.2.1 V ortex ring radius and core radius
The geometry of the v ortex ring is fully defined with t w o parameters:
the v ortex ring radius ( R V R ) and the vortex ring core radius δ 90 as sho wn
in figure 4.2.
W e define the v ortex ring radius ( R V R ( t ∗ ) /D ) as the distance from the
jet axis to the maxim um distribution of I R
I R ( R V R , t ) = 1
U 2 D
∞
Z 0
u 2
R dx, (4.1)
56

Results of jet simulations and exp erimen ts
tak en from P a wlak et al. [20 07]. The maxim um of I R i s asso ciated with the
p eak of radial flow coinciden t with the v ortex centre. The v ortex rin g core
radius ( δ 90 ( t ∗ ) /D ) is defined in t his study as the radial distance from t he
v ortex ring core for which the v orticit y reac hes the 10% of the maxi m um
v orticit y in the v ortex rin g.
Figure 4.2: Geometrical para meters of the v ortex ring: v ortex ring radius ( R VR
)
and v ortex ring core ra dius ( δ 90 ).
Making the assumptio n that the impulse ( I ) of the v ortex ring is con -
stan t, Maxworth y [1972] stated that I ∼ UR 3
VR and hence R VR ∼ t 1 / 3 . The
impulse of the v ortex ri ng w as defined in Maxw orth y [1972] as I = πR 2
VR
Γ ,
where Γ is the circulation of th e v ortex ring.
F or a time interv al denoted as ( t ∗ ) 1 / 3law in table 4.1, starting at ab out
t ∗ =2 , the v ortex ring radius sho ws a dep endence with R VR
( t ∗ ) /D ∼
( t ∗ ) 1 / 3 , as sho wn in figure 4.3.
T able 4.1: V or tex ring non-dimensional radius at the en d of the formation stage.
The ( t ∗ ) 1 / 3 la w reads R VR
/D = a 1 / 3 + b 1 / 3 ( c 1 / 3 + t ∗ ) 1 / 3 , where the co efficie n ts
a 1 / 3 ,b
1 / 3 ,c
1 / 3 are here presen ted. Time interv al for whic h the ( t ∗ ) 1 / 3 la w is v alid
for the differen t cases und er in v estigation.
Case L/D  R VR
D  max a 1 / 3 b 1 / 3 c 1 / 3 ( t ∗ ) 1 / 3la w
1 .a 0 . 4473 0 . 5415 0 . 4578 0 . 0583 − 1 . 8563 [2 − 6]
21 . 1730 0 . 6427 0 . 4837 0 . 0731 − 2 . 5026 [2 − 13 . 5]
3 3 . 8029 0 . 8147 0 . 4380 0 . 1393 − 2 . 3148 [2 . 1 − 21 . 4]
48 . 6047 0 . 8496 0 . 3413 0 . 2108 − 1 . 9296 [1 . 8 − 17 . 5]
5 13 . 5487 0 . 8847 0 . 2813 0 . 2465 − 1 . 5861 [1 . 8 − 16 . 3]
6 28 . 5899 0 . 9201 0 . 1519 0 . 3112 − 0 . 7541 [1 . 8 − 15 . 5]
7 .a →∞ 1 . 0280 0 . 0329 0 . 3684 − 0 . 0435 [1 . 8 − 16 . 3]
V ortex rings gen erated with larger L/D grew fas ter and during a longer
p erio d of time than vortex rings generated with lo w er L/D . Figure 4.4 a
57

Chapter 4. Characterisation of the compressible starting jet
0 5 10 15 20 25 30
0.46
0.48
0.5
0.52
0.54

t ∗
R V R /D

0 5 10 15 20 25 30
0.45
0.5
0.55
0.6
0.65

t ∗
R V R / D

( a ) ( b )
0 5 10 15 20 25 30
0.5
0.6
0.7
0.8

t ∗
R V R / D

0 5 10 15 20 25 30
0.5
0.6
0.7
0.8
0.9
1

t ∗
R V R / D

( c ) ( d )
0 5 10 15 20 25 30
0.5
0.6
0.7
0.8
0.9
1

t ∗
R V R / D

0 5 10 15 20 25 30
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2

t ∗
R V R / D

( e ) ( f )
Figure 4.3: Evolution of the v ortex ring non-dimensional radius. The dashed
line corresp onds to the instan taneous v alue, while the solid blac k line corresp onds
to the prop ortionalit y la w giv en b y the equation R V R /D = a 1 / 3 + b 1 / 3 ( c 1 / 3 +
t ∗ ) 1 / 3 . The parameters a 1 / 3 , b 1 / 3 and c 1 / 3 are in table 4.1. ( a ) Case 1 .a . L/D =
0 . 4473 . ( b ) Case 2 . L/D = 1 . 1730 . ( c ) Case 3 . L/D = 3 . 8029 . ( d ) Case 5 .
L/D = 13 . 5487 . ( e ) Case 6 . L/D = 28 . 5899 . ( f ) Case 7 .a . L/D → ∞ .
compares the R V R ( t ∗ ) /D = a 1 / 3 + b 1 / 3 ( c 1 / 3 + t ∗ ) 1 / 3 trend lines for the cases
in this study .
58

Results of jet sim ulations and exp erimen ts
0 5 10 15 20
0.5
0.6
0.7
0.8
0.9
1

t ∗
R V R / D

0 2 4 6 8 10 12
0.5
0.6
0.7
0.8
0.9

t ∗
R V R / D

( a ) ( b )
Figure 4.4: ( a ) R V R /D = a 1 / 3 + b 1 / 3 ( c 1 / 3 + t ∗ ) 1 / 3 trend lines that follo w the
differen t cases of this study . See legend in table 2.3. a 1 / 3 , b 1 / 3 and c 1 / 3 are the
ones in the captions of figure 4.3. ( b ) Comparison with the literature of the v ortex
ring radius’ prop ortionalit y with ( t ∗ ) 1 / 3 . See legend in table 2.3. • Gao and Y u
[2010],  Didden [1979], N Liess [1978]. The data from Liess [1978] w ere shifted
in time b ecause of a presumed differen t time origin definition.
Although this prop ortionalit y is only found in the w ork of Maxw orth y
[1972], the b eha viour of the v ortex ring radius gro wth agrees with the com-
pared previous w orks, see figure 4.4 b . The time origin do es not agree with
all the previous w orks, presumably due to a differen t time origin definition
based on the formation of the v ortex ring and not on the b eginning of the
fluid flo w.
During the formation of the v ortex ring, its core radius shrank impul-
siv ely (the same w a y that the fluid w as injected), but during the deca y
stage, the vortex ring core radius gro ws with a monotonously decreasing
gro wth rate, figure 4.5 a . In the v ery b eginning, the v ortex w as not com-
pletely formed, and therefore very difficult to trac k. Because of this, the
initial ev olution of the v ortex ring core radius is irregular.
T o directly compare the gro wth of the v ortex ring radius and the core
radius, figure 4.5 b sho ws the v orticit y profile along the radial direction
during the deca y stage of the v ortex ring.
4.2.2 Axial lo cation and propagation v elo cit y
W e define the axial p osition of the v ortex ring as the distance in the
axial direction b et w een the nozzle and the maxim um of I x
I x ( x, t ) = 1
U 2 D 2
nR
Z 0
u 2 RdR, (4.2)
59

Chapter 4. Characterisation of the compressible starting jet
0 150 300 450 600
0.1
0.15
0.2
0.25
0.3

t ∗
δ 9 0 /D

0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
1.2

ω / ( U j /D j )
y/D

( a ) ( b )
Figure 4.5: ( a )Time evolution of δ 90 /D . ( b ) Evolution of the v orticit y profile
along the radial direction. Ligh t grey indicates former times, while dark grey
indicates later times. • maxim um of ev ery v orticit y profile.
P a wlak et al. [2007]. I x is prop ortional to the flux of axial momentum
exiting throughout a disk of arbitrary radius ( nR ), p erp endicular to the
axial direction.
The axial lo cation of the v ortex ring w as prop ortional to x VR /D ∼
( t ∗ ) 1 / 2 , as prop osed b y Witze [1980] and sho wn in figure 4.6. The prop or-
tionalit y constan ts obtained b y least squares in the linear range are in the
captions and also summarised in table 4.2.
T able 4.2: The prop ortionality la w reads x V R /D = a 1 / 2 ( t ∗ ) 1 / 2 + b 1 / 2 . The a 1 / 2
and b 1 / 2 co efficien ts are presen ted here, as well as the dimensionless time for what
the prop ortionalit y starts.
Case L/D a 1 / 2 b 1 / 2 ( t ∗ ) 1 / 2 la w
1 0 . 4473 0 . 8726 − 4 . 0366 92 . 3310
2 1 . 1730 1 . 6633 − 6 . 6136 53 . 1470
3 3 . 8029 2 . 8379 − 10 . 0457 31 . 9937
4 8 . 6047 3 . 9509 − 9 . 0353 12 . 5919
5 13 . 5487 3 . 1999 − 9 . 3714 17 . 4783
6 28 . 5899 3 . 2337 − 9 . 1065 14 . 9599
7 .a → ∞ 3 . 6724 − 10 . 8790 15 . 9968
60

Results of jet sim ulations and exp erimen ts
0 5 10 15 20 25
0
5
10
15

( t ∗ ) 1 / 2
x V R / D

0 5 10 15 20 25
0
5
10
15

( t ∗ ) 1 / 2
x V R / D

( a ) ( b )
0 5 10 15 20 25
0
5
10
15

( t ∗ ) 1 / 2
x V R / D

0 5 10 15 20 25
0
5
10
15

( t ∗ ) 1 / 2
x V R / D

( c ) ( d )
0 5 10 15 20 25
0
5
10
15

( t ∗ ) 1 / 2
x V R / D

0 5 10 15 20 25
0
5
10
15

( t ∗ ) 1 / 2
x V R / D

( e ) ( f )
Figure 4.6: Ev olution of the non-dimensional x -lo cation of the v ortex ring
with ( t ∗ ) 1 / 2 . The solid blac k lines are the instan taneous v alues from the sim-
ulations, while the dashed black lines correspond to the prop ortionality la w giv en
b y x V R /D = a 1 / 2 ( t ∗ ) 1 / 2 + b 1 / 2 . The b eginning of the prop ortionality is indi-
cated with • . ( a ) Case 1 .a . L/D = 0 . 45 . ( b ) Case 2 . L/D = 1 . 17 . ( c ) Case 3 .
L/D = 3 . 80 . ( d ) Case 5 . L/D = 13 . 55 . ( e ) Case 6 . L/D = 28 . 59 . ( f ) Case 7 .a .
L/D → ∞ .
61

Chapter 4. Characterisation of the compressible starting jet
Another prop ortionalit y la ws w ere found in the literature, like the re-
sults from Didden [1979] ( x VR ∼ ( t ∗ ) 3 / 2 ), whic h are not in contradiction
with the results of this study b ecause this study fo cuses on a later time
in terv al, denoted as ( t ∗ ) 1 / 2
start in table 4.2. T o the b est kno wledge of the
authors, there is no men tion in the literature ab out the time for whic h the
( t ∗ ) 1 / 2 prop ortionalit y starts. W e define the b eginning of the prop ortion-
alit y in this study as the non-dimensional time for whic h the x − lo cation
of the v ortex ring and the ( t ∗ ) 1 / 2 prop ortionality differ in a 5% . W e found
that for larger v alues of C , this prop ortionalit y law starts earlier than for
the cases with small C , see figure 4.6.
0 10 20 30 40 50
0
20
40
60
80
100
120
140

( t ∗ ) ( 1 / 2 )
x V R / D

0 2 4 6 8 10
0
2
4
6
8
10

( t ∗ ) ( 1 / 2 )
x V R / D

( a ) ( b )
Figure 4.7: Ev olution of the axial lo cation of the v ortex ring with the square
ro ot of the non-dimensional time. Comparison with exp erimen tal data. ( a ) Witze
[1983] • Re ≈ 1800 ,  Re ≈ 4100 ; Johari et al. [1997] H Re = 5000 , N R e = 15000
and J Re = 20000 . The solid blac k line corresp onds to case 7 .a Re = 5000 and
the dashed blac k line is its linear extrap olation. ( b ) • Gao and Y u [2010] (shifted
in time b ecause of a presumed differen t time origin definition); see the legend in
table 2.3.
When comparing with the literature, figures 4.7 a and 4.7 b , our data
agreed w ell with the previous w ork except for the time origin, this is the
reason wh y some data ha v e b een shifted in time. One exception of go o d
agreemen t are the data from Witze [1983] for R e ≈ 4100 in figure 4.7 a ; the
v ortex ring cannot b e at a p osition of x VR /D ≈ 4 . 6 at a time √ t ∗ ≈ 0 . 008 ,
b ecause it means that the v ortex ring has tra v elled 4 . 6 D in the time that
the injected gas has reac hed only 0 . 008 2 D , and this would mean that the
v ortex ring mo v es with a v elo cit y more than 70 000 times larger than the
c haracteristic v elo cit y with whic h the time w as non-dimensionalised, and
therefore the three first p oin ts of this data series (  ) are not considered in
the comparison. The rest of the data compared with agreed successfully
with our data.
W e no w fo cus on the v ortex ring propagation v elo city . Based on the
62

Results of jet sim ulations and exp erimen ts
Biot-Sa v art la w for the v elo cit y induced near a v ortex, the Kelvin form ula
for the propagation v elo cit y of a circular v ortex ring of small cross section
in a p erfect fluid is
U = Γ
4 π R V R
log  8 R V R
δ 90 − 0 . 25  , (4.3)
tak en from [Saffman, 1971].
10 0 10 1 10 2
0.02
0.05
0.1
0.2
0.5

t ∗
u V R / U j

10 0 10 1 10 2
0.02
0.05
0.1
0.2
0.5
1
2

t ∗
u V R / U j

( a ) ( b )
Figure 4.8: V ortex ring propagation v elo cit y o v er the non-dimensional time
for the differen t cases under study . See the legend in table 2.3. ( a ) V elo cit y
deriv ed from the trac king in the sim ulations. ( b ) V elo cit y feeding the mo del of
Kelvin, equation (4.3) with the data Γ( t ∗ ) , R V R ( t ∗ ) /D and δ 90 ( t ∗ ) /D from the
sim ulations.
Figure 4.8 sho ws a comparison b et w een the v ortex ring propagation
v elo cit y obtained deriving the v ortex ring axial p osition with the time and
the v elo cit y obtained follo wing Kelvin’s form ula (equation (4.3)) with the
curren t v alues of Γ( t ∗ ) , R V R ( t ∗ ) /D and δ 90 ( t ∗ ) /D from the results of this
study . The results are sho wn in figure 4.9 separated for the differen t cases.
Generally , the results agree w ell, esp ecially the laminar cases ( C = 1 , 2 ).
F or the cases with large v alues of C , the rising phase agrees w ell un til the
v ortex rings b ecome turbulen t b ecause they are v ery difficult to trac k and
the assumptions made for the Kelvin mo del do not longer apply .
63

Chapter 4. Characterisation of the compressible starting jet
10 0 10 1 10 2
0.02
0.04
0.06
0.08

t ∗
u V R / U j

10 0 10 1 10 2
0.02
0.04
0.06
0.1
0.15
0.2

t ∗
u V R /U j

( a ) ( b )
10 0 10 1 10 2
0.02
0.05
0.1
0.2
0.4
0.6

t ∗
u V R / U j

10 0 10 1 10 2
0.02
0.05
0.1
0.2
0.5
1
2

t ∗
u V R /U j

( c ) ( d )
10 0 10 1 10 2
0.02
0.05
0.1
0.2
0.5
1
2

t ∗
u V R /U j

10 0 10 1 10 2
0.02
0.05
0.1
0.2
0.5
1
2

t ∗
u V R / U j

( e ) ( f )
Figure 4.9: V ortex ring propagation v elo cit y o v er the non-dimensional time
for the differen t cases under study . The dashed grey line corresp onds to the
estimated v elo cit y feeding Kelvin’s form ula (4.3) with the instan taneous v alue
of Γ( t ∗ ) , R ( t ∗ ) /D and δ 90 ( t ∗ ) /D from the sim ulations and the solid blac k line
corresp onds to the v elo cit y directly deriv ed from the trac king of the v ortex ring
in the sim ulations. ( a ) Case 1 .a . ( b ) Case 2 . ( c ) Case 3 . ( d ) Case 5 . ( e ) Case 6 .
( f ) Case 7 .a .
64

Results of jet sim ulations and exp erimen ts
4.2.3 Effects of compressibilit y
Mo ore [1985] found that the propagation v elo cit y of the v ortex ring
decreases due to compressibilit y effects, and it is giv en b y:
U = Γ
4 π R V R  log  8 R V R
δ 90  − 1
4 − 5
12 M 2
Γ + O ( M 4
Γ )  , (4.4)
where M Γ = Γ / (2 π δ 90 c ∞ ) is the Mac h n um b er based on the v ortex cir-
culation. This is mainly due to the reduction of the densit y in the core
of the v ortex ring, due to the pressure distribution. The con tribution of
compressibilit y is of the form:
∞
Z 0  ρ ( r )
ρ ( ∞ ) − 1  r dr
and it b ecomes negativ e b ecause of the effect of compressibilit y .
W e sim ulated a starting jet with a single v ortex ring and without trail-
ing jet in order to analyse the v ortex ring. Three different configurations
w ere studied (see figure 4.10 a − b ): a lo w compressible case ( 1 .a ), a high
compressible case ( 1 .b ) and a sup ersonic case ( 1 .c ). The parameters of these
sim ulations are summarised in table 4.3.
0 1 2 3 4 5
0
0.25
0.5
0.75
1
1.25
1.5

t ∗
M j

0 20 40 60 80 100 120
0
0.2
0.4
0.6
0.8
1

t ∗
M Γ

( a ) ( b )
Figure 4.10: Ev olution of the fully expanded Mac h n umber ( a ) and the Mac h
n um b er defined from the v ortex ring circulation ( M Γ ) ( b ) for the cases 1 .a (blac k),
1 .b (blue) and 1 .c (red).
In order to estimate the propagation v elo cit y of the v ortex ring us-
ing Mo ore’s form ula, we computed the Mac h n um b er M Γ , whic h are also
summarised in table 4.3. The lo w compressible case (case 1 .a ) has a typical
Mac h n um b er based on the circulation of M Γ = 0 . 1 , while for the compress-
ible cases (high compressible 1 .b and sup ersonic 1 .c ) w e ha ve M Γ ∼ 0 . 4 and
65

Chapter 4. Characterisation of the compressible starting jet
T able 4.3: Main parameters of the sim ulations fo cusing on the analysis of the
v ortex ring.
Case L/D p 0 r /p ∞ max( M j ) M Γ
1 .a 0 . 45 3 . 6 0 . 4 0 . 1
1 .b 0 . 0436 20 . 63 0 . 8267 0 . 4
1 .c 0 . 091 96 . 36 1 . 329 0 . 6
0 . 6 , resp ectiv ely . As the theory predicts, the propagation velocity of cases
1 .b and 1 .c are lo w er than the corresp onding of the case 1 .a , as it can b e
inferred from the slop e of the curv es of figure 4.11 a . The propagation v e-
lo cit y of the case 1 .c is sligh tly lo w er than that of the case 1 .b , although
the Mac h n um b er M Γ is substan tially larger.
0 4 8 12 16 20
0
3
6
9
12
15
18

( t ∗ ) ( 1 / 2 )
x V R / D

0 1 2 3 4 5
0.46
0.5
0.54
0.58
0.62

( t ∗ ) ( 1 / 3 )
R V R / D

( a ) ( b )
0 40 80 120 160
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35

t ∗
δ 9 0 /D

( c )
Figure 4.11: Ev olution of the axial p osition ( a ), the v ortex ring radius ( b )
and the v ortex ring core radius ( c ) of the v ortex ring for cases 1 .a (blac k), 1 .b
(blue) and 1 .c (red). The axial p osition has b een plotted ov er ( t ∗ ) (1 / 2) to sho w its
prop ortionalit y through the straigh t segmen ts. The dimensionless radius has b een
plotted o v er ( t ∗ ) 1 / 3 to show its proportionality through the straight segmen ts.
66

Results of jet sim ulations and exp erimen ts
Figure 4.11 b sho ws the ev olution of the non-dimensional v ortex ring ra-
dius R VR /D with ( t ∗ ) (1 / 3) . In the lo w compressible case, the v ortex ring
gro ws un til a maxim um R VR /D ≈ 0 . 54 and then monotonically decreases.
Ho w ev er, for b oth compressible cases, after the maximum non-dimensional
radius, it do es not decrease monotonically , but it has a lo cal minim um and
then gro ws sligh tly . The maxim um is reac hed earlier for the lo w compress-
ible case, follo w ed b y the high compressible and last the sup ersonic case.
Also related to the geometry of the v ortex ring, the dimensionless core
radius δ 90 /D is sho wn for the three cases in figure 4.11 c . Although the v or-
tex rings w ere generated b y the same nozzle diameter, there are substan tial
differences b et w een the three cases.
4.3 T railing jet formation and dynamics
The existence of a trailing jet after the v ortex ring dep ends mainly on
L/D as rep orted b y Gharib et al. [1998], who termed formation numb er the
limiting v alue of the non-dimensional mass supply that leads to a trailing
jet ( L/D ) lim . Only the starting jets with a larger non-dimensional mass
supply than the formation n um b er L/D > ( L/D ) lim lead to a trailing jet.
x/ D
y / D
0 5 10 15
−2
0
2

x/ D
y / D
0 5 10 15
−2
0
2

( a ) ( b )
x/ D
y / D
0 5 10 15
−2
0
2

x/ D
y / D
0 5 10 15
−2
0
2

( c ) ( d )
Figure 4.12: V orticit y con tour when the v ortex ring is at x/D ∼ 11 . L/D =
0 . 45 , 0 . 0436 and 0 . 091 do not sho w an y trailing jet but L/D = 13 . 55 . The
v orticit y range is the same in all figures [0 , 200] s − 1 . ( a ) Case 1 .a . ( b ) Case 1 .b .
( c ) Case 1 .c . ( d ) Case 5 .
The concept of the formation n um b er also applies for compressible and
sup ersonic flo ws. Figure 4.12 shows the v orticit y con tour for a lo w com-
pressible ( a ), a high compressible ( b ) and a sup ersonic case ( c ) when the v or-
tex ring reac hes x VR /D ∼ 11 . All three cases ha v e a lo w er non-dimensional
mass supply than the formation n um b er and therefore none of them has a
67

Chapter 4. Characterisation of the compressible starting jet
trailing jet. The pressure ratio in all three cases is v ery differen t as sum-
marised in table 4.3, sho wing in this w a y that the pressure ratio do es not
affect the existence of the trailing jet formation. Figure 4.10 a sho ws the
ev olution of the fully expanded Mac h n um b er and figure 4.10 b sho ws the
ev olution of the Mac h n um b er defined from the v ortex ring circulation for
the three cases for comparison. As an example of a sup ersonic starting jet
that leads to a trailing jet w e sho w the case 5 ( L/D = 13 . 5487 ) in figure
4.12 d .
Presumably , the limiting v alue of the non-dimensional mass supply that
leads to a trailing jet in compressible started jets is the same than for the
incompressible ones, but this is an op en question that should b e answ ered
with exp erimen tal w orks rather than n umerical ones due to the high com-
putational costs to carry out suc h a parametric study .
Oscillations of the sho c k w a v es in the trailing jet ha v e b een rep orted
in the literature for con tin uous jets [P anda, 1998]. These oscillations are
visible for a starting jet without deca y in figure 4.13 a . The sho ck w av es
oscillate with resp ect to a fixed p oin t. The larger the distance of the sho ck
w a v e to the nozzle, the larger the amplitude of the oscillations. Figure 4.13 b
sho ws the oscillations of the sho c k w a v es for a case with deca y . The sho ck
w a v es further a w a y from the nozzle oscillate with a larger amplitude than
those close to the nozzle. This sho ck w a v es do not oscillate with resp ect
to a fixed p oin t; due to the deca y stage, they mo v e to w ards the nozzle exit
and they oscillate additionally with resp ect to this mo v emen t.
In this c hapter w e analysed the dynamics of the three main features
of the compressible starting jet: the compression w a v e propagates at the
sp eed of sound for pressure ratios lo w er than 41 . 2 , b eing a pressure w a v e;
for larger pressure ratios the compression w a v e b eha v e as a blast w a v e,
propagating faster than the sp eed of sound.
W e used the definition of the v ortex ring radius based on the w ork of
P a wlak et al. [2007]; we confirmed that for a time in terv al starting at about
t ∗ = 2 , the v ortex ring radius sho ws a dep endence with R V R /D ∼ ( t ∗ ) 1 / 3 .
W e defined the core radius of the v ortex ring as the radial distance from the
v ortex ring core for whic h the v orticit y reac hes 10 % of the maxim um v ortic-
it y in the v ortex ring. Concerning the axial propagation of the v ortex ring
w e confirmed the prop ortionalit y x V R /D ∼ ( t ∗ ) 1 / 2 as prop osed b y Witze
[1980] and w e rep orted the non-dimensional time for whic h this prop ortion-
alit y start. W e also confirmed the effects of compressibilit y predicted b y
Mo ore [1985], reducing the axial propagation v elo cit y of the v ortex ring for
higher compressible v ortex rings.
Concerning the existence of the trailing jet, w e b elieve that the limiting
v alue of the non-dimensional mass supply that leads to a trailing jet in
compressible started jets is v ery similar to the v alue in the incompressible
68

Results of jet sim ulations and exp erimen ts
( a ) ( b )
x/ D
t ∗
0 5 10 15 20
0
20
40
60
80
100

x/ D
t ∗
0 5 10 15 20
0
20
40
60
80
100

( c ) ( d )
Figure 4.13: Characteristic ( x/D − t ∗ ) diagram of the pressure p erturbations
sho wing the dynamics of the sho c k w a ves. ( a ) Case 17 . p 0 ∈ ± 40 000 Pa. ( b ) Case
18 . p 0 ∈ ± 40 000 P a. ( c ) Case 5 . p 0 ∈ ± 25 000 Pa. ( d ) Case 6 . p 0 ∈ ± 37 000 P a.
case. All sim ulations in this study regardless of their compressibilit y agreed
with the criteria of the incompressible case.
W e confirmed the oscillation of the sho c k w a v es of the trailing jet as
rep orted b y P anda [1998]. In the cases with deca y , we sho w ed ho w the
sho c k w a v es mo v e to w ards the nozzle exit and oscillate additionally with
resp ect to this mo v emen t.
69

Chapter 4. Characterisation of the compressible starting jet
70

Chapter 5
Effects of the inflo w
condition
A part of the w ork corresp onding to this section w as presen ted in the
ER COFT A C W orkshop 9 th Direct and Large-Eddy Simulation (DLES9),
[ J.J. P eña F ernández et al., 2013].
This section assesses the influence of the inflo w condition on the flo w
field and the emanated noise in the sup ersonic jet, comparing the jet flo w
for three differen t configurations:
• Laminar h yp erb olic tangen t v elo cit y profile.
• Laminar h yp erb olic tangen t mean v elo cit y profile p erturb ed syn thet-
ically with a tripp ed b oundary la y er ( TBL ) as in Bogey et al. [2011,
2012].
• Laminar compressible Blasius v elo cit y profile with a nozzle using the
v olume p enalisation metho d.
5.1 Laminar inflo w
The inflo w w as mo delled imp osing a h yp erb olic tangen t profile corresp ond-
ing to equation (2.10) and figure 2.6 b . This is the classical inflo w condition
in the jet literature when not sim ulating the nozzle, see F reund et al. [2000]
and Bogey et al. [2003]. The main adv antage of this method is a very goo d
con trol o v er the v elo cit y gradien ts in the shear la y er to pro duce a turbulen t
flo w few diameters a w a y from the inlet when used in com bination with a
Reynolds n um b er in the range 5 000 − 10 000 , as in this study .
71

Chapter 5. Effects of the inflo w condition
5.2 T ripp ed b oundary la y er syn thetic turbu-
lence
0 0.025 0.05 0.075 0.1
−0.5
−0.25
0
0.25
0.5

u 0 / u m a x , c
y/D

Figure 5.1: V elo cit y p erturbations imp osed at the inflo w for the TBL case.
The h yp erb olic tangen t laminar base flo w w as distorted b y sup erp osing
a t ypical turbulence profile from a shear la y er as in Bogey et al. [2011,
2012]. In this case, w e generate a syn thetic turbulen t inflo w that leads to a
fully turbulen t flo w few diameters a w a y from the inlet. The p erturbations
ha v e a p eak of turbulence in tensit y of 10% . Figure 5.1 shows the spatial
distribution of the v elo cit y p erturbations.
5.3 Nozzle flo w
-0.5 -0.25 0 0.25 0.5
0
0.2
0.4
0.6
0.8
1

( a ) ( b )
Figure 5.2: ( a ) Compressible Blasius Mac h n um b er inlet profile. ( b ) Sketc h of
the straigh t nozzle used in case 8 .
A compressible Blasius b oundary la y er profile (figure 5.2 a ) w as imp osed
72

Results of jet sim ulations and exp erimen ts
in com bination with a straigh t nozzle. The nozzle w as implemen ted with
the v olume p enalisation metho d as in section 2.6. A sk etc h of this nozzle
is sho wn in figure 5.2 b .
In the inner side of the nozzle, a turbulen t b oundary la y er is generated
due to the non-slip condition. This b oundary la y er flo w generates some
instabilities, leading to a flo w at the nozzle exit with an in termediate lev el
of turbulence, see Sc h ulze [2011].
5.4 Effects on the flo w field
Figure 5.3 sho ws the instan taneous Mac h n um b er con tours for the laminar,
TBL and nozzle cases, resp ectiv ely . This figure dra ws a general picture of
the three cases.
In the laminar case (figure 5.3 a ), 6 sho ck cells are easy to iden tify from
the inflo w un til appro ximately x/D = 8 . Do wn w ards, the turbulence gro w-
ing in the shear la y er destro ys the sho c k cell structure and the jet b ecomes
subsonic and fully turbulen t.
In the TBL case (figure 5.3 b ), the flo w field is v ery similar to the previous
case; the sho c k cells can b e as w ell iden tified and then the jet b ecomes
turbulen t and subsonic. When lo oking in detail, w e can see some small
p erturbations in the jet core as opp osed to the previous case.
In the nozzle case (figure 5.3 c ), the flo w field after the nozzle is also sim-
ilar to b oth previous cases, but in whic h a faster gro wing of the shear la y er
can b e iden tified. The jet core is shorter, the sho ck cells are sligh tly smaller
and the jet b ecomes turbulen t and subsonic for appro ximately x/D = 10 .
W e are esp ecially in terested in ho w differen t inlet conditions mo dify the
shear la y er and the sho c k cell structure of the jet. Figure 5.4 sho ws the
instan taneous and r.m.s. en trop y con tours for the laminar, TBL and nozzle
cases, resp ectiv ely , giving an idea of ho w this mixing pro cess takes place.
In the laminar case (figure 5.4 a ), the jet core can b e iden tified as a
dark region after the nozzle exit. F rom appro ximately x/D = 8 on, the
com bination of en trainmen t and turbulence enhance the mixing pro cess
and the sho c k cell structure disapp ears.
In the TBL case (figure 5.4 b ), from the nozzle exit, the syn thetic turbu-
lence imp osed at the inlet can b e clearly iden tified as dark and ligh t sp ots
all through the jet core. This do es not mo dify massiv ely the sup ersonic
structure of the jet, whic h can b e still iden tified un til x/D ∼ 8 .
In the nozzle case (figure 5.4 c ), some p erturbations can b e seen inside
the nozzle close to the w all. After the nozzle exit, the structure is v ery
similar to that of the laminar case, with a clear jet core, whic h v anishes
after x/D = 10 due to entrainmen t and turbulent mixing.
73

Chapter 5. Effects of the inflo w condition
( a )
( b )
( c )
Figure 5.3: Instantaneous Mac h n um b er con tours. ( a ) Laminar inflo w (case
10 ). M ∈ [0 , 2 . 64] . ( b ) TBL inflo w (case 9 ). M ∈ [0 , 2 . 3] . ( c ) nozzle inflo w (case
8 ). M ∈ [0 , 2 . 45] .
W e fo cus no w on the jet axis. Figure 5.5 a sho ws the mean Mac h n um b er
along with the jet axis, where the length of the sho c k cell region is clearly
visible where the Mac h n um b er oscillates due to the sho c k w a v es. After
this region, there is subsonic turbulent flo w and the Mac h n um b er decreases
74

Results of jet sim ulations and exp erimen ts
( a )
( b )
( c )
Figure 5.4: Instantaneous en trop y con tours. ( a ) Laminar inflo w (case 10 );
s ∈ [7885 , 8110] m 2 K − 1 s − 2 . ( b ) TBL inflo w (case 9 ); s ∈ [7885 , 8110] m 2 K − 1 s − 2 .
( c ) nozzle inflo w (case 8 ); s ∈ [8110 , 8458] m 2 K − 1 s − 2 .
monotonically . In this figure is also clear that the sho c k cells in the nozzle
case are sligh tly smaller than in the laminar case.
The r.m.s. en trop y p erturbations along the jet axis are represen ted in
figure 5.5 b . They sho w ho w in the laminar and nozzle cases, the entrop y
75

Chapter 5. Effects of the inflo w condition
do es not c hange in the first few diameters. This region corresp onds to the
p oten tial core. The end of the p oten tial core can b e defined as the p oint
in whic h the p erturbations gro wing in the shear la y er reac h the cen terline.
This happ ens at x/D ∼ 4 for the nozzle case, while for the laminar case it
tak es place at x/D ∼ 5 . 6 . The gro wth of the shear la y er in the nozzle case
is v ery similar as in the laminar case, but the nozzle case shows a shorter
p oten tial core than the laminar case as exp ected due to the turbulence
generated in the nozzle.
The p erturbations syn thetically in tro duced in the TBL case deca y for
the first diameter, then it reac hes a lo cal minim um for around x/D = 5
and then it sho ws the same b eha viour as the other t w o cases, with a lo cal
maxim um for around x/D ∼ 10 .
0 5 10 15 20
0
0.5
1
1.5
2
2.5
3

0 5 10 15 20
0
15
30
45
60
75

( a ) ( b )
Figure 5.5: Distributions of mean Mac h n um b er ( a ) and r.m.s. v alues of the
en trop y p erturbations ( b ) along the jet axis. The blue line represents the laminar
inflo w, the blac k line corresp onds to the TBL case and the red line to the nozzle
one.
In this section w e summarised the flo w field c hanges pro duced b y the
nozzle or TBL inlet conditions. As a general trend, for these turbulence
in tensities, the sup ersonic sho ck structure remains although the presence of
strong p erturbations. Lo oking at the r.m.s. v alues of the en trop y along the
jet axis, w e see ho w the p erturbations deca y from the inlet un til x/D ∼ 5 ,
where the p erturbations from the shear la y er start to pla y a role.
5.5 Effects on the radiated acoustics
Since all three cases ha v e the same configuration (con tin uous sup ersonic
jet), w e exp ect the same general b eha viour with sligh tly c hanges in the p osi-
tion and the con tribution of the noise sources dep ending on the in teraction
of the turbulence with the quasi-p erio dic sup ersonic sho c k-cell structure.
76

Results of jet sim ulations and exp erimen ts
F or all three cases, the t ypical sup ersonic noise comp onents w ere observed:
turbulen t mixing noise (TMN) and broadband sho c k noise (BBSN).
Figure 5.6 sho ws the sound pressure lev el sp ectra measured at a radial
distance of 5 D and an angle of 90 ◦ . Concerning the broadband sho c k noise,
all three sp ectra agree in the largest part of the frequency in terv al. The
Strouhal n um b er for the maxim um amplitude of broadband sho c k noise is
sligh tly lo w er than those of the TBL and laminar cases. The developmen t of
the flo w inside the nozzle, sp ecially the b oundary la y er and the w eak sho c ks
inside the nozzle flo w are p ossibly the reason for this differen t b eha viour.
A m uc h more eviden t difference concerns the turbulen t mixing noise.
The frequency con ten t asso ciated purely to turbulen t mixing noise is m uc h
quieter in the case of laminar inlet than the cases TBL and nozzle. As
exp ected, the TBL case has the loudest level o v er the whole frequency
in terv al asso ciated to turbulen t mixing noise. The addition of syn thetic
turbulence to the inlet mo dify the turbulen t mixing noise comp onen t o v er
its whole frequency range.
10 -2 10 -1 10 0
70
80
90
100
110
120
130
laminar
TBL
nozzle

Figure 5.6: Sound pressure lev el sp ectra at 90 ◦ . The laminar case is represen ted
in blue, the T ripped b oundary lay er in blac k and the nozzle case in red. The non-
dimensional time in terv al analysed is t ∗ ∈ [30 , 150] .
The sound pressure sp ectra giv e us man y information ab out the individ-
ual noise comp onen ts, but there is no time lo calisation of the comp onen ts.
F or this purp ose w e p erformed a wa velet analysis: taking as the basis the
complex Morlet w a v elet w e transformed the same time series to analyse
77

Chapter 5. Effects of the inflo w condition
them in the time-frequency domain.
( a )
( b )
( c )
Figure 5.7: Contin uous w a v elet transformation co efficien ts in logarithmic colour
scale for a signal measured at 90 ◦ . ( a ) Laminar inlet. ( b ) TBL inlet. ( c ) Nozzle
inlet.
Figure 5.7 sho ws the w a v elet co efficien ts of the three cases for a signal
measured at 90 ◦ . In all three cases w e can see the trace of the first com-
pression w a v e at the b eginning of the pro cess. After this, the t ypical noise
sources of the sup ersonic con tin uous jet are presen t in all three cases, but
with some differences.
F or the case with a laminar inflo w, figure 5.7 a , the sound pressure level
78

Results of jet sim ulations and exp erimen ts
do es not reac h the maxim um of the scale. All comp onen ts can b e iden tified,
but sp ecially the turbulen t mixing noise has a v ery lo w amplitude.
The case with the TBL inlet, figure 5.7 b , has generally larger amplitudes
than the previous case. The first compression w a v e has a larger amplitude
than in the previous case, but more imp ortan t is the dominan t comp onen t of
the broadband sho c k noise for S t ∼ 3 : the Strouhal n um b er is not constan t
with time, but it oscillates with time; this explains the double p eak in figure
5.6. The high-frequency comp onen t reac hes a higher Strouhal n um b er than
in the previous case. In the turbulen t mixing noise frequency range, m uc h
larger amplitudes than in the previous case are found.
The case with a nozzle inlet, figure 5.7 c , sho ws an in termediate b e-
ha viour. The amplitudes are b et ween the t wo previous cases, with a higher
lev el of turbulen t mixing noise than the laminar case, but not as high as the
TBL case. W e can see the effects on the jet turbulen t mixing noise when
generating an in termediate lev el of turbulence at the inlet.
In this section w e ha v e seen ho w b y increasing the turbulence in ten-
sit y at the inlet, we increase the amplitude of the turbulen t mixing noise
comp onen t. A side effect is the mo dification of the broadband sho c k noise
due to a differen t in teraction b et w een the shear la y er and the sho c k w a v es,
whic h results in a mo dification of the broadband sho c k noise, as c an b e
seen in figure 5.6 with a double maxim um for the broadband sho c k noise
for the TBL case.
5.6 Effect of the nozzle geometry on the start-
ing jet
The laminar inflo w of the previous section w as used in com bination
with a straigh t nozzle and t w o div ergen t nozzles mo delled according to the
v olume p enalisation metho d describ ed in section 2.6. All three nozzles ha v e
a length of one diameter ( D ). The straigh t nozzle corresp onds to the cases
11 .a − h , it is defined b y ( A e / A ∗ = 1 ) and it is sho wn in figure 5.8 a . The
sligh tly div ergen t nozzle is defined b y ( A e / A ∗ = 2 . 36 ), it corresp onds to
cases 12 .a − h and it is sho wn in figure 5.8 b . The div ergen t nozzle is defined
b y ( A e / A ∗ = 4 ), it corresp onds to cases 13 .a − h and it is sho wn in figure
5.8 c .
T able 5.1 summarises the design parameters of the nozzles used in cases
11 − 13 . Assuming one-dimensional isentropic flo w, the exit to critical area
ratio defines the three p oin ts that define the w orking regimes of the nozzle.
This is explained in detail in App endix A for the in terested reader. The
maxim um exit Mac h n um b er when w orking isen tropically in the subsonic
regime is denoted b y M −
e and it defines the b oundary b et w een subsonic
79

Chapter 5. Effects of the inflo w condition
( a ) ( b ) ( c )
Figure 5.8: ( a ) Straight nozzle used in cases 11 .a − h , A e / A ∗ = 1 . ( b ) Slightly
div ergen t nozzle used in cases 12 .a − h , A e / A ∗ = 2 . 36 . ( c ) Divergen t nozzle used
in cases 13 .a − h , A e / A ∗ = 4 . The shaded area represents the w all of the nozzles.
flo w in the whole nozzle and lo cally sup ersonic flo w (just after the critical
area). The same w a y , the exit Mach n um b er when w orking isen tropically
in the sup ersonic regime is indicated with M +
e ; for lo w er Mac h n um b ers
the flo w is o v erexpanded and for larger v alues, the flo w is under-expanded.
F or this conditions, the flo w is said to b e adapte d . A dditionally , the exit
Mac h n um b er when a normal sho c k w a v e is lo cated exactly at the nozzle
exit is denoted b y M +
2 and this conditions sets the b oundary b et w een the
exit subsonic flo w and the o v er-expanded flo w.
T able 5.1: Nozzle design parameters for the cases 11 .a − 13 .h . The exit to critical
area ratio is denoted b y ( A e / A ∗ ), the nozzle subsonic and sup ersonic design Mac h
n um b er are indicated b y M −
e and M +
e , resp ectiv ely and the required pressure
ratio to w ork under the subsonic and sup ersonic design conditions b y ( p 0 r /p −
e )
and ( p 0 r /p +
e ) , resp ectiv ely . More information in app endix A.
Case A e
A ∗ M −
e  p 0 r
p −
e  M +
e  p 0 r
p +
e  M +
2  p 0 r
p +
2 
11 1 1 1 . 893 1 1 . 893 1 1 . 893
12 2 . 36 0 . 255 1 . 046 2 . 38 14 . 1617 0 . 53 2 . 2
13 4 0 . 15 1 . 015 2 . 94 33 . 5717 0 . 48 3 . 38
All these limiting p oin ts collapse for the straigh t-con v ergen t nozzle, this
is, for A e / A ∗ = 1 . Therefore, the straight nozzle of case 11 has a design exit
Mac h n um b er of 1 when w orking with a pressure ratio of ( p 0 r /p ∞ ) d = 1 . 893
and it leads to sonic isen tropic flo w. F or lo w er pressure ratios the nozzle
has isen tropic subsonic flo w and for larger pressure ratios the nozzle w orks
isen tropically un til its exit plane, leading to a sup ersonic expansion just
after. This leads to a sho c k-cell structure as already seen.
F or case 12 a diverg ent nozzle with A e / A ∗ = 2 . 36 w as used. The maxi-
80

Results of jet sim ulations and exp erimen ts
m um exit Mac h n um b er within the isen tropic subsonic flo w is M −
e = 0 . 255 ,
reac hing this condition with a pressure ratio of ( p 0 r /p −
e )=1 . 046 . This
nozzle, when w orking with a pressure ratio b et w een the previous one and
( p 0 r /p +
2 ) = 2 . 2 , will contain normal shock w a v es in the div ergen t part of
the nozzle (stronger sho c k w a v es and lo cated closer to the exit as the pres-
sure ratio increases), increasing the Mac h n um b er until M +
2 = 0 . 53 for
the last condition. F rom this point on, the exit flo w is sup ersonic o v erex-
panded un til the nozzle w orks at the isen tropic sup ersonic conditions, this
is, for a pressure ratio of ( p 0 r /p +
e ) = 14 . 162 , leading to a Mac h n um b er
of M +
e = 2 . 38 . F or larger pressure ratios, the configuration is the same,
but the flo w after the exit plane is underexpanded and m ust through a
sup ersonic expansion ( Prandtl-Meyer ) further expanded.
F or case 13 a div ergent nozzle with A e / A ∗ = 4 w as used. When w orking
within the isen tropic subsonic regime, the maxim um exit Mach n umber is
M −
e = 0 . 15 when applying a pressure ratio of ( p 0 r /p −
e )=1 . 015 . This
nozzle will con tain normal sho c ks inside it when w orking with pressure
ratios b et w een the previous one and ( p 0 r /p +
2 )=3 . 38 , but the exit will
remain subsonic. F or larger pressure ratios the nozzle will b e o v er-expanded
un til the sup ersonic isen tropic flo w, given b y ( p 0 r /p +
e ) = 33 . 5717 . F or
larger pressure ratios, the nozzle will b e under-expanded and a sup ersonic
expansion will tak e place in the vicinit y of the nozzle exit.
0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3

t ∗
M e

0 5 10 15 20 25
0
0.5
1
1.5
2
2.5
3

t ∗
M j

( a ) ( b )
Figure 5.9: T emp oral evolution of the exit (left) and fully expanded (righ t)
Mac h n um b er. ( a − b ) Effect of the geometry . p 0 r /p ∞ = 50 , L/D = 2 . Dot-
dashed: straigh t nozzle ( A/ A ∗ = 1 ), dashed: A/ A ∗ = 2 . 36 , solid: A/ A ∗ = 4 .
Figures 5.9 a − b sho w the effect of the nozzle geometry on the exit and
fully expanded Mac h n um b er. Op erating with a pressure ratio of p 0 r /p ∞ =
50 and a non-dimensional mass supply of L/D = 2 w e see the c hanges when
ha ving a straigh t nozzle ( A/ A ∗ = 1 ) or a divergen t one A/ A ∗ = 2 . 36 and
A/ A ∗ = 4 . F or an early time (serv e as an example t ∗ ∼ 3 ), the pressure
ratio is still large (close to the original v alue of p 0 r /p ∞ = 50 ) and therefore
81

Chapter 5. Effects of the inflo w condition
b oth con v ergen t-div ergen t nozzles deliv er sup ersonic flo w, while the straigh t
nozzle deliv ers sonic flo w. F or a later time ( t ∗ ∼ 7 ), b oth con v ergen t-
div ergen t nozzles deliv er subsonic flo w, b ecause they op erate already in the
o v er-pressured regime, while the straigh t nozzle still op erates in the under-
pressured regime, b eing its exit Mac h n um b er still sonic. F or later times,
as a result of the pressure deca y , the sho c k-cell spacing b ecomes shorter
and all sho c ks tra v el to w ards the nozzle exit. Some of the sho c ks en ter the
nozzle exit. This pro cess leads to the exit Mac h n um b er p erturbation of
figure 5.9 a for t ∗ ∼ 8 .
The fully expanded Mac h n um b er, represen ted in figure 5.9 b , do es not
dep end on the geometry , and should hav e therefore the same v alue for all
three geometries. Apart from the first stage, where the Flow is ev olving
inside the nozzle and therefore w e cannot define the fully expanded condi-
tions, w e can see that all three cases ha v e v ery similar v alues from t ∗ ∼ 6
on.
In this section w e ha v e seen that the exit to critical area ratio is of crucial
imp ortance and it defines the w orking regimes of the nozzle. This, in com-
bination with the op erating conditions of the nozzle define the b eha viour
of the flo w at the nozzle exit and its vicinit y . When op erating a nozzle in
the under-expanded regime, no c hanges can b e observ ed at the nozzle exit
Mac h n um b er, but when the nozzle op erates in the o v er-expanded regime,
the exit Mac h n um b er reduces drastically (ev en reac hing the subsonic flo w).
F or a sp ecific op erating conditions, t w o nozzles with differen t geometries
can op erate one of them in the under-expanded regime and the other in the
o v er-expanded, b eing the exit flow at the former supersonic, while the latter
has subsonic flo w at its exit. Therefore w e can deduce from this section,
that the exit to critical area ratio (mainly) and the op erating conditions
(at some exten t) ha v e a great effect on the nozzle exit flo w.
82

Chapter 6
Pinc h-off definition
A part of this c hapter has b een tak en from [ J.J. P eña F ernández and
Sesterhenn, 2017a].
Pinc h-off is ideally defined as the separation of the v ortex ring from the
rest of the fluid flo w b y a region of zero v orticit y . In turbulen t flo ws, due
to v orticit y diffusion, there is, in fact, no region with zero v orticit y . In
this c hapter, w e discuss the relationship b et w een the v orticit y field and the
pinc h-off.
6.1 Pinc h-off v orticit y threshold ( ω p o /ω v ortex )
W e define the pinc h-off as the separation of the v ortex ring from the
rest of the fluid flo w b y a v orticit y threshold of ω p o /ω vortex . The vorticit y
in the v ortex ring core is denoted b y ω vortex . Cho osing a v alue for this
threshold close to unit y , w e w ould estimate the pinc h-off to o ccur during
the formation of the v ortex ring and c ho osing a v alue of exactly zero, w e
w ould imply that the v ortex ring w ould b e surrounded b y a large region
of almost zero v orticit y , b eing the estimated pinc h-off not represen tativ e
in an y of the t w o cases. Moreo v er, the region of low v orticit y has a flat
distribution and the pinc h-off definition is v ery sensitiv e to ω p o /ω v ortex .
T o c ho ose the prop er pinc h-off v orticit y threshold, we focus now on the
limiting v orticit y v alue that separates the v ortex ring from the nozzle exit
(or the trailing jet) and w e call it sep ar ation vorticity ( ω s ). In other w ords,
v ortex ring and nozzle exit (or trailing jet) are separated b y a region of
v orticit y with a v alue of at least ω s . A lo w er v alue than ω s defines only one
region and cannot distinguish b oth regions, see figure 6.1.
The relativ e separation v orticit y ( ω s /ω v ortex ) decreases with time, see
figure 6.2. The pinc h-off tak es place when this v alue reac hes the pinc h-off
83

Chapter 6. Pinc h-off definition
Figure 6.1: Sk etc h of the pinc h-off v orticity threshold method. The blac k dashed
line corresp onds to the separation v orticit y ( ω s ). Higher v alues of vorticit y define
t w o separated regions, while lo w er v alues define only one region. Case 7 .a is
represen ted.
v orticit y threshold ( ω p o /ω v ortex ). W e defined ω p o /ω vortex = 0 . 1 .
The relativ e separation v orticit y sho ws a lo cal minim um for all cases for
t ∗ ≈ 3 . 4 , as sho wn in figure 6.2. This is follow ed b y differen t local maxima,
whic h are due to v ortices from the shear la y er (compact regions with large
v orticit y v alues) that are tak en in b y the v ortex ring. These v ortices from
the shear la y er ’re-join’ the v ortex ring with the trailing jet and separate
them again when they are con v ected from the shear la y er to the v ortex ring.
This pro cess migh t b e rep eated for the few first Kelvin-Helmholtz v ortices.
This non-monotonic b eha viour of the separation v orticit y mak es the defini-
tion of the pinc h-off difficult. As an example, assuming ω p o /ω v ortex = 0 . 1 ,
the pinc h-off w ould tak e place m ultiple times (for t ∗ = 5 . 2 and 6 . 5 ) for the
case 1 ( L/D = 0 . 45 ). The pinch-off w ould take place as w ell in a ’multiple-
w a y’ for the rest of the cases in this study b ecause the v orticit y pac k ages
ha v e ev en stronger p eaks.
W e therefore define the pinc h-off as the first time that the v ortex ring
separates from the rest of the fluid flo w taking as the pinc h-off v orticit y
threshold ω p o /ω v ortex = 0 . 1 .
F or Reynolds n um b ers in the range 5 000 − 10 000 , the pinc h off tak es
place in a ’m ultiple-w a y’, b ecause there are more Kelvin-Helmholtz vortices
in the shear la y er that are tak en in b y the v ortex ring, leading to the p eaks
84

Results of jet sim ulations and exp erimen ts
2 4 6 8 10 12 14 16
0
0.2
0.4
0.6
0.8

t ∗
ω s / ω vo r t ex

Figure 6.2: Time ev olution of the separation vorticit y . See legend in table 2.3.
The pinch-off vorticity thr eshold ( ω s /ω v ortex = ω p o /ω v ortex = 0 . 1 ) is denoted b y
the horizon tal dashed grey line.
in the separation v orticit y in figure 6.2. F or Reynolds n um b ers b elo w 3 000 ,
the pinc h-off tak es place in a single w a y; the shear lay er is stable and the
separation v orticit y decreases monotonically un til the pinc h-off v orticit y
threshold. There exist a critical Reynolds n um b er that defines the b oundary
b et w een the t w o t yp es of pinc h-off as discussed in 1.3. This is an op en
question that should b e addressed b y future researc h.
The separation v orticit y presen ts a lo cal minim um for all cases for t ∗ ≈
3 . 4 . This minim um can b e in terpreted as an in terruption in the v ortex ring
generation, which in tro duces strong p erturbations in the v ortex ring that
can lead to instabilities that gro w and mak e the v ortex ring transition to
turbulen t.
6.2 Circulation div ergence la w
By computing the circulation ( Γ ) in a semi-plane that contains the jet
axis (hereafter called sagittal semi-plane, sho wn as the ligh t grey region
in figure 6.3) and in the in tersection of this semi-plane with the v ortex
ring (dark grey region in figure 6.3), we sho w that despite injecting more
v orticit y through the nozzle, from a certain p oint on, the v ortex ring did
85

Chapter 6. Pinch-off definition
Figure 6.3: S k etc h of the different methods us ed to compute the circulation.
The ligh t grey shaded area , defined here as the sagittal sem i-plane, is a general
semi-plane that contains the jet axis. The blac k star is th e lo cation of the v ortex
ring follo wing the me tho d prop osed b y P a wlak et al. [2 007]. The dar k grey area
corresp onds to the vortex ring area iden tified b y the flo o d-fill metho d centred in
the v ortex ring lo cation a nd using as a b oundary the con tour line ω =0 . 1 ω v ortex
(plotted here as a solid bla c k line). F o r the cases with turbulen t vortex ring the
most stable metho d to compu te the circulation of the v ortex ring i s a windo w
of 2 D × 1 D (plotted as a dashed black line) defined cen tred in the vortex ring
lo cation.
not absorb an y v ortici t y more, see figure 6.4.
F or the cases with s mall L/D (cases 1 and 2 ), the en tire circulation
con tained in the sagit tal semi-plane w as concen trated i n the v ortex ring
and no trailing jet w as formed , figure 6.4 a . The vortex ring absorb ed the
whole v orticity injected through the nozzle. W e call this situatio n a non-
satur ate d vortex ring . A trailing jet was not formed in cases 1 and 2 .
The small difference b etw een the circulation con tained in the vortex ring
and in the sagittal plane in fi gure 6.4 a is due to the entrainmen t of fluid,
generating a region of p o sitiv e and negativ e v orticity close to the lip of the
nozzle that decays at a differen t rate with time, see figure s 6.5 and 6.6.
This phenomenon tak es place in the rest of the cases, but the amount of
circulation generat ed due to the en trainmen t is neglectable co mpared to
the one in the v ortex rin g.
F or the cases wit h large L/D (cases 3 , 4 , 5 , 6 and 7 ), the qualitati v e
86

Results of jet sim ulations and exp erimen ts
0 10 20 30 40 50
0
0.02
0.04
0.06
0.08
0.1
0.12

t ∗
Γ / ( U j D )

0 10 20 30 40 50
0
0.1
0.2
0.3
0.4

t ∗
Γ / ( U j D )

( a ) ( b )
0 10 20 30 40 50
0
0.3
0.6
0.9
1.2
1.5
1.8

t ∗
Γ / ( U j D )

0 10 20 30 40 50
0
1
2
3
4
5
6

t ∗
Γ / ( U j D )

( c ) ( d )
0 10 20 30 40 50
0
2
4
6
8

t ∗
Γ / ( U j D )

0 10 20 30 40 50
0
3
6
9
12
15

t ∗
Γ / ( U j D )

( e ) ( f )
Figure 6.4: Evolution of the circulation con tained in the fluid flo w for the
starting jet. The solid black line corresponds to the circulation contained in the
x − y sagittal semi plane. The circulation con tained in the in tersection of this
semi-plane with the v ortex ring obtained b y the flo o d fil l metho d is represen ted by
the dashed grey line, while the dot-dashed blac k line corresp onds to the window
metho d. ( a ) Case 1 .a . L/D = 0 . 45 . ( b ) Case 2 . L/D = 1 . 17 . ( c ) Case 3 .
L/D = 3 . 80 . ( d ) Case 5 . L/D = 13 . 55 . ( e ) Case 6 . L/D = 28 . 59 . ( f ) Case 7 .a .
L/D → ∞ .
87

Chapter 6. Pinc h-off definition
x/ D
y / D
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
−0.5
0
0.5
1

Figure 6.5: Relativ e vorticit y to the maxim um v orticit y in the v ortex ring core
( ω /ω vortex ) for the case 1 .a with L/D = 0 . 4473 and for t ∗ ≈ 4 . The v orticit y
generated at the inlet condition due to the en trainmen t affects the circulation
calculated b y in tegrating o v er the whole semi-plane.
b eha viour w as completely differen t. After some time, during the injection of
the v orticit y , the circulation con tained in the v ortex ring did not follo w that
con tained in the sagittal semi-plane and the v ortex ring did not absorb an y
additional v orticit y , as seen in figures 6.4 c , 6.4 d , 6.4 e and 6.4 f . W e call this
situation a satur ate d vortex ring . The ejected circulation not b elonging
to the v ortex ring formed the trailing jet.
When in tegrating the v orticit y in the in tersection of the sagittal plane
and the v ortex ring, the iden tification of the in tegration domain w as done
b y the flo o d fil l metho d, see figure 6.3. While the vortex ring is laminar
or remains in lo w lev els of turbulence, the v ortex is v ery clear and b oth
the iden tification and the in tegration of the v orticit y are straigh tforw ard,
see figure 6.7 a . When the vortex ring becomes turbulent, the region of
high v orticit y is not a simply connected space and c hanges drastically with
time, which mak es the integration m uch more difficult, see figure 6.7 b . This
is the reason of the non-uniform b eha viour of the v ortex ring circulation
(dashed line) in figure 6.4. When the v ortex ring b ecame turbulen t and
the circulation obtained w as not represen tativ e, w e in tegrated the v orticit y
follo wing the most widespread metho d used in the literature: in tegrating
the v orticit y within a 2 D × 1 D window, see figure 6.4. The transv ersal size
of the windo w of 1 D w as c hosen not to cross the jet axis when the leading
88

Results of jet sim ulations and exp erimen ts
x/ D
y / D
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
−0.5
0
0.5
1

Figure 6.6: Relativ e vorticit y to the maxim um v orticit y in the v ortex ring core
( ω /ω vortex ) for the case 1 .a with L/D = 0 . 4473 and for t ∗ ≈ 12 . 5 . After some time
the effects of the en trainmen t disapp ear and the v orticit y calculated b y in tegrating
o v er the whole semi-plane con v erges to the v alue obtained by in tegrating only o v er
the v ortex ring.
x/ D
y/D
0 2 4 6 8
−2
0
2

x/ D
y/D
0 2 4 6 8
−2
0
2

( a ) ( b )
Figure 6.7: V orticity con tour for t w o differen t stages of the case 7 .c . The solid
blac k line corresp onds to ω p o /ω vortex = 0 . 1 . The dark grey region is the v ortex
ring as iden tified b y the flo o d fill metho d. The v ortex ring (follo wing P a wlak
et al. [2007]) is denoted b y ( F ). ( a ) Successful identification of the v ortex ring.
The in tegration of the v orticit y within the dark grey domain leads to a prop er
circulation con tained in the v ortex ring. t ∗ = 12 . 5 . ( b ) Unsuccessful iden tification
of the v ortex ring. The in tegration of the v orticit y do es not giv e us a prop er
circulation v alue b ecause the upp er half of the v ortex has already b een detac hed
from the trailing jet but the lo w er half is still attac hed. F urthermore, due to
turbulence, the v ortex ring is not a simply connected space. t ∗ = 22 .
89

Chapter 6. Pinc h-off definition
v ortex ring is generated, lo cated at r /D = 0 . 5 .
W e can relate this to the pinc h-off pro cess: when the circulation in the
v ortex ring do es not follo w the one con tained in the sagittal semi-plane, tw o
differen t regions with relativ ely high v orticit y w ere created. The div ergence
of these b oth curv es determines the ph ysical separation of the v ortex ring.
This div ergence can b e seen in the differen t cases in figure 6.4 as w ell as
in figure 6.8. In order to predict the pinc h-off, a relationship b etw een the
dimensionless time ( t ∗ ) and the Reynolds n um b er for whic h the pinc h-off
tak es place (Γ / ( U j D )) pinch − off w as established, equation (6.1). Since all
pinc h-off p oin ts are o v er a straigh t line in figure 6.8, an empirical linear
relationship w as indicated as
 Γ
U j D  pinc h − off
= 5 . 0071 − 0 . 3467( t ∗ ) . (6.1)
0 10 20 30 40 50
0
0.5
1
1.5
2
2.5
3
3.5

t ∗
Γ / ( U j D )

Figure 6.8: V ortex ring circulation ov er the non-dimensional time. The pinc h-
off is sho wn b y the p oin ts ( • ) in whic h the circulation of the sagittal semi-plane
and in the v ortex ring div erged. A linear regression ( ) has b een tak en in to
consideration as the relationship that predicts the pinc h-off. Lines as in table 2.3.
90

Results of jet sim ulations and exp erimen ts
6.3 Kelvin-Benjamin v ariational principle
In the case of constan t h ydro dynamic impulse, the Kelvin-Benjamin
v ariational principle states that a steady distribution of v orticit y (relativ e
to a mo ving reference frame) is the state that maximises the total kinetic
energy on an iso-v ortical sheet.
The non-dimensional kinetic energy is defined as α = E /  I 1 / 2 Γ 3 / 2 
where E is the kinetic energy , I is the impulse and Γ is the circulation.
The dimensionless kinetic energy of the starting jet decreases with the
time. In the case of the steady v ortex ring, a constant dimensionless kinetic
energy is exp ected ( α vortex = 0 . 33 ) as generally accepted, Gharib et al.
[1998]. F or the non-dimensional time for whic h the dimensionless kinetic
energy of the starting jet reac hes the constan t limiting v alue for the steady
v ortex ring, the pinc h-off tak es place and the v ortex ring separates from
the rest of the flo w, see figure 6.9.
The limiting v alue α vortex = 0 . 33 is usually deriv ed from the slug mo del,
whic h limits its generalit y to impulsiv ely started jets with a constan t inlet
v elo cit y time distribution and mo derate Reynolds n um b ers, Gao and Y u
[2010]. The cases with lo w C ( C = 1 , 2 ) pinc hed off for α larger than the
limiting v alue of 0 . 33 b ecause the pinch off took place in these cases with
the lip of the nozzle and not with the trailing jet, and therefore they are
not saturated, but the v ortex rings pinc hing off from a trailing jet agreed
v ery w ell with the pinc h-off at α vortex = 0 . 33 . The pinc h-off dimensionless
kinetic energy for the differen t cases in this study is summarised in table
6.1.
T able 6.1: Dimensionless time for whic h the pinc h-off tak es place and dimen-
sionless kinetic energy for the differen t cases under study .
Case ( t ∗ ) pinc h-off α pinch-off
1 .a 6 . 20 0 . 97
2 10 . 06 0 . 69
3 12 . 13 0 . 45
4 12 . 13 0 . 40
5 13 . 60 0 . 34
6 13 . 60 0 . 32
7 .a 11 . 83 0 . 37
In this c hapter w e defined the pinc h-off in a quan titativ e w a y using
a v orticit y threshold as a criterion. The pinc h-off tak es place when b oth
regions (trailing jet and v ortex ring) are separated for the first time through
a region with a v orticit y lo w er than the sp ecified threshold: ω po /ω v or tex =
0 . 1 .
91

Chapter 6. Pinc h-off definition
10 0 10 1 10 2
10 −1
10 0
10 1

t ∗
α
4 6 8 10 12 14
0.4
0.6
0.8
1
1.2

Figure 6.9: Time evolution of the dimensionless kinetic energy for all cases
under study . See the legend in table 2.3. The pinch-off is represen ted b y • . The
horizon tal dashed blac k line corresp onds to the dimensionless kinetic energy of
the steady v ortex ( α = 0 . 33 ).
Through the circulation div ergence la w w e sho w ed that despite injecting
more v orticit y through the nozzle, from a certain p oin t on, the v ortex ring
did not absorb an y v orticit y more. W e called this the saturation of the
v ortex ring. W e related this to the pinch-off process: when the circulation
in the v ortex ring did not follo w the one con tained in the sagittal semi-
plane, tw o different regions of relativ ely high v orticit y w ere created and this
corresp onds to the pinc h-off. Since all pinch-off points are o v er a straigh t
line in figure 6.8 w e dev elop ed an empirical linear relationship to predict
the pinc h-off.
W e applied the Kelvin-Benjamin v ariational principle to predict the
pinc h-off and w e confirmed the v alue of the non-dimensional kinetic energy
of α = 0 . 33 for the starting jet without deca y , but we found that the pinc h-
off tak es place for a higher v alue of al pha for the cases in whic h the v ortex
ring w as not saturated (a trailing jet w as not formed).
92

Chapter 7
In teraction b et w een the
v ortex ring and the trailing
jet
P art of this c hapter has b een tak en from [ J.J. P eña F ernández and
Sesterhenn, 2017a]. Its fluid dynamics part w as presen ted at the Euro-
p ean T urbulence Conference 15 [ J.J. P eña F ernández and Sesterhenn,
2015c] and the acoustic part w as presen ted at the D A GA 2015, [ J.J. Peña
F ernández and Sesterhenn, 2015a].
In b oth the starting and con tin uous stage of the jet there is an in terac-
tion b et w een the v ortices and the sho c k w a v es of the trailing jet, but b oth
of these in teractions tak e place in a sligh tly differen t w a y . In the starting
stage, the vortex ring in teracts with the sho ck w a v es and the v ortices of the
shear la y er, while in the contin uous stage, the sho c k w a v es in teract with the
v ortices of the shear la y er, but not with the v ortex ring. Both in teractions
generate t w o of the three loudest noise source of the compressible starting
jet.
7.1 Sho c k–shear la y er–v ortex in teraction
When the impulsiv ely started jet is only few nozzle diameters long,
the first sho c k w a v e extends un til the v ortex ring core, see figure 7.1 a .
The v ortices from the shear la y er are con v ected to reac h the sho c k w a v e,
corresp onding to the figure 7.1 b . In this p oin t tak es place the sho ck–she ar
layer–vortex in teraction and the sho c k w a v e b ends due to the v orticit y
of the shear la y er.
93

Chapter 7. In teraction b et w een the v ortex ring and the trailing jet
( a ) Straigh t sho c k ( b ) Curv ed sho ck
( c ) Curv ed and reflected sho c k ( d ) Radiated sho ck
( e ) Op en sho c k
Figure 7.1: Stages of the sho c k cell structure during its in teraction with the
v ortex ring. The numerical sc hlieren |∇ ρ | is plotted in the range [0 , 2] . Case 7 .c
is represen ted for ( a ) t ∗ = 5 . 1 , ( b ) t ∗ = 5 . 6 , ( c ) t ∗ = 5 . 9 , ( d ) t ∗ = 6 . 3 and ( e )
t ∗ = 6 . 8 .
94

Results of jet simulations and exp erimen ts
When the first v ortices of the shear la y er pass through the sho c k w a v e,
the v elo cit y gradien ts of the shear la y er b end the affected segmen t of the
sho c k w a v e more and more, and the sho c k w a v e starts to b e reflected in the
shear la y er, as shown in figure 7.1 c .
F or later times, the segmen t of the sho c k w a v e b et w een the shear la yer
and the v ortex ring is rad iated as a strong acoustic w a ve in to the su rround-
ings, figures 7.1 d and 7.1 e .
Figure 7.2: Lo w er half: time-resolv ed colour schlieren image from Kleine et al.
[2010]. Up pe r half: n umerical schlieren corresp onding to case 7 .c .
Figure 7.2 compares vi sually an exp erimen tal schlieren image from Kleine
et al. [2010], with a n umerical ima ge from case 7 .b . The differen t structures
and the sho ck–she ar layer–vortex in teraction ca n b e iden tified at first sigh t
in b oth images. The agreemen t is reasonable. The segment of the sho c k
w a v e extended from the s hear la y er to the v ortex ring is b ent in the same
w a y in b oth cases just b efore the sound radiation. Due to the re solution
of the exp erimental image, the details of the leading v ortex ri ng cannot b e
compared. The re is also a region ahead of the v ortex r ing in whic h the
exp erimen t sho ws some v orticit y con ten t not repro duced in the n umerical
sim ulation. The rest of the vi sible structures are iden tical.
Figure 7.3 sho ws the ev olution of the flo w field during the pro cess. The
in teraction has alrea dy tak en place in the shear la y er and star ts b eing propa-
gated in the surround ings, figure 7.3 a . A t the same time, the front pressure
w a v e arriv es at the numerical prob e (  ) at 5 D . Figure 7.3 b shows the
en trance of the effects o f the in teraction in the region where t he pressure
p erturbations are pl otted. The fron t pressure wa v e has already reac hed the
n umerical prob e. F urther in the pro cess, the fron t pressure w a v e exits the
computational domai n and the sound radiated b y the in teract ion reac hes
3 . 5 D , figure 7.3 c . I n this plot is clear that b efore the sho ck-she ar layer-
95

Chapter 7. In teraction b et w een the v ortex ring and the trailing jet
vortex in teraction there is no noise, as opp osed to after it. This in teraction
states the start of the broadband sho c k asso ciated noise in the starting
jet. Figure 7.3 d sho ws the arriv al of the sound radiated b y the in teraction
to the n umerical prob e. The flow dev elops further un til the sound exits
the computational domain, figure 7.3 e . Still, the broadband sho c k noise is
propagated un til the n umerical prob e, figure 7.3 f .
The time series recorded b y the n umerical prob e (denoted with F in
figure 7.3) during the in teraction is plotted in figure 7.4 a . The iden tification
of the arriv al of the fron t pressure w a v e is straigh tforw ard and the dashed
line indicates the arriv al time of the effects of the in teraction.
In order to sho w that this in teraction generated a noise at least as
loud as the loudest noise sources of the con tin uous jet for the same pa-
rameters, we placed a numerical probe at 5 D from the jet axis. Pressure
fluctuations of appro ximately 600 Pa w ere recorded, figure 7.4 a . T aking
the w a v elet transformation (complex Morlet w a v elet) of the recorded signal
w e analysed its frequency con ten t as a function of time, see figure 7.4 c .
The corresp ondence b et w een the p eak of the fron t pressure w a v e with the
maxima of the co efficien ts for t ∗ ≈ 6 . 67 is straightforw ard and the p eak
corresp onding to the in teraction is also easy to iden tify . The dashed line
in figures 7.4 a and 7.4 c represents the time in whic h the noise generated
b y the in teraction arriv es at the n umerical prob e. The sp ectrum obtained
through the w a v elet transformation for the time of the in teraction is pre-
sen ted in figures 7.4 d and 7.4 b (red), sho wing a maxim um of 118 [dB]. The
same jet during the con tin uous stage generates an SPL sp ectrum sho wn
in figure 7.4 b , sho wing v alues around 100 [dB] for the TMN or the BBSN.
The noise pro duced during the starting of the jet b y the in teraction of the
v ortex ring with the trailing jet generates at least similar sound pressure
amplitudes in the far field to the TMN and BBSN during the con tin uous
stage for the same parameters. The con tin uous case is the w orst scenario to
compare: for starting jets with a finite mass supply , the sound radiated b y
the sho c k-shear la y er-v ortex in teraction is almost as loud as for the infinite
reserv oir (due to the small decrease in the Mac h n um b er), while the TMN
and BBSN are significan tly reduced. If no trailing jet is formed, neither the
sho c k-shear la y er v ortex in teraction tak es place, nor TMN and BBSN are
radiated.
The directivit y of this w a v e has the same pattern as the broadband
sho c k noise, as exp ected due to the similarities in the generation pro cess.
Since this in teraction tak es place b et w een the first Kelvin-Helmholtz
v ortex and the first sho c k, no acoustic asso ciated to sup ersonic flow (BBSN
and screec h) can b e radiated b efore this in teraction. This interaction is the
onset of the acoustics asso ciated with sup ersonic flo w.
96

Results of jet sim ulations and exp erimen ts
( a ) Radiation after the in teraction.
F ront pressure w a v e arriv es at the
n umerical prob e.
( b ) En trance of the effects of the
in teraction in the sub-domain of
in terest.
( c ) Before the in teraction there is no
noise, whereas after the in teraction
starts the broadband sho c k noise.
( d ) Sound radiated b y the in teraction
arriv es at the n umerical prob e.
( e ) The sound radiated b y the
in teraction exits the computational
domain.
( f ) Broadband sho c k noise arriv es to
the n umerical prob e after the
in teraction.
Figure 7.3: Sk etch sho wing the acoustic radiated b y the interaction and mea-
sured b y the microphone at r /D = 5 . Numerical Sc hlieren in blac k and white
con tours in the range [0 , 2] . ( p − p ∞ ) /p ∞ plotted in blue-to-red colour scale in
the range [ ± 0 . 024] . The red line represen ts the acoustic w a v e resulting due to the
sho ck-she ar layer-vortex in teraction. Results from case 7 .c .
97

Chapter 7. In teraction b et w een the v ortex ring and the trailing jet
( a ) ( b )
6 9 12 15 18 21
−0.01
0
0.01
0.02
0.03

t ∗
( p − p ∞ ) / p ∞

10 −2 10 −1 10 0
40
60
80
100
120
St
SP L[ dB ]

( c ) ( d )
30 60 90 120
10 −1
10 0
10 1
10 2

St
SP L[ dB]

Figure 7.4: The noise pro duced b y the in teraction b et ween the v ortex ring and
the trailing jet is at least as loud as the loudest noise source in the con tin uous jet
in terms of sound pressure lev el. ( a ) Pressure history measured at r /D = 5 . The
in teraction is mark ed with a grey dashed line. ( b ) Comparison b et w een the SPL
sp ectrum for the sho c k-shear la y er-v ortex in teraction and for the con tin uous jet
(case 7 .a ), all normalised at 100 D assuming a radial deca y . The shock-shear la y er-
v ortex in teraction is represen ted b y the solid red line. Case 7 .a : prob e lo cated at
θ = 90 ◦ represen ted b y the solid blac k line and the prob e lo cated at θ = 165 ◦ by
the solid grey line. ( c ) W av elet coefficients of the pressure time signal recorded
(sho wn in ( a )). ( d ) Sound pressure lev el sp ectrum for the time of the in teraction
based on the w a v elet analysis.
As sho wn in figure 7.5, the result of man ually trac king the in teraction
w a v e, the in teraction w a v e starts with a subsonic v elo cit y , but after t ∗ ≈ 1
the v elo cit y of the w a v e is appro ximately the sp eed of sound in the un-
b ounded c ham b er where the w a v e propagates.
98

Results of jet sim ulations and exp erimen ts
0 1 2 3 4
0
1
2
3

t ∗ − t ∗
0
r /D

Figure 7.5: Ev olution of the p osition of the in teraction w a v e non-dimensional
radius ( r /D ). As a comparison, the sp eed of sound has b een plotted as a reference
with the grey dashed line. t ∗
0 is an arbitrary start time from whic h the man ual
trac king w as carried out, shortly after the time for which the in teraction tak es
place so that w e can iden tify the w av e.
7.2 Sho c k–shear la y er in teraction
The same in teraction, but in a slightly differen t wa y tak es place when
the v ortices of the shear la y er pass b y ev ery sho c k w a v e. This is the sho ck–
she ar layer inter action .
In a laminar sup ersonic jet, the surface in whic h the sho c k w a v es are
reflected is smo oth and steady . How ever, if the jet is turbulen t, this surface
is only statistically steady and it has man y v ortices. F urthermore, the
v ortices of the shear la y er ha v e a high v orticit y , comparable or even larger
than the maxim um v orticit y in the v ortex ring, that can affect lo cally the
b eha viour of the sho c k w a v e while b eing reflected.
When the sho c k w a v e is reflected in a v ery irregular and time c hanging
surface, the region of the sho c k w a v e close to the shear la y er is b en t due to
the high v orticit y of the shear la y er, and this irregular and time dep ending
in teraction mak es the w a v es not only to reflect but to refract, due to the
c hanging angle to the surface, whic h causes the w a v e to b e radiated in an
unsteady fashion as strong acoustic w a v es in to the surroundings.
Figure 7.6 a sho ws the consequences of the in teraction b et w een the shear
la y er and the sho c k-cell structure: concen tric acoustic w a v es coming from
the in tersection of the sho c k w a v e with the shear la y er, whic h is one of the
strongest acoustic sources in the sup ersonic starting jet. This interaction
is not restricted to the first sho c k cell. Figure 7.6 b sho ws the effect of the
t w o first sho c k-cells.
99

Chapter 7. In teraction b et w een the v ortex ring and the trailing jet
( a ) ( b )
Figure 7.6: The acoustic wa v es radiated when a v ortex passes b y a sho c k w a v e.
( a ) Densit y gradien ts plotted in a logarithmic blac k and white colour scale sho wing
the direction of radiation of the acoustic w a v es generated by the in teraction of the
shear la y er and the sho c k-cell structure. Detailed view. ( b ) Pressure p erturbations
are plotted in the outer region of the domain. 0 < | p − p ∞ | /p ∞ < 0 . 03 . In
the shear la y er region, the vorticit y is plotted in order to iden tify the Kelvin-
Helmholtz instabilities. 0 < ω /ω vortex < 2 . In the core of the jet, densit y gradien ts
are plotted in order to iden tify the p osition of the sho c k w av es. 0 < |∇ ρ | < 20 .
As result from this in teraction, strong acoustic w a v es are radiated ev ery
time that a v ortex from the shear la y er in teracts with one of the sho c k w a v es
presen t in the flo w field. This noise source is also known as broadband shock
noise (BBSN).
7.3 Effect of sho c k-w a v es dynamics on BBSN
Oscillations of the sho c k w a v es in the trailing jet ha v e b een rep orted in
the literature for con tin uous jets, Panda [1998]. The geometry of the sho c k-
cell system is giv en b y p 0 r /p ∞ . In suc h an unsteady problem, the pressure
ratio c hanges drastically with time, and therefore also the geometry of the
sho c k-cell system.
Due to the impulsiv e start, p 0 r /p ∞ changes from zero to the maxim um
v alue m uc h faster than the con v ection v elo cit y of the flo w p erturbations.
The sho c k-cell pattern is created almost with its final geometry . The last
part of the rising stage is no longer impulsiv ely and due to the small and
slo w increase in p 0 r /p ∞ the lo cation of the sho c k w a v es c hanges slightly to
the final lo cation, see figure 7.7.
100

Results of jet sim ulations and exp erimen ts
During the pressure deca y , the lo w er v alues of p 0 r /p ∞ cause the sho c k
w a v es tra v el bac kw ards un til the lip of the nozzle, figure 7.7 c . Since the
in teraction b et w een the sho c k w a v es and the v ortices of the shear la y er pla y
a v ery imp ortan t role in the jet noise, the dynamics of the sho c k w a v es can
b e exp ected to alter the jet noise.
The temp oral ev olution of the pressure p erturbations along the jet axis
is sho wn in figure 7.7 for four represen tativ e cases under study . All these
cases ha v e in common Re D = 5 000 , p 0 r /p ∞ = 3 . 6 and T 0 r /T ∞ = 1 ; they
only differ in the v alue of L/D . Case 3 has a v alue of L/D = 3 . 80 , case
5 L/D = 13 . 55 , case 6 L/D = 28 . 59 and case 7 .a L/D → ∞ . F or t ∗ = 0
the whole jet axis is at rest. The oblique line starting in the origin and
gro wing to the righ t-hand side of the plot is the trace of the v ortex ring.
After the v ortex ring, and close to the lip of the nozzle, the sho c k w a v es
are created only in the cases 5 , 6 and 7 .a . The sho c k w a v es are v ery easy
to iden tify b ecause of the "zebra" pattern in whic h a com bination of a
white and a blac k strip e corresp onds to one sho c k w a v e, due to the p ositive
and negativ e pressure relativ e to the am bien t pressure ( p − p ∞ ). In these
x/D − t ∗ diagrams, the v elo cities can b e v ery easily computed b y calculating
the in v erse of the slop e of the traces. In figure 7.7 a , the trace of the first
pressure w a v e can b e iden tified as an oblique line starting from the origin
with a flatter slop e than the v ortex ring, ha ving a reference of the sp eed of
sound at the temp erature in the un b ounded c ham b er.
Comparing the slop e of the traces of the last sho c k w a v es during the
deca y in figure 7.7 c with the slop e of the first pressure w a v e in figure 7.7 a ,
w e can infer that the last sho c ks are mo ving with a Mac h n um b er close to
unit y .
The BBSN is generated b y the coheren t scattering of the large turbulen t
structures as they pass through the quasi-p erio dic sho c k-cells, T am and
T anna [1982]. Due to the induced v elo cit y of the sho c k w a v es during the
deca y stage, it is exp ected to ha v e a Doppler effect in the sho c k asso ciated
noise. Moreov er, the sho c k-spacing is exp ected to b e reduced, which leads
to a shift in frequencies, including the BBSN p eak frequency .
In this c hapter w e fo cused on the in teraction b et w een the differen t el-
emen ts of the jet flo w in b oth the con tin uous and the starting jet. W e
iden tified the sho ck–she ar layer–vortex in teraction, whic h tak es place in the
starting stage and the sho ck–she ar layer in teraction, whic h tak es place in
the stage where the jet can b e considered as con tin uous. Both in terac-
tions generate t w o of the three loudest noise sources of the compressible
starting jet. Moreo v er, we found that during the deca y stage, with decreas-
ing pressure at the nozzle exit, the sho c k-cell spacing b ecomes shorter and
the asso ciated frequency of the radiated broadband sho c k noise b ecomes
higher. In addition to that, the sho c k w a v es tra v el to w ards the nozzle exit
101

Chapter 7. In teraction b et w een the v ortex ring and the trailing jet
at a Mac h n um b er close to one and it is exp ected to ha v e a Doppler effect
in the sho c k asso ciated noise.
x/ D
t ∗
0 5 10 15 20
0
20
40
60
80
100

x/ D
t ∗
0 5 10 15 20
0
20
40
60
80
100

( a ) Case 3 . p 0 ∈ ± 10000 P a ( b ) Case 5 . p 0 ∈ ± 25000 P a
x/ D
t ∗
0 5 10 15 20
0
20
40
60
80
100

x/ D
t ∗
0 5 10 15 20
0
20
40
60
80
100

( c ) Case 6 . p 0 ∈ ± 37000 P a ( d ) Case 7 .a . p 0 ∈ ± 50000 P a
Figure 7.7: Characteristic diagrams sho wing the dev elopmen t of the first acous-
tic w a v e, vortex ring and shock w a v es for differen t inlet conditions. Sho wn is the
pressure p erturbation ( p 0 ) history along the jet axis ov er time.
102

Chapter 8
A coustics of the starting jet
This c hapter fo cuses on the acoustics of the compressible starting jet.
The three classical noise comp onen ts (see figure 8.1): turbulent mixing
noise, broadband sho c k noise and screec h are already w ell kno wn in the
comm unit y , but they are not the only noise sources in the starting jet. The
noise due to the compression w a v e and the v ortex ring are also describ ed
and analysed in the follo wing sections.
0.01 0.1 1
30
50
70
90
110

St
S P L ( d B )

Figure 8.1: T ypical sound pressure lev el sp ectra for a con tin uous sup ersonic
jet. Exp erimen tal data w ere measured with the following set-up: p 0 r /p ∞ = 2 . 75 ,
T 0 r /T ∞ = 1 , Re D ≈ 2 . 88 × 10 5 , L/D → ∞ .
103

Chapter 8. A coustics of the starting jet
8.1 Compression w a v e
This compression w a v e is generated as a result of the sudden expansion
of an impulsiv ely started jet. The wa ve tra v els at the speed of sound or
faster dep ending on the pressure ratio as explained in section 1.3.3. F or
the set of parameters used in this study , we analysed only pressure w a v es
tra v elling at the sp eed of sound.
By placing a n umerical prob e in the jet axis at a distance of 5 D from
the nozzle w e assessed the pressure profile through the fron t compression
w a v es for the differen t v alues of L/D . The pressure w a v e w as similar for the
differen t cases, figure 8.2 a . By scaling the time axis with the normalisation
factor ( t ∗ ) ( L/D ) 0 . 0061 and the pressure axis with p − p ∞
p ∞ ( t ∗ ) ( L/D ) − 0 . 1186 , the
differen t profiles collapsed in to a single one, figure 8.2 b .
1 2 3 4 5
−0.005
0
0.005
0.01
0.015
0.02
0.025

t ∗
( p − p ∞ ) /p ∞

1 2 3 4 5
−0.005
0
0.005
0.01
0.015
0.02
0.025

( t ∗ ) ( L/ D ) 0 . 0 0 6 1
( p − p ∞ ) / (( t ∗ ) ( L
D ) 0 . 1 1 8 6
p ∞ )

( a ) Original axis ( b ) Similarit y axis
Figure 8.2: Pressure profile through the first wa v e for the cases with differen t
L/D v alues in the original ( a ) and similarit y axis ( b ). See legend in table 2.3.
The time axis origin has b een displaced ( t ∗
0 ) to sho w a con v enien t dimensionless
time range.
Before the pressure w a v e met the n umerical prob e (for t ∗ ∼ 1 ), w e
recorded small but increasing fluctuations. The amplitude of this rising
fluctuations w as not affected b y L/D . After the small pressure fluctuations,
the strong pressure w a v e passed b y the n umerical prob e. The amplitude
of the strong pressure w a v e did not dep end on the v alue of L/D . The
signal after the pressure w a v e did not deca y to the surroundings pressure,
whic h means that the pressure w a v e did not tra v el alone. These p ertur-
bations w ere presumably originated b y the shear la y er v ortices that w ere
leapfrogged b y the main v ortex ring. During the deca y after the strong
pressure w a v e the qualitativ e b eha viour b et w een the curv es with differen t
L/D w as differen t, the cases with smaller L/D tended to deca y faster than
the cases with larger L/D , b ecause the v ortex ring tra v els slo w er in the
104

Results of jet sim ulations and exp erimen ts
cases with lo w er L/D .
The cases with large L/D w ere successfully collapsed in to a single curve.
The cases with L/D = 0 . 4473 and 1 . 1730 w ere the most difficult to scale,
due to the lo w Mac h n um b er relativ e to the others. In these t w o cases
there w as no trailing jet and therefore there w as no signal of the leapfrogged
v ortices from the shear la y er.
8.2 V ortex ring
Here w e ha v e to distinguish b et w een laminar and turbulen t v ortex rings:
L aminar vortex ring : The generated laminar v ortex rings did not
radiate an y acoustic during the whole sim ulation.
T urbulent vortex ring : Ran and Colonius [2009] measured the acous-
tic field around a circle x/R = 16 . 1 and they found that the sound pressure
lev el of the v ortex ring p eaks when the instabilities b ecome non-linear and
the v ortex ring starts to breakdo wn. The pressure signals ha v e a distinct
frequency at this stage. As the v ortex ring transitions and v orticit y b egins
to deca y , b oth the amplitude and the frequency of the pressure distribution
decrease as w ell.
10 −2 10 −1 10 0 10 1
20
40
60
80
100

St
SP L [ dB]

Figure 8.3: SPL sp ectrum of the acoustic radiated b y the v ortex ring in a
reference frame mo ving with the v ortex ring. The dot-dashed light grey line
corresp onds to the acoustic radiated at θ = 90 ◦ with resp ect to the jet axis, the
dashed grey line corresp onds to θ = 135 ◦ and the solid black line to θ = 157 . 5 ◦ .
This results corresp ond to case 1 .a .
105

Chapter 8. A coustics of the starting jet
W e measured the noise radiated b y a turbulen t v ortex ring at different
angles with resp ect to the jet axis and the sound pressure lev el sp ectrum is
sho wn in figure 8.3. The noise is rather lo w-frequen t ( S t ∼ 0 . 1 − 1 ) with a
steep deca y for high frequencies.
8.3 T urbulen t mixing noise
The turbulen t mixing noise (TMN) is the comp onen t of jet noise in
whic h the turbulence of the jet is in v olv ed. This is the dominan t noise
comp onen t in the do wnstream direction of the jet. Both the large coheren t
structures and the fine scale turbulence con tribute to the turbulen t mixing
noise. The terms ’fine scale’ and ’large coherent structures’ turbulence noise
are v ery imprecise and are not w ell defined in the literature.
• Large turbulence noise. Discussed in detail in [T am and Burton,
1994]. Mac h w a v es are generated when the large turbulence structures
propagate do wnstream at a sup ersonic sp eed relativ e to the am bien t
condition. This is the dominan t part of the turbulen t mixing noise
for sup ersonic jets, esp ecially for high sup ersonic. In sup ersonic jets,
it is also called Mach wave r adiation . See figure 8.4.
• Fine-scale turbulence noise. Although it is clear that turbulence
w ould lead to flo w unsteadiness and hence sound generation, y et the
exact pro cess b y whic h fine-scale turbulence pro duces noise remains
not w ell understo o d. This part is resp onsible for the bac kground
noise. This part dominates the turbulent mixing noise for subsonic
jets.
T urbulen t mixing noise app ears in b oth the subsonic and sup ersonic
con tin uous jet cases. Since the in teraction of turbulence tak es place in
b oth regimes, b oth subsonic and sup ersonic jets radiate turbulent mixing
noise, b eing the dominan t comp onen t not the same in b oth cases.
The t ypical radiation diagram of the turbulen t mixing noise has a broad
p eak at Strouhal n um b ers of the order 10 − 1 . In the upstream direction,
the fine scale turbulence noise dominates the sp ectrum and therefore it is
broader than in the do wnstream direction, where the large turbulence noise
dominates.
TMN can b e measured in subsonic jets at all angles. The large-scale
turbulen t noise is dominan t in the forw ard arc, while the fine-scale tur-
bulen t noise in the rear one. F or sup ersonic jets, the sup ersonic jet noise
comp onen ts are dominan t in the rear arc, so that the fine-scale turbulent
noise is more difficult to measure, but in the forw ard arc, the large-scale
turbulen t noise is still dominan t.
106

Results of jet simulations and exp erimen ts
Figure 8.4: Large sca le turbulence noise w a ves visible in a sc hlieren picture of an
under-expanded jet. p 0 r /p ∞ =4 . 1 , Re D ≈ 2 . 88 × 10 5 , T 0 r /T ∞ =1 , L/D →∞ .
8.4 Broadband s ho c k noise
Broadband sho c k noise app ears only in imp erfectly expanded sup ersonic
jets. A quasi-p erio dic sho ck cell structure is built because of the mismatch
of the static pressures inside and outside the jet. The broadband sho c k
asso ciated noise (B BSN) is generated b y the in teractio n b et w een the tur-
bulence structures of the shear la y er and the sho c k w a v es of the sho c k cell
structure. On the one hand, th e sho c k w a v es are quasi stationar y in a con-
tin uous jet and they do not translate. On the other hand, the turbulent
structures of the shear la y er are con v ected do wnstream with the jet flo w.
When the turbulen t scales pass through the p osition where the sho c k w a v e
is lo cated, a h igh amplitude p erturbation is gener ated and radiated in to
the surroundings.
An alternativ e p oint of view of the BBSN generation mec hanism is
through the M =1 line in the shear la y er. As sho wn in figure 8.5 rep-
resen ted in red, the M =1 line is lo cated in the shear lay e r in a sup ersonic
jet. With the ev olution of the K elvin-Helmholtz v ortices, the M =1 line
b ends b ecause of the vorticit y con tained in the v ortices . When the v ortices
of the shear la y er are conv ected along the b oundary of the jet core an d
they tra v el through t he lo cation of a sho c k w a v e, the M =1 line encloses a
small sup ersonic re gion in the subsonic am bient. This p erturbatio n settled
in the subsonic part of the flui d flo w is propagated in to the surrounding s
107

Chapter 8. A coustics of the starting jet
pro ducing the BBSN.
Figure 8.5: Numerical schlieren |∇ ρ | ∈ [0 , 4] represen ted in blac k and white
(white for lo w v alues and blac k for large v alues) ov er a plane con taining the jet
axis. The M = 1 con tour line is represen ted in red. Case 7 .c is shown.
The large range of scales of the turbulence and the randomness of the
in teraction b et w een shear la y er v ortex and sho c k w a v es giv es the BBSN its
broadband prop ert y .
The sho c k cell spacing is the appropriate length scale that determines
the broadband noise p eak, as rep orted b y Norum and Seiner [1982]:
L s
D = 1 . 1 q M 2
j − 1 .
The amplitude of the BBSN in the noise sp ectrum rises rapidly with the
Strouhal n um b er to a w ell-defined p eak and then decreases exp onen tially
at higher Strouhal n um b ers. The Strouhal n um b er of this p eak sho ws a
Doppler shift with the direction of radiation and therefore the p eak Strouhal
n um b er is referenced at 90 ◦ .
The directivit y is found to b e p oin ted in the upstream direction, with
omnidirectionalit y b eing only approac hed at high-pressure ratios. The rel-
ativ e imp ortance of sho c k noise with resp ect to jet mixing noise is found to
b e maxim um near the pressure ratios at whic h a Mac h disk b egins to form
in the jet [Seiner and Y u, 1984]. The relativ e imp ortance of sho c k noise in
comparison with jet noise for a con v ergen t nozzle generally increases with
increasing pressure ratio and increasing angle from the do wnstream jet axis
[T anna, 1977].
108

Results of jet sim ulations and exp erimen ts
8.5 Screec h
Screec h tones are the least understo o d comp onen t of sup ersonic jet
noise. It is generated through a feedbac k lo op: the instabilities from the
shear la y er are detac hed in to acoustic w a v es; th ey tra v el in all directions,
but esp ecially upstream outside the jet to the nozzle lip; there, they trigger
new instabilities that are con v ected do wnstream in the shear la y er to the
place they are detac hed, closing the lo op with a sp ecific frequency . The
screec h tone sho ws a mo de switc hing feature with the fully expanded jet
Mac h n um b er. F rom the three comp onen ts of the feedbac k lo op (sho c k
w a v es, acoustic w a v es and instabilities in the shear la y er of the jet), the
mo de shap e and tone in tensit y are c haracterised b y the prop erties of the
jet instabilities. They are the energy source of the feedbac k lo op.
The screec h frequency can b e predicted with certain accuracy (see equa-
tion (8.1)), but the amplitude of the screec h tones v aries largely without
c hanging the go v erning parameters. Norum [1983] rep orted an increase in
the screec h tone amplitude of 10 dB when a thin lip nozzle was replaced b y
one with a thic k er lip. Moreo v er, Seiner et al. [1986] rep orted a rotation of
almost 90 ◦ in the flapping plane when the same exp erimen t w as rep eated
a mon th later at the same facilit y b y the same in v estigators. F urther w ork
is needed to fully understand the noise generation mec hanisms and the
sensitivit y of this feedbac k lo op.
Both the in tensit y and the frequency of screec h tones generally decreases
with an increase in the jet temp erature:
f s D j
u j
= 0 . 67
q M 2
j − 1 
 1 + 0 . 7 M j
q 1 + γ − 1
2 M 2
j  T 0 r
T ∞  − 1
2 

− 1
(8.1)
Note here the assumption that the con v ectiv e Mac h n um b er is 0 . 7 times the
fully expanded Mac h n um b er. There is not unanimit y ab out the con v ectiv e
Mac h n um b er in the literature, but this form ulation deliv ers go o d enough
results for the purp oses of this thesis.
The fundamen tal screec h tone radiates primarily in the upstream direc-
tion. The principal direction of radiation of the first harmonic is 90 ◦ . The
tone directivit y is not affected b y the temp erature.
8.6 Directivit y of the noise radiated
The directivit y of the differen t jet noise comp onen ts is w ell kno wn in
the comm unit y for the case of the con tin uous jet, see figure 8.6. The TMN
is radiated mainly in the do wnstream direction while the BBSN is radiated
mainly in the transv erse-upstream direction. In the case of the existence of
109

Chapter 8. A coustics of the s tarting jet
Figure 8.6: Dir ectivities of the different components of the jet noise.
screec h tones, they ar e also radiated in the transverse-upstream direction.
W e presen t here the differen t noise sources of the starting jet with their
directivit y in order to a nalyse the angular dep endence of the SP L sp ectra
for the starting jet.
The fron t pressure wa v e is radiated in all directions as a spherica l w a v e,
starting at the lip of th e nozzle and propagating with the sp e ed of sound.
The sp ectrum of the first pressure w a v e w ould b e flat, analogous to a Gauss
pulse.
The noise pro duced b y the v ortex ring is mainly radiated do wnstream,
see figure 8.3. The sp ectrum corresp onding to the point do wnstream has a
larger amplitude for th e complete range of frequencies than fo r the p oin t
upstream.
In the starting jets with a tra iling jet, TMN is ra diated in the do wn-
stream direction. BBSN is on ly radiated when sho c ks are present in the
flo w field, this is, i n the time in terv al when the trailing jet is sup ersonic. In
this case, the BB SN is radiated, as in the contin uous jet, in the transv erse-
upstream direction. The noise radiated b y the sho c k-shear la y er-v ortex
in teraction is also ra diated in the transv erse-upstrea m direction.
W e analysed the SPL sp ectrum in t w o directions ( θ = 165 ◦ and 90 ◦ ) in
order to iden tify the acoustic fo otprin t of the differen t ph ysical phenomena.
Figure 8.7 a shows the stream-wise acoustic radiat ion of the v ortex ring in
the case 1 .a , and the tur bulen t mixing noise in the cases 5 and 7 .a . Except
for the lo w Strouhal num b ers, where there is less in formation, the sp ectra
are qualitatively similar. Figure 8 .7 b shows the acoustic radiation in the
transv erse directio n ( θ = 90 ◦ ). Case 1 .a is qualitativ ely very differen t from
b oth cases 5 and 7 .a . F rom the c urv e of case 7 .a , w e iden tified the frequency
range whic h corresp onds to the BBSN, and w e iden tified as well that the
110

Results of jet sim ulations and exp erimen ts
curv e corresp onding to case 5 has the highest frequencies of the BBSN, but
not the lo w est. The sup ersonic time interv al in case 5 w as not long enough
to radiate the lo w est frequencies of the BBSN.
( a ) ( b )
10 −2 10 −1 10 0 10 1
20
40
60
80
100
St
S P L [ d B ]

10 −2 10 −1 10 0 10 1
20
40
60
80
100

St
S P L [ d B ]

Figure 8.7: SPL sp ectra radiated ( a ) downstream ( θ = 165 ◦ ) and ( b ) in the
transv erse direction ( θ = 90 ◦ ). Only cases 1 .a , 5 and 7 .a are represented.
W e presen t here exp erimen tal results that agree and complemen t the
previous results. Figure 8.8 sho ws the sound pressure lev el sp ectrum of four
exp erimen tal con tin uous jets measured at t w o differen t angles: θ = 90 ◦ and
160 ◦ . All these exp erimen ts w ere p erformed in the anec hoic c ham b er at the
Berlin Institute of T ec hnology (see section 3.2).
The first jet (figure 8.8 a ) w as generated with a pressure ratio of p 0 r /p ∞ =
1 . 8 therefore it is transonic ( M j = 0 . 9562 ). W e see in the sp ectrum at 160 ◦
(in red) the turbulen t mixing noise due to large turbulen t structures with
a maxim um at appro ximately S t = 0 . 1 and in the sp ectrum at 90 ◦ (black)
a broader maxim um in whic h the highest frequencies are not visible b e-
cause of the cut-off frequency of the used microphones ( 20 000 Hz). The
sup ersonic jet noise comp onen ts are not dominan t in this case.
The second jet (figure 8.8 b ) was generated with a pressure ratio of
p 0 r /p ∞ = 2 . 6 , whic h leads to M j = 1 . 2528 . The fundamen tal screec h tone
dominates the sp ectrum with a sharp p eak at appro ximately S t = 0 . 36 .
F or lo w er Strouhal n um b ers in the sp ectrum at 160 ◦ , the large turbulent
scales noise is sho wn. In the sp ectrum measured at 90 ◦ we also see the
screec h tone for the same Strouhal n um b er. F or lo w er Strouhal n um b ers,
the fine-scale turbulence noise comp onen t of the turbulen t mixing noise is
presen t, but for higher Strouhal n um b ers the broadband sho c k noise is not
visible; it is exp ected for higher Strouhal n um b ers than the microphones
w ere able to measure.
The third jet (figure 8.8 c ) was generated with a pressure ratio of p 0 r /p ∞ =
3 . 5 , whic h leads to M j = 1 . 4669 . The fundamen tal screec h tone dominates
the sp ectrum with a sharp p eak at appro ximately S t = 0 . 28 . Concerning
111

Chapter 8. A coustics of the starting jet
0.01 0.02 0.05 0.1 0.2 0.5
40
55
70
85
100
115

0.01 0.02 0.05 0.1 0.2 0.5
40
55
70
85
100
115

( a ) ( b )
0.01 0.02 0.05 0.1 0.2 0.5
40
55
70
85
100
115

0.01 0.02 0.05 0.1 0.2 0.5
40
55
70
85
100
115

( c ) ( d )
Figure 8.8: ( a ) p 0 r /p ∞ = 1 . 8 . ( b ) p 0 r /p ∞ = 2 . 6 . ( c ) p 0 r /p ∞ = 3 . 5 . ( d )
p 0 r /p ∞ = 4 . 1 . The noise sp ectrum is represen ted at θ = 90 ◦ (blac k) and at
θ = 160 ◦ (red). Exp erimental data obtained in the laboratory .
the sp ectrum at 160 ◦ , there is a frequency shift in the differen t noise com-
p onen ts due to the differen t pressure ratio and the amplitude of the large
turbulen t scales noise has increased up to 10 dB. In the sp ectrum at 90 ◦ , the
part of the BBSN with the lo w est Strouhal n um b ers w as measured b y the
microphones and is visible. This sho ws ho w dominan t is the BBSN against
the fine-scale turbulen t noise for this configuration.
The fourth jet (figure 8.8 d ) was generated with a pressure ratio of
p 0 r /p ∞ = 4 . 1 , whic h leads to M j = 1 . 5756 . By some reason, the amplitude
of the screec h tone has b een reduced noticeably , although it is visible in the
sp ectrum at 90 ◦ at appro ximately S t = 0 . 21 . The sp ectrum at 160 ◦ do es
not sho w ma jor differences with resp ect to the previous case. The sp ec-
trum at 90 ◦ sho ws for this configuration the Strouhal n um b er for whic h the
BBSN has the maxim um amplitude, showing as in the previous case ho w
BBSN dominates o v er fine-scale turbulen t noise for this configuration.
112

Results of jet sim ulations and exp erimen ts
8.7 F requency-time domain (w a v elet) analysis
P art of the w ork corresp onding to this section w as presen ted in the
20 th In ternational Congress on Sound and Vibration (ICSV20), [ J.J. P eña
F ernández and Sesterhenn, 2013b].
In this section, we report on the frequency–time lo calisation of the noise
sources in the con tin uous and starting–deca ying jet. Cases 1 .a , 3 , 4 , 5
and 7 .a ha v e b een analysed in whic h the non-dimensional mass supply w as
c hanged, ha ving the case 1 .a the low est v alue and the case 7 .a the largest.
With n umerical prob es, w e recorded the acoustics at three differen t
angles: ( i ) 0 ◦ , at the jet axis at a distance of x/D = 7 . ( ii ) 45 ◦ , at a
distance of 7 D from the nozzle exit cen tre and 45 ◦ with resp ect to the jet
axis. ( iii ) 90 ◦ , at the sideline and at a distance of 7 D with resp ect to the
jet axis.
These signals ha v e b een transformed in to the F ourier space and rep-
resen ted as the sound pressure lev el sp ectra, which is the classical result
sho wn in the jet noise literature. This sho ws the frequency con ten t of the
acoustic signal but there is no time lo calisation, and therefore we performed
as w ell a w a v elet analysis of the radiated acoustics to ha v e a b etter kno wl-
edge ab out the time dep endency .
Starting with the starting con tin uous jet (case 7 .a ) and decreasing the
non-dimensional mass supply v alue w e sho w the obtained results.
0 20 40 60 80 100
−0.01
−0.005
0
0.005
0.01

t ∗
( p − p ∞ ) / ( p r − p ∞ )

10 −2 10 −1 10 0 10 1
40
60
80
100
120
140
160

St
SP L (d B)

( a ) ( b )
Figure 8.9: ( a ) Time series and ( b ) sound pressure lev el sp ectrum of case 7 .a .
The signal w as recorded at 0 ◦ (dot-dashed), 45 ◦ (dashed) and 90 ◦ (solid).In ( a )
the curv e of 0 ◦ (dot-dashed) is scaled by a factor of 0 . 04 to b etter visualization.
Figure 8.9 a sho ws the pressure history measured at the three stations
for the case 7 .a . The quiescence conditions can b e seen in all three curv es
un til t ∗ ∼ 7 . The first large p eak is the trace of the compression w a v e,
and it is follo w ed b y also v ery large oscillations in the prob e at 0 ◦ and
smaller oscillations for the other t w o prob es. The signal do es not deca y ,
113

Chapter 8. A coustics of the starting jet
since w e regard the case with an infinite reserv oir. Figure 8.9 b sho ws the
asso ciated sound pressure lev el sp ectra. A t the jet axis, w e measured only
TMN with v ery large amplitudes. In b oth prob es at 45 ◦ and 90 ◦ w e can
iden tify TMN in the lo w-frequency range, while the BBSN is clear in the
range S r ∼ 0 . 3 − 4 . T w o sharp p eaks are visible for S r ∼ 0 . 5 and 1 , which
corresp ond to the screec h tone and one harmonic.
( a )
( b )
( c )
Figure 8.10: Con tin uous w a v elet transformation co efficien ts in logarithmic
colour scale for case 7 .a . The colour-scale is in the range [60 , 130] [dB]. ( a )
Microphone lo cated at 0 ◦ . ( b ) Signal at 45 ◦ . ( c ) Signal at 90 ◦ .
114

Results of jet sim ulations and exp erimen ts
Figure 8.10 sho ws the w a v elet co efficien ts of the transformed signal mea-
sured at the three stations. A t 0 ◦ the low frequen t TMN dominates after
the first compression w a v e as previously stated. Here w e can observ e ho w
irregular with time is the b eha viour in the high frequencies. A t 45 ◦ , the
first compression w a v e can b e seen follo w ed mainly b y BBSN in the range
S r ∼ 0 . 2 − 1 . A t 90 ◦ the amplitudes are lo w er than at 45 ◦ , but the distri-
bution do es not differ m uc h from the previous plot.
In the follo wing, w e presen t the same results for the cases with a finite
reserv oir, where the jets deca y with time un til quiescence conditions.
Figure 8.11 a presen ts the pressure time series of case 5 . After the large
oscillation due to the first compression w a v e, there are oscillations of lo w er
amplitude generated b y the trailing jet and at some p oin t the oscillations
deca y . In figure 8.11 b is the sound pressure lev el represen ted. A t 0 ◦ only
TMN can b e iden tified, while at 45 ◦ and 90 ◦ also BBSN can b e seen, at
least its largest frequencies. In these plots, no screec h tones w ere observ ed.
0 20 40 60 80 100
−5
−4
−3
−2
−1
0
1
2
3
x 10 −3

t ∗
( p − p ∞ ) / ( p r − p ∞ )

10 −3 10 −2 10 −1 10 0 10 1
20
40
60
80
100
120
140

St
SP L (d B)

( a ) ( b )
Figure 8.11: ( a ) Time series and ( b ) sound pressure lev el of case 5 . The signal
w as recorded at 0 ◦ (dot-dashed), 45 ◦ (dashed) and 90 ◦ (solid). In ( a ) the curv e
of 0 ◦ (dot-dashed) is scaled b y a factor of 0 . 04 to b etter visualization.
Figure 8.12 sho ws the w a v elet co efficien ts of the transformed signal of
case 5 . A t 0 ◦ , w e can recognise the first compression w a v e follo w ed b y
TMN, where the amplitude at lo w frequencies starts to deca y . A t 45 ◦ , after
the compression w a v e, there is a time span un til t ∗ ∼ 90 , where also BBSN
is presen t and then the jet deca ys presumably in the subsonic regime and
the amplitudes deca y as w ell. At 0 ◦ this time in terv al is ev en more clear in
whic h the BBSN is emitted, namely b et w een t ∗ ∼ 30 − 90 .
W e presen t no w the results of case 4 in figure 8.13 for a smaller non-
dimensional mass supply . The pressure time series are qualitatively v ery
similar to the previous case with the only exception to deca y faster. After
the first compression w a v e, we see for 0 ◦ large p erturbations due to pressure
115

Chapter 8. A coustics of the starting jet
( a )
( b )
( c )
Figure 8.12: Con tin uous w a v elet transformation co efficien ts in logarithmic
colour scale for case 5 . The colour-scale is in the range [60 , 130] [dB]. ( a ) Mi-
crophone lo cated at 0 ◦ . ( b ) Signal at 45 ◦ . ( c ) Signal at 90 ◦ .
c hanges at the v ortex ring cen tre. F or 45 ◦ and 90 ◦ , w e can see mainly the
compression w a v e. In the sound pressure lev el sp ectra, w e observ e for the
three directions the first compression w a v e and TMN. Also few of the largest
BBSN frequencies are presen t in the 90 ◦ curve (appro ximately at S r ∼ 0 . 5 ).
Figure 8.14 a sho ws after the first compression w a v e lo w er frequen t con-
116

Results of jet sim ulations and exp erimen ts
0 20 40 60 80 100
−5
−4
−3
−2
−1
0
1
2
3
x 10 −3

t ∗
( p − p ∞ ) / ( p r − p ∞ )

10 −3 10 −2 10 −1 10 0 10 1
20
40
60
80
100
120
140

St
SP L (d B)

( a ) ( b )
Figure 8.13: ( a ) Time series and ( b ) sound pressure level sp ectrum of case 4 .
The signal w as recorded at 0 ◦ (dot-dashed), 45 ◦ (dashed) and 90 ◦ (solid). In ( a )
the curv e of 0 ◦ (dot-dashed) is scaled by a factor of 0 . 04 to b etter visualization.
ten t with time un til t ∗ ∼ 50 and then there is a c hange and higher frequen-
cies are also excited un til t ∗ ∼ 90 . After the first compression w a v e there
is some TMN and BBSN, but then the jet deca ys and these comp onen ts
are no longer presen t. A t t ∗ ∼ 70 − 80 , the noise from the v ortex ring
breakdo wn is radiated and therefore presen t in this plot. A t 45 ◦ and 90 ◦ ,
the three comp onen ts are ev en clearer: the first compression w a v e for early
times, for t ∗ ∼ 30 − 60 we can observ e TMN and BBSN and the v ortex
breakdo wn for t ∗ ∼ 80 .
With lo w er v alues of the non-dimensional mass supply , we presen t here
the results of case 3 . Figure 8.15 a deca ys faster than in the previous case,
as exp ected. F or 0 ◦ b oth the first compression w a v e and the trace of the
v ortex ring are visible, while for 45 ◦ and 90 ◦ only the first pressure wa v e
is observ able. When lo oking at the sound pressure level spectra, w e can
observ e the t ypical sp ectra asso ciated with the first compression w a v e and
for 90 ◦ the pressure p erturbations asso ciated to the v ortex ring. A t 90 ◦
there migh t b e some con ten t asso ciated to TMN at S r ∼ 0 . 8 .
Figure 8.16 sho ws the w a v elet co efficien ts of the transformed signal. A t
0 ◦ , after the compression w a v e, w e observ e lo w frequen t con ten t asso ciated
with the pressure oscillations of the v ortex ring. F or later times w e ob-
serv e lo w-frequency con ten t of m uc h smaller amplitude whic h origin is the
induced flo w after the v ortex ring. A t 45 ◦ and 90 ◦ , we can only see the first
compression w a v e and the p erturbations due to the v ortex ring.
Note here that the p erturbations of the v ortex ring arriv e at the micro-
phone lo cated at 90 ◦ later than at the 45 ◦ one due to their lo cation: all
microphones are lo cated at the same distance from the nozzle exit cen tre
line, but not from the p oin t in whic h the v ortex ring radiates its noise.
117

Chapter 8. A coustics of the starting jet
( a )
( b )
( c )
Figure 8.14: Con tin uous w a v elet transformation co efficien ts in logarithmic
colour scale for case 4 . The colour-scale is in the range [60 , 130] [dB]. ( a ) Mi-
crophone lo cated at 0 ◦ . ( b ) Signal at 45 ◦ . ( c ) Signal at 90 ◦ .
F or the case with the lo w est non-dimensional mass supply in this anal-
ysis (case 1 .a ), we can observ e in figure 8.17 a mainly the compression w a v e
arriving at the three prob es sim ultaneously . A relatively w eak v ortex ring
arriv es at the microphone lo cated at 0 ◦ . Figure 8.17 b shows the associ-
ated sound pressure lev el sp ectra, where only the compression w a ve can be
118

Results of jet sim ulations and exp erimen ts
0 20 40 60 80 100
−5
−4
−3
−2
−1
0
1
2
3
x 10 −3

t ∗
( p − p ∞ ) / ( p r − p ∞ )

10 −3 10 −2 10 −1 10 0 10 1
20
40
60
80
100
120
140

St
SP L (d B)

( a ) ( b )
Figure 8.15: ( a ) Time series and ( b ) sound pressure level sp ectrum of case 3 .
The signal w as recorded at 0 ◦ (dot-dashed), 45 ◦ (dashed) and 90 ◦ (solid). In ( a )
the curv e of 0 ◦ (dot-dashed) is scaled by a factor of 0 . 1 to b etter visualization.
iden tified.
Figure 8.18 sho ws the w a v elet co efficien ts of the transformed signal of
case 1 .a . It is v ery similar for the three angles, sho wing mainly the first
compression w a v e and some trace of the v ortex ring for sligh tly later times.
As a general trend w e ha v e seen in this section ho w starting from a
jet without deca y , with an infinite reservoir, w e can iden tify the differen t
noise comp onen ts b y analysing the time series, the sound pressure lev el
sp ectrum and the w a v elet co efficien ts of the transformed signal. As we
sim ulate smaller and smaller reserv oirs, the sup ersonic stage of the jet is
reduced in time and the BBSN loses imp ortance. The same happ ens to the
TMN as the parameters approac h the limit of trailing jet existence. F or
v ery small reserv oirs, only a compression wa ve and a relativ ely w eak v ortex
ring are presen t in the fluid flo w. This link b et w een the acoustics radiated
and the differen t features of the starting-deca ying jet allo ws us to mo v e
to the next c hapter and predict the go v erning parameters from acoustic
measuremen ts.
119

Chapter 8. A coustics of the starting jet
( a )
( b )
( c )
Figure 8.16: Con tin uous w a v elet transformation co efficien ts in logarithmic
colour scale for case 3 . The colour-scale is in the range [60 , 130] [dB]. ( a ) Mi-
crophone lo cated at 0 ◦ . ( b ) Signal at 45 ◦ . ( c ) Signal at 90 ◦ .
120

Results of jet sim ulations and exp erimen ts
0 20 40 60 80 100
−0.5
0
1
2
3
3.5
x 10 −3

t ∗
( p − p ∞ ) / ( p r − p ∞ )

10 −2 10 −1 10 0 10 1
20
40
60
80
100
120

St
SP L (d B)

( a ) ( b )
Figure 8.17: ( a ) Time series and ( b ) sound pressure lev el sp ectrum of case 1 .a .
The signal w as recorded at 0 ◦ (dot-dashed), 45 ◦ (dashed) and 90 ◦ (solid).
121

Chapter 8. A coustics of the starting jet
( a )
( b )
( c )
Figure 8.18: Con tin uous w a v elet transformation co efficien ts in logarithmic
colour scale for case 1 .a . The colour-scale is in the range [60 , 130] [dB]. ( a )
Microphone lo cated at 0 ◦ . ( b ) Signal at 45 ◦ . ( c ) Signal at 90 ◦ .
122

Chapter 9
Go v erning parameters
prediction from acoustic
and optical measuremen ts
The w ork corresp onding to this section w as presen ted at the In terna-
tional Congress on Theoretical and Computational A coustics (ICTCA),
2015 [ J.J. P eña F ernández and Sesterhenn, 2015d].
Once the b eha viour of the system is clear and w e matc hed the v alues
of the go v erning parameters with the ph ysical phenomena, by linking the
acoustic radiation of ev ery ph ysical phenomenon w e estimated the go v erning
parameters from acoustic parameters.
9.1 Reynolds n um b er
As already describ ed in section 8.3, turbulen t mixing noise is the only
jet noise comp onen t where turbulence is exclusiv ely in v olv ed. Since the
Reynolds n um b er is used to c haracterise turbulence in flo ws, it makes sense
to try to predict the Reynolds n um b er b y analysing the turbulen t mixing
noise.
F rom the t w o comp onen ts of turbulen t mixing noise as prop osed by
T am et al. [1996], the fine-scale similarit y (FSS) sp ectrum sho ws a larger
dep endence on the Reynolds n um b er, see Bailly and Bogey [2006]. The
large-scale similarit y (LSS) sp ectrum is almost indep enden t of the Reynolds
n um b er.
It has b een rep eatedly rep orted that the p eak frequency of the fine-scale
similarit y (FSS) sp ectrum is closely dep enden t on the Reynolds n um b er,
[Bailly and Bogey, 2006; Bogey and Bailly, 2004; Long and Arndt, 1984;
123

Chapter 9. Go v erning parameters prediction from acoustic and optical
measuremen ts
10 3 10 4 10 5 10 6
0.2
0.3
0.4
0.5
0.6
0.7
R e D
S t T MN

Figure 9.1: In the sideline direction, θ ≃ 90 ◦ , the v ariation with the Reynolds
n um b er of the p eak Strouhal n um b er of the FSS sp ectrum is sho wn for jets with
a Mac h n um b er of 0 . 9 ( • ) and for a Mach n um b er of 0 . 6 (  ). Data from Bailly
and Bogey [2006]. F corresp ond to the current study , case 7 .b at R e D = 5 000
and case 7 .c at Re D = 10 000 .
Lush, 1971]. This has b een called the ’lo w Reynolds n um b er effect’. The
authors do not agree whether the correct scaling is through the Strouhal
n um b er S t = f D/u j , based on the fully expanded velocity u j or on the
Helmholtz n um b er H = f D /c ∞ , based on the sp eed of sound. Our results
sho w a b etter agreemen t with a scaling based on the Strouhal n um b er.
In figure 9.1 the p eak Strouhal n um b er of the FSS sp ectra are shown for
differen t Reynolds n um b ers. Cases 7 .b − c of this study are also represen ted
for comparison ( F ) and sho w a go o d agreement.
Visw anathan [2004] rep orted exp erimen tally on free jets that the limit
for the so-called ’lo w Reynolds n um b er effect’ is ab out 400 000 . This means
that from this Reynolds n um b er on, no c hanges in the acoustics of the free
jet are exp ected when increasing this parameter.
124

Results of jet sim ulations and exp erimen ts
9.2 Pressure ratio
W e can easily differen tiate from acoustic measuremen ts whether a start-
ing jet is sup ersonic or subsonic. The existence of broadband sho c k noise
(BBSN) means the existence of sho c k-w a v es in the flo w and therefore a su-
p ersonic flo w. The sup ersonic flo w has to b e generated b y a pressure ratio
larger than the critical v alue ( p 0 r /p ∞ ) ∗ = (( γ + 1) / 2) ( γ / ( γ − 1)) . The p eak
Strouhal n um b er ( f p L s /u j ) of BBSN radiated b y the con tin uous jet based
on the sho c k-cell spacing ( L s ) and the eddy conv ection velocity ( u c = 0 . 7 u j )
w as found to b e unit y , see Norum and Seiner [1982]:
1 = f BBSN L s
u c
, (9.1)
W e w an t to relate this fact to the pressure ratio and for this purp ose, w e
include the necessary information to reac h the desired form ulation:
L s = π q M 2
j − 1 D j
σ 1
(9.2a)
D j = D 1 + 1
2 ( γ − 1) M 2
j
1 + 1
2 ( γ − 1) M 2
d ! γ +1
4( γ − 1)  M d
M j  1 / 2
(9.2b)
M j = 2
γ − 1  p 0 r
p ∞  γ − 1
γ
− 1 !! 1 / 2
(9.2c)
c j
c ∞
= s  1 + γ − 1
2 M 2
j  − 1 T r
T ∞
(9.2d)
where σ 1 ≈ 2 . 404826 is the first ro ot of the zero order Bessel function.
And this leads to the direct relationship b et w een the Helmholtz n um b er
of the maxim um amplitude within the BBSN and the pressure ratio.
f BBSN D
c ∞
= σ 1
π q M 2
j − 1
D
D j
c j
c ∞
, (9.3)
means that the p eak Helmholtz n um b er ( f p D /c ∞ ) based on the BBSN
p eak frequency , the diameter of the nozzle ( D ) and the sp eed of sound for
the un b ounded c ham b er ( c ∞ ) decreases by increasing the pressure ratio, see
figure 9.2 a and equation (9.3). In this w a y , through a measured BBSN p eak
Strouhal n um b er, the pressure ratio ( p 0 r /p ∞ ) of the jet can b e estimated.
This approac h can b e only used in the sup ersonic case but, in order to b e
able to estimate the pressure ratio in the subsonic cases, we presen t another
differen t approac h that regards b oth subsonic and sup ersonic jets.
125

Chapter 9. Go v erning parameters prediction from acoustic and optical
measuremen ts
F or pressure ratios lo w er than ( p 0 r /p ∞ ) ∗ , the jet is purely subsonic
and only TMN is emitted. In this case, w e can analyse how both (LSS
and FSS) sp ectra c hange with the pressure ratio. According to T am et al.
[1996], the p eak sound pressure lev el of the LSS sp ectrum ( SPL LSS ) and
the FSS sp ectrum ( SPL FSS ) c hange with the temp erature ratio reserv oir to
un b ounded c ham b er ( T r /T ∞ ) and the pressure ratio as:
SPL LSS = 35 + 46
 T r
T ∞  0 . 3 + 10 log " M j s T r
T ∞  1 + γ − 1
2 M j  # n LS S
,
(9.4a)
M j = 2
γ − 1  p 0 r
p ∞  γ − 1
γ
− 1 !! 1
2
, (9.4b)
n LS S =10 . 06 − 0 . 495 T r
T ∞
, (9.4c)
and
SPL FSS =43 . 2 + 19 . 3
 T r
T ∞  0 . 62 + 10 log " M j s T r
T ∞  1 + γ − 1
2 M j  # n F S S
,
(9.5a)
M j = 2
γ − 1  p 0 r
p ∞  γ − 1
γ
− 1 !! 1
2
, (9.5b)
n F S S =6 . 4 + 1 . 2
 T r
T ∞  1 . 4 , (9.5c)
resp ectiv ely . This metho d is v alid for b oth subsonic and sup ersonic cases.
Figure 9.2 b sho ws exp erimen tal and analytical data and the go o d agreemen t
with the curren t results.
Another p ossibilit y is to use imaging tec hniques. W e can measure the
relativ e length of the first sho c k-cell to the nozzle diameter ( L s /D ) and use
the existing correlations in the literature to relate it to the fully expanded
Mac h n um b er and therefore to the pressure ratio. W e use the correlation
of Sc h ulze [2011], tak en from T am et al. [1985]:
L s
D = π
σ 1 q M 2
j − 1 D j
D , (9.6a)
D j
D = 1 + 1
2 ( γ − 1) M 2
j
1 + 1
2 ( γ − 1) M 2
d
, ! γ +1
4( γ − 1)
(9.6b)
126

Results of jet sim ulations and exp erimen ts
2 4 6 10 20 40 60 100
0.1
0.2
0.4
0.6
1
2
4
6
10

p 0 r /p ∞
H
B BSN

1.2 1.6 2 2.5 3 3.6 4.5
40
60
80
100
120
p 0 r /p ∞
S P L [ d B ]

( a ) ( b )
Figure 9.2: ( a ) V ariation of the BBSN p eak Helmholtz n um b er with p 0 r /p ∞
for A ∗ / A = 1 . The dashed grey line corresp ond to ( p 0 r /p ∞ ) ∗ for γ = 1 . 4 . 
Norum and Seiner [1982]. ( b ) V ariation of the p eak sound pressure level with the
pressure ratio p 0 r /p ∞ . Data from T am et al. [1996] corresp onding for T r /T ∞ = 1 .
The large-scale similarit y sp ectra w as represen ted b y ( ◦ ) and w as measured at
θ = 160 ◦ . The fine-scale similarit y sp ectra w as represented b y ( M ) and was
measured at θ = 90 ◦ . Cases 7 .b, e are represented b y F : case 7 .b at p 0 r /p ∞ = 3 . 6
and case 7 .e at p 0 r /p ∞ = 4 . 35 .
where σ 1 ≈ 2 . 404826 is the first ro ot of the zero order Bessel function, M d
is the design Mac h n um b er (dep enden t mainly on the critical to exit area
ratio of the nozzle).
Figure 9.3: Sc hlieren picture of an under-expanded jet from a con v ergent noz-
zle ( A ∗ / A e = 1 ). p 0 r /p ∞ = 3 . 6 , T 0 r /T ∞ = 1 , L/D → ∞ , Re D ≈ 2 . 52 · 10 5 .
Exp osure time relativ e to c haracteristic time t exp / ( D /u j ) = 0 . 104 .
Figure 9.3 sho ws a sc hlieren picture where the ratio length of the first
127

Chapter 9. Go v erning parameters prediction from acoustic and optical
measuremen ts
sho c k cell to the nozzle exit is L s /D ≈ 224 / 145 . Us ing the correlations
from the literature w e get a Mac h n um b er of M j = 1 . 4833 and a pressure
ratio of p 0 r /p ∞ = 3 . 5836 . The exp erimen t w as p erformed with a pressure
ratio of p 0 r /p ∞ = 3 . 6 , whic h sho ws a v ery go o d agreemen t.
F or pressure ratios larger than ( p 0 r /p ∞ ) blast , a blas t w a v e is generated.
Blast w a v es propagate faster than the sp eed of sound and faster with larger
pressure ratios, see section 4.1. With the kno wledge of the distance b et w een
the microphone and the nozzle and the time b et w een the release of the
pressure and the reception of the blast w a v e at the measuremen t p oin t,
the Mac h n um b er of the blast w a v e can b e computed with the help of the
Rankine-Hugoniot equations and through it the pressure ratio p 0 r /p ∞ .
9.3 Dimensionless mass supply
The BBSN is radiated during the p erio d in whic h the sho c k-w a v es of the
trailing jet are in teracting with the v ortices of the shear la y er. The highest
frequency that BBSN radiates is related to the frequency with whic h t w o
v ortices are in teracting with a sho c k-w a v e. The smallest frequency that
BBSN radiates can b e related to the duration of the sup ersonic stage, as
explained in detail in section 8.6. F rom the con tin uous jet, w e kno w that
the lo w est frequency of the BBSN is not the duration of the whole analysed
record, so there exist a minim um frequency that the BBSN radiates. If
the duration of the sup ersonic stage in the starting jet under analysis is
long enough to radiate the absolute minim um BBSN frequency , there is no
p ossibilit y with this metho d to estimate the non-dimensional mass supply .
W e exp ect that the maxim um v alue of the non-dimensional mass supply
that w e can estimate with this metho d is appro ximately L/D ∼ 50 .
F or subsonic jets, the previous metho d cannot estimate the dimension-
less mass supply and therefore w e use an alternativ e metho d. As shown
in section 5.4, the fron t pressure w a v e sho ws similarit y , whic h can b e also
used to estimate the non-dimensional mass supply . The normalisation v ari-
ables con tain the constan t C , whic h means that collapsing a giv en pressure
profile to the data here presen ted should lead to the parameter C . With
the help of the estimation of the pressure ratio and assuming a temp oral
distribution of the inlet condition lik e equation (2.10), the equation (D.7)
should giv e a go o d estimation of the non-dimensional mass supply .
9.4 T emp erature ratio
The effects of T 0 r /T ∞ on jet noise are still not clear for the comm unit y .
Often it can b e read in exp erimen tal studies that the temp erature effect on
jet noise is due to c hanges in the v elo cit y when k eeping constan t the pressure
ratio and therefore the Mac h n um b er, see T anna [1977], whic h is clearly a
128

Results of jet sim ulations and exp erimen ts
coupled effect with the Reynolds n um b er. Visw anathan [2004] rep orted
that exp erimen ting with hot jets, the changes found in the literature in
the sound radiated are not t ypically due to temp erature c hanges, but to
c hanges in the Reynolds n um b er, b ecause the exp erimen ts w ere done with
the same nozzle and pressure ratio. They found a similar sp ectral shap e at
aft angles for lo w subsonic jets and highly heated sup ersonic jets. Ho w ev er,
they rep ort as a general trend that the effects of heating are a broadening
of the angular sector for p eak radiation and a broadening of the sp ectral
p eak for aft angles.
In this w ork, w e fo cused on jets with T 0 r /T ∞ = 1 .
Indeed, future w ork needs to b e done to clarify the effects of the tem-
p erature ratio on the jet noise. Due to the coupling b et w een the Reynolds
n um b er with the temp erature ratio and the pressure ratio of the jet, see
equation (9.7) 1 , to find a suitable exp erimental set-up can be complex.
Therefore w e think further n umerical w ork should b e made in the near
future in this direction.
Re D ,j = ρ j u j D
µ j
= ρ ∞ c ∞ D
µ ∞ v
u
u
t 2
γ − 1  p 0 r
p ∞  γ − 1
γ
− 1 !
 p 0 r
p ∞  2 γ
γ − 1  p ∞
p 0 r  γ
γ − 1 T 0 r
T ∞ + T S
T ∞
1 + T S
T ∞  T ∞
T 0 r  2
(9.7)
In this c hapter w e w ere able to predict the v alues of the go v erning
parameters b y linking them to acoustic prop erties.
Starting with the Reynolds n um b er, there is enough evidences in the lit-
erature of the relationship b et w een the p eak frequency of the fine-scale sim-
ilarit y of the turbulen t mixing noise with the Reynolds n um b er, but the au-
thors do not agree in a scaling with the Strouhal or Helmholtz n um b er. Our
results sho w a b etter agreemen t with the Strouhal n um b er. [Visw anathan,
2004] rep orted exp erimen tally the absence of c hanges in the radiated acous-
tics for free jets from a Reynolds n um b er of 400 000 on.
F or sup ersonic jets, Norum and Seiner [1982] found that the p eak Strouhal
n um b er of BBSN based on the sho c k-cell spacing and the eddy con v ection
v elo cit y w as one. W e applied this result to predict the pressure ratio and
the results agreed with data from the literature. F or subsonic jets w e used
the mo del of T am et al. [1996] in order to relate the sound pressure lev el
of b oth fine and large scale comp onen ts of the turbulen t mixing noise with
1 Assuming the classical Sutherland’s la w for the viscosit y with an exp onen t of 2 / 3
and a Sutherland temp erature of T S .
129

Chapter 9. Go v erning parameters prediction from acoustic and optical
measuremen ts
the pressure ratio and the curren t results sho w go o d agreemen t with the
rep orted ones. W e used also imaging tec hniques to predict the pressure
ratio of free jets based on sc hlieren photograph y .
F or sup ersonic jets with a v ery lo w non-dimensional mass supply , we
related the lo w est frequency of the BBSN comp onen t radiated to the du-
ration of the sup ersonic stage of the jet and this to the non-dimensional
mass supply to predict this last parameter. W e exp ect to b e able to predict
accurately L/D for jets under L/D = 50 with Mac h n um b ers of order unit y .
The effects of the temp erature ratio on jet noise are still not clear for
the comm unit y . Visw anathan [2004] rep orted that the c hanges found in
the sound radiated are not due to temp erature c hanges, but to c hanges in
the Reynolds n um b er, b ecause the exp erimen ts in the literature ha v e b een
p erformed with the same nozzle and pressure ratio. Indeed, future w ork is
needed to clarify the effects of the temp erature ratio on the jet noise.
130

Chapter 10
V olcanic jets
The ejection of gas and p yro clasts from v olcano es has t ypically b een
called plume . In a rigorous fluid mec hanics con text, a plume is dominated
b y buo y ancy . In most of the volcanic eruptions, the first stage is dom-
inated b y momen tum, but it deca ys and leads to a stage dominated b y
buo y ancy . This latter stage migh t not b e present when the ash concen tra-
tion is large, leading to the collapse of the jet forming a p yro clasts densit y
curren t. Therefore w e use the term volc anic jets .
Unsteadiness, compressibilit y , turbulence, chemical reactions and m ul-
tiphase pro cesses mak e v olcanic jets a difficult researc h topic. An y of the
t ypical approac hes (analytical, exp erimental and n umerical) are forced to
mak e strong assumptions to face this problem.
Ho w ev er, individual asp ects of the volcanic jet dynamics can be ab-
stracted and scaled for exp erimen ts. V olcanic jets are commonly generated
in the lab oratory using a sho c k-tub e and the main fo cus of the exp erimen-
tal w ork has b een the link b et w een the observ ed b eha viour in nature with
the go v erning parameters of the pro cess. Kieffer and Sturtev an t [1984] used
F reon 12 and 22 among other gases as analog of hea vy and particulate-laden
v olcanic gases. Anilkumar et al. [1993] in v estigated exp erimen tally high-
sp eed t w o-phase v olcanic jets. Recen tly , Cigala et al. [2017] found that
the maxim um v elo cit y of the p yro clasts increases with A e / A ∗ (div ergen t
nozzles) and with the temp erature ratio T r /T ∞ and it decreases with the
reserv oir size ( L/D ) and with the particle size.
W o o ds [1988] deriv ed an analytical mo del for Plinian eruptions, with
separated treatmen t of the momen tum-driv en and buo y ancy-driv en regions;
the mo del w as able to predict general prop erties of the flo w suc h as the max-
im um heigh t as a function of the initial temp erature, relativ e sizes of the
t w o regions (momen tum- and buo y ancy-driv en). The main dra wbac k of
suc h analytical mo dels are the strong assumptions made, t ypically steady
131

Chapter 10. V olcanic jets
flo w, single phase flo w and often incompressible flo w. Nev ertheless, this
mo dels are necessary to isolate some asp ects of the dynamics for under-
standing.
The set of go v erning equations can also b e solv ed n umerically . W e can
relax some of the simplifications usually made b y the theoretical studies.
[V alen tine and W ohletz, 1989] is a v ery go o d example of one of the first
n umerical mo dels used based on this approac h. Curren tly , the fo cus of
the n umerical studies is divided in t w o directions: ( i ) morphology of the
jet in the near-field region and the release of particles in to the atmosphere
dep ending on the go v erning parameters of the eruption, see [Cerminara
et al., 2016] and ( ii ) disp ersion of particles in the atmosphere as a function
of the meteorological conditions, see [Span u et al., 2016].
V olcano es are a p erfect natural example to study the compressible start-
ing jet and due to this a v ery in tense collab oration with the v olcanological
comm unit y has b een tak en place during this pro ject. In the follo wing sec-
tions, w e describ e part of the outcome from the exp erimen tal w ork at the
v olcano es in this pro ject: the lo cation of craters based on an acoustic arra y
and the analysis in the frequency and time domain of v olcanic eruptions.
10.1 Crater lo cation from acoustic measure-
men ts
The w ork corresp onding to this section w as presen ted at the A GU F all
Meeting 2015, [Andronico et al., 2015].
During the field campaign at Moun t Etna 2014 w e measured the acous-
tics of the eruptiv e fissure at the flank of the North-East Crater, see section
3.3.1. W e deplo y ed three microphones at differen t circumferen tial angles
around the craters co v ering an angle of ab out 120 ◦ . The ob jective of this
study w as, kno wing the GPS co ordinates of the three microphones, use the
time of arriv al of ev ery ev en t to the differen t microphones to determine the
exact p osition of the t w o craters and to kno w whic h eruption w as pro duced
b y whic h crater. F urthermore, we w an ted to analyse separately the acous-
tic prop erties of the eruptions of ev ery crater and mak e statistics of the
eruption pattern of the differen t craters.
W e used 3 differen t metho dologies to lo cate the ( 11 758 ) eruptions that
w e recorded during the 2014 campaign:
1. V ectorisation. Assuming a lo cation of the crater and measuring the
dela y of the signal at the differen t microphones, due to the differen t
distance to the crater, from three microphones w e can build t w o v ec-
tors that should in tersect in the crater. W e use an iterativ e metho d
to con v erge to the final solution.
2. Hyp erb olas. The definition of a h yp erb ola is the set of p oints,
132

Results of jet sim ulations and exp erimen ts
suc h that for an y p oin t ( P ) of the set, the absolute difference of
distances to t w o fixed p oin ts (the fo ci F 1 and F 2 ) is constant. | P F 1 | −
| P F 2 | = constant = 2 a . If w e think ab out our ph ysical problem, P
w ould b e the crater and F 1 and F 2 w ould b e tw o microphones. The
constan t 2 a w ould b e the dela y b et w een the t w o signals con v erted
in to a distance with the sp eed of sound, this is, the distance that the
sp eed of sound tra v els within the dela y b et w een the t w o signals. W e
had 3 microphones, so w e can build t w o h yp erb olas and the p oin t in
whic h these h yp erb olas in tersect is the lo cation of the crater.
3. T riangulation. This is a classical m etho d. Assuming that the noise
w a v es are circumferences, w e computed the lo cation of the crater
solving the in tersection of these three circumferences.
Figure 10.1 sho ws the results obtained b y the three metho ds. The p oin t
where w e w ere lo cated is denoted b y ’ st a tion ’ and the three microphones
are denoted with M 1 , M 2 and M 3 . The probabilit y densit y function is
plotted as con tours after lo cating more than 11 000 ev en ts.
The basic information is the same for the three plots, sho wing where is
more probable that the craters w ere lo cated. Some spurious results sho w the
presence of wrong data, mostly due to ec ho es with the flank of the Moun t
Etna itself. Therefore the lo cated as w ell the lo cation of the ec ho es, whic h
can b e esp ecially w ell seen in the v ectorisation and triangulation metho ds.
The t w o red round mark ers are the lo cation of the most in tense eruption
that w e measured, giving a very sharp edge at the measuremen ts and there-
fore v ery accurate results. W e measured the distance in situ b et w een our
station and the summit of the craters with laser-based distometers and w e
obtained 200 m , which is a very close v alue to the ones obtained b y the
acoustic measuremen ts.
W e can conclude that the three metho ds are able to lo cate the craters
from acoustic measuremen ts in a satisfactory manner.
10.2 F requency-time domain (w a v elet and
STFT) analysis
P art of the w ork presen ted in this section w as presen ted in [Stampk a
et al., 2014].
In order to c haracterise the v olcanic eruptions and lo cate the differen t
noise comp onen ts in frequency and time, and therefore ha v e information
ab out the ph ysical pro cesses that led to this noise sources, we performed
a time-frequency analysis of the signals recorded at the v olcano es. W e
133

Chapter 10. V olcanic jets
x (m)
y (m)

0 50 100 150 200 250
−60
−40
−20
0
20
40
60
0.5
1
1.5
2
2.5
3
M1
M3
M2
STATION

x (m)
y (m)

0 50 100 150 200 250
−60
−40
−20
0
20
40
60
0.5
1
1.5
2
2.5
3
3.5
4
M2
M3
M1
STATION

( a ) ( b )
x (m)
y (m)

0 50 100 150 200 250
−60
−40
−20
0
20
40
60
0.5
1
1.5
2
2.5
3
M1
M2
M3
STATION

( c )
Figure 10.1: Crater lo cation results of the eruptions measured during the moun t
Etna campaign in July 2014 . ( a ) V ectorisation metho d. ( b ) Hyp erb olas metho d.
( c ) T riangulation metho d.
p erform b oth main time-frequency analyses: short-time F ourier transform
(STFT) and w a v elet transform.
While the STFT has a uniform resolution in frequency , the w a v elet
transform has an adaptiv e resolution, keeping constan t the relativ e error
in the frequency domain at the exp ense of time resolution. This is as w ell
ho w h uman auditory p erception w orks.
Figure 10.2 sho ws the sp ectrogram of an eruption recorded at Strom b oli
analysed using STFT. The Strouhal n um b er is defined in this case as ( S t =
( f D volcano ) /u exit ), where D volcano = 1 . 5 m is the diameter of the v olcanic
v en t and u exit = 335 m/s is the exit v elo cit y of the gas phase through the
v olcanic v en t. The diameter has b een estimated from drone photographies
and the v elo cit y has b een estimated assuming a sonic v en t for an observ ed
temp erature of the magma with an infrared camera. The Strouhal n um b er
axis is represen ted in logarithmic scale and the fixed resolution can b e
esp ecially w ell seen for lo w frequencies. Figure 10.3 shows the spectrogram
134

Results of jet sim ulations and exp erimen ts
of the same eruption recorded at Strom b oli analysed using the w a v elet
transformation. The frequency axis is represen ted in logarithmic scale and
the adaptiv e frequency resolution can b e seen for lo w frequencies.
In figure 10.2 w e see a w ell-defined Strouhal n um b er, which is the upper
limit of the noise radiated b y this eruption of the v olcano. This corresp onds
to a Strouhal n um b er of ab out 20 . The sound pressure lev el at higher
Strouhal n um b ers is negligible compared to those con tained in the noise
signal. F rom the same figure, w e can infer the duration of the eruption b y
lo oking at the end of the signal where the subsonic turbulen t mixing noise
is no longer radiated.
Figure 10.2: Short time F ourier coefficient con tour of the acoustics recorded at
Strom b oli.
The same signal w as analysed using the w a v elet transform. The corre-
sp onding result is sho wn in figure 10.3. In this figure, w e can iden tify m uc h
b etter the signal con ten t for lo w Strouhal n um b ers. While in figure 10.2,
for Strouhal n um b ers b elo w 0 . 1 the frequency resolution of the STFT is
to o coarse, for the w a v elet metho d (figure 10.3) the signal con ten t can b e
clearly seen. F rom figure 10.3 we can see that for Strouhal n umbers b elow
0 . 1 , the amplitude of the signal decreases. The d ominan t Strouhal n um b ers
are within the range 0 . 1 − 1 .
In this c hapter w e sho w ed three v ery similar metho ds with whic h w e
135

Chapter 10. V olcanic jets
could lo calize the p osition of the craters from acoustic measuremen ts: vec-
torisation, hyperb olas and triangulation. The results w ere quan titativ e v ery
similar to eac h other and o v erall successful although the presence of ec ho es
at the w all of the moun t Etna.
W e analysed the same eruption measured at Strom b oli using the short
time F ourier transform and the w a v elet transform. W e conclude that the
w a v elet metho d is more con v enien t for this application, b ecause it k eeps
constan t the relativ e error in the frequency domain. F or v olcanic applica-
tions the lo w frequen t con ten t is crucial due to the size of the jet flo wing.
Figure 10.3: W a v elet co efficien t con tour of a signal recorded at Strom b oli.
136

Chapter 11
Conclusions and
p ersp ectiv es
Opp osed to the con tin uously blo wn jet and to the incompressible starting-
deca ying jet, the compressible starting-deca ying jet is relativ ely unexplored
and can b e considered a new topic for the scien tific comm unit y . Not only
the scien tific comm unit y will profit from this thesis, but also the industry ,
with applications in extremely differen t fields suc h as pulsejets and pulse
detonation engines in the aerospace industry , air-bag devices or fuel injec-
tion in the automotiv e industry; aquatic lo comotion or the aortic blo o d
flo w in the animal heart in biology; la v a flow and v olcanic jets in geological
sciences and astroph ysical jets in the creation of new stars. This thesis
pro vides the first solid analysis of the compressible starting jet, giving a
detailed description of the c haracterisation of the flo w field together with
the noise sources. The com bination of these t w o asp ects allo ws this w ork
to giv e some insigh ts to the prediction of the go v erning parameters from
acoustic measuremen ts. This analysis has b een made mainly with a n umer-
ical approac h, although exp erimen tal w ork w as also p erformed to supp ort
or complemen t the results where appropriate.
In the first part of the thesis, the n umerical bac kground w as describ ed
with all necessary details. The second part fo cuses on the exp erimental
w ork that supp orts and complemen ts the n umerical results. The third
part is the main b o dy of this thesis. W e examine the fluid flo w and the
acoustics of the compressible starting jet in detail. The most salien t results
are presen ted b elo w.
Concerning the c haracterisation of the compressible jet separated b y
its elemen ts w e can conclude that the compression w a v e propagates at the
sp eed of sound of the un b ounded c ham b er for pressures p 0 r /p ∞ < 41 . 2 and
137

Chapter 11. Conclusions and p ersp ectiv es
it is considered to b e a pressure w a v e. F or pressure ratios larger than this
limit, a blast w a v e is generated and it mov es faster than the sp eed of sound
and faster with increasing pressure ratios. Ab out the vortex ring dynam-
ics, w e can confirm that its axial p osition is a function of ( t ∗ ) 1 / 2 and its
radius v aries as ( t ∗ ) 1 / 3 during a considerable time interv al. Moreov er, it
has b een found that for large pressure ratios the Prandtl-Mey er expansion
mo difies the v ortex ring radius and larger v ortex rings are generated. F ur-
thermore and surprisingly although already rep orted, we confirmed that
compressibilit y mak es the v ortex ring to mo v e slo w er.
Ab out the effects of the turbulence lev el at the inflo w condition, we can
summarise that the jet structure remains although the presence of strong
p erturbations. An increase in the turbulence in tensities at the inlet results
in an increase of the radiated turbulen t mixing noise. A side effect is,
as w ell, a mo dification in the broadband sho c k noise due to a differen t
in teraction b et w een the shear la y er and the sho c k w a v es.
Examining the pinc h-off pro cess, the separation of the v ortex ring from
the trailing jet, this thesis con tributes with a more general and appropriate
definition of the pinc h-off for compressible and turbulen t cases based on
a v orticit y threshold ω po /ω vortex = 0 . 1 . W e prop ose a definition of the
pinc h-off as the creation of a region b et w een the trailing jet and the v ortex
ring with a relativ e v orticit y of ω po /ω vortex = 0 . 1 or lo w er. Examining the
dynamics of the system v ortex ring - trailing jet w e found t w o w a ys in whic h
this system in teracts: the sho ck - she ar layer - vortex in teraction and the
sho ck - she ar layer in teraction. They lead to t w o of the three principal noise
comp onen ts of the starting jet in the sup ersonic case. The former generates
a strong acoustic w a v e few non-dimensional units after the release of the
pressure. The latter is the noise generation mec hanism of the broadband
sho c k noise.
W e also co v ered the acoustics of the starting jet. It has b een found that
only turbulen t v ortex ring radiate noise and they radiate mostly in the
do wnstream direction. Broadband sho c k noise w as found to b e generated
b y the isolation of a sup ersonic flo w region in the subsonic one; this tak es
place esp ecially at the in teraction of the sho c k w a v es and the shear la y er
but it is not exclusiv e.
W e used the kno wledge that the turbulen t mixing noise is closely re-
lated to the turbulence in the jet and therefore to the Reynolds n um b er to
link a prop ert y of the turbulen t mixing noise with the Reynolds n um b er
and b e able to predict the latter from acoustic measuremen ts. W e con-
firm the results of Bailly and Bogey [2006]: the Strouhal n um b er for the
p eak amplitude of the fine-scale comp onen t of the turbulen t mixing noise
increases with the Reynolds n um b er. The same w a y , w e used the result
from Norum and Seiner [1982] that the Strouhal n um b er of the broadband
138

Results of jet sim ulations and exp erimen ts
sho c k noise scales with the sho c k-cell spacing and the con v ectiv e v elo cit y
to establish a dep endency of the Helmholtz n um b er of the p eak amplitude
of the broadband sho c k noise and the pressure ratio. W e also found the
similarit y axes that collapse the pressure profile of the compression w a v e
as a function of the non-dimensional mass supply , which can be used to
predict this parameter.
With an arra y of three microphones distributed around the crater of a
v olcano, w e w ere able to lo cate the crater from the dela y of the arriv al time
b et w een the microphones using three differen t metho ds: vectorisation, h y-
p erb olas and triangulation. The three metho ds w ere successful. Analysing
the noise radiated from v olcanic jets in frequency-time space w e found the
w a v elet transform to b e more appropriate b ecause it k eeps the relativ e er-
ror constan t in the frequency domain and the resolution for lo w frequencies
is m uc h b etter than the short-time F ourier transform metho d. The lo w-
frequency con ten t of v olcanic jet acoustics is crucial for their analysis.
P ersp ectiv es
This thesis has shed ligh t on the fluid flo w prop erties and the acoustics
of the compressible starting jet, but there are still some researc h lines that
require further clarification. The first question is the limiting v alue of the
dimensionless mass supply that leads to a trailing jet for the compressible
case. It has presumably a similar v alue than in the incompressible case,
but is is still to confirm.
W e ha v e seen that for lo w Reynolds n um b ers, the Kelvin-Helmholtz
instabilities of the shear la y er are not fully dev elop ed when reac hing the
first sho c k w a v e, while for high Reynolds n um b ers they do. Because of
this, there is a strong interaction betw een the trailing jet and the v ortex
ring. The second question is the critical Reynolds n um b er for whic h this
in teraction tak es place.
The third and most imp ortan t question still to answ er is the effect of the
jet temp erature ratio on the jet acoustics. F urther w ork in the near future
is needed in this direction and n umerical studies can clarify the c hanges in
the noise generation mec hanisms due to the jet temp erature.
139

Chapter 11. Conclusions and p ersp ectiv es
140

App endices
141

App endix A
Gas dynamics of nozzle
flo ws
A nozzle is a v ery efficien t device to con v ert the in ternal energy of the
flo w in to kinetic energy . The fluid expands and accelerates through the
nozzle isen tropically and hence with high efficiency .
The basic principle of a nozzle is based on ho w c hannel flo ws ev olv e with
a v ariable cross area:
• a subsonic flo w:
– expands and accelerates through a con v ergen t c hannel
– compresses and decelerates through a div ergen t c hannel
• a sup ersonic flo w:
– compresses and decelerates through a con v ergen t c hannel
– expands and accelerates through a div ergen t c hannel
and this is the reason to c ho ose a con v ergen t c hannel to expand the flo w
un til sonic conditions and then a div ergen t c hannel to expand further in to
sup ersonic. F rom the one-dimensional isentropic conserv ation of mass, mo-
men tum and energy (Euler equations), we see that the performance of a
nozzle dep ends mainly on its geometry (exit to critical area ratio 1 ). The
area-Mac h n um b er function can b e deriv ed obtaining:
A
A ∗ =  2
γ + 1  γ +1
2( γ − 1) 1
M  1 +  γ − 1
2 M 2  γ +1
2( γ − 1) (A.1)
1 The critical area A ∗ is the one in whic h through an isentropic process the flow w ould
reac h sonic sp eed ( M = 1 ).
143

Chapter A. Gas dynamics of nozzle flo ws
also plotted for γ = 1 . 4 in figure A.1.
W e can relate through an isen tropic pro cess the pressure and the Mac h
n um b er with:
p
p r
=  1 + γ − 1
2 M 2  − γ
γ − 1
. (A.2)
T o increase the Mac h n um b er in the subsonic flo w ( M < 1 ) in figure
A.1, A ∗ / A has to b e increased. Since A ∗ is fixed, we ha ve to decrease A .
W e find the opp osite b eha viour in the sup ersonic regime: to increase the
Mac h n um b er w e need to decrease A ∗ / A following figure A.1, and since A ∗
is fixed w e need to increase A .
0 1 2 3 4 5 6
0
0.2
0.4
0.6
0.8
1

M
A ∗ / A

Figure A.1: Area - Mach n um b er function for γ = 1 . 4 . See equation (A.1).
The com bination of a con v ergen t subsonic nozzle with a div ergen t su-
p ersonic nozzle is usually called ’ de L aval nozzle’ .
In the div ergen t part of a La v al nozzle, we migh t hav e shock w a v es. The
Mac h n um b er and pressure jump o v er a normal sho c k w a v e is giv en b y:
M +
2 = s ( γ − 1)( M +
e ) 2 + 2
2 γ ( M +
e ) 2 − ( γ − 1) (A.3a)
p +
2
p +
e
= 2 γ ( M +
e ) 2 − ( γ − 1)
γ + 1 . (A.3b)
Minimal example:
Here w e presen t an example with a general geometry (summarised in
table A.1) with whic h w e explain the principles of the La v al nozzle. The
most imp ortan t of these parameters is the exit to critical area ratio A e / A ∗ .
The geometry of the nozzle defines the shap e of the curv es in figure A.2,
but the v alues at the nozzle exit dep end only on the area ratio A e / A ∗ .
Assuming a constan t reserv oir pressure ( p r = const ), we decrease the
pressure in the c ham b er where the nozzle injects the flo w to describ e the
differen t w orking regimes:
144

Results of jet sim ulations and exp erimen ts
T able A.1: Parameters of the nozzle geometry used for the example.
P arameters V alue Description
A 0 / A ∗ 4 Inlet to critical area ratio
∂ D
∂ ( x/L )    0 0 Straigh t inlet
∂ D
∂ ( x/L )    x ∗ /L 0 Straigh t critical section
A e / A ∗ 16 Outlet to critical area ratio
∂ D
∂ ( x/L )    1 0 Straigh t outlet
(a.) p e = p r . Quiescence conditions. This condition is represen ted in
figure A.2 b y the horizon tal blac k dashed line at p/p r = 1 and M = 0 .
(b.) M −
e . The subsonic solution of equation (A.1). It defines the fastest
isen tropic subsonic flo w. Using equation (A.2) w e obtain the nozzle
exit pressure p −
e .
F or this condition, the flo w reac hes M = 1 at the throat for the
first time and the narro w est cross section b ecomes the critical section
A ∗ from this p oin t on. This condition is also called choking , whic h
means that the mass flo w rate through the throat will not increase
b y increasing the exit to reserv oir pressure ratio. This condition is
represen ted in figures A.2 b − c by the solid blac k line.
F or exit pressures b et w een p r and p −
e there is isen tropic subsonic flo w
in the whole nozzle, accelerating from the reserv oir un til the throat,
but not reac hing a sonic sp eed at this section, and then decelerating
from the throat un til the exit b ecause of the div ergen t c hannel in sub-
sonic regime. An example of this condition is represen ted in figures
A.2 b − c b y the solid blue line from the reserv oir to the nozzle exit.
F or exit pressures b elo w p −
e , there is sup ersonic flo w after the critical
section, leading the div ergen t part of the nozzle to a sup ersonic expan-
sion. F or sligh tly lo w er exit pressures than p −
e , a normal sho c k w a v e
is lo cated close to the throat con v erting the flo w in to subsonic and
leading to a subsonic deceleration un til the exit. F or low er exit pres-
sures, the normal sho c k w a v e is lo cated closer to the nozzle exit and
the part of the div ergen t nozzle in sup ersonic flo w is longer, reac hing
larger Mac h n um b ers and leading to stronger sho c k w a v es and larger
total pressure losses. One of these curv es is represented in figures
A.2 b − c b y the com bination of a solid blac k line from the reserv oir
to the critical section, then a red line from the critical section to the
p osition of the sho c k w a v e, the vertical dashed blac k line represen ts
the jump o v er the sho c k w a v e and the blue line as the subsonic com-
pression un til the nozzle exit. The strongest sho c k w a v e is lo cated
145

Chapter A. Gas dynamics of nozzle flo ws
0 0.2 0.4 0.6 0.8 1
2
1
0
1
2

x /L
r/ D ∗

( a )
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1

p −
e
p +
2
p +
e
p ∗
x /L
p / p r

( b )
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4

M +
e
M +
2
M −
e
x /L
M

( c )
Figure A.2: ( a ) Geometry of the La v al nozzle used in this example. ( b ) Pressure
distribution along the nozzle axis. ( c ) Mac h n um b er distribution along the nozzle
axis.
146

Results of jet sim ulations and exp erimen ts
exactly at the nozzle exit and its incoming Mac h n um b er ( M +
e ) can
b e therefore computed from A e / A ∗ .
(c.) M +
e . The sup ersonic solution of equation (A.1). It defines the isen-
tropic sup ersonic flo w. Using equation (A.2) w e obtain the corre-
sp onding nozzle exit pressure p +
e .
F or this condition, the nozzle ejects sup ersonic flo w isen tropically ex-
panded un til the pressure of the surroundings. This is the design con-
dition of the La v al nozzle. This w orking regime is the most efficien t.
This condition is represen ted in figures A.2 b − c b y the com bination
of the solid blac k line from the reserv oir to the critical section and
the solid red line un til the nozzle exit.
F or a pressure at the surrounding sligh tly larger than p +
e , the nozzle
(through its geometry) expanded the flo w ’to o m uc h’ (and therefore
the name over-exp ande d ) and it has to b e compressed bac k to matc h
the surroundings. This compression tak es place through an oblique
sho c k w a v e at the lip of the nozzle, but the flow inside the nozzle
do es not c hange. F or larger surrounding pressures, the angle of the
oblique sho c k w a v e increases, increasing as w ell its strength and the
total pressure losses. The strongest sho ck w a v e that can tak e place is
a normal sho c k lo cated at the lip of the nozzle, b eing the same case
as in the end of the previous p oin t.
F or surrounding pressures b elo w p +
e , the nozzle expanded the flo w
’to o less’ (and therefore the name under-exp ande d ) and it has to
b e expanded further un til reac hing the pressure at the surroundings.
This tak es place through a sup ersonic expansion, this is a Prandtl-
Meyer expansion fan.
Summarising, for a sp ecific v alue of A e / A ∗ in equation (A.1) we get
t w o Mac h n um b ers: M −
e and M +
e , which describe the limits b etw een the
differen t w orking regimes of the La v al nozzle. F or exit Mac h n um b ers b elo w
M −
e the flo w is purely subsonic in the nozzle and for larger ones, there is
a part where there is sup ersonic flo w. M +
e is the isen tropic sup ersonic
exit Mac h n um b er and it defines the limit b et w een the o v er-expanded and
the under-expanded regime. The upp er limit of the ov er-expanded regime
is the Mac h n um b er b ehind a normal sho c k with incoming Mach n umber
M +
e giv en b y equation (A.3a). The low er v alue of the under-expanded flo w
w ould b e to expand in to the v acuum. As it can b e noted in this paragraph,
equation (A.1) pla ys a crucial role in the nozzle flo w dynamics and all the
information ab out its w orking regimes can b e obtained from this equation
and the only parameter A e / A ∗ .
The con v ergen t nozzle is a particular case of all La v al nozzle flo ws
in whic h A e / A ∗ = 1 . This leads to M +
e = M −
e = 1 and p +
e = p −
e =
147

Chapter A. Gas dynamics of nozzle flo ws
(( γ + 1) / 2) γ
( γ − 1) . F or this particular case, b oth solutions collapse and there
is only one limit b et w een the w orking regimes, the isentropic flo w un til
sonic conditions at the nozzle exit. Belo w this exit Mac h n um b er, the flo w
in the whole nozzle is subsonic. Ab o v e this w orking regime, the exit Mac h
n um b er remains at unit y , but the flow is under-expanded and just after the
nozzle exit a Prandtl-Mey er expansion fan tak es place. Note that in the
con v ergen t nozzle, the o v er-expanded flo w cannot tak e place since it w ould
not expand un til sup ersonic flo w, it w ould remain subsonic.
148

App endix B
F ully expanded conditions
As a reference to study jet flo ws w e t ypically use the ful ly exp ande d
c onditions . A compressible flow exiting a nozzle does not alwa ys hav e the
same pressure than in the am bien t. Dep ending on the geometry and the
op erating regime, the pressure at the nozzle exit can b e higher than in the
am bien t (under-expanded) or lo w er (o v er-expanded). When the flo w exits
the nozzle and dev elops further, it reac hes the am bien t pressure at some
p oin t, and w e call this fully expanded conditions, see figure B.1. In this
w a y , w e can merge the study of ov er-expanded and under-expanded nozzle
flo ws taking the same reference in b oth cases. Typically are this conditions
denoted with a subscript ( · ) j .
149

Chapter B. F ully expanded conditions
Figure B.1: F ully expanded conditions in a jet. The Mach n um ber was plotted
in colour and the white lines represen t the am bien t pressure ( p ∞ ). The black star
represen ts the fully expanded conditions.
150

App endix C
Quasi-steady reserv oir
motion
In this c hapter w e describ e the disc harge of a reserv oir b y in tegrating
the con tin uit y equation using a 0 D approach. W e also link the state at the
reserv oir isen tropically to the nozzle exit and the fully expanded conditions
to ha v e a global o v erview of the pro cess.
The con tin uit y equation for the one-dimensional adiabatic motion of a
c harge-disc harge of a reserv oir is giv en b y:
d ( ρ r V )
dt ± ˙ m = 0 (C.1)
where ρ r is the densit y inside the reserv oir, V is the volume of the reserv oir
and ˙ m is the mass-flo w rate through the nozzle. The case with ’ + ’ corre-
sp onds to the disc harge and the case with ’ − ’ corresp onds to the c harge of
the reserv oir.
F rom here on, w e ha v e to differen tiate b et w een t w o differen t cases: sonic
c hok ed nozzle and subsonic nozzle. Of course, the real application will
switc h from one solution to another dep ending on the set-up and the cor-
resp onding parameters.
C.1 Subsonic nozzle
When the nozzle is not c hok ed, the mass-flo w rate is not constan t at the
throat. The mass-flo w rate is not constan t and it dep ends on the pressure
ratio b et w een the reserv oir and the un b ounded c ham b er
˙ m = ρ e u e A throat . (C.2)
151

Chapter C. Quasi-steady reserv oir motion
Relating the nozzle exit v elo cit y with the conditions inside the reserv oir
via the sp eed of sound, w e can write the mass-flo w rate as
˙ m = A throat M e √ γ p r ρ r  1 + γ − 1
2 M 2
e  − γ +1
2( γ − 1)
. (C.3)
Relating the v ariables inside the reserv oir with the stagnation ones w e
get
˙ m = A throat √ γ p 0 r ρ 0 r  ρ r
ρ 0 r  γ +1
2
" 2
γ + 1  p ∞
p 0 r  γ +1
γ  ρ 0 r
ρ r  γ +1 "  p 0 r
p ∞  γ +1
γ  ρ r
ρ 0 r  γ − 1
− 1 ## 1
2
. (C.4)
Plugging the mass-flo w rate in to the con tin uit y equation C.1, w e get
"  ρ r
ρ 0 r  γ − 1
−  p ∞
p 0 r  γ − 1
γ # 1
2
d  ρ r
ρ 0 r  =
= − A throat
V r γ p 0 r
ρ 0 r s 2
γ − 1  p ∞
p 0 r  γ +1
γ  p 0 r
p ∞  γ − 1
2 γ
dt. (C.5)
and in tegrating w e finally get to a relationship for the densit y ratio o v er
time
− 2  ρ r
ρ 0 r  r 1 −  p ∞
p 0 r  γ − 1
γ  ρ r
ρ 0 r  1 − γ
( γ − 3) r  ρ r
ρ 0 r  γ − 1 −  p ∞
p 0 r  γ − 1
γ
2 F 1 1
2 , γ − 3
2( γ − 1) ; 5 − 3 γ
2 − 2 γ ;  p ∞
p 0 r  γ − 1
γ  ρ r
ρ 0 r  1 − γ !
+
2  ρ r
ρ 0 r  initial r 1 −  p ∞
p 0 r  γ − 1
γ  ρ r
ρ 0 r  1 − γ
initial
( γ − 3) r  ρ r
ρ 0 r  γ − 1
initial −  p ∞
p 0 r  γ − 1
γ
2 F 1 1
2 , γ − 3
2( γ − 1) ; 5 − 3 γ
2 − 2 γ ;  p ∞
p 0 r  γ − 1
γ  ρ r
ρ 0 r  1 − γ
initial ! =
= − A
V r γ p 0 r
ρ 0 r s 2
γ − 1  p ∞
p 0 r  γ +1
γ  p 0 r
p ∞  γ − 1
2 γ
t, (C.6)
152

Results of jet sim ulations and exp erimen ts
where 2 F 1 is the ordinary h yp ergeometric function.
Minimal example:
F or the sak e of clarit y , w e sho w the evolution of a specific reservoir with
the set of parameters summarised in table C.1.
T able C.1: Parameters of the example in the subsonic nozzle flo w.
P arameter V alue Description
p 0 r /p ∞ 1 . 5 T otal reserv oir to am bien t pressure ratio
L/D 30 Non-dimensional mass supply
T 0 r /T ∞ 1 T otal reserv oir to am bien t temp erature ratio
W e sho w in figure C.1 the pressure ev olution in the reserv oir for the
example of a subsonic nozzle flo w. The disc harge of the reserv oir tak es
place in t ∗ ≈ 15 . 19 and the end of the pro cess is denoted b y ( b ). A t this
p oin t a pressure p r /p 0 r = 1 / 1 . 5 ≈ 0 . 66 is reac hed.
0 4 8 12 16
0.7
0.8
0.9
1

p r / p 0 r
t ∗

Figure C.1: Dimensionless pressure ev olution in the reserv oir. The end of the
pro cess is denoted b y ( b ).
The exit and fully expanded Mac h n um b ers are sho wn in figure C.2.
Note that b oth Mac h n um b ers are iden tical to eac h other since the subsonic
flo w is p erfectly expanded at the nozzle.
C.2 Chok ed nozzle
In this case, the mass-flo w rate is constan t (critical mass-flo w rate ˙ m ∗ )
and giv en b y the critical section area, the critical density and the critical
sp eed of sound:
˙ m ∗ = ρ ∗ c ∗ A ∗ . (C.7)
153

Chapter C. Quasi-steady reserv oir motion
0 4 8 12 16
0
0.2
0.4
0.6
0.8

t ∗
M e , M j

Figure C.2: Exit and fully expanded Mach n um b er ev olution in the subsonic
nozzle.
Assuming ideal gas ( p = ρR T ) and using the definition of the sp eed of
sound ( c 2 = γ RT ) w e can write the mass-flow rate as
˙ m ∗ = √ γ p ∗ ρ ∗ A ∗ .
Using the one-dimensional isen tropic theory w e can relate the thermo-
dynamic state of the flo w with the stagnation conditions with:
T 0
T =  1 + γ − 1
2 M 2  . (C.8)
W e can relate the differen t thermo dynamic v ariables using the isentropic
relations: p ∼ ρ γ , T ∼ p γ / ( γ − 1) , etc. W e can also write the critical mass-
flo w rate as:
˙ m ∗ = √ γ p r ρ r A ∗  2
γ + 1  γ +1
2( γ − 1)
.
Relating the conditions in the reserv oir ( p r , ρ r ) with the stagnation
reserv oir conditions w e get to:
˙ m ∗ = √ γ p 0 r ρ 0 r A ∗  2
γ + 1  γ +1
2( γ − 1)  ρ r
ρ 0 r  γ +1
2
.
Plugging this in to the con tin uit y equation (C.1) and dividing b y ρ 0 r w e
get  ρ r
ρ 0 r  − ( γ +1)
2
d  ρ r
ρ 0 r  = − A ∗
V r γ p 0 r
ρ 0 r  2
γ + 1  γ +1
2( γ − 1)
dt,
whic h b y in tegration results in
ρ r
ρ 0 r
= " 1 + γ − 1
2
A
V  2
γ + 1  γ +1
2( γ − 1) r γ p 0 r
ρ 0 r
t # − 2
γ − 1
(C.9)
154

Results of jet sim ulations and exp erimen ts
and therefore
p r
p 0 r
= " 1 + γ − 1
2
A
V  2
γ + 1  γ +1
2( γ − 1) r γ p 0 r
ρ 0 r
t # − 2 γ
γ − 1
. (C.10)
W riting this relation in dimensionless v ariables w e get
p r
p 0 r
= " 1 + γ − 1
2
1
L/D  2
γ + 1  γ +1
2( γ − 1)
t ∗ # − 2 γ
γ − 1
, (C.11)
where A/V = 1 /L = D /L 1 /D and t ∗ = tc 0 r /D .
Minimal example:
W e presen t here an example with a sp ecific set of parameters sum-
marised in table C.2.
T able C.2: Parameters of the example in the sup ersonic nozzle flo w.
P arameter V alue Description
p 0 r /p ∞ 3 . 6 T otal reserv oir to am bien t pressure ratio
L/D 30 Non-dimensional mass supply
T 0 r /T ∞ 1 T otal reserv oir to am bien t temp erature ratio
Figure C.3 sho ws the ev olution of the reserv oir pressure. The nozzle
remains c hok ed un til t ∗ ≈ 5 . 96 , which is denoted b y ( • ). The pro cess ends
at t ∗ ≈ 15 . 46 , and this is denoted b y ( b ). At this point, a pressure of
p r /p 0 r = 1 / 3 . 6 ≈ 0 . 278 is reac hed.
0 4 8 12 16
0.25
0.5
0.75
1

p r / p 0 r
t ∗

Figure C.3: Dimensionless pressure ev olution in the reserv oir for the sup ersonic
nozzle flo w. The end of the nozzle c hoking is denoted by ( • ). The end of the
pro cess is denoted b y ( b ).
155

Chapter C. Quasi-steady reserv oir motion
In order to get the conditions of the nozzle exit, w e can relate the
reserv oir conditions with the nozzle exit ones b y means of:
M e = v
u
u
t 2
γ − 1 "  p r
p e  γ − 1
γ
− 1 # , (C.12)
and the same w a y w e can relate the reserv oir conditions with the fully
expanded conditions:
M j = v
u
u
t 2
γ − 1 "  p r
p ∞  γ − 1
γ
− 1 # . (C.13)
Note that the exit Mac h n um b er is giv en b y the isen tropic expansion
un til the pressure at the nozzle exit and the fully expanded Mac h n um b er
is giv en b y the isen tropic expansion un til the pressure at the surroundings.
0 4 8 12 16
0
0.25
0.5
0.75
1
1.25

t ∗
M e , M j

Figure C.4: Exit (solid) and fully expanded (dashed) Mac h n umber evolution
in the c hok ed nozzle.
Figure C.4 sho ws the exit and fully expanded Mac h n um b ers ev olution
for the example of the sup ersonic nozzle flo w. The solid line corresp onds
to the exit Mac h n um b er and it sho ws a constan t M e = 1 un til the nozzle
is no longer c hok ed ( t ∗ ≈ 5 . 96 ). F rom this p oin t on, the nozzle w orks in
the subsonic regime and it expands the flo w p erfectly . This makes the fully
expanded and the exit Mac h n um b er to b e equal to eac h other.
Figure C.5 sho ws the time ev olution of the Prandtl-Mey er angle that de-
fines this sup ersonic expansion. Ha ving a Mac h n um b er of unit y at the noz-
zle exit, w e can compute the angle that the flo w is deflected while expanding
sup ersonically with equation (C.14). The flo w starts in the under-expanded
regime with a Prandtl-Mey er angle of ν PM ≈ 11 . 51 ◦ and it decreases with
time un til the nozzle is no longer c hok ed. In this p oin t, the sup ersonic
expansion do es not tak e place an y longer and the flo w is purely subsonic.
156

Results of jet sim ulations and exp erimen ts
0 4 8 12 16
0
2
4
6
8
10
12

t ∗
ν PM ( ◦ )

Figure C.5: Prandtl-Meyer expansion angle evolution in the c hok ed nozzle.
ν P M ( M ) = r γ + 1
γ − 1 atan  r γ − 1
γ + 1 ( M 2 − 1)  − atan  p M 2 − 1 
(C.14)
0 4 8 12 16
0.4
0.5
0.6
0.7
0.8
0.9
1

t ∗
ρ r / ρ 0 r

0 4 8 12 16
0.7
0.75
0.8
0.85
0.9
0.95
1

t ∗
T r /T 0 r

( a ) ( b )
Figure C.6: Ev olution of the thermo dynamic state in the reserv oir during the
disc harge of the c hok ed nozzle. ( a ) Reserv oir densit y and ( b ) reservoir tempera-
ture.
W e fo cus no w on the thermo dynamics of the nozzle flow. During the
disc harge of the reserv oir is the in ternal energy of the fluid con v erted in to
kinetic energy . Therefore, as the fluid is ejected through the nozzle, the
remaining fluid inside the reserv oir expands and the temp erature decreases
with time, see figure C.6. The same b eha viour is observ ed in the densit y .
Assuming an isen tropic pro cess b et w een the reserv oir and the nozzle
exit, we sho w in figure C.7 that while the nozzle is c hok ed, the temp era-
ture and the densit y remain constan t and during the subsonic stage, they
increase with time un til the end of the pro cess.
157

Chapter C. Quasi-steady reserv oir motion
0 4 8 12 16
0.4
0.5
0.6
0.7
0.8
0.9
1

t ∗
ρ j / ρ 0 r , ρ e / ρ 0 r

0 4 8 12 16
0.7
0.8
0.9
1

t ∗
T j /T 0 r , T e /T 0 r

( a ) ( b )
Figure C.7: ( a ) F ully expanded (dashed) and exit (solid) densit y evolution of
the c hok ed nozzle. ( b ) F ully expanded (dashed) and exit (solid) temperature
ev olution.
F or the fully expanded conditions, b oth the temp erature and the densit y
increase monotonically with time, see figure C.7.
0 4 8 12 16
0
0.4
0.8
1.2
1.6

t ∗
L s /D

0 4 8 12 16
1
1.02
1.04
1.06
1.08

t ∗
D j /D

( a ) ( b )
Figure C.8: Ev olution of the sho c k-cell spacing ( a ) and fully expanded diameter
( b ) during the disc harge of the c hok ed nozzle.
In order to ha v e a b etter idea ab out the flo w c haracteristics, w e sho w in
figure C.8 as w ell the length of the first sho c k cell giv en b y the correlation
from Sc h ulze [2011]:
L s ≈ π q M 2
j − 1 D j
σ 1
(C.15)
where σ 1 is the first ro ot of the zero order Bessel function ( σ 1 ≈ 2 . 4048 ).
The fully expanded diameter D j is related to the fully expanded Mach
158

Results of jet sim ulations and exp erimen ts
n um b er as follo ws:
D j = D 1 + 1
2 ( γ + 1) M 2
j
1 + 1
2 ( γ + 1) M 2
d ! γ +1
4( γ − 1) s M d
M j
. (C.16)
Its time ev olution is also represen ted in figure C.8.
C.3 Effect of the main parameters
In this section, the effect of the main parameters in the pro cess is anal-
ysed. As in the previous sections, a quasi-steady one-dimensional flo w w as
assumed. The previous metho d w as used rep etitiv ely to compute enough
cases to see a trend when c hanging the v alue of the go v erning parameters.
A base flo w w as c hosen, whic h is summarised in table C.3.
T able C.3: Set of parameters chosen as base flo w for this analysis.
P arameter V alue Description
p 0 r /p ∞ 10 T otal reserv oir to am bien t pressure
L/D 10 Non-dimensional mass supply
T 0 r /T ∞ 1 T otal reserv oir to am bien t temp erature ratio
W e sho w the solution of the reserv oir disc harge for this set of parameters
in figure C.9. The maximum fully expanded Mac h num ber is approximately
2 . 16 , decreasing with time un til t ∗ ≈ 1 . 997 , where the nozzle is no longer
c hok ed; this is denoted in figure C.9 a with ( • ). The rest of the pro cess is
subsonic un til the end of the pro cess at t ∗ ≈ 3 . 32 ; also represented in C.9 a ,
this time with ( b ).
Figure C.10 is a con tour plot of the effect of the pressure ratio o v er
the disc harge pro cess. The colour corresp onds to the fully expanded Mac h
n um b er. The subsonic regime is in green, in y ello w w e ha v e the sonic
v elo cities and the red tones represen t the sup ersonic flo w. The blac k dotted
line represen ts the end of the disc harge and the dashed line the end of the
sup ersonic stage.
As exp ected, for reserv oir to am bien t pressure ratios smaller than ( p r /p ∞ ) ∗ =
1 . 893 (for γ = 1 . 4 ), the nozzle cannot b e c hok ed and there is no sup ersonic
flo w, and this is the reason the dashed line to start at 1 . 893 . In this range
of pressure ratios, the discharge tak es place in a relativ ely long time, up
to t ∗ ≈ 5 . 556 for a p r /p ∞ = 2 . 093 , whic h is the longest pro cess for this
set of parameters when c hanging the pressure ratio. In terestingly , this
pressure ratio do es not corresp ond to the longest sup ersonic stage, but for
p r /p ∞ = 5 . 465 . F or this pressure ratio, the sup ersonic stage tak es place in
t ∗ ≈ 2 . 226 and all other pressure ratios lead to shorter sup ersonic stages.
159

Chapter C. Quasi-steady reserv oir motion
0 0.5 1 1.5 2 2.5 3 3.5
0.2
0.4
0.6
0.8
1

p r /p 0 r
t ∗

0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2

M e , M j
t ∗

( a ) ( b )
Figure C.9: ( a ) Ev olution of the pressure in the reservoir. ( b ) Exit (solid) and
fully expanded (dashed) Mac h n um b er ev olution.
The maxim um fully expanded Mac h n um b er of this plot is larger than
the one of the base flo w due to the higher pressure ratios used for the
analysis.
Concerning the non-dimensional mass supply ( L/D ), figure C.11 sho ws
the ev olution of the fully expanded Mac h n um b er. As exp ected, w e see a
linear b eha viour when increasing L/D . A reserv oir with t wice the v alue of
L/D w ould b e the same as disc harging t w o reserv oirs of L/D after eac h
other and it should tak e t wice the time. W e observ e the same b eha viour
with the sonic line. In this case, the maximum fully expanded Mac h num b er
remains the same for all cases since all of them ha v e the same pressure ratio.
Concerning the reserv oir to am bien t temp erature ratio, figure C.12 sho ws
the ev olution of the fully expanded Mac h n um b er for this analysis. It sho ws
longer disc harges when increasing the reserv oir temp erature, but actually
this is only an effect of the normalisation used for the time t ∗ = t c 0 r /D =
t √ γ RT 0 r /D . When re-normalising the time w e find no c hanges with the
temp erature ratio as sho wn in figure C.13. This do es not strike with the
curren t theory when lo oking at equation (C.11): it do es not dep end on the
reserv oir thermo dynamic conditions but p 0 r , whic h w as k ept constan t in
this last analysis.
160

Results of jet sim ulations and exp erimen ts
p r / p ∞
t ∗
10 0 10 1 10 2 10 3
10 −2
10 −1
10 0
10 1
0
0.5
1
1.5
2
2.5

Figure C.10: F ully expanded Mac h n um b er con tour plot o v er the pressure ratio
and the non-dimensional time. The dashed blac k line represen ts the end of the
sup ersonic flo w and the blac k dotted line the end of the pro cess.
161

Chapter C. Quasi-steady reserv oir motion
L / D
t ∗
10 −1 10 0 10 1 10 2
10 −2
10 −1
10 0
10 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2

Figure C.11: F ully expanded Mac h n um b er ev olution with the non-dimensional
time as a function of the non-dimensional mass supply . The dashed blac k line
represen ts the end of the sup ersonic flo w and the blac k dotted line the end of the
pro cess.
162

Results of jet sim ulations and exp erimen ts
T r / T ∞
t ∗
10 −1 10 0 10 1
10 −2
10 −1
10 0
10 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2

Figure C.12: F ully expanded Mac h n um b er ev olution with the non-dimensional
time when c hanging the reserv oir to am bien t temp erature ratio. The dashed black
line represen ts the end of the sup ersonic flo w and the blac k dotted line the end
of the pro cess.
163

Chapter C. Quasi-steady reserv oir motion
T r / T ∞
t ∗ / ( T r /T ∞ )
10 −1 10 0 10 1
10 −2
10 −1
10 0
10 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2

Figure C.13: F ully expanded Mac h n um b er ev olution with the normalised non-
dimensional time t ∗ / ( T r /T ∞ ) when c hanging the reservoir to am bien t temp era-
ture ratio. The dashed black line represen ts the end of the sup ersonic flo w and
the blac k dotted line the end of the pro cess.
164

App endix D
Cylindrical reserv oir
analogy
In order to compare the results obtained in this study with those in
the literature, w e sho w in the follo wing discussion the analogy b et w een
the main non-dimensional parameters of the cylinder-piston device t ypi-
cally used in the literature in the incompressible case with the disc harge
of a pressurised reserv oir in the curren t study . T o mak e this analogy w e
assume that the reserv oir has a cylindrical form with the same diameter
as the nozzle ( D ) in whic h the unkno wn v ariable is the length ( L ) of this
h yp othetical reserv oir. By applying the con tin uit y equation and in tegrating
the mass injected ( m injected ) in to the un b ounded c ham b er with the help of
the temp oral and spatial inlet condition distributions, equations (2.10) and
(2.11), resp ectiv ely , w e compute the mass supplied through the nozzle and
therefore the length of the h yp othetical reserv oir that w ould ha v e supplied
the same mass. In other w ords, using the con tin uit y equation, the mass
con tained in the unkno wn reserv oir at high pressure ( m cyl , 0 ) is equal to the
injected mass plus the mass con tained in the reserv oir at lo w pressure after
the disc harge ( m cyl , f ),
m cyl , 0 = m injected + m cyl , f . (D.1)
W e can in tegrate the inlet condition to kno w the injected mass
m injected =
∞
Z 0
D / 2
Z
− D / 2
D / 2
Z
− D / 2
ρ e ( y , z , t ) u e ( y , z , t ) dy dz dt, (D.2)
w e can express the non-dimensional mass supply ( L/D ) as a function of
the inlet condition parameters. F or this purp ose w e write the densit y and
165

Chapter D. Cylindrical reserv oir analogy
v elo cit y as a function of the main parameters:
u e = M e c e = c e v
u
u
t 2
γ − 1 "  p 0 r
p e  γ − 1
γ
− 1 # (D.3a)
ρ e = p
1
γ
e exp  − s e
C p  (D.3b)
so w e can write the injected mass as a function of the go v erning parameters:
m injected =
∞
Z 0
D / 2
Z
− D / 2
D / 2
Z
− D / 2
p
1
γ
e exp  − s e
C p  c e
v
u
u
t 2
γ − 1 "  p 0 r
p e  γ − 1
γ
− 1 # dy dz dt.
(D.4)
Ha ving the amoun t of injected mass, w e use the con tin uit y equation
(D.1) assuming a cylindrical shap e for the reserv oir:
π  D
2  2
Lρ 0 r = m injected + π  D
2  2
Lρ ∞ , (D.5)
where the initial densit y is the total densit y in the reserv oir ( ρ 0 r ) and the
final densit y is that of the surroundings ( ρ ∞ ). By ordering the terms we
ha v e
π D 2 L
4 RT 0 r
( p 0 r − p ∞ ) =
∞
Z 0
D / 2
Z
− D / 2
D / 2
Z
− D / 2
p
1
γ
e exp  − s e
C p  c e
v
u
u
t 2
γ − 1 "  p 0 r
p e  γ − 1
γ
− 1 # dy dz dt,
(D.6)
where R denotes the sp ecific gas constan t, T r the temp erature of the reser-
v oir, s e the en trop y at the nozzle exit, c p the heat capacit y at constan t
pressure and c e the sp eed of sound at the nozzle exit. Just by re-organising
the differen t terms w e can write the non-dimensional mass supply as a
166

Results of jet sim ulations and exp erimen ts
function of the go v erning parameters and the inlet condition parameters
L
D = 4 RT r
π D 3
p
1 − γ
γ
∞ e
− s e
c p c e
p 0 r
p ∞ − 1 r 2
γ − 1
∞
Z 0
D / 2
Z
− D / 2
D / 2
Z
− D / 2
s  p 0 r /p ∞
NPR  γ − 1
γ
− 1  p 0 r /p ∞
NPR  1
γ
dy dz dt ∗ . (D.7)
167

Chapter D. Cylindrical reserv oir analogy
168

App endix E
Instrumen tation
E.1 A coustic measuremen ts
Microphones:
PCB-378B02: F requency range: 3 . 75 to 20 000 Hz; Dynamic range:
16 . 5 − 138 dB(A)
Figure E.1: F requency response of the microphone with the grid cap at 0 degrees
incidence. The top curve is the corrected free-field curv e and the bottom curve
is the pressure resp onse generated b y the electronic actuator. Copied from the
man ufacturer sp ecifications.
G.R.A.S. 40AZ: F requency range: 0 . 5 to 20 000 Hz; Dynamic range:
14 − 148 dB(A)
169

Chapter E. Instrumen tation
Figure E.2: F requency resp onse of the G.R.A.S. microphone with the grid cap at
0 degrees incidence. The top curv e is the corrected free-field curv e and the b ottom
curv e is the pressure resp onse generated b y the electronic actuator. Copied from
the man ufacturer sp ecifications.
E.2 Sc hlieren comp onen ts
High-sp eed camera Phan tom ® v2512:
• 1 Megapixel sensor ( 1280 × 800 )
• 25 Gigapixel/s throughput
• Minim um exp osure time: 265 ns
• Maxim um fps at ( 1280 × 800 ): 25 700
• Maxim um fps at ( 256 × 32 ): 1 000 000
• Connected with 10GB Ethernet
Lenses Used:
• NIKKOR 18-140mm f/3.5-5.6G D X
• NIKKOR 200-500mm f/5.6E FX
Ligh t source:
High p o w er LED – CREE CXA1830-0000-000N00S430F
• 57 W
• 3060 lm
• Beam angle 115 ◦
170

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