Proceedings

Proceedings of the Combustion Institute 31 (2007) 1385–1392

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of the

CombustionInstitute

Stretch rate effects on displacement speed inturbulent premixed flame kernels in the thin

reaction zones regime

Nilanjan Chakraborty a, Markus Klein b,*, R.S. Cant c

a Engineering Department, University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UKb Technische Universitaet Darmstadt, Institut fuer Energie und Kraftwerkstechnik, Petersenstr. 30,

D-64287 Darmstadt, Germanyc Cambridge University Engineering Department, Trumpington Street, Cambridge CB2 1PZ, UK

Abstract

Direct numerical simulation (DNS) of three-dimensional turbulent premixed flame kernels is carried outin order to study correlations of displacement speed with stretch rate for a range of different flame kernelradii. A statistically planar back-to-back flame is also simulated as a special case of a flame kernel withinfinite radius of curvature. In all the cases the joint pdf of displacement speed with stretch rate showstwo distinct branches, whose relative strength is found to be dependent on the mean flame curvature. Apositive (negative) correlation is prevalent when the flame is curved in a convex (concave) sense towardsthe reactants. The observed non-linearity of the displacement speed-stretch rate correlation, qualitativelyconsistent with previous two-dimensional DNS with complex chemistry, is shown to exist purely due tofluid-dynamical interactions even in the absence of detailed chemistry. It is demonstrated that the com-bined contribution of reaction and normal diffusion is important in the response of flame propagationto stretch rate, and the effect is found to increase with decreasing mean kernel radius. The implicationsof the stretch rate dependence of displacement speed are discussed in detail in the context of the flame sur-face density (FSD) approach to turbulent combustion modelling.� 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Displacement speed; Stretch rate; Surface density function; Curvature; Tangential strain rate

1. Introduction

Premixed flames propagate in the direction oftheir local normal vector with a displacement

1540-7489/$ - see front matter � 2006 The Combustion Institdoi:10.1016/j.proci.2006.07.184

* Corresponding author. Fax: +49 0 6151 166555.E-mail addresses: N.Chakraborty@liverpool.ac.uk

(N. Chakraborty), kleinm@hrzpub.tu-darmstadt.de(M. Klein), rsc10@eng.cam.ac.uk (R.S. Cant).

speed Sd relative to an initially coincident materialsurface [1]. Detailed information about the dis-placement speed is required for modelling of tur-bulent premixed flames using both the flamesurface density (FSD) [2] and G-equation [3] for-mulations. For premixed flames a reaction pro-gress variable c may be defined which increasesmonotonically from zero in fresh reactants tounity in fully burned products and which maybe expressed in terms of the product mass fractionYP as

ute. Published by Elsevier Inc. All rights reserved.

1386 N. Chakraborty et al. / Proceedings of the Combustion Institute 31 (2007) 1385–1392

c ¼ ðY P � Y P0Þ=ðY P/ � Y P0Þ ð1Þwhere subscripts 0 and1 correspond to reactantsand products, respectively. Then the displacementspeed Sd of the isosurface at c = c* is given by

Sd ¼_wþr � ðqDrcÞ

qjrcj

����c¼c�

ð2Þ

where _w is the reaction rate, q is the density and Dis the mass diffusion coefficient. Flame stretch rateis defined as the fractional rate of change of flamesurface area A, given by

K ¼ ð1=AÞdA=dt ¼ aT þ Sdr � ~N ð3Þwhere aT is the tangential strain rate, r � ~N ¼ 2jm

where jm is the local mean flame curvature and~N ¼ �rc=jrcj is the local flame normal vector.In the present convention, flame curvature is posi-tive when the flame is convex to the reactants.Theoretical studies [4–6] suggest that displacementspeed is a linear function of stretch rate K in thelimit of small stretch rates [7]

S�d=SL ¼ 1� K‘=SL ð4ÞHere, S�d is the density-weighted displacementspeed given by S�d ¼ qSd=q0 where q0 is the un-burned gas density, SL is the unstrained laminarflame speed and ‘ is the Markstein length forstretch rate.

The response of displacement speed to strongervariations in stretch rate is important in flameletmodelling [1–3]. Peters [3] has demonstrated thatcurvature effects on displacement speed becomedominant in the thin reaction zones regime. It ispossible to decompose the displacement speed Sd

into three components, namely the reaction com-ponent Sr, normal diffusion component Sn andtangential diffusion component St [3,8]

Sr ¼_w

qjrcj ; Sn ¼~N � rðqD~N � rcÞ

qjrcj ; St ¼�2Djm

ð5ÞFrom Eqs. (3) and (5) it is evident that the contri-bution to the stretch rate K due to tangential dif-fusion is given by Str � ~N ¼ �4Dj2

m which maylead to a non-linear response of the displacementspeed to stretch rate.

Experimental measurements of Sd and stretchrate in a turbulent flame are often difficult to per-form. This has led to measurements of Sd in pla-nar strained flame configurations with smallvalues of strain rate. As a consequence, the effectsof curvature and unsteady strain rates are seldomtaken into account [7]. Previous studies usingdirect numerical simulation (DNS), as well asexperimental results, have indicated that the linearrelationship of Eq. (4) might continue to hold atlarge stretch rates. However, a DNS study intwo dimensions with complex chemistry [7] hasshown significant non-linearity.

Here, the stretch rate effects on Sd are studiedusing three-dimensional DNS of turbulent pre-mixed flames in the thin reaction zones regime. Abroad range of stretch rates is explored by examin-ing spherical flame kernels with different radii. Astatistically planar flame is also considered, as aspecial case of a flame kernel with infinite radius.Stretch rate effects on displacement speed are foundto be non-linear, which is qualitatively in agreementwith the findings of Chen and Im [7]. However, theobserved non-linear behaviour is explained in termsof the response of displacement speed to strain rateand curvature, and is shown to exist purely due tofluid-dynamical interactions between the flameand flow field even in absence of complex chemis-try. The objectives of the paper are as follows:

1. To understand the response of the displace-ment speed to local stretch rate.

2. To understand the effects of the global flamecurvature on this response.

3. To assess the implications of the results for tur-bulent premixed flame modelling, by consider-ing the curvature stretch contribution to FSDtransport in a configuration where the flamehas a finite global curvature.

2. Mathematical background and numerical imple-mentation

The standard conservation equations of mass,momentum and energy are solved in three dimen-sions for compressible reacting flows along with asingle equation for a reaction progress variable[12]. In all cases the global Lewis number is unityand transport properties such as viscosity (l),thermal conductivity (k) and density-weightedmass-diffusivity (qD) are considered to be inde-pendent of temperature. A single step Arrhenius-type reaction mechanism is employed.

Simulations are carried out using the finite-dif-ference DNS code SENGA. All spatial derivativesare evaluated using 10th order central differences[13]. Near the boundaries the spatial differencinggradually reduces to a 2nd order single-sidedscheme. Time advancement is carried out using alow storage third order Runge–Kutta scheme(A.A. Wray, Minimal storage time advancementschemes for spectral methods, unpublished report,NASA Ames Research Center, 1990.). The gridspacing is determined by the flame resolution. Inall cases about 10 points are kept within the ther-mal flame thickness given by: dth ¼ ðT ad � T 0Þ=MaxðjrT jÞ where T is the dimensional tempera-ture, T0 is the fresh gas temperature and Tad isthe adiabatic flame temperature. The velocity isinitialised using an incompressible homogeneousisotropic turbulent field [14]. The thermochemicalfield is initialised using a precomputed laminar

N. Chakraborty et al. / Proceedings of the Combustion Institute 31 (2007) 1385–1392 1387

flame solution. In order to understand the effect ofthe initial flame kernel radius, spherical flames ofdifferent sizes are simulated. In all cases theamount of energy deposited within the laminarflame was sufficient compared with the criticalenergy to ensure self-sustained propagation [15].

All of the kernel simulations were carried out ina cubic domain with non-reflecting boundaries onall sides. A statistically planar back-to-back flamewas also simulated as a special case of a sphericalkernel of infinite radius. Here the transverseboundaries were assumed to be periodic. Boundaryconditions were specified using the Navier–Stokescharacteristic boundary conditions (NSCBC) for-mulation [16]. In order to avoid boundary effectson the flame-turbulence interaction, each side ofthe computational domain was taken to beL � 24dth. The unstrained laminar flame speed SL

and a length equal to 10dth have been taken asthe reference scales for velocity and length.

As with many previous DNS studies [8,17–19],all simulations were carried out for a simulationtime tsim = Max(sc,sf) where sc = dth/SL is thechemical timescale and sf is the initial eddy turn-over time given in terms of the integral length scalel and turbulent velocity fluctuation u 0 as sf = l/u 0.For the purposes of comparison, all data presentedin this paper was taken at a time equal to 3sf whichcorresponds to one chemical timescale sc.

3. Results and discussion

The numerical parameters of the DNS data-base are presented in Table 1. For all of the sim-

Table 1Parameters of the DNS database

Case (grid) r0/dth Cases A–F

A (2303) 1 u 0/SL = 7.5B (2303) 2.91 l/dth = 2.4C (2303) 2.49 Ret = q0u0l/l0 = 47D (2303) 2.10 Ma ¼ SL=

ffiffiffiffiffiffiffiffiffiffifficRT 0p

¼ 0:014E (2303) 1.74 c = Cp/Cv = 1.4F (2303) 1.42 R = Cp � Cv

The initial kernel radius r0 is defined as the radius of theregion in which c = 1.0.

Fig. 1. Contours of reaction progress variable in the x–z(r0/dth = 2.49); (c) case F (r0/dth = 1.42).

ulations, standard values are chosen forZel’dovich number b = 6, heat release parameters = (Tad � T0)/T0 = 3, Prandtl number Pr = 0.7and ratio of specific heats c = Cp/Cv = 1.4. Forthe velocity scale and length scale ratios (u 0/SL

and l/dth) listed, the combustion corresponds tothe thin reaction zones regime [3] where turbulenteddies may enter into the preheat zone of theflame. As a result, the preheat zone becomes dis-torted whereas the reaction zone retains its origi-nal structure. This can be seen from thecontours of reaction progress variable shown inFig. 1 for cases A, C and F. Note that case A(Fig. 1a) corresponds to a statistically planarflame, taken as a kernel of infinite radius. It is evi-dent in all three cases that the contours represent-ing the preheat zone (c < 0.5) are not parallel toeach other, in contrast to the contours in the reac-tion zone (0.6 < c < 0.9) which remain parallel.For all cases, maximum heat release occurs closeto the c = 0.8 isosurface, which is taken to be rep-resentative of the flame surface. The mean positivecurvature of the flame kernels significantly affectsthe propagation behaviour of the flame kernel inresponse to stretch rate, as discussed below interms of displacement speed statistics. Results cor-responding to the cases A, C and F will be pre-sented in detail, and similar qualitative trendsare observed for the other DNS cases listed inTable 2.

The probability density functions (pdfs) of dis-placement speed for cases A, C and F are present-ed in Fig. 2a. In all cases there is a finiteprobability of finding a negative displacementspeed, and this probability decreases with increas-ing flame radius (i.e., going from case F to C toA). It is also apparent from Fig. 2a that the meandisplacement speed is positive in all three caseswith the lowest mean value corresponding to thesmallest kernel (case F). This may be understoodwith reference to the pdfs of the combined reac-tion and normal diffusion components of displace-ment speed (Sr + Sn) presented in Fig. 2b, and ofthe tangential diffusion component of displace-ment speed St presented in Fig. 2c. The pdfs of(Sr + Sn) are similar for all cases. The mean valueof (Sr + Sn) is positive and is of the same order as

plane at y = 0.5. (a) Case A (r0/dth = �); (b) case C

Table 2Correlation coefficients obtained for all DNS cases from the joint pdfs of: (column 2) aT and jm, (column 3) |$c| and aT,(column 4) (Sr + Sn) and jm, (column 5) variation of effective diffusivity Deff with kernel radius

1 2 3 4 5Case Corr. aT � jm Corr. |$c| � aT Corr. (Sr + Sn) � jm Deff/D

A �0.63 0.47 0.095 0.977B �0.66 0.72 0.45 0.682C �0.66 0.68 0.52 0.632D �0.65 0.82 0.62 0.547E �0.63 0.81 0.70 0.274F �0.55 0.64 0.76 �0.073

Fig. 2. Pdfs of displacement speed and its components. (a) Sd; (b) (Sr + Sn); (c) St. Displacement speed and itscomponents are normalised with respect to unstrained laminar flame speed SL. Note that all pdfs are presented for thec = 0.8 isosurface, close to the location of maximum heat release rate.

1388 N. Chakraborty et al. / Proceedings of the Combustion Institute 31 (2007) 1385–1392

the density-weighted displacement speed q0SL/qfor the present case. The pdfs of Sd and (Sr + Sn)are consistent with previous DNS studies usingboth simplified and detailed chemistry [8–10].

Comparing Figs. 2a and b it is evident that theprobability of finding a negative value of (Sr + Sn)is very much smaller than that of finding a nega-tive value of Sd. Obviously the tangential diffusioncomponent St is primarily responsible for this.According to Eq. (5), St is proportional to thenegative of mean flame curvature. Therefore themean value of St becomes increasingly negativewith decreasing kernel radius, and this can be seenclearly in Fig. 2c. These results indicate that forvery small kernel radius the mean displacementspeed may become negative because of the largenegative contribution of St, and this may lead to

Fig. 3. Joint pdfs of displacement speed and cu

shrinkage of the flame kernel and eventually toextinction.

The joint pdfs of displacement speed andcurvature for cases A, C and F are presented inFigs. 3a–c where a somewhat non-linear negativecorrelation is apparent, consistent with previousDNS [7,8,10,20]. It can be seen from Eq. (5) thatSt is negatively and linearly correlated with curva-ture provided that the diffusivity is constant on therelevant isosurface, and this holds true in the pres-ent case of unity Lewis number. Instead, theobserved non-linearity originates from the corre-lation between (Sr + Sn) and curvature. The jointpdfs of (Sr + Sn) and curvature for cases A, C andF are presented in Figs. 4a–c, and again it isevident that the correlation is non-linear. Forthe statistically planar flame (case A, Fig. 4a) it

rvature: (a) case A; (b) case C; (c) case F.

Fig. 4. Joint pdfs of (Sr + Sn) and curvature: (a) case A; (b) case C; (c) case F.

N. Chakraborty et al. / Proceedings of the Combustion Institute 31 (2007) 1385–1392 1389

can be seen that there are two distinct branches inthe joint pdf, and that the positively correlatingbranch becomes stronger with decreasing kernelradius (i.e., moving from case A to C to F, Figs.4a–c).

This can be explained in terms of the curvatureresponse of the progress variable gradient magni-tude r = |$c|, referred to as the surface densityfunction (SDF) [22], which appears in the denom-inator of the expressions for Sr and Sn (see Eq.(5)). In all of the present cases it has been foundthat tangential strain rate and curvature are nega-tively correlated, consistent with previous DNS[8,19,20] and experiment [21] as evident fromcolumn 2 of Table 2. The SDF is found to be pos-itively correlated with tangential strain rate withcorrelation coefficients as presented in column 3of Table 2. This induces flame-thinning effects atpositive or small negative curvatures [10,20], andthe net result is a negative correlation of SDF withcurvature. Conversely, for high negative curva-tures there may be a local thickening of the flamedue to the focussing of heat. This secondary effectproduces a positive correlation between SDF andcurvature for high negative curvatures. For flamekernels, the probability of finding high negativecurvatures decreases with decreasing kernel radi-us, and the positive correlation between SDFand curvature gradually disappears [10,20].

For unity Lewis number flames, the chemicalreaction rate and the density remain uniform ona given c isosurface, and so a decrease in SDFleads directly to an increase in Sr (see Eq. (5)).In the heat-releasing zone, the normal diffusion

Fig. 5. Joint pdfs of displacement speed and stre

rate ~N � rðqD~N � rcÞ is predominantly negative[8,10]. As a result, an increase in SDF will tendto make Sn more negative (see Eq. (5)). Thus,the curvature response of SDF acts to determinethe curvature response of (Sr + Sn). A weak corre-lation between SDF and curvature in the planarflame leads to a weak correlation between(Sr + Sn) and curvature (see Fig. 4a), and anincreasingly negative correlation between SDFand curvature with decreasing kernel radius leadsto an increasingly positive correlation between(Sr + Sn) and curvature as evident from Figs. 4band c, and column 4 of Table 2. This result hassignificant implications for curvature stretch mod-elling in the thin reaction zones regime.

The combined effects on the displacementspeed of stretch rate due to both tangential strainrate and mean curvature are presented in Fig. 5,which shows the joint pdfs of displacement speedand stretch rate for cases A, C and F. There areclearly two branches, consistent with previousDNS in two dimensions with detailed chemistry[7], but in contrast to theoretical studies [4–6]which explain only the branch showing a negativecorrelation. The negative correlations betweendisplacement speed and curvature, and alsobetween tangential strain rate and mean curva-ture, indicate that for highly positively curvedlocations where the displacement speed is nega-tive, displacement speed decreases with stretchrate. The decreasing trend of displacement speedwith decreasing stretch rate in regions of highpositive curvature is responsible for the positivelycorrelating branch observed for the planar flame

tch rate: (a) case A; (b) case C; (c) case F.

Fig. 6. Pdfs of the different contributions to stretch rate: (a) tangential strain rate (aT); (b) curvature stretch rateðSdr � ~NÞ; (c) stretch rate (K).

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case (Fig. 5a). Since the probability of finding neg-ative displacement speed increases with decreasingkernel radius (see Fig. 2), the positively correlat-ing branch becomes stronger with decreasing ker-nel radius, as observed from Figs. 5b and c. Thiseffect is further aided by the increasingly positivecorrelation between (Sr + Sn) and mean curvaturewith decreasing kernel radius.

The pdfs of the different contributions to thestretch rate are presented in Fig. 6. The pdfs oftangential strain rate (Fig. 6a) indicates that inall three cases the mean tangential strain rate ispositive which is consistent with several previousstudies [7,10,20,23,24]. The pdfs of the curvaturecontribution Sdr � ~N (Fig. 6b) in all three casesshow a long negative tail which is a result of thenegative correlation between displacement speedand curvature, and the probability of finding neg-ative values of Sdr � ~N decreases with decreasingkernel radius. The increasingly positive correla-tion between (Sr + Sn) and curvature leads to adecreasing probability of finding negative valuesof the total stretch rate K. Consistent with Figs.6a and b, the pdfs of stretch rate (see Fig. 6c) alsoshow long tails and decreasing probability of find-ing negative stretch rate with decreasing kernelradius, consistent with previous DNS [7,20].

The change in the strength of the correlationbetween (Sr + Sn) and curvature with changingkernel radius (see Figs. 4a–c) has significant impli-cations for FSD modelling. The unclosed FSDtransport equation is given by [2,25]

oRotþ o

oxið~uiRÞ þ

o

oxi½huiiS � ~ui�R

¼ ðdij � N iN jÞoui

oxj

� �S

R� o

oxihSdN iiSR

þ hSdr � ~NiSR ð6Þ

where R is the generalised FSD given by R = Æ|$c|æand ÆQæS is the surface average of a quantity Qgiven by ÆQæS = ÆQ|$c|æ/Æ|$c|æ. In Eq. (6), the finalterm hSdr � ~NiSR is an important source termwhich arises from the generation or destructionof flame area due to the curvature contribution

to stretch rate [2,11,20,25,26]. This term may beexpanded using the decomposition of Eq. (5)

hSdr � ~NiSR ¼ 2hðSr þ SnÞjmiSR� 4D j2m

� �SR

ð7Þ

Hawkes and Chen [11] modelled the curvaturestretch contribution in the thin reaction zones re-gime using the expression

hSdr � ~NiSR ¼ �4Deffhj2miSR ð8Þ

where Deff is an effective diffusivity. An increasingcorrelation between (Sr + Sn) and jm withdecreasing kernel radius implies that, for the sameLewis number and heat release rate, the effectivediffusivity Deff must be dependent on mean kernelradius jm or equivalently on the flame geometry.The terms T 1 ¼ hSdr � ~NiSR ¼ 2 < Sdjmr >and T 2 ¼ �4Dhj2

miSR ¼ �4D < jmjmr > (seeEq. (7)) are plotted in Fig. 7a for cases A, C andF, and it is evident that there is an increasinglylarge difference between these terms with decreas-ing mean kernel radius (i.e., going from case A tocase F). For the statistically planar flame (case A),T1 remains close to T2 throughout the flamebrush indicating that the effective diffusivity is al-most equal to the molecular diffusivity (Deff � D),which is consistent with previous findings [11,20].However, if the model of Eq. (8) is to be applied,then it is evident that there is a decreasing trend ofDeff with decreasing kernel radius. For very smallkernel radius (case F, Fig. 7a) hSdr � ~NiSR mayeven become positive in some parts of the flamebrush, which leads to negative values ofDeff (seecolumn 5 of Table 2). Thus, the model given byEq. (8) predicts hSdr � ~NiSR in a satisfactory man-ner for planar flames with Deff � D, but showsundesirable characteristics when the flame hasnon-zero mean curvature. In order to extend thevalidity of the model, a modification is proposedhere

hSdr � ~NiSR� 2hSdiShjmiSR

¼ �4Dþeff j2m

� �S� hjmiShjmiS

h iR ð9Þ

Fig. 7. (a) Variation of the terms T1 and T2 across the flame brush for cases A, C and F; (b) variation of the terms T3 andT4 across the flame brush for cases A, C and F.

N. Chakraborty et al. / Proceedings of the Combustion Institute 31 (2007) 1385–1392 1391

where Dþeff is a modified effective diffusivity.The terms T 3 ¼ hSdr � ~NiSR� 2hSdiShjmiSR andT 4 ¼ ð�4D½hj2

miS � hjmiShjmiS�RÞ are plotted inFig. 7b, where it is evident that the magnitudeof T3 decreases compared to T4 with decreasingkernel radius, as a result of the positive correla-tion between (Sr + Sn) and jm. This indicates thatDþeff also decreases compared to D with decreasingkernel radius. However, the overall performanceof the model given by Eq. (9) is found to be signif-icantly better than the model given by Eq. (8) pro-vided that ÆSdæS, ÆjmæS and hj2

miS are properlymodelled. A model which explicitly accounts forthe curvature dependence of (Sr + Sn) will certain-ly be preferable. However, building such a modelideally requires three-dimensional DNS data withdetailed chemistry and therefore is beyond thescope of present work.

Eq. (4) can be rewritten after Chen and Im [7]as

‘ ¼ SLð1� qSd=q0SLÞaT þ Sdr � ~N

ð10Þ

It is evident from Figs. 3a–c and 6b that high mag-nitudes of Sd are associated with high magnitudesof curvature. Under these conditions, if the mag-nitude of Sdr � ~N exceeds the magnitude of aT

then the Markstein length ‘ varies as [7]

‘ � �1=ðr � ~NÞ ¼ �1=ð2jmÞ ð11ÞThus for locations of high curvature the Mark-stein length depends on the sign of the local curva-ture, and it is possible for the Markstein length tobecome negative, as observed from Fig. 5. Thisresult is also in agreement with previous two-dimensional DNS using detailed chemistry [7] inwhich the global Lewis number is close to unity,but where some local effects of differential diffu-sion are observed. In the present study, the sameglobal behaviour arises purely from hydrodynamic

and kinematic effects in the absence of detailedchemistry and differential diffusion.

4. Concluding remarks

The effects of stretch rate on flame propagationhave been studied for premixed flame kernelsusing three-dimensional DNS with simple chemis-try. The effects of mean curvature have been dem-onstrated by simulating flame kernels of differentradius, and a statistically planar back-to-backflame has been included as a special case of aflame kernel with infinite radius. In all cases, thejoint pdfs of stretch rate and displacement speedshow two distinct branches. The branch display-ing a negative correlation is in agreement withprevious theoretical studies [4–6], while the otherbranch displaying a positive correlation has beenobserved in a previous 2D DNS study withdetailed chemistry [7]. In the present work, thesame global behaviour has been reproduced withsimplified chemistry. It has been found that thepositively correlating branch becomes strongerwith decreasing kernel radius. This is a combinedeffect of the increasing probability of finding posi-tive curvature at small kernel radius and anincreasingly positive correlation between (Sr + Sn)and curvature. Chen and Im [7] demonstrated thatthe concept of Markstein length is valid over alarger range of stretch rates than originally pro-posed. Here, this conclusion has been found tohold good for planar flames but to fail in the pres-ence of mean flame curvature since the stretch rateresponse of displacement speed is no longer linear.The curvature stretch model of Hawkes and Chen[11] has also been found to require some modifica-tion to the effective diffusivity Deff in order toaccount for the mean flame geometry. Thereremains a need to model the curvature response

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of the combined reaction and normal diffusioncomponents of displacement speed (Sr + Sn) inorder to account properly for curvature stretcheffects. This will be addressed in future, ideallyusing three-dimensional DNS with detailed chem-istry and transport.

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