A MULTI-DIMENSIONAL FLAMELET MODEL FOR IGNITION IN
MULTI-FEED COMBUSTION SYSTEMS
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF AERONAUTICS AND
ASTRONAUTICS
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Eric Michael Doran
March 2011
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/hn380sm8596
© 2011 by Eric Michael Doran. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Heinz Pitsch, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Brian Cantwell
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Gianluca Iaccarino
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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Abstract
The global demand for energy is steadily rising, particularly in the transportation sector
where the combustion of liquid fuels is projected to remain the largest source of energy
during the coming decades. However, due to increasing fuel costs and environmental con-
cerns, new engine designs that offer improved fuel efficiency and lower pollutant emissions
are needed. As a result, there is continued interest in compression ignition engines that are
characterized by high thermal efficiencies, such as modern diesel and Homogeneous Charge
Compression Ignition (HCCI) type designs. In order to achieve clean, efficient, and stable
combustion, these high efficiency engines often inject the fuel using several separate pulses
for improved control of the combustion process. This increases the complexity of the system
and thereby introduces new challenges in the engine design process. Numerical simulations
are a powerful tool that can be employed to help address these challenges and provide in-
sight into the physical phenomena that occur in engines. To enable the use of simulations
in the design of modern engine concepts, this work develops a computational framework
for modeling turbulent combustion in multi-feed systems that can be applied to internal
combustion engines with multiple injections.
In the first part of this work, the laminar flamelet equations are extended to two dimen-
sions to enable the representation of a three-feed system that can be characterized by two
mixture fractions. A coupling between the resulting equations and the turbulent flow field
that enables the use of this method in unsteady simulations is then introduced. Models are
developed to describe the scalar dissipation rates of each mixture fraction, which are the
parameters that determine the influence of turbulent mixing on the flame structure. Fur-
thermore, a new understanding of the function of the joint dissipation rate of both mixture
fractions is discussed.
Next, the extended flamelet equations are validated using Direct Numerical Simula-
tions (DNS) of multi-stream ignition that employ detailed finite-rate chemistry. The results
v
demonstrate that the ignition of the overall mixture is influenced by heat and mass transfer
between the fuel streams and that this interaction is manifested as a front propagation in
two-dimensional mixture fraction space. The flamelet model is shown to capture this be-
havior well and is therefore able to accurately describe the ignition process of each mixture.
To provide closure between the flamelet chemistry and the turbulent flow field, informa-
tion about the joint statistics of the two mixture fractions is required. An investigation of
the joint probability density function (PDF) was carried out using DNS of two scalars mix-
ing in stationary isotropic turbulence. It was found that available models for the joint PDF
lack the ability to conserve all second-order moments necessary for an adequate description
of the mixing field. A new five parameter bivariate beta distribution was therefore devel-
oped and shown to describe the joint PDF more accurately throughout the entire mixing
time and for a wide range of initial conditions.
Finally, the proposed model framework is applied in the simulation of a split-injection
diesel engine and compared with experimental results. A range of operating points and
different injection strategies are investigated. Comparisons with the experimental pressure
traces show that the model is able to predict the ignition delay of each injection and the
overall combustion process with good accuracy. These results indicate that the model is
applicable to the range of regimes found in diesel combustion.
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Acknowledgement
I would like to express my gratitude to my advisor, Prof. Heinz Pitsch, for his support and
guidance in my research, as well as Prof. Brian Cantwell and Prof. Gianluca Iaccarino for
taking the time to participate on my reading committee and for their helpful comments
during the preparation of this work. I would also like to gratefully acknowledge funding
from the Research and Technology Center of Robert Bosch LLC and NASA Ames Research
Center.
I am very fortunate to have had the opportunity to interact with many of my fellow en-
gineering students throughout my studies. This work would not have been possible without
the many discussions and collaboration with colleagues. Thanks to David Cook, Ed Knud-
sen, Vincent Le Chenadec, Varun Mittal, Michael Mueller, Christoph Schmitt, Shashank,
and all the others that I have worked with during my time at Stanford.
Finally, I would like to thank my family and friends for the continual encouragement
and support necessary to allow me to complete this work.
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Contents
Abstract v
Acknowledgement vii
Nomenclature xi
List of Tables xiv
List of Figures xv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Simulation in Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Summary of Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Theory and Model Development 8
2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Turbulent Scales and Averaging Methods . . . . . . . . . . . . . . . 10
2.1.2 Liquid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Laminar Flamelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Definition of Mixture Fraction . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Flamelet Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.3 Scalar Dissipation Rate . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.4 Co-ordinate Transformation . . . . . . . . . . . . . . . . . . . . . . . 28
viii
3 Representative Interactive Flamelets (RIF) 31
3.1 Coupling of Chemistry with Turbulent Flow Field . . . . . . . . . . . . . . 31
3.2 Description of Turbulent Field . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Energy Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.2 Mixing field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Scalar Dissipation Rate Modeling . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Single Mixture Fraction . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.2 Joint Dissipation Rate Model . . . . . . . . . . . . . . . . . . . . . . 40
3.3.3 Independence of Scalar Dissipation Rates . . . . . . . . . . . . . . . 44
3.4 Initialization of a Two-dimensional Flamelet Field . . . . . . . . . . . . . . 46
3.5 Calculation of Mean Quantities . . . . . . . . . . . . . . . . . . . . . . . . . 48
4 Ignition of Multi-Stream Systems 50
4.1 DNS with Finite Rate Chemistry . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.2 Chemical Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . 53
4.2 Multi-dimensional Flamelet Model . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.1 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Validation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Two Feed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.2 Three Feed System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.3 Effect of timing and maximum mixture fraction . . . . . . . . . . . . 70
5 Modeling Joint Scalar Statistics 74
5.1 Model Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Joint Scalar PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 Dirichlet Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.2 Statistically-most-likely Distribution . . . . . . . . . . . . . . . . . . 77
5.2.3 Bivariate Beta Distribution . . . . . . . . . . . . . . . . . . . . . . . 78
5.3 DNS of Two-Scalar Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.1 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . 79
5.3.2 Initial Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Validation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
ix
5.4.1 Symmetric Initial Field . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.4.2 Asymmetric Initial Field . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Application to Split-Injection Diesel Engine 99
6.1 Research Engine Facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.1.2 Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2 Numerical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.2 Spray Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2.3 Chemical Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . . 114
6.3.1 Comparison of Injection Strategies . . . . . . . . . . . . . . . . . . . 114
6.3.2 Variation of Load and EGR . . . . . . . . . . . . . . . . . . . . . . . 116
7 Conclusion 122
A On Bivariate Beta Distributions 125
A.1 Continuation of Appell’s hypergeometric series . . . . . . . . . . . . . . . . 127
A.2 Marginal Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.3 Product Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
x
Nomenclature
Roman Symbols
as Strain rate
cp Specific heat at constant pressure
D Diffusivity
DJS Jenson-Shannon divergence
DKL Kullback-Leibler pseudo-distance
E Energy spectrum function
F2 Appell function of the second kind
F3 Appell function of the third kind
H Total enthalpy
h Specific enthalpy
k Turbulent kinetic energy
Mw Molecular weight
P Probability density function
p pressure
Ru Universal gas constant
Sij Strain rate tensor
T Temperature
t Time co-ordinate
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u Flow velocity
V Volume
X Mole fraction
x Spatial co-ordinate
Y Mass fraction
Z Mixture fraction
B Beta function
pFq Generalized hypergeometric function
m Mass flow rate
q Heat flux
S Source term
Greek Symbols
β Parameters for beta distribution
χ Scalar dissipation rate
∆h◦f Enthalpy of formation
∆ Grid spacing
δij Kronecker delta
ε Dissipation of turbulent kinetic energy
η Kolmogorov length scale
Γ Gamma function
κ Wavenumber
Λ Length scale
λ Taylor microscale
µ Dynamic viscosity
ν Kinematic viscosity
xii
ω Chemical source term
Φ Equivalence ratio
φ Generic Scalar
ρ Density
ρc Correlation coefficient
σij Viscous stress tensor
τ Time scale
τeddy Eddy turnover time
Non-dimensional Numbers
BY Spalding number
Le Lewis number
Pr Prandtl number
Re Reynolds number
Sc Schmidt number
Operators
( · ) Density-weighted (Favre) filter
( · ) Non-density-weighted filter
Abbreviations
CFL Courant-Friedrichs-Lewy number
DNS Direct numerical simulation
EGR Exhaust gas recirculation
EOI End of injection
EVC Exhaust valve close
EVO Exhaust valve open
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IMEP Indicated mean effective pressure
IVC Intake valve close
IVO Intake valve open
LES Large-eddy simulation
PDF Probability density function
RANS Reynolds-averaged Navier-Stokes
r.m.s. root mean square
SML Statistically-most-likely
SOE Start of energization
SOI Start of injection
TDC Top dead center
xiv
List of Tables
4.1 Summary of boundary conditions of each stream for DNS studies. . . . . . . 54
5.1 Turbulent field input parameters . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Turbulent quantities computed from simulation . . . . . . . . . . . . . . . . 81
5.3 Input parameters for scalar field initialization . . . . . . . . . . . . . . . . . 84
5.4 Average Jenson-Shannon divergence, DJS , during mixing . . . . . . . . . . . 95
6.1 Geometry of single cylinder diesel engine . . . . . . . . . . . . . . . . . . . . 100
6.2 Diesel injector characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.3 Diesel engine operating conditions. . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Summary of experimental injection parameters. . . . . . . . . . . . . . . . . 103
6.5 Summary of charge composition used for each engine operating point. The
remaining mass is nitrogen. . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6 Spray model parameters used for KH-RT secondary break-up model. . . . . 113
A.1 Mapping of Appell-beta to F2- and F3-beta . . . . . . . . . . . . . . . . . . 127
xv
List of Figures
1.1 Historical data and projected world marketed energy use by fuel type, 1990-
2035. Liquid fuel represents both conventional and unconventional sources
and renewables include hydroelectric, wind, solar, and biofuels. (Source U.S.
Energy Information Administration (2010).) . . . . . . . . . . . . . . . . . . 2
2.1 One possible configuration for a three-feed system and the corresponding
mapping to mixture fraction space based on Eq. (2.37). . . . . . . . . . . . 17
2.2 Transformation of solution domain for flamelet with two mixture fractions.
Dashed lines show effect on lines of constant Z1 (A,1 − A), constant Z2
(B,1−B), and constant total mixture fraction Z1 + Z2 (C). . . . . . . . . . 30
3.1 Schematic of the RIF concept showing the coupling of the CFD and chemistry
solvers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Marginal distribution of χ1 conditioned on Z1 from mixing of isotropic scalars
in isotropic turbulence for two time instances. Figure (a) shows the initial
distribution and (b) is after 1/4 eddy turnover. The red curve is the mean
computed from the DNS and the filled grey region represents the minimum
and maximum extents of χ1 in the entire domain. Three model distributions
are also plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Distributions of scalar dissipation rates conditioned on (Z1, Z2) as computed
from DNS of two scalar mixing in isotropic turbulence at t ≈ 0. Each mixture
fraction was initialized isotropically in the physical domain. Note that χ0 was
not conditioned from the DNS data, but was rather computed from the data
according to Eq. (2.74). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 Angle of alignment between physical gradients of each mixture fraction. . . 44
xvi
3.5 Scalar dissipation rate χ1 conditioned on Z2 at for scalar fields with unequal
means. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.6 Two-dimensional distribution of χ1 for scalar fields with unequal means. Fig-
ure (a) shows data conditioned from DNS data and (b) shows model distri-
bution using the method of this section. . . . . . . . . . . . . . . . . . . . . 47
3.7 Interpolation used to initialize a 2D solution in mixture fraction space from
an existing 1D solution and a new stream boundary condition. Dashed lines
represent mixing lines along which interpolation is carried out. . . . . . . . 48
4.1 Transfer number computed for thermal and concentration gradients to find
the surface temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Computation of flamelet using data from DNS studies for validation purposes 57
4.3 Initial mixture fraction and temperature fields of the two feed DNS . . . . . 59
4.4 Comparison of pressure and heat release rate computed using DNS and a
one-dimensional flamelet model during ignition and combustion of a single
fuel stream with Φ = 0.789 and Zmax = 0.2. . . . . . . . . . . . . . . . . . 60
4.5 Evolution of temperature during ignition as a function of mixture fraction.
Curves are computed from flamelet model, whereas points are conditioned
from the DNS data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Initial mixture fraction of each fuel stream and the resulting temperature
distribution in the DNS domain for a three-feed system. The second fuel
stream is introduced at a time τ2 = 0.4 ms after initialization of stream 1 (Z1). 62
4.7 Comparison of pressure and heat release rate of single and split fuel stream
systems. The second fuel stream was introduced at a time τ2 = 0.4 ms after
the initialization of the first stream, which contained 20% of the overall fuel
mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.8 Budget of heat release rate contribution from each fuel stream and the inter-
acting region where both fuel streams are present. . . . . . . . . . . . . . . 64
4.9 Temporal evolution of the temperature conditioned on mixture fraction dur-
ing ignition of stream 1. Results are plotted for data along the Z1 axis (Z2 = 0). 65
xvii
4.10 Evolution of DNS properties conditioned on mixture fraction. Time instances
of temperature, OH mass fraction, and computed heat release rate are shown
in each row from top to bottom, with time increasing to the right according
to the axis shown at bottom. . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.11 Temporal evolution of the temperature conditioned on mixture fraction dur-
ing ignition of stream 1. Results are plotted for data along the Z2 axis (Z1 = 0). 67
4.12 Comparison with DNS results of pressure and heat release computed by the
full 2D flamelet and the average solution of a 1D flamelet for each stream
according to Eq. (4.12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.13 Evolution of temperature conditioned on mixture fraction from DNS (top
row) and two-dimensional flamelet model computations (bottom row). Time
increases from left to right according to the axis shown at bottom. . . . . . 69
4.14 Heat release rates for fuel streams with Z1,max = Z2,max = 0.2 and different
delay times (τ2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.15 Comparison of pressure and heat release rates of DNS and flamelet for fuel
streams with Z1,max = Z2,max = 0.2 and different delay times (τ2). . . . . . 72
4.16 Heat release rates for fuel streams with Z1,max = 0.1 and Z1,max = 0.2 and
various delay times (τ2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.1 Schematic indicating the sectors used to determine initial state of scalars.
The three sectors are divided by the two dashed lines, which are determined
from the scalar means according to Eq. (5.25), and the positive φ1 axis. The
circles at (0,0), (0,1), and (1,0) represent the three initial states assigned to
values found in each corresponding sector. . . . . . . . . . . . . . . . . . . . 83
5.2 Planar cross-sections of domain showing typical initial distribution of φ1
and φ2 for different initialization methods (first and second columns, re-
spectively). Scalar field initialization parameters are given in Table 5.3. . . 84
xviii
5.3 Time evolution of joint probability distribution, P (φ1, φ2), for statistically
symmetric isotropic scalar (I01). The top row is the computed disribution
(DNS), with subsequent rows representing the model bivariate beta distribu-
tion (BVB5), statistically-most-likely distribution (SML), and the Dirichlet
distribution, respectively. Each column represents a fixed time increasing
from left to right with instances taken at φ′/φ′0 of 0.95, 0.80, 0.30, and 0.10,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.4 Jenson-Shannon divergence of the bivariate beta (BVB5), statistically-most-
likely (SML), and Dirichlet distributions for symmetric initial distribution
from isotropic scalars with equal means (case I01). . . . . . . . . . . . . . . 90
5.5 Time variation of scalar r.m.s. and correlation coefficient of fields with equal
means using different initialization methods. . . . . . . . . . . . . . . . . . . 90
5.6 Time evolution of joint probability distribution, P (φ1, φ2), for statistically
symmetric isotropic scalar with layered initial field (L05). The top row
is the computed disribution (DNS), with subsequent rows representing the
model bivariate beta distribution (BVB5), statistically-most-likely distribu-
tion (SML), and the Dirichlet distribution, respectively. Each column repre-
sents a fixed time increasing from left to right with instances taken at φ′/φ′0
of 0.90, 0.70, 0.50, and 0.30, respectively. . . . . . . . . . . . . . . . . . . . . 92
5.7 Jenson-Shannon divergence comparison of the bivariate beta (BVB5), statistically-
most-likely (SML), and Dirichlet distributions for symmetric initial distribu-
tion with layered scalars of equal means (case L05). . . . . . . . . . . . . . . 93
5.8 Time evolution of joint probability distribution, P (φ1, φ2), for statistically
asymmetric scalar field initialized with a partially mixed scalar (M10). The
top row is the computed disribution (DNS), with subsequent rows represent-
ing the model bivariate beta distribution (BVB5), statistically-most-likely
distribution (SML), and the Dirichlet distribution, respectively. Each col-
umn represents a fixed time increasing from left to right with instances taken
at φ′/φ′0 of 0.95, 0.80, 0.30, and 0.15, respectively. . . . . . . . . . . . . . . 94
5.9 Jenson-Shannon divergence comparison of the bivariate beta (BVB5), statistically-
most-likely (SML), and Dirichlet distributions for an asymmetric initial field
from a partially mixed scalar (case M10). . . . . . . . . . . . . . . . . . . . 95
xix
5.10 Variation of correlation coefficient over time for layered initial scalar fields
(see Table 5.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.11 Jenson-Shannon divergence comparison of the bivariate beta (BVB5), statistically-
most-likely (SML), and Dirichlet distributions for a layered initial field with
strong positive correlation (L07). . . . . . . . . . . . . . . . . . . . . . . . . 96
5.12 Joint probability distribution, P (φ1, φ2), for layered initialization with posi-
tive correlation (L07) at one time instance with φ′/φ′0 = 0.3 (t/τeddy = 1.2).
The computed distribution is on the far left and the three models are to the
right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.13 Marginal distributions of case L07 shown in Fig. 5.12 at φ′/φ′0 = 0.3 (t/τeddy =
1.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1 Images of the single cylinder direct-injection research engine and test facility
at Robert Bosch, Schwieberdingen, Germany. . . . . . . . . . . . . . . . . . 100
6.2 Volumetric fuel flow rate of typical split-injection computed from AMESim
1D hydraulic model. Dwell time marked as time between pilot EOI and main
SOI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3 Experimental mass flow rates for cases 512MR3 and 513MR8 showing dif-
ferent injection strategies. Injection timings are tabulated in Table 6.4 and
injection fuel mass is given in Table 6.3. . . . . . . . . . . . . . . . . . . . . 104
6.4 Experimental mass flow rates for cases OP10, OP12, and OP13 showing dif-
ferent injection profiles for different loading. Injection timings are tabulated
in Table 6.4 and injection fuel mass is given in Table 6.3. . . . . . . . . . . 105
6.5 Computational mesh of split-injection diesel engine. Top image shows full
3D geometry and bottom image is a section cut at 1 CAD bTDC . . . . . . 106
6.6 Valve timing diagram for M47 experimental engine showing Exhaust Valve
Open/Close (EVO/EVC) and Intake Valve Open/Close (IVO/IVC) with re-
spect to Top Dead Center (TDC). . . . . . . . . . . . . . . . . . . . . . . . 107
6.7 Velocity fields initialized using swirl number after having advanced to just
prior to injection. Cut plane of vector field is halfway between the cylinder
head and piston. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
xx
6.8 Experimental and computed cylinder pressure, p, for the different injection
strategies of operating points 512MR3 and 513MR8 (see Tables 6.3 and 6.4).
Curves are computed results and points are averaged experimental data. The
timing and duration of each injection pulse are represented by the thick solid
lines at the top of the graph, with topmost describing case 512MR3. . . . . 115
6.9 Three-dimensional cylinder temperature of case 512MR3. Each time instance
shows a planar cut normal to the cylinder axis 3 mm below the cylinder head
and a cross-section of the piston bowl. The section locations are indicated in
plot (d) and the black curves represent isocontours of stoichiometric mixture. 117
6.10 Three-dimensional cylinder temperature of case 513MR8. Each time instance
shows a planar cut normal to the cylinder axis 3 mm below the cylinder head
and a cross-section of the piston bowl. The section locations are indicated in
plot (d) and the black curves represent isocontours of stoichiometric mixture. 118
6.11 Experimental and computed cylinder pressure, p, for different engine loads
(IMEP) with no EGR (see Tables 6.3 and 6.4). Curves are computed results
and points are averaged experimental data. The timing and duration of each
injection pulse are represented by the thick solid lines at the top of the graph
for OP13, OP12, and OP10 starting from topmost. . . . . . . . . . . . . . . 119
6.12 Experimental and computed cylinder pressure, p, for different engine loads
(IMEP) with EGR (see Tables 6.3 and 6.4). Curves are computed results
and points are averaged experimental data. The timing and duration of each
injection pulse are represented by the thick solid lines at the top of the graph,
with OP13 the topmost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
xxi
xxii
Chapter 1
Introduction
1.1 Motivation
Although there has been substantial research recently to develop renewable energy sources,
the combustion of fossil fuels will remain a primary energy source for the foreseeable future.
According to the U.S. Energy Information Administration (2010), world energy consump-
tion is projected to increase by 49% from 2007 to 2035, driven primarily by sustained
growth in developing economies. The contribution of different energy sources to overall
use is plotted in Fig. 1.1, where it can be seen that liquid fuels, coal, and natural gas
currently account for approximately 85% of total energy consumption. The market share
of renewables and nuclear energy is estimated to increase from approximately 15% to 30%
over the next three decades, which is insufficient for projected energy demands and thus
the demand for combustion devices will also increase. Increased demand is particularly true
for liquid petroleum fuels, primarily in the transportation sector, where an increase of 1.3%
per year is expected and liquid fuels will continue to dominate the transportation sector
in the absence of significant technological developments. The increasing cost of fossil fuels
and the introduction of ever more stringent future emissions regulations suggests that the
development of improved combustion devices with high fuel efficiency and low emissions
will play a significant role in future energy strategies.
The continued reliance of the transportation industry on liquid fuels has led to sub-
stantial focus in the automotive industry on methods to increase efficiency. Although the
internal combustion engine has been in use for over a century, continual application of new
technology has resulted in a steady increase in overall efficiency. Recently, there has been
1
2 CHAPTER 1. INTRODUCTION
2 CHAPTER 1. TEST
0
50
100
150
200
250
1990 2000 2007 2015 2025 2035
Energy
Consumption(B
tu×10
15)
Year
LiquidsCoal
Natural GasRenewables
Nuclear
Historical Projected
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 1.1: Historical data and projected world marketed energy use by fuel type, 1990-2035.Liquid fuel represents both conventional and unconventional sources and renewables includehydroelectric, wind, solar, and biofuels. (Source U.S. Energy Information Administration(2010).)
increasing focus on the use of compression ignition type engines, as opposed to gasoline
spark-ignition type engines, due to their inherent ability to attain higher thermal efficien-
cies. This class of engines include the traditional Diesel engine, as well as the more recent
Homogenous Charge Compression Ignition (HCCI) technology.
Historically, the diesel engine has been associated with noise and pollution. The combus-
tion in traditional diesels causes considerable emissions of both soot particulate and oxides
of nitrogen (NOx). However, the diesel engine has experienced a steady increase in popu-
larity, especially in Europe, due to its higher overall efficiency when compared with gasoline
engines, coupled with increasing improvements in drivability and significant reductions in
both noise and harmful emissions. Much of the improvement stems from the introduction of
high-pressure common-rail injection systems that allow the fuel to be delivered in multiple
pulses per cycle. Such advanced injection strategies help avoid the conventional Soot/NOx
tradeoff, whereby a change in operating conditions to reduce one pollutant results in a more
favorable condition for creation of the other. For example, since NOx is formed at high tem-
peratures, reducing the temperature below the NOx threshold results in less oxidation of
soot formed in the fuel rich regions. It has been shown experimentally that splitting the
1.1. MOTIVATION 3
fuel injection into pulses helps to split the combustion into phases, resulting in a lower
overall combustion temperature for reduced NOx and fewer fuel rich soot producing re-
gions (Chan et al., 1997; Chen, 2000; Han et al., 1996; Montgomery & Reitz, 2001; Yamane
& Shimamoto, 2002). Additionally, splitting the injection reduces the rate of pressure rise,
helping to alleviate noise and wear on the engine.
Modern engine concepts like HCCI are also being pursued and are a hybrid of the
traditional Diesel and Otto cycles. In pure HCCI, a homogeneous premixed charge of fuel
and air is introduced into the engine. The composition of the mixture is set such that it
will ignite through auto-ignition at the end of the compression stroke of the piston. This
combustion process is attractive as it can provide high efficiencies comparable to those of
a diesel engine, while also achieving very low NOx and soot emissions. The low emissions
are a result of operating the engines at very lean conditions, thus maintaining operating
temperatures below the NOx formation threshold and avoiding fuel rich regions conducive
to soot formation. However, such a configuration lacks the direct ignition timing control
that is present in traditional engines; i.e. a spark for gasoline engines and the fuel injection
for diesel engines. The only control over combustion timing is through mixture preparation,
which is much more difficult to achieve consistently. Thus, widespread introduction of HCCI
engines has been hampered through a limited range of operating conditions resulting from
control difficulties.
Recent research in HCCI engines has focused on introducing control through the use
of multiple injections. For example, the majority of the fuel may be injected early in the
cycle, giving it a long time to reach a homogeneity, but at conditions too lean to auto-ignite
at the end of the compression stroke. A post injection can then be used to control the
start of combustion while retaining the overall lean characteristics of the mixture. Another
problem with HCCI operation is an overly rapid pressure rise during the combustion, leading
to significant noise and eventually engine damage. To prevent this, exhaust gas recirculation
can be used to introduce thermal inhomogeneities into the engine, causing the combustion
in the cylinder to not occur simultaneously and thus damping the pressure rise rate (Epping
et al., 2002).
Another approach is to use blends of multiple fuels in the combustion process. Recent
studies have investigated a multiple injection strategy whereby gasoline is injected using
port fuel injection and diesel is introduced through direct injection later in the cycle to
produce a type of reactivity controlled combustion (Hanson et al., 2010; Splitter et al., 2010).
4 CHAPTER 1. INTRODUCTION
The blending of the different fuels provides regions of charge mixture that have differing
reactivity, which can lead to favorable combustion conditions. The findings demonstrated
that it was possible to extend the duration of combustion and thus control the pressure rise
rate by varying the amount of each of the blended fuels. By controlling the reactivity, it may
also be able to cause the majority of heat release from combustion to occur in the center
of the cylinder, thus helping to reduce heat losses and provide a higher effective thermal
efficiency. Overall, this method has great potential to operate engines at high efficiency
with ultra-low emissions.
From the preceding discussion, it is apparent that modern engines will have multiple
streams contributing to the charge preparation and combustion. These can take the form of
either multiple fuel injections, the use of multiple fuels, or even the introduction of exhaust
gas recirculation. Apart from internal combustion engines, other advanced combustion
devices employ multiple fuel and oxidizer feeds. For example, gas turbines regularly use
secondary air downstream of the combustor to help control maximum turbine inlet temper-
atures. Thus, an understanding of the mixing and interaction of multiple feed systems, as
well as the effect on ignition and combustion, is necessary for future combustion devices.
1.2 Simulation in Design
Further improvement of existing combustion devices for higher efficiency and lower emis-
sions will lead to more complex configurations with additional operational parameters. As
system complexity increases, the role of simulation in aiding the design process becomes
more important. Simulations can provide critical insight into the different mechanisms and
overall behavior of practical devices and while not able to replace experimental develop-
ment entirely, simulations have the potential to considerably reduce the time and cost of
the traditional development cycle.
In order to describe the different physical phenomena present in complex combustion
devices a wide range of physical length and time scales must be accounted for. Even with
modern computing power, it is prohibitively expensive to fully resolve all the relevant phys-
ical scales using Direct Numerical Simulations (DNS), except in simple configurations. To
apply simulations to practical devices, modeling is required. The turbulent flow field may be
partially resolved using Large Eddy Simulation (LES) techniques, where the large, energy
containing scales are numerically resolved, whereas the effect of the small, unresolved scales,
1.2. SIMULATION IN DESIGN 5
must be modeled. Alternatively, the turbulence can be modeled using the Reynolds Av-
eraged Navier–Stokes (RANS) equations, in which flow realizations are ensemble averaged
and additional transport equations for turbulent quantities are solved to provide closure.
By using RANS methods one loses information about various unsteady processes such as
separation, but computations are completed at significantly less cost. The choice of tur-
bulence model should be evaluated based on the importance of the processes that need to
be resolved and the desired turn-around time. For example, parameter studies that require
more qualitative rather than quantitative results can be computed rapidly through the use
of RANS, whereas higher fidelity simulations of specific configurations can be obtained with
LES.
In turbulent reactive flows, it is the mixing of the reactants on the molecular level
through diffusion that enables chemical reactions. Thus, in either RANS or LES based
simulations, the chemical source term must be modeled. For turbulent non-premixed com-
bustion, several models have been proposed to represent the effects of chemical reaction.
One class of models are based on parameterizing the chemistry on the state of mixing of
the reactants through a coupling function, the mixture fraction. Such techniques, known
as moment methods, include the laminar flamelet models (Peters, 1984) and Conditional
Moment Closure (CMC) (Bilger, 1993; Klimenko & Bilger, 1999). Whereas the laminar
flamelet model uses the mixture fraction and its dissipation rate as an input parameter, the
CMC method solves equations for each reaction conditioned on mixture fraction. Moment
methods are typically closed by assuming a shape of the mixture fraction probability density
function (PDF) based on its local mean and variance.
An alternative to moment methods is the transported PDF method proposed by Pope
(1985, 1994). In this model, a transport equation for the PDF of each reactive species is
solved. However, an exact solution of these equations is computationally intractable due to
the high dimensionality of the PDF equation through its dependence on the sample space
of each species. Thus, these methods usually employ Lagrangian or Eulerian particles in a
Monte Carlo type implementation. Furthermore, although the chemical source term appears
in closed form, the diffusive fluxes describing the coupling between mixing and chemistry
must be modeled. Such models are difficult to develop and must employ empirical mixing
coefficients, although some advancements in parameterized models have been made (Meyer
& Jenny, 2006).
6 CHAPTER 1. INTRODUCTION
Finally, Kerstein proposed the use of a one dimensional model, known as the linear
eddy model (LEM), to explicitly solve for a reduced representation of the unresolved chem-
istry (Kerstein, 1992a,b). The method solves the chemistry on a one-dimensional domain
embedded within the turbulence and aligned with the maximum local scalar gradient. The
advantage of the LEM is that it typically fully resolves the interaction of chemistry and tur-
bulence on the small scale and is applicable over all combustion modes. However, the model
also requires the introduction of empirical constants and scaling laws and can sometimes
add considerable computational cost to a simulation.
1.3 Objective
Considering the above discussion, in order for simulations to contribute to the design of
future combustion devices, models for ignition and combustion in multi-feed configurations
are necessary. The objective of this work is to further develop flamelet models for use
in multiple stream systems by extending them to a two-dimensional formulation. The
extended flamelet equations and parameters will be introduced and discussed in Chapter 2
and the method of coupling the equations with existing turbulence models will be described
in Chapter 3.
The approach for the remainder of this work will be to validate the individual compo-
nents of the model framework using DNS studies before applying it to a practical system.
The validity of the flamelet formulation will be investigated using DNS of the ignition of a
multiple-fuel feed system with finite-rate chemistry. The joint scalar statistics of two scalar
mixing will then be studied in isotropic turbulence and a new model for a presumed PDF
to provide closure will be introduced and validated. Finally, the entire model framework
will be applied to a split-injection diesel engine for a range of operating conditions and its
ability to capture ignition and combustion characteristics will be evaluated.
1.4 Summary of Accomplishments
The following list summarizes the accomplishments discussed in this work:
• Fundamental investigation of the effect of multiple fuel streams on ignition character-
istics and identification of the interaction mechanism using finite-rate DNS studies.
Validation of the proposed combustion model is achieved using the DNS results.
1.4. SUMMARY OF ACCOMPLISHMENTS 7
• Provided insight into the scalar dissipation rate of each scalar during multiple scalar
mixing and the corresponding cross-dissipation rate. Developed a method of modeling
the joint scalar dissipation rate in practical applications.
• Introduced a new model for representing the joint statistics of turbulent two-scalar
mixing. Validation is performed using resolved simulations of non-reactive two-scalar
mixing in isotropic turbulence. Comparison to existing models shows that the pro-
posed model provides a substantial improvement.
• Applied the combined model framework to a split-injection diesel engine to test perfor-
mance under realistic applied conditions. The proposed model satisfactorily predicts
the auto-ignition of each fuel injection and is shown to be applicable over wide range
of operating conditions.
Chapter 2
Theory and Model Development
Combustion in practical devices normally occurs in a turbulent mixing field. In fact, tur-
bulence is commonly generated in order to enhance mixing of the fuel and oxidizer to aid
the efficiency of the combustion process. Therefore, a description of the turbulent mixing
field is integral and necessary in order to study combustion devices.
In this chapter, the governing equations describing the gas phase flow field will be
introduced. It is generally not possible to fully resolve these equations in computation, and
therefore the scales of turbulent motion will be defined and time averaged equations with
closure models will be introduced. Then, a description of the liquid phase representation
and its interaction with the gas phase will be given.
The second section of this chapter will develop the additional models that will be used
to account for the reactive aspect of the flow. The flamelet concept will be developed and
an asymptotic analysis of the extension of the flamelet concept to two dimensions will be
given, resulting in equations with two mixture fractions as the independent co-ordinates,
the scalar dissipation rates of each representing the mixing process. A discussion of the
scalar dissipation rates and modeling techniques for two dimensions is given. Furthermore,
some issues regarding the numerical solution of the two-dimensional flamelet equations will
be discussed, including the application of a co-ordinate transformation for computational
convenience.
8
2.1. GOVERNING EQUATIONS 9
2.1 Governing Equations
The equations governing a continuous fluid are the Navier-Stokes equations, which are
a system of coupled, non-linear, partial differential equations. Conservation of mass is
enforced through the continuity equation
∂ρ
∂t+∂ρui∂xi
= ρS (2.1)
where ρ is the gas phase density, ui is the velocity along the co-ordinate xi, and S represents
a source term which can be used to represent the change of overall mass in the system, for
example, from the evaporation of a liquid phase. The rate of change of momentum in the
system is described by
∂ρui∂t
+∂
∂xj(ρuiuj + pδij − σij) = ρfi (2.2)
where p is the pressure, δij is the Kronecker delta, fi are the body forces, and σij is the
viscous stress tensor. For a Newtonian fluid, the stress tensor can be expressed as
σij = 2µ
(Sij −
1
3δij∂uk∂xk
)(2.3)
where µ is the dynamic viscosity and Sij is the strain rate tensor, defined by
Sij =1
2
(∂ui∂xj
+∂uj∂xi
). (2.4)
In addition, the total energy state of the fluid will be accounted for by solving an enthalpy
equation according to
∂ρH
∂t+∂ρujH
∂xj=
∂
∂xj
(JTj +
∑
k
Jk,jhk
)+∂p
∂t− q (2.5)
where H is the total enthalpy, Jqj is the thermal diffusive flux, Jk,j is species mass diffusion
flux, and q are any sources associated with, for example, spray or radiative heat losses. The
total enthalpy is the sum of the individual enthalpies of all chemical species considered, i.e.
H =ns∑
i=1
Yihi (2.6)
10 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
with Yi as the mass fraction of the ith species and hi is the species enthalpy defined by
hi = ∆h◦f (Tref) +
∫ T
Tref
cpidT, (2.7)
where ∆h◦f is the enthalpy of formation at a reference temperature, Tref , and cpi is the
constant pressure specific heat of species i.
Along with the equations describing the turbulent flow, various scalar equations will be
used to represent characteristics of the mixing field, which are required for the combustion
model described in Chapter 3. For example, quantities such as mixture fraction or species
mass fractions will be considered as a transported scalar. The standard transport equation
for a scalar, φ, is defined to be
∂ρφ
∂t+∂ρuiφ
∂xi=
∂
∂xi
(ρDφ
∂φ
∂xi
)+ ρSφ. (2.8)
where Dφ and Sφ represent the diffusivity and source term of the scalar, respectively, and
the scalar is assumed to follow Fick’s diffusion law.
2.1.1 Turbulent Scales and Averaging Methods
A turbulent flow field described by Eq. (2.1) and Eq. (2.2) has a range of associated length
and time scales. The largest integral length scales are of the order of the geometry and
contain most of the energy in the system. The smallest scales, where the energy is dissipated
through viscosity, are characterized by the Kolmogorov scale (Kolmogorov, 1941, 1991) as
η =
(ν3
ε
)1/4
(2.9)
where ν is the kinematic viscosity and ε is the dissipation rate of turbulent kinetic energy.
Another characteristic dissipation scale is the Taylor microscale, defined as
λ =
(15νu′2
ε
)1/2
(2.10)
which represents an upper bound of the dissipation range, with scales greater than λ only
weakly influenced by viscosity. The ratio of the smallest scales, η, to the integral length
2.1. GOVERNING EQUATIONS 11
scales, L0, can be shown to scale with the Reynolds number as
η/L0 ∼ 1/Re3/4 (2.11)
with the corresponding time scales as
τη/τ0 ∼ 1/Re1/2. (2.12)
The above scaling laws indicate that with increasing Reynolds numbers, the separation
between the largest and smallest scales becomes larger. Since the range of scales associ-
ated with Reynolds numbers found in practical applications is typically extremely broad,
it is usually impractical to resolve all length scales exactly. As such, models must be em-
ployed to describe the unresolved features of the flow field. One such technique is to use
the Reynolds Averaged Navier-Stokes (RANS) equations, in which flow realizations are en-
semble averaged, thus only solving directly for the length-scales with ensemble averaged
gradients. This results in a lower computational cost, but of course, one also loses the total
amount of information that can be gained from the system.
Averaging of the equations is accomplished by splitting any variable into two components
according to
φ = φ+ φ′ (2.13)
where φ and φ′ represent the mean and fluctuating components, respectively. The mean is
often taken to be an ensemble average as
φ =1
N
N∑
i=1
φi (2.14)
where N is the number of realizations of the flow field. Since flows with combustion
have variable density, it is convenient to introduce a density-weighted averaging, commonly
known as Favre averaging, according to
ρφ = ρφ+ (ρφ)′ . (2.15)
12 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
If we define Favre variables as
φ =ρφ
ρ, φ′′ =
(ρφ)′
ρ, (2.16)
then the density-weighted averaging can be expressed simply as
ρφ = ρ(φ+ φ′′
)(2.17)
2.1.2 Liquid Phase
The preceding section dealt with the representation of a single phase system. Diesel engines,
as well as modern gasoline or HCCI type engines, inject the fuel as a liquid and consequently
a representation of an additional phase is necessary. There are a number of methods to
represent multiphase systems, which can be broadly characterized by whether each phase
is represented in an Eulerian sense, or whether a mixed Euler-Lagrange model is used.
Solving each phase in an Eulerian system involves solving both phases according to the
standard equations described in Sec. 2.1 while also tracking the interface between the phases
and accounting for any phase change. Techniques for this type of modeling include level-
set (Sussman et al., 1994) or volume of fluid (VOF) (Gueyffier et al., 1999) methods.
Representing diesel fuel injection and atomization in this manner is very challenging.
Diesel sprays have very high density ratios and are injected at high velocity, resulting in
high Reynolds numbers and also high Weber numbers. In practical configurations, sufficient
resolution of the spray, including direct representation of the primary and secondary breakup
processes, is intractable.
In this work, a Lagrangian approximation of the spray will be employed. This class
of methods treats the liquid as a discrete phase represented by particles and is commonly
known as the Discrete Droplet Method (DDM) (Dukowicz, 1980). Each particle can be
considered to be a unique droplet, but typically is taken instead to represent an ensemble of
non-interacting droplets, or parcel, that are assumed to have the same properties or some
specified distribution thereof. Particle trajectories are then solved in a Lagrangian sense
through a force balance on the particle, that is, between particle inertia and the forces
acting on the particle from the surrounding gas phase. The phases are coupled through
the exchange of momentum through the force balance, as well as through mass and energy
transfer that occurs from the evaporation of the liquid.
2.2. LAMINAR FLAMELETS 13
2.2 Laminar Flamelets
Laminar flamelets have been used in the past to represent both premixed and non-premixed
combustion systems. Flamelets represent the chemistry in a thin reactive layer. It is as-
sumed that the thickness of the reaction zone is smaller than the smallest scales of tur-
bulence, i.e. the Kolmogorov scale, since the timescales of the chemical reaction rates are
faster than the mixing timescale. This enables the assumption that the reaction zone can be
taken to be locally laminar. A mixture fraction co-ordinate is introduced across the flame
and the interaction of the flamelet with the small scale turbulent mixing is represented by
a scalar dissipation rate. In this section, the laminar flamelet equation and its extension to
multiple dimensions will be investigated.
We begin by considering the transport equation for a reactive scalar and enthalpy ac-
cording to
ρ∂Yk∂t
+ ρuj∂Yk∂xj
=∂
∂xj(Jk,j) + ρωk (2.18)
ρ∂H
∂t+ ρuj
∂H
∂xj=
∂
∂xj
(Jqj +
∑
k
Jk,jhk
)+∂p
∂t− qr (2.19)
where the subscript k refers to species quantities, Jk,j is the mass diffusion flux, Jqj is
the thermal diffusive flux, and qr represents any heat loss, such as from radiation. It is
sometimes convenient to rewrite the enthalpy equation in terms of temperature in order
that later we may write a coupled set of ordinary differential equations, rather than a
differential-algebraic set of relations. Writing the enthalpy in terms of temperature can be
achieved by recognizing that, with the definition of total enthalpy, Eq. (2.6), we can write
dH = cpdT +∑
k
hkdYk (2.20)
where cp is the specific heat of the mixture, defined as
cp =∑
k
Ykcpk . (2.21)
14 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
Substituting this definition into Eq. (2.19), we first look at the left hand side, giving
ρ
(cp∂T
∂t+∑
k
hk∂Yk∂t
)+ ρuj
(cp∂T
∂xj+∑
k
hk∂Yk∂xj
)= r.h.s.
ρ∂T
∂t+ ρuj
∂T
∂xj+
1
cp
∑
k
hk
(ρ∂Yk∂t
+ ρuj∂Yk∂xj
)=
1
cp(r.h.s.) . (2.22)
Now considering the right hand side, substitution gives
l.h.s. =1
cp
∂
∂xj
(Jqj +
∑
k
Jk,jhk
)
=1
cp
(∂
∂xj(Jqj ) +
∂p
∂t− qr
)+
1
cp
∑
k
∂
∂xj(Jk,jhk)
=1
cp
(∂
∂xj(Jqj ) +
∂p
∂t− qr
)+
1
cp
∑
k
(Jk,j
∂hk∂xj
+ hk∂
∂xj(Jk,j)
). (2.23)
Combining Eq. (2.22) and Eq. (2.23), we can write
ρ∂T
∂t+ ρuj
∂T
∂xj+
1
cp
∑
k
hk
(ρ∂Yk∂t
+ ρuj∂Yk∂xj− ∂
∂xj(Jk,j)
)=
1
cp
(∂
∂xj
(Jqj
)+∑
k
Jk,j∂hk∂xj
+∂p
∂t− qr
)(2.24)
The brackets of the third term on the l.h.s. of Eq. (2.24) can be recognized as Eq. (2.18),
allowing a substitution to give
ρ∂T
∂t+ ρuj
∂T
∂xj+ρ
cp
∑
k
hkωk =1
cp
(∂
∂xj
(Jqj
)+∑
k
Jk,j∂hk∂xj
)(2.25)
Assuming that thermal diffusivity follows Fourier heat conduction, the thermal flux term
can be written as
1
cp
∂
∂xj
(Jqj
)=
1
cp
∂
∂xj
(λ∂T
∂xj
)=
∂
∂xj
(λ
cp
∂T
∂xj
)+λ
c2p
∂cp∂xj
∂T
∂xj. (2.26)
2.2. LAMINAR FLAMELETS 15
Similarly, if it is assumed that the diffusive mass flux follows Fickian diffusion, then
1
cp
∑
k
Jk,j∂hk∂xj
=1
cp
∑
k
ρDk∂Yk∂xj
∂hk∂xj
. (2.27)
Thus, the final transport equations for species mass fractions and temperature are
ρ∂Yk∂t
+ ρuj∂Yk∂xj
=∂
∂xj
(ρDk
∂Yk∂xj
)+ ρωk (2.28)
ρ∂T
∂t+ ρuj
∂T
∂xj=
∂
∂xj
(λ
cp
∂T
∂xj
)− ρ
cp
∑
k
hkωk
+λ
c2p
∂cp∂xj
∂T
∂xj+
1
cp
(ρ∑
k
Dk∂Yk∂xj
∂hk∂xj
+∂p
∂t− qr
).
(2.29)
2.2.1 Definition of Mixture Fraction
In non-premixed combustion, the fuel and oxidizer are initially separated in different streams
and must mix together to react, which will only occur after the fuel and oxidizer intermingle
on the molecular level. This process will often occur on a timescale that is fast relative to the
local mixing timescale. The mixture fraction, Z, represents a co-ordinate that determines
the state of mixing between the fuel and oxidizer streams. There are several definitions of
the mixture fraction, but all are essentially measures of the local equivalence ratio.
Looking at a one-step global reaction of a fuel (F) and oxidizer (O) that combine to
form a product (P),
νFYF + νOYO → νPYP (2.30)
a mixture fraction can be defined in a general sense using elemental mass fractions according
to
Zβ =∑
α
nαβWβ
WαYα (2.31)
where W is the molar mass and nαβ is the number of atoms of element β in the α species
molecule. Applying a linear operator to the above equation and assuming equal diffusivities
of all scalars results in a transport equation for the mixture fraction
∂ρZ
∂t+∂ρujZ
∂xj=
∂
∂Zj
(ρDZ
∂Z
∂xj
)(2.32)
where a linear combination of species has been used to define a coupling function to remove
16 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
the chemical source term and result in a conserved scalar (Peters, 1984). Thus, for a fuel
and oxidizer species a mixture fraction normalized between 0 and 1 can be defined as
Z =YF − (YO − YO,2) (νFWF/νOWO)
YF,1 − YO,2 (νFWF/νOWO), (2.33)
where the indices 1 and 2 indicate the fuel and oxidizer streams, respectively. That is, YF,1
is the fuel mass fraction in the rich stream and YO,2 is the oxidizer mass fraction in the
lean stream. This type of definition is often used for experimental results. For instance, if
the fuel and oxidizer are specified in terms of the basic elements of a general hydrocarbon
reaction (C, H, O), then the above is effectively the definition used by Bilger (1988), which
is convenient as it can be determined by simply measuring the major species. However, the
drawback of this type of definition is that it is unable to account for effects of differential
diffusion.
Alternatively, the mixture fraction can be defined as a co-ordinate that desribes mixing
between streams of different composition by defining it directly from Eq. (2.32) (Pitsch &
Peters, 1998). For a two-feed system, the mixture fraction is then defined as the ratio of
the mass flux of fuel normalized by the total mass flux into the system
Z =m1
m1 + m2. (2.34)
This definition is more convenient for the purposes of this work, as it more easily lends itself
to be extended to multiple streams. In general, an n-feed system will require n− 1 mixture
fractions to fully define the system. Each of these mixture fractions is therefore defined by
Zα =mαn∑
k=1
mk
, α = 1, 2, . . . , n− 1. (2.35)
Since the sum of the mixture fractions is bounded by unity, the mixture fraction for the
final stream is not independent of the others, and can be defined as
Zn = 1−n−1∑
k=1
Zk (2.36)
In this work, a three-feed system will be considered, meaning that two mixture fractions
2.2. LAMINAR FLAMELETS 17
are required and can be defined as
Z1 =m1
m0 + m1 + m2, Z2 =
m2
m0 + m1 + m2(2.37)
with the mixture fraction of the third stream, defined as Z0, found from
Z0 = 1− Z1 − Z2. (2.38)
One possible realization of a three-feed system is shown in Fig. 2.1(a), which shows a
triple counterflow configuration. Although shown in a symmetric configuration, in general
the streams can have any orientation and non-equal mass flow rates. The important aspect
is that there is mixing between each of the three streams individually, as well as mixing of
all three streams together. Another possible three-feed system is a double mixing layer. In
either case, the mapping of the system to mixture fraction space is depicted in Fig. 2.1(b).
The unity constraint of the sum of mixture fractions results in a realizable domain defined
by a unit right triangle. Each side of the triangle represents direct mixing between two
streams and the interior describes all linear combinations of mixture fractions.
m0
m2m1
(a) Triple counterflow
0 1
Z1 +Z2 =
1
1
Z1
Z2
m0 m1
m2
(b) Mixture fraction space
Figure 2.1: One possible configuration for a three-feed system and the corresponding map-ping to mixture fraction space based on Eq. (2.37).
It is useful to be able to relate the mixture fraction to the equivalence ratio, often
quoted in experiments, which represents the ratio of fuel-to-air in the unburnt mixture to
the stoichiometric fuel-to-air ratio. Taking the unburned fuel mass fraction to be YF,1Z and
18 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
the unburned oxidizer as YO,2(1− Z), one obtains
Φ =Z
1− Z1− Zst
Zst. (2.39)
Equation (2.39) confirms the interpretation given earlier of the mixture fraction as a local
normalized equivalence ratio.
2.2.2 Flamelet Approximation
The laminar flamelet equations are a representation of Eqs. (2.28) and (2.29) with the
mixture fraction as the independent variable, first derived by Peters (1984). The flamelet
equations were first obtained by transforming the species and enthalpy transport equations
from physical space into a mixture fraction co-ordinate according to
x, t→ Z(x, t), τ(x, t) (2.40)
where x represents a local co-ordinate system that is aligned with an isosurface of stoichio-
metric mixture fraction. Specifically, x1 is defined to be normal to the isosurface and the
corresponding x2- and x3-directions lay tangent thereto. The transformation rules are given
by
∂
∂t=
∂
∂τ+∂Z
∂t
∂
∂Z
∂
∂x1=∂Z
∂x1
∂
∂Z
∂
∂xk=
∂
∂Zk+∂Z
∂xk
∂
∂Z, k = 2, 3
(2.41)
where it has been taken that Z = x1, Z2 = x2, Z3 = x3, and τ = t. It has been shown
by Peters (1984) that the terms relating to the Z2 and Z3 are of lower order compared to
those with Z, such that applying the above transformation to Eqs. (2.28) and (2.29) results
in one-dimensional flamelet equations for species mass fraction and temperature
ρ∂Yk∂t
= ρχ
2
∂2Yk∂Z2
+ ρωk (2.42)
ρ∂T
∂t= ρ
χ
2
∂2T
∂Z2+ρ
cp
χ
2
{∂cp∂Z
+∑
k
cpk∂Yk∂Z
}∂T
∂Z− ρ
cp
∑
k
hkωk +1
cp
(∂p
∂t− qr
)(2.43)
2.2. LAMINAR FLAMELETS 19
where the scalar dissipation rate,
χ = 2D
(∂Z
∂xj
)2
(2.44)
appears as a parameter that describes the local effect of diffusion on the reaction and is thus
the means of interaction between the turbulence and the flame. This important parameter
is discussed further in Sec. 2.2.3. The above formulation is for unity Lewis number and
neglects the effects of differential diffusion. For cases where such approximations are not
justified, correction terms to take differential diffusion into account were derived by Pitsch
& Peters (1998).
The above derivation of the flamelet equations relied on a local co-ordinate transfor-
mation and boundary layer arguments and is applicable to two-feed systems that can be
represented by a single mixture fraction. When considering a three-feed system, a possi-
ble realization of which is shown in Fig. 2.1, two mixture fractions are necessary to fully
characterize the mixing field. An analogous co-ordinate transformation from physical space
into one defined by two mixture fractions is not obvious. However, Peters (2000) showed
that the flamelet equations can also be derived using a two-scale asymptotic analysis based
on the requirement that the reaction zone thickness be smaller than the smallest turbulent
eddy. Since the flamelet concept is only valid when this criterion is met, it remains valid
regardless of the number of mixture fractions employed. Therefore, the subsequent analysis
follows that of Peters (2000), and furthered by Hasse (2004), to obtain flamelet equations
for two mixture fractions.
As mentioned, the flamelet approximation requires that the reaction zone thickness, lR,
be embedded with the smallest turbulent eddy defined by the Kolmogorov scale, i.e.
lR < η. (2.45)
To perform the asymptotic analysis, another physical length scale characteristic of the flame
surface corrugations is needed. Such a length scale can be related to the mixture fraction
fluctuations and average mixture fraction gradient at stoichiometric mixture according to
Λ = Z ′st
(χst
2Dst
)−1/2
(2.46)
20 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
where Z ′st is the root mean square (r.m.s.) of mixture fraction fluctuations and χst is the
average scalar dissipation rate at stoichiometric mixture, respectively. The length scale Λ
can be thought of as the equivalent of a Taylor scale of the mixture fraction field. A small
parameter for the asymptotic analysis can now be constructed from the above length scales
as
ε =lRΛ. (2.47)
Before applying the expansion to the transport equations they must be non-dimensionalized.
Considering the species transport equation, Eq. (2.28), the independent variables xj , t, and
uj are scaled by the length scale Λ and an associated time scale, tΛ = Λ2/Dst, resulting in
u∗j = xj/Λ, t∗ = t/tΛ, u∗j = ujtΛ/Λ. (2.48)
The species mass fractions, density, and chemical source term are scaled by appropriate
reference values to obtain
Y ∗k = Yk/Yk,ref , ρ∗ = ρ/ρref , ω∗k = ωktΛ/ (ρrefYk,ref) . (2.49)
Furthermore, the diffusivity and mixture fraction are normalized by values at stoichiometric
mixture fraction as
D∗k = Dk/Dst, Z∗/Z ′st. (2.50)
Applying the above transformation to Eq. (2.28) results in the same form,
ρ∗∂Y ∗k∂t∗
+ ρ∗u∗j∂Y ∗k∂x∗j
=∂
∂x∗j
(ρ∗D∗k
∂Y ∗k∂x∗j
)+ ρω∗k (2.51)
and therefore the asterisk will be dropped in the following analysis for convenience.
Now a three-scale asymptotic analysis can be applied to Eq. (2.51). In contrast to the
analysis of Peters (2000), here a short scale must be introduced for each mixture fraction
according to
ζ1 = ε−11 (Z1(xj , t)− Z1,ref) , ζ2 = ε−1
2 (Z2(xj , t)− Z2,ref) . (2.52)
In analogy to the single mixture fraction analysis, if the reference value for each scale is
taken to be the stoichiometric mixture of the system, (Z1 + Z2)st, then each short range
2.2. LAMINAR FLAMELETS 21
scale represents changes in the vicinity of the flame surface. If a flame is in a region of
only Z1 or Z2, then this reduces to the single mixture short range scale, whereas when both
mixture fractions are present, the flame surface will be affected by fluctuations from each
mixture accordingly. Since the fluctuations of each mixture fraction will be of the same
order, the expansion parameter for each will be taken to be the same (ε = ε1 = ε2). As
usual, the flow and mixing field far away from the flame surface are described by the long
range spatial co-ordinates and time, xj and t. In the new co-ordinate system the long range
will be expressed as ξj and a short time scale,
τ = t/ε2 (2.53)
has been scaled to account for rapid temporal changes in the flamelet structure. Thus, the
transformation to express a variable in long and short spatial and temporal variables is
(xj , t)→ (ζ1(xj , t), ζ2(xj , t) , ξj , τ) (2.54)
leading to transformation rules defined as
∂
∂t=∂τ
∂t
∂
∂τ+∂ζ1
∂t
∂
∂ζ1+∂ζ2
∂t
∂
∂ζ2=
1
ε2∂
∂τ+
1
ε
∂Z1
∂t
∂
∂ζ1+
1
ε
∂Z2
∂t
∂
∂ζ2
∂
∂xj=∂ξj∂xj
∂
∂ξj+∂ζ1
∂xj
∂
∂ζ1+∂ζ2
∂xj
∂
∂ζ2=
∂
∂ξj+
1
ε
∂Z1
∂xj
∂
∂ζ1+
1
ε
∂Z2
∂xj
∂
∂ζ2.
(2.55)
Now the species mass fraction can be expanded for the asymptotic analysis according to
Yi = Y 0i + εY 1
i + ε2Y 2i + . . . (2.56)
where only the leading order term will be retained in the following. Thus, applying the
22 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
above transformation to Eq. (2.28) and multiplying by ε2 gives
ρ∂Yk∂τ
=− ε(∂Z1
∂t
∂Yk∂ζ1
+∂Z2
∂t
∂Yk∂ζ2
)− ε2ρuj
∂Yk∂ξj− ερuj
(∂Z1
∂xj
∂Yk∂ζ1
+∂Z2
∂xj
∂Yk∂ζ2
)
+ ε∂
∂ξj
[ρDk
(ε∂Yk∂ξj
+∂Z1
∂xj
∂Yk∂ζ1
+∂Z2
∂xj
∂Yk∂ζ2
)]
+∂Z1
∂xj
∂
∂ζ1
[ρDk
(ε∂Yk∂ξj
+∂Z1
∂xj
∂Yk∂ζ1
+∂Z2
∂xj
∂Yk∂ζ2
)]
+∂Z2
∂xj
∂
∂ζ2
[ρDk
(ε∂Yk∂ξj
+∂Z1
∂xj
∂Yk∂ζ1
+∂Z2
∂xj
∂Yk∂ζ2
)]+ ε2ρωk. (2.57)
Only the leading order terms will be retained, although the chemical reaction rate must
be scaled such that it remains. Additionally, it can be noted that since the gradient of
mixture fraction, ∂Zi/∂xj , does not vary on the short scale, it can be moved across the
∂/∂ζi derivative in the fifth and sixth terms on the r.h.s. of the above, thus we obtain
ρ∂Yk∂τ
=∂
∂ζ1
[ρDk
((∂Z1
∂xj
)2 ∂Yk∂ζ1
+∂Z1
∂xj
∂Z2
∂xj
∂Yk∂ζ2
)]
+∂
∂ζ2
[ρDk
(∂Z1
∂xj
∂Z2
∂xj
∂Yk∂ζ1
+
(∂Z2
∂xj
)2 ∂Yk∂ζ2
)]+ ρωk.
(2.58)
Finally, recognizing that ∂ζj = ∂Zj/ε, the flamelet equation for species mass fraction can
be rearranged and written in dimensional form as
ρ∂Yk∂t
= ρ
(χ1
2
∂2Yk
∂Z12 + χ12
∂2Yk∂Z1∂Z2
+χ2
2
∂2Yk
∂Z22
)+ ρωk. (2.59)
where the scalar dissipation rates of each mixture fraction are defined as
χ1 = 2D
(∂Z1
∂xj
)2
, χ12 = 2D∂Z1
∂xj
∂Z2
∂xj, χ2 = 2D
(∂Z2
∂xj
)2
. (2.60)
2.2. LAMINAR FLAMELETS 23
The same technique can be applied to any reactive variable and application to the temper-
ature transport defined by Eq. (2.29) results in
ρ∂T
∂t= ρ
(χ1
2
∂2T
∂Z12 + χ12
∂2T
∂Z1∂Z2+χ2
2
∂2T
∂Z22
)+
1
cp
(ρ∑
k
hkωk +∂p
∂t− qr
)
+ρ
2cp
∂T
∂Z1
{χ1∂cp∂Z1
+ χ12∂cp∂Z2
+∑
k
(χ1∂Yk∂Z1
+ χ12∂Yk∂Z2
)}
+ρ
2cp
∂T
∂Z2
{χ12
∂cp∂Z1
+ χ2∂cp∂Z2
+∑
k
(χ12
∂Yk∂Z1
+ χ2∂Yk∂Z2
)}.
(2.61)
2.2.3 Scalar Dissipation Rate
The flamelet equations are parameterized by the scalar dissipation rate, which represents
the effect of small scale turbulent mixing on the flame structure. To investigate the scalar
dissipation rate further, consider the mixture fraction equation for constant density and
diffusivity as a first approximation, which can be expressed as
∂Z
∂t+ uj
∂Z
∂xj= DZ
∂2Z
∂xj2(2.62)
A transport equation of the scalar dissipation rate can be obtained by multiplying all
terms of Eq. (2.62) by the gradient of mixture fraction and the gradient operator, i.e.
∂Z/∂xk ∂/∂xk, resulting in
∂Z
∂xk
∂
∂xk
(∂Z
∂t+ uj
∂Z
∂xj
)=
∂Z
∂xk
∂
∂xk
(DZ
∂2Z
∂xj2
)(2.63)
First expanding the terms on the left hand side, the equation can be written
∂Z
∂xk
∂
∂t
(∂Z
∂xk
)+∂Z
∂xk
∂uj∂xk
∂Z
∂xj+ uj
∂Z
∂xk
∂
∂xk
(∂Z
∂xj
)= r.h.s.
1
2
∂
∂t
[(∂Z
∂xk
)2]
+∂Z
∂xk
∂Z
∂xjSij +
uj2
∂
∂xj
[(∂Z
∂xk
)2]
= r.h.s. (2.64)
and recognizing that χ = 2DZ (∂Z/∂xk)2, through substitution Eq. (2.63) becomes
∂χ
∂t+ uj
∂χ
∂xj+ 4DZ
∂Z
∂xk
∂Z
∂xjSij = 4D2
Z
∂Z
∂xk
∂
∂xk
(∂2Z
∂xj2
)(2.65)
24 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
Further, the right hand side can be rearranged according to
l.h.s. = 4D2Z
∂Z
∂xk
∂
∂xk
[∂
∂xj
(∂Z
∂xj
)]
= 4D2Z
∂Z
∂xk
∂
∂xj
[∂
∂xj
(∂Z
∂xk
)]
= 4D2Z
{∂
∂xj
[∂Z
∂xk
∂
∂xj
(∂Z
∂xk
)]− ∂
∂xj
(∂Z
∂xk
)∂
∂xj
(∂Z
∂xk
)}
= 4D2Z
∂
∂xj
{1
2
∂
∂xj
[(∂Z
∂xk
)]2}− 4D2
Z
(∂2Z
∂xk∂xj
)2
= 4D2Z
∂
∂xj
(1
4DZ
∂χ
∂xj
)− 4D2
Z
(∂2Z
∂xk∂xj
)2
= DZ∂2χ
∂xj2− 4D2
Z
(∂2Z
∂xk∂xj
)2
(2.66)
Thus, the transport equation for scalar dissipation rate is
Dχ
Dt︸︷︷︸advection
= DZ∂2χ
∂xj2
︸ ︷︷ ︸diffusion
− 4DZ∂Z
∂xk
∂Z
∂xjSij
︸ ︷︷ ︸production
− 4D2Z
(∂2Z
∂xk∂xj
)2
︸ ︷︷ ︸dissipation
(2.67)
where D/Dt is the material derivative. The first and third terms can be identified as the
transport and destruction of scalar dissipation through molecular diffusion, respectively.
The second term is the production term related to the strain rate fluctuations. Ruetsch
& Maxey (1991, 1992), followed by Overholt & Pope (1996) and Vedula et al. (2001),
investigated the transport of scalar dissipation in the context of isotropic turbulence with
an imposed mean scalar gradient. These studies showed that there exist sheets of intense
scalar gradients which is where the scalar gradient production is large, and thus where the
scalar primarily mixes. The positive scalar production occurs when the scalar gradients
align with the compressive component of the strain, thus the production term in Eq. (2.67)
has a negative coefficient.
Applying the same asymptotic analysis of the previous section to Eq. (2.67), one can
obtain a transport equation for χ with mixture fraction as the independent variable. The
resulting equation reads∂χ
∂t+
1
4
(∂χ
∂Z
)2
− 2asχ =χ
2
∂2χ
∂Z2(2.68)
2.2. LAMINAR FLAMELETS 25
where as appears as the strain rate normal to isocontours of mixture fraction. In simple
flow configurations, a functional dependence of χ on the mixture fraction can be found
directly. For example, Peters (1984, 2000) showed that the scalar dissipation rate for laminar
configurations including a stationary counterflow and the one-dimensional unsteady infinite
mixing layer can be derived as
χ(Z) =asπ
exp{−2[erfc−1(2Z)
]2}. (2.69)
Each of these configurations are often found locally in turbulent mixing fields and thus
Eq. (2.69) is often used in more general turbulent flows. However, the boundary conditions
of both configurations are characterized by an infinite amount of either fuel or oxidizer,
and thus the gradient at both Z = 0 and Z = 1 is zero. For mixing fields that have a
finite amount of fuel, an alternative solution was introduced for a one-dimensional mixing
layer with the fuel originating from the symmetry axis into an infinite oxidizer (Pitsch,
1998; Pitsch et al., 1998). This representation is often used in diffusion flames and spray
combustion as it describes the decay of the maximum mixture fraction as the fuel and
oxidizer mix. The resulting functional dependence of χ in this configuration is found to be
χ(Z) = −2Z2
tln
(Z
Zmax
). (2.70)
In contrast to the profile defined by Eq. (2.69), where χ has zero gradients at both bound-
aries, Eq. (2.70) only has a zero gradient at Z = 0 and a non-zero gradient at Z = 1 resulting
from the finite amount of fuel. It can be shown that both Eq. (2.69) and Eq. (2.70) are
stationary solutions of Eq. (2.68) for a constant strain rate and the appropriate boundary
conditions.
Turning now to the three stream system, a description of the dissipation rate of each
scalar, as well as the joint scalar dissipation rate, is required and must be formulated in
the two-dimensional (Z1, Z2) space. If transport equations for each mixture fraction follow
Eq. (2.62) as∂Zi∂t
+ uj∂Zi∂xj
= DZi
∂2Zi∂xj2
, i = 1, 2 (2.71)
26 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
the same three-scale asymptotic analysis of Sec. 2.2.2 can be applied. The resulting trans-
port equations for each dissipation rate in (Z1, Z2) space are found to be
∂χi∂t
+1
4
(∂χi∂Zi
)2
− 2aiχi =χ1
2
∂2χi
∂Z12 + χ12
∂2χi∂Z1∂Z2
+χ2
2
∂2χi
∂Z22 (2.72)
where each scalar can have a strain rate, ai, that is normal to their respective isocontours.
However, if the mixing of each scalar is governed by Fick’s law, the scalar diffusion can be
argued to be a function of that scalar only and therefore independent of the composition of
the remaining mixture (Girimaji, 1993). Thus, Eq. (2.72) can be reduced to
∂χi∂t
+1
4
(∂χi∂Zi
)2
− 2aiχi =χi2
∂2χi
∂Zi2 (2.73)
which is exactly equivalent to Eq. (2.68) for each mixture. This result implies that the
dissipation rate of each scalar can be modeled according to either Eq. (2.69) or Eq. (2.70),
just as in the single mixture fraction configuration.
The independence of each scalar dissipation rate also suggests that the one-dimensional
profiles will be valid for the entire (Z1, Z2) space. However, further consideration is necessary
to account for the constraint on the sum of the mixture fractions. Figure 2.1(b) represents
the entire realizable domain and therefore the model used for the scalar dissipation rates
must enforce a zero flux condition of any variable normal to the domain boundaries. Since
the profiles assumed for χ1 and χ2 are defined for 0 < Z1 < 1 and 0 < Z2 < 1, respectively,
applying them across the entire domain means that each will have a non-zero value along
the Z1 + Z2 = 1 boundary except at (1,0) and (0,1). This could imply that the profiles
must be scaled according to the local mixture composition, for example defining χ1 over
the range 0 < Z1 < 1 − Z2, however the argument for the independent mixing of each
scalar would no longer be justified. Alternatively, the contribution of χ12, which is still to
be considered, may play a role.
To investigate this further, a model for the joint scalar dissipation rate is required.
Unfortunately, applying the asymptotic expansion to the transport equation for χ12 does
not provide insight into its functional dependence on Z1 and Z2, as the resulting equation
retains physical gradients of mixture fraction that cannot be reduced to scalar dissipation
rates. An additional complication for modeling purposes is that the orientation of the
mixture fraction gradients can be such that χ12 may be either positive or negative.
2.2. LAMINAR FLAMELETS 27
Considering these difficulties, rather than model the joint dissipation rate directly, the
dissipation rate of Z0, as defined by Eq. (2.38), will be considered. Although Z0 does
not appear in the Eqs. (2.59–2.61), all streams must mix in the same fashion and a scalar
dissipation rate for Z0 can be defined as
χ0 = 2D∂Z0
∂xj
∂Z0
∂xj. (2.74)
Substituting the relation for Z0 from Eq. (2.38), the above can be expanded as
χ0 = 2D∂
∂xj(1− Z1 − Z2)
∂
∂xj(1− Z1 − Z2)
= 2D∂
∂xj(Z1 + Z2)
∂
∂xj(Z1 + Z2)
= 2D
[(∂Z1
∂xj
)2
+ 2∂Z1
∂xj
∂Z2
∂xj+
(∂Z2
∂xj
)2]
(2.75)
where the definition of Eq. (2.60) can be used to define χ0 in terms of the dissipation rates
of the other scalars as
χ0 = χ1 + 2χ12 + χ2. (2.76)
In this manner, if the mixing of Z0 is taken to be governed by the same equation as Z1
and Z2, a one-dimensional profile along a line radiating from the origin to the Z1 + Z2 = 1
bound can be found using Eq. (2.69) or Eq. (2.70). This allows χ12 to be modeled indirectly
for any (Z1, Z2) through a linear combination of the one-dimensional profiles for each scalar
as
χ12(Z1, Z2) =1
2[χ0(Z0)− χ1(Z1)− χ2(Z2)] . (2.77)
This result provides some interesting insight into the role of the joint dissipation rate.
As discussed above, the flux of species and energy normal to the boundaries of the domain
defined by Fig. 2.1(b) must be zero. The one-dimensional profiles used for χ1 and χ2 enforce
this condition for the Z2 = 0 and Z1 = 0 boundaries. It can be seen from Eq. (2.76) that
to enforce the same constraint along the Z1 + Z2 = 1 boundary, it is required that χ0 = 0.
This results in an expression for the joint dissipation rate at the boundary
χ12 = −1
2(χ1 + χ2) = −√χ1χ2. (2.78)
28 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
The above indicates that when the sum of the mixture fractions is unity, the gradients of
each scalar in physical space, ∂Z1/∂xj and ∂Z2/∂xj , are aligned and opposite in direction.
That is, the angle between the physical gradients of each mixture is 180◦. It is now clear
that the the transport term with χ12 prevents flux normal to the boundary while χ1 and
χ2 are still non-zero and forces the net flux to be along the boundary between Z1 = 1 and
Z2 = 1. This is an important result, since if χ12 were neglected, χ1 and χ2 would necessarily
be scaled to zero to enforce the boundary constraints. As a result, there would be no mixing
between streams 1 and 2, thus the solution of Eqs. (2.59) and (2.61) would no longer fully
represent a three-feed system.
2.2.4 Co-ordinate Transformation
Due to the unity constraint on the total mixture fraction, the realizable domain of the
two dimensional flamelet equations is a unit right triangle, depicted in Fig. 2.2(a). For
numerical reasons, it is convenient to transform this domain to a unit square. This can
be accomplished in several ways; including a rotation of the hypotenuse of the triangle to
either horizontal or vertical, or transforming both the hypotenuse vertical and the left edge
to the top horizontal. In any transformation considered, one edge of the resulting square
will be a singularity.
In this work, the co-ordinate system used is defined by
ξ = Z1, η =Z2
1− Z1(2.79)
which results in the transformed space depicted in Fig. 2.2(b). Here it can be seen that the
Z1 + Z2 = 1 limit becomes the top boundary and the right edge of the domain is defined
by the point at (1,0). This transformation was chosen primarily due to the fact that the
transformed equations recover the one-dimensional formulation in the limit of χ1 → 0. The
2.2. LAMINAR FLAMELETS 29
resulting transformation rules from Eq. (2.79) are
∂
∂Z1=
∂
∂ξ+
η
1− ξ∂
∂η
∂
∂Z2=
1
1− ξ∂
∂η
∂2
∂Z12 =
∂2
∂ξ2+ 2
η
1− ξ∂2
∂ξ∂η+
η2
(1− ξ)2
∂2
∂η2+ 2
η
(1− ξ)2
∂
∂η(2.80)
∂2
∂Z22 =
1
(1− ξ)2
∂2
∂η2
∂2
∂Z1∂Z2=
1
1− ξ∂2
∂ξ∂η+
η
(1− ξ)2
∂2
∂η2+
1
(1− ξ)2
∂
∂η
Application of Eq. (2.80) to Eqs. (2.59) and (2.61) results in transformed equations for
species mass fraction and temperature as
ρ∂Yi∂t− ρ χξη
1− ξ∂Yi∂η
= ρ
(χξ2
∂2Yi∂ξ2
+ χξη∂2Yi∂ξ∂η
+χη2
∂2Yi∂η2
)+ ρωi (2.81)
ρ∂T
∂τ− ρ χξη
1− ξ∂T
∂η= ρ
(χξ2
∂2T
∂ξ2+ χξη
∂2T
∂ξ∂η+χη2
∂2T
∂η2
)+
1
cp
(ρ∑
k
hkωk +∂p
∂t− qr
)
+ρ
2cp
∂T
∂ξ
{χξ∂cp∂ξ
+ χξη∂cp∂η
+∑
i
cpi
(χξ∂Yi∂ξ
+ χξη∂Yi∂η
)}
+ρ
2cp
∂T
∂η
{χξη
∂cp∂ξ
+ χη∂cp∂η
+∑
i
cpi
(χξη
∂Yi∂ξ
+ χη∂Yi∂η
)}(2.82)
where the transformed scalar dissipation rates are expressed as
χξ = χ1, χξη =χ1η + χ12
1− ξ , χη =χ1η
2 + 2χ12η + χ2
(1− ξ)2 . (2.83)
30 CHAPTER 2. THEORY AND MODEL DEVELOPMENT
0 1
Z1 +Z2 =
1
1
Z1
Z2
A
1−A
B 1−B
C
C(a) Physical Space
0 1
Z1 + Z2 = 11
ξ
η
A
1−A
B1−B
C
C(b) Transformed Space
Figure 2.2: Transformation of solution domain for flamelet with two mixture fractions.Dashed lines show effect on lines of constant Z1 (A,1 − A), constant Z2 (B,1 − B), andconstant total mixture fraction Z1 + Z2 (C).
Chapter 3
Representative Interactive
Flamelets (RIF)
Building on the mathematical framework of Chapter 2, this chapter will describe how the
flamelet representation of chemistry is integrated into the description of a turbulent com-
bustion problem. The coupling between the laminar flamelet equations with the turbulent
flow and mixing field will be detailed. The model equations used to describe the mixing
state of the field will be developed. The model input parameters for the flamelet equations
and the method of closure using a presumed PDF will also be described.
3.1 Coupling of Chemistry with Turbulent Flow Field
The advantage of representing the chemistry through a co-ordinate transformation to mix-
ture fraction is that the mixture fraction can be treated as a non-reactive scalar, allowing
the solution of the turbulent flow field and the chemical reactions to be decoupled. This
allows the use of arbitrarily detailed finite-rate chemistry at minimal additional cost. In this
section, we will be discussing the coupling in the context of a RANS framework, however,
similar concepts would be applicable to LES.
The flow solver computes the three-dimensional Navier-Stokes equations, along with any
turbulence model equations and those for spray modeling. The energy equation is solved
as an averaged total enthalpy so that it can be treated as a conserved quantity in the
presence of chemical reactions. Additionally, equations for the mean and variance of each
mixture fraction are solved according to Eqs. (3.21–3.22), which are discussed in further
31
32 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
detail in the following section and are used as a representation of the mixing state of the
fluid. Furthermore, the mixture fraction moments and turbulence parameters are used to
construct the conditional mean scalar dissipation rates χ1(Z1, Z2) and χ2(Z1, Z2) and the
cross-dissipation, χ12(Z1, Z2), according to the models developed in Sec. 3.3. Since only
a single flamelet is used, the distribution computed in each cell is mass averaged over the
domain. A volumetric pressure is required and computed from the flow field as
p =
∫
VpdV
∫
VdV
. (3.1)
After the flow is advanced from time n to n+ 1, all the inputs required for the flamelet
equations are computed and used to advance the distribution of species, Y (Z1, Z2). The
resulting species distributions in the mixture fraction co-ordinate are then convoluted with a
presumed joint probability density function (pdf) of the mixture fractions that is constructed
from the moment information available in each cell, resulting in mean species mass fractions
at each cell location Y (x, t). Knowing the mean species mass fractions and total enthalpy
in each cell, the temperature can be found through an iteration of Eq. (2.6). Thus, the new
information about the species mass fraction and temperature can be used to update the
density and pressure in the flow field and the solution procedure can begin again.
3.2 Description of Turbulent Field
3.2.1 Energy Equation
The energy state is accounted for by solving an averaged total enthalpy equation. Specifi-
cally, the total enthalpy is solved according to
∂ρH
∂t+∂ρujH
∂xj=
∂
∂xj
(µtPr
∂H
∂xj
)+∂p
∂t− qs − qr + qY (3.2)
where µt is the turbulent viscosity, Pr is the Prandtl number, and qs and qr account for the
heat transfer to the liquid phase and heat losses to the walls, respectively. The final term,
qY , is a source to account for the enthalpy added to the gas phase from the evaporation
of the fuel. Solving this form of the energy equation is advantageous since enthalpies of
3.2. DESCRIPTION OF TURBULENT FIELD 33
CFD
Flameletp (t)
χξ(ξ, η; t), χη(ξ, η; t)
χξη(ξ, η; t)
ξ(x, t), ξ′ 2(x, t),
η(x, t), η′ 2(x, t),
ξ′η′(x, t)
H(x, t)
Y (ξ, η; t)
Yi(x, t) =
∫∫Yi(ξ, η; t)P (ξ, η;x, t)dξdη
H(x, t) =
ns∑
i
Yi(x, t)(T (x, t)
)
Yi(x, t), T (x, t)
Figure 3.1: Schematic of the RIF concept showing the coupling of the CFD and chemistrysolvers.
formation of the mixture are included in the total enthalpy, meaning that it is conserved
across reactions and thus there is no chemical source term.
3.2.2 Mixing field
Non-premixed combustion is often modeled using what is known as a presumed PDF ap-
proach, where the statistics of the mixture fraction field are computed to be used as inputs
to a parameterized probability density function. The statistics required are the average
mean mixture fraction, Z, and its corresponding variance, Z ′ 2. When considering a multi-
stream system, equations must be solved for each mixture fraction and its corresponding
variance, as well as an additional equation to account for the correlation between mixture
fractions. Each mixture fraction can be solved according to a scalar transport equation de-
fine by Eq. (2.8), which after Favre averaging gives the following equation for mean mixture
fraction as a function of time and space
∂ρZi∂t
+∂ρujZ ′ 2i∂xj
= − ∂
∂xj
(ρu′jZ
′i
)+ ρ˜Si (3.3)
34 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
where i = 1, 2 is used to indicate the stream corresponding to the mixture. Although stream
0 will also follow this equation, it is unnecessary to solve a transport equation for it as it can
be related to the other two through Eq. (2.38). The source term, Si, represents the Eulerian
vaporization rate of the fuel from stream i in each cell. This source term has been included
as it has been shown that vaporization of a discrete phase tends to generate fluctuations
of mixture fraction and that these effects are non-negligible (Reveillon & Demoulin, 2007;
Reveillon & Vervisch, 2000; Pera et al., 2006).
From manipulation of Eq. (3.3), the equation for Favre averaged variance of the mixture
fraction is found to be
∂Z ′ 2i∂t
+∂ρujZ ′ 2i∂xj
= − ∂
∂xj
(ρu′jZ
′i
)− 2
(ρu′jZ
′ 2i
)∂Zi∂xj− 2ρD
(∂Z ′ 2i∂xj
)2
+ 2ρ(
1− Zi)Z ′iSi − ρZ ′ 2i Si
(3.4)
for i = 0, 1, 2. Here, three variance equations must be solved to account for the correlation
between the mixture fractions. The fluctuations of stream 0 can be written as
Z ′0 = (1− Z1 − Z2)′ = − (Z1 + Z2)′ (3.5)
from which the variance is
Z ′ 20 = Z ′ 21 + 2Z ′1Z′2 + Z ′ 22 . (3.6)
Thus, the covariance of the mixture is obtained from the variance of each stream according
to
Z ′1Z′2 =
1
2
(Z ′ 20 − Z ′ 21 − Z ′ 22
)(3.7)
Alternatively, one could solve an equation directly for the covariance (Bilger, 1999; Kim,
2002). However, the resulting equations have additional unclosed terms, some of which are
less well understood and must be neglected. Since all streams in a three feed system should
follow Eq. (3.4), computing each variance as such and finding the covariance from Eq. (3.7)
ensures that the approximate models are consistent across all streams.
3.2. DESCRIPTION OF TURBULENT FIELD 35
There are several unclosed terms in Eqs. (3.3) and (3.4), including the transport terms,
the scalar dissipation rate in the variance equation, which will be written
χi = 2D
(∂Z ′ 2i∂xj
)2
, (3.8)
as well as correlations between the mixture fraction fluctuations and the evaporative source
term. For a conserved scalar, the turbulent transport term is typically closed using a
gradient flux approximation, stemming from a balance between production and dissipation,
given as
ujZ ′ = −Dt∂Z
∂xj. (3.9)
However, it has been shown by Peters (2000) that this approximation is not valid for non-
conserved scalars. Although Peters considered reactive scalars and therefore investigated
a scalar sink term, a similar analysis can be performed for a scalar source from droplet
evaporation.
In order to determine the effect of the source on the unclosed terms, information about
the source term is required. Reveillon & Vervisch (2000) investigated evaporation of disperse
liquid droplets mixing in a DNS of isotropic turbulence. It was found that the evaporative
source term conditioned on mixture fraction is a monotonically increasing function that can
be modeled as a power law according to
(S|Z
)= αZ ξ. (3.10)
The parameters of Eq. (3.10) are determined from the local flow properties by assuming
that all droplets within each cell can be represented by a single droplet of equivalent fuel
volume. The exponent accounts for the effect of the turbulence on the droplet evaporation
and is determined from the constraint that the integrated conditional source term must
recover the Eulerian evaporation rate, that is
˜S =
∫
Z∗
˜(Si|Z∗
)P (Z∗)dZ∗, (3.11)
where the Favre probability density function of the mixture fraction, P (Z∗), is assumed to
be a beta function scaled by the local saturation mixture fraction. Through these relations,
36 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
both of the source terms due to evaporation in Eq. (3.4) can be closed using
Z ′S =
∫ Zs
0
(Z∗ − Z
) (S|Z
)P (Z∗)dZ∗, (3.12)
Z ′ 2S =
∫ Zs
0
(Z∗ − Z
)2 (S|Z
)P (Z∗)dZ∗, (3.13)
where Zs is the local saturation mixture fraction.
The remaining unclosed terms are the scalar dissipation rate and the turbulent transport.
The scalar dissipation rate defined by Eq. (3.9) is typically modeled by assuming that it is
proportional to the turbulent time scale, τ = k/ε. One can construct an integral turbulent
time scale for the mixture fraction as
τZ =Z ′ 2
χ(3.14)
Thus, the scalar dissipation rate can be determined from
χ =τ
τZ
ε
kZ ′ 2. (3.15)
For conserved scalars, the time scale ratio is often taken to be τ/τZ = 2.0 (Jones & Whitelaw,
1982). However, the spray also has a direct impact on the scalar dissipation rate. It is found
that the evaporation causes the scalar dissipation rate to be larger in magnitude due to the
increase in gradients caused by a localized source of fuel at the droplet surface (Reveillon
et al., 1998). Neglecting to take this into account will result in an over-prediction of the vari-
ance. Corrsin (1961) investigated a linearly reacting scalar mixing in isotropic turbulence
and derived the power spectrum of the scalar fluctuations. The linear source investigated
is functionally equivalent to that of Eq. (3.10) with unity exponent, i.e. ξ = 1. It has been
found that the exponent is typically only moderately non-linear, with significant departure
from linearity only in regions where there is a large number of droplets and thus the disperse
phase approximation is no longer valid.
Therefore, using Corrsin’s results, the time scale ratio is expressed as
τ
τZ=
3cχαLτ
1− exp (−3αLτ)(3.16)
where cχ is the limit of the time scale ratio for the conserved scalar case, taken to be cχ = 2,
3.2. DESCRIPTION OF TURBULENT FIELD 37
and αL is a coefficient obtained from a linearization about Z of Eq. (3.10) according to Hasse
(2004), giving
αL = α(ξZξ−1 − (ξ − 1)Zξ
). (3.17)
Combining Eqs. (3.15) and (3.16), an expression for the scalar dissipation rate of each scalar
is obtained as
χi =3cχαLτ
1− exp (−3αLτ)
ε
kZ ′ 2i (3.18)
Returning now to the turbulent transport term, a correction factor for the gradient flux
approximation can be obtained to account for the evaporation source. The analysis follows
that of Peters (2000) and furthered by Hasse (2004), where a balance is taken between
production, dissipation, and in this case, evaporation, to give the turbulent transport term
as
u′jZ′ = −Dt
τ/τZτ/τZ − 2αLτ︸ ︷︷ ︸
fD
∂Z
∂xj(3.19)
where the equation has been scaled to recover the conserved scalar formulation. The indi-
cated term, fD, is the correction to the standard gradient flux approximation where τ/τZ
is computed according to Eq. (3.16). Furthermore, the turbulent diffusivity is commonly
expressed as
Dt =µtρSct
(3.20)
where µt is the turbulent viscosity and Sct the turbulent Schmidt number.
Substituting Eqs. (3.19) and (3.20) into Eq. (3.3) gives the closed equation for mean
mixture fraction as
∂ρZi∂t
+∂ρujZ ′ 2i∂xj
=∂
∂xj
(fD
µtSct
∂Zi∂xj
)+ ρ˜Si (3.21)
for i = 1, 2. The final mixture fraction variance equation can be written
∂Z ′ 2i∂t
+∂ρujZ ′ 2i∂xj
=∂
∂xj
(fD
µtSct
∂Z ′ 2i∂xj
)+ 2fD
µtSct
(∂Zi∂xj
)2
− 2ρχ+ ρ ˜S+i − ρ
˜S−i (3.22)
for i = 0, 1, 2 and where χi is defined by Eq. (3.18) and the evaporative source terms ˜S+i
and ˜S−i are computed for each stream according to Eqs. (3.12) and (3.13), respectively.
38 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
3.3 Scalar Dissipation Rate Modeling
The external parameters in the flamelet equations described by Eqs. (2.59) and (2.61) are
the scalar dissipation rates. Considering that these parameters determine the effect of the
turbulent mixing field on the flame structure, it is important to represent them accurately
in the model mixture fraction space. However, except in DNS, the scalar dissipation rates
are not directly available for the small scales and therefore a method of modeling them from
information available in the flow field is necessary.
In this section, methods for obtaining the modeled scalar dissipation rates will be intro-
duced. First, a one-dimensional model for single mixture fractions is developed based on
profiles from canonical cases found in the literature. Then, these models will be extended to
a two-dimensional representation by assuming independence of the scalar dissipation rates
and a method for finding the joint dissipation rate will also be given. Finally, when this
assumption does not hold, a method is developed to account for the dependence of each
scalar dissipation rate on the rest of the mixture.
3.3.1 Single Mixture Fraction
The scalar dissipation rate is modeled as a function of mixture fraction in each grid cell
of the turbulent flow field. Each grid cell is assumed to have a distribution that has a
functional form, f(Z), which is scaled by a value of scalar dissipation rate at a reference
mixture fraction according to
χ(Z;xj) = χref(xj)f(Z). (3.23)
The functional form can be based on either the inverse error function or logarithmic profiles
given by Eq. (2.69) or Eq. (2.70), respectively. However, to remove the time dependence,
the profile will be scaled with respect to the reference mixture fraction corresponding to
χref . Thus, Eq. (2.69) can be used as
fE(Z) = exp{−2[erfc−1 (2Z)
]+ 2
[erfc−1 (2Zref)
]}, (3.24)
and Eq. (2.70) becomes
fL(Z) =Z2
Z2ref
log(Z)
log(Zref). (3.25)
3.3. SCALAR DISSIPATION RATE MODELING 39
In this work, the logarithmic form of Eq. (2.70) will be used for a basis as opposed to the
error function form of Eq. (2.69) since it is more applicable to cases with a finite amount
of fuel, as discussed in Sec. 2.2.3. The reference mixture fraction is sometimes taken as
stoichiometric, however here the local mean, Z, will be used. Finally, the local χ(Z) profile
will be further scaled based on the local maximum mixture fraction estimated from its
variance. Thus, the final profile used in each cell is
f(Z) =Z2
Z2
log(Z/Zmax)
log(Z/Zmax). (3.26)
If the distribution of mixture fraction is known, a mean scalar dissipation rate can be
computed by convoluting Eq. (3.23) with the local average probability distribution, P (Z),
as
χ(xj) = χref(xj)
∫ 1
0f(Z)P (Z;xj)dZ (3.27)
where it is assumed that χref is only a function of the reference mixture fraction, Zref , and
is therefore constant. The local probability distribution is a presumed form parameterized
by the mixture fraction mean and variance and is typically taken to be a beta distribution
as described in Sec. 3.5.
As discussed in Sec. 3.2.2, the mean scalar dissipation rate can also be modeled from the
turbulence by assuming proportionality of the scalar and turbulent time scales. Thus, the
reference scalar dissipation rate can be found by taking the ratio of Eqs. (3.27) and (3.15),
giving
χref(xj) =
τ
τZ
ε
kZ ′ 2
∫f(Z)P (Z;xj)dZ
(3.28)
Substituting the above into Eq. (3.23) results in a closed form for computing χ(Z;xj) in
each grid cell. However, if a single flamelet is used to represent the physical domain, the
spatial dependence is removed by integrating the sub-grid profile over the volume to obtain
a system averaged conditional scalar dissipation rate,
χ(Z) =
∫
Vρχref(xj)f(Z)dV∫
VρdV
. (3.29)
40 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
3.3.2 Joint Dissipation Rate Model
As was discussed in Sec. 2.2.3, the dissipation rate of each scalar can be assumed to be
independent of the other. This assumption makes the modeling of each scalar dissipation
rate fairly straight-forward, as a method analogous to the single scalar case described in the
previous section can be used. Specifically, the distribution for each scalar dissipation can
be written
χi(Z1, Z2;xj) = χref,i(xj)f(Zi), (3.30)
for i = 0, 1, 2 where the same functional form of Eq. (3.26) can be applied to each mixture
fraction. We now require information about the joint statistics of the mixture fractions to
obtain mean scalar dissipation rates from
χi = χref,i(xj)
∫∫f(Zi)P (Z1, Z2;xj)dZ1dZ2. (3.31)
The average joint PDF, P (Z1, Z2;xj), is computed from the local means and variances of
each mixture fraction and methods for modeling it will be discussed in detail in Chapter 5.
The reference scalar dissipation rate can be computed from
χref,i(xj) =
τ
τZ
ε
kZ ′ 2i
∫∫f(Zi)P (Z1, Z2;xj)dZ1dZ2
(3.32)
and the system averaged conditional scalar dissipation rate is then computed for the entire
domain by integrating the sub-grid profile over the volume as
χi(Zj) =
∫
Vρχref,i(xj)f(Zi)dV
∫
VρdV
. (3.33)
Since this method assumes that there is no dependence of each scalar dissipation rate on
the other mixture fraction, the one-dimensional profile computed for Zi is applied uniformly
for all values of the other scalar, Zk 6=i. It is worth investigating the validity of this assump-
tion in further detail, for which we will consider results from a DNS of two scalar mixing
that will be presented in Chapter 5. First, the marginal distribution of χ1 is plotted for
two different mixing times in Fig. 3.2. Here, the mean distribution conditioned on mixture
3.3. SCALAR DISSIPATION RATE MODELING 41
2 CHAPTER 1. TEST
0
20
40
60
80
100
0 0.2 0.4 0.6 0.8 1
χ1
Z1
meanlogarithmic
erfc−1
polynomial
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(a) t/τeddy ≈ 0.0
2 CHAPTER 1. TEST
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
χ1
Z1
meanlogarithmic
erfc−1
polynomial
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(b) t/τeddy = 0.25
Figure 3.2: Marginal distribution of χ1 conditioned on Z1 from mixing of isotropic scalarsin isotropic turbulence for two time instances. Figure (a) shows the initial distribution and(b) is after 1/4 eddy turnover. The red curve is the mean computed from the DNS and thefilled grey region represents the minimum and maximum extents of χ1 in the entire domain.Three model distributions are also plotted.
42 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
fraction from the DNS field is shown along with the minimum and maximum χ1 found in
the domain to serve as an indication of the range of variation. From Fig. 3.2(a) it can be
seen that even though there is initially some range of values, it constitutes very little of the
domain, as the mean value corresponds very closely to the maximum value. After the scalar
field has partially mixed, the mean value is centered between the minimum and maximum
values, but still indicates that there is very little dependence of χ1 on Z2. Note that this
remains true even though the dissipation rate has decreased by an order of magnitude. It
is found that there is significant departure from this behavior only at late times, when the
scalar dissipation rate becomes very small and is therefore not as important. For this case,
each mixture fraction was initially isotropically distributed in the domain and therefore the
results are the same for χ2. Thus, the approximation seems to be valid over the range of
interest.
In Fig. 3.2 the functional forms defined by Eqs. (3.24) and (3.25) are also plotted. These
were computed using the mean mixture fraction in the domain for Zref and the value of χref
was computed by averaging χ1 conditioned on that value. In addition, a simple polynomial
form was also computed according to
f(Zj) =Z(1− Z)
Zref(Zref − 1). (3.34)
The polynomial form was plotted as it appears to be the best representation for this case.
This can be attributed to the fact that the inverse error function and logarithmic profiles
have zero gradients at one or both boundaries, whereas the conditional data from the DNS
has non-zero gradients at both Z1 = 0 and Z1 = 1. These gradients stem from the fact
that the mixture fraction fields for this case were both finite, and thus the approximations
made in obtaining Eq. (2.69) and Eq. (2.70) are not appropriate. The polynomial form of
Eq. (3.34) appears as the result of finite gradient boundary conditions and captures the
non-zero gradients at the boundaries.
Looking at the distributions of each scalar dissipation rate in (Z1, Z2) space, plotted in
Fig. 3.3, one can notice several things. First, it is also here apparent that there is very little
dependence of χ1 on Z2 or χ2 on Z1. Furthermore, as discussed in Sec. 2.2.3, neither χ1
or χ2 is zero along the Z1 + Z2 = 1 boundary and so it was argued that χ0 must be zero
and, as a consequence, χ12 must be negative along this line. The observations confirm both
of these conditions, as can be seen in the Figs. 3.3(c) and 3.3(b). In fact, χ0 as shown in
3.3. SCALAR DISSIPATION RATE MODELING 43
2 CHAPTER 1. TEST
1
Z1
0
1
Z2
0
20
40
60
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(a) χ2
2 CHAPTER 1. TEST
1
Z1
0
1
Z2
-60
-40
-20
0
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(b) χ12
2 CHAPTER 1. TEST
1
Z1
0
1
Z2
0
20
40
60
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(c) χ0
2 CHAPTER 1. TEST
1
Z1
0
1
Z2
0
20
40
60
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(d) χ1
Figure 3.3: Distributions of scalar dissipation rates conditioned on (Z1, Z2) as computedfrom DNS of two scalar mixing in isotropic turbulence at t ≈ 0. Each mixture fraction wasinitialized isotropically in the physical domain. Note that χ0 was not conditioned from theDNS data, but was rather computed from the data according to Eq. (2.74).
44 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
Fig. 3.3(c) was not conditioned directly from the DNS and was computed from the other
dissipation rates according to Eq. (2.74). The angle between the mixture fraction gradients
can also be computed according to
θZ = cos−1
(χ12√χ1χ2
). (3.35)
The computed angle is shown in Fig. 3.4, showing that it is 180◦ at the boundary and
decreases toward the origin.
2 CHAPTER 1. TEST
0
1Z1 0
1Z2
90
135
180
θZ
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
Figure 3.4: Angle of alignment between physical gradients of each mixture fraction.
3.3.3 Independence of Scalar Dissipation Rates
The preceding section showed that scalar dissipation rates of each mixture fraction can be
considered independent of the rest of the mixture. However, it has been observed that some
dependence can arise if the scalar field is far from isotropically distributed or if the scalar
means are not equal. An example of such a field is shown for χ1 in Fig. 3.6(a). It appears
that the dependance arises not from the fact that it has a fully two-dimensional functional
form, i.e. f(Z1, Z2), but rather that the reference value for the one-dimensional functional
form can vary with the mixture composition. For example, if we consider χ1, this can be
expressed by writing
χ1(Z1, Z2) = χref,1(Z2)f(Z1). (3.36)
3.3. SCALAR DISSIPATION RATE MODELING 45
The mean scalar dissipation rate from the model can be computed in the same manner as
previously,
χ1 =
∫∫χref,1(Z2)f(Z1)P (Z1, Z2)dZ1dZ2, (3.37)
except that χref,1 can no longer be taken outside the integral. From the above, Bayes’
theorem can be used to substitute for the joint PDF and obtain
χ1 =
∫χref,1(Z2)P2(Z2)
∫f(Z1)P (Z1|Z2)dZ1dZ2
=
∫χref,1(Z2)F (Z2)P2(Z2)dZ2 (3.38)
where P2(Z2) is the marginal distribution and
F (Z2) =
∫f(Z1)P (Z1|Z2)dZ1. (3.39)
If the actual distribution were known, the mean could also be found directly from
χ1 =
∫∫χ1(Z1, Z2)P (Z1, Z2)dZ1dZ2
=
∫P2(Z2)
∫χ1(Z1, Z2)P (Z1|Z2)dZ1dZ2
=
∫χ1,M2(Z2)P2(Z2)dZ2 (3.40)
where χ1,M2(Z2) is a marginal scalar dissipation rate. Comparing Eq. (3.38) and Eq. (3.40)
and noting that the marginal PDF must be equivalent in both, the marginal scalar dissipa-
tion rate can be written as
χ1,M2(Z2) = χref,1(Z2)F (Z2). (3.41)
Therefore, a reference scalar dissipation rate can be found for each Z2 in the same manner
as Eq. (3.28) by using the conditional p.d.f. as
χref,1(Z2) =χ1,M2(Z2)∫
F (Z2)P (Z1|Z2)dZ1
(3.42)
46 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
The reference mixture fraction required for the profile will also vary with the relevant χref,1
and can be taken to be the conditional mean as
Zref,1(Z2) =
∫Z1P (Z1|Z2)dZ1. (3.43)
However, it is apparent from Eq. (3.42) that we need to model the conditional mean scalar
dissipation rate, χ1,M2(Z2). As the data indicate that this distribution is not uniform, the
simplest solution to recover the mean would be to distribute the marginal scalar dissipation
rate according to the marginal PDF as
χ1,M2(Z2) =χ1
P2(Z2). (3.44)
Unfortunately, this does not provide an accurate representation of the distrbution. Alterna-
tively, it can be assumed that since the distribution of χ1 for any mixture follows the same
functional form, that is χ1(Z1)|Z2 = f(Z1), that the mean marginal distribution will also be
of the same form. Therefore, a marginal distribution, χ1,M1 , can be found using Eq. (3.23)
with mean quantities. A distribution of χ1 in Z2 can then be found by integrating the
marginal scalar dissipation rate with the conditional distribution for each mixture, i.e.
χ1,M2(Z2) =
∫χ1,M1(Z1)P (Z1|Z2)dZ1, (3.45)
to give the functional dependence on Z2. The presumed distributions of χ1 in Z2 are
computed in this manner are shown in Fig. 3.5, where it is observed that the overall trend is
roughly captured. When applied to the two-dimensional domain, the resulting distribution
in (Z1, Z2) is shown in Fig. 3.6 for one of the models along with the data conditioned from
the DNS, demonstrating that the method developed in this section is able to represent the
dependence of χ1 on Z2 reasonably well.
3.4 Initialization of a Two-dimensional Flamelet Field
When considering multi-stream ignition problems, each stream is typically introduced at
different times. For the first stream, a single one-dimensional flamelet will be used for the
solution. When the second stream is introduced, it is necessary to construct a fully two-
dimensional field in mixture fraction based on the existing 1D solution along the ordinate,
3.4. INITIALIZATION OF A TWO-DIMENSIONAL FLAMELET FIELD 47
2 CHAPTER 1. TEST
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1
χ1
Z2
meanlogarithmic
erfc−1
polynomial
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 3.5: Scalar dissipation rate χ1 conditioned on Z2 at for scalar fields with unequalmeans.
2 CHAPTER 1. TEST
1
Z1
01
Z2
0
5
10
15
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(a) χ1 computed
2 CHAPTER 1. TEST
1
Z1
01
Z2
0
5
10
15
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(b) χ12 modeled with parabolic form
Figure 3.6: Two-dimensional distribution of χ1 for scalar fields with unequal means. Figure(a) shows data conditioned from DNS data and (b) shows model distribution using themethod of this section.
48 CHAPTER 3. REPRESENTATIVE INTERACTIVE FLAMELETS
0 1
1
Z1
Z2
Figure 3.7: Interpolation used to initialize a 2D solution in mixture fraction space from anexisting 1D solution and a new stream boundary condition. Dashed lines represent mixinglines along which interpolation is carried out.
and the boundary conditions of the second stream. The most obvious option is to mix each
point along the Z1 axis with the boundary condition of the second stream, i.e. Z2 = 1.
The mixing lines are depicted in Fig. 3.7. Although this solution is an approximation, it is
found that the solution field will adjust itself very rapidly and relax to a consistent solution
within the first time iteration.
3.5 Calculation of Mean Quantities
Since no equations for the mean species mass fractions of the mixture are being solved,
they must be obtained by convoluting the flamelet solution with a PDF. That is, the
instantaneous mean species mass fraction can be computed as
Yi(xj , t) =
∫ 1
0P (Z;xj , t)Yi(Z; t)dZ. (3.46)
In the moment closure methods used here, the shape of the scalar PDF is taken to be
a presumed form that is parameterized by the scalar moments. For a single scalar, the
beta distribution is widely used, as it has been shown to represent closely the mixing of a
conserved scalar in isotropic turbulence (Girimaji, 1991, 1992). The beta distribution is a
3.5. CALCULATION OF MEAN QUANTITIES 49
univariate distribution defined by
P (φ;β1, β2) =1
B(β1, β2)φβ1−1(1− φ)β2−1 (3.47)
for 0 < φ < 1 and β1 > 0, β2 > 0. The distribution is normalized by the beta function,
B(a, b), which can be written in terms of the gamma function according to
B(a, b) =Γ(a)Γ(b)
Γ(a+ b). (3.48)
The shape of the beta distribution is dependent on the parameters β1, β2, and can be U -
shaped (β1 < 1, β2 < 1), uniform (β1 = β2 = 1), unimodal (β1 > 1, β2 > 1), or J-shaped or
reverse J-shaped when either β2 < 1 or β1 < 1, respectively. For more information about
the mathematical properties of the distribution, refer to Gupta & Nadarajah (2004).
For two-dimensional solutions, the problem is more complicated, as a two-dimensional
joint scalar distribution is required such that the mean species can be computed from
Yi(xj , t) =
∫∫P (Z1, Z2;xj , t)Yi(Z1, Z2; t)dZ. (3.49)
The distribution must be parameterized by the second order moments of the mixture frac-
tions, which include a mean and variance for each mixture along with the covariance be-
tween the mixture fractions. The determination of an appropriate joint scalar PDF for use
in Eq. (3.49) is the subject of Chapter 5 and will be discussed in detail.
Chapter 4
Ignition of Multi-Stream Systems
In this chapter, the two-dimensional flamelet representation is investigated using finite-rate
DNS of multi-stream ignition for validation. First, the DNS of constant volume ignition
will be described. Then, the DNS results will be used to evaluate the ability of the flamelet
to capture the ignition and combustion of the multi-stream system
This chapter will first introduce the methods employed to compute the ignition of n-
heptane in isotropic turbulence. The various cases investigated will then be described and
motivated, followed by a description of the procedure and implementation of the flamelet
model to enable a comparison with the DNS.
Of the cases investigated, two will be presented in detail. The first case computed is a
standard single fuel stream system to provide a baseline for comparison. A one-dimensional
flamelet is also computed and the results discussed. A second case with two fuel streams
is then presented. Here, one of the fuel streams is introduced at a specified delay after the
first to represent a split-injection type configuration. The results of the single and split fuel
streams are then compared and the applicability of the flamelet model is discussed. Finally,
the effect of the time delay between the introduction of the streams and the maximum
mixture fraction of each stream are investigated.
4.1 DNS with Finite Rate Chemistry
In order to test whether the flamelet model is capable of representing the mixing and
ignition of a scalar field with multiple streams, a DNS validation was carried out using
finite-rate chemistry in decaying isotropic turbulence in a constant volume. In this section,
50
4.1. DNS WITH FINITE RATE CHEMISTRY 51
the numerical implementation of the finite-rate chemistry DNS will be described, after which
the cases considered will be introduced and motivated.
4.1.1 Numerical Implementation
The turbulent flow field was computed by solving the equations for conservation of mass
and momentum, defined by
∂ρ
∂t+∂ρuj∂xj
= 0 (4.1)
∂ρui∂t
+∂ρuiuj∂xj
=∂p
∂xi+∂σij∂xj
(4.2)
where uj is the velocity vector, p is the pressure, ρ is the density, and σij is the stress tensor.
A low Mach number approximation was used in the solution of the above equations (Muller,
1999; Desjardins et al., 2008). In addition, scalar equations for each species mass fraction
considered, Yk, were solved according to Eq. (2.8) giving
∂ρYk∂t
+∂ρuiYk∂xi
=∂
∂xi
(ρDk
∂Yk∂xi
)+ ρωk. (4.3)
The diffusivity of each species was related to the thermal diffusivity by
Dk =1
Lek
λ
cp(4.4)
where λ and cp are the thermal conductivity and specific heat of the mixture, respectively.
Here, the Lewis numbers of each species, Lek, were taken to be unity. An equation for
temperature is solved as the energy equation according to
∂ρT
∂t+∂ρuiT
∂xj=
∂
∂xj
(λ
cp
∂T
∂xj
)+∑
k
∂
∂xj
(λ
c2p
hk
)∂Yk∂xj
+1
cp
(∂p
∂t+ ρ
∑
k
hkωk
). (4.5)
Alternatively one could represent the energy with the total enthalpy and use Eq. (2.6) as
a coupling with the species transport. Using the temperature equation has the advantage
that the resulting coupling is a set of differential equations, rather than the differential-
algebraic system defined by species mass fractions and total enthalpy, which can be solved
more efficiently.
52 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
The chemical source term for each species, ωk, was obtained by integration of the detailed
multi-step finite rate chemistry using the VODE implementation (Brown et al., 1989) to
advance the coupled set of ordinary differential equations described by Eqs. (4.3) and (4.5).
The total enthalpy was updated using Eq. (2.6) for the new species mass fractions and tem-
perature. The chemistry was defined by the chemical mechanism described in the following
section.
All simulations were carried out on a domain defined by a two-dimensional periodic box
with a side length of 3.1 millimeters. The domain was further considered to be a constant
volume and was discretized with a uniform grid using N2 grid points, resulting in a grid
spacing of ∆ = L/N . A grid resolution of N = 512 was used in the simulations presented
in this section to give sufficient resolution of the reaction zone. Several simulations were
carried out with N = 1024 to confirm that increased resolution does not show different
results. The velocity and scalar fields were solved using the high-order conservative finite
difference method discussed by Desjardins et al. (2008) for variable density flows. In this
work, fourth-order spatial and second-order temporal accuracy were employed. The scalar
fields were solved using a Weighted Essentially Non-Oscillatory (WENO3) formulation (Liu
et al., 1994). Variable time steps were used with a maximum Courant number (CFL)
was limited to 0.8 to keep time stepping errors small. Furthermore, a restriction on the
maximum temperature change during a time step was enforced to ensure that the ignition
process was time-resolved. Thus, during the fast reaction rates of ignition, this became the
time step limiter.
A two-dimensional domain was used in order to keep the overall computational cost
reasonable. Although two-dimensional turbulence is not physically accurate, for the pur-
poses of providing a representation of diffusion to validate the flamelet approximation, it
should be sufficient as long as there is a consistent representation between the DNS and the
flamelet model implementations.
4.1.2 Chemical Mechanism
The reaction in the mixture was modeled using a reduced chemical mechanism for n-heptane.
The mechanism considers a total of 44 species and 185 reactions from Liu et al. (2004). In
order to save computational expense, 19 species with relatively fast chemistry were assumed
to be in steady-state. This enabled solving differential equations for rates and transport
equations only for the 25 non-steady-state species. The mechanism was validated over a
4.1. DNS WITH FINITE RATE CHEMISTRY 53
range of pressures and stoichiometries, as well as for different strain rates, and is found to
represent the ignition delay time of n-heptane-air ignition well.
4.1.3 Initial and Boundary Conditions
The velocity field was initialized by defining a background fluctuating velocity specified
by an r.m.s. value, u′. The initial energy spectrum was then prescribed according to the
Passot-Pouquet distribution, defined by
E(κ) =32
3
√2
π
u′2
κe
(κ
κe
)exp
[−2
(κ
κe
)2]
(4.6)
where κ is the wavenumber, κe is the energetic wavenumber, and u′ is the velocity r.m.s.
In these simulations an initial value of u′ = 0.5 m/s was used.
The mixture fraction fields were initialized isotropically in a manner similar to Sec. 5.3.2.
However, here a maximum value of the mixture fraction was used in order to maintain a
lean overall equivalence ratio in the domain while retaining enough distribution of mixture
fraction for isotropic conditions. This is also justified since in spray combustion applications,
the conditions at the liquid-vapor interface will not be pure fuel, but rather a saturation
mixture fraction determined by the ambient conditions. The maximum mixture fraction
of each stream was therefore defined to be representative of the maximum that would be
obtained from a droplet evaporating under ambient conditions typical of diesel combustion.
Each fuel stream was initialized isotropically in wavespace according to the procedure
described by Eqs. 5.23 and 5.24 in Sec. 5.3.2. The resulting scalar field was then smoothed
by transferring the field back to Fourier space and applying a low-pass filter, defined by
F (κ) =
{1, if κs ≤ κc(κ/κc)
−2 , otherwise.(4.7)
where κc is the specified cutoff wavenumber, which is specified in relation to the wavenumber
of the initialization top-hat function by setting a ratio κc/κs (see Eq. 5.24). Eswaran &
Pope (1988) showed the effect of this ratio is insignificant if κc/κs > 2. In this work, a value
of κc/κs > 8 was used for all mixture fraction fields considered and was found to provide
sufficient resolution of the interface. A typical initial mixture fraction field can be seen in
Fig. 4.3 along with the corresponding temperature distribution.
54 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
Table 4.1: Summary of boundary conditions of each stream for DNS studies.
T (K) YN2 YO2 YH2O YCO2 YC7H16
Oxidizer 960 0.7563 0.1772 0.0453 0.0212 -Fuel 492 - - - - 1.0
For multi-stream simulations, each fuel stream was taken to be a specified percentage
of the total fuel. The first stream was computed as a two feed system and the temporal
evolution of the field was stored. After a specified delay time, τ2, an initial mixture fraction
of the second fuel stream was overlaid onto the partially reacted solution. In this study, the
fuel was split to have 20% of the fuel in the first fuel stream, with the remaining introduced
as the second stream at τ2. An example of the initial fields of each mixture fraction and
the corresponding temperature is shown in Fig. 4.6.
The boundary conditions of the fuel and oxidizer streams were set to be relevant to those
found in an engine, i.e. at elevated temperature and pressure. The initial pressure was set
to 40 bar and the oxidizer stream temperature was chosen to be 960 K. This temperature is
interesting as it is within the low temperature chemistry regime that is found near top-dead-
center (TDC) in a diesel engine, and thus affects the ignition properties. Additionally, the
oxidizer stream was taken to be air plus some combustion products, which could be used in
engines from exhaust gas recirculation (EGR). The overall equivalence ratio of the system
was specified to be Φ = 0.79, which might be found at high load operating conditions in a
diesel engine. The composition of the oxidizer stream is summarized in Table 4.1.
The conditions of the fuel stream were obtained by determining what the surface con-
ditions of an evaporating droplet would be in the ambient environment of the oxidizer.
Although no evaporation is taking place in the simulation, it is desired that the fuel tem-
perature be consistent with one that might be found in a system with liquid fuel evaporation.
The temperature at the surface was found by evaluating the Spalding number based on both
thermal and concentration gradients to drive the evaporation. The transfer number due to
Stefan flow is found from
BT =cpg(T∞ − Ts)
Q(4.8)
where Q = ∆hf + cpl(Ts−Tl,0) is the total energy to heat the droplet from the initial liquid
4.2. MULTI-DIMENSIONAL FLAMELET MODEL 55
to temperature, Tl,0, to Ts. The transfer number due to Fickian diffusion can be found from
BY =Yf,s − Yf,∞
1− Yf,s(4.9)
where Yf is the fuel mass fraction and the subscripts denote surface and ambient properties.
In order to close the problem, a relationship between the surface temperature and surface
fuel mass fraction is required. These quantities can be related by assuming that the droplet
surface is in phase equilibrium, such that the partial pressure can be written
pf,s = psat(Ts) = Xf,sp (4.10)
where Xf,s is the mole fraction of the fuel at the surface. To obtain the saturation pressure,
one could use a standard equation of state such as the Clausius-Clapeyron relation or Peng-
Robinson (Peng & Robinson, 1976). Here, data was taken directly from NIST (Lemmon
et al., 2010) for n-heptane, which uses a twelve parameter equation of state in Helmholtz
energy (Span & Wagner, 2003a,b) resulting in a more accurate representation. Thus, each
transfer number can be computed as a function of surface temperature and is shown in
Fig. 4.1. Note that the critical temperature of n-heptane is 540 K and thus there is no
saturation solution above this point. From the plot, it can be seen that a root for the
saturation condition of a droplet with liquid temperature of 300 K is at a temperature of
Ts = 492 K, which is then taken to be the temperature of the evaporated fuel. However,
since fuel mass fraction at the surface will not be unity, a maximum mixture fraction will
be used. This is taken to be either 0.2 or 0.1, depending on whether the fuel is from the
pilot or main injection. Thus, the actual minimum temperature observed in the domain is
calculated by linear interpolating species and enthalpy between the boundary values and
iterating for temperature at the prescribed mixture fraction.
4.2 Multi-dimensional Flamelet Model
4.2.1 Numerical implementation
The flamelet model was used to compute the ignition of the system described by the DNS
for validation. To do so, Eqs. (2.81) and (2.82) were solved. The solution was advanced
using data obtained from the DNS for the specification of the scalar dissipation rates and
56 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
2 CHAPTER 1. TEST
0
1
2
3
4
5
6
7
300 350 400 450 500 550
B
T (K)
BT
BY
Ts
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.1: Transfer number computed for thermal and concentration gradients to find thesurface temperature
the joint scalar PDF. Each scalar dissipation rate and cross-dissipation rate were computed
according to Eq. (2.60) and were conditioned on the two mixture fractions at each time
step. The joint probability distribution was calculated by creating a histogram of each
scalar corresponding to the grid used for the flamelet computation.
The flamelet was solved on a structured grid with 128×128 points for each mixture
fraction. The solution was initialized at the same pressure and with the same boundary
conditions as the DNS, described by Table 4.1. The conditioned scalar dissipation rate data
from the DNS was then used to advance the solution forward in time, giving an updated
solution for species mass fractions and temperature as a function of mixture fraction. The
time advancement was accomplished by solving Eqs. (2.81) and (2.82) using an Alternating
Direction Implicit (ADI) method (Peaceman & Rachford, 1955). The rows and columns
were solved alternately using the stiff ODE solver CVODE (Cohen & Hindmarsh, 1995).
The time step was restricted using a CFL based on the off-direction explicit derivatives.
An ADI scheme was chosen both because it is easily parallelizable, with each processor
able to solve a subset of the rows/columns for each time step, as well as the fact that
the memory requirements are very low. Alternatively, one could use an inexact Newton
method, which are also available in the same solver suite (Hindmarsh et al., 2005). Using
such a fully implicit method may be advantageous if the CFL restriction from the explicit
4.2. MULTI-DIMENSIONAL FLAMELET MODEL 57
DNS
Flameletχξ(ξ, η; t), χη(ξ, η; t),
χξη(ξ, η; t)
P (ξ, η; t)
Y (ξ, η; t), T (ξ, η; t)
ρ(t) =p∗
Ru
∫∫ ∑i Yi(ξ, η, t)Mwi
T (ξ, η; t)P (ξ, η; t)dξdη
ρ(t) = ρ0 p(t) = p∗
p∗ = p∗ + ∆p
yes
no
Figure 4.2: Computation of flamelet using data from DNS studies for validation purposes
terms becomes overly expensive, however in this work the ADI scheme was found to be
most efficient.
The flamelet solution was combined with the joint PDF conditioned from the DNS to
compute a mean density according to
ρ =p∗
Ru
∫∫ ∑k Yk(ξ, η)Mwk
T (ξ, η)P (ξ, η)dξdη (4.11)
where p∗ is an estimate of the current pressure, Ru is the universal gas constant, and Mwk
are the molecular weights of each species. Since this is a constant volume configuration,
the overall density must remain constant. Thus, an iteration is performed by changing the
pressure until the density at the new time is equal to the initial density. Note that as the
pressure changes, the temperature also changes by the pressure rate source term shown
in equation Eq. (2.61). Once the density has converged, the flamelet solution is advanced
with the new pressure and the scalar dissipation rate obtained from the DNS. This process
is shown schematically in the flow chart depicted by Fig. 4.2.1 and was repeated for all
transient data obtained from the DNS, allowing a direct comparison of pressure and heat
release computed by the flamelet with that of the DNS.
58 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
Independent Stream Solution
In addition to computing the fully two-dimensional flamelet, a solution was also obtained
by solving a single one-dimensional flamelet for each stream. Since each flamelet is only a
representation of a pure mixture fraction, for regions with both mixture fractions the species
mass fractions were obtained through a simple averaging of the 1D solutions according to
Yk(Z1, Z2) = Yk,1(Z1) +Z2
Z1 + Z2(Yk,2(Z2)− Yk,1(Z1)) . (4.12)
This represents the simplest approximation that can be used to represent a three-feed
system in the absence of additional information about interaction between the streams.
Alternatively, a number of one-dimensional flamelets could be solved for each stream with
different boundary conditions reflecting the other mixture, however this would be equivalent
to solving a two-dimensional system while neglecting transport in the direction normal to
the mixture, and will not be considered here.
4.3 Validation Results
4.3.1 Two Feed System
First we will look at the ignition of a standard two feed system with a single fuel and
oxidizer stream to validate the one-dimensional implementation and also provide a basis
for comparison with the effect on the overall characteristics when an additional stream is
introduced. Here, the system is initialized with the same overall equivalence ratio of the
multi-stream system studied in the next section.
The initial mixture fraction field was initialized isotropically with a maximum of Zmax =
0.2. The mixture fraction field and corresponding temperature field of the DNS are shown
in Fig. 4.3.
The flamelet model was initialized with the same boundary conditions as the DNS and
was computed according to the method shown in Fig. 4.2.1. The resulting pressure and
heat release for both the DNS simulations and the flamelet model are shown in Fig. 4.4. It
is observed hat the mixture begins to react after approximately 0.4 ms and starts to ignite
around 0.6 ms. The pressure computed from the flamelet matches that obtained from the
DNS quite closely, with minor deviations in the ignition time and an over-prediction of the
peak heat release. These types of discrepancies can be attributed to the fact that while
4.3. VALIDATION RESULTS 59
Figure 4.3: Initial mixture fraction and temperature fields of the two feed DNS
there can be different regions of the system at various stages of ignition, a single flamelet
lumps all these into a single representation. Thus, regions that are starting to react early
on will be a small contribution in the flamelet, and will auto-ignite simultaneously with the
rest of the mixture at a later time. This is something that is observed in all of the cases
studied here. It is clear that as the number of flamelets used to represent the system is
increased, the solution will approach that of the DNS, with the closest approximation using
a flamelet for each DNS cell.
One can examine the ignition by looking at the temporal evolution of the temperature
solution in mixture fraction space, plotted in Fig. 4.5. The temperature profile is shown at
four time instances over the course of ignition and it can be seen that the ignition is initiated
in the rich regions and propagates towards the oxidizer boundary, eventually reaching a
solution with a peak temperature at stoichiometric. This agrees well with previous studies
of ignition in n-heptane-air mixtures (Peters, 2000). The figure shows both the temperature
computed from the flamelet model as well as the conditioned data from the DNS field. It
can be observed that in general the propagation from the rich region is slower in the model
and that the gradient of the front is typically steeper. This can again be attributed to using
a single flamelet. If a number of flamelet realizations were computed, then each would
60 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS2 CHAPTER 1. TEST
0 0.2 0.4 0.6 0.8 1
0
100
200
300
Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
p(bar)
DNSFlamelet
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.4: Comparison of pressure and heat release rate computed using DNS and a one-dimensional flamelet model during ignition and combustion of a single fuel stream withΦ = 0.789 and Zmax = 0.2.
4.3. VALIDATION RESULTS 61
2 CHAPTER 1. TEST
500
1000
1500
2000
2500
0 0.05 0.1 0.15 0.2 0.25
T(K
)
Z
t (ms) Flamelet DNS
0.600.640.680.70
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.5: Evolution of temperature during ignition as a function of mixture fraction.Curves are computed from flamelet model, whereas points are conditioned from the DNSdata.
have a slightly different propagation speed and slope, thus averaging them would result in
a profile similar to that observed in the DNS.
4.3.2 Three Feed System
We now turn to the effect of splitting the introduction of the fuel into two streams separated
by a time delay. To accomplish this, a single fuel stream with 20% of the desired total
amount of fuel was initialized and computed in the same fashion as described in Sec. 4.3.1.
After the desired delay time had elapsed, a second fuel stream was introduced into the
computed flow field and regions where stream 2 was non-zero were reinitialized.
A number of different cases were investigated by looking at the effect of the maximum
mixture fraction of the first stream, as well as varying the time delay before introduction
of the second stream. First, a case where each stream has a maximum mixture fraction of
0.2 and the delay between the injections is 0.4 ms will be investigated in more detail. This
configuration was chosen as it is representative of the overall characteristics that must be
captured in a multi-stream configuration. The initial field of each stream’s mixture fraction
62 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
2 CHAPTER 1. TEST
Z1 Z2 T (K)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.6: Initial mixture fraction of each fuel stream and the resulting temperature dis-tribution in the DNS domain for a three-feed system. The second fuel stream is introducedat a time τ2 = 0.4 ms after initialization of stream 1 (Z1).
and the system temperature is shown in Fig. 4.6. Notice that there is significantly more
Z2 than Z1 in the domain and that there are interface regions between the two, as well
as regions where each fuel mixture mixes only with oxidizer. The temperature shows that
there is some slight reaction and heat release in the first fuel stream, however full ignition
has hot yet occurred.
In Fig. 4.7, the effect of split fuel streams on overall pressure and heat release rate is
shown. It is immediately apparent that the maximum rate of pressure rise is considerably
lower in the multi-stream configuration. This is seen to be the result of the fact that the
overall chemical heat release rate is spread over a longer time and thus necessarily has a
smaller peak value. This is an effect that is often desired in internal combustion engines, as
it can be used to help reduce noise and wear on the engine. Thus, this configuration shows
that splitting the fuel into two separate streams that are introduced at different times has
a considerable effect on the ignition and combustion characteristics of the overall mixture
and that some of the relevant features that we would like to investigate are captured.
It is worth considering the overall heat release rate of this case in more detail. The heat
release can be budgeted according to its origin; that is, whether it is resulting from the
combustion of stream 1, stream 2, or a mixture of both streams. A plot of this budget is
shown in Fig. 4.8. As can be observed in the figure, the auto-ignition of each fuel stream
4.3. VALIDATION RESULTS 632 CHAPTER 1. TEST
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
0
100
200
300
Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
110
p(bar)
split fuel streamsingle fuel stream
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.7: Comparison of pressure and heat release rate of single and split fuel streamsystems. The second fuel stream was introduced at a time τ2 = 0.4 ms after the initializationof the first stream, which contained 20% of the overall fuel mass.
64 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
2 CHAPTER 1. TEST
0
20
40
60
80
100
120
140
160
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
HeatRelease
(GJ/s)
t (ms)
totalstream 1stream 2
interacting
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
Figure 4.8: Budget of heat release rate contribution from each fuel stream and the inter-acting region where both fuel streams are present.
is apparent in the two peaks shown that represent streams 1 and 2, each of which closely
resemble the auto-ignition behavior of a single mixture as seen in Fig. 4.4. However, during
the ignition of the pilot, the heat release from the interaction between the two streams grows
steadily and eventually represents almost all of the heat release in the time between the
peak heat release of stream 1 and the start of ignition of stream 2. Even during the ignition
of the stream 2, the interacting region accounts for approximately 25% of the overall heat
release. As the streams become more mixed over time, it is obvious that the majority of the
heat release will be from the interacting region as the mixture approaches a homogeneous
state. This observation makes clear that in order to account for the ignition and combustion
of the multi-stream system correctly, the interaction of the different fuel streams must be
taken into account.
The ignition process of each individual stream and the interaction that occurs between
them can be investigated in more detail by considering properties in the mixture fraction co-
ordinate. Looking at the evolution of the temperature profile along the Z1 axis in Fig. 4.9,
we see that the ignition of the first stream is qualitatively similar to that in Fig. 4.5. The
time series shown covers that range of heat release budgeted from stream 1. Between
t = 0.70 ms and t = 0.75 ms, the heat release contribution of stream 1 reaches its maximum
4.3. VALIDATION RESULTS 65
2 CHAPTER 1. TEST
500
1000
1500
2000
2500
0 0.05 0.1 0.15 0.2 0.25
T(K
)
Z1
t (ms) Flamelet DNS
0.600.660.700.730.85
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.9: Temporal evolution of the temperature conditioned on mixture fraction duringignition of stream 1. Results are plotted for data along the Z1 axis (Z2 = 0).
and the temperature profile along the Z1 axis becomes essentially stationary.
From Fig. 4.8 it can be seen that the heat release from a mixture of the streams becomes
the largest component near the peak heat release of stream 1. The source of this heat
release is investigated by looking at properties in the two-dimensional mixture fraction
space. Starting from t = 0.70 ms, Fig. 4.10 shows the evolution of the temperature, OH
mass fraction, and heat release conditioned on both mixture fractions. Looking at the left-
most frame (t = 0.70 ms) one can see that the temperature in the rich regions is similar
to that of the Z1 axis as depicted in Fig. 4.9 across all Z2, however no OH has formed
yet. As the temperature and OH evolves in time, the contour plots shown increasing in
time make it apparent that the interaction of the two streams manifests itself as a front
propagation in mixture fraction space. The front propagates primarily along lines of total
mixture (Z1 + Z2) and is strongest along the line of overall stoichiometric mixture. This
is shown most clearly by looking at OH mass fraction, which appears at (Zst, 0) and the
maximum value propagates along Z1 + Z2 = Zst. This front propagation represents an
exchange of heat and mass between the different fuel streams in physical space. Thus, the
presence of the first fuel stream causes regions of the second fuel stream to ignite earlier
66 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS2 CHAPTER 1. TEST
0 0.2Z1
0
0.2
Z2
t (ms)0.70
0 0.2Z1
0.73
0 0.2Z1
0.77
0 0.2Z1
0
200
400
600
800
Heat Release (GJ/s)
0.85
0
0.2
Z2
0
0.5
1
1.5
2
OH×103
0
0.2
Z2
800
1250
1700
2150
2600
T (K)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
Figure 4.10: Evolution of DNS properties conditioned on mixture fraction. Time instancesof temperature, OH mass fraction, and computed heat release rate are shown in each rowfrom top to bottom, with time increasing to the right according to the axis shown at bottom.
4.3. VALIDATION RESULTS 67
2 CHAPTER 1. TEST
500
1000
1500
2000
2500
0 0.05 0.1 0.15 0.2 0.25
T(K
)
Z2
t (ms) Flamelet DNS
0.850.900.910.921.00
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.11: Temporal evolution of the temperature conditioned on mixture fraction duringignition of stream 1. Results are plotted for data along the Z2 axis (Z1 = 0).
than would occur by autoignition if it were a pure mixture.
At the final time shown in Fig. 4.10 (t = 0.85 ms), regions that contain only Z2 are still
unburned. The ignition of the remaining unburned portion of stream 2 can be seen in the
evolution of temperature along the Z2 axis, shown in Fig. 4.11, where the first time instance
shown is the final time from Fig. 4.10. The solution at 0.85 ms shows that the second stream
has not yet ignited and is just starting to react. As the stream ignites, the temperature
profile is similar to the ignition of stream 1 shown in Fig. 4.9 but with a broader region of
high temperature and shallower gradients as the front propagates towards stoichiometric.
This is interesting since although there is a different mechanism of ignition for the mixed
stream region, the Z2 axis still correctly represents the unmixed portion of stream 2, which
ignites in the same manner as it would in a two feed system. Thus, at the interface between
stream 2 and any mixture with stream 1, any mixing that occurs shifts the solution into the
burning region, leaving the axis to represent the unburned mixture. The burning mixture of
streams 1 and 2 only influences the auto-ignition of regions of pure Z2 through the increase
in pressure in the domain, which will shorten the ignition delay time as pressure rises.
68 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
2 CHAPTER 1. TEST
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
0
100
200
300
Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
p(bar)
DNS2D Flamelet
2 x 1D Flamelet
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.12: Comparison with DNS results of pressure and heat release computed by thefull 2D flamelet and the average solution of a 1D flamelet for each stream according toEq. (4.12).
Next we compare how well the model captures the ignition behavior discussed above.
First, Fig. 4.12 compares the global pressure rise and heat release rate computed from the
flamlet models considered to the results obtained from the DNS. Looking first at the solution
obtained by averaging a 1D solution for each stream, the effect of neglecting the interaction
of the streams is apparent. The pressure trace shows two distinct ignition events with little
pressure rise in between, which is a direct result of the missing heat and mass exchange
between the streams. There is virtually no additional heat release between the ignition
events of the individual streams. Contrast this fact with the heat release computed by the
two-dimensional flamelet, where it can be seen that both the magnitude and especially the
ignition delay time are captured well. As was observed in the single fuel case (Fig. 4.4),
the flamelet captures the ignition and propagation of the first fuel stream well. As seen
in Fig. 4.9, the propagation speed of the ignition in the flamelet from rich regions towards
4.3. VALIDATION RESULTS 69
2 CHAPTER 1. TEST
0 0.2Z1
0
0.2
Z2
t (ms)0.70
0 0.2Z1
0.73
Flamelet
0 0.2Z1
0.77
0 0.2Z1
0.85
0 0.2Z1
0
0.2
Z2
0 0.2Z1
DNS
0 0.2Z10 0.2Z1
800
1250
1700
2150
2600
T (K)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.13: Evolution of temperature conditioned on mixture fraction from DNS (top row)
and two-dimensional flamelet model computations (bottom row). Time increases from leftto right according to the axis shown at bottom.
stoichiometric is quite close to that observed in the DNS. However, once again the slope of
the propagation front is steeper in some stages, which can be attributed to the representation
of the entire DNS domain by a single flamelet.
Figure 4.13 shows the evolution of temperature conditioned on mixture fraction from
the DNS as well as that computed from the two-dimensional flamelet model. The flamelet
is shown to capture the front propagation in mixture fraction space, indicating that repre-
sentation of the scalar dissipation correctly accounts for the turbulent mixing interaction.
Note that since the maximum mixture fraction was initialized as 0.2 in each stream, there is
no conditional data from the DNS outside of the region bounded by Z1 +Z2 < 0.2, whereas
the flamelet is computed over the entire range of Z1 + Z2 < 1
70 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
2 CHAPTER 1. TEST
0
20
40
60
80
100
120
140
160
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
HeatRelease
(GJ/s)
t (ms)
0.400.500.600.700.80
τ2 (ms)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.14: Heat release rates for fuel streams with Z1,max = Z2,max = 0.2 and differentdelay times (τ2).
4.3.3 Effect of timing and maximum mixture fraction
To test the model over a wider range of conditions, the delay between the introduction of
the fuel streams and the effect of the maximum mixture fractions was investigated. First,
considering the configuration discussed in the previous section where the maximum mixture
fraction of each stream is 0.2, the effect of delay time is presented in Fig. 4.14. Five different
delay times are shown ranging from τ2 = 0.4 ms to 0.8 ms, which includes introduction of
the stream 2 before, during, and after the ignition of stream 1. With increasing delay time,
the peak heat release rate (and therefore the maximum pressure gradient) can be seen to
continually decrease. This is a result of the longer time available for the streams to mix
and burn together before primary ignition of the main fuel stream occurs, meaning that
there is less unmixed fuel from stream 2 to ignite simultaneously. For cases where stream 1
had already ignited before stream 2 was initialized, there is a high initial heat release due
to regions of stream 2 that are in the immediate vicinity of high temperature fluid from
combustion of stream 1. These regions will ignite very quickly due to the interaction with the
neighboring high temperature regions, however, the solution returns to the characteristics
of the earlier delays very rapidly. Of course, if the delay time is extended to a time well
4.3. VALIDATION RESULTS 71
after the pilot has completed combustion and is well-mixed with the oxidizer stream, the
peak heat release rate will become larger as the system can essentially be treated as a
two-feed system with oxidizer boundary conditions reflecting the combustion products and
temperature of stream 1.
The pressure and heat release computed by the flamelet are compared to the DNS
results in Fig. 4.15 for each of the delay times plotted in Fig. 4.14. Here it is observed that
the overall agreement between the flamelet and DNS computations is consistent over the
different delay times, with slightly more discrepancy as the delay time becomes large.
The effect of maximum mixture fraction was investigated by changing the value for
stream 1 to Z1,max = 0.1. The second stream was then introduced in the same manner
as before over a range of delay times. By changing the maximum mixture fraction of one
of the streams the magnitude of the dissipation rate for each scalar becomes different.
The resulting heat release rates are plotted in Fig. 4.16 and show that the heat release
rate becomes more gradual compared to Fig. 4.14 and that there is more heat release
before primary ignition of stream 2. As a result, the pressure rise from the combustion
of the regions with both streams is higher, causing the primary ignition of stream 2 to be
slightly advanced. The maximum heat release still decreases with increasing delay time
between the fuel streams, however, by a delay of 0.75 ms it is apparent that the mixture
is already trending upwards showing that a minimum peak heat release may occur with a
delay between 0.6-0.7 ms.
72 CHAPTER 4. IGNITION OF MULTI-STREAM SYSTEMS
0.4 0.6 0.8 1 1.2 1.4
0
100
200
300
Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
110
p(bar)
DNSFlamelet
1(a) τ = 0.5 ms
0.4 0.6 0.8 1 1.2 1.4
0
100
200
300
Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
110
p(bar)
1(b) τ = 0.6 ms
0.4 0.6 0.8 1 1.2 1.4
0
100
200
300
Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
1100.4 0.6 0.8 1 1.2 1.4
p(bar)
1(c) τ = 0.7 ms
0.4 0.6 0.8 1 1.2 1.4
0
100
200
300Heatrelease(G
J/s)
t (ms)
40
50
60
70
80
90
100
110
1200.4 0.6 0.8 1 1.2 1.4
p(bar)
1(d) τ = 0.8 ms
Figure 4.15: Comparison of pressure and heat release rates of DNS and flamelet for fuelstreams with Z1,max = Z2,max = 0.2 and different delay times (τ2).
4.3. VALIDATION RESULTS 732 CHAPTER 1. TEST
0
20
40
60
80
100
120
140
160
180
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
HeatRelease
(GJ/s)
t (ms)
0.400.500.600.700.75
τ2 (ms)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 4.16: Heat release rates for fuel streams with Z1,max = 0.1 and Z1,max = 0.2 andvarious delay times (τ2).
Chapter 5
Modeling Joint Scalar Statistics
To provide closure in the RIF model described in Chapter 3, information about the local
distribution of mixture fraction is required. There are traditionally two classes of methods
used to obtain the necessary joint statistics. The first involves solving a modeled evolution
equation for the joint probability density function (Pope, 1985, 1994). This method has the
advantage that the convection and reaction terms do not require any modeling, however it is
difficult to correctly model the molecular mixing, although some recent work has attempted
to address this problem (Meyer & Jenny, 2006).
The second approach assumes a shape for the joint PDF that is parameterized by its
statistical moments and is the method of interest in this work. This method is typically
applied in the context of either flamelet (Peters, 1984) or conditional moment closure (Bilger,
1993) based techniques, where transport equations are solved for the first and second order
moments of the mixture fraction which are then used to compute the assumed shape.
5.1 Model Requirements
In considering a distribution to use as a model for the scalar PDF, it must possess several
mathematical properties to be a valid candidate. Firstly, any model PDF must be positive
and satisfy the normalization condition, that is
P (φ) ≥ 0 (5.1)∫P (φ)dφ = 1. (5.2)
74
5.2. JOINT SCALAR PDF 75
When considering a joint distribution, since scalars that represent mixture fractions are
bounded, there is an additional requirement on the total mixture given by
∑
i
φi ≤ 1. (5.3)
This defines the domain on which the two-scalar joint distribution must be realizable as a
unit isosceles right triangle, i.e., 0 < φ1 < 1 and 0 < φ2 < 1− φ1.
In addition to the mathematical requirements, physical analysis of the system indicates
that a distribution must be able to represent both initially unmixed states, corresponding to
a multimodal delta type distribution, through the final stages of mixing, where it is known
that the joint PDF decays in a Gaussian manner according to the central limit theorem
and eventually recovers a delta function at the system mean. All of the distributions in this
chapter satisfy the properties listed here.
5.2 Joint Scalar PDF
Whereas the beta distribution has been widely used as a presumed PDF for single or
statistically independent scalars, there have been few investigations of assumed distributions
for the two-scalar mixing case. In this section the distributions that have been proposed in
the literature will be defined and their properties discussed. Next, the proposed model, a
five parameter bivariate beta distribution, will be introduced.
5.2.1 Dirichlet Distribution
Due to the success of the beta distribution at representing single scalar mixing in many cases,
Girimaji (1991) proposed that the Dirichlet distribution, which is the simplest extension of
the beta distribution to the multivariate case, would be suitable for multi-scalar mixing. It
has been applied to a number of turbulent combustion applications, including the work of
Baurle & Girimaji (2003) and Hasse & Peters (2005). The Dirichlet distribution is defined
by (Kotz et al., 2000)
P (φi;βi) =1
B(βi)
K∏
i=1
φβi−1i (5.4)
76 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
for all φ1, ..., φK−1 > 0 satisfying φ1 + · · · + φK−1 < 1, where φK is an abbreviation for
1− φ1 − · · · − φK−1. The normalizing constant is the multinomial beta function
B(βi) =
K∏
i=1
Γ(βi)
Γ
(K∑
i=1
βi
) . (5.5)
It can be noted that for K = 2 the Dirichlet distribution simplifies to the standard beta
distribution. In this work the two-scalar case, corresponding to K = 3, is relevant, for
which the distribution can be written as
P (φ1, φ2) =Γ(β1 + β2 + β3)
Γ(β1)Γ(β2)Γ(β3)φβ1−1
1 φβ2−12 (1− φ1 − φ2)β3−1 (5.6)
where the three parameters, βi, are defined by the moments of the distribution according
to
βi = φi
(1− SQ− 1
)(5.7)
in which a tilde represents the scalar mean, the function S is written as
S = φ 21 + φ 2
2 + (1− φ1 − φ2)2 (5.8)
and the mean turbulent scalar energy is defined by
Q = φ′ 21 + φ′ 22 + ˜(φ1 + φ2)′ 2 (5.9)
where φ′2i indicates the variance of scalar i.
This distribution is attractive from an implementation standpoint since analytical ex-
pressions for the three parameters exist in terms of the moments and only one transport
equation for the variance is required in addition to those for each of the means. However,
since there are only three independent parameters, only three moments of the distribution
can be specified, which typically include the mean (first moment) of each variable and a
quantity representing the overall variance. This means that information about the second
5.2. JOINT SCALAR PDF 77
order moments cannot be conserved, i.e. each scalar variance and the corresponding covari-
ance. Instead, by using the scalar energy parameter, Q, all marginal variance information is
lost. When considering a case such as isotropic turbulence, it is clear that since the means
are constant during all mixing stages the joint PDF at any stage cannot be well represented
by the means alone. Indeed, it will be shown later that this distribution is only acceptable
for very limited initial conditions.
5.2.2 Statistically-most-likely Distribution
The statistically-most-likely (SML) distribution introduced by Pope (1980) for turbulent
flows will also be considered. The SML distribution is based on the principle of maximum
entropy (Jaynes, 1957; Good, 1963), where here the entropy will be maximized for a specified
set of moments (Ramsey & Posner, 1965). The resulting distribution can be shown to be
specified as
P (φ1, φ2) = q(φ1, φ2) exp
(λ∑
n=0
Anφn
)(5.10)
where the coefficients An satisfy the first λ moments and q(φ1, φ2) is an a priori distribution,
which is uniform for conserved scalars. If all first and second order moments are considered
the distribution can be written for the two mixture fraction case as
P (φ1, φ2) = exp(A0 +A1φ1 +A2φ2 +A3φ
21 +A4φ
22 +A5φ1φ2
). (5.11)
The required parameters are obtained by finding the roots of the set of six non-linear
equations that define the first and second order moments, given by
∫∫P (φ1, φ2)dφ1dφ2 = 1 (5.12)
∫∫φiP (φ1, φ2)dφ1dφ2 = φi (5.13)
∫∫(φi − φi)(φj − φj)P (φ1, φ2)dφ1dφ2 = φ′iφ
′j . (5.14)
The SML distribution provides an interesting comparison, as the maximum entropy dis-
tribution can also be interpreted as the distribution that uses the minimum amount of
information. That is, the distribution specifies nothing beyond the moments and therefore
does not imply anything about the shape.
78 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
In general, the primary advantage of the SML joint PDF is its capability of conserving
an arbitrary number of moments. However, the ability to retain higher moments beyond
second order is irrelevant, as information about these higher moments is not practically
available in either RANS or LES modeling contexts. The main drawback of the SML in
comparison to simpler distributions such as the Dirichlet is the increase in computational
cost associated with the non-linear root solve to find the parameters of the distribution,
since there are no analytic expressions for the coefficients An.
5.2.3 Bivariate Beta Distribution
Whereas it can be argued that any information in addition to that used by the SML is
spurious (Pope, 1979), it may be desirable to attempt to include additional characteristics
that are known about the distribution. For instance, it has been observed for a single
conserved scalar that in certain cases a beta distribution can represent the actual PDF more
closely than the SML when only second order moment information is considered (Ihme &
Pitsch, 2008). This indicates that the higher order moments implicit in the shape of the beta
distribution may better represent the mixing process. Given this additional information, we
would like to construct a joint distribution with a similar property, resulting in a bivariate
beta distribution defined as
P (φ1, φ2) = C φβ1−11 φβ2−1
2 (1− φ1)β3−1(1− φ2)β4−1(1− φ1 − φ2)β5−1 (5.15)
where C is a constant to satisfy the normalization condition. Like the Dirichlet distribution,
Eq. (5.15) has a shape related to the beta distribution, but in contrast to the Dirichlet
distribution it has five independent parameters, βi, giving sufficient degrees of freedom to
satisfy all first and second order moments. It is realizable on the correct domain and the
marginal distributions are defined by generalized beta distributions (Libby & Novick, 1982).
It also recovers the univariate beta distribution defined by Eq. (3.47) in the limit of either
φ1 → 0 or φ2 → 0. Thus the asymptotes of the proposed bivariate beta distribution are
consistent with the typical one-dimensional formulation used in flamelet modeling.
This distribution is closely related to the Appell hypergeometric function and is actually
a limit of both the bivariate F2-beta and F3-beta distributions introduced by Nadarajah
(2006a,b). More information about the distribution, its properties, and how to compute
the distribution using hypergeometric series can be found in Appendix A.
5.3. DNS OF TWO-SCALAR MIXING 79
The bivariate beta suffers from the same drawback as the SML in that a non-linear
system of equations must be solved to find the parameters βi. However, analytic expressions
exist for the moments as a function of βi in terms of hypergeometric series. While these
expressions can be used to eliminate the necessity of performing an integration of Eqs. (5.12-
5.14) during each iteration of the root finding algorithm, in practice it was found that
performing the integration was nevertheless more convenient. This is due to the fact that
in practice the distribution is always computed and applied using a discrete grid that is
defined for the desired problem. When βi are evaluated with the analytic series the resulting
parameters are associated with an implicitly infinite discretization. As such, the grid on
which the distribution is subsequently computed may not have sufficient resolution for
accurate representation with the parameters that were found, resulting in an error in the
moments when computed with standard integration techniques. This issue is particularly
relevant for distributions with large variances. Such distributions will tend to delta functions
at the boundaries and any discrete grid would require very fine resolution in the boundary
regions to fully resolve the large values that the analytic solution implies. Thus, it was found
that integration according to Eqs. (5.12-5.14) is a more robust method and was therefore
applied throughout this work.
5.3 DNS of Two-Scalar Mixing
In order to evaluate the model for the presumed joint PDF, DNS was used to compute
the mixing of two passive scalars in stationary isotropic turbulence. This section will first
describe the numerical methods employed and the characteristics of the computed flow
fields. The various initial scalar fields studied will then be introduced.
5.3.1 Numerical Implementation
The turbulent flow field was computed by solving the incompressible equations for conser-
vation of mass and momentum, defined by
∂ui∂xi
= 0 (5.16)
∂ui∂t
+ uj∂ui∂xj
=1
ρ
∂p
∂xi+ ν
∂2ui∂xi∂xj
(5.17)
80 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
Table 5.1: Turbulent field input parameters
Grid R93 R163
N 256 512L 2π 2πν 0.025 0.025Af 2.620 6.602
where u is the velocity vector, p is the pressure, ρ is the density, and ν is the kinematic
viscosity. The evolution of each scalar was computed according to
∂φi∂t
+ uj∂φi∂xj
= Di∂2φi∂xi∂xj
(5.18)
where Di is the diffusivity of the ith scalar.
The equations were solved on a three-dimensional uniform grid using a high-order con-
servative finite difference method, the implementation of which can be found in Desjardins
et al. (2008). The physical domain corresponds to a cube with side L = 2π and was
discretized by N3 grid points, resulting in a grid spacing ∆ = L/N . Periodic boundary
conditions were used for both the velocity and scalar fields. The relevant input parameters
are listed in Table 5.1 and a Schmidt number of 0.7 was taken for all simulations.
In this work, the velocity and scalar fields are solved with fourth-order spatial ac-
curacy and second-order temporal accuracy. The maximum Courant number, CFL =
max(|ui|/∆)∆t was held constant at 0.5 in order to keep the time stepping errors small (Juneja
& Pope, 1996). The scalars were solved using the BQUICK scheme of Herrmann et al. (2006)
to ensure that the scalars remained bounded between 0 and 1.
Simulations were carried out with grid resolutions of either 2563 or 5123 points, result-
ing in approximate average Taylor scale Reynolds numbers of Reλ = 93 and Reλ = 163,
respectively. There are three relevant length scales that characterize the flow field: the
integral length scale,
l =π
2u2
∫ κmax
0
E(κ)
κdκ, (5.19)
representing the energy containing scales, where E(κ) is the energy spectrum function at a
5.3. DNS OF TWO-SCALAR MIXING 81
Table 5.2: Turbulent quantities computedfrom simulation
Grid R93a R163b
u′ 10.00 27.16k 150.247 1107.347ε 686.244 12253.466τeddy 0.2197 0.0908η 0.01230 0.00598κmaxη 1.574 1.532Re 1321 4020Reλ 93.4 163.6
a values averaged over 4.0 τeddy.b values averaged over 2.6 τeddy.
scalar wave number κ; the Kolmogorov scale,
η =(ν3/ε
)1/4, (5.20)
for the dissipation scale, where ε is the turbulent dissipation; and the Taylor microscale,
defined as
λ =
(15νu′2
ε
)1/2
(5.21)
which represents the upper bound of the dissipation range, where u′ is the fluctuating
component of the velocity. Characteristic quantities of the flow field were averaged over the
time of stationary turbulence and are tabulated in Table 5.2 for each grid size.
To achieve statistically stationary turbulence, the velocity field was forced using the
linear forcing scheme proposed by Lundgren (2003) and investigated further by Rosales
& Meneveau (2005). This method of forcing in physical space involves adding an addi-
tional term in the momentum equation linearly proportional to the velocity with a constant
frequency Af , defined as
Af =ε
3u′2. (5.22)
The forcing parameter used for each simulation is given in Table 5.1.
The forced Reynolds numbers were limited on each grid to ensure that the smallest scales
of motion, characterized by the Kolmogorov scale η, were resolved. This was achieved by
82 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
maintaining the relative size of the highest resolved wavenumber, κmax, to be κmaxη ≥ 1.5,
which has been shown to be sufficient for accurate higher-order scalar statistics (Juneja &
Pope, 1996). A larger number of simulations was performed on the coarser grid to determine
the behavior of the joint distribution under different initial conditions and identify a range
of interesting cases, which were then run at the higher resolution to achieve turbulence
above the mixing transition described by Dimotakis (2000).
The conditional joint PDF, P (φ1, φ2), is computed from the DNS at specific times by
discretizing the φ1-φ2 space into 100×100 equal intervals and then calculating the histogram
based on the values of both scalars at every grid point. Statistical errors are expected to
be small due to the number of cells, which is approximately 17 or 134 million for the R93
and R163 simulations, respectively.
5.3.2 Initial Scalar Fields
The model distribution should be able to capture the mixing behavior of two scalars over
a wide range of initial conditions. Therefore, distributions from various initial scalar fields
were computed using two different types of initialization: isotropic and structured layers. In
this section, each method of initialization will be defined and then the initial fields computed
will be described.
Isotropic Scalars
The isotropic fields were initialized according to the method of Eswaran & Pope (1988)
and Juneja & Pope (1996). The method defines two independent fields in Fourier space
with the phase of each coefficient chosen randomly and the amplitude determined to satisfy
a specified scalar energy spectrum function fφ(κ). That is, the scalars are assigned in
wavespace according to
φ(κ) =fφ(κ)
4πκ2exp [2πiθ(κ)] (5.23)
where κ is the wavenumber, κ the wavenumber magnitude, and θ(κ) is a random number
uniformly distributed between 0 and 1. The function fφ(κ) is chosen as a top-hat function
centered on wavenumber κs over a width of κ0, giving
fφ(κ) =
{1, if κs − κ0/2 ≤ κ ≤ κs + κ0/2
0, otherwise.(5.24)
5.3. DNS OF TWO-SCALAR MIXING 83
III
III
φ1
φ2
α1
α2
1
1
Figure 5.1: Schematic indicating the sectors used to determine initial state of scalars. Thethree sectors are divided by the two dashed lines, which are determined from the scalarmeans according to Eq. (5.25), and the positive φ1 axis. The circles at (0,0), (0,1), and(1,0) represent the three initial states assigned to values found in each corresponding sector.
The parameter (κs/κ0)i determines the length scale for each scalar and values for each case
are tabulated in Table 5.3.
Next, each scalar field is transformed to physical space and the value at each point is
assigned its initial state based on where it lies in a composition space, φ1-φ2, defined in
Fig. 5.1. The initial states of each scalar are assigned to approximate a triple-delta function
at the corners of an isosceles right triangle; thus if φ1(x), φ2(x) is found to lie in sector
I, it will be assigned to (φ1(x), φ2(x)) = (1, 0). Likewise, if φ(x) is found in sectors II or
III, it is assigned to (0,1) or (0,0), respectively. The division of the composition space is
determined in proportion to the desired scalar means by defining the sector angle as
αi = 2πφi. (5.25)
This gives approximately the specified mean as the values resulting from Eqs. (5.23) and
(5.24) are uniformly distributed in phase.
Finally, the initial field is smoothed in order for it to be well resolved on the grid
and stable for computation. Instead of the usual method of smoothing, which involves
transforming the physical fields back to Fourier space and applying a low-pass filter, here
the fields are smoothed using a simple moving average method over a specified grid stencil.
84 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
2 CHAPTER 1. TEST
φ1 φ2
(a) Isotropic with equal means (I01)
2 CHAPTER 1. TEST
φ1 φ2
(b) Isotropic with unequal means (I03)
2 CHAPTER 1. TEST
φ1 φ2
(c) Layered with equal means (L05)
2 CHAPTER 1. TEST
φ1 φ2
(d) Mixed with unequal means (M10)
Figure 5.2: Planar cross-sections of domain showing typical initial distribution of φ1 andφ2 for different initialization methods (first and second columns, respectively). Scalar fieldinitialization parameters are given in Table 5.3.
Table 5.3: Input parameters for scalar field initialization
Method ID φ1 φ2 (κs/κ0)1 (κs/κ0)2 τφ2/τeddy
isotropic
I01 0.33 0.33 4 4 0I02 0.33 0.33 4 2 0I03 0.33 0.10 4 4 0I04 0.10 0.10 4 4 0
layeredL05 0.33 0.33 - - 0L06 0.33 0.10 - - 0L07 0.10 0.10 - - 0
mixedM10 0.33 0.21 4 4 1M11 0.33 0.21 4 4 2M12 0.33 0.10 4 4 1
5.3. DNS OF TWO-SCALAR MIXING 85
Thus, φi(xξ, yη, zγ) is averaged to
φiξηγ =1
(2m+ 1)3
ξ+m∑
ξ−m
η+m∑
η−m
γ+m∑
γ−mφiξηγ (5.26)
where m is the number of adjacent grid points to consider, here taken to be 1 in each
direction. This method is used to ensure that the scalars stay bounded in the range 0 <
φi < 1, which is not guaranteed when smoothing in wavespace.
Four isotropic fields were calculated on the R163 grid with various scalar means and
initial length scales, as defined by cases I01-I04 in Table 5.3. Planar cross-sections of two
isotropic fields are shown in Figs. 5.2(a) and 5.2(b).
Layered Scalars
The second type of initialization follows the method of Sawford & de Bruyn Kops (2008),
wherein three stream mixing is represented for each scalar as either a mixing layer or top-hat
profile, resulting in a layered configuration of two scalars. Each scalar is initially constant
in y- and z- directions and is specified by a profile in the x-direction. To avoid a sharp
gradient in the scalar field, the transition between layers was specified as an error function
profile. Thus, the profile for each scalar given the scalar mean can be computed as
φ1(x)|y,z =1
2
{1− erf
[1
δ
(x− L
2φ1
)]}(5.27)
φ2(x)|y,z =1
2
{erf
[1
δ
(L
2(φ1 + φ2)− x
)]− erf
[1
δ
(L
2φ1 − x
)]}(5.28)
for 0 < x < L/2 where L is the box length and δ is the length over which the scalar interface
is smoothed, taken here to be 2∆x. The profile is symmetric about L/2.
Using this type of initialization method is of interest to evaluate the sensitivity of the
modeled distributions to the initial scalar field structure for the same set of scalar moments.
Three cases with layered initial fields were computed on the R163 grid with different initial
means of each scalar. The cases are listed as L05-L07 with the parameters of the initial
field given in Table 5.3 and a planar cut of the initially statistically symmetric layered field
(L05) is shown in Fig. 5.2(c).
86 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
Partially Mixed Scalars
The final initialization method involved generating an isotropic field with one of the scalars
taken from an already partially mixed state. This is representative of more practical situa-
tions, where an additional mixing stream may be introduced at a later time to a two stream
system at various mixing stages. The initialization was accomplished in the following man-
ner. First, a single isotropic scalar field was initialized according to the methods described
in Sec. 5.3.2 and was allowed to mix in a computation. Then, at a specified time τφ2 , a
second isotropic initial field with φ2 = 1 was superimposed onto the partially mixed field of
φ1. Wherever φ2 was non-zero, the φ1 field was reset to ensure that the sum of the scalars
remained less than 1. In φ1-φ2 distribution space, the resulting field has an approximate
delta function at the (0,1) point and an existing distribution along the φ1 axis. Different
fields were tested by varying the mixing time before the introduction of φ2 as well as the
relative proportion taken from each scalar.
Three partially mixed cases were computed on the R163 grid. The scalar means and
initial mixing times for the partially mixed initial fields are listed in Table 5.3 under the
mixed method with identifiers M10-M12. An example of the mixed initial field is shown in
Fig. 5.2(d) for a field with unequal means and φ2 introduced after φ1 has mixed for one
eddy turnover time (M10).
5.4 Validation Results
In this section, a qualitative and quantitative comparison between the computed and mod-
eled distributions will be made for several of the cases considered. First, the baseline with
statistically symmetric initial fields for both isotropic and layered configurations will be
evaluated. Then, an asymmetric initial field with one partially mixed scalar will be inves-
tigated. Finally, a case with high positive correlation will be discussed.
5.4.1 Symmetric Initial Field
The first case investigated is an isotropic initial scalar field with equal means, listed as I01
in Table 5.3 and shown in Fig. 5.2(a). This can be thought of as a statistically symmetric
case as all scalars have the same moments. We also compute this case in order to verify
that our results are in agreement with those obtained in previous studies under the same
conditions by Juneja & Pope (1996).
5.4. VALIDATION RESULTS 87
The behavior of the joint PDF as it evolves in time is shown in Fig. 5.3 for the conditioned
distribution obtained from the DNS and the three models considered. Time instances are
shown for different stages of the mixing process based on the values of the scalar root mean
square (r.m.s.), denoted by φ′, normalized by the r.m.s. value of the initial state, φ′0. As
expected, the initial triple-delta distribution decays to a unimodal distribution after first
mixing inward from the corners along the domain boundaries, passing through a uniform
distribution, and eventually approaching a delta function at the combined scalar mean.
From Fig. 5.3, it can be observed that all three models reflect the same behavior and are
qualitatively similar to the computed distribution. Initially, the SML has a more gradual
slope from the corners than the DNS and the other two models. At later times, the SML
also appears to have greater symmetry about an axis parallel to the φ1 + φ2 = 1 boundary.
In order to quantitatively compare the performance of each model, a metric based on
the relative entropy between the computed and modeled distributions will be used. The
Kullback-Leibler (KL) pseudo-distance is a measure of the relative entropy between two
distributions P (φ) and Q(φ), and is defined by
DKL(P ||Q) =
∫∫P (φ) log
(P (φ)
Q(φ)
)dφ (5.29)
with P (φ) taken to be the distribution computed from the DNS and Q(φ) taken as the
the model distribution. From the above equation, it is obvious that identical distributions
will have DKL = 0, and therefore a KL distance of small magnitude indicates low relative
entropy. However, the KL distance is a non-symmetric measure and is also poorly defined
when one of P or Q is zero. Therefore, an averaged entropy measure will be used instead,
known as the Jenson-Shannon (JS) divergence, which is defined by
DJS(P ||Q) =1
2DKL(P ||M) +
1
2DKL(Q||M) (5.30)
where M(φ) = 1/2(P (φ)+Q(φ)) is the average of the two distributions. The JS divergence
is symmetric and will also be zero when the distributions are equal.
The JS divergence of each model from the DNS was computed as the joint PDF evolved
and the results are plotted in terms of normalized r.m.s. in Fig. 5.4. There, the discrep-
ancy between the distributions becomes more evident and it can be seen that the Dirichlet
distribution has the lowest divergence and the bivariate beta distribution also has good
accuracy. The SML is considerably poorer at earlier times and all models are roughly
88 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
equivalent at later times after the distribution transitions to a unimodal shape (at approx-
imately φ′/φ′0 = 0.5). This is due to the shallower gradients of the SML at the boundary
compared to the DNS, whereas the bivariate beta and Dirichlet distributions have gradients
comparable to the DNS. In fact, the early discrepancy of the SML distribution should be
expected, since the initial fields are an ordered state and therefore cannot have distributions
that maximize entropy.
The symmetric field with equal means can also be initialized using the layered method
described in Sec. 5.3.2. Such an initial field was computed as L05 (see Table 5.3) and is
depicted in Fig. 5.2(c). The main effects of having a structured initial field are the evolution
of the variance and correlation between the scalars, as well as a longer overall mixing time.
A plot of the variance decay for the two initialization methods is shown in Fig. 5.5(a) and
the corresponding correlation coefficient, defined as
ρc = φ′1φ′2/
√φ′ 21 φ′ 22 , (5.31)
is shown in Fig. 5.5(b). There, it can be observed that the overall decay rate of the scalar
r.m.s. is slower for the layered initialization and different for each scalar, in contrast to the
isotropic case. The initial correlation of each case is the same, however the layered case
increases slightly during mixing while still remaining negative.
Another difference of the layered initial field is evident from the time evolution of the
joint PDF, shown in Fig. 5.6. It is observed that the distribution initially mixes only along
the φ1 = 0 and φ1 + φ2 = 1 boundaries, since at first each scalar is in contact with only
one of the other scalars. Therefore, there is a delay of mixing between the two separated
scalars associated with the initial thickness of each scalar layer. For this case, the disjoint
scalars begin to mix with each other at approximately φ′/φ′0 = 0.9.
Qualitatively, both the bivariate beta and the SML distributions capture the correct
behavior of the DNS by mixing first along the appropriate boundaries, whereas the Dirich-
let distribution incorrectly mixes along all boundaries in the same manner as the isotropic
case. However, the initial discrepancy of the Dirichlet distribution is not well reflected in
the JS divergence, shown in Fig. 5.7. Here, even though it is obvious that the distribution
is unphysical, the JS divergence indicates that it compares well. This can be explained by
realizing that the marginal distributions for this case are still very close to beta distribu-
tions. Both the bivariate beta and Dirichlet distributions have beta marginal distributions
5.4. VALIDATION RESULTS 89
2 CHAPTER 1. TEST
P (φ1, φ2)
0 1φ10
1
φ2
10−1 100 101 102
0 1φ1 0 1φ1
Dirichlet
0 1φ1
0
1
φ2
SML
0
1
φ2
BVB5
0.95
(0.10)
0
1
φ2
0.80
(0.23)
0.30
(0.71)
DNS
0.10
(1.28)
φ′/φ′0
(t/τeddy)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.3: Time evolution of joint probability distribution, P (φ1, φ2), for statisticallysymmetric isotropic scalar (I01). The top row is the computed disribution (DNS), withsubsequent rows representing the model bivariate beta distribution (BVB5), statistically-most-likely distribution (SML), and the Dirichlet distribution, respectively. Each columnrepresents a fixed time increasing from left to right with instances taken at φ′/φ′0 of 0.95,0.80, 0.30, and 0.10, respectively.
90 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
2 CHAPTER 1. TEST
0
0.1
0.2
0.3
0.4
0.5
0.10.20.30.40.50.60.70.80.91
DJS
φ′/φ′0
1.51.00.750.50.250t/τeddy
BVB5SML
Dirichlet
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.4: Jenson-Shannon divergence of the bivariate beta (BVB5), statistically-most-likely (SML), and Dirichlet distributions for symmetric initial distribution from isotropicscalars with equal means (case I01).
2 CHAPTER 1. TEST
10−3
10−2
10−1
100
0 0.5 1 1.5 2 2.5
φ′
t/τeddy
isotropic: φ1φ2
layered: φ1φ2
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(a) scalar r.m.s.
2 CHAPTER 1. TEST
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 0.5 1 1.5 2 2.5
ρc
t/τeddy
isotropiclayered
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(b) correlation coefficient (Eq. 5.31)
Figure 5.5: Time variation of scalar r.m.s. and correlation coefficient of fields with equalmeans using different initialization methods.
5.4. VALIDATION RESULTS 91
for this case, and therefore capture the marginal distributions well. In order for the Dirich-
let distribution to have the correct marginal while also being non-zero along the φ2 = 0
boundary, the Dirichlet distribution must necessarily be less than the actual distribution
along the other two boundaries. Thus, where P > 0 and Q > 0, Q < M < P meaning that
the first term in the JS divergence is negative, whereas when P = 0 and Q > 0, the second
term is positive. Therefore, in this case, the net contribution to the Dirichlet JS divergence
is not significantly more than the other models, even though it is clearly incorrect. At late
times, both visual inspection of Fig. 5.6 and the JS divergence in Fig. 5.7 show the inade-
quacy of the Dirichlet distribution for such a non-isotropic case. At that time the bivariate
beta distribution seems to capture the general shape of the PDF best and represents, for
instance, the absence of φ1 = 0 correctly, in contrast to the SML distribution.
5.4.2 Asymmetric Initial Field
Since multiple mixing streams in practical applications rarely have perfectly symmetric
initial conditions, here statistical asymmetry is introduced into the initial distributions.
This is accomplished by varying the initial means, variances, and correlation. In this section,
a fully asymmetric case will be considered that was initialized with unequal scalar means
and with one scalar in a partially mixed state. The partially mixed scalar was computed
for one eddy turnover time before the second scalar was introduced. The initial parameters
are listed as M10 in Table 5.3 and the initial field is shown in Fig. 5.2(d).
The evolution of the joint PDF, given in Fig. 5.8, indicates clearly that the bivariate
beta distribution captures the features of the computed distribution closely, whereas the
SML shows the same behavior, but is substantially different during early stages of mixing.
Also, it is clear that the Dirichlet distribution fails completely to represent the distribution
due to the missing marginal information.
These observations are noticeable in the comparison of the JS divergence computed
for each model, plotted in Fig. 5.9, where the bivariate beta distribution shows the least
divergence over the entire mixing time. The SML distribution is once again seen to be
poorer in the initial mixing stages.
Strong Positive Correlation
In the cases seen so far, the bivariate beta has been shown to be the most accurate model.
This is also true for the majority of the other cases investigated, which is evident from
92 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
2 CHAPTER 1. TEST
0 1φ10
1
φ2
10−1 100 101 102
P (φ1, φ2)
0 1φ1 0 1φ1 0 1φ1
Dirichlet
0
1
φ2
SML
0
1
φ2
BVB5
0
1
φ2
0.90
(0.39)
0.70
(0.75)
0.50
(1.15)
DNS
0.30
(1.50)
φ′/φ′0
(t/τeddy)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.6: Time evolution of joint probability distribution, P (φ1, φ2), for statistically sym-metric isotropic scalar with layered initial field (L05). The top row is the computed dis-ribution (DNS), with subsequent rows representing the model bivariate beta distribution(BVB5), statistically-most-likely distribution (SML), and the Dirichlet distribution, respec-tively. Each column represents a fixed time increasing from left to right with instancestaken at φ′/φ′0 of 0.90, 0.70, 0.50, and 0.30, respectively.
5.4. VALIDATION RESULTS 93
2 CHAPTER 1. TEST
0
0.1
0.2
0.3
0.4
0.5
0.10.20.30.40.50.60.70.80.91
DJS
φ′/φ′0
2.52.01.51.00.50t/τeddy
BVB5SML
Dirichlet
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.7: Jenson-Shannon divergence comparison of the bivariate beta (BVB5),statistically-most-likely (SML), and Dirichlet distributions for symmetric initial distribu-tion with layered scalars of equal means (case L05).
Table 5.4, where the average Jenson-Shannon divergence over the entire mixing process is
tabulated. However, notice that of the ten cases studied, one was poorly represented by the
bivariate beta distribution. The specific case used a layered initialization with relatively
small and equal means of scalars 1 and 2 (L07), which results in a strong positive correlation
between the two scalars.
The correlation over time is shown in Fig. 5.10, where it is also compared with the
correlation for the other layered cases. It can be seen that the scalar field is slightly positively
correlated in early mixing stages and at late times it approaches near perfect positive
correlation. By examining the JS divergence over time in Fig. 5.11, it is observed that the
bivariate beta distribution starts to diverge considerably from the DNS at approximately
φ′/φ′0 = 0.5, where the correlation coefficient is approximately ρc = 0.75.
Contour plots of the distributions are shown at φ′/φ′0 = 0.3 in Fig. 5.12. The strong
positive correlation can be seen in that the distribution is oriented along the bisector of
the φ1 + φ2 = 1 boundary. Here, only the SML distribution retains the character of the
computed distribution, whereas the bivariate beta forms a bimodal type function. The
second peak is not visible in the contour plot since it lies along the φ1 + φ2 = 1 boundary
94 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
2 CHAPTER 1. TEST
P (φ1, φ2)
0 1φ10
1
φ2
10−1 100 101 102
0 1φ1 0 1φ1
Dirichlet
0 1φ1
0
1
φ2
SML
0
1
φ2
BVB5
0.95
(0.08)
0
1
φ2
0.80
(0.22)
0.30
(0.75)
DNS
0.15
(1.13)
φ′/φ′0
(t/τeddy)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.8: Time evolution of joint probability distribution, P (φ1, φ2), for statistically asym-metric scalar field initialized with a partially mixed scalar (M10). The top row is thecomputed disribution (DNS), with subsequent rows representing the model bivariate betadistribution (BVB5), statistically-most-likely distribution (SML), and the Dirichlet distri-bution, respectively. Each column represents a fixed time increasing from left to right withinstances taken at φ′/φ′0 of 0.95, 0.80, 0.30, and 0.15, respectively.
5.4. VALIDATION RESULTS 95
2 CHAPTER 1. TEST
0
0.1
0.2
0.3
0.4
0.5
0.10.20.30.40.50.60.70.80.91
DJS
φ′/φ′0
1.51.00.750.50.250t/τeddy
BVB5SML
Dirichlet
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.9: Jenson-Shannon divergence comparison of the bivariate beta (BVB5),statistically-most-likely (SML), and Dirichlet distributions for an asymmetric initial fieldfrom a partially mixed scalar (case M10).
Table 5.4: Average Jenson-Shannon divergence, DJS , during mixing
DJSa
Method ID BVB5 SML Dirichlet
isotropic
I01 0.020 0.071 0.014I02 0.025 0.081 0.072I03 0.020 0.073 0.030I04 0.037 0.088 0.062
layeredL05 0.044 0.074 0.116L06 0.071 0.092 0.238L07 0.260 0.135 0.293
mixedM10 0.017 0.052 0.244M11 0.044 0.076 0.421M12 0.027 0.055 0.311
a Averaged over 0.05 < φ′/φ′0 < 1.0.
96 CHAPTER 5. MODELING JOINT SCALAR STATISTICS
2 CHAPTER 1. TEST
-1
-0.5
0
0.5
1
0 0.5 1 1.5 2 2.5
ρc
t/τeddy
L05L06L07
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.10: Variation of correlation coefficient over time for layered initial scalar fields (seeTable 5.3).
2 CHAPTER 1. TEST
0
0.1
0.2
0.3
0.4
0.5
0.10.20.30.40.50.60.70.80.91
DJS
φ′/φ′0
2.01.51.00.50t/τeddy
BVB5SML
Dirichlet
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 5.11: Jenson-Shannon divergence comparison of the bivariate beta (BVB5),statistically-most-likely (SML), and Dirichlet distributions for a layered initial field withstrong positive correlation (L07).
5.4. VALIDATION RESULTS 97
2 CHAPTER 1. TEST
0 1φ10
1
φ2
DNS
0 1φ1
BVB5
0 1φ1
SML
0 1φ110−1
100
101
102Dirichlet
P (φ1, φ2)
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
Figure 5.12: Joint probability distribution, P (φ1, φ2), for layered initialization with positivecorrelation (L07) at one time instance with φ′/φ′0 = 0.3 (t/τeddy = 1.2). The computeddistribution is on the far left and the three models are to the right.
and is therefore best seen by examining the marginal distributions, shown in Fig. 5.13.
The bivariate beta assumes a bimodal shape and will approach a double-delta function
along the hypotenuse bisector. This appears as the distribution must satisfy the restraints
set by the moments while also imposing the character of the beta distribution. It should
be noted that even though the bivariate beta and SML distributions differ in shape, they
both have identical moments. It can also be noted here that the Dirichlet cannot represent
any case with positive correlation, as it is negatively correlated by construction.
In light of the behavior of the bivariate beta for distributions with strong positive cor-
relation, one might consider using a hybrid model where the SML is used above a certain
threshold for ρc. There is no clear relation for this threshold, however numerical exper-
imentation indicates that ρc < 0.5 − 0.6, generally does not exhibit the distinct bimodal
behavior.
In summary, this chapter has presented a new bivariate beta distribution as a model for
the joint probability distribution of two scalars. The distribution was validated using DNS
of two scalar mixing and was found to match the computed distribution well and provide
a closer representation than either the SML or Dirichlet distributions. To investigate and
validate the distribution further, one could apply the model to different flow conditions, such
as one with a mean shear flow, and investigate whether the influence on the scalar variance
results in a distribution that is not represented well by the bivariate beta distribution.
98 CHAPTER 5. MODELING JOINT SCALAR STATISTICS2 CHAPTER 1. TEST
0
500
1000
1500
2000
0 0.2 0.4 0.6 0.8 1
P(φ
1|φ
2)
φ1
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(a) P (φ1|φ2)
2 CHAPTER 1. TEST
0
500
1000
1500
2000
0 0.2 0.4 0.6 0.8 1
P(φ
2|φ
1)
φ2
BVB5SML
DirichletDNS
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.
(b) P (φ2|φ1)
Figure 5.13: Marginal distributions of case L07 shown in Fig. 5.12 at φ′/φ′0 = 0.3 (t/τeddy =1.2).
Chapter 6
Application to Split-Injection
Diesel Engine
In this chapter the model framework developed in the previous chapters will be applied to
the computation of ignition and combustion in a split-injection diesel engine. First, the
experimental setup and methods of the research engine will be described and the operating
conditions considered are then described and motivated. Then, the numerical setup is
detailed and the simulation results are compared with experimental data and discussed.
6.1 Research Engine Facility
6.1.1 Experimental Setup
The experimental measurements were obtained using a single cylinder research engine devel-
oped by BMW and located at the research facility of Robert Bosch GmbH in Schwieberdin-
gen, Germany. The engine is pictured in Fig. 6.1(a) and the test facility with the engine
installed is shown in Fig. 6.1(b). The engine was refitted with modified ω-type research
bowl piston with a maximum diameter of 51.5 mm, a maximum depth of 15.0 mm, and
a volume of approximately 24.5 cm3. Further details of the engine geometry are listed in
Table 6.1.
The intake air was turbocharged by an air supply system capable of a maximum pressure
of 4 bar and temperature conditioning up to 390 K, allowing the investigation of various
boosting scenarios. The intake flow induced swirl in the chamber and the resulting swirl
99
100 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
(a) Single cylinder research engine (b) Engine installed in test facility
Figure 6.1: Images of the single cylinder direct-injection research engine and test facility atRobert Bosch, Schwieberdingen, Germany.
Table 6.1: Geometry of single cylinder diesel engine
Bore 84.7 mmStroke 90.0 mmConnecting Rod Length 136.0 mmCam Radius 45.0 mmDisplacement 0.5 LCompression Ratio 16:1
Piston Bowl ω-type
6.1. RESEARCH ENGINE FACILITY 101
numbers were determined from a flow bench experiment at a constant intake pressure over
a range of valve lift heights by using a paddle wheel to measure the rotational velocity of
the cylinder charge. The swirl numbers for the operating conditions considered are listed in
Table 6.3. The engine was water cooled and the oil temperature was kept constant at 85 ◦C
by a heat exchanger with the cooling water. Indicated pressure measurements were taken
using a water-cooled Kistler pressure transducer. The pressure traces used in this work are
an ensemble average over 25 combustion cycles.
Injection of the fuel was accomplished using a high-pressure Bosch common-rail injection
system with a maximum injection pressure of 2000 bar achieved through the use of a radial
piston pump and the injector is capable of a hydraulic flow rate of 965 cm3/min at 100 bar.
The common-rail injection system allows for multiple injections per cycle. The spray nozzle
has seven uniformly spaced radial holes with diameter of 0.140 mm and a spray cone angle
of 160◦. The details of the injector are summarized in Table 6.2.
Table 6.2: Diesel injector characteristics
Injector Bosch CRI3.0/CRI3.2Injection system Common railNozzle diameter 0.140 mmNumber of holes 7Spray cone angle 160◦
The injector was characterized by Bosch Corporate Research using an experimental test
bench to determine methods for retrieving the experimental mass flow rate for the injection
profile. The volumetric injection rate, rail pressure, needle lift, and solenoid current are
measured to provide a characterization of the injector. The mass flow rate profile can then
be found for different operating conditions using a one-dimensional hydraulic model in the
commercial code AMESim. The validation and computation of the AMESim simulations
were also carried out at Bosch Corporate Research. A typical computed volumetric flow,
Vf , for a split-injection profile is shown in Fig. 6.2. The flow rate is referenced to start of
energization (SOE) of the injector (t = 0). A time delay is seen to occur between SOE
and the actual start of injection (SOI). After the end of the pilot injection (EOI), there is a
dwell time before SOI of the main injection occurs. Note that there are small fluctuations in
the volumetric flow rate about zero which are neglected as they are unphysical. In order to
102 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
2 CHAPTER 1. TEST
-5
0
5
10
15
20
25
30
35
40
45
0 0.5 1 1.5 2 2.5
SOI1 SOI2
Vf(m
m3/ms)
t (ms)
pilot main
τdwell
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 6.2: Volumetric fuel flow rate of typical split-injection computed from AMESim 1Dhydraulic model. Dwell time marked as time between pilot EOI and main SOI.
construct a usable injection profile for simulations, the volumetric flow rate is converted to a
mass flow rate using an average fuel density computed from the integrated volume over the
injection and the total mass injected. Furthermore, the SOE must be referenced to crank
angle from the measured solenoid current. Injection profiles for the timings investigated are
given in the next section.
6.1.2 Operating Conditions
A number of different operating cases were investigated in order to evaluate model perfor-
mance over a range of conditions. The majority of cases were run at an engine speed of 2000
RPM, excepting one low load case at a speed of 1500 RPM. Three different engine loads
were considered and the indicated mean effective pressure (IMEP) of each case is listed in
Table 6.3. The table also lists the mass of fuel injected per cycle and the resulting overall
equivalence ratio. Several cases also used exhaust gas recirculation (EGR) and although
most cases were at moderate swirl numbers of 2.3, case OP10 had double the swirl.
Each case has an injection timing and rail pressure determined by the operating point.
All cases except 513MR8 used a classic split injection scenario, where a small amount of
the fuel is introduced before top-dead-center (TDC), with the majority of the fuel injected
6.1. RESEARCH ENGINE FACILITY 103
Table 6.3: Diesel engine operating conditions.
IDRPM IMEP Swirl no. EGR mf Φ
(1/min) (bar) (%) (mg)
512MR3 2000 8.0 2.3 26.4 22.9 0.59513MR8 2000 8.0 2.3 28.3 22.9 0.61OP10 1500 4.3 2.8 0.0 11.9 0.30OP12 a 2000 8.0 4.5 0.0 21.4 0.45
b 2000 8.0 4.5 32.8 21.9 0.68OP13 a 2000 14.8 2.8 0.0 40.1 0.56
b 2000 14.8 2.8 24.1 42.3 0.79
shortly after TDC. The case 513MR8 has a different injection strategy, whereby the pilot has
a larger percentage of the overall fuel and is injected later at a time after TDC, resulting in
a much shorter dwell time between the pilot and main injection. The rail pressure, injection
timing, and mass ratio of each case investigated are tabulated in Table 6.4.
Table 6.4: Summary of experimental injection parameters.
IDprail mpilot SOIpilot SOImain τdwell
(bar) (%) (CAD aTDC) (CAD aTDC) (CAD)
512MR3 1500 5.0 -9.6 4.4 12.1513MR8 1500 18.9 3.6 7.8 1.0OP10 700 8.6 -8.2 3.3 10.0OP12 900 4.9 -10.6 2.4 11.2OP13 1000 3.0 -12.7 3.1 14.0
The mass rate injection profiles for each case are shown in Figures 6.3 and 6.4 referenced
to the correct crank angle based on the engine speed and experimental SOE. A direct
comparison of the different injection strategies is shown by plotting the injection profiles of
512MR3 and 513MR8 together in Fig. 6.3. These cases are at similar operating conditions
except for the injection timing. The injection profiles for the remaining cases are shown in
Fig. 6.4. As the engine load increases, the timing of the pilot is moved more forward. The
amount of fuel injected in the pilot remains roughly constant, meaning that the percentage
104 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
2 CHAPTER 1. TEST
0
10
20
30
40
50
-10 -5 0 5 10 15 20
mfuel(m
g/m
s)
CAD aTDC
512MR3513MR8
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 6.3: Experimental mass flow rates for cases 512MR3 and 513MR8 showing differentinjection strategies. Injection timings are tabulated in Table 6.4 and injection fuel mass isgiven in Table 6.3.
of total fuel injected decreases with increasing engine load due to the overall increase in fuel
per cycle.
6.2 Numerical Setup
The commercial code Fluent was used to solve the unsteady RANS equations for mass
and momentum. The turbulence closure employed a two equation k − ε model with the
realizable formulation of Shih et al. (1995), which is expected to give improved results over
the standard model for flow with considerable rotation. Modifications to the standard code
were as follows. The standard energy equation was turned off and a total enthalpy equation
was implemented to enable coupling with the flamelet chemistry. The total enthalpy was
initialized according to Eq. (2.6) using the initial temperature and species mass fractions
defined in Sec. 6.2.1. This enthalpy equation included source terms accounting for heat loss
to the walls and spray evaporation, as well as the pressure rate term due to the compression
of the closed volume. The boundary heat flux was computed assuming constant temperature
walls of 440 K. Furthermore, major species were considered to compute the density of
the gas mixture, however mean mass fractions of each species were updated according to
6.2. NUMERICAL SETUP 105
2 CHAPTER 1. TEST
0
10
20
30
40
50
-15 -10 -5 0 5 10 15 20
mfuel(m
g/m
s)
CAD aTDC
OP10OP12OP13
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 6.4: Experimental mass flow rates for cases OP10, OP12, and OP13 showing differentinjection profiles for different loading. Injection timings are tabulated in Table 6.4 andinjection fuel mass is given in Table 6.3.
Eq. (3.46) and thus no transport equations were required. Additional transport equations
for two mixture fractions and three variances were solved according to Eqs. (3.21) and
(3.22), respectively. An interface was developed to couple the combustion code, described
in Sec. 4.2.1, with Fluent and provide the updated species mass fractions and temperature
from reaction according to the method described by Fig. 3.1.
The system was solved using a pressure based solver where mass conservation of the ve-
locity field is enforced using a pressure correction. Second-order upwind spatial discretiza-
tion was used for all momentum and scalar quantities and the pressure-velocity coupling
employed a Pressure-Implicit with Splitting of Operators (PISO) method (Issa, 1986). A
reduction in order may be present in cells adjacent to moving boundaries defined by the
piston motion. A first-order implicit time integration scheme was employed and a time-step
size convergence study was carried out which showed that time-steps smaller than 0.25 CAD
(2×10−5 s at 2000 RPM) provided sufficient resolution. A time step of 0.25 CAD was used
during the compression stroke until just prior to SOIpilot, after which a time-step of 0.1
CAD was used for the combustion phase.
106 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
To perform the CFD simulations a hexahedral mesh of the closed volume chamber
geometry was created. The full three-dimensional geometry was used rather than a radial
section due to the asymmetries caused by the spacing of the seven nozzle holes relative
to the intake and exhaust valves. The average mesh spacing was approximately 1 mm,
resulting in approximately 440,000 computational cells at intake valve close (IVC) and
50,000 computational cells at TDC. The grid is shown one crank angle prior to TDC in
Fig. 6.5, where a section cut is also shown to illustrate the interior mesh configuration. The
Figure 6.5: Computational mesh of split-injection diesel engine. Top image shows full 3Dgeometry and bottom image is a section cut at 1 CAD bTDC
amount of grid refinement was constrained by the Lagrangian spray model, described in
Sec. 6.2.2, which relies on the assumption that the liquid phase is disperse. If further grid
6.2. NUMERICAL SETUP 107
refinement is employed, the liquid volume fraction in each cell approaches the gas volume
fraction and therefore violates this assumption. Thus the maximum grid refinement in
the core region was limited by the spray model. Nonetheless, gas exchange simulations of
similar operating conditions have shown that prior to fuel injection the mean flow does not
contain any under-resolved features. This is partly because this engine has relatively low
swirl and tumble velocities and small scale structures in the mean flow caused by the gas
exchange largely disappear during the compression stroke.
In this study, the turbulent fields were initialized at intake valve close (IVC), which
can be seen to occur at -134 CAD aTDC from the valve timing diagram in Fig. 6.6. The
EVO
EVC
48◦
28◦IVO
IVC
18◦
46◦
TDC
Intake: 244◦Exhaust: 256◦
Figure 6.6: Valve timing diagram for M47 experimental engine showing Exhaust ValveOpen/Close (EVO/EVC) and Intake Valve Open/Close (IVO/IVC) with respect to TopDead Center (TDC).
velocity was initialized as a swirling flow by assuming that the velocity field is defined by
solid body rotation about the cylinder axis with a magnitude determined according to the
swirl number listed in Table 6.3. Although in reality the flow field is more complex and
the observed structures are not two-dimensional, it has been found that as the flow in the
cylinder is compressed, the approximation of the swirling flow by solid body rotation is
reasonable (Heywood, 1988; Hill & Zhang, 1994). Initializing the flow at IVC also allows
for the development of a boundary layer in the velocity profile, and the asymmetries of the
valve cutouts also change the flow structure. In Fig. 6.7, the flow field before injection is
108 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
shown for a low swirl and high swirl case. The results for tangential velocity profiles are
qualitatively similar to those found experimentally in a typical engine (Crnojevic et al.,
1999; Nishida et al., 1984). The initial turbulence intensity and dissipation of the field
(a) OP10, swirl no. 2.8 (b) OP12, swirl no. 4.5
Figure 6.7: Velocity fields initialized using swirl number after having advanced to just priorto injection. Cut plane of vector field is halfway between the cylinder head and piston.
was obtained from one-dimensional models using GT-Power that were conducted by Bosch
Corporate Research.
From IVC, the flow field was computed during the compression stroke without chemistry
until a point just before the pilot injection event (SOIpilot). At that time in the simulation,
a one-dimensonal flamelet field is initialized based on the average charge temperature for
the oxidizer stream and the fuel according to the following section.
6.2. NUMERICAL SETUP 109
6.2.1 Boundary Conditions
Oxidizer Stream Boundary Condition
The boundary conditions of the oxidizer in each case were specified according to the amount
of EGR quoted in Table 6.3. The burned and unburned mixture compositions were com-
puted by assuming that the fuel in the previous cycle is fully burned and the composition is
frozen during compression (Heywood, 1988). First, the burned gas composition is calculated
based on the amount of fuel injected and air inducted during each cycle, thus giving the
equivalence ratio, Φ, which is tabulated in Table 6.3. Then, the burned gas composition is
mixed with the charge composition according to the EGR(%), giving the resultant composi-
tion of the charge for the compression stroke. The mass fractions of the major components
of the oxidizer stream are listed in Table 6.5 for the cases considered, where nitrogen is the
remaining mass not listed.
Table 6.5: Summary of charge composition used for each engine operating point. Theremaining mass is nitrogen.
ID YO2 YH2O YCO2
512MR3 0.1965 0.0139 0.0295513MR8 0.1925 0.0153 0.0327OP10 0.2286 0.0014 0.0031OP12 a 0.2268 0.0021 0.0045
b 0.1726 0.0230 0.0491OP13 a 0.2254 0.0027 0.0057
b 0.1860 0.0178 0.0381
Note that the cases listed with no EGR still have small amounts of combustion products.
Even though there is no external EGR, there can be some residual gas that is trapped in
the cylinder during each cycle. Using the methods of Fox et al. (1993) and Yun & Mirsky
(1974), the residual gas fraction was estimated based on the geometry and cylinder pressure.
It was found to range between 5-6% in the configurations investigated, thus this amount of
burned gas was used to compute the composition of the cylinder charge.
The enthalpy of the oxidizer boundary is computed by evaluating Eq. (2.6) using the
above species mass fractions and the average cylinder temperature at the time of initial-
ization. The initial temperature distribution in mixture fraction space is found by linear
110 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
interpolating the enthalpy and species mass fractions between the values at the oxidizer and
the fuel boundaries and then using Eq. (2.6) to iterate for the local temperature, resulting
in a non-linear distribution in mixture fraction.
Flamelet Fuel Boundary for Liquid Injection
Since there is a source term for the enthalpy equation due to the evaporation of the spray,
this must be taken into account in the flamelet equations. However, rather than computing
this source and providing it as an input to the flamelet, here another approach will be
used to account for the additional heat loss which involves modifying the fuel boundary
condition based on the phase change of the fuel. This is accomplished by carrying out an
energy balance between the pure fuel liquid and vapor phases. The energy balance can be
written in terms of enthalpy according to
hl(T lref) +
∫ T l
T lref
clpdT = hv(T vref) +
∫ T v
T vref
cvpdT (6.1)
where the superscripts l and v denote liquid and vapor properties, respectively. The enthalpy
of vaporization for a specific temperature T ∗ can be expressed as
∆hvap(T ∗) = hv(T ∗)− hl(T ∗). (6.2)
Taking a common reference temperature, Tref = T lref = T vref , and adding and subtracting
enthalpies at T ∗, the energy balance can be rewritten in the form
hl(Tref)− hl(T ∗) + hl(T ∗) +
∫ T l
Tref
clpdT = hv(Tref)− hv(T ∗) + hv(T ∗) +
∫ T v
Tref
cvpdT. (6.3)
Substituting Eq. (6.2) and converting the remaining enthalpy differences to integrals of
specific heat gives
∫ Tref
T ∗clpdT +
∫ T l
Tref
clpdT = ∆hvap(T ∗)−∫ T ∗
Tref
cvpdT +
∫ T v
Tref
cvpdT
∫ T l
T ∗clpdT = ∆hvap(T ∗) +
∫ T v
T ∗cvpdT (6.4)
6.2. NUMERICAL SETUP 111
Rewriting the above, we can approximate
clp(Tl − T ∗) = ∆hvap(T ∗) + hv(T v)− hv(T ∗). (6.5)
Taking that the temperature of the liquid and its properties are known and that an enthalpy
of vaporization is available for a temperature T ∗, the only property unknown in Eq. (6.5)
is the enthalpy of the vapor at T v. Thus rearranging the equation gives
hv(T v) = clp(Tl − T ∗)−∆hvap(T ∗) + hv(T ∗) (6.6)
After obtaining hv(T v), the fuel boundary temperature can be determined by iterating
Eq. (2.6) for total enthalpy.
The preceding method can be thought of as taking the liquid fuel and adjusting it to
a temperature at which phase change properties are known, then applying the enthalpy
of vaporization and readjusting the resulting vapor phase temperature back to the same
energy level of the initial liquid phase. Although this process conserves energy, it obviously
violates the second law of thermodynamics and results in low fuel boundary temperatures.
However, as long as this method is consistently applied it is taken here to provide a better
approximation to the fuel boundary than just the liquid temperature.
Another possible approach would be to consider evaporation of a single droplet, as
described in Sec. 4.1.3, and use the surface temperature of the droplet obtained from a
phase equilibrium assumption. The difficulty with this approach is that the fuel mass
fraction at a droplet surface will not be unity, and thus the temperature obtained is not
at the boundary of mixture fraction space, but rather somewhere within the domain. As
a result, extrapolation is required in order to determine the fuel boundary temperature.
Since the droplet surface conditions are dependent on the ambient environment, which will
be changing in time, it is difficult to formulate a consistent method to ensure a correct
boundary condition in this manner.
6.2.2 Spray Modeling
The injection of the fuel as a liquid must be accounted for to develop the mixture fraction
field. In this work, the liquid fuel spray is modeled as a discrete phase represented by
particles that are tracked in a Lagrangian manner, know as the Discrete Droplet Method
(DDM). The injection mass flow rate was determined from the experimental measurements
112 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
combined with a one-dimensional hydraulic model, AMESim, as described in Sec. 6.1.2.
The spray here was modeled as a solid cone injection with a half angle of 7◦.
An ensemble of particles representing fuel droplets are introduced from the nozzle loca-
tion with a range of diameters and a total mass consistent with the prescribed mass flow
rate. The particle sizes are determined from a Rosin-Rammler distribution with a maximum
diameter corresponding to the nozzle diameter (0.140mm) and the mean diameter half the
nozzle diameter. The particle velocities, up, are computed from the variable mass flow rate,
diesel liquid properties, and injector geometry as
up =minj
CdρlAnoz(6.7)
where Anoz is the nozzle area and an average liquid density, ρl, and a discharge coefficient
of Cd = 0.8 were used. The particle temperature is that of the liquid fuel, which is 300 K
for this case.
Once introduced, the particle trajectories are solved according to standard equations
of motion with full coupling between the phases through exchange of mass, momentum,
and energy. The particle drag was computed dynamically to account for droplet shape
distortion (Liu et al., 1993). During the evaporation process, the fuel droplets are heated
by the surrounding gas and the evaporation rate is determined from the saturated vapor
pressure of the fuel at the droplet surface until the boiling temperature is reached, at
which point a boiling rate law is applied (Kuo, 1986). The saturation curve was taken as
a piecewise function of the available data, which is unfortunately sparse. Evaporation of
fuel from each injection is used to represent the source term equations for mixture fraction
defined by Eqs. (3.21) and (3.21).
Representing the breakup of the liquid jet is an important aspect of spray modeling.
Two modes of breakup occur; primary breakup in the near nozzle region, and secondary
breakup further downstream. The physics of primary breakup are difficult to represent using
a disperse droplet method, but since ignition in the near nozzle region does not occur until
long after primary breakup is complete, here only secondary breakup will be considered.
This is accomplished through a hybrid of the models based on either Kelvin-Helmholtz
(KH) (Reitz, 1987) or Rayleigh-Taylor (RT) (Su et al., 1996) instabilities, known as the
KH-RT model. This model accounts for both stripping of droplets through high shear
between between the liquid and gas phase, as well as breakup at lower Weber numbers.
6.2. NUMERICAL SETUP 113
The KH model is used until a break-up length specified is reached (Levich, 2000), after
which the Taylor Analogy Breakup (TAB) model is used (O’Rourke & Amsden, 1987).
Each of these models has coefficients that need to be specified for the respective regimes.
Unfortunately, no experimental spray data was available for the engine configuration con-
sidered in this section. However, spray modeling for diesel fuel was investigated using
experimental results from a spray chamber at similar ambient conditions, mass flow rate
and nozzle geometry to that of the engine experiments studied in this work (Waidmann
et al., 2006; Nentwig, 2007; Cook, 2007). The model parameters used were found to provide
reasonably accurate results for characteristics like spray penetration and droplet distribu-
tion. For combustion problems, spray penetration is an important quantity to represent
correctly, as it directly influences the combustion model parameters such as mixture frac-
tion. The parameters used here were found to predict the mixture fraction field quite well.
Since there are only a few parameters in the spray model, these were varied over a physically
realistic range and it was found that there was not a strong influence on the overall com-
bustion behavior, thus the same set of parameters were applied to the engine investigated
in this work. The spray model parameters used in the KH-RT model are summarized in
Table 6.6.
Table 6.6: Spray model parameters used for KH-RT secondary break-up model.
Cone angle (◦) B0 B1 CL C1
7 0.61 40 30 0.5
6.2.3 Chemical Mechanism
The simulations used n-heptane chemistry as a surrogate for diesel fuel. This is often
assumed since n-heptane has a similar cetane number and approximately the same C/H
ratio as typical diesel, thus resulting in similar auto-ignition characteristics. The n-heptane
mechanism was based on that of Liu et al. (2004) and used a total of 36 species and 71
reactions. The mechanism was validated over a range of pressures and stoichiometries,
as well as for different strain rates, and is found to represent the ignition delay time of
n-heptane-air ignition well.
114 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
6.3 Comparison with Experimental Data
In this section, the results of the simulation will be compared with the data available from
the experiments. First, the injection strategies are investigated by varying the amount of
fuel in each pulse as well as the timing between the injections. Then, the model will be
tested under different engine load conditions. Finally, the effect of EGR will be investigated.
6.3.1 Comparison of Injection Strategies
To effectively use simulations to compare different injection strategies, the model must
be able to predict the effect of different injection timings, durations, and mass ratios on
the ignition behavior and resulting combustion. Therefore, in this section two operation
points will be considered that have similar engine load (IMEP), equivalence ratio, and EGR
percent, but have substantially different injection profiles. The two cases 512MR3 and
513MR8 will be used for this comparison, for which the operating conditions and injection
properties can be found in Tables 6.3 and 6.4, respectively. Looking at the experimental
injection profiles in Fig. 6.3, one can see the main differences between injection strategies.
Case 512MR3 is a commonly used split-injection strategy and employs a small percentage
of the overall fuel injected as a pilot approximately 10 CAD bTDC, with the majority of
the fuel introduced several CAD aTDC. The small amount of fuel in the pilot evaporates
and ignites quickly, thereby pre-conditioning the combustion chamber to reduce the ignition
delay of the main injection and help reduce noise and NOx formation in the early phases
of combustion (Baumgarten, 2006). Alternatively, injection strategies with shorter dwell
times as in case 513MR8 have been developed through optimization to provide an overall
improvement with regard to noise, emissions, and performance for a given operating point.
The experimental and computed cylinder pressure are compared in Fig. 6.8. Although
the experimental pressured was measured at an single point in the combustion chamber, here
a comparison will be made with the volume average of the computed pressure, as the spatial
variation of pressure is typically a small percentage of the mean. The overall agreement
between the computed and experimental pressure is good during the entire combustion
process. However, there are several features that depart from the experiment that are
worth closer inspection. First, the ignition events of both the pilot and main injections are
captured with quite good accuracy. This is significant as it shows that the representation
of mixing through the scalar dissipation rates is effective and that the influence of the
6.3. COMPARISON WITH EXPERIMENTAL DATA 115
pilot injection on overall ignition of the mixture is properly accounted for in the model.
After ignition, both cases have a pressure rise rate that is steeper than observed in the
2 CHAPTER 1. TEST
20
30
40
50
60
70
80
-30 -20 -10 0 10 20 30
p(bar)
CAD aTDC (◦)
512MR3
513MR8
sim. exp.
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 6.8: Experimental and computed cylinder pressure, p, for the different injectionstrategies of operating points 512MR3 and 513MR8 (see Tables 6.3 and 6.4). Curves arecomputed results and points are averaged experimental data. The timing and duration ofeach injection pulse are represented by the thick solid lines at the top of the graph, withtopmost describing case 512MR3.
experimental data. This discrepancy can be attributed to the use of a single flamelet
representation for the entire physical domain. Since the scalar dissipation rates input to
the flamelet are averaged over the volume, even regions of high scalar dissipation rate that
would normally not have reached ignition, such as the near-nozzle region, will burn early.
To reduce this discrepancy, multiple flamelets could be employed in an Eulerian Particle
Flamelet Model (EPFM) approach as described by Barths et al. (2000). There it was shown
that solving multiple flamelet histories for regions of similar dissipation rate improved the
prediction of the initial pressure rise. It should also be noted that since the experimental
data shown are averaged over 25 combustion cycles, any variations in the start of ignition of
each cycle will cause the averaged pressure rise rate to appear damped. In both cases, the
computed pressure during expansion is slightly higher than the experimental observations
even though the pressure rate is approximately the same. This could be a consequence of
116 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
errors in computed heat transfer to the cylinder walls, as it was assumed that the cylinder
wall temperature remained constant during combustion and expansion.
Further differences between the injection strategies can be observed in the temperature
distribution in the combustion chamber, shown for various crank angle degrees for cases
512MR3 and 513MR8 in Figs 6.9 and 6.10, respectively. The classic configuration (512MR3)
shows that the pilot injection does not penetrate far into the bowl and after ignition the
combustion products stay in the vicinity of the nozzle. This region of higher temperature
along with the corresponding increase in cylinder pressure contributes to a reduced ignition
delay of the main injection, which can be seen after ignition in Fig. 6.9(b). The pressure
peaks at approximately 17 CAD aTDC, shown in Fig. 6.9(b), at which point the majority
of the fuel is burned.
In Fig. 6.10(a), the cylinder temperature is shown at 10.3 CAD aTDC, just after first
ignition of the fuel from the pilot. Here it can be seen that the additional mass in the pilot
caused the fuel to penetrate further into the combustion chamber and that unburned fuel
from the main injection is already present in the system. At 13.7 CAD aTDC, just prior to
the second ignition event, it can be seen that the fuel originating from the main injection
has not fully ignited but has begun to react in regions where it interfaces with burning
fuel from the pilot. Peak pressure is achieved at approximately the same crank angle as
512MR3, but is slightly lower from the resulting lower temperature in the cylinder.
6.3.2 Variation of Load and EGR
In order to ensure that the model is applicable over a range of typical operating points,
different engine loads were computed using the classic pilot injection strategy. First, the
load conditions for OP10, OP12, and OP13 as described in Table 6.3 are considered without
any EGR. The resulting pressure traces are shown in Fig. 6.11. It is seen that the results
for all loads are in good agreement with the experiments, especially with regard to the
ignition timing of each mixture. The mid and high load cases show either an over or under-
prediction of the peak pressure, but note that a point was made of using the experimental
values quoted for initial conditions, for which there can be uncertainty in quantities such
as charge temperature and overall equivalence ratio. The difference in peak pressure is
less than 3%. Overall, it is seen that the pressure rise after ignition of the main mixture
is captured well and that the slope changes with increasing load in the same manner as
observed in the experiment. This results from the higher sustained scalar dissipation caused
6.3. COMPARISON WITH EXPERIMENTAL DATA 117
(a) 0 CAD aTDC (b) 10 CAD aTDC
(c) 17 CAD aTDC (d) 29 CAD aTDC
Figure 6.9: Three-dimensional cylinder temperature of case 512MR3. Each time instanceshows a planar cut normal to the cylinder axis 3 mm below the cylinder head and a cross-section of the piston bowl. The section locations are indicated in plot (d) and the blackcurves represent isocontours of stoichiometric mixture.
118 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
(a) 10.3 CAD aTDC (b) 13.7 CAD aTDC
(c) 18 CAD aTDC (d) 29 CAD aTDC
Figure 6.10: Three-dimensional cylinder temperature of case 513MR8. Each time instanceshows a planar cut normal to the cylinder axis 3 mm below the cylinder head and a cross-section of the piston bowl. The section locations are indicated in plot (d) and the blackcurves represent isocontours of stoichiometric mixture.
6.3. COMPARISON WITH EXPERIMENTAL DATA 119
2 CHAPTER 1. TEST
0
20
40
60
80
100
120
-30 -20 -10 0 10 20 30
p(bar)
CAD aTDC (◦)
OP10OP12 aOP13 a
sim. exp.
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 6.11: Experimental and computed cylinder pressure, p, for different engine loads(IMEP) with no EGR (see Tables 6.3 and 6.4). Curves are computed results and points areaveraged experimental data. The timing and duration of each injection pulse are representedby the thick solid lines at the top of the graph for OP13, OP12, and OP10 starting fromtopmost.
120 CHAPTER 6. SPLIT-INJECTION DIESEL ENGINE
by the additional fuel in the injections at higher load which causes a higher variance in the
mixture fraction field and therefore less of the charge is near stoichiometric.
Finally, we consider the effect of EGR on the combustion. EGR is commonly used to
help reduce NOx emissions by reducing the peak combustion temperature. Such a strategy
results in overall lower peak cylinder pressure for the same loading conditions and this is
observed in the pressure traces plotted in Fig. 6.12. The simulations with EGR match the
2 CHAPTER 1. TEST
20
40
60
80
100
120
-30 -20 -10 0 10 20 30
p(bar)
CAD aTDC (◦)
OP12 ab
OP13 ab
sim. exp.
Figure 1.1: need some text here to see what the caption will look like. It will also show me if themargins are correct.Figure 6.12: Experimental and computed cylinder pressure, p, for different engine loads(IMEP) with EGR (see Tables 6.3 and 6.4). Curves are computed results and points areaveraged experimental data. The timing and duration of each injection pulse are representedby the thick solid lines at the top of the graph, with OP13 the topmost.
behavior of the experimental runs very well. This resulting match in pressure indicates
that the effect of the combustion products in the oxidizer is correctly accounted for in the
chemistry, which is an advantage of have a detailed chemistry representation. Unfortunately,
emissions data for these experimental cases are not available, however the close agreement
of the pressure traces both with and without EGR indicates that effect on peak combustion
6.3. COMPARISON WITH EXPERIMENTAL DATA 121
temperature is correct and therefore it would be expected that the trend of NOx reduction
would also be well represented.
It is worth noting that there a number of experimental uncertainties that can contribute
to variations in the cylinder pressures. Quantities such as intake and fuel mass flow rates,
intake temperature and EGR fraction can fluctuate from cycle to cycle and result in cor-
responding variations in cylinder pressure during the combustion phase. It is difficult to
quantify exact error bars, but data suggest that the range of variation around the peak
pressure is on the order of ±2-3%. The choice of PDF used in the flamelet model has a
moderate influence on cylinder pressure, but will ultimately be of more importance for the
estimation of species concentrations for emissions. Unfortunately, the emissions data was
not available for these configurations. It is important to note that the simulations presented
in the previous section used the experimentally quoted or estimated initial conditions. If
closer agreement with the experimental data were desired, one could adjust some of the
initial parameters within the range of experimental uncertainty, but here it is interesting to
observe the level of agreement without this type of adjustment.
In summary, the results in this section indicate that the two-dimensional flamelet model
presented is able to predict combustion in split-injection configurations for a wide range
of operating points. The results agree both qualitatively and quantitatively using initial
conditions taken directly from experiments. Thus, the flamelet model framework has great
potential to be applied in the design and optimization of multiple injection engines.
Chapter 7
Summary and Conclusions
The ability to use numerical simulations to aid in the design and optimization of combus-
tion devices will be increasingly important. With the advent of high-pressure common-rail
injection systems, many modern engine designs use multiple fuel injections to help reduce
emissions. Thus, this work has focused on developing models to represent the mixing,
ignition, and combustion in multiple feed systems.
In order to correctly represent the various length and time scales in turbulent reacting
flows, an extension of the flamelet model was used as it is found to capture the coupling
between the small scale flow structures and the chemistry. Asymptotic analysis showed
that the traditional flamelet equations can be extended to two independent variables to
represent the mixing in a three-feed system. The associated scalar dissipation rates of
each mixture fraction were investigated to correctly parameterize the flamelet equations.
It was found that the scalar dissipation rate of each mixture can generally be considered
independent of the other, and that the joint scalar dissipation rate is the mechanism by
which the two-dimensional flamelet equations represent full three-stream mixing. Methods
were developed to model all required scalar dissipation rates, including models for cases
where the assumption of independence is not valid.
To validate the two-dimensional flamelet formulation, a fundamental investigation of
auto-ignition in multi-feed systems was carried out using DNS of n-heptane ignition with
detailed finite-rate chemistry. It was found that the two fuel streams interact through heat
and mass transfer to cause regions where both mixtures are present to ignite earlier than
would occur in a two feed system. The interaction in physical space appears as a front
propagation in two-dimensional mixture fraction space that is governed by the magnitude
122
123
of the scalar dissipation rates. The combustion of the mixture of both fuel streams becomes
the primary source of heat release shortly after the ignition of the first fuel and reduces the
ignition delay time of the unmixed second fuel through the resulting increase in pressure. It
was found that the front propagation in mixture fraction space is correctly described by the
two-dimensional flamelet equations and therefore the model is able to accurately represent
the interaction of the multiple fuel streams.
To provide closure for the coupling between the flamelet chemistry and the turbulent
flow field, information about the joint statistics of the two mixture fractions is required.
This work used a presumed PDF approach, where a joint distribution is used that can
be parameterized by the second order moments of the two mixture fractions. Several ex-
isting model distributions were investigated, including the Dirichlet distribution and the
statistically-most-likely distribution, and were found to be unable to correctly represent
the higher order moments of the mixing field. Thus, a new five parameter bivariate beta
distribution was developed in order to provide an improved description.
The new model distribution was validated by comparing it with joint distributions com-
puted from two-scalar mixing in stationary isotropic turbulence at a Taylor-scale Reynolds
number of Reλ = 165. A range of different initial scalar fields were investigated, including
isotropically distributed scalars as well as mixing layer type profiles. The effect of different
means and mixing states were investigated to ensure that the model could capture fully
asymmetric initial conditions. The results were compared both qualitatively and quantita-
tively, showing that the new bivariate beta distribution exhibits the best overall performance
in the majority of configurations. It was also observed, however, that the bivariate beta
is unable to represent scalar fields with strong positive correlation well, and so for such
configurations the statistically-most-likely distribution should be considered.
Finally, the overall model framework was applied in the simulation of a split-injection
diesel engine. Two different piloted injection strategies were considered, whereby distri-
bution of fuel and dwell time between the two injection pulses was varied. It was found
that the experimentally observed ignition characteristics of both strategies were correctly
represented by the two-dimensional flamelet model. A range of different engine loads was
considered and it was shown that the pressure rise rate was predicted well in all cases.
Furthermore, the computed effect of exhaust gas recirculation was compared with the ex-
periments and found to be in good agreement.
124 CHAPTER 7. CONCLUSION
There is also considerable potential for further development of the methods presented
in this work. Whereas here the case of two distinct injection pulses was considered, modern
injector technology and control systems are capable of additional injections and new engine
designs are employing more complex strategies to further increase operational efficiency and
reduce emissions. Therefore, methods for handling three or more injections will be necessary.
Depending on the configuration, it may be possible to extend the method presented here
to three injections if the timing is such that the first injection is well mixed by the time the
third is introduced. In such a scenario, the mixing in one of the mixture fraction directions
could be neglected and a series of two-dimensional flamelets could be used to represent the
system. Such an approach is attractive as it does not increase the dimensionality of the
coupled system of equations. Furthermore, as new injection strategies for HCCI engines
are developed, methods to account for three reactant streams as well as variations in the
temperature of the system may be required. Overall, flamelet based methods provide a
promising framework for addressing such future modeling problems.
Appendix A
On Bivariate Beta Distributions
In this section, a discussion of various bivariate distributions related to the beta distribu-
tion is given and properties of the bivariate beta distribution introduced in Sec. 5.2.3 are
described. In the statistics literature, a general review of the Dirichlet distribution can
be found in Kotz et al. (2000) and a review of distributions with beta conditionals can be
found in Arnold et al. (1999). More recently, a reference on beta distributions was edited
by Gupta & Nadarajah (2004). Gupta & Wong (1985) presented three and five parameter
bivariate beta distributions, the three parameter being the Dirichlet and the five parameter
being a Morgenstern type system defined by the cumulative distribution functions (CDF).
A four parameter distribution which has scaled beta conditionals such that it is supported
on a unit triangle defined by {0 < x ≤ 1 − y, 0 < y ≤ 1} was presented by James (1975).
Nadarajah & Kotz (2005) present three bivariate beta distributions, two which are sup-
ported on a unit triangle such that {0 < x ≤ y, 0 < y ≤ 1} and one distribution supported
on the unit square. Olkin & Lui (2003) proposed a four parameter bivariate beta distribu-
tion also supported on the unit square. The most recent distribution to be proposed is that
of Nadarajah (2007), which is a four parameter distribution supported on a unit square that
has generalized beta marginal distributions following Libby & Novick (1982) (alternatively
see Gupta & Nadarajah (2004) chapter 5, section IX).
Here, we investigate a distribution that can be written in the form
f(x, y) = C xβ1−1yβ2−1(1− x)β3−1(1− y)β4−1(1− x− y)β5−1 (A.1)
for βi > 0, 0 < x < 1, and 0 < y < 1 − x. This appears as a limit of both the so-called
125
126 APPENDIX A. ON BIVARIATE BETA DISTRIBUTIONS
F2-beta and F3-beta distributions introduced by Nadarajah (2006a,b).
The F2-beta is defined as
f2(x, y) =C2 x
β−1yβ′−1(1− x)γ−β−1(1− y)γ
′−β′−1
(1− ux− vy)α(A.2)
for 0 < x < 1, 0 < y < 1, −∞ < α <∞, γ > β > 0, γ′> β
′> 0, 0 ≤ u < 1, and 0 ≤ v < 1,
where the normalizing constant is found to be
1
C2=
Γ(β)Γ(β′)Γ(γ − β)Γ(γ
′ − β′)Γ(γ)Γ(γ′)
F2
(α, β, β
′, γ, γ
′;u, v
)(A.3)
with the Appell function of the second kind defined as
F2
(α, β, β
′; γ, γ
′;x, y
)=
∞∑
m=0
∞∑
n=0
(α)m+n(β)m(β′)n
(γ)m(γ′)n
xmyn
m!n!(A.4)
and (φ)k = Γ(φ+ k)/Γ(φ) is the Pochhammer symbol.
The F3-beta distribution is
f3(x, y) =C3 x
β−1yβ′−1(1− x− y)γ−β−β
′−1
(1− ux)α(1− vy)α′ (A.5)
for 0 < x < 1, 0 < y < 1, 0 < x + y < 1, α > 0, α′> 0, β > 0, β
′> 0, γ > β + β
′,
−1 < u < 1, and −1 < v < 1, where the normalizing constant is found using
1
C3=
Γ(β)Γ(β′)Γ(γ − β − β′)Γ(γ)
F3
(α, α
′, β, β
′; γ;u, v
)(A.6)
with the Appell function of the third kind defined as
F3
(α, α
′, β, β
′; γ;x, y
)=
∞∑
m=0
∞∑
n=0
(α)m(α′)n(β)m(β
′)n
(γ)m+n
xmyn
m!n!. (A.7)
It is interesting to note that for the case u = v = 0, Eq. (A.2) is the product of two
independent beta distributiona and Eq. (A.5) is the bivariate Dirichlet distribution. It
should also be noted that the case u = v = 1 is not strictly included in either Eqs. (A.2)
or (A.5), as the Appell function is singular at (1,1). However, in the following section it
will be shown that in the vicinity of (1,1), an expression that is absolutely convergent may
A.1. CONTINUATION OF APPELL’S HYPERGEOMETRIC SERIES 127
be found and that Eqs. (A.2) or (A.5) are equivalent and therefore may be represented by
Eq. (A.1).
A.1 Continuation of Appell’s hypergeometric series
It is noted that the F2 series is absolutely convergent for |u| < 1, |v| < 1 and the F3 series
converges for |u| + |v| < 1. Thus, the case u = v = 1 is not absolutely convergent and
an analytic continuation for this point is necessary. One such continuation in the vicinity
of (1,1) has been given by Hahne (1969). Sud & Wright (1976) express the F2 series as a
sum of four F3 series and provide continuations that are rapidly convergent when one of the
parameters u,v is near unity. The series can also be used near (1,1), but one of the series
will converge more slowly. Tarasov has also shown the reflectional symmetry of the F2 and
F3 series at (1,1) and discussed their representations as 3F2 series (Tarasov, 1993, 1995).
Table A.1: Mapping of Appell-beta to F2- and F3-beta
F2-beta F3-beta
α 1− β5 1− β3
α′
- 1− β4
β β1 β1
β′
β2 β2
γ β1 + β3 β1 + β2 + β5
γ′
β2 + β4 -
Using the mapping between the bivariate beta for unity argument given in Table A.1,
the F2-beta (A.2) and F3-beta (A.5) distributions can be recast as
f2(x, y) = C2xβ1−1yβ2−1(1− x)β3−1(1− y)β4−1(1− ux− vy)β5−1 (A.8)
f3(x, y) = C3xβ1−1yβ2−1(1− ux)β3−1(1− vy)β4−1(1− x− y)β5−1 (A.9)
where
1
C2=
Γ(β1)Γ(β2)Γ(β3)Γ(β4)
Γ(β1 + β3)Γ(β2 + β4)F2 (1− β5, β1, β2;β1 + β3, β2 + β4;u, v) (A.10)
1
C3=
Γ(β1)Γ(β2)Γ(β5)
Γ(β1 + β2 + β5)F3 (1− β3, 1− β4, β1, β2;β1 + β2 + β5;u, v) (A.11)
128 APPENDIX A. ON BIVARIATE BETA DISTRIBUTIONS
Using the definition of the beta function
B(α, β) =Γ(α)Γ(β)
Γ(α+ β)
the normalising constants can be rewritten
1
C2= B(β1, β3)B(β2, β4)F2 (1− β5, β1, β2;β1 + β3, β2 + β4;u, v) (A.12)
1
C3= B(β1, β2, β5)F3 (1− β3, 1− β4, β1, β2;β1 + β2 + β5;u, v) (A.13)
From Hahne (1969) Eq. (12) only the first and third terms are non-zero at (1,1) and so
can be written as
F2(α, β, β′; γ, γ
′; 1, 1) =
Γ(γ)Γ(γ′)Γ(γ − β − γ′ + β
′)Γ(γ
′ − β′ − α+ β)
Γ(β′)Γ(γ − β)Γ(γ′ − β′ + β)Γ(γ − α)e±iπ(α+β
′−γ′ )
× 3F2
[α+ 1− γ + β − β′ , γ′ − β′ − α+ β, γ
′ − β′ ; 1
γ′ − β′ + β, 1− γ + β + γ
′ − β′
]
+Γ(γ)Γ(γ
′)Γ(γ
′ − β′ − γ + β)Γ(γ − β − α+ β′)
Γ(β)Γ(γ′ − β′)Γ(γ − β + β′)Γ(γ′ − α)e±iπ(α+β−γ)
× 3F2
[α+ 1− γ′ + β
′ − β, γ − β − α+ β′, γ − β; 1
γ − β + β′, 1− γ′ + β
′+ γ − β
](A.14)
where 3F2 is the generalized hypergeometric function
qFp
[α1, α2, . . . , αp
β1, β2, . . . , βq;x
]=∞∑
n=0
(α1)n(α2)n . . . (αp)n(β1)n(β2)n . . . (βq)n
xn
n!(A.15)
for q = 3, p = 2. When q = p + 1 the series is absolutely convergent for |x| < 1 and also
|x| = 1 in the case that
R
q∑
i=1
βi −p∑
j=1
αj
> 0.
In the case of the F2-beta, this requires simply β > 0, β′> 0. Buhring (1987) provides an
expression for use near the unit argument as
3F2
[a, b, c
e, f; 1
]=
Γ(e)Γ(f)Γ(s)
Γ(a+ s)Γ(b+ s)Γ(c)3F2
[e− c, f − c, sa+ s, b+ s
; 1
](A.16)
A.2. MARGINAL DISTRIBUTIONS 129
where s = e + f − a − b − c. The advantage of this reformulation is that the series on the
r.h.s. converges for |x− 1| < 1.
It some case, using the F3-beta distribution gives more stable results. A relation for F3
has been given by Vidunas (2009) as
F3
(α, α
′, β, β
′; γ;x, 1
)=
Γ(γ)Γ(γ − α′ − β′)Γ(γ − α′)Γ(γ − β′) 3F2
[α, β, γ − α′ − β′
γ − α′ , γ − β′;x
]. (A.17)
This relation appears to approximate the F3 series well, and when combined with Eq. (A.16)
it will be absolutely convergent. Namely, we can obtain
F3
(α, α
′, β, β
′; γ; 1, 1
)=
Γ(γ)Γ(γ − α− β)
Γ(γ − α)Γ(γ − β)3F2
[β′, α′, γ − α− β
γ − β, γ − α; 1
]. (A.18)
Therefore, it is often more convenient to use the F3-beta. The problem is that this expression
was specified around x = 0, so it may not be as valid for x = 1.
A.2 Marginal Distributions
The marginal distributions of each scalar can be represented in terms of Appell hyperge-
ometric functions of either the second or third kinds from Eqs. A.7 and A.4, respectively.
First, using the Appell F2 function, the marginals can be written
P (x|y) = C2B(β2, β4) 2F1
(β2, 1− β5;β2 + β4;
1
1− x
)xβ1−1(1− x)β3+β5−2 (A.19)
P (y|x) = C2B(β1, β3) 2F1
(β1, 1− β5;β1 + β3;
1
1− y
)yβ2−1(1− y)β4+β5−2 (A.20)
where C2 is obtained from Eq. (A.3) and 2F1 is the Gauss hypergeometric function defined
by Eq. (A.15) for q = 2, p = 1. Alternatively, the Appell F3 function can be used to write
the marginals according to
P (x|y) = C3B(β2, β5) 2F1 (β2, 1− β4;β2 + β5; 1− x)xβ1−1(1− x)β2+β3+β5−2 (A.21)
P (y|x) = C3B(β1, β5) 2F1 (β1, 1− β3;β1 + β5; 1− y) yβ2−1(1− y)β1+β4+β5−2 (A.22)
with C3 defined by Eq. (A.6).
130 APPENDIX A. ON BIVARIATE BETA DISTRIBUTIONS
A.3 Product Moments
The product moment of the F2-beta can be expressed as
E2(XmY n) =C2Γ(m+ β)Γ(n+ β
′)Γ(γ − β)Γ(γ
′ − β′)Γ(m+ γ)Γ(n+ γ′)
× F2(α,m+ β, n+ β′;m+ γ, n+ γ
′;u, v)
=(β)m(β
′)n
(γ)m(γ′)n
F2(α,m+ β, n+ β′;m+ γ, n+ γ
′;u, v)
F2(α, β, β′ ; γ, γ′ ;u, v)(A.23)
The product moment of the F3-beta can also be expressed as
E3(XmY n) =C3Γ(m+ β)Γ(n+ β
′)Γ(γ − β − β′)
Γ(m+ n+ γ)
× F3(α, α′,m+ β, n+ β
′;m+ n+ γ
′;u, v)
=(β)m(β
′)n
(γ)m+n
F3(α, α′,m+ β, n+ β
′;m+ n+ γ;u, v)
F3(α, α′ , β, β′ ; γ;u, v)(A.24)
If we consider the equation at the unit argument for F3, we can obtain
E3(XmY n) =(β)m(β
′)n(γ − α− β)n
(γ − β)n(γ − α)m+n
3F2
[β′+ n, α
′, γ − α− β + n
γ − β + n, γ − α+m+ n; 1
]
3F2
[β′, α′, γ − α− β
γ − β, γ − α; 1
] (A.25)
A.3. PRODUCT MOMENTS 131
The preceding expressions can be used to find the moments of the distribution using ei-
ther the F2 or F3 Appell function. For example, the second-order moments can be expressed
from E3 as
x =β
γ
F3(α, α′, β + 1, β
′; γ + 1)
F3(α, α′ , β, β′ ; γ)
y =β′
γ
F3(α, α′, β, β
′+ 1; γ + 1)
F3(α, α′ , β, β′ ; γ)
x′ 2 =β(β + 1)
γ(γ + 1)
F3(α, α′, β + 2, β
′; γ + 2)
F3(α, α′ , β, β′ ; γ)− x2
y′ 2 =β′(β′+ 1)
γ(γ + 1)
F3(α, α′, β, β
′+ 2; γ + 2)
F3(α, α′ , β, β′ ; γ)− y2
x′y′ =ββ′
γ(γ + 1)
F3(α, α′, β + 1, β
′+ 1; γ + 2)
F3(α, α′ , β, β′ ; γ)− (x)(y)
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