Effects of Damkohler number on flame extinction and reignition in turbulentnonpremixed flames using DNS

David O. Lignella,∗, Jacqueline H. Chenb, Hans A. Schmutza

aChemical Engineering Department, 350 CB, Brigham Young University, Provo, UT 84602bReacting Flow Research Department, Combustion Research Facility, Sandia National Laboratories, Livermore, CA 94551

Abstract

Results from a parametric study of flame extinction and reignition with varying Damkohler number using direct nu-merical simulation are presented. Three planar, nonpremixed ethylene jet flames were simulated at a constant Reynoldsnumber of 5,120. The fuel and oxidizer stream compositions were varied to adjust the steady laminar extinction scalardissipation rate, while maintaining constant flow and geometric conditions. Peak flame extinction varies from approxi-mately 40% to nearly global blowout as the Damkohler number decreases. The degree of extinction significantly affectsthe development of the jets and the degree of mixing of fuel, oxidizer, and combustion products prior to reignition. Theglobal characteristics of the flames are presented along with an analysis of the modes of reignition. It is found thatthe initially nonpremixed flame undergoing nearly global extinction reignites through premixed flame propagation in ahighly stratified mixture. A progress variable is defined and a budget of key terms in its transport equation is presented.

Keywords: DNS, turbulent nonpremixed flames, extinction, reignition, progress variable, ethylene, partially premixedcombustion

1. Introduction

Flame extinction and reignition are important processesin nonpremixed combustion under highly turbulent con-ditions. Nonpremixed flames occur at the stoichiometricinterface between fuel and oxidizer streams, and the rateof combustion is limited by the rate of diffusive mixing ofthe streams. Turbulence acts to increase flame surface areaand scalar gradients, which increases the rate of diffusivemixing. These high mixing rates allow smaller combustionvolumes and higher heat release rates. However, as mixingrates increase, finite rate chemical kinetic effects becomeimportant, and flame extinction may occur if rates of dif-fusive heat loss exceed the heat release rate of combustion.Local flame extinction results in flame holes through whichunburned fuel may escape, reducing combustion efficiencyand allowing fuel emission. High rates of flame extinc-tion may result in unstable combustion, flame liftoff, andif excessive, global flame blowout, posing operational andsafety hazards in combustion equipment.

Flame extinction occurs in regions of high turbulentstrain, e.g., in the near field of jet flames. As turbulentstrain rates decrease, extinguished regions may reignite.The mechanisms and modes of reignition, and the develop-ment of turbulent combustion models that can accuratelyand reliably capture such processes, are the subjects ofongoing research. Reignition of extinguished fluid parcelsmay occur through several possible mechanisms, including:

∗Corresponding Author; Tel.: (801) 422-1772; Fax: (801) 422-0151; Email: davidlignell@byu.edu

• Auto-ignition, possibly coupled with ignition frontpropagation. This mechanism is expected to occurin regions for which the homogeneous ignition delaytime is low and temperature gradients are not toolarge [1, 2].

• Edge flame propagation along the (nominally) sto-ichiometric surface of the extinguished region. Bythis mechanism a flame hole is essentially healed asburning portions of the stoichiometric surface propa-gate into extinguished regions under favorable strain[3]. A review of edge flames is given by Buckmaster[4].

• Turbulent flame folding. Reignition may occur throughturbulent flame folding, where burning regions or hotcombustion products are folded onto extinguished re-gions, which reignite through diffusive heat and masstransfer [3, 5].

• Premixed flame propagation. If the extent of mixingis sufficient during periods of extinction and reig-nition, a premixed flame may develop and propa-gate. This flame propagation occurs in a partiallypremixed mode through a nonhomogeneous mixtureof varying stoichiometry and temperature, both ofwhich impact the flame speed. Subsequent reactionbetween rich and lean combustion products may alsobe important. This mode is expected under condi-tions of severe and pervasive extinction, but short offlame blowout.

Preprint submitted to Elsevier October 20, 2010

While many experimental investigations of diffusionflames with extinction phenomena have been performed(see e.g., [6], references in [3]), the most detailed informa-tion available are two-dimensional images of the three di-mensional flow. Direct numerical simulation (DNS) is cur-rently the only approach that provides detailed temporallyand spatially resolved information of the flow field andscalar state space (though at relatively modest Reynoldsnumbers). Earlier DNS studies have been limited to rel-atively simple configurations (homogeneous decaying tur-bulence) and simple chemistry, e.g., [7, 8, 9]. Pantano [3]simulated a three-dimensional methane jet flame with afour-step reduced chemical mechanism. Recently, simula-tions with detailed combustion chemistry have been per-formed in a methane jet in two dimensions [10]. Hawkeset al. [5, 11] performed a parametric study of extinctionand reignition in planar, temporally-evolving CO/H2 jetflames with detailed chemistry and varying Reynolds num-ber. In that study, analysis of isosurfaces of a reactivescalar showed the dominant mode of flame reignition tooccur via edge flame propagation and turbulent flame fold-ing [5]. Edge flame propagation was found to dominate inregions of lower scalar dissipation rate, whereas the high-est rates of reignition were found to occur by flame foldingat higher scalar dissipation rates.

Here, we present DNS results of a parametric study ofextinction and reignition in nonpremixed, planar, tempo-rally evolving ethylene jet flames. The configuration andflow parameters are chosen to match those of the mediumReynolds number simulation of Hawkes et al. [11]. Inthis study, rather than vary the Reynolds number at fixedDamkohler number, as in [11], the Damkohler number isvaried at fixed Reynolds number. It is expected that theextinction and reignition properties of ethylene are sig-nificantly different than those of CO/H2. Hawkes et al.observed monomodal PDFs of reactive scalars (e.g., OH)during extinction and reignition, whereas we have observedbimodal behavior of PDFs of reactive scalars for the ethy-lene flame during extinction. These results are consistentwith the experimental studies of Masri and Bilger, whoconsidered nonpremixed flame extinction with H2, CO/H2,and CH4 flames [12]. CO/H2 has a broader reaction zonewidth in the mixture fraction coordinate and a lower acti-vation energy than ethylene, making CO/H2 a more robustfuel in terms of its extinction and reignition characteristics.Important differences in the flame structure and resultingreignition processes may result.

In this paper, we present results of the DNS simulationsto quantify the effect of the varying Damkohler numberon the evolution of the jet flame and the combustion pro-cesses. These results include extent of extinction, scalardissipation rate profiles, and flame surface area. Non-premixed and premixed modes of flame reignition are con-sidered using a flame index. A progress variable is consid-ered [13], and dissipation terms and reaction terms of itstransport equation are investigated a-priori.

2. Numerical Implementation

The following sections describe the DNS code imple-mentation, along with the configuration of the cases andthe chemical mechanism employed.

DNS Code Description

The DNS code used in the simulations is called S3D[14]. The code is written in Fortran 90 and parallelizedwith MPI. S3D solves the compressible, reacting Navier-Stokes equations given by the following equations (in indexnotation) for continuity, momentum, energy, and speciesmass fractions, respectively:

∂ρ

∂t= −∂(ρvi)

∂xi, (1)

∂(ρvi)

∂t= −∂(ρvjvi)

∂xi− ∂P

∂xi+∂τi,j∂xi

, (2)

∂(ρeo)

∂t= −∂(ρeovi)

∂xi− ∂(Pvi)

∂xi

+∂(τi,j · vj)

∂xi− ∂qi∂xi

, (3)

∂(ρYk)

∂t= −∂(ρYkvi)

∂xi− ∂(jk,i)

∂xi+ ωk. (4)

In these equations, ρ is density, vi is velocity, t is time,xi is position, P is pressure, τi,j is the viscous stress ten-sor, Yk is the mass fraction of species k, and ωk is thespecies reaction rate. In addition, eo, qi, and jk,i are thespecific total energy, heat flux vector, and species diffu-sion flux vector, respectively. The total energy is given byeo = vivi/2 + h − P/ρ, and the heat flux vector is givenby qi = −λ∇T +

∑k hkjk,i, where λ is the thermal con-

ductivity, T is temperature, and h and hk are the specificenthalpies of the mixture and species, respectively. Thetransport equations are closed with constitutive relationsfor the species diffusivities, the stress tensor, a thermo-dynamic relation linking enthalpy and temperature, h =h(T, Yk), and the ideal gas equation of state P = ρRT/M ,where M is the gas mixture molecular weight, and R isthe universal gas constant. The viscous stress tensor is

computed as τi,j = µ(∂vi∂xj

+∂vj∂xi− 2

3δi,j∂vl∂xl

). A mixture-

averaged formulation based on a form of Fick’s law is usedfor the species diffusion flux jk,i = −ρDk

∂Yk

∂xi− ρDkYk

M∂M∂xi

,where Dk is a mixture-averaged diffusion coefficient com-puted using the Chemkin Transport package [15]. Thediffusion flux of nitrogen is computed as one minus thesum of the diffusion fluxes for the other species in order toenforce a null sum of diffusion fluxes.

S3D integrates the reacting flow equations, given above,using a fourth-order, six-stage explicit Runge-Kutta method[16]. Spatial derivatives are approximated on a finite dif-ference grid using eighth order central differences. A tenth-order spatial filter is applied at each timestep to removehigh wavenumber content and reduce aliasing errors [17].Nonreflecting outflow boundary conditions are employed

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at open boundaries using the formulation of Sutherlandand Kennedy [18].

Combustion Chemistry

The present study considers flames with intense finiterate chemistry associated with turbulent extinction andreignition under a wide range of stoichiometric conditions,and flame modes (both nonpremixed and premixed). Thecombustion mechanism used should be accurate under theseconditions, but also minimize computational costs associ-ated with the number of chemical species transported andthe cost of computing reaction rates. These costs are as-sociated with small chemical timescales which dictate thetimestep in an explicit integration scheme and lengthscalesof reactive intermediates, the resolution of which dictatesthe computational grid size.

The chemical mechanism used for this parametric studyhas been described previously [19]. The mechanism isbased on a validated detailed ethylene mechanism [20], inwhich the directed relation graph method and sensitivityanalysis are used to remove species to create a skeletalmechanism, which is further reduced by defining quasi-steady state (QSS) species using computational singularperturbation. The resulting reduced mechanism consistsof the following 19 transported species: H2, H, O, O2, OH,H2O, HO2, H2O2, CH3, CH4, CO, CO2, CH2O, C2H2,C2H4, C2H6, CH2CO, C3H6, and N2. In addition, thefollowing 10 QSS species are included in the mechanism:C, CH, CH2, CH∗2, HCO, CH3O, C2H3, C2H5, HCCO,CH2CHO. This mechanism was validated extensively againstthe detailed mechanism for ignition delay time, PSR ex-tinction residence time, laminar premixed flame speed, andspecies profiles in premixed laminar flames and opposed jetdiffusion flames.

DNS Configuration

The flow configuration is a temporally-evolving, planarslot-jet consisting of a central fuel core surrounded by oxi-dizer. The flow is periodic in the streamwise and spanwisedirections, and open in the cross-stream direction, result-ing in a constant bulk pressure as the flow evolves. Figure1 shows a schematic of the configuration. The computa-tional costs of DNS preclude the use of very large domains,and the temporal configuration allows the study of the de-velopment of the jet within the domain (in effect maxi-mizing the residence time of flames and fluid structures inthe domain by following the mean convective fluid evolu-tion). The jet provides one direction of mean shear (inthe cross-stream direction), and two statistically homoge-neous directions in the streamwise and spanwise directions,maximizing the data available for statistical analysis. Thisconfiguration has been used previously in the study of tur-bulent soot formation [21].

The parameters of the configuration are similar to thoseused in a study by Hawkes et al. [11], who simulated ex-tinction and reignition in a CO/H2 jet at three Reynolds

Figure 1: Schematic of the DNS configurations.

numbers, but at a constant Damkohler number. Here,we study a representative hydrocarbon fuel, ethylene, at afixed Reynolds number of 5,120, with three varying Damkolernumbers. The flow conditions are those of the mediumReynolds number case of Hawkes et al.

The velocity and composition fields are initialized asfollows. The streamwise velocity field is set with ∆U/2in the central fuel stream, and −∆U/2 in the surroundingoxidizer streams, for a velocity difference of ∆U betweenthe streams. A mixture fraction ξ profile is set with ξ = 1in the fuel core, and ξ = 0 in the surrounding oxidizer.Hyperbolic tangent profiles with transition thickness δ areused as smooth transitions between the fuel and oxidizerstreams, and between the positive velocity in the jet coreand the negative velocity surrounding the core. Table 1gives the geometric parameters used in the three paramet-ric cases. H and Hξ are the velocity jet height and thefuel core height, respectively. That is, Hξ is the thick-ness of the central fuel core, and H is the thickness of thevelocity jet. δ is the transition thickness of the velocityprofile, and δξ is the transition thickness of the mixturefraction profile. Lx, Ly, and Lz are the domain lengths inthe x, y, and z directions, respectively. The x, y, and zdirections correspond to the streamwise, cross-stream, andspanwise directions, respectively. ∆x is the grid spacing(uniform and equal in all three directions), and τjet is thejet timescale defined as H/∆U . A homogeneous, isotropicturbulent velocity profile is overlaid on the streamwise ve-locity profile in the region of the velocity core (H) to trip

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Table 1: Simulation parameters of the three parametric DNS cases.Repeated values (e.g., on Ly) refer to values for DNS Cases 1, 2, and3, respectively.

H (mm) 0.96 Lx/H 12∆U (m/s) 196 Ly/H 15, 17, 19Rejet 5120 Lz/H 8Hξ (mm) 1.5 ∆x (µm) 17δu (mm) 0.19 δξ (mm) 0.74u′/∆U (init) 5% τjet (ms) 0.0049H/L11 (init) 3 τrun/τjet 74,87,140

Figure 2: Initial profile of temperature (increasing white to black),with z-velocity contours and horizontal lines demarcating the jet corefor velocity (inner lines), and mixture fraction (outer lines).

the shear layers. The turbulence intensity (u′) and integrallengthscale (L11) parameters used in the turbulence spec-ification are also shown in the table. The turbulence in-tensity and integral timescale (L11/u

′) are small comparedto the shear velocity, and jet timescale. The shear layersdevelop rather quickly, and velocity fluctuations rise sub-stantially within 7τjet. Figure 2 shows the initial conditionat a streamwise and cross-stream (x-y) plane of the flow.Smoothed temperature contours are shown in grayscale.The z-velocity field from the initial turbulence is shownas contours in the jet core. Four horizontal lines are alsoshown. The inner two lines indicate the velocity core H,and the outer two lines indicate the fuel core Hξ. Notethat the temperature peaks outside the initial turbulencefield at the stoichiometric point.

The gas composition is specified through the mixturefraction profile. A steady laminar flamelet solution (as-suming unity Lewis numbers) is mapped to the domainwith a consistent scalar dissipation rate profile betweenthe flamelet solution and the mixture fraction profile inthe DNS [19]. The profile width δξ corresponds to 50% ofthe steady laminar extinction width.

It was desired to match as many flow/flame parame-ters as possible between the previous CO/H2 simulation[11], and the present cases. The parameters of interestare the stoichiometric mixture fraction, the density ra-tio of the reactants and products, the kinematic viscos-

Table 2: Comparison of fuel parameters.

CO/H2 C2H4

Case 1 Case 2 Case 3ξst 0.422 0.422 0.17 0.17 0.17To (K) 500 550 550 550 550ρr/ρp 4.2 4.3 5.2 5.0 4.8ν (cm2/s) 0.416 0.421 0.35 0.36 0.37χq (1/s) 2380 2380 4774 3587 2380Tq (K) 1296 1700 1953 1896 1822Tad (K) 2376 2345 2721 2660 2569

ity, and the extinction scalar dissipation rate. Table 2shows a comparison of these parameters between the fu-els. The density ratio reported is for reactants and adia-batic equilibrium products of a stoichiometric mixture ofthe fuel and oxidizer streams. The kinematic viscosity isreported for the fuel stream. The scalar dissipation rateχq shown is the stoichiometric value at flame extinctionfor an adiabatic, steady laminar flamelet solution usingthe same chemical mechanism as in the DNS, but withunity Lewis numbers. The corresponding stoichiometricflame temperature at the extinction limit Tq is also shown.The flamelet solutions were obtained solving the unsteadyproblem to steady state in the mixture fraction coordi-nate, and beginning each calculation with the solution ata lower scalar dissipation rate as the extinction dissipa-tion rate was approached. The first two columns in thetable compare CO/H2 and ethylene with the same valueof the stoichiometric mixture fraction. The stoichiomet-ric extinction scalar dissipation rate for these two cases(as well as Case 3) is the same. This was achieved in theethylene cases by varying the total N2 in the system. Thatis, the stream compositions were varied so that, at the sto-ichiometric point, the relative concentration of nitrogen tofuel was changed. The stoichiometric mixture fraction wasvaried by moving N2 from the oxidizer stream to the fuelstream [22]. Note the remarkable agreement between theparameters of these two cases. The quench temperature ofthe ethylene case is much higher than that of the CO/H2

case.In test simulations (run at lower grid resolution to re-

duce computational costs), the ethylene case at ξst = 0.422resulted in global extinction, and the value of ξst had to bereduced to 0.17. This effectively moves the flame furtherfrom the highest shear regions of the jet. Figure 3 showsthe moles of N2, O2, and fuel (C2H4, or CO/H2) at thestoichiometric point for the three cases considered here,along with the CO/H2 case, and that for an ethylene airmixture. The graph is on a basis of one mole of fuel, with0.5 and 3 moles O2 for the CO/H2 and ethylene cases, re-spectively. The moles of N2 in the graph are 2.1667, 6.870,7.927, 9.516, and 11.28 moles N2 for the CO/H2 fuel, Cases

4

0  

2  

4  

6  

8  

10  

12  

14  

CO/H2   Case  1   Case  2   Case  3   C2H4/Air  

Stoichiometric  Moles   N2  

O2  

Fuel  

Figure 3: Stoichiometric composition in terms of moles on a one molefuel basis for several cases.

Table 3: Stream compositions (mole fractions) for the three cases.

Case 1 Case 2 Case 3ξ = 0 O2 0.33516 0.30525 0.26914

N2 0.66484 0.69475 0.73086ξ = 1 C2H4 0.52105 0.47642 0.47205

N2 0.47895 0.52358 0.57795

1, 2, 3, and ethylene/air, respectively. The correspondingstream compositions for Cases 1-3 are given in Table 3.The mole ratio of CO to H2 for the syngas fuel is 5:1 asreported in [11].

For the three simulation cases 1-3, the density ratios,kinematic viscosities, and quench and adiabatic tempera-tures vary by only 3-4% from their respective means. This,together with the observation that the combustion ratesare dominated by turbulent mixing rates up to the pointof extinction, results in a sole parameter, χq, determiningthe degree of extinction, reignition, and flow evolution forthe three cases studied.

In Table 2, the configuration parameters for the threecases studied are shown. Each case is initialized with thethe same ξst, but the stoichiometric extinction scalar dis-sipation rate (steady laminar value) is varied by adjust-ing the nitrogen concentration of the streams as notedabove. This results in Damkohler numbers Da of 0.023,0.017, and 0.011 for Cases 1, 2, and 3, respectively, whereDa = χqτjet. Case 1 experiences the least extinction, andCase 3 the most. This definition of Damkohler numberwas used by Hawkes et al. [11]. Definition of an intrinsicchemical timescale is difficult for detailed chemical mecha-nisms. Here, the intrinsic chemical timescale is defined asthe inverse of the stoichiometric steady laminar flameletextinction scalar dissipation rate. For steady flamelets atextinction, the mixing rate, characterized by χ, is balancedby the chemical reaction rate, and increases in χ result inflame extinction. Hence, 1/χq is a reasonable estimate ofthe chemical timescale, especially considering the presentemphasis on flame extinction.

3. Simulation Results

In the following sections, results of the three parametricsimulations are presented in terms of general behavior andobservations of the flame structure, heat release rates, thedegree of extinction, followed by an analysis of the flamereignition mechanisms for the three cases, and analysis ofa flame progress variable.

3.1. Extinction Effects on Jet Evolution

The purpose of the parametric simulations is to varythe degree of flame extinction achieved under nearly iden-tical flow conditions (constant Re), through variation inthe Damkohler number. The Damkohler number variesover a factor of two and a wide range of flame extinctionis observed in the three cases with Case 1 exhibiting amodest degree of extinction, Case 2 experiencing signif-icant extinction, and Case 3 experiencing near blowoutconditions. Three regimes exist in the development of theflow:

1. Shear layer development and flame-turbulence inter-action.

2. Flame extinction with mixing of fuel, oxidizer, andcombustion products.

3. Reignition of the turbulent flame.

Figure 4 shows contour plots of temperature at the pointof maximum extinction for the three cases (t=0.14, 0.18,and 0.38 ms, for Cases 1, 2, and 3, respectively). The sto-ichiometric isocontour of mixture fraction is overlaid. Thefigures show a highly contorted stoichiometric surface withregions of burning and extinguished flame zones. The de-gree of flame extinction will influence the degree of mixingbetween fuel, air and combustion products prior to reigni-tion, which may have a significant effect on the rate andmode of reignition. In Case 3, the extinction is nearlycomplete, with all but a single, hot reacting flame kernelsurviving the straining velocity field. As the scalar dissipa-tion rate relaxes after flame extinction, the single kernel,shown in the lower center region of the plot for Case 3,grows to reignite the turbulent flame.

Figure 5 shows mean temperature profiles conditionedon the mixture fraction as a function of mixture frac-tion and time for the three cases, along with the con-ditional mean standard deviation of temperature. In allthree cases, the temperature is high initially, then de-creases to a minimum at the time of maximum extinction,then rises again as the flame strain relaxes and reignitionoccurs. The degree of extinction is again shown to bemuch higher for Case 3 as observed from these conditionalmean temperature profiles. The conditional standard de-viations for Cases 2 and 3 are similar. The values are lowprior to the onset of extinction, then rise to a maximumaround the level of maximum extinction, and subsequentlydecrease again as the flame reignites and the scalar dissi-pation rate decreases. In contrast, the conditional meanstandard deviation of temperature for Case 3 begins low,

5

Figure 4: Temperature contours for three cases.

rises to a maximum as the flame is extinguished, then de-creases again as the flame nearly disappears and the mix-ture is homogenized somewhat, followed by an increase indeviation through reignition. Reignition for Case 3 is in-complete as noted by the relatively low conditional meantemperature for this case at the end of the simulation.If the simulation were carried out longer, the conditionalstandard deviation would likely decrease again as the flamefully reignites. Figure 5 also shows a continual decrease inthe peak value of the mixture fraction as the fuel core ismixed out. By the end of the simulation, the peak mixturefraction in the domain is approximately 0.7, 0.6, and 0.3for Cases 1, 2, and 3, respectively.

The conditional mean stoichiometric scalar dissipationrate as a function of time for the three cases is shown inFig. 6. Each of the curves begins at the initial scalar dissi-pation rate. The curves then decrease slightly as the flamesrelax, prior to the onset of turbulence-flame interactions.As the turbulence developes, the mean scalar dissipationrate increases, reaches a maximum, then decreases as theturbulence intensity relaxes. While the curves for the threecases are qualitatively similar, they show significant quan-titative differences even though the flow parameters (veloc-ity, geometric properties) are the same for the three cases.Early on, the three curves are nearly identical. The peakvalues for Cases 1 and 2 are close in value, while the peakfor Case 3 is significantly higher than the other cases. Thedashed lines in Fig. 6 correspond to the steady laminar ex-tinction values for the three cases. The peak mean dissipa-tion rate for Case 1 occurs at approximately the same valueas the steady laminar extinction value, while the mean ofCase 2 is somewhat above the value and Case 3 is morethan twice as high as the laminar extinction value. Thewidth of the profiles in time increases for Cases 1, 2, and3, so that Case 3 experiences a higher dissipation rate fora longer time. The increasing width of the profiles resultsin cases with greater extinction residing above the lami-nar extinction values for longer times, delaying the onsetof the reignition processes. This increased scalar dissipa-tion rate is due to the increased level of flame extinction.These results are consistent with a local increase in theReynolds number, hence scalar dissipation rate, associatedwith decreased viscosity as the stoichiometric temperaturedecreases through flame extinction. In addition, increasedflame extinction results in decreased flow dilatation anddensity ratios due to reduced global heat release rates,contributing to higher scalar dissipation rates. Pantanoalso found higher scalar dissipation rates for DNS withoutheat release, compared to DNS with heat release throughcombustion [23]. The resulting increased scalar dissipa-tion rate with increasing extinction results in a positivefeedback causing higher levels of extinction. The result isan increased sensitivity to extinction, which can lead toglobal flame blowout in extreme cases. The conditionalroot mean square of the stoichiometric scalar dissipationrate for the three cases is also shown in Fig. 6. These pro-files are very similar in both shape and magnitude to the

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Case 1

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<σ

T|ξ

>Case 2

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Figure 5: Conditional mean temperature (left plots) and conditional mean standard deviation of temperature (right) plots for the three DNScases.

7

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

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|ξst>

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/s)

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st>

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Figure 6: Conditional means and standard deviations of stoichiomet-ric scalar dissipation rate versus time for the three cases, along withthe steady laminar extinction values (dashed lines).

mean values and indicate a relatively high level of condi-tional fluctuations in the stoichiometric scalar dissipationrate.

The PDF of the scalar dissipation rate is important inmodeling of turbulent reacting flows. Figure 7 shows thePDF of log10 χ conditioned on stoichiometric mixture frac-tions. The PDFs are presented on log and linear scales forthe three cases. Three times are shown for each case rep-resenting the time of peak flame extinction, the end of thesimulation after flame recovery, and an intermediate time.Also shown is the normal distribution. These PDFs whereconstructed by extracting the stoichiometric isosurface ofmixture fraction and area-weighting the resulting mixturefractions obtained on a triangular grid; 70 bins are used.The PDFs shown in the figure have been centered andscaled on the abscissas and scaled by the standard devia-tion on the ordinates. In this way, the shape of the PDFsare observed on a consistent basis. Values of the mean,variance, and skewness of log10 χ are provided in Table4, where skewness is defined as 〈(logχ− 〈logχ〉)3〉/σ3

logχ.The general shape of the PDFs adhere to the lognormaldistribution. However, there is some negative skewness inthe profiles, which is most clearly shown on the log scale asa positive deviation from lognormal at low dissipation anda negative deviation from lognormal at high dissipation.These results are consistent with the experimental stud-ies of Su and Clemens [24] who performed experimentsof scalar mixing in inert, planar, turbulent gaseous jets.

Table 4: Mean, variance, and skewness of log10(χ) for three cases atfour times.

Time (ms) 〈logχ〉 σ2logχ Skewness

Case 1 0.09 3.4 0.31 -0.220.15 3.2 0.27 -0.140.21 3.8 0.34 -0.220.36 2.3 0.32 -0.37

Case 2 0.09 3.4 0.25 -0.330.18 3.1 0.38 -0.520.27 2.6 0.43 -0.360.42 2.0 0.52 -0.38

Case 3 0.09 3.4 0.29 -0.370.39 2.4 0.46 -0.470.54 2.0 0.50 -0.510.63 1.7 0.47 -0.49

These authors concluded that the negative skewness islikely a property of the scalar dissipation rate PDF andnot an artifact of experimental uncertainty or moderateReynolds number. The DNS of Hawkes et al. [11] (whichare similar to the present simulations, but with syngas asthe fuel and varying Reynolds number), show the samebehavior as that shown here for jet Reynolds numbers be-tween 2510 and 9079.

The stoichiometric surface area as a function of timeis shown in Fig. 8. The stoichiometric surface area isreported since reactions nominally occur on this surface,but specifically, it is reported here simply as a measure ofthe influence of the extent of combustion on the mixingcharacteristics of the flow. The surface area increases sig-nificantly as the degree of extinction increases. In Cases1 and 2, the surface area increases as the flame extin-guishes, reaches a maximum, then decreases again. InCase 1, the peak surface area occurs at 0.18 ms, slightlylater than the peak flame extinction at 0.14 ms. Case 2 issimilar, but the time between the peak surface area andthe peak extinction is longer: 0.3 and 0.18 ms, respec-tively. These results indicate a competition between theabsence of flame, which tends to increase flame area, andthe presence of flame which suppresses the stoichiometricarea, as the flame reignition begins on average at the pointof maximum extinction. Interestingly, the peak in the sto-ichiometric surface area coincides with the inflection pointfraction of the burning stoichiometric surface (see Fig. 9,below). In contrast, Case 3 exhibits a monotonic increasein the stoichiometric surface area over the whole simula-tion time, though it does appear to level off somewhataround the time of peak flame extinction (0.38 ms), thenrises through the reignition process.

Figure 9 shows the total heat release rate of the threecases as a function of time along with the fraction of the to-tal burning stoichiometric surface. The latter is defined asthe fraction of the stoichiometric surface with an OH massfraction above 50% of the steady laminar extinction value.

8

0

0.1

0.2

0.3

0.4t=0.15 mst=0.21 mst=0.36 ms

0

0.1

0.2

0.3

P(l

og

(χ))

σlo

g(χ

) t=0.18 mst=0.27 mst=0.42 ms

-4 -2 0 2 4(log(χ)−<log(χ)>)/σ

log(χ)

0

0.1

0.2

0.3

t=0.39 mst=0.54 mst=0.63 ms

Case 1

Case 2

Case 3

0.001

0.01

0.1

1

t=0.15 mst=0.21 mst=0.36 ms

0.001

0.01

0.1

1

P(l

og

(χ))

σlo

g(χ

)

t=0.18 mst=0.27 mst=0.42 ms

-4 -2 0 2 4(log(χ)−<log(χ)>)/σ

log(χ)

0.001

0.01

0.1

1

t=0.39 mst=0.54 mst=0.63 ms

Case 1

Case 2

Case 3

Figure 7: Normalized PDFs of stoichiometric scalar dissipation rate on log and linear scales. For each case the time of peak extinction, theend of the simulation and an intermediate time are shown. The bold solid line is a log-normal distribution.

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

0

5

10

15

ξ st S

urfa

ce A

rea

(cm

2 )

Case 1Case 2Case 3

Figure 8: Stoichiometric surface area as a function of time for thethree cases.

This is a reasonable criterion as the transition from a burn-ing to a quenched state is relatively sharp for most hydro-carbon fuels including ethylene. For Case 3, the burningfraction of the stoichiometric surface is shown as a dashedline during the reignition process after the peak extinctionas the reignition does not occur as a nonpremixed flame(as discussed later). Hence, the steady laminar extinc-tion value of the OH mass fraction is not consistent withthe flame state. The given definition holds over the wholeflame, however. The fraction of the burning stoichiomet-ric surface gives a quantitative measure of the degree ofextinction of the flame. Cases 1 and 2 experience approx-imately 40%, and 70% extinction by this measure, whileCase 3 is almost completely extinguished. Additionally,Cases 1 and 2 experience nearly complete reignition of theflame, while incomplete flame reignition occurs for Case 3by the end of its simulation.

The total heat release rate shown in Fig. 9 highlightsthe competition between increased heat release rate withincreasing scalar dissipation rate, and decreasing heat re-lease rate with increasing extinguished flame area. In Case1, the total heat release rate rises to a maximum slightlylater than the point of maximum extinction (the mini-mum in the fraction of the burning surface curve), which isslightly after the peak in the scalar dissipation rate in Fig.6. As the dissipation rate decreases the heat release ratealso decreases. In Case 2, the heat release rate initially

9

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

0

500

1000

1500

2000

Tot

al H

eat R

elea

se R

ate

(J/s

)

Case 1Case 2Case 3

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

0

0.2

0.4

0.6

0.8

1

Frac

tion

Bur

ning

ξst S

urfa

ce

Case 1Case 2Case 3

Figure 9: Total heat release rate (top) and fraction of the total burn-ing stoichiometric surface (bottom) versus time for the three cases.

increases as the scalar dissipation rate increases, but asthe burning flame area is decreasing in this region, the re-duction in the burning area competes with and dominatesthe increased heat release through dissipation, resultingin the small peak at a time of approximately 0.1 ms. Asthe fraction of the burning surface reaches a minimum andbegins to increase, the total heat release rate rises sharplyat a time of 0.2 ms. This occurs even as the dissipationrate is decreasing (Fig. 6) through a combination of theincreasing flame surface area and the increasing burningfraction of that area. At a time of 0.3 ms, a peak in thetotal heat release rate occurs and the value decreases asthe scalar dissipation rate and the stoichiometric surfacearea decrease, while the fraction of the burning surfacearea continues to increase. Case 3 follows a similar trendto Case 2, but the burning surface area is so severely re-duced that the heat release rate is low for most of thesimulation before rising again through reignition at a timeof approximately 0.4 ms, nominally corresponding to therise in the heat release rate profile of Case 2 at 0.2 ms.

The location of the flame, extinction, and reignitionprocesses in the mixture fraction coordinate is quantifiedthrough the heat release occurring in given regions of themixture fraction. The heat release weighted PDF of themixture fraction PHR(ξ) is shown in Fig. 10 for Cases 1, 2,and 3 at six times through the evolution of each flame. Thequantity PHR(ξ)dξ is the fraction of the total heat releaseoccurring in the mixture fraction range of ξ to ξ + dξ.PHR(ξ) is defined as

PHR(ξ) =〈HR|ξ〉P (ξ)

〈HR〉. (5)

Note, however, that the weighting function (heat releaserate) may be positive or negative, so that the correspond-ing PHR(ξ), may also be negative, and hence calling thisquantity a PDF is not strictly correct. However, if onemultiplies PHR(ξ)dξ by the total heat release rate, shownin Fig. 9, the result is the heat release in the range ξ toξ + dξ, so that PHR(ξ) may be regarded as a normalizedheat release density in the mixture fraction coordinate. Ineach of the three plots in the figure there are two graphs,and time increases for each curve from the lower graph tothe upper graph, and from the dash-dotted, to the dashed,to the solid curves. The stoichiometric mixture fraction isdenoted by the vertical dotted line in each plot. For eachcase, the peak heat release rate occurs rich of stoichio-metric. The shape of the profiles in Cases 1 and 2, aresimilar, and relatively stationary in the mixture fractioncoordinate. The peak in the heat release rate at ξ = 0.25is reduced at the point of maximum extinction in Cases1 and 2, and rises steadily through reignition. In Case3, the heat release is initially as for Cases 1, and 2. Af-ter extinction in the mixing phase prior to reignition (0.39ms), the heat release is positive with significant fractionsof the heat release at very rich mixture fractions. The to-tal heat release is very small during this period, however,because of the high degree of extinction. As reignition

10

-2

0

2

4

6

8

10

PH

R(ξ

) t=0.36 mst=0.30 mst=0.24 ms

0 0.2 0.4 0.6 0.8 1

ξ

-2

0

2

4

6

8

PH

R(ξ

)

t=0.18 mst=0.12 mst=0.06 ms

Case 1

-2

0

2

4

6

8

10

PH

R(ξ

) t=0.42 mst=0.36 mst=0.30 ms

0 0.2 0.4 0.6 0.8 1

ξ

-2

0

2

4

6

8

PH

R(ξ

)

t=0.24 mst=0.18 mst=0.06 ms

Case 2

-2

0

2

4

6

8

10

PH

R(ξ

) t=0.63 mst=0.51 mst=0.39 ms

0 0.2 0.4 0.6 0.8 1

ξ

-2

0

2

4

6

8

PH

R(ξ

)

t=0.30 mst=0.18 mst=0.06 ms

Case 3

Figure 10: Heat release weighted PDFs of mixture fraction for thethree cases. Time increases from the lower to the upper graphs,from dash-dot, to dashed, to solid curves in each graph. The verticaldotted line denotes the stoichiometric mixture fraction.

occurs, the peak heat release rate migrates progressivelytowards lower values of mixture fraction. Note that thepeak mixture fraction in the domain at the final time is 0.3through mixing of the fuel jet with surrounding oxidizer.This mixing effectively squeezes all reaction towards lowermixture fraction at the same time as the reignition processoccurs.

Flame extinction tends to occur in regions of the sto-ichiometric surface with high scalar dissipation rate. Ina turbulent flow in which the flame structure is strainedand contorted and wrinkled the scalar dissipation rate willlikely be correlated with the turbulent flow structures andthe geometric properties of the stoichiometric surfaces.Figure 11 shows a top view of the stoichiometric surface(view of the jet from the oxidizer stream) in which thesurface contours are OH mass fraction. The figures corre-spond to the times of peak flame extinction. Black regionsare extinguished. The white contour lines shown in thefigures correspond to 50% of the steady laminar extinc-tion value of OH mass fraction and delineate burning andextinguished regions. In Case 1, there is a clear visualcorrelation between the sign of the surface curvature andthe presence of extinguished regions. The flames tend toextinguish first in regions where the center of curvature isin the fuel stream. The surface mean curvature is definedas κ = ∇·~n, where ~n is the surface normal vector directedwith the defining scalar gradient. The scalar dissipationrate tends to be higher in these regions, and this corre-lation between scalar dissipation rate and curvature hasbeen previously reported [21]. This is due primarily to thestoichiometric mixture fraction below 0.5 having higherdissipation rates when the center of curvature is on thefuel side than on the air side. Case 2 experiences a higherdegree of extinction than Case 1, and the correlation be-tween the orientation of curvature and extinction is lessobvious at the point of maximum extinction. Initially, theextinction occurs in regions of curvature centered in thefuel stream, but as the dissipation rate increases in time,more and more of the flame extinguishes and flame holesspread over larger areas, with less correlation of extinctionwith curvature.

As the stoichiometric scalar dissipation rate decreaseswith time (after its peak), the flame holes heal as the flamereignites. In Case 2, the rate of this healing is decreasedbecause of the larger regions of extinction and correspond-ingly smaller quenched/burning interface relative to thedegree of extinction. The larger regions of flame extinc-tion in Case 2, compared to Case 1, will result in moremixing of fuel, oxidizer, and combustion products prior toreignition. This mixing complicates the description andmodeling of the flame, as well as the mode of reignition. Ifsufficient mixing occurs prior to reignition, the reignitionprocess may occur through a premixed flame propagationin a heterogeneous mixture. Indeed, the extreme extinc-tion that occurs in Case 3, results in the reignition of Case3 as a predominantly premixed flame.

11

Case 1 Case 2

Figure 11: View from flow outlet down on the stoichiometric mixture fraction iso-surface (y-directed view), colored by the OH mass fractionon a scale from 0 to 0.015 (dark to light or black to red). Specular highlights added to enhance surface rendering. The nominal extinctionisosurface is shown as a white line at 50% of the steady laminar extinction OH mass fraction.

3.2. Reignition Mode

The mode in which the flame reignites can be investi-gated by considering the degree of alignment of fuel andoxidizer gradients through a flame index [25] defined as

GFO =∇YF · ∇YO2

|∇YF ||YO2|. (6)

The flame index ranges from -1 to 1, with negative valuescorresponding to opposing gradients of fuel and oxidizer, asoccur in nonpremixed flames; positive values correspond toaligned gradients of fuel and oxidizer, as occur in premixedflames. In the calculation of the flame index, the fuel istaken as all CxHy species. Figure 12 shows the cumulativeheat release weighted PDF of the flame index over threeranges of the flame index. The heat release weighted PDFof flame index Pγ(GFO) multiplied by dGFO is the fractionof the heat release in the system between a flame index ofGFO and GFO + dGFO. The cumulative PDF is takenas∫ G2

G1Pγ(GFO)dGFO, where G1 and G2 are the bounds

of the integration. The three regions shown in the figurecorrespond to integration over angles between gradients offuel and oxidizer aligned within 45o, opposed within 45o,and alignments intermediate between these regions. Thesethree regions are termed R1, R2, and R3, respectively.

Initially, all three cases show perfectly opposed gradi-ents of fuel and oxidizer consistent with the initial con-ditions and the nonpremixed nature of the flames. Asextinction occurs, the degree of alignment increases, re-sulting in more of the heat release occurring in regions R3and R1, while flames in region R2 decrease. As reignitionof the flames occurs, Cases 1 and 2 reverse their trends,and the reaction in region R2 increases, while that in re-gions R1 and R3 decreases. Cases 1 and 2 are essentiallynonpremixed flames whose fuel/oxidizer gradients becomemore aligned through the highly turbulent processes dur-ing extinction, with greater opposition of the gradients inthe initial flow and during the reignition processes. The

curves for Case 1 show less deviation from opposed flowthan Case 2, as this case experiences less extinction. Incontrast, Case 3 begins as a nonpremixed flame with allof its heat release occurring in region R2 (as for Cases 1and 2), but as extinction and reignition occur, there is asteady shift from region R2 to region R1. Evidently, thereignition process occurring in Case 3 proceeds througha premixed flame mode and results in burning premixedflames. The heat release in the intermediate region R3 in-creases during extinction, then decreases during reignitionfor Case 1. Cases 2 and 3 are similar to Case 1 in regionR1, but the decrease in the curves occurs prior to the onsetof the mean reignition process.

We note that this analysis presents an integrated pic-ture of the reignition process in terms of the fraction ofthe flames existing as premixed or non-premixed flamesduring the flame evolution using the flame index. Thespecific process of reignition is complicated in that non-premixed flame reignition through edge flames cannot bedistinguished locally from premixed flames due to the pre-mixed nature of the flame tip in edge flames [26], unlessa local flame structure analysis is performed. For exam-ple, Hawkes et al. [5] examined local alignments of OH (aflame indicator) and mixture fraction gradients. Alignedgradients indicate flame folding or premixed propagation,whereas non-aligned gradients are indicative of edge-flamepropagation. The rise in the fraction of the flame indexin regions R1 and R3 in Fig. 12 may be due in part tothe presence of the flame holes (illustrated in Fig. 11) andcorresponding edge flames. The rise in the curves in theseregions is heightened by the relative increase in alignedand intermediate regions as the non-premixed opposed re-gions are extinguished. As the flames reignite, however,we observe the clear trend that Cases 1 and 2 recover toburning nonpremixed flames, while Case 3 recovers to aburning premixed flame. This is indicated by the flameindex in Fig. 12, and also by 9 where the total heat re-

12

Region R1–Gradients Aligned

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

0

0.2

0.4

0.6

0.8

1

Pγ(F

I >

0.7

07)

Case 1Case 2Case 3

Region R2–Gradients Opposed

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

0

0.2

0.4

0.6

0.8

1

Pγ(F

I <

-0.

707)

Case 1Case 2Case 3

Region R3–Gradients Intermediate

0 0.1 0.2 0.3 0.4 0.5 0.6Time (ms)

0

0.2

0.4

0.6

0.8

1

Pγ(-

0.70

7 <

FI

< 0

.707

) Case 1Case 2Case 3

Figure 12: Heat release weighted PDF of flame index integrated overthree regions of the flame index coordinate. Region 1 correspondsto gradients of fuel and oxidizer aligned within 45o, Region 2 corre-sponds to gradients opposed within 45o. Region 3 is in between.

lease has recoved to a high value in all three cases.The determination of the specific mechanisms of flame

reignition and their statistics will require investigation intothe topology of the reignition zones of individual flames.That investigation is planned for a future study, especiallyrelating to Cases 1 and 2. For Case 3, as noted above, thereignition proceeded from a single reaction kernel. Thatkernel was tracked and analysed through ignition delaycomputations, and surface displacement analysis discussedbelow, and apparently reignites through premixed flamepropagation through a stratified mixture. However, thatdoes not preclude the presence of edge flame structuresduring the extinction processes prior to reignition.

A surface displacement speed analysis was performedto confirm the reignition of Case 3 as a premixed flame.The transport equation of a simple reacting scalar φ isgiven by

Dφ

Dt= −1

ρ∇ · jφ +

ωφρ. (7)

The surface velocity of an isosurface of φ, say φc is givenby

sd =Dφ/Dt

|∇φ|

∣∣∣∣φ=φc

, (8)

where the sign convention is defined such that sd is pos-itive when the isosurface is moving towards values of φless than φc. During the period of intense reignition, avalue of YCO2 = 0.035 was found to closely track the peakheat release rate in the domain and this value is used asφc. This value does not track the peak heat release forthis case during the initial period when the nonpremixedflame is extinguishing, and a general scalar that can cap-ture both regions was not identified. However, our primaryfocus here is on the reignition process. The surface veloc-ity, sd, as defined, was computed and then normalized ass∗d = ρ

ρusd, where ρu is local unburnt density computed

from the local mixture fraction and enthalpy. This defini-tion permits direct comparison of the local isosurface ve-locity to the corresponding local one-dimensional, steadylaminar flame speed sL. Note that the definition of sd isgeneral, and applies to both premixed and nonpremixedflames. Comparison of the normalized value to the steady,laminar one-dimensional flame speed gives a direct mea-sure of the degree to which the reignition mode occurs asa premixed flame. One-dimensional laminar flame speedswere computed using the Chemkin Premix code [27] andtabulated as a function of mixture fraction and upstreamtemperature. The DNS mixture fraction and enthalpy per-mit calculation of the corresponding unburnt (upstream)temperature and the local sL is interpolated from the ta-ble.

Figure 13 shows the area-weighted mean surface dis-placement speed s∗d as a function of time during the periodof extinguished mixing and the reignition process. Thecorresponding average sL is also shown in the figure. Ini-tially, the curve for s∗d is negative as there is little reactionat the φc surface and the CO2 field experiences strong

13

0.2 0.3 0.4 0.5 0.6 0.7Time (ms)

-200

-100

0

100

200

300

Mea

n Y

CO

2 S

urf

ace

Spee

d (

cm/s

)

SD

*

SL

ReignitionPeak

ExtinctionStrong

Figure 13: Mean normalized surface velocity of YCO2 = 0.035 andcorresponding laminar flame speed.

scalar mixing. The displacement speed increases to posi-tive values at the beginning of the reignition process afterpeak extinction at approximately 0.4 ms. At times laterthan 0.5 ms, significant reignition and flame propagationare occurring and the mean s∗d and sL are very close, in-dicating the premixed nature of the flames. Discrepen-cies between the curves in this region arise from unsteadyflame propagation, multi-dimensional transport effects, ornon-burning regions on the φc surface. At times prior toreignition, much of the CO2 surface is not associated withreaction, being a product of the nonpremixed flames un-dergoing scalar mixing. Late in the simulation, nearly allof the YCO2 = 0.035 coincides with the peak heat releaserate. The mean sL curve is initially very small as theYCO2 = 0.035 surface prior to extinction resides at mix-ture fractions of 0.04 and 0.72, which are very lean andvery rich compared to stoichiometric (0.17), so that thecorresponding laminar flame speed is small.

3.3. Progress Variable

The DNS presented exhibit a high degree of extinctionallowing for the possibility of partial premixing prior toreignition. Partial premixing can be important in combus-tion processes including lifted flames, liquid fuel sprays,and combustion in engines and gas turbines. Stratifiedmixtures consisting of reactants mixed with products areused as a NOx control strategy (exhaust gas recirculation).A progress variable approach to modeling has been appliedto represent partially premixed combustion [13, 28, 29].This is done by specifying a reaction progress variable andits associated transport equation. This equation has termsthat are often neglected, or require closure in LES model-ing. Here, a progress variable is defined and reported forthe DNS simulations, along with key terms in its transport

0 0.2 0.4 0.6 0.8 1ξ

0

0.2

0.4

0.6

0.8

1

Mas

s Fr

actio

n, P

rog.

Var

.

YCO+CO

2

Prog. Var.

Case 1Case 2Case 3

Figure 14: CO+CO2 mass fraction (lower curves) for equilibriumproducts (dashed lines) and products of complete combustion andcombustion to CO products (solid lines). The upper plots are theequilibrium progress variable with increasing progress variable forincreasing case number.

equation.There are many ways to define the progress variable.

Typically, a given species, either a reactant or productspecies, or combination of species is chosen. The definitionof the progress variable used here is given by

c =Yi − Y mixi

Y ∗i − Y mixi

, (9)

where Yi is some linear combination of species mass frac-tions, Y ∗i is a reference state (ideally the equilibrium stateof Yi), and Y mixi is Yi for pure mixing defined as

Y mixi = Yi, ξ=1 · (ξ) + Yi, ξ=0 · (1− ξ). (10)

Here, we define Yi = YCO+YCO2. This combination is cho-sen because CO is the dominant product species for veryrich compositions, CO2 is the dominant product speciesfor lean compositions, and both species are present in in-termediate regions. These three regions are defined asRegion 1: 0 ≤ ξ < ξst; Region 2: ξst ≤ ξ < ξ∗st; andRegion 3: ξ∗st < ξ ≤ 1. Here, ξst is the stoichiomet-ric mixture fraction for products of complete combustion,ξ = 0.17, and ξ∗st is the stoichiometric mixture fraction forCO products: C2H4+O2→2CO+2H2, with ξ∗st = 0.3805.The lower curves of Fig. 14 show Yi at equilibrium for thethree cases. Also shown are Yi under conditions of com-plete combustion products in Region 1; CO combustionproducts in Region 3; and linear profiles for CO and CO2

in Region 2, where CO is zero at ξst and CO2 is zero at ξ∗st.Under these conditions (used as a reference state), Yi is de-noted Y ∗i , and the equilibrium state is denoted Y eqi . Notehow closely Y ∗i approximates Y eqi . The progress variableis defined using Y ∗i since this gives piecewise-linear pro-files of Yi(ξ), and is close to the equilibrium profile. Forreference, the equilibrium progress variable, that is, the

14

progress variable when Yi = Y eqi in Eq. (9), is also shownin Fig. 14 in the upper three curves for the three cases,with the progress variable increasing with case number.

A transport equation for the progress variable has beenderived by Bray et al. [13] using the following transportequations for the mixture fraction (assuming unity Lewisnumbers) and Yi:

∂ρξ

∂t+∇ · (ρ~vξ) = ∇ · (ρD∇ξ), (11)

∂ρYi∂t

+∇ · (ρ~vYi) = ∇ · (ρD∇Yi) + ωi. (12)

Using Yi = Yi(c, ξ), with c defined in Eq. (9), the transportequation for c is derived:

∂ρc

∂t+∇ · (ρ~vc) = ∇ · (ρD∇c)+ (13)

1

∂Yi/∂c

[ωi +

∂2Yi∂c2

ρχc +∂2Yi∂ξ2

ρχξ +∂2Yi∂c∂ξ

ρχξ, c

].

Here,

χc = D∇c · ∇c, (14)

χξ = D∇ξ · ∇ξ, (15)

χξ, c = D∇ξ · ∇c. (16)

In Eq. (13), the second and third terms in brackets arezero because Yi is linear in both c and ξ, as defined in Eq.(9). The third term in brackets is infinite at the stoichio-metric points where the piecewise-linear profiles have dis-continuous slopes. Bray et al. [13] (using and equilibriumreference state) indicate that this term is approximatelyzero except at the stoichiometric point. This motivatesthe definition of c in terms of Y ∗i , as we have done here,which is piecewise linear with mixture fraction. The re-maining terms in brackets are the reaction source termand the cross dissipation term. The cross dissipation termis a measure of partial premixing and accounts for mixturefraction variations across a reaction front [29] in the senseof a propagating premixed flame, or variations of progressvariable along gradients of mixture fraction in a straineddiffusion flame. While definition in terms of Y eqi may seemmore physical since that is the end state of reaction, usingY ∗i is reasonable as a scaling factor, and is close to Y eqi .One implication of this definition is that c will not varyfrom zero to unity as reaction progresses from unburnt toequilibrium products, but rather from zero to the valueof c shown in Fig. 14, at a given mixture fraction. Inpractice, the reactions are far from equilibrium anyway, asshown below. The present definition of c in terms of Y ∗iis advantageous as there are fewer terms in the c trans-port equation, and numerical differentiation (first and sec-ond derivatives) of the equilibrium mass fraction with re-spect to mixture fraction is avoided. Figure 15 shows theprogress variable for steady laminar flamelet solutions forthe stream compositions of Case 2 for two values of thestoichiometric scalar dissipation rate. Also shown in the

0 0.2 0.4 0.6 0.8 1ξ

0

0.2

0.4

0.6

0.8

1

Mas

s F

ract

ion

, P

rog

. V

ar.

YCO+CO2

Prog. Var.

χst=935 (1/s), t=0.18 ms

0 0.2 0.4 0.6 0.8 1ξ

0

0.2

0.4

0.6

0.8

1

Mas

s F

ract

ion,

Pro

g. V

ar.

YCO+CO2

Prog. Var.

χst= 95 (1/s), t=0.42 ms

Figure 15: Progress variable and CO+CO2 mass fractions for Case2 at two times. DNS (solid) and steady laminar flamelet (dashed)curves are shown. Y∗ (dash-dot) is shown for reference.

15

figure are the progress variables for the DNS at the corre-sponding scalar dissipation rate, along with the DNS con-ditional mean profiles and steady laminar flamelet profilesof CO+CO2 mass fraction. The higher scalar dissipationvalue is at the time of the peak flame extinction, and thelower value is at the end of the simulation when the tur-bulence is relaxing, scalar dissipation decreasing, and theflames are largely reignited. At the later time shown inthe figure, the DNS and flamelet results are in good agree-ment, while, as expected, the DNS and flamelet resultsdiffer widely at the time of peak extinction. In both cases,the progress variable is significantly below unity, indicat-ing the flames are far from equilibrium. The shape of theprogress variable profiles is similar when c is scaled withequilibrium products instead of ideal combustion products,but the initial dip in the progress variable, while present,is somewhat smoother when using equilibrium products.

The conditional mean progress variable for the threeDNS cases is shown in Fig. 16 at several times duringthe simulations. In each case the progress variable beginsat a high value, decreases through the peak extinctiontime (center time shown for each case), then rises againthrough the reignition process. Outside the reaction zone,the progress variable is relatively flat, then decreases asextinction and mixing occur. As the flame reignites, theseregions gradually begin to recover due to mixing from thereaction zone, but this process is incomplete. The lowprogress variable at ξ > ξ∗st is due to a much higher con-centration of C2H4 (at the expense of CO and CO2) thanoccurs at equilibrium.

The profiles of the conditional mean progress variableinclude intermittent effects (such as burning and extin-guished regions). Low values of progress variable maybe more reflective of intermittency than a diminished ap-proach to equilibrium in burning regions. These effectsare not present, however, at the early times due to theinitial condition, or at the later times (for Cases 1 and 2)as shown by the close agreement between the DNS andflamelet profiles in Fig. 15 at the lower dissipation rate.The progress variable is significantly below unity (espe-cially in rich regions) at both these early and late times.

For Case 3, the lowest value of the progress variableoccurs at the time of peak extinction, and occurs near thestoichiometric point, while the progress variable is muchhigher in regions of higher mixture fraction at that time.This is due to the high degree of extinction and the ex-cess oxidizer. Larger values of mixture fractions consistof a fuel/product mixture, but under fuel-limited condi-tions, whereas stoichiometric and lean regions are contin-ually mixed with fresh, unreacted oxidizer.

A comparison of the conditional mean reaction rate(RR) and cross dissipation (CD) terms in Eq. (13) isshown in Fig. 17. Two times are shown for each of thethree cases: the time of peak extinction and end of thesimulation at flame recovery. Cases 1 and 2 show similarbehavior. The RR and CD terms are higher at the ear-lier times where mixing and hence burning rates are most

0 0.2 0.4 0.6 0.8 1

ξ

0

0.2

0.4

0.6

0.8

1

<c | ξ

>

0.03 ms

0.09 ms

0.15 ms

0.21 ms

0.36 ms

Case 1

0 0.2 0.4 0.6 0.8 1

ξ

0

0.2

0.4

0.6

0.8

1<

c | ξ

>

0.03 ms

0.09 ms

0.18 ms

0.30 ms

0.42 ms

Case 2

0 0.2 0.4 0.6 0.8 1

ξ

0

0.2

0.4

0.6

0.8

1

<c | ξ

>

0.03 ms

0.12 ms

0.39 ms

0.51 ms

0.63 ms

Case 3

Figure 16: Progress variable for the three DNS cases at various times.

16

0 0.2 0.4 0.6 0.8 1ξ

-200

0

200

400

600

CD

, R

R t

erm

s (k

g/m

3/s

)

CD, t=0.15 msRR, t=0.15 msCD, t=0.36 msRR, t=0.36 ms

Case 1

0 0.2 0.4 0.6 0.8 1ξ

-200

0

200

400

CD

, R

R t

erm

s (k

g/m

3/s

)

CD, t=0.18 msRR, t=0.18 msCD, t=0.42 msRR, t=0.42 ms

Case 2

0 0.2 0.4 0.6 0.8 1ξ

-40

0

40

80

120

CD

, R

R t

erm

s (k

g/m

3/s

)

CD, t=0.39 msRR, t=0.39 msCD, t=0.63 msRR, t=0.63 ms

Case 3

Figure 17: Plots of the cross dissipation and reaction terms in theprogress variable transport equation. Two times are shown for eachcase, the point of peak extinction and a final recovery time at flamerecovery.

intense. The reaction term is active only below a mix-ture fraction of 0.25, peaking at the stoichiometric mix-ture fraction of 0.17. At both times shown, the RR termis dominant below mixture fractions of 0.25, whereas theCD term exhibits several peaks and valleys with a strongnegative peak at a mixture fraction of 0.35. The magni-tude of the RR and CD terms are higher in Case 1 than inCase 2 owing to the higher Damkohler number and earliersimulation times reported for Case 1. In Case 3, the reac-tion rate term at the early time is very small as the flameis nearly extinguished. The peak at t=0.63 ms is similarto that of Cases 1 and 2, but at a significantly lower mag-nitude. In addition, the difference between the RR andCD term at the final time for Case 3 is greater than thatfor Cases 1 and 2 as the flow field is more homogeneous inCase 3 owing to the greater mixing between the fuel andoxidizer streams and products during reignition for Case 3.This suggests a dominance of reaction over cross dissipa-tion in the progress variable for partially premixed com-bustion proper in which predominantly premixed flamesare propagating in a vitiated flow. In contrast, while theRR term is greater than the CD term in the primarily non-premixed flames, both terms, RR and CD, are significant.These results are in agreement with assertions in [13, 28].

4. Conclusions

A series of three parametric direct numerical simula-tions of turbulent planar ethylene jet flames has been per-formed at three values of the Damkohler number at a fixedReynolds number. These simulations exhibit high levels offlame extinction followed by reignition. The level of extinc-tion ranged between 40% and nearly 100 %. It is shownthat, for fixed flow and geometric parameters, the degreeof extinction has a significant effect on the developmentof the flow including the scalar dissipation rate, stoichio-metric surface area, and heat release rate evolution. Com-petition between flame surface area and scalar dissipationrate for overall heat release is shown. Flame extinction iscorrelated with the sign of curvature of the stoichiomet-ric mixture fraction isosurface, with extinction occurringpreferentially in regions of the stoichiometric surface withcenter of curvature in the fuel stream. The flame reig-nition mode is examined in terms of a cumulative heatrelease-weighted PDF of the flame index, which is used todelineate between nonpremixed and premixed flames. Thetwo cases that exhibit smaller degrees of extinction, Case1, and Case 2, exist predominantly as nonpremixed flamesthrough reignition, whereas the reignition mode of Case3 is dominated by premixed flame propagation as a re-sult of the high degree of extinction followed by mixing offuel, oxidizer, and combustion products prior to reignition.Comparison of a progress variable iso-surface displacementspeed to laminar premixed flame speeds confirmed the pre-dominance of reignition through premixed flame propa-gation in Case 3. In Case 3, reignition commences fuel-rich of stoichiometric and migrates towards leaner values

17

with peak heat release rates favoring fuel-rich conditionsat all times for all three cases. A progress variable basedupon CO+CO2 was defined and analysed. The behavior ofthis progress variable in time is consistent with the extinc-tion and reignition phenomena. Combustion processes donot approach equilibrium here, resulting in relatively lowmagnitudes of the progress variable, which is comparedbetween equilibrium, flamelets, and DNS. Comparison ofcross dissipation and reaction source terms in the progressvariable transport equation shows dominance of the reac-tion source term. The cross dissipation term is small forCase 3 and more important in Cases 1 and 2.

5. Acknowledgments

Simulations presented in this work were supported bythe U.S. Department of Energy, Office of Basic EnergySciences, Division of Chemical Sciences, Geosciences, andBiosciences. Simulations were performed on the Cray XT4Jaguar supercomputer at Oak Ridge National Laborato-ries.

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