Review articles on the study of combustion. Combustion Theory and Modelling. Volumes Part 12 1-13

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Nonlinear cellular dynamics of the idealized detonation model: Regular
cells
G. J. Sharpe a; J. J. Quirk
a School of Mechanical Engineering, University of Leeds, Leeds, UK
First published on: 08 May 2007
To cite this Article Sharpe, G. J. and Quirk, J. J.(2008) 'Nonlinear cellular dynamics of the idealized detonation model:
Regular cells', Combustion Theory and Modelling, 12: 1, 1 — 21, First published on: 08 May 2007 (iFirst)
To link to this Article: DOI: 10.1080/13647830701335749
URL: http://dx.doi.org/10.1080/13647830701335749

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CombustionTheoryandModelling
Vol.12,No. 1,February2008,1–21
Nonlinear cellular dynamics of the idealized detonation
model: Regular cells
G.J.SHARPE∗†andJ.J.QUIRK‡
†SchoolofMechanicalEngineering,UniversityofLeeds,LeedsLS29JT,UK
‡LosAlamos,NewMexico,NM87544,USA
(Received27September2006;infinalform6March2007)
High-resolutionnumericalsimulationsofcellulardetonationsareperformedusingaparallelizedadap-
tivegridsolver,inthecasewherethechannelwidthisverywide.Inparticular,thenonlinearresponse
ofaweaklyunstableZeldovich–Neumann–Doering(ZND)detonationtotwo-dimensionalperturba-
tionsisstudiedinthecontextoftheidealizedone-stepchemistrymodel.Forrandomperturbations,
cellsappearwithacharacteristicsizeingoodagreementwiththatcorrespondingtothemaximum
0 growthratefromalinearstabilityanalysis.However,thecellsthengrowandequilibrateatalarger
1
0 size.Itisalsoshownthatthelinearanalysispredictswelltheratioofcelllengthstocellwidthsof
2
y thefullydevelopedcells.Theevolutionarydynamicsofthegrowtharenonethelessquiteslow,inthat
ar thedetonationneedstorunoftheorderof1000reactionlengthsbeforethefinalsizeandequilibrium
u
an stateisreached.Forsinusoidalperturbations,itisfoundthatthereisalargebandofwavelengths/cell
J sizeswhichcanpropagateoververylongdistances(∼1000reactionlengths).Byperturbingthefully
2
2 developedcellsofeachwavelength,itisfoundthatsmallercellsinthisrangeareunstabletosymmetry
36 breaking,whichagainresultsincellulargrowthtoalargerfinalsize.However,arangeoflargercell
:
18 sizesappeartobenonlinearlystable.Asaresultitisfoundthatthefinalcellsizeofthemodelis
: non-unique,evenforsuchaweaklyunstable,regularcellcase.Indeed,inthecasestudied,theequilib-
t
A riumcellsizevariesby100%withdifferentinitialconditions.Numericaldependenciesofthecellular
ed dynamicsarealsoexamined.
d
a
o
l
wn Keywords: Detonation;Numericalsimulation;Instabilities
o
D
1. Introduction
Detonationsaresupersoniccombustionwaveswhichcanpropagatethroughreactivematerials.
Ingases,thesewavespropagateas‘cellular’detonations,wherethefrontismulti-dimensional
andtime-dependent[1].Ifthewallsofashocktubearecoatedwithsoot,then,asitpropagates
down the tube, the detonation etches diamond shaped patterns (known as detonation cells)
in the soot (see [2] for a large collection of soot records from different mixtures and initial
conditions).Inthiscellularconfiguration,transverseshockwaveswhichextendbackintothe
reactionzoneintersectthedetonationshockfrontatso-calledtriplepoints,dividingthefront
itself into weaker (incident) and stronger (Mach stem) shock regions. It is the tracks of the
triple points, which are regions of high vorticity, that produce the cell patterns in the soot
films[1].
∗Correspondingauthor.E-mail:[e-mail]
CombustionTheoryandModelling
ISSN:1364-7830(print),1741-3559(online)(cid:3)c 2008Taylor&Francis
http://www.tandf.co.uk/journals
DOI:10.1080/13647830701335749
2 G.J.SharpeandJ.J.Quirk
The cell patterns in experimental soot traces are loosely classified as either regular or
irregular(e.g.[2–5]).Infuelmixtureswithregularcells,thereisasinglecharacteristiclength-
scaleofthecells,whichishighlyreproduciblebetweenexperimentsunderthesameconditions.
Forirregularmixtures,however,thereisrangeofdifferentlength-scalesforthecellsthatappear
on the soot trace. For such mixtures, an ‘average’ cell size may be determined, but there is
someambiguityindefiningsuchaquantity(M.Radulescu,J.Austin,privatecommunications).
Nevertheless,theconceptofacellsizecharacteristictoagivenmixture(andinitialstate)is
averyusefulone,duetoseveralempiricalobservationsofphenomenawhichappeartoscale
with this characteristic size. These correlations include the minimum values of (i) the tube
diameter in which the detonation can propagate in narrow, smooth walled tubes, (ii) the
critical tube diameter for a detonation to survive diffraction out of the end of the tube into
open space, (iii) the critical tube diameter for propagation in porous walled tubes, and (iv)
thesourceenergyrequiredfortheestablishmentofadetonationfromdirect(blast)initiation
(see [2, 4–6] for reviews). The correlations obtained differ between regular and irregular
mixtures.Forexample,inthedetonationdiffractionproblemthecriticaltubewidthisfound
tobe∼13λforregularmixtures,but∼40λforirregularmixtures,whereλisthecharacteristic
cellsizemeasuredfromexperimentalsootrecords[4,5].
Theoreticalprogressinunderstandingdetonationcellsingaseshasbeenachievedbycon-
sideringthecellularnatureofthefronttobetheresultofaninstabilityoftheunderlyingsteady,
0
1
0 one-dimensional (planar) wave solution. This approach includes linear stability analyses
2

ry [7–9],weaklynonlineartheories[10,11]anddirectnumericalsimulations[9,12–17].Dueto
a
nu theaboveempiricalcorrelationswithcellsize,amajorobjectiveoftheoreticalornumerical
a
J
studies is to predict and understand the dependence of cell size and regularity on the fuel
2
2
mixtureandinitialconditions.Muchofthisworkemploysanowstandardidealizedmodelas
6
3
8: definedbyErpenbeck[18],whichconsistsofanidealgas(polytropic)equationofstate,anda
1
: one-stepchemistrymodelwithArrheniuskinetics.Numericalsimulationsusingtheidealized
t
A
modelhaveshownthatthemodelappearstocontainmanyofthefeaturesobservedexperi-
d
e
ad mentally,includingcellsofvaryingregularityandsize,complextransversewaveinteractions,
o
nl andtheformationofunburntpocketsoffuel[9,12–17,19–21].
w
o
D However,determiningcellsizesfromnumericalsimulationsisproblematic.Firstly,ithas
been shown that obtaining detonation dynamics and cell sizes which are grid independent
requireshighresolutionofthedetonationreactionzone[16,17,22].Secondly,ifthenumerical
channelwidthisnotsufficientlylarge,thenthecomputedcellsizeisdeterminedbythechannel
widthitself,ratherthanbeingthenaturalorpreferredcellsizeofthemixture,duetomode-
locking(asisalsothecaseinexperimentsinnarrowtubes[2]).Thusifoneisinterestedin
determiningthe‘natural’cellsize,achannelwidthmanytimeslargerthanthissizeisneeded
toensuretheresultisindependentofthechannelwidth.However,thecharacteristiccellsize
istypicallyanorderofmagnitudelargerthantheone-dimensionaldetonationreactionzone
length.Thusthenumericaldomainwidthneedstobeabouttwoordersofmagnitudelarger
thanthereactionzonelength.Combinedwiththefactthathighresolutionofthereactionzone
is required, this makes accurate and reliable calculations of cell sizes computationally very
expensive,evenfordynamicallyadaptivemeshrefinementcodes.Indeed,evenforthesimple
idealizedmodel,calculationsofthecellulardynamicshaveonlybeenperformedusingeither
widechannelsbutlow(possiblyinsufficient)resolution[19]orhigherresolutioncalculations
in narrower channels [9, 15, 16, 21]. As discussed above, neither of these are sufficient to
provide reliable intrinsic cell sizes without higher resolution, wide channel calculations to
compare with. A notable exception is a case presented in [15], using a domain sufficiently
widesuchthatseveralcellsformedacrossit,togetherwithmoderateresolution.However,it
is unclear whether the run time of this calculation was sufficient to ensure that the cellular
evolutionwasnotstillinatransitorystate(seebelow).
Nonlinearcellulardynamicsoftheidealizeddetonationmodel 3
Sinceresolved,longtimecalculationsinverywidechannelshavenotbeenperformedeven
in two-dimensions, a number of even more fundamental questions exist about the idealized
model itself. For example, one question is whether the concept of a well defined natural or
preferredcellsizeevenexistsfortheidealizedmodel,i.e.forgiveninitialconditionsdoesthe
cellulardynamicseverreachafinalequilibriumstateatlargeenoughtimes?Calculationsin
moderately wide channels indicate that the cell size can keep changing between larger and
smallervaluesevenafterverylongtimes[15,16].Itisunclearwhetherthisissimplydueto
the constraints on the cellular dynamics from the presence of the walls in narrow channels
(i.e.amodeclosetothepreferredvalueisnotavailableinthesetypesofchannels)[16].Even
ifthecellsizedoesreachanequilibriumvalueinwiderchannels,doesthisequilibriumvalue
dependmarkedlyontheinitialperturbation?Also,howlongdoesthedetonationhavetorun
fortoensurethatatrulyequilibriumvaluehasbeenachieved,inotherwordshowsloworfast
arethedynamicsofcellularevolution?Furthermore,manydifferentnumericalschemeshave
been used to compute cellular detonations with the idealized model [9, 12, 14–16, 19–21].
Hencefundamentalnumericalquestionsalsoexistsuchaswhetherthecellulardynamicsare
sensitivetotheflavourofthenumericalmethodused.
The purpose of this paper is to begin to examine and understand these issues with the
idealized model, by using a parallel and adaptive mesh refinement approach to solve the
idealizedmodelinwidechannels.Inthisfirstpaper,weconsideraweaklyunstablecasewith
0
1
0 highlyregularcells,sothatitisclearwhetherawelldefinedequilibriumcellsizeexistsfor
2

ry theidealizedmodelandhowthefinalcellsizedependsoninitialperturbations,i.e.avoiding
a
nu the complication of multiple cell length-scales as in irregular cell cases. Furthermore, such
a
J
weakly unstable cases avoid the necessity of using computationally prohibitive resolutions
2
2
inordertocalculatethecellulardynamicscorrectly.Indeed,forirregulardetonationcases,it
6
3
8: appearsthatverysmallscalesmaybeimportantintherealphysicalpropagationmechanisms
1
: (e.g.enhancedratesofburningduetoturbulentmixingandsmall-scalewrinklingattheedges
t
A
ofunburnedpocketsoffuelandatsliplines)[4,5,23],anditwouldbecurrentlyunfeasible
d
e
ad tocapturesuchscalesinadirectnumericalsimulation.
o
nl Hereweonlyconsiderthetwo-dimensionalcase,sinceourpurposeistoexaminetheprop-
w
o
D erties of the idealized model system, rather than to model a real experiment. Clearly, it is
importanttothoroughlyunderstandthenonlinearpropertiesofthetwo-dimensionalsolutions
beforemovingontothree-dimensions.Furthermore,resolvedthree-dimensionalcalculations
would be currently very computationally challenging for the reasons outlined above. It is
worth noting that two-dimensional (so-called ‘planar’) detonations have been observed in
experiments of very regular mixtures in rectangular tubes [1, 24], but are very rare unless
the narrower dimension is much smaller than the characteristic cell size, although then the
detonationbecomesmarginal[24].Inmostinstances,thestructureisthree-dimensional(‘rect-
angular’detonations)withorthogonalfamiliesoftransversewaves,producingcellpatternson
bothsetsofwalls,whicharetypicallyoutofphase[24].However,Strehlow[24]arguesthatthe
rectangularcasesconsistsoftwoindependentplanarmodes,inwhichcasetwo-dimensional
studiesarealsorelevanttothesethree-dimensionaldetonations.
Anotherquestiononemayaskabouttheidealizedmodeliswhetherornottherangeand
typesofcellulardynamicsitpredictsdifferfromthosefoundfromusingmorerealistic,com-
plex chemistry models. It is well known that the idealized model cannot properly describe
theheatreleasestructuresofsomechain-branchingmixtures(e.g.[3,22]).Nevertheless,one-
dimensional calculations of pulsating detonations with one-step and more realistic kinetic
models[22,25]insomecasesshowqualitativelysimilardynamics,e.g.limitcycleoscilla-
tions with period doubling bifurcations as the wave becomes more unstable (note that one
advantage of the one-step model is that more complex models may actually require signifi-
cantlyhigherresolutioninordertoobtainwellconvergedsolutionsforthesetypesofdynamics
4 G.J.SharpeandJ.J.Quirk
[22,25]).However,astheinstabilityincreases,theone-stepmodelappearstohavespurious
featureswhicharenotpresentforthemorerealisticmodels,suchasthelackofawelldefined
one-dimensional detonability limit. Similarly, one can ask whether the idealized model can
essentially describe the full complement of experimentally observed cellular dynamics, by
suitablechoicesoftheparameters.While,wedonotattempttoanswerthisquestiondirectly
here,anotherpurposeofthecurrentstudyistoprovideresultsusingtheidealizedmodelwhich
canthenbecomparedandcontrastedwithfuturestudiesusingmorecomplexmodels.
Insummary,itistobestressedherethatthispaperisconcernedwithpreliminaryevaluations
ofthemathematicalpropertiesoftheidealizedmodelitself,andnotwithtryingtodetermine
the cell size of any real mixtures at this stage. The structure of the paper is as follows: the
idealizedmodelisdescribedin§2;thenumericalstrategyisdiscussedin§3;theresultsare
givenin§4,while§5containsconclusionsandadiscussion.
2. Themodel
The idealized model [18] consists of a single reaction step with an Arrhenius form of the
reaction rate which releases all the heat, together with an ideal gas equation of state. The
0 non-dimensionalgoverningequationsarethus:
1
0
2
y ∂ρ ∂(ρu) ∂(ρv) ∂(ρu) ∂(p+ρu2) ∂(ρuv)
ar + + =0, + + =0,
nu ∂t ∂x ∂y ∂t ∂x ∂y
a
J

22 ∂(ρu) ∂(ρuv) ∂(p+ρv2) ∂E ∂(Eu+ pu) ∂(Ev+ pv)
6 + + =0, + + =0,
:3 ∂t ∂x ∂y ∂t ∂x ∂y
8
1
: ∂(ρλ) ∂(ρuλ) ∂(ρvλ)
At + + =−Kρλexp(−Ea/T), (1)
d ∂t ∂x ∂y
e
d
a
lo whereuandvarethecomponentsofthefluidvelocityinthex and ydirectionsrespectively,
n
Dow ρ thedensity, pthepressure,
p 1
E = + ρu2−ρQ(1−λ),
γ −1 2
isthetotalenergyperunitvolume,λthereactionprogressvariable(withλ=1forunburnt
andλ=0forburnt),T = p/ρthetemperature, Qtheheatofreaction,γ the(constant)ratio
ofspecificheats, E theactivationtemperature,and K aconstantratecoefficient.
a
Theseequationshavebeenmadenon-dimensionalbychoosingthecharacteristicscalesfor
pressureanddensity(andhencetemperature)tobethoseintheinitial,quiescent(upstream)
fuelandK issetsuchthatunitlengthisthestandardhalf-reactionlength,denotedhereafterby
l1/2[18].Theseequationsadmitasolution,knownastheZNDdetonation,inwhichthewave
issteady(initsownrestframe)andone-dimensional.Fortheself-sustaining(or‘Chapman–
Jouguet’) detonations of interest here, the idealized model has three adjustable parameters,
Q,γ and E .Intermsofrepresentingaspecificmixture, Q andγ canbeset,forexample,
a
bysimultaneouslyfixingthedetonationspeedandratiooftemperaturesacrosstheshockin
theZNDwavetobeequaltothosefoundfromcomplexkineticcalculations(e.g.[23]).The
activation energy E can then be determined via the sensitivity of homogeneous induction
a
timestotemperaturesneartheZNDshocktemperature[2,4,23].HoweverE alsodetermines
a
theprofilesoftheZNDheatreleasestructure.Asdiscussedabove,theidealizedmodelcannot
capturetherealreactionzonestructureofsomefuels.Henceintermsofcellulardynamics,
the choice of E as a tunable parameter would perhaps be better determined by obtaining
a
Nonlinearcellulardynamicsoftheidealizeddetonationmodel 5
Figure1. (a)Pressure(solidline)andtemperature(dashedline)and(b)λprofilesforZNDsolutionwhenEa=20,
Q=11.5,γ =1.54.
agreementwithexperimentalcellrecords.Again,itisnotclearwhethertheidealizedmodel
hassufficientcomplexitytobeabletoaccomplishthisforallfuels.
As we will see, the nonlinear response of the ZND to two-dimensional perturbation is
veryrichandhenceinthisfirstpaperweconsiderasingleparametersetindetail.Although
0
1 wearenotattemptingtosimulatearealfuelinthispaper,sincewewishtostudyaweakly
0
y 2 unstable,regularcellcase,wewereguidedinourchoiceof Q andγ bythevaluesobtained
r
nua forthoseofamixture2H2+O2+70%Ar,whichhasextremelyregularcells.Fittingthevalues
Ja of Q and γ in the way described above to the results of a detailed chemistry calculation

22 gives Q =11.5,γ =1.54(M.Radulescu,privatecommunication).Notethatalargeγ hasa
6
:3 significantstabilizingeffect[8],sothisisindeedconsistentwithaweaklyunstablecase.The
8
1
activationenergycanthenbechosentoenhancetheregularityofthecells,sincedecreasing
:
d At Ea alsohasasignificantstabilizingeffect[7,8].Hereweuse Ea =20.Figure1showsthe
de ZNDprofilesforthisparameterset.
a
o
l Forrealisticparameters,linearstabilityanalysesshowthattheZNDwaveisalwaysunstable
n
w
Do to at least a band of wavelengths of multi-dimensional perturbations of the normal modes
formexp(σt)exp(iky)[7–9],wherek =2π/W isthewavenumberofthedisturbanceinthe
transversedirection(orwavelengthW),Re(σ)isthegrowthrateandIm(σ )isthefrequency.
Figure2showsthelineargrowthrateandfrequencyversusthewavelengthofperturbationfor
ourparameterset,calculatedusingthemethoddescribeddescribedin[7].Inthiscase,there
isasingleunstablemode,whichisunstabletoabandofwavelengthsbetween1.43and28.2.
The growth rate is maximum for W =6.5, where Re(σ)=0.184. However, the maximum
in the dispersion relation is rather flat, with wavelengths in the range 6< W <7.5 having
growthrateswithinabout1%ofthemaximumvalue.Thefrequencydecreaseswithincreasing
wavelength.
3. Numericalmethodandinitialandboundaryconditions
In order to perform the simulations with sufficiently high resolution of the reaction zone,
but indomains which are much larger than the characteristic reaction zone length-scale,
we employ a parallelized adaptive mesh refinement strategy [26], within Quirk’s Amrita
computational environment [27, 28]. The mesh refinement scheme employs a hierarchical
systemofmeshpatches.Thecoarsestgrid,whichcoversthewholedomain,ischosentohave
ameshspacingcorrespondingtotwogridpointsperl1/2.Refinementwasemployedsoasto
trackandresolvethemainfeaturesoftheflow(reactionzones,shocksandsliplines).Unless
6 G.J.SharpeandJ.J.Quirk
Figure2. (a)Growthrateand(b)frequencyfromalinearstabilityanalysis,asfunctionsofthewavelengthof
perturbationforEa=20,Q=11.5,γ =1.54.
otherwisespecified,thefinestgridwaschosensoastogivearesolutionof32pointsperl1/2
intheseregions.Resolutionstudies(see§4.2)showedthatthiswassufficienttoobtaingrid
independentconclusionsandcellsizesfortheweaklyunstablecaseconsidered.Acalculation
wasalsoperformedsuchthatthegridintheentireregionbehindthereactionzonewaskept
0
1
20 to a minimum of eight points per l1/2, inorder to ensure that the derefinement of the grid

ry behind the reaction zone does not influence the solution. The cellular dynamics/numerical
a
nu soottracewereunchanged.AnumericalschemebasedonRoe’slinearizedRiemannsolver
a
J
wasemployedusingaminmodlimiterfunction[29],althoughtheeffectofdifferentchoices
2
6 2 forthelimiterfunctionandsolverisinvestigatedin§4.3.
3
8: Since we are concerned with intrinsic cell sizes, unless otherwise specified, the domain
1
t: width was taken to be 240l1/2. This is a very wide channel as compared to most previous
A
simulations,andissufficientlywidetobeanorderofmagnitudelargerthanthecharacteristic
d
e
ad cellsizes.Severalofthecaseswerealsoranforadomainwidthhalfthissize,andonecasewas
o
wnl re-calculatedwithadomainofwidth480l1/2(see§4.1)toensurethatthischoiceissufficiently
o
D widethattheconclusionsareindependentofthewidth.Wealsouseaverylongdomain(3000
inthex-direction),toensurethatthewavehassufficienttimeforanyinitialtransientstohave
decayedbeforethewavereachestheendofthechannelat x =3000.Itisimportanttonote
that,whileingeneralin§4weonlyshowpartsofthesootrecord/domainthatarepertinentto
thediscussion,ineachcasethecalculationwasalwaysallowedtorununtilthedetonationhad
reachedtheendofthedomainatx =3000.Thisistoensurethatanequilibriumconfiguration
has been reached (typically this means that the detonation has propagated for many tens of
cellcyclessubsequenttothefinalcellsizehavingbeenreached).Physicalreactionlengthsare
typicallyoftheorderofmillimetrestocentimetres,sothischoiceofdomainsizecorresponds
to a shock tube of at least 0.24 m by 3 m. We stress here that our purpose is not simply to
perform bigger and better simulations (in terms of domain sizes and numerical resolution)
thanpreviouscalculations,butitistoperformsufficientlylargecalculationssuchthatdefinite
conclusionscanforthefirsttimebereachedaboutcellsizes,freefromanydomainsizeand
resolutiondependencies.
The boundary conditions on y =0 and y =240 are symmetry conditions, so that these
boundariesrepresentsolid,reflectingwalls.Theflowstateatx =0 isprescribedtobethatof
thepost-detonation(CJ)state,whiletheboundaryatx =3000isafree-flow(linearextrapo-
lation)conditions.
The initial conditions are given by placing the ZND solution onto the grid, such that the
detonationisrunninglefttorightinthex-directionasinfigure1,whiletheinitialpositionofthe
shockischosentobex =100.Thisissufficientlyfaraheadoftherearboundaryatx =0that
Nonlinearcellulardynamicsoftheidealizeddetonationmodel 7
thisboundarydoesnotinfluencethecalculationduringtheruntime,sincethepost-detonation
flowisonaveragesonicwithrespecttothefrontforChapman–Jouguetdetonations.Notethe
boundaryatx =3000isinactiveduringthecalculation(i.e.whiletheshockremainswithin
thedomain).
Inthesimulations,theZNDsolutionisthenperturbedbyplacingaweakperturbationacross
thechanneljustaheadofitsinitialposition.Hereweconsidertwotypesofperturbation.The
firstisaweakrandomperturbationoftheform
λ=1−β r , |x −x |≤β /2 (2)
1 1 c 2
whereineachgridcellr takesarandomvaluebetweenzeroandone,andtheperturbation
1
is applied in a band of width β in the x-direction centred on x = x . Here we have used
2 c
x =108,andβ =0.1,andthenvariedβ .
c 1 2
Thesecondtypeofperturbationisoftheform
λ=1−α (1−cos(2πy/W))exp(−α (x −x )2), (3)
1 2 c
i.e. a sinusoidal perturbation in the y-direction with wavelength W, and a smooth (Gaus-
sian)distributioninthe x-direction.Herewefixα =0.1,α =2and x =108(sothatthe
1 2 c
0
1 Gaussianiscentredatadistanceof8aheadoftheinitialshockposition),andthenstudythe
0
2
y dependenceofthecellularevolutionontheperturbationwavelength W.Notethatthereare
r
ua onlydiscretewavelengthscompatiblewithourchoiceofdomainwidthandboundarycondi-
n
Ja tions,namely W =240/n,n =0.5,1,1.5,...,where n correspondstothenumberofcells

22 acrossthechannel.Thelinearstabilityanalysisthuspredictsthatn =37isthemostunstable
6
:3 mode,withmodesn =32to40allhavinglineargrowthrateswithin 1%ofthemaximum
8
1
value.
:
t
A Several methods have been used previously to produce numerical versions of the exper-
d
de imental soot records as a diagnostic for observing the cellular dynamics from simulations.
a
lo Theseincluderecordingthemaximumpressurereachedateachgridpointofthenumerical
n
w
o domain(e.g.[19–21]),thetotalheatreleasedateachpoint(e.g.[16,30]),ortheamplitudeof
D
theshockfront(e.g.[9,13]).Sincehighvorticityatthetriplepointsisbelievedtoberesponsi-
blefortheetchingofthetracksintheexperimentalsootfilms,herewehaveaoptedtorecord
themaximumvorticityreachedineachgridcellduringthecalculation[31].Notethat,once
the detonation front has passed by, the vorticity record is only stored on the coarsest grid,
sincetheadaptivemeshcoversonlythefront.Thevorticityrecordisplottedonalineargrey-
scaleinthispaperwithwhitecorrespondingtoavalueofzero,andblackforvaluesof12or
above,choseninordertoshowthecellboundariesclearly.
4. Simulationresults
WefirstconsidertheresponseoftheZNDdetonationtorandomperturbationoftheform(2).
Allmodesarepresentinsuchaperturbation,andhencethisavoidspreferentiallystimulating
specific wavelengths. We would thus expect that cells will appear with a wavelength (cell
width)closetothatcorrespondingtothemaximumlineargrowthrate[16].Theearlypartof
thenumericalsootrecordisdisplayedinfigure3forthecaseβ =4,whichshowsthatthe
2
cellsdoappearwithacharacteristicsize.Initially,thereareabout35cellsacrossthechannel,
ingoodagreementwiththelinearstabilityprediction.
Figure4(a)showstheanumericalschlieren[9,12,14,17,27]pictureofpartofthedetonation
frontduringthisinitialstagecorrespondingtoasnapshotatt =56.60.Thenumericalschlieren
8 G.J.SharpeandJ.J.Quirk
Figure3. Numericalsootrecordforrandomperturbationwithβ2=4.
consistsofalineargrey-scaleplotofthemagnitudeofthetotaldensitygradient,
(cid:1)(cid:2) (cid:3) (cid:2) (cid:3) (cid:4)
∂ρ 2 ∂ρ 2 1/2
+ ,
010 ∂x ∂y
2

y
r withwhitecorrespondingtozeroandblackcorrespondingtoavalueof4orabove,chosensoas
a
u
an tobestshowthemainfeaturesofthecellularstructure(shockfront,transversewavesandslip
J
2 linesseparatingmaterialwhichpassedthroughincidentshockandMachstempartsoffront).
2

6 Figure4(a)showsthatatthisstagethespacingsbetweentransversewavesarenon-uniform
3
:
18 atagiventime,andthesewavesarealsoofvariablestrength.Theschlierensarereminiscent

t: ofthosein[9,13]foraweaklyunstablecasewithlowactivationenergy,whogiveadetailed
A

d descriptionofthistypeofobservedstructure.
e
d
a
o
l
n
w
o
D
Figure4. Numericalschlierensofpartofdetonationfrontwithin40l1/2 fromtopwall,forrandomperturbation
withβ2=4,attimes(a)t=56.60and(b)t=469.26.Numericalresolutionofdetonationfrontis32points/l1/2.
Nonlinearcellulardynamicsoftheidealizeddetonationmodel 9
0
1
0
2

y
r
a
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Figure5. Numericalsootrecordforrandomperturbationwithβ2=4.
However, after the appearance of the cells, figure 3 also shows that the average cell size
subsequently begins to grow as the detonation propagates down the channel (cf. [16]). The
mechanismofthegrowthisduetotheinitialtransversewaveshavingdifferentstrengthsand
speedsandsomeoftheoriginaltransversewavesthenweakeningandeventuallydisappearing,
whileothersstrengthenorcatchupandmergewiththeneighbouringtransversewavemoving
inthesamedirection.
Figure5showsthesootrecordatsubsequenttimes,intheremainderofthedomain.The
numberofcellsacrossthechannelcontinuestogrowinitially,buteventuallybegintosaturate
ataconstantvalueof21cellsacrossthechannel.However,thecellsarenotallofthesame
sizeatthisstage.Furthermore,thedetonationhastorunaconsiderabledistance(∼1500l1/2)
beforethefinal,equilibriumnumberofcellsisreached.Thisdemonstratestheneedforvery
longtimecalculationsifoneistoensureonehasobtainedthefinalcellsizes.Indeed,thisrun
distanceislongerthanthoseusedinmanyprevioussimulations,e.g.[9,15].Ifthedetonation
wasonlyallowedtorunforacoupleofhundredl1/2 say,onewouldpredictapreferredcell
sizeclosertothatofthesmallerinitialcells(aspredictedbythelinearstability)thantothe
actualfullynonlinearcellsize.