Review articles on the study of combustion. Combustion Theory and Modelling. Volumes Part 9 1-13

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CombustionTheoryandModelling
Vol.9,February2005,1
Combustion Theory and Modelling
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CombustionTheoryandModelling
ISSN:1364-7830(print),1741-3559(online)(cid:1)c 2005Taylor&FrancisGroupLtd.
http://www.tandf.co.uk/journals
DOI:10.1080/13647830500168317
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A method for the transverse modulation of reactive flows with application
to combustion instability
S. Ducruix a; C. Rey a; S. Candel a
a EM2C Laboratory, CNRS, Ecole Centrale Paris, 92295 Chatenay-Malabry Cedex, France.
To cite this Article Ducruix, S., Rey, C. and Candel, S.(2005) 'A method for the transverse modulation of reactive flows
with application to combustion instability', Combustion Theory and Modelling, 9: 1, 5 — 22
To link to this Article: DOI: 10.1080/13647830500051950
URL: http://dx.doi.org/10.1080/13647830500051950

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distribution in any form to anyone is expressly forbidden.
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will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses
should be independently verified with primary sources. The publisher shall not be liable for any loss,
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CombustionTheoryandModelling
Vol.9,February2005,5–22
A method for the transverse modulation of reactive flows
with application to combustion instability
C.REY,S.DUCRUIX∗andS.CANDEL
EM2CLaboratory,CNRS,EcoleCentraleParis,92295Chatenay-MalabryCedex,France
(Received16April2003;infinalform11March2004)
Amethodallowingthetransverseexcitationofnon-reactiveorreactiveflowsisdescribed.Themethod
reliesonacharacteristicwavemodulationappliedonthelateralsidesofthecomputationaldomain.It
isshownthatthisprocedurecanbeusedtoinduceatransversesloshingmotionintheregionofinterest.
Twotwo-dimensionalgeometriesarestudied:inthefirst,theflowfeaturesoneortwowakesembedded
inahigh-speedstream;thesecondconfigurationinvolvesapremixedreactivejetflame.Theexcited
flowstructurecalculatedinthislastcaseisfoundtobesimilartothatobservedinanexperimentcarried
outpreviously.Asthesimulationsareperformedintwodimensions,theycannotdescribemanyofthe
0
01 processestakingplaceinaturbulentflow.Theyare,however,valuablewhentheflowisdominatedby
2
alarge-scaleorganizedmotioninducedbyatransverseacousticmodulation.Thepresentcalculations
y
ar indicatethatthemethodincombinationwithalargeeddysimulationflowsolvercouldbeusedtostudy
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n combustionresponsetotransverseacousticperturbations.Withadditionaldevelopmentsthismightbe
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A 1. Introduction
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nl High-frequencyinstabilitiesariseinpropulsionsystemsandare,inmostcases,theresultof
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Do strong interactions between combustion and transverse acoustic modes of the chamber. In
liquidpropellantrocketmotors,themostunstablemotionsarecoupledbytangentialorradial
modes(seeforexample[1,2]).Thetransversemodeproducesasloshingmotionofthemulti-
pleflamesoriginatingfromthechamberbackplane.Theseflamesresponddifferentiallytothe
acousticmodalstructureandthismayproducelocalizedregionsofintensecombustion[3,4].
The propellant streams injected through a ‘showerhead’ arrangement are so closely packed
thatonecansuspectthattheircollectiveinteractionsmaybeoneofthemechanismsdriving
instability in the motor. In standard configurations using coaxial injectors the inner flow is
usuallyliquidandatlowspeed,whiletheouterstreamisgaseousandflowsathighspeed.The
inner stream behaves like a wake with respect to the surrounding propellant. Neighbouring
wakes (or jets) may interact with one another generating local spots of heat release and
combustion fluctuations. If these reaction rate perturbations occur with the proper phase
andaresuitablylocatedwithrespecttotheacousticmode,theRayleighcriterionisfulfilledand
energyisfedintotheoscillationleadingtoamplificationofthemotion[5,6].Studiesofsingle
∗Correspondingauthor.E-mail:[email protected]
CombustionTheoryandModelling
ISSN:1364-7830(print),1741-3559(online)(cid:2)c 2005Taylor&FrancisGroupLtd.
http://www.tandf.co.uk/journals
DOI:10.1080/13647830500051950
6 C.Reyetal.
andmultiplejetsorwakesresponsetotransversewavescanhelpintheunderstandingofthe
couplingmechanism.
Tocalculatesuchcomplexunsteadyturbulentflows,itislogicaltouselargeeddysimulation
(LES) techniques. In such formulations the larger structures of the flow, which often cause
combustioninstabilities[7],areresolvedonthegridwhilefine-grainturbulenteddiesaremod-
elled[8–11].Oneimportantissueistodeviseasuitabledescriptionofthecombustionprocess
intheLESframework.Manystudieshavebeencarriedoutonthistopicandanumberofwell
establishedcombustionmodelsexistsuchasthelineareddymodel[12],theflamewrinkling
model[13]andtheflamesurfacedensitymodel[14].Thisissuewillnotbeexploredinthis
paper, which focuses on the question of the acoustic forcing methodology. The simulations
relyontheartificialthickeningapproach[15–18]todescribetheflamedynamics.Thismethod
is already well explained in these references and it is only briefly exposed in what follows.
Thispaperdescribesqualitativevalidationsoftheacousticmethodology,leavingmoreprecise
comparisonsoftheresponseofdifferentturbulentcombustionsubgridmodelstotransverse
acousticforcingforfuturework.
Instudyingcombustioninstabilities,twotypesofsimulationscanbeexplored.Inthefirst,
the motion is self-sustained while the second corresponds to an external modulation of the
flow [19]. In the case of a simulation of self-sustained oscillations, the coupling between
combustionandacoustics,whichcontrolstheunstableprocess,hastoreachaself-excitedstate
0
01 duringthecalculations[20,21].Oneissueinthiscaseistodefinesuitableboundaryconditions
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y [22]. When these conditions are not well handled or when flow non-uniformities reach the
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nu boundaries,spuriouswavesarecreatedthatcanperturbthesimulations.Thisoftenprecludes
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J studiesofcombustionoscillations.Thealternativestrategyistoexaminetheflowdynamics
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by modulating the computational domain at a given frequency [23–25]. Such calculations
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0: provide the system response to the external excitation. In these simulations, non-reflecting
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: boundary conditions can be used to let waves exit the computational domain. This reduces
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issuesassociatedwithreflectionsattheboundaries,butalsochangesthenatureoftheproblem.
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ad Whiletheflameresponsetoexternalperturbationsiseffectivelyobtained,thefeedbackprocess
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nl is not represented and there can be no resonance [26, 27]. This paper presents a method
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Distinct from most previous simulations of modulated flames which were carried out by
injectinglongitudinalwaveperturbations[23–25],thecaseoftransverseacousticexcitation
is investigated in this article. Numerical simulation tools are described in sections 2 and 3.
Thelattercontainsadetailedaccountofthetransversemodulationmethod.Sections4and5
describeresultsoftwo-dimensionalcalculationsintwogenericconfigurations.
(cid:1)
Non-reactiveflowsofnon-premixedsingleordouble-wakecompoundstreamsareconsid-
eredinsection4.Inthedouble-wakecase,thedistancebetweenthetwowakesischosen
smallenoughtoenablecollectiveinteractionsunderexternalmodulation.Thissimulationis
avalidationtestofthemethodology.Itisalsoafirstattempttolookatcollectiveinteractions
betweentwoneighbouringwakes.
(cid:1)
A premixed propane–air configuration of a single reactive jet is examined in section 5.
This case has been investigated experimentally [28]. Experiments consisted of exciting a
combustorwithatransverseacousticmodegeneratedintheupstreamduct(usingamode
selectiveexcitationmethod(MSEM)).Thismodulationinducesalternateshearperturbations
ofthejetflamegivingrisetowelldefinedvortexpatterns.Intheexperiment,theinletplane
comprisesfiveinjectionslots,eachonehavingarectangularcross-sectionextendingover
the whole depth of the chamber. The combustor geometry can then be viewed as two-
dimensionalandtheflowcanbeassumedtobetwo-dimensional,atleastintheupstream
part where most of the acoustic interaction takes place and where organized large-scale
Transversemodulationofreactiveflows 7
motiondominatestheflow.Thispointisdiscussedfurtherinsection5.Fromexperimental
results given in reference [28] one can deduce that, close to the injection plane, the jets
behave as if they were isolated (the separation between adjacent jets is five times the jet
initial thickness). We also know that the dynamics in the initial region are not controlled
bycollectiveeffects,whichcantakeplaceinthefarfield,andthesimulationofasinglejet
maybecarriedout.Twocalculationswereperformed.First,thenaturalconfigurationofthe
singleflamewasinvestigatedusingsimplifiedlateralsymmetryconditions.Theinteraction
betweenasinglejetandatransverseacousticmodewasthenconsidered.Thejetwasforced
atthesamefrequencyasintheexperimentstoallowcomparisonsbetweennumericalresults
andexperimentalvisualizationsoftheflow.Performingthesimulationinatwo-dimensional
configurationreducedthecomputerresourcerequirementsandconstitutedavalidationcase
fortheacousticmodulationmethod.
While the two-dimensional assumption may be questionable, these simulations indicate
thatthemethodincombinationwithaLESflowsolvercanbeusedtostudytheresponseof
flamestotransverseacousticperturbations.Moreoverinthecaseofcoherentmotionsstudied
here,thetwo-dimensionalsimplificationcanberegardedasafirstapproximation.
10 2. Combustionmodellingforlargeeddysimulation
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ua To calculate combustion instabilities in which the larger structures of the flow often play a
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J keyrole,itislogicaltouselargeeddysimulationtechniques.Thecalculationspresentedin

3
2 thispaperarecarriedoutwithAVBP,aLESflowsolverdevelopedbyCERFACS[29].This
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:4 parallelcodeintegratestheNavier–Stokescompressibleequationsintwoandthreedimensions
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onstructuredorunstructuredgrids.Thefilteringoperationiscarriedoutimplicitly,thatisto
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d At say no analytical filter is used. It is performed by the characteristic dimension (cid:1)grid of the
de mesh.Tomodelsubgridterms,afilteredSmagorinskyapproachisused[30,31],whichenables
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Do method is based on a finite element scheme Two step Taylor Galerkin version C (TTGC),
whichfeaturesreduceddissipationanddispersion[32].
The reactive numerical simulations performed in section 5 consisting of a propane–air
premixedjetflamerequireacombustionmodel.Amongtheestablishedturbulentcombustion
models for LES the formulation used in the present paper is based on the thickened flame
approachasdescribedinreferences[15,16].Inthismodel,theflamethicknessisartificially
increasedbyafactorF andtheinnerflamestructurecanberesolvedonthecoarsemeshused
inthesimulationwhilethelaminarburningvelocityiskeptconstant.Thethickeningoperation
decreasestheDamko¨hlernumber,aneffectcorrectedbyanefficiencyfunctionE.Thisisused
torepresentthesubgridscaleturbulenceinfluenceonthelocalrateofreactionintheflame
[17,18].
ThetransportequationofanymassfractionY iswrittenintermsofFavreaverages:ρY =
ρ¯Y˜,
∂ (ρ¯Y˜)+∇·(¯ρu˜Y˜)=∇·(ρ¯FED∇Y˜)+∇·(−ρ¯(u(cid:1)Y −u˜Y˜))+ Eω˙
∂t F
whereuisthevelocityvector,ω˙ isthemodelledreactionrategivenbelow,Fisthethickening
factor equal to 1 outside the flame front and greater than 1 in the flame. E is the efficiency
function. The term u(cid:1)Y − u˜Y˜ is the subgrid scale term modelled with a standard gradient
approach:
−(u(cid:1)Y −u˜Y˜)= D ∇Y˜
t
8 C.Reyetal.
with D =ν /Sc (Sc =0.7)andν thesubgridviscosity,whichismodelledusingafiltered
t t t t t
Smagorinskyformulation:
(cid:4)
(cid:2) (cid:3)
ν = C (cid:1) 2 2HP(S˜ )HP(S˜ )
t SF grid ij ij
whereC istheconstantofthemodelequalto0.37,S˜ arethefiltereddeformationratesand
SF ij
HPisahigh-passfilter[31].
The chemical reaction between propane and air is represented by a single-step global
chemicalkineticsasdescribedinreference[24]:
C H +5O →3CO +4H O
3 8 2 2 2
withareactionrategivenby:
ω˙ =[C H ]α[O ]βAexp(−E /RT)
3 8 2 a
where[C H ]and[O ]arethemolarconcentrationsofreactants,αandβ thecorresponding
3 8 2
exponents, Athepre-exponentialfactor, E theactivationenergy, R theperfectgasconstant
a
and T the absolute local gas temperature. From the results of [24, 33] one takes α =1.0,
β =0.5, A =1.6109 (cgsunits), E =1.4104 (calmol−1).Thelaminarburningvelocityis
a
0 equal to 0.25 m s−1 for an equivalence ratio (cid:8)=0.75. This model is very simple but it is
1
0
2 convenientheresinceonlyperfectlypremixedcombustionisconsidered.
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ar ThethickeningfactorFistakenequalto4inordertoresolvetheflamefrontonthegridused
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n
a inthecalculations.Itiscrucialtokeepthevalueofthisparametersmallsothattheefficiency
J

23 function E does not take large values. In this case, one can expect a good response of the

41 flame to acoustic waves. An examination of the efficiency function fields obtained during
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1 the calculations confirms this point. Further investigations are necessary on this important

:
At issue.

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ow 3. Theacousticmodulationtechnique
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Thetransversemodulationofacomputationaldomainisacomplexproblemwhichhasnotbeen
envisaged in previous studies but raises some new computational problems. The transverse
motionmaybeinducedbythefollowing.
(cid:1)
IntroducingsourcetermsintheNavier–Stokesequations[34]asadistributionoverlocalized
regionsonthelateralsidesofthedomain(seefigure1(a)).
(cid:1)
Placing lateral channels and applying characteristic wave modulation at the end of these
channels(seefigure1(b)).
(cid:1)
Modulatingtheincomingacousticwavesonthelateralsides(seefigure2).
Systematic tests indicate that the two first techniques are not suitable because spurious
modes appear at the initial instant and perturb the simulations [27]. It is also found that a
suitablemodulationoftheincomingacousticwaveenteringthecomputationaldomainonthe
lateralsidesdoesnottriggerthesemodes[26,27].Themethodusescharacteristicboundary
conditions (the Navier–Stokes characteristic boundary conditions as developed in reference
[35])whichrelyonthedecompositionofthevariationsofconservativevariables(ρ,ρu,ρe)
into a set of waves travelling normally to the boundary [36, 37]. The main variables ρ, u
and e are the local density, velocity and energy. It is convenient at this point to introduce
basic notations for the different characteristic waves. In the 2D case, the vector of varia-
tions comprises four elements δW =[δW1,δW2,δW3,δW4]T which may be cast in the
n n n n n
Transversemodulationofreactiveflows 9
Figure1. Twopossiblesolutionsforthetransversemodulation;(a)twoadditionalsourceregionsinthecomputa-
tionaldomain;(b)useoftwoauxiliarylateralchannelsexcitedwithcharacteristicwavemodulation.
form[38]:
1
δW1 =δρ− δp
n c2
0
201 δWn2 =s·δu
y
r
a 1
nu δW3 =+n·δu+ δp
Ja n ρc

3
2
41 δW4 =−n·δu+ 1 δp
0: n ρc
1

:
At wherendesignatesthenormalunitvectorattheboundarypointinginwards,s aunitvector

ed tangenttotheboundary.Intheseexpressionscisthespeedofsoundand pthepressure.The
d
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o firsttwovariationscorrespondtoentropyandtransverseshear(vorticity)perturbationswhile
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ow thelasttwocomponentsrelatetoacousticwaves.Thesedifferentperturbationspropagatein
D
thenormaldirectionwiththefollowingvelocities:[n·u,n·u,n·u+c,n·u−c]T.Depending
onthesignofn·u,δWi correspondstoanincomingoroutgoingwavevariation.
n
Inthetransverseacousticmodulationdevisedinthisarticle,thevariationsδW3arespecified
n
asfunctionsoftimeonthetwolateralsidesofthecomputationaldomain(uppersidesubscript
Figure2. Typicalcomputationaldomain.Thelateralboundariesareusedtomodulatetheflow.