on relation of the conditional moment closure and unsteady...

On relation of the Conditional Moment Closure andunsteady Flamelets

A.Y.Klimenko†

(Published: Combustion Theory and Modelling, 5(3), pp. 275-294, 2001)

Abstract. We consider the relation of the Conditional Moment Closure (CMC)and the unsteady Flamelet Model (FM). The CMC equations were originallyconstructed as global equations while FM was asymptotically derived for a thinreaction zone. The recent tendency is to use FM-type equations as globalequations. We investigate the possible consequences and suggest a new versionof FM – Coordinate-Invariant FM (CIFM). Unlike FM, CIFM complies withconditional properties of the exact transport equations which are effectivelyused in CMC. We analyse the assumptions needed to obtain another globalversion of FM – Representative Interactive Flamelets (RIF) – from original FMand demonstrate that, in homogeneous turbulence, one of these assumptions isequivalent to the main CMC hypothesis.

†Dept. of Mechanical Engineering, the University of Queensland, Qld 4072,Australia

1. Introduction

Apart from the stochastic pdf simulations [1], the most common models used inturbulent non-premixed combustion are the Fast Chemistry Model (FCM) [2], theFlamelet Model (FM), different versions of which were independently suggested byKuznetsov [3, 4] and Peters [5, 6, 7], and also the Conditional Moment Closure whichwas independently suggested by Klimenko [8] and Bilger [9, 10]. Existence of a linkbetween these models is quite obvious: all of these models involve some form ofthe diffusion term in the conserved scalar space multiplied by the conserved scalardissipation and a reaction term. This similarity is observed due to the fact that thesemodels belong to the same class: the models which effectively use the conservationproperties of the mixture fraction appearing in the Shvab-Zeldovich formulation ofthe problem (see [11] for details) while it is expected that the mixture fraction is animportant parameter for combustion processes. Similarity of some terms, however,does not mean that the physical assumptions behind these models are the same.

In FCM, it is assumed that the reactive scalar Y is linked to the mixture fraction(or conserved scalar) ξ by a deterministic function Y = Y (ξ) in the whole flow or ina large part of it. Practically, this situation may occur when reactions are fast andclose to their equilibrium state Y = Ye(ξ) (including piece-wise linear dependence ofY on ξ as considered by Burke and Schumann [12]) or, in a trivial case, when Y isa linear function of ξ. The FM asymptotic analysis is performed for a thin reactionzone. The advantage of this model is that, within this zone, Y may be different fromits equilibrium value and also may evolve in time. The main idea behind CMC is to

On relation of the Conditional Moment Closure and unsteady Flamelets 2

construct a FCM- and FM-like model which can describe non-equilibrium, transientand non-homogeneous reaction processes in the whole turbulent field (rather thanin a thin reaction zone) while properly taking into account the turbulent transport.The potential benefits of constructing such models are quite obvious and do not needto be advocated. It was found that this goal can be achieved by using conditionalexpectations of the reactive scalars and that the equations for these expectations cannot simply be derived from the transport equations and a closure is needed. In CMC,the closure is based on the theory of the inertial interval of turbulence. Some ofterms in CMC equation are similar (but not identical) to the terms in FCM and FMequations. Other terms appear only in CMC equations.

We should emphasize that the Eulerian-type time derivative is present in theoriginal derivation of Peters’s version of FM [6]. While the steady version of FM was(and is) widely used due to its simplicity, the unsteady version of FM was effectively(and, probably, unduly) forgotten (although unsteady FM was used occasionally [13]).A new wave of interest to the unsteady version of FM [14, 15, 16] was raised whenCMC, which involves the time derivative and the convective terms, appeared. Inthe present work, we treat FM as unsteady when investigate its relation to CMC.Haworth et al. [17] analysed the effect of Lagrangian evolution of the scalar fields ina thin reaction zone on formulation of steady FM. The ability of FM to achieve itssteady state was also considered by Mell et al. [18].

Mauss et al. [13] solved the unsteady flamelet equations for a thin reaction zonein vicinity of the stoichiometric surface. The calculated solutions were compared toa jet flame by estimating the average Lagrangian residence time. A more recentwork of Pitsch et al. [19] indicates definite Lagrangian understanding of FM. In[14], a new version of unsteady FM – Representative Interactive Flamelets (RIF) –was applied to the whole domain of the mixture fraction values but not to a thinreaction zone. The chemical processes considered are essentially transient and non-equilibrium everywhere in the flow which is generally not homogeneous. This problemis similar to the combustion kinetics that CMC has been designated for. We do notexpect, of course, that CMC goals can not be achieved by constructing models whichare different from CMC or, may be, which are similar to CMC but involve somemodifications. Unfortunately, Pitsch et al. [14] did not give any comparison with theCMC approach. As it can be inferred from [14], the major differences are a) RIFneglects the convective terms; b) RIF does not define the diffusion coefficient in themixture fraction space as the conditional expectation of the scalar dissipation andleaves a greater freedom in choosing this coefficient; and c) the model is formulatedfor the instantaneous value of Y, which is similar to FM, but not for the conditionalexpectation of Y , which would be similar to CMC.

In his comment, Peters [16] concluded that if the specific form of the CMCequations formulated for homogenous turbulence looks similar to the unsteady FMand RIF, then FM, RIF and CMC should have the same physical basis. The readeris left to determine himself what this common physical hypothesis could be and howit can be validated. Apart from the obvious common feature – the effective use of theconservation properties of the mixture fraction while expecting that the correlationof Y and ξ is significant – the other possible candidates are: a) assumption of athin reaction zone and b) the CMC closure. The assumption of a thin reaction zoneis not particularly good for CMC since CMC is not expected to be valid for verythin reaction zones (smaller than the Kolmogorov length scale) [8, 20] and the CMCclosure is not particularly good for FM because of the same reasons. Unless the


hypotheses, which are used in mixture-fraction-based modelling, are clearly defined,their validation is problematic. In the present work, our goal is in investigation ofthe links between the models as well as in identifying major assumptions which areneeded for their derivations. We do not attempt to give a verdict concerning validityof the assumptions but determine if and under which conditions the assumptionsare equivalent. Considering that unsubstantiated statements may bring even greaterconfusion into the area, we believe that the rather formal style of this paper is justified.

2. FM, RIF and CMC: comparative analysis

The instantaneous equations governing turbulent transport of the conserved scalar (orthe mixture fraction) and the reactive scalars are given by

dξ

dt≡ ∂ξ

∂t+ v ·∇ξ = D∇2ξ (1)

dY

dt≡ ∂Y

∂t+ v ·∇Y = D∇2Y +W (2)

respectively. The density ρ and the diffusion coefficient D are assumed constant forsimplicity but, of course, the equations considered here can be generalized for moregeneral ρ and D.

2.1. Flamelet Model

The flamelet theory is an asymptotic theory. This means that FM is obtained asthe result of flamelet asymptotic analysis which is conducted in a thin vicinity of thestoichiometric surface. The conventional flamelet analysis is limited to the leadingorder terms. As any asymptotic equation, FM allows for some variations of its terms.Indeed, small changes, whose order is higher than the leading order, do not affectthe validity of the leading order terms. This point of view corresponds to genericunderstanding of FM — any model whose terms are in agreement with the flameletasymptotic analysis is a flamelet model. We believe that the conventional FM shouldbe interpreted as generic. Such interpretation of FM is given in most recent workof Peters [7] (which is considered further in this section). The differences betweenthe flamelet models can be related to different choices of the flamelet coordinates. Inthe present work, we need exact (i.e. not only the leading terms) specification of theterms of the flamelet models. Hence, we have to use coordinate-invariant notation(i.e. notation whose meaning is precise and does not depend on choice of the systemof coordinates).

The unsteady FM equation can be written in the from(∂Y

∂t

)FM

−N(∂2Y

∂ξ2

)FM

= W (3)

where

N ≡ D (∇ξ)2 (4)

is the conserved scalar dissipation. The subscript ”FM” indicates that these derivativesshould be interpreted as special flamelet derivatives which are discussed below. (Thepartial time derivatives, which are not indexed, are evaluated for constant x.).Conventional derivations of FM involve introduction of a new system of coordinates(ξ, z2, z3, t) (that is z1 ≡ ξ) linked to the surfaces ξ = const while neglecting the scalar


gradients along the surface in comparison with the scalar gradients across the surface.The first step is an equivalent mathematical transformation while the second step isbased on a geometric assumption: the region under consideration has a characteristicscale across the surface ξ = const much smaller than the characteristic scale alongthe surface. Physically, this means that the reaction zone is thin. Different choicesof coordinates z2 and z3 are possible. Kuznetsov [3] prefers the local orthogonalcoordinates while Peters [5, 6] tends to choose the Cartesian coordinates: z2 = x2 andz3 = x3, which form a non-orthogonal system with ξ (with exception of some pointswhere ∇ξ has the same direction as the axis x1). The conventional flamelet analysisdoes not consider the terms of higher order and, consequently, does not give anyjustified algorithm for choosing the ”best flamelet coordinates”. All basic flameletapproaches are equivalent from point of view of the flamelet asymptotic analysis.The generic interpretation of FM does not pay attention to the small differences indifferent formulations of FM and declares all these formulations equivalent (indeed, inhomogeneous turbulence, the choice of z2 = x2 and z3 = x3 is no better than z2 = x1

and z3 = x2). The coordinate-invariant vector-gradient formulations of the flameletoperators (...)FM are now considered.

We demonstrate below that the flamelet operator (∂/∂ξ)FM, which represents thederivative evaluated for constant z2, z3 and t, can be written in the form(

∂

∂ξ

)FM

=D

N(∇ξ ·∇) + s ·∇ (5)

where s · ∇ξ = 0. The first term represents the derivative in the direction whichis normal to the surfaces of ξ = const . The second term appears in (5) when thesystem of coordinates (ξ, z2, z3) is not orthogonal. Indeed, for a given t, the Cartesiancoordinates can be expressed as functions of the flamelet coordinates x = x(ξ, z1, z2).The derivative ∂/∂ξ, applied to Y = Y (x(ξ, z1, z2)) and evaluated for constant z2

and z3, represents, by definition, the flamelet derivative. Chain differentiation yields(∂Y/∂ξ)FM = ∇Y · eξ where eξ ≡ (∂x/∂ξ)z1,z2 is evaluated for constant z2 and z3.This vector (shown in Figure 1a) has direction of the lines of constant z2 and z3 andsatisfies eξ ·∇ξ = 1 since, obviously, (∂ξ/∂ξ)FM = 1. Comparison with equation (5)determines

s =(

eξ−D

N∇ξ

)(6)

This equation requires that the vector s lays on the surface of ξ = const since s ·∇ξ =1 − D (∇ξ)2 /N = 0. Another property of the vector s is that s = 0 when the linesof constant z2 and z3 are orthogonal to the surfaces of ξ = const. Indeed, in thiscase, eξ has the same direction as ∇ξ so that |eξ| = 1/ |∇ξ| = |∇ξ|D/N . The exactdefinition of the flamelet derivative (∂Y/∂ξ)FM in (5) involves two terms. The firstterm is the major, leading term of the flamelet asymptotic analysis while the secondterm is coordinate-dependent and its selection is not considered by the conventionalflamelet analysis. According to FM logic, this term (as well as any derivative alongthe surface ξ = const) is small and can be neglected. The definitions given above andthe flamelet analysis itself are, obviously, not valid at singular points where N = 0but these points are not specifically considered in conventional flamelet modelling.

The time derivative in (3) can be written as(∂

∂t

)FM

=∂

∂t+ (U ·∇) (7)


where

U ≡ v + u + w, u ≡ − ∇ξ

N/D

dξ

dt= −D

2∇2ξ

N∇ξ (8)

and w is an arbitrary vector satisfying w · ∇ξ = 0. This operator represents asubstantial-type derivative evaluated on the surfaces of ξ = const . Indeed, the flamelettime derivative must satisfy (∂ξ/∂t)FM = 0 since Y = ξ, W = 0 is solution of (2). Weseek operator (∂/∂t)FM in the form of ∂/∂t + U ·∇ where U is unknown a priori.Substitution of (1) into (∂ξ/∂t)FM = 0 yields equation (U − v) ·∇ξ + D∇2ξ = 0whose solution is sought in the form U = v + u + w where u is determined by (8).The equation takes the from w ·∇ξ = 0 which specifies the vector w as arbitraryvector laying on the surface ξ = const. Let us determine the term w for Eulerian(∂/∂t)FM as defined by Peters [6]. Chain differentiation of Y (ξ, x1, x2, t) with respectto t yields ∂Y/∂t = Yt + Yξ∂ξ/∂t. The value Yt, defined as time derivative evaluatedfor constant ξ, x1 and x2, represents the Eulerian (∂/∂t)FM. Substitution of ∂ξ/∂tdetermined by (1) and Yξ determined by (5) indicates that(

∂

∂t

)FM

=∂

∂t+(v ·∇ξ −D∇2ξ

)(DN

(∇ξ) + s)·∇ =

=∂

∂t+ (V⊥ + u + s◦) ·∇

where

V⊥ ≡ (v ·∇ξ)D

N∇ξ, s◦ ≡ s

(v ·∇ξ −D∇2ξ

)The vector V⊥, which represents the velocity component which is perpendicular tothe surface ξ = const, is shown in Figure 1b. Comparing with (7) determines thevectors w and U

w = −V + s◦, U = V⊥ + u + s◦ (9)

were V ≡ v − V⊥ is the component of velocity laying on the surface ξ = const(see Figure 1b). Obviously, w is a coordinate-dependent term: interpretation of thederivative (∂/∂t)FM depends on the frame of reference used (i.e. moving or stationarysystem of coordinates) and also it involves vector s which is coordinate-dependent.The stationary frame of reference, which is used in [6], corresponds to w definedby (9) and such FM is called Eulerian. Flamelet models linked to moving frame ofreferences are called Lagrangian (or Lagrangian-type) [19]. Lagrangian FM wouldhave a different w (whose exact values depend on how the frame of reference moves).From point of view of conventional flamelet analysis, the terms involving w are smallsince they are linked to the scalar gradients evaluated along the surfaces of ξ = const .In a special case of Lagrangian FM defined by w =0, the velocity v+u in (7) coincideswith the velocity of isoscalar surfaces introduced by Gibbson [21]. This definition ofthe flamelet derivative (∂/∂t)FM is used in some recent publications [22, 23, 24].

When this paper was being revised, Peters suggested a new derivation of FM inhis new book [7]. The derivation is based on representation Y = Φ(∆ξ/ε,x, t/ε2) =Φ(∆ξ/ε, x1, x2, x3, t/ε2), where ∆ξ/ε ≡ (ξ − ξst)/ε ∼ 1 within the reaction zoneand ε is a small parameter indicating that the reaction zone is thin. In general,since ξ = ξ(x, t), this representation is not unique. The dependence of Y on ξ(x, t)(which specifies short-range variations of Y ) is distinguished from the dependence ofY on x and t (which specifies the long-range variations of Y ) only asymptotically,


at the limit of ε → 0. The short-range and long-range variations are distinguishedat the leading order but these variations overlap at higher orders. The asymptoticanalysis [7] is generic since it specifies only the leading order terms of FM and leavesa certain freedom in exact interpretations of the higher order components of theflamelet derivatives. This point can be illustrated by the following consideration.Let us represent Y by the equation Y = Φ◦(ξ/ε,x, t/ε2) + φ(x) where φ(x) isan arbitrary function of the coordinates x. This representation is functionallyequivalent to the representation of [7] since we can always introduce a new function:Φ◦(ξ/ε,x, t/ε2) = Φ(ξ/ε,x, t/ε2) − φ(x). The derivative (∂Y/∂ξ)FM = Φ◦ξ/ε can bedefined using Φ◦ξ - the partial derivative of Φ◦ with respect to ∆ξ/ε for constant xand t. Since φ(x) is an arbitrary function we can put φ(x) =ξ(x,t0). At the momentt = t0, when Y = Φ◦(ξ/ε,x, t0/ε2) + ξ(x,t0), the flamelet derivative can be equallydefined as (∂Y/∂ξ)FM = Φ◦ξ/ε + 1. To the leading order (∼ 1/ε), this definition isthe same as Φ◦ξ/ε but, as one can see, the higher order terms can be interpreted indifferent ways.

The discussion of the flamelet operators within the reaction zone can now besummarised. We gave a coordinate-independent, vector-gradient formulation of theflamelet derivatives and distinguished two types of terms. The terms of the first typeare the coordinate-invariant terms which correspond to the leading terms of flameletanalysis. The terms of the second type (these involving s, w and specifying gradientsalong the surface ξ = const) are the terms of higher order which are expected tobe small in flamelet modelling. These terms are either dependent on the system ofcoordinate (or frame of reference) used in flamelet analysis or, as in generic FM, arenot uniquely specified. Since the conventional flamelet analysis does not give anyalgorithm (or justification) how these terms should be selected, we will follow theflamelet logic and neglect the terms of the second type when needed. At this point,we note that these terms may become significant when FM applied outside the formallimits of its validity.

The flamelet analysis is conventionally applied only to the vicinity of thestoichiometric surface where the reactions are most intense [6, 11, 13]. It should beemphasised that FM, which is applied within a thin zone needs boundary conditions.These conditions can be obtained only by matching with the solutions outside the thinzone which can be referred to as the outer solutions. Obtaining the outer solutions canbe rather difficult but, without these solutions, FM does not form a closed problem andis practically useless. Kuznetsov [3, 4] assumed that the reactive scalar concentrationis close to the chemical equilibrium state Ye(ξ) as specified by FCM. Peters [5]considered two possibilities for the outer zone: 1) near-equilibrium solution and 2)a model represented by quasi-steady, linear functions (∂2Y/∂ξ2 = 0) combined withthe negligible reaction rates. In the following work, Peters [6] mentioned possibility ofunsteady evolution outside the reaction zone, where the reaction rates are negligible,but concluded that, with exception of the ignition and extinction processes, theunsteady development in the outer zone can be neglected. Kuznetsov and Sabelnikov[4] supplemented FM by an approximate integral model which allows for analysis ofunsteady effects outside the reaction zone (specifically, CO transport on lean side of theflame). Peters [25] reasonably noted that if the reactions outside the stoichiometricsurface are fast or if ∂2Y/∂ξ2 = 0 and W = 0 then the solution of any FM-typeequations in the flow would give the correct answer. Hence, in this case, (3) canbe formally solved in the whole domain of the mixture fracture values. It shouldbe remembered, however, that if, outside the reaction zone, Y deviates from the


solutions mentioned above this consideration is no longer applicable. In the latestwork on flamelet modelling [7], the flamelet asymptotic analysis is extended to includechemically inert zones wider than the reaction zone. Such approach seems attractivebut it should be remembered that the precision of the asymptotic analysis decreasesas the zone under consideration widens. As soon as the width of the zone becomescomparable with the corresponding macroscale, the asymptotic precision would belost completely. Whatever the reaction rate may be, the nature of the flameletasymptotic analysis is that it should be applied to a thin zone in vicinity of thesurface with a constant value of mixture fraction (consideration of the zones widerthan the corresponding Kolmogorov scale requires a so-called filtering procedure whichis discussed later in the paper).

2.2. Representative Interactive Flamelets

In the most recent version of FM – the Representative Interactive Flamelets (RIF)– the unsteady flamelet equation was used to describe a strongly transient processof reactions in a diesel engine [14]. The species considered are not close to thereequilibrium states. The model which can be written in our notations as

∂YRIF

∂t−NRIF

∂2YRIF

∂ξ2= W (10)

is assumed to represent properties of the reactive scalar for the domains in the physicalspace with similar averaged parameters. It is not assumed that the reaction zone isthin. The model interacts with the CFD code through NRIF which is assumed tobe independent of z2 and z3. Several approximations for NRIF, deterministic andstochastic, are considered.

The main difference between FM and RIF is that RIF is applied to essentially non-equilibrium process with a wide reaction zone and, at the same time, it is assumedthat YRIF = YRIF(ξ, t). Of course, we need to specify which properties of the realreactive scalar Y are represented by the solution of (10) denoted here as YRIF todistinguish YRIF from Y . The interpretation that YRIF, the model prediction, selectedas a function of current value of ξ, is close to the instantaneous value of reactive scalarY seems reasonable and agrees with Peters’s comment [16]. That is

Y = YRIF(ξ, t) + y (11)

where y is a stochastic field whose amplitude is relatively small.

2.3. Comparison with CMC

CMC is the model which is formulated for the expectation of the reactive scalar Yconditioned on a fixed value η of the conserved scalar ξ so that Q ≡ 〈Y |ξ = η〉 or,using a more brief notation, Q ≡ 〈Y |η〉. Since the CMC equations, formulated forthe problem considered in [14], would involve the convective terms, it would be mostappropriate to compare CMC, FM and RIF in homogeneous isotropic turbulence wherethe structure of the models is most similar. In homogeneous turbulence, ensembleaveraging 〈Y 〉 can be replaced by volume averaging 〈Y 〉V if the volume is sufficientlylarge. The same ergodic principle applies to conditional expectations of Y so that (11)yields

Q = 〈Y |ξ = η〉V = YRIF(η, t) + 〈y|ξ = η〉V


since, obviously, 〈YRIF|ξ = η〉V = YRIF(η, t). Since y is uniformly small, so must be itsvolume average and YRIF is not much different from Q. Hence, Q should also satisfy(or approximately satisfy) equation (10)

∂Q

∂t−NRIF

∂2Q

∂ξ2= W (12)

and we also can write Y = Q(ξ, t) + y where y is small. Since Q is a deterministicfunction, NRIF must be also deterministic. Evaluation of the unconditional mean of〈Y 〉 , while taking into account the exact equation for pdf P (η) [20], indicates that

∂ 〈Y 〉∂t

=∂

∂t

∫QP (η)dη =

∫(NRIF −Nη)P (η)

∂2Q

∂η2dη +

∫WP (η)dη

where Nη ≡ 〈N |η〉. Here, the term Q∂2(NηP (η))/∂η2 is integrated by parts two times(see [20] for details). The source term in (12) should be understood as the conditionalaverage of the reaction rate. The conservation integral is preserved for any Q only ifNRIF = Nη. Equation (12) coincides now with the CMC equation for homogeneousturbulence. Thus we proved the following:

Proposition 1 A model – which, in homogeneous isotropic turbulence, a) isformulated in form of equation (10), b) approximates the reactive scalar in the sense ofequation (11) in the whole flow and c) preserves the conservation integrals – coincidesor approximately coincides with the homogeneous CMC equation.

Peters [16] does not consider the differences between homogeneous CMC andRIF models as significant. In the rest of this section, the homogeneous CMC equationand RIF are treated as the same model (denoted as CMC/RIF) involving Nη asthe diffusion coefficient. Now we analyse if (and how) the CMC/RIF model can beobtained a) from FM (13) and b) from transport equation (2). The conserved scalarequation (1) can be taken into account when needed for both derivations. It shouldbe remembered that y is assumed to be small in RIF but it is not necessarily small inCMC.

Since CMC/RIF is a global model, we also have to apply FM to the whole fieldof the reactive scalar. This means that the result of substitution of (5), (7) and (9)into (3)

∂Y

∂t+(

V⊥ −∇2ξD2

N∇ξ

)·∇Y −D∇ξ ·∇

(D

N∇ξ ·∇Y

)= W (13)

is assumed to be valid everywhere in the flow. This step, of course, constitutes anassumption: the small terms neglected in local FM may potentially be accumulatedinto large differences. The value of reaction rate term W can be quite small in someregions and quite large in other regions. We consider FM as being Eulerian-typebut neglect the coordinate-dependent term in (5). Equation (13) effectively replacestransport equation (2) by an approximate model and it is not easier to solve (13) than(2). Applying FM globally is the first assumption needed to derive CMC/RIF fromFM.

Substitution of Y = Q(ξ, t)+y into equations (13) and (2), followed by conditionalaveraging, yields

∂Q

∂t−Nη

∂2Q

∂ξ2+ E = W (14)


EFM =⟨∂y

∂t+(

V⊥ −∇2ξD2

N∇ξ

)·∇y −D∇ξ ·∇

(D

N∇ξ ·∇y

)|η⟩

ECMC =⟨∂y

∂t+ v ·∇y −D∇2y −D∇2Q−D

(∇ξ ·∇∂Q

∂η

)|η⟩

The values of EFM and ECMC are expected to be small if (12) is valid. The assumptionECMC ≈ 0 is the major assumption used in conventional derivation of CMC fromthe transport equations [10]. The assumption EFM ≈ 0 allows for derivation ofthe CMC/RIF equation from FM. Even if y is small, the conditional expectation ofderivatives of y which determine the order of E may still be large and the additionalassumption E ≈ 0 is needed. This point can be illustrated by the following well-knownexample [20]:

⟨∇2ξ

⟩= ∇2 〈ξ〉 ∼ ξ but

⟨∇2ξ|η

⟩∼⟨(∇ξ)2|η

⟩/ξ ∼ ξRe� ξ. First, we

consider the term 〈V ·∇y|η〉 where V ≡ v −V⊥ is the velocity component tangentialto the surfaces ξ = const. This term, which specifies transport of y along the surfacesξ = const in homogeneous turbulence, is traditionally neglected in flamelet modelling.We use now the average of the mathematical identity

Dψ(∇2y −∇ξ ·∇G−G∇2ξ

)= D∇ · (ψ∇y − ψG∇ξ)

which takes the form⟨D∇2y|η

⟩P (η)−

⟨D∇ξ ·∇

(D

N∇ξ ·∇y

)|η⟩P (η)

−⟨D2

N(∇2ξ)(∇ξ ·∇y)|η

⟩P (η) =

= D∇· (P (η) 〈∇y −G∇ξ|η〉) (15)

for G defined by G = (∂y/∂ξ)FM = (D/N)(∇ξ ·∇y). Here, ψ is the so called fine-grained pdf ψ ≡ δ(ξ − η) and δ is Dirac’s delta function. The averages involving ψare convenient for evaluations of conditional expectations 〈(·)ψ〉 = 〈(·)|η〉P (η) [20].Equation (15) is used to evaluate the difference

EFM − ECMC = D∇2Q+D 〈∇ξ|η〉 ·∇∂Q

∂η+

+D

P (η)∇ · (P (η) (〈∇y|η〉 − 〈G∇ξ|η〉))→ 0 as Re→∞ (16)

which appears to be small for large Reynolds numbers (see [20] for Reynolds-basedestimations of conditional averages). It has been shown that

Proposition 2 When turbulence is homogeneous and isotropic and the Reynoldsnumber is large, the CMC hypothesis is necessarily and sufficient for obtainingCMC/RIF equation from FM (assuming that the coordinate-dependent terms of FMare neglected).


3. Coordinate-invariant and CMC-compliant formulation of FM

The consideration of the previous section leads us to a new, coordinate-invariantformulation of FM which appears to be more consistent with CMC equations thanthe standard FM. The Flamelet Model, as it was introduced by Peters [6], is notcoordinate-invariant. The exact physical meaning of the terms involved are dependenton the choice of the surface coordinates z2 and z3. In the case of generic FM[7], only the leading terms are specified exactly and this leaves a certain freedomin interpreting the flamelet operators. From the point of view of FM asymptoticanalysis [6], non-invariance of FM with respect to the choice of z2 and z3 does notmatter: the differences in interpretations are small. However, from CMC point ofview, these differences may be accumulated into the terms of the leading order. Therecent tendency in flamelet modelling is in applying FM-style unsteady equationsglobally [14]. The features, which are not so significant for a thin reaction zone,become significant when FM is formally applied to the whole flow. Physically, it isobvious that any term which is not uniquely defined in FM should be consistentlyneglected. Here, we introduce the Coordinate-Invariant Flamelet Model (CIFM), byputting s = 0 and w = 0 in (5) and (7). We should emphasise that, in CIFM unlike inconventional FM, these terms are not neglected but are required to be exactly zeros.This requirement imposes certain restrictions on how the flamelet variables can beintroduced. The invariant operators of FM (3) take the from(

∂

∂ξ

)FM

=D

N(∇ξ ·∇) (17)(

∂

∂t

)FM

=∂

∂t+ (U ·∇) (18)

where

U ≡ v + u = V− ∇ξ

N/D

∂ξ

∂t, V ≡ v −V⊥, V⊥ ≡

v ·∇ξ

N/D∇ξ (19)

the velocity components V⊥ and V are shown in Figure 1b while u is defined in(8). Equation (17) evaluates derivative in the direction perpendicular to the surfacesof ξ = const . This definition of the derivative is similar to the definition used byKuznetsov [3] for stationary FM. Equation (18) defines the Lagrangian-type flamelettime derivative with the assistance of U — the velocity of the surfaces of ξ = constintroduced by Gibbson [21]. A similar definition of (∂/∂t)FM is used by Pitsch[22, 24] and Klimenko [23]. The invariant definition of (∂/∂t)FM is linked to the localinstantaneous velocity but not to the velocity of the frame of reference. The FlameletModel given by (3), (17) and (18) is specified by coordinate-invariant operators and themodel has exactly the same meaning in different systems of coordinates (or differentframes of reference). It should be noted that CIFM is a Flamelet Model: i.e. thedifferences between the models are asymptotically small from point of view of theflamelet analysis. The differences, however, become apparent when the conditionaltechniques are applied. In the next section, CIFM is shown to possess a specialconsistency with CMC.

If the model is applied to the whole flow, CIFM replaces the exact transportequation (2) by the approximate (but fully coordinate-invariant) model

∂Y

∂t+ U ·∇Y − D

N∇ξ ·∇

(D

N∇ξ ·∇Y

)= W (20)


When the model is applied to the regions which are not thin, the model performancecan not be guaranteed by the flamelet asymptotic analysis which needs the zone underconsideration to be thin. Under these conditions, the difference between CIFM andFM becomes essential. We also note that using the conventional FM coordinate systemξ, x1, x2 as global coordinates is problematic because of the complicated topologicalstructure of the surfaces ξ = const. Equation (20) defines the flamelet model in termsof the invariant gradient operators without using the FM coordinates.

3.1. Conditional averaging of CIFM.

In this section we demonstrate that the specific definitions in CIFM are well-chosen sothat this model complies with conditional averages of the transport equations (whichis most important for CMC). In order to explore the links between CIFM and CMC,we apply conditional averaging to CIFM. This procedure is not trivial since CMCinvolves derivatives of conditional means while conditional averaging of CIFM yieldsconditional expectations of the derivatives. Unlike for unconditional expectations, theconditional averaging and differentiation do not commute [20]:⟨

∂Y

∂t|η⟩6= ∂ 〈Y |η〉

∂t

Commuting these operators would constitute a significant mathematical error whichmay result in losses of some fundamental properties of the transport equations. Properaveraging procedure is given below. First, we introduce the so called fine-grained pdfψ ≡ δ(ξ−η) where δ is Dirac’s delta function. Second, we evaluate the time derivativeof the product Y ψ

∂Y ψ

∂t−Wψ = ψ

∂Y

∂t− ∂

∂η

(Y∂ξ

∂tψ

)= −∇ · (vψY )−D∂Y ψ∇

2ξ

∂η+

Dψ∇2ξ

(∂Y

∂ξ

)FM

+Dψ

(∇ξ ·∇

(∂Y

∂ξ

)FM

)(21)

where ∂ξ/∂t is evaluated from (1); ∂Y/∂t is evaluated from (20) and the continuityequation ∇ ·v = 0 is taken into account (see [20] for details). Obviously, the transportequation (2) is not to be used in the derivations since this equation is replaced by CIFMwhile the derivation of CMC from the exact transport equations is well-known (see[20]). Ensemble averaging is applied to (21). The equation

〈ψD(∇G ·∇ξ)〉+⟨GψD∇2ξ

⟩=∂ 〈GψN〉

∂η(22)

formulated for an arbitrary scalar, G, can be obtained by averaging the identity

D∇ · (Gψ∇ξ) = Dψ(∇G ·∇ξ) +DGψ∇2ξ − ∂GψN

∂η

and neglecting the term on the left-hand side which is small for large Reynolds numbers(see [20]). Equation (22) is used to modify the three last terms on right-hand sideof (21) by a) putting G = (∂Y/∂ξ)FM for the two last terms in (21) and b) lettingG = Y and differentiating (22) with respect to η for the remaining term (the derivativeG = (∂Y/∂ξ)FM is defined by (17)). Equation (21) takes the form

∂QP (η)∂t

+ ∇ · (〈vY |η〉P (η)) =∂J

∂η+ 〈W |η〉P (η) (23)


where

J ≡⟨ψN

(∂Y

∂ξ

)FM

⟩+ 〈D(∇ξ ·∇Y )ψ〉 − ∂ 〈Y Nψ〉

∂η=

= 2 〈D(∇ξ ·∇Y )|η〉P (η)− ∂ 〈Y N |η〉P (η)∂η

(24)

These equations represent the unclosed CMC equations which can be strictly derivedfrom the transport equations (1) and (2) for large Reynolds numbers. If w was non-zero in (18), then equation (23) would involve an additional term, ∇ · (〈wY |η〉P (η)) ,which does not appear in the exact unclosed CMC equation. Since w is determined bythe velocity and conserved scalar fields and should be independent of Y, we concludethat w must be zero. In FM, the convective terms are of smaller order and their exactrepresentation is not important. In conditional modelling these terms are of the leadingorder and are inherently present in the equations. We also note that the conditionalaveraging of the term involving u, which may seem to be small, generates the termsof the leading order in the final equation. If the coordinate-invariant formulation isnot used, FM involves another coordinate-dependent term – the term involving s in(7). This term would redefine J in (24) and add 〈Ds ·∇ (∂Y/∂ξ)FM |η〉P (η), whichshould not be present in the equation, to (23). The results obtained in this sectiondemonstrate that:

Proposition 3 The version of FM that a) is not affected by the choice of the flameletcoordinates and b) is consistent with the consequence of the transport equations in thehigh-Reynolds number flows – the unclosed CMC equation – is represented by CIFMwhich is defined by equations (3), (18) and (17).

3.2. CIFM in the vicinity of the stoichiometric surface

We introduce new curvilinear coordinates, ξ, z2, z3, t, instead of x1, x2, x3, t. In orderto preserve the conventional structure of FM and avoid appearing the derivatives ∂/∂z2

and ∂/∂z3 in the diffusion term, we must require that the lines z2 = const, z3 = constare orthogonal to the surfaces ξ = const (equations (5) and (6) demonstrate that theoperator ∂/∂ξ corresponds to s = 0 only when the orthogonality condition is satisfied).The time derivative is transformed according to(

∂

∂t

)FM

=(∂

∂t

)x

+ (U ·∇) =

=(∂

∂t

)zi,ξ

+(∂zi

∂t

)x

∂

∂zi+((

∂ξ

∂t

)x

−D∇2ξ

)D

N(∇ξ ·∇) + (v ·∇) =

=(∂

∂t

)zi,ξ

+(∂zi

∂t

)x

∂

∂zi+((

v−(v ·∇ξ)D

N∇ξ

)·∇)

=

=(∂

∂t

)zi,ξ

+(∂zi

∂t

)x

∂

∂zi+ (V ·∇)

Here we use equations (17), (18), the definition of u in (8), the definition of V in (19)and the conserved scalar transport equation (1). In this section, the superscript indexi = 2, 3 denotes the two surface contravariant components (laying on the surfaces ofξ = const) of the corresponding vectors in (ξ, z2, z3) and sum is taken over i when the


index is repeated. The subscripts of the derivative ∂/∂t specify the variables whichare kept constant when the derivative is evaluated. In the new system of coordinates,equation (20) takes the form(

∂Y

∂t

)zi,ξ

+(∂zi

∂t

)x

∂Y

∂zi+ vi

∂Y

∂zi−N ∂2Y

∂ξ2= W (25)

The tensor notations are used here since (ξ, z2, z3) is a curvilinear system ofcoordinates. Note the following equations vi = V i and vi∂Y/∂zi = V · ∇Y fori = 2, 3. Equation (25) represents full replacement of coordinates. For example,∂Y/∂t is evaluated for constant z2, z3 and ξ. Two geometric terms involving ∂zi/∂tappear in (25). We emphasize that, unlike for conventional FM, the meaning of (25)does not depend on choice of the coordinates.

The orthogonality condition, generally, does not allow to cancel the termsinvolving ∂zi/∂t in (25) everywhere in the flow but it appears to be possible to nullifythese terms at a given surface, say ξ = ξst, by selecting a proper system of coordinates(

∂Y

∂t

)zi,ξ,st

+ (V ·∇Y )st −(N∂2Y

∂ξ2

)st

= (W )st (26)

The subscript index ”st” indicates that the terms are evaluated at the stoichiometricsurface or any other surface of ξ = const. We use the ”stoichiometric notation”here since flamelet models are most likely to be applied to the stoichiometric surface.Rewriting CIFM in the from of (26) requires new coordinates Zi instead of zi,i = 2, 3. Indeed, let ai ≡ ∂zi/∂t taken on the surface ξ = ξst be non-zero. Thenew coordinates on this surface are sought in the form Zi = Zi(z2, z3, t). We use thechain differentiation and require ∂Zi/∂t, calculated for x = const, to be zero(

∂Zi

∂t

)x,st

=(∂Zi

∂t

)zi,st

+(aj∂Zi

∂zj

)st

= 0 (27)

The value ai is given and can be treated as a ”velocity”. Assuming Zi = zi att = 0, we can solve (27) by marching in time. Leaving apart the possibility ofunrestricted increase of the gradients and formation of singularities, we note that(27) provides a system of coordinates on the surface ξ = ξst as it is required in (26).Note that, generally, equation (27) can not be satisfied in the whole domain (besidesthe stoichiometric surface) since the stoichiometric values Zist and the orthogonalityto the surfaces of ξ = const restrict the values of Zi in the rest of the volume.

Practically, equation (26) can be assumed to be valid not only on the surfaceξ = ξst but also in its vicinity where fields are sufficiently smooth and ∂zi/∂t are small.The characteristic thickness of this vicinity must be smaller than the Kolmogorovlength scale, LK . If ξ = ξst is a stoichiometric surface and reactions are intense onlyin the near-stoichiometric region while outside this region the reactions are at theirequilibrium state Y = Ye(ξ), then solving FM in the near-stoichiometric region formsa closed problem. Equation (26) represents unsteady CIFM formulated in the vicinityof stoichiometric surface for a given realizations of the velocity, conserved scalar andscalar dissipation fields as functions of time. Equation (27) specifies the topologicallink between corresponding points on the surface ξ = ξst at different time moments.Since (V)st is independent of ξ, equation (26) can be written in the original form ofFM (

∂Y

∂τ

)st

−(Nc

∂2Y

∂ξ2

)st

= (W )st (28)


where the Lagrangian time τ calculated along the characteristics of the convectiveoperator determined by the equation ∂Zic/∂t = V i or, using the conventionalcoordinates, ∂xc/∂t = U. The subscript ”c” indicates values evaluated along thecharacteristics. The characteristics, obviously, lay on the surface ξ = ξst. The solutionY = Y (t, ξ, [Nc(t)]) appears to be a function of t and ξ and a functional of Nc(t)evaluated along characteristics. The averages of the reactive scalar can be found bycalculating Y for a sufficient ensemble of realizations of Nc(t) and xc(t) and selectingrealizations of xc(t) going through a certain point at a certain time moment.

4. FM and CMC in shear flows

In turbulent shear flows the general CMC equation can be simplified [8, 10, 26, 27, 20]

U∗∂Q

∂x1−N∗ ∂

2Q

∂η2= W (29)

In this equation the coefficients U∗ and N∗ can be evaluated in different ways

U∗(η, x1) =⟨v1⟩, N∗(η, x1) = 〈N∗〉 taken at 〈ξ〉 = η (30)

or

U∗ =

∫ ⟨v1|η

⟩P (η)dx2dx3∫

P (η)dx2dx3, N∗ =

∫〈N |η〉P (η)dx2dx3∫

P (η)dx2dx3(31)

The asymptotic analysis [26], using the conventional small parameter ε defined as theratio of characteristic scales across the flow and along the flow, demonstrates that bothequations are asymptotically correct, although the definitions (31) would preserve theconservation integrals and also would be better near the centre line of the flow. Infact, any reasonably selected approximations of U∗ and N∗ by average values of v1

and N linked to fixed values of the conserved scalar would generally comply with theasymptotic analysis, although some approximations can be, in a certain sense, betterthan the other.

The first attempt of using unsteady FM in shear flows was made by Mauss etal. [13] who compared the libraries of unsteady FM calculations for a thin reactionzone with reactions in a jet flow by estimating the Lagrangian residence time. Thisestimation is very approximate. In more recent publications [19, 22], the time τ of aLagrangian Flamelet Model (LFM) which can be written as

∂YLFM

∂τ−NLFM

∂2YLFM

∂ξ2= W (32)

is calculated by integrating the equation

dx1

dτ= ULFM (33)

In [19], ULFM is⟨v1⟩

evaluated at the surface 〈ξ〉 = ξst while in [22] the definition ULFM

is similar to U∗ defined by (31) and taken at η = ξst. The value NLFM is linked to N∗st– the integral average of the stoichiometric value of Nst which is similar to (but not thesame with) N∗(ξst) defined by (31). The definition of NLFM, given in [19], allows forthe following functional representation: NLFM = N∗st(x

1)Fξ(ξ). The value of YLFM canbe assumed: 1) to be equal to Q or 2) to approximate real Y by Y = YLFM(ξ, x1) + ywhere y is small. If YLFM is deterministic, the later definition is not much differentfrom the former: Q(η, x1) = YLFM(η, x1) + 〈y|η〉 ≈ YLFM(η, x1).


Unsteady FM and CIFM on one side and CMC on the other side representthe polar cases in FM/CMC modelling: FM deals with local and instantaneousvalues while CMC operates with global and averaged characteristics. The place forLagrangian Flamelet Models is somewhere in-between. On one hand, LFM effectivelyoperates with averages or conditional averages. On the other hand, LFM preservesthe traditional structure of the unsteady FM equation.

Result of comparison of LFM and the CMC shear flow equation depends on howLFM is applied to combustion problems. LFM can be treated as a local model whichspecifies evolution of YLFM in a vicinity of the stoichiometric surface. If, in the restof the flow, the evolution of YLFM is not significant, the model can be formally solvedin the whole domain of the mixture fraction. Another approach to LFM is based onthe assumption that the model is valid globally. In [13], the model is referred to asthe model of a thin reaction zone while the character of the models in [19] and [22] isnot clarified. First, we treat LFM as a local model. In vicinity of the stoichiometricvalue of the mixture fraction, the coefficients U∗ and N∗ can be approximated by theirstoichiometric values U∗st and N∗st which do not depend on η. Hence, equation (29) canbe easily transformed into (32) by introducing the Lagrangian time dx1/dτ = U∗st (thedefinitions of ULFM and NLFM(ξst) are reasonably similar to U∗st and N∗st). However,this operation can not generally be performed in the whole domain of the mixturefraction since U∗ = U∗(η, x1) depends on η. As a global model, LFM is not accuratein specifying the streamwise velocities far from the stoichiometric surface by treatingthese velocities as being the same with U∗st. This consideration can be summarised inthe following proposition:

Proposition 4 The CMC shear flow equation and LFM are the same or similar(depending on exact definitions of the coefficients) when used locally, in the vicinity ofthe stoichiometric value of the mixture fraction. These models are, generally, differentin the rest of the mixture fraction domain.

There are some other differences of CMC and Flamelet Models which should bementioned. First, the general CMC equation involves the term ∇ ·(〈vy|η〉P (η)) /P (η)which does not seem to appear in LFM. Second, the original FM model operates withinstantaneous (or, as considered in the next section, filtered) values and its Lagrangianversion should be similar to (28). In LFM, the choice was made in favour of usingaveraged (or effectively averaged) values which is similar to CMC. LFM potentiallyallows for comparison of instantaneous and averaged models under conditions whenthe equations solved have exactly the same structure. A conclusion about the effectof the fluctuations of scalar dissipation on combustion, drawn from the comparison,would be most interesting. Although it seems that sufficient attention has not beenpaid to this point in recent Flamelet developments, Pitsch and Steiner [22] give someinteresting comparison of instantaneous and conditionally averaged evaluations of thestreamwise velocity.

5. High-frequency filtering

The applicability conditions of the original asymptotic FM, which uses instantaneousstoichiometric value of the scalar dissipation, are quite restrictive. The characteristicscales of the thin flamelet region can be justifiably applied to the gradients only if thefields are smooth in this region. The turbulent fields are smooth only in the regionssmaller then the Kolmogorov length scale LK . The ideas that FM can be subject to


a certain averaging procedure aimed to extend applicability of the model were aroundfor some time. The main difficulty with this averaging is that FM should not be usedas the starting point for mathematical derivations when the flame thickness, LF , islarger than LK and the model is not valid. Nevertheless, some ideas of extendingFM by averaging are worthwhile to be considered. Kuznetsov and Sabelnikov invoke”averaging across the flame zone” [4]) while Peters [6] speaks about ”filtering out” ofthe high-frequency part of the turbulent spectrum. Actual meaning of these procedurescan be understood from the following consideration.

Let us assume that the Reynolds number is too large and the Kolmogorov lengthscale LK is too small in comparison with the flame thickness LF � LK . Thenwe consider another flow with the same macro-configuration and a lower Reynoldsnumber so that LF � LK2 is satisfied and FM is applicable to the second flow. Dueto self-similarity of the large-scale motions in developed turbulence with respect tochanging the Reynolds number, it is expected that the difference between the flows isonly in the high-frequency part of the spectrum. The high-frequency filtering actuallymeans that the first field is smoothed over the smallest scales and the small-scaledifferences between the two fields are neglected. The smoothed field would have alarger effective diffusion coefficient and a lower effective Reynolds number. Sincethe amplitude of the smoothed high-frequency fluctuations is relatively small, it isassumed that the chemical kinetics having a larger characteristic scale, LF , is notsignificantly affected by the high-frequency fluctuations which are filtered out in themodel. If FM is good for the second field, it can also be approximately applied tothe first field having the higher Reynolds number. There is, of course, price to bepaid. While FM represents an exact asymptotic limit when condition LF � LKis satisfied, the ”smoothed” FM is an approximate model. Indeed, we can dividethe continuous turbulent spectrum of the scalar fields of the first flow into large-scale fluctuations (� LF ), intermediate-scale fluctuations (∼ LF ) and small-scalefluctuations (� LF ). The large-scale motions are assumed to be the same as inthe second flow, the small-scale fluctuations are responsible for appearance of thelarger effective diffusion coefficient while the intermediate-scale fluctuations have tobe approximately neglected.

It should be noted that the ”filtered” type of FM is not obtained by filteringthe FM equation (3) for given instantaneous ξ (we can not filter something whichis not valid under specified conditions). The instantaneous field ξ is replaced byanother filtered field and this smoothed field is actually used as the basis for the FMcoordinates. Filtering does not alter the requirement that the flame width (or widthof any zone under consideration) must be small in comparison with the macroscalesof turbulence. High-frequency filtering has the same affect on CIFM as on FM. SinceCMC, unlike FM, is a Reynolds-independent model, decreasing the effective Reynoldsnumber by high-frequency filtering affects CMC only by its side-effect: reducingprecision of the terms involved.

6. Discussion of the closures

In this section, we compare the major physical assumptions needed to derive FCM,FM and CMC equations. First, we discuss what is actually assumed and what isnot assumed in CMC. In conventional derivation of the closed CMC equations [8],the process of diffusion in the conserved scalar space is assumed to be similar to aMarkov process. In this case, J must be represented by a diffusion approximation


J = AQ+ ∂(BQ)/∂η where the diffusion coefficient B and the drift coefficient A arenot known a priori. The values of B and A are limited by certain properties of thetransport equations and this results in the unique closure

J = NηP (η)∂Q

∂η−Q∂NηP (η)

∂η(34)

The assumption of analogy with a Markov process is based on the theory of the inertialinterval of turbulence (in other words, within the inertial interval, particle diffusion inthe conserved scalar space is something similar to Brownian motion) and, of course,for this assumption it is not significant whether Y = Y (ξ, t) or not. In CMC, thereaction zone is not assumed to be thin. On the contrary the CMC closure maynot be valid if the reaction zone is very thin (thinner than the Kolmogorov lengthscale, LK) [20]. The equivalent CMC equation can be alternatively derived by thedecomposition Y = Q(ξ,x, t) + y and making certain assumptions with respect to y.These assumptions are certainly valid in the trivial case of y = 0 but, as they wereintroduced in [10], explicitly allow for large y. It is found that, in shear flows, y issmaller than the unconditional fluctuations of the reactive scalar Y´≡ Y − 〈Y 〉. Thisresult is 1) not assumed in CMC but derived by asymptotic analysis which is similarto the boundary layer approach [26, 27, 20] and 2) y is still quite significant and thismay potentially affect the results of conditional averaging when averaging is appliedto the derivatives of y.

The derivation of the conventional unsteady FM [6, 13] involves two majorsteps: a) transforming the transport equations into the flamelet coordinates and b)asymptotically neglecting some of the terms for a thin zone. While step (a) is amathematical operation transforming the transport equations from one equivalentform to another, the physics of the model is represented by step (b). Originalformulation of FM does not comply with the assumption Y = Y (ξ, t) (or Y = Y (ξ)for steady flamelets). The FM solutions Y = Y (ξ, t, [Nst(t)]) (or Y = Y (ξ,Nst) forsteady flamelets) do not specify unique mapping of Y on ξ for the whole turbulent fieldsince Nst is different in different physical locations (although Y may be a function of ξoutside the reaction zone if FCM is effectively valid there). In fact, discarding globalmapping of Y on ξ is the major advantage of FM over FCM. In FM, local dependenceof Y and ξ is not presumed but derived in the vicinity of a selected point inside thereaction zone and this dependence involves another parameter – the local value of N .

Physically, the assumption Y = Y (ξ, t) can be valid in very special cases: 1) fastreactions Y = Ye(ξ) or 2) in the trivial case when ∂2Y/∂ξ2 = 0. In other cases wecan assume only that Y = YM(ξ, t) +y where YM is the modelled value (which is quitesimilar to Q when YM is deterministic) and y could be small under certain conditions.Substitution of Y = YM(ξ, t) + y into the transport equation (2) generates the spatialand temporal derivatives of ξ and y which, generally, represent the largest terms in theequation. The derivatives of ξ (except for N) disappear due to the transport equation(1) while the derivatives of y must be small in order to be neglected. In turbulentflows, smallness of y does not guarantee that the derivatives are small [20] (note that,in turbulent flows ∇y � y ).

In order to investigate the relationship between FM and CMC we formally applyFM to the whole flow. Obviously, this step is an assumption but not a strict result:equations, which have been derived for a local region, may not be valid when appliedglobally. The Flamelet Model is often referred to as the model of a thin reactionzone. The flamelet asymptotic analysis is based on certain geometrical properties of


the zone under consideration rather than on a specific values of the reaction rates.The role of high reaction rates within a thin reaction zone is in giving a physicaljustification for considering a thin zone around the stoichiometric surface. However,one can choose a zone near any surface of ξ = const (which may have a negligiblereaction rate) and formally apply flamelet asymptotic analysis to this surface. Whilethe flamelet analysis is formally valid in vicinity of any surface of ξ = const, this doesnot justify application of FM to the whole flow and, by itself, does not allow to builda closed model (similarly, a smooth function can be approximated by a constant invicinity of any point but this would not allow to assume that the function is constantin a finite region). In general, the zone under consideration must be thinner than theKolmogorov scales but, when the high-frequency filtering is used (as it is discussedin the pervious section), the zone can be widened provided it remains much thinnerthan the corresponding macroscales of turbulence. The values of reaction rates donot affect this requirement. Thus, application of FM to the whole flow represent amodelling assumption rather than a strict asymptotic result.

Coordinate-Invariant FM (CIFM) is similar to FM when used in the vicinity ofa point but gives different results when applied to the whole flow. Since the globalversion of CIFM is consistent with the exact unclosed CMC equation (23), its closurewould require the same hypothesis and the same hypothesis would have the sameaffect as in CMC. The situation with the global version of FM is more problematicsince its conditional average does not coincide with the unclosed CMC equation. Theconsistency with CMC can be achieved by neglecting the coordinate-dependent termsor/and by considering special cases when the differences are not important. In theanalysis of section 2.3, which demonstrates the equivalence of the hypotheses used toderive RIF from FM and the major CMC hypothesis, we neglected the term involvings in (7), restricted our consideration to homogeneous turbulence and neglected thetransport along the surfaces ξ = const. Pitsch et al. [14] nominated a differenthypothesis for this derivation: assuming that NRIF(ξ, t) is independent of z2 and z3.Realistic field of the scalar dissipation N is, of course, dependent on z2 and z3. If notstrictly then effectively, NRIF(ξ, t) is a conditional average of N over z2 and z3 with ξand t fixed. If NRIF(ξ, t) is an effective conditional average over z2 and z3, then what isrepresented by YRIF? It should be also some type of effective conditional average of thereal value of Y over z2 and z3 which is, physically, similar to Q. Since in [14] YRIF is notcalled as the conditional average, it should be something close to the real value of thereactive scalar: Y = YRIF(ξ, t) + y. This representation needs the level of conditionalfluctuations to be small – something what is not assumed in CMC and could have beeneasily avoided in RIF by labelling YRIF as the conditional expectation of Y . Similarconsideration, which is based on the decomposition Y = YLFM(ξ, x1)+y, can be appliedto LFM and lead us to the same conclusions. The possibility to write the unsteadyFM equation in terms of effective conditional expectations needs assumptions whichare similar to the closure used in CMC. It is remarkable that this physical similaritycan be proved strictly as it is done by the analysis of section 2.3.

7. Conclusions

We suggest a new formulation of FM – CIFM (Coordinate-Independent FM). Thedifferences of CIFM and FM are small when considered from the point of view of localflamelet modelling (although CIFM, unlike FM, is independent of the choice of theflamelet coordinates). The advantage of CIFM can be observed when the models are


formally applied to large regions rather than to thin zones for which FM is derived.In this case, the small differences between the models are accumulated into the termsof the leading order. CIFM complies with the unclosed CMC equation while a generalFM does not. For large Reynolds numbers, the unclosed CMC equation is the directconsequence of the transport equations. The convective terms present in CIFM specifyLagrangian-type derivatives. It is shown that, using the Lagrangian-type time whoseexact expression is introduced in the model, CIFM can be written in the conventionalFM from but this can be done only in the vicinity of a chosen surface of constantmixture fraction. In order to form a practical global model, the conditional averageof CIFM needs closures which are similar to ones used in CMC.

It is shown that, in homogeneous isotropic turbulence, the terms of another newversion of FM – RIF – must coincide with the terms of CMC equation and, thus,RIF represents a model equivalent to CMC. In inhomogeneous turbulence, CMC isnot equivalent to RIF; the CMC equation involves the convective terms while RIFdoes not. It is shown that the derivation of RIF from the original formulation of FMrequires an assumption which is, for homogeneous turbulence, equivalent to the CMCclosure when the Reynolds number is high.

The most recent versions of FM/CMC-type models – the Lagrangian FlameletModels (LFM) – are similar to the CMC shear flow equation (a special case of thegeneral CMC equation) near the stoichiometric value of the mixture fraction but,generally, the models are different in the rest of the flow. The major source of thisdifference is in the LFM assumption that the streamwise velocities in the whole floware the same as their stoichiometric values.

8. Acknowledgment

The author thanks Prof. Bilger and Dr. Pitsch for useful comments. This work issupported by the Australian Research Council.

[1] S.B. Pope. Pdf methods for turbulent reactive flows. Progress in Energy Combustion Sci.,11:119–192, 1985.

[2] R.W. Bilger. Turbulent flows with nonpremixed reactants. In P.A. Libby and F.A. Williams,editors, Turbulent Reacting Flows. Springer Verlag, 1980.

[3] V.R. Kuznetsov. The influence of turbulence on formation of large superequilibriumconcentrations of atoms and free radicals in diffusion flames. Fluid Dynamics, 17:815–820,1982.

[4] V.R. Kuznetsov and V.A. Sabelnikov. Turbulence and combustion. Hemisphere, 1989.[5] N. Peters. Local quenching due to flame stretch and non-premixed turbulent combustion. Comb.

Sci. and Tech., 30:1–17, 1983.[6] N. Peters. Laminar diffusion flamelet models in non-premixed turbulent combustion. Progr.

Energy and Combust. Sci., 10:319–340, 1984.[7] N. Peters. Turbulent Combustion. Cambridge University Press, 2000.[8] A.Y. Klimenko. Multicomponent diffusion of various scalars in turbulent flow. Fluid Dynamics,

25:327–334, 1990.[9] R.W. Bilger. Conditional moment methods for turbulent reacting flow using crocco variable

conditions. Report TN F-99, Dept. Mech. Eng., University of Sydney, 1991.[10] R.W. Bilger. Conditional moment closure for turbulent reacting flow. Phys. Fluids A, 5:436,

1993.[11] F.A. Williams. Combustion theory. Addison-Wesley, 1985.[12] S.P. Burke and T.E.W. Schumann. Diffusion flames. Ind. Eng. Chem., 20:988–1004, 1928.[13] F. Mauss, D. Keller, and N. Peters. A Lagrangian simulation of flamelet extinction and re-

ingition in turbulent jet diffusion flames. In XXIII International Combustion Symposium,pages 693–698. The Combustion Institute, 1990.


[14] H. Pitsch, Y.P. Wan, and N. Peters. Numerical investigation of soot formation and oxidationunder diesel engine conditions. SAE Paper 952 357, 1995.

[15] H. Pitsch, H. Barths, and N. Peters. Three-dimensional modeling of nox and soot formation indl-diesel engines using detailed chemistry based on the interactive flamelet approach. SAEPaper 962 057, 1996.

[16] N. Peters. Comment. Combust Flame, pages 675–676, 1999.[17] D.C. Haworth, M.C. Drake, S.B. Pope, and R.J.Blint. The importance of time-dependent flame

structures in stretched laminar flamelet models for turbulent jet diffusion flames. In XXIIInternational Combustion Symposium, pages 589–597. The Combustion Institute, 1988.

[18] W.E. Mell, V. Nilsen, G. Kosaly, and J.J. Riley. Investigation of the closure models fornonpremixed turbulent reacting flows. Phys. Fluids, 6:1331–1356, 1994.

[19] H. Pitsch, M. Chen, and N. Peters. Unsteady flamelet modelling of turbulent hydrogen-airdiffusion flames. In XXVII International Combustion Symposium, pages 1057–1064. TheCombustion Institute, 1998.

[20] A.Y. Klimenko and R.W. Bilger. Conditional moment closure for turbulent combustion.Progress in Energy and Combustion Science, 25:595–687, 1999.

[21] C.H. Gibson. Fine structure of scalar fields mixed by turbulence. Phys. Fluids, 11:2305–2315,1968.

[22] H. Pitsch and H. Steiner. Large-eddy simulation of a turbulent piloted methane/air diffusionflame (Sandia flame D). CTR Annual Research Briefs, Center for Turbulence Research,Stanford University,, 1999.

[23] A.Y.Klimenko. The coordinate-invariant formulation of unsteady flamelet model. Report2000/04, Dept. Mech. Eng., University of Queensland, 2000.

[24] H. Pitsch and H. Steiner. Large-eddy simulation of a turbulent piloted methane/air diffusionflame (Sandia flame D). Phys. Fluids, 12:2541–2554, 2000.

[25] N. Peters. Private communication, 1999.[26] A.Y. Klimenko. Note on conditional moment closure in turbulent shear flows. Phys. Fluids,

7:446, 1995.[27] A.Y.Klimenko and R.W.Bilger. Some analytical and asymptotic results for pdfs and conditional

expectations in turbulent shear flows. In 10th Symposium on Turbulent Shear Flows,volume 3, pages Sess.31, 25–29. Pennsylvania State University, USA, 1995.


FIGURE CAPTURE

FIGURE 1. Schematic of the vectors used in the definitions of the flameletoperators.

��const

z ,z =const2 3

��const

�

�

�

�

V�

v

s

e

V

u

u

w

U

�

�

�

a)

b)

1

on relation of the conditional moment closure and unsteady...

Documents