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Princeton-CI Summer SchoolJune 21-25, 2021
1
Lectures onDynamics of Combustion Waves
in Premixed Gases
Professor Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)
Copyright 2019 by Paul ClavinThis material is not to be sold, reproduced or distributed
without permission of the owner, Paul Clavin
Orders of magnitude
2
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
3
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
4
P.Clavin IV
5
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
6
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
7
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�
8
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
9
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
10
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
11
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
12
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
13
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
14
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
15
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
16
Princeton Summer School on CombustionJune 20 - June 26, 2021
1
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)
Copyright 2021 by Paul ClavinThis material is not to be sold, reproduced or distributed
without permission of the owner, Paul Clavin
Aix-en-ProvenceMarseille
2
3
P.Clavin I
Contents
Combustion Waves and Fronts in FlowsP. Clavin and G. Searby
Cambridge University Press 2016
Lecture 1: Orders of magnitude
Lecture 2: Governing equations
Lecture 3: Thermal propagation of flames
Lecture 4: Hydrodynamic instability of flames of flames
Lecture 5: Thermo di�usive phenomena
Lecture 6: Thermal quenching and flammability limits
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
s
Combustion Waves and Fronts in FlowsFlames, Shocks, Detonations, Ablation Fronts and Explosion of Stars
Paul Clavin and Geo! Searby
Orders of magnitude
5
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
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
front
Hot burned gas (equilibrium)
Propagation velocity U
Davy 1830Cold (frozen) Reactive mixture
Euler 1738
6
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
7
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
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)
8
P.Clavin I
Maxwell 1867 Einstein 1905
Back to the kinetic theory of gases
Bolzmann 1877
9
P.Clavin I
Equilibrium state. The Maxwell-Boltzmann distribution<latexit sha1_base64="NdI2GzI4JzKEbAmq52Qzw6mUMKA=">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</latexit><latexit sha1_base64="rQgnknMWqSg9nNZ5IBA/MS5am8I=">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</latexit><latexit sha1_base64="rQgnknMWqSg9nNZ5IBA/MS5am8I=">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</latexit><latexit sha1_base64="GtOVttJCov0p5qlTiYFtLrl5GoE=">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</latexit>
f (eq)(v, n, T ) d3v d3r = nm3/2
(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=">AAADJ3icjVFLaxRBEK4ZH4lJ1FWPXhoXwcOyzHiJ5hTixWOEbBLIhqWntzYO24+hH8HNsj/IHyPeRMGLN/0XVvfOgBpEe2Dmm+9RVFdVjaydL4ovWX7j5q3bG5t3trZ37t6733vw8NiZYAWOhJHGnlbcoaw1jnztJZ42FrmqJJ5U81dRP7lE62qjj/yiwXPFL3Q9qwX3RE16l2pv7PGdXyruHDMzpoxEESS61YCNB+MB03ts7dBBVWjZFLWr/aKTj9Z5tvSoGrTcB4udNp8cdOEDI/2V4loz4fxwNen1i2GRDrsOyhb0oT2HpvcBxjAFAwICKEDQ4AlL4ODoOYMSCmiIO4clcZZQnXSEFWxRNpALycGJndP7gv6WhA15DPlXwOAp6YaclnCszJIeUpXI/r0Op55iHwv6Vm0tRayHt8T+K9c5/zd3RqyHGbxIt61pFk1i4lREWyWkCcTO2S+38lShIS7iKemWsEjJbqYsZVy6e5wjT/r35Ixs/BetN8CP2CUts/xzddfB8fNhWQzLN0V//2W71k14DE/gGe1uF/bhNRzCiGp/zfJsO9vJ3+cf80/557U1z9rMI/jt5N9+Ajjqtsg=</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=">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</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=">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</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=">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</latexit>
⌧coll : collision time ⇡ relaxation time<latexit sha1_base64="dXE8mrX2FUNmasi2m8DqKvyvCb0=">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</latexit><latexit sha1_base64="9jAvEIAwnl2ayf3sbSnB/0KfXTg=">AAAC83icjVHLThRBFD00oIiI42NHTCoSElaTbjaoK6Ibl5g4QMKQSXVTQIXuqU51tRky6S9xZ1iw4UNc6Fbda6JfgadqmgQlRmqS6VPncXPr3rTMdeXi+NtUND0ze+v23J35uwv3Fu93HjzcqkxtM9XLTG7sTiorleuh6jntcrVTWiWLNFfb6fErr2+/U7bSZvjWnZRqr5CHQ32gM+lIDTov+07WgzEL5Y140Xdq5MJF+4RwulCN6MuytGYkJqpVuRyFdJBFM+gsx904HHEdJC1Y3li8+PHxyePvm6bzCX3swyBDjQIKQzjiHBIVf7tIEKMkt4cxOUukg67QYJ7Zmi5FhyR7zP9D3sbEhh5DfwOBFeqGTkvsK4ug16GKZ/9dR7In38cJv2lbqyDrcET2f7lL501zu2QdDvAsvFZzFmVg/FSytkodJuA7F1de5VihJOfxPnVLnIXk5UxFyFTh7X6OMug/g9Oz/p613hq/fJdcZvL36q6DrbVuEneTN9zqc0zOHJbwFKvc3To28Bqb6LH2GT7jC75GdfQ++hCdTqzRVJt5hD9OdP4bb0GrZg==</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 =�
Tb/Tu
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
10
P.Clavin I
Dimensional analysis
[qm] = (velocity)2
[D] = (velocity)2 � time
D � 10�3m2/s
Orders of magnitude
11
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
Energy
Progress variableBin
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Inelasticcollision
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kBTb� 8
Collision in gases
Overall reaction rate. Arrhenius factorI � 3)
12
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
Flame
Unburnt mixture at rest
Burnt gas
Zoom
Tb
TudL
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=">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</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 )
=1
�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
13
P.Clavin I
UL Ub - UL
Flame
Unburnt mixture at rest
Burnt gas
Zoom
Tb
TudL
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)
=1
�colle�E/kBTb
Orders of magnitude
14
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
Methane-air flame
Hydrocarbon/air flamesI � 4)
� =NF /NO2
⇥+F /⇥+
O2
�+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
UL dL
� = 0.65near to the flammability limit
”thicker flame”
15P.Clavin I
Orders of magnitude
16
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
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
18
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
19
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
21
P.Clavin I
Gravity
Propane flame propagating upwards Slow downwards propagating flame
1
Structure and Dynamics of
Combustion Waves in Premixed Gases
Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)
Lecture 1IGoverning equations
Princeton Summer School on CombustionJune 20 - June 26, 2021
Copyright 2021 by Paul ClavinThis material is not to be sold, reproduced or distributed
without permission of the owner, Paul Clavin
2
P.Clavin I
Contents
Combustion Waves and Fronts in FlowsP. Clavin and G. Searby
Cambridge University Press 2016
Lecture 1: Orders of magnitude
Lecture 2: Governing equations
Lecture 3: Thermal propagation of flames
Lecture 4: Hydrodynamic instability of flames of flames
Lecture 5: Thermo di�usive phenomena
Lecture 6: Thermal quenching and flammability limits
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
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”
3
!internal structure of shock waves
P.Clavin II
4
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
(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
V
⇤(�a)/⇤t = ��.Ja + ⇥a
⇥(�a)/⇥t = ��.Jaconserved scalar
conserved quantities(mass, momentum and energy)
: no production terms �a = 0
a(r, t)
V
�
dA/dt =���
[⇥(�a)/⇥t] d3r = (dA/dt)1 + (dA/dt)2
(dA/dt)1 = ���
n.Jad2�
conservation equation V fixed
�
V
n
Ja
(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
V
P.Clavin II
6
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
(simplified form, see de Groot et Mazur (1962) or Williams (1985) for more details)
⇥�/⇥t = ��.(�u)J � �u
Conservation of mass: continuity equationII � 2)
v � 1/�1v
DvDt
= �.u
7
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
DDt
� �
�t+ u.�material (convective) derivative
mass is a conserved scalar
��/�t = ��.J
1�
D�
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/��
i
Yi = 1
inert mixture mass fraction of species is a conserved scalar�
i
J�i = 0�DYi/Dt = ��.J�
i
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)
8
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) =�
k
Yk(t)eik.r
dYk(t)/dt = �(Dk2)Yk(t) Yk(t) = Yk(0)e�Dk2t
k = |k|
Fourier anaysis
Fourier 1824
9
P.Clavin II
P.Clavin II
10
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
(simplified form, see de Groot et Mazur (1962) or Williams (1985) for more details)
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)
11
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
12
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
(simplified form, see de Groot et Mazur (1962) or Williams (1985) for more details)
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
12�
DDt
|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
�(j)i miW
(j),
conservation equation for the species
echem ��
i
hiYi
equation for the chemical energy
enthalpy of formation per unit of mass of species i
Q(j) =n�
i=1
(�(j)+i � �(j)�
i )mihi(To),
sum over the species
heat of the j th reaction
�Dechem/Dt = ��
i
�.(hiJ�i) �
�
j
Q(j)W (j)
Reactive flows
�DYi
Dt= �.(�Di�Yi) +
�
j
�(j)i miW
(j)(T, p, ..Yk..)
sum over the reactions
J0i = �⇢DirYi
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⇥cvDT/Dt = �.(��T ) � p�.u +�
j
Q(j)W (j)
p = (cp � cv)�T.
�p�.u =p
�
DDt
� =DDt
p � �DDt
[(cp � cv)T ]
ideal gas law
continuity �
15
P.Clavin II
�eT = cv�T cv � cst. (for simplicity, can be easily removed)
⇥cpDT/Dt = Dp/Dt + �.(��T ) +�
j
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 ��
i
hiYi �Dechem/Dt = ��
i
hi�.J�i �
�
j
Q(j)W j
� D(eT + echem)/Dt = ��.J�q � p�.u
Jq � J�q �
�
i
hiJ�i Jq = ���Theat flux
Thermal balance of inviscid flow of reactive gas
Governing equations for inviscid flows of reactive gas1�
D�
Dt= ��.u, �
DuDt
= ��p, p = (cp � cv)�T,
⇥cpDT
Dt=
Dp
Dt+ �.(��T ) +
�
j
Q(j)W (j),
�DYi
Dt= �.(�Di�Yi) +
�
j
⇥(j)i miW
(j).
stoichiometric coe�cient
progress variable
�qmD⇥/Dt ��
j
Q(j)W (j), ⇥ � [0, 1]
heat released per unit mass
16
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 +
�
i
hiJ�i = J�
etot� pu
di�usive flux of total energyconvective flux of enthalpy
�(�etot)/�t = ��.[�uetot + up + Jq +�
i
hiJ�i] = ��.[�u(etot + p/�) + Jq +
�
i
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
17
P.Clavin II
⇥�
⇥t= �⇥(�u)
⇥x⇥(�u)
⇥t= � ⇥
⇥x
�p + �u2 � µ
⇥u
⇥x
�
⌅(⇥etot)⌅t
= � ⌅
⌅x
�⇥u(cpT + u2/2 � qm⇤) � �
⌅T
⌅x� µu
⌅u
⌅x
�
µ � 4�/3 + ⇥
one-dimensional compressible flow
(simplest form of heat flux)
P.Clavin II
18
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
(simplified form, see de Groot et Mazur (1962) or Williams (1985) for more details)
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
TDs
Dt=
DeT
Dt+ p
D(1/�)Dt
��
i
µiDYi
Dt
ws = J�q.�
�1T
��
�
i
J�i.�
�µi
T
�� 1
T� : (�u)(s)
s(�, T, ..Yi, ..)
(s � so)cv
= ln�
p/��
po/��o
�ideal gas
T �s = �eT + p�v ��
i
µi�Yi
19
P.Clavin II
Js = ⇥us � �
T
⇤T
⇤x, ws =
µ
T
�⇤u
⇤x
�2
+�
T 2
�⇤T
⇤x
�2
µ > 0, � > 0
inert mixture in one-dimensional geometry
Js
⇥TDs
Dt=
⇤
⇤x
��
⇤
⇤xT
�+ µ
�⇤u
⇤x
�2�Ds
Dt=
�
�x
��
T
�T
�x
�+ �s
1
Lecture I1IThermal propagation of flames
Structure and Dynamics of
Combustion Waves in Premixed Gases
Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)
Copyright 2021 by Paul ClavinThis material is not to be sold, reproduced or distributed
without permission of the owner, Paul Clavin
Princeton Summer School on CombustionJune 20 - June 26, 2021
2
P.Clavin I
Contents
Combustion Waves and Fronts in FlowsP. Clavin and G. Searby
Cambridge University Press 2016
Lecture 1: Orders of magnitude
Lecture 2: Governing equations
Lecture 3: Thermal propagation of flames
Lecture 4: Hydrodynamic instability of flames of flames
Lecture 5: Thermo di�usive phenomena
Lecture 6: Thermal quenching and flammability limits
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
Quasi-isobaric approximation. LowMach number
�(u.�)u � ��p � �p � ⇥u�u
p � �a2 � �p/p � u2/a2 � M2
III � 1)
3
P.Clavin III
⇥cpDT/Dt = Dp/Dt + �.(��T ) +�
j
Q(j)W (j)
(in open space)
slow evolution �/�t � u.⇤ ⇥ a |⇤|
�T = �oTo
⇥cpDT/Dt = �.(��T ) +�
j
Q(j)W (j)(T, ..Yk..)
�DYi/Dt = �.(�Di�Yi) +�
j
�(j)i miW
(j)(T, ..Yk..),
very subsonic flow+ M2 � 1 � �p/p � �T/T = O(1)
����1p
Dp
Dt
���� �����1T
DT
Dt
����
p
�cpT= O(1)
�����Dp
Dt
���� ������cp
DT
Dt
����
� �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=">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</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=">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</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>
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◆UL
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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=">AAACxHicjVHBShxBFCxHTYwmcaPHXBoXIaelW4K73oRA8Gggq4KK9IytDvZuDzM9SZZl/RWv8WP8AP9A/yLVnVnQg2gPzNRU1Stev5cWNq+8lHczyezc/Ju3C+8Wl95/+Ljc+rSyV7m6zEw/c9aVB6mujM2Hpu9zb81BURo9SK3ZTy+/BX3/lymr3A1/+lFhjgf6fJif5Zn2pE5aq0fe/PHj39paob0oTeUnJ6227EgplVIiANXdlARbW70N1RMqSDxtNGfXtW5xhFM4ZKgxgMEQnthCo+JzCAWJgtwxxuRKojzqBhMssramy9ChyV7yfc6/MbGjx9E/gcA6dUdnSRySRdTrmBLY53M0ewp9jPhNm6wBWY8Lsi/VTZ2vrTsk63GGXrxtzlkUkQlTyZqUOk4gdC4e3cozoSAX8Cn1kjiLldOZilhTxbuHOeqo30dnYMN/1nhrPIQuuczpxsTzYG+jo2RH/ZDt7a/NWhfwGWv4wt11sY0d7KLP7BGu8Rc3yffEJlVS/7cmM03NKp6c5OofjhiTIw==</latexit>
reference frame of the lab<latexit sha1_base64="vBlEi2HhrA/Ao6ZIqjMnpDKvg4M=">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</latexit><latexit sha1_base64="vBlEi2HhrA/Ao6ZIqjMnpDKvg4M=">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</latexit><latexit sha1_base64="vBlEi2HhrA/Ao6ZIqjMnpDKvg4M=">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</latexit><latexit sha1_base64="vBlEi2HhrA/Ao6ZIqjMnpDKvg4M=">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</latexit>
burned gas<latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit><latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit><latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit><latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit>
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u = Ub � UL<latexit sha1_base64="mjgKSRpk8iAFnSJLbPUVTzG8ZQo=">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</latexit><latexit sha1_base64="mjgKSRpk8iAFnSJLbPUVTzG8ZQo=">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</latexit><latexit sha1_base64="mjgKSRpk8iAFnSJLbPUVTzG8ZQo=">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</latexit><latexit sha1_base64="mjgKSRpk8iAFnSJLbPUVTzG8ZQo=">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</latexit>
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◆UL
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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=">AAACxHicjVHBShxBFCzHaIxGs9FjLk0WIaelW8RdTwoBydFAVgUV6RlbHezdHmZ6Epdl8x/xkmvyCX6EH5A/0L9IdbsLyUH0DczUVNUrXr9OC5tXXso/U8n0i5nZl3Ov5hdeLy69abxd3qtcXWammznryoNUV8bmfdP1ubfmoCiN7qXW7KeXH4O+/9WUVe76X/ygMMc9fd7Pz/JMe1InjZUjb6788Ju2VmgvSlP50UmjKVtSSqWUCEC1NyTB5mZnTXWEChKruXXzI9T1rmvc4gincMhQoweDPjyxhUbF5xAKEgW5YwzJlUR51A1GmGdvTZehQ5O95Pucf0NiR4+jfwSBVeqOzpI4JIuo1zElsI/naM4U5hjwm46zemQ9Lsg+1TdxPrfvkKzHGTrxtDl3UUQmbCUbp9RxA2Fy8c+pPBMKcgGfUi+Js9g52amIPVU8e9ijjvpddAY2/Gdjb437MCUvc3Jj4nGwt9ZSsqU+y+b2Oh5qDu/wHh94d21s4xN20WX2AD/xC7+TncQmVVI/WJOpcc8K/qvk+18/55ey</latexit><latexit sha1_base64="jzdn0NR923xSlkmq23je1RMgkYk=">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</latexit>
burned gas<latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit><latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit><latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">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</latexit><latexit sha1_base64="+lZ/eE+qvR54iSXxJUg4COdJRQI=">AAACwnicjVHLSsNAFD2Nr1pfqS7dBIvgqiQi6LKgC5cV7APaUibptMammZhM1FL7KW71a/wA/0D/wjtjCmoRnUDmzLnnHu7DjQI/kbb9mjMWFpeWV/KrhbX1jc0ts7hdT0Qae7zmiUDETZclPPBDXpO+DHgzijkbuQFvuMNTFW/c8jjxRXgpxxHvjNgg9Pu+xyRRXbPYlvxeTtw0DnnPGrBk2jVLdtnWx5oHTgZKyE5VmC9oowcBDylG4AghCQdgSOhrwYGNiLgOJsTFhHwd55iiQLkpqTgpGLFD+g/oNSEsSCNIP4WFfYoLUsaElbOl46l2UezvPoxqUnWM6XYzrxGxElfE/pU3U/43r0WsRB8nulufZhFpRk3Fy1xSPQFVufWlK0kOEXEK9ygeE/Z05mymls5JdO9qjkzH37RSsertZdoU76pKWqbzc3XzoH5Yduyyc3FUqpSzteaxiz0c0O6OUcE5qqiR9x0e8YRn48y4Nm6M5FNq5LKcHXw7xsMHPAKSLw==</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>Ub = (Tb/Tu)UL
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⇢u⇢b
=Tb
Tu=
Ub
Uu> 1
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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=">AAAC4nicjVHPShtBHP5ctbVRa6rHXoYGwVPY9VIVCoFePCo0RjAis5NJsri7s8zMSkPIC/RWevDSB+hLeBMLgg/QN2iPfYP+ZrKCGko7CzvffL/v9/H7ExdpYmwY/pgL5hcWnz1felFbXll9uVZ/tX5kVKmFbAuVKn0ccyPTJJdtm9hUHhda8ixOZSc+f+/inQupTaLyD3ZUyNOMD/KknwhuiTqrR10rP9qxULmR+sKTTPVZxo3ZY5OuHirWZu+8iI2Fsc3JWb0RNkN/2CyIKtBobfy+3r7tfDtQ9Tt00YOCQIkMEjks4RQchr4TRAhREHeKMXGaUOLjEhPUKLcklSQFJ/ac/gN6jQkr0ijST8CwSXFFSk3YOTMfL72LY//uw6kmV8eI7rjyyoi1GBL7r7x75f/mnRBr0ceO7zahWRSecVMRlUvpJ+AqZw+6suRQEOdwj+KasPCZ9zNlPsf43t0cuY//9ErHureotCV+uSppmdHT1c2Co+1mFDajQ9rqLqZnCa/xBlu0u7doYR8HaJP3Ja5wg+9BL/gUfA6+TKXBXJWzgUcn+PoHn2ijBA==</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=">AAAC3HicjVHBThRBEH0MCIiKqxy9dCAmJiabGS/gbRMuHjFhgQSQ9MwW0GF2ejLTY9xs9ubNcODCr3CFxKsf4B/o0T/gdTMkKDFak/S8fvXqpboqLXNTuzj+PhVNzzyYnZt/uPDo8ZPFp51nz7dq21SZ9DOb22on1bXkppC+My6XnbISPUxz2U5P1n1++6NUtbHFphuVsj/UR4U5NJl2pA46r/ecfHJjW4iSYqBMrbLc1jJQmjd3LMryqJQtpZgcdFbibhxC3QdJC1Z6S7++9gBs2M437GEAiwwNhhAUcMQ5NGp+u0gQoyS3jzG5isiEvGCCBdY2VAkVmuwJzyPexsSWGkv9BAovmbdUVsTeWYV8E1w8+3cfzZ58HyP+09ZrSNbhmOy/6m6V/1u3S9bhEGvhtYazKAPjp5K1Lk2YgO9c3XmVo0NJzuMB8xVxFipvZ6pCTR3e7ueoQ/5HUHrW37NW2+Cn75LLTP5c3X2w9aabxN3kPbf6FjcxjxdYxivubhU9vMMG+vQ+wwUucRV9iD5HX6LTG2k01dYs4beIzq8BAAGeng==</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>
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equations m?
5
P.Clavin III
m � �uUL = �bUb, Ub/UL � Tb/Tu, � 4 � 8
D/Dt = md/dx
Flame
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>
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
6
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
mdY
dx� d
dx
��D
dY
dx
�= ��W
mcpdT
dx� d
dx
��
dT
dx
�= ⇥qRW qR = energy released per unit of mass of R
m � �uUL unknown
Velocity and structure of the planar flame
7
Arrhenius law
�W = �bY
⇥r(T )1
�r(T )� e�E/kBT
�coll
1�rb
� e�E/kBTb
�coll
1�r(T )
=1
�rbe�
TbT �(1�⇥) � � E
kBTb
�1 � Tu
Tb
�� � T � Tu
Tb � Tu� [0, 1]
P.Clavin III
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
8
Unity Lewis number and large activation energyIII � 3)
9
md�
dx� ⇥DT
d2�
dx2= ⇥
W
Y1u, m
d⇤
dx� ⇥DT
Led2⇤
dx2= �⇥
W
Y1u,
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
d2�
dx2= ⇥
W
Y1u,
x = �� : � = 0, x = +� : � = 1,
⇥W
Yu= ⇥b
(1 � �)⇤rb
e�TbT �(1�⇥)
� � 1
P.Clavin III
(a) (b)
0 1
⇥W
Yu= ⇥b
w�(�)⇤rb
w�(�) � (1 � �)e��(1��)
(reaction rate is non negligible only when T � Tb)
� � E
kBTb
�1 � Tu
Tb
�
�W
Yu= �b
⇤
⇥rbe�
TbT �(1�⇥)
1�rb
� e�E/kBTb
�coll
10
w�(�) � (1 � �)e��(1��)
Zeldovich, Frank-Kamenetskii asymptotic analysisIII � 4)
md�
dx� ⇥DT
d2�
dx2= ⇥
W
Y1u,
P.Clavin III
origin x = 0 : location of the reaction zone � = 1
W � 0preheated zone
matching condition
m?
1�rb
� e�E/kBTb
�coll
(a) (b)
0 1
Zeldovich 1938
� � � � � E
kBTb
�1 � Tu
Tb
�
11
(a) (b)
0 1
⌅bDTbd2⇤
dx2� ⌅bDTb
⇥⇤
d2r
� ⌅bDTb
d2r
1�
⌅b
⇧rbw�(⇤) � ⌅b
⇧rb⇥⇤ � ⌅b
⇧rb
1�
md�/dx � ⇥DT d2�/dx2 =⇥b
⇤rb(1 � �)e��(1�⇥)
dimensional analysisUL �
�DTb/�rb
�DTb
2ddx
�d�
dx
�2
� 1⇥rb
(1 � �)e��(1�⇥) d�
dx
� = �(1 � �) ⇤rbDTb
2
�d⇥
dx
�2
�� 1
⇥(1 � ⇥)e��(1�⇥)d⇥ =
1�2
� �(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
�m
d�
dx� m
��
dr
m = �DT /dL�⇥⇤ = O(1/�)
� �bDTb
drdL
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=">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</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
kBTb
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Tb
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Ub =
s2
�2
DTb
⌧rb<latexit sha1_base64="q/rXHIO9hz1eOlLOEoeT7JhYUSg=">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</latexit><latexit sha1_base64="LCED17Fn7VOO0PzYZ4bKhDaO5Y4=">AAAC5XicjVHdStxAGD3G/vjTn9Ve9iZUCt50SRahLSgK9aK9U3BVcDVM4uw2mN1JJxNhCXkE76QX3vRVeqvgA/QN6lv0zGyEtlLqBDJnzne+w/cT51lamCD4MeVNP3j46PHM7Nz8k6fPnrcWFncLVepEdhOVKb0fi0Jm6Uh2TWoyuZ9rKYZxJvfikw82vncqdZGq0Y4Z5/JwKAajtJ8mwpCKWivdKF7rFV+0qXp9LZKqU1e9WBpx1KknxGZU7cQ1WSPKqNKEddRaCtqBO/5dEDZgaWPtU77+ZnW8pVrX6OEYCglKDCExgiHOIFDwO0CIADm5Q1TkNFHq4hI15phbUiWpEGRP+B/wVRErahT1NXy8ZlxRqYmts+/ipXOx7L99BGuydYx5x43XkKzBZ7L/y7tV3jfvgKxBH+9ctylnkTvGTiVpXEo3AVu5/1tXhg45OYuPGdfEicu8nanvcgrXu52jcPGfTmlZ+04abYkbWyWXGf69urtgt9MOg3a4za2+x+TM4CVeYZm7e4sNfMQWuvS+wHdc4sobeGfeufd1IvWmmpwX+ON4334BxzqkFA==</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=">AAACz3icjVHNTttAGJy4hUJKIbQnVKmyiCpxIbK5AJcqUjn0CBJJkEhqrZ0NtXCylr0ujaKgHrjwKlzhAfoCfYC+QfsWnd04UluE6Fryzs433+j7CdMkzrXn/ag4T54uLD5bWq4+X3mxulZbf9nOVZFFshWpRGUnochlEo9kS8c6kSdpJsUwTGQnPH9v4p3PMstjNTrW41T2huJsFA/iSGhSQe3NQTA5DqddkaaZ+uKKIPy409WiCCb0TqZBre41PHvc+8AvQb25sfdt4d12eKhq39FFHwoRCgwhMYImTiCQ8zuFDw8puR4m5DKi2MYlpqgyt6BKUiHInvN/xteEWFGjqJ/CxVvGFZUZsXF2bbywLoZ92EewJlPHmHdYeg3Janwi+1jeXPm/eadkNQbYs93GnEVqGTOVqHQp7ARM5e4fXWk6pOQM7jOeEUc2cz5T1+bktnczR2HjP63SsOYdldoCv0yVXKb/7+rug/ZOw/ca/hG3uo/ZWcJrbGKLu9tFEx9wiBa9r3CDW9w5R86Fc+l8nUmdSpnzCn8d5/o3KmyZpw==</latexit><latexit sha1_base64="lAlviWC6rfS074XuPjcp584oVFA=">AAACz3icjVHLSsNAFD2Nr1pfVZeCBIvgqiRu1F1BFy5b6AvaGibpVINpJyQTtZSKCzf+ilv9DT/AP9C/8M6YglpEJ5A5c+65h/tww8CPpWW9ZoyZ2bn5hexibml5ZXUtv75Rj0USebzmiUBETZfFPPAHvCZ9GfBmGHHWdwPecC+PVbxxxaPYF4OqHIa802fnA7/ne0wS5eS3T5xR1R23WRhG4sZkjnu235YscUbkHYydfMEqWvqY08BOQQHpKYv8C9roQsBDgj44BpCEAzDE9LVgw0JIXAcj4iJCvo5zjJGj3IRUnBSM2Ev6n9NrRFiQRpB+DBO7FBekjAgrZ1PHE+2i2N99GNWk6hjS7aZefWIlLoj9K2+i/G9ei1iJHg51tz7NItSMmoqXuiR6Aqpy80tXkhxC4hTuUjwi7OnMyUxNnRPr3tUcmY6/aaVi1dtLtQneVZW0TPvn6qZBfb9oW0W7YhVKR+las9jCDvZodwco4RRl1Mj7Ho94wrNRMa6NW+PuU2pk0pxNfDvGwwefBJeE</latexit>
Ub
ab=
1
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p2 e�E/kBTb
<latexit sha1_base64="TAZnNQw1bBch1eRaeazbQjDjF+o=">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</latexit><latexit sha1_base64="yhw/s4ST4vFMQ/KFik5rQ13dIx0=">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</latexit><latexit sha1_base64="yhw/s4ST4vFMQ/KFik5rQ13dIx0=">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</latexit><latexit sha1_base64="omtkJsKqqw5cmsI31Ac/ac0u3bs=">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</latexit>
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kBTb� 1 :
<latexit sha1_base64="pBFSX7xKKWWROY222FJsa8XMt48=">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</latexit><latexit sha1_base64="ui4H6IpzMQeggxLz90KqGp6OaHE=">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</latexit><latexit sha1_base64="ui4H6IpzMQeggxLz90KqGp6OaHE=">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</latexit><latexit sha1_base64="NaQpmQXl8zeT/iejGzRrYalbXFU=">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</latexit>
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=">AAAC5nicjVHNahRBGKwdfxJj1FUPHrw0BiGXLDOBYE4hEBDBS4RsEkxC6JntTZrMTg/dPcGw7CvkJh68+CpeFR/AN9DLPkOqOxNQg2gPzNTUV1V8/X15XWrn0/R7J7lx89btmdk7c3fn791/0H34aNuZxhaqX5jS2N1cOlXqSvW99qXara2So7xUO/nJRqjvnCrrtKm2/FmtDkbyqNJDXUhP6rC7su/VOz9++/K1sMo1pRdDY4UUplJLzqtaDLV1fsnYgbJUyCLYJofdhbSXxiOug6wFC+tPplMBYNN0v2EfAxgUaDCCQgVPXELC8dlDhhQ1uQOMyVkiHesKE8zR21ClqJBkT/g+4t+Y2FBjqJ9A4DnrhkpLHJJFrDcxJbB/z5HsKfRxxm/eZo3IehyT/ZfvSvm/vj2yHkOsxttqzqKOTJhK0aY0cQKhc/HLrTwTanIBD1i3xEV0Xs1URI+Ldw9zlLH+IyoDG/6LVtvgZ+iSy8z+XN11sL3cy9Je9oZbXcPlmcVTPMMid/cC63iFTfSZ/RGf8QVfk+PkPHmffLiUJp3W8xi/neTTBUZ8ov4=</latexit><latexit sha1_base64="zE9cTWY67I+qMKJpgUuYXUjX0pw=">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</latexit><latexit sha1_base64="zE9cTWY67I+qMKJpgUuYXUjX0pw=">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</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=">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</latexit><latexit sha1_base64="uDR+IcKkicyz1cFrSMZOiDWuDOc=">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</latexit><latexit sha1_base64="mkVFzRQ/j7RJ5rnZhmo2KU299Ig=">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</latexit>
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
13
14
non-dimensional form
III � 5) Reaction diffusion wavesP.Clavin III
��
�t� �2�
�x2= w(�) � � 0 � � [0, 1]
t = 0
�
1
0
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)
15
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
Con
cent
ratio
n of
sca
lar
Upstreamunstable state
Downstreamequilibrium state
ZFK flame model: � > 0, case (II)
� � [0, 1]
Y = µdX
d�k± = µr±
�dY
dX
�
±= k±
16
P.Clavin III
Con
cent
ratio
n of
sca
lar
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
Non
dege
nera
tePart
icular
trajec
tory
Degenerate
Infinite numbers of solutionsOne particular solution
r+ > 0, r� < 0r+ > r� > 0� � �� � � �
��� < 0��
� > 0
case (II)A� = 0
17
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⇥�
ot2
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
Non
dege
nera
tePart
icular
trajec
tory
Degenerate
t = 0
�
1
0
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 �(�)
1
Structure and Dynamics of
Combustion Waves in Premixed Gases
Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)
Lecture IVHydrodynamic instability of flames
Princeton Summer School on CombustionJune 20 - June 26, 2021
Copyright 2021 by Paul ClavinThis material is not to be sold, reproduced or distributed
without permission of the owner, Paul Clavin
2
P.Clavin I
Contents
Combustion Waves and Fronts in FlowsP. Clavin and G. Searby
Cambridge University Press 2016
Lecture 1: Orders of magnitude
Lecture 2: Governing equations
Lecture 3: Thermal propagation of flames
Lecture 4: Hydrodynamic instability of flames of flames
Lecture 5: Thermo di�usive phenomena
Lecture 6: Thermal quenching and flammability limits
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
P.Clavin IV
3
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
4
Jump across an hydrodynamic discontinuityIV � 1)
UL Ub - UL
Flame
Unburnt mixture at rest
Burnt gas
Zoom
Tb
TudL
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
x
uu w
z
�u
�b
2 dim
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
5
�u �b
UL Ub
�u > �b
Flame front
Burnt gasat rest
Fresh mixture
UL Ub - UL
Flame
Unburnt mixture at rest
Burnt gas
Zoom
Tb
TudL
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
Ub
UL=
�u
�b=
Tb
Tu
u�|B > UL
u�|A < UL
flame motion
flame motion
instability mechanism� � dL, dL/� � 0
� � UL/�
dL
”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
nf =
�
� 1�1 + ��2
y
, ���
y�1 + ��2
y
�
� , un � uf .nf =�uf � ��
ywf
�/�
1 + ��2y
wtg =�wf + ��
yuf
�/�
1 + ��2y
�t � ⇥�/⇥t Df =�t�
1 + ��2y
normal velocity of the front
equation of the perturbed front
��U�n = �+U+
n
⇥��u�
f � �t � ��yw�
f
�= ⇥+
�u+
f � �t � ��yw+
f
�
conservation of mass
[Wtg]+� = 0
�p + �U2
n
�+� = 0
p�f + ⇥�
�u�
f � �t � ��yw�
f
�2
1 + ��2y
= p+f + ⇥+
�u+
f � �t � ��yw+
f
�2
1 + ��2y
�w�
f + ��yu�
f
�=
�w+
f + ��yu+
f
�
conservation of momentum
uf = (uf , wf )flow velocity at the frontP.Clavin IV
6
Wtg = wtgUn ⇥ un �Df =�uf � �t � ��
ywf
�/�
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
7
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
Linearised Euler equations of an incompressible fluid IV � 2)
a = a + �amf = ��u�
f = �+u+f
�⇤2
⇤x2+
⇤2
⇤y2+
⇤2
⇤z2
��⇥± = 0
⇥
⇥x�u± +
⇥
⇥y�w± = 0,
�⇤±
⌅
⌅t+ mf
⌅
⌅x
��u± = � ⌅
⌅x�⇥±
�⇤±
⌅
⌅t+ mf
⌅
⌅x
��w± = � ⌅
⌅y�⇥±,
�± � p± � ⇥±g(t)x,
x � �� : no disturbances, �u� = 0x � +� : disturbances remain finite,
8
⇥±�
⇤
⇤t+ u± ⇤
⇤x
�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±
R
�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
9
P.Clavin IV
�u±R
�t+ u± �u±
R
�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) = �⇥��
ddt
+ 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
�u+
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
10
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
�
u�f = d�/dt �
Equation for the front
(�� + �+)d2�
dt2+ 2mfk
d�
dt�
�(�� � �+)g(t)k + (u+ � u�)mfk2
�� = 0
elimination of u+p
11
�du+
r
dt= �
du+p
dt+
d2�
dt2
x < 0 :
���
��
u�(x, t) = u�f (t)e+kx,
k��(x, t) = �⇥��
ddt
+ 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
�u+
p (t)e�kx,x > 0 :
� mf
�1
u+
ddt
� k
�u+
p + mf
�1
u�ddt
+ k
�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) +
1u+
du+r (t)dt
+ ku�f (t) = mf
�1
⇥+ � 1⇥�
�k2�(t)�
Tangential momentum⇥
⇥y
�w�
f + ��yu�
f
�=
⇥
⇥y
�w+
f + ��yu+
f
�� �
kd�
dt
d�
dt
notationaf (t) � a(x = 0, t)
a(x, t)
�
�1
u+
ddt
� k
�u+
p = �mf
�1
⇥+ � 1⇥�
�k2� +
1u+
d2�
dt2+ ku�
f
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
�= ⇥+
�u+
f � �t � ��yw+
f
�= 0
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P.Clavin IV
12
(⇥� + ⇥+)d2�
dt2+ 2mfk
d�
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)
� =�
Atgk
�(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
13
Stabilisation at small wavelength, � � dL
⇥�
⇥t= BDT
⇥2�
⇥y2. �di� ⇥ 1/⇥di� = �BDT k2 = �BULk(dLk)
(⇥� + ⇥+)d2�
dt2+ 2mfk
d�
dt� (u+ � u�)mfk2� = 0
mf = ��u� = �+u+g = 0
�
ULk=
11 + ⇥�1
b
��1 ±
�1 + ⇥b � ⇥�1
b
�� = 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
14
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
⇥u
�d2�
dt2+ 2ULk
d�
dt=
�⇥u
⇥b� 1
�k
�� ⇥b
⇥u|g| + U2
Lk
�1 � k
km
���
1km
� 2BdL
equation for the flame front
curvature e�ect(⇥u+⇥b)
d2�
dt2+2mfk
d�
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 � �
�m
��= 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 � �
�m
��= 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 � �
�m
��= 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 =
kcdL
2, kc =
km
2, ULc =
�
2�b
�u
|g|kc
OK with experiments by Boyer Quinard and Searby (1982)
15
Flames propagating upwards: bubble flames
(1 + ��1b )s2 + 2�s � (�b � 1)�
��Go + �
�1 � �
�m
��= 0s = ��L
g = 0, G0 = 0
G0 = G0cG0 > G0c
1
Structure and Dynamics of
Combustion Waves in Premixed Gases
Paul Clavin Aix-Marseille UniversitéECM & CNRS (IRPHE)
Lecture VThermo-diffusive phenomena
Princeton Summer School on CombustionJune 20 - June 26, 2021
Copyright 2021 by Paul ClavinThis material is not to be sold, reproduced or distributed
without permission of the owner, Paul Clavin
2
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
3
P.Clavin V
Two mechanisms modify the inner flame structureV-1) Flame stretch and Markstein numbers
�u �b
TbTu
dL
tran
sver
seve
loci
ty
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
M
(U�n � UL)/UL = �M(�L/�s)
1/�s
U�n = u�
n �Df , u�n ⇥ nf .u�
f
U�n
transverse di�usion
Unburnt gas Burnt gas
θ = 0
θ = 1
ψ = 0
ψ = 1
ξ = 0ξ = -√Le
θ, ψ
ξ
A
B
P.Clavin V
4
Stretch rate, strain and curvature of a flame
d�⇥/dt = nf . [ue(rf + �⇥ nf ) � ue(rf )]
1⇥s
⇥ 1�2s
ddt
�2s = ⇤.ue|f � nf .⇤ue|f .nf .
�coordinate normal to the front1
�3r
ddt
�3r =1
�2s
ddt
�2s +1�⇥
ddt
�⇥
1�3r
ddt
�3r = �.ue|felement of volume �3r = �2s �⇥
continuity
1⇥s
=1
�2s
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��
ddt
�� = nf .�ue.nf
5
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
Flame
Str
ea
ml in
e
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 +
l
2D
(�b � 1)lean hydrocarbon air mixtures M ⇥ 1 � 4
J =� 1
0
(⇤b � 1)⇥(�)1 + (⇤b � 1)�
d�, D = �� 1
0
(⇤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
6
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
R > 0
R < 0
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
dL
tran
sver
seve
loci
ty
U�n �= UL U+
n �= Ub
�bU�n �= �bU
+n
M+fc �= M�
fc M+sr �= M�
sr
numerical and experimental data
U�n � UL
UL= �2(M�
fc �M�sr)
dL
Rf
U+n � Ub
Ub= �2M+
fc
dL
Rf
Df = 0, U�n = u�
nu+
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
dL
R+ M+
sr�Lnf .⇥u+|f .nfUb
UL=
Tb
Tu
Flame
Unburntmixture
Burntgas
P.Clavin V V-2) Thermo-diffusive instabilities of planar flames
instability mechanism �= hydrodynamic instability
� � E
kBTb
�1 � Tu
Tb
�� � 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+
1Le
d2�
d�2= 0
reaction layer
7
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
2⌅e��(1��) µ
d⇤
d⇥� 1
Led2⇤
d⇥2= ��2
2⇤e��(1��)
?
Unburnt gas Burnt gas
µ =�
Le
matching(first order reaction rate)
�T = �oTo⇥cpDT/Dt = �.(��T ) +
�
j
Q(j)W (j)(T, ..Yk..)equations + fluid mechanics
�DYi/Dt = �.(�Di�Yi) +�
j
�(j)i miW
(j)(T, ..Yk..),
d�/d⇥|⇤=0� = e��(1�⇥f )/2
valid up to 1st ordervalid at the leading order
�d�
d�+
1Le
d�
d�
�0+
0�= 0
jump conditions across the reaction layer
Joulin Clavin 1979
P.Clavin V
8
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+
1Le
d2�
d�2= 0 � = non-dimensional normal coordinate � � x
dL(Le = 1)
P.Clavin V Linear equations
9
�dd�
� d2
d�2
��(�) +
�� + �2
��(�) =
�� + �2
� d�
d��
dd�
� 1Le
d2
d�2
��(�) +
�� +
�2
Le
��(�) =
�� +
�2
Le
�d�
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
��2
���� =
��a��
� �2a��2
�d�
d�
��
��+
�
��� 1
Le
��2
��2+
�2
��2
���� =
��a��
� 1Le
�2a��2
�d�
d�
external equations
10
P.Clavin V Analysis�
dd�
� d2
d�2
��(�) +
�� + �2
��(�) =
�� + �2
� d�
d��
dd�
� 1Le
d2
d�2
��(�) +
�� +
�2
Le
��(�) =
�� +
�2
Le
�d�
d�
external equations
⇥ = d⇥/d� ⇥ = d⇥/d�
��
= e� ��
= 1 � eLe��+
= 1 �+
= 0
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
�� +
�2
Le
��1Le
s2 � s ��
� +�2
Le
�= 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
11
d�/d⇥|⇤=0� = e��(1�⇥f )/2
valid up to 1st ordervalid at the leading order
�d�
d�+
1Le
d�
d�
�0+
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 (⇥,�)
� l
2
�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�
⇥y2
P.Clavin V
12
� l
2
�1 �
�1 + 4(⇥ + �2) + 2⇥
�=
�1 �
�1 + 4(⇥ + �2)
� �1 + 4(⇥ + �2)
�
� = 0 : �(�) = 0
⇤�
⇤t=
�⇥(Le � 1) + 2
2
�DT
⇤2�
⇤y2
⇤�
⇤t=
�⇥(Le � 1) + 2
2
�DT
�⇤2�
⇤y2+
⇤2�
⇤z2
�= [⇥(Le � 1) + 2]
DT
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 +
l
2D
(�b � 1)J =
� 1
0
(⇤b � 1)⇥(�)1 + (⇤b � 1)�
d�, D = �� 1
0
(⇤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
13
P.Clavin V
Stable
Unstable
M < 0
M > 0
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