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THESIS

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MICHIGAN-lIB S
STATE UNIVERSITY
EAST LANSING, MICH 48824-1048

This is to certify that the
thesis entitled

Numerical Simulations of Partially Premixed Methane
Combustion in Laminar and Turbulent Flows

presented by

Hyderuddin Mohammad

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MS. degree in Mechanical Engineering
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Numerical Simulations of Partially Premixed
Methane Combustion in Laminar and Turbulent
Flows

By

Hyderuddin Mohammad

A THESIS

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Mechanical Engineering

2004

ABSTRACT
Numerical Simulations of Partially Premixed Methane

Combustion in Laminar and Turbulent Flows

By

Hyderuddin Mohammad

The effects of partial premixing of methane fuel with air on the structure of a laminar
counterﬂow ﬂame were studied via various reduced and full chemistry models. Our
results indicate that the mechanisms containing both the Cl and 02 species perform
much better than those involving only 01 species. Results obtained with the 10-step
and 12-step mechanisms compare very well with those of the full GRI mechanisms
for all the values of strain rate and premixing, and generate the same ignition and
extinction characteristics in all simulated ﬂames. However, the mechanisms with
lesser number of species such as the 4-step, 5—step and 6-step perform well in some
of the conditions, they were not fully accurate. The performance of these reaction
mechanisms were found to deteriorate as the strain rate is decreased or the level of

fuel premixing with air is increased.

The 10-step mechanism was used in a direct numerical simulation (DNS) of
isotropic turbulent ﬂow interacting with non-premixed and partially-premixed
ﬂames. DNS data were used to understand the behavior of the ﬂames in more
realistic ﬂow situations. Our results show signiﬁcant changes in the ﬂame structure
embedded in a high intensity turbulent ﬁeld due to the imposed variable strain ﬁeld.
Deviation from the initial (laminar) ﬂame structure in compositional space has been
observed and was accompanied with signiﬁcant local extinction at the locations
where the strain rate was high. At early times, signiﬁcant vorticity dissipation was
observed due to volumetric expansion and increased viscosity near the ﬂame. In the

later times, vorticity was generated by the so called baroclinic term near the ﬂame.

Copyright © by
Hyderuddin Mohammad
2004

To my parents and my brother

iv

ACKNOWLEDGEMENTS

I would like to express my gratitude to my thesis advisor Dr. Farhad A. Jaberi for
his constant support and brilliant guidance. He sincerely dedicated his time and
effort to help me accomplish my goal. Without his incessant encouragement and

appreciation this thesis would not be ‘complete’ in its true sense.

This work was sponsored by the ofﬁce of Naval Research under Grant N00014-01-
1-0843 and National Science Foundation. I am thankful to them for their support.
I am also thankful to Dr. Gabriel D. Roy of the ofﬁce of Naval Research for his

continuous support and encouragcn‘ient during the course of this research.

I would also like to thank my cormnittec members, Dr. Z. J. Wang and Dr.
Mei Zhuang, and Dr. Indrek Wichman. They took their time out of their busy
schedule and gave me insightful comments and suggestions. I would also like to
thank Dr. Sunil James, Mr. Bharatwaj, and Mr. Kian for their valuable help

throughout my research.

Most importantly, I would like to thank my parents, brother and my sisters
for their love and unconditional support throughout my life. I am grateful to my
friends for their help and sharing memorable times during my stay at Michigan State

University.

TABLE OF CONTENTS

LIST OF TABLES
LIST OF FIGURES

1 Laminar Flames

1.1 Introduction ..................................
1.2 Mathematical and Numerical Formulation .................
1.2.1 Equations for Gaseous Combustion ................
1.2.2 Flamelet Formulation ........................
1.2.3 Counterﬂow Flames .........................
1.2.4 Numerical Method ..........................
1.3 Results and Discussion ............................
1.3.1 Temperature Field ..........................
1.3.2 Species Field .............................
1.3.3 Autoignition ..............................

2 'Ihrbulent Flames

2.1 Introduction ..................................
2.2 Governing Equations for Reacting Flows ..................
2.2.1 Conservation Equations .......................
2.2.2 Transport Models ...........................
2.2.3 Chemistry Models ..........................
2.3 Numerical Solution ..............................
2.3.1 Methodology .............................
2.3.2 Simplifying Assumptions ......................

vi

viii

ix

11
14
17
19
20
24
30

64

69

69
72

2.4 Simulations ................................... 86

2.4.1 Initialization .............................. 86
2.4.2 Results ................................. 89
3 Conclusions 120
BIBLIOGRAPHY 124

vii

1.1

2.1
2.2
2.3
2.4
2.5
2.6

2.7

LIST OF TABLES

Reaction Mechanisms Assessed ........................ 19
Characteristic scales in turbulent non-premixed ﬂames ......... 66
Differencing sequence foe thended Maccormack scheme ........ 79

CPU time requirements for subroutines before and after approximation 82

Percentage of CPU time in stress subroutine ............... 83
Characteristics of developed turbulence ﬁeld ............... 87
Dimensional characteristics of turbulence ﬁelds .............. 88
Simulation cases studied ........................... 88

viii

1.1

1.2

1.3

1.4

1.7

1.8

1.9

LIST OF FIGURES

Temperature structures in physical and compositional space for various

domain sizes, (a) 0.4 cm, (b) 1.0 cm, (c) 2.0 cm. ............. 33

Temperature structure in mixture fraction space with GRI-Mech 2.11
(a) for different strain rates and premixing of 75% air in the fuel
stream, (1)) for different premixing levels of the fuel stream and strain
of 25(1/s). Vertical lines indicate stoichiometric mixture fraction. . . . 34

Compositional temperature structure obtained with various mecha-
nisms for 75% premixing of fuel stream and different strain rates, (a)
10 (1/s) (b) 50 (l/s) (c) 150 (l/s) (d) 300 (1/s). Vertical lines indicate

the stoichiometric mixture fraction ...................... 35

Compositional temperature structure obtained with different mecha—
nisms for different premixing of fuel stream and a strain rate of 25
(1/s), (a) 0% (b) 25% (c) 50% (d) 75%. Vertical lines indicate the

stoichiometric mixture fraction ........................ 36
Production rates of CH4, 02, and C02. (a) 5-step mechanism ..... 37
[contd...] Production rates of 0114,02, and 002. (b) 10—step mechanism. 38

Percentage mean of temperature difference from GRI 3.0 for different
mechanisms vs. (a) strain rate for premixing of 75% air in fuel stream,
(1)) premixing level of fuel stream with air for strain rate of 25(1/s). . 39

Maximum temperature deviation from GRI 3.0 predictions for different
mechanisms and a strain of 25 (1 /s) ..................... 40

Temperature difference from GRI 3.0 for different mechanisms vs. mix-
ture fraction for strain rate of 25( 1/s) and (a) 75% air in the fuel
stream, (b) 85% air in the fuel stream .................... 41

Effect of partial premixing on YCH4 structure. —<>—, 4-Step; — x —,

5-Step; — =0: —, 6—Step; —A—, 10-Step; —D—, 12-Step; -— O -—, OR] 2.11;
-*—, GRI 3.0 .................................. 42

ix

1.9

1.10

1.10

1.11

1.12

1.12

1.13

1.13

1.14

1.14

1.15

1.16

1.16

[contd...] Effect of strain rate on YC H 4 structure. See Figure 1.9(a) for
labeling .....................................

Effect of partial premixing on Y02 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction. . .

[contd...] Effect of strain rate on YO» structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction. . .

Reaction rate of the global reactions in (a) 4-step mechanism. .....

[contd...] Reaction rate of the global reactions in (b) 5-step mechanism.

Effect of partial premixing on 1’002 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction. . .

[contd...] Effect of strain rate on 1’002 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction. . .

Effect of partial premixing on YH2 structure. (a) A% = 0, (b) A% =
25, (c) A% = 50, and (d) A% = 75. See Figure 1.9(a) for labeling.
Vertical litres indicate the stoichiometric mixture fraction. .......

[contd...] Effect of strain rate on YH2 structure. (a) strain 2 10 l/s,
(1)) strain 2 50 1/s, (c) strain = 150 1/s, and (d) strain 2 300 Us
See Figure 1.9(a) for labeling. Vertical lines indicate the stoichiometric
mixture fraction. ...............................

Effect of partial premixing on YCO structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction. . .

[contd...] Effect of strain rate on YCO structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction. . .

C02 production rate and mass fraction compared for 0% partial pre-
mixing and a strain rate of 25 1/s 4-step mechanism. ..........

[contd...] C02 production rate and mass fraction compared for 0%
partial premixing and a strain rate of 25 l/s 5—step mechanism. . . . .

Effect of partial premixing on YH structure. (a) A% = 0, (b) A% =
25, (c) A% = 50, and (d) A% = 75. See Figure 1.9(a) for labeling.
Vertical lines indicate stoichiometric mixture fraction ...........

[contd...] Effect of strain rate on Y” structure. (a) strain 2 10 l/s,
(b) strain 2 50 US, (c) strain = 150 l/s, and ((1) strain = 300 Us
See Figure 1.9(a) for labeling. Vertical lines indicate stoichiometric
mixture fraction. ...............................

43

44

45
46

47

48

49

50

52

54

55

56

1.17 Effect of partial premixing on YNO structure. - x -, 5-Step; —— O —,
GRI 2.11; —*—, GRI 3.0. (a) A% = 0, (b) A% = 25, (c) A% = 50, and
(d) A% = 75. Vertical lines indicate stoichiometric mixture fraction. .

1.17 [contd...] Effect of strain rate on YNO structure. — x —, 5—Step; — O —,
GRI 2.11; —*—, GRI 3.0. (a) strain = 10 l/s, (b) strain = 50 l/s,
(0) strain = 150 l/s, and (d) strain 2 300 1/s.Vertica1 lines indicate
stoichiometric mixture fraction ........................

1.18 (DF) vs. strain rate for different mechanisms and premixing of 75%
air in fuel stream. (a) CH4, (b)CO ......................

1.19 (DF) vs. partial premixing level of fuel with air for different mecha-
nisms and strain rate of 25(1/s). (a) CH4, (b) CO. ...........

1.20 Maximum YCO difference between the GRI 3.0 results with those ob—
tained with other mechanisms. .......................

1.21 Ignition temperature vs. strain rate for various premixing levels of the

fuel with air using GR] 3.0 ...........................
2.1 Combustion regimes for non-premixed ﬂame ...............
2.2 Computational Domain ............................

2.3 Sequence of temperature contours for turbulence case I and ﬂame No.
3 with time; (a) t/n = 0.0, (b) t/Tt = 1.0, (c) t/rt = 1.5, (d) t/Tt = 2.0. .

2.4 Mixture Fraction contours for turbulence case I and ﬂame No. 3 with
time: (a) t/Tt = 0.0 (b) t/Tt = 1.0 (C) t/Tt = 1.5 (d) t/Tt = 2.0 ......

2.5 YOH contours for turbulence case I and ﬂame No. 3 with time: (a)
13/11 = 0.0 (b) t/Tt = 1.0 (c) t/Tt = 1.5 (d) t/Tt = 2.0 ............

2.6 YCO contours for turbulence case I and ﬂame No. 3 with time: (a)
t/Tt = 0.0 (b) t/Tt = 1.0 (C) t/Tt = 1.5 (CI) t/Tt = 2.0 ............

2.7 Y-averaged plots at different times for (a) Temperature and (b) Y0 H .

2.8 Flame normal analysis at a burning region of the ﬂame (a) Mixture
fraction contours showing the line of section (b) Temperature (c) Strain
rate .......................................

2.9 Flame normal analysis at a about-to—be—queuched region of ﬂame and
for t/Tt = 1.0 (a) Mixture fraction contours showing the line of section
(b) Temperature (c) Strain rate .......................

xi

58

r

59

60

63

78

95

96

97

98
99

101

2.10

2.11

2.12

2.13

2.14

2.15

2.16

2.17

2.18

2.19

2.20

2.21

2.22

Flame normal analysis at a quenched region of ﬂame and for t /Tt = 1.5
(a) Mixture fraction contours showing the line of section (b) Temper-
ature (c) Strain rate ............................. 102

Scatter plots in mixture fraction space for turbulence case I and ﬂame
No. 1 at one eddy turn-over time. (a) Temperature (1)) YCO (c) YO H
(d) YH- Dashed lines shows the initial laminar profile .......... 103

Scatter plots in mixture space for turbulence case I and ﬂame N o. 2 at
one eddy turn-over time. (a) Temperature (b) YCO (c) Y0 H ((1) Y”.
Dashed lines shows the initial laminar proﬁle ............... 104

Scatter plots in mixture fraction space for turbulence case I and ﬂame
No. 3 at one eddy turn-over time. (a) Temperature (b) YCO (c) Y0 11
(d) YH- Dashed lines shows the initial laminar proﬁle .......... 105

Temperature scatter plots for turbulence case I and ﬂame No. 3 for (a)
t/Tt = 0.0 (b) t/rt = 1.0 (c) t/rt = 1.5 (d) t/rt = 2.0 in mixture fraction
space. Dashed line shows the initial laminar proﬁle. ........... 106

Y0 H scatter plots for turbulence case I and ﬂame No. 3 for, (a) t/rt =
0.0, (b) t/n = 1.0, (c) t/rt = 1.5, (d) t/Tt = 2.0 in mixture fraction
space. Dashed line is the initial laminar proﬁle. ............. 107

YCO scatter plots for turbulence case I and ﬂame No. 3 for (a) t/Tt :-
0.0, (b) t/Tt : 1.0, (c) t/n = 1.5, (d) t/rt = 2.0 in mixture fraction
space. Dashed line is the initial laminar proﬁle. ............. 108

Sequence of vorticity images for turbulence case I and ﬂame No. 3 with
time: (a) t/Tt : 0.0, (b) t/Tt = 1.0, (c) t/Tt = 1.5, (d) t/Tt = 2.0 ...... 109

Sequence of enstrophy images for turbulence case I and ﬂame No. 3
with time: (a) t/Tt = 0.0, (b) t/Tt = 1.0, (C) t/Tt = 1.5, (d) t/Tt = 2.0. . . 110

Sequence of turbulent kinetic energy images for turbulence case I and
ﬂame No. 3 with time: (a) t/rt = 0.0, (b) t/Tt = 1.0, (c) t/Tt = 1.5, (d)
t/Tt = 2.0 ..................................... Ill

Sequence of strain rate images for turbulence case I and ﬂame No. 3
with time: (a) t/Tt : 0.0, (b) t/Tt:1.0, (c) t/Tt = 1.5, (d) t/n = 2.0. . . 112

Y-averaged vorticity plots at four eddy turn-over times. (a) t/rt = 0,
(b) t/Tt : 1.0, (c) t/Tt = 1.5, and (d) t/Tt : 2.0 .............. 113

Y-averaged enstropy plots at four eddy turn—over times. (a) t/rt = 0,
(b) t/Tt 21.0, (c) t/Tt 21.5, and (d) t/Tt = 2.0 .............. 114

xii

2.23 Sequence of vorticity images for turbulence case I and ﬂames of various
premixing levels. (a) Initial vorticity, (b) A% = 0 & t/n = 1.0, (c) A%
= 25 & t/Tt =1.0, (d) A% = 75 & t/Tt=1.0 ................ 115

2.24 Sequence of enstrophy images for turbulence case I and ﬂames of var-
ious premixing levels. (a) Initial enstrophy, (b) A% = 0 & t/Tt = 1.0,
(C) A% = 25 & t/Tt=1.0, (d) A% = 75 & t/Tt 2: 1.0. .......... 116

2.25 Sequence of turbulent kinetic energy images for turbulence case I and
ﬂames of various premixing levels. (a) Initial TKE, (b) A% = 0 &
t/Tt=1.0,(C)A% = 25 & t/Tt = 1.0, (d) A% = 75 &t/n=1.o. . . . . 117

2.26 Y—averaged vorticity for different partial premixing. (a) Initial proﬁle,
and (b) At t/Tt=1.0 .............................. 118

2.27 Y—averaged enstrophy for different partial premixing. (a)lnitial proﬁle,
and (b) At t/rt = 1.0 .............................. 119

xiii

CHAPTER 1

Laminar Flames

1 . 1 Introduction

Many of the applications, where combustion occurs, involve turbulent ﬂow in a
complex geometry. Turbulent ﬂow analysis which is itself computationally expensive
will be much more expensive with the inclusion of combustion with detailed chemical
kinetics model. To circumvent the problems arising in numerical simulations of
such ﬂows, researchers have either resorted to generic description of combustion
or to the simpliﬁed geometries so that the total number of numerical operations
become manageable. While the former is prone to serious errors due to the empirical
parameters [58] involved in global representation of the fuel combustion, the latter is

a serious constraint in the study of realistic configurations.

Reduced chemistry models can be a remedy for the high computational cost
problem. They reduce the computational effort in numerical calculations of ﬂames
by replacing a number of differential equations for intermediate species by algebraic
relations. Apart from reducing the computational expense in numerical simulations,
they allow studying the ﬂame structure by asymptotic methods and by that help
in identifying the important parameters that inﬂuence the global properties of

the ﬂame. Typically, a reduced mechanism is reduced from a skeletal mechanism,

which in turn is reduced from a full mechanism. The skeletal mechanism will be
formed from the full mechanism by eliminating some of the reactions, which are not
inﬂuential in predicting a solution (characteristic quantity of interest). Sensitivity
analysis of the reactions in full mechanism cart be performed to identify the reactions,

which are not inﬂuential and thus to be eliminated.

Methods of reducing mechanisms were well-known to the chemists from long
time especially, using partial equilibrium and steady state species eliminations. The
potential of which was not realized until they were applied in reducing mechanisms
for hydrocarbon fuel combustion. Since the work of Peters and Williams [40], many
have come up with reduced mechanisms, which represent the combustion process in
special situations. Even though such reduced mechanisms can represent combustion
fairly well in the assumed situations, their performance can deteriorate in other
conditions than the assumed. In practice, where there are lots of variations in the
operating conditions, these chemistry models can perform poorly. Examples of the
variations in conditions include lean and rich mixtures, premixed and non-premixed,
various strain and pressure conditions, various levels of preheating of fuel and/or
oxidizer. In a practical combustion system there exists a wide range of conditions at
different locations in space of the same system. It is quite natural to question the ap-

plicability of the reduced mechanisms that are proposed for some simpliﬁed situations.

The discrepancy in the numerical simulation results and experimental results
in turbulent reacting ﬂows can be attributed to three factors: one is from the
modeling of the unclosed terms, the second due to chemistry and transport models,
and the third is the turbulence-chemistry interaction. The effects of initial / boundary
conditions and numerical error can also be important. When comparing the results
for practical combusting system, it is difficult to differentiate the discrepancy caused
by each of the reasons individually. In this chapter the effect of chemistry and
transport models of the different reduced reaction mechanisms of methane/air

combustion will be studied in a simple geometry: laminar counter ﬂow geometry. In

next chapter the effect of turbulence on methane ﬂames and vice versa will be studied.

Counter-ﬂow geometry has been widely used for the study of diffusion and
partial premixed ﬂames over the past three decades. It consists of an oxidizer
jet (or stream) and a fuel jet (or stream) that are issued from two different
nozzles opposing each other. A stagnation plane will be established in between
the two nozzles at a place depending on the momentum of the two streams. This
makes an interesting conﬁguration to study the ﬂame experimentally as a ﬂat
planar ﬂame will be established between the two jets. Also the two-dimensional
problem can be transformed into a one-dimensional problem which is easier to
study analytically and numerically. Counterﬂow diffusion ﬂame is ideal for study-
ing chemistry models and molecular transport since the ﬂame in this conﬁguration

often exhibits very similar scalar structure as “ﬂamelets” in a turbulent ﬂow field [35].

According to the laminar ﬁamelet concept, the turbulent diffusion ﬂame can
be viewed as an ensemble of laminar diffusion ﬂamelets. A steady laminar ﬂamelet is
a solution of steady one-dimensional diffusion reaction partial differential equations.
The reaction zone is thin sheet near the stoichiometric value. A look-up library of
laminar ﬂamelet solutions can be generated and stored that can be used in simula-
tions of turbulent combustion. F lamelet tables consists of the temperature, species
proﬁles in mixture fraction space parameterized by mixture fraction dissipation rate

at stoichiometric [35, 37].

The interest in ﬂamelets is due to its importance in turbulent modeling of
combustion. Lilian [29] studied the asymptotic structure of temperature for counter-
ﬂow diffusion ﬂames for one-step reaction with large activation energy and identiﬁed
four regimes. In this classic study of high activation-energy asymptotics (AEA), it is
predicted that the departure of the proﬁles from the equilibrium proﬁles is small for
all the strains up to the extinction. AEA predicts signiﬁcant fuel leakage through

the reaction zone but negligible oxygen leakage as extinction is approached. This is

contrary to what has been observed in experiments and numerical results. According
to AEA, in mixture fraction space the reactive-diffusive zone lies in between the two
convective-diffusive zones. The one-step reaction that represents the combustion

occurs in the reactive-diffusive zone.

It is always not possible that fuel and oxidizer are separated before burning in
the turbulent ﬁeld; sometimes fuel and oxidizer are mixed before they burn. Partial
premixed ﬂames are observed in non-premixed combustion at the base of lifted
ﬂame zone where the fuel and oxidizer are mixed before ignition. They are also
observed when there is a local extinction and consequently mixing takes place
without reaction. The partially premixed mode of burning produces less amount of
pollutants, and hence it is a preferred mode of combustion. These situations can be
modeled as an ensemble of laminar, partial premixed diffusion ﬂamelets. Domingo et
al. [17] have used partial premixed ﬂamelets in simulation of non-premixed turbulent

combustion using large eddy simulation.

Peters [36] analyzed the structure and extinction of partial premixed diffusion
ﬂamelets in initially unpremixed turbulent ﬂows with a one—step irreversible reaction
and large activation energy assumption. Situation where the local premixing occurs
has been considered and the outer ﬂame structure has been derived with a diffusion
ﬂame located around the stoichiometric mixture fraction and transition layers on rich
and lean sides, which are characterized by premixed ﬂame. When the Damkohler
number is reduced the premixed regions merge with the diffusion ﬂame and can lead
to extinction because of increased sensitivity to ﬂame stretch. And it was also shown
that quenching couldn’t occur in the diffusion ﬂamelet as long as the transition

layers are separated from it and thus, at extinction a merged ﬂame structure develops.

Tanoff and Smooke [54] have also studied the structure of partially premixed
ﬂames in physical space for a wide range of fuel premixing and strain rates in

methane-air vs. air counterﬂow ﬂames. They observed that the structure of their

partially premixed ﬂame changes rather signiﬁcantly as the fuel stream premixing
changes slightly around a critical equivalence ratio of (13 = 1.5 — 1.4. For 45 > 1.5 - 1.4,
a single merged ﬂame in the vicinity of stagnation plane was observed and for 45 < 1.4
a double ﬂame consisting of a premixed ﬂame and a diffusion ﬂame is formed. The
diffusion ﬂame was formed primarily due to the oxidation of CO and the premixed
ﬂame mainly involves the oxidation of the fuel and the production of H2 species.
The exact value of the critical equivalence ratio was observed to be dependent on
the strain rate - ﬂames close enough to the extinction remained merged even at the

lowest equivalence ratio.

Xue and Aggarwal [60] studied the effects of reaction mechanisms on the physical
structure and the extinction characteristics of a partially-premixed methane-air ﬂame
over a range of strain rates and for different equivalence ratios. They considered the
12—step, GRI-Mechs 2.11 85 3.0, and Cl and 02 mechanisms of Peters and Rogg [39].
Premixed ﬂame speed calculated using C2 and GRI-2.11 were in good agreement
with experimental results, whereas Cl mechanism showed signiﬁcant differences.
Diffusion ﬂame structure was found to be similar for all the mechanisms with
some differences in the peak temperatures. Results for Partially premixed ﬂames
showed a quantitative disagreement for all the ﬁve mechanisms, though they were
qualitatively similar, reproducing the double ﬂame structure associated with these
ﬂames. Compared to Cl mechanism 02 was more sensitive to the equivalence ratio.
Also for lower equivalence ratios the reaction zones merged more for 02 mechanism
than that for CI mechanism. Temperature gradients increased as the strain rates
were increased. However, the extinction strain rates for partial premixed ﬂames
were higher for 02 and GRI-Mech 2.11 mechanisms compared to those using Cl
mechanism. Xue et al. [61] compared the numerical results for temperature and
species with the experimental data for moderate strain rates an found that the C2

mechanism predictions are in better agreement with the experiment than those of

GRI-Mech 3.0 mechanism.

Turbulent non—premixed or partial premixing combustion can be modeled us-
ing laminar ﬂamelets. A ﬂamelet library is compiled using the solutions of laminar
counter-ﬂow ﬂames prior to the actual simulation of the complex ﬂows. Full mech-
anisms can be utilized for these laminar ﬂow calculations, since the computational
requirement is not stringent. In turbulent simulations with ﬁnite rate chemistry
reduced mechanisms are used due to the computational cost. A comparison of
turbulent simulation results via these two methods will not differentiate the effects
of chemistry and thus a simple conﬁguration is chosen here, where there are no
other modeling assumptions, to study the chemistry effects. In this present work
a comprehensive study of various chemistry models for varying strain and partial
premixing level of fuel stream with air will be done. The results of different reduced

mechanisms will be compared to the prediction of “full” mechanism (GRI-Mech 3.0).

The organization of this chapter is as follows. First, the general governing
equations for gaseous combustion are given, then the ﬂamelet equations are derived.
A brief description of counter-ﬂow diffusion ﬂames along with the governing equa-
tions and the numerical methods in solving the equations will be outlined next. In
the results section,the temperature structure in mixture fraction as predicted by
various mechanisms will be presented. Effects of strain and partial premixing on
the temperature structure will also be discussed. The structure of ﬂames obtained
with various mechanisms is compared with that of the GRI-Mech 3.0 for differ-
ent strain rates and different partial premixing. Finally, Major species and minor

species structures are presented and discussed and will be followed by the conclusions.

1.2 Mathematical and Numerical Formulation

A continuous reacting ﬂow system is described by the conservation equations of mass,
momentum, energy and the species transport equations along with the boundary
conditions, initial conditions, and the constitutive relations. In the following sections,
the general three-dimensional equations for the gaseous combustion system are ﬁrst
presented along with the models for chemical processes and molecular transport, after
that the ﬂamelet equations and the equations for counterﬂow (opposed diffusion)
ﬂame, which represents the ﬂamelet equations in physical space will be presented.
A mapping will then be deﬁned to transform the physical space counterﬂow ﬂame

solutions to mixture fraction space ﬂamelet solutions.

1.2.1 Equations for Gaseous Combustion

The conservation of mass law applied to a ﬂuid passing through an inﬁnitesimal, ﬁxed

control volume yields the following equation of continuity

0p 3(pU-i)
__ __ = 1.1
at + 3:121: 0 ( )

where p is the ﬂuid density and u,- is the ﬂuid velocity in ith direction.

The momentum balance applied to the same inﬁnitesimal, ﬁxed control volume

yields in the ith direction,

 

0(p111) 8(p’ll.z"ll.f) 0077
2 + . J = J +pfz' (1-2)
at dmj axj
where f, .—_—. g, is the gravitational body force per unit mass applied on the ﬂuid
element and oij is the stress tensor. The total stress tensor 0,-3- represents the surface

forces per unit volume on the ﬂuid element.

For a Newtonian ﬂuid, the stress tensor (023') will be given, in tensorial nota-

tion, by:

 

811.- {)u- 2 an.
(70' = ‘P‘Sij + ”(033; + 55%) - —A(5ij (1.3)

T 3 Lack
where u is the coefﬁcient of viscosity, p is the pressure and 6,7- is the Kronecker delta

function. In Eq. (1.3) the bulk viscosity is assumed to be negligible.

The application of energy conservation law to the control volume yields the

total energy equation,

313, (NEW) _ 0Q __ 0‘12? . . ' 0(0ijuj)
(9t + 0:17., _ (9t 8232. + pgykfkamt + Vim) + 81:,- (1.4)

 

where Et is the total energy per unit volume given by

”I ”I

 

Et : p(e + + potential energy + ...)

and e is the internal energy per unit mass, Yk is species It mass fraction, fk’, is the
body force in 2th direction of species k, and Vk.i is the diffusion velocity of species I:
in 2th direction. Q is the heat produced per unit volume by the external sources and
can be assumed to be zero and the ﬁrst term gives the rate of heat produced per unit
volume, second term gives the rate of heat ﬂux exchanged and third and fourth term
are respectively the work done per unit volume by body forces and surface forces. q,-

of the second term is the heat ﬂux in 2th direction and it is given by

0T
(12- : —k-(;)-;; + ngk,iYkllk + (IRJ' + (1.5)

where k is the coefﬁcient of thermal conductivity and T is the temperature. The
ﬁrst term in Eq. (1.5) is the heat transfer by heat conduction, the second term is

heat ﬂux due to mass diffusion and the third term represents the heat ﬂux due to

radiation. Dufour effect is neglected.

The species mass fractions in a reacting mixture are determined by the species

continuity equations which are given by

Bpk 3[pk(“j + Vii-4)]
T— + r ‘
()t ()Jij

 

=5]. k=1,2,...,K (1.6)

where pk (2 [211) is the partial density of species k, 5,, is the production rate of
species k due to chemical reactions and K is the total number of species. Yk is the
mass fraction of the species k. In the above equation ( Eq. (1.6)), I: does not obey

the summation rule. This equation can also be written in terms of mass fractions as

==wk k==1,2pn,Kf (L7)

peg art)+8(pYr1ft.j)
, .

, + u -—
dt 3 OJ: j 03:
where the mass ﬂux of species 1' due to the diffusion can be approximated using the

Fick’s law:

, Dk BY):
w...- = ‘7?
k I2

Dk being the diffusion coefﬁcient of species k.

The rate of production of each species 5,, is evaluated by an appropriate chemistry
model. A chemistry model consisting N reactions and K reactant species can be

symbolically represented as

K K
E: 14154;; :3 Z kaAk n : 1,2, N
kzl k~1

where A}; is the chemical symbol of reactant species A: and ”ka and ”ffk are the
stoichiometric coefﬁcients of species It in reaction 72. The chemical source term wk

is a consequence of the production and destruction of species 1: due to each of the

reaction step, i.e.

N
wk 2 Z wnk

n=1
and wnk IS given by:
II

7 K I I{
W . I V II V
- II I I. . ~ 5 ‘,
wnk : (Vnk _ Vnk)Wn 'I‘f’npp' I I Yk "k — A'bnpp I I Yknk (18)
kzl k2]

where p’ : 25:1 V7216, p” :2 25:] fok and kfn and kbn are respectively forward and
backward reaction rates for reaction n. Wk is the molecular weight of species k and

Wu is

K
= z w.
k: 1

The rate constants in Eq. (1.8) can be calculated by the following equations:

/ —E
kfn Z AnT‘Jn exp (ﬁg) (1.9)
k-
[Tb—’1‘ = anT)
fn

where An is the pre—exponential factor, [in is the temperature exponentJ-t is the
universal gas constant, En is the activation energy characterizing the reaction n and

Kn is the equilibrium constant for reaction 72..

The ﬁnal equation that closes the above described equations by relating the
thermodynamic variables (p, p, T,e,h) is the state equation which is given for a

perfect gas by:

pszT (1.10)

10

where W is the mixture-averaged molar mass given by

1.2.2 Flamelet Formulation

\Nhen the chemistry is fast as compared to transport processes (convection and
diffusion), combustion can be shown [35, 37] to occur in laminar like thin layers
within the turbulent ﬂow ﬁeld. These thin layers are called ‘ﬂamelets’ and the regime
of turbulent combustion where such an assumption is valid is known as ﬂamelet
regime. There has been a great amount of study on this concept, and extensive
literature is available since most of the practical applications fall in to this regime.
For non-premixed ﬂame, the one-dimensional ‘ﬂamelet equations’ will be derived in

this section from Eqs. (1.1)-(1.7) via a coordinate transformation.

In the non-premixed ﬂamelet, all the scalars (temperature, concentration of
species and density) are uniquely related to a single chemistry independent tracking

scalar variable called “mixture fraction” (Z) as,

Yk = lit-(Z) T = T(Z) p = plz)

The mixture fraction describes the level of mixedness of the fuel and oxidizer and can
be interpreted (in a two-feed system) as the amount of mass originating from the fuel
stream in the total mass at a particular location in space. It was ﬁrst introduced by

Burke and Schumann [12] for a one—step global chemical reaction
VFF + V00 —+ VpP,

where F, O and P refer to fuel, oxidizer and product respectively. If .2 be set to an

11

operator deﬁned as

0 8 0 0
3’ - ”a ”“107; ‘ 500%,)»

where D is the diffusivity of fuel and oxidizer assumed to be equal. Then the transport

equations for the species mass fractions becomes
sop/F) : —upwpo’

20/0) :3 —VOI’VFti)/.

Manipulation of the above equations to eliminate (reaction rate) (12’ leads to

 

 

_ YF Y0 _
$03) 2 gfupwp _ VOWO) _ O

The above function 5 can be normalized to yield the deﬁnition of mixture fraction
with a value of unity in fuel stream (feed 1) and a value of zero in the oxidizer stream

(feed 2) as
_ {3...[32 _ VYF—YO+YO,2

z _ 7 _
51 — (32 VYF,1 + Y0,2

 

 

where V is given by,
Vol/170

_ 1!}: IV}:

This deﬁnition of mixture fraction based on the fuel and oxidizer mass fractions is
not valid for more general chemistry models. Since the intermediates are formed from
the reactants over a broad region in space, it is possible that the mass fractions of
fuel and oxidizer may be small in this region and therefore will have a constant value
of mixture fraction according to the above deﬁnition. This situation can be remedied

by using the deﬁnition given by Bilger [7] based on the element mass fractions

2(ZC—ZC,0) + 0-5(ZH_ZH.O) _ (Zo—Zoc)

 

 

 

Z: ”'5 W” ”0 (1.11)
2(ZC,F-ZC,0) 0-5(ZH.F-ZH.0) _ (Zo.F-ZO.0)
WC + W” W0

12

where Z, is the element mass fraction of element 2' and W,- is the atomic weight of

the element 2T.
Mixture fraction transport is governed by the equation given by

02 (92 (9 (D OZ) (1.12)

P— +p“j—”T = ““1" P 2““.

0t 0x] (9.15] 0:13]

where Z is the mixture fraction and D Z is the diffusion coefﬁcient of mixture fraction.
This is exactly same equation as the species equation (Eq. (1.7)) with Fick’s law for

species diffusion and no chemical source/sink terms.

The balance equations for species and temperature can be written from Eq. (1.7) and
Eq. (1.4) respectively. In Eq. (1.7), Fick’s law is assumed to be valid for diffusion
velocities. In Eq. (1.4), the heat production term by external forces is neglected].
Viscous terms and pressure gradient terms are also neglected based on the low mach
number approximations and q, can be substituted by Eq. (1.5). Consequently, the

species and energy equation reduces to, after some manipulations

 

BY), OYk a OYk _ , _
0T 8T 6 er 1 K 1 0cm .-
p —+u»,—— —+— pD——— =—— hkwk—-— ’J (1.14)
(dt J 011-) OIJ-( Oxj) cp k=1 6,; 0x]-

where D is assumed to be same for all the species and is equal to the thermal
diffusivity of the mixture at that point (equal diffusivity assumption) implying a
Lewis number of one (Le = 1) for all the species and Cpl: is assumed constant and

equal to GP for all species.

In ﬂamelet models, the above equations are written in a coordinate system
that is attached to the surface of stoichiometric mixture fraction in which the 2:1

coordinate is replaced by mixture fraction Z. With a Crocco—type transformation

13

Eqs. (1.13) and (1.14) are written as

 

2 2

pa.- 01:,- 622+”
K

(9T 82 262T h . 10m,

——-—pD( )_ 2:..1. __

”07‘ E 322—, (1'16)

w!
1 0;; cp 011:,-

where r = t and 2:2 & .723 are directions perpendicular to Z. All the higher order
terms in the (thin) ﬂamelet reaction zone have been neglected (see [35] for details).
This formulation leads to a one-dimensional ﬂame structure normal to the surface
of stoichiometric mixture. The inﬂuence of the ﬂow ﬁeld in the ﬂamelet equations,
Eqs. (1.15) and (1.16), is introduced through the instantaneous scalar dissipation rate
X = 2D(%)2
An essential objective of the ﬂamelet formulation is to decouple the chemical kinet-
ics from the turbulence. One of the simple laminar ﬂow conﬁguration that can be
representative for ﬂamelets and may be used for building a ﬂamelet library is the
counter-ﬂow ﬂame. A ﬂamelet library consists of the proﬁles of the temperature and
mass fractions of the reactive species as a function of the mixture fraction Z & x and

is used in turbulent ﬂame calculations.

1 .2.3 Counterﬂow Flames

In counterﬂow ﬂames, a stagnation plane ﬂame is established in the middle of the two
opposing nozzles at a place, which is dependent on the momentum of two streams.
One of the streams, which is called fuel stream, contains the partially premixed fuel
and the other, called oxidizer stream, contains the air. The mass conservation equa-

tion for this counter-ﬂow geometry is given by

14

807a) +1807”)
03: 7‘ 0r

 

 

=0

where u and v are the axial and radial velocity components and p is the mass density.
With a karman similarity transform, the dependent variables depends only on the
axial coordinates. Assuming velocity is a function of r only, the above equation can

be transformed to

 

dF(;r)
, : 1.1
cos) d, ( 7)
where
C(23) = —— and F(:r) : p711
The radial pressure gradient is constant,
10p ., ,.
ff)? — H — constant, (1.18)
and the radial momentum equation is governed by
d PG 302 d d C
H—2—(——- _ —[-—— _]= 1.19
dd: p)+ p +(1:1: 'udx(p) O ( )
The transport equations for energy and species are
(1T 1 d dT p (1T 1
v.———— A— — .Y.V.— —- h..' .20 1.20
uda: c-da:( (1.1:)+cp§k:cpA 1‘ kdr+cp; kwk ( )
dY)c d
—— — Y-V. — ' . '17.: k: 1.21
pads: +d:r(p’” I.) “lift 0 1,K ( )

where the diffusion velocity is either given by the multi-component formulation

15

K T
1 dX- D. ldT

V.=___ W-D.-—J——"—— 1.22

f Xij; J “(1.2; kade ( l

or the mixture-averaged formulation

v 1 D ka DZ.“ 1 dT D 1— Y),
k 2 - . km , — ——_,‘, km. 2 K
A}; d]: ka T (1.1: Ejaék Xj/Djk

 

(1.23)

where ij, ka, Djk and D; are the multi-component, mixture averaged, binary,

and thermal diffusion coefﬁcients, respectively.

The boundary conditions for the fuel (subscript F) and oxidizer (subscript O)

streams at the nozzles are

Atx=02 F=KP§E,G=0,T=TF, p’erk+kaVk=(pqu)F (1.24)
Atx:L: F=p—02'E,G=0,T=TO, pqu+kaVk=(pqu)0 (1.25)

In this chapter, when diffusivities are equal, we use the mixture-averaged formula for
diffusion velocity. Further, we assume that the thermal diffusion coefficient D; is
zero, and the mixture-averaged diffusion coefﬁcient is replaced by thermal diffusivity
(A/pcpymix). In effect it will be reduced to the F ick’s law with unit Lewis number. If
Eq. (1.23) with the above mentioned modiﬁcations is used in Eq. (1.21), will give the
ﬂamelet equation in physical space. By using the deﬁnition given in Eq. (1.12) the
mixture fraction can be calculated at each point in the physical space then the solution
of Eq. (1.21) can be mapped to be the solution of the ﬂamelet equation (Eq. (1.15)).
Similarly the equation (1.20) can be transformed to the ﬂamelet equation (1.16) and
the strain rate, which is uniquely related to the scalar mixture fraction dissipation [35],

can be approximately computed as [26, 49]

16

 

2[1’0] [Uflvfjf
— 1 + ______.. 126

Here, L is the distance between the two co-axial nozzles, 22 is the velocity and p is the

density at the nozzle exit. Subscripts o and f denote the oxidizer and fuel.

1 .2.4 Numerical Method

The above set of equations for counterﬂow ﬂame Eqs. 1.17-1.25 will be solved to
determine the temperature and species ﬁelds. Then each grid point in the physical

space is mapped into a point in mixture fraction space using the Eq. (1.11).

The counter-ﬂow diffusion and partial premixed ﬂame is simulated using the
Oppdif [30] application of Chemkin II [24]. The differential equations, Eqs. (1.17).—
(1.23) along with the boundary conditions (Eqs. (1.24)—(1.25)) given above will form
a two-point boundary value problem for the dependent variables (F, G, H, T, Yk).
These equations are discretized with conventional ﬁnite differencing techniques for
non—uniform mesh spacing. Diffusive terms use central differences, with truncation
error that is second order in the mesh spacing. Convective terms use the upwind
differencing, which uses the sign of the velocity to choose which direction the spatial
difference will go. The truncation error of this approximation is ﬁrst-order in the
mesh spacing, leading to “artiﬁcial diffusion”, which will avoid unwanted oscillations
during the solution on the coarse mesh. Alternatively, the convective terms can be

central differenced with second order accuracy in mesh spacing.

Oppdif, incorporates all the features described above and solves for the tem-
perature, species mass fractions, axial and radial velocity components and radial
pressure gradient. Chemical, thermodynamic and transport properties are evaluated
in the model with the use of Chemkin and transport packages [23]. All the

simulations conducted in this work uses mixture averaged transport properties.

17

Some selected test cases are run using the multi-component and mixture—averaged
approximations for the transport properties. The difference in temperature predicted

obtained by these two methods is insigniﬁcant.

The code is modiﬁed to include the equal diffusivity assumption in some of
the simulations. This was achieved by setting the diffusion coefﬁcient for all species
equal to thermal diffusivity of the mixture at that location. Mixture fraction is
calculated using the elements C, H and O and the formulation suggested by Bilger
for the fuel containing C, H and O (see Eq. (1.11)). Since the governing equations
given above does not contain the strain rate explicitly, the characteristic strain rate
of the ﬂow may be approximated by a formula based on the velocities and densities
of the oxidizer and fuel streams and is given by Eq. (1.26). Velocities of both the
fuel and the oxidizer jet streams are always taken to be equal and varied for different
strain rates along with the distance between the nozzles. Initial temperature for the

fuel and oxidizer stream is 300K in all simulations, unless stated otherwise.

18

1.3 Results and Discussion

All the mechanisms considered for this study, along with the number of species,
reaction steps, and the corresponding reference are tabulated in Table 1.1. Results
obtained by simulating the laminar counterﬂow ﬂame for various cases are discussed
in this section. The effects of strain and partial premixing are studied by analyzing
the structure of temperature and species in physical and compositional (mixture
fraction) space. The performance of reduced chemistry models are assessed by
comparing the results generated with these mechanisms with these via GRI—Mech
3.0. The sensitivity of the ignition temperature to strain rate, premixing and

chemistry model will also be discussed.

Table 1.1. Reaction Mechanisms Assessed.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mechanism Species Reactions Reference
4-step 8 4 Peters N. [38]
5-step 9 5 Hernanth P. Mallampalli et a1 :33]

6-step 10 6 W.-C. Chang and C.-Y. Chen :14:

10-step 14 10 W.-C. Chang and C.-Y. Chen :14]

12-step 16 12 C. J. Sung et a1 [53]

Skeletal 18 35 M. D. Smooke and V. Giovangigli [51]
Gri-Mech 1.2 32 177 M. Frenklach et a1 [19]
Gri-Mech 2.11 49 279 C. T. Bowman et al [11]
Gri-Mech 3.0 53 325 Gregory P Smith et a1 [50]

 

 

When the diffusivities of all the species in the system are taken to be equal and
the Lewis number is taken to be unity, the structure produced by counterﬂow
diffusion/partially-premixed ﬂame is dependent only on mixture fraction and the
stoichiometric scalar dissipation rate [6, 35, 37]. However, the results could be depen-
dent on the computational domain size (the distance between the two stream nozzles)
when the domain size is not sufﬁciently large. In this case, a zero gradient inlet con-

ditions are not guaranteed and there will be a heat loss from the reaction zone to the

19

nozzle due to conduction [60]. This may cause a decrease in predicted peak tempera-
ture. To ensure the accuracy of results, various testings were performed. For example,
in Figure 1.1(a), 1.1(b) and 1.1(c) show the ﬂame (temperature structure) for three
different domain sizes, 0.4, 1 and 2 centimeters respectively, a strain rate of 100(1/5),
and 75% partial premixing of the fuel. For the cases with 1 and 2 cm domain size
(cases b and c) the ﬂame is suﬂiciently far from both the fuel and oxidizer nozzles,
and for the case with 0.4 centimeters domain size the ﬂame hits the nozzles creating
a nonzero temperature gradient at the boundary, which causes the heat to ﬂow out of
the domain [60]. Figure 1.1(d) compares the temperature structure, produced for the
three different cases discussed above, in mixture fraction space. Whereas in case (b)
and (0) temperature proﬁles are almost identical with negligible difference [6], case
(a) temperature proﬁle is signiﬁcantly different from the other two. In case (a) there
is a heat loss to the nozzles due to non-zero gradients at the boundary. In this study
care has been taken to contain the temperature proﬁle in between the nozzles and

cases such as (a) which display a disparity with others have been avoided.

1.3. 1 Temperature Field

In this section the temperature structure is studied in both physical and compo-
sitional space as predicted by different reduced mechanisms. Figure 1.2 shows the
typical effects of strain rate and partial premixing of the fuel on the temperature
structure. Temperature is plotted in compositional space as predicted by GRI-Mech
2.11 mechanism for constant premixing and varying strain rates (Figure 1.2(a)), and

constant strain rate and varying levels of partial premixing (Figure 1.2(b)).

Figure 1.2(a), which is plotted for constant fuel premixing of 75% and a strain
that is varying from 10400 (Us), is in agreement with the earlier analytical,
experimental, and numerical studies. The peak temperature is seen to be decreasing
with increasing strain rate (or scalar dissipation rate, which is directly related to

the strain rate). At the lowest strain rate of 10 1/s, a slight hump is seen on the

20

rich side of the ﬂame, which is a precursor to the famous double ﬂame structure. As
the strain rate (or scalar dissipation rate) is increased, the premixed ﬂame on the
rich side will be pushed into the diffusion ﬂame leading to a merged ﬂame as noted

by asymptotic analysis of Peters [36] and numerical studies of Tanoff and Smooke [54].

Figure 1.2(b) shows the temperature plot in mixture fraction space for a con-
stant strain rate of 25 Us and ﬁve different premixing levels of the fuel stream with
air (%A =0, 25, 50, 75, 85). Premixing level has been given in terms of volume or
mole percentage of air in the fuel stream. As the premixing level is increased from 0
(pure diffusion ﬂame) to 85 percentage of air in fuel stream, the peak temperature
shifts to the higher mixture fraction value (so does the stoichiometric mixture
fraction value). The vertical lines in the plot show the stoichiometric mixture
fraction values, from left to right, for each of the increased premixing level. A
premixed ﬂame is formed in the fuel stream as the premixing level is increased, which
is due to the availability of enough oxidizer to the fuel in the fuel stream. A diffusion
ﬂame is formed from the products of premixed ﬂame and oxidizer stream [10, 54].
The separation of the premixed ﬂame and diffusion ﬂame will lead to a double ﬂame
structure [28, 36, 47, 52, 62]. Plots similar to the above, with variable diffusivity
effects included (not shown here) indicate trends similar to those of equal diffusivity
case, even though there are some quantitative disagreements. Interestingly, the peak
adiabatic ﬂame temperature does not signiﬁcantly change with partial premixing of

the fuel with air, but it decreases slightly with strain rate as expected.

The temperature proﬁles in mixture fraction space as predicted by different
mechanisms at various strain rates and various partial premixing levels are shown in
Figures 1.3 and 1.4. Figure 1.3 shows the plots for strain rates of 10, 50, 150 and 300
l/s for a fuel stream premixed with 75% air. Here the 5-step and 6-step mechanisms
over—predict the temperature in the fuel-rich region, but the 4-step one under-predicts
the temperature in the same region. However, the difference in the temperatures

predicted by various mechanisms is less for the lower strain rates (compare 1.3(a)

21

and 1.3(b)). However, for all the values of strain rates 5—step over-predicts the
peak temperature whereas 4-step under—predicts it and the 10-step and 12-step
mechanisms predictions are in good agreement with the GRI mechanisms. Even
though the difference between the GRI-Mechs and the reduced mechanisms decrease
with the strain rate, 4-step, 5-step and 6—step mechanisms, which contains only the

Cl chain species, are not able to fully reproduce the results of GRI-mechanisms [60].

The temperature over-prediction of 5—step, 6—step mechanisms (as compared to
GRI-Mechs and 10, 12 steps mechanisms) can be partially explained by considering
the proﬁles of reactant production rates in mixture fraction space. Figure 1.5
shows the production rates of CH4, 02, and 002 species for 5-step and 10-step
mechanisms. For 5-step mechanism the consumption of CH4 and 02 has one of
the peaks in the fuel stream (near Z = 1). The early peak in the fuel stream for
consumption of CH4 for 5-step maybe due to the representation of combustion in
reactions. Oxidation of CH4 involves 02 and H20 which are readily available in
partially premixed fuel stream. In 10-step mechanism, radicals like H and CH20 are

involved in the oxidation of methane.

Figure 1.4 is plotted for various levels of partial premixing and at ﬁxed strain
rate of 25 US for different mechanisms. The temperature is over-predicted on the
rich side at lower premixing levels for 4-step mechanism, but is under-predicted for
higher partial premixing. And, for 5-step and 6—step mechanisms their prediction
deteriorates as the level of premixing is increased from 0% air to 75% as the air
available in the fuel stream is sufficient to cause the reactions to occur in the fuel
stream. Again, the 10-step and 12-step mechanisms perform well and predict the
temperature structure as the full mechanisms with high accuracy due to better

representation of fuel consumption reactions by radicals near the diffusion ﬂame zone.

To better quantify the differences in the predictions of temperature by differ-

ent mechanisms, a parameter called mean relative percentage difference is deﬁned

22

here, which represents the difference between GRI-Mech 3.0 (which is assumed to be

the best mechanism tested here) and the mechanism in question. It is given as

 

(T -T A,1)><100
(PD): Ref TR
Ref

 

where the T Re f is the temperature obtained by the reference mechanism (here,
GRI-Mech 3.0) and TR M is the temperature obtained by the reduced mechanism.
Averaging is performed over all the values in mixture fraction space on the rich
side of stoichiometric value which is represented by the bar. This parameter has
been plotted in Figure 1.6 for all of our simulations. Figure 1.6(a) shows the mean
percentage difference versus strain rate for different reduced mechanisms for 75%
premixing. This ﬁgure shows that the percent difference is reducing with strain rate
for all the mechanisms. 4-step, 5-step, 5-step (Modiﬁed version), 6-step reduced
reaction mechanisms perform badly for all cases and specially for low strains (refer
to the explanation of Figure 1.3). 10-step, 12-step results lie within i2% for all
the strain rates. While the 4—step, 10—step and 12-step results show positive values
indicating that the temperature predicted by these mechanisms are lesser than that
of the reference mechanism GRI-Mech 3.0, 5-step, modiﬁed 5-step and 6—step values
are all negative and thus over-predicting temperature on rich side. It can also be

observed that (PD) is nearly constant for higher strain rates.

Figure 1.6(b) shows the variation of (PD) with respect to percentage of air in
fuel stream for all the mechanisms. The relative mean difference for 4-step, 5-step
and 6-step increases as the percentage of air in fuel stream is increased. 10-step
and 12-step are predicting the temperature with good accuracy. One important
point to be noted is that for 4-step mechanism the error seems to decrease ﬁrst and
then increase as more air is added to the fuel stream. However, the mean error
plotted in this ﬁgure is absolute and since 4—step moves from over-predicting at lower

premixing levels to under-predicting at higher premixing levels, it appears that the

23

error decreases ﬁrst and then increases. It can also be seen that the mean difference
decreases when percentage of air increases from 75% to 85%. The reasons for this
behavior are explained below. Figure 1.7 show the maximum temperature difference
between reduced and full (GRI-Mech 3.0) mechanisms in mixture fraction space
versus the partial premixing. It can be seen that the maximum temperature increases
with increasing premixing of fuel stream indicating that indeed the increasing levels

of premixing worsens the temperature prediction for all the reduced mechanisms.

Figure 1.8(a) and Figure 1.8(b) shows the plot of temperature difference in
mixture fraction space for 75% premixing and 85% premixing, respectively. It can be
seen in Figure 1.8(a) that the width of region in mixture fraction space where there
is signiﬁcant temperature difference is much more than that for the 85% premixing
even though the difference itself is small indicating that the premixed reaction zone
is well separated from the diffusion ﬂame in the 85% case. Since the prediction is
bad in the premixed region than the non-premixed, and the average values are taken
over the richer side of stoichiometric mixture fraction region, we see in Figure 1.6(b)

that this average is lower for 85%.

1.3.2 Species Field

In addition to temperature, the other variables calculated and modeled in the
ﬂamelet methods include the species mass fractions, which are functions of mixture
fraction alone under equal diffusivity and fast chemistry assumptions. So, it becomes
important that the reduced reaction mechanisms should predict the species proﬁles
accurately. The results for some important species are discussed in this section.
First, the results for the major species CH4, 02 and C02 will be considered, and
comparison will be made between the structures of these species as obtained by
different mechanisms in compositional space. Then the minor species such as
H, H2,C0 and NO will be discussed. Similar to that of temperature analysis, a

parameter will be deﬁned for each species, which will represent the deviation of

24

predicted results via various mechanisms from GRI-Mech 3.0 results. The variation

of this parameter with strain and partial premixing will be plotted.

The effect of partial premixing and strain rate on methane and oxygen are
considered in ﬁgures 1.10 and 1.10, respectively. Figures 1.9(a) and 1.10(a) shows
the mass fractions of CH4 and 02 in mixture fraction space as produced by different
mechanisms. Each plot contains four subplots at different premixing levels (0,
25, 50, 75 percent) of fuel stream. Evidently, the compositional structure of both
YCH4 and Y02 as computed by 4-step, 5-step and 6—step deviate more from the
GRI-Mech 3.0 as the level of partial premixing is increased for a constant strain
rate. 4-step over-predicts the mass fractions (or under-predicts the consumption)
on the rich ﬂame side, where as the 5-step and 6-step under-predicts the mass
fractions. 10-step and 12-step follow closely the trends of full mechanisms. These
trends are consistent with the trends in temperature structure (see ﬁgure 1.4).
Higher temperatures are predicted at the regions where the consumption rates
for reactants are predicted higher. A slight deviation is seen in the predictions
of 4-step mechanism. At lower premixing levels even though the consumption
rates of methane and oxygen for 4-step mechanism on the rich side of the ﬂame
are higher compared to other mechanisms, there is also a slightly higher predic-
tion of temperature. This may be due to the fact that the 4-step mechanism does

not involve a lot of other endothermic reactions as compared to the fuller mechanisms.

The effect of strain rate on YCH4 and Y02 for various mechanisms is shown in
Figures 1.9(b) and 1.10(b). The four subplots in these ﬁgures are for different
strain rates at a constant premixing of 75%. Again, similar to the temperature
structure (see Figure 1.3), the reduced reaction model predict results closer to
those of GRI-Mech 3.0 when the strain rate is increased. In both the ﬁxed strain
rate and ﬁxed partial premixing plots, the proﬁles predicted by 4-step mechanism
are piece-wise linear with a discontinuity in the gradient near the stoichiometric

mixture fraction, where the reaction zone is located. This can be explained by the

25

non-existent of premixed ﬂame zone in 4-step ﬂames (double ﬂame structure) for
any of the cases considered (Figure 1.11). For the 5-step and 6-step mechanisms, at
the lower strain rates and higher partial premixing levels one can ﬁnd a pronounced
double ﬂame structure as the reaction mechanisms predict a longer separation in the

diffusion and premixed reaction zones as shown in Figure 1.5 and Figure 1.11.

The compositional structures of C02, H2 and C0, as predicted by different
mechanisms, and the effect of strain rate and partial premixing on them, are studied
through ﬁgures 1.12, 1.13, and 1.14. Subplots (a) and (b) show the effects of varying
partial premixing and strain rates, respectively. Interestingly in Figure 1.12(a) for
002, unlike for the other species, it is seen that the disagreement is more pronounced
in lower premixing levels than the higher premixing especially for 4-step and 6—step.
This is explained in Figure 1.15, where the molar production rate of 002 along with
its structure as predicted by 4-step and 5—step mechanisms are shown. The sharp
peak in the mass fraction proﬁle is not predicted very well by 4-step mechanism
because of the sharp gradient in the production rate proﬁle. In 4—step mechanism

the only reaction that involves 002 is the reaction

C0+0H.—_-CO.Z+H2

whereas, in 5-Step mechanism it is combination of three reactions

3H2 + 02 + 002 = 3HQO + CO

ffg + C02 2 H20 + CO
3H2 + 002 + 2N0 = ZHQO + CO + N2

where only the ﬁrst two reactions are prominent in deciding the production of
C02. Prediction via 5-step at lower premixing levels compare well with GRI—Mechs,
but it deviates at higher partial premixing levels, which is due to the incorrect

representation of the premixed ﬂame in fuel stream. The effect of strain rate in

26

Figure 1.12 (b), shows similar trends of betterment with increased strain rate.

Figures 1.13 and 1.14 show the structures of H2 and CO respectively. Similar
to other species, the deviations between GRI-Mech 3.0 and other mechanisms
increases as the strain rate is decreased, but the differences are much more pro-
nounced for H2 and CO than for the 002 and 02. An interesting observation is
the similarities in the reduced reaction mechanisms behavior. For example, while
5-step and 6-step mechanisms have same behavior and predict higher mass fraction,
4-step predicts lower, and 10—step and 12-step predict much closer to the proﬁle
predicted by GRI-Mechs. This is due to the way these mechanisms are developed;
5-step and 6—step are derived from GRI-Mech 1.2, and contains only the 01 species;
10—step and 12-step are also derived from GRI-Mech 1.2, but in addition to 01
species they also incorporate the effects of C2 species reactions. 4-step mechanism
is derived from a skeletal reaction mechanism of 25 reactions and contain only 01
species. As also observed earlier (Refer to [60]), mechanisms that took only 01
reactions into consideration tend to be generally less accurately. The effect of partial
premixing, on all the mechanisms, has been the reduction in peak mass fraction,

and movement of the position of its occurrence towards higher mixture fraction values.

The results for H2 and CO in Figures 1.13(b) and 1.14(b) indicates that the
maximum mass fraction predicted by 4-step increases as the strain rate increases
whereas, for the other mechanisms the opposite is true. This gives an erroneous view
of the effect of strain on CO and H2 structures if 4-step mechanism is used [15].
10-step and 12-step compares well with the GRI-Mechs, especially the GRI-Mech
2.11, even though there are some deviations from GRI-Mech 3.0 at lower strain rates.
In ﬁgure 1.14(a), it is shown that the peak of CO increases as the partial premixing
is increased. This may seem surprising as we expect a reduction in CO emissions as
more complete combustion occurs due to increased amount of oxidizer. The reason
for this behavior is that in partially premixed ﬂames, production of CO will start

in fuel stream due to the oxidation of methane [10], which has to be transported to

27

the CO/H2 oxidation diffusion ﬂame. Therefore, the increase in CO production is
due to increase in partial premixing and increase in the distance between methane

oxidation zone and CO/H2 oxidation zone.

Figures 1.16(a) and 1.16(b) shows the variation of mass fraction of H in mix-
ture fraction space. The plots needed to be zoomed to show the details and
consequently, the scales on individual plots are different. As the premixing is
increased the H structure is broadened in mixture fraction space, a behavior that is
different for different mechanisms. The peak mass fraction is decreasing for all the
mechanisms with an increase in premixed air except for 4-step mechanism for which
it remains nearly same. 5-step mechanism does not contain H, and the deviation
from the full mechanisms for the 6—step mechanism is seen to increase initially
then reduces. However, as the strain rate is increased the peak of H mass fraction
structure is increased slowly for all the kinetic models except for the 4-step one.
4-step structure tends to deviate more from the full mechanisms as the strain rate is
increased. Overall, predictions of H radical by 4-step is inaccurate for the cases (see

Figure 1. of [8]).

Figures 1.17(a) and 1.17(b) Show of structure of NO in compositional space
for different partial premixing and different strain rates. There has been a great
detailed study about the formation of N O; in counterﬂow ﬂames [10]. The mecha-
nisms considered here that contain N 0 species are the 5-step, GRI—Mech 2.11 and
GRI-Mech 3.0. The prediction by GRI-Mech 3.0 is very different from the other two
as was noted earlier and 5-step prediction is close to GRI-Mech 2.11. Nevertheless,
in all cases the peak production of NO occurs near the stoichiometric value of
mixture except in the ﬂame with 75% premixing and low strain. In this ﬂame it
seems to shift slightly to the right of the stoichiometric value. This is because of the
production of N O by two different mechanisms - F inimore (Prompt) and thermal
N 0;; mechanisms [54]. Increasing premixing decreases the peak N 0 production as

can be seen from Figure 1.17(a) [10, 54] and also increasing strain rates sees the

28

decrement in production of N O.

A parameter, similar to that which was deﬁned for temperature will be de—
ﬁned below for species mass fractions to study the deviations of results obtained
by various mechanisms from those of reference “full” mechanism. The reference
mechanism is again taken to be GRI-Mech 3.0. Instead of relative mean difference,

we deﬁne the mean difference (mean is represented by overbar or 0), (DP), to be

 

(DP) = (YRef — YM)

where Y denotes the mass fraction of species and the subscripts Ref and M stands

for reference (which is GRI-Mech-3.0) and mechanism in question respectively.

Figures 1.18(a) and 1.18(b) show the temperature difference between GRI-
Mech 3.0 and other mechanisms, averaged over the mixture fraction values greater
than the stoichiometric value, with respect to the strain rate for CH4 and CO,
respectively. It can be observed in both the ﬁgures that as the strain is increased,
(DF) decreases for almost all the mechanisms. The results for 4—step mechanism are
somewhat different; for this mechanism ﬁrst, (DF) decreases, passes through zero
line, and then increases on other side (Values above 0 implies species mass fraction
is under-predicted). There is a point in both ﬁgures for which the 4-step mechanism
crosses the zero line which is where the 4-step values matches to that of GRI-Mech
3.0 in an average sense. But, it has to be noted that the value of the strain for
different species is different, and thus one can not conclude that 4-step accurately

predict all the variables (temperature and species mass fraction).

Figures 1.19(a) and 1.19(b) show the plot of (BF) versus percentage of air in
fuel stream for CH4 and CO. The trend for 5-step and 6—step mechanisms is similar
to that observed for temperature; it ﬁrst increases with increasing the premixing

then drops at 85% premixing. 4-step mechanism shows relatively large deviations

29

as compared to other mechanisms for CH4. The drop in (DF) in Figure 1.19(b)
when premixing air is increased from 75% to 85% can be attributed to lesser width
of difference zone in mixture fraction domain. This is shown in Figure 1.20, where
the maximum difference in YCO predictions by 5-step and 6—step mechanisms from

those of GRI-Mech 3.0 increases with premixing.

4-step, 5-step and 6—step mechanisms, which contain only the Cl species, does
not perform satisfactorily for any of the strain rates and for different partial pre-
mixing though they predict within tolerance limits in some situations. 10—step and
12-step which contain both Cl and Cg species are able to reproduce the temperature
structure within 2 percent difference from the reference GRI-Mech 3.0 for all the
strain rates and premixing levels that are considered in this study. The temperature
structure of 10 and 12 step mechanisms are more close to GRI-Mech 1.2 and 2.11
than GRI-Mech 3.0, which can be explained by the fact that they both are reduced
from GRI-Mech 1.2. Considering that 10—step mechanism contains two species and
two reactions steps less than that of 12-step mechanism, it can be said that it will

be computationally less expensive to use 10-step mechanism.

1.3.3 Autoignition

In the past several decades the phenomena of ﬂame extinction has been widely
studied using analytical, experimental, and numerical approaches. Equally important
ignition phenomena has also been the subject of extensive study in the recent years.
Non-premixed mode of ignition takes place usually in diesel engines and supersonic
combustors where the fuel is injected into heated air, which acts as a source of
ignition. This type of ignition can be studied through the counterﬂow (opposed
jet) ﬂame model, where the cold fuel is ignited by an opposed hot air jet. Another
ﬂame ignition process which can be modeled by opposed jets is the ﬂamelet ignition
in turbulent jet ﬂames. Ignition by this procedure has been studied extensively

for methane/ air mixtures of various dilutions with nitrogen [18]. The method that

30

was employed to determine the ignition temperature was to increase the oxidizer
temperature incrementally until ignition occurs and a converged steady state solution

is obtained.

An equally important situation that occurs in turbulent ﬂames is the re-ignition
of the strain extinguished ﬂamelets, which can not be studied using cold fuel/ hot
air conﬁguration. It is more likely, in these situations, that the fuel and air will
both be at the same high temperature. For example quenched ﬂamelet may contain
both fuel and oxidizer at same temperature rather than the fuel being at 300K and
oxidizer being at “ignition temperature” [18]. Ignition temperature obtained by this
procedure may not be accurate for modeling the re-ignition of ﬂamelets as they may

over-predict the actual values.

In this work we determined the ignition temperature with the same fuel and
oxidizer jet temperatures, and the initial temperature variation is taken to be a
straight line in the domain. The magnitude of initial temperature is gradually
increased in equal increments and a steady state solution is sought. If, a steady state
solution is obtained at some temperature then the incremental step is halved and
solution is sought in other direction (i.e., temperature is reduced with a halved step
from the point where it was found to ignite). This procedure is repeated until the
ignition temperature is determined with in a prescribed range. This whole procedure
is automated and the ignition temperature can be determined with an accuracy of

10 degrees Kelvin within 10—15 iterations.

Figure 1.21 is plotted for the ignition temperature with respect to the strain
rate for various partial premixing levels of fuel stream with air. It can be seen that
the ignition temperature increases with strain rate for all premixing levels then it
reaches a constant value for strain rate values higher than 300 Us The constant
value it reaches at higher strain rate is dependent on the the level of premixing,

and it decreases with increasing partial premixing. At strain rates below 100, the

31

added premixed oxidizer to fuel stream does not seem to have signiﬁcant effect on

the ignition temperature.

These results indicate that in a turbulent ﬂow, where signiﬁcant variations in
strain rate and partial premixing of the fuel/oxidizer/products, the ignition, ex-
tinction and re—ignition of the ﬂame is a very complex phenomena and may not be
accurately modeled by simple laminar steady opposed jet data in the form of ﬂamelet
tables. The ﬂamelet tables need to incorporate the level of partial premixing and

mixture temperature as well as variable strain rate effects into account.

2200

_L —L _L
O A 00
O O O
O O 0

Temperature [K]

200 l
0

2200

 

(a)

600'

 

0"

 

0:1

0:2

 

073 0.4

Distance from fuel nozzle [cm]

 

1000‘

Temperature [K]

600'

_L —L
‘b 00
O O
O O

 

200

 

-----d

 

0

(15

1

1.5 2

Distance from fuel nozzle [cm]

2200 ‘

(b)

 

Temperature [K]
a) " I; 63
8 § 8 8

 

 

200

 

0 0:5 1
Distance from fuel nozzle [cm]

2200 ‘ ‘ ‘ ‘
(d)

 

 

- - 2.0 cm
----- 0.4 cm
— 1.0 cm

 

 

  

N
8
c?

 

1800'

Temperature [K]

 

 

1600
(12

 

0.3 0:4 0.5 0.6
Mixture Fraction [—]

Figure 1.1. Temperature structures in physical and compositional space for various
domain sizes, (a) 0.4 cm, (b) 1.0 cm, (c) 2.0 cm.

33

Temperature [K]

Temperature [K]

2200 ‘ ‘ ‘ ‘

-9— Strain = 10(1/8)
+ Strain = 25(1/s)
—- Strain = 100(1/s)
, + Strain = 200(1/3)
“ ,. -9- Strain = 400(1/s) _

 

 
     
  

1800‘

v

1600‘

1400‘

1200‘

 

 

 

o 0.2 0:4 03 0.8 i
Mixture Fraction {-1

(a) Varying strain rate

 

1400'

12004 "

 

 

600 5': a : ' o/OA = 0
_ ,‘ °/oA = 25
400 >. _ 7 ; °/oA = 50
' i %A = 75
200 , %A = 85

 

 

o 0.2 0:4 0:6 0.8 i
Mixture Fraction {-1

(b) Varying fuel stream premixing

Figure 1.2. Temperature structure in mixture fraction space with GRI-Mech 2.11 (a)
for different strain rates and premixing of 75% air in the fuel stream, (b) for different
premixing levels of the fuel stream and strain of 25(1/s). Vertical lines indicate
stoichiometric mixture fraction.

34

 

 

 
 
    
     
   
   

 

 

 

 

       

 

 

 

 

 

  
 
   
   
   
     

  

 

 

 

 

 

 

2000*
$1500-
2
a
g
8. 1000‘ '
E 4-Step
+9 + 5-Step
-~— 6-Step
500‘ + 10-Step
+ 12-Step
+ GRI 2.11
G + GRI 3.0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
2000, , 2000- (d) . . . ~, '
$1500 » 1500‘ '
9
3
t‘.’
81000‘ 1000- i
E —o— 4-Step
.2 + 5-Step + S-Step
’ —-— 6-Step j -«-— 6-Step
500‘ , . -«- 10—Step ~ ’ 50° + 10-Step
( . + 12-Step + 12-Step
+ GRI 2.11 . + 6812.11
0 - i :- GR|3;0 - _ 0' T v —9: (3513.0 _ .
0 0.2 0.4 0 6 0.8 1 O 0.2 0.8 1

. 0.4 0.6
Mixture Fraction {-1 Mixture Fraction [-1

Figure 1.3. Compositional temperature structure obtained with various mechanisms
for 75% premixing of fuel stream and different strain rates, (a) 10 (1 /s) (b) 50 (1 /s)
(c) 150 (1/s) (d) 300 (1 /s) Vertical lines indicate the stoichiometric mixture fraction.

35

 

   
 
  
   
  

 

2000' + 4-Step

+ S-Step
—~— 6-Step
+ 10-Step

$1500 + 12-Step

g .‘ —+— GRI 2.11

a GRI 3.0

S .

31000 .

E

o

l-

T

v

 

 

 

    
 

c. - - -
0 0 2 0.4 0 6 0 8
2000‘ + 4-Sfep
+ 5—Step
—-- 6—Step
.— , —+— 10-Step
£1500 ...— 12-Step
2 GRI 2.11
P.
S
310001
E
a:
i-

 

 

o . . f -
0 0.2 0.4 0.6
Mixture Fraction {-1

ole

 

A A A

 

  
  
 
 
   

~o— 4-Step
+ 5—Step
—-— 6—Step
+ 10-Step
+ 12-Step
+ GRI 2.11
+ 6813.0

1500‘

1000*

 

 

 

 

 
   
     
    

  

 

 

 

 

0 0 2 0.4 0 6 0 8
2000]
1500'
1000-
4—Step
+ S-Step
-— 6-Step
500‘ + 10-Step
‘ + 12-Step
(d) , + GFlI 2.11
c. f 1 -e-: GRI 3.0 1
0 0.2 0.8

0.4 0.6
Mixture Fraction {-1

Figure 1.4. Compositional temperature structure obtained with different mechanisms
for different premixing of fuel stream and a strain rate of 25 (1 /s), (a) 0% (b) 25%
(c) 50% (d) 75%. Vertical lines indicate the stoichiometric mixture fraction.

36

-4

x10

 

0
0'1
I
N

 

O

l
.0
01

 

I
...L
I

 

Production Rate [Moles/CC-Sec]

I
_.L
U"

I
~

 

 

 

0 0.2 0.4 0.6 0.8
Mixture Fraction H

(a)

Figure 1.5. Production rates of CH4,02, and 002. (a) 5-step mechanism

37

—5

 

 

 

 

 

 

 

1
8x 0 l T I I
'— 6' ..
g ‘2.
3 CO

C’.’ 4' / 2 ‘
0 -
Q 2)- -l
(D
9
O 1
a 0
a: "
H ‘ ..
(U '2' '\
CE .
g _4. .
S o
'0 ...6P 2 II
9
0..

—8- I

_10 l l l l
O 0.2 0.4 0.6 0.8 1

Mixture Fraction H

(b)

Figure 1.5. [contd...] Production rates of CH4,02, and C02. (b) lO-step mechanism.

38

 

 

 

(PD)

     

 

 

 

 

‘15 -B- 4-Step ’
—-l— 5-Step
-20. ->- Sn-Step .
-A— 6-Step
'V" 10—Step
”25‘ -9- 12-Step
-B- GRI 2.11
-30 v v . . . , ,
O 50 100 150 200 250 300 350 400
Strain Rate [1/s]
(a) (PD) Vs. Strain rate
18 - - : L ‘ + ‘ 4' .
16‘ -*- 4-Step _
-9— 5-Step
—x— 6-Step
14‘ -A— 10—Step r
-9- 12-Step
, -B- GRI 2.11 _
12
A 10- ~ 1
O o
n.
V 81 .
’I
6' .
4d I
2 6 p

 

 

 

% Air in Fuel Stream

(1.)) (PD) Vs. Fuel stream premixing

Figure 1.6. Percentage mean of temperature difference from GRI 3.0 for different
mechanisms vs. (a) strain rate for premixing of 75% air in fuel stream, (b) premixing
level of fuel stream with air for strain rate of 25(1/s).

 

 

 

 

 

1200 ‘ ‘ ‘ ‘ ‘ ‘ ‘
4?- 5—Step
@— 6—Step
1000' -G- 1 —Step '
—a— 12-Step
—B— GRI 2.11
800- .
E
g 600-
I—
<1
400- -
200- ~
0%: r , , H

 

 

 

O 10 20 30 40 50 60 70 80 90
% Air in Fuel Stream

Figure 1.7. Maximum temperature deviation from GRI 3.0 predictions for different
mechanisms and a strain of 25 (1 / s).

40

 

 

 

 

 

 

 

 

 

 

500 ‘ ‘ L ‘
,'\
---- S—Step i X
- - 6—Step ” '\
400‘ - — 10-Step ." '1 b
12-Step -' ‘-
.-.- GRI2.11 ,’ ‘
\
I: ‘-
,_, i ‘.
E i ‘.
,_ 200- _i x\ ‘. '
<1 ’ ’ f‘ ’
I I’ \ ‘.
I I \\ ‘-
100. IX ‘ ‘- ’
.C’ \\ “
.’/ \‘-
I’ll 0,
o - , “ w‘ ' ‘
"géw($(ff7' “a ~ i- t“~'~‘-'— tura-W'5“-‘- Tm,
-100 . . - .
0 0.2 0.4 0.6 0.8 1
Mixture Fraction {-1
(a) 75% premixing
1200 ‘ ‘ ‘ ‘
---- S-Step
1000. ' " 6-Siep I I
—— 10-Step f-
. . 12-Step i:
800, - - GRI2.11 1,.
'i
-'!
p. 600- [i'
s. i!
II
*2 *-
400- -' "’
' I
I
1' ll
‘ i
200 i i“
p l
i
—200 - . - '
0 0.2 0.4 0.6 0.8 1

Mixture Fraction {-1
(b) 85% premixing
Figure 1.8. Temperature difference from GRI 3.0 for different mechanisms vs. mixture

fraction for strain rate of 25( l/s) and (a) 75% air in the fuel stream, (b) 85% air in

the fuel stream.

41

l

A

 

0.8-

0.6-

YCH4

0.4-

0.2-

 

°/o Air in Fuel = 0

 

 

02

A

Y

0.4

A

0.6

 

0.15-

0.1-

0.05-

 

1
|

%]Air in Fuel = 50
!

!

 

 

 

034 ore
Mixture Fraction {-1

A A A

 

0.61

0.5-

0.44

0.3-

0.2-

 

 

3% Air in Fuel = 25

 

 

0.2 04 0.6 0.8 1

 

0.04-
0.02-

 

% Air in Ft[el = 75

 
 

i
l

 

 

 

0.2 04 0:6 0:8 1
Mixture Fraction [-]

Figure 1.9. Effect of partial premixing on YCH4 structure. —<>—, 4-Step; — x —,
5-Step; — =1: -—, 6—Step; —A—, 10—Step; —D—, 12-Step; — O —, GR] 2.11; —*—, GR13.0.

42

0.18

 

0.161
0.14«
0.12-

. 0.1‘

>0 0.081
0.06-
0.04-
0.02-

 

Strain = RIO/s)

 
 

 

 

0.18

 

0.16‘
0.14‘

0.12‘

>0 0.08-
0.06-
0.041
0.021

 

Strain = 1510(1/3) r

 

 

 

Ii

 

Figure 1.9.

labeling

1i

.2 6.71 (I; 0.8 1
Mixture Fraction {—1

O

[contd...] Effect of strain rate on YCH4 structure. See Figure 1.9(a) for

43

4

 

0.18
0.161
0.141
0.121

0.11
0.084
0.06-
0.04'
0.021

 

3mm=xmm)

 

 

 

 

0.18
0.161
0.14‘
0.12‘

0.1'
0.08‘
0.06'
0.04%

0.021

 

Strain = 300(1/3)

 

 

wvv

0.2

0.4

0.6

Mixture Fraction {—1

0.8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

   

 

0.25- . 4 ° 025' A A ‘ A
% Air in Fuel = 25
0.2 % Air in Fuel = 0 . 0-2 ’
015' l 0.15“ P
N
O
>.
o 1 r 0.1 ‘ r
0.05‘ - 0.05‘ b
o._Lmr“‘vevv‘-°° ;‘ A ‘ ATL ~ ° 2*“: 01 . ' r
0 0.2 0.4 0.6 0.8 1 o 0.2 0.4 0.6 0.8 1
0.25‘ A ° 4 025‘
a . . ' °/. Air in Fuel = 75
0 2‘ l °/o Alr ln Fuel: 50 , 0,2.
0151 0.151
N
O
>.
0.1‘ 0.11
0.051 0.05-
o - c . . v
0 0.2 0.3 1 0 0.2 0.8 1

074 0T6 0:4 0.6
Mixture Fraction {-1 ”WWW ”30150" H

(a)

Figure 1.10. Effect of partial premixing on 1’02 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.25 - . - - 0'25! A ‘ A A
‘ Strain = 10(1/3) Stfaln = 50(1/3)
0.2- 0.2'
0.15‘ 0.15‘
N
o
>.
0.1. 0.1*
0.05‘ 0.05“
0 0
0
0.25 025‘
0.2‘ 0.2‘
0.15« 0.15<
N
O
>-
01 4 0.1-
0.051 0.05‘
0 ' . 0 . . . .
0 0.2 0.4 0.6 0.8 1 0 0.2 ' 0.4 0.6 0.8 1
Mixture Fraction {—1 Mlxture Fraction {-1

(b)

Figure 1.10. [contd...] Effect of strain rate on Y02 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

45

x 10'3

9°
01
d
I

 

03

17'
l

_ l
02 + 3H2 <—> 2H + ZHZO.

CH4 + 2H + H20 <=> C0 + 4H2

/

l
i

N
01
I

N

l
l
I
|
l
l

.

—L
r

C0 + H20 <=> C02 + H

o
01
I
/
74

Reaction Rate [Moles/CC.Sec]
O 01

   

l
.0
01

I

/

/H+H+M<=>H +M
/ 2

 

l
—L
I

 

 

0:4 ole 0.8
Mixture Fraction H

o
.0
re

(a)

Figure 1.11. Reaction rate of the global reactions in (a) 4-step mechanism.

46

-4

 

 

 

 

 

 

 

 

x10
8 l l I T
6' u
H2+20H=2H20 '.
8 4. 3H2+oz+coz=3H20+cow \1 q
0? _"-.\
8 .' g" (3H2+002+2NO=3H20+C0+N2)' 103
a 2' -"'-.'\. '
£3 l l -'
9 O . . _
<0
(I
C n u
.9 ‘2
‘6
<0
0)
c1: -4- .
-6- .
_8 l I l 1
0 0.2 0.4 0.6 0.8 1

Mixture Fraction H

(b)

Figure 1.11. [contd...] Reaction rate of the global reactions in (b) 5-step mechanism.

47

 

0.12 ‘ ‘ ‘ L

 

% Air in Fuel = 25

°/. Air in Fuel = 0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l

0.12‘ l % Air in Fuel = 75 ’

0.11 l ’

0.081 ? ’

0.061 l '

0.041 ’

0.021 L

or i . . . T c 1 i 1 F -

o 0.2 0.8 1 0 0.2 0-8 1

0.4 0.6 0.4 0.6
Mixture Fraction {—1 Mixture Fraction [—]

Figure 1.12. Effect of partial premixing on Y002 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

48

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

042‘ ’ Strain = 50(1/s)
0.11 Strain = 10(1/s)
l
0.081 ; r
N l
8 l
> 0.061 l t
0.041 r
|
i
0.021 i r
l
0 Y . f
o 0 2 0.4 o 6 0 8 1
0.121 .
0-12 ’ Strain = 300(1/3)
Strain = 150(1/s)
01‘ _ 0.11 r
0.08- r 0'08‘ '
N
8
>. 0.061 ' 006‘ '
0.041 3 . 0.041 . l
l
0.021 r 0-02‘ ‘
0 j 0
0 0.2 0.4 0.6 0.8 1 o 0.2 . 0.4 0.6 0.8 1
Mixture Fraction {-1 MIXthG Fractlon ["1

(b)

Figure 1.12. [contd...] Effect of strain rate on YCO2 structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

49

x10-3 _ x 10‘3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

         

 

 

-3 —3
1
8x10 8X 0
(d)
6 ‘ 6' ’
>% 4 4 4 1 l
21 l 2'
0 - ' ‘¢::,‘> 0 . _. T ‘~":"
0 0.5 1 0 0.5 1
Mixture Fraction {—1 Mixture Fraction ['1

(a)

Figure 1.13. Effect of partial premixing on YH2 structure. (a) A% = 0, (b) A70 = 25,
(c) A% = 50, and (d) A% = 75. See Figure 1.9(a) for labeling. Vertical lines indicate

the stoichiometric mixture fraction.

50

x 10’3 ' x 10'3
(b)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

         

 

 

-3 -3
8 x 10 - 8 x 10
(C) (d)
61 l 6' '
>% 4 l 4 J l
2' 2‘ ’
0 - o i -
0 0.5 1 0 0.5 1
Mixture Fraction [-] Mixture Fraction H

(b)

Figure 1.13. [contd...] Effect of strain rate on YH2 structure. (a) strain 2: 10 l/s, (b)
strain 2 50 l/s, (0) strain = 150 l/s, and ((1) strain 2 300 Us See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

51

 

0.12

0.11

0.081

0.041

0.021

 

% Air in Fuel = 0

 

0.2 0.4 0.6

0.8

 

 

 

Figure 1.14.

% Air in Fuel = 50

 

 

 

V

0.2 0T4 0f6

Mixture Fraction [-]

0:8

 

0.12

 

0.11

0.08 1

0.061

0.04 1

0.021

 

°/o Air in Fuel = 25

 

 

0.12

0.11

0.08 1

0.06 1

0.041

0.02 1

 

% Air in Fuel = 75

    

 

 

0.2 ' 0:6

0.4
Mixture Fraction H

V

0.8

Effect of partial premixing on YCO structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

52

 

 

0.12 ‘ L 4

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.12 1 1 1 .
0... Strain =: 10(1/s/ _ 0.1. Straln 50(1/s)
0.081 , 0.081
8 0.061 0.061
).
0.041 0.041
0.021 0.021
0‘ 04
0.12 - - 1 . 0-12 ‘ ‘
0... Strain = 150(1/6) _ 0.11 Straln = 300(1/8)
0.081 0.081
8 0.061 0.061 .
>-
0.041 0.041
0.021 0.021
01 1 - - . 01 . . - -
0 0.2 0.4 0.6 0.8 1 0 0.2 . 0.4 0.6 0.8
Mixture Fraction {-1 Mlxture Fractm {-1
(b)

Figure 1.14. [contd...] Effect of strain rate on YCO structure. See Figure 1.9(a) for
labeling. Vertical lines indicate the stoichiometric mixture fraction.

53

 

      
    

 

 

 

 

0.14 ‘ ‘ ‘ ‘
0.12- f: (a) 12
YC02
0.11 l -1
No.08-r 1; ------------------------ 0
8 v ‘
>. I
0.061 : ~-1
: C02 Production rate
0.04. : F_2
l
I
0.02 : ~—3
l
o. . . . . -4
O 0.2 0.4 0.6 0.8 1

Mixture Fraction

Figure 1.15. C02 production rate and mass fraction compared for 0% partial premix-
ing and a strain rate of 25 l/s 4—step mechanism.

w:
o

I
.p

Production rate [moles/cc-sec]

 

0.14 - - - 1 X410

012- (b)

002

ch

Production rate [moles/cc-sec]

       

   

0.11

rx'a

0.08-

YC02
‘ .1

0.061-

 

 

 

 

0.04‘ ._1
" CO2 Production rate
0.02 '-2
o . . . . . -3
0 0.2 0.4 0.6 0.8 1
Mixture Fraction
Figure 1.15. [contd...] 002 production rate and mass fraction compared for 0%

partial premixing and a strain rate of 25 1 /s 5-step mechanism.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' ‘VW
0.15 0.2 . ,
Mixture Fraction {—1 Mlxture Fraction {-1

Figure 1.16. Effect of partial premixing on YH structure. (a) A% = 0, (b) A% = 25,
(c) A% = 50, and (d) A% = 75. See Figure 1.9(a) for labeling. Vertical lines indicate
stoichiometric mixture fraction.

56

x 10‘3 . . x 10‘3
(b)

 

 

 

 

 

 

 

 

 

 

 

x 10‘3 x 10”3
(c) (d)

 

 

 

 

 

         

 

 

 

0 I 05 1 0 0:5 1
Mixture Fraction [-] Mixture 930130" {-1

(b)

Figure 1.16. [contd...] Effect of strain rate on YH structure. (a) strain = 10 l/s, (b)
strain = 50 l/s, (c) strain = 150 l/s, and ((1) strain = 300 Us See Figure 1.9(a) for
labeling. Vertical lines indicate stoichiometric mixture fraction.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 J 1 , 0-5_
Mixture Fraction [-] Mlxture FVaCt'O" H

(a)

Figure 1.17. Effect of partial premixing on YNO structure. — x —, 5-Step; — O —,
GRI 2.11; —*-, GR13.0. (a) A% = 0, (b) A% = 25, (c) A% = 50, and (d) A% = 75.
Vertical lines indicate stoichiometric mixture fraction.

58

 

 

 

 

 

 

 

 

 

 

 

(c) 1")

 

 

 

 

 

 

 

 

O 0.5 1
Mixture Fraction [-]

 

Figure 1.17. [contd...] Effect of strain rate on YNO structure. — x —, 5-Step; — O —,
GRI 2.11; —*—, GRI 3.0. (a) strain 2 10 l/s, (b) strain 2 50 l/s, (c) strain = 150
1/s, and ((1) strain = 300 1/s.Vertical lines indicate stoichiometric mixture fraction.

59

 

0.015 t r . v . . .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.01
0.005%
0
-0.005
3 -0.01
-0.015
-0.02
-0.025
-0.03
_0.o35 l I I I I I I
0 50 100 150 200 250 300 350 400
Strain Rate (1/s)
(1") CH4
0.06 I f ﬁr 1' I I I
0.05 ~ .
+ 4-step .
0.04 b + 5_ step
—9— 5n-step
0.03 p + 6_step ‘
—v— 10-step
0.02» —G— 12-step '
A - - 902.11
E 0.01 l- .
V
0
-0.01
—0.02
-0.03
_o.04 I I I I I I I
0 50 1 00 1 50 200 250 300 350 400
Strain Rate (1/s)
(b) CO

Figure 1.18. (DF) vs. strain rate for different mechanisms and premixing of 75% air
in fuel stream. (a) CH4, (b)CO.

60

 

0.018

I
at
‘
I
1
I
I
:1

0.016
0.014
0.012
(C 0.01

D <
V0008 - .

 

0.006

0.004

1
I

0.002

I
l

 

 

 

 

0 I l I I
0 10 20 30 40 60 70 80 90

°/o of Air in Fuel [-]

8-111

 

 

(a) CH4
0.03 V I I I I I I r
—e— 101-Step
0.025 - + 12-Step .
—e- 4-Step
—~— 5-Step
—1~— 6—Step
002' + GRI2.11 1
g 0.015 - 11
1
0.01 '1 1
t
0.005 " -l
- l 1‘: :—l=—.——-¢
C ‘J

 

 

 

0102030405060708090
°/. of Air in Fuel H

(1)) )CO

Figure 1.19. (DF) vs. partial premixing level of fuel with air for different mechanisms
and strain rate of 25(1/8). (a) CH4, (b) CO.

61

 

 
  
     

 

 

 

 

 

 

 

0.08 r t . t .
0.07 ' —+—— 10-step 1'
—v— 12—Step
-6- 5—step ‘
0'06 ' —e— 6-step
431- GR12.11
x 0.05 ' 1
to
AS
0 . .r
O 0.04
>.
3
0.03 ’ .
0.02 " .
0.01 - 1
O T L A E» ' r ‘%
20 30 40 50 60 70 80 90

% of Air in Fuel [—]

Figure 1.20. Maximum YCO difference between the GR] 3.0 results with those obtained
with other mechanisms.

62

 

 

 

 

 

 

 

1350 . . . . .
1300 '- n
'2? '-
h‘ 1250 t- 1
CD
8.
:3
'8
5 1200 - -
Q .
0E)
1.. 1150 - A/l .
c .’ ‘7'
.9 l’
.1: / —a- A°/o = O
1- 19’ Ill
511“)" ., + on.=25
— + A% = 50
‘/ + A°/. = 75
1050 - g .
1000 L I I l l
0 1 00 200 300 400 500 600

Strain rate [1/s]

Figure 1.21. Ignition temperature vs. strain rate for various premixing levels of the
fuel with air using GRI 3.0.

63

CHAPTER 2

Turbulent Flames

2. 1 Introduction

Turbulent ﬂows are characterized by a large number of length scales and time scales
depending on the Reynolds number of the ﬂow. For each dimension that exists in.
the flow, scale separation grows proportional to Re3/4. Combustion occurring in
turbulent ﬂows further adds chemical scales to the flow scales. Depending on the
interactions between flow and chemical scales, a diagram of combustion regimes
can be constructed, and the combustion can be modeled using various methodolo-
gies [43, 27, 35, 44]. A diagram illustrating the different regimes of non-premixed
combustion in “Ret — Da” space is shown in Figure 2.1 [56]. D0 is Damkéhler
number which represents the ratio of turbulent to chemical time scales, and Rat is
the turbulence Reynolds number, which compares turbulent transport to molecular

transport.

Non-premixed turbulent combustion occurs in many of the practical applications
where the fuel and oxidizer enter the combusting region separately. Combustion rate
is enhanced by strong turbulent mixing, where more efficient mixing of reactants
is possible [45]. Many of the industrial applications use non-premixed combustion
since unmixed fuel and oxidizer are safer to handle. Consequently, there has been

considerable interest in the study of non-premixed turbulent combustion. A vast

64

   
   

Laminar

Quenching

 

 

all-O-I-I- —I-I-.-.

Re

Figure 2.1. Combustion regimes for non-premixed flame

amount of literature has accumulated on this subject which includes analytical,

numerical and experimental studies.

A complete description of regimes in non-premixed combustion is difficult since
it does not exhibit any intrinsic characteristic scales [43]. A description based on
the work of Cuenot and Poinsot [16] will be given here. Two limiting Damkiihler

numbers, Da L F A and Daext, can be identified from their flame/vortex interaction

studies, which will be used to describe the combustion regimes. W'hen D0* is greater
than De L F A: the ﬂame front is assumed to be a steady laminar ﬂame which is not

affected by the vortices. Here, Da“ represents the Damkéhler number for laminar

L
XstTc

Tc is the chemical time scale. The ﬂame front will be quenched when Da“ 3 Daerrt-

diffusion ﬂame (Da* 2: ), where Xst is the dissipation rate at stoichiometry and

Unsteadiness is observed in the intermediate range.

These regimes can also be explained based on different scales that exist in the
reactive ﬂows. Reaction zones are embedded inside the mixing layer of fuel and
oxidizer, and the mixing region length is roughly the order of turbulent integral
length scale (It z I; z IVEI). Diffusive thickness, controlled by Kolmogorov scales,
can approximated to depend on diffusion coefﬁcient (Dst) and conditional mean of
)1/2

scalar dissipation rate, conditioned upon Z = Zsti 1d x (£12:
3

scale can be deﬁned using the conditioned scalar dissipation rate as: 1f z 321—; z Tk.‘
3

z 771." Flow time

A summary of different scales of reactive layer along with the diffusive layer (which

is controlled by the Kolmogorov scales) is given in Table 2.1.

Table 2.1. Characteristic scales in turbulent non-premixed ﬂames

 

 

 

 

 

 

 

 

Length scale Time scale Velocity scale
. . ~ D 1/2 ... 1
D1ffu51ve layer Id ~ (751:) T ~ f; ld/Tf
Reactive layer Ir = ld(Da)_1/a [29] Tc : 1/(Da)23t) lr/Tc

 

At lower Reynolds numbers, we will have laminar ﬂames. If the Reynolds number
is high and when the chemistry is fast, the chemical time scale ('rc) is small and
Damkéihler number large, which implies lr << Id 2 77k- In this case the ﬂame is thin
and laminar (‘ﬂamelet’), and it will not be effected by turbulence. For larger values
of Tc, the length scale of reactive layer will be comparable to the Kolmogorov length
scale, which makes the turbulence affect the reaction zone. For further larger values

of TC or lower Damkéihler numbers, the chemistry can not withstand the turbulent

66

ﬂuctuations and extinctions occur.

Numerical simulations of ﬂows that can resolve all the scales in complex ge-
ometries with complex chemistry seem to be impractical in the near future [45].
Combustion modeling approaches that utilize the simplifying characteristics of
different regimes have been devised to simulate such ﬂows. But, it is not always the
case that all the points in the physical space of interest obey the same combustion
regime characteristics. Direct numerical simulations (DNS) has recently emerged as a
important tool in validating existing models and improving them, and understanding
fundamentals of turbulent combustion [55]. In DNS of turbulent reacting ﬂows,
equations of hydrodynamic ﬁeld and scalar ﬁeld are solved on a ﬁne grid, which

resolves all ﬂow and chemical scales.

Due to formidable requirements of computational power, direct numerical simé
ulations are performed for various simplifying situations depending on the interest of
study. In the choice for solving hydrodynamics equations, depending on applicability,
incompressible, low Mach number or full compressible equations can be chosen.
Along with the simpliﬁcations mentioned for ﬂow ﬁeld, different degrees of simpliﬁ—
cation in describing chemistry, from one-step global description to full mechanisms,
and computing the transport properties can be utilized [20]. Another valuable
simpliﬁcation that seems to be valid in reacting ﬂows is performing simulations in
two—dimensions. Conﬁrmation is obtained from three-dimensional direct simulations
that ﬂame structures exist in two-dimensions with a higher probability even though

the ﬂow itself is three-dimensional [4, 41, 31].

The advantage in performing simulations in two-din‘lensions is signiﬁcant as the com-
putational costs associated with them are much less compared to three-dimensional
simulations. Jaberi and James [22] have studied the interactions between chemical
reaction and turbulence in compressible two—dimensional homogeneous ﬂows. Both

isothermal and exothermic non—premixed combustion reactions are considered for

67

their study. Signiﬁcant increase in the ﬂuctuations of density, pressure, temperature
and dilatational term is seen due to non-uniform heat release. Angoshtari [1] studied
two-dimensional premixed methane ﬂames using three different reduced reaction
mechanisms, l-step, 5-step and l2—step, for strain and curvature effects on the
ﬂame. She concluded the 12-step reduced reaction mechanism to perform better in
predicting the tear-offs and pocket formations. Maas and Thévenin [31] extracted
data from two—dimensional DNS of turbulent non-premixed hydrogen-air ﬂame for

determining local Intrinsic Low Dimensional Manifolds (ILDMS).

In this work we perform direct simulations of two-dimensional turbulence in-
teracting with initially laminar methane ﬂame. Effects of turbulence on chemical
reactions and vice versa are studied for different ﬂame thickness, premixing levels of
fuel, and turbulent intensities. Laminar solution is obtained using the counterﬂow
diffusion ﬂame code for a very low strain rate. Chemistry is modeled using the
lO-step reduced reaction mechanism of methane combustion which has been reduced
from GRI-Mech 2.11. It represents methane combustion with ten reaction steps
involving 14 species. Laminar studies have been used to determine that this
mechanism represents the combustion accurately at various strain rates and various
levels of premixing with minimum number of species (see chapter 1). This will
greatly reduce the computational costs as the number of species transport equations

to be solved have reduced without sacriﬁcing much in accuracy.

Following this introduction, section 2 describes the governing equations that
are required to solve in turbulent reacting ﬂows along with the thermodynamic,
transport, and chemistry models to close them. Section 3 brieﬂy describes the
numerical methodology used in this study. Section 4 describes the laminar ﬂames
that are required for the initialization of the direct numerical simulations and

followed by a description of turbulent ﬂames and DNS results.

68

2.2 Governing Equations for Reacting Flows

In this section the governing differential equations for turbulent reacting ﬂows will
be brieﬂy described. For complete description of these equations the reader is
referred to the textbooks on the subject [43, 59]. Description of thermodynamic,
transport and chemistry models will be provided that are needed to close the system
of “hydrodynamic” and scalar equations. The equations will be solved for p, two
components of velocity, ui, total energy, species mass fractions, Yk, and pressure.
Continuity equation gives p, momentum equations in two directions give the two
velocity components, total energy and species equations give total energy and species
mass fraction, and pressure can be obtained from equation of state. All equations

are coupled.

2.2.1 Conservation Equations

The equation representing the conservation of mass is:

QB 3(pui)_
8t+ 6x.- _0 (2.1)

where p is the density of mixture and u,- is the component of velocity.

The equation of momentum is similar to Navier-Stokes equation for non-reacting

C886:

06111) WWW) 3:) 370' -
Z —— ‘ : 2.2
at + ailij 0131' + axj 1‘sz (l 1,2) ( )

where p is the pressure, 135]- is the viscous stress tensor components, and f, is the

 

 

external force in ith direction. Even though explicit reaction terms does not appear
in these equations, combustion will modify the hydrodynamic ﬁeld by changing the

temperature dependent viscosity it, which will appear in the closure model for Ti]-

69

ll

 

and by changing the density. The typical variation of these quantities will be in a
ratio of 1:8. As a consequence the local Reynolds number varies more compared to

the non-reacting case.

There are many variations for energy equation depending upon the variable
chosen for solving (enthalpy, internal energy, total energy, etc.). A complete list of
variables that are solved for and their derivation is given in reference [43]. Here in

this study we use the total energy as our variable and the equation is:

D(p€t) We.) 905060 _ 3% +0001”

K
D, — a, ij “07,- ac, +ee+pgiykfmuj+vw>1 (2.3)

 

where qj is the component of heat ﬂux vector in jth direction, Q6 is the external
source of energy (like spark ignition source), 0,31 is the stress tensor, and ka is.
the body force on species I: in jth direction. The last term represents the power

generated due to the volume force fk on species Is.

Different simpliﬁcations for this equation is possible depending on the assump-
tions like constant pressure (low Mach number), equal heat capacities for all species.

In this study we solve the full equation without any simpliﬁcations.

For each species present in the system we need to solve a species balance

equation given by:

8(ka) away-Yr.) _ amt/km) ‘ _

where K is the total number of species present, Vk.j is the jth component of diffusion
velocity for species 1:, and 0,, is the rate of mass poduction/destruction per unit

volume for species 1:.

The above system of partial differential equations given in Eqs. 2.1 - 2.4 solve

70

for density p, the velocity components uj, the K species mass fractions Yk, and the
total energy et. This system of equations are not closed in this form and relations are
needed to express species enthalpies and the relation between pressure and mixture
composition. Expressions for viscous stress tensor, the heat ﬂux vector, and diffusion
velocities are also needed. And most importantly, we need a closure model for the
rate of mass production, wk. In this section we give the expressions that are required

for closing the above set of equations.

In this study we assume that the gas obey the perfect gas law. This assump-
tion is valid in most of the combustion applications which assume low pressure and
high temperatures. The gas law gives a relation between pressure, temperature, and

local mixture composition as:

p = per (2.5)

where R is the universal gas constant whose value if R = 8.314Jmol‘1K—1 and W is

the mixture averaged molar mass given by

K -1
-f _ Yk
li _ ( W1.) (2-6)

k=l

 

The variant of energy variable that is solved in this study can be expressed as:

1
et = h - P/P + 57111711 (217)

where h is the enthalpy of the mixture and the last term is the kinetic energy. The

mixture enthalpy It can be evaluated from the species enthalpies by:

K
h. = Z thk (2.8)
I~.=1

The speciﬁc enthalpy of species k is given by two parts: sensible and chemical. The

71

chemical part is the enthalpy that is required for the formation of the species at
some reference temperature (usually taken to be 298.15 K). The sensible part will
be computed from the specific heat of the species. The complete relation for species

enthalpy is:

T
11km = [T Cpk(T)dT+Ah‘}’k (2.9)
0

where Cpk is the speciﬁc heat capacity of species k and the second term is the chemical
part of the enthalpy. The values of speciﬁc enthalpy of formation are available in tables
and the dependence of speciﬁc heat capacities on temperature are available in the
form of polynomial functions. In this study we used the CHEMKIN thermodynamic
database [24] for their values. C k is given in the NASA chemical equilibrium format:

p

Cpk 2 3 11 '
If 2 (11k + aszk + (13ka + 04ka + (15ka (2.10)

 

where all; — “5!; are the coefﬁcients and Tk is the temperature of the species It.

2.2.2 Transport Models

The ﬂuids encountered in most of the practical combustion applications are Newto—
nian. In this study we assume the ﬂuid to be Newtonian also and thus the viscous

stress tensor used in Eq. 2.2 can be approximated as:

 

0n, adj 2 Bu 1.-
: . __ - _ _ —_ . 1
TU ,u [81']- + 8%] + (8 3p) 81:):6” (21 )

where p is the coefficient of viscosity, K2 is the bulk viscosity and 6,-3- is the Kroenecker
delta. Bulk viscosity can be neglected on the basis of low-Mach number approxima-
tion. Usually the stress tensor as it appears in Eq. 2.3 can be written as the sum of

pressure term and the viscous stress term:

72

Uij = -p5ij + Tij (2.12)

The coefficient of strain term it that appears in Eq. 2.11 is the mixture viscosity and
it is calculated from the pure species viscosity by VVilke’s formula [57] combined with

the modiﬁcation of Bird [9] as:

 

K XH’A
,. = X K .. (2.13)

where ,uk is the pure species viscosity, X k is the mole fraction of species I: and (I’kj

will be given by:

.7

— 1 ”/k “A: 1/2 Wj 1/4
‘ij _ \/§(1+ W) (1+ (“1) (ll—I) (2.14).
Pure species viscosity pk is obtained from [23]:

I .k T
_ 3mm (2.15)

“k ‘ 16 WIZQCL'ZV
where 0k is the Lennard-Jones collision diameter, ’"k is the molecular mass, K B is
the Boltzman constant, and T is the temperature. The collision integral 0912)”! is
determined by a quadratic interpolation of the tables based on Stockmayer potentials
given in Monchick and Mason [34]. A fourth degree polynomial is obtained as
a function of temperature and the coefficients are tabulated for the pure species

viscosity based on the description given above.

The heat ﬂux in 1th direction is given by:

8T 1"
(12' = ‘48:; + Igl/t’lkyirvkil (2115)

here /\ is the thermal coruluctivity of the mixture. The first term is the Fourier

73

heat conduction term and the second term is the heat transport due to differential
diffusion of species. In the above equation heat ﬂux due to radiation and Dufour

effect has been neglected.

The mixture-averaged thermal conductivity is calculated using the formula:

1
)1:—
2

K K X —-1
Z XkAk + (2 if) ] (2.17)
k=1

1021
here Ak is the pure species thermal conductivity of species k which are calculated

 

and the temperature dependent polynomial ﬁt coefficients are tabulated similar to

that of viscosity.

The complete general form of expression for diffusion velocity of the species 1'

can be given as:

K K
VP 10
v,- = _ E : 0,3,, vxk + (X), — Yp— +5 2:11.336,- — 13,) — D?"
k=1 i=1

2:

T (2.18)

where DH; is the multi-component diffusion coefﬁcient, which is dependent on all the
state variables. The term involving the thermal diffusion coefﬁcient of species 17 (D?)
is known as Soret effect (thermodiffusion). This term gives the diffusion of species
due to thermal gradients. This expression for the diffusion velocity is computationally
very intensive and will not be used here. A computationally less intensive yet enough

accurate formula will be used in this study. Diffusion velocity is then obtained from:

OXk
Vki : ”ka

" 03:2;

(2.19)

where the diffusion coefﬁcient is for species I: in to the mixture, and it will be calcu-

lated from the mixture-averaged formula (Zeroth order approximation) [21]:

74

l—Yk
Zj<=1,j3£k Xj/ij

 

ka : (220)

where ij is the diffusion coefﬁcient for species I: in species j. The diffusion velocity

calculated by this formula does not obey the global mass conservation equation,

K
E: Ykaﬂ' # 0 (2.21)
7:21

To overcome this problem, a correction term will be added for all the species, which

is given by:

Vc,k = ‘Yk Z Vk,i (222)

In order to calculate the binary coefﬁcients a fourth degree polynomial was ﬁtted to
the data obtained from CHEMKIN for all the species involved in the simulation in

the temperature range from 250K - 5000K.

2.2.3 Chemistry Models

Let the K species in the system react amongst themselves through R reactions. Then,

the the reactions can be represented in a compact form as:

Z UrkAk: Z VTIkAk T =1,2,...,R (2.23)

where AA. is the chemical symbol for the species k, u’ k and u’ [k are the molar stoi-
chiometric coefﬁcients of species I: in reaction r. The conservation of mass for each

reaction implies that the stoichiometric coefﬁcients satisﬁes the following relation

K K
kzl k=l

75

The progress rate for the rth reaction will be given as:

K I K I!
Y. . Y. .
w. = a II 1%“ — 1,, H [pa/ff“ (we
k=1 k=1

where kfr and kbr are the forward and backward reaction rates for reaction r, and

are expressed by Arrhenius law:

E .

- _ ﬂ (Afr
kf?‘ — AfT'T fr exp ( — *R—T') (2.26)
here A fr is the pre—exponential coefﬁcient, ﬂfr is the temperature exponent, Ea, fr is
the activation energy for the forward reaction r, and R is the universal gas constant.

Forward and backward reaction rates are related through the equilibrium constant of

that reaction

Ke’r : '7— (2.27)

and

 

K
Pa 231.21va A89 AH"
6” RT exp R RT ( )
where pa is the atmospheric pressure, AS? and AH? are the changes in the entropy and
enthalpy respectively during the transformation of reactants to products for reaction

r, and ”In 2 1x1” — Vzr. The mass reaction rate of species I: is then

R
wk = Wk 2 Vkrwr (2.29)

r=1

and the conservation of mass forces the equation:

K
2 wk 2 0 (2.30)
15:1

76

For evaluating the sources term in Eq. 2.4 for each species we need all the reactions
that describe the mechanism of conversion of reactants to products, and also need all
the constants that are mentioned here for each of such reaction. The most commonly

used format to describe the reaction mechanism is the CHEMKIN format.

77

2.3 Numerical Solution

The numerical implementation of the governing equations explained in section 2 will
be given here along with the tests performed to improve the performance of some of
the computationally intensive subroutines. A representation of computational domain

along with the flame and the boundary conditions used is given in Figure 2.2.

 

 

 

 

 

 

 

 

 

NSCBC
air
5 5
3:
U ‘c’
C
o 8
O
2‘ E
3 2
g a
3 ﬂame .3
g 2
2;. E
g 3
Y
L x fuel
NSCBC

Figure 2.2. Computational Domain

78

2.3. 1 Methodology

Equations 2.1 — 2.4 are integrated using the extended Maccormack method [25] with
periodic boundary conditions on the boundary perpendicular to the ﬂame and non-
reﬂecting outﬂow boundary conditions [42, 48] on boundaries parallel to the ﬂame. If

the general equation, which represents any of the four governing equations is:

21.1.21: 29

at 81: + 6y +S (2.31)

then symbolically the Maccormack scheme can be represented as:

Predictor pass:

At

At
3:: r1__ (1) n__ (2) n_ n
Uz U, AxA F AyA G AtS (2.32)
and the corrector pass:
n+1_l n 1‘__A_t (1) *__A_t (2) *_ *
U, _ 2[U, +U, MA F Aya G ms] (2.33)

where A115, Ag, and At represents the grid size in space 2:, y, and time t-directions, A“)
and Am are the forward or backward fourth order compact difference (Carpenter [13])
operators which are repeated in the solution procedure as given in Table 2.2. Time
step At will be given by minimum of the chemical time step and time step calculated

through CFL criteria.

Table 2.2. Differencing sequence foe thended Maccormack scheme

 

 

 

 

 

 

 

 

 

 

 

Step Predictor Corrector
A0) A9) AU) N2)
1 Forward Forward Backward Backward
2 Backward Backward Forward Forward
3 Forward Backward Backward Forward
4 Backward Forward Forward Backward

 

79

 

The stress terms in F and G are evaluated using second—order and fourth-order central
differences. The ﬁrst derivative of velocity in viscous term and temperature derivative
in heat ﬂux calculation are calculated using second-order central differences, and the
mass fraction gradient in the diffusion term is calculated using fourth-order central

difference.

2.3.2 Simplifying Assumptions

As discussed in the introduction of this chapter, Direct Numerical Simulations are
computationally intensive and valid simpliﬁcations are to be made in order to compute
the solution in meaningful amount of time. A brief description of simpliﬁcations used

in this thesis will be given along with the justifications in using them.
2D Turbulence

Direct simulations of three-dimensional turbulent reacting ﬂows is a very expensive
task even with the computational power available now or in the near future [45]. As
the scale separation is dependent on the Reynolds number and resolution of reaction
zone on Damkohler number (Da). The space in “Re-Da” plot that is accessible by
3D-DNS is limited to lower left corner, and this domain can be extended a little
bit. more by resorting to 2D-DNS (See Figure 4 of reference [55]). But, proper

justifications are to be given before one can use two-dimensional turbulence.

The energy and enstrophy conservation equations for two-dimensional turbu-

lence can be given as:

do") (IQ
(1t — —2VQ, 'd—t — “—7) (2.34)

 

where the total energy, enstrophy and enstrophy dissipation rate are given by

0° 1
011 = / E(k) (1k : -2'<’U.1"Ui> (2.35)
0
Q —— / k EUC) (if: = 5(wiwi) (2.36)
0
00
7) : 211/ k4E(k)dk = l/(IV X wlz) (2.37)
0

and the integral and microscale length scales and Reynolds numbers based on these

length scales are deﬁned, analogous to the three-dimensional turbulence as:

1 1
L = 03/723, 1 = (WM)? (238)

and

l 1 .
ReL : al./(mﬂ), Rel = (102)1/11 (2.39)

The differences in two-dimensional turbulence and three-dimensional turbulence are
that the second law in Eq. 2.34 implies that the enstrophy can not be generated
by non-linear effects and are only dissipated by viscosity. Therefore, the rate of
energy dissipation in decaying 2D turbulence is bounded by its initial value. In other
words, there is no vortex stretching mechanism in two—dimensional turbulence and
the cascade of energy transfer from large to small scales is not complete. Thus, small
scales eddies are not as inﬂuential as they are in three-dimensional turbulence and
they do not follow the Kolmogorov’s scaling law. However, Poinsot et al [41] has
shown using direct simulations of three-dimensional turbulent premixed combustion
that the ﬂame exists as locally two-dimensional sheets with more probability than
as three-dimensional structure. So it is somewhat justified to use two-dimensional

turbulence with combustion simulations [43].

81

Simpliﬁcations in evaluating molecular properties

It is observed in turbulent combustion simulations that most part of the computa-
tional time is spent in calculating molecular properties like diffusion velocity, viscosity
and thermal conductivity [1, 20]. In this thesis all the pure species properties were
approximated to a fourth-degree polynomial rather than being computed from the
basic equations. The resulting coefﬁcients of the polynomial are tabulated and read

from a file during simulations, which has reduced a considerable amount of time.

Table 2.3. CPU time requirements for subroutines before and after approximation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No. Subroutine Function (70 CPU ‘70 CPU
Name time time

(Before) (After)
1 Continuity Solves Continuity 0.020 0.081
2 Y-Momentum Solves Y momentum 0.013 0.084
3 Energy Solves energy equation 0.015 0.067
4 Species Eq. Solves species equation 0.225 1.183
5 Cale. Interm. Updates intermediate 0.056 0.230

values
6 Avg. Mol. W t. Calculates average 0.033 0.140
molecular weight

7 Cale. Temp Calculates temperature 0.174 0.972
8 Pressure Calculates pressure .002 0.010
9 B. Values Imposes B. Cs 0.150 0.705
10 Stress Calculates thermal and 86.024 8.036

transport properties
11 Chem. Kin. Calculates species 13.287 88.492
reaction rates

 

 

Evaluating the diffusion velocities is the most expensive task as it involves equa-
tion 2.18 and solution of a system of linear equations of 3K2 unknowns for each
grid point and at each time step. A simplification to this equation is to solve the
zeroth-order approximation (or mixture-averaged) Eq. 2.20 along with Eq. 2.19. The

approximation of evaluating mixture-averaged diffusion coefficient will significantly

82

reduce the computational time.

The other important transport property which requires huge amount of CPU
time is the mixture-averaged viscosity. An acceptable simpliﬁcation for mixture-

averaged viscosity is:

K
X./l..
u = Z —F-—k——l‘—— x Z kalk (2.40)

this approximation has been tested for premixed turbulent ﬂames extensively and
found to give good results. Before the approximation, the double summation used to

take up a huge amount of time.

Table 2.3 shows the percentage of time spent in each of the subroutine in our
program before and after making the simpliﬁcations in calculating the diffusion
velocity and viscosity. About 86% of total computational time is spent in the
subroutine “stress”, and 87% of time in stress subroutine is used to calculate the

diffusion coefficients as shown in Table 2.4.

Table 2.4. Percentage of CPU time in stress subroutine

 

No. Subroutine Function % CPU % CPU
part time time
(Before) (After)

1 Temp & Vel. Der. Calculates velocity and 0.005 0.257
temperature derivatives

 

 

 

 

 

 

 

 

 

 

 

 

 

2 Mass Frac. Der. Calculates mass 0.126 10.195
fraction derivatives

3 Enthalpy Calculates Enthalpy 0.074 4.161

4 Viscosity Calculates viscosity 12.192 3.114

5 Diffusion Vel. Calculates diffusion 87.193 78.008
velocity

6 Cond. Heat flux Calculates Conductive 0.408 8.265
heat flux

 

83

Reduced Chemistry

Performing simulations using full chemistry is very expensive. Researchers have de-
veloped different strategies to overcome this problem. Most important among them
are tracking the chemistry with one or two tracking variables, reducing the chemistry
based on some assumptions, tabulation of the possible chemical states, etc. In this
thesis we use a reduced mechanism that will predict the combustion mechanism very
well in many different situations. The chemical reaction mechanism that is used in this
study is the 10-step reduced reaction mechanism [14] of methane, which represents
the combustion using ten reactions among fourteen species. It has been reduced from
a skeletal mechanism which itself is derived from GRI-Mech 2.11 [11]. A validation

of this mechanism is given at length in the previous chapter on laminar ﬂames.

Reactions steps are:

2H + 2011.20 :1 02 + 20113 (

2H + CHQO = OH + CH3 (
H2+CH20=OH+CH3 (

H + H20 + CHQO .2 20H + CH3 (

H + 2CH20 .2 H20 + CH3 + CO (RV
H + CH4 + CH20 :2 OH + 20H3 (RVI)
H+C0+CH20.—_~CH3+002 (
H+H20+CQH2-.:H2+CH3+CO (
CHQO + C2H4 := 20113 + co (
C1720 + 0sz : OH + CH3 + 02H4 (

After applying the simplifications mentioned in the previous section for molecular
properties, maximum CPU time is required in evaluating the reaction rate of the

species (See Table 2.3). The reaction rates of the species for 10-step is evaluated by

84

a CHEMKIN subroutine CKWYP. Each time this subroutine is called it performs
iterations to find the mass fractions of steady-state species in the skeletal mechanism,
which contains more than 20 species with 150 reactions. This makes the CKWYP
subroutine expensive and evaluating reaction rates at every grid point and at each
time step will be unnecessarily expensive, especially when the state of the mixture
does not change very much. In this study of non-premixed ﬂames, we have air on
one side and the fuel on the other side with constant mixture composition, where the
temperature does not vary very much. At these locations if we do not compute the
reaction rate for every iteration, it can save a huge of computational time. We choose
a criteria to calculate the reaction rate in the fuel stream and oxidizer stream only
when the temperature is more than 500K. Reaction rates are evaluated on the whole

domain only once in every 50 iterations.

2.4 Simulations

In modeling non-premixed and partially-premixed turbulent combustion, it is often
assumed that the ﬂame exists as a thin laminar sheet its inner structure is not
affected by the turbulence. Such a laminar ﬂame termed as ‘ﬂamelet’ is stretched
and strained by the turbulent ﬁeld. Study of such assumption along with the
general behavior of the ﬂame in turbulent ﬁelds is important to the understanding
of combustion phenomena. Flame itself will also affect the turbulence in several
other ways. Heat release from the ﬂame will cause a volumetric expansion, which
will smooth out the vorticity ﬁeld. It also increases the viscosity, which is dependent
on the temperature, and thus reduces the local Reynolds number. The reduced
local Reynolds number increases the dissipation of turbulent kinetic energy. In this
section the results obtained from the direct numerical simulations of 2D-turbulence
interacting with initially laminar ﬂame will be presented. Different statistics are

studied which help the understanding of combustion in turbulent ﬂows.

2.4. 1 Initialization

The one—dimensional ﬂame obtained from the low-strain laminar opposed—jet simula-
tions is projected into two-dimensions (see Figure 2.2) and is used for initialization
of scalar (temperature and mass fraction) ﬁelds. A two-dimensional well developed
and well resolved isotropic decaying turbulence is then imposed on this ﬂame at the
initial time. The resulting turbulent non—premixed/partially-premixed ﬂame will be
allowed to develop according to the governing equations. Simulations were performed

long enough to remove slight “inconsistency” between the initial turbulence and ﬂame.
Initial turbulence ﬁeld was generated using integration of the non-dimensional

governing equations via the Fourier pseudo-spectral method with periodic boundary

conditions. Explicit second-order Adams—Bashforth scheme is used to advance in

86

time. A random solenoidal, 2D ﬁeld with zero mean and Gaussian spectral density

function is used for initialization of the velocity ﬁeld [22],

2
E(k) 2 C134 exp]: — 2(5) ] (2.41)

where C is calculated based on initial turbulent kinetic energy and k0 is the wave
number at peak in energy spectrum. In our simulations the values k0 = 5 and C = 0.8
were taken. The pressure ﬁeld is initialized by solving the Poison equation using
velocity ﬁeld. The initial velocity and pressure ﬁeld are then allowed to decay for
sufficiently long time such that the enstrophy spectrum at small scales is well devel-
oped. The developed ﬁeld has the characteristics given in Table 2.5. The deﬁnitions
of the parameters in Table 2.5 are given in Eqs. 2.35 - 2.39.

Table 2.5. Characteristics of developed turbulence ﬁeld

 

av Q n l L Rel ReL
0.359 8.94 2.75 0.06 0.427 9.71 230.32

 

 

 

 

 

 

 

 

 

 

The above discussed ﬁeld is non-dimensional, which has the advantage in providing
different turbulent ﬁelds of different characteristics when multiplied by appropriate
scale factors. This will be helpful in generating the desired dimensional turbulence
ﬁelds from the same set of turbulence data. In the simulations for this thesis three
dimensional turbulence ﬁelds with characteristics given in Table 2.6 were used. In
the table, u’ represents the rms of velocity ﬂuctuations, L and I represent the integral
and microscale length scales, Re L and Re, are the Reynolds numbers based on L and
1, respectively and rt is the eddy turn-over time. Eddy turn-over time is deﬁned as
the time that a ﬂuid particle traveling with rms velocity requires to cover the integral
length distance. Reference viscosity is calculated based on the air viscosity at initial
temperature (294 K), which is the minimum viscosity in the domain. Length of the

domain is 5 centimeters in either direction.

87

Table 2.6. Dimensional characteristics of turbulence ﬁelds

 

Case 11’ L 1 Re L Rel Tt

(cm/s) (cm) (6111) (sec)
I 6000 0.34 0.048 12022 1781 5.6665
II 4800 0.34 0.048 10102 1420 7076-5
111 3000 0.34 0.048 6311 891 1.13e—4

 

 

 

 

 

 

 

 

 

 

 

 

Very low strain ﬂames produced by three different partial premixing of fuel are used
in these simulations. Partial premixing of 0% (pure diffusion ﬂame), 25%, and 75%
air in fuel stream are considered. Some of the parameters for these ﬂames and the
simulations performed in this thesis are given in Table 2.7. In the table, 6 represents
the ﬂame thickness and L is the integral length scale of turbulent ﬂow ﬁeld. The ﬂame
thickness can be deﬁned in many ways based on the gradient of mixture fraction, the
zone temperature and the species production rate. We chose the ﬂame thickness to

be the width of zone whose end-point temperature is 500 K.

Table 2.7. Simulation cases studied

 

 

 

 

No. Premixing (%) 6 (cm) 6 / L Turbulence cases
1 0 0.693 2.04 I
2 25 0.94 2.76 I
3 75 0.82 2.41 I, II, III

 

 

 

 

 

 

 

All of our turbulent ﬂame simulations were performed on Sun 4 processors worksta-
tion with parallelized code. Extended MacCormack scheme is used to time march
the variables. Results in the form of contours, statistics and images are presented in

the next section. Images in this thesis are presented in color.

88

2.4.2 Results

DNS results for the cases described in the previous section will be discussed here.
Analysis of the data will be performed after at least one eddy turn-over time so
as to give the ﬂame sufﬁcient time to adjust itself to the turbulence. We will
aim to study the effects of turbulence on the inner structure of initially laminar
non-premixed and partially-premixed ﬂames and thus the validation of the use
of laminar ﬂamelet models (libraries) in turbulence modeling. These effects can
be studied through various means as explained in reference [55]. First the effects
on the general structure in physical space is studied through contour plots and
y-averaged (y is the homogenous direction) statistics of different variables. Analysis
is performed normal to the surface of the stoichiometric line at the ﬂame zone for
both burning and locally quenched ﬂame, and proﬁles of various variables in this
(ﬂamelet) direction are compared to the corresponding laminar proﬁles. Turbulence
effects on the ﬂame inner structure can also be studied via the scatter plots of the

desired variable in the mixture fraction space (deﬁned in the previous chapter).

Flame Structures

Contour plots of temperature, mixture fraction, and 0H & C0 mass fractions at
four different times are shown in Figures 2.3-2.6 for 75% partially premixed ﬂame
interacting with turbulence of case I. As the time progresses the ﬂame is wrinkled,
distorted and stretched. The turbulence ﬁeld imposes a highly varying strain at
some regions of the ﬂame making the local tangential stretch so high that the ﬂame
will extinguish locally. Structures similar to edge ﬂames can be observed on either
side of the extinction region [5]. Later in time, pockets of burning ﬂame regions can
be seen, which are separated by the quenched ﬂame regions. This can be clearly seen
in OH mass fractions contours in Figure 2.5. The maximum temperature location
in Figure 2.3, which approximately represents the ﬂame sheet, follows exactly the

stoichiometric mixture fraction line of Figure 2.4. It has been argued that OH radical

89

tracks the ﬂame-front more precisely [1, 46]. Figure 2.5 shows the contours of mass
fractions for OH. It can be seen from the contours that the OH proﬁle is close to
the stoichiometric mixture fraction location as long as the ﬂame exists.In Figure 2.6

contours of CO mass fraction are plotted, which show similar trends to others proﬁles.

Temperature and the OH mass fraction averaged in y-direction are plotted in
Figure 2.7 at four different eddy turn-over times. It is observed that the averaged
values of both the temperature and OH widens in x-direction and the peak values
drops with time. As the results in Figures 2.3—2.6 suggests, the widening of the
averaged values due to stretching and wrinkling of the ﬂame. The drop in the peak
value is due to the fact that averaging has been performed in periodic direction
rather than along the ﬂame. These y-averaged plots cannot be compared with the
laminar structures, and the reasonable turbulent counterpart to laminar proﬁles are

the proﬁles in the direction normal to the ﬂame.

In ﬂamelet models of turbulent non-premixed combustion, it is assumed that
a laminar ﬂame is located surrounding the stoichiometric mixture fraction value.
It’s state (burning or quenched) will depend on the scalar dissipation rate at
stoichiometric (Xst)- If Xst > My then the ﬂame is extinguished, and if Xst < xq
it is assumed to be burning, where xq is the quenching limit of scalar dissipation
rate. Normal to the stoichiometric mixture fraction line the ﬂame structure will
be laminar-like. Proﬁles for different variables in the mixture fraction space are
extracted at various locations and compared with the laminar proﬁles. The data
obtained for 75% partially-premixed methane ﬂame and turbulence case I are used

for this analysis.

Figure 2.8(a) shows the contours of mixture fraction and the location where a
section of the ﬂame is taken for analysis. Figures 28(1)) and 2.8(c) show the temper-
ature and the strain rate versus mixture fraction in this section. The temperature

plot in mixture fraction space is identical with the low strain laminar temperature

90

proﬁle. This is because the section considered in Figure 2.8 is in a low strain region
of the ﬂow that is characterized with a fully burning ﬂame. This region can be seen
to sustain a complete combustion at later times (see Figure 2.5(d)). Figure 2.9 shows
similar plots for a section of the ﬂame where the reaction will eventually extinguish
at long time. The strain rate plot Figure 2.9(c) shows that at this section the strain
rate is much higher than that was in earlier case. Consequently, the maximum
temperature is less than 1900 K and the temperature proﬁle in the mixture fraction
domain is much different from the initial laminar proﬁle. Figure 2.10 shows similar
plots for a quenched ﬂame region. The strain rate is high for the ﬂame at this

location and ﬂame is quenched with maximum temperature near 900 K.

Scatter Plots

Scatter plots of temperature, mass fractions and other important ﬂame quantities in
mixture fraction space show in one plot the various ﬂame-turbulence states existing
in the entire domain. Usually, the temperature peaks near the stoichiometric mixture
fraction value for a burning ﬂamelet and follow the laminar proﬁles depending on
the value of tangential strain rate (or scalar dissipation rate) at that location. At
a location where the tangential strain is more than the quenching limit, the ﬂame
should normally extinguish and the maximum temperature will be lower than the
laminar values. Scalar mixture fraction dissipation represents the extent of turbulent
mixing. For low levels of strain or scalar dissipation rate, the reaction rate is
limited by the mixing. When the scalar dissipation rate is increased, the reaction
rates increase. At high scalar dissipation rates the heat is transported away from
the reaction zone with a rate faster than the rate it is produced by the chemical
reactions. This effectively cools down the ﬂame and in a sustained high strain ﬁeld

will result in low temperature and ﬂame extinction.

Figures 2.11—2.13 show the scatter plots for various variables in the mixture

91

fraction space for three different types of ﬂames. These ﬂames are produced by
0% air in fuel stream (pure diffusion ﬂame, ﬂame no. 1), 25% air in fuel stream
(ﬂame No. 2), and 75% air in fuel stream (ﬂame no. 3) and they all are affected
by turbulence case I and are plotted at one eddy turn-over time. Initial laminar
proﬁles (shown as dashed lines) are also plotted in these plots for comparison. For
all the three ﬂames, the variables show signiﬁcant scattering even at shorter times.
This scatter is due to the enhanced turbulent mixing because of varying strain ﬁeld
of turbulence, and departure from the ﬂamelet behavior. For laminar ﬂames, as
the strain rate is increased, the ﬂame temperature decreases (see chapter 2). The
temperature values much lower than the initial laminar values can be attributed to
the strong strain or extinction. The over-prediction of radicals like OH, H, and CO
as observed here are also seen in earlier studies. Experimental data of Barlow et
al [3] and three-dimensional direct numerical simulations of Mahalingam et al [32]
show similar scatter for the radicals. For OH a simple step change in the scalar
dissipation rate for laminar calculations also showed similar over-prediction [2]. This
over-prediction can be attributed to the unsteady strain rate that exists in the
turbulence ﬁeld. The behavior of CO in turbulent ﬂame is different on different
side of stoichiometric line as the turbulence mixing enhances its production on the
oxidizer side and increases its consumption on the fuel side. Scatter plots for 25%

premixing and 75% premixing ﬂame also show similar trends.

Next three Figures 2.14-2.16 show the scatter plots of temperature, OH mass
fraction, and CO mass fraction for the 75% partially premixed ﬂame interacting with
turbulence of case I at three different eddy turn-over times. As the time advances the
scattering increases due to the local ﬂame extinction. At two eddy turn-over times
the flame has extinguished at many places and has been cooled down to temperatures
as low as 600 K. The extinguished ﬂamelet will be convected and carried in the ﬂow
until the scalar dissipation rate is low enough for it to re—ignite. Sometimes, even
though the scalar dissipation rate becomes low, the ﬂamelet will not re-ignite due

to the lack of radical species or source of energy. This shows that the re—ignition of

92

the quenched ﬂamelet depends on many parameters of the ﬂow and chemistry and
its modeling/ prediction is very complicated. The CO and OH mass fraction scatter
plots show similar trends in time. Extinction is marked with increased scatter at
stoichiometric value of mixture fraction. The scatter plots are also similar for the
ﬂames with other premixing levels of fuel stream provided that the turbulence ﬁeld
remain similar.At lower turbulence intensity levels (turbulent cases II and III) the

amount of scatter and ﬂame extinction are less signiﬁcant as expected.

Turbulence Statistics

Figures 2.17-2.18 show the vorticity, the enstrophy, and the turbulent kinetic energy
images for case I and ﬂame No.3. Four subplots in each ﬁgure show the images at
four different eddy turn-over times. Vorticity images show an isotropic turbulence
ﬁeld with no preferred directional dependency at the initial time. However, as the
simulation progresses, the vorticity ﬁeld is affected by the ﬂame and exhibits peaks
near the ﬂame regions. Vorticity will be affected mainly by three mechanisms in
reacting ﬂows: (1) due to heat released from the chemical reactions volumetric
expansion occurs, which smoothes out the vorticity ﬁeld, (2) the temperature
dependence of viscosity results in higher viscosity near to the ﬂame (this local high
viscosity will dissipate the vorticity near the ﬂame), (3) the generation of vorticity
due to the baroclinic term and misalignment of the pressure and density gradients.
The latest effect is clearly shown in Figures 2.17 and 2.18 where it is observed
that there is signiﬁcant increase in vorticity and enstrophy close to the ﬂame. The
volumetric-averaged values of the enstrophy (not shown here) shows an early decay
and a sudden increase at later times. The decay is due to viscous dissipation which
is more signiﬁcant in exothermic reacting systems, whereas, the increase is due
to baroclinic generation. There is in fact a two-way coupling between ﬂame and
turbulence. Vorticity is generated because of the baroclinic effect, which causes the

increase in strain rate at. the ﬂame location (see Figure 2.20). The increased strain

93

rate will then cause the ﬂame to extinguish as discussed previously.

To better understand the ﬂame effect on turbulence, the y-averaged statistics
(denoted by subscript avg) of various turbulent / ﬂame variables are computed. The
y-direction is the homogenous (periodic) direction. Figure 2.21 shows the y—averaged
vorticity as a function of normalized x direction. After one eddy turn-over time
(UT; 2 1.0), it can be seen that the vorticity ﬁeld is decayed due to the ﬂame. Heat
release and viscous dissipation effects are important up to this time as explained in
the previous paragraph. At later times (t /Tt = 1.5 and t/Tt = 2.0), there is an increase
in the magnitude (in the negative direction) near the ﬂame. This can be observed
more clearly in the y-averaged enstrophy plots, which are shown in Figure 2.22.
There is a slight decrease near the ﬂame region at t/rt = 1.0 and at t/Tt = 1.5, 2.0

there is a signiﬁcant increase in the magnitude of enstrophy.

Figures 2.23-2.25 show the vorticity, enstrophy, and turbulent kinetic energy
contours at t = 0 and t/Tt = 1.0 for various fuel partial premixing levels with air.
There is little difference between different fuel premixing results except that the
magnitude is larger for 75% case. The trend being same for all the variables. The
y-averaged plots (Figures 2.26 and 2.27) also show only small differences near the
ﬂame region. These differences can be attributed to the premixing level of fuel
stream with air or the ﬂame thickness, which are different in ﬂames with different

partial premixing (see Table 2.6).

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.3. Sequence of temperature contours for turbulence case I and ﬂame No. 3
with time; (a) t/T, = 0.0, (b) t/T,=1.0, (c) i/Tt :15. (d) t/T, : 2.0.

95

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 0.5
x/Lx x/Lx

Figure 2.4. Mixture Fraction contours for turbulence ease I and ﬂame No. 3 with
time: (a) UT, 2 00(1)) t/T,:1.0(<') f/Tt:1.5 ((1) t/T, : 2.0

96

 

 

 

 

 

 

 

 

 

 

 

y/Lx

 

 

 

 

 

 

 

Figure 2.5. YOH contours for turbulence case I and ﬂame No. 3 with time: (a)
t/r, : 0.0 (b) t/r,=1.0 (c) t/r,=1.5 (d) t/r, = 2.0

97

 

y/Lx

 

 

 

 

 

 

 

 

 

 

 

 

F igurc 2.6. Ypo contours for turbulence case I and ﬂame No. 3 with time: (a)
f/T, : 0.0 (l)) {/77 21.0 (() f/T, : 1.5 ((l) t/Tt: 2.0

98

 

 

 

 

 

 

 

 

 

 

 

 

 

2500 ‘ * ‘ ‘
— Urt=0.0
--- 3:10
2000‘ (a) ..... t/Tt = 1.5
g ........ t/rt=2.0
91500' ‘
3
9
3
E10001
0)
'—
SOO‘ 55,,
0o 02 0'4 06 08 1
' x/Lx
—3
35x10
— t/rt=0.0
3. (b) 0151.0 »
.-.-. Vttz1'5
2.5‘ ........ t/‘Ct=2.0 '
I 2
>9
1i?
1.
(15* ,
0 .
O (18 1

 

Figure 2.7. Y-averaged plots at different times for (a) Temperature and (b) YO H

99

(a)

100

x-points

300

400

 

 

 

 

512
100 200 _ 30
y—pounts
2000 (b) .. .
g .3 .
.1500 .
e 5 ..
3. a! “
E1000 ii)
.— c
500. “.3
'1
o 0.25 0 0.75 1

.5
Mixture Fraction

0 400 512
5
2_=x10
E 2 (C)
151.5
9
IE 1
5
0.5
unuufuovucgﬁc— :a viruﬂ
G
0 0.25 0.5 0.75 1

 

 

 

 

 

 

 

Mixture Fraction

Figure 2.8. Flame normal analysis at a burning region of the ﬂame (a) Mixture
fraction contours showing the line of section (1)) Temperature (c) Strain rate

100

 

 

 

 

 

 

 

 

 

100 200 _ 300 400 512
y-pomts
5
2 E x 10
2000 (b) 2 (C)
a: A E
E D I; n C 3
E1500 (_ “it n $1.5 CD (1 D
a c “
_ u . ._ 1 :1 a D
$11000 5 .. ” 2..-
a)
*— ‘1 0.5 item
500 = “a
F 1 G
0 02 0 0.25 0.5 0.75

5 0.5 o 75 . .
Mixture Fraction Mlxture Fraction

Figure 2.9. Flame normal analysis at a about—tt>-be-quenched region of ﬂame and for

t/r, : 1.0 (a) Mixture fraction contours showing the line of section (b) 'I‘emperature
(c) Strain rate

101

 

 

 

1 00 200 300 400 51 2

 

 

 

 

 

 

y—points
5
2000 2'5 10
(b) . (C)
E‘ 2
E 1500 3 o u
.3 1 D 31.5
2 000 “ e i:
5 ‘ . 1 .
I— :1 ‘7) n u
500 a 0.513 a, U
f :1
o 0.25 0.5 0.75 1 C

Mixture Fraction

 

 

O 0.25 0 5 0.75 1

Mixture Fraction

Figure 2.10. Flame normal analysis at a quenched region of ﬂame and for t/r, : 1.5
(a) Mixture fraction contours showing the line of section (b) Temperature (c) Strain

rate

102

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

0.07
2000 (a) 0.06
g 0.05
m 1500
5 00.04
g o
8 >' 0.03
E 1000
,9 0.02
500 0.01
0
o 0.2 0.4 0.6 0.8 1 0 0-2 0~4 0-6 0~8 1
Mixture Fraction Mixture Fraction
-3
-4
6x 10 x 10 .
1.5 31
(c) (d)
4 -. 1
I .
o I
>- 53; . >3:
2 'f 0.5
0 -..... ....
0.05 0.1 0.15 0.2 0 Q1 0.2 0,3
Mixture Fraction Mixture Fraction

Figure 2.11. Scatter plots in mixture fraction space for turbulence case I and ﬂame
No. 1 at one eddy turn-over time. (a) Temperature (b) YOU (0) Y0 H (d) YH- Dashed
lines shows the initial laminar proﬁle

103

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

0.07
2000
(a) 0.06
_. (b)
25.
an 1500 0.05
‘5
.6 o 0.04
g >-
E 1000 . 0.03
(D
[— f 0.02
500 i 0.01 \\
G \
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Mixture Fraction Mixture Fraction
-3
X 10 .4
6 x 10
1.5
41"; (d)
213
1 if
>-I J1}.
if!
0.5
O 0.05 0.1 0.15 0.2 0 0.1 0.2 0.3
Mixture Fraction

Mixture Fraction

Figure 2.12. Scatter plots in mixture space for turbulence case I and ﬂame No. 2 at
one eddy turn—over time. (a) Temperature (b) YC’O (c) Y0 H ((1) Y”. Dashed lines

shows the initial laminar proﬁle

104

 

 

 

 

 

 

 

 

 

 

 

 

2000 0.08
._. 0.07
E.
93 1500 0.06
.3 )8 0.05
“én’iooo 0'“
19 0.03
0.02
500 0.01
0‘ .
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Mixture Fraction Mixture Fraction
—3
x 10 -4
6 x 10
c
f l 1.5 (d)

 

 

 

 

 

 

Mixture Fraction Mixture Fraction

Figure 2.13. Scatter plots in mixture fraction space for turbulence case I and ﬂame
No. 3 at one eddy turn-over time. (a) Temperature (b) YCO (c) Y0 H (d) YH. Dashed
lines shows the initial laminar proﬁle

105

 

 

 

 

 

2000 (a) / ~\
I ‘ ~
I 8 ~
I \
r—o , ‘
i .’ 9‘
9 1500 ,I t‘ o
a .’ ‘. 5
S I \ ‘6
a ’I \‘ a
I \
E 1000 1 1 ‘5‘
o ,' X o
I‘— I \ I'-
I \
I \
I \
500 I ‘~
I \
I \
1
O 0.2 0.4 0.0 0.8 1
Mixture Fraction
2000

Temperature [K]

 

 

S

d

Temperature [K]

 

 

0 0.2 0.4 0.6

Mixture Fraction

0.8 1

 

S

S

8

 

 

 

 

0 0.2 0.4 0.6

0.8 1
Mixture Fraction

 

 

 

 

 

0 0.2 0.4 0.6

0.8 1
Mixture Fraction

Figure 2.14. Temperature scatter plots for turbulence case I and ﬂame No. 3 for (a)

t/Tt = 0.0 (b) t/Tt = 1.0 (c) t/Tt = 1.5 (d) t/Tt = 2.0 in mixture fraction space. Dashed
line shows the initial laminar proﬁle.

106

—3
6x10

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

-3
6x10
(8 (m
4.
I
o A
>' 1' 1‘
2‘ i. ’
0
0 05 1
Mixture Fraction Mixture Fraction
3 -3
6 x 10 - 6 x 10 -
(0
. 41
I
o
>.
2.
i ,. =7 -. .. .~ 13f.
0‘ , ~ 4' -. .I ’_
1 0 05
Mixture Fraction Mixture Fraction

the initial laminar proﬁle.

Figure 2.15. Y0 H scatter plots for turbulence case I and ﬂame No. 3 for, (a) t/rt = 0.0,
(b) t/Tt = 1.0, (c) t/rt = 1.5, (d) t/Tt = 2.0 in mixture fraction space. Dashed line is

107

 

0.08 (a)

0.06

O

o
>' 0.04

0.02

 

 

 

 

0.08 (b)

0.06

O

>0 0.04

0.02

 

 

 

0.5

Mixture Fraction

 

0.08

 

0.08

0.06

O

o
>' 0.04

 

0.5

 

Mixture Fraction

the initial laminar proﬁle.

0

>-°0

0.06
.04

0.02

 

 

0.5
Mixture Fraction

 

Figure 2.16. YCO scatter plots for turbulence case I and ﬂame No. 3 for (a) t/Tt — 0 0
(b) t/rt : 1.0, (c) t/r, : 1.5, (d) t/‘rt = 2.0 in mixture fraction space. Dashed line is

 

M
O
0

Grid points
(a)
o
O

4;
O
O

 

(’1
O
O

100 200300400500 100 200 300 400500
Grid points Grid points

Grid points
8 8 a
O O O

A
O
O

 

500

 

100 200 300 400 500 100 200 300 400 500
Grid points Grid points

Figure 2.17. Sequence of vorticity images for turbulence ease I and ﬂame No. 3 with
time: (a) t/T, = 0.0, (b) t/T, = 1.0. (c) t/Tt = 1.5. (d) t/rt = 2.0.

109

Grid points 3?
Q N —L v
o o O
o o o

&
O
O

 

8

100200300400500 100200300400500
Grid points Grid points

A
O
V

m —L
O O
O 0

Grid points
0)
O
O

400

   

100200300400500 100200300400500
Grid points Grid points

Figure 2.18. Sequence of enstrophy images for turbulence case I and flame No. 3 with
time: (a) f/Tt : 0.0, (b) t/T,:1.0, (c) i/T,:1.5. (Mt/1": 2.0.

110

   

   

(a) (b)

100 100
2 2
.E 200 - .E 200
8 ‘ 8
33 300 E 300
(D ,. (D

400 , 400

500 500

100 200 300 400 500 100 200 300 400 500
Grid points Grid points
a a
x 10 x 10
(c) (d)

100 . 4 100 . 8
.9 .9
g 200 3 g 200 6
.2 300 ' 2 E 300 4
(D . ‘ (D

400 1 400 2

500 500

100 200 300 400 500 100 200 300 400 500
Grid points Grid points

Figure 2.19. Sequence of turbulent kinetic energl images for turbulence case I and
ﬂame No. 3 with time: (a) t/Tt = 0.0, (b) t/Tt = 1.0, (c) t/T‘t = 1.5, (d) t/rt : 2.0.

111

A
m
v
A
U"
v

8 5

Grid points
8
0

Grid points
8 ‘8 8 s

b
8

   

8

100200300400500 100200300400500
Grid points Gn'd points

Grid points §
00 N —t
8 8 8

A
O
O

   

01
O
O

100 200 300 400 500 100 200 300 400 500
Grid points Grid points

0.1

Figure 2.20. Sequence of strain rate images for turbulence case I and ﬂame No.
with time: (a) t/T, = 0.0, (b) t/T, = 1.0, (c) t/Tt = 1.5, (d) t/Tf = 2.0.

112

 

 

 

 

 

 

 

 

 

3x104
6)
2
1
9’
1001
8
_‘|1 1
-2 1
-3 ‘
0 0.5 1
x/L
104 x
X
c) 4 -
2 . ..1 7'. '
AH. 5;", I If!!! I
so "-. or iii. 11.1 !i!,w
«1 -' 41." 1-21- 15v
3 f 1 1'1““! 1,-3.’
-2. s
I
-4 - ,
0 0.5 1

 

 

 

 

 

 

 

 

x10
b) ,
2’ 1‘ g I
11 :i I" ‘ 11“ I ‘
It
9 "ii:"" run"
<00 l “'0‘,‘ ‘ I ‘ 'I‘ "I
8 . " . "U IplI‘ '
_1 2 I 1‘, "' \l‘
-2 1 i
|
-3 ‘
0 0.5 1
x/L
x10“ x
d) 3 ' =
1 r 9:53 : ":2 ‘51:!
C)” 5’ f 3 I 33.25 I?"
350L513; 3’ g,«
-1 £5 2?: E" ' 35 .5: . iii;
4.; ' "i
-3 ‘ f
0 0.5 1

Figure 2.21. Y-averaged vorticity plots at four eddy turn-over times. (a) t/Tt = 0, (b)
t/Tt 21.0, (C) t/Tt = 1.5, and (d) t/Tt = 2.0

113

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x10
3 -
a) 1"
2.5' ' l. I ,I‘
' I 0
"hi" I.
92’ ‘ I I l |" I ' I
0}“, 1 ll 5 I: .
1.5 1 u- I .1
1 1
0.5 ‘
0 0.5 1 1
X/L
109 x
x
c) 12 g
9
1O 2 1.5’ ; ‘
81 .‘_ ‘ E
9 5 1 3
w 6’ ‘ II ‘ :2
C} .9. ii 3 a
4' r - l‘ ' O}
". 'I -! 05 Z;
2 JV, 'Vi ' 1 ° 1 ' 5 5' ....- 3 .. ‘1
thorﬂ.’ 4., 3-"'c\.l-‘ 3' 5- , '5". ~ :5..- '1
o - 0’
0 05 1 O 0.5 1
x/L x/Lx

Figure 2.22. Y-averaged enstropy plots at four eddy turn-over times. (a) t/Tt = O, (b)
t/Tt = 1.0, (c) t/Tt = 1.5, and (d) t/n = 2.0

114

Grid points
8 M
o 8

A
O
0

§

 

100200300400500 100200300400500
Grid points Grid points
x10

100200300400500 100200300400500
Grid points Grid points

N

N
O
O
0

Grid points
(.0
o
O

#
8

01
O
O

 

Figure 2.23. Sequence of vorticity images for turbulence case I and ﬂames of various
premixing levels. (21) Initial vorticity. (b) A‘Z = 0 & t/T, = 1.0. (C) A% : 25 &
i/Tp =1.0. ((1) A%= 75 git/77:10.

115

  
 

   

(a) 25 (b)
100 - 100
9 2 .‘e
.E 200 .5 200
8 15 8
2300 72300
o 1 <5
400 05 400
500 500
100200300400500 100 200300400500
Grid points Grid points

   

100200300400500 100200300400500
Grid points Grid points

Figure 2.24. Sequence of enstrophy images for turbulence case I and ﬂames of various
premixing levels. (a) Initial enstrophy, (b) A% = 0 85 l/Tt = 1.0, (c) A% = 25 $1.
t/Tt = 1.0, (d) A% = 75 & t/TI = 1.0.

116

(a)

   

   

100
.‘2
.E 200
a
E 300
(D
400
500
100 200 300 400 500 100 200 300 400 500
Grid points Grid points .
x10!a
(C)
2
£2
E 1.5
O
D.
jg 1
(D
0.5
100 200 300 400500 100 200300400500
Grid points Grid points

Figure 2.25. Sequence. of turbulent kinetic energy images for turbulence case I and
ﬂames of various premixing levels. (a) Initial TKE, (b) A% = 0 & t/T, = 1.0, (c) A%
= 25 & t/Tt 21.0, (d) A% = 75 & t/Tt = 1.0.

117

 

 

 

 

 

 

 

o 0.2 0:4 0.6 0:8 1
lex

x104

 

 

(b) 25%

 

 

 

 

 

 

-3 - - - _
O (12 (l4xnxd16 (18 1

Figure 2.26. Y-averaged vorticity for different partial premixing. (a) Initial proﬁle,
and (b) At t/r, : 1.0.

118

x10

 

(a)
2.5‘

avg

1.5'

 

 

 

 

 

 

 

0.5-
x/Lx

x10

0.8 1

 

(b)

 

 

 

 

 

0%
25% ,
75%

ﬂ .

 

 

 

 

O-
0 0.2 0.4 x/Lx 0.6

Figure 2.27. Y-averaged enstrophy for different partial premixing. (a)Initial profile,

and (b) At t/Tt = 1.0.

119

0.8 1

CHAPTER 3

Conclusions

Numerical simulations of turbulent combustion involving detailed chemical kinetics
mechanisms are computationally very expensive. Even the simplest hydrocarbon
fuel, methane needs about 50 species and 300 reactions for a “complete” modeling
of its combustion with air. Reduced reaction mechanisms would be a partial remedy
for the high computational requirements in such simulations and are usually derived
systematically from the original “full” mechanism. These simpler mechanisms
should represent the combustion chemistry accurately in all the possible situations
of practical importance. In this work, a detailed assessment of methane-air reduced
reaction mechanisms is performed in a laminar counterﬂow ﬂame conﬁguration for
various ﬂow/ chemistry conditions. The effects of partial premixing of the fuel with
air on the compositional and physical structure of the ﬂame are studied via variety

of reduced and full reaction mechanisms.

Temperature and species profiles produced by different mechanisms were com-
pared to those obtained by the GRI-Mech 3.0 for wide range of strain rates and
partial premixing levels of fuel stream with air. Based on the simulations performed,
it was found that the reaction mechanisms that contain reactions involving both the
C1 and C2 species perform much better than the mechanisms with only Cl species.
Results obtained with 10-step and 12—step mechanisms compare very well with those
of the full GRI mechanisms for all values of strain rate, and premixing, and generate

the same ignition and extinction characteristics. It is concluded that the 10-step

1‘20

reaction mechanism with only 14 species and 10 reactions, is a good candidate for
modeling of the the chemistry in ﬁnite-rate simulations. However, other mechanisms
with lesser number of species such as the 4-step, 5—step and 6—step, although perform
well in some of the conditions, are not fully accurate. The performance of these
reaction mechanisms are found to deteriorate as the strain rate is decreased or the
level of fuel premixing with air is increased. Whereas, 4-step mechanism fail to
predict the rich premixed ﬂame zone in partially premixed ﬂames, 5-step and 6—step
mechanisms predict a premature premixed ﬂame zone. The worst affected species by
these improper premixed ﬂame prediction are CO and H2 species. In fact the trends
observed for these species as the strain rate is varied are exactly opposite to what is

predicted by all the other mechanisms.

The 10—step mechanism, which was found to be an accurate model in different
ﬂow/ ﬂame conditions, was used for prediction of ignition temperatures. Prediction
of ignition temperature is important in the modeling of quenched ﬂamelet re—ignition
process. Our laminar opposed-jet results indicate that the ignition temperatures are
increased as the strain rate increase, which is expected. However, the effect that the
fuel premixing has on the ignition temperature is not very clear. At low strain rates,
the effect is small but at high strain rate the ignition temperature tend to increase

as partial premixing decreases.

The 10-step mechanism is also used in a direct numerical simulation (DNS) of
isotropic turbulent flow interacting with a diffusion/partially—premixed ﬂame. DNS
were conducted to understand the behavior of the ﬂames in realistic ﬂow situations
of highly varying unsteady strain ﬁeld. The effect of the ﬂame on the turbulence ﬁeld
is also studied. Our results show signiﬁcant changes in the ﬂame structure embedded
in a high intensity turbulent ﬁeld due to the imposed variable strain ﬁeld and
significant ﬂame stretching and distortion. Deviation from the initial (laminar) ﬂame
structure in compositional space has been observed accompanied with signiﬁcant

local extinction at the locations where the strain rate was too high for the ﬂame

121

to be sustained. These effects were reﬂected in the enhanced scatter in the scatter
plots for the temperature and the species mass fractions. At early times, signiﬁcant
vorticity dissipation was observed due to increased viscosity near the ﬂame and due
to the volumetric expansion. In the later times, vorticity is seen to be generated by

the so called baroqlinque term near to the ﬂame.

As the list of reduced reaction mechanism for different fuels are not exhaus-
tive, detailed assessment of available mechanisms including variable diffusivity
and radiation models can be performed as a future work. Since it is essential to
understand various characteristics of ﬂame-turbulence interactions for combustion
modeling, a detailed study with various ﬂame thicknesses, premixing and using
different levels of approximations to the transport models can be pursued as an
extension to this work. Finally, as the small scale behavior in the two-dimensional
turbulence is different from the realistic three-dimensional turbulence, simulations
using three-dimensional DNS can be used to understand the ﬂame/turbulence

interactions in more realistic ﬂows.

122

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