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References 33

The value of an experiment will be greatly increased if several different ﬂow proper-

ties, preferably including velocity, can be measured simultaneously, with good space

and time resolution. A sufﬁciently large database of this kind will allow co-variances

to be estimated and related to ﬂow conditions.

DNS aims to provide a fully resolved numerical solution to the complete system

of equations of a chemically reactive ﬂow, largely avoiding the need for empirical

models. The output from such a simulation can be used to evaluate the unknown

statistical quantity that is the subject of a speciﬁc submodel, together with the combi-

nation of mean variables forming the submodel, allowing the validity of the submodel

to be tested. Many comparisons of this kind can be found in the literature, from which

it is clear that DNS comparisons provide a very powerful tool for model develop-

ment and testing. Examples of the use of DNS data in this way may be found in

various sections of Chapter 2. Nevertheless, the limitations of DNS must be kept in

mind when it is used for model testing: The Reynolds number is usually very low

in comparison with those of practical interest, the ﬂow geometry is greatly simpli-

ﬁed, and the chemical kinetic mechanism is often simpliﬁed to make the calculation

feasible.

Users of RANS or LES simulation codes to predict ﬂow ﬁelds involving turbu-

lent combustion frequently need to be reminded that their codes contain numerous

empirical submodels that have, at best, been tested in a few corners of their multi-

dimensional parameter space. Model developers must do more to test their models

over a wider range of conditions. They must work towards a goal to quantify the

range of validity of their models with greater accuracy and minimise, if not eliminate,

the empiricism in current modelling.

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40 Fundamentals and Challenges

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2 Modelling Methods

2.1 Laminar Flamelets and the Bray, Moss, and Libby Model

By K. N. C. Bray

This section is concerned with premixed turbulent combustion in circumstances in

which the heat release reactions can be considered ‘fast’, a situation that is common in

practical combustion systems. Several dimensionless parameters can help to identify

the meaning of fast chemistry in this context. A Damk

ohler number, Da = τ

/τ

compares a characteristic turbulent ﬂow time scale τ

k/˜ with a laminar ﬂame

time τ

= δ/S

. As explained in Chapter 1, when Da  1, combustion occurs in thin

wrinkled interfaces separating unburned reactants from fully burned products. If also

a Karlovitz number Ka = τ

/τ

is sufﬁciently small, where τ

is the Kolmogorov

time characterising the smallest eddies in a turbulent ﬂow, these reacting interfaces

will resemble unstretched laminar ﬂames. It is then possible to divide a Reynolds-

averaged Navier–Stokes (RANS) simulation or large-eddy simulation (LES) into

two separate parts: a chemical kinetic analysis of a laminar ﬂame and a subsequent

ﬂuid ﬂow calculation. A third dimensionless quantity is

N = τS

rms

, where τ =

) − 1 is a heat release parameter and T

and T

are the temperatures in

unburned reactants and fully burned combustion products, respectively. As explained

in Chapter 1, gradient scalar transport expressions are found [1] to fail unless

N ≤

O(1). Mura and Champion [2] showed that the velocity gradient in a laminar ﬂame

is greater than the typical velocity gradient that is due to turbulence if

N > Da

−1/2

;

dissipative effects that are due to gradients in laminar ﬂames can then become

signiﬁcant and should be taken into account.

Second-moment or Reynolds stress turbulence models [3, 4] were ﬁrst developed

in order to avoid the gradient transport and near-isotropic turbulence assumptions

of k– models in non-reactive ﬂow calculations and were subsequently applied to

turbulent combustion. The ﬁrst such combustion model to fully exploit the limit

Da  1 in premixed turbulent combustion was proposed by Bray, Moss, and Libby

(BML) [5–8], and is reviewed here. The BML model has been adapted to predict

combustion processes in complex engineering ﬂows [9], in which its performance is

similar to that of other second-moment models. However, by partitioning the prob-

ability density function (PDF) into separate contributions from reactants, products,

42 Modelling Methods

and burning zones, it also provides a mechanism for taking account of the fact that

burning occurs in thin ﬂamelets, leading to very sudden changes in gas density and

other ﬂow properties.

This model, which is discussed in Subsection 2.1.2, provides an intuitive frame-

work for describing such systems and for identifying interactions between ﬂow and

combustion. ‘Intermittency’ effects that are due to the passage of thin-ﬂame reac-

tion zones past a ﬁxed point are found to be important: A measurement at such a

location within the ﬂame brush would reveal the alternate appearance of unburned

and fully burned gases separated by brief periods of chemical change that are due

to the passage of a thin ﬂamelet. Application of the BML model to combustion in a

ﬂow approaching a stagnation point is presented in Subsection 2.1.3 as an example.

This is followed in Subsection 2.1.4 by a review of information concerning gradi-

ent and counter-gradient scalar transport, and Subsection 2.1.5 discusses laminar

ﬂamelet models. A simple laminar ﬂamelet mean-burning-rate model is developed

in Subsection 2.1.6. The aim is to illustrate relevant physical processes rather than

to develop a detailed and accurate method of prediction, which can be found in

[11] and other references subsequently cited. This section is concerned with RANS

calculations. However, it should be noted that some of the issues identiﬁed here also

arise in LES closures; see, for example, the LES ﬂame surface density (FSD) analysis

of Chakraborty and Cant [10].

2.1.1 The BML Model

Several reviews of the basic BML model are available [8, 11], and only a brief

summary is presented here, followed by a review of some more recent developments.

The essential features are as follows.

(1) THERMOCHEMISTRY. An adiabatic, low-Mach-number ﬂow is considered, char-

acterised by a Lewis number of unity. A progress variable c(x, t) is introduced and

is deﬁned here as a normalised temperature, so c(x, t) = [T (x, t) − T

]/(T

− T

Alternatively, c can be deﬁned in terms of the total mass fraction of combustion

products. An essential assumption is that all thermochemical variables, including

the rate of heat release, can be expressed as functions of c alone, and this requires

changes in pressure to be small enough to be thermodynamically negligible. Addi-

tionally the BML model assumes constancy of mean molecular weight and speciﬁc

heat, and the thermodynamic state of the mixture can then be reduced to

ρ(x, t) =

1 + τc(x, t)

. (2.1)

However, more realistic equations of state can be accommodated without difﬁculty,

so long as the state remains as a function of c alone. It should also be noted that

non-unity Lewis number effects, which are not considered here, can be important in

premixed combustion: see, for example, Chakraborty and Cant [12].

(2) MODELLED EQUATIONS. The Favre mean ﬁrst-moment transport equations (see

Sections 1.5 and 1.7) for an assumed adiabatic low Mach number ﬂow, at a high