Computations of Turbulent
Premixed Flames Using
Conditional Moment Closure
Shokri Amzin
Department of Engineering
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
February 2012
2
I would like to dedicate this Doctoral
dissertation to my parents and my family for
their moral support and continued love for my
entire life. Most importantly, to my beautiful
wife Catharina and my gorgeous son Shaheer
for their love and patience.
Declaration
I hereby declare that the work presented in this dissertation is the result of my
own work includes nothing which is the outcome of work done in collaboration
except where specifically indicated in the text by a reference. No part of this
work has been submitted for any other degree or diploma.
This dissertation contains approximately 38000 words and 56 figures.
Hopkinson Laboratory, Cambridge Shokri Amzin
28th February 2012
ii
Acknowledgements
It has been an honor to work with Dr. N. Swaminathan during my
PhD. I am very grateful for his guidance, advice and supervision from
the initial to the final level which, enabled me to develop an under-
standing of turbulent combustion modelling.
I am also heartily thankful to all those who assisted me in the prepara-
tion and completion of this study. I mention here, Dr. Jim Rogerson
and Dr. Hemanth Kolla for their continuous support.
I also wish to extend my sincere gratitude to Professors Nondas Mas-
torakos, Stewart Cant, Nick Collings and Simone Hochgreb for their
assistance and guidance through the combustion courses.
I owe my deepest gratitude to Mr. Peter Benie for his IT support and
Mrs. Kate Graham for her assistance. Sincere thanks to my friends
and colleagues members of Heat gallery and Hopkinson Labs.
Finally, the generous financial support of Cambridge Commonwealth
Trust, Siemens and EPSRC is acknowledge gratefully. This disserta-
tion would not have been possible without their supports.
Publications
The results of this thesis have been partially presented and published
in:
1. S. Amzin, N. Swaminathan, J. W. Rogerson, J. H. Kent. Condi-
tional moment closure for turbulent premixed flames. presented
in 23rd ICDERS, California, Irvine , 2011.
2. S. Amzin, N. Swaminathan, J. W. Rogerson, J. H. Kent. Condi-
tional moment closure for turbulent premixed flames. Combus-
tion Science and Technology, 2011,(accepted).
3. S. Amzin, N. Swaminathan. Conditional moment closure for
turbulent lean premixed flames. Combustion Science and Tech-
nology, 2011,(in preparation).
Abstract
Lean premixed combustion is at present one of the most promising
methods to reduce emissions and to maintain high efficiency in com-
bustion systems. As the emission legislation becomes more stringent,
modelling of turbulent premixed combustion has become an important
tool for designing efficient and environmentally friendlier combustion
systems. However, in order to predict these emissions reliable pre-
dictive models are required. One of the methods used for predicting
pollutants is the conditional moment closure (CMC), which is suit-
able to predict pollutants with slow time scales. Despite the fact that
CMC has been successfully applied to various non-premixed combus-
tion systems, its application to premixed flames is not fully tested
and validated. The main difficulty is associated with the modelling
of the conditional scalar dissipation rate (CSDR) of the condition-
ing scalar, the progress variable. In premixed CMC, this term is an
important quantity and represents the rate of mixing at small scales
of relevance for combustion. The numerical accuracy of the CMC
method depends on the accuracy of the CSDR model. In this study,
two different models for CSDR, an algebraic model and an inverse
problem model, are validated using two different DNS data sets. The
algebraic model along with standard k-ε turbulence modelling is used
in the computations of stoichiometric and very lean pilot stabilized
Bunsen flames using the RANS-CMC method. A first order closure
is used for the conditional mean reaction rate. The computed non-
reacting and reacting scalars are in reasonable agreement with the
experiments and are consistent with earlier computations using flam-
lets and transported PDF methods for the stoichiometric flames, and
transported PDF methods for the very lean flames. Sensitivity to
chemical kinetics mechanism is also assessed.
Contents
Declaration ii
Contents vii
List of Figures x
Nomenclature xvi
1 Introduction 1
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Background on Turbulent Premixed Combustion 8
2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Fluid Flow Equations . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Reaction Rates . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Three Simulation Paradigms . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 LES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 RANS Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1 Turbulent Reynolds Stress Model . . . . . . . . . . . . . . 22
2.4.2 Turbulent Scalar Flux Model . . . . . . . . . . . . . . . . 24
2.5 Premixed Combustion Sub-Models . . . . . . . . . . . . . . . . . 25
2.5.1 Eddy Break-Up Model . . . . . . . . . . . . . . . . . . . . 29
2.5.2 Bray-Moss-Libby Model . . . . . . . . . . . . . . . . . . . 30
vii
CONTENTS
2.5.3 Flame Surface Density Model . . . . . . . . . . . . . . . . 33
2.5.4 The Probability Density Function . . . . . . . . . . . . . 34
2.5.5 G-Equation Model . . . . . . . . . . . . . . . . . . . . . . 37
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Conditional Moment Closure 39
3.1 CMC Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Premixed CMC Method . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Premixed CMC Sub-Models . . . . . . . . . . . . . . . . . . . . . 45
4 Scalar Dissipation Rate 49
4.1 Mean Scalar Dissipation Rate . . . . . . . . . . . . . . . . . . . . 50
4.2 Conditional Mean Scalar Dissipation Rate . . . . . . . . . . . . . 51
4.2.1 Algebraic Model . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2 Inverse Model . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 DNS Database . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . 58
5 Mathematical Model 61
5.1 Discretisation of the CMC Equation . . . . . . . . . . . . . . . . . 61
5.2 Implementation for Premixed Flames . . . . . . . . . . . . . . . . 66
5.3 Computer Model and Sequence . . . . . . . . . . . . . . . . . . . 68
6 Stoichiometric Premixed Flame Calculation 70
6.1 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2.1 Non Reacting Flow . . . . . . . . . . . . . . . . . . . . . 75
6.2.2 Conditional Scalar Dissipation Rate . . . . . . . . . . . . . 79
6.2.3 Terms in CMC Transport Equation . . . . . . . . . . . . . 82
6.2.4 Comparison to Experimental Results . . . . . . . . . . . . 85
7 Lean Premixed Flame Calculation 102
7.1 Flames Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . 105
viii
CONTENTS
7.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.1 Conditional Mean Mass Fractions . . . . . . . . . . . . . . 110
7.3.2 Comparison to Experimental Results . . . . . . . . . . . . 111
8 Summary and Conclusions 135
8.1 Conclusion From This Work . . . . . . . . . . . . . . . . . . . . . 135
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Appendix A 141
Bibliography 145
ix
List of Figures
2.1 Structure of a laminar diffusion flames. . . . . . . . . . . . . . . . 10
2.2 Structure of a laminar premixed flames. . . . . . . . . . . . . . . . 10
2.3 The premixed combustion regime diagram. . . . . . . . . . . . . . 29
3.1 Scatter plots of instantaneous and conditional averages for the tem-
perature and OH mass fraction at x/D =20 and r/D = 1.7 (◦) ;
1.4 (¤); 1.1 (4) [1]. Solid line represents fully burning strained
laminar flame result. . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1 The variation of the Favre averaged PDF of the progress variable,
p˜(ζ), at θ = 0.5 inside the flame brush of flames R2a and R2c. . . 56
4.2 The variation of the Favre averaged PDF of the progress variable,
p˜(ζ), at θ = 0.6 inside the flame brush of flames VA and VB. . . . 57
4.3 The premixed combustion regime diagram showing the selected
DNS flames. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 The typical variation of the conditional scalar dissipation rate,
〈N |ζ〉+, calculations obtained by, Eq. (4.7) and Eq. (4.11) with
ζ in flames R2a and R2c. The values are non-dimensionalised us-
ing SoL and δ
o
L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.5 The typical variation of the conditional scalar dissipation rate,
〈N |ζ〉+, obtained by, Eq. (4.7) and Eq. (4.11) with ζ in flames VA
and VB at two different axial locations in the flame; x1 = 16.7δ
◦
L
and x2 = 27.9δ
◦
L. These values are non-dimensionalised using S
o
L
and δoL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
x
LIST OF FIGURES
5.1 Control volume in a finite volume method. . . . . . . . . . . . . . 62
5.2 Interaction of CMC and RANS equation in the simulation. . . . . 68
6.1 Schematic diagram of the burner setup [2] and the computational
domain along with BCs. . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 The premixed combustion regime diagram showing flames F1, F2
and F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3 The normalised mean velocity (lines) from non-reacting flow are
compared to the the experimental measurements (symbols) for
flames F1, F2 and F3. . . . . . . . . . . . . . . . . . . . . . . . . 77
6.4 The normalised mean turbulent kinetic energy (lines) from non-
reacting flow are compared to the experimental measurements (sym-
bols) for flames F1, F2 and F3. . . . . . . . . . . . . . . . . . . . 78
6.5 Computed radial variation of Z˜ in cold flow simulation at five axial
locations for the conditions of flame F1. . . . . . . . . . . . . . . . 79
6.6 The typical variation of the conditional scalar dissipation rate,〈N |ζ〉+,
obtained in CMC calculations by, Eq. (4.7), with ζ in flame F1 at
different locations inside the flame brush at x/D = 4.5. The values
are non-dimensionalised using SoL and δ
o
L. . . . . . . . . . . . . . . 80
6.7 The typical variation of the conditional scalar dissipation rate,〈N |ζ〉+,
obtained in CMC calculations by, Eq. (4.7), with ζ in flame F1 at
different locations inside the flame brush at x/D = 8.5. The values
are non-dimensionalised using SoL and δ
o
L. . . . . . . . . . . . . . . 81
6.8 Typical variation of different terms of the CMC transport equation,
Eq. (6.2) , with ζ in flame F1 at θ˜ = 0.55. The results are shown
for few representative scalars. . . . . . . . . . . . . . . . . . . . . 83
6.9 The variation of the conditional mean mass fraction with ζ for
major and minor species in flame F1 at θ˜ = 0.55 and x/d = 4.5. . 84
6.10 The variation of the conditional mean mass fraction with ζ for
major and minor species in flame F1 at θ˜ = 0.55 and x/d = 8.5. . 84
6.11 The variation of the conditional mean mass fraction with ζ for
major and minor species in flame F1 at θ˜ = 0.55 and x/d = 10.5. 85
xi
LIST OF FIGURES
6.12 The variation of the Favre averaged PDF of the progress variable,
p˜(ζ), at three locations in the flame brush of flame F1. . . . . . . 86
6.13 Contours of the mean progress variable and its variance for flame F1 87
6.14 Contours of the mean mixture fraction variable and temperature
for flame F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.15 The normalised mean velocity and turbulent kinetic energy from
the CMC calculations (lines) at five axial locations are compared
to experimental results (symbols) for flame F1. . . . . . . . . . . . 90
6.16 The normalised mean temperature and CH4 mass fraction × 100,
from the CMC calculations (lines) at five axial locations are com-
pared to experimental results (symbols) for flame F1. . . . . . . . 92
6.17 The mean mass fractions of H2O, CO2 and O2 from the CMC calcu-
lations (lines) at five axial locations are compared to experimental
results (symbols) for flame F1. . . . . . . . . . . . . . . . . . . . . 93
6.18 The mean mass fractions of CO× 10, OH× 75 and H2× 100 from
the CMC calculations (lines) at five axial locations are compared
to experimental results (symbols) for flame F1. . . . . . . . . . . . 94
6.19 The normalised mean velocity from the CMC calculations (lines) at
five axial locations are compared to experimental results (symbols)
for flames F2 and F3. . . . . . . . . . . . . . . . . . . . . . . . . 96
6.20 The normalised mean turbulent kinetic energy from the CMC cal-
culations (lines) at five axial locations are compared to experimen-
tal results (symbols) for flames F2 and F3. . . . . . . . . . . . . 97
6.21 The normalised mean mass fraction of CH4× 100 from the CMC
calculations (lines) at five axial locations are compared to experi-
mental results (symbols) for flames F2 and F3. . . . . . . . . . . . 98
6.22 The normalised mean temperature from the CMC calculations
(lines) at five axial locations are compared to experimental results
(symbols) for flames F2 and F3. . . . . . . . . . . . . . . . . . . . 99
6.23 The mean mass fractions of H2O (4,−), CO2 (◦, · · ·) and O2 (•,−·
−) from the CMC calculations (lines) at five axial locations are
compared to experimental results (symbols) for flames F2 and F3. 100
xii
LIST OF FIGURES
6.24 The mean mass fractions of CO× 10 (◦, · · ·), OH× 75 (4,−) and
H2× 100 (•,− · −) from the CMC calculations (solid lines) at five
axial locations are compared to experimental results (symbols) for
flames F2 and F3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.1 Schematic diagram of the premixed Bunsen burner from [3; 4]. . . 103
7.2 The premixed combustion regime diagram showing the stoichio-
metric and lean flames. . . . . . . . . . . . . . . . . . . . . . . . . 105
7.3 Schematic diagram of the burner setup and the computational do-
main along with BCs. . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 The calculated (lines) normalised mean axial velocity and rms ve-
locity from non-reacting and reacting flows are compared to exper-
imental data (symbols) for flame PM1-200. . . . . . . . . . . . . . 109
7.5 The calculated (lines) normalised mean axial velocity and rms ve-
locity from non-reacting and reacting flows are compared to exper-
imental data (symbols) for flame PM1-50. . . . . . . . . . . . . . 110
7.6 The variation of the conditional mean mass fraction with ζ for
major and minor species for flame PM1-200 at θ˜ = 0.55 and x/D
= 4.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.7 The variation of the conditional mean mass fraction with ζ for
major and minor species for flame PM1-200 at θ˜ = 0.55 and x/D
= 8.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.8 The variation of the conditional mean mass fraction with ζ for
major and minor species for flame PM1-200 at θ˜ = 0.55 and x/D
= 10.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
7.9 The typical variation of the conditional scalar dissipation rate,〈N+θ |ζ〉,
obtained in the CMC calculation for flames PM1-50 and PM1-200
at θ˜ = 0.55, x/D = 4.5 . The values are non-dimensionalised using
SoL and δ
o
L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.10 The typical variation of the conditional scalar dissipation rate,〈N+θ |ζ〉,
obtained in the CMC calculation for flames PM1-50 and PM1-200
at θ˜ = 0.55, x/D = 8.5 . The values are non-dimensionalised using
SoL and δ
o
L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
xiii
LIST OF FIGURES
7.11 The typical variation of the conditional scalar dissipation rate,〈N+θ |ζ〉,
obtained in the CMC calculation for flames PM1-50 and PM1-200
at θ˜ = 0.55, x/D = 10.5 . The values are non-dimensionalised
using SoL and δ
o
L. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.12 The variation of the Favre PDF of the progress variable, p˜(ζ), at
selected locations of the flame brush of flame PM1-50. . . . . . . . 115
7.13 The variation of the Favre PDF of the progress variable, p˜(ζ), at
selected locations of the flame brush of flame PM1-200. . . . . . . 116
7.14 Computed radial variation of Z˜1 from the hot flow simulation at
six axial locations for the conditions for flame PM1-200. . . . . . 117
7.15 Computed radial variation of Z˜1 from the hot flow simulation are
compared to PDF calculations [5] for flame PM1-200. . . . . . . . 118
7.16 Computed radial variation of Z˜2 from the hot flow simulation at
six axial locations for the conditions for flame PM1-200. . . . . . 118
7.17 Computed radial variation of Z˜3 from the hot flow simulation at
six axial locations for the conditions for flame PM1-200. . . . . . 119
7.18 Contours of Z˜3 for flame PM1-200 . . . . . . . . . . . . . . . . . . 119
7.19 Contours of Z˜1 and Z˜2 for flame PM1-200 . . . . . . . . . . . . . 120
7.20 Contours of the mean progress variable and its variance for flame
PM1-200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.21 The mean velocity from the CMC calculations at six axial locations
are compared to the experimental data [4] and PDF calculations,
k-ε and k-ω [6] for flame PM1-200. . . . . . . . . . . . . . . . . . 125
7.22 The mean velocity from the CMC calculations at four axial loca-
tions are compared to the experimental data [4] and PDF calcula-
tions, k-ε and k-ω [6] for flame PM1-50. . . . . . . . . . . . . . . 126
7.23 The mean rms velocity from the CMC calculations at six axial
locations are compared to the experimental data [4] and PDF cal-
culations, k-ε and k-ω [6] for flame PM1-200. . . . . . . . . . . . . 127
7.24 The mean rms velocity from the CMC calculations at four axial
locations are compared to the experimental data [4] and PDF cal-
culations, k-ε and k-ω [6] for flame PM1-50. . . . . . . . . . . . . 128
xiv
LIST OF FIGURES
7.25 The mean temperature from the CMC calculations at four axial
locations are compared to the experimental data [4] and PDF cal-
culations, k-ε and k-ω [6] and VFJPDF [5] for flame PM1-200. . . 129
7.26 The mean temperature from the CMC calculations at four axial
locations are compared to the experimental data [4] and PDF cal-
culations, k-ε and k-ω [6] and VFJPDF [5] for flame PM1-50. . . 130
7.27 The mean C˜O× 103 from the CMC calculations at four axial loca-
tions are compared to the experimental data [4] and PDF calcula-
tions, k-ε and k-ω [6] and VFJPDF [5] for flame PM1-200. . . . . 131
7.28 The calculated mean C˜O × 103 mass fraction from CMC calcu-
lations at four axial locations are compared to the experimental
data [4] and PDF calculations, k-ε and k-ω [6] and VFJPDF [5]
for flame PM1-50. . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.29 The calculated mean OH× 106 from CMC calculations at five ax-
ial locations are compared to the experimental data [4] and PDF
calculations, k-ε and k-ω [6] for flame PM1-200. . . . . . . . . . . 133
7.30 The calculated mean OH × 106 from CMC calculations at four
axial locations are compared to the experimental data [4] and PDF
calculations, k-ε and k-ω [6] for flame PM1-50. . . . . . . . . . . . 134
xv
Nomenclature
Latin
A frequency factor
c progress variable
c Reynolds averaged c
c˜ Favre averaged c
c′′ Favre fluctuation of c
C²1 constant in k-ε model
C3, C4 model parameters, Eq. (4.1)
D molecular diffusivity
Ea activation energy
h specific enthalpy
hα total enthalpy of species α
hs sensible enthalpy
k˜ Favre average turbulent kinetic energy
K∗c constants in Eq. (4.1)
N scalar dissipation rate
p˜(c) Favre PDF of c
P pressure
Q conditional average
R characteristics gas constant
R universal gas constant = 8.314 JK−1mol−1
xvi
SoL unstrained planar laminar flame speed
t time
T temperature
Ta activation temperature
Tad adiabatic flame temperature
Tu temperature of unburnt mixture
ui velocity component in direction i; i = 1, 2, 3
Wα molecular weight of species α
WR radiative heat loss
Y mass fraction
Z mixture fraction
Greek
α thermal diffusivity, a species in the mixture
β′ model constant
δij Kronecker delta; δij = 1 if i = j and 0 otherwise
δ◦L thermal laminar thickness
∆x grid spacing
² turbulence eddy viscosity
²˜ Favre average turbulent kinetic energy dissipation rate
ηk Kolmoforov length scale
Λ integral length scale
λ thermal conductivity
µ dynamic viscosity
µt turbulent viscosity
ν kinematic viscosity, stoichiometric coefficient
ω˙ mean reaction rate
φ equivalence ration
ρ density of mixture
Σ flame surface density
σij stress tensor
τ heat release parameters
xvii
τij shear stress
τc chemical time scale
τk Kolmogrov time scale
τt integral time scale
²˜c Favre averaged scalar dissipation rate of c
ζ sample space variable for c
Symbols
φ Reynolds average of φ
φ˜ Favre (density weighted) average of φ
φ′ fluctuating component, φ′ = φ− φ˜
φ′′ conditional fluctuation, φ′′ = φ− 〈φ|η〉
φ˜′′2 Favre variance of φ
Dimensionless Numbers
Da τt/τc Damko¨hler number
Le SC/Pr = λ/(ρcpD) Lewis number
Pr µcp/λ Prandtl number
Sc µ/(ρD) Schmidt number
Re ρUL/µ Reynolds number
Pe Re.Pr Peclet number
Subscripts
1 Oxidant stream
2 Fuel stream
f Fuel
u Unburnt mixture
t Turbulent
xviii
Acronyms
BML Bray-Moss-Libby
CFD Computational Fluid Dynamics
CMC Conditional Moment Closure
CSDR Conditional Scalar Dissipation Rate
DNS Direct Numerical Simulations
EBU Eddy Break-Up
LES Large Eddy Simulation
LHS Left Hand Side
PDF Probability Density Function
RANS Averaged Navier Stokes Simulation
RHS Right Hand Side
RPV Reaction Progress Variable
RSM Reynolds Stress Models
xix
Chapter 1
Introduction
Energy dates back to the beginning of the first human. Primitive man used the
wood in the fire for the purpose of heating and cooking. In the later stages, he
domesticated animals to help him in agriculture and transport, as well as using
other types of energy such as wind and water power for the purpose of maritime
transport and irrigation.
Western European countries have witnessed, during the eighteenth and nine-
teenth centuries, the emergence of the industrial revolution, which shortly there-
after spread around the globe. As a consequence, scientific and technical changes
have emerged, especially in industry and transportation sectors. The actual start
of the industrial revolution with the invention of the steam engine by James Watt,
in 1769 at the University of Glasgow. The basic idea was to transform the steam
thermal energy to mechanical energy using coal as a source of energy. Steam en-
gines were used in pumping water and powering trains and ships. The invention
of electricity, by Thomas Edison in 1880, followed, by the invention of internal
combustion engine, by Rudolf Diesel in 1898, was the start to the shift to the use
of fossil fuel in its current forms. Later on, the rapid development in industry
1
1. Introduction
and transportation sectors increased the demand for fuel.
Fossil fuels are not renewable energy sources and have an impact on the envi-
ronment and mankind. Combustion can significantly disturb the natural carbon
cycle between the earth and atmosphere [7]. For example, the major products
from combustion are carbon dioxide (CO2) and water (H2O), these gases are
known as greenhouse gases. They trap the heat in the atmosphere and as re-
sult of that, the average global temperature of earth rises. This phenomenon
is widely known as global warming [8]. Also, primarily pollutant such as sulfur
dioxide (SO2) and nitric oxides (NOx) emitted by the use of fossil fuels, would
often reach the upper levels of the atmosphere and drift into areas where natu-
ral rain clouds are regularly formed and fall to the ground in the form of acid
rains. Acid rain causes severe damage to the environment and humans [9]. As
a consequence, the environmental legislation became more stringent to minimise
the environmental impact of combustion. For instance, the Clean Air Act 1956
which was introduced after London Smog in 1952. Later on, in 1993, the Charter
was updated in its current version to protect the general public from exposure to
airborne contaminants that are known to be hazardous to human health.
Although, rising oil prices and growing awareness of the global warming in
recent years has led to increase interest in bio-fuel and other forms of non-
combustible renewable energy such as solar, wind and hydro power. However, ac-
cording to European Commission statistics, renewable energy sources accounted
for 8.4 % of the EUs consumption in 2008 [10]. Certainly, it is highly unlikely
to replace fossil fuel with renewable energy in the near future. Thus, the chal-
lenge remains how to reduce emissions and their impact without compromising
the efficiency of combustion systems. This is can be achieved by developing new
2
1. Introduction
combustion technologies that meet environmental and efficiency demands. To
accomplish this we must first understand the complex physics of the combustion
process.
Combustion is traditionally classified into non-premixed and premixed com-
bustion. In non-premixed combustion, the fuel and the oxidiser enter the com-
bustion zone from two separate streams, but in the premixed situation, the fuel
and oxidiser are mixed homogeneously before entering into the combustion zone.
However, in practice neither of these two modes occur on their own, but a com-
bination of them is common. The fuel lean premixed combustion is known to
have potential to meet the environmental demands without compromising the ef-
ficiency by reducing combustion temperature [11; 12; 13; 14; 15; 16; 17]. However,
the fuel lean premixed flames are susceptible to combustion oscillations, stability
and extinction issues. For instance, lean premixed combustion is sensitive to the
quality of the mixing in unburned mixture. If the unburned mixture is not per-
fectly mixed some regions will be richer than others and form high temperature
pockets which leads to NOx formation [18]. Further more, the stability of lean
premixed flames can reduce the efficiency and the life time of the combustion de-
vice [19]. Thus, the original equipment manufacturers of combustion systems are
in constant search to develop methods, which could be used to identify avenues
to improve the stability of turbulent lean premixed flames.
Computational fluid dynamics (CFD), is embraced more in industries to iden-
tify potential ways to achieve the desired behaviour of combustion systems, since
it is cost-effective and can produce detailed information that are some times diffi-
cult and expensive to obtain from experiments. Although modelling of turbulent
combustion has been pursued for many years, a precise theoretical description to
3
1. Introduction
make the combustion models rigorous for industrial use is still evolving due to the
complexity of this subject [20]. In premixed flames, there are strong interactions
among turbulence and chemical reaction. Turbulence contains a wide range of
length and time scales [21]. The chemical reaction process also contains a wide
range of length and time scales and determining them over all reaction rates is
not a straight forward task for turbulent flames [22]. Thus, modelling turbulent
lean premixed flames is challenging. However, combustion models which are ca-
pable to address these interactions accurately with ability to predict emissions
are required in the design and development of next generation of lean combustion
systems. This requirement gives motivation for this study.
Turbulent flows are simulated using three approaches [22; 23]; direct numeri-
cal simulation (DNS), large-eddy simulation (LES) and Reynolds averaged Navier
stokes simulation (RANS). In DNS, all the length and time scales for turbulence
and chemical reactions are captured and the conservation equations (discussed
in the next chapter) are solved without any modelling using highly accurate nu-
merical methods. This method is prohibitively expensive to simulate practical
flames, because of the geometrical complexities and the range of scales involved.
Thus, this method is mainly used as a research tool to gain fundamental insights
from turbulent combustion in simple geometries or flows. In LES, the energy con-
taining scales of turbulence are captured explicitly using approximate governing
equations. Since chemical reactions occur at small scales, some sort of models
are required to represent combustion and its interaction with turbulence. These
models are known [22] as sub-grid scale combustion models. However, the nu-
merical resolution requirement for LES and the uncertainty in the sub-grid scale
modelling currently limits the use of LES for industry scale flames.
4
1. Introduction
In RANS, described in some details in chapter 2, all the scales of turbulence
and reactions are averaged and thus, the computational costs are lower compared
to the previous two approaches. However, the interaction of scales and their ef-
fects are to be included using turbulence and combustion models, discussed in
the next chapter. Since this method is developed in the past and many computa-
tional tools are readily available, the use of RANS is more common in industries,
but this may change in the next decade or so. However, as noted earlier, the
currently available combustion models are not as good as one would like to have
although RANS simulation is commonly used to model the turbulent reacting
flows since it is computationally cheaper approach compared to DNS and LES.
The main objective of turbulent combustion modelling is to provide a clo-
sure for the mean reaction rate, ω˙α, which appears in the mean species transport
equation. The reaction rate is a highly non-linear function of temperature and
species concentration, evaluating its mean value using average temperature and
scalar concentrations is known to be erroneous [24]. Flamelet based methods are
common for turbulent premixed flames [17] and these methods consider the tur-
bulent flame as a collection of laminar flames. This view is acceptable if the flame
scales are much smaller than the turbulence scales and thus the laminar flame
structure is not disturbed by turbulence. Even if the turbulence scales become
smaller than the thermal thickness of the laminar flame, the flamelet concept
can be used if the reaction zone thickness is smaller than the turbulence scale.
However, if the reaction time scale for a scalar become larger than the typical
time scale for smaller turbulence eddies then the use of flamelet based ideas to
compute that particular scalar becomes an issues [25; 26]. Pollutants such as
carbon monoxide, nitric oxides are good examples for this. Alternative methods
5
1. Introduction
such as conditional moment closure (CMC) [27; 28] and transported probability
density function (transported PDF) [29] have been proposed and developed. The
CMC, described in chapter 3 is of specific interest to this study for the following
reason. Although the CMC has been developed and successfully applied to vari-
ous non-premixed combustion systems such as hood fires [30], lifted flames [31],
bagasse-fired boiler [32], bluff-body flames [33; 34], spray ignition [35] and soot
formation [36], its application to premixed flames is not fully tested and validated
yet.
1.1 Objectives
The prime objective of this work is to study if the pollutants emitted from lean
premixed combustion can be calculated using CMC and to validate this calcula-
tions using published experimental results.
The specific objectives are:
1. To further develop CMC sub-models for premixed combustion.
2. To develop a new model for the conditional scalar dissipation rate required
in the CMC for premixed flames.
3. To validate the performance of this model with DNS data.
4. To implement the CMC sub models for premixed flames into a computa-
tional code.
5. To validate the simulation results using two different experimental data sets
[2; 3].
6
1. Introduction
The outline of this thesis is as follows.
Chapter 2 presents a brief review of the theory and background of turbulent
premixed flames. Chapter 3 describes the CMC methodology for turbulent pre-
mixed flames and chapter 4 discuses the modelling of mean and conditional scalar
dissipation rates used in this study. Chapter 5 describes the numerical methods
and the computational tool used in this study. The chosen test cases are described
briefly in chapters 6 and 7 along with the simulation results. The conclusions of
this study are summarised in the final chapter along with some suggestion for
future directions for the CMC modelling of turbulent premixed flames.
7
Chapter 2
Background on Turbulent
Premixed Combustion
The scope of this chapter is to present a comprehensive but a concise, overview of
turbulent premixed combustion modelling that appears in the literature [16; 22;
23; 25; 26; 37; 38; 39]. The well known modes of combustion are; non-premixed
and premixed combustion, the second type is the focus of this study, as noted in
chapter 1.
Non-premixed flames are also called diffusion flames, the oxidiser and the
fuel streams are completely separated as shown in Fig 2.1. The fuel and oxidiser
diffuse simultaneously towards the reaction zone by molecular diffusion to sustain
the combustion. The classical examples for non-premixed combustion include the
compression ignition engines, furnaces and fires.
In premixed combustion the fuel and oxidiser are homogeneously mixed in a
desired proportion prior to combustion and a suitable ignition source, e.g. spark,
provides sufficient energy needed to initiate combustion. Exothermic reactions
8
2. Background on Turbulent Premixed Combustion
release energy in the flame front which propagates normal to itself to consume
the available reactant mixture with a flame propagation speed S◦L. The burnt
gases and fresh gases are separated by a thin flame with a thickness, δ◦L, of about
0.5 mm. The structure of a laminar premixed flame is presented in Fig 2.2.
However, in practical combustion devices the combustion occurs in mixed mode
having characteristics of both non-premixed and premixed combustion due to in-
complete mixing between fuel and oxidiser causing large local fluctuations in the
equivalence ratio. This mode of combustion is called partially premixed combus-
tion [26].
The occurrence of turbulence in practical combustors is inevitable and thus
one must consider turbulent combustion in modelling. The major challenge in
turbulent combustion modelling is due to the interaction of turbulence and chem-
ical reactions. This has been studied in the past and these studies, specifically
on turbulent premixed flames, are reviewed briefly in the rest of this chapter.
This chapter is organised as follows. Since the governing equations are the
starting point for the modelling studies, the instantaneous conservation equa-
tions for mass, momentum, energy and species mass fractions are discussed in
section 2.1 along with other constitutive relations required. The modelling of
instantaneous reaction rate is noted briefly in section 2.1.2. The three simulation
paradigms are briefly discussed to present their essential features, advantages and
disadvantages in section 2.2. Since RANS methodology is used in this study, this
methodology and related turbulence modelling are discussed in sections 2.3 and
2.4 respectively in some detail. The background studies on turbulent premixed
combustion and its modelling are discussed in section 2.5.
9
2. Background on Turbulent Premixed Combustion
Figure 2.1: Structure of a laminar diffusion flames.
Figure 2.2: Structure of a laminar premixed flames.
2.1 Governing Equations
2.1.1 Fluid Flow Equations
These equations are given by the conservation of mass, momentum, species mass
fractions and energy in Cartesian coordinates of a reacting system with N species
and I reactions [22; 23; 37; 38].
• Conservation of mass
10
2. Background on Turbulent Premixed Combustion
∂ρ
∂t
+
∂(ρui)
∂xi
= 0. (2.1)
The first and second terms on the LHS of Eq. (2.1) are local temporal change of
ρ and its convective transport respectively inside a control volume.
• Conservation of momentum
∂(ρui)
∂t
+
∂(ρuiuj)
∂xj
= − ∂p
∂xi
+
∂τij
∂xj
+Fi. (2.2)
The first and second terms on LHS of Eq. (2.2) are the unsteady and advective
terms respectively. The first and third terms on the RHS represent the
contribution of the pressure gradient and body force in the ith coordinate
direction respectively to the force balance. The second term on the RHS is due
to the viscous forces, where τij denotes the viscous stress tensor,
τij = µ
(
∂ui
∂xj
+
∂uj
∂xi
)
− 2
3
µ
∂uk
∂xk
δij, (2.3)
where µ is the dynamic viscosity and δij is the Kronecker delta.
• Conservation of mass fraction of species α
∂(ρYα)
∂t
+
∂(ρuiYα)
∂xi
= − ∂J
α
i
∂xi
+ ω˙α (α = 1, 2, ..., N). (2.4)
11
2. Background on Turbulent Premixed Combustion
The first and second terms on the LHS of Eq. (2.4) are local rate of change of Yα
and convective transport respectively. The first and second terms on the RHS are
molecular diffusion flux of species α in the ith coordinate direction and the total
rate of mass production by chemical reaction respectively. The diffusion flux is
represented by Fick’s first law as
∂Jαi
∂xi
= − ∂
∂xi
(
ρDα
∂Yα
∂xi
)
= − ∂
∂xi
(
µ
Scα
∂Yα
∂xi
)
(2.5)
where Dα is the molecular diffusivity of species α relative to other species and
Scα is the Schmidt number of species α.
• Conservation of energy
The enthalpy, h, of a reactive mixture is defined as the sum of the specific
enthalpies hα of species α
h =
n∑
α=1
Yα hα. (2.6)
The absolute enthalpy, hα, is equal to the sum of enthalpy of formation, h
◦
f,α, of
species α at a reference temperature and sensible enthalpy, hs, so that
hα = h
◦
f,α + h
s
α. (2.7)
12
2. Background on Turbulent Premixed Combustion
The sensible enthalpy, hs, of an ideal gas is equal to
hsα =
∫ T
Tref
cpα(T ) dT, (2.8)
where T is the absolute temperature. By assuming low Mach number flow, the
simple form of the energy conservation equation can be written as [22]
∂(ρh)
∂t
+
∂(ρuih)
∂xi
=
∂p
∂t
+
∂
∂xi
(
λ
cp
∂h
∂xi
)
+ SR. (2.9)
The two terms on the LHS have their usual meaning. The first term on the RHS
represents the pressure gradient while the second and third terms represent the
diffusion of enthalpy and the source or sink term due to radiative heat exchange
respectively.
• State equation
The pressure, p, is obtained from the equation of state as
p = ρ R T
n∑
α=1
Yα
Wα
. (2.10)
It is common to use two additional equations, the reaction progress variable and
mixture fraction equations in turbulent combustion modelling and simulation
studies. These two equations are presented next.
• Progress variable
The progress variable, c, is required for turbulent premixed combustion mod-
elling which measures the reaction progress in premixed combustion. The progress
13
2. Background on Turbulent Premixed Combustion
variable can be defined based on temperature, T , or using fuel mass fraction, Yf ,
[17] as
CT =
T − Tu
Tb − Tu , (2.11)
where Tu and Tb respectively denote the temperature of unburnt and burnt mix-
ture in adiabatic laminar flames, and
Cf = 1− Yf
Y uf
. (2.12)
If the Lewis numbers, define as the ratio of thermal diffusivity to mass diffusivity,
of the mixture is unity then Cf = CT [22]. An alternative choice using the sensible
enthalpy is also possible [40]:
c =
hs − hsu
hsb − hsu
. (2.13)
The subscripts b and u denote the burnt and unburnt mixtures respectively. For
this study, c based on fuel mass fraction is used and it is governed by
∂(ρc)
∂t
+
∂(ρuic)
∂xi
− ∂
∂xi
(
ρDc
∂c
∂xi
)
=ρ ω˙c. (2.14)
in the usual nomenclature.
• Mixture fraction
The mixture fraction is a passive scalar which describes the stoichiometry of
the reacting mixture. It takes a value of 0 in the oxidiser stream and 1 in the fuel
14
2. Background on Turbulent Premixed Combustion
stream. The instantaneous transport equation for Z is given by
∂(ρZ)
∂t
+
∂(ρuiZ)
∂xi
− ∂
∂xi
(
ρDZ
∂Z
∂xi
)
= 0. (2.15)
2.1.2 Reaction Rates
Consider a set of I elementary reactions, which can be expressed in symbolic form
as
N∑
α=1
ν ′α Mα ⇐⇒
N∑
α=1
ν ′′α Mα. (2.16)
The net rate of chemical reaction k is given by
ω˙k = kfk
N∏
α=1
(
ρYα
Wα
)ν′αk
− kbk
N∏
α=1
(
ρYα
Wα
)ν′′αk
, (2.17)
where kfk and kbk are the forward and backward rate coefficients respectively and
Wα is the molecular weight of species α. The exponent ν
′
αk and ν
′′
αk denote the
forward and backward stoichiometric coefficients respectively. The forward and
backward rate coefficients are expressed using Arrhenius form as
k = AT n exp (−Ea/RT ) , (2.18)
where A is the pre-exponential factor and Ea is the activation energy. It should
also be noted that the backward rate is related to the forward rate through the
equilibrium constant. Now, the chemical reaction rate, on a mass basis, for species
α is given by
15
2. Background on Turbulent Premixed Combustion
ω˙α = Wα
I∑
k=1
ω˙k (ν
′′
αk − ν ′αk). (2.19)
2.2 Three Simulation Paradigms
In the simulation of turbulent premixed flames, the above conservation equations
are solved with or without additional models depending on the methodology to
be followed. As noted in chapter 1, there are three methods used commonly, and
these methods are DNS, LES and RANS. The essential features, advantages and
disadvantages of these methodologies are briefly discussed below.
2.2.1 DNS
In direct numerical simulation, the full set of the instantaneous Navier-Stokes
equations discussed in section 2.1 are solved without any turbulence models.
It resolves all spatial and temporal scales in turbulent flows and simulates the
problem in detail with specified initial and boundary conditions. Thus, DNS
can produce detailed information which may not be available from experimental
measurements due to the cost involved.
However, unfortunately there are restrictions in applying the method to real-
istic turbulent flows, due to the non-linearity in the governing equations and the
wide range of length and time scales in such flows [16]. For instance, to resolve
all the spatial and temporal length scales, the integral length Λ and Kolmogorov
length ηk scales must be captured by the simulation grid. This is can be achieved
16
2. Background on Turbulent Premixed Combustion
only when
L = N. ∆x > Λ and ∆x = L
N
6 ηk, (2.20)
where L is the domain size, ∆x is the grid spacing and N is the number of
grid points in each direction in the physical space. Since the turbulent Reynolds
number, Ret, is related to the turbulent length scales by [22]
Re
3/4
t =
Λ
ηk
, (2.21)
then the maximum number of grid points used in DNS simulation must obey
N > Re
3/4
t [22]. Therefore, the DNS simulation is prohibitively expensive for
realistic flows with Reynolds number of order of 106 [16].
For turbulent reacting flows this situation is even more complex, because
chemical reactions introduce further length and timescales. In some cases, these
scales are smaller then the turbulence scales. This situation requires further
resolution to resolve all the range of the scales involved in the combustion process.
Therefore, DNS is used only in laboratory scale for research purposes to study the
physics of combustion [41; 42; 43], or for a flow with low Reynolds number [44].
The chemistry used in DNS, has an influence on the complexity of the DNS by
increasing the number of differential equations needed to be solved. Thus, DNS
of turbulent combustion is simplified by using a single global chemical reaction
or limited sets of elementary reactions.
17
2. Background on Turbulent Premixed Combustion
2.2.2 LES
LES is considered to be an intermediate tool between DNS and RANS. In this
approach, only the large scale eddies are resolved using the filtered form of the
governing equations shown in section 2.1, while the effect of small scales eddies,
which are the most expensive computational wise, are represented by models
[45; 46; 47; 48; 49]. This technique reduces the restriction in the domain size
and the computational expenses in DNS simulation that makes LES applications
to practical systems possible. The filter retains eddies with lengths greater than
the filter width, ∆, to be resolved on the grid such that ∆x ≈ ∆ . On the other
hand, the small scales are removed and modelled by sub-grid scales (SGS). The
filtered quantity φ is defined as
φ(x, t) =
∫
φ(x, t) F [(x− x′);∆] dx′, (2.22)
where F is the filtering function and it can have a different prescribed shapes
[22]. The values of F is close to zero when x− x′ exceeds the filter size ∆.
2.2.3 RANS
Since this approach is used for this study, the next section describes this method
in some detail.
2.3 RANS Methodology
In this method, the instantaneous governing equations are averaged and then
solved. Since the governing equations are non-linear, the averaging process in-
18
2. Background on Turbulent Premixed Combustion
troduces additional unknown terms. This can be seen clearly in the following
example. By decomposing an instantaneous quantity φ as its mean and fluctuat-
ing parts, φ(x, t) = φ(x, t) + φ′(x, t), the mass conservation equation gives
• Conservation of Mass
∂ρ
∂t
+
∂
∂xi
(
ρui + ρ′u′i
)
= 0, (2.23)
where the over bar denotes an averaging procedure, which can be either spatial or
temporal depending on the flow characteristics. The averaging procedure yields
additional term (ρ′u′i), which arises from the correlations between the velocity
and the density fluctuations and it needs to be modelled. To avoid these kind
of correlations terms involving density fluctuations, Favre or density weighted
averaging is used for flows involving large density variation, such as combustion.
The instantaneous quantity is decomposed as φ(x, t) = φ˜(x, t) + φ′′(x, t), where
φ˜ = ρφ/ρ and φ˜′′ = 0. It is well known that applying Favre averaging to the
governing equations in section 2.1, gives additional terms, such as u˜′′i u
′′
j , u˜
′′
i Y
′′
α ,
etc. Applying Favre averaging to the instantaneous governing equations gives
• conservation of mass
∂ρ
∂t
+
∂ρu˜i
∂xi
= 0, (2.24)
• conservation of momentum
∂(ρu˜i)
∂t
+
∂(ρu˜iu˜j)
∂xi
=− ∂p
∂xj
+
∂
∂xi
(
τij − ρu′′i u′′j
)
+ Fi, (2.25)
19
2. Background on Turbulent Premixed Combustion
where, ρu′′i u
′′
j is the Reynolds stress and its closure is discussed in section 2.4.
• conservation of mass fraction of species α
∂(ρY˜α)
∂t
+
∂(ρu˜iY˜α)
∂xi
=
∂
∂xi
(
ρDα
∂Yα
∂xi
− ρu′′i Y ′′α
)
+ ω˙α (α = 1, 2, ..., N). (2.26)
Closures for the turbulent fluxes, ρu′′i Y ′′α , and mean reaction rate, ω˙α, are dis-
cussed later in sections 2.4 and 2.5 respectively.
• conservation of energy
∂(ρh˜)
∂t
+
∂(ρu˜ih˜)
∂xi
=
∂p
∂t
− ∂
∂xi
(ρu˜′′i h′′)+
∂
∂xi
(
λ
cp
∂h˜
∂xi
). (2.27)
• progress variable equation
∂(ρc˜)
∂t
+
∂(ρu˜ic˜)
∂xi
=
∂
∂xi
(
ρDc
∂c
∂xi
− ρu′′i c′′
)
+ ω˙c, (2.28)
A closure for the turbulent scalar flux, ρu′′i c′′, and the mean reaction rate are
discussed later.
• mixture fraction equation
∂(ρZ˜)
∂t
+
∂(ρu˜iZ˜)
∂xi
=
∂
∂xi
(
ρDZ
∂Z
∂xi
− ρu′′iZ ′′
)
. (2.29)
20
2. Background on Turbulent Premixed Combustion
In addition to the governing equations for the mean progress variable and mix-
ture fraction, transport equations for their variances must also be solved. These
equations are [16; 22]
∂(ρc˜′′2)
∂t
+
∂(ρu˜ic˜′′2)
∂xi
=
∂
∂xi
(
ρDc
∂c′′2
∂xi
− ρu′′i c′′2
)
− 2ρu′′i c′′
∂c˜
∂xi
−2ρ²˜c + 2c′′ω˙c, (2.30)
and
∂(ρZ˜ ′′2)
∂t
+
∂(ρu˜iZ˜ ′′2)
∂xi
=
∂
∂xi
(
ρDZ
∂Z ′′2
∂xi
− ρu′′iZ ′′2
)
− 2ρu′′iZ ′′
∂Z˜
∂xi
−2ρ²˜Z . (2.31)
The contribution of the turbulent fluxes, dissipation rates and chemical reactions
in the above two variance equations are to be modelled. The modelling of turbu-
lent fluxes are discussed in the next section and the modelling of reaction related
terms are discussed in chapter 3. The dissipation rate of the mixture fraction
variance is given by ρ²˜Z = ρDZ(∂Z ′′/∂xi)(∂Z ′′/∂xi) and it is modelled using the
classical model [22] given by
²˜Z ≈ CD( ε˜
k˜
) Z˜ ′′2, (2.32)
where CD is a model parameter which takes a value of about 0.9 to 1.0. The
modelling of the dissipation rate, ρ²˜c = ρDc(∂c′′/∂xi)(∂c′′/∂xi), is discussed in
details in chapter 4.
21
2. Background on Turbulent Premixed Combustion
2.4 Turbulence Models
2.4.1 Turbulent Reynolds Stress Model
The transport equations presented in the previous section are unclosed and mod-
els are required for Reynolds stresses, turbulent transport, mean reaction rates,
and dissipation rates. In this section, the turbulence modelling is reviewed very
briefly and the k− ε models used in this study for turbulent premixed flames are
discussed in some detail [22; 23]. The Reynolds stress, ρu˜′′i u
′′
j , are modelled using
the turbulent viscosity hypothesis, and according to this hypothesis the Reynolds
stress is proportional to the mean rate of strain [22; 23]:
ρ u˜′′i u
′′
j = − µt
(
∂u˜i
∂xj
+
∂u˜j
∂xi
− 2
3
δij
∂u˜k
∂xk
)
+
2
3
ρ k˜ δij, (2.33)
involving the turbulent viscosity, µt, which needs to be modelled. Many ap-
proaches [50] have been proposed in the past for this modelling. In the sim-
plest possible approach proposed by Prandtl, the eddy viscosity is given by
µt = ρ l
2
mix |S˜|, where S˜ is the mean stress tensor defined as S˜ij =
(
∂u˜i
∂xj
+
∂u˜j
∂xi
)
/2,
and lmix is the mixing length. This model is known as mixing length or zero equa-
tion model.
In another approach [50], the eddy viscosity is modelled as µt = ρ Cµ lm k˜
1
2 ,
where k˜ is the turbulence kinetic energy obtained from the solution of its transport
equation and Cµ is a model parameter.
It is known from many past studies [51] that the eddy viscosity obtained using
the solution of k˜ and its dissipation rate, ε˜, transport equations give an acceptable
level of accuracy for a range of flows. This approach, is widely used in RANS
22
2. Background on Turbulent Premixed Combustion
simulation and is called the k - ε [51] model. The turbulent viscosity is given by
µt = ρ Cµ (k˜
2/ε˜). (2.34)
The transport equations for k˜ and ε˜ are written as [52]
∂(ρk˜)
∂t
+
∂(ρu˜ik˜)
∂xi
=
∂
∂xi
[
(µ +
µt
σk˜
)
∂k˜
∂xi
]
+ Pk − ρε˜. (2.35)
and
∂(ρε˜)
∂t
+
∂(ρu˜iε˜)
∂xi
=
∂
∂xi
[
(µ +
µt
σε
)
∂ε˜
∂xi
]
+ Cε1
ε˜
k˜
Pek − Cε2ρε˜
2
k˜
. (2.36)
where the source term Pk is given as Pk = −ρu˜′′i u′′j (∂u˜i/∂xi)−u′′i ∂p/∂xi+p′u′′i /∂xi.
The Reynolds stresses, ρu˜′′i u
′′
j , are determined from equation Eq. (2.33) and the
standard model constants [16] are Cµ = 0.09, σk = 1, σε = 1.3, Cε1 = 1.44 and
Cε2 = 1.92. The pressure related terms are to be modelled and their contributions
can be significant for turbulent premixed flames in enclosed geometries [53] and
can be ignored for open flames. In this study, these terms are neglected.
In another approach, known as the Reynolds stress method (RSM), transport
equations for the individual components of the Reynolds stress are solved with
additional models [54; 55; 56]. This model is expected to provide improved
accuracy for complex flows but with relatively high computational cost. Also, the
Reynolds stress equations are stiff causing difficulties while finding their numerical
solution [22].
23
2. Background on Turbulent Premixed Combustion
2.4.2 Turbulent Scalar Flux Model
To solve the progress variable, c˜, transport equation shown in section 2.1, the
turbulent scalar flux, u˜′′i c′′, needs to be modelled, where the symbols, u
′′
i and
c′′, are the Favre fluctuation of velocity vector and progress variable, respectively
[57]. This term is often modelled using the classical gradient transport hypothesis
based on the eddy viscosity [58] given as
ρ u˜′′i c′′ = ρu
′′
i c
′′ = − µt
Scc
∂c˜
∂xi
, (2.37)
where µt is the turbulent eddy viscosity and Scc is the turbulent Schmidt number.
According to the gradient assumption, turbulent flux transport is analogues to
molecular transport. However, theoretical analysis [59; 60], experimental [61; 62;
63; 64; 65] and DNS [57; 66; 67] studies pointed out the existence of both gradient
and counter gradient fluxes in turbulent premixed flames. The transition from
gradient type to counter gradient type is influenced by the ratio u′/S◦L and the
heat release factor τ [66], where u′ is the rms of turbulence velocity fluctuations
and S◦L is the laminar flame speed. According to the DNS analysis in [66], the
counter gradient transport occurs when the flow field is dominated by thermal
expansion due to the heat release, while the gradient transport occurs when
the flow field is dominated by turbulence, ie., when the ratio u′/S◦L is large.
The physical mechanism responsible for counter-gradient transport in turbulent
premixed flames is explained by Bray-Moss-Libby model [59], where the turbulent
scalar flux is given by
ρ u˜′′i c′′ = ρc˜(1− c˜)(uib − uiu) (2.38)
24
2. Background on Turbulent Premixed Combustion
where, uib and uiu are the conditional mean velocity for burnt and unburnt gases,
respectively. Since the density of the burnt gases, ρb, is lower compared to un-
burnt gases, ρu, thus, the pressure gradient due to the heat release accelerates
preferentially more the burnt gases compared to the unburnt gases that makes
uib > uiu. This situation will promote counter gradient flux which contradicts the
gradient assumption given in Eq. 2.37. A transport equation for the turbulent
flux can be solved to account for the occurrence of gradient and counter gradient
fluxes as has been done in [68] and explained in [16]. A simple algebraic model
has also been proposed in an earlier study [66] to include the gradient and non-
gradient scalar flux transports in premixed flames. However, it is not uncommon
to use gradient flux model in calculations of high Reynolds number turbulent
premixed flames and this model is used in this work.
2.5 Premixed Combustion Sub-Models
The main objective of turbulent combustion modelling is to provide models for
the mean reaction rate term ω˙α which appears in the species transport equation.
The average reaction rate ω˙α cannot be easily expressed as a function of averaged
mass fractions Y˜f , Y˜o, mean density ρ and mean temperature T˜ due to the non-
linearity associated with this term. If one assumes a single step reaction then the
mean reaction rate can be expressed as [22].
ω˙f = −A ρ2 T˜ n Y˜f Y˜o exp
(
−Ta
T˜
) 1 + (Ta
T˜
)2
T˜ ′′2
T˜ 2
+
Y˜ ′′f Y ′′o
Y˜f Y˜o
+ ....
 .(2.39)
25
2. Background on Turbulent Premixed Combustion
This expression also involves a number of higher order correlations which must
be retained, at least up to 20th order, to maintain accuracy [24] and this is
impractical. Furthermore this will introduce a large set of modelling parameters.
In another words, trying to find a closure model for the mean reaction rate using
this approach is not practical. Hence, alternative approaches have been proposed
in the past and have been discussed elaborately in [17; 20; 22; 52]. It is not
possible to review all of these approaches here. Furthermore, the interest of
this work is on turbulent premixed combustion and thus the related combustion
modelling are reviewed in this section.
The mean reaction rate models involve statistical relationships among fluctu-
ating quantities in turbulent flames. These relationships, or the models, depend
on the structure of the small scales lost due to averaging. Furthermore, the
models depend on the relativity between the scales of turbulence and that of the
flame. Damko¨hler [69] identified two limiting scenarios; flamelet and non-flamelet
combustion.
In flamelet combustion, the flame scales are much smaller than the turbulence
scales and vice-verse for non-flamelet combustion. These concepts are explained
better using a combustion regime diagram.
The turbulent scales range from the Kolmogorov length scale, ηk, with a char-
acteristic velocity u′k to the integral length scale, Λ, with characteristic velocity
u′. Thus, the integral time scale, τt, and the Kolmogorov time scale, τk, are
defined respectively as
τt =
Λ
u′
, τk =
ηk
u′k
. (2.40)
26
2. Background on Turbulent Premixed Combustion
The combustion scales are typically represented by flame front thickness, δ, and
speed, SoL. Thus, the chemical time scale, τc, is defined as
τc =
δ
SoL
. (2.41)
The ratio of the integral turbulence time scales to the chemical time scale is
defined as the Damko¨ler number
Da =
τt
τc
=
Λ/δ
u′/SoL
. (2.42)
The Karlovitz number is defined as the ratio of the chemical time scale to the
small-scale turbulence time scale
Ka =
τc
τk
=
δ2
η2k
. (2.43)
The turbulence Reynolds number can also be written [70] as
Re =
u′/SoL
δ/Λ
, (2.44)
if ν = δSoL. The relationship among these three non-dimensional numbers are
commonly represented using the combustion regime diagram [16; 70; 71], which
is used to classify the turbulent premixed combustion. Such a diagram is shown
in Fig. 2.3.
If the thickness of the flame front is smaller than the integral eddies then
chemical time scale, τc, is faster compared to the integral turbulence time scale,
τt, in high Re turbulence. In this situation, turbulence cannot enter the flame
27
2. Background on Turbulent Premixed Combustion
front and its structure remains close to laminar flame. This flame front is simply
wrinkled by turbulence. This situation is denoted on the regime diagram by
Da > 1 and is known as flamelet combustion regime. When the thickness of the
flame front is larger than the size of integral eddies then turbulence time scale, τt,
is faster compared to chemical time scale, τc, disturbing its internal structure if
the turbulence level is large. The turbulent eddies can penetrate the flame front
and continuously mixing reactants with products. This regime is denoted on the
regime diagram by Da < 1 and is known as perfectly stirred reactor regime.
Karlovitz number is used to further subdivide the flamelets regime depend-
ing on the role played by the small-scale eddies. When the flame scales are
much smaller than the turbulence small-scales then the internal structure of the
flamelets are not disturbed by the small scales of turbulence also. This situa-
tion is simply denoted by Ka < 1 in the regime diagram and Ka = 1 is called
the Klimov-Williams line signifying the δ = ηk situation. When Ka > 1 but
Ka < 100, the reaction zones of the flamelets are intact but the small eddies
disturb the preheat zone structure which enhances the heat and mass transport
in that zone. This situation is known as thin reaction zones combustion [70] as
marked in Fig. 2.3. Most of the practical combustion is expected to span between
corrugated and thin reaction zones regime. The review by Driscoll [13] suggests
that the evidence for non-flamelet behaviour is sparse. With these understand-
ings, a brief review of the existing modelling for premixed turbulent combustion
is presented in the following subsections.
28
2. Background on Turbulent Premixed Combustion
0.1
1
10
100
1000
0.1 1 10 100 1000
u
′
/s
o l
Λ/δ
R
e
=
1
Laminar
R
e
=
100
R
e
=
1000
D
a
=
1
Ka
= 1
Ka
= 1
00
wrinkled
corrugated
thin reaction zones
distributed
Figure 2.3: The premixed combustion regime diagram.
2.5.1 Eddy Break-Up Model
As proposed by Spalding [72] the Eddy Break Up (EBU) model is used for flames
with high Re and Da numbers. The basic idea of the model that the reaction
zone is described as pockets of unburnt and burnt gases and the turbulent eddies
will mix these pockets. Correspondingly, the reaction rate is mainly controlled
by this turbulent mixing rate and it is given by
ω˙ = CEBUρ
√
c˜′′2
τt
, (2.45)
where, CEBU is the model constant and τt is the turbulent time scale which is
equal to k˜/ε˜. An estimation for c˜′′2 based on the assumption of infinitely thin
flame yields c˜′′2 = c˜(1 − c˜). The EBU model is typically limited to one-step
chemistry.
29
2. Background on Turbulent Premixed Combustion
2.5.2 Bray-Moss-Libby Model
Bray-Moss-Libby model (BML) is based on a statistical approach using the prob-
ability density function (PDF) of the progress variable, c, which is defined to be
zero in the reactant and 1 in the product [73; 74; 75]. In order to simplify the
analysis the combustion is taken to occur at constant pressure with unity Lewis
numbers under adiabatic conditions and the mixture is made of ideal gases. The
turbulent premixed flame is taken to be a collection of thin flamelets separating
reactants from products implying that at a specific location inside the flame brush
and time, reactants are detected for a while and then products are detected for
the rest of the time [70; 71]. This view allows us to express the PDF of c at a
given location to be two delta functions, one at c = 0 and another at c = 1 with
respective weights of α and β, and a contribution from the burning parts. Thus,
p(c = ζ;x, t) = α(x, t)δ(ζ) + β(x, t)δ(1− ζ) + γ(x, t)f(ζ), (2.46)
where, ζ is the sample space variable for c and α + β + γ = 1. For high Re
and Da numbers, the flame front is thin and the probability of finding burning
gases is very low compared to finding unburnt or burnt mixture therefore, the last
term in Eq. (2.46) is neglected. The coefficients α and β can be determined from
the Favre mean progress variable which is obtained from its transport equation,
Eq. (2.28) as follows:
ρc˜ =
∫ 1
0
ρ(ζ) ζ p(ζ) dζ = ρbβ, (2.47)
30
2. Background on Turbulent Premixed Combustion
after using Eq. (2.46) with γ ' 0. The coefficients α and β can now be written
as [74]
α =
1− c˜
1 + τ c˜
; β =
(1 + τ)c˜
1 + τ c˜
, (2.48)
where τ = (ρu/ρb) - 1. Now, the mean density is given [58] by ρ = ρu/(1 + τ c˜)
and c = (1− τ)c˜/(1 + τ c˜). Since the burning mode part of the PDF is neglected
(γ = 0) in the BML approach, an alternative methodology is required to close
the mean reaction rate. Bray [75; 76] showed that
ω˙c = 2
ρε˜c
2cm − 1 , (2.49)
in the limit of γ → 0 by noting the relationship between c˜ and c˜′′2. The model
constant cm is given by
cm =
cω˙c
ω˙c
=
∫ 1
0
ζω˙c p(ζ) d(ζ)∫ 1
0
ω˙c p(ζ) d(ζ)
, (2.50)
and its typical values vary from 0.7 to 0.8. The mean scalar dissipation rate, ²˜c,
in Eq. (2.49) is defined as
ρ²˜c = ρD
∂c′′
∂xi
∂c′′
∂xi
. (2.51)
Models such as ²˜c = c˜′′2/τt have been used in the past studies, where τt = k˜/ε˜.
Recent developments of this approach for non-negligible γ have been discussed
elaborately by Bray in reference [16].
Another approach to close the mean reaction rate is based on the flame cross-
31
2. Background on Turbulent Premixed Combustion
ing frequency analysis. Bray and Libby [77] observed that the mean reaction rate
at a given location depends on the frequency at which the flame front crosses this
location in turbulent flow. In this analysis, the mean reaction rate is expressed as
the product of the flame crossing frequency, fc, and the reaction rate per flame
crossing, ω˙f , [77] as .
ω˙c = ωffc. (2.52)
The flame crossing frequency, fc, can be estimated as
fc = 2
c(1− c)
τc
, (2.53)
where τc is the mean period of a telegraphic signal, which is used to represent
the instantaneous c. This period can be estimated from a turbulence time scale
τt = k˜/ε˜. The reaction rate per flame crossing, ω˙f , is modelled as
ω˙f =
ρuS
0
L
δ0L/tt
, (2.54)
where tt is the transit time and is defined as the time required to cross the flame
front [77].
Bradley [78] suggested that
ω˙c =
∫ ∫
ω˙(c, a) p(c, a) dc da, (2.55)
by assuming that the local flamelets are stretched by turbulent eddies. The
stretching effects include the influences of straining and bending of the flame front
by turbulent eddies. The symbol a denotes the turbulent stretch and ω˙(c, a) rep-
32
2. Background on Turbulent Premixed Combustion
resents the reaction rate in stretched flamelets. Recently, an equivalent approach
using the scalar dissipation rate, instead of the stretch rate, has been developed
[79; 80].
2.5.3 Flame Surface Density Model
The flame surface density (FSD) model was originally proposed [81] for non-
premixed flames and was developed later [82; 83] for premixed flames. This
approach assumes that the flame propagates locally as a laminar flame and thus
the mean reaction rate is expressed as
ω˙c = ρuSLΣ, (2.56)
where Σ is the flame surface area per unit volume and SL is the local laminar
flame speed accounting for the turbulent stretch. This flame speed is typically
expressed as SL = I0S
0
L, where I0 is known as the stretch factor [84] and its
typical values range from 0.9 to 1.
One can estimate the flame surface density Σ either from algebraic expressions
or by solving exact balance equation for Σ with suitable closure. A balance
equation for Σ was first proposed by [81] for non-premixed flames followed by more
concrete mathematical derivation based on geometrical [82; 85] and statistical
analysis [83; 86]. Pope [83] showed that Σ associated with a specific iso-surface
of progress variable, c = c∗, as
Σ(c∗; x, t) =
〈
| 5 c||c = c∗
〉
p(c∗; x, t), (2.57)
33
2. Background on Turbulent Premixed Combustion
where,
〈
| 5 c||c = c∗
〉
is the conditional average of | 5 c| for c = c∗ and p(c∗; x, t)
is the PDF.
A balance equation for Σ can be written [82; 83; 85] as
∂Σ
∂t
+
∂
∂xi
(〈ui〉s Σ) + ∂
∂xi
(〈sdni〉s Σ) =
〈
(σij − ninj)∂ui
∂xi
〉
s
Σ +
〈
sd
∂ni
∂xi
〉
s
Σ,
(2.58)
where sd is the flame surface displacement speed and is defined as
sd =
(
1
| 5 c|
Dc
Dt
)
c=c∗
(2.59)
(σij − ninj) is the tangential strain rate and ni is the ith component of the
unit vector normal to the flame surface. The modelling of the various terms in
Eq. (2.58) has been the subject of many past studies and the important results
have been summarised in the books [16; 20; 22; 84] and review paper [17].
2.5.4 The Probability Density Function
The Probability Density Function based approach is a statistical description of
the problem. For a given location, x, and time, t, in the flame, the PDF measures
the probability to find a variable ψ within [ψ − 4ψ/2, ψ +4ψ/2] and satisfies
the condition.
∫
ψ1,ψ2,..ψN
p(ψ1, ψ2, ..ψN) d(ψ1, ψ2, ..ψN) = 1, (2.60)
34
2. Background on Turbulent Premixed Combustion
where ψ1, ψ2, ..ψN are the sample space variables for the density, temperature,
mass fraction, velocity, etc. The mean reaction rate can be written as
ω˙c =
∫
ψ1,ψ2,..ψN
ω˙(ψ1, ψ2, ..ψN) p(ψ1, ψ2, ..ψN) d(ψ1, ψ2, ..ψN), (2.61)
where ω˙(ψ1, ψ2, ..ψN) is the functional form of the instantaneous reaction rate.
The mean value of, say, ψ1 can be simply obtained as
ψ1 =
∫
ψ1,ψ2,..ψN
ψ1 p(ψ1, ψ2, ..ψN) d(ψ1, ψ2, ..ψN). (2.62)
The challenge now is how to determine the joint PDF, p. Two different approaches
are commonly used.
The first approach is to solve a transport equation for the joint probability
density function [29; 87]. The advantage of this approach is that the chemical
reaction does not require any modelling. However, one major drawback is the
computational expense in solving the PDF transport equation, which is written
[29; 87] as
∂ρ p˜ (ψ)
∂t
+
∂
∂xi
[
ρ u˜i p˜ (ψ)
]
+
n∑
α=1
∂
∂ψα
[
1
ρ
ω˙α(ψ) ρ p˜(ψ)
]
=
− ∂
∂xi
[〈
u′′i |ψ
〉
ρp˜(ψ)
]
+
n∑
α=1
∂
∂ψα
[〈
1
ρ
∂Ji,α
∂xi
| ψ
〉
ρ p˜ (ψ)
]
. (2.63)
The first, second and third terms on the LHS of Eq. (2.63) are respectively the
rate of change of p˜ (ψ), convective transport and the flux of the PDF in the
sample space due to chemical reaction. The first and the second terms on the
RHS represent turbulent convection and molecular diffusion effects respectively.
35
2. Background on Turbulent Premixed Combustion
The vector ψ represents (Y1, Y2, ...YN , T ). The terms on the LHS are closed.
The turbulent flux is usually closed using gradient assumption. The molecular
diffusion term requires a closure, several mixing models are proposed in past
studies. The details of this approach has been reviewed recently by Haworth
[88] and Haworth & Pope [89]. The application of this approach for turbulent
premixed flames has been discussed in detail by Lindstedt [90].
In the second approach, a shape for the PDF is presumed and it is usually for
the marginal PDF of the progress variable. A presumed shape for the joint PDF
has also been attempted in the past [91] but involves a number of uncertainties.
The BML PDF in section 2.5.2 is a good example for the presumed PDF approach.
It is also typical to use a β−function [16] for given values of mean and variance.
The Favre β-PDF is given by
p˜(ζ) =
ζa−1 (1− ζ)b−1
β(a, b)
, (2.64)
where,
a = c˜
(
1− g
g
)
, b = (1− c˜)
(
1− g
g
)
, (2.65)
where g = c˜′′2/c˜(1− c˜) is the variance parameter and
β(a, b) =
∫ 1
0
ζa−1 (1− ζ)b−1 dζ. (2.66)
The values of c˜ and c˜′′2 are usually obtained by solving their respective transport
equations. The Favre PDF, p˜, is related to the Reynolds PDF through pρ = ρp˜
[16].
36
2. Background on Turbulent Premixed Combustion
2.5.5 G-Equation Model
The G-equation model is based on the flamelet modelling assumption, whereby
the chemical reaction occurs within the flame front, which separates the unburned
and burned mixtures and moves with a burning velocity normal to the front ST .
In this approach, the flame propagation is described as the propagation of an
iso-surface of G(x, t). A transport equation for G is derived [24] to track the flame
propagation by considering an arbitrary iso-scalar value G0 with G < G0 repre-
senting the unburnt mixture and G > G0 representing the burnt mixture. This
method has been developed and discussed elaborately by Peters [70]. Basically,
a transport equation for the mean G-field, written as,
∂ρG˜
∂t
+
∂ρu˜iG˜
∂xi
= ρST |∂G˜
∂xi
| − ρDtκ|∂G˜
∂xi
|, (2.67)
is solved with models for ST , Dt and κ along with a transport equation for G˜′′2.
The detail of this method is discussed by Peters [70] and Peters and Bray [92].
This formulation is valid only for the corrugated flamelets regime of combustion
[93].
2.6 Summary
All the methods described above are used in modelling turbulent premixed com-
bustion but choosing the right method is essential. For example, in the EBU
model the reaction rate is controlled by mixing and finite rate chemistry effects
are not typically included in the model. The laminar flamelet models are based
on the assumption that the flame scales are smaller than the turbulence scales.
37
2. Background on Turbulent Premixed Combustion
This assumption becomes invalid if one considers some minor species and pollu-
tants. The transported PDF models depend on many variables that make the
method computationally expensive.
Conditional moment closure [27; 40] is an alternative approach showing a
good potential to predict minor and pollutant species of engineering interest with
moderate computational cost as one shall note from this study. This method has
been used widely for non-premixed flames and it is development and application
to premixed flames is sparse. Hence, this method is chosen for this study and is
described in detail in the next chapter.
38
Chapter 3
Conditional Moment Closure
This chapter presents a review of the main features of the conditional moment
closure (CMC). The first section begins by presenting the concept and fundamen-
tals of this method . The second and third sections focus on CMC for premixed
combustion followed by premixed CMC sub-models.
3.1 CMC Methodology
As shown in section 2.5, an accurate closure for the mean reaction rate cannot
be found using the conventional moment method. This is because, the fluctu-
ations of various scalar concentrations and temperature over the mean are very
large. This large fluctuation along with the strong non-linearity in the reaction
rate mainly renders the moment method to be of less use to obtain a closure for
the mean reaction rate. However, as Bilger [27] pointed out, the philosophy of
the moment method can be adopted if one uses conditional moments rather than
the unconditional moments. This is because the fluctuations over the conditional
39
3. Conditional Moment Closure
mean are small compared to the unconditional fluctuation level. This point is
depicted in Fig. 3.1, showing the scatter plot of temperature and OH mass frac-
tion in mixture fraction space. The corresponding conditional averages are also
shown on the left. The solid lines represent the solution of fully burning strained
laminar diffusion flame in counter-flow configuration with a strain rate parameter
of 5 s−1. It is seen in the scatter plot that the fluctuations in T , YOH and Z are
large. If one constructs conditional averages, to be defined below, then the fluc-
tuations levels become small as shown on the right hand side of Fig. 3.1. Here the
main hypothesis of CMC is that the fluctuations of temperature and scalar mass
fraction values are closely associated to the fluctuation in one or two key scalar
quantities. Thus, if one constructs averages conditional upon a particular value
for the key scalars then there is a hope to find an accurate closure for the mean
reaction rate using the concept of moment methods. The mixture fraction and
progress variable are good conditioning variables for non-premixed and premixed
flames respectively [40].
In CMC, transport equations for the conditional averages are derived and
solved along with other moment equations subject to appropriate initial and
boundary conditions. First, the instantaneous value of, say mass fraction of
scalar α, is decomposed into its conditional mean,Qα , and fluctuation, y
′′
α, over
Qα as
Yα(x, t) = 〈Yα|Z = η〉+ y′′α(x, t) = Qα(η; x, t) + y′′α(x, t), (3.1)
where the angled brackets denote ensemble averaging subject to the condition on
the right of the vertical bar and this average is defined as conditional average.
Substituting this decomposition into Eq. (2.4) and then taking the conditional
40
3. Conditional Moment Closure
average of the resulting equation gives a transport equation for Qα. The detail
of this derivation is presented clearly by Bilger [27] and Klimenko and Bilger
[40]. An appropriate joint PDF transport equation, see for example Eq. (2.63),
can also be used to obtain the transport equation for Qα as shown by Klimenko
[28]. The similarities and differences between the decomposition and joint PDF
methods to obtain Qα transport equation are discussed in detail by Klimenko
and Bilger [40] and interested readers are referred to that study. However, both
of these methods led to almost the same transport equation for Qα, which will
be presented in the next section. As one shall see in that section, this transport
equation involves some terms, primarily the conditional averages of velocities,
reaction rates, and scalar dissipation rate of the key conditioning scalar, requiring
closure models. Many models have been developed in past studies and the CMC
have been successfully applied to various non-premixed combustion systems such
as bagasse-fired boiler [32], hood fires [30], bluff-body stabilised[33; 34] and lifted
jet flames [31], spray autoignition [35] and soot formation [36]. The application of
CMC to premixed flames is not fully tested and validated yet. Despite some initial
results reported in [94; 95; 96], the main difficulty in applying CMC to turbulent
premixed flames is associated with modelling of conditional scalar dissipation
rate, 〈Nc|c = ζ〉 = 〈Dc(∇c · ∇c)|ζ〉, of the key conditioning scalar, the progress
variable c.
Many studies [16; 79; 97; 98; 99] have attempted to shed more light on this
dissipation rate and its modelling, which encourages to explore CMC for premixed
flames.
41
3. Conditional Moment Closure
Figure 3.1: Scatter plots of instantaneous and conditional averages for the tem-
perature and OH mass fraction at x/D =20 and r/D = 1.7 (◦) ; 1.4 (¤); 1.1 (4)
[1]. Solid line represents fully burning strained laminar flame result.
42
3. Conditional Moment Closure
3.2 Premixed CMC Method
As noted earlier, the CMCmethod is based on the hypothesis that the fluctuations
of species mass fractions, temperature and enthalpy are associated only with the
fluctuation of a key scalar [40], which is the progress variable, c, for premixed
combustion. The progress variable is a reactive scalar and it can be defined
based on temperature, Eq. (2.11), or using the fuel mass fraction, Eq. (2.12) [17].
An alternative definition using sensible enthalpy is also possible [40]. Here, it
is defined as θ = Yf/Y
u
f using the fuel mass fraction, where Y
u
f is the unburnt
fuel mass fraction value. This is a progress variable and is used as conditioning
variable. The value of θ = 1 implies that the mixture is unburnt and θ = 0
implies that the mixture is completely burnt.
Transport equations for conditional mean scalar values, Qα, are derived by
substituting, as noted earlier in Eq. (3.1)
Yα(x, t) = Qα(ζ;x, t) + y
′′
α(x, t), (3.2)
in the transport equation for the instantaneous scalar value Yα, Eq. (2.4) [27]. It is
to be noted that density weighted conditional averaging [40] must be used, which
is given by Qα(ζ;x, t) = 〈ρYα|ζ〉/〈ρ|ζ〉, where ζ is the sample space variable for
θ. A full derivation of the CMC transport equation is given in the Appendix A.
It is to be noted that the decomposition method is used because of its simplicity,
although one can follow the joint PDF approach of Klimenko [28]. Both of these
approaches essentially yield the same transport equation for Qα which is written
as [40; 96]
43
3. Conditional Moment Closure
〈ρ|ζ〉∂Qα
∂t
+ 〈ρui|ζ〉∂Qα
∂xi
− Leθ
Leα
〈ρNθ|ζ〉∂
2Qα
∂ζ2
= 〈ω˙α|ζ〉 − 〈ω˙θ|ζ〉∂Qα
∂ζ
−1
p˜
∂
∂xi
[〈ρu′′i y
′′
α|ζ〉〈ρ|ζ〉p˜] + eQα , (3.3)
where Leα is the Lewis number of species α and p˜ is the Favre PDF of c. The
physical meaning of various terms in Eq. (3.3) is as follows. The first and second
terms of Eq. (3.3) respectively denote the unsteady and convective changes of
Qα. The third term represents the diffusion of the conditional average in the
sample space ζ, which is modulated by the non-unity Lewis number effects. The
fourth term is the conditional chemical reaction rate for species α. Since θ is a
reactive scalar, its influence on the evolution of Qα is given by the fifth term in
Eq. (3.3). The sixth term represents the contribution of conditional fluctuation
y′′α to Qα evolution, which comes from the convective term of eyα given in Eq. 16
of Appendix-A. The last term represents contributions of molecular diffusion in
the physical space and differential diffusion effects. The explicit from of eQα is
given in Eq. 17 of the Appendix-A.
The CMC transport equations are to be solved along with other governing
and modeling equations discussed in sections 2.3 for the RANS methodology.
One must also not forget that appropriate initial and boundary conditions are
required for the CMC transport equation and these conditions will be discussed
while presenting the numerical methods in chapter 5. In the next section closures
for the unclosed terms in Eq. (3.3) are discussed.
44
3. Conditional Moment Closure
3.3 Premixed CMC Sub-Models
The quantities 〈u′′i y′′α|ζ〉, p˜, 〈ui|ζ〉, 〈ω˙α|ζ〉, 〈ω˙θ|ζ〉, eQα , 〈ρ|ζ〉 and 〈Nθ|ζ〉 require
suitable models. The PDF, p˜, is modelled using a presumed shape with a Beta
function, Eq. (2.64).
The conditional turbulent scalar flux, 〈u′′i y′′α|ζ〉, is generally closed using gradi-
ent transport hypothesis for non-premixed flames [40], although it is also possible
for this conditional flux to be counter-gradient [100] and new modelling methods
are to be developed to account for this. However, the adequacy of the gradi-
ent flux model for this study can be judged from the results to be presented in
chapter 6, and it is written as
〈u′′i y′′α|η〉 = −
µt
Scα
∂Qα
∂xi
, (3.4)
where Scα is the turbulent Schmidt number for species α, which is typically taken
to be 0.7. This model is reasonable for high Re flows and it has been used in the
past for RANS and LES simulations [32; 80] and [101].
The conditional mean reaction rate, 〈ω˙α|ζ〉, for species α in Eq. (3.3) is closed
using a first order CMC closure [40]. According to this closure, the conditional
mean reaction rate has the functional dependence on the conditional mean as the
instantaneous reaction rate has on the instantaneous scalar values. Hence,
〈ω˙α|ζ〉 = ω˙α(〈ρ|ζ〉,Qα, QT ), (3.5)
45
3. Conditional Moment Closure
where QT is the conditional mean temperature. This first order closure is shown
[40] to be accurate for non-premixed flames when there is no local extinction or
ignition events present in the flow. A prior analyses [96] using DNS data showed
that this closure is also accurate for premixed flames and thus it is used in this
study. Since the transport equations for Qα are solved along with appropriate
initial and boundary conditions, an arbitrarily complex chemical kinetics can be
included in the CMC.
The closure of 〈ω˙θ|ζ〉 is based on the definition of θ. Since θ is defined based
on the fuel mixture fraction
〈ω˙θ|ζ〉 = 〈ω˙f |ζ〉
Y uCH4
. (3.6)
The conditional mean velocity components are closed using a linear model
[40], which has been used for non-premixed flames. This model is given by
〈ui|ζ〉 = u˜i + u˜
′′
i θ
′′
θ˜′′2
(ζ − c˜), (3.7)
where, u˜i is the unconditional Favre mean velocity, u˜′′i θ′′ is the correlation be-
tween the velocity fluctuations and the progress variable fluctuations, θ˜′′2 is the
variance of the progress variable fluctuations. This model showed good agreement
with DNS results [96] for turbulent premixed flames. Although other modellings
are possible [96], this linear model is used because it is simple and sufficiently
accurate.
46
3. Conditional Moment Closure
From the Eq. 17 of Appendix A,
eQα ≡
〈
∂
∂xi
(
ρDα
∂Qα
∂xi
)
+ ρDα
∂θ
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
|ζ
〉
+〈
∂Qα
∂ζ
∂
∂xi
[(
1− Leα
Leθ
)
ρDα
∂θ
∂xi
] ∣∣∣ζ〉 , (3.8)
The molecular diffusion contributions are negligible for high Reynolds number
flows and the differential diffusion effect, the third term in Eq. (3.8) remains as
the dominant term [96]. Thus,
eQα =
(
1− Leα
Leθ
)
∂Qα
∂ζ
[
ρDα
∂θ˜
∂xi
]
(3.9)
eQα =
(
1− Leα
Leθ
)
∂Qα
∂ζ
1
p˜
∂〈Nθ|ζ〉p˜
∂ζ
. (3.10)
The second part of Eq. (3.10) is obtained after using the inter-relationship be-
tween the conditional diffusion and conditional dissipation in high Re turbulent
flames [40; 96]. It is apparent that the contribution of eQα depends on Leα, if
Leα/Leθ = 1, this contribution is zero. It is to be noted that this closure re-
quires a model for the conditional dissipation rate 〈Nθ|ζ〉, which is discussed in
chapter 4.
The conditional density, 〈ρ|ζ〉, is obtained using the state equation and the
conditional temperature, QT , through
〈ρ|ζ〉 = P
R QT
, (3.11)
47
3. Conditional Moment Closure
where R is the characteristics gas constant.
As noted earlier in section 3.1, the modelling of the conditional and uncondi-
tional dissipation rate of the progress variable offer significant challenges, since
the progress variable is a reactive scalar. This is discussed in the next chapter.
48
Chapter 4
Scalar Dissipation Rate
The effect of turbulence on premixed flames is to increase the flame surface area
by wrinkling the flame front [24]. In premixed flames, mixing of cold reactants
and hot products on the flame surface, is crucial to sustain the combustion. This
mixing phenomenon is characterised by the scalar dissipation rate, Nθ, which
may be broadly defined as the rate of mixing at a molecular level. This term
appears in many turbulent combustion models such as flamelets [102], probability
density function [29] and conditional moment closure [40] and requires an accurate
model. A considerable amount of experimental and modelling literature has been
published on the scalar dissipation rate for non-premixed [17; 103; 104; 105; 106]
and for premixed [79; 97; 98; 107; 108] combustion. Nevertheless, in premixed
flames, a sensible model for the scalar dissipation rate must account for the
interaction between turbulence, chemical reaction and molecular diffusion and
thus using turbulent time scale alone proved to be insufficient [97; 109].
Recent development in the modelling of the mean dissipation rate is explored
briefly in section 4.1. The modellings of unconditional dissipation rate are dis-
49
4. Scalar Dissipation Rate
cussed in section 4.2. The validation of these models are presented in section 4.3,
before choosing a suitable model for the CMC calculation.
4.1 Mean Scalar Dissipation Rate
An exact transport equation for the dissipation rate has been derived and analysed
to alleviate these issues [97]. This analysis and the insights obtained using this
transport equation in a number of studies have been summarised by Chakraborty
et al., [110]. For the purpose of this study, the final result, an algebraic model
for the mean scalar dissipation rate, ²˜θ = ρDθ(∇θ′′ · ∇θ′′)/ρ, is employed. This
algebraic model [98] is obtained by analysing the closure models proposed in
[99] for the leading order terms of the mean scalar dissipation rate transport
equation, when the Damko¨ler number is large. This model, validated using DNS
and experimental data [99], is written as
²˜θ =
1
β′
[
(2K∗c − τC4)
S0L
δ0L
+ C3
ε˜
k˜
]
θ˜′′2 , (4.1)
where, β′ = 6.7, K∗c = 0.85τ for hydrocarbon-air flames, C4 = 1.1/(1 + Ka)
0.4
and C3 = 1.5
√
Ka/(1 +
√
Ka). The Karlovitz number is defined as
Ka =
{
1
2(1 + τ)0.7
(
u′
S◦L
)3(
δ◦L
Λ
)}0.5
. (4.2)
The Favre averaged turbulence kinetic energy and its dissipation rate are denoted
by k˜ and ε˜ respectively.
50
4. Scalar Dissipation Rate
4.2 Conditional Mean Scalar Dissipation Rate
The conditional mean scalar dissipation rate, 〈N |θ〉, is related to the uncondi-
tional mean dissipation rate through
²˜θ =
∫ 1
0
〈Nθ|ζ〉 p˜(ζ) dζ. (4.3)
Thus, a closure for 〈Nθ|ζ〉 can be obtained from ²˜θ using two different approaches;
one is based on physical understanding and the other one is by directly solving the
integral equation in Eq. (4.3). Both of these are explored below, before choosing
a model for the CMC calculations.
4.2.1 Algebraic Model
It is well known that the turbulent stretch influences the local reaction rates in
premixed flames. Thus, the local scalar gradients will also be clearly influenced
by the stretch rate since the scalar gradients are produced predominately by
chemical reactions in premixed flames. If one considers the local flame front as
stretched flamelets then Kolla and Swaminathan [79] showed that the dissipation
rate in the stretched flamelets, subject to a stretch rate a, can be expressed as
N(θ, a) = N(θ∗, a)f(θ), where θ is the conditioning variable and θ∗ is its value at
the inner reaction zone. The inner reaction zone is defined as the location where
the fuel molecule will be attacked by a hydrogen atom and the typical value of
θ∗ is about 0.3 for common hydrocarbon fuels.
If one considers the turbulent flame-brush as a collection of stretched laminar
51
4. Scalar Dissipation Rate
flames then the mean dissipation rate is given by
²˜θ =
∫ 1
0
∫ ∞
0
N(θ, a) p(θ, a) da dθ, (4.4)
²˜θ =
∫ 1
0
〈N |θ〉 p˜(θ) dθ, (4.5)
²˜θ = 〈N |θ∗〉
∫ 1
0
f(θ) p˜(θ) dθ. (4.6)
Replacing 〈N |θ∗〉 = 〈N |θ〉/f(θ) one gets
〈N |θ〉 = ²˜ f(θ)∫ 1
0
f(θ′)p˜(θ′)d(θ′)
, (4.7)
as a model for the conditional dissipation rate [79]. The function f(θ) obtained
from the unstrained laminar premixed flame is shown to be a very good approx-
imation by Kolla and Swaminathan [79]. If one knows θ˜ and θ˜′′2 then 〈N |θ〉 can
be obtained by presuming the PDF, p˜, as in Eq. (2.64), and using Eq. (4.1) for
²˜θ. The validation of this model will be discussed in section 4.3.
4.2.2 Inverse Model
The second method to obtain the conditional scalar dissipation rate is based on
finding a solution [111] for the ill-posed integral equation in Eq. (4.3). Ill-posed
problems arise in many science and engineering applications, which have a non-
unique and/or an unstable solution. To explain this, let’s consider a Fredholm
integral equation of the first kind as
F (y) =
∫ b
a
K(x, y)G(x)dx, c ≤ y ≤ d, (4.8)
52
4. Scalar Dissipation Rate
where, G(x) is unknown function, F (y) and the kernel K(x, y) are approximately
known functions. It can be written in matrix form as
KG = F, (4.9)
where K is the n× n matrix, F is n× 1 matrix and G is n× 1. The solution for
Eq. (4.9) would be given by
G ' K−1 F, (4.10)
and if the problem is well-posed then a unique solution will exist. But these prob-
lems are characterised as ill-posed system when the solution G is very sensitive to
small changes or errors in F (y) or K(x, y), because the matrix K may have local
singularities. For these situations, a regularisation method is adopted usually
to find a stable and meaningful solution. Out of several regularisation method
available Tikhonov regularisation algorithm [111] is used in this study to find an
approximate solution for the conditional scalar dissipation rate from Eq. (4.3).
This regularisation method translates the problem into a minimisation prob-
lem written as
min
{|p˜Nαθ − ²˜θ|2 + α|(Nαθ −Nα−1θ )|2)} , (4.11)
where, α ≥ 0 is the regularisation parameter. The notation Nαθ implies the value
of Nθ obtained with regularisation parameter α. The value of this parameter
evolves as αj = e αj−1 while solving the above minimisation using an iterative
technique, where j is the iteration index and 0 < e < 1. Thus, the final converged
53
4. Scalar Dissipation Rate
solution for Nθ will strongly depend on its initial guess value and the behaviour
of the kernel p˜.
4.3 Model Validation
4.3.1 DNS Database
The above two models are validated using two different sets of DNS database
having different thermo-chemical and turbulence conditions. A brief discussion
of these databases is presented first before discussing the validation results.
Direct numerical simulations of turbulent premixed V-flames of methane-air
mixture of equivalence ratio 0.58 with preheated reactant temperature of 600K
[112]. The V-flame simulations were carried out using the fully compressible,
three-dimensional DNS code SENGA2. The chemical kinetics was simulated [112]
using a single irreversible reaction with kinetic parameters giving a value of about
0.6 m/s for the unstrained laminar flame speed and about 0.4 mm for the thermal
thickness. The Lewis number was taken to be unity.
The domain is cubic with sides Lx = Ly = Lz = 29.7δ
◦
L. Non-reflecting out-
flows are specified on the downstream and transverse faces, and the remaining
direction is periodic. Turbulence is supplied at the inlet from a pre-computed
frozen solution of fully developed, homogeneous isotropic turbulence at the ap-
propriate turbulence intensity. Once inside the V-flame domain, the turbulence
freely decays as it interacts with the flame and is convected downstream with
the mean velocity. The flame is stabilised at the flame holder by imposing prod-
uct mass fractions over a small cylindrical region at x = 3.48δ◦L from the inlet
54
4. Scalar Dissipation Rate
boundary; equivalent to a catalytic wire aligned in the periodic direction.
Once initial transients have decayed, the simulations are run for one complete
flow-through time, τD = Lx/u¯in where u¯in is the mean inlet velocity, during which
data are saved at regular intervals. All mean quantities are then time and space
averaged from ensembles of the saved snapshots and over all points in the periodic
direction. To capture the spatial development of the flame, two locations down-
stream from the flame holder are considered: X1 = 16.7 δ◦L and X2 = 27.9 δ
◦
L, at
which the influence of the flame holder is known to be negligible. The relevant
parameters of these two V flames are given in table 4.1 and the combustion is in
the corrugated flamelets and thin reaction zones regime of turbulent combustion
as noted in Fig. 4.3.
Flame u′/S◦L Λ/δ
◦
L Re Da Ka t
∗
an
R2a 0.85 78.0 106.8 91.8 0.2 1.0
R2c 3.40 19.5 106.2 5.7 3.2 2.5
VA 2.0 3.62 37 1.81 1.49 1.0
VB 6.0 3.43 92 0.57 7.94 6.0
Table 4.1: Inlet parameters for the selected DNS flames. t∗an is in terms of the
initial eddy turnover for R2a & R2c flames, it is in term of flow through time for
V flames.
The second set of DNS data considered includes turbulent premixed flames of
H2-air turbulent premixed flames propagating in three-dimensional homogeneous
isotropic turbulence developed at Tokyo Institute of Technology [113; 114] are
considered. In this DNS, stoichiometric premixed H2-air flames with preheated
reactants at Tu = 700K were simulated using a multiple-step detailed kinetic
mechanism involving 27 reactions and 12 reactive species. These two simulations
denoted as R2a and R2c were in the wrinkled flamelets and thin reaction zones
55
4. Scalar Dissipation Rate
0
2
4
6
0 0.2 0.4 0.6 0.8 1
P˜
(ζ
)
ζ
θ˜ = 0.5
R2a
R2c
Figure 4.1: The variation of the Favre averaged PDF of the progress variable,
p˜(ζ), at θ = 0.5 inside the flame brush of flames R2a and R2c.
regimes of turbulent combustion as shown in Fig. 4.3. The relevant parameters
of these two data sets are listed in table 4.1.
The last column of table 4.1 shows the time of analysis for the H2-air data
set and the sampling time for the V flames. This time is normalised using the
initial large eddy turn over time of the turbulence for H2-air flames and the flow
through for the V flames. These data sets are analysed to validate the models for
the conditional dissipation rate discussed in the previous section. Note that the
results from V flames are obtained at two different axial locations in the flame;
x1 = 16.7δ
◦
L and x2 = 27.9δ
◦
L.
Figs. 4.1 and 4.2 show the Favre PDF of the progress variable θ˜ for flames
R2a and R2c at θ = 0.5 and flames VA and VB at θ = 0.6.
56
4. Scalar Dissipation Rate
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
P˜
(ζ
)
ζ
VA flame
θ˜ = 0.6
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1
P˜
(ζ
)
ζ
VB flame
θ˜ = 0.6
Figure 4.2: The variation of the Favre averaged PDF of the progress variable,
p˜(ζ), at θ = 0.6 inside the flame brush of flames VA and VB.
57
4. Scalar Dissipation Rate
0.1
1
10
100
1000
0.1 1 10 100 1000
u
′
/s
o l
Λ/δ
R2a
R2c
VA
VB
R
e
=
1
Laminar
R
e
=
100
R
e
=
1000
D
a
=
1
Ka
= 1
Ka
= 1
00
wrinkled
corrugated
thin reaction zones
distributed
Figure 4.3: The premixed combustion regime diagram showing the selected DNS
flames.
4.3.2 Results and Discussion
The code used to determine the solution for the minimal least squares norm,
Eq. (4.11), is developed at Moscow State University [115]. The regularization
parameter set to be 1.001 to give the smallest value of residual with 560 number
of iterations. The initial solution is specified from the results obtained using
algebraic model, Eq. (4.7).
Figures 4.4 and 4.5 show the variation of the conditional mean scalar dissipa-
tion rate, 〈Nθ|ζ〉+, with sample space, ζ, at θ˜ = 0.5 for flames R2a and R2c and
θ˜ = 0.6 for V flames. The results are obtained using algebraic model, Eq. (4.7),
and inverse model, Eq. (4.11), and compared to DNS results. The results ob-
tained are normalised using SoL and δ
o
L. As shown in the figures, both the inverse
model Eq. (4.11) and the algebraic model Eq. (4.7) have predicted the peak of
conditional mean scalar dissipation rate. The values of 〈Nθ|ζ〉+ obtained by the
58
4. Scalar Dissipation Rate
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
flame R2a
c˜ = 0.5
Algebraic model
DNS data
Inverse
0
0.02
0.04
0.06
0.08
0.1
0.12
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
flame R2c
c˜ = 0.5
Algebraic model
DNS data
Inverse
Figure 4.4: The typical variation of the conditional scalar dissipation rate,
〈N |ζ〉+, calculations obtained by, Eq. (4.7) and Eq. (4.11) with ζ in flames R2a
and R2c. The values are non-dimensionalised using SoL and δ
o
L.
algebraic model are in good agreement with the DNS data for flame R2c and
slightly under predicts flame R2a, while values obtained by the inverse model
are over-prediction. As noted earlier, the inverse model requires an initial esti-
mate for the conditional mean scalar dissipation rate and the model was found
to be very sensitive to the initial estimate. For example if the initial estimate is
increased by 30% then the model will deviate drastically from the DNS value.
59
4. Scalar Dissipation Rate
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.6
VA flame
X1
Algebraic model
DNS data
Inverse
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.6
VA flame
X2
Algebraic model
DNS data
Inverse
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.6
VB flame
X1
Algebraic model
DNS data
Inverse
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.6
VB flame
X2
Algebraic model
DNS data
Inverse
Figure 4.5: The typical variation of the conditional scalar dissipation rate,
〈N |ζ〉+, obtained by, Eq. (4.7) and Eq. (4.11) with ζ in flames VA and VB at two
different axial locations in the flame; x1 = 16.7δ
◦
L and x2 = 27.9δ
◦
L. These values
are non-dimensionalised using SoL and δ
o
L.
60
Chapter 5
Mathematical Model
This chapter presents a summary of the numerical methods used to simulate
turbulent flows, since these methods are well established in the literature now.
The discretisation of the CMC equation using the finite volume methodology
is described in some detail, and this methodology follows an earlier study [32]
where the CMC code for non-premixed flames was developed. The modifications
to the code needed for premixed flames are described in section 5.2. The solution
methodology and the interaction of RANS and CMC equations are described in
section 5.3.
5.1 Discretisation of the CMC Equation
The stationary form of the governing equations given in section 2.3 for the mean
fields are discretised by finite volume methodology of Patankar [116] using a power
law scheme [116] for the spatial derivatives. In Cartesian coordinates (x, y, z), the
conservative form of CMC transport equation is discretised [32] over a physical
61
5. Mathematical Model
Figure 5.1: Control volume in a finite volume method.
space control volume as shown in Fig. 5.1. The point P is at the center of the
control volume and the neighbouring cells are designated as west, east, south,
north, up and down. The walls of the cells are represented by lower case letters.
In the progress variable space, the neighbouring points are designated as left and
right. The conservative form of Eq. (3.3) is written as
∂
∂xi
(
γ〈ui|ζ〉Qα − γDt∂Qα
∂xi
)
− Leθ
Leα
γ〈Nθ|ζ〉∂
2Qα
∂ζ2
= γ〈ω˙α|ζ〉 −
∂
∂ζ
[γ〈ω˙θ|ζ〉Qα] , (5.1)
where, γ = 〈ρ|ζ〉 p˜(ζ). Integrating equation Eq. (5.1) over the control volumes in
real and progress variable spaces and dividing by γV∆ζ yields
γeAe
γV
(
〈ui|ζ〉Qα −Dt∂Qα
∂x
)
e
− γwAw
γV
(
〈ui|ζ〉Qα −Dt∂Qα
∂x
)
w
62
5. Mathematical Model
+
γnAn
γV
(
〈uj|ζ〉Qα −Dt∂Qα
∂y
)
n
− γsAs
γV
(
〈uj|ζ〉Qα −Dt∂Qα
∂y
)
s
+
γuAu
γV
(
〈uk|ζ〉Qα −Dt∂Qα
∂z
)
u
− γdAd
γV
(
〈uk|ζ〉Qα −Dt∂Qα
∂z
)
d
−Leθ
Leα
〈Nθ|ζ〉
∆ζ
[(
∂Qα
∂ζ
)
r
−
(
∂Qα
∂ζ
)
l
]
=
γV
γV
〈ω˙α|ζ〉 − γV 〈ω˙θ|ζ〉
γV∆ζ
[
Qr −Ql
]
(5.2)
The face values of the conditional quantities, Qα, in the advection terms of
Eq. (5.2) are written as
Qe = feQE + (1− fe)QP (5.3)
and
Qw = fwQW + (1− fw)QP , (5.4)
where the subscripts e and w represent the east and west walls of the cell and
the subscripts E and W represent the east and west neighbouring cells. The
weighting factor f is giving by
f ≡ 1
Pe
(
1− (1− 0.1Pe)5
)
, (5.5)
according to the power-law scheme of Patankar [116]. Accordingly, the total flux
across each face in the positive and negative x-directions are
Je =
γeAe
γV
[
〈ui|ζ〉eQe − Dt(QE −QP )
(δx)e
]
=
γeAe
γV
[
(QE −QP )
(
fe〈ui|ζ〉e − Dt
(δx)e
)
+ 〈ui|ζ〉eQP
]
, (5.6)
63
5. Mathematical Model
and
Jw =
γwAw
γV
[
〈ui|ζ〉wQw − Dt(QP −QW )
(δx)w
]
=
γwAw
γV
[
(QW −QP )
(
fw〈ui|ζ〉w − Dt
(δx)w
)
+ 〈ui|ζ〉wQP
]
, (5.7)
Defining the coefficients, aE, Fe, aW and Fw as
aE =
γeAe
γV
(
−fe〈ui|ζ〉e + Dt
(δx)e
)
, (5.8)
Fe =
γeAe〈ui|ζ〉e
γV
, (5.9)
aW =
γwAw
γV
(
fw〈ui|ζ〉w + Dt
(δx)w
)
, (5.10)
and
Fw =
γwAw〈ui|ζ〉w
γV
. (5.11)
The total fluxes in the positive and negative directions in the physical space are
written as
Je = aE(QP −QE) + FeQP , (5.12)
Jw = aW (QW −QP ) + FwQP . (5.13)
The same principle is used for the other faces.
In progress variable space, the advection terms of Eq. (5.2) are written as
Qr = frQR + (1− fr)QP , (5.14)
64
5. Mathematical Model
Ql = flQL + (1− fl)QP . (5.15)
where the subscripts L and R represent the left and right neighbouring grid points
in the progress variable space ζ. Accordingly, the total fluxes are defined as
Jr =
Leθ
Leα
〈Nθ|ζ〉
∆ζ(δx)r
(QR −QP )− 〈ω˙θ|ζ〉
∆ζ
Qr, (5.16)
=
(QR −QP )
∆ζ
(
−fr〈ω˙θ|ζ〉+ Leθ
Leα
〈Nθ|ζ〉
(δx)r
)
− 〈ω˙θ|ζ〉
∆ζ
QP ,(5.17)
and
Jl =
Leθ
Leα
〈Nθ|ζ〉
∆ζ(δx)l
(QL −QP )− 〈ω˙θ|ζ〉
∆ζ
Ql, (5.18)
=
(QL −QP )
∆ζ
(
−fl〈ω˙θ|ζ〉+ Leθ
Leα
〈Nθ|ζ〉
(δx)l
)
− 〈ω˙θ|ζ〉
∆ζ
QP .(5.19)
By defining the coefficients, aR and aL as
aR =
1
∆ζ
(
−fr〈ω˙θ|ζ〉+ Leθ
Leα
〈Nθ|ζ〉
(δx)r
)
, (5.20)
aL =
1
∆ζ
(
−fl〈ω˙θ|ζ〉+ Leθ
Leα
〈Nθ|ζ〉
(δx)l
)
. (5.21)
The total fluxes in the positive and negative directions of ζ co-ordinate are written
as
Jr = aR(QR −QP ) + 〈ω˙θ|ζ〉QP
∆ζ
, (5.22)
65
5. Mathematical Model
and
Jl = aL(QP −QL) + 〈ω˙θ|ζ〉QP
∆ζ
. (5.23)
Now the total flux balance equation is written as
Je − Jw + Jn − Js + Ju − Jd = (Jr)− (Jl) + 〈ω˙α|ζ〉
+QP (Fe − Fw + Fn − Fs + Fu − Fd) . (5.24)
Substituting the total fluxes in Eq. (5.24), and after some arrangement yields the
final discretised CMC transport equation, which is written as
∑
anbQP −
∑
anbQnb = 〈ω˙α|ζ〉. (5.25)
A first order closure of the chemical source term is obtained by linearising it with
respect to Q. This algebraic equation is solved in the CMC part of the computer
code.
5.2 Implementation for Premixed Flames
The original version of the research code was designed for non-premixed flames
with two separate streams [117], fuel and oxidiser, with their mixing being de-
scribed by solving a transport equation for Favre averaged mixture fraction, which
is a passive scalar, see Eq. (2.29). In contrast, the pilot stabilised premixed flames
involve more than two streams, the premixed reactant stream, the pilot stream
and the ambient air stream. Note that, some cases can have more than one pilot
stream, as noted in chapter 7. For these reasons, the code has been modified
66
5. Mathematical Model
to include more than two streams along with appropriate transport equations to
track the fluids emerging from these various streams. The transport equations
for the Favre averaged progress variable θ˜ and its variance θ˜′′2, Eqs. (2.28) and
(2.30) are also included. Furthermore, one must also include additional variables
which are passive scalars, to account for the mixing among streams. The simple
mixing rules used in this study are explained while presenting results for the test
flames in the following chapters. The source term, ω˙θ, in Eq. (2.28) is closed in
a manner consistent with CMC methodology as
ω˙θ =
∫ 1
0
〈ω˙θ|ζ〉 p dζ. (5.26)
The dissipation rate, ²˜θ, of the variance θ˜′′2 in Eq. (2.30) is closed using Eq. (4.1)
and the chemical source term is modelled using
c′′ω˙θ =
∫ 1
0
ζ〈ω˙θ|ζ〉 p dζ − θ˜
∫ 1
0
〈ω˙θ|ζ〉 p dζ. (5.27)
The solution to CMC equations requires initial conditions. For this purpose,
steady flamelet equations written as
Leθ
Leα
ρNθ
∂2Yα
∂ζ2
+ ω˙α − ω˙θ ∂Yα
∂ζ
= 0, (5.28)
with appropriate boundary conditions at ζ = 0 and ζ = 1 for Yα, are solved.
This equation is also implemented in the code for this purpose by comparing
its solution with that obtained using Chemkin for a freely propagating laminar
premixed flame. The variation of Nθ with θ obtained from Chemkin was supplied
to this flamelet equation and the chemical kinetics was modelled using GRI-mech.
67
5. Mathematical Model
Figure 5.2: Interaction of CMC and RANS equation in the simulation.
3.0. The premixed CMC transport equation is closed using the models discussed
in section 3.3
5.3 Computer Model and Sequence
The Favre averaged, continuity, momentum and enthalpy equations are solved
along with transport equations for θ˜, θ˜′′2 , Z˜, k˜ and ε˜.
The flow chart shown in Fig. 5.2 describes the computational sequence and
the interaction between RANS and CMC equations. A SIMPLER approach [116]
is used to couple the velocity and pressure fields inside the computational domain.
These discretised equations are solved using an iterative algorithm using under
relaxation factors, given in table 5.1. The mean flow and turbulence quantities
obtained from the fluid dynamics solver are passed to the CMC solver. A steady
68
5. Mathematical Model
P u v w k ε h µ ρ T θ˜ θ˜′′2 Z˜ Z˜ ′′2
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.1 0.1 0.1 0.1 0.1
Table 5.1: Under-relaxation factors.
flamelet solution is used to specify the initial and the boundary conditions for
the CMC equations as noted in section 5.2, and the CMC equations are solved to
get a converged solution. Then the mean density, ω˙θ and θ′′ω˙θ calculated using
the respective conditional means obtained from the CMC solver are fed back to
the fluid dynamics part. These processes are iterated, until convergence criteria
are met for all quantities of interest as shown in Fig. 5.2. This convergence
criteria is set to be 5E-5 for the fluid dynamics and CMC. Further detail on the
computational tool used in this study can be found in [32; 117] as that study
validated this tool for bagasse-fired boiler involving a non-premixed flame.
From the converged solution, the various mean quantities, required for com-
parison with experimental measurements, are obtained as
Y˜α =
∫ 1
0
Qα p˜ dζ. (5.29)
The computational results obtained using the numerical methods described
above are compared to experimental measurement of pilot stabilised turbulent
premixed flames for range of thermochemical and turbulence conditions. These
comparisons are discussed in the following chapters.
69
Chapter 6
Stoichiometric Premixed Flame
Calculation
6.1 Computational Details
The test flames used in this chapter are the pilot stabilised turbulent Bunsen
flames of Chen et al. [2], which is shown schematically in Fig. 6.1 along with the
computational domain and boundary conditions used. These flames have been
investigated in earlier studies to validate various premixed turbulent combustion
models such as transported PDF model [68; 118], flame surface density approach
[119], strained flamelets [79] and G-equation [120]. Recently, these flames have
also been computed using large eddy simulation with the thickened flame model
[121]. Stoichiometric methane-air flames with a fuel nozzle diameter D = 12 mm
and a pilot diameter dp = 68 mm were considered. Three flames designated as
F1, F2, and F3, were investigated in the experiments. The bulk mean velocity,
Uo, at the exit of the fuel nozzle is 65, 50 and 30 m/s for flames F1, F2 and F3
70
6. Stoichiometric Premixed Flame Calculation
respectively. The Reynolds number, Re, based on D and Uo, for these flames are
52000, 40000 and 24000 respectively. The hot stoichiometric products from the
pilot flames were flowing with a uniform velocity of 0.2 m/s, and this value is
obtained from the reported [2] volumetric flow rate at the burner exit.
As shown in the turbulent combustion regime diagram, Fig. 6.2, flames F1 and
F2 are in the thin reaction zones regime and flame F3 is on the border between the
thin reaction zones and the flamelets regimes. In the current study these flames
are computed using the RANS-CMC approach, with a specific interest on carbon
monoxide since the suggested [40] contribution of the CMC for premixed systems
is expected to be in the prediction of species with slow time scales including
pollutants.
Since no explicit assumptions on the influence of turbulent eddies on the flame
front structure were made while deriving the CMC equation, Eq. (3.3), the CMC
may hold in all regimes of premixed combustion although its successful application
would be determined by the accuracy of the closure models and the assumptions
used to obtain them. The conditional scalar dissipation rate, 〈Nc|ζ〉, is important
from this view-point since the progress variable gradient is strongly determined by
chemical reaction, molecular diffusion and their interaction in premixed flames.
However, as [40] noted, the contribution of CMC for premixed combustion would
be on the prediction of species with slow time scales, including the effects of
turbulence on the chemical structure. Using one time or length scale related
to the fast chemical process, typically used to construct the regime diagram,
may be insufficient to argue the applicability of the CMC in various premixed
combustion regimes. Careful studies using DNS data with complex chemical
kinetics in three dimensions and laser diagnostics data on the progress variable
71
6. Stoichiometric Premixed Flame Calculation
Figure 6.1: Schematic diagram of the burner setup [2] and the computational
domain along with BCs.
72
6. Stoichiometric Premixed Flame Calculation
0.1
1
10
100
1000
0.1 1 10 100 1000
u
′
/s
o l
Λ/δ
R
e
=
1
Laminar
R
e
=
100
R
e
=
1000
D
a
=
1
Ka
= 1
Ka
= 1
00
wrinkled
corrugated
thin reaction zones
distributed
F1
F2
F3
Figure 6.2: The premixed combustion regime diagram showing flames F1, F2 and
F3.
gradient information and multi-species measurements are very much required to
address this open question.
The Favre averaged, continuity, momentum and enthalpy equations are solved
along with transport equations for θ˜, θ˜′′2 , Z˜ and Z˜ ′′2 . Although the mixture
fraction variance is included, for the calculation of partially premixed flames in
future, its solution is not used in this study. The mean mixture fraction must be
seen as a marker for the reacting mixture from the main jet and pilot, and it is
set to be unity in these two streams at the burner exit. This marker is used to
account for the mixing between the jet fluid and surroundings air as discussed by
[79]. The value of the total enthalpy computed using its transport equation at a
grid point is written as
h˜ = cp,mix
(
T˜ − To
)
+∆hof,mix, (6.1)
73
6. Stoichiometric Premixed Flame Calculation
where cp,mix = Z˜
∑
Y˜α θp,α + (1 − Z˜)θp,air, and ∆hof,mix = Z˜
∑
Y˜α ∆h
o
f,α + (1 −
Z˜)∆hof,air, to obtain the average temperature T˜ . The enthalpy of formation for
the air is taken to be zero.
The turbulence is modeled using a standard k-ε model with changes to C²1
(from 1.44 to 1.52) to account for round jet anomaly [122]. The boundary condi-
tions reported by [2] for the mean flow and turbulence quantities are used at the
burner exit (see Fig. 6.1) for the calculations reported in this study. A small uni-
form velocity of 0.2 m/s is specified to account for the ambient air entrainment.
The mean progress variable is specified to be one in the main jet and zero in the
pilot and ambient air streams. The variance of the progress variable is specified
to be zero in all the streams entering the computational domain. The conditions
at other computational boundaries are marked in Fig. 6.1.
A computational domain of size 1000 mm in the axial direction, x, and 270 mm
in the radial direction, r, is used with refined grid near the fuel exit. The smallest
cell size is 0.5×1.0 (in mm) and the CFD computational domain has 80×80×120
cells in y, z, x directions, for turbulence and mean quantities. For the CMC
calculations, 500 non-uniform cells are used in ζ space to resolve the very strong
gradients near the hot side (ζ = 0 in Figs. 6.8 and 6.9 to be discussed later) and
to obtain meaningful and smooth variations of conditional averages of major and
minor species involved in the chemical mechanism. Two cells in each physical
direction of the main grid for the fluid dynamics variables are combined to create
the physical space grid for the CMC equations, since the conditional averages are
expected to vary slowly in the physical space. These grids gave solutions with
negligible sensitivity to any further changes in the physical or CMC grids. Thus,
the results reported here are for the above grids.
74
6. Stoichiometric Premixed Flame Calculation
In order to assess the influence of chemical kinetics mechanism and eQα term
on the reacting flow results, three simulations were performed for flame F1 using
two different chemical mechanisms for methane-air combustion. In the first case,
the GRI-mechanism [123] is used, which consists of 325 reactions and 53 species.
In the second case, the skeletal mechanism of [124] involving 25 elementary chem-
ical reactions and 16 species is used. The effect of chemical kinetics mechanism
is studied by comparing these two cases. To evaluate the influence of eQα , see
Eq. (3.3), on the evolution of Qα, a third simulation is conducted using the GRI-
mech and including eQα given in Eq. (3.10). The other two flames, F2 and F3,
are simulated using the GRI-mechanism excluding the contributions from eQα .
A steady flamelet solution is used to specify the initial and the boundary con-
ditions for the CMC equations as noted in section 5.2. A single three-dimensional
simulation took about 336 hours wall clock time in a 3.5 GHz, 8GB RAM, intel
Xeon desktop computer to get a converged solution. This computational time can
be reduced significantly by using parallel computation techniques, which are yet
to be implemented in this computer code. The computed solutions are compared
to experimental data reported by [2]. These comparison are discussed next.
6.2 Results and Discussion
6.2.1 Non Reacting Flow
The non-reacting flow were simulated first to assess the turbulence models and
the flow boundary conditions used. Figures 6.3 and 6.4 show the normalised mean
velocity and turbulence kinetic energy results in non-reacting flow for flames F1,
75
6. Stoichiometric Premixed Flame Calculation
F2 and F3. The turbulence kinetic energy, k˜o, at the centre line of the fuel
nozzle exit are 12.7, 10.8 and 3.82 m2/s2 for flames F1, F2 and F3 respectively.
These results are shown for five axial locations, the symbols are the experimental
measurements and the lines are the computational results. The good agreement
shown here for the mean velocity suggests that the spread rate of the cold jet
is captured in the simulations and the boundary conditions used describe the
experimental conditions well.
The normalised mean turbulence kinetic energy profiles for flame F1 are in
excellent agreement with the measurements at the centre line for locations close
to the nozzle exit, while further down-stream, it is slightly over predicted. In the
radial direction, the computed k˜ is slightly over predicted at all locations in flame
F1.
Almost similar levels of agreement in the mean velocity and turbulent kinetic
energy are observed for flames F2 and F3 as shown in figures 6.3 and 6.4. The
normalised mean turbulent kinetic energy agrees well with the measurements at
the center line for all axial locations. In the radial directions, it is in good agree-
ment with the measurements for all axial locations except, at location x/d = 10.5,
it is slightly over predicted in the radial direction. In general, the results from
the cold flow are typical for k-ε model, and they are satisfactory and acceptable.
The radial variation of Z˜ at different axial distances is shown in Fig. 6.5
for flame F1. Unfortunately, there is no experimental data for this comparison,
however, the spread of this marker variable is as expected in this type of flow.
Since Z˜ = 1 in the fuel jet and pilot, the center-line value does not drop from
unity for the stream-wise locations noted.
76
6. Stoichiometric Premixed Flame Calculation
0
0.5
1
8.5
0
0.5
1
e U
U
o
6.5
0
0.5
1
4.5
0
0.5
1
0 0.4 0.8 1.2 1.6
r/D
2.5
0 0.4 0.8 1.2 1.6
r/D
0 0.4 0.8 1.2 1.6 2
r/D
0
0.5
1
1.5
F1 x/D = 10.5 F2 F3
Figure 6.3: The normalised mean velocity (lines) from non-reacting flow are com-
pared to the the experimental measurements (symbols) for flames F1, F2 and
F3.
77
6. Stoichiometric Premixed Flame Calculation
0
5
10
8.5
0
5
10
e k k
o
6.5
0
5
10
4.5
0
5
10
0 0.4 0.8 1.2 1.6
r/D
2.5
0 0.4 0.8 1.2 1.6
r/D
0 0.4 0.8 1.2 1.6 2
r/D
0
5
10
15
F1 x/D = 10.5 F2 F3
Figure 6.4: The normalised mean turbulent kinetic energy (lines) from non-
reacting flow are compared to the experimental measurements (symbols) for
flames F1, F2 and F3.
78
6. Stoichiometric Premixed Flame Calculation
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
Z˜
r/D
x/D = 2.5
4.5
6.5
8.5
10.5
Figure 6.5: Computed radial variation of Z˜ in cold flow simulation at five axial
locations for the conditions of flame F1.
6.2.2 Conditional Scalar Dissipation Rate
The typical values of conditional dissipation rate obtained in the CMC calcula-
tion of flame F1 are shown in Figs. 6.6 and 6.7. Note that the model used in
this calculation is the algebraic model, Eq. 4.7. These values obtained at axial
locations x/D = 4.5 and 8.5 for different θ˜ inside the flame brush. these values
are non-dimensionalised using SoL and δ
o
L. As shown in the figure, the charac-
teristics variation of 〈N |ζ〉+ with ζ is observed in the CMC calculation. This
is because the chemical reactions occur towards the hot side of the flame front,
which is denoted by low value of ζ, the progress variable. Unfortunately, there is
no experimental data for this comparison, however, if one compares the values of
〈N |ζ〉+ of flame F1 to the DNS flames shown in chapter 4, these values appear to
be larger because the turbulence level in flame F1 is large compared to the DNS
flames. Also as seen from the results the variation of 〈N |ζ〉+ with different θ˜ and
axial locations are very small.
79
6. Stoichiometric Premixed Flame Calculation
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.45
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.55
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.6
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.65
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.75
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.85
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.9
Figure 6.6: The typical variation of the conditional scalar dissipation rate,〈N |ζ〉+,
obtained in CMC calculations by, Eq. (4.7), with ζ in flame F1 at different loca-
tions inside the flame brush at x/D = 4.5. The values are non-dimensionalised
using SoL and δ
o
L.
80
6. Stoichiometric Premixed Flame Calculation
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.45
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.55
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.6
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.65
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.75
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.85
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1
<
N
|ζ
>
+
ζ
c˜ = 0.9
Figure 6.7: The typical variation of the conditional scalar dissipation rate,〈N |ζ〉+,
obtained in CMC calculations by, Eq. (4.7), with ζ in flame F1 at different loca-
tions inside the flame brush at x/D = 8.5. The values are non-dimensionalised
using SoL and δ
o
L.
81
6. Stoichiometric Premixed Flame Calculation
6.2.3 Terms in CMC Transport Equation
After neglecting the unsteady term in Eq. (3.3), the steady CMC transport equa-
tion can be written as
Conv.−Diff.− reac.+ reacθ − eQα = Imb. (6.2)
The typical variation of these terms with ζ for a few species is shown in Fig. 6.8
using GRI-mech. The results are shown for few representative scalars from flame
F1 at θ˜ = 0.55 and x/d = 4.5. Note that ζ = 0 is the burnt side and ζ = 1 is the
unburnt side. Since the values shown in Fig. 6.8 do not include the algebraic sign
in Eq. (6.2), a positive value indicates a source and a negative value implies a sink
for the terms related to the reactions and eQα . Since the four species shown in
Fig. 6.8 are produced by the chemical reactions in methane-air flames, the values
of the “reac.” term is positive. The reactive nature of the conditioning variable is
making a sink contribution to these scalars. Also as observed in Fig. 6.8, there is a
predominant balance between the convective and the diffusive terms and reaction
rate of species α is balanced by the “reacθ” and eQα terms in the reactive zone.
The Lewis number for the conditioning variable is unity. The magnitude of eQα is
dictated by ∂Qα/∂ζ and the value of Lewis number for species α, see Eq. (3.10).
However, if Leα is close to unity then one can see that eQα = 0 unless ∂Qα/∂ζ
is singular. The negligible value of eQα is noted in Fig. 6.8 for CO, since LeCO
= 1.084. The contribution of eQα for H2O is smaller then the reactive terms
but non-negligible because LeH2O = 0.78. However, for the atomic and molecular
hydrogen the contribution of eQα is as big as the reactive term in Eq. (3.10) since
the Lewis number for these species are small (LeH = 0.172,LeH2 = 0.293). Finally
82
6. Stoichiometric Premixed Flame Calculation
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
C
O
+
ζ
Conv.
reacc.
reac.
Diff.
eQ
α
Imb.
-0.02
-0.01
0
0.01
0.02
0 0.2 0.4 0.6 0.8 1
H
+
ζ
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 0.2 0.4 0.6 0.8 1
H
+ 2
ζ
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1
H
2
O
+
ζ
Figure 6.8: Typical variation of different terms of the CMC transport equation,
Eq. (6.2) , with ζ in flame F1 at θ˜ = 0.55. The results are shown for few repre-
sentative scalars.
83
6. Stoichiometric Premixed Flame Calculation
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
H2O−GRI
H2O− Smook
O2 −GRI
O2 − Smook
CH4 −GRI
CH4 − Smook
CO2 −GRI
CO2 − Smook
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
CO-GRI
CO-Smook
10 ∗ H2 −GRI
10 ∗ H2 − Smook
5*OH-GRI
5*OH-Smook
Figure 6.9: The variation of the conditional mean mass fraction with ζ for major
and minor species in flame F1 at θ˜ = 0.55 and x/d = 4.5.
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
H2O−GRI
H2O− Smook
O2 −GRI
O2 − Smook
CH4 −GRI
CH4 − Smook
CO2 −GRI
CO2 − Smook
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
CO-GRI
CO-Smook
10 ∗ H2 −GRI
10 ∗ H2 − Smook
5*OH-GRI
5*OH-Smook
Figure 6.10: The variation of the conditional mean mass fraction with ζ for major
and minor species in flame F1 at θ˜ = 0.55 and x/d = 8.5.
the numerical errors represented by the “imb.” in Eq. (6.2) are observed to be
very close to zero in Fig. 6.8 indicating that the equations are solved with small
errors.
The typical variation of conditional mean mass fractions of representative
major and minor species is shown in Figs. 6.9, 6.10 and 6.11 for flame F1. The
results are shown for θ˜ = 0.55, the middle of flame brush at x/d = 4.5, 8.5
and for the two kinetics mechanisms used. It is clear that the influence of the
chemical kinetics mechanism on the computed values of the major species is
84
6. Stoichiometric Premixed Flame Calculation
0
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
H2O−GRI
H2O− Smook
O2 −GRI
O2 − Smook
CH4 −GRI
CH4 − Smook
CO2 −GRI
CO2 − Smook
0
0.005
0.01
0.015
0.02
0.025
0.03
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
CO-GRI
CO-Smook
10 ∗ H2 −GRI
10 ∗ H2 − Smook
5*OH-GRI
5*OH-Smook
Figure 6.11: The variation of the conditional mean mass fraction with ζ for major
and minor species in flame F1 at θ˜ = 0.55 and x/d = 10.5.
minimal. However, as one would expect the intermediate and the minor species
values are strongly influenced by the kinetics mechanism. The results suggest
that the skeletal mechanism gives lower values of Qα compared to the GRI-mech.
This is because of the additional chemical pathways available for the formation
of the intermediate species in the GRI-mech. As seen from the calculated results
that they have identical values at different axial locations x/d = 4.5, 8.5 and
10.5. This is supported by the CMC hypothesis that conditional mean quantities
have weaker spatial dependence than unconditional quantities. Thus, CMC has
a coarse grid compared to RANS, as indicated in section 6.1 .
6.2.4 Comparison to Experimental Results
To compute the mean values using Eq. (5.29), one requires the marginal PDF of
the conditioning variable. As noted earlier, this PDF is obtained by presuming
its shape as a Beta function, see Eq. (2.64), and using the computed values of θ˜
and θ˜′′2. The PDF obtained thus in the CMC calculation is shown in Fig. 6.12 for
flame F1. The typical bimodal shape, a well known characteristics of turbulent
85
6. Stoichiometric Premixed Flame Calculation
0
10
20
30
40
50
60
0 0.2 0.4 0.6 0.8 1
p˜
(ζ
)
ζ
c˜ = 0.55
c˜ = 0.7
c˜ = 0.9
Figure 6.12: The variation of the Favre averaged PDF of the progress variable,
p˜(ζ), at three locations in the flame brush of flame F1.
premixed flames, is seen in this figure for all locations shown.
Typical contours for the calculated Favre averaged θ˜, θ˜′′2, Z˜ and T˜ are shown in
Figs. 6.13 and 6.14 for flame F1 for visual inspection and the expect behaviour
is seen clearly. The results from the reacting flow calculations are shown in
Figs. 6.15 to 6.24 for up to five axial locations, the symbols are the experimental
measurements and the lines are the computational results.
Figure 6.15 shows the radial variation of the normalised mean velocity and
turbulent energy for five axial locations for flame F1. The results are shown
for GRI-mech and Smooke’s skeletal mechanism. The influence of eQα is also
shown. In general, the computed mean velocity agrees well with the measured
values. The values computed using GRI-mech are closer to the experimental
data, compared to the values obtained using Smooke’s mechanism. The differen-
tial diffusion effects denoted by eQα do not seem to influence the mean velocity
values. However, the computed values of the turbulent kinetic energy show some
86
6. Stoichiometric Premixed Flame Calculation
Figure 6.13: Contours of the mean progress variable and its variance for flame
F1
87
6. Stoichiometric Premixed Flame Calculation
Figure 6.14: Contours of the mean mixture fraction variable and temperature for
flame F1
88
6. Stoichiometric Premixed Flame Calculation
sensitivity to the chemical kinetics mechanism and the differential diffusion ef-
fects. At x/d = 2.5, the Smooke mechanism seem to work very well while the
GRI-mech over predicts the turbulent kinetic energy. However, the contribution
of eQα seem to correct this over prediction. For all the axial locations studied in
Fig 6.15, the turbulent kinetic energy computed using Smooke mechanism and
GRI-mech with eQα compares well with the measured values up to r/d ' 0.5 for
flame F1. By comparing the results obtained using GRI-mech with and without
eQα contribution, one notes that the effects of eQα is large up to r/d = 0.8 in the
near field and this region started to move slowly inward (toward the jet center) as
x/d increases. This suggests that the differential diffusion effects are important
near the central core region. This is consistent with our expectation, since the
turbulence is produced by the shear production mechanism and the turbulence
in the jet is small.
The computed values of mean mass fraction of methane and normalised mean
temperature are compared to the experimental data in Fig. 6.16 for flame F1. The
agreement in the mean mass fraction of methane is very good and its sensitivity to
the chemical kinetics mechanism is negligible. The mean temperature is over pre-
dicted in the near field and the agreement improves with down-stream distance.
Except for x/d = 2.5, the computed mean temperature is weakly sensitive to
the chemical kinetics mechanism and differential diffusion effects. There is some
uncertainty in the temperature boundary condition for the pilot stream as has
been noted in many previous studies [68; 79; 119; 120; 121] using this experiment
and thus one must be careful while interpreting the results for the mean temper-
ature at x/d = 2.5. However, the general agreement between the computed and
measured values is satisfactory and the computed values show negligibly small
89
6. Stoichiometric Premixed Flame Calculation
0
0.5
1
1.5
0 0.4 0.8 1.2 1.6 2
U U
o
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
5
10
15
0 0.4 0.8 1.2 1.6 2
k k
o
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.5
1
1.5
0 0.4 0.8 1.2 1.6 2
U U
o
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
5
10
15
0 0.4 0.8 1.2 1.6 2
k k
o
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.5
1
1.5
0 0.4 0.8 1.2 1.6 2
U U
o
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
5
10
15
0 0.4 0.8 1.2 1.6 2
k k
o
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.5
1
1.5
0 0.4 0.8 1.2 1.6 2
U U
o
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
5
10
15
0 0.4 0.8 1.2 1.6 2
k k
o
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.5
1
1.5
0 0.4 0.8 1.2 1.6 2
U U
o
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
0
5
10
15
0 0.4 0.8 1.2 1.6 2
k k
o
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
Figure 6.15: The normalised mean velocity and turbulent kinetic energy from
the CMC calculations (lines) at five axial locations are compared to experimental
results (symbols) for flame F1.
90
6. Stoichiometric Premixed Flame Calculation
sensitivity to the chemical mechanism and the eQα contribution. This observa-
tion also applies to the major species mass fraction reported in Fig. 6.17. The
agreement for H2O and O2 is better than for CO2 as shown in Fig. 6.17 and these
relative behaviours (between the computations and measurements) is similar to
these reported in earlier studies using other combustion modelling methods.
The radial variation of mean mass fractions of CO, OH and H2 are compared
to the experimental measurements in Fig. 6.18 for five axial locations for flame
F1. As one would expect the computed values are sensitive to the kinetics scheme,
however, the influence of eQα is observed to be small as in Fig. 6.18. The peak
values of the averaged mass fraction of these species are larger for GRI mechanism
compared to Smooke mechanism and the values obtained using Smooke mecha-
nism are closer to the experimental data. The agreement between the computed
and measured values are good for OH and H2 as shown in Fig. 6.18. However
for CO, there seems to be large discrepancies and the reason for this is unclear
at this stage. The peak values and the radial variation of mean mass fraction of
CO computed using CMC in this study are similar to those reported earlier using
other modelling approaches for premixed flames.
The flames, F2 and F3, are simulated using the GRI-mechanism excluding the
contributions from eQα . Figures 6.19 and 6.20 show the radial variation of the
normalised mean velocity and turbulent kinetics energy for flames F2 and F3. The
level of agreement between the computed and measured values of these quantities
for flames F2 and F3 is similar to that shown for flame F1. At the center line,
the computed mean velocity and turbulent energy are in excellent agreement
with the experimental measurements for all axial locations for both flames F2
and F3. In the radial direction, the computed mean velocity agrees well with
91
6. Stoichiometric Premixed Flame Calculation
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 2
T
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
2
4
6
8
10
0 0.4 0.8 1.2 1.6 2
Y
C
H
4
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 2
T
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
2
4
6
8
10
0 0.4 0.8 1.2 1.6 2
Y
C
H
4
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 2
T
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
2
4
6
8
10
0 0.4 0.8 1.2 1.6 2
Y
C
H
4
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 2
T
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
2
4
6
8
10
0 0.4 0.8 1.2 1.6 2
Y
C
H
4
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 2
T
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
0
2
4
6
8
10
0 0.4 0.8 1.2 1.6 2
Y
C
H
4
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
Figure 6.16: The normalised mean temperature and CH4 mass fraction × 100,
from the CMC calculations (lines) at five axial locations are compared to exper-
imental results (symbols) for flame F1.
92
6. Stoichiometric Premixed Flame Calculation
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
O
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
2
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
O
2
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
O
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
2
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
O
2
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
O
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
2
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
O
2
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
O
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
2
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
O
2
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
O
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
2
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
O
2
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
Figure 6.17: The mean mass fractions of H2O, CO2 and O2 from the CMC
calculations (lines) at five axial locations are compared to experimental results
(symbols) for flame F1.
93
6. Stoichiometric Premixed Flame Calculation
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.4 0.8 1.2 1.6 2
Y
O
H
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
r/D
x/D = 10.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.4 0.8 1.2 1.6 2
Y
O
H
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
r/D
x/D = 8.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.4 0.8 1.2 1.6 2
Y
O
H
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
r/D
x/D = 6.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.4 0.8 1.2 1.6 2
Y
O
H
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
r/D
x/D = 4.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
C
O
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.4 0.8 1.2 1.6 2
Y
O
H
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
0
0.05
0.1
0.15
0.2
0.25
0 0.4 0.8 1.2 1.6 2
Y
H
2
r/D
x/D = 2.5
GRI
eQ
α
Smook
Exp.
Figure 6.18: The mean mass fractions of CO× 10, OH× 75 and H2× 100 from
the CMC calculations (lines) at five axial locations are compared to experimental
results (symbols) for flame F1.
94
6. Stoichiometric Premixed Flame Calculation
the measurement values for flame F2, while flame F3 shows some discrepancies
particularly at r/d > 0.8. The turbulent energy is slightly over predicted in the
near field and the agreement improves with down stream distance for flames F2
and F3.
The computed values of mean mass fraction of methane and normalised mean
temperature of flames F2 and F3 are compared to the experimental data in
Figs. 6.21 and 6.22. The level of agreement between the computed and mea-
sured values of the mean temperature and mass fraction of CH4 improves slightly
as turbulence Reynolds number decreases (ie. from flame F1 to F3). As shown
in the figure, the mean mass fraction of methane agrees well with experimen-
tal values and the mean temperature is over predicted in the near field and the
agreement improves gradually in down stream direction. The computed values
of the mean mass fraction of the major species of flames F2 and F3 are shown
in Fig. 6.23. The major species are in good agreement in the near field and the
agreement improves with down stream distance. The computed values of mean
mass fractions of CO, OH and H2 are compared to the experimental values in
Fig. 6.24 for flames F2 and F3. The results are shown as radial variations at a
few down stream locations. The relative behaviours between the calculated and
measured values are similar to that for flame F1 for all species studied. The peak
CO values measured for a given x/d location seem to be weakly sensitive to the
changes in the Reynolds and Damko¨hler numbers (compare the peak value, for
example for x/d = 8.5 among F1, F2 and F3 flames), but the CMC results shows
a gradual drop in the peak CO value as the Damko¨hler number increases. This
is consistent with our expectation.
95
6. Stoichiometric Premixed Flame Calculation
0
0.5
1
8.5
0
0.5
1
e U U
o
6.5
0
0.5
1
4.5
0
0.5
1
0 0.4 0.8 1.2 1.6 2
r/D
2.5
0
0.5
1
1.5
x/D = 10.5
F2
0
0.5
1
e U U
o
0
0.5
1
0
0.5
1
0 0.4 0.8 1.2 1.6 2
r/D
0
0.5
1
F3
Figure 6.19: The normalised mean velocity from the CMC calculations (lines) at
five axial locations are compared to experimental results (symbols) for flames F2
and F3.
96
6. Stoichiometric Premixed Flame Calculation
0
5
10
8.5
0
5
10
e k k
o
6.5
0
5
10
4.5
0
5
10
0 0.4 0.8 1.2 1.6 2
r/D
2.5
0
5
10
15
x/D = 10.5
F2
0
5
10
e k k
o
0
5
10
0
5
10
0 0.4 0.8 1.2 1.6 2
r/D
0
5
10
15
F3
Figure 6.20: The normalised mean turbulent kinetic energy from the CMC calcu-
lations (lines) at five axial locations are compared to experimental results (sym-
bols) for flames F2 and F3.
97
6. Stoichiometric Premixed Flame Calculation
0
2
4
6 4.5
0
2
4
6
Y
C
H
4
6.5
0
2
4
6
0 0.4 0.8 1.2 1.6 2
r/D
2.5
0
2
4
6 8.5
0
2
4
6
8
x/D = 10.5F2
0
2
4
6
0
2
4
6
Y
C
H
4
0
2
4
6
0 0.4 0.8 1.2 1.6 2
r/D
0
2
4
6
8
F3
Figure 6.21: The normalised mean mass fraction of CH4× 100 from the CMC
calculations (lines) at five axial locations are compared to experimental results
(symbols) for flames F2 and F3.
98
6. Stoichiometric Premixed Flame Calculation
0
0.2
0.4
0.6
0.8 4.5
0
0.2
0.4
0.6
0.8
T
6.5
0
0.2
0.4
0.6
0.8
0 0.4 0.8 1.2 1.6 2
r/D
2.5
0
0.2
0.4
0.6
0.8 8.5
0
0.2
0.4
0.6
0.8
1
x/D = 10.5F2
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
T
0
0.2
0.4
0.6
0.8
0 0.4 0.8 1.2 1.6 2
r/D
0
0.2
0.4
0.6
0.8
1
F3
Figure 6.22: The normalised mean temperature from the CMC calculations
(lines) at five axial locations are compared to experimental results (symbols) for
flames F2 and F3.
99
6. Stoichiometric Premixed Flame Calculation
0
0.1
0.2
8.5
0
0.1
0.2
Y
i
6.5
0
0.1
0.2
4.5
0
0.1
0.2
0 0.4 0.8 1.2 1.6 2
r/D
2.5
0
0.1
0.2
0.3
x/D = 10.5F2
0
0.1
0.2
Y
i
0
0.1
0.2
0
0.1
0.2
0 0.4 0.8 1.2 1.6 2
r/D
0
0.1
0.2
0.3
F3
Figure 6.23: The mean mass fractions of H2O (4,−), CO2 (◦, · · ·) and O2 (•,− ·
−) from the CMC calculations (lines) at five axial locations are compared to
experimental results (symbols) for flames F2 and F3.
100
6. Stoichiometric Premixed Flame Calculation
0
0.1
0.2
8.5
0
0.1
0.2
Y
i
6.5
0
0.1
0.2
4.5
0
0.1
0.2
0 0.4 0.8 1.2 1.6 2
r/D
2.5
0
0.1
0.2
0.3
x/D = 10.5F2
0
0.1
0.2
0
0.1
0.2
Y
i
0
0.1
0.2
0 0.4 0.8 1.2 1.6 2
r/D
0
0.1
0.2
0.3
F3
Figure 6.24: The mean mass fractions of CO× 10 (◦, · · ·), OH× 75 (4,−) and
H2× 100 (•,− ·−) from the CMC calculations (solid lines) at five axial locations
are compared to experimental results (symbols) for flames F2 and F3.
101
Chapter 7
Lean Premixed Flame
Calculation
7.1 Flames Details
The second test case used in this study is the pilot stabilised lean premixed
flames of Dunn et al. [3], which is shown schematically in Fig. 7.1. This jet
burner (PPJB) used in the experimental study [3] consists of a central jet of
a lean premixed methane-air mixture with equivalence ratio of 0.5 and initial
temperature of 300 K, and a pilot of a hot stoichiometric premixed methane-
air products. The central jet and the hot pilot have diameters of 4 and 23.5
mm, respectively. These two streams are surrounded by a hot co-flow from lean
premixed laminar H2-air flames with equivalence ratio of 0.43. The diameter
for this hot co-flow stream is 197 mm. The pilot and the hot co-flow streams
were flowing with small uniform velocities of 0.8 and 0.7 m/s, respectively [3].
Four flames were considered in the experimental study [3] designated as PM1-
102
7. Lean Premixed Flames Calculation
Figure 7.1: Schematic diagram of the premixed Bunsen burner from [3; 4].
103
7. Lean Premixed Flames Calculation
Stream D (mm) U0 (m/s) T (K) φ mixture
Jet 4 50 - 200 300 0.5 CH4 − air (reactant)
Pilot 23.5 0.8 2280 1 CH4 − air (products)
Co-flow 197 0.7 1500 0.43 H2 − air (products)
Table 7.1: The properties of the streams of flames PM1-50 and PM1-200.
50, PM1-100, PM1-150 and PM1-200 with central jet bulk mean velocities, Uo,
of 50, 100, 150 and 200, respectively. The Reynolds number, Re, based on the
central jet velocity, diameter, D, are 12500, 25000, 37000 and 50000, respectively.
A small uniform velocity of 0.2 m/s is specified to account for the ambient air
entrainment. The characteristics of these streams are summarized in table 7.1.
As shown in the turbulent combustion regime diagram, Fig. 7.2, these flames
are in the distributed combustion regime. In this regime, scales of the chemical
length and time scales are larger than the turbulence. Hence, applying flamelets
based methods to these flames may be inappropriate. Note that the values of Λ/δ
for flames PM1-50 and PM1-200 are 2.4 and 1.8, respectively and the values of
u′ are 15 and 54, respectively as reported by Dunn et al.[125]. The laminar flame
speed is SoL = 0.051 m/s. These flames have been used in earlier studies to validate
premixed turbulent combustion models such as a transported PDF models [6] and
joint velocity-turbulence frequency-composition VFJPDF method [5] and large
eddy simulation [126]. In the current study, these flames are computed using the
RANS-CMC approach discussed in earlier chapters of this thesis.
104
7. Lean Premixed Flames Calculation
0.1
1
10
100
1000
0.1 1 10 100 1000
u
′
/S
o L
Λ/δ
R
e
=
1
Laminar
R
e
=
100
R
e
=
1000
D
a
=
1
Ka
= 1
Ka
= 1
00
wrinkled
corrugated
thin reaction zones
distributed
F1
F2
F3
PM1-200
PM1-50
PM1-100PM1-150
Figure 7.2: The premixed combustion regime diagram showing the stoichiometric
and lean flames.
7.2 Computational Details
A schematic of the computational domain for this flame is shown in Fig 7.3.
It must be noted that the three streams have different composition and equiv-
alence ratios as given in table 7.1, and thus more than one passive scalar (fluid
marker variable) is required to account for the mixing of these streams. Thus,
in addition to the Favre averaged transport equations used for the stoichiometric
premixed flames discussed in chapter 6, additional transport equations for two
passive scalars, Z˜2 and Z˜3, are implemented to track the fluids emerging from the
hot pilot and hot co-flow streams and to account for the effects of mixing of these
streams with the main jet and with the entrained air. Thus, each stream has its
own designated markers, Z˜1, Z˜2 and Z˜3 and their values are set to be unity at
the inlet of appropriate streams as shown in Fig 7.3. These markers are used in
105
7. Lean Premixed Flames Calculation
the following manner to obtain the average temperature T˜ . The total enthalpy
computed using its transport equation at a given grid point in the physical space
is written as
h˜ = hs +∆hof,mix, (7.1)
=
∑
i
[
Z˜1Y˜α + Z˜2Y˜α,2 + Z˜3Y˜α,3 +
(
1− Z˜1 − Z˜2 − Z˜3
)
Y˜α,air
]
θpi(T˜ − T0)
+
∑
i
(
Z˜1Y˜α + Z˜2Y˜α,2 + Z˜3Y˜α,3
)
∆hofα , (7.2)
where Y˜α is the mass fraction of species α at the grid point. Y˜α,2 and Y˜α,3 are
the mass fractions of species α in the streams 2 and 3 at the inlet. ∆hofα is the
enthalpy of formation for species α and the enthalpy of formation for air is taken
to be zero. From the above equation, one can easily verify that the temperature
of the incoming streams are recovered by using the boundary values for Z˜1, Z˜2
and Z˜3 marked in Fig 7.3.
The inlet velocity profile is specified using the bulk mean value as u(r)/U0 =
(1− r/R)1/7, where R is the radius of the main jet port. Uniform velocities of 0.8
and 0.7 m/s are specified as initial values for the pilot and hot co-flow streams. A
small uniform velocity of 0.2 m/s is specified for the surroundings air. A uniform
value for the mean turbulent kinetic energy, k˜o, at the center line of the fuel
nozzle exit, is obtained from I = u′/U0 =
√
2k/3/U0. The axial rms velocity,
u′, for the cold and reacting flows for both selected flames are reported in the
experiment [3], see u′, Figs. 7.4 and 7.5. The initial value for dissipation rate, ²˜o,
106
7. Lean Premixed Flames Calculation
Figure 7.3: Schematic diagram of the burner setup and the computational domain
along with BCs.
107
7. Lean Premixed Flames Calculation
at the center line of the fuel nozzle exit, is obtained using ²˜o = Cuk˜
3/2
o /Λ.
The computational domain is 600 mm long in the axial direction, x, and
300 mm in the radial direction, r. The CFD computational domain has 150 cells
in the axial direction and 100 cells in the radial direction. The domain is refined
non uniformly near the fuel, pilot and co-flow streams, where the smallest cell
size is 1 × 1.6 (in mm). As for the stoichiometric flame, the CMC grid has 500
non-uniform cells in ζ space, which were required to resolve the strong gradients
in the progress variable space. Two cells in each direction of the physical space
grid for the RANS equations are combined to create the physical space grid for
the CMC equations. The sensitivity of the computed solution to the grid size is
tested by doubling and halving the size of the smallest cell, while retaining the
grow the rate for the cell size. The solution showed a weak sensitivity and thus,
the grid resolution noted above is used for the results reported in this chapter.
The values of various model parameters in the turbulence and other models
discussed in earlier chapters are kept to be the same as those used for stoichio-
metric flame studied in chapter 6, except for C²1 = 1.6. The chemical kinetics is
modelled using the GRI-mechanism [123] and a steady flamelet solution is used as
an initial solution and to specify the boundary conditions for the CMC equations.
The computed solutions are compared to the experimental data reported in [4; 6]
in the following discussion.
7.3 Results and Discussion
Turbulence models and boundary conditions used in this calculation are assessed
first by comparing the non-reacting flow results for both flames, PM1-200 and
108
7. Lean Premixed Flames Calculation
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 0  10  20  30  40  50  60  70
U~
/U
0
Axial Direction, x/D 
  N-CMC
N-EXP.
 R-CMC
R-EXP.
 0
 0.05
 0.1
 0.15
 0.2
 0  10  20  30  40  50  60  70
u
'/U
0
Axial Direction, x/D 
 N-CMC
 N-EXP.
 R-CMC
R-EXP.
Figure 7.4: The calculated (lines) normalised mean axial velocity and rms ve-
locity from non-reacting and reacting flows are compared to experimental data
(symbols) for flame PM1-200.
PM1-50 selected for this study. Figures. 7.4 and 7.5 compares the calculated mean
and rms axial velocity. The values are normalised using the bulk mean velocity,
U0, at the jet exit. The non-reacting and reacting flows are denoted by the letter
N and R, respectively in these figures. The normalised mean axial velocity for
both of these flames are in excellent agreement with the measurements. A very
good agreement for the normalized rms velocity is observed. These comparison
clearly indicated that the boundary conditions used in the simulations mimic the
experimental conditions well. Note that the u′ is obtained from the calculated k˜
as explained in the previous section, while the experimental results is obtained
using
u′ =
√∑i=n
i=1 [u(ti)− U ]2∆ti∑i=n
i=1 ∆ti
, (7.3)
where n is taken to be the number of realisations and ∆ti is the gate period in
the measurements for the ith realisation of the velocity.
109
7. Lean Premixed Flames Calculation
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 1.4
 0  10  20  30  40
U~
/U
0
Axial Direction, x/D 
  N-CMC
N-EXP.
 R-CMC
R-EXP.
 0
 0.1
 0.2
 0.3
 0  10  20  30  40
u
'/U
0
Axial Direction, x/D 
 N-CMC
 N-EXP.
 R-CMC
R-EXP.
Figure 7.5: The calculated (lines) normalised mean axial velocity and rms ve-
locity from non-reacting and reacting flows are compared to experimental data
(symbols) for flame PM1-50.
7.3.1 Conditional Mean Mass Fractions
The conditional mean mass fraction, Qα, is obtained by solving the CMC equa-
tion, Eq. (3.3). The typical variation of conditional mean mass fractions for some
selected representative major and minor species is shown in Figs. 7.6, 7.7 and
7.8 for flame PM1-200. The results are shown for θ˜ = 0.55, the middle of flame
brush at axial location x/D = 4.5, 8.5 and 10.5. ζ = 0 is the product side and
ζ = 1 is the reactant side. Note that, the values of H2 and OH are multiplied by
40 and 20, respectively for plotting purpose. The unstrained laminar premixed
flame is computed using CHEMKIN-II package along with GRI-mech. The vari-
ation of conditional averages of major species in the sample space is very close to
those found in the unstrained laminar flame. However, the representative minor
species variation with ζ in the CMC calculation is similar to those observed in
the laminar flame, except for the molecular hydrogen. In the laminar flame, the
hydrogen mass fraction varies little with a sharp drop near ζ = 0 and ζ = 1, but
110
7. Lean Premixed Flames Calculation
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
H2O− CMC
H2O− Laminar
O2 − CMC
O2 − Laminar
CH4 − CMC
CH4 − Laminar
CO2 − CMC
CO2 − Laminar
0
0.003
0.006
0.009
0.012
0.015
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
40×H2 − CMC
40× H2 − Laminar
20×OH− CMC
20×OH− Laminar
CO− CMC
CO− Laminar
Figure 7.6: The variation of the conditional mean mass fraction with ζ for major
and minor species for flame PM1-200 at θ˜ = 0.55 and x/D = 4.5.
the conditional mean value for the hydrogen shows a peak near ζ ≈ 0.16. For the
PM1-50 flame, the variation of conditional means are similar to those shown in
Fig. 7.6. As seen from the calculated results that they have identical values at
different axial locations x/d = 4.5, 8.5 and 10.5. This is supported by the CMC
hypothesis that conditional mean quantities have weaker spatial dependence than
unconditional quantities.
7.3.2 Comparison to Experimental Results
The typical variation of the conditional dissipation rate for the flames, PM1-
200 and PM1-50 is shown in Figs. 7.9, 7.10 and 7.11 at θ˜ = 0.55 and axial
locations x/D = 4.5, 6.5 and 10.5. The turbulence level for flame PM1-200
is higher compared to flame PM1-50, and thus the mean dissipation rates are
larger in flame PM1-200. The variation of 〈Nθ|ζ〉+ shown in Figs 7.9 , 7.10
and 7.11 are similar to that observed in Fig 6.6 for the stoichiometric flame and
the magnitudes are different because of the difference in the thermo-chemical
111
7. Lean Premixed Flames Calculation
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
H2O− CMC
H2O− Laminar
O2 − CMC
O2 − Laminar
CH4 − CMC
CH4 − Laminar
CO2 − CMC
CO2 − Laminar
0
0.003
0.006
0.009
0.012
0.015
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
40×H2 − CMC
40× H2 − Laminar
20×OH− CMC
20×OH− Laminar
CO− CMC
CO− Laminar
Figure 7.7: The variation of the conditional mean mass fraction with ζ for major
and minor species for flame PM1-200 at θ˜ = 0.55 and x/D = 8.5.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
H2O− CMC
H2O− Laminar
O2 − CMC
O2 − Laminar
CH4 − CMC
CH4 − Laminar
CO2 − CMC
CO2 − Laminar
0
0.003
0.006
0.009
0.012
0.015
0 0.2 0.4 0.6 0.8 1
Q
s
ζ
40×H2 − CMC
40× H2 − Laminar
20×OH− CMC
20×OH− Laminar
CO− CMC
CO− Laminar
Figure 7.8: The variation of the conditional mean mass fraction with ζ for major
and minor species for flame PM1-200 at θ˜ = 0.55 and x/D = 10.5.
112
7. Lean Premixed Flames Calculation
θ˜ θ˜′′2 Z˜1 Z˜2 Z˜3
0.9 0.027 0.971 0.026 0.0025
0.8 0.063 0.931 0.061 0.006
0.7 0.097 0.894 0.094 0.011
0.6 0.129 0.857 0.126 0.015
0.5 0.161 0.821 0.158 0.020
0.4 0.193 0.784 0.190 0.024
0.2 0.145 0.635 0.308 0.055
Table 7.2: Values of θ˜, θ˜′′2, Z˜1, Z˜2 and Z˜3 in flame PM1-50 for the PDFs shown
in Fig. 7.12.
and turbulence conditions of these flames. Also it obvious from the results that
the conditional scalar dissipation rate changes slowly in the axial direction. Its
magnitude is increased about 5% in the down stream direction.
The marginal PDF of the progress variable, obtained using the presumed
Beta function for the given Favre mean and variance is shown in Figs. 7.12 and
7.13 for flames PM1-50 and PM1-200, respectively. The Favre mean, θ˜, and
variance, θ˜′′2, are obtained from their respective transport equations. Unlike
for the stoichiometric flames discussed in chapter 6, these PDFs are not bimodal
across the flame brush. These PDFs are shown for seven different radial locations,
denoted by θ˜, at a given axial location x/D = 4.5. The values of θ˜′′2, Z˜1, Z˜2 and
Z˜3 at these locations are given in tables 7.2 and 7.3 respectively for flames PM1-
50 and PM1-200. The value of the Favre variance for the bimodal PDF limit
given by θ˜′′2 = θ˜(1− θ˜) is observed only for low values of θ˜. At location very close
to the burnt side of the flame brush, ie. θ˜ ≤ 0.15, bimodal PDFs are observed for
both flames PM1-50 and PM1-200. The mono-modal behaviour of this marginal
PDF is typical of distributed combustion regime.
The radial variations of Z˜1, Z˜2 and Z˜3 at six axial locations in flame PM1-200
113
7. Lean Premixed Flames Calculation
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
<
N
θ
|ζ
>
+
ζ
PM1 − 50
PM1 − 200
Figure 7.9: The typical variation of the conditional scalar dissipation rate,〈N+θ |ζ〉,
obtained in the CMC calculation for flames PM1-50 and PM1-200 at θ˜ = 0.55,
x/D = 4.5 . The values are non-dimensionalised using SoL and δ
o
L.
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
<
N
θ
|ζ
>
+
ζ
PM1 − 50
PM1 − 200
Figure 7.10: The typical variation of the conditional scalar dissipation
rate,〈N+θ |ζ〉, obtained in the CMC calculation for flames PM1-50 and PM1-200
at θ˜ = 0.55, x/D = 8.5 . The values are non-dimensionalised using SoL and δ
o
L.
114
7. Lean Premixed Flames Calculation
0
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
<
N
θ
|ζ
>
+
ζ
PM1 − 50
PM1 − 200
Figure 7.11: The typical variation of the conditional scalar dissipation
rate,〈N+θ |ζ〉, obtained in the CMC calculation for flames PM1-50 and PM1-200
at θ˜ = 0.55, x/D = 10.5 . The values are non-dimensionalised using SoL and δ
o
L.
0
3
6
9
12
15
0 0.2 0.4 0.6 0.8 1
p˜
(ζ
)
ζ
θ˜ = 0.2
0.4
0.5
0.6
0.7
0.8
0.9
Figure 7.12: The variation of the Favre PDF of the progress variable, p˜(ζ), at
selected locations of the flame brush of flame PM1-50.
115
7. Lean Premixed Flames Calculation
0
3
6
9
12
15
0 0.2 0.4 0.6 0.8 1
p˜
(ζ
)
ζ
θ˜ = 0.2
0.4
0.5
0.6
0.7
0.8
0.9
Figure 7.13: The variation of the Favre PDF of the progress variable, p˜(ζ), at
selected locations of the flame brush of flame PM1-200.
θ˜ θ˜′′2 Z˜1 Z˜2 Z˜3
0.9 0.030 0.964 0.0181 0.017
0.8 0.051 0.925 0.035 0.039
0.7 0.094 0.883 0.054 0.062
0.6 0.143 0.848 0.069 0.081
0.5 0.141 0.809 0.087 0.103
0.4 0.205 0.761 0.101 0.136
0.2 0.155 0.620 0.126 0.252
Table 7.3: Values of θ˜, θ˜′′2, Z˜1, Z˜2 and Z˜3 in flame PM1-200 for the PDFs shown
in Fig. 7.13.
116
7. Lean Premixed Flames Calculation
0
0.2
0.4
0.6
0.8
1
0 2 4 6
Z˜
1
r/D
x/D 2.5
5
15
25
35
45
Figure 7.14: Computed radial variation of Z˜1 from the hot flow simulation at six
axial locations for the conditions for flame PM1-200.
are shown in Figs. 7.14, 7.15, 7.16 and 7.17 respectively. Since these fields are non-
reactive, convective and diffusive fields, they are expected to spread with down
stream distance. The results in Figs. 7.14 to 7.17 clearly shown this behaviour.
However, Z˜3 does not seem to decrease below one for 10 ≤ r/D ≤ 20 for the axial
locations shown in Fig. 7.17, this is because x/dp2 = 0.91 for x/D = 45. To see
any significant drop from unity for 10 ≤ r/D ≤ 20, one must go beyond 100D
axial distance. This point is clearly from Fig. 7.18, showing a colour map of Z˜3.
The spatial evolution of Z˜1 and Z˜2 is shown in Fig. 7.19. From these results,
one observes that the mixing of these three streams is captured well. Figure 7.20
shows the spatial variation of θ˜ and θ˜′′2. Since the variance θ˜′′2 is produced by
the chemical reaction and 5θ˜, one will observe a sharp increase in θ˜′′2 inside the
flame brush. This behaviour is clearly in Fig.7.20
The variation of the center-line value of the mean and rms values of the axial
velocity is shown in Figs. 7.4 and 7.5 respectively. These values are normalised
117
7. Lean Premixed Flames Calculation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 2 4 6
Z˜
1
r/D
CMC 30
VFJPDF. 30
EXP. 30
CMC 45
VFJPDF. 45
EXP. 45
Figure 7.15: Computed radial variation of Z˜1 from the hot flow simulation are
compared to PDF calculations [5] for flame PM1-200.
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6
Z˜
2
r/D
x/D 2.5
5
15
25
35
45
Figure 7.16: Computed radial variation of Z˜2 from the hot flow simulation at six
axial locations for the conditions for flame PM1-200.
118
7. Lean Premixed Flames Calculation
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40
Z˜
3
r/D
x/D 2.5
5
15
25
35
45
Figure 7.17: Computed radial variation of Z˜3 from the hot flow simulation at six
axial locations for the conditions for flame PM1-200.
Figure 7.18: Contours of Z˜3 for flame PM1-200
119
7. Lean Premixed Flames Calculation
Figure 7.19: Contours of Z˜1 and Z˜2 for flame PM1-200
120
7. Lean Premixed Flames Calculation
Figure 7.20: Contours of the mean progress variable and its variance for flame
PM1-200
121
7. Lean Premixed Flames Calculation
using the bulk mean value at the main jet inlet, U0. The computed values are
compared to the measurements in these figures. The agreement in the mean
values at locations close to the fuel nozzle is excellent. Further down stream,
particularly for x/D > 20, the computed values are over-predicted for flame
PM1-200, while the agreement is improved for flame PM1-50 which has a lower
Re number. The rms values for both flames show some discrepancies at all axial
locations.
In the radial direction, the computed mean velocity using the CMC for flame
PM1-200 is over predicted for all axial locations as shown in Fig. 7.21. A good
agreement is observed for x/D = 5 and 55. However, for flame PM1-50, a good
agreement with the experimental measurements is observed for all axial loca-
tions as shown in Fig. 7.22. The rms velocity for flame PM1-200 is shown in
Fig. 7.23, and the CMC values are in reasonable agreement with the experimen-
tal measurements. While, for flame PM1-50 there are some discrepancies with the
experimental measurements as in Fig. 7.24. These are typical for k-ε turbulence
model used in the calculations. These behaviours for the computed velocities are
very similar to those reported by Dunn et al. [6] and Dunn [4] using transported
PDF modelling. This comparison is also shown in Figs 7.21 to 7.24.
Figures 7.25 and 7.26 show the radial variation of mean temperature from the
CMC calculations of flames PM1-200 and PM1-50, respectively. The influence
of the pilot gases on the main jet mixture is well captured. The adiabatic flame
temperature u1480 K, is predicted well inside the flame brush at r/D = 0.7 at
x/D = 2.5, and it is consistent with these reported in the experiment [6]. The
high temperature between r/D = 1 to 2.5, indicate that the reaction of the main
jet is influenced mainly by the hot pilot temperature. Also, the effect from the
122
7. Lean Premixed Flames Calculation
lean hot co-flow stream is well captured at r/D = 2.8 and beyond. This effect
is captured as a drop in the temperature between the stoichiometric pilot and
the lean co-flow streams. The drop in the mean temperature to a value of about
1300K near r/D = 3 for x/D = 2.5 could be due heat loss to the burner, which is
not included in the CFD model. This heat loss effect is shown clearly by Rowinski
and Pope [5], however, it is also noted in that study that the sensitivity of the
computed results at locations beyond x/D = 25 is small.
The radial variation of the mean temperature shown here for PM1-200 is very
similar to that shown by Rowinksi and Pope [5], Dunn et al. [6] and Dunn
[4] computed using different approaches and modellings for transported PDF
methodology. The radial variation of the calculated mean temperature is shown in
Fig 7.26 for flame PM1-50 and the agreement with the measured values is observed
to be very good, when compared to the PM1-200 flame. This is not surprising
because the turbulence model used in this study seems to predict the turbulence
quantities better for PM1-50 flame as noted earlier. Since the CMC sub-models
feed on the predicted turbulence quantities, one observes a good prediction for
flame PM1-50. The relative behaviour of the computed and measured mean
temperature for flame PM1-50 is very similar to that reported by Rowinksi and
Pope [5], Dunn et al. [6] and Dunn [4]. This comparison is also shown in Figs 7.25
to 7.26
The radial variation of the mean mass fractions of CO and OH computed
using the CMC are compared to the experimental measurements and PDF com-
putations [4; 6] in Figs. 7.27, 7.28, 7.29 and 7.30. Note that the values of CO
and OH are multiplied by 103 and 106, respectively. Except for the near field
location, x/D = 2.5, the mean CO and OH mass fractions are over predicted for
123
7. Lean Premixed Flames Calculation
flame PM1-200 at all axial location as in Fig 7.27 and 7.29. The radial variation
of the mean CO and OH mass fractions are captured well for flame PM1-50. The
mean CO mass fraction for PM1-50 is in very good agreement with the exper-
imental measurements except for location x/D = 25, while the mean OH mass
fraction is in good agreement only in the near field location, x/D = 2.5. The
comparisons between the CMC and measured values are similar to those observed
for transported PDF calculations, as noted in Figs 7.27 to 7.30
The comparison of computed and measured CO shown in Fig 7.28 for flame
PM1-50 is slightly better than those shown by Rowinksi and Pope [5], Dunn et al.
[6] and Dunn [4] from calculations using the PDF methodologies. However, the
comparison shown here for flame PM1-200 is somewhat similar to that reported
in [5] and [4; 6]. It is interesting to note a similar behaviour for the reactive scalar
mean mass fractions and temperatures, despite a vast difference in the modelling
methods used. There is gross over prediction of mean CO and OH mass fractions
and temperature for flame PM1-200. Rowinksi and Pope [5] showed that this is
because the reaction rates in flame PM1-200 are reduced and the reaction rates
have to be slowed down by a factor of 5 in their PDF calculation to obtain a
good match with measured mean mass fraction values and temperature. This
observation seem to apply for the CMC calculations reported here, but further
calculations are required to verify this.
124
7. Lean Premixed Flames Calculation
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
e U U
o
r/D
x/D = 45
CMC
Exp.
PDF, k-ε
PDF, k-ω
0
0.2
0.4
0.6
0.8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
e U U
o
r/D
55
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
e U U
o
r/D
25
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
e U U
o
r/D
35
0
0.2
0.4
0.6
0.8
1
1.2
0 0.4 0.8 1.2 1.6 2
e U U
o
r/D
5
0
0.2
0.4
0.6
0.8
1
1.2
0 0.4 0.8 1.2 1.6 2
e U U
o
r/D
15
Figure 7.21: The mean velocity from the CMC calculations at six axial locations
are compared to the experimental data [4] and PDF calculations, k-ε and k-ω [6]
for flame PM1-200.
125
7. Lean Premixed Flames Calculation
0
0.2
0.4
0.6
0.8
1
1.2
0 0.4 0.8 1.2 1.6 2
e U U
o
r/D
x/D = 25
CMC
Exp.
PDF, k-ε
PDF, k-ω
0
0.2
0.4
0.6
0.8
1
0 0.4 0.8 1.2 1.6 2
e U U
o
r/D
35
0
0.2
0.4
0.6
0.8
1
1.2
0 0.4 0.8 1.2 1.6 2
e U U
o
r/D
5
0
0.2
0.4
0.6
0.8
1
1.2
0 0.4 0.8 1.2 1.6 2
e U U
o
r/D
15
Figure 7.22: The mean velocity from the CMC calculations at four axial locations
are compared to the experimental data [4] and PDF calculations, k-ε and k-ω [6]
for flame PM1-50.
126
7. Lean Premixed Flames Calculation
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5 3
e u
′
U
o
r/D
x/D = 45
CMC
Exp.
PDF, k-ε
PDF, k-ω
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
e u
′
U
o
r/D
55
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5 3
e u
′
U
o
r/D
25
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5 3
e u
′
U
o
r/D
35
0
0.05
0.1
0.15
0.2
0 0.4 0.8 1.2 1.6 2
e u
′
U
o
r/D
5
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5 3
e u
′
U
o
r/D
15
Figure 7.23: The mean rms velocity from the CMC calculations at six axial
locations are compared to the experimental data [4] and PDF calculations, k-ε
and k-ω [6] for flame PM1-200.
127
7. Lean Premixed Flames Calculation
0
0.05
0.1
0.15
0.2
0 0.4 0.8 1.2 1.6 2
e u
′
U
o
r/D
x/D = 25
CMC
Exp.
PDF, k-ε
PDF, k-ω
0
0.05
0.1
0.15
0.2
0 0.4 0.8 1.2 1.6 2
e u
′
U
o
r/D
35
0
0.05
0.1
0.15
0.2
0 0.4 0.8 1.2 1.6 2
e u
′
U
o
r/D
5
0
0.05
0.1
0.15
0.2
0 0.4 0.8 1.2 1.6 2
e u
′
U
o
r/D
15
Figure 7.24: The mean rms velocity from the CMC calculations at four axial
locations are compared to the experimental data [4] and PDF calculations, k-ε
and k-ω [6] for flame PM1-50.
128
7. Lean Premixed Flames Calculation
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
x/D = 45
CMC
Exp.
PDF, k-ε
PDF, k-ω
VFJPDF
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
30
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
15
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
7.5
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
2.5
Figure 7.25: The mean temperature from the CMC calculations at four axial
locations are compared to the experimental data [4] and PDF calculations, k-ε
and k-ω [6] and VFJPDF [5] for flame PM1-200.
129
7. Lean Premixed Flames Calculation
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
x/D = 15
CMC
Exp.
PDF, k-ε
PDF, k-ω
VFJPDF
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
25
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
2.5
300
700
1100
1500
1900
2300
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
T˜
(K
)
r/D
7.5
Figure 7.26: The mean temperature from the CMC calculations at four axial
locations are compared to the experimental data [4] and PDF calculations, k-ε
and k-ω [6] and VFJPDF [5] for flame PM1-50.
130
7. Lean Premixed Flames Calculation
0
5
10
15
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
x/D = 30
CMC
Exp.
PDF, k-ε
PDF, k-ω
VFJPDF
0
3
6
9
12
15
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
45
0
5
10
15
20
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
2.5
0
5
10
15
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
15
Figure 7.27: The mean C˜O×103 from the CMC calculations at four axial locations
are compared to the experimental data [4] and PDF calculations, k-ε and k-ω [6]
and VFJPDF [5] for flame PM1-200.
131
7. Lean Premixed Flames Calculation
0
5
10
15
20
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
x/D = 15
CMC
Exp.
PDF, k-ε
PDF, k-ω
VFJPDF
0
5
10
15
20
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
25
0
5
10
15
20
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
2.5
0
5
10
15
20
0 1 2 3 4 5
Y˜
C
O
∗
1
0
3
r/D
7.5
Figure 7.28: The calculated mean C˜O × 103 mass fraction from CMC calcula-
tions at four axial locations are compared to the experimental data [4] and PDF
calculations, k-ε and k-ω [6] and VFJPDF [5] for flame PM1-50.
132
7. Lean Premixed Flames Calculation
0
200
400
600
800
1000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
x/D = 45
CMC
Exp.
PDF, k-ε
PDF, k-ω
0
200
400
600
800
1000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
30
0
200
400
600
800
1000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
15
0
500
1000
1500
2000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
7.5
0
500
1000
1500
2000
2500
3000
3500
4000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
2.5
Figure 7.29: The calculated mean OH× 106 from CMC calculations at five axial
locations are compared to the experimental data [4] and PDF calculations, k-ε
and k-ω [6] for flame PM1-200.
133
7. Lean Premixed Flames Calculation
0
500
1000
1500
2000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
x/D = 15
CMC
Exp.
PDF, k-ε
PDF, k-ω
0
300
600
900
1200
1500
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
25
0
500
1000
1500
2000
2500
3000
3500
4000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
2.5
0
500
1000
1500
2000
2500
3000
0 1 2 3 4 5
Y˜
O
H
∗
1
0
6
r/D
7.5
Figure 7.30: The calculated mean OH× 106 from CMC calculations at four axial
locations are compared to the experimental data [4] and PDF calculations, k-ε
and k-ω [6] for flame PM1-50.
134
Chapter 8
Summary and Conclusions
8.1 Conclusion From This Work
The over all aim of this study was to explore the conditional moment closure
method for turbulent lean premixed flames. Specifically, to predict pollutants
such as NOx, CO in turbulent lean premixed flames. These species have slow
time scales and thus flamelet type approach may be insufficient. The CMC and
the transported JPDF are alternative approaches. In this study, the CMC is
considered, which was used and validated mainly for non-premixed flames in the
past studies [30; 31; 32; 33; 34; 35; 36]. The main difficulties in applying CMC
to premixed flames is due to the conditional scalar dissipation rate in the CMC
transport equations, see Eq. 3.3. This term is crucial in the CMC method and the
accuracy of the method depends on accurate modelling for this term. The scalar
dissipation rate signifies the local mixing rate of reacted and unreacted mixtures
inside the turbulent flame brush and this mixing rate will dictate the chemical
reaction rate.
In this study, two methods to find a closure for the conditional scalar dissi-
pation rate was tested in chapter 4. One method, see Eq. 4.11, is based on the
135
8. Conclusions
mathematical techniques for finding a solution for ill-posed problem and involves
an integral equation relating the mean dissipation rate to the conditional dissi-
pation rate. The second method uses a simple algebraic model, proposed in an
earlier study [79]. These two models were validated using two different DNS data
sets; one for stoichiometric and another one for lean turbulent premixed flames.
The results showed that the simple algebraic model was in better agreement with
the DNS data and the results obtained using the first method showed a high
sensitivity to the initial values for the conditional dissipation rate. Thus, the
algebraic model is chosen as a closure for the conditional scalar dissipation rate
in this study.
The CMC sub-models for the premixed flames, discussed in chapter 3, were
implemented along with the conditional scalar dissipation rate model in a three-
dimensional CFD code. The CFD code is based on the RANS approach using
the standard k-ε turbulence modelling. This code, originally developed for non-
premixed combustion [127], was modified for the premixed flames by including
transport equations for the progress variable, c˜, and its variance c˜′′2 along with
suitable closures for the source/sink terms, ω˙c, c′′ω˙c and the mean scalar dissipa-
tion rate ²˜c. The closures used for the terms related to the chemical reactions are
consistent with the CMC methodology as described in section 5.2. The RANS-
CMC equations are solved using finite volume methodology. The details are
presented in chapter 5.
The RANS-CMC method is used to compute stoichiometric [2] and lean pilot
[3] stabilized Bunsen flames, five flames in total. A first order closure is used
for the conditional mean reaction rate. The PDF of the progress variable is
obtained using a presumed shape with a Beta function. The GRI-mech. 3.0 and
136
8. Conclusions
the skeletal mechanism of Smooke are used for modelling the chemical kinetics
for the stoichiometric flames to study the effects of chemical mechanisms on the
computed results. The results are presented in chapter 6. The GRI-mech 3.0 is
used for the lean flames and the results are discussed in chapter 7. The turbulence
models and the boundary conditions used are assessed first by simulating the non-
reacting flow where the computed mean velocity and turbulent kinetic energy
agree reasonably well with the measurements for all the flames studied. As for
the reacting flow, the mean velocity and mean turbulent kinetic energy were
satisfactory and acceptable for both stoichiometric and lean flames. The influence
of chemical kinetic mechanism and eQα term on the reacting flow were assessed
using three simulations for the flame F1 of Chen et al.[2]; one using the skeletal
mechanism, the second simulation using the GRI-mech and, the third simulation
using GRI-mech and including eQα term. The results showed that the effect of
eQα was in general negligible and the effect of chemical mechanism was significant
only for minor species.
For the stoichiometric flames reported in [2], the computed mean mass frac-
tion of methane and normalised temperature are in reasonable agreement with
measurements for all the three flames. The predicted major species mass fractions
are also in reasonable agreement with the experimental data. However, the level
of agreement improves as the overall Damko¨hler number increases. As for the mi-
nor species reported in the experiments are concerned, CO is over predicted, H2
is under predicted and OH is observed to agree quite well with the measurements
at all axial locations for all the three flames in [2]. The reason for the discrepan-
cies observed for CO mass fraction is unclear, although the level of disagreement
is similar to that reported in earlier studies [68; 79; 118; 119; 120; 121] using
137
8. Conclusions
other combustion modelling approaches for these experimental flames. This gives
confidence on the models and methodology used in this study.
A comparison of peak CO values at a given axial location show a weak sensi-
tivity to the turbulence Reynolds and Damko¨hler numbers, at least for the range
considered in the experimental studies, but the CMC results suggest a drop in
this value as the Damko¨hler number increases, which is in agreement with our
expectation.
Although the sensitivity of CO to the choice of the progress variable was
shown to be small in an earlier study using flamelets based modelling [79], the
effects of changing the conditioning variable definition on the CMC results are yet
to be explored. The effects of other modelling, for example the counter-gradient
scalar flux, on the CMC prediction need to be studied.
For the lean flames, the computed mean temperature is in good agreement
for the flames PM1-50 for all axial locations. As for the flame PM1-200, the
agreement was better near the fuel exit, while in the down stream distance, the
agreement was less. However, the influence of the pilots gases and the adiabatic
temperature were well captured. The predicted minor species CO and OH mass
fractions were in reasonable agreement in the flame PM1-50, and the agreement
was less for the flame PM1-200. The observed agreement for the PM1-200 flame
was similar to those observed in earlier studies using transported joint PDF ap-
proach [3; 4] and [5].
138
8. Conclusions
8.2 Future Work
There are several points, which can be explored in future studies, some of these
points are:
1. The unconditional and conditional turbulent scalar flux is modeled using
the classical gradient transport hypothesis in this study. As theoretical
analysis and experimental studies pointed out the existence of both gradient
and counter gradient fluxes and their co-existence in turbulent premixed
flames. It is worthwhile to explore CMC with scalar flux models which can
include, both gradient and counter gradient fluxes.
2. The progress variable based on fuel mass fraction is used as the conditioning
variable in this study. Thus, the effects of changing the conditioning variable
would be of some interest.
3. The standard k-ε turbulence model is used in this study. Hence, it is worth
to explore the effects of other turbulence modelling, such as RNG k-ε [128],
k-ω SST, Reynolds stress equation, etc., on the CMC prediction.
4. The effects of radiative heat loss on the CMC prediction.
5. Alternative models for the unconditional and conditional mean scalar dis-
sipation rate for the conditioning variable.
6. Effects of alternative shapes for the PDF of the conditioning variable.
7. Effects of flame geometries on the CMC prediction.
139
8. Conclusions
8. This study examined the premixed CMC for laboratory scale flames. Since,
this gave encouraging results, it is worth to explore this methodology for
practical combustion systems such as stationary gas turbine combustors.
9. Finally, implementing the premixed CMC in LES framework could prove
to be worthwhile, since LES-CMC has been used in many previous studies
for non-premixed flames.
Availability of mean species, temperature and velocity measurements with
clearly specified errors and uncertainties from fully characterised premixed
burners would be instrumental in validating and further developing models
for turbulent lean premixed flames, and to support or unsupport the obser-
vation on the applicability of the CMC across the combustion regimes.
140
Appendix A
CMC equation for premixed combustion
In this section the conditional moment closure equations are derived using the
decomposition method [40]. The instantaneous mass fraction of species α is
decomposed into conditional mean, Qα, and a fluctuation, y
′′
α, as
Yα(xi, t) = Qα(ζ;xi, t) + y
′′
α(xi, t), (1)
where the conditional average is defined as, Qα(ζ;xi, t) = 〈Yα|c = ζ〉, the ensem-
ble average subject to the condition c = ζ. The instantaneous progress variable
is denoted as c, which is governed by
ρ
∂c
∂t
+ ρui
∂c
∂xi
=
∂
∂xi
(
ρDc
∂c
∂xi
)
+ ω˙c (2)
in the usual notation. The transport equation for the instantaneous scalar value,
Yα, in the usual notations is
ρ
∂Yα
∂t
+ ρui
∂Yα
∂xi
=
∂
∂xi
(
ρDα
∂Yα
∂xi
)
+ ω˙α. (3)
141
Substituting Eq. (1) into Eq. (3) gives
∂Yα(xi, t)
∂t
=
∂Qα
∂t
+
∂Qα
∂ζ
∂c
∂t
+
∂y′′α
∂t
, (4)
∂Yα(xi, t)
∂xi
=
∂Qα
∂xi
+
∂Qα
∂ζ
∂c
∂xi
+
∂y′′α
∂xi
, (5)
∂
∂xi
(
ρDα
∂Yα
∂xi
)
=
∂
∂xi
[
ρDα
(
∂Qα
∂xi
+
∂Qα
∂ζ
∂c
∂xi
+
∂y′′α
∂xi
)]
(6)
=
1︷ ︸︸ ︷
∂
∂xi
(
ρDα
∂Qα
∂xi
)
+
2︷ ︸︸ ︷
∂
∂xi
(
ρDα
∂Qα
∂ζ
∂c
∂xi
)
+
3︷ ︸︸ ︷
∂
∂xi
(
ρDα
∂y′′α
∂xi
)
(7)
The second term can written as
2︷ ︸︸ ︷
∂
∂xi
(
ρDα
∂Qα
∂ζ
∂c
∂xi
)
=
4︷ ︸︸ ︷
∂Qα
∂ζ
∂
∂xi
(
ρDα
∂c
∂xi
)
+
5︷ ︸︸ ︷(
ρDα
∂c
∂xi
)
∂
∂xi
∂Qα
∂ζ
(8)
The fifth term can written as
5︷ ︸︸ ︷(
ρDα
∂c
∂xi
)
∂
∂xi
∂Qα
∂ζ
=
6︷ ︸︸ ︷
ρDα
(
∂c
∂xi
∂c
∂xi
)
∂2Qα
∂ζ2
+
7︷ ︸︸ ︷
ρDα
∂c
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
(9)
Now the diffusion term in Eq. (3) is the sum of the terms 1, 3, 4, 6 and 7
∂
∂xi
(
ρDα
∂Yα
∂xi
)
=
1︷ ︸︸ ︷
∂
∂xi
(
ρDα
∂Qα
∂xi
)
+
4︷ ︸︸ ︷
∂Qα
∂ζ
∂
∂xi
(
ρDα
∂c
∂xi
)
+
142
6︷ ︸︸ ︷
ρDα
(
∂c
∂xi
∂c
∂xi
)
∂2Qα
∂ζ2
+
7︷ ︸︸ ︷
ρDα
∂c
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
+
3︷ ︸︸ ︷
∂
∂xi
(
ρDα
∂y′′α
∂xi
)
(10)
Substituting Eqs. (4), (5) and (10) into Eq. (3) yields
ρ
∂Qα
∂t
+ ρui
∂Qα
∂xi
− ∂
∂xi
(
ρDα
∂Qα
∂xi
)
− ρDα
(
∂c
∂xi
∂c
∂xi
)
∂2Qα
∂ζ2
−
ρDα
∂c
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
+
∂Qα
∂ζ
[
ρ
∂c
∂t
+ ρui
∂c
∂xi
− ∂
∂xi
(
ρDα
∂c
∂xi
)]
+
ρ
∂y′′α
∂t
+ ρui
∂y′′α
∂xi
− ∂
∂xi
(
ρDα
∂y′′α
∂xi
)
− ω˙α = 0. (11)
After some arrangement, the progress variable equation, Eq. (2), is rewritten as
ρ
∂c
∂t
+ ρui
∂c
∂xi
− ∂
∂xi
(
ρDα
∂c
∂xi
)
=
∂
∂xi
[
ρ (Dc −Dα) ∂c
∂xi
]
+ ω˙c. (12)
Substituting Eq. (12) into Eq. (11) yields
ρ
∂Qα
∂t
+ ρui
∂Qα
∂xi
− ∂
∂xi
(
ρDα
∂Qα
∂xi
)
− ρDα
(
∂c
∂xi
∂c
∂xi
)
∂2Qα
∂ζ2
−
ρDα
∂c
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
+
∂Qα
∂ζ
[
∂
∂xi
[
ρ (Dc −Dα) ∂c
∂xi
]
+ ω˙c
]
+
ρ
∂y′′α
∂t
+ ρui
∂y′′α
∂xi
− ∂
∂xi
(
ρDα
∂y′′α
∂xi
)
− ω˙α = 0. (13)
Taking conditional average of Eq. (13) yields,
〈ρ|ζ〉∂Qα
∂t
+ 〈ρui|ζ〉∂Qα
∂xi
−
〈
ρDα
∂c
∂xi
∂c
∂xi
∣∣∣ζ〉 ∂2Qα
∂ζ2
=〈[
∂
∂xi
(
ρDα
∂Qα
∂xi
)
+ ρDα
∂c
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
− ∂Qα
∂ζ
∂
∂xi
(
ρ (Dc −Dα) ∂c
∂xi
)] ∣∣∣ζ〉
143
−
〈[
ρ
∂y′′α
∂t
+ ρui
∂y′′α
∂xi
− ∂
∂xi
(
ρDα
∂y′′α
∂xi
)] ∣∣∣ζ〉 + 〈ω˙α|ζ〉 − ∂Qα
∂ζ
〈ω˙c|ζ〉. (14)
It is to be taken that density weighted conditional averaging [40] is used and
the rules of differentiating the conditional averages as ex-positional in [129] are
employed. Rearranging the above equation gives
〈ρ|ζ〉∂Qα
∂t
+ 〈ρui|ζ〉∂Qα
∂xi
− Lec
Leα
〈ρNc|ζ〉∂
2Qα
∂ζ2
= 〈ω˙α|ζ〉 − 〈ω˙c|ζ〉∂Qα
∂ζ
+
eyα + eQα , (15)
where
eyα ≡ −
〈[
ρ
∂y′′α
∂t
+ ρui
∂y′′α
∂xi
− ∂
∂xi
(
ρDα
∂y′′α
∂xi
)] ∣∣∣ζ〉 (16)
and
eQα ≡
〈
∂
∂xi
(
ρDα
∂Qα
∂xi
)
+ ρDα
∂c
∂xi
∂
∂xi
(
∂Qα
∂ζ
)
|ζ
〉
+〈
∂Qα
∂ζ
∂
∂xi
[(
1− Leα
Lec
)
ρDα
∂c
∂xi
] ∣∣∣ζ〉 , (17)
The conditional dissipation rate is given by Nc = 〈Dc(∇c · ∇c)|ζ〉 and 〈ρNc|ζ〉 =
〈ρ|ζ〉〈Nc|ζ〉 [130]. The Lewis number of species α is denoted by Leα.
144
Bibliography
[1] A. R. Masri., R. W. Dibble., and R. S. Barlow. The structure of turbulent
non-premixed flames of methanol over a range of mixing rates. Combust.
Flame, 89(2):167–185, 1992. x, 42
[2] Y. C. Chen, N. Peters, G. A. Schneemann, N. Wruck, U. Renz, and M. S.
Mansour. The detailed flame structure of highly stretched turbulent pre-
mixed Methane-air flames. Combust. Flame, 3:223–226, 1996. xi, 6, 70, 71,
72, 74, 75, 136, 137
[3] M. J. Dunn, A. R. Masri, and R. W. Bilger. A new piloted premixed
jet burner to study strong finite-rate chemistry effects. Combust. Flame,
151(1-2):46 – 60, 2007. xiii, 6, 102, 103, 106, 136, 138
[4] M. J. Dunn. Finite-rate chemistry effects in turbulent premixed combustion.
PhD thesis, The University of Sydney, Australia, 2008. xiii, xiv, xv, 103,
108, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 138
[5] D. H. Rowinski and S. B. Pope. PDF calculations of piloted premixed jet
flames. Combust. Theory Modelling, 15(2):245–266, 2011. xiv, xv, 104, 118,
123, 124, 129, 130, 131, 132, 138
145
BIBLIOGRAPHY
[6] M. J. Dunn, A. R. Masri, R. W. Bilger, and R. S. Barlow. Finite rate
chemistry effects in highly sheared turbulent premixed flames. Flow Turb.
Combust., 85:621–648, 2010. xiv, xv, 104, 108, 122, 123, 124, 125, 126, 127,
128, 129, 130, 131, 132, 133, 134
[7] R. A. Houghton. The contemporary Carbon cycle. In D. H. Holland and
K. K. Turekian, editors, Treatise on Geochemistry, pages 473 – 513. Perg-
amon, Oxford, 2003. 2
[8] M. Maslin. Global warming: Causes, effects and the future. LLC and
Voyageur Press, Minneapolis, USA, 2007. 2
[9] C. N. Lane. Acid rain: Overview and Abstracts. Nova Science Publisher,
inc, New York, USA, 2003. 2
[10] European Commission Statistics. Renewable energy statistics, September
2010. 2
[11] S. M. Correa. A review of NOx formation under gasturbine combustion
conditions. Combust. Sci. Tech., 87:329362, 1992. 3
[12] J. B. Heywood. Pollutant formation and control in sparkignition engines.
Prog. Energy Combust. Sci., 1:135164, 1976. 3
[13] J. F. Driscoll. Turbulent premixed combustion: Flamelet structure and its
effect on turbulent burning velocities. Prog. Energy Combust. Sci., 34(1):91
– 134, 2008. 3, 28
[14] B. S. Brewster, S. M. Cannon, J. R. Farmer, and F. Meng. Modeling of lean
146
BIBLIOGRAPHY
premixed combustion in stationary gas turbines. Prog. Energy Combust.
Sci., 25(4):353 – 385, 1999. 3
[15] Y. Huang and V. Yang. Dynamics and stability of lean-premixed swirl-
stabilized combustion. Prog. Energy Combust. Sci., 35(4):293 – 364, 2009.
3
[16] N. Swaminathan and K. N. C. Bray. Turbulent premixed flames. Cambridge
University Press, Cambridge, UK, 2011. 3, 8, 16, 17, 21, 23, 25, 27, 31, 34,
36, 41
[17] D. Veynante and L. Vervisch. Turbulent combustion modelling. Prog.
Energy Combust. Sci., 28(3):193 – 266, 2002. 3, 5, 14, 26, 34, 43, 49
[18] D. Dunn Rankin. Lean combustion technology and control. Elsevier, Cali-
fornia, USA, 1st. edition, 2008. 3
[19] T. Lieuwen, H. Torres, C. Johnson, and B. T. Zinn. A mechanism of
combustion instability in lean premixed gas turbine combustors. J. Eng.
Gas Turb. Power, 123(1):182–189, 2001. 3
[20] T. Echekki and E. Mastorakos. Turbulent Combustion Modeling: Advances,
new trends and perspectives. Springer, New York, USA, 2010. 4, 26, 34
[21] H. Tennekes and J. L. Lumley. A first course in turbulence. The Mas-
sachusetts Inst. of Technology, Boston, USA, 1972. 4
[22] T. Poinsot and D. Veyante. Theoretical and numerical combustion. West-
view Press, Philadelphia, USA, 2nd. edition, 2005. 4, 8, 10, 13, 14, 17, 18,
21, 22, 23, 25, 26, 34
147
BIBLIOGRAPHY
[23] S. B. Pope. Turbulent flows. Cambridge University Press, Cambridge, UK,
2000. 4, 8, 10, 22
[24] F. A. Williams. Turbulent combustion. In J. D. Buckmaster, editor, The
mathematics of combustion, pages 97–132. Society for Industrial and Ap-
plied Mathematics (SIAM), Philadelphia, USA, 1985. 5, 26, 37, 49
[25] R. W. Bilger. Future progress in turbulent combustion research. Prog.
Energy Combust. Sci., 26:367–380, 2000. 5, 8
[26] R. W. Bilger, S. B. Pope, K. N. C. Bray, and J. F. Driscoll. Paradigms in
turbulent combustion research. Proc. Combust. Inst., 30:21–42, 2005. 5, 8,
9
[27] R. W. Bilger. Conditional Moment Closure for turbulent reacting flow.
Phys. Fluids A, 5(2):436–444, 1993. 6, 38, 39, 41, 43
[28] A. Yu. Klimenko. Multi-component diffusion of various admixtures in tur-
bulent flow. Fluid Dynamics., 25:327–334, 1990. 6, 41, 43
[29] S. B. Pope. PDF methods for turbulent reactive flows. Prog. Energy Com-
bust. Sci., 11(2):119 – 192, 1985. 6, 35, 49
[30] M. J. Cleary, J. H. Kent, and R. W. Bilger. Prediction of Carbon Monoxide
in fires by Conditional Moment Closure. Proc. Combust. Inst., 29:273–279,
2002.Adelaide, Australia,. 6, 41, 135
[31] I. S. Kim and E. Mastorakos. Simulations of turbulent lifted jet flames
with two-dimensional conditional Moment Closure. Proc. Combust. Inst.,
30(1):911 – 918, 2005. 6, 41, 135
148
BIBLIOGRAPHY
[32] J. W. Rogersson. Measurements and modelling of a Bagasse-fired boiler.
PhD thesis, The University of Sydney, Sydney, Australia, 2006. 6, 41, 45,
61, 69, 135
[33] S. H. Kim, K. Y. Huh, and L. Tao. Application of the elliptic Conditional
Moment Closure model to a two-dimensional non-premixed methanol bluff-
body flame. Combust. Flame, 120:75–90, 1999. 6, 41, 135
[34] S. Sreedhara and Kang. Y. Huh. Modelling of turbulent two-dimensional
non premixed Methane-Hydrogen flame over a bluff-body using first and
second order elliptic Conditional Moment Closure. Combust. Flame,
143:119–134, 2005. 6, 41, 135
[35] Y. M. Wright, G. De Paola, K. Boulouchos, and E. Mastorakos. Simulations
of spray autoignition and flame establishment with two-dimensional CMC.
Combust. Flame, 143(4):402 – 419, 2005. 6, 41, 135
[36] Y. Yunardi, R. M. Woolley, and M. Fairweather. Conditional Moment
Closure prediction of soot formation in turbulent non-premixed ethylene
flames. Combust. Flame, 152:360–376, 2008. 6, 41, 135
[37] F. A. Williams. Combustion theory. Perseus Books, USA, 2nd. edition,
1985. 8, 10
[38] J. Warnatz, U. Maas, and R. W. Dibble. Combustion physical and chemical
fundamentals, modelling and simulation, experiments, pollutant formation.
Springer, New York, USA, 4th. edition, 2006. 8, 10
149
BIBLIOGRAPHY
[39] E. Mastorakos and R. S. Cant. An introduction to turbulent reacting flows.
Imperial College Press, London, UK, 2008. 8
[40] R. W. Bilger and A.Y. Klimenko. Conditional Moment Closure for turbu-
lent combustion. Prog. Energy Combust. Sci., 25:595–687, 1999. 14, 38, 40,
41, 43, 45, 46, 47, 49, 71, 141, 144
[41] A. Neophytou, E. Mastorakos, and R.S. Cant. DNS of spark ignition and
edge flame propagation in turbulent droplet-laden mixing layers. Combust.
Flame, 157(6):1071 – 1086, 2010. 17
[42] P. Schroll, A. P. Wandel, R. S. Cant, and E. Mastorakos. Direct Numerical
Simulations of autoignition in turbulent two-phase flows. Proc. Combust.
Inst., 32(2):2275 – 2282, 2009. 17
[43] M. Klein, N. Chakraborty, and R. Cant. Effects of turbulence on self-
sustained combustion in premixed flame kernels: A Direct Numerical Sim-
ulation DNS study. Flow Turb. Combust., 81:583–607, 2008. 17
[44] J. G. and Wissink. DNS of separating, low Reynolds number flow in a
turbine cascade with incoming wakes. Int. J. Heat and Fluid Flow, 24(4):626
– 635, 2003. 17
[45] A. Triantafyllidis, E. Mastorakos, and R. L. G. M. Eggels. Large Eddy
Simulations of forced ignition of a non-premixed bluff-body Methane flame
with Conditional Moment Closure. Combust. Flame, 156(12):2328 – 2345,
2009. 18
[46] M. W. A. Pettit, B. Coriton, A. Gomez, and A. M. Kempf. Large Eddy
150
BIBLIOGRAPHY
Simulation and experiments on non-premixed highly turbulent opposed jet
flows. Proc. Combust. Inst., 33(1):1391 – 1399, 2011. 18
[47] V. D. Sarli, A. D. Benedetto, and G. Russo. Sub-grid scale combustion
models for Large Eddy Simulation of unsteady premixed flame propagation
around obstacles. J. Hazardous Materials, 180(1-3):71 – 78, 2010. 18
[48] Y. Y. Wu, C. K. Chan, and L. X. Zhou. Large Eddy Simulation of an
ethyleneair turbulent premixed v-flame. J. Computational Applied Math.,
235(13):3768 – 3774, 2011. 18
[49] P. Wang and X. S. Bai. Large Eddy Simulation of turbulent premixed
flames using level-set G-equation. Proc. Combust. Inst., 30(1):583 – 591,
2005. 18
[50] W. C. Reynolds. Computation of turbulent flows. Annual Review Fluid
Mech., 8(1):183–208, 1976. 22
[51] W. P. Jones and B. E. Launder. The prediction of laminarization with a two-
equation model of turbulence. Int. J. Heat and Mass Transfer, 15(2):301 –
314, 1972. 22, 23
[52] W. P. Jones. Turbulence modelling and numerical solution methods for
variable density and combusting flows. In P. A. Libby and F. A. Williams,
editors, Turbulent reacting flows, pages 309 – 368. Academic Press, London,
1994. 23, 26
[53] O. R. Darbyshire. Modelling of turbulent stratified flames. PhD thesis, The
University of Cambridge, Cambridge, UK, 2011. 23
151
BIBLIOGRAPHY
[54] B. E. Launder, G. J. Reece, and W. Rodi. In the development of a Reynolds
stress turbulence closure. J. Fluid Mech., 41:537–566, 1975. 23
[55] C. G. Spezilae and T. B. Gatski. Modelling the pressure-strain correlation
of turbulence:an invariant dynamical systems approach. J. Fluid Mech.,
227:245–272, 1991. 23
[56] D. B. Taulbee. An improved algebraic Reynolds stress model and corre-
sponding non-linear stress model. Phys. Fluids A, 4:2555–2560, 1992. 23
[57] N. Swaminathan, R.W. Bilger, and B. Cuenot. Relationship between turbu-
lent scalar flux and conditional dilatation in premixed flames with complex
chemistry. Combust. Flame, 126(4):1764 – 1779, 2001. 24
[58] K. N. C. Bray. Turbulent transport in flames. Proc. Royal Soc. London.
Series A: Math. Phys. Sci., 451(1941):231–256, 1995. 24, 31
[59] K.N.C. Bray, Paul A. Libby, and J.B. Moss. Unified modeling approach
for premixed turbulent combustionpart I: General formulation. Combust.
Flame, 61(1):87 – 102, 1985. 24
[60] K. N. C. Bray, M. Champion, and P. A. Libby. Premixed flames in stag-
nating turbulence part IV: A new theory for the Reynolds stresses and
Reynolds fluxes applied to impinging flows. Combust. Flame, 120(1-2):1 –
18, 2000. 24
[61] K. N. C. Bray, P. A. Libby, G. Masuya, and J. B. Moss. Turbulence produc-
tion in premixed turbulent flames. Combust. Sci. Tech., 25(3-4):127–140,
1981. 24
152
BIBLIOGRAPHY
[62] J. B. Moss. Simultaneous measurements of concentration and velocity in
an open premixed turbulent flame. Combust. Sci. Tech., 22(3-4):119–129,
1980. 24
[63] R. K. Cheng and I. G. Shepherd. The influence of burner geometry on
premixed turbulent flame propagation. Combust. Flame, 85(1-2):7 – 26,
1991. 24
[64] I. G. Shepherd, J. B. Moss, and K. N. C. Bray. Turbulent transport in a
confined premixed flame. 19th Symp Int Combust., 19(1):423 – 431, 1982.
24
[65] J. H. Frank, P. A.M. Kalt, and R. W. Bilger. Measurements of conditional
velocities in turbulent premixed flames by simultaneous OH PLIF and PIV.
Combust. Flame, 116(1-2):220 – 232, 1999. 24
[66] K. N. C. Bray D. Veynante, A. Trouve and T. Mantel. Gradient and
counter-gradient scalar transport in turbulent premixed flames. J. Fluid
Mech., 332:263–293, 1997. 24, 25
[67] N. Swaminathan, R. W. Bilger, and G. R. Ruetsch. Interdependence of the
instantaneous flame front structure and the overall scalar flux in turbulent
premixed flames. Combust. Sci. Tech., 128(1-6):73–97, 1997. 24
[68] R. P. Lindstedt and E. M. Vaos. Transported PDF modeling of high-
Reynolds number premixed turbulent flames. Combust. Flame, 145:495–
511, 2006. 25, 70, 89, 137
[69] N. Swaminathan and K. N. C. Bray. In N. Swaminathan and K. N. C.
153
BIBLIOGRAPHY
Bray, editors, Turbulent premixed flames, pages 1–33. Cambridge University
Press, Cambridge, UK, 2011. 26
[70] N. Peters. Turbulent combustion. Cambridge University Press, Cambridge,
UK. 27, 28, 30, 37
[71] R. S. Cant and E. Mastorakos. Introduction to turbulent reacting flows.
Imperial College Press, London, UK, 2007. 27, 30
[72] D. B. Spalding. Mixing and chemical reaction in steady confined turbulent
flames. Proc. Combust. Inst, 13:649–657, 1970. 29
[73] K. N. C. Bray and P. A. Libby. Interaction effects in turbulent premixed
flames. Phys. Fluids, 19:1687–1701, 1976. 30
[74] K. N. C. Bray and J. B. Moss. A unified statistical model of the premixed
turbulent flame. Acta Astronautica, 4:291–319, 1977. 30, 31
[75] K. N. C. Bray. Turbulent flows with premixed reactants. In P. A. Libby and
F. A. Williams, editors, Turbulent Reacting Flows, page 115183. Springer-
Verlag, New York, 1980. 30, 31
[76] K. N. C. Bray. The interaction between turbulence and combustion. 17th
Symp. Int. Combust. The Combustion Institute, Pittsburgh, 17(1):223–223,
1979. 31
[77] K. N. C. Bray, P. A. Libby, and J. B. Moss. Flamelet crossing frequencies
and mean reaction rates in premixed turbulent combustion. Combust. Sci.
Tech., 41:143–172, 1984. 32
154
BIBLIOGRAPHY
[78] D. Bradley. How fast can we burn? 24th Symp. Int. Combust. The Com-
bustion Institute, University of Sydney, Australia, 24(1):247 – 262, 1992.
32
[79] H. Kolla and N. Swaminathan. Strained flamelets for turbulent premixed
flames I: Formulation and planar flame results. Combust. Flame, 157:943–
954, 2010. 33, 41, 49, 51, 52, 70, 73, 89, 136, 137, 138
[80] H. Kolla and N. Swaminathan. Strained flamelets for turbulent premixed
flames II: Laboratory flame results. Combust. Flame, 157:12741289, 2010.
33, 45
[81] F. E. Marble and J. E Broadwell. The coherent flame model for turbu-
lent chemical reactions. Project SQUID, technical report TRW-9-PU, 1977.
Purdue University. 33
[82] S. M. Candel and T. J. Poinsot. Flame stretch and the balance equation
for the flame area. Combust. Sci. Tech., 70:1–15, 1990. 33, 34
[83] S. B. Pope. The evolution of surfaces in turbulence. Int. J. Eng. Sci.,
26(5):445 – 469, 1988. 33, 34
[84] K. N. C. Bray and P. A. Libby. Recent development in the BML model of
premixed turbulent combustion. In P. A. Libby and F. A. Williams, editors,
Turbulent reacting flows, pages 115 – 147. Academic Press, London, 1994.
33, 34
[85] A. Trouve and T. Poinsot. The evolution equation for the flame surface
density. J. Fluid Mech., 278:1 – 31, 1994. 33, 34
155
BIBLIOGRAPHY
[86] E. V. Kalmthout and D. Veynante. Direct Numerical Simulations analysis
of flame surface density models for non premixed turbulent combustion.
Phys. Fluids, 10(9):2347, 1998. 33
[87] E. E. OBrien. The probability density function PDF approach to reacting
turbulent flows. Turbulent Reacting Flows, pages 185–218, 1980. 35
[88] D. C. Haworth. Progress in probability density function methods for tur-
bulent reacting flows. Prog. Energy Combust. Sci., 36(2):168 – 259, 2010.
36
[89] D. C. Haworth and S. B. Pope. Transported Probability Density Func-
tion Methods for Reynolds-Averaged and Large-Eddy Simulations. In
Tarek Echekki and Epaminondas Mastorakos, editors, Turbulent Combus-
tion Modeling, volume 95, pages 119–142. Springer, New York, USA, 2011.
36
[90] R. P. Lindstedt. Transported probability density function methods for pre-
mixed turbulent flames. In N. Swaminathan and K. N. C. Bray, editors,
Turbulent premixed flames, pages 102–133. Cambridge University Press,
Cambridge, UK, 2011. 36
[91] S. S. Girimaji. Assumed β-PDF model for turbulent mixing: Validation and
extension to multiple scalar mixing. Combust. Sci. Tech., 78(4-6):177–196,
1991. 36
[92] K. N. C. Bray and N. Peters. Laminar flamelets in turbulent flames. In
P. A Libby and F. A. Williams, editors, Turbulent Reacting Flows, pages
63–113. Academic, London, UK, 1994. 37
156
BIBLIOGRAPHY
[93] F. A. Williams. Some recent studies in turbulent combustion. In A. Yoshida,
editor, Smart control of turbulent combustion, pages 1–12. Springer-Verlag,
Tokyo, Japan, 2001. 37
[94] S. M. Martin. An improved Conditional Moment Closure model. pre-
sented at the Third Joint Meeting of the U.S. Sections of the Combus.
Inst., Chicago, 2003. 41
[95] S. M. Martin, J. C. Kramlich, G. Kosaly, and J. J. Riley. The premixed
Conditional Moment Closure method applied to idealized lean premixed
gas turbine combustors. J. Eng. Gas Turb. Power, 125(4):895–900, 2003.
41
[96] N. Swaminathan and R. W. Bilger. Analysis of conditional Moment Clo-
sure for turbulent premixed flames. Combust. Theory Modelling, 5:241–260,
2001. 41, 43, 46, 47
[97] N. Swaminathan and K.N.C. Bray. Effect of dilatation on scalar dissipation
in turbulent premixed flames. Combust. Flame, 143(4):549 – 565, 2005. 41,
49, 50
[98] H. Kolla, J. W. Rogerson, N. Chakraborty, and N. Swaminathan. Scalar dis-
sipation rate modelling and its validation. Combust. Sci. Tech., 181(3):518–
535, 2009. 41, 49, 50
[99] N. Chakraborty, J. W. Rogerson, and N. Swaminathan. A prior assessment
of closures for scalar dissipation rate transport in turbulent premixed flames
using Direct Numerical Simulation. Phys. Fluids, 20(4):045106, 2008. 41,
50
157
BIBLIOGRAPHY
[100] E. S. Richardson, N. Chakraborty, and E. Mastorakos. Analysis of Di-
rect Numerical Simulations of ignition fronts in turbulent non-premixed
flames in the context of Conditional Moment Closure. Proc. Combust.
Inst., 31(1):1683 – 1690, 2007. 45
[101] A. Simon and E. Mastorakos. Conditional Moment Closure/Large Eddy
Simulation of the Delft-III natural gas non-premixed jet flame. Flow Turb.
Combust., 88:207–231, 2012. 45
[102] N. Peters. Laminar diffusion flamelet models in non-premixed turbulent
combustion. Prog. Energy Combust. Sci., 10:319–339, 1984. 49
[103] S. Sreedhara, Y. Lee, Kang Y. Huh, and D.H. Ahn. Comparison of sub
models for conditional velocity and scalar dissipation in CMC simulation
of piloted jet and bluff-body flames. Combust. Flame, 152(1-2):282 – 286,
2008. 49
[104] P. A. Libby and F. A. Williams. Fundamental aspects and a review. In
P. A. Libby and F. A. Williams, editors, Turbulent reacting flows, pages 1
– 62. Academic Press, London, 1994. 49
[105] Y. C. Chen and M. S. Mansour. Measurements of scalar dissipation in
turbulent Hydrogen diffusion flames and some implications on combustion
modeling. Combust. Sci. Tech., 126(1-6):291–313, 1997. 49
[106] S. H. Starner, R. W. Bilger, M. B. Long, J. H. Frank, and D. F. Marran.
Scalar dissipation measurements in turbulent jet diffusion flames of air di-
luted Methane and Hydrogen. Combust. Sci. Tech., 129(1-6):141–163, 1997.
49
158
BIBLIOGRAPHY
[107] F. O’Young and R. W. Bilger. Measurement of scalar dissipation in pre-
mixed flames. Combust. Sci. Tech.., 113(1):393–411, 1996. 49
[108] N. Swaminathan and R. W. Bilger. Scalar dissipation, diffusion and dilation
in turbulent H2-air premixed flames with complex chemistry. Combust.
Theory Model., 5(1):429–446, 2001. 49
[109] T. Mantel and R. W. Bilger. Some conditional statistics in a turbulent
premixed flame derived from Direct Numerical Simulations. Combust. Sci.
Tech., 110(1):393–417, 1995. 49
[110] N. Chakraborty, M. Chapion, A. Mura, and N. Swaminathan. Scalar dissi-
pation rate approach. In N. Swaminathan and K. N. C. Bray, editors, Tur-
bulent premixed flames, pages 74–102. Cambridge University Press, Cam-
bridge, UK, 2011. 50
[111] A. N. Tikhonov, A. Goncharsky, V. V. Stepanov, and A. G. Yagola. Nu-
merical method for the solution of ill-posed problems. Kluwer Academic
Publishers, Moscow, Russia, 1990. 52, 53
[112] T. Dunstan, N. Swaminathan, K. N. C. Bray, and S. R. Cant. Geometrical
properties and turbulent flame speed measurements in stationary premixed
v-flames using Direct Numerical Simulation. Flow Turb. Combust., 87:237–
259, 2011. 54
[113] J. W. Rogerson, N. Swaminathan, M. Tanahashi, and N. Shiwaku. Analysis
of progress variable variance equations using DNS data. Proc. European
Meeting, 2007. 55
159
BIBLIOGRAPHY
[114] Y. Nada, M. Tanahashi, and T. Miyauchi. Effect of turbulence character-
istics on local flame structure of H2-air premixed flames. J. Turb., 5:16,
2004. 55
[115] A. N. Tikhonov, A. V. Goncharsky, and A. G. Yagola. Algorithms and pro-
grams for the solution of linear ill-posed problems.Moscow State University-
Faculty of Physics. http://foroff.phys.msu.ru/Illposed/eng/, 1995. 58
[116] S. V. Patankar. Numerical heat transfer and fluid flow. Hemisphere Pub-
lishing Corporation, Washington, USA, 1980. 61, 63, 68
[117] J. W. Rogerson, J. H. Kent, and R. W. Bilger. Conditional Moment Closure
in bagasse-fired boiler. Proc. Combust. Ins., 31:2805–2811, 2007. 66, 69
[118] A. Mura, F. Galzin, and R. Borghi. A unified PDF-flamelet model for
turbulent premixed combustion. Combust. Sci. Tech., 175(9):1573–1609,
2003. 70, 137
[119] R. O. P. Prasad and J. P. Gore. An evaluation of flame surface density
models for turbulent premixed jet flames. Combust. Flame, 16:1–14, 1999.
70, 89, 137
[120] M. Herrmann. Numerical simulation of turbulent bunsen flames with a level
set flamelet model. Combust. Flame, 145:357–375, 2006. 70, 89, 137
[121] G. Wang, M. Boileau, and D. Veynante. Implementation of a dynamic
thickened flame model for large Eddy Simulations of turbulent premixed
combustion. Combust. Flame, 158(11):2199 – 2213, 2011. 70, 89, 137
160
BIBLIOGRAPHY
[122] S. B. Pope. An explanation of the turbulent round-jet/plane-jet anomaly.
AIAA J., 16(3):279 – 281, 1978. 74
[123] G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriarty, B. Eiteneer,
M. Goldenberg, C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, Jr.,
V. V. Lissianski, and Z. Qin. Gri-mech 3.0. The Gas Research Inst., 2011.
GRI-Mech 3.0 available at: http://www.me.berkeley.edu/grimech. 75, 108
[124] M. D. Smooke. Lecture notice in physics. pringer-Verlag, 1991. 75
[125] M. J. Dunn, A. R. Masri, R. W. Bilger, R. S. Barlow, and G. H. Wang. The
compositional structure of highly turbulent piloted premixed flames issuing
into a hot co-flow. Proc. Combust. Inst., 32(2):1779–1786. 104
[126] C. Duwig, K. J. Nogenmyr, C. K. Chan, and M. J. Dunn. Large Eddy
Simulations of a piloted lean premix jet flame using finite-rate chemistry.
Combust. Theory Modelling, 15(4):537–568, 2011. 104
[127] J. W. Rogerson. Measurements and Modelling of a Bagasse-Fired Boiler.
PhD thesis, The University of Sydney, Australia, 2006. 136
[128] V. Yakhot, S. A. Orszag, S. Thangam, T. B. Gatski, and C. G. Speziale.
Development of turbulence models for shear flows by a double expansion
technique. Phys. Fluids A: Fluid Dynamics, 4(7):1510–1520, 1992. 139
[129] A. Yu. Klimenko and R. W. Bilger. Conditional Moment Closure for tur-
bulent combustion. Prog. Energy Combust. Sci., 25:595–687, 1999. 144
[130] N. Swaminathan and R. W. Bilger. Scalar dissipation, diffusion and dilata-
161
BIBLIOGRAPHY
tion in turbulent H2-air premixed flames with complex chemistry. Combust.
Theory Model., 5(3):429–446, 2001. 144
162