Swaminathan N., Bray K.N.C. (eds.) Turbulent Premixed Flames

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2.4 Transported Probability Density Function Methods 103

for turbulent reacting ﬂows [91, 227]. The terms relating to chemical reaction appear

in closed form as convection in scalar space, and, accordingly, kinetically controlled or

inﬂuenced phenomena can in principle be readily included. A particularly attractive

consequence is that independently developed chemical mechanisms can be applied

without further simpliﬁcation. The formulation of more complete closures is non-

trivial as inconsistencies in submodels may lead to accuracy or realisability problems

(e.g. [17, 228, 229]), be it in the context of LES- or moment-based methods. The

transported PDF approach has the potential to overcome difﬁculties related to the

coupling of chemical reaction and spatial transport (turbulent convection) and may

be used as subﬁlter-scale models for LES by means of ﬁltered density function

(FDF) methods. The coupling between turbulent transport and chemical reaction

becomes signiﬁcant in view of the delicate balance of gradient and non-gradient

scalar transport in turbulent premixed ﬂames [7, 17, 44, 228 ].

Most previous work on transported PDF closures for ﬂames featured non-

premixed combustion in which, typically, scalar transport is more readily addressed.

For example, work on piloted diffusion ﬂames (e.g., the Sandia ﬂame series [230])

enabled stable ﬂames and those undergoing extinction or reignition to be studied

through a combination with augmented reduced [231–234] or detailed [235] chemi-

cal reaction mechanisms. Studies featuring more complex ﬂow ﬁelds were also per-

formed. For example, Muradoglu et al. [236] modelled a bluff-body stabilized ﬂame

using a joint scalar approach by applying time averaging after a statistically station-

ary solution was obtained. Kuan and Lindstedt [237] performed computations of a

bluff-body ﬂame [238, 239] by using a hybrid Monte Carlo–Finite-volume algorithm

with the velocity ﬁeld closed at the second-moment level. The thermochemistry was

computed by means of a systematically reduced mechanism featuring 300 reactions,

20 solved, and 28 steady-state species. Studies of ﬂames featuring auto-ignition [235,

240], signiﬁcant local extinction [232–234], or both, suggest that ignition tends to

occur in a partially premixed mode and that the transported PDF approach can

reproduce experimental data with encouraging accuracy. Bidaux et al. [241]useda

scalar PDF approach and a particle-interaction mixing model with the mean reaction

rate determined from the PDF solution to predict a partially premixed Bunsen ﬂame.

The attraction of extending moment-based methods to include ﬁnite-rate chem-

istry effects through the application of transported PDF methods to premixed tur-

bulent ﬂames is of increasing practical interest. For example, ﬂows found in burners,

gas turbine combustion chambers, and reciprocating engines cover a wide range of

Damk

ohler (Da) and Karlovitz (Ka) numbers and are increasingly characterised by

strong interactions between turbulence and ﬁnite-rate chemistry because of changes

in burning modes [242–244]. One implication is the need to extend the knowledge

gained from the application of high-Damk

ohler-number-based approaches to en-

compass an extended range of conditions. The latter may include (re)ignition phe-

nomena, dilution effects, local or global ﬂame extinction, non-adiabatic conditions,

and the presence of a multitude of chemical time-scales (e.g., the fast heat release

vs. the relatively slow formation of pollutants).

Comparatively few studies of premixed turbulent combustion were performed

with transported-PDF-based methods. Pope and Cheng [245] considered the evo-

lution of spherical ﬂame kernels by using a simpliﬁed Langevin model [91] to ac-

count for the effects of viscous dissipation and the ﬂuctuating pressure gradient.

The mean pressure gradient acted upon the ﬂuid irrespective of the instantaneous

104 Modelling Methods

density, and thus preferential acceleration effects could not be observed. Anand and

Pope [66, 226] extended the work and applied a joint velocity–reaction progress-

variable formulation to calculate steady premixed ﬂames in the ﬂamelet regime of

combustion. The viscous dissipation of velocity and the ﬂuctuating pressure gradient

were incorporated by means of a particle-interaction model featuring stochastic

reorientation [91]. The chemical source term incorporated the coupled effects of

chemistry and molecular diffusion. The work showed that the transported PDF ap-

proach can reproduce key features of the BML ﬂamelet approach developed by

Bray et al. [7], as reviewed in earlier sections.

ulek and Lindstedt [246] computed steady and transient 1D ﬂames for a range

of turbulent Reynolds (Re

) numbers by using an elliptic formulation closed at

the joint velocity–scalar level. The work extended past efforts through the applica-

tion of the comprehensive Haworth and Pope [247, 248] and Speziale, Sarkar, and

Gatski [249] closures for velocity statistics combined with the binomial Langevin

model [250] for molecular mixing. It was shown that the approach can reproduce con-

ditional velocities of burned and unburned gases as well as the change in anisotropy

across the turbulent ﬂame brush. It was also shown that the predicted slip velocity

− u

) at the leading edge (

c → 0) compared reasonably well with the theoretical

analysis of Bray [147]. The impact of alternative formulations for the contributions

made by small-scale diffusion processes was analysed, and it was found that direct

extraction of source terms from laminar ﬂamelet computations provided a suitable

means of predicting the heat release at moderate turbulence intensities. It was also

shown that the conventional model for the SDR leads to the expected linear scaling

of the turbulent burning velocity with turbulence ﬂuctuations.

Lindstedt and V

aos [251] subsequently investigated the effects of mixing mod-

els on the relationship between computed turbulent burning velocities and ﬂame

structures. The closures considered included the Linear–Mean–Square–Estimate

(LMSE) model [252], the nonlinear integral binomial sampling [253], and modi-

ﬁed Curl’s [254] models as well as a binomial Langevin-based formulation [250, 255].

It was shown that the modiﬁed Curl’s and binomial Langevin models can be expected

to produce agreement with experimental data whereas the other models suffered

from signiﬁcant shortcomings.

More recently, Lindstedt and V

aos [211] extended their past work through an

investigation of the effects of closure approximations for the SDR on premixed

turbulent ﬂame structures and on the scaling relationship between turbulence ﬂuc-

tuations and the turbulent burning velocity. It was shown that a multiscale extension

to the conventional closure was capable of reproducing turbulent burning velocities

across a wide range of turbulence intensities. The study also considered the piloted

methane–air ﬂames investigated experimentally by Chen et al. [256], with the most

turbulent ﬂame featuring a Re  52500 and a Da ∼ 1. The transported PDF ap-

proach was closed at the joint scalar level with the velocity ﬁeld computed at the

second-moment level [249]. The thermochemistry was closed using an augmented

systematically reduced C–H–O mechanism featuring 141 reactions, 15 solved, and

14 steady-state species. The extended multiscale mixing time-scale closure [211]was

also applied by Lindstedt et al. [257] in the context of partially premixed ﬂames at

high Reynolds numbers. It was f ound that the standard time scale expression had a

tendency to produce excessive extinction.

2.4 Transported Probability Density Function Methods 105

Few studies featuring more complex ﬂow ﬁelds were pursued. However,

Cannon et al. [258] computed a bluff-body stabilised lean premixed methane–air

ﬂame by using an elliptic calculation procedure combined with the k-ε model for

the velocity ﬁeld and with the molecular mixing term closed by use of the LMSE

model. Sabel’nikov and Soulard [259] applied a stochastic Eulerian ﬁeld method to

compute a stoichiometric methane–air ﬂame stabilised in a backward-facing step

conﬁguration using the k–ε closure coupled with the LMSE model for molecular

mixing and a one-step chemical reaction. In summary, the majority of past studies

focussed on the application of simpliﬁed chemistry combined with different levels

of closure for the velocity and scalar ﬁelds. In the subsequent subsections, some of

the key closure aspects are reviewed and their potential impact assessed.

2.4.1 Alternative PDF Transport Equations

The brief outline just presented suggests that work remains on formulating ap-

proaches for the treatment of terms appearing as part of the application of the trans-

ported PDF approach to premixed turbulent ﬂames. It is evident that additional

complexities associated with turbulent transport arise for premixed ﬂames and that

in many cases very well-established closures and simple approximations for the

modelling of chemical reaction and molecular mixing have been applied. The sub-

sequent discussion considers the impact of closure assumptions on premixed and

partially premixed ﬂames at low and high Da in the context of joint s calar and joint

velocity–scalar methods. The impact of turbulent transport, molecular mixing mod-

els, unclosed pressure-related terms, and different levels of accuracy in the applied

thermochemistry are considered.

The JPDF of composition and velocity in low-Mach-number ﬂows is considered

for premixed and partially premixed ﬂows. In the latter case, the composition is

described by two (c, Z) scalars, where Z denotes a conserved and c a chemically

reacting scalar. The pressure is assumed to be thermochemically and thermodynam-

ically constant. Under such conditions the evolution equation for the mass density

function (MDF) F

zcu

(ξ, ζ, U, x; t) = ρ(ξ, ζ)f

zcu

(ξ, ζ, U; x, t), where f

zcu

is the JPDF of

composition and velocity, can be derived from basic conservation laws [91]:

∂F

zcu

∂t

∂u

zcu

∂x

ρ(ξ, ζ)

∂

∂x

∂F

zcu

∂U

−

∂S

(ξ, ζ)F

zcu

∂ζ

∂

∂U



ρ(ξ, ζ)



−

∂τ

k

∂x



∂p



∂x



ξ, ζ, U



zcu



∂

∂ξ



ρ(ξ, ζ)



∂J

∂x



ξ, ζ, U



zcu



∂

∂ζ



ρ(ξ, ζ)



∂J

∂x



ξ, ζ, U



zcu



. (2.144)

In the MDF equation, p is the pressure, τ

k

is the viscous stress tensor, S

is the chem-

ical reaction source term of the scalar c, and J

and J

are the molecular-diffusion

ﬂux vectors of the scalars Z and c, respectively. The notation



A|Q



expresses the

106 Modelling Methods

expected value of A conditioned on the event Q, and the prime



indicates the dif-

ference from the expected value (i.e., φ



= φ −





). Mean and ﬂuctuating quantities

based on Favre averaging are denoted as φ =



φ + φ



= ρφ/ρ + φ



. The quantities

ξ, ζ, and U are sample-space counterparts of the actual values Z, c, and u, respec-

tively. The MDF conservation equation contains, in closed form, the effects of mean

and turbulent convection, the mean pressure gradient acceleration, and chemical re-

action. Unclosed terms express the molecular mixing of scalars, viscous dissipation

of momentum, and the ﬂuctuating pressure gradient. In addition, the mean pressure

ﬁeld has to be solved in a manner that guarantees consistency of the MDF with mass

conservation. The corresponding MDF equation for a fully premixed case is [the

notation is the same as applied in Eq. (2.144)]

∂F

∂t

∂u

∂x

ρ(ζ)

∂

∂x

∂F

∂U

−

∂S(ζ)F

∂ζ

∂

∂U



ρ(ζ)



−

∂τ

k

∂x





c = ζ, u = U





∂

∂U



ρ(ζ)



∂p



∂x



c = ζ, u = U





∂

∂ζ



ρ(ζ)



∂J

∂x



c = ζ, u = U





. (2.145)

The application of the preceding equations to different cases is reported in order to

explore the importance of different closure elements. The further simpliﬁcation of

the approach is also considered through the application of the corresponding joint

composition PDF equation in combination with alternative closures for the velocity

ﬁeld. The joint composition PDF results from integration of Eq. (2.145) over the

full velocity space and the decomposition of convection in physical space into an

unconditional mean part and a conditional intensity part:

∂

P(!)

∂t

∂

ρ ˜u

P(!)

∂x



α=1

∂

∂!



ρS

(!)

P(!)



=−

∂

∂x



ρ<u



|ϕ = !>

P(!)





α=1

∂

∂!



∂J

l,α

∂x

ϕ = !







P(!)



. (2.146)

At this closure level, a model is required for the turbulent transport of the PDF

in physical space and, typically, a gradient-diffusion-type closure is used,

u



|ϕ = !

P(!) =

∂

P(!)

∂x

, (2.147)

where ν

is the ‘standard’ kinematic eddy-viscosity coefﬁcient and σ

is the turbulent

Prandtl or Schmidt number. It is also possible to formulate a second-moment-based

closure for the transport of the PDF in physical space. Dopazo [260] suggested a clo-

sure in which the conditional intensity terms are approximated with a deterministic

LMSE-type model:

u



|ϕ = !

P(!) =





− ˜ϕ

)





P(!). (2.148)

2.4 Transported Probability Density Function Methods 107

This model was also proposed in the context of moment-based closures by Borghi

and Dutoya [164] and Libby [261], with possible extensions outlined by Pope [227].

For a single scalar, the conditional expectation may be written in the familiar form as

u



|c = ζ

P(ζ) =





(ζ −





P(ζ). (2.149)

Swaminathan and Bilger [262] evaluated the preceding linear model against DNS

data obtained for lean hydrogen ﬂames with u



 30. Two alternative formula-

tions based on gradient and PDF models were also considered. The work showed

that the preceding model captures DNS data comparatively well, whereas the alter-

native formulations produced less accurate results – possibly because of insufﬁcient

samples. The application of the model in the context of transported PDF closures

was very limited and is explored in the next subsection. The impact of the treatment

of molecular mixing is also discussed further in the context of speciﬁc examples.

2.4.2 Closures for the Velocity Field

The velocity ﬁeld may be obtained through LES or at the equivalent second-moment

or eddy-viscosity levels. In view of the anisotropy typical of premixed turbulent

ﬂames, the latter is considered mainly for comparison purposes, and the treatment

of pressure-related terms has to be consistent across scales. Analysis of the Reynolds

stress equations show [263] that, in variable-density, low-Mach-number turbulent

ﬂows, the effects of the anisotropic part of the ﬂuctuating pressure gradient and tur-

bulence dissipation are redistributive. In other words, the terms are traceless in ten-

sorial form and therefore do not appear in the turbulent kinetic energy (



k =





/2)

conservation equation. The latter condition is automatically satisﬁed by the gener-

alised Langevin model (GLM) of Haworth and Pope [264], and the velocity statis-

tics (viscous dissipation of momentum and ﬂuctuating pressure gradient) presented

here are based on their approach and other suitably modiﬁed closures, such as the

Lagrangian version [265] of the Speziale–Sarkar–Gatski (LSSG) model [249]. How-

ever, the intention is not to evaluate speciﬁc second-moment methods, but to study

the impact of closures at the equivalent ﬁrst- and second-moment levels alongside

other model aspects. The k-ε and GLM models are thus used as representative

examples of each class.

In the presence of volumetric changes, the so-called ‘pressure–dilatation’ term





∂u



/∂x



appears. There is some evidence that pressure–dilatation may not be

important when the turbulence intensity u

rms

(and thus the turbulent burning ve-

locity u

) is much larger than the laminar burning velocity u



[266]. Furthermore,

issues with respect to consistency and realizability remain [17], and the matter is

subsequently treated separately. These considerations lead to a Lagrangian model

of the form

(p )



−

∂





∂x

+ G

k

(p )



−u



)



dt +(C

ε)

1/2

, (2.150)

where the superscript

(p )

denotes the properties of a Lagrangian particle (solution

methods are discussed in Subsection 2.4.5), W

is an isotropic Wiener process [267],

ε is the dissipation rate of



k, and C

is assumed to be a constant C

= 2.1[265].

108 Modelling Methods

The latter value was discussed by Pope [268], and it has been shown, by compar-

ison with DNS data, that some uncertainties prevail. However, in this section the

‘standard’ value is retained, and the tensor of modelling coefﬁcients G

k

is given in

Appendix 2.A.

A modiﬁcation of the ‘standard’ Lagrangian velocity statistics model is also in-

vestigated in order to assess the importance of solving the joint composition–velocity

statistics by introducing a modiﬁcation so that the acceleration by the mean pres-

sure gradient depends on the mean rather than on the local instantaneous density.

Equation (2.150) then changes to

(p )



−





∂





∂x

+ G

k

(p )



−u



)



dt +(C

ε)

1/2

. (2.151)

This formulation is equivalent to a second-moment closure in which (1) the pressure–

work term –







∂





/∂x



in the Reynolds stress conservation equations – and

(2) the terms –







∂





/∂x

and –







∂





/∂x

– in the turbulent scalar ﬂux

conservation equations are neglected. The terms are mainly responsible for the

observed ‘counter-gradient’ (or non-gradient) turbulent transport in ﬂames [44], and

comparisons of the result obtained with models (2.150) and (2.151) should indicate

the importance of this effect in the context of premixed and partially premixed

ﬂames, as s ubsequently discussed.

2.4.3 Closures for the Scalar Dissipation Rate

The closure for the mixing frequency is directly related to the closure for the SDR

(ε

). Fox [269] argues that in premixed turbulent ﬂames a strong dependence between

the joint SDR 

|! and the chemical source term can be expected and that the

majority of mixing models omit this feature. The customary scaling relationship that

has a direct impact on the treatment of the molecular mixing term is given by

−1

ε



2

ε



−1

. (2.152)

This relationship provides a direct link to moment-based closures for premixed tur-

bulent ﬂames. Bray [147] showed that any SDR model corresponds to a speciﬁc

mean-reaction-rate model at the high-Damk

ohler-number limit. Hence the assump-

tion of a constant value for the velocity–scalar time-scale ratio results in an EBU-type

expression for the mean reaction rate, as suggested by Spalding [270]:





∝





ε





−1



2





−1



2





−1



2

. (2.153)

The assumption of a bimodal PDF (



2



c(1 −



c)) leads to the standard BML

form [147] with values of C

4[17]. Transport-based closures for the SDR are typi-

cally based on non-reacting ﬂows and have been derived by Jones and Musonge [167]

and Borghi and Mantel [165], among others. A detailed review of such approaches

can be found in Section 2.3 of this book. The scaling is typically exclusively based on

large-scale (integral) properties with added terms featuring a dependency on lami-

nar ﬂame properties. Kuan et al. [210] used a fractal-based approach [271] to derive

2.4 Transported Probability Density Function Methods 109

a multiscale expression for the time-scale ratio to alleviate the former limitation:

−1

ε



2

∝





ε



. (2.154)

Yeung et al. [272] investigated the effects of ε

on material surfaces for cases in

which u

 v

and suggested that the Kolmogorov velocity provides an appropriate

scaling. Bray et al. [273] considered closures for the chemical reaction rate and

concluded that the majority of algebraic and transport-equation-based models could

not be calibrated to perform satisfactorily irrespective of the choice of modelling

constants. The simple form of Eq. (2.154) was found to be an exception. The model

implies a monotonic increase of the u

ratio with the Da number in agreement

with the experimental study of O’Young and Bilger [157], which also indicated that

the velocity–scalar ratio of time scales increases by almost an order of magnitude as

Da →∞. A disadvantage is that τ

−1

becomes zero for a passive scalar, and Lindstedt

and V

aos [211] proposed a simple modiﬁcation as

−1

ε



2



1.0 + C

∗







ε



. (2.155)

The form of Eq. (2.155) ensures a consistent scaling behaviour of turbulent burning

velocities for ﬂames in the ﬂamelet regime of combustion. The expression also

complies with the standard approach for the limiting case of a passive scalar. The

constant C

∗

was assigned a value of 1.2 [210]. An extensive review of alternative

algebraic expressions was presented in Section 2.3, though these have not as yet been

evaluated in the context of serving as closure elements for transported PDF models.

2.4.4 Reaction and Diffusion Terms

The terms responsible for reaction and diffusion may be expressed in the customary

form as

∂

∂ψ



− ρ(ψ)S

(ψ) +

∂J

∂x







∂

∂ψ



(ψ)f



. (2.156)

The quantity h

(ψ) appears in closed form for laminar ﬂamelets. In general, two

different aspects need to be considered in the modelling of the conditional expecta-

tion of the molecular scalar ﬂux term in Eq. (2.156), as discussed by Haworth [275].

The ﬁrst is transport in physical space, and the second is mixing in composition

space. At high Reynolds numbers and for Sc and Pr numbers of the order of unity,

the term describing transport in physical space is typically small. Furthermore, the

implicit coupling present in the laminar ﬂamelet assumption will weaken at lower

Damk

ohler numbers and can be expected to break down in the well-stirred reactor

regime. However, for premixed combustion at high Damk

ohler numbers the term

can be important and the effect may be included by means of the addition of a

random-walk term in the equation for the particle position in physical space [275].

The approach provides a correction for the mean scalar composition at the expense

of a spurious production term in the scalar variance equation [275, 276]. An alter-

native approach, which does not lead to such difﬁculties [275], is to add an explicit

deterministic molecular-transport term in the manner implied by Eq. (2.156). The

impact of such approximations at the leading edge of the ﬂame was explored by

110 Modelling Methods

Lindstedt et al. [277], and McDermott and Pope [278] provided an extensive analysis

of diffusion approximations in the context of FDF methods.

2.4.5 Solution Methods

Solution techniques for transported PDF methods were recently reviewed by

Haworth [275], and the following discussion is conﬁned to the approach applied

to obtain the illustrative examples shown. The PDF evolution equations were solved

with a Lagrangian-particle-based Monte Carlo approach combined with fractional

step method [91] featuring a ﬁrst-order (Euler) time integration. The integration of

particle position and velocity follows the procedure of Haworth and Pope [279]. The

masses of modelling particles were modiﬁed at each time step to maintain a uniform

particle density. For axisymmetric ﬂows the mass of each particle (p ) is therefore

approximately proportional to the value r

(p )

[r is the distance from the symmetry

axis and ρ

(p )

is the local instantaneous density].

The mean momentum and mass conservation equations were solved with a

Pressure Implicit with Split Operator (PISO)-based [17, 280] ﬁnite-volume solver,

and the governing equations are given in Appendices 2.B and 2.C. The time dis-

cretisation was ﬁrst-order implicit and second-order central differencing was used

for the diffusive terms. The convective terms were discretised by van Leer’s to-

tal variation diminishing (TVD) scheme with a limited diffusion antiﬂux modiﬁed

for implicit time discretisation [281 ]. The stochastic and ﬁnite-volume parts of the

algorithm were coupled as follows. The scalar statistics obtained from the Monte

Carlo PDF solution were used to evaluate the mean density (i.e., to close the mean

equation of s tate) in the pressure–momentum solver. Depending on the method

used (ﬁrst- or second-moment closure), the PDF solution was used to provide

the unclosed correlations in the momentum as well as



k and  equations (







and







). The ﬁnite-volume solution for the mean velocity ﬁeld was then used in

the Monte Carlo part to provide the mean pressure gradient and turbulence time

scales.

Following each time step, the particle velocities were adjusted so that the mean

velocity and



k in the ﬁnite-volume and Monte Carlo solutions were consistent [282].

To decrease the statistical error resulting from the use of a ﬁnite number of stochas-

tic particles, time averaging was used in evaluating mean proﬁles such as



c,the

anisotropy tensor





,or





. The number of particles used in the simulations, ≥ 60

per cell combined with the additional time-averaging procedure, resulting in excess

of an order-of-magnitude increase in effective particle numbers, is arguably sufﬁcient

to reduce statistical errors to an acceptable level [231, 283]. For cases featuring joint

velocity–scalar closures, the mean proﬁles obtained from the PDF solution were

approximated with tensor-product cubic smoothing splines [284] with the smoothing

coefﬁcient obtained with a cross-validation technique [285]. Reynolds stress proﬁles

are not suitable for ﬁtting with smooth interpolants because values vary by orders

of magnitude within the ﬂow ﬁeld and because sharp peaks may be observed in

shear layers and similar regions. Therefore, instead of ﬁtting the Reynolds stresses

directly, the smoothing splines were ﬁtted to the Reynolds stress anisotropy tensor.

The anisotropy is of the order of 10

−1

, and the proﬁles (typically), do not feature

sharp gradients.

2.4 Transported Probability Density Function Methods 111

01234

tu'/L

(m/s)

Figure 2.19. The computed evolution of

[211] for stoichiometric CH

–air mix-

tures ( = 20 mm) with u

rms

= 0.5m/s

(bottom), u

rms

= 1.0 m/s (middle), and

rms

= 2.0 m/s (top). Transported PDF

calculations feature the modiﬁed Curl’s

model [254]andEq.(2.155) with C

= 4

and C

∗

= 1.2.

2.4.6 Freely Propagating Premixed Turbulent Flames

The impact of the mixing frequency closures given by Eqs. (2.152) and (2.155)

on predictions of the turbulent burning velocity was evaluated by Lindstedt and

aos [211] by using freely propagating planar ﬂames. It was shown by Hakberg and

Gosman [286], through the application of a KPP-type [214] analysis, that the govern-

ing equations should yield a linear scaling of the turbulent burning velocity with the

turbulence intensity. The same result is obtained numerically with moment-based

methods, provided that the intrinsic length and time scales corresponding to the

governing equations are well resolved [287], as conﬁrmed by Sabel’nikov et al. [288].

For transient propagating ﬂames, the ﬂame thickness scales on the integral length

scale of turbulence. The need to resolve the intrinsic length and time scales inher-

ent in the solution of laminar reacting ﬂows and in DNSs is well established, and

it is unsurprising that similar considerations apply in the context of model-based

simulations of turbulent ﬂames. For transported PDF approaches, the functional

form of Arrhenius-type reaction-rate expressions at the leading edge of the ﬂame

(dS

/dc → 0, c → 0) obviates the need to apply a Heaviside function to prevent the

growth of numerical round-off errors [287]. On the other hand, the solution proce-

dure features an explicit and ﬁrst-order-accurate (overall) fractional step method,

and it was shown by Lindstedt and V

aos [211] that computed values of the turbulent

burning velocities are consistent, provided the time step does not exceeds a certain

value (here  10

−5

s). Examples of the temporal evolution of turbulent burning

velocities are shown for methane ﬂames at different Reynolds numbers in Fig. 2.19.

The chemical source term was extracted from a laminar ﬂame computation, and the

modiﬁed Curl’s model [254] was used to represent mixing in scalar space. The trans-

port of the PDF in physical space was closed by use of the gradient approximation

given in Eq. (2.147).

Hulek [283] showed that the interaction between the mixing model and the

reaction rate is mainly conﬁned to a region close to the “cold” boundary and that

the width of the region is inversely proportional to the Damk

ohler number. In this

region, the mixing model induces transport of the discrete PDF in scalar space so that

the reaction rate becomes effective. Accordingly, the predicted burning velocities will

112 Modelling Methods

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.5

1.0

Normalised Reaction Rate

Figure 2.20. Normalised mean-reaction-

rate proﬁles in scalar space [211] along

with the probability of reaction (nor-

malised crossing frequency) measure-

ments (◦) of Cheng and Shepherd [35].

Lines: (—) modiﬁed Curl’s model [254]

and Eq. (2.155) with C

= 4andC

∗

= 1.2;

(– – –) modiﬁed Curl’s model [254]and

Eq. (2.155) with C

= 4andC

∗

= 0. In

both cases u

rms

= 0.5 m/s and  = 20 mm.

depend, ﬁrst, on the structure of the model for turbulent transport in scalar space

(i.e., the structure of the prescribed mixing transition density function) and, second,

on the magnitude of the relevant (mixing) time scale. Examples of computed mean-

reaction-rate proﬁles obtained with both the standard Eq. (2.152) and extended

Eq. (2.155) algebraic closures for the SDR are shown in Fig. 2.20. The proﬁles

were normalised by the maximum mean reaction rate to facilitate comparisons and

essentially show the probability of reaction. A comparison with the experimental

measurements by Cheng and Shepherd [35], obtained for a stoichiometric ethylene–

air mixture (u

 0.67 m/s) with u

rms

∼ 0.5 m/s, reveals that the computed shape

is in reasonable agreement with experimental data for the crossing frequency. The

statistical noise visible in Fig. 2.20 does not inﬂuence the computed turbulent burning

velocities strongly, as is evident from examples of the temporal evolution of u

shown

for u

rms

= 0.5, 1.0, and 2.0 m/s in Fig. 2.19.

The customary (near-) linear relationship [66, 246]ofu

rms

is not satisfactory

at the high-Damk

ohler-number limit [17]. Comparison with the experimental data of

Abdel-Gayed et al. [274], presented in Fig. 2.21, indicates that the extended closure

for the SDR given in Eq. (2.155) results in signiﬁcantly improved agreement for

1 < u

rms



< 7. The closure results in a burning velocity scaling behaviour that

rms

0.0

1.0

2.0

3.0

4.0

5.0

T rms

Figure 2.21. Predicted turbulent burn-

ing velocities for stoichiometric CH

–

air mixtures ( = 20 mm) [211]. Trans-

ported PDF calculations feature (—)

the modiﬁed Curl’s model [254]and

Eq. (2.155) with C

= 4andC

∗

= 1.2;

(– – –) modiﬁed Curl’s model [254]

and Eq. (2.155) with C

= 4andC

∗

= 0.

Experimental burning velocity data (◦)

from Abdel-Gayed et al. [274].