Proceedings

Proceedings of the Combustion Institute 31 (2007) 1377–1384

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of the

CombustionInstitute

Low-dimensional manifolds in direct numericalsimulations of premixed turbulent flames

J.A. van Oijen *, R.J.M. Bastiaans, L.P.H. de Goey

Mechanical Engineering, Technische Universiteit Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Abstract

Direct numerical simulation (DNS) is a very powerful tool to evaluate the validity of new models andtheories for turbulent combustion, but the application of detailed chemistry is limited. In this paper, adimension-reduction technique called the flamelet-generated manifold (FGM) method is considered. In thismethod a manifold is created by solving a set of one-dimensional flamelet equations. The use of low-dimen-sional FGM’s in DNS of premixed turbulent flames in the thin reaction zones regime is investigated. Athree-dimensional (3D) DNS is performed of a spherically expanding, premixed, turbulent, methane–airflame. 1D and 2D FGM’s are created and used in simulations of flamelets which are subjected to stretchand curvature effects derived from the 3D DNS results. The results are compared with results from flameletsimulations with detailed chemistry. The results show that deviations from the 1D FGM due to stretch andcurvature effects are significant, but they appear to be embedded in a 2D manifold. This 2D manifold cor-responds well with 2D FGM’s that are created in different ways, but it shows large differences with a 2Dmanifold based on chemical kinetics alone. This indicates that an attracting low-dimensional manifoldexists which is not solely determined by chemical kinetics. As a consequence, the results of the flamelet sim-ulations using 2D FGM’s are more accurate than when a 1D FGM is applied: the mean error in the burn-ing velocity is almost an order of magnitude smaller.� 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Manifolds; Flamelets; Turbulent premixed flames; Direct numerical simulation

1. Introduction

Direct numerical simulation (DNS) is a verypowerful tool to test the validity of new modelsand theories for premixed turbulent combustion(see e.g., [1–3]). The disadvantage of using three-dimensional (3D) DNS is that the application ofdetailed chemical kinetic mechanisms is still limit-ed. It is well recognized that methodologies areneeded, which radically decrease the computation-

1540-7489/$ - see front matter � 2006 The Combustion Institdoi:10.1016/j.proci.2006.07.076

* Corresponding author. Fax: +31 40 2473445.E-mail address: j.a.v.oijen@tue.nl (J.A. van Oijen).

al burden imposed by the use of detailed chemicalkinetics. Very successful are dimension-reductiontechniques, which are reviewed in [4]. The mostcommonly used dimension-reduction methodsare based on the quasi steady-state assumption(QSSA). In this methodology, the n chemical spe-cies are divided in ns ‘‘slow’’ species (or controllingvariables) and nf = n � ns ‘‘fast’’ species. For eachfast species a QSSA is invoked and the differentialequation describing its evolution is replaced by analgebraic one. In the n-dimensional species orcomposition space, these nf algebraic equationsdefine an ns-dimensional manifold. Since all com-positions in a reactive system are assumed to lie

ute. Published by Elsevier Inc. All rights reserved.

1378 J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384

on this manifold, the system can be described byns controlling variables and a dimension reductionis achieved. A well-known method is the intrinsiclow-dimensional manifold (ILDM) method [5],which uses an eigenvalue analysis of the jacobianof the chemical reactive system to identify fast andslow species.

Recently, manifold methods have been intro-duced that are not only based on reaction kinetics.These methods, which are called the flamelet-gen-erated manifold (FGM) method [6] and flame pro-longation of ILDM (FPI) [7], combine the ideas ofthe flamelet and manifold approach. In thesemethods, a manifold is formed by the composi-tions found in a 1D premixed flame. This hasthe advantage that the main effects of convectiveand diffusive transport are taken into account inthe manifold as well. FGM with only a single con-trolling variable (apart from additional control-ling variables to take variations in enthalpy andstoichiometry into account) has proven to be suc-cessful for partially premixed laminar flames[8–10]. In [11], FGM has been applied in DNSof turbulent flames in the corrugated flameletregime [12].

In this paper, we study the use of FGM inDNS of premixed turbulent flames in the thinreaction zones regime. In such highly turbulentflames, deviations from 1D flamelet behaviordue to flame stretch and curvature are impor-tant and a 1D manifold may not suffice. Exten-sion of FGM to more dimensions is thereforeinvestigated and the most important questionsare: does a low-dimensional manifold exist forthese premixed turbulent flames? And if so,can it be represented by a low-dimensionalFGM? To answer these questions, we firstintroduce a flamelet description for premixedflames, which is the basis for the FGMmethod. In Section 3, it is explained how mul-ti-dimensional FGM’s are created. The flameletdescription is also used to investigate flamestretch and curvature effects in 3D DNS ofspherically expanding turbulent methane–airflames. A similar analysis has been carried outin [11], but then for turbulent flames in thecorrugated flamelet regime.

The coupling of flame stretch and curvaturewith non-unity Lewis number transport processesis an important issue in premixed turbulent flames(see e.g., [13]). In previous publications [8,9] wehave shown that the combined effect of flamestretch and preferential diffusion can be modeledwith FGM. In this paper, however, a unity Lewisnumber approach is adopted to simplify the anal-ysis. Furthermore, it is expected that preferentialdiffusion effects are negligible in the stoichiometricmethane–air flames studied here, because theLewis number of methane is close to unity. Thishas been demonstrated recently by de Swartet al. [14].

The 3D DNS is performed by using FGM,because the use of detailed kinetics is currentlytoo costly from a computational point of view.As a result, a direct comparison between FGMand detailed chemistry in DNS is not possiblein this paper. Nevertheless, to compare theFGM method with detailed kinetics, 1D flam-elets subjected to stretch and curvature fieldsderived from the 3D DNS results are simulatedwith detailed kinetics. The compositions inthese detailed flamelet simulations are analyzedin composition space and they are comparedwith 2D ILDM and FGM’s. Furthermore, 1Dand 2D FGM’s are used to compute the sameflamelets and results for the mass burning rateare compared with results using detailedchemistry.

2. Flamelet description

The flamelet description for premixed flamesintroduced by De Goey and Ten Thije Boonk-kamp [15] has been derived in a systematic wayby decomposing the system of combustion equa-tions in three parts: (1) a flow and mixing partwithout chemistry, (2) a kinematic equation forthe flame motion, including internal flame dynam-ics and (3) a flamelet part describing the innerstructure and propagation speed of the flamestructure. The flame, including internal structure,is described in terms of iso-surfaces of a progressvariable Y, which can be any linear combinationof species mass fractions. The motion of eachiso-surface of Y is described by the kinematicequation [16]

oY

otþ~vf � rY ¼

oY

otþ~v � rY � sLjrYj ¼ 0; ð1Þ

where~vf ¼~vþ sL~n is the local velocity of a flamesurface being the sum of the fluid velocity ~v andthe local burning velocity sL~n and where the unitnormal vector~n on the flame surface, directed to-wards the unburnt mixture, can be written as~n ¼ �rY=jrYj. Note that Eq. (1) is introducedfor each iso-surface in the complete ‘flame’ regionwhere 0 < Y < 1. As a result ~vf and sL are fieldvariables, which vary throughout the flame re-gion. In some other publications (e.g., [2]) sL is re-ferred to as displacement speed.

The stretch field K inside the flame zone isdefined as the relative rate of change of the massM in a small part of the flame, enclosed by asmall, ‘flame’ volume V, moving with velocity~vf :

K ¼ 1

MdMdt

with M ¼Z

V ðtÞq dV : ð2Þ

From this definition of K and the kinematicEq. (1) the following set of flamelet equationsfor the conservation equations of mass and Y

J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384 1379

can be derived rigorously in a orthogonal flame-adapted coordinate system [15]:

o

osðrmÞ ¼ �rqK; ð3Þ

o

osrmY � rqDY

oY

os

� �¼ r _xY � rqKY; ð4Þ

where s is the arc length through a flamelet, locallyperpendicular to the iso-planes of Y and r the sur-face area through which the transport processes inthe flame-adapted system take place. Note thatthe local curvature j of the flame surfaces is relat-ed to the surface area r by

j ¼ r �~n ¼ � 1

roros: ð5Þ

In Eq. (3) the mass burning rate m = qsL is intro-duced with q the density of the gas mixture. Di

and _xi are the diffusion coefficient and chemicalsource term of species i, respectively. The diffusivefluxes are assumed to be proportional to the gra-dient of the species mass fraction. The equationsfor the other species mass fractions Yi have a sim-ilar form as Eq. (4)

1

ro

osrmY i � rqDi

oY i

os

� �� _xi

¼ �qKY i þ Qi; ð6Þ

but with an extra term Qi defined as

Qi ¼dY i

dsþr � ðqDirkY iÞ; ð7Þ

with $i the gradient operator in directionparallel to the flame surface only. The first termon the right-hand side accounts for unsteadyeffects in the flame-adapted reference frame: o

os ¼oot þ ~vf � r. The second term accounts for diffusivetransport processes due to the fact that iso-planesof different species are not parallel. These extraterms are small and neglected inside the laminarflamelet regimes [15]. The set of flamelet Eqs.(3), (4) and (6) together with a similar equationfor the enthalpy, describes the internal flamestructure, dynamics and the eigenvalue for themass burning rate m(s) for a flamelet with aparticular Qi(s), stretch field K(s), and curvaturefield j(s).

3. Flamelet-generated manifolds

The flamelet equations derived in the previoussection are used to generate FGM’s. If we assumethat all perturbations from local 1D flat flamebehavior can be neglected, then we can useK = 0, j = 0 (or equally r = 1), and Qi = 0. Theset of equations then describes an unstretched,1D, flat flamelet: the most elementary premixed

flame structure. Its solution Yi(s) can be consid-ered as a 1D manifold in composition space. This1D FGM can be parameterized by a single con-trolling variable Y1. In [6] it is shown that the1D FGM corresponds to the 1D ILDM at hightemperatures, where chemistry is dominant. Atmoderate temperatures, molecular transport isimportant as well and the FGM is closer to com-positions found in a premixed flame.

In turbulent flames large deviations from the1D manifold may arise due to flame stretch andcurvature effects. To take this into account thedimension of the manifold can be increased. Alarger degree of freedom makes a better descrip-tion of the composition possible and therefore itincreases the accuracy of the model. A multi-dimensional FGM is generated by computing aseries of flamelets, which form together a multi-dimensional surface in composition space. Thereare two ways to compute the different flamelets.First, the boundary condition at the unburnt side(s fi �1) can be changed. This method has beenintroduced by Pope and Maas [17] to generatemulti-dimensional manifolds and it was used in[6] to create a 2D FGM. In this paper, we use a2D FGM generated by converting a part of theinitial CH4 and O2 into H2 and CO, while keepingthe element mass fractions and the enthalpy of themixture constant.

Second, a multi-dimensional FGM can be gen-erated by including terms in the flamelet equa-tions, which have been neglected in the 1DFGM approach. In this way more physics is takeninto account, which results in a more accuratedescription of the composition. Effects of flamestretch, for instance, can be included by modelingit with a constant stretch rate K = const in theflamelet equations. By applying a range of stretchrates a 2D manifold is created. Differently, flamecurvature can be taken into account by assumingconstant curvature j = const. In this paper, thesetwo obvious choices are considered, but manyothers could be thought of as well. In Section 6,these two 2D FGM’s and the one created bychanging the inlet condition will be compared.It’s worth mentioning, that all three 2D manifoldsin this paper are parameterized in the same way,i.e., by two controlling variables Y1 and Y2, andnot by using the stretch rate or curvature.

4. Direct numerical simulation with FGM

Freely expanding flames are modeled in a tur-bulent flow field by using DNS. The fully com-pressible combustion equations are solved in acubic 3D domain with a length of 5.0 mm and127 grid points uniformly distributed in eachdirection. All boundaries of the domain are mod-eled as outlet boundaries to prevent pressurebuild-up in the domain. The initial field is

–2 –1 0 1 2

–2

–1

0

1

2

Fig. 1. Cross sections through the center of the flame attu 0/‘T = 1. The solid lines are isotherms correspondingto the unburnt side, the inner layer and the burnt side ofthe flame (T = 305, 1698 and 1849 K). The vectorsrepresent the gas velocity (u, v). Six flamelets (thick graylines) are projected on the plane z = 0. The spatialcoordinates are given in mm.

1380 J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384

obtained from detailed simulations of laminarspherically expanding flames with the 1D flamecode CHEM1D [18]. A stoichiometric methane–air mixture is considered at atmospheric pressurewith an unburnt temperature Tu = 300 K. The ini-tial flame is a sphere with a radius of approxi-mately 1 mm at 1698 K and its center located atthe center of the computational domain. This ini-tial field is superimposed onto a turbulent flowfield, which is modeled to be homogeneous andisotropic. This yields ‘T/dL = 1 with dL = 0.5 mmthe laminar flame thickness based on the maxi-mum gradient of the temperature. The turbulentvelocity scale u0=s0

L ¼ 4:1 with s0L the burning

velocity of a flat stretchless flame with respect tothe unburnt mixture. The turbulent Reynoldsnumber is Re ¼ u0‘T=s0

LdL ¼ 4:1. In this case,combustion takes place in the thin reaction zonesregime.

The DNS is performed using a 1D FGM basedon the GRI reaction mechanism 3.0 [19]. The useof detailed kinetics is currently not feasible due tothe extremely high computational demands.Applying a multi-dimensional FGM is unneces-sary, because a direct comparison with detailedchemistry is not possible anyway. The mass frac-tion of carbon dioxide, which is monotonouslyincreasing, is used as single controlling variableY1 ¼ Y CO2

=Y burntCO2

. Since pressure, enthalpy andelement mass fractions are constant in theseflames, they are not needed as additional control-ling variables. When a 1D FGM is applied, trans-port equations do not have to be solved for allspecies mass fractions. Instead a differential equa-tion is solved for the controlling variable only.Since the reaction layer for this, ‘slowly changing’variable is thicker than for radicals such as CH, arelatively coarse grid is sufficient to resolve thestructure of the flame. A more detailed descriptionand a validation of the numerical method are giv-en in [20].

A cross section through the center of the flameat t = ‘T/u 0 is shown in Fig. 1. After this time nosignificant changes in the statistics are found andeffects of the artificial initial condition have disap-peared. Velocity vectors and isotherms corre-sponding to the position of the unburnt side, theinner layer and the burnt side of the flame(T = 305, 1698 and 1849 K) are displayed. Theinner layer is defined as the position where thechemical source term of the progress variable_xY1

reaches its maximum value along s. The tem-perature at the burnt side corresponds to the posi-tion where _xY1

has decreased to 10% of itsmaximum value. The distance between the iso-therms T = 1698 and 1849 K gives a good indica-tion of the thickness of the reaction layer. Thepreheat zone of the flame is the region betweenthe isotherms T = 305 and 1698 K. It can be seenthat the average radius of the flame ball hashardly changed during one eddy turn-over time.

However, the local flame front is clearly distortedby the turbulent flow. Turbulent eddies have dis-torted the preheat zone of the flame, but the reac-tion layer remains intact. This behavior isexpected in the thin reaction zones regime.

5. Flamelet analysis

To perform the flamelet analysis, 1D flamepaths~xðsÞ are reconstructed from the DNS results,by integrating d~x ¼~ndn in the direction normalto the iso-surfaces of the progress variable. Anumber of these flame paths (or flamelets) isshown in Fig. 1. The flamelets are projected onthe plane z = 0 through the center of the flameand cross the plane of projection at the positionof their inner layer. Since the flame paths arecurved, the other parts of the flamelets do notlie in this plane and therefore they do not seemto reach the unburnt boundary. Once the flamepaths are constructed, the profiles of the relevantvariables along these paths can be computed. InFig. 2 the profiles of the temperature T, thedimensionless flame stretch rate KdL=s0

L and cur-vature jdL are shown for the different flameletsdisplayed in Fig. 1. The arc-length s is scaled withthe flame thickness dL. The profiles of T in thepreheat zone are changed significantly by the flow,but they are almost undisturbed near the innerlayer s = sil. The gradient of T at the inner layeris nearly constant. The dimensionless stretchrates are O(1), locally reaching values up to 10.

300

500

700

900

1100

1300

1500

1700

1900

–4

–2

0

2

4

6

8

10

–1 –0.8 –0.6 –0.4 –0.2 0 0.2

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

Fig. 2. Profiles of temperature T, the dimensionlessflame stretch rate KdL=s0

L and curvature jdL along theflame paths displayed in Fig. 1.

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Fig. 3. Scatter plot of the scaled mass burning ratemil=m0

il at the inner layer computed with 1D FGM (y-axis) and detailed chemistry (x-axis).

J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384 1381

This means that the stretch rates are comparableto the laminar flame time scale and thus not neg-ligible. The same holds for curvature, because thedimensionless curvature jdL of the flame surfaceis also O(1).

It is expected that such high stretch and curva-ture rates cause significant deviations in composi-tion space from the 1D manifold approach. Thiscannot be checked directly because the DNS iscarried out with a 1D FGM, which does not allowsuch deviations. Therefore, the influence of flamestretch and curvature on the kinetics is investigat-ed by computing flamelets with detailed chemistrythat are subjected to the stretch and curvatureeffects occurring in the turbulent flame. In thesesimulations the flamelet Eqs. (3), (4) and (6) aresolved for the stretch and surface fields (K(s)and r(s)) taken from the DNS data as describedabove. The term Qi in (6) is neglected, which

makes the system steady and 1D. Extinction andre-ignition phenomena are therefore not includedin these flamelet simulations. The set of equationsis solved similar to a flat, stretchless, 1D flame,but now with prescribed K and r field. The simu-lations are carried out using a 1D FGM anddetailed chemistry, which makes a direct compar-ison at turbulent conditions possible without per-forming a DNS with detailed kinetics. The massburning rate mil at the inner layer of O(103) flam-elets is determined for both models and the resultsare compared in Fig. 3. The results from the sim-ulations with FGM agree quite well with thosecomputed using detailed chemistry. The differenceis larger at small values of mil, because these flam-elets experience the highest stretch rates. The aver-age deviation is 5%.

6. Manifold analysis

The accuracy of the FGM model can beimproved by increasing the dimension of the man-ifold. In order to assess whether this approach canbe successful, the dimension of the accessed spacein composition space is investigated for the flam-elet simulations with detailed chemistry as pre-sented in the previous section. As a first step weconstructed scatter plots of species mass fractionsversus Y1 ¼ Y CO2

=Y burntCO2

and compare the result tothe 1D FGM. The top row of Fig. 4 shows thatthe species mass fractions Yi (i = H2, O and CO)remain clustered around the 1D FGM, but thatdeviations of �10% occur. The deviationsDY i ¼ Y i � Y 1D

i of the species mass fractions fromthe 1D FGM are not random, but appear to bestrongly coupled. This can be seen in the secondand third row of Fig. 4, in which scatter plots ofDYi versus DY2 are shown conditioned at

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10–3

–0.05 0 0.05 0.1 0.15–0.08

–0.06

–0.04

–0.02

0

0.02

0.04

–0.02 –0.01 0 0.01 0.02–0.03

–0.02

–0.01

0

0.01

0.02

0.03

0 0.2 0.4 0.6 0.8 110

–5

10–4

10–3

10–2

10–1

100

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

x 10–3

–0.05 0 0.05 0.1 0.15–0.2

–0.1

0

0.1

0.2

0.3

–0.02 –0.01 0 0.01 0.02–0.06

–0.04

–0.02

0

0.02

0.04

0.06

0 0.2 0.4 0.6 0.8 110

–5

10–4

10–3

10–2

10–1

100

0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

–0.05 0 0.05 0.1 0.15–0.03

–0.02

–0.01

0

0.01

0.02

0.03

0.04

–0.02 –0.01 0 0.01 0.02–5

–4

–3

–2

–1

0

1x 10

–6

0 0.2 0.4 0.6 0.8 110

–5

10–4

10–3

10–2

10–1

100

Fig. 4. Top row: scatter plots of species mass fraction Yi versus first controlling variable Y1. The solid line is the 1DFGM. The dashed line denotes the scaled chemical source term of Y1. Second and third row: scatter plots of variations inthe species mass fraction DYi versus variations in the second controlling variable Y2 at Y1 ¼ 0:6 and 0.8, respectively.The lines are the 2D ILDM (solid) and the 2D FGM created by including stretch (dashed). Bottom row: mean error ofdifferent manifold approaches as function of Y1. Dash–dotted, 1D FGM; solid, 2D ILDM; dashed, 2D FGM (stretch);solid gray, 2D FGM (curvature); dashed gray, 2D FGM (initial condition); left column, H2; middle column, O; rightcolumn, CO.

1382 J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Fig. 5. Scatter plot of the scaled mass burning ratemil=m0

il at the inner layer computed with 2D FGM (y-axis) and detailed chemistry (x-axis).

J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384 1383

Y1 ¼ 0:6 and 0.8, respectively. Y1 ¼ 0:6 corre-sponds to the point at which _xY1

is maximum(see top-left graph of Fig. 4), while Y1 ¼ 0:8 ismuch closer to chemical equilibrium. The massfraction of OH is used as second controlling vari-able Y2 ¼ Y OH and the deviations are scaled withthe maximum value of the variable in the 1DFGM. In these scatter plots, cross sections of2D manifolds are shown as well. The 2D ILDMshown in these plots is in fact the 2D planespanned by the right eigenvectors of the Jacobianmatrix J ij ¼ o _xi=oY j corresponding to the twoslowest chemical processes [5]. At Y1 ¼ 0:8 thescatter corresponds well with the 2D ILDM, butat Y1 ¼ 0:6 relatively large deviations occur. The2D FGM created by applying a constant stretch,agrees well with the scatter from the detailedchemistry simulations at both values of Y1. Theother 2D FGM’s are not shown because theyalmost coincide with the 2D FGM created bystretch and the plots then become cluttered.

A quantitative analysis is performed by com-puting the mean deviation between the scatterand the different manifolds. These results are plot-ted as function of Y1 in the bottom row of Fig. 4.The 2D ILDM improves the 1D manifold resultsonly a little at high values of Y1. The 2D FGM’screated by applying stretch and curvature arealmost the same and result in a significant reduc-tion of the error in the main part of the domain,where chemical source terms are important. AtY1 ¼ 0:27 a small peak arises, because Y2 cannot parameterize the 2D manifold there:DY2 � DY i. Small variations in Y2 then result inlarge changes in Yi. Another choice of controllingvariable(s) might solve this problem. The 2DFGM created by changing the inlet condition,gives comparable results, although somewhatpoorer than the other 2D FGM’s. Near theunburnt mixture, this method results in a largeerror for H2 and CO, again because Y2 fails toparameterize the 2D manifold there. Note thatthis 2D FGM has been created by adding H2

and CO to the inlet composition, while YOH

remained zero.Scatter plots for other species show similar

behavior than those shown in Fig. 4. This a priorianalysis demonstrates that the accuracy of FGMcan be increased significantly by increasing thedimension of the manifold to two. Three or moredimensions are not investigated here because theerror for a 2D FGM is already in the same orderof numerical errors due to interpolation. In orderto investigate whether the use of a 2D manifoldactually improves the accuracy of flame simula-tions, a posteriori analysis is performed. To thatend the flamelets introduced before are computedwith the 2D FGM generated by including stretch.In this case, transport equations similar to Eq. (4)are solved for both controlling variables. Theresults are shown in Fig. 5 as a scatter plot of

mil computed with the 2D FGM versus mil com-puted with detailed chemistry. The mean devia-tion is now 1%, while it was 5% for a 1D FGM.Similar results are achieved when using the other2D FGM’s.

7. Conclusions

The results in this paper show that deviationsfrom ideal 1D flamelet behavior due to stretchand curvature effects in premixed turbulent flamescan be significant in the thin reaction zonesregime. These deviations are not random, butappear to be embedded in a 2D manifold for thecase studied here. This is in favor of using low-di-mensional manifold methods like ILDM andFGM. However, the deviations are not imbeddedin the 2D ILDM for a large range of Y1. On theother hand, all three 2D FGM’s are able to cap-ture the composition variations quite accurately.It is interesting to note that the way the 2DFGM is created, has only little influence on thefinal accuracy. The manifolds created by includingstretch or curvature are nearly the same: the differ-ence between these manifolds is smaller than thedifference with results from detailed simulations.This indicates that for the flames studied here,an attracting low-dimensional manifold exists,which is not governed by chemical kinetics only.

The DNS is carried out with a 1D FGM. As aresult, the stretch and curvature fields obtainedfrom the DNS results are not exactly the sameas they would appear in a DNS with detailedchemistry. However, they will also be not muchdifferent, because the error introduced by FGMis approximately 5%. Furthermore, the couplingwith the flow field is via the density, which is hard-ly influenced by changes in the species composi-tion and therefore well described by the 1DFGM. Another consequence of using FGM in

1384 J.A. van Oijen et al. / Proceedings of the Combustion Institute 31 (2007) 1377–1384

the DNS is that extinction and re-ignition phe-nomena are not included in the current analysis.Although they are not likely to occur at the condi-tions studied in this paper, they will be relevant athigher turbulence levels. To model these phenom-ena with FGM, a manifold with more than twodimensions might be needed. A DNS usingdetailed chemistry can clarify these aspects.

Acknowledgment

The financial support of the Dutch technologyfoundation STW is gratefully acknowledged.

References

[1] J.B. Bell, M.S. Day, J.F. Grcar, Proc. Combust.Inst. 29 (2002) 1987–1993.

[2] E.R. Hawkes, J.H. Chen, Proc. Combust. Inst. 30(2005) 647–655.

[3] D. Thevenin, Proc. Combust. Inst. 30 (2005) 629–637.

[4] J.F. Griffiths, Prog. Energy Combust. Sci. 21 (1995)25–127.

[5] U. Maas, S.B. Pope, Combust. Flame 88 (1992) 239–264.

[6] J.A. van Oijen, L.P.H. de Goey, Combust. Sci.Technol. 161 (2000) 113–138.

[7] O. Gicquel, N. Darabiha, D. Thevenin, Proc.Combust. Inst. 28 (2000) 1901–1908.

[8] J.A. van Oijen, F.A. Lammers, L.P.H. de Goey,Combust. Flame 127 (2001) 2124–2134.

[9] J.A. van Oijen, L.P.H. de Goey, Combust. TheoryModelling 6 (2002) 463–478.

[10] J.A. van Oijen, L.P.H. de Goey, Combust. TheoryModelling 8 (2004) 141–163.

[11] J.A. van Oijen, G.R.A. Groot, R.J.M. Bastiaans,L.P.H. de Goey, Proc. Combust. Inst. 30 (2005)657–664.

[12] N. Peters, Turbulent Combustion, Cambridge Uni-versity Press, 2000.

[13] A.N. Lipatnikov, J. Chomiak, Prog. Energy Com-bust. Sci. 31 (2005) 1–73.

[14] J.A.M. de Swart, G.R.A. Groot, J.A. van Oijen,J.H.M. ten Thije Boonkkamp, L.P.H. de Goey,Combust. Flame 145 (2006) 245–258.

[15] L.P.H. de Goey, J.H.M. ten Thije Boonkkamp,Combust. Flame 119 (1999) 253–271.

[16] M. Matalon, Combust. Sci. Technol. 31 (1983) 169–181.

[17] S.B. Pope, U. Maas, Simplifying chemical kinet-ics: Trajectory-generated low-dimensional mani-folds, Tech. Rep. FDA 93-11, CornellUniversity (1993).

[18] CHEM1D, A one-dimensional laminar flame code,Eindhoven University of Technology. URL<http://www.combustion.tue.nl/chem1d/>.

[19] G.P. Smith, D.M. Golden, M. Frenklach, N.W.Moriarty, B. Eiteneer, M. Goldenberg, C.T. Bow-man, R.K. Hanson, S. Song, W.C. Gardiner Jr.,V.V. Lissianski, Z. Qin. URL <http://www.me.berkeley.edu/gri_mech/>.

[20] R.J.M. Bastiaans, J.A. van Oijen, S.M. Martin,L.P.H. de Goey, H. Pitsch, CTR Annual ResearchBriefs (2004) 257–268.