Combustion and Flame 157 (2010) 465–475

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Combustion and Flame

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A filtered tabulated chemistry model for LES of premixed combustion

B. Fiorina a,*, R. Vicquelin a,b, P. Auzillon a, N. Darabiha a, O. Gicquel a, D. Veynante a

a EM2C – CNRS, Ecole Centrale Paris, 92295 Châtenay Malabry, Franceb GDF SUEZ, Pôle CHENE, Centre de Recherche et d’Innovation Gaz et Energies Nouvelles, 93211 Saint-Denis la Plaine, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 May 2009Received in revised form 24 July 2009Accepted 23 September 2009Available online 24 November 2009

Keywords:Large Eddy SimulationTurbulent premixed combustionTabulated chemistry

0010-2180/$ - see front matter � 2009 The Combustdoi:10.1016/j.combustflame.2009.09.015

* Corresponding author. Fax: +33 1 47028035.E-mail address: benoit.fiorina@em2c.ecp.fr (B. Fior

A new modeling strategy called F-TACLES (Filtered Tabulated Chemistry for Large Eddy Simulation) isdeveloped to introduce tabulated chemistry methods in Large Eddy Simulation (LES) of turbulent pre-mixed combustion. The objective is to recover the correct laminar flame propagation speed of the filteredflame front when subgrid scale turbulence vanishes as LES should tend toward Direct Numerical Simu-lation (DNS). The filtered flame structure is mapped using 1-D filtered laminar premixed flames. Closureof the filtered progress variable and the energy balance equations are carefully addressed in a fully com-pressible formulation. The methodology is first applied to 1-D filtered laminar flames, showing the abilityof the model to recover the laminar flame speed and the correct chemical structure when the flame wrin-kling is completely resolved. The model is then extended to turbulent combustion regimes by includingsubgrid scale wrinkling effects in the flame front propagation. Finally, preliminary tests of LES in a 3-Dturbulent premixed flame are performed.

� 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

Flame ignition and extinction or pollutant predictions are cru-cial issues in LES of premixed combustion and are strongly influ-enced by chemical effects. Unfortunately, despite the rapidincrease in computational power, performing turbulent simula-tions of industrial configurations including detailed chemicalmechanisms will still remain out of reach for a long time. A com-monly-used approach to address fluid/chemistry interactions at areduced computational cost consists in tabulating the chemistryas a function of a reduced set of variables. Some techniques, suchas Intrinsic Low Dimensional Manifold (ILDM) developed by Massand Pope [1], are based on a direct mathematical analysis of the dy-namic behavior of the chemical system response. Alternative ap-proaches are Flame Prolongation of ILDM (FPI) [2,3] or FlameletGenerated Manifold (FGM) [4]. Both techniques assume that thechemical flame structure can be described in a reduced phase sub-space from elementary combustion configurations. For instance,the chemical subspace of a turbulent premixed flame can beapproximated from a collection of 1-D laminar flames. In such sim-ple situations, all thermo-chemical quantities are related to a sin-gle progress variable.

To couple tabulated chemistry with turbulent combustion,mean quantities can be estimated with presumed probability den-sity functions. This approach, that does not require prohibitive re-sources, has been developed for Reynolds Averaged Navier–Stokes

ion Institute. Published by Elsevier

ina).

(RANS) computations in the past [5,6]. Unfortunately, the exten-sion of RANS turbulent combustion models to LES is not straight-forward. Indeed, the primary recurrent problem is that the flamethickness is typically thinner than the LES grid size. As the progressvariable source term is very stiff, the flame front cannot be directlyresolved on practical LES grid meshes, leading to numerical issues.To overcome this difficulty, dedicated models have been developedunder simplified chemistry assumptions. A solution to propagate aflame on a coarse grid is to artificially thicken the flame front bymodifying the diffusion coefficient and pre-exponential constant[7,8]. Following a different strategy and under simplified chemistryassumptions, Boger et al. [9] and more recently Duwig et al. [10]have introduced a filter larger than the mesh size to resolve the fil-tered flame structure. An opposite alternative is to solve a largescalar field where a given iso-surface is identified to the instanta-neous flame front position. In such technique, called G-equationmodel, the inner layer is tracked using a level-set technique.Initially developed in a RANS context [11], the G-equation has beenreformulated for LES [12–14]. However, as level-set techniquesprovide information only on the thin reaction zone position andnot on the filtered flame structure, the coupling with the flowequations is challenging. In particular the knowledge of the tem-perature field is required for taking into account heat expansion.As recently proposed by Moureau et al. under simplified chemistryassumption [15], a solution is to solve an additional progress var-iable equation to ensure a consistent coupling with a LES flowsolver.

The FPI–PCM (Presumed Conditional Moment) model [16],developed to introduce tabulated chemistry effects in LES, com-

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466 B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475

bines presumed Probability Density Functions (PDF) and FPI tablesto describe the chemical reaction rate of the filtered progress var-iable accounting for interactions between turbulence and chemis-try at the subgrid scale level. However, as will be shown further,this formulation does not guarantee a proper prediction of regimeswhere the subgrid scale flame wrinkling vanishes. This regime, ob-served when the subgrid fluctuations are lower than the laminarflame speed, is encountered in practical LES of premixed combus-tion [15,17]. Additionally LES should tend toward DNS when thefilter size becomes lower than the Kolmogorov scale. Hawkes andCant [18] extensively discussed realizability in premixed combus-tion LES.

In the present work, it is first demonstrated that the b-PDF for-malism applied in the context of premixed combustion LES doesnot guarantee a proper description of a filtered laminar flame front.Therefore an alternative is proposed to include tabulated chemis-try in LES approach ensuring the correct propagation speed ofthe filtered laminar flame front. The resolved flame structure ismapped from 1-D filtered laminar premixed flames. The idea oftabulating filtered quantities has already been introduced [19]but unresolved convective and diffusive terms where neglected.As it will be demonstrated further, these assumptions do not allowa proper description of the filtered flame structure and propaga-tion. Here, closure of filtered flow and progress variable equationsare first carefully addressed in regimes where the flame wrinklingis fully resolved. One-dimensional computations are performed toinvestigate the capability of the proposed model to reproduce thecorrect propagation speed and the filtered flame structure. Themodel is then extended to turbulent combustion regimes takinginto account subgrid scale flame wrinkling. Finally, simulations ofa turbulent swirled premixed flame are performed and comparedto experimental data.

2. Coupling tabulated chemistry and LES: filtered equations

Low-dimensional trajectories in composition space are identi-fied in FPI framework from the knowledge of the complex chemicalstructure of 1-D laminar flames [2]. For premixed combustion sys-tems, a 1-D freely propagating flame is first computed using de-tailed chemical schemes. Thermodynamical and chemicalquantities are then tabulated as a function of a unique monotonicprogress variable c related to temperature or to a combination ofchemical species, where c ¼ 0 corresponds to fresh gases andc ¼ 1 to fully burnt gases. The chemical database is then coupledto the flow field by adding the progress variable balance equationto the Navier–Stokes equations. The progress variable reaction rateand heat release are extracted from the chemical database. For LES,under unity Lewis numbers assumption, these equations are fil-tered leading to the following system:

@�q@tþr � ð�q~uÞ ¼ 0 ð1Þ

@�qeu@tþr � ð�q~u~uÞ ¼ �rP þr � �s�r � ð�qfuu � �q~u~uÞ ð2Þ

@�q~c@tþr � ð�q~u~cÞ ¼ r � ðqDrcÞ � r � ð�qfuc � �q~u~cÞ þ �q ~_xc ð3Þ

@�qeE@tþr � ð�q~ueEÞ ¼ �r � ðPudÞ þ r � ðsuÞ � r � ð�qfuE � �q~ueEÞ

þ r � ðqDrhsÞ þ �q e_xE ð4ÞP ¼ �qreT ð5Þ

where q is the density, u the velocity vector, P the pressure, d theunit tensor, s the laminar viscous tensor, E ¼ H � P=q with H the to-tal non-chemical enthalpy, hs the sensible enthalpy, D is the diffu-sivity, _xc and _xE, respectively, the progress variable and energy

source terms. r ¼ R=W , where R is the ideal gas constant and Wthe mean molecular weight. The overbar denotes the spatial filter-ing operation,

�/ðxÞ ¼Z Z Z

Fðx� x0Þ/ðx0Þdx0 ð6Þ

where / represents reactive flow variables and velocity componentsand F the filtering function. The tilde operator denotes the density-weighted filtering defined by �qe/ ¼ q/.

The subgrid scale terms, �r � ð�qfuu � �q~u~uÞ and �r � ð�qguu��q~u ~uÞ, where u denotes c or E quantities, the pressure term Pu,as well as the filtered laminar diffusion terms qDru and the fil-tered source terms e_xu, require closure models. The model con-straints are both to ensure a correct flame propagation and torecover the chemical structure of the filtered flame under two pos-sible situations: (1) the flame wrinkling is fully resolved at the LESfilter size and (2) wrinkling occurs at the subgrid scale and affectsthe filtered flame speed.

Different strategies exist to model the filtered progress variablereaction rate e_xc. An approach that does not require extensive CPUresources is to presume the shape of progress variable PDF, gener-ally by a b function. This formalism has been applied to LES of tur-bulent premixed combustion [16] but, to our knowledge, theability of the method to reproduce the propagation speed of fil-tered flame front has not yet been investigated. In the followingsection the influence of the PDF shape on the filtered flame proper-ties is discussed when the flame wrinkling is resolved at the LES fil-ter scale, i.e. when the subgrid scale flame front remains laminarand planar. The use of a b function is found to introduce errors inthe filtered flame front propagation speed. A new modeling alter-native based on the tabulation of filtered premixed flame elementsis then proposed to correct this drawback.

3. A priori testing of presumed b-PDF formalism in the laminarregime

An unstretched 1-D filtered laminar premixed flame is consid-ered in this section. If no wrinkling occurs at the subgrid scale,the propagation speed SD of the filtered flame front is identical tothe laminar flame speed S0

l . The following relation then needs tobe satisfied:

q0SD ¼Z þ1

�1�q e_xcðxÞdx ¼

Z þ1

�1q _xcðxÞdx ¼ q0S0

l ð7Þ

where q0 is the fresh gases density and x is the spatial dimension.The ability of presumed b-PDF to satisfy this property is inves-

tigated by conducting a priori tests on a 1-D stoichiometric freelypropagating laminar premixed propane/air flame computed withPREMIX [20] using a modified version of the GRI 3.0 mechanism[21]. The progress variable c is plotted as a function of the spatialcoordinate x in Fig. 1a. The laminar flame thickness, defined bydl ¼ 1=maxðjdc=dxjÞ is approximately equal to 0:4 mm. IntroducingeP , the mass weighted PDF defined by �qeP ¼ qP, the progress vari-able filtered reaction rate reads:

e_xcðxÞ ¼Z 1

0

_xcðcÞePðx; cÞdc ð8Þ

Assuming that c follows a b distribution [22]:

ePðx; cÞ ¼ cac�1ð1� cÞbc�1R 10 cac�1ð1� cÞbc�1dc

ð9Þ

where parameters ac and bc are determined from ~c and the segrega-tion factor Sc ¼ ð ecc � ~c~cÞ=ð~cð1� ~cÞÞ:

Fig. 1. A priori test of the b-PDF formalism in laminar regime. Left (a): progress variable c (solid line) and filtered progress variable ~c (bold line) profiles as a function of thespatial coordinate x. Dashed line is the subfilter progress variable segregation factor Sc . Right (b): a priori computations of the filtered progress variable propagation speed fordifferent values of filter size. The filtered progress variable reaction rate is modeled by a b-PDF (squares) or by a Gaussian filter (triangles).

B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475 467

ac ¼ ~c1Sc� 1

� �bc ¼ ac

1ec � 1� �

ð10Þ

The knowledge of the first and second moment of the progress var-iable provides the filtered reaction rate e_xc ¼ e_xcð~c; ScÞ. For the con-figuration considered here, ~c and Sc profiles across the filteredlaminar flame front are computed by applying a 1-D Gaussian filterF of size D defined by:

FðxÞ ¼ 6pD2

� �1=2

exp �6x2

D2

� �ð11Þ

on the detailed chemistry laminar flame solution.Favre-filtered progress variable and the segregation factor are

shown in Fig. 1a for a filter size of D ¼ 20dl. According to Eq. (9),the presumed b-PDF, ePðx; cÞ, is deduced from these two quantities.The reaction rate e_xc across the filtered flame front is then esti-mated from Eq. (8). Finally, the integration of the filtered reactionrate according to Eq. (7) gives an a priori estimation of the filteredflame front propagation speed SD. The ratio SD=S0

l (square symbols)is plotted as a function of the ratio D=dl in Fig. 1b. When D=dl < 1the effect of the b-PDF on the flame structure is moderate andthe propagation speed is correctly reproduced. However whenthe filter size is larger than the flame front, as in LES practical sit-uations, the propagation speed of the filtered progress variable islargely over-estimated by the presumed b function up to a factorof 2.5. In fact, the b-PDF is not relevant when subgrid scale wrin-kling is resolved.

A solution to propagate a flame front at the correct speed is toartificially thicken the reaction zone. In the thickened flame modelfor LES (TFLES) [7,8], both reaction rate and diffusion fluxes are af-fected in order to ensure a correct propagation of the flame front.However the structure of the thickened flame front does not corre-spond to the filtered flame front.

An alternative to presumed PDF formalism and TFLES is to di-rectly employ a normalized filter function FðxÞ to estimate the fil-tered reaction rate. Then the filtered reaction rate reads:

e_xcðxÞ ¼1�q

Z þ1

�1qðxÞ0 _xcðx0ÞFðx� x0Þdx0 ð12Þ

Since by definition, FðxÞ satisfiesRþ1�1 FðxÞdx ¼ 1, Eq. (7) is then al-

ways satisfied:

q0SD ¼Z þ1

�1�q e_xcðxÞdx ð13Þ

¼Z þ1

�1

Z þ1

�1qðx0Þ _xcðx0ÞFðx� x0Þdx0dx ð14Þ

¼Z þ1

�1qðx0Þ _xcðx0Þ

Z þ1

�1Fðx� x0Þdx

� �dx0 ð15Þ

¼Z þ1

�1qðx0Þ _xcðx0Þdx0 ð16Þ

¼q0S0l ð17Þ

This property is verified in Fig. 1b, where the propagation speed SD

of the filtered flame, is a priori computed using Eqs. (11) and (12).By taking advantages of this property, a model is proposed in

Section 4 to ensure the correct propagation of filtered laminarflame front. The closure of unknown terms is carefully addressedand the model is tested on 1-D filtered flame configurations. Thisapproach is extended to turbulent regimes where subgrid flamewrinkling occurs at the subgrid scale level in Section 5 by the intro-duction of the subgrid flame wrinkling factor.

4. Filtered laminar premixed flames modeling

4.1. Modeling

The flame structure in the direction n normal to the flame frontis assumed identical to the structure of a planar 1-D freely propa-gating premixed laminar flame. A detailed chemical mechanismwith Ns species is considered. From this reference flame structureand using the filter operators introduced in Section 2, the a priorifiltered flame structure is determined. For instance, for a given fil-ter size D, any filtered fluxes or filtered thermo-chemical quantitiesof a planar filtered laminar flames are known.

As an example, a 1-D laminar stoichiometric premixed propane/air flame is computed taking into account detailed chemistry ef-fects. The PREMIX [23] solver is combined with a modified versionof the GRI 3.0 mechanism [21] involving Ns ¼ 70 species and 463elementary reactions. The filtered operator given by Eq. (11) isthen applied to the 1-D laminar flame solution. Fig. 2 shows allbudget terms of the ~c balance equation in a steady 1-D laminarpremixed flame remapped in the ~c coordinate system for differentvalues of D. Using these results, a modeling procedure based on thetabulation of the filtered 1-D laminar flame structure is proposedin the following sections. The closure of each unclosed terms iden-tified in Eqs. (2)–(4) is first carefully addressed in the situationwhere no flame wrinkling occurs at the subgrid scale level.

4.1.1. Filtered chemical reaction ratesThe filtered source terms for c and the energy equations are di-

rectly given by the filtered database:

e_xu ¼ e_x�u½~c;D� ð18Þ

where u denotes c or E quantities and the * superscript denotesquantities issued from a 1-D unstretched laminar premixed flame.

Fig. 2. Budget terms (in kg m�3 s�1) as a function of ~c of the filtered progress variable balance equation of a steady 1-D laminar planar filtered premixed flame for differentvalues of filter size D: � : @ð�q~u�~c�Þ=@x�: r : @ðqD@c�=@x�Þ=@x�: j : �q0S0

L@ðc� � ec� Þ=@x� : N : �q e_x�c : � : @ðqD@ ec�=@x�Þ=@x�: Terms are plotted in the ~c coordinate for differentvalues of filter size D.

468 B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475

The notation /½~c;D� means that the variable / is tabulated in a 2-Dlook-up table with coordinates ~c and D. Fig. 2 shows that the filteroperator affects dramatically both the amplitude and the shape ofe_xc (triangles symbols) profiles.

4.1.2. Filtered laminar diffusion terms r � ðqDrcÞ and r � ðqDrhÞThese terms are usually neglected or approximated as:

r � ðqDruÞ � r � ðqDr~uÞ: ð19Þ

This approximation is very rough and may introduce large er-rors. Indeed, in Fig. 2 the exact laminar diffusion fluxes@@x� qD @c�

@x�

� �(filled diamonds) and the approximation by

@@x� qD @ec�

@x�

� �(empty diamonds) are shown for different values of

the filter size D. When the filter size is smaller than the laminarflame thickness dl, approximation by Eq. (19) remains valid. How-ever as soon as the filter size D becomes larger than dl, important

differences are observed between @@x� qD @c�

@x�

� �and @

@x� qD @ec�@x�

� �. As

shown further, these errors impact dramatically the prediction ofthe propagation speed. In the present work, the filtered diffusionterm for the c equation is modeled by:

r � ðqDrcÞ ¼ �r � ðqDjrcjnÞ ð20Þ

¼ �r � qD@c�

@x�

���� ����n !

ð21Þ

By introducing a corrective factor acð~cÞ, one can write:

r � ðqDrcÞ ¼ r � ðac½~c;D��qDr~cÞ ð22Þ

The normal to the flame front n ¼ �r~c=jr~cj points into the freshreactants. The correction factor acð~cÞ is defined as:

ac½~c;D� ¼qD @c�

@x�

�� ��qD @ec�

@x�

��� ��� : ð23Þ

The quantity ac½~c;D� is estimated from the 1-D filtered flame solu-tion and is tabulated as a function of ~c for a given value of filter sizeD.

Similarly, the energy-filtered laminar diffusion term is writtenas:

r � ðqDrhsÞ ¼ r � ðaEð½ec;D�Þ�qDr~hsÞ ð24Þ

where the correction factor aE½ec;D� is defined as:

aE½~c;D� ¼qD @h�s

@x�

��� ���qD @~h�s

@x�

��� ��� ð25Þ

The correction factors ac½~c;D� and aE½~c;D� are plotted in Fig. 3 for dif-ferent values of filter size D. For small values of D, as ac½~c;D� remainsconstant and close to 1, effects on the laminar diffusion fluxes mod-eling will be negligible. However, the profiles present strong varia-tions in terms of ec when the filter size D is larger than dl.

Fig. 3. Diffusion correction factor ac (left) and aE (right) as a function of ec for different values of D. Dashed dotted lines: D ¼ 0:2dl . Dashed lines: D ¼ 1dl . Dashed dotted lines:D ¼ 5dl . Solid lines: D ¼ 25dl .

B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475 469

4.1.3. Unresolved convection terms �r � ð�qguu � �q~u euÞThe displacement speed sd, measuring the flame front local

speed relative to the flow, i:e: the difference between the absoluteflow velocity u and the absolute flame front speed w, is firstintroduced:

u ¼ wþ sd ð26Þ

The filtered flame front speed w remains constant across the flamebrush ð ~w ¼ �w ¼ wÞ, therefore after replacing the flow velocity byrelation (26), the subgrid scale convection term then reads:

�r � ð�qguu � �q~u ~uÞ ¼ �r � ð�qgsdu � �q~sd ~uÞ ð27Þ

In a 1-D laminar premixed flame the laminar flame speed S0l and the

fresh gas mixture density q0 are related to the displacement speedthrough the following relation:

q0S0l ¼ �qs�d ð28Þ

Therefore, under the assumption that the flame remains planar atthe subgrid scale level, the unresolved convection terms are directlyestimated from the reference laminar 1-D detailed chemistry pre-mixed flame:

�r � ð�qguu � �q~u ~uÞ ¼ � @

@x��q gs�du� � �qes�d ~u�� �

ð29Þ

¼ � q0S0l@u�@x�� @

fu�@x�

� �ð30Þ

¼ Xu½~c;D� ð31Þ

The term Xc½~c;D� ¼ �q0S0L

@@x� ð�c � ~cÞ is plotted in Fig. 2 for differ-

ent values of filter size D (squares). For D < dl, unresolved convec-tive fluxes are very small compared to other fluxes. However, whenD P dl, these fluxes become important and are counter-gradienttype. Note that this result is in agreement with recent experiments[24]. The quantity Xu½~c;D�, estimated from the 1-D filtered flamesolution, is then tabulated as a function of ~c and D. In practice, asthe unresolved convective terms are modeled as a source term,only the sum Ru½~c;D� ¼ Xu½~c;D� þ e_xu½~c;D� is stored in the filtereddatabase where / denotes c or E quantities.

4.1.4. Pressure termIn a similar way, the pressure term in the energy equation (Eq.

(4)) is written as:

�r � ðPudÞ ¼ �r � ðP~udÞ � ðr � ðPudÞ � r � ðP~udÞÞ ð32Þ

¼ �r � ðP~udÞ � ðr � ðqgrTudÞ � r � ðqfrT ~udÞÞ ð33Þ¼ �r � ðP~udÞ þXp½~c;D� ð34Þ

with

Xp½~c;D� ¼ �q0S0l@ðrT�Þ@x�

� @ðgrT� Þ@x�

!: ð35Þ

4.1.5. Momentum equationsUnclosed terms in the filtered momentum equations may be

modeled following the same approach. The subgrid scale convec-tion term is written as:

�r � ð�qfuu � �q~u~uÞ ¼ @

@x��qgs�ds�d � �q esd

� esd�

� �n ð36Þ

¼ q0S0l@sd

�

@x�� @

esd�

@x�

� �n ð37Þ

¼ Xu½~c;D�n ð38Þ

The strain tensor is expressed by:

r � �s ¼ r � ðau½ec;D�~sÞ with au½ec;D� ¼ �s�es� ð39Þ

where ~s is defined as:

~s ¼ �l r~uþ ðr~uÞT � 23ðr � ~uÞd

� �ð40Þ

However, as shown further, the influence of these terms is moder-ate and can be neglected.

4.2. Summary of the model equations

The momentum, the progress variable and the energy equationsare modeled as:

@�q~u@tþr � ð�q~u~uÞ ¼ �rP þr � ðau½~c;D�~sÞ þXuð~cÞn ð41Þ

@�q~c@tþr � ð�q~u~cÞ ¼ r � ðac½~c;D��qDr~cÞ þ Rc½~c;D� ð42Þ

@�qeE@tþr � ð�q~ueEÞ ¼ �r � ðPeudÞ þXp½~c;D� þ r � ðes~uÞ

þ r � ðaE½~c;D��qDr~hsÞ þ RE½~c;D� ð43Þ

These equations are implemented in the compressible LES codeAVBP [25]. The third-order finite element scheme TTGC [26] isused. Boundary conditions are prescribed using Navier–StokesCharacteristic Boundary Conditions [27].

The sum of filtered chemical reactions rates and the subgridscales fluxes Ru ¼ Xu þ �q e_xu and the diffusion fluxes correction

470 B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475

factors au are estimated after filtering a 1-D laminar stoichiometricpremixed propane/air flame. These quantities are stored in a look-up table as a function of ~c and D.

4.3. 1-D laminar premixed flame simulations

Filtered steady 1-D laminar flames are computed to verify theability of the present model to reproduce both the correct flamefront propagation speed and the filtered flame structure. Computa-tions are performed on uniform meshes with a grid spacing of Dx. Aparametric study is conducted for different filter sizes relative tothe laminar flame thickness. For each case, a reference solution isobtained by filtering the 1-D laminar premixed flame detailedchemistry solution. The simulations are initialized with the refer-ence solution and the overall physical time for each run istrun ¼ 50d~c=S0

l , where d~c ¼ 1=maxðj @~c@x jÞ is an estimation of the fil-

tered flame thickness.A comparison between the numerical solutions on uniform

mesh (solid lines) and the reference solution (dashed line) withd~c=Dx ¼ 50 and for different values of D=dl is first shown in Fig. 4.The predicted filtered progress variable profiles match the refer-ence solution for all the filter size values. Fig. 5a shows that thepredicted filtered front propagation speed SD (square symbols) re-mains very close to the reference laminar flame speed for variousvalues of D=dl. The triangular symbol in Fig. 5a represents simula-tion results with the approximation given by Eq. (19), i.e., au ¼ 1.This rough assumption leads to an under-prediction by a factorof 3 of the flame front propagation speed.

An important information for premixed combustion LES is theminimal number of grid points required to capture the filteredflame front without introducing numerical artifacts. The filteredflame front propagation speed is plotted as a function of the meshresolution Dx in Fig. 5b. The flame speed is recovered with a goodapproximation for d~c=Dx P 5. Below this limit, numerical errors be-come important and the filtered flame front does not propagate atthe correct speed. Then, for numerical reasons, the filter should beat least five times larger than the mesh size. Note that even ap-proaches based on level-set transport that use sophisticatednumerical methods to track the flame front position also requireto filter the flame front at a scale larger than the mesh size in orderto resolve density gradients [15].

Finally, a simulation has been performed without consideringthe filtering effect on the momentum equations (Eq. (41)), i.e., withau ¼ 1 and Xu ¼ 0 and is compared with the complete model solu-tion in Fig. 6. For both simulations, density as well as velocity pro-files match perfectly. In fact, the induced differences aretransferred to the pressure that becomes a macro-pressure. As this

Fig. 4. Filtered 1-D premixed flame solutions. Filtered progress variable (solid)compared to the reference solution (dashed) for D=dl ¼ 2, 10 and 20.

macro-pressure remains very close to the static pressure, effects onthe thermodynamic state are very limited. Then, in order to sim-plify the model implementation in 3-D configurations, the contri-bution corresponding to the filtering of a laminar flame in themomentum equation will be neglected.

5. Filtered turbulent premixed flames modeling

In practical LES of turbulent combustion, turbulence may causeflame front wrinkling at the subgrid scale level. Here, a strategy isproposed to extend the previously described model to suchsituations.

5.1. Modeling

Turbulent structures induce flame wrinkling that increases theflame surface area at the subgrid scale. As a consequence the fil-tered flame front propagates at a subgrid scale turbulent flamespeed St [22] related to the laminar flame speed through the flamewrinkling factor N ¼ St=S0

l .The model developed here ensures that the filtered flame front

propagates at the turbulent flame speed St . The filtered flamethickness is assumed to be only related to the filter size D and isnot altered by small-scale eddies.

Then, the filtered progress variable turbulent reaction rate ismodeled by:e_xct ¼ N: e_x�c ½~c;D� ð44Þ

and the turbulent diffusion term is expressed as follows:

Xct ¼ �ðr � ð�qfuc � �q~u~cÞÞt¼ NXc½~c;D� þ ðN� 1Þr � ðac½~c;D�qDr~cÞ ð45Þ

The first term on the r.h.s corresponds to the thermal expansion andthe second one models the unresolved turbulent fluxes. This formu-lation corresponds to multiply diffusion and source terms by theflame wrinkling factor in the laminar flame balance equation andthen ensures that the unstretched filtered flame front propagatesat the turbulent flame speed St ¼ NS0

l in the normal direction.

5.2. Summary of the model equations

To summarize, momentum, progress variable and energy equa-tions for this new model called Filtered Tabulated Chemistry forLES (F-TACLES) can be written as follows:

@�q~u@tþr � ð�q~u~uÞ ¼ �rP þr � ~sþr � �st ð46Þ

@�q~c@tþr � ð�q~u~cÞ ¼ Nr � ðac½~c;D��qDr~cÞ þ NRc½~c;D� ð47Þ

@�qeE@tþr � ð�q~ueEÞ ¼ �r � ðP~udÞ þ NXp½~c;D� þ r � ð~s~uÞ

þ Nr � ðaE½~c;D��qDr~hsÞ þ NRE½~c;D� ð48Þ

Note that here the effect of the flame filter D on the momentumequations is neglected and the subgrid scale turbulent fluxesr � �st are modeled using the Smagorinsky model. Different alterna-tives exist to estimate the subgrid flame wrinkling factor that ap-pears in Eqs. (46) and (47). It can be either estimated fromanalytical models [8,14,28,29] or from the solution of a surface den-sity balance equation [30,31].

5.3. Large Eddy Simulation of a swirled premixed burner

The proposed method is applied to the simulation of the com-plex PRECCINSTA swirled burner experimentally investigated by

Fig. 5. Predicted flame speed as a function of D=dl (left) and d~c=Dx (right). Square symbols are the complete model solution and the triangle symbol is the solution with au ¼ 1.

Fig. 6. Filtered 1-D premixed flame solutions. Effects of the flame filter in themomentum equation. Solid: au ¼ 1 and Xu ¼ 0. Symbols: auðecÞ and XuðecÞ from thefiltered database.

Fig. 7. LES of PRECCINSTA with F-TACLES turbulent combustion model. Thecomputational domain features the plenum, the swirl-injector and the combustionchamber. An instantaneous view of the filtered flame front iso-surface ð~c ¼ 0:8Þ isshown.

B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475 471

Meier et al. [32]. The geometry, shown in Fig. 7, derives from anaeronautical combustion device. It features a plenum, a swirl-injector and a combustion chamber. Details of the burner geometryand of the measurement can be found in Ref. [32]. Different mod-eling strategies for LES have been used to numerically investigatethis configuration: an LES of the combustor using the thickenedflame model and a two-step mechanism has been first performedby Roux et al. [33]. Moureau et al. [34] used this configuration tovalidate a new level-set algorithm to track the flame front position.Recently, Galpin et al. [16] performed the LES of this lean premixedburner by using a presumed b-PDF to couple a thermo-chemicallook-up table with the filtered flow equations.

The operating conditions chosen in the present study corre-spond to an air mass flow rate of 12.2 g/s and to a methane massflow rate of 0.6 g/s. In the experiment, air and methane are injectedseparately in the swirler inlet, however in the present simulationthe mixing is assumed to be fast enough to burn a perfect mixingof oxidizer and fuel in the combustion chamber. Thus methaneinjection is not taken into account and a methane/air mixture char-acterized by an equivalence ratio of 0:83 is injected at the plenuminlet. These conditions correspond to a stable regime where laserRaman scattering has been performed, allowing comparison be-tween predicted and measured thermo-chemical quantities suchas temperature and species mass fractions.

The boundary conditions and the computational geometry havebeen already described in [33]. The mesh used to perform the com-

putation is unstructured and made of 12.7 millions elements. Thethird-order finite element scheme TTGC [26] is retained. For build-ing-up the chemical look-up table, a 1-D laminar methane/airflame is first computed for an equivalence ratio equal to 0.83 usingthe GRI 3.0 mechanism [21]. Then, according to the modeling pro-cedure discussed previously, this laminar flame solution is filteredby the Gaussian function defined by Eq. (11).

Note that, as the mesh considered here is almost uniform in thefiltered flame front region, an unique filter width D is considered.In order to ensure a sufficient meshing of the filtered flame front,the filter width has been set to D ¼ 20dl. The progress variable isdefined by c ¼ YCO2=Yeq

CO2, where Yeq

CO2is the equilibrium CO2 mass

fraction in the fully burnt gases. The filtered quantities requiredby the model: Rc½~c; D�; ac½~c; D�; Xp½ec; D�; RE½~c; D� and aE½~c; D�are then tabulated as a function of ~c for D ¼ 20dl. For stronglynon-uniform meshes this procedure is not optimized and couldlead to over-refined or under-refined flame front regions. Then,an additional coordinate, the filter width, can be easily consideredwhen computing the look-up table.

Following the system of Eqs. (46)–(48), this new modelF-TACLES has been implemented into the compressible LES codeAVBP [25]. The subgrid flame wrinkling factor N is estimated fromthe analytical model developed by Colin et al. [8]. Mean and re-solved Root Mean Square (RMS) quantities are computed by timeaveraging LES solutions over a physical time that correspond to 6flow-through times based on the fresh gas inlet velocity. Meantemperature and CO2 mass fractions are plotted in Fig. 8 (top)and Fig. 9 (top), respectively. A very good agreement is observedbetween experimental and numerical profiles, which demon-strated that the correct flame angle and mean flame thicknessare reproduced by the model. Because heat losses have not beenconsidered when generating the chemical database and in the

472 B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475

numerical simulation, the LES slightly over-estimates the temper-ature profiles close to the combustion chamber wall, in the outerrecirculation zone for x < 20 mm and at a distance larger than20 mm from the jet axis. Note that heat losses effects on the flamestructure can be taken into account with the addition of the enthal-py as a control parameter of the chemistry tabulation [3,6].

Figs. 8 (bottom) and 9 (bottom) show a comparison between re-solved LES RMS and measured RMS of the temperature and the CO2

mass fraction, respectively. As the plotted LES RMS does not in-clude the subgrid scale RMS, conclusions regarding the model per-formance in terms of flame turbulence interactions are moredifficult. However, it is observed that LES RMS remains lower thanmeasured RMS, as expected from theory.

As all thermo-chemical variables are related to ~c, the post-pro-cessing of the filtered progress variable solution with the filteredchemical database allows to access all chemical species. As anexample, Fig. 10a shows 2-D contours of ~c used to estimate HCOmass fraction plotted in Fig. 10b.

Finally, Fig. 11 indicates the flame position in the Pitsch LESregime diagram for turbulent premixed combustion [17], wherethe ratio D=dl is expressed as a function of the Karlovitz numberKa in logarithmic scale. The Karlovitz number is related in LES tothe subgrid velocity fluctuations v 0D and laminar flame scales[17]:

Ka2 ¼ dl

S0l

3 e ¼ v 0DS0

l

dl

Dð49Þ

x = 15mm2000

-40

-20

0

20

40

x = 2

-40

-20

0

20

40

x = 10mm2000

-40

-20

0

20

40

x = 6mm

Distancefromaxis(mm)

2000-40

-20

0

20

40

x = 15mm500

-40

-20

0

20

40

x = 10mm0 500

-40

-20

0

20

40

x = 2

-40

-20

0

20

40

x = 6mm

Distancefromaxis(mm)

0 500-40

-20

0

20

40

Fig. 8. Mean (top) and RMS (bottom) of temperature, case / ¼ 0:83. Symbols: me

where e is the kinetic energy transfer rate. The subgrid velocity fluc-tuations are computed as follows:

v 0D ¼lt

�qCkDffiffiffiffiffiffiffiffi3=2

p ð50Þ

where the turbulent viscosity lt is estimated from Smagorinskymodel. For Ka < 1, combustion takes place in the corrugated flameregime while the thin reaction zone regime is observed whenKa > 1. Computational nodes located in the filtered flame frontare considered, i.e. for 0:01 < ~c < 0:99, and are plotted in the LESdiagram (horizontal thick solid black line in Fig. 11). As a unique fil-ter width D is considered in the present simulation, the scatter plotreduced to the line D=dl ¼ 20. The smallest size of the flame wrin-kling is given by the Gibson length [11]:

DlG¼ v 0D

S0l

ð51Þ

The substitution of Eq. (51) into Eq. (49) shows that D ¼ lG conditioncorresponds to D=dl ¼ Ka�2 represented by a line of slope �2 in theLES diagram (Fig. 11). In the corrugated flame regime, when the fil-ter width becomes smaller than the Gibson length, the subgridvelocity fluctuation v 0D is smaller than the laminar flame speed S0

l .In such cases, the flame wrinkling is fully resolved at the LES filterscale. At the opposite, on the right side of the lG ¼ D line, subgridscale wrinkling exists and will impact the filtered flame front prop-agation speed SD. The node distribution versus the Karlovitz number

0mm2000

x = 30mm2000

-40

-20

0

20

40

x = 40mm2000

-40

-20

0

20

40

x = 60mm2000

-40

-20

0

20

40

x = 30mm0.05 0.1

-40

-20

0

20

40

0mm500

x = 30mm500

-40

-20

0

20

40

x = 40mm500

-40

-20

0

20

40

x = 60mm500

-40

-20

0

20

40

asurements. Lines: simulation with F-TACLES. x = 0 matches the swirler exit.

x = 6mm

Distancefromaxis(mm)

0.05 0.1-40

-20

0

20

40

x = 10mm0.05 0.1

-40

-20

0

20

40

x = 15mm0.05 0.1

-40

-20

0

20

40

x = 20mm0.05 0.1

-40

-20

0

20

40

x = 30mm0.05 0.1

-40

-20

0

20

40

x = 30mm0.05 0.1

-40

-20

0

20

40

x = 40mm0.05 0.1

-40

-20

0

20

40

x = 60mm0.05 0.1

-40

-20

0

20

40

x = 30mm0.05 0.1

-40

-20

0

20

40

x = 10mm0 0.02 0.04

-40

-20

0

20

40

x = 15mm0 0.02 0.04

-40

-20

0

20

40

x = 60mm0 0.02 0.04

-40

-20

0

20

40

x = 30mm0 0.02 0.04

-40

-20

0

20

40

x = 6mm

Distancefromaxis(mm)

0 0.02 0.04-40

-20

0

20

40

x = 20mm0 0.02 0.04

-40

-20

0

20

40

x = 40mm0 0.02 0.04

-40

-20

0

20

40

Fig. 9. Mean (top) and RMS (bottom) of CO2 mass fraction, case / ¼ 0:83. Symbols: measurements. Lines: simulation with F-TACLES. x = 0 matches the swirler exit.

B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475 473

is plotted in Fig. 12. First, it can be observed that most of the pointsare located in the corrugated flame regime ðKa < 1Þ. The chemicalflame structure remains therefore laminar as assumed in the pres-ent model. Secondly, for a substantial area of the flame surface(about 30%), the Gibson length lG is larger than the filter width

Fig. 10. 2-D instantaneou

and consequently the flame wrinkling is fully resolved. With futureincrease of computational power, as meshes will be finer, this trendshould be emphasized. It demonstrates the crucial need of ensuringa proper propagation of the laminar flame front when deriving aturbulent combustion model.

s view of ec and eY HCO .

Fig. 12. Node distribution versus the Karlovitz number. Only nodes located into the filtered flame front have been considered, i.e. for 0:01 < ~c < 0:99.

Fig. 11. LES regime diagram for turbulent premixed combustion. The thick solid black line represent the range covered by the PRECCINSTA flame simulation.

474 B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475

6. Conclusion

A new modeling strategy called Filtered Tabulated Chemistryfor LES (F-TACLES) has been developed to introduce tabu-lated chemistry methods in premixed combustion LES. A filtered1-D laminar premixed flame is used to build a filtered chemi-

cal look-up table. The model performances are demonstratedon 1-D filtered laminar flame computations. Finally the pro-posed strategy has been applied to perform a 3-D simulationof a swirled turbulent premixed flame. Good agreementbetween the numerical simulation and the experiments isobserved.

B. Fiorina et al. / Combustion and Flame 157 (2010) 465–475 475

Acknowledgments

This work was supported by the ANR-07-CIS7-008-04 Grant ofthe French Ministry of Research. We are grateful to the CERFACS(Toulouse, France) combustion team for providing the PRECCINSTAburner geometry. The authors warmly acknowledge the support ofthe 2008 Summer Program of the Center for Turbulence Research(Stanford University – NASA Ames) during which this work wasinitiated. This work was granted access to the HPC resources ofIDRIS under the allocation 2009-i2009020164 made by GENCI(Grand Equipement National de Calcul Intensif).

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