Advances in the analytical study of the dynamics of gaseous detonation waves

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Advances in the analytical study of the dynamics of

gaseous detonation waves

Paul Clavin

To cite this version:

Paul Clavin. Advances in the analytical study of the dynamics of gaseous detonation waves. C. R.

Mecanique, 2019, 347, pp.273 - 286. �10.1016/j.crme.2019.03.008�. �hal-02557143�

Contents lists available atScienceDirect

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Rendus

Mecanique

www.sciencedirect.com

Patterns and dynamics: homage to Pierre Coullet / Formes et dynamique : hommage à Pierre Coullet

Advances

in

the

analytical

study

of

the

dynamics

of

gaseous

detonation

waves

Paul Clavin

Aix–MarseilleUniversité,CNRS,CentraleMarseille,IRPHEUMR7342,49,rueFrédéric-Joliot-Curie,13384Marseille,France

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received29November2018

Acceptedafterrevision20January2019 Availableonline28March2019 Keywords:

Nonlinearhyperbolicequations Asymptoticanalysis

Integralequation

Recent theoretical results on the dynamics of gaseous detonations are presented. An asymptotic analysis is performed, retaining the physical mechanisms controlling the modifications to the inner structure of the detonation. As a result, the system of hyperbolic equations for the compressible fluid mechanics coupled with a detailed chemical kinetics of heat release is reduced to a single integral equation for the propagation velocity of the combustion wave versus time. Concerning the direct initiation of spherical detonations by a blast wave, curvature effects are shown to be responsible for a critical condition of initiation. Near criticality, the role of the unsteadiness of the inner structure is pointed out. The whole complexity of the critical dynamics is reproduced and explained by the integral equation. The necessary background knowledge in gaseous detonation is recalled in the two first sections of the article in order to facilitate the reading by non-specialists.

©2019 Published by Elsevier Masson SAS on behalf of Académie des sciences. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Detonationsaresupersoniccombustionwaves.Major advancesintheunderstandingofthecomplexdynamicsofthese waveshaveresulted fromasymptoticanalysesthat are reviewedina 2017article[1] and ina recentbook[2],inwhich an historyof the topic with an extensive list of references of the pioneering workscan be found so that they are not included in the list of references of the presentpaper. In gaseous mixtures of fuels and oxygen, the propagation Mach number atordinary conditionsis in between4 and 8. Detonationswere discovered during the last quarter of the 19th century,halfacenturyafterthefirstexperimentsonpremixedflames,whicharesubsonicwaves(propagationMachnumber

≈

10−3–10−2). A propagation velocity (a few km/s) of reaction waves in gaseous mixtures much faster than the mean velocityofmoleculeshasbeenintriguingforalongtime,eventhoughshockwaveswereknowntoexistininertgases.The steadyinnerstructureofplanegaseousdetonationswas understoodin1940, morethanhalfacenturyafterthediscovery ofdetonations. In themeantime, experiments showedthe multidimensionalgeometry andthe complexdynamics ofthe detonation fronts involving irregular and strongly unsteady cellular structures delimited by lines of singularities (triple points)propagating inthetransverse direction.The explanationofthecellular structurehasbeenelusive foralong time andthe major steps in the understanding are recent.This is alsothe casefor the initiationmechanism ofa detonation by a blast wave. Forexample, thedeflagration to detonationtransition isnot yetunderstood. The complex dynamicsof detonationfrontsisgovernedbytheunsteadinessoftheirinnerstructure.

E-mailaddress:[email protected]. https://doi.org/10.1016/j.crme.2019.03.008

1631-0721/©2019PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

The releaseof chemical heatresults fromthedifference ofbinding energyof themolecules inthe freshmixtureand inthecombustionproducts,thelatterbeingmorestablethantheformer.Combustionproceedsthroughacomplex chem-ical network ofhundreds ofelementaryreactions involvingtens ofintermediate specieswitha widerangeof timescales (10−3–10−8s).Twodistinctperiodsareidentifiedintheoverallrateofheatrelease,aninductionperiodinwhichtheheat releaseisnegligible,followedbyanexothermicperiod.Theinductiondelayishighlysensitivetotemperature. Thereisno exothermic reactionin gaseous mixturesatatmosphericpressure foratemperaturebelow 500 K(compositionfrozen far fromchemicalequilibrium).Theinductiondelayvariesfromafewsecondsat800Ktolessthan10−5 sat1200K.Above 1000K,thedurationoftheexothermicperiodisofthesameorderofmagnitudeastheinductiondelay.Thestrongrelease ofchemicalenergyinducesnonlinearphenomena intheflow(as,forexample,shockwaves)retroactinguponthechemical kineticscontrollingtheoverallrateofheatrelease.

The strategydevelopedatIRPHEtostudysuchcomplexphenomenaistheoppositeofthemostpopularmethodbased today onhuge numericalcodes includingthe wholedetails ofthe physicalandchemical mechanisms characterizedby a bunchofparameters.Instead,weconsiderlimitsofparametersforwhichthecomplexityofthebasicsystemofconservation equations is reduced sufficiently to be solved analytically. The key point is to find out the relevant limits stressing the essentialmechanismsgoverningthedynamics.Thecorrespondingasymptoticanalysesprovideuswiththephysicalinsights thatarenecessarytoputexperimentsanddirectnumericalsimulationsonarighttrack.

Inthesamespiritasinmyconferenceatthe2017meetinginNiceinhonorofPierreCoullet,thisarticleisaddressedto physicistswhoarenotspecialistsofcombustiontheoryandwhoareinterestedinnonlinearphenomenainfluidmechanics. Theobjectiveisneithertowriteanotherreviewarticleondetonationsnortopresentthedetailsoftheanalyses.Thepurpose is toexplain thephenomena inphysicaltermsasthey are enlightenedbyasymptoticanalyses [3] and [4].Forsimplicity, attentionwillbe focused onone-dimensionalproblemsdescribing thecriticalcondition ofthedirectinitiationprocess of detonationbylocaldepositionofenergy.Thenecessarybackgroundknowledgeisrecalledinthenextsection.

2. Physicalmechanismsatworkingaseousdetonations

2.1. Background

Consider apistonsetinmotion(subsonicvelocity)attheclosedendofaninfinitelylongtubeinwhichaninertgasis enclosed.Soonafterthepistonreachesaconstantsubsonicvelocity,ashockwaveafewmeanfreepathsthickisformedand propagatesaheadofthepistoninthequiescentgasataconstantandsupersonicvelocity. Theflowvelocityofthecolumn ofgasdelimitedbythepistonandtheshockhasauniformvelocityequaltothatofthepiston.Thelengthofthiscolumn increaseslinearlywithtimeatarateequaltothedifferenceofpropagationvelocitiesbetweenthesupersonicshockandthe subsonicpiston.Thisself-similarsolutioninplanargeometryistheresultofthewave-breakingmechanismincompressible flows,describedinoneofthemasterpiecesofB.Riemann(1860).Conservationofmass,momentum,andtotalenergyacross the steady inner structure ofthe shock wave leads to the jump conditions of W.J.M. Rankine(1870) andP.H. Hugoniot (1889). ConsidernowasimilarGedankenexperiment forareactivegaseous mixture.Theincrease oftemperatureacrossthe shockignitesthechemicalreactionandthechemicalenergyisreleasedinthecompressedgasaftertheinductiondelay.For asufficientlylargevelocityofthepiston,areactivelayerafewmillimetres thickinthecompressedgasstaysattachedtothe inertshock.Thispiston-supportedexothermicwaveiscalledoverdrivendetonation.Theseparationoflengthscalebetween theleadshockintheinertgasandthereactivelayerisaconsequenceofawell-knownphenomenonincombustionpointed out byYa.B.Zel’dovichandD.A.Frank-Kamenetskii(1938):theelementaryreactionsinvolvedintheinductiondelayresult frominelasticcollisions that arelessfrequentthan theelasticcollisions controllingtheinner structureofthe leadshock. This isbecause the reactive collisions requirea sufficientlyhighenergy forbreaking themolecules inthe freshmixture. Thebindingenergyofthemoleculesismuchlargerthanthethermalenergy,explainingwhythecompositionofareactive mixturecanstayfrozenatordinaryinitialconditions(T

≈

300 K,p

≈

1 atm).Evenathightemperature(T

≈

1000–2500 K), thereactivecollisionsconcernsthetailoftheMaxwell–Boltzmanndistribution.ThisleadstotheArrheniuslawwithalarge activation energy

E/

kBT



1 describingthehighthermalsensitivityandthethermalrunawaythatareresponsibleforthe sudden andviolent nature of some combustion phenomena. In simple words, assoon asthe temperature is sufficiently hightoinitiate theexothermicreaction,theincrease oftemperatureby thereleaseofchemical heatfurtherincreasesthe reactionrate.

Tosummarise,theinnerstructureofagaseousdetonationconsistsina(non-reactive)shockwavethatcanbeconsidered asanhydrodynamicdiscontinuity,followedbyamacroscopicreactionlayer(induction

+

exothermiczone).Inthelaboratory frame, the velocity of the compressed gas is oriented in the direction of propagation. In the reference frame attached to thelead shock, the flow velocity is subsonic, of the order ofthe local speed of sound, and orientedin the opposite direction, seeFig. 1in §4.This velocity is sufficientlylarge forneglecting heatconduction andviscosity (largeReynolds number, see below). The quasi-steadyinner structure of a detonationis controlled by the balance between thereaction rateandtheLagrangian motionoffluid particlesinthereferenceframe ofthelead shock(balancebetweenreactionand convection). Denoting au the soundspeed inthe initial mixtureandt−r1 thereaction rateatthecompressed side ofthe leadshock(Neumannstate),thedetonationthicknessisoforderd

=

autr.Thereactionratebeingsmallerthanthecollision frequency t−_coll1 (Arrhenius law:tcoll

/

tr

≈

e−E/kBT



1), the Reynolds numberis large, Re

=

a2utr

/

ν



1, where

ν

≈

a2utcoll is themolecular diffusion.Therefore,thedynamicsoftheinner structure isdescribed bythereactive Eulerequationsfor

inviscid compressible fluid complemented by the Rankine–Hugoniotconditions at the lead shocklinking the conditions (pressure,temperature, anddensity)inthecompressedgasjustbehind theshock(Neumannstate)totheshock’svelocity. Thedynamicsofthelead shockisa freeboundaryproblemofanhyperbolic nature,whichisclosedbytheconditionsat theexitofthereactionzone.

Ten years after the discovery of the gaseous detonation, a young Russian scientist, V.A. Mikhel’son, reported in his 1893PhDthesisatMoscowUniversitythattheconservationofmass,momentum,andtotalenergy(includingthebinding energy ofmolecules) across the steady-state structure ofa plane detonation wave leads to two families ofsolutions for a givensupersonicpropagationvelocity.Theydiffer bythe velocityofthe burntgasrelative totheshockwave, subsonic witha lead shockin one familyand supersonicwithout lead shockin the other. The formeris the solution mentioned above,whilethelatterisnot relevantbecauseignitioncannotoccur atordinarytemperature. Thetwosolutions mergeat aminimumvelocityofpropagation

D

CJ,forwhichthevelocityoftheburntgasrelativetotheshockwave issonic.There isno planarsupersonicwave havingasteadyinner structureandpropagating ataconstant velocity

D

smallerthan

D

CJ, sothatoverdrivendetonationsarecharacterisedby

D > D

CJ

∝

√

qm,qm beingthechemicalenergyperunitofmassofthe reactive mixture.The marginal solution

D = D

CJ iscalledthe CJsolution inhonor oftheworks ofChapman (1899) and Jouguet(1904)(itwouldhavebeenbettercalledMikhel’sonsolution).

Moreover,CJwavesareself-sustained(propagatingatconstantvelocitywithoutthesupportofapiston).Thesonic con-ditionattheendofthereactionzoneprotectstheinnerstructureofthedetonationfrombeingdampedbytherarefaction wave.Thelatterdevelopssystematicallyintheburntgasbetweentheflowattheexitofthereactionzoneandthe down-streamboundary condition (zerovelocity atthecloseendofa tubeoratthe centreforsphericaldetonations expanding freelyinopenspace).Thepropagationvelocityofoverdrivendetonationsdecreasesdownto

D

CJ undertheinfluenceofthe rarefactionwaveassoonasthesupportingmechanismissuppressed.

2.2. Theproblemofdirectinitiationofdetonation

The directinitiationprocess of detonationrefers tothe formationof aself-sustained detonationinopen space inthe decayofastrongblast waveproduced byaconcentrated energysource.Theenergyisdepositedquasi-instantaneouslyin ahotspotoftinnysizesothatthedensityofenergyisinitiallymuchlargerthanthedensityofchemicalenergyavailable inthegaseous mixture.Therefore,theinitialcondition correspondstotheSedov(1946)–Taylor(1950)self-similarsolution for a strong blast wave in an inertgas. Soon after, overdriven detonations are generated witha decreasingpropagation velocitywhiletheradiusoftheleadshockincreases.Theexperimentsshow thatasphericalCJwave isformed atafinite distancer∗_f fromthesource onlyiftheamount ofenergyliberated issufficientlylarge E

>

E∗.Aroughevaluationofthis radius corresponds to r∗_f

≈ (

E∗

/

ρ

uqm

)

1/3 where

ρ

u is theinitial density ofthe reactive mixture.Initiation ofdetonation failsforE

<

E∗ andthereisnosphericalCJdetonationwithasmallerradius.Pioneeringnumericalsolutionswereobtained around 1970underthe approximation consideringthe detonationwave asa discontinuity acrosswhich theplanar jump conditionsaresatisfied.Thisapproximationdoesnothaveacriticalenergy:incontrastwithexperiments,suchanumerical solutionpredictsthattheoverdrivenwaverelaxessystematicallytoaCJdetonation,nomatterhowsmallthevalueofE is.

Thisindicatesthatthecriticalconditionforinitiationisassociatedwiththefinitethicknessofthedetonationwave.Afirst criterionforthedirectinitiationwasproposedbyZeldovichetal.(1956),assuming thatthetimetakenfortheblastwave velocitytodecreaseto

D

CJ

∝

√

qm mustnotbeshorterthanthereactiontime.Thiscriterionleadstoacriticalradiusofthe orderofthethicknessoftheCJwaveandtoacriticalenergysmallerthantheexperimentaldatabyafactor10−5 _to10−6 whenrelevantvaluesofthereactiontimeareused,

E∗

ρ

uqm

=

2

(

γ

+

1

)

4

(

γ

−

1

)

2d 3 oCJ

,

r ∗ f

=

21/3

(

γ

+

1

)

4/3

(

γ

−

1

)

2/3doCJ

≈

8 doCJ (1)

wheredoCJ isthedetonationthicknessoftheplanarCJwaveand

γ

≡

cp

/

cv istheratioofspecificheat.

Afurtherstepwasachievedfortyyearslaterbyconsideringthesmallmodificationoftheinnerstructurebythecurvature ofthewave[5].ThisanalysisofacurvedCJdetonationwasperformedinthelimitofalargeactivationenergy

E/

kB



1, assumingthattheinnerstructureisinsteadystate.UsingacrudemodelfortheinnerstructureoftheplaneCJwave,namely thesquare-wavemodelforwhichthechemicalenergyisreleasedinstantaneouslyaftertheinductiondelay,thisnonlinear analysisincorporatesasmallcurvaturetermwhoseeffectisamplified bythelarge activationenergy.An extensionofthe analysisto a realisticinner structure ispresented in§6.2. Theanalysisin [5] leads to anon-linear relationbetweenthe propagationvelocity of thecurvedCJ detonation

D

CJ andthe curvaturedoCJ

/

rf, rf denoting theradius ofthe lead shock. DenotingtheCJvelocityoftheplanewave

D

oCJ,therelationbetween

D

CJ

/

D

oCJ andrf

/

doCJ presentsaturningpointinthe phase-space“propagationvelocity–radius”. There isno quasi-steadysolutiontospherical CJwavesbelowa criticalradius

r∗_f, whichismuch largerthan doCJ,essentially becauseofthe large activationenergy

E/

kBT



1.The criticalradiusr∗f is typically 102–103 largerthan thatin(1).Therefore,theenergyvarying liker3,the orderofmagnitudeofE∗ observed in experiments is recovered by the theoretical analysis [5]. The numerical simulations ofHe [6] using a detailed chemical scheme for the combustion of hydrogen–oxygen mixtures showed results in satisfactory agreement with the theoretical prediction, even though unsteady effects that are neglected in [5] are non-negligible in the numerical simulation. The

importance of unsteadiness in the direct initiation process of gaseous detonation was also observed later in numerical simulations usinga simpleone-step exothermalreactiongovernedbyan Arrheniuslaw[7].Theunsteadytermsarefound to beevenlargerthanthegeometricaltermsdescribingthecurvatureeffect.However, surprisingly,thecriticalradiuswas not muchdifferentfromthat predictedin[5],theratioofthe numericalradius tothetheoreticalradiusbeingbetween2 and4.Consideringthedifferenceofmodelsofinnerstructurein[5] andin[7],theagreementisquitesatisfactoryindeed.

The purposeof theanalytical study[4] isprecisely to investigatetherole ofunsteadiness indirect initiationof deto-nation, especially nearthe criticalradius. An overviewofthe asymptoticmethod andofthe resultsare presented in§5

and §6. Afirst step consistsin analyzingthe decay tothe CJ regime in planar geometrywhen the supporting piston is suddenly arrested. Thishyperbolic problem isold andhas beensolved under theapproximation ofa detonation consid-eredasa discontinuity(detonation frontwithoutmodificationoftheinner structure)followingChandrasekhar(1943) and Friedrichs (1948) forthe decayofapure shockwave. Theanalytical solutiontakingintoaccount theunsteadiness ofthe innerstructureofthedetonationhasbeenobtainedonlyrecently[3] andispresentedin§5.2.

The mathematical formulation is given in §3. Physical insights are presented in §4 followed by the analyses of the detonationdecayintheplanarandsphericalgeometryin§5and§6.

3. Generalformulation

3.1. Constitutiveequations

Insphericalgeometry,

∇·

u

= ∂

u

/∂

r

+

2u

/

r,thereactiveEuler’sequationsare 1

ρ



∂

∂

t

+

u

∂

∂

r



ρ

+

∂

u

∂

r

+

2 u r

=

0

,

ρ



∂

∂

t

+

u

∂

∂

r



u

= −

∂

p

∂

r (2)



∂

∂

t

+

u

∂

∂

r

 

ln T

−

(

γ

−

1

)

γ

ln p



=

qm cpT

˙

w

(

Y

,

T

,

p

)

tr

,



∂

∂

t

+

u

∂

∂

r



Y

=

w

˙

(

Y

,

T

,

p

)

tr (3) where

ρ

, p andu are respectivelythe density, thepressure, andthe radial velocity in thelaboratory frame and

γ

,qm,

T ,Y ,trand w

˙



0 arerespectivelytheratioofspecificheat

γ

≡

cp

/

cv

=

cst.,thechemical heatreleaseper unitmassof

mixture,thetemperature,theprogressvariable(Y

=

0 intheinitialmixtureandY

=

1 intheburnedgas),thereactiontime attheNeumannstate oftheCJwave andthenon-dimensional heat-releaserateexpressedintermofthethermodynamic variables.Thefirstequationin(3) istheconservationofenergywrittenintheentropyform.Theentropyproductionresults fromtherateofheatrelease,heat conductionandmoleculardiffusionbeingnegligiblebehindtheleadshock. Thesecond equation in(3) isashortnotation foracomplexchemicalkineticsofcombustion.The pressure p andthesoundspeeda

aregivenbytheidealgaslaw

p

=

γ

−

1

γ

cp

ρ

T

,

a

=



γ

p

ρ

(4)

Usingthemassconservationin(2) andtheequationofstatein(4) foreliminating

ρ

andT ,theenergyequationin(3) can bewrittenintermsofp andu intheform

1

γ

p



∂

∂

t

+

u

∂

∂

r



p

+

∂

u

∂

r

+

2 u r

=

qm cpT

˙

w tr (5) Equationsfortheconservationofmassandmomentumin(2) canbeputintheformoftwohyperbolicequationsforu and p whentheequationforconservationofmomentumin(2) multipliedbya

/(

γ

p

)

=

1

/(

ρ

a

)

isaddedtoorsubtractedfrom (5) 1

γ

p



∂

∂

t

+ (

u

±

a

)

∂

∂

r



p

±

1 a



∂

∂

t

+ (

u

±

a

)

∂

∂

r



u

=

qm cpT

˙

w tr

−

2u r (6)

When(4) isusedandwhentheexpressionofthereactionrateintermofthethermodynamicvariablesw

˙

(

T

,

p

,

Y

)

isknown, equations(3) and(6) form aclosedsetfor p,u, T , andY .Equations (3) describetheentropywave propagatingwiththe velocity of theflow. Equation (6) is theextension ofthe usual characteristicequationsto the caseofa reactinggaseous mixtureinsphericalgeometry.Theydescribecompressiblewaves

C

₊and

C

₋propagatingintwooppositedirectionsatthe speed ofsoundrelativelytothefluid particles.Whentheright-hand sideissetequaltozero,thelinearizedversionof(6) representsthesimplewavesofplanar acoustics

δ

p

= ±ρ

a

δ

u.Thedivergenceofthesphericalflow2u

/

r intheright-hand sideof(6) istheonlydifferencewiththeplanargeometry.

3.2. Boundaryconditions

Introducing the instantaneous radius and velocity of the shockfront r

=

rf

(

t

)

,

D(

t

)

≡

drf

/

dt

>

0, it is convenient to considerthecoordinateattachedtotheleadshock

x

≡

r

−

rf

(

t

)

⇒ ∂/∂

r

→ ∂/∂

x

,

∂/∂

t

→ ∂/∂

t

−

D

(

t

)∂/∂

x (7)

Foran expandingspherical detonation,drf

/

dt

>

0, u



0,theinitialmixture andthe compressedgascorrespondto x

>

0 andx



0 respectively.Theboundaryconditionsatthecompressedgassidex

=

0−oftheshockfront(calledNeumannstate anddenotedbythesubscriptN)taketheform

x

=

0−

:

Y

=

0

,

w

˙

= ˙

wN

(

TN

) >

0

,

p

=

pN

(

t

),

T

=

TN

(

t

),

u

=

uN

(

t

)

(8) where,usingtheRankine–Hugoniotconditions, pN

(

t

)

,TN

(

t

)

,anduN

(

t

)

areexpressed intermof

D(

t

)

andthe thermody-namicvariables pu andTu ofthequiescentmixture,frozenintheinitialstatedenotedbythesubscriptu.Thepropagation velocity

D(

t

)

is determinedwhenarear boundaryconditionattheexitofthereactionzone isappliedtothesolutionto thehyperbolicequations(3) and(6) satisfying(8).Therearboundaryconditionofaweaklycurveddetonationrf

/

aNtr



1 takes a simple form ifthe length scale ofthe external flow in the burntgas uext

(

r

,

t

)

(rarefaction wave) is larger than thedetonationthickness, lext



aNtr,1

/

lext

≡ |(

1

/

uext

)∂

uext

/∂

r

|

r=rf(t).Then, introducingthe non-dimensionalcoordinate

ξ

attachedtothemovingfrontoftheleadshock(radiusrf),

ξ

≡

x autr

=

r

−

rf

(

t

)

autr

(9) anddenotingtheendofthereactionbyasubscriptb(

ξ

= ξ

b

<

0)

|ξ

b

| =

O

(

1

),

ξ

 ξ

b

: ˙

w

(

Y

=

1

,

T

)

=

0

,

ξ

b

< ξ



0

: ˙

w

(

Y

,

T

) >

0 (10) therearboundaryconditionoftheinnerstructuretakestheform

ξ

= ξ

b

<

0

:

u

=

ub

(

t

),

ub

(

t

)

=

uext

(

rf

(

t

),

t

)

(11) wheretheflowfield ofburntgasuext

(

r

,

t

)

is solutiontoanexternal problem(inertrarefactionwave).Thesolutionto (3) and(6) satisfying(8) and(11) yieldstheexpressionof

D(

t

)

asafunctionalofub

(

t

)

.

AnalyticalsolutionstothedecaytotheCJregimecannotbeobtainedwithoutfurthersimplification.Notonlytheintrinsic dynamicsoftheinnerstructureisatoughproblem,butalsotheexternalflow uext

(

r

,

t

)

isararefactionwavethatdepends on the dynamicsof its leading edge so that the boundary condition ub

(

t

)

in(11) dependsin fact on the solution.This difficultyisovercomein§6.1thankstothequasi-transoniccharacteroftheflowattheexitoftheexothermiczonewhen approachingtheCJregime.Thedynamicsoftheinnerstructureissolvedin§5.

4. Physicalinsights

Thepurposeofthissectionistoidentifythemainphysicalmechanismscontrollingthedynamicsinordertodetermine thelimit ofparameters tobe usedintheasymptoticanalysis.From nowon,weconsiderthereferenceframeattachedto theleadshockofadetonationpropagatingwithapositivevelocity

D >

u

>

0.Theflowvelocityintheinnerstructureofa detonationissubsonicrelativelytotheshockanditsabsolutevalueincreases0

< (D −

u

)

<

a withtheamountofreleased heat from theNeumann state at x

=

0− to the end ofthe reaction at x

=

xb

<

0,

(D −

uN

)

< (D −

ub

)

, see Fig. 1.The soniccondition isverifiedonlyattheexitofthereactionzoneoftheCJwave,

(

D

CJ

−

ub

)

=

ab while

(

D −

ub

)

<

ab inthe overdrivenregimes.

4.1. Newtonianapproximation

Acrosstheinnerstructureofusual gaseousdetonations,adiabaticcompressionalheating isnotimportant.Itseffecton temperatureis relativelysmallcompared to thetemperatureincrease dueto thereleaseof chemical heat.Therefore,the pressuretermcanbeneglectedinEqs. (3) governingthedownstreamrunningentropywave,



∂

∂

t

+ (

u

−

D

)

∂

∂

x



T

≈

qm cp

˙

w

(

T

,

Y

)

tr

,



∂

∂

t

+ (

u

−

D

)

∂

∂

x



Y

=

w

˙

(

T

,

Y

)

tr (12) x

=

0−

:

T

=

TN

(

t

),

Y

=

0 (13)

Thiscorresponds towhatiscalledtheNewtonian approximation 0

< (

γ

−

1

)



1.The compressiblephenomenaare fully retainedintheRankine–Hugoniotconditionsattheleadshockandalsointhecharacteristicequations(6) ofthe compress-ible modes propagating in thetwo directionsacross the inner structure of the detonation,see Fig. 1. For usual gaseous

Fig. 1. Sketchoftheinnerstructureofagaseousdetonation.Left:theprofileoftherateofheatreleaseisplottedwiththeabsolutevalueoftheflow velocityinthereferenceframeoftheleadshock,(D−u)>0.Thelatterhasashapesimilartothatofthetemperaturedistribution.Theupstreamand downstreamrunningmodes(compressiblesimplewaves),respectivelyC+andC−,areshownontop,upstream,anddownstream,referringtotheflowin thereferenceframeoftheleadshock.Thedownstreamrunningentropywavethatpropagateswiththefluidparticlesattheflow’svelocityisalsoshown. Right: plotoftheflowvelocityinthereferenceframeattachedtotheleadshock, (u− D) <0.Itsprofileissimilartothatofthereducedflowfield

μ(ξ,τ)>0 usedintheasymptoticanalysis,μ

∈ [

μb,1],0<μb1,seedefinition(23).

detonations,theactivationenergyislargeandthevariationofthereactionratewiththepressurecanbeneglectedinfront ofitsvariationwithtemperatureandwiththeprogressvariable,w

˙

(

T

,

Y

)

.

The solution to (12)–(13) is easily obtainedif the variationof

(

u

−

D)

withtime is negligible.Introducing the delay associated

_

with a fluid particle issued fromthe lead shock to reach the point ata distance

|

x

|

from the shock,

(

x

)

=

0

x dx

/(

D −

u

)

>

0, the instantaneous distributionof the rateofheat release w

(

x

,

t

)

= ˙

w

(

T

,

Y

)

obtainedfrom (12)–(13)

thentakestheform w

(

x

,

t

)

=

w



x

,

TN

t

− (

x

)



(14)

where w

(

x

,

TN

)

isthe steady-statesolution associatedwith theboundary condition x

=

0

:

T

=

TN

,

Y

=

0. Equation(14) means that the value ofthe instantaneous distributionof thereaction rate w

(

x

,

t

)

ata distance

|

x

|

fromthelead shock is relatedto itsvalue inthesteadystate,corresponding toaNeumanncondition definedatatime shiftedinthepastby thedelay

(

x

)

.The temperatureTN

(

t

)

attheNeumannstate isrelatedtothepropagationvelocityoftheleadshock

D(

t

)

throughtheRankine–Hugoniotconditions,sothatthedistributionoftherateofheatrelease(14) isexpressedintermsof the propagationvelocity atan earlytime.Solving (6) withtheboundaryconditions(8) and(11) is stillatoocomplicated hyperbolicproblemforageneralanalyticalsolution

D(

t

)

tobeobtained.

4.2. Twotimescalesinthefeedbackloops

According to the Rankine–Hugoniotconditions,the variation ofthe propagationvelocity

D(

t

)

generatesperturbations of p andu at the Neumannstate. Theyare transported towardthe reactinggas by the entropywave with theabsolute velocity

(D −

u

)

>

0 andalsobythedownstreamrunningcharacteristics

C

₋withtheabsolutevelocitya

+ (

D −

u

)

>

a.Part oftheresultingdisturbancesofthesourcetermsintheright-handsideof(6) aresentbacktotheshockbytheupstream runningmode

C

₊propagatingwiththevelocitya

−(

D −

u

)

>

0,seeFig.1.Whentheseupstreamrunningdisturbancesreach the Neumannstate,they modify

D(

t

)

throughtheRankine–Hugoniotconditions.Thereforetheinstantaneous propagation velocity ofthelead shock,

D(

t

)

isdeterminedbythecumulative effectsofacontinuoussetoffeedbackloops,illustrating thecomplexityofthehyperbolicproblem.Forafixedconditionattheexitofthereactionzone,theseloopscanleadtoan intrinsicinstabilityoftheinnerstructurestudiedbyasymptoticanalysesin[3,8] and[9],brieflyrecalledinplanargeometry in§5.2.

Ifthesubsonicvelocity oftheflow

D −

u,a

> (

D −

u

)

>

0,issufficientlyclosetothespeed ofsound(quasi-transonic flow), 0

<

[

a

− (

D −

u

)

]



a,theproblemofthedynamicsoftheinner structureisoneoftwotimescalessincethe prop-agationvelocity ofthedownstreamrunningmodes(entropywave andcharacteristics

C

₋)becomes muchlargerthanthat oftheupstreamrunningmode

C

₊,

(

D −

u

)

 [

a

− (

D −

u

)

]

anda

+ (

D −

u

)

 [

a

− (

D −

u

)

]

.Therefore,thetransittimeof thedisturbancesthatarepropagatedbythedownstreamrunningmodesismuchshorterthanthosepropagatedby

C

₊.The delay ineach feedbackloop iscontrolled bythe slowest mode,namely

C

₊,theeffectofthedownstream runningmodes beingquasi-instantaneous.Thereforethecharacteristictimeoftheoveralldynamicsoftheinnerstructureislargerthanthe transittimeofafluidparticleacrossthedetonationthicknessand,toleadingorderinatwotimescalesanalysis,thedelay

[

a

− (

D

−

u

)

] 

a

⇒

w

(

x

,

t

)

≈

w

(

x

,

TN

(

t

))

(15) Equation (15) is valid for anycomplex chemical scheme. Moreover, the effects of small variations of TN

(

t

)

in (15) are amplifiedbyahighthermalsensitivity.

Unfortunately,thequasi-transonicapproximation

[

a

− (

D −

u

)

]



a isnotuniformlyvalidintheinnerstructureofreal gaseous detonations, even for overdriven regimesclose to the CJ regime,

(

D −

ub

)

≈

ab. In the inductionlayer, one has typically

D −

uN

=

0

.

3aN.The quasi-transonicapproximation isverified everywherein theinner structureofdetonations closetotheCJregimeonlyinthelimitofsmallheatrelease.Thislimitprovidestheframeworkforasystematictheoretical analysis of the unsteady inner structure of detonations. Even though the limit of small heat release is not realistic for real detonations,it is a convenientapproximation provided that the lead shockis still considered asa discontinuity. All themechanismsinvolvedinthedynamicsoftheinner structurearewell keptandthetechnicaldifficulty associatedwith thevariation ofthesoundspeedthat doesnot play asignificantrole issuppressed.Thisyields analyticalresultsthat are qualitativelyrelevantandquantitativeagreementwithrealdetonationscanbeobtainedsimplybyrescalingtheasymptotic results,seetheendof§5.2.

5. Asymptoticanalysisoftheunsteadyinnerstructure

Theanalysespresentedbelowwerecarriedoutinthelimitofsmallheatrelease 2

_≡

_q

m

/

cpTu



1,usingtheNewtonian approximation(

γ

closetounity)inordertosuppressthecompressionalheating,

≡ (

MoCJ

−

1

)



1

,

(

γ

−

1

)/



1 (16)

whereMoCJ

≡

D

oCJ

/

au

≈

1

+



qm

/

cpTuistheMachnumberoftheplanarCJwaveinsteadystatepropagatingatthevelocity

D

oCJ,au beingthespeedofsoundintheinitialmixture.Asufficientlylargethermalsensitivitythenensuresthatthesmall fluctuationsthatareproducedinthelimit(16) produceasubstantialeffectontheoveralldynamics,

δD(

t

)/D =

O

(

1

)

.

5.1. Formulationintheasymptoticlimit

Onthe basis of theresults of[5], anticipatingthat the criticalradius of thelead shockis larger than thedetonation thickness,anon-dimensionalcurvature

κ

(

τ

)

oforderunityinthelimit(16) isintroducedby

autr rf

(

τ

)

=

κ

(

τ

),

κ

=

O

(

1

)

(17)

Forconditions closeto the CJregime ofplane detonations we introduce the followingdimensionless quantities oforder unityinthelimit(16),

μ

(ξ,

τ

)

,

π

(ξ,

τ

)

and

α

˙

τ

(

τ

)

for,respectively,theflowvelocityrelativetotheleadshock,thepressure,

andtheinstantaneouspropagationvelocityoftheleadshock

D(

t

)

=

drf

/

dt, u

−

D

oCJ au

≡ −

1

+

μ

(ξ,

τ

),

D

−

D

oCJ au

≡

α

˙

τ

(

τ

),

1

γ

ln



p pu



≡

π

(ξ,

τ

)

(18)

where

τ

is the reduced time of order unity describing the dynamics, see (25) below. The objective of this subsection is to show that, in the limit (16), the non-dimensional shockvelocity

α

˙

τ

(

τ

)

is obtainedby solving a single (nonlinear)

partial–differentialequationforthenon-dimensionalflowfield

μ

(ξ,

τ

)

,

∂

μ

∂

τ

+ [

μ

− ˙

α

τ

(

τ

)

]

∂

μ

∂ξ

=

w

(ξ,

τ

)

2

− (

1

+

μ

)

κ

(

τ

)

(19)

ξ

=

0

:

μ

=

1

+

2

α

˙

τ

(

τ

),

ξ

= ξ

b

(

τ

)

:

μ

=

μ

b

(

τ

)

(20)

where

μ

b

(

τ

)

isagivenfunctionobtainedfromtheexternalflowintheburntgas.

Inthelimitofsmallheatrelease(16),thespeedofsounda

/

au

=

√

T

/

Tu andthecurvaturetermrf

/

r are, accordingto (9) and(17),almostconstantacross theinnerstructureofthedetonations

ξ

=

O

(

1

)

,a

/

au

=

1

+

O

(

)

,rf

/

r

=

1

/

[

1

+

κ

ξ

]

. Whentheseterms,whichareofordersmallerthan 2,are neglected,Eqs. (6), writteninthereferenceframe attachedto theleadshock(7),usingthenotations(9) and(18),takethenon-dimensionalform



tr

∂

∂

t

+

[

−

2

+

(

μ

− ˙

α

τ

)

]

∂

∂ξ



(

π

−

μ

)

=

2w

˙

−

2

2

(

1

+

μ

)

κ

(21)



tr

∂

∂

t

+

(

μ

− ˙

α

τ

)

∂

∂ξ



(

π

+

μ

)

=

2w

˙

−

2

2

(

1

+

μ

)

κ

(22)

obtainedbyusing(16)–(18) intheform u au

=

(

1

+

μ

),

(

u

−

D

)

au

=

(

μ

− ˙

α

τ

)

−

1

,

u r

=

2

_κ

₍

₁

₊

_μ

₎

rf r (23)

UsingtheRankine–Hugoniotrelations,theboundaryconditionsattheNeumannstatetaketheform,

ξ

=

0

:

μ

=

μ

N

(

τ

)

= (

1

+

2

α

˙

τ

)

+

O

(

),

π

=

π

N

(

τ

)

=

2

(

1

+ ˙

α

τ

)

+

O

(

)

(24)

The two-timescalenature ofthe dynamicsinthe limit (16) isrevealed by thecomparisonof (21) and(22). The velocity ofthe simplewave(21), issuedfromtheleadshock(

ξ

=

0) andpropagating towardtheexitofthereactionzone (inthe negative

ξ

direction)is larger(bya factorof1

/

) thanthe velocityofthe simplewave (22) propagating intheopposite directionforclosingthe feedbackloops. Therefore,toleading order inthelimit (16), thepropagationmechanismin(21) isinstantaneous andthedynamicsoftheinnerstructureiscontrolled bythesimplewave(22).Theresultingdynamicsis slow atthescale ofthetransit timetr of afluid particlepropagating fromthe leadshockup tothe endofthe reaction zone.Therefore,thereducedtimescale

τ

oforderunityis

τ

≡

t tr

,

∂

∂

t

=

tr

∂

∂

τ

(25)

Theleadingorderof(21),

∂(

π

−

μ

)/∂ξ

=

0,showsusing(24) thatthequantity

π

−

μ

isconstant,

(

π

−

μ

)

≈

1.Expressedin termsofthereducedtime(25),theleadingorderof(22) inthelimit



1,afterdivisionofthetwosidesby 2,takesthe form(19). Skippingforthemomentthematchingdifficulty mentionedattheendof§3.2,thefunction

μ

b

(

τ

)

isgivenby theexternalsolutionintheburntgas,exceptfortheCJregime,forwhichthedynamicsoftheinnerstructureisdecoupled fromtheflowofburntgasbythesoniccondition,

CJ wave:

μ

b

(

τ

)

= ˙

α

τ

(

τ

)

(26)

(

α

˙

τ

=

0 intheplanarCJwave,

κ

=

0).

The unsteady distribution of the reaction rate (15) requires to compute the inner structure of a family of steady overdriven detonations w

(

x

,

TN

)

for different propagation velocity

D

,

D

, and TN, being in one-to-one correspondence (Rankine–Hugoniotrelation).Numericalsimulationsofoverdrivendetonationsofhydrogen–oxygenmixtures[10] showthat

w

(

x

,

TN

)

canbewellapproximatedfromthedistributionoftheCJdetonationinthesteadystate, woCJ

(ξ )



0 byrescaling thelengthscalewiththetime-dependentinductionlength,yielding

w

(ξ,

τ

)

=

ebα˙τ(τ)_w oCJ

(ξ

e bα˙τ(τ)

_),

_with 0

−∞ woCJ

(ξ )

d

ξ

=

1

⇒

0

−∞ w

(ξ,

τ

)

d

ξ

=

1 (27)

wheretheparameterb characterizesthermalsensitivity, b

=

2

(

γ

−

1

)

E

kBTu

(28) and

E

is theactivation energyof theArrhenius law controllingthe variation ofthe inductionlength withthe Neumann temperature. The normalizationcondition in(27) correspondsto a referencetimescaletr in(3)–(9) equal tothe reaction time at theNeumannstate of theCJ wave,so that its non-dimensionalthicknessis equalto unity,

ξ

 −

1

:

woCJ

(ξ )

=

0,

−

1

< ξ



0

:

woCJ

(ξ )

>

0.Therefore,theendofthereactionzoneintheunsteadystructureislocatedat

ξ

= ξ

b

(

τ

)

= −

e−bα˙τ(τ)

,

ξ

 ξ

b

(

τ

)

:

w

(ξ,

τ

)

=

0 (29) Tosummarize,when(27) isinsertedinto(19),thehyperbolicproblem(2)–(11) isreducedintheasymptoticlimit(16) tosolve(19) withtheboundaryconditions(20) for

ξ

b in(29).Iftheflowofshockedgasiskeptsubsonicrelativelytothe leadshock,asisthecaseinthesteadystate,theterminthebracketontheleft-handsideof(19),computedfrom(18),

ξ

b

< ξ



0

: [

μ

− ˙

α

τ

] =

[au

− (

D

−

u

)

]

/

au

>

0 (30) is positiveeverywhere acrossthe innerstructure andincreasesfromtheendoftheheat release(

ξ

= ξ

b

<

0) tothe lead shock (

ξ

=

0), like theflow velocity u increasesin thelaboratory frame fromub to uN undertheeffect ofthe chemical heat release,seeFig.1(right). Therefore,the wave-breakingmechanismbythenonlinearterm

(

μ

− ˙

α

τ

)

∂

μ

/∂ξ

cannotbe

producedinthereactingflowbehindtheleadshock.

5.2. Dynamicsofplanardetonations

Thesolutionto(19)–(20) with(29) leadstoanintegralequationfortheinstantaneouspropagationvelocity

α

˙

τ

(

τ

)

.Before

considering thedirectinitiationofdetonations,thedynamicsofplanar detonationsisworth recalling.Theequation tobe solvedis(19) for

κ

=

0 andthestabilityofplanardetonationsagainstplanardisturbancesisperformedforaconstantvalue oftheflowvelocityattheendoftheexothermicreaction,

μ

b

=

μ

b

=

cst.in(20).Foranypositivevalue

μ

b

>

0,thereare twosteady-statesolutions,onlyonedescribinga(weakly)overdrivenwave,

[μ

(ξ,

μ

b

)

− ˙

α

τ

(

μ

b

)

]

>

0,

α

˙

τ

(

μ

b

)

>

0,

y

/

b

= (

1

+

μ

b

)

−1

μ

b2

/

2

,

μ

(ξ,

y

)

−

y

/

b

=



(

μ

b

−

y

/

b

)

2

+

μ

2oCJ

(ξ

e

y

₎

₍₃₁₎

where thenotation y

≡

b

α

˙

τ hasbeen used andwhere themarginal CJ solution

μ

oCJ

(ξ )

(

μ

b

=

0, y

=

0) is a increasing functionfrom0 atthe endofthereaction

ξ

= −

1 to 1 attheNeumannstate

ξ

=

0.Thelinearversion tothehyperbolic equation(19) issolvedin[3] and[8] foraparameterb oforderunityinthelimit(16),correspondingtoalargeactivation energy,

E/

kBTu



1.Introducingthedecomposition y

(

τ

)

=

y

+ δ

y

(

τ

)

,thevariationoftheshockvelocity

δ

y

(

τ

)

isfoundto besolutiontoanintegralequation,

b

=

O

(

1

),

κ

=

0

:

2

(

1

+

2 y

/

b

) δ

y

(

τ

)

=

0

−∞ g

(ξ )

y

(

τ

+ ζ (ξ))

d

ξ

(32) where g

(ξ )

≡



_b 2

∂

∂

y



eywoCJ

(ξ

e y

₎



y=y

+

d

μ

d

ξ



,

ζ (ξ )

≡ −

0

ξ d

ξ

μ

(ξ

)

<

0

,

ζ

b

≡ −

0

ξb=−e−y d

ξ

μ

(ξ

)

(33)

where

|ζ

b

|

<

∞

isthetotaltransittimeoftheperturbationtransportedupstreambythecompressiblemode

C

+toreachthe leadshockfromtheendofthereactionand

|ζ (ξ)|

<

|ζ

b

|

isthetransittimefromtheposition

ξ

.Thephysicalinterpretation of(32) isstraightforward:thefunction g

(ξ )

y

(

τ

− |ζ|)

istheperturbationoftheshockvelocityattime

τ

resultingfromthe localmodificationat

ξ

ofthereactionrateproducedatearliertimebythedownstreampropagatingmodes(entropywave and

C

₋),thetimelag

|ζ (ξ)|

beingthedelaytakenby

C

₊ forsendingtheperturbationbacktotheleadshock.Theintegral termin(32) correspondstothecumulativeeffectsoftheloopsassociatedwithallthepointsintheinternalstructure.The lower boundoftheintegral in(32) isreplacedby

−∞

because thekernel g

(ξ )

vanishes for

ξ < ξ

b.Thestability limitis obtainedbylookingforasolutionto(32) intheform

δ

y

(

τ

)

=

eσ τ yieldingatranscendentalequationforthelineargrowth rate

σ

(acomplexnumber)correspondingtotheLaplacetransformofg,

0

−∞

˜

g

(ζ )

eσζd

ζ

=

2

(

1

+

2 y

/

b

)

where

˜

g

(ζ )

≡

μ

oCJ

(ξ(ζ ))

g

(ξ(ζ ))

(34)

thefunction

ξ(ζ )

beingobtainedby inversionof(33), correspondingto aone-to-one relationbetween

ξ

and

ζ

.Equation (34) hasadiscretesetofcomplexroots

σ

i,i

=

1

,

2

...

.Foratemperaturesensitivitysufficientlysmallandforadistribution

woCJ

(ξ )

sufficientlysmooth, thedetonationisstableagainstplanar disturbances;the realpartofalltherootsis negative, corresponding todamped oscillatory modes(Im

σ

i

=

0, Re

σ

i

<

0

∀

i). An oscillatory instabilityoccurswhen b is slightly

increasedabovetheinstabilitythresholdbc atwhichoneoftheoscillatorymodes

σ

jbecomesneutral,b

=

bc:Re

σ

j

=

0,Im

σ

j

=

0 oforderunity,b

>

bc:Re

σ

j

>

0.Whenb isfurtherincreased,manyunstableoscillatorymodesdevelop.Thevalue

bcdependsontheshapeofwoCJ

(ξ )

.FortypicaldistributionswoCJ

(ξ )

ofrealdetonations,theparameterbc attheinstability thresholdandthereducedfrequencyofthemarginalmodeare oforderunity. Thestiffer is woCJ

(ξ )

thesmallerisbc and thelargeristhefrequency.

Anonlinearextensiontomarginallyunstabledetonationinplanargeometryhasalsobeenobtained[3]

2 y

(

τ

)

=

y

+

0

−∞ G



ξ,

y

(

τ

+ ζ (ξ))



d

ξ,

G

(ξ,

y

)

≡

W

(ξ,

y

)

+

d

μ

oCJ d

ξ

y (35) where W

(ξ,

y

)

≡

b 2



eywoCJ

(ξ

e y

₎

₋

_w oCJ

(ξ )



,

0

−∞ W

(ξ,

y

)

d

ξ

=

0

,

0

−∞ G

(ξ,

y

)

d

ξ

=

y (36)

and2y

/

b has been neglected in frontto 1 in theboundary condition at

ξ

=

0. The numericalsolution to (35) showsa supercritical bifurcation.Nonlinear oscillationsdevelopsforb slightlylarger thanbc,followedby atransitionto achaotic signal y

(

τ

)

throughperioddoublingwhenb isfurtherincreased.

The delay

|ζ

b

|

increaseswhenapproaching the CJregime (

μ

b

→

0+) and, foran usual reactionrate, itdiverges atCJ. Thisdoes notchange the stability analysis, because g

˜

(ζ )

decreases sufficiently quicklyto zerowhen

ζ

increases.In that respect,themarginalcharacteroftheCJregime doesnotplayaparticularrole.Thisisnotthecaseforthelinearresponse todisturbancesoftheflowattheexitofthereactionzone

δ

μ

b

(

τ

)

.Foraslightlyoverdrivendetonation

μ

b

>

0 inthestable domainb

<

bc,

|δμ

b

|

<

μ

b,onegets

2

δ

α

˙

τ

(

τ

)

=

0

−∞

Whenthetimescaleoftheforcing term

δ

μ

b

(

τ

)

islargerthanthatoftheinnerresponse,thetimedelayscanbeforgotten, leading to a quasi-steadyresponse. The linear response (37) has no meaning for a CJ wave since

μ

b

=

0. The relevant problemistodeterminethedecayofthepropagationvelocityofadetonationassociatedwithavelocityoftheburntgas relaxingtoits CJvalueintheplanarcase

μ

b

(

τ

)

→

0+.Thisproblemcanbesolved inthelimit(16) whenthesupporting pistonissuddenlyarrested[3].ForastableCJdetonation(b

<

bc),theendoftherelaxationtowardsthemarginalregimeis describedby

μ

b

(

τ

)

→

0+

:

2 y

(

τ

)

=

0

−∞ g

(ξ )

y

(

τ

+ ζ (ξ))

d

ξ

+

μ

2_b

(

τ

− |ζ

b

|)/

2 (38)

thesquareintheforcingtermbeingthesignatureofthemarginalityoftheCJwave.Using



_−∞0 g

(ξ )

d

ξ

=

1 thequasi-steady approximationof(38)

α

˙

τ

(

τ

− |ζ

b

|)

≈ ˙

α

τ

(

τ

)

yields,

y

≈

b

μ

2_b

(

τ

)/

2

⇔ ˙

α

τ

(

τ

)

≈

μ

2b

(

τ

)/

2 (39)

recoveringtheclassicalquadraticrelationbetweenthepropagationvelocityofslightlyoverdrivendetonationandthe veloc-ityoftheburntgasinthelimit

μ

b



1.ThedecaytotheCJregimeofamarginallystableorunstabledetonation(b

≈

bc) isdescribedbyanonlinearequation[3],

κ

=

0

,

μ

b

(

τ

)

→

0+

:

2 y

(

τ

)

=

0

−∞ G



ξ,

y

(

τ

+ ζ (ξ))



d

ξ

+

b

μ

2 b

(

τ

)/

2 (40)

writtenforaforcing termevolvingslowly,

μ

−_b1d

μ

b

/

d

τ

 |ζ

b

|

−1 sothat

μ

2b

(

τ

− |ζ

b

|)

→

μ

2b

(

τ

)

.Forstabledetonations,the quasi-steadyrelaxation(39) iseffectivelyobservedwhilenonlinearoscillationsofy

(

τ

)

withaperiodofoscillationoforder unityaresuperimposedtothequasi-steadyrelaxationformarginallyunstabledetonations.

It isworthmentioningthatan integralequation similarto(35),butwithadifferenttimelag,wasobtainedpreviously [10] intheoppositelimitofsmallheatrelease,namelyforstronglyoverdrivendetonationsandlargeheatreleaseM2oCJ



1. Inthislimit,incontrasttowhatissaidbelow(24) inthelimit(16),thecompressiblemodes

C

₊ and

C

₋ propagatefaster than the entropy wave. The integral equation is the same as(35) with the difference that the delay is now associated with the downstream running entropy wave, the loop being closed quasi-instantaneously by the upstream running

C

₊. A description of real detonations is obtainedwith a good quantitative accuracy by (35) into which thesum of the two delays(oftheentropywaveandofthecompressiblemode

C

₊)isintroduced,theroleofthecompressiblemode

C

₋being limitedtorelatethefluctuationsofpressureandvelocityasisindicatedbelow(25).Thisindicateshowusefulareasymptotic analyses todescribethedynamicsofrealdetonations,atleastqualitatively,evenwhen theycorrespondtolimitingvalues oftheparametersthatarenotwellrepresentativeofrealsituations.

6. AsymptoticanalysisofthedecaytotheCJregimeinsphericalgeometry

6.1. Conditionattheexitofthereactionzone

Thefirstdifficultymentionedattheendof§3.2concernstheevolutionoftheflowattheexitofthereactionzone.When approachingtheCJregime,thisflowisquasi-transonicinrealdetonations.Followingthepioneeringworksconcerningpure shockwaves,theproblemiseasilysolvedinplanargeometrywhendetonationsareconsideredasdiscontinuities.Thanksto themarginalcharacteroftheCJregime,theperturbationsoftheflowofburntgas,introducedbytheentropywaveandthe

C

− mode,arenegligible.Theentropybeingnot disturbed,therarefactionwave intheburntgas(inertflow)issimplythe continuation ofthecentred rarefactionwave(isentropicself-similarsolution)developingassoonasthepistonissuddenly arrested. This isalso truewhen theresponse ofthe inner structure ofthe detonationis takenintoaccount [3], yielding inthelimit(16),

μ

b

(

τ

)

=

α

(

τ

)/

τ

,sothatthe1967relaxationlaw

D − D

oCJ

∝

1

/

t

2 _{isrecoveredwhentheresponseofthe} innerstructureisneglected,

α

˙

τ

≈ (α

/

τ

)

2

/

2.

Insphericalgeometry,retainingtheterm

−

2u

/

r in(6),thedecayisdifferent.Undertheapproximationofadetonation considered asa discontinuity,there isno curvature-inducedmodification ofthe innerstructure. However, thedivergence of theflow

−

2u

/

r plays an importantrole;incontrasttothe planarcase, theCJ velocityisreachedata finitetime and ata finiteradius.In anenlighteningpaper,A. Liñanetal.(2012) [11] deciphered thetransitionbetweentwoself-similar solutions, namely the Sedov–Taylor solution to a strongnon-reactive blast wave andthe Zeldovich (1942)–Taylor (1950) solutiontoasphericalCJdetonation.Theyalsoshowedthat,thankstothetransoniccharacteroftheburntgasflowatthe detonation front, thefinal stage of thedecayis a localphenomenon. Thiscan also be viewedon(19) at theexitof the reactionzone w

=

0 for

μ



1 and

α

˙

τ



1,

Fig. 2. Sketchofthetrajectoriesinthephasespace( y–x)“propagationvelocity–radiusoftheleadshock”.Theboundaryofthedashedregionrepresentsthe sphericalCJwaves(45) whoseinnerstructureisinasteadystate,thebluesolidlinebeingtheupperbranch.Theblacksolidlinesarethequasi-steady trajectories(48).Line1representsasuccessoftheinitiationprocesswhileline2isafailure.Typicalresultsofdirectnumericalsimulations[5–7] forstable ormarginallyunstabledetonationsaresketchedbythethindarklines.Line(i)isasuccessoftheinitiationprocesswhilelines(ii)and(iii)arefailures. Thedifferencebetweenthenumericalresultsandlines1and2illustratestheunsteadinessoftheinnerstructure.

Thisrelationbetweenthetime derivativeoftheflowvelocity andtheradiusisthesameastheoneobtainedby A.Liñan etal.(2012)[11] foradetonationfrontofzerothickness(nomodificationtotheinnerstructure)andinalimitoppositeto (16),namelyforalargepropagationMachnumberMoCJ



1.

Asmentionedin§2.2,smallmodificationstotheinnerstructureofacurveddetonationhaveadrasticeffectifthe induc-tionishighlysensitivetotemperaturevariations.Asaresult,initiationofaCJdetonationbecomesacriticalphenomenon; dependingontheinitialconditions,successorfailure oftheinitiationprocessisproduced.Thiswas clearlypointedout in thelimit MoCJ



1 using thesquare-wavemodelofdetonationwhenunsteadiness intheinnerstructure isneglected, re-tainingonlythegeometricalmodificationsduetothecurvatureofthedetonationwave[5].Here,thepurposeistotakeinto accounttheunsteadyeffectsthatareexpectedtobeessentialatthecriticalcondition,asisshownbythedirectnumerical simulationssketchedinFig.2.Thiscanbeachievedanalyticallyforanychemicalkineticsinthelimitofsmallheatrelease byinvestigatingthesolutionto(19)–(20) using(27) andtheboundarycondition(41) attheexitofthereactionzone.We will justoutlinethe analysis;the details aregiven ina companionarticle[4].It isworth revisitingfirstthe sphericalCJ waveswhentheinnerstructureisassumedtobeinasteadystate.

6.2. SphericalCJwavesinasteadystate.CJpeninsula

Considerthesteadyversion of(19) for

κ

=

0,neglectingtheunsteadyterm

∂

μ

/∂

τ

,andlookforthesolutionsatisfying theboundaryconditions(20)–(26) with,accordingto(29),

ξ

b

= −

e−

bα˙τCJ_{.Thesteady-statesolutions willbedenoted with} an overbar.When theparameter b issufficiently large,thevelocity oftheCJ sphericalvelocityin steadystate,expressed intermsofthecurvature,

α

˙

τCJ

(

κ

)

,presentsaturningpointandtaketheformofa peninsulainthephasespacevelocity– radius,seeFig.2.ThesteadyplanarCJwave

μ

oCJ

(ξ )

=



ξ

−1

ω

oCJ

(ξ

)

d

ξ

in(31) issolutionto(19) for

κ

=

0 and

∂/∂

τ

=

0. Anticipatingthattheturningpoint(

κ

=

κ

∗) correspondsto

κ

∗ oforder1

/

b inthelimitb



1,theunknown

μ

infrontof

κ

intheright-handsideof(19) canbereplacedby

μ

oCJ

(ξ )

neartheturningpoint,

b



1

:

κ

∗

=

O

(

1

/

b

)

;

κ

/

κ

∗

=

O

(

1

)

⇒

μ κ

→

μ

oCJ

κ

(42)

Integratingthe so-modifiedversion of (19) fromthe endof thereaction

ξ

= −

e−bα˙τCJ _{to theNeumann state}

_ξ

=

_{0 then} yieldsthesolutionintheformpropagationvelocityversusradius

α

˙

τCJ

(

κ

)

.Introducingtheparameter

λ

∈ [

1

,

2

]

,

λ

≡

1

+

0

−1

μ

oCJ

(ξ )

d

ξ

(43)

withthereducedvelocityandradiusoftheshockfront, y andx,oforderunityneartheturningpoint,

b



1

:

y

≡

b

α

˙

τ

=

O

(

1

),

yCJ

≡ (

b

α

˙

τCJ

)

=

O

(

1

),

1

/

x

≡ (

b

κ

)λ

=

O

(

1

)

(44) theequationofthepeninsulainthephasespacevelocity–radius y–x forthesphericalCJdetonationswhoseinnerstructure isinsteadystate yCJ

(

x

)

takestheform