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2001

Proceedings of the Combustion Institute, Volume 29, 2002/pp. 20012008

LARGE-EDDY SIMULATION OF PREMIXED TURBULENT COMBUSTIONUSING A LEVEL-SET APPROACH

H. PITSCH andL. DUCHAMP DE LAGENESTE

Center for Turbulence ResearchStanford University

Stanford, CA 94305-3030, USA

In the present study, we have formulated the G equation concept for large-eddy simulation (LES) ofpremixed turbulent combustion. The developed model for the subgrid burning velocity is shown to cor-rectly reflect Damkohlers limits for large- and small-scale turbulence. From the discussion of the regimediagram for turbulent premixed combustion, it is shown that given a particular configuration of flow pa-rameters, changes in the LES filter width result in changes along constant Karlovitz number lines. The

Karlovitz number is chosen as the horizontal axis to construct a new regime diagram for LES of premixedturbulent combustion, where changes in the filter width are represented by vertical lines. In addition, somenew regimes appear in the new diagram, which are related to the numerical treatment. An importantconclusion is that changes in the filter width cannot result in changes of the combustion regime amongthe corrugated flamelets, thin reaction zones, and broken reaction zones regimes. This is a consistencyrequirement for the model, since the choice of the filter width cannot change the fundamental combustionmode. With decreasing filter width, changes from corrugated to wrinkled flamelets, or in general to alaminar regime, are possible. In applying the model in a numerical simulation of a turbulentBunsen burnerexperiment, it is shown that the results predict the mean flame front location, and thereby the turbulentburning velocity, and the influence of the heat release on the flow field in good agreementwithexperimentaldata.

Introduction

Large-eddy simulation (LES) of premixed turbu-lent combustion is now considered to be a promisingfield. It has the potential to improve predictions ofreacting flows over classical Reynolds-averaged Na-

vier-Stokes (RANS) approaches, which lack preci-sion, especially in complex geometry or, for instance,for swirling flows with recirculation.

Since the reaction zone thickness of premixedflames in typical engineering devices is thin, with acharacteristic length scale much smaller than a typ-ical LES filter width, chemical reactions occur only

on the subgrid scales. Hence, these must be mod-eled entirely. Different methods have been pro-posed to model premixed combustion in LES.

One approach is to artificially thicken the flame byincreasing the diffusivity [13]. The flame can thenbe resolved on the LES grid and a subgrid chemicalclosure is not necessary. To correct the effect on theflame propagation speed, the chemical source termmust be modified by the use of so-called efficiencyfunctions obtained from direct numerical simula-tions (DNS) [4]. This method has already been ap-plied with some success. However, the main draw-back of this method is that it can change the

principle underlying physical process from a trans-port-controlled to a chemistry-controlled combus-tion mode.

The basic idea in flamelet models is that of a thinflame sheet. This assumption leads to two importantconsequences. First, if the flame, or at least the re-action zone, is thinner than the small turbulentscales, the structure of the flame or the reaction zoneremains laminar. Second, within the thin flame, thegradients are very high, and hence, the chemical re-actions are balanced by diffusive transport. As animplication of the latter, in a turbulent flow, theseterms cannot be modeled independently. In theflamelet model, the flame is replaced by an interfacemoving with a laminar burning velocitysL, which can

be expressed in terms of the molecular diffusivityD and the characteristic chemical time scale tc assL (D/tc)

1/2, thereby capturing the balance of mo-lecular transport and the chemistry.

In a turbulent flow, the local mass burning ratecan then be modeled if the laminar burning velocityand the local flame surface area per unit volume areknown. Hawkes and Cant [5], for instance, have pro-posed a flamelet model for LES based on a reactionprogress variable. In this model, a transportequationfor the subgrid flame surface density is solved. Sincethis formulation relies on the laminar burning veloc-ity, the model can only be applied in the corrugated

flamelet regime.A different formulation of flamelet models is the

level-set method, where the instantaneous flame


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2002 TURBULENT COMBUSTIONLarge Eddy and Direct Numerical Simulations

front is treated as an interface, represented by anisosurface of a scalar field G, defined byG G0.This isosurface is convected by the velocity field u,

while it propagates normal to itself with the laminarburning velocity. An equation describing the evolu-

tion of this scalar field, the so-called Gequation, hasbeen proposed by Williams [6]. For turbulent flows,this equation is only valid if the flame thickness,comprising the inert preheat zone and the thin re-action zone, is small compared to the small turbulentscales. This is commonly called the corrugatedflamelets regime. For this regime, a closure for thetransport equations of the Reynolds-averagedGandits variance has been provided by Peters [7].

If the flame thickness is larger than the small tur-bulent scales, but the reaction zone remains thinner,the reaction zone structure still remains laminar, andonly the transport in the preheat zone is changed by

the turbulence. This has been called the thin reac-tion zones regime [8].

Different formulations of flamelet models for LESbased on the G equation have been proposed, forthe corrugated flamelets regime, for instance, byChakravarthy and Menon [9] and for the thin reac-tion zones regime by Kim and Menon [10].

Peters [8] has provided a transport equation forG,valid in both the corrugated flamelets and the thinreaction zones regime as

Gq qu G qs |G| qDj|G| (1)L

twheretis the time, qis the density,uis the velocityvector in the unburned mixture, and j is the cur-vature of an isosurface ofG. In this equation, thefirst term on the right-hand side is the propagationterm involving the laminar burning velocity, whichis dominant in the corrugated flamelets regime, andthe second term is the curvature term, most impor-tant in the thin reaction zones regime. Since the lam-inar burning velocity is defined only at the flame sur-face, equation 1 is only valid at G0, and the G fieldoutside of this surface must be defined differently.

Note that since equation 1 has no diffusion term, thesolution atG0does not depend on the definition ofthe remaining field.

A Reynolds-averaged form of equation 1 has beenderived by Peters [8]. Also an analytic model for theturbulent burning velocity, which appears as an un-closed term in the mean G equation, is providedfrom a transport equation for the flame surface arearatio.

In the following, using arguments similar to thoseof Peters [8], a model for LES of premixed com-bustion based on the G equation and valid for boththe corrugated flamelets and the thin reaction zones

regime will be presented. The model will be vali-dated using data from a turbulent Bunsen burnerexperiment by Chen et al. [11]. Numerical results

from the LES are compared with measured velocityand temperature data for the cold and the reactiveflow.

Regime Diagram in Large-Eddy Simulation

Regime diagrams for premixed turbulent combus-tion in terms of velocity and length scale ratios havebeen proposed by Borghi [12] and other authors. Arecent version by Peters [8] introduces the thin re-action zones regime, limited by the criterionKad 1,

whereKadis Karlovitz number based on the reactionzone thickness. This considerably extends the flame-let regime beyond the Klimov-Williams criterion, for

which flamelets exist only if the turbulent KarlovitznumberKa is smaller than unity.

In addition to the Karlovitz number, which is theratio of the time scales of the laminar flame and theKolmogorov eddies, the turbulent Reynolds num-ber,Re, and the turbulent Damkohler number,Da,appear in the discussion of premixed turbulent com-bustion and should therefore be introduced here.For LES, these numbers can be defined with thecharacteristic subgrid velocity fluctuationvD andtheLES filter width D as

2v D s D lD L FRe , Da , Ka (2)D D 2s l v l gL F D F

wherelFis the laminar flame thickness and g is theKolmogorov length scale. It is important to note that,

while changes of the filter width result in changes inthe Reynolds and the Damkohler number, the Kar-lovitz number is independent of the filter width.Changes in the filter width therefore proceed at con-stant Karlovitz number.

The regime diagram by Peters [8] is given as v/sLas a function ofl/lF, wherev is the turbulent veloc-ity fluctuation at the integral length scalel.In LES,the integral length scale must be replaced by a char-acteristic subgrid length scale, which can be givenby the filter width. It follows that the location in theregime diagram depends on the LES filter size. Theobjective here is to discuss the influence of the filtersize on the location in the regime diagram and todevelop a new regime diagram for LES, which al-lows to clearly distinguish between changes of theturbulence/chemistry interaction and the numericaltreatment in an LES.

The transfer rate of turbulent kinetic energye isinertial range invariant. Hence, the scaling for e interms of a velocity and a length scale can be em-ployed at any length scale within the inertial sub-range. The appropriate relation is given here at theintegral length scale and at the filter scale

3 3 3v v (s l )D L Fe (3)4l D g

The third relation in equation 3 is the scaling at the


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LES OF PREMIXED TURBULENT COMBUSTION 2003

0.01

0.1

1

101

102

103

0.01 0.1 1 101 102 103

/lF

Ka = ((v'/sL)3 . lF/)

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laminar flameletsG-Eq. DNS

wrinkledflamelets

Re= 1

Ka = 1 Ka= 1

brokenreactionzones

thin reaction zones

corrugatedflamelets

effect of

decreasingfilter size

DNS

lG

Fig. 1. Regime diagram for LES of turbulent premixedcombustion.

Kolmogorov scale e m3/g4, where the kinematicviscositym with the assumption of constant Schmidtnumber has been replaced bym D sL lF. Com-bining equation 3 and the Karlovitz number in equa-tion 2 leads to

3l v lF D F2Ka e (4)3 s s DL L

Equation 4 has two important implications. The firstrelation in equation 4 shows that the Karlovitz num-ber only depends on the laminar flame scales andthe turbulent kinetic energy transfer rate e and is

hence independent of the filter width. Because ofthis, the second part of equation 4 provides a relationforvD/sLas a function of the filter width. Since theKarlovitz number is not affected by the filter width,it demonstrates how variations in the filter width in-fluence the subgrid velocity fluctuation. Equation 4can be inserted into equation 2 to obtain for theReynolds number and the Damkohler number

4/3 4D vD2/3 2Re Ka Ka andD l sF L

2/3 2D vD2/3 2

Da

Ka

Ka (5)D l sF LSince the only quantity independent of the filterwidth is the Karlovitz number, it seems convenientto use the Karlovitz number instead of the lengthscale ratio to construct a regime diagram for LES.As a second axis either the velocity ratio vD/sL orthe length scale ratio D/lFcan be chosen. Here we

will use D/lF, since this clearly reveals the influenceof the filter width.

The new regime diagram can be constructed fromthe relations given in equation 5 and is shown in Fig.1. It includes the same physical information as Pe-

ters diagram [8], but in addition it reveals some newregimes, which are related to the numerical treat-ment. This diagram also has the advantage that the

consequences of changing the filter width becomeevident. This is indicated in Fig. 1, showing, for in-stance, that a change in the filter width cannotchange the combustion regime across the Klimov-

Williams criterion (Ka 1) or the Peters criterion

(Kad

1). Therefore, in the new regime diagram,changes in the horizontal direction correspond tochanges in the turbulence/chemistry interaction atconstant filter width, while changes in the verticaldirection correspond to changes in the filter widthat constant physical parameters.

Starting from the corrugated flamelets regime, asmaller filter width can decrease the subgrid fluc-tuations such that vD sL when the filter widthbecomes smaller than the Gibson scale lG, therebychanging the modeled part of the combustion pro-cess to the wrinkled flamelet regime. The corruga-tion of flamelets would then occur at the resolved

scales and would not appear in the subgrid model-ing. Although the subgrid part of the flame in thisregime has a completely laminar structure, the flowfield still reveals subgrid turbulence. If from here,or from the thin reaction zones regime, the filter

width is decreased even further, also the subgrid partof the flow field becomes laminar when D g. Inthis so-called laminar flamelets regime, the turbu-lence is resolved by the simulation, but the reactionzone thickness is still smaller than the filter width.However, if theG equation according to equation 1

would be used in a simulation, all relevant lengthscales would be resolved in this regime. This regimecan therefore also be referred to as the G equationDNS regime. If in this regime or starting from thebroken reaction zones regime the filter width is fur-ther decreased to a value smaller than the reactionzone thickness d, then also the length scale of thereaction zone is resolved and the simulation consti-tutes a DNS, also if a temperature or progress vari-able equation rather than theG equation is solved.

Governing Equations

We consider the low Mach number approximation

of the Navier-Stokes equations. To achieve a properdescription of the flow field, the filtered equationsfor mass and momentum conservation must besolved. The filtered density appearing in these equa-tions has to be provided by the combustion model.

Filtered G Equation

Following a procedure similar to that of Peters[13], an equation for the filteredGequation valid inboth the corrugated flamelets and the thin reactionzones regime can be derived as

G q qu G qs |G| qD j|G| (6)T tt

Here, the bar denotes a spatial filtering and the tilde


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a spatial Favre filtering. The double primeappearingbelow stands for the Favre fluctuations. This equa-tion is only valid at the flame surface, outside of

whichGmust be defined differently. Here, for con-venience, the flame is assumed to be at G0 0 and

the remaining part of theGfield will be determinedto be a signed distance function, negative in the un-

burned and positive in the burned gases.The turbulent burning velocity sT appearing in

equation 6 has been introduced as

( qs ) |G| (qs ) r (7)T L

where is the flame surface area ratio, which is yetrunknown and must be modeled. Modeling of theconditional velocity appearing in equation 6 is de-scribed in Ref. [13].

Turbulent Burning Velocity

Similarly to Peters [14], an equation for the sub-grid variance ofG corresponding to equation 6 canbe derived as

2G 2 2q qu G ( quG )

t2 2 qD (G) qx qv (8)t

Since equation 6 is only defined to be valid forG G0 and the variance can only be defined with themean, also equation 8 can only be used at G G0.

The fluctuationG represents the departure of theinstantaneous flame front from the mean flame frontlocation. In equation 8, the turbulent productionterm has been modeled using a gradient transportassumption, where Dt is the turbulent diffusivity.The last two terms denote the kinematic restorationand the scalar dissipation, respectively. Both resultin the attenuation of the variance ofG and act onthe small scales. Therefore, the scaling for theseterms developed by Peters [8] in a Reynolds-aver-aged context can also be used here. In particular, ithas been shown that the kinematic restorationshould be proportional to the mean propagation andindependent of the small scales [7], and the scalardissipation must involve a molecular diffusivity [8].This leads to

C Dm2 2x c (s r) and n c D r (9)2 L 3vD

Here,c2andc3are proportionality factors. The sub-grid time scale has been determined from the sub-grid velocity fluctuation and the sub-grid lengthscaleCmD, where Cmis the Smagorinsky coefficient deter-mined by a dynamic model as described in Moin etal. [15]. Assuming that in the scalar variance equa-

tion the production term balances the attenuationterms leads with equations 9 and 7 to an analyticexpression for the turbulent burning velocity

s s D/DT L t b3

2s b C D sL 3 m L1 2b Sc l v1 t F D

vD Da /ScD D b (10)3

2s bL 31 DaD2b Sc1 DThe laminar burning velocity has been added toachieve the correct limit for laminar flows. The con-stantsc2andc3have been determined such that theexpression for the turbulent burning velocity forCmD/lF kvD/sLorDaD k 1 results in Damkohlerslarge-scale limit

s

s v

T L D b (11)1s sL L

and forCmD/lF KvD/sLorDaD K 1 in Damkohlerssmall-scale limit [16]

s s DT L t b (12)3s DL

The resulting expressions are c2 and22/(Sc b )D 1

c3 where the constantsb1andb3have been22/b ,3

taken from Peters [14] to beb1 2.0 andb3 1.0.The subgrid diffusivity, the subgrid Schmidt numberScD, and the subgrid velocity fluctuation can all bedetermined by dynamic models. Here, following thesuggestion of Pitsch and Steiner [17], a constantSchmidt number ofScD 0.5 is assumed. The sub-grid diffusivity is determined fromDt CmDvD/ScD.

Since the subgrid burning velocity is a function ofthe filter size, it is interesting to discuss the respec-tive dependency. Inserting equation 4 into equation11 shows that (sT sL)/sL (D/lF)

1/3, implying thatin the large-scale limit a doubling in the filter sizeonly leads to an about 25% increase in the burning

velocity. For the small-scale limit, equations 5 and12 lead to (sT sL)/sL (D/lF)

2/3, which would

result in an approximately 50% change, if the filtersize was doubled.As shown in the discussion above, the subgrid

burning velocity essentially includes two parts, thekinematic restoration, important in the corrugatedflamelets regime, and the dissipation, important inthe thin reaction zones regime. As equation 10shows, the transition depends on the Damkohlernumber and thereby on the filter width. This seemsto be in contrast with the earlier finding that thecombustion regime is independent of the filter

width. In the corrugated flamelet regime, the tur-bulent length scale is much larger than the flame

thickness, and hence, the dissipation, representingcurvature effects, is present, but cannot be impor-tant. However, the curvature becomes important,


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when the turbulent length scale becomes of the or-der of the flame thickness. In the corrugated flame-let regime, decreasing the filter size reduces the larg-est scales which must be modeled. When the filter

width is reduced such that it becomes comparable

to the flame thickness, the effects of curvature be-come important for the subgrid burning velocity.However, it is easy to see from equation 5 that theDamkohler number cannot become smaller thanunity in the corrugated or the wrinkled flamelet re-gime. Thereby, even if curvature effects become in-creasingly important, sT still remains dominated bythe kinematic restoration.

Evaluation of the Filtered Density

To solve the momentum equations, the filtereddensity must be computed from the combustion

model. Peters [14] has suggested an algorithm in theReynolds-averaged context for adiabatic simulations.The algorithm used in this study is similar, but ex-tended to account for possible heat losses to theburner, which have been found to be important inthe experiment considered for the present simula-tions. The present formulation also accounts for themixing of the postflame region with the coflowingair.

If the flame thickness is small compared to thefilter width, the temperature can be assumed to bediscontinuous at the flame, changing from the con-stant unburned to the adiabatic flame temperature.

This assumption can then also be made for the den-sity.

To account for heat losses, an equation for the fil-tered enthalpy is solved. The enthalpy is definedhto include the heat of formation and is hence a con-served quantity, except for heat losses at walls andradiative heat losses, the latter of which have beenneglected here. To account for inert mixing with thecoflowing air, also an equation for a filtered mix-ture fractionZis solved. The mixture fraction is de-fined to be zero in the coflowing air and unity in thepremixed jet. Both of these equations have no chem-

ical source terms and, hence, do not require anychemistry closure model.The density is assumed to be a function of the

mixture fraction, the enthalpy, and G only. The in-stantaneous density can then be determined as

q (Z) ifG 0uq(Z, h , G) (13)1 q (Z, h ) otherwiseb 1where in the unburned gases, the density is deter-mined by inert mixing given byqu (Z) [Z/q1 1 Z/q2]

1, and q2 and q1 are the densities in atZ 0 andZ 1, respectively. In the burned gases,the density is given by the solution of a steady-state

flamelet library, describing the mixing of hot stoi-chiometric gases with cold coflowing air. For thesimulation of steady-state flamelets, the enthalpy

varies linearly with the mixture fraction. For the testcase considered below, heat losses occur only in thepilot flame, whereZ 1. Therefore, onlyh1 appearsas a parameter in the flamelet equations, and thedensity can be described as q qb(Z, h1).

The evaluation of the filtered density then followsthe presumed pdf approach as described in Peters[14].

Numerical Simulation

Experimental Setup

As a test problem for the model presented in theprevious section, the so-called flame F3, experimen-tally investigated by Chen et al. [11], has been cho-sen. The experimental setup consists of a stoichio-

metric premixed methane/air flame stabilized by alarge pilot flame. Both incoming streams, the main

jet and the pilot, have the same composition and aresurrounded by a coflowing air stream. The nozzlediameterD of the main stream is 12 mm. The pilotstream issues through a perforated plate (1175 holesof 1 mm in diameter) around the central nozzle withan outer diameter of 5.67 D. The main stream isturbulent with a jet Reynolds number based on theinner nozzle diameter and a bulk velocity ofU0 30 m/s of Re 23,486. Based on the estimatedcharacteristic length and time scale given in Chen etal. [11], the velocity and length scale ratios can bedetermined to beu/SL 11.9 and l/lF 13.7. Thisleads to a Karlovitz number ofKa 11.1, indicatingthat flame F3is located well within the thin reactionzones regime, but still far from the broken reactionzones regime.

Large-Eddy Simulation

The computer code used in this study has beendeveloped at the Center for Turbulence Researchby Pierce and Moin [18,19]. The filtered low Machnumber approximation of the Navier-Stokes equa-

tions is solved in cylindrical coordinates on a struc-tured staggered mesh. The numerical method is aconservative, second-order finite-volume scheme.Second-order semi-implicit time advancement isused, which alleviates the CFL restriction in regions

where the grid is refined. Details of the numericalmethod can be found in Pierce [19] and Akselvolland Moin [20]. The flamelet library has been com-puted using the GRI-MECH 2.11 chemical reactionmechanism [21].

To ensure sufficient regularity of theGfield awayfrom theG 0 surface,G is defined in the whole do-main as a distance function. The constraint |G| 1

needs to be enforced at each time step by a so-calledreinitialization procedure. The numerical procedureused here has first been described in Sussman et al.


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x/D=0.25 2.5 6.5

T

U

Fig. 2. Instantaneous flame position (gray surface), temperature distribution (top part), and axial velocity distribution(bottom part)

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

U/U0,c

y/D

x/D = 0.25

Fig. 3. Time-averaged temperature and axial velocity atx/D 0.25. Lines, simulation; symbols, experimental data.

[21] and was later extended by Russo and Smereka[22].

Grid and Boundary Conditions

The computational domain extends to 20Ddown-stream of the nozzle and 4Din the radial direction.The LES grid is 296 64 64, corresponding toapproximately 1.2 million cells. At the inflow bound-ary, instantaneous velocities extracted from a sepa-rate LES of a fully developed pipe flow are pre-scribed. Convective conditions [19] are prescribed

at the outflow boundary, while traction-free condi-tions [23] are imposed on the lateral boundaries toallow entrainment of fluid into the domain. The en-thalpy in the inflow streams has been prescribed tomatch the experimental temperature at the first ex-perimental station, which is very close to the nozzleatx/D 0.25.

Results and Discussion

An example of the instantaneous flame surface isshown in Fig. 2. The acceleration of the burned

gases behind the flame front caused by the heat re-lease is illustrated by the instantaneous field of theaxial velocity given in the bottom part of the figure.

The top part of Fig. 2 shows the corresponding tem-perature field. The highest temperatures can be ob-served in the region behind the instantaneous flame

front.Due to the consumption of the unburned mixture,

the mean flame front position is converging towardthe centerline at farther downstream positions. Thecomputed, time-averaged axial position, where thefuel is completely consumed, is at approximately

x/D 9.0, which is in good agreement with theexperimental value ofx/D 8.5 given in Chen etal. [11]. This result shows that the model for thesubgrid turbulent burning velocitysTleads to a rea-sonable prediction of the averaged flame location.

Quantitative results are compared to experimentaldata in Figs. 36. To judge the performance of thecombustion model and also to discuss the effect ofheat release, results from a cold flow simulation willbe discussed first. Figures 4 and 5 show radial pro-files of the time averaged axial velocity at x/D 2.5and 6.5. The turbulent kinetic energy at these posi-tions is shown in Fig. 6. The agreement for all datais very good, providing confidence in the ability ofLES to describe this flow correctly and particularlyin the proper treatment of the inflow conditions pro-

vided from the experiments.The mean radial profiles of temperature and axial

velocity very close to the nozzle atx/D 0.25 are

compared to experimental data in Fig. 3. The agree-ment is quite good and is shown here only to dem-onstrate that the inflow conditions have been chosenappropriately. In the adiabatic case the maximumnon-dimensional temperature, defined as H (T T1)/(Tb T1), whereTbis the adiabatic flametemperature, should beH 1.0. However, becauseof the strong heat losses to the pilot nozzle, the max-imum temperature in the pilot flame is onlyH 0.8.

The mean radial profiles of temperature and axialvelocity for the reactive case are also shown in Figs.4 and 5. Significant differences can be observed in

the velocity predictions compared to the cold flowcase, for which the mean axial velocity at the cen-terline decreases much faster than for the reactive


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0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

U/U0

,c

y/D

x/D = 2.5

Reactive

Non-reactive

Fig. 4. Time-averaged temperature and axial velocity atx/D 2.5. Lines, simulation; symbols, experimental data.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

U/U0,c

y/D

x/D = 6.5

Reactive

Non-reactive

Fig. 5. Time-averaged temperature and axial velocity at

x/D

6.5. Lines, simulation; symbols, experimental data.

0

0.005

0.01

0.015

0.02

0.025

0

0.005

0.01

0.015

0.02

0.025

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

k/U0

2 k/U0

2

y/D

x/D = 2.5

Reactive

Non-reactive

Non-reactive

Reactive

x/D = 6.5

Fig. 6. Turbulent kinetic energy at x/D 2.5 and 6.5.Lines, simulation; symbols, experimental data.

case. This also reveals much broader radial profilescompared to the cold flow case. The main reason forthis is the acceleration induced by the heat release.This effect is captured by the simulation in goodagreement with the experimental data.

The temperature profiles generally overestimate

the experimental data slightly, but the agreement isstill reasonable. The strong cooling influence of thebroad pilot flame can be observed atx/D 2.5. At

x/D 6.5, the cold coflowing air stream also be-comes important.

Figure 6 shows the radial profiles of the turbulentkinetic energyk for the cold and the reactive simu-lation. As for the mean velocity, the evolution ofkat

the centerline is found to be very different in bothcases. For the cold flow case, the turbulent kineticenergy is increasing in the downstream direction andreaches a maximum at x/D 8.5, which is notshown here, whereaskstays approximately constantfor the reactive case. This result confirms that com-bustion strongly influences transport of turbulencetoward the centerline by preventing radial transportof the turbulent kinetic energy produced in theshearlayers. Although some discrepancies remain, this im-portant result, also observed in the experiments [11],is well reproduced by the simulation. In the reactivecase, the turbulent kinetic energy profiles reveals

two maxima, one on the burned side and one on theunburnt side of the mean flame position. The radiallocation of both peaks is well predicted. However,the maximum values are generally overpredicted.

Conclusions

In conclusion, the G equation model has beenshown to be well suited for applications in LES. Themodel for the subgrid burning velocity presentedhere correctly represents Damkohlers large-scaleand small-scale turbulence limits. The influence of

the LES filter width on the model predictions hasbeen discussed in terms of the regime diagram forturbulent premixed combustion. It has been shownthat the Karlovitz number is independent of the LESfilter width. The implication is, for instance, thatchanges of the filter width cannot change the com-bustion regime, but can account for the correct tran-sition of the subgrid burning velocity to the laminarburning velocity when the filter width becomessmaller than the Kolmogorov length. Although thechoice of the filter width can potentially change thesubgrid burning velocity from the Reynolds-aver-aged turbulent to the laminar burning velocity, the

dependence has been shown to be weak. This is abenefit for LES, since then the exact choice of thefilter width becomes less important.

In the application of the model to a turbulent Bun-sen burner experiment, reasonable agreement withthe experimental data indicates that the turbulentburning velocity and the influence of the heat releaseon the flow field are correctly represented by themodel.

Acknowledgments

The authors gratefully acknowledge funding by Snecma

Moteurs and by the US Department of Energy within theASCI program. We also thank C. D. Pierce for supplyingthe LES solver and N. Peters for comments and discussion.


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COMMENTS

Yung-Cheng Chen, University of Sydney, Australia. Howdoes the prediction compare with the experimental datafor the turbulent flame brush thickness for flame F3 (Ref.[11] in the paper)? The initial velocity data for the pilotflame have also been measured and are available for com-parison from the first author of Ref. [11]. Is there any planin the future to compute the flame F2 in Ref. [11], whichhas a high turbulence level and a different behavior in theaxial evolution of the turbulent flame brush thickness?

Authors Reply.We have not looked into the data for theturbulent flame brush thickness yet. We are currently sim-ulating flame F2 and will then possibly move on to flameF1 to assess the applicability of the flamelet model nearthe limits of its formal validity. In the comparison of F2and F3, we will discuss the predictions of the flame brushthickness.

Andrei Lipatnikov, Chalmers University of Technology,Sweden. Let us consider the simplest case of a statisticallyplanar, one-dimensional, fully developed, turbulent flame

that propagates in a statistically uniform and stationary mix-ture. Then, in the framework linked with the flame, equa-tion 6 from your paper is reduced to

dG dGqu qSt dx dx

if the curvature term j 0. Taking into account the massconservation equation, qu quSt instead of in theqStright-hand side of equation 6 appears to be more appro-priate. What do you think? Is this correction of impor-tance? In addition, what is the value ofq at the flame sur-face associated withG(x,y,z,t) G0?

Authors Reply.The equation you are suggesting would

be correct in the Reynolds averaged sense. Equation 6from the present paper is a spatially filtered equation forLES. Even if the flow is steady and planar in the mean, ina turbulent flow, these assumptions will not be instanta-neously met, and all terms in equation 6 remain. The G-equation can be written in different forms, as long as theturbulent burning velocity specifies the propagation speed

with respect to the location where the convective velocityis evaluated. It is evident that the G-equation as given byequation 6 is a kinematic relation independent of the den-sity, which would not be the case if the turbulent burning

velocity was multiplied with the unburned rather than thelocal filtered density. However, the velocity appearing in

the convective term is actually an average conditioned onthe instantaneous position ofG0. The modeling of this con-ditional velocity is described in the paper.

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