Developments of Efficient Numerical Methods for Combustion ...€¦ · Combustion modeling with...

Developments of Efficient Numerical Methods for Combustion

Modeling with Detailed Chemical Kinetics

Weiqi Sun

A DISSERTATION

PRESENTED TO THE FACULTY

OF PRINCETON UNIVERSITY

IN CANDIDACY FOR THE DEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FOR ACCEPTANCE BY

THE DEPARTMENT OF

MECHANICAL AND AEROSPACE ENGINEERING

Adviser: Yiguang Ju

June 2020

iii

Abstract

Combustion modeling with detailed chemical kinetics has been challenging for

decades. Even with the availability of supercomputing capability at petascale and beyond,

direct numerical simulation (DNS) with detailed chemical kinetics remains infeasible. The

major difficulty of utilizing detailed chemistry in combustion modeling comes from the

high dimensional, nonlinear, and stiff ODE system which governs chemical reactions.

Additionally, detailed calculations of transport properties at local thermodynamic

conditions also consume significant computation powers and make combustion modeling

with detailed chemistry prohibitive. In this study, advanced numerical methods are

developed to improve the efficiency of combustion modeling from three aspects. (1) A

correlated dynamic adaptive chemistry (CO-DAC) method is developed to provide

on-the-fly chemical reductions with negligible computational costs, so that the ODE

system can be significantly simplified by locally reduced chemical mechanisms. (2) A

correlated dynamic adaptive chemistry and transport (CO-DACT) method is presented. It

performs detailed calculations of the mixture-averaged model in correlated groups

according to similarities of thermodynamic states in phase space. Redundant calculations

of transport properties can be avoided in the CO-DACT method, which improves the

computational efficiency of detailed transport properties by 2 orders of magnitude. (3) An

adaptive analytical Jacobian (AAJ) method is proposed to apply locally reduced

mechanisms in an analytical Jacobian method with sparse matrix solver. The computational

cost of the analytical Jacobian method is reduced by half with the adaptive chemistry. More

importantly, the AAJ method is stable, so that it can utilize large integration time steps to

iv

accelerate overall simulations. Moreover, a multi-scale adaptive reduced chemistry solver

(MARCS) is developed by integrating the CO-DACT and AAJ methods into a

multi-dimensional full-speed fluid solver to perform efficient combustion simulations in

practical geometries. The advanced numerical methods developed are demonstrated to be

efficient, accurate, and robust. They extend the capability of high-fidelity combustion

simulations with detailed chemistry and transport. Finally, numerical studies of the

dynamics and ignition to flame transitions in stratified mixtures are conducted to

investigate the knocking-like acoustic wave formations at NTC (negative temperature

coefficient) conditions. It is found that n-alkanes with rich low temperature chemistry

promote knocking formations in stratified mixtures.

v

Acknowledgements

My PhD study has been a long journey with many precious memories that I’ll never

forget in my life. I would like to thank people that helped and advised me in this journey.

First and foremost, I extend my deepest gratitude to my advisor Professor Yiguang

Ju. Many thanks for his endless support and advice throughout my PhD study. It is my best

fortune to have Professor Yiguang Ju as my academic advisor at Princeton who took every

possible step to guide me and help me grow. I learned a lot from him. I learned how to be

professional and how to think as a researcher. I enjoyed all the inspiring discussions with

Professor Yiguang Ju. Thanks for everything.

I thank Professor Chung K. Law and Professor Luigi Martinelli for their service on

my PhD committee. I appreciate the outstanding teaching of Combustion Theory from

Professor Law. It served as the corner stone of my research works. The inspiring

discussions with Professor Martinelli during committee meetings are also greatly

appreciated. Thanks also go to Professor Chung K. Law and Professor Peng Zhao for

reading my thesis and providing thorough feedbacks which greatly helped me to produce

a high quality dissertation and to Professor Luigi Martinelli and Professor Michael Mueller

for taking time to be my examiner. I would also like to thank Professor Michael Mueller

for the outstanding teaching. I was deeply inspired by his passion of teaching when I was

his teaching assistant.

I will never forget my friends and colleagues in Ju’s Combustion Lab: Wenting Sun,

Sang-Hee Won, Joseph K Lefkowitz, Jeffrey S. Santner, Hao Zhao, Chris Reuter, Aric

Rousso, and Tianhan Zhang. Thank them for inspiring me during day-to-day work and

vi

collaboration. My experience in this lab is by all means remarkable. I’ll certainly miss the

days that the team have lunch together in the meeting room. Besides, thanks also go to the

MAE staff for their excellent professional work and especially to Jill Ray for administrative

help during my graduate study.

Finally, my deepest love and gratitude goes to my parents, Zhanjiang Sun and

Xiangru Hou, my wife, Yakang Xing, and my sister, Xiabing Sun. I couldn’t finish this

long journey without their support and encouragement.

This dissertation carries T#3346 in the records of the Department of Mechanical

and Aerospace Engineering.

vii

Table of Contents

Abstract ............................................................................................................................. iii

Acknowledgements ........................................................................................................... v

Table of Contents ............................................................................................................ vii

Chapter 1 Introduction..................................................................................................... 1

1.1 Overview ............................................................................................................ 1

1.2 DNS Challenges with Detailed Chemical Kinetics ......................................... 3

1.2.1 Multi-scale Nature of Combustion Processes ................................................. 3

1.2.2 Detailed Chemical Kinetics ............................................................................ 5

1.2.3 Detailed Diffusion Coefficients Calculation ................................................... 9

1.3 Efficient Chemical Solvers and Model Reduction Methods ....................... 12

1.3.1 Stiffness Reduction Methods ........................................................................ 12

1.3.2 Tabulation Methods ...................................................................................... 14

1.3.3 Cell Clustering Methods ............................................................................... 17

1.3.4 Linear Scaling ODE Solvers ......................................................................... 18

1.3.5 Skeleton Mechanism Reduction ................................................................... 21

1.3.6 Dynamic Adaptive Chemistry....................................................................... 24

1.4 Motivation and Objectives ............................................................................. 25

1.5 Organization of the Dissertation.................................................................... 27

Chapter 2 Mathematical Formulation and Numerical Approach ............................. 29

2.1 Conservation Equations ................................................................................. 29

2.1.1 Mass Conservation Equation ........................................................................ 30

2.1.2 Momentum Conservation Equation .............................................................. 30

2.1.3 Energy Conservation Equation ..................................................................... 31

2.1.4 Species Conservation equation ..................................................................... 31

2.2 Closure Equations ........................................................................................... 32

2.2.1 Stress Tensor ................................................................................................. 32

2.2.2 Internal Energy.............................................................................................. 32

2.2.3 Heat Flux ....................................................................................................... 33

2.2.4 Species Diffusion Velocity ........................................................................... 33

2.2.5 Chemical Reactions ...................................................................................... 36

2.2.6 Ideal Gas Equation of State........................................................................... 37

2.3 Numerical Representation of Transport Properties and Chemistry ......... 38

2.3.1 Transport Properties Representation ............................................................. 38

2.3.2 Chemistry Representation ............................................................................. 41

2.4 Numerical Implementations ........................................................................... 44

2.4.1 Adaptive Cell Discretization ......................................................................... 44

2.4.2 Fractional Step .............................................................................................. 45

2.4.3 Finite Volume Method .................................................................................. 48

viii

2.4.4 Numerical Schemes ...................................................................................... 49

2.4.5 ASURF Program ........................................................................................... 53

Chapter 3 Correlated Dynamic Adaptive Chemistry (CO-DAC) Method ................ 55

3.1 Introduction and Motivation ......................................................................... 55

3.2 Numerical Methods ......................................................................................... 57

3.2.1 Mathematical Formulation of CO-DAC Method .......................................... 58

3.2.2 Kinetic Model and Fuel Mixtures ................................................................. 66

3.3 Results and Discussion .................................................................................... 66

3.3.1 CO-DAC with the first generation PFA reduction ....................................... 68

3.3.2 CO-DAC with the Second generation PFA reduction .................................. 72

3.4 Conclusion ....................................................................................................... 75

Chapter 4 Correlated Dynamic Adaptive Chemistry and Transport (CO-DACT)

Method ............................................................................................................................. 86

4.1 Background and Objectives ........................................................................... 87

4.2 Numerical Methods ......................................................................................... 88

4.2.1 Concept of CO-DACT method ..................................................................... 88

4.2.2 Selection of Phase Parameters ...................................................................... 90

4.2.3 Numerical Identification and Implementation of Correlated Groups ........... 91

4.2.4 Numerical Error Analysis ............................................................................. 95

4.2.5 Numerical Setup............................................................................................ 98

4.3 Results and Discussion .................................................................................... 99

4.3.1 Premixed Outwardly Propagating Spherical Flames .................................... 99

4.3.2 Diffusion Ignition........................................................................................ 103

4.3.3 Unsteady Ignition and Spherical Flame Formation with LTC.................... 105

4.3.4 Numerical Error Analysis ........................................................................... 107

4.4 Conclusion ..................................................................................................... 111

Chapter 5 Adaptive Analytical Jacobian (AAJ) Method .......................................... 125

5.1 Background and Objectives ......................................................................... 125

5.2 Numerical Methods ....................................................................................... 127

5.2.1 Analytical Jacobian (AJAC) Method with Sparse Solver........................... 127

5.2.2 Adaptive Analytical Jacobian (AAJ) Method ............................................. 130

5.3 Results and Discussion .................................................................................. 132

5.3.1 Stability Comparisons between AJAC Method and HMTS Method .......... 133

5.3.2 Validations of AAJ method ........................................................................ 135

5.4 Conclusion ..................................................................................................... 140

Chapter 6 Multi-scale Adaptive Reduced Chemistry Solver (MARCS) .................. 149

6.1 Motivations and Objectives .......................................................................... 149

6.2 Numerical Implementations ......................................................................... 150

6.2.1 Multi-dimensional Parallelized Fluid Solver .............................................. 150

ix

6.2.2 Implementations of MARCS ...................................................................... 151

6.3 Examples of Multi-dimensional Turbulent Simulations ........................... 152

6.3.1 Turbulent Stretch Ignition ........................................................................... 152

6.3.2 Detonation Formation in a Turbulent Stratified Mixture ............................ 155

6.4 Conclusion ..................................................................................................... 156

Chapter 7 Dynamics and Ignition to Flame Transitions in High Pressure Stratified

Mixtures ......................................................................................................................... 163

7.1 Motivation and Objectives ........................................................................... 163

7.2 Numerical Setup ............................................................................................ 165

7.3 Results and Discussion .................................................................................. 166

7.3.1 Dynamics and Combustion Regimes in Stratified n-Heptane/Air Mixture 166

7.3.2 Dynamics and Combustion Regimes in Stratified Toluene/Air Mixture.... 170

7.3.3 Dynamical Analysis of Ignition to Shockwave Formation ......................... 172

7.4 Conclusions .................................................................................................... 174

Chapter 8 Conclusions and Future Work .................................................................. 181

8.1 Conclusions .................................................................................................... 181

8.2 Future Work and Recommendations .......................................................... 184

References ...................................................................................................................... 185

1

Chapter 1

Introduction

1.1 Overview

The utilization of fire by early humans was one of the first innovations (James et al.

1989) which distinguishes Homo sapiens from primitive animals. Since then, flame has

been the major energy source in human civilization for heating, lighting, cooking, smelting,

transportation and power. Until now, combustion of fossil fuels still contributes more than

80% of the total energy consumption on this planet (Newell, Raimi, and Aldana 2019).

While fire is useful and powerful, it can also be harmful and sometimes deadly. The

Australian wildfires in the early 2020 have burnt more than 18 million hectares and

destroyed many buildings, with extensive loss of lives, both human and animal. (Burton

2020). In addition, combustion of fossil fuels generates nearly all the anthropogenic

pollutions, including sulfur dioxide (SO2), nitrogen oxides (NOx), carbon monoxide (CO)

and fine particulate matter (PM 2.5), which are harmful to human beings and the

environment. Therefore, the study of combustion is of great importance to help understand

and predict combustion phenomena more accurately, so that we can utilize such great

powers not only more efficiently, but also in a way that is cleaner and safer for our society.

Since the mid-1970s, there have been substantial accomplishments in the

fundamental combustion discipline. Sophisticated combustion theories, advanced

experimental facilities, and high-fidelity numerical modeling methods have been

developed. Unfortunately, even with the help of modern combustion techniques, the world

2

today is still facing energy sustainability and global warming issues. Energy sustainability

efforts can be greatly reduced if we can utilize energy more efficiently. For example, more

than 60 percent of the energy is lost in the form of exhaust gas from the internal combustion

engines currently employed in motor vehicles. A major culprit of global warming is

greenhouse gases, a majority of which comes from the carbon dioxide emitted during

combustion processes. These situations urge researchers to seek after innovative concepts

of high-performance engines with improved energy conversion efficiency and reduced

emissions.

Recently, tremendous efforts have been devoted to the development of more

efficient and lower emission internal combustion engines working at low temperatures and

high pressure conditions. These advanced engines include the homogeneous charge

compression ignition (HCCI) engine (Tanaka et al. 2003; Zhao et al. 2003), the twin

annular premixing swirler (TAPS) engine (Dodds 2005), the reactivity controlled

compression ignition (RCCI) engine (Kokjohn et al. 2011), and the gasoline direct-

injection compression-ignition (GDCI) engine (Sellnau et al. 2014). However, the control

of fuel injection times, ignition timings, and heat release rates at different engine loads

requires advanced understanding of turbulence-chemistry-transport coupling, especially in

the negative temperature coefficient (NTC) regime, ranging from low temperature (700 K)

to intermediate temperature (1100 K) for practical engine pressures (1 atm - 50 atm).

Unfortunately, modeling turbulence-chemistry-transport interactions in this temperature

and pressure range requires significantly large chemical kinetics with detailed descriptions

of both low and high temperature chemistry. Even with the availability of supercomputing

3

capability at petascale and beyond, direct numerical simulation (DNS) with detailed

chemical kinetics in a multi-dimensional geometry remains infeasible.

This study focuses on the development of advanced computational methods to

improve the simulation efficiency and accuracy, and hence to extend the limit of

combustion modeling with detailed chemical kinetics.

1.2 DNS Challenges with Detailed Chemical Kinetics

Conducting high-fidelity combustion simulation with detailed chemical

mechanisms is extremely challenging. The challenges primarily come from three aspects,

namely the multi-scale nature of combustion processes, the large and stiff ODE system that

govern chemical reactions, and the expensive calculation of local transport properties. The

details of these challenges are discussed in this section.

1.2.1 Multi-scale Nature of Combustion Processes

Direct numerical simulation gets expensive when it involves multi-scale

computation. Unfortunately, a combustion system can behave multi-scale in both spatial

and temporal coordinates. The characteristic lengths and times in a combustion process

span multiple orders of magnitude.

In terms of the length scale, flame is a thin layer within which rapid chemical

reactions occur with massive heat release and significant temperature increase. The flame

thickness, 𝑙𝐹, can be estimated by the thermal conductivity, 𝜆, specific heat, 𝑐𝑝, chemical

reaction rate, 𝜔, and the Zel’dovich number, 𝑍𝑒, of the mixture:

4

p

F

cl Ze

(1.1)

A typical value of the above estimation in a typical combustion chamber at 10 atmosphere

pressure is 0.1 mm. In a direct numerical simulation, in order to accurately capture the

detailed reaction processes and flame structures in the thin layer, the size of the

computational grid should be at least one order of magnitude smaller than the flame

thickness, which results in a grid size ∆𝑥 = 0.01 𝑚𝑚.

On the other hand, the domain of interest in a combustion simulation, such as the

internal combustion chamber, is on the scale of 10 cm. Considering the grid size, it leads

to 10,000 grids in one dimension, and a total of 1 trillion (1012) computational cells in a

three dimensional computation space. The multi-length scale nature requires a huge

number of cells in a combustion simulation, which dramatically increases the

computational cost.

Furthermore, the multi-length scale nature introduces the multi-time scale nature in

combustion modeling. The computational time step, ∆𝑡𝑐, in any explicit PDE solver to

numerically integrate the convection term is limited by the Courant-Friedrichs-Lewy (CFL)

number, grid size (∆𝑥), and the characteristic speed (the acoustic speed, 𝑎, in a typical

combustion simulation):

CFL ctax

=

(1.2)

The CFL number must be equal or smaller than 1, otherwise the numerical viscosity would

become negative, making the numerical scheme unstable. Therefore, the size of the

integration time step has to satisfy the following constraint:

5

c

xt

a

(1.3)

With ∆𝑥 = 0.01 𝑚𝑚 and 𝑎 ~ 1000 𝑚/𝑠 in a typical combustion system, the time step to

integrate the convection flux must be set below 10−8 𝑠.

Apart from the convection time step, the diffusion time scale is also limited by the

grid size and mixture diffusivity, 𝐷,:

2

d

x

D

= (1.4)

A typical value of 𝐷 in a combustion simulation at 10 atm and 1000 K is 10−4 𝑚2/𝑠, which

results in 𝜏𝑑 = 10−6 𝑠.

Due to the limit of the convection time step and diffusion time scale, the time step

that can be utilized in a simulation needs to be smaller than the minimum of the above two

values, which is 10−8 𝑠 in an explicit PDE solver. Considering 0.1 𝑠 to be the typical

physical time duration in a combustion simulation, the overall three dimensional

computation requires solving detailed governing equations for

107 timestep × 1012 cells (1.5)

The small time step resulting from the above analysis hasn’t even take into

consideration of the chemical time scales yet. An analysis that includes the chemical

reaction time scales (shown in the next sub section) results in an even smaller time step

limits described above. Therefore, without optimization and advanced numerical methods,

high dimensional combustion simulations are extremely expensive.

1.2.2 Detailed Chemical Kinetics

Apart from the multi-scale nature, the large and detailed chemical kinetic also

makes high-fidelity combustion modeling infeasible. The increasing interest in cool flames

6

and the discovery of distributed ignition regimes due to the low temperature chemistry

necessitate the development of large hydrocarbon fuel kinetics. For example, a detailed n-

heptane mechanism can have 1034 species and 4236 reactions (Curran et al. 2002). A semi-

detailed jet fuel surrogate model, which contains four different fuel molecular structures,

including n-alkanes, iso-alkanes, cyclo-alkanes and aromatics, has more than 2000 species

and 8000 reactions (Dooley et al. 2012), while a detailed diesel fuel kinetic mechanism

with lightly branched alkanes and straight chain alkanes contains approximately 7200

species and 31400 reactions (Sarathy et al. 2011). Theses chemical kinetics can accurately

predict combustion properties in various temperature and pressure ranges. Unfortunately,

combustion modeling becomes unaffordable when such large chemical kinetics are

involved.

The difficulty of utilizing large chemical mechanisms arises from the large, non-

linear, and stiff ordinary differential equation (ODE) system that governs the chemical

reactions. Species conservations are described by the following ODE in a 0-dimentional

reaction system:

( )1, ...,i

i s

d Yi N

dt

= = (1.6)

where 𝜌 is the mixture density, 𝑌𝑖 the mass fraction of the 𝑖_𝑡ℎ species, 𝑁𝑠 the total number

of species, and 𝜔𝑖 the mass production rate of the 𝑖_𝑡ℎ species, defined as:

( ), ,' ''

, , , ,

1 1 1

'' '

j k j ks sr N NN

j j

i i i k i k f k b k

k j jj j

Y YW

W W

= = =

= − −

(1.7)

with 𝑊𝑖 being the molecular weight of the 𝑖_𝑡ℎ species, 𝑁𝑟 the total number of reactions,

𝜐𝑖,𝑘′ and 𝜐𝑖,𝑘

′′ the stoichiometric coefficients of species 𝑖 appearing in reaction 𝑘 as a

7

reactant and product, respectively, and 𝜅𝑓,𝑘, 𝜅𝑏,𝑘 the forward and backward reaction rate

constants, respectively. Due to the multiplication of mass fractions in Equation (1.7) and

the exponential temperature dependence of rate constants (discussed in Section 2.2.5),

Equation (1.6) becomes a non-linear ODE system.

Moreover, the stiffness of the ODE system comes from two folds: rapid chain

branching and chain propagation reactions produce active radicals with significantly short

lifetimes; the relatively slow chain initiation and chain termination reactions

consume/generate major species with much longer lifetimes. The characteristic time, 𝜏𝑖, of

species 𝑖 can be estimated as:

1

ii

iY

−

=

(1.8)

where φi is the destruction rate of the 𝑖_𝑡ℎ species. In the early stage of ignition or in areas

that are far away from flames, time scales of species can be as large as seconds due to the

slow reactions. However, after the thermal runaway, active radicals can be produced and

destructed rapidly within picoseconds.

The significant disparities of characteristic times in a reaction system, ranging from

seconds to sub-nano-seconds, lead to a strong stiffness in the ODE system. In order to

numerically solve the stiff ODE system, implicit ODE solvers with the backward

differentiation formula (BDF) are applied traditionally, such as the differential/algebraic

system solver (DASSL) (Petzold 1982), the variable-coefficient ODE solver (VODE)

(Brown, Byrne, and Hindmarsh 1989), the Livermore solver for ordinary differential

equations (LSODE) (Radhakrishnan and Hindmarsh 1993), the Livermore solver for ODE

with general sparse Jacobian matrices (LSODES) (Hindmarsh 1983), and the variant of

8

VODE solver with Preconditioned Krylov methods (VODPK) (Brown and Hindmarsh

1989). These implicit methods generally require Jacobian matrix calculation as well as the

matrix decomposition and inversion. Based on the decomposed Jacobian matrix, the stiff

and nonlinear ODE system can be linearized and solved independently. The implicit

solvers can be fast and robust if the dimension of the ODE system is small and if one can

use large integration time steps.

However, in a combustion simulation, the number of equations in the ODE system

equals the number of species, 𝑁𝑠, in the reaction system. When detailed chemical kinetics

are involved, the dimension of the ODE system becomes huge. Unfortunately, the

computational complexity of matrix decomposition and inversion is 𝑂(𝑁𝑠3). Even with the

optimized Goppersmith-Winograd algorithm (Gall 2014), which reduces the inversion

complexity to 𝑂(𝑁𝑠2.373), the computational costs of traditional implicit ODE solvers with

large chemical kinetics are still unaffordable. For instance, a direct integration using the

VODE solver on the detailed diesel mechanism, which has more than 7000 species,

requires 20 hours for just a single time step. Moreover, since the integration time step in a

combustion simulation not only is determined by the ODE solver, but is also limited by the

CFL number and the diffusion time scale, the time step is usually very small (10−8 𝑠 based

on the earlier discussion). The small time step diminishes the robustness of the implicit

ODE solver. Therefore, the huge computational costs of matrix operations together with

the small time steps cause the combustion modeling with detailed chemical kinetics

infeasible.

9

1.2.3 Detailed Diffusion Coefficients Calculation

Besides the expensive implicit ODE solvers, one side effect introduced by the

detailed chemical kinetics in combustion modeling is the time-consuming calculations of

local transport properties, specifically the mass diffusion coefficients.

In a combustion simulation, transport properties need to be evaluated on-the-fly,

since they are sensitive to the local thermodynamic conditions. To evaluate transport

properties, the most rigorous model is the Maxwell-Stefan multi-component diffusion

model (Curtiss and Hirschfelder 1949; Furry 1948; Williams 1958) derived from the

Boltzmann’s equation of kinetic theory. The multi-component model is recently revised

(Lam 2006). In the multi-component diffusion model, the mass diffusion coefficient, Dij,

of species 𝑖 with respect to species 𝑗 are calculated by the diffusion matrix in the following

formula:

( )1 116

25ij i ij ii

j

TWD X L L

pW

− −= − (1.9)

where 𝑋𝑖 is the mole fraction of the 𝑖_𝑡ℎ species, 𝑇 the temperature, 𝑝 the pressure, �� the

mean molecular weight and 𝐿 the diffusion matrix, defined as:

( ) ( )( )*1

161

25

Kk

ij j j ik i j ij jk

k i ik

XTL W X W X

p W D

=

= − − − (1.10)

where 𝐷𝑖𝑘∗ is the binary diffusion coefficient between species 𝑖 and 𝑘 . Equation (1.9)

involves an inversion operation of a square matrix 𝐿, which has the dimension of the

mechanism size. Therefore, similar as the implicit ODE solver, the multi-component

diffusion model becomes expensive and impractical when large and detailed chemical

kinetics are involved.

10

To improve the computational efficiency of the multi-component model, a reduced

multi-component diffusion model (Xin et al. 2015) is developed, which only conducts the

multi-component calculations for important species and applies coarse estimations for

other unimportant species. More recently, Ambikasaran & Narayanaswamy (2017)

proposed an innovative algorithm to perform the computations of the multi-component

model. This algorithm is developed under the assumption that the diffusion coefficient

matrix has low rank, which enables an efficient computational approach with the low rank

matrix representation. In this method, the computational time scales linearly with the

number of species.

Another widely used diffusion model is the mixture-averaged model (Mathur,

Tondon, and Saxena 1967; R. Byron Bird, Warren E. Stewart 1960; Wilke 1950), which is

an asymptotic solution of the Boltzmann equation by first order perturbation. In the

mixture-averaged model, the mass diffusion coefficient of species 𝑖 with respect to the gas

mixture, 𝐷𝑖𝑚, is defined as:

*

s

s

N

j jj i

im N

j ijj i

X WD

W X D

=

(1.11)

The mixture-averaged model significantly improves the computational efficiency

of diffusion coefficients by avoiding the detailed diffusion matrix inversion. The

computational complexity of this model is quadratically dependent on the number of

species in a chemical mechanism. Compared with the multi-component model, the

mixture-averaged model offers good accuracy with substantially lower computational cost

for most combustion systems. Recently, a comparison between these two diffusion models

in turbulent combustions (Bruno et al. 2015) concludes that except for the extreme cases

11

such as supercritical combustion, numerical errors introduced by the mixture-averaged

diffusion model are minor with respect to the multi-component model. Therefore, the

mixture averaged diffusion model is widely used in combustion modeling and it is regarded

as the de facto standard (Smooke 2013).

Unfortunately, even with the mixture-averaged model or the multi-component

model with the linear scaling algorithm, detailed calculations of transport properties at

every computational cell and time step are still too inefficient in a reacting flow simulation

with large kinetic mechanisms. In order to reduce the computational cost, many studies

have been conducted. One approach is to use the assumption of the constant Lewis number

to simplify the species diffusion (Pitsch and Peters 1998). However, in a reacting flow

simulation, the Lewis number is far from constant due to the significant variations of the

temperature and compositions. The constant Lewis number assumption may result in

incorrect molecular diffusions and, thereby, affect reaction pathways and lead to incorrect

flame structures. Thus, a high-fidelity modeling requires detailed transport properties.

Another approach to accelerate transport property calculations is species bundling (Xin et

al. 2015). By grouping species with similar binary diffusivities, species bundling method

can reduce the number of species in a transport property calculation. However, it requires

a pre-reduction with a specified temperature range. Since transport properties are sensitive

to temperature, if the temperature in a calculation is beyond the specified temperature range,

species bundling may fail. Therefore, developing advanced methods to drastically reduce

the computational cost of transport properties without sacrificing the accuracy is urgently

needed in combustion modeling.

12

In summary, detailed chemical kinetics with large number of species and reactions

introduce difficulties into combustion modeling due to the nonlinear reaction system, the

stiffness in the species conservation equation, and the expensive transport property

calculation. Without advanced numerical methods, combustion modeling with detailed

chemistry is prohibitive.

1.3 Efficient Chemical Solvers and Model Reduction Methods

In order to facilitate the utilization of detailed chemical kinetics in high-fidelity

combustion simulations, numerous efforts have been expended to developing model

reduction methods and efficient chemical solvers over the last two decades. These methods

can be divided into 6 categories, to be discussed in this section.

1.3.1 Stiffness Reduction Methods

Due to the strong stiffness in the reaction system, the ordinary differential equations

are difficult to be solved numerically. This motivates development of methods that can

systematically reduce the stiffness. The basic idea of the stiffness reduction can be

described as follows: according to the chemical time scales, rapid chain

brunching/propagation reactions and fast exhausted species can be estimated by the quasi-

steady-state approximation (QSSA), the partial equilibrium (PE) approximation or the low

dimensional manifold; Then, the ordinary differential equations of radicals with short

lifetimes can be replaced by algebra equations with the combination of slow reactions, so

that the stiffness in the ODE system can be reduced.

The fundamental approach in this category is to simplify a reaction system by

empirically identifying the partial equilibrium reactions and the quasi-steady state species

13

(Chen 1988; Peters and Williams 1987). However, these methods require sophisticated

identification of PE reactions and QSSA species, which cannot be achieved without the

deep understanding of chemical kinetics. Moreover, the simplified mechanisms are

sensitive to the calculation conditions, so that it is hard to transfer them to simulations with

different conditions.

Fortunately, the time scales of reactions and species can be computed dynamically

from detailed reaction rates and stoichiometric coefficients, so that the fast exhausted

reactions and quasi-steady state species can be identified numerically. For example, in the

computational singular perturbation (CSP) method (Lam and Goussis 1994), species time

scales are defined as the inverse of absolute eigenvalues of the corresponding Jacobian

matrix. By selecting species with time scales below the integration time step as the fast

species, one can decouple the reaction system into a fast subspace projection and a slow

subspace projection. The species conservation equations then can be estimated by the slow

subspace projection together with a corresponding radical correction. Therefore, by

avoiding direct numerical integrations of fast species, the CSP method removes the

stiffness in the ODE system, which provides the possibility to integrate the ODE system

by explicit methods.

Based on the CSP method, Valorani & Paolucci (2009) developed a G-Scheme

method. It further splits a reaction system into a fast subspace, an active subspace, and a

slow subspace based on the eigenvalues of the Jacobian matrix. The species conservation

equations in different subspaces are integrated differently, and direct ODE integrations are

applied to the active subspace. Since only species with moderate time scales, which are

close to the integration time step, are included in the active subspace, one can apply either

14

explicit or implicit methods to a substantially smaller ODE system. Then, the species in

the fast subspace are estimated by the CSP radical correction mentioned above. Finally,

linear extrapolations are applied to update the species in the slow subspace.

Another study that applies the similar concept as in the CSP method but with a

different approach is the intrinsic low-dimensional manifolds (ILDM) method (Maas and

Pope 1992). Similarly, the ILDM method calculates species time scales by the inverse of

the absolute eigenvalues of the Jacobian matrix. But this method doesn’t use time scales to

directly reduce the stiffness of the ODE system. Instead, it recognizes fast exhausted

reactions and identifies the major reaction trajectories in the composition space, which are

called the low-dimensional manifolds. With the main reaction trajectories in the

composition space, the ILDM method can reduce the possible states that a reaction system

can reach, so that the system can be tabulated according to a smaller subspace for

subsequent uses.

The methods mentioned in this category can all simplify a chemical reaction system

by identifying and eliminating fast reactions and species. However, computing the time

scales in these methods involves the calculation of Jacobian eigenvalues, which is a 𝑂(𝑁𝑠3)

operation in terms of the time complexity. Therefore, these methods become

computationally expensive when large mechanisms are involved.

1.3.2 Tabulation Methods

Since the stiffness reduction methods are expensive, the direct numerical

integration of the large reaction system still remains infeasible. Therefore, tabulation

methods are developed to avoid the direct chemical integration. These tabulation methods

15

store pre-calculated integration results in high dimensional tables, so that the chemical

solver can provide local ODE solutions by retrieving and interpolating the tabulated results.

The basic tabulation approach was proposed in the probability density function

(PDF) study (Taing, Masri, and Pope 1993). In this study, chemical integration results are

prepared and tabulated in a high dimensional table. This table is a representation of

discretized states in the composition space, which is a subset of the species space with only

linearly independent components. Pre-calculations of chemical integrations are conducted

at the thermochemical states on a mesh that covers the entire composition space. Then, the

integration results are stored in the high dimensional table. In the simulation, direct ODE

integrations are replaced by an on-the-fly table lookup and linear interpolation. The

interpolation error can be adjusted by changing the mesh size in the composition space.

This approach is efficient with extremely small skeleton mechanisms, such as the H2/CO2

mechanism reported in the study with 3 linearly independent compositions. However, the

computational cost of this tabulation method grows exponentially with the dimension of

the composition space. The space and time complexity to store the tabulation results and

to perform linear interpolations are 𝑂(𝑀𝑁𝑐) and 𝑂(2𝑁𝑐) , respectively, where 𝑀 is the

number of mesh nodes in one dimension of the composition space and 𝑁𝑐 the dimension

of the composition space. Considering a relatively small chemical mechanism with 20

degrees of freedom (𝑁𝑐 = 20) and a mesh with 100 nodes in one dimension of the

composition space, this tabulation requires the storage of 1040 chemical integration results.

Moreover, a single table-lookup operation requires 220 ≈ 106 linear interpolations.

Therefore, the basic tabulation approach in this category is computationally unachievable

with detailed chemistry.

16

To make the tabulation approach affordable for combustion modeling with detailed

chemistry, Pope (1997) proposed an in situ adaptive tabulation (ISAT) method. This

method dramatically reduces the computational cost of the basic tabulation approach based

on the assumption that only a small sub region, called the ‘accessed region’, in the entire

composition space can be accessed in a reaction system. The accessed region can be

represented by the major reaction trajectories, which can be produced by the intrinsic low-

dimensional manifold (ILDM) method. Assuming reaction pathways merge to the major

trajectories quickly, tabulating only the accessed region, instead of the whole composition

space, is sufficient to the modeling. Moreover, a gradient matrix is tabulated together with

the chemical integration result to represent the local tabulation error. This gradient matrix

is used to replace the multi-linear interpolations to provide table-lookup corrections. In

addition, the tabulation process in ISAT method is conducted on-the-fly, so that only the

integration results at necessary thermochemical states are calculated and stored. Therefore,

the ISAT method significantly reduces the memory and CPU time consumptions in a

tabulation process, which makes the tabulation method affordable.

Another similar approach in this category is the piecewise reusable implementation

of solution mapping (PRISM) method (Tonse et al. 1999). The composition space in the

PRISM method is divided into hypercubes with relatively coarse resolutions. Instead of

tabulating the exact integration result and the corresponding interpolation error at a given

state, the PRISM method parametrizes integration solutions by a set of polynomial

equations constructed from other state points in the specific hypercube. Thus, the PRISM

method is efficient since it not only can reduce the number of tabulated states, but it also

avoids interpolations in the subsequent lookup operations.

17

These tabulation methods improve the computational efficiency of chemical

solvers by replacing direct ODE integrations with table lookups. Because of the efficiency

of the ISAT and PRISM method, direct numerical simulations with small or medium sized

chemical mechanisms are affordable. However, the memory consumption still grows

exponentially with the dimension of the composition space. For a large chemical

mechanism, the tabulation methods become expensive. Furthermore, due to the high

sensitivity of ignition and chemistry-transport interactions in the low and intermediate

temperature ranges, the interpolation and polynomial fitting errors can be unbounded.

1.3.3 Cell Clustering Methods

In order to overcome the huge memory consumption difficulty in the tabulation

methods while reducing the number of detailed ODE integrations required in a combustion

simulation, the cell clustering strategy is developed. Methods in this category are

developed based on the concept that during a combustion simulation, a particular set of

thermochemical states can occur repeatedly at different cell locations. Unlike the tabulation

methods, the clustering methods in this category don’t perform result storages and

retrievals. Instead, the clustering methods conduct dynamic partitions of computation cells

at each time step. Detailed ODE integrations are performed in each partition, instead of

each computation cell, to minimize the computational cost of solving chemical reactions.

For example, in the cell agglomeration (CA) algorithm (Goldin, Ren, and Zahirovic

2009), computational cells are agglomerated to pre-defined hypercubes in the composition

space based on their thermochemical states. The size of hypercubes determines the relative

error in the agglomeration algorithm. Similarly, in the dynamical multi-zone (DMZ)

partitioning scheme (Liang, Stevens, and Farrell 2009b), computational cells are also

18

clustered based on their positions in the composition space. But unlike the CA method, cell

partitions in the DMZ method is dynamically computed. This method initializes the

composition space with just one partition, and it recursively splits partitions until

compositional distances between any two cells in the same partition are below a given

threshold. Another method in this category is the high-dimensional unsupervised cell

clustering method (Perini 2013), which utilizes a grid-like unsupervised technique to

perform the cell partition. There are a few other similar methods in this category, with

slightly different partition strategies, such as the multi-zone chemistry mapping method

(Jangi, Yu, and Bai 2012) and the chemistry coordinate mapping method (Shi et al. 2010).

These clustering methods share the similar fundamental principle. Thus, they also

share the same strength and weakness. Compared with tabulation methods, the cell

clustering methods avoid the storage of pre-calculated chemical integration results, thus

eliminate memory consumption by the high-dimensional table. However, since the cell

clustering and ODE integrations are performed at every time step, there may be duplicate

calculations if two time steps have similar states. More importantly, due to the nonlinear

behavior of the ODE system, the validity of the chemical integration mapping in a partition

is strongly dependent on the complexity of chemistry as well as flame regimes. In some

scenarios, the results mapping in these methods may fail due to unbounded errors.

1.3.4 Linear Scaling ODE Solvers

The tabulation and cell clustering methods discussed above reduce the

computational cost of chemical solvers by reducing the number of direct ODE integrations

in combustion simulations. However, the interpolation error in these methods may be

unbounded. Conducting direct ODE integrations at local conditions is still preferred in

19

high-fidelity combustion simulations. Therefore, tremendous efforts have been committed

to developing efficient chemical solvers which can perform direct ODE integrations with

linear scaling. The time scale splitting and the sparse matrix computation are the two major

strategies in this category.

Due to the significant disparities of time scales in a reaction system, the ODE

system is stiff, which makes the linearly scaled explicit integration infeasible. However, if

the small time scales in the ODE system can be removed or can be handled separately,

explicit methods with substantially large time steps will become practical in combustion

simulations. Based on this principle, a multi-time scale (MTS) method and a hybrid multi-

timescale (HMTS) method (Gou et al. 2010) are developed. In the MTS and HMTS

methods, species are separated into fast, intermediate, and slow groups according to their

time scales estimated by Equation (1.8). Based on the assumption that species in the fast

group converge to their steady states quickly, the MTS method uses adaptive ODE

integration time steps. The time scale of the fastest species is utilized at the beginning to

perform integrations with the explicit Euler method. After species in the fast group reach

their steady states, the integration time step is increased to be the smallest time scale in the

intermediate group. By applying this strategy repeatedly, the integration time step increases

gradually until reaching the global integration time step. Similarly, the HMTS method also

integrates different groups separately. But unlike in the MTS method, the fast species in

the HMTS method are integrated by the implicit Euler method with the global integration

time step. The rest of the intermediate and slow groups in the HMTS method are integrated

by the same process as in the MTS method. In the MTS and HMTS methods, since species

conservation equations are integrated independently without any matrix operation, the

20

computation cost is linearly dependent on the size of mechanisms. Therefore, the MTS and

HMTS methods are efficient in combustion modeling with large mechanisms.

Another linear scaling method based on the time scale splitting strategy is the

dynamic stiffness removal method (Lu et al. 2009). In this method, time scales of species

are also estimated by Equation (1.8). For the species with time scales smaller than the

integration time step, the quasi-state-state and partial equilibrium assumptions are applied

to replace the ODEs with algebra equations. For the rest of slow species, the explicit

integration method is utilized. Similarly, a dynamic adaptive method for hybrid integration

(AHI) (Gao et al. 2015) is proposed and developed. In this method, coupling of chemistry

and transport is considered. The diffusion flux is considered as one of the source terms in

species conservation equations and solved jointly with chemical reactions. With this source

term modification, the AHI method integrates fast species implicitly and non-stiff species

and diffusion terms explicitly. The linear scaling performance is also reported in these

methods.

Besides the time scale based integration strategy, sparse matrix algebra is also

widely involved in the development of linear scaling solvers. In a detailed chemical

reaction system, since only limited number of species can participate in one elementary

reaction, the Jacobian matrix of species conservation equations is mostly sparse. By

utilizing efficient sparse matrix operations, the computational complexity of implicit ODE

solvers can be reduced to 𝑂(𝑁𝑠) . Perini et al. (2012) proposed an efficient chemical

integration method based on the sparse ODE solver, LSODES, together with an analytical

Jacobian matrix. Since a chemical reaction system can be explicitly described by Equations

(1.6) and (1.7), the proposed method computes the analytical form of the Jacobian matrix

21

prior to the simulation. During the subsequent simulation, the value of the Jacobian matrix

is estimated by the analytical form and the local thermochemical states. The analytical

Jacobian is used to replace the Jacobian matrix calculated by the numerical perturbation in

an implicit ODE solver. Therefore, the computational efficiency is significantly improved.

This proposed method is further accelerated by replacing the LSODES solver with a Krylov

iterative sparse solver (Perini, Galligani, and Reitz 2014). Alternatively, a sparse solver

with adaptive preconditioners method (McNenly, Whitesides, and Flowers 2015) is

developed. In this method, the sparse Jacobian matrix is used as the preconditioner of the

backward differentiation formula (BDF) in the implicit ODE solver to achieve the high

computational efficiency. Moreover, an extrapolation-based stiff ODE solver (Imren and

Haworth 2016) is reported. It is also an implicit ODE solver based on the sparse Jacobian

matrix, but the BDF operator in this solver is improved by using the extrapolated time step

from the previous integration.

Both the time scale based integration methods and the sparse matrix based implicit

ODE solvers have demonstrated significant improvements of computational efficiency in

combustion modeling. Although numerical simulations with large chemical kinetics are

still expensive, the linear scaling methods discussed in this category are of great potential

to provide affordable combustion modeling in practical geometries when combining with

other chemical reduction methods.

1.3.5 Skeleton Mechanism Reduction

While the linear scaling solvers discussed above are efficient, they will be more

powerful if the chemical system can be simplified so that only a subset of the ODE system

needs to be solved. Therefore, various methods have been developed to reduce the size of

22

the detailed chemical mechanisms. These methods simplify the detailed mechanism by

generating a skeleton representation, which can reproduce the main reaction pathways and

major combustion properties at certain conditions.

The most practical methods in this category utilize the graphic based model

reduction strategy. For example, in the directed relation graph (DRG) method (Lu and Law

2005), the chemical reaction system is represented by a weighted directed graph (digraph),

with nodes being species and edges being chemical reactions that take the upstream nodes

as reactants and the downstream nodes as products. The weight of an edge is represented

by the reaction rate of the underlying reaction. The DRG method simplifies chemical

mechanisms by keeping only important nodes on the graph. To perform the reduction,

source species need to be specified as the initial set of selected nodes. Then a breadth-first

search is performed iteratively to identify all the nodes that are connected to the selected

nodes with edge weights above a pre-defined threshold. In the end of each iteration, the

identified nodes are added to the set of selected nodes. The DRG method repeats this

process until there is no additional nodes need to be included in the selected set. The nodes

in the final selected set represent the species in the reduced skeleton mechanism. With this

approach, the DRG method can efficiently identify important reaction pathways and

remove unimportant species from the detailed chemical kinetics.

Based on the DRG method, a DRG with error-propagation (DRG-EP) method

(Pepiot and Pitsch 2008) is proposed to restrict the reduction error by not only the edge

weight threshold, but also a few interested global properties, such as the ignition delay time

and flame speed, so that the reduced mechanism can accurately reproduce both local

reaction pathways and targeted global properties within certain tolerant error.

23

Besides the DRG method, the path flux analysis (PFA) method (Sun et al. 2010)

also incorporates a similar concept of the graphic representation of the chemical reaction

system. It can be multi-generation when calculating the edge weight between two nodes,

meaning that not only the directly connected edges, but also the edges with other by-pass

nodes are considered.

Another model reduction method in this category is the global pathway selection

(GPS) method (Gao, Yang, and Sun 2016). It is also based on the graph set up of a reaction

system, but it considers the pathways of elements instead of species. It identifies important

species that act as ‘hubs’ in the element flux transfer. Then reduced global pathways are

generated by selecting a set of shortest pathways from reactants to final products with

bypassing the hub species.

Methods in this category can produce globally reduced skeleton mechanisms prior

to the actual simulation to reduce the computational cost. However, there are a few

drawbacks. To generate a skeleton mechanism that can represent the chemical system in

the actual simulation, one has to estimate all the possible thermochemical states that the

simulation can encounter. Usually homogeneous ignition and steady premixed flames are

calculated prior to the actual simulation with expected initial conditions. The snapshots of

the thermochemical states during ignition and flame calculation are sampled to generate

the reduced mechanism. However, simple ignition and steady flame calculation are not

sufficient to produce all the possible thermochemical states in a practical simulation,

especially when turbulence is involved. Since reaction pathways are strongly coupled with

local thermochemical conditions, the reduced mechanism may fail if the condition in a

calculation exceeds the specified range during the model reduction. Furthermore, since the

24

skeleton mechanism is produced to globally represent the detailed chemistry, it is not

optimized for a local thermochemical condition. There are still redundant species and

reactions can be removed in a specific computation cell at a particular time step. Therefore,

the skeleton mechanism is not optimal in local chemical solvers.

1.3.6 Dynamic Adaptive Chemistry

As discussed above, globally reduced skeleton mechanisms suffer local

optimization and validity problems. Therefore, methods have been developed to reduce

chemistry dynamically based on the local thermochemical conditions. Various chemical

reduction methods from previous categories are incorporated to produce the dynamic

adaptive chemistry.

Løvås et al. (2002) and Banerjee & Ierapetritou (2006) proposed methods to apply

the QSSA locally to produce the adaptive chemistry. Schwer et al. (2003) performed a

study on the adaptive chemistry with the on-the-fly selection of pre-reduced models.

Valorani et al. (2006) developed an automatic procedure to simplify the local chemistry

based on the CSP method. Liang et al. (2009) proposed a dynamic adaptive chemistry

method by using DRG to produce locally reduced mechanisms, and Gou et al. (2013)

developed a dynamic adaptive chemistry method with error controls.

These dynamic adaptive chemistry methods can produce locally optimized

chemical reductions, and hence greatly reduce the computational cost of chemical solvers.

However, producing the adaptive chemistry on-the-fly is also time consuming. The

acceleration of the chemical solver with locally reduced mechanisms can be diminished by

the computational cost of the on-the-fly chemical reduction.

25

The major approaches to enable detailed chemical kinetics in combustion modeling

have been reviewed - from the traditional implicit ODE solvers to the dynamic adaptive

chemistry with efficient model reduction methods, demonstrating that significant efforts

have been expended and the computational costs with detailed chemical kinetics have been

reduced by orders of magnitude. With these advanced methods, combustion modeling with

relatively large mechanisms has become affordable in low-dimensional or small scale

simulations. However, high-dimensional simulations with complex geometries still remain

expensive, especially when turbulent conditions are involved. More efficient numerical

methods are still urgently needed in practical engine simulations. Therefore, the motivation

of this dissertation is to develop more efficient numerical methods for combustion

simulations in practical geometries.

1.4 Motivation and Objectives

The goal of this dissertation study is to develop advanced numerical methods for

multi-dimensional and high-fidelity combustion modeling with large and detailed chemical

kinetics. The advanced numerical methods presented in this study improve the

computational efficiency from three aspects: providing on-the-fly chemical reduction with

negligible computational costs; reducing the computational complexity of direct ODE

integrations; and accelerating calculations of detailed transport properties. The detailed

objectives of this study are stated below:

(1) As discussed, the globally reduced skeleton mechanism suffers the drawback

that the reduced mechanism is not locally optimized. On the other hand, the time

consumption of the on-the-fly chemical reduction diminishes the acceleration of the

26

dynamic adaptive chemistry. To take advantages of both approaches and eliminate their

disadvantages, a correlated dynamic adaptive chemistry (CO-DAC) with on-the-fly PFA

reduction method is developed in this study. By utilizing the similarities of reaction

pathways in both temporal and spatial coordinates, the CO-DAC method provides an

innovative approach to produce locally reduced mechanisms on-the-fly with negligible

computation cost.

(2) Based on the concept of the correlated space in the CO-DAC method, a

correlated dynamic adaptive chemistry and transport (CO-DACT) method is further

proposed as the second objective. The CO-DACT method aims to dramatically accelerate

the calculation of detailed transport properties without sacrificing the accuracy. The

mixture-averaged model is utilized to compute the transport properties in a correlated space,

so that redundant calculations can be avoided.

(3) With the locally reduced mechanisms from the CO-DAC/CO-DACT method,

the computational efficiency of direct ODE integrations is further improved by an adaptive

analytical Jacobian (AAJ) method, which applies the adaptive chemistry to a sparse

analytical Jacobian solver (Perini et al. 2012). With the on-the-fly chemical reduction, the

dimension of both the ODE system and the Jacobian matrix are reduced. Therefore, the

simulations can be accelerated by only solving a subset of species equations in the chemical

reaction system.

(4) Since the CO-DAC, CO-DACT, and AAJ methods dramatically increase the

computational efficiency and extend the limit of combustion modeling with detailed

chemical kinetics, the fourth objective is to implement them into a multi-dimensional full-

27

speed fluid solver (Fu and Wang 2013). This will allow us to conduct efficient combustion

modeling in practical geometries.

(5) Finally, the advanced methods developed are applied to a numerical study of

flame dynamics and ignition to flame transitions in high pressure stratified

n-heptane/toluene mixtures. The objective of this numerical study is to investigate how

ignition to flame transitions in stratified fuel mixtures affect the knocking-like acoustic

wave formations in typical HCCI engine conditions and how the kinetic difference between

n-alkanes and aromatics affects ignition to flame transitions.

The above motivation and objectives provide the basic guideline for the dissertation

research. The structure of this dissertation is presented next.

1.5 Organization of the Dissertation

Chapter 2 presents the mathematical formulation and numerical approaches.

Governing equations, numerical methods, numerical schemes, and the mesh discretization

are discussed in detail.

Chapter 3 is on the development of the correlated dynamic adaptive chemistry (CO-

DAC) method, with detailed methodologies, model analysis and validations.

Chapter 4 presents the development of the correlated dynamic adaptive chemistry

and transport (CO-DACT) method, with the algorithm implementation, the mathematical

error analysis, and detailed validations.

Chapter 5 is on the adaptive analytical Jacobian (AAJ) method, with the analytical

form of Jacobian matrix, the matrix sparsity, the reduced ODE system, and the solver

stabilities.

28

Chapter 6 presents the MARCS method by implementing the CO-DACT and AAJ

methods in a high-dimensional full-speed fluid solver, incorporating the numerical

implementations and examples of high-dimensional simulations with turbulent conditions.

Chapter 7 is on the numerical study of dynamics and ignition to flame transitions

in stratified fuel mixtures, analyzing and comparing the different behavior of n-alkanes and

aromatics in the knocking-like acoustic wave formations.

Chapter 8 summarizes the developed numerical algorithms and numerical studies.

Recommendations for future works are also proposed.

29

Chapter 2

Mathematical Formulation and Numerical Approach

In this chapter, the flow governing equations and the numerical approach applied

in this study will be presented. The details of chemistry and transport representations, the

adaptive grid, the discretization method and the numerical scheme will be discussed.

2.1 Conservation Equations

About a century ago, the French mechanical engineer and physicist Claude-Louis

Navier and the Anglo-Irish physicist and mathematician George Gabriel Stokes derived the

differential equations to describe the motion of viscous fluid substances, known as the

Navier-Stokes equations. No general analytical solutions have been found for the non-

linear equation system. Therefore, the subject of computational fluid dynamics (CFD) has

been extensively investigated to simulate fluid systems computationally. The mass

conservation, momentum conservation and energy conservation are described by the

Navier-Stokes equations. The specific equations can be written in different formats,

including the conservative form and non-conservative form, where the independent

variables can be either the conservative variables or the primitive variables.

In this study, the Navier-Stokes equations are presented in the conservative form

with the conservative variables. Furthermore, since multiple species exist in a combustion

system, the species equations are included, in addition to the Navier-Stokes equations, to

describe the unsteady, compressible, viscous, and reactive flow system. The governing

equations derived using an Eulerian approach, based on the continuity assumption, are

30

listed in this section. The derivation of governing equations can be found in general text

books (Law 2006).

2.1.1 Mass Conservation Equation

The mass conservation is described by the continuity equation

( ) 0t

+ =

v (2.1)

where 𝜌 is the fluid density and 𝐯 the vector of convection velocity. The mass conservation

equation describes that the density change rate in a control volume equals the mass flux

passing through the surface.

2.1.2 Momentum Conservation Equation

The conservation of momentum is given by the following equation

( )( )

1

sN

i

i

Yt

=

+ = +

i

vvv σ f (2.2)

where 𝛔 is the stress tensor, 𝑌𝑖 the mass fraction of species 𝑖, and 𝐟i the vector of unit body

force on species 𝑖, such as the gravity. The mass fraction is defined by the ratio of the

density of species 𝑖 in a mixture to the total density, 𝑌𝑖 = 𝜌𝑖 𝜌⁄ . This is a vector equation

with three dimensions in a Cartesian coordinate. The first term in this equation is the rate

of the momentum change in a control volume. The second term is the momentum

convection. The third term represents the surface force, including viscos force and pressure.

And the last term in the equation is the total body force applied in the control volume. This

equation is the representation of the Newton’s second law. Since this study focuses on the

numerical simulation of gas phase mixtures, the gravity is negligible compared with surface

forces. Assuming no other body forces are applied, the last term is ignored. Furthermore,

The fluid in this study is assumed Newtonian, which means the viscous forces, indicated

31

by the divergence of stress tensor 𝛔, are linearly proportional to the velocity gradients. The

definition of the stress tensor will be shown in the next section.

2.1.3 Energy Conservation Equation

The energy conservation is described by the first law of thermodynamics

( )( ) ( ) ( )

1

sN

i s

i

ee Y Q

t

=

+ = − + + + +

'

i iv q v σ v v f (2.3)

where 𝑒, 𝐪, 𝐯𝐢′, and 𝑄𝑠 are the internal energy per unit mass, the heat flux, the diffusion

velocity of species 𝑖, and the heat source, respectively. This equation represents that the

rate of total energy change per volume (the first term) and the net energy flux passing

through the control volume surface (the second term) equals the summation of the inbound

head flux (the third term), the work done by the surface forces (the fourth term), the total

work done by the body forces (the fifth term) and the energy generated from sources (the

last term). The energy source is ignored in this study and 𝐟𝐢 ≡ 0. Thus, the last two terms

will not be included in the following discussions. The definitions of 𝑒 and 𝐪 will be

introduced in the next section.

2.1.4 Species Conservation equation

In a chemical reaction system, each species is described by the following

conservation equation

( )( )( ) , 1, ,i

i i s

YY i N

t

+ + = =

'

iv v (2.4)

where 𝜔𝑖 , 𝐯𝐢′,and 𝑁𝑠 are the mass production rate of species 𝑖, the diffusion velocity of

species 𝑖 , and the total number of species in the reaction system, respectively. The

formulation of the diffusion velocity will be discussed in the next section. There are 𝑁𝑠

species equations. However, only 𝑁𝑠 − 1 equations are independent because the

32

summation of species equations is identical to the mass conservation equation, which

results in the following constraints:

1 1 1

1, 0, 0s s sN N N

i i i

i i i

Y Y = = =

= = = '

iv (2.5)

2.2 Closure Equations

There are 𝑁𝑠 + 2 independent equations (with one vector equation), which is less

than the number of unknowns, including 𝜌, 𝐯, 𝛔, 𝑒, 𝐪, 𝑌𝑖, 𝐯𝐢′, and 𝜔𝑖. Additional equations

need to be included to close the equation system.

2.2.1 Stress Tensor

In a Newtonian fluid, the stress tensor can be calculated from the gradient and

dyadic product of the velocity:

( ) ( ) ( )( )2

3

Tp p

= − + − + + = − +

σ v I v v I τ (2.6)

where 𝑝 is the hydrostatic pressure, ξ the bulk viscosity, 𝜇 the dynamic viscosity, and 𝐈 the

identity matrix. 𝛕 represents the stress tensor excluding the pressure term. In this study, the

stress tensor is simplified by taking out the bulk viscosity. The dynamic viscosity, 𝜇, is a

fluid property, which can be represented by a mixture averaged model (introduced in the

next section). Equation (2.6) closes the stress tensor 𝛔, but it introduces another unknown,

𝑝.

2.2.2 Internal Energy

The internal energy per unit mass can be denoted by:

33

21

2e p v h = − + + (2.7)

where 𝑣2 is the inner product of the velocity vector 𝐯 and ℎ the total enthalpy per unit mass.

The total enthalpy is the average of the species enthalpies weighted by their mass fractions:

( ) ( ) ( ) ( )0

0

0 0 0 ,

1

,, ; ;sN

Ts s

i i i i i i p iT

i

h Yh h h T h T T h T T c dT=

= = + = (2.8)

where ℎ𝑖 is the enthalpy of species 𝑖 in mass unit, ℎ𝑖0 and ℎ𝑖

𝑠 the formation and sensitive

enthalpy of species 𝑖, respectively, 𝑇0 the reference temperature of the formation enthalpy,

𝑇 the temperature, and 𝑐𝑝,𝑖 the specific heat capacity at constant pressure of the 𝑖-th species.

Similar as the dynamic viscosity, the formation enthalpy and the specific heat can be

interpolated from a pre-defined data set (discussed in the next section). While internal

energy 𝑒 is closed by Equation (2.7), a new unknown, 𝑇, is introduced.

2.2.3 Heat Flux

The heat flux in Equation (2.3) is given by:

1

sN

i i

i

Yh T =

= − '

iq v (2.9)

where 𝜆 is the thermal conductivity, which is a function of temperature 𝑇 and the mixture

compositions. The heat flux due to mass gradient, known as the Dufour effect, is not

included in this study since it is usually negligible. The value of 𝜆 will also be estimated

by the mixture averaged model. Equation (2.9) closes the heat flux.

2.2.4 Species Diffusion Velocity

The diffusion velocity of the 𝑖 -th species, 𝐯𝐢′ , can be solved by the following

equation system:

34

( ) ( ) ( ) , ,

1 1 1

s s sN N Ni j i j T j T i

i i i i j

j j jij ij j i

X X X X D Dp TX Y X YY

D p p T D Y Y

= = =

= − + − + − + −

' '

j i i jv v f f (2.10)

where 𝑋𝑖 is the mole fraction of species 𝑖, 𝐷𝑖𝑗 the binary diffusion rate of species 𝑖 respect

to species 𝑗, and 𝐷𝑇,𝑖 the thermal diffusion rate of species 𝑖. The first term on the RHS

represents the relative species diffusion due to the different diffusion velocities. The second

term denotes the mole fraction gradient induced by the hydrostatic pressure gradient. The

third term on the RHS represents the gradient of the mole fraction contributed by the

variations of unit body forces. And the last term represents the relative species diffusion

due to thermal diffusions, known as the Soret effect, which is the mass diffusion induced

by the temperature gradient. The mole fraction of the 𝑖-th species, 𝑋𝑖, can be calculated

from the corresponding mass fraction 𝑋𝑖:

/i i iX YW W= (2.11)

where 𝑊𝑖 and �� are the molecular weight of the 𝑖-th species and the mean molecular

weight, respectively. The species molecular weight is a constant value determined by the

components of the species molecule. And the mean molecular weight can be calculated by

either mole fractions or mass fractions as ( )1

1 1

s sN N

i i i ii iW X W Y W

−

= == = .

Equation (2.10) is an equation system with the dimension of 𝑁𝑠 . It is

computationally expensive to solve the exact solution. To reduce the computational cost,

simplified models are developed to estimate the diffusion velocity. The mixture averaged

model (Mathur et al. 1967) is used in the present study. It solves the species diffusion

velocity by the Fickian formula in the following manner:

35

( )1 1 1

,im iim i i i

i i

DD T X X Y p

X X T p

= − − = + − '

i,m i iv d d (2.12)

where 𝐯𝐢,𝐦′ is the diffusion velocity of the 𝑖-th species calculated by the mixture averaged

mode, 𝐷𝑖𝑚 the diffusion coefficient of species 𝑖 relative to the gas mixture, and Θ𝑖 the

thermal diffusion ratio. The first item represents the mass diffusion due to the mole fraction

gradient. The second term describes the thermal diffusion due to the temperature gradient.

Note that the mixture averaged formulas are an approximation, and they are not

constrained to satisfy that the net species diffusion flux is zero. Therefore, the mass

conservation is not preserved. A systematical correction term must be invoked in the

diffusion velocity:

= +' ' '

i i,m cv v v (2.13)

where 𝐯𝐜′ is the correction speed. To satisfy Equation (2.5):

1 1 1 1

0s s s sN N N N

i i i i

i i i i

Y Y Y Y= = = =

= + = + ' ' ' ' '

i i,m c i,m cv v v v v (2.14)

Therefore,

1

sN

i

i

Y=

= −' '

c i,mv v (2.15)

Equation (2.13) together with (2.12) and (2.15) provide the representation of the

species diffusion velocity. It can be solved by the mass fraction, pressure, temperature and

diffusion coefficients. Again, the coefficients 𝐷𝑖𝑚 and Θ𝑖 are pre-defined in a data set,

which can provide run time polynomial fitting on the temperature. The details will be

included in the next section.

36

2.2.5 Chemical Reactions

In Equation (2.4), the mass production rate, 𝜔𝑖 , need to be evaluated for every

species. A chemical reaction system is composed by a set of elementary reactions which

consume reactant species and produce product species. For a given elementary reaction k,

it can be represented by the following formula:

, ,

1 1

' ''s sN N

i k i i k i

i i

M M = =

(2.16)

where 𝜐𝑖,𝑘′ and 𝜐𝑖,𝑘

′′ are the stoichiometric coefficient of species 𝑖 appearing in reaction 𝑘 as

a reactant and product, respectively. The net production rate for the 𝑖-th species is the

summation of the production rate (appearing as a product) subtracting the destruction rate

(appearing as a reactant) in each elementary reaction, given by

( ), ,

1

'' 'rN

i i i k i k k

k

W =

= − (2.17)

where 𝑁𝑟 is the total number of reactions in a chemical reaction system and Ωk the progress

rate of reaction 𝑘, defined as

, ,' ''

, ,

1 1

i k i ks sN N

i ik f k b k

i ii i

Y Y

W W

= =

= −

(2.18)

where 𝜅𝑓,𝑘 and 𝜅𝑏,𝑘 are the forward and backward reaction rate constants, respectively.

The reaction rate constant is primarily a function of temperature. Only one of the forward

and backward reaction rate constants need to be determined, since the net reaction rate is

zero at equilibrium. In this study, the forward reaction rate constant is modeled by the

Arrhenius law:

37

, expkB kf k k

EA T

RT

= −

(2.19)

where 𝐴𝑘 , 𝐵𝑘 , 𝐸𝑘 are the rate coefficient constant, the temperature exponent, and the

activation energy of reaction k, respectively, and 𝑅 the universal gas constant. The

backward reaction rate constant can be obtained from the forward reaction rate constant

and the equilibrium constant, 𝐾𝑐,𝑘, by

, ,1'' '

,

, , , 0

,

1,

Nsi k i ki

f k

b k c k p k

c k

K KK R T

=−

= =

(2.20)

where 𝐾𝑐,𝑘 and 𝐾𝑝,𝑘 are the equilibrium constant of the 𝑘-th reaction in concentration unit

and pressure unit, respectively. The value of 𝐾𝑝,𝑘 can be determined by the Gibbs free

energy change that occurs in passing completely from reactants to products in reaction 𝑘.

It is a function of temperature and the detailed calculation will be introduced in the next

section.

The mass production rate of species 𝑖, 𝜔𝑖, can be fully represented by Equation

(2.17) to (2.20). It is primarily a function of temperature and species mass fractions.

2.2.6 Ideal Gas Equation of State

To further close the conservation equations, the ideal gas law is applied to

constraint pressure by density and temperature. The partial pressure of species 𝑖 is given

by:

i ii

i i

RT Y RTp

W W

= = (2.21)

where 𝜌𝑖 is the density of species 𝑖 in the mixture. The total pressure, 𝑝, can be obtained

by the summation of the partial pressures:

38

1 1

s sN N

ii

i i i

Y RTp p RT

W W

= =

= = = (2.22)

With the additional equations specified in this section, the independent variables

reduce to 𝜌, 𝐯, 𝑇, and 𝑌𝑖 (𝑖 = 1, ⋯ , 𝑁𝑠−1). The number of unknowns is 𝑁𝑠 + 2 (with one

vector variable 𝐯), which is identical to the number of conservation equations. Thus, the

equation system is complete and can be solved numerically with specific initial and

boundary conditions.

2.3 Numerical Representation of Transport Properties and Chemistry

In Section 2.2, additional equations are presented to close the conservation

equations. However, these equations also introduce new thermodynamic variables, such as

the diffusion coefficient, the thermal conductivity, the viscosity, the specific heat capacity,

the enthalpy, the equilibrium constants, and the reaction rate coefficients. The numerical

representations of these thermodynamic variables are discussed in this section.

2.3.1 Transport Properties Representation

To calculate the transport properties, the most rigorous method is the Boltzmann’s

equation of kinetic theory. However, the exact solution is costly and computationally

prohibitive in a large chemical system. Therefore, this study uses the mixture-averaged

model, which is an asymptotic solution of the Boltzmann equation by the first order

perturbation. A TRANSPORT package (Kee et al. 1986) developed in Fortran

programming language is applied to obtain the mixture-averaged transport properties in a

multi-species reaction system.

39

The dynamic viscosity 𝜇 in the stress tensor described by Equation (2.6) is given

by the following mixture-averaged approximation:

21 1 1

2 2 4

11

1, 1 1

8

s

s

Nji i i i

ijNi j j ij ijj

WX W

W WX

−

==

= = + +

(2.23)

where 𝜇𝑖 is the pure species viscosity. The mixture averaged viscosity can be modeled by

the pure species viscosities and mole fractions with the above equation.

Similarly, the mixture-averaged thermal conductivity, 𝜆 , in Equation (2.9) is

modeled by:

11

1 1

2

s

s

N

i i Ni i ii

XX

=

=

= +

(2.24)

where 𝜆𝑖 is the pure species conductivity of species 𝑖.

Base on the mixture-averaged model, the diffusion coefficient of species 𝑖 relative

to the mixture, 𝐷𝑖𝑚, in Equation (2.12) is given by

*

s

s

N

j jj i

im N

j ijj i

X WD

W X D

=

(2.25)

where 𝐷𝑖𝑗∗ is the binary diffusion coefficient of species 𝑖 relative to species 𝑗. Once the

binary diffusion coefficient is obtained, Equation (2.25) can be applied to each species to

calculate the mixture averaged diffusion coefficients.

The thermal diffusion ratio of the 𝑖-th species, Θ𝑖, in Equation (2.12) is modeled by

40

sN

i ij i j

j i

X X

= (2.26)

where 𝜃𝑖𝑗 is the binary thermal diffusion ratio of species 𝑖 relative to species 𝑗. The thermal

diffusion that induced by the Soret effect is modeled by this thermal diffusion ratio.

The above mixture-averaged transport properties of species 𝑖 require the

species-wise parameters, including the pure species viscosity, 𝜇𝑖 , the pure species

conductivity, 𝜆𝑖, the binary diffusion coefficient, 𝐷𝑖𝑗∗ , and the binary thermal diffusion ratio,

𝜃𝑖𝑗. The detailed calculations of these properties involve fundamental molecular dynamic

parameters, such as the Lennard-Jones collision diameter, the collision integrals, the

Lennard-Jones potential well and the dipole moments. Instead of the exact solution, the

TRANSPORT package performs a polynomial fitting on temperatures with pre-measured

properties and produces the corresponding fitting coefficients. During runtime, the package

calculates these properties by the fitting coefficients. Since the pure species viscosity, pure

species conductivity, and binary diffusion coefficient are sensitive to temperature, they are

fitted to the logarithm of temperature. And the binary thermal diffusion ratio is just fitted

against temperature since it is weakly dependent on temperature. The polynomial fitting is

performed with the following format:

41

( )

( )

( )

1

1

1

1

1*

,

1

1

,

1

ln ln

ln ln

ln ln

ft

ft

ft

ft

Nn

i ni

n

Nn

i ni

n

Nn

ij n ij

n

N

n

ij n ij

n

a T

b T

D c T

d T

−

=

−

=

−

=

−

=

=

=

=

=

(2.27)

where 𝑁𝑓𝑡 indicates the number of polynomial terms in the fitting fictions, including the

zero order fitting coefficients. A third-order fitting is used in this study, thus 𝑁𝑓𝑡 = 4.

Based on Equation (2.27), all the transport properties involved in the conservation

equations can be calculated on-the-fly with given local temperature.

2.3.2 Chemistry Representation

The chemical reaction system needs to be numerically represented to provide the

chemical reaction information and the thermal properties used in Equation (2.17)-(2.20).

These parameters can be obtained from detailed chemical mechanisms together with the

CHEMKIN Fortran package (Kee, Miller, and Jefferson 1980).

A detailed chemical mechanism includes three pieces of information: (1) atom

elements involved in the reaction system; (2) a complete list of species involved in the

elementary reactions and their corresponding molecular weights, 𝑊𝑖 ; (3) a set of

elementary reactions. This reaction information provides the stoichiometric coefficients,

𝜐𝑖,𝑘′ and 𝜐𝑖,𝑘

′′ , for each species involved in a given reaction and the corresponding forward

reaction rate parameters, 𝐴𝑘, 𝐵𝑘 and 𝐸𝑘, indicated in Equation (2.19).

Besides the detailed chemical mechanism, CHEMKIN package requires an

additional data set to calculate the species thermal properties. Similar as the TRANSPORT

42

package, the CHEMKIN package utilizes a polynomial fitting on temperature to provide

the thermal properties. The fitting formulas are described in the following equations. Note

that CHEMKIN performs the polynomial fitting on the mole based thermal properties. In

the following discussion, the mole based thermal properties are indicated by uppercase

letters, and the mass based thermal parameters are denoted by lowercase letters.

For a given species 𝑖 , the specific heat capacity at constant pressure, 𝐶𝑝,𝑖0 , is

provided by:

( )0

, 1

1

fcNp i n

ni i

n

CT

R

−

=

= (2.28)

where the superscript 0 denotes the thermal properties at the reference pressure, which is

1 atm, 𝑁𝑓𝑐 the order of the polynomial fitting for thermal properties, and 𝛼𝑛𝑖 the 𝑛-th

fitting coefficient of species 𝑖.

Based on the specific heat capacity, the species enthalpy can be calculated by

Equation (2.8) with 0 as the reference temperature:

( )101,0

,0

1

1 fci fc

N nT N ii ni i

p i

ni i i

H TC dT

RT RT n T

−

+

=

= = + (2.29)

It requires an additional parameter 𝛼𝑁𝑓𝑐+1,𝑖 to close the integration constant.

Similarly, the species entropy can be obtained by the integration of heat capacity

divided by the temperature:

( ) ( )0 0 10,

1 2,298

2

2981ln

1

fci

fc

N nT p i ii ni i

i i N i

n

C SS TdT T

R R T R n

−

+

=

= + = + +−

(2.30)

where 𝑆𝑖0(298) is the 𝑖-th species entropy per mole at 298 K and 1 atm, and 𝛼𝑁𝑓𝑐+2,𝑖𝑅 is

the evaluation of the standard-state entropy at 298 K.

43

In CHEMKIN, a fifth order polynomial fitting is applied to the heat capacity, which

indicates 𝑁𝑓𝑐 = 5. Therefore, there are 7 parameters tabulated for each species in the

thermal data set. Based on the value at the reference state, the species thermal properties at

any conditions and the corresponding mass based properties can be calculated by the

following equations (given the specific heat and enthalpy are independent of pressure):

,0

, , ,

0

0

,

,

ln ,

p i

p i p i p i

i

ii i i

i

i ii i i

atm i

CC C c

W

HH H h

W

p SS S R s

p W

= =

= = = − =

(2.31)

where 𝑝𝑎𝑡𝑚 is the value of the pressure at 1 atm. All the other thermal properties can be

calculated from parameters in Equation (2.31). For example, the equilibrium constant, 𝐾𝑝,𝑘,

in Equation (2.20) can be determined by the Gibbs free energy change that occurs in

passing completely from reactants to products in reaction 𝑘:

( ) ( )0 0 0 0

, , , , ,

1 1

exp exp '' ' '' 's sN N

k k i ip k i k i k i k i k

i i

S H S HK

R RT R RT

= =

= − = − − −

(2.32)

According to the fitting procedures described in this section, all the transport

properties, thermal properties and chemistry information can be numerically calculated

from the pre-defined data sets based on the local thermodynamic conditions. Therefore, all

the parameters in the conservation equations can be obtained numerically, and hence the

equation system is ready to be solved.

44

2.4 Numerical Implementations

With the complete set of governing equations and the numerical representations of

thermodynamic parameters, the conservation equations are ready to be solved numerically.

Based on the assumptions of zero body force, zero heat source and Newtonian fluid, the

conservation equations can be simplified to the following PDE system:

( ) ( ) ( )t

+ + =

UF U G U Q U (2.33)

where 𝐔 is the conservative variables, 𝐅 the convection fluxes, 𝐆 the diffusion fluxes, and

𝐐 the source terms. They are vector variables defined as:

( )

'

1 11 1 1

', , ,

0

0

s s s s sN N N N N

YY Y

Y Y Y

p

e e p

= = = = + − − +

vv

U F v G v Q

v vv I τ

q v τv

(2.34)

Note that the 𝑁𝑠 species equations are used in this study to replace the continuity equation

with the constraint given by Equation (2.5). In this section, the numerical implementations

to solve this PDE system will be discussed.

2.4.1 Adaptive Cell Discretization

In order to integrate Equation (2.33) numerically, the spatial domain is discretized

to computation cells to solve the convection, diffusion, and source terms. Once the

gradients of transport fluxes and the source terms are evaluated, discretized time steps are

performed for time integrations to update the conservative variables.

The size of the computation cells and time steps are limited by the smallest

phenomenon that a direct numerical simulation (DNS) needs to resolve. As discussed in

45

Chapter 1, in a combustion system, computational cells need to be one order of magnitude

smaller than the flame thickness in order to capture detailed flame structures in the thin

reaction zone. Moreover, in a turbulent combustion system, computational cells are also

required to be smaller than the Kolmogorov scale in order to capture the small vortexes.

On the other hand, the overall geometry in a practical simulation are orders of magnitude

larger than the smallest phenomenon. This leads to a multi-length scale nature in

combustion.

In a multi-length scale system, uniformly discretized computational cells are

largely wasted in regions where the gradients of local thermodynamic states are small.

Therefore, in this study, an adaptive cell discretization strategy is applied to allow the

numerical method to adapt cell resolutions based on the local length scale. The cell splitting

or merging is triggered by the first-order and second-order gradients of the local density,

temperature, velocity, and species concentrations. Upon a cell refinement, the dimension

of the cell is reduced to half, meaning that one cell is equally spitted from the center to 2

sub cells in 1D simulations and to 4 sub cells in 2D simulations.

The conservative variables, 𝐔, are solved at the center of cells. They are used to

evaluate the other terms in Equation (2.33) at a given time step.

2.4.2 Fractional Step

The source term, 𝐐 , in Equation (2.34) is determined by the species mass

production rates from chemical reactions. Due to the rapid chain brunching and chain

propagation reactions, characteristic times of fast reactions can be sub-nano seconds, which

introduces a strong stiffness in this source term. Therefore, the time integration of the

46

source term, 𝐐, is usually performed separately from the transport terms. This procedure is

called the fractional step (Kim and Moin 1985), which is described below.

In the fractional step procedure, the time derivation of the conservative variables,

𝐔, are separated into two sources:

( ) ( ) ( )( )( ) , ( )t

= + = − +

UT U Q U T U F U G U (2.35)

where 𝐓 indicates the change of conservative variables due to the transport effect. In the

first fractional step, only 𝐓 is included in the PDE to solve the system as a non-reactive

system based on the initial conditions 𝐔𝑛 at the 𝑛-th time step:

( ) n

t

= =

UT U U U (2.36)

The numerical integration of the above PDE with given time step, ∆𝑡, produces an

intermediate solution of the conservative variable, indicated by 𝐔∗ . This intermediate

solution will be passed as the initial condition to the second fractional step to solve the

chemical reactions. Note that the chemical reaction is a local process, which only depends

on the local thermodynamic states. Therefore, the equations are reduced to an ODE system,

written as:

*( )d

dt= =

UQ U U U (2.37)

The value of conservative variables at time step 𝑛 + 1, 𝐔𝑛+1, can be obtained by

the numerical integration of Equation (2.37). The fractional step procedure can be

represented by the following notation:

47

( )1n t t n+ =U Q T U (2.38)

where 𝐓∆𝑡 and 𝐐∆𝑡 represent the numerical integration of Equation (2.36) and (2.37) for a

given time step, ∆𝑡, respectively.

The fractional step procedure can decouple the stiff chemical reaction terms from

the transport equations, thus simplifies the computation process. However, this process can

be mathematically demonstrated to be first order accurate on the time step ∆𝑡. A similar

splitting method with the second order accuracy is given by the Strang splitting (Strang

1968):

( )1 2 2n t t t n+ =U Q T Q U (2.39)

Based on the above splitting process, the conservative variables can be updated numerically

step by step.

The time integration of the reaction source term, Q, described in the ODE (2.37)

remains as the major challenge in combustion modeling with detailed chemical kinetics

due to the large dimension of the stiff and non-linear reaction systems. This present study

primarily aims to reduce the computational cost of this ODE integration. The methods

developed to handle the ODE integration will be discussed in the following chapters. In

the rest of this section, the discussion focuses on the numerical solution of the PDE system

described in Equation (2.36). In this study, the PDE system is discretized by a classic finite

volume method for the numerical integration. The finite volume discretization is described

in the next sub section.

48

2.4.3 Finite Volume Method

To solve the PDE of transport terms, the finite volume method applies volume

integral on Equation (2.36) over a computation cell, given by the following integral form

of the conservation equation:

0i i iv v v

dv dv dvt

+ + =

U

F G (2.40)

where vi is the volume of the computation cell 𝑖 . Using Gauss’s theorem, the volume

integral of a divergence term can be converted to the surface integral. Thus, the integral of

convection flux, 𝐅, can be written as:

∫ ∇ ⋅ F dv𝑣𝑖

= ∮ F ⋅ n ds𝑠𝑖

= ∑ F𝑘,𝑖 𝑆𝑘,𝑖𝐾𝑘=1 (2.41)

where 𝑆𝑖 is the surface of the 𝑖-th cell, 𝐧 the surface normal vector, 𝐾 the total number of

surfaces on the discretized cell 𝑖, F𝑘,𝑖 the convection flux evaluated on the 𝑘-th surface of

cell 𝑖 (positive sign for the outbound flux and negative sign for the inbound flux), and S𝑘,𝑖

the surface area of the 𝑘-th surface of cell 𝑖. The same conversion also applies to the

diffusion flux, 𝐆. Equation (2.40) can be written as:

, , , ,

1 1

1 10

K Ki

k i k i k i k i

k ki i

dS S

dt v v= =

+ + = U

F G (2.42)

where ��𝑖 is the cell averaged conservative variables, defined as:

1

i

iv

i

dvv

= U U (2.43)

The conservations of mass, momentum and energy are naturally preserved in the

finite volume method since the flux entering a given cell is identical to that leaving the

adjacent cell.

49

Equation (2.42) involves the evaluation of convection and diffusion fluxes on the

cell surfaces. However, the conservative variables are calculated at the center of cells.

Therefore, numerical schemes are required to construct the numerical flux on cell surfaces.

To simplify the discussion of numerical schemes, we consider a one-dimensional domain.

The second and third term can be simplified to the following expressions:

, , 1 1

1 2 2

, , 1 1

1 2 2

1 1

1 1

K

k i k ii i

ki i

K

k i k ii i

ki i

Sv x

Sv x

+ −=

+ −=

= −

= −

F F F

G G G

(2.44)

where ∆𝑥𝑖 is the size of the 𝑖-th grid. And 𝑖 +1

2 and 𝑖 −

1

2 subscripts denote the right and

left boundary of the 𝑖-th grid, respectively. The numerical scheme to construct fluxes on

grid boundaries will be discussed in the next sub section.

2.4.4 Numerical Schemes

The construction of numerical fluxes on a grid boundary is a typical Riemann

problem, described as: evaluating the value of a function 𝑓(𝑥) = 𝑓(𝑢(𝑥)) at 𝑥 = 𝑥𝑖+1 2⁄ ,

given the discontinuity of 𝑢(𝑥), where

( )1 2

1 2

,

,

L i

R i

u x xu x

u x x

+

+

=

(2.45)

In combustion modeling, upwind numerical schemes are usually applied to solve

the Reimann problem, such as the monotonic upstream-centered scheme for conservation

Laws (MUSCL) scheme (van Leer 1979) and the Harten-Lax-van Leer-contact (HLLC)

solver (Toro, Spruce, and Speares 1994). In this study, a third-order upwind weighted

essential non-oscillatory (WENO) method (Jiang and Shu 1996; Shu 1999) is utilized to

construct the flux on grid boundaries. The WENO method involves averaging numerical

50

stencils weighted by the smoothness of the flux. It can adjust the non-linear weights

automatically based on the local measurements of smoothness. Thus, the WENO scheme

performs well in the regions where the change of flux is slow without losing the capability

of capturing sharp discontinuities, such as shock waves and detonations.

However, the WENO scheme is mostly used in the form of uniform grid size, which

doesn’t apply to this study since the local adaptive cells are involved. Therefore, a WENO

scheme based on the non-uniform grids is derived in this study from the fundamental

WENO reconstruction. The scheme is described below.

In a third-order WENO scheme, the numerical flux is obtained by the weighted

average of two second-order ENO stencils:

( ) ( )0 1

1 0 1 1 1

2 2 2

ˆ ˆ ˆi i i

f w f w f+ + +

= + (2.46)

where 𝑓𝑖+

1

2

is the numerical flux on the boundary between grid 𝑖 and 𝑖 + 1, 𝑓𝑖+

1

2

(0) and 𝑓

𝑖+1

2

(1)

the two ENO stencils, and 𝑤0 and 𝑤1 the corresponding weights of the ENO stencils. The

hat (^) in the expression indicates the value is numerically constructed.

In non-uniform grids, the two ENO stencils are given by:

( )

( )

0 11 1

1 12

1 11 1

1 12

ˆ

2ˆ

i ii i

ii i i i

i i ii i

ii i i i

x xf f f

x x x x

x x xf f f

x x x x

++

++ +

−−

+− −

= + + +

+ = − +

+ +

(2.47)

where 𝑓��, 𝑓𝑖+1 , and 𝑓𝑖−1

, are the cell averaged flux in cell 𝑖, 𝑖 + 1, and 𝑖 − 1 respectively.

They can be estimated by the values at the center of cells.

The weights in Equation (2.46) can be calculated in the following manner

51

( )1 2

0

ˆˆ, , 0,1

ˆ

r rr r

rk

k

w dw w r

w

=

= = =+

(2.48)

where ��𝑟 , 𝑑𝑟 , and 𝛽𝑟 are the unnormalized weight, the weight coefficient, and the

measurement of the smoothness of the 𝑟-th ENO stencil, respectively. 휀 is a small and

positive real number to avoid the denominator being zero. In this study, the value 10−6 is

chosen.

The weight coefficients, 𝑑𝑟, are defined by the cell sizes:

10

1 1

11

1 1

i i

i i i

i

i i i

x xd

x x x

xd

x x x

−

− +

+

− +

+ = + +

=

+ +

(2.49)

The smoothness of the stencils, 𝛽𝑟 , are measured by the variations of the cell

averaged fluxes:

( )

( )

2

0 1

1

2

1 1

1

2

2

ii i

i i

ii i

i i

xf f

x x

xf f

x x

+

+

−

−

= −

+

= −

+

(2.50)

When the computational cells are uniform (∆𝑥𝑖−1 = ∆𝑥𝑖 = ∆𝑥𝑖+1), this scheme is

reduced to the normal 3rd WENO scheme:

52

( ) ( )

( ) ( )

0 1

1 1 1 1

2 2

0 1

2 2

0 1 1 1

1 1 1 3ˆ ˆ,2 2 2 2

2 1,

3 3

,

i i i ii i

i i i i

f f f f f f

d d

f f f f

+ −+ +

+ −

= + = − +

= = = − = −

(2.51)

The above equations describe the 3rd order upwind WENO scheme with the positive

wind direction. If the wind direction is negative, the numerical flux 𝑓𝑖+

1

2

can be obtained

as the exact mirror image with respect to the boundary 𝑥𝑖+

1

2

. In this study, we perform a

smooth flux splitting on 𝑓(𝑢) to obtain the positive and negative fluxes:

( ) ( ) ( )f u f u f u+ −= + (2.52)

where 𝑓+(𝑢) and 𝑓−(𝑢) are the positive and negative wind fluxes, respectively, which

satisfy 𝜕𝑓+(𝑢) 𝜕𝑢⁄ ≥ 0 and 𝜕𝑓−(𝑢) 𝜕𝑢⁄ ≤ 0. A Lax-Friedrichs flux splitting method is

applied to separate the positive wind flux from the negative wind flux:

( ) ( )( )1

2f u f u au = (2.53)

where 𝑎 is the largest characteristic velocity across all computational cells:

( )maxua f u u= (2.54)

This completes the description of the 3rd order upwind WENO scheme for a scalar

equation. However, the PDE in Equation (2.36) is an equation system with vector variables

and functions. The scalar discretization cannot be directly applied. Fortunately, In the

hyperbolic system described in Equation (2.36), there is a complete set of left and right

eigenvectors and the eigenvalues of the Jacobian matrix 𝜕𝐅(𝐔) 𝜕𝐔⁄ are all real. It allows

us to obtain the flux in the characteristic form by applying the WENO scheme on each

53

scalar field in the characteristic transformation. Therefore, in this study the numerical

fluxes are constructed in the characteristic field, followed by the transformation to the

physical field by multiplying the right eigenvectors.

Based on the 3rd order WENO scheme, the transport term in Equation (2.36), 𝐓(𝐔),

can be numerically calculated. To solve the time integration of the conservative variables,

𝐔, a 3rd order total variation diminishing (TVD) Runge-Kutta method is utilized. Given the

cell averaged value ��𝑖𝑛 in cell 𝑖 at time step 𝑛 and the differential equation:

( )n

nii

d

dt=

UT U (2.55)

The cell averaged value at time step 𝑛 + 1 is calculated by the following steps:

( ) ( )( ) ( ) ( )( )

( ) ( )( )

1

2 1 1

1 2 2

3 1 1

4 4 4

1 2 2

3 3 3

n n

i i i

n

i i i i

n n

i i i i

t

t

t+

= +

= + +

= + +

U U T U

U U U T U

U U U T U

(2.56)

where ��𝑖(1)

and ��𝑖(2)

are two intermediate integration values of the conservative variable.

The Runge-Kutta method is an explicit time integration, thus the CFL condition must be

satisfied for the integration stability:

CFL 1t

ax

=

(2.57)

where 𝑎 is the largest characteristic velocity defined in Equation (2.54).

2.4.5 ASURF Program

In this study, an in-house adaptive simulation of unsteady reactive flow (ASURF)

program (Chen 2009) is utilized as the codebase and test bed for the development of

advanced numerical methods. It features the adaptive cells, fractional splitting steps, finite

54

volume discretization, and the Runge-Kutta method. Without specific mentioning, the

numerical methods developed in the following chapters are implemented in ASURF.

55

Chapter 3

Correlated Dynamic Adaptive Chemistry (CO-DAC) Method

Based on the detailed discussion in previous chapters, the major challenge of

utilizing detailed chemical kinetics in combustion simulations is introduced by the

non-linear and stiff reaction source term in the ODE system described in Equation (2.37).

To simplify the ODE system, an innovative correlated dynamic adaptive chemistry

(CO-DAC) method is developed in this chapter to provide on-the-fly chemical reductions

with negligible computational costs. A concept of correlated space in both spatial and

temporal coordinates is proposed by using a few key phase parameters which govern the

low, intermediate, and high temperature chemistry. Locally reduced mechanisms are

generated dynamically in each correlated group, instead of in each computational cell, to

accelerate the on-the-fly chemical reduction. The proposed CO-DAC method not only

provides the flexibility and accuracy of the locally reduced kinetic models but also avoids

redundant model reductions in time and space when the chemistry is frequently correlated

in phase space. The detailed methodology and validations are presented in this chapter.

3.1 Introduction and Motivation

Recently, significant amount of work has been conducted to develop detail

chemical mechanisms including the low temperature chemistry in order to model the

turbulence-chemistry interaction in low temperature (700 K) to intermediate temperature

(1100 K) range with practical engine pressures (1 atm - 50 atm). These detailed chemical

kinetics of real transportation fuels are usually large, which involve hundreds of species

56

and thousands of reactions. Unfortunately, the large number of species and the strong

stiffness in the chemical kinetics result in a great challenge in combustion modeling. Even

with the availability of supercomputing capability at petascale and beyond, numerical

simulations with such large kinetic mechanisms remain infeasible.

In order to utilize the large and detailed chemical mechanisms in combustion

modeling, many model reduction methods have been developed in the last two decades.

These methods have been discussed in Section 1.3. Among them, the graphic reduction

based dynamic adaptive chemistry (DAC) methods can automatically conduct model

reduction on-the-fly and provide locally reduced mechanisms, which greatly reduce the

complexity of chemical integrations.

Unfortunately, the computational complexity of the graphic based model reduction

method with the first order accuracy is 𝑂(𝑁𝑠 × 𝑁𝑟). Thus, when a kinetic mechanism is

large, the graphic reduction based DAC method becomes computationally expensive. On

the other hand, the previously developed DAC methods utilize the traditional implicit ODE

solvers, such as the VODE method, to conduct chemical integrations based on the locally

reduced mechanisms. Since the computational cost of the traditional implicit ODE method

is proportional to the cubic of number of species in the reaction system, the direct chemical

integration is still expensive even with the help of locally reduced mechanisms.

Therefore, the goal of this study is to develop a correlated dynamic adaptive

chemistry (CO-DAC) method by utilizing the similarities of reaction pathways in both

temporal and spatial coordinates to dramatically increase the efficiency of the on-the-fly

model reduction while retaining the high accuracy of chemical integrations. At first, a

concept of correlated space is proposed by using temperature, equivalence ratio, and a few

57

key intermediate species. Base on the correlated space, a correlated model reduction in

temporal and spatial coordinates is conducted on-the-fly by the multi-generation path flux

analysis (PFA) method (Sun et al. 2010), which enables both the first and second order

accuracy of the path flux. To further increase the computational efficiency, the hybrid

multi-timescale (HMTS) method (Gou et al. 2010) is applied to replace the VODE method

for the chemical integration based on the locally reduced chemistry. The proposed

HMTS/CO-DAC method is validated against the VODE, VODE/DAC, HMTS, and

HMTS/DAC methods in simulations of ignitions and unsteady flame propagations with

real jet fuel surrogate mechanisms. Finally, the computational accuracy and efficiency are

examined.

3.2 Numerical Methods

The mathematical formulation and numerical implementation of the CO-DAC

method is presented in this section. To mathematically define the CO-DAC method, the

ODE system that governs chemical reactions need to be specified. According to the

fractional step procedure discussed in Section 2.4.2, chemical reactions are solved

independently of the transport terms. By ignoring the transport terms, Equation (2.37) can

be written in the weak conservative format for species:

, 1, ,i is

dYi N

dt

= = (3.1)

And the energy conservation equation becomes:

58

( )

1 1 1 1

10

s s s sN N N Ni i i i

i i i i

i i i iv

d Ye dY dede dT dTe Y e

dt t dt dT dt dt c

= = = =

= = = + = −

(3.2)

where 𝑒𝑖 is the internal energy of species 𝑖 per unit mass, and 𝑐�� the mean specific heat

capacity at constant volume. They are defined as:

i i

i

RTe h

W= − (3.3)

, , ,

1

,sN

v i v i v i p i

i i

Rc Yc c c

W=

= = − (3.4)

where ℎ𝑖 and 𝑐𝑝,𝑖 can be obtained from Equation (2.31).

Equation (3.1) together with Equation (3.2) form the ODE system, which need to

be solved in the chemical fractional step. The dimension of this ODE system is 𝑁𝑠 + 1.

The computational cost to integrate this ODE system is strongly dependent on the total

species number, 𝑁𝑠 . The goal of the CO-DAC method is to generate locally reduced

mechanisms in the correlated space to minimize the number of species, i.e. the dimension

of the ODE system, so that the chemical integration can be performed more efficiently

without generating large computational overhead for the on-the-fly model reduction.

3.2.1 Mathematical Formulation of CO-DAC Method

As stated above, the computational cost to integrate the ODE system highly

depends on 𝑁𝑠 . And the total computational cost of chemical integrations, 𝑡𝑐ℎ𝑒𝑚 , in a

numerical simulation can be estimated by the complexity of the ODE solver, the total time

steps, 𝑁𝑡, and the total number of cells 𝑁𝑐:

chem s t ct N N N (3.5)

where 𝛼 is the exponential order of the solver complexity, which is 1 for the linear scaling

solvers and 3 for the traditional implicit ODE solvers. In a combustion simulation, the total

59

number of time steps and computational cells are controlled by the domain of interest as

well as the temporal/special resolutions, which are more or less constant. Therefore, to

accelerate chemical integrations, it is critical to reduce the reaction system on-the-fly so

that a smaller 𝑁𝑠 can be used in the local chemical solver.

To generate the locally reduced chemical mechanism on-the-fly, the graphic based

reduction methods, such as the DRG and PFA method, are often used. In such methods,

locally reduced mechanisms are generated by selecting a set of important species and

reactions from the original detailed chemical mechanism. The total computational time to

generate the dynamic adaptive chemistry, 𝑡𝐷𝐴𝐶, is proportional to

m

DAC o o t ct N I N N (3.6)

where 𝑁𝑜 and 𝐼𝑜 are the total number of species and reactions, respectively, in the original

detailed chemical mechanism, and 𝑚 the exponential scaling factor of the chemical

reduction method, which is 1 for the first order accurate reduction and 2 for the second

order accurate reduction. The number of 𝑁𝑜 can be much larger than the size of the locally

reduced mechanism, 𝑁𝑠. Therefore, in a large chemical reaction system, it is possible that

the computational cost of the DAC reduction can exceed the computational cost of the

chemical integration. Especially when a high-order model reduction approach is employed

together with a linear scaling ODE solver. In this case, the DAC method is computationally

inefficient to be used in combustion modeling with detailed chemical mechanisms.

Therefore, in order to apply the DAC method in practical combustion simulations,

a correlated dynamic adaptive chemistry (CO-DAC) method is proposed in this study to

improve the efficiency of the on-the-fly chemical reduction. The basic principle of the

CO-DAC method is described as following. It is well known that in a given fuel mixture,

60

the major reaction pathways depend on the local temperature, pressure, and species

compositions. In an unsteady, multi-dimensional reaction system, similar thermodynamic

conditions may occur repeatedly in the special and temporal coordinates. Thus, it is a great

waste to generate reduced mechanisms on-the-fly at every time step and every

computational cell. Instead, only a few correlated chemical reductions are needed to be

generated at unique thermodynamic conditions. And these reduced mechanisms can be

reused in other local reductions if the local thermodynamic states are correlated. As such,

the total computational cost of the on-the-fly model reduction in the CO-DAC method,

𝑡𝐶𝑂−𝐷𝐴𝐶, will be reduced to

( )( )m

CO DAC o o t t c ct N I N N − (3.7)

where 𝛿𝑡 and 𝛿𝑐 are correlation ratios in the temporal and spatial coordinate, respectively.

The values of 𝛿𝑡 and 𝛿𝑐 depend on the degree of the chemical similarities in the physical

coordinate. Since in a chemical reaction system the reaction pathways converge to a low

dimensional manifold, there are massive duplications of the thermodynamic states in the

compositing space. Therefore, 𝛿𝑡 and 𝛿𝑐 are two small numbers, which lead to a much

lower computational cost compared to Equation (3.6).

The basic idea of the CO-DAC method is schematically shown in Figure 3.1. The

local reaction system is represented by a set of phase parameters, which define the

correlations between different computational cells in time and space. A correlated space is

defined by the span of the phase parameters. Then a set of correlation thresholds of the

phase parameters are defined to discretize the correlated space. Each discretized hypercube

in the correlated space represents a correlated group. Only one chemical reduction is

required for all the computational cells that belong to the same correlated group to avoid

61

redundant calculations. The procedure to conduct the correlated chemical reduction is

stated as following: if the phase parameters of a given computational cell are correlated to

another cell at the same time step (space correlation) or to the same cell at the previous

time step (time correlation) where a reduced chemical mechanism has been generated, then

the same reduced mechanism will be reused for the local ODE integration; Otherwise, if

neither time nor space correlation is found, a locally reduced mechanism will be generated

and stored as a new correlated group in the phase space.

The success of the CO-DAC method depends on the definition of the phase

parameters. In order to construct an appropriate phase space, we need to understand which

parameters are important and govern the reaction pathways in low, intermediate, and high

temperature ranges. First, due to the Arrhenius law, temperature is always a dominant

factor in a chemical reaction system. Second, the radial pool is strongly affected by the fuel

concentration. Third, with a given temperature and fuel concentration, the ignition

transitions from low temperature to intermediate temperature, and from intermediate

temperature to high temperature are governed by concentrations of a few key intermediate

species and radicals. The typical reaction pathway of a hydrocarbon fuel with low

temperature chemistry are shown in Figure 3.2. It shows that CH2O and HO2 are the key

species dominate the low temperature chemistry. And OH is an important radical in the

entire temperature range. In fact, a recent study shows that the low temperature ignition of

various jet fuel surrogate mixtures can be correlated to OH and HO2 concentrations (Won

et al. 2014). Therefore, we choose temperature and the mass fractions of fuel, OH, CH2O

and HO2 as the key phase parameters to construct the correlated space.

62

As mentioned, the correlated group in time and space can be identified by using a

set of user specified threshold values, 𝛆 = (휀𝑇 , 휀𝐹𝑢𝑒𝑙, 휀𝑂𝐻, 휀𝐶𝐻2𝑂 , 휀𝐻𝑂2)

𝑇. The correlated

space is discretized to hypercubes with the size of 𝛆. If two computational cells fall into

the same hypercube, the local chemical kinetics in these two cells are considered to be

correlated and the same reduced mechanism will be reused.

In the CO-DAC procedure, the time correlation will be examined and applied first:

we compare the local phase parameters at the current time step, 𝑛, to the phase parameters

stored at the previous time step, 𝑛 − 1, in each computation cell and define the variation

as:

2 2

2

2 2

2

0

1

0

, , 1

0

, , 1

0

, , 1

0

, , 1

ln ln

ln ln

ln ln

ln ln

n n

T

Fuel n Fuel n

Fuel

OH n OH n

OH

CH O n CH O n

CH O

HO n HO n

HO

T T

Y Y

Y Y

Y Y

Y Y

−

−

−

−

−

−

− − =

−

−

(3.8)

where the 0 superscript denotes the phase parameters stored at the previous time step. If

1

, the local chemistry is considered time correlated, and we can simply pass the

reduced mechanism at time step 𝑛 − 1 to time step 𝑛.

For the rest of the cells which are not time correlated, the space correlation is

applied. The total number of space correlated groups in spatial coordinate, 𝐌 =

(𝑀𝑇 , 𝑀𝐹𝑢𝑒𝑙, 𝑀𝑂𝐻, 𝑀𝐶𝐻2𝑂 , 𝑀𝐻𝑂2)

𝑇, and the correlated group index of the 𝑖 -the cell,

63

𝐦𝒊 = (𝑚𝑇𝑖 , 𝑚𝐹𝑢𝑒𝑙

𝑖 , 𝑚𝑂𝐻𝑖 , 𝑚𝐶𝐻2𝑂

𝑖 , 𝑚𝐻𝑂2

𝑖 )𝑇, can be obtained by the following equations,

respectively.

max minmax min

minmin

ln ln1, 1

ln ln1, 1

j j

T j

T j

iij ji i

T j

T j

Y YT TM M

Y YT Tm m

−−= + = +

−−= + = +

(3.9)

where 𝑗 = 𝐹𝑢𝑒𝑙, 𝑂𝐻, 𝐶𝐻2𝑂, and 𝐻𝑂2. According to the group index 𝐦𝑖 , existing space

correlated groups will be recorded by a hash table. The key in the hash table is the

correlated group index and the corresponding value, 𝑣, in the table is the physical cell index

of the first cell in this correlated group. If the correlated group index of a computational

cell already exists, this cell will reuse the local reduced mechanism in cell 𝑣. Otherwise the

reduced mechanism will be generated on-the-fly, followed by the insertion of the

corresponding key-value mapping in the hash table.

The algorithm to generate the space correlated groups in phase space can be

schematically represented by the following pseudo code:

HashTable correlated = new HashTable<int[], int>();

for int i = 1 to max cell Nc

int[] m = getGroupIndex(Y(i), T(i))

if correlated.containsKey(m)

int v = correlated.get(m)

reduced mechanism at cell i = reduced mechanism at cell v

else

calculate reduced mechanism at cell i

correlated.add(m, i)

64

The average and amortized time complexity of the hash table operations, including

the insertion and lookup, is 𝑂(1). Therefore, it is negligible compared with the detailed

calculation of the chemical reduction, which is 𝑂(𝑁𝑜𝑚𝐼𝑜) given by Equation (3.6). The

space complexity is the size of the hash table. It equals the number of correlated groups

𝛿𝑐𝑁𝑐 . Since the correlation rate 𝛿𝑐 is a number less than one, the space complexity is

bounded by the number of cells 𝑂(𝛿𝑐𝑁𝑐) ≤ 𝑂(𝑁𝑐) . Therefore, we can construct the

correlated groups on-the-fly without generating expensive overheads.

The threshold values used in this study are: 20 K for 휀𝑇 and 0.05 for 휀𝑗. Based on

the sensitivity analysis, the most sensitive threshold value is 휀𝑇. When the threshold values

locate in a reasonable range, where 휀𝑇 < 50 and 휀𝑗 < 0.2, variations of the computational

accuracy are negligible.

Compared with the cell clustering methods discussed in Section 1.3.3, which also

partition computational cells based on their thermodynamic states in the phase space, the

major differences and advantages of the proposed CO-DAC method are: (1) unlike the cell

clustering methods, the CO-DAC method agglomerates cells with similar phase parameters

to correlated groups in order to obtain the correlated reduced chemical mechanism, instead

of conducting the numerical mapping of the local integration results. The subsequent ODE

integration in the CO-DAC method is performed directly at each physical cell based on the

locally reduced mechanism. All the important reaction pathways related to the phase

parameters are retained and integrated by the HMTS method accurately. Therefore, the

CO-DAC method avoids the backward mapping error, which is the major drawback of the

cell clustering methods, and guarantees the conservation of mass and energy. (2) In the

CO-DAC method, the phase parameters (T, Fuel, CH2O, HO2 and OH) cover the low,

65

intermediate, and high temperature chemistry by the set of key species and radicals. While

in the cell clustering methods, the phase parameters do not include the key radicals and

intermediate species at the low temperature region. The cell clustering methods may work

reasonably well for high temperature flames. However, they will become problematic when

the low temperature chemistry is involved since the solution can be non-monotonic and

multi-valued, and hence cannot be simply extrapolated linearly or nonlinearly. Therefore,

the proposed CO-DAC method can avoid the difficulty of nonlinear extrapolations by

directly integrating the ODE system with the locally reduced chemistry.

Furthermore, compared with the ISAT method described in Section 1.3.2, which

performs the on-the-fly storage and retrieval of the chemical integration results, the

proposed CO-DAC method just simply passes the on-the-fly reduced mechanism from the

phase space to the physical domain. And the rest of the ODE integrations are locally

conducted by the chemical solver. Therefore, the advantages of the CO-DAC method are:

(1) no interpolation errors in the result retrievals , (2) no CPU time required to perform the

high-dimensional table lookup and interpolation, and (3) no memory required to store the

ODE integration results and the local mapping gradients.

The differences between the cell clustering methods, the ISAT method and the

proposed CO-DAC method are summarized in Figure 3.3. Moreover, the main purpose of

the CO-DAC method is to provide an efficient, flexible, and robust way to conduct the

chemical reduction on-the-fly. It can be integrated with any efficient ODE solvers such as

the HMTS method.

66

3.2.2 Kinetic Model and Fuel Mixtures

The mathematical formulation and numerical implementation of the CO-DAC

method have been discussed above. To validate the proposed CO-DAC/HMTS method,

direct numerical simulations with large chemical mechanisms are carried out. A Real Fuel

2 mechanism (Dooley et al. 2012), which is a kinetic model for a real jet fuel surrogate

consisting of four component fuels (40% n-dodecane, 30% iso-octane, 23% n-propyl

benzene, and 7% 1,3,5-trimethyl by mole fraction), is utilized in the validation. Both the

detailed Real Fuel 2 mechanism (2051 species and 8428 reactions) and the

comprehensively reduced one (425 species and 2275 reactions) are employed in the

calculation to test the performance of the CO-DAC/HMTS method. Numerical simulations

of homogeneous ignitions and unsteady outwardly propagating spherical flames are carried

out to demonstrate the accuracy and robustness of the proposed algorithm. The validation

results are shown in the next section.

3.3 Results and Discussion

In order to validate the algorithm and test its performance, the proposed HMTS/CO-

DAC method is compared to the VODE, VODE/DAC, HMTS, and HMTS/DAC method

to examine the computation accuracy and efficiency for both detailed and comprehensively

reduced mechanisms. The homogeneous ignition of Real Fuel 2 at different initial

temperatures is modeled by the methods mentioned above to evaluate the proposed method

in the zero-dimensional simulation without transport terms. To further test the effect of the

computational dimension on the algorithm efficiency, calculations of an outwardly

propagating premixed spherical flame in the stoichiometric Real Fuel 2/air mixture with

67

and without CO-DAC are carried out to validate the CO-DAC method in the

one-dimensional combustion simulation. Moreover, the comparisons of the first and

second order PFA reductions are presented.

Figure 3.4 shows the detailed histories of active species as well as the accumulated

mechanism reduction time calculated by the CO-DAC and DAC method during an auto

ignition process of the stoichiometric Real Fuel 2/air mixture using the comprehensively

reduced mechanism. The initial pressure and temperature are 10 atm and 800 K,

respectively. This figure clearly shows that the number of active species varies significantly

at different ignition stages. Initially, only half of the total species are involved in the

reduced mechanisms. However, when the first stage ignition occurs at 1 ms, the number

of active species increases rapidly due to the low temperature chemistry. After the second

stage ignition at 3.2 ms, the system approaches equilibrium state and only a few species

are active. This result demonstrates clearly that large locally reduced mechanisms are only

required around the low and high temperature ignitions. When the computational cell is far

away from the thermal runaway or is at equilibrium state, a significantly small local

mechanism is sufficient to represent the reaction system. Therefore, the gobally reduced

kinetic mechanism is not locally optimized because it needs to keep a large set of species

and reactions in order to guarantee the accuracy of the mechanism at different local

conditions. Moreover, compared with the DAC method (dash lines), the proposed

CO-DAC method is more than two orders of magnitude faster in terms of the on-the-fly

model reduction. Due to the time correlation, the CO-DAC method can reuse the reduced

mechanism if the phase parameters of the adjacent time steps fall into the same correlated

group, while the DAC method needs to do mechanism reduction at every time step even

68

when the chemical equilibrium is achieved. Therefore, the CO-DAC method is much more

efficient than the DAC method without correlation, especially in the equilibrium or near

equilibrium region. As such, a combination of the CO-DAC method with efficient ODE

solvers, such as the HMTS method, is a computationally efficient way for direct numerical

simulations.

3.3.1 CO-DAC with the first generation PFA reduction

In this section, the CO-DAC method only includes the first generation PFA

reduction, meaning that when evaluate the reaction path flux between species, only the

directly connected reactions are considered. The comparison between the first and second

generation PFA reduction will be discussed later. Figure 3.5 and Figure 3.6 show the

comparisons of ignition delay times of the jet fuel surrogate mixture at 10 atm and

stoichiometric condition with different initial temperatures for the reduced and detailed

kinetic mechanisms, respectively. It is seen that the CO-DAC method can predict the

ignition times accurately in both low and high temperatures with both reduced and detailed

mechanisms. The maximum discrepancy of the predicted ignition delay times in all initial

conditions happens between the VODE and the HMTS/CO-DAC method, which has the

value of 2%. The small discrepancy at both low and high temperatures suggests that the

proposed parametric space using OH, HO2, and CH2O as the key intermediate species to

represent reaction pathways in low and high temperatures is very effective.

Figure 3.7 and Figure 3.8 plot the detailed histories of temperature and species mass

fractions, including CH2O, HO2 and OH, initialed at 800 K using the reduced and detailed

Real Fuel 2 mechanisms, respectively. There is a horizontal time shift of the profiles in the

high temperature ignition region. In the calculations using the reduced mechanism, the

69

maximal shift happened between the HMTS and HMTS/CO-DAC method with the value

of 0.021 ms. Compared with the averaged ignition delay time calculated by all these five

methods, 3.17 ms, the relative error in time is as small as 0.67%. Similarly, the maximal

shift of in the calculations with detailed mechanism is 0.032 ms and it also happened

between HMTS and HMTS/CO-DAC method. Considering the averaged ignition delay

time, 5.46 ms, the maximal relative error in time is 0.59%. Besides the time shift, other

properties calculated by different methods are identical. The excellent agreement of

temperature and species profiles demonstrates that the accuracy of CO-DAC method is

good enough to provide correlated chemical reductions in a large temperature range. Also,

the agreement in the two-stage ignition profiles demonstrates that the CO-DAC method

has the capability to capture the low temperature chemistry accurately.

In order to further demonstrate the accuracy of the CO-DAC method, the profiles

of peroxides (C12H25O2) and H2O2 radicals at the same calculations are plotted in Figure

3.9 (reduced mechanism) and Figure 3.10 (detailed mechanism). These two figures

demonstrate that besides the species included in the phase parameters, other radicals can

also be predicted accurately by the CO-DAC method. Therefore, the locally reduced

mechanisms generated in the 5-dimensional phase space (T, Fuel, CH2O, HO2 and OH)

can capture the important reaction pathways and predict the ignition properties accurately.

Figure 3.11 and Figure 3.12 show the CPU time comparisons in the homogeneous

ignition calculations between the five compared methods using the reduced and detailed

kinetic mechanisms, respectively. The traditional implicit VODE method serves as the

benchmark of the comparison. Thus, the CPU times plotted in these two figures are

normalized by the CPU times of the VODE method. It is seen that the integration of DAC

70

with the VODE method (red) can improve the computational efficiency by 40 to 90 percent.

However, the DAC method is not computationally efficient when it is integrated with the

HMTS method. The computational cost of the HMTS method (blue) can be increased with

the integration of the DAC method (green). The increase of the computational cost can be

explained by Equation (3.3) and (3.4). Since the HMTS method is an efficient linear scaling

solver, when chemical mechanisms become larger, the computational cost of the DAC

reduction grows much faster than the computational cost of the HMTS method, so that the

CPU time to perform the on-the-fly chemical reduction becomes comparable to or even

larger than the chemical integration time in the HMTS method. Therefore, it is

demonstrated that the benefit of the locally reduced mechanism generated by the traditional

DAC method without correlation is diminished by the slow on-the-fly chemical reduction.

More importantly, Figure 3.11 and Figure 3.12 also demonstrate that by applying the

proposed CO-DAC method in the HMTS method (purple), the on-the-fly chemical

reduction time can be significantly reduced and the HMTS/CO-DAC method increases

computational efficiency by a factor of 2 in a broad temperature range. Therefore,

conducting the chemical reduction in the correlated space, as proposed in the CO-DAC

method, is an essential improvement when dealing with large and detailed chemical

kinetics.

The computational accuracy and efficiency of the CO-DAC method has been

demonstrated successfully in the 0D auto ignition simulations. However, flames have much

different species distributions, radical pools and reaction pathways, which may affect the

effectiveness of the phase parameters selected in the CO-DAC method. To prove the

applicability of the CO-DAC method in a broad ignition and flame regimes, it needs to be

71

tested in both unsteady and steady flame propagations. Therefore, numerical simulations

of the unsteady, outwardly propagating spherical flame are carried out to validate the both

unsteady and quasi-steady flame structures and propagation speeds. Figure 3.13 shows the

time-dependent outwardly propagating spherical flame trajectories calculated by HMTS

and VODE with and without DAC or CO-DAC. Here, only the comprehensively reduced

kinetic mechanism of the real fuel surrogate mixture with 425 species is applied. The

stoichiometric fuel/air mixture is initiated at 1 atm and 400 K. The mixture is ignited at the

center by a hot spot with 2 mm radius and a uniform temperature of 1600 K. It is seen from

Figure 3.13 that the present CO-DAC method shows excellent agreement compared with

other methods even when the transport terms are involved. The largest relative error in the

flame trajectory is less than 2%, which is far below the experimental accuracy of flame

speed measurements.

Figure 3.14 shows the predicted flame structures of the spherical flame. The

distributions of temperature and selected species calculated by different methods are

compared. It is seen that the CO-DAC method has the ability to not only predict the

propagating trajectories accurately, but also reproduce the detailed flame structures

precisely. Therefore, the CO-DAC method is demonstrated to be accurate and robust for

both unsteady and quasi-steady flame propagations.

Figure 3.15 is the comparison of the CPU times consumed by different methods in

the premixed spherical flame simulations. The black section represents the CPU time

consumed by the calculations of transport properties, diffusion flux, and convection flux.

The summation of the red and blue sections represents the total computational cost to solve

the chemical reactions, including the on-the-fly PFA reduction (red) and chemical

72

integration (blue). This figure clearly shows that compared to the DAC method, the

CO-DAC method can dramatically decrease the on-the-fly model reduction time and makes

it negligible even when a large kinetic mechanism is involved. It is worth mentioning that

by using the proposed HMTS/CO-DAC method, the integration of chemical reactions for

a large chemical mechanism is no longer the most time consuming part in the simulation.

Instead, the computation of the transport terms is now the major cost of the CPU time. It

should be noted that as discussed in Equation (3.7), the CO-DAC method will be more

effective in multi-dimensional simulations due to the increased mechanism similarity in

high-dimensional domains. As such, the CO-DAC method coupled with the HMTS solver

can significantly increase the computational efficiency in combustion modeling with

detailed chemical kinetics.

3.3.2 CO-DAC with the Second generation PFA reduction

The above results show that the CPU time of the first generation PFA reduction in

the CO-DAC method is negligible. Therefore, the CO-DAC method may have the

capability to enable higher-order PFA model reductions. In this section, the second

generation PFA reduction is utilized in the CO-DAC method. Compared with the first

generation reduction, which calculates the path flux between two species directly, the

second generation PFA reduction searches all the species in the mechanism as the

intermediate species to get the path flux between two specified species. Therefore, the

second generation reduction is 𝑁𝑠 times slower than the first generation reduction, where

𝑁𝑠 is the number of species in a mechanism.

In order to test the performance of the CO-DAC method with the second generation

reduction, the same reduced Real Fuel 2 mechanism with 425 species and 2275 reactions

73

is applied in the validation. The comparison between the CO-DAC method with 1st and 2nd

generation reduction is conducted in both homogeneous auto-ignition and unsteady flame

propagation calculations.

The green dash lines in Figure 3.5 and Figure 3.13 show the ignition delay time and

flame trajectory, respectively, calculated by the CO-DAC method with the 2nd generation

PFA reduction. The difference between 1st and 2nd generation reduction is too trivial to be

noticed. The reason for such a trivial discrepancy is that the threshold value in the PFA

reduction is as small as 0.005. With such a small threshold value, both 1st and 2nd generation

PFA reduction methods are equally accurate.

Figure 3.16 shows the ignition delay time as well as the PFA reduction time versus

the threshold value used in the PFA reduction. Both ignition delay time and PFA reduction

time are plotted in logarithm coordinates. It is seen that both the 1st and 2nd generation PFA

reductions are accurate when the threshold value is small. When the threshold value

increases, the ignition delay times calculated by the reduced mechanisms deviate from the

exact value. But the error introduced by the 2nd generation PFA reduction is always smaller

than that introduced by the 1st generation PFA reduction. On the other hand, the ratio of

the computational time between the 2nd and 1st generation PFA remains constant, which is

close to the number of species (425). This figure demonstrates that the 2nd generation

reduction is more accurate than the 1st generation reduction, especially when the threshold

value is large. However, the 2nd generation PFA is 𝑁𝑠 times slower than the 1st generation

PFA reduction. Therefore, in combustion simulations with large chemical kinetics, the 1st

generation PFA reduction with a small threshold value is more computationally efficient.

74

Figure 3.17 is the CPU time comparison between the CO-DAC method with the 1st

and 2nd generation PFA reductions in the spherical flame propagation calculation. A small

threshold value, 0.005, is applied in both the 1st and 2nd generation PFA reductions. It is

seen that the HMTS method in the two calculations consumes identical CPU times due to

the similar active species produced by the PFA reductions. However, the CPU time of the

2nd generation PFA reduction is 462 times higher than the 1st generation reduction. Again,

this number is close to the number of species (425) in the mechanism. This result further

demonstrates that the 2nd generation reduction is 𝑁𝑠 times slower than the 1st generation

reduction.

However, even if the 2nd generation PFA reduction is much slower than the 1st

generation one, it may be utilized in higher dimensional simulations due to the increased

mechanism similarity in high-dimensional domains. Table 3.1 shows the HMTS time and

the 2nd generation PFA time as well as the ratio of PFA reduction time to HMTS time in

the 0D auto-ignition and 1D spherical flame propagation calculations. The ratio of PFA

reduction time to HMTS time decreases by factor of 4 when the calculation goes from 0D

to 1D. The reason of the decreased PFA/HMTS ratio in a higher dimensional calculation

is that in the 0D calculation, only the time correlation is effective. But in the 1D calculation,

not only the time correlation but also the space correlation is utilized to reduce the number

of chemical reductions. In higher dimensional calculations, the space correlation will be

more efficient, and hence the PFA/HMTS ratio will further decreases. Therefore, the

CO-DAC method with the 2nd generation PFA reduction has the possibility to be applied

in higher dimensional simulations.

75

0D 1D

HMTS Time 13.1 min 11.7 hr

2nd Generation PFA Time 274.0 min 61.8 hr

PFA/HMTS Time Ratio 20.9 5.3

Table 3.1: Computation time comparison of HMTS, 2nd generation PFA and PFA/HMTS

between 0D and 1D calculation

3.4 Conclusion

In this chapter, a correlated dynamic adaptive chemistry (CO-DAC) method is

developed by utilizing the similarity of reaction pathways in phase space. The results show

that the CO-DAC method can significantly increase the computational efficiency while

keeping excellent accuracy. The simulations of homogeneous ignitions of the Real Fuel

2/air mixture with different initial temperatures show that the chemical reduction time in

the CO-DAC method is two orders of magnitude shorter than that in the DAC method. In

addition, the results show that the present HMTS/CO-DAC method can improve the

efficiency of the HMTS solver by a factor of 2. Additionally, the numerical results of the

unsteady spherical propagating flames demonstrate that the computation efficiency of the

CO-DAC method is further increased in a higher dimensional computation. The CPU time

consumed by the on-the-fly model reduction in the CO-DAC method becomes negligible.

Moreover, the HMTS/CO-DAC method can also accurately predict the flame speeds and

detailed structures. Furthermore, comparisons between the 1st and the 2nd generation PFA

reductions show that the 2nd generation PFA has a better accuracy but is much more

computationally expensive than the 1st generation PFA. The ratio of the PFA reduction

time to the chemical solver time decreases with the increased number of physical

dimensions. Therefore, the promising results in the present study indicate that with the

76

presented CO-DAC method, large chemical mechanisms can be efficiently handled in

multi-dimensional combustion simulations. With the CO-DAC method, the chemical

solver is not the most time consuming part anymore in a combustion simulation. Instead,

the calculations of the detailed transport properties dominate the computational cost, which

will be addressed in the following chapter.

77

Figure 3.1: Schematic of time and space correlation. Black cells are time correlation and

red, blue and green cells are space correlation groups.

Figure 3.2: The typical reaction pathways of a conventional hydrocarbon fuel at different

temperature range

78

Figure 3.3: Schematic of the differences between cell clustering method, ISAT method

and the proposed CO-DAC method.

Figure 3.4: Comparison of number of active species and the accumulated on-the-fly

mechanism reduction time by using CO-DAC and DAC methods in a homogeneous

ignition process with a stoichiometric Real Fuel 2/air mixture at 10 atm. The solid line is

calculated by CO-DAC method and the dash line is calculated by DAC method.

79

Figure 3.5: Comparison of ignition delay times predicted by HMTS and VODE solvers

with DAC and CO-DAC at 10 atm and stoichiometric condition of the reduced Real Fuel

2/Air.

Figure 3.6: Comparison of ignition delay times predicted by HMTS and VODE solvers

with DAC and CO-DAC at 10 atm and stoichiometric condition of the detailed Real Fuel

2/Air.

80

Figure 3.7: Species mass fraction profiles calculated by different methods at 10 atm,

stoichiometric condition and initiated at 800 K of the reduced Real Fuel 2/Air.

Figure 3.8: Species mass fraction profiles calculated by different methods at 10 atm,

stoichiometric condition and initiated at 800 K of the detailed Real Fuel 2/Air.

81

Figure 3.9: Mass fraction profiles of O, H and H2O2 calculated by different methods at

10 atm, stoichiometric condition and initialed at 800 K of the reduced Real Fuel 2/Air.

Figure 3.10: Mass fraction profiles of O, H and H2O2 calculated by different methods at

10 atm, stoichiometric condition and initialed at 800 K of the detailed Real Fuel 2/Air.

82

Figure 3.11: CPU time comparison between HMTS and VODE solvers with and without

DAC or CO-DAC at 10 atm and stoichiometric condition of the reduced Real Fuel 2/Air.

Figure 3.12: CPU time comparison between HMTS and VODE solvers with and without

DAC or CO-DAC at 10 atm and stoichiometric condition of the detailed Real Fuel 2/Air.

83

Figure 3.13: Flame trajectories of stoichiometric reduced Real Fuel 2/air mixture at 1 atm

and 400 K.

Figure 3.14: Flame structure of stoichiometric reduced Real Fuel 2/air mixture at 1 atm

and 400 K.

84

Figure 3.15: CPU time comparison between HMTS and VODE method with and without

DAC or CO-DAC of the 1D flame propagation case with stoichiometric reduced Real

Fuel 2/air mixture at 1 atm and 400K.

Figure 3.16: Comparison of ignition delay time and PFA reduction time between CO-

DAC method with 1st and 2nd generation PFA reduction for different threshold values

calculated at 10 atm, stoichiometric condition, and initiated at 800 K.

85

Figure 3.17: CPU time comparison between CO-DAC method with 1st and 2nd

generation PFA reduction in a spherical flame propagation calculation of stoichiometric

reduced Real Fuel 2/air mixture at 1 atm and 400K.

86

Chapter 4

Correlated Dynamic Adaptive Chemistry and Transport (CO-

DACT) Method

In the previous capture, a CO-DAC method was developed to significantly reduce

the on-the-fly model reduction time, so that it can provide the locally reduced mechanisms

with negligible computational costs. Together with the efficient HMTS method, solving

the chemical reaction term described by the ODE (3.1) and (3.2) is not the most time

consuming part anymore in a combustion simulation. Therefore, it is of great importance

to improve the computational efficient of the transport terms, which are governed by the

PDE (2.36). In this chapter, a correlated dynamic adaptive chemistry and transport (CO-

DACT) method is developed based on the concept of the correlated space to further

accelerate the calculation of detailed transport properties in a reactive flow simulation. The

correlated transport properties in phase space are dynamically updated by the

mixture-averaged diffusion model with user-specified correlation thresholds. In the present

method, the transport properties are only computed once for all the computational cells in

the same correlated group. Therefore, without sacrificing the accuracy, the CO-DACT

method can eliminate redundant transport property calculations in temporal and spatial

coordinates due to the similarities of thermodynamic states in the phase space. The detailed

methodology, error analysis, and validations of the CO-DACT method are presented in this

chapter.

87

4.1 Background and Objectives

It is well-known that transport properties are strongly dependent on temperature

and species and have a significant impact on the turbulent energy transport, fuel/air mixing,

flame speed, flame instability, extinction limit and even knocking formation. In

combustion modeling, the transport properties vary significantly across temporal and

spatial coordinates due to the rapid species and temperature variations caused by chemical

reactions. Therefore, in a high-fidelity computation with detailed chemical kinetics, the

transport properties have to be accurately evaluated during calculations according to the

local thermodynamic states. However, based on the discussion in Section 2.3.1, even with

the efficient mixture-averaged diffusion model, the computational cost of evaluating

transport properties is still proportional to the square of the mechanism size. Thus, the

calculation of the detailed transport properties in a reacting flow with large chemical

kinetics is expensive. This problem has become the limiting issue when a computationally

efficient chemical solver, such as the HMTS/CO-DAC method described in the previous

chapter, is used. Therefore, the development of advanced methods to drastically reduce the

computational cost of transport properties while maintaining the accuracy in CFD is

urgently needed in combustion modeling in order to utilize large and detailed chemical

mechanisms.

The goal of this study is to develop and validate a correlated dynamic adaptive

chemistry and transport (CO-DACT) method by taking advantages of similarities of

transport properties in temporal and spatial coordinates. At first, the methodology of the

CO-DACT method, including the basic concept, the numerical implementation, the

selection of phase parameters, and the error analysis are introduced. Then, the validity of

88

the CO-DACT method is investigated by numerical simulations of unsteady premixed

propagating spherical flames and diffusion ignitions in jet fuel surrogate (Dooley et al.

2012) mixtures. The same jet fuel surrogate kinetic model as used in the previous chapter

is applied in the simulations, which has 425 species and 2275 reactions. Moreover, an

additional validation is conducted by modeling an unsteady spherical propagating diffusion

flame with a detailed DME mechanism (Cheung, Liu, and Iglesia 2004), which has 55

species and 290 reactions. Finally, the detailed numerical error studies at different

pressures, equivalent ratios and threshold values are carried out.


The concept of the CO-DACT method will be presented first in this section. Then,

the selection of phase parameters will be introduced. This is followed by the numerical

implementation of the correlated group. Finally, a mathematical analysis of the numerical

error in the proposed CO-DACT method will be presented.

4.2.1 Concept of CO-DACT method

As mentioned in the introduction, transport properties have to be evaluated locally

in high-fidelity combustion simulations. Thus, in a traditional combustion modeling

method the transport properties are calculated in every computational cell at each time step,

which results in a huge computational cost when the size of the chemical mechanism is

large. However, conducting the transport property calculation in every existing cell is a

great waste considering the fact that similar thermodynamic states can occur repeatedly in

spatial and temporal coordinates. For instance, in a laminar premixed flame propagation

process, all cells in front of (unburned side) or after (burned side) the flame front have

89

similar transport properties due to the similarities of their thermodynamic states. Therefore,

the computational complexity of the transport properties can be significantly reduced if the

detailed calculation results are reused is similar cells. Based on this idea, a correlated

dynamic adaptive chemistry and transport (CO-DACT) method is proposed. The

CO-DACT method is inherited from the CO-DAC method developed in the previous

chapter, which agglomerates similar computational cells to correlated groups for efficient

on-the-fly chemical reductions. On top of the correlated chemical reduction, the proposed

CO-DACT method further extends the concept of the correlated space to the transport

property calculations. The concept of the correlated space for transport properties is

described as following. We assume that the local transport properties are primarily

determined by a few key parameters in the phase diagram. Therefore, a correlated space

for transport properties can be constructed by the span of the key parameters, known as the

phase parameters. This correlated space is a subspace of the composition space with a much

lower dimension. Then the correlated space is discretized to hypercubes by a user specified

threshold value. Computational cells that fall into the same hypercube form the correlated

group, and they are considered correlated in the transport property calculation, so that the

result of the detailed transport property calculation can be reused by the other cells in the

same correlated group.

It is worth noting that the proposed CO-DACT method includes both the correlated

adaptive chemistry (CO-DAC) and the correlated transport properties (CO-T). However,

the correlated space for transport properties is different from the correlated space for

on-the-fly chemical reductions, because reaction pathways and transport properties are

dominated by different phase parameters, and hence their correlated spaces are independent.

90

Since the CO-DAC method has been discussed extensively in the previous chapter, the

following discussions in this study will only focus on the correlated transport properties.

4.2.2 Selection of Phase Parameters

Based on the concept of the correlated space, the validity of the CO-DACT method

highly depends on the selection of the phase parameters, which form the basis of the

correlated space. The selected phase parameters should be able to govern the transport

properties in a wide thermodynamic condition range. To choose the phase parameters, the

formulation of the mixture-averaged model needs to be examined. The mixture-averaged

viscosity, conductivity, and diffusivity are given by Equation (2.23), (2.24), and (2.25),

respectively. According to these equations, the transport properties in the mixture-averaged

model are functions of molar fractions and species-wise diffusion properties, including the

pure species viscosity, pure species conductivity, binary diffusion coefficient, and binary

thermal diffusion ratio.

The species-wise diffusion properties are given by the polynomial fitting of

temperature in the TRANSPORT program described in Equation (2.27). Since temperature

is the only parameter in Equation (2.27), it must be included in the phase parameters. Once

the species-wise diffusion properties are computed, the mixture-averaged transport

properties can be obtained by the combination of the species-wise diffusion properties

weighted by the species molar fraction. Moreover, the equations also suggest that the

species with larger molar fraction has larger impact on transport properties. Therefore, the

phase parameters for transport properties should also include the major species. In a typical

combustion system, the total molar fraction of N2, O2, Fuel, H2O, H2, CO2 and CO are often

more than 95%. Besides those species, the OH is a critical radical in combustion processes

91

and the CH2O is a dominating intermediate specie in low temperature fuel oxidations. For

other minor species and radicals, the maximal error in total they can introduce to the

mixture-averaged model is limited by 5%, which is much smaller than the uncertainty of

transport property measurements. Therefore, the phase parameters for transport properties

are chosen as (T, XN2, XO2, XFuel, XH2O, XH2, XCO2, XCO, XOH, XCH2O)T. Moreover, the

proposed CO-DACT method has the flexibility to add or remove species to/from the phase

parameters. If a non-typical combustion system is involved and the molar fraction of a

minor species becomes high, it can be added into the phase parameters straightforwardly.

The current selection of the phase parameters will be validated in the Results and

Discussion section by a sensitivity analysis.

4.2.3 Numerical Identification and Implementation of Correlated Groups

The previous sub section provides the detailed analysis of the selection of the phase

parameters. In this sub section, the numerical identification of the correlated group and the

corresponding implementation are introduced based on the selected phase parameters.

To simplify the following discussion, the phase parameters will be represented by

a vector, 𝒛, with the dimension of M, where 𝒛 = (T, XN2, XO2, XFuel, XH2O, XH2, XCO2, XCO,

XOH, XCH2O)T and M = 10. Based on the chosen phase parameters, the phase space, S, can

be constructed by taking the logarithm of the phase parameters:

( ) ( ) ( )( )1 2ln , ln , , lnT

Mz z z=S (4.1)

In order to avoid zero in the logarithm, a small positive number, σ, is added to each zi. In

this study, σ is set to be 10-30.

In the constructed phase space, a distance, d, between two points, S1 and S2, is

defined as:

92

d

= −1 2S S (4.2)

If S1 and S2 are close to each other, the i-th component of the above equation becomes:

( ) ( ) ( ) ( )1 2 1 2 2 2ln ln ln ln 1i i i i i i i i i iz z z z z z z z z z− = = + = (4.3)

where Δzi = zi1 – zi

2, zi = zi2 ≈ zi

1 and Δzi/zi << 1 because S1 and S2 are very close. Thus, in

this case, the definition of d is equivalent to:

1 2 1 2

1 2 1 2

, ,..., max , ,...,

T

M M

M M

z z z z z zd

z z z z z z

= =

(4.4)

Based on the above definition, the phase space can be discretized to hypercubes by

a user specified threshold value ε. Each discretized hypercube represents a correlated group

with a volume of (ε)M in the phase space. Therefore, the maximal distance between any

pair of computational cells in a correlated group is less than or equal to ε. According to the

group size, the total number of correlated groups in the i-th direction of the phase space, Li,

is:

( ) ( )( )_ max _ minln ln 1i i iL z z = − +

(4.5)

Where zi,max and zi,min are the maximal and minimal value of the 𝑖-th phase parameter in the

computational domain, respectively. And the square bracket represents the round-up to the

nearest integer. Thus, the maximal number of correlated groups in the phase space is 1

M

i iL= ,

which is a huge number. However, since there are only Nc cells in the computational

domain, the maximal number of existing correlated groups is no more than Nc at a certain

time step. Therefore, the correlated space is largely sparse.

According to the definition of the correlated space, computational cells in the

physical domain can be mapped to correlated groups by the time correlation and space

93

correlation procedures. The schematic of the time and space correlations are plotted in

Figure 4.1. Note that since the correlated chemical reduction is inherited in the CO-DACT

method, this figure also includes the correlation procedure of the chemical reductions (left

side) for the completeness. However, the following discussion will only focus on the

correlation procedure of the transport properties (right side). The time and space correlation

procedures are described below.

Time correlation: The time correlation procedure is conducted locally in each

specific cell. The thermodynamic states of a particular cell at two adjacent time steps are

compared. If the phase parameters in the current time step and in the previous time step

fall into the same correlated group, this particular cell at the adjacent time steps are time

correlated and the transport properties at the previous step will be reused in the current time

step. Here we only consider the time correlation between two adjacent time steps to avoid

the memory consumptions by data storage.

Space correlation: For other cells which are not time correlated, we apply the space

correlation procedure. A set of group indexes, (l1, l2, …, lM) will be assigned to each

computational cell, where

( ) ( )( ) ( )_ minln ln 1, 1, 2, ,i i il z z i M = − + =

(4.6)

Based on the correlated group index, the same hash table implementation described in

Section 3.2.1 is used to map the computational cell to the representation of the correlated

group for the transport property calculation. The average and amortized time complexity

of the hash table operations, including insertion and lookup, is 𝑂(1). It is negligible

compared with the detailed transport property calculation, which is 𝑂(𝑁𝑠2). Therefore, the

correlated groups can be constructed on-the-fly without generating expensive overheads.

94

Based on the time and space correlation procedures, the detailed calculation of

transport properties can be significantly accelerated. The computational cost of a traditional

combustion modeling method and the proposed CO-DACT method are compared in Figure

4.2. Again, the correlated chemical reduction is included in the figure, but the discussion

will only focus on the correlated transport properties.

In a conventional method, the mixture-averaged model is commonly applied to

calculate the detailed transport properties. The CPU time, 𝑡𝑚𝑖𝑥 , consumed by the

mixture-averaged method is proportional to 𝑁𝑠2, where 𝑁𝑠 is the number of species in the

chemical mechanism. Considering the total number of computation cells, 𝑁𝑐, the total CPU

time to calculate transport properties for all cells, 𝑡𝑡𝑟𝑎𝑛|𝑐𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙, is given by

|tran conventional mix ct t N= (4.7)

Since 𝑡𝑚𝑖𝑥 is proportional to 𝑁𝑠2 , it becomes time-consuming when a large chemical

mechanism is involved.

On the other hand, in the proposed CO-DACT method, the mixture-averaged model

is conducted only once in each correlated group instead of in each computational cell. As

mentioned above, the time and space correlation procedure are 𝑂(1) for each cell.

Therefore, the CPU cost to completely construct the correlated groups, 𝑡𝐶𝑂−𝑇 , at a

particular time step is 𝑂(1 × 𝑁𝑐) = 𝑂(𝑁𝑐). It is negligible compared with 𝑡𝑚𝑖𝑥. Thus, the

total CPU time to calculate transport properties in the CO-DACT method can be

approximated as:

( )( )| | |co T tran CO DACT tran CO DACT mix s c t s t tran conventionalt t t t N t − − −+ = = (4.8)

where 𝜉𝑡 and 𝜉𝑠 are transport correlation ratios in temporal and spatial coordinates,

respectively. In a combustion system with highly correlated thermodynamic states, 𝜉𝑡 and

95

𝜉𝑠 will be two small positive numbers that are far less than one. Therefore, the CO-DACT

method can significantly reduce the computational cost of transport properties by the factor

of 𝜉𝑡𝜉𝑠.

4.2.4 Numerical Error Analysis

The concept of the CO-DACT method and the corresponding numerical

implementation have been discussed. To apply the CO-DACT method in high-fidelity

combustion simulations, the numerical error of the correlated transport properties has to

been examined carefully. In this sub section, a detailed mathematical analysis of the

numerical error is carried out.

According to Equation (2.27), the numerical errors in the species-wise diffusion

properties, including μi, λi, Dij, and θij, come from the temperature variation, ΔT, in a

correlated group. By using Taylor expansion theory, the maximal variation of μi in a

correlated group can be calculated as following:

( )4

1 2

1

1(ln )

ln

i n

ni

ni

na T O

T

−

=

− = +

(4.9)

where T and μi are the values at the center of the correlated group. Because n≤4, i.e. n-1≤3,

thus:

( )23 ln

ln

i i

i

OT

+ (4.10)

In a typical combustion system, T ~ O(103) K and μi ~ O(10-3) g/cm/s, therefore

96

3 ln~ (1)

ln

i OT

(4.11)

( )~i

i

O

(4.12)

By applying similar analyses to the pure species thermal conductivity λi, the binary

diffusivity Dij, and the binary thermal diffusion ratio θij, the same order of accuracy can be

obtained:

( )

( )

( )

( )

~

~

~

~

i

i

i

i

ij

ij

ij

ij

O

O

DO

D

O

(4.13)

Thus, the species-wise diffusion properties are first-order accurate in the

CO-DACT method. According to Equation (2.24), the mixture-averaged thermal

conductivity, λ’, of an arbitrary computational cell in a correlated group, and the

corresponding value at the center of the correlated group, λ, can be written as:

( ) ( )( ) ( )

11

11

1 1

2

1 1' 1 1

2 1 1

s

s

s

s

N

k k Nk k kk

NX

i i i i N Xi i i i ii

XX

XX

==

==

= +

= + + + + +

(4.14)

where Xi and λi are the molar fraction and pure species conductivity of the 𝑖-th species at

the center of the correlated group, respectively. And δiX and δi

λ represent the discrepancies

of Xi and λi between the value in the computational cell and the value in the center of the

97

correlated group, respectively. The discrepancy in mass fraction, δiX, is bounded by the

user specified threshold value ε, i.e. | δiX | ≤ ε. And the discrepancy in the pure species

conductivity, δiλ, is also bounded by ε based on Equation (4.13). Therefore,

( )( ) ( )( )2 21 2 ' 1 2O O − + + + (4.15)

2 ~ ( )O

(4.16)

By applying the similar analysis for viscosity, mass diffusivity, and thermal

diffusion coefficient, we have:

~ ( )

~ ( )

~ ( )

~ ( )

im

im

i

i

O

O

DO

D

O

(4.17)

Therefore, it is mathematically demonstrated that proposed CO-DACT method is

first-order accurate and the numerical errors in transport properties are on the order of the

user-specified threshold value for discretizing the correlated space. In addition, the first-

order numerical error, O(ε), can be eliminated by considering a linear variation of

temperature and molar fractions within the correlated group. Thus, the accuracy of the

CO-DACT method can be extended to second order by applying a linear extrapolation of

the transport properties.

98

4.2.5 Numerical Setup

The numerical efficiency and accuracy of the proposed CO-DACT method have

been analyzed mathematically in this section. To validate it numerically, direct numerical

simulations of various combustion processed are conducted using the CO-DACT method.

Similar as the previous chapter, the HMTS method is applied to perform the chemical

integration based on the locally reduced mechanism. The performance of the proposed

CO-DACT/HMTS method is compared with other benchmark methods, including the

VODE, HMTS and HMTS/CO-DAC methods. The major differences between these

methods are listed in Table 4.1. A jet fuel surrogate mechanism (Dooley et al. 2012), which

contains four fuel components (40% n-dodecane, 30% iso-octane, 23% n-propyl benzene,

and 7% 1,3,5-trimethyl by mole fraction), and a detailed DME mechanism (Cheung et al.

2004) are applied in the following combustion simulations. The jet fuel surrogate

mechanism has 425 species and 2275 reactions. And the DME mechanism contains 55

species and 290 reactions. In this study, unless specified the minimum cell size is 7.5 μm

and the threshold values for both correlated chemical reduction and correlated transport

properties are 0.05. Based on this setup, various combustion processes are simulated and

compared in the next section.

Table 4.1: Comparisons of details between different methods

Method Chemical

reduction

Chemical

integration

Transport

properties

VODE - VODE Mixture-averaged

HMTS - HMTS Mixture-averaged

HMTS/CO-DAC Correlated PFA HMTS Mixture-averaged

HMTS/CO-DACT Correlated PFA HMTS Correlated mixture-averaged

99


To validate the proposed CO-DACT method, numerical simulations of unsteady

outwardly propagating spherical premixed flames and diffusion ignitions in a jet fuel

surrogate mixture are carried out, which cover premixed and non-premixed as well as

stretched and un-stretched reactive flow conditions with strong chemistry-transport

coupling. Moreover, an additional validation is conducted by modeling an unsteady

spherical propagating diffusion flame in a DME mixture. Finally, the numerical error of

the CO-DACT method will be examined with different threshold values, pressures, and

equivalence ratios.

4.3.1 Premixed Outwardly Propagating Spherical Flames

The premixed spherical propagating flame is ignited by a hot spot with 2000 K and

2 mm in radius at the center (X=0 cm) of the spherical coordinate. The homogeneous

stoichiometric jet fuel surrogate mixture is initialized at 10 atm and 300 K. The overall size

of the computational domain is 5 cm. And the boundary conditions are reflective at X = 0

and transmissive at X = 5 cm. The simulation results are compared between the VODE,

HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods.

Figure 4.3 shows the flame trajectories of the premixed spherical propagating flame

calculated by the four methods mentioned above. It is difficult to see the difference between

the four predicted results because they are mostly identical, which demonstrates the

excellent agreement between these methods. The maximum relative error along the flame

trajectory among all these methods is less than 0.5%, which is much smaller than the

experimental uncertainty of transport properties and flame speed measurements (~5%).

100

Figure 4.4 plots the spatial distributions of five phase parameters, including

temperature, O2, Fuel, H2O and CO2, at physical time t = 1.6 ms. The flame surface is

located around the location X = 0.495 cm. It is worth noting that this figure is plotted

according to the result calculated by the HMTS/CO-DACT method. For other methods, the

results are horizontally shifted to compare with the HMTS/CO-DACT method. The amount

of the horizontal shift between different methods are listed in Table 4.2. It is seen that these

horizontal shifts are trivial compared to the flame radius. Besides the horizontal shift, the

structures calculated by different methods are identical, which demonstrate that the

HMTS/CO-DACT method is accurate to capture the major structure of a premixed flame,

including temperature and major species. It should be noted that all other major species are

also follow the same accuracy. But in order to keep the figure tidy only temperature and

the selected species are plotted as demonstrations.

VODE to

HMTS/CO-DACT

HMTS to

HMTS/CO-DACT

HMTS/CO-DAC to

HMTS/CO-DACT

Shift distance (mm) 2 0.5 0.7

Ratio to flame radius 0.4% 0.1% 0.14%

Table 4.2: The amount of horizontal profile shift for different methods and the ratios of

the shifts to the flame location.

The mass fraction distributions of the selected minor species and radicals, including

OH, H, HO2 and H2O2, at the same physical time t = 1.6 ms are plotted in Figure 4.5 to

examine and demonstrate the accuracy of the HMTS/CO-DACT method on capturing those

species which are not included in the phase parameters. Similar as the previous figure,

Figure 4.5 is plotted by following the same horizontal shift shown in Table 4.2. This figure

shows that the minor species follow the same accuracy as the major species and the

discrepancies between different methods are trivial. Therefore, it successfully

demonstrates that the proposed HMTS/CO-DACT method can accurately predict the

101

distributions of minor species and radicals even they are not included in the phase

parameters. It is worth mentioning that the H radical is the most diffusive radical in a

combustion system, which may introduce the largest error in the transport properties. If the

proposed HMTS/CO-DACT method can accurately predict the H radical, it implies that

this method is able to predict other radicals precisely as well.

Figure 4.6 shows the comparisons of the CPU time consumed by the transport

property calculation between different methods. The CPU times are plotted in logarithm

coordinate with the unit of hour [hr]. The purple, black and red lines, which represent the

VODE, HMTS and HMTS/CO-DAC methods, respectively, overlap together because the

transport properties in these methods are computed by the same mixture-averaged diffusion

model, as shown in Table 4.1. Thus, the CPU time of transport property calculations in the

first three methods should be the same, which is demonstrated in Figure 4.6. By comparing

the HMTS/CO-DACT method with other methods, it shows that the proposed CO-DACT

method (blue line) is 230 times faster in terms of the transport properties calculation, which

is a dramatic increase of the computational efficiency.

Figure 4.7 is the overall CPU time comparisons between VODE, HMTS,

HMTS/CO-DAC and HMTS/CO-DACT methods at physical time t = 1.6 ms. The green,

purple and red sections represent the CPU time consumed by convection flux, diffusion

flux and transport properties calculations, respectively. And the blue section is the

computational time consumed by chemical integrations. It is seen from this figure that

compared to the first three columns, the proposed HMTS/CO-DACT method can

significantly reduce the computational cost of the transport properties and, most important,

makes it negligible even when a large chemical mechanism with several hundreds of

102

species is involved. With the present CO-DACT method, the overall computation of the

flame propagation case can be accelerated by factor of three compared with the original

HMTS method and becomes more than one order of magnitude faster than the VODE

method.

In order to further investigate the effect of the threshold value on the number of

correlated groups and the computational cost of transport properties, simulations of the

same premixed flame with different threshold values are conducted and compared in Figure

4.8 and Figure 4.9. Five threshold values are compared: ε = 0 (black lines), 0.0001 (red

lines), 0.0005 (blue lines), 0.005 (green lines) and 0.05 (purple lines). Figure 4.8 shows the

ratio of the number of correlated groups (Ngroup) to the total number of computational cells

(Ncell) during the numerical simulation. Since the adaptive mesh is applied in this study and

the total number of cells varies from time to time, it is more reasonable to plot the ratio

instead of the actual number of correlated groups. In the ε = 0 case, the volume of the

correlated group in the phase space is 0 and there are no correlations can be constructed.

Therefore, it is identical to the methods which conduct the detailed calculations of transport

properties in every cell and time step. In the CO-DACT method, only Ngroup times, instead

of Ncell times of transport property calculations are required. Therefore, the inverse ratio of

Ngroup to Ncell indicates the speed up of the transport property calculation. It is seen that with

larger threshold values, the number of the correlated groups reduces significantly, so that

the calculation of transport properties becomes more efficient. In the ε = 0.05 case, which

is used in rest of studies in this chapter, the averaged ratio of Ngroup to Ncell is as small as

0.6%. Therefore, the speed up is more than two orders of magnitude.

103

Figure 4.9 plots the actual CPU Time consumed by the transport property

calculations during the flame simulation. The CPU Time ratio between the five cases is

approximately 1 : 0.3 : 0.1 : 0.02 : 0.005, which is close to the Ngroup/Ncell ratio in Figure

4.8. Therefore, it confirms that the speed up of transport property calculations can be

indicated by the inverse ratio of the number of correlated groups to the number of cells.

4.3.2 Diffusion Ignition

In order to comprehensively validate the proposed CO-DACT method, a 1D

diffusion ignition study is carried out to demonstrate the performance of the method in a

diffusion controlled system. This diffusion ignition is initiated at 1 atm and 1200 K with

the pure air in the left half domain and pure fuel in the right half domain. A planar 1D

domain with 5 cm is used and the reflective boundary conditions are applied on both sides.

The results of the diffusion ignition are compared between HMTS, HMTS/CO-DAC and

HMTS/CO-DACT methods.

Figure 4.10 shows the trajectory of the maximum heat release rate location. This

figure indicates that, this system auto-ignites around t = 1.0 ms and transits to a diffusion

flame after the ignition. Before ignition, the fuel and air mixed at the contact surface. Then

an ignition wave front was formed due to the variation of the ignition delay time caused by

the gradient of the equivalence ratio in the mixing layer. This ignition wave front induces

an acoustic wave and generates a flow oscillation in the domain. The period of the

oscillation is around 0.2 ms. Considering the twice of the domain size, 10 cm, the

propagation speed of this oscillation is around 500 m/s, which is close to the acoustic wave

speed in the mixture at 1000 K. Moreover, the comparison shows that the maximum heat

104

release rate trajectories predicted by these three methods are identical and the relative error

is trivial.

Figure 4.11 and Figure 4.12 respectively show the detailed distributions of the

selected phase parameters, including temperature, O2, Fuel, H2O and CO2, and the selected

radicals, including OH, H, HO2 and H2O2, at t = 3.5 ms. Similar as the spherical

propagating flame calculation, all the species, radicals and temperature follow the same

accuracy regardless of whether they are included in the phase parameters or not. Again,

only the selected species are plotted in order to keep the figures tidy. The maximal

deviation among these three methods is only 0.21% in space, which happens between

HMTS and HMTS/CO-DACT method around the location X = 1.1 cm on both figures. The

numerical error is trivial comparing with the uncertainty in transport properties and kinetic

mechanisms. Therefore, these two figures demonstrate the excellent accuracy of the

proposed CO-DACT method even in the diffusion dominated system.

Figure 4.13 is the comparison of the CPU time consumed by the transport property

calculations during the 1D diffusion ignition calculation. The black, red and blue lines are

computed by the HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods, respectively.

The lines from HMTS and HMTS/CO-DAC methods overlap since the transport properties

are calculated by the same mixture-averaged model. Compared with the other two methods,

the HMTS/CO-DACT method is at least 200 times faster in terms of the transport property

calculation. It can be also seen from this figure that around t = 1.0 ms, there is a

discontinuity in the slope of the CPU time consumption in the HMTS/CO-DACT method.

This discontinuity comes from the auto-ignition of the diffusion flame started at t = 1.0 ms.

The transport properties change rapidly after the onset of the ignition due to the significant

105

variations of temperature and concentration distributions, which results in the increased

number of the time and space correlated groups in order to update the transport properties

more frequently.

Figure 4.14 plots the overall CPU time comparisons at t = 3.5 ms. Similar as Figure

4.7, it shows that the computational cost of transport properties can be reduced dramatically

and becomes trivial in the HMTS/CO-DACT method.

4.3.3 Unsteady Ignition and Spherical Flame Formation with LTC

To validate the CO-DACT method for the unsteady ignition as well as the low and

high temperature diffusion flame formation under the stretched and non-premixed

condition, a numerical study of the transient combustion involving fuel injection into a hot

air within a spherical chamber is conducted. The numerical configuration is shown in

Figure 4.15. The spherical chamber is 20 cm in diameter with an adiabatic wall boundary

condition. At the center of the chamber, there is a porous sphere with 2 cm in diameter

allowing fuel injection into the chamber. Initially, the gas in the chamber is pure quiescent

air at 20 atm and 700 K. At time t = 0, a fuel mixture starts to be injected into the chamber

from the porous sphere surface with an injection speed fixed at U0 = 1 m/s. The inlet fuel

mixture contains 10% DME and 90% N2 in mole fraction. On the porous surface, the

pressure and temperature of the inlet gas are recovered from the mixture on the surface.

The boundary condition on the porous surface can be described as by the following

equation:

,0

0 0 ,0 0 0 ,0 0

i

i i i

dYU Y U Y D

dx +

+ + + += − (4.18)

where ρ0, U0 and Yi,0 are, respectively, the density, velocity and mass fraction of the i-th

species in the inlet mixture. And ρ0+, U0+ and Yi,0+ are the corresponding values of the gas

106

mixture on the porous surface inside the chamber. Di is the mass diffusivity of the i-th

species. This equation describes that the summation of the convection flux and diffusion

flux on the porous surface in the chamber equals the total convection flux injected into the

chamber. It represents the species mass conservation on the porous surface.

With the continuously injection of the inlet fuel, the fuel and oxidizer mixes at the

contact surface due to the molecular diffusion. After the mixture is auto-ignited at the

contact surface, a diffusion flame will be formed and propagate to the outside wall.

Figure 4.16 plots the histories of the maximum temperatures (solid lines) and the

corresponding locations of the maximum temperature (dash lines) in the unsteady ignition

and flame formation process. The black, red and blue lines are the results calculated by the

HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods, respectively. Due to the high

pressure and elevated temperature, a two-stage auto-ignition and the cool diffusion flame

formation can be observed clearly in this figure. The low temperature ignition happens

around 4 ms. Then, a cool diffusion flame is induced with temperature at 910 K. The cool

flame lasts for 3 ms. After that, a hot ignition occurs at 7 ms and the cool flame becomes a

hot diffusion flame and propagates outwardly. Note that the ignition, cool flame formation,

and cool flame to hot flame transition processes in the simulation are strongly governed by

molecular diffusion. These processes are not in quasi-steady states. This figure

demonstrates that the proposed CO-DACT method together with the HMTS method can

accurately capture the maximum temperature and the corresponding locations for both low

and high temperature ignitions and diffusion flames. The maximum relative error among

different methods is less than 1%.

107

In order to validate the accuracy of the diffusion flame structure that calculated by

the HMTS/CO-DACT method, the profiles of the selected phase parameters, including

temperature, fuel, O2, CO2 and H2O, and the minor radicals including H, OH, H2O2 and

HO2, at 48 ms are plotted and compared in Figure 4.17 and Figure 4.18, respectively. These

two figures demonstrate that the HMTS/CO-DACT method can not only capture the

temperature and major species included in the phase parameters, but also predict the minor

radicals outside of the phase parameters precisely.

Figure 4.19 shows the comparisons of CPU times of the transport property

calculations in the unsteady diffusion flame simulation. It demonstrates that the transport

property calculation can be speed up by 190 times with the CO-DACT method. The

improvement of the computational efficiency can be seen more clearly from the total CPU

time comparisons, which are plotted on Figure 4.20. Similar as the premixed flame and

diffusion ignition cases, the CPU time consumed by the calculation of transport properties,

shown as the red section on Figure 4.20, becomes negligible in the CO-DACT method.

The overall computational efficiency can be improved by more than 3 times compared to

HMTS method. Therefore, the comparisons demonstrate that the proposed

HMTS/CO-DACT method can significantly accelerate the calculation without sacrificing

accuracy.

4.3.4 Numerical Error Analysis

In order to validate the selection of the phase parameters, which form the basis of

the correlated space, the sensitivity analysis of transport properties, including conductivity,

viscosity and mass diffusivities, is conducted in a premixed flame. Figure 4.21 shows the

sensitivities of the thermal conductivity in the unburned region, flame surface and

108

equilibrium region. The molar fractions, X, and temperature, T, are perturbed by a small

factor and the corresponding error of thermal conductivities, dλ, are computed. The

sensitivity is defined by:

d dSensitivity or

dX X dT T

= (4.19)

This figure plots the top 14 sensitive parameters of the thermal conductivity. It is

seen that temperature is the most sensitive parameter in the transport properties. Besides

temperature, N2 has the largest impact on transport properties among all the species and

radicals since it has the largest molar fraction in the mixture. The sensitivities of other

major species vary at different regimes. H2O, CO2, CO, OH and H2 have higher sensitivities

on and after the flame surface since they are produced by the flame. O2 has higher

sensitivity in the unburned mixture and on the flame surface because it is consumed after

the flame. According to the sensitivity analysis, the temperature and major species,

including H2O, O2, CO2, CO, OH and H2 should be included in phase parameters. Note that

OH, O, and H only have high sensitivities in the burned region.

All the species below H2 have the sensitivities smaller than 0.03%. But in

simulations conducted in this study, the species nC12H26, iC8H18, C6H5C3H7 and CH2O are

also included in the phase parameters since the first three species are the fuel components

and CH2O is a critical radical in low temperature chemistry. Besides these phase

parameters, the error of transport properties caused by the variation of other species in one

correlated group is negligible. Therefore, the sensitivity analysis validates the selection of

the phase parameters proposed in the CO-DACT method.

109

In order to investigate the effect of the threshold value on the numerical error of

transport properties, the thermal conductivity at different regions of the premixed spherical

flame are calculated and the compared in Figure 4.22. The relative error is defined as:

0

0

'

−= (4.20)

where λ0 is the conductivity computed at the center of the correlated group and λ’ is the

value at the boundary of the corresponding group.

Figure 4.22 shows the relative error of the thermal conductivity as the function of

threshold value at 10 atm and stoichiometric condition. The black, red and blue lines

represent the relative error calculated in the unburned gas, on the flame surface and in the

burned gas, respectively. The gradients of the linear regression of the three curves in

logarithm coordinate are 0.976, 0.981 and 0.947, respectively. It demonstrates that the

relative error is linearly dependent on the threshold value. Therefore, this figure confirms

the statement made in Section 4.2.4: the proposed CO-DACT method is first-order accurate

and the numerical error in transport properties is on the same order of the user-specified

threshold value for discretizing the correlated space.

Besides the local quantities, the numerical error of the flame speed, which is a

representation of the global combustion properties, is also studied. The speeds of the

premixed spherical flames at different conditions are compared between HMTS/CO-DAC

and HMTS/CO-DACT method. The relative errors, δ, plotted in Figure 4.23 and Figure

4.24 are defined as:

110

CO DAC CO DACT

CO DAC

S S

S − −

−

−= (4.21)

where SCO-DACT and SCO-DAC are the flame speeds calculated by the HMTS/CO-DACT and

HMTS/CO-DAC method, respectively. The only difference between the

HMTS/CO-DACT and the HMTS/CO-DAC method are the correlated transport properties.

Therefore, the relative error in Equation (4.21) is purely induced by the correlated transport

properties.

The red dash line in Figure 4.23 represents the linear fitting with gradient of 1 in

the logarithm plot. From Figure 4.23 we can see that when the threshold value is below 0.3,

the relative error in the flame speed is smaller than 1%, which is far below the extrapolation

uncertainties in flame speed measurements. The increment of the relative error follows the

red dash line, so that the relative error grows linearly with the threshold value. Thus, the

results prove that in the CO-DACT method, not only local quantities, but also global

properties are first-order accurate and the relative error is linearly dependent on the

threshold value of phase parameters. It is worth mentioning that the relative error in flame

speed is much smaller than the threshold value, because the threshold value indicates the

numerical error in transport properties and not directly represents the error in flame speed.

The error in transport properties will be diminished by chemical reactions and convections

and it results in a smaller numerical error in flame speed. This figure also demonstrates that

the threshold value we used in this study, 0.05, is good enough to accurately predict the

flame speed.

Figure 4.24 shows the effect of pressure on the relative error of flame speed at

different equivalence ratios with the threshold value of 0.05. The red, green, and blue lines

represent the results of the equivalence ratio at 0.5, 1.0 and 1.5, respectively. The black

111

dash line is the base line where the relative error is zero. At the fuel lean condition where

equivalence ratio is 0.5, the relative errors are negative, which means that the

HMTS/CO-DAC method predicts slightly smaller flame speeds than the

HMTS/CO-DACT method. And the relative errors are positive at the stoichiometric and

fuel rich conditions. These three lines give similar trends: the relative error is weakly

dependent on the pressure. When the pressure increases, the absolute value of the relative

error slightly decreases. This trend can be explained by the formulation of the transport

properties. The viscosity and heat conductivity only depend on temperature and species

mole fractions. Thus, the change of pressure will not directly affect these two parameters.

On the other hand, the mass diffusivity is proportional to the inverse of pressure, 1/P.

Therefore, higher pressure results in smaller mass diffusivities, which leads to a smaller

numerical error in diffusion coefficients.

4.4 Conclusion

In this chapter, a correlated dynamic adaptive chemistry and transport (CO-DACT)

method is developed to drastically accelerate the calculation of transport properties in an

unsteady reactive flow simulation with large chemical kinetics. In this method, the

correlated group in temporal and spatial coordinates for transport properties are identified

successfully by selecting a set of phase parameters. The proposed CO-DACT method is

demonstrated, mathematically and numerically, to be first-order accurate. The numerical

error in transport properties is on the same order of the user-specified threshold value for

discretizing the correlated space. The accuracy of the CO-DACT method can be extended

to 2nd order with a linear extrapolation of the transport properties. Validations in outward

112

propagating spherical premixed flames, diffusion ignitions and spherical propagating

diffusion flames illustrate that the proposed HMTS/CO-DACT method can accurately

reproduce flame speeds, ignitions, flame structures, and species distributions. Moreover,

the calculation of transport properties in the CO-DACT method is 200 times faster than the

mixture-averaged method. Numerical simulations with different threshold values,

pressures, and equivalence ratios show that the method is rigorous in a very broad range.

Therefore, the present HMTS/CO-DACT method provides an innovative and

computationally efficient way for combustion modeling with large chemical kinetics and

detailed transport properties.

It is worth mentioning again that the CO-DACT method is inherited from the

CO-DAC method. This chapter focuses on the discussion of the correlated transport

properties. However, in the following chapters, when specifying the CO-DACT method, it

refers to both the correlated chemical reduction and the correlated transport properties.

113

Figure 4.1: Schematic of time and space correlations in chemical reductions and transport

properties.

Figure 4.2: Comparison of the algorithm procedures between the conventional method

and the HMTS/CO-DACT method.

114

Figure 4.3: Flame trajectories comparison between VODE, HMTS, HMTS/CO-DAC and

HMTS/CO-DACT methods of the stoichiometric jet fuel surrogate mixture initiated at 10

atm and 300 K.

Figure 4.4: Distributions of temperature and major species (O2, Fuel, H2O and CO2) in

the stoichiometric premixed flame at 1.6 ms.

115

Figure 4.5: Distributions of minor radicals (OH, H, HO2 and H2O2) in the stoichiometric

premixed flame at 1.6 ms.

Figure 4.6: Comparison of CPU time for the transport properties calculation between

VODE, HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods of the stoichiometric

premixed flame.

116

Figure 4.7: CPU time comparison between VODE, HMTS, HMTS/CO-DAC and

HMTS/CO-DACT methods at 1.6 ms of the stoichiometric premixed flame.

Figure 4.8: Comparison of Ngroup/Ncell ratio for the threshold value equals 0 (black),

0.0001 (red), 0.0005 (blue), 0.005 (green) and 0.05 (purple) respectively.

117

Figure 4.9: Comparison of CPU time for the transport properties calculation with

threshold value equals 0 (black), 0.0001 (red), 0.0005 (blue), 0.005 (green) and 0.05

(purple) respectively.

Figure 4.10: Comparison of the maximum heat release rate trajectories between VODE,

HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods of the 1D diffusion ignition.

118


the diffusion ignition at 3.5 ms.

Figure 4.12: Distributions of minor radicals (OH, H, HO2 and H2O2) in the diffusion

ignition at 3.5 ms.

119


HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods of the diffusion ignition.

Figure 4.14: CPU time comparison between HMTS, HMTS/CO-DAC and HMTS/CO-

DACT methods at 3.5 ms of the diffusion ignition.

120

Figure 4.15: Schematic of the unsteady ignition and flame formation in a spherical

diffusion flame configuration.

Figure 4.16: Comparisons of the predicted time history of the maximum temperatures and

corresponding locations between HMTS, HMTS/CO-DAC and HMTS/CO-DACT

methods of the unsteady diffusion ignition and flame formation.

121


the unsteady diffusion flame at 48 ms.

Figure 4.18: Distributions of minor radicals (OH, H, HO2 and H2O2) in the unsteady

diffusion flame at 48 ms.

122


HMTS, HMTS/CO-DAC and HMTS/CO-DACT methods of the unsteady diffusion

flame.

Figure 4.20: CPU time comparison between HMTS, HMTS/CO-DAC and HMTS/CO-

DACT methods at 48 ms of the unsteady diffusion flame.

123

Figure 4.21: Sensitivity analysis of thermal conductivity in the stoichiometric jet fuel

surrogate mixture initiated at 10 atm and 300 K.

Figure 4.22: Relative error of thermal conductivity versus threshold values in unburned

gas (black), on flame surface (red) and in burned gas(blue).

124

Figure 4.23: Relative error of flame speed versus threshold values at 10 atm and

stoichiometric condition.

Figure 4.24: Relative error of flame speed versus pressure at equivalence ratio equals 0.5

(red line), 1.0 (green line) and 1.5 (blue line) with 0.05 threshold value.

125

Chapter 5

Adaptive Analytical Jacobian (AAJ) Method

In previous chapters, the CO-DAC and CO-DACT methods are integrated with the

hybrid multi-time scale (HMTS) method to perform chemical integrations, which is an

efficient linear scaling ODE solver. However, it suffers the stability issue when reaction

modes are strongly coupled during the thermal runaway. In this situation, a very small time

step needs to be applied in the HMTS integration, which can be much smaller than the CFL

limitation. The unnecessarily small time step significantly slows down the overall

simulation process and makes the combustion modeling inefficient. To overcome this

difficulty, an adaptive analytical Jacobian (AAJ) method is developed to perform efficient

and stable chemical integrations by applying the locally reduced chemical mechanism from

the CO-DACT method to an analytical Jacobian method. In this chapter, the principle of

the analytical Jacobian method, the integration of the locally reduced chemistry with the

analytical Jacobian method, and comparisons between the AAJ and HMTS methods will

be presented.

5.1 Background and Objectives

In order to solve the stiff and non-linear ODE system that governs chemical

reactions, traditional implicit ODE solvers with the backward differentiation formula (BDF)

are usually applied, which linearize the ODE system by the Jacobian matrix decomposition.

However, the computational complexity of these implicit ODE solvers is proportional to

the cubic of the number of species in a reaction system. When a large and detailed chemical

126

mechanism is involved, the computation cost grows drastically and becomes unaffordable.

Therefore, linear scaling ODE solvers play important roles in combustion modeling with

detailed chemical kinetics. According to the discussion in Section 1.3.4, the two major

approaches of the linear scaling solvers are 1) time scale based integration, such as the

hybrid multi-timescale (HMTS) method (Gou et al. 2010), the dynamic stiffness removal

method (Lu et al. 2009), and the adaptive method for hybrid integration (AHI) (Gao et al.

2015), and 2) sparse matrix based ODE solvers, such as the analytical Jacobian (AJAC)

method (Perini et al. 2012), the sparse solver with adaptive preconditioners method

(McNenly et al. 2015), and the extrapolation-based stiff ODE solver (Imren and Haworth

2016).

The HMTS method has been extensively evaluated for computational accuracy and

efficiency. Therefore, it has been integrated with the CO-DAC and CO-DACT method in

Chapter 3 and Chapter 4 to conduct chemical integrations based on the locally reduced

chemical mechanisms. However, since the species and energy equations are decoupled and

integrated iteratively in the HMTS method, the solver becomes unstable when reaction

modes are strongly coupled. Due to this stability issue, the chemical integration time step

is required to be small, which can be much smaller than the CFL limitation. Therefore, the

overall computational efficiency is limited by the small time step. This stability issue

generally exists in the time scale based integration approach of the linear scaling solvers.

To overcome this stability issue while still keeping the linear scaling behavior, the

sparse Jacobian matrix based implicit ODE solvers can be utilized to replace the HMTS

method for chemical integrations with large time steps. Among the sparse matrix based

ODE solvers, the analytical Jacobian (AJAC) method is an efficient alternative.

127

However, even with the efficient analytical form of the Jacobian matrix and the

sparse matrix algorithms, the computational cost of a single time step integration in the

AJAC method is still more expensive than the computational cost in the HMTS method.

Moreover, the AJAC method takes the full dimension of the species equation, meaning that

it cannot benefit from the locally reduced mechanisms generated by the CO-DACT method.

Therefore, the objective of this study is to develop an adaptive analytical Jacobian (AAJ)

method to utilize the local dynamic adaptive chemistry in the AJAC method to accelerate

the chemical integration while maintaining the stability. In the following sections, the

concept of the AJAC method and the integration of the AAJ method will be introduced.

Then the stability of the AJAC and HMTS method will be compared, followed by the

detailed validations of the proposed AAJ method.


5.2.1 Analytical Jacobian (AJAC) Method with Sparse Solver

The ODE system which governs chemical reactions are described in Equation (3.1)

and (3.2). They can be written in the vector format:

( )' =y f y (5.1)

where the ’ superscription represents the time derivation. The variable vector y and the

vector function f (y) are defined as:

128

( )

1

1 1,

s

ss

N

i i vi

N N

T e c

Y

Y

= − = =

y f y (5.2)

In the above equation, the analytical formula of the internal energy, 𝑒𝑖 = 𝑒𝑖(𝑇), the mean

specific heat capacity at constant volume, 𝑐�� = 𝑐��(𝑇, 𝐘), and the net production rate, 𝜔𝑖 =

𝜔𝑖(𝑇, 𝐘), can be obtained from Equation (3.3), (3.4), and (2.17), respectively.

The Jacobian matrix of this ODE system can be written as:

( )' = =

f yyJ

y y (5.3)

which can be divided into four parts:

'

'

'

'

T T

T

T

Y

Y

Y Y

(5.4)

where Y = (Y1, …, YN)T. In Equation (5.2), the dependencies of f (y) on y are explicitly

given by 𝑒𝑖, 𝑐��, and 𝜔𝑖. Therefore, the partial derivatives in Equation (5.3) and (5.4) can

be obtained analytically.

In a combustion simulation, the analytical form of the Jacobian matrix can be

calculated and memorized prior to the detailed simulation according to the chemical

reactions. In the following ODE integrations, the value of Jacobian matrix at each time step

can be evaluated by the analytical form together with the local thermodynamic states. This

analytical Jacobian matrix can be utilized to replace the numerical Jacobian matrix, which

is calculated by the expensive numerical perturbation, in implicit ODE solvers to

dramatically improve the solver efficiency.

129

Moreover, since the number of species involved in an elementary reaction is limited,

the Jacobian matrix in Equation (5.3) is mostly sparse. Examples of the Jacobian matrix

are visualized and plotted in Figure 5.1, where the block node represents non-zero matrix

element. In this figure, four chemical kinetic mechanisms with different sizes and different

low and high temperature chemistries are evaluated: a hydrogen mechanism (Burke et al.

2012) with 10 species and 21 reactions (Figure 5.1 (a)), a methane mechanism (Wang et al.

2007) with 30 species and 184 reactions (Figure 5.1 (b)), a DME mechanism (Cheung et

al. 2004) with 55 species and 290 reactions (Figure 5.1 (c)), and a n-heptane mechanism

(Dooley et al. 2012) with 170 species and 962 reactions (Figure 5.1 (d)). The detailed

mechanism information and the sparsity of the corresponding Jacobian matrix are listed in

Table 5.1. It is seen that with the increase of the size of the chemical mechanism, the

sparsity of Jacobian matrix increases quickly. Therefore, efficient sparse matrix algorithms

can be applied to conduct the Jacobian matrix decomposition and inversion when the large

and detailed chemical kinetics are involved.

Mechanism Number of Species Number of Reactions Jacobian Sparsity

Hydrogen 10 21 24.0%

Methane 30 184 28.9%

DME 55 290 62.0%

n-Heptane 170 962 84.2%

Table 5.1: Detailed mechanism information and Jacobian matrix sparsity

Based on the analytical form of the Jacobian matrix and the efficient sparse matrix

algorithms, an analytical Jacobian (AJAC) method is developed (Perini et al. 2012).

Chemical mechanisms in the CHEMKIN format are utilized to calculation the analytical

form of the Jacobian matrix. Based on the analytically computed Jacobian matrix, a

Livermore solver for ODE with general sparse Jacobian matrices (LSODES) (Hindmarsh

1983) is applied to conduct the ODE integration with efficient sparse matrix algorithms.

130

The AJAC method is an efficient linear scaling ODE solver, which is implicit and stable.

Therefore, large integration time steps can be applied to accelerate the overall numerical

simulation.

5.2.2 Adaptive Analytical Jacobian (AAJ) Method

Although the AJAC method is efficient and stable, the computational cost of a

single time step integration in the AJAC method is still more expensive than the

computational cost in the HMTS method. More importantly, the AJAC method doesn’t

have the flexibility to adjust the dimension of the species equation on-the-fly, so that it

cannot benefit from the locally reduced mechanisms generated by the CO-DACT method.

Therefore, an adaptive analytical Jacobian (AAJ) method is proposed by integrating

the adaptive chemistry generated by the CO-DACT method with the analytical Jacobian

method. With the locally reduced mechanisms, the dimension of the Jacobian matrix and

the ODE system can be significantly reduced, so that the computational efficiency of the

analytical Jacobian method and be improved. Moreover, the reduced number of elementary

reactions in the chemical system can improve the efficiency of the reaction rate calculation.

The correlated dynamic adaptive chemistry from the CO-DACT method contains

two sets of information: a list of active species, AS, and a list of active reactions, AR. All the

reactions associated with the inactive species are considered frozen, which have absolute

0 forward and backward reaction rates. Thus, the production, destruction and net

production rates of the inactive species remain zero:

0,i Si A (5.5)

Thus, the mass fraction of the inactive species remain constant:

131

,i i SY C i A (5.6)

Therefore, they become independent of any system evolution, i.e. for any given system

properties η:

0, S

i

i AY

(5.7)

With the reduced chemical mechanism, the ODE system is simplified to:

1'

' ,

S

i i

i Av

i i S

T ec

Y i A

= −

=

(5.8)

Therefore, the dimension of the ODE system is reduced from Ns to the number of species

in AS. According to Equation (5.6), the rows of the inactive species in the Jacobian matrix

become,

'0

,'

0

i

Si

j

Y

Ti A

Y

Y

=

=

(5.9)

And according to Equation (5.7), the columns of the inactive species in the Jacobian matrix

are simplified to

'0

,'

0

i

S

j

i

T

Yi A

Y

Y

=

=

(5.10)

The above equations indicate that all the rows and columns that associated with the inactive

species remain 0 in the Jacobian matrix. Therefore, the dimension of Jacobian matrix can

be reduced from Ns to the number of species in AS.

132

On the other hand, the list of active reactions, AR, can be applied to simplify the

reaction rates calculations. According to Equation (2.17), by ignoring inactive reactions,

the net production rate of the 𝑖-th species can be simplified to:

( ), ,'' 'R

i i i k i k k

k A

W

= − (5.11)

Equation (5.11) indicates that only the progress rates of the active reactions, Ωk, need to be

calculated. Therefore, the reaction rate calculations can be substantially accelerated.

Based on the above discussion, the flowchart of the AAJ method is plotted in Figure

5.2. At the beginning of a integration step, the CO-DACT method is applied to produce the

locally reduced chemistry with a list of active species, AS, and a list of active reactions, AR.

The active species are used to reduce the dimension of the ODE system and the dimension

of the Jacobian matrix, while the active reactions are applied to accelerate the reaction rate

calculations in the Jacobian matrix and ODE equations. Then, according to the reduced

analytical form of Jacobian matrix and the reduced reaction rates, the value of the Jacobian

matrix is evaluated for this particular integration step. Finally, the LSODES method is

involved to conduct the ODE integration based on the reduced equation system and the

sparse Jacobian matrix.


To validate the proposed AAJ method, a detailed n-heptane mechanism (Dooley et

al. 2012) with 170 species and 960 reactions is utilized in the following simulations. In this

section, the stabilities of AJAC and HMTS methods are compared at first, followed by the

detailed validations of the AAJ method with homogeneous auto-ignition and planar

unsteady premixed flame calculations.

133

5.3.1 Stability Comparisons between AJAC Method and HMTS Method

To compare the stability of the chemical integration in the AJAC and HMTS

method, a homogenous auto-ignition calculation is carried out in a stoichiometric

n-heptane/air mixture initiated at 10 atm and 1200 K.

Figure 5.3 plots the CPU time consumption per integration time step during the

ignition calculation. The integration time step used in this Figure is 2 nanoseconds (2 ×

10-9 s). It is seen that the HMTS method (blue line) is extremely efficient in the pre-ignition

region and post-ignition region, where the chemical system is near the equilibrium state.

In these regions, the HMTS method can be one-order of magnitude faster than the AJAC

method. However, while the chemical system approaching the thermal runaway, the

computational cost of the HMTS method grows dramatically by more than two orders of

magnitude. The reason of this growth is that during the thermal runaway, the concentration

of radicals and species evolve quickly due to the rapid chain branching and propagation

reactions. Therefore, the species equations and timescales are strongly coupled. However,

the HMTS method integrates the fast species using piece-wise implicit Euler solver, which

treats each species equation independently. Thus, the implicit Euler solver needs more

iterations to converge to the final state. The large number of the iteration steps during

thermal runaway introduces the stability issue in the HMTS method. On the other hand,

the AJAC method (red line) is very stable. The CPU time per integration step is mostly flat

during the ignition process in the AJAC method. Therefore, it demonstrates that the AJAC

method is much more stable than the HMTS method at a large time step.

As mentioned above, the high computational cost during the thermal runaway in

the HMTS method is caused by the large number of iteration steps to converge the reaction

134

system. This argument can be demonstrated by Figure 5.4, which plots the number of

iteration steps in the HMTS method during the ignition calculation. It shows that only a

few iterations are required in the near equilibrium regions. It is worth noting that in the

equilibrium state after ignition, only 1 iteration step is required to converge the system,

which makes the HMTS method extremely efficient. However, around the ignition point,

the number of the iteration steps grows significantly. The maximal iteration steps at the

ignition point reaches 748. The slow convergence during thermal runaway diminishes the

high efficiency in the near equilibrium regions. More importantly, the larger the integration

time step is, the harder the system can converge. When larger time steps are used, the

piece-wise implicit integration of the fast species in the HMTS method will eventually fail

and cause the solver to diverge, which can be demonstrated in Figure 5.5.

Figure 5.5 shows the number of maximal iteration steps in the HMTS method

versus the value of the integration time step. It is seen that when the integration time step

is small, the HMTS method converge easily with a small amount of iteration steps. The

HMTS integration can be converged within 26 iterations when a 10-10 second time step is

applied. However, when the integration time step becomes large, the number of maximal

iteration steps increases quickly and the HMTS method becomes unstable. Any time step

beyond 2 nanoseconds will cause the solver to diverge. This upper bound of the time step

in HMTS method could be much smaller than the CFL limitation. Therefore, this stability

issue limits the time step that can be utilized in combustion simulations with the HMTS

method. The small time step limitation reduces the overall computation efficiency.

Moreover, the stability issue not only restricts the integration time step, but also

reduces the computational efficiency of the HMTS method directly. Figure 5.6 plots the

135

comparisons of averaged CPU time per time step between the HMTS (blue line) and AJAC

(red line) method in the ignition calculation with different time steps. It shows that when

small time steps are applied, the HMTS method is much faster than the AJAC method.

However, with the increase of the time step, the large number of iterations slows down the

HMTS method. With a 2 ns time step, the computational cost of the HMTS method exceeds

the AJAC method. On the other hand, the performance of the AJAC method is stable. The

CPU time consumptions in AJAC method are mostly independent of the time step. More

importantly, the AJAC method can tolerance much larger time steps, which makes it

preferable to the HMTS method in practical simulations.

It is worth mentioning that the upper bound of the integration time step in HMTS

method is 2 ns in the above ignition calculation. However, the system can be unstable when

using this time step in different initial conditions. With a higher initial temperature, such

as 1600 K, the HMTS method fails to converge at the 2 ns time step. Therefore, in the

following validations and comparisons, a smaller time step, 1 ns, is used as the benchmark

tests.

5.3.2 Validations of AAJ method

The AJAC method has been demonstrated to be more stable than the HMTS method.

However, it is slower than the HMTS method when a relatively small time step is used. To

accelerate the AJAC method, the adaptive analytical Jacobian (AAJ) method is developed

in this study by applying the locally reduced chemical mechanism generated by the CO-

DACT method to the AJAC method. The accuracy and performance of the AAJ method is

validated in this sub section.

136

The HMTS/CO-DACT method is included as the benchmark of the validations.

The HMTS/CO-DACT method, the AJAC method, and the AAJ method are compared in

a homogeneous auto-ignition calculation and a planar unsteady premixed flame calculation

with the n-heptane mechanism. The integration time step is 1 ns in both calculations for

the basic comparisons. Moreover, since the AAJ method is more stable and can tolerate

larger integration time steps, the AAJ method with the 10 ns time step is also included in

the comparisons. The details of the compared methods is listed in Table 5.2.

Chemical

Solver

Adaptive

Chemistry

Correlated

Transport

Integration

Time Step

HMTS/CO-DACT HMTS Yes Yes 1 ns

AJAC AJAC No No 1 ns

AAJ AJAC Yes Yes 1 ns

AAJ (Δt = 10 ns) AJAC Yes Yes 10 ns

Table 5.2: Comparisons of details between different methods

Homogenous Auto-ignition

The homogenous auto-ignition calculation is conducted in the stoichiometric

n-heptane and air mixture initiated at 10 atmosphere pressure. Figure 5.7 plots the ignition

delay times at different initial temperatures. It is seen that the HMTS/CO-DACT, AJAC,

AAJ with 1 ns time step and AAJ with 10 ns time step predict identical ignition delay times

across the low, intermediate, and high temperature ranges. The maximal relative error

among all the compared cases are below 0.3%. Therefore, it demonstrates that the AAJ

method is accurate when predicting the global combustion properties, even when a much

larger time step is involved.

Figure 5.8 compares the detailed ignition histories initiated at 800 K. Temperature

and the mass fractions of important radicals, including CH2O, OH, and HO2 are included

in the comparison. The results show that the AAJ method can precisely produce the species

and temperature histories, as well as the two-stage ignitions in the negative temperature

137

coefficient (NTC) region. Thus, this figure proves that the proposed AAJ method can not

only predict the global combustion properties, but also simulate detailed evolutions of the

reaction system accurately. Therefore, the accuracy of the AAJ method is successfully

validated.

Figure 5.9 shows the CPU time comparison between the four methods in the

ignition calculation initiated at different temperatures. The CPU times are normalized by

the CPU time of the HMTS/CO-DACT method. It is seen that at a small time step, such as

1 ns, the AJAC method is about one order of magnitude slower than the HMTS method.

With the benefit of the locally reduced mechanisms in AAJ method, the computational

costs of the AJAC method are reduced by 40%-60% depends on the initial temperature.

More importantly, by taking the advantage of the larger integration time step of 10 ns, the

AAJ method becomes 30%-70% more efficient than the benchmarked HMTS/CO-DACT

method. Therefore, it is demonstrated that with the benefit of large integration time steps,

the AAJ method can effectively reduce the computational cost without sacrificing accuracy.

Planar Unsteady Premixed Flames

To validate the proposed AAJ method in combustion simulations with

chemistry-transport coupling, a one-dimensional planar unsteady premixed flame

calculation is carried out. The length of the computation domain is 5 cm. Adaptive grids

are applied with the smallest grid sizing being 16 μm. The integration time step is 1 ns for

all different methods with an additional 10 ns time step calculation for the AAJ method.

Since the acoustic speed in a combustion system is on the order of 1000 m/s, the time step

of 10 ns satisfies the CFL constraint. Moreover, this integration time step is also below the

limit of the diffusion timescale defined as Δx2/DH = 1.062 × 10-6 s, where DH =

138

2.41 × 10-4 m2/s is the mixture-averaged diffusion coefficient of the most diffusive species,

H, at 3000 K. A reflective boundary condition and an outlet boundary condition are applied

on the left and right boundaries, respectively. The initial mixture temperature and pressure

are 300 K and 1 atm, respectively. A 2 mm hot spot is located at the left boundary with the

temperature of 2000 K.

The flame trajectories calculated by the HMTS/CO-DACT, AJAC, AAJ, and AAJ

with Δt = 10 ns are plotted and compared in Figure 5.10. It shows excellent agreements

between different methods. Both of the maximal relative errors in flame locations and in

flame speeds are below 1%, which are between the HMTS/CO-DACT method and AAJ

method with Δt = 10 ns. This figure demonstrates the accuracy and robustness of the AAJ

method in the flame speed prediction.

To further validate the accuracy of the AAJ method in the prediction of the detailed

flame structures, the distributions of temperature and selected species, including fuel and

HO2, at 2.5 ms are plotted in Figure 5.11. The distributions are zoomed in around the flame

surface to illustrate the detailed flame structures. It is seen that the flame structure predicted

by the AAJ method agrees well with the one predicted by the HMTS/CO-DACT method,

even when a much larger time step is used in the AAJ method. The relative error in

locations between the HMTS/CO-DACT method and the AAJ method with 10 ns time step

is less than 1%. The excellent agreements in both flame trajectories and flame structures

demonstrate the accuracy of AAJ method in flame calculations.

The CPU time consumptions in the flame calculation at 2.5 ms are plotted and

compared in Figure 5.12 between the four methods mentioned above. The computational

cost of chemical solvers are shown as the red bars and the CPU time of solving the transport

139

PDEs are shown as the black bars. Figure 5.12 shows that the original AJAC method is 8.1

times slower than the HMTS/CO-DACT method in term of the chemical solver. Moreover,

the calculation of transport terms in the AJAC method is also slower than the

HMTS/CO-DACT method because the correlated transport properties are not involved in

the AJAC method. In the AAJ method, with the locally reduced chemistry and correlated

transport properties, computational costs of both the chemical solver and transport terms

are reduced by half compared with the AJAC method. By further increasing the integration

time step by 10 times, the AAJ method becomes 8.6 times faster and the overall

computational efficiency exceeds the HMTS/CO-DACT method by 54%. Therefore, it is

demonstrated that the AAJ method can be more efficient and robust than the

HMTS/CO-DACT method by taking the advantage of the large integration time step.

The flame calculation also shows the drawback of the HMTS method due to the

stability issue at a large time step. In this calculation, a small time step, 1 ns, has to be

applied in the HMTS method in order to avoid the failure of the chemical integration.

Assuming the acoustic speed, a, in this calculation is on the order of 1000 m/s, the CFL

number based on the time step is:

9

6

1000 10CFL 0.0625 1

16 10

a t

x

−

−

= = =

(5.12)

This CFL number is far below the limit required by the explicit time integration, such as

the Runge-Kutta method and the Euler method, in the transport PDE system. Moreover,

the 1 ns time step is also much smaller than the limit of the diffusion timescale, given by:

140

( )2

62 26 9

4

16 101.062 10 s 10 s

2.41 10d

H

x x

D D

−

− −

−

= = =

(5.13)

where DH is the value of the diffusion coefficient of the most diffusive species, H, at 3000 K.

Therefore, it is demonstrated that in order to guarantee the convergence of the HMTS

method, an unnecessarily small time step need to be applied in a combustion modeling,

which significantly slows down the overall simulation.

5.4 Conclusion

In this chapter, an adaptive analytical Jacobian (AAJ) method is developed by

integration the correlated dynamic adaptive chemistry and transport method (CO-DACT)

with the analytical Jacobian (AJAC). The proposed AAJ method is validated using a

n-heptane kinetic model for ignitions and flame propagations. The locally reduced

chemical mechanisms in the AAJ method can not only reduce the dimension of the

Jacobian matrix and the ODE system but also accelerate the calculation of reaction rates.

The accuracy and efficiency of the AAJ method are validated in a homogenous auto

ignition calculation and a planar unsteady premixed flame calculation. It is demonstrated

that the AAJ method can reduce the computational cost by 30% - 70% compared with the

AJAC method without sacrificing accuracy. Moreover, the AAJ method is demonstrated

to be more stable than the benchmarked HMTS/CO-DACT method, which allows larger

integration time steps to be used in a combustion modeling. Compared with the

HMTS/CO-DACT method, the overall computational efficiency can be improved by 55%

in the AAJ method with a 10 times larger time step. Therefore, the proposed AAJ method

141

is demonstrated to be an efficient, accurate, stable, and robust chemical solver for

combustion modeling with detailed chemical kinetics.

142

Figure 5.1: Jacobian matrix visualization of hydrogen mechanism (a), methane

mechanism (b), DME mechanism (c), and n-heptane mechanism (d).

143

Figure 5.2: Flowchart of AAJ method

144

Figure 5.3: Comparison of CPU time per integration step in a homogeneous auto ignition

calculation of stoichiometric n-heptane/air mixture initiated at 10 atm and 1200 K.

Figure 5.4: Iteration Steps of the piece-wise implicit ODE integration in HMTS method

during the calculation of homogeneous ignition of stoichiometric n-heptane/air mixture

initiated at 10 atm and 1200 K.

145

Figure 5.5: Maximal HMTS iteration steps at different integration time steps in the

homogeneous auto ignition calculation of stoichiometric n-heptane/air mixture initiated at

10 atm and 1200 K.

Figure 5.6: Averaged CPU time per step of HMTS method (blue) and AJAC method

(red) at different integration time step in the homogeneous auto ignition calculation of

stoichiometric n-heptane/air mixture initiated at 10 atm and 1200 K.

146

Figure 5.7: Ignition delay time comparisons between HMTS/CO-DACT method, AJAC

method, AAJ method and AAJ method with 10 ns time step in stoichiometric n-

heptane/air mixture initiated at 10 atm.

Figure 5.8: Temperature (solid lines) and selected species (dash line: CH2O, dot line: OH,

dash dot line: HO2) profile comparisons in the homogeneous ignition calculation of

stoichiometric n-heptane/air mixture initiated at 10 atm and 800 K.

147

Figure 5.9: CPU Time comparisons in the ignition calculation normalized by the CPU

Time of HMTS/CO-DACT method.

Figure 5.10: Flame trajectory comparisons between HMTS/CO-DACT method, AJAC

method, AAJ, method, and AAJ method with 10 ns time step in the premixed flame

calculation of stoichiometric n-heptane/air mixture initiated at 1 atm and 300 K.

148

Figure 5.11: Temperature (solid lines) and selected species (dash line: 50 times of HO2,

dot line: fuel) distribution comparisons in the planar flame calculation at 2.5 ms.

Figure 5.12: CPU time comparisons in the planar flame calculation at 2.5 ms.

149

Chapter 6

Multi-scale Adaptive Reduced Chemistry Solver (MARCS)

The advanced numerical methods developed in previous chapters are efficient and

robust. However, they are limited to the one-dimensional numerical studies. To conduct

efficient combustion modeling in multi-dimensional geometries with chemistry-turbulent

interactions, a multi-scale adaptive reduced chemistry solver (MARCS) is developed in

this chapter by integrating the CO-DACT and AAJ methods to an in-house

multi-dimensional CFD program. The detailed numerical methods and implementations

will be discussed, followed by examples of MARCS applications in high-dimensional

turbulent combustion simulations.

6.1 Motivations and Objectives

In previous chapters, efficient numerical methods, including the CO-DAC method,

the CO-DACT method, and the AAJ method, have been developed to significantly improve

the computational efficiency of the on-the-fly chemical reduction, the transport properties

calculation, and the chemical integration. They are implemented in an adaptive simulation

of unsteady reactive flow (ASURF) program (Chen 2009) for validations and applications.

ASURF is a combustion modeling program with the adaptive grid, finite volume

discretization, fractional splitting steps, and Runge-Kutta time integration strategy. By

utilizing a 3rd order WENO scheme to capture sharp discontinuities and integrating with

the CO-DACT and AAJ methods, ASURF is further updated to ASURF+ (Sun et al. 2015;

Sun and Ju 2017). ASURF+ is efficient and light weighted when conducting combustion

150

simulations with detailed chemical kinetics. However, it is a one-dimensional combustion

modeling program, which lacks a few key features for large scale combustion simulations

with turbulent and chemistry interactions, such as the parallel computing and

three-dimensional computation domain.

Therefore, the objective of this study is to develop a multi-scale adaptive reduced

chemistry solver (MARCS) by integrating the advanced algorithms developed in previous

chapters to a high-dimensional CFD program for efficient combustion modeling in

practical geometries.

6.2 Numerical Implementations

6.2.1 Multi-dimensional Parallelized Fluid Solver

To conduct high dimensional combustion simulations in practical geometries, an

in-house parallelized program with the full speed fluid solver is utilized as the code base

for the integration of the advanced numerical methods developed in previous chapters.

In the in-house fluid solver, three dimensional governing equations are solved with

structured computational cells. The finite volume method is used to discretize the

computation domain. The convection term in the Navier-Stokes equations, described in

Equation (2.34), is constructed by a 3rd order advection upstream splitting method with

pressure wiggles (AUSMPW+) scheme (Kim, Kim, and Rho 2001). And the diffusion term

is discretized by a regular central difference scheme. To improve the stability of the time

integration, a modified fully-implicit all-speed lower-upper symmetric Gauss-Seidel (LU-

SGS) scheme (Yoon and Jameson 1988) with the Newton-like sub-iterations in pseudo

time is taken as the time marching method to solve the Navier-Stokes equations. To

151

accelerate the numerical simulation in large computational domains, the message passing

interface (MPI) technique is taken to parallelize computations by dividing computational

domains to multiple CPUs.

This in-house fluid solver is efficient and has been extensively validated (Fu and

Wang 2013). The detailed comparisons between the in-house fluid solver and the ASURF+

are listed in Table 6.1. Compared with the ASURF+ method, the in-house fluid solver has

the abilities to conduct high-dimensional calculations with parallel computing technique.

However, this fluid solver is developed for unsteady, compressible, but non-reactive flow

simulations. It doesn’t solve the species equations, species binary diffusions, and chemical

reactions. Thus, it doesn’t have the capability to conduct combustion simulations with

chemical reactions.

In-house Fluid Solver ASURF+

Dimension of Equations 3 1

Parallel Computing MPI -

Discretization Finite volume Finite volume

Convection Flux 3rd order AUSMPW+ 3rd order WENO

Diffusion Flux Central difference Central difference

Time Integration Fully-implicit LU-SGS Explicit 3rd order Runge-Kutta

Transport Properties - Correlated transport

Chemical Solver - HMTS or AAJ

Table 6.1: Comparisons of solver details between the in-house fluid solver and ASURF

method.

6.2.2 Implementations of MARCS

To enable the calculation of chemical reactions in the fluid solver, the basic

governing equations are implemented into the program, including the species conservation

equation (Equation (2.4)), the species mass diffusion (Equation (2.10)), the head flux

introduced by the diffusion of species (Equation (2.9)), the enthalpies contributed by

species in the internal energy (Equation (2.8)), the chemical reactions (Equation (2.17)),

and the mixture averaged transport properties (Equation (2.23-2.25)).

152

Moreover, the CO-DACT method and AAJ method developed in Chapter 4 and

Chapter 5, respectively, are integrated based on the complete set of governing equations to

improve the computational efficiency of the on-the-fly chemical reduction, chemical

integration, and transport properties calculation.

By fully implementing chemistry related governing equations, the CO-DACT

method, and the AAJ method into the in-house fluid solver, a multi-scale adaptive reduced

chemistry solver (MARCS) is developed, which is able to conduct efficient combustion

modeling with detailed chemical kinetics in multi-dimensional computational domains and

practical geometries.

6.3 Examples of Multi-dimensional Turbulent Simulations

In order to demonstrate the ability of the multi-dimensional combustion modeling,

two examples of numerical simulations with turbulent-chemistry interactions are carried

out by the presented MARCS method with detailed chemical kinetics. The details of the

simulations are discussed in this section.

6.3.1 Turbulent Stretch Ignition

A 2-dimensional simulation of the turbulent stretch ignition is conducted by the

MARCS method. An n-decane mechanism with 45 species and 112 reactions are utilized

in the simulation. The computational domain is set as a square with the length of 2 cm in

both x any y directions. The uniform computational cell with Δx = Δy = 20 μm are applied

to discretize the computational domain. Since MARCS is a 3-dimensional solver, 3 layers

of computational cells are included in the z direction. Therefore, it ends up with

1000 × 1000 × 3 = 3 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 computation cells. The integration time step is 5 ns. The

153

non-slip wall boundary conditions are applied on the top and bottom boundaries in the y

direction. And the periodic boundary conditions are applied in x and z directions. With the

parallel computing, the entire simulations are equally split into 400 CPUs. Therefore, there

are 7500 cells calculated on each CPU.

The n-decane/air mixture is initiated at 500 K, 1 atm, and fuel lean condition with

an equivalent ratio of 0.7. The mixture is stretched by the top and bottom walls, where the

top wall moves at 5 m/s towards the x direction and the bottom wall also moves at 5 m/s

but in the opposite direction. The initial velocity in x direction is linearly stretched with a

gradient of 500 s-1. Thus, the initial velocity, u, in the flow field is:

( ) ( )0.01 500u y y= − (6.1)

assuming that the origin point of the cartesian coordinate is at the bottom left corner of the

computational domain.

The initial conditions are shown in Figure 6.1, where the normalized properties are

plotted. The temperature (T) and density (R) are normalized by the initial values. And the

velocity in x direction (U) is normalized by 5 m/s. The n-decane/air mixture is ignited by a

round ignition kernel at the center of the computational domain. The radius of the ignition

kernel is 2 mm and the temperature in the ignition kernel is 2000 K. An turbulence is

introduced in the initial condition, with the fluctuations of velocity, temperature, and

equivalent ratio being 0.97 m/s, 15 K, and 0.1, respectively. The integral scale and

Kolmogorov scale are 1.24 mm and 28.4 μm, respectively.

Figure 6.2 shows the flow properties at 650 μs, including temperature (T), density

(R), velocity in x direction (U), and velocity in y direction (V), where V is also normalized

by 5 m/s. It is seen that the mixtures inside and around the ignition kernel have been ignited.

154

And the radius of the ignition kernel grows to an averaged value of 3.8 mm. The

temperature at the center of the ignition kernel is increased to around 2800 K due to the

exothermic reactions during ignition. The shape of the ignition kernel is stretched in x

direction and its boundary is perturbed by the turbulent flow field. Therefore, this figure

demonstrates that the MARCS method can predict chemical reactions as well as turbulent

effects in a high dimensional combustion modeling.

Figure 6.3 and Figure 6.4 plots the mass fractions of selected major species,

including fuel (a), oxygen (b), nitrogen (c), and water (d), and selected radicals, including

OH (a), CO (b), CH2O (c), and H2O2 (d), in the ignition process at 650 μs, respectively. It

is seen that fuel is completely consumed inside the ignition kernel with leftover oxygen,

since the initial mixture is in the fuel lean condition. H2O is generated as the product of

chemical reactions inside the ignition kernel. Additionally, OH and CO are also produced

after ignition due to the equilibrium of forward and backward reactions. On the other hand,

CH2O and H2O2 only exist on the ignition front. The mass fractions of species on the

ignition front are fluctuated by the turbulence flow field. This figure clearly shows that the

proposed MARCS method can reproduce the detailed species profiles in the ignition front

with chemistry-turbulent interactions.

The entire calculation consumed 87 thousand CPU hours. By splitting it to 400

CPUs, it takes 217 hours of the wall time. Therefore, this multi-dimensional simulation can

be completed with affordable computational cost even with the detailed chemical

mechanism and turbulence are involved.

155

6.3.2 Detonation Formation in a Turbulent Stratified Mixture

A study of detonation formation in a 2-dimensional stratified mixture with turbulent

perturbations are carried out by using MARCS method. A detailed DME mechanism with

55 species and 290 reactions are involved in the calculation. The computational domain is

a rectangle with 1 cm in x direction and 1 mm in y direction. Uniform square computational

cells are applied with the size of 10 μm. Similar as the previous example, since MARCS is

a 3-dimensional program, 3 layers of computation cells are included in the z direction. The

total computation cells in this calculation is 300 thousand. 500 CPUs are allocated for this

calculation, results in 600 cells per CPU. The left and right boundaries in the x direction

are adiabatic walls. And the periodic boundary conditions are applied in the y and z

directions.

The DME/air mixture is initiated at 920 K, 40 atm, and stoichiometric condition.

The detonation is induced by an ignition kernel with a constant fuel concentration gradient

in the left end of the computational domain. The size of ignition kernel is 1mm. And the

distribution of equivalate ratio, ϕ, in the ignition kernel is:

1 ( )il xx

= − −

(6.2)

where li is the ignition kernel size and x = -0.033 mm-1. An initial turbulent

fluctuation of velocity is introduced with the integral scale of 1 mm, Kolmogorov scale of

20 μm, and turbulent Reynolds number of 4000.

Figure 6.5 shows the temperature and pressure contour plots in the mixture at 385

μs. It is seen that the ignition is initiated from the ignition kernel on the left boundary. And

the ignition front is developed and propagated towards the end gas on the right. It is worth

mentioning that the shape of the ignition kernel is deformed due to the turbulent disturbing.

156

The flame surface is wrinkled, and it propagates faster on the top and bottom boundaries.

Thus, it is demonstrated that the interactions between turbulence and chemistry can be

captured successfully.

The temperature and pressure contour plots at 386 μs are shown in Figure 6.6. By

comparing it with Figure 6.5, it clearly shows the development of the ignition kernel, the

movement of the flame front and the propagation of the pressure wave. A detonation wave

is formed, and the wave front propagates with a speed of 1700 m/s. Therefore, it is

demonstrated that the MARCS method can successfully capture the rapid combustion

phenomenon, such as the detonation, in a turbulent flow.

The total computational cost of this 2-dimensional detonation formation calculation

is 40 thousand CPU hours. Considering that there are 500 CPUs involved in the calculation,

the whole simulation costs 80 hours wall time in total. Therefore, the 2-dimensional

simulation of detonation formations in a turbulent mixture can be efficiently completed

with a relatively cheap computational cost, even when the detailed chemical mechanism is

included.

6.4 Conclusion

In this chapter, a multi-scale adaptive reduced chemistry solver (MARCS) is

developed by implementing the CO-DACT method, the AAJ method, and the chemistry

related governing equations, such as the species equations, chemical reaction terms, and

binary species diffusions, into an in-house full-speed fluid solver. The MARCS method

features the 3-dimensional governing equations, finite volume discretization, fully-implicit

time integrations, MPI parallel computing technique, locally reduced chemical kinetics,

157

stable analytical Jacobian chemical solver, and correlated transport properties. A

calculation of turbulent stretch ignition with a n-decane mechanism and a calculation of

detonation formation in a turbulent stratified DME mixture are carried out by the proposed

MARCS method. The results indicate that the MARCS method is efficient and has the

ability to conduct combustion simulations in multi-dimensional computational domains.

Therefore, DNS of high dimensional combustion modeling in practical geometries with

detailed chemical kinetics becomes affordable with the presented MARCS method.

158

Figure 6.1: Initial conditions of temperature (a), density (b), velocity in x direction (c),

and fuel mass fraction (d) in the 2D turbulent stretch ignition calculation.

159

Figure 6.2: Temperature (a), density (b), velocity in x direction (c), and velocity in y

direction (d) at 650 μs in the 2D turbulent stretch ignition calculation.

160

Figure 6.3: Mass fractions of fuel (a), oxygen (b), nitrogen (c), and water (d) at 650 μs in

the 2D turbulent stretch ignition calculation.

161

Figure 6.4: Mass fractions of OH (a), CO (b), CH2O (c), and H2O2 (d) at 650 μs in the 2D

turbulent stretch ignition calculation.

162

Figure 6.5: Temperature (top) and pressure (bottom) contour plot at 385 μs in the 2D

turbulent detonation calculation.

Figure 6.6: Temperature (top) and pressure (bottom) contour plot at 386 μs in the 2D

turbulent detonation calculation.

163

Chapter 7

Dynamics and Ignition to Flame Transitions in High Pressure

Stratified Mixtures

In previous chapters, advanced numerical methods have been developed to

accelerate the combustion modeling with detailed chemistry kinetics. In this chapter, a

numerical study of ignition to flame transitions in stratified fuel mixtures is carried out by

utilizing the advanced numerical methods. This study attempts to understand how the

kinetic difference between n-alkanes and aromatics leads to different behaviors of the

knocking-like acoustic wave formations at low temperature and engine pressure conditions.

The motivation and objectives of this study will be presented, followed by the numerical

setup. Then the dynamics and ignition to flame transitions in stratified fuel mixtures will

be discussed in detail. Finally, the conclusion will be drawn.

7.1 Motivation and Objectives

Recently, there is increasing interest in developing more efficient and lower

emission internal combustion engines, operating at the higher compression ratio and ultra-

lean conditions. These advanced engines include the homogeneous charge compression

ignition (HCCI) engine, the reactivity controlled compression ignition (RCCI) engine, and

the next generation twin annular premixing swirler (TAPS) engine. Unfortunately, there

are still several technical challenges for HCCI and RCCI engines due to the increased

ignition sensitivity to fuel chemistry at low and intermediate temperature ranges. Stochastic

164

engine knocking at high engine load and at high compression ratio is one of the major

challenges. The control of ignition timing and heat release rate via fuel stratification or fuel

additives (adding aromatics or cetane number enhancers) has been found to be an effective

way to control engine knocking at different engine loads (Reitz 2013). However, the

understanding of combustion processes and fundamental mechanism of ignition to

knocking transitions for stratified fuel mixtures with both n-alkanes and aromatics at

engine conditions is still limited due to the complexity of the low temperature chemistry

(LTC) (Reitz 2013) with negative temperature coefficient (NTC) regime and the

complicated multiple flame regimes (Ju et al. 2011).

Several numerical studies for HCCI engines have been carried out to understand

the details of ignition to flame transitions. Unfortunately, due to the limitation of

computational power, most of studies have focused on small hydrocarbons (Chen et al.

2006; Hawkes et al. 2006). Recently, a multi-scale modeling of n-heptane/air mixtures in

a one-dimensional spark assisted HCCI engine was conducted, identifying four different

flame regimes (Ju et al. 2011). More recently, the auto-ignition of DME/air mixtures with

temperature and fuel stratifications accounting for the exhaust gas recirculation (EGR) was

modeled by using direct numerical simulations (El-Asrag and Ju 2014). However, these

studies were conducted in small computational domains with short time periods.

Consequently, they were not able to capture the transition from ignition to flames.

Therefore, it can be regarded that the ignition to flame transitions at NTC conditions for

stratified large hydrocarbons have not been numerically investigated. Considering the

strong chemical kinetic couplings between alkanes and aromatics (Won, Sun, and Ju 2010),

which is common in real fuels (Dooley et al. 2010), it is also of interest to investigate how

165

the ignition to flame transitions occur differently between alkanes and aromatics. More

importantly, the relation between engine knocking and ignition to flame transitions at NTC

conditions is not well-understood either.

Therefore, the goal of this study is to answer following three questions: (1) How

does auto-ignitions transit to flames when there is a fuel stratification at NTC condition?

(2) How does the ignition to flame transition affect combustion oscillations and

knocking-like acoustic wave formations? (3) How does the kinetic difference between

n-alkanes (with NTC) and aromatics (no NTC) affect ignition to flame transitions and

flame dynamics? Firstly, the unsteady ignition to flame transition and combustion

dynamics in one-dimensional stratified n-heptane and toluene mixtures at a high pressure

are modeled using the correlated dynamic adaptive chemistry and transport (CO-DACT)

method coupled with a hybrid multi-timescale (HMTS) method. The formations of

different ignition and flame regimes are examined and compared. The dynamics of the

formation of flame oscillations and strong pressure waves are analyzed by using a

simplified Burgers’ equation.

7.2 Numerical Setup

In order to simplify the complicated ignition to flame transitions and combustion

dynamics of stratified alkane/aromatics-air mixtures in a large domain at engine-like

conditions, we model a stratified mixture with a given equivalence ratio (ϕ) gradient in a

one-dimensional constant volume domain.

The distribution of the equivalence ratio is to mimic a simplified fuel stratification

caused by the pulsed fuel injections in engines. Although real engines are turbulent, the

166

laminar assumption can still provide insights of the fundamental phenomena. The length

of the domain is 5 cm. In order to understand the kinetic effect of the low temperature

chemistry on ignition to flame transition, two fuel molecules are chosen; n-heptane exhibits

rich low temperature reactivity with NTC behavior, whereas toluene follows the simple

Arrhenius-type reactivity without NTC behavior. A comprehensively reduced chemical

kinetic model (170 species and 962 reactions) for the n-heptane/toluene mixture produced

by the multi-generation path flux analysis method from the detailed model is used in this

study. Unless otherwise stated, the initial pressure and temperature is 40 atm and 800 K,

respectively. The reflective and zero gradient boundary conditions are applied, respectively,

for velocity and all other scalar variables.

To utilize large chemical kinetic models efficiently in computations, the correlated

dynamic adaptive chemistry and transport (CO-DACT) method developed in Chapter 4 is

applied together with the hybrid multi-timescale (HMTS) method. In the modeling of this

study, the smallest grid size after mesh adaption is 16 μm. And the threshold values for

both correlated chemical mechanism and correlated transport properties in the CO-DACT

method are chosen as 0.05.


7.3.1 Dynamics and Combustion Regimes in Stratified n-Heptane/Air Mixture

Prior to the detailed computations, in order to envision the dependency of low and

high temperature ignitions on the equivalence ratio, the ignition delay times of homogenous

n-heptane/air and toluene/air mixtures at different equivalence ratios are calculated firstly

at the initial condition of 40 atm and 800 K. The results are summarized as the blue and

167

red curves in Figure 7.1. In case of n-heptane/air mixtures, two distinctive ignition

processes are observed, known as two-stage ignition. The first-stage ignition is controlled

by the radical pool initiation predominantly by the low temperature chemistry, so called

low temperature ignition (LTI). After the onset of LTI, the HO2 radical plays important

role in the fuel destruction, while accumulating the H2O2 with the gradual increase of

temperature. When the mixture temperature reaches the critical condition to initiate the

thermal decomposition of H2O2 to form two OH radicals, then the second-stage ignition

occurs, known as high temperature ignition (HTI). The calculated results for n-heptane/air

mixtures show that the LTI is weakly dependent on the equivalence ratio, whereas the HTI

exhibits the strong sensitivity to the fuel equivalence ratio. Particularly, with the increase

of equivalence ratio, the time interval between LTI and HTI decreases dramatically. The

faster transition from LTI to HTI at the richer equivalence ratio indicates that the fuel

stratification might induce the unexpected chemistry/transport coupling or interaction

during the ignition to the flame transition processes. Compared to the n-heptane/air mixture,

the toluene/air mixture shows relatively simpler ignition behaviors, demonstrating only the

HTI process for all computed equivalence ratios. The HTI delay times of toluene/air

mixture is about two orders of magnitude slower than those of n-heptane/air mixture.

The computations in this study aim to simulate the circumstance of multiple fuel

injection strategy, where the small amount of fuel is pre-injected into the chamber to form

lean homogeneous fuel/air mixture prior to the main fuel injection. Thus, initial fuel

distribution in the chamber is formulated as below for both n-heptane and toluene cases.

168

( )5.9 3 0.3, 3 cm

0.3, 3 cm

x x

x

− + =

(7.1)

where ϕ is the equivalent ratio and x the spatial location in the unit of centimeter. The

distribution of fuel is depicted in the insert of Figure 7.1. The right half domain with

constant equivalence ratio as 0.3 represents the typical HCCI engine working condition. In

order to incorporate the fuel injection and subsequent fuel/air mixing process, a large

gradient of equivalence ratio is adopted as an initial condition on the left half domain.

Consequently, the initial boundary condition on the left is set to the equivalence ratio of

18.

Figure 7.2 shows the trajectories of the locations of local maximum heat release

rates of the stratified n-heptane/air mixture in a function of time, t, with the initial condition

at 40 atm and 800 K. In order to differentiate ignition wave fronts and flame fronts observed

in this computation, the reaction terms and diffusion terms in governing equations have

been traced along the trajectories of maximum heat release rates. When the reaction terms

dominate over diffusion terms, the wave front is defined to be controlled by ignition

characteristics (ignition wave front). On the other hand, it is defined as flame front when

the diffusion term becomes comparable to the reaction term, as a typical configuration of

flame structure.

Although the LTI delay time is a weak function of equivalence ratio, it still has the

shortest delay time at the fuel rich condition. Thus, the LTI occurs at the fuel richest

location (x = 0 cm) at t = 0.62 ms. After achieving the LTI at the fuel rich side, the LTI

wave front propagates quickly to the direction of lower equivalence ratio, indicated by the

steep gradient right after the onset of LTI. The estimated propagation speed of the LTI

169

wave front is about 1400 m/s, indicating that the propagation of LTI wave front can be

regarded as sequential event of LTI controlled by the weak dependency of LTI on the fuel

stratification, shown in Figure 7.1. Consequently, this fast propagation of LTI wave will

lead to the formation of a strong pressure wave (shock). This LTI wave behavior is similar

to the concept of “spontaneous” wave introduced by Zeldovich (Zeldovich 1980). Both this

LTI wave and the “spontaneous” wave exhibit the characteristics of strong dependence of

propagation rate on initial conditions, as well as its independence of heat conductivity and

the sound speed of the gas. At t = 0.66 ms, a new HTI ignition occurs at x = 2 cm. This

HTI wave front propagates at the speed of 350 m/s to the same direction of LTI wave front.

At x = 3 cm, the LTI wave front disappears because of failing to initiate LTI at the

excessively lean condition (ϕ = 0.3) prior to achieving HTI. HTI wave front continues to

propagate toward the fuel lean side. The large pressure rises due to the large heat release

after the HTI wave front results in a strong oscillation of the ignition front at t = 0.72 ms.

At t = 0.80 ms, it is interesting to note that the HTI wave front splits into a premixed

flame front and a diffusion flame front, determined by the balance between reaction and

diffusion terms in governing equations. The premixed flame front propagates into the

region with uniform equivalence ratio (insert of Figure 7.1) at an average speed of 30 m/s

while continues to oscillate. However, the average advection speed of diffusion flame front

speed is very low, stagnantly stabilized at x = 3.2 cm, but the oscillation continues with a

constant frequency about 8300 Hz even after the disappearance of the premixed flame front.

This oscillation is caused by acoustic wave reflection on the chamber walls. Similar

pressure oscillations have been observed in engines when a knock occurred. The result in

Figure 7.2 demonstrates that there exist four different combustion regimes during the

170

ignition to flame transition for n-heptane/air mixture, which has high cetane number (low

octane number), a low temperature ignition (LTI) regime, a high temperature ignition (HTI)

regime, a premixed flame (PF) regime, and a diffusion flame (DF) regime. In addition, the

fast propagation of LTI and HTI wave fronts result in strong oscillations of the subsequent

diffusion and premixed flame fronts.

To understand the dynamics and structure of different wave fronts, the temperature

and heat release rates at three sampling times at the LTI, HTI and PF and DF regimes

marked on Figure 7.2 are further analyzed in a spatial coordinate. Figure 7.3 shows the

spatial distributions of temperature (a) and the heat release rate (b) at three sampling times,

respectively. It is seen that at t = 0.64 ms (sample 1 in Figure 7.2), the LTI wave front is

located at x = 2.3 cm (peak heat release) and temperature rises modestly after the LTI. At

t = 0.78 ms (sample 2 in Figure 7.2), a very sharp heat release wave is located at x = 3.4

cm, leading to a large temperature rise (2700 K). At 0.99 ms (sample 3 in Figure 7.2), two

heat release zones are observed, where the left is a diffusion flame reaction zone and the

right is a premixed flame front. The premixed flame leads a lower temperature rise in the

fuel lean zone (ϕ = 0.3) but the diffusion flame further increases the flame temperature at

the interface of the fuel lean and fuel rich mixture (Figure 7.1). The wave front structures

in Figure 7.3 confirm the observation of different ignition and flame regimes in Figure 7.2.

7.3.2 Dynamics and Combustion Regimes in Stratified Toluene/Air Mixture

Figure 7.4(a) shows the trajectories of the locations of maximum heat release rates

in the toluene/air mixture initiated at 40 atm and 1000 K. Since the toluene oxidation

chemistry does not exhibit the LTI behavior, there are only two combustion regimes

existing in the mixture evolution histories, a HTI regime and a premixed flame regime.

171

The onset of the HTI in toluene/air mixture occurs at t = 3.3 ms and x = 0. After the

onset of the HTI, the HTI wave front begins to propagate toward the fuel lean side. Note

that the speed of the HTI wave front in toluene/air mixture is significantly lower (~10 m/s)

than that observed in n-heptane/air mixture. The slow ignition transition of toluene/air

mixture during the HTI wave propagation is clearly demonstrated in the top plot of Figure

7.4(b) by plotting the temperature time history along the trajectory of maximum heat

release rate. In case of n-heptane/air mixture shown in the bottom plot of Figure 7.4(b), the

ignition transition happens immediately within 0.15 ms for both LTI and HTI, whereas the

transition of HTI in toluene/air mixture takes much longer (~2.5 ms) to achieve the

maximum temperature (~1600 K) at t = 5.925 ms, where the flame structure begins to form,

onset of flame. This slow ignition transition of toluene/air mixture can be attributed to the

slower ignition delays, as well as a strong dependence of HTI delays on the equivalence

ratio as the HTI wave front propagates toward fuel lean side, depicted in Figure 7.1.

Figure 7.5 shows the spatial profiles of temperature (a) and the heat release rate (b)

at two sampling times marked in Figure 7.4(a). After completing the HTI transition at t =

5.925 ms (~1600 K), thus achieving the onset of flame, two premixed flame fronts are

formed at x = 2.2 cm and propagate in opposite directions with the propagation speed of

20 m/s (toward fuel rich side) and 40 m/s (toward fuel lean side), respectively. Comparing

to the n-heptane case where DF and PF branches are followed after very fast LTI and HTI

transitions, the slow HTI ignition transition provides extra time for flow to be further mixed.

Consequently, two premixed flames propagating toward fuel rich and lean sides can be

seen by plotting spatial profiles of temperature (Figure 7.5(a)) and heat release rates (Figure

172

7.5(b)). Although not shown in the figure, the completion of HTI consumes entire oxygen

near x = 2.2 cm, thus prohibiting the formation of diffusion flame.

The comparison between the combustion regimes of the stratified n-heptane/air

mixture and toluene/air mixture clearly demonstrated that the low temperature chemistry

is the key factor of formation of the four-regime behavior in n-heptane/air mixture. The

weak dependence of LTI to the equivalence ratio results in a sharp, fast propagated low

temperature ignition front. After the HTI, the increase of overall temperature amplifies the

pressure variation across the ignition front and leads to the strong oscillation afterward.

7.3.3 Dynamical Analysis of Ignition to Shockwave Formation

Figure 7.6 shows the profiles of pressure (a) and velocity (b) in the n-heptane/air

mixture, respectively, at the three sampling times marked in Figure 7.2. As mentioned

above, in the LTI regime, the supersonic ignition front (1400 m/s) can induce a shock wave.

The flame front oscillation frequency shown in Figure 7.2 is about 8300 Hz. Figure 7.6(a)

clearly demonstrates that at t = 0.64 ms, a sharp pressure jump is caused by the LTI ignition

front. The pressure increase after the LTI is more than 20 atm. In the HTI regime of t =

0.78 ms, the amplitude of pressure rise is increased to 40 atm due the larger heat release.

This large pressure rise may lead to large velocity fluctuation as shown in Figure 7.6(b)

and the possible transition to knocking, occurring in a real internal combustion engine. In

order to understand the effect of the fuel stratification on the shock (knocking) formation,

a modified Burgers’ equation analysis is conducted. The Burgers’ equation represents a

typical development of front discontinuities (e.g. shockwave). Consider the Burger’s

equation coupled with pressure gradient, neglecting the viscous terms in a one dimensional

flow, we have:

173

1u u pu

t x x

+ = −

(7.2)

where u is the velocity, ρ the density and p the pressure. We define the local velocity

gradient as A:

uA

x

=

(7.3)

And further assume the constant density, we get

( )2

2

2

/pDAA

Dt x

= − −

(7.4)

If there is no pressure gradient due to the heat release in the flow, the development

of 𝐴 only depends on −𝐴2 , which is always negative. Thus, when 𝐴 is less than zero

initially, it will always develop to a shock. If temperature/pressure gradient exists due to

LTI or HTI in a concentration gradient, the second term on the right hand side will modify

the transition to shock wave. Figure 7.7 shows the influence of pressure gradient (𝑃1 =

200𝑥2 𝑃𝑎; 𝑃2 = −400𝑥2 + 1600 𝑃𝑎) to the shock formation in a one dimensional flow

with two open ends and initiated with constant negative velocity gradient. It is found that

the pressure gradient can either promote or retard the shock formation, depending on the

pressure gradient and the sign of the initial velocity gradient.

In the study of the stratified n-heptane/air mixture in Section 7.3.1, the heat release

from low temperature combustion affects the pressure variation in the flow. The

equivalence ratio gradient leads to the heat release rate gradient and, therefore, provides a

favorable pressure gradient, which can accelerate the formation of knocking or shock-lets.

174

7.4 Conclusions

The ignition to flame transitions and combustion dynamics of stratified n-heptane

and toluene mixtures are modeled by a correlated dynamic adaptive chemistry and

transport method coupled with a hybrid multi-timescale method. The results demonstrate

that in the n-heptane mixture, there are four combustion regimes, a LTI regime, a HTI

regime, a premixed flame regime, and a diffusion flame regime. It is found that in the LTI

regime, a supersonic ignition front coupled with a strong pressure wave is formed due to

the weak dependence of LTI delay time on equivalence ratio. Increased heat release in HTI

further increases the strength of pressure wave and led to strong oscillations in pressure,

velocity, and premixed and diffusion flame fronts. On the other hand, for toluene mixture,

which does not have low temperature chemistry, only two combustion waves, a HTI regime

and a premixed flame regime, are observed. No pressure waves and flame oscillations are

observed. Therefore, the differences in chemical kinetics of alkanes and aromatics have

different impact on flame regimes and pressure wave formations. The present results show

that alkanes with rich LTC promote knocking. The dynamical analysis of ignition to shock

formation in a stratified mixture shows that fuel stratification can either accelerate or

decelerate shock (knock) formation, depending on the directions of velocity gradient and

the concentration gradient. The present study provides insights to understand the

combustion dynamics of stratified combustion and the mechanism of knocking.

175

Figure 7.1: Calculated HTI and LTI delay times for zero dimensional homogeneous n-

heptane/air (blue lines) and toluene/air (red line) mixtures in a function of equivalence

ratio at the initial condition of 40 atm and 800 K. Insert shows the initial condition of fuel

distribution used in one-dimensional simulations. (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.)

Figure 7.2: Trajectories of maximum heat release rate in the stratified n-heptane/air

mixture at the initial condition of 40 atm and 800 K.

176

Figure 7.3: Profiles of maximum temperature (a) and maximum heat release rate (b) for

n-heptane/air mixture at three sampling points, t = 0.64, 0.78 and 0.99 ms, marked in

Figure 7.2.

177

Figure 7.4: (a) Trajectories of maximum heat release rate in the stratified toluene/air

mixture at the initial condition of 40 atm and 1000 K. (b) Temperature time history along

the trajectory of maximum heat release rate during ignition stages of toluene (top plot)

and n-heptane (bottom plot).

178

Figure 7.5: Profiles of maximum temperature (a) and maximum heat release rate (b) for

n-toluene/air mixture at two sampling points, t = 5.925 and 6.080 ms, marked in Figure

7.4.

179

Figure 7.6: Profiles of pressure (a) and velocity (b) for n-heptane/air mixture at three

sampling points, t = 0.64, 0.78 and 0.99 ms, marked in Figure 7.2.

180

Figure 7.7: Pressure gradient effects on the acceleration/deceleration of shock formation,

analyzed with Burgers’ equation.

181

Chapter 8

Conclusions and Future Work

8.1 Conclusions

In this dissertation, efficient numerical methods are developed and combined into

a multi-scale adaptive reduced chemistry solver (MARCS) to perform high-fidelity

combustion simulations with detailed chemistry and transport. These advanced

computational algorithms dramatically accelerate combustion simulations by dynamic

chemical reductions, correlated transport property calculations, and efficient chemical

integrations. The high dimensional combustion modeling with detailed chemical kinetics

becomes affordable with these advanced algorithms.

Specifically, in Chapter 3, a correlated dynamic adaptive chemistry (CO-DAC)

method is presented to conduct the on-the-fly chemical reduction with negligible

computational cost. A concept of the correlated group is proposed by grouping cells with

similar thermodynamic states in the phase space. The path flux analysis (PFA) method is

performed to produce the locally reduced mechanism in each correlated group, instead of

in each computational cell, to avoid redundant calculations. The on-the-fly chemical

reduction becomes two orders of magnitude faster in the proposed CO-DAC method. It is

demonstrated that the CO-DAC method can efficiently provide locally reduced chemical

kinetics while keeping accurate predictions of detailed ignition profiles and flame

structures. Therefore, the presented CO-DAC method significantly reduces the

182

computational cost of chemical integrations in a combustion simulation without sacrificing

the accuracy.

In Chapter 4, a correlated dynamic adaptive chemistry and transport (CO-DACT)

method is developed to improve the computational efficiency of detailed transport

properties. Based on the CO-DAC method, the CO-DACT method further accelerates the

transport property calculations according to the similarity of thermodynamic states in the

correlated phase space. The CO-DACT method performs the mixture-averaged diffusion

model in correlated groups to avoid redundant calculations due to similarities. The

proposed CO-DACT method is demonstrated, mathematically and numerically, to be first-

order accurate. The numerical error in transport properties is on the order of the user-

specified threshold. The results show that the efficiency of the transport property

calculations in the CO-DACT method is improved by more than 200 times. Therefore, the

CO-DACT method provides an innovative and computationally efficient way to conduct

combustion simulations with detailed chemistry and transport.

In Chapter 5, an adaptive analytical Jacobian (AAJ) method is proposed to apply

the dynamic adaptive chemistry in the analytical Jacobian solver. The locally reduced

chemical kinetics generated by the CO-DAC method not only reduce the dimension of the

Jacobian matrix, but also accelerate the calculation of detailed reaction rates. With the

locally reduced chemistry, the computational cost of the analytical Jacobian method is

reduced by half. Moreover, the AAJ method is demonstrated to be more stable than the

hybrid multi-timescale (HMTS) method. Therefore, it can tolerate much larger integration

time steps, and hence significantly improves the overall computational efficiency in a

combustion simulation.

183

In Chapter 6, a multi-scale adaptive reduced chemistry solver (MARCS) is

presented by integrating the advanced numerical methods developed in this study to an in-

house multi-dimensional full-speed fluid solver to conduct efficient combustion modeling

in practical geometries. The in-house program features the 3-dimensional computation

domain, finite volume discretization, fully-implicit time integration, and MPI parallel

computing. The CO-DACT method, the AAJ method, and the chemistry related governing

equations, such as species equations, chemical reaction terms, and binary species diffusions,

are integrated to the multi-dimensional in-house program. With the help of the on-the-fly

chemical reduction, the stable chemical solver, and the efficient transport property

calculation, the high dimensional combustion simulation with detailed chemical kinetics

becomes affordable with the presented MARCS method.

In Chapter 7, a numerical study of flame dynamics and ignition to flame transitions

in stratified n-heptane and toluene mixtures are conducted to investigate the relation

between engine knocking and ignition to flame transitions at NTC conditions. It is

demonstrated that the low temperature ignition (LTI) in the n-heptane mixture produces a

supersonic ignition front coupled with a strong pressure wave due to the weak dependence

of the LTI delay time on fuel stratifications. And the subsequent heat release in the high

temperature ignition (HTI) further increases the strength of the pressure wave and leads to

strong oscillations in pressure and flow velocities. On the other hand, no pressure waves

and flame oscillations are observed in the toluene mixture due to the lack of the LTI.

Therefore, compared with aromatics, n-alkanes with rich low temperature chemistry

promote knocking formations in stratified mixtures.

184

8.2 Future Work and Recommendations

The advanced numerical methods presented in this study dramatically increase the

computational efficiency of combustion modeling with detailed chemical mechanisms.

These methods can become more robust with the following improvements.

In the CO-DACT method, transport properties are calculated at the center of each

correlated group. It is demonstrated that this method is first order accurate. The accuracy

of this method can be extended to second order by applying linear extrapolations of the

thermal dynamic states in the transport property calculations.

In both the CO-DAC and CO-DACT method, the phase parameters are pre-defined

based on the important parameters in reaction pathways and transport properties,

respectively. However, these methods can be more generic by identifying the phase

parameters dynamically based on local compositions, such as the high concentration

species.

Finally, chemical integrations in MARCS are conducted by the AAJ method, which

is more stable and more efficient than the HMTS method during the thermal runaway.

However, the HMTS method is more than one order of magnitude faster than the AAJ

method when the system is near equilibrium. Therefore, a binary classification model can

be trained based on the local thermodynamic states to select more efficient chemical solver

on-the-fly. A logistic regression or a feed-forward neural network based learning process

can be applied to produce the binary classification.

185

References

Ambikasaran, Sivaram and Krithika Narayanaswamy. 2017. “An Accurate, Fast,

Mathematically Robust, Universal, Non-Iterative Algorithm for Computing Multi-

Component Diffusion Velocities.” Proceedings of the Combustion Institute

36(1):507–15.

Banerjee, I. and M. Ierapetritou. 2006. “An Adaptive Reduction Scheme to Model Reactive

Flow.” Combustion and Flame 144(3):619–33.

Brown, Peter N., George D. Byrne, and Alan C. Hindmarsh. 1989. “VODE: A Variable-

Coefficient ODE Solver.” SIAM Journal on Scientific and Statistical Computing

10:1038-1051.

Brown, Peter N. and Alan C. Hindmarsh. 1989. “Reduced Storage Matrix Methods in Stiff

ODE Systems.” Applied Mathematics and Computation 31(C):40–91.

Bruno, Claudio, Vaidyanathan Sankaran, Hemanth Kolla, and Jacqueline H. Chen. 2015.

“Impact of Multi-Component Diffusion in Turbulent Combustion Using Direct

Numerical Simulations.” Combustion and Flame 162(11):4313–30.

Burke, Michael P., Marcos Chaos, Yiguang Ju, Frederick L. Dryer, and Stephen J.

Klippenstein. 2012. “Comprehensive H2/O2 Kinetic Model for High-Pressure

Combustion.” International Journal of Chemical Kinetics 44(7):444–74.

Burton, Jesinta. 2020. “‘It Was a Line of Fire Coming at Us’: South West Firefighters

Return Home.” Busselton-Dunsborough Mail. Retrieved February 21, 2020

(https://www.busseltonmail.com.au/story/6620313/it-was-a-line-of-fire-coming-at-

us-firefighters-return-home/).

Chen, J. Y. 1988. “A General Procedure for Constructing Reduced Reaction Mechanisms

with Given Independent Relations.” Combustion Science and Technology 57(1–

3):89–94.

Chen, Jacqueline H., Evatt R. Hawkes, Ramanan Sankaran, Scott D. Mason, and Hong G.

Im. 2006. “Direct Numerical Simulation of Ignition Front Propagation in a Constant

Volume with Temperature Inhomogeneities: I. Fundamental Analysis and

Diagnostics.” Combustion and Flame 145(1–2):128–44.

Chen, Zheng. 2009. “Studies on the Initiation, Propagation, and Extinction of Premixed

Flames.” PhD dissertation, Department of Mechanical and Aerospace Engineering,

Princeton University.

Cheung, Patricia, Liu, and Enrique Iglesia. 2004. “Kinetics and Mechanism of Dimethyl

Ether Oxidation to Formaldehyde on Supported Molybdenum Oxide Domains.” The

Journal of Physical Chemistry B 108(48):18650–58.

186

Curran, HJ, P. Gaffuri, WJ Pitz, and CK Westbrook. 2002. “A Comprehensive Modeling

Study of Iso-Octane Oxidation.” Combustion and Flame 129(3):253-280.

Curtiss, Charles F. and Joseph O. Hirschfelder. 1949. “Transport Properties of

Multicomponent Gas Mixtures.” The Journal of Chemical Physics 17(6):550–55.

Dodds, Will. 2005. “Twin Annular Premixing Swirler (TAPS) Combustor.” in The Roaring

20th Aviation Noise & Air Quality Symposium.

Dooley, Stephen, Sang Hee Won, Marcos Chaos, Joshua Heyne, Yiguang Ju, Frederick L.

Dryer, Kamal Kumar, Chih Jen Sung, Haowei Wang, Matthew A. Oehlschlaeger,

Robert J. Santoro, and Thomas A. Litzinger. 2010. “A Jet Fuel Surrogate Formulated

by Real Fuel Properties.” Combustion and Flame 157(12):2333–39.

Dooley, Stephen, Sang Hee Won, Joshua Heyne, Tanvir I. Farouk, Yiguang Ju, Frederick

L. Dryer, Kamal Kumar, Xin Hui, Chih-Jen Sung, Haowei Wang, Matthew a.

Oehlschlaeger, Venkatesh Iyer, Suresh Iyer, Thomas a. Litzinger, Robert J. Santoro,

Tomasz Malewicki, and Kenneth Brezinsky. 2012. “The Experimental Evaluation of

a Methodology for Surrogate Fuel Formulation to Emulate Gas Phase Combustion

Kinetic Phenomena.” Combustion and Flame 159(4):1444–66.

El-Asrag, Hossam A. and Yiguang Ju. 2014. “Direct Numerical Simulations of NOx Effect

on Multistage Autoignition of DME/Air Mixture in the Negative Temperature

Coefficient Regime for Stratified HCCI Engine Conditions.” Combustion and Flame

161(1):256–69.

Fu, Song and Liang Wang. 2013. “RANS Modeling of High-Speed Aerodynamic Flow

Transition with Consideration of Stability Theory.” Progress in Aerospace Sciences

58:36–59.

Furry, W. H. 1948. “On the Elementary Explanation of Diffusion Phenomena in Gases.”

American Journal of Physics 16(2):63-78.

Gall, François Le. 2014. “Powers of Tensors and Fast Matrix Multiplication.” Proceedings

of the International Symposium on Symbolic and Algebraic Computation, ISSAC 296–

303.

Gao, Xiang, Suo Yang, and Wenting Sun. 2016. “A Global Pathway Selection Algorithm

for the Reduction of Detailed Chemical Kinetic Mechanisms.” Combustion and

Flame 167:238–47.

Gao, Yang, Yufeng Liu, Zhuyin Ren, and Tianfeng Lu. 2015. “A Dynamic Adaptive

Method for Hybrid Integration of Stiff Chemistry.” Combustion and Flame

162(2):287–95.

Goldin, Graham, Zhuyin Ren, and Selma Zahirovic. 2009. “A Cell Agglomeration

Algorithm for Accelerating Detailed Chemistry in CFD.” Combustion Theory and

Modelling 13(4):721–39.

187

Gou, Xiaolong, Zheng Chen, Wenting Sun, and Yiguang Ju. 2013. “A Dynamic Adaptive

Chemistry Scheme with Error Control for Combustion Modeling with a Large

Detailed Mechanism.” Combustion and Flame 160(2):225–31.

Gou, Xiaolong, Wenting Sun, Zheng Chen, and Yiguang Ju. 2010. “A Dynamic Multi-

Timescale Method for Combustion Modeling with Detailed and Reduced Chemical

Kinetic Mechanisms.” Combustion and Flame 157(6):1111–21.

Hawkes, Evatt R., Ramanan Sankaran, Philippe P. Pébay, and Jacqueline H. Chen. 2006.

“Direct Numerical Simulation of Ignition Front Propagation in a Constant Volume

with Temperature Inhomogeneities.” Combustion and Flame 145(1–2):145–159.

Hindmarsh, Alan C. 1983. “ODEPACK, A Systematized Collection of ODE Solvers.” Pp.

55–64 in Scientific Computing. North-Holland, Amsterdam.

Imren, A. and D. C. Haworth. 2016. “On the Merits of Extrapolation-Based Stiff ODE

Solvers for Combustion CFD.” Combustion and Flame 174:1–15.

James, Steven R., R. W. Dennell, Allan S. Gilbert, Henry T. Lewis, J. A. J. Gowlett,

Thomas F. Lynch, W. C. McGrew, Charles R. Peters, Geoffrey G. Pope, Ann B. Stahl,

and Steven R. James. 1989. “Hominid Use of Fire in the Lower and Middle

Pleistocene: A Review of the Evidence [and Comments and Replies].” Current

Anthropology 30(1):1–26.

Jangi, M., R. Yu, and X. S. Bai. 2012. “A Multi-Zone Chemistry Mapping Approach for

Direct Numerical Simulation of Auto-Ignition and Flame Propagation in a Constant

Volume Enclosure.” Combustion Theory and Modelling 16(2):221–49.

Jiang, GS and CW Shu. 1996. “Efficient Implementation of Weighted ENO Schemes.”

Journal of Computational Physics 228(126):202–28.

Ju, Yiguang, Wenting Sun, Michael P. Burke, Xiaolong Gou, and Zheng Chen. 2011.

“Multi-Timescale Modeling of Ignition and Flame Regimes of n-Heptane-Air

Mixtures near Spark Assisted Homogeneous Charge Compression Ignition

Conditions.” Proceedings of the Combustion Institute 33(1):1245–51.

Kee, R. J., G. Dixon-Lewis, J. Warnatz, M. E. Coltrin, and J. A. Miller. 1986. “A Fortran

Computer Code Package for the Evaluation of Gas-Phase Multicomponent Transport

Properties.” SAND-86-8246, Sandia National Labs., Livermore, CA (USA).

Kee, R. J., J. A. Miller, and T. H. Jefferson. 1980. “Chemkin: A General-Purpose, Problem-

Independent, Transportable, Fortran Chemical Kinetics Code Package.” SAND-80-

8003, Sandia National Labs., Livermore, CA (USA).

Kim, J. and P. Moin. 1985. “Application of a Fractional-Step Method to Incompressible

Navier-Stokes Equations.” Journal of Computational Physics 59(2):308–23.

188

Kim, Kyu Hong, Chongam Kim, and Oh Hyun Rho. 2001. “Methods for the Accurate

Computations of Hypersonic Flows. I. AUSMPW+ Scheme.” Journal of

Computational Physics 174(1):38–80.

Kokjohn, S. L., R. M. Hanson, D. a. Splitter, and R. D. Reitz. 2011. “Fuel Reactivity

Controlled Compression Ignition (RCCI): A Pathway to Controlled High-Efficiency

Clean Combustion.” International Journal of Engine Research 12(3):209–26.

Lam, S. H. 2006. “Multicomponent Diffusion Revisited.” Physics of Fluids 18(7):073101.

Lam, S. H. and D. A. Goussis. 1994. “The CSP Method for Simplifying Kinetics.”

International Journal of Chemical Kinetics 26(4):461–86.

Law, Chung K. 2006. Combustion Physics. Cambridge University Press.

van Leer, Bram. 1979. “Towards the Ultimate Conservative Difference Scheme. V. A

Second-Order Sequel to Godunov’s Method.” Journal of Computational Physics

32(1):101–36.

Liang, Long, John G. Stevens, and John T. Farrell. 2009a. “A Dynamic Adaptive

Chemistry Scheme for Reactive Flow Computations.” Proceedings of the Combustion

Institute 32(1):527–34.

Liang, Long, John G. Stevens, and John T. Farrell. 2009b. “A Dynamic Multi-Zone

Partitioning Scheme for Solving Detailed Chemical Kinetics in Reactive Flow

Computations.” Combustion Science and Technology 181(11):1345–71.

Løvås, T., F. Mauss, C. Hasse, and N. Peters. 2002. “Development of Adaptive Kinetics

for Application in Combustion Systems.” Proceedings of the Combustion Institute

29(1):1403–10.

Lu, Tianfeng and Chung K. Law. 2005. “A Directed Relation Graph Method for

Mechanism Reduction.” Proceedings of the Combustion Institute 30(1):1333–41.

Lu, Tianfeng, Chung K. Law, Chun Sang Yoo, and Jacqueline H. Chen. 2009. “Dynamic

Stiffness Removal for Direct Numerical Simulations.” Combustion and Flame

156(8):1542–51.

Maas, U. and S. B. Pope. 1992. “Simplifying Chemical Kinetics: Intrinsic Low-

Dimensional Manifolds in Composition Space.” Combustion and Flame 88(3–

4):239–64.

Mathur, S., P. K. Tondon, and S. C. Saxena. 1967. “Thermal Conductivity of Binary,

Ternary and Quaternary Mixtures of Rare Gases.” Molecular Physics 12(6):569–79.

McNenly, Matthew J., Russell A. Whitesides, and Daniel L. Flowers. 2015. “Faster Solvers

for Large Kinetic Mechanisms Using Adaptive Preconditioners.” Proceedings of the

Combustion Institute 35(1):581–87.

189

Newell, Richard G., Daniel Raimi, and Gloria Aldana. 2019. “Global Energy Outlook 2019:

The Next Generation of Energy.” National Bureau of Economic Research

(https://media.rff.org/documents/GEO_Report_8-22-19.pdf).

Pepiot, P. and H. Pitsch. 2008. “An Efficient Error-Propagation-Based Reduction Method

for Large Chemical Kinetic Mechanisms.” Combustion and Flame 154(1–2):67–81.

Perini, Federico. 2013. “High-Dimensional, Unsupervised Cell Clustering for

Computationally Efficient Engine Simulations with Detailed Combustion Chemistry.”

Fuel 106:344–56.

Perini, Federico, Emanuele Galligani, and Rolf D. Reitz. 2012. “An Analytical Jacobian

Approach to Sparse Reaction Kinetics for Computationally Efficient Combustion

Modeling with Large Reaction Mechanisms.” Energy & Fuels 26(8):4804–22.

Perini, Federico, Emanuele Galligani, and Rolf D. Reitz. 2014. “A Study of Direct and

Krylov Iterative Sparse Solver Techniques to Approach Linear Scaling of the

Integration of Chemical Kinetics with Detailed Combustion Mechanisms.”

Combustion and Flame 161(5):1180–95.

Peters, N. and F. A. Williams. 1987. “The Asymptotic Structure of Stoichiometric

Methaneair Flames.” Combustion and Flame 68(2):185–207.

Petzold, Linda R. 1983. “Description of DASSL: A Differential/Algebraic System Solver”

Pp. 65–68 in Scientific Computing. North-Holland, Amsterdam.

Pitsch, H. and N. Peters. 1998. “A Consistent Flamelet Formulation for Non-Premixed

Combustion Considering Differential Diffusion Effects.” Combustion and Flame

114(1–2):26–40.

Pope, S. B. 1997. “Computationally Efficient Implementation of Combustion Chemistry

Using in Situ Adaptive Tabulation.” Combustion Theory and Modelling 1(1):41–63.

R. Byron Bird, Warren E. Stewart, Edwin N. Lightfoot. 1960. Transport Phenomena.

edited by W. Anderson. John Wiley and Sons, New York.

Radhakrishnan, Krishnan and Alan C. Hindmarsh. 1993. “Description and Use of LSODE,

the Livermore Solver for Ordinary Differential Equations.” LLNL report

UCRL-ID-113855 (https://computing.llnl.gov/casc/nsde/pubs/u113855.pdf).

Reitz, Rolf D. 2013. “Directions in Internal Combustion Engine Research.” Combustion

and Flame 160(1):1–8.

Sarathy, S. M., C. K. Westbrook, M. Mehl, W. J. Pitz, C. Togbe, P. Dagaut, H. Wang, M.

A. Oehlschlaeger, U. Niemann, K. Seshadri, P. S. Veloo, C. Ji, F. N. Egolfopoulos,

and T. Lu. 2011. “Comprehensive Chemical Kinetic Modeling of the Oxidation of 2-

Methylalkanes from C7 to C20.” Combustion and Flame 158(12):2338–57.

190

Schwer, Douglas a., Pisi Lu, and William H. Green. 2003. “An Adaptive Chemistry

Approach to Modeling Complex Kinetics in Reacting Flows.” Combustion and Flame

133(4):451–65.

Sellnau, Mark, Matthew Foster, Kevin Hoyer, Wayne Moore, James Sinnamon, and Harry

Husted. 2014. “Development of a Gasoline Direct Injection Compression Ignition

(GDCI) Engine.” SAE International Journal of Engines 7(2):835–51.

Shi, Yu, Long Liang, Hai-Wen Ge, and Rolf D. Reitz. 2010. “Acceleration of the

Chemistry Solver for Modeling DI Engine Combustion Using Dynamic Adaptive

Chemistry (DAC) Schemes.” Combustion Theory and Modelling 14(1):69–89.

Shu, Chi-Wang. 1999. “High Order ENO and WENO Schemes for Computational Fluid

Dynamics.” Pp. 439–582 in High-order methods for computational physics. Springer,

Berlin, Heidelberg.

Smooke, Mitchell D. 2013. “The Computation of Laminar Flames.” Proceedings of the

Combustion Institute 34(1):65–98.

Strang, Gilbert. 1968. “On the Construction and Comparison of Difference Schemes.”

SIAM Journal on Numerical Analysis 5(3):506–17.

Sun, Weiqi, Xiaolong Gou, Hossam a. El-Asrag, Zheng Chen, and Yiguang Ju. 2015.

“Multi-Timescale and Correlated Dynamic Adaptive Chemistry Modeling of Ignition

and Flame Propagation Using a Real Jet Fuel Surrogate Model.” Combustion and

Flame 162(4):1530–39.

Sun, Weiqi and Yiguang Ju. 2017. “A Multi-Timescale and Correlated Dynamic Adaptive

Chemistry and Transport (CO-DACT) Method for Computationally Efficient

Modeling of Jet Fuel Combustion with Detailed Chemistry and Transport.”

Combustion and Flame 184:297–311.

Sun, Wenting, Zheng Chen, Xiaolong Gou, and Yiguang Ju. 2010. “A Path Flux Analysis

Method for the Reduction of Detailed Chemical Kinetic Mechanisms.” Combustion

and Flame 157(7):1298–1307.

Taing, S., A. R. Masri, and S. B. Pope. 1993. “PDF Calculations of Turbulent Nonpremixed

Flames of H2 CO2 Using Reduced Chemical Mechanisms.” Combustion and Flame

95(1–2):133–50.

Tanaka, Shigeyuki, Ferran Ayala, JC Keck, and JB Heywood. 2003. “Two-Stage Ignition

in HCCI Combustion and HCCI Control by Fuels and Additives.” Combustion and

Flame 132:219–39.

Tonse, Shaheen R., Nigel W. Moriarty, Nancy J. Brown, and Michael Frenklach. 1999.

“PRISM: Piecewise Reusable Implementation of Solution Mapping. An Economical

Strategy for Chemical Kinetics.” Israel Journal of Chemistry 39(1):97–106.

191

Toro, E. F., M. Spruce, and W. Speares. 1994. “Restoration of the Contact Surface in the

HLL-Riemann Solver.” Shock Waves 4(1):25–34.

Valorani, M., F. Creta, D. Goussis, J. Lee, and H. Najm. 2006. “An Automatic Procedure

for the Simplification of Chemical Kinetic Mechanisms Based on CSP.” Combustion

and Flame 146(1–2):29–51.

Valorani, M. and S. Paolucci. 2009. “The G-Scheme: A Framework for Multi-Scale

Adaptive Model Reduction.” Journal of Computational Physics 228(13):4665–4701.

Wang, Hai, Xiaoqing You, Ameya V. Joshi, Scott G. Davis, Alexander Laskin, F. N.

Egolfopoulos, and Chung K. Law. 2007. USC Mech Version II. High-Temperature

Combustion Reaction Model of H2/CO/C1-C4 Compounds.

Wilke, C. R. 1950. “A Viscosity Equation for Gas Mixtures.” The Journal of Chemical

Physics 18(4):517.

Williams, F. A. 1958. “Elementary Derivation of the Multicomponent Diffusion Equation.”

American Journal of Physics 26(7):467–69.

Won, Sang Hee, Wenting Sun, and Yiguang Ju. 2010. “Kinetic Effects of Toluene

Blending on the Extinction Limit of N-Decane Diffusion Flames.” Combustion and

Flame 157(3):411–20.

Won, Sang Hee, Bret Windom, Bo Jiang, and Yiguang Ju. 2014. “The Role of Low

Temperature Fuel Chemistry on Turbulent Flame Propagation.” Combustion and

Flame 161(2):475–83.

Xin, Yuxuan, Wenkai Liang, Wei Liu, Tianfeng Lu, and Chung K. Law. 2015. “A Reduced

Multicomponent Diffusion Model.” Combustion and Flame 162(1):68–74.

Yoon, Seokkwan and Antony Jameson. 1988. “Lower-Upper Symmetric-Gauss-Seidel

Method for the Euler and Navier-Stokes Equations.” AIAA Journal 26(9):1025–26.

Zeldovich, Y. 1980. “Regime Classification of an Exothermic Reaction with Nonuniform

Initial Conditions.” Combustion and Flame 39(2):211–14.

Zhang, Hui Lu, Le Xing Yu, Wen Yang, Liang Tang, Yan Lin, Han Wu, Bo Zhai, Ye

Xiong Tan, Lei Shan, Qiong Liu, Hai Yang Chen, Rong Yang Dai, Bi Jun Qiu, Ya

Qin He, Chao Wang, Long Yi Zheng, Yu Qiong Li, Fu Quan Wu, Zhong Li, He Xin

Yan, and Hong Yang Wang. 2012. “Profound Impact of Gut Homeostasis on

Chemically-Induced pro-Tumorigenic Inflammation and Hepatocarcinogenesis in

Rats.” Journal of Hepatology 57(4):803–12.

Zhao, Fuquan, Thomas N. Asmus, Dennis N. Assanis, John E. Dec, James A. Eng, and

Paul M. Najt. 2003. “Homogeneous Charge Compression Ignition (HCCI) Engines.”

SAE Technical Paper PT-94.

Developments of Efficient Numerical Methods for Combustion ...€¦ · Combustion modeling with...

Documents

Transcript of Developments of Efficient Numerical Methods for Combustion ...€¦ · Combustion modeling with...