lectures on dynamics of combustion waves in premixed gases · 2021. 6. 21. · inner structure of a...

Princeton-CI Summer SchoolJune 21-25, 2021

Lectures onDynamics of Combustion Waves

in Premixed Gases

Professor Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)

without permission of the owner, Paul Clavin

Orders of magnitude

Lecture 1:

1-1: Overall combustion chemistry

1-2: Combustion waves in gaseous mixtures

1-3: Arrhenius law

1-4: Hydrocarbon/air flames

1-5: Instabilities of flames

P.Clavin II

Lecture 2: Governing equations

2-1. Conserved extensive quantities2-2. Continuity

2-3. Fick’s law. Di�usion equation2-4. Conservation of momentum2-5. Conservation of total energy

Thermal equation

Inviscid flows in reactive gases

Conservative forms

One-dimensional inviscid and compressible flow

2.6. Entropy production

Lecture 3: Thermal propagation

3-1. Quasi-isobaric approximation (Low Mach number)

3-2. One-step irreversible reaction3-3. Unity Lewis number and large activation energy

3-4. Zeldovich & Frank-Kamenetskii asymptotic analysisPreheated zoneInner reaction layer

Matched asymptotic solution

3-5. Reaction di�usion wavesPhase space

Selected solution in an unstable medium

P.Clavin III

P.Clavin IV

Lecture 4 : Hydrodynamic instability of flames

4-2. Linearized Euler equations of an incompressible fluid

4-1. Jump across an hydrodynamic discontinuity

4-3. Conditions at the front

4-4. Dynamics of passive interfaces

4-5. Darrieus-Landau instability

4-6. Curvature e�ect: a simplified approach

P.Clavin V

Lecture 5: Thermo-diffusive phenomena

5-1. Flame stretch and Markstein numbers

5-2. Thermo-di�usive instabilitiesPlanar flames for Le �= 1

Jump conditions across the reaction layer

Linear equations and linear analysis

Cellular instability (Le < 1)

Oscillatory instability (Le > 1)

Passive interfaces

One-step flame model

The second Markstein number

P.Clavin VI

Lecture 6: Thermal quenching and flammability limits

6-1. Extinction through thermal loss

6-2. Basic concepts in chemical kinetics

6-3. Flame speed near flammability limits

Combustion of hydrogen

Two-step model. Crossover temperature

One-step model with temperature cuto�

P.Clavin VIILecture 7: Flame kernels and quasi-isobaric ignition

7-1. Introduction7-2. Zeldovich critical radius7-3. Critical radius near the flammability limits

7-4. Dynamics of slowly expanding flames

7-5. Quasi-steady dynamic of thin flamesSemi-phenomenological modelOpened-tip Bunsen flames

P.Clavin VIIILecture 8: Thermo-acoustic instabilities

Lecture 8-1. Rayleigh criterion

Lecture 8-2. Admittance & transfer function

Lecture 8-3. Vibratory instability of flames

Acoustic waves in a reactive mediumSound emission by a localized heat sourceLinear growth rate

Flame propagating in a tubePressure couplingVelocity and acceleration coupling

Acoustic re-stabilisation and parametric instability (Mathieu’s equation)Flame propagating downward (sensitivity to the Markestein number)Bunsen flame in an acoustic field

P.Clavin IX

Turbulent flamesLecture 9 :

9-1. Introduction

Einstein-Taylor’s di�usion coe�cient9-2. Turbulent di�usion

Rough model of turbulent transportWell-stirred flame regime

Kolmogorov’s cascade

Gibson’s scaleElements of fractal geometrySelf similarity of strongly corrugated flames

Co-variant laws

Monopolar sound emission

Sound generated by a turbulent flameBlow torch noise

9-3. Strongly corrugated flammelets regime

9-4. Turbulent combustion noise

P.Clavin X

Lecture 10 : Supersonic waves

10-1. BackgroundModel of hyperbolic equations for the formation of discontinuity

Riemann invariants

Rankine-Hugoniot conditions for shock waves

Mikhelson (Chapman-Jouguet) conditions for detonations

10-2. Inner structure of a weak shock waveFormulation

Dimensional analysisAnalysis

10-3. ZND structure of detonations

10-4. Selection mechanism of the CJ wave

Initiation of detonationLecture 11:

11-1. Direct initiation

11-2. Spontaneous initiation and quenching

11-3. Deflagration-to-detonation transition

P.Clavin XI

Flow of burnt gas in spherical CJ detonationsPoint blast explosionsZeldovich criterionCritical energy

Initiation at high temperatureSpontaneous quenching

Basic ingredientsExperimentsRunaway phenomenonMechanisms of DDT

P.Clavin XII

Lecture 12 : Galloping detonations

12-1. Physical mechanismsInstability mechanismTwo limiting cases

Constitutive equations12-2. General formulation

12-3. Strongly overdriven regimes in the limit (� � 1) � 1

12-4. CJ detonations for small heat release

Strong shock in the Newtonian approximation

Distinguished limitIntegral-di�erential equation for the dynamicsOscillatory instability

Reactive Euler equations in 1-D geometryNear CJ regimes for small heat release. Transonic reacting flowsSlow time scaleAsymptotic model for CJ or near CJ regimesResults for simplified chemical kinetics

P.Clavin XIII

Lecture 13 :

13-1. Acoustic waves and entropy-vorticity wave

Stability analysis of shock waves

13-2. Analyses

Linearized Euler equationsLinearized flow field

Dispersion relation for general materials

Classification of normal modes

Spontaneous emission of sound and instability

Stability of reacting shocks

Stability of shocks in ideal gases

P.Clavin XIV

Nonlinear dynamics of shock wavesMach stem formation

Lecture 15 : Cellular detonations

15-1. Cellular detonations at strong overdrive

15-2. Cellular instability near the CJ condition

Order of magnitude. ScalingFormulationOuter flow in the burnt gasInner structureMatchingLinear growth rateWeakly nonlinear analysis

FormulationScalingModel for CJ or near CJ regimesMultidimensional stability analysis

P.Clavin XV

Princeton Summer School on CombustionJune 20 - June 26, 2021

Theoretical analyses (analytical studies of simplified models) + laboratory experiments

Structure and Dynamics of

Combustion Waves in Premixed Gases

Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)

Aix-en-ProvenceMarseille

P.Clavin I

Contents

Combustion Waves and Fronts in FlowsP. Clavin and G. Searby

Cambridge University Press 2016

Lecture 1: Orders of magnitude

Lecture 3: Thermal propagation of flames

Lecture 4: Hydrodynamic instability of flames of flames

Lecture 5: Thermo di�usive phenomena

Lecture 7: Flame kernels and quasi-isobaric ignition

Lecture 8: Thermo-acoustic instabilities. Vibratory flames

Lecture 10: Supersonic wavesLecture 9: Turbulent flames

Lecture 11: Initiation of detonationsLecture 12: Galloping detonations

Lecture 13: Stability analysis of shock waves

Lecture 14: Nonlinear dynamics of shock waves. Mach stem formation

Lecture 15: Cellular detonations

Combustion is a fascinating phenomenon coupling complex chemistry to transport mechanisms and nonlinear !uid dynamics. "is book provides an up-to-date and comprehensive presentation of the nonlinear dynamics of combustion waves and other non-equilibrium energetic systems. "e major advances in this #eld have resulted from analytical studies of simpli#ed models performed in close relation with carefully controlled laboratory experiments. "e key to understanding the complex phenomena is a systematic reduction of the complexity of the basic equations.

Focusing on this fundamental approach, the book is split into three parts. Part I provides physical insights for physics-oriented readers, part II presents detailed technical analysis using perturbation methods for theoreticians, and part III recalls the necessary background knowledge in physics, chemistry and !uid dynamics. "is structure makes the content accessible to newcomers to the physics of unstable fronts in !ows, whilst also o$ering advanced material for scientists who wish to improve their knowledge.

Paul Clavin is Professor Emeritus at Aix-Marseille Université and is an honorary member of the Institut Universitaire de France (Chair of Mécanique Physique 1993-2004). In 1995 he founded a renowned research institute, the Institut de Recherche sur les Phénomènes Hors Équilibre (IRPHE) and has received major awards from the Société Française de Physique (Plumey 1988), French Academy of Sciences (Grand Prix 1995) and from the Combustion Institute (Zeldovich Gold medal, San Francisco, August 2014).

Geo! Searby is retired Director of Research at the Institut de Recherche sur les Phénomènes Hors Équilibre (IRPHE). He is a renowned specialist of the physics of thermo-acoustic instabilities in combustion chambers and rocket motors, and his experiments have made major contributions to the understanding of the dynamics of !ame fronts. In 2004 he obtained a major award from the French Academy of Sciences.

Clavin and Searby. 9781107098688. PPC. C M Y K

Cover illustration: TBC

Clavin and SearbyCom

bustion Waves and Fronts in Flow

Combustion Waves and Fronts in FlowsFlames, Shocks, Detonations, Ablation Fronts and Explosion of Stars

Paul Clavin and Geo! Searby

Orders of magnitude

Lecture 1:

1-3: Arrhenius law

reactants � products + heat release Overall combustion chemistry

T < 500K : �r(T ) � � (frozen mixture of reactants)reaction time �r(T ) extremely sensitive to temperature:

T � 2500K : �r(T ) � 10�6s.

Lavoisier 1777I � 1)

thermal feedback � combustion waves

Ignition Reaction

Hot burned gas (equilibrium)

Propagation velocity U

Davy 1830Cold (frozen) Reactive mixture

Euler 1738

P.Clavin I

binding energy of small molecules � a few eV� Tb � Tu � 2000 K1eV/molecule � 23 kcal/mole

(Cp � 10 cal/mole/K)

Orders of magnitude

Lecture 1:

1-3: Arrhenius law

John H.S. Lee 1990

Fast deflagrations : � 100 m/s, �p/p � �10�1

Turbulent propagation

Berthelot, Vielle 1884Detonations : � 2000 m/s, �p/p � +30

Cellular structureShchelkin 1960

Flames : 10 cm/s � 10 m/s, �p/p � �10�5

Bec BunsenJ. Quinard 2000

Laser TomographyL.Boyer 1980

Mallard, Le Chatelier 1883

Laminar propagation

acethylen/oxygen

Combustion waves in premixed gaseous mixturesI � 2)

P.Clavin I

Maxwell 1867 Einstein 1905

Back to the kinetic theory of gases

Bolzmann 1877

P.Clavin I

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(2⇡kBT )3/2e�m|v|2/(2kBT )d3v d3r

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m : mass of molecules, n : number density, T : temperature, kB : Boltzmann cst.<latexit sha1_base64="4CNzxmQYA4vN6ldmjXG1EJX9LRo=">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</latexit><latexit sha1_base64="Y/NmmmulfNcDGYwn0BgD4NIy6p4=">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</latexit><latexit sha1_base64="Y/NmmmulfNcDGYwn0BgD4NIy6p4=">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</latexit><latexit sha1_base64="bHBZkrAdrqmlSAdNI2OLnOewdKI=">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</latexit>

velocity of molecules<latexit sha1_base64="SqOUvJMgj2jJZPvdPA86XixwcXU=">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</latexit><latexit sha1_base64="HNY/IOIzl+eTqeniOZj02duagmc=">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</latexit><latexit sha1_base64="HNY/IOIzl+eTqeniOZj02duagmc=">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</latexit><latexit sha1_base64="jBoWPGVcYe4rIcrUolSHveOgmT8=">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</latexit>

element of volume<latexit sha1_base64="74oitxFpKgY8Gplcb47vR2VFPYM=">AAACyXicjVHLSsNAFD3GV31X3QhugkVwVRI36q7gRnCjYG2hLSVJpzp0konJRKylbvwVt/oVbv0A/0CX/oF3xhR8IDqBzJlzzz3chx8LnirHeR6zxicmp6YLM7Nz8wuLS8XlldNUZknAqoEUMqn7XsoEj1hVcSVYPU6YF/qC1fzevo7XLlmSchmdqH7MWqF3FvEuDzxFVLu41lTsSg2YYCGLlC279qUUWciG7WLJKTvm2D+Bm4NSZfXtsQLgSBaf0EQHEgEyhGCIoAgLeEjpa8CFg5i4FgbEJYS4iTMMMUu5GakYKTxie/Q/o9eAsCSNJP0QNjYpLkmZENbOtolnxkWzv/t4VJOuo0+3n3uFxCqcE/tX3kj537wGsQpd7JpuOc0iNoyeSpC7ZGYCunL7U1eKHGLiNO5QPCEcmMzRTG2Tk5re9Rw9E38xSs3qd5BrM7zqKmmZ7vfV/QSn22XXKbvHtNU9fJwC1rGBLdrdDio4wBGq5H2DO9zjwTq0Lqwr6/pDao3lOav4cqzbd6Sfl4A=</latexit><latexit sha1_base64="McQ2A+cTDbb3LIWjNFgOSpu8s14=">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</latexit><latexit sha1_base64="McQ2A+cTDbb3LIWjNFgOSpu8s14=">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</latexit><latexit sha1_base64="4bTpiZLc9VE5Vq0ppiXwdwZcqHU=">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</latexit>

< distance >= l

< velocity >= V ��

3kBT/m

< time >= �coll � l/V

mean free path

Binary collisions of molecules<latexit sha1_base64="KEBQaZPchVqXA1+AXLYpY4N4HPI=">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</latexit><latexit sha1_base64="5Q4b2SCi8ePmDFCJgpMzxjhzoDI=">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</latexit><latexit sha1_base64="5Q4b2SCi8ePmDFCJgpMzxjhzoDI=">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</latexit><latexit sha1_base64="l5bN/kOGRo0iBnNc/wWRdLTQ5Z8=">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</latexit>

Molecular di�usion � Random WalkD = lV = l2/�coll � a2�collspreading :

1(4�Dt)3/2

e�r2/4Dt

Di↵usion coe�cient<latexit sha1_base64="8mwfXZEFsmuX8RSkhti3+tULyYE=">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</latexit><latexit sha1_base64="T8oVOEF63RmrwfubhaAx3C9tr5E=">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</latexit><latexit sha1_base64="T8oVOEF63RmrwfubhaAx3C9tr5E=">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</latexit><latexit sha1_base64="mtSXLha6DAlMjx/40sEu7o2qVsM=">AAACzXicjVHLTsJAFD3UF+ILdamLRmLiirRu1B2JLtyJiYAJENKWKU4onaadGgniwo2/4lb/ww/wD/QvvDOWRCVGp0l7es65J3fudaOAJ9KyXnPGzOzc/EJ+sbC0vLK6VlzfqCcijT1W80Qg4kvXSVjAQ1aTXAbsMoqZM3AD1nD7x0pvXLM44SK8kMOItQdOL+Q+9xxJVKe43ZLsRo5OuO+nymR6gvkkcxbKcadYssqWPuY0sDNQQnaqoviCFroQ8JBiAIYQknAABwk9TdiwEBHXxoi4mBDXOsMYBapNycXI4RDbp3eP/kaEBXkE+ccwsUu6IGdMWCWbWk91imJ/z3GoJ9XHkL5uljUgVuKK2L/qJs7/1jWJlfBxqG/LaRaRZtRUvCwl1RNQnZtfbiUpISJO4S7pMWFPV05mauqaRN9dzdHR+pt2Klb9e5k3xbvqkpZp/1zdNKjvl22rbJ9bpcpRttY8trCDPdrdASo4RRU1yr7HI57wbJwZqXFr3H1ajVxWs4lvx3j4AGZ1lw8=</latexit>

speed of sound: a ⇡ V<latexit sha1_base64="IjawFMorvzgSfFN0uzUdSL3uBzY=">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</latexit><latexit sha1_base64="2ogfQAGe8jQiwNl2uWfMTFAfXvY=">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</latexit><latexit sha1_base64="2ogfQAGe8jQiwNl2uWfMTFAfXvY=">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</latexit><latexit sha1_base64="SObFEoMDbXV1MM/ImO4ihqjCeAE=">AAAC1HicjVHLbtNAFD11H4TSlhSW3YyIkFhFNhseq0rddBkk8pCaqhrbk9Sq4xnZY9QqzQqxYMOvsKUfwQf0D8pf9MzgSi0RgrHkOXPuuUf3EZs8q2wYXq8Eq2vrG49ajzefbG3vPG3vPhtUui4T1U90rstRLCuVZ4Xq28zmamRKJWdxrobx2YGLDz+pssp08dFeGHU8k9Mim2SJtKRO2h0xturcziujVCr0RNC5SN+LhZBjaUypz8WAqrAb+iOWQdSADprT0+2fGCOFRoIaMygUsMQ5JCp+R4gQwpA7xpxcSZT5uMICm8ytqVJUSLJn/E/5mhNrajT1Cwi8ZFxTWRI7Z+HjtXdx7N99JGtydVzwjhuvGVmLU7L/yrtT/m/eEVmLCd76bjPOwnjGTSVpXGo/AVe5uNeVpYMh53DKeEmc+My7mQqfU/ne3Rylj994pWPdO2m0NX65KrnM6M/VLYPB624UdqMPYWf/XbPWFvbwAq+4uzfYxyF66NP7K77jB66CQXAZfA6+/JYGK03Oczw4wbdbAlOYxQ==</latexit>

⌧coll : collision time ⇡ relaxation time<latexit sha1_base64="dXE8mrX2FUNmasi2m8DqKvyvCb0=">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</latexit><latexit sha1_base64="9jAvEIAwnl2ayf3sbSnB/0KfXTg=">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</latexit><latexit sha1_base64="9jAvEIAwnl2ayf3sbSnB/0KfXTg=">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</latexit><latexit sha1_base64="8uKs+WaK9KjM25TTxHRv827Uy6c=">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</latexit>

sound speed, ab/au =�

molecular and thermal di�usion coe⇥cients D � DT

chemical energy/unit mass qm � Tb/Tu = 5 � 10

reaction rate 1/�r(Tb)

detonation: D � ⇥qm � ab

� 1000 m/s

laminar flames: UL ��

D/�r(Tb)� 1 m/s

propagation velocity supersonic

subsonic

D/au > 1

UL/au < 1

Dimensional parameters

P.Clavin I

Dimensional analysis

[qm] = (velocity)2

[D] = (velocity)2 � time

D � 10�3m2/s

Orders of magnitude

Lecture 1:

1-3: Arrhenius law

Energy

Progress variableBin

Rectants

Products

Rectants

Inelasticcollision

kBTb� 8

Collision in gases

Overall reaction rate. Arrhenius factorI � 3)

P.Clavin I

sound speed

UL/a ��

e�E/kBTb � 1subsonic

laminar flame velocity

UL ��

DT /�r

elastic collision rate 1/�coll � 109s�1

� 1/�r(Tb) � 3 � 105s�1

Tb/Tu = 8 � �r(Tu) � 1010 years !!

e�E/kBTb � 3 � 10�4

Tb/Tu = 8 � e�E/kBTu � 1.6 � 10�28

mean free pathflame thickness

dL � l�

eE/kBTb � lmacroscopic structure

UL Ub - UL

Unburnt mixture at rest

Burnt gas

Temperature

dL � DT /UL

Kinetic theory of gases � Flame properties

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sound speed<latexit sha1_base64="sueleeOpUR8v6rmTB6PVSAzMmlA=">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</latexit><latexit sha1_base64="JufZ9Uz5cgaDK+kDYqV1G4bhhy8=">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</latexit><latexit sha1_base64="JufZ9Uz5cgaDK+kDYqV1G4bhhy8=">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</latexit><latexit sha1_base64="eVSK6hcKQBdqnQWBvoHFqyhUCjc=">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</latexit>

men free path<latexit sha1_base64="IZ+7Rg1cw3jTmLjRgUUWa0gOsYU=">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</latexit><latexit sha1_base64="dtUq8qvKT3/xfJtVMwtuMJEbcuk=">AAACxXicjVHLSsNAFD1NfdT6qo+dm2ARXJVEF7oUXOiygn2AikziVAeTTJhMRC3FX3Gr4D/4A36Af6B/4Z0xBbWITiBz5txzD/cRpJHItOe9lpzy2PjEZGWqOj0zOzdfW1hsZzJXIW+FMpKqG7CMRyLhLS10xLup4iwOIt4JLndNvHPFVSZkcqhvUn4Ss/NE9ETINFGnteVjza91P+aJ21OcuynTF4PTWt1rePa4o8AvQH1nrvn8VH59aMraC45xBokQOWJwJNCEIzBk9B3Bh4eUuBP0iVOEhI1zDFCl3JxUnBSM2Ev6n9OrT1iSRpJ+ABdrFJekVISNs2vjuXUx7O8+jGoyddzQHRReMbEaF8T+lTdU/jfviFiNHrZtt4JmkVrGTCUsXHI7AVO5+6UrTQ4pcQafUVwRDm3mcKauzcls72aOzMbfrNKw5h0W2hzvpkpapv9zdaOgvdHwvYZ/QFvdxOepYAWrWKfdbWEH+2iiRd63uMcDHp09J3a0c/UpdUpFzhK+HefuA4NYltI=</latexit><latexit sha1_base64="dtUq8qvKT3/xfJtVMwtuMJEbcuk=">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</latexit><latexit sha1_base64="AY3MsnnHuhK7dEHeu7fWl8lRXPA=">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</latexit>

MB distribution � e�12

mv2kBT �

Arrhenius factor

1�r(T )

�colle�E/kBT

Kinetic theory of gases ) Arrhenius law<latexit sha1_base64="sZ1fGfUexj8OV9VjTB09HghJCqY=">AAAC8HicjVHNShxBGCxHkxjzt9FjLo1LIKdlJgSSQyAGL4FcVFwVXJGesXe3cXZ66OmJWZZ9D28hh1zyKuJND75B8hZWtyNoJCQ9MFNTX1Xx9felZa4rF8cXM9Hs3L37D+YfLjx6/OTps9bzxa3K1DZT3czkxu6kslK5LlTXaZerndIqOUpztZ0ervr69hdlK22KTTcu1d5IDgrd15l0pPZbH3pOfXWTz7Q7nQk3VMaOhemLAU MrMe1t6MHQSWvNkbiSfrR2qApdVyKXR9P9VjvuxOGIuyBpQBvNWTOtU/RwAIMMNUZQKOCIc0hUfHaRIEZJbg8TcpZIh7rCFAv01lQpKiTZQ74H/JsQG2oM9VMIvGTdUGmJfbII9TqkePbvOZI9+T7G/KZN1oisw5Dsv3zXyv/17ZJ16ONduK3mLMrA+KlkTUodJuA7Fzdu5ZhQkvP4gHVLnAXn9UxF8FTh7n6OMtR/BaVn/X/WaGv89l1ymcmfq7sLtl53kriTrL9pr7xv1jqPF1jGK+7uLVbwCWvoMvsHTnCG88hGx9G36PuVNJppPEu4daKfl6X0pcs=</latexit><latexit sha1_base64="sZ1fGfUexj8OV9VjTB09HghJCqY=">AAAC8HicjVHNShxBGCxHkxjzt9FjLo1LIKdlJgSSQyAGL4FcVFwVXJGesXe3cXZ66OmJWZZ9D28hh1zyKuJND75B8hZWtyNoJCQ9MFNTX1Xx9felZa4rF8cXM9Hs3L37D+YfLjx6/OTps9bzxa3K1DZT3czkxu6kslK5LlTXaZerndIqOUpztZ0ervr69hdlK22KTTcu1d5IDgrd15l0pPZbH3pOfXWTz7Q7nQk3VMaOhemLAU MrMe1t6MHQSWvNkbiSfrR2qApdVyKXR9P9VjvuxOGIuyBpQBvNWTOtU/RwAIMMNUZQKOCIc0hUfHaRIEZJbg8TcpZIh7rCFAv01lQpKiTZQ74H/JsQG2oM9VMIvGTdUGmJfbII9TqkePbvOZI9+T7G/KZN1oisw5Dsv3zXyv/17ZJ16ONduK3mLMrA+KlkTUodJuA7Fzdu5ZhQkvP4gHVLnAXn9UxF8FTh7n6OMtR/BaVn/X/WaGv89l1ymcmfq7sLtl53kriTrL9pr7xv1jqPF1jGK+7uLVbwCWvoMvsHTnCG88hGx9G36PuVNJppPEu4daKfl6X0pcs=</latexit><latexit sha1_base64="sZ1fGfUexj8OV9VjTB09HghJCqY=">AAAC8HicjVHNShxBGCxHkxjzt9FjLo1LIKdlJgSSQyAGL4FcVFwVXJGesXe3cXZ66OmJWZZ9D28hh1zyKuJND75B8hZWtyNoJCQ9MFNTX1Xx9felZa4rF8cXM9Hs3L37D+YfLjx6/OTps9bzxa3K1DZT3czkxu6kslK5LlTXaZerndIqOUpztZ0ervr69hdlK22KTTcu1d5IDgrd15l0pPZbH3pOfXWTz7Q7nQk3VMaOhemLAU MrMe1t6MHQSWvNkbiSfrR2qApdVyKXR9P9VjvuxOGIuyBpQBvNWTOtU/RwAIMMNUZQKOCIc0hUfHaRIEZJbg8TcpZIh7rCFAv01lQpKiTZQ74H/JsQG2oM9VMIvGTdUGmJfbII9TqkePbvOZI9+T7G/KZN1oisw5Dsv3zXyv/17ZJ16ONduK3mLMrA+KlkTUodJuA7Fzdu5ZhQkvP4gHVLnAXn9UxF8FTh7n6OMtR/BaVn/X/WaGv89l1ymcmfq7sLtl53kriTrL9pr7xv1jqPF1jGK+7uLVbwCWvoMvsHTnCG88hGx9G36PuVNJppPEu4daKfl6X0pcs=</latexit><latexit sha1_base64="sZ1fGfUexj8OV9VjTB09HghJCqY=">AAAC8HicjVHNShxBGCxHkxjzt9FjLo1LIKdlJgSSQyAGL4FcVFwVXJGesXe3cXZ66OmJWZZ9D28hh1zyKuJND75B8hZWtyNoJCQ9MFNTX1Xx9felZa4rF8cXM9Hs3L37D+YfLjx6/OTps9bzxa3K1DZT3czkxu6kslK5LlTXaZerndIqOUpztZ0ervr69hdlK22KTTcu1d5IDgrd15l0pPZbH3pOfXWTz7Q7nQk3VMaOhemLAU MrMe1t6MHQSWvNkbiSfrR2qApdVyKXR9P9VjvuxOGIuyBpQBvNWTOtU/RwAIMMNUZQKOCIc0hUfHaRIEZJbg8TcpZIh7rCFAv01lQpKiTZQ74H/JsQG2oM9VMIvGTdUGmJfbII9TqkePbvOZI9+T7G/KZN1oisw5Dsv3zXyv/17ZJ16ONduK3mLMrA+KlkTUodJuA7Fzdu5ZhQkvP4gHVLnAXn9UxF8FTh7n6OMtR/BaVn/X/WaGv89l1ymcmfq7sLtl53kriTrL9pr7xv1jqPF1jGK+7uLVbwCWvoMvsHTnCG88hGx9G36PuVNJppPEu4daKfl6X0pcs=</latexit>

Maxwell 1867 Einstein 1905Bolzmann 1877

too smalltoo large

hydrocarbon/air10 � 50 cm/s

1 � 10�1 mm

UL � 8.6 m/se�E/kBTb � 3 � 10�4

a � 500m/sl � 10�7 m dL � 0.6 � 10�5 m} �

Limitations of the dimensional analysis

Molecular di�usion � Random Walk

< distance >= l

< velocity >= V ��

3kBT/m

< time >= �coll � l/V

a � V

mean free path

spreading :

D = lV = l2/�coll � a2�coll

1(4�Dt)3/2

e�r2/4Dt

P.Clavin I

UL Ub - UL

Burnt gas

Temperature

UL/a ��

e�E/kBTb � 1

dL/l � UL�r/l ��

D�r/l ��

eE/kBTb � 1

Flame structureUL �

�D/�r(Tb)

1�r(Tb)

�colle�E/kBTb

Orders of magnitude

Lecture 1:

1-3: Arrhenius law

Methane-air flame

Hydrocarbon/air flamesI � 4)

� =NF /NO2

⇥+F /⇥+

�+F F + �+

o O2 � P

Equivalence ratio

� = 1 : stoichiometry� > 1 : fuel rich� < 1 : fuel lean

Semenov 1934

Chain reactions in combustion

dL � UL�r(Tb)

��

DT �r(Tb)� DT

��r(Tb)/DT

� DT /UL

� = 0.65near to the flammability limit

”thicker flame”

15P.Clavin I

Orders of magnitude

Lecture 1:

1-3: Arrhenius law

Instabilities of flamesI � 5)

Intrinsic instabilitiesStable Linearly stableLinearly unstable

Nonlinearly unstablePlanar flames are linearly unstable:

�u > �b- hydrodynamic instability of the flame front

induced flow Cusp formationHuygens construction

Propane lean flame

Propane rich flame

- di�usional-thermal instability of the inner flame stucture DT < D

Unstable inner structure17

P.Clavin I

System instability (combustion in a cavity)

The coupling of flames with acoustics can be unstable

Lord Rayleigh 1878

Thermo-acoustic instabilities (Rayleigh criterion)

Rocket engineCombustion chambers

Gas turbines

P.Clavin I

Vibratory instability of flames in tubes

G. Searby IRPHE 2006

Lean methane-air flame

� = 0.73

� = 0.8

UL = 23 cm/s

UL = 30 cm/s

P.Clavin I

Tomography cut: L. Boyer 1980

Effect of geometrical parameters on thermo-acoustic instability of downward propagating flames in tubes

Ajit Kumar Dubey, Yoichiro Koyama, Nozomu Hashimoto, Osamu FujitaDivision of Mechanical and Space Engineering, Hokkaido University, Kita 13 Nishi 8 Kita-ku, Sapporo, Hokkaido, 930-8555, Japan

Received 1 December 2017; accepted 19 June 2018 Available online

Symposium on combustion (2018)

E�ect of acceleration

equipotential surfacein the absence of flame

in the presence ofan axial acoustic field

slightly faster

Methane rich Bunsen flame� = 1.5

E�ect of an acoustic field on a Bunsen flame

P.Clavin I

Gravity

Propane flame propagating upwards Slow downwards propagating flame

Lecture 1IGoverning equations

P.Clavin I

Contents

in local equilibriumGaseous mixtures in normal conditions � continuum medium

mean free path � macroscopic length

relaxation time towards equilibriumof fluid particles macroscopic time scale�

many microscopic particles � a fluid ”particle”

!internal structure of shock waves

P.Clavin II

Thermal equation

Conservative forms

(simplified form, see de Groot et Mazur (1962) or Williams (1985) for more details)

Conserved extensive quantitiesII � 1)

extensive quantities mass weighted distribution a(r, t)mass density �(r, t)

AV =��

�ad3r

⇤(�a)/⇤t = ��.Ja + ⇥a

⇥(�a)/⇥t = ��.Jaconserved scalar

conserved quantities(mass, momentum and energy)

: no production terms �a = 0

a(r, t)

dA/dt =��

[⇥(�a)/⇥t] d3r = (dA/dt)1 + (dA/dt)2

(dA/dt)1 = ��

n.Jad2�

conservation equation V fixed

(dA/dt)1 = ��

�.Ja d3r

Vvector field Ja(r, t)

⇥(�a)/⇥t = ��.Jaa(r, t)conserved vectortensor field Ja(r, t) 5

P.Clavin II

(dA/dt)2 = ��

�a(r, t) d3r

P.Clavin II

Thermal equation

Conservative forms

⇥�/⇥t = ��.(�u)J � �u

Conservation of mass: continuity equationII � 2)

v � 1/�1v

= �.u

P.Clavin II

�Da/Dt = ��.J�a

conserved scalar:

⇥(�a)/⇥t = ��.Ja

�Da/Dt = ��.J�a + ⇥a

Ja � �au + J�a

di�usion flux

convection fluxLagrangian form of conservation equations

⇤(�a)/⇤t = ��.Ja + ⇥a

� �

�t+ u.�material (convective) derivative

mass is a conserved scalar

��/�t = ��.J

Dt= ��.u

continuity equation

(classical mechanics)

!(definition of the di�usion flux in the equation for energy is slightly di�erent)

see slide 11

Fick’s law. Diffusion equationII � 3)

mass fraction Yi = �i/��

Yi = 1

inert mixture mass fraction of species is a conserved scalar�

J�i = 0�DYi/Dt = ��.J�

Kinetic theory of gas (binary di�usion in an abundant species)

Fick’s law : J�i = ��Di�Yi

�DYi/Dt = �.[�Di�Yi]Di > 0

archetype of irreversible phenomenon

�Di � cst.

di�usion equation

�Yi/�t = Di�Yiu = 0

(random walk)

P.Clavin II

�Y/�t = D�Y D > 0 [D] = (length)2/time

Di�usive damping. Dissipative phenomenon

�n/�t = D�n

n(r, t) = NG(r, t) n(r, t) =��

n(r�, 0)G(r � r�, t)d3r�

number density

G(r, t) =1

(4�Dt)3/2e�r2/4Dt

��G(r, t)d3r = 1

�G/�t = D�G

t = 0 : G(r, 0) = �(r) r = |r|

Green function

probability distribution of the test particle

Self-similar solution

Y (r, t) =�

Yk(t)eik.r

dYk(t)/dt = �(Dk2)Yk(t) Yk(t) = Yk(0)e�Dk2t

k = |k|

Fourier anaysis

Fourier 1824

P.Clavin II

Thermal equation

Conservative forms

II � 4) Conservation of momentum

Euler equations�Du/Dt = ��p

non dissipative equations

⇤Du/Dt = ��(p + ⇤gz) + ��u + (⇥ + �/3)�(�.u)

Navier Stokes equations gravity

thermodynamic pressure (isotropic)

P.Clavin II

⇤ � �2�(�u)(s) � I(⇥ � 2�/3)�.uViscous stress tensor

��gez,

gravity (body force)� = pI + �surface force (stress tensor)

Dvis = �/⇥Viscous shear di�usivity

�Du/Dt = ��.�

Momentum is a conserved vector (isolated system)

P.Clavin II

Thermal equation

Conservative forms

II � 5)

eint = eT + echeminternal energy,

thermal energy, chemical energy (chemical bonds),

etot = |u|2/2 + eT + echem + ...total energy

⇥(�etot)/⇥t = ��.Jetot � Detot/Dt = ��.J�etot

J�etot

� Jetot � �etotu

total energy is a conserved scalar

thermal di�usivity�eT = cV �T DT � �/⇥cV [DT ] = (length)2/time13

P.Clavin II

inert material echem = cst.no flow u = 0

�T/�t = DT �T

Fourier equation

(kinetic + rotational & vibrational energy)

(Additive in a gas when interactions are neglected)

J�q = ��T,

Fourier law

thermal conductivity

(simplest form of heat flux)

Conservation of total energy

|u|2 = �u.⇥p = �⇥.(pu) + p⇥.uEuler equation �

(first law of thermodynamics)Inviscid and inert flows

p�.u = �p(D��1/Dt)� D(eT + echem)/Dt = ��.J�

q � p�.uwork doneheat

J�q � J�

etot� pu

heat flux, ?

�+1 A1 + . . . + �+

n An � ��1 A1 + . . . + ��

n An + Q.

elementary reaction

W (j) � (J (j)+ � J (j)

� ) and �(j)i � (��(j)

i � �+(j)i ), stoichiometric coe�cientreaction rate

nb/(volume � time)

�DYi/Dt = ��.J�i +

�(j)i miW

conservation equation for the species

echem ��

equation for the chemical energy

enthalpy of formation per unit of mass of species i

Q(j) =n�

(�(j)+i � �(j)�

i )mihi(To),

sum over the species

heat of the j th reaction

�Dechem/Dt = ��

�.(hiJ�i) �

Q(j)W (j)

Reactive flows

�DYi

Dt= �.(�Di�Yi) +

�(j)i miW

(j)(T, p, ..Yk..)

sum over the reactions

J0i = �⇢DirYi

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⇥cvDT/Dt = �.(��T ) � p�.u +�

Q(j)W (j)

p = (cp � cv)�T.

�p�.u =p

� =DDt

p � �DDt

[(cp � cv)T ]

ideal gas law

continuity �

P.Clavin II

�eT = cv�T cv � cst. (for simplicity, can be easily removed)

⇥cpDT/Dt = Dp/Dt + �.(��T ) +�

Q(j)W (j)

compression chemistryconduction

thermal equation of an invicid fluid

heat released by the jth reaction rate of the jth reaction(number per unit time and unit volume)

echem ��

hiYi �Dechem/Dt = ��

hi�.J�i �

Q(j)W j

� D(eT + echem)/Dt = ��.J�q � p�.u

Jq � J�q �

hiJ�i Jq = ��Theat flux

Thermal balance of inviscid flow of reactive gas

Governing equations for inviscid flows of reactive gas1�

Dt= ��.u, �

= ��p, p = (cp � cv)�T,

⇥cpDT

Dt+ �.(��T ) +

Q(j)W (j),

�DYi

Dt= �.(�Di�Yi) +

⇥(j)i miW

stoichiometric coe�cient

progress variable

�qmD⇥/Dt ��

Q(j)W (j), ⇥ � [0, 1]

heat released per unit mass

P.Clavin II

etot + p/� = cpT + |u|2/2 � qm⇥

⇥(�etot)/⇥t = ��.JetotJetot = J�

etot+ �etotu

Conservative form of the energy equation (inviscid approximation)Jq +

hiJ�i = J�

etot� pu

di�usive flux of total energyconvective flux of enthalpy

�(�etot)/�t = ��.[�uetot + up + Jq +�

hiJ�i] = ��.[�u(etot + p/�) + Jq +

hiJ�i]

Viscous flow� = pI + �

⇤ � �2�(�u)(s) � I(⇥ � 2�/3)�.u

⌅(⇥etot)/⌅t = �⇥.[⇥u (cpT + |u|2/2 � qm⇤) + J�q + u.�]

bulk viscosityshear viscosity

P.Clavin II

⇥�

⇥t= �⇥(�u)

⇥x⇥(�u)

⇥t= � ⇥

�p + �u2 � µ

⌅(⇥etot)⌅t

= � ⌅

�⇥u(cpT + u2/2 � qm⇤) � �

⌅x� µu

µ � 4�/3 + ⇥

one-dimensional compressible flow

(simplest form of heat flux)

P.Clavin II

Thermal equation

Conservative forms

Entropy productionII � 6)

2nd law of thermodynamics

entropy is a function of state that is not a conserved quantity

dissipative e�ects �

⇥(�s)/⇥t = ��.Js + ws,

ws � 0

D(1/�)Dt

��

µiDYi

ws = J�q.�

��

J�i.�

�µi

�� 1

T� : (�u)(s)

s(�, T, ..Yi, ..)

(s � so)cv

= ln�

p/��

po/��o

�ideal gas

T �s = �eT + p�v ��

µi�Yi

P.Clavin II

Js = ⇥us � �

⇤x, ws =

�⇤u

�⇤T

µ > 0, � > 0

inert mixture in one-dimensional geometry

⇥TDs

��

�+ µ

�⇤u

�2�Ds

��

�+ �s

Lecture I1IThermal propagation of flames

P.Clavin I

Contents

Quasi-isobaric approximation. LowMach number

�(u.�)u � ��p � �p � ⇥u�u

p � �a2 � �p/p � u2/a2 � M2

III � 1)

P.Clavin III

⇥cpDT/Dt = Dp/Dt + �.(��T ) +�

Q(j)W (j)

(in open space)

slow evolution �/�t � u.⇤ ⇥ a |⇤|

�T = �oTo

⇥cpDT/Dt = �.(��T ) +�

Q(j)W (j)(T, ..Yk..)

�DYi/Dt = �.(�Di�Yi) +�

�(j)i miW

(j)(T, ..Yk..),

very subsonic flow+ M2 � 1 � �p/p � �T/T = O(1)

��1p

�� 1T

��

�cpT= O(1)

��Dp

�� cp

��

� �u � u

1-D flame in a tube<latexit sha1_base64="0hOW0ryTFP8iWJpNDzp4HPzg9R8=">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</latexit><latexit sha1_base64="0hOW0ryTFP8iWJpNDzp4HPzg9R8=">AAACy3icjVHLSsNAFD3Gd31V3elmsAhuLIkI6k7QhRuhglWhlTJJpzqYZEIyEWstuPFX3OqP+AH+gf6Fd8YIPhCdQObMuece7sNPQplp130ecAaHhkdGx8ZLE5NT0zPl2bmjTOVpIOqBClV64vNMhDIWdS11KE6SVPDID8Wxf7Fj4seXIs2kig91NxGnET+LZUcGXBPVKi80tbjSPW91l3VCHgkmY8aZzn3Rb5UrbtW1h/0EXgEqKE5NlZ/QRBsKAXJEEIihCYfgyOhrwIOLhLhT9IhLCUkbF+ijRLk5qQQpOLEX9D+jV4+wIo0ifR8MyxRXpEwJG2dm47l1MezvPpxqMnV06fYLr4hYjXNi/8r7UP43r0GsRgebtltJs0gsY6YSFC65nYCpnH3qSpNDQpzBbYqnhAOb+TFTZnMy27uZI7fxF6s0rHkHhTbHq6mSlul9X91PcLRW9dyqd7Be2d4q1jqGRSxhhXa3gW3soYY6ed/iHg94dPadzLl2bt6lzkCRM48vx7l7A951lMg=</latexit><latexit sha1_base64="0hOW0ryTFP8iWJpNDzp4HPzg9R8=">AAACy3icjVHLSsNAFD3Gd31V3elmsAhuLIkI6k7QhRuhglWhlTJJpzqYZEIyEWstuPFX3OqP+AH+gf6Fd8YIPhCdQObMuece7sNPQplp130ecAaHhkdGx8ZLE5NT0zPl2bmjTOVpIOqBClV64vNMhDIWdS11KE6SVPDID8Wxf7Fj4seXIs2kig91NxGnET+LZUcGXBPVKi80tbjSPW91l3VCHgkmY8aZzn3Rb5UrbtW1h/0EXgEqKE5NlZ/QRBsKAXJEEIihCYfgyOhrwIOLhLhT9IhLCUkbF+ijRLk5qQQpOLEX9D+jV4+wIo0ifR8MyxRXpEwJG2dm47l1MezvPpxqMnV06fYLr4hYjXNi/8r7UP43r0GsRgebtltJs0gsY6YSFC65nYCpnH3qSpNDQpzBbYqnhAOb+TFTZnMy27uZI7fxF6s0rHkHhTbHq6mSlul9X91PcLRW9dyqd7Be2d4q1jqGRSxhhXa3gW3soYY6ed/iHg94dPadzLl2bt6lzkCRM48vx7l7A951lMg=</latexit><latexit sha1_base64="0hOW0ryTFP8iWJpNDzp4HPzg9R8=">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</latexit>

Propagation towards the closed end<latexit sha1_base64="QGHYd+GIzCr1vtLQ23bAq01UWFw=">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</latexit><latexit sha1_base64="71ukwvBtkO0meLas3i5/hpEK7PE=">AAAC3HicjVG7btRAFD1xeIQQYHl0NCMipEhIq5kUySZVJBrKRWKTSHlpbE82VrweazyGRKvt6BAFDb9Ch6CjyQfwB/AXnJnsSlBEZCzZx+dxdefetC6Lxkv5cy6Zv3Hz1u2FO4t3l+7df9B5+Gi7sa3LzCCzpXW7qW5MWVRm4Atfmt3aGT1KS7OTnr4M+s5b45rCVm/8eW0ORnpYFcdFpj2po86LfW/OmBv3na31MLLC23fa5Y3wJ0ZkpW1MLkyVT446y7IrpVRKiQDU+pok2NjoraqeUEHiWd568nW+vvjR9m3nAvvIYZGhxQgGFTxxCY2Gzx4UJGpyBxiTc0RF1A0mWGS2pcvQocme8j3k35jY0mPpn0DgOXVLpyMOlUXU21glsFfX0ewp9HHObzqtNSLrcUL2f7mZ87q5PbIex+jF2xacRR2ZMJVsWqWNEwidi79u5VmhJhdwTt0RZzE5m6mImSbePcxRR/1XdAY2/GdTb4vfoUsuc7YxcTXYXu0q2VWvudVNXJ4FPMUzrHB369jCK/QxYO1P+IJv+J4cJu+TD8nHS2syN808xj8n+fwHRjmg/A==</latexit><latexit sha1_base64="71ukwvBtkO0meLas3i5/hpEK7PE=">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</latexit><latexit sha1_base64="oEY/Gi7xnhETpq8ZtcOyOnjLEyg=">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</latexit>

Propagation from the closed end<latexit sha1_base64="D6DbzRhi6HWWqbXoE9Xrk8V+kYM=">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</latexit><latexit sha1_base64="NnFKc2VwtUCfLvI47RzqnoUzf1o=">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</latexit><latexit sha1_base64="NnFKc2VwtUCfLvI47RzqnoUzf1o=">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</latexit><latexit sha1_base64="KxmcYGTMNb58XUncm9vYz3sOo9Q=">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</latexit>

Propagation from left to right<latexit sha1_base64="rOdGpJT3jON/e6F+BeFexD7g4mc=">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</latexit><latexit sha1_base64="4bIaxmAHlVoXV5xoE7U4gg5boyQ=">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</latexit><latexit sha1_base64="4bIaxmAHlVoXV5xoE7U4gg5boyQ=">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</latexit><latexit sha1_base64="U2U0+JC3KYK8RT4tYiA3oDbK03E=">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</latexit>

UL<latexit sha1_base64="tNbxzPgEmxDKR4E61K3icFf4yX4=">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</latexit><latexit sha1_base64="lovE9HpXRdQuT+CO6Ep4Ew2Rzz4=">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</latexit><latexit sha1_base64="lovE9HpXRdQuT+CO6Ep4Ew2Rzz4=">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</latexit><latexit sha1_base64="Lgy3f3rUzmziuagC6+9XxQK9cUg=">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</latexit>

u = 0<latexit sha1_base64="8nt2kRPKYPWjygUwpkhCnyu87UY=">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</latexit><latexit sha1_base64="8nt2kRPKYPWjygUwpkhCnyu87UY=">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</latexit><latexit sha1_base64="8nt2kRPKYPWjygUwpkhCnyu87UY=">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</latexit><latexit sha1_base64="8nt2kRPKYPWjygUwpkhCnyu87UY=">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</latexit>

ub = Ub � UL<latexit sha1_base64="O4k534TCfBvEVTMbMMuAHan7vsc=">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</latexit><latexit sha1_base64="O4k534TCfBvEVTMbMMuAHan7vsc=">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</latexit><latexit sha1_base64="O4k534TCfBvEVTMbMMuAHan7vsc=">AAACunicjVHLSsNAFD2Nr1pfVZdugkVwY0lEUEGh4MaFiwqmFmoJSTqtQ5NMSCZCKf0Mt/pFfoB/oH/hnTEFtYhOIHPmnHMv9+EnIc+kZb2WjLn5hcWl8nJlZXVtfaO6udXKRJ4GzAlEKNK272Us5DFzJJchaycp8yI/ZLf+8ELptw8szbiIb+QoYd3IG8S8zwNPEtXJXf/ccf0Dx71yqzWrbuljzgK7ADUUpymqL7hDDwIBckRgiCEJh/CQ0deBDQsJcV2MiUsJca0zTFCh2JxcjBwesUP6D+g1JizII8g/gYk90gU5U8Iqs6n1XGdR7O95PKpJ1TGi2y9yRcRK3BP7V9zU+d+4DrESfZzobjnNItGMmkpQZMn1BFTl5peuJGVIiFO4R3pKONCR05maOibTvas5elp/007FqndQeHO8qyppmfbP1c2C1mHdtur29VGtcVqstYwd7GKfdneMBi7RhKMn/ognPBtnhm9wY/hpNUpFzDa+HUN+ADSMjo8=</latexit><latexit sha1_base64="O4k534TCfBvEVTMbMMuAHan7vsc=">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</latexit>

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wall at rest<latexit sha1_base64="zShSqzck7OWWPL1LPby0jttWWyA=">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</latexit><latexit sha1_base64="e9XIJQ+iAKwklVOZL8mYoLaHhvM=">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</latexit><latexit sha1_base64="e9XIJQ+iAKwklVOZL8mYoLaHhvM=">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</latexit><latexit sha1_base64="jzdn0NR923xSlkmq23je1RMgkYk=">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</latexit>

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Ub<latexit sha1_base64="A+odXD5GGwHqW7b2jyw+vK1+MEI=">AAACsnicjVHLSsNAFD2Nr1pfVZdugkVwVRIpqLuCG5cVTVuspUym0xqadEIyEUrpJ7jVr/ID/AP9C++MKahFdAKZM+eee7gPPw6DVDnOa8FaWl5ZXSuulzY2t7Z3yrt7zVRmCRcel6FM2j5LRRiMhacCFYp2nAgW+aFo+aMLHW89iCQN5PhGTWLRjdhwHAwCzhRR117P75UrTtUxx14Ebg4qyE9Dll9whz4kODJEEBhDEQ7BkNLXgQsHMXFdTIlLCAUmLjBDiXIzUglSMGJH9B/Sa0pYkkaSfgYbRxSXpEwIa2fbxDPjotnffRjVpOuY0O3nXhGxCvfE/pU3V/43r0OswgBnptuAZhEbRk+F5y6ZmYCu3P7SlSKHmDiN+xRPCHOTOZ+pbXJS07ueIzPxN6PUrH7zXJvhXVdJy3R/rm4RNE+qrlN1r2qV+nm+1iIOcIhj2t0p6rhEAx55D/GIJzxbNevWYhb/lFqFPGcf344VfgDsUIuf</latexit><latexit sha1_base64="A+odXD5GGwHqW7b2jyw+vK1+MEI=">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</latexit><latexit sha1_base64="A+odXD5GGwHqW7b2jyw+vK1+MEI=">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</latexit><latexit sha1_base64="A+odXD5GGwHqW7b2jyw+vK1+MEI=">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</latexit>

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Tu� 1

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reference frame of the flame<latexit sha1_base64="KgLwbtakZvmbeCCKwPlvkwMNMhk=">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</latexit><latexit sha1_base64="KgLwbtakZvmbeCCKwPlvkwMNMhk=">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</latexit><latexit sha1_base64="KgLwbtakZvmbeCCKwPlvkwMNMhk=">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</latexit><latexit sha1_base64="KgLwbtakZvmbeCCKwPlvkwMNMhk=">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</latexit>

⇢u⇢b

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quasi-isobaric approximation: �T � cst.

conservation of mass: ⇢U = cst.<latexit sha1_base64="YhV+kojnzqFEyqUWMZ0Hyzf/kGk=">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</latexit><latexit sha1_base64="U+B296DWYk44uiC8JU0xjBXns0c=">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</latexit><latexit sha1_base64="U+B296DWYk44uiC8JU0xjBXns0c=">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</latexit><latexit sha1_base64="Y+9Ie811hFa7v4YBhWTnJSiniNA=">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</latexit>

Quasi-steady inner structure: Velocity of the front / Fresh mixture UL = cst.<latexit sha1_base64="sV+7yBpktxgojoRZ/m9TMT8PzT0=">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</latexit><latexit sha1_base64="sV+7yBpktxgojoRZ/m9TMT8PzT0=">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</latexit><latexit sha1_base64="sV+7yBpktxgojoRZ/m9TMT8PzT0=">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</latexit><latexit sha1_base64="nm05qU8VWyeysrWIVwd9QMvdHOw=">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</latexit>

one end is closed and the other open<latexit sha1_base64="o5UZWMJtagnEEhmqXSfD9yAq/1o=">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</latexit><latexit sha1_base64="LjRINJBaYGFH2bL4D3WxvTJCoRM=">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</latexit><latexit sha1_base64="LjRINJBaYGFH2bL4D3WxvTJCoRM=">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</latexit><latexit sha1_base64="PjtNz2JK4X622PveAIPat/ArLxQ=">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</latexit>

dx� d

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Q(j)W (j)(T, ..Yi..)

dx� d

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⇥(j)i MiW

(j)(T, ..Yi� ..),

equations m?

P.Clavin III

m � �uUL = �bUb, Ub/UL � Tb/Tu, � 4 � 8

D/Dt = md/dx

Unburntmixture

Burntgas

Planar flame reference frame of flame

mass flux across the planar flame

quasi-isobaric approximation: �T � cst.

boundary conditionsfrozen stateequilibrium statex = +1 : dT/dx = 0, Yi = Yib, W (j) = 0

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x = �1 : T = Tu, Yi = Yiu, W (j) = 0<latexit sha1_base64="x+jYwkDSaDKgpQe778rHtGwcBkE=">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</latexit><latexit sha1_base64="pxh945HnryLtYvlmJ2a0rYs019I=">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</latexit><latexit sha1_base64="pxh945HnryLtYvlmJ2a0rYs019I=">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</latexit><latexit sha1_base64="jB1mXfW/82wRAGCUJ1G+CNY7vXY=">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</latexit>

axis oriented towards the burned gas<latexit sha1_base64="LVYPKHICt4yMNf3oiq5SdgZudtY=">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</latexit><latexit sha1_base64="oEhquQYeUTicsWfjMRv+8nB13NU=">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</latexit><latexit sha1_base64="oEhquQYeUTicsWfjMRv+8nB13NU=">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</latexit><latexit sha1_base64="jo9Cjh3rCvgxKu9Lmxh0lircYdE=">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</latexit>

P.Clavin III

R � P + Q

One-step irreversible reactionIII � 2)

R in an inert ; Y = mass fraction of R

x � +� : Y = 0x � �� : Y = Yu, T = Tu

mYu =� +�

��Wdx

cp(Tb � Tu) = qm � qRYu

dx� d

��D

�= ��W

dx� d

��

�= ⇥qRW qR = energy released per unit of mass of R

m � �uUL unknown

Velocity and structure of the planar flame

Arrhenius law

�W = �bY

⇥r(T )1

�r(T )� e�E/kBT

�coll

1�rb

� e�E/kBTb

�coll

1�r(T )

�rbe�

TbT �(1�⇥) � � E

�1 � Tu

�� T � Tu

Tb � Tu� [0, 1]

P.Clavin III

Unity Lewis number and large activation energyIII � 3)

dx� ⇥DT

dx2= ⇥

Y1u, m

dx� ⇥DT

Led2⇤

dx2= �⇥

x = �� : � = 0,⇥ = 1, x = +� : � = 1,⇥ = 0

� � T � Tu

Tb � Tu� [0, 1] � � Y

Yu� [0, 1]

Reduced temperature and mass fractionLe � DT /D

Le = 1

� = 1 � � md�

dx� ⇥DT

dx2= ⇥

x = �� : � = 0, x = +� : � = 1,

Yu= ⇥b

(1 � �)⇤rb

e�TbT �(1�⇥)

� � 1

P.Clavin III

(a) (b)

Yu= ⇥b

w�(�)⇤rb

w�(�) � (1 � �)e��(1��)

(reaction rate is non negligible only when T � Tb)

� � E

�1 � Tu

Yu= �b

⇥rbe�

TbT �(1�⇥)

1�rb

� e�E/kBTb

�coll

w�(�) � (1 � �)e��(1��)

Zeldovich, Frank-Kamenetskii asymptotic analysisIII � 4)

dx� ⇥DT

dx2= ⇥

P.Clavin III

origin x = 0 : location of the reaction zone � = 1

W � 0preheated zone

matching condition

1�rb

� e�E/kBTb

�coll

(a) (b)

Zeldovich 1938

� � � � � E

�1 � Tu

(a) (b)

⌅bDTbd2⇤

dx2� ⌅bDTb

⇥⇤

� ⌅bDTb

⇧rbw�(⇤) � ⌅b

⇧rb⇥⇤ � ⌅b

md�/dx � ⇥DT d2�/dx2 =⇥b

⇤rb(1 � �)e��(1�⇥)

dimensional analysisUL �

�DTb/�rb

�DTb

�d�

� 1⇥rb

(1 � �)e��(1�⇥) d�

� = �(1 � �) ⇤rbDTb

�d⇥

�� 1

⇥(1 � ⇥)e��(1�⇥)d⇥ =

� �(1�⇥)

0�e��d��

⇥bDTbd�/dx|�=1 = m� � 1 :downstream entrance of the external zone

P.Clavin III

� dr/dL = O(1/�)

dr ��

DTb�rb

dx� m

��

m = �DT /dL�⇥⇤ = O(1/�)

� �bDTb

1� dr � dL

Inner reaction layer

Asymptotic solution � � �upstream exit of the inner layer �(1 � ⇥) � � : DTbd⇥/dx �

�(2/�2)DTb/⇤rb

� �

0�e��d� = 1

m = ⇥b

�(2/�2)DTb/⇤rb, UL = m/⇥u,

matching

Flames = quasi-isobaric reaction-di↵usion waves<latexit sha1_base64="I0beAZBAlEVNyxwP9BeQcvs8E3Q=">AAAC6nicjVHLThRBFD20Dx6+Rl3ioiIxccOk2426ICGaGJaQOEAChFTXVGOF7qm2qxolk/kJd8aFGz7CH2AJrN251L/wVNEkKjFanXSdOvfck/vI69I4n6Zfp5IrV69dn56Znbtx89btO72799adbRulB8qWttnMpdOlGemBN77Um3WjZZWXeiPffxniGwe6ccaOXvvDWu9Ucm9kCqOkJ7Xbe77t9XufF+NXpay0E0vibSudWTTO5rIxStBMBeni0BRFG3zEO3mg3WS3t5D203jEZZB1YGH5wZe1bwBWbe8E2xjCQqFFBY0RPHEJCcdvCxlS1OR2MCbXEJkY15hgjrktVZoKSXaf/z2+xsSWGkv9BAKPGLdUNsTBWcR4G10C+3cfyZpCHYe8886rIuvxhuy/8i6U/5u3RdajwLPYreEs6siEqajOpY0TCJWLX7rydKjJBTxkvCFWMfNipiLmuNh7mKOM8e9RGdjwVp22xY9QJZeZ/bm6y2D9ST9L+9kat/oC52cG83iIx9zdUyxjBasY0PszjnGKs6RMPiQfk0/n0mSqy7mP305y9BPGXqUh</latexit><latexit sha1_base64="seLQiMWtAUmO/NWl+raUE8NqP0A=">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</latexit><latexit sha1_base64="seLQiMWtAUmO/NWl+raUE8NqP0A=">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</latexit><latexit sha1_base64="2ikqGOi1gzS1sP5i/9BLQJFqPoU=">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</latexit>

� � E

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�1�rb

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⌧rb<latexit sha1_base64="q/rXHIO9hz1eOlLOEoeT7JhYUSg=">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</latexit><latexit sha1_base64="LCED17Fn7VOO0PzYZ4bKhDaO5Y4=">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</latexit><latexit sha1_base64="LCED17Fn7VOO0PzYZ4bKhDaO5Y4=">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</latexit><latexit sha1_base64="dkaofhdLGcxC7zd555q0ijr1F+E=">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</latexit>

DTb ⇡ a2b⌧coll<latexit sha1_base64="/GlrL0/LG7/qpEZbVEvxYZMEV/4=">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</latexit><latexit sha1_base64="DgAbVoUJwU7oTYXfyPOvPnNG1Ec=">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</latexit><latexit sha1_base64="DgAbVoUJwU7oTYXfyPOvPnNG1Ec=">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</latexit><latexit sha1_base64="lAlviWC6rfS074XuPjcp584oVFA=">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</latexit>

p2 e�E/kBTb

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kBTb� 1 :

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markedly subsonic wave<latexit sha1_base64="5m5dWG98LJJm8DwVr1uRgRBT/+4=">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</latexit><latexit sha1_base64="wXWJEAK6vgnNTn167eS+NFKLW7Y=">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</latexit><latexit sha1_base64="wXWJEAK6vgnNTn167eS+NFKLW7Y=">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</latexit><latexit sha1_base64="8wOfDYmM1bqsFFlOWzAwIZzTsgQ=">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</latexit>

ZFK result for a one-step first-order reaction<latexit sha1_base64="3tb2pNSf2pIawLZ0zGJNsbCMDNw=">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</latexit><latexit sha1_base64="zE9cTWY67I+qMKJpgUuYXUjX0pw=">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</latexit><latexit sha1_base64="zE9cTWY67I+qMKJpgUuYXUjX0pw=">AAAC5nicjVHNahRBGKxMosb4kzUePHhpDIKXLDOCJAeRgCCClwjZJJiE0DPbmzSZnR66e8Sw7Ct4Czl48R18Aq9KHiAPIMRLniHVnQmoQbQHZmrqqyq+/r68LrXzaXoykUxOXbt+Y/rmzK3bd+7Odu7NrTnT2EL1ClMau5FLp0pdqZ7XvlQbtVVymJdqPd9/Gerr75V12lSr/qBW20O5W+mBLqQntdN5tuXVBz969+qNsMo1pRcDY4UUplILzqtaDLR1fsHYvrJUyCLYxjud+bSbxiOugqwF88sPzs5mn3/5sWI6x9hCHwYFGgyhUMETl5BwfDaRIUVNbhsjcpZIx7rCGDP0NlQpKiTZfb53+TciNtQY6scQeMy6odISh2QR601MCezfcyR7Cn0c8Ju3WUOyHntk/+W7VP6vb5OsxwBL8baas6gjE6ZStClNnEDoXPxyK8+EmlzAfdYtcRGdlzMV0ePi3cMcZayfRmVgw3/Rahv8DF1ymdmfq7sK1p52s7SbveVWX+DiTOMhHuEJd7eIZbzGCnrM/oSv+IbvyV7yMTlMji6kyUTruY/fTvL5HADHpME=</latexit><latexit sha1_base64="FIp+zyTZsMD4yyDLi9iTxW3BagA=">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</latexit>

SUMMARY<latexit sha1_base64="/9LlJ9L9VgVNYzd3blWLAdSkFyA=">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</latexit><latexit sha1_base64="uDR+IcKkicyz1cFrSMZOiDWuDOc=">AAACwXicjVHLSsNAFD3GV31HXboJFsFVSdzozpZu3BTqo1VRkSROdDDphMlELKHgh7jVT3HnB/gH+hfemUbwgegNZM6ce+7h3jtBGvNMue7LiDU6Nj4xWZmanpmdm1+wF5e6mchlyDqhiIU8CvyMxbzHOoqrmB2lkvlJELPD4Lqp84c3TGZc9A5UP2VniX/Z4xEPfUXUuW2fKnargqjY77Rajb3jwblddWuuCecn8EpQ3X6KTLSF/YxTXEAgRI4EDD0owjF8ZPSdwIOLlLgzFMRJQtzkGQaYptqcVIwUPrHX9L+kW0FYkEaQfgAHa5QXpJSEtbNj8rlx0ezvPj71pPvo0xmUXgmxClfE/lX3ofxv3QmxChG2zLScdpEaRm8lLF1yswHdufNpKkUOKXEaX1BeEg5N5cdOHVOTmdn1Hn2TfzVKzep7WGpzvOku6TG970/3E3Q3ap5b83bdar1xBxMVrGAV6/R2m6hjB210yPsG93jAo9W0uJVacii1RoYnlvElrOIdauyVOA==</latexit><latexit sha1_base64="uDR+IcKkicyz1cFrSMZOiDWuDOc=">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</latexit><latexit sha1_base64="mkVFzRQ/j7RJ5rnZhmo2KU299Ig=">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</latexit>

P.Clavin III

non-dimensional form

III � 5) Reaction diffusion wavesP.Clavin III

��

�t� �2�

�x2= w(�) � � 0 � � [0, 1]

t � � t � �

µ = µminiµ = µmini

propagation of steady state � = 1 into steady state � = 0

stability of steady statesstable: dw/d� < 0,

unstable: dw/d� > 0,

steady state: � = 0

� = 1, w = 0 stable steady state(equilibrium state)

two di�erent cases depending on the property of the initial state � = 0

(I) : � = 0, w = 0 metastable steady state(II) : � = 0, w = 0 unstable steady state

initial state:(out of equilibrium)(less stable)

Metastable state Stable equilibrium state Unstable state Stable equilibrium state

d�/d� = �w(�)

(I) (II)

µ unknown, number of solutions ?

P.Clavin IIIpropagating planar wave at constant velocity

��

�t� �2�

�x2= w(�)

µd�

d⇥� d2�

d⇥2= w(�)

�/�t = µd/d�

�/�x = d/d�� x + µt

�(�)

traveling wave solution (from right to left)

Metastable state Stable equilibrium state Unstable state Stable equilibrium state

d�/d� = �w(�)

(II)(I)

� = �� : � = 0, w = 0,initial state

� = +� : � = 1, w = 0final (equilibrium) state

µ is an eigenvalue of the problem

Upstreamunstable state

Downstreamequilibrium state

ZFK flame model: � > 0, case (II)

� � [0, 1]

Y = µdX

d�k± = µr±

±= k±

P.Clavin III

Upstreamunstable state

Downstreamequilibrium state

�X = A+e� r+ + A�e� r� , �Y = k+A+e� r+ + k�A�e� r�

Two eigenvalues r+and r� and two eigendvectors k+and k�

µd�X

d�� d2�X

d�2= ��

��X µr � r2 � �� = 0 2r± = µ ±

�µ2 � 4��

(equilibrium state)Linearisation about X = 0, Y = 0 and X = 1, Y = 0

� � ��

� < 0

Unstable state

FocusSaddle

Equilibrium state

No solution (� � 0)

r+ > 0, r� < 0Im r� �= 0, Im r+ �= 0� � ��

� < 0��

� > 0

Re r± > 0

dY/d� = µ [Y � �(X)]

Saddle

Metastable orequilibrium state

Saddle

Equilibriumstate

One solutionr� < 0, r+ > 0 r+ > 0, r� < 0

� � ��

� < 0�� < 0

case (I)

Number of solutions ? phase space, phase portrait� � 0

µ ?µdX

d�� d2X

d�2= �(X)X � �,

dX/d� = Y/µ

Y � µd�/d�

second order system

fixed points: dX/d� = dY/d� = 0Y = 0, dY/d� = 0

dY /dX = µ2[Y � w(X)]/Y

dY/dX = 0/0

Unstable state

Saddle

Equilibrium state

tePart

icular

trajec

Degenerate

Infinite numbers of solutionsOne particular solution

r+ > 0, r� < 0r+ > r� > 0� � ��

�� < 0��

� > 0

case (II)A� = 0

Unstable medium

⌅�/⌅t � ⌅2�/⌅⇥2 = ⇤�o�, where ⇤�

o ⇥ ⌅w/⌅�|�=0 > 0

�(⇥, t) � Z(⇥, t)e+��ot ⇥Z/⇥t � ⇥2Z/⇥�2 = 0

Z = e��2/4t/�

t � � exp[�⇥2/4t + ⇤�ot � ln(t)/2] �2 � 4⇥�

the lower bound solution is selected µmini � 2�

dw/d�|�=0

Kolmogorov 1937

µflame �

2� 1

0w(�)d��=

Clavin, Linan 1984

OK for a soft term �(�)Wrong for a sti� term �(�)

The lower bound solution changes of nature when �(�) get sti�er

Soft �(�) = �(1 � �)

Sti� w(⇥, �) = (�2/2)⇥(1 � ⇥)e��(1��), � � 1

P.Clavin III

2r± = µ ±�

µ2 � 4��

µ wave velocity

µmini

continuous spectrum with a lower bound

Unstable state

FocusSaddle

Equilibrium state

Unstable state

Saddle

Equilibrium state

tePart

icular

trajec

Degenerate

t � � t � �

µ = µminiµ = µmini

r+ = r� = µ/2 k+ = k�lower bound :µmini � 2�

dw/d�|�=0

soft case � collapse of the 2 eigenvaluessoft nonlinear term �(�)

Lecture IVHydrodynamic instability of flames

P.Clavin I

Contents

P.Clavin IV

Jump across an hydrodynamic discontinuityIV � 1)

UL Ub - UL

Burnt gas

Temperature

[�u]+� = 0 �u = �uUL = �bUb

�p + �u2

�+� = 0

[w]+� = 0�u �= 0 �

jump relations (reference frame of the flame)

flame considered as a discontinuityflame thickness and curvature neglected

P.Clavin IV

⇥�/⇥t = �⇥(�u)/⇥x � ⇥(�w)/⇥z,

⇥(�u)/⇥t = �⇥(p + �u2)/⇥x � ⇥(�uw)/⇥z,

⇥(�w)/⇥t = �⇥(�uw)/⇥x � ⇥(p + �w2)/⇥z

r = (x, z), u = (u, w)reference frame of the flame

tilted planar front

�a(x, y, t)dx = 0lim

dL � 0dL

if a(r, t) is regular

flame � surface of zero thickness separating two incompressible flows

�⇥�/⇥t = ��.(�u) �Du/Dt = ��p ⇥(�u)/⇥t = ��.(pI + �uu)Low Mach nb approx + inviscid approx: Euler eqs

� � dL

�u �b

�u > �b

Flame front

Burnt gasat rest

Fresh mixture

UL Ub - UL

Burnt gas

Temperature

Piston e�ectreference frame of the flame front

Inclined flame front

Burnt gas

Fresh mixture

deviation of the stream lines

tilted front

P.Clavin IV

u�|B > UL

u�|A < UL

flame motion

instability mechanism� � dL, dL/� � 0

� � UL/�

”instantaneous” modification of the flow field, both upstream and donstream(low Mach nb approx: the speed of sound is infinite, a � �)

conservation of mass + isobaric approx

Unburned gas Burned gas

�� +

x = �(y, t)

��y � ⇥�/⇥y

� 1�1 + ��2

, ��

y�1 + ��2

� , un � uf .nf =�uf � ��

�/�

1 + ��2y

wtg =�wf + ��

�/�

1 + ��2y

�t � ⇥�/⇥t Df =�t�

1 + ��2y

normal velocity of the front

equation of the perturbed front

��U�n = �+U+

⇥��u�

f � �t � ��yw�

�= ⇥+

f � �t � ��yw+

conservation of mass

[Wtg]+� = 0

�p + �U2

�+� = 0

p�f + ⇥�

�u�

f � �t � ��yw�

1 + ��2y

= p+f + ⇥+

f � �t � ��yw+

1 + ��2y

�w�

f + ��yu�

f + ��yu+

conservation of momentum

uf = (uf , wf )flow velocity at the frontP.Clavin IV

Wtg = wtgUn ⇥ un �Df =�uf � �t � ��

�/�

1 + ��2y

flow velocity relative to the perturbed front

normal component tangentail component

( reference frame !)

reference frame of the planar unperturbed flame(x = 0)

P.Clavin IV

Linearised Euler equations of an incompressible fluid IV � 2)

a = a + �amf = ��u�

f = �+u+f

�⇤2

⇤x2+

⇤y2+

��⇥± = 0

⇥x�u± +

⇥y�w± = 0,

�⇤±

⌅t+ mf

��u± = � ⌅

⌅x�⇥±

�⇤±

⌅t+ mf

��w± = � ⌅

⌅y�⇥±,

�± � p± � ⇥±g(t)x,

x � �� : no disturbances, �u� = 0x � +� : disturbances remain finite,

⇥±�

⇤t+ u± ⇤

�u±(x, t) = ±|k|�±

f (t)e�|k|x�u±

�x+ ik.w± = 0flow velocity

⇥2�±

⇥x2� |k|2�± = 0 �±(x, t) = �±

f (t)e�|k|xpressure

general solution to the homogeneous equation + particular solution

u±(x, t) = u±R(x, t) + u±

P (x, t)

�u±R

�t+ u± �u±

�x= 0, ⇥±

�ddt

� u±k

�u±

p (t) = ±k�±f (t)

u±P (x, t) = u±

p (t)e�kx,

P.Clavin IV

⇥a(x,y, t) = a(x, t)eik.y �(y, t) = �(t)eik.y

transverse coordinates

P.Clavin IV

�u±R

�t+ u± �u±

�x= 0, u±

R = u±r (t � x/u±), u�

R = 0,

3 unknown functions: u�f (t), u+

p (t), u+r (t)

x < 0 :

��

u�(x, t) = u�f (t)e+kx,

k��(x, t) = �⇥��

+ u�k

�u�

f (t)e+kx,

��

u+(x, t) = u+p (t)e�kx + u+

r (t � x/u+),

k�+(x, t) = ⇥+

�ddt

� u+k

p (t)e�kx,x > 0 :

u±(x, t) = u±R(x, t) + u±

P (x, t)

ik.w�(x, t) = �ku�(x, t), ik.w+(x, t) = � �

�xu+(x, t).

2 for the conservation of normal and tangential momentum

4 boundary conditions at the flame front involving the additional unknown �(t)

�m�f = �m+

f = 02 for the conservation of mass (inner flame structure not modified)

m � �(u � ��/�t)

u+R = u+

r (t � x/u±),vorticity of the burnt gas flow

u+P = u+

p (t)e�kxk � |k| = 2�/� u� = u�P = u�

f (t)ekx

potential flows

P.Clavin IV

u�f = d�/dt �

Equation for the front

(�� + �+)d2�

dt2+ 2mfk

�(�� +)g(t)k + (u+ � u�)mfk2

�� = 0

elimination of u+p

�du+

dt= �

x < 0 :

��

u�(x, t) = u�f (t)e+kx,

k��(x, t) = �⇥��

+ u�k

�u�

f (t)e+kx,

��

u+(x, t) = u+p (t)e�kx + u+

r (t � x/u+),

k�+(x, t) = ⇥+

�ddt

� u+k

p (t)e�kx,x > 0 :

� mf

p + mf

u�ddt

�u�

f = (⇥� � ⇥+)kg(t)�(t)

Normal momentum

⇥p�f + 2⇤�u�f (⇥u�

f � �t) = ⇥p+f + 2⇤+u+

f (⇥u+f � �t)

IV-3) Conditions at the front

ku+p (t) +

du+r (t)dt

+ ku�f (t) = mf

⇥+ � 1⇥�

�k2�(t)�

Tangential momentum⇥

�w�

f + ��yu�

f + ��yu+

��

notationaf (t) � a(x = 0, t)

a(x, t)

p = �mf

⇥+ � 1⇥�

�k2� +

dt2+ ku�

elimination of du+r /dt

ik.w�(x, t) = �ku�(x, t), ik.w+(x, t) = � �

�xu+(x, t).

��y/�y � �k2�(t)

⇥a(x,y, t) = a(x, t)eik.y �(y, t) = �(t)eik.y

transverse coordinates

⇥+f � ⇥�

f = (⇤� � ⇤+)g(t)�(t)�hydrostatic presure

�± � p± � ⇥±g(t)x,

u�f (t) = u+

p (t) + u+r (t) = d�/dt

⇤��⇥u�

f � �t

�= ⇤+

�⇥u+

f � �t

�= 0 ⇥u�

f = ⇥u+f = �t�

Mass ⇥��u�

f � �t � ��yw�

�= ⇥+

f � �t � ��yw+

�= 0

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P.Clavin IV

(⇥� + ⇥+)d2�

dt2+ 2mfk

dt� k[(⇥� � ⇥+)g(t) + (u+ � u�)mfk]� = 0

IV-4) Dynamics of a passive interfacemf = 0

Rayleigh

Rayleigh-Taylor instability

g = cst. (�� +)g > 0 �2 � Atkg = 0, At > 0

Ububble = 0.361�

2gRg > 0, At �� +

�� + �+> 0

Rayleigh-Taylor bubble (upwards propagation)

� =�

�(t) = � ei�t � = B�

gk B �

�(�+ � ��)(�+ + ��)

g = cst.

Gravity waves

(�� +)g < 0� � Im � �= 0

Faraday (parametric) instability. Mathieu’s equationd2�

dt2+ ⌅2

o [1 + ⇥ cos(⌅⇤)] � = 0g(t) oscillating

linearly unstableRe(�) < 0 :

�(t) = � e�tFourier mode

linearly stable Re(�) > 0 :

�(y, t) = �(t)eik.y

ok with dimension

P.Clavin IV IV-5) Darrieus-Landau instability of flames

shorter is the wavelength stronger is the instability !?k = 2�/�however the analysis is valid only in the limit dL/� � 0

Stabilisation at small wavelength, � � dL

⇥�

⇥t= BDT

⇥2�

⇥y2. �di� ⇥ 1/⇥di� = �BDT k2 = �BULk(dLk)

(⇥� + ⇥+)d2�

dt2+ 2mfk

dt� (u+ � u�)mfk2� = 0

mf = ��u� = �+u+g = 0

11 + ⇥�1

��1 ±

�1 + ⇥b � ⇥�1

�� = AULk, A > 0u� � UL

⇥b � ��/�+ = u+/u� > 1

�u � �b : ⇥ =�

UbUL k (�u � �b)/�u � 1 : ⇥ = (Ub � UL)k/2

� � ULkno length scale in the problem; dimensional analysis �dL/� � 0 :

(�� + �+)⇥2 + 2mfk⇥ � (u+ � u�)mfk2 = 0�(t) = � e�t

Darrieus 1938Landau 1944

Landau Darrieus

Unstable

Stable

B > 0 ?first order correction

kdL < 1 : �/UL = Ak � Bk2dL + . . .

P.Clavin IV

VI-6) Curvature effect: a simplified approachmodification to the inner flame structure

⇥m�f /⇤� ⇥ (⇥u�

f � �t) = �BDT ⌅2�/⌅y2

m�f (t)/mf � BdLk2�(t) DT = ULdL

flame propagating downwards g < 0�

1 +⇥b

�d2�

dt2+ 2ULk

�⇥u

⇥b� 1

�� ⇥b

⇥u|g| + U2

�1 � k

��

� 2BdL

equation for the flame front

curvature e�ect(⇥u+⇥b)

dt2+2mfk

dt(1 + BkdL) = k�(⇥u�⇥b) [g(t) + UbULk (1 � 2BkdL)]

(correction due to curvature, finite thickness e�ect kdL �= 0)

Stability limits of flames propagating downwards � = 0marginal wavenumber

��Go + �

�1 � �

��= 0,

Normal momentum ⇥p�f + 2⇤�u�f (⇥u�

f � �t) = ⇥p+f + 2⇤+u+

f (⇥u+f � �t)

⇥f+ � ⇥f� = �2mf

�1⇤b

� 1⇤u

�mf (t) + (⇤u � ⇤b)g(t)�(t)(flame notations: �+ � �b, �� u, �u > �b)

non-dimensional parameters ⇥b � ��/�+ = u+/u� > 1� � kdL G0 � (�b/�u)Fr�1 Fr�1 � |g|dL/U2

L�m � 1/(2B)s = ��L

(1 + ��1b )s2 + 2�s � (�b � 1)�

��Go + �

�1 � �

��= 0

�m�f = �m+

f �= 0?

first order in perturbation analysis dL/� � 1

P.Clavin IV Stability limits of flames propagating downwards� � kdL G0 � (�b/�u)Fr�1 Fr�1 � |g|dL/U2

L�m � 1/(2B)non-dimensional parameters

� = 0marginal wavenumber��Go + �

�1 � �

��= 0,

curvature stabilizes the small wavelengths

gravity stabilizes the large wavelengths of slow propagating flame

gravity stabilizes the large wavelengths of slow propagating flame UL < 10cm/s

UL � 10cm/sinstability thresholdGoc =

2, kc =

2, ULc =

OK with experiments by Boyer Quinard and Searby (1982)

Flames propagating upwards: bubble flames

(1 + ��1b )s2 + 2�s � (�b � 1)�

��Go + �

�1 � �

��= 0s = ��L

g = 0, G0 = 0

G0 = G0cG0 > G0c

Lecture VThermo-diffusive phenomena

P.Clavin V

Lecture 5: Thermo-diffusive phenomena

5-1. Flame stretch and Markstein numbers

5-2. Thermo-di�usive instabilitiesPlanar flames for Le �= 1

Jump conditions across the reaction layer

Linear equations and linear analysis

Cellular instability (Le < 1)

Oscillatory instability (Le > 1)

Passive interfaces

One-step flame model

The second Markstein number

Unburnt gas Burnt gas

P.Clavin V

Two mechanisms modify the inner flame structureV-1) Flame stretch and Markstein numbers

�u �b

transverse convection

One-step model R � P + Q

rate of stretch of flame surface

A single scalar: the Markstein number

normal flame velocity in the fresh mixture

(U�n � UL)/UL = �M(�L/�s)

1/�s

U�n = u�

n �Df , u�n ⇥ nf .u�

transverse di�usion

θ = 0

θ = 1

ψ = 0

ψ = 1

ξ = 0ξ = -√Le

θ, ψ

P.Clavin V

Stretch rate, strain and curvature of a flame

d�⇥/dt = nf . [ue(rf + �⇥ nf ) � ue(rf )]

⇥ 1�2s

�2s = ⇤.ue|f � nf .⇤ue|f .nf .

�coordinate normal to the front1

�3r =1

�2s +1�⇥

�⇥

1�3r

�3r = �.ue|felement of volume �3r = �2s �⇥

continuity

d�2s

dt�2selement of surface area drf/dt = ue(rf )

Passive interface

Flame (first order correction) dL/� � 1ue(rf ) = u�

f � ULnf

1/�s = �UL⇥.nf + ⇥.u�|f � nf .⇥u�|f .nfnf .�n|f .nf = 0

nf .nf = 1

ue (rf + �⇥ nf ) � ue (rf ) + �⇥ nf .�uefirst order correction in dL/L � 11��

�� = nf .�ue.nf

P.Clavin V Flame (first order correction) dL/� � 1

1/�s = �UL⇥.nf + ⇥.u�|f � nf .⇥u�|f .nf

��.nf = 1/R � (1/R1 + 1/R2)di�erential geometry

1/�s = UL/R � nf .⇥u�|f .nf

incompressibility

front curvature strain rateBurnt gas

Fresh gas

Solid wallImploding flame

Quiescent fresh gas

Outflowing burnt gas

(U�n � UL)/UL = �M(�L/�s)

R � P + Qstudy of structure of the one-step flame modelClavin, Williams 1982 Clavin, Garcia 1983

M =�b

�b � 1J +

(�b � 1)lean hydrocarbon air mixtures M ⇥ 1 � 4

J =� 1

(⇤b � 1)⇥(�)1 + (⇤b � 1)�

d�, D = �� 1

(⇤b � 1)⇥(�) ln �

1 + (⇤b � 1)�d�,

reduced activation energy Lewis number Le = DT /D�⇥b � �u/�b > 1 � � (T � Tu)/(Tb � Tu) heat conductivity �(�)l � �(Le � 1)

gas expansion � hydrodynamics kinetics + di�usion

Clavin Garcia 1983

P.Clavin V The second Markstein numberClavin Grana-Otero 2011

flow strain rate

multiple-step flame model

(U�n � UL)/UL = �Mfc(dL/R) + Msr(�Lnf .⇥u�|f .nf )

Mfc �= Msr

front curvatureUnburned gas Burned gas

Flame front

Burnt gasFresh gas

Flame front

Streaml ine

Msr varies with the position inside the flame structure

di�culty with the finite thickness:

Msr(�Lnf .�u�|f .nf ) = cst.

�u �b TbTu

U�n �= UL U+

n �= Ub

�bU�n �= �bU

M+fc �= M�

fc M+sr �= M�

numerical and experimental data

U�n � UL

UL= �2(M�

fc �M�sr)

U+n � Ub

Ub= �2M+

Df = 0, U�n = u�

n = 0 U+n = �Df

Markstein numbers in the burned gasU+

n ⇥ u+n �Df u+

n � nf .u+(rf )

(U+n � Ub)Ub

= �M+fc

sr�Lnf .⇥u+|f .nfUb

Unburntmixture

Burntgas

P.Clavin V V-2) Thermo-diffusive instabilities of planar flames

instability mechanism �= hydrodynamic instability

� � E

�1 � Tu

�� T � Tu

Tb � Tu� [0, 1] 1

�rb� e�E/kBTb

�coll� � Y/Yu

Thermo-di�usive flame model for a one-step kinetics (� � 1)

⌅�

⌅t� DT �� =

⇥rbe��(1�⇥) ⇤⇥

⇤t� D�⇥ = � ⇥

�rbe��(1�⇥)

x = �� : � = 0,⇥ = 1 x = +� : � = 1,⇥ = 0� = cst.

�d2�

d�2� �2

2�e��(1��) d2�

d�2+

d�2= 0

reaction layer

Joulin Clavin 1979Sivashinsky 1977

Planar flame for Le � DT /D �= 1� � x

dL(Le = 1)

µ � UL

UL(Le = 1)

µd⇥

d⇤� d2⇥

d⇤2=

2⌅e��(1��) µ

d⇥� 1

Led2⇤

d⇥2= ��2

2⇤e��(1��)

µ =�

matching(first order reaction rate)

�T = �oTo⇥cpDT/Dt = �.(��T ) +

Q(j)W (j)(T, ..Yk..)equations + fluid mechanics

�DYi/Dt = �.(�Di�Yi) +�

�(j)i miW

(j)(T, ..Yk..),

d�/d⇥|⇤=0� = e��(1�⇥f )/2

valid up to 1st ordervalid at the leading order

�d�

0�= 0

jump conditions across the reaction layer

Joulin Clavin 1979

P.Clavin V

Flame temperature of curved flame for Le �= 1Le �= 1 � �f �= 1

Le � 1 = O(1/�) � (⇥f � 1) = O(1/�)� � 1

�1 = ⇥1/Le 1�2

d2�1

d�2=

12e��(1��f )⇥1e��1

integration and matching d�/d⇥|⇤=0� � Le1/2 e��(1�⇥f )/2

�f ?

Preheated zone

⇥ � x/dL, � = y/dL, ⇤ � t/⇤L⇧�

⇧⇤+

⇧�

⇧⇥� �� = 0,

⇧⌅

⇧⇤+

⇧⌅

⇧⇥� 1

Le�⌅ = 0

non-dimensional equations in the reference frame attached to the unperturbed flame

boundary conditions: jump conditions and ⇥ = �� : � = 0,⇤ = 1, ⇥ = � : � = 1,⇤ = 0.

⇥ = ��1/� + ..⇥ = ⇥f � �1/� + ..

�(⇥f � 1) = O(1) d⇥/d⇤|0+ = O(1/�)

the thin reaction layer of curved flame is quasi-planar�d2�

d�2� �2

2�e��(1��) d2�

d�2+

d�2= 0 � = non-dimensional normal coordinate � � x

dL(Le = 1)

P.Clavin V Linear equations

�dd�

� d2

��(�) +

�� + �2

��(�) =

�� + �2

� d�

d��

� 1Le

��(�) +

�� +

��(�) =

�� +

�d�

boundary conditions: jump conditions and � = �� : ⇥ = 0, ⇤ = 0, � = +� : ⇥ = 0, ⇤ = 0

⇤ = ⇤(⇥) + �⇤, ⇤f = 1 + �⇤f , ⌅ = ⌅(⇥) + �⌅linearization

frame of reference attached to the reaction sheet (�, ⇥, ⇤)� � ⇤ � a(⇥, ⌅)

⇧⇤=

⇧�,

⇧⇥� ⇧

⇧⇥� ⇧a

⇧⇥

⇧�,

⇧⌅� ⇧

⇧⌅� ⇧a

⇧⌅

⇧�

� = 0 : reaction sheet

a(⇥, ⌅) = ae(i⇥�+⌅⇤), �⇤f (⇥, ⌅) = ⇤f ae(i⇥�+⌅⇤)

⌅ � ⇥⇤L, � � kdL �⇤ = ⇤(⇥)ae(i��+��)�⇤ = ⇤(⇥)ae(i��+��)

harmonic analysis(normal modes)

� ?reduced linear growth rate

��

��+

��

��2

��2+

��2

�� =

��a��

� �2a��2

�d�

��

��+

�� 1

��2

��2+

��2

�� =

��a��

� 1Le

�2a��2

�d�

external equations

P.Clavin V Analysis�

� d2

��(�) +

�� + �2

��(�) =

�� + �2

� d�

d��

� 1Le

��(�) +

�� +

��(�) =

�� +

�d�

external equations

⇥ = d⇥/d� ⇥ = d⇥/d�

��

= e� ��

= 1 � eLe��+

= 1 �+

particular solutions

� and �f ?

�(� = 0) = �f

boundary conditions� � ±� : � = 0

temperature in the external zones

� given

�� = d��/d� �

�d�

�/d�

��=0

�es��

s� =Le2

�1 +

1 +4Le

�� +

��1Le

s2 � s ��

� +�2

�= 0

�(� = 0) = 0� � �� : � = 0 mass fraction in the preheated zone zone � > 0 : � = � = 0

r± =12

�1 �

�1 + 4(⇥ + �2)

�⇥± = d⇥

±/d� +

�⇥f � d⇥

±/d�

��=0

�er±�

r2 � r � (� + �2) = 0general solution tothe homogeneous equation

P.Clavin V

d�/d⇥|⇤=0� = e��(1�⇥f )/2

valid up to 1st ordervalid at the leading order

�d�

0�= 0jump conditions � and �f ?

�f (r+ � r�) = (s� � r�) + (1 � Le)2nd condition(r+ � r�) = �

�1 + 4(⇥ + �2)

Le � 1 : s� � r� � 0

��f = 1 +�

1 + 4(� + �2)

� � 1 : Le = 1 + l/�, l � �(Le � 1) = O(1)asymptotic analysis

�(1 � ⇥f ) = O(1) �⇥f = O(1)

dispersion relation (⇥,�)

�1 �

�1 + 4(⇥ + �2) + 2⇥

�1 �

�1 + 4(⇥ + �2)

� �1 + 4(⇥ + �2)

�f � (Le � 1)2

�1�

1 + 4(� + �2)� 1 +

2� + 4�2

1 + 4(� + �2)

�to leading order in small values of (Le-1) = O(1/�)

d⇤�/d⇥|�=0 = �⇤f/2 1 � r� = r+ = �⇥f/2linearized 1stconditionvalid to leading order using �f = O(1/�)

M < 0⇥�

⇥t� �DT

⇥2�

P.Clavin V

�1 �

�1 + 4(⇥ + �2) + 2⇥

�1 �

�1 + 4(⇥ + �2)

� �1 + 4(⇥ + �2)

� = 0 : �(�) = 0

⇤�

�⇥(Le � 1) + 2

⇤2�

⇤�

�⇥(Le � 1) + 2

�⇤2�

⇤y2+

⇤2�

�= [⇥(Le � 1) + 2]

R, 2/R = 1/R1 + 1/R2

M = �(Le � 1) + 2

�f = (Le � 1)�2 ⇥ = �(l + 2)�2/2

Weakly curved limit. Small wavenumber expansion � � kdL � 1 |�| � |�|�L ⇥ 1

l � �(Le � 1) < �2l � �(Le � 1) > �2 stable unstable� > 0� < 0: :

M =�b

�b � 1J +

(�b � 1)J =

(⇤b � 1)⇥(�)1 + (⇤b � 1)�

d�, D = �� 1

(⇤b � 1)⇥(�) ln �

1 + (⇤b � 1)�d�,

Cellular instability for Le � DT /D < 1

⇥ = �(l + 2)�2/2 � 8�4 unstable disturbances

Turing type of instabilityTuringZeldovich

P.Clavin V

Stable

Unstable

Hydrodynamics + di�usion

Propane rich flameM < 0

hydrodynamic + cellular instabilitiesshorter wavelengths are unstable

Oscillatory instability Le � DT /D > 1Im(�) �= 0

l � �(Le � 1) = l� : Re(⇤) = 0, ⇥� �= 0Poincare-Andronov bifurcation l� � 10

l�� 11 : planar pulsation. OK for solid combustion Cellularinstability

Stable

Oscillatinginstability

Adiabatic

Extinction

e�ect of heat lossesJoulin Clavin 1979

Propane lean flameM > 0

hydrodynamic instability only��

��H(�) �⇥� +

12|⇤�|2 = 0Sivashinsky eq. 1977

geometrical term: Huygens constructionnonlinear equation

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