1-s2.0-S000925090400315X-main.pdf

Chemical Engineering Science 59 (2004) 34773490www.elsevier.com/locate/ces

PDF modelling of turbulent non-premixed combustionwith detailed chemistry

Haifeng Wang, Yiliang ChenDepartment of Thermal Science and Energy Engineering, University of Science and Technology of China, Heifei,

Anhui 230027, Peoples Republic of China

Abstract

Although the signi/cant advantage for the probability density function (PDF) methods of the exact treatment of chemical reactions inturbulent combustion problems, a detailed chemistry mechanism (e.g., the GRI mechanism) has not been implemented in the practicalcalculations by now due to the prohibitive computation of PDF methods. In this work, a detailed mechanism (GRI-Mech 3.0, consistingof 53 species and 325 elemental reactions) is /rstly incorporated into the PDF calculation of a turbulent non-premixed jet 6ame (SandiaFlame D). The 6ow is formulated in the boundary layer form. The joint composition PDF closure level is applied and a multiple-time-scale(MTS) k turbulence model is combined for the closure of turbulent transport terms. The molecular mixing process is modelled bythe Euclidean minimum spanning tree (EMST) mixing model. The solutions are obtained by using the space marching algorithm forturbulence equations and node-based Monte Carlo particle method for PDF evolution equation. The chemical reaction source terms areintegrated directly. Extensive comparisons between the predictions and the measurements are made, which involve radial pro/les of meanand rms (root mean square), conditional mean, scatter plots of scalars and conditional PDF distribution etc. The 6ame structures are wellrepresented by the present calculation, including intermediate species (e.g. CO and H2) mass fractions, pollutant NO emission and localextinction.? 2004 Elsevier Ltd. All rights reserved.

Keywords: Turbulent non-premixed combustion; PDF modelling; Detailed chemistry; Multiple-time-scale k model; EMST mixing model

1. Introduction

Turbulent non-premixed combustion takes place widelyin chemical engineering and industrial applications. Akey issue arising from combustion problems is the stronginteractions between turbulent 6uctuations and chemicalreactions, which make the failure of conventional momentclosure to calculate turbulent combustion problems. Duringthe past two decades, several closure methods have been ad-vanced to treat such interactions, e.g. the probability densityfunction (PDF) methods (Pope, 1985, 1994; Dopazo, 1994),steady and unsteady 6amelet models (Peters, 1984; Pitschet al., 1998), conditional moment closure (CMC)(Klimenko, 1990; Bilger, 1993) and one dimensional tur-bulence (ODT) model (Hewson and Kerstein, 2002) etc.In the aspect of experimental investigations, several

kinds of burners have been designed and measured in TNF

Corresponding author. Tel.: +86-551-3601650;fax:+86-551-3606459.

E-mail address: [email protected] (H. Wang).

workshop (1996), e.g. piloted jet 6ames, bluF-body stabi-lized jet 6ames and swirling jet 6ames, to investigate thein6uence of turbulence on 6ame structures. These 6ameshave become standard test cases for the combustion mod-els and have been simulated extensively (see proceedingsof TNF workshop, 1996). Piloted jet 6ames are geometri-cally simplest, while it still re6ects strong interactions be-tween turbulence and /nite rate chemistry. Therefore, theyare preferred for the validation of combustion models. Oneof the piloted jet 6ames (Sandia 6ame D, Barlow and Frank(1998), experimental data sets are available from the web-site of TNF workshop, 1996) is mainly concerned with inthe present work.The piloted jet 6ame (6ame D) has been extensively sim-

ulated, including 6amelet simulations (Pitsch et al., 1998;Coelho and Peters, 2001; Pitsch, 2002), CMC simulations(Roomina and Bilger, 2001) and PDF simulations (Lindstedtet al., 2000; Tang et al., 2000; Xiao et al., 2000; Xu andPope, 2000) etc (more simulations can also be found in theproceedings of TNF workshop, 1996). Finite rate chemistrycan all be handled by above three models, while detailed

0009-2509/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2004.05.015

3478 H. Wang, Y. Chen / Chemical Engineering Science 59 (2004) 34773490

mechanisms are only easily incorporated into the 6ameletmodels and CMC. For PDF methods, it is not tractable toperform arbitrary detailed mechanism because of its heavycomputational cost. Reducing computational time is a mainresearch task of PDF methods. Usually reduced mecha-nisms, e.g. for methane oxidation, reduced four-step mecha-nism (Peters and Kee, 1987; Bilger et al., 1990), are appliedin the PDF calculations (Chen et al., 1989; Jones and Kakhi,1997, 1998). Saxena and Pope (1999) incorporate a skele-tal mechanism in the PDF methods by using the in situadaptive tabulation (ISAT) algorithm (Pope, 1997) whicheKciently decreases the computational time. In Sung et al.(1998), it is recognized that, with the increasing compu-tational capacity, conventional reduced mechanisms (four-or /ve-step) are not necessary, and eForts to develop aug-mented reduced mechanisms (ARMs) are worthwhile tobe made. An ARM for methane oxidation consisting of 16species and 12-step reactions is generated by Sung et al.(1998) from GRI-Mech 1.2 (GRI Website, 1995) andARMs containing nitrogen chemistry are then developedbased on GRI-Mech 3.0 (GRI Website, 1995) by Sung et al.(2001). Xu and Pope (2000) and Tang et al. (2000) incor-porate ARM in their PDF calculations of 6ame D by usingISAT algorithm (Pope, 1997; Saxena and Pope, 1999). The6ame structures are well predicted by their calculations, in-cluding the pro/le of CO in the fuel-rich region. Usually theCO concentration in the fuel-rich region is under-predictedby conventional reduced mechanisms (Chen et al., 1989)and skeletal mechanism (Saxena and Pope, 1999), whichshows the superiority of ARMs. Similar improvement isalso reported by Lindstedt et al. (2000), where a compre-hensive reduced mechanism containing 16 independent, 4dependent and 28 steady-state species is adopted.Although the progress of reduced mechanisms and im-

proved predictions of PDF methods with developed reducedmechanisms, a detailed mechanism has not been imple-mented in the practical PDF calculations and the examina-tion of detailed mechanisms performance in the PDF simula-tions is desired. Chen (Proceedings of third TNF workshop,1996) simulated 6ame D using joint scalar PDFmethod witha detailed mechanism (GRI-Mech 1.2) on a parallel cluster,while poor results were obtained, which may be attributedto the small number of particles (50 particles/cell) used forthe Monte Carlo simulation (Pope, 1981). In the presentwork, we concentrate on the implementation of a detailedmechanism (GRI-Mech 3.0) in the PDF modelling of 6ameD within acceptable computational time.In the following sections, the experimental conditions for

6ame D are outlined /rstly. The modelling methods aredescribed subsequently, including turbulence model, scalarjoint PDF evolution equation and small-scale mixing modeletc. Then the numerical solution methods and speci/cationsof boundary conditions are discussed. The numerical re-sults are presented and compared with experimental datain the next section, and conclusions are drawn in the /nalsection.

D

Fuel

jet

Pilo

ted

flam

e

Air c

oflo

w

DP

Fig. 1. Burner geometry of Sandia 6ame D.

2. Sandia Flame D

The piloted methane/air turbulent non-premixed jet 6ame(Sandia 6ame D of Barlow and Frank, 1998) is chosen asnumerical test case. The geometry of the burner is illustratedin Fig. 1. The jet 6ows consist of three parts, the fuel jet, thepiloted 6ame and the air co-6ow. The inner diameter of thefuel jet nozzle is D=7:2103 m and the outer diameter ofthe annular piloted 6ame is DP=18:4103 m. The fuel isa mixture of air and methane with the ratio 3:1 by volume.The temperature of the fuel is 294 K, and the bulk velocityof the fuel jet equals 49:6 m=s (2 m=s). The annular piloted6ame is a lean mixture ( = 0:77) of C2H2, H2, air, CO2and N2 with the same nominal enthalpy and equilibriumcomposition as methane/air at the same equivalence ratio.The bulk velocity of the pilot is 11:4 m=s (0:5 m=s). Theair co-6ow temperature is 291 K, and the velocity is 0:9 m=s.The velocity /elds of 6ame D are measured by Schneider

et al. (2003) and the scalar /elds are measured by Barlowand Frank (1998). The measured scalars include tempera-ture, mixture fraction, CH4, O2, CO2, H2O, CO, H2, OH andNO. The mixture fraction is de/ned as

=0:5(YH YH;2)=WH + 2(YC YC;2)=WC0:5(YH YH;2)=WH + 2(YC YC;2)=WC : (1)

The stoichiometric mixture fraction for 6ameD is st=0:351.

3. Modelling methods

Due to the high expense of PDF methods in dealing withdetailed chemical reaction mechanism, modelling methodsshould be well designed to accommodate the computational

H. Wang, Y. Chen / Chemical Engineering Science 59 (2004) 34773490 3479

cost. In this work, the boundary layer approximation isadopted to reduce the 6ow to parabolic type 6ow. Thejoint scalar closure level PDF method is employed to re-duce the dimension of the joint PDF and to facilitate theusage of node-based Monte-Carlo algorithm (Pope, 1981)which makes the solution easier. For the model closure, amultiple-time-scale (MTS) k turbulence model of Kimand Chen (1989) is introduced to calculate the turbulence/eld, and the Euclidean minimum spanning trees (EMST)mixing model developed by Subramaniam and Pope (1998)is employed to model the small-scale mixing term appearingin the PDF evolution equation. Chemical reactions are de-scribed by the newest version of GRImechanism, GRI-Mech3.0 (GRI Website, 1995), which consists of 53 species and325 elemental reactions and contains nitrogen chemistry.

3.1. Parabolized NavierStokes equations

Parabolic type 6ow can be described by the parabolizedNavierStokes (PNS) equations (Rubin and Tannehill,1992) which omit the turbulent transport along the streamdirection. Since the PNS equations can be solved by thespace marching algorithm which only needs to scan thecomputational domain once to obtain the solutions anddoes not need iteration, the computational time will begreatly saved in contrast to general iteration method. Hencethe PNS equations are preferred in the present models. Inthe cylindrical coordinate system, the PNS equations arewritten as

@( Mu)@x

+1r@(r Mv)

@r= 0; (2)

@( Muu)@x

+1r@(r Muv)

@r=

1r

@@r

(rt

@u@r

)+ ( M)g: (3)

The transport equation for radial momentum is not givenhere since it is not necessary for the numerical solution (seeSection 4).The turbulent eddy-viscosity t can be modelled by stan-

dard models, such as k model and Reynolds stress model(RSM), while model coeKcients should be adjusted to ob-tain correct jet spreading rate, e.g. the c2 is adjusted from1.92 to 1.8 (Lindstedt et al., 2000). Here, an alternative tur-bulence model, multiple-time-scale (MTS) k model (Kimand Chen, 1989), is introduced to the present modelling ofturbulent axisymmetric jet non-premixed 6ame which doesnot need coeKcient adjustment.

3.2. MTS k turbulence model

In the conventional k model, only the production anddissipation of turbulent kinetic energy are accounted for,whereas in the MTS k model of Kim and Chen (1989),the in6uence of cascade of turbulent kinetic energy is also

involved except for the production and dissipation. The con-cept of the MTS k model is that the turbulent kinetic en-ergy spectrum is partitioned into two regions, productionrange and dissipation range. The total turbulent kinetic en-ergy k contains two parts, the energy of large eddies inproduction range kp and the energy of /ne-scale eddies indissipation range kt (k = kp + kt). The large eddies turbu-lent energy is generated by the mean 6ow instability andcascades to /ner eddies. The energy transfer rate p is in-troduced to describe the eddies cascade. The /ne-scale ed-dies turbulent energy is dissipated into thermal energy bythe viscous forces, which are characterized by the turbulentkinetic energy dissipation rate t (or ). The modelled trans-port equations for kp, kt , p and t are as follows (Kim andChen, 1989).

@( Mukp)@x

+1r@(r Mvkp)

@r

=1r

@@r

(r

tkp

@kp@r

)+ M(P p); (4)

@( Mup)@x

+1r@(r Mvp)

@r

=1r

@@r

(r

tp

@p@r

)

+Mkp

(cp1P2 + cp2Pp cp32p); (5)

@( Mukt)@x

+1r@ (r Mvkt)

@r

=1r

@@r

(rtkt

@kt@r

)+ M

(p t

); (6)

@( Mut)@x

+1r@(r Mvt)

@r

=1r

@@r

(rtt

@t@r

)+

Mkt(ct12 + ct2pt ct32t ); (7)

where P = t= M(@u=@r)2. The model constants in Eqs. (5)(7) are kp=kt=0:75, p=t=1:15, cp1=0:21, cp2=1:24,cp3 = 1:84, ct1 = 0:29, ct2 = 1:28, ct3 = 1:66.The turbulent eddy viscosity is de/ned as

t = Mcfk2=p; (8)

where cf = 0:09. Based on the MTS turbulence model, atime scale for turbulent kinetic energy dissipation can bede/ned from large-eddy quantities as

= kp=p: (9)


3.3. Joint scalar PDF transport equation

Suppose there are -dimension scalar vector (x; t) ={i(x; t); i = 1; : : : ; } in the 6ow /eld, and at position xand at time t, the scalar joint PDF is f( ; x; t), where ={ i; i = 1; : : : ; } is the sample space vector correspondingto (x; t). Using the standard technology of Pope (1985),the evolution equation for f( ; x; t) can be derived as

@@x

( Muf) +1r

@@r(r Mvf)

= 1r

@@r(rv|= Mf) @

@ i( MSif)

+@@

{M1@(rJ ir)r@r

= f} : (10)The /rst term on the right-hand side of Eq. (10) denotesthe transport of f( ; x; t) along the radial direction in thephysical space that is induced by turbulent 6uctuations. Thegradient transport assumption (Pope, 1985; Lindstedt et al.,2000) is often used to model this term as

1r

@@r(v|= Mf) = 1

r@@r

(rtf

@f@r

); (11)

where f = 0:9.The second term on the right-hand side of Eq. (10) de-

notes the change of f( ; x; t) in sample space due to chem-ical reaction. Obviously, the chemical reaction term is inclosed form, which enables the exact treatment of arbitrarilycomplicated chemical reaction mechanisms. In this work,the chemical reaction is described by GRI-Mech 3.0 formethane oxidation and nitrogen oxides formation.The third term on the right-hand side of Eq. (10) is the

conditional mean of molecular transport, namely small-scalemixing term. It is not closed and has become one of themain eForts for the study of PDF method. The often usedmixing model are IEM (interaction by exchange with themean) (IEM)model (Pope, 1985) Curls model and its mod-i/cations (Pope, 1985; Lindstedt et al., 2000) etc. Recently,Subramaniam and Pope (1998) developed a new mixingmodel based on EMST which will be used in the presentsimulation.The radiative heat losses are not taken into account in

Eq. (10) due to their relative small impact on the tempera-ture /eld of gaseous 6ames. Meanwhile the existing radia-tion models, such as the optically thin limit model (Barlowet al., 2001), are too simple to describe the interactions be-tween turbulence and radiation (Li and Modest, 2002) andthe implementation of such radiation model will spend toomuch extra CPU time. Therefore, for simplicity, the radia-tion losses are ignored in the present 6ame temporarily.The PDF evolution equation (10) is usually solved by

Monte Carlo particle method (Pope, 1981, 1985). The PDFf( ; x; t) is represented by an ensemble of N particles interms of j (j=1; : : : ; N ). Each particle evolves in physical

and sample space and the Favre average of scalars can beestimated from the ensemble average of the particles as

(x; t) =1N

Nj=1

j (x; t): (12)

3.4. EMST mixing model

One common drawback of popular mixing models, suchas the above-mentioned IEM model and Curls models, isthat they do not ful/ll the localness principle of mixing mod-els (Subramaniam and Pope, 1998). With them, the particleswill mix homogeneously with each other in each grid celllike a homogeneous volume reaction system. When they areapplied to diFusion combustion problem, the fuel and theoxidizer might penetrate the reaction zone, which results inthe non-physical mixing of cold fuel and cold oxidizer. How-ever, in fact, only those particles in the physical space neigh-borhood can in6uence the local small-scale mixing behav-ior, non-adjacent particle should not aFect each other. Sincethe scalar /eld is continuous in physical space, the principleof localness in physical space is equivalent to the localnessin scalar space. In order to remedy the drawback of conven-tional mixing models, Subramaniam and Pope (1998) devel-oped an EMST based mixing model, where the localness ofmultiple scalars is de/ned through the Euclidean minimumspanning tree constructed in scalar space, and the princi-ple of localness is ful/lled through interactions of neighborparticles in scalar space. The good performance of EMSTmodel has been shown by Xu and Pope (2000) in their PDFcalculations of turbulent non-premixed 6ames.A complete description of the EMST model can be found

in Subramaniam and Pope (1998). Here, we brie6y describethe implementation of the model. At any time, given theensemble of N particles in the grid cell, a subset of NTparticles is chosen from the particle ensemble for mixing,according to an age property associated with each particle.An EMST is constructed on this subset of NT particles. Thistree containsNT particles (nodes) andNT1 edges, and eachparticle is connected with at least one neighbor particle. Themixing is simulated through the evolution of each particlej (j = 1; : : : ; NT ) as

wjdjdt

=(NT1)=1

B)

{(j n)),jm) + (j m)),jn)}; (13)where the )th edge of the tree connects the particle pair(m); n)). The speci/cation of the model constants ( andB) is described in Subramaniam and Pope (1998). In thedetermination of (, a mixing time-scale is needed whichis usually related to the turbulence time-scale (Eq. (9))as

= =c; (14)


where c is the mechanical to scalar time-scale ratio. Exper-iments (Beguier et al., 1978; Panchapakesan and Lumely,1993; Dai et al., 1995; Wang and Tong, 2002; Andersonand Bremhorst, 2002) and DNS (Eswaran and Pope, 1988;Johansson and Wikstrom, 1999) show that c is nearly aconstant. The standard value 2.0 is usually used for c(Spalding, 1971). Sometimes it performs better with smalltuning of the standard value, e.g. with 1.5 by Xu and Pope(2000) and with 2.3 by Lindstedt et al. (2000). However,actually, c is 6ow dependent, to account for the c vari-ation in the 6ow /eld, Hidouri et al. (2003) have evertried a variable c method in the modelling of turbulent jet6ame, while no substantial improvement is shown. There-fore the constant method is still adopted in the presentcalculation and the standard value c = 2:0 is chosen forsimplicity.

4. Numerical method

The space marching algorithm is employed to solve theproblem. The axial coordinate x is viewed as time-likecoordinate. The governing equations for mean velocity(Eq. (3)) and turbulence (Eqs. (4)(7)) are discretizedby FV (/nite-volume) method, namely upwind diFerencefor convection terms, central diFerence for diFusion termsand implicit scheme for axial convection. The obtainedalgebraic equations are solved by using iteration method.The radial momentum equation is not solved and the radialvelocity is deduced from the continuum equation (6). Thecomputational domain along the radial direction is dividedinto 50 cells, most of which are laid out around the cen-terline. The step size along axial direction is /xed to beQx=D=0:05 within the whole computational domain exceptthe beginning, where Qx=D=0:005 is used temporarily.Total about 2000 steps are marched from nozzle to down-stream x=D 80. Since the implicit scheme is used for theturbulence equations discretization, there is no step sizerestriction for turbulence /elds calculation. The time-likestep size can be speci/ed as Qt =Qx=u.The PDF evolution equation (10) is solved by the

node-based Monte Carlo particle method (Pope, 1981)which /xes the particle ensemble on grid cells. The deriva-tives with respect to physical space coordinate in Eq. (10)are /rstly discretized by FV method with explicit schemefor axial convection. Through a time-splitting scheme,the evolution of the PDF can be separated into three pro-cesses, convection and diFusion process C, small-scalemixing process M , and chemical reaction process R. Sincethe explicit scheme is used for Eq. (10), the step sizeQx for three processes is restricted. To match the stepsize speci/ed in the turbulence /elds, some sub-steps areinvolved inside these processes according to CFL condi-tions, e.g. nR sub-steps for R, nM sub-steps for M , and nCsub-steps for C. Therefore the discrete form of Eq. (10) is

written as

f|x+Qx =nRi=1

(I +QtiR) nMj=1

(I +QtjM)

nCk=1

(I +QtkC) f|x; (15)

wherenR

i=1 Qti=nM

j=1 Qtj=nC

k=1 Qtk =Qt. In each gridcell, 400 particles are allocated, and total 400 50=20000particles are located in the stream-cross section.The boundary conditions need to be speci/ed. Most

of them are determined according to experimental con-ditions (Schneider et al., 2003; Barlow and Frank, 1998;Xu and Pope, 2000). The turbulent kinetic energy k atthe inlet boundary is estimated from measured Reynoldsnormal stress. The turbulent kinetic energy of large eddiesin production range kp is assumed to be 80% of k. Thedetermination of the inlet condition for p and t (or )needs special attention since experimental data for themis hardly available, and Merci et al. (2002) found that thecomputational results are very sensitive to inlet conditionof . We assume that p is equal to t at the inlet boundaryand estimate their radial pro/les through mixing length lmas p= t = c

3=4f k3=2=lm, where lm is given following Merci

et al. (2002) as:

lmDh

=

{1 exp

[2 106

(yDh

)3]}

[115

(12 y

Dh

)4]: (16)

The thermodynamic, chemical kinetic parameters and trans-port properties are provided by CHEMKIN-II package(Kee et al., 1989). The particle evolution equations dueto chemical reaction are solved by subroutine LSODE(Radhakrishnan and Hindmarsh, 1993). EMST mixingmodel is implemented using Fortran package from Renet al. (2002), where the numerical weight wj (j=1; : : : ; N )of particles is set to be unity.The calculation begins at x=D = 0. At /rst, The particle

evolutions of convection and diFusion, mixing, and reac-tion process are solved in turn to obtain the reactive scalar/elds. The mean density is calculated from the obtained tem-perature and composition concentrations through ideal gasstate relation, and is substituted into the turbulence /elds tocompute mean velocity and turbulent eddy viscosity. Whenthe velocity and scalar /elds are all obtained, the compu-tational plane will be advanced by Qx along axial direc-tion and new computations continue. The cycle will not stopuntil it reaches downstream x=D 80. The required CPUtime for present calculation is about a week on a PC (AMDAthlon(tm) XP 2400+, 2:0 GHz, 512 MB memory).


5. Results and discussion

In this section, the numerical predictions of 6ame D arepresented and compared to the measurements of Barlowand Frank (1998) and Schneider et al. (2003) extensively.The explored results involve radial pro/les of mean andrms (root mean square), conditional mean, scatter plots ofscalars and conditional PDF distribution, etc. Due to spacelimitation, only the results at three diFerent axial locations,viz. x=D = 7:5, 15 and 30, are presented.The predicted and measured radial pro/les of mean ax-

ial velocity, mean mixture fraction and rms mixture frac-

tion ( 21=2) are compared in Fig. 2. The predictions

of mean axial velocity and mixture fraction are in goodagreement with experimental data. The jet spreading rate iswell predicted. The general over-prediction of the spread-ing rate of round jet by conventional k is remedied bythe adopted MTS k model, which shows the superior-ity of the MTS model. The overall agreement between thepredicted and measured rms mixture fraction is also good,while at x=D=7:5, the rms mixture fraction is over-predictedby 40% and at x=D = 30, it is under-predicted by 30%.The evolution of scalar variance is mainly determined bysmall-scale mixing which is not closed in the PDF evolu-tion equation. Therefore, the large discrepancy between thepredicted and measured rms mixture fraction is put downto the mixing model. The modelling of mixing is the mostdiKcult task for PDF methods, and by now no existing mix-ing model can ful/l all the performance criteria of mixingmodels (Subramaniam and Pope, 1998). Moreover, the onlysingle scalar time-scale for all scalars and the unwarranted

r/D

0

1

2x/D=7.5

r/D

0

2

4x/D=15

u (m/s)

r/D

0 500

3

6 x/D=30

~

r/D

0

1

2x/D=7.5

r/D

0

2

4x/D=15

r/D

0 0.5 10

3

6 x/D=30

~

r/D

0

1

2x/D=7.5

r/D

0

2

4x/D=15

r/D

0 0.20

3

6 x/D=30

"

Fig. 2. Predicted (solid lines) and measured (symbols) radial pro/les ofmean axial velocity, mean mixture fraction and rms mixture fraction.

x/D=7.5 ;r=6mm

1

x/D=7.5 ;r=9mm

CPDF

0 0.25 0.5 0.75

0.25 0.5 0.75

0.25 0.5 0.75 0.25 0.5 0.75 0.25 0.5 0.75

0.25 0.5 0.75 0.25 0.5 0.75

0.25 0.5 0.75 0.25 0.5 0.750

5

10x/D=7.5 ;r=3mm

x/D=15 ;r=6mm

1

x/D=15 ;r=12mm

CPDF

00

5

10x/D=15 ;r=2mm

x/D=30 ;r=9mm

CPDF

00

5

10x/D=30 ;r=3mm

1

x/D=30 ;r=21mm

Fig. 3. Predicted (solid lines) and measured (dashed lines) mixture fractionPDFs conditional on particular physical positions.

constant mechanical to scalar time-scale ratio c also causeunpredictable error. Based on these situations, the observeddiscrepancy between predicted and measured rms mixturefraction is not strange and the error is acceptable. The pre-diction of the radial position corresponding to the peak rmsmixture fraction is same as the measurement, which alsoindicates the accurate prediction of the jet spreading rate.The evolution trend of the predicted rms mixture fraction issimilar to the prediction of Xu and Pope (2000) in whichthe same mixing model, viz. EMST model, was used. Fig. 3compares the predicted and measured mixture fraction PDFsat diFerent axial and radial locations. At the same axial dis-tance, from inner to outer, the peak of mixture fraction PDFmoves from low mixture fraction to higher one. The predic-tions reasonably represent this trends and the overall agree-ment between the prediction and measurement is good. The6ow and mixing /eld of 6ame D is well predicted by presentmodels, which provides the prerequisite to the accurate pre-diction of the reactive scalar /elds.The radial pro/les of mean temperature and mass frac-

tions of reactive species in 6ame D are shown in Figs. 46.The predicted mean temperature in Fig. 4 and mass fractionsof major species , such as CH4, O2 in Fig. 4 and CO2, H2Oin Fig. 5, are in good agreement with the experimental dataof Barlow and Frank (1998), including the magnitude andthe radial positions corresponding to peak values. The goodprediction of the temperature indicates that, for the present6ame, the impact of radiative heat losses is minor and theomission of that is acceptable. The predicted mass fractionsof radicle OH in Fig. 5 and intermediate species CO, H2

H. Wang, Y. Chen / Chemical Engineering Science 59 (2004) 34773490 3483r/D

0

1

2x/D=7.5

0

2

4x/D=15

T(K)0 1000

0

3

6 x/D=30

~

r/D0

1

2x/D=7.5

0

2

4x/D=15

YCH4

0 0.10

3

6 x/D=30

~

r/D

0

1

2x/D=7.5

r/Dr/Dr/D

r/Dr/D

0

2

4x/D=15

YO2

r/D

0 0.1 0.20

3

6 x/D=30

~

Fig. 4. Predicted (solid lines) and measured (symbols) radial pro/les ofmean temperature and mean mass fractions of CH4 and O2.

r/Dr/D

r/D

r/Dr/D

r/D

r/Dr/D

r/D

0

1

2x/D=7.5

0

2

4x/D=15

YCO2

0 0.10

3

6 x/D=30

~

0

1

2x/D=7.5

0

2

4x/D=15

YH2O

0 0.10

3

6 x/D=30

~

0

1

2x/D=7.5

0

2

4x/D=15

YOH

0 0.0020

3

6 x/D=30

~

Fig. 5. Predicted (solid lines) and measured (symbols) radial pro/les ofmean mass fractions of CO2, H2O and OH.

in Fig. 6 are also in reasonable agreement with measure-ments. Whereas the pollutant NO emission is over-predictedby 30% in Fig. 6. The omission of radiative heat losses maylead to the over-prediction of NO emission, while, for thepresent 6ame, this in6uence may be ignored due to the goodprediction of temperature. Barlow et al. (2001) found thatthe NO formation in laminar opposed-6ow partially pre-mixed methane/air 6ames is also over-predicted by using

r/Dr/D

r/D

r/Dr/D

r/D

r/Dr/D

r/D

0

1

2x/D=7.5

0

2

4x/D=15

YCO

0 0.050

3

6 x/D=30

~

0

1

2x/D=7.5

0

2

4x/D=15

YH2

0 0.0020

3

6 x/D=30

~

0

1

2x/D=7.5

0

2

4x/D=15

YNO

0 5E-050

3

6 x/D=30

~

Fig. 6. Predicted (solid lines) and measured (symbols) radial pro/les ofmean mass fractions of CO, H2 and NO.

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

x/D=7.5

x/D=15

T(K)0 1000 2000

x/D=30

x/D=7.5

x/D=15

YCH4

0 0.1

x/D=30

x/D=7.5

x/D=15

YO2

0 0.2

x/D=30

Fig. 7. Predicted (solid lines) and measured (symbols) conditional meantemperature and mass fractions of CH4 and O2 versus mixture fraction.

GRI-Mech 3.0 and including the radiative heat losses, whichimplies that the over-prediction of NO formation is a com-mon drawback of GRI-Mech 3.0. Excluding this drawback,we may say that the NO emission from 6ame D is well pre-dicted by present models.To further explore the turbulencechemistry interactions

directly, the predicted conditional mean temperature andspecies mass fractions versus mixture fraction are comparedwith the experimental data in Figs. 79. The agreements be-tween the predictions and measurements are very good for


0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

x/D=7.5

x/D=15

YCO2

0 0.1

x/D=30

x/D=7.5

x/D=15

YH2O0 0.1

x/D=30

x/D=7.5

x/D=15

YOH0 0.003

x/D=30

Fig. 8. Predicted (solid lines) and measured (symbols) conditional meanmass fractions of CO2, H2O and OH versus mixture fraction.

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

00.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

0

0.3

0.6

0.9

x/D=7.5

x/D=15

YCO0 0.05

x/D=30

x/D=7.5

x/D=15

YH2

0 0.003

x/D=30

x/D=7.5

x/D=15

YNO0 0.0001

x/D=30

Fig. 9. Predicted (solid lines) and measured (symbols) conditional meanmass fractions of CO, H2 and NO versus mixture fraction.

conditional mean temperature in Fig. 7 and major species(CH4, O2 in Fig. 7 and CO2, H2O in Fig. 8) mass fractions.The predicted shape of radicle OH conditional mean massfraction is somewhat narrower than the experimental data,and the peak value of OH at x=D=7:5 is a bit over-predicted,while the overall agreements between predictions and mea-surements are still satisfying. The intermediate species CO,H2 in Fig. 9 are also well predicted. It has been reportedthat the intermediate species in the fuel-rich region (st)tend to be over-predicted by both steady 6amelet model and/rst-order CMC (Barlow et al., 2001; Coelho and Peters,2001; Roomina and Bilger, 2001). The reaction progressin the region st seems to be over-estimated by steady

Fig. 10. Scatter plot of measured (left) and predicted (right) temperatureversus mixture fraction. Lines: stretched laminar 6amelet (a= 100 s1).

6amelet model and /rst-order CMC, while it is well repre-sented by PDF model. The conditional mean mass fractionof NO is also over-predicted like its radial pro/le shown inFig. 6. With regard to the mentioned drawback of GRI-Mech3.0, the NO emission is also well represented by presentmodels. The fairly good agreements between the predictedand measured conditional means in mixture fraction spacedemonstrate the capacity of PDF model to represent theturbulencechemistry interactions. In the physical space,the agreements between the predictions and measurements(shown in Figs. 46) is similar to those in mixture fractionspace. This bene/ts from the good predictions of 6ow andmixing /eld except for the capacity of PDF model.In the node-based Monte Carlo algorithm, the scalar joint

PDF f( ; x; r) is represented by an ensemble of particleslocated at position (x; r). Each particle can be viewed as arealization of the turbulent reactive 6ow at (x; r). Therefore,the particle ensemble is comparable to the instantaneousmeasurements of 6ame D. It is informative to compare thescatter plot of the particle ensemble with the instantaneousexperimental data of scalars, although the comparisons havelimitations (Xu and Pope, 2000).Fig. 10 shows the scatter plot of measured and predicted

temperature versus mixture fraction at three axial locationsx=D= 7:5, 15 and 30. For the sake of reference, a stretchedlaminar 6amelet pro/le under a moderate strain rate (a =100 s1) is also plotted in the /gure. The 6amelet pro/leis obtained by solving the steady 6amelet equations (Peters,1984). The overall agreement between the experimental data


Fig. 11. Scatter plot of measured (left) and predicted (right) CH4mass fraction versus mixture fraction. Lines: stretched laminar 6amelet(a = 100 s1).

and the prediction is comparatively good. Most of the pointsdistribute around the 6amelet line, while some points de-parture far away from the 6amelet line, e.g. the temperaturenear st drops to 1000 K. The suppressed temperature in thereaction zone indicate the 6ame local extinction (Barlow andFrank, 1998). This is the re6ection of the strong turbulencechemistry interactions. The amount of local extinction canbe estimated from the number of points that departure awayfrom the 6amelet line. According to the experimental data(left in Fig. 10), the local extinction shows low probabilityat x=D=7:5 and gets more at x=D=15, while the amount oflocal extinction decreases at x=D= 30, which indicates thatsome extinguished samples return to the 6amelet state again,namely re-ignition process. The local extinction process iswell represented by present models. While the amount oflocal extinction seems to be a bit over-predicted at x=D=30and the re-ignition process is postponed to downstream. Inthe center of the scatter plot of predicted temperature, Thereis a void hole surrounded by the extinguished samples andby samples near the 6amelet line, where almost no sam-ples locate. The experimental data does not show this phe-nomenon. This may come from the stranding problem ofEMST mixing model (Subramaniam and Pope, 1998).Fig. 11 shows the scatter plot of measured and predicted

CH4 mass fraction versus mixture fraction. In the fuel-leanregion (st), the CH4 mass fraction of 6amelet is almostzero, while some measured instantaneous CH4 mass fractionhas non-zero value. The oxidizer of 6ame D is air, so the

Fig. 12. Scatter plot of measured (left) and predicted (right) COmass fraction versus mixture fraction. Lines: stretched laminar 6amelet(a = 100 s1).

fuel existing in the fuel-lean region is penetrated from thefuel side through the reaction zone. These samples must beextinguished locally and are transported through moleculardiFusion to the oxidizer side (If not, they will be consumedwithin the reaction zone). The prediction of scatter plot ofCH4 mass fraction is in good agreement with the experiment.The number of predicted non-zero samples in the fuel-leanregion are more than the experiment at x=D = 30, which isconsistent with the over-predicted amount of local extinctionat the same axial location.The scatter plot of measured and predicted CO mass frac-

tion versus mixture fraction is illustrated in Fig. 12. Themeasured scatter plot of CO mass fraction shows much moredispersion than that of temperature (see Fig. 10). Most sam-ples departure from the 6amelet line. The prediction repre-sents this property fairly well. However the prediction showsconglomeration of some samples at some places. This is notre6ected by the experimental data and may result from thestranding problem of EMST mixing model.Fig. 13 shows the scatter plot of measured and predicted

NO mass fraction versus mixture fraction. The strain rateof referenced stretched laminar 6amelet here is diFerentfrom the former, viz. a = 800 s1 instead of 100 s1, be-cause the NO pro/le of 6amelet under a = 1001 is muchhigher than the NO emission in 6ame D. The NO emissionis over-predicted, and the reason has been discussed in theconditional mean of NO mass fraction in Fig. 9. There isalso a void hole in the center of the predicted scatter plotof NO just like the scatter plot of temperature in Fig. 10.


Fig. 13. Scatter plot of measured (left) and predicted (right) NOmass fraction versus mixture fraction. Lines: stretched laminar 6amelet(a = 800 s1).

The experiment of NO does not exhibit such structure andthe stranding phenomenon of EMST mixing model is alsothought to be responsible for this.Other species such as O2, CO2, H2O, H2 and OH scatter

plot are also well represented by present models (/gures notshown).Fig. 14 shows the CPDFs of temperature and CO2, H2O

mass fractions at three diFerent axial locations x=D=7:5, 15and 30. The mixture fraction interval used for the estimationof CPDFs is also shown in the /gure for diFerent scalars, e.g.(0:3 0:4) for temperature. From x=D= 7:5 to 30, theshape of measured temperature CPDF become thinner andthe peak value shifts toward higher temperature. This trendsare well predicted by present models. However, the predictedCPDF shapes are more thinner than measurements, whichresults in a higher peak value of CPDF. The distributionsof CPDFs of CO2 and H2O in Fig. 14 are similar to that oftemperature.The measured and predicted CPDFs of CO, H2 and OH

mass fractions are illustrated in Fig. 15. The moving trendof the peak value of the measured CPDFs of CO and H2is same as that of temperature in Fig. 14, and it is wellrepresented by the predictions. The predicted shapes of theCPDFs of CO and H2 are also thinner than measurements. Atx=D=30, the peak value of CO and H2 CPDFs is very high,which is in accordance with the conglomeration of samplesin the scatter plot of CO in Fig. 12. At this axial location,a bimodal shape of the CO and H2 CPDFs is observed,which is not re6ected by the measurements. The bimodal

0.05 0.1 0.15

PDF (0.3


YCO0.05 0.1

PDF (0.43


prediction of Flame F is very sensitive to the mechanicalto scalar time-scale ratio c in Xu and Pope (2000). Itseems that c, or broadly the mixing, is very crucial to themodelling of the 6ames with intense interactions betweenturbulence and chemistry, especially Flame F. More eFortsare desired on this issue, such as the sensitivity of PDFpredictions to mixing models (Ren and Pope, 2004) andc, the determination of c and about variable c in the6ow /eld etc. Moreover the performance of detailed mech-anism GRI-Mech 3.0 in these 6ames will also need to beexamined in the future.

6. Conclusions

In this paper, the detailed reaction mechanism, GRI-Mech3.0, is incorporated into the PDF modelling of turbulentnon-premixed jet 6ame (Sandia 6ame D) for the /rst time.The adopted models contain theMTS kmodel, joint scalarPDF method and EMST small-scale mixing model. The 6owis reduced to a parabolic type problem and is describedby parabolized NavierStokes (PNS) equations. The spacemarching algorithm is adopted to solve this problem, whichreduces the computational cost greatly. The large /xed stepsize for turbulence /elds calculation is adopted and adaptivesub-steps are used for scalar /elds calculation. The requiredCPU time for the calculation is about a week on a modernPC, which is acceptable for present computational capacity.The numerical results are presented in detail and are com-

pared to the experimental data extensively. The predictionsare in fairly good agreement with the measurements, includ-ing the radial pro/les of mean and rms, conditional mean,scatter plots and conditional PDFs. The shown and com-pared quantities involve axial mean velocity, mean and rmsmixture fraction, temperature and mass fractions of speciesCH4; O2; CO2; H2O, CO, H2, OH and NO. The NO emis-sion is over-predicted by GRI-Mech 3.0. Due to the limita-tion of EMST model, some phenomena are resulted in, suchas the void hole in the center of the scatter plots and con-glomeration of samples, which is not re6ected by measure-ments. Sophisticated mixing models are desired to improvethe capacity of PDF model further.

Notation

a strain rateB) EMST model constantcf constant coeKcient in Eq. (8)cpj, ctj model constants in Eqs. (5) and (7),

respectively (j = 13)c mechanical to scalar time-scale ratioD diameter of fuel jet nozzleDP diameter of annular piloted 6ameC convection and diFusion processDh hydraulic diameter

f( ; x; t) scalar joint probability density func-tion (PDF)

g acceleration of gravityJ ir molecular transport 6ux of ith scalar

along radial directionkp turbulent kinetic energy of large ed-

dies in production rangekt turbulent kinetic energy of

/ne-scale eddies in dissipationrange

k turbulent kinetic energy, =kp + ktlm mixing lengthM small-scale mixing processnC;M ;R number of sub-steps for process C,

M and R, respectivelyN number of particlesNT number of particles in the EMSTP production rate of turbulent kinetic

energyR chemical reaction processSi reaction rate of ith scalart timeu axial velocityv radial velocityw numerical weight of particleWX molecular or atomic weight of

species or element Xx coordinate vectorx, r cylindrical coordinatey normal distance from the nearest

solid boundaryYX mass fraction of species or element

X mathematical expectations

Greek letters

dimension of scalar vector( model parameter controlling the

variance decay rate, Kronecker deltaQt time step sizeQx step size along axial directionp energy transfer rate from production

range to dissipation range, t dissipation rate of turbulent kinetic

energyt turbulent eddy-viscosity mixture fraction densitykp, kt , p, t , f turbulent Prandtl number for kp, kt ,

p, t and f, respectively time scale of turbulent kinetic en-

ergy dissipation mixing time-scale


equivalence ratioi ith scalar i sample space variable of ith scalar scalar vector (={i; i = 1; : : : ; }) scalar vector of particle ensemble sample space scalar vector

(={ i; i = 1; : : : ; })

Subscripts

1 fuel jet 6ow2 oxidizer 6owst stoichiometric condition in/nity

Superscripts conventional average Favre average 6uctuation from Favre average; rms

(root mean square)

Acknowledgements

This work was supported by the Special Funds forMajor State Basic Research Projects (G1999022207), PRChina and the National Natural Science Foundation ofChina (50206021). We are grateful to Dr. R.S. Barlow andDr. A. Dreizler to provide us the experimental data forscalar /elds and velocity /elds of 6ame D, respectively.

References

Anderson, S.M., Bremhorst, K., 2002. Investigation of the 6ow /eld of ahighly heated jet of air. International Journal of Heat and Fluid Flow23, 205219.

Barlow, R.S., Frank, J.H., 1998. EFects of turbulence on species massfractions in methane/air jet 6ames. In: 27th Symposium (International)on Combustion. The Combustion Institute, Pittsburgh, PA,pp. 10871095.

Barlow, R.S., Karpetis, A.N., Frank, J.H., Chen, J.-Y., 2001. Scalarpro/les and NO formation in laminar opposed-6ow partially premixedmethane/air 6ames. Combustion and Flame 127, 21022118.

Beguier, C., Dekeyser, I., Launder, B.E., 1978. Ratio of scalar and velocitydissipation time scales in shear 6ow turbulence. Physics of Fluids 21,307310.

Bilger, R.W., 1993. Conditional moment closure for turbulent reacting6ow. Physics of Fluids A 5, 436444.

Bilger, R.W., Starner, S.H., Kee, R.J., 1990. On reduced mechanismsfor methane-air combustion in nonpremixed 6ames. Combustion andFlame 80, 135149.

Chen, J.-Y., Kollmann, W., Dibble, R.W., 1989. PDF modeling ofturbulent nonpremixed methane jet 6ames. Combustion Science andTechnology 64, 315346.

Coelho, P.J., Peters, N., 2001. Unsteady modelling of a piloted methane/airjet 6ame based on the Eulerian particle 6amelet model. Combustionand Flame 124, 444465.

Dai, Z., Tseng, L.K., Faeth, G.M., 1995. Velocity/mixture fractionstatistics of round, self-preserving, buoyant turbulent plumes. Journalof Heat Transfer-Transactions of the ASME 117, 918926.

Dopazo, C., 1994. Recent developments in PDF methods. In: Libby, P.A.,Williams, F.A. (Eds.), Turbulent Reacting Flows. Academic, NewYork (Chapter 7).

Eswaran, V., Pope, S.B., 1988. Direct numerical simulations of theturbulent mixing of a passive scalar. Physics of Fluids 31, 506520.

GRI-Mech website, 1995. http://www.me.berkeley.edu/gri mech/.Hewson, J.C., Kerstein, A.R., 2002. Local extinction and reignition in

nonpremixed turbulent CO=H2=N2 jet 6ames. Combustion Science andTechnology 174, 3566.

Hidouri, A., Gazzah, M.H., Ticha, H.B., et al., 2003. Dynamic and scalarturbulent 6uctuation in a diFusion 6ame of an-axisymmetric methanejet into air. Computational Mechanics 31, 253261.

International Workshop on Measurement and Computation ofTurbulent Nonpremixed Flames, 1996. http://www.ca.sandia.gov/TNF/abstract.html.

Johansson, A.V., Wikstrom, P.M., 1999. DNS and modelling of passivescalar transport in turbulent channel 6ow with a focus on scalardissipation rate modelling. Flow, Turbulence and Combustion 63,223245.

Jones, W.P., Kakhi, M., 1997. Application of the transported PDFapproach to hydrocarbon-air turbulent jet diFusion 6ames. CombustionScience and Technology 129, 393430.

Jones, W.P., Kakhi, M., 1998. PDF modeling of /nite-rate chemistryeFects in turbulent nonpremixed jet 6ames. Combustion and Flame115, 210229.

Kee, R.J., Rupley, F.M., Miller, J.A., 1989. CHEMKIN-II: a fortranchemical kinetics package for the analysis of gas-phase chemicalkinetic. Sandia Report, SAND89-8009.

Kim, S.-W., Chen, C.-P., 1989. A multiple-time-scale turbulence modelbased on variable partitioning of the turbulent kinetic energy spectrum.Numerical Heat Transfer, Part B 16, 193211.

Klimenko, A.Y., 1990. Multicomponent diFusion of various admixturesin turbulent 6ow. Fluid Dynamics 25, 327334.

Li, G.N., Modest, M.F., 2002. Application of composition PDF methodsin the investigation of turbulenceradiation interactions. Journal ofQuantitative Spectroscopy and Radiative Transfer 73, 461472.

Lindstedt, R.P., Louloudi, S.A., Vaos, E.M., 2000. In: Proceedings ofthe Combustion Institute. The Combustion Institute, Pittsburgh, PA,Vol. 28, pp. 149156.

Merci, B., Dick, E., Vierendeels, J., Langhe, C.D., 2002. Determinationof at inlet boundaries. International Journal of Numerical Methodsfor Heat and Fluid Flow 12, 6580.

Panchapakesan, N.R., Lumely, J.L., 1993. Turbulence measurements inaxisymmetric jets of air and helium. Part 2. Helium jet. Journal ofFluid Mechanics 246, 225247.

Peters, N., 1984. Laminar diFusion 6amelet models in non-premixedturbulent combustion. Progress in Energy and Combustion Science 10,319339.

Peters, N., Kee, R.J., 1987. The computation of stretched laminarmethane-air diFusion 6ames using a reduced four-step mechanism.Combustion and Flame 68, 1729.

Pitsch, H., 2002. Improved pollutant predictions in large-eddy simulationsof turbulent non-premixed combustion by considering scalar dissipationrate 6uctuations. In: Proceedings of the Combustion Institute. TheCombustion Institute, Pittsburgh, PA, Vol. 29, pp. 19711978.

Pitsch, H., Chen, M., Peters, N., 1998. Unsteady 6amelet modelingof turbulent hydrogen-air diFusion 6ames. In: 27th Symposium(International) on Combustion. The Combustion Institute, Pittsburgh,PA, pp. 10571064.

Pope, S.B., 1981. A Monte Carlo method for the PDF equations ofturbulent reactive 6ow. Combustion Science and Technology 25,159174.

Pope, S.B., 1985. PDF methods for turbulent reactive 6ows. Progress inEnergy and Combustion Science 11, 119192.

Pope, S.B., 1994. Lagrangian PDF methods for turbulent 6ows. AnnualReview of Fluid Mechanics 26, 2363.


Pope, S.B., 1997. Computationally eKcient implementation of combustionchemistry using in situ adaptive tabulation. Combustion Theory andModelling 1, 4163.

Radhakrishnan, K., Hindmarsh, A.C., 1993. Description and use ofLSODE, the Livermore solver for ordinary diFerential equations.Livermore National Laboratory Report, UCRL-1D-113855.

Ren, Z., Subramaniam, S., Pope, S.B., 2002. Implementation of the EMSTmixing model. http://mae.cornell.edu/laniu/emst.

Ren, Z.Y., Pope, S.B., 2004. An investigation of the performance ofturbulent mixing models. Combustion and Flame 136, 208216.

Roomina, M.R., Bilger, R.W., 2001. Conditional moment closure (CMC)predictions of a turbulent methane-air jet 6ame. Combustion and Flame125, 11761195.

Rubin, S.G., Tannehill, J.C., 1992. Parabolized/reduced NavierStokescomputational techniques. Annual Review of Fluid Mechanics 24,117144.

Saxena, V., Pope, S.B., 1999. PDF simulations of turbulent combustionincorporating detailed chemistry. Combustion and Flame 117,340350.

Schneider, C.H., Dreizler, A., Janicka, J., Hassel, E.P., 2003. Flow /eldmeasurements of stable and locally extinguishing hydrocarbon-fuelledjet 6ames. Combustion and Flame 135, 185190.

Spalding, D.B., 1971. Concentration 6uctuations in a round turbulent freejet. Chemical Engineering Science 26, 95107.

Subramaniam, S., Pope, S.B., 1998. A mixing model for turbulent reactive6ows based on Euclidean minimum spanning trees. Combustion andFlame 115, 487514.

Sung, C.J., Law, C.K., Chen, J.-Y., 1998. An augmented reducedmechanism for methane oxidation with comprehensive globalparametric validation. In: 27th Symposium (International) onCombustion. The Combustion Institute, Pittsburgh, PA, pp. 295304.

Sung, C.J., Law, C.K., Chen, J.-Y., 2001. Augmented reduced mechanismfor NO emission in methane oxidation. Combustion and Flame 125,906919.

Tang, Q., Xu, J., Pope, S.B., 2000. Probability density functioncalculations of local extinction and NO formation in piloted-jetturbulent methane/air 6ame. In: Proceedings of the CombustionInstitute. The Combustion Institute, Pittsburgh, PA, Vol. 28,pp. 133139.

Wang, D., Tong, C., 2002. Conditionally /ltered scalar dissipation,scalar diFusion, and velocity in a turbulent jet. Physics of Fluids 14,21702185.

Xiao, K., Schmidt, D., Maas, U. 2000. A study of local 6ame structures inpiloted jet diFusion 6ame using a probability density function method.In: Proceedings of the Combustion Institute. The Combustion Institute,Pittsburgh, PA, Vol. 28, pp. 157165.

Xu, J., Pope, S.B., 2000. PDF calculations of turbulent nonpremixed6ames with local extinction. Combustion and Flame 123, 281307.

PDF modelling of turbulent non-premixed combustionwith detailed chemistryIntroductionSandia Flame DModelling methodsParabolized Navier--Stokes equationsMTS k--epsilon turbulence modelJoint scalar PDF transport equationEMST mixing model

Numerical methodResults and discussionConclusionsAcknowledgementsReferences

1-s2.0-S000925090400315X-main.pdf

Documents

Transcript of 1-s2.0-S000925090400315X-main.pdf