direct numerical simulation of ignition front propagation...

Combustion and Flame 145 (2006) 128–144www.elsevier.com/locate/combustflame

Direct numerical simulation of ignition front propagationin a constant volume with temperature inhomogeneities

I. Fundamental analysis and diagnostics

Jacqueline H. Chen a, Evatt R. Hawkes a,∗, Ramanan Sankaran a,Scott D. Mason b, Hong G. Im c

a Reacting Flow Research Department, Combustion Research Facility, Sandia National Laboratories, P.O. Box 969 MS 9051,Livermore, CA 94551-0969, USA

b Lockheed Martin Corporation, Sunnyvale, CA 94089, USAc Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2125, USA

Received 18 December 2004; received in revised form 20 August 2005; accepted 25 September 2005

Available online 5 January 2006

Abstract

The influence of thermal stratification on autoignition at constant volume and high pressure is studied by directnumerical simulation (DNS) with detailed hydrogen/air chemistry with a view to providing better understandingand modeling of combustion processes in homogeneous charge compression-ignition engines. Numerical diag-nostics are developed to analyze the mode of combustion and the dependence of overall ignition progress oninitial mixture conditions. The roles of dissipation of heat and mass are divided conceptually into transport withinignition fronts and passive scalar dissipation, which modifies the statistics of the preignition temperature field.Transport within ignition fronts is analyzed by monitoring the propagation speed of ignition fronts using the dis-placement speed of a scalar that tracks the location of maximum heat release rate. The prevalence of deflagrativeversus spontaneous ignition front propagation is found to depend on the local temperature gradient, and may beidentified by the ratio of the instantaneous front speed to the laminar deflagration speed. The significance of pas-sive scalar mixing is examined using a mixing timescale based on enthalpy fluctuations. Finally, the predictions ofthe multizone modeling strategy are compared with the DNS, and the results are explained using the diagnosticsdeveloped. 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Ignition; Direct numerical simulation; HCCI; Multizone model

* Corresponding author. Fax: +1 925 294 2595.E-mail address: [email protected]

(E.R. Hawkes).

0010-2180/$ – see front matter 2005 The Combustion Institute.doi:10.1016/j.combustflame.2005.09.017

1. Introduction

Homogeneous charge compression-ignition (HC-CI) engines are considered a viable concept as analternative to diesel engines [1]. In contrast to con-ventional compression-ignition (CI) engines, HCCIengines exploit a lean intake charge that is well mixed

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J.H. Chen et al. / Combustion and Flame 145 (2006) 128–144 129

prior to combustion. HCCI engines may thus pro-vide efficiency gains over spark-ignition engines, andlower emissions as compared with CI engines.

At present, the combustion mode in HCCI en-gines is not well understood, as both volumetric andfrontlike combustion modes can occur [2]. A majorchallenge posed by this method of combustion is tocontrol the heat release rate, and in particular, to pro-vide a means to spread it out over several crank angledegrees, suppressing the occurrence of rapid rate ofpressure rise. One possible control strategy is to intro-duce inhomogeneities in the temperature or mixturecomposition in order to produce the desired heat re-lease rate [1,3]. Moreover, incomplete turbulent mix-ing and temperature stratification between the bulkgases and the cylinder wall lead to spatial nonunifor-mities that also contribute to a range of combustionmodes that are distinct from homogeneous autoigni-tion. Therefore, a better understanding of autoignitionof an inhomogeneous charge at constant volume iscrucial in the development of control strategies forHCCI engines.

The overall objectives of the present study areto understand the influence of temperature inhomo-geneities on characteristics of ignition and subsequentcombustion in an HCCI-like environment and to pro-vide information relevant to the modeling of HCCIcombustion. The evolution of autoignition from aninitial spectrum of “hot spots” at high pressure is sim-ulated using two-dimensional direct numerical simu-lations (DNS) with a detailed hydrogen/air reactionmechanism. Although turbulence is inherently three-dimensional, the prescribed two-dimensional randomtemperature distribution can still provide useful in-formation regarding the chemical induction and frontpropagation behavior in response to different statis-tics of the imposed scalar and velocity fields. Dueto the computational expense of resolving extremelythin ignition fronts at high pressure throughout theirtemporal evolution, the chemical kinetics consideredare necessarily oversimplified to a hydrogen/air sys-tem. Hydrocarbon fuels, which build on hydrogen asan important subset of the mechanism, will be consid-ered in future studies.

One of the primary objectives in this study isto develop systematic and rational methods to iden-tify and analyze various regimes of ignition and sub-sequent combustion processes. The theoretical basisstems from an earlier study by Zel’dovich [4], whodescribed ignition of a nonuniform mixture in severaldistinct regimes of propagation, including deflagra-tion, spontaneous ignition front propagation, and det-onation. He further showed that the spontaneous igni-tion front speed is inversely proportional to the initialtemperature gradient, suggesting a characterizationof ignition regimes based on the speed of the igni-

tion front. For the parametric conditions applicable toHCCI engines, formation of a detonation wave is un-likely and thus the first two regimes are consideredrelevant in the present study. The concept has beenapplied to identifying ignition regimes in various one-dimensional configurations [5–8]. The present workextends the idea into more realistic two-dimensionalignition problems, where details of multidimensionalaspects including wrinkling and mutual interaction offronts are fully accounted for.

The second main objective is to provide informa-tion relevant to the modeling of HCCI. In the case ofpurely homogeneous combustion, the modeling prob-lem reduces to simple homogeneous ignition calcu-lations. However, stratification of the charge, whetherin terms of temperature or mixture composition, intro-duces a range of ignition delay times into the cylinder,which must be accounted for in a model. At low lev-els of stratification or long stratification length scales,typical of HCCI engine conditions, molecular mixingeffects can be neglected [4,9]. Presuming that lengthscales are not so long so that compressible flow ef-fects are important, a reasonable approximation isthen that different locations in the cylinder interactonly by pressure-work. These assumptions are em-ployed in a modeling strategy referred to as the mul-tizone model, which has been advanced by Aceveset al. [10–12]. The model has been demonstrated togive good predictions of engine pressure traces andheat release rates at low computational cost in typi-cal HCCI engine conditions. There is a need to un-derstand why these predictions accurately reproducethe pressure and heat release rate traces and to de-velop techniques for estimating a priori the region ofapplicability. Using the DNS as a numerical experi-ment, we evaluate the performance of the model, andprovide understanding of its limitations. In order todevelop methods of determining the model’s validity,we will deliberately apply it in some regions outsidethe valid regime for which it was intended. Also, themultizone model serves as a useful foil to help vali-date the tools developed to characterize the diffusivetransport processes precisely because the model doesnot include these processes and hence any discrepan-cies between the model and the DNS give an indepen-dent measure of their importance.

Our previous DNS study Sankaran et al. [13] in-vestigated the characteristics of turbulent ignition byconsidering three initial temperature fields character-ized by different skewness of the fluctuation. In thisstudy, a heuristic temperature gradient cutoff condi-tion was used to quantify the relative dominance be-tween spontaneous ignition and deflagration regimes.A criterion to predict the relative dominance of thetwo ignition regimes was also proposed.

130 J.H. Chen et al. / Combustion and Flame 145 (2006) 128–144

The present study is composed of two parts. Thispaper, Part I, extends the previous work [13] to ex-plore the fundamental issues with an emphasis placedon front propagation aspects of the phenomena. Themain objective is to develop numerical diagnostics toidentify and analyze various regimes of ignition andsubsequent combustion processes. We also presentcomparisons of the DNS results with the multizonemodel, and identify the reasons for discrepancies. Us-ing the tools developed in this paper, in Part II [14]a parametric study is performed on the influence ofthree key controlling parameters: the initial stratifica-tion amplitudes, the stratification length scales, andthe turbulence intensity. Comparisons with the multi-zone model are presented for each case.

In particular in the present paper, the utility of ascalar isosurface to track the propagation speed andbehavior of the ignition fronts is assessed. The role ofmolecular diffusive effects including heat conductionand mass diffusion on ignition propagation is identi-fied. Molecular diffusive effects are divided concep-tually into passive scalar mixing and diffusion withinpropagating ignition fronts, with the latter controllingthe transition between the regimes identified by Zel’-dovich. Ignition front propagation evolution duringconstant volume combustion is monitored by evalu-ating the surface mean of the ratio of the local frontspeed to the local deflagration speed. This local mea-sure is also used to distinguish front propagation inthe different regimes. The results are compared witha volumetric measure based on a critical temperaturegradient [13]. Predictions of the multizone model arepresented and the reasons for discrepancies are ex-plained.

2. Numerical implementation and initialconditions

The present study is focused primarily on the ef-fect of temperature inhomogeneities on autoignitionin the presence of compression heating resulting fromcombustion heat release rate due to ignition frontpropagation. In an HCCI engine, these effects are im-portant near top dead center. It is assumed that theisentropic compression stage, where turbulent mix-ing of inert mixtures prevails, has already occurred.Therefore, the present simulations focus on autoigni-tion of a lean hydrogen–air mixture in a closed vol-ume in the presence of an inhomogeneous temper-ature field. A uniform hydrogen/air mixture with afuel-air equivalence ratio of 0.1 is chosen. This mix-ture has been chosen such that the pressure rise is suf-ficient for studying ignition under compression heat-ing conditions, but is insufficient for shock wave for-mation. It is known that as the volumetric heating

increases there exists a potential for the coalescenceof ignition fronts to form shock waves [6,7], whichthe present numerical grid cannot resolve. In all casespresented, the mean initial temperature is 1070 K, andthe uniform initial pressure is 41 atm. These condi-tions correspond to a compression ratio of 15:1 start-ing from 1 atm and 400 K. At this pressure and tem-perature an insignificant amount of heat release ratehas occurred for the hydrogen/air mixture. The pri-mary source of heat release rate is through the ther-mal dissociation of hydrogen peroxide which is neg-ligible at this pressure until the temperature exceeds1100 K.1 Hence, it is valid for the present H2/airmixture to decouple the initial isentropic compres-sion stage from the combustion-induced compressionheating stage.

The ignition behavior is first analyzed by a sim-ple one-dimensional model in order to provide a ref-erence case to help interpret the two-dimensionalsimulation results. The 1D constant-volume caseswere initialized from a sinusoidal temperature pro-file. Boundary conditions are periodic, such that heatrelease rate in the computational domain leads to apressure rise and compression heating of the reactantsas in the 2D calculations. Parametric studies are con-ducted by varying the length scale and amplitude ofthe temperature fluctuation.

In the latter part of the paper, two-dimensionalsimulation results, previously studied as Case A inRef. [13], are analyzed further. In the two-dimensionalturbulent ignition case, the turbulent flow field is pre-scribed by an initial turbulent kinetic energy spectrumfunction by Passot–Pouquet [15],

(1)E(k) = 32

3

√2

π

u′2

ke

(k

ke

)4exp

[−2

(k

ke

)2],

where k is the wave number magnitude, ke is the mostenergetic wavenumber, and u′ is the RMS velocityfluctuation. A random temperature field is superim-posed on a mean temperature field. The temperaturespectrum, which is similar to the kinetic energy spec-trum, is used to specify the characteristic scales of theinitial exothermic spots shown in Fig. 1.

Turbulent mixing modulates the scalar gradientsimposed by the initial inhomogeneities, affecting thelocal dissipation rates of key radicals and thermal en-ergy. Therefore, the specification of the initial tem-

1 However, this would not be the case for a multistageignition of a hydrocarbon fuel such as n-heptane wherelow-temperature reactions involving the thermal dissociationof ketoheptylhydroperoxide generate significant heat releaserate at temperatures greater than approximately 750 K at thispressure.


Fig. 1. Initial temperature spectrum for 2D DNS. Tempera-tures vary between 1106 K (blue) and 1203 K (red).

perature and turbulence fields affects both the ignitiontiming and the subsequent heat release rate.

The initial autocorrelation integral scale of the ve-locity fluctuations is 0.34 mm, and the most energeticlength scale is 1.0 mm. This length scale correspondsto the wavelength in the spectrum where the kineticenergy is maximum. The velocity RMS is 0.5 m/s.These parameters were selected so that the turbulenceintegral time scale based on the most energetic lengthscale is comparable to the homogeneous ignition de-lay of the mixture at the same mean temperature andpressure, thus promoting strong interaction betweenthe turbulent mixing and ignition chemistry. The ho-mogeneous ignition delay is τ0 = 2.9 ms. The tem-perature field has an RMS fluctuation of 15.0 K, theautocorrelation length scale is 0.15 mm, and the mostenergetic length scale is 1.32 mm. These parametersare comparable to those in an engine operating atlow load, with the exception that the length scale ofthe exothermic regions and the velocity fluctuationsare approximately five times smaller than observedin an engine [2]. The resulting turbulence integraltime scale and ignition delay are comparable to thosetimescales in an engine.

The conservation equations for a compressible re-acting flow are solved in the DNS using an expliciteighth-order accurate finite differencing scheme [16]and a fourth-order accurate Runge–Kutta scheme fortime advancement [17]. The boundary conditions areperiodic and lead to compression heating of the re-actants as the mixture ignites. The mixture-averagedthermal conductivity is temperature-dependent [18],and the individual species specific heats are obtainedusing the Chemkin thermodynamic database [19].The diffusion coefficients are obtained by prescrip-tion of Lewis numbers for individual species, fit-ted from mixture-averaged transport coefficients [19].The pressure-dependent hydrogen–air chemistry isbased on a detailed chemical mechanism [20] with 8

reactive species and 21 reversible reactions. The do-main is 4.1 mm in each of the two spatial directions.A uniform grid spacing of 4.3 µm was required to re-solve the ignition fronts at this pressure.

3. Ignition front tracking and species budget

According to Zel’dovich [4], the speed of the ad-vancing combustion wave, given an inhomogeneousinitial condition for temperature, is related to the gra-dients of temperature and to the mode of combustion.Zel’dovich [4] identifies the following main regimesin order of decreasing speed and increasing initialtemperature gradient, which are also discussed in [6]:(i) a nearly instantaneous thermal explosion, (ii) anautoignitive spontaneous ignition front propagatingat speeds greater than the normal detonation rate,(iii) a developing detonation, (iv) a subsonic sponta-neous ignition front propagation, in which moleculardiffusion effects ahead of the front are unimportant,and (v) a normal deflagration, in which mass and heattransport at the molecular level control the propaga-tion. The focus of the present study is on the transitionbetween the normal deflagration regime and the sub-sonic spontaneous ignition front propagation regime.In the subsonic spontaneous ignition front regime, thefront speed sig may be related to the gradient of theignition delay time τ [4] as follows:

(2)sig = 1

|∇τ | .

Note that, in the constant volume case, the reactivityof a mixture depends not only on its initial state, butalso on the amount of compression heating it receivesduring the progress of ignition. Therefore, τ dependsnot only on the local mixture condition but also onthe distribution of mixture conditions in the volumewhich affect the amount of pressure work. When thecomposition is uniform, the gradients of ignition de-lay are solely due to temperature variations, and theequation may be expanded as

(3)sig = 1∣∣ dτdT0

∣∣|∇T0| ,

where T0 is the initial temperature. For low initialtemperature gradients, it is evident that the sponta-neous ignition front speed becomes large. In this sit-uation there is insufficient time for molecular trans-port toward the reactants to be significant, and thefront speed is well approximated by Eq. (3). How-ever, as the temperature gradient is increased, thefront speed slows down to the point where the mole-cular transport to the fresh gases becomes significant.The observed speed of the propagating front is then


increased relative to the prediction of Eq. (3). For suf-ficiently high temperature gradients, the propagationis controlled by a reactive–diffusive balance and issimilar to a normal laminar premixed flame, althoughthe flame is propagating into an unsteady autoignit-ing mixture. These theoretical predictions underlinethe importance of both the local temperature gradient,and the front speed as indicators of the combustionregime. In this study, we explore the utility of thesequantities as delineators of the regime.

To obtain the speed of the advancing combus-tion wave, a surface tracking technique is employed.While it is clear that in these conditions reaction isnot always confined to a thin surface, it does providea very convenient means for analyzing the DNS re-sults.

If an ignition delay time is defined for each fluidparcel as the time taken for a scalar φ to reach a crit-ical level φc, then the speed sd of the ignition frontmay be identified with the displacement speed [21]given by

(4)sd = ±Dφ/Dt

|∇φ|∣∣∣∣φ=φc

,

where D/Dt is the material derivative, and the signconvention depends on whether the scalar increases(+) or decreases (−) as the critical value is ap-proached. We have developed analytic front speedexpressions for a temperature isosurface and a scalarisosurface based on the fuel mass fraction. While bothdefinitions give very similar conclusions, the temper-ature evolution equation contains multiple terms rep-resenting different physical effects. Therefore, in thisstudy only results from the mass fraction isosurfaceare presented. A density-weighted speed can be de-fined as s∗

d = ρsd/ρ0, where the ρ0 is a representativedensity of the reactants. This is calculated from the lo-cal enthalpy and unburnt mixture condition assumingpressure and enthalpy do not vary as the front passes.Statistics were extracted from the YH2 = 8.5 × 10−4

isocontour, which corresponds approximately to thelocation of maximum heat release rate through mostof the simulation.

A nominal measure of the front speed of a dif-fusion controlled deflagration is required in order tocompare with the speed of the propagating front.Here, we employ the steady premixed flame speed atthe local conditions. It is recognized that because ofthe highly transient conditions with temperature gra-dients and autoigniting reactants, this is simply anapproximation, however it will be later shown thatit provides a good estimate of the transition betweenregimes that is consistent with several other measures.Laminar flame speeds are obtained with the PREMIXcode [22], and are tabulated as a function of the reac-tants enthalpy and pressure.

In the following sections, the budget terms in thespecies continuity equation for hydrogen are exam-ined to determine the relative importance of diffu-sion to reaction in determining the front propagationregime. The evolution of the hydrogen mass fraction,YH2 , is given by

(5)DYH2

Dt= 1

ρ

∂

∂xiρDH2

∂YH2

∂xi+ ω̇

ρ,

where the first term on the right-hand side representsdiffusion, and the second term represents the net re-action rate of hydrogen. DH2 is the diffusivity of hy-drogen, ω̇ is the reaction rate, and ρ is the density.

4. Multizone model

The multizone model [10–12] has been demon-strated to accurately predict pressure traces and heatrelease rates in engine-derived HCCI conditions, witha very low computational overhead. It is an approxi-mation that is applicable in the subsonic spontaneousignition front propagation regime. It ignores effectsassociated with molecular diffusion and with the fi-nite speed of sound. An initial spatial distribution ofmixture states obtained from a prior nonreacting com-putational fluid dynamics calculation which fully in-cludes turbulent mixing is approximated by a finitenumber of zones such that a joint probability den-sity function (PDF) of temperature and species massfractions approximately matches that of the originaldistribution. These zones are then treated as homoge-neous reactors that are coupled together only by pres-sure, which is assumed to equilibrate instantaneously.In HCCI engine conditions, the pressure coupling isexpected to be the most significant interaction be-tween different parts of the cylinder.

In the constant volume case, the evolution equa-tions for the species mass fraction and temperature ineach zone may be summarized as follows,

(6)dYi,j

dt= ωi,j

ρj,

(7)dTj

dt= 1

ρj cp,j

(Aj +

∑Nzk=1 AkBk

1 − ∑Nzk=1 Bk

),

where the subscript j refers to the zone to which theequations apply, the subscript i refers to the chemi-cal species, the subscript k refers to any zone, Nz isthe total number of zones, Yi,j is the mass fraction,ωi,j is the reaction rate, ρj is the density, Tj is thetemperature, and cp,j is the specific heat at constantpressure. For each zone the quantities Aj and Bj aredefined as follows,


(8)

Aj =Ns∑i=1

(Nz∑k=1

(ρk

Vk

VRiTk

dYi,k

dt

)− ρjhi,j

dYi,j

dt

),

(9)Bj = Vj

V

(1 − 1

γj

),

where Ns is the number of chemical species, Vj is the

volume of a zone, V = ∑Nzk=1 Vk is the constant total

volume of all zones, γj is the ratio of specific heats,hi,j is the species enthalpy, and Ri is the species gasconstant. The uniform pressure is determined fromthe equation of state applied to all zones, and the in-dividual zone volumes are determined from the equa-tion of state applied to each zone.

The multizone model is initialized from the DNSsolution at a specified time by sampling a number ofpoints. In the 1D cases, each DNS grid point is treatedas a zone, whereas in the 2D case a reduced numberis employed (57,600). The equations are integratedwith the identical numerical time integration method,reaction mechanism and thermodynamic approxima-tions as in the DNS. It is noted here that fewer zones,typically around 10–20, are found to be sufficient tosatisfactorily reproduce experimental pressure traces.Thus, multizone calculations have a low computa-tional cost which allows the treatment of very detailedchemical kinetic models. Here, a very large numberof zones was used in all cases simply so that resultsare ensured to be independent of the number of zonesand to allow an exclusive focus on the physical issuesrather than those of numerical convergence. Yet thecost of the multizone model calculations was negligi-ble when compared to the complete DNS.

5. One-dimensional test cases

To understand the combustion process in the pres-ence of temperature inhomogeneities, 1D test calcu-lations were performed. These test cases provide areference case and a basis for understanding the morecomplicated 2D turbulent DNS results.

In a first parametric study, the RMS temperaturefluctuation is fixed at 15.0 K, matching the two-dimensional Case A. To help determine the role oftemperature gradient and the wavelength of the ini-tial fluctuations, cases were run with different wave-lengths (4.1, 1.5, 0.75, 0.56, and 0.38 mm) represent-ing the spectrum of length scales observed in the 2Dsimulations. These cases will be referred to hence-forth by these wavelengths. In a second parametricstudy, the wavelength is fixed at 3.0 mm, and the RMStemperature fluctuation is varied. RMS temperaturefluctuations of 3.75, 7.5, 15.0, and 30.0 K were em-ployed. These cases will be henceforth referred to bythese amplitudes. Comparison of the results of the two

Fig. 2. (a) Temperature versus distance for a sequence oftimes; (b) heat release rate normalized by the maximum heatrelease rate of the homogeneous ignition at the mean tem-perature (HR0) versus distance. The equally spaced timesequence is numbered from 1 to 10, starting at 1 ms withan increment of 0.25 ms.

Table 1Reference values corresponding to homogeneous ignitiondelay at 1070 K and 41 atm for a hydrogen/air mixture atequivalence ratio 0.1

τ0 (ms) HR0 (MJ/m3/s)

2.9 2.9408 × 104

parametric studies will help delineate between molec-ular diffusion effects occurring in ignition fronts, andthe effects of passive scalar dissipation.

The overall progression and character of the igni-tion and combustion process in this 1D configurationis illustrated in Fig. 2. Fig. 2a shows the tempera-ture profile and Fig. 2b shows the heat release rateat several instants in time. All results presented in thepresent paper are normalized by their correspondingreference values for the maximum heat release rate,HR0, and ignition delay, τ0, corresponding to homo-geneous ignition at the mean temperature, 1070 Kand initial pressure of 41 atm summarized in Table 1.The time increments in these figures are 0.250 ms,or approximately 0.1 times the homogeneous igni-tion delay τ0. Combustion in this configuration pro-ceeds as follows. Ignition occurs first at the locationof highest temperature, in the middle of the domain,moderated by the effects of scalar dissipation in theignition kernel. Subsequently, a propagating combus-tion wave emanates from this location and travels tothe left and right of the domain. As fronts propagate,the remaining charge is heated by compression, accel-erating the ignition of the end-gas. Finally the end-gas


Fig. 3. (a) Front speed (s∗d ) and (b) |∇T | versus time for wavelengths ranging from 0.56 to 4.10 mm in the 1D test cases.

is consumed as the front arrives at the boundaries ofthe domain. The presence of a propagating front doesnot imply that this is a normal deflagration. The na-ture of the front and the role of molecular diffusionneed to be identified.

Fig. 3a shows the extracted front speed s∗d ver-

sus time for the various wavelengths. Also shown isa representative deflagration speed sL that has beendetermined based on a freely propagating premixedflame [22] for the enthalpy and pressure conditionsat the front surface. Fig. 3b shows the correspondingevolution in the magnitude of the temperature gradi-ent at the surface location, |∇T |. Comparison of thetwo figures indicates that the “U-shape” behavior inthe front speed is due to the influence of the gradi-ent term in Eq. (3). Both the initial ignition and finalconsumption points have a zero initial temperaturegradient and the speed becomes unbounded.2

Comparison of the different wavelength casesshows that the initial ignition time is delayed forshorter wavelengths due to diffusive losses from theignition kernel. On the other hand, the final consump-tion occurs earlier for shorter wavelengths, due todiffusive gains in the end gas parcel. In the longerwavelength limit where the spontaneous ignition frontspeed is much larger than the deflagration speed (butis still subsonic), diffusion is insignificant and the ig-nition delay and combustion duration are governedsimply by constant-volume homogeneous ignition,moderated by the effects of compression heating.

2 It is worth noting that while the speed does become un-bounded, this does not mean that consumption of reactantsis unbounded in the present case. In fact, the description ofcombustion as a front breaks down at the early stage of ig-nition, when combustion occurs in a kernel, and at the finalstage, when the fronts annihilate. Unbounded front speedsare theoretically possible in the case of a stratification occur-ring over an unbounded length scale. Practically, of course,this would never occur.

In the other limit of small wavelengths, the initialtemperature fluctuations dissipate before significantreaction occurs and the entire domain again igniteshomogeneously according to the mean temperature.Therefore, spontaneous ignition front propagation isexpected for both very long and short wavelengths.Between the two limits, a situation may exist wherethe local temperature gradients at the ignition time aresufficient to develop a deflagration wave whose prop-agation speed is higher than the spontaneous ignitionfront speed at this local condition. Therefore, for anintermediate hot-spot size, the ignition process mayundergo transition from a spontaneous propagation todeflagration, and then back to spontaneous propaga-tion again.

It is important to emphasize that diffusive effectscan be conceptually divided into two different phe-nomena: (1) passive scalar dissipation—essentiallythe modification of the initial distribution of temper-ature within the domain, independently of reaction,and (2) diffusive effects occurring within the front,as in a normal premixed deflagration. The two effectsare governed by different nondimensional parame-ters. Passive scalar dissipation may be important if thescalar dissipation timescale is comparable to the ig-nition delay, introducing a Damköhler number Da =τmix/τ0. On the other hand, the effects of diffusion inthe front become important when sig/sL ≈ O(1). Inthe present 1D parametric study, the RMS tempera-ture fluctuation is held fixed while the wavelength isvaried. This results in a change in both the Damköhlernumber and the front speed ratio. Due to the range ofwavelengths used in the 1D cases, both effects wereobservable.

The wave-propagation stage of the combustionprocess occurs near the bottom of the U-curve inFig. 3a. For the largest wavelength of 4.1 mm, thetemperature gradient never becomes large enoughto develop a deflagration wave, such that the entireprocess occurs in the spontaneous propagation regimeat a substantially higher speed. As the wavelength


Fig. 4. (a) YH2 budget terms and (b) temperature gradientin fronts, 1D test cases with 4.1 and 0.75 mm wavelengths.Dashed line: reaction. Solid line: diffusion. Dotted line: tem-perature gradient.

is decreased, the temperature gradient increases andthe deflagration regime starts to occur, as evidencedby the existence of the minimum propagation speed.Note that further decrease of the wavelength from1.5 to 0.75 mm does not result in a decreasedfront speed. Even further decrease in the wavelength(0.56 mm) approaches the small-wavelength limit be-havior, where passive scalar dissipation dominates,and the speed increases again. The lower limit of thefront speed corresponds to the critical initial tempera-ture gradient for which diffusion becomes important,and the minimum front speed can be identified witha deflagration speed. This speed is approximately50 cm/s. The transition evidently occurs near the 1.5-mm wavelength, which corresponds approximatelyto the most energetic wavelength of the initial tem-perature spectrum in the DNS, indicating that bothdeflagration and spontaneous ignition front propaga-tion regimes are expected in the 2D simulations. Fur-thermore, it is observed that the heuristic temperaturegradient cut-off of 1800 K/mm used in Sankaran etal. [13] is consistent with this transition. Note that thespeed independently obtained from PREMIX agreeswith the lower limit. The quantitative agreement sug-gests that it can serve as a nominal measure of thereference deflagration speed to be used in the analysisof the 2D results, even under highly transient condi-tions.

The above observation is further supported by ex-amining the structure of the fronts in the differentignition regimes. Fig. 4 shows the reaction and diffu-sion budgets for the H2 mass fraction given by Eq. (5)along with the temperature gradient for the combus-

Fig. 5. Mean temperature versus time for 1D test cases andmultizone model (MZ 0%). MZ 0% refers to the multizonemodel initialized by the initial DNS field at 0% pressure rise.

tion wave at the time of minimum propagation speed.The 4.1-mm case at this point is propagating as aspontaneous ignition front, and diffusion is nearly in-significant. For the 0.75-mm case, for which the frontspeed is limited from below by the effects of dif-fusion, a greater influence of diffusion is observed.These results suggest that the relative magnitude ofdiffusion is an alternative indicator to delineate com-bustion in the deflagrative and spontaneous ignitionregimes.

6. Comparison of 1D cases with multizone model

To understand the roles of passive scalar dissi-pation and in-flame diffusion, we have performed anumber of the multizone model calculations initial-ized from the 1D DNS cases presented in Section 5.A sufficiently large number of zones has been used sothat results are independent of the number of zones.The comparison between the 1D DNS results in theprevious section and the multizone model results willthus reveal the effect of molecular mixing and dissi-pation during the ignition event.

Fig. 5 shows the mean temperature in the domainfor various wavelength cases of the 1D DNS and theprediction from the multizone model initialized fromthe initial field of the 1D DNS (namely 0% pressurerise). Since diffusive transport between the zones isnot considered, the results of the multizone modelfor all wavelengths are identical because the initialtemperature PDF is identical. It is clearly seen thatthe DNS results for longer wavelengths converge tothe multizone model predictions. As the wavelengthis decreased, however, the DNS results deviate sig-nificantly from the multizone model results becausethe enhanced molecular mixing of the scalar field de-creases the burn duration and increases the maximumrate of heat release rate. It is interesting to note thatfor the shortest wavelength cases (0.38 and 0.56 mm),


Fig. 6. Mean temperature versus time for 1.50- and 0.56-mm1D test cases and multizone model. MZ 0% and MZ 10%represent the multizone models initialized by the DNS fieldat 0% and 10% pressure rise, respectively.

although the front speeds are larger than the deflagra-tion speed, they still do not agree with the multizoneprediction. These cases exemplify the significant roleof passive scalar mixing in the induction phase ofignition, thereby strongly modifying the mean tem-perature evolution. Therefore, it is evident that themultizone model can adequately predict the ignitionof stratified mixtures only if the length scales of thestratification are sufficiently large.

To better understand the role of passive scalardissipation, alternative multizone model calculationswere made by using the initial condition of the 1DDNS solutions at the point of 10% of the final pres-sure rise (called MZ 10%). Fig. 6 shows such re-sults for two different wavelength cases, 0.56 and1.50 mm, compared with the 1D DNS results. TheMZ 0% result shown in Fig. 5 is also overlaid. Forboth wavelengths, a much better agreement betweenthe DNS and multizone model results is found, sug-gesting that a substantial portion of passive scalar dis-sipation occurs during the earlier phase prior to the10% pressure-rise point. Comparing with the MZ 0%result, it is also clearly seen that the dissipation effectis much stronger for shorter wavelengths.

Fig. 7. RMS temperature fluctuation T ′ versus time for 1Dcases and multizone model (MZ 0%).

Fig. 7 shows the RMS temperature fluctuation T ′as a function of time for various wavelength cases,along with the multizone model. This further demon-strates that the short wavelength cases exhibit signif-icant dissipation prior to ignition. The extent of thiseffect is represented by the Damköhler number de-fined earlier.

To unravel the distinct effects of the in-flame trans-port in contrast to the passive scalar dissipation, a sec-ond parametric study is conducted in which the lengthscale is fixed at 3.0 mm but the amplitude of temper-ature fluctuations is varied. The choice of the 3.0 mmwavelength is made because the passive scalar dissi-pation timescale is long compared with the ignitiondelay. Therefore, any discrepancies with the multi-zone model may be attributed to transport within theignition front. Specifically, Da = τmix/τ0 ≈ 8, wherethe mixing time scale is defined as

(10)τmix = h′2

2α|∇h|2 ,

in which h is the mixture enthalpy, h′ is the RMSenthalpy fluctuation, and α is the thermal conduc-tivity. Enthalpy is employed rather than temperaturebecause it is less sensitive to changes during reaction.Discounting pressure work and differential diffusioneffects, enthalpy would be a conserved scalar. Notethat changes in the amplitude of temperature fluc-tuation, T ′, have a negligible effect on the mixingtimescale since its contribution in the numerator anddenominator cancels.

Fig. 8 shows the comparison of the multizonemodel and DNS results for cases with T ′ = 7.5, 30,and 60 K. There is near-perfect agreement in the7.5 K case, but the agreement degrades as the am-plitude of temperature fluctuation increases. In thissituation, where the passive scalar dissipation is notimportant, the spontaneous front propagation speed isslow enough due to the increase in the temperaturegradient (see Eq. (4)), such that ignition occurs in the

Fig. 8. Mean temperature versus time for rms temperaturefluctuation between 7.5 and 60 K in the 1D test cases andmultizone model (MZ).


deflagration regime where heat and mass transport tothe front becomes important.

In summary, the comparison of the 1D and themultizone predictions reveals that both passive scalardissipation and molecular diffusion in deflagrationfronts may play an important role in constant volumeignition of an inhomogeneous mixture. Moreover, thenew diagnostics, based on the front speed, mixingtimescales and species evolution budgets, provide aconsistent delineation between regimes.

7. 2D DNS results

In this section, the utility of the ignition frontspeed and species budget to delineate between defla-grative versus spontaneous ignition front propagationis further assessed using 2D DNS data of constantvolume ignition in the presence of initial temperatureinhomogeneities.

The simulation data for Case A in our previ-ous study [13] is used. Although details of the igni-tion processes can be found in [13], they are brieflysummarized here for completeness. After an initialignition delay, which is governed by the competi-tion between reaction and diffusion in several can-didate kernels, the strongest ignition kernels igniteand form combustion fronts. The thickness of thesefronts varies significantly and determines their prop-agation speed and the importance of diffusive trans-port. As fronts propagate, the remaining charge isheated by compression, accelerating the ignition oflater-igniting kernels. Near the peak heat release rate,fronts begin to merge and annihilate, and eventu-ally the end-gas is consumed in a more volumetricprocess.

Instantaneous solution fields are analyzed to inves-tigate the characteristics of the front propagation. The1D test cases showed that the front speed in the de-flagration regime is very close to the laminar flamespeed at the corresponding local enthalpy and pres-sure conditions. Therefore, we define a cut-off cri-terion, s∗

d = 1.1sL, to be a reasonable boundary todistinguish between the two ignition regimes. Fig. 9shows the heat release rate (color) and YH2 = 8.5 ×10−4 (white/black lines) isocontours at 0.83τ0 whenignition kernels are forming and ignition fronts havebegun to propagate. This instantaneous field is rep-resentative of the events during the interval of majorheat release rate. The figure shows that this isocon-tour selection provides good tracking of the maximumheat release rate location. Similar results are obtainedfrom the inception of front propagation through theburnout of the end-gas. Along the YH2 isocontourlines, the black and white segments denote the defla-gration (s∗ � 1.1sL) and spontaneous ignition regime
d
Fig. 9. Heat release rate isocontours (rainbow color scale—red denotes maximum heat release rate), YH2 surface isocon-tour (white lines: s∗

d > 1.1sL, black lines: s∗d � 1.1sL), and

location of representative front structures. (A) Spontaneousignition front; (B,C) deflagration fronts.

Fig. 10. Ignition front length (dashed line) and volumetricmean normalized heat release rate (solid line) versus timefor 2D DNS.

(s∗d > 1.1sL), respectively, thereby indicating the rel-

ative importance of the two regimes at a given instantin time.

The evolution of the ignition front length and thevolumetric mean heat release rate (normalized by themaximum heat release rate of the homogeneous igni-tion at the mean temperature) is presented in Fig. 10.The ignition front length is defined simply as thelength of the hydrogen mass fraction isoline repre-senting approximately the location of maximum heatrelease rate. After the initial ignition delay, both theheat release rate and front length begin to rise, reachthe maximum at approximately the same time, and di-minish as remaining parcels of end-gas are consumed.While it should not be inferred that heat release rateoccurs only on an isolated surface, the two quantitiesdo show a qualitative match, validating the use of thisisosurface to study the rate of combustion. Note that


Fig. 11. (a) Structure for YH2 budget terms and (b) tempera-ture gradient for locations A, B, and C in Fig. 9. Propagationis to the left. Dashed line: reaction. Solid line: H2 diffusion.Dotted line: |∇T |.

early in time, heat release rate increases during ther-mal runaway in ignition kernels, but no surface exists.Note that the two curves do not collapse exactly anddeviate from each other in the earlier phase. This isattributed to the fact that a larger fraction of the to-tal heat release rate comes from the ignition kernelsduring the initial ignition process.

To underscore the importance of the local temper-ature gradient in determining the propagation regime,Fig. 11 shows the reaction–diffusion budget for theevolution of H2 mass fraction. Cut A in Fig. 9 repre-sents a spontaneous ignition front, having a speed ofapproximately 2.2sL, with a relatively low tempera-ture gradient. These data correspond to a spontaneousignition front, as in the 4.1 mm 1D case in Fig. 4.The effects of diffusion are found to be much smallercompared to reaction, as was observed in the 1D testcase. Cut B, on the other hand, shows a front whichhas a speed approximately equal to sL. In this casethe temperature gradient and diffusion–reaction bal-ance are comparable to the 0.75-mm 1D case (Fig. 4),which was found to be diffusion-controlled. Finally,cut C shows an area with a much larger tempera-ture gradient. At this location, where s∗

d ≈ sL, diffu-sion nearly counterbalances reaction, and the front be-haves much like a normal deflagration. For the meanand RMS temperature fluctuation considered in the1D test cases, such a high gradient could not be ob-tained, pointing to a possible influence of turbulencein modifying the local temperature gradient, as willbe clarified later in this study.

The evolution of the ignition front speed and themagnitude of the temperature gradient on the front ismonitored by tracking s∗ and |∇T |, averaged on the
d
Fig. 12. Density weighted ignition front speed (s∗d , thick

lines) and |∇T | (thin lines) for 2D DNS (solid lines) andthe 1D 1.5-mm reference case (dashed lines).

YH2 = 8.5 × 10−4 isocontour. A comparison of thefront propagation for the 2D DNS and the 1.5-mm 1Dtest case is presented in Fig. 12. It is first noted thatthe 2D case ignites earlier. This is due to higher initialtemperatures that occur owing to the random natureof the initialization. The front speed curve for the 2Dand 1D test cases both show the same qualitative U-shape, with the 2D case showing a longer durationof combustion because of a wider distribution in thetemperature. The 2D case attains higher mean speedscompared to the 1D test case. This is expected since,throughout the duration of combustion, new kernelsform and ignite, and fronts merge and annihilate as theignition front density reaches a maximum at the timeof volumetric peak heat release rate. Both ignition andannihilation events involve very high speeds [23], sothe mean will also be higher. Finally, the tempera-ture gradients extracted at the isosurface show that the2D case shows a higher average gradient relative tothe 1D test case, indicating the influence of turbulentstraining, as will be addressed later in this paper.

Fig. 12 suggests that the variation in the instanta-neous displacement speed is correlated with the localtemperature gradient. To confirm this observation, theinstantaneous solution field for the 2D DNS at 0.9τ0is analyzed and sd versus |∇T | at all points along theisosurface is plotted in Fig. 13 on a log scale. A powerfit to these data reveals an exponent of −1.005, veryclose to the value of −1. This is analogous to the the-ory by Zel’dovich [4], where spontaneous front prop-agation speed is inversely proportional to the temper-ature gradient. The dependence of s∗

d on |∇T | wasalso confirmed by the 1D test case study in the sponta-neous ignition regime with moderate values of |∇T |.The results suggest that the inverse linear correlationbetween the two variables is a good indication of thespontaneous propagation regime.


Fig. 13. Displacement speed (sd) shown on a log scaleagainst |∇T | (inset: (s∗

d /sL) plotted on a regular scale), andpower-law fit at 0.9τ0 of the 2D DNS data.

On the other hand, the inset in Fig. 13 furtherreveals that at approximately |∇T | = 1800 K/mmand above [13], s∗

d does not decrease indefinitely butrather levels off to a minimum value, consistent withthe 1D test result (recall the 1.5- and 0.75-mm cases).This is attributed to the reactive–diffusive balance thatis intrinsic to a laminar deflagration wave. Note thatthe spontaneous front speed is determined by the ig-nition delay differential along neighboring mixtureparcels, which is directly related to the local tem-perature gradient. In a deflagration wave, however,an increase in scalar gradient leads to an increase indiffusive transport and subsequent reaction rate dueto the reaction–diffusion balance mechanism. There-fore, the front can self-adjust and the speed remainsrelatively constant over a wide range of temperaturegradients, unless perhaps an extreme amount of trans-port can cause flame extinction. The weaker sensi-tivity of the front speed to the temperature gradientappears to be another distinct characteristic of the de-flagration front.

To determine what fraction of the volume is un-dergoing combustion in the deflagration versus spon-taneous ignition regimes, the local normalized frontspeed is used as a conditioning variable for the frontlength and the rate of production of burnt gases.Fig. 14 shows the fraction of the ignition front area ex-hibiting a speed comparable to the deflagration speed,specifically having a front speed <1.1sL. The choiceof the value of 1.1 is meant to give a qualitative senseof the transition from deflagration to spontaneous ig-nition. In fact, transition occurs more gradually. Thisfigure also shows the fraction of production rate ofarea of “burnt” gases (having YH2 < 8.5 × 10−4) dueto deflagration for this case. If dl is an incremen-tal segment length associated with a given contoursegment, the rate of production of burnt gases is cal-culated by summing the quantity sd dl for isosurfacesegments flagged as deflagrations, divided by the totalfor all segments.

Fig. 14. Fraction of ignition front length undergoing defla-gration (solid lines) and fraction of burnt gas area productiondue to deflagration (dashed lines).

Fig. 15. Evolution of volumetric mean absolute value of dif-fusion (solid lines) and reaction (dashed lines) for the 2DDNS data.

The fraction of ignition front length initially be-gins at zero, when ignition kernels dominate, and in-creases as combustion in fronts becomes important,reaching a maximum of around 40%. Eventually, thefraction decreases again during the burnout of theend-gas parcels. The production of area of burnt gasesis perhaps a more relevant quantity to consider thanthe fraction of ignition front length, if heat releaserate characteristics are of interest. This quantity isless than the length fraction, simply because the speedof the deflagration wave is lower. A comparison canbe made of the production of area of burnt gases inthis figure with volumetric heat release rate of Fig. 6in [13], attained from a different measure based ona critical temperature gradient of 1800 K/mm. Thecomparison yields good qualitative agreement sup-porting the use of both measures.

To further assess the global effect of diffusionwithout confining attention to a particular isosurfacechoice, the evolution of the volume average of the ab-solute value of the YH2 budget terms for the 2D DNSin Fig. 15. The use of the absolute value is neces-sary since diffusion is conservative in the mean. The


Fig. 16. Curvature-induced stretch (sd∇ · N, solid lines) andcurvature (∇ · N, dashed lines) versus time for the 2D DNSdata.

figure shows that, for the 2D DNS data under study,diffusion is not negligible. At the time of maximumreaction, it is approximately 27% of reaction, roughlyconsistent with the previous discussion in relation tothe front speeds. Qualitatively similar results are ob-tained considering the balance terms in the energyequation for temperature, showing that diffusion atmaximum heat release rate is approximately 17% ofreaction.

The nature of the ignition front is further examinedbased on the flame stretch due to the combined effectof curvature and propagation, sd∇ · N, where ∇ · Nis the isosurface curvature, and N is the normal vec-tor to the surface, pointing toward the unburned mix-ture. (Curvature-induced stretch dominates over tan-gential strain at these conditions.) The mean curvatureof the surface and the associated stretch are plottedagainst time in Fig. 16. Initially, very large values ofstretch are observed as the first kernels ignite: curva-ture and flame speed are both large. As fronts subse-quently propagate outward, the curvature reduces andthe mean speed also reduces as higher temperaturegradients are encountered. Near the maximum heatrelease rate time (≈ 0.9τ0), the front length reaches amaximum and stretch and curvature both switch sign.Subsequently fronts propagate inward, and finally an-nihilate leading to large negative stretch.

It is of interest to examine what fraction of thestretch is due to spontaneous ignition fronts and de-flagrations waves. The fraction of stretch is defined asratio of the sum of the absolute values of stretch asso-ciated with deflagration fronts to that with the entirefronts. In Fig. 17, the fraction of front length markedas deflagration is presented along with the fraction ofstretch. Interestingly, while a significant fraction offront length is in the deflagration mode, there is prac-tically no contribution to the total stretch rate—themaximum is less than 5%. To understand this result

Fig. 17. Fraction of ignition front length (solid line) andcurvature-induced stretch (sd∇ · N, dashed line) due to de-flagration for 2D DNS at 0.9τ0.

Fig. 18. Magnitude of the temperature gradient (|∇T |)against curvature (∇ · N) for 2D DNS at 0.9τ0.

further, the local temperature gradient, on which thespeed has been shown to be highly dependent in theignition front regime, is plotted versus curvature forthe time 0.9τ0 in Fig. 18. The data have been av-eraged over discrete intervals of curvature for clearpresentation. The figure shows that low curvaturesare typically associated with high temperature gra-dients, and therefore promote deflagrations, whereashighly curved elements typically have lower tempera-ture gradients, and propagate as ignition fronts. Notealso that, because of the low temperature gradients,speeds are very high in these regions, leading to verylarge stretch rates and the result observed in Fig. 17.This result could have also been qualitatively inferredfrom Fig. 9, where it was evident that the flame sec-tions marked as deflagrations typically did not oc-cur in highly curved elements. The increased likeli-hood of low gradients in highly curved elements canbe explained by considering the effects of turbulentstraining. Compressive strain, which decreases thetemperature gradient, leads to curved fronts, whereasextensive strain, which increases temperature gradi-ents, tends to straighten the fronts. This is verified inFig. 19, showing the positive correlation of tempera-


Fig. 19. Magnitude of the temperature gradient (|∇T |)against strain (atτ0) for 2D DNS at 0.9τ0.

Fig. 20. Mixing timescale for 2D DNS as defined by Eq. (9).

ture gradient with tangential strain rate on the surface.This also explains why larger temperature gradientsare observed in the 2D turbulent case relative to the1D case.

Thus far, discussion has focused on the front prop-agation characteristics, which mostly involve the in-flame diffusion processes. However, it is also impor-tant to examine the effect of passive scalar dissipation.Fig. 20 shows the temporal evolution of the mixingtime scale (Eq. (10)) for the 2D data. Both quantitiesare normalized by the homogeneous ignition delaytime, τ0. It is observed that the mixing time scale isinitially comparable to the ignition delay, and sub-sequently reduces as turbulence increases the scalargradients. Therefore, it is expected that turbulent mix-ing is important in changing the distribution of initialtemperatures in the period prior to ignition.

8. Comparison of 2D DNS with multizone model

To assess the effect of passive scalar mixing on theprediction of overall heat release rate, two multizonemodel simulations were conducted using a large num-ber of zones having the thermochemical state sampled

Fig. 21. Mean heat release rate against time for the 2D DNS(solid line) and the multizone model started from the 2DDNS initial condition (MZ 0%, dash–dot line) and at 10%pressure rise (MZ 10%, dashed line).

from points in the 2D DNS data at 0% and 10% pres-sure rise, denoted as MZ 0% and MZ 10%, respec-tively. Fig. 21 shows the mean heat release rate historyobtained from the DNS and the multizone model cal-culations. Due to the complete neglect of the initialmixing process and diffusive transport in fronts, theMZ 0% results overestimated the duration of burningby 78% and underestimated the peak heat release rateby 55%. Significantly better predictions are obtainedfrom the MZ 10% case, where the burn duration isoverpredicted by 12%, and the peak heat release rateis underpredicted by 16%. This demonstrates the im-portance of the passive scalar mixing during the earlyphase of ignition when a considerable level of tur-bulence is present. The remaining errors in the MZ10% cases are mainly attributable to transport withinfronts. The error values are qualitatively consistentwith the fraction of burnt gas production that wasidentified to occur in deflagrations (cf. Fig. 14).

The primary contribution of the present work isto identify the relevant parameters to determine theregime of validity of the models, rather than to makean absolute statement regarding the validity of themodel across the range of possible engine operatingconditions. The results show that it is possible us-ing the diagnostics developed to explain the modelperformance. Indeed the results can be used to in-fer why good performance has been obtained to datewith the multizone model. At typical operating condi-tions, temperature gradients in the bulk gases are low,leading to rapid ignition front propagation relative toslower deflagration waves, and turbulent mixing oc-curs on longer timescales than the range of ignitiondelays within the cylinder. However, near walls andcrevices, the thermal gradients are likely to be largerthan in the bulk gases, and hence, the possibility ofincreased mixing influences and deflagrative propa-


gation exists. The diagnostics developed here will beuseful in assessing near-wall model performance.

From the modeling perspective, the present resultsare quite encouraging in that, even though diffusiveeffects are found to be important, the main influenceappears to be due to passive scalar mixing—that is,modifying the PDF of initial temperatures. In actualmultizone model implementations, these effects arealready largely accounted for in the CFD calculation,which includes mixing, that occurs prior to the start ofthe multizone calculation. Subsequent turbulent mix-ing may or may not be important, and depends on theDamköhler number. In the case where mixing is im-portant, it is worth noting that modeling of turbulentscalar mixing is relatively well understood and therealready exist a number of viable strategies. For ex-ample, the unsteady laminar flamelet approach [25]accounts for passive mixing by using a time-varyingPDF of mixture fraction fluctuations. The transportedPDF approach [24] also adopts the mixing time-scaleas a key component of the model. In addition, the pas-sive scalar mixing effect can be easily incorporatedin the multizone model. This could be achieved byreweighting the mass in the different zones, similarlyto the typical implementation of an unsteady laminarflamelet model. Indeed, a hybrid approach using CFDcomplete with standard turbulent mixing models tocontinuously advance the mixing fields coupled witha multizone model to calculate reaction rates [26]has been shown to provide improved predictions ofhydrocarbon and CO emissions compared with theoriginal multizone model. In contrast, if deflagra-tions were found to be significant, modeling wouldbecome very challenging since mixing may be con-trolled through deflagration rather than through theturbulence time scale. Moreover, the deflagration can-not be adequately modeled through the usual flameletapproaches for premixed flames because it occurs inthe highly unsteady state corresponding to large tem-perature gradients. Which type of diffusive effects ifany will be important in HCCI applications dependson the turbulence time scales relative to the ignitiontime under practical operating conditions.

9. Conclusions

Direct numerical simulations of lean hydrogen–air ignition at high pressure at constant volume inthe presence of temperature inhomogeneities havebeen conducted to begin to understand the combus-tion process in HCCI engines.

The behavior of the speed and structure of an ig-nition front was demonstrated in simple one-dimen-sional constant-volume calculations, illustrating theimportance of the initial temperature gradient and

temperature fluctuation length scale on the propa-gation. The effects of diffusion were shown to betwofold. The first effect is in limiting the minimum at-tainable front speed, resulting in propagation in a de-flagration wave. Based on the theory of Zel’dovich thenondimensional controlling parameter for this tran-sition is identified as the ratio of front speed to thedeflagration speed s∗

d/sL. The second diffusive effectis in the dissipation of initial temperature gradients,which on the other hand promotes spontaneous ig-nition front propagation rather than deflagration. Inturbulent flows the latter effect may be curtailed ow-ing to the expected continuous cascade from larger tosmaller scales. The importance of this effect can beascertained with a Damköhler number τmix/τ0, rep-resenting the ratio between the mixing timescale andthe ignition delay time.

The performance of the multizone model in pre-dictions of the constant volume combustion was eval-uated in one-dimensional cases. By testing the modelboth inside and outside of its expected valid regimewe are able to better understand what factors controlits validity. The levels of transport by passive scalarmixing and transport in ignition fronts were foundto be controlling factors in the performance of themodel. Good agreement was obtained in the expectedvalid regime in which mixing effects are unimpor-tant, and it was noted that the multizone model alwayspredicts a longer burn duration and lower peak heatrelease rate than the one-dimensional simulations.

Two-dimensional simulations were performedwith a random temperature distribution and the util-ity of the front speed as an indicator of the transitionbetween deflagration and spontaneous ignition frontpropagation was assessed. Analysis of local frontstructures and global averages showed that this cri-teria is consistent with the observed importance ofdiffusion. The case considered here was found tocorrespond to mixed mode propagation, with a sig-nificant fraction of the front length propagating as adeflagration wave. However, since deflagration wavespropagate slower, they result in a lower fraction ofburnt gas production. It is noted that if the local con-ditions are below the lean limit, then no importanceof deflagrations is expected according to this criteria,since the laminar flame speed is zero.

An analysis of the flame stretch showed that adominant fraction of flame stretch is associated withthe spontaneous ignition fronts due to the strainingmechanism by turbulence. The conceptual picture thatemerges is of rapidly propagating ignition fronts mov-ing along lines of low temperature gradient and leav-ing behind longer surfaces that show more importantdiffusive effects, but consume less fuel owing to theirlower propagation speed.


The comparison between the multizone model and2D DNS results revealed that the most significant dif-fusive effects were due to passive scalar dissipationduring the early phase of ignition. Transport withinignition fronts was relatively less important but stillsignificant. The explanation of the model predictionsdemonstrated the utility of the tools developed to un-derstand the combustion process for these conditions.Application of these tools to actual engine conditionswill help explain why previous good predictions havebeen obtained with this model, and help ensure thatfuture applications will be within the expected validregime.

Part II of this study [14] applies the understandingand methods developed here to parametric studies re-garding the influence of the initial temperature distri-bution and turbulence parameters on the combustionmode and the predictions of the multizone model.

Future work needs to extend the present study tomore realistic chemistry and to engine-relevant turbu-lence parameters. Fuels exhibiting multistage ignitionand negative temperature coefficient regimes may re-sult in different conclusions to those presented here.More information is needed regarding the character-istics of scalar and turbulence fluctuations occurringin real engines operating in HCCI mode in order todesign numerical experiments that will be representa-tive. It is anticipated that larger initial integral lengthscales of the temperature fields more representative ofobservations in HCCI engines [2] will tend to favorthe mode of spontaneous ignition front propagationmore than observed here. Considering the wide spec-trum of length scales in turbulence, however, theremay be some small regions in which diffusive trans-port is important. Further studies are needed to de-termine whether they are important for the overallcombustion process, including the duration of heat re-lease rate and emissions characteristics.

Acknowledgments

The work at UM was supported by the Consortiumon HCCI Engine Research directed by the UM andfunded by the Department of Energy, and also by De-partment of Energy, Office of Basic Energy Sciences,SciDAC Computational Chemistry Program.

Sandia National Laboratories (SNL) is a multi-program laboratory operated by Sandia Corporation,a Lockheed Martin Company, for the United StatesDepartment of Energy under Contract DE-AC04-94-AL85000. The work at SNL was supported by theDivision of Chemical Sciences, Geosciences and Bio-sciences, the Office of Basic Energy Sciences, theU.S. Department of Energy. Calculations were per-formed at SNL and at the National Energy Research

Scientific Computing Center, which is supported bythe Office of Science of the U.S. Department of En-ergy under Contract DE-AC03-76SF00098. The HighPerformance Computing and Networking departmentat SNL provided access to a 256 processor Infini-band testbed. The authors acknowledge fruitful dis-cussions with Drs. John Hewson, John Dec, and Mag-nus Sjöberg.

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direct numerical simulation of ignition front propagation...

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