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Modelling of Premixed Laminar Flames using FlameletGenerated ManifoldsJ.A. VAN OIJEN

a& L.P.H. DE GOEY

a

aDept. of Mechanical Engineering , Eindhoven University of Technology , PO Box 513,

Eindhoven, MB, 5600, The Netherlands

Published online: 27 Apr 2007.

To cite this article:J.A. VAN OIJEN & L.P.H. DE GOEY (2000)Modelling of Premixed Laminar Flames using Flamelet-GeneratManifolds, Combustion Science and Technology, 161:1, 113-137, DOI: 10.1080/00102200008935814

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Combwt. Sci. and Tuh. 2000, Vol. 161, pp. 113-137

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Published by license under the

Gordon and BreachScience Publishers imprint.

Printedin Malaysia

Modelling of Premixed Laminar Flames

using Flamelet Generated Manifolds

J A VAN OIJEN and L P H DE GOEY

Eindhoven University

of

Technology, Dept.

of

Mechanical Engineering, PO Box

513 56

MB Eindhoven, The Netherlands

RevisedJune21,2000

In order to reduce the computational cost of flame simulations, several methods have been developed

during the last decades. which simplify the description of the reaction kinetics. Most of these meth

ods are based on par ti al -equil ibri um and steady- st at e assumpt ions, assuming t hat most chemi cal

processes have a much smaller time scale than the flow time scale. These assumptions, however, give

poor approximations in the colder regions of a flame, where transport processes are also important.

The method presented here, can

be consi dered as a combi nati on of t wo approaches t o simplif y

flame calculations, i.e. a flamelet and a manifold approach. The method, to which we will refer as the

Flamelet Generated Manifold

FGM) method. shares the idea with f1amelet a pp ro ac he s that a

multi-dimensional flame may be considered as an ensemble of one-dimensional flames. The imple

mentation, however, is typical for manifold methods: a low-dimensional manifold in composition

space is constructed, and the thermo-chemical variables are stored in a database which can be used in

subsequent flame simulations. In the FGM method a manifold is constructed using one-dimensional

flamelets. Like in other manifold methods. the dimension of the manifold can be increased to satisfy

a desired accuracy. Although the method can

be

applied to different kinds of flames. only laminar

premixed flames are considered here.

Since the major parts of convection and diffusion processes are present in one-dimensional flame

lets. the FGM is more accurate in the colder zones of premixed flames than methods based on local

chemical equilibria. Therefore, less controlling variables are sufficient to represent the combustion

process. Test results

of

one and two-dimensional premixed methane/air flames show that detailed

computati ons are r eproduced very well wi th a FGM consi st ing of onl y one progr ess variabl e apart

from the enthalpy to account for energy losses.

Keywords: Premixed laminar flames; low-dimensional manifolds; flamelets

C or respondi ng aut hor: J. A. van Oij en E- mail: J.A .v.O ij en@w tb.t ue.nl Telephone: . ..31 40

2473286 Fax: ... 3 4 2433445

113

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4

INTRO U TION

I

VAN OIJEN and L P H DE

GOEY

Recently, detailed numerical simulations of flames have become within reach

due to the still increasing speed of modem super) computers. Results of

multi-dimensional time-dependent flame calculations have been reported by sev

eral authors, see e.g.

[1 5].

However, computational times of these simulations

often exceed days, prohibiting an extensive study on the effect of different

parameters. Moreover, detailed simulations of practical systems such as in

engines or furnaces will still be impossible for a long time.

In order to reduce the computational expense, several methods have been devel

oped which simplify the description of the reaction kinetics. One way of reducing

the chemical reaction mechanism is the use of ad hoc few-step global mecha

nisms. In these chemical models the combustion process is described by only a few

reactions, representing the conversion of fuel and oxidizer into products. Although

global mechanisms bave been applied in many combustion simulations see e.g.

[6-9]), their lackof accuracy causes them to

be useful for global studies only.

More accurate reduction techniques are based on the observation that a typical

combustion system contains many chemical processes with a much smaller time

scale than the flow time scales. These fast processes can be decoupled, reducing

the stiffness of the system and thus increasing the computational efficiency. The

most prominent reduction methods are the systematic reduction technique by

Peters and co-workers [10], the computational singular perturbation CSP)

method from Lam and Goussis

]

and the intrinsic low-dimensional manifold

ILDM) approach

of Maas and Pope [12]. Each of these methods uses a different

way to identify the fast chemical processes. In the systematic reduction technique

partial-equilibrium and steady-state assumptions are invoked for particular reac

tions and species which are associated with fast processes. A detailed investiga

tion of the reaction paths and the time scales in the chemical reaction mechanism

should give information about which species are closest to steady state. For

higher hydrocarbon fuels with reaction mechanisms consisting

of

hundreds of

reactions this is a complicated task. Moreover, the steady-state assumptions

invoked are also applied to ranges of compositions and temperatures in flames

where they provide poor approximations.

In the CSP and ILDM method the time scale analysis is performed automati

cally, based on the local Jacobian

of

the chemical source term in the balance

equations. The CSP method uses a computational singular perturbation to iden

tify the fast time scales. Since this is done dynamically, the number and identity

of the steady-state species may vary during the combustion process. Although

this technique is the most accurate, its applicability on complex simulations is

questionable due to its high computational cost.

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FL MELET GENER TED M NIFOLD METHOD

The ILDM method identifies the fast processes using an eigenvalue analysis of

the local Jacobian of the chemical source term. The processes represented by the

eigenvectors associated with small time scales are assumed to be in steady state.

The eigenvectors which represent the slow chemical processes are used to con

struct a low-dimensional manifold in composition space. This manifold is para

metrized by a small number of variables which evolve slowly during the

combustion process. For application in flame calculations the manifold is tabu

lated in a look-up table as a function of the parametrizing or controlling) varia

bles. This is done once in a pre-processing step for a given mixture and,

subsequently, the look-up table can be used in a series of combustion calcula

tions. The dimension of the look-up table is determined by the number of con

trolling variables which is needed to describe the slow chemical processes. For

simple fuels like O H

z

mixtures, two controlling variables are sufficient [13,

14]. However, for an accurate reproduction of the burning velocity of premixed

methane/air flames at least three degrees of freedom are needed [IS]. Together

with variations in enthalpy, pressure and element composition which can occur in

combustion systems, this results in a high-dimensional look-up table. Therefore,

the application of the ILDM method seems to be very difficult for higher hydro

carbon fuels, especially in case of premixed flames where the burning velocity

plays an important role.

In this paper a different method is presented to create a manifold. This method

shares the idea with so-called flamelet approaches [16, 17] that a multi-dimen

sional flame can be considered as an ensemble of one-dimensional I D) flames.

This implies that the path followed in composition space in case of multi-dimen

sional flames will be close to the path found in ID flames. Therefore, the chemi

cal compositions in ID flames are used to construct a manifold in this method.

The resulting manifold, to which we will refer as lamelet Generated Manifold

FGM), will be a better approximation of the mixture composition in colder

zones of laminar premixed flames than the lLDM. As a result of this, less con

trolling variables are sufficient to represent the combustion process. Therefore,

the dimension of the look-up table can be kept low, making it an interesting man

ifold method.

The main goal of this paper is to present the FGM method and to explain the

basic ideas behind

it. Although the technique can be applied to different kinds of

flames, we focus here on atmospheric premixed laminar flames. A FGM will be

constructed for a hydrogen/air mixture and it will be compared with the corre

sponding lLDM. For a methane/air mixture a FGM has also been generated,

which has been used in several test simulations. The results of these simulations

will be compared with results of detailed computations.

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116 J.A. VAN OllEN and L P H DE GOEY

The outline of this paper is as follows.

the next section the so-called flamelet

equations are derived starting from the set of three-dimensional instationary con

servation equations describing the system. How a FGM is constructed for a

premixed laminar flame using the set of flamelet equations, is discussed in the fol

lowing section. The main differences with the ILDM method are discussed briefly.

The implementation of the FGM method in numerical simulations of laminar

flames is demonstrated, subsequently. The main issues which are discussed are the

implementation of the look-up table and the method used to solve the equations.

Test results are reported for both one and two-dimensional 2D) laminar meth

ane/air flames in the subsequent section. A discussion is given in the final section.

FL M L T QU TIONS

The evolution of chemically reacting flows is described by a set of differential

equations representing the conservation of mass, momentum, species mass frac

tions, and energy. The equations are the continuity equation

ap

+ ~ . pv) = 0, 1)

with p the mass density and v the flow velocity, the momentum equations

P 7 f t p v , ~ v = - ~ P ~ T ,

2)

where p denotes the pressure and r the viscous stress tensor, the conservation

equations for the N species mass fractions Y

j

apY; ) 1 A

+

~ . pvYi ~ - - ~ y .

-w-:-

. Le c

p

,

i I N

3

4)

and the balance equation for the enthalpy

h

aph + pvh) - ~ ~ h

=

t h; -

~ ~ Y ;

.

Cp = Le, c

p

Here, the thermal conductivity A the specific heat

c

p

and the Lewis numbers

Lei = VpDicp> with

D;

the diffusion coefficient of species

i,

have been intro

duced. The chemical source term is divided in a production part w+ and a con

sumption part

w

consisting of the positive and negative contributions,

respectively. The pressure is given by the perfect gas law

N

p=pRT LY;/M;,

5)

l

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FLAMELET GENERATED MANIFOLD METHOD

7

where

R

is the universal gas constant and

T

the absolute temperature. The spe

cific enthalpy follows from

N

h L) ih;,

l

hi T) h? +

r

T

cp,i r)dr,

lTD

6)

M;

c

p

.;

h,

and

h?

being the molar mass, the specific heat, the specific enthalpy

and the enthalpy of formation of species

i,

respectively.

The numerical solution of this set of equations is computationally expensive.

Due to the large number

o

species which is involved in general, a large set of

equations arises, which is strongly coupled by the chemical source terms. Fur

thermore, the different contributions to the chemical source terms describe proc

esses with a wide range of time scales. This causes the set of equations to be stiff,

which implies that special techniques have to be used to solve the system. In

order to tackle this problem, the chemical reaction models can be simplified

using reduction techniques. Most of these reduction methods are based on the

assumption that a large number of species or linear combinations of them), is in

steady state. This implies that the left hand side of Equation 3) can be neglected

for these species, leaving a balance between chemical production and consump

tion:

wi - wi = 0, i = 1, , M,

7)

assuming the first

M

species to be the steady-state species. The solution of this

set of algebraic equations 7) forms a manifold which can be stored in a look-up

table with dimension

N M.

Due to the application of steady-state relations, both

the number and the stiffness of the differential equations have been decreased.

These steady-state relations give, however, poor approximations in the colder

regions of a flame where convection and diffusion play an important role. In the

cold region of a flame, where chemical reactions are negligible, bad manifold

data will not hurt too much. However, there is an important region in flames

where reaction and diffusion are in balance. A better approximation of the mix

ture composition in these regions can be found if we take the most important

transport processes also into account. Since the major parts of convection and

diffusion processes are also present in ID flames, the compositions in ID flames

will be representative for the compositions in more general flames as well.

In order to explain this idea, we con sider a part of a premixed flame as dis

played in Fig. 1, and follow a curve

x s)

through the flame, locally perpendicular

to isosurfaces of a certain species mass fraction

Y

and parametrized by the arc

length s. The evolution of all species mass fractions

Y;

along this curve may be

described approximately by a one-dimensional equivalent of Equation 3)

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118

J.A. VAN OIJEN and L.P.H. DE GOEY

un u t

urnt

Y

j

= o st

FIGURE I Schematic Representation of a premixed flame with the curve xis through the flame front

8

=

1,

N

y 0 1

x

oy; + _

m jj

-

s

Lei

cp jj ;

= Wi

-

Wi

+ Pi

with m taken to be a constant mass-flow rate. Since x s) is perpendicular to the

isosurfaces of

j

the most important

10

transport and diffusion phenomena are

represented by the left-hand side of Equation 8). Multi-dimensional and tran

sient effects - due to flame stretch, i.e. variations of the mass-flow rate along the

curve, and curvature - are gathered in the perturbation term

Pj s,

t This pertur

bation includes terms due to flame stretch, i.e. variations of the mass-flow rate

along the curve, and curvature effects. For instance, a most likely small) diffu

sion term associated with the curvature of the flame is included in Pj. Further

more, the isosurfaces of Yj ,j generally don t coincide with the isosurfaces of j

resulting in extra transport terms which are included in Pj j-

It is expected that in most situations

P

j

is small compared to the other terms in

Equation 8), although this might not be justified under extreme circumstances,

such as near local flame extinction and in regions with strong flame stretch.

Therefore, the next step is to neglect the perturbation

Pj,

yielding a set of

one-dimensional differential equations

y

0 1 + _

m jj -

s

Lei cp jj ; =

Wi - W i i

= 1,

. .. N

9)

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FLAMELET-GENERATED MANIFOLD METHOD

119

This resulting balance equation 9) between convection, diffusion and reaction

can be considered as a steady-state relation just like Equation 7). Note that in

conventional reduction techniques not only Pi is neglected, but also the left-hand

side

of

Equation 8). The set of equations 9), completed with a similar

one-dimensional equation for the enthalpy, is referred to as a set of flamelet

equations.

A solution of these equations Yj s),

s

is called a flamelet and forms a curve

in composition space which can be regarded as a one-dimensional manifold par

ametrized by

s

order to be able to use the manifold in further calculations it

can be parametrized by a single controlling variable Y

ey

This controlling varia

ble or progress variable) may be the mass fraction of any linear combination of)

species which results in a unique mapping Y,{Y

cy

for all i. This implies that

Yey s

should be either a continuously increasing or decreasing function of

s

FGM S FOR PREMIXED FL MES

this section it is described how the set of flamelet equations 9) is used to cre

ate a manifold. As in the ILDM method a distinction is made between variables

which are conserved by chemical reactions element mass fractions Zj pressure p

and enthalpy

h)

and variables which are changed by reactions: the species mass

fractions. Note that the chemical composition of the burnt mixture is determined

by the conserved variables, while the combustion process from unbumt to burnt

is described by the reactive controlling variables, which are often called progress

variables. Following Maas and Pope [l2l we will first consider the case of con

stant pressure, enthalpy and element mass fractions.

order to construct a manifold for premixed flames, the set of flamelet equa

tions 9) can be solved treating the system as a freely-propagating premixed

flame. This implies that the boundary conditions at the unbumt side are of

Dirichlet type

Y; s

- 0 = Y

u

i

,

h s - 0 = h,

and at the burnt side of Neumann type

8Y;

a s 0

=

0,

8

s s

0

=

0,

10)

11

12)

13)

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120 JA V N

OUEN and L P H DE GO Y

and that the mass-burning rate m is an eigenvalue of the system. The resulting

curve Y; s) is determined by its starting point

u,;,

h

u)

The one-dimensional

manifold in composition space is simply the flamelet starting at the point which

represents the unburnt mixture for which the manifold is created. In this way the

manifold connects the two most distinguished points in composition space: the

point corresponding to the unburnt mixture and the equilibrium point.

As an example we have shown the one-dimensional ILOM and FGM for a sto

ichiometric hydrogen/air mixture in Fig. 2 together with a scatter plot of the

chemical state at different positions in a

stationary premixed Bunsen flame

stabilized on a cold burner rim computed using detailed chemical kinetics. The

detailed reaction mechanism is a subset of the mechanism used in the Maas and

Pope papers and includes 7 species and 7 reversible reactions. The mass fraction

of

H

is used as controlling variable, because it is continuously increasing dur

ing the combustion process from

YH o

=

0

at the unbumt mixture to the chemi

cal equilibrium value

YH o

0 24

Generally, Lewis-number effects cause

local enthalpy and element mass fraction variations in flames and thus also in the

FGM database. In the ILOM method, however, the element composition and the

enthalpy of the mixture are constant throughout the manifold and variations in

the conserved variables can only be accounted for using additional controlling

variables. Therefore, to make a fair comparison of the methods in Fig. 2, unit

Lewis numbers are assumed in the computation of both the FGM and the

flame so that the element mass fractions and the enthalpy are constant in both

manifolds. An example of a FGM computed using non-unit Lewis numbers, is

presented in the following section. Note that the composition is determined by

chemical processes in the high temperature range

YH O

>

0.19,

T

>

1600 K

and that the results of the ILOM and FGM method are therefore equivalent.

However, in the colder zone

YH O

3

2

1

0

0 1

0

FIGURE 8 Profiles of Y H O and Y

H

in a one-dimensional burner-stabilized flame with

m

=

0.030 g1cm

2s

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FLAMELET GENERATED MANIFOLD METHOD

131

0.58

13

Detailed

0.12

0.56

Reduced

0.54

11

0.10

0.52

0.09

S

0.50

v

0.08

< >

f.,.

48

7

0.46 0.06

0.44

0.05

0.42

0.04

5

1 2

4

m g/cm

2s

FIGURE 9 Arrhenius plot for

and the stand offdistance 8 as function of the mass flow rate m

found in premixed laminar flames such as flame cooling. stretch and curvature

are present. We have simulated a methane/air flame with an equivalence ratio

of 0.9 which stabilizes on a 2D slot burner in a box. The burner configuration is

shown in Fig. 10. The burner slot is 6mm wide while the box is 24mm wide. The

burner and box walls are kept at a constant temperature of T

burner

300 K. The

velocity profile at the inlet is parabolic with a maximum velocity of

u

max

=1.0mls. Isocontours of and the mass fractions of CH

2

0 and H are

shown in Fig.

l ion

a portion of the computational domain for both the detailed

and reduced computations.

It

may be seen that the results obtained with the FGM method are in excellent

agreement with the detailed computations: not only the position of the flame

front is predicted very well but the absolute values of the mass fractions are

reproduced as well. Flame cooling governing the stabilization of the flame on the

burner is captured very well by the FGM although one can hardly speak of

flamelets in this cold region. Also in the flame tip where stretch and curvature

are very important the reduced computations appear to coincide with the

detailed calculations.

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132

l VAN OUEN and L.P.H. DE GOEY

Outlet

2r

a

c.1 1 e

s

e

> C

c

o

U

Inlet

FIGURE 10 Numerical configuration used for the 2D computation. Results will

be

shown for the

region enclosed

by

the dotted line

omput tion time

Besides the accuracy the efficiency is another important aspect of reduction

methods. In order to give an indication of the efficiency of the FGM method the

computation times of detailed and reduced simulations are compared. For both

models we determined the time needed to perform a time dependent ID flame

simulation for a period of

10

3

seconds under the same conditions. To solve the

equations we used a fully implicit solver with varying time steps. The FGM

method has also been used with an explicit time stepper using constant time steps.

The computations are performed on a Silicon Graphics workstation and the com-

putation times are shown in Table I

The CPU time per time step reduces approxi-

mately a factor 8 when the FGM is applied. This speed up is caused by the

reduction of the number of differential equations to be solved and a faster evalua

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FLAMELET-GENERATED MANIFOLD METHOD

133

a

0.5

b

0.5

c

0.5

FIGURE II 1socontours of a b Y H O and c Y

H

computed using left a detailed mecha

nism and right a FGM. The spatial coordinates are given inem

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4 J.A. VAN OIJEN and L.P.H. DE GOEY

tion of the chemical source terms. Another advantage of the reduced model is that

larger time steps can be taken because the smallest time scales have been elimi-

nated. Therefore the total CPU time of the reduced computation is 20 times less

than the detailed simulation. An even higher efficiency is reached if an explicit

solver is used for the reduced computations. For more complex reaction mecha-

nisms and multi dimensional systems the speed up will be even larger.

TABLE I Computation time per time step. Istep and the total simulation time ItotaJ. of a

time-dependent ID meth ne ir flame simulation

Kinetics Solver

sup ms)

ltotal s

Detailed

Implicit

247 94

FGM Implicit

32 9

FGM Explicit 2

5

The computation of the FOM database used in this paper involved approxi-

mately 30 minutes which is quite long compared to the CPU times mentioned in

Table Obviously it is not efficient to construct a FOM for a single flame

simulation. However the computation time which can be gained using a FOM in

a series of multi dimensional flame simulations is orders of magnitude larger

than the time needed to construct the database.

IS USSION

In this paper a new method has been presented to create low dimensional mani-

folds and it has been applied to premixed laminar flames. Since in this method a

manifold is constructed using one dimensional flamelets it can be considered as

a combination of a manifold and a flamelet approach. The FOM method shares

the assumption with flamelet approaches that a multi dimensional flame may be

considered as an ensemble of one dimensional flames. The implementation

however is typical for a manifold method which means that the reaction rates

and other essential variables are stored in a look up table and are used to solve

conservation equations for the controlling variables. Therefore the local

mass burning rate follows from the balance between chemical reaction and

multi dimensional convection and diffusion. In classical flamelet approaches

however a kinematic equation for the scalar G is solved [17]. In this so called

G equation the burning velocity enters explicitly and the influence of flame

stretch and curvature on the mass burning rate has to be modelled. Moreover

while the conservation equations for the controlling variables are valid through

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FLAMELET-GENERATED MANIFOLDMETHOD

135

out the complete domain, the G-equation is only valid at one value of G

O

which denotes the position of the flame sheet. Everywhere else G is simply

defined as the distance to the flame sheet, resulting in a flame of constant thick

ness. In the FGM method the flame thickness follows from the conservation

equations and is in general not constant.

Another advantage of the FGM method is that the number of progress variables

is not limited to one as in existing flamelet approaches. Although the test results

of laminar premixed burner-stabilized methane/air flames show that one progress

variable and the enthalpy are sufficient to reproduce detailed simulations very

well, more progress variables can be added to increase the accuracy of the

method. Addition of progress variables might cause problems in the look-up pro

cedure as described in this paper, because the different flamelets rapidly con

verge to form a lower-dimensional manifold see Fig. 3 . However, this problem

can be solved using a modified storage and retrieval technique, based on the

lower-dimensional manifold.

There is no difficulty in adding further conserved controlling variables. For

instance, to treat non-premixed flames the manifold can easily be extended so

that variations in the mixture fraction can be accounted for. order to generate a

manifold for non-premixed flames, the flamelet equations are solved for different

stoichiometries and an element mass fraction can be used as extra conserved con

trolling variable.

In turbulent flames the perturbations unsteady effects. flame stretch and cur

vature will probably not be small compared to the other terms in the governing

equations

l

convection, diffusion and reaction . Therefore, the dimension of

the manifold should be increased in such way that the perturbation vector

Plies

in the manifold. This can be done by adding an additional progress variable as

described earlier, but also by including more of the physics in the flamelet equa

tions. For instance, if flame stretch is expected to be important, a con

stant- s tretch term can be included in the flamelet equations 9 . Then the

equations are solved for different stretch rates, which results in an extra dimen

sion for the manifold.

The enormous reduction of computation time due to application of a FGM

allows us to perform more extensive studies of realistic flames. More tests and

experience will clarify the influence of unsteady effects, flame stretch and curva

ture.

knowledgements

The financial support of the Dutch Technology Foundation STW is gratefully

acknowledged.

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136

J A

VANOIJEN and L.P.H. DE GOEY

eferen es

[I D. Thevenin,

F

Behrendt, U. Maas, B. Przywara

1.

Warnatz, Development of a parallel

direct simulation code to investigate reactive flows, CompoFluids 25,

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