The application of Flamelet Generated Manifolds in modelling of turbulentpartially-premixed flames

W.J.S. Ramaekers∗, B.A. Albrecht, J.A. van Oijen and L.P.H. de GoeyDepartment of Mechanical Engineering

Eindhoven University of Technology, The NetherlandsR.G.L.M. Eggels

Rolls-Royce Deutschland Dahlewitz, Germany

AbstractTo reduce harmful emissions numerical models are developed to simulate combustion processes in engineering appli-cations. In this paper a model for partially-premixed combustion used in Reynolds Averaged Navier-Stokes Simula-tions (RANS) is presented. A flamelet approach combined with a Probability Density Function (PDF) closure methodfor the chemical source term is used to describe turbulence-chemistry interaction. The laminar flamelet database isgenerated using the Flamelet Generated Manifold (FGM) chemistry reduction technique; it is assumed that mixing andchemistry are fully described by mixture fraction and a reaction progress variable respectively. A look-up databasefor turbulent combustion is constructed by PDF-averaging the laminar flamelet database. For the PDF a β-functionis assumed. In this paper the FGM/PDF model as implemented in FLUENT is described and validated for a welldocumented turbulent jet flame (Sandia Flame D). Results are presented and compared with a reference model andmeasurements.

Keywords: Turbulent combustion; Partial premixing; Reduced chemistry; Flamelets

Nomenclature

Latinc Reaction progress variabled Fuel inlet diameterD Diffusion coefficientk Turbulent kinetic energyL Characteristic length combustorMi Molar mass species ip PressureP (x) Probability Density FunctionR Universal gas constantsL Laminar burning velocityT Temperature~u Velocity vectorU Absolute velocityXi Molar fraction species iYi Mass fraction species iZ Mixture fraction

Dimensionless numbersDa Damkohler numberKa Karlovitz numberRe Reynolds numberSc Schmidt number

∗Corresponding author: giel.ramaekers@gmail.comAssociated Web site: http://www.combustion.tue.nl

GreekδL Laminar flame thicknessε Dissipation rate of kµ Meanν Molecular kinematic viscosityξ Variance of Zρ Densityσ2 Varianceφ Variance of cχ Destruction rateω Species source term

Sub-/superscriptL LaminarT Turbulent′ Reynolds fluctuation′′ Favre fluctuation¯ Reynolds averaged˜ Favre averagedax Axialmax Maximumst Stoichiometric

1 IntroductionTo gain more insight in combustion processes in

engineering applications models are developed withthe ultimate goal of optimizing fuel-efficiency andminimizing harmful emissions. Many engineeringdevices, aero-engines for example, operate in a turbulent

regime and fuel and oxidizer are not fully mixed beforecombustion takes place. The interaction between tur-bulence and chemistry is still not fully understood andmodels are developed for the simulation of turbulent,partially-premixed combustion processes.

Turbulent combustion can take place in many differentcombustion regimes which all have a different interac-tion between chemistry and turbulence. When turbulenceintensity is low turbulent eddies, both macroscopic andmicroscopic, will not be able to distort the flame whichwill thus exhibit a quasi-laminar character. With increas-ing turbulence intensity microscopic eddies will be ableto intrude the reaction layer. Macroscopic eddies will beable to deform the flame more severe with increasing tur-bulence intensity. Borghi [5] defined different regimesand their characteristics. To determine the turbulent com-bustion regime under aero-engine conditions estimationshave to be made for main parameters describing the flow.The mean gas velocity U in a aero-engine combustor isestimated to be of the order

U =√

~u · ~u = O(102)

m/s ,

and the turbulence intensity I , being the kinetic energyassociated with gas velocity fluctuations U ′ =

√~u′ · ~u′

divided by the kinetic energy associated with mean gasvelocity:

I =U ′ U ′

U U= 10−2 .

This leads to:

U ′ = O(101)

m/s .

The characteristic length scale of the combustion cham-ber L is estimated to be

L = O(10−1

)m .

It is assumed that the integral length scale is equal to thecharacteristic length scale of the combustion chamber.For the laminar burning velocity sL a small range for theorder of magnitude is estimated:

sL = O(10−1

)− O

(100)

m/s .

For the molecular kinematic viscosity ν the order of mag-nitude well known:

ν = O(10−5

)m2/s .

The assumption is made that diffusion of mass takes placeas fast as the diffusion of momentum (ν ≈ D). The orderof magnitude for the laminar flame thickness δL can befound by assuming that reaction and diffusion of speciesare in equilibrium in the reaction layer:

δL ∼ D

sL

= O(10−5

)− O

(10−4

)m .

These estimations translate to the turbulent Reynoldsnumber which is the ratio of the inertial and viscousforces:

Re =U ′L

ν= O

(105)

.

The Damkohler number is the ratio of macroscopic tur-bulent timescale and the chemical timescale. SmallDamkohler numbers (Da � 1) imply a strong deforma-tion of the flame by macroscopic eddies since chemistryis relatively slow. High Damkohler numbers (Da � 1)imply a very thin flame in which chemical processes takeplace very fast. For the estimations made the Damkohlernumber is of the order:

Da =

(τt

τc

)≈ L

U ′

sL

δL

= O(101)− O

(103)

.

The Karlovitz number is the ratio of the chemicaltimescale and the microscopic turbulent timescale: theKolmochorov timescale. The Kolmochorov lengthscaleis the lengthscale at which inertial forces are equal to vis-cous forces; dissipation of turbulent kinetic energy takesplace at this scale. Small Karlovitz numbers (Ka �)imply that turbulent eddies can not intrude the flamewhich will exhibit a laminar character. With increasingKarlovitz numbers (1 < Ka < 100) turbulent eddies willbe able to intrude the pre-heat zone of the flame. TheKolmochorov length scale is still larger that the reactionzone of the flame and turbulent eddies can not intrude thereaction zone of the flame. For large Karlovitz numbers(Ka � 1) no laminar structure can be identified anymore.The Karlovitz number for aero-engine combustor condi-tions is estimated to be:

Ka =

(τc

τk

)≈

[δL

L

(U ′

sL

)3]0.5

= O(10−1

)−O

(101)

.

For turbulent premixed combustion these values for Re,Da and Ka correspond to the corrugated flamelet regimeand the thin reaction zones regime and for turbulentnon-premixed combustion these values correspond to theflamelet regime with local extinction. For these regimesthe flame thickness is small compared to other length-scales describing the flame. This disables the possibil-ity of describing the chemistry with a perfectly-stirred-reactor (PSR) model. On the other hand the flame thick-ness can not be assumed to be infinitely thin which dis-ables the possibility of describing the chemistry using aBray-Moss-Libby (BML) [19], a Eddy-Break-Up (EBU)[19] or a similar model which is based on the assumptionthat only unburnt and burnt gasses can be present.A turbulent flame can be seen as an ensemble of thin,laminar, locally one-dimensional flames, called flamelets,embedded within the turbulent flow field. For the influ-ence of turbulence an appropriate closure has to be ap-plied. In reality variables like species concentrations andtemperature are a function of all other variables. However

2

the flamelet assumption states that most variables can beassumed to be dependent on a small number of controlvariables which are relevant for the flamelets which theturbulent flame is thought to be composed of. Turbulentcombustion processes have often been modelled assum-ing that chemistry is much faster than mixing. Mixturefraction, which defines the mass fraction of fuel in the un-burnt gas mixture, is one of the main variables describingnon-premixed combustion. Flamelets for non-premixedcombustion can be generated using a geometry consistingof opposed, axisymmetric fuel and oxidizer jets. As thedistance between the jets is decreased and/or the velocityof the jets increased, the flame is strained and increas-ingly departs from chemical equilibrium until it is even-tually extinguished. The use of non-premixed flameletshas become quite common in modelling of turbulent com-bustion, see [16, 11, 14] for example. Flamelets for pre-mixed combustion can be generated using a geometrywith only one inlet in which species are assumed to beperfectly mixed. Chemical reactions will take place asthe gas mixture is convected and the mixture composi-tion changes towards chemical equilibrium composition.Partially-premixed combustion is defined as a combustionprocess in which species are not perfectly mixed beforecombustion takes place although species are better mixedthan in a pure diffusion flame. Partially-premixed com-bustion can thus be interpreted as a combination of non-premixed and premixed combustion.Van Oijen [15] showed in simulations of laminarpartially-premixed triple flames that a chemical databaseconsisting of premixed flamelets is appropriate for mod-elling partially-premixed combustion when the lengthscale affiliated with the gradient in mixture fraction Z islarger than the flame thickness δL:

(~∇Z · ~∇Z

)−0.5

� δL . (1)

For highly turbulent flows mixing occurs very fast andgradients in mixture fraction, ~∇Z, will be small exceptnear inlets when fuel and oxidizer are inserted sepa-rately. For such cases it can be assumed that premixedcombustion dominates non-premixed combustion. Thisimplies that combustion can be better modelled by usingpremixed flamelets instead of non-premixed flamelets.For now it is assumed that partially-premixed com-bustion can be described using an Flamelet GeneratedManifold (FGM) consisting of premixed flamelets withmixture fraction covering the entire range between theflammability limits.The use of premixed flamelets enables the use of areaction progress variable c [6] for the description ofnon-equilibrium chemistry, which defines the localchemical state quantitatively between unburnt and burnt.In case of normalized mass fractions or temperature thereaction progress variable ranges between zero and unity.If mass fractions or temperature are not normalized

the reaction progress variable ranges between an initialand final value, which do not have to be equal to zeroor unity. For now the common used approach of anormalized reaction progress variable is adopted. It isinvestigated whether a reaction progress variable, whichis directly coupled to chemistry, is a better variableto describe non-equilibrium chemistry than the strainrate in non-premixed flamelets. The strength of FGMreduction technique is that the number independentcontrol variables, which is now chosen to be equal to 2(Z and c), can be increased straightforward for increasedaccuracy.When the flamelet based reduction methods like FGMare compared to conventional reduction methods whichtake only chemical kinetics into account, like the IntrinsicLow Dimensional Manifold (ILDM) reduction technique[8] for example, they prove to be more accurate in regionswhere chemistry is not dominant (since diffusion is takeninto account) and as accurate as conventional reductionmethods in regions where chemistry is dominant [15].

The turbulence-chemistry interaction is accountedfor by describing variables in a stochastic way insteadof a deterministic way: locally a variable is describedby a Probability Density Function (PDF) defining theprobability of occurrence for several states instead ofonly one fixed state that can occur. The PDF P (x) canbe thought of as the fraction of time the fluid spends instate x.For diffusion flames, in which the mixture fraction Z canbe assumed to be the main parameter, the PDF approachis well-known and it has already been implementedin commercial CFD codes like FLUENT [9]. For theintroduced reaction progress variable c an analogousapproach is applied to describe the turbulence-chemistryinteraction. The only difference is that the transportequation for c contains a chemical source term while thetransport equation for Z contains no source term.

The objective of this paper is to investigate theFGM/PDF method using premixed flamelets by compar-ison with existing models and experimental results fora partially-premixed flame, the Sandia flame D. Exist-ing methods that will be considered are the equilibriumchemistry approach and the use of a diffusion flameletwith a fixed strain rate in a flamelet model.

2 Models for turbulent aero-thermochemistryThree main simulation strategies can be distinguished:

Direct Numerical Simulation (DNS), Large Eddy Simu-lation (LES) and Reynolds Averaged Navier-Stokes Sim-ulation (RANS). In a DNS simulation all turbulent scalesdown to the Kolmochorov scale are resolved which im-plies that computational costs are very high. A compar-ison of the integral length scale with the Kolmochorovlength scale and the integral time scale with the Kol-

3

mochorov time scale shows that, for non-reacting flows,computational requirements CRNR scale with:

CRNR ∼ Re2.25

︸ ︷︷ ︸space

·Re0.5

︸ ︷︷ ︸time

= Re2.75 .

For turbulent reacting flows the reaction layer has to becaptured properly and computational requirements CRR

will not only be determined by the Reynolds number butby the Damkohler number as well [19]:

CRR ∼ (Re Da)1.5

︸ ︷︷ ︸space

· Da︸︷︷︸time

=[Re3Da5

]0.5.

The most severe requirement (CRNR or CRR) deter-mines the total required computational facilities forsimulations of turbulent reactive flows. The Reynoldsnumber and/or Damkohler number for which DNS sim-ulations can be run are thereby limited by the availablecomputational facilities.The LES approach reduces the requirements on comput-ers by only simulating the largest turbulent eddies andmodelling small eddies. This approach is justified by thefact that the large eddies are far more important for theturbulent diffusion of species, momentum and heat thansmall eddies [17]. Since the flame can still be thinnerthan the numerical cell size, for turbulent combustionmodels usually can not be as simplified as the subgridturbulence model.In RANS simulations turbulent eddies are not resolved.In both RANS and LES simulations models have tobe used to account for the eddies that are not resolved.This is the so called ”closure-problem” for the unknowncorrelations that occur in the PDE’s describing turbulentreacting flows. For the modelling of turbulent combus-tion under aero-engine conditions the RANS approachhas been chosen because the complex geometry of anaero-engine combustion chamber would make LES andDNS simulations too time-consuming.

To model turbulence, the realizable k, ε-model [14, 9]which solves equations for the turbulent kinetic energyk and the turbulent kinetic energy dissipation rate ε isused. Dimensional analysis leads to an expression foran additive turbulent viscosity which is a function of thegeometry; it is not a property of the working fluid. Therealizable k, ε-model is more suitable for axisymmetricjet flames than the standard k, ε-model [9].

Chemistry is represented by the GRImech 3.0 mech-anism [23] which contains 325 elementary reactions be-tween 53 species with hydrocarbons up to propane; it isimportant to include higher hydrocarbons than present inthe fuel to be able to describe hydrocarbon recombina-tion in rich regions. This is why a relative simple re-action mechanism like the Smooke reaction mechanism[24] will not suffice in methane flames containing richregions.

2.1 Partially-premixed combustion parametersTo describe partially-premixed combustion two control

parameters describing non-premixed (mixture fractionZ) and premixed combustion (reaction progress variablec) are important, as explained in section 1.

The mixture fraction is a conserved scalar which de-scribes the conservation of elements. According to Bilger[4] the mixture fraction is defined as [3]:

Z =0.5[

YH−YH,2

MH

]+ 2.0

[YC−YC,2

MC

]

0.5[

YH,1−YH,2

MH

]+ 2.0

[YC,1−YC,2

MC

] , (2)

in which the subscript H denotes hydrogen, the subscriptC denotes carbon, the subscript 1 denotes the fuel inletand the subscript 2 denotes the oxidizer inlet. MH rep-resents the element mass of hydrogen and MC representsthe element mass of carbon. When it is assumed that dif-fusion coefficients are equal for all species, the transportequation for the mixture fraction is described by [19]:

~∇ ·[ρ~uZ

]

︸ ︷︷ ︸convection

− ~∇ ·[ρ (D + DT ) ~∇Z

]

︸ ︷︷ ︸diffusion

= 0 , (3)

which is derived by applying the single-perturbation the-ory and subsequently Reynolds averaging the equationdescribing the transport of Z in laminar flows. The trans-port by convection and diffusion has been marked in thisequation for clarity; they will not be marked in follow-ing equations. Mixture fraction has no source term sincemixture fraction is a conserved scalar. Molecular diffu-sion is modelled by Fick’s diffusion law with diffusioncoefficient D and turbulent diffusion is modelled in asimilar way using the Boussinesq approximation whichintroduces a turbulent diffusion coefficient DT . The useof Fick’s diffusion law and an analogous approach for theinfluence of turbulence implies that preferential diffusionand counter-gradient diffusion can not occur.A random Favre averaged, or mass-averaged, variable ϕis defined as:

ϕ =ρϕ

ρ,

in which ρ is the Reynolds averaged density. Favre aver-aged variables are introduced to prevent terms containingdensity fluctuations for which closure assumptions haveto be made.

For the reaction progress variable c an indicator hasto be chosen which discriminates between unburnt, burntand intermediate stages. For c the requirement mustbe posed that it is monotonous from the initial state tochemical equilibrium in order to facilitate an unambigu-ous determination of dependent variables as a function ofc. Straightforward choices for c could be mass fractionsof reactants or products, a linear combination of species

4

mass fractions or gas temperature. The choice made for cin this study will be presented in section 2.4.The transport equation for c is derived analogous to thetransport equation for Z; the only difference is the pres-ence of a chemical source term on the right hand side. Itis described by

~∇ ·[ρ~uc − ρ (D + DT ) ~∇c

]= ωc . (4)

Both laminar diffusion of c and diffusion of c due toturbulent fluctuations is modelled similar to diffusion ofthe mixture fraction.

In all transport equations molecular diffusion is not as-sumed to be much smaller than the redistributive flux dueto turbulent fluctuations since in very hot regions withlow turbulence intensities molecular diffusion can play asignificant role in diffusion of species, momentum andheat. For both Z and c molecular diffusion coefficientsare related to molecular viscosity using laminar Schmidtnumbers Sc and turbulent diffusion coefficients are re-lated to turbulent viscosity using turbulent Schmidt num-bers ScT according to:

D =ν

Sc; DT =

νT

ScT

. (5)

Turbulent Schmidt numbers can be expected to take val-ues closer to unity than laminar Schmidt numbers sincetransport by turbulent fluctuations can be expected to bemore equal for momentum and species than transport bymolecular processes. For the mixture fraction in litera-ture [9, 13] laminar and turbulent Schmidt numbers arefound equal to 0.7 and 0.85 respectively; these Schmidtnumbers will also be used in this study. For the reactionprogress variable the same Schmidt numbers are taken.

2.2 PDF closure methodAs explained in section 1 turbulence influences the

combustion chemistry by distorting the flame. A stochas-tic description of variables is appropriate for the regimesas mentioned in section 1. All variables are describedas an ensemble of different realizations each with a cer-tain probability of occurrence. Using the flamelet ap-proach it is assumed that all variables are only a func-tion of mixture fraction Z and reaction progress variablec: ϕ = ϕ (Z, c). Variables are from here on describedstochastically by mass-weighted PDF’s, P (Z, c) and aFavre averaged variable is calculated according to:

ϕ =

∫ 1

0

∫ 1

0

ϕ (Z, c) P (Z, c) dZdc .

For the Reynolds averaged density and Reynolds aver-aged source term for c a conversion has to be introducedto be able to use the same mass-weighted PDF’s for Favreaveraged variables. The Reynolds averaged density ρ is

given by:

ρ =

[(1

ρ

)]−1

=

[∫ ∫P (Z, c)

ρ (Z, c)dZdc

]−1

. (6)

The Reynolds averaged source term ω can be written as:

ω = ρ

(ω

ρ

)= ρ

(ω

ρ

)

= ρ

∫ ∫ω (Z, c)

ρ (Z, c)P (Z, c) dZdc . (7)

The shape of the joint mass-averaged PDF, P (Z, c), isnot known but can be computed by using expensive meth-ods like Monte-Carlo simulations [21] or the use of trans-ported PDF’s [5, 14, 21]. A more simple approach is Pre-sumed PDF approach. The assumption that Z and c arestatistically independent is used allowing the joint PDFP (Z, c) to be written as the product of its two marginalPDF’s:

P (Z, c) = P (Z) P (c)

Subsequently known shapes that can be described by asmall number of parameters are presumed for the twomarginal PDF’s. In this study it is assumed that the shapeof each of the two marginal PDF’s can be described by amean and a variance implying that all variables becomea function of mean mixture fraction Z, mean reactionprogress variable c, variance of mixture fraction ξ andvariance of progress variable φ. Since Z and c can bedecomposed to a mass-averaged mean, Z respectively c,and a fluctuation, Z ′′ respectively c′′, for ξ and φ can bewritten:

ξ =(Z − Z

)2

= Z ′′2 and φ = (c − c)2 = c′′2.

A transport equation for a variance can be derived bymultiplying a standard transport equation for a variablewith the fluctuation of that variable and subsequently ap-ply Reynolds averaging. The transport equation for thevariance of the mixture fraction ξ reads:

~∇ ·[ρ~uξ − ρ (D + DT ) ~∇ξ

]

= 2C1ρDT

[~∇Z]2

− 2C2ρχξ , (8)

in which C1 and C2 are modelling constants. In thisderivation the assumption that ξ and its destruction, χξ ,scales linear to the turbulent kinetic energy k and its de-struction rate ε according to:

χξ =( ε

k

)ξ (9)

In this study C1 and C2 have been have been given thevalues used in FLUENT [9]: C1 = 1.215 and C2 = 1.This is done for reasons that will be explained later.

5

The transport equation for the reaction progress vari-able variance φ is described by:

~∇ ·[ρ~uφ − ρ (D + DT ) ~∇φ

]

= 2C3ρDT

[~∇c]2

+ 2C4c”ω − 2C5ρ( ε

k

)φ (10)

in which C3, C4 and C5 are modelling constants.For thedestruction of φ the same coupling to the k, ε turbulencemodel as for ξ, as stated in equation 9, has been applied.Equation 10 contains an additional source term c”ω whencompared to equation 8 for the variance of the mixturefraction. This additional source term occurs due to thesource term in equation 4 and is defined as:

c”ω = cω − cω. (11)

ω is defined according to equation 7 and cω is, analogousto equation 7, defined as:

cω = ρ

∫ ∫cω (Z, c)

ρ (Z, c)P (Z) P (c) dZdc.

For the variance of the mixture fraction the laminar andturbulent Schmidt numbers are assumed to be equal to0.7 and 0.85, respectively [13]. For the reaction progressvariable variance the same Schmidt numbers are as-sumed, similar to the assumption made for the reactionprogress variable. Modelling constants C3 to C5 are keptequal to unity in this study.The laminar viscosity, which is used to calculate molec-ular diffusion coefficient D according to equation 5, isdetermined using the Sutherland law for air. Since thelaminar viscosity can be assumed to be quasi-linear de-pendent on temperature it can be determined using theFavre-averaged temperature T . Hereby a complex PDFintegration like for the averaged source term (equation 7)is circumvented. The expression for the laminar viscosityreads:

ν =µ0

ρ

(T

T0

)1.5 [T0 + A

T + A

]

in which the modelling constants µ0, T0 and A have beentaken equal to those for air [9]:

µ0 = 1.7894 · 10−5 ; T0 = 273.11 ; A = 110.56

The choice to use the same modelling constants as for airis funded on the assumption that laminar viscosity willonly play a role in hot regions with a low turbulence in-tensity. This typically is the burnt-out region in whichburnt gases are further mixed with excess air implyingthat gas mixture composition will resemble the composi-tion of air relatively well.

2.3 Assumed PDF shapeSince mixture fraction and progress variable are

bounded variables a PDF with a bounded domain is pre-

ferred over PDF’s with a non-bounded domain. The β-distribution is a bounded PDF for x ∈ [0, 1] and accord-ing to FLUENT [9] it most closely represents experimen-tally observed PDF’s. The β-distribution is described by:

P (x; p, q) =

[Γ(p + q)

Γ(p)Γ(q)

]xp−1 [1 − x]

q−1 (12)

in which the gamma-function Γ is defined as [1]:

Γ(p) =

{ ∫∞

0tp−1e−tdt if p is continuous

(p − 1)! if p is discrete (13)

The two parameters p and q in the β-distribution (statedin equation 12) are related to the mean, µ, and variance,σ2, according to:

p = µ

[µ(1 − µ)

σ2− 1

]> 0 (14)

q = (1 − µ)

[µ(1 − µ)

σ2− 1

]> 0 (15)

When the PDF for Z is considered µ and σ2 in equation14 and 15 are:

µ = Z ; σ2 = ξ = Z ′′2 ∈[0, Z

(1 − Z

)]

and for the PDF for c:

µ = c ; σ2 = φ = c′′2 ∈ [0, c (1 − c)]

It can be concluded that the PDF for mixture fraction is afunction of mean mixture fraction, Z, and mixture frac-tion variance:

P (Z) = P(Z; Z, Z ′′2

)= P

(Z; Z, ξ

).

The PDF for the reaction progress can be written analo-gous:

P (c) = P(c; c, c′′2

)= P (c; c, φ) .

2.4 Determination of the laminar source term ωTo be able to use the FGM/PDF method one must be

able to compute the Reynolds-averaged source for the re-action progress variable, ωc, which depends on the lam-inar variable ωc (Z, c) according to equation 7. Accord-ing to the flamelet assumption, as explained in section 1,laminar variables can be retrieved from laminar flameletswhen Z and c are known. A schematic comparison be-tween the use of non-premixed and premixed flameletsis given in figure 1 in which cmax is the maximum valuefor c that can be achieved for non-normalized variablesat stoichiometric combustion; if normalized variables areconcerned c ranges between zero and unity. It can beseen that diffusion flamelets contain chemical informa-tion beyond flammability limits which is not included ina database consisting of premixed flamelets. On the other

6

hand non-equilibrium chemistry is expected to be betterrepresented using premixed flamelets since then the en-tire range for the reaction progress variable c is included;diffusion flamelets reduce to a pure mixing problem be-yond the maximum strain rate.

For the reaction progress variable c still an appropriatechoice has to be made and several options are available:a major reactant mass fraction or major product massfraction would be a straightforward option. A morerefined option would be a linear combination of multiplespecies mass fractions to ensure a monotonous reactionprogress variable in both lean and rich regions; a possiblecombination could include CO2, H2O, CO and H2

[2] of which the first two are typical products for leancombustion and the latter two are typical products forrich combustion.The reaction progress variable ranges from zero tounity; species mass fractions or a linear combination ofmultiple species mass fractions will typically not be inthe same range! Scaling a species mass fraction or alinear combination of multiple species mass fractionsbetween zero and unity will introduce additional terms inequation 4 [12, 6] (due to the dependence of cmax on Z):scaling of c is assumed to reduce the dependence of c onZ [12].

In this study the mass fraction of CO2 is chosen asa reaction progress variable: c = YCO2

. It satisfiesthe requirement to be monotonous increasing betweenthe flammability limits given by Cashdollar et al [7],

0 0.5 10

0.5

1

Z [−]

c/c m

ax [−

]

LFL UFL

χmax

χ0

Figure 1: Schematic comparison between non-premixedflamelets (blue lines) and premixed flamelets (blacklines). The red line, also indicated by χ0, representschemical equilibrium corresponding to a non-premixedflamelet with a strain rate equal to zero. The blue line in-dicated by χmax represents the non-premixed flamelet atits extinction limit. LFL denotes the lower flammabilitylimit and UFL denotes the upper flammability limit.

XCH4∈ [0.05, 0.15], when the GRImech 3.0 reaction

mechanism is used. The maximum value for c is afunction of Z but in this study it is assumed that c and Zcan nevertheless be considered statistically independentand a non-scaled reaction progress variable is used toavoid additional terms in equation 4.

The laminar chemical database consisting of premixedflamelets is generated using the flamelet code CHEM1D[10] which solves all PDE’s describing laminar combus-tion [19]. Heat transfer by means of radiation is not in-cluded. There can be no interaction between flamelets.A flamelet is represented by a large number of discretepoints and an adaptive grid makes sure that the reactionlayer is captured well. It must be noted that boundaryconditions for dependent variables must correspond toboundary conditions encountered in simulations in whichthe laminar chemical database will be used.

2.5 Numerical proceduresTo decrease computational costs the use of a structured

pre-integrated chemical database and an interpolationroutine in the CFD code is preferred over an integrationroutine inside the CFD code [9].In order to generate a pre-integrated chemical databasetwo steps have to be taken: first a laminar flameletdatabase has to be generated and subsequently the lami-nar flamelet database has to be integrated over multiple,different PDF’s for Z and c.As stated in section 2.2 in the laminar flamelet databaseall variables are defined as a function of mixture fractionZ and reaction progress variable c: ϕ = ϕ (Z, c). Asan example the laminar source term for CO2, which isrequired to compute the Reynolds averaged source termfor c, ωc, is shown in figure 2.The PDF integration routine integrates the laminarflamelet database with a shape for the PDF’s of Z and cdefined by Z, ξ, c and φ. All Favre-averaged variablesare defined by a combination of mean mixture fractionZ, mixture fraction variance ξ, mean reaction progressvariable c and reaction progress variable variance φ:ϕ = ϕ

(Z, ξ, c, φ

). This procedure is performed for

multiple combinations of Z, ξ, c and φ and averagedvariables ϕ are stored in a structured database so thatthey can be retrieved as a function of Z, ξ, c and φ. Afast look-up procedure to retrieve chemical data duringa CFD computations is possible since the locationof the requested data in the integrated database canbe determined directly due to the known structureof the database. Figure 3 shows the dependence ofthe Reynolds averaged source term for c, ωc, on thenormalized mixture fraction variance and normalizedreaction progress variable variance at Z = 1.03Zst andc = 0.53cmax(Zst). These values for Z and c correspondto the region in the laminar chemical database wherethe laminar source term for c, ωc, reaches its maximum

7

value as can be seen from figure 2.

To compute the incomplete β-function as defined byequation 12, Numerical Recipes [20] recommends a poly-nomial for the Γ-function described in equation 13. Forthe integral of the β-distribution P (x; p, q), as described

40

90

140Sourcec

[kg/m3s]

0.20.3

0.40.5

Z [-]0.050.11c [-]

Source c: 0 20 40 60 80 100 120 140

Figure 2: FGM for the GRImech 3.0 mechanism display-ing the source for the reaction progress variable as a func-tion of mixture fraction (Z) and reaction progress vari-able (c). Z ranges between the flammability limits. Themanifold consists of 101 flamelets each discretized in 80points.

40

90

140

Sourcec

[kg/m3s] 0

0.250.5

0.751 Scaled variance Z [-]

00.25

0.50.75

1

Scaled variance c [-]

Source c: 0 20 40 60 80 100 120 140

Figure 3: Source term for c as a function of the mixturefraction variance and the reaction progress variable vari-ance at Z = 1.03 Zst and c = 0.53 cmax(Zst). Both vari-ances have been normalized with their maximum value.

in equation 12, a continued fraction approach is recom-mended [20]. For optimal numerical efficiency the PDFintegration routine contains three innovative routines:

1. Gridpoints are clustered in area’s where the PDFhas relatively large values using an adaptive grid.During integration the domains for Z ∈ [0, 1] and(c/cmax) ∈ [0, 1] are divided in multiple subdo-mains; it is made sure that the integrated probabilityof each subdomain is equal by using a Newton iter-ation containing the PDF and the cumulative prob-ability density function (CDF) to find the boundingcoordinates of each subdomain.

2. Integration points are placed in the middle of thebounding coordinates of each subdomain whichhave been determined by the adaptive gridding rou-tine. This can cause the real mean and varianceto differ slightly from the imposed mean and vari-ance. To reconstruct the imposed mean and variancethe real mean and variance, which can be computedfrom the integration points and the PDF, are modi-fied by scaling the PDF height and introducing twoDirac functions at the extremes of the domain (forthe mixture fraction Z = 0 and Z = 1 and for thereaction progress variable c = 0 and c = cmax).

3. The PDF-averaged variables are calculated on aninitial grid and subsequently on a more refined gridwith twice the number of gridpoints in both direc-tions (Z and c). The the PDF-averaged variablescomputed on the finer grid are compared to the val-ues on the previous, more coarse grid. This processis repeated until a convergence criterion is reachedwhich is based on the Reynolds averaged densityρ and the Reynolds averaged source term for theprogress variable ωc and is set to 10−3. The max-imum number of subdomains in both directions wasset to 160.

3 ResultsThe FGM/PDF model will be compared to an estab-

lished model for turbulent combustion to prove its con-cept. The comparison will be made for a well-knowntest case to be able to compare numerical results for bothmodels with experimental data in order to point out ad-vantages and disadvantages of the FGM/PDF concept.

3.1 Description test caseThe Sandia Flame D was chosen as a test case. It

is a piloted jet diffusion flame. This flame is chosenfrom a series of flames, ranging from laminar (FlameA) to highly turbulent (Flame F). Flame D (Re=22.400based on fuel inlet diameter) is a turbulent flame forwhich measurement data of both the flow (velocitiesand turbulence) [22] and species [3] is available. Sincethe turbulence-chemistry interaction is only moderate

8

Figure 4: Configuration of Sandia flame D. All sizesare in mm. The sketch is not to scale. Courtesy ofH.A.J.A. van Kuijk.

there is no clear advantage for the FGM/PDF modelbeforehand. Because the fuel is premixed with airrecombination of hydrocarbons and the production ofsoot are reduced. This allows the use of the GRImech3.0 mechanism [23] which includes hydrocarbons up topropane.

The fuel inlet is a cylindrical tube with a diameter ofd = 7.2 mm and the fuel consists of XCH4

= 0.25 andXair = 0.75. It is surrounded by a pilot inlet tube, havinga diameter of 18.2 mm. In the pilot mixture fraction Zequals 0.77 Zst and the gas mixture composition equalsthe chemical equilibrium composition for this value ofZ. The burner is placed in a wind tunnel blowing onlyair. The computational geometry for the Sandia FlameD is shown in figure 4. A structured numerical gridconsisting of 40.000 cells with refinements on the inletsand along the axis of symmetry, the y-axis, is used forcomputations on the geometry shown in figure 4. Theradial coordinate is indicated by r. The smallest cell hasa surface of 0.26 mm2, the largest cell has a surface of1.7 · 103 mm2.

In the inlets at y = 0 (air, pilot and fuel) the ax-ial velocity, turbulent kinetic energy and turbulent ki-netic energy dissipation rate inlet profiles are prescribed

by measured profiles [3]. For mixture fraction and re-action progress variable values see table 1. The valuefor c in the pilot inlet corresponds to the chemical equi-librium value of c for the value of Z in the pilot inletwhich implies that all values in the pilot inlet are equalto chemical equilibrium values. For the tube walls a no-slip boundary condition is prescribed. For the outlet aconstant pressure, dictated by the Sandia Flame D de-scription [3], which is equal to the inlet pressure is pre-scribed: p = 1.0062·105Pa. Under the assumption that atr = 300 mm radial velocities approximately equal zero,a zero-shear wall has been used as a far-field boundarycondition.

Z cFuel inlet 1.00 0.00Pilot inlet 0.27 0.1156Air inlet 0.00 0.00

Table 1: Mixture fraction (Z) and reaction progress vari-able (c) inlet conditions.

3.2 Accuracy integrated databaseThe laminar chemical database generated using the

FGM reduction method as described in section 2.4 iscreated for the Sandia Flame D fuel inlet boundaryconditions [3]: T = 294 K and p = 1.0062 · 105 Pa. Forthe domain between the flammability limits the valuesXCH4

∈ [0.05, 0.15] have been used which correspond tovalues for the mixture fraction equal to Z ∈ [0.18, 0.56].The laminar chemical database contained 101 flameletseach discretized in 80 points using an adaptive grid tocapture the reaction layer. The adaptive grid made surethat the increase of CO2 between two subsequent pointswas equal over the entire domain.

During the integration procedure values for Z outsidethe flammability limits variables are interpolated. For val-ues for Z below the lower flammability limit variables areinterpolated between the oxidizer stream and the lean-est flamelet. When values for Z are beyond the upperflammability limit variables are interpolated between therichest flamelet and the fuel stream.For the integrated table 26 uniformly distributed mixturefraction values have been taken to properly capture theexpected maximum value for c and T , which lies a fewpercent on the rich side of Zst = 0.351 in the lami-nar chemical database. The reaction progress variablec is represented by 20 linearly distributed values. Forboth the mixture fraction variance ξ and reaction progressvariable variance φ 10 values are taken. Values for vari-ances are clustered more dense for small values since in-tegrated variables show a strong dependence on variancesfor small values of variances. Clustering of values for

9

0.2 0.3 0.4 0.51400

1700

2000

2300

Z [−]

T [K

]

Figure 5: Temperature at chemical equilibrium as a func-tion of mixture fraction for the laminar chemical database(circles) and the PDF-integrated database (solid line).

0 0.2 0.4 0.6 0.8 1300

800

1300

1800

2300

c/cmax [−]

T [K

]

Figure 6: Temperature as a function of reactionprogress variable at stoichiometric conditions for the lam-inar chemical database (circles) and the PDF-integrateddatabase (solid line).

variances is visible in figure 3: it can be seen that val-ues for the mixture fraction variance are clustered moresevere than values for the reaction progress variable vari-ance.In figure 5 the temperature at chemical equilibrium as a

function of mixture fraction is shown for both the FGMand as it is retrieved from the integrated table using linearinterpolation: it shows that the temperature is capturedwell using linear interpolation in combination with thechosen distribution for Z values. When the interpolationerror ε is defined as ε =

[TL−TT

TL

], in which subscript L

denotes data from the laminar database and subscript Tdenotes data from the integrated database, the maximumerror is reached at Z = 0.98 Zst and is less than 1%.In figure 6 the temperature at stoichiometric mixture frac-tion as a function of the reaction progress variable isshown for both the FGM and as it is interpolated fromthe integrated table. Since the profile does not have anysharp peaks, it is reproduced very well using linear inter-polation between 20 values for the reaction progress vari-able c. There is a very small region (c/cmax ∈ [0, 0.05])

where errors up to 10% occur. Outside this region errorsare always less than 0.5%. The same definition for theinterpolation error ε is used.

3.3 Comparison of results for the FGM/PDF methodTo show the importance of the inclusion of both re-

action progress variable and the turbulence-chemistry in-teraction for the reaction progress variable three versionsFGM/PDF models are compared:

1. Full FGM/PDF model: equations for Z, c, ξ andφ are solved. This model contains non-equilibriumchemistry and the interaction between turbulenceand chemistry for both Z and c.

2. Reduced FGM/PDF model: only equations for Z,c and ξ are solved. The equation for the variancefor the reaction progress variable φ, equation 10, isexcluded from the set of equations and φ is takenequal to zero. This exclusion implies that the PDFfor c is always represented by a Dirac δ-function atc.

3. Chemical equilibrium FGM/PDF model: both equa-tions for c and φ are excluded from the set of equa-tions. This implies that convection and diffusion forc are not accounted for. Chemistry is assumed to beinfinitely fast (c = cmax) and mixing is rate-limiting.

As reference model a diffusion flamelet from a coun-terflow flame with a strain rate of 100 s−1 has beenused [11]. This moderate strain rate has been chosenbecause the diffusion flamelet should exhibit only littlelocal extinction corresponding to experimental data. Thislaminar chemical database has been integrated with aβ-PDF using PRE-PDF included in FLUENT.

The results of the model consist of variables of theflow and the combustion chemistry. First two mainproperties of the flow, the axial velocity and the turbulentkinetic energy k, are discussed because they determinemixing processes. Numerical results for all FGM/PDFmodels and the diffusion flamelet model are compared toexperimental data of Schneider et al [22].In figure 7 the axial velocity Uax on the symmetry axis ispresented for the all FGM/PDF models and the diffusionflamelet model. All models predict approximately thesame values for the velocity magnitude in the entiredomain; for 35 < y/d < 75 predictions from theFGM/PDF model are slightly closer to measurements.In figure 8 predictions of the turbulent kinetic energyalong the symmetry axis show that the maximum valuepredicted by all versions of the FGM/PDF model isapproximately 10% higher than predicted by the dif-fusion flamelet model . The shapes of both curves aresimilar. Both models predictions only correspond toexperimental data in the downstream part (y/d > 40):the broad plateau for 20 < y/d < 40 is not predicted by

10

0 25 50 75 1000

25

50

75

y/d

U−ax

ial [

m\s

]

Figure 7: Axial profile (r = 0) of the axial velocityUax for the full FGM/PDF model (solid line), the re-duced FGM/PDF model (dotted line), the chemical equi-librium FGM/PDF model (dash-dotted line) and the dif-fusion flamelet model (dashed line) together with mea-surements (circles).

0 25 50 75 1000

20

40

60

y/d

k [m

2 /s2 ]

Figure 8: Axial profile (r = 0) of the turbulent kineticenergy k for the full FGM/PDF model (solid line), the re-duced FGM/PDF model (dotted line), the chemical equi-librium FGM/PDF model (dash-dotted line) and the dif-fusion flamelet model (dashed line) together with mea-surements (circles).

either model.

Now that the main properties of the flow are describeda closer look is taken at the combustion process. Pre-dicted values for mixture fraction and combustion scalarsare compared to experimental data of Barlow et al [3].

Thus figure 9 presents the mean mixture fractionZ on the symmetry axis for all models together withexperimental measurements. It shows that all FGM/PDFmodels predict similar values. For 20 < y/d < 60the diffusion flamelet model predicts values for Z thatare slightly higher than the FGM/PDF models. This iscaused by predicted values for k which are lower for thediffusion flamelet model than for the FGM/PDF modelsfor 20 < y/d < 40 as has been shown in figure 8. Thereis no clear advantage of the FGM/PDF models over the

0 25 50 75 1000

0.25

0.5

0.75

1

y/d

Z [−

]

Figure 9: Axial profile (r = 0) of the mean mixture frac-tion Z for the full FGM/PDF model (solid line), the re-duced FGM/PDF model (dotted line), the chemical equi-librium FGM/PDF model (dash-dotted line) and the dif-fusion flamelet model (dashed line) together with mea-surements (circles).

0 25 50 75 1000

0.005

0.01

0.015

0.02

y/d

Z−va

rianc

e [−

]

Figure 10: Axial profile (r = 0) of the mixture frac-tion variance ξ for the full FGM/PDF model (solid line),the reduced FGM/PDF model (dotted line), the chemicalequilibrium FGM/PDF model (dash-dotted line) and thediffusion flamelet model (dashed line) together with mea-surements (circles).

diffusion flamelet model or vice versa for predictions ofZ.In figure 10 can be seen that the shape of the curve forthe variance of the mixture fraction (ξ) is similar forall FGM/PDF models and the diffusion flamelet modelalthough the position at which the maximum value forξ is reached is better predicted by the diffusion flameletmodel. All FGM/PDF models overpredict the maximumvalue for ξ; the diffusion flamelet model underpredictsthe maximum value for ξ. Predictions made by the diffu-sion flamelet model correspond better to measurementsthan predictions made by any FGM/PDF model. For allFGM/PDF models modelling constants in equation 8have been taken equal to constants used by FLUENT [9]to facilitate a fair comparison with the diffusion flameletmodel.

11

Figure 11 presents the mean reaction progress variablec on the symmetry axis for all models together withexperimental measurements. In the upstream regiony/d < 60 (d denotes the fuel inlet diameter) there is aclear discrepancy between all FGM/PDF models on onehand and the diffusion flamelet model and experimentaldata on the other hand. In this region the flame isnot in equilibrium. Predictions made by the diffusionflamelet model correspond better to measurements thanpredictions made by any FGM/PDF models. In thedownstream region y/d > 60 the reduced- and chemicalequilibrium FGM/PDF model predictions convergetowards the diffusion flamelet predictions.For 30 < y/d < 40 a clear distinction is visible betweenthe chemical equilibrium FGM/PDF model on one handand the reduced- and full FGM/PDF model on the otherhand: the chemical equilibrium underpredicts valuesfor c. This is explained by the fact that in the chemicalequilibrium model convection and diffusion of c is notincluded. Since there is a large radial gradient in c at thesymmetry axis for 20 < y/d < 40, as can be seen infigure 12, convection and diffusion will play an importantrole in the transport of c towards the symmetry axis.These transport phenomena are included in the reduced-and full FGM/PDF models and therefore these modelsare assumed to be more accurate. For 60 < y/d < 80the predictions for c made by the full FGM/PDF modelcorrespond better to experimental data than all othermodels. This can be explained by equation 1: in regionswith small gradients in Z, like the downstream part ofthe computational domain, chemistry can be representedbetter using premixed flamelets than using non-premixedflamelets. In contrary, the upstream part y/d < 60,where gradients in Z are large, combustion can be betterrepresented by using non-premixed flamelets. Thisexplains why the diffusion flamelet model yields betterresults in the upstream part of the computational domain.

The reaction progress variable variance φ predicted bythe full FGM/PDF model is shown in figure 13 togetherwith the mean reaction progress variable c. They areplotted together to point out that the location of thewiggle in φ (y/d ≈ 45) corresponds to the location ofthe maximum value for c. According to equation 10the production of φ scales with the square of the spatialgradient of c; when a local maximum (or minimum) ofc is reached spatial gradients of c equal zero and therewill be no production of φ. If the influence of convectionand diffusion is assumed to be constant in the regionclose to the peak in c the value for φ will decrease sincedestruction is a non-zero value. The predicted trend issimilar to experimental data but the full FGM/PDF modeloverpredicts φ for 30 < y/d < 55 and underpredicts φfor 55 < y/d < 80.

In figure 14 temperature predictions made by allFGM/PDF models are compared to predictions madeby the diffusion flamelet model and measurements. Inthe upstream part y/d < 40 predictions made by thereduced- and chemical equilibrium FGM/PDF modelscorrespond almost exactly to predictions made by thefull FGM/PDF model. In the downstream part the pre-dictions made by the reduced- and chemical equilibriumFGM/PDF resemble the predictions of the diffusionflamelet model better than the predictions made by thefull FGM/PDF model. In the upstream part y/d < 55the diffusion flamelet approach corresponds better tomeasurements; in the downstream part (y/d > 55) thefull FGM/PDF model using premixed flamelets predictsvalues closer to measurements.The small wiggle at y/d ≈ 45 in the temperature profilepredicted by the full FGM/PDF model is most probablycaused by the wiggle in the predicted profile for φ as

0 25 50 75 1000

0.04

0.08

0.12

0.16

y/d

c [−

]

Figure 11: Axial profile (r = 0) of the mean reac-tion progress variable c for the full FGM/PDF model(solid line), the reduced FGM/PDF model (dotted line),the chemical equilibrium FGM/PDF model (dash-dottedline) and the diffusion flamelet model (dashed line) to-gether with measurements (circles).

0 2 4 6 80

0.05

0.1

0.15

r/d

c [−

]

Figure 12: Radial profile of the mean reaction progressvariable c for the full FGM/PDF model at y/d = 20(solid line), y/d = 30 (dashed line) and y/d = 40 (dottedline). Convection and diffusion of c is included in the fullFGM/PDF model.

12

shown in figure 13. This conviction is confirmed by thefact that predictions made by the reduced- and chemicalequilibrium FGM/PDF model, in which no equation forφ is solved and φ is set to zero, do not exhibit this wiggle.

In figure 15 the methane mass fraction, YCH4, predic-

tions made by the full FGM/PDF model and the diffusionflamelet model together with measurements are shown.The diffusion flamelet model predictions correspondbetter to measurements than predictions made by the fullFGM/PDF model.Figure 16 shows the oxygen mass fraction, YO2

, pre-dictions made by the full FGM/PDF model and thediffusion flamelet model together with measurements.Similar to figure 11 the diffusion flamelet model yieldsbetter predictions in the upstream region y/d < 60 andthe full FGM/PDF model gives better predictions in the

0 25 50 75 100

0.08

0.16

y/d

c [−

]

0 25 50 75 100

0.001

0.002c−

varia

nce

[−]

0 25 50 75 100

0.001

0.002

Figure 13: Axial profile (r = 0) of the mean reactionprogress variable c (solid line) together with measure-ments of c (circles) and the reaction progress variablevariance φ (dashed line) together with measurements ofφ (triangles).

0 25 50 75 1000

750

1500

2250

y/d

T [K

]

Figure 14: Axial profile (r = 0) of temperature for thefull FGM/PDF model (solid line), the reduced FGM/PDFmodel (dotted line), the chemical equilibrium FGM/PDFmodel (dash-dotted line) and the diffusion flamelet model(dashed line) together with measurements (circles).

0 25 50 75 1000

0.04

0.08

0.12

0.16

y/d

Mas

s fra

ctio

n CH

4 [−]

Figure 15: Axial profile (r = 0) of the methane massfraction (YCH4

) for the full FGM/PDF model (solid line)and the diffusion flamelet model (dashed line) togetherwith measurements (circles).

0 25 50 75 1000

0.05

0.1

0.15

0.2

y/d

Mas

s fra

ctio

n O

2 [−]

Figure 16: Axial profile (r = 0) of the oxygen mass frac-tion (YO2

) for the full FGM/PDF model (solid line) andthe diffusion flamelet model (dashed line) together withmeasurements (circles).

downstream region 60 < y/d < 80.

Figure 17 shows radial profiles for the mean mixturefraction Z and the mixture fraction variance φ predictedby the full FGM/PDF model and the diffusion flameletmodel together with measurements. Radial profiles havebeen taken at three heights from the inlets: y/d = 15,y/d = 45 and y/d = 75. Similar to predictions at thesymmetry axis shown in figure 9 there is no model ofwhich its predictions correspond better to measurementsthan other models. For predictions of the mixture frac-tion variance the diffusion flamelet model correspondsslightly better to measurements than predictions made bythe full FGM/PDF model.In figure 18 radial profiles for the mean reaction progressvariable c and temperature T are shown at the same dis-tances from the inlets as figure 17. The radial profiles forc at y/d = 15 and y/d = 45 show that the full FGM/PDFoverpredicts values for c. This is in correspondence to theaxial profile for c predicted by the full FGM/PDF modelas shown in figure 11.

13

0

0.5

1Z [−]

0

0.02

0.04Z−variance [−]

0

0.5

1

0

0.02

0.04

0 5 100

0.5

1

r/d0 5 10

0

0.02

0.04

r/d

y/d=15

y/d=45

y/d=75

Figure 17: Mean mixture fraction Z and mixture fraction variance ξ as a function of radius at three heights above theinlet plane: y/d = 15, y/d = 45 and y/d = 75. The full FGM/PDF model (solid line) and the diffusion flamelet(dashed line) are shown together with measurements (circles).

0

0.05

0.1

0.15c [−]

0

750

1500

2250T [K]

0

0.05

0.1

0.15

0

750

1500

2250

0 5 100

0.05

0.1

0.15

r/d0 5 10

0

750

1500

2250

r/d

y/d=15

y/d=45

y/d=75

Figure 18: Mean reaction progress variable c and temperature T as a function of radius at three heights above the inletplane: y/d = 15, y/d = 45 and y/d = 75. The full FGM/PDF model (solid line) and the diffusion flamelet (dashedline) are shown together with measurements (circles).

14

The diffusion flamelet model yields better predictionsfor c not just at the symmetry axis for y/d < 60 but alsoin radial direction at y/d = 15 and y/d = 45. For thetemperature T predictions made by the full FGM/PDFmodel do not deviate significantly from predictions madeby the diffusion flamelet model for y/d = 15 and y/d =45. At y/d = 75 predictions made by the full FGM/PDFcorresponds slightly better to measurements than predic-tions of the diffusion flamelet model.

4 DiscussionThe Sandia Flame D has been used as a first test case

to prove the concept of the FGM/PDF approach for thesimulation of turbulent partially-premixed combustion.The innovation of the FGM/PDF method compared toothers is that it takes non-equilibrium into account byusing detailed chemistry information from premixedflamelets. In highly turbulent flows it is assumedthat information from premixed flamelets describeschemistry better than information from non-premixedflamelets since species are mixed very fast by turbulenteddies. The influence of turbulence has been modelledusing the realizable k, ε turbulence model. To test theturbulence-combustion interaction of the full FGM/PDFmodel it has been compared with a reduced FGM/PDFversion, a chemical equilibrium FGM/PDF version, adiffusion flamelet model and experimental data fromliterature. The reduced FGM/PDF model includes atransport equation for the reaction progress variable c butcontains no turbulence-chemistry interaction for c. Thechemical equilibrium FGM/PDF model only describedthe transport and turbulence-chemistry interaction for themixture fraction Z.

The predictions of axial velocity and the mixturefraction made by all FGM/PDF versions agreed well,both qualitatively and quantitatively, with the diffusionflamelet model and measurements. All versions of theFGM/PDF model and the diffusion flamelet model didnot predict the magnitude of the turbulent kinetic energycorrectly. This is most probably caused by shortcomingsof the (realizable) k, ε turbulence model used by allversions of the FGM/PDF model and the diffusionflamelet model. A more refined turbulence model like theReynolds Stress Model (RSM), or using LES [18] insteadof RANS (with the k, ε turbulence model) simulations,could yield better results since turbulence is generallypredicted better.For the mixture fraction variance ξ the diffusion flameletmodel gives a better prediction than the full FGM/PDFmodel. The overprediction of ξ by all versions of theFGM/PDF model is most probably caused by a largergradient in Z in the region 20 < y/d < 60.

The FGM/PDF models overestimate the mean reactionprogress variable c by as much as 20% compared to

measurements in the region 40 < y/d < 55. Thisoverestimation is probably due to the use of premixedflamelets instead of diffusion flamelets to describenon-premixed combustion. The sharp peak of c producesa wiggle in the reaction progress variable variance φdue to applied modelling assumptions for the productionterm of φ.

In the upstream region of the computational domain(y/d < 60) where mixture fraction gradients are largethe diffusion flamelet model gives a better prediction forthe temperature than the full FGM/PDF model. In thedownstream region (y/d > 60) where mixture fractiongradients are relatively small, and thus premixed com-bustion phenomena will dominate, the full FGM/PDFmodel using premixed flamelets gives a better predictionthan the diffusion flamelet model. The wiggle in φ aty/d ≈ 45 causes all profiles for interpolated variables,like temperature and species concentrations, to exhibit asimilar wiggle.

It is important to notice that the full FGM/PDF andthe diffusion flamelet model corresponded better toeach other than any of them with the experiments. Theshortcomings in modelling of influences from turbulenceseem to be more limiting than shortcomings in thepredictions of chemistry using either the full FGM/PDFmodel or the diffusion flamelet model. Results canbe improved slightly without using another turbulencemodel by fine-tuning modelling constants in equations 8and 10. In this study for equation 8 modelling constantshave been taken equal to constants used in FLUENT [9]to facilitate a fair comparison between the FGM/PDFmodels and the diffusion flamelet model.

The results presented in this work show that a dif-fusion flame with a moderate turbulence-combustioninteraction like the Sandia Flame D can be describedapproximately as accurate using the full FGM/PDFmodel as by a diffusion flamelet model. In the vicinityof the separate fuel- and oxidizer inlets (y/d ≤ 30),where predicted turbulence intensity is relatively low, thefull FGM/PDF model using premixed flamelets does notpredict variables as good as the diffusion flamelet model.This was expected since due to the separate inlets andthe low turbulence intensity gradients in mixture fractionare large and non-premixed combustion will dominatepremixed combustion.The conclusion can be drawn that in a non-premixedregion with a moderate deviation from chemical equilib-rium the full FGM/PDF model does not perform as wellas a diffusion flamelet model although a reaction progressvariable takes this deviation from chemical equilibriuminto account. The fact that the full FGM/PDF modeldoes not perform as good as the diffusion flamelet modelis probably caused by the fact that there is no interaction

15

between flamelets in the full FGM/PDF model; in non-premixed regions there is a strong diffusive interactionbetween different Z levels.

It can be concluded that the full FGM/PDF modelyields results approximately as good as established mod-els like the diffusion flamelet model for a test case witha moderate turbulence-chemistry interaction. It is ex-pected that the full FGM/PDF model will give better pre-dictions than the diffusion flamelet model for turbulentflames with a higher turbulence intensity. Gradients in Zwill then be smaller and non-premixed combustion willnot dominate premixed combustion anymore. With in-creasing turbulence intensity larger regions can then bedescribed properly using the full FGM/PDF model.

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