PDF-Modeling and Simulation of Turbulent Spray Combustion

PDF-Modeling and Simulation of
Turbulent Spray Combustion
Dissertation
zur
Erlangung des Grades
Doktor–Ingenieur
der
Fakultät für Maschinenbau
der Ruhr–Universität Bochum
von
Seung Jin Baik
aus Seoul, Südkorea
Bochum, 2016

Abstract
In this dissertation, based on comprehensive studies of the two-phase PDF method, a
two-dimensional reactive-spray code has been developed and applied. Given two-phase
spray PDF transport equation is solved by monte-carlo particle method and corresponding
models. The Langevin model is employed to model the gasphase velocity. The molecular
mixing term is closed through mixing models, i.e., Interaction by Exchange with the
Mean (IEM) model, Curls mixing model and Euclidian Minimum Spanning Tree (EMST)
model. The exchange terms between two phase are modeled by evaporation models, i.e.,
D 2 -law model and film model. Since we will select velocity model and mixing models that
include the turbulent kinetic energy k and the turbulent dissipation rate ɛ, the standard
k-ɛ turbulence model is employed.
For discretization of the governing equations, the unstructured mesh and correspondingly
formed control volumes are defined. In each control volume, the number of particles is
monitored and controlled by particle cloning and annihilation scheme to keep the statistical
errors approximately uniform in the overall computational domain.
The two-dimensional reactive-spray code has been applied to three different experiments,
i.e., a turbulent hydrogen-air flame, a non-premixed, turbulent methane-air counterflow
flame with water droplets, and a turbulent ethanol-air spray flame. To reduce the computing
costs, three different parallelization schemes, i.e., OpenMP, MPI, and the MPI-
OpenMP hybrid scheme, has been introduced and applied to the code. The numerical
results obtained in the present thesis show good agreement with experimental data available
in the literature.
Keywords: turbulent spray flow, Monte Carlo method, two-phase PDF method, parallelization
i

Acknowledgements
The present dissertation represents the culmination of my work at the chair of fluid
mechanics of Ruhr-University Bochum.
First and foremost I would like to thank my supervisor professor Rogg for providing me
with opportunity to work on such a great research team, and for his continues encouragement,
motivation, and suggestions during my PhD time.
Secondly, I would like to thank for my committee member professor Scherer who friendly
took the review of my dissertation and gave me many fruitful suggestions. I would also
like to thank professor Abramovici for being the head of my exam committee and for his
questions and comments during my exam.
I owe many thanks to all my colleagues in the group of Prof. Rogg, who have given the
pleasant research atmosphere and the countless support. I also give a special thanks to
my friend Dr. Sang-Ho Na who has given me lots of comment and help for computation
technical point of view.
Last but not the least, I would like to thank my parents, family, and especially my wife
Nalye Hong. Without her selfless support, great patience and constant encouragement,
this work can not be finished.
Finally, I acknowledge Deutsche Forschungsgemeinschaft (DFG) for the financial support
through grant no RO2249/3-1.
ii

Contents
Abstract
Acknowledgements
ii
iii
1 Introduction 1
1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Outline of the Present Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Governing Equations for Two-Phase Flows 8
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Deterministic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Gas-Phase Conservation Equations . . . . . . . . . . . . . . . . . . 9
2.2.2 Liquid-Phase Conservation Equations . . . . . . . . . . . . . . . . . 10
2.2.3 Phase Exchange Terms . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.4 Phase-Indicator Function . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.5 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Probabilistic Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Two-Phase Joint PDF . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.2 Gas-Phase DF-Transport Equation . . . . . . . . . . . . . . . . . . 17
2.3.3 Liquid-Phase DF-Transport Equation – Williams’ Spray Equation . 17
2.4 Pressure-Correction Equation . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Computational Approach 21
3.1 Two-Phase Particle Method . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 Particle Method for Liquid Phase . . . . . . . . . . . . . . . . . . . 22
3.1.2 Particle Method for Gas Phase . . . . . . . . . . . . . . . . . . . . 24
3.2 Expectations and Moving Averages . . . . . . . . . . . . . . . . . . . . . . 26
3.2.1 Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Moving Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Models 28
iii

4.1 Velocity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Mixing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 IEM-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.2 Curl’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.3 EMST-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Evaporation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.1 D 2 ´ law Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.2 Film Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Discretization 39
5.1 Domain of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Eulerian Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2.1 Poisson Equation for Pressure . . . . . . . . . . . . . . . . . . . . . 41
5.2.2 Turbulence Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Lagrangian Particle Equations . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Particle Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.5 Particle Controlling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.6 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 48
5.6.1 Initial and Boundary Conditions for Particles . . . . . . . . . . . . 48
5.6.2 Boundary Conditions for Eulerian Equations . . . . . . . . . . . . . 48
5.7 Overall Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6 Parallelization 52
6.1 Parallelization Architecture and Strategies . . . . . . . . . . . . . . . . . . 52
6.1.1 Parallel Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.1.2 Distributed Memory System and Shared Memory System . . . . . . 54
6.1.3 Parallelization Strategies . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2 Domain and Particles Decomposition . . . . . . . . . . . . . . . . . . . . . 56
6.2.1 Schur-Complement Method . . . . . . . . . . . . . . . . . . . . . . 57
6.2.2 Particles Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Functional Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4 MPI-OpenMP Hybrid Approach . . . . . . . . . . . . . . . . . . . . . . . . 66
6.4.1 Performance and Limitation . . . . . . . . . . . . . . . . . . . . . . 67
6.4.2 Hybrid Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.5 Heterogeneous Computing with GPU . . . . . . . . . . . . . . . . . . . . . 69
6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.5.2 GPU Acceleration with OpenCL . . . . . . . . . . . . . . . . . . . . 70
iv

7 Turbulent Jet Diffusion Flames (DLR H3 Flame) 75
7.1 Experimental and Computational Setup . . . . . . . . . . . . . . . . . . . 75
7.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.3 Speedup by OpenMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8 Turbulent Counterflow Flames with Water Droplets 82
8.1 Experimental and Computational Setup . . . . . . . . . . . . . . . . . . . 82
8.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.3 Speedup by MPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
9 Turbulent Jet Diffusion Spray Flames (Sydney flame) 90
9.1 Experimental and Computational Setup . . . . . . . . . . . . . . . . . . . 90
9.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.3 Speedup by the MPI-OpenMP Hybrid Approach . . . . . . . . . . . . . . . 97
10 Conclusions and Perspectives 98
Appendices 100
A Derivation of the Eulerian Equations 100
A.1 Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
A.2 Turbulent Kinetic Energy and Dissipation Rate . . . . . . . . . . . . . . . 103
B Chemical Reaction Mechanism 107
C Mixing Models 109
Bibliography 114
v

List of Figures
2.1 Schematic diagram of two-phase spray. . . . . . . . . . . . . . . . . . . . . 12
4.1 Euclidian minimum spanning tree in two dimension composition space. . . 32
5.1 Section of an unstructured triangular mesh with control volumes. . . . . . 39
5.2 An unstructured triangle with physical coordinates and the mapped into
natural coordinate ξ and η. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 A mapping to determine control volume which the particle is located. . . . 46
5.4 Statistical error and speed down in a control volume as a function of the
number of particles in a control volume. . . . . . . . . . . . . . . . . . . . 47
5.5 Schematic of the fully stochastic particle method. . . . . . . . . . . . . . . 50
5.6 Flowchart of the overall computation algorithm. . . . . . . . . . . . . . . . 51
6.1 Classification of parallel computer architectures by Flynn [22]. . . . . . . . 53
6.2 Schematic of distributed memory system. . . . . . . . . . . . . . . . . . . . 54
6.3 Schematic of shared memory system. . . . . . . . . . . . . . . . . . . . . . 54
6.4 An example of executing time segments. . . . . . . . . . . . . . . . . . . . 55
6.5 Schematic of MPI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.6 Schematic of domain decomposition. . . . . . . . . . . . . . . . . . . . . . 58
6.7 Schematic of particle exchange. . . . . . . . . . . . . . . . . . . . . . . . . 64
6.8 Partial speedup by OpenMP on calculating chemical source term. . . . . . 66
6.9 Partial speedup by OpenMP on expectation values. . . . . . . . . . . . . . 66
6.10 Overall speedup by OpenMP and MPI on sprays flame. . . . . . . . . . . . 67
6.11 Schematic of MPI-OpenMP hybrid approach. . . . . . . . . . . . . . . . . 68
6.12 Comparing overall speedup between Pure OpenMP and Hybrid MPI-OpenMP
(MPI 8 ˆ OpenMP 4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.13 Schematic of CPU(left) and GPU(right). . . . . . . . . . . . . . . . . . . . 70
6.14 Schematic of implementation GPU accelerating with OpenCL. . . . . . . . 71
6.15 Sample calculation for premixed methane flame on 1D code. . . . . . . . . 72
6.16 Comparing speedup of parallelized target functions between GPUs; 48 cores
(left), 384 cores(center) and 24 cores(right). . . . . . . . . . . . . . . . . . 73
vi

6.17 Schematic of MPI-OpenCL hybrid concept. . . . . . . . . . . . . . . . . . . 74
7.1 Schematic of the DLR burner [56] and of the computational domain. . . . . 76
7.2 Axial profile of the mass fraction of H 2 along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65]. . . . . . 76
7.3 Axial profile of the mass fraction of O 2 along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65]. . . . . . 77
7.4 Axial profile of the mass fraction of H 2 O along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65]. . . . . . 77
7.5 Axial profile of the mass fraction of OH along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65]. . . . . . 78
7.6 Axial profile of temperature along the centerline. Solid line: computational
result; symbols: experimental data from [1, 21, 65]. . . . . . . . . . . . . . 78
7.7 Temperature as a function of mixture fraction at x{D “ 5. Red points:
computational result; blue points: experimental data from [1, 21, 65]; line:
chemical equilibrium from [1, 21, 65]. . . . . . . . . . . . . . . . . . . . . . 79
7.8 Temperature as a function of mixture fraction at x{D “ 20. Red points:
computational result; blue points: experimental data from [1, 21, 65]; line:
chemical equilibrium from [1, 21, 65]. . . . . . . . . . . . . . . . . . . . . . 79
7.9 Surface plots – from top to bottom – of the mass fraction of H 2 O, the mass
fraction of OH, and the temperature, respectively. . . . . . . . . . . . . . . 80
7.10 Marginal density function of the mass fraction of OH at position x{D “ 5. 81
8.1 Schematic of the counterflow burner used in the experiments [107]. . . . . . 83
8.2 Schematic of computational domain and boundaries. . . . . . . . . . . . . 84
8.3 Profile of the mean axial velocity component of water droplets along the
symmetry line. Solid line: computational result; symbols: experimental
data from [107]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.4 Radial profile of the mean axial velocity component of water droplets at
the axial location z “ 1.4 mm. Solid line: computational result; symbols:
experimental data from [107]. . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.5 Phase-plane identifying regimes of extinction and stable burning. Stars:
computational results, other symbols: experimental data from [107]. . . . . 86
8.6 Maximum mean flame temperature as a function of the water mass concentration
Y d for inlet mole fractions X O2 “ 0.21 and X CH4 “ 0.23. . . . . . 87
8.7 Surface plots of mean temperature with different Y d . A,C: without seeding
of water drops; B: stable flame with Y d “ 0.1%; D: extinction with Y d “ 0.9%. 87
8.8 Surface plot of mean temperature with streamlines for a case with Y d “ 0.1%. 88
vii

9.1 Schematic of the spray burner used in the Sydney experiments [39, 55]. . . 91
9.2 Schematic of computational domain and boundaries. . . . . . . . . . . . . 92
9.3 Radial profile of the mean axial velocity component of fuel droplets along
at axial station z/D=10. Solid line: computational result; symbols: experimental
data from [39, 55]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.4 Radial profile of the mean axial velocity component of fuel droplets along
at axial station z/D=20. Solid line: computational result; symbols: experimental
data from [39, 55]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.5 Radial profile of temperature at axial location z/D=10. Solid line: computational
result; symbols: experimental data from [39, 55]. . . . . . . . . . . 93
9.6 Radial profile of temperature at axial location z/D=20. Solid line: computational
result; symbols: experimental data from [39, 55]. . . . . . . . . . . 94
9.7 Surface plots – from top to bottom – of the axial velocity component, the
mass fraction of H 2 O, and the temperature, respectively. . . . . . . . . . . 95
9.8 Surface plots – from top to bottom – of the axial velocity component, and
the radial velocity component, respectively. . . . . . . . . . . . . . . . . . . 95
9.9 Profile along the jet center line of the marginal density function f W pR l q as
a function of droplets radius. . . . . . . . . . . . . . . . . . . . . . . . . . . 96
9.10 Droplets radius represented by the liquid Monte-Carlo particles along the
symmetry line of the jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.1 Integrating points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
C.1 The convergence comparing between three mixing models. . . . . . . . . . 110
C.2 The convergence shapes of IEM mixing model. . . . . . . . . . . . . . . . . 111
C.3 The convergence shapes of modified Curl mixing model. . . . . . . . . . . . 112
C.4 The convergence shapes of EMST mixing model. . . . . . . . . . . . . . . . 113
viii

List of Tables
8.1 Inlet composition for both the experiment [107] and the present computations. 82
9.1 Properties of liquid ethanol. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.1 Reduced chemical mechanism of methane [24]. The activation energies E a
are in cal/mole and the pre-exponential constants in cgs units. . . . . . . 108
ix

Nomenclature
Roman Symbols
B M
B T
C D
Spalding mass transfer number
Spalding heat transfer number
drag coefficient
c p specic heat capacity [J/kg K]
D
F D
diffusion coefficient [m 2 /s]
drag force
ɛ dissipation rate of turbulent kinetic energy [m 2 {s 3 ]
K
number of species
k turbulent kinetic energy [m 2 {s 2 ]
m
n l
P
p
p
P g
P l
R
S
particle mass [kg]
droplet number density
number of particles
particle index
pressure [Pa]
number of gasphase particles
number of liquid phase particles
radius of particle [m]
source term
x

T
t
u
V
v
V k
w
w α
X α
Y α
Y d
Nu
P r
Re
Sc
Sh
temperature [K]
time [s]
velocity component along x [m/s]
velocity [m/s]
velocity component along y or r [m/s]
kth control volume
velocity component along z [m/s]
reaction rate of species α
mole fraction of species α
mass fraction of species α
mass concentration of water droplets
Nusselt number
Prandtl number
Reynolds number
Schmidt number
Sherwood number
Greek Symbols
α
Γ
index of species
interface between two control volumes
λ thermal conductivity [J/m ¨ s ¨ K]
µ viscosity [kg/m ¨ s]
Ω
domain
ρ density [kg/m 3 ]
V
finite volume
xi

φ
ψ
general scalar variable
scalar variable – for PDF transport equations
Mathematical Symbols
x P A x is an element of A
x :“ A x is defined as A
Superscript
ˆ
sample properties
9 time rate
r
Favre average
Subscript
e
g
l
m
v
energy related
gasphase
liquid phase
mass related
momentum related
Abbreviations
CFD
CFL
Computational Fluid Dynamics
Courant–Friedrichs–Lewy condition
GPGPU General-Purpose Computing on Graphics Processing Unit
LHF
Locally Homogeneous Flow models
CDM Continuous Droplet Model
CFM Continuum Formulation Model
DDM Discrete Droplet Model
DF
Density Function
xii

DNS
DSF
LES
MPI
PDF
Direct Numerical Simulation
Deterministic Separated Flow
Large Eddy Simulation
Message Passing Interface
Probability Density Function
RANS Reynolds Averaged Navier-Stokes
SSF
Stochastic Separated Flow
CUDA Compute Unied Device Architecture
OpenCL Open Computing Language
OpenMP Open Multi-Processing
SF
Separated Flow models
xiii

Chapter 1
Introduction
Notwithstanding that environmental contamination due to pollutants emissions through
combustion has been on the rise for several decades, fossil fuels still cover more than 85
percent of the world’s total energy resource requirement. Nuclear resources, which were
considered a substitute for fossil fuels, have continuously shown to give rise to several
critical risks such as radiation leaks due to the disposal of the nuclear fuel. After the
Fukushima Daiichi nuclear disaster in Japan in 2011, radioactivity there still is released
into the sea and into the air, and hence – as a consequence of new political aims – the
proportion of nuclear systems is in decline. Renewable energy resources from nature
are, definitely, the way to proceed in the future, however their exploitation is not yet
sufficiently developed, neither in terms of economy nor in terms of efficiency. Therefore,
usage of fossil fuels will be unavoidable for decades, and hence it is necessary to optimize
the implementation of those fuels to be both efficient in energy generation and with respect
to environmental aspects.
Combustion, however, is a complex phenomenon as it is couples chemistry, fluid dynamics,
and heat and mass transfer. Moreover, combustion also contains many phenomena that
up to date have not been fully investigated. In contrast to laminar combustion, turbulent
combustion is even more complex due to the inherently large number of turbulent time and
length scales. Further complexities arise through complicated geometries of combustion
chambers, say. Consequently, the analysis of combustion phenomena requires a deep
theoretical understanding of the fundamental physical processes and the development of
both physical and simulation methods that are particularly powerful.
1

1.1 Historical Background
Spray combustion plays an important role in many combustion systems such as furnaces,
diesel engines, gas turbines etc. For instance, in internal combustion engines, an implementation
of the concept of feeding liquid fuels in the form of a spray is well known
to result in increased combustion efficiency and reduced pollutant formation. Although
combustion of liquid fuel sprays up to date has been widely used in praxis, it is still a
challenging field because it inherits all the difficulties from gaseous combustion, and, in
addition, encompasses injection of liquid fuel, evaporation of atomized fuel droplets, as
well as fluid dynamical interaction and heat transfer between the liquid and the gaseous
phase.
A spray, reacting or non-reacting, can be defined as a special case of two-phase flow, that
is composed of a dispersed liquid phase in the form of droplets or ligaments and a continuous,
gaseous or carrier phase [45, 86, 97]. After injection through a nozzle into a gaseous
environment, the liquid fuel splits up into many droplets which are of variable size and,
have variable velocity as a consequence of the atomization process. During the injection
process, depending on the environmental conditions, droplets break up continuously or
collide with one or several other droplets, or with a wall. Depending on the underlying
atomization process, sprays flow can be divided into dense sprays, which are characterized
by a large liquid volume fraction near the exit of a nozzle, and dilute sprays, which have
a much smaller liquid volume fraction there. After atomization or even during that process,
droplets evaporate into the gaseous flow, and chemical reaction occurs when proper
ignition conditions are met. Even though natural spray flow contain variable interactions
between the liquid and the gaseous phase, as well between the droplets themselves such as
break-up and collision, many previous researches assumed that spray flow has been fully
diluted to circumvent the difficulties of spray analyse under dense condition.
Existing numerical methods to solve dilute-sprays problems can be widely separated into
two categories as Locally Homogeneous Flow models (LHF) [17, 18, 19, 45, 97] and Separated
Flow models (SF) [45, 84, 85, 86, 97]. LHF models are based on the fundamental
assumption that the flow field locally exists as a homogeneous mixture of liquid and gas.
Under this assumption, both phases have the same velocity and temperature everywhere
in the flow field. The main advantage of LHF models is that they require relatively little
information on initial conditions at the injection nozzle, because initial droplet size and
velocity distribution play no role in the computations. Moreover, since computations for
sprays flow with LHF models are basically identical to single-phase flow computation,
traditional solution methods and codes can be used. LHF models, however, can be ap-
2

plied only when the spray flow consists of infinitely small droplets because the error due
to fundamental assumptions underlying these models cannot be acceptable [17, 18, 19].
In separated flow or SF models, spray flow is separated basically into a liquid phase (dispersed
phase) and a gasphase (carrier phase). Therefore, finite-rate exchange of mass,
momentum and energy between the two phases can be implemented. Generally, the SF
approach for reacting spray flow can be classified into three sub-models, viz., the Continuum
Formulation Model (CFM), the Continuous Droplet Model (CDM) and the Discrete-
Droplet Model (DDM) [17, 45, 84, 85, 86, 97]. CFM employs a continuum formulation of
the conservation equations for both phases and also for evaporation. Since the governing
equations for both phases are similar, numerical implementation is relatively efficient;
however, this formulation has drawbacks for establishing some critical characteristics of
reacting spray flow such as effects of droplet heat-up, turbulent stresses and its transport.
The second sub-model in the separated flow approach is the Continuous Droplet Model
(CDM) as described by Williams [45, 97]. In this model, the properties (e.g. radius,
position, velocity, concentration and temperature) of the liquid phase are described by
a statistical distribution function, the so-called Williams spray equation. The governing
equations for the gasphase involve suitable source terms for interaction effects due to the
presence of droplets in the flow. However, most combustion studies have not allowed for
a continuous droplet distribution function. Instead, a finite number of different, discrete
droplets has been employed in seeking computer solutions of the conservation equations
[45, 97]. In the last model, called the Discrete-Droplet Model (DDM), the liquid phase is
divided into representative samples of discrete droplets whose motion and transport are
tracked through the flow field based on Lagrangian formulations. This procedure corresponds
to a statistical approach, the so-called Monte Carlo Method, for liquid properties
because a finite number of particles is used to represent the entire liquid phase. In fact,
CDM and DDM only differ in their approach to the liquid phase in the spray flow. If continuous
distribution or density functions are approximated by a finite number of sample
particles in a finite volume, both methods yield approximately identical results [16, 19].
As mentioned above, the liquid or dispersed phase can be subjected to various different,
alternative models to describe the characteristics of the droplets. On the other hand, the
gaseous carrier phase of a reactive turbulent spray flow can, often, be described sufficiently
accurately by the usual governing equations of continuum fluid mechanics with added
source terms that describe the interaction between the droplets and the carrier phase.
Direct numerical simulation (DNS), large eddy simulation (LES), and simulations based
on Reynolds Averaged Navier-Stokes (RANS) equations for the carrier phase, combined
3

with a probability density function (PDF) method to cater for chemistry are widely used
to obtain numerical solutions for the continuous carrier phase.
The respective status of studies on spray combustion, both in theory and application, has
been reported in [2, 17, 18, 19, 84, 85, 86, 97]. Fundamental models for combustion of
individual droplets were established in the 1950s by Godsave [27], Spalding [88] and Goldsmith
[28] under ideal conditions, e.g, assuming spherically symmetric of the droplets. In
the early stages of droplet and spray research, experiments and investigations on single
droplets helped to understand single droplet combustion. For instance, Godsave determined
experimentally the burning rate for benzene, ethyl alcohol, and heptane drops
under atmospheric pressure, and estimated the diameter of flame surrounding the liquid
drop [27]. Spalding [88] introduced the so-called mass and heat transfer number B to
express the fluxes of mass and heat, respectively, at the droplet surface. Modeling of
multi-component fuels began with the work reported by Landis and Mills [76] in 1970s.
They studied the evaporation of heptane-octane two-component drops by means of a
spherically symmetric model. Sirignano and Law [87] focused on internal heat, mass,
and momentum transport and other effects related to evaporation. The extreme cases
are the rapid regression or zero-diffusivity limit [49], that were interestedly investigated
in the early 1980s. In [49], the classical evaporation model, so-called D 2 -law [27, 97],
for droplet vaporization and combustion which is formulated for a single-component fuel,
was extended for multi-component fuels. Some noteworthy reviews of turbulent spray
flows have been reported by Faeth [17] in late 1980s. He compared three types of twophase
models for turbulent flows, i.e., locally homogeneous flow (LHF), deterministic
separated flow (DSF), and stochastic separated flow (SSF). He recommended the SSF
model for practical dilute sprays. This model considers finite interphase transport rates
and uses random-walk computations to simulate turbulent dispersion for the dispersed
phase. Crowe et al. [9] reviewed both time-averaged and time-dependent free-shear flows
of two-phase flows. Their time-dependent numerical methods captured the instantaneous
flow and let to better prediction of the particle trajectories. In the time-averaged method,
a steady flow with gradient diffusion is usually considered. In 1990s, Crowe et al. [10]
reviewed computational approaches to turbulent two-phase flows.
In the early 2000s, Zhu et al. [108] derived a two-phase PDF transport equation encompassing
all gaseous phase and liquid phase random variables. A phase-indicator function
[42] was used to capture the interface of the multiphase flow. Rumberg and Rogg [79, 80]
suggested to describe the multiphase flow with two separate density function. Both phase
density functions and their transport equations were defined in the overall two-phase flow
field. Effects of interfacial surface interaction, including heat and mass transfer, on the
4

overall flow were taken into account by source terms in the transport equation. A few
years later, Naud [60] adopted a joint velocity-composition PDF for the continuous phase
and the joint PDF of droplet velocity, droplet temperature, droplet mean gas velocity,
and droplet mean gas composition for dispersed phase. After the year 2000, the stochastic
separated-flow (SSF) model [29, 45, 97], which considers the effects of turbulent fluctuations
on particle motion, based on the Discrete-Droplet Model or DDM has been widely
employed to numerically simulate turbulent reactive sprays [12, 33, 37, 95].
Following Rumberg and Rogg [79, 80], dissimilar to the traditional models CDM and
DDM, in the present work the gasphase is described by a PDF transport equation rather
than by Eulerian equations. Since the PDF method has the main advantage that chemical
source terms are closed and hence do not require modelling, it is widely used to solve
reacting flow problems. Lundgren [54] studied nonhomogeneous turbulent flow by solving
the velocity PDF transport equation. Although the application was limited initially to
simple inert gaseous flow, that is the first well-known application for implementing the
PDF approach to turbulent flow. After the PDF concept was extended to the reactive
flow problems by Hill [34], Dopazo and O’Brien [13] also solved for composition variables,
which is a set of scalar quantities, usually encompassing mass fractions and temperature,
joint PDF transport equation for describing turbulent reactive flows.
The relationship between Monte Carlo particle methods and PDF methods was worked
out by Pope [67, 68], through whom’s work, Monte Carlo particle methods have become
the key approach to solving PDF transport equations. Since Pope’s review article in 1985
[69], numerous applications of the PDF method to analyse turbulent reactive flow have
contributed to the popularity of the PDF concept. The Monte-Carlo framework needs
a reasonably large number of statistical particles, that means it causes relatively high
computing costs. During the development of PDF methods, to mitigate the computing
costs, a combination of the PDF method and the traditional Eulerian approach, e.g.,
simulations based on Reynolds Averaged Navier-Stokes (RANS), or, large eddy simulation
(LES) have been introduced [4, 30, 59]. Early steps to implement the hybrid concept were
taken by Anand [4] in 1989. However, the early work reported that the hybrid concept
had poor efficiency, robustness, and accuracy due to lacking numerical consistency [30].
Since the successful resolution of the consistency problem by Muradoglu [59], the hybrid
concept has been widely implemented for a number of flows [30, 59, 74, 78, 92, 105].
Overview of the PDF method relating to turbulent single-phase combustion can be found
in [13, 30, 43].
For turbulent spray combustion, the PDF methods play an important role. From a
conceptional point of view, methods for reacting spray flows can be broadly separated
5

into two categories. These categories can be identified by employing the PDF method for
only the dispersed phase or for both the dispersed phase and the carrier phase. Approach
of the first category is similar to employing the Discrete-Droplet Model (DDM) which uses
William spray equations for modelling of the droplet phase and the traditional Eulerian
approach for the carrier phase [17, 45, 84, 85, 86, 97]. Approaches of the second category,
in which Monte-Carlo particles are used for either phase, can be subdivided further into
two classes. To the first class belong all approaches that, in a so-called fractional step,
use gaseous particles exclusively to describe or model the combustion chemistry but use
traditional formulations such as RANS or LES for the fractional solution step regarding
the accumulative, convective and diffusive terms as well as the phase-interaction terms in
the Eulerian governing gasphase equations for overall mass, species mass, momentum and
energy. Examples are the formulations in, e.g., [12, 26, 60]. To the second class belong
all approaches that use a full PDF description for the overall combustion system such as
the approach taken in the present thesis, and the approaches presented in [79, 80, 108].
Several models or techniques, respectively, have been developed that help reducing the
costs for turbulent, purely gaseous flames or, alternatively, reactive sprays computed by
PDF methods. For instance, so-called flamelet models were developed according to which
the local and instantaneous combustion is computed a-priori to the computation of the
turbulent reactive flow [50, 64, 97]. Another possibility to reduce computing costs in the
application of PDF-methods is to employ the so-called in-situ adaptive tabulation (ISAT)
developed by Pope [72]. Using this kind of tabulation, for mots particles the composition
variables are obtained by interpolation/extrapolation in a binary-tree database based on
previously stored solutions. In the present work, apart from the above techniques, the
computing costs will be reduced further by applying suitable parallelization schemes.
1.2 Outline of the Present Thesis
In the present work, turbulent spray combustion has been modelled and – for spatially
two-dimensional geometries – simulated by a full PDF approach. Here the adjective full
points to the fact that not only the droplet phase but also the gasphase was subjected to
a full joint-PDF formulation that encompasses both phases.
Based on comprehensive studies of the two-phase PDF method carried out in the framework
of the present research, a two-dimensional reactive-spray code has been developed
and applied. The remainder of the present thesis is structured as follows.
In Chapter 2, the full, overall two-phase PDF transport equation encompassing both the
6

gaseous and the liquid phase as well as marginal PDF transport equations for either spray
phase are derived from the general two-phase governing equations.
In Chapter 3, the Monte-Carlo particle method is introduced for either phases. In Chapter
4, models are presented and discussed that are suitable for closure of the various originally
unclosed terms in the governing equations presented in Chap. 2.
In Chapter 5, the unstructured mesh and correspondingly formed control volumes are
defined to enable discretization of the governing equations. To keep the statistical errors
approximately uniform in the overall computational domain, the number of liquid and
gaseous Monte-Carlo particles in each control volume is monitored and controlled by
suitable methods.
In Chapter 6, the two-dimensional reactive-spray code is parallelized by applying the
two parallelization methods MPI and OpenMP, respectively. 1 An MPI-OpenMP hybrid
scheme is suggested, and also implemented in the code. Furthermore, the so-called CPU-
GPU hybrid computing approach 2 is tested to estimate how this so-called heterogeneous
approach to computing can be helpful in applications of computational fluid dynamics or
CFD.
In Chaps. 7 to 9, the computer code is applied to three different turbulent problems.
Specifically, in Chap. 7, a turbulent, purely gaseous, hydrogen-air flame is computed to
validate the code, in the first instance, for pure gasphase combustion.
In Chaps. 8 and 9, turbulent sprays are computed. Specifically, in Chap. 8 the computer
code is applied to a non-premixed, turbulent methane-air counterflow flame with a water
mist added to the oxidizer jet. In this study, also the extinction behavior of the flame, in
dependence of the amount of water added, is studied.
In Chap. 9, the computer code is applied to a turbulent ethanol-air spray flame in which
the fuel, ethanol, is provided in liquid form.
Finally, in Chap. 10, a summary summary of the research work and results is given, an
and possibly future work on the research subject is outline.
At the end of the thesis, three appendices provide additional information.
1
MPI stands for Message Passing Interface, OpenMP for Open Multi-Processing.
2
CPU stands for Central Processing Unit, GPU for Graphics Processing Unit.
7

Chapter 2
Governing Equations for Two-Phase
Flows
2.1 Notation
In the present work, two-phase flows and flames are considered that are special in that
they comprise a gasphase (phase 2 or phase g) and a liquid phase (phase 1 or phase l).
Specifically sprays are considered, i.e., the gasphase is the so-called carrier phase in which
a huge number of small liquid drops – also called droplets – are carried.
Drops are taken as single component drops. The gasphase is taken as a multicomponent,
chemically reactive ideal-gas mixture. In particular, the gasphase comprises K g chemical
species that participate in I chemical reactions. The mass fractions are summarized in a
vector pY 1 , ..., Y K q.
Independent variables are the time, t, and the the three components x i of the position
vector x “ px 1 , x 2 , x 3 q.
Be the aggregate state either gaseous or liquid, a mass density will be denoted by ρ, and
a velocity component by V i , i “ 1, 2, 3. Generally, tensor notation will be used for vectors
and 2nd-order tensors. Temperature will be denoted by T .
In the entire formulation, effects of gravity will be neglected.
8

2.2 Deterministic Formulation
2.2.1 Gas-Phase Conservation Equations
Gasphase overall-mass continuity is governed by
Bρ g
Bt ` B`ρ g V g,i˘
Bx i
“ S m , (2.1)
where S m represents the mass per unit volume and time gained by the gasphase from the
liquid phase, or vice versa.
Gasphase species-mass conservation is
ρ g
DY g,α
Dt
“ ´Bj g,α,i
Bx i
` w g,α ` S m,α , (2.2)
where α “ 1, ..., K g . In (2.2) and below, j α,i represents the diffusion flux of chemical
component α in the i-th coordinate direction, w g,α its net rate of production due to
chemical reaction, and S m,α its mass per unit volume and time gained due to evaporation,
or lost by condensation.
Gasphase-momentum conservation is
ρ g
DV g,j
Dt
“ ´ Bp
Bx i
` Bτ g,ij
Bx j
` S v , (2.3)
where S v denotes the momentum exchange term described in detail further below, in
Chap. 2.2.3, and where p and τ g,ij are the pressure and the viscous part of the gas-phase
stress tensor, respectively.
Gasphase-energy conservation can be written as
c g,p ρ DT g
Dt “ ´Bq g
Bx i
´
K
ÿ g
α“1
h g,α w g,α ` S e , (2.4)
where S e denotes the energy exchange term described in detail further below, in Chap.
2.2.3, and where h g,α and w g,α denote the specific enthalpy and mass rate of production,
respectively, of species α. Dufour effect and radiative heat loss [45, 97] are assumed to be
negligibly small. As a consequence, the heat flux q g can be expressed as
q g,i “ ´λ g
BT g
Bx i
`
9
K
ÿ g
α“1
h g,α j g,α,i , (2.5)

with – assuming Fick’s law – j g,α “ ´ρ g D g,α ∇Y g,α . Note that the second term on the
r.h.s. of (2.5) represents the energy transported by species mass diffusion.
2.2.2 Liquid-Phase Conservation Equations
Liquid phase overall-mass continuity is governed by
Bρ l
Bt ` B`ρ l V l,i˘
Bx i
“ ´S m , (2.6)
where the r.h.s term represents the mass lost per unit volume an time to the gas phase
due to the evaporation.
Liquid phase species-mass conservation equation is
ρ l
DY l,α
Dt
“ ´ B
Bx i
pρ l D l,α q BY l,α
Bx i
´ S m,α . (2.7)
Liquid phase-momentum conservation is
ρ l
DV l,i
Dt
“ ´ Bp
Bx i
` Bτ l,ij
Bx j
´ S v , (2.8)
where the last term of r.h.s represents the momentum lost due to evaporation per unit
volume and time and F D represents the drag force per unit volume and time exerted from
the liquid phase on the gas phase derived as Eq. (2.15).
Liquid phase-energy conservation can be written as
c p,l ρ l
DT l
Dt “ ´ Bq l
Bx i
´ S e , (2.9)
where q l denotes the heat flux for liquid phase and S e is expressed in Eq.(2.16).
2.2.3 Phase Exchange Terms
After having presented and defined in Chaps. 2.2.1 and 2.2.2 the governing equations
for a general two-phase flow, respectively, and the associated physical quantities, we now
specify the phase exchange terms S m , S v and S e for the special case that the two-phase
flow is a spray.
10

The exchange of overall mass between the two phases, S m , is is related to the exchange
of species mass, S m,α , by
S m :“
Kÿ
S m,α , (2.10)
α“1
where – for species-mass exchange – S m,α is given by
S m,α “ 9m α
V . (2.11)
Here, and below, V denotes a finite volume that is sufficiently small for a differential
formulation of the underlying process – here overall mass conservation – to be applied.
The quantities 9m α denote the mass gained per unit time by the gasphase from the liquid
component α, or vice versa, in the finite volume V. Specifically, if 9m α ą 0, evaporation
of species α takes place at the liquid surface, and if 9m α ă 0, condensation.
For the special case of a single-component liquid fuel – this is the case considered in the
present thesis – the overall mass per unit volume and time gained by the gasphase from
the liquid phase can be expressed as
S m “ 9m V “ ´ 1
V
ÿN l
d“1
9m d . (2.12)
Here 9m denotes the mass gained per unit time by the gasphase from the liquid phase in
the finite volume V. In the second equality of (2.12), and below, N l denotes the number
of real, physical drops in volume V, i.e.,
N l “ n l V (2.13)
where n l denote the local and instantaneous droplet number density.
Momentum exchange between the two phase is described by the exchange term S v which
is the sum of the momentum gained by the gasphase from the liquid phase per unit volume
and time, 9mV l {V, the drag force per unit volume and time exerted from the liquid phase
on the gas phase, F D {V, and of a term ´ 9mV g {V that results by formulating momentum
conservation in terms of a primitive variable rather than a conservative variable, i.e.,
S v “ pV l ´ V g q 9m V ´ FD
V , (2.14)
where 9m{V is given by (2.12). The quantity F D denotes the drag force experienced by
11

the gasphase, exerted from the liquid phase, in the finite volume V. In terms of the drag
force experienced by the droplets in V,
ÿN l
d“1
F d D ,
F D can be written as
F D “ ´
ÿN l
d“1
F d D . (2.15)
Energy exchange between the two phase is described by the exchange term S e which
represents the energy gained by the gasphase from the liquid phase. Specifically, for a
spray S e can be written as
S e “ ´
ÿN l
d“1
where 9 Q d is the heat gained by a drop in finite volume V. 1
9Q d
V , (2.16)
2.2.4 Phase-Indicator Function
In separated-flow models of two-phase flow, the instantaneous location of the interface
is conveniently described by level surfaces [82] βpx, tq is constant with the particular
surface βpx, tq “ β I corresponding to the instantaneous location of the interface. For the
two-phase spray which is the special case from general two-phase flow, Fig. 2.1 shows
schematically. Those constant value of β I is used to distinguish the fluid phases with
Figure 2.1: Schematic diagram of two-phase spray.
1
Thus, the heat lost by the d-th drop is ´ 9Q d .
12

espect to time and physical space. At certain time and position, the condition of β px, tq ă
β I represents liquid phase and β px, tq ą β I for gas phase. At the interface, where β “ β I ,
the level surface is governed by The governing equation for phase-indicator function θ k
for each phase k – in the present work, k “ 1 denotes liquid phase and k “ 2 denotes
gasphase – can be written as
Bθ k
Bt ` V k ¨ ∇θ k “ Π k , (2.17)
where Π k is defined as
Π k “ `´1˘k`1
Vk,r a , (2.18)
where V k,r denotes the normal component of the relative velocity V k,r :“ `V I ´ V k˘
¨ n1
and interfacial area concentration a is defined as
a :“ |∇β|δ`β ´ β I˘ . (2.19)
With Eq.(2.17), two-phase overall-mass continuity is governed by
and two-phase momentum conservation is
and two-phase energy conservation is
B
Bt pθ kρ k q ` B pθ k ρ k V k,j q “ ρ k Π k , (2.20)
Bx j
B
Bt pθ kρ k V k,i q ` B
Bx j
pθ k ρ k V k,i V k,j q
“ θ k
ˆBτk,ij
Bx j
B
Bt pθ kc p ρ k T k q ` B
Bx j
pθ k c p ρ k V k,j T k q
and two-phase species conservation is
“ θ k
ˆBqk
Bx i
˙
´ θ k
Kÿ
´ Bp ˙
k
` ρ k V k,i Π k , (2.21)
Bx j
α“1
B
Bt pθ kρ k Y k,α q ` B
Bx j
pθ k ρ k V k,j Y k,α q
h k,α w k,α ` ρ k c p T k Π k , (2.22)
α
˙ ˆBJk,i
“ ´θ k ` θ k ρ k w k,α ` ρ k Y k,α Π k . (2.23)
Bx i
13

In the first term of r.h.s of conservation equation, θ k is located outside of differential term.
For example, the first term of r.h.s of momentum conservation equation can be expressed
as
θ k
ˆBτk,ij
Bx j
2.2.5 Field Equations
´ Bp ˙
k
“ Bθ kτ k,ij
Bx j Bx j
´ Bθ kp k
Bx j
´ pτ k,ij ´ p k q Bθ k
Bx j
. (2.24)
Heretofore flow field of both phases have been considered separately with phase indicator
θ k with same definition as k “ 1 for liquid phase and k “ 2 for gasphase. Along with
both phases flow field, whole flow field needs to be defined as
φ :“
2ÿ
θ k pβq φ k . (2.25)
With this general form, the field variables at any `x, t˘ are also defined as
and
ρ :“
V :“
ψ :“
k“1
2ÿ
θ k pβq ρ k , (2.26)
k“1
2ÿ
θ k pβq V k , (2.27)
k“1
2ÿ
θ k pβq ψ k
. (2.28)
k“1
In terms of ρ, V and ψ defined in Eqs.(2.26) - (2.28), the conservation equations for
two-phase flow can be deterministically described as following.
Now overall-mass continuity can be simply derived that summation of Eq.(2.20) thorough
k “ 1 and k “ 2 is expressed as
And momentum conservation is
B
Bt pρV iq ` B pρV i V j q
Bx i
“ Bτ ij
Bx j
´ Bp
Bx j
´
B
Bt ρ ` B pρV j q “ 0 . (2.29)
Bx j
2ÿ
k“1
"
pτ k,ij ´ p k q Bθ k
Bx j
´ ρ k V k,i Π k
*
, (2.30)
where τ ij :“ ř 2
k“1 θ kτ k,ij and p :“ ř 2
k“1 θ kp as defined at Eq.(2.25).
14

The energy conservation is
B
Bt pc pρT q ` B pc p ρV j T q
Bx j
“ Bq
Bx i
´
Kÿ
h α w α ´
α“1
2ÿ
k“1
"ˆBqk
Bx i
˙ Bθk
Bx j
´ ρ k c p T k Π k
*
. (2.31)
And the species conservation is
B
Bt pρ kY α q ` B pρ k V j Y α q
Bx j
“ ´BJ α i
Bx i
` ρw α ´
2ÿ
k“1
2.3 Probabilistic Formulation
2.3.1 Two-Phase Joint PDF
Following Rumberg [79, 108], we introduce the joint random vector
"ˆBJ
α
k,i
Bx i
˙ Bθk
Bx j
´ ρ k Y k,α Π k
*
. (2.32)
Z :“ pV , ψq , (2.33)
where V denotes the 6-dimensional velocity random vector pV 1 , V 2 q, and ψ the 2 pK ` 1qdimensional
scalar random vector pψ 1
, ψ 2
q where the phase subscription 1 denotes liquid
phase and 2 denotes gasphase. The two-phase joint PDF f Z,β of Z and β can be written
as [79, 108] 2 ˆρ f Z,β pẐ, ˆβ; x, tq “ ρ 1 g˚ppẐ, ˆR, ˆβ; x, tq ` G 2 pẐ, ˆβ; x, tq . (2.34)
In (2.34) and below, the subscript 1 identifies the liquid phase, the subscript 2 the gaseous
phase. On particular, where the subscript p is used the indicate that the liquid phase is
present in form of a spray. 3 In (2.34), G 2 is the gas-phase density function which, in terms
2
ˆρ f Z,β pẐ, ˆβ; x, tq “ G 1 pẐ, ˆβ; x, tq ` G 2 pẐ, ˆβ; x, tq
“ ρ 1 g˚1 pẐ, ˆr, ˆβ; x, tq ` G 2 pẐ, ˆβ; x, tq
“ ρ 1 g˚p pẐ, ˆR, ˆβ; x, tq ` G 2 pẐ, ˆβ; x, tq
3
A spray is one type of two-phase flow. It involves a liquid as the dispersed or discrete phase in the
form of droplets or ligaments and a gas as the continuous carrier phase – see, e.g., [86], pp. 1.
15

of gas-phase averaged density
¯ρ 2 px, tq “ 1
xθ 2 y
8ż
8ż
´8 ´8
θ 2 p ˆβqρpẐ, ˆβqfpẐ, ˆβq dẐd ˆβ “ xθ 2ρy
xθ 2 y , (2.35)
the void fraction
¯θ 2 px, tq “
8ż
8ż
θ 2 p ˆβqfpẐ, ˆβq dẐd ˆβ (2.36)
´8 ´8
and the gas-phase Favre and phase PDF
rf 2 pẐ, ˆβ; x, tq “ ρpẐ, ˆβq θ 2 p ˆβq
¯ρ 2 px, tq ¯θ 2 px, tq fpẐ, ˆβq (2.37)
is defined as
G 2 :“ ¯ρ 2 ¯θ2 r f2 . (2.38)
The gasphase density function G 2 is governed by [108]
BG 2
Bt ` ˆV BG 2
i “ ´ B `@
DZ{Dt| Ẑ,
Bx ˆβ D G 2˘
i BẐi
´ B `@
Dβ{Dt|
B ˆβ
Ẑ, ˆβ D G 2˘
` δp ˆβ ´ β I q @ Dβ{Dt|Ẑ, ˆβ D G 2 . (2.39)
The function g˚p denotes the liquid-phase DF. In particular, R denotes the drop radius
which is a random variable with corresponding sample-space variable ˆR.
It is important to note that (2.34) is valid only for a single-component liquid. 4 In case of
a multi-component liquid, a suitable generalization of (2.34) will have to replace (2.34).
4
If, physically, the liquid consists of only a single component, mathematically this situation corresponds
to K liquid components where those K ´ 1 components that, in addition, are present in the
gaseous phase in non-zero concentrations attain strictly zero concentrations in the liquid phase.
16

2.3.2 Gas-Phase DF-Transport Equation
At first, we consider gas-phase for k “ 2 with (2.39).
BG 2
Bt ` ˆV BG 2
i “ ´ B `@
DZ{Dt| Ẑ,
Bx ˆβ D G 2˘
i BẐi
´ B `@
Dβ{Dt|
B ˆβ
Ẑ, ˆβ D G 2˘
Integration of (2.40) over the ˆβ for k “ 2 yields
` δp ˆβ ´ β I q @ Dβ{Dt|Ẑ, ˆβ D G 2 . (2.40)
G c pẐq “ ż 8´8
G 2 pẐ, ˆβq dβ , (2.41)
the gas-phase marginal-DF transport equation can be derived as
BG c
Bt ` ˆV i
BG c
Bx i
“ ´ B
B ˆV i
`@
DVi {Dt| ˆV , ˆψ D G c˘
´ B `@
B ˆψ
Dψα {Dt| ˆV , ˆψ D G c˘
α
` ĝ M G c , (2.42)
where
ρ1
ĝ M “ ´
ρ 2 p ˆψq
ż 8
0
B F
DR
Dt | ˆV , ˆψ, ˆR 4π ˆR
` 2 f W ˆR| ˆV L , ˆψ L˘d ˆR . (2.43)
The density function f W occurring in (2.43) is defined in (2.50).
2.3.3 Liquid-Phase DF-Transport Equation – Williams’ Spray
Equation
5
Starting from a suitably defined liquid-phase density function g˚1 pẐ, ˆβ, ˆr; x, tq – see [79,
5
For a single-component, ideally incompressible liquid, the density is a true constant that is denoted
by ρ 1 . Hence,
G 1 :“ ρ 1 g 1 pẐ, ˆβq , (2.44)
where g 1 :“ ¯θ 1
r f1 . According to Rumberg [108], G k :“ ¯ρ k ¯θk r fk and g k :“ ¯θ k
r fk , i.e. g 1 “ ¯θ 1
r f1 .
17

108] for details of that definition 6 – and defining
g˚p pẐ, ˆβ, ˆR; x s , tq :“
ż 8
0
g˚1 pẐ, ˆβ, ˆr; x, tq δpˆr ´ ˆRq dˆr , (2.46)
where R denotes droplet radius (a random variable whose sample-space variable is denoted
by ˆR), the DF-transport equation
Bg˚p
Bt ` ˆV i
Bg˚p
Bx i
“ ´ B
"B DZj
BẐj Dt
"B F
Dβ
|Ẑ, ˆβ, ˆR
Dt
"B F
DR
|Ẑ, ˆβ, ˆR
Dt
´ B
B ˆβ
´ B
B ˆR
F *
|Ẑ, ˆβ, ˆR g˚p
*
g˚p
g˚p
*
(2.47)
can be derived. Integration of (2.47) over all possible ˆβ yields
Bg p
Bt ` ˆV Bg p
i “ ´ B "B F *
DVi
Bx i B ˆV i
Dt | ˆV , ˆψ, ˆR g p
´ B "B F *
Dψα
B ˆψ α
Dt | ˆV , ˆψ, ˆR g p
´ B
B ˆR
"B DR
Dt | ˆV , ˆψ, ˆR
F
g p
*
,
(2.48)
where
g p pẐ, ˆR; x s , tq :“
ż
g˚p
8´8
pẐ, ˆβ, ˆR; x s , tq d ˆβ “ ¯θ 1 px s , tqf r p pẐ, ˆR; x s , tq . (2.49)
The second equality in (2.49) is derived in detail in [108, 79].
Upon multiplying g p with the local and instantaneous number density npx, tq, we obtain
6
This DF is governed by the DF-transport equation
"B F *
Bg˚1
Bt ` ˆV
Bg˚1 B DZj
i “ ´
Bx i BẐj Dt |Ẑ, ˆβ, ˆr g˚1
´ B "B F *
Dβ
B ˆβ Dt |Ẑ, ˆβ, ˆr g˚1
´ B "B F *
Dr
Bˆr Dt |Ẑ, ˆβ, ˆr g˚1
B F B F
´ δp ˆβ Dβ
´ β I q
Dt |Ẑ, ˆβ, Dr
ˆr g˚1 `
Dt |Ẑ, ˆβ, Bˆθ 1
ˆr g˚1
Bˆr .
(2.45)
18

the density function
f W p ˆV , ˆψ, ˆR; x, tq :“ npx, tq g p p ˆV , ˆψ, ˆR; x, tq (2.50)
that obeys the so-called William’s Spray Equation – see [97], pp. 449 – viz.,
where
and
Bf W
Bt
` ˆV Bf W
i “ ´ B "B F *
DVi
Bx i B ˆV i
Dt | ˆV , ˆψ, ˆR f W
´ B "B F *
Dψα
B ˆψ α
Dt | ˆV , ˆψ, ˆR f W
´ B "B F *
DR
B ˆR Dt | ˆV , ˆψ, ˆR f W
` Q ` Γ , (2.51)
Q “ 9n b
n f W , (2.52)
Γ “ 9n c
n f W (2.53)
represent the time rate of change of formation/desctruction 7 and droplet collision, 8 respectively;
9n b and 9n c denote the corresponding time rates of change of the number density
n.
2.4 Pressure-Correction Equation
To calculate pressure, a SIMPLE-like pressure correction algorithm [79, 80] is proposed
and taken. The original SIMPLE (Semi-Implicit Method for Pressure Linked Equations)
method is proposed by [63]. Original algorithm is the purpose that the Eulerian pressure
field is corrected through combination of momentum equation and continuity equation.
The derivation of a pressure correction equation is based on the overall mean mass conservation
equations as
where
Bxρy
Bt
` BxρV iy
Bx
“ 0 , (2.54)
xρy “ xθ l yxρ l y ` xθ g yxρ g y , (2.55)
xρV i y “ xθ l yxρ l V l,i y ` xθ g yxρ g V g,i y . (2.56)
7
A droplet may be destroyed, e.g., by coalescence of drops. Formation of droplets may occur, e.g.,
by droplet breakup or by droplet nucleation.
8
Collision of a droplet may occur with one or several other droplets or with a wall.
19

Detailed derivation of a pressure equation will be presented below in Chaps. 5.2.
xpy n`1 “ xpy n ` α p`p1n ´ p 1 n´1˘
(2.57)
where α p is a relaxation factor which have been taken in the range 0.1 ď α p ď 0.3 [79, 80].
For a fixed time step pressure corrector p 1 can be separated as
p 1 “
`p
1˘
` `p 1˘S (2.58)
M
where M denotes correction term for unsatisfied mass conservation as usually contained
original SIMPLE method and S for statistical error. In the course of the iterations for the
time step, p 1 M approaches zero. In contrast, p1 S hardly varies between successive iteration
1˘n 1˘n´1
1˘n´1
and time steps. With sufficiently large time step n,
`p «
`p and
`p « 0,
S
S
M
therefore p 1n ´ p 1n´1 1˘n
«
`p . The pressure correction specified in (2.58) thus corrects
M
towards satisfaction of mass conservation and, at the same time, filters out the influence
of statistical fluctuations between successive iteration and time steps [79, 80].
20

Chapter 3
Computational Approach
3.1 Two-Phase Particle Method
For either the gaseous and the liquid phase, we distinguish between molecules, fluid particles
and particles. In addition, since in the present work a spray is considered that
represents a special form of two-phase flow in which the liquid phase is represented exclusively
by a very large number of drops, also drops – which due to there small size are
also termed droplets – need to be considered,
The numerical solution method selected in the present is a so-called particle method. In
the context of particle methods, a particle is representative for a huge number of fluid
particles having – in a numerical sense – nearly the identical properties, Specifically, for the
spray problems considered in the present thesis, both gasphase particles and liquid-phase
particles are considered. Thus, a gasphase particle is representative of a huge number
of gaseous fluid particles having nearly identical velocity and scalar properties such as
temperature and concentrations. Similarly but yet differently, a liquid-phase particle is
representative of a huge number of liquid drops having nearly identical velocity and scalar
properties such as temperature, concentrations and droplet size.
In addition to carrying a velocity (3 components), temperature and K concentrations,
a gasphase particle also carries a size. Specifically, since we assume that particles are
spherical, the particle size is taken as a radius. This radius, together with the density
attached to a particle, is used to define the mass attached to a particle. For instance, the
mass attached to to gasphase particle number p is
ż
ż R
p
g
m p gptq “ ρ p gptq dV “ 4 π pRgq p 2 ρ p gptq dR , (3.1)
Vg
p 0
21

and ρ p g denote the radius, volume and density of pasphase particle p, re-
where Rg, p Vg
p
spectively.
Whilst it seems natural to attach to a liquid-phase particle a size (because a droplet has
a size), on first glance its seems surprising to attach a size also to a gasphase.
The essence of such a method consists in [69] and it is limited for gasphase.
The number of liquid particles is denoted P l , the number gaseous particles P g , p “ 1, ..., P l
and p “ 1, ..., P g . While the number of gaseous particles can be extended up to infinity to
represent every single gaseous molecule, the number of liquid particles has upper bound
as number of physical drops.
3.1.1 Particle Method for Liquid Phase
Here and below, for a liquid particle p, the particle mass is denoted by m p l . Assuming
spherical particles,
m p l
“ 4 3 π ρp l Rp 3 l
. (3.2)
Accordingly, the time rate of change of the mass of liquid particle p is
9m p l
“ 4π ρ p l Rp 2 dR p l
l
dt
, (3.3)
where R p l
and ρ p l
denote the radius and density of liquid phase particle p, respectively.
In terms of the particle masses 9m p l , we can write the interfacial mass exchange term S m
specified earlier in Eq. (2.12) as
S m “ ´ 1
V
ÿN l
d“1
9m d « ´ 1
V
ÿP l
p“1
9m p l . (3.4)
Here the first, exact equality has been taken from (2.12), and the second, approximate
equality stems by replacing physical drops by liquid particles.
With 9m p l
, the mass of species is written as
S m,α “
ÿP l
p“1
Y p
l,α 9mp l , (3.5)
In terms of droplet radius rather than diameter, the so-called D 2 -law [28, 49, 93, 97] can
be written as
dR p l
dt
“ ´Kp
R p l
22
, (3.6)

where the evaporation constant K p is given by
K p “
8λ g
ρ p l c ln pB T ` 1q , (3.7)
p,g
where λ g , c p,g and B T denote the thermal conductivity, the specific heat capacity and the
Spalding heat transfer number, respectively. Here B T is ,
B T “ c p,g pT g ´ T boil q
L
, (3.8)
where L denotes is the latent heat of vaporization at mean temperature pT g ` T boil q {2.
The interfacial momentum exchange term S v specified earlier in Eq. (2.14) can be written
as
With
S v “ 1 V
ÿN l
d“1
S d v « 1 V
ÿP l
p“1
S p v . (3.9)
S p v “ pxV l,i y ´ xV g,i yq 9m p l ´ F p D , (3.10)
where
with the drag coefficient
and the Reynolds number
F p D “ 3 p
ρ g C D
8 ρ p l
R |xV g,iy ´ V p
p l,i |`xV g,i y ´ V p
l,i
˘mp
l , (3.11)
C p D “ 24
Re p l
˜
Re p l
“ 2xρ g yR p l
1 ` Rep l
6
2{3
¸
, (3.12)
|xV g,i y ´ V p
l,i |
µ g
. (3.13)
In terms of the liquid particle, we can write the interfacial energy exchange term S e
specified earlier in Eq. (2.16) as
S e “ 1 V
ÿN l
d“1
9Q d « 1 V
ÿP l
p“1
9Q l
p
, (3.14)
with 9 Q l
p
is defined as
9Q l
p
“ 9m
p
l
˘
p
#¯c p,g`Tg ´ T p
l
B p T
˘+
´ L`T p
l
. (3.15)
23

˘
Here B p p
T
denotes the Spalding heat transfer number and L`Tl
is the latent heat of
vaporization at droplet surface temperature T p
l
. The overbar indicates that the value of
the specific heat is to be calculated according to the so-called 1/3 rule [2, 36, 102] – see
Chap. 4.3.2.
The equation of motion for the p-th liquid particle can be written as
dV p
l,i “ ´ 1
ρ p l
˙ ˆBxpy
dt ´ 1
Bx i ρ p Svdt p , (3.16)
l
where ρ p l
denote the density of liquid phase particle p and S p v has been defined in Eq.
(3.10).
If we assume that the liquid particle has uniform temperature and only single component,
the energy equation for the p-th liquid particle can be reduced as
dT p
l
dt
“
9Q l
p
4
3 πρp l c p,l p R p 3 , (3.17)
l
where the energy exchange term 9 Q l
p
has been defined in Eq. (3.15).
3.1.2 Particle Method for Gas Phase
After formulating the overall exchange terms S m , S v and S e in terms of liquid-particle
properties, the task now is to define the corresponding gasphase-particle exchange terms
such that in each finite volume V evaporation or condensation, respectively, obey mass,
momentum and energy conservation. This task is accomplished by allocating the overall
exchange quantities S m , S v and S e to the gasphase-particles, i.e., by defining
S p g,m “ ´ 1
P g V
Sg,m,α p “ 1 ÿP l
P g
p“1
ÿP l
p“1
9m p l , (3.18)
Y p
l,α 9mp l , (3.19)
S p g,v “
S p g,e “
1
P g V
1
P g V
ÿP l
p“1
ÿP l
p“1
S p v , (3.20)
9Q p , (3.21)
24

where the subscript g identifies gasphase particles.
In terms of the distributed mass exchange term, the species-mass conservation for the
p-th gas particle can be written as
dY
ρ p g,α
p
g
dt
“ ´Bjp g,α,i
Bx i
` w p g,α ` S p g,m,α , (3.22)
where α “ 1, ..., K g . In (3.22), w p g,α denotes net rate of production due to chemical reaction
for the p-th gas particle and the diffusion flux j p g,α,i will be modelled by suitable model
represented in Chap. 4.2. And Sg,m,α p has been defined in Eq. (3.19).
In terms of the distributed momentum exchange term, The equation of motion for the
p-th gas particle can be written as
dV p
g,i “ ´ 1 Bp
ρ p dt ` 1 Bτ p g,ij
g Bx i ρ p dt ` 1
g Bx j ρ p Sg,vdt p , (3.23)
g
where the viscous part of the gas-phase stress tensor for the p-th gas particle τ p g,ij will be
modelled by suitable model represented in Chap. 4.1. And Sg,v p has been defined in Eq.
(3.20).
The energy conservation for the p-th gas particle can be written as
dT
c p g,p ρ p g
p
g
dt
“ ´Bqp g
Bx i
´
K
ÿ g
α“1
h p g,αw p g,α ` S p g,e , (3.24)
where q p g, h g,α and w p g,α denote the heat flux, the specific enthalpy and mass rate of
production for the p-th gas particle, respectively. And S p g,e has been defined in Eq. (3.21).
25

3.2 Expectations and Moving Averages
3.2.1 Expectations
In [30, 69, 79] Reynolds averaged expectations and Favre averaged expectations were
defined for single-phase flows. 1
For two-phase flows, for any physical quantity φ, Reynolds averaged expectations can be
written as [79, 80]
xφy “
8ż
8ż
8ż
ˆφf` ˆV , ˆψ, ˆβ˘d ˆV d ˆψd ˆβ
´8 ´8 ´8
“ xθ l yxφ|β ă ˆβ I y ` xθ g yxφ|β ą ˆβ I y
« 1 ÿP l
xθ l yφ p l
P ` 1
l
p“1
P
ÿ g
xθ g yφ p g . (3.27)
P g p“1
1
For single-phase flow, for any physical quantity φ, Reynolds averaged expectations xφy can be
calculated as [30, 69, 79]
xφy “
8ż
« 1 P
8ż
´8 ´8
ˆφf` ˆV , ˆψ˘d ˆV d ˆψ
Similarly, Favre averaged expectations can be calculated as [30, 69]
rφ “
“
8ż
8ż
Pÿ
φ p . (3.25)
p“1
´8 ´8
8ż
« 1 P
8ż
´8 ´8
Pÿ
p“1
ˆρ ˆφ
xρy f` ˆV , ˆψ˘d ˆV d ˆψ
ˆφ r f` ˆV , ˆψ˘d ˆV d ˆψ
ρ p φ p
xρy . (3.26)
26

Similarly, Favre averaged expectations for two-phase flows, can be written as
rφ “
8ż
8ż
8ż
´8 ´8 ´8
ˆρ ˆφ
xρy f` ˆV , ˆψ, ˆβ˘d ˆV d ˆψd ˆβ
“ xθ ly
xρ l y xρφ|β ă ˆβ I y ` xθ gy
xρ g y xρφ|β ă ˆβ I y
« 1 ÿP l
P l
3.2.2 Moving Averages
p“1
xθ l yρ p l φp l
xρ l y
` 1
P g
P
ÿ g
p“1
xθ l yρ p gφ p g
xρ g y
. (3.28)
Generally a distinction is made between ensemble averaging to obtain stochastic mean
values, time averaging to obtain temporal mean values, and spatial averaging to obtain
spatial mean values. Spatial averaging leads to LES-like 2 formulations of turbulent flows
and hence are not considered in the present work.
For flows that are steady in the
temporal mean, ie., for which the mean values do not vary with time, ensemble averaging
and temporal averaging leads to identical mean values provided that, in practice, the
number of particle as well as the number of time steps is sufficiently large.
Since in the current work in the temporal mean unsteady numerical simulations are performed
to approach a statistically and hence temporally steady solution, it is advantageous
to monitor that approach by considering a so-called moving average. This averaging procedure
is similar to time averaging but rather than averaging over a large (theoretically
infinite) number of time steps averaging is carried out backwards from timestep n only
over the last n back steps, i.e., over the backward period t pnq ´ t pn´n backq . As a numerical
solution evolves, taking moving averages smoothes out temporal fluctuations thereby facilitating
both the display of graphical or even animated temporal solutions as well as the
identification of approaching an in the mean steady state, see, e.g., [7, 35]
For instance for a non-constant time step size,
∆t pnq “ t pnq ´ t pn´1q (3.29)
a moving average is calculated according to
¯φ pnq “
ř n
i“n´n back
xρy piq ∆t piq φ piq
ř n
i“n´n back
xρy piq ∆t piq (3.30)
In the present work, a typical value is n back “ 100.
2
LES stands for Large-Eddy-Simulation.
27

Chapter 4
Models
To evaluate the various conditional means occurring on the right-hand sides of (2.42) and
(2.51), respectively, certain local and instantaneous terms require modeling. Specifically,
these are the terms DV g,i {Dt, Dψ α {Dt, and DR l {Dt, giving rise to a so-called velocity
model, mixing model, and evaporation model, respectively. Since we will select a velocity
model that includes the turbulent kinetic energy k and the turbulent dissipation rate ɛ, a
suitable two-equations turbulence model will also be required.
Since in the computations a Lagrangian approach will be adopted in which so-called
Monte-Carlo particles are tracked, the Eulerian derivatives following the motion of a
fluid particle, viz., DV g,i {Dt, Dψ g,α {Dt, and DR l {Dt, are replaced by corresponding Lagrangian
derivatives following individual Monte-Carlo particles, viz., DV ˚
g,i{Dt, Dψ˚g,α{Dt,
and DR˚l {Dt.
4.1 Velocity Model
In the present work, as model for DV ˚
g,i{Dt the so-called Langevin equation
dVg,i ˚ “ ´ 1 B xpy
dt `
ρ g Bx rV BV r ´
g,j
r,j dt ` G˚ij Vg,j ˚ ´ g,j¯
i Bx rV dt ` B ij dW j , (4.1)
i
where mean slip velocity r V r “ r V l ´ rV g and G˚ij are given by [58]
«
G˚ij “ ´ 1 δ
T ˚L,K
ij ´
1
T ˚L,‖
ff
´ 1 r
T ˚L,K
i r j , (4.2)
28

where the integral timescales T L , T ˚L,K and T ˚L,‖
T L “
ˆ1
2 ` 3
˙´1
4 C k
0
ɛ , (4.3)
T ˚L,K “
T ˚L,‖ “
T L
b
, (4.4)
1 ` 4β 2 V r r
2
2k{3
T L
b , (4.5)
1 ` β 2 V r r
2
2k{3
with r i “ U r,i {| r V r |.
And B ij are given by
B ij “
d
ɛ
ˆ
˙
C 0 b i
r 2 k{k `
3 pb r ik{k ´ 1q , (4.6)
with b 1 “ T L {T ˚L,‖ and b 2 “ T L {T ˚L,K .
In the present work, the effect of slip velocity has been neglected. Hence, (4.1) reduces to
dVg,i ˚ “ ´ 1 ˆBxpy
ˆ1
´ S v˙
dt `
ρ g Bx i
2 ` 3 ˙ ɛ
´
4 C 0 Vg,i ˚ ´ g,i¯
k
rV dt ` pC 0 ɛq 1{2 dW i , (4.7)
which was used earlier for [69]. 1 Here, in addition to the quantities introduced or defined
above, C 0 is a model constant with numerical value 2.1 [3, 69]. The quantity W i denotes
a so-called Wiener process [69, 73], i.e., is a stochastic process, normally distributed with
mean zero and variance dt, and with the property
dW i ptq “ W i pt ` dtq ´ W i ptq , (4.9)
i “ 1, 2, 3. The quantities k and ɛ were introduced above – see page (36). In the present
work, the fields of k and ɛ will be computed by use of a so-called k-ɛ turbulence model –
see Chap. 4.4.
1
The Langevin equation (4.7) is a special form of the more general Langevin equation
dV ˚
g,i “ ´ 1
ρ g
B xpy
Bx i
dt ` G ij`V ˚
g,j ´ rV g,j˘dt ` pC0 ɛq 1{2 dW i . (4.8)
For details and discussions related to (4.8), references [32, 69] should be consulted.
29

4.2 Mixing Models
For the gaseous phase of two-phase flow, locally and instantaneously, species mass conservation
is
Dψ α
Dt
“ ´Bj α,i
Bx i
` Spψ α q ` S m,α , (4.10)
α “ 1, ..., K. Assuming in (4.10) Fick’s law of diffusion with a constant diffusion coefficient
D α for species α, the composition PDF
Df ψ
Dt “ ´ B
B ˆψ α
ˆB
D α
B 2 ψ α
Bx 2 i
F
| ˆψ f ψ˙
`
B
B ˆψ α
`Sp ˆψα qf ψ˘
`
B `A
B ˆψ
S m,α | ˆψ
E
f ψ˘
α
(4.11)
is obtained [73]. Since the chemical source term is closed,
AS`ψ α˘|ψα “ ˆψ
E
α “ S` ˆψα˘
. (4.12)
However, the molecular-diffusion term
B
F
B 2 ψ α
D α |
Bx ˆψ 2 α
i
(4.13)
is not closed and hence requires modelling by a so-called mixing model. Various mixing
models are available. Specifically, in the present work the so-called IEM model, Curl’s
model, Curl’s modified model, and the so-called EMST-model have been employed. Subsequently,
these models are presented and discussed.
4.2.1 IEM-Model
IEM stands for Interaction by Exchange with the Mean. In the framework of turbulent
combustion, IEM models have been used extensively, see – e.g. – references [13, 94]. In
the present context, the IEM model can be written as
B
D α
B 2 ψ α
Bx 2 i
F
| ˆψ α “ 1 2 C ɛ `
ψ ˆψ ´ xψy˘
. (4.14)
k
Here k and ɛ have their usual meaning, and C ψ is a model constant with numerical value
2.0, see [73]. With (4.14), the gasphase particle mass conservation equation becomes
Dψ˚α
Dt
“ ´1
2 C ɛ
ψ
`ψ˚α ´ xψ α y˘ ` Spψ˚αq ` S m,α . (4.15)
k
30

4.2.2 Curl’s Model
Curl’s mixing model [11] is based on – from a uniform distribution randomly and independently
selected – particles p and q, 1 ď p, q ď N p , where N p denotes the number of
gasphase particles. For these particles, Curl’s mixing model can be written
ψ˚ppq
α pt ` dtq “ ψ˚pqq
α pt ` dtq “ 1 `ψ˚ppq
α ptq ` ψ˚pqq
α ptq˘ . (4.16)
2
The so-called modified Curl model [38] replaces the constant 1/2 on the r.h.s. of (4.16) by
a randomly selected weight parameter ξ, 0 ď ξ ď 1, such that the mixing model becomes
ψ˚ppq
α pt ` dtq “ p1 ´ ξqψ˚ppq
α ` ξ 1 `ψ˚ppq
α ptq ` ψ˚pqq
α ptq˘ , (4.17)
2
ψ˚pqq
α pt ` dtq “ p1 ´ ξqψ˚pqq
α ` ξ 1 `ψ˚ppq
α ptq ` ψ˚pqq
α ptq˘ . (4.18)
2
The number of randomly selected pairs pp, qq of particles is N mix ,
N mix “ ΩP dt , (4.19)
where Ω denotes the mixing frequency,
Ω “ C ψ
ɛ
k . (4.20)
For Curl’s model, the numerical value of the model constant C ψ is 2.0; for the modified
Curl model it is 2.3 [51].
4.2.3 EMST-Model
EMST stands for Euclidian Minimum Spanning Tree. EMST-Model [89] is an extension
work for multi-dimension scalar mixing based on the mapping closure model [70] which is
implemented for a single scalar mixing. In this model, particle composition is changed by
particle interactions through the edges of a tree which is constructed to have Euclidean
minimum total length. In order to develop the Euclidean Minimum Spanning Tree, it is
required to connect every particle by an edge to at least one neighboring particle. The
spanning property would come by a connection between every particle in the set. And
the sum of Euclidean lengths of the edges in the tree should be minimum.
31

1
2
0
-1
-1 0 1
Figure 4.1: Euclidian minimum spanning tree in two dimension composition space.
1
Euclidean distance between particle p and q in scalar composition is defined as
g
fÿ
dpp, qq “ e K ´
α“1
ψ˚ppq
α
In the present context, the EMST model can be written as
´ ψ˚pqq
¯2
α . (4.21)
dψ˚ppq
α
w ppq
dt
“ ´γ
Nÿ
p´1
ν“1
B ν tpψ˚ppq
α
´ ψ˚pnνq
α q δ p,nν ` pψ˚ppq
α ´ ψ˚pmνq
α q δ p,mν u , (4.22)
where the νth edge of the tree connects the particle pair (m ν , n ν ).
δ represents the
Kronecker delta, therefore δ p,mν and δ p,nν to be zero except when p “ m ν or n ν that they
are equal to one.
The left hand side of (4.22) is evaluated as follows.
1. For each particle p, p “ 1, ..., N p , specify a weight w ppq . For instance, a popular
choice is w ppq “ 1{N p . 2
2 The original meaning of w is importance weight. In EMST-model, w is represented to particles
weight attributed from particles mass ∆m. A classical way is that ∆m is set as constant [23, 69], then
32

2. For node p, determine a tree consisting of all N p nodes and N p´1 edges, ν “ 1, ..., N p .
The edge coefficients B ν are calculated by carrying out the following steps [89].
(a) Assume that edge ν connects particles m and n. Then the tree is partitioned
into three subtrees, namely, the trivial subtree tνu, the subtree T mν
and the
subtree T nν . The subtrees T mν and T nν are defined such that they result from
the overall tree by removing edge ν.
(b) First calculate the sums
W Tmν :“ ÿ
and then obtain the weight of wedge ν,
kPT mν
w pkq , (4.23)
W Tnν :“ ÿ
kPT nν
w pkq , (4.24)
w ν “ minpW Tmν , W Tnν q . (4.25)
(c) Take the edge coefficient B ν as
B ν “ 2w ν . (4.26)
3. The model parameter γ controls the rate of variance decay of the scalars. It is
obtained through an iterative procedure [89] by solving the following equations.
Here γ is given by
¯M pq “ ´ 1
Nÿ
p´1
B ν tδ p,mν δ q,nν ` δ q,mν δ p,nν u . (4.27)
γ
ν“1
1
γ “ τ ř Np
ř Np
ψ k“1
α
ř K
β“1 C ββ
l“1 w pkq2 ψ˚pkq1
¯Mkl ψ˚plq1
α
, (4.28)
where the C αβ , 1 ď α, β ď, K denote the covariances
C αβ “ x pψ α ´ xψ α yq pψ β ´ xψ β yq y (4.29)
and with τ ψ “ τ{C ψ followed as standard modeling assumptions [69]. C ψ is taken
to be 2.0 [89].
all particles importance becomes equal, w ppq “ 1{N since sum of w ppq should be unity in EMST-model
[89, 103].
33

4.3 Evaporation Models
4.3.1 D 2 ´ law Model
One of the classic evaporation models is so-called D 2 ´ law [28, 49, 97]. In this model,
the droplet temperature is spatially uniform and the temperature is assumed to be the
boiling point of the fuel, T d “ T boil [93]. The instantaneous diameter of k-th droplet Dl kptq
has a correlation
2
ptq “ D
k2 l pt0 q ´ Kpt ´ t 0 q . (4.30)
D k l
Here Dl k ptq is the instantaneous diameter of a droplet that entered physical space at time
to and since then has moved along its trajectory. The constant K is given by
K “
8k g
ρ k L c ln pB T ` 1q , (4.31)
p,g
with Spalding heat transfer number B T
B T “ c p,G pT g ´ T boil q
L
, (4.32)
where L is the latent heat of vaporization at mean temperature 1 2 pT g ` T boil q. Differentiation
of (4.30) yields, in terms of the instantaneous droplet radius R l ,
BR k l
Bt
“ ´ K
R k l
. (4.33)
4.3.2 Film Model
Alternatively, to determine the mass vaporization rate of droplet, the so-called film model
is provided by [2, 84]. In film model, it is assumed that the mass and heat exchange
only occurs at thin layer (so-called film) between the droplet surface and the gas. The
instantaneous droplet radius R l which has rate of change, 9R l is written as
dR k l
dt
“ ´
9m k l
4πρ k l Rk l
2 . (4.34)
According to film model [2, 84], 9m k l
is defined as
9m k l “ ´2π ¯ρ k ¯Dk Rl k Sh k˚ ln`1 ` BM˘ k , (4.35)
where D denotes the binary diffusion coefficient. The overbar indicates that the value
of the specific heat is to be calculated according to the so-called 1/3 rule [2, 36, 102].
For example, the averaged density ¯ρ k and the averaged specific heat capacity ¯c k p can be
34

obtained as
¯ρ k “ ρ k l ` 1 ˘ `xρg y ´ ρ k l , (4.36)
3
¯c k p “ c k p,L ` 1 ˘ `cp,g ´ c k p,l . (4.37)
3
To complete Eq. (4.35), the modified Sherwood number Sh˚ is defined as
Sh k˚ “ 2 ` `Sh k 0 ´ 2˘ {F k M , (4.38)
with
and
where
FM k “ `1 ` BM
k ˘0.7 lnp1 ` BM k q , (4.39)
B k M
Sh k 0 “ 2 ` 0.552 Re k1{2 Sc k1{3 , (4.40)
Re k “ 2xρ g y ˇˇV k
l ´ xV g yˇˇ Rk l {µ g , (4.41)
and Schmidt number Sc
Sc k “
The spalding mass transfer number B M is defined as
¯µk g
. (4.42)
¯ρ
k ¯D
k
B k M “ Y F,S ´ Y F,g
1 ´ Y F,S
, (4.43)
where Y F,S denotes the fuel mass fractions of vapor at the droplet surface and calculated
by Clausis-Clapeyron equation [2, 93] as
Y F,S “
M F
M F ` `xpy{p k F ´ 1˘ ¯Mg
, (4.44)
where M F is the molecular weight of the fuel and ¯M g is the mean molecular weight of gas
phase and the partial pressure of the fuel vapor p F is given by
p k F “ A exp`´B{ ¯T k˘ , (4.45)
where
¯T k “ Tl k ` 1 ˘ `Tg ´ Tl
k , (4.46)
3
and the constants A and B are obtained from the Clausis-Clapeyron equation [93].
35

The heat penetrating into liquid phase Q l can be defined as
Q k l “ 9m k l
# ˘
¯cp k`T g ´ Tl
k
BT
k
˘+
´ L`Tl
k , (4.47)
with the spalding heat transfer number B T
B k T “ `1 ` B k M˘φ
´ 1 , (4.48)
where
And the modified Nusselt number Nu˚ is
φ “ c¯
P k ¯ρ k ¯Dk Sh k˚
¯λ k Nu k˚ . (4.49)
Nu k˚ “ 2 ` pNu 0 ´ 2q {F k T , (4.50)
with
¯ Nu 0 “ 2 ` 0.552 Re k1{2 P r k1{3 , (4.51)
where Prandtl number P r defined
P r k “ ¯µk ¯c p
k
¯λ k . (4.52)
Similarly to F M – see Eq. (4.39)
FT k “ `1 ` BT
k ˘0.7 lnp1 ` BT k q . (4.53)
˘
And L`Tl
k is the latent heat of vaporization at droplet surface temperature T
k
l .
Finally, the rate of change of droplet temperature can be written as
B k T
dT k
l
dt
“
Q k l
4
3 πρk l c p,l k R k l
3 , (4.54)
where c p,l is the specific heat for droplet.
4.4 Turbulence Model
Since the mixing model and Langevin models are affected by turbulent effect through
the turbulent time scale τ ” k{ɛ, a suitable approach is required to determine turbulent
kinetic energy k and its dissipation rate ɛ. For simple flow, τ can be specified by assuming
36

it to be uniform [69]. However, most flows needs a general determination of τ. In present
work,the standard k-ɛ model [40, 47, 48] is selected to obtain k and ɛ. The k-ɛ model is
one of the most widely applied two-equations turbulence model, in which model transport
equations are solved for two turbulence quantities k and ɛ. Since the k-ɛ model is widely
used solve turbulence flow problems, it has been improved and modified to overcome its
well known shortcomings, e.g., poor predictions for swirling and rotating flows and certain
unconfined flows, valid only for fully developed turbulent flows and inaccurate near the
wall. Representative variants are The RNG k-ɛ [99] model and Realizable k-ɛ model [83].
Instead of variants, the standard k-ɛ model [40, 48, 47] and its model constants are
implemented in present work with transport equations for k as
and for dissipation rate ɛ
B
Bt pρ Gkq ` B pρ G kV G,i q “ B „ˆ ˙ j
µ ` µt Bk
Bx i Bx j σ k Bx j
B
Bt pρ Gɛq ` B pρ G ɛV G,i q “ B „ˆ ˙ j
µ ` µt Bɛ
Bx i Bx j σ ɛ Bx j
Modelled turbulent viscosity µ t is defined as
And production of k is defined as
`
µ t “ ρ G C µ
k 2
` P k ` P b ´ ρ G ɛ ´ Y M , (4.55)
C 1ɛ
ɛ
k pP k ` C 3ɛ P b q ´ C 2ɛ ρ G
ɛ 2
k . (4.56)
ɛ . (4.57)
P k “ ´ρ G VG,i 1 V G,j
1 BV G,j
Bx i
“ µ t S 2 , (4.58)
with the modulus of the mean rate-of-strain tensor S defined as
S ” a 2S ij S ij , (4.59)
where the mean strain rate S ij
S ij “ 1 2
ˆBVG,j
Bx i
` BV ˙
G,i
. (4.60)
Bx j
37

In the present k-ɛ formulation, effects of buoyancy P b and effects of compressibility on
Turbulence Y m are neglected. And the model constants [48] for standard k-ɛ model are
provided as
C 1ɛ “ 1.44, C 2ɛ “ 1.92, C µ “ 0.09, σ k “ 1.0, σ ɛ “ 1.3
38

Chapter 5
Discretization
5.1 Domain of Integration
In the present work, both Lagrangian particle equations and Eulerian equations are solved
employing the so-called method of fractional steps [69, 100, 101]. To this end, herein the
spatial computational domain is first triangulized, and then control volumes are constructed
from the triangles [63, 75]. Specifically, in the present work control volumes were
constructed on the basis of a Delaunay triangulation [77, 96], T , consisting of N T triangles
as schematically shown in Fig. 5.1.
Figure 5.1: Section of an unstructured triangular mesh with control volumes.
As an example, shown in Fig. 5.1 is a triangle with the vertices i, j, and k. Point C marks
the center of gravity of this triangle. It is seen that the three straight lines running from
39

C to the midpoints of the three triangle edges constitute parts of the control volumes V i ,
V j , and V k around nodes i, j and k, respectively. The whole control volumes V i , V j , and
V k are then made up of “pieces of cake” stemming from triangles centered about i, j and
k, respectively [63, 75].
Following the principle of construction of control volumes just outlined, control volumes
centered about boundary points of the domain can be constructed. Naturally, parts of
such control volumes coincide with particular triangle edges.
5.2 Eulerian Governing Equations
Governing equations in Eulerian form are the gasphase equations governing the pressure
p, and the equations of the k-ɛ turbulence model. The approach to the discretization of
these equation is through linear triangle shape functions. Specifically, each dependent
variable φpx, tq – where φ “ p, k, ɛ, ũ, ṽ, ... – is written in terms of shape functions α φ , β φ
and γ φ , viz.,
φpx, tq “ α φ ptq ` β φ ptqx ` γ φ ptqy , (5.1)
where
α φ ptq “
β φ ptq “
γ φ ptq “
3ÿ
α k φ k ptq , (5.2)
k“1
3ÿ
β k φ k ptq , (5.3)
k“1
3ÿ
γ k φ k ptq , (5.4)
k“1
and where φ k denotes the value of variable φ at triangle vertex k, k “ 1, 2, 3.
The coefficients α k , β k and γ k depend only on the triangle geometry and hence are inde-
40

pendent of time; they are given by
α 1 “ px 2 y 3 ´ x 3 y 2 q{det , (5.5)
α 2 “ px 3 y 1 ´ x 1 y 3 q{det , (5.6)
α 3 “ px 1 y 2 ´ x 2 y 1 q{det , (5.7)
β 1 “ py 2 ´ y 3 q{det , (5.8)
β 1 “ py 3 ´ y 1 q{det , (5.9)
β 1 “ py 1 ´ y 2 q{det , (5.10)
γ 1 “ px 3 ´ x 2 q{det , (5.11)
γ 1 “ px 1 ´ x 3 q{det , (5.12)
γ 1 “ px 2 ´ x 1 q{det , (5.13)
where
det “ px 2 y 3 ´ x 3 y 2 q ` px 3 y 1 ´ x 1 y 3 q ` px 1 y 2 ´ x 2 y 1 q . (5.14)
From (5.1), the partial derivatives with respect to x and y, are obtained as
Bφ
Bx “ βφ ptq “
Bφ
By
“ γ φ ptq “
3ÿ
β k φ k ptq , (5.15)
k“1
3ÿ
γ k φ k ptq . (5.16)
k“1
5.2.1 Poisson Equation for Pressure
Presented in App. A.1 is the detailed derivation of the discretized Poisson equation for
pressure. In the appendix, first a differential form of the Poisson equation for pressure is
is derived, i.e.,
∆t B2 p 1
Bx 2 i
which in the appendix is labelled (A.9).
“ Bρ
Bt ` BρV i
˚
, (5.17)
Bx i
The result of the derivation of the discretized Poisson equation for p is
˘ ÿ ÿ`An p 1 n “ Bn , (5.18)
which in the appendix is labelled (A.21). In (5.18), the summation on both the l.h.s. and
the r.h.s. extends over all neighboring triangles of the control-volume defining node n.
The coefficients A n and B n are defined in App. A.1.
41

5.2.2 Turbulence Model
Presented in App. A.2 is the detailed derivation of the discretized equations for turbulent
kinetic energy k and dissipation rate ɛ.
The result of the derivation of the discretized equations for k and ɛ are
ÿ`Apkq
ÿ
n k n˘
“ B
pkq
n , (5.19)
ÿ`Apɛq n ɛ n˘
“
ÿ
B
pɛq
n , (5.20)
which in the appendix are labelled (A.34) and (A.35) respectively. In (5.19) and (5.20),
the summation on both the l.h.s. and the r.h.s. extends over all neighboring triangles of
the control-volume defining node n. The coefficients A pkq
n , Bn
pkq , A pɛq
n and Bn
pɛq are defined
in App. A.2.
5.3 Lagrangian Particle Equations
With particle method, the density functions derived in Chap. 2.3.2 and 2.3.3 is represented
by a finite number of stochastic particles. Each particle has a set of properties. The
properties are mass m p , position x p i , velocity V p
i , mass fraction Y p α , temperature T p .
From a given initial condition at t “ t 0 , the particle properties are varied in time with
the reasonably small time-steps ∆t. To determine the time-step size has used a suitably
formulated Courant–Friedrichs–Lewy (CFL) condition [8] as
„? j
AT
∆t “ C CFL ¨ min , (5.21)
|V i |
where A T denotes area of triangle T and V i contains both gasphase and liquid phase
velocity. In present work, the courant constant C CFL is taken as 0.7. The restricted
time-step size indicates that a fluid particle can not move more than one triangle in single
time-step. The particle properties are updated simultaneously by several different process,
but the method of fractional steps [69, 100, 101] is used so that the effects of some of these
processes can be treated sequentially.
The gas particle position and velocity are updated as
x p g,i
`t ` ∆t˘
“ x
p
`t ` ∆t˘
“ V
p
V p
g,i
g,i
g,i
`t˘
` V
p
g,iptq∆t (5.22)
`t˘
` ∆V
p
ptq∆t . (5.23)
g,i
42

∆V p
g,i ptq “ ´ 1
`
ρ p g
„ˆ Bp
Bx i
˙p
ˆ1
2 ` 3
4 C 0
j
p
´ S g,v ∆t
˙ ɛ p
`V g,i
k ´ xV g,iy p˘ ∆t ` pC 0 ɛq 1{2 ∆W i , (5.24)
with S v p denotes momentum exchange between the two phase specific at particle p as
S g,v
p
“
`
1
VP g
ÿP l
p“1
˜
1 ÿ Pl
VP g
p“1
pxV l,i y p ´ xV g,i y p q 9m p l
(5.25)
1
2 ρp l
`xVg,i y p ´ V p
l,i
˘| xVg,i y p ´ V p
l,i |C DA p l
¸
, (5.26)
where mean quantities pBp{Bx i q p , xV g,i y p and xV l,i y p are interpolated into particle position
x p g,i as
ˆ Bp
Bx ` Bp
˙p
By
“ β p ` γ p (5.27)
xV g,i y p “ α xV g,iy ` β xV g,iy x p g ` γ xV g,iy y p g (5.28)
xV l,i y p “ α xV l,iy ` β
xV l,iy x
p
l ` γxV l,iy y
p
l , (5.29)
where α φ , β φ and γ φ are defined in Chap. 5.2. And the mass exchange rate 9m p g is written
as
9m p l
“ 4π ρ p l Rp 2 dR l
l
dt , (5.30)
where the particle radius time rate of change dR l {dt has been modelled suitable evaporation
model considered in Chap. 4.3.
The species mass of gas particles is evaluated by mixing models – see Chap. 4.2 – ,
chemical source term and phase exchange term as
∆ψg,α p “ ´1
2 C ɛ
ψ
`ψp
k g,α ´ xψ g,α y˘∆t ` Spψg,αq∆t p ` Sg,m,α∆t p , (5.31)
where
S p g,m,α “ 1 P g
ÿP l
p“1
Y p
l,α 9mp l . (5.32)
Chemical source term Spψ p g,αq does not need any model as discussed in Chap. 4.2. To
calculate chemical reaction term, the software package COSILAB [77] and its API libraries
43

has been used.
The temperature of gas particles is evaluated as
with
˜ ¸
Kÿ
∆Tg p “ ´1
2 C ɛ p
ψ
`T
k g ´ xT g y p˘∆t ´ 1
c p ρ p h p αwα p ` Sg,e
p ∆t , (5.33)
g
α“1
˜
Sg,e p “ 1 1
V
ÿP l
Q p l
P g p“1
where the heat exchange term Q p l
is addressed in Chap. 4.3.2.
Similarly, the liquid particle position and velocity are updated as
x p l,i
`t ` ∆t˘
“ x
p
l,i
`t ` ∆t˘
“ V
p
V p
l,i
l,i
¸
, (5.34)
`t˘
` V
p
l,iptq∆t (5.35)
`t˘
` ∆V
p
ptq∆t , (5.36)
l,i
with
∆V p
l,i ptq “ ´ 1 „ˆ ˙p j
Bp
ρ p p
´ S v ∆t . (5.37)
g Bx i
Since the liquid particles has single component in the present work, contrary to gas particles,
the species mass of liquid particles per volume is constant during computation. The
radius of liquid particles is evaluated by suitable evaporation model, for example with
D-square law [28, 49, 93, 97] as
where the evaporation constant K is defined in Chap. 4.3.1.
liquid particles is evaluated as
5.4 Particle Tracing
∆R p l
“ ´ K
R p ∆t , (5.38)
l
And the temperature of
∆T p
l
“ 1
c p,l ρ p Se p ∆t . (5.39)
l
On their way through the computational domain, particles move through a number of
control volumes and hence triangles. Obviously, the numerical-solution method “needs
to know” where, at any time, each particle is located. To this end, a suitable method of
particle tracing is required.
44

In the present work, a particle-tracing method consisting of two subsequent steps has
been developed. The first step was devised by Löhner [52, 53].
In the first step, for any particle p P t1, 2, ..., P u at time t pn`1q , the host triangle T j ,
j “ 1, ..., N T , is sought. If at time t pnq the host triangle was T i , than at t pn`1q either
T j “ T i or T j P tTn i
i1
, ..., , Tn i
iP
u, where n iP denotes the number of direct neighbors of
triangle T i , provided a suitably formulated Courant–Friedrichs–Lewy (CFL) condition
has been used in the determination of the time-step size ∆t :“ t pn`1q ´ t pnq –see Eq.
(5.21).
Specifically, consider the left picture of Fig. 5.2, and assume – for the moment – that the
Figure 5.2: An unstructured triangle with physical coordinates and the mapped into
natural coordinate ξ and η.
picture applies to a particle p located at time t pnq at point P “ px, yq inside the depicted
triangle. To establish whether the particle at time t pn`1q is still located at a point P 1
inside this triangle, the coordinates of P 1 “ px p , y p q pn`1q are expressed in terms of the
natural coordinates
ξ “
η “
1 `yki x pn`1q
p
x ji y ki ´ y ji x ki
1 `´yji xp
pn`1q
x ji y ki ´ y ji x ki
˘
´ x ki yp
pn`1q , (5.40)
˘
` x ji yp
pn`1q , (5.41)
with the notation x ij “ x i ´ x j . The particle is located inside the triangle, if and only if
the shape functions [52, 53]
N 1 “ 1 ´ ξ ´ η , N 2 “ ξ , N 3 “ η (5.42)
satisfy
max`N 1 , N 2 , N 3˘
ď 1 and min`N1 , N 2 , N 3˘
ě 0 . (5.43)
45

If based on this criterion it is found that the particle has left the triangle, all neighbouring
triangles have to be investigated as possible new host-triangle; the search is stopped once
the new host triangle has been found. That a host triangle is indeed a member of set
tT i
n i1
, ..., , T i
n iP
u is ensured by formulating and applying a suitably formulated Courant–
Friedrichs–Lewy (CFL) condition in the determination of the time-step size ∆t :“ t pn`1q ´
t pnq .
In the second step, for each particle p the new host-triangle T j found in the first step
is investigated to find out to which of the possible 3 control volumes P belongs at time
Figure 5.3: A mapping to determine control volume which the particle is located.
t pn`1q . Referring to the right picture in Fig. 5.3, this is accomplished by evaluating the
angle θ between vector ÝÝÝÑ C 1 H jk and ÝÝÑ C 1 P 1 . Specifically, if the cross product ÝÝÝÑ C 1 H jk ˆ ÝÝÑ C 1 P 1 is
positive, then θ is measured counterclockwise, otherwise clockwise. Obviously, if θ is in
the range of ?H jk C 1 H ki and θ is measured counterclockwise, then particle p is located in
control volume k. Similarly, if θ is in the range of ?H jk C 1 H ki and θ is measured clockwise,
then p is located in j.
5.5 Particle Controlling
Since the local and instantaneous (turbulent) expectation values of the various physical
quantities are calculated from a finite number of particles in each control volume, the
statistical error in forming the expectations is of the order P ´1{2 [71, 79, 98] Here P
stands for either P g or P l , respectively. As can be seen from Fig. 5.4, with increasing P
the statistical error decreases, however, the efficiency of the numerical algorithm expressed
in terms of required CPU time (speed-down time) increases. Referring specifically to the
numbers displayed Fig. 5.4, the optimum number of particles lies at the cross-over of the
two curves, i.e., at P « 28.
46

1.0
Statistical Error
Speed down
50
40
Statistical Error
0.5
30
20
10
Speed down
0.0
10 30 50 70 90 110
Number of Particles
0
Figure 5.4: Statistical error and speed down in a control volume as a function of the
number of particles in a control volume.
In the course of a computation, the number of particles evolves dynamically. As a
consequence, – assuming that initially particles were distributed spatially in a quasihomogenously
manner 1 – with increasing time certain areas of the computational domain
may be thinned out of particles whilst in other areas particles may cluster.
To keep during a computation the number of particles per unit finite volume approximately
constant, and hence to ensure that the statistical error remains approximately
uniformly distributed throughout the physical domain, in the present work the particle
cloning and annihilation method [31, 91, 105] is employed. According to this method, particles
in the finite control volumes V are cloned or annihilated such that in each control
volume P varies between an upper and lower bound, i.e.,
P min ď P ď P max . (5.44)
Thus, after monitoring each control volume, particles are cloned in a control volumes
having P ă P min by splitting a particle located in the identical control volume.
most popular cloning procedure is that a particle is selected randomly for cloning (with
preference given to high-mass particles), and then is divided into two particles having the
same properties as the original particle except particle mass. If the mass of the original
1
In this context quasi-homogenous means that for all finite volumes V the number of particles per
unit finite volume is approximately the same.
The
47

particle is m, then the mass of each cloned particle is m{2. 2
cloning does not affect overall mass conservation.
Hence it is ensured that
Particles are annihilated in control volumes having P ą P max by merging three particles
selected at random (with preference given to low-mass particles) into two particles
having exactly preserved particle mass and mean particle properties. 3 Since this annihilation
method leads to artificial mixing, it does not exactly preserve the particle property
distribution, however it conserves the properties of particles at the microscopic level [105].
5.6 Initial and Boundary Conditions
5.6.1 Initial and Boundary Conditions for Particles
At the inlet, the Monte-Carlo particles are created and activated according to the inlet
flow properties. The total mass of the new gas particles M P is set to be the mass flux
during the current time step,
M P “ P new m p “ ρ inlet V x,inlet A inlet ∆t . (5.45)
When the particle moves across the axis of symmetry, the particle is reflected from the
boundary without change in their properties except for the velocity and position in the
direction normal to the boundary. When the particle moves across the open boundary,
the particle is discarded and at the wall, the velocity is forced to zero.
5.6.2 Boundary Conditions for Eulerian Equations
The flows under consideration in present work require boundary conditions to be prescribed
along the entire boundary line or surface surrounding the solution domain. No-slip
conditions are applied for the velocities and zero normal gradients for scalar variables at
wall boundary. At the symmetry axis, the velocity components normal to the boundary
are set to zero. At the outlet boundary, the pressure is fixed as ambient pressure 1 bar.
At the open boundary, the flow is allowed to adjust its direction into the compute domain
2
Note that after cloning the position of the two new particles is identical to the position of the
particle cloned.
3
Note that after annihilation the position of the two remaining particles could be taken as unchanged
or, alternatively, as the mean position of the original three particles. Since in the computations herein
it was found that, within temporal and spatial discretization errors neither instantaneous, local values
nor instantaneous expectation values were affected, the first alternative was selected because in terms of
computational operations it is more efficient.
48

or out of it. At inlet boundary, the velocity, composition, and turbulent properties of
gasphase and liquid phase are fixed to the initial values.
For the turbulence models also require the specification of certain variable values at the
boundaries. There are several methods used for turbulence specification at the boundaries,
and normally those methods need the additional information of the turbulence intensity,
turbulence length scale or the hydraulic diameter.
In present work, the turbulent quantities at the boundaries, especially inlet, are specified
as [66]
k “ 3 `
¯Vi,inlet I˘2 , (5.46)
2
ɛ “ 0.1643k1.5
l
. (5.47)
The turbulence intensity, I, is defined as the ratio of the root-mean-square of the velocity
fluctuations, V 1
i , to the mean velocity, xV i y as
I “ V i
1
xV i y . (5.48)
For internal flows the value of turbulence intensity can be fairly high with values ranging
from 1% - 10% being appropriate at the inlet and for external flows the value of turbulent
intensity can be as low as 0.05%.
5.7 Overall Solution Algorithm
In the present work, we combine the advantage of a full Monte-Carlo stochastic description
for chemistry, which does not require modeling, with the advantage of a full Monte-Carlo
stochastic description of both gaseous and liquid phase flow. Fundamental idea and one
dimensional spray simulation studies was done by Rumberg [79, 80], and in the present
work, it has been extended to two-dimensional reactive spray code with various selection of
fuel mechanism, e.g., detailed chemistry, reduced chemistry or global single-step chemistry
reaction mechanism. In contrast to PDF-FVM hybrid concept algorithm [4, 26, 59, 90],
most flow fields are obtained from expectation of particle properties except the pressure
p, the turbulent kinetic energy k, and the dissipation rate ɛ as schematically shown in
Fig. 5.5. This Figure shows also relationship between three main variable fields. Each
particle has own velocity and composition properties. From those particle properties, the
expectations are obtained stochastically. On the other hand, xpy, k and ɛ are calculated
49

Figure 5.5:
Schematic of the fully stochastic particle method.
by solving Eulerian governing equations – see Chap. 5.2. Unlike general FVM methods,
velocity components and compositions are known quantities – expectations from particles
– to solve the governing equation. for xpy, k and ɛ. Moreover, the particle properties are
updated by those Eulerian fields.
In Fig. 5.6, the flowchart of overall algorithm is schematically reported. The overall
algorithm can be divided up two region, i.e., Lagrangian and Eulerian part. Detailed for
each task in the algorithm has been considered in the present thesis, e.g., calculate time
step size, calculate the Eulerian variables, trace particles, and control number of particles
have been introduced Chap. 5.3, App. A, Chap. 5.4 and Chap. 5.5, respectively.
50

Figure 5.6:
Flowchart of the overall computation algorithm.
51

Chapter 6
Parallelization
The numerical simulation of a complex flow field requires a high computing power, especially
for a combustion problem. Since the efficiency of computing is mainly dominated by
the performance of the central processing units – commonly such a unit is referred to as
CPU – ongoing development of CPUs affects the implementation of numerical algorithms.
For more than two decades, factually, the density of transistors – which is related directly
to the performance of a CPU – has doubled every two years. Since the mid 2000s,
however, increasing computing power through more dense transistors had certain reached
limitations and, therefore, most of the major CPU manufacturers began putting multiple
processors on a single circuit for increasing performance by parallelism. Because of the
changed roadmap of CPU manufacturers, parallel algorithms have become an essential
point in numerical simulations.
In this chapter, first fundamental parallelization theories, strategies and applications are
discussed. Then suitable parallelization methods are selected and applied to parallelize
the Monte-Carlo particle method employed in the present work.
6.1 Parallelization Architecture and Strategies
6.1.1 Parallel Architecture
According to the so-called Flynn’s taxonomy [22], see Fig.6.1 – classically parallel computer
architectures can be categorized according to the number of instruction streams and
the number of data streams which the system can handle at the same time. For example,
a computer program may be executed by following a single series of programming statements
(single instruction stream) for one processor or, alternatively, it may be executed by
52

Figure 6.1: Classification of parallel computer architectures by Flynn [22].
splitting the overall programming statements into multiple series of programming statements,
one series for each processor (multiple instruction streams). On the other hand,
input data for a computer program may be provided as a whole to one processor (single
data stream) or, alternatively, overall input data may be split up into multiple input data
(multiple data streams) such that each processor gets its own input data. As specified by
Flynn, the classical von Neumann system has a single instruction stream and single data
stream and, therefore, can be classified as SISD.
SIMD (single instruction, multiple data) execute identical instructions for all multiple
data sets synchronously. In other words, each processor has the same task with different
data streams. Main characteristics of SIMD should execute the same instruction whether
it has some data to handle or not can lower the overall performance of the system, therefore
SIMD is difficult to apply on general purpose. However, SIMD concept is ideal for
parallelizing simple loops that operate on large arrays of data so called data-parallelism,
and with this strength, vector processors which is an application of SIMD concept was
used dominantly for supercomputer during the 1990s [62].
One of many other SIMD applications is the graphics processing unit (GPU) which was
originally designed to exclusively manipulate computer graphics and image processing.
Recently, GPUs have been developed that have many cores which can execute individual
instructions and, therefore, GPUs now are also utilized for general high performance
computing. A GPU that also serves such generalized purposes is called a General-Purpose
Computing on Graphics Processing unit (GPGPU).
Different from SIMD systems, multiple instruction, multiple data (MIMD) systems carry
multiple concurrent instructions which operate on multiple data streams. Depending
upon how memory is accessed, MIMD systems can be classified with distributed memory
system and shared memory systems as discussed in the next section.
53

6.1.2 Distributed Memory System and Shared Memory System
MIMD systems are most popular in parallelization concepts because of their versatility.
There are two principal subtypes of MIMD systems, viz., distributed memory systems
and shared memory systems.
Figure 6.2: Schematic of distributed memory system.
Figure 6.3: Schematic of shared memory system.
As shown schematically in Fig. 6.2, in a distributed memory system, each processor basically
has its own memory space, and consequently can access it independently of the
other processors. Thus, each processor modifies data locally, and data exchanged between
processors occur explicitly via an interconnection network.
The so-called Message Passing Interface (MPI) is a communication protocol for programming
of distributed memory systems. As its name implies, with MPI processor-memory
pairs can communicate by message passing: one pair sends the message (data) by calling a
54

“send” function and an other pair or several pairs receive the message (data) by calling a
“receive” function. There are several MPI implemented libraries for C, C++ and Fortran,
e.g., MPICH, OpenMPI and MSMPI.
In a shared memory system, a group of processors is connected to a common memory system
via an internal network as schematically shown in Fig 6.3. Principally, the processors
communicate by accessing common memory with equal priority.
An example of sharing memory is Open Multi-Processing (OpenMP) which is a very
popular applications programming interface (API). It can be used in programs written in
C, C++ and Fortran, that are intended for use on shared memory systems.
6.1.3 Parallelization Strategies
To obtain reasonable parallelization effects in a CFD code in terms of speedup, say,
individual code sections need to be identified and analyzed with respect to their suitability
Figure 6.4: An example of executing time segments.
for parallelization. Schematically shown in Fig. 6.4 are sections of a CFD code such as
– ordered according to their serial CPU time requirement – solution of source terms,
solution of pressure equation, preparation of initial data, and so on.
For a variety of reasons, not every section in a computer program is suitable to be subjected
to parallelization. Thus, prior to code parallelization, suitable sections need to be
identified. After identification and selection of meaningful code sections, suitable parallelization
approaches and parallel program models should be considered. To this end a
very effective and simple approach is to break the most CPU-time intensive section of
55

the code – in the example of Fig. 6.4 this is the section “solution of source terms” –
into temporally or spatially discrete segments that can be handled by multiple processes.
Depending on the partitioning target, basically two parallelization methods are available,
i.e., the so-called domain decomposition and the so-called functional decomposition.
In the present work, these two methods and respective programming models have been
implemented, namely,
ˆ the overall computational domain and the total number of particles have been decomposed
by employing MPI,
ˆ the most repetitively used functions have been decomposed by employing OpenMP.
Further details will be discussed in the following sections.
6.2 Domain and Particles Decomposition
In the domain decomposition approach to a problem partitioning, we first seek to decompose
the data associated with the problem. If possible, we divide these data into small
pieces of approximately equal size. Next, we partition the computation that is to be performed,
typically by associating each operation with the data on which it operates. This
partitioning yields a number of tasks, each comprising some data and a set of operations
on that data. An operation may require data from several tasks. In this case, communication
is required to move data between tasks. In the present work, this requirement is
handled by MPI.
MPI is an API library containing functions for communicating so-called messages, or
data, between processors. These exchange messages can be between just two processors
or, alternatively, messages can be collectively sent or received by several processors. MPI
defines a so-called Communicator, which has associated with it a Processor Group and a
Context. A Processor Group consists of a subset of the processors available to the system,
and a Context is a tag given by system, whose aim is to distinguish the processors. The
MPI library offers many functions for common or complicated computing assignment.
The most basic functions are
1. MPI INIT, initializing MPI,
2. MPI FINALIZE, finishing MPI,
3. MPI COMM SIZE, get the number of available processors,
56

Figure 6.5: Schematic of MPI.
4. MPI COMM RANK, get the identity of a processor,
5. MPI SEND, send a message,
6. MPI RECV, receive a message.
Since the overall program is divided up into subtasks by MPI, a certain processor should
control the overall procedure. This processor is so-called “master” processor and the
others are so-called “slave” processors. A master processor can, eventually, in addition
also work as a slave shown as schematically in Fig. 6.5.
6.2.1 Schur-Complement Method
The Schur-complement method describes a mathematical algorithm that is not limited
to applications in parallel solution of CFD methods. However, in the present work we
will apply it to a particular class of parallelization methods, viz., the so-called class of
domain decomposition methods. As the name says, the fundamental idea underlying
domain-decomposition methods is, that the overall domain of solution of a CFD problem
is split up into subdomains such that (usually) each single processor solves (nearly) the
underlying problem for a single subdomain.
To be specific in the present example, let us assume that we have carried out a discretization
for N “ 9 grid points, i.e., that we have a mesh
M “ tx L “ x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 “ x R u , (6.1)
57

Figure 6.6: Schematic of domain decomposition.
where x L and x R denote the left boundary and the right boundary. Let P denote the
number of processors which are available to us for simultaneous, parallel use to solve the
problem. To be specific, in the present example let us assume P “ 3. Hence we decompose
the computational domain into P “ 3 parts, e.g. into
M p1q “ tx L “ x 1 , x 2 , x 3 , x 4 u ,
M p2q “ tx 4 , x 5 , x 6 , x 7 u , (6.2)
M p3q “ tx 7 , x 8 , x 9 “ x R u .
Obviously, domain 1 and 2 have node x 4 in common, and domains 2 and 3 node x 7 . At
the nodes x L and x R we apply the physical, or natural, boundary conditions mentioned
above. Since node x 4 represents a right boundary to domain 1 and a left boundary to
domain 2, x 4 is termed a logical boundary node or, alternatively, a ghost node. For similar
reasons, also node x 7 is a logical boundary or ghost node. Thus the set of ghost nodes is
M pGq “ tx 4 , x 7 u . (6.3)
6.2.1.1 Linear System for Subdomain p – General Formulation
For each subdomain p, the vector of unknowns is written as
X ppq “
˜
X ppq
N
X ppq
G
¸
. (6.4)
Here, and below, the subscripts N and G denote the (still unknown) solution vector at a
non-ghost and ghost node, respectively. The superscript p indicates, that the respective
quantity refers to processor p, 1 ď p ď P . Analogous sub and super scripting applies to
various vectors and matrices to be introduced below.
58

For subdomain p we obtain the linear system
A ppq X ppq “ B ppq , (6.5)
which, after introducing the overall ghost-node vector
X G “ ppX pGq
1 q T , ..., pX pGq
N G
q T q T (6.6)
can be partitioned as
˜
A ppq
NN
A ppq
GN
Appq NG
A ppq
GG
¸ ˜
X ppq
N
X G
¸
“
˜
B ppq
N
B ppq
G
¸
. (6.7)
Here N G denotes the number of ghost nodes. Recall that for our specific example N G “ 2,
with X pGq
1 “ δ 4 and X pGq
2 “ δ 7 ; the example-specific form of the various matrices and
vectors is left as a homework or exercise.
6.2.1.2 The Linear System for the Overall Domain – General Formulation
Assembly of the linear system for the overall domain yields – using the principle of superposition
–
¨
diagpA p1q
NN , Ap2q NN , ¨ ¨ ¨ , ApP q
NN q ˚
˝
X p1q
N
.
X pP q
N
˛
‹
‚`
Pÿ
p“1
¨
A ppq
NG X G “ ˚
˝
B p1q
N
.
B pP q
N
˛
‹
‚ (6.8)
and, for 1 ď p ď P ,
A ppq
GN Xppq N ` Appq GG X G “ B ppq
G . (6.9)
Inspection shows that (6.8) and (6.9) can be combined to give
¨
A p1q
NN 0 ¨ ¨ ¨ 0 0 ˛
Ap1q NG
0 A p2q
NN ¨ ¨ ¨ 0 0 ¨
Ap2q NG
. .
. . .
pP ´1q
pP ´1q
0 0 ¨ ¨ ¨ ANN 0 A ˚
NG
˝
˚
˝ 0 0 ¨ ¨ ¨ 0 A pP q
NN
A pP q ‹
NG ‚
A p1q
GN
A p2q
pP ´1q
GN
¨ ¨ ¨ AGN
A pP q
GN
A GG
X p1q
.
X pP q
X G
˛ ¨
‹
‚ “ ˚
˝
B p1q
.
B pP q
B G
˛
‹
‚ , (6.10)
59

where for brevity of notation we have defined
and where
X ppq :“ X ppq
N , (6.11)
B ppq :“ B ppq
N , (6.12)
A GG :“
B G :“
Pÿ
p“1
Pÿ
p“1
A ppq
GG , (6.13)
B ppq
G . (6.14)
Note that the various matrices and vectors introduced so far have the following dimensions:
1. matrix A ppq
NN
is of dimension N
ppq
N
ˆ N ppq
N ;
2. matrix A ppq
ppq
NG
is of dimension NN ˆ N G;
3. matrix A ppq
GN is of dimension N G ˆ N ppq
N ;
4. matrix A ppq
GG is of dimension N G ˆ N G ;
5. vectors X p1q to X pP q are of dimension N ppq
N ;
6. vectors B p1q to B pP q are of dimension N ppq
N ;
7. vectors B p1q
G
q
to BpP
G
, and hence vector B G, are of dimension N G
8. vector X G is of dimension N G .
6.2.1.3 LU Decomposition of the Matrix of Eq. (6.10)
It is readily shown that the matrix appearing on the left-hand side of (6.10) has the
LU-decomposition given by
¨
A p1q
NN 0 ¨ ¨ ¨ 0 0 0
˛
0 A p2q
NN ¨ ¨ ¨ 0 0 0
. .
. . .
L “
pP ´1q
0 0 ¨ ¨ ¨ ANN 0 0
˚
˝ 0 0 ¨ ¨ ¨ 0 A pP q ‹
‚
A p1q
GN
A p2q
GN
¨ ¨ ¨ A
pP ´1q
GN
NN
0
A pP q
GN
I
(6.15)
60

¨
U “
˚
˝
I 0 ¨ ¨ ¨ 0 0 A p1q
0 I ¨ ¨ ¨ 0 0 A p2q
NN
.
.
0 0 ¨ ¨ ¨ I 0 A
0 0 ¨ ¨ ¨ 0 I A pP q
NN
0 0 ¨ ¨ ¨ 0 0 S
.
.
.
´1 p1q
NN A
NG
´1 p2q A
NG
pP ´1q´1 pP ´1q
NN A
NG
´1 pP q A
NG
˛
. (6.16)
‹
‚
Also straightforward to show is that
Pÿ
p“1
A ppq ´1
GN Appq ppq
NN A
NG ` S “ A GG “
Since the latter equation can be written as
S “
Pÿ
p“1
Pÿ
p“1
´
¯
A ppq
GG ´ ´1 Appq GN Appq ppq
NN A
NG
A ppq
GG . (6.17)
, (6.18)
it is plausible to define
with
Pÿ
S :“ S ppq (6.19)
p“1
in the second equation of (6.20),
S ppq :“ A ppq
GG ´ Appq GN pAppq NN q´1 A ppq
NG
“ A ppq
GG ´ Appq GN Cppq NG ; (6.20)
C ppq
NG
:“ pAppq q´1 NN
A ppq
NG . (6.21)
The usual way to solve a linear system of the form
LUX “ B (6.22)
is to use two sweeps in succession, viz., to solve first
LY “ B (6.23)
and then
UX “ Y . (6.24)
61

Here Y “ ppY p1q q T , ..., pY pP q q T , pY G q T q T ; similar definitions apply to B and X – see
(6.12) and (6.14), respectively. Due to the special form of L, (6.23) can be solved independently
for each sub-vector Y ppq , where p “ 1, ..., P . Specifically, formally the solution
Y ppq can be written as
and hence Y pGq as
Y G “ B G ´
Y ppq “
Pÿ
p“1
´
A ppq
NN¯´1
B ppq , (6.25)
A ppq
GN Y ppq “
Pÿ
p“1
´
ppq¯
B ppq
G ´ Appq GN Y
. (6.26)
If (6.25) and (6.26) are substituted into (6.27), after minor manipulation the linear system
¨
I 0 ¨ ¨ ¨ 0 0 A p1q
NN
0 I ¨ ¨ ¨ 0 0 A p2q
NN
. . . . .
0 0 ¨ ¨ ¨ I 0 A NN
˚
˝ 0 0 ¨ ¨ ¨ 0 I A pP q
NN
0 0 ¨ ¨ ¨ 0 0 S
results. The last block in this system,
´1 p1q A
NG
´1 p2q A
NG
pP ´1q´1 pP ´1q A
NG
´1 pP q A
NG
˛
¨
˚
˝
‹
‚
X p1q
.
X pP q
X G
˛ ¨
‹
‚ “ ˚
˝
Y p1q
.
Y pP q
Y G
˛
‹
‚
(6.27)
S X G “ Y G , (6.28)
can be solved first for X G . Then the solution X ppq of the remaining blocks are obtained
by solving for each p independently and simultaneously
X ppq ` pA ppq
NN q´1 A ppq
NG X G “ Y ppq . (6.29)
or, alternatively,
A ppq
NN Xppq “ B ppq ´ A ppq
NG X G . (6.30)
6.2.1.4 Summary of the Steps for the Schur-Complement Method
Here is a summary of the individual steps of the Schur-complement method to obtain
a solution to the Schur-complement linear system (6.10). It is assumed that to each
slave p ě 1 all its interior nodes and the totality of ghost nodes are known, as well as a
solution from the previous Picard-iteration step at these nodes. Furthermore, we assume
that processor p “ 1 represents the master, and that the master additionally acts as an
ordinary slave.
62

‚ Step 0, in a subroutine or function solve 0 parallel: For each slave p ě 1,
(a) allocate the matrices A ppq
NN , Appq GN , Appq NG , Appq GG , Cppq NG , Sppq and S;
(b) allocate the vectors B ppq , B ppq
G , Y ppq , X ppq , D ppq ,
(c) only for the master p “ 1 allocate the vector X global .
‚ Step 1, in a subroutine or function solve 1 parallel: For each slave p ě 1,
(a) assemble the matrices A ppq
NN
(b) solve the linear system (6.25) for Y ppq ,
and Appq
NG , and the vector Bppq ,
(c) compute the matrix
C ppq
NG “ pAppq NN q´1 A ppq
NG
(6.31)
by solving number of ghost nodes linear systems,
(d) assemble the matrices A ppq
GG
and Appq
GN
, and the vector Bppq
G ,
(e) compute the difference vector
D ppq :“ B ppq
G
then communicate it to the master,
´ Appq GN Y ppq , (6.32)
(f) compute the matrix S ppq from (6.20), then communicate it to the master.
‚ Step 2: The master, p “ 1,
(a) waits until S ppq and D ppq have arrived from all processors p,
(b) evaluates S “ ř P
p“1 Sppq ,
(c) evaluates Y pGq “ ř P
p“1 Dppq ,
(d) solves (6.28) for X pGq ,
(e) communicates X pGq to all slaves.
‚ Step 3: Each slave p ě 1,
(a) waits until X pGq has arrived from the master, then
(b) computes A ppq
NG XpGq , and then
(c) solves (6.30) to obtain X ppq .
(d) sends X ppq to the master,
(e) the master then assembles the overall solution X global .
63

‚ Step 4: Eventually repeat several times the sequence of steps 1, 2 and 3
‚ Step 5: Deallocate all arrays allocated in step 0, except, eventually, the global
solution vector X global .
6.2.2 Particles Exchange
In present work, Monte carlo particles method has been implemented. Particles are seeded
in the computation domain and they travel through the domain. A simple possibility at
the implementation aspect is that all particles are initially distributed into each processors,
and they belong to their pre-introduced processor permanently. However, since the
expectation values, e.g., mean velocity, mean mass fraction of species and temperature,
are calculated from particles which locate in correspond control volume, the particles
distribution implied above occurs critical difficulties.
For instance, let us imagine a particle P which is initially located in the sub-domain
1 handled by processor 1. While this particle travels in the sub-domain 1, processor 1
can use this particle’s data for getting expectation value for corresponded node. If the
location of particle P exceed physical neighbour sub-domain 2, processor 1 does not need
anymore the information of particle P but processor 2 needs it. Since each processor has
its own memory space, processor 2 can not access the particle P . Therefore, constantly
distribution of particles in each processor is simple and efficient only at the beginning of
calculation.
Presented difficulty can be solved by introducing particles transferring. As shown schematically
in Fig. 6.7, when any particle meets a logical boundary, all properties of the particle
Figure 6.7: Schematic of particle exchange.
64

is sent to neighbour processor by MPI, and that particle becomes inactivated in previous
processor. Then each processor owns only the particles involved in correspond
sub-domain.
6.3 Functional Decomposition
A decomposition at the functional level – also termed a functional decomposition – focuses
on target functions which usually demand high computing power.
Implementation of
functional decomposition into a computer code appears to be simpler for OpenMP than
for MPI. However, to use OpenMP or any other method of the shared memory type,
the code needs to be carefully designed. Besides, functional decomposition is especially
suitable for particle methods such as the ones employed in the present work, since each
particle employs an identical but independent procedure to update particle properties
whilst taking time steps.
OpenMP is provided with most Fortran and C++ compilers, and it is implemented into
a respective computer code by employing compiler directives, library routines and environmental
variables. For instance, a loop structure in a serial code can be parallelized
simply by placing suitable OpenMP compiler directives. The OpenMP library routines
are meant to serve as a control and query tool for the parallel execution environment,
which the programmer can use from inside its program. Therefore, the OpenMP runtime
library contains a set of external procedures with clearly defined interfaces. For the
detailed description of OpenMP directives and library routines, [57] should be consulted.
In the present work, OpenMP has been implemented into the numerical solution procedure
that evaluates the chemical source terms for each gasphase particle, and into the numerical
solution procedure that evaluates the nodal expectation values from both gasphase
and liquid-phase particles properties. To test the effects of parallelization by OpenMP,
the two-dimensional reactive-spray code has been used in a 32-processors environment
system. The resulting speedup due to parallelization is shown in Figs. 6.8 and 6.9, respectively.
Specifically, to measure parallelization effect solely on the target part of the
whole program, an indicator for increasing the performance so-called speedup is defined
as
speedup “
execution time of the serial target functions
execution time of the parallel target functions . (6.33)
Interestingly, defining number of processors increasing beyond 32 slows the code down
again. The reason for this phenomenon probably is that parallelization overheads have
increased to a point where no overall gain in speed can be made.
65

15
speedup
10
5
0
1 2 4 8 16 32 64
number of processors
Figure 6.8: Partial speedup by OpenMP on calculating chemical source term.
10
speedup
5
0
1 2 4 8 16 32 64
number of processors
Figure 6.9: Partial speedup by OpenMP on expectation values.
6.4 MPI-OpenMP Hybrid Approach
In this section, a comparison between MPI and OpenMP is done by employing alternatively
each of the two parallel schemes in the code.
66

6.4.1 Performance and Limitation
The parallelized code was executed at a PC-cluster installed at in the IT Center of RWTH
(Rheinisch-Westfälische Technische Hochschule) Aachen [81] in order to investigate the
effect of the number of processors on speedup. To this end, the maximum number of
processors was 32.
To compare different parallelization effects on the serial code, the overall speedup has
been calculated by an indicator similar to that presented in Chap. 6.3, however, now
measuring total computing time instead of target function computing time. As shown
in Fig 6.10, generally the performance of MPI is better than that of OpenMP. This is
easily understandable because OpenMP is applied to certain target parts of the whole
program, whilst MPI affects the overall code. However, such performance phenomena
may not be general but rather specific to the underlying computer code. Specifically, the
present two-dimensional simulation code woks with particle methods that obtain bulk of
variables statistically from particles. Therefore, domain and particles decomposition by
MPI results in large parallelization effects.
3 OpenMP
MPI
speedup
2
1
1 2 4 8 16 32 64
number of processors
Figure 6.10: Overall speedup by OpenMP and MPI on sprays flame.
As considered two different parallel schemes, the main dissimilarity is on a way of communication
between processors. While distributed memory programming represented by MPI
involves a communication through message sending and receiving explicitly, shared mem-
67

ory represented by OpenMP relies on implicit communication which utilizes the shared
memory space as a passage.
Generally, MPI is often limited in the number of processors that can be used, and the
application on lager number of processors is often expensive or hard to tune for communication
between processors. On the other hand, OpenMP has strong advantages when
it is applied to loop level parallelization. However, applying OpenMP gets reasonable
performance only on the single computing environment which allows data transferring
implicitly in hardware base.
To overcome above limitation of single parallelization scheme, a hybrid concept will be
discussed in the following sections.
6.4.2 Hybrid Approach
A hybrid (MPI-OpenMP) scheme is that MPI and OpenMP are combined to compensate
the limitations of both of parallel scheme. Specifically, in present work, the hybrid
scheme is designed by adding looplevel OpenMP parallelization into the existing domain
decomposition MPI parallelized code as shown schematically in Fig. 6.11.
Figure 6.11: Schematic of MPI-OpenMP hybrid approach.
For running the hybrid parallelized two-dimensional code, identical computing environment
as introduced in Chap. 6.4.1 has been used. As shown in Fig. 6.12, the performance
with hybrid algorithm is 2 times better than pure OpenMP algorithm used identical CPU
resources.
68

Figure 6.12: Comparing overall speedup between Pure OpenMP and Hybrid MPI-
OpenMP (MPI 8 ˆ OpenMP 4).
6.5 Heterogeneous Computing with GPU
6.5.1 Introduction
In the previous section, the parallelization methods are considered under the homogeneous
computing environment such as CPU. Obviously, better clocks and more core integrated
system or cluster will give a guarantee against its performance. However, as discussed in
the beginning of previous section, raising performance of CPU has reached its limitations,
moreover, not every people can access a cluster system due to the financial reason.
One of the potential solutions is to distribute computing load by using GPU. A GPU
was designed originally for handling the image creations for output to display and it has
been still rapidly developing with high demanding of complicated graphics. As shown
schematically in Fig. 6.13, especially GPU has highly parallel structure, therefore for the
purpose of manipulating graphics GPU is much efficient than CPU which is designed for
general task. With such unique characteristics of GPU, the effort for using GPU at more
general purpose has been arisen by many researchers [14, 44, 46] so called GPGPU.
GPGPU stands for General-Purpose computation on Graphics Processing Units. From
an implementation of a matrix multiplication whihc was one of the first applications
of non-graphical computations on a GPU [14, 46], many attempts of using GPU for
general computations is proceeding actively. Since basically GPU should be controlled by
CPU, using GPU for general purpose means a computing on CPU-GPU heterogeneous
environments. Controlling between different devices needs specific program language or
69

Figure 6.13: Schematic of CPU(left) and GPU(right).
API which allow to port a applications to GPU for return values.
CUDA(Compute Unified Device Architecture) is a parallel computing platform and programming
model to use a GPU for general purpose processing invented by NVIDIA. A
main drawback is that CUDA can be used only on NVIDIA’s devices. That means GPUs
of Intel and AMD which share over then 70 percent of the GPUs are not available to use
the CUDA platform.
On the other hand, OpenCL(Open Computing Language) is a general interface for generating
programs that are able to run across heterogeneous devices maintained by the
non-profit technology consortium Khronos Group. OpenCL itself provides only standard
protocols for parallel computing on heterogeneous computing environments. OpenCL
conformant implementations have been provided by multiple vendors, e.g., AMD, Apple,
Intel, NVIDIA. While OpenCL generally has an advantage that it can be adopted many
other platforms, handling and writing a parallel program is more complicate than CUDA
programming due to cover much wide range of devices.
In the present work, relatively simple 1D code is modified for applying OpenCL to know
how a heterogeneous computing is able to be adaptable on computational fluid dynamics
fields and how efficient it is.
6.5.2 GPU Acceleration with OpenCL
In principle, OpenCL has been proposed as an open standard, therefore each GPU vendors
has implemented theirs own way. Nevertheless, OpenCL API is general enough to
run on significantly different architectures while being adaptable enough that each hardware
platform can still obtain high performance. Using the core language and correctly
70

following the specification, any program designed for one vendor can execute on anothers
hardware. Because of the above characteristics, OpenCL is practical for GPUs performance
comparison.
The OpenCL API is a C with a C++ Wrapper API that is defined in terms of the C API.
The code that executes on an OpenCL device, which in general is not the same device as
the host CPU, is written in the OpenCL C language. OpenCL C is a restricted version
of the C99 language with extensions appropriate for executing data-parallel code on a
variety of heterogeneous devices [25].
Figure 6.14: Schematic of implementation GPU accelerating with OpenCL.
In the present work, an OpenCL API which is provided by Intel is used for applying GPU
acceleration on existing one-dimensional reacting code (written in Fortran90). Obviously
the implementation OpenCL one-dimensional code for the reacting flows given here is
straightforwardly extended to the general two or three-dimensional case. Since directly
using OpenCL feature with Fortran is limited to certain Fortran compiler 1 , it is necessary
to divide 1D code into several parts and generate DLL (Dynamic-link library) for using
in C++ program as schematically shown in Fig. 6.14.
To construct CPU-GPU combined parallel code with OpenCL, there are certain restrictions,
e.g., calling external functions are not allowed, but functions defined in host side
can be used and variable length arrays and structures are not supported. As discussed in
Chap. 6.3, most computing work concentrated part of present whole code is the calculating
chemical source term. However, since external functions or subroutines are not able
1 Only the PGI, CAPS and Cray compiler fully support OpenACC for Fortran and partial support
from GNU Fortran. And another possibility, CUDA Fortran is the Fortran version of CUDA C and is
only supported by the PGI compile.
71

to call from outside of device, naturally any subroutine corresponded to chemical source
term can not be used from divided .dll directly. Therefore, relatively simple function has
been implemented on device.
In the present work, the reactive flow simulation code has been implemented by Monte-
Carlo particle method which widely used to solve PDF transport equations. Since the
accuracy of simulation carried by the Monte-Carlo particle method is affected strongly
the number of statistical particles. Each particle has own properties, and those properties
are updated through computing time steps. Because those particle properties updating is
simple and needs to execute many time, this function is reasonable choice of parallelization
target function. To validate the parallelized one-dimensional reactive code by OpenCL,
Figure 6.15: Sample calculation for premixed methane flame on 1D code.
simple premixed flame test case has been constructed. Gasphase combustion is described
by a 10-step reduced chemical mechanism for methane/air systems [41] consisting of 14
chemical species and 10 chemical reactions. Simulated results is reported in Fig. 6.15,
and the results between original Fortran code, ported C++ code and C++ with OpenCL
code are identical. Shown in Fig. 6.16, three different GPUs were compared. It should be
72

noted that the speedup shown in figure is measured by computing time of only targeting
functions as same strategy in Chap. 6.3. It is interesting to note that the number of core
does not guarantee correspond speedup.
Figure 6.16: Comparing speedup of parallelized target functions between GPUs; 48 cores
(left), 384 cores(center) and 24 cores(right).
In this chapter, CPU-GPU heterogeneous computing with OpenCL has been introduced
and considered as an alternative parallelization strategy of CFD codes. The general
purpose using on GPUs so-called GPGPU is now developing rapidly on the strength of
increasing GPU performance. Because of that reason, although applying CFD codes
to GPUs has some critical limitations, the potential of using GPUs for executing CFD
codes can not be judged at the moment. When the programm technics for computing
on the heterogeneous device environments becomes mature enough, the CFD code can
be parallelized efficiently. For example, host device – multi core CPU – has a roll for
decomposed computational domain and particles by MPI, and devices – GPU or GPUs –
separate the massive loop functions to small loop by OpenCL as schematically shown in
Fig. 6.17.
73

Figure 6.17: Schematic of MPI-OpenCL hybrid concept.
74

Chapter 7
Turbulent Jet Diffusion Flames
(DLR H3 Flame)
7.1 Experimental and Computational Setup
To validate the computer code developed in the present work for pure gasphase combustion,
the code has been applied to a turbulent hydrogen-air flame investigated at DLR
Stuttgart in 1996 [56, 65]. This flame is non-premixed and it was operated without a stabilizing
pilot burner. Experiments were carried out with the burner shown schematically
at the left of Fig. 7.1. The burner consists of a vertical tube with a nozzle diameter of
8 mm. The co-flowing air had an average flow velocity of 0.2 m/s with an exit diameter
of 140 mm. The fuel jet consisted of 50 vol.% nitrogen and 50 vol.% hydrogen; it was
issued at an average flow velocity of 34.8 m/s through a nozzle into the ambient air. The
Reynolds number of the fuel jet at the burner exit was 10000, the temperature there was
300 K. The pressure was 1 bar. Details on the experimental arrangement are given in
[56, 65] and experimental data for temperature and species concentrations are reported
in [1, 21, 65]. On the right of Fig. 7.1, the computational domain is shown. Since for
this computation only OpenMP was tested, a domain decomposition was not employed.
Roman numbers I to V indicate the different parts of the domain boundary. Specifically,
I indicates the jet inlet, II the co-flow inlet, III the open boundary where entrainment of
air occurs, IV the outlet boundary, and V the symmetry boundary. 1
Simulations are performed on the unstructured mesh shown in Fig. 7.1. The computational
domain extends to 10 and 90 diameters in the radial- and axial direction, respectively.
At the inlet plane the velocity, composition, and turbulent properties of the gas
flow are fixed to the respective boundary values. At the symmetry axis, V, the velocity
1
Note that the problem is axisymmetric.
75

Figure 7.1: Schematic of the DLR burner [56] and of the computational domain.
components normal to the boundary are set to zero. At the open boundary, III, the flow
is allowed to adjust its direction into the compute domain or out of it.
In the computations 195441 particles were used, and the mesh consisted of 3409 nodes
and 6484 triangles.
Gasphase combustion was described by a detailed chemical mechanism of hydrogen [61]
consisting of 10 chemical species and 21 chemical reactions.
7.2 Results and Discussion
Shown in Figs. 7.2 to 7.6 are profiles in physical space of various quantities as obtained
0.06
experiment
simulation
H 2
mass fraction
0.04
0.02
0.00
0 20 40 60 80 100
x/D
Figure 7.2: Axial profile of the mass fraction of H 2 along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65].
76

0.2
experiment
simulation
O 2
mass fraction
0.1
0.0
0 20 40 60 80 100
x/D
Figure 7.3: Axial profile of the mass fraction of O 2 along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65].
0.15 experiment
simulation
H 2
O mass fraction
0.10
0.05
0.00
0 20 40 60 80 100
x/D
Figure 7.4: Axial profile of the mass fraction of H 2 O along the centerline. Solid line:
computational result; symbols: experimental data from [1, 21, 65].
in experiments performed by Neuber et al. [1, 21, 65] and from the numerical simulations
carried out in the present work. Specifically, shown in Fig. 7.2 and 7.3, respectively, is
the mass fraction of hydrogen and oxygen on the centerline of the turbulent jet.
It can be seen that generally the agreement between the experimental and numerical data
is good. The same is the case for the mass fractions of water vapor, whose centerline
profile is shown in Fig. 7.4. Results showing the same good agreement are shown for the
mass fraction of the hydroxyl radical in Fig, 7.5, and for the temperature in Fig. 7.6.
77

0.002
experiment
simulation
OH mass fraction
0.001
0.000
0 20 40 60 80 100
x/D
Figure 7.5: Axial profile of the mass fraction of OH along the centerline.
computational result; symbols: experimental data from [1, 21, 65].
Solid line:
Figure 7.6: Axial profile of temperature along the centerline. Solid line: computational
result; symbols: experimental data from [1, 21, 65].
Plotted in Figs. 7.7 and 7.8 is the temperature as a function of the mixture faction at two
axial positions, x{D “ 5 and x{D “ 20, respectively. In both, the experiments [1, 21, 65]
and the computations performed herein, the mixture fraction Z was calculated using the
Bilger’s definition [5], viz.,
Z “
2Y C {W C ` 1
2 Y H{W H ` `Y O,air ´ Y O˘{WO
2Y C,fuel {W C ` 1
2 Y H,fuel{W H ` `Y O,air ´ Y O,fuel˘{WO
, (7.1)
78

where Y α and W α denote the mass fractions and molecular weights, respectively, for
carbon, C, atomic hydrogen, H, and atomic oxygen, O.
Figure 7.7: Temperature as a function of mixture fraction at x{D “ 5. Red points: computational
result; blue points: experimental data from [1, 21, 65]; line: chemical equilibrium
from [1, 21, 65].
Figure 7.8: Temperature as a function of mixture fraction at x{D “ 20. Red points:
computational result; blue points: experimental data from [1, 21, 65]; line: chemical equilibrium
from [1, 21, 65].
79

It can be seen from the figures that the agreement between experimental data, equilibrium
data and computational results is good, although in both the rich and the lean part of
mixture-fraction space the computational results are somewhat closer to the equilibrium
results than to the experimental data.
Shown in Fig. 7.9 are surface plots of the expectations of the mass fractions of H 2 O and
OH, and of the temperature as obtained by the computations. It can be seen that the
field of all three quantities look reasonable, both in values and shape, as can be expected
due to the reasonable agreement between experimental data and computational results
shown for the centerline profiles in Figs. 7.2 to 7.6. The partial lack of smoothness in
the fields is due to the statistical errors that are – despite forming moving averages, see
Chap. 3.2.2, – inherently present in Monte Carlo simulations due to the finite number of
particles employed.
Figure 7.9: Surface plots – from top to bottom – of the mass fraction of H 2 O, the mass
fraction of OH, and the temperature, respectively.
80

Shown in Fig. 7.10 is the computed marginal density function G c – see Eq. (2.41) – of the
hydroxyl mass fraction as a function of the radial coordinate r at the non-dimensional
axial position x{D “ 5. It can be seen that from the centerline up to approximately
r “ 0.015 m practically no gasphase particle carries OH. The mass fraction of OH then
increases in the radius range 0.015 m ď 0.03 m up to a value of, approximately, 0.003,
however, for not too many gasphase particles. From r “ 0.03 m on, the mass fraction of
OH then decreases again, taking for many gasphase particles frequently very small values
at the edge of the jet, i.e., upon approaching r “ 0.07 m.
Figure 7.10: Marginal density function of the mass fraction of OH at position x{D “ 5.
7.3 Speedup by OpenMP
The results presented and discussed in Chap. 7.2 were obtained by carrying out two
computations on an 8-processor computer. The first computation was carried with a serial
version of the code, the second was carried with a version of the code parallelized by use of
OpenMP. For none of the two computations domain decomposition was employed. Both
computations produced identical results. However, with the parallelized version of the
code a speedup of approximately 1.8 could be achieved.
81

Chapter 8
Turbulent Counterflow Flames with
Water Droplets
8.1 Experimental and Computational Setup
To validate the computer code developed in the present work for turbulent sprays, the
code has been applied to a non-premixed, turbulent methane-air counterflow flame with
water droplets, which was investigated at Cambridge University in 1997 [107]. Specifically,
in the Cambridge experiments the counterflow burner shown schematically in Fig. 8.1 was
used. This burner consisted of two opposed nozzles each with an inner exit diameter of
25 mm. Both nozzles had a honeycomb-flow straightener to ensure a uniform gas flow at
the respective exit plane. Also for both nozzles, the average exit flow velocity was 0.543
m/s. For both nozzles the inlet composition is specified in Table 8.1.
upper nozzle lower nozzle
gaseous phase X CH4 = 0.23 X O2 = 0.21
X N2 = 0.77 X N2 = 0.79
liquid phase Y d =0.906%
Table 8.1: Inlet composition for both the experiment [107] and the present computations.
In the table, X α denotes the mole fraction for gasphase species α, α = CH 4 , O 2 , and N 2 ;
Y d denotes the mass concentration of water droplets,
Y d :“
9m d
9m d ` 9m air
, (8.1)
where 9m d and 9m air denote the mass flow rate of water droplets and the mass flow rate
of the carrier gas, respectively. From the definition (8.1) it can be seen that in the
experiments, and hence also in the present computations, the water drops were added to
82

Figure 8.1: Schematic of the counterflow burner used in the experiments [107].
the oxidizer, not to the fuel.
In the experiments and hence the computations, each nozzle was surrounded by an external
nozzle through which nitrogen was fed to form a curtain that reduced the influence of
the ambient air currents on the flame. The separation distance between the nozzle exits
was 14 mm [106]. The temperature and the pressure there at the nozzle exits were 300 K
and 1 bar, respectively. Details on the experimental arrangements and experimental data
are reported in [107].
83

Figure 8.2: Schematic of computational domain and boundaries.
On the left side of Fig. 8.2 a schematic of the Cambridge burner is shown which helps
to understand the particular choice for the computational domain, which – because of
axisymmetric symmetry – has been taken as a half plane as shown on the right side of the
figure. Since for this computation MPI was tested, the domain has been decomposed into
8 sub-domains that in the pdf-version – not the in the printed version – of the present
thesis each carry a different background color. Roman numbers I to V denote the different
parts of the domain boundary. Specifically, I indicates the upper inlet, II the lower inlet,
III the wall boundary, IV the open boundary where entrainment of air occurs, and V the
symmetry boundary.
Simulations are performed on the unstructured mesh shown in Fig. 8.2. At the inlets
to the computational domain, labelled I and II, respectively, velocity, composition, and
turbulent properties of the gas flow are fixed to the respective boundary values. At the
symmetry axis, V, the velocity components normal to the boundary are set to zero. At
the open boundary, IV, the flow is allowed to adjust its velocity value and direction into
or out of the computational domain.
In the computations 22400 particles were used, and the mesh consisted of 4666 nodes and
9088 triangles.
Gasphase combustion was described by a 2-step reduced chemical mechanism for methane/air
systems [24] consisting of 6 chemical species and 2 chemical reactions.
84

8.2 Results and Discussion
Shown in Figs. 8.3 and 8.4 are profiles in physical space of various quantities as obtained
in experiments performed by Zheng [107] and from the numerical simulations carried out
Figure 8.3: Profile of the mean axial velocity component of water droplets along the
symmetry line. Solid line: computational result; symbols: experimental data from [107].
Figure 8.4: Radial profile of the mean axial velocity component of water droplets at the
axial location z “ 1.4 mm. Solid line: computational result; symbols: experimental data
from [107].
85

in the present work. Specifically, shown in Fig. 8.3 is the mean axial velocity component
of water droplets along the symmetry line of the counterflow geometry.
It can be seen that generally the agreement between the experimental and numerical data
is good. The same is the case for the mean axial velocity component of water droplets as
a function of radius taken at axial station z=1.4 mm, which is shown in Fig. 8.4.
Shown in Fig. 8.5 is a phase plane spanned by the methane mole fraction in the fuel jet
lower Jet: O 2
mole fraction
0.22
0.21
0.20
extinction
stable burning
sim. Y d = 0.4%
exp. Y d = 0.0%
exp. Y d = 0.4%
exp. Y d = 0.9%
increasing the mass
concentration of
water droplets
0.19
0.20 0.23 0.26 0.29 0.32 0.35
upper Jet: CH 4mole fraction
Figure 8.5: Phase-plane identifying regimes of extinction and stable burning. Stars:
computational results, other symbols: experimental data from [107].
and the oxygen mole fraction in the oxidizer jet. In this phase plane, regimes of extinction
and stable burning are identified for three different values of the water mass concentration
Y d . Stars denote computational results obtained herein, the other symbols denote
experimental results [107]. It can be seen that the agreement between the experimental
and the computational results is very good. The computed extinction points shown in
this figure have been obtained from plots such as the one shown in Fig. 8.6. Plotted in
the latter figure is the maximum mean flame temperature as a function of the water mass
concentration Y d as obtained by computations, with inlet mole fractions as specified in
Table 8.1. It can be seen from the graph that extinction has occurred at Y d « 0.4%.
Shown in Fig. 8.7 are two surface plots on top of each other of the mean gasphase temperature
as obtained by computations. In each of the two surface plots to the left and the
86

2000
maximum temperature (K)
1500
1000
500
0
0.0 0.2 0.4 0.6 0.8 1.0
mass concentration of water droplets Y d(%)
Figure 8.6: Maximum mean flame temperature as a function of the water mass concentration
Y d for inlet mole fractions X O2 “ 0.21 and X CH4 “ 0.23.
Figure 8.7: Surface plots of mean temperature with different Y d . A,C: without seeding of
water drops; B: stable flame with Y d “ 0.1%; D: extinction with Y d “ 0.9%.
right of the symmetry line different conditions – these are labelled A, B, C and D – apply.
Specifically, in both the top and the bottom surface plot, to the left of the symmetry line,
i.e., under conditions A and C, respectively, temperature surfaces are displayed that have
been obtained for a purely gaseous counterflow flame, i.e., a flame which was not seeded
with water drops. Naturally, since conditions A and C are identical, so are the corresponding
temperature surface plots. The two surface plots to the right of the symmetry
87

line, labelled B and D, respectively, have been obtained as follow. The plot labelled B
is for a stable burning flame that was seeded with water drops with the under-critical
water mass concentration Y d “ 0.1%. Therefore, this flame pertains to the stable-burning
regime shown in Fig. 8.5. The plot labelled D is for an extinguished flame that was seeded
with water drops with the over-critical water mass concentration Y d “ 0.9%. Therefore,
the latter flame pertains to the extinction regime shown in Fig. 8.5.
Shown in Fig. 8.8 is a surface plot of the mean gasphase temperature and of streamlines
as obtained by a computation for an under-critical case with Y d “ 0.1%. Since the flame
is axisymmetric, so are the temperature surfaces and the streamlines shown in the figure.
The temperature surfaces are as expected and already discussed in the context of Fig.
8.7. In the vicinity of the symmetry axis, between the nozzles, the streamlines show a
stagnation point and an approximate stagnation plane. Just outside the lower nozzle
through which gaseous oxidizer and droplets are supplied, a relatively strong ring-shaped
vortex due to air outflow, caused by expansion, at the bottom of the computational
domain can be identified. Outflow occurs everywhere at the computational boundaries
except at the solid outer walls of the nozzles where no-slip velocity boundary conditions
apply. Also, several smaller vortices can be identified in the computational domain shown
in the figure.
Figure 8.8: Surface plot of mean temperature with streamlines for a case with Y d “ 0.1%.
88

8.3 Speedup by MPI
The results presented and discussed in Chap. 8.2 were obtained by carrying out two
computations on an 8-processor computer. The first computation was carried with a serial
version of the code, the second was carried with a version of the code parallelized by use
of MPI. Both computations produced identical results. However, with the parallelized
version of the code a speedup of approximately 2.3 could be achieved.
89

Chapter 9
Turbulent Jet Diffusion Spray
Flames (Sydney flame)
9.1 Experimental and Computational Setup
To validate the computer code developed in the present work for turbulent spray combustion,
the code has been applied to a turbulent ethanol-air spray flame investigated at the
University of Sydney in 2009 [39, 55]. Specifically, in the Sydney experiments, the spray
burner was assisted by a stabilizing pilot burner, as shown schematically in Fig. 9.1.
The central jet nozzle diameter was 10.5 mm. The outer diameter of the annulus was 25.0
mm. The pilot flame holder was fixed 11 mm from the centre. A co-flow of diameter 104
mm surrounded the burner and had an average flow velocity of 4.5 m/s with temperature
298 K. The pilot stream carried fully burnt combustion products of a stoichiometric
mixture of ethylene/hydrogen and air, with an adiabatic flame temperature of 2493 K.
The bulk velocities of the pilot stream and jet were 11.6 m/s and 24 m/s, respectively.
Details on the experimental arrangements and experimental data are reported in [39, 55].
The liquid phase properties used in the experiments and the present simulations are listed
in Table 9.1.
property and unit value
density ρ l (g/cm 3 ) 0.789
latent heat L (kJ/kg) 846
boiling temperature (K) 351.1
Table 9.1: Properties of liquid ethanol.
The computational domain is shown in the Fig. 9.2. Since for this computation MPI-
OpenMP hybrid parallelization was tested, the domain has been decomposed into 8 sub-
90

Figure 9.1: Schematic of the spray burner used in the Sydney experiments [39, 55].
domains that in the pdf-version – not the in the printed version – of the present thesis
each carry a different background color. Roman numbers I to VI indicate the different
parts of the domain boundary. Specifically, I indicates the jet inlet, II the pilot flame
inlet, III the co-flow inlet, IV the open boundary where entrainment of air occurs, V the
outlet boundary, and VI the symmetry boundary. 1
Simulations are performed on the unstructured mesh shown in Fig. 9.2. The computational
domain extends to 8 and 40 diameters in the radial and axial direction, respectively.
At the inlets to the computational domain, labelled I, II and III, respectively, velocity,
composition, and turbulent properties of the gas flow and liquid flow are fixed to the
1
Note that the problem is axisymmetric.
91

Figure 9.2: Schematic of computational domain and boundaries.
respective boundary values. At the symmetry axis, VI, the velocity components normal
to the boundary are set to zero. At the open boundary, IV, the flow is allowed to adjust
its direction into or out of the computational domain.
In the computations 505072 particles were used, and the mesh consisted of 5156 nodes
and 9924 triangles.
Gasphase combustion was described by a global single-step chemical mechanism for ethanol/air
systems [15] comprising of 5 chemical species.
9.2 Results and Discussion
Shown in Figs. 9.3 and 9.4 are profiles in physical space
of the mean axial velocity
Figure 9.3: Radial profile of the mean axial velocity component of fuel droplets along at
axial station z/D=10. Solid line: computational result; symbols: experimental data from
[39, 55].
92

Figure 9.4: Radial profile of the mean axial velocity component of fuel droplets along at
axial station z/D=20. Solid line: computational result; symbols: experimental data from
[39, 55].
component w as obtained in the experiments performed by Gounder et al. [39, 55], and
from the numerical simulations carried out in the present work. Specifically, Fig. 9.3
refers to the axial station z{D “ 10. It can be seen that generally the agreement between
the experimental and numerical data is good. The same is the case for the mean axial
velocity component of fuel droplets as a function of radius taken at non-dimensional axial
station z{D “ 20, which is shown in Fig. 9.4. Shown in Fig. 9.5 is the radial profile
of mean temperature at z{D “ 10. It can be seen that for small radii the agreement
Figure 9.5: Radial profile of temperature at axial location z/D=10. Solid line: computational
result; symbols: experimental data from [39, 55].
93

is good, but that for radii greater than approximately unity temperatures are greatly
over-predicted by the simulation. A similar over-prediction by the simulation can be
seen in Fig. 9.6, which shows the same radial profile of temperature but at z{D “ 20,
i.e., further downstream. Since on the Sydney flame further experimental data than the
one shown here are not available, the cause of the disagreement between experimental
and computed radial temperature profiles can only be speculated about. For instance,
the observed disagreement may, perhaps, be due to an over-prediction of the transversal
velocity component or to an over-prediction of the rate of turbulent mixing. Since in the
turbulent mixing-model employed the ratio ɛ{k appears, either an over-prediction of the
turbulent dissipation rate ɛ or an under-prediction of the turbulent kinetic energy k could
lead to a mixing rate that is too large.
Figure 9.6: Radial profile of temperature at axial location z/D=20. Solid line: computational
result; symbols: experimental data from [39, 55].
Shown in Fig. 9.7 are surface plots of the expectations of the mean axial velocity component,
the mass fraction of H 2 O, and of the temperature as obtained by the computations.
The plots show surfaces of the three quantities as it is to be expected on physical grounds.
Specifically, in the plot of mean axial velocity, w, the local acceleration of the flow in the
vicinity of the symmetry axis due to heat release and hence expansion is obvious as is, as
a consequence of overall mass conservation, the observed slow-down and homogenization
sufficiently far downstream. Close to the nozzle exit, the surface plots of the mean H 2 O
mass fraction and of the temperature both clearly show the effect of the pilot flame.
Shown in Fig. 9.8 are surface plots of the expectations of the mean axial velocity component
of droplets, w l , and at the mean radial velocity component of droplets, v l , as
94

Figure 9.7: Surface plots – from top to bottom – of the axial velocity component, the
mass fraction of H 2 O, and the temperature, respectively.
Figure 9.8: Surface plots – from top to bottom – of the axial velocity component, and the
radial velocity component, respectively.
95

obtained by the computations. It can be seen that the fields of both velocity components
look reasonable, both in values and shape, as can be expected due to the reasonable agreement
between experimental data and computational results shown for the radial profiles
in Figs. 9.3 and 9.4. From axial station z{D « 8, droplets are strongly accelerated into
the radial direction by expansion due to heat release.
Shown in Fig. 9.9 is the computed marginal density function f W pR l q of droplet radius
– see Eq. (2.50) – along the jet center line as a function of the non-dimensional axial
coordinate z{D. It can be seen that from the nozzle up to approximately z{D “ 5
practically most droplets keep formation as distributed at inlet. The droplet radius R l
then decreases along the axial line to a value of, approximately, 7 µm, however, droplets
has been distributed wider than between z{D “ 0 and 5. This phenomena shown in this
figure can be confirmed by plotting each single droplet as shown in Fig. 9.10.
Figure 9.9: Profile along the jet center line of the marginal density function f W pR l q as a
function of droplets radius.
Shown in Fig. 9.10 is a scatter plot of droplet radius R l along the jet symmetry axis
versus axial distance z from the nozzle exit plane obtained by plotting a point pR l , z{Dq
for each liquid Monte-Carlo particle that – at statistical steady state of the reactive flow
– instantaneously is located somewhere on the symmetry axis. It can be seen that the
scattering is relatively small close to the nozzle exit plane and that it is increasing with
increasing distance from the nozzle. This is to be expected on physical grounds because
96

Figure 9.10: Droplets radius represented by the liquid Monte-Carlo particles along the
symmetry line of the jet.
the initial droplet size distribution, which in the simulations was assumed to be Gaussian,
widens downstream continuously due to differential effects of droplet evaporization.
9.3 Speedup by the MPI-OpenMP Hybrid Approach
The results presented and discussed in Chap. 9.2 were obtained by carrying out two
computations on a 32-processor cluster. The first computation was carried with a serial
version of the code, the second with a version of the code parallelized by use of the MPI-
OpenMP hybrid scheme. Both computations produced identical results. However, with
the parallelized version of the code a speedup of approximately 4.2 could be achieved. It
is interesting to note that computations for this case that were carried out applying only
OpenMP – i.e., without MPI – resulted in a speedup of approximately only 2.
97

Chapter 10
Conclusions and Perspectives
The present thesis has essentially aimed to extend the early research done by Rumberg
[79]. Rumberg’s fully stochastic approach to the formulation and solution of a turbulent
spray flame in one-dimensional geometry with simplified variables representing evaporation,
mixing, and chemical reaction, is now extended to a fully two-dimensional reactive
spray formulation and code.
For the special case of two-phase spray flows, for both the gaseous phase and the liquid
phase PDF-transport equations are given in which the terms concerning the local and
instantaneous interaction between the two phases, and certain turbulence terms require
modelling.
To solve the two-phase spray PDF transport equation, for both the gaseous and the liquid
phase, a particle method has been employed. Specifically, gasphase particles and liquidphase
particles, have simultaneously been used to represent the two coexisting phases.
For closure of various terms, a velocity model, mixing models and evaporation models are
employed. To obtain the turbulent kinetic energy k and the turbulent dissipation rate ɛ,
the standard k-ɛ turbulence model has been used. The latter has been implemented and
solved for in the code in Eulerian form, as was the Poisson equation for pressure.
To validate both the computer code developed and the various physical models for turbulent
spray combustion, the code has been applied to three different experiments, i.e., a
turbulent hydrogen-air flame, a non-premixed, turbulent methane-air counterflow flame
with water droplets, and a turbulent ethanol-air spray flame.
In the present thesis, the two-dimensional reactive spray code has been parallelized by
three different parallelization schemes, i.e., OpenMP, MPI, and the MPI-OpenMP hybrid
scheme. The parallelized code was executed on a 32-processor cluster. Since OpenMP
and MPI have not only strengths but also limitations, the MPI-OpenMP hybrid scheme
98

has been introduced and implemented to combine the advantages and, simultaneously,
mitigate the disadvantages. The computational results in Chaps. 7 to 9 were obtained
with the three parallelization schemes. Specifically, in Chap. 7, the code parallelized by
OpenMP achieved a speedup of 1.8. In Chap. 8, the code parallelized by MPI achieved
a speedup of 2.3. In Chap. 9, the code parallelized by MPI-OpenMP hybrid scheme
achieved a speedup of 4.2.
The present work has shown that the two-phase PDF method is feasible for the simulation
of turbulent reactive sprays. There are, naturally, still points that require improvement
to get more accurate results. For instance, using a generalized Langevin model instead of
the simplified version, would help to predict better the effects of turbulence on velocity.
In addition, a more powerful turbulence model could be implemented. Furthermore, the
present work could be extended to a spray simulation with multi-component liquid fuels.
99

Appendix A
Derivation of the Eulerian Equations
A.1 Pressure Equation
In the present work, the pressure is calculated by SIMPLE-like algorithm which is modified
from SIMPLE [6, 20, 63]. The main point of SIMPLE or SIMPLE liked algorithms is
obtaining the pressure correction field p 1 through combination with the mass continuity
equation and the momentum conservation equation.
Start from mass continuity equation which is written as
And applying separated velocity fields leads
Bρ
Bt ` BρV i
Bx i
“ 0 . (A.1)
Bρ
Bt ` BρV i
˚
“ ´BρV i
1
, (A.2)
Bx i Bx i
where V ˚ denotes a velocity prediction for next time step and V 1 denotes a velocity
correction related as
V n`1
i
Similar assumption applies to the pressure as
“ V ˚
i ` V 1
i .
(A.3)
p n`1 “ p˚ ` p 1 .
(A.4)
The pressure correction p 1 is related to the velocity correction by approximate momentum
equation [6, 20].
ρ BV i
Bt “ ´ Bp
Bx i
(A.5)
100

ρ`V n`1
i
ρ`V ˚
i
˘
´ Vi
n Bp n`1
“ ´∆t
Bx i
˘
´ Vi
n Bp˚
“ ´∆t
Bx i
ρV 1
i “ ´∆t Bp1
Bx i
.
Substituting (A.8) into (A.2), we obtain the pressure-correction Poisson equation.
∆t B ˆ ˙ Bp
1
“ Bρ
Bx i Bx i Bt ` BρV i
˚
Bx i
(A.6)
(A.7)
(A.8)
(A.9)
Integrating over the domain Ω, (A.9) yields
With applying Green’s theorem,
ż
Ω
∆t B2 p 1 ż
Bρ
dΩ “
Bx 2 i
Ω Bt
żΩ
dΩ `
BρV ˚
i
Bx i
dΩ (A.10)
ż
Γ
ż
∆t Bp1
Bρ
¨ n i dΓ “
Bx i Ω Bt
żΓ
dΩ ` ρVi ˚ ¨ n i dΓ , (A.11)
where n denotes normal vector which is perpendicular to interface Γ between two control
volumes.
As shown Fig. A.1, each fragments of triangle has an integral point for surface(ips) and
for volume(ipv).
The first r.h.s term of (A.11) is
where
ż
Ω
ρ ipv “
Bρ
Bt dΩ “
2ÿ
ipv“1
ˆρipv ´ ρ˝ipv
∆t
˙
ipv
Ω ipv ,
3ÿ
pα k ρ k ` β k ρ k x ipv ` γ k ρ k y ipv q .
k“1
(A.12)
(A.13)
The second r.h.s term of (A.11) is
101

Figure A.1: Integrating points
ż
Γ
ρV ˚
i ¨ n i dΓ “
“
2ÿ
ips“1
2ÿ
ips“1
ρ ips V ˚
i ips ¨ n iipsL ips
ρ ips
`u˚ips b ips ` v˚ipsa ips˘
,
(A.14)
where
pn x ` n y q L “ b ` a ,
(A.15)
where L denotes the length of Γ. And
3ÿ
V i,ips “ pα k V i,k ` β k V i,k x ips ` γ k V i,k y ips q .
k“1
(A.16)
with the shape functions α, β and γ introduced in Chaps. 5.2.
The l.h.s term of (A.11) is
ż
Γ
∆t Bp1
Bx i
¨ n i dΓ “
2ÿ
ips“1
1
˙
ˆBp
∆t
Bx b ips ` Bp1
By a ips , (A.17)
102

with
We obtain
ż
Γ
∆t Bp1
Bx i
¨ n i dΓ “
Bp 1
Bx “ βp1 “
Bp 1
By “ γp1 “
2ÿ
ips“1
“ ∆t
3ÿ
β k p 1 k ,
k“1
3ÿ
γ k p 1 k .
k“1
˜ 3ÿ
∆t β k p 1 kb ips `
k“1
3ÿ
2ÿ
k“1 ips“1
3ÿ
γ k p 1 ka ips¸
k“1
pb ips β k ` a ips γ k q p 1 k
(A.18)
(A.19)
(A.20)
Finally, the result of the derivation of the discretized Poisson equation for p is derived as
˘ ÿ ÿ`An p 1 n “ Bn ,
(A.21)
In (A.21), the summation on both the l.h.s. and the r.h.s. extends over all neighboring
triangles of the control-volume defining node n. Specifically,
2ÿ
A n “ ∆t pb ips β n ` a ips γ n q
ips“1
(A.22)
and
B n “
`
2ÿ
ipv“1
2ÿ
ips“1
ˆρipv ´ ρ˝ipv
∆t
˙
Ω ipv
ρ ips
`u˚ips b ips ` v˚ipsa ips˘
,
(A.23)
A.2 Turbulent Kinetic Energy and Dissipation Rate
Turbulent kinetic energy k and its dissipation rate ɛ are calculated by standard k-ɛ turbulence
model [40, 47, 48] and the transport equations of k and ɛ are
B
Bt pρkq ` B pρkV i q “ B „ˆ ˙ j
µ ` µt Bk
Bx i Bx j σ k Bx j
` P k ´ ρɛ (A.24)
103

and
B
Bt pρ Gɛq ` B pρ G ɛV G,i q “ B „ˆ ˙ j
µ ` µt Bɛ
Bx i Bx j σ ɛ Bx j
`
C 1ɛ
ɛ
k P k ´ C 2ɛ ρ G
ɛ 2
k ,
(A.25)
with neglecting the effects of buoyancy and compressibility on turbulence.
Firstly discretized transport equation of k is derived from the equation A.24. Similarly
to discretization of pressure equation, integrating over the domain Ω, Eq. (A.24) yields
ż
Ω
B
Bt
żΩ
pρkq dΩ ` B
pρkV i q dΩ “
Bx i
`
ż
ż
Ω
Ω
„ˆ ˙ j
B
µ ` µt Bk
dΩ
Bx j σ k Bx j
pP k ´ ρɛq dΩ
(A.26)
With applying Green’s theorem on the second l.h.s term of (A.26) and the first r.h.s term
of (A.26), these terms can be expressed as
ż
Ω
ż
B
pρkV i q dΩ “ ρkV i ¨ n i dΓ ,
Bx i Γ
(A.27)
ż
Ω
„ˆ ˙ j ż ˆ ˙
B
µ ` µt Bk
dΩ “ µ ` µt Bk
¨ n i dΓ ,
Bx j σ k Bx j Γ σ k Bx j
(A.28)
where n i , Ω and Γ are introduced previous discretization in A.1.
With further discreization executed at integration point ipv and ips – see Fig. A.1, Eq
(A.26) can be expressed as
2ÿ
ipv“1
ˆρkipv ´ ρk˝ipv
∆t
˙
Ω ipv
`
“
`
2ÿ
ips“1
2ÿ
ips“1
2ÿ
ipv“1
pρkq ips
`u˚ips b ips ` v˚ipsa ips˘
ˆ
µ ` µt
σ k
˙ips
pP k ´ ρɛq ipv
Ω ipv ,
˙ „ˆBk
b ips `
Bx
˙ j ˆBk
a ips
By
(A.29)
104

where
Bk
3ÿ
Bx “
Bk
By
“
k“1
β k k ,
3ÿ
γ k k .
k“1
(A.30)
(A.31)
Production of k is
P k “ µ t S 2
“ ρC µ
k 2
ɛ .
(A.32)
Recall linear interpolation variable {phi at point ips or ipv as
φpx, tq “ α φ ptq ` β φ ptqx ` γ φ ptqy .
(A.33)
The discretized equation for k can be written as
and discretized equation for ɛ is also derived similarly
ÿ`Apkq
ÿ
n k n˘
“ B
pkq
n , (A.34)
ÿ`Apɛq n ɛ n˘
“
ÿ
B
pɛq
n . (A.35)
In (A.34)and (5.20), the summation on both the l.h.s. and the r.h.s. extends over all
neighboring triangles of the control-volume defining node n.
Specifically,
A pkq
n “
`
´
2ÿ
ipv“1
2ÿ
ips“1
2ÿ
ips“1
pα n ` β n x ipv ` γ n y ipv q ρ nΩ ipv
∆t
pα n ` β n x ips ` γ n y ips q ¨ pu n b ips ` v n a ips q ρ n
pα n ` β n x ips ` γ n y ips q ¨ pβ n b ips ` γ n a ips q ¨
ˆ ˙
µ n ` µt,n
σ k,n
(A.36)
105

and
B pkq
n “
`
2ÿ
ipv“1
2ÿ
ipv“1
pα n ` β n x ipv ` γ n y ipv q ρk˝nΩ ipv
∆t
pα n ` β n x ipv ` γ n y ipv q ¨ pP k ´ ρɛq n
Ω ipv
(A.37)
are for k, and
and
are for ɛ.
A pɛq
n “
`
´
B pɛq
n “
`
2ÿ
ipv“1
2ÿ
ips“1
2ÿ
ips“1
2ÿ
ipv“1
2ÿ
ipv“1
pα n ` β n x ipv ` γ n y ipv q ρ nΩ ipv
∆t
pα n ` β n x ips ` γ n y ips q ¨ pu n b ips ` v n a ips q ρ n
pα n ` β n x ips ` γ n y ips q ¨ pβ n b ips ` γ n a ips q ¨
pα n ` β n x ipv ` γ n y ipv q ρɛ˝nΩ ipv
∆t
ˆ
˙
ɛ
pα n ` β n x ipv ` γ n y ipv q ¨ C 1ɛ
k P ɛ 2
k ´ C 2ɛ ρ G
k
ˆ ˙
µ n ` µt,n
σ ɛ,n
Ω ipv
n
(A.38)
(A.39)
106

Appendix B
Chemical Reaction Mechanism
As discussed in Chap. 4.2, the chemical source term is closed form in gasphase PDF transport
equation, therefore, the chemical reaction affects directly in the particle composition
increment [60]. To obtain a composition increment due to a reaction, the reaction rates
should be consulted. In 1889, Svante Arrhenius proposed a formula that can describe the
rate of a reaction as a function of temperature. This formula is so-called an Arrhenius
equation. Specifically, the Arrhenius equation describes the temperature dependence of
reaction rate constant k by
ˆ
k “ A exp ´ E ˙
, (B.1)
RT
where A is the pre-exponential factor, E the activation energy, R the universal gas constant,
and T the temperature. Equation (B.1) was extended to describe high-temperatures
gasphase kinetic systems. The extended Arrhenius equation describes the temperature
dependence of the rate coefficient as
ˆ
k “ A T n exp ´ E ˙
. (B.2)
RT
Chemical reaction of fuel and oxidizer will lead to several intermediate species and require
several or even many reaction steps until final combustion products are obtained.
The production and consumption reaction-steps of reactants and intermediate species are
called elementary reactions, and set of elementary reactions is called a detailed kinetic
mechanism. Generally, detailed kinetic mechanisms are developed to cover wide ranges
of operating conditions in a variety of different applications [45, 97, 104]. Thus detailed
kinetic mechanisms contain hundreds to thousands of chemical species and reactions. In
order to use kinetic mechanisms in CFD, detailed mechanisms should be reduced to a
few numbers of species and chemical reactions, or even reduced to a single-step global
reaction.
107

In the present thesis, mainly three chemical mechanisms have been used. Pure gaseous
combustion in Chap. 7 has been described by a detailed chemical mechanism of hydrogen
comprising of 10 chemical species and 21 chemical reactions. Detailed reaction information
should be consulted from [61]. For the spray flame with counterflow geometry in Chap.
8, gasphase combustion has been described by a 2-step reduced chemical mechanism for
methane/air systems [24]. The reactions and conditions are reported in Table B.1. The
Reaction A E a
1 CH 4 + 1.5O 2 Ñ CO + 2H 2 O 2E15 35000
2 CO + 0.5O 2 Ø CO 2 2E9 12000
Table B.1: Reduced chemical mechanism of methane [24]. The activation energies E a are
in cal/mole and the pre-exponential constants in cgs units.
reaction order n CH 4
ford and nO 2
ford
for first reaction, are 0.9 and 1.1, respectively and nCO ford , nO 2
ford
and n CO 2
rord
for second reaction are 1, 0.5 and 1, respectively. Abbreviation ford and rord
mean forward and reversed order, respectively. For the jet diffusion spray flame in Chap.
9, gasphase combustion has been described by a global single-step chemical mechanism
for ethanol/air systems [15] comprising of 5 chemical species. According to [15] The mole
based global single-step reaction rate 9w for ethanol/air, can be written as
˙
ˆ´1.256 ˆ 10
9w “ 1.55 ˆ 10 10 8
exp
rFs 0.15 rOs 1.6
8314.3 ˆ T
with the stoichiometric condition as
kmol
m 3 s ,
(B.3)
C 2 H 2 ` 2H 2 ` 3.5 pO 2 ` 3.76N 2 q ÝÑ 2CO 2 ` 3H 2 O ` 13.16N 2 .
(B.4)
108

Appendix C
Mixing Models
In the present work, effects of the molecular diffusion are taken into account through
the mixing models – see Chap. 4.2. In a reactive flow, especially non-premixed flame,
suitable mixing effect is important to get reasonable and stable burning, hence the need
to test three mixing models, i.e., IEM mixing model, modified Curl mixing model, and
EMST mixing model, consulted in the present thesis. These tests have applied on a single
control volume. In order to observe the effect of molecular mixing, the chemical reaction
has been eliminated and the velocity components in both directions were also neglected so
that the particles remain in the single control volume, therefore, the mixing results would
be not influenced by the velocity. In the tests have used ensemble particles comprising two
composition scalars, i.e., temperature T and the mass fraction of O 2 . Initially particles
have been distributed with random properties, for this case, T and Y O2 . The value of
Y O2
has been distributed between zero and unity, and Temperature T has been set to be
randomly distributed between 300´600 Kelvin. For the Standard test, Ensemble number
of particles has been considered to be P “ 1000 and the time step is fixed as ∆t “ 10´6 .
Also the turbulent timescale τ (κ{ε) has been set to the value of 1.0.
To compare three different mixing model, the so-called convergence parameter σ α for the
component scalar α has been defined as
d řP
p“1 pψp α ´ xψ α y 2 q
σ α “
P
, (C.1)
where p and P denote particle index and number of particles, respectively. This convergence
of σ α indicates that value of the composition α of particles relaxing towards the
mean and therefor the decay of the variance. Figure C.1 is shown the convergence of all
models in comparison to each other. It can be seen that IEM model and EMST model
with approximately the same rate, have the fastest convergence among all models. Then,
109

modified Curl model converge, although more slowly. In case of modified Curl model, it
can be observed that the convergence have not occurred completely. The reason is in the
nature of this model, where a certain number of the particles are chosen in each time step
to engage in mixing. Therefore, in some cases, in the final steps one or more particles
would remain out of the selection causing a slight derivation from the presumed mean
value and also an incomplete convergence. This phenomena could be seen in the last
panels in Fig. C.3. As expected in Chap. 4.2, two mixing models IEM and Modified Curl
have straightforward and simple implementations and approaches. Although they would
not produce results with the highest accuracy in case of complicated problems, they have
the advantage of the speed of computation. This two models would require the least and
the most effortless calculations, therefor time and resources in order to converge. This
test have been performed over the single control volume in several different conditions for
all mixing models mentioned above. It was observed that the change in the number of
particles P in the control volume would not alter the results significantly. Although the
standard condition for this test includes 1000 particles, obviously this number would not
be a logical choice for further more complicated tests as discussed in Chap. 5.5.
Figure C.1: The convergence comparing between three mixing models.
110

Figure C.2: The convergence shapes of IEM mixing model.
111

Figure C.3: The convergence shapes of modified Curl mixing model.
112

Figure C.4: The convergence shapes of EMST mixing model.
113

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