Simulation of turbulent flames
relevant to spark-ignition engines
Irufan Ahmed
Hughes Hall
University of Cambridge
A dissertation submitted for the degree of
Doctor of Philosophy
January 2014
Declaration
This dissertation is the result of my own work and includes noth-
ing which is the outcome of work done in collaboration except where
specifically indicated in the text. This dissertation has not been sub-
mitted in any form for another qualification to this or any other uni-
versity. Information derived from the published and unpublished work
of others has been referenced in the text.
This dissertation contains approximately 46,000 words and includes
52 figures.
Irufan Ahmed
April 14th, 2014
Keywords: computational fluid dynamics, turbulent combustion, spark-
ignition engines
To my parents
Acknowledgements
I would like to express my gratitude towards my supervisor, Professor
N. Swaminathan, for introducing me to the fascinating world of com-
putational combustion. His patience, encouragement and guidance
throughout the course of this work contributed to the successful com-
pletion of this thesis. I would also like to thank Professors Forman
Williams, Ken Bray and Stewart Cant for their insightful comments
regarding this work.
Special thanks to Dr Yuri Wright, Ms Ste´phanie Schlatter and Mr
Jann Koch from ETH Zu¨rich for the STAR-CD setup and for provid-
ing the computational mesh used to simulate the internal combustion
engine. I am also thankful for the assistance I received from Dr He-
manth Kolla regarding the in-house CFD code, and Mr Karl Bass
from CD-adapco for his technical support on STAR-CD.
Many others have contributed to the success of this work, including
colleagues from the Hopkinson laboratory with whom I have had nu-
merous interesting and fruitful discussions. In addition, the comput-
ing help I received from Mr Peter Benie and the staff at the Cambridge
High Performance Computing Service is gratefully acknowledged.
This work would not have been possible without the generous finan-
cial support from my uncle, Mr Hussain Abdullah. I would also like
to acknowledge the conference grants I received from the Cambridge
University Engineering Department, the Combustion Institute and
Hughes Hall. The financial support I received from Tokyo Institute
of Technology to participate in the ACEEES forum is gratefully ac-
knowledged.
Abstract
Combustion research currently aims to reduce emissions, whilst im-
proving the fuel economy. Burning fuel in excess of air, or lean-burn
combustion, is a promising alternative to conventional combustion,
and can achieve these requirements simultaneously. However, lean-
burn combustion poses new challenges, especially for internal combus-
tion (IC) engines. Therefore, models used to predict such combustion
have to be reliable, accurate and robust.
In this work, the flamelet approach in the Reynolds-Averaged Navier-
Stokes framework, is used to simulate flames relevant to spark-ignition
IC engines. A central quantity in the current modelling approach is
the scalar dissipation rate, which represents coupling between reaction
and diffusion, as well as the flame front dynamics.
In the first part of this thesis, the predictive ability of two reaction rate
closures, viz. strained and unstrained flamelet models, are assessed
through a series of experimental test cases. These cases are: spheri-
cally propagating methane- and hydrogen-air flames and combustion
in a closed vessel. In addition to these models, simpler algebraic clo-
sures are also used for comparison.
It is shown that the strained flamelet model can predict unconfined,
spherically propagating methane-air flames reasonably well. By com-
paring spherical flame results with planar flames, under identical ther-
mochemical and turbulence conditions, it is shown that the turbulent
flame speed of spherical flames are 10 to 20% higher than that of pla-
nar flames, whilst the mean reaction rates are less influenced by the
flame geometry.
Growth of the flame brush thickness in unsteady spherical flames have
been attributed to turbulent diffusion in past studies. However, the
present analyses revealed that the dominant cause for this increase is
the heat-release induced convective effects, which is a novel observa-
tion.
Unlike methane-air flames, hydrogen-air flames have non-unity Lewis
numbers. Hence, a novel two degrees of freedom approach, using
two progress variables, is used to describe the thermochemistry of
hydrogen-air flames. Again, it is shown that the strained flamelet
model is able to predict the experimental flame growth for stoichio-
metric hydrogen-air flames. However, none of the models used in this
work were able to predict lean hydrogen-air flames. This is because
these flames are thermo-diffusively unstable and the current approach
is inadequate to represent them.
When combustion takes place inside a closed vessel, the compression
of the end gases by the propagating flame causes the pressure to rise.
This is more representative of real IC engines, where intermittent
combustion takes place. The combustion models are implemented in
a commercial computational fluid dynamics (CFD) code, STAR-CD,
and it is shown that both strained and unstrained flamelet models are
able to predict the experimental pressure rise in a closed vessel.
In the final part of this work, a spark-ignition engine is simulated in
STAR-CD using the flamelet model verified for simpler geometries. It
is shown that this model, together with a skeletal mechanism for iso-
octane, compares reasonably well with experimental cylinder pressure
rise. Results obtained from this model are compared with two models
available in STAR-CD. These models require some level of tuning to
match the experiments, whereas the modelling approach used in this
work does not involve any tunable parameters.
Contents
List of Figures x
Nomenclature xii
1 Introduction 1
1.1 Combustion research . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Combustion modelling . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Objectives of this study . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Background on turbulent premixed combustion 8
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.1 Thermochemistry in premixed combustion . . . . . . . . . 11
2.2 Numerical paradigms . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Direct numerical simulation(DNS) . . . . . . . . . . . . . . 12
2.2.2 Large-eddy simulation (LES) . . . . . . . . . . . . . . . . 13
2.2.3 Reynolds-averaged Navier-Stokes equations (RANS) . . . . 13
2.3 Characteristics of laminar premixed flames . . . . . . . . . . . . . 17
2.3.1 Flame thickness . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Flame speed . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Flame stretch . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Regimes in turbulent premixed combustion . . . . . . . . . . . . . 21
2.5 Premixed combustion sub-models . . . . . . . . . . . . . . . . . . 23
2.5.1 Eddy Break-Up (EBU) model . . . . . . . . . . . . . . . . 23
2.5.2 Bray-Moss-Libby (BML) modelling . . . . . . . . . . . . . 24
2.5.3 Flame surface density (FSD) modelling . . . . . . . . . . . 27
2.5.4 The level set approach (G-equation) . . . . . . . . . . . . . 28
2.5.5 Conditional moment closure (CMC) . . . . . . . . . . . . . 30
2.5.6 Transported pdf approach . . . . . . . . . . . . . . . . . . 31
vi
Contents
2.5.7 Presumed pdf approach . . . . . . . . . . . . . . . . . . . 33
2.5.8 Scalar dissipation rate (SDR) based modelling . . . . . . . 34
3 Numerical setup for spherical flame simulation 38
3.1 Governing equations and modelling . . . . . . . . . . . . . . . . . 39
3.2 Reaction rate closures . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Flamelet library generation . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Unstrained flamelet model . . . . . . . . . . . . . . . . . . 44
3.3.2 Strained flamelet model . . . . . . . . . . . . . . . . . . . 45
3.4 Computational approach . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 Initial and boundary conditions . . . . . . . . . . . . . . . 48
4 Spherical methane-air flames 50
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Test Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Validation case . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 Flames for further analysis . . . . . . . . . . . . . . . . . . 54
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3.1 Model validation . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 Spherical and planar flames comparison . . . . . . . . . . . 60
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Spherical hydrogen-air flames 80
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2.1 Reaction rate model . . . . . . . . . . . . . . . . . . . . . 83
5.2.2 Accounting for non-unity Lewis number . . . . . . . . . . 84
5.3 Test flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3.1 Experimental flame . . . . . . . . . . . . . . . . . . . . . . 86
5.3.2 Test cases for further analyses . . . . . . . . . . . . . . . . 87
5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4.2 Comparison of hydrogen- and methane-air flames . . . . . 92
vii
Contents
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Combustion in a closed vessel with swirl 101
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2 Experimental test case . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3.1 Flamelet table generation . . . . . . . . . . . . . . . . . . 103
6.3.2 Model implementation . . . . . . . . . . . . . . . . . . . . 109
6.3.3 Computational details . . . . . . . . . . . . . . . . . . . . 110
6.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.1 Effect of ignition energy . . . . . . . . . . . . . . . . . . . 115
6.4.2 Effect of turbulence model . . . . . . . . . . . . . . . . . . 115
6.4.3 Flame propagation . . . . . . . . . . . . . . . . . . . . . . 116
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Spark-ignition engine simulation 120
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7.2 Engine measurements . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Numerical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3.1 Flamelet library generation . . . . . . . . . . . . . . . . . 125
7.3.2 Computational mesh . . . . . . . . . . . . . . . . . . . . . 132
7.3.3 Initial and boundary conditions . . . . . . . . . . . . . . . 133
7.3.4 Computational details . . . . . . . . . . . . . . . . . . . . 134
7.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 134
7.4.1 Auto-ignition simulation . . . . . . . . . . . . . . . . . . . 138
7.4.2 Combustion inside the cylinder . . . . . . . . . . . . . . . 139
7.4.3 Heat release rate . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.4 Effect of turbulence model . . . . . . . . . . . . . . . . . . 144
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8 Conclusions and future work 147
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 Recommendations for future work . . . . . . . . . . . . . . . . . . 149
viii
Contents
A Spherically symmetric equations 152
A.1 Radial momentum equation . . . . . . . . . . . . . . . . . . . . . 152
B Code validation 156
B.1 Diffusion test case . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
B.2 Convection test case . . . . . . . . . . . . . . . . . . . . . . . . . 160
C List of Publications 161
References 163
ix
List of Figures
2.1 Regime diagram for turbulent premixed combustion . . . . . . . . 22
3.1 Plots from laminar flame calculations of stoichiometric methane-
air flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Variation of normalised reaction rate with normalised scalar dissi-
pation rate for methane-air flames at two equivalence ratios . . . . 47
4.1 Methane-air flames simulated in this work plotted on the combus-
tion regime diagram . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Model validation for spherically propagating methane-air flames . 57
4.3 Flame propagation comparison between planar and spherical flames 61
4.4 Propagation speed of different iso-contours of the progress variable
plotted for planar and spherical flames . . . . . . . . . . . . . . . 63
4.5 Variation of the propagation speed across the flame brush for pla-
nar and spherical flames . . . . . . . . . . . . . . . . . . . . . . . 65
4.6 Spatial variation of the flame induced velocity for planar and spher-
ical flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.7 Evolution of flame brush thickness with time for planar and spher-
ical flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.8 Schematic diagram showing the difference in propagation mecha-
nisms between planar and spherical flames . . . . . . . . . . . . . 68
4.9 Reaction and turbulent flux contributions to displacement speed
for planar and spherical flames . . . . . . . . . . . . . . . . . . . . 69
4.10 Variation of normalised mean reaction rate across the flame brush
for both planar and spherical flames . . . . . . . . . . . . . . . . . 71
4.11 Flame speeds of planar and spherical flames plotted against the
turbulent Reynolds number . . . . . . . . . . . . . . . . . . . . . 73
4.12 Flame speeds of planar and spherical flames plotted against the
turbulent Reynolds number . . . . . . . . . . . . . . . . . . . . . 74
x
List of Figures
5.1 Comparison of computed and measured unstretched laminar flame
speeds for hydrogen-air flames at various equivalence ratios . . . . 83
5.2 Cross plot of two progress variable definitions and the change in
molecular weight across the flame . . . . . . . . . . . . . . . . . . 85
5.3 Regime diagram showing the flames simulated in this study . . . . 88
5.4 Comparison of computed turbulent burning velocities for stoichio-
metric and lean hydrogen-air flames . . . . . . . . . . . . . . . . . 91
5.5 Flame propagation for comparison between methane- and hydrogen-
air flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.6 Temporal variation of normalised flame radius for methane- and
hydrogen-air flames . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.7 Flame speeds of methane- and hydrogen air flames plotted against
the turbulent Reynolds number . . . . . . . . . . . . . . . . . . . 95
5.8 Evolution of flame brush thickness for stoichiometric hydrogen-air
flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.9 Comparison of normalised mean reaction rates across the flame
brush for different mixtures and geometries . . . . . . . . . . . . . 98
6.1 Comparison of computed and measured unstretched laminar flame
speeds for stoichiometric propane-air flames at various pressures . 104
6.2 Spatial plots of normalised reaction rates and the progress variable
for stoichiometric propane-air flames at various pressures . . . . . 105
6.3 Normalised reaction rate variation with pressure and temperature
for stoichiometric propane-air flames . . . . . . . . . . . . . . . . 106
6.4 Variation of normalised reaction rate with normalised scalar dissi-
pation rate for stoichiometric propane-air flames at two different
pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.5 Plots from laminar flame calculations of propane-air flames at dif-
ferent pressures and temperatures . . . . . . . . . . . . . . . . . . 108
6.6 Pressure rise prediction using three different combustion models . 112
6.7 Plot of the mean scalar dissipation rate and reaction rate across
the flame brush using strained and unstrained flamelet models . . 114
6.8 Pressure rise prediction with different ignition energy definitions . 115
xi
List of Figures
6.9 Pressure rise prediction using two different turbulence models . . 116
6.10 Flame propagation comparison between experimental and numer-
ical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.11 Flame induced velocity prediction using three different combustion
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.1 Engine geometry and boundary conditions . . . . . . . . . . . . . 124
7.2 Comparison of computed and measured unstretched laminar flame
speeds for stoichiometric iso-octane – air flames at various pres-
sures and temperatures . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Variation of computed laminar flame speeds with pressure . . . . 128
7.4 Pressure-temperature plot showing auto-ignition and flame regions
for stoichiometric iso-octane – air mixtures . . . . . . . . . . . . . 130
7.5 Ignition delay time variation with both temperature and pressure
for iso-octane – air . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.6 Events sequence in the engine showing the valve movements with
crank angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.7 Flow motion inside the cylinder . . . . . . . . . . . . . . . . . . . 135
7.8 Pressure trace prediction using combustion models . . . . . . . . . 136
7.9 Pressure and unburnt mixture temperature variation with crank
angle during the simulation. . . . . . . . . . . . . . . . . . . . . . 137
7.10 Combustion progress within the engine near spark-timing . . . . . 140
7.11 Combustion progress at 10◦ CA after TDC . . . . . . . . . . . . . 140
7.12 Reaction progress variable is visualised at different crank angles
during combustion. . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.13 Calculated heat release rate as a function of crank angle. . . . . . 142
7.14 Burnt mass and volume fractions . . . . . . . . . . . . . . . . . . 143
7.15 Relation between mass fraction burnt and volume fraction burnt . 144
7.16 Cylinder pressure predictions using two different turbulence models 145
B.1 Code validation for diffusion and convection problems . . . . . . . 159
xii
Nomenclature
Roman Symbols
A – Flame surface area
– Flame speed coefficient for sT
c Progress variable
C3, C4, K
∗
c Constants in the model for ˜c
tc Chemical time
Qk Conditional mean of species k
Cp Mixture specific heat capacity
Cpk Specific heat capacity of species k
D Molecular diffusivity
Da Damko¨hler number
Dc Diffusivity of the progress variable
V k Diffusion velocity of species k
x Spatial vector
sd Displacement speed of c˜ iso-contour
s′d Displacement speed of c iso-contour
Ei Ignition energy
E Total energy
q Energy flux
hk Total enthalpy of species k
hs,k Sensible enthalpy of species k
rf Flame radius
K Stretch factor
G Non-reacting scalar used in the level set approach
Q Heat release
I Identity matrix
k Turbulent kinetic energy
Ka Karlovitz number
xiii
Nomenclature
Le Lewis number
DL Markstein diffusivity
m Mass
mf Fuel mass
Yk Mass fraction of species k
M – Mixture molecular weight
– Number of reactions
N Number of species
Nc Instantaneous scalar dissipation rate of c
n Unit normal vector to the flame
p∞ Atmospheric pressure
p Hydrostatic pressure
Qj Progress rate of reaction j
QLHV Lower heating value
r Radial distance
Ret Turbulence Reynolds number
Ru Ideal gas constant
Sc Schmidt number
s0L Unstrained laminar flame speed
a Flame strain rate
sij Strain rate tensor
T Absolute temperature
Dth Thermal diffusivity
t Time
T Transport vector in turbulent kinetic energy equation
st Flame propagation speed
st Turbulent flame speed
ut Turbulent burning velocity
u Velocity vector
fk Volume force acting on species k
Wk Atomic weight of species k
Calligraphic Symbols
L Markstein length
xiv
Nomenclature
Pk Turbulence production term
R Flame surface radius
Greek Symbols
β′ Constant in ˜c model
δ0L Laminar thermal thickness
δt Turbulent flame brush thickness
c Scalar dissipation rate of c
ε Dissipation rate of turbulent kinetic energy
κc Flame curvature
κ Flame stretch rate
γ Specific heat ratio
ηk Kolmogorov length scale
δ Chemical length scale, Zeldo¨vich flame thickness
Λ Turbulence integral length scale
λ – Heat diffusion coefficient
– air-to-fuel ratio
µ Dynamic viscosity
µt Eddy viscosity
ν Kinematic viscosity
νt Turbulent diffusivity
Ω Angular velocity
Φ Equivalence ratio
Ψ General scalar property
ψ Sample space variable for N
ρ Fluid density
Σ Flame surface density
τ Heat release parameter
τk Kolmogorov time scale
τ Viscous stress tensor
τt Turbulence time scale
θ Crank angle
τi Ignition delay time
u′k Kolmogorov velocity scale
xv
Nomenclature
ω˙ Specific dissipation rate
ω˙c Mass reaction rate of c
ω˙T Combustion heat release rate
ζ Sample space variable for c
Superscripts
′′ Favre fluctuation
˜ Favre mean
+ Normalised using planar laminar flame quantities
′ rms of Reynolds fluctuation
Subscripts
b Burnt mixture value
j Refers to a particular reaction
k Refers to a particular species
L Laminar quantity
t Turbulent quantity
u Unburnt mixture value
Acronyms
BDC Bottom dead centre
BMEP Brake mean effective pressure
BML Bray-Moss-Libby
CA Crank angle
CFD Computational fluid dynamics
CFM Coherent flame model
CMC Conditional moment closure
DISI Direct-injection spark-ignition
DNS Direct numerical simulation
EBU Eddy break-up
ECFM Extended CFM
EGR Exhaust gas recirculation
FSD Flame surface density
FVM Finite volume method
GDI Gasoline direct injection
HCCI Homogeneous charge compression ignition
xvi
Nomenclature
HRR Heat release rate
IC Internal combustion
IVO Intake valve opening
KPP Kolmogorov-Petrovskii-Piskunov
LES Large-eddy simulation
LHS Left hand side
LHV Lower heating value
MARS Monotone advection and reconstruction scheme
pdf Probability density function
PFI Port fuel injection
RANS Reynolds-averaged Navier-Stokes
RHS Right hand side
rms Root mean square
rpm Revolutions per minute
SDR Scalar dissipation rate
SI Spark-ignition
TDC Top dead centre
URANS Unsteady RANS
xvii
1. Introduction
Combustion is mankind’s oldest technology and it is one of the main contributors
to the technological progress that we see today. Combustion is ubiquitous, since
majority of energy conversion for transportation, power generation and domestic
heating is from the burning of a fuel in the presence of an oxidiser. Most com-
bustion processes use fossil fuels, which contributed to 81% share of the global
energy consumption in 2010 (IEA, 2011).
Fossil fuels are particularly attractive for the transportation sector, with 93%
of all fuels used for transport in 2010 being based on fossil fuels and it is predicted
to account for 83% share in 2035 (IEA, 2011). Most motor vehicles today use
reciprocating internal combustion (IC) engines, which burn fossil fuels to convert
chemical energy stored in the fuel to mechanical energy.
Combustion of fossil fuels for energy conversion is not without problems. Pol-
lutants such as unburnt hydrocarbons, oxides of nitrogen, carbon monoxide, sul-
phur oxides and soot are formed as unwanted by-products when fossil fuels are
burnt. In addition, combustion is the major anthropogenic contributor to CO2
concentrations in the atmosphere; combustion of fossil fuels in industrial, domes-
tic, energy and transportation sectors contributed to 71% of the total greenhouse
gas emissions in 2004 (IPCC, 2007). These issues are compounded by the fact
that fossil fuel resources are finite. Therefore, it is imperative that we improve
combustion efficiency and reduce emissions.
Conventional combustion technologies are unable to simultaneously increase
efficiency and reduce emissions. Burning fuel in the presence of excess air, known
as fuel-lean combustion, can achieve both requirements (Dunn-Rankin et al.,
2008; Swaminathan and Bray, 2011), but it poses new challenges, especially for
IC engines (Urata and Taylor, 2011). In the last half-century, there have been
a drive to reduce emissions and to achieve better fuel efficiency. This is an
area where improvements are still being made, despite having a long history
of development. For example, new engine types, such as the homogeneous charge
compression ignition (HCCI) engine, which burn lean, have been shown to provide
improvements in fuel economy and reduced emissions (Drake and Haworth, 2007).
1
1.1. Combustion research
Such improvements are only possible when fundamental processes that take
place during combustion are clearly understood. In the last few decades, combus-
tion science has shifted from a field based on empiricism to one that is quantitative
(Bilger et al., 2005). These fundamental studies are used to develop combustion
models that can be used to design and develop efficient and clean combustion
devices.
1.1 Combustion research
It is convenient to classify gaseous combustion according to the extent of mixing:
premixed or non-premixed. In premixed flames, fuel and oxidiser are mixed homo-
geneously before combustion. Lean-burn gas turbines for power generation and
spark-ignition (SI) IC engines are typical examples of this type of combustion.
In non-premixed or ‘diffusion flames’, the fuel and oxidiser are transported sep-
arately into the reaction zones by diffusion. Aero-engine gas turbines and diesel
engines are typical examples of non-premixed combustion. It is often the case in
practical systems that fuel and air is not completely mixed, leading to partially
premixed combustion. For example, fuel is injected directly into the combustion
chamber in a gasoline direct-injection (GDI) engine, which does not give sufficient
time and space for it to mix with air completely.
These subdivisions of combustion systems are important for modelling pur-
poses, since the flame behaviour in these conditions are radically different. Only
premixed combustion is considered in this work; however, due to the multifaceted
nature of combustion research, it is impractical to present a complete background
review on premixed combustion research. Instead, a general introduction to re-
search relevant to this work is provided.
Virtually all combustion in practical devices takes place in turbulent flows.
Turbulence is extremely complex even in non-reacting situations and the inclusion
of chemical reactions introduces additional complexities in studying such flows.
The greatest challenge associated with turbulent combustion is understanding
the interaction between turbulence and combustion over a broad range of time
and length scales (Bray, 1996). Thus, to a large extent, our fundamental under-
standing of combustion phenomena remains limited, which makes it difficult to
2
1.1. Combustion research
develop robust combustion models that can be used to design new combustion
technologies.
Damko¨hler (1940) explained in his pioneering work that, in turbulent pre-
mixed flames, large-scale eddies wrinkle the flame surface without altering the
internal flame structure, whereas small-scale eddies affect the reactive-diffusive
structure. This understanding of the effect of turbulence on the flame structure
lead to the identification of various regimes of turbulent combustion, where the
turbulence time and length scales are used to obtain non-dimensional numbers
to separate different regimes. A description of relevant background theory on
turbulent premixed flames, including the combustion regime diagram of Borghi
(1985) is given in section 2.4.
The regime diagram is useful for combustion modelling because it can identify
relative scales for flow and combustion and thus the validity, in a broad sense, of
a particular combustion model invoking some assumptions among these scales.
The local flame structure that can be determined using the regime diagram can
in turn give useful information on the turbulence-chemistry interactions.
One of the popular flame configurations for experimental investigation of pre-
mixed combustion is the outwardly propagating spherical flames. These exper-
iments are usually conducted in spherical bombs (Andrews et al., 1975) or in
wind tunnels (Hainsworth, 1985), where turbulence is generated using fans and
grids respectively. These flames are representative of flames in SI engines and
have been used to increase our fundamental understanding of the ignition pro-
cess (Shy et al., 2008). Furthermore, turbulent spherical flames can be used to
study accidental explosions of vapour clouds in the atmosphere (Baker et al.,
1983).
An important quantity for turbulent premixed flames is the turbulent flame
speed. On a practical level, flame speed determines how quickly a mixture will be
burnt. However, due to flame wrinkling being geometry dependent, it may not be
possible to get a geometry-independent turbulent flame speed, which explains the
large scatter for measured flame speeds (Driscoll, 2008). Turbulent flame speed
can be also be influenced by thermodiffusive effects of the mixture, which can be
explained using the flame stretch theory (Law and Sung, 2000). Furthermore,
unequal thermal and mass diffusivities could make the flame thermodiffusively
3
1.2. Combustion modelling
unstable. An example of one such flame is the lean hydrogen-air flame, which
according to the stretch theory, is unstable when it is positively stretched (Law
and Sung, 2000).
Three distinctly different numerical paradigms exist for turbulent combustion
simulation; namely direct numerical simulations (DNS), large-eddy simulations
(LES) and Reynolds-averaged Navier-Stokes (RANS) simulations. There are ad-
vantages and disadvantages associated with each of these simulation paradigms
and the choice depends on the level of detail required from them. Essentials of
these different paradigms of computational fluid dynamics (CFD) for reacting
flows are presented in section 2.2.
1.2 Combustion modelling
Direct solution of the equations describing reacting flows is computationally ex-
pensive and it is necessary to use either averaged (RANS) or filtered (LES) forms
of the non-linear governing equations for practical applications. Simulation of
reacting flows using both RANS and LES require combustion modelling, which
makes it an active area of research. The main objective of combustion modelling
is to develop closure models for the mean chemical source terms that appear in
the averaged/filtered equations.
These source terms cannot be described using simple averaging techniques,
and physical analysis based on the comparison between chemical and turbulent
time scales have to be used (Veynante and Vervisch, 2002). In a review article,
Veynante and Vervisch (2002) described that most combustion models are de-
veloped according to three different approaches: geometrical analysis, turbulent
mixing and one-point statistics. A challenge in all these approaches is to close
unknown terms, such as the flame surface-density, scalar dissipation rate, and the
probability density functions (Poinsot and Veynante, 2005).
Combustion models used in LES are similar to those developed earlier for
RANS (Pitsch, 2006). The models developed for premixed combustion can be
broadly categorised into flamelet and non-flamelet approaches. Most models cur-
rently in use are based on the flamelet concept. In this concept, the flame is
assumed to be thin compared with the smallest scales of turbulence and it can
4
1.3. Objectives of this study
be thought of as an interface between hot products and cold reactants. Then one
can assume that the turbulent flame is made up of an ensemble of laminar flames
(Williams, 1975). For premixed flames, the reaction diffusion balance is well
represented by this flamelet approach and are computationally more economical
than non-flamelet approaches (Cant and Mastorakos, 2008).
Combustion models have already been used for a wide range of turbulent
premixed applications in the RANS context; the applications mentioned here are
not intended to be exhaustive. For example, the well-known Bray-Moss-Libby
(BML) model (Bray et al., 1985) and its variants have been used for applications
ranging from stagnation flames (Bray et al., 1998) to SI engines (Abu-Orf and
Cant, 2000). The Coherent Flame Model (CFM) has been used for Bunsen burner
flames (Prasad and Gore, 1999) as well as in IC engines (Boudier et al., 1992).
The level set approach or G-equation has also been used to simulate these flames
(Schneider et al., 2005; Ewald and Peters, 2007). The transported probability
density function (pdf) approach has been used to simulate spherical flames (Pope
and Cheng, 1986) and Bunsen burner flames (Hack and Jenny, 2013). More
recently, the conditional moment closure (CMC) method, which was originally
developed for non-premixed flames, has been used to simulate Bunsen burner
flames (Amzin et al., 2012; Amzin and Swaminathan, 2013). A brief description
of these models is given in section 2.5 and detailed description can be found in
the review articles of Peters (1986) and Veynante and Vervisch (2002).
1.3 Objectives of this study
Combustion models implemented in CFD codes are now used as engine design
tools. However, the desired results of such simulations rely heavily on the accu-
racy, fidelity and robustness of the combustion model used. These models should
be able to handle a range of engine operating conditions that may change the
combustion characteristics. In this regard, combustion models based on the un-
derlying physics of the turbulence-chemistry interaction have a clear advantage
over empirical or semi-empirical models.
The overall aim of this work is to use one such modelling framework, devel-
oped by Kolla and Swaminathan (2010a), to simulate intermittent combustion in
5
1.3. Objectives of this study
the RANS context. Kolla and Swaminathan (2010b) have validated this approach
for continuous combustion. This modelling framework is based on the flamelet
concept, where scalar dissipation rate (SDR) that measure the decay rate of scalar
fluctuations by turbulent micromixing, is a central quantity. Kolla and Swami-
nathan (2010a) introduced two closure models for the reaction source terms, viz.
unstrained and strained flamelet models. For this approach to be useful in design-
ing IC engines, it needs to be accurate, robust, and computationally economical.
Furthermore, the model must be easily implemented in multi-dimensional CFD
codes. Therefore, the objectives of this study are:
1. To validate the SDR based models for spherically propagating flames that
are formed in spark-ignition engines. Validation is to be done for methane-
and hydrogen-air flames, by comparing with available experimental data.
These fuels have different thermochemical properties that influence their
dynamic behaviours. In order to simulate spherically propagating flames,
an in-house CFD code originally written in Cartesian coordinates is to be
modified for spherical coordinates.
2. To identify the effect of flame geometry on flame propagation, by comparing
spherical flame results with those of planar flames. The objective here is to
determine the effect of global mean curvature on flame propagation.
3. To study the effect of turbulence on flame propagation for both methane-
and hydrogen-air flames. Two key quantities that are used to analyse this
effect are the turbulent flame speed and thickness.
4. To implement the combustion model in a commercially available CFD code,
STAR-CD and to validate the model for intermittent combustion. The
experimental case to be used for validation is a closed combustion vessel
with swirl. Unlike spherically symmetric cases above, this closed volume
leads to a large pressure rise due to combustion, and is more representative
of IC engines.
5. To use the modelling framework to simulate combustion in a practical SI
engine.
6
1.4. Thesis outline
1.4 Thesis outline
Chapter 2 of this thesis describes the background theory on turbulent premixed
flames relevant for this work. This includes the governing equations for turbulent
reacting flows, description of various numerical paradigms, brief description of
laminar premixed flames and combustion submodels used for turbulent premixed
flames.
Chapter 3 presents details of the in-house CFD code used to simulate out-
wardly propagating spherical flames. The governing equations written in spherical
coordinates are presented, along with the numerical details for simulating such
flames. The CFD code presented here is used to address objectives 1, 2 and 3
listed in the previous section.
Model validation for methane-air spherical flames is presented in Chapter 4.
In addition, geometry and turbulence effects on flame propagation, noted in ob-
jectives 2 and 3, are investigated. Similarly, details of hydrogen-air flames are
presented in Chapter 5. The influence of turbulence on flame propagation is
explored for both spherical methane- and hydrogen-air flames.
Implementation of the combustion models in STAR-CD and the simulation
of intermittent combustion are presented in Chapter 6. This chapter addresses
objective 4 outlined in the previous section, where differences between the mod-
elling approach used for this case and the constant pressure cases simulated in
the previous chapters are also highlighted. Finally in Chapter 7, the modelling
framework is used to simulate a practical SI engine. This thesis concludes with
Chapter 8, listing the main outcomes from this work and recommendations for
future work
7
2. Background on turbulent
premixed combustion
This chapter presents background information on turbulent premixed combustion
relevant to this work. Governing equations used to describe turbulent reacting
flows are described first. This is followed by a description of laminar premixed
flames, which is an important pre-requisite for turbulent premixed flame mod-
elling. The next section describes various combustion modelling techniques used
to simulate turbulent premixed flames. This chapter ends with the modelling
approach used in this work, which is based on the scalar dissipation rate.
2.1 Governing equations
The instantaneous governing equations for a multi-component reacting flow of
ideal-gas mixtures with N species and M reactions are given by [for example see
Williams (1985a)]:
Mass:
∂ρ
∂t
+∇ · (ρu) = 0. (2.1)
Momentum:
∂ρu
∂t
+∇ · (ρuu) = −∇p+∇ · τ + ρ
N∑
k=1
Ykfk. (2.2)
where Yk is the mass fraction of species k and fk is the volume force acting on
species k. The viscous stress tensor in the momentum equation is given by:
τ = µ
[∇u+ (∇u)T]− 2
3
µ(∇ · u)I, (2.3)
where µ is the dynamic viscosity and I is the 3× 3 identity matrix.
8
2.1. Governing equations
Energy:
∂ρE
∂t
+∇ · (ρuE) = ω˙T +∇ · (τ · u)−∇ · q + ρ
N∑
k=1
Ykfk · (u+ V k) , (2.4)
where the total non-chemical internal energy, E, is given by
E = es +
1
2
(u · u) , (2.5)
which includes the sensible energy, es. The energy flux, q, in Eq. (2.4) is given
by:
q = −λ∇T + ρ
N∑
k=1
hs,kYkV k, (2.6)
where λ is the heat diffusion coefficient, hs,k is the sensible enthalpy of species k,
Yk is the mass fraction of species k and V k is the diffusion velocity of species k.
The combustion heat release, ω˙T , appearing in Eq. (2.4) is given by:
ω˙T = −
N∑
k=1
∆hof,kω˙k, (2.7)
where ∆hof,k is the mass enthalpy of formation of species k at temperature T0. The
total rate of mass production by chemical reactions is given by: ω˙k =
∑M
j=1 ω˙kj =
Wk
∑M
j=1 νkjQj, with Wk being the atomic weight of species k, νkj = ν
′′
kj − ν ′kj
denotes the difference in molar stoichiometric coefficients of species k in reaction
j and Qj is the reaction j progress rate. The energy equation given in Eq. (2.4)
can also be written in terms of total sensible enthalpy:
∂ρh
∂t
+∇ · (ρuh) = ∂p
∂t
+ ω˙T + τ :∇u−∇ · q + ρ
N∑
k=1
Ykfk · (u+ V k) . (2.8)
9
2.1. Governing equations
where
h =
∫ T
T0
Cp dT +
1
2
(u · u) and Cp =
N∑
k=1
CpkYk. (2.9)
Species:
∂ρYk
∂t
+∇ · [ρ (u+ V k)Yk] = ω˙k, k = 1, . . . , N, (2.10)
where chemical reaction sources, ω˙k, are obtained using chemical kinetics as func-
tions of species concentrations, temperature and pressure. The Eqs. (2.1), (2.2),
(2.4) and (2.10) constitute a system of equations for a reacting flow problem with
N + 5 dependent variables. Rest of the variables can be related to these using
the state equation:
p = ρRuT
N∑
k=1
Yk
Wk
. (2.11)
In principal, the governing equations give above can be solved numerically, how-
ever, for practical simulations certain simplifications have to be made. In this
work, the following phenomena are neglected to simplify these equations, which
have been used in many previous studies (Pierce, 2005).
• flame acoustic interactions
• heating due to viscous dissipation
• diffusion due to pressure gradients
• volume forces
• thermal radiation
The governing equations given in this section are valid for premixed and non-
premixed combustion. If one uses detailed chemical kinetic mechanisms, which
typically involve tens of species with hundreds of reactions, the dimensionality
10
2.1. Governing equations
of the problem can be large. Furthermore, these equations are numerically stiff
owing to the disparity between the timescales associated with different chem-
ical reactions. Only premixed combustion is considered in this work, and the
simplification described in the following section is often made to simplify the
thermochemistry.
2.1.1 Thermochemistry in premixed combustion
For premixed flames, instead of solving for all the reactive species, the problem
can be made tractable by using a reaction progress variable, c. This progress
variable can then represent all the concentrations and temperature in the system.
For adiabatic flows with unity Lewis number (Le = λ/ρCpD, i.e. equal mass and
temperature diffusivities), the progress variable can be related to temperature by
c =
T − Tu
Tb − Tu , (2.12)
where the subscript u and b denote the initial conditions in the unburnt and
burnt mixtures respectively. Alternatively, the progress variable can be defined
in terms of mass fractions. It can be seen that c = 1 in burnt products and
c = 0 in unburnt gases. For adiabatic flows, assuming uniform pressure with
various gas species having similar molecular weights, one can use the state equa-
tion [Eq (2.11)] to write density and temperature in terms of the instantaneous
progress variable (Bray et al., 1985)
ρ =
ρu
1 + τc
, T = Tu(1− τc), (2.13)
where ρu and Tu are the density and temperature within the unburnt gases re-
spectively and τ = Tb/Tu − 1 = ρu/ρb − 1 is the heat release parameter. One can
obtain the following transport equation for the instantaneous progress variable
by taking the temperature transport equation and substituting the definition of
the progress variable given in Eq. (2.12).
∂ρc
∂t
+∇ · (ρuc) = ∇ · (ρDc∇c) + ω˙, (2.14)
11
2.2. Numerical paradigms
where Dc and ω˙ are the diffusivity and reaction rate of the progress variable, c.
If all the species are assumed to have the specific heat of the mixture, Cp, then
Dc is equivalent to the thermal diffusivity, Dth. The reaction rate of the progress
variable is given by
ω˙ =
−∑Nk=1 ∆hkω˙k
Cp (Tb − Tu) . (2.15)
Note that in Eq. (2.14) the transient pressure term, Dp/Dt, that appears in the
temperature equation has been neglected. However, this term must be retained
for reciprocating engine applications.
2.2 Numerical paradigms
Most flows of practical importance are turbulent, which is caused by instabilities
in the flow at large Reynolds numbers (Tennekes and Lumley, 1972). It is not
possible to obtain analytical solutions to the instantaneous governing equations
presented in section 2.1, and numerical methods have to be used. Various numer-
ical paradigms exist for turbulent combustion simulations, and they are classified
according to the level of detail they provide as described below.
2.2.1 Direct numerical simulation(DNS)
The direct approach of solving the turbulent flow field is called direct numerical
simulation. All the relevant temporal and spatial scales are resolved in DNS,
which is often referred to as numerical experiments since it does not involve any
empirical constants and can reveal physical information on various processes tak-
ing place within a turbulent flow field. However, the high resolution required in
DNS makes it computationally very expensive; and the computational cost in-
creases as the cube of the flow Reynolds number (Pope, 2000). In order to reduce
computational costs, earlier DNS studies were performed for 2-D problems with
single-step chemistry. Three-dimensional DNS with complex chemistry have only
recently become feasible for moderate Reynolds number flows (Chen, 2011). Even
with the current rate of increase in computational capabilities, it is unfeasible to
12
2.2. Numerical paradigms
simulate combustion in practical engines using DNS for the foreseeable future.
2.2.2 Large-eddy simulation (LES)
The second level for solving turbulent reacting flows numerically is the large-eddy
simulations, where only the larger turbulent structures are resolved directly, while
the small-scale structures are modelled. In terms of computational expense, LES
lies between DNS and RANS (explained in the next section) and the computa-
tional cost of LES is proportional to Re9/4 (Hirsch, 2007). LES equations are
obtained using a filtering operation, where the governing equations are averaged
over the part of the spectrum that is not resolved. The general filtering operation
(Leonard, 1974) is defined as
φ (x, t) =
∫
φ (x, t) G(x− x′) dx′, (2.16)
where G is the spatial filter function. The filtered governing equations for LES
are similar to RANS equations. However, unlike Reynolds decomposition, the
filtered residual in LES is non-zero, i.e. φ′ 6= 0 (Pope, 2000). LES require
combustion models since combustion takes place at the unresolved scales. It
is particularly attractive for combustion applications, since it can give accurate
results on complex flow fields that involve swirling flows, recirculation zones and
vortical structures (Pitsch, 2006). In addition, LES can be used to study flows
with inherent unsteadiness, such as in IC engines to simulate the cycle-to-cycle
variability (Haworth, 1999; Rutland, 2011).
2.2.3 Reynolds-averaged Navier-Stokes equations (RANS)
The next level of numerical simulation is the Reynolds-average Navier-Stokes
methodology, where the instantaneous governing equations are averaged. This
averaging procedure leads to unclosed terms that need to be modelled. RANS
simulations are computationally the least expensive out of the three approaches,
but it only provides averaged quantities. This leads to a loss of information re-
garding the fluid dynamic and chemical timescales. This is acceptable for most
engineering applications, since it is the statistical moments that are mostly of
13
2.2. Numerical paradigms
interest, rather than the instantaneous quantities. It is the RANS methodology
that is used for the present thesis, and in this section, the governing equations for
RANS are described along with the models used to close various terms in these
equations. Hence, combustion modelling becomes an integral part of this frame-
work and various combustion models have been devised to numerically simulate
combustion.
RANS governing equations
It is conventional to use Favre (density-weighted) averaging for turbulent com-
bustion, since Reynolds averaging of variable density flows introduces products of
fluctuations that require correlations for closure. However, Favre averaging allows
the product ρu to be selected as a fundamental variable instead of u, in which
case a product of fluctuations is not obtained (Williams, 1985a). Favre-average
of a quantity f is defined as: f˜ = ρf/ρ, where the overbar is the usual Reynolds
average. The instantaneous quantity f can be decomposed into its mass-averaged
part, f˜ , and its fluctuating part, f ′′, i.e. f = f˜ + f ′′. From this definition, f ′′ 6= 0
but f˜ ′′ = 0. Averaging Eqs. (2.1) and (2.2) and neglecting the body forces leads
to:
∂ρ
∂t
+∇ · (ρu˜) = 0, (2.17)
∂ρu˜
∂t
+∇ · (ρu˜u˜) = −∇p−∇ · (ρu′′u′′) +∇ · τ , (2.18)
where the term ∇ · (ρu′′u′′) is the Reynolds stress tensor. The Favre-averaged
viscous stress tensor is given by:
τ = µ
[∇u˜+ (∇u˜)T]− 2
3
µ(∇ · u˜)I. (2.19)
The progress variable, c, is used to make the problem tractable and the transport
equation for Favre-averaged progress variable, c˜, is given by
∂ρc˜
∂t
+∇ · ρu˜c˜ = ∇ · (ρDc∇c˜− ρu′′c′′)+ ω˙, (2.20)
14
2.2. Numerical paradigms
where the molecular diffusion term is usually neglected for high Reynolds number
flows.
Turbulent kinetic energy
Appearance of Reynolds stresses in the Favre-averaged momentum equation [Eq.
(2.18)] leads to the closure problem (Pope, 2000), which requires closure models
if one were to solve the above equations. Most turbulence models are based on a
equation for the turbulent kinetic energy.
Turbulent kinetic energy per unit mass, k, is defined as: k = (1/2) (u′ · u′),
where u′ denotes the three-dimensional velocity fluctuations. It is known that
turbulence is statistically isotropic at small scales, which implies that velocity in
the three coordinate directions are equal. Hence, the turbulent kinetic energy
is given by: k = (3/2)u′2. For compressible flows, the exact Favre-averaged
turbulent kinetic energy equation is given by [for example see Wilcox (1993)]:
∂ρk˜
∂t
+∇ · (ρu˜k˜) = −∇ ·T + Pk − ρε˜− u′′∇p+ p′∇ · u′′, (2.21)
where the production term Pk is given by
Pk = −ρ(u′′u′′):∇u˜. (2.22)
and the transport term, T is given by
T =
1
2
ρu′′ (u′′ · u′′) + p′u′′ − 2ρνu′′ · s, (2.23)
where the strain-rate tensor is given by
s =
1
2
[
∇u˜+ (∇u˜)T
]
. (2.24)
and the dissipation rate is given by
ε˜ = ν (∇u′′ : ∇u′′). (2.25)
The fifth and sixth terms on the RHS of Eq. (2.21) are the pressure work and
15
2.2. Numerical paradigms
pressure dilatation terms which do not have an analog in the incompressible
equation for turbulent kinetic energy. The production, dissipation and turbulent
transport terms in Eq. (2.21) are unclosed. In addition, closure models need to
be provided for the pressure work and pressure dilatation terms.
Turbulent scales
One of the characteristic features of high Reynolds number flows is the existence
of different length scales. Richardson (1922) first introduced the idea of the
energy cascade where kinetic energy that enters the largest scales of motion are
transferred to successively smaller scales until it is dissipated at the smallest scales
by the action of viscous processes.
This dissipation of turbulent kinetic energy occurs at a rate determined by
large-scale dynamics of turbulence. An important estimate of this dissipation
rate is given by (Taylor, 1935a)
ε ∼ u
′3
Λ
, (2.26)
where Λ is the integral length scale. Therefore, the small-scale structures of
turbulence can be parametrised using this dissipation rate per unit mass, ε, and
the kinematic viscosity, ν, which can be used to obtain the following Kolmogorov
microscales :
ηk =
(
ν3
ε
)1/4
, τk =
(ν
ε
)1/2
, u′k = (νε)
1/4 . (2.27)
It can be shown that these Kolmogorov scales correspond to the smallest eddies
in a turbulent flow (Pope, 2000). Before proceeding to a discussion of turbulent
combustion models, it is helpful to look at some results from laminar premixed
flames, which are described in the next section.
16
2.3. Characteristics of laminar premixed flames
2.3 Characteristics of laminar premixed flames
Study of laminar premixed flames is a necessary pre-requisite for turbulent pre-
mixed flames. Various characteristics of laminar premixed flames are presented
in this section, which are essential for the turbulent combustion model used in
this work.
2.3.1 Flame thickness
Laminar flame thickness, δ, is a characteristic quantity for laminar premixed
flames. It is also important from a modelling perspective, since it can be used as
a reference length scale for the flame and can control the mesh resolution (Poinsot
and Veynante, 2005). From scaling laws, the thickness of the laminar premixed
flame can be obtained as (Go¨ttgens et al., 1992):
δ =
λu
ρuCps0L
, (2.28)
where all the values are evaluated in the unburnt gases, s0L is the unstretched
laminar flame speed, which is used as a reference speed for all combustion studies
(Poinsot and Veynante, 2005). If the Prandlt number, Pr = ν/α, and the Schmidt
number, Sc = ν/D, are assumed to be unity and the reactive scalars have the
same diffusion coefficient, D, then the flame thickness given in Eq. (2.28) can be
written as
δ =
D
s0L
. (2.29)
Laminar flame thickness is often referred to as the Zeldovich thickness. The flame
thickness based on the temperature profile, called the thermal thickness, is defined
as (Spalding, 1955):
δ0L = (Tb − Tu)/(∂T/∂x)max, (2.30)
which can only be determined if the temperature gradient is known. A thickness
close to the thermal thickness has been defined using correlations (Blint, 1986)
17
2.3. Characteristics of laminar premixed flames
as
δbL = 2δ(1 + τ)
0.7 ' δ0L. (2.31)
2.3.2 Flame speed
Propagation is the most important characteristic of premixed flames, since it de-
termines fuel consumption rate and heat release. Propagation rate of laminar
premixed flames is determined by the laminar burning velocity, s0L, which is the
velocity with which the flame moves relative to the unburnt reactants. The direc-
tion of propagation is locally normal to the flame itself and towards the reactants.
This laminar flame speed can also be used to parameterise premixed flame phe-
nomena such as extinction, flashback, blowoff and turbulent flame propagation
(Law, 2006).
Laminar flame speed measurement
A number of experimental techniques are used to measure s0L. For example,
Bunsen-type burners can be used to determine the flow velocity at the burner
exit, usually using various imaging techniques of the flame surface. A useful re-
search tool in studying laminar premixed flames is the flat-flame burner (Williams,
1985a), where the flow is stabilised over a porous plate in a flow. Stagnation point
flames are a type of flat-flame and is often used to study laminar flames (Poinsot
and Veynante, 2005). Furthermore, spherical flames, obtained either by igniting
a combustible mixture in a closed chamber or by using the soap-bubble method
(Williams, 1985a) are also used. Most experimental techniques used to measure
the laminar flame speed suffer from stretch, which needs to be subtracted from
the measured laminar flame speeds to determine a stretch-free flame speed (Law,
2006).
2.3.3 Flame stretch
Non-uniformities in the flow can cause the flame front to be strained and curved.
The resulting fractional rate of change of the flame surface area is collectively
18
2.3. Characteristics of laminar premixed flames
called the flame stretch, which is defined as (Williams, 1985a)
κ =
1
A
dA
dt
, (2.32)
where A is the flame surface element. Candel and Poinsot (1990) derived the
following expression for the flame stretch in terms of strain rate and curvature.
κ = −nn:∇u+∇ · u+ s′d(∇ · n), (2.33)
where u is the local flow velocity, s′d is the speed with which the flame front
propagates normal to the background fluid, called the displacement speed and is
given as (Jenkins et al., 2006)
s′d =
ω˙ +∇ · (ρD∇c)
ρ|∇c|
∣∣∣∣∣
c=c∗
. (2.34)
The unit normal vector to the flame front is given as n in Eq. (2.33). This unit
normal vector points towards the unburnt gases, which can be defined in terms
of the reaction progress variable as
n = − ∇c|∇c| , (2.35)
where the negative sign is included because the convention is for n to be positive
when it is pointing towards the burnt side (Law, 2006), i.e. a positive curvature
is when it is convex to the fresh gases. Thus, an outwardly propagating spherical
flame has positive curvature. The three terms on the RHS of Eq. (2.33) are: nor-
mal strain rate tensor, volumetric expansion of the fluid and the flame curvature,
respectively. The curvature term can be written in terms of the principal radii of
curvature as (Rutland and Trouve´, 1993)
s′d∇ · n = ±s′d
(
1
R1
+
1
R2
)
, (2.36)
where ± is used to indicate outwardly (+) and inwardly propagating (−) flames
and R1 = R2 for a sphere. As the flame propagates the stretching of an expanding
19
2.3. Characteristics of laminar premixed flames
flame becomes weaker; whereas it becomes stronger for an imploding flame (Law,
2006).
Effect of flame stretch
Flame stretch is important in practical flames; examples include the negatively
stretched (κ < 0) Bunsen burner flame and the positively stretched (κ > 0)
outwardly propagating spherical flame. Since flame stretch is a characteristic
property of the flame it should be related to other physicochemical characteristics
such as flame speed and temperature.
Matalon and Matkowsky (1982) used the method of matched asymptotic ex-
pansions to show the response of the flame speed to stretch. It was found that for
moderate rates of strain and curvature with large activation energy, the normal
flame speed is directly proportional to the flame stretch. Studies such as those
by Matalon and Matkowsky (1982) and Clavin and Williams (1982) obtained the
following expression for sL
sL = s
0
L − Lκ, (2.37)
where L is the Markstein length, which is proportional to the laminar flame
thickness, δ (Peters, 2000). This equation implies that the flame speed changes
linearly with stretch. Thus, Markstein length can also be used to characterise a
laminar flame (Clavin, 1985). However, if the stretch was intensified, the above
Markstein relation would not hold. Therefore, one cannot always use the flame
stretch to parameterise the flame structure.
Experimental and numerical studies have also been conducted to determine
the effect of flame stretch. Hassan et al. (1998a) carried out experimental inves-
tigations to determine the properties of outwardly propagating spherical laminar
premixed flames and compared the results with numerical simulations. The re-
sults of their study showed that there is a significant effect of the stretch for the
flames considered.
20
2.4. Regimes in turbulent premixed combustion
2.4 Regimes in turbulent premixed combustion
In his pioneering work, Damko¨hler described how large-scale turbulence acts to
wrinkle the flame without changing the internal structure, whereas small-scale
eddies affect the transport processes within the flame. Therefore, in order to
build a model for turbulent combustion it is necessary to consider the physical
characteristic of the flame as well as the various time and length scales of the
turbulent combustion process. For example, a flame with holes will not be treated
in the same way as a continuous flame front (Meneveau and Poinsot, 1991). Such
an analysis can be used to obtain a combustion regime diagram that shows the
regimes as function of various non-dimensional numbers.
The turbulence Reynolds number, Ret, is used to characterise the ratio of
inertial to viscous effects of the turbulent flow. The turbulence Damko¨hler num-
ber, Dat, compares the turbulent large-eddy turn-over timescale to the chemical
timescale of the laminar flame. In addition, turbulent Karlovitz number, Ka, is
also defined for premixed combustion, which is a ratio of laminar flame chemi-
cal timescale to the Kolmogorov timescale. These non-dimensional variables are
given as:
Ret =
u′
s0L
Λ
δ
, Dat =
s0L
u′
Λ
δ
, Ka =
u′k
s0L
δ
ηk
. (2.38)
where ηk is the Kolmogorov length scale and u
′
k is the Kolmogorov velocity scale.
Definitions of the dissipation rate given in Eq. (2.26), the Kolmogorov length scale
and unity flame Reynolds number can be used to obtain the following equivalent
definitions for the Karlovitz number (Poinsot and Veynante, 2005):
Ka =
(u′/s0L)
3/2
(Λ/δ)1/2
=
(εδ)1/2
s0L
3/2
=
(
δ
ηk
)2
=
(ε/ν)1/2
s0L/δ
. (2.39)
The regime diagram introduced by Peters (1999), which is shown in Figure 2.1,
includes five different regimes of premixed combustion. The dimensionless groups
used to plot the diagram are: u′/s0L and Λ/δ, which respectively represents the
velocity scales in the flow and the flame, and the size of the turbulent eddies that
21
2.4. Regimes in turbulent premixed combustion
0.1
1
10
100
1000
0.1 1 10 100 1000 10000
u
′ /
s0 L
Λ/δ
wrinkled flameletslaminar flames
corrugated flamelets
thin reaction zones
broken reaction zones
Ka = 1
Ka = 100
Da = 1
Ret = 1
Figure 2.1: Regime diagram for turbulent premixed combustion (Peters, 1999).
interacts with the flame.
Laminar flames exist for low Reynolds numbers (Re < 1) and two laminar
flamelet regimes exist when Re 1, Da 1 and Ka 1. Laminar flamelet
concept means that the turbulent flame can be thought of as an ensemble of
stretched laminar flamelets (Peters, 2000), i.e. the local flame structure is the
same as that of a laminar flame. This allows the separation of chemistry from
turbulence, where laminar flame theory can be used to determine the reaction rate
and transport. The global properties of the flow can then be computed using usual
turbulence modelling techniques. Most models developed for turbulent premixed
combustion use this existence of laminar flamelets (Cant and Mastorakos, 2008).
In the flamelet regimes, fast-chemistry is assumed and the flow consists of
a burnt phase and an unburnt phase, separated by a flame surface made up of
laminar flamelets. In the wrinkled flamelets regime, the laminar flame propaga-
tion overwhelms the flame front corrugation by turbulence. The flame surface
becomes more wrinkled when the turbulence intensity increases, creating pockets
in the flame surface. Flames in this region are called corrugated flamelets.
22
2.5. Premixed combustion sub-models
Unlike the flamelet regimes, in the thin reaction zone, the turbulent eddies of
size ηk enter the preheat zone (since ηk < δ). However, these eddies do not enter
the inner layer as ηk is still larger than the inner layer thickness δ (Peters, 2000).
Thus, the reaction zone still has the laminar flame structure, and the laminar
burning velocity remains well defined.
The above description of various combustion regimes explains why the pre-
mixed laminar flame theory is needed for turbulent combustion models. Even
though such diagrams are highly qualitative, it provides a useful reference for
comparison between modelled and computed flames. In the broken reaction
zones, the turbulent eddies are smaller than the inner layer thickness and can
therefore influence the chemical reactions in the inner layer causing local flame
extinctions. The resulting flame does not have a laminar flame structure and will
not be considered further in this study. The commonly used premixed combus-
tion models are described in the following section, including the model used in
the present work.
2.5 Premixed combustion sub-models
The main objective of turbulent combustion modelling is to provide closure for
the mean reaction rate term, ω˙, appearing in Eq. (2.20). A detailed description
of these combustion models is beyond the scope of this work [see for example
Veynante and Vervisch (2002)], and only a brief overview of Reynolds-averaged
sub-models for turbulent premixed combustion is given here.
2.5.1 Eddy Break-Up (EBU) model
This classical model for the mean reaction rate, first proposed by Spalding (1971),
is based on phenomenological analysis, assuming Re 1 and Da 1. The
reaction zone is described as pockets of burnt and unburnt gases and the mixing
of these pockets by turbulent eddies controls the reaction rate. The reaction rate
23
2.5. Premixed combustion sub-models
according to this model is given as:
ω˙ = −CEBUρ ˜
k˜
√
c˜′′2, (2.40)
where the model constant CEBU is of the order unity. Large values of Re and Da
imply that the combustion is in the flamelets regime, with the flame infinitely
thin and the reaction assumed to be fast. Thus one could estimate the variance as
c˜′′2 = c˜(1− c˜). The biggest limitation of the Eddy Break-Up model in its original
form is that chemical kinetics is not taken into account. Even though EBU-like
models are one of the simplest reaction rate models, they do not represent the
physics of the problem and the fast chemistry assumption leads to overprediction
of reaction rates. A variant of this approach, called the eddy dissipation concept,
is developed to include complex chemical kinetics.
2.5.2 Bray-Moss-Libby (BML) modelling
This flamelet model is based on statistics of the progress variable for thermochem-
ical closure (Bray et al., 1985). The main assumption for this model is that the
turbulent flame is thin and its structure is not disturbed by turbulent structures
in the surrounding fluid. The probability density function (pdf) of the progress
variable is then a sum contributions from fresh gases, burnt mixtures and reacting
mixtures, written as:
p(c;x, t) = α(x, t)δ(ζ) + β(x, t)δ(1− ζ) + γ(x, t)f(ζ;x, t), (2.41)
where α(x, t), β(x, t) and γ(x, t) respectively denote the probability of finding
the reactants and products and reacting mixture, at position x and at time t.
Dirac δ functions δ(ζ) and δ(1− ζ) are used to denote the fresh (c = 0) and fully
burnt (c = 1) gases respectively. When the flame front is taken to be thin (i.e.
Re 1 and Da 1), the probability of finding the reacting gas is negligible
(i.e. γ 1). When the normalisation condition is then applied to the pdf in
24
2.5. Premixed combustion sub-models
Eq. (2.41), the following condition is obtained:
α(x; t) + β(x; t) = 1. (2.42)
The coefficients α and β can be determined as a function of the Favre-averaged
progress variable, c˜, as follows:
ρc˜ =
∫ 1
0
ρ(ζ) ζ p (ζ) dζ = ρbβ, (2.43)
where ρb is the burnt gas density. Using the heat release parameter, τ , one obtains
ρu = ρ(1 + τ c˜) and the probabilities α and β become:
α =
1− c˜
1 + τ c˜
, β =
(1 + τ)c˜
1 + τ c˜
. (2.44)
This means that the thermochemical state of the mixture can be determined by
c˜, which can be obtained from its transport equation. When γ is neglected one
could obtain c˜′′2, this result is used in several models for premixed combustion,
including the EBU model. The consequence of this being that only the conser-
vation equation for the mean progress variable is needed together with the fluid
dynamics equations in solving a turbulent premixed combustion problem.
A gradient transport hypothesis is sometimes used to close the turbulent scalar
flux term, u˜′′c′′, appearing in Eq. (2.20). BML formulation can be used to express
the turbulent fluxes as
ρu˜′′c′′ = ρc˜(1− c˜)(ub − uu), (2.45)
where uu and ub are the conditional mean velocities in unburnt and burnt gases
respectively. For a steady planar flame the mean velocity increases towards the
burnt gas (ub > uu) and ρu˜′′c′′ > 0. This contradicts with the classical gradient
transport assumption:
ρu˜′′c′′ = − µt
Scc
∇c˜. (2.46)
Therefore, the BML model is able to predict the countergradient diffusion phe-
25
2.5. Premixed combustion sub-models
nomena, which has been observed both experimentally (Libby and Bray, 1981)
and numerically (Veynante et al., 1997). Countergradient gradient diffusion yield
a negative turbulent diffusivity, which mainly arises when the local pressure forces
accelerate burnt and unburnt mixtures differentially due to their density differ-
ence.
The BML formulation can also be used to obtain relations between Reynolds-
averaged and Favre-averaged values of c as (Bray, 1980):
c =
(1 + τ)c˜
1 + τ c˜
. (2.47)
The above analysis cannot be used to determine the mean reaction rate term,
ω˙. This is because the flamelet model assumption makes the probability of find-
ing reacting gas negligible and alternative modelling approaches are required to
estimate this term.
One such estimation proposed by Bray (1979) was obtained by analysing
transport equations for c˜ and c˜′′2 to show that the mean reaction rate is propor-
tional to the mean scalar dissipation rate, ˜c = ραc (∇c′′ · ∇c′′)/ρ.
ω˙ =
2
2Cm − 1ρ˜c, (2.48)
where Cm is given by
Cm =
∫ 1
0
ζ ω˙(ζ) p(ζ) dζ∫ 1
0
ω˙(ζ) p(ζ) dζ
, (2.49)
which typically varies between 0.7 and 0.8 for hydrocarbon-air flames (Swami-
nathan and Bray, 2005) and values of Cm for hydrogen-air mixtures at various
equivalence ratios have been reported by Rogerson and Swaminathan (2007).
Since Eq. (2.48) shows a direct link between the reaction rate and the scalar dis-
sipation rate, physically correct modelling of the scalar dissipation rate will enable
one to determine the reaction rate. However, the closure for the mean scalar dis-
sipation rate is challenging for premixed flames (Kolla and Swaminathan, 2010a).
Using the classical model for the mean scalar dissipation rate, ˜c ' c˜′′2/τt, where
the turbulence time scale is estimated as τt = k˜/ε˜, will recover the EBU model.
26
2.5. Premixed combustion sub-models
Another approach to close the mean reaction rate term is based on the crossing
frequency of the flame front at a given point (Bray et al., 1984). The mean
reaction rate is then expressed as the product of the flame crossing frequency,
νf , and the reaction rate per flame crossing, ω˙f : ω˙ = ω˙fνf . By treating the
progress variable signal as a telegraphic signal one can write: νf = 2c(1− c)/Tˆ ,
where the Tˆ is the mean period of a telegraphic signal, which can be estimated as
the turbulence time scale, τt. The Reynolds-averaged progress variable, c can be
obtained from Eq. (2.47). The reaction rate per flame crossing can be expressed
as (Bray et al., 1984): ω˙f = ρus
0
L/|σf |, where |σf | is a flamelet orientation factor.
2.5.3 Flame surface density (FSD) modelling
This widely used flamelet approach is based on the earlier coherent flame model
(CFM) of Marble and Broadwell (1977). In this model, the mean reaction rate
is taken as the product of reaction rate per unit flame area and the flame surface
area per unity volume, Σ, which is known as the flame surface density (Marble
and Broadwell, 1977): ω˙ = ρu〈sc〉sΣ, where sc is the flame consumption speed
and 〈 〉s denotes averaging over the flame surface. The stretching of the flame
surface by turbulent eddies can influence the flame propagation speed and it is
useful to write 〈sc〉s = s0LI0 (Bray, 1990), where I0 is a stretch factor that takes
into account the effects of stretch on sL. This approach is attractive because the
turbulence-combustion interaction is modelled by the flame surface density, Σ,
which can be separated from complex chemistry effects incorporated into s0L and
I0.
Two main approaches are used to close Σ; in one method an algebraic expres-
sion is used (Bray et al., 1984), and in the other method a modelled balanced
equation (Pope, 1988; Candel and Poinsot, 1990) is solved. A simple algebraic
model is given as (Bray and Swaminathan, 2006)
Σδ0L '
2CDc
(2Cm − 1)
ρ
ρu
(
1 +
2
3
Cεcs
0
L
1√
k˜
)(
1 +
CDε˜δ
0
L
CDc k˜s
0
L
)
c˜′′2, (2.50)
where the three model parameters are of order unity. An exact balance equation
for Σ written after Favre decomposition is given as (Pope, 1988; Candel and
27
2.5. Premixed combustion sub-models
Poinsot, 1990):
∂Σ
∂t
+∇ · 〈u〉sΣ = Σ〈∇T · u˜+∇T · u′′ +∇T · (s′dnˆ)〉s
− Σ∇ · 〈u′′〉s − Σ∇ · 〈s′dnˆ〉s, (2.51)
where s′d is the displacement speed of the surface relative to the unburnt mixture
and nˆ is the unit normal vector to the flame front pointing towards the fresh gases.
This balance equation for the flame surface density is not closed and modelling
is required to close various terms. Such modelling has been the subject of many
past studies [see Veynante and Vervisch (2002) for a summary of such studies].
One such modelled transport equation is given as (Boudier et al., 1992)
∂Σ
∂t
+∇ · (uΣ)−∇ ·
(
νt
σΣ
∇Σ
)
= ακΣ− β
ωL
(
1 + a
√
k/s0L
)
ρY f
Σ2, (2.52)
where ωL is the rate of fuel per unit flame area, Y f is the mean fuel mass fraction
and α, a, β and σΣ are model parameters.
2.5.4 The level set approach (G-equation)
In this approach, the flame surface is represented by a level set of a non-reacting
scalar, G, at G(x, t) = G0, where G0 is the value at the flame front. This approach
is suitable to model corrugated flamelets and thin reaction zone regimes (Peters,
2000). The original form of the G-equation introduced by Williams (1985b) is
given as
∂G
∂t
+ u · ∇G = s′d|∇G|. (2.53)
The above equation is modified by first including the stretch effects through the
displacement speed (Matalon and Matkowsky, 1982):
s′d = s
0
L − s0LLκc − aL, (2.54)
28
2.5. Premixed combustion sub-models
where a = −n · ∇u · n is the strain rate and κc = ∇ · n is the flame curvature.
The flame normal vector is given by n = −∇G/|∇G|. Substitution of Eq. (2.54)
into Eq. (2.53) gives the modified G-equation:
∂G
∂t
+ u · ∇G = s0L|∇G| −DLκc|∇| − aL|∇G|, (2.55)
where DL = s
0
LL is the Markstein diffusivity. This equation is only valid for
corrugated flamelets regime. The G-equation for thin reaction zones regime is
given by Peters (1999)
∂G
∂t
+ u · ∇G = (sn + sr)|∇G| −Dκc|∇G|, (2.56)
where sr and sn are the reaction and normal diffusion component to the dis-
placement speed s′d and −Dκc is the tangential diffusion component. Transport
equations for G˜ and G˜′′2 have been developed (Peters, 1992, 1999) for RANS
simulations of turbulent combustion, where the transport equation for G˜ is given
by
∂G˜
∂t
+ u˜ · ∇G˜ = sT |∇G˜| −Dκc|∇G˜|. (2.57)
This equation requires a model for the turbulent flame speed, sT , which is usually
of the general form (Peters, 1999)
sT = s
0
L
[
1 + A
(
u′/s0L
)n]
, (2.58)
where n is around 0.7 (Williams, 1985a).
This level set approach removes the complexities involved with counter-gradient
diffusion since turbulent transport terms normal to the flame front do not appear
in the balance equations for G. In addition, this approach does not require a
closure model for a chemical source term since, by definition, G is a non-reactive
scalar. However, it has been noted by Williams (2001) that this approach is
strictly valid for wrinkled flamelets.
29
2.5. Premixed combustion sub-models
2.5.5 Conditional moment closure (CMC)
The models presented so far are based on conventional averages. In turbulent
reacting flows, when conventional averaging is used, the large spatial and tempo-
ral fluctuations of scalar quantities makes it difficult to obtain accurate closures
for the mean reaction rate (Borghi, 1974). To overcome this issue, Klimenko
(1990) and Bilger (1993), independently derived a model for non-premixed tur-
bulent combustion, where the averages of the reactive scalars are conditioned on
the mixture fraction. This conditional averaging makes the fluctuations around
the mean smaller. Klimenko and Bilger (1999) later proposed an extension of
this conditional moment closure (CMC) method to premixed combustion, where
conditioning is done with the progress variable, c. Recently, Amzin et al. (2012)
successfully applied the CMC method to premixed combustion of a pilot stabilised
Bunsen flame.
The conditional mean of a scalar, k, is given as: Qk ≡ 〈Yk|c = ζ〉 ≡ 〈Yk|ζ〉,
where the angled brackets denote ensemble averaging, ζ is the sample space vari-
able of c and Yk is a reactive scalar. Transport equations for the conditional mean
scalar values, Qk are obtained by substituting Yk(x, t) = Qk(c;x, t)+yk(x, t) into
the instantaneous equation for the reactive scalars [for example Eq. (2.10)] (Bil-
ger, 1993). Alternatively, one could obtain the same equation by using the joint
pdf transport equation (Klimenko, 1990). For premixed flames the conditional
moment closure equation can be written as (Klimenko and Bilger, 1999; Swami-
nathan and Bilger, 2001a)
〈ρ|ζ〉∂Qk
∂t
+ 〈ρu|ζ〉 · ∇Qk − Lec
Lek
〈ρNc,k|ζ〉∂
2Qk
∂ζ2
= 〈ρω˙k|ζ〉 − 〈ρω˙|ζ〉∂Qk
∂ζ
− 1
p˜ (ζ)
∇ · [〈ρu′′Y ′′k |ζ〉 p˜ (ζ)] + eQk , (2.59)
where Lek is the Lewis number of species k, p˜ is the Favre pdf of c and ω˙k and
ω˙ are given by Eqs. (2.10) and (2.14) respectively. The instantaneous scalar
dissipation rate of species k is defined as Nc,k = Dk (∇c · ∇c). The symbol eQk is
30
2.5. Premixed combustion sub-models
defined as (Swaminathan and Bilger, 2001a)
eQk ≡ ∇ · (ρDk∇Qk) +
〈
ρDk∇c · ∇∂Qk
∂ζ
〉
+
〈
∂Qk
∂ζ
∇ · [(1− Lek) ρDk∇c|ζ]
〉
. (2.60)
The terms 〈ρu′′Y ′′|ζ〉, p˜(ζ), 〈ρu|ζ〉, 〈ω˙k|ζ〉 and 〈Nc,k|ζ〉 need to be modelled
and various closure models have been discussed by Klimenko and Bilger (1999)
and Swaminathan and Bilger (2001a). The closure of the conditional mean scalar
dissipation rate, 〈Nc,k|ζ〉, is linked to the unconditional mean scalar dissipation
rate, ˜c (Swaminathan and Bilger, 2001b). Modelling of this unconditional mean
scalar dissipation rate is discussed in section 2.5.8.
The sound theoretical basis for CMC equations means that, when compared
with closure models based on phenomenology or physical analyses, it is expected
to give improved predictions for species with slow time scales, such as pollutants
(Klimenko and Bilger, 1999). In addition, CMC may be valid for all regimes of
premixed combustion since no explicit approximations are made on the influence
of turbulent eddies on the flame structure (Amzin et al., 2012). However, one of
the drawbacks of CMC is that it is more expensive than the flamelet methods
described earlier, since the number of transport equations that need to be solved
are equal to (N+1) times the number of points in the conditioning variable space.
Note that here N refers to the number of species and is not to be confused with
the instantaneous scalar dissipation rate Nc.
2.5.6 Transported pdf approach
In this approach, statistical methods based on the stochastic nature of turbulence
are used to calculate turbulent reacting flows. A pdf can be used to determine the
likelihood of finding a particular solution, given certain physical conditions of the
flow at a given time. A complete statistical description of a turbulent reacting
flow can be obtained by defining the joint pdf for the velocity components and
the thermochemical variables (Pope, 1985). If the pdfs of the thermochemical
31
2.5. Premixed combustion sub-models
variables are known, the mean reaction rate can be obtained from
ω˙ =
∫
ψ
ω˙ p˜(ψ;x, t) dψ, (2.61)
where ψ = (Y1, Y2, . . . , YN , T ) is the scalar field vector that represent the species
mass fractions of N species in addition to temperature and p˜(ψ;x, t) is the
Favre-averaged joint pdf of this scalar field, ψ. This statistical description of
the stochastic flow field is applicable to premixed, non-premixed and partially
premixed combustion (Borghi, 1988).
The following exact evolution equation for the joint pdf p˜(ψ) (for ease of
notation the dependence of x and t is omitted) has been derived by Dopazo and
O’Brien (1974) and Pope (1976) [the details of this derivation can be found in
O’Brien (1980)].
∂
∂t
[
ρp˜(ψ)
]
+∇ · [ρu˜p˜(ψ)]+ N∑
α=1
∂
∂ψα
[
ω˙α(ψ)ρp˜(ψ)
]
= −∇ · [ρ〈u′′|ψ〉p˜(ψ)]+ N∑
α=1
∂
∂ψα
[〈∇ · (Jα|ψ)〉p˜(ψ)] , (2.62)
where ω˙α and Jα respectively represent the reaction per unit volume and the
molecular diffusive flux of species α. The unsteady, convection and chemical
source production terms (the three terms on the LHS of the equation) are closed.
This exact form of the chemical source term implies that finite chemistry can be
implemented in the calculations without the need to define the flame structure,
which is the main reason why the transported pdf approach is attractive for
reacting flows (Jones, 2002). The turbulent flux term and the micromixing term
on the RHS of this equation are unclosed.
Jones (2002) notes that the classical closure problem of turbulent combustion
has been transferred mathematically from the mean reaction term in moment
closures to the micromixing terms in the pdf closures. The turbulent transport
term can be modelled using gradient transport assumptions and the standard k−ε
model has been used to obtain the turbulent flow field (Pope, 1985). Alternatively,
one could use the velocity-composition joint pdf, p(u, ψ;x, t), which makes the
32
2.5. Premixed combustion sub-models
turbulent diffusion term exact and no turbulence model is required. However, this
introduces further unknowns into the pdf transport equation (Pope, 1985). Many
models have been proposed for the micromixing term [see for example Haworth
and Pope (2011) for a discussion of these models].
Solving the pdf transport equation by conventional numerical methods is ex-
pensive and alternatives such as the Monte-Carlo method applied for turbulent
reacting flows (Pope, 1981) have to be used. In such methods the computa-
tional cost of the problem increases only linearly with the dimensionality of the
pdf. However, the transported pdf methods are still more complex and time-
consuming compared to the moment methods and its practical use in industrial
applications is limited (Poinsot and Veynante, 2005).
2.5.7 Presumed pdf approach
In general, the pdf, p˜(ψ;x, t), in Eq. (2.61) can take any shape. However, in
some combustion problems, certain similarities can be seen in the pdf functions
(Poinsot and Veynante, 2005). This is the motivation behind presuming a certain
shape for the pdf. In this approach the pdf shapes are parametrised using mo-
ments, which are obtained from their balance equations (Borghi, 1988). It can be
easily envisaged that this method is much less time-consuming than solving the
pdf transport equation, and therefore more suitable for three-dimensional flow
calculations of industrial problems. An important point to note here is that the
presumed pdf approach is only applicable to cases where the scalar fluctuations
are small, therefore, its applicability is only limited to premixed combustion or
non-premixed flames with fast chemical reactions (Borghi, 1988).
The BML formulation described in section 2.5.2 is one such example of a
presumed pdf model where a bimodal pdf is used. The shape of the presumed
pdf determines the number of scalar moments balance equations that need to
be solved. For the BML formulation only the first moments, c˜, is required. In
contrast the popular pdf shape, β-function pdf, requires the first two moments,
c˜ and c˜′′2.
Presumed pdf approach can also be used for the level set formulation. For
example, if the non-reactive scalar G is assumed to be well-defined and well-
33
2.5. Premixed combustion sub-models
behaved outside the surface, G(x, t) = G0, then a pdf, p(G;x, t), can be defined
for G. According to the presumed pdf approach, the first two moments of G
obtained from their balance equations can be used to parametrise and calculate
p(G;x, t) (Peters, 2000). In comparison with the transported pdf approach, both
the BML and G-equation formulations have: i) reduced the dimensionality of the
problem by considering only the progress variable (in case of the BML model) and
the scalar G (in G-equation formulation) to describe the thermochemical state of
the problem and ii) used a presumed pdf approach, which is not only less costly
computationally but also separates the complex chemistry from fluid dynamics.
In premixed flames, the reaction rate can be closed using the presumed pdf
approach as
ω˙ =
∫
ω˙(ζ) p(ζ) dζ, (2.63)
where the function ω˙(ζ) is obtained using canonical laminar flame calculations
having the same thermochemical attributes as that of the turbulent flame. In
the above formulation, it is assumed that the flame structure is undisturbed
by turbulent eddies. The fluid dynamic stretch effects can be included using
(Bradley, 1992)
ω˙ =
∫
p(κ) dκ
∫
ω˙(ζ) p(ζ) dζ, (2.64)
where statistical independence of ζ and κ has been assumed. Recently, Kolla and
Swaminathan (2010a) used the scalar dissipation rate to parametrise flamelet
stretch effects. This method is described in the following section.
2.5.8 Scalar dissipation rate (SDR) based modelling
In order to understand the physical significance of scalar dissipation rate, it is
helpful to look at the transport equation of the progress variable variance, c˜′′2.
These fluctuations represent inhomogeneities and intermittencies (Veynante and
Vervisch, 2002). One can obtain a transport equation for the scalar variance
as follows: first subtract the Favre-averaged transport equation for the progress
34
2.5. Premixed combustion sub-models
variable [Eq. (2.20)] from its instantaneous equation [Eq. (2.14)] to obtain an
equation for c′′. Then multiply this equation by 2c′′ and average the result to
obtain the following exact transport equation for c˜′′2 [see for example Veynante
and Vervisch (2002)]
∂ρc˜′′2
∂t
+∇ · (ρu˜c˜′′2) = ∇ ·
(
ρDc∇c′′2
)
+ 2c′′∇ · (ρDc∇c˜)︸ ︷︷ ︸
molecular diffusion
− ∇ · (ρu′′c′′2)︸ ︷︷ ︸
turbulent transport
− 2(ρu′′c′′) · ∇c˜︸ ︷︷ ︸
production
+ 2ω˙′′c′′︸ ︷︷ ︸
reaction
− 2ρDc(∇c′′ · ∇c′′)︸ ︷︷ ︸
dissipation
. (2.65)
Note that in the above equation, the ‘production term’ may take negative values
(i.e. act as a sink term) in the case of countergradient flames. The last term in
the above equation is unclosed and it is known as the mean scalar dissipation
rate ˜c. It can be written as
ρ˜c = ρDc∇c′′ · ∇c′′. (2.66)
Scalar dissipation rate measures the decay rate of scalar fluctuations by turbulent
micromixing. It can be used to determine the mixing rate at the molecular level,
since a well-mixed fluid does not have any fluctuations (Cant and Mastorakos,
2008). By taking the average of the instantaneous dissipation rate, N , it can be
shown that (Veynante and Vervisch, 2002)
ρN = ρDc∇c · ∇c = ρDc∇c˜ · ∇c˜+ 2ρDc∇c′′ · ∇c˜+ ρDc∇c′′ · ∇c′′. (2.67)
The assumption, ρN ≈ ρ˜c, can be made since gradients of the fluctuations are
much larger than the gradients of the mean progress variable.
It is the micromixing between burnt products and unburnt reactants that
sustains combustion in premixed flames. For example, Eq. (2.65) clearly shows
the coupling between turbulent mixing (˜c) and chemical reaction (ω˙′′c′′). Since
turbulent mixing also leads to dissipation of scalar fluctuations, the scalar dissi-
pation rate is an important quantity in turbulent premixed combustion (Libby
and Bray, 1980) and appears directly or indirectly in most combustion submodels
(Veynante and Vervisch, 2002).
35
2.5. Premixed combustion sub-models
In addition to having unclosed reaction rate terms, ω˙ and ω˙′′c′′, the mean
scalar dissipation rate, ˜c, term is also unclosed. This poses an additional chal-
lenge in using Eq. (2.65) to solve premixed combustion problems. It is known
that the modelling of ˜c is challenging for premixed flames (Mantel and Borghi,
1994; Mantel and Bilger, 1995; Swaminathan and Bray, 2005). Such a model
should include the interaction between turbulence, chemical reaction and molec-
ular diffusion, and the earlier models that only used the turbulence time scale
were inadequate in capturing the correct physics (Swaminathan and Bray, 2005).
Mean scalar dissipation rate closure
Swaminathan and Bray (2005) derived the following exact equation for the mean
scalar dissipation rate, ˜c
ρ
∂˜c
∂t
+ ρu˜ · ∇˜c = ∇ · (ρDc∇c)︸ ︷︷ ︸
D1
− 2ρDc2 [∇ (∇c′′)]2︸ ︷︷ ︸
D2
+ T1 + T2 + T3 + T4, (2.68)
T1 = −∇ · (ρu′′c)︸ ︷︷ ︸
T11
− 2ρDcu′′ · ∇c′′(∇(∇c˜))︸ ︷︷ ︸
T12
,
T2 = 2ρc∇ · u,
T3 = −2(ρDc∇u′′ · ∇c′′)∇c˜︸ ︷︷ ︸
T31
− 2ρDc∇c′′ · ∇u′′ · ∇c′′︸ ︷︷ ︸
T32
− 2(ρDc∇c′′ · ∇c′′)∇ · u˜︸ ︷︷ ︸
T33
,
T4 = 2Dc∇c′′ · ∇ω˙′′,
when the diffusivity Dc has a weak dependence on temperature. The LHS rep-
resents unsteady and convective terms. Molecular diffusion and dissipation are
denoted by D1 and D2 respectively. Turbulent transport of ˜c is denoted by T1.
The dilatation term is denoted by T2, which was shown to be significant at all
Da using an order of magnitude analysis by Swaminathan and Bray (2005). This
term was absent in the earlier exact equation derived by Borghi and co-workers
(Borghi, 1990; Mantel and Borghi, 1994; Mura and Borghi, 2003), where they used
a constant density approximation. Influences of the interaction between turbu-
36
2.5. Premixed combustion sub-models
lence and scalar concentration is denoted by T3, while the influence of chemical
reactions is denoted by T4.
Details of this transport equation as well as the results obtained from a number
of studies using this equation were summarised by Chakraborty et al. (2011).
This exact equation, together with necessary DNS data validations, facilitated
the derivation of an accurate algebraic model for the mean scalar dissipation
rate, ˜c, by Kolla et al. (2009). In deriving this model, Kolla et al. (2009) used
closure models proposed by Chakraborty et al. (2008) for the leading order terms
of Eq. (2.68), when the Damko¨hler number is large. This model is written as
˜c ' 1
β′
[
(2K∗c − τC4)
s0L
δ0L
+ C3
ε˜
k˜
]
c˜′′2, (2.69)
Various model parameters are: β′ = 6.7, C3 = 1.5
√
Ka/(1 +
√
Ka) and C4 =
1.1/(1+Ka)0.4. The model constant, K∗c , depends on the thermochemistry of the
mixture, and it is chosen such that its sensitivity to internal flame front struc-
ture is small (Kolla et al., 2009). Values of K∗c for hydrocarbon and hydrogen-
air flames at different equivalence ratios have been calculated by Rogerson and
Swaminathan (2007). These parameters are specified to satisfy certain physical
aspects of turbulence-flame interaction (Kolla et al., 2009; Kolla and Swami-
nathan, 2011) and cannot be changed arbitrarily. In this work three different
reaction rate closures that uses the mean scalar dissipation rate model will be
used. These models are given in section 3.2.
37
3. Numerical setup for spherical
flame simulation
One of the objectives of this work, as outlined in section 1.3, is to validate com-
bustion models based on the scalar dissipation rate for spherically propagating
flames. These flames are of both fundamental and practical relevance, and can
be found in a number of engineering devices as well as in accidental explosions of
vapour clouds released into the atmosphere. This chapter discusses the numerical
method used to simulate such flames.
In this work, an in-house CFD code that was previously used to study freely
propagating planar flames, is modified to study spherically propagating flames.
Governing equations written in spherical coordinates are implemented in the code.
These spherical equations are given in section 3.1. The algebraic model for ˜c
given in Eq. (2.69) is used in this work. A modification to this model is proposed
in section 3.2, which includes the mean curvature effects for a spherical flame.
Spherical flames simulated in this work are computed using three different
models: strained, unstrained and algebraic flamelet models. These models were
explained in section 2.5 and some additional details are discussed here. Both
unstrained and strained flamelet models require a flamelet library that will be
accessed during the CFD simulations. The generation of these flamelet libraries
is described in section 3.3.
Description of the CFD code, including the numerical schemes, initial and
boundary conditions are given in section 3.4. The CFD code developed here
is used to simulate spherically propagating methane- and hydrogen-air flames,
which are presented in Chapters 4 and 5 respectively. Note that the description
of the reaction rate models and the flamelet library generation is relevant not
only to these spherical flames, but also to other cases simulated in this work.
38
3.1. Governing equations and modelling
3.1 Governing equations and modelling
The unsteady RANS (URANS) approach is used to simulate spherical turbulent
explosions. These flames are assumed to be spherically symmetric, which re-
sults in considerable simplification, since only the radial terms in the governing
equations written in (r, θ, φ) coordinates need to be retained.
The Favre-averaged, RANS equations given in section 2.2.3 can be written for
a spherically symmetric flow field (see Appendix A on how the spherically sym-
metric radial momentum equation is obtained). Continuity and radial momentum
conservation are then given by
∂ρ
∂t
+
1
r2
∂r2ρu˜r
∂r
= 0, (3.1)
∂ρu˜r
∂t
+
1
r2
∂
∂r
[
r2 ρu˜2r
]
= −∂p
∂r
+
1
r2
∂
∂r
[
r2
(
τ rr − ρu′′r2
)]
−
(
τ θθ − ρu′′θ2 + τφφ − ρu′′φ2
)
r
, (3.2)
where τ rr, τ θθ and τφφ denote the normal components of the viscous stress tensor
in the respective directions. The centrifugal forces per unit volume arising from
the Reynolds stresses in θ and φ directions are ρu′′θ
2/r and ρu′′φ
2/r respectively,
which do not vanish even in the spherically symmetric case. Thus, they must
be retained as their contributions are significant in the earlier period of flame
development (small r).
The progress variable equation [Eq. (2.20)] can be written for high Reynolds
number flows as
∂ρ c˜
∂t
+
1
r2
∂
∂r
(
r2ρ u˜rc˜
)
=
1
r2
∂
∂r
{
r2
[(
αc +
µt
Scc
)
∂c˜
∂r
]}
+ ω˙, (3.3)
where ω˙ is the mean rate of production of c˜ per unit volume, which is modelled
using the flamelet models to be explained in section 3.2. The following transport
39
3.1. Governing equations and modelling
equation for the progress variable variance, c˜′′2
∂ρ c˜′′2
∂t
+
1
r2
∂
∂r
(
r2ρu˜rc˜′′2
)
=
1
r2
∂
∂r
{
r2
[(
αc +
µt
Scc
)
∂c˜′′2
∂r
]}
+ 2
µt
Scc
∂c˜
∂r
2
− 2ρ˜c + 2ω˙′′c′′, (3.4)
is also included in the simulation. The mean scalar dissipation rate, ˜c, is closed
using the model given in Eq. (2.69). In the next section, a modification to this
model is proposed to simulate spherical flames. The mean density is calculated
using the equation of state written as ρ = ρu/ (1 + τ c˜).
Uncertainties related to turbulence modelling is minimised in these spherical
flame simulations by using the k˜-ε˜ equations (Jones and Launder, 1972) given by
∂ρ k˜
∂t
+
1
r2
∂
∂r
[
r2 ρ u˜rk˜
]
=
1
r2
∂
∂r
{
r2
[(
µ+
µt
Sck
)
∂k˜
∂r
]}
− ρu′′r2
(
∂u˜r
∂r
)
−
(
ρu′′θ
2 + ρu′′φ
2
) u˜r
r
− u′′r
∂p
∂r
+ p′
1
r2
∂ (r2u′′r)
∂r
− ρε˜, (3.5)
∂ρ ε˜
∂t
+
1
r2
∂
∂r
[
r2(ρ u˜rε˜)
]
=
1
r2
∂
∂r
{
r2
[(
µ+
µt
Scε
)
∂ε˜
∂r
]}
− Cε1 ε˜
k˜
[
ρu′′r
2
(
∂u˜r
∂r
)
+
(
ρu′′θ
2 + ρu′′φ
2
) u˜r
r
− u′′r
∂p
∂r
]
− Cε2ρε˜
2
k˜
, (3.6)
where µ and µt represent the molecular and eddy viscosities respectively. The
model constants are C1 = 1.44, C2 = 1.92 and Sck = Sc = 1. The compressible
form of these equations are written above (Wilcox, 1993), where the second and
third terms appearing on the RHS of Eq. (3.5) represent the production of k˜ by
the gradients of mean velocity. The next two terms respectively represent the
effects of mean pressure gradient and pressure-dilatation. The dissipation of k˜ is
represented by the last term of Eq. (3.5). Reynolds stresses are modelled using
40
3.2. Reaction rate closures
the eddy-viscosity hypothesis as
ρu′′r
2 = −2µt∂u˜r
∂r
+
2
3
µt
[
1
r2
∂
∂r
(
r2u˜r
)]
+
2
3
ρk˜, (3.7)
ρu′′θ
2 = ρu′′φ
2 = −2µt u˜r
r
+
2
3
µt
[
1
r2
∂
∂r
(
r2u˜r
)]
+
2
3
ρk˜. (3.8)
If one uses an anisotropic turbulence model then ρu′′θ
2 and ρu′′φ
2 will be dif-
ferent. The pressure work and pressure-dilatation terms are often neglected
or combined with the diffusive term in reacting flow simulations, while these
are modelled explicitly in this study. The pressure-dilatation is modelled as
p′(1/r2) [∂(r2u′′r)/∂r] = 1/2c˜ (τs
0
L)
2
ω˙ (Zhang and Rutland, 1995). The average of
u′′ in the pressure work term is modelled (Libby, 1985) as u′′ = u˜′′c′′τ/ (1 + τ c˜),
where the turbulent scalar flux u˜′′c′′ is modelled using the classical gradient trans-
port. It is well known that this scalar flux can be counter-gradient in premixed
flames, which can be included in simulations using second order closures. How-
ever, the gradient model is used in this work for the sake of simplicity and its
validity can be evaluated from the experimental comparisons to be shown in
Chapters 4 and 5. Although it is ideal to include the pressure-dilatation effect in
both k˜ and ε˜ equations it is included only in k˜ equation following many previous
studies (Bray et al., 1985; Jones, 1994; Kolla and Swaminathan, 2010a). Also,
the effects of these terms may be small for open flames (Swaminathan and Bray,
2011).
3.2 Reaction rate closures
The mean reaction rate, ω˙, is modelled using the scalar dissipation rate based
approach described in section 2.5.8. Three different reaction rate closures that
uses the mean scalar dissipation rate modelling approach are described in this
section.
First is the algebraic model of Bray (1979), which was given in Eq. (2.48).
For this model, the source term in the progress variable variance, c˜′′2, equation
is given as: ω˙′′c′′ = (Cm − c˜)ω˙. This model does not involve complex chemical
kinetics and, therefore, finite rate chemistry effects cannot be taken into account.
41
3.2. Reaction rate closures
Note that the mean scalar dissipation rate model of Kolla et al. (2009) [given in
Eq. (2.69)] is used in this study because it is simple and satisfies the realisability
condition (˜c ≥ 0).
The second model is an unstrained flamelet model, where the reaction rate
closure is given by Eq. (2.63). The pdf, p (ζ), in this equation is usually presumed
to be a β-function, which requires both c˜ and c˜′′2. Detailed chemistry can be used
to obtain, ω˙ (ζ) from laminar flame calculations. This model does not involve ˜c
explicitly, however, in RANS calculations involving the use of this model, one
also need to solve the variance equation, which includes ˜c as a source term. For
this model, the unclosed source term ω˙′′c′′ in the c˜′′2 is given by
ω˙′′c′′ =
∫ 1
0
ζω˙0(ζ)p(ζ) dζ − ω˙c˜. (3.9)
For given values of c˜ and c˜′′2 one can build a look-up table or flamelet library
for ω˙ and other required quantities using laminar flame solutions. This look-up
table is accessed during turbulent flame simulations to obtain the source terms
required for Eqs. (2.20) and (2.65).
The third model is a strained flamelet model proposed by Kolla and Swami-
nathan (2010a). In this model, the flamelets are parametrised using the scalar
dissipation rate. This model is described briefly here and elaborate detail can
be found in Kolla and Swaminathan (2010a). The flamelets, which are freely
propagating laminar flames and those established in opposing flows of reactant
and product, are parametrised using ˜c. The mean reaction rate is given by
ω˙ =
∫ 1
0
[∫ N2
N1
ω˙(ζ, ψ) p(ψ|ζ) dψ
]
p(ζ) dζ, (3.10)
where ζ and ψ are the sample space variables for c and the instantaneous scalar
dissipation rate, Nc, respectively. The reaction rate of flamelets, ω˙(ζ, ψ), and
the integration limits N1 and N2 are obtained using results of fully burning and
almost extinguished flamelets. As with the unstrained flamelet model, complex
chemistry can be used in the look-up table generation.
The presumed shapes for the pdfs, p(ζ) and p(ψ|ζ), are specified using the β
and lognormal functions respectively. Lognormal function for p(ψ|ζ) requires the
42
3.2. Reaction rate closures
conditional mean and variance of the natural logarithm of the conditional scalar
dissipation rate, i.e. ln(Nc|ζ). The log-normal pdf is given by
p(ψ|ζ) = 1
(ψ|ζ)σ√2pi exp
{
1
2σ2
[
ln (ψ|ζ)− µ2]} (3.11)
where the mean and variance of ln (Nc|ζ) are denoted by µ and σ2 respectively.
These quantities are related to the conditional mean, 〈Nc|c = ζ〉 and conditional
variance, G2N , of the scalar dissipation rate via 〈Nc|ζ〉 = exp (µ+ 0.5σ2) and
G2N = 〈Nc|ζ〉2 (expσ2 − 1). These dissipation related quantities are obtained fol-
lowing the method described by Kolla and Swaminathan (2010a), where they used
the assumption that effects of strain rate on the progress variable gradient are
primarily felt in the inner reaction zone (denoted by ζ∗). This enables one to ap-
proximate the scalar dissipation rate for a given strain rate, a, as: Nc (ζ, a) f (ζ).
Kolla and Swaminathan (2010a) showed that f (ζ) curves for various strain rates
collapse well in the region of the flame where chemical reactions dominate. Using
these approximations, they obtained the following expression for the conditional
mean scalar dissipation rate
〈Nc|ζ〉 ≈ ˜cf(ζ)∫ 1
0
f(ζ)P˜ dζ
. (3.12)
The source term, ω˙′′c′′, using the strained flamelet model is given by
ω˙′′c′′ =
∫ 1
0
ζ〈ω˙|ζ〉p(ζ) dζ − ω˙c˜. (3.13)
where the conditional reaction rate, 〈ω˙|ζ〉 is given by the integral in the square
brackets of Eq. (3.10). The look-up table built for this model is three-dimensional,
where the source terms are a function of c˜, c˜′′2 and ˜c.
All three models require an algebraic closure for the mean scalar dissipation
rate, ˜c. The model proposed by Kolla et al. (2009) [Eq. (2.69)] includes the
effects of curvature induced stretch on flamelets and various important effects
of turbulence and its interaction with chemical reaction and molecular diffusion.
The parameter β′ specifically represent the flamelet curvature induced effects.
43
3.3. Flamelet library generation
However, a spherical flame brush also experiences stretch due to its mean cur-
vature, which is absent in a planar case. This particular effect is not included
in Eq. (2.69), and thus an additional correction can be included based on the
analysis of Chakraborty et al. (2010). This revised model written as
˜c ' 1
β′
{[
2K∗c − τC4
(
1− Du
s0L
∇ · n
)]
s0L
δ0L
+ C3
ε˜
k˜
}
c˜′′2, (3.14)
is obtained through a leading order balance analysis, similar to Kolla et al. (2009),
using the models proposed by Chakraborty et al. (2010). The normal vector in
Eq. (3.14) is defined as n = −∇c˜/|∇c˜|. The major difference between the models
in Eqs. (2.69) and (3.14) is the contribution of flame brush curvature ∇·n. Note
that the revised model in Eq. (3.14) is unconditionally realisable for explosion
but the realisability condition imposes a minimum radius for implosion. Both
models in Eqs. (2.69) and (3.14) are used in this study to understand the extent
of influence of ∇ · n.
Both the unstrained and the strained flamelet models require flamelet li-
braries, which are generated from laminar flame calculations. The generation
of these libraries is described in the next section.
3.3 Flamelet library generation
Details of flamelet library generation for both strained and unstrained flamelets
are described in this section. Note that a similar procedure is used to generate
all the flamelet libraries used in this work.
3.3.1 Unstrained flamelet model
Chemistry can be separated from fluid mechanics in flamelet models and the
flamelet libraries are generated a priori. This makes it computationally eco-
nomical to use detailed chemistry in turbulent combustion simulations. Before
proceeding with the flamelet library generation it is important to ensure that the
chemical mechanism can predict laminar flame speeds found in the literature.
For the unstrained flamelet model given in Eq. (2.63), one needs to calculate
44
3.3. Flamelet library generation
freely propagating laminar premixed flames for a particular fuel-air mixture at a
given equivalence ratio and thermodynamic conditions. These laminar flames are
calculated using CHEMKIN’s PREMIX code (Kee et al., 1985), which gives the
instantaneous reaction rate, ω˙(ζ), as a function of the progress variable, c. Since
the unstrained flamelet library is two-dimensional, a bilinear interpolation with c˜
and c˜′′2 is used during the CFD simulations to obtain source terms, ω˙ and ω˙′′c′′.
3.3.2 Strained flamelet model
As described in section 3.2, the strained flamelet model requires the calculation
of both freely propagating and reactant-to-product (RtP) counterflow laminar
flames. These counterflow flames are calculated using CHEMKIN’s OPPDIF
code (Lutz et al., 1997).
Figure 3.1(a) shows the normalised reaction rate of the progress variable,
ω˙+ = ω˙ (δ0L/ρus
0
L), for various normalised strain rates, a
+, plotted against the
normalised distance from the stagnation plane. The plot shown in this figure
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-8.0 -6.0 -4.0 -2.0 0.0 2.0
ω˙
+
x/δ0L
a+ = 0.14
0.56
1.69
2.14
(a)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.0 0.2 0.4 0.6 0.8 1.0
f
(ζ
)
ζ
(b)
Figure 3.1: Reactant-to-product, stoichiometric methane-air flame: (a) Reaction
rate of the progress variable, ω˙+ for various strain rates, a+, plotted against the
distance from the stagnation plane. All quantities in this figure are normalised;
(b) curves of f(ζ) up to the point the reaction zone reaches the stagnation plane
(unstrained, freely propagating flame is shown as a solid line).
is for a stoichiometric methane-air flame. The values of the unburnt mixture
45
3.3. Flamelet library generation
density, ρu, laminar flame thermal thickness, δ
0
L, and the unstrained laminar
flame speed, s0L, for this mixture are obtained from freely propagating laminar
flame calculations.
In this figure, the location at which x = 0 is defined as the stagnation plane,
where the region x < 0 corresponds to reactants and x > 0 corresponds to
products. This figure shows that increasing the strain rate reduces the reaction
rate and moves the flame towards the product stream. Also, the flame reaches
the stagnation plane at a strain rate of roughly 1602 1/s. Further increasing the
strain rate reduces the reaction until extinction takes place.
Figure 3.1(b) shows curves for f (ζ) ≈ Nc (ζ) /Nc (ζ∗) for this flame at various
rates of strain (solid line in this figure is for the unstrained, freely propagating
flame). Here the inner reaction zone, ζ∗ ≈ 0.7, and a good collapse of these curves
is obtained in the region where the chemical reactions dominate. As already
described in section 3.2, this collapse in f (ζ) helps to simplify the calculation of
the conditional mean scalar dissipation rate given in Eq. (3.12).
In order to calculate the mean reaction rate using the strained flamelet for-
mulation, one needs to understand the response of the reaction rate to strain
through changes in the scalar dissipation rate. Figures 3.2(a) and 3.2(b) show
the plots of ω˙+ vs. N+c for methane-air flames with Φ = 1.0 and Φ = 1.1 respec-
tively. Note that the superscript + is to denote a quantity that is normalised
using laminar flame values (i.e. laminar flame speed and thickness). Each sym-
bol on this figure corresponds to a particular laminar flame calculation, with the
maximum reaction rate value given by the unstrained, freely propagating flame.
It is interesting to note that even though these flames have Lewis numbers close
to unity, the response of the reaction rate to strain is considerably different. For
the stoichiometric case the variation of ω˙+ vs. N+c is single-valued whereas for
Φ = 1.1 it is multi-valued. Kolla and Swaminathan (2010a) described how the
integral given in Eq. (3.10) could be evaluated for a multi-valued function, such
as the one shown in Figure 3.2(b). This multi-valued behaviour of ω˙+ with N+c
is an indication that there exist a higher and a lower burning rate in the domain
ψ ∈ |N1, N2|, here the integration limits, N1 and N2, corresponds to unstrained
flame value and the extinction limit respectively, while the intermediate value is
given by N0. Unlike the unstrained flamelet model, the look-up table for this
46
3.4. Computational approach
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
ω˙
+
N+c
N1
N2
ζ = 0.7
ζ = 0.6
ζ = 0.5
(a)
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
ω˙
+
N+c
N1
N2
N0
ζ = 0.7
ζ = 0.6
ζ = 0.5
(b)
Figure 3.2: Curves of ω˙+ vs. N+c conditioned on the progress variable, ζ for
two cases: (a) stoichiometric methane-air flame and (b) methane-air flame with
Φ = 1.1.
model is three-dimensional involving c˜ and c˜′′2 and ˜c. Therefore a tri-linear
interpolation is used during the CFD simulations.
3.4 Computational approach
The governing equations given in section 3.1 can be written in the following
standard form for a transported scalar property, Ψ
∂
∂t
(ρΨ) +
1
r2
∂
∂r
(
r2ρu˜rΨ
)− 1
r2
∂
∂r
(
ΓΨ
∂Ψ
∂r
)
= SΨ. (3.15)
The values of Ψ, ΓΦ and SΦ for the modelled governing equations are given in
Table 3.1 (refer to Appendix A to see how the radial momentum equation is cast
in the form given in this table).
These equations are discretised using the finite volume method (FVM). Power
law scheme of Patankar (1980) is used for spatial discretisation and the implicit
backward Euler method is used for time-stepping. The pressure-velocity coupling
is through the SIMPLER approach of Patankar (1980) and the set of discrete
algebraic equations are solved iteratively.
47
3.4. Computational approach
Table 3.1: Diffusion coefficients and source terms in the generalised conservation
law for all the governing equations
Ψ ΓΨ SΨ
1 0 0
u˜r
4
3
(µ+µt)
∂p
∂r
− ∂
∂r (
2
3
ρk˜)− 1r
[
∂ΓΨ
∂r
+2
ΓΨ
r
]
u˜r
k˜ µ+
µt
Sck
− 2
3(
∂u˜r
∂r
+2ur
r )ρk˜+
4
3
µt( ∂u˜r∂r − u˜rr )
2−ρε˜+ µt
ρScc
∂c˜
∂r (
τ
1+τc˜)
∂p
∂r
+ 1
2
c˜(τs0L)
2ω˙
ε˜ µ+
µt
Scε
−Cε1 ε˜
k˜
[
2
3(
∂u˜r
∂r
+2ur
r )ρk˜− 43µt( ∂u˜r∂r − u˜rr )
2− µt
ρScc
∂c˜
∂r (
τ
1+τc˜)
∂p
∂r
]
−Cε2ρ ε˜2
k˜
c˜ αc+
µt
Scc
ω˙
c˜′′2 αc+ µtScc 2
µt
Scc
( ∂c˜∂r )
2−2ρc+2ω˙′′c′′
3.4.1 Initial and boundary conditions
Length of the computational domain depend on the ratio of turbulence integral
length scale, Λ, to the Zeldovich thickness, δ, so that the simulated flames remain
within the domain for the simulation period. The number of grid points are chosen
such that there are at least 10 points inside min(Λ, δt) for a uniform grid spacing.
The turbulent flame brush thickness is given by δt ≡ 1/|∂c˜/∂r|max. The size of
time-step is chosen to be 0.1 µs, which ensures the resolution of reaction, diffusion
and convection time scales and satisfies the numerical stability conditions for the
chosen grid, turbulence and thermochemical conditions.
The initial spatial variation of c˜ having 0 in the unburnt and 1 in the burnt
mixtures is chosen after few tests to minimise the initial transients for the given
turbulence and thermochemical conditions. This saves computational time re-
quired to attain a “steady propagation” state. The initial u˜, ρ and p are specified
to be consistent with the initial c˜ variation. It is straightforward to specify
the boundary conditions for the planar flames as has been done in many earlier
studies (Corvellec et al., 1999, 2000; Swaminathan and Bray, 2005; Kolla and
Swaminathan, 2010a). Assuming that the flame propagates radially outward the
48
3.4. Computational approach
following boundary conditions are obtained at r = 0:
u˜r(0, t) =
∂c˜
∂r
(0, t) =
∂c˜′′2
∂r
(0, t) =
∂k˜
∂r
(0, t) =
∂ε˜
∂r
(0, t) = 0. (3.16)
Similarly, the boundary conditions applied at r = R (at the end of the domain)
are:
∂u˜r
∂r
(R, t) =
∂k˜
∂r
(R, t) =
∂ε˜
∂r
(R, t) = 0, c˜(R, t) = c˜′′2(R, t) = 0, p(R, t) = p∞.
(3.17)
The values of turbulent kinetic energy, k˜, and its dissipation rate, ε˜, obtained
using the chosen value of u′ and Λ are used to specify their initial conditions. As
noted in the beginning of this chapter, governing equations written in spherical
coordinates are implemented in an existing in-house CFD code. Before proceeding
with turbulent combustion simulations, it is necessary to validate this code for
simple, non-reacting flow cases; this is discussed in Appendix B.
49
4. Spherical methane-air flames
Simulation of statistically spherical, methane-air flames using the scalar dissipa-
tion rate based modelling approach is presented in this chapter. The predictive
ability of the unstrained and strained flamelet models described in section 3.2 is
assessed by comparing with experimental data. In addition, a simple algebraic
closure for the mean reaction rate is also used for comparison. Simulations of
spherical and planar flames at various turbulence intensities are used to obtain
the effect of geometry and turbulence on flame propagation.
4.1 Introduction
Expanding statistically spherical flame in turbulent environment is a canonically
important configuration and its investigation helps us to enhance our understand-
ing of combustion in practical devices such as the spark ignited IC engine, modern
stratified charge engines and accidental explosions of fuel vapour cloud. Although
it is a classical problem our current understanding is not fully satisfactory and
complete.
When a combustible mixture cloud is ignited at the centre, a laminar flame
kernel is initiated and it develops into a turbulent spherical flame. During this
evolution, the flame front is stretched due to its time varying curvature and flow
straining acting on it. In addition to these effects on the flame front, the flame
brush experiences stretch due to its curvature in this geometry. The effects of
stretch on laminar flame speed was explained in section 2.3.3.
Practical combustion systems involve turbulence invariably and hence turbu-
lent spherical flames have been studied using various experimental configurations
such as fan-stirred bombs involving stationary turbulence (Andrews et al., 1975;
Abdel-Gayed et al., 1984; Bradley et al., 1994; Lawes et al., 2012), bombs with
decaying grid turbulence (Checkel and Thomas, 1994) and wind tunnels with
grid turbulence (Hainsworth, 1985; Renou et al., 2002) to address the influence of
turbulence on spherical flame propagation. Beretta et al. (1983) and Hainsworth
(1985) have shown that the turbulent spherical flames initially expand as a lam-
50
4.1. Introduction
inar flame and then it is exposed gradually to a wide range of length and time
scales of turbulence, resulting in flame wrinkling thereby leading to an increase
in the burning velocity that is larger than the laminar value (Abdel-Gayed et al.,
1987). Additional flame wrinkling can arise in thermo-diffusively unstable flames
(of reactant mixtures with negative Markstein number). The flame wrinkling
was shown to increase with pressure and for mixtures with negative Markstein
numbers (Haq et al., 2002). The tendency to greater flame wrinkling, resulting
in faster flame propagation and high flame front curvature for mixtures with low
Lewis number (thermo-diffusively unstable mixtures), is also known (Renou et al.,
2000).
As a spherical flame brush expands its thickness increases, with significant
amount of unburnt gas inside the flame brush (Beretta et al., 1983; Abdel-Gayed
et al., 1988). This poses a challenge to define the turbulent burning velocity since
its definition relies on the correct choice of an associated flame radius. One way to
define this radius is to equate the volume of unburnt gas inside the flame brush to
that of burnt gas outside the flame brush (Bradley et al., 2003). The mass burning
velocity defined using this radius is equal to the velocity of turbulent entrainment
of unburnt gas into the flame brush. The flame propagation model using this
entrainment concept has been developed in several past studies (Blizard and Keck,
1974; Tabaczynski et al., 1980; Groff, 1987; Bradley et al., 1994). Alternatively,
flame area enhancement due to turbulence has also been considered using a vortex
tube model (Ashurst et al., 1994) and an exponential growth of flame surface area
(Ashurst, 1995) to study expanding spherical flames. These studies treated the
flame surface to be a passive surface which is not fully satisfactory. An analogy
to the laminar flame theory has also been used to study turbulent spherical flame
growth rate involving a turbulent Markstein number (Lipatnikov and Chomiak,
2004). These studies have helped to develop some understanding of spherical
flame propagation within the scope defined by the assumptions used in their
development.
The three numerical paradigms described in section 2.2 have been used to
simulate spherical flames. DNS studies were initially aimed to address ignition
related issues (Baum and Poinsot, 1995; Poinsot et al., 1995) using a single irre-
versible reaction in two-dimensional turbulence. Some of these limitations were
51
4.1. Introduction
relaxed in later DNS studies on spherical flames (Kaminski et al., 2000; Jenkins
and Cant, 2002; Jenkins et al., 2006; Klein et al., 2006, 2008; Albin and D’Angelo,
2012; The´venin et al., 2002; The´venin, 2005; van Oijen et al., 2005) and these
studies predominantly addressed flame surface density related modelling issues.
LES has recently been used to study ignition and propagation of turbulent
spherical flames (Nwagwe et al., 2000; Tabor and Weller, 2004; Fureby, 2005;
Colin and Truffin, 2011; Lecocq et al., 2011). Combustion models based on sub-
grid scale wrinkling factor (Nwagwe et al., 2000; Tabor and Weller, 2004; Fureby,
2005) and flame surface density (FSD) transport equation (Colin and Truffin,
2011) have been used in conjunction with simplified chemistry in the past to
compute spherical flames of the Leeds bomb experiments. These studies showed
a good comparison with experimental data. Recently, a combustion modelling
approach combining the FSD and presumed pdf concepts has been used (Lecocq
et al., 2011) to calculate the spherical flame propagation in weak turbulence
(Renou et al., 2000) showing a good comparison.
RANS calculations of spherical turbulent flames of Hainsworth (1985) were
done by Schmid et al. (1998) using a turbulent flame speed closure. A similar
approach was also used by Lipatnikov and Chomiak (2000) to study turbulent
spherical flames in various configurations. A transported joint velocity-scalar pdf
approach was used by Pope and Cheng (1986) to compute the spherical flames of
Hainsworth (1985) and showed a very good agreement with the measurements.
In this work, the RANS methodology is used to study the propagation of
turbulent premixed spherical and planar flames. Reaction rate closure is provided
by the models described in section 3.2, which are based on the scalar dissipation
rate of a progress variable. The progress variable is calculated using temperature,
given by Eq. (2.12). The main objectives of this work are:
1. To assess the predictive ability of the various scalar dissipation rate based
models described in section 3.2; secondly,
2. To contrast flame propagation mechanisms in spherical and planar cases
and to elucidate the underlying physics.
Numerical method, including the combustion models have been described
in Chapter 3. The detailed chemical mechanism, GRI-Mech 3.0 (Smith et al.,
52
4.2. Test Flames
accessed 10th November 2013), is used for combustion kinetics of methane-air
mixture. This chapter is organised as follows: the experimental test case for val-
idation and various computational cases considered are described in section 4.2.
The simulation results are presented and discussed in section 4.3. The main
conclusions from this study are summarised in the last section.
4.2 Test Flames
The numerical method described in Chapter 3 is used to study the influence of
turbulence and thermochemical conditions on the evolution of expanding spher-
ical flames. Simulation results are also used to elucidate the difference in the
propagation of planar and spherical flames. Before discussing the conditions of
the flames considered here, an experimental case used to validate the numerical
models is described briefly.
4.2.1 Validation case
Spherical flames established in wind tunnel turbulence by Hainsworth (1985)
are considered for model validation purpose. This wind tunnel turbulence gener-
ated using perforated plates was homogeneous and isotropic, and the methane-air
mixture having equivalence ratios of Φ = 1.1 and 0.8 were experimentally investi-
gated. For reasons to be discussed in section 4.3.1, Φ = 1.1 mixture is considered
for this study and its thermochemical characteristics along with the experimen-
tal conditions at ignition are given in Table 4.1. The flame was ignited using
a spark downstream of the perforated plate and it was convected downstream
by the mean flow as it evolves in an approximately spherical shape. Tempo-
ral changes of position and radius of this flame were recorded using high-speed
schlieren movies and it has been suggested that this flame is representative of
combustion in spark-ignition engines (Pope, 1987). This flame was also consid-
ered in earlier computational studies (Pope and Cheng, 1986; Schmid et al., 1998;
Lipatnikov and Chomiak, 2000).
53
4.2. Test Flames
Table 4.1: Experimental conditions for Φ = 1.1 flame of Hainsworth (1985)
Parameter Value
s0L 0.43 m/s
δ0L 0.0408 cm
δ 0.00565 cm
rf,0 0.15 cm
τ 5.25
u′ 1.93 m/s
Λ 0.838 cm
p0 0.1 MPa
T0 298 K
4.2.2 Flames for further analysis
Spherical flames propagating outwardly in nearly homogeneous isotropic turbu-
lence field in an unconfined space are considered. Boundary conditions discussed
earlier in section 3.4 describe this problem. The influences of combustion on
turbulence are also included in the simulation by solving the k˜-ε˜ equation. A
stoichiometric methane-air mixture at 298 K and atmospheric pressure is con-
sidered for these flames. Since this mixture has unity Lewis number, the influ-
ence of turbulence on the flame propagation can be studied without the added
complexity of differential diffusion, which could amplify the stretch-induced ef-
fects. Furthermore, this mixture was considered in an earlier study addressing
the turbulence effects on the propagation of statistically planar flames (Kolla and
Swaminathan, 2010a). Thus, the behaviour of spherical flames can be compared
directly to planar flames to understand the geometry effects. The thermochemi-
cal characteristics of this mixture are, s0L = 0.4 m/s, δ
0
L = 0.41 mm, τ = 6.48 and
δ = 0.047 mm.
The turbulent combustion conditions of 8 flames simulated in this study are
shown in Figure 4.1. Two different values for the stretch factor, as defined by
Abdel-Gayed et al. (1987), K = 0.157 (u′/s0L)
2
Re−0.5t = 0.157 and 1 are consid-
ered. The turbulence Reynolds number was given earlier in Eq. (2.38). The flames
with the smaller stretch value have the Karlovitz number, Ka = (u′/s0L)
2
Re−0.5t ,
of unity and they are located at the upper limit of the corrugated flamelets regime.
54
4.3. Results
0.1
1
10
100
1000
0.1 1 10 100 1000 10000
u
′ /
s0 L
Λ/δ
wrinkled flameletslaminar flamelets
corrugated flamelets
thin reaction zones
broken reaction zones
Ka = 1
Ka = 100
Da = 1
Ret = 1
Figure 4.1: Regime diagram of turbulent premixed combustion (Peters, 1999)
with the flame conditions considered for this study. The filled symbols correspond
to two flame stretch parameters: K = 0.15 (•) and K = 1.0 () and the open
triangle (4) is for the experimental case of Hainsworth (1985).
The other case with larger K value is in the thin reaction zones regime as shown
in Figure 4.1. These particular values for K are chosen so that the combustion
conditions remain the same for the current spherical and planar flames of Kolla
and Swaminathan (2010a). For the three flames with K = 0.15, the values of
u′/s0L are 5, 6 and 8, and these values are 12, 16, 18, and 20 for the other cases
with K = 1. It is also to be noted that the experimental flame of Hainsworth
(1985) is in the corrugated flamelets regime for the conditions noted in Table 4.1.
4.3 Results
Computational results of spherically expanding flames under a wide range of tur-
bulence conditions are analysed in this section. Validation of the computational
models are discussed first. Then, the influence of turbulence on the propagation
of spherical flames are explored using the seven test flames and they are compared
with corresponding planar flames to understand the geometry effects.
55
4.3. Results
4.3.1 Model validation
Figure 4.2(a) shows the temporal variation of the flame brush radius measured
using high speed schlieren techniques for Φ = 1.1 mixture in two sets of experi-
ments (Hainsworth, 1985). Since schlieren images show the burnt side and mark
the regions with strong density gradients, the flame radius reported in the exper-
iment is taken to be the leading edge of the flame brush (Bradley et al., 2000,
2011). For comparison purposes, the location at which c˜ = 0.05, is taken to be
the leading edge in the simulated flames. As one expects, this radius grows with
time as in Figure 4.2(a), where radius is normalised using its initial value, rf,0
in Table 4.1, and time is normalised using the laminar flame time or chemical
time, tc = δ
0
L/s
0
L, (see Table 4.1). The bottom two curves, marked for laminar
flames, represent the evolution of the initial flame kernel if it evolves as a laminar
spherical flame. This laminar evolution can be computed simply by considering
mass conservation, dmb/dt = ρu sLA, for the burnt gas mass, mb, inside the
kernel having a surface area of A. This simplifies to drf/dt = (ρu/ρb)sL for a
spherical kernel. If one takes sL = s
0
L, ignoring stretch effects on the laminar
flame propagation, then rf grows linearly with t and this line passes through
the experimental data for t+ ≤ 2. This suggests that the initial evolution is
laminar and it may be uninfluenced by the stretch effects induced by flow strain-
ing and curvature. This is supported by the result shown for stretched laminar
flame in Figure 4.2(a) (the bottom-most curve). Stretch effects are included in
the above mass conservation equation by using Eq. (2.37), where stretch rate,
κ = 2(d ln rf/dt), and L is the Markstein length scale for the chosen mixture.
This length scale is computed as 0.89 mm using Eq. (2.109) of Poinsot and Vey-
nante (2005), which is close to the value reported by Bradley et al. (1996). The
comparison of unstretched and stretched laminar flame results to the experimen-
tal data suggests the following: it was clearly noted by Hainsworth (1985) that
the mixture of Φ = 1.1 is thermodiffusively stable and there are no cell formation
on the flame surface. Hence, the increase in the burning rate is purely due to tur-
bulence. As the initial laminar flame grows, it is exposed to progressively wider
range of scales, which would increase the surface area through flame wrinkling.
This results in increased burning rate as has been noted in earlier studies (Beretta
56
4.3. Results
0
5
10
15
20
25
30
0 1 2 3 4 5 6 7 8
r f
/r
f
,0
t+
Exp. set 1
Exp. set 2
Laminar
Stretched laminar
Str. flamelet [Eq. (2.69)]
Str. flamelet [Eq. (3.14)]
Unstr. flamelet
Algebraic
(a)
0
5
10
15
20
25
30
2 4 6 8 10 12 14
(d
r f
/d
t)
/s
0 L
rf/rf,0
Exp.
Str. flamelet [Eq. (2.69)]
Str. flamelet [Eq. (3.14)]
Unstr. flamelet
Algebraic
(b)
Figure 4.2: Comparison of numerical and experimental (Hainsworth, 1985) re-
sults: (a) radius as a function of time and (b) flame propagation speed as a
function of radius.
57
4.3. Results
et al., 1983; Hainsworth, 1985). It is possible that this increase is compensated
by the stretch-induced negative effect, resulting in a growth rate similar to that
of freely propagating spherical laminar flame for about t+ ≤ 2. Beyond this time,
the effects of flame wrinkling produced by turbulence overwhelms the stretch ef-
fect, producing a smooth departure from the freely propagating spherical laminar
flame as seen in Figure 4.2(a).
When the flame kernel is much smaller than the turbulence integral length
scale, it will simply be convected by the large-scale eddy and the flame-turbulence
interaction is limited to a small part of the wide spectrum of turbulence scales and
thus RANS combustion modelling may not hold. This leads to some ambiguity in
using turbulent combustion modelling to simulate the transition from laminar to
turbulent growth using this approach. This has been recognised by Pope (1987),
and so, a joint velocity-scalar pdf approach was chosen by Pope and Cheng (1986)
to simulate this experimental flame from t+ = 0, showing a good comparison with
measurements over the whole period of the experiment. Thus, the experimentally
measured flame radius at t+ ≈ 2.5 obeying this condition is chosen as the initial
flame radius for the simulation. This flame radius departs from the laminar result
by about 5% as shown in Figure 4.2(a). Thus, the turbulent flame results start
from t+ ≈ 2.5 in this figure.
The URANS approach along with the k˜-ε˜model was also used in earlier studies
employing an empirical mean reaction rate model based on turbulent flame speed
closure (Schmid et al., 1998) and a time dependent mean reaction rate closure
with a laminar-like source (Lipatnikov and Chomiak, 2000). The later study also
excluded momentum equation in the analysis.
The difficulties noted previously, due to the relative size of the flame and tur-
bulence integral length scale and further reasonings given below, means that only
the Φ = 1.1 flame from the experiments of Hainsworth (1985) can be simulated
in this study. Also, the lean methane-air mixture is thermo-diffusively unstable
(weakly) and this effect must be taken into account in the combustion modelling
approach. It is unclear at this time how to include these effects into RANS com-
bustion modelling. As shown in Figure 4.1, the conditions of the experimental
flames are in the corrugated flamelets regime, whether one can ignore the ther-
modiffusive instabilities, however weak they may be, and their influence on flame
58
4.3. Results
propagation is an open question.
The turbulent flame results are shown for three different combustion mod-
els in Figure 4.2(a). The algebraic model in Eq. (2.48) overpredicts the flame
growth as one would expect because this model assumes fast chemistry resulting
in faster burning. The unstrained flamelet model in Eq. (2.63) includes finite rate
chemistry effects but assumes the flame front to be a freely propagating laminar
flame and thus excludes the local stretch effects on the flame front. Thus, the
flame growth rate is overpredicted by this model as well, but the level of over-
prediction is reduced when compared with the algebraic model case. The values
of rf computed using the strained flamelet model given by Eq. (3.10) agree well
with the measured values as shown in Figure 4.2(a) for the following reason. In
premixed flames, the local balance among reaction, diffusion and fluid dynamic
effects determines the local scalar gradient magnitude which is directly related
to the scalar dissipation rate. The stretch effects from turbulence are due to
straining and curvature and both of these will directly influence the scalar gradi-
ent. Thus, using the scalar gradient to parametrise the flamelets seems prudent
for spherical flames as it has been shown earlier for planar flames (Kolla and
Swaminathan, 2010a). The relative behaviour of the three combustion models
shown here for the statistically spherical flame is similar to the observation of
Kolla and Swaminathan (2010a) for statistically planar flames. Also, the use of
equation (3.14) to include the curvature of the flame brush shows negligible effect
on the growth of the flame as in Figure 4.2(a) and for this reason Eq. (2.69)
will be used for further analyses of spherical flames to be discussed below, unless
mentioned otherwise.
There is some uncertainty in choosing the initial flame radius for the compu-
tations, as noted earlier. Thus, the variation of normalised propagation speed,
(drf/dt) /s
0
L, with the normalised radius is shown in Figure 4.2(b). The computa-
tional results are about 12% larger than the values derived from the experimental
results, and this level of difference is acceptable. A best-fit cubic curve for the
two sets of experimental data for t > 2.4 ms given in Figure 4.2(a) is used to
calculate drf/dt for the experimental result.
59
4.3. Results
4.3.2 Spherical and planar flames comparison
The flame geometry effect on the propagation and consumption speeds of turbu-
lent premixed flames is investigated in this section using the results of spherical
and planar flames simulated in this study. The planar flame results computed in
this study were observed to be very close to those reported by Kolla and Swami-
nathan (2010a). All the flames investigated in this section are simulated for a
period of about 8 ms. Typical time evolution of these two, planar and spherical,
flames is shown in Figure 4.3 by plotting the spatial variation of c˜ at various
times. The spatial position, x′, shown in this figure is a Galilean transformed,
x′ = x − u˜bt, because the burnt side velocity, u˜b, is different in the planar and
spherical cases. This allows a direct comparison between these two flames. The
burnt side velocity is zero in the spherical case and it is negative in the planar
case.
The initial variation is shown by dashed lines and the profiles are shown
for a period of 8 ms (t+ ≈ 7.8) at an interval of 2 ms. These flames have
u′/s0L = 6 and K = 0.15, and the same thermochemical parameters because they
are stoichiometric CH4-air mixture. These flames propagate from left to right
in Figure 4.3 and they are computed in the Cartesian and spherical coordinate
systems respectively. This flame pair is used to demonstrate the flame geometry
effects because the relative behaviours shown and discussed in this section hold
for other cases considered, unless noted otherwise.
In Figure 4.3 the profile at 8 ms is plotted using symbolled lines to show the
grid resolution. For all the cases the grid resolution was defined such that there
are at least 10 points within the integral length scale. This figure also shows that
the planar flame reaches a nearly steady propagation speed after some initial
transients, but the spherical flame does not seem to suggest a steady value for
its propagation speed [shown in Figure 4.3(b) by the increasing gap between
consecutive iso-contour profiles]. As the spherical flame grows outwardly, the
leading surface area increases, resulting in increased burning rate, which can be
seen clearly by plotting the temporal variation of the propagation speed of an
iso-value, c˜ = c1. This speed is extracted from the computed time variation of
60
4.3. Results
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.05 0.10 0.15 0.20 0.25
c˜
x′ (m)
Planar
t
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.05 0.10 0.15 0.20 0.25
c˜
r′ (m)
Spherical
t
(b)
Figure 4.3: Spatial variation of c˜ at five different times, 0 to 8 ms at an interval
of 2 ms, in (a) planar and (b) spherical flames having u′/s0L = 6 and K = 0.15.
Dashed lines show the variation at t = 0.
61
4.3. Results
the spatial position x (c1) or r (c1) through
dx(c1)
dt
· n = u˜ · n+ sd, (4.1)
where u˜ is the fluid velocity and sd is the displacement speed of the c˜ iso-level in
its normal, n, direction due to its relative movement created by combined effects
of mean reaction rate, turbulent flux and molecular diffusion. Note the difference
between this displacement speed and the one defined in section 2.3.3, which was
defined for instantaneous c. For the rest of this work, the displacement speed
will be defined with respect to Favre-averaged progress variable, c˜. The effect
of molecular diffusion can be neglected in high-Reynolds number turbulent flows
and the displacement speed can then be written as
sd =
1
r2
∂
∂r
[
r2
(
µt
Scc
∂c˜
∂r
)] / (
ρ
∣∣∣∣∂c˜∂r
∣∣∣∣)︸ ︷︷ ︸
sTd
+ ω˙
/(
ρ
∣∣∣∣∂c˜∂r
∣∣∣∣)︸ ︷︷ ︸
srd
, (4.2)
using the mean progress variable equation, Eq. (3.3). It is understood that all the
quantities on the right hand side of Eq. (4.2) must be evaluated at c˜ = c1. A corre-
sponding equation can also be written in the Cartesian system. The displacement
speed of the leading edge, sd (c˜ = 0.05), is referred as the turbulent flame speed
in the latter part of this section. The equality in Eq. (4.1) is verified using the
computational results since the three terms can be evaluated individually.
Figure 4.4 shows the temporal variation of the propagation speed, sp, with
respect to the burnt mixture computed from dx′/dt for the iso-levels. The results
with low and high turbulence levels are shown respectively in Figures 4.4(a) and
4.4(b). This propagation speed is normalised using the unstrained planar laminar
flame speed and the time is normalised using the respective integral time scale of
the turbulence in the reactants, te. This normalised time is related to t
+ through
t∗ = t+ (tc/te). After going through some initial transients for t+ ∼ 2 to 2.5, all
the iso-levels converge to a nearly constant propagation speed that depends on
the value of u′/s0L for the planar flames, and a small decrease with t
∗ suggest
the persistence of the initial transients. On the other hand, the propagation
speed increases with t∗, and different iso-levels are travelling at different speeds
62
4.3. Results
-10
0
10
20
30
40
50
60
70
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
s p
/s
0 L
=
(d
x
′ /
d
t)
/s
0 L
t∗
Planar
Spherical
c˜ = 0.05
c˜ = 0.50
c˜ = 0.80
(a)
-10
0
10
20
30
40
50
60
70
80
90
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
s p
/s
0 L
=
(d
x
′ /
d
t)
/s
0 L
t∗
Planar
Spherical
c˜ = 0.05
c˜ = 0.50
c˜ = 0.80
(b)
Figure 4.4: Temporal variation of propagation speed of c˜ iso-levels in planar and
spherical turbulent flames for (a) u′/s0L = 6 and K = 0.15, and (b) u
′/s0L = 18
and K = 1. Results are shown for three different iso-levels.
63
4.3. Results
in spherical flames. The larger values seen in the early period for the planar flame
is because of the high u˜b. The continuous growth in the spherical cases is because
of the continuous increase in the burning surface area. Further discussion on this
point is postponed until section 4.4.
The iso-levels with lower c˜ values are moving more quickly compared to those
with greater values in the spherical cases. This relative behaviour can be seen
clearly by plotting the variation of the propagation speed in Eq. (4.1) across
the flame brush at a given instant. This variation is shown in Figure 4.5(a) for
both planar and spherical flames at t = 5 ms. The results are shown for two
combustion conditions, u′/s0L = 20 and K = 1 and u
′/s0L = 6 and K = 0.15. A
gradual decrease of the normalised propagation speed across the spherical flame
brush is seen, and this decrease is about 9 to 12%, depending on the value of
u′/s0L (larger decrease for greater u
′/s0L). It is to be noted that the values of the
propagation speed are divided by 2 for u′/s0L = 20 cases to fit within the scale
shown in Figure 4.5(a). The statistically planar flames (open symbols) do not
show any decrease across their flame brushes, except for the sharp change near
the burnt side, which is for an obvious reason. The large scatter seen at c˜ = 0 is
due to sharp variation of sd over a small range of c˜ near the unburnt side of the
flame brush.
Typical variations of the two components, u˜/s0L and sd/s
0
L, across the flame
brush are shown in Figure 4.5(b) for the u′/s0L = 6 case at t = 5 ms. The results
for planar and spherical flames are shown respectively with dashed and solid lines
and using the corresponding symbol in Figure 4.5(a). The displacement speed
is calculated using Eq. (4.2). The following observations can be made from this
figures: (1) The normalised u˜ and sd have the same sign in the spherical case,
whereas they have opposite signs in the planar case. This implies that the fluid
and c˜ iso-level move in opposite directions in planar flames, unlike in spherical
flames. (2) The values of sd/s
0
L and u˜/s
0
L near the leading edge of the spherical
flame is much larger than in the planar flame. On the burnt side, the planar flames
have larger values. It is to be noted that the fluid velocity shown here is because
of heat-release effects, since u˜ (c˜ = 0) and the velocity gradient at the burnt side
are specified to be zero for the planar flames as noted earlier in Subsection 3.4.
Thus, the flow acceleration across the flame brush gives a large flow velocity on
64
4.3. Results
0
10
20
30
40
50
60
0.0 0.2 0.4 0.6 0.8 1.0
(s
d
+
u˜
)/
s0 L
c˜
(a)
-60
-40
-20
0
20
40
60
80
0.0 0.2 0.4 0.6 0.8 1.0
N
or
m
al
is
ed
sp
ee
d
c˜
(b)
Figure 4.5: Variation of propagation speed across the flame brush is shown in
(a) for planar (open symbols and dashed lines) and spherical (closed symbols and
solid lines) flames with u′/s0L = 6 and K = 0.15 (circle), and u
′/s0L = 20 and
K = 1 (square) at t+ = 4.88 (t = 5 ms). The two components, sd (symboled
lines) and u˜ (lines), are shown in (b) for the u′/s0L = 6 case.
65
4.3. Results
the burnt side of the planar flame brush as it is well known and this is clear in
Figure 4.5(b).
In the spherical case, the burnt mixture and the unburnt mixture at large
radial distance are at rest and thus the flame-induced velocity has to decay to zero
on both sides of the flame brush. These behaviours, especially on the unburnt side,
are unclear in Figure 4.5(b). Thus, the spatial variation of (u˜/s0L) at t = 5 ms
is shown in Figure 4.6, where the distance is normalised using the turbulence
integral length scale, Λ. The peak flow velocity occurs near the leading edge of
-60
-50
-40
-30
-20
-10
0
10
20
30
40
50
0 5 10 15 20 25
u˜
/s
0 L
Normalised distance
c˜ = 0.95
0.05
0.05
0.95
Planar
Spherical
r−2
Figure 4.6: Spatial variation of u˜/s0L at t
+ = 4.88 for the cases with u′/s0L = 6
and K = 0.15. The locations of c˜ = 0.05 and 0.95 in the respective flames are
marked to indicate the flame brush sizes. The r−2 decay of mean velocity is also
plotted.
the spherical flame brush and it decays to zero as r−2 in the unburnt mixture.
Also, the flame brush thickness as marked roughly in Figure 4.6 is relatively
smaller for the spherical case compared to the planar flame.
The time evolution of the flame brush thickness normalised by the laminar
flame thermal thickness, δt/δ
0
L, is plotted in Figure 4.7. The flame brush thick-
ness is defined in two ways, one is using the maximum gradient of c˜ (shown as
66
4.3. Results
10
20
30
40
50
60
70
80
0 1 2 3 4 5 6 7 8
δ t
/δ
0 L
t+
Planar
Spherical
0.4δ√
c˜′′2
Taylor’s theory
Figure 4.7: Evolution of flame brush thickness with time in flames with u′/s0L = 18
and K = 1.
symboled lines in Figure 4.7) and another one is using the variance, c˜′′2 (shown
as a solid line). This second thickness is defined as the thickness over which rms
value drops to 5% of its maximum value to be consistent with Taylor’s theory
of turbulent dispersion (Taylor, 1935b). Note that this variance thickness is also
scaled to fit in the scale shown in Figure 4.7. These results are discussed fully
in section 4.4. The planar flame reaches a steady value, dictated by the turbu-
lence and thermochemical conditions, after t+ ≈ 2.6, whereas there is no such
steady-state value for the spherical flame, and its thickness keeps growing with
time, which is well known in the literature. This relative behaviour is the same
in other flames investigated in this study.
From the results discussed so far, it seems that this continuous growth is
because the burnt side of the flame is advancing slowly compared to the leading
edge. This difference can be seen clearly for Figure 4.4 for c˜ = 0.05 and 0.8
[also see Figure 4.5(a)]. It is obvious from the discussion that the fluid velocity
at the leading edge is larger as shown in Figures 4.4-4.6 and it acts together
67
4.3. Results
with the displacement speed in the spherical case. Based on these results, a
simple schematic diagram can be drawn as shown in Figure 4.8 to represent the
difference in the physical mechanisms influencing the propagation of statistically
planar and spherical flames. Note the difference in directions of flow, u˜, and flame
displacement speeds, sd, between spherical and planar cases.
Figure 4.8: Schematic diagrams showing the propagation mechanism in a statis-
tically planar and spherical flames. Dashed and solid arrows represent the flow
and iso-level displacement directions respectively.
Behaviour of sd
From the discussion in the previous section, it is evident that δt will influence
the flame displacement speed. As noted in Eq. (4.2), the sd has two (reaction
and turbulent flux), components and their typical variations across the flame
brush are shown in Figure 4.9 for two instances, t+ = 4.88 in Figure 4.9(a) and
8 in Figure 4.9(b). The reaction rate contribution can be written as (srd/s
0
L) =
ω˙
+
(1 + τ c˜) /|∂c˜/∂r+|. Thus, the behaviour of (srd/s0L) with c˜ is expected to be
approximately linear according to (1 + τ c˜) because the variations of ω˙
+
and
|∂c˜/∂r+| with c˜ would be similar. This observation explains the variations of
(srd/s
0
L) shown in Figure 4.9. The difference between the planar and spherical
flames predominantly comes from 1/|∂c˜/∂r+|, which is related to δt shown in
68
4.3. Results
-20
-10
0
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1
N
or
m
al
is
ed
s d
co
nt
ri
bu
ti
on
c˜
Reaction rate (planar)
Turbulent flux (planar)
Reaction rate (spherical)
Turbulent flux (spherical)
(a)
-10
0
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1
N
or
m
al
is
ed
s d
co
nt
ri
bu
ti
on
c˜
Reaction rate (planar)
Turbulent flux (planar)
Reaction rate (spherical)
Turbulent flux (spherical)
(b)
Figure 4.9: Reaction rate and turbulent flux contributions to (sd/s
0
L) for both
planar and spherical flames with u′/s0L = 6 and K = 0.15 at: (a) t
∗ = 1.14
(t+ = 4.88) and (b) t∗ = 1.87 (t+ = 8).
69
4.3. Results
Figure 4.7. The planar flame is thicker at t+ = 4.88 and thus (srd/s
0
L) is larger
compared to the spherical flame, and the value of this displacement speed is about
the same at t+ = 8 because δt is nearly equal for these two flames.
The difference in the mean reaction rate variation is observed to be small in
Figure 4.10 and this relative behaviour is also observed in other flames considered
for this study. Also,the inset shows that the maximum value of the normalised
mean reaction rate does not vary much over the wide range of turbulence condi-
tions of both planar and spherical flames considered in this study. The values of
ω˙
+
max differ by a small amount between the planar and spherical flames. These
behaviours of ω˙
+
is observed to hold after the initial transient. The reduced
sensitivity of ω˙
+
to the turbulence level and the flame geometry has also been
reported in a direct numerical simulation study (Dunstan et al., 2012) by con-
sidering oblique and planar turbulent premixed flames established in a range of
turbulence conditions.
The turbulent flux contribution, sTd , to (sd/s
0
L) decreases across the flame
brush as shown in Figure 4.9. This is an expected behaviour for the planar flames.
To understand its behaviour in spherical flames and for the difference seen near
the leading edge, one can expand the first term of Eq. (4.2). This will identify
an extra term of 2µt(∂c˜/∂r)/(r Scc) in spherical flames, which will increase as c˜
increases in outwardly propagating flames. Since (∂c˜/∂r) is negative for these
flames, this extra term contributes negatively leading to a decrease of sTd as c˜ in-
creases. The flux contribution near the leading edge is larger in the spherical case
because of the additional increase in ∂2c˜/∂r2 resulting from the flame geometry.
Thus, the difference in the sd of spherical and planar flames comes predominantly
from the turbulent scalar flux. This is seen clearly in Figure 4.9, specifically at
the leading edge. The influence of turbulence on the leading edge displacement
speed and the consumption speed is discussed in the next subsection.
Turbulent flame speed comparison
The displacement speed of flame brush leading edge is defined as the turbu-
lent flame speed, st. This quantity is of interest for theoretical investigation of
turbulent flames and the influence of flame geometry on this quantity is of consid-
70
4.3. Results
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
ω˙
/ω˙
m
a
x
c˜
0.00
0.25
0.50
5 10 15 20
ω˙
+ m
a
x
u′/s0L
Planar
Spherical
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
ω˙
/ω˙
m
a
x
c˜
0.00
0.25
0.50
5 10 15 20
ω˙
+ m
a
x
u′/s0L
Planar
Spherical
(b)
Figure 4.10: Variation of normalised mean reaction rate across the flame brush
for both planar and spherical flames, with u′/s0L = 6 and K = 0.15 at t
+ = 8,
at (a) t+ = 4.88 and (b) t+ = 8, and the insets show the variation of ω˙
+
max with
(u′/s0L) for the various flames considered in this study, where ω˙
+
= ω˙δ0L/(ρu s
0
L).
71
4.3. Results
erable interest for turbulent combustion modelling (Driscoll, 2008). It has been
suggested recently that st is weakly sensitive to the flame geometry among freely
propagating planar, strained planar and rod stabilised oblique turbulent premixed
flames (Dunstan et al., 2012). It must be noted that there is no “stationary value”
for st in the spherical flames as for the planar flames.
The variation of st/s
0
L with the turbulence Reynolds number is shown in Fig-
ures 4.11(a) and 4.12(a), and its variation with u′/s0L is shown as the inset. The
turbulent flame speed of the spherical flames is always larger than the correspond-
ing planar flame value for a given turbulence condition by about 10 to 20% and
the greater value is due to the turbulent scalar flux contribution at the leading
edge as noted in the previous subsection. The results in the inset clearly suggests
an approximate relation st ∼ u′ when the value of the stretch factor, K is kept
constant. This relation was noted by Bray (1990) in his theoretical analysis using
the Bray-Moss-Libby model. Lipatnikov and Chomiak (2002) analysed a number
of experimental flames to show that st ∼ u′q, with 0.5 ≤ q ≤ 1. This increase in
st with u
′ is generally believed to be due to turbulent eddies increasing the flame
surface area by stretching and contorting it (Lipatnikov and Chomiak, 2002).
The increase in st with u
′ is sharp for the high Damko¨hler number cases (low
K) and the values of the slopes, obtained using the best linear fit, are about
6.3 and 5.1 respectively for the spherical and planar cases. These values become
three time smaller for the cases with high stretch, however the relative difference
in st between the spherical and planar flames remains almost the same. The
computed variations with Ret shown in Figures 4.11(a) and 4.12(a) suggest a re-
lation st/s
0
L ' B Rent with 0.57 ≤ n ≤ 0.58. The curves of least square fits shown
in the figure for both the spherical and planar flames suggest that B = 0.20
for the spherical and 0.18 for the planar flames. The approximate square root
dependence on the turbulence Reynolds number observed here is similar to that
reported by Chaudhuri et al. (2012) for the propagation speed of spherical and
Bunsen flames of methane-air mixtures which are thermodiffusively stable and
do not include Darrieus-Landau instability. The approximate square root depen-
dence observed in this study is consistent with the classical analysis of Damko¨hler
for the thin reaction zone combustion through a hypothesis st ∼
√
Dt/tc , where
Dt is the turbulent diffusivity, which is similar to s
0
L ∼
√
D/tc , where D is the
72
4.3. Results
5
10
15
20
25
30
400 900 1400 1900 2400 2900 3400 3900 4400 4900
s t
/s
0 L
Ret
y = 0.2015 x0.58, R2 = 0.9929
y = 0.1782 x0.57, R2 = 0.9962
10
20
30
5 10 15 20
s t
/s
0 L
u′/s0L
(a)
4
6
8
10
12
14
16
18
20
400 900 1400 1900 2400 2900 3400 3900 4400 4900
s c
/s
0 L
Ret
y = 0.1349 x0.57, R2 = 0.9941
y = 0.1368 x0.57, R2 = 0.9952
5
10
15
20
5 10 15 20
s c
/s
0 L
u′/s0L
(b)
Figure 4.11: Planar (open symbols) and spherical (closed) flame speeds for the
cases simulated in this study are plotted against the turbulence Reynolds number.
The inset shows the variation with u′/s0L. All flame speeds are taken at t
+ = 8 (t =
8.2 ms) in (a) normalised turbulent flame speed and (b) normalised consumption
speed. The symbols correspond to those shown in Figure 4.1.
73
4.3. Results
5
10
15
20
25
30
400 900 1400 1900 2400 2900 3400 3900 4400 4900
s t
/s
0 L
Ret
y = 0.2126 x0.57, R2 = 0.9958
y = 0.1698 x0.58, R2 = 0.9895
10
20
30
5 10 15 20
s t
/s
0 L
u′/s0L
(a)
4
6
8
10
12
14
16
18
20
400 900 1400 1900 2400 2900 3400 3900 4400 4900
s c
/s
0 L
Ret
y = 0.1204 x0.58, R2 = 0.9897
y = 0.1433 x0.56, R2 = 0.9958
5
10
15
20
5 10 15 20
s c
/s
0 L
u′/s0L
(b)
Figure 4.12: Planar (open symbols) and spherical (closed) flame speeds for the
cases simulated in this study are plotted against the turbulence Reynolds number.
The inset shows the variation with u′/s0L. All flame speeds are taken at t
+ =
5.86 (t = 6 ms) in (a) normalised turbulent flame speed and (b) normalised
consumption speed. The symbols correspond to those shown in Figure 4.1.
74
4.4. Discussion
molecular diffusivity, in the laminar flame theory (Peters, 1999). The results of
this study suggests that the flame geometry does not impart influence on this
scaling relation for turbulent flames.
Figure 4.11(b) and 4.12(b) show the consumption speed variation at t+ = 4.88
and 8 respectively. This speed is defined as
sc =
1
ρu
∫ 1
0
ω˙
|∂c˜/∂r| dc˜ =
∫ 1
0
srd
(1 + τ c˜)
dc˜. (4.3)
The latter part is obtained using srd defined in Eq. 4.2 and thus the consump-
tion speed is the reactive component of the density-weighted displacement speed
integrated across the flame brush. As noted in the previous subsection, the dif-
ference between the planar and spherical flames comes predominantly through
∂c˜/∂r and the mean reaction rate is less influenced by the flame geometry. This
gives the variation of sc/s
0
L with Ret similar to st/s
0
L; however the magnitude of
sc is smaller than st as shown in Figures 4.11 and 4.12. The difference in the
consumption speeds of the planar and spherical flames at t+ = 8 is very small
because the flame brush thickness is nearly equal, as shown in Figure 4.7.
4.4 Discussion
The results discussed in the previous section suggest that the flame brush leading-
edge displacement speed, which will be referred to as the turbulent flame speed in
this work, is larger for spherical flames compared to planar flames. The propaga-
tion speed, which is the sum of fluid velocity and the displacement speed, of the
leading edge grows continuously with time in spherical flames, while it reaches
a nearly constant value in planar flames. The increasing difference between the
propagation speeds of the leading and trailing edges in spherical flames yields a
continuous growth of its flame brush thickness. This growth is usually attributed
to turbulent diffusion in the past studies, which is different from the physical
explanation. The aim of this section is to shed more light on these observations.
The Kolmogorov-Petrovskii-Piskunov (KPP) analysis applied to multidimen-
sional premixed in high-Reynolds number turbulent flow gave an expression for
75
4.4. Discussion
the turbulent flame speed as (Kolla et al., 2010)
st = 2
√
νt
ρuScc
(
∂ω˙
∂c˜
)
c˜=0
+
(
νt
RScc
)
c˜=0
, (4.4)
where νt is the eddy viscosity and R is the radius of the leading edge. Figure 4.10
shows that the quantities
(
∂ω˙/∂c˜
)
c˜=0
in the planar and spherical flames are al-
most identical and the influence of flame brush curvature, R−1, is responsible
for the larger value of st observed in Figures 4.11 and 4.12. One expects that
this contribution will decrease as R becomes very large and st of the spherical
flame will reach the planar flame value eventually. This limiting behaviour is
not observed in the simulation studied here because of their finite domain size
and computational time. One requires a much larger computational domain than
those considered in this study.
The Favre-averaged fluid velocity at the leading edge of a turbulent spherical
flame is larger than at its trailing edge as shown in Figure 4.8. The maximum
value of this velocity will increase with time because of a continuous increase in
mass burning rate resulting from the growth of the leading surface area. This
increase, indeed observed in this study, results in a continuous acceleration of the
leading edge of spherical flames unlike in planar flames. Thus, a transition from
turbulent deflagration to a detonation can occur eventually if the conditions are
right. This transition is aggravated if the spherical flame propagates in a closed
vessel under appropriate conditions.
The spatial or temporal variation of flame brush thickness has been studied
in many earlier investigations and the results are summarised by Lipatnikov and
Chomiak (2002, 2005) and an increase in the thickness with time or distance has
been observed in these studies. Furthermore, Lipatnikov and Chomiak (2005)
showed that the evolution of the measured flame brush thickness is well predicted
by Taylor’s theory of turbulent diffusion for a passive scalar (Taylor, 1935b). This
theory predicts a linear growth in t for the rms displacement of a fluctuating
passive scalar iso-surface when t is smaller than the turbulent eddy turn over time
te, and this growth becomes
√
t when t is very much larger than te. Analysis using
direct numerical simulation date of turbulent “V” flames offered good support
76
4.5. Summary
for this (Minamoto et al., 2011; Dunstan et al., 2012) theory, suggesting that the
turbulent diffusion plays a predominant role on the growth of the flame brush
thickness. The applicability of this theory to the spherical flames studied here
is tested in Figure 4.7, which is typical for the flames studied here. As noted
earlier in section 4.3.2, the solid line denotes the temporal variation of δ√
c˜′′2
, a
thickness over which
√
c˜′′2 drops to 5% of its maximum value. This variation is
similar to δt as shown in Figure 4.7. The values predicted by Taylor’s theory are
also shown in that figure. The gap in the theoretical curve is intentional to mark
some transition from linear to square root dependence. This result suggests that
the variations of δt and δ√
c˜′′2
do not follow the turbulent diffusion theory, except
for a very short initial period. The relative gap between the theoretical curve and
δt in the spherical flame is controlled by the propagation mechanisms governed
by chemical reaction, convection and turbulent diffusion. The role of turbulent
diffusion for the growth of δt seems secondary compared to the convection because
of the fluid velocity induced by the chemical reaction.
4.5 Summary
Spherically expanding and statistically planar turbulent premixed flames of methane-
air mixtures are simulated using the URANS approach. The mean chemical re-
action rate is modelled using strained and unstrained flamelet models and the
algebraic model of Bray (1979). The unstrained flamelet model requires c˜ and c˜′′2
for its computation, while the strained flamelet model (Kolla and Swaminathan,
2010a) uses the mean scalar dissipation rate, ˜c, in addition to c˜ and c˜′′2. The
values of c˜ and c˜′′2 are obtained by solving their transport equations and the
mean dissipation rate is obtained using two algebraic models. These models are
obtained by balancing the leading order terms of the transport equation for the
mean scalar dissipation rate. One of this algebraic model was proposed in an
earlier study (Kolla et al., 2009) for statistically planar flames and the second
model includes the effects of mean curvature. The turbulence is modelled using
the k-ε equations.
These models are first validated by computing a spherical methane-air flame
77
4.5. Summary
investigated experimentally in an earlier study (Hainsworth, 1985). A good com-
parison between the computed and measured flame ball growth rate is observed
for the strained flamelet model and the other two combustion submodels yield a
faster growth.
Statistically planar and spherical flames, fourteen flames in total, experienc-
ing low and high turbulence stretch rates are computed using strained flamelet
model and these flames are analysed in detail to understand the influence of
geometry on their propagation. For the conditions investigated in this study,
including curvature corrections in the algebraic model for ˜c did not influence the
flame propagation. Detailed analyses of the computed flames showed that the
advancement of the leading edge is aided by the local fluid velocity in the spheri-
cal case. In the planar flames, the directions for the fluid flow and the advancing
leading edge are opposite. The planar flame showed a steady propagation once
a balance between the local flow and displacement speeds is achieved for a given
turbulence conditions. The spherical flames accelerated continuously because of
the compounded effects of flow and leading edge displacement. This continuous
acceleration cause the heat-release induced convective effects to be dominant for
the growth of the flame brush thickness.
The flame geometry is observed to influence the magnitude of turbulent scalar
flux at the leading edge, spherical flames showing larger magnitude compared to
the planar flames for a given turbulence and thermochemical conditions. The
mean reaction rate is found to be less influenced by the flame geometry. Thus,
the influence of flame geometry on the turbulent flame speed, leading edge dis-
placement speed, is observed to result from the contribution of the turbulence
scalar flux. The turbulent flame speed, st, of the spherical flames is observed to
be 10 to 20% greater than the corresponding planar flame values for the condi-
tions investigated in this study. For a constant value of turbulence stretch rate,
st ∼ u′ as noted by Bray (1990) and this scaling is observed for both planar and
spherical flames. The values of st, normalised by the laminar flame speed, for
the 14 flames computed in this study scales as Rent with 0.57 ≤ n ≤ 0.58. This
scaling is consistent with the classical analysis of Damko¨hler. The consumption
speed also shows a similar scaling with Ret. The results presented in this chapter
is encouraging in using scalar dissipation rate based models to simulate turbulent
78
4.5. Summary
combustion in spark-ignition engines, which involve expanding flame balls.
The numerical method used in this chapter is used to simulate hydrogen-air
spherical flames in the following chapter.
79
5. Spherical hydrogen-air flames
Results from the simulation hydrogen-air spherical flames are presented in this
chapter. The numerical method described in Chapter 3 is used to simulate these
flames. Application of this numerical method for hydrocarbon flames has been
explained in the preceding chapter. The main aim of this chapter is to assess the
predictive ability of the scalar dissipation rate based combustion models in simu-
lating hydrogen-air flames, which have non-unity Lewis numbers. In addition, the
effect of turbulence on the propagation of hydrogen-air flames is investigated, and
results obtained for hydrogen-air flames are compared with those of methane-air
flames obtained in Chapter 4.
5.1 Introduction
The envisaged depletion of fossil fuel resources and a need to reduce pollutants
emission from combustion have led to a surge in finding alternative energy sources.
Hydrogen is considered as a potential future energy carrier with many benefits
over the current hydrocarbon fuels (DeLuchi, 1989; Ogden, 1999; Balat, 2008). In
particular, good combustion characteristics of hydrogen make it an attractive fuel
for internal combustion engines (White et al., 2006; Verhelst and Wallner, 2009).
Hydrogen has certain favourable combustion properties such as wide flammable
range and large burning velocity, which render hydrogen as an ideal additive to im-
prove combustion characteristics of new and bio-derived hydrocarbon fuels (Bauer
and Forest, 2001). Also, fundamental understanding of hydrogen combustion is
important from safety view points; for example, generation and accumulation of
hydrogen in nuclear reactors (Stohl et al., 2012) and rupturing of a pressurised
hydrogen storage tank can lead to explosions.
A spherically expanding flame is commonly used to investigate fundamental
characteristics of hydrogen-air combustion from various view points and these
studies related to internal combustion engines (Verhelst and Wallner, 2009) and
safety aspects (Kumar et al., 1989; Molkov et al., 2007) have been reviewed in the
past. In earlier experimental studies the influence of fluid dynamic stretch, κ, on
80
5.1. Introduction
the laminar flame speed, s0L, were ignored. This influence is given in Eq. (2.37)
for small values of κ and the Markstein length scale L can be positive or negative
(Law and Sung, 2000). For a spherical flame with radius rf , the stretch rate is
defined as: κ = (2/rf )(drf/dt) (Law and Sung, 2000). Later studies (Dowdy
et al., 1990; Kwon et al., 1992a; Aung et al., 1998; Kwon and Faeth, 2001; Tse
et al., 2000; Kwon et al., 2002; Verhelst, 2005; Verhelst et al., 2005) showed that
the stretch effects must be included in the analysis to explain the presence of
cellular instabilities observed in experiments of lean hydrogen-air spherical flames.
The additional flame area resulting from this instability led to an increase in sL,
implying a negative L for thermo-diffusively unstable lean hydrogen-air mixtures.
The stoichiometric and rich mixtures showed positive L.
Since the thermo-diffusive instabilities result from differential and/or preferen-
tial diffusion phenomena, the Lewis number is typically used to identify thermo-
diffusively unstable mixtures. Lewis number is typically less than unity for lean
hydrogen-air mixtures (Law and Sung, 2000) and these flames are more suscep-
tible to cellular instability (Tse et al., 2000; Verhelst et al., 2005; Bradley et al.,
2007; Kitagawa et al., 2008). A review of these studies on thermo-diffusive insta-
bilities can be found in Matalon (2007).
Similar to methane-air flames considered in Chapter 4, turbulent spherical
hydrogen-air flames have been investigated using fan-stirred bombs (Wu et al.,
1990; Kwon et al., 1992b; Aung et al., 2002; Kido et al., 2002; Kitagawa et al.,
2008) and wind tunnels with grid turbulence (Renou et al., 1998, 2000) to ad-
dress the role of turbulence. These studies showed that the turbulent burning
velocity is increased when the reactant mixture yields thermo-diffusively unstable
flames and this effect is pronounced when the turbulence is weak. Accounting
for thermo-diffusive instability effects in turbulent combustion modelling is a
challenging task and is still an open question although some attempts has been
made in the past to include hydrodynamic instability effects (Paul and Bray,
1996). One way to account for thermo-diffusive instability effects is to use an
effective Lewis number to modify the turbulent burning velocity expressions as
has been done in Muppala et al. (2009) for hydrocarbon flames. An alternative
approach is to include the instability effects in laminar flame speed correlations
obtained using spherically expanding laminar flames and use them as input to
81
5.2. Numerical setup
turbulent combustion models based on turbulent burning velocity or flame sur-
face density (Gerke, 2007). These approaches were shown to yield a satisfactory
comparison with measurements for laboratory scale flames (Muppala et al., 2009),
a single-cylinder compression machine (Gerke, 2007) and spark-ignited internal
combustion engines (Rakopoulos et al., 2010). Following the second approach,
the thermo-diffusive effects are expected to be included when s0L calculated with
detailed chemistry and transport is used. This philosophy of including thermo-
diffusive effects, however, does not seem to be adequate for a test case considered
here as one shall see later in section 5.4.1.
The simulations are performed using the numerical approach outlined in Chap-
ter 3. The specific objectives of this work are:
1. To assess the scalar dissipation rate based models given in section 3.2 for
turbulent spherical hydrogen-air flames by comparing measured (Kitagawa
et al., 2008) and computed results;
2. To compare and contrast turbulent spherical flame propagation character-
istics of hydrogen- and methane-air mixtures having the same equivalence
ratio and turbulence conditions.
This chapter is organised as follows. The numerical problem setup is described
in the next section. Test cases used for model validation are described in sec-
tion 5.3. Results from these simulations are discussed in section 5.4 along with a
comparison of different reaction rate closures. The final section summarises the
main learnings from these simulations.
5.2 Numerical setup
The numerical setup described in Chapter 3 is used to simulate spherical hydrogen-
air flames. Application of this numerical method to simulate methane-air flames
has been described in Chapter 4. In this section, the combustion modelling chal-
lenges that are unique to hydrogen-air flames are discussed.
82
5.2. Numerical setup
5.2.1 Reaction rate model
Details of the modelling technique have been described in section 2.5.8 and the
flamelet library generation was given in section 3.2. For hydrogen-air flames,
K∗c in the algebraic models for the mean scalar dissipation rate [Eqs. (2.69) and
(3.14)] takes a value of 0.73τ for φ = 0.4 and 0.66τ for φ = 1.0 (Rogerson and
Swaminathan, 2007).
0
50
100
150
200
250
300
0.0 0.2 0.4 0.6 0.8 1.0
s0 L
(c
m
/s
)
φ
Westbrook (1982) △
Miller et al. (1982) ▽
Conaire et al. (2004) ©
Li et al. (2004)
Burke et al. (2011) ♦
EXP
MECH
Wu & Law (1984)
Egolfopoulos & Law (1990)
Dowdy et al. (1990)
Vagelopoulos et al. (1994)
Aung et al. (1997)
Tse et al. (2000)
Kwon & Faeth et al. (2001)
Lamoureux et al. (2003)
Verhelst et al. (2005)
Huang et al. (2006)
Tang et al. (2008)
Kitagawa et al. (2008)
Hu et al. (2009)
Pareja et al. (2010)
Kuznetsov et al. (2012)
Figure 5.1: Comparison of computed and measured unstretched laminar burning
velocity for various equivalence ratios. The experimental values are obtained from
15 earlier studies.
Comprehensive chemical kinetic mechanisms can be used to calculate laminar
flames used in the reaction rate modelling technique described in section 3.2; a
number of such mechanisms are available for hydrogen-air combustion. Figure 5.1
compares unstretched laminar burning velocity, s0L, computed in this study em-
ploying several such mechanisms (Westbrook, 1982; Miller, 1982; O´ Conaire et al.,
2004; Li et al., 2004; Burke et al., 2011), denoted as MECH in Figure 5.1, with
measured values available in the open literature (Wu and Law, 1984; Egolfopoulos
and Law, 1990; Dowdy et al., 1990; Vagelopoulos et al., 1994; Aung et al., 1997;
83
5.2. Numerical setup
Tse et al., 2000; Kwon and Faeth, 2001; Lamoureux et al., 2003; Verhelst et al.,
2005; Huang et al., 2006; Tang et al., 2008; Kitagawa et al., 2008; Hu et al., 2009;
Pareja et al., 2010; Kuznetsov et al., 2012).
These experiments, denoted as EXP in Figure 5.1, were conducted at atmo-
spheric pressure and reactant temperature of about 300 K using various flame
configurations such as outwardly propagating spherical, stagnation point and
counter-flow flames. These results are shown in Figure 5.1 only for the equiva-
lence ratio spanning from lean to stoichiometric range. There is a large scatter in
the experimental data and the computational results for various chemical mech-
anisms used here agree quite well for lean mixtures but the values of s0L obtained
using the mechanisms of Westbrook (1982) and Miller (1982) are beyond the
experimental scatter. The values obtained using the mechanisms of O´ Conaire
et al. (2004), Li et al. (2004) and Burke et al. (2011) are in good agreement with
the experimental data for φ = 1 mixture. The mechanism of Li et al. (2004) is
used in this study to calculate turbulent spherical flames of hydrogen-air mixture
having φ = 0.4 and 1. In addition, the mechanism of Westbrook (1982) is used
for the strained flamelet in order to find the effect of the chemical mechanism.
Unlike the methane-air flame simulated in Chapter 4, hydrogen-air flames
typically have non-unity Lewis numbers and the method used in this study to
account for this effect is described next.
5.2.2 Accounting for non-unity Lewis number
One needs at least two degrees of freedom to describe the thermochemical state of
a mixture with non-unity Lewis number, since the temperature and mass fraction
evolve differently. Thus, two progress variables are used in this study to account
for this. One is based on H2O mass fraction normalised by its burnt side value in
an unstrained laminar flame, defined as c = cH2O = YH2O/Y
b
H2O and another one
is based on temperature cT = (T − Tu)/(Tb − Tu). In this work, water vapour is
used instead of H2 for the progress variable because the Lewis number of H2O is
close to unity. It is well-known that non-unity Lewis number flames can display
super-adiabatic values of temperature (Rutland and Trouve´, 1993; Chakraborty
and Cant, 2005); therefore, it is questionable to use temperature based progress
84
5.2. Numerical setup
variable for such cases. Indeed, such super-adiabatic behaviour was observed for
the lean hydrogen-air flame simulated in this work.
The variation of c with cT in the laminar strained and unstrained flames is
shown in Figure 5.2(a). If c and cT are identical then the variation in Figure 5.2(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
c
=
c H
2
O
cT
(a)
0.98
1.02
1.06
1.10
1.14
1.18
0.0 0.2 0.4 0.6 0.8 1.0
M M
u
cH2O
(b)
Figure 5.2: Variation of (a) c = cH2O with cT and (b) mixture molecular weight,
M , normalised by its unburnt side value, across unstrained (solid line) and
strained (dashed lines) flamelets.
should fall on the diagonal line and this result clearly show that the difference
between these two progress variables is large on the burnt side. This is as expected
and a similar observation has been made in turbulent flames as well (Chakraborty
and Cant, 2009; Minamoto et al., 2011). The mixture molecular weight also
changes considerably, nearly 18%, across the flamelets as shown in Figure 5.2(b).
For these reasons, the mean density is calculated as
ρ
ρu
=
(
M
Mu
)
1
(1 + τ c˜T )
, (5.1)
using the equation of state. The non-unity Lewis number effects on the chemi-
cal source terms is presumed to be included at the flamelets level since detailed
transport and chemical kinetics are used to compute the necessary laminar flames.
This is akin to the methodology used in earlier studies as noted in the introduc-
tion. Thus, two Favre-averaged progress variables, c˜ and c˜T , and c˜′′2 equations
85
5.3. Test flames
in addition to other conservation equations are solved for the turbulent flames.
Note that c˜ is used as an independent variable in the flamelet library generation
and the chemical source terms ω˙c, ω˙cT and ω˙
′′c′′ are obtained using the method
described in section 3.2. As before a single variance equation corresponding to
c˜ is solved as well. Since the molecular diffusivities are smaller than turbulent
diffusivities, the difference between the c˜ and c˜T transport equations is only in
their respective source terms.
The algebraic model for the mean scalar dissipation rate, ˜c is influenced by
Lewis number (Chakraborty and Swaminathan, 2011). Using the closure models
proposed by Chakraborty and Swaminathan (2011), one could modify the closure
model given in Eq. (2.69). A significant difference in the results were not observed
during initial tests using this modified version of ˜c, hence it is not considered
further.
5.3 Test flames
The numerical model of an outwardly propagating turbulent spherical flame de-
scribed in the previous section is used to investigate turbulence effect on the
propagation of hydrogen-air spherical flames. The results of these simulations
will be compared to spherical methane-air flames presented in Chapter 4. Before
discussing these test cases, experimental flames, used to assess the validity of
scalar dissipation rate based combustion models for hydrogen-air mixtures, are
described.
5.3.1 Experimental flame
Spherically expanding turbulent hydrogen-air flames inside a fan-stirred combus-
tion vessel investigated by Kitagawa et al. (2008) are considered for the assessment
of the reaction rate models. In these experiments, turbulent flame propagation
was reported for two equivalence ratios (φ = 0.4 and 1) at two different ini-
tial pressures, 1 and 5 atm and an initial temperature of 300 K. Out of these
cases, only the atmospheric flames are considered here for computational rea-
sons. Thermo-chemical characteristics of these two flames are given in Table 5.1
86
5.3. Test flames
and τ 6= ρu/ρb − 1, specifically for φ = 1 case, due to the variation of mixture
molecular weight across the flame front as noted earlier.
Table 5.1: Thermo-chemical conditions of experimental flames (Kitagawa et al.,
2008) considered for this study
φ s0L (m/s) δ
0
L (mm) δ (mm) ρu/ρb τ
0.4 0.21 0.67 0.09 4.31 3.66
1.0 1.98 0.35 0.01 6.84 6.97
Two fans, continuously running during the experiments, were used to mix the
reactants inside the vessel and to generate non-decaying isotropic turbulence. Ex-
periments were conducted at two turbulence intensities and the higher turbulence
case with rms of velocity fluctuations, u′, of 1.59 m/s is simulated in this study.
The experimentally determined turbulence integral length scale, Λ, was 10.3 mm.
Since turbulence in the experiment is non-decaying, values of k˜ and ε˜ were frozen
for the simulations. As with the methane-air experiments of Hainsworth (1985)
described in Chapter 4, the propagation of turbulent flames was recorded using
high speed schlieren photography in the experiments of Kitagawa et al. (2008)
and as before the flame radius, rf , is defined using c˜ = 0.05 in the simulations.
5.3.2 Test cases for further analyses
Outwardly propagating turbulent spherical flames in an unconfined space are con-
sidered and the influences of heat release on turbulence are also included in these
simulations by solving for k˜ and ε˜ unlike in the above experimental cases. Only
stoichiometric hydrogen-air mixture at 300 K and atmospheric pressure are con-
sidered. The hydrogen-air flame results are to be compared with stoichiometric
methane-air flames under the same initial pressure, temperature, and turbulence
(u′/s0L and Λ/δ) conditions to understand their relative behaviour. The thermo-
chemical characteristics of stoichiometric methane-air flame are: s0L = 0.4 m/s,
δ0L = 0.41 mm, δ = 0.047 mm, τ = 6.48 and ρu/ρb = 7.54.
The conditions of the three stoichiometric hydrogen-air flames used for com-
parative analyses are depicted in the combustion regime diagram shown in Fig-
ure 5.3. These flames have the same stretch factor, K = 0.157 (u′/s0L)
2
Re−0.5t
87
5.4. Results and discussion
0.1
1
10
100
1000
0.1 1 10 100 1000 10000
u
′ /
s0 L
Λ/δ
wrinkled flameletslaminar flamelets
corrugated flamelets
thin reaction zones
broken reaction zones
Ka = 1
Ka = 100
Da = 1
Ret = 1
Figure 5.3: Turbulent premixed combustion regime diagram (Peters, 1999) show-
ing the conditions of test cases for further analyses () and the experimental case
of Kitagawa et al. (2008) (4) considered in this study.
= 0.157 (Abdel-Gayed et al., 1987), where the turbulent Reynolds number is de-
fined as Ret = u
′Λ/ν with ν as the kinematic viscosity of the reactant mixture.
For these flames, u′/s0L values are 5, 6 and 8 and the stoichiometric methane-air
flames also have the same combustion conditions as those for H2-air flames shown
in Figure 5.3. All of these flames are in the border between corrugated flamelets
and thin reaction zones regime. The experimental flames of Kitagawa et al.
(2008) are also shown in this figure – the stoichiometric flame is near the border
between wrinkled and corrugated flamelets while the lean flame with φ = 0.4 is
near Da = 1 line, where Da = (Λ/δ)/(u′/s0L) is the Damko¨hler number.
5.4 Results and discussion
The computational results of spherically propagating hydrogen-air flames under
a range of turbulence conditions are analysed in this section. Validation of the
computational models is discussed first before presenting comparative analyses of
hydrogen- and methane-air flames experiencing the same turbulent combustion
conditions.
88
5.4. Results and discussion
5.4.1 Validation
The amount of heat released by burning mf amount of fuel is simply given by
Q = mf QLHV, where QLHV = 120 MJ/kg is the lower heating value of hydrogen
(Verhelst and Wallner, 2009). The amount of fuel consumed by the spherical
turbulent flame when its leading edge moves from a radius of rf1 = 4.2 cm to
rf2 = 7.74 cm over a period of ∆t = 1.7 ms is mf = ρu Yf,u4pi
(
r3f2 − r3f1
)
/3,
where Yf,u = 0.028 is the fuel mass fraction in the stoichiometric hydrogen-air
mixture. The leading edge radius, rf , is defined using c˜ = c˜1 = 0.05 iso-level in
the simulation and the above radii and ∆t are taken from one of the simulations
chosen arbitrarily. The theoretical value for the heat release is 4.68 kJ and this
value calculated from the results of numerical simulation using
Q = 4pi (Tb − Tu)
∫ t+∆t
t
[∫ rf2
rf1
Cp(r, t) ω˙cT (r, t) r
2 dr
]
dt, (5.2)
agrees within 6% of the above value. The symbol Cp(r, t) is the mean mixture heat
capacity at constant pressure. This level of agreement is acceptable in the light
of various approximations made in the modelling of turbulence and combustion.
As noted earlier, two atmospheric flames, φ = 1 and 0.4, of Kitagawa et al.
(2008) are simulated to assess the validity of scalar dissipation rate based models
for hydrogen-air combustion. The variation of turbulent flame propagation speed,
ut, with rf was reported in Kitagawa et al. (2008) and this will be used for the
validation purpose. This speed is defined as (Andrews and Bradley, 1972):
ut =
ρb
ρu
drf
dt
, (5.3)
where ρb is the mean density of the burnt gases. As with the methane-air flame
results given in section 4.3, the numerical simulations give c˜(r, t), and thus it is
straightforward to compute ut. Figure 5.4 compares the computed and measured
ut variation with rf for the stoichiometric, Figure 5.4(a), and lean, Figure 5.4(b),
hydrogen-air flames. The computational results are shown for three combustion
models described in section 3.2. The chemical kinetics mechanism of Li et al.
(2004) is used for the unstrained flamelets model in Eq. (2.63). The mechanisms
89
5.4. Results and discussion
of both Li et al. (2004) (noted as mech 1 in Figure 5.4(a)) and Westbrook (1982)
(mech 2) are used for the strained flamelets model in order to assess the influence
of chemical kinetics. Both the algebraic and the unstrained flamelet models yield
larger ut values.
The strained flamelet model with the mechanism of Li et al. (2004) is able
to capture the measured variation reasonably well, whereas the values of ut com-
puted using the mechanism of Westbrook (1982) are smaller as in Figure 5.4(a).
This is because the mechanism of Westbrook (1982) underpredicts the laminar
burning velocity, s0L, as was shown in section 5.2.1. This highlights the importance
of choosing a chemical mechanism that gives accurate laminar flame characteris-
tics required for the strained flamelets model. The maximum error between the
measured and computed ut values is about 5% for the strained flamelets model
and this error is well within the experimental scatter of Kitagawa et al. (2008).
The variation of ut shown in Figure 5.4 suggests that the burning velocity for
the spherical flame is not constant as has been observed in many earlier studies
(Kwon et al., 1992b; Aung et al., 2002). In turbulent flames, the mean reaction
rate depends on the fuel, the equivalence ratio and local turbulent strain rate
(Cant and Bray, 1989). In these flames, local fluid dynamic strain induced by
turbulent eddies acts to reduce the burning rate. Amongst the models used in
this study, the strained flamelets model seems to describe both the complex chem-
istry and the local fluid dynamic effects quite well for stoichiometric hydrogen-air
flame. This flame is thermo-diffusively stable whereas the lean flame having
φ = 0.4 is thermo-diffusively unstable as noted by Kitagawa et al. (2008). Fig-
ure 5.4(b) compares the measured and computed ut values for φ = 0.4 flame. All
the three models severely underestimate the turbulent burning velocity because
these models do not include the influences of thermo-diffusive instability on the
propagation of turbulent flame leading edge. Including these effects in turbulent
combustion modelling is a challenging task and one way to include them may
be through the use of Markstein numbers but this approach is strictly valid for
small stretch rates κ+ ≡ κδ0L/s0L 1 (Poinsot and Veynante, 2005). In the
remainder of this chapter only stoichiometric hydrogen-air flames simulated us-
ing the strained flamelet model [Eq. (3.10)], will be considered for further and
comparative analyses.
90
5.4. Results and discussion
0
1
2
3
4
5
0 10 20 30 40 50 60 70
u
t
(m
/s
)
rf (mm)
φ = 1.0
Experiment
Str. flamelet (mech 1)
Str. flamelet (mech 2)
Unstr. flamelet
Algebraic
(a)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 10 20 30 40 50 60
u
t
(m
/s
)
rf (mm)
φ = 0.4
Experiment
Str. flamelet
Unstr. flamelet
Algebraic
(b)
Figure 5.4: Comparison of computed and measured variation of ut with rf in (a)
stoichiometric and (b) lean hydrogen-air flames.
91
5.4. Results and discussion
5.4.2 Comparison of hydrogen- and methane-air flames
Propagation characteristics
The stoichiometric hydrogen- and methane-air spherical flames experiencing the
same combustion conditions are compared in this section. Typical evolution of
these flames is shown in Figure 5.5 for u′/s0L = 6 case. Both of these flames
evolve from the same initial variation of c˜ shown as dashed lines and the results
are shown for a period of 3 ms at an equal interval of 1 ms. The symbols used for
3 ms indicates typical spatial resolution used in simulations. The stoichiometric
hydrogen flames propagate faster than the methane flames, which is expected
since the laminar burning velocity for the hydrogen-air mixture is larger than for
the methane-air mixture.
Variation of flame radius, normalised by its initial value, with time is shown
in Figure 5.6 for all 6 (3 H2- and 3 CH4-air) flames considered in this study. For a
given combustion condition denoted by u′/s0L and Λ/δ, the hydrogen flames prop-
agate significantly (nearly 4 to 5 times) faster compared to methane-air flames.
This result also shows that the increase in u′, for a constant K or Ka, results in
faster flame propagation for both of these fuel-air mixtures. This increase in flame
propagation speed agrees with many previous studies summarised by Lipatnikov
and Chomiak (2002).
Turbulent flame speed and consumption speed
The importance of the turbulent flame speed, st, has been described in sec-
tion 4.3.2. As described in this section, st is defined to be the displacement
speed of the flame brush leading edge, marked using c˜ = 0.05. The displacement
speed, sd, is given in Eq. (4.1). Methane-air flame simulations in Chapter 4 have
already shown that sd is influenced directly by the mean reaction rate, turbu-
lent flux and molecular diffusion. Note the difference in definitions of st and the
turbulent burning velocity, ut, given in Eq. (5.3).
Figure 5.7(a) shows the variation of normalised turbulent flame speed, st/s
0
L,
with turbulent Reynolds number, Ret. The results are shown for a normalised
time of t+ = 8 and the results for other earlier times are similar to those shown
92
5.4. Results and discussion
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
c˜
x (m)
CH4/air
t
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
c˜
x (m)
H2/air
t
(b)
Figure 5.5: Spatial variation of c˜ at four instances from 0 to 3 ms at an interval
of 1 ms in (a) CH4-air and (b) H2-air flames having u
′/s0L = 6.
93
5.4. Results and discussion
0
5
10
15
20
25
0 1 2 3 4 5 6 7 8
r f
/r
f
0
t (ms)
u′/s0L = 5
u′/s0L = 6
u′/s0L = 8
Figure 5.6: Temporal variation of normalised flame radius of CH4-air (open sym-
bols) and H2-air (closed) turbulent flames at various combustion conditions.
here, as noted in section 4.3.2 for methane-air flames. The inset shows the varia-
tion of st/s
0
L with u
′/s0L, which again suggests st ∼ u′. The slopes of st/s0L versus
u′/s0L curves are 6.3 and 4.4 respectively for the methane- and hydrogen-air flames
considered here. It was observed earlier in section 4.3.2 that this slope became
steeper when the stretch factor, K, is increased.
These results suggest the same scaling relation, st/s
0
L ' BRent , is applicable
for both flames, with n ≈ 0.5. The least squares fit shown in the figure suggests
B = 0.21 for the methane- and 0.16 for hydrogen-air flames. The variation of
consumption speed, sc [defined in Eq. (4.3)] with Ret is shown in Figure 4.11(b)].
The scaling of sc/s
0
L with Ret is similar to that of st/s
0
L, albeit with a lower
magnitude. As with st/s
0
L, the computed values of sc/s
0
L are about 20% lower for
hydrogen-air flames when compared with methane-air flames; this is due to the
larger s0L value.
Turbulent flame brush thickness
Figure 5.8 shows the temporal variation of flame brush thickness for the stoi-
chiometric H2-air flame with u
′/s0L = 8. The flame brush thickness is normalised
using the laminar flame thermal thickness and time is normalised using the flame
94
5.4. Results and discussion
5
10
15
20
25
30
500 1000 1500 2000 2500 3000 3500 4000 4500
s t
/s
0 L
Ret
y = 0.21 x0.58, R2 = 1.0
y = 0.16 x0.57, R2 = 0.9997
0
15
30
5 6 7 8
s t
/s
0 L
u′/s0L
(a)
5
10
15
20
25
30
500 1000 1500 2000 2500 3000 3500 4000 4500
s t
/s
0 L
Ret
y = 0.21 x0.58, R2 = 1.0
y = 0.16 x0.57, R2 = 0.9997
0
15
30
5 6 7 8
s t
/s
0 L
u′/s0L
(b)
Figure 5.7: Variation of normalised turbulent (a) flame and (b) consumption
speeds of H2-air (closed symbols) and CH4-air flames (open) with turbulence
Reynolds number. The inset shows the variations with u′. The results are shown
for t+ = 8.
95
5.4. Results and discussion
20
30
40
50
60
70
80
90
0 5 10 15 20
δ t
/δ
0 L
t+
1/|∂c˜/∂r|max
0.54δ√
c˜′′2
Taylor’s theory
Figure 5.8: Temporal variation of normalised flame brush thickness for H2-air
flame with u′/s0L = 8.
time, tc = δ
0
L/s
0
L. As in section 4.3.2, the turbulent flame brush thickness is cal-
culated using the maximum gradient of c˜ and the variance, c˜′′2 (which is scaled
in Figure 5.8). After going through some initial transients, both thickness grow
with time. If the turbulent diffusion plays the central role for this growth then
one would expect to see a growth similar to that shown for Taylor’s theory (also
plotted). This theory suggests a linear variation for t+ Da and a square root
dependence for t+ Da. It is apparent that the computed thicknesses do not
follow these variations as it has been observed for methane-air flames (see sec-
tion 4.3.2). Thus, it is concluded here the growth of the flame brush thickness
in the stoichiometric hydrogen-air flames studied here is governed by the differ-
ential propagation of the leading and trailing edges of the flame brush as for
the methane-air flames, i.e. the leading edge propagates faster compared to the
trailing edge.
According to the concept of turbulent premixed combustion regimes, two
flames with identical values for Λ/δ, u′/s0L and τ are expected to have simi-
96
5.4. Results and discussion
lar st/s
0
L. Although these quantities are kept to be identical for the hydrogen-
and methane-air flames investigated in this work, there is a significant difference
in st/s
0
L. The well known KPP analysis (Zeldovich et al., 1985) shows that st
strongly depends on the rate of change of ω˙c with respect to c˜, as c˜ → 0. This
quantity not only depends on the turbulence-chemistry interaction but also on
the combustion kinetics. The turbulence-chemistry interaction is expected to be
predominantly the same if u′/s0L and Λ/δ are kept the same. To gain an under-
standing of
(
∂ω˙c/∂c˜
)
c˜→0, the variation of ω˙c with c˜ is studied next.
Mean reaction rates
Figure 5.9(a) shows a comparison of the mean reaction rate variation across the
flame brush for planar and spherical hydrogen-air flames. The mean reaction
rates are normalised by the respective maximum value inside the flame brush. It
was shown earlier in Figure 4.10 for methane-air flames that the normalised mean
reaction rates are insensitive to geometry. The plots in Figure 5.9(a) shows that
this is true even for hydrogen-air flames. Only the flame at u′/s0L = 6 is shown
in this figure but it is expected that this will hold for other flames as well.
Figure 5.9(b) shows a comparison of the normalised mean reaction rates for
spherical hydrogen- and methane-air flames. The top inset in this figure, shows
the maximum value of the normalised mean reaction rate, ω˙
+
max, for different
turbulence conditions considered for this study. As was shown previously for
methane-air flames, these results show that, the turbulence level does not have a
significant effect on ω˙
+
max for hydrogen-air flames as well. However, there is some
difference in the variation of normalised reaction rate across the flame brush
between the methane- and hydrogen-air mixtures. The mean reaction rate is
more uniform inside the flame brush for hydrogen compared to methane. The
difference between these two flames looks small for low c˜ values, however the
bottom inset clearly shows that there is a substantial difference. The value of(
∂ω˙c/∂c˜
)
c˜→0 for the hydrogen flame can be two orders of magnitude larger than
for the methane flame. This is because of the low activation temperature for
hydrogen combustion. Obviously, this parameter is related to chemical kinetics
of the fuel.
97
5.4. Results and discussion
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
ω˙
/ω˙
m
a
x
c˜
Planar
Spherical
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
ω˙
/ω˙
m
a
x
c˜
0.00
0.25
0.50
4 6 8 10
ω˙
+ m
a
x
u′/s0L
0.0
0.1
0.2
0 0.02 0.04
ω˙
/ω˙
m
a
x
c˜
H2/air
CH4/air
(b)
Figure 5.9: Comparison of normalised mean reaction rate for u′/s0L = 6 at t
+ = 8
between (a) planar and spherical H2-air flames and (b) H2- and CH4-air flames.
The top inset shows the variation of ω˙
+
max, normalised as ω˙maxδ
0
L/ (ρus
0
L), with
u′/s0L.
98
5.5. Summary
5.5 Summary
Numerical simulations of turbulent spherical hydrogen-air flames have been per-
formed using RANS methodology employing three different reaction rate closures
based on the scalar dissipation rate: an algebraic model of Bray (1979), strained
and unstrained flamelets. Since the hydrogen-air mixture has non-unity Lewis
number, a two progress variable approach is used in this work to account for dif-
ferential evolution of temperature and mass fraction in hydrogen-air combustion.
This modelling approach is assessed using the experimental data of Kitagawa et al.
(2008). The measured variation of propagation speed of flame brush leading edge
with flame radius is compared with the computed values for stoichiometric and
a lean, φ = 0.4, flames. This comparison demonstrate that the stretch effects on
flamelets must be included to capture experimental measurements. The strained
flamelets model is able to capture the experimental variations quite well while
the unstrained flamelets and the algebraic models give faster flame propagation.
Also, the results also showed that the chemical kinetics mechanism to be used in
the calculation must capture the laminar flame characteristics such as burning
velocity, flame thermal thickness, flame structure, etc., of the corresponding mix-
ture well. The use of the strained flamelets modelling approach is justified for
stoichiometric hydrogen-air flames as these flames are thermo-diffusively stable.
The propagation speed of the lean flame, which is thermo-diffusively unstable, is
underestimated by all three combustion models used in this study. It seems that
the approach of including the thermo-diffusive effects in the laminar flamelets is
inadequate and an alternative methodology need to be found.
The results of stoichiometric hydrogen- and methane-air spherical flames ob-
tained using strained flamelets model are analysed comparatively to understand
the relative effects of turbulence on the propagation of these flames. It is ob-
served that for a constant value of turbulence stretch rate, st ∼ u′ for hydrogen-
air flames, which was shown earlier for methane-air flames. Furthermore, the
normalised turbulent flame speed, st/s
0
L, and consumption speed, sc/s
0
L, scale
as Rent , with n ≈ 0.5 for both mixtures. However, the magnitudes of these
speeds are observed to be substantially different for the stoichiometric hydrogen-
and methane-air mixtures despite the fact that these H2- and CH4-air flames
99
5.5. Summary
have identical combustion conditions in terms of u′/s0L and Λ/δ, implying sim-
ilar turbulence-flame interactions. It is observed that this difference is related
to behaviour of
(
∂ω˙c/∂c˜
)
c˜→0, which is controlled not only by turbulence and its
interaction with the flame but also by chemical kinetics. This gradient value is
observed to be nearly two orders of magnitude larger for the H2-air flame com-
pared to the CH4-air flame. The predominant role of differential propagation
between the leading and trailing edges of the flame brush on the growth of the
flame brush thickness is also observed for the hydrogen flames.
Now that the scalar dissipation rate based models have been validated for
spherically propagating flames of two different fuel-air mixtures, the aim is to
use these models to simulate intermittent combustion, found in internal combus-
tion engines. Before proceeding with a simulation of a practical spark-ignition
engine, a simple validation test case for intermittent combustion is carried out as
explained in the next chapter.
100
6. Combustion in a closed vessel
with swirl
Flames simulated in Chapters 4 and 5 can be considered to be unconfined and
combustion is continuous, where experimental measurements were taken before
compression effects due to combustion are felt by the flame. In contrast, inter-
mittent combustion take place in IC engines due to compression of the end gases
by the propagating flame. The scalar dissipation rate based models is used in
this work to simulate combustion inside a closed vessel, where turbulence is gen-
erated through the swirling in-flow of premixed fuel-air mixture. This is used as
simple test case for a SI engine. The simulations are performed in a commercially
available CFD code called STAR-CD.
6.1 Introduction
Computations of lean flames require reliable and robust combustion models. Eddy
dissipation rate based models, commonly used for internal combustion engine
simulations, have a limited predictive ability as they do not include the effects of
chemical kinetics. The scalar dissipation rate based flamelet models described in
section 3.2 have shown to be robust for continuous combustion systems (Chap-
ters 4 and 5).
This modelling approach is used in this chapter to simulate the experiments
of Hamamoto et al. (1988). Combustion inside the closed vessel caused the pres-
sure to increase by 10 folds in these experiments. This provides an additional
challenge for the models, which were previously only used for constant pressure
combustion simulations. As before, a reaction progress variable is used to de-
scribe the thermochemical system. The main objective of this work is to validate
the flamelets based approach described in section 2.5.8 for combustion in closed
vessel with complex flows.
101
6.2. Experimental test case
6.2 Experimental test case
The experiment of Hamamoto et al. (1988) is used to assess the performance
of strained and unstrained flamelet models that were used to simulate spherical
flames in the preceding chapters. This experiment is more representative of real
spark-ignition combustion since the compression effects caused by combustion
are taken into consideration. Hamamoto et al. (1988) conducted experiments in
a cylindrical combustion chamber, where swirling flow was produced by charging
the fuel-air mixture tangentially in to the combustion chamber. Swirl increases
the turbulence intensity, which leads to an increase in the turbulent flame burning
velocity. The mixture was then spark ignited at the centre to create a propagating
turbulent flame.
The combustion chamber had a diameter of 125 mm and a width of 35 mm.
It was initially at a pressure of 50 kPa and was connected to a tank containing
stoichiometric propane-air mixture. This fuel-air mixture in the tank was ini-
tially at room temperature, while two initial pressure conditions at 300 kPa or at
400 kPa were investigated. Only the case of 300 kPa is simulated in this work. In
this case, the pressure inside the combustion chamber increased from 50 kPa to
243 kPa, once the fuel-air mixture was discharged in to the combustion chamber
by opening a valve. This discharge also initially increased the temperature inside
the combustion chamber, which reduced gradually due to heat loss from the walls.
Hamamoto et al. (1988) conducted experiments with various flow fields by
spark igniting the mixture at different times after the valve closure. The swirling
flow decayed with time, which reduced the turbulence intensity. Using the radial
distribution of the mean tangential velocity an angular velocity, Ω, was calculated.
In this work, Ω = 139.1 rads/s case is simulated, which corresponded to tv =
10 ms (i.e. the time between valve closure and spark ignition). At this time the
temperature inside the combustion chamber was at 325 K.
Flame propagation was measured using high-speed schlieren photography and
an ion probe. A two-dimensional laser Doppler anemometer was used to measure
flow velocity. Combustion inside the vessel caused the pressure to increase due to
compression and a pressure transducer was used to measure the pressure inside
the combustion chamber.
102
6.3. Numerical setup
6.3 Numerical setup
Numerical setup used to simulate the experimental is described in this sec-
tion. The simulations are performed using STAR-CD v4.18, which is a multi-
dimensional CFD code. The URANS approach is used in this work, where the
equations solved in the code are given in section 2.2.3. The numerical method
used in this chapter is similar to the setup described in Chapter 3 and only the
differences are highlighted in this section.
6.3.1 Flamelet table generation
The look-up table generation for the flamelet models is described in this section.
The method described here is similar to the one outlined in section 3.3, how-
ever, unlike the constant pressure calculations described in Chapters 4 and 5,
additional dimensions have to be included to account for the change in mixture
thermochemistry due to compression.
Chemical mechanism
As with the spherically symmetric cases described in Chapters 4 and 5, a detailed
chemical mechanism is used to calculate freely propagating and strained laminar
flames. In this work, the detailed chemical mechanism of Sung et al. (1998) is used
for propane-air combustion. Figure 6.1 shows a comparison of the unstretched
laminar flame speed calculated using this chemical mechanism with stoichiometric
propane-air flame experimental data of Vagelopoulos et al. (1994), Hassan et al.
(1998b) and Jomaas et al. (2005), at different pressures with Tu = 298 K. The
comparison is reasonably good for the pressures considered in these experiments.
Effect of pressure
Compression effects from combustion cause the unburnt mixture temperature,
Tu, to change with pressure, p. Therefore, a number of laminar flames at various
temperatures and pressures have to be computed to cover the range of reaction
rates, ω˙, experienced by the turbulent flame. Furthermore, the values of s0L and
103
6.3. Numerical setup
0.0
0.1
0.2
0.3
0.4
0.5
1.0 2.0 3.0 4.0 5.0 6.0
s0 L
(m
/s
)
p (bar)
Computations
Jomaas et al. (2005)
Hassan et al. (1998)
Vagelopoulos et al. (1994)
Figure 6.1: Comparison of unstretched laminar flame speed of stoichiometric
propane-air mixtures from different experiments with the values predicted nu-
merically using the detailed chemical mechanism of Sung et al. (1998).
δ0L appearing in Eq. (2.69) are functions of both temperature and pressure of the
unburnt reactants and will need to be tabulated.
Figure 6.2(a) shows normalised burning rates, ω˙+, of steady one-dimensional
planar laminar flames, plotted against distance for different pressures. Reaction
rates are normalised using ρu, s
0
L and δ
0
L and the plots are shifted by x
∗, the loca-
tion of the peak heat release rate. This figure shows that increasing the pressure
intensifies the reaction and drastically reduces the flame thickness. This figure
also shows the non-linear effect of pressure on the laminar burning rate. This
non-linear response of the flame to pressure means that simple scaling arguments
cannot be used to extrapolate laminar flame results from a lower pressure to a
higher pressure. The implication of this to flamelet library generation is that
laminar flames have to be calculated for a number of initial pressures and tem-
peratures, which increases the pre-processing time for the simulations. The effect
of pressure on the laminar flame structure can be seen in Figure 6.2(b), where
the spatial profiles of c are plotted for various pressures. It is evident that the
gradient of the progress variable becomes steeper with increasing pressure. Due
104
6.3. Numerical setup
0.0
0.5
1.0
1.5
2.0
2.5
3.0
-0.4 -0.2 0 0.2 0.4
ω˙
+
x− x∗ (mm)
1 bar
5 bar
20 bar
(a)
0.0
0.2
0.4
0.6
0.8
1.0
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
c
x− x∗ (mm)
1 bar
5 bar
20 bar
(b)
Figure 6.2: (a) Normalised reaction rate of progress variable, ω˙+ = ω˙δ0L/ρus
0
L
plotted against distance for different pressures and Tu = 300 K. (b) Spatial profiles
of c for different pressures. In these plots, the peak heat release location is denoted
by x∗.
105
6.3. Numerical setup
to reasons to be explained later, the progress variable is defined based on water
vapour mass fraction.
Figure 6.3 shows the variation of the normalised maximum reaction rates,
ω˙+max, of laminar flames for different initial pressures and unburnt mixture tem-
peratures. This figure shows that for a given unburnt mixture temperature, in-
creasing the pressure increases the reaction rate. However, for a given pressure,
increasing the unburnt mixture temperature reduces ω˙+max. The reason for this
decrease is because, s0L, which is used to normalise the reaction rate, increases
with temperature.
300 350 400 450 500 550 6000
5
10
15
20
25
1.6
1.8
2.0
2.2
2.4
2.6
2.8
ω˙+max
Tu (K)
p0 (bar)
1.6
1.8
2.0
2.2
2.4
2.6
2.8
Figure 6.3: Normalised reaction rate of progress variable, ω˙+max = ω˙maxδ
0
L/ρus
0
L,
variation with pressure and temperature.
Strained flamelet
As described in section 3.3.2, response of the reaction rate due to strain need
to be determined for the strained flamelet model. Figs. 6.4(a) and 6.4(b) show
the plots of ω˙+ vs. N+ for stoichiometric propane-air flames at two different
pressures at unburnt mixture temperatures of 300 K. Similar plots are obtained
for other temperatures, therefore, these figures show that the variation of ω˙+ vs.
N+ is multi-valued and that the general shape of these curves do not change when
temperature or pressure is changed and thus remains a function of the chemical
properties of the fuel-air mixture.
106
6.3. Numerical setup
0.0
0.5
1.0
1.5
2.0
2.5
0.5 1.0 1.5 2.0 2.5 3.0 3.5
ω˙
+
N+
ζ = 0.7
ζ = 0.6
ζ = 0.5
(a)
0.0
0.5
1.0
1.5
2.0
2.5
1.0 1.5 2.0 2.5 3.0 3.5
ω˙
+
N+
ζ = 0.7
ζ = 0.6
ζ = 0.5
(b)
Figure 6.4: Curves of ω˙+ vs. N+ conditioned on the progress variable, ζ, for an
unburnt mixture temperature of 300 K at two different pressures: (a) p = 2 bar
and (b) p = 5 bar.
107
6.3. Numerical setup
The collapse of f (ζ) curves for stoichiometric propane-air flames subjected to
various strain rates and at different unburnt mixture temperatures and pressures
are shown in Figs. 6.5(a) and 6.5(b). As with the methane-air flame shown in
Figure 3.1(b), the chemical reactions dominate in the region closer to the burnt
side, with ζ∗ ≈ 0.6.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0
f(ζ)
ζ
(a)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.2 0.4 0.6 0.8 1.0
f(ζ)
ζ
(b)
Figure 6.5: Curves of f (ζ) for stoichiometric propane-air mixture at (a) Tu =
300 K and p = 11 bar and (b) Tu = 400 K and p = 8 bar. The solid curve in
these figures show the freely propagating unstrained flame.
Similar to the simulations described in Chapters 4 and 5, unstrained and
strained flamelet models can be used to build a library for ω˙ and other required
quantities. Due to the above mentioned effect of pressure on the reaction rates,
this look-up table becomes four and five dimensional respectively for unstrained
and strained cases.
Choice of progress variable
In this work the progress variable is defined in terms of the mass fraction of water
vapour, YH2O, as c = YH2O/Y
b
H2O
. This is because a progress variable based on
temperature will over-define the system of equations, since an enthalpy equation
is already solved. Furthermore, defining the progress variable in terms of species
mass fraction removes the complexity involved in implementing the Dp/Dt term
that exists in the transport equation for temperature. Here c based on fuel
108
6.3. Numerical setup
mass fraction is not used because of the poor resolution of ω˙T with c. Another
possibility is to use the mass fraction of the combination, CO + CO2.
Data given in the look-up are interpolated to obtain values required during
the CFD calculations. Unburnt mixture temperature, Tu, is obtained from the
simulation by assuming that the mixture is adiabatically compressed by the flame
front. The look-up table is generated with a resolution of 0.01 in c˜, 0.02 in g, 50 K
in temperature and 1 bar in pressure. For the strained flamelet, the additional
dimension, c, had a resolution of 5 1/s.
6.3.2 Model implementation
Strained and unstrained flamelet models are implemented in STAR-CD. Note
that the combustion models available in STAR-CD are de-activated during the
simulations. Therefore, in the current combustion model implementation, the
temperature rise during combustion is achieved by including the source term due
to combustion heat release, ω˙T , in the following Favre-averaged enthalpy equation
∂ρh˜
∂t
+∇ ·
(
ρh˜u˜
)
=
∂p
∂t
−∇ ·
(
q + ρu˜′′h′′
)
+ τ : ∇u˜+ τ : ∇u′′ + ω˙T . (6.1)
This source term is also calculated using strained and unstrained flamelet models.
Similarly, the mean heat capacity at constant pressure, Cp, is calculated using
these flamelet models and its value is specified for each cell at every time step
during the simulations.
As shown in section 3.1, the pressure-dilatation term, p′∇ · u′′, that appears
in the turbulent kinetic energy is modelled in this study and this term is included
in both k˜ and ω˜ equations. Since there is no spatial gradient of pressure in the
domain, the pressure work term is ignored. The flow and turbulence quantities
necessary for combustion modelling are obtained from the CFD code. Additional
transport equations for the progress variable, c˜ [Eq. (2.20)], and its variance, c˜′′2
[Eq. (2.65)], are implemented using user subroutines.
109
6.3. Numerical setup
6.3.3 Computational details
In this work, the k − ω SST model is chosen for turbulence closure, where ω is
the specific dissipation rate. The effect of using this turbulence model is analysed
in section 6.4, where the k − ω SST model is compared with k − RNG model.
Standard values are used for the turbulence model coefficients, which are given
in tables 6.1 and 6.2 for k − ω SST and k − RNG models respectively.
Table 6.1: Coefficients of the k − ω SST model
σωk1 1.176
σωk2 1.0
σωω1 2.0
σωω2 1.168
β1 0.075
β2 0.0828
β∗1 = β
∗
1 0.09
κ 0.41
Table 6.2: Coefficients of the k − RNG model
Cµ 0.085
σk = σ 0.719
σh = σm 0.9
C1 1.42
C2 1.68
C3 0 or 1.42
C4 -0.387
κ 0.4
E 9.0
η0 4.38
β 0.012
Since the combustion chamber is axisymmetric, only a one degree segment of
the cylindrical domain is modelled to save computational costs. Furthermore, as
spark ignition occurs at the centre of the chamber, only the top half of the cylinder
is considered. Cyclic boundary conditions are imposed on the two geometrically
identical faces and a symmetry boundary condition is applied at the mid-plane
110
6.3. Numerical setup
where the cylinder is halved. Wall boundary conditions are applied for the top
and side walls. The wall boundary is fixed at a temperature of 288 K since there
was heat loss at the walls in the experiment.
The segment is one-cell thick and the grid spacing is uniform in both the
axial and radial directions. The grid is refined until the solution did not show
a significant change in the results and the spatial resolution in both axial and
radial directions is about 0.18 mm. The size of the time-step is chosen to be 5 µs,
which ensures the resolution of reaction, diffusion and convection time scales.
A certain number of cells are selected for spark ignition, which correspond to
the spark gap and an ignition source term is included in the enthalpy equation. In
this work the temperature is fixed at the burnt temperature, Tb, at these ignition
cells during the spark duration. The ignition energy, Ei, can then be calculated
using
Ei = ρuViCp (Tb − Tu) , (6.2)
where Vi is the volume of ignition cells. Thus by changing the burnt temperature
of the ignition cells and the volume of the cells, one could alter the ignition energy.
Different ignition energy values are tested since experimental ignition data was
not available. Note that the spark duration is set to one time step size. In
the progress variable based approach used in this work, in addition to supplying
energy for spark ignition, hot products and cold reactants need to be specified
for ignition. Here values of c˜ = 1 and c˜′′2 = 0.25 are prescribed for the ignition
cells and c˜ = c˜′′2 = 0 is prescribed for the rest of the domain.
The initial temperature and pressure inside the combustion chamber are 325 K
and 243 kPa respectively. Radial profiles of turbulence intensity, TI, and swirling
velocity, V , obtained from the experimental measurements of Hamamoto et al.
(1988) are used to define the initial flow field. The integral length scale, Λ, is taken
as 12.5 mm, which is 10% of the vessel diameter. The simulation results does not
change significantly when this initial value of Λ is halved. Radial distribution of
TI is used to calculate the turbulent kinetic energy and dissipation rate as
k˜ =
3
2
TI2 and ε˜ =
TI3
Λ
. (6.3)
111
6.4. Results and discussion
The governing equations are solved in STAR-CD using FVM. In this work, PISO
method is chosen for the pressure-velocity coupling. The second-order MARS
scheme (Asproulis, 1994) is used to discretise convective terms in the momen-
tum and modelled turbulence equations. First-order upwind scheme is used to
discretise the convective terms in enthalpy and c˜ and c˜′′2 equations. Accuracy of
temporal discretisations lie between first- and second-order, in which the discreti-
sation scheme is based on the fully-implicit Euler scheme and explicit deferred
correctors.
6.4 Results and discussion
In this section, simulation results obtained using the strained and unstrained
flamelet models are compared with the experimental data of Hamamoto et al.
(1988). Figure 6.6 shows these predictions, where the results from the EBU
model available in STAR-CD is plotted as well. This figure shows that in the
2
4
6
8
10
12
14
16
18
20
22
24
0 5 10 15 20 25
p
(b
ar
)
time (ms)
exp
strained
unstrained
EBU
Figure 6.6: Pressure rise prediction using three different combustion models.
numerical simulations, pressure reaches a peak and then drops slightly. This
drop is due to heat loss at the walls. This figure shows that the prediction from
strained and unstrained flamelet models are reasonably good when compared
112
6.4. Results and discussion
with the EBU model. In addition, these models are able to predict the temporal
gradient of pressure, ∂p/∂t, reasonably well. The predicted pressure rise from
the unstrained flamelet is 5% higher than the experimental value, while the EBU
model overpredicts the pressure rise by 12%.
The time taken to generate the look-up table for the strained flamelet model is
considerably longer than that for the unstrained flamelet model, since a number of
strained laminar flames have to be calculated for each temperature and pressure
condition considered. The plots in Figure 6.6 indicate that the unstrained flame
is adequate for this problem and the improvements from the strained flamelet
model are marginal. Therefore, only the results obtained using the unstrained
flamelet model are discussed in the rest of this chapter.
It is important to note that the strained flamelet model gave considerably im-
proved results when compared with the unstrained flamelet model for the flames
simulated in Chapters 4 and 5. Even though the experimental flames simulated
in Chapter 5 were conducted in a spherical bomb, they can be considered to be
unconfined, since the experimental measurements were made before a consider-
able pressure rise was observed and the diameter of the combustion vessel was
more than three times larger than the one considered here. The experiments of
Hamamoto et al. (1988) were performed in a closed vessel with pressure rise due
to combustion. It is believed that confinement changes the turbulence stretch-
ing effect on the flame, since it restricts the entrainment of air and changes the
turbulence response of the flame.
Figure 6.7 shows the comparison of the mean scalar dissipation rate, ˜c, and
the mean reaction rate, ω˙, across the flame brush using simulations from both
strained and unstrained flamelet models. These quantities are taken across the
centre line of the computational domain at a simulation time of 10 ms. These
figures show that the difference in the quantities, ˜c and ω˙, obtained from the
strained and unstrained flamelet models are small, which could explain the small
difference in the predictions from these two models for the confined flame simu-
lated in this work.
113
6.4. Results and discussion
0
50
100
150
200
250
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ǫ˜ c
(1
/s
)
c˜
unstrained
strained
(a)
0
100
200
300
400
500
600
700
800
900
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
ω˙
(k
g
/m
3
s)
c˜
unstrained
strained
(b)
Figure 6.7: Comparison of flame quantities obtained using strained and un-
strained flamelets. These quantities are taken along the centreline of the compu-
tational domain at a simulation time of 10 ms; (a) mean scalar dissipation rate,
and (b) mean reaction rate.
114
6.4. Results and discussion
6.4.1 Effect of ignition energy
In section 6.3 it was described that ignition is induced by depositing energy
in some computational cells during the spark duration. Figure 6.8 shows the
sensitivity of the ignition energy in predicting the pressure rise during combustion.
This figure shows that increasing the ignition energy, Ei, to 5 mJ leads to faster
burning.
2
4
6
8
10
12
14
16
18
20
22
0 5 10 15 20 25
p
(b
ar
)
time (ms)
exp
Ei = 2 mJ
Ei = 5 mJ
Figure 6.8: Pressure rise prediction using different ignition energy definitions.
6.4.2 Effect of turbulence model
Figure 6.9 shows the prediction using two different turbulence models, with other
modelling parameters unchanged. It can be seen that flame propagation is faster
when k− ε RNG model is used. Applicability of the k− ε model and its variants
to swirling flows is questionable (Hogg and Leschziner, 1989) and could explain
the poor agreement with experimental data here.
115
6.4. Results and discussion
2
4
6
8
10
12
14
16
18
20
22
0 5 10 15 20 25
p
(b
ar
)
time (ms)
exp
k-ω SST
k-ε RNG
Figure 6.9: Pressure rise prediction using two different turbulence models.
6.4.3 Flame propagation
Hamamoto et al. (1988) used ion probe measurements to determine the position
of the flame front, where experiments were repeated with the ion probe fixed at
several locations. Measurements were also made using schlieren photographs and
Hamamoto et al. (1988) reported that the two measurements coincided with each
other. Therefore, similar to the previous experimental flames simulated in Chap-
ters 4 and 5, the flame profiles reported in these experiments are taken to be the
leading edge of the flame brush. For comparison purposes, the location at which
c˜ = 0.05 is taken to be the leading edge in the simulated flames. As described
earlier, the progress variable, c˜, is defined in terms of water vapour mass frac-
tion. Lewis number of H2O is roughly 0.88 for stoichiometric propane-air mixture,
which implies that the mass diffusivity of H2O is higher than the thermal diffu-
sivity. Therefore, the flame leading edge is taken as c˜ = 0.06 for the simulation
using unstrained and strained flamelet models. For the computation using the
EBU model, c is taken as the normalised temperature, c = (T − Tu)/(Tb − Tu),
and the leading edge as c˜ = 0.05. Figure 6.10 shows the comparison of flame
propagation obtained from different models with the experimental data at seven
116
6.4. Results and discussion
different time instants. It can be seen that all models predict a slower flame
0
10
0 10 20 30 40 50 60
z
(m
m
)
x (mm)
4.2 ms 6.4 8 9 10 11 12
Figure 6.10: Flame propagation comparison between experimental data (dashed
lines with symbols) and the predicted results from the unstrained (red lines),
strained (green) and EBU (blue) models at seven different time instants.
propagation. This figure also shows that the flame brush becomes distorted at
later times for the EBU model. Calculations using the unstrained and strained
flamelet models are similar; however, c˜ contours obtained using these two models
resemble the experimental flame brush more closely than with the EBU model.
These contours show that the flame is initially spherical but becomes cylindrical
due to the presence of the side walls.
Gas expansion within the flame induces a flow velocity. Figure 6.11 shows the
time-variation of the induced radial velocity, u˜, for the experimental measure-
ments and the predictions using combustion models. These quantities are taken
at a radial location of x = 50 mm. Note that the comparisons are approximate as
the experimental data were reported as ensemble-averaged velocities but the re-
sults from the numerical simulation are density-weighted averaged. Nevertheless,
such comparisons can be found in the literature (Roomina and Bilger, 2001). The
models are able to qualitatively predict the increase in the mean radial velocity
and the time at which the peak velocity was encountered in the experiments is
in agreement with the predicted values. The increase in flame velocity at the
measuring location is due to the approaching flame. Once the flame passes the
measuring location the velocity drops. This figure appears to show that both
models are able to predict the flame arrival time, which contradicts with the
117
6.5. Summary
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 2 4 6 8 10 12 14 16
u˜
(m
/s
)
t (ms)
exp
unstr. flamelet
str. flamelet
EBU
Figure 6.11: Flame induced radial velocity, u, variation with time. Comparison
is made between the two combustion models and the experimental data.
results shown in Figure 6.10 for the experimental part.
6.5 Summary
Predictive capability of strained and unstrained laminar flamelet models are
tested inside a closed vessel with swirl. Compression effect due to combustion
means that the change in the thermochemistry of the mixture needs to be in-
cluded in the flamelet library generation. Therefore, compared with constant
pressure cases simulated in earlier chapters, the dimensionality of the look-up ta-
bles has to be increased to include both temperature and pressure dependencies
The combustion models are implemented in STAR-CD.
It is shown that both the unstrained and strained flamelet models are able to
predict the experimental pressure rise reasonably well. In addition, these results
are an improvement from the results of the EBU model already implemented in
STAR-CD. None of the models used here are able to predict the flame propagation
reported in the experiments. However, these models are able to predict the flame
arrival time at a particular radial location, which appears to show a mismatch
118
6.5. Summary
between the results reported in the experimental study.
Results from this chapter show that the unstrained flamelet is adequate in sim-
ulating combustion of confined flames. It is believed that these results are similar
because of the restriction in the air entrainment and the change in the turbulence
characteristics of the flow when the flame is confined. The pre-processing time for
the unstrained flamelet model is significantly shorter than the strained flamelet
model, since the strained flamelet model requires the calculation of a number of
strained laminar flamelets. Therefore, in the next chapter, the unstrained flamelet
model is used to simulate combustion in a practical SI engine.
119
7. Spark-ignition engine
simulation
Results from the previous chapter have shown that the unstrained flamelet model
given in Eq. (2.63) is adequate to simulate intermittent combustion in closed
vessels. In this chapter, this reaction rate closure is used to simulate combustion in
a practical spark-ignition engine. These simulations are performed using STAR-
CD and the model implementation is largely similar to the description given in
Chapter 6. An overview of various closure models used in the past to simulate
SI engine combustion is presented first. The results from the unstrained flamelet
model are compared with two models available in STAR-CD.
7.1 Introduction
Three-dimensional CFD has become an indispensable tool in engine design, which
can provide detailed information on flow and combustion inside IC engines. RANS
methodology has been the industry standard, but with increasing computational
power, LES is now becoming feasible. The predictive capability of both of these
computational frameworks depends largely on the combustion model. This work
will focus on spark-ignition gasoline engines, where the models outlined in sec-
tion 2.5 are used in combustion simulation.
The widely used eddy break-up model (Spalding, 1971) is known to predict
unphysically high reaction rates near the wall (Abu-Orf and Cant, 2000) and
it is necessary to adjust the empirical model coefficients for different operating
conditions. By simulating a research engine, Abu-Orf and Cant (2000) showed
that a flamelet model, based on the BML formalism, does not suffer from this
problem of flame acceleration near the walls. Drake et al. (2005) also used a
modified version of the BML model to simulate stratified combustion in a spray-
guided direct-injection spark-ignition (DISI) engine.
Zhao et al. (1993) used the coherent flame model (CFM) to simulate SI en-
gine combustion. They showed that the surface production term constant in the
120
7.1. Introduction
model needs to be calibrated for each engine. The original form of CFM has
been extended to study both homogeneous and stratified combustion (Baritaud
et al., 1996). This extended coherent flame model (ECFM) was used by Duclos
and Zolver (1998) to investigate combustion in a DISI engine at both homoge-
neous and stratified conditions. In their approach, the model coefficients had to
be calibrated based on previous engine data, which gave good agreement with
experimental pressure measurements. Henriot et al. (1999) used ECFM, with
the same model constants, to predict the experimental results in a different DISI
engine. This model has been adapted to hydrogen combustion by Knop et al.
(2008), where they validated the model for both port fuel injection (PFI) and
DISI engines. More recently, Galloni et al. (2012) used ECFM to simulate com-
bustion in a downsized SI engine. As with previous studies, the model constants
had to be tuned depending on the engine configuration.
Weller et al. (1994) proposed a flame surface wrinkling model, in which a
wrinkling factor was introduced into the transport equation of a regress variable.
The model is supposed to capture the evolution of an initial laminar flame kernel
into a fully turbulent flame and its subsequent propagation by using either a
spectral model or a semi-empirical model for the local wrinkle factor. Using this
model, Weller et al. (1994) simulated combustion in a research engine with a
square piston, and was able to obtain a good match with the measured pressure
histories. For these calculations, an orifice-type equation was used with some
tuning to account for the leakage past the piston rings. Heel et al. (1998) used
the same model to show reasonably good agreement for the same research engine
for certain operating conditions. They attributed the discrepancies in some of
the results to the combustion model used. Kech et al. (1998) extended this
wrinkling model for DISI engines, where they showed that the model can be used
for stratified combustion.
Ewald and Peters (2007) validated the G-equation modelling approach for
a homogeneous charge SI engine and Tan and Reitz (2006) used this model to
simulate combustion in SI engines with exhaust gas recirculation (EGR). Peters
and Dekena (1999) applied the model to a hypothetical DISI engine, while Liang
and Reitz (2006) validated the model for both SI and stratified DISI engines.
The transported PDF method has also been used to simulate IC engines.
121
7.1. Introduction
For example, Taut et al. (2000) used a RANS/PDF method to simulate com-
bustion in a two-stroke SI engine. However, the results only showed qualitative
agreement with experimental measurements for the flame speed and a pressure
history comparison was not shown. Haworth (2010) noted that these calculations
show that the transported PDF method is feasible in simulating such complex
three-dimensional combustion processes.
Most IC engine simulations in the past were done in the RANS framework, but
LES is now gaining popularity. The choice between LES or RANS for IC engine
simulations depends on the objectives of the study. The interaction between
turbulence and combustion are more accurately captured in LES when compared
with RANS. LES is particularly useful in determining cycle-to-cycle variations
found in IC engines (Haworth and Jansen, 2000) but RANS is adequate if one is
only interested in the statistics. The combustion models used in LES are usually
extensions of the ones used in RANS. For example, Vermorel et al. (2009) used
the ECFM with LES to simulate a practical SI engine. Banaeizadeh et al. (2008)
used a LES/PDF method to simulate a research DISI engine.
Almost all of the above mentioned spark-ignition engine simulations were
performed using single-step or reduced chemistry, which does not allow the pre-
diction of pollutant formation (Heywood, 1988). One exception is the study of
Liang and Reitz (2006), who used detailed chemical mechanism for propane to
simulate the SI engine. They also simulated the gasoline DISI engine using a
21-species, 42-reaction iso-octane mechanism. Complex chemistry in CFD sim-
ulations is important for lean burn engines in which the turbulence-chemistry
interactions are expected to be stronger. Chemical kinetics also play a central
role in HCCI engines, and unlike the SI engine simulations above, a number of
simulations with detailed chemistry can be found in the literature (Bhave et al.,
2005; Zhang et al., 2005).
It is also noted that the studies mentioned above used some amount of model
tuning to obtain a good match between measured and computed in-cylinder pres-
sure with crank angle. However, if the model parameters are closely tied to the
physics of the problem, then one could eliminate model tuning. Furthermore,
these models should include complex kinetics that can be used to calculate pol-
lutants such as CO and NOx.
122
7.2. Engine measurements
The objective of this work is to use the reaction rate closure model given
in Eq. (2.63) to simulate combustion in a single-cylinder, four-stroke SI engine,
experimentally investigated in ETH Zu¨rich (Wright, 2013). As with the previous
simulations performed in this thesis, a reaction progress variable is used to track
the reaction zones. In addition, a transport equation for the progress variable
variance is also solved, which includes the mean scalar dissipation rate as a sink
term. In terms of combustion modelling, the homogeneous charge SI engine
is simpler than the stratified combustion taking place in a DISI engine (Colin
et al., 2003). However, complex turbulent-chemistry interactions take place inside
the cylinder and it is a good first test for the application of the combustion
model discussed here. Note that unlike many previous studies used to simulate
SI engine combustion there are no tunable parameters in this modelling approach
and detailed chemistry is included in this study.
This chapter is organised as follows: the test SI engine is described in sec-
tion 7.2. This is followed by a description of the numerical setup, which includes
the flamelet library generation. CFD simulations are performed using STAR-
CD. In the results section, computations using the reaction rate closure given in
Eq. (2.63) and two models available in STAR-CD are compared with experimental
measurements. This chapter ends with the main conclusions of this investigation.
7.2 Engine measurements
The engine used for this work is a 250cc single cylinder, four-stroke SI engine,
which was designed for Kart racing (Wright, 2013). Figure 7.1 shows the engine
geometry simulated (boundary conditions shown in this figure are described in
section 7.3.3), and the engine parameters are shown in Table 7.1. The spark plug
is located in the middle of the four valves and is slightly off centre, while the
cylinderhead has a pent-roof arrangement. Fuel is injected using a port injection
system; however, fuel injection is not modelled in this work.
Engine measurements were made by researchers in ETH Zu¨rich, where the in-
cylinder pressure was measured at every 0.2◦ CA using a piezo-electric pressure
sensor, while absolute pressure values inside the intake and exhaust ducts were
measured using pressure transducers (Wright, 2013).
123
7.2. Engine measurements
(a)
Combustion chamber
Intake
Exh
aust
T=470 K
T=470 K
T=400 KT=400 K
T= 315 K
p= p(θ)
c=c''2=0
~ ~ T = 1050 K
p = p(θ)
∇c = ∇c''
2
= 0~ ~
(b)
Figure 7.1: Engine geometry showing (a) top view with the symmetry axis marked
and (b) side view showing the specified boundary conditions, where outlet bound-
ary conditions are shown for a case without backflow.
124
7.3. Numerical setup
Table 7.1: Engine parameters
Engine configuration Single-cylinder, four-stroke, SI engine
Engine displacement 249.6 cm3
Bore/Stroke 75/56.5 mm
Cylinderhead Four valves per cylinder, pentroof
Injection Port injected
Fuel Standard gasoline (RON 95)
Geometric compression ratio 12.5:1
The engine was run in a range of operating conditions and 144 cycles were run
for each operating point. The operating point corresponding to an engine speed
of 3500 rpm with a torque of 20 Nm is considered for this work. This corresponds
to a brake mean effective pressure (BMEP) of 9.9 bar and is close to engine full
load. All operating conditions have a stoichiometric (i.e. air-to-fuel ratio, λ = 1),
homogeneous mixture and the engine was spark ignited at 30.4◦ crank angle (CA)
before top dead centre (TDC).
7.3 Numerical setup
Combustion simulations inside the engine are performed using STAR-CD v4.18,
together with the module es-ice, which handles the moving grid for IC engine
applications. The numerical setup used for the engine is similar to that of the
simple geometry described in section 6.3, and additional issues related to IC
engine simulations are highlighted here.
7.3.1 Flamelet library generation
Fuel-air mixture and chemical mechanisms
Gasoline fuel used in the engine is a complex mixture of several hydrocarbons,
without a standard mixture composition (Ihracska et al., 2013) and currently
it is not possible to represent detailed chemistry of gasoline (Pitz et al., 2007).
For the computational studies carried out in this work, iso-octane is used as a
substitute for gasoline, which is a single component fuel that can be used as a
125
7.3. Numerical setup
gasoline surrogate (Pitz et al., 2007).
A number of detailed chemical mechanisms for iso-octane can be found in the
literature. However, these detailed mechanisms involve a large number of species.
For example, the detailed iso-octane mechanism of Curran et al. (2002) has 857
species and 3606 reactions. These species are also associated with large differences
in their time scales, making the problem numerically stiff (Lu and Law, 2006).
Therefore, for an IC engine simulation that requires a wide range of temperatures
and pressures, it is unfeasible to calculate freely propagating laminar flames using
these detailed mechanisms.
In order to reduce the computational effort, one could use skeletal mechanisms
that have been derived from detailed mechanisms using various reduction tech-
niques. In this work, two such mechanisms are considered. First is the skeletal
mechanism of Hasse et al. (2000), which consists of 29 species and 49 reactions.
The second skeletal mechanism is that of Pepiot-Desjardins and Pitsch (2008a),
which is considerably larger as it has 109 species and 393 reactions. Note that
these mechanisms are respectively denoted as mech-1 and mech-2 in this chapter.
It is necessary for these mechanisms to be able to predict experimental flame
speeds for a wide range of conditions. Figure 7.2 shows the comparison of cal-
culated stoichiometric iso-octane – air laminar flame speeds, s0L, for different un-
burnt mixture temperatures and pressures with the experimental measurements
of Bradley et al. (1998); Lawes et al. (2005); Mandilas et al. (2007); Jerzem-
beck et al. (2009); Kelley et al. (2011) and Galmiche et al. (2012). CHEMKIN’s
PREMIX module (Kee et al., 1985) is used to calculate unstrained laminar flame
speeds, s0L. In this figure, solid lines represent flame speeds computed using the
mechanism of Hasse et al. (2000), while dashed lines represent values obtained
using the mechanism of Pepiot-Desjardins and Pitsch (2008a).
Figure 7.2 shows that even though these mechanisms are able to predict ex-
perimental trends, there are significant differences between computed and mea-
sured flame speed values, especially at higher pressures. Galmiche et al. (2012)
noted that current iso-octane chemical mechanisms need to be improved. It is
also important to note that there are significant discrepancies in experimentally
measured flame speeds. For the purposes of the present simulations, these dif-
ferences between measured and computed flame speeds using the mechanisms of
126
7.3. Numerical setup
Hasse et al. (2000) and Pepiot-Desjardins and Pitsch (2008a) are considered to
be acceptable.
0.0
0.2
0.4
0.6
0.8
5 10 15 20 25
s0 L
(m
/s
)
p (bar)
Tu = 373 K
0.2
0.4
0.6
0.8
s0 L
(m
/s
)
Tu = 360 K
0.2
0.4
0.6
0.8
s0 L
(m
/s
)
Tu = 353 K
2 4 6 8 10
p (bar)
Tu = 473 K
Tu = 450 K
Tu = 400 K
Jerzembeck et al. (2009)
Galmiche et al. (2012)
Bradley et al. (1998), Tu = 358 K
Lawes et al. (2005)
Mandilas et al. (2007)
Kelley et al. (2011)
Galmiche et al. (2012)
Bradley et al. (1998)
Bradley et al. (1998)
Figure 7.2: Flame speeds computed using the mechanism of Hasse et al. (2000)
(solid lines, mech-1) and Pepiot-Desjardins and Pitsch (2008a) (dashed lines,
mech-2) compared with experiments at different pressures and temperatures.
The flamelet library is generated with the progress variable defined based on
water vapour mass fraction, c = YH2O/Y
b
H2O
, due to reasons outlined earlier in
section 6.3. It was shown in section 6.4 that the reaction rate closure expression
given by Eq. (2.63) is adequate to simulate combustion in enclosed chambers
127
7.3. Numerical setup
with intermittent combustion; therefore, only this closure model is compared
with models implemented in STAR-CD.
Issue of auto-ignition
Beyond certain temperatures and pressures it is difficult to obtain a converged
solution in PREMIX when the chemical mechanism of Pepiot-Desjardins and
Pitsch (2008a) is used. This difficulty in convergence is also observed when
the mechanism of Kelley et al. (2011) is used, which is another large skeletal
mechanism. A similar observation was made by Martz et al. (2011), who showed
that a steady flame solution could not be obtained at high temperatures and
pressures, since at these conditions the ignition delay times, τi, were of the order
of the characteristic flame time (tc = δ
0
L/s
0
L).
Another surprising observation made from these laminar flame calculations is
the increase in computed flame speeds with pressure, for certain unburnt mixture
temperatures. This contradicts the well known negative dependence of laminar
flame speed with pressure (Turns, 2000). Figure 7.3 shows the variation of com-
0
50
100
150
200
250
300
1 5 10 15 20 25 30 35 40 45
s0 L
(c
m
/s
)
p (bar)
Tu = 600 K
675 K
700 K
800 K
900 K
Figure 7.3: Variation of laminar flame speed with pressure for different unburnt
mixture temperatures. Computations are made using the skeletal mechanism of
Pepiot-Desjardins and Pitsch (2008a).
128
7.3. Numerical setup
puted laminar speeds with pressure for different unburnt mixture temperatures.
This figure shows that beyond a particular pressure for a given Tu, s
0
L increases
with pressure and this value of p reduces when Tu is increased. For example, s
0
L
starts to increase after 25 bar when Tu = 700 K, while it starts to increase after
20 bar when Tu = 800 K. Note that this increase in s
0
L with p was only observed
for Tu ≥ 675 K. For Tu below this value, s0L decreased with p as expected (for
example, Tu = 600 K curve in Figure 7.3). The calculations for which the flame
speed starts to increase with pressure are also the ones that were difficult to con-
verge. Therefore, it is believed that this positive dependence of s0L with p is due
to the auto-ignition of the fuel-air mixture, in which case there is no flame speed
eigenvalue.
Indeed, a recent study by Habisreuther et al. (2013) showed a similar re-
sult for stoichiometric methane-air flames, where beyond a certain temperature,
the flame speed increased with pressure. They explained that this behaviour is
caused by flame structure changes, induced by the ignition delay time, which is a
function of Tu. They used different detailed chemical mechanisms to verify that
their results are not a spurious effect of the chemical kinetic mechanism. There-
fore, PREMIX calculations with the mechanism of Pepiot-Desjardins and Pitsch
(2008a), are used to estimate ignition pressure and temperature of iso-octane,
i.e. the temperature at which an increase in pressure leads to an increase in the
laminar flame speed. This auto-ignition temperature variation with pressure is
shown in Figure 7.4, which shows that the ignition temperature increases as the
pressure is decreased. Also shown in this figure is the curve corresponding to
the temperature rise resulting from adiabatic compression of a mixture initially
at 300 K and 1 bar. These two curves do not intersect since auto-ignition was
not observed for Tu < 675 K. A straight dotted line is drawn from this point to
the adiabatic compression curve in Figure 7.4 to separate auto-ignition and flame
regions.
With regards to the flamelet library generation, the region left of the adiabatic
compression curve is of no interest as it is not possible to have such conditions
inside the cylinder when the engine is running. Below the ignition temperature
curve one could use a similar procedure as outlined in section 6.3.1 to generate
the flamelet library since the eigenvalue of s0L exists. The region bounded by
129
7.3. Numerical setup
1
10
20
30
40
50
60
70
80
500 600 700 800 900 1000 1100
p
(b
ar
)
Tu (K)
Flame (mech-2)
Auto-ignition (mech-2)
Fl
am
e
(m
ec
h-
1)
Adiabatic compression
Ignition
Figure 7.4: Curves showing the variation of temperature for adiabatic compres-
sion (dashed line) and the ignition temperature at a particular pressure obtained
from laminar flame calculations.
the two curves in the upper right hand corner is the region where auto-ignition is
expected, and in this region flame solutions cannot be found using the mechanism
of Pepiot-Desjardins and Pitsch (2008a). The homogeneous constant-pressure
module (CONP) from CHEMKIN is used to obtain the reaction rates in this
region. Note that the same reaction rate closure expression, given by Eq. (2.63),
is also used to build the look-up table for the auto-ignition region.
The constant-pressure reactor can also be used to determine the ignition delay
time, τi, which is defined as the time corresponding to the maximum temperature
gradient with respect to time, dT/dt. Figure 7.5 shows the variation of iso-octane
ignition delay times with both temperature and pressure. In this figure, compar-
ison is made between the values calculated numerically using the mechanism
of Pepiot-Desjardins and Pitsch (2008a), experimental data on iso-octane auto-
ignition delay times and the correlation of He et al. (2005). This figure shows
that there are some discrepancies between different experiments as well as the
numerical values, even though they all show the expected general trend with
temperature and pressure.
An important feature of the combustion modelling approach used in this work
130
7.3. Numerical setup
0.01
0.1
1
10
100
0.75 0.8 0.85 0.9 0.95 1 1.05
τ i
(m
s)
1000/T (1/K)
10 bar
20 bar
30 bar
40 bar
50 bar
p=10 bar, numerical
p=30 bar, numerical
p=50 bar, numerical
Fieweger et al. 13 bar
Fieweger et al. 33 bar
Fieweger et al. 45 bar
Shen et al. 10 bar
Shen et al. 25 bar
Shen et al. 50 bar
Davidson et al. 10 bar
Davidson et al. 50 bar
Figure 7.5: Ignition delay time variation with both temperature and pressure for
iso-octane – air mixtures. Straight lines show the correlation of He et al. (2005).
is the use of the progress variable variance, c˜′′2, equation [Eq. (2.65)]. This equa-
tion contains the mean scalar dissipation rate, ˜c, as a sink term, which has so
far been modelled using Eq. (2.69). This model contains s0L and δ
0
L, which are
meaningless for auto-ignition since there is no flame. Therefore, whenever condi-
tions inside the cylinder correspond to auto-ignition as depicted by Figure 7.4, a
linear relaxation model of the form, ˜c '
(
ε˜/k˜
)
c˜′′2, is used for ˜c.
It is important to note that the mechanism of Hasse et al. (2000) cannot be
used for ignition studies. This mechanism yields flame solutions even in the auto-
ignition region shown in Figure 7.4. Therefore, ˜c model given in Eq. (2.69) is
used in CFD simulations involving this mechanism. Table 7.2 shows the three
reaction rate closure models used in this work together with various model pa-
rameters. Note that the model parameters used in the algebraic equation for
˜c [Eq. (2.69)] cannot be changed arbitrarily and are left unchanged in these
simulations. G-equation and CFM models are available in STAR-CD and only
single-step chemistry is used for these two models. The model parameters in
131
7.3. Numerical setup
these two models have to be tuned for a particular engine geometry and operat-
ing condition.
Table 7.2: Combustion models used in this work together with various model
parameters.
Model Parameter and value
M1: G-equation coefficient A in Eq. (2.58)
vary between 3 to 4
M2: CFM a α β in Eq. (2.52)
2.1 0.1 1.0
M3: Model given in Eq. (2.63) β′ K∗c in Eq. (2.69)
6.7 0.85τ
7.3.2 Computational mesh
The engine geometry consists of the intake port (including the inlet valve), the
combustion chamber (cylinder) and the exhaust port (including the exhaust
valve). Only half of the geometry is used for meshing by considering the geo-
metrical symmetry of the engine [see Figure 7.1(a)]. Computational mesh used
in this work consists of unstructured hexagonal cells (Koch, 2013). The move-
ment of the valves and the piston is handled using STAR-CD’s es-ice module. To
verify whether the results are independent of the grid resolution, a coarse and a
fine mesh are used. Table 7.3 shows the number of cells in these two meshes at
both TDC and BDC. These computational meshes were provided by researchers
in ETH Zu¨rich (Wright, 2013).
Table 7.3: Cell count of the two computational meshes.
Coarse Fine
Cell count at TDC 175,000 280,000
Cell count at BDC 655,000 1,080,000
Average cell size 1.2 mm 0.7 mm
132
7.3. Numerical setup
7.3.3 Initial and boundary conditions
The engine events sequence is shown in Figure 7.6. Simulations are started just
before the intake valve opening (IVO), which is at 420◦ CA before TDC. At
this crank angle, cylinder and exhaust ducts contain burnt products; therefore, a
temperature of 900 K is specified initially inside the cylinder. Mean pressure trace
measured experimentally from 144 cycles is used to define the initial pressure
inside the cylinder. Koch (2013) used these experimentally measured pressure
0
2
4
6
8
10
-400 -300 -200 -100 0 100 200 300
V
al
ve
lif
t
(m
m
)
Crank angle position ATDC (◦CA)
Init. TDC BDC TDC BDC
intake compression power
exhaust
spark
exhaust
intake
Figure 7.6: Events sequence in the engine, which shows the valve movement
plotted against the crank angle.
traces as inputs to a two-zone model to calculate the heat release. Since it is
difficult to measure wall temperatures in real engine setups, a sensitivity analysis
was performed to determine the effect of wall temperatures on the calculated heat
release rate. Koch (2013) used the two-zone model to show that the pressure trace
is insensitive to wall temperatures. The assumed wall temperatures are given in
Table 7.4, which are fixed during the simulations. In addition to temperature,
no-slip, isothermal conditions are applied at the walls.
Pressure boundary conditions are specified for both inlet and outlet. Due
to significant differences in the measured inlet and outlet pressures in a given
cycle, the specified pressure boundary conditions are crank angle, θ, dependent.
These crank angle dependent pressure values were obtained from the mean of 144
133
7.4. Results and discussion
measured cycles (Koch, 2013). Furthermore, constant average temperatures at
the inlet and outlet are prescribed, which defines the trapped mass within the
cylinder. Dirichlet boundary conditions are used for the scalar variables and tur-
bulence quantities at the inlet. For the outlet, Neumann boundary conditions are
applied, where the gradients of T , k˜, ˜, c˜ and c˜′′2 are set to zero. However, in the
case of backflow, the quantities such as temperature and turbulence variables are
also specified at the outlet. These boundary conditions are shown in Figure 7.1.
Note that symmetry conditions are applied at the plane of symmetry.
Table 7.4: Wall temperatures
Wall Temperature (K)
Cylinder 400
Piston crown 470
Combustion dome 470
7.3.4 Computational details
Discretisation schemes used in these simulations are similar to the ones used in
Chapter 6. As before, PISO algorithm (Issa, 1986) is used for pressure-velocity
coupling. A constant time step size of 0.1◦ CA is chosen, which resolves the
reaction, convection and diffusion time scales. The k-ε RNG model (Yakhot
et al., 1992; Han and Reitz, 1995) with the wall-function of Angelberger et al.
(1997) is used for turbulence closure and the ignition treatment is similar to the
one described in section 6.3.3, with an ignition radius of 0.5 mm and an ignition
duration of 0.1◦ CA. Note that the same ignition radius and duration are used
for the three models given in Table 7.2.
7.4 Results and discussion
Flow motion inside the cylinder at 200◦ and 50◦ crank angles before TDC are
given in Figures 7.7(a) and 7.7(b) respectively. Note that 2-D cut-planes are
shown in these figures to highlight the flow features. The inlet valve is already
134
7.4. Results and discussion
(a)
(b)
Figure 7.7: Flow motion inside the cylinder at (a) 200◦ and (b) 50◦ CA before
TDC.
135
7.4. Results and discussion
open at 200◦ (see Figure 7.6) and Figure 7.7(a) shows that a strong annular jet is
created when the inlet valve is open; and two vortical structures can be identified
in this figure. These vortices are compressed and quenched by the piston crown
and the combustion dome as shown in Figure 7.7(b), where both valves are closed
at this crank angle.
0
20
40
60
80
100
120
-40 -30 -20 -10 0 10 20 30 40
C
yl
in
de
r
pr
es
su
re
(b
ar
)
ATDC ◦ CA
Exp (mean)
M1, A = 3.3
M1, A = 4.0
M2
M3 (mech-1, coarse)
M3 (mech-1, fine)
M3 (mech-2, coarse)
Figure 7.8: Comparison of computed results with experiment. Experimental
pressure trace shown here is the mean of 144 cycles. Mech-1 and mech-2 refer
to the chemical mechanisms of Pepiot-Desjardins and Pitsch (2008a) and Hasse
et al. (2000) respectively. Two different values of the flame speed coefficient, A
[see Eq. (2.58)], are shown for model M1 (Wright, 2013).
Computed and measured pressure traces are shown in Figure 7.8. This figure
shows the results obtained using the reaction rate closure expression given in
Eq. (2.63), which is denoted as M3 in Table 7.2, with chemical mechanisms of
Hasse et al. (2000) and Pepiot-Desjardins and Pitsch (2008a) (denoted as mech-1
and mech-2 respectively). It can be seen that the simulation using the mechanism
of Pepiot-Desjardins and Pitsch (2008a), in which auto-ignition is assumed to
take place inside the cylinder, is unable to predict the experimental pressure rise.
Computed results using model M3 and the mechanism of Hasse et al. (2000) gives
a reasonable comparison with experimental results, but the initial pressure rise
136
7.4. Results and discussion
shortly after ignition is underpredicted. Combustion is through flame propagation
when this mechanism is used. Results from the fine mesh using this mechanism
is also shown for comparison. Difference in computed peak pressures between the
fine and coarse mesh is about 5%, which is considered to be acceptable, and only
the coarse mesh is considered in the rest of this work. Also shown in this figure
are the results from G-equation (M1) and CFM (M2). Note that the flame speed
coefficient, A, in model M1 has been tuned to fit the experimental pressure curve,
whereas no parameters are tuned for CFM model. This figure shows that CFM
model, with default model parameters set in STAR-CD, is unable to predict the
cylinder pressure rise, where the fuel-air mixture is burnt very rapidly.
Both intake and exhaust valves are closed when the mixture is spark-ignited
(see Figure 7.6). Pressure traces in Figure 7.8 show that the cylinder pressure
increase continuously and reaches a maximum after TDC. This is due to com-
pression of the unburnt gas mixture by the gas-expansion from combustion. The
pressure then starts to drop as the cylinder volume is increased in the expansion
stroke. Therefore, pressure and temperature of the unburnt gas mixture changes
during the combustion process.
1
10
20
30
40
50
60
70
-40 -30 -20 -10 0 10 20 30 40
550
600
650
700
750
800
850
900
C
yl
in
de
r
pr
es
su
re
(b
ar
)
Te
m
pe
ra
tu
re
(K
)
ATDC ◦ CA
Pressure
Unburnt mixture temperature
Figure 7.9: Pressure and unburnt mixture temperature variation with crank angle
during the simulation.
137
7.4. Results and discussion
Figure 7.9 shows the unburnt mixture temperature, Tu, and pressure variation
with crank angle during the simulation. The unburnt mixture temperature is
necessary to interpolate the reaction source terms from the flamelet library. It
is calculated using a volume average of the unburnt cell temperatures inside the
combustion chamber. The plots in Figure 7.9 are shown for model M3 using the
chemical mechanism of Hasse et al. (2000). It can be seen that the maximum
unburnt mixture temperature reaches around 885 K and the mixture becomes
fully burnt around 20◦ CA after TDC.
Difference between results obtained using the chemical mechanisms of Hasse
et al. (2000) and Pepiot-Desjardins and Pitsch (2008a) shown in Figure 7.8 is
explained in the next section. It was described in section 7.3.1 that the look-up
table generation is different for these two mechanisms since the mechanism of
Pepiot-Desjardins and Pitsch (2008a) predicts auto-ignition.
7.4.1 Auto-ignition simulation
The engine is spark-ignited at 30.4◦ CA before TDC, where unburnt mixture
temperature, Tu, near the spark location is approximately at 647 K with a pres-
sure of 10.9 bar (see Figure 7.9). According to Figure 7.4, this corresponds to a
flame solution and the flamelet library region corresponding to freely propagating
laminar flame is used to obtain the source terms. However, around 15◦ CA before
TDC, the mechanism of Pepiot-Desjardins and Pitsch (2008a) predicts that the
mixture will be auto-ignited and from this point onwards the auto-ignition part
of the flamelet library is used to calculate the source terms. When auto-ignition
takes place, the scalar dissipation rate model given in Eq. (2.69) is replaced by
the classical algebraic model for ˜c. On the contrary, the mechanism of Hasse
et al. (2000) cannot predict ignition, and the model given in Eq. (2.69) is used
for simulations using this mechanism, which assumes that combustion proceeds
through flame propagation.
Auto-ignition of the end gases is usually (but not always) followed by knock-
ing. No knocking behaviour was observed during engine experiments (Wright,
2013), which makes it questionable to use auto-ignition for these simulations.
The assumption that iso-octane can be used as a surrogate fuel for gasoline needs
138
7.4. Results and discussion
to be considered carefully. For example, the auto-ignition behaviour of these two
fuels should be similar. More complex surrogates that match the auto-ignition
delay times of gasoline may need to be considered (Pitz et al., 2007; Mehl et al.,
2011). Another important point to note is whether the skeletal mechanism of
Pepiot-Desjardins and Pitsch (2008a) can predict the correct auto-ignition be-
haviour of a detailed mechanism. Pepiot-Desjardins and Pitsch (2008a) reported
an average error of 10.86% for the ignition delay times. It was noted by Pepiot-
Desjardins and Pitsch (2008b) that the error in the ignition delay time is not a
monotonic function of the size of the skeletal mechanism and that a smaller error
could still lead to different dynamics when compared with a detailed mechanism.
In this work, laminar flame reaction rates are computed using a constant-
pressure homogeneous reactor. It is unclear whether this method can be used to
obtain correct combustion characteristics for turbulent combustion. The mixture
is homogeneous and turbulent mixing and micro-mixing are assumed to occur
instantaneously in the constant-pressure reactor. A possible alternative is to use
a partially stirred reactor (PaSR), which can handle additional non-linearities of
chemistry-turbulence interactions and can serve as a good model for turbulent
combustion (Correa, 1993).
These uncertainties regarding the modelling approach used for auto-ignition
could explain why the closure expression given in Eq. (2.63) with the chemi-
cal mechanism of Pepiot-Desjardins and Pitsch (2008a) is unable to predict the
correct peak pressure inside the cylinder.
7.4.2 Combustion inside the cylinder
Combustion taking place inside the cylinder at two different crank angles are
shown in Figures 7.10 and 7.11, where the top cylinder view is shown with the
ducts removed. These plots are obtained using model M3 and mech-1 where the
progress variable (defined using water vapour mass fraction) and the mean reac-
tion rates of water vapour are plotted. One could see how the flame propagates
inside the cylinder from these two figures. Combustion can also be visualised by
plotting the iso-contours of c˜ as shown in Figure 7.12. In this figure, the leading
edge of the flame with c˜ = 0.05 iso-contours are plotted at different crank angles.
139
7.4. Results and discussion
(a) (b)
Figure 7.10: Combustion at TDC: (a) progress variable, c˜, (b) mean reaction rate,
ω˙. Top view of the chamber surface is shown with holes marking the positions of
the removed ducts.
(a) (b)
Figure 7.11: Combustion at 10◦ CA after TDC: (a) progress variable, c˜ (b) mean
reaction rate, ω˙.
7.4.3 Heat release rate
The apparent heat release rate can be calculated from the cylinder pressure. It
is equal to the difference between heat released during combustion and the heat
transfer from the systems, which can be written as (Heywood, 1988)
dQa
dt
=
γ
γ − 1p
dV
dt
+
1
γ − 1V
dp
dt
, (7.1)
140
7.4. Results and discussion
Figure 7.12: Reaction progress variable is visualised at different crank angles
during combustion.
where dQa/dt is the difference between heat transfer rate due to combustion,
dQc/dt, and the wall heat-transfer rate, dQw/dt, γ is the ratio of specific heats
and V is the cylinder volume. During CFD simulations, the heat release rate can
be calculated using:
Q =
∫
V
ω˙FQLHV dV, (7.2)
where ω˙F is the mean reaction rate of fuel and QLHV is the lower heating value of
fuel, which is 44.3 MJ/kg for iso-octane (Broustail et al., 2011). Note that Q is
calculated in terms of kJ/◦CA. The heat release rate, HRR, variation with crank
angle is shown in Figure 7.13. This is given as a percentage of the total heat
release rate if the fuel inside the cylinder is completely burnt:
HRR =
Q
mfQLHV
× 100. (7.3)
where mf is the mass of fuel inside the cylinder. The numerical results shown in
Figure 7.13 are obtained using model M3 and mech-1.
This figure shows that there are considerable differences in the heat release
141
7.4. Results and discussion
-1
0
1
2
3
4
5
-40 -30 -20 -10 0 10 20 30 40
H
ea
t
re
le
as
e
ra
te
(%
/◦
C
A
)
ATDC ◦ CA
Exp, Eq. (7.1)
Numerical, Eq. (7.1)
Numerical, Eq. (7.2)
Figure 7.13: Calculated heat release rate as a function of crank angle.
rate computed using Eq. (7.1) for experimental and numerical pressure traces.
Also shown on this plot is the direct calculation of the heat release rate using
the CFD simulations (shown as a solid line). Unlike the results obtained using
Eq. (7.1), the heat release rate obtained using Eq. (7.2) is smaller. Difference
between these HRR calculations for the numerical case is because Eq. (7.1) does
not use any heat-transfer models. However, even more complete methods used
to calculate HRR contain several empirical approximations (Heywood, 1988) and
cannot be used to compare directly with the HRR calculated numerically in CFD
simulations.
Figure 7.14(a) and Figure 7.14(b) respectively show the burnt mass fraction,
xb = mb/m, and burnt volume fraction, yb = Vb/V , for the models used in this
work as a function of the crank angle. These figures correspond to results shown
in Figure 7.8 where it was shown that the fuel-air mixture is rapidly burnt when
model M2 is used. Plots in Figure 7.14 show that with model M3, the mixture
is completely burnt around 20◦ CA after TDC, which agrees with plot shown in
142
7.4. Results and discussion
0.0
0.2
0.4
0.6
0.8
1.0
-40 -30 -20 -10 0 10 20 30 40
B
ur
nt
m
as
s
fr
ac
ti
on
,x
b
ATDC ◦CA
M1
M2
M3
(a)
0.0
0.2
0.4
0.6
0.8
1.0
-40 -30 -20 -10 0 10 20 30 40
B
ur
nt
vo
lu
m
e
fr
ac
ti
on
,y
b
ATDC ◦CA
M1
M2
M3
(b)
Figure 7.14: (a) Mass fraction burnt and (b) volume fraction burnt as a function
of the crank angle.
143
7.4. Results and discussion
Figure 7.9. The relation between xb and yb is given by (Heywood, 1988)
xb =
[
1 +
ρu
ρb
(
1
yb
− 1
)]−1
, (7.4)
which has a universal form (Krieger and Borman, 1966). The plot of xb against yb
obtained using different models and the expression in Eq. (7.4) is shown in Fig-
ure 7.15. Note that the ratio ρu/ρb = 4 is used in Eq. (7.4), which is close to the
value in most spark-ignition engine operating conditions (Heywood, 1988). Fig-
ure 7.15 shows that even though all models show the general trend as expected,
none of the models follow the theoretical value. It is to be noted that for inter-
mittent combustion, the ratio ρu/ρb will change depending on the temperature
and pressure inside the cylinder.
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
B
ur
nt
m
as
s
fr
ac
ti
on
,x
b
Burnt volume fraction, yb
M1
M2
M3
Eq. (7.4)
Figure 7.15: Relation between mass fraction burnt and volume fraction burnt for
the three models simulated in this work and the expression given in Eq. (7.4).
7.4.4 Effect of turbulence model
Most CFD simulations of internal combustion engines are performed using some
version of the widely-used k-ε model (Gosman, 1999). The standard k-ε model
144
7.5. Summary
(Jones and Launder, 1972) has been used by a number of researchers (Taut et al.,
2000; Brandl et al., 2005). The k-ε RNG model (Yakhot et al., 1992; Han and
Reitz, 1995), has also been widely used for engine flows (Cao et al., 2008; Bohbot
et al., 2009). The results presented so far have been obtained using this turbulence
model. Figure 7.16 shows the sensitivity of the turbulence model in obtaining the
pressure trace, where the k-ε realisable model is shown to significantly change the
pressure rise. This shows the importance of having correct turbulence description,
especially near the spark location. The more complex Reynolds-stress model have
also been used for engine flow simulations (Yang et al., 2005), however it has not
been widely used, indicating that the benefits are not significant (Gosman, 1999).
0
10
20
30
40
50
60
70
-40 -30 -20 -10 0 10 20 30 40
C
yl
in
de
r
pr
es
su
re
(b
ar
)
ATDC ◦ CA
Exp (mean)
k-ε RNG
k-ε realisable
Figure 7.16: Cylinder pressure predictions using two different turbulence models.
7.5 Summary
A spark-ignition engine is simulated using the URANS methodology, with the
combustion model based on the flamelet approach. In this approach, the chemi-
cal reaction rates are decoupled from CFD calculations. Two skeletal mechanisms
145
7.5. Summary
for iso-octane with different sizes are used in this work. The larger mechanism
of Pepiot-Desjardins and Pitsch (2008a) predicts auto-ignition inside the cylin-
der after certain pressures and unburnt mixture temperatures and the modelling
technique used had to account for this, whereas the smaller mechanism of Hasse
et al. (2000) predicts flame propagation, where the modelling is similar to one
described in previous chapters.
Results from this work show that the reaction rate closure model given in
Eq. (2.63) (M3) with the mechanism of Hasse et al. (2000) gives a reasonable
agreement with experimental pressure measurements. However, the mechanism
of Pepiot-Desjardins and Pitsch (2008a), which predicts auto-ignition, is unable
to give the correct pressure rise. Engine experiments did not show any knocking
behaviour, which is an indication that there was no auto-ignition inside the cylin-
der and could explain why the auto-ignition simulation is unable to predict the
experimental pressure rise. Results from model M3 are compared withG-equation
and CFM models available in STAR-CD. As expected the model parameters for
these two models need to be tuned for this engine operating condition, whereas
the model M3 does not involve any tunable parameters. This is the first appli-
cation of the modelling approach described in this thesis to internal combustion
engines, and the results obtained in this chapter are encouraging to apply this
flamelet based modelling approach to more complex IC engines such as DISI
engines.
146
8. Conclusions and future work
8.1 Conclusions
A computationally economical approach is used to simulate turbulent flames rel-
evant for spark-ignition engine combustion. This is a flamelet based approach,
where the turbulent flame is assumed to be an ensemble of laminar flamelets and
the chemical time scales are shorter than the turbulence time scales. A central
parameter in the current modelling approach is the mean scalar dissipation rate,
˜c, which appears either directly or indirectly in all turbulent combustion models.
In the current work, an algebraic model for ˜c is used, which was derived based
on the physics of the combustion problem (Kolla et al., 2009).
Two different closure models for the mean reaction rate, ω˙, are tested: un-
strained and strained flamelet models. Strained flamelet model accounts for the
flame straining due to turbulence eddies. Detailed chemical mechanisms are used
to generate flamelet libraries. Mean scalar dissipation rate term appears directly
in the strained flamelet model, while both unstrained and strained modelling ap-
proaches require a variance equation of the progress variable, which includes ˜c
as a sink term.
In the first part of this work, outwardly propagating spherically symmetric
flames are simulated for both methane- and hydrogen-air mixtures. Spherical
flames are found in spark-ignition engines as well as accidental explosions of
vapour clouds. These simulations are performed using an in-house CFD code
after modifying it for spherical coordinates.
For methane-air flames, it is shown that the strained flamelet is able to predict
the experimental flame propagation of Hainsworth (1985), which were carried out
in a wind tunnel with decaying turbulence. The strained flamelet model is then
used to compare planar and spherical flames having the same turbulence and
thermochemical conditions. It is observed that the propagation of the flame
leading edge is aided by the local fluid velocity at that location for spherical
flames. Whereas for planar flames the direction of fluid flow and the advancing
147
8.1. Conclusions
leading edge are in opposite directions. A novel observation from these spherical
flame simulation is that the growth of the turbulent flame brush thickness is
mainly due to heat-release induced convective effects and not due to turbulent
diffusion. The role of turbulent diffusion for this thickness growth has been
advocated in many past studies [see the review article by Lipatnikov and Chomiak
(2002)].
Experiments of Kitagawa et al. (2008) are used to validate the combustion
models for hydrogen-air spherical flames. These experiments were performed in
a spherical bomb with fans used to generate turbulence. It is assumed that tur-
bulence does not decay for the bomb experiments since the fans were left running
during the experiments. Unlike the methane-air flames previously simulated,
hydrogen-air flames do not have unity Lewis numbers. In order to overcome this
issue, a two-progress variable approach is used, with one based on water vapour
mass fraction and the other on temperature. As before, it is shown that the
strained flamelet model is able to predict the experimental results for stoichio-
metric hydrogen-air flames. However, the predictions for lean hydrogen-air flames
is poor, because these flames are thermo-diffusively unstable and the current nu-
merical methodology is inadequate to deal with such flames. Further analysis of
turbulent hydrogen-air flames showed that the flame response to turbulence are
similar to that of methane-air flames.
Strained and unstrained flamelet models are then used to simulate intermit-
tent combustion taking place inside a closed vessel. Here the experiments of
Hamamoto et al. (1988) are used for validation. In these experiments turbulence
was generated using swirling inflow of premixed fuel and air, where stoichio-
metric propane-air mixture was used. This is a more realistic test case for IC
engine combustion and the simulations are performed using STAR-CD. Pressure
rise measured in the experiment is predicted reasonably well using both strained
and strained flamelet models. It is believed that flame confinement restricts the
entrainment of air, which reduces the flame straining due to turbulence. This
could explain why the differences in the results between unstrained and strained
flamelet models are small for this case.
Therefore, only the unstrained flamelet model is used for a practical IC engine
simulation, since the strained flamelet look-up table generation took significantly
148
8.2. Recommendations for future work
longer compared with the unstrained one with a marginal gain. A single-cylinder,
four-stroke SI engine experimentally investigated in ETH Zu¨rich is simulated
using STAR-CD, with iso-octane used as a gasoline surrogate. Two chemical
mechanisms are tested in this work, one predicts auto-ignition for the condi-
tions inside the cylinder whereas the other predicts flame propagation. Results
obtained using the mechanism that predicts flame propagation compares reason-
ably well with experimental data, whereas the simulation using auto-ignition is
unable to predict the correct pressure rise. It is to be noted that no knocking
was observed during experimental measurements, therefore, it is unlikely that
the mixture auto-ignites, which could explain why the auto-ignition simulation is
unable to predict the experimental pressure rise. Unlike the combustion models
available in STAR-CD, the modelling approach used in this work does not involve
any tunable parameters.
8.2 Recommendations for future work
The work carried out in this study has identified the following areas for further
investigations.
Spherical flame propagation: Results presented in Figure 4.4 clearly show
that the propagation speed, sp, of the spherical flame progress variable iso-
contours approach that of planar flame. The domain length used in these simula-
tions are not sufficient to simulate these flames for a longer time. It is generally
believed that spherical flames will eventually propagate at the same speed as pla-
nar flames, which can be verified by running the simulation for a much longer
time. This will give further fundamental insight into spherical flame propagation.
Self-similarity in spherical flames: The flame radius variation with time is
given by
rf (t) = R0 + At
n ≈ Atn, (8.1)
149
8.2. Recommendations for future work
where A and R0 are empirical constants. Note that R0 does not correspond
to the ignition radius since self-similarity is expected to start at a later time
(Gostintsev et al., 1988). The exponent, n, has been reported to fall in the
range 1.25–1.5 (Gostintsev et al., 1988; Bychkov and Liberman, 1996; Pan and
Fursenko, 2008). Experiments of Jomaas and Law (2009) showed that the n is
around 4/3. They also showed that this value was independent of the fuel-air
mixture, i.e. the chemical composition does not affect the power-law given in
Eq. (8.1). By simulating spherical methane- and hydrogen-air flames given in
Chapters 4 and 5 for a longer time it will be possible to determine the exponent
n, which can then be compared with published data.
Accounting for thermo-diffusive effects: Results from this work indicate
that including detailed chemistry in the laminar flame calculations is inadequate
to simulate turbulent flames that are thermo-diffusively unstable. These effects
need to be included in the turbulence models and it appears that the governing
equations have to be modified to account for thermo-diffusive effects in turbulent
flames.
Determining the effect of Lewis number remains an active area of combustion
research. For example, Chakraborty and Cant (2011) used DNS to investigate the
effect of Lewis number on flame surface density transport and proposed closure
models for various terms. In addition, Lewis number effects for the progress
variable and its variance equations for scalar dissipation rate based modelling
have been studied and closure models have been proposed (Chakraborty and
Swaminathan, 2010, 2011).
Using DNS of the G-equation, Dandekar and Collins (1995) showed that the
thermo-diffusive instability analyses of Sivashinsky (1977) can be used to de-
scribe the evolution of a passive but propagative scalar G. Peters (1999) noted
that the kinematic G-equation, which contains the Markstein diffusivity, must be
modified to study thermo-diffusive effects. Recently, Regele et al. (2013) studied
laminar flames with non-unity Lewis numbers using a progress variable equa-
tion together with an equation for the mixture fraction that includes non-unity
Lewis numbers. Ranga Dinesh et al. (2013) studied non-premixed hydrogen-air
flames using DNS, where they included a mixture fraction equation and two dif-
150
8.2. Recommendations for future work
ferent progress variable equations; one based on unity Lewis number assumption
and the other based on non-unity Lewis numbers. These studies indicate that
the non-unity Lewis numbers have to be explicitly implemented in the turbulent
simulations.
Auto-ignition of iso-octane: In this work, unburnt mixture temperature and
pressure conditions for auto-ignition are determined using freely propagating
flame calculations using the skeletal mechanism of Pepiot-Desjardins and Pitsch
(2008a). These calculations can be repeated in a future study using other skele-
tal mechanisms that can predict ignition, to determine the effect of chemical
mechanism.
Partially-stirred reactor for flamelet generation: A number of popular
combustion technologies for IC engines, such as HCCI, use controlled auto-ignition.
It is crucial that the laminar flame configuration used to generate the flamelet
library is representative of turbulent auto-ignition. In this case a homogeneous
constant-pressure reactor is used, however, one could use a partially stirred reac-
tor which can serve as a good model for turbulent combustion.
Simulation of gasoline direct injection: Gasoline engines are moving to-
wards direct injection, which provides a number of added benefits. However, in
GDI engines the mixture is only partially premixed and the modelling approach
used in this work has to be changed to account for that.
For partially premixed flames, closure models for the scalar dissipation rate
of both passive and reactive scalars are required. Furthermore, one also needs to
model the cross scalar dissipation rate. Algebraic models for partially premixed
flames have been proposed by Ribert et al. (2005); Robin et al. (2006); Mura
et al. (2007); Malkeson and Chakraborty (2010, 2011); Ruan et al. (2012). One
could implement these algebraic closures in STAR-CD to simulate combustion in
a spark-ignited GDI engine.
151
A. Spherically symmetric
equations
Spherically symmetric equations for RANS can be obtained from the governing
equations described in section 2.2.3. Only the momentum equation is shown here
since the derivation of the rest of the equations follow a similar procedure.
A.1 Radial momentum equation
The momentum equation [Eq. (2.2)] can be written in spherical coordinates [see
for example Bird et al. (2002)]. Since a spherically symmetric system is considered
in this work, only the radial momentum equation needs to be considered. The
Favre-averaged radial-momentum equation in conservative form can be written
as
∂ρu˜r
∂t
+
1
r2
∂
∂r
[
r2(ρu˜2r)
]
+
1
r sin θ
∂
∂θ
(ρu˜ru˜θ sin θ) +
1
r sin θ
∂
∂φ
(ρu˜ru˜φ)− ρu˜θ + u˜φ
r
= −∂p
∂r
+
1
r2
∂
∂r
[
r2
(
τ rr − ρu′′r2
)]
+
1
r sin θ
∂
∂θ
[
sin θ(τ θr − ρu′′θu′′r)
]
+
1
r sin θ
∂
∂φ
(
τφr − ρu′′φu′′r
)− (τ θθ − ρu′′θ2 + τφφ − ρu′′φ2)
r
, (A.1)
where u˜r, u˜θ and u˜φ are the Favre-averaged velocities in the r, θ and φ direc-
tions respectively. For a spherically symmetric system, the following additional
assumptions can be made
u˜θ = u˜φ = 0, u˜r = u˜r(r),
∂
∂θ
=
∂
∂φ
= 0, (A.2)
152
A.1. Radial momentum equation
to obtain the radial momentum equation [Eq. (3.2)], which is given below for
convenience.
∂ρu˜r
∂t
+
1
r2
∂
∂r
[
r2(ρu˜2r)
]
= −∂p
∂r
+
1
r2
∂
∂r
[
r2
(
τ rr − ρu′′r2
)]
− (τ θθ − ρu
′′
θ
2 + τφφ − ρu′′φ2)
r
. (A.3)
The non-vanishing viscous stress tensor appearing in the above equation is given
by Eq. (2.3). From this equation, the three non-vanishing stress tensors in spher-
ical coordinates are given as
τ rr = 2µ
∂u˜r
∂r
− 2
3
µ
[
1
r2
∂
∂r
(r2u˜r)
]
, (A.4)
τ θθ = τφφ = 2µ
u˜r
r
− 2
3
µ
[
1
r2
∂
∂r
(r2u˜r)
]
. (A.5)
Similarly the Reynolds stress terms using Boussinesq approximation are given
by Eqs. (3.7) and (3.8). Substitution of these stress terms into Eq. (A.3) and
term-by-term expansion gives
τ rr − ρu′′r2 = 2(µ+ µt)
∂u˜r
∂r
− 2
3
(µ+ µt)
(
1
r2
∂
∂r
(r2u˜r)
)
− 2
3
ρk˜, (A.6)
1
r2
∂
∂r
[
r2
(
τ rr − ρu′′r2
)]
=
1
r2
∂
∂r
{
r2
[
2(µ+ µt)
∂u˜r
∂r
− 2
3
(µ+ µt)
(
1
r2
∂
∂r
(r2u˜r)
)]}
− 1
r2
∂
∂r
(
r2
2
3
ρk˜
)
. (A.7)
The first term on the RHS of Eq. (A.7) can be simplified as
1
r2
∂
∂r
{
r2
[
2(µ+ µt)
∂u˜r
∂r
− 2
3
(µ+ µt)
(
1
r2
∂
∂r
(r2u˜r)
)]}
=
1
r2
∂
∂r
{
r2
4
3
(µ+ µt)
(
∂u˜r
∂r
− u˜r
r
)}
. (A.8)
153
A.1. Radial momentum equation
Therefore
1
r2
∂
∂r
[
r2
(
τ rr − ρu′′r2
)]
=
1
r2
∂
∂r
{
r2
4
3
(µ+ µt)
(
∂u˜r
∂r
− u˜r
r
)}
− 1
r2
∂
∂r
(
r2
2
3
ρk˜
)
. (A.9)
Expand the last term in equation (A.3) to obtain:
(τ θθ − ρu′′θ2 + τφφ − ρu′′φ2)
r
= 4(µ+ µt)
u˜r
r2
− 4
3
(µ+ µt)
(
1
r3
∂
∂r
(r2u˜r)
)
− 4
3
ρk˜
r
.
(A.10)
Now let Γ = [4 (µ+ µt)]/3. By substituting Eqs. (A.9) and (A.10) in Eq. (A.3)
to obtain:
∂ρu˜r
∂t
+
1
r2
∂
∂r
[
r2(ρu˜2r)
]
= −∂p
∂r
+
1
r2
∂
∂r
(
r2Γ
∂u˜r
∂r
)
− 1
r2
∂
∂r
(
r2Γ
u˜r
r
)
− 3Γ u˜r
r2
+
Γ
r3
∂
∂r
(r2u˜r) +
4
3
ρk˜
r
− 1
r2
∂
∂r
(
r2
2
3
ρk˜
)
. (A.11)
The terms on the RHS of Eq. (A.11) can be simplified further as follows:
4
3
ρk˜
r
− 1
r2
∂
∂r
(
r2
2
3
ρk˜
)
= − ∂
∂r
(
2
3
ρk˜
)
, (A.12)
and
− 1
r2
∂
∂r
(
r2Γ
u˜r
r
)
= −∂Γ
∂r
u˜r
r
− Γ
r
∂u˜r
∂r
− Γ u˜r
r2
(A.13)
Γ
r3
∂
∂r
(r2u˜r) =
Γ
r
∂u˜r
∂r
+ 2Γ
u˜r
r2
. (A.14)
154
A.1. Radial momentum equation
Hence
− 1
r2
∂
∂r
(
r2Γ
u˜r
r
)
− 3Γ u˜r
r2
+
Γ
r3
∂
∂r
(r2u˜r)
4
3
ρk˜
r
− 1
r2
∂
∂r
(
r2
2
3
ρk˜
)
= − ∂
∂r
(
2
3
ρk˜
)
− ∂Γ
∂r
u˜r
r
− 2Γ u˜r
r2
. (A.15)
The momentum equation can finally be written as
∂ρu˜r
∂t
+
1
r2
∂
∂r
[
r2(ρu˜2r)
]
= −∂p
∂r
+
1
r2
∂
∂r
(
r2Γ
∂u˜r
∂r
)
− ∂
∂r
(
2
3
ρk˜
)
− 1
r
(
∂Γ
∂r
+
2Γ
r
)
u˜r. (A.16)
155
B. Code validation
The modified in-house CFD code is validated for simple, spherically symmetric
diffusion and convection problems. For the sake of simplicity, it is assumed that
there is no reaction (i.e. ω˙ = 0) and turbulence is assumed to be frozen.
B.1 Diffusion test case
Convection is assumed to be zero in order to validate the code for pure diffusion.
Then Eq. (3.3) can be written as the familiar diffusion equation
∂c˜
∂t
= D
1
r2
∂2c˜
∂r2
, (B.1)
where diffusivity, D = αc + µt/Scc and c˜ defined in terms of temperature as in
Eq. (2.12). This parabolic PDE in r and t can be solved for a sphere with radius,
R, using the method of separation of variables, assuming a solution of the form
c˜(r, t) = R(r)T (t). Substituting this into Eq. (B.1), and taking the separation
constant as −λ2 gives the following two equations;
∇2R + λ2R = 0, dT
dt
+ λ2DT = 0. (B.2)
The simple solution to the time-dependent equation above is: T (t) = Ae−λ
2Dt,
where A is a constant to be determined later. The first equation in Eq. (B.2) is
the Helmholtz’s equation, which for spherical symmetry (no φ or θ dependence)
can be written as
1
r2
d
dr
(
r2
dR
dr
)
+ λ2R = 0. (B.3)
156
B.1. Diffusion test case
When there is spherical symmetry it is useful to make the substitution u = rR.
Now Eq. (B.3) can be written as:
d2u
dr2
+ λ2u = 0, (B.4)
which has the solution: u(r) = B sin(λr) + C cos(λr). Hence
R(r) =
1
r
[B sin(λr) + C cos(λr)] . (B.5)
Boundary and initial conditions have to be applied to obtain the constants A,B
and C. Only spherical flames that propagate radially outward (explosions) are
considered in this work.
Now consider a sphere with 0 < r < b, where b is the radius of the sphere.
Assuming that one is not interested in the case where an explosion started at the
origin reaches the outer walls of the sphere, the temperature at the wall, at all
instances of interest, can be set to Tu, which gives the boundary condition
c˜ = 0 at r = b. (B.6)
In addition, as a result of symmetry and since there is no heat influx at the centre
of the sphere, an additional condition
dc˜
dr
= 0 at r = 0. (B.7)
can be set at the origin to solve the problem. The initial condition is a particular
initial profile of c˜:
c˜(r, 0) = c˜0(r). (B.8)
Since c˜ has to be finite at the origin, the constant C in the solution needs to be
zero. In addition, applying the boundary condition (B.6) yields the eigenvalues,
157
B.1. Diffusion test case
λn = (npi)/b, and the eigenfunction
Rn(r) =
An
r
sin
(npir
b
)
, n = ±1,±2,±3, . . . (B.9)
A particular solution to Eq. (B.3) is then
c˜(r, t) = Rn(r)T (t) =
An
r
sin(λnr)e
−λ2Dt. (B.10)
Since the diffusion equation is linear, the principle of superposition yields the
following general solution
c˜(r, t) =
∞∑
n=1
An
r
sin(λnr)e
−λ2Dt. (B.11)
Note that only the positive values of n have been taken since negative values will
lead to exponential grown when n → ∞. The term, n = 0 is omitted since it
is identically zero. The coefficients An can now be chosen to satisfy the initial
condition (B.8) to yield
c˜(r, 0) = c˜0(r) =
∞∑
n=1
An
r
sin(λnr). (B.12)
When both sides of Eq. (B.12) are multiplied by r, it can be seen that this is
half-range Fourier series of c0(x) between 0 and b. Therefore An is given by the
Fourier coefficient
An =
2
b
∫ b
0
c˜0(r
′) sin
(
npir′
b
)
r′ dr′. (B.13)
Finally, the full solution to the problem is given by
c˜(r, t) =
∞∑
n=1
[
2
b
∫ b
0
c˜0(r
′) sin
(
npir′
b
)
r′ dr′
]
1
r
sin
(npir
b
)
e−λ
2Dt. (B.14)
158
B.1. Diffusion test case
If the initial profile, c˜0(r
′), is taken as [see Figure B.1(a) for the values of r1 and
r2]
c˜0(r
′) =
1 for 0 ≤ r′ < r1(
r−r2
r1−r2
)
for r1 ≤ r′ ≤ r2
0 for r2 < r
′ ≤ R
(B.15)
then the exact solution obtained after integration is
c˜(r, t) =
∞∑
n=1
{
2
λ2R
[
sin(λnr1) +
1
r1 − r2
(
r2 sin(λnr2)− (2r1 − r2) sin(λnr1)
+
2
λn
(cos(λnr2)− cos(λnr1))
)]}
× 1
r
sin
(npir
R
)
e−λ
2Dt. (B.16)
Figure B.1(a) shows that this analytical result at t = 1 s can be predicted nu-
merically using the method outlined in this section.
0.0
0.2
0.4
0.6
0.8
1.0
0 50 100 150 200 250
c˜
r (mm)
r1
r2 R
Initial profile
Analytical
Numerical
(a)
0.0
0.2
0.4
0.6
0.8
1.0
0 50 100 150 200 250
c˜
r (mm)
Initial profile
Analytical
Numerical
(b)
Figure B.1: Validation of numerical code for: (a) diffusion at t = 1 s and (b) con-
vection at t = 0.1 s.
159
B.2. Convection test case
B.2 Convection test case
In the case of pure convection without heat release, Eq. (3.3) can be written as
∂c˜
∂t
+
1
r2
∂
∂r
(
r2u˜rc˜
)
= 0, (B.17)
which is the linear advection equation in spherical coordinates. Since ω˙ = 0 , the
heat release parameter, τ = 0, thus, the convection velocity, u˜r, is not a function
of c˜. One can then think of c˜ as being a convected quantity, with its value at
time, t, given by
c˜(r, t) = c˜0(r − u˜rt). (B.18)
Note that if Eq. (B.17) is solved together with the momentum and pressure Pois-
son equation [essentially what is done in SIMPLER method (Patankar, 1980)],
then the velocity u˜r will not remain constant due to the change in radius as the
profile is convected. Therefore, in order to maintain a constant convection ve-
locity, the pressure equation is not solved for this particular test problem. The
results of this convection problem is shown in Figure B.1(b), which shows good
agreement with the analytical result.
160
C. List of Publications
Journal publications
• Ahmed, I., and Swaminathan, N. 2013. Simulation of spherically expanding
turbulent premixed flames. Combust. Sci. Technol., 185, 1509–1540.
• Ahmed, I., and Swaminathan, N. 2014. Simulation of turbulent explosion
of hydrogen-air mixtures. Int. J. Hydrogen Energ., 39, 9562–9572.
• Ahmed, I., Swaminathan, N., Schlatter, S., and Wright, Y. M. Simulation
of combustion in a closed vessel with swirl (in preparation).
• Ahmed, I., Swaminathan, N., Koch, J., and Wright, Y. M. Spark-ignition
engine simulation using a flamelet based combustion model (in preparation).
Invited talks
• Swaminathan, N., and Ahmed, I. 2013. Investigation of turbulent spher-
ical flames. American Physical Society, 66th Annual Meeting of the APS
Division of Fluid Mechanics, November 24-26, Pittsburgh, PA, USA.
Conference publications
• Ahmed, I., and Swaminathan, N. 2011. Simulation of spherical turbulent
premixed flames. Proceedings of the 5th European Combustion Meeting,
June 28-July 1, Cardiff, UK.
• Ahmed, I., and Swaminathan, N. 2011. Effects of mean curvature on flame
propagation. Proceedings of the 23rd International Colloquium on the Dy-
161
namics of Explosions and Reactive System, July 24-29, Irvine, California,
USA.
• Ahmed, I., and Swaminathan, N. 2012. Simulation of spherical methane-air
flames. Proceedings of the 1st International Education Forum on Environ-
ment and Energy Science, December 14-18, Hawaii, USA.
• Ahmed, I., Swaminathan, N., Schlatter, S., and Wright, Y. M. 2013. Simu-
lation of swirl combustion in a closed vessel. Proceedings of the 6th European
Combustion Meeting, June 25-28, Lund, Sweden.
162
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