Center for TurbulenceResearch

Annual Research Briefs

Incorp orating realistic

into direct numerical simulations of

turbulent nonpremixed combustion

By W K Bushe R W Bilger AND G R Ruetsch

Motivation and ob jectives

Combustion is an imp ortant phenomenon in many engineering applications Com

bustion of hydro carb ons is still by far the most common source of energy in the

world In many devices of interestsuch as in furnaces diesel engines and gas

turbinesthe combustion takes place in what is known as the nonpremixed

regime The fuel and oxidizer are initially unmixed and in order for chemical reac

tion to take place they must rst mix together In this regime the rate at which

fuel and oxidizer are consumed and at which heat and pro duct sp ecies are pro duced

is therefore to a large extent controlled by mixing

In virtually all engineering applications of combustion pro cesses the ow in which

the combustion takes place is turbulent Furthermore the combustion pro cess it

self is usually describ ed byavery large system of elementary chemical reactions

These chemical kinetic mechanisms are usually extremely sti and involve for longer

chain hydro carb on sp ecies hundreds of chemical sp ecies The governing equations

describing the chemical comp osition are closely coupled to those describing the

turbulent transp ort Also the chemical reaction rates are nonlinear and strongly

dep end on the instantaneous comp osition and temp erature

Modeling turbulent combustion

In order to mo del turbulent combustion one must circumvent what is known as

the chemical closure problem The chemical source term in the Reynolds averaged

sp ecies transp ort equation must b e mo deled Several mo dels have b een prop osed

to achievechemical closure but many of these are only applicable to limited ow

or chemistry regimes For example in the fast chemistry limit the chemistry is

assumed to b e innitely fast in comparison to the turbulent mixing pro cess Bil

ger which completely neglects the inuence of nite rate chemistry on the

combustion pro cess Laminar amelet mo dels Peters are only applicable in

what is known as the amelet regime where the chemical reactions take place

along an interface which is thinner than the smallest turbulent length scale The

PDF mo del Pop e where the transp ort equation for the joint probability

density function of the comp osition vector is solved is only practical for systems

with very simple chemical kinetic mechanismssuch as reduced chemical kinetic


The University of Sydney Australia

W K Bushe R W Bilger G R Ruetsch

A new metho d for closing the chemical source term was recently prop osed in

dep endently by Klimenko and Bilger a b In the Conditional

Moment Closure CMC metho d the transp ort equations are conditionally aver

aged with the condition b eing some variable on which the chemical reaction rates

are known to dep end For nonpremixed combustion an appropriate conditioning

variable is the fraction This is a conserved scalar suitably dened to

haveavalue of zero in pure oxidizer and unity in pure fuel

The average of the mass fraction Y of a particular sp ecies I conditional on the


mixture fraction Z having some value is

Q x t hY x tjZ x t i

I k I k k

Foraowinwhich the velo city and mixture fraction elds are b oth isotropic and

homogeneous the conditionally averaged transp ort equation for Y b ecomes Smith




h jZ i hD jZ i


t x x

i i

the righthand side of which has two unclosed terms the conditionally averaged

reaction rate and a mixing term in which app ears the conditionally averaged scalar


dissipation hD jZ i

x x

i i

There are several mo dels available for the scalar dissipation such as presumed

PDF mo dels Mell et al and mapping closure mo dels Bushe Clo

sure of the reaction term can b e achieved through the rst order CMC hyp othesis

that the conditional average of the chemical source term of some sp ecies I which

is a function of the comp osition vector Y and the temp erature T can b e mo d


eled byevaluating the chemical reaction rates using the conditional averages of the

comp osition vector Q and temp erature hT jZ iThus


h Y T jZ i Q hT jZ i



Various renements to the closure hyp othesis for the chemical reaction term have

b een prop osed using either a second conditioning variable Bilger Bushe

or a second moment Li Bilger Smith which are intended to

extend the validity of the closure hyp othesis to account for ignition and extinction

phenomenon and to improve the p erformance of the mo del for chemical reactions

where the activation energies are very large

Validation of turbulent combustion models

Work attempting to improve and validate mo dels for turbulent combustion has

b een hamp ered byalack of adequate exp erimental results Only recently have

exp erimental techniques b een devised whichmightprovide the necessary insight

these exp eriments metho ds are still quite limited in the information they provide

and are also extremely exp ensive and dicult to p erform

As an alternative to exp eriments Direct Numerical Simulation DNS of the

governing equations can b e p erformed however to date suchsimulations have b een

Incorporating realistic chemistry into DNS of combustion

limited byavailable computer resourcesand by the complexity and stiness of the

asso ciated equationsto simple chemical kinetic mo dels Vervisch

With the advent of new techniques for the systematic reduction of chemical ki

netic mechanisms new reduced kinetic mechanisms are nowavailable which are still

relatively simple but which retain sucient complexity from the original mecha

nism to provide go o d predictions of ame structure and reaction rates In a previous

study which implements such a reduced mechanism in DNS Swaminathan and Bil

ger a b the owwas assumed to b e incompressible so that eects of

heat release on the owwere neglected While the results of this study havebeen

encouraging validation of the CMC metho d against this constant prop erty DNS

data is not completely convincing There is clearly a need to obtain DNS data

using realistic chemical kinetics in turbulence where eects of the heat release on

the ow are included

In the present study a reduced kinetic mechanism has b een incorp orated into a

fully compressible DNS co de The results of the simulations will b e used for the

validation and hop efully improvement of current combustion mo dels such as the

CMC mo del describ ed ab ove



Original kinetic mechanism

The chemical kinetic mechanism that was used in the simulations is one repre

sentative of the oxidation of a methanenitrogen mixture byanoxygennitrogen

mixture There are three reactions in the mechanism the rst two represent the

oxidation of the methane Williams and the third represents the formation

of nitric oxide The reactions are

Fuel Oxi Int Prod I

Int Oxi Prod I I

N Oxi NO I I I


where Fuel is CH Oxi is O Int is H CO andProd is H O CO

Rates for reactions I and I I expressed in terms of mass fractions are given



mol cm K

T exp Y Y

I Fuel H

g s K T


K cm


II Oxi H M

g s

The mass fraction of which app ears in the b oth of these reaction rate

expressions is given by the steady state approximation







exp K T Y e


Y Prod

W K Bushe R W Bilger G R Ruetsch




F uel

exp K T exp



The inuence of the enhanced third b o dy M app earing in the rate expression for

reaction I I is


M F uel Oxi Int Prod

The rate expression for reaction I I I is obtained by placing the oxygen free radical

in the simple Zeldovichmechanism



in steady state The of the oxygen free radical is estimated by assum

ing it is in partial equilibrium with the hydrogen and hydroxyl free radicals The

resulting rate expression is


mol cm Y Y Y K

Prod N




g s T Y


The rate expressions in Eqs and eachgive the reaction rates in

units of mol g s The rates of change of mass fractions can b e calculated by

mixtur e

multiplying the reaction rates by each participating sp ecies molecular weight


F uel I

g mol


g mol

I II Int

g mol

Prod I II

g mol


The rate of change of energy due to chemical reaction is calculated bymultiplying

the reaction rates byeach reactions enthalpy of formation

k J mol k J mol k J mol


Simplifying the mechanism

In order to reduce computational costs and to make the mechanism more tractable

for mo deling purp oses the reaction rate expressions were simplied

Incorporating realistic chemistry into DNS of combustion

The equation for Eq contains the function


f T T exp


which over a temp erature range from to K iswell approximated as

b eing constantas seen in Fig a This was taken to b e which over that

temp erature range predicts Eq within

The expression for reaction I Eq contains the function


g T T exp


This function can b e approximated by


g T exp


as shown in Fig b also to within o ver the range of K

The expression for the hydrogen freeradical mass fraction b ecomes



F uel

Y exp








exp K T



and the expression for the reaction rate of reaction I b ecomes


mol cm K

exp Y Y

I Fuel H

g s T

Nondimensionalizing the mechanism

The DNS co de for which the mechanism was b eing mo died uses the constant

pressure sp ecic heat C the ratio of sp ecic heats and the sp eed of sound to


nondimensionalize the governing equations Thus C and are implicitly assumed


to b e constant It is also implicitly assumed that the molecular weights of the fuel

and oxidizer streams are equal

In order to remain consistent with previous implementations of this mechanism

Swaminathan Bilger a b it was decided that the fuel stream would

consist of methane b y mass balance nitrogen and the oxidizer stream would

consist of o xygen balance nitrogen These have molecular weights

of gmol and g mol resp ectively The constant molecular weight of b oth

streams for the purp oses of the DNS co des calculations was taken to b e gmol

The initial temp erature of b oth streams was taken to b e K

W K Bushe R W Bilger G R Ruetsch


25.5 a 25



T 23.5 f 23




21 1200 1300 1400 1500 1600 1700 1800 1900 2000

11 x 10 14

12 b


8 T g 6




1200 1300 1400 1500 1600 1700 1800 1900 2000


Figure Comparison of simplied functions in kinetic mechanism

a f T T exp f T



g T T exp g T exp b


In cho osing a constantvalue for the sp ecic heat care had to b e taken to ensure

that the maximum temp erature would b e appropriate for the ame b eing mo deled

The adiabatic ame temp erature for a stoichiometric mixture of the two streams

describ ed would b e K However the sp ecic heat is an increasing function with

increasing temp erature and in a nonpremixed ame the maximum temp erature

is limited by the diusion of heat away from the reaction zone Thus cho osing a

Incorporating realistic chemistry into DNS of combustion

constant C such that the adiabatic ame temp erature would b e matched would


result in an underprediction of the maximum ame temp erature in the nonpre

mixed ame A value for C of Jg K was chosen which yields an adiabatic


ame temp erature of K With the ideal gas constant

Jmol K

R Jg K

g mol

this gives Cho osing an initial temp erature and a value for the sp ecic

heat xes the reference temp erature density and sp eed of sound which are used to

nondimensionalize the quantities in the reaction rate expressions

Onedimensional simulations

The chemical kinetic mechanism describ ed ab ovewas incorp orated into a DNS

co de which solves the governing equations for fully compressible turbulentow

Ruetsch based on the algorithms of Lele and Poinsot and Lele

In order to test the implementation of the mechanism and to provide initial con

ditions for simulations with turbulentow elds the co de was run for a simple

onedimensional problem

Initial and boundary conditions

In order to ensure that the pressure in the domain remains constant uid must

b e allowed to leave the domain An additional constraint is that the reaction rates

at the b oundaries must b e zero otherwise the b oundary conditions are illp osed

Also b ecause the chemical kinetic rates dep end on the hydrogen free radical concen

tration the mechanism cannot autoignite if the ow is initially unreactive it will

remain so therefore the elds must b e initialized such that at least some chemical

reaction is already underway

Partially nonreecting outow b oundary conditions Poinsot Lele were

chosen for b oth b oundaries in the onedimensional simulations

The sp ecies mass fractions were initialized by rst dening the mixture fraction

as a linear combination of mass fractions such that the chemical source term in its

transp ort equation is zero


F uel Oxi Prod NO


The mixture fraction was initialized with the analytical solution to the diusion

equation for a semiinnite slab of fuel mixing with a semiinnite slab of oxidizer




Z x t erf

D t

at an arbitrary time chosen such that the reaction zone would b e suciently re

solved with the available numb er of grid p oints Mass fractions for each sp ecies

were then calculated by assuming that an arbitrary fraction of moles for eachof

W K Bushe R W Bilger G R Ruetsch

reaction I and I I had reacted to completion This assumption also allowed for the

calculation of the heat released as a function of mixture fraction from which the

temp erature eld can b e calculated The initial pressure was assumed constantat

atmwhich corresp onds to nondimensional pressure unitsfrom which

the density could b e calculated The initial velo citywas zero

Broadening the reaction zone

The region in which the chemical reactions are signicantreferred to as the

reaction zonefor the mechanism describ e ab ovewas found to b e very narrow

In order to make the reaction zone broader reaction rate I and the exp onent inof

Eq are divided by a constant of This has b een shown in previous studies

Swaminathan Bilger a b to broaden the reaction zone suciently

to allow the reactions to b e easily resolved by DNS without substantially altering

the structure of the ame This is shown in Fig where the results of a very

well resolved simulation without the broadened reaction zone are compared to those

from a simulation with the broadened reaction zone The twosimulations had the

same initial conditions and were each run for nondimensional time units

Despite the changes to the mechanism it can b e seen in Figs a and b that

the proles of the mass fractions of the Fuel Intermediate and Pro duct are only

slightly mo died from those given by the original mechanism Only the fraction of

Oxidizer that leaks through the reaction zone to the rich side of the ame is signif

icantly altered by the mo dication Figure c compares the estimated hydrogen

radical It is clear that without the mo dications to the mechanism

it would b e very dicult to resolve the sharp drop in Hydrogen radical mass fraction

at the stoichiometric mixture fraction of However with the mo dications that

sharp drop disapp ears and the hydrogen free radical mass fraction can b e resolved

with far fewer p oints The signicant dierence in magnitude of the reaction rate

for reaction I shown in Fig d is almost entirely attributable to that reaction

having b een slowed by a factor of It should b e noted that this reaction as

with the hydrogen free radical mass fraction would severely constrain the resolution

of the ame if the mo dications were not includedso much so that DNS includ

ing turbulence would hardly seem p ossible without using the mo dications The

inuence of the mo dications on the reaction rates of reactions I I and I I I shown

in Figs e and f is primarily to also broaden the region of mixture fraction

in which they are signicant and they also seem to ease resolution constraints by

making the rates more smo oth functions of mixture fraction

Twodimensional simulations

Once tests of the newly implemented kinetic mechanism had b een completed the

addition of turbulence in twodimensions was undertaken This was seen primarily

as b eing a means of establishing what turbulence parameters would b e appropriate

to provide adequate threedimensional simulation results for mo deling purp oses

however it was anticipated that these twodimensional tests would also provide

results from which direct insight could b e gained

Incorporating realistic chemistry into DNS of combustion

0.3 0.25 b 0.25 a

0.2 Prod 0.2 0.15

0.15 Oxi Y Y

F uel 0.1

0.1 Int 0.05 0.05

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

−3 −5 x 10 x 10

1 3 d 0.8 c 2.5 2 0.6 I H

1.5 Y 0.4 1

0.2 0.5

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

−6 −12 x 10 x 10 3.5 2


e f 1.5 2.5

2 II III 1 1.5

1 0.5 0.5

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1


Figure Comparison of original mechanism to mo died mechanism a Fuel

and oxidizer mass fractions b Intermediate and pro duct mass fractions c Hy

drogen free radical mass fraction d Rate of reaction I e Rate of reaction I I f

Rate of reaction I I I original mechanism mo died mechanism

W K Bushe R W Bilger G R Ruetsch

Initial and boundary conditions

The same constraints on the initial and b oundary conditions describ ed for a one

dimensional domain in Section apply to two and threedimensional domains

This means that at least one of the b oundaries in a two or threedimensional domain

must allow for outow Also the reaction rate at an outow b oundary must b e zero

The mass fraction temp erature density and velo city elds were initialized by

using the stabilized onedimensional ame solution describ ed in Section By

placing a stable ame in the middle of the twodimensional domain the time until

the reaction zone the region of the ow in whichchemical reaction takes place

reached a b oundary could hop efully b e maximized

Initial turbulentvelo city uctuations were obtained by using a pseudosp ectral

co de to solvethegoverning equations for incompressible ow and forcing a p erio dic

threedimensional ow eld on a grid from quiescence until its statistics b ecame

stationary Ruetsch Ferziger For the twodimensional simulations a slice

of the threedimensional owwas extracted and the comp onents of the velo city that

did not satisfy continuitywere discarded Two identical slices were placed next

to each other to ll out the domain At the b oundaries the turbulent

uctuations were ltered to zero to avoid p otential generation of unphysical vorticity

at the b oundaries The turbulent uctuations were divided by the densitysoasto






This rescaling of the turbulent uctuations was p erformed to avoid the generation

of large unphysical pressure waves in the ame It should b e noted that this means

the vorticity inside the ame is initially much higher than elsewhere however since

the viscosity in that region is also much higher the vorticity inside the ame decays

very quickly The resulting turbulentvelo city eld was added to the onedimen

sional ame velo city eld

The initial cold velo city eld had a Taylor Reynolds numb er of The dis

sipation length scale was b ox lengths and the dissipation Reynolds number

was In order to avoid forcing the turbulence through the reaction zone the

turbulence was allowed to decay


In extracting statistics such as PDFs or ensemble averages from DNS results

it is usually necessary to make use of isotropy in the turbulence In an isotropic

ow ensemble statistics can b e approximated byaveraging p oints along directions

of isotropyFor the results that follow it was necessary to neglect the eect that

the anisotropy in the mixture fraction eld mighthave in order to obtain converged

statistics Previous DNS Mell et al and exp erimental Bilger b stud

ies of mixing layer ows such as the one used in this studyhave shown that the

inhomogeneity in the mixture fraction has only a slight eect on conditional statis


The evolution of the Probability DensityFunction PDF of the mixture fraction

as a function of nondimensional time is shown in Fig The PDF represents

Incorporating realistic chemistry into DNS of combustion



15 Z

P 10

5 0

50 0

0 0.2 0.4 100 Time

0.6 0.8 1


Figure Evolution of P Z intime

the entire ow eld The delta functions at Z and Z decay with time

as one would exp ect There is no evidence of a p eak in the PDF b etween the two

delta functions at Z and Z as has b een seen for similar ows Broadwell

Mungal This is likely b ecause there is no velo city dierence b etween

the fuel and oxidizer streams whichwould lead to coherent structures created by

shear b etween the streams In the absence of such coherent structures the eect of

turbulence on the mixing pro cess would app ear to b e essentially random

In Fig a the PDF of mixture fraction at nondimensional times is com

pared to the b eta PDF Co ok Riley evaluated using the measured mean

and variance of mixture fraction the b eta PDF is

a b

a b






a and b a Z Z a


The b eta PDF compares well to the PDF measured from the DNS with two

notable exceptions For Z and Z there app ears to b e some kind

of structure in the DNS PDF This could b e a result of there b eing an inadequate

number of points in the domain to obtain converged statistics however it is curious

that these structures app ear at these values of Z there are considerably more

p oints with these values of Z than there are for Z where the DNS

PDF do esnt exhibit such structure The second dierence in the two PDFs is more

subtle The PDF given by the DNS seems to decrease from Z to Z

Z This may b e indicative of the The b eta PDF reaches a minimum at

inuence of variable density on the PDF of mixture fraction

W K Bushe R W Bilger G R Ruetsch

7 1

a b 6 0.8 5 0.6 4 Z 3 P 0.4 2 0.2 1

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1


Figure a Comparison of PZ at time units to b eta PDF with similar

means and variances DNS b eta PDF b Comparison of density

proles b etween the initial laminar ame and the conditional average in the turbu

lent ame after time units D laminar ame hjZ i in D

turbulent ame

In Fig b the conditional average of the density for the entire domain is com

pared to the density prole in mixture fraction of the onedimensional ame that

was used to initialize the twodimensionalsimulation Variations from the condi

tional average of the densitywere quite smallthe conditional variance of the den

sitywas of the order nevertheless there is a discernible dierence b etween

the two curves This is likely caused by the straining of the ame bythevelo city

eld pro ducing regions of lo cal extinctionregions along the ame front where the

rate of diusion of pro ducts and heat away from the reaction zone surpasses the

reaction rates and the reactions are eectively quenched

The b eta PDF in Fig a seems to cross the DNS PDF at a mixture fraction

of ab out At ab out this same value of mixture fraction the density reaches a

minimum This app ears to supp ort the notion that the dierence in the two PDFs

is attributable to the variation in density

Scatter plots of the Intermediate Pro duct and NO mass fractions as functions

of the mixture fraction after time units are shown in Fig Also shown is the

temp erature Figure a provides further evidence that the ame has undergone

lo cal extinction Where at a mixture fraction of around Y has fallen b elow


it seems likely that reaction I has essentially stopp ed providing fresh new

Intermediate and reaction I I has then depleted the remaining Intermediate and

likely stopp ed as well Figures b and d show not only how the temp erature

isavery strong function of the Pro duct mass fraction for unity Lewis numb er

but also how the temp erature and pro duct mass fractions are aected by the lo cal

extinction phenomenon evident in Fig a Figure c reveals just howchallenging

the chemical closure problem can b e when the activation energy of a participating

chemical reaction is very large as is the case with reaction I I I There is clearly

Incorporating realistic chemistry into DNS of combustion

0.03 0.25

a b 0.025 0.2

0.02 0.15 0.015 Int Prod Y

Y 0.1 0.01 0.05 0.005

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

−8 x 10 2


c d

1.5 1500

NO 1

K 1000 Y T

0.5 500

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1


Figure Scatter plots of mass fractions and temp erature as functions of mixture

fraction after time units a mass fraction of Intermediate b mass fraction

of Pro duct c mass fraction of NO d temp erature

a great deal of scatter in the NO mass fraction around its conditional average

considerably more than in the temp eratureso that the single conditional average

would b e a p o or approximation for the NO mass fraction

Contour plots of the three reaction rates are shown in Fig At this time a region

of lo cal extinction is just b eing reignited with reactions I and I I having p eaks at

x y and Reaction I I I in the region that has b een extinguished

has essentially stopp ed This is a result of the strong temp erature dep endence of

the reaction rate

In Fig the conditionally averaged reaction rates are compared to those pre

dicted byevaluating the reaction rates with the conditionally averaged mass frac

tions temp erature and density This is a test of the validity of rst order CMC

hyp othesis given by Eq The reaction rates for reactions I and I I I are predicted

to within and that for reaction I I is predicted to within It would app ear

that rst order closure would b e adequate for predicting the mean reaction rates

This is esp ecially surprising in the case of reaction I I I which has an extremely large

activation energy and would b e exp ected to require some correction for uctuations

in the temp erature around its conditional average

W K Bushe R W Bilger G R Ruetsch


7 a



4 y




0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


7 b



4 y




0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


7 c



4 y





0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


Figure Contour plots of reaction rates at time units a reaction I b reaction I I c reaction I I I

Incorporating realistic chemistry into DNS of combustion

−5 −6 x 10 x 10

2.5 7 b

a 6 2 5

1.5 4 I II 1 3 2 0.5 1

0 0

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1


−12 x 10 1.4

1.2 c


0.8 III 0.6




0 0.2 0.4 0.6 0.8 1


Figure Comparison of conditionally averaged reaction rates to reaction rates

predicted using the rst order CMC approximation at time units

a h Y TjZ i Q hT jZ i


h Y TjZ i Q hT jZ i b


c h Y TjZ i Q hT jZ i


Future work

The simulation results presented in Section app ear to indicate that the imple

mentation of the chemical kinetic mechanism has b een successful Unfortunately

the conditional statistics extracted from the small twodimensional domain are in

adequately converged to b e of signicant use in the validation of the CMC mo del

It has b ecome clear that results in the full threedimensional domain will haveto

b e obtained

The eect of heat release from a premixed ame on a turbulentow is more

signicant than in a nonpremixed ame since the heat is released in a thin front

which propagates through the uid The use of constantproperty DNS for validation

of mo dels of premixed combustion is thus even more questionable than for mo dels

W K Bushe R W Bilger G R Ruetsch

of nonpremixed combustion For this reason a new pro ject will also b e undertaken

in whichsimulations of a premixed ame will b e p erformed using the same co de for

the purp ose of providing validation for mo dels of premixed turbulent combustion


The authors would like to thank J Ferziger K Mahesh N Swaminathan and T

Poinsot for helpful discussions and suggestions R W B gratefully acknowledges

the nancial supp ort of the Australian Research Council The simulations were

p erformed at the NAS facility of the NASA Ames ResearchCenter


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