a mixing model for turbulent reactive flows based on ... · mixing term in particular...

A Mixing Model for Turbulent Reactive Flows based onEuclidean Minimum Spanning Trees

S. SUBRAMANIAM* and S. B. POPESibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853-7501, USA

In modeling turbulent reactive flows based on the transport equation for the joint probability density function(jpdf) of velocity and composition, the change in fluid composition due to convection and reaction is treatedexactly, while molecular mixing has to be modeled. A new mixing model is proposed, which is local incomposition space and which seeks to address problems encountered in flows with simultaneous mixing andreaction. In this model the change in particle composition is determined by particle interactions along the edgesof a Euclidean minimum spanning tree (EMST) constructed in composition space. Results obtained for themodel problem of passive scalars evolving under the influence of a mean scalar gradient in homogeneousturbulence are found to be in reasonable agreement with experimental findings of Sirivat and Warhaft (1983).The model is applied to the diffusion flame test model problem proposed by Norris and Pope (1991) and itsperformance is found to be superior to that of existing models. The essential feature of the new EMST mixingmodel, which accounts for its success in the diffusion flame test, is that mixing is modeled locally in compositionspace. © 1998 by The Combustion Institute

NOMENCLATURE

A accelerationB model coefficientC convex region in composition spaceCf model constantCbg covariance matrixD number of composition variables$ diffusion coefficient in Langevin

equationfUf, f joint velocity–composition

probability density function (pdf)F cumulative distribution function

(cdf) of compositionFZ cdf of particle ageG standardized normal cdfH Heaviside functionI indicator functionk turbulent kinetic energyM, M interaction matrixn decay exponent in grid turbulenceN number of particlesp probability, pressureRl Taylor-scale Reynolds numbers scaled timeS reaction ratet timeT, T transformation matrix

7 inverse drift termu fluctuating velocity field, uj 5 Uj 2

^Uj&U, U velocity8 uniformly distributed random

variableV sample space variable of Uw importance weightwn edge-weight of nth edgeWn cumulative edge-weight of nth edgex Cartesian vector in physical spaceX mappingX particle positionY reaction progress variablez sample space variable of ZZ age property

Greek Symbols

a model parameter controlling thevariance decay rate

G composition diffusivitye mean dissipation rate of turbulenceh sample space variable of GaussianQ mixing model terml matrix eigenvaluen edge indexj mixture fractionP mixing model matrixr scalar–velocity correlation¥ variance function

*Corresponding author. Address: T-3, M.S. B216, LosAlamos National Laboratory, Los Alamos, NM 87545.

COMBUSTION AND FLAME 115:487–514 (1998)© 1998 by The Combustion Institute 0010-2180/98/$19.00Published by Elsevier Science Inc. PII S0010-2180(98)00023-6

t time-scale, turbulence time-scalew fluctuating composition field, wb 5

Fb 2 ^Fb&f, f, F compositionf* fluctuating compositionxbg joint-dissipation tensorc sample space variable of fV particle exchange coefficient^v& mean turbulent frequency

Subscripts

0 initial condition, state 01 state 1MC mapping closure quantityT quantity associated with the treei, j, m, n particle index in ensemble, index of

Cartesian coordinateI corresponding to the IEM modelk index of Cartesian coordinatel lean, lower value of intervalr richs stoichiometricss statistically stationary valueu upper value of intervalw width of intervalb index of composition variablen edge indexf quantity associated with

composition

Superscripts

i particle index in ensembleb index of composition variable* transformed quantity9 fluctuating quantity, standard

deviation thereof

INTRODUCTION

In turbulent reacting flows the fluid compositionfield at any given physical location changes dueto convection, molecular mixing, and reaction.Probability density function (pdf) formulationsare particularly suitable for the modeling ofturbulent combustion problems since the effectsof reaction and convection are in closed form[1]. Furthermore, in turbulent combustion,where the reaction rate exhibits a highly nonlin-ear dependence on temperature, the pdf ap-

proach is capable of predicting important ex-tinction phenomena more accurately thanmoment closures which rely on questionableclosure assumptions for the mean reaction rate.Pdf calculations of piloted jet diffusion flames[2] show good agreement with experimentalmeasurements. In spite of the success of pdfmethods in predicting phenomena such as ex-tinction [2], studies [3] have revealed seriousdeficiencies in the modeling of the molecularmixing term in particular circumstances. Theobjective of this paper is to develop an im-proved mixing model which describes the mix-ing of an arbitrary number of reactive scalars ininhomogeneous flows.

Modeling mixing in pdf methods involvesprescribing the evolution of stochastic particlesin composition space such that they mimic thechange in the composition of a fluid particle dueto mixing in a turbulent flow. In addition to theclosure and realizability problems common toall turbulence modeling, there are issues such aslocalness (described later in this section) whichare specific to modeling mixing. Some of thedifficulties inherent in developing a mixingmodel for multiple reactive scalars are nowdescribed.

The rate of scalar variance decay in highReynolds number turbulence is controlled bythe large scales. In the absence of a model forthe scalar gradients or energy transfer in thescalar spectrum, it is reasonable to assume thatthe scalar variance decay rate is proportional tothe mean turbulent frequency. The evolution ofthe scalar pdf, however, is determined by turbu-lent motions down to the smallest scales. There-fore, the scalar pdf evolution is difficult tomodel without explicit characterization of thesmall scales. Accounting for the effect of thesmall scales is also the natural way to includethe effects of species diffusivity and differentialdiffusion. However, the description of theseprocesses necessitates a higher level of closureand more computational expense.

Despite these observations, there is scope forimproving mixing models without resorting to ahigher level of closure. This is achieved byrequiring mixing models to satisfy modelingprinciples that have a physical or mathematicalbasis. One principal drawback of many mixingmodels in current use (notably the interaction

488 S. SUBRAMANIAM AND S. B. POPE

by exchange with the mean (IEM) mixing model[4]) is that they are not local in compositionspace. In nonpremixed combustion with fastchemistry and initial equilibrium condition, thereaction is confined to reaction sheets and thisdrawback can result in the unphysical mixing ofcold fuel and oxidizer across the reaction sheetas demonstrated by Norris and Pope [3] in adiffusion flame test problem. Such behavior isobserved because the IEM (and other) mixingmodels violate the physics of mixing: namely,that it is the composition field in the neighbor-hood (in physical space) of a fluid particle thatinfluences its mixing. Since the compositionfields encountered in reality are smooth, thisneighborhood in physical space corresponds toa neighborhood in composition space. It followsthat if a mixing model is to perform satisfacto-rily in the diffusion flame test problem, which isrepresentative of the coupling of reaction andmixing in nonpremixed combustion, then themodel should reflect the fact that the change incomposition due to mixing is influenced by theneighborhood in composition space. This moti-vates inclusion of a new principle which mixingmodels should satisfy, namely localness in com-position space.

For any turbulent reactive flow there exists aregion in composition space called the realiz-able region within which any point correspondsto a possible composition value that a fluidparticle may attain. Points in composition spacethat lie outside the realizable region correspondto compositions that cannot occur and have nophysical meaning (for instance, if the composi-tion variables represent species mass fractionsthese may correspond to negative mass fractionsor mass fractions greater than unity). Conse-quently mixing models should preserve theboundedness of compositions. The goal of thismodeling effort is to provide a phenomenolog-ical description of mixing (without explicit rep-resentation of the physical processes associatedwith the small scales), while satisfying the im-portant principles of boundedness and local-ness.

In this paper a new mixing model (referred toas the Euclidean minimum spanning tree[EMST] mixing model) that is local in compo-sition space is described. This particle interac-tion mixing model is an extension of the map-

ping closure particle model to multiple scalars.For the purposes of validating the mixingmodel, an inert flow is first considered. Theproblem of scalar mixing under the influence ofan imposed mean scalar gradient has beenstudied experimentally [5]. A model problembased on this flow is posed and results arepresented. Comparisons are made with the ex-perimental data and direct numerical simula-tions (DNS) data of Overholt and Pope [6]. Themodel performance in reactive flows is testedusing the diffusion flame test model problemposed by Norris and Pope [3]. Results arecompared with those obtained using the IEMmixing model. An earlier version of the EMSTmixing model has also been applied to thepiloted jet diffusion flame [7]. Qualitative re-sults support the hypothesis that the model’slocalness property is responsible for an im-proved description of mixing at various axiallocations in the flow field.

MIXING MODELS

In this section the EMST mixing model isplaced in the context of existing modeling ef-forts by first describing a class of mixing modelsto which it belongs. A list of performancecriteria for mixing models is then presented. It isshown that the class of mixing models underconsideration naturally satisfy some of theseperformance criteria.

In Monte Carlo simulations of inhomoge-neous flows the solution domain in physicalspace is discretized into a number of cells forthe purpose of extracting local mean quantitieswhich appear in the particle evolution equa-tions. Properties of particles drawn from thesame cell are local in physical space. Further-more, to a first approximation, the propertyfields in a cell can be assumed to be statisticallyhomogeneous. Within each cell at any giventime t, the joint pdf of velocity and compositionis represented by an ensemble of N particles. Ifthe number of compositions is D, let wi repre-sent the importance weight (such that the par-ticle weights sum to unity) and fbi (b 5 1, . . . ,D) represent the composition of the ith particle(i 5 1, . . . , N) at time t. Also let X(i) represent

489MIXING MODEL FOR TURBULENT REACTIVE FLOWS

the position and U(i) represent the velocity ofthe ith particle.

The evolution equations of these particleproperties can be written as:

dX~i!

dt5 U~i! (1)

dU~i!

dt5 A~i! (2)

dfbi

dt5 Qb

~i! 1 Sb~@fi#!, (3)

where A(i) represents a model for the particleacceleration, Qb represents the mixing modeland Sb([fi]) represents the reaction rate of thebth scalar, and [fi] represents the vector ofcompositions corresponding to the location ofthe ith particle in composition space. Each par-ticle’s position evolves according to its velocityand the velocity evolves by the model for parti-cle acceleration. Mixing models describe theevolution of particle compositions in composi-tion space which represent (in a stochasticallyequivalent sense) the composition change of afluid particle under the effect of molecularmixing.

The evolution equation for the joint pdf ofvelocity and composition fUf(V, c; x, t) (denot-ed f for brevity) implied by the particle equa-tions is

ft

1 Vj

f xj

5 2

Vk@^ AkuV, c&f#

2

cb

@^QbuV, c&f#

2

cb

@Sb~@c#! f#. (4)

Particle Interaction Mixing Models

Particle interaction mixing models are charac-terized by

Qb~i! 5 2

1w~i!

Mij~b!fbj

. (5)

The matrix Mij(b) represents the interaction be-

tween particles i and j for the bth scalar (inpractice the same matrix describes the interac-

tion for all the scalars but this is not required).Summation is implied over repeated indicesunless they are in parentheses. Thus, in Eq. 5there is summation over the suffix j, but notover i.

The specification of the interaction matrixcoefficients cannot be arbitrary. In fact, severalrestrictions arise from principles that guide themodeling of mixing. It is useful to summarize aprovisional list of these principles, which alsoserve as performance criteria on the basis ofwhich the merit of different mixing models canbe evaluated.

Performance Criteria for Mixing Models

Conservation of Means

The exact evolution equations of the composi-tion fields and their moments imply specificconditions that mixing models (Qb) must satisfy.For particle interaction mixing models this im-poses conditions on the matrix interaction coef-ficients (Mij). In deriving these exact equationsit is convenient to omit the terms due to reac-tion (since reaction is in closed form) thoughthe mixing models themselves are valid for bothinert and reactive flows.

Consider a set of D conserved, inert, passivecomposition fields, Fb(x, t), b 5 1, . . . , D,each with the same diffusivity G, evolving fromarbitrary initial conditions in constant densityturbulence. Assuming Fickian diffusion, theirevolution is described by:

Fb

t1 Uj

Fb

xj5 G

2Fb

xk xk, (6)

where Uj(x, t) is the turbulent velocity field.The evolution of the mean composition fields,

^Fb&, is given by

^Fb&

t1 Ûj&

^Fb&

xj1

ûjwb&

xj5 G

2^Fb&

xk xk,

(7)

where wb 5 Fb 2 ^Fb& and uj 5 Uj 2 Ûj&. Thecorresponding mean equation implied by themodeled pdf equation (Eq. 4) can also bederived. Multiplying Eq. 4 by cb and integratingover the whole (V, c) sample space results inthe mean evolution equation


^fb&

t1 Ûj&

^fb&

xj1

ûjwb&

xj5 ^Qb&. (8)

In high Reynolds number turbulence, the termG¹2^Fb& is negligible compared to the otherterms in Eq. 7 and can be omitted. This givesthe requirement (Eq. 7, cf. Eq. 8)

^Qb& 5 0. (9)

From the above it is clear that the mixing modeldoes not directly affect the mean scalar fields.

The implication of this requirement on thestructure of the particle interaction matrix isnow derived. The ensemble scalar mean is de-fined as

^fb&N ; Oi51

N

wifbi. (10)

The evolution of the particle compositions (Eq.5) assuming the same interaction matrix for allthe compositions is

dfbi

dt5 2

1w~i!

Oj51

N

Mijfbj, (11)

where the summation over other particles isexplicitly written out. From these equations theevolution of the ensemble scalar mean can bewritten as:

d^fb&N

dt5 2 O

i51

N Oj51

N

Mijfbj (12)

5 2 Oj51

N H Oi51

N

MijJfbj

5 0 if Oi51

N

Mij 5 0.

This special structure—namely that each col-umn of Mij sums to zero—automatically guar-antees conservation of the ensemble scalarmean. The requirement ¥i51

N Mij 5 0 is clearlya sufficient condition for conservation of theensemble scalar mean. In fact, unless Mij de-pends on fbj in some contrived way, the deri-vation of Eq. 12 reveals that ¥i51

N Mij 5 0 is alsoa necessary condition for conservation of the

ensemble scalar mean (since Eq. 12 cannot besatisfied by Mij that violates this condition and isalso independent of fbj).

Decay of Variances

The exact evolution of the covariance of thecomposition fields, ^wbwg& is

^wbwg&

t1 Ûj&

^wbwg&

xj1

ûjwbwg&

xj

1 ^wbuj&^Fg&

xj1 ^wguj&

^Fb&

xj

5 2 ^xbg& 1 G2^wbwg&

xk xk, (13)

where

xbg ; 2Gwb

xk

wg

xk,

and ^xbg& is the symmetric positive semidefinite“joint” dissipation tensor. The diagonal ele-ments of this tensor are always non-negativeand they cause the respective scalar variances todecay. In high Reynolds number turbulence theterm G¹2^wbwg& is negligible compared to theother terms in Eq. 13 and can be omitted.

The corresponding scalar covariance equa-tion implied by the modeled pdf equation is

^f9bf9g&

t1 Ûj&

^f9bf9g&

xj1

ûjf9bf9g&

xj

1 ^f9buj&^fg&

xj1 ^f9guj&

^fb&

xj

5 ^f9bQg& 1 ^f9gQb&, (14)

where [u(i) 5 U(i) 2 Û&(X(i))] denotes thevelocity fluctuations and [f*i 5 fi 2 ^f&(X(i))]represents the scalar fluctuations. Details of thisderivation may be found in Ref. [8]. Comparingthe terms in Eq. 14 with those in Eq. 13 it isclear that in the high Reynolds number limit^xbg&M [ 2[^f9bQg& 1 ^f9gQb&] is a model for^xbg&.

From the properties of the “joint” dissipationtensor it follows that ^xbg&M should also be asymmetric positive semidefinite tensor. In otherwords, for any vector xb we require


Ob

Og

xbxg^xbg&M $ 0. (15)

This in turn implies that the ensemble averagedcounterpart of ^xbg&M, which is defined as

^xbg&NM ; 2FKf9b

df9gdt L

N

1 Kf9gdf9bdt L

NG

5 2d^f9bf9g&N

dt, (16)

should also be positive semidefinite. For parti-cle interaction models, ^xbg&N

M can be expressedin terms of the interaction matrix M as [8]:

^xbg&NM 5 O

i51

N Oj51

N

Mij~f9gif9bj 1 f9bif9gj! (17)

5 Oi51

N Oj51

N

~Mij 1 Mji!f9bif9gj.

The second line in Eq. 17 follows simply bycommuting i and j in the first term, i.e.

Oi51

N Oj51

N

Mijf9gif9bj 5 Oj51

N Oi51

N

Mjif9gjf9bi

5 Oi51

N Oj51

N

Mjif9bif9gj.

Thus the condition in Eq. 15 can be written as:

Ob

Og

xbxg^xbg&NM 5 O

b

Og

xbxg Oi51

N Oj51

N

~Mij

1 Mji!f9bif9gj

5 Oi51

N Oj51

N

~Mij 1 Mji! yiyj,

where yi [ ¥b xbf9bi. The requirement for thedecay of all variances is then:

Oi51

N Oj51

N

~Mij 1 Mji! yiyj $ 0,

which is satisfied for all x (and therefore for ally), if and only if the symmetric part of M (Ms [12 [M 1 MT]) is positive semidefinite.

This mathematical condition is automatically

satisfied by pairwise-exchange mixing models,which are defined as having a symmetric inter-action matrix with nonpositive off-diagonal ele-ments:

Mij 5 Mji # 0 ; i Þ j,

and diagonal elements given by

M~i!~i! 5 2 Oj51, jÞi

N

Mij

5 2 Oj51, jÞi

N

Mji.

This definition of the diagonal elements guar-antees the conservation of the means (Eq. 12).In these models the interaction between the ith

and jth particles is given by

w~i!

dfbi

dt5 2V~i!~ j!~fbi 2 fbj!,

and

w~ j!

dfbj

dt5 2V~i!~ j!~fbj 2 fbi!,

where V(i)( j) 5 2Mij $ 0. Clearly this ex-change conserves the mean (cf. Eq. 12) anddecreases (or leaves unchanged) the differenceufbi 2 fbju which contributes to the variance.

It follows from the Gershgorin circle theorem[9] (p. 341) that the eigenvalues l(M) of Mij liein the interval

M~i!~i! 2 Oj51, jÞi

N

uMiju # l~M!

# M~i!~i! 1 Oj51, jÞi

N

uMiju

which is

0 # l~M! # 2M~i!~i!,

and so indeed Mij is positive semidefinite forpairwise-exchange models.

Boundedness

For every physical problem, a surface can bedefined in composition space that is the bound-


ary of the region of realizable compositions. Fornonreactive scalars with equal diffusivities, thecomposition values are restricted to an allowedregion (a subset of the realizable region anddependent on the initial compositions), which isconvex and decreases with time. The initialensemble of particles with scalar propertiesfbi(t 5 0), i 5 1, . . . , N is sampled from aspecified initial jpdf and must lie in this allowedregion. These particle locations define a convexhull in composition space which is denoted byC(0). At later times the convex hull formed byfbi(t) is denoted C(t). It follows from theconservation equation (Eq. 6) that the convexhull can only shrink; i.e. for t2 . t1, C(t2) #C(t1). Boundedness is guaranteed if the evolu-tion of particle properties is such that fbi(t2)lies in C(t1), thus ensuring C(t2) # C(t1), forall t1, t2 satisfying t2 . t1 $ 0.

Any composition fb which can be expressedas a weighted sum of compositions, i.e.

fb 5 uifbi~t!, i 5 1, . . . , n (18)

with ui $ 0, ¥i51n ui 5 1, lies within the convex

hull C(t).Starting at any given time t, the particle

compositions after an infinitesimal time intervaldt evolving according to Eq. 11 can be written as

fbi~t 1 dt! 5 2dtw~i!

Oj51,jÞi

N

Mi~j!fbj~t! 1 F1

2dtw~i!

M~i!~i!Gfbi~t!. (19)

Using the fact that pairwise exchange modelssatisfy Eq. 12, Eq. 19 can be rewritten as

fbi~t 1 dt! 5 2dtw~i!

Oj51,jÞi

N

Mi~j!fbj~t! 1 F1

1dtw~i!

Oj51,jÞi

N

MijGfbi~t!,

where the right hand side is a weighted sum ofthe form in Eq. 18. In the first term, thecoefficient 2dtMi(j)/w(i) is a non-negative infin-itesimal; and in the second term the coefficientof fbi(t) is infinitesimally less than unity.Hence, every composition fbi(t 1 dt) lies

within C(t), and so C(t 1 dt) # C(t) thusestablishing boundedness for pairwise-exchangemodels.

Linearity and Independence

Linearity. The set of governing equations forthe evolution of scalar fields in turbulence (Eq.6) is linear with respect to the scalar fields. Forequal diffusivities of all scalars, it follows thatsubjecting the set of scalars (f 5 {fb}, b 51, . . . , D) under consideration to an arbitrarylinear transformation T (which may be singu-lar), such that

f*g 5 Tgbfb, (20)

results in the same set of governing equationsfor the transformed set of scalars (f*) [10]. Theimplication of this for particle models of multi-scalar mixing is that the evolution equation forparticle properties of the scalars (including allmodel constants) should transform unchangedwhen the scalars are subject to an arbitrarylinear transformation. For moment models theimplication is that the models should only con-tain those statistics that transform linearly un-der linear transformations of the scalar set. Forthe case of unequal diffusivities the linear trans-formation is restricted to diagonal stretchings ofthe scalar fields. One of the implications of thisprinciple is that there is no intrinsic set of scalesin composition space for the set of scalars.Independence. If we consider the mixing of a setof D conserved passive scalars denoted by{fb}, b 5 1, . . . , D, then the evolution of anyone of these scalar fields, say fb(x, t), is unaf-fected by any of the other scalar fields fg (g Þb). This is the independence principle proposedby Pope [10]. The implication of this for particlemodels of multiscalar mixing is that the evolu-tion of the particle property of one scalar fb

should not depend on the particle properties orstatistics of any of the other scalars fg (g Þ b).For moment models of multiscalar mixing, theimplication is that the evolution of any statisticinvolving one scalar fb should not depend onstatistics involving any of the other scalars fg (gÞ b).

It may be noted that contrary to earliernotions, independence is a more fundamentalcondition that must be satisfied for the linearity


principle to hold. To illustrate this point, con-sider the class of mixing models represented by

Qb 5 Pbgfg, (21)

which includes all particle interaction models(Eq. 5). While this is not the most generalrepresentation of a mixing model, it includes amajority of mixing models in current use barringthose that have a diffusion term. Models thathave nonzero off-diagonal entries in the matrixP violate the independence principle. The cor-responding model for the set of scalars obtainedby the transformation given by Eq. 20 is

Q*b 5 Pbgf*g. (22)

Noting that the transformation law requiresthat

df*dt

5 Tdf

dt5 TPf (23)

and also

df*dt

5 Q* 5 PTf,

it is evident that the linearity principle requires

PT 5 TP, (24)

for any T. Any model P with only diagonalentries (satisfying the independence principle)automatically satisfies the linearity conditionexpressed in Eq. 24. It is also easy to see that thecondition is not satisfied by all nondiagonalmatrices. This demonstrates that for mixingmodels expressible in the form of Eq. 21, inde-pendence is a necessary and sufficient conditionfor linearity. It is also noteworthy that for thecase of unequal diffusivities the linearity condi-tion is restricted to diagonal (stretching) trans-formations (T). In this situation there could bemodels of the form of Eq. 21 that violateindependence but satisfy the linearity condition(Eq. 24) and here independence is sufficient butnot necessary to satisfy linearity.

Relaxation to Gaussian

In homogeneous isotropic turbulence, both ex-periments [11] and direct numerical simulations[12] indicate that the scalar pdf relaxes to aGaussian after sufficient time has elapsed for

the solution to become independent of theinitial conditions. It is desirable that a mixingmodel reproduce this behavior, though it is wellknown that this is not a necessary condition forsatisfactory performance of the model in inho-mogeneous flows.

Localness

The concept of localness was motivated by thediffusion flame test which is now described.Consider an idealized diffusion flame in homo-geneous isotropic turbulence with infinitely fastchemistry. The fluid composition is representedby two variables: mixture fraction j and a reac-tion progress variable Y. The initial conditionfor mixture fraction is chosen to be a b-distri-bution with mean (0.1) and variance (0.01)which correspond to a typical experiment. Theprogress variable is initially at equilibrium de-noted by Ye(j). The one-step, irreversible reac-tion is confined to a region R in compositionspace as shown in Fig. 1. The region R iscentered at the stoichiometric mixture fraction(js 5 0.05) and extends from the lean mixturefraction limit of jl 5 0.03 to the rich mixturefraction limit of jr 5 0.07, for Y . 0.6.

As the flow evolves from this homogeneousinitial condition, reactants enter the reactionzone by mixing and are instantly reacted to formproducts resulting in flame sheet combustion

Fig. 1. Initial condition for the diffusion flame test: mixturefraction j is distributed according to a b-pdf with mean 0.1and variance 0.01; progress variable Y is initially at equilib-rium Ye(j); thermochemistry is infinitely fast in the reactionzone R which is centered at the stoichiometric mixturefraction (js 5 0.05) and extends from jl 5 0.03 to jr 50.07.


[3]. The correct behavior expected of mixingmodels in this test is that the compositionsremain along the equilibrium line for all time.For satisfactory behavior in this limit, a mixingmodel must be local in composition space. Forparticle interaction models this means that par-ticles should mix with other particles in theirimmediate neighborhood in composition space.The reason for the failure of mixing models inthe diffusion flame test is that they are not localin composition space, resulting in mixing ofparticles across the reaction zone.

The principle of localness can also be relatedto the structure of the interaction matrix for thecase of a single scalar. Let the ensemble of Nparticles be ordered by nondecreasing scalarvalue such that

f1 # f2 # · · · # fN,

where the b subscript indicating scalar dimen-sion has been dropped since the compositionspace is one-dimensional. Since proximity incomposition space of particle i to particle j (andhence, localness) is directly related to the dif-ference between their indices, it follows that thebandwidth of the interaction matrix determinesthe localness of the particle interaction mixingmodel. The relation between localness and sim-ple ordering holds only in a one-dimensionalcomposition space.

Dependence on Re

The effect of flow Reynolds number (for suffi-ciently large Reynolds number) on scalar mixingis to change the scalar spectra in the range ofwavenumbers greater than those correspondingto the inertial subrange. While including a de-pendence on Reynolds number in the mixingmodel allows for a better description of mixing,this does not seem appropriate at the currentlevel of closure.

Differential Diffusion

When modeling practical combustion problemswith different chemical species which may havedifferent diffusivities (e.g., H2 and N2 in hydro-carbon–air flames), it is desirable that the mix-ing model take these effects into account. This

requires modeling the effect of the small scalesof turbulence.

Dependence of Length Scales of ScalarFields

Experiments of Warhaft and Lumley [13] showthat the variance decay rate of a single con-served passive scalar in grid turbulence showsdependence on the initial length scales of thescalar field. The dependence of scalar mixing onthe initial lengthscales of the scalar field hasalso been studied using direct numerical simu-lations of turbulent mixing of single [12] andmultiple scalars [14]. The important conclusionfrom these sources is that the smaller the initiallengthscale of the scalar field the faster themixing (and hence the scalar variance decay).This effect can be captured only if the mixingmodel contains some representation of thelength scales associated with the scalar field.

Flamelet Combustion

In flamelet combustion the scalar gradients (ofthe reactive scalars) are enhanced by reaction.As a consequence the assumption that the diag-onal elements of the “joint” dissipation tensorare determined solely by the large scales of theturbulence is no longer valid. Extension ofcurrent mixing models to the flamelet limitrequires incorporating the effect of steepeningof scalar gradients due to reaction.

Examples of Pairwise Exchange Models

It is convenient to view the locations of thenonzero entries in the interaction matrix asrepresenting “edges” connecting particles incomposition space, along which particles inter-act in a model of the mixing process. The valuesof these nonzero entries can be thought of ascoefficients associated with these edges, whichcontrol the rate at which the two particles ateither end of the edge are attracted to eachother (note that for a symmetric interactionmatrix the off-diagonal entries must all be neg-ative). Different choices of edges, and differentvalues of their coefficients result in differentmixing models.

Models like Curl’s model [15] and its variants,IEM [4], and the mapping closure particle


model for a single scalar are all pairwise ex-change models (or can be expressed as such). InCurl’s model mixing is achieved by subjectingthe ensemble to the following stochastic processover a sequence of small time steps Dt(Dt^v&,, 1, where ^v& 5 e/k is the mean turbulentfrequency in the cell, k and e being the turbulentkinetic energy and its dissipation rate respec-tively). Particle pairs are selected at randomfrom the ensemble, with Np pairs selected (Np isthe minimum number of particle pairs such thatthe sum of the weights of selected particlesexceeds Dt^v&). If the two members in the pairare denoted n and m, mixing is performed bychanging the compositions to

fbn~t 1 Dt! 5 f*b, (25)

fbm~t 1 Dt! 5 f*b,

where,

f*b 5w~n!fbn 1 w~m!fbm

w~n! 1 w~m!

. (26)

This mixing process is performed for each of theNp pairs selected to produce the ensemble attime t 1 Dt. For particles not selected, theircompositions are unaltered. While in Curl’smodel particle composition values change dis-continuously in time (whereas Eq. 11 representsa continuous change of composition), Curl’smodel can still be represented by a discrete timeversion of Eq. 11, namely

Dfbi 5 21

w~i!Oj51

N

MijfbjDt,

with interaction matrix M such that the mth rowcorresponding to that member in the particlepair has entries:

M~m!~m! 5w~m!w~n!

w~n! 1 w~m!

Mmn 5 2w~m!w~n!

w~n! 1 w~m!

Mmk 5 0, k Þ m, k Þ n.

The apparent discrepancy in the dimension ofM arises from the fact that the fraction ofparticle pairs selected results in an inverse timeterm in the pdf evolution equation [16]. Curl’smodel satisfies the conservation of means and

decay of variance criteria by virtue of being apairwise-exchange model. It also satisfiesboundedness and independence principles.However, it does not satisfy the localness prin-ciple since the particle pair selected for mixingmay be drawn from anywhere in compositionspace. As noted previously [16], since the com-position changes discontinuously in Curl’smodel, it is incapable of generating a differen-tiable scalar cumulative distribution function(cdf) from an initially nondifferentiable cdf.

For the IEM model [4], the ith particle’scompositions evolve as

dfbi

dt5 2

12

Cf^v&~fbi 2 ^fb&!, (27)

where Cf is a model constant chosen to be 2.0to yield the desired scalar variance decay rate.Since ^fb& is usually estimated by simple aver-aging of the particle compositions within a cell,the IEM model can also be cast in the interac-tion matrix form with M now given by,

M~i!~i! 512 Cf^v&~1 2 w~i!!w~i!

Mij 5 2 12 Cf^v&w~ j!w~i!, i Þ j.

The IEM model also satisfies the conservationof means and variance decay criteria by virtue ofbeing a pairwise-exchange model. It also satis-fies the boundedness and independence princi-ples. However, it does not satisfy the localnessprinciple since the particle composition mixestoward the mean composition value: Mij isnonzero for all i and j.

Even the computational stencil for directnumerical simulations can be cast in the particleinteraction form. Mixing models such as thebinomial Langevin mixing model [17] and theFokker-Planck closure model [18] are examplesof models that are not members of this class.The mixing model to be described in this paperalso belongs to the class of particle interactionmixing models.

Implications for the EMST Model

It is a formidable task to construct a mixingmodel which performs well with respect to all ofthe above performance criteria. Moreover someof these criteria (scalar length scale depen-


dence, flamelet combustion) cannot be directlyaddressed at the current level of closure (name-ly, one-point, one-time Eulerian pdf of scalaramplitude). However, attempting a higher levelof closure raises several issues such as: (i)consistent extension of the velocity model tothis level of closure, (ii) difficulties in modelingdue to lack of information (either from experi-ments or DNS) about the unclosed terms at thenew level of closure, (iii) computational expenseand numerical issues. In this paper, an attemptis made to satisfy selectively those criteria whichwe believe are most important in reactive flowapplications; namely localness and bounded-ness, at the one-point, one-time scalar ampli-tude level of closure, in a computationally trac-table model.

Since the EMST mixing model is an extensionof the form of the mapping closure particleequations to multiple scalars using the Euclid-ean minimum spanning tree in compositionspace, it is advantageous to first consider the1-D mapping closure model.

MAPPING CLOSURES

Mapping closures for turbulent mixing wereproposed by Chen et al. [19] and a particleimplementation of this model for a single scalarwas described by Pope [20]. The salient aspectsof this model which are pertinent to the EMSTmixing model are briefly reviewed here.

In the mapping closure formalism, a statisti-cally homogeneous, isotropic, time-independentGaussian random field (with standardized nor-mal cumulative distribution function (cdf)G(h)) is mapped through a mapping X(h, t) toa statistically homogeneous scalar field (with cdfF(c; t)) evolving in a turbulent flow, such that

F~X~h, t!; t! 5 G~h!. (28)

While the evolution equation of the scalar cdf(or pdf) contains unclosed terms, these can beexpressed in terms of the known statistics of theGaussian reference field and properties of themapping. This model has been applied to thecase of a single inert, passive, statistically homo-geneous scalar field evolving in isotropic turbu-lence and the analytic model solution [20] forthis problem shows remarkably close agreementwith the direct numerical simulations of Eswa-ran and Pope [12]. The mapping closure for a

single scalar yields a mixing model that is localin composition space (as is shown later in thissection) and also satisfies the performance cri-teria of mean conservation, variance decay,boundedness, and relaxation to a Gaussian.However, there are difficulties in extendingmapping closures to multiple reactive scalarssince the mappings are nonunique and expen-sive to compute. Hence, an alternative approachto generalizing the mapping closure mixingmodel to multiple scalars is considered via itsparticle implementation with the aim of retain-ing as many of its desirable properties listedabove.

In a particle method solution of the pdftransport equation, the pdf f(c; t) at time t of ahomogeneous scalar field f(x, t) (c being thesample space variable of f) is indirectly repre-sented by an ensemble of N particles withcomposition fi and weight w(i), i 5 1, . . . , N.Since the condition for uniqueness of the map-ping requires that it be a nondecreasing func-tion of its argument, this requires that theparticles are first sorted in nondecreasing orderof composition value such that

f1 # f2 # . . . # fN.

The discrete representation of the mappingequation (Eq. 28) is expressed by the correspon-dence between the points hi11/ 2 (in the samplespace of the Gaussian reference field) andci11/ 2 (in the sample space of the scalar)through the cdf’s F and G:

F~ci11/ 2; t! 5 F~X~hi11/ 2, t!; t!

5 G~hi11/ 2!

5 Oj51

i

w~ j!, (29)

see Fig. 2. From the mapping evolution equa-tion [20], the evolution of each particle’s com-position can be derived with the only approxi-mation being a finite-difference approximationfor the mapping gradient in sample space. Theevolution equation for the particle compositionis

w~i!

df~i!

dt5

1tMC

@2Bi11/ 2~fi 2 fi11!

2 Bi21/ 2~fi 2 fi21!#, (30)


where Bi61/ 2 are model coefficients given by

Bi11/ 2 ;g~hi11/ 2!

@hi11 2 hi#, i 5 1, . . . , N 2 1.

(31)

In Eq. 31 g(h) represents the standardizednormal density function and hi 5 (hi21/ 2 1hi11/ 2)/ 2. The factor tMC in Eq. 30 is thecharacteristic time associated with the mappingclosure model and is a function of the scalardiffusivity, and properties of the mapping andreference field [20].

The mapping closure particle model is also apairwise-exchange model and hence satisfiesmean conservation and variance decay. It alsosatisfies the boundedness principle. Each parti-cle evolves by interaction with its neighborparticles (in composition space) and the rate ofattraction is given by Bi61/ 2 (with B1/ 2 5 0,BN11/ 2 5 0). Clearly, this model also satisfiesthe localness principle.

The mapping closure model can be expressedas a particle exchange model with interactionmatrix coefficients:

M~i!~i! 5 ~Bi11/ 2 1 Bi21/ 2!/tMC

M~i!~i11! 5 2Bi11/ 2/tMC

M~i!~i21! 5 2Bi21/ 2/tMC, i 5 1, . . . N.

Extension of this model to higher scalar di-mensions requires (a) a definition of localness

that is valid in any dimension, and (b) general-ization of the form of the particle compositionevolution equation (Eq. 30). Definitions of lo-calness will be discussed in subsequent sections,but the issue of generalizing Eq. 30 is consid-ered here.

For the case of a single scalar each particleevolves by interaction with its two adjacentneighbors (except for the extremum scalar val-ues which have only one neighbor each). Fortwo or more scalars there is no reason to restrictthe number of neighbors to two and the evolu-tion equation needs to be modified to reflectthis. Further the specification of the interactioncoefficients needs to be generalized for anynumber of scalar dimensions.

With this generalization in mind it is conve-nient to associate the interaction coefficientsBi61/ 2 with the particle pairs whose evolutionthey affect. Denoting the unordered particlepair (mn, nn) to be the nth edge, the coefficientassociated with this edge is denoted Bn. Nowthe evolution equation for the particle compo-sition can be rewritten as

w~i!

df~i!

dt5 2tMC

21 On

Bn$~f~i! 2 fnn!dimn

1 ~f~i! 2 fmn!dinn

%, (32)

where the nth edge connects the particle pair(mn, nn) and d represents the Kronecker delta.Summing over the list of edges, the term inbraces is nonzero when the particles mn and nn

are connected by the nth edge. Since the edge isan unordered particle pair, either combinationneeds to be accounted for by the two terms withKronecker d’s. This model is also a pairwiseexchange model since the same coefficient Bn

appears in the evolution equation of mn as wellas in nn.

Of course for the 1-D mapping closure eachparticle with index 1 , i , N, has exactly twoedges incident on it. The extremum particles i 51 and i 5 N have only one edge incident onthem. Given the list of particle indices orderedby composition value, an edge list is triviallygenerated by defining the nth edge as connectingparticles n and n 1 1, for 1 # n # N 2 1. SeeFig. 2.

For the single scalar case there exists a one-

Fig. 2. Cumulative distribution function Ff(c) (discreterepresentation) of a single scalar f and mapping of ci11/ 2

to hi11/ 2 through the Gaussian cdf G(h). Also shown arethe 1-D edge list and the particle and cumulative edgeweights.


to-one relationship between an edge in the edgelist defined by this procedure and the location ofthe corresponding particle pair connected bythe edge in composition space. This relationshipis established by associating a cumulative edge-weight with each edge. For the nth edge this canbe defined as

Wn 5 Oi51

n

w~i!. (33)

The cumulative edge-weight is a minimum atthe minimum scalar value of the distributionand increases monotonically until we approachthe maximum scalar value where it takes themaximum value of 1. As an example (assumingeven N and equal particle weights w(i) 5 1/N),the first edge (with cumulative edge-weight1/N), connects f1 to f2, and the edge withindex N/ 2 (cumulative edge weight 1/2) con-nects particles with index N/ 2 and N/ 2 1 1which lie near the median of the distribution.

For the 1-D mapping closure the coefficientsBn are solely a function of the cumulativeedge-weight. In the limit of N3 ` this functionhas an analytic form (expressible in terms of thestandardized Gaussian density) and is symmet-ric with respect to the cumulative edge-weight(Wn) value of 1/2 [20].

Exploiting this symmetry, a quantity calledthe edge-weight wn can be associated with thenth edge such that

wn 5 min ~Wn, W# n! (34)

where W# n 5 Oi5n11

N

w~i!.

The edge-weight defined in this way is a mini-mum at the extremal values of the distributionand increases as we approach the median whereit takes the maximum value of 1/2. Note thatedges corresponding to different particle pairscan have the same edge-weight. For the singlescalar case exactly two such edges will have thesame edge-weight. In higher dimensions morethan two edges can have the same edge-weight.

Since the coefficients Bn are symmetric withrespect to the cumulative edge-weight (Wn),they can be expressed as a function of only theedge-weight wn. See Fig. 3. This functional

dependence is generalized in the EMST modelwhich is described in the following section.

EMST MIXING MODEL

In modeling the mixing of a single scalar, wherethe composition space is one-dimensional, asimple ordering of the particle scalar valuesprovides the definition of localness. For this casethe mapping closure particle model provides anadequate description of the evolution of a par-ticle’s composition by interaction with its neigh-bor particles in composition space. In modelingthe mixing of multiple scalars, where the com-position space is of a dimension higher thanone, a particle interaction mixing model that islocal requires a definition of neighboring parti-cles. The Euclidean minimum spanning tree(described in the following subsection) con-structed on the ensemble of particles in compo-sition space of arbitrary dimension provides onesuch definition of neighbor particles. A parti-cle’s composition then evolves as per the gener-alized version of the mapping closure evolutionequation (Eq. 32).

However, results obtained using this modelfor the problem of two statistically identicalinitially joint-normal scalars evolving under theinfluence of imposed linear mean scalar gradi-ents in isotropic turbulence (described in thenext section) show that the model producesunphysical scalar joint pdf’s which are no longerjoint-normal. In this problem the fluctuatingscalar fields are homogeneous. In the simulation

Fig. 3. Coefficient Bn/N vs. edge-weight wn for the 1-Dmapping closure.


particle properties evolve from the specifiedinitial conditions but since no new particlesenter (or leave) the domain, the ensemble ofparticles remains unchanged. As a consequencethe EMST in the 2-D composition space isformed on the same ensemble of particles. Thisresults in repeated attraction of any given par-ticle to the same restricted set of neighborparticles (which is not consistent with the phys-ical process of mixing where a fluid particle’scomposition changes as a result of interactionwith the entire composition field in its neighbor-hood), and hence the ensemble of particlescollapses along “strands” in composition space(see Fig. 4).

In order to alleviate this “stranding” problem,an intermittency feature is introduced into theEMST mixing model. At any given time aparticle is in one of two states: a mixing statewhere its composition changes due to mixing, ora nonmixing state where its composition isunchanged due to mixing. A nondimensionalage property is associated with each particlewhich determines how long a particle mixes andhow long it is in the nonmixing state. The timea particle spends in both the mixing and non-mixing states is bounded. With the addition ofthis feature the EMST is now formed based onlyon those particles that are in the mixing state atthat instant of time. Results obtained for the

EMST mixing model using this feature aresatisfactory and are described in the next sec-tion.

The following subsections are devoted todescribing the EMST mixing model with theintermittency feature. This is followed by adescription of model parameters and coeffi-cients. The section concludes with a discussionof the various model features and its properties.

EMST Model Implementation

Consider an inhomogeneous flow where thesolution domain in physical space is discretizedinto a number of cells. To a first approximationthe property fields in a cell can be assumed to bestatistically homogeneous and the joint pdf ofcompositions is represented by an ensemble ofN particles fbi, b 5 1, . . . , D, i 5 1, . . . , N,where D is the dimension of the compositionspace. In some of the illustrations that follow,we consider two scalars (D 5 2). An ensembleof particles drawn from the specified initialcomposition jpdf (in this case chosen to be anuncorrelated joint-normal) can be representedas a scatter plot in composition space as shownby the circles (open and filled) in Fig. 5.

At any given time only a subset (referred to asthe mixing subset) of the total ensemble of Nparticles participates in the mixing process.Whether or not a particle belongs to the mixingsubset is determined by a nondimensional par-ticle age property, Z(i), associated with eachparticle. The initialization and evolution of thisage property is now described.

Intermittency

If the particle age property is positive Z(i) . 0(referred to as state 1), the particle is in themixing subset, otherwise if Z(i) # 0 (referred toas state 0), then the particle is not in the mixingsubset. The time a particle spends in the mixingand nonmixing states is required to scale withthe timescale of turbulence in the cell. If k is theturbulent kinetic energy and e is the meandissipation rate, then ^v& 5 e/k represents themean turbulent frequency (the reciprocal of theturbulence timescale). A scaled time can now bedefined in terms of the mean turbulent fre-quency (^v&) and the physical time t by

Fig. 4. An example of “stranding”: scatter plot of particlesin 2-D composition space with EMST superimposed at t 50.5tf in the mean scalar gradient test using EMST mixingwithout intermittency; initial distribution was joint-normal.


s ; s0 1 E0

t

^v&~t! dt,

where s0 is the origin of the scaled time whichcan be fixed arbitrarily.

Each particle’s age property Z(i) is chosen tobe a realization of a stochastic process Z(s)(called the age process), which evolves in s, thescaled time. Figure 6 shows a sketch of the ageprocess Z(s). When the particle age is positiveit decreases linearly until it reaches zero whereit discontinuously jumps to a negative value. Atany instant a positive particle age represents the

excess life (in scaled time) the particle has toparticipate in the mixing process. A negativeparticle age represents the waiting time beforethe particle mixes again.

The evolution of the age process can bewritten as follows. For infinitesimal positive ds,

Z~s 1 ds! 5 Z~s! 2 ds, for Z~s! . ds (35)

Z~s 1 ds! 5 Z~s! 1 ds, for Z~s! , 2ds

Z~s 1 ds! 5D

81~ z!, 0 # Z~s! # ds

Z~s 1 ds! 5D

80~ z!, 2ds # Z~s! # 0

where 81( z) and 80( z) are uniform randomvariables in the intervals [Z1l

, Z1u], and [Z0l

,Z0u

] respectively.For stationary homogeneous turbulence the

mean turbulent frequency is constant and thescaled time is linearly related to the physicaltime. For this case we require the age process toadmit a stationary solution. It can be shownfrom the theory of renewal processes [21] thatthis is achieved if the initial age distribution,G( z), is specified as follows:

G~ z! 5 ~1 2 p0! 1 p0H 2zZ1l

1 Z1u

J , 0 , z # Z1l

5 ~1 2 p0! 1 p0H 2Z1l

Z1l1 Z1u

1@2Z1u

~ z 2 Z1l! 2 ~ z2 2 Z1l

2 !#

Z1u

2 2 Z1l

2 J , Z1l, z # Z1u

5 ~1 2 p0!H1 22uzu

Z0l1 Z0u

J , 2Z0l, z # 0

5 ~1 2 p0!H1 22Z0l

Z0l1 Z0u

2@2Z0u

~uzu 2 Z0l! 2 ~ z2 2 Z0l

2 !#

Z0u

2 2 Z0l

2 J , 2Z0u, z # 2Z0l

,

Fig. 5. Euclidean minimum spanning tree constructed onthe mixing subset of an ensemble of particles in 2-Dcomposition space (open circles represent particles in thenonmixing state with Z , 0); N 5 512 particles, joint-normal distribution.

Fig. 6. The age process Z(s) versus scaled time s.


where p0 is the fraction of the total ensemble ofparticles which is initially assigned to the mixingstate (Z(0) . 0). In other words

P@Z~0! . 0# 5 1 2 G~0! 5 p0. (36)

If I(s) is the indicator function for the processZ(s) being positive (i.e., the particle is in amixing state), then it can be shown that forstationary homogeneous turbulence

lims3`

1s E

0

s

I~u! du 5^Z1&

^Z0& 1 ^Z1&, (37)

where the absolute value of the positive andnegative jumps in Z(s) each time the processhits zero are represented by the random vari-ables Z1 and Z0 respectively. In other words Eq.37 states that time a particle spends on averagein the mixing state is given by the ratio of theaverage positive jump to the sum of the averagepositive and negative jumps taken by the ageprocess Z(s). The specified initial distributionwill be a stationary solution to the age processevolution equation only if p0 is chosen such that

p0 5^Z1&

^Z0& 1 ^Z1&. (38)

For the general inhomogeneous problem ^v&varies from cell to cell and could also be afunction of time. However, since the age processis defined in terms of nondimensional quantitiesits initialization and evolution remain un-changed.

If there are N particles in a cell at any giventime t, the NT particles that make up the subsetof this ensemble that participate in the mixingprocess is determined based on the age-prop-erty of each particle:

NT~t! 5 Oi51

N

H~Z~i!~s!!, (39)

where H( z) is the Heaviside function.

Definition of the EMST

Loosely speaking, constructing the EMST onthe mixing subset of NT particles requires con-necting every particle with at least one neighborparticle through an edge (an unordered pair of

particles). Since every particle is connected inthe EMST it possesses the spanning property. Atree is a set of edges that contains no closedloops or cycles; i.e., starting from a particle,tracing an alternating sequence of edges andparticles terminating in a particle, does not leadback to the initial particle. Finally since it is aEuclidean minimum spanning tree, the sum ofthe Euclidean lengths of these edges is a mini-mum. A formal graph-theoretic definition of aminimum spanning tree (MST) and algorithmsfor constructing an MST can be found in theliterature [22], [23].

The EMST in composition space constructedon the mixing subset of the ensemble of parti-cles representing two scalars joint-normally dis-tributed is shown in Fig. 5. It can be shown thata tree constructed on NT particles has NT 2 1edges [24]. The EMST on NT particles is thusdefined by the set of NT particles and the NT 21 edges that connect them.

The temporal behavior of the EMST con-structed on the mixing subset of particles in aparticular computational cell in an inhomoge-neous flow can be qualitatively described asfollows: Particles enter and leave the cell as theymove through physical space according to theirparticle velocity. Depending on their age prop-erty they are divided into mixing and nonmixingsubensembles and the EMST is formed on themixing subset. Consequently, in the generalinhomogeneous case, the EMST changes dis-continuously in time due to particles that are inthe mixing state entering/leaving the cell, anddue to particles in the cell entering/leaving themixing state.

Particle Composition Evolution

Each edge in the EMST is assigned an edge-weight which is defined analogously to the edge-weight definition in the mapping closure parti-cle implementation. Let n denote the edge thatconnects particles mn and nn. If this edge wereremoved from the set of EMST edges, then theremaining set of edges define two subtrees oneof which (denoted Tmn

) contains particle mn,and the other (denoted Tnn

), particle nn. Eachof these subtrees consists of a set of edges andthe particles that are connected by these edges.Each of these subtrees Tmn

and Tnnis assigned a


weight ~WTmn

and WTnn

respectively) which isdefined as the sum of the weights of the parti-cles that belong to that subtree. Mathematically,

WTmn

5 Ok[Tmn

w~k!

WTnn

5 Ok[Tnn

w~k!.

The edge-weight of edge n is now defined as

wn 5 min ~WTmn

, WTnn

!. (40)

Since the particle weights sum to unity, theedge-weight ranges from the minimum particleweight to 1/2.

An edge coefficient Bn is associated with thenth edge analogous to the mapping closureparticle model. The edge-coefficient is chosento be a linear function of the edge-weight:

Bn 5 2wn.

The choice of this edge-coefficient is discussedlater in this section. Since the edge-weight ismaximized by 1/2, the edge-coefficient isbounded by unity.

As in the mapping closure particle model, aparticle’s composition evolves by interactionwith its neighbors in composition space. Theneighbors of a particle are defined by the edgesin the EMST that are incident on the particle.Using these definitions the evolution equationfor the vector of particle compositions f(i) 5fb(i), b 5 1, . . . D, i 5 1, . . . NT can bewritten as (cf. Eq. 32)

w~i!

df~i!

dt5 2a O

n51

NT21

Bn$~f~i! 2 fnn!dimn

1 ~f~i! 2 fmn!dinn

%, (41)

where the nth edge connects the particle pair(mn, nn) and d represents the Kronecker delta.This model is also a pairwise exchange modelsince the same coefficient Bn appears in theevolution equation of mn as well as nn, and thiscoefficient is a function solely of the edge-weight (which by definition is the same foreither particle belonging to the edge). Theparameter a is determined by requiring that thescalar variances decay at a prescribed rate and itis described later in this section.

The matrix form of EMST model evolution(Eq. 41) is

dfbi

dt5 2

1w~i!

Mijfbj, i 5 1, . . . , NT, (42)

with the interaction matrix elements given by

Mij 5 2a On51

NT21

Bn$dimndjnn

1 djmndinn

%, j Þ i

M~i!~i! 5 a Oj51

NT On51

NT21

Bn$dimndjnn

1 djmndinn

%.

It is convenient at this point to also define

M# ij ;1a

Mij.

Model Parameters

In this subsection the choice of model parame-ters and coefficients is described and modelsensitivity to variation of these parameters isdiscussed qualitatively.

Model Coefficients

The choice of B coefficients is crucial in deter-mining the evolution of the composition jpdf.These edge coefficients are assumed to be afunction of only the edge-weight in the spirit ofthe mapping closure model. Their choice isdetermined by the criterion that in the meanscalar gradient test problem, an initially joint-normal composition jpdf (dimension of compo-sition space D 5 2) remains joint-normal. Afternumerical experimentation with several differ-ent functions of edge-weight, a linear functionBn 5 2wn was chosen to be the model specifi-cation. This specification resulted in an evolu-tion to composition jpdf’s that are close to (butnot exactly) joint-normal for the mean scalargradient test case with two scalars. Comparableresults were also obtained for five and 10 scalars.

The following general observations can bemade about the B coefficients. Since particlesthat lie at the extrema of the distribution aremore likely to have only one edge incident onthem (such particles are called leaves), theycorrespond to the lowest edge-weight. (Notethat the converse is not true). Hence, a relativeincrease in the B coefficient values at lower


edge-weights will cause the tails of the compo-sition jpdf to be drawn in faster.

For a given distribution of scalars, the distri-bution of edge-weight changes as the dimen-sionality of the scalar space (D) is increased. Inan effort to keep the model as simple as possi-ble, there is no dependence of the B coefficientson D. As the dimensionality of the compositionspace increases, the probability of finding leavesincreases and it is reasonable that the B coeffi-cient values at the lowest and highest edge-weight predominantly govern the model evolu-tion. Consequently, as the number ofcompositions increases the model is expected toshow decreased sensitivity to the choice of B inthe intermediate range of edge-weights.

Determination of a

The model parameter a controls the rate ofvariance decay of the scalars, and its specifica-tion is described in this subsection.

Consider a single conserved passive scalarfield with mean ^f& and variance ^f92& evolvingin constant-density homogeneous turbulence.The mean is unaffected by the mixing, hence

d^f&

dt5 0.

The scalar variance evolution can be written as

d^f92&

dt5 2

^f92&

tf

,

such that the scalar variance decays exponen-tially with a scalar timescale tf. Following stan-dard modeling assumptions [1], tf can be re-lated to the timescale of the turbulence t([k/e) by

tf 5 t /Cf,

where the empirical constant Cf is taken to be2.0. The parameter a is chosen such that theensemble variance evolution implied by theEMST model evolution equation obeys the samedecay law. This is accomplished by first expressingthe variance decay rate implied by the EMSTmodel equation in terms of a and a function of theensemble composition values. This EMST modelvariance decay rate is then equated to the desiredvariance decay rate and the value of a that satis-fies this equality is solved for.

The ensemble scalar variance evolution isgiven by

d^f92&N

dt5 O

i51

NT

w~i!2f9~i!df9~i!

dt(43)

5 2a Oi51

NT

w~i!2f9~i!M# ijf9j.

Equating this to the desired decay rate andsolving for a yields

a 5^f92&N

tf Oi51NT w~i!2f9~i!M# ijf9j

. (44)

For multiple scalars the choice of a is not asstraightforward. In general this parameter is tobe chosen such that it yields a prescribed behav-ior for some function of the covariance matrixwhich is called the variance function. Since eachscalar does not evolve independently in theEMST model, requiring that one of the scalarsobey a specific decay law imposes a decay law onthe other scalars (which may not be consistentwith standard modeling). One way to circum-vent this problem is to require that a be basedon the trace of the covariance matrix. Thevariance function is defined to be the trace ofthe covariance matrix

O 5 Cbb.

If each scalar decays with the same scalar time-scale tf, then the variance function obeys theevolution equation

d¥dt

5 2¥

tf

5 2Cf¥

t. (45)

The EMST model evolution in matrix form formultiple scalars is

w~i!

dfb~i!

dt5 2a O

j51

NT

M# ijfbj, i 5 1, . . . NT, (46)

Similar to the single scalar case, the variancefunction evolution can be written as

d¥dt

5 Oi51

NT

w~i!2f9b~i!

df9b~i!

dt

5 2a Oi51

NT

w~i!2f9b~i!M# ijf9bj.


Equating this to the desired decay rate andsolving for a yields

a 5¥

tf ¥i51NT w~i!2f9b~i!M# ijf9bj

. (47)

Though the expression for a in continuoustime is written explicitly in terms of knownquantities, in practice the value of a has to bedetermined at each time step in the MonteCarlo simulation such that the variance functionsatisfies the prescribed decay law. Since thevariance function is a nonlinear function oftime, a root finding technique is needed tocompute the parameter a. The root finder typ-ically converges in 2 or 3 iterations.

Age Process Parameters

In the age process specification there are fivemodel parameters, namely p0, Z1l

, Z1u, Z0l

, andZ0u

. In view of Eq. 38, only four of theseparameters are independent. In the model thesefour parameters are specified indirectly by spec-ifying p0, ^Z0&, Zw0

, and Zw1, where

Z0l5 ^Z0& 2 Zw0

Z0u5 ^Z0& 1 Zw0

Z1l5 ^Z1& 2 Zw1

Z1u5 ^Z1& 1 Zw1

.

The specification of each of these parameters isnow considered. As p0 approaches unity theeffect of intermittency decreases until there isno intermittency at p0 5 1. As p0 approacheszero the effect of intermittency increases, but p0cannot be decreased arbitrarily in view of Eq.47. As the number of particles that participatein the mixing process at initial time NT(0) (andat all times for the homogeneous test caseswhere the age process is stationary) decreases,their compositions need to decay at faster ratesin order to yield the same decay rate for thevariance function based on the total ensembleof particles. In practice the maximum decay rateof the composition of mixing particles is limitedby a threshold value which is a multiple (chosento be 2.5) of the mixing frequency ^vf&, where^vf& 5 1/tf 5 Cf^v&. If the variance functiondecay timescale is specified by tf 5 t /Cf, thengiven a maximum decay rate threshold, there

exists a minimum p0 above which a solution toEq. 47 exists. In the limit of N 3 ` and for thevalues considered (Cf 5 2.0 and maximumdecay rate 5 2.5 3 ^vf&) this gives p0min

51/2.5 5 0.4. The fraction of the total ensembleof particles which is initially assigned to themixing state p0 is chosen to be 1/2.

The parameter ^Z0& determines the averagetime that a particle spends in the nonmixingstate after it changes from a mixing state. This ischosen to be 1/6 based on numerical experi-ments performed on the mean scalar gradienttest problem with the objective of producingscalar jpdf’s that remain joint-normal whenstarted from a joint-normal initial condition.This together with p0 5 1/ 2 determines ^Z1& tobe 1/6. The width of the Z0 interval Zw0

ischosen to be zero. This forces the particles todeterministically spend the time correspondingto ^Z0& in the nonmixing state. The final param-eter in the age process Zw1

is also chosen bynumerical experimentation. Finally, the ageprocess model parameters are:

p0 5 1/ 2

Z0l5 1/6

Z0u5 1/6

Z1l5 0.0176

Z1u5 0.3157

Discussion

Choice of EMST as the Definition ofLocalness

Given a set of N points in a D-dimensionalcomposition space, there are several definitionsof proximity. Notable among them are nearestneighbor pairs, the Delaunay trianguation (DT)(and its dual the Voronoi diagram), and theMST [23]. Any of these definitions of localnesscan be used to define a set of edges that connecteach particle to a neighbor and subsequentlymixing models can be constructed based onparticle–neighbor interactions that are local incomposition space. It is advantageous to modelthese particle–neighbor interactions as pair-wise-exchange models for the reasons adducedearlier in “Conservation of Means” and “Decay


of Variances.” The choice of proximity defini-tion will depend on the model performance withrespect to the list of criteria listed in “Perfor-mance Criteria for Mixing Models.” In practice,apart from the ability of the model to producereasonable composition jpdf’s in test problems,an overriding concern is the computational ex-pense incurred by the model.

The nearest neighbor proximity definitiondoes not yield a satisfactory particle exchangemixing model since it collapses each nearestneighbor pair into a single composition valueand the fully mixed state is never reached. In thecase of the DT (or Voronoi diagram) the matrixdescribing the particle interaction mixing modelis considerably more complex and it is difficultto predict the model behavior a priori. However,since the number of edges incident on a particlein the MST is a subset of the number of edgesincident on the same particle in the DT, it isguaranteed that the particle interaction matrixfor the DT-based mixing model will be lesssparse than that for the MST-based one. Theparticle interaction matrix for the DT-basedmixing model has no special structure and solv-ing the corresponding evolution equation ateach time step in a Monte Carlo simulationwould be computationally expensive. On theother hand, the MST-based particle interactionmatrix has a very special structure (owing to theunderlying tree structure) that permits an 2(N)direct solution of the particle composition evo-lution equation. This alone makes the MSTdefinition of localness attractive for the con-struction of mixing models that are local incomposition space.

Intermittency

The reason for introducting the intermittencyfeature via the age process is the formation ofstranding patterns observed in the mean scalargradient test problem. This intermittency mayalso be interpreted, and to some extent justified,on the following physical grounds. Regions ofhigh scalar dissipation rate xf (xf 5 G¹f. ¹f,where G is the molecular diffusivity of scalar f)are confined to a small fraction of the total fluidvolume in physical space. Experimental evi-dence for large Schmidt number turbulent flowsshows that 90% of the total scalar dissipation is

contained in just over 45% of the volume [25].While the actual numerical values are very likelyReynolds number dependent, the “spottiness”in the fine structure of the scalar dissipationfield is presumably characteristic of scalar mix-ing in high Reynolds number turbulent flows.Consequently, even though the EMST model isintended for flows with Sc ' 1, this experimen-tal evidence can be taken as qualitative justifi-cation for introducing the intermittency featurein the EMST model. The choices of p0 5 1/ 2(the fraction of particles that participate in themixing process being half) and the values for theresidence times in states 0 and 1 for the ageprocess have no direct support from physicalevidence, but are made mainly with a view toenhancing the model performance.

Model Properties

Since the EMST model is a pairwise-exchangemodel it satisfies the conservation of means,decay of variances, and boundedness principles.The EMST mixing model depends on the Eu-clidean norm in composition space and hencedoes not satisfy the independence principle. Asa direct consequence, the EMST model doesnot satisfy the linearity principle (for equal orunequal) diffusivities either. The EMST modelis invariant to rotational transformations of theset of scalars (i.e., for T being an orthogonalmatrix). For the homogeneous problem, evenwith joint-normal initial conditions, the EMSTmixing model does not yield acceptable jpdf’s asshown in Fig. 4. The model exhibits a phenom-enon termed “stranding” as shown in Fig. 4. Theintroduction of the intermittency feature wasmotivated by the necessity to alleviate this par-ticular deficiency in the model. The strandingproblem is alleviated by the intermittency fea-ture, which is independent of both the numberof particles per cell and the number of scalars.

The EMST model is inherently a particlemodel. In contrast to IEM or Curl’s model, thepdf evolution equation corresponding to theEMST model’s particle evolution equation isconsiderably more difficult to derive. In spite ofthe difficulty in mathematically analyzing theasymptotic behavior and convergence proper-ties of the EMST model with respect to thenumber of particles N, numerical tests for the


mean scalar gradient test case (for two scalarsand for 10 scalars), indicate that all the mo-ments, as well as the cdf’s, converge as thenumber of particles is increased from 512 to8192.

The EMST model is also asymptotically localin composition space. As the number of parti-cles per cell (N) is increased, the characteristicedge-length (in composition space) associatedwith the EMST model decreases, thereby in-creasing localness.

The EMST model has no Reynolds numberdependence and has no dependence on thesmall scales or the molecular diffusivity, thoughfor equal diffusivities this can be easily intro-duced in the particle position evolution equa-tion. The model assumes that the scalar dissipa-tion is determined by the large scales which isvalid in high Reynolds number turbulence. Thecurrent implementation of the model assumesequal diffusivities and has no feature to accountfor differential diffusion effects. Since themodel is based on a one-point one-time pdfclosure, it has no dependence on compositiongradients or lengthscales of scalar fields. Themodel is currently incapable of accounting forthe steepening of composition gradients due toreaction as seen in flamelet combustion.

INERT FLOW TEST CASE

The EMST model is validated in a simple inertscalar mixing problem. One of the simplestscalar mixing problems is an inert, passive scalardecaying from specified initial conditions indecaying isotropic turbulence. Experimentalstudies of scalar decay in grid turbulence [13]show that the decay rate of the scalar fluctua-tions shows a strong dependence on the length-scales of the initial scalar field. It is also foundthat there is no equilibrium value for the scalarvariance decay rate. In addition there is noevidence of relaxation of the mechanical-to-scalar timescale ratio to an equilibrium value.Given that the EMST model is a closure at thelevel of the scalar pdf with no information ofscalar gradients or scalar lengthscales, and not-ing that the model assumes a constant mechan-ical-to-scalar timescale ratio, the scalar decay

test problem does not seem appropriate formodel validation.

The evolution of an inert, passive scalar fieldevolving in grid turbulence under an imposedlinear mean scalar gradient has also been stud-ied experimentally by Sirivat and Warhaft [5]. Inthis experiment the scalar field was temperatureand measurements were made of the tempera-ture variance and heat flux. This mean scalargradient (MSG) test problem is preferable tothe test case of an inert, passive scalar decayingin grid turbulence since in the MSG test the flowtends to an equilibrium where the scalar vari-ance growth rate reaches a stationary value.Experimental evidence also shows that the ratioof mechanical-to-thermal lengthscales evolvesto an equilibrium value, whereas in the decayingscalar problem this ratio does not change withtime. Furthermore in the MSG test, comparisonof the scalar flux values with experiment pro-vides a test of the performance of the mixingmodel in conjunction with the model for parti-cle velocity. Finally, direct numerical simula-tions [6] have also been performed for the MSGproblem in stationary turbulence, providing ad-ditional detailed information about the scalarmixing.

A mean scalar gradient model problem isposed based on the Sirivat and Warhaft exper-iment [5], which is then used to test the EMSTmodel. For a single scalar, the MSG modelproblem is the evolution of an inert passivescalar f1, in homogeneous isotropic turbulencewith an imposed linear gradient of ^f1& in the x1

direction. The velocity field in the simulationcan be either decaying or stationary. For decay-ing turbulence the results are compared withthe experimental data of Sirivat and Warhaft[5]. For the stationary case comparison with theDNS results is appropriate.

Since the EMST mixing model is intended foruse in flows with multiple scalars, it is useful tostudy the model performance for two or morescalars. The MSG model problem extends con-ceptually to multiple scalars (say D in number)provided the dimension of physical space isincreased to have D coordinate directions alongwhich the mean gradients for each scalar com-ponent are imposed. In the simulation results tobe presented, the mixing of two scalars is con-


sidered first and subsequently the mixing of 10scalars.

Moment Equations

The scalar fields can be decomposed into meanfields and fluctuating fields fb 5 ^fb& 1 f9b. Ifthe fluctuating scalar fields are initially homo-geneous then they remain so for all time. Theequation for the fluctuating scalar in homoge-neous, isotropic turbulence with zero mean ve-locity can be written as

f9bt

1 uj

^fb&

xj1

xj~ujf9b 2 ûjf9b&!

5 G2f9b

xk xk, (48)

where G is the scalar diffusivity (assumed equalfor all scalars). The scalar variance evolves asfollows:

12

t^f9b

2& 5 2^f9buj&^fb&

xj2 G Kf9b

xk

f9b xk

L ,

(49)

with the terms on the right hand side being theproduction of scalar variance due to the im-posed mean scalar gradient and the scalar dis-sipation respectively. The terms correspondingto turbulent and molecular transport are absentdue to homogeneity, and the summation con-vention is not applied to Greek indices.

Estimates for the growth rate of the scalarvariance in Eq. 49 can be obtained by simplescaling arguments. For simplicity a single scalarwith mean scalar gradient in x1 is considered.Assume the scalar dissipation scales like ^f91

2&/tf, where tf is the scalar timescale which to afirst approximation can be assumed to be pro-portional to the turbulence timescale t. Furtherassuming gradient modeling for the scalar flux^f91u1& 5 Gt^f1&/ x1, the production termscales as Gt(^f1&/ x1)2, where the eddy diffu-sivity Gt is assumed to scale like u9l (u9 and lbeing the characteristic turbulence velocity andlength scales respectively). If the empirical re-lations for the evolution of the turbulent scalesin decaying isotropic grid turbulence are used(l ; t12n/ 2, u9 ; t2n/ 2, t ; t, where n is theexponent in the velocity variance decay law forgrid turbulence), then the production scales like

(^f1&/ x1)2t12n and the dissipation like^f91

2&t21. The growth rate of the scalar variancecan become stationary if

^f912& , S^f1&

x1D2

l2 , t22n < t0.7.

The scalar–velocity correlation coefficient isdefined as

rbj 5 ^f9bu~ j!&/Î^f9b2&û9~ j!

2&

where there is no sum over bracketed indices.The equation for the scalar flux evolution is

tûif9b& 5 2ûiuj&

^fb&

xj1 ûjf9b&

Ûi&

xj

2 ~G 1 n!Kf9b xk

ui

xkL

1 Kp9

r

f9b xi

L , (50)

where statistical homogeneity is used to simplifythe dissipation and pressure-scrambling terms,and to neglect the convection and triple corre-lation terms. In the absence of homogeneousshear the second term on the right hand side iszero. It is customary to assume that in highReynolds number turbulence the scalar fluxdissipation can be neglected if local isotropyprevails. DNS results of Overholt and Pope [6]show that this term may not be negligible for Rl

, 350. For sufficiently high Reynolds numberthis term may be neglected and the scalar fluxevolution reaches an equilibrium state when thepressure scrambling term ^p9/r 3 f9b/ xk&balances the production of scalar flux throughthe mean scalar gradient.

Particle Evolution Equations

Since the velocity and fluctuating scalar fieldsare homogeneous, the particles in the simula-tion need have only velocity and scalar proper-ties associated with them. The velocity andfluctuating scalar values are scaled as follows:

u*i 5 ui/u9 (51)

f*b 5 f9b/Su93

e

^f~b!&

x~b!D (52)


where u9 is the standard deviation of the veloc-ity and e is the dissipation rate of turbulence.The particles are evolved in scaled time s givenby:

s ; E0

t

^v& dt, (53)

where ^v& is the mean turbulent frequency.The particle velocity evolves by the simplified

Langevin model (SLM):

du*i 5 217

u*ids 1 $dWi, (54)

where the drift (21/7) and diffusion ($) coef-ficients are chosen appropriately for stationaryor decaying turbulence. These values are givenin Table 1.

The particle fluctuating compositions evolveby

df*bds

5 Q 232 u*b, (55)

where Q represents the mixing model, and thesecond term in u*b represents the effect of themean scalar gradient.

EMST Model Results

Decaying Turbulence

The EMST model is applied to the MSG modelproblem with two scalars. There is an imposedlinear gradient of ^f1& in the x1 direction and of^f2& in the x2 direction respectively. The velocityfield is chosen to be decaying isotropic turbu-lence. The simulations are performed with N 58192 particles in the ensemble, initially joint-normally distributed in velocity–compositionspace. All the velocities and scalars have zeromean. The velocity–composition covariancematrix is diagonal. The velocity variances are all

unity, and the scalar variances are set to 10212.The particle properties are incremented overtime steps of Ds 5 0.005. Scalar–velocity cor-relations attain nonzero stationary values forthe velocity components along which the scalarhas a mean gradient. The scaled scalar variancesand nonzero scalar–velocity correlations evolveto their stationary values. Stationary values forall the moments are determined by time aver-aging. The time averaging commences when theratio of scalar variance production to scalardissipation reaches 92% of its stationary valuewhich is unity. The stationary values of themean and skewness are shown in Table 2. Theevolution of the cumulative distribution func-tion (cdf) of scalar 2 is shown in Fig. 7. Theother scalar is statistically identical.

Experimental Data

Sirivat and Warhaft [5] performed experimentsto study the evolution of temperature varianceand heat flux under the influence of a passivecross-stream temperature gradient in grid tur-bulence. The mean temperature gradient wasgenerated using two different heating arrange-ments (referred to as the mandoline and thetoaster). Even for this relatively simple flow it isdifficult to reconcile the measurements of scalarvariance growth rate and the stationary valuesof the scaled variance obtained by the twomethods [8]. The principal reasons are: (i)uncertainty whether the toaster flow hadreached a truly stationary state, (ii) uncertaintywhether the Reynolds number was high enoughfor the measurements to be Reynolds numberindependent, (iii) influence of the different ini-tial conditions in the two experiments. TheEMST model predicts a stationary scaled scalarvariance which lies closer to the toaster (relativeerror in scalar standard deviation is 16%) thanthe mandoline data (Table 3). The stationaryscalar–velocity correlation predicted by theEMST model (relative error of 30%) is alsoshown in Table 3.

Stationary Turbulence

The scaled scalar variances and nonzero scalar–velocity correlations for stationary turbulencealso evolve to stationary values. The stationaryvalues of the mean and skewness for the station-

TABLE 1

Values of Drift and Diffusion Coefficients in the SLMModel for Stationary and Decaying Turbulence

Drift (21/7) Diffusion ($)

Stationary 23C0/4 =3C0/2Decaying 2(1/2 1 3C0/4) =3C0/2


ary turbulence case are tabulated in Table 2.The plot of the cumulative distribution function(cdf) of scalar 2 after 10 mixing timescales isvery similar to that obtained in the decayingturbulence case. DNS results of the same flowby Overholt and Pope [6] indicate a strongdependence of the scaled scalar variance onReynolds number. The DNS results actuallyshow a decrease in the scaled scalar variancewith increasing Reynolds number. The DNSresults also indicate that the mechanical-to-scalar timescale ratio varies from 1.8 to 3.0 andincreases with Reynolds number. On the otherhand in the EMST model this ratio is heldconstant with the choice of Cf 5 2.0. As aconsequence the model (which does not incor-porate a Reynolds number dependence) is in-

capable of reproducing the range of scalar vari-ance values obtained from the DNS. The onlyreasonable (and weak) conclusion that can bedrawn from this comparison is that the modelpredictions are within the range of the DNSdata. The relevant values are tabulated in Table4.

Multiple Scalars in Stationary Turbulence

The EMST model is applied to the MSG modelproblem with 10 scalars. The numerical param-eters are identical to those in the previousresults. However, for this case the initial scalarvariances are all chosen to be unity. Since all thescalars are statistically identical, results are pre-sented only for the tenth scalar. The stationaryvalues of the mean and skewness of the tenthscalar for the stationary turbulence case are

TABLE 3

Comparison of Stationary Values of the Scaled ScalarVariance and Scalar–Velocity Correlation from

Experiment with the EMST Model Results for the MSGTest in Decaying Turbulence

^f*2& r

Mandoline 1.10 [1.06, 1.18]a 20.7Toaster 0.75 [0.68, 0.81] 20.68EMST

ModelScalar 1 0.53 [0.51, 0.55] 20.48 [20.51, 20.46]Scalar 2 0.51 [0.49, 0.52] 20.49 [20.51, 20.47]

a Corresponding minimum and maximum values in thetime interval over which averaging was performed areshown in square brackets.

TABLE 2

Values of the Stationary Scalar Mean and Skewness for the MSG Test Problem Usingthe EMST Mixing Model

Scalar Mean Scalar Skewness

DecayingScalar 1 24.6 3 10210[21.4 3 1028, 1.3 3 1028]a 20.008[20.06, 0.03]Scalar 2 23.9 3 10210[21.4 3 1028, 1.2 3 1028] 0.02[20.05, 0.08]

StationaryScalar 1 5.1 3 10210[21.9 3 1028, 1.4 3 1028] 23.8 3 1022[28.1 3 1022, 1.7 3 1022]Scalar 2 7.6 3 10210[22.2 3 1028, 1.2 3 1028] 5. 3 1022[6.5 3 1024, 0.1]

StationaryScalar 10 28.5 3 10210[23.8 3 1029, 2.9 3 1029] 20.22[20.32, 20.15]

a Corresponding minimum and maximum values in the time interval over which averaging was performed are shown insquare brackets.

Fig. 7. Evolution of the cdf of scalar 2 in the mean scalargradient test (decaying turbulence) using the EMST model;— initial cdf (s 5 0), - - - s 5 5.0.


tabulated in Table 2. The cumulative distribu-tion function (cdf) of the tenth scalar after 10mixing timescales is shown in Fig. 8. While theseresults are not identical to those obtained fromthe simulation for two scalars, the departuresare small enough to not invalidate the use ofthis model for a wide class of reactive flows. Thedependence on dimensionality of the composi-tion space arises from the variation in thestructure of the EMST with dimension of thecomposition space, and this dependence is ex-pected to become weaker with increasing scalardimension.

IEM Model Results

The evolution equation for f*b corresponding tothe IEM model for mixing is

df*bds

5 2aIf*b 2 u*b, (56)

where aI is a model constant. The relationbetween aI and Cf can be deduced from Eq. 16to be

aI 5Cf

2. (57)

The corresponding scalar variance and scalarflux evolution equations are (dropping the bsubscript since all the scalars are statisticallyidentical)

dû*f*&

ds5 u9*Srbb

df*ds

1 f*drbb

ds D (58)

5 2SaI 117D û*f*& 2 û*2& (59)

d^f*2&

ds5 2aI^f*2& 2 û*f*& (60)

Setting the time derivatives to zero for station-ary solutions, we obtain the stationary scalar–velocity correlation and scalar variance valuesto be

rss 5 2ÎaI7/~1 1 aI7! (61)

^f*2&ss 5 2rssu9*

aI. (62)

The stationary values of the scalar–velocity cor-relation and scaled scalar variance for stationaryand decaying turbulence cases as predicted bythese formulae (using the values for the modelconstants given in Table 1) are tabulated inTable 5. It is seen that the IEM model predic-tions are closer to the experimental results forthe decaying case, whereas in the stationary casethe large variation with Reynolds number in theDNS results precludes any meaningful compar-ison.

Since the model for scalar dissipation in thescalar variance equation (Eq. 49) is the same forboth the IEM and EMST models, the differ-ences in the stationary scalar variance resultsmust arise from the production term. This canbe explained as arising from differences in thenonzero scalar flux terms ^f9buj&. Recently,Pope [26] has shown that since the scalar flux is

TABLE 4

Comparison of Stationary Values of the Scaled ScalarVariance and Scalar–Velocity Correlation from DNS Data

with the EMST Model Results for the MSG Test inStationary Turbulence

^f*2& r

DNSRl 5 28 1.01 20.60Rl 5 52 0.69 20.56Rl 5 84 0.59 20.53Rl 5 185 0.31 20.46EMST Model

Scalar 1 0.62 20.53Scalar 2 0.60 20.52

Fig. 8. Evolution of the cdf of scalar 10 after ten mixingtimes in the mean scalar gradient test (stationary turbu-lence) using the EMST model; — s 5 0, - - - s 5 5.0.


independent of the diffusivity at high Pecletnumber, the modeled scalar flux ^f*bu*j& shouldbe independent of the mixing model (i.e., itshould be determined from the drift coefficientsin the particle velocity model). The IEM modelas described in this paper has a spurious sourceterm in the modeled scalar flux equation. Acorrection to this model has also been suggestedby Pope [26] but numerical implementation ofthis may not be straightforward. It should benoted that the EMST model also violates thediffusivity independence principle as expressedin terms of the modeled scalar flux equation’sindependence of the mixing model coefficients.Any two mixing models that satisfy the diffusiv-ity independence principle would yield identicalresults for the stationary scaled variance (in theMSG model problem) and scalar–velocity cor-relation, provided the same velocity model wasused. Of course there would be differences inthe scalar cdf’s which would also be manifest inthe higher scalar moments.

DIFFUSION FLAME TEST RESULT

The EMST mixing model is applied to thediffusion flame test which is an idealized reac-tive flow test problem. The objective is to verifythe hypothesis that a mixing model that is localin composition space will yield the correct phys-ical behavior for this particular test case. Torecapitulate, consider an idealized diffusionflame in homogeneous isotropic turbulence withinfinitely fast chemistry. The fluid compositionis represented by two variables: mixture fractionj and a reaction progress variable Y. The initialcondition (t 5 0) for mixture fraction is chosen

to be a b-distribution with mean and variancewhich correspond to a typical experiment. Theprogress variable is initially at equilibrium (de-noted Ye). The one-step, irreversible reaction isconfined to a region in composition space asshown in Fig. 1.

As the flow evolves from this homogeneousinitial condition, reactants enter the reactionzone by mixing and are instantly reacted to formproducts resulting in flame sheet combustion[3]. Since the reaction is infinitely fast comparedto the mixing and is also irreversible, the onlystable solution for t . 0 corresponds to com-plete combustion, i.e., all the particles lie on theequilibrium line.

The evolution of the composition jpdf iscomputed using a Monte Carlo particle methodwhere the discrete jpdf of composition, which isan ensemble of delta functions in the two-dimensional composition space, is representedby an ensemble of 4096 particles with twoproperties: mixture fraction j and progress vari-able Y. At any time t in the simulation, particleproperties are advanced over a time step Dtwhich is chosen to be a user-defined fraction(4 3 1023 for the diffusion flame test) of theminimum physical timescale in the problem. Forthe diffusion flame test with infinitely fast chem-istry the chemical time scale is zero. Conse-quently the minimum physical timescale is theonly other timescale in the problem, which is themixing timescale tf. Both mixture fraction andprogress variable properties of the particles areadvanced using the EMST mixing model. Theeffect of infinitely fast, irreversible reaction isaccounted for by setting Y to the correspondingequilibrium value. The scatter plot in composi-tion space at time t 5 0.8tf (Fig. 9) shows thatthe EMST model reproduces the expected phys-ical behavior. It should be noted, however, thatlocalness is affected by the number of particlesNT which are used to form the EMST.

The same simulation is performed with theIEM mixing model, and the corresponding scat-ter plot in composition space at time t 5 0.8tf

is shown in Fig. 10. It is clear that particles donot all lie on the equilibrium line and the modelfails to reproduce the expected physical behav-ior. Particles corresponding to composition val-ues outside the reaction zone relax to the meancomposition and are drawn away from their

TABLE 5

Comparison of IEM and EMST Model Predictions ofStationary Scaled Scalar Variance and Scalar–Velocity

Correlation for the Mean Scalar Gradient Model Problemfor Stationary and Decaying Turbulence

IEM EMST

Stationary^f*2&ss 0.873 0.62rss 20.62 20.53

Decaying^f*2&ss 0.73 0.53rss 20.57 20.48


initial condition on the equilibrium line. Parti-cles in the reaction zone are of course instantlyreacted back to their equilibrium values at eachtime step. This accounts for the piecewise linearscatter plot in Fig. 10 [7].

CONCLUSION

A new model for the mixing of multiple reactivescalars has been developed. This pairwise ex-change model is based on Euclidean minimumspanning trees constructed in compositionspace and possesses the important property thatit is local in composition space. It also satisfiesthe mean conservation and variance decayproperties. Boundedness of compositions is pre-served by the EMST model. However, this

model does not satisfy the independence andlinearity properties.

The model is applied to an inert scalar mixingtest problem based on the imposed mean tem-perature gradient experiment of Sirivat andWarhaft [5]. Reasonable results are obtainedfor the scalar variance and scalar–velocity cor-relation but the agreement with experimentaldata (which has a high variability depending onthe heater configuration used) is not close.The model results are also compared with theDNS data of Overholt and Pope [6] for asimilar mean scalar gradient problem in sta-tionary turbulence. Unfortunately the com-parison is not conclusive since there is astrong Reynolds number dependence in theDNS results, whereas the model incorporatesno Reynolds number dependence. It is as-sumed in the modeling that the Reynoldsnumber is very large.

Finally the model is tested in a reactive flowtest problem called the diffusion flame test.Excellent results are obtained using the EMSTmodel, confirming the hypothesis that the local-ness principle forms a sound basis for reproduc-ing the expected physical behavior in this prob-lem. These are contrasted with the incorrectresults obtained using the IEM model. Similardeficiencies have been documented with Curl’smodel also [3].

The EMST model has been applied to pilotedjet diffusion flames and preliminary qualitativeresults are encouraging [7]. Further quantitativeresults are anticipated and will provide a furthertest for the model. The model is also currentlybeing applied to a variety of reactive flow testproblems which will provide detailed informa-tion concerning the model behavior in variousflow situations.

This work was supported by the Air Force Officeof Scientific Research, Grant F49620-94-1-0098.

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4. Dopazo, C., Phys. Fluids 18:397–404 (1975).

Fig. 9. Scatter plot of reaction progress variable Y vs. j:EMST model result for the diffusion flame test at t 5 0.8tf.

Fig. 10. Scatter plot of reaction progress variable Y vs. j:IEM model result for the diffusion flame test at t 5 0.8tf.


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Received 1 April 1997; accepted 13 February 1998


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