The influence of chemical mechanisms on PDF calculations of ... · The influence of chemical mechanisms on PDF calculations of nonpremixed piloted jet flames Renfeng Richard Cao∗,

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Combustion and Flame 143 (2005) 450–470www.elsevier.com/locate/combustflam

The influence of chemical mechanisms on PDF calculatof nonpremixed piloted jet flames

Renfeng Richard Cao∗, Stephen B. Pope

Mechanical and Aerospace Engineering, Cornell University, 245 Upson Hall, Ithaca, NY 14853, USA

Received 9 April 2005; received in revised form 28 August 2005; accepted 31 August 2005

Available online 10 October 2005

Abstract

Seven different chemical mechanisms for methane are used in PDF model calculations of the Barlow anflames D, E, and F in order to investigate the ability of these mechanisms to represent the local extinctiontion, and other chemical phenomena observed in these nonpremixed piloted jet flames. The mechanismrange from a 5-step reduced mechanism to the GRI3.0 mechanism which involves 53 species. As in severecent studies, we use the PDF method based on the joint probability density function of velocity, turbfrequency, and composition. Extensive tests are performed to ensure the numerical accuracy of the calcto relate them to previous calculations based on the same model, and to reexamine the sensitivity of thetions (especially of flame F) to uncertainties in the pilot temperature and the treatment of radiation. As hobserved in other studies of laminar and turbulent nonpremixed flames, we find that the GRI3.0 mechanipredicts the levels of NO, typically by a factor of 2. Apart from this, the GRI3.0 and GRI2.11 mechanismcomparably good agreement with the experimental data for all three flames, including the level of localtion and the conditional means of major and other minor species. Two augmented reduce mechanism (ARARM2) based on GRI2.11 and containing 16 and 19 species are slightly less accurate; while the 5-stepmechanism and twoC1 skeletal mechanisms containing 16 species display significant inaccuracies. An extion of the autoignition and laminar-flame behavior of the different mechanisms confirms (with some exceexpected trends: there is an association between long ignition delay times, small extinction strain rates,levels of local extinction. This study again demonstrates the ability of the joint PDF method to represent acthe strong turbulence–chemistry interactions in these flames, and it clarifies the necessary level of descthe chemical kinetics. 2005 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords:PDF methods; Turbulent flames; Detailed chemistry; Mixing models

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This paper is accompanied by Supplementary maial which is available at doi:10.1016/j.combustflame.20008.018.

* Corresponding author. Fax: +1 (607) 255 1222.E-mail address:[email protected](R.R. Cao).

0010-2180/$ – see front matter 2005 The Combustion Institutdoi:10.1016/j.combustflame.2005.08.018

1. Introduction

In this work we use PDF methods to study tperformance of seven different chemical mechanisin the calculation of turbulence–chemistry interations in nonpremixed turbulent flames. The calcutions are compared to the experimental data of Bar

e. Published by Elsevier Inc. All rights reserved.

http://www.elsevier.com/locate/combustflame

http://dx.doi.org/10.1016/j.combustflame.2005.08.018

mailto:[email protected]



R.R. Cao, S.B. Pope / Combustion and Flame 143 (2005) 450–470 451

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and Frank[1] which were obtained using the Syney burner. This burner consists of a central fueland a substantial annular pilot, and it is surroundby a coflowing air stream. It is most fitting to dscribe these results in this special issue of Combtion and Flame honoring Bob Bilger. The Sydnburner was developed 20 years ago by StärnerBilger [2], with the aim of creating strong turbulencechemistry interactions in a stable flame with relativsimple fluid mechanics and turbulence structure[3].The demonstration of local extinction and reignitiin these flames earned Masri and Bilger[4] the silvermedal of the Combustion Institute in 1988. Singpoint laser diagnostics were then applied to thflames (as reviewed by Masri et al.[5]), culminatingin the experiments of Barlow and Frank[1] which arethe focus of the current work. The subsequent liimaging measurements of Karpetis and Barlow[6]yielded, in 2004, a second silver medal for work baon the Sydney burner.

The flow parameters and the pilot temperaturethe nonpremixed piloted jet methane–air flamesE, and F are listed inTable 1. The fuel, consistingof 25% methane and 75% air, with a temperature294 K, forms the inner fuel jet with a diameterD = 7.2 mm. The flame is stabilized using a pilwith a diameter ofDp = 18.2 mm. The pilot is aburnt lean mixture of C2H2, H2, air, CO2, and N2,chosen to have the same elemental compositiomethane/air at 0.77 equivalence ratio. The cofloing air stream has a temperature of 291 K. Flamhas a small degree of local extinction, while flameand F have significant and increasing amount of loextinction, with flame F being quite close to globextinction. (The jet velocity in flame F is over 90%the estimated blowoff velocity[7].)

In 2000, fifteen years after the developmentthe Sydney burner, the first modeling studiespeared[8–10] which convincingly and quantitativeldescribed local extinction and reignition in these npremixed piloted jet methane flames. These twoof calculations from Imperial College[10] and fromCornell[8,9] also raised questions about the two moeling ingredients at the core of turbulence–chemisinteractions, namely, the chemical mechanisms

the turbulent mixing model. The two sets of calcutions use different mechanisms and different mixmodels. The EMST mixing model[11] with modelconstantCφ = 1.5 is used in [8,9], whereas themodified Curl model[12,13] with Cφ = 2.3 is usedin [10].

Some recent investigations[14–16] have shedlight on the relative performance of different miing models, although our understanding remainscomplete. In general, the calculated amount of loextinction decreases with increasingCφ , and EMSTyields less local extinction than modified Curl (fthe same value ofCφ ). The present study aimsadvancing our understanding of the issues relatechemical mechanisms.

There are some recent studies of the BarlowFrank flame D using PDF methods with detailchemistry[17,18]. Raman et al.[17] calculated themean profiles and conditional means in flame Ding the joint velocity–composition PDF method withe detailed GRI mechanisms (GRI3.0 and GRI2.and a 16-species reduced mechanism. In this wwe present PDF calculations of flames D, E, anusing seven different mechanisms. These range fa 5-step reduced mechanism[19,20], to the GRI3.0detailed mechanism[21] which involves 53 specieand 325 reactions. The principal results conside(which are compared to the experimental data[22,23]) are the burning index[8] and means of temperature and species mass fractions conditional on mixfraction.

In previous work[8,10], it has been found thathe calculated level of local extinction (particularin flame F) is sensitive to the value of the mixinmodel constantCφ . The base case considered husesCφ = 1.5, the value used in conjunction withe EMST[11] model in the previous studies of theflames[8,9]. The present calculations, using the mcomprehensive detailed methane mechanismsGRI2.11 and GRI3.0), verify that this value ofCφ isappropriate. We also investigate the sensitivity of Pcalculations using different chemical mechanismsthe mixing model constantCφ .

Previous calculations[9] have revealed that somflames exhibit a strong sensitivity to the temperat

Table 1Flow parameters of flames D, E, and F

Flame Rejet Uj,b(m/s)

Up,b(m/s)

Uc(m/s)

Tp(K)

Local extinction

D ∼22,400 49.6 11.4 0.9 1880 LittleE ∼33,600 74.4 17.1 0.9 1880 ModerateF ∼44,800 99.2 22.8 0.9 1860 Severe

Uj,b is the bulk velocity for the fuel jet;Up,b is the bulk velocity for the pilot;Uc is the coflow velocity;T p is the pilottemperature.

452 R.R. Cao, S.B. Pope / Combustion and Flame 143 (2005) 450–470

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of the pilot (which is an imposed boundary conditioand to the treatment of radiation. Here we investigthese sensitivities more comprehensively by perfoing calculations of all three flames with the inclusiand neglect of radiation, and with different valuesthe pilot temperature.

The behavior of the chemical mechanisms injoint PDF calculations are related to their behaviorvery simple test cases, i.e., autoignition and lamiopposed-flow nonpremixed flame calculations (pformed using the code OPPDIF[24]).

In the next section, the submodels used in the joPDF calculations are briefly introduced. Then thelution domain and boundary conditions are givenSection3 where the results of tests are reportedestablish and quantify the numerical accuracy ofcalculations. The numerical method is outlined athe numerical parameters used in these calculatare given in Section4. The current calculations arcompared to previous calculations in Section5 andthe “base case” is defined and investigated. Detacomparisons of all seven mechanisms are presein Section6 based on the results obtained from tPDF calculations, autoignition tests, and the OPPcalculations. Results on the effect to radiation, setivity to the change of pilot temperature, sensitivitythe change of reaction rates, and the effect of the ming model constant are also presented in Sectio6.The final section provides a summary of the work, aconclusions are drawn.

This paper is accompanied by a file of Supplemtary material which contains many more results thcan be included in this paper. Specifically, burningdices and profiles of conditional and unconditionmeans and rms’s are shown for 50 joint PDF callations, corresponding to the base configurationeach flame and each mechanism as well as to pebations to the pilot temperature and to the treatmof radiation. The Supplementary material is availaat doi:10.1016/j.combustflame.2005.08.018.

2. Joint PDF method and chemical mechanisms

There are several different kinds of PDF meods[25] depending on the set of variables whose joPDF is considered. The simplest is the compositPDF method[26–28], in which the modeled equation for the joint PDF of composition is solved: thmean flow and turbulence fields are obtained fra separate model calculation. A second PDF metis based on the joint PDF of velocity and compotion [17]: in this case a separate model is requifor the time or length scale of the turbulence. Attetively, a complete PDF method is based on the joPDF of velocity, turbulence frequency, and compo

Table 2Model constants in the joint PDF models

Constant Value Used in

C0 2.1 SLMCΩ 0.6893 Definition of the mean frequencyΩCω1 0.65 Turbulence frequency modelCω2 0.9 Turbulence frequency modelC3 1.0 Turbulence frequency modelC4 0.25 Turbulence frequency modelCφ 1.5a EMST mixing model

a Note that the effect ofCφ is studied in Section6.3.4byusing a range of values, i.e., 1.2, 1.4, 1.5, 2.0, and 3.0.

tion [31–35]. This is the method used in the currework, and is hereafter referred to as the joint Pmethod.

From the Lagrangian viewpoint, the joint PDmethod requires models for velocity, turbulence fquency, and composition following a fluid parcle [25,36]. The simplified Langevin model (SLM) iused for the evolution of the particle velocity. The schastic frequency model of Van Slooten et al.[30] isused for the turbulence frequency of particles, whprovides the time scale of turbulence. These moare the same as those used in many previous stuusing the joint PDF method, e.g.,[8,9,31–35], and arefully described in[36]. The values of the model constants are shown inTable 2, and are the same as thoused in[31–34]. The only difference in the constanused in earlier calculations of these flames[8,9] isthat thereCω1 is set to 0.56. A detailed comparisoof the current calculations and previous calculatiois presented in Section5, where this difference is discussed.

In PDF methods, the effect of molecular diffusioon the composition is represented by a mixing moIn the present work, the EMST[11] mixing model isused with the mixing model constantCφ set to 1.5(for the base case) following the works of Xu aPope[8], and Tang et al.[9]. The impact on the calculations of flame F of changing the value ofCφ isexamined in Section6.3.4.

The seven chemical mechanisms consideredthis paper are listed inTable 3. The GRI detailed ki-netic mechanisms[21] provide the most comprehensive and standardized set of mechanisms for methcombustion. The detailed versions of 2.11 and 3.0investigated and denoted as GRI2.11 and GRI3.0spectively.

The 12-step augmented reduced mechanism[20]without NO obtained from GRI2.11 is denotedARM1. This mechanism has been successfully uin the joint PDF calculations of flames D, E, andperformed by Xu and Pope[8]. The corresponding15-step reduced mechanism which includes NO isnoted as ARM2, and has been successfully use



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Table 3Chemical mechanisms used in the joint PDF calculations

Mechanism No. of species No. of steps NO species Refer

GRI3.0 53 325 With NO [21]GRI2.11 49 277 With NO [21]ARM1 16 12 Without NO [20]ARM2 19 15 With NO [20]S5G211 9 5 With NO [19,20]Skeletal 16 41 Without NO [37]Smooke 16 46 Without NO [38,39]

Table 4Revised rates in[39] the current Smooke mechanism (k = AT b exp(−E/RT ))

Reactions A (K−b mol/(cm3 s)) b E (cal/mol)

CH4 + H = CH3 + H2 2.2× 104 3.0 8750.0H + O2 = OH + O 2.0×1014 0.0 16800.0

H + O2 + M = HO2 + M 2.1×1018 −1.0 0.0Enhanced third body coefficientsH2O/21.0/ CO2/5.0/ H2/3.3/

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the joint PDF calculations of flames D, E, and F pformed by Tang et al.[9]. The 5-step reduced mechnism [19,20] obtained from the GRI2.11 mechanisis denoted as S5G211. A detailed description ofskeletal mechanism is provided by James et al.[37].The Smooke mechanism is described in[38], butthree reactions have been updated[39] and are shownin Table 4. Both the skeletal and Smooke mechanisdo not include any species with more than one caratom.

An optically thin limit radiation model is used fothe calculations of radiative heat loss[40]. Four gas-phase emitting species H2O, CO2, CO, and CH4 areincluded in this model, and their Planck mean asorption coefficients are calculated by RADCAL[41].This is the same radiation model as used in[9]. Be-cause absorption is neglected in the optically tlimit, the present radiation model somewhat overtimates the radiative heat loss from the flames[22,42,43]. On the other hand, adiabatic calculations (wthe complete neglect of radiation) obviously undestimate the radiative heat loss. In Section6.1 weexamine the impact on the calculations of the trement of radiation.

3. Solution domain and boundary conditions

The flames considered in this paper are the seof nonpremixed piloted methane/air jet flames invegated by Barlow and Frank[1], termed Sandia flameD, E, and F. The flow conditions of these three flamare listed inTable 1. These flames are statistical

steady 2D axisymmetric, and nonswirling. A polacylindrical (z, r) coordinate system is used with thorigin at the center of the fuel jet at its exit plane. Tcomputational solution domain is rectangular, oftent (0,80D) in the axial (z) direction, and (0,20D)in the radial (r) direction, whereD is the diameter othe jet (D = 7.2 mm).

At the inlet plane (z = 0), the joint PDF of ve-locity is taken to be joint normal, with the meavelocities and Reynolds stresses obtained fromcently updated measured inlet profiles[22,44], whichare different from those used in previous calcutions [8–10]. (These differences, however, are fouto be inconsequential.) The Reynolds normal str〈w2〉 in the circumferential direction, which was nmeasured, is taken to be equal to the radial norstress〈v2〉. The turbulence frequency is specifieda gamma distribution and is independent of the veity. The ratio of production to dissipation of turbulekinetic energy is specified as unity which, togethwith the other specified profiles, determines the inprofile of mean turbulence frequency. The tempeture, composition, and density are specified as beuniform in each stream in accord with the experimtally determined values[22]. The coflow boundary(r = 20D) is treated as a perfect-slip wall. Symmtry conditions are applied on the centerline (r = 0).At the exit plane, in the finite-volume solution of thmean conservation equations, the mean densitythe mean axial and radial velocities are extrapolafrom the interior, and the pressure is specified touniform.


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4. Numerical method and accuracy

There are several implementations of particmesh methods to solve the modeled joint PDF eqtions. The computations presented here use a cnamed HYB2D [31–34] which implements a hybrid finite-volume/particle algorithm. In the hybrialgorithm, the PDF/particle method (particle part)coupled with a finite volume solver (FV part). ThFV part solves the mean conservation equationsmass, momentum, energy and the mean equatiostate, and the particle part solves the modeled trport equation for the fluctuating velocity–turbulenfrequency–composition PDF. The FV part providmean fields of velocity, density, and pressure toparticle part and obtains the turbulent fluxes and retion source terms from the particle part.

The in situ adaptive tabulation (ISAT) algorith[45] is used to implement the chemistry given by tdifferent mechanisms. A parallel algorithm for thparticle part of HYB2D, named domain partitioninof particles[31], is implemented using MPI. Thestwo methods greatly facilitate the calculations psented in this paper.

Systematic tests have been performed on themerical parameters which affect the accuracy ofcalculations to determine appropriate values forin the current calculations. The procedures to testeffect of these parameters are the same as thosscribed in previous publications[31,35]based on calculations using HYB2D. The only exception is thadifferent method is used to test the effect of the ISerror toleranceεtol, which is specified to control therror incurred in retrieving from the ISAT table.

It has been shown by Liu and Pope[46] that theglobal error due to ISAT varies linearly with the speified error toleranceεtol. Following the procedure developed by Liu and Pope[46], calculations of flamesE and F were performed (using the skeletal meanism) for a range of values ofεtol. Fig. 1 shows,plotted againstεtol, the conditional mean temperture and the conditional mean mass fraction of Oconditional on the mixture fractionξ being close tostoichiometric (specifically 0.34 ξ 0.36). Suchconditional means are found to be particularly ssitive to ISAT errors[46], much more so than unconditional means. As may be seen fromFig. 1, for thecalculations of flame E, the value ofεtol of 2× 10−5

results in a global error of substantially less than 2However, for the calculations of flame F, the sameror tolerance (i.e.,εtol = 2× 10−5) results in a globaerror of about 5% in the conditional mean tempeture and 18% in the conditional mass fraction of O

Flame F is quite close to global extinction, whican be brought about by increasing the jet velocdecreasing the pilot temperature or (in the calcu

-

Fig. 1. Effect of the ISAT error tolerance on calculatconditional means using the skeletal mechanism andEMST mixing model withCφ = 1.5 for flames E and F az/D = 15. Solid symbols represent calculations for diffent flames: triangles, flame E; circles, flame F. Solid liare linear fits through the left three points. Dashed lines±2% error and dotted lines are±5% error relative to the extrapolated (εtol = 0) values.

tions) by decreasing the value ofCφ , as is studiedin Section6.3.4. As global extinction is approachethe calculated flame exhibits increasing sensitivitythese physical and numerical parameters, and indat the point of extinction the sensitivities are infiniAs a consequence it is difficult to ensure the numcal accuracy of calculations of flame F when thelution is close to global extinction.

To confirm this interpretation, the calculationsflame F were repeated with the valueCφ increasedfrom 1.5 to 2.0, which moves the calculated flame fther from global extinction (see Section6.3.4). Thischange reduces the error in the conditional mean mfraction of minor species from 18 to 5% (for the samvalue of the ISAT error tolerance,εtol = 2× 10−5).

In the subsequent calculations the ISAT error terance is set toεtol = 2 × 10−5. The above resultshow that the resulting errors are quite small (lthan 2%) for calculations far from global extinctio(flames D and E); but, unavoidably, these errorscome larger for flame F as global extinction is aproached.

The numerical parameters used in the currentculations are as follows: (i) the ISAT error toleranεtol is 2×10−5, (ii) the number of cells in the domaiis 96× 96, (iii) the nominal number of particles pecell is 100, and (iv) time averaging is performed ov


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Table 5Ranges inz of bins used to evaluate conditional means

Nominal axiallocationz/D

1 2 3 7.5 15 30 45 60 75

zlow/D 0.50 1.51 2.55 7.48 14.7 29.8 44.5 59.8 74.9zup/D 1.01 2.03 3.07 8.05 15.4 30.6 45.5 60.9 76.2

Table 6Lower (ξl ) and upper (ξu) limits of the mixture-fraction range, and reference values used in the definition of BI based on diquantities

H2O CO2 CO OH H2 T

ξl 0.35 0.30 0.43 0.28 0.48 0.30ξu 0.45 0.40 0.53 0.36 0.58 0.40Y|ξ (or T|ξ ) 0.1278 0.1127 0.05745 4.527× 10−3 3.639× 10−3 2023

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at least 2000 time steps after the calculations rea statistically stationary state (using a moving tiaverage[33]). Extensive numerical tests were peformed of flame F using the skeletal mechanismwith Cφ modified toCφ = 2.0 in order to examine thnumerical errors in a calculated flame not too closglobal extinction. These test cases were performegrids with 32× 32, 48× 48, 72× 72, and 96× 96cells; with 50, 75, and 100 particles per cell; and wtime averaging over at least 2000 steps. The resof these tests show that, with the values of themerical parameters stated above, the numerical eare generally no greater than 2% (with respect topeak value) for the conditional mean temperaturemajor species, and 5% for the conditional meannor species, at all investigated locations. It is reasable to suppose that this level of error—2% for maspecies and temperature, 5% for minor speciesrepresentative of the numerical errors in the calcutions of flames that are not close to global extinct(i.e., flames D and E, and flame F with some meanisms and larger value ofCφ ). But for flames closeto global extinction (e.g., flame F using the skelemechanism andCφ = 1.5), larger errors are likely, ahas already been observed inFig. 1.

Results are reported below of various means cditional on mixture fraction. At a given nominal axilocation,z, these conditional means are formed froall particles in a rectangle inz–r space, extendingfrom z = zlow to z = zup and fromr = 0 to r = 20D.The values ofzlow and zup for the nine output lo-cations are given inTable 5. The conditional meanand rms’s are then formed in 50 equal-sized bin mixture-fraction space. At downstream locatio(z/D 30) there are significant statistical fluctutions in some conditional rms’s due to the relativesmall sample size in some bins (see, e.g.,Fig. 12).

Another quantity reported below is the burniindex (BI). Following the previous work of Xu an

Pope[8], BI is defined as the ratio of the conditionmean (conditioned on mixture fraction being in trangeξl < ξ < ξu) to a reference value obtained usia laminar flame calculation with strain rate 100 s−1.The lower (ξl ) and upper (ξu) limits of the mixturefraction range, and the reference values (Y|ξ andT|ξ )used in the definition for BI are listed inTable 6.Burning index is a parameter used to quantifylevel of local extinction. Generally speaking, smalvalues of BI indicate more local extinction, and BIessentially zero for a globally extinguished flame.

5. Comparison with previous calculations

It is appropriate to compare the current resuwith those obtained by Xu and Pope[8] and Tanget al. [9], since the same models are used. Firstnote the following differences between the three sof calculations.

• Here the mechanisms used include ARM1 aAMR2, whereas only ARM1 is used in[8] andonly ARM2 is used in[9].

• In all calculations, the pilot temperature is setTp = 1880 K, except that for flame F Xu anPope use the measured valueTp = 1860 K, andhere we consider both values,Tp = 1860 K andTp = 1880 K (for flame F).

• Radiation is neglected in[8], while here andin [9] calculations are performed both with awithout radiation.

• Here the ISAT error tolerance is 2× 10−5

whereas in[8,9] the value 5× 10−5 is used (andin addition, an earlier version of ISAT is used[8,9]).

• In all three studies, the models and model cstants used are identical, with one exceptiHere, following[31–34], we use the valueCω1 =


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0.65 for this constant in the turbulence frequenmodel, whereas in[8,9], the valueCω1 = 0.56 isused. This difference is discussed below.

• Updated inlet profiles of mean velocity anReynolds stresses[44], which are different fromthose used in[8,9], have been used for the curent calculations. The effect of this difference hbeen tested and found to have a negligible efon the conditional means.

• A stand-alone particle method implementedthe code PDF2DV was used for previous cculations[8,9]; the hybrid finite-volume/particlemethod implemented in the code HYB2D[33]is used for the current work. The hybrid algrithm and HYB2D were developed to reduce tbias error observed in PDF2DV, and indeed coparative tests show that (for the same grid snumber of particles, etc.) HYB2D is substantiamore accurate[32].

• A different solution domain and a different number of grid cells have been used in the currand previous calculations. A computational dmain of 20D × 80D with 96× 96 cells is usedhere, in contrast to the computational domain12.5D × 60D with 60× 60 cells used in the previous calculations[8,9]. Grid refinements havbeen performed separately to make sure that bsets of calculations are numerically accurate.

• In the present calculations, time averaging hbeen performed for a longer time than in the pvious calculations.

The different values ofCω1 used in the differ-ent calculations require further comment. This moconstant (analogous toCε1 in the k–ε model) sensi-tively controls the calculated spreading rates of jeIn all three sets of calculations the spreading ratcalculated accurately, as evidenced by the accumean velocity and mixture-fraction profiles. On tother hand, in the current calculations, ifCω1 is de-creased to 0.56, then markedly inaccurate meanfiles result. Even though both here and in[8] greatcare is taken to quantify and control the numerierrors, it is difficult to explain these inconsistencwithout invoking numerical error. Given the numecal parameters used here compared to[8,9], and giventhe results of comparative tests of the two codes[32],there are good reasons to suppose that the currenculations based on HYB2D are more accurate.

This issue notwithstanding, it is reassuring to fithat in general there is a reasonably good leveagreement between the present and the previousculations. For example, inFig. 2 we compare theburning indices obtained in the current and previocalculations of flames D, E, and F. All of these calclations use either the ARM1 or the ARM2 mechani

Fig. 2. Burning indices versus jet velocity atz/D = 15 forflames D, E, and F. Solid circles, measurements[1,22]. Solidline with plus, the current ARM1 calculations; dashed linthe current ARM2 calculations; dash-dotted line, the preous ARM1 calculations (Xu and Pope[8]); dotted line, theprevious ARM2 calculations (Tang et al.[9]). All calcula-tions using EMST withCφ = 1.5; radiation is omitted; thepilot temperature is set to 1880 K except the flame F calation of Xu and Pope, in which it is set to 1860 K.

and the EMST mixing model withCφ = 1.5. The pi-lot temperatureTp is set to 1880 K except for thflame F calculation of[8], whereTp is set to 1860 K:the effect of radiation is neglected in all calculationThe following observations concerning the curreand previous calculations[8,9] are made based on thresults shown inFig. 2.

It is clear that the current calculations usiARM1 and ARM2 are very close to each other,both major species and minor species. Due to theof the hybrid method and longer time averaging,statistical error is less than 2% in these calculatioAnd these calculations are in excellent agreemwith the experimental data.

The calculations in[8,9] differ from each otheronly in the mechanisms used (ARM1 and ARMand in the pilot temperature for flame F (1860 a1880 K, respectively). Since the only differencestween ARM1 and ARM2 is that the latter includes Nchemistry, Tang et al.[9] were surprised by the larg


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Fig. 3. Burning indices versus jet velocity atz/D = 15 ob-tained using EMST withCφ = 1.5, and including radiationfor flames D, E, and F. Solid circles, measurements[1,22].Solid line with plus, GRI2.11; dashed line with triangleGRI3.0 (flames E and F only). The pilot temperature isto 1880 K for flames D and E while it is set to 1860 K fflame F.

difference observed for major species. For flameat least part of the observed differences between[8]and[9] can be attributed to the different pilot tempeatures used. For flames D and E it is plausible tthe difference is due to statistical error which in thecalculations is estimated to be about 10% at this lotion [8,9].

Given the level of statistical error in the previocalculations, and given the sensitivity of flame Fnumerical errors, we regard the level of agreemas satisfactory between the current and the prevcalculations. Furthermore, none of the conclusidrawn (in Section7) from the present calculations ain conflict with those drawn in[8,9].

The value of the mixing model constantCφ = 1.5was determined by Xu and Pope[8] as that whichyields the measured level of local extinction. Sin

Fig. 4. Burning indices of flames D, E, and F. Symbomeasurements[1,22] for different flames; lines are PDF caculations for different flames. Triangle and solid line, flaD; cross and dash-dotted line, flame E; diamond and daline, flame F. The gray lines are obtained using the GRI2mechanism while the black lines for flames E and F aretained using the GRI3.0 mechanism (GRI3.0 calculationsnot reported for flame D). EMST withCφ = 1.5 is used andradiation is included in these calculations. The pilot tempature is set to 1880 K for flames D and E calculations wit is set to 1860 K for flame F calculations.

the current calculations are numerically more acrate, and the GRI mechanisms can be supposebe more accurate than the reduced mechanismsin [8,9], we verify here that indeedCφ = 1.5 is theappropriate value.

Figs. 3 and 4show the calculations of flameD, E, and F using the GRI2.11 and GRI3.0 mecnisms. Radiation is taken into consideration andpilot temperatureTp is set to the measured valu(Tp = 1880 K for the flames D and E,Tp = 1860 Kfor flame F). The EMST mixing model withCφ = 1.5is used for all of these calculations.

As may be seen, for GRI3.0 there is excelleagreement with the experimental data. For GRI2


dif-We

n-

isealcul is;al-for.”

enen

else-

ried.an-ex-ns,nalcalu-ing

nde ofnt

esting

c-

fectec-are,on-thethehe

u-

le,

byx-

ef-m-

3.0,es

tesso-

rme,ia-thith

ity

3.0,ne,

the agreement is in general good, although someferences are evident for BI based on CO and OH.take these results as verification thatCφ = 1.5 is theappropriate value for the EMST mixing model costant.

Based on the discussion above, unless otherwstated, the parameters used for the subsequent clations are set as follows: the EMST mixing modeused with constantCφ = 1.5; radiation is includedthe pilot temperature is set to the experimental vues, i.e., 1880 K for flames D and E, and 1860 Kflame F. We refer to this as the “base configuration

6. Results and discussion

More than 100 joint PDF calculations have beperformed for flames D, E, and F using the sevdetailed methane mechanisms listed inTable 3. Theeffects of radiation, pilot temperature, mixing modconstant, and reaction rates are investigated by aries of test cases in which these parameters are vaThe calculated unconditional and conditional qutities have been extensively compared with theperimental data. However, due to space limitatiothe results shown in this section focus on conditiomeans in flame F, which has the most significant loextinction and hence it is the most difficult to calclate. For all three flames, we also examine the burnindex, which reveals the level of local extinction asuccessfully characterizes the overall performancdifferent mechanisms and the sensitivity to differeparameters. The key information of 10 or more tcases can be summarized in one plot using burnindices atz/D = 15, where the greatest local extintion is observed.

In Section6.1, plots of burning indices atz/D =15 versus jet velocity are used to investigate the efof radiation and the pilot temperature. Then in Stion 6.2, the performance of all seven mechanismsinvestigated. In Section6.3, concentrating on flame Fthe measured and computed unconditional and cditional quantities are shown, and we investigatesensitivity of these calculations to the changes tomixing model constant and to the reaction rates. Tcalculations of NO are shown in Section6.4. Finally,the joint PDF calculations are related with the atoignition and OPPDIF calculations in Sections6.5and 6.6.

6.1. Effect of radiation and pilot temperature

It has been shown in previous studies[9,40] thatradiation and the pilot temperatureTp can signifi-cantly affect the calculations of flame F. For examp

-

for the calculations with ARM2 in[9], using a pi-lot temperature ofTp = 1880 K yielded a burningsolution; while decreasing the pilot temperature20 K (to Tp = 1860 K) resulted in an essentially etinguished solution.

In the current work, to understand better thefect of radiation and the pilot temperature, a systeatical study has been performed using the GRIGRI2.11, ARM1, and ARM2 mechanisms for flamD, E, and F. The results are summarized inFigs. 5and 6which show the burning indices (based onT

and H2) versus jet velocity atz/D = 15 for these fourmechanisms. A burning index equal to zero indicathat the calculation yields a globally extinguishedlution.

With the exception of GRI3.0 for flame D, foeach of the tested mechanisms and for each flathree different calculations are shown: first, adbatic calculations (i.e., with radiation omitted) wiTp = 1880 K; second, radiative calculations wTp = 1880 K; third, radiative calculations withTp =1860 K.

Fig. 5. Burning indices of temperature versus jet velocat z/D = 15. Circles, measurements[1,22]. Lines in thesuccessive plots are PDF calculations using the GRIGRI2.11, ARM1, and ARM2 mechanisms. Dash-dotted licalculations without radiation andTp = 1880 K; solid line,calculations with radiation andTp = 1880 K; dashed lineand cross, calculations with radiation andTp = 1860 K(mostly indistinguishable from the solid line).


s isutof

hef at

he1

urn--are

ula-

pi-ct,in-cu-soelycan

ss

2try,tly—

t

ces

d for

isfor

O,ing

ies,

sanc-

ieldheex-

edle,

For flames D and E, it is clear fromFigs. 5 and 6that the difference between the three calculationrelatively small: compared to the calculations withoradiation, including radiation results in a decreaseat most 4 and 10% in the BI ofT and H2, respectively;while for calculations with radiation, decreasing tpilot temperature by 20 K results in a decrease omost 2 and 5% in these BIs.

Turning now to flame F, it may be seen that for tcalculations with the GRI3.0, GRI2.11, and ARMmechanisms, including radiation decreases the bing index ofT and H2 by at most 5 and 17% compared to the adiabatic solution. These changessomewhat larger than those observed in the calctions of flames D and E.

With GRI3.0 and GRI2.11, decreasing thelot temperature by 20 K has no discernible effewhereas it causes the ARM1 calculation to extguish. This reconfirms the fact that, in these callations, flame F is very close to global extinction,it can display extreme sensitivities, and a relativsmall change in a physical or numerical parameterlead to extinction.

For the calculations with ARM2, the heat lofrom radiation (withTp = 1880 K) is sufficient tocause global extinction. Compared to ARM1, ARMcontains additional species related to NO chemisand these species depress the temperature slighapparently enough to cause extinction.

Fig. 6. Burning indices of H2 versus jet velocity az/D = 15. Symbols and lines, same asFig. 5.

6.2. Comparison of different mechanisms

The measured and calculated burning indibased on temperature and CO atz/D = 15 are plottedagainst the jet velocity inFigs. 7 and 8for all sevenmechanisms. The base case parameters are usethese calculations, i.e., EMST withCφ = 1.5, radia-tion is included, and the value of pilot temperatureset to 1880 K for flames D and E and to 1860 Kflame F.

One can see fromFigs. 3 and 4that burning in-dices obtained fromT , CO2, and H2O have similarbehavior, and the burning indices obtained from CH2, and OH have similar behavior. So, the burnindices based onT and CO, shown inFigs. 7 and 8,are characteristic of these major and minor specrespectively.

It is clear fromFig. 7that the S5G211 calculationyield significantly larger values of burning index ththe measurements, indicating too little local extintion. On the other hand, the Smooke calculations ysignificantly smaller values of burning index than tmeasurements, corresponding to too much localtinction.

Fig. 7. Burning indices versus jet velocity atz/D = 15.Solid circles, measurements[1,22]. Lines with open symbolsare PDF calculations. Solid line with plus, GRI2.11; dashline with diamond, skeletal; dotted line with right triangS5G211; dash-dotted line with star, Smooke.Cφ = 1.5, withradiation,Tp = 1880 K for flames D and E andTp = 1860 Kfor flame F.


ed1;

exheed toeldsra-

3.0a-mithl

owthe

alsoesItthethe

es-in-

m-ata.me-foran-ri-ntlythehet its

s.

asest

ionea-lityent

al-toain-

byasn

in)x-eso-n

ula-ro-

alu-

m aten-

t ofre-iteeat-outis

ndi-

Fig. 8. Burning indices versus jet velocity atz/D = 15.Solid circles, measurements[1,22]. Lines with open symbolsare PDF calculations. Solid line with plus, GRI2.11; dashline with triangle, GRI3.0; dotted line with square, ARMdash-dotted line with cross, ARM2.Cφ = 1.5, with radia-tion, Tp = 1880 K for flames D and E andTp = 1860 K forflame F.

It is also clear fromFig. 7 that in flames D and Ethe skeletal calculations overpredict the burning indof CO by almost a factor of 2. This is also true for tburning indices of H2 and OH which are shown in thSupplementary material. Nevertheless, comparethe experimental data, the skeletal mechanism yiquite accurate calculations of BI based on tempeture and CO for flame F.

It may be seen fromFig. 8 that both GRI mech-anisms yield burning indices forT and CO in goodagreement with the experimental data, with GRIbeing marginally superior. The two ARM mechnisms yields results almost indistinguishable froeach other. As already observed, for flame F (wradiation andTp = 1860 K) they both lead to globaextinction. For flames D and E they generally shgood agreement with the experimental data, butBI for CO in flame D is 17% below the data.

The performance of all seven mechanisms hasbeen investigated for adiabatic calculations of flamD, E, and F (i.e., with the neglect of radiation).is found that the omission of radiation changesagreement with the experimental data slightly, but

relative performance of different mechanisms issentially the same, no matter whether radiation iscluded or not.

In summary, the two GRI mechanisms yield coparably good agreement with the experimental dFor flames D and E the ARM mechanisms are sowhat less accurate and they yield extinctionflame F. The remaining mechanisms yield substtial errors (up to 100%) compared to the expemental data, with S5G211 and Smooke consisteunderpredicting and overpredicting, respectively,amount of local extinction. The large errors for tskeletal mechanism in flames D and E suggest thaaccurate calculations of BI in flame F are fortuitou

6.3. Calculations of flame F

In this section we focus on flame F which hthe highest jet velocity, and hence the strongturbulence–chemistry interactions. The informatcontained in the comparison of calculations and msurements of flame F is very useful to test the abiof different turbulent combustion models to represthese complex turbulence–chemistry interactions.

6.3.1. Sensitivity of flame FBecause flame F is close to global extinction, c

culations of it can display extreme sensitivitiesnumerical and model parameters, and to uncertties in the boundary conditions (primarilyTp). Thesensitivity to numerical parameters is exemplifiedthe sensitivity to the ISAT error tolerance which,shown in Fig. 1, is much stronger in flame F thain flame E. The results inFig. 5 show that differ-ent radiation models (i.e., adiabatic or optically thcause calculations with ARM1 to be burning or etinguished, and similarly different pilot temperatur(1880 and 1860 K) yield burning or extinguished slutions with ARM2. Useful conclusions can be drawfrom the comparison of measurements and calctions of flame F only if these sensitivities are apppriately accounted for.

Referring again toFig. 5, we have observed thatsmall decrease in the pilot temperature or the incsion of radiation can cause an abrupt transition froburning solution to an extinguished solution. But noalso that the burning solutions exhibit very little sesitivity to the pilot temperature and to the treatmenradiation. And this conclusion is confirmed by thesults (Figs. 9–11) later in this section. Hence, despthe uncertainties in the pilot temperature and the trment of radiation, conclusions can be drawn abthose models which yield a burning solution. Thisdone in the next two subsections, based on uncotional and conditional means, respectively.


and

ithd

di-s,

ra-cesallatethe

si-of

-ndm-ownandons

andare

a-

ndonleswn.0

iblyal,theer-

he

ndu-mall

Fig. 9. Computed and measured radial profiles of meanrms axial velocity in flame F. Circles, measurements[44];lines, PDF calculations using the EMST mixing model wCφ = 1.5. Solid line, GRI3.0 calculations with radiation anTp = 1860 K; dashed line, GRI3.0 calculations with raation andTp = 1880 K; dotted line, GRI3.0 calculationwithout radiation andTp = 1880 K; gray dash-dotted lineS5G211 calculations with radiation andTp = 1860 K. (Mostlines are indistinguishable at most locations.)

Because of the strong sensitivity to model pameters, it could be that a particular model produinaccurate calculations of flame F, and yet a smperturbation to the same model could yield accursolutions. Perturbations to the reaction rates and tomixing model constant are examined in Sections6.3.3and 6.3.4, respectively, and by doing so it is posble to draw firm conclusions about the inaccuracysome models.

Figs. 9–11show radial profiles (at different axial locations) of the unconditional Favre mean arms velocity, mixture fraction, and temperature. Copared to the experimental data, the calculations share of the base configuration using the GRI3.0S5G211 mechanism, and also GRI3.0 calculati

Fig. 10. Computed and measured radial profiles of meanrms mixture fraction in flame F. The symbols and linesthe same as inFig. 9.

with Tp increased by 20 K, both with and without rdiation.

The correct representation of the velocity amixture-fraction fields is essential in the calculatiof nonpremixed turbulent combustion. Radial profiof the Favre mean and rms axial velocities are shoin Fig. 9. As may be seen, the three different GRI3calculations are indistinguishable, and are negligdifferent from the S5G211 calculations. In generthe agreement among all of the calculations andexperimental data is quite good. The largest diffences are for the rms downstream (z/D = 60) wherethe calculated rms is typically 40% lower than tmeasurements.

Fig. 10shows the radial profiles of the mean arms mixture fraction obtained from the four calclations noted above. For the mean there are s


andthe

ns.r-anree-if-ich

on-ons

ndula-

ichntalr-

theibiten-try

Ft-

ex-are

nssee–

een

ha-seand

OHnd,msthered

di-the

re-talaretalted

thend

alOH,l and

fons

er-od

rueal-

theile

Fig. 11. Computed and measured radial profiles of meanrms temperature in flame F. The symbols and lines aresame as inFig. 9.

but discernible differences between the calculatioDownstream (z/D = 30) and close to the centeline, the calculations are typically 30% greater ththe measurements; but otherwise there is good agment. For the rms, there are again negligible dferences among the three GRI3.0 calculations whagree well with experimental data up toz/D = 45,but thereafter yield twice the measured values. In ctrast, large inaccuracies in the S5G211 calculatiare evident atz/D = 15 andz/D = 30, where thereis most local extinction.

Fig. 11 shows the unconditional Favre mean arms temperature obtained from the same four calctions as shown inFigs. 9 and 10. There is again littledifference among the three GRI3.0 calculations whgenerally show good agreement with the experimedata up toz/D = 45; but there are discrepancies fu

ther downstream, especially for the rms. For bothmean and the rms, the S5G211 calculations exhsignificant discrepancies compared to the experimtal data in the region of strong turbulence–chemisinteractions (7.5 z/D 30).

In summary, the results shown inFigs. 9–11con-firm that the burning GRI3.0 calculations of flameare insensitiveto the pilot temperature and the treament of radiation; and that, up toz/D = 45, theGRI3.0 calculations are accurate compared to theperimental data, whereas the S5G211 calculationssignificantly inaccurate.

In the reminder of the paper we focus on meaand rms’s conditional on mixture fraction. Thequantities are more revealing of the turbulencchemistry interactions and of the differences betwthe mechanisms.

6.3.2. Calculation of conditional quantitiesWith the use of base case configuration (i.e.,Tp =

1860 K, the inclusion of radiation, andCφ = 1.5),the GRI3.0, GRI2.11, skeletal, and S5G211 mecnisms yield burning solutions for flame F. For themechanisms, the conditional mean temperaturethe conditional mean mass fractions of CO andare examined in this subsection. On the other hathe ARM2, the ARM1, and the Smooke mechanisyield globally extinguished solutions for flame F withe base case configuration, and so are not considhere.

Fig. 12shows the measured and computed contional mean and rms temperature obtained usingfour mechanisms yielding burning solutions. Thesults obtained with the GRI3.0, GRI2.11, and skelemechanisms are very similar to each other andgenerally in good agreement with the experimendata, although the conditional rms is underpredicfor rich mixtures (ξ > 0.5) at z/D = 7.5 and 15. Onthe other hand, S5G211 substantially overpredictsconditional mean temperature—by up to 550 K—aunderpredicts the rms.

Similar observations apply to the conditionmeans and rms’s of the mass fractions of CO andwhich are shown inFigs. 13 and 14. Here, howeversome differences are evident between the skeletathe GRI mechanisms (e.g., the rms of CO atz/D = 30and the mean of OH atz/D = 45), with the results othe skeletal mechanism showing greater deviatifrom the experimental data.

In summary, for the results shown inFigs. 12–14, the GRI3.0 and GRI2.11 calculations are genally similar to each other and are in reasonably goagreement with the experimental data. This is tfor other species (except NO), and also for the cculations of flames D and E (which are shown inSupplementary material). On the other hand, wh


rms

s.a-ele-

ableH,tor-

ele-ula-hat-ldsch-ilot. As

dit isel

rmsms.

dub-tesctorou-y a

the

tor(as

K.arees.m-

hatalex-

Fig. 12. Measured and computed conditional mean andtemperature obtained in the flame F calculations (Cφ = 1.5,Tp = 1860 K, with radiation) using different mechanismMeasurements[22] are shown by solid circles and mechnisms used are: GRI3.0 (solid line), GRI2.11 (cross), sktal (dashed line), and S5G211 (dotted line).

the skeletal mechanism yields generally reasoncalculations of the conditional means of CO and Othese quantities are typically overpredicted by a facof 2 in flames D and E (seeFig. 7and the Supplementary material).

6.3.3. Sensitivity to reaction ratesWe have seen that the GRI3.0, GRI2.11, and sk

tal mechanisms yield reasonably accurate calctions of flame F in the base configuration, and tARM1 and ARM2 do so with perturbed pilot temperature or radiation treatment. But S5G211 yieinaccurate calculations, while with the Smooke meanism extinguished solutions are obtained for all ptemperatures and radiation treatments considereddiscussed in Section6.1, for mechanisms that yielinaccurate solutions (i.e., S5G211 and Smooke)important to determine if a slightly perturbed modcan yield accurate solutions.

Fig. 13. Measured and computed conditional mean andmass fraction of CO in flame F using different mechanisThe symbols and lines are the same as inFig. 12.

To this end, inFig. 15 we show results obtainewith the S5G211 and Smooke mechanisms with sstantial perturbations. Specifically, all reaction rain the S5G211 mechanism are decreased by a faof 10, while those in the Smooke mechanism are dbled and tripled. (In practice, changing all rates bfactorα is achieved by changing the time stept inthe reaction fractional time step toαt .)

It may be seen fromFig. 15 that the standardS5G211 mechanism substantially overpredictsconditional mean temperature atz/D = 15, in fact, byup to 550 K. Reducing the reaction rates by a facof 10 reduces the conditional mean temperatureexpected); but it is still overpredicted by up to 350

For the Smooke mechanism, burning solutionsobtained by doubling and tripling the reaction ratBut with them doubled, the conditional mean teperature is overpredicted by up to 350 K atz/D =15. It is natural to seek a smaller perturbation tyields a burning solution with a lower conditionmean temperature—in better agreement with the


rmss.

sedinghisnsi-

ac-vedthe

b-cu-of

l

m-wither-

theper-

lid),nismh re-

ease

inare

hn-

alin

11

hs by11tionstantthis

on.erof

fer-

ith-

g

Fig. 14. Measured and computed conditional mean andmass fraction of OH in flame F using different mechanismThe symbols and lines are the same as inFig. 12.

perimental data. But with the reaction rates increaby a factor of 1.9 we are unable to obtain a burnsolution. Note the strong sensitivity to decreasing tfactor from 2.0 to 1.9, compared to the modest setivity observed inFig. 15to increasing the factor from2 to 3.

It can clearly be concluded, therefore, thatcurate calculations of flame F cannot be achieby a small perturbation to either the S5G211 orSmooke mechanisms.

6.3.4. Sensitivity to the mixing model constantCφ

With the same motivation as in the previous susection, we examine here the sensitivity of the callations using the different mechanisms to the valuethe mixing model constantCφ . This is the only modeconstant directly affecting the compositions.

Fig. 16shows the conditional mean and rms teperature obtained using the S5G211 mechanismthe valuesCφ = 1.2, 1.4, 1.5, and 2.0. Several obsvations can be made. First, decreasingCφ results in

Fig. 15. The effect of reactions rates investigated usingmeasured and computed conditional mean and rms temature in flame F. Circles, measurements[1,22]; lines, jointPDF calculations (adiabatic,Cφ = 1.5, Tp = 1880 K) usingthe Smooke mechanism with doubled reaction rates (sotripled reaction rates (dashed), and the S5G211 mechawith the standard reaction rates (dash-dotted) and tentaction rates (dotted).

a decrease in the conditional means and an incrin the rms’s, as has previously been observed[8,16].Second, withCφ = 1.2 the results are generallygood agreement with the experimental data, andvery similar to those obtained with ARM2 (witCφ = 1.5). Third, the calculations become more sesitive to changes inCφ when they are closer to globextinction. Fourth, compared to the results shownFig. 15, starting from the same case (the S5G2mechanism and EMST withCφ = 1.5), a 20% changein the value ofCφ (from 1.5 to 1.2) yields a muclarger difference than decreasing the reaction ratea factor of 10. This indicates that (for the S5G2mechanism and the parameters used), the calculaare much more sensitive to the mixing model consthan to the chemical reaction rates. The reason foris not evident and is deserving of further investigati

Similar tests have been performed for all othmechanisms. The maxima (over mixture fraction)the conditional mean and rms temperatures for difent values ofCφ are shown inFig. 17. One may seethat the change of the maximum temperatures wthe change ofCφ has a similar trend for all mechanisms. First, decreasingCφ results in decreasin


eanha-of

114

nale tox-helueal

ndi-

onnalthe

nin-p-

retideistryrlyand

ofm-ssentula-

tionent

mer-ha-

ntls

ler-ionnd

dothise ofanc-

ismtion

Fig. 16. Effect of the mixing model constantCφ investi-gated using the measured and computed conditional mand rms temperatures in flame F using different mecnisms and EMST mixing model with different valuesCφ . Circles, measurements[1,22]; lines, joint PDF calcula-tions (adiabatic,Tp = 1880 K) using the ARM2 mechanismwith Cφ = 1.5 (dashed line with triangle), and the S5G2mechanism with the value ofCφ set to 1.2 (dashed), 1.(dash-dotted), 1.5 (dotted), and 2.0 (solid).

the conditional means and increasing the conditiorms’s. Second, the calculations are more sensitivthe change ofCφ when they are closer to global etinction. Third, for several of the mechanisms (tSmooke, ARM2, and skeletal mechanisms), the vaof Cφ yielding the measured value of the conditionmean also yields the measured value of the cotional rms.

The finding is, therefore, that, with a perturbatito Cφ , accurate calculations of the peak conditiotemperature in flame F can be obtained usingS5G211 mechanism (Cφ = 1.2) and with the Smookemechanism (Cφ = 2.0).

What is to made of this finding? For a givemixing model (here EMST), there should be a sgle value ofCφ : there is no physical basis for suposing that a different value ofCφ is appropriatefor use with different chemical mechanisms. Theare several reasons (see, e.g.,[47]) to suppose thathe GRI mechanisms—although not perfect—provan adequate representation of the C–H–O chemin these flames; certainly more so than the overeduced S5G211 mechanism, and the skeletal

Fig. 17. Sensitivity of joint PDF calculations to the valueCφ in flame F: maximum conditional mean and rms teperature atz/D = 15 againstCφ . Horizontal dashed lineare the experimental data and lines with symbols repredifferent mechanisms. All calculations are adiabatic calctions withTp = 1880 K.

Smooke mechanisms which have no representaof C2 and C3 species. Hence we take the presresults fromFig. 17 for GRI3.0, GRI2.11, ARM1,and ARM2 as reconfirmation thatCφ = 1.5 is theappropriate value for EMST. UsingCφ = 1.2 withS5G211 orCφ = 2.0 with the Smooke mechanisshould therefore be viewed as introducing anror to compensate for deficiencies in these mecnisms.

6.4. Calculations of NO

Accurate calculations of NO are very importafor the application of turbulent combustion modeto pollutant control.Fig. 18 shows the conditionamean and rms of the mass fraction of NO at diffent axial locations in flame E for base-configuratcalculations using the ARM2, GRI2.11, GRI3.0, aS5G211 mechanisms. (The other three mechanismnot represent NO chemistry.) We use flame E foraspect of the investigation so that the performancall four mechanisms which include NO chemistry cbe compared: for flame F, ARM2 yields global extintion (for the base case configuration).

One can easily see that the S5G211 mechanoverrepresents the conditional mean mass frac


rmsase

sings),

oI3.0eat

iss Dell

er-ble

truehatethe

er--

ef-anNO

s

omthethe

thetesttests

ix-e,c-i-

al

offig-DTs

esthe

ini-1,ach

hatRIarem-

iningr-inernalfor

for

Fig. 18. Measured and computed conditional mean andmass fraction of NO obtained in flame E using the bcase configuration. Solid circles are measurements[1,22]and others lines and symbols are PDF calculations udifferent mechanisms: GRI3.0 (solid line), GRI2.11 (crosARM2 (dashed line), and S5G211 (dotted line).

of NO, typically by a factor of 2 compared tthe measurements. The calculations using GRyield significantly higher levels of NO than thmeasurements—higher by a factor of up to 2.4z/D = 45. This substantial overprediction of NOalso observed in the current calculations of flameand F (shown in the Supplementary material), as was in previous calculations of flame D[17] and oflaminar nonpremixed flames[47].

The ARM2 and GRI2.11 calculations are genally very close to each other and have reasonaagreement with the measurements. This is alsofor the calculations of flames D and F (except tARM2 yields global extinction for flame F with thbase case configuration). This is consistent withlaminar flame studies performed by Barlow et al.[47]and the composition PDF calculations of flame D pformed by Raman et al.[17] and the joint PDF calculations of Tang et al.[9].

While the pilot temperature has a negligiblefect on the calculations of NO, including radiation cdecrease the peak value of the conditional mean

Fig. 19. Ignition delay times (IDTs) of different mechanismat different initial temperatures,T .

by up to 25%. The relative tendencies obtained frincluding or neglecting radiation are essentiallysame for flames D, E, and F (as may be seen inSupplementary material).

6.5. Autoignition test

In this and the next subsection, we comparerelative performance of the mechanisms in simplecases, and relate the observed behavior in theseto that in the calculations of flames D, E, and F.

Autoignition tests have been performed for a mture of the Barlow and Frank fuel (25% methan75% air) and air at the stoichiometric mixture fration (ξst = 0.351) for all seven mechanisms. The igntion delay times (IDTs) obtained using different inititemperatures are shown inFig. 19. The IDTs of theARM1 mechanism are almost identical to thosethe ARM2 mechanism and are not shown in theure. The S5G211 mechanism has much shorter Ithan the other mechanisms, e.g., 13 to 820 timshorter than those of GRI3.0. On the other hand,Smooke mechanism has the longest IDTs for alltial temperatures. The IDTs of the ARM2, GRI2.1and GRI3.0 mechanisms are generally close to eother, although the IDTs for GRI3.0 are somewlonger at low temperatures. Compared to the Gmechanisms, the IDTs of the skeletal mechanismshorter at low temperatures, but longer at high teperatures.

The relative behavior of different mechanismsthe joint PDF calculations (characterized by burnindices,Figs. 7 and 8, and conditional mean tempeature,Fig. 12) can be related to their IDTs (shownFig. 19). In general, the shorter the IDT, the largthe burning indices, and the higher the conditiomean temperature. This is consistently the casethe S5G211 mechanism (with the shortest IDTs),


Table 7Parameters for the laminar opposed-flow nonpremixed flame calculations with nominal strain ratea ≡ (Ufu +Uox)/D = 50 s−1

Mole fraction Temperature Velocity Pressure DistanceD

CH4 N2 O2 (K) (cm/s) (atm) (cm)

Fuel 0.25 0.5925 0.1575 300 50 (Ufu) 1 2Oxidizer 0.0 0.79 0.21 300 50 (Uox) 1

nd-ave

oreerne to

m-hile-esneis

tureofela-

t

ha-m-themsedop-

ispa-

thatelyra-ly

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the Smooke mechanism (with the largest IDTs), afor the GRI3.0, GRI2.11, ARM1, and ARM2 mechanisms, considered together as a group (which hintermediate IDTs).

The behavior of the skeletal mechanism is mcomplex and does not conform simply to the pattobserved above for the other mechanisms. Relativthe GRI mechanisms, we observe fromFig. 19 thatthe skeletal mechanism has a short IDT at low teperatures and a long IDT at high temperatures, wwe observe fromFig. 7 that it has higher burning indices for flames D and E, but lower burning indicfor flame F. To maintain the consistent pattern, owould have to contend that a low-temperature IDTrelevant to flames D and E, and a high-temperaIDT to flame F; whereas, if anything, the physicsthe problem suggests the opposite—given the rtively lower temperatures in flame F.

6.6. Opposed-flow nonpremixed laminar flame tes

In order to relate the performance of these mecnisms in turbulent combustion to that in laminar cobustion, calculations have been performed usingGRI2.11, GRI3.0, skeletal and Smooke mechanisfor a steady, laminar, axisymmetric nonpremixflame between two opposed jets of equal (butposite) velocity. The OPPDIF code[24] coupled inthe commercial software package CHEMKIN 3.7used for these calculations. The configuration andrameters used for these tests are listed inTable 7. Thecomposition of the fuel stream is the same asin the Barlow and Frank turbulent flames, nam25% methane and 75% air. While the configution is, therefore, partially premixed, as previousobserved[1,47], the combustion occurs in a nopremixed mode. The mixture-averaged formulaused for diffusion velocities. (Calculations using treduced mechanisms ARM1, ARM2, and S5G2cannot readily be performed using this version of OPDIF.)

The maxima of the temperature, and of the mfractions of CO, OH and NO obtained using diffeent values of nominal strain rate are shown inFig. 20.The right-most points for different mechanisms reresent the nominal extinction strain rates, which250, 350, 350, and 375 s−1 for the Smooke, GRI2.11GRI3.0, and skeletal mechanisms, respectively

strain rates of 25 s−1 greater than these values tflames are extinguished.

For the Smooke and GRI mechanisms, the ptern observed above is naturally extended: relativthe GRI mechanism, the Smooke mechanism hlonger IDT, a smaller extinction strain rate, and crespondingly lower burning indices and conditionmean temperatures in the turbulent flames.

The relative behaviors of the skeletal and Gmechanisms are the same in the laminar flamesturbulent flames in the following respects. In the lainar flames, the peak temperatures are comparableast for strain rates above 50 s−1), as are the BIsbased on temperature (seeFig. 7). On the other handthe skeletal mechanism yields significantly highlevels of CO and OH both in the laminar flam(Fig. 20) and in the turbulent flames, as revealed

Fig. 20. Peak values of temperature, and mass fractionCO, OH, and NO against the nominal strain ratea obtainedin laminar opposed-flow diffusion flame calculations.


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the BIs based on CO shown inFig. 7 and on OH(shown in the Supplementary material).

It has been shown in previous studies[47] that cal-culations of laminar opposed-flow partially premixmethane/air flames using GRI3.0, GRI2.11, ARMand ARM1 mechanisms yield similar results for mjor species, while there may be significant differenfor minor species. The current calculations are incord with these results. It is clearly shown inFig. 20that the GRI3.0 calculations yield about two timhigher levels of NO when compared with the GRI2.calculations. This is consistent with the joint PDF cculations shown inFig. 18.

7. Summary and conclusions

A comprehensive study has been conducted onperformance of seven different chemical mechanisused in joint PDF model calculations of the Balow and Frank[1] nonpremixed piloted jet flames DE, and F. The seven mechanisms (GRI3.0, GRI2ARM2, ARM1, S5G211, skeletal, and Smooke) ranfrom the 53-species GRI3.0 mechanism to a 5 sreduced mechanism (S5G211): details are givenTables 3 and 4. As in many previous studies[8,9,31–35], the PDF model is based on the joint PDof velocity, turbulence frequency, and compositionuses the simplified Langevin model[29] for velocity;the Jayesh–Pope model[30] for turbulence frequencyand the EMST mixing model[11] for the composi-tions, all with the standard model constants givenTable 2. The Supplementary material contains mamore results of these PDF calculations than canshown here.

The modeled joint PDF equation is solved uing the hybrid finite-volume/particle method implmented in the code HYB2D[31,33,35], and the chemistry is implemented using the ISAT algorithm[45].The results of numerical tests are reported (varythe grid size, number of particles, ISAT error toleance, and time-averaging period) to establish themerical accuracy of the calculations. For flamesclose to global extinction, the numerical errors in coditional means of temperature and major speciesgenerally no greater than 2%, and for minor specno greater than 5%. The large number of numericaaccurate PDF calculations reported here demonsthat this PDF/ISAT methodology can be effectiveapplied to turbulent flames using chemical mecnisms with of order 50 species.

The performance of the seven different chemimechanisms is examined through comparison ofPDF model calculations with the experimental daincluding unconditional and conditional means a

rms’s, and the burning index (BI). The principal coclusions are as follows.

(1) For temperature and species (excluding NO),two GRI mechanisms yield comparably goagreement with the experimental data forthree flames.

(2) For flames D and E, the two ARM mechanismyield results very similar to each other, wislightly worse agreement with the experimendata compared to GRI. For flame D, for exaple, the ARM calculations of BI based on CO a17% below the measurement (atz/D = 15).

(3) For flame F (and the base configuration) bARM mechanisms yield global extinction. However, with the relatively small perturbations ofincreasing in 20 K of the pilot temperature athe neglect of radiation, both ARM mechanismyield burning solutions in good agreement wthe experimental data.

(4) The ARM2 mechanism and the GRI2.11 meanism on which it is based provide reasonaaccurate calculations of NO (seeFig. 20). How-ever, consistent with previous observations[17,47], the GRI3.0 mechanism overpredicts Ntypically by a factor of 2.

(5) The other three mechanism (S5G211, sktal, and Smooke) display significant inaccucies. Fig. 7, for example, shows that S5G21grossly underestimates the level of local extintion; whereas the Smooke mechanism overpdicts local extinction in flames D and E anyields global extinction for flame F. The skelemechanism shows fortuitous agreement for flaF, but is quite inaccurate for flames D and E.

(6) Moderate perturbations to the Smooke aS5G211 mechanisms are incapable of yieldagreement with the experimental data. Evenall rates in the S5G211 mechanism are decreaby a factor of 10, local extinction is still grossoverpredicted (seeFig. 15).

(7) The predicted level of local extinction is sentive to the value specified for the EMST miing model constantCφ , with this level increasingwith decreasingCφ (seeFig. 17). The calcula-tions with the GRI and ARM mechanisms recofirm that Cφ = 1.5 is the appropriate value, afirst determined by Xu and Pope[8] and consis-tently used thereafter. WithCφ = 1.2 andCφ =2.0 the S5G211 and Smooke mechanisms,spectively, yield the correct level of local extintion in flame F. However, these changes toCφ

should be regarded as inappropriate additioncompensating errors, to compensate for deficcies in the mechanisms.


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(8) To an extent there is a consistent pattern amthe ignition delay times, laminar flame proerties, and the turbulent flame calculationsthe different mechanisms. Compared to the Gand ARM mechanisms, the Smooke mechanhas a long IDT, a small extinction strain raand it consistently overpredicts local extinctiin the turbulent flames. Conversely, S5G2has, comparatively, a very short IDT, andgrossly underpredicts local extinction. Comparto GRI2.11, in the laminar flames GRI3.0 ovepredicts NO (typically by a factor of 2), anthe skeletal mechanism significantly overprediCO and OH. These same discrepancies areobserved in the conditional means in the turblent flames.

It should be appreciated that the strength andliability of the conclusions drawn above dependstudying the mechanisms over the range of flamesfor perturbed conditions. A study of flame F alomight conclude that the skeletal mechanism is safactory, whereas the results for flames D and E cleshow this not to be the case. Or, if only the base cfiguration were studied it might be concluded thARM1 is unsatisfactory in that it yields global etinction for flame F. But as shown inFig. 5, with a20 K increase in the pilot temperature—well with texperimental uncertainty of 50 K—the ARM1 mecanism yields a burning solution, in good agreemwith the experimental data.

These calculations again demonstrate the vaof the piloted jet flames (using the Sydney burngeometry) in the study of turbulence–chemistry intactions. The experiments performed on these flahave been central to the advances made overlast decade in our abilities to model accuratturbulence–chemistry interactions in such flames.

Acknowledgments

This paper is dedicated to Bob Bilger on the ocsion of his seventieth birthday. The authors are grful to Profs. A.R. Masri and D.A. Caughey for valuable discussions and suggestions during the coof the research. This work is supported by Air ForOffice of Scientific Research under Grant F-49600-1-0171 and the Department of Energy under GDE-FG02-90ER. The computations were conducusing the resources of the Cornell Theory Cenwhich receives funding from Cornell University, NeYork State, federal agencies, foundations, and corate partners.

Supplementary material

Supplementary data for this article may be fouin the online version at doi:10.1016/j.combustflame2005.08.018.

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The influence of chemical mechanisms on PDF calculations of ... · The influence of chemical mechanisms on PDF calculations of nonpremixed piloted jet flames Renfeng Richard Cao∗,