A linear eddy model for steady-state turbulent combustion

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Copyright ©1996,American Institute of Aeronautics and Astronautics,Inc.
AIAA Meeting Papers on Disc,January 1996
A9618480,AIAA Paper 96-0519
A linear eddy model for steady-state turbulent combustion
GrahamM.Goldin
Georgia Inst.of Technology,Atlanta
Suresh Menon
Georgia Inst.of Technology,Atlanta
AIAA34th Aerospace Sciences Meeting and Exhibit,Reno,NVJan 15-18,1996
A newapplication of the linear eddy model to steady-state turbulent combustion is presented.The model constructs the
single point joint scalar pdf fromLEMsimulations of scalar decay in homogeneous turbulence.The physical
mechanisms of convection by all turbulent scales and molecular diffusion are taken into account,allowing for the
inclusion of differential diffusion and extinction affects.By integrating the pdf,mean scalar quantities,in particular the
mean chemical reaction rate,are tabulated once and for all as a function of the lower moments.Model pdfs are
compared with conventional pdfs using a two variable chemical mechanism.The model is then applied to a turbulent
jet flame and the predictions are compared with nonequilibriumexperimental results.(Author)
Page 1
AIAA-96-0519
A LINEAR EDDY MODEL FOR STEADY-STATE
TURBULENT COMBUSTION
Graham M. Goldin *
Suresh Menon"
School of Aerospace Engineering
Georgia Institute of Technology
Abstract
A new applicatio n of the Linear Eddy Model to steady-stat e
turbulen t combustio n is presented. The model construct s the
single point joint scalar pdf fro m LEM simulation s of scalar
decay in homogeneou s turbulence. The physical mechanisms of
convection by all turbulen t scales and molecula r diffusio n are
taken into account, allowing for the inclusion of differentia l
diffusio n and extinctio n affects. By integratin g the pdf, mean
scalar quantities, in particula r the mean chemical reaction rate,
are tabulate d once and for all as a functio n of the lower
moments. Model pdf s are compare d with conventiona l pdf s
using a two variabl e chemical mechanism. The model is then
applied to a turbulent jet flam e and the prediction s are compared
with non-equilibriu m experimenta l results.
1. Introductio n
The two prominen t approache s to solve the steady-stat e Navier -
Stokes (NS) equation s for turbulen t reacting flow s are the
probabilit y densit y functio n (pdf ) evolution equation method'
and the moment equation method. The latter include the
classical Reynolds averaged NS (RANS) 2, as well as the recentl y
proposed conditione d averages1. All these methods requir e
closure, and their success is determine d by the accuracy of the
model s they employ.
In the RANS method, the unclosed higher moment s are usuall y
modeled as function s of the availabl e information, namel y the
solved lower order moments. These unclosed terms can be
broadl y divided into two groups; one comprisin g of velocit y
correlations, such as the Reynolds stress tensor, and the other
comprisin g of scalar averages, such as the mean reaction rate.
The velocit y correlatio n fluxe s are invariabl y modeled by mean
gradient diffusion, and the present consensu s is that despit e the
known limitation s of gradient diffusion, the major source of error
lies in the estimat e for the scalar averages, especiall y the mean
reaction rate4-2.
Existing model s for the scalar averages assume the statistica l
distributio n of the scalar fiel d (which is specified by the single
point joint scalar pdf) using simpl e analytica l function s without
taking into account the underlyin g unsteady turbulen t dynamic s
and molecula r mixing. These model s include the assumed shape
pdf 5 and the Laminar Flamele t model 6, which may employ
Beta7* and (clipped) Gaussian 5 functions. Assumed shape
method s have obtained successfu l predictio n of flow s in
* Graduat e Researc h Assistant, Studen t Membe r AIAA.
** Associat e Professor. Senior Member AIAA
Copyrigh t C 1996byQ.M. OoldinmdS. Menon
Publishe d by the America n Institut e of Aeronautic s and Astronautics,
Inc. wit h permission.
chemical equilibrium5 and flow s with moderat e departur e fro m
equilibrium * for simpl e geometries. However, since the
influenc e of all turbulenc e scales, as well as molecular diffusion,
is implicitl y assumed in the pdf, Reynolds (Re) and Schmidt (Sc)
numbe r effect s cannot be discerned, and more complex
phenomen a such as extinctio n and differentia l molecula r
diffusio n are preclude d due to the difficult y of prescribin g the
pdf. In addition, owing to the highl y non-linea r instantaneou s
reaction rate term, the mean reaction rate is sensitive to the pdf
shape, so that large errors in the finite-rate-kinetic s flo w solution
may accrue fro m smal l errors in the pdf. This work proposes a
new model for the scalar pdf as a functio n of the lower order
moments. Direct account of the unsteady physical processes of
turbulen t transpor t and molecula r diffusio n is taken, overcomin g
the aforementione d limitations. Althoug h the model formulatio n
is general, it is applied in this paper to non-premixe d
combustion.
The scalar pdf is constructe d fro m simulation s of the Linear
Eddy Model (LEM)'-">. One of the key features of the LEM, in
contrast to most other model s such as gradient diffusion, is that
all turbulenc e scales are resolved so that the distinct processes of
turbulen t advectio n and diffusion-reactio n at the smallest scales
are captured. This distinctio n is critical since chemical reaction
proceeds at the molecula r level when reactants diffus e into
contact, and the rate of diffusion (hence chemical conversio n for
fas t chemistry), is strongl y affecte d by the scalar gradient s
induced by the smal l scale turbulen t motion. The central LEM
assumptio n is that the evolution of the scalar field at the smal l
scales can be adequatel y captured by a simplifie d statistical
description in one spatial dimension. A ID formulatio n allows
tractabl e simulation s at high Re and Sc, a capabilit y impossibl e
in Direct Numerica l Simulatio n (DNS).
Previous application s of the LEM include isotherma l mixing in
spatiall y developin g turbulen t mixing layers"' plumes9 and
jets12, and reacting jets with reduced11 and detailed14 chemical
kinetics. The predicted Re and Sc dependencie s are in accord
with turbulenc e scaling laws and experimenta l data. In a furthe r
application, which has particula r significanc e in this work, the
LEM tempora l simulatio n of isotherma l scalar decay in
homogeneou s turbulenc e obtained quantitativ e agreemen t with
DNS15. Although the diverse range of flo w geometrie s is
achieved by altering the initial and boundar y conditions, the
fundamenta l LEM kinemati c structur e is unchange d in all cases,
mixing model.
2. Model formulatio n
Since a detailed derivatio n of the LEM is availabl e in the
literature9-10, only a brief overview is presented in section 2.1.
The crux of this work, which is a formulatio n of the LEM to the
steady-stat e momen t equatio n method, is introduce d in section
2.2 for genera l chemistry. Section 2.3 is an applicatio n of the
mode l to a reduce d chemica l mechanism. Henceforth,
conventiona l averagin g is denote d by fla t overbars, and Favr e
(density weighted) averaging is denoted by tilde overbars.
2.1 The Linear Eddy Model
The LE M is a spatio-tempora l simulatio n of the evolutio n of a
scalar fiel d in ID, with turbulent advection treated distinctl y
fro m molecula r diffusio n an d reaction, dow n to th e smalles t
turbulen t time and lengt h scales. Since all scales are resolved,
the ID Fickia n species equatio n
£5
dt
\_jd_
' pdx
Wk] 
^r"k'
(1)
is solve d withou t averagin g and subsequen t modeling. In
implementation, equatio n (1) is solve d numericall y by centra l
difference s on a domai n discretized into scalar elements of equal
size, S. Typicall y six LEM element s are require d to resolve the
smalle r of the Batchelo r and Kolmogoro v lengt h scales so that
sub-elemen t fluctuation s are absen t and the simulatio n is gri d
independent.
The ingenious aspect of the LEM is the mechanism of advection
by turbulen t eddies, whic h is modele d in ID as a stochasti c re-
arrangemen t of a portio n of the scalar element s alon g the line.
The re-arrangemen t templat e is terme d the triplet map, and is
illustrate d in figur e 1 for an initia l linear scala r fiel d on the
discret e LEM domain. Triple t mappin g does not chang e the
element concentration, consistent with the pictur e that eddies
transpor t flui d elements and the element concentration can onl y
chang e by diffusio n wit h neighborin g elements. The scalar
gradien t is howeve r increased, analogou s to the effec t of a
simila r sized microscal e turbulen t edd y on a physica l scala r
field. Triple t mapping, whil e not a uniqu e re-arrangemen t
template, ha s capture d mos t of the principa l spectra l scalin g
regimes'
1 0
The LEM mapping s hav e three stochasti c parts. First, the size of
an eddy (denoted / in figur e 1) is sampled randoml y fro m an
eddy-siz e pdf, denote d /(/) . Second, the locatio n center of the
mapping s ar e chose n randomly, constraine d by the LE M
boundar y conditions. Third, the mapping s are implemente d as a
Poisson process in time wit h mea n rat e paramete r A . The edd y
size pdf /(/), as wel l as the Poisso n rat e paramete r /t , are
derived by equating the turbulen t diffusivit y of the LEM re-
arrangement s to Kolmogoro v cascad e diffusivit y laws of a scalar
in a hig h Re flow. The resultin g governin g equation s ar e
5 I
r-8/3
54 vRe, (L/>jf'3-l
5 L 3 1-
4/3
(2)
(3)
(4)
wher e L, rj. Re, and v are the mode l integra l and
Kolmogoro v lengt h scales, turbulen t Reynold s numbe r and
kinemati c viscosit y respectively. Velocit y does not appear
explicitly in the LEM, and the turbulenc e fiel d must be specified
a priori by providin g L and Re,.
A LEM simulation, on a discret e initia l scala r field, progresse s
as follows. The species equatio n is continuousl y advance d at a
time step sufficientl y smal l for solution independence. This
smoothin g process is interrupte d by instantaneou s triplet
mappin g event s whic h occur stochasticall y at a mea n Poisson
frequenc y E = JA< & . The mappin g size / , boun d betwee n the
limits L and rj , is sample d fro m the pdf /(/) and centere d at
a rando m location with probabilit y X./E . Like all Monte-Carl o
methods, the LEM statistics mus t be average d ove r a large
ensembl e of simulation s to wash out stochasti c error.
The affec t of LEM heat release, which is assumed to occur at
constan t pressure, is twofold. Th e firs t i s a chang e in the
viscosit y an d diffusivit y transpor t co-efficient s wit h the
temperatur e increase. The secon d is due to the correspondin g
densit y decreas e whic h conserve s mas s by volum e expansion.
Volum e expansion, in turn, causes a decreased vorticit y by
conservin g angula r momentum 16. Thi s reductio n in the
turbulence is modeled on the linear domai n as follows. The
kinemati c viscosit y v is calculate d as a volum e averag e ove r the
scalar field, and 1 7 is adjuste d fro m equation (4) by holding the
fluctuatio n velocit y «' an d integra l lengt h scal e L constant.
The net effec t is an increase in tj and a decrease in the mappin g
frequenc y as heat is released. Since differen t LEM element s
expan d to differen t sizes, whil e triplet mappin g require s equa l
sized elements, the scalar domai n is re-gridde d afte r heat release
at the new require d resolution, as depicte d in figur e 2. The
overlapping scalar fiel d in each LEM element after re-gridding is
mixed withou t reaction. Not e that thi s spurious mixin g is
inconsequentia l since it occur s below the Batchelo r lengt h scale
wher e scala r fluctuation s are rapidl y smoothed.
To illustrat e the effec t of heat releas e on produc t formation, the
evolutio n of the produc t mas s fractio n Y f of a binar y mixtur e in
a homogeneou s turbulen t fiel d is presented in figur e 3. The LEM
geometry, describe d in section 2.2, has L « Is and Re, = 90 ,
identica l to the simulatio n of McMurtr y et. al.15. The reactant s
hav e constan t transpor t co-efficient s an d infinit e rat e irreversibl e
chemistr y wit h an adiabati c temperatur e rise of twic e the
reactan t initia l temperature. Hea t releas e is seen to decreas e the
rate of produc t formation 1*.
2.2 The LEM for steady-state turbulent combustion
Befor e presentin g the new model, it is expedien t to restat e the
objective. The momen t equatio n approac h requires calculatio n of
mean scalar quantities, such as the mea n reactio n rate
wk = wk(p,T,Y k) , fro m th e solve d lowe r moments, suc h as th e
means p, T , Y k and Re, (p, T and Y k are the flui d density,
temperatur e and /tth species mass fraction, respectively). This is
conventionall y modele d by assumin g a pdf shap e P(p,T,Y k) ,
and integratin g ove r all states as
(5)
0 00
The objectiv e is to calculat e a mos t likel y for m of mi s pdf,
satisfyin g the give n lowe r moments, as opposed to assumin g it.
Note that the single point joint scalar pdf can be constructed
fro m the scala r time trac e at the poin t in question.
The LEM is configure d to the geometr y of a decayin g scala r in a
stationar y homogeneou s turbulen t field, as in the stud y by
McMurtr y et. al.15. This entails a periodi c domain, with unifor m
spatia l probabilit y of a mappin g event. The domai n size is set
equa l to the integra l scale L, and mappin g epochs occu r at a
fixed frequenc y E = XL. The initial LEM scalar distribution is
assigned, for non-premixed combustion, as the complet e
segregatio n of fue l and oxidize r as illustrate d in figur e 4(a),
correspondin g to a binar y mixtur e at the maximu m variance. For
clarity, a singl e scalar, viz. the mixtur e fractio n /, is shown,
althoug h the metho d is readil y extende d to multipl e species wit h
differen t molecula r diflusivities.
Unde r the actio n of molecula r diffusio n and turbulen t stirring
(mappin g events), the scalar varianc e g = ( f - f ) 2 decay s
towar d zero. Figur e 4(b) is a plot of the spatia l distributio n of /
at some later time. The centra l concep t of thi s wor k is to
approximat e the actua l scalar time trace at a poin t in a turbulen t
flo w by the scala r distributio n in a LEM simulation. The LEM
spatial axis (the x/L axi s in figur e 3) is transforme d to a time
axis whic h may be interprete d as a Taylo r hypothesi s (also
referre d to as a froze n turbulenc e approximation), and holds
exactl y for isotropi c turbulence. From this time trace the single
point joint scalar pdf can be constructe d by binnin g ove r al l
states.
The implementatio n in a RAN S solve r is as follows. Durin g the
solutio n of th e modele d transpor t equation s fo r th e lowe r
moment s (see sectio n 3 below), assume tha t the numerica l
value s ~p, T, Y t and Re, ar e attaine d at one poin t in the flow -
field. In the metho d proposed above, a separate, full-kineti c
LEM simulatio n is conducte d wit h the appropriat e initial
conditions and turbulent scales, as detailed later, such that the
LEM pdf has lowe r moment s p"", f"", £"" and Re?"
tha t exactl y matc h thos e in the flow-fiel d (viz. p"" = p,
fLEU = f t yLEH = j? ^ RgLEM = ^ ) ^ ^^ ^ m
the RANS solution is then assumed to have the LEM pdf and
scalar means such as ti> * and RT (R i s the gas constant ) ar e
obtained fro m equation (5). Althoug h onl y firs t moment s have
been matche d in the abov e example, highe r moment s and
correlations can be equated to increase the accuracy, provide d
tha t their modele d RAN S transpor t equation s are accurate.
Th e fac t tha t th e LE M simulatio n i s time accurat e i s of no
consequenc e here since the physica l time trac e satisfyin g give n
highe r moment s i s used, independen t of the trac e history.
Reflectin g this, the pdf contain s no time information.
As oppose d to performin g the LEM simulation s and pdf
integrations during the solution of the RANS, the calculation s
mar be perfbrme d once and for a H over the ^ entire^ rang e of
Ip  T  YL  Re. space an d th e scalar mean s tabulate d a s
(6)
wher e p, T, Y k and Re, are the tabl e parameters. The curl y
braces denot e a tabula r format, and are not to be confuse d wit h
the approximatio n of the mea n reactio n rate by its instantaneou s
(laminar ) expression. Durin g the numerica l solutio n of the
RAN S equations, the tabl e is searche d and interpolate d as a
functio n of the solution variables ~p, T, Y k and Re,.
Owin g to the computationa l econom y of the LEM, tables for
comple x chemistr y at high Re can be affordabl y assembled.
However, for a large numbe r of chemical species, the table size
ma y becom e intractabl y larg e if al l mea n species 7 * ar e
parametized. As an alternative, the tabl e may be parametize d by
an appropriat e reduce d subse t of mea n species, wit h some loss in
generality.
Sc effect s ar e implici t in the model. For example, two LEM
simulation s at differen t Sc that have the same mean s p, T, Y t
an d Re, will hav e differen t wk  Differentia l diffusio n ma y be
incorporate d withou t difficult y int o the LEM simulation, and
since the ful l scalar pdf P(p,T,Y k) i s constructed, ignitio n and
extinction can be modeled in a RANS solution. These issues will
be addresse d in a futur e study. An additiona l advantag e of the
mode l is that, unlik e mos t assume d shape methods, moment s
highe r tha n firs t orde r are not require d to specif y the pdf. One
limitatio n of the presen t mode l is the inabilit y an d apparen t
difficult y in addressing the fluctuatin g pressure field.
The LEM triplet map, whil e an excellen t mode l for the smal l
scale motion, is a somewha t crude approximatio n for the integra l
scales, and is a poor mode l for the anisotropi c coheren t
structures. The mode l is hence incapabl e of capturin g certai n
large scale features. However, the large scale effec t of
intermittenc y can be capture d (intermittenc y occur s whe n the
time trace is locate d at / = 0 or / = 1, such as in figur e 4(a)).
In thi s work, the integra l scal e is assume d to be the larges t eddy,
and is set equal to the jet nozzl e diameter d . Althoug h the size
of the coheren t structure s in a turbulen t jet increase downstream,
and may be thus muc h large r than d, this is not considere d to
hav e an extreme affec t because the larg e scales, governe d by the
eddy-size pdf /(/) , occur infrequently.
Anothe r issue whic h require s justificatio n is the assumptio n of
an initial binar y mixtur e (see figur e 4(a)), as opposed to a plug -
flo w typ e initia l distribution, whic h ha s evenl y space d 'slabs' of
fue l and oxidizer. The latter case decays faste r due to the
increased scalar gradients. McMurtr y et al.13 performe d LEM
simulation s in an identica l geometr y for a range of initial scalar
distributions, and showe d that the fiel d became independen t of
the initia l scalar lengt h scal e afte r a shor t transient.
2.3 Application to a reduced chemical mechanism
The model is demonstrated using a two-variable (f,n) reduced
chemica l mechanis m for H2-a.a combustion 17. Unde r the
assumption s of unit y Lewis numbe r for ail species, constant
pressure, low Mach number, low bouyanc y and adiabaticity,
nine species and temperatur e are completel y specifie d by the
conserve d mixtur e fractio n /, and a progress variabl e n.
Since the allowabl e domai n in f-n space is tri-angular, the
progress variabl e is normalize d as
(7)
(13)
for graphical presentatio n on a rectangula r f -n domain. The
limits / = 0 and / = 1 correspond to oxidizcr and fuel, and at
a fixe d / , n* = 0 and n = 1 indicat e the equilibriu m and the
mixed withou t reactio n limit s respectively.
Compariso n is mad e wit h two conventiona l models. The firs t is
the assume d shape pdf * whic h require s RAN S equation s for the
firs t two. moment s of / (mea n / and varianc e g), and the
mean reaction progress, «*. The joint pdf of / and «* is
invariabl y modele d as*
(8)
wit h Pi(J) assume d as a Bet a functio n and P 2(n") as a Dirac
delt a functio n at the mean, viz. P 2(w* ) = J(»*-n"). I t i s
common 1 to mode l a n* varianc e equatio n and modif y P2(/T)
by Dirac delt a function s at n* = 0 an d « * = 1. Since the LEM
pdf does not requir e the varianc e of n', this pdf for m is
discarde d fo r fai r comparison. Th e mea n specie s an d mea n
reaction rate w,, at a point in an assumed shape pdf RANS
solutio n are function s of / , g and it' at the point.
The second conventiona l metho d compare d is the Lamina r
Flamele t model 6, whic h consider s a turbulen t flam e brus h as an
ensembl e of unstead y diffusion flamelets. Each flamele t has the
structur e of a stead y strained lamina r diffusio n flam e (Tsuj i
flame). The Tsuj i flam e is characterize d by the singl e paramete r
Xf~f m > which is the scalar dissipation, define d as
(9)
at the point of stochiometri c mixtur e fraction. In a turbulen t
diffusio n flam e calculation, the mea n species at a poin t are
obtaine d fro m the statistica l distributio n of the flamele t
ensemble,
? = J J W,*/-/.. )P(f,Zf.fJdXdf. (10)
o o
The pdf P(f>Z/-/M ) > * commonl y assumed uncorrelated as
wit h /*,(/) assume d as a Beta function, andP 2(/j-) as a log-
norma l distributio n wit h a varianc e of 2,
wher e k an d "e ar e the mea n turbulen t kineti c energ y and its
dissipation, and the constant Cx=2. The mean scalar
dissipation, whic h is a measur e of the intensit y of local mixing,
ma y be interprete d as a Re, parameter. For a give n flamele t
librar y iJ(/,jf/»/ ) , the mea n species at a poin t in a Lamina r
Flamelet RANS solution are function s of / , g , k and ~e at
the point.
The abov e two method s assume the statistical distributio n of the
fluctuatin g scalar fiel d (equation s (8) and (11)) at a poin t
withou t accountin g for the physica l passage of turbulen t eddies,
or the molecula r diffusio n propertie s of the scalar.
The LEM specie s equatio n (1), reduce d to the (/,») chemistr y
is
dt
and
(14)
(15)
The mea n scalar dissipatio n is modele d in the RAN S solutio n as
The discrete value s of / and n in each LEM cell evolv e
subjec t to the deterministi c equation s (14) and (IS), and the
stochastic stirring events governed by equations (2) through (4).
Result s fro m an arbitrar y LEM simulatio n ar e show n in figure s
(4) to (6) for a fue l of 22 % Argo n (mol ) in Ht, whic h ha s a
stochiometri c mixtur e fractio n fM = 0.161. Since / is a
conserved scalar, the Favre averaged mixtur e fractio n /,
remain s constan t throughou t the simulations, and is set equa l to
fllo here. Figur e 4(a) is the initial scalar field, whic h has
maximu m varianc e g « / - /2. Molecula r diffusio n causes the
scalar varianc e to deca y in time as the simulatio n proceeds. The
rate of scalar deca y increases wit h Re, since the mor e frequen t
mappin g event s amplif y the scalar gradients, leading to an
increased diffusio n rate. In turn, the departur e fro m chemica l
equilibrium increases as the diffusio n time scale approaches the
chemica l time scale.
For L = 0.052m, correspondin g to the jet diamete r of the
experimenta l geometry 1* chosen, and Re, = 60, the
instantaneou s scalar fiel d at g = 0.01 is plotted in figure s 4(b)-
(d). Thes e sho w the spatia l distribution s of /, « * and z
respectively, whic h have spatial means / = fM  n* = 0.0633
and z - 2l7s~l. The LEM equivalen t of the scala r dissipatio n
(equatio n (9)) is
(16)
By transformin g the spatial axis to a time axis, scatter plots
(created by placin g a prob e in the time trace) and pdf plot s
(created by binning the time trace) for a point in a turbulen t
diffusio n flam e wit h f-f*,, £ = 0.01, « * = 0.0633 and
£ = 2175'' may be constructed. Figur e 5(a) present s a scatter
plot of temperatur e in mixtur e fractio n space using 2500 points
fro m a n ensembl e of th e abov e describe d LE M simulations.
Figur e S(b) is a correspondin g experimenta l plot11', and the
similarit y is evident. Bot h plots display sub-equilibriu m
temperatures, wit h the greates t deviatio n fro m equilibriu m
occuring around stochiometri c mixture fraction.
Figur e 6 present s LEM pdf plots constructe d fro m an ensembl e
of 10000 LEM realizations, as wel l as thei r correspondin g
conventiona l pdf models. Figur e 6(a) is the singl e poin t joint
scala r pdf P(f,n') wit h lowe r moment s /=/«,,, g * 0.01 and
n' = 0.0633 . Not e tha t the pdf peak has been truncate d for
clarity. Element s closest to stochiometri c mixtur e (fM = 0.161)
ar e a t greate r non-equilibriu m (n * > 0) tha n off-stochiometri c
elements, whic h is synonymou s wit h the experimenta l dat a in
figur e 5(b). The conventiona l assumed shape pdf, equatio n (8),
wit h the same lower moment s / , g and n * is plotte d in figur e
6(b). The mode l assumes that all flui d element s passing ove r the
point hav e the same (normalized ) departur e fro m equilibrium,
whic h is clearl y a big assumption.
Figur e 6(c) is a plot of the singl e poin t joint scalar pdf P(f,z)
with lower moment s /=/«,, g = 0.01 and ^ =2L7*"', and
wit h the pdf peak truncate d at a heigh t of 5. The conventiona l
Laminar Flamelet pdf, equation s (11) and (12), with the same
lower moment s / , g and # is plotte d in figur e 6(d). The
LEM time trac e has larg e period s wit h % near zero, and shor t
burst s of strained diffusio n layer s wit h larg e % (see figur e
4(d)). Recallin g that bot h plot s hav e the same mea n scalar
dissipatio n J, the LEM trac e has greate r probabilit y of low£
and concentrate d regions of ver y large %, wit h a dependenc e
stronge r tha n log-normal. The LEM pdf does not sho w
significan t correlatio n betwee n / and %, in agreemen t wit h the
conventiona l assumptio n of / and % independence. The
present approac h is vali d in the connecte d reaction (partial
premixed ) zone where flamelet s are no longe r distinc t laminar
flame s fro m / = 0 to / = 1, and the classical Lamina r Flamele t
mode l break s down. Also, i t ca n be use d t o predic t loca l
extinctio n withou t resortin g to heuristi c Percolatio n theory 4.
By performin g simulation s over a rang e of / and Re, (henc e
n and #), the entire discretize d f-g-n and f-g-"z
space can be tabulated. Mean scalars ( wm, Y k etc.) are stored in
a look-u p tabl e as function s of the parameter s / and g , and n
or "%. The f-g-n tabl e is used in a RANS solver with a
transpor t equatio n for n , wherea s the / - g - r tabl e is used
in a Lamina r Flamele t typ e application. It is emphasize d tha t for
a give n chemistry, the LEM tabl e is constructe d onc e and for all.
The run-time cost in a RANS solver is equal to the conventiona l
assume d shape approac h if the latte r is implemente d as a look -
up table.
In summary, the LEM time trace wit h lower orde r mean s
{/;«;«} or {f;g;"%} i s used to calculat e require d mea n scala r
quantitie s in a RAN S solver wit h transpor t equation s for the
equivalen t mean s {f;g;n} or {f;g\z}.
3. Applicatio n in a RANS solver
The RANS are closed in this work with a conventiona l k-s
turbulenc e model19. The dimensional conservation equations
with the (f,n) reduced chemistry are:
(17)
(18)
(19)
(20)
(21)
c,\ Sc Sc, ac t
wher e ^y=~^~"(
= pRT.
(22)
(23)
(24)
(25)
(26)
(27)
The mode l constant s C^.C^.Qj.a^.Cji.C^.Sc and Sc, are
0.09,1.44,1.92,1.3,2.0,2.0,0.7 and 1.0 respectively.
The system of equation s (17X27) is solved in an axi-symmetri c
fram e usin g the finit e volum e metho d of Jameso n et. al.20.
Invisci d fluxe s are calculate d wit h a third order MUSCL-TVD 21
AUSM 22 scheme, and marche d to a steady-stat e wit h a five-stag e
Runge-Kutt a algorithm 23. At stead y state, the tempora l terms on
the LHS of equations (17X23) approach zero.
Simulation s wer e conducte d using bot h f-g-n and
f-g-% look-up tables with a size of 50x25x15. Tables were
appropriatel y clustere d aroun d / -f^, and g = 0. Bot h tables
contai n the scalar means Y k, f and RT, whil e the f-g-n
table has the additiona l quantit y w m tabulated. Durin g solutio n
of the RAN S equations, tri-linea r interpolatio n is use d to
approximat e the mea n scalars betwee n tabl e node points.
4. Result s and discussio n
The NS code validatio n follow s in section 4.1. A compariso n of
the new theor y wit h experimenta l data is presente d in section
4.2. Section 4.3 present s futur e mode l prospects.
4.1 Code validation
The baseline NS solver and k-e mode l wer e validate d for an
axi-symmetri c isotherma l turbulen t jet (result s not presente d
here). Althoug h the mea n velocit y decay rate was over-predicted,
in accordanc e wit h k-e theory", perfectl y self-simila r and
grid-independen t mea n velocity, turbulenc e intensit y an d
Reynolds stress profiles were attained.
Experimental 2'4 an d predicte d mea n mixtur e fractio n an d
temperatur e radial plot s at axia l location s x/</ = 23, 90 and
135 for a turbulen t H 2 jet flam e ar e show n in figur e 7.
Reasonable agreement is obtained, except for the far-fiel d
temperature. Thi s i s du e to an under-estimat e of the mea n
mixtur e fractio n deca y (figur e 7(c)) and, to some extent, the
neglec t of flam e radiation.
4.2 Application to a non-equilibrium flame
The LEM pdf approac h was applie d to the experimenta l Hl
flame of Drake et. al.25. Measured OH radical concentratio n
profile s ar e compare d agains t mode l prediction s usin g the
f-g-n and f-g-'z approaches in figure s 8 and 9
respectively. The ope n circles are the measure d mea n OH
concentrations, and the plus signs are the hypothetica l measure d
mea n OH equilibriu m concentrations, calculate d fro m the majo r
species. The solid curv e is the LEM predicte d mea n OH
concentration, and the dashe d line is the correspondin g LEM
mean equilibriu m OH concentration.
In bot h cases the near fiel d OH is under-predicted. Since the
equilibriu m concentration s are reasonabl y wel l predicted, this
implie s eithe r an erro r i n bot h th e non-equilibriu m variable s n
and "%, or a chemistr y error du e to the reduce d mechanis m
assumptions.
Simulations using the conventiona l assumed shape /7-pdf
equatio n (8) are indicated by the dotted line in figur e 8. The
conventiona l results are essentially identical to the LEM pdf, so
the superiorit y of th e LE M mode l i s no t conclusivel y
demonstrate d in this example. Analysi s of the LEM and assume d
shape f-g-n tables reveal s that althoug h thei r pdf s are
considerabl y differen t (figure s 6(a) and (b)), the pdf mean s
Yon(f>g>n} and *{/,«."} are similar.
4.3 Future prospects
The LEM pdf mode l can be tractabl y extende d to genera l
chemistr y for premixe d as wel l as non-premixe d steady-stat e
combustion. The joint pdf P(p,T,Y k) can be constructe d as a
functio n of it s lowe r moments, allowin g ignitio n an d extinctio n
modelin g i n a RAN S solver. Differentia l diffusio n ca n be
include d i n th e LE M simulatio n an d it s effec t on P(p,T,Y k)
studied. These issues are unde r investigatio n and will be
reporte d in the near future.
Conclusion s
A Linea r Edd y Mode l applicatio n to steady-stat e turbulen t
combustio n has been formulated. The mode l is based on the
assumptio n that the actua l scalar time trace at a poin t in a
turbulen t flo w can be well approximate d by the spatial scalar
distributio n in a LEM simulation. By matchin g the lowe r orde r
moment s in a RAN S solver wit h the LEM lower order moments,
look-u p tables for require d scala r mean s ma y be constructed.
Althoug h the LEM table is mor e expensive to construc t tha n
conventiona l pdf tables, the LEM tabl e is calculate d onl y once,
and has equal look-u p cost in a RAN S solver. The mode l offer s
potentiall y increase d accurac y at no additiona l computationa l
expense.
The mode l has been applied to a RANS solution of a turbulen t
jet flam e using a two variabl e chemica l mechanism. Conclusiv e
superiorit y ove r conventiona l method s has not yet been
demonstrated, althoug h the metho d allows extensio n to genera l
chemistr y wit h differentia l diffusion.
Reference!
1. Pope, S.B. (1985 ) "PDF Method s for Turbulen t Reactiv e
Flows", Prog. Energy Comb. Sci. 11,119
2. Libby, P.A. and Williams, FA (1980) "Fundamenta l
Aspects", in Turbulent Reacting Haws (Libby, P. A. and
Williams, F.A. ed) Springer-Verla g
3. Bilger, R.W. (1992 ) "Conditiona l Momen t Closur e for
Turbulen t Reacting Flows", Phys. Fluids A S, 436
4. Bilger, R.W. (1989) "Turbulen t Diffusio n Flames", Ann.Rev.
Fluid. Mech. 21,101
5. Lockwood, F.C. and Naguib, A.S. (1975) "The Predictio n of
the Fluctuation s in the Propertie s of Free, Round-jet,
Turbulent, Diffusio n Flames", Comb, an d Flame 24,109
6. Peters, N. (1984) "Laminar Diffusio n Flamelet Models in
Non-Premixe d Turbulen t Combustion", Prog. Energy Comb.
Sci. 10,319
7. Girimaji, S. S. (1991) "Assume d fl -pdf Mode l for
Turbulen t Mixing: Validatio n and Extensio n to Multipl e
Scalar Mixing", Comb. Sci. and Tech. 78,177
8. Janicka, J. and Kollmann, W. (1979) "A Two-Variable s
Formalis m for the Treatment of Chemical Reactions in
Turbulen t H1 -Ai r Diffusio n Flames", Seventeenth Symp.
(Int.) on Comb., The Combustio n Institute, 421
9. Kerstein, A.R. (1988) "Linear Edd y Model of Turbulen t
Scalar Transpor t and Mixing", Comb. Sci. and Tech. 60,391
10. Kerstein, A.R. (1991) "Linea r Edd y Modelin g of Turbulen t
Transport. Par t 6. Microstnictur e of Diffusiv e Scalar Mixin g
Fields", J. Fluid Mech. 231,361
11. Kerstein, A.R. (1989) "Linear Eddy Modeling of Turbulen t
Transpor t n: Applicatio n to Shear Laye r Mixing",
Comb, andFlame,r75y 397
12. Kerstein, A.R. (1990 ) "Linea r Edd y Modelin g of Turbulen t
Transport. Par t 3. Mixing and Differentia l Molecula r
Diffusio n in Roun d Jets", J. Fluid Mech. 216,411
13. Kerstein, A.R. (1992 ) "Linea r Edd y Modelin g of Turbulen t
Transport. Part 4. Diffusion-Flam e Structure", Comb. Sci.
and Tech. 81, 75
14. Calhoon, W.H., Menon, S. and Goldin, G. (1995 )
"Compariso n of Reduce d an d Ful l Chemica l Mechanism s for
Non-Premixe d Turbulen t Hydrogen-Ai r Jet Flames", Comb.
Sci. and Tech. 104,115
15. McMurtry, PA, Oansuage, T.C., Kerstein, A.R. and
Krueger, S.K. (1993) "Linear Edd y Simulation s of Mixing
in a Homogeneou s Turbulen t Flow", Phys. Fluids A 5,1023
16. McMurtry, P.A., Riley, 3.J. and Metcalfe, R.W (1989)
"Effect s of heat release on the large-scal e structur e in
turbulen t mixin g layers",/. Fluid. Mech. 199,297
17. Chen, J.Y. and Kollman, W. (1990) "Chemica l Model s for
Pdf Modelin g of Hydrogen-Ai r Nonpremixe d Turbulen t
Flames", Comb, and Flame 79,75
18. Magre, P. and Dibble, R. (1988 ) "Finit e Chemica l Kineti c
Effect s in a Subsoni c Hydroge n Flame" Comb, and Flame
73,195
19. Launder, B.E. and Spalding, D.B. (1974) "The Numerica l
Computatio n of Turbulen t Flows", Camp. Meth. i n App.
Mech. andEng. 3,269
20. Jameson, A., Schmidt, W. and Turkel, E. (1981) "Numerica l
Solutio n of the Euler Equation s by Finite Volum e Methods
Using Runge-Kutt a Time Steppin g Schemes", AIAA-81-1259
21. Anderson, W.K., Thomas, J.L. and Van Leer, B. (1986)
"Comparison s of Finit e Volum e Flux Splitting s for the Eule r
Equations", AIAA J. 9,1453
22. Liou, MS. and Steflen, C.J. (1993) "A New Flux Splitting
Scheme", J. Camp. Physics 107,23
23. Martinelli, L. and Jameson, A. (1988) "Validatio n of a
Multigri d Metho d for the Reynold s Average d Equations",
AIAA-88-0414
24. Barlow, R.S. and Carter, C.D. (1994 ) "Nitri c Oxid e Forma -
tion in Hydroge n Jet Flames", Comb, and Flame 97,261
25. Drake, M.C., Pitz, R.W., Fenimore, C.P., Lucht, R.P.,
Sweeney, D.W., an d Laurendea u M.M. (1984 )
"Measurement s of Superequilibriu m Hydroxy l
Concentration s in Turbulen t Non-Premixe d Flame s Usin g
Saturated Fluorescence", Twentieth Symp. (Int.) on
Comb., The Combustio n Institute, 327
I I I
0
l
J
Figur e 2:
Schemati c illustratio n of volum e expansion and rc-gridding.
(a) Origina l LEM fiel d wit h elemen t size 6* befor e heat release.
(b) LEM Gel d afte r heat release and volume expansion.
(c) LEM fiel d re-gridde d and mixe d at new resolutio n 5* .
Ir
0 1 23
Non-dimensional time
Figur e 1:
Schemati c illustration of a triplet mappin g of an initial unifor m
scalar gradient. The re-arrangemen t make s three compresse d
images of the origina l segmen t by taking ever y thir d scalar
dement, then replaces the original segmen t by the three images
with the middl e fiel d inverted.
Figur e 3: LEM evolutio n of the tota l produc t mas s fractio n
(a)
1.0
0.8
0.6
0.4
0.2
n n
-
-
-

-
-
-
-
f
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
. 0.6
n
0.4
0.2
0.0
200
150
X 10 0
50
A/v
(b)
(c)
2500
2000
1500
1000
500
(a)
LEM point
Equilibrium
0.0
0.2 0.4 0.6 0.8
1.0
(d)
M
0.0 0.2
0.4 0.6
x/L
0.8 1.0
Figure 4: Instantaneou s 1^^ scalar distributio n
(a) Initial binar y mixtur e fraction, / = /^  0.161
(b) Mixtur e firactio n at a later time, /-/^.g-O-Ol
(c) Nonnalize d progres s variable, f =f M>, g = 0.01, w" * 0.063 3
(d) Scalar dissipation, /=/,* 8 - 0.01, £ = 21.75'1
Ftgure-5:Terapcrature-seatter-plotsinraixtare^aettKi^aec
(a) LEM points
(b) Experimental 11
fV.ii')
0.1
Figur e 6(a): LEM pdf P(/V)
J = /*,«"* 0.01, «*« 0.0633
02S
Figur e 6(b): Conventiona l assume d shap e pdf P(f,n')
- J ~f*.,g = 0.01, w* = 0.0633
40 0
Figure 6(c): LEM pdf
Figur e 6(d): Conventiona l assume d shap e pdf P(f,x)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
(a) x/d=22.5
LEMj-g-H
QExp.
0.12
0.10
0.08
0.06
0.04
0.02
0.00
(c)x/d=13 5
8
r/d
4 8 12 16
r/d
(b) x/d=9 0
12 16
2000
(d) x/d=22.5
2500
2500
8
r/d
(e) x/d=90
12 16
(f ) x/d=135
Figure?
Experimental 14 and predicte d J and f in «turbulen t //2 jet flam e
10
(a) x/d=10
K
O
50
40
•*!
1 3°
-? 20
§
~ 10
0
QU
40
30
10
n
'
o
- 1 -
  LEMf-g-n
- - - LEM equil.
oExp*
+ Exp. equil.
........ 3pdf
-
-
2 4 6
r/d
(b) x/d=50
(c)x/d=150
2 4 6
r/d
8 10 12
8 10 12
Figures
Experimental 75 and predicted (wit h a J-f-n table)
mea n OH concentratio n in a turbulen t tf, jet flam e
50
40
30
(a) x/d=1 0
ffi
io
50
40
"i
1 3°
r 20
" 10
50
40
*t
I 3°
"o
£20
O
~ 10
0
  LEM/-g-x
- - - LEM equil.
oExp.25
+ Exp. equil.
0 2 4 6 8 10 12
r/d
(b) x/d=50
0 2 4 6 8 10 12
r/d
(c) x/d=150
0 2 4 6 8 10 12
r/d
Figur e 9
Experimental 25 and predicte d (wit h a / - g - "x table )
mea n OH concentratio n in a turbulen t #2 jet flam e
11