Analysis of transported probability density function mixing models for premixed turbulent combustion using direct numerical simulation

School of Mechanical and Manufacturing Engineering
Faculty of Engineering
The University of New South Wales
Analysis of transported probability density
function mixing models for premixed turbulent
combustion using direct numerical simulation
by
Bradley Alderton
Thesis submitted as a requirement for the degree of
Bachelor of Engineering in Mechanical Engineering
Submitted: 26th
October 2015
Supervisor: A/Prof. Evatt Hawkes
Student ID: z3417835

Analysis of transported probability… Bradley Alderton

ii

Certificate of Originality
I, Bradley Alderton, hereby declare that this submission is my own work and to the best of
my knowledge it contains no materials previously published or written by another person, or
substantial proportions of material which have been accepted for the award of any other
degree or diploma at UNSW or any other educational institution, except where due
acknowledgement is made in the thesis. Any contribution made to the research by others,
with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis.
I also declare that the intellectual content of this thesis is the product of my own work,
except to the extent that assistance from others in the project’s design and conception in
style, presentation and linguistic expression is acknowledged.
Signed ………………………………
Date ………………………………

Bradley Alderton Analysis of transported probability…

iii

Abstract
Reducing pollutant emissions and improving fuel economy from combustion-based energy
systems would be much less costly if improved computational models of turbulent premixed
combustion were available. This modelling is difficult because it involves nonlinear dynamics
spanning a huge range of length and timescales. Normally, equations are formulated for
averaged quantities, but this introduces closure problems for nonlinear terms. Transported
probability density function methods circumvent these difficult closure problems by solving for
the complete one-point joint probability density functions (PDFs). To solve the PDF transport
equations, a stochastic Lagrangian Monte Carlo approach is adopted in which an ensemble of
notional particles are evolved according to stochastic differential equations. Formulated in this
way, modelling is required for the effects of molecular mixing. Data from direct numerical
simulations (DNS) is used as both an input source, and a benchmark with which to assess the
mixing models. Use of a DNS eliminates all of the uncertainties associated with using turbulent
models or other methods to generate input, allowing an exclusive focus on the mixing models.
Previous work has successfully applied this approach to model non-premixed flames, but
premixed flames have not been considered. In this context, the aim of this thesis is to test three
mixing models against a DNS database modelling hydrogen-air combustion. The results will
inform future selection of PDF models and thus contribute to improved predictions of turbulent
premixed combustion systems.
The results from the TPDF modelling showed that the EMST model most accurately predicts
the temperature profiles, while the IEM and MC fail to do this. This is most likely due to neither
the IEM nor MC mixing model enforcing Locality in Composition Space. The conditional PDFs
were also evaluated, with all models unable to predict the conditional mean of OH or H2O2. A
sensitivity study was completed in order to verify the results, and this showed that there was
sensitivity to cell number, and no sensitivity to the number of particles per cell. There was
however statistical noise generated for lower numbers of particles per cell.


iv

Acknowledgements
I would firstly like to thank A/Prof. Evatt Hawkes for providing the TPDF code, and DNS data
input for use in this thesis. I would like to thank Joshua Tang for monitoring my progress throughout
this paper. He has provided me with valuable knowledge on TPDF theory and the computational
modelling involved. I would like to thank him for answering my many questions on every aspect of
thesis writing, and for encouraging me to keep my head up when things weren’t going to plan.
Thanks to Joshua’s knowledge, wisdom, and patience, I was able to complete this thesis on time with
a level of quality that I would otherwise have not been able to meet. I would like to thank Austin
Kong for his help with both the TPDF code, and Matlab processing. Austin helped me solve a
number of very important problems with little effort. He was also willing to help me at short notice,
and spend significant amount of time going through and solving problems with me. Finally, I would
like to thank my girlfriend Yen Yi. I would like to thank her for helping me strategise, show me how
to use my time wisely, and how to get the most work out of my days. Yen Yi was supportive, and
understanding of the time I spend on my thesis, even when it subtracted from the time we spent
together.


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Abbreviations
C-TPDF Composition Joint Transported Probability Density Function
CFD Computational Fluid Dynamics
CV-TPDF Composition-Velocity Joint Transported Probability Density Function
DNS Direct Numerical Simulation
EMST Euclidian Minimum Spanning Tree
FDA Finite Difference Algorithms
FEA Finite Element Analysis
HCCI Homogeneously Charged Compression Ignition
IEM Interaction by Exchange with the Mean
LES Large Eddy Simulation
NS Navier-Stokes
MC Modified Curl
ODE Ordinary Differential Equation
PDE Partial Differential Equation
PDF Probability Density Function
RANS Reynolds-Averaged Navies-Stokes
TPDF Transported Probability Density Function


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Mathematical Symbols
𝑒! Total Internal Energy
𝑓 Probability Density Function
𝑓!!,!
Body Force
𝑔 Gravity Force
𝐻 Jet Height
ℎ Some Space-step
𝑘 Turbulent Kinetic Energy
𝑁!"## Number of cell used in simulation
𝑁!"#$%&'(),! Number of particles in cell 𝑘
𝑁!" Initial number of particles per cell
𝑁!"#,! Cell mixing number in cell 𝑘
𝑃 Pressure
𝑝′ Fluctuating component of Pressure
𝑞 Heat Flux
𝑅 Gas Constant
𝑆 Chemical Source Term
𝑇 Temperature
𝑡 Time
𝑡! Jet Time
𝑈 Mean Velocity
𝑢 Velocity
𝑢! Kolmogorov Velocity Scale
𝑢′ Fluctuating Velocity
𝑉 Sample-space velocity corresponding to U
𝑥 Position
𝑌 Chemical Species Mass Fraction
𝜂 Kolmogorov Length Scale
𝜖 Rate of Turbulent Kinetic Energy
Dissipation
𝜙 Represents any scalar quantity
𝜇 Dynamic Viscosity
𝜙 𝑜𝑟 〈𝜙〉 Mean component of variable
𝜙′ Fluctuating component of variable


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Table of Contents
1
Introduction 4

2 Literature Review 6

2.1 Introduction ..........................................................................................................................6

2.2 Basics of Turbulent Flows ................................................................................................6

2.2.1 Governing Equations ..................................................................................................7

2.3 Turbulent Premixed Flames..............................................................................................8

2.3.1 Canonical representation............................................................................................8

2.3.2 Progress variable..........................................................................................................9

2.3.3 Turbulent premixed combustion regimes .............................................................9

2.3.4 Effects of turbulence on premixed flame speed............................................10

2.4 Modelling Methods ..........................................................................................................12

2.4.1 Direct Numerical Simulation..................................................................................13

2.4.2 Large Eddy Simulation.............................................................................................14

2.4.3 Reynolds Averaged Navier Stokes........................................................................14

2.5
Transported Probability Density Function Methods................................................16

2.5.1 Basic Concepts...........................................................................................................17

2.5.2 Composition TPDF Model ......................................................................................18

2.5.3 Solution Procedure - Monte Carlo Methods........................................................20

2.6.1 Role of mixing models .............................................................................................22

2.6.2 Criteria for mixing models ......................................................................................23

2.6.3 Interaction by Exchange with the Mean (IEM) ..................................................24

2.6.4 Modified Curl (MC)..................................................................................................24

2.6.5 Euclidian Minimum Spanning Tree (EMST)......................................................25

3 Modelling Methods 29

3.1 Simulation Scenario .........................................................................................................30


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3.2 Modelling the C-TPDF transport equations................................................................31

3.3 Pre-processing....................................................................................................................32

Mixing Frequency ...................................................................................................................32

Turbulent Diffusivity..............................................................................................................33

Mean Drift Term......................................................................................................................33

3.4 Implementing the splitting scheme ...............................................................................35

3.4.1 Transport (T)...............................................................................................................35

3.4.2 Mixing (M)..................................................................................................................36

3.4.3 Reaction (R)................................................................................................................39

4 Results and Discussion 40

4.1 Mean and Variance...........................................................................................................41

4.1.1 N2 Mass Fraction.......................................................................................................42

4.1.2 Temperature................................................................................................................42

4.1.3 Radical OH Species ..................................................................................................45

4.2 Conditional PDFs..............................................................................................................46

4.2.1 OH mass fraction conditioned on progress variable..........................................47

4.2.2 H2O2 mass fraction conditioned on progress variable.......................................49

5 Validation and Verification 51

5.1 Sensitivity Analysis..........................................................................................................51

5.1.1 Cell Number ..............................................................................................................51

5.1.2 Particles Numbers ...................................................................................................54

6 Conclusion 58

Bibliography 60


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List of Figures
2.1: Canonical version of a turbulent premixed flame. .........................................................................9
2.2: Turbulent premixed combustion regimes.....................................................................................10
2.3: Diagram of Flame Wrinkling Factor. ...........................................................................................11
2.4: Variations of the turbulent flame speed with RMS turbulent speed.............................................12
2.5: The difference between various modelling techniques by order of resolution.............................13
2.7: A two dimensional TPDF simulation where colour denotes mixture fraction.............................17
2.8: Mixture fraction of a piloted jet methane flame...........................................................................22
2.9: Euclidian minimum spanning tree for two composition variables...............................................26
2.10: An example of “stranding”.........................................................................................................27
2.11: Temporal evolution of the age variable......................................................................................28
3.1: Schematic of DNS configuration..................................................................................................30
3.2: Mixing Frequency input for the DNS data set..............................................................................33
3.3: Turbulent diffusivity input for the DNS data set..........................................................................34
3.4: Mean drift term input for the DNS data set..................................................................................34
4.1: Mean and RMS spatial N2 mass fraction profile comparison.......................................................43
4.2: Mean spatial temperature profile comparison ..............................................................................43
4.3: Mean and RMS spatial temperature profile comparison..............................................................45
4.4: Mean and RMS spatial OH radical profile comparison ...............................................................46
5.1: IEM mixing model cell sensitivity analysis at 23 tj .....................................................................52
5.2: MC mixing model cell sensitivity analysis at 23 tj ......................................................................53
5.3: EMST mixing model cell sensitivity analysis at 23 tj..................................................................54
5.4: IEM mixing model particles per cell sensitivity analysis at 23 tj ................................................55
5.5: MC mixing model particles per cell sensitivity analysis at 23 tj..................................................56


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Chapter 1
Introduction
HE rise in concern for greenhouse gas emissions and fuel efficiency has put demand on power
generation and transportation services to be cleaner for the environment, and make better use of
non-renewable resources including fossil fuels. Over 100 countries have pledged to reduce their
global warming limit to 2°C [1], and the European Commission have set a goal to reduce global
emissions by at least 60% below 2010 levels by 2015 [2]. By releasing strict emissions laws, a
demand for highly efficient, low emission combustion engines has been generated, with diesel
turbocharged engines being a common solution. A new and promising technology is the
Homogeneous Charge Compression Ignition (HCCI) engine, whereby fuel and air are
homogeneously mixed prior to ignition. These type of engines promise low emissions and high
efficiency however have been difficult to control. Computational models are a very promising
method that can be used to further develop the HCCI combustion engine, with their ability to
produce rapid computational prototypes, industry can test prototypes without having to physically
build them, thus reducing the time taken to develop fully operational engines. Computational Fluid
Dynamics (CFD) can be used to model the combustion events that happen inside a combustion
engine and thus accurately predict combustion, mixing, soot production, and ignition. With a
thorough awareness of these phenomena, engine designers are able to develop engines that are
capable of being highly efficient, while producing low emissions.
Unfortunately, modelling turbulent reactive flows is difficult as it involves non-linear events on a
large range of length and time scales, thus direct computation of the governing equations is
unaffordable for practical purposes. In common CFD methods, the governing Navier Stokes
equations are formulated for averaged quantities such as temperature and species concentration in
order to decrease the computational power required, although this introduces closure problems for
T


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nonlinear terms such as turbulent transport and chemical reaction. One method to solve this closure
problem is through use of Direct Numerical Simulation (DNS) where the smallest time and length
scales are solved numerically without the use of averaging operations or a turbulence model [3].
DNS produce highly detailed, highly accurate data, although due to their need to solve for all time
and length scales they require large computational resources and are only able to solve simple and
idealised flows. Due to the large computational expense, DNS are not viable for use in industry.
A second method to computationally analyse turbulent reactive flows is transported probability
density function (TPDF) methods. TPDF methods offer an accurate and computationally tractable
way to model CFD, although this comes at the cost of fidelity, as the small turbulent effects are not
directly solved and need to be modelled. TPDF methods take a statistical approach to solve the fluid
flow, by discretizing the flow into an ensemble of notional particles, and by representing the flow
with a large enough sample of particles the flow can be accurately modelled [4]. These notional
particles are point samples of the fluid that carry information that define the flow, including species
concentrations, temperature and velocity. The main drawback for the TPDF method is that the
molecular mixing needs to be modelled, and as molecular mixing directly influences critical
combustion phenomena, whose occurrence determines the production of combustion products
including NOx, soot, and CO2, a highly accurate mixing model is necessary for accurate results [5].
There are a number of mixing models that have the potential to accurately model premixed
turbulent reactive flows. The focus of this paper is to evaluate the comparative accuracy between
three mixing models, in predicting turbulent premixed flames. The mixing models to be examined
are the Interaction by Exchange with the Mean (IEM) [6], Modified Curl’s (MC) [7], and Euclidian
Minimum Spanning Tree (EMST) [8] model. A DNS database modelling a statistically one-
dimensional configuration of premixed hydrogen-air combustion will be used for both data source
input, and to verify the accuracy of the mixing models [9]. The results will inform future selection of
PDF sub-models and thus contribute to improved predictions of turbulent combustion systems.
The remainder of this thesis will be presented in the following way:
• Chapter 2 contains a literature review summarising the relevant research already
undertaken in this field
• Chapter 3 discusses the methodology used to run the C-TPDF simulations
• Chapter 4 presents the results of the C-TPDF simulations and discusses the implications
• Chapter 5 presents the validation and verification of the results
• Chapter 6 concludes the key findings of the study and discusses the potential for future
work in the field of premixed turbulent flame modelling


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Chapter 2
Literature Review
2.1 Introduction
This chapter provides an overview of turbulent premixed reacting flows and the modelling
techniques used to analyse them. Difficulties encountered in modelling these flames are
outlined, motivating the need for accurate and computationally tractable models. Key
challenges in producing such models are then discussed. An overview of the dominant
modelling paradigms is presented. TPDF modelling techniques are introduced and contrasted
against conventional modelling techniques. Advantages of the C-TPDF method in the
context of turbulent premixed reacting flows are discussed to motivate their use in this thesis.
2.2 Basics of Turbulent Flows
Turbulence is a highly complex phenomena that is not easily described by any one theory or
equation. Computational modelling and improved measurement techniques have however
empowered researchers to better understand turbulent events. Pope reflected “A century of
experience has shown the ‘turbulence problem’ to be notoriously difficult, and there are no
prospects of a simple analytic theory” [10]. Bilger et al (2005) reviewed some 50 years of
combustion theory [11], noting that even though significant progress has been made, there
still remain many fundamental concepts undeveloped, specifically, the interaction between


7

the chemistry and the turbulent flow of the system.
2.2.1 Governing Equations
All flows of practical importance can be fully described by the following set of Partial Differential
Equations (PDEs), jointly referred to as the Navier-Stokes (NS) Equations, presented here in
Cartesian Tensor form. These equations are derived from the Navier-Stokes (differential) equations,
coupled with the first law of thermodynamics (conversation of energy), a chemical species balance,
and the ideal gas law.
𝜕𝜌𝑢!
𝜕𝑡
+
𝜕 𝑢! 𝑢! 𝜌
𝜕𝑥!
= −
𝜕𝜌
𝜕𝑥!
+ 𝜌 𝑓!!,!
𝑌!
!"
!!!
+
𝜕 2𝜇𝑠!,!
𝜕𝑥! (2.1)
𝜕𝜌
𝜕𝑡
+
𝜕𝜌𝑢!
𝜕𝑥!
= 0
(2.2)
𝜕𝜌𝑌!
𝜕𝑡
+
𝜕 𝜌𝑢! 𝑌!
𝜕𝑥!
= −
𝜕 𝜌𝑌! 𝑉!,!
𝜕𝑥!
+ 𝑆!
(2.3)
𝜕𝜌𝑒!
𝜕𝑡
+
𝜕𝑢! 𝜌𝑒! + 𝜌
𝜕𝑥!
=
𝜕𝜏!,! 𝑢!,!
𝜕𝑥!
−
𝜕𝑞!
𝜕𝑥!
+ 𝜌 𝑓!!,!
𝑌!
!"
!!!
∙ 𝑉!,! + 𝑢!
(2.4)
𝑃 = 𝜌𝑅𝑇 (2.5)
Equation 2.1: Navier-Stokes equations for conservation of momentum
Equation 2.2: Conservation of mass statement, otherwise known as the continuity equation
Equation 2.3: Conservation of mass species, 𝑌!
Equation 2.4: Conservation of energy
Equation 2.5: Ideal gas law
These equations provide all the relevant information necessary to describe turbulent combustion,
and they remain the cornerstone of fluid dynamics. Unfortunately, analytical solutions to these
equations do not exist for flows that are of practical importance, ultimately due to the non-linear
convective terms and the chemical source term 𝑆 in Equation 2.3. In order to get around this, the NS
equations can be solved using numerical methods, such as DNS. A second way to apply the NS
equations is through a statistical approach. Turbulence can be viewed as a random process, in the
sense that the properties of the fluid at a particular location and time cannot be predicted with any
level of certainty. Take two experiments whose boundary and initially conditions may be nominally


8

the same although not identical - a small vibration in one experiment, a small impurity in the
mixture, and a small difference in the air distribution inlet – and these all get magnified by the
turbulence and result in two differing solutions. As the conditions can never be controlled with
sufficient detail to determine the exact evolution of the flow, random variables are appropriate [4].
2.3 Turbulent Premixed Flames
Premixed flames are an important type of flame to study, as they appear frequently in the
commercial combustion engine. In the HCCI engine, air and fuel are homogenously mixed prior to
combustion. This premixing of air and fuel means that these flames will be fundamentally different
from non-premixed flames. In order to better understand premixed flames, a few of the basic
concepts will be covered, as well as the effect that turbulence has on them. Two noteworthy concepts
will be discussed; the canonical representation of premixed flames, and the measure commonly used
to characterize the progress of a premixed flame. Two effects of turbulence on premixed flames are
also considered; the different regimes of turbulent premixed flames, and the effect that turbulence
has on flame speed.
2.3.1 Canonical representation
A premixed flame is a flame where the oxidant and fuel are homogeneously mixed prior to
interacting with the flame front [22], such as in a diesel engine. Premixed flames are different from
non-premixed flames in a number of ways, and the implications that this has on how to model them
is important for accurate modelling. The simplest version of a turbulent premixed flame is the
canonical version, which is statistically one-dimensional. The canonical turbulent premixed flame is
depicted in Figure 2.1, with variation along the horizontal axis only. It is simply a flame progressing
through a field with burnt gases on one side of the flame, and fresh gases on the other side.


9

Figure 2.1: Canonical version of a turbulent premixed flame [29], where 𝑆! is the velocity of the
flame.
2.3.2 Progress variable
In premixed flames, the progression of the flame front can be tracked by a reaction progress variable,
called the progress variable 𝑐. This is equal to some measure of the burnt nature of the mixture;
temperature is a suitable measure, but hydrogen and oxygen mass fraction can be used as well [9,
29]. In this study, temperature is chosen, thus we define the progress variable as a non-dimensional
parameter as shown in Equation 2.6, where subscripts 𝑓 and 𝑏 are for fresh and burnt temperatures
respectively.
𝑐 ≡
𝑇 − 𝑇!
𝑇! − 𝑇!
Equation 2.6: Definition of Progress Variable
(2.6)
Thus, anywhere in the reactants 𝑐 = 0, and in the products 𝑐 = 1. The flame front can then be
identified when 𝑐 = 0.5 [22]. In Figure 2.1, 𝑐 can be assigned to different locations of the flow. On
the left hand side of the flame, in the fresh gas region, it takes the value of 0, and on the right hand
side of the flame, in the burnt gas region, it takes the value of unity. Progress variables are very
useful for tracking the location of the flame front, especially in thin flame fronts as 𝑐 will mostly take
values of either 0, or 1, and only in the thin flame sheet region take the value 0.5 [27].
2.3.3 Turbulent premixed combustion regimes


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There are many different ways to characterize turbulent premixed flames. In this analysis four
different turbulent premixed combustion regimes will be considered: thickened flame, thickened-
wrinkled flame, wrinkled flamlet, and laminar flames, as seen in Figure 2.2. These flames differ in
thickness due to both the Karlovitz number (𝐾𝑎) and the Damköhler number (𝐷𝑎). When 𝐾𝑎 < 1
and 𝐷𝑎 ≫ 1, the flame is in the wrinkled flamelet region where the flame is thinner than all turbulent
scales [28]. For other values of 𝐷𝑎 and 𝐾𝑎, the flame will be in other regions. However there are
limitations to this categorization of flames due to a number of significant reasons [29]. Limited as
this categorization is, it is useful for visualizing the boundary cases. For example, for high values of
𝐷𝑎, the flame will be in the laminar combustion region. In this region, typical turbulent combustion
modelling techniques may not be utilized, as these flames are governed by different conditions.
Figure 2.2: Turbulent premixed combustion regimes [28]
2.3.4 Effects of turbulence on premixed flame speed
The effects of turbulence on the flame speed are significant and should be noted. The flame speed of
a turbulent premixed flame will be higher than that of a laminar premixed flame due to the increase
in the total flame surface, leading to a higher consumption rate for the same cross-section [28].
An important ratio in characterizing turbulent premixed flames is given by the available flame
surface divided by its projection in the propagating flame direction. This is diagrammatically


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represented in Figure 2.3. This ratio is known as the flame-wrinkling factor and is defined as
Ξ = 𝐴!/𝐴, where 𝐴! is the flames available surface, and 𝐴 is the flames projection in the direction it
is moving. The flame-wrinkling factor increases with the Reynolds number [28], as the Reynolds
number is correlated to the flame speed: an increase in flame speed will lead to an increase in the
flames available surface. This ratio is the main mechanism controlling turbulent premixed flames
[30].
Figure 2.3: Diagram of Flame Wrinkling Factor [28].
For higher Reynolds numbers, as the flame becomes increasingly turbulent, the flame speed will
increase up to some quenching limit, as summarized by Figure 2.4 [28]. Experimentally true as this
is, it is of little assistance to us in predicting the quenching limit of a turbulent premixed flame, as it
is almost impossible to precisely determine the turbulent RMS velocity that will correspond to the
quenching limit, as it is a condition that is more closely dependent on the initial and boundary
conditions than the turbulent RMS velocity [28]. As the flame-wrinkling factor increases with 𝑅𝑒,
the turbulent flame speed increases with 𝑅𝑒 also [36-38].


12

Figure 2.4: Variations of the turbulent flame speed with RMS turbulent speed [28]
2.4 Modelling Methods
Equations 2.1 → 2.5 are theoretically able to describe completely the fluid flow of a system,
although analytical solutions are often not readily available. Numerical methods are thus necessary
to solve these equations, however difficulties arise due to the presence of a wide range of time and
length scales, and this complicates the solution process as the discretization scheme needs to capture
both large-scale events and small-scale events. To solve using a statistical approach creates
additional variables due to the non-linear terms, with equations for higher moments involving terms
of successively higher moments [13]. As no additional equations are created from each process, a
model is needed to close the system of equations. The difference in modelling techniques can be
represented by Figure 2.5. With the determining factor being the degree to which the fluid flow has
been modelled, or inversely, resolved.
According to Kolmogorov (1941) [14] the smallest scales of time, length, and velocity can be
calculated numerically, giving an indication as to what length a DNS has to go to fully resolve a fluid
flow. Kolmogorov posed that the smallest scales (Kolmogorov scales) in any fluid are only based on
two parameters – the dissipation rate (𝜖) and the kinematic viscosity (𝜈). The Kolomogorov scales
can be calculated using Equations 2.7 → 2.9.
𝜂 =
𝜈!
𝜖
!/!
(2.7)
𝜏! =
𝜈
𝜖
!/!

(2.8)
𝑢! = 𝜈𝜂 !/!
(2.9)


13

Equation 2.7: Kolmogorov scale for length
Equation 2.8: Kolmogorov scale for time
Equation 2.9: Kolmogorov scale for velocity
Figure 2.5: The difference between various modelling techniques by order of resolution [33].
2.4.1 Direct Numerical Simulation
DNS are the easiest method to understand, as they simply brute force their way through the NS
equations, to arrive at a highly detailed solution. They resolve all fluctuations over all time, length,
and velocity scales down to the Kolmogorov scales, and as such, require no modelling. This is
achieved through either finite volume or finite difference approximations for Equations 2.1 → 2.5.
This requires all cells to be small enough to resolve the Kolmogorov scales. As well as this, the
domain must be large enough so as to capture the largest scales that are of the order of magnitude of
the flow geometry itself. Due to this need to capture both the large and small scales, DNS are
extremely computationally expensive. The computational expense is found to be proportional to
𝑅𝑒!
[10, 15], and thus, there is another limitation, as the flow simulation is not only limited by the
size of the geometry, but also by the Reynolds number of the flow. Apart from these limitations,
DNS is a valuable tool as it can be used to study the behavior of unclosed variables in RANS
methods, and thus assist in building more robust RANS methods. Another advantage of DNS is that


14

the error in simulation is solely attributed to the numerical discretization, and thus easily measured,
and accounted for [15].
2.4.2 Large Eddy Simulation
In order to decrease the computational power required to solve the NS equations for a fluid flow
system, it is possible to limit the size of scales that are resolved. If the smallest scales are not
resolved, but modelled, the computational power required to solve the flow greatly reduces. It is
apparent that over 99% of the computational power is used to resolve the smallest scales [13], thus,
by modelling these scales, and resolving the larger ones, the computation power is dramatically
reduced. This method is referred to as a Large Eddy Simulation (LES), and in this method, a model
is used to predict the smaller unresolved scales. In order to separate the large eddies from the smaller
eddies, a filtering operation is used which spatially averages the flow features that are smaller than
the filter size. Mathematically, the central difference operator used in an LES can be represented by
Equation 2.10 [16].
𝑢 𝑥 + ℎ − 𝑢 𝑥 − ℎ
2ℎ
=
𝑑
𝑑𝑥
1
2ℎ
𝑢 𝜉 𝑑𝜉
!!!
!!! (2.10)
By simulating the large scale features of the flow, the influence of the flow geometry is captured,
as it is the large scales that are influence by the geometry [3]. The smaller scales can be modelled
using a RANS (Section 2.4.3) approach or a PDF (Section 2.4.4) approach. Combining LES
modeling with RANS modelling has proved to be a promising method giving results with a high
level of agreement with DNS at a fraction of the computational expense [35].
2.4.3 Reynolds Averaged Navier Stokes
To model a flow with an even lower amount of computational power than DNS and LES, a Reynolds
Averaged Navier Stokes method can be taken. This method applies the Reynolds averaging
operation to Equations 2.1 and 2.2 to simplify the solution process. The underlying theory is based
on the premise that any variable can be broken down into its mean component 𝜙 , and its
fluctuating component 𝜙′. Thus, some variable 𝜙 can be broken down as shown in Equation 2.11.


15

𝜙 = 𝜙 + 𝜙!
(2.11)
One of the key definitions of Reynolds averaging is that the average of the fluctuation is zero.
Solving equations by this method removes the problem of having a wide range of length and time
scales. However, it introduces another problem, namely, the Reynolds stress tensor term. Taking the
time average and Reynolds decomposition of the momentum equation (Equation 2.1) for a non-
reacting flow yields Equation 2.12.
𝜌
𝜕 𝑢!
𝜕𝑡
+ 𝜌 𝑢!
𝜕 𝑢!
𝜕𝑥!
= −
𝜕 𝑠!,!
𝜕𝑥!
+
𝜕
𝜕𝑥!
2𝜇 𝑠!,! − 𝜌 𝑢!
!
𝑢!
! (2.12)
Formulated in this way, the additional term shown in Equation 2.13 is generated; commonly
know as the Reynolds Stress Tensor.
𝜏!,! = 𝜌 𝑢!
!
𝑢!
!
(2.13)
The Reynolds stress tensor introduces 6 additional terms to the momentum equation, rendering
the system of equations unclosed. In order to solve this, modelling is required for the Reynolds stress
tensor, which proves to be the fundamental shortcoming of the RANS approach [16]. There are
numerous ways to model the effects of the Reynolds stress tensor, but the most common is to use a
Boussineq Turbulent velocity hypothesis, and generate a two equation model – such as the 𝑘 − 𝜖 or
𝑘 − 𝜔 model – to solve two additional transport equations (one for 𝑘 and one for 𝜖 or 𝜔) [16].
RANS modelling remains one of the most widely used approaches to turbulence modelling, as it
incurs only slight to moderate computational cost compared to LES or DNS, while it is able to
accurately predict most features of the mean flow for a variety of scenarios [16]. The major
shortcoming of the RANS method is its inability to describe the underlying turbulent features, as it
uses an averaging operation that removes the turbulent fluctuations. As well as this, one of the key
assumptions of the RANS method is that the fluctuations and the mean are separated by orders of
magnitude, and in the scenarios that this condition is not met then RANS is not suitable for use. As a
method of comparison, a visual representation of the three modelling techniques covered thus far is
shown in Figure 2.6. The level of detail in the DNS (a), is visually more detailed than both the LES
(b) and the RANS method (c), while they all show the same average structure.


16

Figure 2.6: Visualisation of a diffusion flame with a DNS (a), an LES (b), and a RANS simulation
(c) [34].
2.5 Transported Probability Density Function
Methods
Transported Probability Density Functions (TPDF) methods are the most unique of all the methods
covered thus far. TPDF methods take a statistical approach to modelling the fluid flow by
discretizing it into an ensemble of notional particles, while the other mentioned methods discretize
the flow into control volumes. These notional particles contain all the one point statistics
(temperature, composition, velocity) necessary to fully define the flow at each point.
By using a large ensemble of particles, and tracing them through space and time, a description of
the flow can be attained, thus critical combustion events, such as jet spreading, can be studied in
great detail. As shown in Figure 2.7 a finite number of particles can be used to generate a complete
representation of the flow.
A complete explanation of TPDF methods is given by Pope (1985) [4], as only a brief
explanation will be given here.


17

Figure 2.7: A two dimensional TPDF simulation where colour denotes mixture fraction [10].
2.5.1 Basic Concepts
In order to better understand the context of TPDF methods, the basic statistical instruments will be
briefly described here. Due to the random nature of turbulence any flow property, 𝜙, can be treated
as a random variable. Since 𝜙 is a random variable we denote its PDF as 𝑓!(𝜓) where 𝜓 is the
sample space corresponding to 𝜙. This PDF can then be used to extract useful information about 𝜙.
For example, the probability that 𝜙 lies between 𝜙! and 𝜙! is evaluated by integrating the PDF
between these two values: 𝑃 𝜙! < 𝜙 < 𝜙! = 𝑓!(𝜓) 𝑑𝜓
!!
!!
.
Other important statistics can be extracted from the PDF including the mean, 𝜙 , and the
variance, 𝜙!"
as shown in Equations 2.14 and 2.15 respectively.
𝜙 = 𝜓𝑓!(𝜓) 𝑑𝜓
!
!!
(2.14)
𝜙!"
= 𝜓 − 𝜙 !
𝑓!(𝜓) 𝑑𝜓
!
!!
(2.15)


18

If a PDF has more than one random variable it is called a joint PDF, where the joint PDF of 𝜙!
and 𝜙! is given by 𝑓!!!!
𝜓!, 𝜓! . An important feature of joint PDF’s is that they contain all the
information about both 𝜙! and 𝜙!, as well as any joint statistics between the two. The single PDF of
either of these variables can be extracted by integrating over the other variable by using the
definition 𝑓!!
(𝜓!, 𝜓!) = 𝑓!!!!
𝜓!, 𝜓! 𝑑𝜓!
!
!!
. The concept of joint PDF’s can be extended to any
number of random variables, with the total number of random variable being referred to as the
dimension of the PDF.
Conditional statistics are a further extension of these concepts. Conditional statistics can be
thought of as “The probability of Event B happening given that Event A has already happened”.
Conditional statistics are often encountered when dealing with PDFs and are a central concept to
TPDF methods [4]. The probability of 𝜙! = 𝛷! given that 𝜙! = 𝛷! is given by Equation 2.16.
𝑃 𝜙! = 𝛷! 𝜙! = 𝛷! =
𝑃 𝜙! = 𝛷!, 𝜙! = 𝛷!
𝑃 𝜙! = 𝛷!
(2.16)
The corresponding PDF is given by Equation 2.17.
𝑓!!|!!
(𝜓!, 𝜓!) =
𝑓!!!!
𝜓!, 𝜓!
𝑓!!
𝜓!
(2.17)
When using PDF methods one must pose the model, this means that in order to use PDF methods,
the joint PDF of velocity and composition (or just composition) needs to be determined. There are a
few ways of doing this. The first is that the shape of the PDF can be assumed. This approach is
seldom used, as often it is impossible to know the shape of the PDF beforehand [4]. The second
method is more common and will be the one used in this study. This involves determining the PDF
shape by solving a modelled transport equation [4].
2.5.2 Composition TPDF Model
The main feature of PDF methods is their ability to solve the two main problems found in turbulent
reactive flows; the non-linear chemical source term, and the unclosed Reynolds stress tensor.


19

As mentioned in Section 2.5.1, a model needs to be posed, and the most common way to do this
is to solve a modelled transport equation for the joint composition-velocity PDF. The equation for
the Joint Composition-Velocity Transported Probability Density Function (CV-TPDF) describes the
evolution of the joint PDF through physical, composition, and velocity space, and is show in
Equation 2.18 [3]. For a derivation of this equation see Pope (1985) [4].
𝜌 𝜓
𝜕𝑓
𝜕𝑡
+ 𝜌 𝜓 𝑉!
𝜕𝑓
𝜕𝑥!
+ 𝜌𝑔! −
𝜕 𝑝
𝜕𝑥!
𝜕𝑓
𝜕𝑉!
+
𝜕
𝜕𝜓!
𝜌 𝜓 𝑆! 𝜓 𝑓
=
𝜕
𝜕𝑉!
−
𝜕𝜏!,!
𝜕𝑥!
+
𝜕 𝑝!
𝜕𝑥!
𝑉, 𝜓! 𝑓 +
𝜕
𝜕𝜓
𝜕𝐽!
!
𝜕𝑥!
𝑉, 𝜓! 𝑓
Equation 2.18: Joint Velocity-Composition PDF transport equation.
(2.18)
The solution to this transport equation comes in the form of a PDF, 𝑓(𝑉, 𝜓, 𝑥, 𝑡), which contains
all the information for velocity and composition. From this PDF statistical measures can be extracted
including the mean and variance, by Equations 2.14 and 2.15 respectively.
The advantages and difficulties associated with the TPDF method become clear after examining
Equation 2.18. The terms on the left had side of the equation can be determined exactly (appear in
closed form), and thus do not need to be modelled, while the terms on the right hand side contain
conditional expectations, and thus need some level of modeling [13]. The first two terms on the left
hand side represent transport in physical space and can be interpreted as the change in 𝑓 along a
density weighted streamline [18]. The third and fourth term represent the change in 𝑓 in velocity
space by gravitational forces and the mean pressure gradient, respectively. The fifth term represents
the change in 𝑓 in composition space by chemical reaction.
The first two terms on the right hand side represent transport in velocity spare for 𝑓 due to
viscous stresses and the fluctuating pressure gradient respectively. The final term represents the
change of 𝑓 in composition space by molecular diffusion, and it is this term that is the key modelling
difficulty for TPDF methods.
The ability of the CV-TPDF method to treat as exact the advection (transport of 𝑓 in physical
space), body forces, mean pressure gradient, and variable density effects makes it a very powerful
modelling tool. However, the key advantage of the CV-TPDF method is its ability to determine the
chemical source term exactly as this term constitutes a significant portion of the computational
power used.
The mean chemical source term, 𝑆!(𝜓)𝑓 , can be determined from the chemical source term (as
it is a known function of only scalar quantities [4]) by the following formulation [22]:


20

𝑆!(𝜓)𝑓 = 𝑆! 𝜓 𝑓!" 𝑉, 𝜓; 𝑥, 𝑡 𝑑𝑉𝑑𝜓 = 𝑆! 𝜓 𝑓! 𝜓; 𝑥, 𝑡 𝑑𝜓
(2.19)
If the PDF of the CV-TPDF is restricted to only scalar quantities (composition and enthalpy),
then the computational expense can be reduced. This is the rationale behind the Composition
Transport Probability Density Function (C-TPDF) method, and is derived by integrating Equation
2.18 (the CV-TPDF equation) over velocity space [4].
𝜌 𝜓
𝜕𝑓
𝜕𝑡
+
𝜕
𝜕𝑥!
𝜌 𝜓 𝑉 𝜓! 𝑓 +
𝜕
𝜕𝜓!
𝜌 𝜓 𝑆! 𝜓 𝑓 =
𝜕
𝜕𝜓
𝜕𝐽!
!
𝜕𝑥!
𝜓! 𝑓
Equation 2.20: Joint Composition PDF transport equation.
(2.20)
This method retains the strength of the CV-TPDF formulation in that is can still treat reaction
without approximation. This can be seen in that the reaction term (the third term on the left of
Equation 2.20) has not changed after integration. The composition PDF method does, however,
introduce a difficulty - as it contains no information on velocity. This information on velocity must
be introduced in order to achieve mathematical closure. Often this information can be obtained by
introducing additional transport equations, to model the effect of the velocity flow field [13], with
the 𝑘 − 𝜖 turbulence model being the most widely used.
In order to minimize error sources, this paper will obtain velocity information from the DNS. The
Scalar dissipation, turbulent diffusivity, and the mean drift term fields will be extracted from the
DNS in order to further reduce modelling errors. By reducing the errors sources, any errors in the
simulation can be solely attributed to the mixing models. The reliance on either computationally
expensive DNS data or inferior 𝑘 − 𝜖 models [4] for velocity information in the C-TPDF method
reduces its flexibility, its relative simplicity has motivated its use in other studies [8, 26].
2.5.3 Solution Procedure - Monte Carlo Methods
The major advantage of the TPDF method is in closing the mean chemical source term, but in order
to utilize this, different solution procedures are required compared to traditional CFD techniques.
New formulations are needed due to the high dimensionality of the joint composition PDF, which is
equal to the total number of scalar variables in the PDF – the number of chemical species, and
enthalpy. Typical CFD simulations use Finite Element Analysis (FEA) or Finite Difference
Algorithms (FDA) in order to solve the transport equations. Finite difference methods work very


21

poorly for solving PDFs, as the computational expense rises exponentially with the dimension of the
PDF [4].
Pope (1985) [4] outlined a Monte Carlo method, which can be used to solve the PDF transport
equation (Equation 2.20), which scales linearly with the dimension of the PDF. This is the best that
can be achieved currently, and is much less computationally expensive than the Finite Difference
equivalent, making Monte Carlo methods the preferred solution method for TPDF methods. Figure
2.8 shows a comparison between a Finite Difference Algorithm and a Monte Carlo method for a two-
dimensional methane jet. The two methods show little difference to one another, thus showing that
Monte Carlo methods provide a tractable way to solve PDF transport equations.
A commonly used solution method is to use a hybrid mesh-particle domain. In such a system, the
domain is discretized into grids, and a nominal number of particles are distributed within the domain,
with each particle uniquely residing in one cell. Each of these nominal particles represent a zero-
dimensional point within the flow, which sample the local fluid properties at that point [20]. The C-
TPDF equations are solved for each particle, and each particle is free to move around the domain
when acted upon by various physical phenomena. This freedom allows for transport in physical
space due to the mean and turbulent flow fields, and transport in composition space due to chemical
reaction and mixing. Particles are permitted to cross cell boundaries, and even exit the domain
(depending on boundary conditions) [21]. The particles are evolved through time using both a
stochastic process, where their properties are altered by sampling from a Gaussian distribution, and a
deterministic process, by application of the transport equations. By using a large sample of particles,
the mean and variances can be estimated and the joint PDF of composition can be formed as a
solution to Equation 2.20 [3].
A useful property of Monte Carlo methods is their scalability on multi-node/processor systems.
With particle partitioning Monte Carlo methods exhibit excellent scalability on distributed
computing platforms (ie, cluster computers) [3]. For all the advantages mentioned, Monte Carlo
methods have become the standard solution procedure for TPDF methods [13].


22

Figure 2.8: Mixture fraction of a piloted jet methane flame for Finite Difference and Monte Carlo solution
methods [19].
2.6 Mixing Models
2.6.1 Role of mixing models
As mentioned in Section 2.5.2 the molecular diffusion term in Equation 2.20 is not closed, therefore
a model for this term is required in order to use TPDF methods. These models are known as “mixing
models”. The difficulty with accurately modelling the diffusion term is the coupling between several
different phenomena; turbulence, chemistry and diffusion [13]. Chemical reaction greatly affects the
concentrations (and concentration gradients) within a flow, thus influencing the molecular diffusion
[13]. Similarly, turbulence strongly influences molecular diffusion, as the transport via turbulent
fluctuations affects the local concentrations of burned and unburned fuel in a premixed combustion
system [13]. This coupling means that accurately modelling the diffusion term is a non-trivial
problem.


23

Mixing models are responsible for the exchange of compositional information between particles
within the fluid domain. The mixing models to be evaluated in this study are the IEM [6], MC [7]
and EMST [8] models.
2.6.2 Criteria for mixing models
Mixing models are an active area of research with many new models being created, tested and
validated. In order to evaluate the merit of these models on a theoretical level there needs to be a set
of criteria that determines how valid the mixing model is when applied to physical phenomena. A
condensed version of Subramaniam and Pope’s (1998) [8] extensive list of important characteristics
will be addressed here.
1. Decay of Variance:
The mixing model should cause the variance of scalar quantities to decrease with time, this means
that over time the composition of each fluid particle will approach the mean composition. That is, the
mixing model tends to create a homogeneous mixture [13].
2. Preservation of the Mean:
The mixing model only exchanges composition information between particles, therefore there can be
no loss or gain of scalar quantities. This means the mixing model only redistributes the scalar
variables and therefore the mean composition of the fluid cannot be altered [13].
3. Boundedness:
The mixing model must maintain physical compositions when particles interact, for instance
negative mass fractions must not be permitted. Additional constraints must also be enforced such as
the sum of the mass fractions must equal one, as in Equations 2.21 and 2.22. This criterion is one of
the most important in order to ensure consistency in particle compositions [13].
0 ≥ 𝜙! ≥ 1 (2.21)
𝜙! = 1 (2.22)
4. Locality in Composition Space:
This criterion restricts the mixing of particles so that only particles of similar chemical composition
can mix. This restriction is to reduce non-physical mixing phenomena [13].


24

One example of a non-physical mixing event in premixed flames is mixing across the reaction
zone. Particles cannot cross the reaction zone without their composition changing therefore it would
create a non-physical mixing pair if the mixing model allowed this to happen. If the reaction zone
bisected a single computational cell, the mixing models may create mixing pairs across the reaction
zone creating non-physical compositions [13].
While it is not a necessity to satisfy all four conditions, it is desirable as this should produce
results that more closely resemble physical reality [13]. The EMST model satisfies all four criteria
while the IEM and MC models do not satisfy the Locality in Composition Space criterion. In the
following sections each of the mixing models will be described in detail.
2.6.3 Interaction by Exchange with the Mean (IEM)
The Interaction by Exchange with the Mean is the simplest mixing model of the three [26], originally
proposed by Villermaux and Devillon (1972) [6]. It involves particles changing their composition by
interacting with the mean composition. The model advances particle composition using Equation
2.23.
𝜙 𝑡 + 𝑡!"#$ = 𝜙 𝑡 + 𝑐 𝜙 𝑡 − 𝜙 𝑡 (2.23)
Where c is evaluated by Equation 2.24.
𝑐 = 1 − exp −𝛺! 𝑡!"#$ (2.24)
The model pulls the particles composition, 𝜙 𝑡 , towards the mean composition, 𝜙 𝑡 . The
mean composition is usually taken as the mean composition within a single computational cell. The

IEM model satisfies the first three criteria but fails to satisfy the final one. Due to this, the IEM is
fundamentally flawed, as there are no restrictions on non-physical mixing. Despite its
shortcomings, the IEM model has remained a common mixing model in TPDF methods mainly due
to its relative simplicity.
2.6.4 Modified Curl (MC)


25

The Modified Curl’s mixing model is a minor improvement on the IEM mixing model. While in the
IEM mixing model particles interacted with the arithmetic mean composition of a cell, in this model
particles are chosen at random to interact with each other. First the two particles are chosen to
interact, and then these particles exchange information regarding their scalar quantities.
For each timestep, a certain number of particles, 𝑁!"#,!, forms a set of particles from the total
particles, 𝑁!"#$%&'(),!, in each computational cell, 𝑘. The amount of particles that mix per cell is
given by Equation 2.25.
𝑁!"#,! = 2 × 1 + 1.5 𝑡!"#$/2 𝛺!,! 𝑡 𝑁!"#$%&'(),!∆𝑡 (2.25)
Where 𝛺!,! 𝑡 is the cell centre mixing frequency. From this equation it can be seen that for
higher values of 𝛺!,! 𝑡 , more particles will be mixed together, mirroring physical reality. Two
certain particles from the 𝑁!"#,! set are then selected to mix with one another. For particles 𝑖 and 𝑗 of
the set 𝑁!"#,! the particle compositions are advanced according to Equations 2.26 and 2.27, where ℎ
is a random number between 0 and 1.
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
𝑡 +
1
2
ℎ 𝜙!
!
𝑡 − 𝜙!
!
𝑡
(2.26)
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
𝑡 +
1
2
ℎ 𝜙!
!
𝑡 − 𝜙!
!
𝑡
(2.27)
Similarly to the IEM mixing model, the MC mixing model satisfies the first three mixing model
criteria but fails to satisfy the final one. The pairwise exchange of particles scalar information is a
more realistic method of mixing, than the IEMs. The lack of enforcement of Locality in Composition
space still means that this model has no ability to avoid non-physical mixing pairs.
2.6.5 Euclidian Minimum Spanning Tree (EMST)
Subramaniam and Pope (1998) [8] developed a new mixing model to address the deficiencies
of other mixing models, and most importantly, to develop a model that would satisfy the
fourth criteria – Locality in Composition Space.
The basis of this model is that particles that are similar composition are more likely to mix
than those that are of vastly different compositions [8]. This is a way to ensure that non-
physical mixing pairs are created. From looking at scalar quantities, such as temperature, it


26

can be seen that they vary smoothing, without any sharp discontinuities. This smooth
transition of scalar quantities justifies the idea that particles of similar composition are more
likely to mix with one another [21].
A tree is a mathematical construct which connects a set of elements to all other elements
by forming one or more branches between them, without forming a circuit. A Euclidian
Minimum Spanning Tree (EMST) joins pairs of particles with branches, such that the sum
length of all the branches is a minimum. In the EMST model, the branch distance is measured
in composition space, rather than physical space [13]. For example, assume there are three
particles, A, B, and C, with compositions 2, 5, and 1 respectively, where A and B are 1m
apart, B and C are 2m apart, and C and A are 3m apart. In composition space A and C are the
closest, even though in physical space they are the furthest apart. Once the tree is formed,
pairs of particles joined by a branch exchange information. This formulation means that
particles will not be allowed to mix if separated by large composition spaces, but small
physical spaces, thus non-physical mixing pairs will not be created. An example of an EMST
for two composition variables is shown in Figure 2.9
Figure 2.9: Euclidian minimum spanning tree for two composition variables, with intermittency [8]. The tree
joins all variables in the mixing state (black).
Similarly to the MC model, a subset of the total particles is selected to be involved in the
mixing. This is particularly important in the EMST model, as it stops the effect of “stranding”.


27

“Stranding” occurs after multiple particle interactions if one set of particles is used to form the
EMST. When the first EMST is formed, each pair of particles that interact will have similar
composition. As the EMST is reformed and the compositions are exchanged multiple times,
the compositions converge - not to a mean value, but to a mean network. This mean network
is not a good representation of physical mixing, and thus needs to be avoided. Figure 2.10
shows the effect of “stranding”.
Figure 2.10: An example of “stranding”: scatter plot of particles in two-dimensional composition space with
an EMST superimposed [8]
The feature that inhibits stranding is called an intermittency feature. In the EMST model
developed by Subramaniam and Pope (1998) [8] a particle can either be in the mixing state, or
in the non-mixing state. When it is in the mixing state, its composition is altered by mixing
with other particles, and when it is in the non-mixing state, it does not mix with particles and
thus mixing has no effect on its composition. A non-dimensional age value is applied to each
particle to determine the duration that the particle resides in either state.
When the particle’s age is positive, it is considered in the mixing state, and when it is
negative it is in the non-mixing state. If the particles age is positive, it decays linearly down to
zero, then jumps discontinuously to a randomly assigned negative age value. The particles age
then decays towards zero, and then jumps discontinuously to a randomly assigned positive age
value. This process repeats for all particles. Figure 2.11 shows this process for one particle.


28

Figure 2.11: Temporal evolution of the age variable [8].
Particle information is exchanged in a similar manner to the MC method. For particles 𝑖
and 𝑗 Equation 2.28 and 2.29 are used to exchange composition information.
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
𝑡 + 𝑏𝐵! 𝛿𝑡 𝜙!
!
𝑡 − 𝜙!
!
𝑡 (2.28)
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
!
𝑡 − 𝜙!
!
𝑡 (2.29)
In these equations 𝑏 characterizes the mixing frequency of the system [31], and 𝐵!
represents the weight of each branch, where the weight of the branch is determined by its
distance from the centre of the tree. Thus branches at the very centre will have the largest
weights [13].


29

Chapter 3
Modelling Methods
The code used, which shall be called “The TPDF code”, is a Fortran-90 program designed to use
Monte Carlo methods (as outlined in Section 2.5.3) to simulate reacting flows. This program has the
flexibility to be run in parallel or series modes to enable a reduction in the computational time
required. The Message Passing Interface (MPI) implementation OpenMPI was used to allow the
code to run in parallel.
Two computer-cluster resources provided by The University of New South Wales Faculty of
Engineering were made available for this study. Leonardi, a medium-sized High Performance
Computing cluster, and Fyr. Only one node was used per simulation in order to avoid the bottleneck
of requiring large amounts of data to be moved over slower node interconnects.
The TPDF code takes several inputs from the DNS simulation in order to direct the source of
error solely to the mixing models. For the initial conditions the complete thermo-chemical
description was used, with chemical mass fractions for each species involved 𝑘 (𝑌!!
), temperature
(𝑇!), and density (𝜌). As mentioned previously in Section 2.5.2, the C-TPDF formulation requires
input for the mean and turbulent velocity fields, thus it takes the following inputs for all time steps of
the simulation: mixing frequency (𝛺!), the turbulent diffusivity (𝛤!), and the one-dimensional mean
velocity plus mean drift term (𝑉 + 1/𝜌 𝛻(𝜌𝛤!)). Each of these terms will be fully defined in
Section 3.3.
The TPDF code finds a PDF (𝑓) that satisfies the C-TPDF transport equation (Equation 2.20). In
order to do this the TPDF code solves Equations 3.1 → 3.2, which describe the coupled phenomena
of mixing, advection and reaction. In reality these three processes occurs simultaneously, however to
solve these equations in practice, a time splitting scheme is used. The time splitting scheme
subdivides each timestep into smaller substeps, and within each substep one of these processes is


30

solved. The details of the splitting scheme will be explained in detail in Section 3.4.
The TPDF code produces binary output data which is exported and analysed in Matlab. The
binary data is processed into insightful information by generating plots that can then be analysed and
interpreted.
3.1 Simulation Scenario
The simulation scenario is based on a large Direct Numerical Simulation performed by Professor
Hemanth Kolla et al. [9]. This simulation takes a high-velocity rectangular jet of unburnt lean
hydrogen–air mixture of equivalence ratio 0.7 flowing in a quiescent fluid of adiabatic burnt
products of the same equivalence ratio. See Figure 3.1 for a schematic diagram of the simulation
scenario. The simulations were run using a program called S3D, developed by Sandia National
Laboratories. This program solves the compressible Navier Stokes equations using high order
numerical methods. For a complete description of the program and its performance see [17].
Figure 3.1: Schematic of DNS configuration [9].
The simulations run by Professor Kolla consisted of three cases, the Damköhler number was


31

varied between the cases while maintaining a constant Reynolds number of 𝑅𝑒 = 10,000. The cases
‘Da-’, ‘Baseline’, and ‘Da+’ correspond to Damköhler numbers 0.13, 0.27, and 0.54 respectively. As
mentioned in Section 2.3.3, the higher the Damhöhler number of a premixed flame, the closer it is to
the laminar flame regime. As different methods are used to model laminar flames, it is expected that
the TPDF code will not predict the ‘Da+’ case very well. Due to constraints on time and computing
resource this study will only concern itself with case ‘Da-’ (𝐷𝑎 = 0.13).
The set up of these simulations allows all the random variables to be ensemble averaged in both
the 𝑥 and 𝑧 directions since the probabilities do not vary as you move in either of these directions.
This means there is only a statistical dependence on the 𝑦 direction and time, 𝑡. This flow is said to
be statistically one-dimensional, since the probabilities do not depend on the position in both the 𝑥
and 𝑧 directions [4].
3.2 Modelling the C-TPDF transport equations
Solving the C-TPDF transport equation reduces to solving two equations with represent each
notional particles transport in physical and composition space. Equation 3.1 represents the transport
in physical space, while Equation 3.2 represents the transport in composition space.
𝑑𝑥 𝑡 = 𝑈 +
𝛻𝜌𝛤!
𝜌
𝑡!"#$ + 2𝛤!/𝑡!"#$ 𝑑𝑊
(3.1)
𝑑𝜙 𝑡 = 𝑀 𝑡!"#$ + 𝑆 𝑡!"#$ (3.2)
Equation 3.1 is composed of a deterministic part and a stochastic part. The first part of the
equation is the deterministic part, and as this represents the particles movement by the bulk flow of
the fluid there is no fluctuating component. The second part of the equation is the stochastic part and
is described as a Weiner process, which is analogous to Brownian motion. The first part of Equation
3.2 represents the particles evolution through composition space due to the effects of molecular
mixing. This is where the mixing models are implemented into the solution method. The second part
of Equation 3.2 represents the particles evolution through composition space due to the effects of
chemical reaction. 𝑆 is made up of a set of Ordinary Differential Equation’s (ODE) which relate
the chemical production rates to the species mass fractions. These ODE’s are the same ones that have
been used in the DNS, which involves nine species and 19 elementary reactions. The ODE’s have
been verified by Li et al. [25].


32

These equations represent the phenomena of mixing, advection, and reaction. Due to coupling,
these mechanisms are difficult to solve as they influence each other closely. A splitting scheme is
necessary to solve them separately, which involves splitting up each timestep into substeps, where
each phenomena can be evaluated in isolation. The splitting scheme is called a Strang splitting
scheme and is denoted TMRMT [26].
The scheme is symmetric, where T represents transport in physical space (Equation 3.1), M
represents molecular mixing (Equation 3.2), which are both evaluated twice, and R is reaction
(Equation 3.2), which is only evaluated once. By only evaluating the reaction once per timestep the
computational expense is greatly reduced, as solving the chemical source term usually dominates the
total computational expenditure [17].
3.3 Pre-processing
The DNS data, including the conditional velocity term, mixing frequency, turbulent diffusivity and
initial conditions for temperature, density and species mass fractions, first must be processed to be
used in the C-TPDF code. The C-TPDF code that is used in this study only requires statistically one-
dimensional data, and as the DNS simulation was run in three spatial dimensions, the data must be
averaged and collapsed from three dimensions into one. This process creates some statistical noise,
thus, this averaged data must then be smoothed. The smooth data is then interpolated onto a fine grid
to remove future interpolation errors. Both the smoothing and interpolation is completed using
Matlab’s 𝑐𝑠𝑎𝑝𝑠 function, which produces a cubic spline 𝑓. The future interpolation errors are those
that arise when interpolating the cubic spline onto the PDF grid. By using a very fine grid, this
reduces the interpolation error. The DNS inputs of mixing frequency, turbulent diffusivity, and
velocity/mean drift term will be defined below.
Mixing Frequency
The mixing frequency is a way of quantifying the rate of mixing, and as all mixing models depend
on this value, its accurate modelling is central to TPDF methods. Due to the availability of the DNS
data set, the mixing frequency will be calculated by taking the ratio of the Favre averaged (density
weighted) scalar dissipation 𝜒!!to the scalar variance 𝜙!!!
, as shown in Equation 3.3. Figure 3.2
represents the values of the mixing frequency throughout the computational domain.


33

𝛺! =
𝜒′′
𝜙′′2
(3.3)
Figure 3.2: Mixing Frequency input for the DNS data set
Turbulent Diffusivity
The turbulent diffusivity (𝛤!) is a parameter used to account for the effects that turbulence has on the
fluid transport equation. The effects of molecular diffusion become less significant compared to the
transport caused by turbulence. From Equation 3.1, 𝛤! is an important quantity for the stochastic
element of the modeled C-TPDF transport equation. Figure 3.3 represents the values of 𝛤! throughout
the domain of the simulation.
Mean Drift Term
The first part of the particles spatial evolution (Equation 3.1) is the part controlled by a deterministic
process. This part is composed of the mean drift term and the velocity, 𝑈. The mean drift term
!!!!
!
is
evaluated by calculating the gradients of 𝛤!. Figure 3.4 represents the values of the mean drift term
throughout the computational domain.


34

Figure 3.3: Turbulent diffusivity input for the DNS data set
Figure 3.4: Mean drift term input for the DNS data set


35

3.4 Implementing the splitting scheme
The C-TPDF method is solved using the TMRMT Strang Splitting Scheme. T, the physical transport,
R, chemical reaction, and M, molecular mixing, are all handled within a Fortran-90 code, provided
by A/Prof. Evatt Hawkes.
3.4.1 Transport (T)
The transport of particles is evaluated every half timestep - once at the beginning, and once at the
end. From Equation 3.1 (reproduced below), it can be seen that there are three parameters that need
to be evaluated for the transport in physical space to be calculated.
𝑑𝑥 𝑡 = 𝑈 +
𝛻𝜌𝛤!
𝜌
𝑡!"#$ + 2𝛤!/𝑡!"#$ 𝑑𝑊
The steps for evaluating transport through physical space are given below:
1. Evaluate the deterministic component (𝑈!)
The DNS data, 𝑈 +
!!!!
!
, is interpolated onto each particle location within the computational
domain.
2. Evaluate the Stochastic component (𝑈!)
The scaling factor, 2𝛤!/𝑡!"#$, is interpolated onto each particle location within the computational
domain. The stochastic velocity is then calculated by multiplying this scaling factor by 𝑑𝑊, which is
a random number taken by sampling from a Gaussian distribution.
3. Update particle locations
Using the velocity calculated in steps 1 and 2, the particles location is updated by a simple forward
Euler finite difference scheme, as shown in Equation 3.4.


36

𝑥 𝑡 + 𝑡!"#$ = 𝑥 𝑡 +
𝑡!"#$
2
𝑈! + 𝑈!
(3.4)
As 𝑥 𝑡 is updated twice per timestep, the half timestep is used to calculate the value each time, thus
the effective equation per whole timestep is shown in Equation 3.5:
𝑥 𝑡 + 𝑡!"#$ = 𝑥 𝑡 +
𝑡 𝑠𝑡𝑒𝑝
2
𝑈 𝐷1
+ 𝑈 𝑠1
+
𝑡 𝑠𝑡𝑒𝑝
2
𝑈 𝐷2
+ 𝑈 𝑠2
(3.5)
Where subscripts 1 and 2 represent the first and second half timestep. In practice Equation 3.4 is
used, while Equation 3.5 is given for the readers conceptual understanding.
4. Pass particles between processors
As the TPDF code is run on a multi-processor system, different processors handle different
computational cells. If a particles updated position coincides with a portion of the computational
domain that is computed on another processor, then the particles information needs to be passed to
that processor. This is completed using the OpenMPI software package.
3.4.2 Mixing (M)
Similar to the method of computing transport, mixing is also evaluated every half timestep. From
Equation 3.2 (reproduced below), it can be seen that mixing is one of the two parameters that need to
be evaluated for transport in composition space to be calculated.
𝑑𝜙 𝑡 = 𝑀 𝑡!"#$ + 𝑆 𝑡!"#$ (3.2)
As there are three mixing models that are being evaluated, the mixing model selected by the code
is chosen based on a user defined input file. The steps for evaluating mixing, by each mixing model,
are given below.
IEM
The IEM mixing model is a simple model that mixes particles with the arithmetic mean composition
of each computational cell.
The process by which the model is implemented into the TPDF code is shown below.


37

1. Interpolate the c scaling factor
First the DNS data for the mixing frequency is interpolated onto each particle location. Then using
Equation 2.24, the c scaling factor is computed for each particle
2. Evaluate the local mean
The local mean is computed by taking the arithmetic mean of the composition of each particle within
each computational cell.
3. Mix particles with the cell mean
Using Equation 2.23, reproduced below, the new composition of each particle is calculated by
mixing it with the local mean.
𝜙 𝑡 + 𝑡!"#$ = 𝜙 𝑡 + 𝑐 𝜙 𝑡 − 𝜙 𝑡
MC
The MC is another simple model that mixes particles by exchanging their particle information by
creating pairs within a computational cell.
1. Calculate the Cell Centre Mixing Frequency
The DNS mixing frequency is calculated then interpolated onto the centre of each cell using bilinear
interpolation.
2. Calculate cell mixing numbers
Using Equation 2.25, reproduced below, the cell mixing number for each cell is calculated
𝑁!"#,! = 2 × 1 + 1.5 𝑡!"#$/2 𝛺!,! 𝑡 𝑁!"#$%&'(),!∆𝑡
This function ensures that at least two particles are selected each iteration, and that an even
number of particles is selected to mix. The brackets on the left are evaluated and then converted to an
integer before being multiplied by two to ensure that 𝑁!"#,! is an integer.
3. Calculate the mixing probability
The probability that a mixing pair in cell 𝑘 will undergo the mixing process (𝑃!"#,!) is evaluated by
Equation 3.2.


38

𝑃!"#,! =
1.5× 𝑡!"#$/2 𝛺!,! 𝑁!"#$%&'(),!
𝑁!"#,!
(3.2)
4. Create mixing pairs
Mixing pairs for cell 𝑘 are created by selecting two particles from the total 𝑁!"#,! particles by way
of a uniform random number generator: if 𝑃!"#,! is greater than the random number selected, then
the particle is selected. The random number generator used is an inbuilt function of the Fortran-90
language. Pairs are created for all of the 𝑁!"#,! particles, with the restriction that particles may not
mix with themselves.
5. Mix particle pairs
Once the mixing pairs have been formed, the species mass fractions and enthalpies between the
paired particles 𝑖 and 𝑗 are mixed by Equations 2.26 and 2.27, reproduced below.
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
𝑡 +
1
2
ℎ 𝜙!
!
𝑡 − 𝜙!
!
𝑡
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
𝑡 +
1
2
ℎ 𝜙!
!
𝑡 − 𝜙!
!
𝑡

Where
ℎ is a random number between 0 and 1 that is determined by sampling from a uniform
distribution on the interval [0,1]. The sampling process uses a random number generator that is an
inbuilt function of the Fortran-90 language
EMST
The EMST mixing models is a complex particle interaction model that restricts the mixing pairs by
enforcing locality in composition space.
1. Calculate the Cell Centre Mixing Frequency
Similarly to the MC, the cell centre mixing frequency (𝛺!,!) is calculated then interpolated from the
DNS onto each cell centre using bilinear interpolation.
2. Determine particle age
The particles age (𝜉) is a representation of its status of whether it is in the mixing state or whether it
is in the non-mixing state. If its age is positive then it is in the mixing state, and its composition will


39

be altered once it is mixed with another particle. If its age is negative, then it is in the non-mixing
state, and it will not be included in any mixing pairing, nor will its composition will be altered
through mixing. Each particle in the cell has an age randomly assigned to it.
The age of particles in the mixing state linearly decay towards zero, whereby it is assigned a
random negative age. If a particle is in the non-mixing state, its age decays towards zero, and it is
assigned a random positive age. The age of a particle cannot be altered outside the mixing step,
therefore each particles age will have to be stored for the next mixing substep.
3. Form the Euclidian Minimum Spanning Tree (EMST) in Composition Space
The EMST is formed from all the particles in the mixing stage (𝜉 > 0). The EMST is based on
forming a tree whose branches are of the minimum distance in composition space, which means the
particles of close composition will be linked together by branches of the tree.
4. Mix particles along the branches of the EMST
Mixing occurs over each branch 𝐵! of the EMST. Each branch joins two particles, for particles 𝑖 and
𝑗, their compositions will interact by Equations 2.28 and 2.29, reproduced below.
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
!
𝑡 − 𝜙!
!
𝑡
𝜙!
!
𝑡 + 𝛿𝑡 = 𝜙!
!
!
𝑡 − 𝜙!
!
𝑡
3.4.3 Reaction (R)
The reaction time step is the only substep that occurs once per step. This is due to the large amount
of computational time required to calculate the chemical source term. By only calculating this step
once the computational time required is greatly reduced. In this study the chemical mechanism is
identical to the DNS, which includes 9 chemical species and 19 elementary chemical reactions. By
having the same chemical mechanicism, the source of error can be more precisely located. As well as
this, the chemical mechanism has already been verified by
Li et al. [25]. The chemical species that
are accounted for are 𝐻!, 𝐻, 𝑂, 𝑂!, 𝑂𝐻, 𝐻! 𝑂, 𝐻𝑂!, 𝐻! 𝑂!, and the inert species 𝑁!,. For a full
description of the mechanism used in this study see Li et al. (2004) [25].
The system of ODEs that govern the chemical kinetics are solved using a 6th stage, 4th order
Runge-Kutta Method [32]. An iterative procedure using Newton’s Method is also performed using
the updated composition to calculate the updated temperature of the system [13].


40

Chapter 4
Results and Discussion
This chapter will present the results of the TPDF simulations undertaken in this study. The data
presented is cell averaged, with each data point representing the arithmetic mean of a cell in the
simulation. The mean of each variable is taken over each of the particles (𝑁!"#$%&'(),!) within a cell.
This method creates a discontinuous data set due to the piecewise averaging process of the smooth
particle data. Methods for producing smooth cell interpolated are available, but are considered
unnecessary for this study [13]. Unless otherwise stated the parameters for each simulation are as
follows; 𝑁!" = 1000, 𝑁!"## = 320, 𝑡!"#$ = 1 × 10!!
[𝑠].
This study will use the scalar quantities of 𝑁! mass fraction, temperature, 𝑂𝐻 mass fraction, and
𝐻! 𝑂! mass fraction in order to assess the performance of the mixing models against the DNS. 𝑁!
mass fraction (𝑌!!
) is a good quantity to assess the physical transport model and it can be used to
track the mixing in the system as it is an inert species and thus traces the net movement of particles
through physical space. 𝑌!!
is unaffected by chemical reaction (apart from in some high temperature
applications) [24], thus it can be accurately predicted even if there are incorrect predictions for the
temperature and other phenomena.
Temperature is used to characterize the thermochemical behavior of the flow. It is a non-
conserved scalar property as it depends on the properties of the flow, the properties of mixing, and
chemical reaction. Thus, temperature can be used to assess the effects of mixing on the properties of
the flow.
𝑂𝐻 mass fraction (𝑌!") is a good quantity to assess the timescales of the system, as this radical
species exists for a very short time during one chemical reaction stage of the combustion process. If
the 𝑌!" is predicted accurately, then it is reasonable to say the timescales of the system are accurate.


41

𝐻! 𝑂! mass fraction (𝑌!!!!
) is a good quantity to assess the timescales of the system for similar
reasons.
In order for the magnitudes of the results to be meaningful, the spatial coordinates are normalized
over the jet height (𝐻), taken from the DNS, of value 2.7 × 10!!
[𝑚] [9]. Thus, the distance
variable, y, is divided by the jet height, H. Any point of interest can be described by some multiple of
jet heights away from the origin. The computational domain size is 5.4 × 10!!
[𝑚], and as this is
symmetric about the transverse midplane, only half of this needs to be computed. Thus, our
computational domain size is 2.7 × 10!!
[𝑚], yielding a maximum normalized domain size of 10 𝐻.
Due to the nature of the simulation, all variables reach near constant values after 5 𝐻, thus no
additional information is gained by plotting past here. To assist in clarity, and size constraints, the
figures have been clipped at 5 𝐻. The time component of the simulation is normalized over the jet
time (𝑡!), with a value of 8.64 × 10!!
[𝑠]. Jet time is defined by 𝑡! = 𝐻/𝑈!, where H is the jet
height, and 𝑈! is the peak velocity of the jet, thus 𝑡! can be understood as the time taken for one
particle to pass from one side of the domain to the other, if travelling at the peak jet velocity. Times
of interest are described in multiples of 𝑡! with the duration of the simulation 2.36 × 10!!
[𝑠],
yielding a maximum normalized time of 27.35 𝑡! [9].
4.1 Mean and Variance
The mixing models will be evaluated based on how close they come to the mean spatial profiles and
the RMS (root mean square) profiles of 𝑌!!
, temperature, and the radical 𝑌!" of the DNS. The spatial
profiles will be plotted at 14.33 𝑡!, and at 23.15 𝑡!. For clarity and size constraints the figures show
𝑡! rounded to zero decimal places.
The mean profiles characterize how capable the simulation is at capturing the macroscopic
effects. In order to characterize how capable the simulation can capture the microscopic effects
another variable is necessary.
RMS values measure the variance of the flow, and thus are capable of capturing the microscopic
events. RMS values are affected by physical transport (velocity and turbulent diffusion) as well as
scalar dissipation, which thus characterize the microscopic events of the flow [13]. Hence, the RMS
values can be used to assess the mixing model and molecular diffusion effects. By studying the mean
and RMS spatial profiles a quantitative measure of the performance of each mixing model can be
evaluated.


42

4.1.1 N2 Mass Fraction
𝑌!!
is a good parameter to test the physical transport model, and it can be used to track the mixing in
the system. The mean 𝑌!!
profiles are predicted with a similar level of accuracy by all the models at
14 𝑗! as shown in Figure 4.1. All three models over predict from 0.5 → 1.1 𝐻, then under predict
from 1.1 → 2 jet heights. After 2 𝐻 all models recover, and predict the mean profile accurately.
While IEM and MC predict smaller values at 1.2 𝐻, EMST comes closer to predicting the profile.
The three mixing models perform almost identically at 23 𝑗!, over predicting from 0 → 2.2 𝐻, under
predicting from 2.2 → 3.9 𝐻, and recovering from 3.9 𝐻 onwards. Again, EMST predicts the 𝑌!!
profile with a noticeable improvement at 3 𝐻.
The lower part of Figure 4.1 shows that the RMS 𝑌!!
profiles of the mixing models vary
significantly from the DNS, but not from one another, with the EMST mixing model producing the
results farthest from the DNS for both 14 and 23 𝑗!. At 14 𝑗! for distances less than 2.5 𝐻, all three
models under predict the RMS profile significantly, but recover from 2.5 𝐻 onwards. At 23 𝑗!, all
models under predict the RMS profile again, recovering soon after 4 𝐻.
4.1.2 Temperature
Temperature characterizes the thermo-chemical behavior of the flow [13]. As it is affected by the
coupled interactions of mixing, transport, and reaction it will reveal differences in the predictive
ability of the mixing models. All three models perform with varying degrees of success with
predicting the temperature profiles. An overall qualitative representation of the temperature is shown
in Figure 4.2. From this figure, it can be seen that all the models capture most of the features of the
temperature profile, albeit differences in timing, and location. MC and EMST are able to capture the
jet spreading with high accuracy, while IEM fail to capture the jet spreading and results in an overall
lower temperature. It is concluded that the MC and EMST mixing models perform noticeably better
in terms of qualitative temperature predictions.


43

Figure 4.1: Mean and RMS spatial N2 mass fraction profile comparison. The left column is for 14 𝑡!,
while the column on the right is for 23 𝑡!. The top row shows mean temperature, while the bottom row
shows RMS temperature. Each simulation was run using 𝑁!"## = 320, 𝑁!" = 1000, and 𝑡!"#$ =
1×10!!
Figure 4.2: Mean spatial temperature profile comparison. Each simulation was run using 𝑁!"## =
320, 𝑁!" = 1000, and 𝑡!"#$ = 1×10!!


44

Figure 4.3 shows a quantitative measure of the temperature prediction performance, mean and
RMS temperature profiles at 14 and 23 𝑡!. The upper portion of this figure clearly illustrates that
EMST predict the mean temperature profiles for all jet times significantly better than IEM and MC.
Figure 4.3 also clearly shows that MC comes closer than IEM at predicting the temperature profiles,
while both still under predict the profiles at 23 𝑡!.
The lower part of Figure 4.3 shows the RMS temperature profiles for the four mixing models at
four different times. This figure shows that for 14 and 27 𝑡! the RMS profile is predicted noticeably
better than for the 17 and 23 𝑡!. Overall it can be seen that the IEM mixing models predicts the RMS
temperature profile with the lowest accuracy, albeit no mixing model performs significantly better
than all others.
One possible explanation for the IEM and MC’s inability to predict the temperature profiles
accurately may be due to non-local mixing. As the flame propagates, the unburnt gas first needs to
warm in order to ignite. From a Lagrangian frame of reference this would mean that the hot particles
must move to colder regions. As this process isn't instantaneous, initially only a few hot particles will
migrate over. These hot particles mix normally with cooler particles and warm them slightly. After
the first few, another few hot particles might migrate. In physical reality, these hot particles should
then mix with the particles that are the hottest, rather than the cooler particles. In the IEM and MC
mixing models, this won’t happen, as there is no enforcement of the Locality in Composition Space
property. Instead, the enthalpy of these hot particles will spread out over the whole cell producing
warm particles don’t have sufficient energy to ignite. In the EMST mixing model, the Locality in
Composition Space is enforced, thus we see a highly accurate temperature profile.


45

Figure 4.3: Mean and RMS spatial temperature profile comparison. The left column is for 14 𝑡!,
while the column on the right is for 23 𝑡!. The top row shows mean temperature, while the bottom row
1×10!!
4.1.3 Radical OH Species
Plotting a radical species, such as OH, reveals information about the timescales in the flame, as
these species should exist for very small durations, for they exist only as an intermediate species in
the combustion process. If the timescales on the radical species are correct, it confirms that the
timescales for all processes are correct, especially the mixing time scale.
Figure 4.4 presents the mean and RMS 𝑌!" profiles. The mixing models mean profiles are
predicted with a high level of accuracy for 14 𝑡!. Thus the timescales are captured and predicted
well for all the mixing models for the first portion of the simulation. At 23 𝑡! the IEM mixing
model fails to predict the 𝑌!" profile for values lower than 1.5 𝐻, while the other mixing models
are able to predict the 𝑌!" profile with a high level of accuracy. The lower portion of Figure 4.4
shows that the RMS 𝑌!" profiles are predicted with a moderate level of accuracy at 14 𝑡!. At 23 𝑡!
all three mixing models fails to predict the 𝑌!" profile for values lower than 1.5 𝐻, while they all
are able to predict its profile after 1.5 𝐻 with a moderate level of accuracy.


46

From the spatial temperature profiles, the EMST mixing model significantly outperforms the
other models. This is likely because of the enforcement of the Locality in Composition Space
property. The flame sheet in premixed combustion is thin [22], models without this property will
allow hot and cold particles to mix across the flame creating non physical mixing pairs. It is likely
that these cold particles will not have a high enough temperature to ignite. The IEM or MC models
don’t possess this property and so cannot achieve the correct temperature profile, as this relies on
particles achieving the correct temperature for ignition and combustion to occur [13].

Figure 4.4: Mean and RMS spatial OH radical profile comparison. The left column is for 14 𝑡!, while
the column on the right is for 23 𝑡!. The top row shows mean temperature, while the bottom row
1×10!!
4.2 Conditional PDFs
For further insight into the performance of the mixing models the conditional PDFs as well as the
conditional means can be studied. For this thesis the PDFs of scalar variables 𝑌!", and 𝑌!!!!
conditioned on 𝑐 (progress variable) are considered, where 𝑐 is defined by Equation 2.6.


47

The PDF of variable 𝜃 conditional on variable 𝛩 is found by first calculating the joint PDF of 𝜃
and 𝛩. To do this, take the paired data (𝜃, 𝛩) and place them in bins. An exact number of bins isn’t
necessary, but a sufficient number is required to fully resolve the important features of the PDF
[13].
Once the data has been binned the PDF, 𝑃(𝜃!, 𝛩!), is formed by taking the probability of 𝜃 at
each value of 𝛩 and normalising them such that they integrate to unity at all values of the 𝛩 axis,
𝑛!,! 𝑃(𝜃!, 𝛩!)
!
!
!
! = 1, where 𝑛!,! is the bin width. For each value of 𝛩! the values of 𝜃! are
normalised using 𝜃! = 𝜃!× 1/ 𝑛!,! 𝑃 𝜃!, 𝛩!
!
! , such that 𝑛!,! 𝑃 𝜃!, 𝛩!
!
! = 1 for all 𝑗 in the
domain 𝐽. The function 𝑃(𝜃!, 𝛩!) now represents the conditional PDF, 𝑝 𝜃! 𝛩! [13].
The conditional mean is obtained by further discretizing the 𝛩 space into 𝐽 bins, and then
finding the mean of 𝜃 in each bin. The conditional mean, 𝜃 𝛩 , represents the mean of 𝜃 given that
the conditional variable has taken on the value 𝛩 [13].
4.2.1 OH mass fraction conditioned on progress variable
Figure 4.6 and 4.7 show the conditional PDF and means for 𝑌!" conditioned on 𝑐 at 14 and 23
𝑡!. All three mixing models perform similarly, with thin profiles that over predict between 𝑐 = 0
and 𝑐 = 0.9, and under-predict at values of 𝑐 greater than 0.9. All three profiles are the incorrect
shape, with the 𝑌!" peak value occurring significantly further to the left than the DNS profile
predicts.
The PDF created by the EMST mixing model is both thinner than the DNS, and the IEM and
MC profiles, which is due to the restricted mixing of particles in the EMST model. As the model
only mixes particles that are close in composition space, the variation of possible particle
compositions is thus small. This behavior is to be expected of the EMST mixing model and has
been observed in other studies [23]. At 23 𝑡! (Figure 4.7), the EMST mixing model shows highly
irregular behavior, over predicting the conditional mean by 0.012. It is unclear why the EMST
mixing model has behaved like this.
It can be seen in Figures 4.6 and 4.7 that the variation in the IEM mixing model is larger than
EMST, but smaller than MC. This is due to the IEM mixing model seeking to bring all particles to
the mean composition. In this model, there is no real randomness in the mixing.
The MC mixing model at both 14 and 23 𝑡! (Figure 4.6 and 4.7) shows higher variation in the
𝑌!" profile than the other two mixing models. The MC mixing model produces the most variation,
as it is the least constrained mixing model of the three. Any two particles in a cell can mix, and thus
there is potential to get very large fluctuations from the mean composition.


48

Figure 4.6: PDF of 𝑌!" conditioned on progress variable at 14 𝑡!. Each simulation was run using
𝑁!"## = 320, 𝑁!" = 1000, and 𝑡!"#$ = 1×10!!
Figure 4.7: PDF of 𝑌!" conditioned on progress variable at 23 𝑡!. Each simulation was run using
𝑁!"## = 320, 𝑁!" = 1000, and 𝑡!"#$ = 1×10!!


49

4.2.2 H2O2 mass fraction conditioned on progress variable
The conditional probabilities of 𝑌!!!!
have been included to complement the conditional
probabilities of OH. Figure 4.8 shows the conditional PDF and means for 𝑌!!!!
conditioned on 𝑐 at
14 𝑡!. All three mixing models perform similarly, with thin profiles that over predict between
𝑐 = 0.3 and 𝑐 = 0.7, and under-predict at values of c greater than 0.7. All three profiles for both 𝑡!
are incorrect, with the peak 𝑌!!!!
values occurring significantly further to the left than the DNS
profile.
Similar to the 𝑌!" profile, the PDFs created by the EMST mixing model at both 14 (Figure 4.8)
and 23 𝑡! (Figure 4.9) are the thinnest, followed by the IEM, with the MC profile showing the most
variation. Interestingly, the variation in the MC profile at 23 𝑡! is higher than the variation of 𝑌!!!!
in the DNS at the corresponding 𝑡!.
An interesting feature at 23 𝑡! of both the 𝑌!" profile (Figure 4.7), and the 𝑌!!!!
(Figure 4.9) is
that the largest peak of the EMST profile is shifted the furthest back. In the 𝑌!" profile the peak
occurs at 𝑐 = 0.3 while the other mixing models predict a peak at 𝑐 = 0.3, with the DNS peak
occurring at 𝑐 = 0.9. In the 𝑌!!!!
profile the peak occurs at 𝑐 = 0 while the other mixing models
predict a peak at 𝑐 = 0.5, with the DNS peak occurring at 𝑐 = 0.9. Further analysis is necessary to
find the cause of this phenomenon.
The conditional PDFs and conditional means do not shed much light on the performance of the
models. It can be seen that the PDFs generated by the EMST model do not match the physical
reality. All mixing models PDFs are overly thin and do not vary far from the conditional mean plot.
The MC model tends to mix particles in all regions creating a less defined PDF. The IEM model
performs very similarly to the MC model, apart from the IEM models reduction in variation.


50

Figure 4.8: PDF of 𝑌!! 𝑂2
conditioned on progress variable at 14 𝑡!. Each simulation was run using
𝑁!"## = 320, 𝑁!" = 1000, and 𝑡!"#$ = 1×10!!
Figure 4.9: PDF of 𝑌!! 𝑂2
conditioned on progress variable at 23 𝑡!. Each simulation was run using
𝑁!"## = 320, 𝑁!" = 1000, and 𝑡!"#$ = 1×10!!

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