A multi-dimensional flamelet model for ignition in multi ...hn380sm8596/... · a multi-dimensional flamelet model for ignition in multi-feed combustion systems a dissertation submitted

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



http://purl.stanford.edu/hn380sm8596

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


Brian Cantwell


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.

iii

iv

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.

vi

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.

vii

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

xi

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

xiii

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


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


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


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


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).


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).


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)


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)


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)


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


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


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


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)


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)


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


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


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)


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)


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)


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)


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.


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)


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


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)


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)


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


= − ∂

∂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.


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,


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,


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


=∂

∂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


=∂

∂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.


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)


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


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


(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.


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


2 CHAPTER 1. TEST

1

Z1

0

1

Z2

0

20

40

60


(a) χ2

2 CHAPTER 1. TEST

1

Z1

0

1

Z2

-60

-40

-20

0


(b) χ12

2 CHAPTER 1. TEST

1

Z1

0

1

Z2

0

20

40

60


(c) χ0

2 CHAPTER 1. TEST

1

Z1

0

1

Z2

0

20

40

60


(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).


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 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)


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)


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


(a) χ1 computed

2 CHAPTER 1. TEST

1

Z1

01

Z2

0

5

10

15


(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.


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.


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


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.


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.


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


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.


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 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


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 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.


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.


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


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


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


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.


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.


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)


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


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


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)


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.


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


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).


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).


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


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


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.


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


(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


(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.


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


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.


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


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.


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.


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).


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 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


(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


(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


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


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.


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


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


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.


6.2.1 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


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)


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


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.


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.


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


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


(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.


(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.


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.


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


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)


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).


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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