Direct Numerical Simulation of hydrogen combustion at auto-ignitive conditions: Ignition, stability and turbulent reaction-front velocity

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ContentslistsavailableatScienceDirect

Combustion and Flame

journalhomepage:www.elsevier.com/locate/combustflame

Direct Numerical Simulation of hydrogen combustion at auto-ignitive conditions: Ignition, stability and turbulent reaction-front velocity

Andrea Gruber

^a^,^b^,^∗

, Mirko R. Bothien

^c^,^b^,^d

, Andrea Ciani

^d

, Konduri Aditya

^e

, Jacqueline H. Chen

^f

, Forman A. Williams

^g

aThermal Energy Department, SINTEF Energy Research, Trondheim, Norway

bDepartment of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim, Norway

cInstitute of Energy Systems and Fluid Engineering, Zurich University of Applied Sciences, Winterthur, Switzerland

dAnsaldo Energia Switzerland, Baden, Switzerland

eDepartment of Computational and Data Sciences, Indian Institute of Science, Bangalore, India

fCombustion Research Facility, Sandia National Laboratories, Livermore, CA, USA

gDepartment of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA, USA

a rt i c l e i nf o

Article history:

Received 29 September 2020 Revised 20 February 2021 Accepted 21 February 2021

Keywords:

Hydrogen

Spontaneous ignition Reheat combustion Flame pulsation Turbulent ﬂame velocity Direct Numerical Simulation

a b s t r a c t

Direct Numerical Simulations(DNS) areperformed toinvestigate the processof spontaneousignition ofhydrogenflamesatlaminar,turbulent, adiabaticand non-adiabaticconditions.Mixturesofhydrogen and vitiated airattemperatures representinggas-turbine reheatcombustion areconsidered. Adiabatic spontaneousignitionprocessesareinvestigatedfirst,providingaquantitativecharacterizationofstable andunstableflames.Resultsindicatethat,inhydrogenreheatcombustion,compressibilityeffectsplaya keyrole inflamestabilityand thatunstableignitionand combustionareconsistentlyencounteredfor reactanttemperaturesclosetothemixture’scharacteristiccrossovertemperature.Furthermore,itisalso foundthatthecharacterizationoftheadiabaticprocessesisalsovalidinthepresenceofnon-adiabaticity duetowallheat-loss.Finally,aquantitativecharacterizationoftheinstantaneousfuelconsumptionrate withinthereactionfrontisobtainedandofitsability,atauto-ignitiveconditions,toadvanceagainstthe approachingturbulentflowofthereactants,forarangeofdifferentturbulenceintensities,temperatures andpressurelevels.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Hydrogen-ﬁring ofstationary gas turbines isemerging asone of the most robust approachesto reduce carbon emissions from large-scalepowergeneration.Thisequallyapplies,inaconvenient synergy, to power generation schemes that can utilize a steady streamofhydrogenfromlarge-scalereformingofnaturalgaswith carbon captureand storage (CCS)[1]or, exploiting excesspower fromnon-dispatchablerenewable energyresources (windandso- lar), an unsteady stream ofhydrogen produced from water elec- trolysiscoupledto large-scaleenergystoragesolutions (power-to- H₂-to-power)[2].

However, state-of-the-art gas-turbine technology does not presently allow,without importantperformance compromises,for

∗Corresponding author at: SINTEF Energy Research, Thermal Energy Department, 7465 Trondheim, Norway.

E-mail address: andrea.gruber@sintef.no (A. Gruber).

combustion of pure (undiluted) hydrogen. This fuel notoriously poses important burner design challenges with respect to flame stabilityandNOx emissionsthat are conventionallysolved by di- lutionof the hydrogenfuel with largequantities of steam orni- trogen [3]. The main reason for these problems is due to hydrogen’s higher reactivity compared to naturalgas, the standard gaseousfuelforgasturbines.Hydrogen’shighreactivityintroduces severechallengesinsimultaneouslyachievinglowemissionperfor- mancetogetherwithstaticanddynamicflamestability(i.e.avoid- ing flashback[4]andthermo-acoustics instabilities[5,6]), andre- mainsoneofthemainobstaclesforlarge-scale,cleanandefficient utilizationofhydrogeningasturbines.

In this context, multi-stage combustion systems seem to of- fer the most promising solution for power plants that, in to- day’s changing powermarket, have to ensure highturndown ratios, part-load efficiency, and fuel flexibility (including hydrogen firing), while keeping pollutant emissions low. A two-stage pre- mixedsystem,inwhichthetwocombustionstagesaredistributed longitudinally in a sequential arrangement and separated by an

https://doi.org/10.1016/j.combustﬂame.2021.02.031

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ofa so-calledreheat flamestabilized(mainly)by spontaneousignition in a sequential combustor [8]. In Ansaldo’s longitudinally- staged combustionsystem, thefirst combustionstage serves asa hot-gas generator while the predominant energy conversion occurs in the second stage, posing new and interesting challenges duetotheunconventionalcombustionconditionsandrate-limiting processesthat characterizethesereactiveflows.Other gasturbine manufacturersarepursuingsimilarlongitudinalfuelstagingstrate- giesalthoughthesearetypicallycharacterizedbydifferentstaging ratios,inwhichmostofthefuelisconsumedinthefirststage,ul- timatelyresultingindifferentcombustionandoperationalbehavior [9,10].

In principle, the reheat combustion scheme, due to its re- lianceon spontaneousignitiontoachieve flamestabilization[11], iswell-suited toprovideintrinsicallystableandcleancombustion ofhydrogen-richfuelmixtures,asrecentlydemonstrated[12].This result is achieved through an operational strategy for hydrogen- firingthatimplementsareductionoftheflametemperatureinthe firststage,throughlean(er)operationofthepropagation-stabilized flame, achieving flashback and NOx control locally and, simultaneously, ensuring a beneficial increase inthe ignition delaytime of the(now colder)reactants’ mixture entering thesecond stage.

There,inthesequentialcombustor,thereactants’inlettemperature represents the main rate-controlling parameter that controls, to leadingorder,thestabilizationlocationofthereheatﬂamethrough the process of spontaneousignition. Therefore,thisis one ofthe keyquantitiesfocuseduponinthepresentpaperwhereasthefuel- oxidantequivalenceratioplays itselfaminorrole withrespectto ﬂame stability(withinthe relevantoperationalrange ofinterest).

The reliance on spontaneous ignitionrather than flame propaga- tiontoachievestabilizationoftheflameinthesecond(main)com- bustionstagehasseveraladvantages.Theprincipalconsequenceof thisstrategyisthathighbulkvelocitiescanbeutilizedwithinthe sequential combustorflow path toreduce NOx formation (dueto short residence time) and diminish the propensity for flashback (due to highflow velocity). Moreover, increasing the fuel supply inthesecondstage,whilenotaffectingflamestabilization(princi- pally controlledby thereactant’s temperature),fullycompensates forthereducedfueladditioninthefirststage,maintainingthetar- getflametemperatureinthesecondstageandminimizingorelim- inatingde-ratingoftheengine[12].

Depending onthe boundaryandoperatingconditionsin prac- ticalcombustionapplications,itisreasonabletoexpect thatcom- plex mixed combustion modescan occur in the sequential combustor[13].Thesearecharacterizedbythesimultaneousexistence of deﬂagration andspontaneous ignition fronts, eitherpresentin different localities or co-located, andtheir interaction makes the physical process of reheat combustion more complex to under- stand and predict. The local behaviour of propagating ﬂames or spontaneousauto-ignitingfrontsisaffectedbytheirsurroundings.

As a result, the balance of combustion modes is able to affect global combustor behaviour through feedback mechanisms with thevelocityandacousticﬁelds(leadingtoﬂashbackortothermo- acousticinstabilities).

Therehavebeenrelativelyfew paststudiesoncombustionun- der reheat conditions,i.e.vitiatedoxidant, elevated pressures (up to ∼25bar) and high reactant temperatures (>1000K). One of the earlyresearch effortswasconductedby theInstituteofCom- bustion ResearchofDLR.Pressurizedlaboratoryexperimentswere performedonascaled,geometricallysimpliﬁed versionofthese- quential combustor ﬁred with hydrocarbon fuels [14–16]. How-

searchers performedstate-of-the-art LargeEddy Simulation (LES) withtheDynamicallyThickenedFlame[17] toinvestigatetheoc- currence ofdeflagration andspontaneous ignitioninmethane-air flamesandtheirrelativeimportanceinflamestabilization[18–22]. InthenumericalmodellingstudybyKrisman[23],thedesignation ofauniquely-defined,quantitativereferencespeedforlaminarpre- mixedflamesatauto-ignitiveconditionhasbeenproposedforthe firsttime.Other modellingstudiesspecificallyfocused onthedy- namic response of auto-igniting flames, using LES to extract the flametransferfunction (FTF)basedonpressureandvelocity fluctuations [24,25] while the most recent investigations have high- lightedtheimportanceofinlettemperaturefluctuationsthatmust beaccountedforina3×3flametransfermatrix(FTM)[26–28].

Beyond theaforementioned effortsmostly focusing on hydrocarbon fuels, to date, only a handful of studies have investigated the characteristic features of hydrogen combustion at reheat conditions.The earliest among thesewere zero-dimensional (0D) andone-dimensional (1D) reactor modellingstudies charac- terizingignitionandpropagationtimescales,complementingfull- scale,high-pressureexperiments [29–31]. Onlyveryrecently full- fledged,three-dimensionalDirect NumericalSimulations (DNS)of turbulentpremixedhydrogen–aircombustionatreheatconditions (albeit atmospheric pressure) have been performed in conjunc- tionwithdetailedchemicalkineticsandChemicalExplosiveMode Analysis[32]toquantifytherelativeimportanceofflamepropaga- tionversusspontaneousignitionforarangeofturbulenceintensi- tiesinstatisticallyplanarflames [33]andinthepresence of wall heatlossinasemi-realisticcombustorgeometry[34].

The presenteffortbuilds upon the aforementionedDNS studies and deploys one- two- and three-dimensional DNS to investigate the conditions leading to steady or unsteady ignition and combustion in premixed hydrogen–air reheat flames under laminar, turbulent, adiabatic and non-adiabatic conditions. The first partofthisstudyfocusesontheunsteadinessrelatedtothespon- taneous ignition process itself that takesplace when combustion arisesinamixtureofpreheatedreactants.The occurrenceofself- excitedflameinstabilities,emergingwellaftertheirinitialignition andinducedbyavariationofthereactanttemperature,isreported (Section3).Thesecondpartofthepresentstudyreportstheeffect oftheturbulenceintensitycharacterizingtheapproachingreactant flow ontheturbulent reheat flame-brush meandisplacementve- locity(Section 4). Section 2briefly describesthephysical process that is the objective of thisinvestigation and the numericaltool chosen forthestudy(Sandia’sS3DDNScode) whilethemainre- sultsaresummarizedinSection5,whichalsopresentsanoutlook aboutfurtherworkonthetopic.

2. Physicalproblemandnumericaltool 2.1. Spontaneousignitionofhydrogenreheatﬂames

Forhydrogen-richreactant mixtures, theignition delay(or in- duction time),

τ

ig, is a key quantity at preheated reheat conditions thatis stronglydependenton initialtemperature(T₀), pressure(P₀)andcomposition(Y_k_,₀).Theignitiondelayexhibitsalarge rateofchangenearthecrossovertemperaturewherethereaction rateof theelementary H +O₂ branching step equals that of the recombination step H + O₂ + M [35]. This characteristic behavior of hydrogen–air systems hasbeen extensively investigated in the past and it can be accurately reproduced through advanced and elegantly simple kinetic models [36]. It has been observed

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that, whentransitioning frommethane-airmixtures tohydrogen- enriched methane-air mixtures and ﬁnally to hydrogen–air mixtures, the temperature dependence of

τ

ig at conditions relevant to reheat combustion increases considerably with hydrogen con- tent [37].Therefore, forhydrogen–airmixtures,in whichthefuel consistsofpure hydrogenundilutedby hydrocarbons,nitrogenor steam, it is reasonable to expect a spontaneous ignition behavior that largely departs from the behavior typically observed in hydrocarbon–airpremixedﬂames.

Nearly 40 years ago Zel’dovich postulated the existence of a

“spontaneous propagation” regime ofrelevanceforcombustion at auto-ignitive conditions that is characterized by an intermediate ignition-front velocity between the deﬂagration and the detona- tion velocities[38].Spontaneouspropagationoftheignitionfront occurs if the inverseof the magnitude ofthe local ignition-time gradient is largerthan the deﬂagration velocity andsmallerthan thedetonationvelocity,i.e.Ssp=[(^d

τ

ig/dT₀)·

|∇

^T0

|

^]⁻¹^for^the^case

S_L<Ssp<c<S_D,wherecisthespeedofsoundandS_LandS_D are the “conventional” deflagration anddetonation velocities, respectively.Basedon Zel’dovichcriterion,thetransitionbetweendefla- gration and spontaneous propagation of ignition fronts is highly sensitive to spatial gradients intemperature presentin thereactant mixture andto the temperature dependence ofthe ignition delay.Consequently, itisreasonabletoassume thatcompressibil- ityeffects(e.g.compressionheating)canplayakeyroleincontrol- lingthebehaviorofreactiveflowsinthespontaneous-propagation regime. The above assumption has been already validated nu- mericallythroughpreviousDNSstudies,focusingoncompression- ignition engine conditions [39,40], that illustrated a dependency betweenthepropagationspeedofthefrontandthermal-orcom- binedthermal-andcomposition-gradients,modulatedbyturbulent mixing and isentropiccompression heating. These early observations are expected to be of particular relevance to hydrogen–air systemsduetothelargevaluesthatcharacterized

τ

ig/dT₀ intem- peraturerangesnearcrossover.

2.2. DirectNumericalSimulationcode

From the aforementioned discussion it is important to cap- turethecouplingbetweenpressure,densityandtemperatureﬁelds for the reactive ﬂows under investigation, in addition to a de- tailedrepresentationofthechemicalreactionkinetics.Tothisend, thecompressiblereactingDirectNumericalSimulation(DNS)code, S3D, originally developed at SandiaNational Laboratories [41],is employedforallcalculationsdescribedinSections3and4.

S3D is written in FORTRAN90 and uses the Message Passing Interface (MPI) forinterprocesscommunicationin parallel execu- tion.Inthepresentapplication,thealgorithmimplementedinS3D solvestheNavier–Stokesequationsforacompressibleﬂuidincon- servative formon structured,Cartesian meshesinone-,two- and three-dimensional computational domains to simulate premixed combustion ofH₂-airﬂames atauto-ignitiveconditionsrepresen- tative of a reheat combustion system. The present simulations are mostlylimitedto atmosphericpressuredueto computational cost with increasing pressure. All DNS use a spatial resolution of at least

δ

^s=

δ

^x=

δ

^y=

δ

^z=25μm (

δ

^s=10μm in 1-D and2-D calculations) that is sufficient to resolve all spatialscales of the reactive flows investigated at atmospheric pressure. The spatial derivatives arecomputed withan eighth-order,explicit,centered, finite-difference scheme (third-order one-sided stencils are used atthe domainboundariesinthenon-homogeneousdirections)in conjunctionwithatenth-order,explicit,spatialfilter,assuggested by Kennedy and Carpenter [42],to remove highfrequency noise andreducealiasingerror.Afourth-order,six-stage,explicitRunge–

Kutta scheme,describedin[43],is usedfortime integration and thetimestepissetto

δ

^t=4nsforallreactiveﬂowsinvestigated.

Thermodynamic properties are modelled as polynomial func- tionsoftemperatureandtransportcoeﬃcientsasdescribedinthe CHEMKIN and TRANSPORT packages, respectively [44]. Radiative heattransferisneglectedbecauseofthemodestopticalthickness of hydrogenﬂames. The chemical reactions in the gas phase are described by a detailed mechanism for hydrogen combustion in air[45]. Thismechanismconsistsof9speciesand19elementary reaction steps. Nitrogen is assumed to be inert such that NOx- formationreactionsarenotconsidered.

Inﬂow and outﬂow boundary conditions are implemented following the Navier–Stokes Characteristic Boundary Conditions (NSCBC) methodology andare based on the original formulation of [46], incorporating the later improvements described in [47–

49] that include source and transverse terms. Wall boundaries, wherepresent,aretreatedasno-slip,isothermal,smoothsolidsur- faces and are implemented followingthe methodology described in[50]and[51]fornon-porous,impermeablematerials,suchthat the wall-normal mass ﬂux of all chemical species is identically zero.

3. Spontaneousignitionofhydrogenreheatﬂames

Inthis section,we utilizeone- andtwo-dimensional DNScal- culationsinordertoinvestigatetheignitionandstabilizationchar- acteristics of hydrogen ﬂames at reheat conditions. The effect of the following parameters is studied: 1) the domain size (Lx); 2) thefuel-oxidantequivalenceratio(

φ

=f(^Yk)^);³⁾^theînlet^velocity (U_in);4)theinlettemperature(Tu);5) thewall heatlossandflow confinementbywalls(aty=0andy=Ly).

3.1. Generalfeaturesoftheinitialignitionprocess

The spontaneous ignition process is studied adopting an idealized (and simplified) representation of the reactive flow of interest.Figure1illustratesasketchofthecomputationaldomain that is characterized by a (main) longitudinal dimension Lx and, whenpresent(e.g.in2-D and3-Dcalculations),bythetransverse dimensions Ly and Lz (the latter is not shown and extends in the direction normal to the page, it is always a periodic/cyclic boundary). The reactant mixture consists of hydrogen and the vitiated oxidant stream originating from the first combustion stage. The latter is assumedto be a hot-gas generator that pro- videstheproducts of hydrogen–aircombustion atan equivalence ratio

φ

=0.43 and reactants’ temperature of Tu=773K, mixed withadditional air. Accordingly, the nominaltarget conditions of interestforthehydrogenreheatﬂamearedeﬁned,similarto[34], asTu=1100K and

φ

=0.35(mass-fractions:0.008 H₂; 0.183O₂; 0.052 H₂O; 0.757 N₂) resulting in nominal values for the ignition delay time,

τ

ig∼0.15ms, and adiabatic ﬂame temperature, T_ad∼1800K (from homogeneous reactor calculations). These are compatible with high eﬃciency and low emission in a typical gas-turbinecombustionsystem(premixed).

Inthe presentinvestigation, to ﬁndhow the combustionmay varyforconditionsinthe vicinityofthe normaltarget condition, a parametricvariation is introducedin theinitial (fresh-reactant) temperatureTu andcompositionY_k(i.e.equivalenceratio

φ

⁾^of^the

reactant mixturethat initially ﬁlls the entiredomain atthe start ofeachsimulation.Enteringintoacontinuousstreamfromtheleft (NSCBCinlet)boundaryatx=0withameanvelocityU_in,thereac- tantmixtureisadvecteddownstreamuntiltheinitialspontaneous ignitionoccursafteraconvectiveresidencetimet=t_res∼

τ

ig,cor- respondingtospatiallocationx∼L_ig_,_init∼U_in·

τ

ig.Thereacted,par- tiallyreactedorunreacted(dependingonspeciﬁclocalconditions) gasesexitthe computational domain fromtheright (NSCBCout- let) boundary at x=Lx. Note that, when ignition occurs, due to thewaythesimulationsareinitialized,theentiredomainbetween

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Fig. 1. Sketch of the computational domain represented in the DNS calculations: L xis the longitudinal dimension, L yis the transverse dimension, when present, and L igis the ignition length estimated as L ig∼τig·U in, where τigis the ignition time and U inis the mean velocity imposed at the upstream inlet boundary.

Fig. 2. Temperature (left y -axis) and pressure (right y -axis) proﬁles across the 1-D domain. Initial “explosive” phase (a) followed by “relaxation” phase (b) for φ= 0 . 12 and T u= 1100 K . (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

x=L_ig_,_init andx=Lx simultaneously ignites.This impliesthatthe domain size affects the initial transient of the unsteady ignition process.

The initialtransientphaseofthe spontaneousignitionprocess is describedbelowfor an exemplar(1-DDNS withthefollowing parameters:

φ

=0.12, Tu=1100K, Lx=10cm andU_in=200m/s) as it is qualitatively similar for all cases investigated; however, quantitative differences emerge,dependingon thespeciﬁc conditions, andlead todifferentsolutionsthat canremainunsteady or approachsteady-state.

At a time t∼

τ

ig after the start of the simulation the spontaneous ignition process, in its initial phase that can be deﬁned as “mildly explosive”, leads to a sudden temperature increase throughoutthedownstreamportionofthecomputationaldomain for L_ig,init<x<L_x. Locally, the temperature increase is accompa- nied by thesimultaneous pressure increaseandexpansion ofthe gas mixture that isundergoing chemical reaction. Thisis graphi- cally illustrated inFig. 2(a)by the dash–dot linesof thetemper- ature (black)and pressure(red) proﬁles corresponding to theignition time t=

τ

ig∼0.2ms. At later timesduring this initial ex- plosivephase,thepressurewavegeneratedbythespontaneousig- nition processpropagatesdownstream (exiting thecomputational domainthroughtheNSCBCoutlet)andupstreamtowardsthefresh reactantsandthedomaininlet(exiting thecomputationaldomain through the NSCBC inlet). The latter, upstream-propagating pressure wave, however, as opposed to the downstream-propagating one, increasesin amplitude asit movestowards the domain inlet(Fig.2(a)coveringthetimeintervalt=0.25−0.35ms)because it effectively represents an adverse pressure gradient forthe approaching reactant ﬂow (asan immaterial “piston” actingagainst it).Thecharacteristicstrengthofthe“pistoneffect” isdirectlypro- portional to the domain size (amount of reactants that ignite), equivalence ratio (temperature increase due to ignition) and re- actantinletﬂowvelocity (steepnessofadversepressuregradient).

The pressureincrease thatoccursinthefreshreactantsalsoleads toa localtemperatureincrease(uptoalimitingvalueTu,max)and to a shortening of the ignition time (down to a limiting value

τ

ig,min and a corresponding L_ig_,_min) that, in turn,causes the pro-

gressiveupstream displacementof thespontaneousignition front (blacklinesinFig.2(a)),strengtheningthe“piston” effect.Theob- served upstream combustionfront displacement,although slower thanthespeedofsound(itlagsthepressurewave),occursagainst ameanﬂowU_in∼200m/s.Accordingly,itismuchlargerthanthe

”normal” flamedeflagration velocity S_L. Therefore,it can be con- cluded that the unsteady ignition phenomenon fits the criterion proposedbyZel’dovichforaspontaneouspropagationregime.

Following the ﬁrst,“explosive” phase of the unsteady spontaneousignition,oncethemainpressurewavegeneratedbytheini- tialﬂuidexpansion leavesthecomputationaldomainthrough the left boundary(NSCBC inlet) andthe upstream propagationofthe combustionfrontstops(dueto

τ

ig(^T^u,max)>tres),the“piston” effect abruptlyendsandthisleadstoasecond,“relaxation” phaseofthe unsteadyprocess. Duringthe“relaxation” phase, thepressurede- creasesrapidly throughoutthe computational domainalong with thetemperatureinthefreshreactantsthatdecreasestoitsoriginal value T_u (set by the boundarycondition).Spontaneousignition is nolongersustainedatalllocationsx<L_ig_,_init andthecombustion frontis displaceddownstreamto its“natural” position wherethe solutionreachessteady-state.Thisrelaxationprocessisillustrated by the broken lines in Fig. 2(b); please note that the solid lines represent the steady-state solution at the ﬁnal (and much later) timet=0.01s.

3.2. Effectsofthedomainsizeandequivalenceratio

Whilethe phenomenologicalpicturedescribedintheprevious sectionqualitativelyappliestoallcasesinvestigated,atleastinre- spectto thefirst “explosive” phase, thetransitionto andthe be- haviorofthesecond, “relaxation” phaseisstronglydependent on the quantitative characteristicsof thefirst phase, i.e.on thespe- cificconditionsof thesimulated case. Inthe presentsection, the effectsof domainsize andfuel-oxidantequivalence ratioare discussed, while, in the next sections,an analysis of the effects of inlet-temperature variations is presented and the discussion ex- tended to thecaseofconfined flows withheat lossto thewalls.

Althoughtheeffectofthereactantinletﬂowvelocityontheigni-

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Fig. 3. Maximum pressure value recorded across the computational domain as a function of time for a range of domain sizes ( L x= 10 , 20 and 30 cm ) and equivalence ratios ( φ= 0 . 12 , 0 . 17 , 0 . 26 , 0 . 35 ) (a). Maximum hydrogen mass fraction and maximum ﬂame temperature as a function of time for a gradual increase of φfrom 0.17 to 0.35 (b).

(For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

tion processisnotexplicitlyillustrated anddescribedhere,it has beenthoroughlyinvestigated,anditisqualitatively similartothe effects ofdomain size andequivalence ratiodescribedbelow, i.e.

thestrengthofupstreampropagatingpressurewavesincreasesfor increasinginletvelocities.

Figure 3 (a) illustrates, for eight different 1-D ignition cases, the time history of the maximum value of pressure recorded at each time step throughoutthe computational domain.A striking qualitativesimilarityisobservedforallcurves:aprimarypressure peak due to the initial spontaneous ignition, followed by a sec- ondary pressure peak due to the upstream-propagating pressure wave that exhibitsahighervalue comparedwiththeformer(the notable exceptions being the

φ

=0.12 cases with the larger domains, Lx=20cmand30cm).The magnitudeofthe dual-peaked maximum-pressuretimehistoryshowsaweakdependenceonthe domainsize,i.e.largerdomains giveslightlyhigherpressurepeak (compare the black, violet and pink lines in the ﬁgure), and a strong dependence on the equivalence ratio with the maximum pressure valuesincreasing fromPmax∼1.2atmto∼1.6atmforan increaseoftheequivalenceratiofrom

φ

=0.17to0.35(blue,green orangeand redlines). Thisis consistent withtheincreasing heat release withincreasing equivalenceratio.It should be notedthat for

φ

>0.3 the displacement of the spontaneous ignition front, trailing the upstream-propagating pressure wave, rapidly reaches the one-dimensional domain inlet boundary and the NSCBC implementation isunable tohandle the ensuing interaction, result- inginthesimulationscrashingatt∼0.0006s.Ontheotherhand, for

φ

<0.3,followingtheinitial unsteadytransient(consisting of theexplosiveandrelaxationphases),allﬂamesreachsteady-state, with the spontaneous-ignition front positioned at the expected stabilization location L_ig=U_in·

τ

ig. Finally, before concluding the present section, it is important to mention that it is possible to stabilize hydrogenreheat ﬂames athigher equivalence ratios(i.e.

thetargetvalue

φ

=0.35)byinitializingthecalculationwithare- actant mixtureat lower equivalence ratio i.e.

φ

=0.17and, after theinitial transientiscompleted andtherelaxationhasoccurred, increasing the amount of fuel introduced at the inlet boundary.

The effect of thisprocedure is illustrated in Fig.3(b) that shows agradual,smoothincreaseintheﬂametemperaturefollowingthe increaseinthehydrogenmassfractionimposedattheinletbound- ary. Note that the maximum-pressure time historyfor this case, shown by the dashed blue line in Fig. 3(a), is virtually indistin- guishable from the case where no fuel increase is implemented, shownbythesolidblueline.

3.3. Effectofinitialreactanttemperature

Figure4illustratesthetypicalde-stabilizingeffectonthespon- taneous ignition process of hydrogen reheat ﬂames observed for values of the reactant mixture initial (inlet) temperature in the temperaturerangenear crossover980K<Tu<1080K (see Fig.5 in[35]).AlthoughonlyresultsobtainedforT_u=1000Kareshown in Fig. 4,the same trend is observed in all spontaneous-ignition testsconducted below a value of Tu∼1080K.The ﬁrst explosive phase of the spontaneous-ignition process is consistent with the description in the previous section for Tu=1100K, see Fig. 4(a).

The relaxation phase, however, differs considerably. Here, a self- excitedinstabilityoftheﬂameemergesdisplacingthespontaneous ignition front back andforth in a nearly periodic fashion. A vast parametricstudyconductedintheframeworkofthepresentwork for980K<Tu<1080Kandatatmosphericpressureconditionsin- dicatesthattheamplitudesofthepressureandtemperatureﬂuc- tuationsremainnearlyconstant(forthetimedurationoftheDNS) or decrease for a reactant mixture equivalence ratio lower than

φ

∼0.2, see Fig. 4(c). Conversely, forequivalence ratios

φ

>0.2, the amplitudes of the pressure and temperature ﬂuctuationsex- hibit non-monotonic and non-linear growth rates, see Fig. 4(d).

Note that ﬂames are always stablefor T_u=1100K, as evidenced inFig.4(c)and(d).

3.3.1. Responseofhydrogenreheatﬂamestoinlettemperature variations

At this point it is important to clarify the fact that the self- excitedflameinstabilityobservedfor980K<Tu<1080Kisrelated notonly totheunsteady initialignitionprocess. Evenafterstabi- lizationofthespontaneous-ignitionfrontisreachedatthecharac- teristic locationL_ig,the combustion process always canbe desta- bilizedby areduction oftheinlettemperaturebelowT_u∼1080K. This is illustrated in Fig. 5(a) showing that, as soon as the inlet temperature(green curve) is reducednearthe crossover temperature,a spatialoscillation ofthespontaneous ignitionfront,with growingamplitude,occurs(inthiscase,causingacrashofthesim- ulation). Onthe other hand,duringan increase ofthe inlettem- peraturefromT_u=1100KtoT_u=1150K(awayfromcrossover),no self-excitedflameoscillationisobserved,asshowninFig.5(b).Ad- ditionally,thefigure alsoillustrates that alater inlettemperature reduction from Tu=1150K back to Tu=1100K does not have a destabilizingeffect,confirmingthattheobservedself-excitedcom- bustioninstability isnotsimply relatedtoa genericreduction (in

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Fig. 4. Top: temperature (left y -axis) and pressure (right y -axis) proﬁles across the 1-D computational domain during the initial “explosive” phase (a) followed by the

“relaxation” phase (b) at φ= 0 . 12 and T u = 10 0 0 K . Bottom: maximum pressure (red lines, delta symbols) and minimum temperature (black lines, gradient symbols) as a function of time for cases with T u = 10 0 0 K and T u= 110 0 K , constant φ= 0 . 12 (c) and φincreasing from 0 . 12 → 0 . 35 (d). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 5. Time history of the maximum pressure and temperature and of the minimum temperature recorded throughout the computational domain. Gradual reduction of the inlet temperature T u1100 K → 1050 K (a) and gradual increase and successive reduction of the inlet temperatureT u1100 K → 1150 K → 1100 K (b). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

itself) of the inlet temperature, but rather is induced by its de- creasebelow1080K,towardsthecrossovervalue.

3.3.2. Temporalcharacterizationofself-excitedﬂameinstabilities The time scale ofthe observed periodic spatial oscillations of the ﬂame,resulting froma self-excitedintrinsicinstability ofthe

flame inthe computational domain, is relatedto the convective- acousticfeedbackmechanismdescribedbyWilliams,seep.207in [52].Althoughtheoriginalreferenceisconcernedwithsupersonic flowsofreactingmixtures thatinvolveshockwaveswithexother- mic, finite-rate, temperature-sensitive chemistry occurring in the subsonicflowbehindtheshock,oneparticularandveryprevalent

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mechanismdescribedthereinseemsrelevanttothepresentsitua- tion.Adisturbanceinthetemperatureoftheincomingflowalters theauto-ignitiontimeofthemixture,therebyperturbingtheposi- tionatwhichignitionoccurs.Theperturbationintheignition-front locationproducesapressurepulse,whichtravelsupstreamacous- tically,modifying theinlettemperature(justbehindtheshock, in theoriginalconfiguration),which,inturn,furtheraffectstheauto- ignition time of the fluid element convected downstream. This thenresultsinaself-sustainedconvective-acousticfeedbackmech- anism,theperiodofwhichistwicethesumofthetimeforafluid elementtobetransportedfromtheinlettotheignitionpoint(convective timetc) andthetime foran acousticwave totravel from that point back tothe inlet(acoustic time ta). Twicebecausethe oscillationinvolvesacompressionasthefrontmovesupstreamfol- lowed bya front-generatedrarefactionasthe frontmovesdown- stream.

Forthespeciﬁcexampleoftheﬂameinstabilityobservedwhen Tu=1000Kand

φ

=0.12andrepresentedinFig.4(a-c),theflame front oscillates with an observed period T_f_,_obs∼2ms between the twospatialpositionsx_f,₁∼10cmandx_f,₂∼15cm.Thetime- averagedmeanflowvelocityisU_in∼195m/s(time-dependentde- viations from the target value U_in=200m/s due to the NSCBC formulation), and the speed of sound is c∼640m/s. Assuming that the flame instability canbe characterized by two convective trips between theinlet andthe flame (atits extremespatial positions), one corresponding to the explosive (compression) phase and the other to the relaxation (rarefaction) phase of duration t_c,₁₋₂=x_f_,₁₋₂/U_in and by two acoustic trips back to the inlet of duration t_a,₁₋₂=x_f_,₁₋₂/(^c−U_in), then the present interpretation oftheconvective-acousticmechanismpredictsthattheoscillation periodT_W canbeapproximatedas:

TW=

(

^xf,1+x_f_,₂

)

^·^[

(

¹^/Uin

)

⁺¹^/

(

^c⁻^Uin

)

^] ⁽¹⁾

resultinginapredictedT_W∼1.8ms.Thepredictionagreesqualita- tivelywiththeobservedoscillationperiodT_f,obs∼2ms.

Theshorteningoftheoscillationperiod(i.e.frequencyincrease) that is visiblein Fig. 4(d) forthe ﬂame subjected to an increasing equivalenceratio is dueto the movement of the meanloca- tion of the unstable ﬂame during its oscillation cycle to spatial positions increasingly closerto the inlet boundary, such that the length scales that appear in the mechanismdecrease with time.

The heat releaseincrease with time associated withthe increasing equivalenceratiocontributesto strengtheningofthepressure waveswhichthenareresponsiblefortheobservedincreaseofthe amplitudeovertime.Thisisobservedintheﬁguretobedisrupted whentheﬂame(temporarily)reachestheinlettwice.

In concluding the present section, it is important to high- light the fact that the self-excited instability of the ﬂame presented above isalsoobserved to take placein multi-dimensional conﬁgurations and in the presence of turbulence modulation.

This is discussed in Section 3.4 for 2-D non-adiabatic configurations with quasi-realistic turbulent velocity fluctuations andin Section 4.3.2 for3-D configurations witha realistic turbulent ve- locityfield.

3.4. Effectofwall-conﬁnementandheat-losstothewall (non-adiabaticity)

The aim of the present section is to investigate whether the spontaneous ignition and flame stabilization processes described above for adiabatic conditions are significantly affected by the presence of non-adiabatic conditions.It isfound that, in general, the observations presented previously are qualitatively valid also forthecaseofconfinedflowswithwallheatloss,i.e.stablespon- taneous ignition is achieved for

φ

∼0.17 (initial equivalence ra-

tio)andTu∼1100Kwhileunstableignitionbehaviourisobserved above

φ

∼0.18andforTu<1080K.

Following thesame wall boundary conditions implementation previously used in DNS studies performed with S3D [51,53–56], twoisothermal,no-slip, smoothwallsareplacedoppositetoeach other separated by a distance of 1.5cm in the transverse y- direction and kept at a fixed temperature T_w=750K (i.e. lower than the fluid temperature) to form a 15cm long straight chan- nelflowwherethespontaneousignitionprocess occurs.Therela- tively“cold” temperatureoftheisothermalchannelwalls(actingto confinethebulkflowof“hot” reactants)andtheensuingthermal boundarylayersareintendedtoprovideasimplifiedmodelrepre- sentingtheeffectofwall-flushingbycompressorair.Moreover,in ordertorepresenttheeffectofturbulentconvectiveheat transfer betweentheisothermalwallsandthebulkflowwithintheintrin- siclimitationsofatwo-dimensionalconfiguration,arandom flow field with a prescribed Passot–Pouquetenergy spectrum (charac- terizedbyarmsvelocityfluctuationofu=25m/sandanintegral length scaleof L_T=0.5cm) issuperimposed onto themean flow accordingto a well-establishedprocedure [57].The meanflow is describedbyacharacteristicturbulentchannelmeanvelocitypro- file with a centerline (inlet) velocity Uc=U_in=200m/s. The velocity fluctuationsentering the domain from the inlet boundary, inconjunctionwiththewallheatloss,induceatemperaturevari- anceinthereactantmixture,actingtodissipateheatandradicals thatare formedintheprocess thatultimatelyleadstoadelayin spontaneousignition.Thisisillustrated inFig.6(a)bya compari- son ofthelongitudinal temperatureprofilefortheadiabatic one- dimensional laminar case (black line) with the two-dimensional instantaneous (bluelines) andpointwise,time-averagedtempera- tureprofiles inthe bulk flow (red lines). The wall heat lossand thetemperaturevarianceintroducedbytherelativelycoldwallsin the two-dimensional channel-flow configuration affects the non- adiabatic spontaneous ignition process that is displaced down- streambyapproximately3mmcomparedtotheadiabaticprocess in the one-dimensional configuration. Note that, in the plots of Fig.6(a),the two 2mmthick regions ofthe flow thatare imme- diately adjacent to the isothermal channel walls are not consid- ered.Thisfindingconfirmsearlierobservationsabouttherolethat temperature andcompositional inhomogeneities play in delaying spontaneousignition[58].

Importantly,theoccurrenceofthecharacteristicself-excitedintrinsic instability described in the previous section is also observed atnon-adiabatic conditionsforopportunely chosen values of Tu and

φ

^. Â ^typical êxample îs illustrated in Fig. 6(b) showing the time history of the maximum pressure and temperature recordedwithinaLx=15cmlong,wall-boundedductcrossedbya Uc=200m/smeanflowforTu=1000Kand

φ

∼0.185.Thesecon- ditionsresultinanominalinductiontime

τ

ig∼0.75msthatisap- proximately equal tothe channel residencetime, tres=Lx/Uc. Ac- cordingly, thenominal ﬂame stabilizationlocation isclose to the downstreamendoftheduct.Notealsothattheﬁrstignitionevent, takingplaceatt_ig∼1.3ms,isdelayedwithrespecttothenominal inductiontime forthemixture (

τ

ig∼0.75ms) duetothe gradual increaseoftheequivalenceratiofrom

φ

=0att=0to

φ

=0.185 at t=0.5ms. The maximum temperature and pressure traces in Fig.6(b)clearlyindicatetheoccurrenceofacyclicprocesscharac- terizedbydistinctivetimescales,whilethesequenceofplotspre- sentedinFigs.7and8illustratestheunderlyingspatialoscillation ofthereactionfrontandtheintermittentspontaneousignitionand extinctionevents.Thelatterarecausedbytheﬂameintermittently exitingthedownstreamendofthecomputationaldomain.

Thetwo-dimensionalsimulationsconfirmtheimportantroleof compression heating, caused by the upstream-propagating pressure wave acting on the approaching reactant flow, during the first explosive phase of the spontaneous-ignition process. This is

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Fig. 6. Longitudinal fluid temperature profiles for a stable flame (a): 1-D adiabatic laminar case (black line) vs 2-D turbulent case with wall heat-loss, instantaneous and pointwise, time-averaged temperature profiles. Self-excited, unstable flame in 2-D turbulent case with wall heat-loss (b): time history of the maximum pressure and temperature recorded throughout the computational domain (blue and red lines, respectively). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Ignition of hydrogen reheat flame in a two-dimensional channel flow with heat loss to isothermal, no-slip walls: temperature field and centreline pressure profile (left); temperature field and the ratio between the local mixture ignition time, τig, and the channel convective residence time, t res(right).

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Fig. 8. Blow-out and re-ignition of a hydrogen/air reheat flame in a two-dimensional channel flow with heat loss to isothermal, no-slip walls: temperature field and centreline pressure profile (top), refer to Fig. 7 for colorscale legend. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

clearlyillustratedbythetemperaturedistributionsandthecenter- linepressureproﬁlesrepresentedintheplotsontheleft-handside ofFig.7.Thecompressionheatingofthereactantsresultsinapro- gressiveshortening ofthelocalignitiontime

τ

ig,loctovaluescom- parablewiththelocalconvectiveresidencetimet_res_,_loc therebyre- sulting intheupstream displacementofthespontaneousignition front.Thisisquantitativelyillustratedbytheratioof

τ

ig,loc/t_res_,_loc∼ 1 in the plots on the right-hand side of Fig. 7, where the local value ofthe ignition time

τ

ig,loc iscalculated by the analytic ex- pression providedin [35] forhydrogen–airmixtures, whilet_res_,_loc is estimated as x/Uc. The upstream displacement of the sponta- neousignitionfront haltsoncetheratio

τ

ig,loc/t_res,loc (andits spatial gradient)becomes toolarge to be overcome by the effect of compression heating, see Fig. 7(h). Figure 8 focuses on the sub- sequent relaxationphase, in whichthe reactant mixture temperature decreases (lightgreen region immediatelyupstream ofthe spontaneous ignitionfront)because ofthe simultaneouslocalde- creaseinpressure,whichinturn,isduetotheabruptinterruption of the upstream displacement of the spontaneous ignition front.

Auto-ignitioncannotbesupportedanylongeratsuchanadvanced location dueto thelow(er) temperatureofthe approachingreac- tants. Thereaction frontisdisplaced downstreamand,ultimately, ﬂushedout ofthechannel(in thepresentconﬁguration)beforea newignitioncyclecommences.Thisprocessresultsinintermittent spontaneous ignitioncycles,alternating betweenignitionandup- streamadvancementfollowedbyrecessionandextinction.

The characteristic time scales of the cyclic process observed in thesetwo-dimensional DNSwith wall heat lossare consistent withtheconvective-acousticfeedback mechanismofthecombus- tion instabilitydescribedinSection3.3.2 fortheone-dimensional adiabatic conﬁgurations. Approximating x_f_,₁∼5cm, x_f_,₂∼15cm, U_in∼200m/sand c∼650m/sinEq.(1),a value ofT_W∼1.44ms is obtainedwhichprovides satisfactoryagreement withthevalue T_f_,_obs∼1.6ms observed in Fig. 6(b) as the time period between eachignitionandextinctionevent(thewidthofthered“bumps”).

4. Reaction-frontvelocityandinstabilitiesofturbulent hydrogen/airﬂamesatauto-ignitiveconditions 4.1. Backgroundandrationale

Understandingthemeanvelocitiesofthereactantsthataresuf- ﬁcient tostabilizea turbulent reactionfront resultingfrom ahy-

bridcombustionmodeofpropagationandauto-ignitionisimpor- tant tothedesign ofreheat combustionsystems[59].Thisis be- causemixedcombustionmodestransitioningfrompredominantly spontaneousignitiontopredominantlyflamepropagation,are be- lieved to limit engine operation with hydrogen-based fuels [12]. Therefore, using three-dimensional turbulent DNS, the main rationale forthepresentsection is toobtain quantitative estimates of the ability of the turbulent reaction front of hydrogen reheat flamestobalancethemeanvelocityofanapproaching flowofre- actants. Moreover, it is alsoshown that the self-excitedintrinsic instabilities ofthe reactionfront, describedin Section 3for one- andtwo-dimensional configurations,can occurin themodulating presenceofathree-dimensionalturbulentvelocityfield ifspecific conditionsaremet.

Although the present investigation builds upon the work by Savard etal. [33], it substantially differs fromtheir earlier work in several respects. Firstly, Savard imposes upon the mean flow andthroughoutthecomputationaldomainhomogeneousisotropic turbulencesupportedby artificial(numerical)forcingofthelarge- scaleturbulentmotionsacrosstheflamebrush.Thismethodology wasoriginally developed forisotropic, incompressible turbulence [60]anditsapplicabilitytocombustionDNSremainscontroversial becauseoftheimplications ofartificialforcingontheturbulence- chemistry interaction dynamics, particularly on the burnt side of the flame which is not well understood as of yet. Secondly, Savard characterized the transitionbetweenspontaneous ignition andflamepropagation utilizingrelatively longcomputational do- mainsthatpermit residencetimesoftheorderrequiredforspon- taneous ignition but, through the useof a low-Mach approxima- tion, neglect compressibility effects and compression heating on the spontaneousignition process. Onthe basis ofthe results describedin Section3 ofthepresentstudy,thereis compelling ev- idence suggesting that compressibility plays a significant role in hydrogen-reheatflamestabilization.ThepresentDNSinvestigation usesrelativelyshortcomputationaldomainsand,placingthereac- tionfrontatadistancefromtheinletboundarythat theoretically can notsupport spontaneousignition (initially),focuseson propagating turbulent reaction fronts and on their ability to balance theapproachingupstreamflowofthereactantmixture.Here,com- pressiblity effects are fullycaptured andable to detect eventual transition to mixed propagation/auto-ignition combustion modes duetolocal effectsofcompressionheating, shortening of

τ

ig and subsequentearlyspontaneousignition.

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L T(cm ) 0.3 0.3 0.3 0.3

L 11(cm ) 0.31 0.17 0.14 0.11

ηk(cm ) 0.028 0.01 0.009 0.006

τT = L T/u (s ) 1.00e-03 3.00e-04 2.00e-04 1.20e-04 τ11 = L 11/u (s ) 1.02e-03 1.68e-04 9.14e-05 4.25e-05

τk(s ) 2.11e-04 3.46e-05 1.88e-05 8.75e-06

Re t 22 75 112 188

Ka = τR/τk 0.27 1.63 3.0 6.43

Da = τT/τR 17.8 5.33 3.55 2.13

t end/τT 4 13.3 20 33.3

T u(K ) 1100 1100 1100 1100

P(atm ) 1 1 1 1

4.2. 3-DDNSconﬁguration

The approach chosen in the present DNS study employs an idealized reheat combustion configuration corresponding to a statisticallyplanarturbulentpremixedflameplacedinunbounded turbulent flows of the target reactant mixture injected at the domain inlet boundary and characterized by auto-ignitive and adiabatic conditions (no heat loss). The three-dimensional computational domain has physical dimensions Lx=2cm, Ly=1cm andLz=0.5cminthestreamwise(non-homogeneous)x-direction, transverse (periodic/cyclic) y- and spanwise z-direction, respectively. The domain is discretized on a 800×400×200 Cartesian (uniform)mesh providing25μm spatialresolution.Thesimulated three-dimensionalturbulenthydrogenreheatflamesrepresentthe target conditionspreviously describedinSection 3.1(Tu=1100K,

φ

=0.35 and T_ad∼1800K at normal atmospheric pressure). For the present auto-ignitive conditions a unique reference laminar flamespeed,S_R,isdefinedastheinletflow velocityatwhichthe rateofchangeofthepositionoftheflamefromtheinletwithinlet velocity is ata maximum [23].This referencespeed corresponds to the inlet flow velocity above which the steady-state laminar flame detaches from the domain inlet in a one-dimensional DNS configuration [23]. The reference speed and the associated thickness of an unstrained adiabatic laminar premixed flame in the limit of an unreacted upstream composition are S_R∼24m/s and l_R∼1.35mm (estimated by the maximum-thermal-gradient method). Thesevaluesresultinareferencechemical(flame)time scale,

τ

R=l_R/S_R=5.63e⁻²ms. Notably, S_R differs from the conventional “laminar-flame-speed” definition, S_L, not applicable at auto-ignitiveconditions,byaccountingfortheroleoftheresidence time(owingtotherelevanceofignition)ontheflamespeed.

In themainparametric investigation,thereactantflow issub- jectedtodifferentlevelsofturbulenceintensity(Section4.3)while inasecondparametricsweepadifferentinduction-timehistoryof the flammablemixture (Section 4.4) is investigated. The initially planar flames are subjected to different inlet turbulence intensities u=3 (Case A), 10 (Case B), 15 (Case C) and 25m/s (Case D).Theseturbulenceintensities arespecifiedasrandom 3-Dflow fields withaprescribed Passot–Pouquetenergyspectra,following a well-established procedure describedin [57].The velocity fluctuations are superimposed onto the mean flow that is advected intothe domainfromthe upstreamboundarywithavelocityU_in. The chosenconditionscorrespond toturbulentReynoldsnumbers Ret=22, 75,112 and188,respectively. Assuminga size limit for thelargesteddiesintheflowequaltoL_T=0.3cm(wellwithinthe smallest transverse domain dimension, Lz=0.5), thecorrespond- ing longitudinal integral length scales, L₁₁, lie between 0.11 and 0.31cm.Table1summarizesthephysicalscalesoffluidmotion(al-

domainthroughaprogress-variablemappingfromthecorrespond- inglaminar1-Dsolutionandplacedatadistancex_f=0.5cmfrom the domain inlet boundary. The progress variable C is a scalar parametrizationofthereactiveflowfield,basedforthepresentim- plementationon thehydrogenfuelmassfraction,thatis equalto zerointhefreshreactants(C=0)andunityintheburntproducts (C=1). An initial mean velocity U₀_,_in=25m/s isimposed atthe domaininletboundaryandthroughoutthecomputationaldomain forallcasesdescribedinthefollowingsections.Forthedurationof thesimulation,themeanflowvelocityimposedattheinletbound- aryU_inisadjustedateachtimestepsuchthatthetotalamountof fuelthatinstantaneouslyentersthedomainmatchesthevolumet- ricfuelconsumption rateofthedeficientreactant,hydrogen.This procedureassuresthatthemeanflowvelocityU_in∼Umisapproxi- matelyequaltothedisplacementvelocityS_t oftheturbulentflame reactionfront,therebyensuringthatthelatterremainswithinthe computational domain at all times. This simple method is able to retainthe meanflame position(approximately) inthe vicinity oftheinitializationlocationx_f,withonlymarginal upstreamdis- placements.Thisisimportantforthefollowingreasons:1)thetur- bulentvelocityfluctuationsimposedattheinletboundaryareable tointeractwiththeflamefrontbeforetheyaredissipated;and2) thereactant-mixtureresidencetimetres∼x_f/Umbetweentheinlet boundaryandtheflamepositionremainssmallerthantheignition delaytime,

τ

ig∼0.15ms,ofthetargetmixture,therebypreventing apurelyauto-ignitioncombustionregimefromoccurring.

The typicaltime evolution ofthe DNSsolutionis qualitatively illustrated in Fig.9, whichprovides a graphical representationof the turbulent (statistically planar) flame, at auto-ignitive conditions, asit respondsto the approaching turbulent flow. The flow conditionsrepresentedinFig.9,asan example,correspondtothe turbulenceintensityu=15m/s(Case C)andexhibit considerable wrinklingoftheflamefront(representedbytheisosurfaceoftem- perature atT=1500K) while, at the lowest turbulenceintensity level u=3m/s (Case A), only a very mild wrinklingis observed (notshown).At theonset ofthesimulation, theinitially flatsur- face marking the reaction-front location is wrinkled by the un- derlyingturbulent flowfield, anditrapidlyaccelerates, advancing in the upstream direction towards the inlet boundary. However, theproceduredescribedaboveautomaticallyadjuststhemeanin- let velocity accordingly and ensures that the reaction front re- mainsata(statisticallyconstant)distancefromtheinletboundary plane.

4.3. Effectoftheturbulenceintensity

Themainparametricinvestigationconductedquantiﬁestheef- fect of the turbulenceintensity ofthe approaching reactant ﬂow ontheturbulentreaction-frontvelocityatauto-ignitiveconditions.

Hydrogen flames at reheat conditions are characterized by values of the referenceflame speed S_R that are considerably larger than those found at “conventional” premixed-combustion condi- tionsand,forthepresentconfiguration,S_R∼24m/s.

4.3.1. Globalanalysisofthereaction-frontvelocity

Theglobalreaction-frontvelocities,St,estimatedfromtheDNS cases,arepresentedinFig.10intermsofnon-scaledvalues.Scaled valuesaresummarizedinFig.13(b)below.Timehistoriesofthein- stantaneous(ﬂuctuating)valuesofSt (redlines)forturbulent hydrogen reheat ﬂames, subjected to inlet turbulence intensities u

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Fig. 9. Initial conditions and time evolution of the turbulent statistically planar hydrogen reheat ﬂame (Case C). The pink isosurface ( T = 1500 K ) demarcates the reaction front while the streamwise velocity component and the temperature ﬁelds are shown on the xy - and xz -planes respectively.

equal to 3,10, 15 and25m/s, are shownin Fig.10(a). The solution fora one-dimensional laminarconfigurationis alsoincluded for reference (black line). After an initial transientwhich occurs untiltimet∼0.5msthereaction-frontvelocity,S_t,relaxestowards itsmeanvaluesof32,34,36and42m/s,respectively.TheDNSare discontinued once amoving time windowcorresponding to 1ms doesnotchangemorethan1%fromthemeanvalueofSt.Because theinstantaeousfuelconsumptionrateinaturbulentflameis,in- herently, a fluctuating quantity, in order to obtain a meaningful estimate for St using this approach, the averaging must be con- ducted over a sufficiently long period.As discussed in[33], it is unclear how longthisperiod should beand, while previous DNS results suggest this periodmay be on the orderof 10–100

τ

T,11, for shorteraveraging periods an uncertainty ofapproximately5–

20%onthecalculatedvaluesofSt isproposedbySavardetal.(see AppendixBof[33]).

4.3.2. Self-excitedinstabilityofturbulenthydrogenreheatflames Although the rationale for performing the three-dimensional DNSprincipallyconcernstheestimationoftheturbulentreaction- frontvelocity, inthissection exploratoryDNSare performedout- sideofthetarget(stable) conditionstoinvestigatetheoccurrence of self-excited flame instabilities in the presence of a realistic, three-dimensionalrepresentationoftheturbulentvelocityfield.As discussed in Section 3, one- and two-dimensional DNS indicate that,atatmosphericpressure, self-excitedflame instabilitiestypi- callyariseatareactantstemperatureTu∼1000Kduetotheprox- imity to the crossover temperature. Therefore, an additional 3-D DNS, otherwiseidenticaltoCase B(i.e.u=10m/s),isperformed atTu=1000Kandatmosphericpressure(named CaseB2). More- over,threeadditional“variants” ofCaseBareperformedforpres- surized conditions,corresponding to P=5bar, atthree reactants’

temperature equal to Tu=1000K,1100K and 1135K (named B3,

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Fig. 10. Time evolution of the turbulent reaction-front velocity S testimates for turbulent hydrogen reheat ﬂames subject to different turbulence intensities (a), reactants’

temperatures and pressures (b). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

Fig. 11. Longitudinal proﬁles of average temperature for the different turbulence intensities (a). Scatter plots of the H-atom mass fraction and net reaction rate of the HO 2

radical (b), of the heat-release rate (c) and of the temperature versus the progress variable C.

B4 and B5 respectively). The 3-D DNS configuration for the ele- vatedpressurecasesisformallyidenticaltotheatmosphericpres- surereferenceCaseB(i.e.samedomain size,turbulenceintensity etc)exceptforthefinermeshresolution(8.3μm)thatisrequiredto resolvethesmallerlengthscalesofmotion,diffusionandreaction ofthereactiveflowatelevatedpressure.Thisimplies,ofcourse,a considerableincreaseinthecomputationalcostofthepressurized casesand,therefore,thesecalculationsareintegratedforashorter timeinterval(2msinsteadof4ms).

The global turbulent reaction-front velocities, St, estimated fromtheseadditionalDNSdatasets, “variants” ofCase B, are pre- sentedinFig.10(b)andcomparedtoCaseB,suggestingthefollow- ing:

1. Onset of instability – At atmospheric pressure, the turbulent reaction-front velocity St develops a distinct oscillation pat- ternwithonlyminorspatialdisplacement(notshown)atTu=

1000K, Case B2 (black dashed line, delta symbols). This is shown by the almost sinusoidal pattern of the St ﬂuctuation andby a visibly larger oscillation amplitudecompared to the referenceCase B for which only small stochastic ﬂuctuations canbeobserved(blacksolidline,gradientsymbols).

2. Self-excitedinstability– Atpressurizedconditions,theturbulent reactionfrontdevelopsastrongself-excitedinstabilitywithsig- niﬁcantupstream anddownstreamspatialdisplacementofthe reactionfront(seeSupplementarymaterial)atT_u=1135K,Case B5(red dashed line,diamond symbols).It is clearthat the St

ﬂuctuationexhibitshigherorderfrequencyharmonics.Thisisa signofnon-linearphenomenacausedbythenon-linearsatura- tionoftheﬂameathighamplitudes.

3. Noinstability – At pressurized conditions but lower reactants temperatures,Tu=1000K (Case B3) and 1100K(Case B4), no self-excited ﬂame instability is observed (solid and dash–dot