o

!

INSTABILITIES AND SUBHARMONIC RESONANCES OF

SUBSONIC HEATED ROUND JETS

by

Lian Lai Ng

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINNERING

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

WITH A MAJOR IN AEROSPACE ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

1990

(NASA-CR-18605_) INSTABILITIES ANO

SUBHA_MONTC REsnNANCES nF SUBSONIC HEATEO

ROUND JETS, VnLUME 7 Ph.D. Thesis Final

Report (_rizon_ Univ.) 186 p C_Ck 200

G3134

N90-22017

Unclas

0280243
=

!
INSTABILITIES AND SUBHARMONIC RESONANCES OF

SUBSONIC HEATED ROUND JETS

by

Lian Lai Ng

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINNERING

In Partial Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

WITH A MAJOR IN AEROSPACE ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

1990

THE UNIVERSITY OF ARIZONA

GRADUATE COLLEGE 2

As members of the Final Examination Committee, we certify that we have read

the dissertation prepared by Lian Lai Ng

entitled INSTABILITIES AND SUBHARMONIC RESONANCES IN SUBSONIC

HEATED ROUND JETS

and recommend that it be accepted as fulfilling the dissertation requirement

Doctor of Philosophy

for the Degree of

T_omas F. Balsa

Chuan F. Chen

Edward Ker _en

Henry Perkins

4..

Date _" '

z5

Date

Date

Date

Final approval and acceptance of this dissertation is contingent upon the

candidate's submission of the final copy of the dissertation to the Graduate

College.

I hereby certify that I have read this dissertation prepared under my

direction and recommend that it be accepted as fulfilling the dissertation

requirement.

_'._ _'t_'_ '_z" Dat/_' I_/qO

Dissertati°n T-rect°r Thomas F. Balsa

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to

be made available to borrowers under rules of the library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for

extended quotation from or reproduction of this manuscript in whole or in part may

be granted by the head of the major department or the Dean of the Graduate College

when in his or her judgement of the proposed use of the material is in the interests of

scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

ACKNOWLEDGMENTS

I would like to acknowledge my advisor, Professor Thomas Balsa, for his invaluable

guidance, assistance and support throughout my research. The author also wishes to

express his sincere appreciation to other members of the committee, Professors C. F.

Chen, Edward Kerschen, Henry Perkins, and Arne Pearsltein, for their availability and

constructive comments.

Special thanks must go to the stability and transition team (A. J. Kumar, Gordon

Erlebacher, Michele Macaraeg, Bart Singer, Craig Streett, and To m Zang) at NASA Langley who constantly encouraged me to finish writing my dissertation while I was working

at NASA Langley. I have also benefited greatly from the many discussions with Eckart

Laurien and fellow researchers Bart Singer and Gordon Erlebacher. The support and

friendship of Sylvain Yapo, Xasan Ayanle, Yen-Ming Chen and Y. Jarrah are also appreciated.

Finally, I would like to thank my wife, Wendy Weiner, for her infinite patience, endless love and encouragement. This work is also dedicated to my family in Malaysia and.

Singapore.

This work was principally supported by NASA Lewis Grant NAG3-485, and partially

by NASA Langley Grant NASI-18599.

5

TABLE OF CONTENTS

Page

LIST OF FIGURES .................................... 8

ABSTRACT ........................................ 14

1 INTRODUCTION

1.1

1.2

1.3

16

Motivation & Technical Importance ...................... 16

Historical Background .............................. 21

Scope of Present Research ........................... 20

2 GOVERNING EQUATIONS

LEM

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

AND PRIMARY STABILITY PROB29

Governing Equations .............................. 29

Nondimensionalizing the Governing Equations and Simplifying Assumptions 30

The Base flow and Perturbation Equations .................. 34

Primary Linear Stability Analysis ....................... 36

Kinematic Boundary Conditions on the Linearized Disturbances ...... 40

Asymptotic Behavior of a Disturbance on a Uniform Base Flow ...... 41

The Behavior of Disturbances in the Critical Layer .............. 43

Summary ..................................... 47

3 FORMULATION OF THE SECONDARY INSTABILITY PROBLEM 49

3.1 Method of Multiple Scales ........................... 51
4

3.2 Kinematic Resonance Conditions........................ 53

3.3 Amplitude Equation ............................... 61

3.4 Normal Mode Analysis .............................. 65

NUMERICAL METHODS

4.1

4.2

4.3

4.4

73

SpectralMethod ................................. 73

Some Remarks on SpectralMethods ...................... 85

Fourth-Order Finite-DifferenceMethod .................... 86

Some Remarks on the Linear InviscidCalculation .............. 87

5 LINEAR STABILITY OF PARALLEL JETS

5.1

5.2

5.3

5.4

97

ViscosityEffect ................................. 97

Mach Number Effect .............................. 100

Heating Effect .................................. 105

Mode Number and Shear Layer Thickness Effects .............. 111

6 SUBHARMONIC INSTABILITY MODES 127

6.1 Resonance Conditions .............................. 132

6.2 Comparison of Approaches ........................... 134

6.3 Compressibility Effect .............................. 137

6.4 Heating Effect ................................... 139

6.5 Shear Layer Thickness Effect .......................... 140

6.6 Mode Number Effect .............................. 142

6.7 Influence of A on the Eigenfunction of Subharmonic Modes ......... 143

7 SUMMARY & CONCLUSIONS 159

APPENDIX A ....................................... 164

APPENDIX B ....................................... 166
7

APPENDIX C ....................................... 168

APPENDIX D ....................................... 170

APPENDIX E ....................................... 174

APPENDIX F ....................................... 176

REFERENCES ...................................... 178

LIST OF FIGURES

1.1

2.1

4.1

4.2

4.3

4.4

4.5

The flow field of a jet ............................ 28

Jet mean profiles:solidlines(0 = 1/5,Mj = 0.0,T. -- 2), dot (0 :

1/15, Mj = 0.0, T. = 2), chaindot (0 = 1/30, Mj = 0.0, T, = 2), dash

(0= 1/15,Mj = 0.8,T, = I)........................ 48

A set of odd and even Chebyshev polynomials, Tt(z) ........... 88

(a) and (b) z as a function of the radial variable r for various n (c)

collocation points on r. The number of collocation points is 35 using

Mapping-A with 11 = 7 and 12 - 3. Crosses are the collocation points. 89

(a) z as a function of the radial variable r for various stretching parameter r (b) collocation points on r. The number of collocation points is 21

using Mapping-B. Crosses are the collocation points. Bullets (r = 5),

circles (r = 10), and squares (r - 1) .................... 90

Distributions of the leigenfunctionl of a near maximally amplified disturbance for Re'- 1.0E+6,T, = 1.O,M i = 0.1,0 = 1/15, c_ -- 3.5,8 =

1, and the complex phase velocity C -- (0.577,0.189) ........... 91

Distributions of the phase of the eigenfunction of a near maximally

amplified disturbance for Re = 1.0E + 6, T. = 1.O,M i = 0.1,0 =

1/15, a = 3.5,/_ = 1, and the phase velocity C = (0.577,0.189) ..... 92
4.6

4.7

4.8

4.9

Distributions of the leigenfunctionl of a near maximally amplified disturbance for Re = 1.0E+ 3,T, = 1.O,M 1 = 0.1,0 = 1/15, c_= 3.5,/_ =

2, and the complex phase velocity C = (0.557,0.142) ........... 93

Distributions of the phase of the eigenfunction of a near maximally

amplified disturbance for Re = 1.0E + 3,7". = 1.O,M 1 = 0.1,0 =

1/15, a = 3.5, _ = 2, and the phase velocity C = (0.557,0.142) ..... 94

Distributions of the leigenfunctionl of a near neutral disturbance for

Re = 5.0E + 3, T. = 1.0, M 1 = 0.8, 0 = 1/15, c_= 6.55, fl = 0, and the

complex phase velocity C = (0.569,0.001) ................. 95

Distributions of the phase of the eigenfunction of a near neutral disturbance for Re= 5.0E+3,T, = 1.0, Mj = 0.8,0 = 1/15, a = 6.55,/_ = 0,

and the complex phase velocity C = (0.569,0.001) ............ 96

5.1 The effect of Re on the spectrum of linear, parallel flow, temporal

eigenvalues using 31 collocation points for 0 = 1/15, M 1 = 0, T, =

1, a = 3.66, and/9 = 0 ........................... 115

5.2 The effect of Re on the spectrum of linear, parallel flow, temporal

eigenvalues using 31 collocation points for 0 = 1/15, M i = 0, T, =

2, a = 3.66, and _ = 1........................... 116

5.3 The effect of Re on the spectrum of linear, parallel flow, temporal

eigenvalues using 31 collocation points for 0 = 1/15, Mj = 0.8, T. =

1, a = 3.66, and/_ = 0 ........................... 117

5.4 (a) Growth rate and (b) phase velocity of unheated, axisymmetric

modes for various Re: Re - oo (solid lines), Re -- 5000 (dot), Re

= 500 (dash). Calculations are for 0 = 1/15, M i = 0, T. = 1 ...... 118
5.5

5.6

5.7

5.8

5.9

5.10

5.11

5.12

I0

(a) Grow_th race and (b) phase velocity of heat;ed, axisymmetric modes

for various Re: Re -- co (solid lines), Re = 5000 (dot), Re = 500

(dash). Calculations are for 0 = 1/15, Mj = 0, 7', = 2 ......... 119

(a) Growth rate and (b) phase velocity of heated, _ = 1 modes for

various Re: Re = co (solid lines), Re = 5000 (dot), Re =500 (dash).

Calculations are for 8 = 1/15, Mj = 0, 2", --- 2 .............. 120

Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

1_ = 0 and 8 = 1/15 for various Mj and Tj. Mj = 0, T° = 1 (solid

lines). Mj = 0.8, T, = 1 (chaindot). Mj = 0, T, = 2 (dash) ...... 121

Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

= 1 and 0 = 1/15 for various Mj and Tj. Mj = 0, 3", = 1 (solid

lines). M i = 0.8, T, = I (dot). Mj = 0, 7". = 2 (dash) ......... 122

(a) Amplificationrate (Ci) and (b) phase velocity(CR) of a vortex

sheet for /_ = 0 and /9 = 1 modes. /_ = 0, 7",= I (solidlines).

= 0, 7".= 2 (dot)._ = 1,T, = I (dash)./_= I, T, = 2 (chaindash).

Ms = MI = 0................................ 123

Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

- 0 and /_ = I for 8 = 1/15 and Mj = 0. /_ = 0, T, -- I (solid

lines)._= 0, T, =2 (dot).fl= 1, T. = 1 (dash). fl= 1, T, = 2

(chaindash).Mj = ACt = 0 .......................... 124

Inviscid(a)growth rateand (b)phase velocityof the azimuthal modes

= 0 (solidlines),fl--1 (dot),and fl--2 (dash).Calculationsare for

0 = 1/15, Mj = 0, T, = I ......................... 125

Inviscid(a)growth rateand (b)phase velocityofthe azimuthal modes

= 0 (solid lines), _ = 1 (dot), and/9 = 2 (dash). Calculations are for

0=1/5, Mj=0, T,=I .......................... 126
11

6.1

6.2

6.3

6.4

6.5

Inviscidphase velocityforvariousmean profilesat 0 = 1/15. Solid

lines: Mj = 0.0, T, = 1. Dot: Mj = 0.0, T. = 2. Dash: Mj ----

0.8,7',= 1. (a) Helicalmode: _ = 1. (b) Axisymmetric mode: _ = 0. 146

(a)Growth rateand (b)IrelativefrequencyIas a functionof the amplitude oftheperiodicwave with a near maximally amplifiedwavenumber.

Calculationsare for Re = 5000,0 = 1/15,Mj = 0.1,T. = 1,/_i= 0,

a2 = 1.7,/32= 1. Solidlines:multiplescales.Circles:normal mode

analysis................................... 147

(a) Growth rate and (b) [relative frequency I as a function of the amplitude of the periodic wave with a near maximally amplified wavenumber.

Calculations are for Re = 5000,0 = 1/15,Mj = 0.1, T. = 1,/31 : 0,

a2 = 1.7,/32 = 1. Solid lines: multiple scales. Circles: normal mode

analysis ................................... 148

(a) Growth rate and (b) Irelative frequency[ as a function of ithe amplitude of the periodic wave with a near neutral wavenumber. Calculations are for Re = 500,0 = 1/15, Mj = 0.1, T, : 2,_I = 0,

a2 ----3.2, _2 = 1. Solid lines: multiple scales. Circles: normal mode

analysis ................................... 149

(a) Growth rate and (b) Irelative frequency I as a function of the amplitude of the periodic wave with a near neutral wavenumber. Calculations, using normal analysis, are for Re = 5000, 0 = 1/15, T. =

1,_1 = 0. Dot: a2 = 3.275,_ = 0, Mj = 0.8. Dash: a2 = 3.275,_2 =

I,Mj : 0.8. Chaindot: a2 = 3.66,/_2= 0,Mj = 0.I. Chaindash:

a2 = 3.66,/_2= 1,Mj = 0.1............... •......... 150
12

6.6

6.7

6.8

6.9

6.10

(a) Growth rate and (b) IrelativefrequencyI as a functionof the amplitudeof the periodicwave with a near neutralwavenumber. Calculations,using normal analysis,are for0 --1/15,Airi---0.1,/_I--0.Dot:

Re- 500, a2 : 3.2, /_2 : 0, T, = 2. Dash: Re: 500, a2 : 3.2,j32 :

1,T, : 2. Chaindot: Re = 5000, a2 --- 3.66,/_2 : 0,T. : 1. Chaindash:

Re = 5000, a2 : 3.66,_2 _. l,T. ----1 ................... 151

(a) Growth rate and (b) [relative frequency[ as a function of the amplitude of the periodic wave with a near neutral wavenumber. Calculations, using normal analysis, are for Re = 5000, Mj = 0.1, T. : 1, fll ---

0. Dot: a2 = 7.3,_s = 0,0 = 1/30. Dash: as "- 7.3,/_s : 1,0 = 1/30.

Chaindot: as = 3.66,/_2 = 0,0 = 1/15. Chaindash: as -- 3.66,/_2 -

1,8 : 1/15. Solid lines: as = 1.7,_s : 1,0 = 1/15 ........... 152

(a) Growth rate and (b) ]relative frequency I as a function of the amplitude (unrestricted) of the periodic wave with a near neutral wavenumbet. Calculations, using normal analysis, are for Re : 5000, 8 -

1/15, Mj = 0.1,T, = 1,/_1 = 0, a2 = 3.66. Solid lines: /_2 = 0. Dot:

/_s = 1. Dash: _s = 2 ............................ 153

(a) Growth rate and (b) Irelative frequency I as a function of the (unrestricted) amplitude of the periodic wave with a near neutral wavenumber. Calculations, using normal analysis, are for Re -- 5000, P : 1/15,

M i = 0.1,T, = 1,_1 : 0, a2 = 3.66. Solid lines: /_2 = 3. Dot: /_2 = 4. 154

Distributions of the [eigenfunction I of a subharmonic disturbance as a

function of the radial variable r. Calculations are for Re -- 5000, T. :

1.0, M i : 0.1,8 = 1/15,_1 = 0, a2 = 1.7, fl2 = 1. Solid lines: A = 0.

Dot: A - 0.05. Chaindot: A : 0.1 .................... 155
6.11

6.12

6.13

13

Distributions of the phase of a subharmonic disturbance as a function

of the radial variable r. Calculations are for Re = 5000, T. = 1.0, Mj =

0.1,#= 1/15,/91= 0, a2=1.7,/_2=1. ×: A=0. e: A=0.05. o: A

= 0.1 .................................... 156

Distributions of the [eigenfunction I of a near maximally amplified subharmonic disturbance as a function of the radial variable r. Calculations are for Re = 5000,T, = 1.0, Mj = 0.1,g = 1/15,/_1 -: 0, c_2 =

3.66,/_2 = 1. Solid lines: A = 0. Dot: A = 0.05. chaindot: A = 0.1. 157

Distributions of the phase of a near maximally amplified subharmonic

disturbance as a function of the radial variable r. Calculations are for

Re = 5000, 7', = 1.0, M i = 0.1, 0 : 1/15,/_l : 0, a2 = 3.66, f12 : I. x:

A : 0. *: A : 0.05. o: A = 0.1 ...................... 158

1
14

ABSTRACT

When a jet is perturbed by a periodic excitation of suitable frequency, a large-scale

coherent structure develops and grows in amplitude as it propagates downstream. The

structure eventually rolls up into vortices at some downstream location.._We_pproximate

the "wavy flow _ associated with the roll-up of a coherent structure by a parallel mean flow

and a small, spatially periodic, axisymmetric wave whose phase velocity and mode shape

are given by classical (primary) stability theory. The periodic wave acts as a parametric

excitation in the differential equations governing the secondary instability of a subharmonic disturbance.

The (resonant) conditions for which the periodic flow can strongly destabilize a subharmonic disturbance are derived. When the resonant conditions are met, the periodic

wave plays a catalytic role to enhance the growth rate of the subharmonic. The stability characteristics of the subharmonic disturbance, as a function of jet Mach number, jet

heating, mode number and the amplitude of the periodic wave, are studied via a secondary

instability analysis using two independent but complementary methods : (i! method of

multiple scales and (ii) normal mode analysis. We found that the growth rates of the

÷

subharmonic waves with azimuthal numbers _. = 0 and fl = 1 are enhanced strongly,

but comparably, when the amplitude of the periodic wave is increased. Furthermore,

compressibility at subsonic Mach numbers has a moderate stabilizing influence on the

subharmonic instability modes. Heating suppresses moderately the subharmonic growth

rate of an axisymmetric mode, and it reduces more significantly the corresponding growth

rate for the first spinning mode (i.e,/3 --- 1). Our calculations also indicate that while the
15

presence of a finite-amplitude periodic wave enhances the growth rates of subharmonic

instability modes, it minimally distorts the mode shapes of the subharmonic waves.
16

Chapter 1

INTRODUCTION

1.1 Motivation & Technical Importance

It isnow generallyaccepted that large-scalecoherent structures,in addition to

fine-scalechaoticmotions,existwithin allturbulentfreeshear flows.Free shear flows--

such as mixing layers,wakes and jetsm are a classofflowscharacterizedby the existence

of an inflectionpoint in the strearnwisebase velocityprofile.For compressibleflows,the

classicalinflectionpoint isreplacedby a generalizedinflectionpoint which isdetermined

by the cross-spacederivativesof the base velocityand density. These free shear flows

are highlyunstable.Small perturbationsof suitablefrequencies(orwavenumbers) in the

flow willgrow rapidlythrough the so-calledKelvin-Helmholtzinstabilitymechanism. The

instabilitywaves willevolveinto a finiteamplitude disturbancewhich ischaracterizedby

streamwise-periodicregionsof concentratedvorticity-- the so calledKelvin cat'seyes.

Although the precisemechanisms of the spreading of turbulent flowsare presentl.ystill

under study,itisgenerallyacceptedthat the coherentstructuresplay a considerablerole

in the dynamics of turbulent flows;the relativeimportance of the largeand finescale

structuresisstillnot understood allthat well and the effectof the latteron the former

has not been quantifiedexperimentallywith any degree of precision.

Distinguishableevidenceof thesewave-likecoherentstructuresin fullyturbulentsubsonic(unheated)round jetsisconvincinglydisplayedinthe schlierenphotographs ofAhuja
17

et al.(1982).They concluded that theselarge-scalestructures,whose dimension iscomparable to the shear layerthickness,play an important rolein controllingthe dynamics

of turbulentflows.For exarnple,when they exciteda jetby an upstream acousticsignal

at a strouhalnumber, Se = _ D}/U_ = 0.5 (C isthe excitationfrequency,D} isthe jet

diameter,and U_ isthe jetvelocity),the jetplume was widened considerably.By using a

photographicenhancement process,the large-scalevorticeswere clearlyidentifiedatsome

distancedownstream. Further,they found that ifthe excitationfrequency was doubled,

a differentinstabilitymode was excited,the sizeof the coherent structurewas reduced

and the spacing between successivestructureswas halved. These resultsconfirmed the

sensitivityofthenaturallyexistinglarge-scaleturbulencestructureto the frequencyofthe

imposed disturbance.The adjacentvorticalstructuresmay amalgamate atvariousdegrees

ofregularityas they convectdownstream, therebygeneratinglargervortices.This pairing

ofvorticescan be seen from Figure 1 ofWiUe's (1963)work. When he injectedsmoke into

the wall boundary layerofa circularjetnozzle,the smoke rolledup intoringsofvortices

and proceeded to pair with theirneighborsfartherdownstream. The coalescenceof the

largescalestructureswas alsodemonstrated by Winant and Browand (1974).By injecting

a filamentof dye into a shear layer,they found that the dye roiledup intodiscretetwodimensional (vortical)structures;fartherdownstream, they observed that two adjoining

structuresinteractto form a single,largervortex.Together with Laufer,Kaplan & Chu

(1973),they conjecturedthat the sequentialmergings of these structuresis a possible

mechanism forthe mixing and spreading of allfreeshear flows.

Over the years,experiments have establishedthat externalexcitationscan produce

changes in largescaleand small scalemotions injetflows,although the effectoftheseon

the the radiated noisefieldisnot entirelyclear.Ffowcs Williams ancl Kempton (1978)

found that the increasein jet noise,due to excitation,isa directconsequence of the
18

amplified large-scale structures, while the fine-scale chaotic turbulence plays a relatively

minor role in the noise generation mechanism. On the other hand, Morris and Tam (1977)

argued that, although the large-scale structures are responsible for producing considerable

changes in the fine-scale turbulence, the actual noise generation mechanism is a result of

the latter. They provided two explanations. First, since the phase speed of an excited

large-scale instability wave is subsonic relative to the ambient fluid, the large scale structures are inefficient in directly generating the sound waves. Secondly, if the noise increase

is directly caused by the large-scale structures, there would be a distinct frequency band

centered around the frequency of an excited, most amplified, instability wave. Instead,

jet-noise is broad-band. Nevertheless, irrespective of the mechanism through which noise

is produced, the large-scale structures play an important part in the noise generation and

the dynamics of turbulent shear flows.

A distinguishingfeatureof coherent structuresisthat they are relativelypermanent

entities-- in thesensethat they persistforlongtimes and distances-- which co-existwith

the seemingly completely random turbulentscales.This isin contrastto the completely

chaoticsmall-scalemotions which quicklylosetheiridentities.This perseveranceof the

largescalestructuresoffersa hope that one can manipulate the development ofturbulent

flowsby modifying the evolutionof the largescalestructures,especiallyin the initialregion ofjetsand mixing layers.There isnow considerableexperimentalevidence that the

large-scalestructures,and hence,the globalfeaturesofturbulentflowscan be organizedby

artificialexcitationsuch asoscillatingflaps,vibratingribbons and acousticdevices(Wygnanski and Petersen 1987). By externallyexcitinga jetwith a loud speaker upstream of

the jet nozzle,Ahuja et al.found that not only did the radiatedsound increase,but the

strengthand regularityof the coherentstructureswere augmented. Ahuja et al. (1982)

presenteda seriesofmeasurements forsubsonicjetsexcitedat differentfrequencies.They
19

have concluded that higher Mach number unheated jetsrequirehigher excitationlevels

to produce noticeablechanges in both large-scaleturbulenceand small-scaleturbulence

intensities.One of the objectivesof thisresearchisto investigateone mechanism (i.e.,a

secondary instabilitytheory)that might explainthisexperimental observation.

The effectofexcitationlevelson jetmean flowvelocityand turbulenceintensitieswere

demonstrated by Ahuja et al..They found that when the excitationlevelwas increased,

there was a rapid decay of the mean-velocitywith downstream distance. Petersen has

shown that the mean velocityprofileand the spreadingrateof a turbulent,axisymmetric,

unheated low speed jet can be alteredby introducinga controlledacousticexcitation.

This observationisimportant because the increaseof jet spreading and the decreaseof

the mean velocityimpinging on the flapsof a short takeoffand landing aircraftwith an

under-the-wing or over-the-wingexternallyblown flapsystem can lead to a reduction of

flap loads and, therefore,a decreasein structuralfatigueproblems. By decreasingthe

aerodynamic and thermal loads impinging on the ground through increasedmixing and

spreading rates,the ground effectsforverticaland short takeoffand landingaircraftcan

similarlybe reduced.

Stone & Mckinzie (1984) concluded that %ontrol offlowsby intentionalexcitationof

naturalflow instabilitiesinvolvesnew and largelyunexplored phenomena and offersconsiderablepotentialforimproving component performance." They stressedthat deliberate

acousticexcitation-- which circumvents the use of intrusivemethods such as suction

and blowing -- of these coherent structuresoffersa promising new means of controlling

turbulent shear flows. Most importantly,they (seealso,Ahuja et al. 1986) have also

emphasized the factthat,todate,most researchwork has been centeredaround unheated

jetflows,usuallyatlow speed. Therefore,futureresearchwork should be directedtowards
20

the understanding and control of coherent structures in heated high speed jets.

Controlof turbulentflowsviacoherentstructureshas obvioustechnologicalsignificance

forapplicationto a considerablenumber ofengineeringdevicesthatdepend on mixing such

as ejectors,internalmixers,combustors,jetsenginesand high liftdevices,to name a few.

One of the main culpritsofpoor ejectorperformance isan incomplete mixing; therefore,

ejectorperformance may be improved by enhanced mixing. Enhanced mixing in internal

mixer nozzlescan increasethrust,improve fuel economy, and reduce engine noise. An

increasein fuel/airmixing in combustors can lead not only to shortercornbustors,but

alsoto improved pollutioncontrol.In supersonicjetengines,the interactionoflargescale

structureswith shocks isthe sourceofshock noise(Tam & Jackson 1983);by excitingthe

jetshear layerwith a high frequency,the formation ofcoherent structureswilloccur at a

smallerscale,thereby reducingthe intensityofthe radiatedjetnoise.

Having establishedthe practicalimportance ofmanipulating the downstream development of a jetshear layerviacontrollingthe evolutionoflarge-scalestructures,we theorize

that the effectof manipulating these large-scalestructures(orsynonymously, instability

waves) may be predictedand understood without resortingto the usual empiricalrequirement forthe descriptionof the fine-scaleturbulence(seealso,Gaster,Kit & Wygnanski

1983).To reinforcethispoint,the traditionalapproach to turbulentflowshas been known

to be more successfulin describingexistingdata rather than predictingnew insights.

Therefore,the goal of thisresearchisto understand, at leastin qualitativeterms, how

certaincharacteristicsofheated and compressiblejets-- such as spreadingrate-- can be

alteredby externalexcitationwithout relyingon the empiricalmodeling of the fine-grain

turbulence.We willdo thisby a systematicexamination of the stabilityand resonances

of the coherentstructures(assuming that they are instabilitywaves) ofsubsonic heated
jets.

21

1.2 Historical Background

When a flowpassesthrough a nozzle,the boundary-layernear the inletduct at the

jet exitquicklyevolvesinto a freeshear layer(seeFigure 1) which separatesthe potentialcore from the unperturbed ambient. When thisshear layerisexcitedby a wave-like

disturbanceof suitablefrequency,an instabilitywave develops in the flow. In the initial

stages of development, where the disturbance amplitude isstillsmall compared to the

jetexitvelocity,thiswave grows in amplitude as itpropagatesdownstream. The characteristicsof thisinstabilitymode isdescribedwell by linearstabilitytheory (Michalke &

Hermann 1982, Petersen& Samet 1988). Farther downstream, where the amplitude of

the disturbanceisof appreciablesize,say a smallpercentageof the mean flow,nonlinear

effectsbecome important,and the wave (orcoherentstructure)reachesa finite-amplitude

saturation.An important objectiveof thiswork to understand and to promote the (resonance) conditionsunder which thisinstabilitywave can destabilizea subharrnonicmode.

Physically,thisisimportant sincea subharmonic disturbancecan generatevortexpairing;

thispairingis,at leastpartially,responsibleforjetspreading and mixing.

The existenceof large-scaleinstabilitywaves has now been acknowledged by an overwhelming number of observations;an extensivereview of coherent structuresin excited

shear flowsisgivenby Wygnanski & Petersen(1987).Crow and Champagne (1971) provided the firstcomprehensive work on the responseof a circularincompressibleunheated

jet to a controlledaxisymmetric excitationof a certainamplitude-and frequency. By

measuring the velocityfluctuationsalong the jetaxis,they found a "preferred"mode at

which an excitedinstabilitywave reached a maximum amplitude. This preferredmode
22

has a frequency of fp ----0.3U_/D* where U_ is the jet exit speed, and D* is the diameter.

They concluded that this mode is an instability wave most capable of reaching a large

amplitude before saturating. The significance of this preferred mode lies in the fact that

it is the most dominant and frequently occurring of all large scale structures at the end

of the potential core. Michalke (1971) determined that the measured phase velocity of

the coherent structure, found by Crow & Champagne, agreed reasonably well with that

obtained from a linear stability theory based on a turbulent mean flow. By using phase

locked measurements of controlled excitations in an axisymmetric cold jet, Petersen &

Samet (1988) concluded that the preferred mode is, in fact, a Rayleigh instability mode,

provided that the stability analysis is based on the local shear layer thickness and the

measured mean velocity profiles. As a result of this finding, they concluded that the

preferred mode in, both excited and naturally occurring, jets is a shear layer instability

mode rather than a global instability mode involving the entire jet column. This lends

considerable credibility to the use of stability theory to understand the nature and the

evolution of the quasi-deterministic large scale motions associated with the appearance of

coherent structures in both unexcited and excited jets (see also Gaster, Kit & Wygnanski

1985).

There have recently been a number of theoretical works on the inviscid stability characteristics of axisymmetric jets, although studies of either compressible or heated jets remain

scarce. Batchelor & Gill (1962) pioneered a theoretical analysis on the inviscid stability

characteristics of a top hat velocity profile which characterizes the mean flow close to the

jet exit. They found that the top hat velocity profile is unstable to a small disturbance for

all axial and azimuthal wavenumbers. Their analysis is supported by Plaschko (1979) and

Cohen (1986). Stability theory has now been successfully extended in various ways and

applied for jets at near ambient temperatures and at low jet Mach numbers (for example,
23

Cohen 1986).

The stabilitycharacteristicsfor hot jets,which are of technologicalinterest,have received relativelylittleattention.Michalke (1971) and Nfichalke& Hermann (1982) are

among the few who investigatedthe stabilitycharacteristicsof subsonic heated jetsby

the linear,inviscidand parallelflow stabilitytheory. They found that the localgrowth

rate of spatiallygrowing waves increasesas the totaltemperature of the jet isincreased.

This means that forheated jetsthereisa more unstablecoherent structure,resultingin

the widening of the mean jetvelocityprofile.Recently,Sohn (1986) found that cold jets

are convectivelyunstable,whereas hot jetswith an exittemperature 1.5 times the ambient temperature are absolutelyunstable. By using a _quasi-linear_ model, Ahuja et. al.

(1982)concluded that the small-scaleturbulence,which isinduced by the passage of excitedlarge-scalestructures,isresponsibleforthejetspreadingand mixing forboth heated

and unheated jets.Their theoreticaland experimentalstudieshave shown that both jet

noise amplificationand jet mixing decrease as the Mach number increases.The effect

of heating ispredictedto increasethe turbulenceintensityand to widen the jet width.

They pointedout thattheirtheoreticalresultsforthe heatingeffecton jetsareinconsistent

with limitedexperimentaldata and, therefore,areinconclusive.The _quasi-linearmodel,_

which requiresempiricalconstants and certainturbulenceclosureassumptions,provides

littlephysicsgoverning the roleof coherent structuresin turbulentjet flows.Hence, the

mystery and intricaciesofcoherentstructuresin heated jetshave yet tobe systematically

unravelled.

Over the past few years,ithas been establishedthat linearinviscidstabilitytheorycan

predictwith surprisingaccuracy the phase velocityand the passage frequency ofcoherent structures,as wellas the lateraldistributionof the perturbationsinexcitedjets (e.g.,
24

Crighton & Gaster 1976, Plaschko 1979, Strange & Crighton 1983). The convincing agreement obtained by Gaster, Kit & Wygnanski (1985) between experimental data and the

linear inviscid stability theory for slowly diverging shear layers reconfirmed the preceding

remark. Consequently, coherent structures can be viewed as (inviscid) instability waves

of the mean flow, and fine-grain turbulence plays a passive role in the development of the

coherent structures. They stressed that from the theoretical point of view, the large-scale

structures offered a hope that certain characteristics -- such as mixing enhancement and

suppression -- may be predicted and understood without resorting to the empirical requirement for the description of the fine-grain turbulence.

Although the linear parallel flow instability theory can describe accurately the local

stability characteristics of the initial stages of the development of a large-scale instability

wave, it is unable to predict the overall streamwise amplification of the disturbance. By

retaining weakly nonparallel terms and using the method of multiple scales, Crighton &

Gaster (1976) gained substantial improvement in the total streamwise growth, although

there still exists discrepancies between their numerical results and experimental data. Petersen & Wygnanski (1987) attributed these differences to the neglect of the nonlinear

terms in a linearanalysis.

Another deficiency of linear (parallel flow) theory is that it is unable to predict the

growth rate of a subharmonic disturbance in a periodic base flow. In the case of a planar

mixing layer, this growth rate was first calculated by Kelly (1967) through the so-called

linear secondary stability analysis. His temporal analysis predicted that a periodic fundamental component superimposed on a parallel mean flow destabilizes a subharmonic disturbance whose phase velocity matches that of the fundamental. Purely numerical work

on the instability of a periodic flow has been carried out by Pierrehumbert and Widnall
25

(1982)who studied the linear (temporal) secondary instability of a row of Stuart vortices.

Their results are in reasonable agreement with Kelly's analysis. Cohen (1986) extended

Kelly's analysis to spatially growing waves in a planar mixing layer. While Kelly decoupled

the predominant fundamental wave from the growth of the subharmonic, Cohen considered the exchange of energy between a fundamental disturbance and its subharmonic as

both waves propagate downstream. Additionally, Cohen established subharmonic and azimuthal resonance conditions in low speed and unheated jets. In particular, he has shown

that the nonlinear interaction between two azimuthal modes can produce a third mode

which is initially absent in the flow. His theoretical results are in close agreement with

his experimental studies. More recently, Monkewitz (1988) generalized Kelly's analysis

and examined vortex pairing in more detail in incompressible mixing layers. By incorporating explicitly the weakly nonlinear interaction terms between two instability modes in

his method of multiple scales analysis, he found that the development of the subharmonic

depends crucially on its phase relation with the fundamental. He has also shown that a

critical threshold fundamental amplitude is required for the subharmonic to become phase

locked with the fundamental; this presence of the fundamental leads to an enhancement

of a subharmonic growth rate.

It is perhaps appropriate at this point to mention briefly that the linear secondary

stability mechanism is also important in the studies of the laminar-turbulent transition

in walled-bounded boundary layer flows. By investigating the temporal instability of 3-D

disturbances in a spatially periodic base flow consisting of a Blasius velocity profile and a finite amplitude 2-D Tollmien-Schlichting wave, Herbert (1984) found that the subharmonic

resonance mechanism is the principal route to transition in low disturbance environments;

hence the designation H-type breakdown. His numerical results agree closely with the

experimental data of Saric (1983).
26

1.3 Scope of Present Research

The purpose of this research is to examine in detail the stabilities and resonances

of coherent structures -- assuming that they are synonymous with instability modes --

in heated subsonic round jets. We choose temporal stability theory in favor of the spatial

theory because the validity of a spatial theory was unclear at the initiation of this work (see

remarks on convective instabilities of heated jets in Chapter 2). We believe that a temporal

analysis of heated jets will contain all the essential physics and qualitative trends, although

it is certainly true that the corresponding spatial analysis may provide a better agreement

with experimental data. Of course, spatial instability arises naturally in convectively

unstable flows when periodic excitation is present. The point is that the dispersion relation

(i.e., the eigenvalue relation) is an analytic function of the complex wavenumber, and

spatial instability is merely the analytic continuation of temporal instability. In addition,

there are a number of conceptual di_culties associated with spatial instability -- such as

the calculation of the mean flow distortion -- which we wish to circumvent through the

use of temporal analysis. The present investigation is divided into the following two tasks:

1. Although the linear parallel flow instability theory of incompressible flows is presently

understood and documented, the stability analysis of compressible and heated round

jets is still rather sketchy. In the first part of this work, we focus on the systematic

determination of the individual role of Niach number and temperature ratio in _fecting jet instabilities. A good understanding of the primary instability characteristics

is also a prelude to secondary instability calculations. In this work, an analytic mean

flow (tanh shape) which provides a good local approximation to that obtained from

measured experimental data is used in our stability calculations.

2. The principal task of this work is to study the linear subharmonic instability of a periodic flow consisting of the steady mean flow and a small but finit_.__ee amplitude primary
27

wave.In orderto makethe physicsgoverning this instability more transparent, we

first establish the conditions -- the so-called resonance conditions -- which provide

an effective mechanism for the destabilization of an instability wave. The resonant

interactions between two waves allow us to determine the relevant parameters to be

used as input forthe secondary instabilitycalculations.

We carryout the %econdary" instabilityanalysisusingthe methods ofmultiplescales

and a generalizationof the normal mode analysison a streamwise periodicflow.It

may be noted that the secondary instabilityof round jetsis,to date,stillnew, in

the presence of heating and compressibility.The growth rate of the subharmonic

disturbance as a function of the amplitude of the primarywave for differentjet

parameters -- such as Mach number, temperature ratio,shear layerthicknessand

modal numbers -- are determined.

In summary, this research will provide a deeper understanding of the relevant parameters (e.g., wavenumber of excitation, modal number, Mach number and jet temperature)

that affect the secondary instability and the physical mechanisms governing vortex mergings in subsonic heated round jets.
t,l

28

Fig. 1.1 The flow field of a jet.
29

Chapter 2

GOVERNING EQUATIONS AND PRIMARY

STABILITY PROBLEM

2.1 Governing Equations

The governing equationsfora viscous,heat conducting and compressibleidealgas

in dimensionalform (characterizedby a superscriptasterisk)are the conservationof mass,

momentum, energy,and the thermodynamic equation ofstate.They are

ap__"+ V" .(fv') = 0 (2.1)

cgt*

,D'v*

P D't* =-V*P*+P*g*+V*'r* (2.2)

p*C_ D'T* • v* D,t---- T = -p'V* + V*. (k*V*T*) + _* (2.3)

p* =p'_*T* (2.4)

where p*, v*, p* and T* represent the density, velocity, static pressure and the temperature,

respectively. A point in space is denoted by the Cartesian coordinate x*, V* = 0/ax*

and t* stands for time. The convective derivative is defined by

D* 0

- + v'.v', (2s) D't* at*

g* isthe constant gravitationalacceleration,C_ isthe specificheat capacityat constant

volume and _* isthe gas constant.In (2.3),Fourier'sLaw ofheat conduction isassumed;

the heat fluxisgivenby (-k*V*T*). These equationsmay be found in Bird,Stewart and

Lightfoot (1960).
3O

Dissipative processes in the fluid are due to the viscosity, #*, and the thermal conductivity k*. For a Newtonian fluid, the shear stress tensor, r*, and the Udlssipation

function _ , _*, are defined by

2 , ,

r" = 2#'s" - _v (v •v*)I (2.6)

2 ,--, V,)2 0"=r*:S*=2#*S °:S*-_# (V • (2.7)

where

- _ tt

s'= 5iv v + (v'v') T] (2.8)

S* is the rate of strain tensor, I is the idemtensor and (.)r denotes the transpose of a

tensor or matrix.

2.2 Nondimensionalizing the Governing Equations and

Simplifying Assumptions

Our principal interest is in the study of certain types of instabilities associated with

compressible and heated round jets. For this purpose, the governing equations (2.1) - (2.4)

can be made dimensionless by using the centerline valueE at the jet exit of the base flow

(also called the mean flow or the unperturbed flow). We nondimensionalize all length,

velocity and time scales by R', U] and R*/U], respectively -- where R* is the radius of

the jet. The dimensionless physical variables become:

V* . p*

"-" w

• p

T* p*

T=_-j,, p=_p_

f$* k*

#= .._, k=

#j k'f'
31

and with the above normalizations, the governing equations (2.1) - (2.4) become

ap

a-_+V.pv= 0

D_ iv _

p--_=-Vp+ Re .r+ Fr

DT

p-_-7=-('_-l)_/M]pV-v+ "/V.kVTprRe -t-

"IM]p = pT

"y(_/- 1)M]

Re

(2.9)

(2.10)

(2.11)

(2.12)

where

s = 1⁄2[vv + (vv)T]

2

r = 2_s - g_(v. v)Z

(2.13)

(2.14)

= r:S (2.15)

and k is the unit vector in the direction anti-parallel to the gravity. The Reynolds number,

Re, Mach number, Mj, Froude number, Fr, and Prandtl number, Pr, are given by

Re- p_U] R* Mi= U_,

la_ ' aj .

v;' .iv;

Fr -- Pr -- ,

g'R* ' k_

where a_ is the unperturbed sound speed at the jet exit, C_ is a constant representing the

specific heat at constant pressure, _/= constant = C_/C*, and g* is a constant representing

the magnitude of the gravitational acceleration. The nondimensional convective derivative

is given by

D a

+v.V (2.16) Dt cOt

In order to make any progress with the relevant equations, a large number of assumptions must be made which are discussed in books on stability theory (Lin 1955, Betchov

and Criminale 1967, Drazin and Reid 1981). To begin with, the last term in (2.10) is

ignored because gravitational effects are unimportant at large Froude numbers; this is a
32

standard assumption in aerodynamics, for example, when R* -- 5 cm, U_ = 5000 cm/sec,

p; --- 1.3× 10-s gm/cm s, p; = 1.8 × 10-' dynes sec/cm 2 (air at normal conditions), Fr -

5 × 103, Re = 1.4 × l0 s, Pr -- O(1). In instability theory, it is convenient to decompose

the dependent variables into two parts: the base flow and the perturbations. At high

Reynolds numbers, the base flow (i.e., unperturbed flow) will satisfy the boundary layer

equations so that the unperturbed pressure will be a constant (approximately) throughout

the flow field, and the jet will "diverge" with downstream distance on a long length scale

, 1

of (R Re_). Since the characteristic wavelength of the instability wave is on the order

of R*, the divergence of the base flow may be ignored for the purpose of calculating the

instability wave. In other words, the base flow is assumed to be locally parallel. The

precise form of this flow will be specified in the next section.

At high Reynolds number (say, above 10s), the effects of viscosity and heat conductivity are negligible for the instability wave, except possibly for neutral modes and their

derivatives in the cross-space. For this reason, we shall retain a highly simplified version

of the viscous and conduction terms: these terms are only important in a thin layer of

thickness h = 0 (Re-i/s), centered around the generalized inflection point. In this layer,

¢

the so-called the critical layer, p and k are constants to a high degree of approximation

so that V. r may be replaced by/JcV_v and V.(kVT) by kcV2T, where the subscript

c denotes the value of a quantity in the critical layer. As a result of this approximation,

the effects of heat conductivity and viscosity are correctly represented only in the critical

layer where they may be important. Outside the critical layer, these effects are incorrectly

represented in our analysis, but this is immaterial because they are unimportant.

Further, when the disturbances are small, the largest terms in ¢, in (2.11), are proportional to the product of the velocity gradient [which is O(1)] and the perturbation velocity

gradient [which is at most O (8/h), where 6 << 1 is the characteristic magnitude of the

disturbance]. On the other hand, the perturbation temperature gradients in the conduc-
33

tion term of the energy equation scale as 6/h 2 because of the presence of the Laplacian

and are, therefore, much more important in the critical layer than _. Consequently, the

viscous dissipation, _, will be ignored here.

In summary, viscosity and heat conduction are included, but only to the extent needed

to eliminate any possible singularity in the instability modes arising from the reduced

inviscid and nonconducting equations. Although the viscosity and conductivity of a fluid

vary with temperature, these variations will be negligible across the thin critical layer

where they play an appreciable diffusive role. Consequently, the viscosity and conductivity

are assumed to be uniform throughout the fluid. It may be remarked that outside the

critical layer, the viscous and the conduction terms are negligible relative to the convection

terms. With the preceding assumptions, the governing equations are approximated by

ap

8_ + V-pv = 0 (2.17)

Dv

p-_-=-Vp% -_--1 V2v (2.18) Rec

DpD"_ = -_pV. v + PrcRe---'-_ '_ V2(p) (2.19)

where Rec and Pro are the Reynolds and Prandtl numbers based on the viscosity and

thermal conductivity in the critical layer. The temperature has been eliminated in the

energy equation in favor of the pressure via the equations of state and continuity. The

Prandtl number and _/(ratio of specific heats) are taken to be 0.7 and 1.4, respectively. In

the rest of this thesis, the subscript c on Re and Pr are dropped, with the understanding

that these quantities are based on _, and kc. The assumptions that lead to (2.17) - (2.19)

are discussed more fully in Moore (1964; see the article by S. F. Shen).
34

2.3 The Base flow and Perturbation Equations

We shallnow formallyseparatethe disturbancefrom the base flowby writing

P--* Po-t- P

v --,Vo + v (2.20)

P--'Po +P

where the subscript0 hereafterrefersto the base flow,and p, v, p are perturbation

quantities.After substituting(2.20)into (2.17)- (2.19)and collectingterms, we obtain

a set of equations for the base flow and for the perturbations. The latterset is given

vectoriallyas

Dop

-F "/poV .v Dt

Do.__pp + poV" v -{-v. Vp0 = -V. (pv)

Dt

po--_- D0v + pov. Vvo + Vp - _ V2v = -p-_.-Dv

v_(£ pop

PrRe Po P_ ) = -v . Vp - "IpV "v

{_ _1 w2 / PoP2 PP

p-_v _ p--_-o- _ +...)

(2.21)

(2.22)

(2.23)

where,

Do_ c3 O (2.24)

Dt cgt + u°_x'

and the dots stand for terms for the product of perturbationsthat are cubic and higher.

In deriving (2.21) - (2.23), we have made the assumption that the base flow is parallel;

thisimpliesthat

vo = [uo(y,z),o,o] (2.25)

po= [po(y,z),o,o)] (2.26)

Po = constant (2.27)
35

where x = (z, F, z) is a cartesian coordinate system with z pointing along the axis of

the jet. Although the numerical calculations and the theoretical developments are carried

out in a cylindricalcoordinatesystem (z,r,_) with y = •cosC, z = r sin_, itismore

compact towriteclownthe perturbationequationsina Cartesiansystem. Note that Vp0 ---

(0,apo/cgy,Opo/CgZ)so that v- Vp0 does not involvethe z-component ofthe perturbation

velocity.Similarremarks hold true for other terms of this type. In order to generate

specificresults,the mean flow istaken as

where

an d,

uo = uo(•)= 1{1- tanh[(,- 1)140])

p0= p0(•)=

T.

1+ iT. - 1)-o+ - ,,o)

1

P0 = constant --

0 O0 is the dimensionless momentum thickness = u0(1 - uo)dr

T. is the temperature ratio = T0(0)

T0(oo)

(2.28)

(2.29)

(2.30)

To is the base temperature = To(V, z) = To(r)

Note that the steady unidirectional velocity profile (2.28) is not an exact solution of the

Navier-Stokes equation. It is, however, a good local approximation to that obtained from

the measured experimental data. This mean velocity profile, which has also been used by

other investigators (e.g., Michalke & Hermann 1982), is capable of representing the tophat profile at the jet nozzle exit and the fully developed Gaussian profile at around the

end of the potential core as well as the profiles between these two extremes. This can be

done by varying the parameter O (momentum thickness) from (near) zero to a value which

is a substantial fraction of unity (i.e., the nondimensional jet radius). The parameter 0

can thus be used to characterize different velocity profiles at different axial locations.
36

For compressible ideal gas flows, the mean density profile, (2.29), is obtained from

the mean veiocity profile via the Busemann-Crocco law, which is valid for high Reynolds

number flows and when the Prandtl number is unity (Schlichting 1979, p. 330). For a

given velocity profile, u0, the mean density distribution is governed by three dimensionless

parameters: Mach number Mj, temperature ratio T., and the isentropic exponent _. In

this work, the isentropic exponent _/= 1.4, and T. is taken to be greater than one, which

means that the jet is hotter than the quiescent environment. For example, when T, = 2,

the jet is twice as hot as the environment.

Figure (2.1a) shows the dependence of velocity profiles on 0 while the effect of 0, Mj,

and T, on the mean density distributions are shown in Figure (2.1b). It is interesting

to note that in Figure (2.1b), T, has a much greater influence than the (subsonic) Mach

number M i upon the base density profile, and therefore, is expected to produce a greater

effect on the instability.

2.4 Primary Linear Stability Analysis

Although our principal goal is the study of the secondary instabilities and subharmonic resonances in a compressible and heated jet, we begin our discussion with the

classical linear instabilities of a jet. A knowledge of these linear instabilities is needed in

order to study the secondary instabilities and resonances. Perhaps it is worthwhile, first,

to say a few words about the parallel flow assumption for the mean flow in view of some

relatively recent developments.

Petersen & Samet (1988) have found that the local stability characteristics (e.g, eigenfunctions, phase velocity , Reynold stress, etc.) of an instability mode developing on a

jet column can be predicted accurately from linear viscous stability theory based on the

measured mean velocity profiles, which are again assumed to be locally parallel. They
37

foundthat the precedingobservationis valid even when the local streamwise velocity disturbance reaches a level as high as 24% of the jet speed, when the flow is unquestionably

nonlinear. The last remark lends considerable credibility to the generally accepted claim

that even in _nonlinear" flows, the mode shape of the fundamental is given quite accurately by the linear mode. A large body of literature has been built on this assumption,

beginning with the pioneering work of Ko, Kubota and Lees (1970).

In fact, this finding of Petersen & Samet is important in the instability analysis of

a jet which has a finite-amplitude wave-like structure developed on its column. This

structure is approximately both spatially and temporally periodic. In order to represent

the fundamental component of this periodic flow, we shall use a linear instability mode.

This makes the study of secondary instabilities much easier because we do not have to

compute a finite amplitude primary disturbance. In our secondary instability analysis,

this amplitude is a given quantity; the mode shape is given by linear theory.

Unfortunately, a shortcoming of linear instability based on parallel flow assumption

is that it predicts poorly the total streamwise amplification of an instability mode. By

retaining weakly nonparallel terms in an axisymmetric round jet, Crighton & Gaster (1976)

[see also Gaster, Kit and Wygnanski 1985] gained a significant improvement on the total

growth rate, although there still exists some unsatisfactory discrepancies between their

results and experimental measurements. According to Wygnanski & Petersen (1987),

these discrepancies are attributed to the neglect of the nonlinear interaction terms in the

linear parallel flow analysis of a single wave train. Petersen et al. have, in fact, dispelled

the skepticism surrounding the predictive capability of a parallel flow stability analysis

for the mode shape. While there is no doubt that an accurate determination of the total

growth rate still requires further studies on nonlinear analysis, it is not a concern in this

work. The quantities of interest here are the local values of the phase velocity, growth

rate and the shape of a specific mode, and that can be described well by linear parallel
38

flow analysis.

Because of the effectsof compressibilityand heating in a cylindricalgeometry, the

algebraicaspectsof the analysisin thisthesisare very lengthy.In order not to burden

the reader,we use a compact operatornotation;the specificelements of theseoperators

may be found in the variousappendices.

To recall,we use a cylindricalpolarcoordinate(x,r,@),where x isalong the axisofthe

jetand (r,_) arethe radialand azimuthalcoordinates;thecomponents ofthe perturbation

velocityarev = (u,_,w) in thiscylindricalsystem. To representthe entireperturbation

field,we use the five-dimensionalvector

F = 6 fb= 6 (u,v,_,p,p)_

After substitutingthe above equationintothe (nonlinear)perturbationequations,(2.21)-

(2.23),and neglectingallquadraticand higherorderterms ind_,we arriveatthe linearized

disturbanceequations

Abfb= 0 (2.31)

Here Ab isa linearoperator whose actionon fb isdefinedin detailin Appendix A. The

variousterms inAb arefunctionsofthe base flowalone;Ab has variablecoefficientswhich

depend only on the radialcoordinates.Here 6 isa small parameter which measures the

magnitude of a typicaldisturbancewith respectto the base flow. Since Ab depends on

r alone,we may extractthe dependence ofthe disturbanceon the other coordinatesvia

Fourier (space)and Laplace (time)transforms. Because of the linearity,we considera

"singlemode _ (orwave component) and write

fb : f(r)e_(az+_)e_t+ complex conjugate; i :

(2.32)

fCr)= T

where a and/_ arethe wavenumbers inthe axialand azimuthal directions,respectively,and

isthe complex growth rate.The disturbanceiscompletelydescribedby a, /_,# and the
39

complex amplitude function (also known as the mode shape) f(r). The modal form (2.32)

is simply a single mode of a Fourier-Laplace component representing the wavenumber and

frequency component of an arbitrary disturbance. This means that each individual mode

evolves independently in the course of time, according to the the linearized equations.

Because of the periodicity in ¢, _ must be an integer (0, 1,2 ... etc), and it is also called

the mode number. In temporal theory, a is real, and a is complex. The real part of a

determines the temporal growth rate of a disturbance. The imaginary value of a is the

radian frequency. In spatial stability, a is usually written as -iw where w is the given

real frequency, but a is complex. The spatial growth rate of a disturbance is given by

the imaginary value of a. An approximate relation between the growth rates of these

two forms of instability can be obtained by Gaster's transformation (Gaster 1962) under

certain restrictive assumptions -- such as for small growth rates.

Although there still exists some questions on the application of temporal or spatial

theory (especially for nonlinear flows), it is now generally accepted that temporal analysis

is meaningful for absolutely unstable flows while spatial analysis has physical meaning

for convectively unstable flows. The basic concepts of absolute and convective instability

are discussed in detail by Huerre & Monkewitz (1985). Suppose a jet is perturbed by

a disturbance which is localized in space and is impulsive in time. Some time after the

triggering of this disturbance, a wavetrain of finite extent evolves on the jet. If both the

leading and trailing edges of this wavetrain propagate downstream, the flow is said to

be convectively unstable. Otherwise, the flow is absolutely unstable. In an absolutely

unstable flow, the trailing edge of the wavetrain propagates upstream and the group

velocity is negative in some region of the wave. Because of this, disturbances which are

generated downstream of the nozzle in an absolutely unstable flow can actually interact,

through instability waves, with the nozzle to produce "feedback" or "resonance' loops.

Temporal stability is also applicable to flows which are required to be periodic in space.
4O

Such flows may not actually exist in practice, but are at least fairly close approximations

to those that do. Once a flow is forced to be periodic in space, its natural spatial evolution

is inhibited and the (linear) instability will appear as an exponential growth in time. For

this restricted class of flows, temporal instability is always valid. Since at the beginning of

this study, the concepts of convective and absolute instabilities were not clarified, we opted

for a study which is based on temporal instabilities and spatial periodicity. In retrospect,

this was a good choice because it is now known that hot jets with 7". > 1.5 may be

absolutely unstable (Sohn 1986); for such flows a study based on spatial instability would

be meaningless.

The modal representation (2.32) of a disturbance transforms the system of linearized

partial differential equations, (2.31), into a system of ordinary differential equations (hereafter ODE's) in r. These equations can be conveniently put into the matrix form

Aif = aBlf (2.33)

where the elements, which depend on the wavenumbers (a,/9), of A1 and 131 are given in

Appendix B. The principal task is to solve (2.33) for a given (a,/_) and suitable boundary

conditions in order to obtain the eigenvalue from the dispersion relation a = a(a,/_). We

now discuss the boundary conditions associated with (2.33).

2.5 Kinematic Boundary Conditions on the Linearized

Disturbances

The appropriate boundary conditions for a disturbance superimposed on a subsonic

jet require that f remains finite on the jet axis and vanishes at infinity. They are,

f --* 0 as r --+ oo (2.34)

f remains finite at r = 0 (2.35)
41

In fact, some of the boundary conditions to be satisfied at the origin are purely "kinematic"

in nature, and therefore, are independent of the viscosity (Batchelor and Gill 1962). In

order for a disturbance to remain single-value at the origin, these kinematic boundary

conditions yield

_,(0)= p(o) = p(o) = 0 for_ # 0 (2.36)

and,

_(0)= to(0)= 0 for_ _ 1 (2.3z)

Other boundary conditionsat the origin,which cannot be determined from kinematic

considerationalone, are derived from theirasymptotic behavior developed in the next

section;note that not allthe boundary conditionsare given by kinematic considerations.

2.6 Asymptotic Behavior of a Disturbance on a Uniform

Base F]ow

The numerical solutionsofhydrodynamic stabilityproblems in an unbounded domain, such as in round jets,are often solved over a finitedomain. Consequently, the

analyticalstructureofa disturbanceforlarger and small r isrequired.Because the base

flow,u0 and P0, approaches a constant value,the asymptotic behavior can be obtained

explicitlyby solvinga seriesof Bessel-likeequations.In fact,for • --*0,we can write

U = ClUp "Jr-C2U v (2.38)

t_= Clt_p Jr-C2t)U (2.39)

tO = Clto p + C2to v (2.40)

p : Clpp (2.41)

p : ClPp (2.42)
42

where

{'' }

,_p - _ ir_(_,-)

PP- KJ poi_(uo - c) ,_p

o¢2

p,,= poCT,,-

a[1 - po(uo - C)'M_ + iaCuo - C)M_/Re] 1

[1 + iaCu0 - C)M]i/Rel1⁄2

O_

A,j = [a 2 + iapo(uo - C)Re]1⁄2

where c, and c2 are constants, the primes denote differentiation with respect to the argument in parenthesis, and the subscripts p and v refer to the so-called "pressure mode" and

"viscous mode" , respectively. Here I_ is the modified Bessel function whose order is the

azimuthal wavenumber, fl = integer, and C = ia/a is the complex phase speed. Clearly,

the kinematic boundary conditions are contained in the above asymptotic expressions. In

order to maintain a highly accurate finite difference numerical procedure, the asymptotic

boundary conditions are enforced. This not only fulfills the requirement of the value of a

specific quantity at a specific point, but also the manner it behaves (e.g., in powers of r).

Similarly, as r --* oo, the same asymptotic results hold provided that I_ is replaced

by the modified Bessel function of the second kind, K_ (Abramowitz & Stegun 1972),

and u0 and P0 are replaced by their values at r : oo. Note that in all likelihood, the
43

viscoussolution is unimportant for large r because the argument of KZ is large (i.e.,

Avr ,-_ (aRe)]r >> 1).

2.7 The Behavior of Disturbances in the Critical Layer

Although we have retained the most important effects of viscosity and heat conductivity in the critical layer, it is instructive to examine the nature of the flow in this layer

when (formally) Re = oo in equation (2.33). It is known that under certain conditions

to be made more precise momentarily -- the instability modes will become singular at

the critical point. It is desirable to know the "strength" of this singularity for numerical

purposes, and to see under what conditions at least some of this singularity may be eliminated, even when Re = oo. The latter condition gives us a generalization of the Rayleigh

inflection point criterion to compressible and heated flows in round jets. We emphasize

that, strictly speaking, there will be no singularity in the modes because the "diffusive"

effects of viscosity and conductivity have been retained. However, at large Re, we will

have a "near" singularity that does affect the accuracy of the numerical solutions.

By substituting Re = oo and a = -iaC in equations (2.33), the linearized, inviscid

and nonconducting disturbance equations can be obtained as

poiC_(uo - C)u + pou_ov + ic_p ----0 (2.43)

pOia(uo - C)v + p' = 0 (2.44)

poia(uo - C)w + i___pp= 0 (2.45)

r

_ i_w iau + _ + ¢ + _ + i_(_o - C)p = o (2.46)

r r

ia(uo - C)p -t- PloV"t- po( v + v' -t- iflWr ÷ i_u) : 0 (2.47)

where primes denote differentiation with respect to r. After some algebraic manipulations

in favor of the pressure, the above equations can be reduced to a second order ODE with
44

variable coefficients (essentially a Rayleigh equation) of the form

d_p 1 2u'o p'o)dP - --- (2.48) d,'+ (, "o- C Po_ + [_'("o- C)2poM] _,, _2],= 0

When the instability wave is neutral, the differential equation (2.48) has a regular singular

point at the point r -- rc where u0(rc) -- C (C is real for a neutral wave); the subscript c

hereafter refers to the value of a variable at the critical point. In the neighborhood of re,

the coefficients of the differential equation can be expanded in terms of powers of r - re.

For example,

_0 = c + _'_(r- ro)+ _"(r - r_)2/2+...,

(249)

p0 = P0+ P'0Cr- r0)+ p"(r- ro)'/2 +-..

where u'c = u_(rc) etc. Substituting (2.49) into the differential equation (2.48), we obtain

the approximate differential equation around the critical point for p, i.e.,

P" [r 2 +D,+D2(r-rc)+'"]p'+[EI+E2(r rc)+..-]p=0

rc

(2.50)

where

and

m2--

pcuc 1

D1 =

p°u_ re

2u_.. - ,,2 . ,2

u_ P° + e°

3,'+ 2u,2 po p_ q

_2

E1 =-(c_ 2 + _'2)

E_= 2j_[

r_

According to the method of Frobenius (Bender & Orszag 1978), equation (2.50) has two

linearly independent solutions of the form r = re,

p°=(r-ro)391(r-ro) (2.51)

Pb -- g2( r -- re) + Kp= log(r - r0) (2.52)
45

where

glCr- ro)= 1+ _lCr- r_)÷ _,C,"- ,.o)2+...

g2(r - re): 1 _-b1(r- re)_-b2(r- re)2 _-""

The coefficients a's, b's and K are found by substituting the Frobenius series (2.51) - (2.52)

into the differential equation (2.50) and equating like powers of (r - re):

3DI

al = --4

a2 =

3D2 + 4aiD1 - E1

10

bl :0

E1

b2 -- 2

DIE,- E2

K=

3

The general solution of the differential equation (2.50) is of the form

p = Alpa + A2pb (2.53)

where A1 and A2 are constants. The solution Pa is analytic, while Pb is a multi-valued

function because of the logarithmic singularity unless K = 0. For the particular case of

an unheated jet where P0 is a constant, the condition K = 0 reduces to

d ru_o

dr(_2 ¥_2r_) = 0 at r = to. (2.54)

Condition (2.54), which was established by Batchelor & Gill (1962) to be a necessary condition for the existence of an amplified inflectional type of disturbance in an incompressible

unheated round jet, is, in fact, valid even in the presence of compressibility. Furthermore,
46

the familiargeneralizedinflectionpoint theorem forthe existenceof instabilitymodes in

planar mixing layers,derivedby Lees & Lin (1946),i.e.,

-_)=0, (2.55)

at y = Yc where y isthe transversecoordinate,can be recoveredby letting1/re --*0 in

the conditionK = 0.

The behavior of the velocityand densityperturbationsin the vicinityof the critical

point isobtained by substituting(2.51)and (2.52)into equations (2.43)- (2.47).The

correspondingtwo independent solutionsnear the criticalpoint,r = re,are

_, _ u'_(' 3; _ ,,)+ [,3po_, "+2-_,p_ (6p'0- 8aip°)u'] j (r-re)'+""

2ib_.____2 +"" + Kv, log(r - re)

v_ = ap_u_

3 (3p',_- 4axp_)(,_ ,_) + ...

ua-" _2pcu_ + rv2Pc2."%t

p_

+'" + Kualog(r - re) "_= d_poe(r - re) (2.56)

t/)b --

/_ r re) 2 -tWa = c_rScu; ( ....

#

+"" + Kwalog(r - re)

,,'poro_(r- re)

3r, u_ - (4azrc + 3)u_

J[- • • •

Pa "- 0¢2rc ulc3

- --_u'c_---rc) +"" + Kpalog(r - re)

The above equations (2.56) show that the logarithmic term is present in all perturbations

unless the generalized inflection point criterion is satisfied; that is K = 0. In addition, the

leading order term in the pressure fluctuation and velocity perturbations of compressible
47

flowshas the same analyticalbehavior asin incompressibleflow.For example, the leading

term foru and w has a simple pole unlessthe azimuthal mode number iszero,irrespective

ofcompressibility.The densityfluctuation,or the temperature fluctuation,which isa new

quantitydue tocompressibility,has generallya simple poleatleadingorder.Consequently,

for a neutralmode, the p and v perturbationsare finite,the u and w modes becomes

infinitefor_ _ 0, and the p perturbationisgenerallyinfinite.Itispreciselythe singular

nature of the perturbationsat the criticalpoint that createsdifficultiesin the numerical

calculationsfor neutral modes. With the inclusionof dissipativeeffects,some of this

di$culty disappears,although a very finegridisneeded inthe vicinityofthe criticallayer

to resolvethe fluctuations.

2.8 Summary

In summary, the ODE's, which issymbolicallyexpressedby (2.33),issolvedby two

independent numerical schemes (tobe discussedlater)with suitableboundary conditions

on the jetaxisand inthe quiescentmedium at infinity.Nontrivialsolutionsexistonly for.

a certainset(a,/_,a),or equivalentlyin the more usualnotation,

The above equation iscalledthe dispersionrelationwhose realpart (ifpositive)indicates

instabilityand whose imaginary part isthe frequency of the wave.
I

2.5

2.0

1.5

...............

1.0-

0.5

i i i I

0.0 0.2 0.4 o.e o.e 1.0

Mean Velocity Profile,uo

0.0

48

0.8 1.0 1.2 1.4 1.e 1.8 2.0

Mean Density Profile ,po

Fig. 2.1 Jet mean profiles: solid lines (0 -- 1/5, M i = 0.0, T. -- 2),

dot (0 = 1/15, M i = 0.0, T. = 2), chaindot (0 = 1/30, Mi = 0.0, T. = 2),

dash (8 = 1/15, M i ---0.8, T. = 1).
49

Chapter 3

FORMULATION OF THE SECONDARY INSTABILITY

PROBLEM

We begin our discussionin thissectionwith a physical descriptionof how an

instabilitywave evolveson a jetand how the flow fieldmay be approximated at a couple

ofdiameters downstream of the jetnozzle.

When a jet isperturbed by an externalexcitationof a suitablewavenumber or frequency, an instabilitywave develops in the jet shear layer. For clarity,this instability

wave will be calledthe fundamental wave or the primary wave. In this work, we will

use the words "primary" and "fundamental" interchangeably.During the initialstages

of development, where the amplitude of the disturbanceisstillsmall,the stabilitycharacteristicsof thiswave are described quite accuratelyby the primary stabilityanalysis

developed in chapter 2. However, as thiswave propagates downstream, itsamplitude

grows and the mean flow diverges.The mean flow divergenceresultsin the reductionof

the mean flow vorticityand hence, contributesto a decreasein the localgrowth rate of

thiswave. Fartherdownstream, at approximately one or two diameters from the jetnozzle,thiswave reachesa finite-amplitude(i.e.,saturation)as a resultof nonlineareffects.

The flow fieldarisingfrom the presenceofthe fundamental wave may (roughly)be viewed

as convectingdiscretevortex rings.This flow isperiodic(approximately)in time and in

space.

A principalobjectiveof thiswork isto determine the sensitivityof thisperiodicflow

to variouscontrollablejetparameters such as wavenumber of excitation,azimuthal mode
5O

number, Mach number and jet temperature ratio.

The investigation is carried out via a secondary instability analysis. The instability

analysis of small disturbances -- which for clarity, will be called secondary disturbances

-- superimposed on a base flow consisting of the finite-amplitude fundamental wave and

a long time averaged mean profile is termed secondary instability analysis. In order to

make the secondary instability analysis as simple as possible, we invoke the following

assumptions:

1. The locally parallel mean profile and the shape of the fundamental mode is minimally

affected by the nonlinear self-interactions of the fundamental (i.e., shape assumption).

In physical terms, this fundamental wave can be thought of as a coherent structure

or a large scale instability wave.

2. The amplitude A of the fundamental wave, subject to an appropriate normalization,

remains approximately constant during the evolution of the secondary instability

disturbance.

3. The mean flow remains unchanged and the fundamental mode keeps its (initial) shape

and phase velocity during the evolution of the secondary instability disturbance.

4. The magnitude of the secondary disturbance is kept sufficiently small to ensure that

a linear secondary instability theory is valid.

The justification of the first assumption has been discussed in the primary stability analysis. Because of the nonlinear terms in the governing equations, a single linear instability

mode can never exist, by itself, in the flow. It will interact with itself to produce higher

harmonics and a mean flow distortion. The effects of the higher harmonics and of the mean

flow distortion in subsonic round jets were recently investigated in detail by Jarrah (1989).

His results indicate that the distortion of the mean flow due to wave Reynolds stresses is

quite small locally and is probably unimportant for the secondary instability. The small
51

change in A during the course of its nonlinear evolution relative to an anticipated strong

convective growth of a subharmonic disturbance justifies the use of assumption (2). This

assumption is further supported by experimental studies on planar mixing layers; for example, Ho & Huang (1982) found that when discrete vortices develop on the flow, the

subharmonic mode grows rapidly while the amplitude A remains approximately constant.

The recent work of Petersen & Samet (1088) supports assumption (3), especially on the

insensitivity of the mode shape to nonlinearities, although no detailed experimental results

exist for heated and compressible round jets in this regard. Furthermore, in subsequent

analysis, we find that the growth rate of the subharmonic wave is indeed very much larger

than that of the primary wave; this finding will, in fact, justify the use of assumptions (2)

and (3).

In order to have confidence in our results, two different approaches for secondary instability analysis are used here. The first is based on the method of multiple scales, while

the second is done by generalizing the normal mode analysis from a parallel base flow to

a periodic base flow.

3.1 Method of Multiple Scales

Kelly (1967) was the first to use the method of multiple scales to describe theoretically the strong growth of a subharmonic disturbance in an incompressible planar

mixing layer. His results are consistent with the the experimental observations of the

vortex-pairing phenomenon found by Sato (1956). By using full numerical simulations on

the instability of an array of Stuart vortices in a mixing layer, Pierrehumbert & Widnall

(1982) confirmed that the two-dimensional subharmonic modes responsible for the coalescence of vortices are indeed the most dominant instability modes. Recently, Monkewitz

(1988) generalized Kelly's work and examined vortex pairing in more detail in incompres_
52

ible mixing layers.

The formulation of the secondary instability problem is based on Kelly's ideas, although

a compact notation is used here. The finite-amplitude of the fundamental wave is regarded

as a moderately small perturbation parameter 6 [say, 6 = O(A) _- 0.1]. The physical

variable, which describes the perturbations, F, is assumed to have a general asymptotic

expansion of the form

F = 6F1 + eF: + 62F11 + $61_12 -_- 0(g2), 0 < _ << 6 << 1 (3.1)

where F1 is the fundamental (or primary) wave, F= is the secondary disturbance (which

will later be taken as a subharmonic disturbance), Fll denotes the quadratic nonlinear

interactions of the fundamental wave with itself, and F12 represents all the quadratic

interactions between F1 and F2. As the F1 mode propagates downstream, it travels into

a shear flow whose transverse length scale continuously increases, the growth rate of this

wave continuously decreases and the wave eventually becomes neutral. In shear layers, a

strong interaction between the fundamental wave and its subharmonic (i.e., one half the

fundamental frequency) occurs where the fundamental becomes neutral and its amplitude

is a constant. This neutral state, which arises from the spreading of the shear layer and

nonlinear effects, is really of no importance in this study. What really matters is that

there is a region in the jet where the amplitude of the fundamental (or primary) wave is a

constant and the growth rate of the subharmonic is enhanced from that given by parallel

flow stability theory; of course, this amplitude is _unknown" to us, although experimental

data suggests that it can be as large as 10% of the mean flow, and the cross-space structure

of this wave may be represented by the local linear instability mode of the jet (Petersen &

Samet 1988). In this region, the primary wave interacts with itself at 0(62) and higher.

This interaction is assumed to have a negligible effect on the mean flow and the primary

wave. Consequently, the Fll term in (3.1) can be ignored. Of course, the nonlinear

interactions of primary wave will eventually lead to the appearance of higher harmonics,
53

but they play no significantrolein the generationof the subharmonic at O(_6) because

they occur at O(52e) and higher .

When the primary wave saturates,the characteristicmagnitudes of the primary and

secondary disturbancesare 5 and e respectively.The focus of this work liesat order

_5 where the primary wave F1 interactswith the subharmonic disturbanceF2 to form

sum and differencemodes. Ifthe axialwavenurnbers,frequencies,and azimuthal mode

numbers satisfythe so-calledresonance conditions,which willbe discussedmomentarily,

then the growth rateofthe disturbanceF2 may be enhanced considerablyfrom that given

by linear parallel flow analysis.

3.2 Kinematic Resonance Conditions

We now illuminatethe physicsgoverning resonantinteractionsbetween two linear

waves of the form

where

FR : fl(r)E1+ c.c.

F2 = f2(r)E2+ c.c.

(3.2)

(3.3)

El = expf(alZ+ _1_)expo'it

E2 = expi(a2z+ _2_)expc2t

and c.c.denotes the complex conjugatesofthe preceding terms.

After substituting(3.1)- (3.3)into(2.17)- (2.19)and equating coefficientsat various

orders to zero,we obtain the corresponding disturbance equations. The zeroth order

solutionscome from the 0(5) and 0(e) terms. They are the eigenmodes of linearparallel

flow stabilityanalysis(2.33),and consequentlyneed no furtherexploration.
54

The equation which governs the quadratic nonlinear interaction between two linear

waves, obtained at 0(eS), is of the form

where

L12(F)12 = E÷R+ + E_R_ + c.c. (3.4)

E+ -- exp[i(_l + ot2)x -i- (/_I _-/_)_b] exPC_rl + ¢r2)t (3.5)

E_ ----exp[i(¢_l - ¢_2)x % (/_I -/_2)_] exPC_rl + _2)t, (3.6)

R+ and R_ are given in Appendix C, L12 is the linearized operator (given in Appendix

A) which governs the evolution of small disturbances in a parallel base flow, the tilde

denotes the complex conjugate of a quantity, and c.c. is the complex conjugate of the

terms preceding c.c..

Since L12 is a linear operator, the solution F12 consists of the superposition of the

homogeneous solution and the particular solution due to the forcing function, namely the

right hand side of (3.4). The homogeneous problem satisfies

L12(F12)ho,n = 0 (3.7)

and the particular problem is determined from

L12(F12)pat = E+R+ + E_R_ + c.c. (3.8)

where the subscripts pat and horn respectively denote the particular and the homogeneous

solutions of the inhomogeneous problem (3.4). Therefore, the physical variables can be

written, up to order _6, as

F = 6171 + eV2 + e6[(F12)ho,n + (Vl2)pa,] (3.0)

Since the homogeneous solution (F12)ho,_ has an initial magnitude of _6 and is described

by linear eigenmode analysis, it will always continue to be smaller than F2 by an order 6,
55

and hence, can be neglected.The interactionsbetween the two waves of the form given

by (3.2)and (3.3)willproduce the following four sets:

(al + a2, _1 + ,02,_1 + o,2), (-al + a2, -81 + ,02,_l + _2),

(at- a2,8,- 82,_I+ 02), (-al- _2,-81- 82, _i+ _2) (3.x0)

As long as these four setsdo not satisfythe dispersionrelation(c_,8,a), a solutionto

(3.4)may be found, in principle,in a straightforwardmanner by the method of variation

of parameters. In such cases,(3.4)representsan inhomogeneous equation for which a

particularsolutionmay be sought in the form

(F12)pat = E+FI+2(r) + E-F1-2(r) + c.c. (3.11)

Equation (3.11) shows that the growth or decay of (F12)#at is the same as that given by

linear parallel flow analysis. Further, in order to satisfy the resonance conditions, which

will be discussed momentarily, the disturbance F2 is required to be the most unstable wave

given by parallel flow theory. This implies that, for the case where resonance conditions

are not met, the (F12)#atwillremain smallerthan F2 by 0(6), and itcan thereforebe

similarlyneglected.

On the other hand, when one of the four sets of a, 8, and a satisfiesthe dispersion

relation,the operator L12, with the appropriateboundary conditions,admits nontrivial

solutionswhich are, in fact,the linearinstabilitymodes of the base flow. This leads

to the appearance of secularterms in (3.4)which resultsin the occurrenceof resonance

interactions.Accordingly,a solutionispossibleonly ifthe righthand sideof (3.4),which

contains the resonance terms, satisfiesthe so-calledsolvabilitycondition (or Fredholm

alternative).In general,thisconstraintcannot be satisfiedusing a regularperturbation

expansion. Before we proceed to considerhow thisconstraintcan be determined from a

singularperturbationmethod, letus now explorethe resonance interactionsa step deeper.

To do this,we considerthe linearizedinviscidprimary disturbanceequation (2.48)and its
56

conjugate;they axe,respectively,

_p • I 2u_ p_ dp

dr 2 q-('r uo-i_r/ol po)'_r -{-[c_2(u°-C)2p°M_ ;92r_ a_]p : 0 (3.12)

db I 2._ _ _ + [_2(u0-b)2.0M_ _2

+ (r - uo + i_r/a Po" dr _-_ a_]l_ : 0 (3.13)

where C is the complex phase velocity, # : -iaC, the primes denote differentiation with

respect to r, and the tilde denotes the complex conjugate of a quantity. The complex

phase velocity of an unstable mode which satisfies (3.12) and (3.13) can be expressed as

c(_,a) = cR(l_l, lal)+ sgn(_);c,(l_l, I_1), (3.14)

and the corresponding dispersion relation is

_(_,_): I_lC,(l_l,I_1)- i_cRClal,I_1) (3.1s)

where C = CR + iCi, and sgn(a) means the sign of a. In view of (3.10),the complex

equation (3.15)can be separatedintotwo realequations-- a growth rate equation and a

frequency equation-- which governthe resonanceconditionsbetween two unstablew_ve_.

These equations are,respectively,

I_'liCz(lall,I_ll)+ 1_21c,(1_21,1_21) -- 1_2+ c_11c,(1_2-1-C_ll,I/_1+_21) (3.16)

_aCR(l_ll,I_11)+ _20R(1_21,1_21) -- (_2+ c_I)CR(]c_2 + _11,l_x+ _21) (3.17)

where the e's and #'s can be positive or negative, and are so far arbitrary. It may be

noted that when/91 = 0, the resonance conditions are valid independent of the direction

to which a wave is propagating; this is because the inviscid equations (3.14) and (3.15) do

not distinguish/9 from -#.

In order to satisfy the resonance conditions, the wavenumbers and mode numbers of

two unstable waves are restricted to the following choices:

1. The fundamental mode is axisymmetric and has an axial wavenumber close to a

neutral wavenumber.[i.e.,_ ----0, C,(1_11,0)--_0].
57

2. The secondary disturbance,whether axisymmetric orhelical,isan almost maximally

amplifiedsubharmonic wave. [a2--ai/2].

Itmay be remarked that although choices(1)and (2)arenecessary,they do not guarantee

resonance because when the waves aredispersive,condition(2)placesa restrictionon the

phase velocity,CR.

When the resonanceconditionsare (nearly)met, a subharmonic wave interactseffectively

with the fundamental to produce an instabilitywave whose complex growth rate,_, is

(nearly)the same as that of the subharmonic. It iseffectivebecause resonant interactionsproduce secularterms which lead to a particularsolutionof (3.4)that grows more

rapidlythan the corresponding homogeneous solutionby at leasta factort. Of course,

the wavenumber and the mode number of the wave produced isalways the sum of the

wavenumbers and mode numbers of the two interactingwaves. Consequently,an axisymmetric subharmonic which interactswith itsfundamental willreproduce itself.On the

other hand, a helicalsubharmonic mode which interactswith the fundamental willnot

reproduce itselfbut willproduce a mode that propagates at an equal but opposite angle

to the z-dlrection.For example, an axisymmetric mode with the wavenumber 2a and"

the mode number 0 -- denoted by (2a,0) -- interactswith a helicalmode i-a,/_) to

excitea mode (a,_). Similarly,the mode (2a,0) interactswith mode (-c_,-_) resultsin

the mode (cf,-_).Therefore,an interactionbetween mode (2a,0) with modes i-a, +_)

resultsin modes (a,+_). Because a helicalmode which interactswith the fundamental

mode does not reproduce itself,itisnecessary to considerthe secondary disturbanceto

be comprised of a pair ofhelicalwaves spinningat an equal but oppositedirectionto the

x-axis.Consequently,ifa non-axisymmetric subharmonic isinvolved,we need to assume

that the secondary disturbanceisof the form

F2 : F_, + F2_ + c.c. (3.18)
58

where

F2r -----f2r (r) exp i(cx2z -{-_2_b) exp a2rt + c.c. (3.19)

F2Z = f2l(r) exp i(-ot2z -}-/92_b)exp a2,t -{-c.c., (3.20)

where the subscripts r and I denote a helical wave propagating in the direction of positive

and negative _, respectively; for clarity, we call these waves right and left spinning (or

propagating) waves, respectively. Since the terms involving the product of e are truncated in the expansion (3.1), the left and right spinning waves of F2 do not interact with

each other. Each will, however, independently interact with the primary wave Ft. For

example, F2t interacts with F1, producing F2r, and F2r interacts with FI, producing F2_.

Consequently, I_2 is again reproduced. Furthermore, since both the mean flow and the

axisymmetric FI are independent of the azimuthal angle, we can assume that the growth

rate of the right and left spinning waves, as a result of resonant interactions between F2

(given by 3.18) and an axisymmetric primary wave, are modified exactly the same. In

other words, there is no physical preference for the growth rates between the left and the

right spinning waves. In fact, a second approach -- which will be discussed in section 3.5

shows that the growth rates of F2r and F2t in a frame of reference that moves with the

phase velocity of FI are indeed the same.

When the resonance conditions are met, the subharmonic disturbance interacts strongly

with the fundamental to reproduce itself and therefore to increase its growth rate. The

modified subharmonic growth rate, as a result of resonant wave interactions, will be accommodated by a set of slow scales.

We now proceed to obtain the governing equation for the modified growth rate using

the method of multiple scales. The method of multiple scales is discussed in several books

[e.g., Nayfeh 1973]. Accordingly, we can write

F = _F1 + eF2 + e_F12 + 0(52), 0 < e << _ << 1 (3.21)

where
59

F1 -- Fl(x,t,_,r)

F_2 = F12(x, xl, t, tl, _, _, _)

(3.22)

(3.23)

(3.24)

and tl ---- 8t,_bl ---- 8_b, Xl ---- 8Z are the slow scales of time, azimuthal angle, and axial

direction, respectively. It may be remarked that although this set of slow scales eliminates

the nonuniformity caused by the secular terms in equation (3.4), it does not determine a

uniform zeroth order expansion. This is because the amplitudes of the mean flow and the

fundamental wave are assumed to remain unchanged with time, while the subharmonic

wave, chosen from the resonance conditions, is strongly amplified. The series expansion

_!3.2I) becomes di__r.deredwhenever the total ...................... magnitude of the growing subharmonic:_ __

exceeds the preceding term; uniform validityonly holds,at most, on a time scale8-I.

This setof slow scalesiscompatible with (3.21)where the 62 term which arisesfrom the

thirdorder interactions(i.e.,52e) istruncated.Itisused to examine the influenceof the

fundamental wave on itssubharmonic to linearorder in terms ofthe amplitude 6. In the

followingequation (3.27),we shallassume that the shape of the primary subharmonic is

unalteredby the appearance of the fundamental. This assumption ismore fullyjustified

by the second approach, tobe discussedinthe next section,which allowsthe shape of the

subharmonic inthe presenceof itsfundamental to be determined.

Sincethe physicalvariableF isa functionoffastand slow scales,the partialderivatives

in t,x, and @ become

a a a

a_a aa..a_

_ + ax_

a a a

a_ a_ a_

(3.25)
6O

As in the parallelflow analysis,we assume the normal mode concept and write

where

FI= f1(r)El+ c.c.

= B(,I,tl, + c.*.

(3.26)

(3.27)

El = exp i(alz +/_1¢) expalt

Ez = exp i(a2x + _2¢) exp a_t

and c.c. denotes complex conjugate of the preceding terms. It should be noted that in

(3.27), we have omitted the left propagating helical component -- with the understanding

that the left propagating helical wave which interacts with an axisymmetric primary wave

produces and, therefore, enhances the growth of a right propagating helical wave; similarly,

the left propagating helical wave can be reproduced through the interaction between a right

propagating helical wave and an axisymmetric wave.

The amplitude function B is slowly varying in space and time, and will be determined

at 0(e_) by means of a solvability condition; the mode shape f2 is that given by the parallel

flow theory.

It is now perfectly clear from equation (3.27) how the growth rate of the subharmonic is

modified. Because of the departure of the growth rate of F2 from the parallel flow analysis,

a set of slow scales is needed to reflect this change. For example, the slow scale 5x implies

a long length scale of O(R*/8), in addition to the usual length scale R*. This means that

as long as the evolution of the subharmonic is on a length scale that is much smaller than

O(R*/$), its local growth is given by parallel flow theory. However, if the growth of the

disturbance is on a distance that is comparable to O(R*/8), the change in the amplitude

function B will then be of the same order as that provided by the parallel flow analysis.

In temporal theory, it is clear that this change, which may be interpreted as an additional
61

(positive or negative) growth rate for the disturbance, is physically meaningful only over

a time scale .that is of Oil ).

3.3 Amplitude Equation

We now proceed to derive the amplitude equation of the subharmonic disturbance

when the resonance conditions are met. The amplitude equation will be derived using

two independent methods: the first is based on an _improved" inviscid analysis and the

second is based on a viscous theory. These derived amplitude equations will be called

an inviscid amplitude equation and a viscous amplitude equation, respectively. Here, the

word "improved" signifies that the primary instability modes are actually obtained from

viscous considerations. The inviscid amplitude equation is derived from a single ODE,

while the viscous counterpart is based on a system of ODE's. Since the wavenumber of

a subharmonic wave is near a maximally amplified wavenumber, the growth rate of the

subharmonic obtained from an inviscid analysis will provide a good approximation to that

obtained from a fully viscous analysis. We now proceed to derive the inviscid amplitude

equation, while the viscous amplitude equation is given in Appendix D.

For the particular case where the product of F1 and F2 reinforces F2, the disturbance,

F12, can assume the form:

where

F12 : fl2(Zl, tl, ¢1, r)E2 -_- c.c. -_- . . .

f12 = (u12, v12, w12,p12,p12) r,

E2 = exp i(a2x +/_2¢) exp azt

and the dots stand for the non-resonating terms that are unimportant.

(3.28)
62

We note that the linearizedprimary disturbanceequationsforan inviscidcompressible

jet at O(e6) can be manipulated to a singlepressureperturbationpl2(r)of the form

L(p12) -- _ "P UO -- C2 P0J _ -I- - r2

(3.29)

where

1

hi = -ia2pou2-I8B 8B 8B

,hs = hl_- 1 + h2_-_-z_+ hs_- 1 + h12/_ (3.301

2t_ d

+ -';'-](pov=) - i_=pow=/r + ia=(uo - C=)poM]p=

uo - Ct Po at-- -

I 2u_ p_

h2=-/o_2P0u0u2-[; uo-C p0

d

+ _](uopov2) - i_2pouow2/r

+;_2(uo- c2)poMfluop2+ "_u2)

h, = ;_(_o - C2)PoC,,,2/r + _2rp2)

1 2ub pb+ d -

h,2=;,_2P,,-+[; uo-C2 ,'0 2";]F_-+i_2P_'-/r-;_2C"°-C2)p°M]F"-

In the above equations, the tilde denotes the complex conjugate of a quantity, the prime.

denotes a differentiation with respect to r, C_ is the complex phase velocity of the subharmonic mode, while _bu_, _bv_, F,_ and F__ are given in Appendix E. Note that althoygh

(3.30) is derived based on an inviscid analysis, its coefficients, h's, are evaluated using

viscous parallel flow instability theory. Further, hn represents the particular set of the

interaction terms between F1 and Fz that give rise to a subharmonic disturbance, while

hi, h2, and hs come from the dependence of F2 on the slow scales.

We multiply (3.29) by r/{p0(u0 - C2) 2} to make it self-adjoint. The equation in the

self-adjoint form becomes

L(pn) = _o(UO - C2)' L(pn) = rhs { _o(UO - C2)2 } (3.31)
63

In orderfor (3.31) to possess a nontrivial solution, its right hand side must be orthogohal to all _'s such that _k - 0 where _ is the adjoint operator and _ is the adjoint solution.

The orthogonality requirement, or solvability condition, is necessary and sufficient for the

existence of solutions at order 5_. We note that since the operator £ is se!f-adjoint [i.e,

--- £] and the boundary conditions on _ are the same as those of the homogeneous solution

[i.e., £(P12) = 0], the adjoint solutions are identical with the homogeneous solution.

The orthogonality requirement becomes

0B aB aB

ql_l + qz_-_x1 + qs_l -t- q4/_ = 0. (3.32)

where

ql _oo • 1 2_ p' ----" P2 pO(UO- 02) 2 {ia2pOU, Jr- (-_ uo -- C p: )pov2

+i_2po_2/r- ! f. i,,2(uo- c2)po_fp2}- ' _p2__,) po_2d 'po(uo- c2)2

fooo r_ fi_2P.-+ (1 2uL _)po_ q4 = P2p0(t,0 C2) 2 r u0 - C2 po

+;_2_.-/r - i¢¢2(u0- O2)PoM2_-}-_ _p2__r)'P__dr, 'po(,,o- c2)2

wherethe (.)' denotesa dia'erentiation ofits argumentwithrespectto r, and¢, containsthe quadratic resonant wave interactions. Here q2 and qs are included to allow the

resonance to be manifested spatially. Their explicit expressions are not given here because

this thesis focuses only on temporally growing waves. The integrands in the q's are evaluated from linear, viscous, parallel flow instability analysis. It may be noted that if we

include the Fll term in the series expansion (3.21), an amplitude equation for the modified growth rate of the fundamental can similarly be obtained at order 52 (see Monkewitz

1988). This modified growth rate, which will be coupled into equation (3.32), provides a

relationship for the mutual feedback between the subharmonic and the fundamental. In
64

accordance with our previous discussion, we shall not obtain the amplitude equation for

the primary wave because this amplitude is assumed to be a constant. The constant amplitude A assumes a catalytic role in the enhancement of the growth rate of an infinitesimally

small subharmonic secondary disturbance. Therefore, we called our instability analysis

a linear "secondary" instability analysis instead of a "weakly" nonlinear analysis for two

interacting waves.

The equation (3.32) together with its complex conjugate yields

a2 02 a2

Iqll2bi_2+ [q212_-_z21 + Iqsl2_-_l+

a2 02 02

P_al[2_lq2atlax---I7+ 2_qsat_a_ + 2q2qsa_--_ ]B = Iq*12B (3.33)

where the tildestands for the complex conjugate of a quantity. The above equation is

vMid for temporal or spatialanalysisbecause the amplitude B isa functionofspatialand

temporal variables.Sincethe coei_icientsin (3.33)are independent ofthe slow scales,we

can write

B -_exp(AlZl + A2tl+ A3_1) (3.34)

and substitutingthe above into(3.33)yields

Iq_12A_ + Iq212Al + Iqsl2A_ + (2_lq2)RAIA2 + (2_tqs)RA2As + (2_2qs)RAIA3 = Iq, I2 (3.35)

The A's are in general complex and ( • )/z denotes the real part of a complex number.

Each real part of A1, A2, A3 modifies the streamwise, temporal and azimuthal growth rate,

respectively. Similarly, each imaginary part of A's modifies its corresponding frequency (or

wavenumber). Running parallel to Kelly's investigation of the mixing layer, our investigation is based on a disturbance of a fixed spatial wavenumber growing in time. Therefore,

A1 and As are zero, while A2 is real. (3.35) can therefore be simplified to

I Iq4] (3.36)

A2 -----"ri_1_.
65

The secondary subharmonic disturbancecan be expressed in terms of the originaltime

variableas

IP2 --f2exp f(a2z -{-/3z_)exp(¢2 q-A28)t (3.37)

The dependence ofthisgrowth rateas a functionofvariousjetparameters isdiscussedin

Chapter 6. Clearly,the growth rate of the subharmonic wave ischanged by A26 clueto

parametric resonance.

3.4 Normal Mode Analysis

Our previous approach models the secondary instabilitiesof a round jet,which

resultsinvortex pairing,by a perturbationanalysisoftwo instabilitywaves, F1 and F2,

of differentwave vectorsundergoing interactions.When theirproduct reproduces with

F2, the growth rateofF2 ismodified. While thismodel providesthe physicsof resonant

interactions,ithas at leastthree shortcomings.

1. The finitesmplitude of the fundamental wave isregarded as a small perturbation

parameter. Thereforethe perturbationtheory,can providean accuratesolutiononly

forthose valuesof 6 forwhich the perturbationseriesisan accuraterepresentation

of the exact solution.As usual,the region of validityof an asymptotic analysisis

not known a priori,and remains to be determined from a more complete analysis.

2. There is no mechanism for the determination of the threshold amplitude of the

fundamental where the phase-lockingwith the subharmonic occurs.Itisnow known

that thisphase-lockingisa mechanism forthe onset of secondary instability.

3. As previouslymentioned,thisperturbationapproach failstodetermine the correction

to the shape of a secondary subharmonic mode; itonly providesinformationon its
66

growth rate.

These shortcomingscan be circumventedby an alternativeapproach developed by Herbert (1983).He pioneeredthe development ofthe normal mode analysisofthe secondary

instabilityarisingfrom the presenceof TS waves in plane channel flows.His theory predictswith reasonableaccuracy the growth rate of a 3-d secondary disturbanceup to the

stagespriorto laminar breakdown and transition.In his subsequent analysisof the (linear) secondary instabilityof flatplateboundary layers,he has shown that thismodel is

capable ofexplainingallthe essentialflow physicsofstaggered lambda vortex formation

(1984) and peak-valley splitting (1985) in Blasius flows.

Motivated by Herbert's success, we will adapt his approach to subsonic heated round

jets. In this approach, a su_ciently small disturbance F2 is superimposed on a periodic

base flow consisting of the steady unidirectional mean flow F0 and a finite amplitude

axisymmetric primary fundamental wave Fl. We assume that the mean flow as well as

the shape of F1 is minimally affected by the nonlinear self-interactions ofF1. The choice of

an axisymmetric wave is based on the resonance conditions established previously. Using

a frame of reference moving downstream with the phase velocity of F1, it is appropriate

to decompose the perturbation vector field F as

F(z,r,4?,t)= AF1(z,r) ÷ eF2(z,r,@,t) (3.38)

where

FI = fl(r)e _az ÷ c.c. , (3.39)

A, as discussed earlier, is assumed to be a constant, and z now refers to the streamwise

coordinate of the moving frame of reference. In (3.38), A is usually small but finite (say,

A -_ 0.1), not a perturbation parameter, while e is assumed to be suf_ciently small to allow

linearization about a steady and spatially periodic base flow. Of course, the range of e for

which linear theory is valid remains to be determined by comparing it with future work,
67

say,directnumerical simulations.The finitevalue of A willremove the usual accuracy

issueassociatedwith an asymptoticanalysis.Further,sincethe amplitude A has a physical

meaning provided the eigenfunctionflissuitablynormalized,we choose,forconvenience,

max [ul(r)[ = 1/2 (3.40)

O_<r_<oo

so that A is the maximum magnitude of the streamwise velocity perturbation. This norrealization, which is also used in our multiple scales approach, is not unique and is rather

subjectivebecause fl isan eigenfunction.Herbert normalized A so that itisthe maximum rms value ofthe streamwise velocityfluctuation.The magnitudes of eigenfunctions

of Cohen (1986) are normalized such that the experimentaland theoreticalprofileshave

identicalareas.At thispoint,itisimportant to note that thissecondary instabilityanalysisisstillinear,as isthe classicalinstabilityanalysis.In the latter,the flowislinearized

about a locallyparallelflow.The coefficientsinthe resultantequationsdepend solelyon r

sincethe (z,_b)and t dependences of the (primary) disturbanceare triviallyextractedby

Fourier-Laplacetransforms.This leadstothe mode shape fl(r).In the present(secondary

instability)analysis,linearizationismade about a flowwhich isperiodicin thestreamwise

coordinatex. This streamwise periodicflow becomes independent of time in a reference

frame that moves with the phase velocityofthe fundamental wave, and isindependent of_b

0

sinceF1 isaxisymrnetric.Therefore,the _band t dependence ofthe secondary disturbance

can again be extractedby Fourier- Laplace transforms,but the z dependence requiresa

more carefulconsiderationand additionalassumptions,to be discussedshortly.We can

now apply the normal mode analysis in _band t and write

F2 = f2(z,r)ei/32¢'e a2t + c.c., (3.41)

where b2 isthe complex growth rate,and/_2 isthe azimuthal mode number. Aftersubstituting(3.38),(3.39)and (3.41)intothe governingequations(2.17)-(2.19),and neglecting

quadraticand higherorderterms ine,we obtaina system oflinearpartialdifferentialequa-
68

tionswhich govern the instabilityof the linearizedsecondary disturbance,F2. Apart from

a complicated r dependence and numerous other terms,theseequations are essentiallyof

the classicalMathieu type because the coefficientsare (streamwise)periodic.Itisknown

that,accordingto Floquet theoryoflinearperiodicdifferentialequations,a Mathieu equation whose coefficientsare x-periodichas two important classesof Floquet solutions:a

x-periodicand a 2x-periodicfunction.With guidance from Floquet theory (Ince,1956),

we can thereforeexpressthe secondary disturbancein the Floquet form as

f2(z,r)-* e_ffif_2(z,r) (3.42)

where #ffiis the characteristic exponent and _2 is periodic in z. The solutions with period A1

-- where A1 is the wavelength of the primary wave -- are called the primary (fundamental)

resonance, while the solutions with period 2A1 are known as the principal parametric

(subharmonic) resonance. The instability modes which arise from these resonances can,

in general, be expressed in terms of Fourier series as

co

h-- _ f2j(r)e _ja_ffi/2 (3.43)

j_--co

The fundamental modes are the harmonic series whose period is the same as the streamwise

wavelength of the primary wave and are given by j = even. The subharmonic modes have

odd j, and therefore, constitute a subharmonic series whose basic wavelength is twice

that of the primary wave. The fundamental and subharmonic modes are completely

uncoupled since the wavelength of the basic flow is different from the basic wavelength of

the subharmonic series. In this work, we shall investigate only the parametric resonance

case.

We now digress briefly to present some known results in incompressible boundary layer

flows. The fundamental modes produce the peak-valley splitting of the lambda-shaped

vortical structures, while the subharmonic modes cause the staggered formation of lambda

vortices. By approximating the Fourier series (3.43) to consist of merely two oblique waves
69

propagating at equal and opposite angles to the x direction, Herbert (1984) has shown that

these two oblique subharmonic modes, which provide a reasonably accurate approximation

to the streamwise structure of the subharmoni_ disturbance, are sufficient to give rise to

the formation of staggered lambda vortices.

We return to equatic, n (3.43). Its substitution into the linearized disturbance equations yields an infinite set of ODE's for the complex growth rate _r2 and the mode shape

f2j. Numerically, the streamwise structure of the secondary disturbance is expanded in

terms of a finite Fourier modes whose stability is governed by a system of Na ODE's --

where Art is 5 times the number of Fourier modes used. Numerically, these (stability)

equations are approximated by a complex algebraic elgenvalue problem whose computation requires O(N_) storage and O(N s) operations (see Chapter 4) -- where N'b equals

Na times the number of grid points in the computation domain. In order to be practical in the numerical evaluation of the secondary stability problem and in order to make

a direct comparison with the previous perturbation approach, simplification of equation

(3.43) is deemed desirable. The justification of this simplification will now be discussed.

In controlled jet experiments, a primary instability wave, which is often generated by

an acoustic source or through vibrating ribbons, produces a vortex structure that is responsible, at least partially, for the mixing and spreading of jet shear layers. In order to

control the global spreading rate via the manipulation of this vortex structure, a discrete

subharmonic disturbance can be excited simultaneously with the primary (or fundamental) frequency. In accordance with our discussion in section 3.3, when the subharmonic

disturbance is axisymmetric, it is assumed to be a subharmonic wave; when the subharmonic is helical, it is comprised of a pair of helical waves whose propagation angles are

equal but opposite with respect to the mean flow direction. As dictated by the resonance conditions, these excited discrete modes are sufficient to promote a strong growth

of a subharmonlc disturbance, in order to compare the two approaches, we assume that

the (secondary) subharmonic growth rate is minimally affected by the unexcited higher
7O

terms inthe subharmonic series.In otherwords, we assume that the (two) Fouriermodes

(al//2and -al/2) are reasonablysufficientto representthe the streamwise structureof

the subharmonic disturbance.

With the preceding assumptions,we can approximate the subharmonic disturbanceby

F2 = eb=Ze_2tci_@(f2,1 eial/2 q- f2,-1 c-/al/2) q- c.c. , (3.44)

where

and,

f2,1 = (U2,1, V2,1, ltt2,1,1l_,l,P2,1),

f2,-1 = (U2,-1: I/2,-1, W2,-1, _,-1, P2,-I)

As in primary stabilityanalysis,we employ temporal theory,which impliesthat b= = 0

and &2 _ 0 isthe complex eigenvaiueto be solved. Thus, in a temporal theory,we can

write

F2 = e#2teiB2_(f2,1eiat/2 + f2,-le -iaa/2) + c.c. (3.45)

Note that in the moving frame of reference, (3.45) shows that the complex growth rates

b2 of a pair of helical modes, whose propagation angles (with respect to the =-axis) are

equal and opposite, are exactly the same.

To recall, n main result of the multiple scales analysis is that the secondary disturbance

can interact with its fundamental to produce a wave whose wavenumber and frequency are

the same as the disturbance itself and can therefore reinforce it. Since such a destabilization of the subharmonic requires the matching of its phase velocity with the fundamental,

our calculations focus on the particular ease where the subharmonie becomes phase-locked

with the fundamental and b2 becomes purely real. The modal representation of F2, given

by (3.45), will transform the linearized secondary disturbance equations from a system

of partial differential equations into a system of ODE's. These ODE's are obtained after

substituting (3.38), (3.39) and (3.45) into (2.17)- (2.19), and by neglecting quadratic and
71

higherorderterms in e. The resulting equations become

(Ao + AAI + A2A_ + ASA3)g = b2_lg (3.46)

where

g = (U2,1, U2,-1,1)2,1, t)2,-1, iV2,1, W2,- 1, P2,1, P2,- 1, P2,1, P2,- 1) T (3.47)

Recall that when the secondary disturbance is axisymmetric, we need only to consider

one subharmonic wave. In this case,/'2,-1 = 32,1 where tilde is the complex conjugate of a

quantity. In (3.46) the precise elements of A0, A1 and ]3 can Be extracted from Appendix

F, the A 2 term arises from the cubic interaction, A2e, while the A s term originates from

the quartic interaction, ASe. The quartic term sterns from the V2(p/p) term of the energy

equation; of course, this term would have disappeared if the temperature had been used

as a dependent variable. Note that for incompressible and unheated flows, A 2 and A s do

not exist because the nonlinearity in the governing equations is quadratic. Further, since

our representation of the flow arising from the finite-amplitude saturation of the primary

wave is approximated only to order A (see Chapter 6), we ignored, for consistency, the A 2

a_d A 3 terms. Hence, the secondary disturbance equations become

(_0 + AAI)g : _2]_g (3.48)

We should recognize that when A _ 0 in (3.48), the parallel jet mean flow is slightly

distorted by the presence of the fundamental mode. This leads to a modified growth rate

of the subharmonic from that obtained through a parallel flow analysis.

At this point, it is crucial to realize that the difference between our two different

approaches lies in the treatment of the terms representing the interactions between two

waves. When using the normal mode approach, the interaction terms appear explicitly as

the coefficients in the disturbance equations (3.48). In contrast, these terms, when using

the method of multiple scales, appear as inhomogeneous terms in the equation (3.4).
72

The boundary conditions at r = 0 and r ---- co for the secondary disturbances are

identical with those of the primary stability analysis. The system of ODE's (3.48) together

with the appropriate boundary conditions constitute an eigenvalue problem that yields,

for a given mean flow and primary eigenmodes, a dispersion relation of the form

= (3.49)

The evaluation of the eigenvalues and their corresponding eigenfunctions is described in

Chapter 4, and the secondary instability results are discussed in Chapter 6.
73

Chapter 4

NUMERICAL METHODS

This section deals with the numericM formulation of the eigenvalue problem which

arises from the analysis of the stability of subsonic round jets. The primary stability

problem can be cast in a system of 5 coupled ODE (2.33), while the normal mode analysis

of the secondary (subharmonic) instability problem is approximated by a system of 10

coupled ODE (3.48). The numerical solution of these systems of equations is obtained by

a spectral collocation method with Chebyshev polynomials, and the primary instability

problem is checked by a finite difference method. For both methods, a function, say f(z),

is approximated by a discrete set of values of f(z) on a set of nodal points z_. The discrete

pointwise approximation of the function f(z) will eventually convert a linear differential

equation into a system of algebraic equations which can be solved by several techniques.

We begin by outlining an application of spectral method to the instabilities of shear layer

generated from a jet emanating from an axisymmetric nozzle.

4.1 Spectral Method

The spectralmethod has now been successfullyimplemented in many types of

hydrodynamic stabilityproblems such as boundary layers,wakes and mixing layers.A

detailedand extensivediscussionofthe theoryand applicationofspectralmethods, particularlywith regard to fluiddynamic problems, can be found inthe monograph of Canuto,
74

Hussaini,Quarteroni, and Zang (1987). The startingpoint of a spectralmethod is to

expand a functionu = u(z) in an infinitecomplete setof orthogonalfunctionsof the form

CO

u(z) = __, akTt(z) (4.1)

k=O

where the Tt(z) are the known basis functions and the coefficients at (k = 0, 1,..-) are

called the spectrum of u(z). Numerically, the function u(z) is expanded in terms of a

truncated set of the basis functions Tt. For example,

)/

u _s un(z) = _ akTk(z) (4.2)

k=0

where un(z) is an approximation to u(z) for each A/ E 0. For a smooth function with

periodic boundary conditions, the natural choice of the basis functions is sines and cosines.

This leads to the usual Fourier series and its finite version. On the other hand, for non

periodic boundary conditions, the most widely used basis functions are the Chebyshev

polynomials. In any expansion, the task is either to find the set of values of u at the nodal

gridsthrough an appropriateinterpolationor the spectralcoefficientsat (k = 0,1,...3/).

Intuitively,when the number ofgridpointsissufficientlyarge,they are equivalentto each

other.

The Chebyshev polynomial ofdegree k, Tk(z),isdefinedby

Tk(z) = cos(kcos -1 z) -1 < z < 1. (4.3)

Therefore, To = 1, T1 = z, and the higher polynomials can be generated by the recursion

relation,

2zTk : Tk+x(z) + T_-l(z), k >_ 1 (4.4)

Details of various identities, orthogonal properties, and numerical quadratures can be

found in the book by Gottlieb and Orszag (1977). The Chebyshev polynomials are even

for even k and odd for odd k, as illustrated in Figures (4.1a) and (4.1b). They are the
75

solutionsof the Sturm-Liouvilledifferentialequation

+ k2Tk = 0 (4.5)

which are bounded at z = -1 and z = 1.

It is known that if a smooth function is expressed in terms of the eigenfunctions of

a suitable Sturm-Liouville problem, "spectral accuracy _ is obtained. This means that

the spectrum ak, according to this basis function, decays more rapidly than any power

of k. To he more specific, the Fourier cosine series for an even function f(x) is used for

illustration. The appropriate expansion is

cO

f(_): _ _cos(k_)o_<• _<_ (4.6)

k=O

where

ak : --2/0-f(x) cos(kz)dz, k : 0, 1,2,... (4.7)

If

If a periodic function f(x) has 3{ - 1 continuous derivatives, and the 34th derivative is

integrable,then,afterintegrating(4.7)by parts (34- 1) times,we find that

2 f

ak= _ f0 /_)(x)cosCx_/2 - kx)d_ (4.8)

where X = 1 when 34 is even and 0 otherwise, and the f(_) is the 34-th derivative of f. It

is worth noting that the integral in equation (4.8) is bounded by some constant which is

independent of k but may depend on 34. Therefore, the truncation error, say e, using the

3/+ 1 modes to represent an infinitely smooth function by (4.2) can be estimated to be

lel < eonstant(34)n _ , for any 34 (4.9)

This is because each of the coefficients a_ [for k _> J¢] satisfies (4.8) and the neglected

series expansion for f,

cO

k=Jl
76

willbe of the order givenby the righthand sideof (4.9).[More preciseerrorestimatesfor

the eigenfunctionexpansion in terms of Chebyshev polynomialsisdiscussedin Canuto et

al].Itmay be remarked that a finitedifferencemethod with )4 gridpointsusuallyhas an

errorestimate of )4-P where _ isthe order of the accuracy of a particularscheme used;

for example, a second order method has p = 2. Thus, ifa functionisinfinitelysmooth,

then, according to (4.8),the spectrum ak decays fasterthan any finitepower of/c,or,

simply,itdecays "spectrally'.An important consequence ofthisspectralaccuracy isthat

accuratenumerical resultscan be obtained by usingrelatively"few" gridpoints,and itis

preciselythisaccuracy that makes spectralmethods an emerging powerful computational

tool.

Further,aftermaking the substitutionz = cos(z)into(4.5),we findthat

d2T 

dx2 + k2Tt --0 (4.10)

so that the Chebyshev polynomialsare the usual trigonometricfunctionsof the argument

k= -- kcos -1 z. From (4.3) and (4.6), we obtain

oo

f(z) = _ akTk(z), (4.11)

k=0

where z = cos-X(z). Therefore,according to equations (4.11)and (4.1),the Chebyshev

serieshas the same convergencepropertiesas the Fouriercosineseries,sincethe former is

reallythe latterin disguise.

Ifwe expand an arbitraryfunction u in (A/+ 1) Chebyshev polynomials,(4.2)will

contain (_ + 1) unknown coef_cients,ao,al,...,an. In order to evaluate these using

collocationmethods, the approximate expansion of u(z) issatisfiedexactlyat a set of

(,_-l-1) points,say z0,zl,..,z#. These points are calledcollocationpoints;they are

arbitrarilysubjected to a rather qualitativeconstraint.In order to see how to choose

thesecollocationpoints,we go back to (4.8)and note that the requiredcoefficientsak(k =

0,1,..,A/)are given by a trigonometricintegral.This integralcan be evaluatedextremely
77

accurately by a trapezoidal formula because, in fact, this scheme is equivalent to an 9Vpoint Gaussian quadrature rule. Thereforel an optimum Way is to divide the interval

0 < z < z"into equal increments:

Xj---- _"

j = 0, 1,...,._ (4.12)

or in terms of the original variable, z,

zj -- cos -_-- j = O, 1,...,.Y (4.13)

The set {zj} is called the Gause-Lobatto points where dT_/dz = T_ = 0. Since Tk(z) is

a polynomial in z of degree k, u)_ is an R-polynomial approximation to u(z).

In order to represent the derivative of the function at the collocation points, Lagrange

interpolating polynomials of degree R are used, i.e.,

A'

(4.14)

2)d2 + 1

Doo = ---- D_,_

6

cj (-1) t+j

D 't- forj#k

Dit -zj for j= k

2(1-

(4.17)

t=0

where the interpolant At(z) can be expressed in an explicit form:

AtCz) = (1 - z 2) T'_(z) (__l)t+ 1 (4.15)

2

where co = cg = 2 and ct = 1 for 0 < k < R. Of course, this Lagrangian polynomial

representation provides perfect accuracy for all functions with polynomials of degree _< A/.

The pth derivative of u(z) at the jth collocation point, in terms of the values of u at, all

the collocation points, is given as

amu -_

•._zpl i = _ D_.tut, (4.16)

t=0

where ut = u(zt), the Dlt are the elements of the derivative matrix given in Canuto, et

a]. (1987) as
78

and,

D_jk : (Dj_)'

With approximate representations (4.14) and (4.16) for a function u(z) and its derivatives,

itispossibleto convert any ODE into a system of algebraicequations for the (A/-{-I)

unknowns u(zo),u(zl),.,u(z#), where zo, Zl...z# are the collocationpoints. These

(linear)algebraicequations may be solved by severalpowerful methods [seeWilkinson

1965],although a principaldifficultyisthat the matrix associatedwith theseequationsis

not very sparse.

Clearly,inorderto apply a spectralmethod with the Chebyshev polynomials,the physicalsemi-infinitedomain 0 < r < co must firstbe transformed onto the finiteChebyshev

domain, [z[_< 1. For boundary layerflows,the two mappings which have been successfullyemployed in transformingthe infinitedomain (0,co) onto the half intervalof the

Chebyshev domain (0,1) are the exponentialmapping (Laurien & Kleiser1985) and the

algebraicmapping (Herbert 1984). These mappings, which clusterthe collocationpoints

near the _wall"at r - 0 and become more widely spaced away from the wall,areinefficient

insubsonicround jetswhere steepgradientsoccur ina very thin criticallayerofthickness

O(Re-_) locatedinthe neighborhood ofthe jetradius,not on the jetcenterline.The thin

criticallayerin jetsrequiressufficientlydense collocationpoints in thislayer.In fact,a

difficultnumerical task that arisesfrom the coupled system of ODE isthat any numerical

scheme must be capable of capturingaccuratelythe eigenfunctionsand theirderivatives

up to second order,simultaneously.The viscousand conduction terms, which contribute

to the second derivatives,axe includedto eliminateany singularbehavior ofinviscidneutraleigenmodes. We utilizetwo differentmapping methods for the stabilitycalculation

ofcompressiblesubsonicjets.The firstmethod maps the infinitephysicaldomain onto a

finiteChebyshev domain. The second approximates the unbounded physicaldomain by

a finitedomain which ismapped onto the Chebyshev domain. We now outlinethese two
mappings.

Mapping A

The firstmapping maps from r = 0,co into z = 0,1 via

11 1 r n

g +

[ ]

2(r+ll) ]expL-v J

79

k=0

the truncation error, e, which arises from using the )4 -f 1 modes to represent r, is

O0

/_=A/

This error e must be so exponentiallysmall that it will not affectthe accuracy of a

spectralmethod. In (4.18),the pointr = 0 ismapped onto z = 1,while the point r = co

The parameters 1112and n are adjustableconstantswhich controlthe amount ofstretching

inthe r direction.Note that when n = 1 and Ii= O,the mapping reducesto the classical

exponential mapping. For a given 11and 12,n controlsthe number of gridpoints in the

neighborhood ofthe criticalpoint (nearr = 1)When the valueof n increases,more nodal

points are clusteredaround the criticallayer.The parameters 12and n redistributesome

of nodal pointsnear r = 0 to the regionwhere strong gradientsexist(nearr = 1). For a

given number of collocationpoints,the concentrationof nodes in the physicaldomain is

governed by the metric coefficient:

7.n

z' -- dz _ 11 nrn-1 exp[---] (4.20)

dr 2(r ÷ Ix) 2 l_ t /2 J

The higher the value [zt[ in the neighborhood of the critical layer, the greater the concentration of nodal points. As a caveat, in order not to decrease the exponential convergence

ratesassociatedwith a spectralmethod, a mapping, r --r(z),which maps the physical

domain onto the Chebyshev domain must not be too complicated [Canuto et al.].In other

words, ifwe expand

)/

(4.18)
8O

is mapped onto z - 0. The point z -- 0 is not used as a collocation point, and we used odd

polynomials expansion so that the boundary conditions at this point are automatically

satisfied.

In general, a function and its higher order derivatives which are regular at infinity in

the physical domain r might not necessarily be regular in the computational domain z

(Spalart 1984). In order not to destroy a spectral accuracy, the function and all its higher

order derivatives at z = 0 must be sufficiently well behaved so that the formula (4.8) for

the spectral decay of the spectrum ak is valid. We now examine the behavior of u(z) as

z -_ 0. As r _ oo, z vanishes algebraically according to the first term on the right hand

side of (4.18). Since a disturbance decays exponentially (2.38) - (2.42), we can conclude

that all the derivatives of the disturbance --_ 0 as z --* 0, i.e.,

dSu

lim _ 0 for all j (4.19)

Although no studies have been done here, we suspect that when a function is properly

resolved, mapping A will preserve the rapid convergence rates associated with a spectral

method. Further, the algebraic term of this mapping provides an adequate resolution of

nodal points far away from the origin.

The cylindrical geometry of a round jet inherits a coordinate singular point at r = 0

which requires a little care. To remove this singularity, we imposed explicitly the boundary

conditions at • - 0. These boundary conditions, which are independent of the viscosity,

can be obtained from (2.38) - (2.42). For/_ -- 0 or/_ =- an even integer, they are

and, for/_ = odd integer

du dp dp

= = d-; = d-7= d-7"= 0 (4 21)

dv dw

u = p "- p : d-"r= d'_ : O. (4.22)

Mapping B
81

For a disturbance which is Confined to thecritical layer at r - 1 and which vanishes

rapidly away from the origin, an alternate approach is to transform a truncated finite

physical domain 0 < r __ Rmax onto the computational interval 0 < z __ 1 according to

Z _ COS 0

sinh[r(y - B)]

(4.23)

where

l+(e" + 1)(ro/Rm )J'

r is the concentration parameter which varies from zero for no stretching to high values

which concentrate more grid points near re, and rc is the approximate location of the

critical layer. Figure (4.3) shows the distribution of collocation points for various values

of r. [See Anderson et al. 1984 for a part of this mapping]. Since the slope, dz/dr, is

large in the region of the critical layer, the mappings (4.23) provide high resolution for a

suitable value of r. Unfortunately, because dz/dr vanishes at Rmax, any disturbance under

this mapping has zero slope at Rmax, unless the asymptotic behavior of this disturbance

is implemented appropriately (see remarks in section 4.1) into the discretization. As a

consequence, this mapping will work only for the case where a disturbance is confined

mainly in a narrrow region and decays rapidly. Extensive sensitivity studies were performed to determine the effect on the eigenmodes due to the zero slope approximation of

a disturbance at Rm_. We found that when aRm_, is sufficiently large, typically 4 to 7,

the stability characterlstics are rather _nsensitive to the:value of Rm_ as long as there is

an adequate clustering of points in the critical layer. This insensitivity of the numerical

results can be attributed to the sufficiently rapid decay of a disturbance. According to

-t/2 the asymptotic solutions (2.38)- (2.42), a disturbance is of order P_= exp(-aRm_,) at

r = R,_z, which issmall enough to be approximated by zero derivative whenever _R,_z

is sufficiently large. In this work, r is between 5 to 10, while rc ranges from 0.95 to 1.2.
82

In eithermapping, as a resultof using eitherodd or even polynomials over the half

interval,the spectralderivativeoperators (4.16)need to be modified accordingly.The

matrix elements of odd and even derivativeoperatorsbecome

D_jk(odd) -- D_jk(full ) - D_j,n_k(full )

(4.24)

D_j_(even)--- D_t(full)+ D_j,_,_k(full )

(4.25)

where

)4+1

N--

2

j--O,I,...,N

k =0,1,...,N

The superscriptp denotes the pth derivative,)4 isan odd integer,and N isthe actual

number of collocationpointsin the halfChebyshev domain. The derivativesin the physicaldomain can be expressed in the computational domain by using the chain rule of

differentiation.For example,

u(r)=

du du dz

dr dz dr

--_= dr dz 2 dr + dzdr-T

(4.26) "

where

dz dz dO d'Cj

dr dO d_ dr

(4.28)

d2z dz td'g_,d2g d2z fd'_32(dO.2 dz dg d2y (4.29)

In an implementation of a spectral collocation method, the spectral derivative operators

are replaced by full matrices, non-constant coefficient terms are represented by diagonal

matrices, and the boundary conditions (when exists) are imposed by replacing the appropriate rows of matrix. The spectral matrix representation of the stability equations
83

convertsthem intothe generalizedalgebraicproblem

AX ----o'BX (4.30)

where A and B are complex general square matrices of size N'; N ° is 5x N for the primary

stability problem, 10x N for the secondary subharmonic analysis, and N is the number of

collocation points in the computational domain.

There are generally two different classes of numerical approaches for computing the

eigenvalues a of (4.30), global and local methods. Global methods, which require no

initial guess on the eigenvalues and compute all the eigenvaiues of the discretized system,

are based on the QZ algorithm (Wilkinson 1965). For a matrix of dimension N*, it

requires 0(N *_) storage and 0(N *s) operations. Local methods, however, require a starting

guess for the eigenvalue and an iteration scheme based on an accelerated inverse Rayleigh

iteration method (Wilkinson). The global algorithm requires much more computer time

and storage than the local method since they compute the whole spectrum of eigenvalues

of the discretized system. Therefore, in order to maximize the numerical efficiency, it

is desirable to minimize the number of Chebyshev polynomials used in a global method.

On the other hand, the main shortcoming of a local method is that it converges to the

true eigenvalue only if an initial guess is sufficiently close to it. As a rule of thumb, in

the absence of any prior guidance on the eigenvalue, we select the physically desirable

eigenvalue from a global method and use it as input to a more refined local method

(using more collocation points) to obtain a more accurate eigenvalue and its corresponding

eigenfunction.

For the tanh mean profiles of (2.28) - (2.30), we found that the eigenvalues obtained

from the global method and the local method agreed to at least six decimal places when

the same number of grid points were used. The typical rapid convergence of complex phase

velocity C corresponding to the most unstable eigenvalue of a near maximally amplifying

disturbance is reflected in the values below:
84

N CR Cr

15 0.57644 0.18860

21 0.57656 0.18866

31 0.57656 0.18866

41 0.57656 0.18866

The above results, obtained using Mapping B, are for a = 3.5,8 = 1,M i = 0.1, Re =

106, T* = 1 and 0 = 1/15. The corresponding value of C obtained by an independent

finite difference code (see Section 4.2) using 200 grid points is C = 0.57656 +

0.18867i.

For a near neutral wave the convergenceof the unstable eigenvalue with N is observed

to be much slower than that of a maximally amplified wave. For example, the least stable

eigenvalue for a near-neutral wave of a heated and compressible jet [a = 5.8,/9 = 0, M i =

0.8, Re = 5000,T* = 2 and 0 = 1/15] is found, using Mapping A, to be

N CR Cz

35 0.4821 0.0096

45 0.4840 0.0109

65 0.4840 0.0100

This slow convergence is attributed to the near singular behavior of the eigenmodes in

the vicinity of a critical point at high Reynolds number flows (see section 2.7). In fact,

as heating and compressibility increase for high Reynolds numbers flows, we have found

that the structures of the near-neutral eigenmodes become increasingly more difficult to

resolve.

Figures (4.4) - (4.9) exhibit the typical transverse distributions of the shape and phase

of the most amplified and near neutral eigenfunctions. These eigenfunctions are normalized

such that the maximum magnitude of the eigenfunction of the streamwise velocity is 0.5.

The pressure eigenfunction is seen to exhibit the "smoothest" structure. This behavior is

sensible since the pressure perturbation has the weakest form of singularity at the critical

point. Numerically, it is precisely this relatively smooth behavior of the pressure that
85

makes itthe most usefulvariableto work with in connection with a Rayleigh equation

(when one exists).Itisinterestingto note that fora near maximally amplifiedwave, the

strearnwiseperturbationu has a typicaldistinctpeak and the radialcomponent v has a

plateau-likestructureat the criticallayer.For a near neutralwave, the peak structureofu

perturbationchanges drasticallyto a shape with two localmaxima and isaccompanied by

a significantchange of the phase,while theplateaufeatureof the v component transforms

to a smooth shape with a singlemaximum point.Note alsothat the amplitude ofa highly

amplifiedu perturbationislargeat thejetliplineand becomes progressivelysmall as the

wave approaches the neutralpoint.

4.2 Some Remarks on Spectral Methods

I. The impositionof the asymptotic boundary conditions(seesection2.6)forbidsthe

use of a globalmethod because the eigenvaluea then entersnonlinearlyinto the

eigenvalueproblem through the boundary conditions.In addition,sincethe analytic

far fieldbehavior of a disturbancein a periodicbase flow isstillunknown, itis

suitableto use the setofmappings (4.18)and (4.23)with a sufficientnumber of grid

points to resolvethe disturbancestructurein the criticallayer.

2. The extension to spatial stability analysis can be done without much difficulty. Since

the basic instability mechanism is essentially inviscid, the viscous terms which contribute to a quadratic nonlinearity in a, can be neglected in a global method for

spatial stability problems. This is because a 2 ,,., c92/c9x 2 and in the critical layer

(where the effects of viscosity and heat conductivity are small), this derivative is

negligible compared to c92/8y 2 (i.e., the boundary layer assumption). This inviscid

spatial analysis can be reduced to a generalized eigenvalue problem for a. The physically meaningful a, usually the most unstable mode selected from a global scheme,
86

is usedasa starting guess for the eigenvalue of a more refined viscous local code.

4.3 Fourth-Order Finite-Difference Method

In order to have confidence in the spectral method, the eigensolutions of the viscous

problem as viscosity vanishes, or equivalently as the Reynolds number becomes infinite,

were checked by an independent fourth order inviscid finite difference method. Eigensolutions of the linear stability problem of unidirectional round jets have been obtained using

finite difference methods by Paragiri (1985) for inviscid analysis and by Jarrah (1989)

for viscous modes. For completeness, a brief description of the method is discussed here.

In the absence of viscosity (and conductivity), the governing disturbance equation for an

inviscid compressible parallel flow can be reduced to a single Rayleigh-type second order

ODE for the pressure disturbance (2.48)

Since u_ and p_ vanish as r --* 0 and as r --* oo, the asymptotic solutions are

p= - M]po(,,o- c)')

p= - M/po(,,o- C)')

(2.48)

as r ---*_ (4.32)

where I_ and K# are modified Bessel functions of the first and second kind, respectively,

and C1 and C2 are constants. Numerically, the asymptotic far field boundary condition is

satisfied at approximately r -- Rma= = 4.0 and the boundary condition on the jet axis is

satisfied at the first two grid points where constant mean flow prevails to a good degree

of approximation. Details and some sample calculations may be found in Paragiri (1985).

A fourth order central difference scheme is employed which translates the ODE (2.48)

together with the boundary conditions (4.31) and (4.32) into a pentadiagonal system of

as r --* 0 (4.31)
87

complex algebraicequations.The complex eigenvalueC isthen found by a localmethod

based on the Newton-Raphson scheme where iterationson the eigenvvdueare performed

Untila prescribedconvergence criterionismet. In thiswork, the eigenvaluesare iterated

to fivedecimal places for highly amplifiedmodes, and to three decimal placesfor near

neutralmodes. Note that a physicaleigenvaluecan easilybe distinguishedfrom a spurious

(unphysical)one because the latterdoes not converge asthe number ofcomputational grid

pointsisincreased.

4.4 Some Remarks on the Linear Invlscld Calculation

The inviscidpressureequation (2.48)has a regularsingularpoint whenever the

mean flow velocityequals the phase velocityof a neutral wave. This singularityoften

createsnumerical difficulties. For sufficientlyamplifiedmodes, the imaginary part of the

complex phase velocityC removes the singularityfrom the differentialequation along

the real axis and hence poses no numerical problem. However, for eigenmodes that are

merely slightlyunstable,the almost singularnature of the differentialequation requiressome care. To make matters worse, the secondary instabilitycalculationsrequirean

accuratedeterminationfor allthe primary disturbances(not just the pressure)and th'eir

higherorderderivatives.As shown from section(2.7),compressibleflowsintroducedensity

fluctuationswhose leadingorder behavior in the neighborhood ofthe criticalpoint is

- { ,a2r_uT(r - re) }+"" + Kp_ log(r - re) (2.56)

This singularity, which is a pole of order one, may create a change in density fluctuation

across the critical layer that is too rapid for an accurate determination of an inviscid

near-neutral eigenmode. Therefore, we conclude that although the instability is predominantly inviscid, the inclusion of viscosity is mandatory in secondary instability calculations

because the primary wave is very nearly neutral.
PRECEDING PAGE BLANK NOT FILMED

8O

0.(]

0.0 I .0

Oo O O

O

'e e O

ge

rl--6

°8t 

0.4 _,, •

0.0 _ ' '

40

! I I

80 120

rt

6

4

2

li, OOlli- _ m_ _ m_

llmlrllr3DOOOOOO o o o o o o

(-] I i i _. I I l i

0.0 1.0 2.0 5.0 4.0

I"

Fig. 4.2 (a) an(] (b) z _ a fonction of the radlal variable r rot v_riou_ ,. Cro,_e_

are the collocation points. (c) collocation points on r. The number of collocation

points is 35 u_ing Mepping.A with II - 7 and Is - 3.
Z

1.0

0.0 w ! I I

0.0 1.0 2.0 $.0 4.0

r

90

10.0 0-- _ _--_"_,,,_ 0 _ _ v" D

1" 6.0'

0.0

(b)

@ ,_@,j. ....... :,_A .,. . A : ,_. ,_. |

I I I I

0.0 1.0 2.0 3.0 4.0

r

Fig. 4.3 (a) z as a function of the radial variable r for various stretching parameter r

(b) collocation points on r. The number of collocation points is 21 using Mapping-B.

Crosses are the collocation points. Bullets (r = 5), circles (r = 10), and squares (r

-- 1).
3.0

2.0-

t

! t

91

2.0

1.0 1 0

0.0 010

0.00 0.26 0.60 0.711 1.00 0.00 0.26 0.60 0.76 1.00

lul IvJ

,'1.0 3.0

2.0 2.0 -

¢0

1.0!_ 1.0

O0 -i'

000 026 0_60 0.76 1.00 0.0

0 0 7B I O0

lwl lwl

Fig. 4.4 Distributions of the leigenfunctionl of a near maximally amplified disturbance for Re = 1.0E + 6,1', = 1.0,My = 0.1,0 = 1/15, ot = 3.5,8 = 1, and the

complex phase velocity C = (0.577,0.189).
92

3.0 ¢

0

2.0

0

0

0

0

1.0

O.O

-3.14

!

0.00

phue of u

0%

0

0

0

3.14

T

3.0

2.0

1.0

0.0

-3.14

_'3

0

0

0

0

0

0

0

I

0.00

phase of v

1.14

r

3.0 o

0

2.0- o

0

0

1.0- o6_

0.0

-3.14

3.0

O

0

0

0

i

3.14

2.0-

P

1.0-

0

0

0

0

0

0

0

0

, 0.0 I

0.00 -3.14 0.00

phase of w phase of p

Fig. 4.5 Distributions of the phase of the eigenfunction of a near maximally amplified disturbance for Re = 1.0E + 6, T. = 1.O,M i = 0.1,6 = 1/15,_x = 3.5,/_ = 1,

and the complex phase velocity C = (0.577,0.189).

3.14
93

3.0

2.0

1.0

0.0 I ! I

0.00 0.25 0.50 0.78 1.00

I,,I

,0 i i

2.0-

0.0

n 7K 1.00

r

3.0_'

2.0

3.0

2.0-"

!

f

1.0 1.0

0.0 0.0

0.000.25 0.60 0.75 1.00 0.000._6

I_I I;'I

I II

I I

0.80 0.75 1.00

Fig. 4.6 Distributions of the JeigenfunctionJ of a near maximally amplified disturbance for Re - 1.0E + 3, T, = 1.0,M# = 0.1,0 = 1/15, cz = 3.5,B = 2, and the

complex phase velocity C = (0.557,0.142).
3.0

2.0

1.0

0.0

-3.14

o

0

0

o

0

0

0

0

0

o

o

I

I"

3.0

, 0.0

0.00 3.14 -

phase of u

1.14

0

o

0

0

0

0

o

0

I

0.00

phase of tt

94

3.14

3.0 0

0

2.0-

0

0

0

1.0- o°8

0.0

-3.14

r

3.0

2.0

_0o T 1.0

oI

0

0

3.14

o

0

0

0

o

o

0

0

', 0.0 ,

0_00 -3.14 0.00

phase of to phase of p

3.14

Fig. 4.7 Distributions of the phase of the eigenfunction of a near maximally amplified disturbance for Re = 1.0E + 3, T, = 1.0, M# = 0.1,0 -- 1/15, a -- 3.5,/_ - 2,

and the complex phase velocity C = (0.557,0.142).
_.0

2.0

I I • $.0

2.0

g5

1.00

3.0

2.0

3.0

2.0'

• p

1.0 1.0

o.o o.ooo., o.so o.7. ,.oo o.o o.ooo.2s o..o 0.7, ,.oo

Ipl II,I

Fig. 4.8 Distributions of the lelgenfunctlonl o! a near neutral disturbance for

Re -- 5.OE % 3, T, -- 1.0, Mj - 0.8,# - 1/15, a - 6.55,/3 - 0, and the complex

phase velocity C - (0.569,0.001).
9fl

3.0;- ' 3.0

0

2.0 2.0- o

0

1" r 0

,0 o_ '° i

! ! o°

O

0.0 -M_,_,_,I,_,__ 0.0

"-. 0.00 $.14 $.14 0!00 "' .4

phas_ of u ph_e of v

2.0

f,

1.0

0.0

-3.14 0.00

ph_me of p

_J.14

2.0

1.0

0.0

-_.14 0.00 3.14

ph_e of p

Fig. 4.9 Distrib,tions of the phase of the elgenf, nction of a near nelltral di_t,Thance

for Re -- 5.OE _ 3,7. = I.O, Mj - 0.8,8 - 1/15,_ -- 6.,55,d = O, and the complex

phase velocity C --- (0.569,0.001).
97

Chapter 5

LINEAR STABILITY OF PARALLEL JETS

The linear stability of a parallel jet is important in its own right. In order to be

able to manipulate the downstream evolution of jet shear layers, the early stages of jet

instability must be understood. We reiterate that linear analyses can reveal accurately

not only the stability characteristics of the initial stages of development of the large-scale

instability wave, but also, rather surprisingly, it can predict the cross-space structure of the

perturbation, even when the local disturbance level is as high as 24_ of the jet exit velocity

(Petersen & Samet 1988). This linear analysis, being the first cascade of instability, is

used as input to fully nonlinear numerical simulations, weakly nonlinear analysis, and

secondary instability theory. This section considers some features of the linear stability of

subsonic heated round jets. The different parameters affecting jet instability mechanism

axe discussed separately in order to elucidate their individual influences. The important

parameters are the jet Mach number Mj, the temperature ratio T0, the jet momentum

thickness 0, wavenumbers (a,/_), and the Reynolds number Re.

5.1 Viscosity Effect

There are now a number of investigations on the inviscid stability characteristics of

round jets, although studies on the viscous instability analysis remain scarce. Batchelor

& Gill (1962) pioneered a theoretical framework on the inviscid stability of a top hat
98

velocityprofilewhich characterizesthe mean flow closeto the jet exit. This top hat

velocity,which ignoresthe presence ofthe jet nozzle,as iscustomary in allparallelflow

analysis,provides a closedform solutionof the stabilitycharacteristicsof a cylindrical

vortex sheet. By usingmore realisticprofiles,which depend continuouslyon r,Michalke

(1971) and Mattingly & Chang (1974) obtained a good agreement with the experimental

results of Crow & Champagne (1971). Morris (1976) went further to investigate the effect

of viscosity on the instability of various jet mean velocity profiles for different values of the

jet parameters. He found that for jet velocity profiles with thin shear layers, i.e., close to

the jet nozzle, viscosity has a slight stabilizing effect on the growth rate for flows above a

Reynolds number of 1000, based on jet centerline velocity and jet nozzle radius. In other

words, viscosity for instability modes in jets is predominantly passive in character, and

can be ignored for flows at high Reynolds number (except for neutral waves).

Typical results exhibiting the influence of Reynolds number on the linear (temporal)

instability of subsonic heated round jets with the tanh base profiles of (2.28) - (2.30) are

summarized in Figures (5.1) - (5.6). In particular, the effect of Reynolds number on the

full spectrum of eigenvalues using 31 Chebyshev polynomials for a = 3.66 and 0 = 1/15 are

plotted for various jet parameters in Figures (5.1) - (5.3). We caution the reader that the

discretized eigenvalues in these figures, except for the unstable ones, are only "roughly"

determined since their associated eigenfunctions did not adequately satisfied the far field

boundary conditions. Nevertheless, these eigenvalues may provide a rough guide on the

the full spectrum of the primary instability problem. The salient features that can be

extracted from these figures are

1. As the viscosity tends to zero (Re ---*oo), the spectrum consists of a string of neutral

modes and a complex conjugate pair of eigenvalues.

2. The string of eigenvalues, which probably constitutes part of the continuous spectrum, generally becomes increasingly stable with decreasing Reynolds number. The
99

inclusionofviscosity(athigh Reynolds number) only slightlystabilizesthe unstable

(discrete)eigenvalue,and breaks down the symmetry of an inviscidcomplex conjugate pair. In otherwords, viscosityhas a considerableeffecton damped waves but

exertsonly a slightlystabilizingeffecton unstable (inflectional)waves.

3. The %ontinuous modes", whether viscousor inviscid,have phase velocitieslying

between the minimum (freestream value)and the maximum (jetcenterlinevalue)

of the jetmean flow.

Observation (1)isperfectlyconsistentwith the well-known propertiesofthe reduced inviscidequation (2.48).Sincethisequationadmits a complex conjugate pairof eigenvalues,

ithas eitherneutralmodes or growing modes which have corresponding Udamped" modes

moving with the same phase velocity.Furthermore,sincethe mean velocityprofileused in

thiswork has only one generalizedinflectionpoint,the spectrum isthus restrictedto an

assembly ofa singleunstablemode, itsconjugate damped mode, and the remaining neutralmodes. The factthat an inviscidequation describesthe same instabilitymechanism

as the viscousproblem in the limitof vanishingviscosityisreinforcedby (2). Although

the continuousspectrum offreeshearflowsisstillnot completely understood,observation

(3)suggeststhat the complex eigenvaluesof the highlydamped continuous modes move

toward the realaxis.This observationisagain consistentwith the theoreticalresultsof

the continuousspectrum of boundary layers(Grosch & Salwen, 1978); they found that

the phase velocityofthe highlydamped continuousmode matches the mean flowvelocity.

In view ofthe physicalinsignificanceofthe damped waves compared to the rapidlyamplifiedunstablewaves of round jets,the continuousspectrum willnot be pursued furtherin

thiswork. Nevertheless,the continuousspectrum obtained from a numerical discretization scheme can influencethe resultsof an unstablemode. The inclusionof viscositywill

generallyincreasethe rate ofnumerical convergenceof the unstable mode by moving the

continuousspectrum away from it.This impliesthat viscositywillplay an important role
100

in the convergence of a near neutralunstableeigenvalue(seeSantos,1987).

In Figures(5.4)- (5.6),the growth rateand phase velocityas functionsofwavenumber

axe plottedfor0 = 1/15. These figuresshow the typicalinfluenceof Reynolds number on

the unstablemode ofsubsonicheated round jets.Note that the eigenvaluesasymptotically

approach the inviscidlimitat a Reynolds number ofabout 5000. In contrast,the inviscid

limitof a planar mixing layeroccurs at a Reynolds number of about 200 (Jarrah 1989).

This means that viscosityplays a more important rolein the instabilityof round jets

than in mixing layers.This isbecause in a mixing layer,where there are two external

streams,the viscoussolution(seesection2.6)iscompletelynegligible;but thissolutionis

not negligibleon the jetaxisunlessthejetmomentum thicknessisextremely small.Thus

the viscoussolutionwillaffectthe eigenvaluethrough the boundary conditionnear the jet

axis,even when the Reynolds numbers ison the order of severalthousand. For jetflows

at Re __5000,viscositynot only slightlyreducesthe range of unstablewavenumbers but

alsothe growth rate.Viscosityisalsoseen tohave virtuallyno effecton the phase velocity

of unstabledisturbances,even though the Reynolds number can be as low as 500. These

resultsagreed perfectlywith the viscouscalculationsofJarrah (1989),and are consistent

with the numerical resultsfor the spatialand viscousinstabilitycalculationsof Morris

(1976).

5.2 Mach Number Effect

The new quantityin compressibleflowsisthe densityfluctuation,which occurs

simultaneouslywith the velocityand pressurefluctuations.The new parameters which

willgenerallyinfluencecompressibleflowsarethe Mach number Mj, the temperature ratio

7",,the Prandtlnumber Pr, and theratioofthe heat capacitycoei_icients,_/= C_/C_. In

thissection,the influenceof Mach number on the stabilityofjetshear layersisgiven,the
101

effect of temperature ratio will be discussed in next session, while the last two parameters

are assumed to be constants [q¢= 1.4, Pr = 0.72], and hence, will not be addressed in

this work. The Mach number can influence the instability via the base density profiles

as well as through the disturbance equations. For example, in the unheated jet, the jet

Mach number appears explicitly in the disturbance equations as well as in the base density

distribution.

Although the stability of incompressible shear layers has been studied extensively, a

systematic determination of the role of Mach number on the instability of jet shear layers

is still unavailable. In the case of mixing layers, on the other hand, there are now several

fairly complete studies [see Gropengiesser 1967, Jackson & Grosch 1988]. In his spatial

stability calculations, Michalke (1971) found that increasing Mach number reduces the

range of unstable waves as well as stabilizes the flow. Furthermore, he found that the

phase velocity is sensitive to Mach number only for lower frequencies (or, wavenumbers)

while in contrast, the growth rates become almost independent of Mach numbers. His

results also show that the stabilizing effect due to Mach number is slightly more significant

for an axisymmetric mode than for the first helical mode.

Eckart (1963) extended the Rayleigh stability criterion and Howard's (Howard 196.1)

semi-circle theorem to a compressible, adiabatic plane jet. By using a different approach,

Blumen (1970) found that the influence of aM i on the complex wave velocity of an unstable

mode is the same as that of the Richardson number in Howard's results. He concluded

that compressibility reduces the range of unstable wavenumbers and stabilizes the mixing

layer. We will show that Howard's semicircle analysis can similarly bound the complex

wave speed and illuminate the role of compressibility in jet shear layers. We will do this

by generalizing Blumen's approach to a round jet.

The linear disturbance equations for an inviscid compressible jet can be reduced to a
102

singleequation for the pressure perturbation p(r):

L(p)= -g'jr2 + uo-C ,,oJ_r+ _2("°-C)2p°M]_2 _2 p=O. (2.48)

We multiply the above by r/{po(Uo - 0) 2} to make it self-adjoint. The equation in the

self-adjoint form becomes

p0(u0--- C) 2 _r -_" ¢lP ---- 0 (5.1)

whereql = r [,_2(uo- C)2poM2 - _2/r2- ¢¢2]/{.o(uo- C)2}. After multiplying (5.1) by

the complex conjugate of p and integrating over the range of r, we find that

/o°[ c)21p'l2- ql]p]2]dr = 0. (5.2)

Separating (5.2) into the real and imaginary parts, we obtain

2c,Z(uo- CR)rq2dr= 0 (5.4)

where

q2 = p0lu° L_OI4[IP']=+ (_2 + )lpl2]

C = CR + iC_.

For unstable waves (Cz > 0) which vanish at infinity, (5.3) and (5.4) can be simplified to

// // uoq_dr = Ca q2dr (5.5)

If we let a and b be, respectively, the maximum and minimum of the base velocity [i.e.,

a < -0(Y) -< b], then (5.5) and (5.6) can be reduced to

// 0 >_ (uo - a)(uo - b)qzdr (5.71
103

The above expression can be simplified to the equation governing the Howard's semicircle

theorem:

[CR- +b)]+ c /_ t co

Hence, the stability of inviscid compressible round jets, here a --- 0 and b = 1, can be

reduced to exactly the same form as in Howard's investigation of the stability of variable

density parallel shear flows.

Therefore, the complex wave velocity for an unstable mode of jet shear layers is also

restricted to the semicircle which has the range of u0 for diameter;just as in unbounded

planar mixing layers. Furthermore, since equations (5.5) and (5.6), which govern the

instability of round jets, are exactly the same as equations (3.4) and (3.5) in Howard's

results, we can conclude that compressibility stabilizes jet instability modes.

For the tanh profile, with 0 = 1/15, the typical role of subsonic Mach number in jet

instability for the axisymmetric and the first helical modes is illustrated in Figures (5.7)

and (5.8), respectively. Compressibility is found to be stabilizing and reduces the range of

unstable wavenumbers. For example, when the jet Mach number, Mj, is increased from 0"

to 0.8, the maximum growth rate of the axisymmetric and the first helical modes for an

unheated jet is reduced from 0.68 to 0.55, and from 0.66 to 0.55, respectively. Compressibility also reduces, but only slightly, the phase velocity of waves whose wavenumbers are

smaller than the maximally amplified wavenumber, and has no apparent effect on higher

wavenumbers. In other words, the phase velocities of disturbances, whether axisymmetric

or helical, are only weakly dependent on the subsonic Mach number.

In the limiting case as a --* O, these figures also show that long waves are unaffected

by compressibility, i.e., waves with axial symmetry travel with the base flow velocity on

the center line, whereas helical modes travel with half that velocity. Our calculations

are consistent with the results of Jarrah (1989) and are in reasonable agreement with
104

the spatial stability calculations of Michalke & Hermann (1982). Physically, according to

Blumen (1970), the mean flow energy, which provides the energy for an unstable mode, is

diminished by working against the elastic force associated with a compressible medium,

and consequently the growth rate of an unstable disturbance is reduced.
105

5.3 Heating Effect

In the previous section, we have discussed the effects of jet Mach numbers on

the instability. An extension of the Howard semicircle theorem provided some analytical

understanding for the reduction of the growth rates with increasing compressibility. We

would now like to gain some understanding of the effects of heating. To do this, we shall

begin by generalizing the vortex sheet problem of Batchelor & Gill (1962) to a hot jet. In

order to obtain simple results, we set M i = 0 in the final expressions.

As mentioned in section (5.2), the presence of density fluctuations in a zero Mach

number flow is caused by a variable base density profile which is a function of 0 and

T,. Although stability calculations have now been successfully carried out for unheated

jets, studies on the influence of heating on jet instability characteristics remain scarce.

Michalke (1971) and Michalke & Hermann (1982) are among the few who investigated the

stability characteristics of subsonic heated round jets by using linear spatial (parallel flow)

stability theory. They found that the local growth rate of the spatially growing instability

wave in a zero Mach number flow (i.e., Mj = 0) increases as the total temperature is

increased.

In order to shed light on the effect of heating and to increase the understanding of the

stability characteristics of a viscous heated round jets with continuous velocity and density

profiles, the solution for a cylindrical inviscid and heated vortex sheet is investigated here.

The stability analysis of a vortex sheet has a closed form analytical solution, and has

been given by Batchelor & Gill (1962) for the incompressible case. We now consider the

corresponding compressible case.

The base profiles of a compressible vortex sheet are

u0-Ua=constant, p0=Pa---constant forr < 1

(5.10)

uo'-O, Po=Po----constant for r > 1

On both sides of the vortex sheet, given by r - 1, where irrotational flows persist, the
106

pressureperturbationsare

p. ----Ehl_(arar) for r < 1 (5.11)

Pb = DhK_(ar) for r > 1 (5.12)

where

Ta= V/1- (Ua - C)'M]pa

where I_ and K_ are Bessel functions, and Eh and Dh are arbitrary constants. The corresponding radial velocity can be obtained from the linearized radial momentum disturbance

equation. They are

Eh/'.(aT.r)

_.= -. (5.13) ,_p°(uo - c)

thK',(ar) (5.14)

Vb:- ic_pbC

where primes denote differentiation with respect to the argument in parenthesis. Let _/be

the radial displacement of the vortex sheet which assumes a normal mode representation

of the form

r/: Ah exp i[c_(x - Ct) + _¢] -f c.c. (5.15)

where Ah is a free constant which denotes the arbitrary amplitude.

The kinematic boundary condition that the interface ,r is a material surface (i.e., the

interface is formed by the same fluid particles at all times) can be expressed as

DI

D--'t"= 0 for r = 1+ (5.16)

DI

=0 for r= 1- (5.17)

Dt

where

D

Dt

0

- Ot + (v + vo). v

and,

I = I(x,r,¢,t)
107

Substituting equations (5.10) - (5.15) into (5.16) -

disturbance equations, we obtain

(5.17) and after linearizing the

Ehr_(_T_) (5.15) i_(Uo - C)A,, = po(U_- C)i,_

iaCAh = DhK'p("y) (5.19)

pbCia

Furthermore, the dynamic condition that the pressure is continuous at the interface yields

EhI_(aTa)- nhK/_(a) (5.20)

(5.18) - (5.20) provide the the equation for the determination of Ah, Eh and Dh. In order

to have a nontrivial solution, these equations yield

(Ua_ C) s = -Lp(Taa) pb (5.21) Pa

where

L_(T.a) = K_(a)I'_(T.a)

K'_(a)Ip(Taa)

The complex wave velocity in equations (5.21) cannot be solved analytically for a nonzero

Mach number. In order to investigate the individual influence of heating, the solution for

an incompressible heated jet is examined. In this case, an explicit analytical expression

for the real and imaginary parts of the complex wave velocity can be obtained as below:

OR : So (5.22)

I+ T.L_(_)

Cl = Uav/T*Lp(ct) (5.23)

1 + T,L_(c*)

where T, = Pb/Pa.Sincethejetcenterlinevelocityisused inthe normalizationinthiswork,

Ua = 1. Thus, the incompressiblevortexlayerisunstableto alldisturbancewavenumbers,

as in the classicalKelvin-Helmholtzproblem. Sincevery shortwaves forrealflowswith a

finiteshear layerthicknessare stable,thisvortex sheetmodel isapplicableto disturbances
108

whose wavelength ismuch longer than the shear layerthickness. This willbe further

elucidatedin subsequent discussion.Since the wavelength of a disturbanceisinversely

proportionalto itswavenumber, long wave approximations are, therefore,characterized

by small wavenumbers and thus low frequencies.

Using (5.22) and (5.23), the typical influence of heating on the value of Cx and CR as

functionsof the axialwavenumbers forfl ----0 and/9 --i are plottedin Figures (5.9a)and

(5.9b),respectively.Itisfound that,except forlong axisymmetric waves, heatingreduces

the phase velocitysubstantially;forexample, the phase velocityof an axisymmetric wave

with axialwavenurnber 7.5 isreduced from 0.53 to 0.36.For waves with a approximately

lessthan 1.5,this reductionismore pronounced for the firsthelicalmode than for the

axisymmetric mode.

In the limitingcase,the values of C for long wave approximation (a --*0) can be

obtained from (5.22)-(5.23).They are

CR=I, C!=O, forfl=O (5.24)

1

CR= 1+---_,' Cz-- I+To for/_0 (5.25)

Therefore, a heated vortex sheet is always stable to long axisymmetric modes, but is

unstable to all non-axisymmetric modes whose growth rates are reduced by heating. For

a cold jet (T. = 1), long waves with axial symmetry travel with the speed of the center

of the jet, whereas all helical modes travel with half that speed (Batchelor & Gill 1962).

The phase velocity of long axisymmetric waves is independent of heating, whereas that of

non axisymmetric waves is substantially reduced by heating.

The stability characteristics of a continuous base velocity profile are in general not

expressible in closed form, and, therefore, must be determined numerically. The phase

velocity and growth rate as functions of wavenumbers for the tanh velocity profile with 0

: 1/15 are displayed in Figure (5.7) for fl : 0 and in Figure (5.8) for/_ ---1. Note that

axisymmetric disturbances are more unstable in heated jets than in cold jets only at small
109

axial wavenumbers, and that heating reduces the range of unstable waves. Except for very

small axial wavenumbers, heating generally decreases significantly the phase velocity; for

example, the phase velocity of a near neutral axisymmetric disturbance is decreased from

0.562 to 0.476. While the phase velocity is rather insensitive to Mach number, it is, except

for long axisymmetric waves, substantially reduced by heating.

By using a slightly different velocity profile (profile 1 of Michalke 1984), we found that

these stability characteristics are virtually indistinguishable from those of profile (2.28) -

(2.30). This insensitivity of the stability characteristics to the details of the (two slightly

different mean velocity profiles) convinces us that our results are representative of what

might be obtained for a wide class of mean flows with reasonable shapes.

By comparing Figures (5.9 a-b) and Figures (5.10 a-b), the phase velocities obtained

from a tanh velocity profile with 0 = 1/15 and the vortex sheet model are seen to exhibit

very similar behavior. In contrast, the quantitative values of the amplification factors from

these two profiles are considerably different (except at small wavenumbers); this shows the

true limitationofa vortexsheet theory.

Further,the long wave resultsfor0 -- 1/15 are consistentwith Jarrah's(1989) calculationson mean velocityprofileswith a differentmomentum thickness,namely 0 -- I/5.

For shear flows,as previouslymentioned, the stabilitybehavior of a disturbancewhich

has a largewavelength in comparison to a characteristicprofilelength (e.g.,shear layer

thickness)isunaffectedby the form of velocityprofiles.The validityof the preceding

statement needs some clarification, though. For round jets,we found that thisstatement

istrue foraxisymmetric modes but not strictlyvalidforhelicalmodes. To be precise,the

phase velocitiesof long waves (i.e.,a --40) in a tanh velocityprofilewith a finiteshear

layerthickness,whether axisymmetric or helical,agreed with that obtained from simple

vortex models. For the correspondinggrowth rates,thisagreement isfound in axisymmetric modes but not in helicalmodes. This disagreement isexacerbated as the .helical
110

mode number and the shear layerthicknessisincreased.
111

5.4 Mode Number and Shear Layer Thickness Effects

Since the evolution of instability modes in an axisymmetric jet is affected by

the shear layer thickness and the mode number, these effects are concurrently discussed

in this section. The length scale characterizing the effect of shear layer thickness is the

dimensionless momentum thickness $ which, in this work, is normalized with the jet radius.

There have now been a number of theoretical studies on the stability characteristics of

incompressible round jets. In their theoretical work, Batchelor & Gill (1962) found that

a top-hat jet velocity profile (i.e, $ --, 0), which represents the mean flow very close to jet

exit, is unstable to a small disturbance for all axial and azimuthal wavenumbers. Their

analysis is supported by Plaschko (1979) who found that the thin jet shear layer (i.e.,

small 0) for a slowly diverging jet is unstable to a large number of discrete azimuthal

modes. The importance of the parameter 0 in the initial evolution of an axisymmetric jet

is further explored in detail by Cohen (1986). By using a scaling analysis, he found that

all azimuthal modes will amplify at identical rates whenever 0 << 1 and (Az/At_) 2 << 1;

where A. and A_ are the streamwise and the azimuthal wavelengths. This implies that

the results obtained from the thin jet shear layer are equivalent to the planar mixing layer

where the stability characteristics depend only on a single parameter a.

Although the shear layer near the jet nozzle exit is unstable to a large number of azimuthal modes, only the first few are considered since these are the modes that dominate

the coherent structures experimentally (Cohen 1986). In fact, the geometrical design of

most laboratory jets, which consists of a small nozzle emanating from a large chamber

where an acoustic excitation signal is introduced, cuts off higher azimuthal modes. It is

for this reason that azimuthal excitation of a jet is usually accomplished by a series of circumferentially placed acoustic "drivers". To be more precise, a linearized monochromatic

pressure wave traveling inside a plane duct, where for simplicity the mean flow in the duct

is set to zero, before exiting downstream to the jet nozzle is used for illustration. This
112

wave is of the form

e,<,> (-,<<,.,)""":""

cost--_] expi Laooi _ j z (5.26)

where w_ is the excitation frequency, h is the duct height, aoo is the sound speed, y

is the cross stream direction, z is the downstream direction, and nd is the duct mode

number. Equation (5.26) shows that the fundamental mode (nd = 0) will always propagate

sinusoidally downstream. However, if the excitation frequency w, is small such that we <

ndXaoo/h, then the behavior of the acoustic mode along the x axis is purely exponential,

and the wave is said to be cut-off. Therefore, at a given frequency, only a finite number

of modes can propagate out of the jet nozzle exit. In order to excite the instability wave

whose frequency is on the order of the ratio of the dimensional jet exit speed (Uj) and

the shear layer thickness (0), the excitation frequency of acoustic waves corresponding to

most laboratory experiments is often quite low. Consequently, only the fundamental plane

acoustic wave survives through the nozzle exit. This implies that if the nozzle is round,

then an axisymmetric low frequency acoustic wave, which accentuates an axisymmetric

instability wave, will be observed at the nozzle exit. Fartherdownstream where the rollingup of coherent structure occurs, the axisymmetric and the first azimuthal components have

been experimently observed to predominate the flows. Therefore, the stability calculations

are focused mainly on these modes.

The parameter 0 appears explicitly in the velocity profile (2.28) - (2.30). If the flow

is compressible, 8 also appears as a parameter in the base density via the coupling with

the velocity profile. In order to single out the effect of the shear layer thickness, the base

density profile is taken to be constant (i.e., Mj = 0 and T, - 1). In this work, the tanh

velocity profiles for two different axial locations are investigated, although the analysis

can be carried out to include profiles at an arbitrary location, provided that the mean

profiles are known a priori. The values of momentum thickness which characterize the

two locations are chosen to be 8 = 1/15 and 8 = 1/5. Crighton & Gaster (1976) reported
113

that these values of 0 correspond to axialpositions0.5 and 2.7 diameters downstream

of the nozzleexitin the experimentalmeasurements of Crow & Champagne (1971). Of

Course,the functionalrelationbetween 0 and the streamwisedirection,which depends on

A_j and T., willvary from experiment to experiment.

The growth rate and phase velocityofthe firstthreeazimuthal disturbancesfora cold

and incompressiblejet for# = 1//15and # = 1/5 are summarized in Figure (5.11)and

Figure (5.12),respectively.In Figure (5.11),the jet shear layeris dispersiveto waves

longerthan the most amplifiedwave and relativelynondispersiveforshorterwaves; _ --1

produces the largestrange of (approximately)nondispersivewaves. The phase velocityof

axisymmetric modes decreaseswith a from unity to0.56.In contrast,the phase velocities

of helicalmodes (_ = 1, 2) increaseswith a from 0.5 to 0.56. In particular,the pha_e

velocityfor a neutralwave isapproximately 56 % of the jet centerlinevelocity,and it

isindependent of the mode numbers when the shear layerisvery thin. An important

consequence ofthe almost nondispersivecharacterof azimuthal modes over a broad range

ofwavenumbers isthat itallowswaves to interactresonantly.We shallshow in Chapter 6

that thisresonance wave interactionphenomenon is,in fact,a mechanism forthe strong

growth of a subharmonic disturbance.Further,the axisymmetric (/_= 0) mode has the

highestmaximum growth rateof0.680,the _ = 1 mode amplifiesat a slightlylower _ate

of 0.663,while the most unstablerateof _ = 2 is0.595.This isreasonable,sinceas # -_

0, the resultsof the tanh profilestend to that obtainedfrom vortex sheet analysiswhere

allazimuthal modes amplify at the same rate at largeenough axialwavenumbers.

It isinterestingto recallthat the phase velocityfor allazimuthal waves fora simple

vortex sheet (i.e.for 9 = 0) isat approximately 50% of the jet centerlinevelocity.This

shows that although the vortex layerunder-estimatesthe phase velocityofthe tanh velocityprofilesat0 --1/15 by about 11%, itcorrectlypredictsthat allazimuthal disturbances

with high axialwavenumbers propagate at the same phase velocity.
114

Farther downstream at 0 = 1/5, Figure (5.12a) shows the _ = 1, D = 2, and/_ = 0,

are respectively the most, the second, and the third unstable modes. The maximum

growth rates of the disturbances with _ = 1, /_ =0, and _ -- 2 are 0.173, 0.134 and

0.039, respectively. Therefore, if both the axisymmetric and helical modes have the same

initial amplitudes, the helical mode will eventually dominate downstream of the end of the

potential core. The emergence of/_ = 1 mode as the dominant instability mode towards

the end of the potential core is also observed by Michalke & Hermann (1982), Cohen

(1986) and Miles & Raman (1988). Batchelor & Gill (1962) theoretically demonstrated

that the _ fully developed" jet, after the end of a potential core, is unstable only to the

= 1 mode. For the tanh profiles, the phase velocities of axisymmetric modes are seen

to decrease with wavenumbers from unity to 0.71. In contrast, the phase velocities of

modes _ - 1 and _ = 2 increase with wavenumbers from 0.5 to 0.67 and from 0.5 to 0.56,

respectively. Owing to a greater value of 0, the near-neutral azimuthal modes now travel

with different phase velocities. :..........

By comparing Figure (5.11) and Figure (5.12), we found that as the shear layer thickness grows, not 0niy does the wavenumber of the maximum local growth rate and the range

of unstable waves becomes smaller, but the range of nondispersive waves is significantly

reduced. It should be noted that, as pointed out by Petersen (1978), each coalescence

of coherent structures will result in a bigger value of 0, making the layer more dispersive

to unstable waves . These results are consistent with the numerical results of Petersen

(1978), Michalke & nermann (1982) and Cohen (1986) which are based on spatial stability

calculations.
115

Isothermal, Incompressible jet

>.,

L)

©

_)

q)

50

C)

__C:

CL

1.0

0.8

0.6

0.4

0.2

0.0

stable _ I---_ unstable

o:/l

xx |

0 o 0 X_XO C_

0

co _ ®0

0

0

c_ •

0

0

0

-0.2 i I I I I

-0.8 -0.4 0.0 0.4

Imaginary Part, cr

• Re=5.E+5

x Re=5000

o Re=500

Fig. 5.1 The effect of Re on the spectrum of linear, parallel flow, temporal eigenvalues using 31 collocation points for 0 = 1/15, Mj = 0, T. = 1, a = 3.66, and

/_=0.
116

Heated, Incompressible jet

0

0

©

0

EEL

1 . U

0.8

0.8

0.4

0.2

0.0

_9

0

stable _ [_ unstable

o

0

B

0

e

°

0

Ol

o _ S_ll

o_

co_

-0.2 I :! I l I l

-0.8 -0.4 0.0 0.4

Imaginary Part, ¢_

• Re-5.E+5

x Re= 5000

o Re=500

Fig. 5.2 The effect of Re on the spectrum of linear, parallel flow, temporal eigenvalues using 31 collocation points for 0 = 1/15, M s = 0, 7". = 2, c_ = 3.66, and

/_=1.
117

Isothermal, Compressible jet

,m

O

O

©

>

©

C)

c-

_L

1.0

0.8

0.6

0.4

0.2

0.0

stable -_ unstable

o _ t

o o o _@ So cm

0

0

0

°

0

cb •

O

0

0

-0.2 r _ I j I I

-0.8 -0.4 0.0 0.4

Imoginory Port ,cl

• Re=5.E+5

 : OOOo

Fig. 5.3 The effect of Re on the spectrum of linear, parallel flow, temporal elgenvalues using 31 collocation points for 0 = 1/15, M s = 0.8, T, -" 1, c_ --- 3.66, and

/_=0.
G

reco

t=,

0.8

0.6

0.4

0.2

0.0

Isothermal, Incompressible jet

I I I I

0.0 1.5 3.0 4.B e.o 7.5

Wave Number, a

118

1.0

" 0.8

.>.

P,

_" O.e

°.1

0.2

0.0

i

I I I I

1.5 3.0 4.6 e.O 7.5

Wave Number,

Fig. 5.4 (a) Growth rate and (b) phase velocity of unheated, axisymmetric modes

for various Re: Re = oo (solid lines), Re = 5000 (dot), Re =500 (dash). Calculations

are for 0 = 1/15, M i = O, T. = 1.
G)

¢0

fI:

cO

L--

0.6

0,5-

0.4

0.2

0.0

Heated, Incompressible jet

I I i !

0.0 1.5 3.0 4.5 5.0

Wave Number ,

7.5

119

4_

On

(D

>

¢tJ

0.

1.0

0.8  ̧

0.4

0.2 I I I I

0.0 1.5 3.0 4.5 5.0 7.5

Wave Number,

Fig. 5.5 (a) Growth rate and (b) phase velocity of heated, axisymmetric modes for

various Re: Re = c_ (solid lines), Re = 5000 (dot), Re =500 (dash). Calculations

are for 8 = 1/15, Mj = 0, T. = 2.
G

¢o

O:

.fz

O

L_

¢,D

0.8

0.8-

0.4-

0.2-

0.0

Heated, Incompressible jet

i

I I I I

0.0 1.5 3.0 4.5 8.0

Wave Number, a

7.5

120

a_

O

O

m

>

¢o

0.

1.0

0.8

0.8

0.4

0.2

(b)

i |

I I I I

0.0 1.s 3.0 4.8 8.0 7.5

Wave Number, a

Fig. 5.6 (a) Growth rate and (b) phase velocity of heated,/_ = 1 modes for various

Re: Re = oo (solid lines), Re -- 5000 (dot), Re =500 (dash). Calculations are for

0=1/15, Mj =0, T.=2.
0.8

0._

(D

(o

rr

r- 0.4

0

0.2-

0.0

Axlsymmetric Mode

I ! t !

0.0 1.5 3.0 4.5 e.O

Wave Number,

7.5

121

4_

O

>

¢o

rQ..

1.0

0.8

0.6

0.4

0.2

| iiii ii

%% -'_i,

I I I I

o.o 1.5 3.0 4.5 e.o 7.6

Wave Number, a

Fig. 5.7 Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

/_ = 0 and # = 1/15 for various M i and T 1. M 1 = 0, 7', = 1 (solid lines).

M i = 0.8, T0 = 1 (chaindot). M i = 0, T, = 2 (dash).
0.8

First Azimuthal Mode

122

tO

¢O

re"

¢-

O

0.6'

0.4

0.2

0.0

"'°"

I

0.0 1.5

"%

"°,

o,

"o I

• 0 _

; I

•. •

! ! l

3.0 4.5 6.0 7.5

Wave Number, _

1.0

>,

0

!

>

tQ.

0.8

0.6

0.4'

0.2

(b)

J

J

I I I I

0.0 1.5 3.0 4.6 6.0 7.6

Wsve Number, a

Fig. 5.8 Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

/3 = 1 and 0 = 1/15 for various M i and Tj. M i = 0, 7". = 1 (solid lines).

M i = 0.8, T. = 1 (dot). M i = 0, 7", = 2 (dash).
(D

4._

¢o

eyeO

o0

Ogi

Ni

e_

E

<

o

Vortex Sheet

123

0.0 1.5 3.0 4.5 6.0 7.5

Wave Number, o_

>.

,i

O

O

(D

>

(D

¢D

J:

13_

1.0

0.8

0.6

0.4

0.2

(b)

.°

• °.

"'"°"°%,.°,o°,

Fig. 5.9 (a) Amplification rate (C,) and (b) phase velocity (Cn) of a vortex sheet

for l_ = 0 and/_-- 1 modes. /_-- 0, T. = 1 (solid lines). /_-- 0, T. -- 2 (dot).

_/= 1, T. -- 1 (dash). /_ -- 1, 7'. = 2 (chaindash). M s = M= --0.
0.6

Tanh Profile

124

G)

OC 0.4-... I

o

/

o.2 -

0.0

0.0 1.5 3.0 4.5 e.o "z.6

Wave Number,

1.0

_. 0.8

0.8

rn

0.4

\. (b)........

°*""°t°l°,°°.,,,.11 .... o ,

0.2- I I I I I

0.0 1.5 3.0 4.5 6.0 7.5

Wave Number, o_

Fig. 5.10 Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

/_ = 0 and _ = 1 for 0 = 1/15 and M i = 0. /_ = 0, To = 1 (solid lines}. /? = 0, T. --- 2

(dot). /_ = l, T, = 1 (dash). /_ = 1, T. = 2 (chalndash). M i = Mo =0.
Variation of Azimuthal Mode Number

08. .....

125

ca

frO

0._ -

0.4

0.2  ̧

0.0 |

I I • I

0.0 1.5 3.0 4.6 e.O 7.5

Wave Number,

1.0

_. O.O

O.a

¢o

O.

0.4-

0.2 ii

I ! I I

o.o 1.5 3.0 4.5 e o 7.5

Wave Number, a

Fig. 5.11 Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

/_ -- 0 (solid lines), /7 - 1 (dot), and /_ = 2 (dash). Calculations are for

0=1/15, M i =0, To--- 1.
a)

(a

12:

J=

o

L_

Variation of Azimuthal Mode Number

0.20 n .................

0.18 "" "-.

o," ,,

*" %,

0.12

0.08

0.04 -

0.00 ! i I i

0.0 O.G 1.0 1.6 2.0 2.6

Wave Number,

126

G)

¢o

(v

O.

1.0

0.8

o.,::::::::::::::::::::::::::::::::::::: .............................

0.4

I !

0.0 0.6 2.0 2.5

0.2

(b)

t I

1.0 1.8

Wave Number

Fig. 5.12 Inviscid (a) growth rate and (b) phase velocity of the azimuthal modes

/? = 0 (solid lines), /_ = 1 (dot), and _ = 2 (dash). Calculations are for

0=1/5, Mj=O, T.=I.
127

Chapter 6

SUBHARMONIC INSTABILITY MODES

As discussed in Chapter 3, a principal effect of excitation is to generate a primary

wave which grows in amplitude, and then reaches a finite amplitude (i.e., saturation) as it

propagates downstream. In turbulent flows, this primary wave is usually called a coherent

structure. An objective of this work is to investigate, via a secondary instability theory,

the conditions under which this primary wave can destabilize a subharmonic mode. Since

subharmonic disturbances generate vortex pairing (and vice-versa), in a physical sense,

the present chapter focuses on the conditions, as a function of Mach number, temperature

ratio etc, which enhance, or at least influence, vortex pairing. This process of merging

(or pairing or amalgamation), is generally believed to be a_i important mechanism in the

spreading of jet flows.

Although secondary instability theory in round jets is presently still new, its predictive

capabilities are, for the most part, undisputed in incompressible planar mixing layers and

boundary layer flows. We now digress momentarily to review some of these known results,

and outline some distinct features of round jets that are absent in mixing layers and

boundary layers.

By using a secondary instability analysis, Kelly (1967) argued that the sequential merging of vortices in planar mixing layers can be attributed to the subharmonic instability

modes in a periodic base flow. These modes, which absorb energy from the mean flow via

the catalytic role of a periodic fundamental mode, predominate in the flow at approximately the location where vortex pairing occurs. Ho & Huang (1982) also demonstrated
128

that aftereach pairingof coherentstructures,the peak frequency ofthe energy spectrum

was halved, and the merging processwas found to be at the locationwhere the subharmonic modes become the dominant instabihtymodes. Their findingsreinforcedthe

predictivecapabilityofa secondary instabilitytheory.

Further,Herbert (1983,1984, 1985)demonstrated that a secondary instabilityanalysis

can unravelsome of the fundamental routesto transitionin incompressiblewall-bounded

flows. In a particularroute, he has shown that the essentialflow physics associated

with the formation of staggeredlarnbdavorticesin boundary layers,although seemingly

formidable,can be revealedby a linearsecondary instabilitytheory.Itmay be remarked

that boundary layerflowsare fundamentally differentfrom shear flows. This difference

ismarked by the presence of a solidwall which constitutesan additionalviscous noslipconditionon the velocityperturbations.Consequently, the growth of an instability

mode isinduced by viscosity,and hence, the growth rateissmall.This means that when

the viscosityisturned offin wail-bounded boundary layerflows,there are no instability

modes. In contrast,the instabilitymodes in shear flowsare predominantly inviscid,and

the growth rates axe generallylarge;at leastin the sense that the amplitude of the

fundamental changes by a significantamount in one period or wavelength.

Jets and planar mixing layersbelong to the same classofflows,which arecharacterized

by the existenceof a generalizedinflectionpoint in the parallelflow profiles.However,

the former have a few distinctfeaturesthat are absent in the latter.The qualitiesthat

discriminatejetsfrom mixing layersare:

1. The radiusof a jetnozzleconstitutesa lengthscale,which isused in our normalization,in additionto the shearlayerthickness.Thus the shearlayerthicknessbecomes

a bona-fideparameter for a jet;thisquantitycan be scaledout fora mixing layer.

2. The vortexstretchingterm,which representsthe extensionor contractionofvorticity

linesby the strainrate ofthe flow,isnonzero forazimuthal disturbances.
129

Because of (1) above, the symmetry of the base vorticity about the critical point of a jet is

broken, and, as a result, the waves are dispersive so that waves with different wavenumbers

are convected with the flow at different speeds. Furthermore, the criterion for the existence

of an inflectional mode is now a function of the mean flow and the mode number, 8, of

the instability mode (Batchelor & Gill 1962). In contrast, the necessary and sufficient

condition for the existence of an instability mode in a planar mixing layer depends on the

base flow alone.

We will employ a generalization of the current incompressible secondary instability

theories to gain a better understanding of the mechanisms governing the amalgamation

of coherent structures in compressible round jets. High speed jets, whether for heating or

for nonzero Mach number, introduce density perturbations. These result in quadratic and

cubic interactions between the density perturbations and other types of perturbations.

This cubic interaction results in considerable algebra in the secondary instability analysis,

and therefore necessitate some simplifications which we will now discuss.

Recall that a deficiency of linear stability analysis is that it cannot determine a unique

amplitude A. In this work, the amplitude A is a small arbitrary prescribed quantity

which corresponds to the maximum amplitude of the streamwise velocity perturbation. A

consequence of this normalization is that if all other types of perturbations are at most

the same order as A, then the quadratic interaction between F1 and F2 will play a more

important role than the cubic interaction between FI and itself and F2. This is because,

for small A, the quadratic interaction is of order Ae while the cubic interaction is of order A2e. There are, however, possible situations where this definition of A might cause

inconsistencies with our analysis. These happen in strongly heated jets where the parallel

stability analysis provides a density eigenfunction that is much greater than the streamwise velocity eigenfunction. As a result, the magnitude of the density perturbation (Apl)

will no longer be small compared to the unperturbed density (P0). This will invalidate the
130

present stability analysis where the amplitudes of all types of perturbations are assumed

to be small compared to the mean flow. Nevertheless, these possibilities can be avoided

by normalizing A so that it corresponds to the maximum amplitude of the density perturbation. This implies that when A is small, the amplitude of all types of perturbations

will be small too. Therefore, in this work, we shall focus on the cases where A z and higher

order terms can be neglected. The preceding remark is consistent with our representation

of the flow arising from the finite-amplitude growth of the primary disturbance associated

with the parallel base flow. Formally, this flow [say, _ ] may be expanded as

_C_,_, t)= _o(_)+ A_-o' + ...+ A{_x(_)+ A_-I(_)+-..} exp(i_,_)

+ {A',_+...} exp(2i_,)+-..+_._ (6.11

where the superscript 1 denotes a correction due to the nonlinear self-interactions of the

primary wave, and the finite amplitude A can, in principle, be obtained from the socalled Landau's equation (Jarrah 1989). Since the periodic base flow we have modeled

(i.e., u0 + (Aul exp(ialx) + c.c.)) is formally valid to O(A), on the basis of the expansion

(6.1), any quantity involving A 2 and higher order terms in the secondary stability analysis

should be truncated. Further, as noted by Kelly, we also assume that the terms of O(A)

and higher in (6.1) do not eliminate the generalized inflection point associated with the

parallel mean flow, as otherwise, the perturbation analysis will be invalid.

Fully developed coherent structures (or roughly, vortices) can be viewed as manifestations of a primary instability wave that propagates and grows in the downstream direction.

Although the quantitative experimental data on the vortex-pairing scenario in round jets

is presently still very scant, the evidence of the pairing phenomenon can be seen in the

experimental work of Wille (1963) and Reynolds (1988). In our study, the merging of

coherent structures is idealized as being composed of two externally excited disturbances:

an axisymmetric fundamental wave and its subharmonic. The fundamental wave is chosen

to be axisymmetric for two reasons. First, the axisymmetric geometry of a jet nozzle tends
131

to accentuate an axisymmetric wave near the nozzle. Secondly, the resonance condition

on mode numbers, which has been established in Chapter 3, can be satisfied exactly for

a subharmonic mode with an arbitrary mode number. Further, the merging of two ringlike coherent structures is idealized as the resonant interaction between two axisymmetric

waves. And, any appearance of helical motions can be attributed to a helical subharmonic

resonanting with its axisymmetric fundamental. This idealized model, which assumes

that other frequencies and fine scale motions play a minor role in the overall flow physics

describing the merging of coherent structures, provides a blueprint for future studies and

comparisons.

To recap, the assumptions used in the secondary instability analysis are:

1. The magnitude of the subharmonic disturbance is sufficiently small to allow linearization about the primary wave and the parallel base flow.

2. The mean flow profiles are kept unchanged and the mode shape (i.e., the cross-space

structure) of the primary wave is given by a linear theory even though the amplitude

of this wave may be quite large (say, A -- 0.10). Some justification of this assumption

lies in the work of Petersen & Samet (1988).

3. We shall assume that A is a prescribed constant, although in experiments we find that

during and after saturation, while the subharmonic is growing, A is a slow function

of x. In our temporal theory, this translates into A -- A(t); we ignore this slow

dependence on time.

A consequence of these assumptions is that the stability characteristics of the fundamental

mode can be approximated by the linear modes of parallel flow analysis. It may be

reiterated that the disturbances are normalized such that A is the amplitude corresponding

to the maximum of the streamwise velocity perturbation of the primary wave, and we

consider a disturbance to be an infinite wave train that is spatially periodic but temporally
growing.

132

6.1 Resonance Conditions

In the method of multiple scales, we considered two small amplitude waves of

arbitrary wavenumbers and growth rates evolving temporally in a parallel shear flow. The

lowest order nonlinear interaction can occur only through quadratic wave interactions.

In a perturbation scheme, the flow field at this order may be thought of as the response

of a linear oscillator to a small forcing function comprised of the quadratic wave (or,

oscillation) interaction terms. The response of the oscillator to this small forcing function

is small unless there is a resonant wave interaction. This happens when the growth rate,

mode number and wavenumber of a forced wave, generated by the forcing term, are the

same as that of an instability wave of the parallel flow. A consequence of the quadratic

wave interactions is an enhancement of the transfer of energy from the mean shear flow

to the forced wave, which after a sufficient period of time, grows to an amplitude that is

comparable in size to a primary mode associated with the roll-up of a coherent structure.

Recall that in order to have resonant wave interactions, the two unstable wave modes

must fulfill the following resonance conditions:

I_,_lcdl,_l,l/3_t)+ l,_lcdl_l,l_21) = l_,_+ _llC,(Io,_+ o,,l,I/_+ _l) (316)

,_2cR(I,:,,11,1_11) + o_.,cR(l_,l, I_..,1)= (_ + at)CR(la2+ '_,.I,I_1+ Z_I) (3.,7)

where the a's and fl's can be positive or negative, and are so far arbitrary. These conditions, upon close examination, can be satisfied by selecting all of the following restrictions:

(6.1) The wavelength of the subharmonic disturbance is twice that of the primary

(or, fundamental) wave [i.e, al -- -2a,].
133

(6.2) The sum of the azimuthal mode numbers of the fundamental and its subharmonic is equal to the azimuthal mode number of the subharmonic alone.

(6.3) The phase velocity of the fundamental equals that of the subharmonic.

(6.4) The sum of the growth rates of the fundamental and the subharmonic is the

same as the growth rate of the subharmonic alone.

The condition (6.2) can be satisfied exactly by taking the fundamental to be axisymmetric, while the subharmonic can be axisymmetric or helical. The condition (6.3) requires

the waves to be nondispersive and this can only occur when the jet momentum thickness

is small (i.e., near the jet nozzle). In round jets, where the waves are dispersive, this condition is met only approximately. In contrast, the phase velocities in planar mixing layers

are independent of the wavenumbers and hence guarantee this condition exactly. For the

tanh velocity profile, the matching of phase velocity can be approximately satisfied in two"

different ways. The first way represents a near neutral fundamental mode interacting with

a near maximally amplified subharmonic, while the second way involves a near maximally

amplified fundamental interacting with a subharmonic whose growth rate lies between

zero for a -- 0 and the most amplified value. For clarity, we called these two ways route I

and route H to subharmonic instabilities, respectively. Route I is physically more meaningful since a fundamental wave attains a finite-amplitude saturation when it is within the

vicinity of a neutral point. As revealed in Figure (6.1), an axisymmetric subharmonic can

interact with the fundamental via route I, while the matching of phase velocities between

a helical subharmonic and the fundamental can occur through either route I or route II;

note that the fundamental is axisymmetric. In Figure (6.1), the subscripts f and s denote
134

the phase velocities of the fundamental and the subharmonic, respectively; symbolically,

route I is represented by an interaction between I! and I,, while route II is represented

by an interaction between II I and II,.

The condition (6.4) can approximately be fulfilled by choosing an almost maximally

amplified subharmonic and a fundamental wave that is near neutral. Strictly speaking,

this condition will preclude route II, although sometimes this condition is abandoned

altogether under the assumption that growth rates are generally Usmall'. It is worth

noting that the wave resonant interactions in the bulk of Kelly's work (1967) on planar

mixing layers in achieved via route II. A consequence of this choice is that the condition

on the growth rate is no___t_t met. The effect on the secondary subharmonic modes for not

imposing the condition (6.4) will be examined in round jets in subsequent sections.

In summary, in order for all the resonance conditions to be satisfied simultaneously,

the fundamental must be a near neutral axisymmetric mode, while the subharmonic can

have an arbitrary azimuthal mode number and is nearly maximally amplified.

6.2 Comparison of Approaches

In view of the absence of any previous studies on the secondary instabilities of

subsonic heated round jets, two distinct yet complementary approaches are used. One of

these is a perturbation method and the other is a generalization of normal mode approach

to a periodic base flow. Typical results from the two approaches for 0 = 1/15 and various

jet parameters are compared in this section. In Figures (6.2) - (6.4), the solid lines

represent the method of multiple scales, while the open circles indicate the normal mode

analysis. In the normal mode analysis, the Irelative frequency[ = Ib2t[- the absolute value

of the imaginary part of b2 defined in (3.45)-- refers to the frequency of the subharmonic in

a frame of reference that moves with the phase velocity of the fundamental wave. When
135

this IrelativefrequencyI iszero,the relativephase velocityin the streamwise direction

between the fundamental and itssubharmonic becomes zero.

In Figures (6.2a-b),the growth rate of the secondary wave (i.e.,the realpart of b2 =

_2R) and itsIrelativefrequencyl,as a functionof the amplitude,A, of a near maximally

amplifiedfundamental mode areplotted.These figuresrepresentthe casewhere the phase

velocityapproximately satisfiesthe resonance condition(6.3),whereas the growth rates

do not adequately fulfillcondition (6.4). A couple of interestingobservationscan be

extracted from these figures.First,the normal mode analysisproduces a "dominant"

subharmonic mode whose growth rate increasesat a rate proportionalto the amplitude

A of the fundamental and a second mode which suffersa decay as A increases.In the

perturbationmethod, we focuson the subharmonic whose growth rate increaseswith A.

The growth rateof the dominant mode isincreasedby about 3°_ when A changes from 0

to 0.02. At a largervalue of A, the decaying mode isphysicallyinsignificantrelativeto

the dominant mode and, itwillnot be pursued here.Secondly,when A ranges from 0 to

0.02,the growth rateof the dominant subharmonic mode agreeswell with that obtained

from the method of multiplescales.Finally,the phase velocityof the subharmonic wave

remains synchronizedwith itsfundamental when A becomes finite.Note that in allof

thesecalculations,A isstillvery small;because ofthe presenceof the (periodic)primary

wave, the subharmonic possessesa growth rate which islargerthan what itwould have

in the parallelbase flow.

The previousresultsareextended tolargeramplitudes,A, inFigure (6.3).In comparing

the two approaches,itcan be seen that when A islessthan 0.03,they are in agreement.

However, when A increasesfrom 0.03 to 0.1,the perturbationmethod overestimatesthe

growth rate. We conjecturethat this disagreement arisesfrom the inaccuracy of the

low order perturbationmethod forlarge A, which ispresumably caused by the limited

accuracy of the truncated asymptotic expansion (3.21) - (3.24). An obvious suggestion is
136

tointroducea second slow time scale7"2= 52tinorderto absorb the parabolicdependence

ofthe growth rateon A forlargeA. Rather than extending the analyticalapproach inthis

way, we chose to explorethe normal mode method whose range of validityissomewhat

largerthan that ofthe asymptotic method and which alsoprovidesan independent check.

Once the primary wave attainsa finiteamplitude, the secondary instabilityof the jet,

as manifested by the growth rate of the subharmonic, increases.We believethat this

instabilitywillresultin the pairingof the vortex ringsassociatedwith the primary wave.

Figures(6.4a-b) show the growth rateand the ]relativefrequency]as a functionof the

amplitude of a near neutralfundamental mode fora heated jet. Itcan be observed that

when A increasesfrom 0 to 0.02,the growth rateof the subharmonic mode isaugmented

by about 7%. Further,for thisrange of A, the two approaches are in good agreement,

and the phase velocityofthe subharmonic mode matches that ofthe fundamental.

Itisworth noting that the periodicflow destabilizesthe subharmonic mode of Figure

(6.4)more dramaticallythan that of Figure (6.2).This can be attributedto the factthat

the resultsin Figure (6.2)do not adequatelyfulfillthe growth raterestriction(6.4).These

resultsarerathertypicalofour calculations,and therefore,we can conclude that inorder

to achieve optimum enhancement of a subharmonic instabilitymode, itswavenumber,

phase velocityand growth ratemust satisfythe resonance conditionssimultaneously.

Additionally,in subsequent sectionsof thischapter,itwillbe found that in order to

compare these approaches,itisessentialthat the phase velocityof the primary subharmonic disturbancematches (or,almost matches) the fundamental. This isbecause when

the linearwaves are nondispersive,thereisno mechanism in the low order perturbation

analysisto guarantee the destabilizationof a subharmonic mode by the periodicflow by

means of phase-lockingwith the fundamental.
137

6.3 Compressibility Effect

In this section, the influence of bifj on the secondary subharmonic instability in an

unheated compressible jet is presented. It should be remembered that as the jet biach

number changes, the shape of the primary mode also changes. Therefore, both this change

in the primary waves as well the explicit dependence of the secondary wave on Mj will

alter the instabilities of the subharmonic. [For reference, compressibility stabilizes the

primary eigenmodes]. The axial wavenumbers of the subharmonic and the fundamental

are chosen to correspond to a near maximally amplified wavenumber and a near neutral

wavenumber, respectively. In accordance with the discussion in the previous section, only

the "dominant" subharmonic is shown.

A typical influence of Mj on the growth rate and the Irelative frequency I of subharmonic

instability modes (f12 = 0) and (f12 = 1) in the presence of a finite amplitude primary mode

is exhibited in Figure (6.5). With reference to Figure (6.5b), an interesting feature is that

when the waves are (initially at A =0) slightly dispersive, a finite threshold (fundamental)

amplitude Ath is required for the matching of phase velocities to be established. After this

matching is established, the dominant subharmonic mode always travels with the speed

of the fundamental, and the growth rate of the subharmonic mode increases substantially

as A increases (see Figure 6.5a). This threshold amplitude generally decreases as the

difference (at A = 0) between the phase velocities of the fundamental and subharmonic

decrease. This observation is important because it implies that for small amplitudes of A

which are less than the threshold value, the low order perturbation series (3.21), which

assumes that the matching of phase velocities is independent of A, might not be able to

correctly predict the growth rate of the secondary disturbance when the primary waves

are dispersive. This is sensible, on physical grounds, since the matching of phase velocity

means that the two waves travel in unison. Therefore, there is a possibility for the effective

transfer of energy from the mean flow to the subharmonic mode via the catalytic role of
138

the finite-amplitudeprimary wave; this is because there is no "phase cancellation" by the

motion of the subharmonic relative to the fundamental.

In order to have a representative idea of the effect of compressibility, it is sensible to use

the total increment of the growth rate of a subharmonic mode from parallel flow analysis

due to the effect of the primary wave. Let R represent this increment when A changes

from 0 to 0.1. Conveniently, some of the more important results pertaining to Figure (6.5)

can be tabulated as below:

/_ M_ Ath

0 0.I 0.011 0.236

0 0.8 0.004 0.204

I 0.I 0.006 0.205

I 0.8 0.011 0.180

The above calculations are for Re = 5000, 0 = 1/15, T,= 1, and fll = O. The value of Ath

is only an estimated one because we did not have enough runs in the neighborhood of Ath.

For an incompressible jet, when the amplitude A changes from 0 to about 0.1, the growth

rates of the/_2 = 0 and f12 = 1 are augmented by 0.236 and 0.205, respectively. This is

about a 40 % increase over their values at A - 0 and suggests that subharmonic instabilities"

play a major role in the development of a jet flow. Furthermore, when A is greater

than the threshold value, compressibility has a moderate stabilizing effect on secondary

subharmonic instability modes so that vortex pairing, which results from subharmonic

instability, is expected to be less important in compressible flows. For example, for the/_2

= 0, the increment, bt, is reduced from 0.236 to 0.204 due to compressibility.
139

6.4 Heating Effect

In this section, the influence of T, on the secondary subharmonic growth rates in a

heated (T. > 1) and low Mach number (Mj = 0) jet is discussed. The comparison between

heated and unheated flows is again rather subjective since the primary eigenmodes also

depend on T,. As noted in Chapter 5, heating is found to have a slight stabilizing effect

on primary modes with axial wavenumbers larger than the most amplified wavenumber.

As before, the axial wavenumbers of the subharmonic and the fundamental are chosen to

correspond to a near maximally amplified wavenumber and a near neutral wavenumber,

respectively (i.e., via Route I).

Figure (6.6) shows the influence of T. on the growth rate and the Irelative frequency I

of subharmonic instability modes (/_2 = 0 and /_2 = 1) as a function of the amplitude

A of the fundamental mode. In this figure, 0 = 1/15, My -- 0.1, and /_1 = 0 are used.

Again, for clarity, t{ refers to the increment in the growth rate of the subharmonic mode

when A changes from 0 to 0.1, and Ath represents an approximate threshold amplitude for

the onset of a secondary subharmonic instability mode. The more significant information

pertaining to Figure (6.6) can be tabulated as below:

_2 To Ath I_

0 1 0.011 0.236

0 2 0.014 0.206

1 1 0.006 0.205

1 2 0.001 0.122

Several interestingfeaturescan be seen in this figure,using the above table as an

aid. First,the growth rate of subharmonic instabilitymodes in a heated jet, again,

increasesas A increases.Secondly,while heatingslightlysuppressesthe growth rate of

an axisymmetric secondary mode (by about 15%), itdecreasesthe growth rate of/_ = 1

mode considerably(by about 40%). This leadsus to believethat heatingtends to inhibit
140

vortex-pairing in round jets, and this inhibition is substantially more pronounced for the

first azimuthal subharmonic mode with/9 = 1. Finally, heating increases the threshold

amplitude required for the onset of a secondary subharmonic with/_ -- 0, but it reduces

the corresponding threshold amplitude for ;9 = 1 mode. To conclude, vortex pairing in a

heated, incompressible jet is likely to be an axisymmetric phenomenon.

6.5 Shear Layer Thickness Effect

We recallthat jet instabilityiscaused by the induction of the vorticity,which

is contained within a narrow region in jet shear layers. The physical mechanism for

an amplifieddisturbancecan be explained in terms of a vortex sheet in a homogeneous

incompressiblefluid(Drazin& Reid). According to the Biot-Savartlaw,each vortex in an

inviscidfluidiscarriedwith the fluidand inducesa circulatingflow proportionalto the

strengthof the vortex line.Specifically,the velocityinduced at any point by the vortices

willamplifythe sinusoidaldisplacementofa vortex=sheet.

In a given parallelvelocityprofile,the mean vorticityisinverselyproportionalto the

thicknessofthe shear layer.This vorticityofthe steady mean flowcan be interpretedas a

source term from which energy istransferredto the unsteady perturbationvorticity.This

point can be illustratedby consideringa two-dimensionalincompressibleinviscidfluid

with a steady mean vorticityno and an unsteady small perturbationvorticityfl_. The

equationsgoverningthe dynamics ofthe totalvorticityand the perturbationvorticityare,

respectively,

D

_(no + n') = o (6.2)

o a d_

(_ + u0_z)n' = -v dr (6.3)
141

where

D a

- +v.V,

Dt cgt

and u0 and v are the mean flow velocity and the radial perturbation velocity, respectively.

These equations reveal that while the total vorticity of a material element carried with

the fluid is conserved, the perturbation vorticity, which is being convected by the mean

flow u0, changes along the mean flow because of the "source term" -vdflo/dr.

Although in secondary instability analysis, the presence of a finite-amplitude fundamental mode will restructure the vorticity distribution, the growth rate of a subharmonic

mode is, nevertheless, expected to increase as $ becomes smaller. This growth is revealed

in Figure (6.7), which is plotted for Re = 5000, Mj = 0.1, 7", = 1,/_1 = 0 and two different

values of 0:0 = 1/30 and a = 1/15. It may be noted that the flow is taken to be unheated

(T. = 1 ) and almost incompressible(Mj = 0.I)inorder toisolatethe effectofthe parameter0,which appears explicitlyin the velocityprofile.Itisshown that,when A changes

from 0 to0.1,the totalincrement,R,ofthe growth ratesofthe axisymmetric subharmonic

modes for0 = 1/30 are approximatelytwicethat for0 = 1/15. These resultsindicatethat

although the presenceofa primary wave redistributesthe vorticityofthe parallelflow,the

growth rate of a secondary subharmonic mode isseen to increaseproportionallyto 1/0,

just as in the primary stabilityanalysis.This impliesthat ifvortex pairingisa resonant

interactionbetween thesestructuresand a small amplitude subharmonic mode, the former exertsthe greatestinfluenceinreinforcingthe latterwhen the shearlayerthicknessis

small (i.e.,near thejetnozzle).This means that each successivecoalescenceof a coherent

vortex structuresnot only makes the primary wave more dispersivebecause of the larger

shear layerthickness,but alsomakes itlessconducive to pairing.

In Figure (6.7),itmay alsobe noted that the solidlinesrepresentthe case where the

resonance conditionon the growth rate (6.4)isnot met, and subsequently,the increase

in the subharmonic growth rate islesspronounced than in the case where the condition
(6.4) is fulfilled.

142

6.6 Mode Number Effect

In this section, we will discuss the influence of the finite amplitude primary axisymmetric mode on the growth rate and the Irelative frequency I of a subharmonic mode

for various mode numbers. In Chapter 5, we found that the tanh velocity profile with

small values of 0, which characterize the mean flow close to jet exit, is unstable to a

large number of azimuthal modes. When 0 increases, the fl = 1 mode increasingly becomes the dominant instability mode. Further, it is found that the periodic flow is most

unstable to subharmonic disturbances whose phase velocity is the same as that of the

fundamental. This matching of the phase velocity only occurs when the jet momentum

thickness is sufficiently small, since otherwise, the wave speed cannot be matched because

of dispersiveness. The value 0: -- 1/15 is chosen because at this value of #, not only can

resonant wave interactions occur, but also thestreamwise location implied by this value

is sufficiently far from the jet nozzle to ensure that a finite-amplitude saturation of the

fundamental mode can be attained. Some evidence of this particular value of # can be

found in the work of Cohen (1986).

In Figures (6.8) and (6.9), the subharmonic growth rate and the [relative frequency I for

various azimuthal numbers are plotted as a function of the amplitude of the fundamental

mode. The results in these figures are based on Re = 5000, 0 = 1/15,/_1 = 0, 7", = 1,

Mj ----0.1, a2 -- 3.66, al = 7.32. These figures show that for a low Mach number jet,

the overall increment of the subharmonic growth rate of the azimuthal modes/_ = 0, 1

and 2 increase as A increases. The secondary growth rates of/_ =_0 and/_ = 1 are of

comparable size, while the corresponding quantity for/_ = 2 is relatively small. It is rather

surprising to note that the threshold amplitude Ath of _ = 3 mode is rather high, and the
143

phase-lockingphenomenon is never established for fl = 4 mode. The secondary growth

rate of/_ = 3 is in fact smaller than the growth rate obtained from parallel flow analysis.

These results are of interest because they indicate that the approximate fulfillment of

the resonance condition on the phase velocity does not automatically lead to an increase

in the (temporal) growth rate of a subharmonic mode. Physically, the suppression of

the growth rate of the subharmonic, and hence the vortex-pairing, means the energy is

pumped from the subharmonic to the mean flow. It would be interesting to see whether

these observations could be verified experimentally or by full numerical simulations.

These results suggest that in order to achieve maximum mixing enhancement in round

jets via the amalgamation of coherent structures, it is most effective to excite a subharmonic mode with _ = 0 or _ = 1 modes.

6.7 Influence of A on the Eigenfunction of Subharmonic

Modes

In the previous sections,the secondary growth rate and the IrelativefrequencyI

in the presence of a base flow consistingof a parallelmean flow and a finite-amplitude

fundamental mode are discussed.This sectionconsidersthe influenceofthe fundamental f

mode on the eigenfunctionsof the subharmonic. For clarity,the eigenfunctionsof the

fundamental and itssubharmonic are written,respectively,as

f_ = Ifll expi(01) (6.4)

f2,1= If2,11exp1"(92,1) (6.5)

f2,-1 _-_ If2,-11expi(92,-1) (6.6)

where the amplitudes Ifll, If_,ll and If2,-_l as well as the phases9,, 92,1and 9_,__are

functions of the radial variable r.
For a real 82, the substitution of (6.4) - (6.6) into (3.39) and (3.45) yields

144

rl = 2Alfl[ cos(alx + 01) (6.7)

1% = 26 exp(a2t){lf2,il cos(a=x +/_2¢ + 02,1) 0t- Jf2,-ll cos(_=x -/_2¢ + 02,-1)} (6.8)

Recall that when 82 is purely real, the lrelative frequency I is zero. Further, the eigenfunctions are normalized in such a manner that A and e becomes the maximum amplitudes of

the streamwise velocity perturbations.

In Figures (6.10) - (6.11), the amplitude If3,1] and the phase 02,1 as a function of

the radial variable, r, are plotted for three different values of A: A = 0.0, A = 0.05

and A = 0.1. [For reference, the corresponding growth rate and lrelative frequency I are

plotted in Figures (6.2) - (6.3).] As discussed before, these figures characterize the case

where the resonance condition on phase velocity (6.3) is approximately met, whereas the

resonance condition on the growth rate is not satisfied. Except for small changes in the

amplitudes of the radial and azimuthal velocity components as A becomes larger, the

secondary eigenfunctions are generally unaffected by the size of A. This lends credibility

to the fact that the mode shape of the secondary instability wave or that of the primary

during saturation is given reasonably well by the parallel flow results.

Figures (6:12) :{6'13) show{he amplitude and the phase of a near maximally amplified subharmonic disturbance (/_ = 1) for an unheated low Mach number jet. The jet

parameters are identical with the previous case. [For reference, the corresponding growth

rate and "relative frequency" are plotted in Figure (6.4).] These figures represent the

case where the resonance conditions on phase velocity (6.3) and growth rate (6.4) are

adequately satisfied. Again, with the exception of a slight decrease of the eigenfunction

amplitudes in the vicinity Of the critical layer as A increases, the amplitude functions of

the secondary subharmonic modes are minimally affected by the magnitude of A.

In summary, we conclude that while the finite-amplitude fundamental mode can sub-
145

stantiaUy enhance the growth rate of its subharmonic, it has only a small to moderate

effect on the mode shape of the subharmonic.
Routes of Resonant Interactions: I & II

146

10

@

o_

e0

ta.

o.6

0,6

04

02

(a)

.,.,,,.-'°" .... ..,,o ..... o .... ..,o,. .... ..°,o,,o,,°°..,,,,...,°°.,.°o

°°"

! I ! I

0.0 1.5 3.0 4.6 e.O

Wove Number,

7.6

1.0

0.9

:_ 0.0

t_

0,.

0,4-

0.2

0.0

.... ....... lll I1

I ! I !

1.a 3.0 4.a e.o 7.5

Wove Number, _

Fig. 6.1 Inviscid phase velocity for various mean profiles at 0 = 1/15.

Solid lines: M_ = 0.0, 71. = 1. Dot: Mj = 0.0, 7", = 2. Dash: Mj = 0.8, T. = 1.

(a) Helical mode: /_ = 1. (b) Axisymmetric mode: /_ = 0.
147

Comparison of Approaches: Incompressible, Isothermal Jet

0.7-

0.e

rv-

(_ 0.G

0.4

0.000

o o o o o

| ! t1

0.008 o.oto o.ot8

Amplitude of Periodic Wave, A

0.020

0

c

a)

@

d_

0)

.,_

(o

4)

0.07

0.00

-0.07 -

(b)

0 0 0 0 0 0 0

| ii i i i1_1 i I

0.000 0.006 0.OtO 0.016 O.t

Amplitude of Periodic Wave, A

)2O

Fig. 6.2 (a) Growth rate and (b) lretative frequency I as a function of the amplituoe

of the periodic wave with a near maximally amplified wavenumber. Calculations

are for Re = 5000,0 "- 1/15, My --- 0.1,7', -- 1,/_l "- O, cr_ -" 1.7,/_2 "- 1.

Solid lines: multiple scales. Circles: normal mode analysis.
148

Comparison of Approaches: Incompressible, Isothermal Jet

0.6 '

0.4

a

0.0

| i m

O

O

(a)

O

O

i,

I I I I

0.00 0.02 0.04 o.oa o.os 0.10

Amplitude of Periodic Wave, A

0.05

t_ 0.00

@

>

.m

¢B

i

@

n,-

-0.0|

(b)

<X)O00000 0 0 0 0

' I II ' I I

o.oo o_o_ 0.04 o.oe o.os o._o

Amplltude of Perlodlc Were, A

Fig. 6.3 (3) Growth rate and (b) Irelative frequency] as a function of the amplitude

of the periodic wave with a near maximally amplified wavenumber. Calculations

are for Re = 5000,0 = 1/15, Mj = 0.1,7'. = 1,/_1 -- 0, _ = 1.7,/_2 = 1.

Solid lines: multiple scales. Circles: normal mode analysis.
Comparison of Approaches: Incompressible, Hot Jet,

149

0.7

0.6-

tr"

r0

¢J 0.5

0.4

o

o

o

o

0

o

o

I I I

0.000 0.005 0.010 0.016 0.020

Amplitude of Periodic Wave, A

O

c

O"

(D

tl:

°_

ee0.07

0.00

-0.07

(b)

o 0 0 0 o 0 0

I I I

0.000 0.005 0.010 0.016 0.020

Amplitude of Periodic Wave, A

Fig. 6.4 (a) Growth rate and (b) Irelative frequency I as a function of ithe amplitude

of the periodic wave with a near neutral wavenumber. Calculations are for

Re = 500,0 = 1/15, Mi = 0.1,T. = 2,/_i = 0, _2 = 3.2,/_ = 1.

Solid lines: multiple scales. Circles: normal mode analysis.
Compressibility Effect: Isothermal jet

150

O.g[ /'_ (_) ,,,._1' ....

"? 0,J ./J" ................ ::

o

o.oo o.oz 0.04 o.oe o.oe o.lo

Amplitude of Periodic Wave, A

--¢;-

,_¢_

0

C

ID

O"

m

II)

I£

0.07

0.00

mm

"'XI/ (b)

-/

ttl

_ I I I I

o.oo o.02 o.04 o.oe o.oe o.lo

Amplitude of Periodic Wave , A

Fig. 6.5 (a) Growth rate and (b) Irelative frequency I as a function of the amplitude

of the periodic wave with a near neutral wavenumber. Calculations, using normal

analysis, are for Re = 5000,0 = 1/15, To = 1,_z = 0. Dot: a2 = 3.275,/_= = 0, M*1 =

0.8. Dash: a2 = 3.275,_3 = 1,M# = 0.8. Chaindot: a2 = 3.66,/_: = 0,M 1 = 0.1.

Chaindash: a= = 3.66,/_: = 1,M# = 0.1.
Heating Effect: Incompressible Jet

0.g , , ,, , T,

0"8 1

.. _v

o. ..................

O.l I ' I I w

0.00 0.02 0.04 o.oa o.oe 0.10

Amplitude of Periodic Wave, A

151

C:

¢D

Or

¢D

¢D

>

,m

4-J

¢0

rt0.07

0.00

\

\

I"

0.00

(b)

! 1 ! " I

o.o= 0.04 o.oe o.oe

Amplitude of Periodic Wave, A

0.10

Fig. 6.6 (a) Growth rate and (b) lrelativefrequency Ias a function of the amplitude

of the periodic wave with a near neutral wavenumber. Calculations, using normal

analysis, are for 0 = 1/15, Mj = 0.1,/_l= 0.Dot: Re = 500, c_ = 3.2,#2 = 0,T, = 2.

Dash: Re = 500, c_ = 3.2,/_2= 1,T, = 2. Chaindot: Re = 5000, as = 3.66,

/_ = 0,T, = 1. Chaindash: Re = 5000,a2 = 3.66,/_ = 1,7',= 1.
Shear Layer Thickness Effect: Incompressible, Cold Jet

:1.6

1.0

0.8

--" I TI

........... _?_'.

(a)

0.00 f' v ; ";'"

0.02 oo4 o oe o oe

Amplitude of Periodic Wsve, A

0.10

152

0.07

o.oooo!" o'0. o'o; o o

Amplltude of Perlodlo Wave , A

Fig. 6.7 (a) Growth rate and (b) Jrelatlve frequency I as a function of the amplitude

of the periodic wave with a near neutral way, number. Calculations, using normal

analysis, are for Re = ,5000, Mj -- 0.1,7. - 1,_n - 0. Dot: a2 -- 7.3,_2 - 0,

0 -- 1/30. Dash: a2 - 7.3,_s = 1,0 - 1/30. Chalndot: ct_ - 3.6e,/_ - 0,# = 1/15.

Chaindash: _2 = 3.e6,/_z = 1,tt = 1/15. Solid lines: _2 = 1.7,_z = 1,0 = 1/15.
Mode Number Effect: Incompressible, Isothermal Jet

c_

t_

Ix:

o

t..

t9

1.0

0.8  ̧

0.8-

0.4-

0.2 -I

0.0

0.00

,.o0,O0.O0.o,o.o0.,,,,,..,,°_,°.61,t" ''''°'.'°* .... 0.'''0°_' ....

! l ! I

0.02 0.04 0.06 0.08 0.10

Amplitude of Periodic Wave , A

153

0.07

-_ (b)

t_ 0.00

-0.01

0.00 0.02 0.04 O.Oe O.OII 0.10

Amplltude of Periodic Wave, A

Fig. 6.8 (a) Growth rate and (b) Irelatlve frequency I as a function of the (unrestrited) amplitude of the periodic wave with a near neutral wavenumber. Calculations,

using normal analysis, are for Re = 5000, 0 = 1/15,M 1 = 0.1,T. = 1,/_z = 0,

as = 3.66. Solid lines: /_2 = 0. Dot: /_2 = 1. Dash: /_= = 2.
=

Mode Number Effect: Incompressible, Isothermal Jet

154

0._ =

,_ 0.4-

0.3-

0.2

o,1

0.0 -

I I I I

0.00 0.02 0.04 0.08 0.08 0.10

Amplitude of Periodic Wave , A

0.6

w

>.

0

==

I_- 0.0-

>

¢o

-O.e

,o.°O.°,,'"°"

.°..,'

(b)............................................

..... "°" *''" °"°""'°'°°4....°°.....,,,,.., °°,

°'.....,

°'"°°°O°o°.,.

*.,°.°

°°°O.o

I I I ' I

o.oo 0.02 0.04 o.oe o.oe O,lO

Amplitude of Periodic Wave , A

Fig. 6.9 (a) Growth rate and (b) [relative frequency I as a tunction of the (unrestricted) amplitude of the periodic wave with a near neutral wavenumber. Calculations, using normal analysis, are for Re = 5000,# = 1/15, M 1 = 0.1,T. --- l,

/3, = 0, c_2 -- 3.66. Solid lines: /32 = 3. Dot: /32 -- 4.
Effect on IEigenfunction I

Incompressible, Isothermal Jet

155

3.0-

2.0-

1.0

0.0

2.0-

1.0-

0.0-

1

! I I I

0.00 0.26 0.60 0.7§ 0.00 0.26 0.50 0.76

lu,,ll Iv,,ll

3.0

2.0-

, [

1.0

0.0

0.00

3.0 -

2.0-

r

1.0

0.0

0.00

I I I !

o.2s o.so 0.76 0.26 0.60 0.75

I_,,,I Ip,.,I

Fig. 6.10 Distributions of the JelgenfunctlonJ of a subharmonic disturbance as a

function of the radial variable r. Calculations are for Re = 5000, T. = 1.0, M i -- 0.1,

8 = 1/15,/31 = 0,_2 = 1.7,/32 = 1. Solid lines: A = 0. Dot: A -- 0.05.

Chaindot: A = 0.1.
Effect on Phase of E|genfunctlon

Incompressible, Isothermal Jet

156

3.0 3.0

4i 4K

l( el

2.0- t 2,0 - u

e /" e

T e II(

41(

1.0- % 1.0-

" %

:o

0.0 w 0.0

-3.14 0.00 3.14 -3.14 •

phase of u_,l phase of v2,1

01: i 01:

r 2.0- i r 2.0-1 i

iii i'i1' "} !

-3.14 0.00 3.14 -3.14 0.00

phase of w2,1 phase of P2,_

3.14

3.14

Fig. 6.11 Distributions of the phase of a subharmonic disturbance as a function

of the radial variable r. Calculations are for Re = 5000, T. -- 1.0, Mi -- 0.1,

8 = 1/15,/_1 = 0, a_ = 1.7,/_2 -- 1. ×: A = 0. o: A -- 0.05 . o: A = 0.1.
Effect on IEigenfunct|onl

Incompress|ble, Isothermal .let

157

3O- $.0-

2.0- 2.0-

r r

1.0

0.0-| , , 0.0- !

0.00 0.26 O.BO 0.76 0.00 0.28 0.60 0.78

lu,,,I I_,,,I

3.0

2.0

r

1.0

0.0

3.0-

2.0

1.0

0.0

0,00

I I l

ooo o2s o so o.Ts o:,s o.so o.7s

Fig. 6.12 Distributions of tile {eigenfunction I of a near maximally amplified subharmonic disturbance as a function of tile radial variable r. Calculations are for

Re = 5000, T, = 1.0, AIi = 0.1,0 = 1/15,17t = 0, ct2 = 3.66,172 = 1.

Solid lines: A = O. Dot: A = 0.05. chaindot: A = 0.1.
Effect on Phase of Eigenfunctions

Incompressible, Isothermal Jet

3.0

4<

2.0

I(

1.0

0.0

-3.14

3.0

2.0

1.0'

41<

gk(

0(

_4_

_4D >41)

i 0.0 I

0.00 3.14 -3.14 0.00

phase of u_,l phase of v2,1

3.14

158

3.0 3.0-_

_K

41<

2.0 2,0

0(

0.0 0.0

-3.14 0.00 3.14

phase of w2,1

Q(

Q(

_D

I

3.14 o.0o

phase of p_,l

3.14

Fig. 6.13 Distributions of the phase of a near maximally amplified subharmonic disturbance as a function of the radial variable r. Calculations are for

Re = 5000, To = 1.O,M i = 0.1,0 = 1/15,81 = 0, a2 -- 3.66,]_2 -- 1. ×: A -- 0.

• : A = 0.05. o: A = 0.1.
159

Chapter 7

SUMMARY & CONCLUSIONS

This researchconcerns the (temporal)instabilities and resonancesofthe shear layer

generated by a subsonic heated jet emanating from an axisymmetric nozzle.When the

shear layerisexcitedby a harmonicallyoscillatingdisturbanceof a suitablefrequency (or

wavenumber), an instabilitywave developsin the flow. For clarity,thisinstabilitywave

istermed a primary wave or fundamental wave. In the initialstages of development --

where the amplitude ofthe disturbancerelativeto the mean flow issmall-- the stability

characteristics(e.g.,growth rate,phase velocityand mode shape) of the primary wave

are describedquite accuratelyby linearinstabilitytheory. We begin our researchwith

the systematicinvestigationof the linearinstabilitycharacteristicsof a small disturbance

superimposed on a prescribedparallelmean flowapproximated by the hyperbolictangent

function. By generalizingHoward's semi-circletheorem to compressibleround jets,we

have shown that compressibilityreduces the range of unstable wavenumbers and has a

stabilizingeffect.The role of subsonic Mach number, Mj, in affectingjet instability

propertiesisnumericallyexamined forthe tanh velocityprofilewith 0 = 1/15. We have

concentratedour researchon thisvalue of0 because itprovidesa good representationof

the velocityprofileat a downstream locationwhere the shear layerissufficientlythin to

ensurethat resonanceinteractionsbetween two waves (i.e.,fundamental and subharmonic)

can occur.Hence, subsequentdiscussions,unlessotherwiseindicated,pertainonly to these

mean profileswith 0 = 1/15.

Calculationsshow that the most unstabletemporal growth ratesofa (2-d)axisymmet-
160

ricmode and a (3-d)firsthelicalmode (i.e.,/5--1)inan incompressibleflow are approximately equallydominant, and compressibility(inthe subsonicrange) reducesthe growth

ratesofthe axisymmetric mode and the helicalmode by approximately 19_ and 17_, respectively.Compressibilityalsodecreasesslightlythe phase velocitiesof those instability

modes whose wavenumbers are smallerthan the maximally amplifiedwavenumber, and

has no apparent effecton the phase velocitiesofthosewith higherwavenumbers. In order

to singleout the effectof heating,the stabilitycharacteristicsof the tanh profilewith a

temperature ratioT, = 2 -- but with Mach number M_- = 0 -- arenumericallyexamined.

The resultsindicatethat whilethe phase velocityisratherinsensitiveto Mach number, it

is,except forvery long axisymmetric waves, substantiallydecreased by heating.Heating

also appears to reduce the range of unstable wavenumber and to slightlydecreasethe

growth rate of the most amplifiedwavenumber. The stabilitybehavior of an instability

wave which has a largewavelength relativeto the radiusofthe jetnozzleiscompared to

that obtained from a heated vortex sheet analysis.The resultsindicatethat the phase

velocitiesof long waves (i.e.,a --,0) -- irrespectiveof the azimuthal mode number --

in a tanh velocityprofileagreed wellwith those obtained from the vortex sheet. For the

correspondinggrowth rates,thisagreement isfound only in axisymmetric modes but not

in helicalmodes. In fact,the disagreement becomes progressivelyworse as the helical

mode number and the shearlayerthicknessisincreased.

When the primary wave propagates downstream, itsamplitude will-- as a resultof

the exponentialgrowth ratepredictedby the linearinstabilitytheory -- quicklygrow to

an appreciablesize.Because of the mean flow divergence(causedby viscousspreading)

and nonlinear effects,thiswave eventuallyreachesa finite-amplitudeequilibriumstate

(i.e.,saturation)at some downstream location.We approximate the nearlyperiodicflow

which arisesfrom the saturationof the primary disturbanceby a parallelmean flow of

the hyperbolic-tangenttype and a periodiccomponent whose phase velocity,wavenumber,
161

mode number and mode shape are the same as those associatedwith the primary wave.

The principaltask of thisresearchisto determine the resonancesand instabilities of this

periodicflow. The conditionsforresonant interactionbetween two arbitraryinstability

modes are derived.The resonantinteractionbetween two modes allowsan effectivemechanism for the transferof energy from the mean flow to an instabilitymode. In order

to satisfythe resonance conditions,we have shown that the wavenumbers and the mode

numbers ofthe two unstablemodes, a primary and a secondary,must be restrictedto the

followingchoices:

1.The primary mode isaxisymmetric and has an axialwavenumber closeto a neutral

wavenumber.

2. The secondary wave, which can have an arbitrarymode number, isalmost the maximally amplifiedsubharmonic wave; the wavelength of the subharmonic istwicethat

ofthe primary wave.

A key resultfrom the resonantconditionsindicatesthat resonantinteractionbetween two

modes can occur only when the jetmomentum thicknessissumcientlysmallsince,otherwise,the wave speeds cannot be matched because of dispersiveness.When the resonance

conditionsare met, a subharmonic mode interactsstronglywith the primary to produce

an instabilitymode whose wavenumber and mode number isthe sum of the wavenumbers

and mode numbers ofthe two interactingwaves. Ifthe subharmonic mode isaxisymmettic,itwillinteractwith the fundamental to reproduce itself.The interactioninvolvinga

helicalmode with mode number _ will,however, produce a mode with mode number -fl,

and viceversa.Therefore,whenever an interactioninvolvesa subharmonic mode which is

helical,we shallassume that the secondary disturbanceiscomprised of a pair of helical

waves whose propagating anglesare equaland opposite.As a result,the secondary disturbance, whether axisymmetric or helical,when interactingwith an axisymmetric primary

wave willreproduce itselfand thereforeincreaseitsgrowth rate. We have obtained this
162

(modified) growth rate of the subharmonic as a function of the amplitude of the primary

wave -- for different jet Mach numbers and jet temperatures -- using two independent

methods: the method of multiple scales and a generalization of the normal mode analysis

in a spatially periodic base flow.

We found that when the resonance conditions are met almost exactly and when the

amplitude A of the primary mode is less than 3 % of the mean flow, the two approaches

are in good agreement. However, as A increases from 3 % to 10%, the perturbation

method appears to progressively over-estimate the subharmonic growth rate. It is also

shown that when the amplitude of the primary increases from 0 to 2%, the growth rate

of the subharmonic mode is augmented by about 7%. This leads us to believe that the

secondary instability -- as manifested by the increase of the growth rate of the subharmonic -- will enhance the pairing of the vortex rings associated with the primary wave.

Our results indicate that the secondary (subharmonic) growth rates for a (3-d) instability

wave with azimuthal number/_ = 1 and a (2-d) axisymmetric wave are equally dominant,

while those associated with higher spinning numbers are relatively insignificant. Furthermore, compressibility has a moderate stabilizing effect on the secondary instability modes

(whether axisymmetric or helical) so that vortex pairing, which results from subharmonic

instability, is expected to be less pronounced in compressible flows.

In order to have a representative idea of the effects of compressibility and heating,

we use the total increment, denoted by R, of the growth rate of a subharmonic mode

when the amplitude of the primary wave changes from 0 to 10%. Compressibility reduces

the increment, R, of an axisymmetric mode and a helical (/_ = 1) mode by about 15%

and 12%, respectively. While heating seems to suppress moderately the increment, R,

of the secondary growth rate of an axisymmetric mode (by about 15%), it reduces the

corresponding increment of the _8 = 1 mode significantly (by about 40%). This leads

us to conjecture that heating tends to inhibit vortex-pairing in an axisymmetric jet, and
163

this inhibition is most pronounced for the fl = 1 mode; we infer that vortex-paring in a

heated, incompressible jet is most likely to be a (2-d) axisymmetric phenomenon. Our

calculations also show that the mode shapes of the subharmonic instability modes are

generally unaffected by the presence of the (finite) amplitude primary wave. Finally,

when we calculated secondary instability for other values of 0, we found that the growth

rate of the subharmonic mode increased in proportion to 1/0. This implies that vortexparing is more likely to occur near the jet nozzle. (of course, we also have the requirement

of phase speed matching which is most perfect for small 0).

In summary, the principal effect of excitation is to generate a saturated large-scale

structure whose vortices may merge because of subharmonic (secondary) instabilities.

Since the secondary growth rates of the subharmonic instability modes -- which are at

least partially responsible for jet spreading and mixing -- are suppressed by heating and

compressibility, we conjecture that it will be more difficult to control vortex-mergings in

a subsonic heated jet than in a low speed cold jet.
APPENDIX A

164

The Primary Stability Equations

The stability equations for the evolution of linear disturbances superimposed on a

parallel mean flow are of the form

Abfb = 0 (2.31)

where Ab is a 5 × 5 matrix whose elements, aiy, are

a. = _e1 (a_+10r+ r 0__ + c9_) - poOzuo-Ot

a12 ---- --pOU_)

a13 : 0

a14 --'----C_z

a15 = 0

a21 = 0

1

a22 : all Rer2

-2a_

a_3 : Rer2

a24 --'=--C_r

a25 : 0

a31 = 0

a32 ---- --a23

a33 --" a22
165

a44 = -- azUO

"/1_

a45 = PrRe

-0÷

a34 --

r

a35 "- 0

a41 --" --%t;_)Oz

a. = --_m(_ + a,)

a43 -- "_

r

'7 t'po_ 2p,zo p_ 0._. ,z O_, )

J

+ "Y a=,- a,

PrRepo

-_ _p'o+ _ + p_+ pg,'] P,Re -4o"

aBl ---- --POOz

,,. = -p'o- po(1 + a,)

-poO_

a53 -----

r

a54 -- 0

ass -- --uOOz - c3t

where

a n

a_ =

O"

a_=--

az n

o÷

a_= o,--- _

8

a, = a_'

the primes denote differentiation with respect to r, and the subscript 0 refers to the parallel

base flow.
APPENDIX B

166

Stability Equations for the Primary Eigenmodes

The elements of A1 and B1 in equation (2.33) are

all = _e1(o,-}-1o_,,) r -Er -- a2 --

a12 -- --POU_)

a13 = 0

a14 = --ic_

a15 : 0

a21 = 0

1

a22 = all Rer 2

a23- Rer2

a24 = -- D

a25 : 0

a31 : 0

a32 = --a23

a33 = a22

a34 --

r

a3$ : 0

poi_uo
167

a44 ---- --{_tl, o

a45 ----

PrRe

a41 = --"}'pOi_

a42 -- --'TP0( 1 + D)

--_i_

a4 3 --

r

Pr Repo

a54 = 0

a55 ------uOiO_

bll : PO

b22 : PO

b33 : PO

b44 : 1

bss = 1

bik=O forj _-k

where

d"

dr n

and the primes denote differentiation with respect to r.

0 2
APPENDIX C

168

Governing Equations for the Secondary Disturbance

Perturbation Method

The governing equation for the secondary (subharmonic) disturbance -- obtained at O (6e)

-- is of the form

L12(F)12 = E+R+ + E_R_ + c.c. (3.4)

where

It+ = [F,,,F_,F,_,Fp,rp]T,

R_ = [f._,f._,f.._,r,,_,fp_] T,

and the components of R+ and It_ are found to be:

-_-S'/_2t/'2 wr'_ -_- I'OtlU2Ul "_ iOL2UlU2] + U_)(t_2Pl + "JlP2)

-_-U0(P2$'C_lUl _- plS'_2U2) -J- pl_r2U2 _- p2ul_rl}

+uo(p2ialVl + plia2V2) % pla2V2 -_ p2Vlal}

r_ = -{p0[_ + _ + i_ + ;_2_2

+uO(p_ialWl + plia2w2) + P2alWl + pla2W2}

Fv = -(v2p_ + vlp_ + u2ic_lpl + ulp2(ot_

+_,liZ_ + _,_i_2 + ,_p_(_+ ,/_+
169

+p2,.l;_Ir + (.i + Pllr),_2+ P2'4+ ("_+ p2/r)Vl

+plf_2u2 + u2falpl + p2f_lul

+p2f_2ul}

Fu- = -{po[_2U'l + vl_ + l'_lUlWr -J"

-_uO(p2ialUl -- Plia2u2) + P152U2 + p2Ulal}

F__ = -{p0[_2v i + vl_ +ial_2Vl

+tLo(_21"_lt_l -- plt'_2_2) + Plff2U2 + p2Ulffl}

Fw- = -{po[_2w; + ttl_2 + I'O_lU2tOl - 1"o_2tLli_)2

r r f" • •

"_-U0(p2iO:lU)l -- pliO_2t_)2) + p2qltUl "_ plO'2_)2}

+_p_;_ - _,-" , _,_2 + u_(_ + _i+

F,_ = -{p1.'_1.:.,2/r - _2.'&_l/r+ ,_ - ple2;Z2/r

+_2.-_1/r+ (p_+ p_/r)_2+ _2'4+ (_'2+ _/r)"l

--pliO_2_2 -[- U2iOelfll -_- _25"O_1Ul -- _21"O_2Ul}

In the above equations, the primes denote differentiation with respect to r, and the

tilde represents the complex conjugate of a quantity.
APPENDIX D

170

Amplitude Equation Derived from Viscous Analysis

Here we consider the solvability condition for a system of ODE's. The approach here

follows closely from Nayfeh (1980). In analyzing the stability problem of a compressible

and viscous jets, the lowest inhomogeneous problem occurs at order 5e. This problem

governs the quadratic interaction between two linear waves: F1 and F2. The appropriate

governing equations for the case where the interaction between F1 and F_ reinforces 102

are

(§ - a2po)un - poUloVn - ia2pn = R_ (D.1)

(§ - a2po - )v12 - R--_r2 2s'& w12 - Dpn = R_, (D._)

where

2,'#2 1 )wn iflZpn = R_ (D.3) R----_r2v,2 + (§ - a2po - Rer------ f - ---;-

-Tpoia2un - 7po( 1 + D)vn 7poi_2• wn + (-iauo - a2)P,2 = R_ (D.4)

§=_-;e 1 ( D2+ 1Dr f_] D- a]) poia2u°

R_ = P0

OB aB

+ uo_'x,)u2 +P2#z ,

2 . 8B

Re ('a2 _ + _

i,82 8B

r2 o-%7) u2} - hpu_
171

R_ = po(-_t8B + uo-_xtxt)w, aB - r--f-_Re-_tv2 2 OB - -_eeL, 2 ,. a,-_x, 8B + i/92 r---_" a¢---_)w2 8B + r-_zP2 8B I -B_',_-

+ u°_=7=_) 0B 0¢, +u=nT=_)}- 0B _"-

where Fu-, Fu-, Fw- and Fp_ are given in Appendix E.

The boundary conditions at r = 0 are

u,2(0)= p,2(o)= _2(0) = wh(0)

u%2(0)= p_2(o)= _2(0) = w_2(0)

for/92 = odd (D.5)

for/9 ¢ odd (D.6)

and at r --* co, the disturbances vanish. The inhomogeneous problem has a solution

provided that a solvability condition is satisfied. To determine this condition, we multiply

D.1 by r_l, D.2 by r_2, D.3 by r_s and D.4 by r_4. After adding the resulting equations

and integrating the results by parts, we obtain

0a°{L,}ul_ + {L2}v12 + {Ls}wl_ + {L4}P12 dr + {Boundary Terrns}_ °

= r(R. I_)dr (D.7)

where

R = (R_,R_,R_,P4)

The adjoint equations are defined by setting each of the coefficients of u12, v,2, w12 and

P12 in the integrands of D.7 equals to zero. They are

(D.8)

R_ 2i/92 . -pou'o_l + (§ - a2po p)_2 + _er2ts + "lPoD_, = 0

(D.9)
172

2i/32

Rer2 _2 + (§ - a2po -

1

)_3 - "_"'7Po_4 = 0 (D.10) Rer 2 r

1

"_'_s (ia2uo + ¢2)_, 0 (D.11) --S'Ot2_l-]- (_ -_- D)_2 - r - =

To determine the boundary conditions on the _ts, we set the boundary terms in D.7 =

0, and after invoking the homogeneous boundary conditions (D.5) and (D.6), the adjoint

boundary conditions are found to be

_i(0): _4(0)= _(0) = _(0) = 0 for_z = odd (D.12)

_(0) : _(0) = _2(0)= _s(O)= 0 for_2 ¢ odd (D.13)

It may be noted that if we replace _1 by u12, _2 by -v12, _8 by w12 and _/p0_4 by p12 in

(D.8) - (D.12), the asymptotic equations where contant flows prevail are identical with

(D.1) - (D.4). Comparing the adjoint problem with the original homogeneous problem,

we find that although their boundary conditions are identical, the differential equations

are different, and hence, the homogeneous problem is not self-adjoint. The solvability

condition, which obtained from equation (D.7), is given as

fo ° rR . =dr = 0 (D.14)

The above equation can be simplified to

OB U rOB v _OB _

+ q2-b-llz,+ q oT;- + =o. (D.15)

q[ = f0°°rpo {u2 _I + v2 _9.-4-w2 _3 + p2 ,} _ ) dr

q_ = _0°° -r {ibu_ _1 + lbv_ _, A- _'w- _s + Fp- _4} dr
173

where the superscript v indicates that a quantity is obtained from the viscous analysis, and

the tilde denotes the complex conjugate. Here q_, q_ and q_ arise from the dependence

of the le2 on the slow scales, q_ contains the particular set of interactions between FI

and F2 that reproduces F2, while q_, q_ are included in the above equation to allow the

resonance to be developed spatially; their explicit expressions are not given here because

our analysis is based on temporal theory.

Again, as in equation (3.32), the same exponential dependence of B v on the slow scales

is assumed, i.e.,

B v _ exp(A_xl + A_tl + A_¢1)

The above equation, together with (D.15) and its complex conjugate, yields

(D.16)

v2 v2 v2 v2 v2 v2 ~v v v v ~v v v v -v v v v

Iqll A1 nt-lq2l 12 "4-lq3l A3 +(2qlq2)RAIA2+(2qlq3)RA2As+(2q2q3)RA1As = Iq_t 2 (D.17)

The AU's are in general complex and ( • )R denotes the real part of a complex number.

We consider a subharmonic disturbance with fixed wavenumber and it is allowed to grow

only in time, (D.17) can then be simplified to

(D.18)

The physical interpretation of A_ has been discussed in Chapter 3, and will not be repeated

here.
174

APPENDIX E

Resonant Wave Interactions Terms

The governing equation for the linearized disturbance equation for pressure --

obtained at O(Se) -- is of the form

d2p12

where

uo - C2

OB aB OB

P12 = rhs

(3.29)

(3.30)

• . x 2u_

hi2 = ,a2F.- + [r uo - C2 Po

In the above equations, all variables, except for Fu-, /_-, _'w- and /_p_, have been

defined in Chapter 3. The undefined variables are

_B tt l " _V--2

= --fpO[tt_2U_. + 1U-- 2 "_ I_lU 1 "_"

+i_2u_2_ + icqu_2m - ia2u_u_2] + u_(v_2pl + v_p-2)

+uO(p-ZialUl -- pliot2u-2) + plaZU-2 + p-2Ulal)

£- = -fpo[,_-2,d+,,1,,'2+ ;_1u-2,,1

vlw-2 /r_V_2tO1 21o_2_oi.] --iot2UlV-2 -t- ifll 7 + r ",

+uo(p-zialVl - pxi_zv-2) + pla2V-2 + p-2VlO'l)
175

' +ic{lu-2Wl - {e2ulw-2

-7- + ,_2---v-+ -T--+ _-_]

Jruo(P_21"_lU,,l -- pll'CX2t/J-2) Jr p-20"IWl nL P10"2W-2}

_p_ = -{v-2p_ + vlp__,+ u-2i_iPl - u,p-2;_2

+_pl;_l+ _p-2;_2+ _p-2(_+ _+

t_--2 I

In accordance with the discussionin Chapter 3.2,the eigenfunctionsu-2,v-2,w-2,p-2

and p-2 are foran unstablemode with a wavenumber -a2 and a mode number/3:.
176

APPENDIX F

Secondary Instability Equations Using Normal Mode

Analysis

The shape function of the subharmonic disturbance g is written as

g = (u2, u-2, vz, v-2, w2, w-2, P2,P-2, P2, P-2) T (3.47)

The stability of subharmonic disturbance superimposed on a periodic flow consisting of a

finite amplitude primary wave and a parallel mean flow is governed by the following set

of 10 coupled linear ODE's:

§u2 - poulov_ - {c_2p2 - AFu- : b2(pou2 -[- Aplu-2) (F.1)

(§ Re__)v 2 Rer2 2ifh w2 - Dp2 - A]Pv- = b2(pov2 + Aplv-2) (F.3)

2i_2

,_2+ (§ - _2po _j)w2 - --

"P2 - A_'w- = b2(pow_ + Aplw-_) (F.5)

T

--ypoia2u, - -ypo(1 + D)v2 "_poi/_2r w2 -{- a44P2 -{-a4sp2 -- A(_'p_ -[- t_con) ---- b2 (p2 v,_Apl / PO)

(F.7)

where

-p0ia2u2 - pb_2- p0(_ + D)v2 p0i_2rw2 - uoia2p2 - A_'p_ = b2p2 (F.9)

§=-Re 1( 92+ 1D l_ a_) poiot2uo 'r -_
177

After replacing the subscript 2 of the components of the shape function g by -2 (and

-2 by 2) and by substituting the coefficients (with the exception of i_2) by their complex

conjugates in equations (F.1), (F.3), (F.5), (F.7), and (F.9), equations (F.2), (F.4), (F.6),

(F.8), and (F.10) are obtained, respectively.

In equations (F.1) - (F.10), the prime denotes a differentiation with respect to r, u0

refers to the mean velocity in a moving frame of reference, a44 and a4s are given in

Appendix B, i_u_, Fv-, F_-, Fp- and Fp_ are given in Appendix E, and ¢_on, which arises

from the quadratic interactions in the conduction term of the energy equation, is found

to be

where

Coon -- - _¢i (F.11)

j=l

pla22p_2 pla2_p_2 4poc_lvl2plp-_ 2alc_2plp-22 POPlP_2 PlP_2

¢' = p_ p'o p_ + p_ + p------_-o +

POPlP_2 4-YlPoP-2 '' PlP-2 2'__' PlP-2 POPl_P-2 4POPlP-2 ''

pl_22p-2 4pOPlP-2_2_1 2_1_2P-ZP1 pOa_PlP-Z

¢3= (,_p_o) (,2p_) + (,2p0_) po_

pOp'_p-2 _ poPllp-2 Po_PlP-2 _PlP_P-2 , PlPIoP-2 .2PIlPIoP--2 2PtlP--2 , 2_p1P-2

¢' = __ (_-T;_p0) (,'po_) _ p--VFo-_(_p-T;;_0) T (p0_) (%_-0)T ('2po_)

2cI_plp-2 21]_P-2 P-2c_21Pl 2p-2Pl_l_2 P-2P7 4p-2PrOPll

¢5= p_o _ p_ + p_ + _ p_o +

P-2P_ 2f12fllp-2Pl P-2P_Pl P-2PIoPl 2p_PIoP.___________I

¢6= (,2p_) + (,.2p_o)+ p------Fo+ (,.p_) + p_o

2a_p-2pl 2p_2Pl

¢' : p_, -- p--T-o

The appropriate boundary conditions at • --, 0 are

P-2Pfl _ 2Pn-2P_

2p-2pl 2_22P-2Pl

(,p_) + (,2p_)

u2(0) = p2(0) = p2(0) = v_(0) = w_(0) for/92 = odd (F.12)

u_(0) = p_(0) -- p_(0) -- v2(0) -- w2(0) for _ ¢ odd (F.13)
u_2(o) = p_2(o) = p-2(o) = ¢2(0) = ,1,'2(0)

u'2(o) = p'_2(o)= p'_2(o)= ,__2(0)= w_2(o)

and at r --* co, the subharmonic disturbances vanish.

178

for & = odd (F.14)

for _ _t odd (F.15)

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