Dynamics of shock induced ignition in Fickett's model with chain-branching kinetics: influence of $χ$

0.94 MB·English

by  Justin Tang

Dynamics of shock induced ignition in Fickett’s model with chain-branching
kinetics: influence of χ
J.Tang&M.I.Radulescu
DepartmentofMechanicalEngineering,UniversityofOttawa,OttawaCanadaK1N6N5
Abstract
2 The problem of shock induced ignition by a piston is addressed in the framework of Fickett’s model for reactive
1 compressible flows, i.e., the reactive form of Burgers’ equation. An induction-reaction two-step chain-branching
0 modelisusedtostudythecouplingbetweentheenergyreleaseandthecompressiblehydrodynamicsoccurringduring
2
the shock ignition transient leading to a detonation. Owing to the model’s simplicity, the ignition and acceleration

n mechanismisexplainedusingthetwofamiliesofcharacteristicsadmittedbythemodel. Theenergyreleasealongthe
a
particlepathsprovidestheamplificationofforward-travellingpressurewaves.Thesewavespre-compressthemedium
J
intheinductionlayeraheadofthereactionzone,thereforechangingtheinductiondelaysofsuccessiveparticles. The
3
variationoftheinductiondelayprovidesthemodulationoftheamplificationoftheforwardtravellingpressurewaves
1
by controlling the residence time of the pressure waves in the reaction zone. A closed form analytical solution is

] obtained by the method of characteristics and high activation energy asymptotics. The acceleration of the reaction
h
zonewasfoundtobeproportionaltotheproductoftheactivationenergy,theratiooftheinductiontoreactiontimeand
p
- theheatrelease. Thisfindingprovidesatheoreticaljustificationfortheprevioususeofthisnon-dimensionalnumber
h tocharacterizetheignitionregimesobservedexperimentallyindetonationsandshockinducedignitionphenomena.
t
a Numericalsimulationsarepresentedandanalysed. Bothsubsonicandsupersonicinternalflamepropagationregimes
m areobserved,consistentwithexperimentandpreviousreactiveEulermodels.
[
Keywords: Shockinducedignition,Fickettmodel,methodofcharacteristics,activationenergyasymptotics,

1 amplificationmechanism,coherentpressurewaveamplification,Clarkeequations
v
2
1
9 1. Introduction
2
. The present study focuses on the problem of shock induced ignition by a piston suddenly accelerated into a
1
reactive medium. This problem has attracted significant interest in recent years due to its universality in capturing
0
2 thecanonicalprocessesoccurringinreactivemediainthepresenceofgas-dynamicdisturbances,i.e. pressurewaves
1 [1,2,3,4,5,6,7,8,9,10]. Theproblemisalsoequivalenttoshockinducedignitionbyareflectedshock[11].
: Of particular interest to us is to determine the parameter that controls the different one-dimensional ignition
v
i regimes that can be realized experimentally [11]. For example, Meyer and Oppenheim first suggested that the sen-
X
sitivityoftheinductiontimetotemperaturefluctuations,withrespecttothedurationoftheexothermicpowerpulse
r controlstheignitionprocess[12]. Mathematically,thiscanbewrittenastheproductofthereducedactivationenergy
a
controllingtheinductionreactionsandtheratioofinductiontoreactiontimes[13],namely,
t E
χ= ind a (1)
t RT
reac s
Thesameparameterwasalsoidentifiedinthestabilityanalysesofself-propagatingdetonationwaves[14,15,16]and
has also been used for correlating the ignition regimes and stability of cellular detonations [13]. Typically, a value
ofχ ≈ 10correspondstotheonsetofstrongpressurewavesdrivenduringignitionbehindtheshocksandthelossof
stabilityofdetonationwaves. However,thephysicalmechanismresponsibleforchangesoftheshock-ignitionregime
whenthisparameterisvariedremainunclear.
PreprintsubmittedtoProceedingsoftheCombustionInstitute January16,2012
Toaddresstherelativeimportanceoftheinductionzonesensitivityandthedurationoftheexothermicstage,the
presentstudyassumesthemedium’sdecompositionisviaatwostepchainbranchingmodel,withdistinctinduction
andreactionstages. Thisistypicalofmostreactivegasesandcanbeshowntorationallyderivefromagenericthree
stepchainbranchingmodel[17,18]. TheinvestigationthusbuildsuponthestudiesofDoldandKapila[6], Sharpe
[10]andBlytheetal.[4],whohaveaddressedthisprobleminthepastusingthereactiveEulerequationscoupledwith
twoandthree-stepchain-branchingmodels.
Althoughsignificantinsighthasbeengainedinthesepaststudies,thepresentworkfocusesonthemuchsimpler
Fickettmodelforcompressiblereactivemedia[19,20].ThismodelisanextensionofBurgers’equation,aprototypical
model for inert compressible hydrodynamics, to a reactive medium. Due to its simplicity, this model can serve to
studythebasicmechanismwithoutthecomplicationsofthehighlynon-linearreactiveEulerequations. Forexample,
this simple model has allowed us in the past to clarify the instability mechanism of pulsating detonations and has
reproducedtheuniversalperioddoublingroutetochaosintravellingdetonations[21]. Usingthismodel,weattempt
todetermineinthesimplestformpossiblehowtheχparametercontrolsthedynamicsoftheignitionprocessfollowing
shockcompression.
2. Fickett’sreactionmodel
Fickett’sanaloguemodeltoone-dimensionalinviscidreactivecompressibleflowisgivenby[20,21]:
∂ρ+∂ p=0 (2)
t x
The model is formulated in Lagrangian coordinates, x denotes a material coordinate and t represents time [19, 20].
Thevariableρhasthemeaningofdensityinthemodel. Theterm pin(2)hasthemeaningofpressure,see[20]. For
simplicity,wechoosethegenericequationofstateproposedbyFickett,thatis
1(cid:16) (cid:17)
p= ρ2+λ Q (3)
r
2
where λ is a reaction progress variable ranging from 0 (unreacted) to 1 (reacted). The model has the property that
r
pressureincreaseswithincreasingdensity,orreleaseofenergyatconstantdensity. Notethatonerecoverstheinviscid
Burgers’equationbysettingQ=0. AsimilarsimplifiedreactiveflowmodelwasalsoproposedbyMajda[22],with
slightdifferencesinformulation.
In the present work, we adopt the two step reaction model proposed by Radulescu and Tang [21] in order to
capture the main features of chain-branching reactions [23]. Following the leading shock, we assume the existence
of a thermally neutral induction zone, whose duration depends on the medium’s local compression. Following the
inductionstage, weassumeanexothermicreactionthatproceedsatastate-independentconstantrate. Theresulting
genericinduction-reactionmodelweproposeisthus:
∂tλi =−H(λi)e(cid:16)ρ−(cid:15)1(cid:17) (4)
∂λ ≡r= K(1−H(λ))(1−λ )ν (5)
t r i r
The variables have been non-dimensionalized by the characteristic state behind the initial shock. Time is thus
non-dimensionalizedbytheinductiondelaybehindtheinitialshock. Theparameter(cid:15)istheinverseactivationdensity
of the induction reactions. The parameter K, through our non-dimensionalization, is the ratio of the induction and
reactiontimes. TheHeavisidefunctionH(·)controlsthetimingoftheonsetofthesecondexothermicreaction,which
startswhentheinductionvariableλ reaches0. Aheadoftheshock,λ =1andλ =0. Wearealsoassumingthatthe
i i r
inductionreactionisactivatedbythepassageoftheinertleadingshock. Thesystemtobesolvedisthus(2),(4)and
(5)withtheequationofstategivenby(3).
Forthepistoninitiationproblemofinterest,thepistonpathcorrespondsto x = 0(seeFig. 1). Inordertomodel
a steady piston, we set ρ(x = 0,t) = 1. Without loss of generality, we assume the strong shock limit and take the
quiescent gas ahead of the shock as ρ(x,t = 0) = 0, λ(x,t = 0) = 1 and λ (x,t = 0) = 0. For a non-reactive
i r
medium, Burgers’ equation with the prescribed initial and boundary conditions result in a constant strength shock,
2
Figure1:Characteristict−xplotofshockinitiationforK=0.2,(cid:15)=0.2;theredlinerepresentsthemainshockfrontts(x),thegreenlinerepresents
thefiretrajectoryt∗(x);thesolidlinesareC+characteristics;thebrokenlineistheanalyticalpredictionforthefiretrajectorygivenby(26)for
t=O(1).
propagating with speed 1/2, followed by a constant state ρ = 1 [24]. When the medium is exothermic, the energy
which is deposited in the medium alters the medium’s mechanical response; we wish to analyse this mechanical
responsebelow.
3. Amplificationmechanism
WefirstsolvedtheabovesystemnumericallybyafinitevolumetechniquedescribedbyRadulescu&Tang[21].
Inthepresentstudy,agridresolutionof1600gridpointsperunitlengthwasused. WealsosetQ=1.
Figure1showsanexampleofthesolutionobtainednumericallyfor K = 0.2and(cid:15) = 0.2. Thevariationofρis
shown in the accompanying Video 1, but can be deduced directly, as it is given by the slope of the pressure waves
(seebelow). Thetrajectoriesoftheleadingshock(theredlineinthefigure), theonsetofenergyrelease, whichwe
willhenceforthcallfire(greenline),andtheinternalpressurewaves(blacklines)areshowninthet-xdiagram. The
pressurewaveemittedattheonsetofreactionsat(x = 0,t = 1)markstheendoftheinertsolution. Thesubsequent
solutionconsistsofanacceleratingshockandfire,whicheventuallyformadetonationwave.
ThephysicalmechanismresponsiblefortheamplificationshowninFig. 1isbestexplainedbyfirsttransforming
thegoverningequationsincharacteristicformandintegrating(4)fromthetimeaparticlecrossestheshockatt=t (x)
s
tothetimeatwhichtheexothermicreactionsbeginatthefiret=t∗(x). Weobtain:
dp 1 dx
= rQ along =ρ (6)
dt 2 dt
(cid:90) t∗ (cid:32)ρ−1(cid:33)
1= exp (7)
(cid:15)
ts
dλ dx
r =r along =0 (8)
dt dt
By inspection of (6), we deduce that the pressure waves propagating forward with acoustic speed dx/dt = ρ
amplifywhenevertheytravelthroughareactivefield. ThisisthecaseofthepressurewavesinFig. 1originatingat
3
Figure2:Space-timediagramillustratingtherun-awayprocessatearlytimes.AC+pressurewaveoriginatingatthepiston(x=0)amplifieswhile
travellingthroughthereactionzone(R).Itsemergenceintotheinductionzonemodifiesthepostshockstatefrom(S)toa(D).Thisshortensthe
ignitiondelay(t∗−ts).TheshorterignitiondelayinturnprovidesamorecoherentamplificationofC+pressurewavesinthereactionzone.Zones
OandPdenotetheinitialandproductstates,respectively.
thepistononcethereactionshavebegunatt=1. Therateofchangeofthepressurewave’samplitudevarieswiththe
heatadditionQattherater.Indeed,Figure1showsthesecharacteristicsacceleratingforward.Foraconstantreaction
rate,thepressurewavesamplificationisproportionaltotheirresidencetimeinthereactionzone. Oncethesewaves
exitthereactionzone,theypropagateintotheinductionzone,compressingtheunreactedmaterialandstrengthening
theleadingshock.
Theotherkeyelementintheamplificationstageisthedependenceoftheinductiondelayonthispre-compression,
via(7).Forlow(cid:15)(i.e.,highactivationenergy),theinductiondelaycanbesignificantlyreduced.Thetendencytowards
loweringtheignitiondelaypermitsthepressurewavestoremaininthereactionzoneforlongertimes,ascanbeseen
from Fig. (1). This provides a more coherent amplification of these pressure waves, which in turn enhances the
compressionintheinductionzoneandthatoftheshock. Thefeedbackmechanismisthusbetweenthepressurewaves
propagatingforwardandthemodificationoftheonsetofreaction, whichmodulatestheamplificationamount. This
mechanismwasalsofoundcentralincontrollingthedetonationwavestabilitybyRadulescuandTang[21].
4. Asymptoticanalysis
The early transient dynamics illustrated in Fig. 1 and discussed above have been summarized in Figure 2. The
arguments presented can now be posed mathematically in an asymptotic analysis, which assumes that the inverse
activationenergyissmall(i.e.,(cid:15) << 1). TheframeworkissimilartothatusedbySharpe[10]forthereactiveEuler
equations. Theanalysisaimstocapturethecouplingbetweentheamplificationofforwardtravellingpressurewaves
andtheresultingshorteningoftheignitiondelaydiscussedabove. Weassumethattheenergyreleaserateisslow,i.e.
K
ζ ≡ =O(1) (9)
(cid:15)
4
Thismakesthesolutionaccurateontimescalesoforderunity(i.e. inductiontimescales). Theanalysisisalsovalid
forfastheatrelease, whichcanbeobtainedby rescalingthetimevariableforfastertimescales[10], restrictingthe
resulttoearliertimes. Inbothcases,whatweareinterestedinistheaccelerationofthefire. Wewillalsorestrictour
analysistotheearlyamplificationoccurringbeforetheleadshockhasbeenmodifiedbythearrivalofapressurewave
originatinginthereactionzone. Weexpandthedensityandreactionprogressvariableintheform
ρ(x,t)=ρ (x,t)+(cid:15)ρ (x,t)+O((cid:15)2) (10)
1 2
λ (x,t)=λ (x,t)+(cid:15)λ (x,t)+O((cid:15)2) (11)
r r1 r2
t∗(x)=t∗(x)+O((cid:15)) (12)
1
Replacing(10),(11)and(12)inthegoverningequationsandboundaryconditions,weobtaintoleadingorder:
ρ (x,t)=1 (13)
1
λ (x,t)=0 (14)
r1
t∗(x)=1+2x (15)
1
Thiscorrespondstotheinertsolutionofapistondrivenshockwave.
WiththeO(1)solutionobtained,theO((cid:15))problemtobesolvedis:
d (cid:18) Q(cid:19) 1 dx
ρ +λ = ζQ along =1 (16)
dt 2 r22 2 dt
dλ dx
r2 =ζ along =0 (17)
dt dt
ρ (x=0,t)=0 (18)
2
λ (x,t<t∗)=0 (19)
r2
(cid:90) t∗(x)
eρ2dt=1 (20)
ts(x)
The resulting problem is one of linear acoustics, which is very similar in form to what is obtained for the reactive
Eulercase,i.e.,theClarkeequations[5,10]. Accordingto(16),theforwardfacingpressurewavesallpropagateatthe
constantspeed(ofunity),butamplifyatconstantrateinthereactionzoneduetoenergyrelease. Thesolutioncanbe
obtainedbythemethodofcharacteristics. Inthereactionzone,fort > t∗ (zoneRinFigure2),thereactionprogress
variablecanbeimmediatelyobtained.
λ (x,t>t∗)=ζ(t−t∗(x)) (21)
r2
Substituting(21)into(16)andusingtheboundarycondition(18), weobtainthevariationofdensityinthereaction
zone:
ζQ
ρ = (t∗(x)−1) (22)
2
2
Oncethedensityisknowninthereactionzone,thevalueofdensityinthedisturbedregionD(seeFigure2)canbe
foundbyextendingtheC+ characteristicsintothisregion. InregionD,λ =0andtheC+ characteristicrelation(16)
2
requiresthatdensityisconstantalongacharacteristic. Theconstantisevaluatedusing(22)attheintersectionofthe
C+characteristicwiththepatht=t∗(x). Denotingthisreferencepositionx (x,t)(seeFigure2),weget:
ref
ζQ(cid:16) (cid:17)
ρ (x,t)= t∗(x (x,t)−1 (23)
2 ref
2
withx givenimplicitlyfromthetrajectoryoftheC+characteristic,i.e.,
ref
x−x
ref =1 (24)
t−t∗(x )
ref
5
Region S is bounded by the shock t (x) = 2x and the first C+ disturbance originating at (x = 0,t = 1), given by
s
t = 1+ x. In this region, using the method of characteristics, we get ρ (x,t) = λ (x,t) = 0. The solution is now
2 2
complete,andtheintegralrelation(20)cannowbewrittenas
(cid:90) t∗(x)
eQ2ζ(t∗(xref(x,t)−1)dt−x=0 (25)
1+x
Thisexpression,alongwith(24),givethetrajectoryt∗(x)implicitly. AdoptingSharpe’siterativestrategy[10]tofind
t∗(x), we substitute the first order approximation t∗(x) in the integrand and solve for the correction appearing in the
1
integral’supperbound. Fromthefirstiteration,wefindthefollowingapproximation:
1
t∗(x)≈1+x+ log(1+Qζx) (26)
Qζ
ThisexpressionisshowninFigure1;itreproducesquiteaccuratelytheaccelerationofthereactionpath. Evaluating
theaccelerationattheorigin,weobtain:
Qζ 1K 1
a= = Q= χQ (27)
8 8 (cid:15) 8
Thisconcludesouranalysis, whichshowsquantitativelythefeedbackmechanismdiscussedintheprevioussection.
Moreimportantly,however,itshowsexplicitlythattheaccelerationofthereactionzoneisgivenastheproductofχ
andtheheatreleaseQ.
5. Influenceofχandignitionregimes
To illustrate the effect of varying χ on the process of ignition and detonation formation, we have conducted
numericalsimulationsbyvaryingboth(cid:15) andK. Thispermitsustodeterminetheevolutionoftheflowfieldonlonger
timescales,whentheanalysispresentedabovebreaksdown(i.e. χ >> 1). Keeping(cid:15) = 0.2asaboveandincreasing
K toavalueof2providesamorerapidamplification, asshowninFigure3. TheaccompanyingVideo2showsthe
evolutionoftheρfield. Thiscorrespondstoχ=10. Ascanbeseeninthefigure,ashockwavenowformsinsidethe
inductionzonefromtheconfluenceoftheforwardtravellingpressurewaves. Thisisduetothestrongeramplification
ofthepressurewavesinthereactionzone. Thissubsequentmotionoftheinternalshockandfirearenowinphase,
andpropagatequasi-steadily. Themodelthusrecoverstheinternalreactionwavedynamicsobservedexperimentally
[11] and for the reactive Euler equations [10], where both subsonic and supersonic waves can be established. The
supersonicwavesoccurwhentheinitialaccelerationofthefireissufficientlylarge.
Furtherincreaseintherateofenergyreleaserate K leadstoamorepromptamplificationofthepressurewaves.
Figure4showstheevolutionoftheflowfieldwhenK =5,i.e.χ=50.TheaccompanyingVideo3showstheevolution
oftheρfield. Theinternalshockwavenowformsatanearliertime. Thestrongercompressionintheinductionzone
now gives rise to a significant reduction in the induction delay. The onset of energy release now becomes in phase
with the internal shock motion, and the two slowly accelerate. The arrival of this internal shock to the lead shock
transformstheleadshockintoadetonationwave.
This interior wave takes the form of a quasi-steady weak internal detonation. It is not self-sustaining, as the
expiration of the induction delay essentially sets its trajectory. For reference, a Chapman-Jouguet internal wave
propagating into the induction zone medium (ρ = 1) would have a speed of 2 [20]. Instead, we find that the speed
ofthisinternaldetonationwaveisapproximately1.5whenitencounterstheleadingshock. Forevenlargervaluesof
K,weseethesameevolution,buttheinternalweakdetonationstakeonvelocitieshigherthanthecorrespondingCJ
values.
Wehavefurtherexploredtheeffectofvaryingboth(cid:15) and K inordertodetermineifauniquevalueofχcanbe
used to characterize the acceleration process. While this is so for χ = O(1) through the analysis presented above,
larger values of χ yield internal shock waves, invalidating the analysis once the shock has formed. Figure 5 shows
threedifferentcaseswhereboth(cid:15)andKarevariedsoastokeepaconstantχ=20. Forallthreecases,weseethatthe
formationoftheinternalshockandsubsequentamplificationareverysimilar,inspiteoflargedifferencesinreaction
rates. Thissuggeststhatindeedtheentireaccelerationprocesscanbecharacterizedbytheuniqueparameterχ. The
analysisoftheaccelerationofthefireinthepresenceoftheinternalshockisleftforfuturestudy.
6
Figure3:Characteristict−xplotofshockinitiationforK=2,(cid:15)=0.2;samelegendaswithFig.1,additionallywithbluelinerepresentingtheend
ofthereactionlayer.
Figure4:Characteristict−xplotofshockinitiationforK=5,(cid:15)=0.2;samelegendasforFig.3.
7
8
Figure5:Characteristict−xplotofshockinitiationforχ=20,a)K=2,(cid:15)=0.1.b)K=4,(cid:15)=0.2.c)K=8,(cid:15)=0.4.
6. Concludingremarks
Modelling the shock induced ignition using Fickett’s model permitted us to clearly illustrate the mechanism of
accelerationoftheinducedhotspot: theenergyreleasealongtheparticlepathsprovidesanamplificationofforward-
travellingpressurewaves. Thesewavespre-compresstheinductionlayeraheadofthereactionzone,thereforechang-
ing the induction delays. The variation of the induction delay then provides a modulation of the amplification of
forwardtravellingpressurewaves. Themodelsimplicityalsoallowedustoobtainananalyticalsolutionfortheac-
celerationofthereactionzone,whichwasfoundproportionaltotheproductoftheactivationenergy,theratioofthe
inductiontoreactiontimeandtheheatrelease. Thisfindingprovidesatheoreticaljustificationfortheprevioususeof
thisnon-dimensionalnumbertocharacterizetheignitionregimesobservedexperimentallyindetonationsandshock
inducedignitionphenomena.
Acknowledgements
We wish to thank the financial support of NSERC through a Discovery Grant to MIR and the support of the
H2CANNSERCStrategicNetworkofExcellence.
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