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NUMERICAL ANALYSIS OF TURBULENT REACTING
FLOWS IN COMPLEX COMBUSTION SYSTEMS

presented by

MURAT YALDIZLI

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NUMERICAL ANALYSIS OF TURBULENT REACTING FLOWS IN
COMPLEX COMBUSTION SYSTEMS

by
MURAT YALDIZLI

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
MECHANICAL ENGINEERING
2009

ABSTRACT
NUMERICAL ANALYSIS OF TURBULENT REACTINC FLOWS IN
COMPLEX COMBUSTION SYSTEMS
BY
MURAT YALDIZLI

The main objective of this study is to investigate hydrocarbon ﬂames when they
are strongly affected by turbulence and/ or fuel spray via numerical analysis of chem-
ically reacting turbulent flows in simple and complex combustion systems.

First, direct numerical simulations (DNS) are conducted to study the struc-
ture of partially-premixed and non-premixed methane ﬂames in high-intensity two—
dimensional isotropic turbulent ﬂows. The results obtained via “ﬂame normal anal-
ysis” show local extinction and reignition for both non-premixed and partially-
premixed ﬂames. Dynamical analysis of the ﬂame with a Lagrangian method in-
dicates that the time integrated strain rate characterizes the ﬁnite—rate chemistry
effects and the ﬂame extinction better than the strain rate. It is observed that the
ﬂame behavior is affected by the“pressure-dilatation” and “viscous-dissipation” in
addition to strain rate. Consistent with previous studies, high vorticity values are
detected close to the reaction zone, where the vorticity generation by the “baro-
clinic torque” was found to be signiﬁcant. The inﬂuences of (initial) Reynolds and
Damkohler numbers, and various air-fuel premixing levels on ﬂame and turbulence
variables are also studied.

For more complex systems, the subgrid-scale (SGS) ﬁltered mass density function
(FMDF) model is employed for large eddy simulations (LES) of partially-premixed
methane jet ﬂames and two-phase turbulent reacting flows in a spray—driven axisym—
metric dump combustor, respectively. The F MDF is the joint probability density

function (PDF) of the scalars and is determined via the solution of a set of stochas—

tic differential equations (SDE).

In the analysis of partially-premixed methane jet ﬂames, the single phase LES/-
FMDF is implemented using a highly scalable, parallel hybrid Eulerian-Lagrangian
numerical scheme and its applicability and the extent of validity for SGS closure
of turbulent hydrocarbon combustion are further established by simulating different
methane-air jet ﬂames with the “ﬂamelet” and “ﬁniterate” kinetics models such
as the 1—step global and 12-step reduced chemistry mechanisms. The higher heat
release by the ﬂamelet or 1-step models are observed to suppress the strong strain-
ing/ extinction effects of turbulence on the ﬂame at high jet speeds, while the 12—step
model successfully predicts the extend of ﬂame extinction at higher jet speeds. Var-
ious formulations of the linear mean square estimation (LMSE) model are employed
for the conditional mixing of the scalars in the FMDF equation.

For turbulent spray combustion in the axisymmetric dump combustor, a new two-
phase LES model based on a Lagrangian-Eulerian—Lagrangian numerical scheme is
introduced. The model is applied here to high Reynolds number compressible turbu-
lent reacting ﬂows in a spray—controlled lean premixed dump combustor. The velocity
ﬁeld is obtained by a high-order compact ﬁnite difference method in Eulerian frame
work, where as the spray is modeled with a Lagrangian mathematical/ computational
method that allows two-way mass, momentum and energy coupling between phases.
The subgrid gas-liquid combustion is modeled with a PDF-based two—phase subgrid
combustion model, termed the ﬁltered mass density function (FMDF). Several cases
are simulated by using different initial values of Sauter mean diameter, spray and
injection angles, liquid fuel to gas mass loading ratio, droplets injection velocity, pul-
sative spray injection frequency, carrier gas equivalence ratio, inlet temperature and

turbulence forcing amplitude.

To my beloved wife Sinem

iv

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to the National Science Foundation (NSF)
and the Ofﬁce of Naval Research (ONR) for giving me the opportunity to complete
this work. I owe my deepest gratitude to my supervisor Dr. Farhad A. Jaberi, Pro-
fessor and Director of Computational Fluid Dynamics Laboratory, ﬁrst introduced
me to the fascinating ﬁeld of turbulent combustion research. His excellent guidance
in the numerical analysis of turbulent reacting flows has been essential for the accom-
plishment of this work. I am also grateful to my committee members and professors:
Dr. Charles Petty, Dr. Nikolai Priezjev, Dr. Indrek Wichman, and Dr. Mei Zhuang
for their valuable comments and constructive criticism on this dissertation. Finally,
my loving thanks go to my beloved wife Sinem, who has shown great understanding,

patience and continuous support towards my PhD research as well as everything else

I do.

TABLE OF CONTENTS

ABSTRACT .................................... ii
LIST OF TABLES ................................. viii
LIST OF FIGURES ................................ ix
1 THE STRUCTURE OF PARTIALLY-PREMIXED METHANE FLAMES
IN HIGH-INTENSITY TURBULENT FLOWS ............... 1
1.1 Introduction ................................ 1
1.2 Governing Equations ........................... 3
1.3 Numerical Solution ............................ 5
1.4 Results and Discussion .......................... 7
1.4.1 Physical Flame and Turbulence Structures ........... 8
1.4.2 Turbulence and Flame Statistics ................. 9
1.4.3 Compositional Flame Structure ................. 11
1.4.4 Flame Normal Analysis ...................... 12
1.4.5 Lagrangian Flame Analysis ................... 17
1.4.6 Effects of Various Flow and Flame Parameters ......... 21
1.5 Summary and Conclusions ........................ 24
1.6 Tables and Figures ............................ 27
2 LARGE—EDDY SIMULATIONS OF TURBULENT METHANE JET FLAMES
WITH FILTERED MASS DENSITY FUNCTION .............. 50
2.1 Introduction ................................ 50
2.2 Formulation ................................ 53
2.2.1 The Filtered Transport Equations ................ 53
2.2.2 The Filtered Mass Density Function (FMDF) ......... 55
2.3 Numerical Solution ............................ 59
2.3.1 The Lagrangian Monte Carlo Method .............. 60
2.3.2 Consistency ............................ 61
2.3.3 Parallelization ........................... 62
2.3.4 Chemistry ............................. 65
2.3.5 Jet Parameters and Boundary Conditions ........... 66
2.4 Results and Discussion .......................... 67
2.4.1 Turbulence and Flame Statistics ................. 68
2.4.2 Compositional Flame Structure ................. 70
2.4.3 Subgrid-Scale Mixing ....................... 74
2.5 Summary and Conclusions ........................ 78
2.6 Tables and Figures ............................ 80

vi

3 LARGE-SCALE SIMULATIONS OF TURBULENT SPRAY COMBUS-

TION IN A DUMP COMBUSTOR VIA TWO-PHASE FMDF ....... 100
3.1 Introduction ................................ 100
3.2 T wo—Phase LES/FMDF Model Equations ................ 105
3.2.1 Eulerian Gas-Phase LES Equations ............... 105
3.2.2 Lagrangian Liquid/Droplet-Phase Equations .......... 107
3.2.3 Two—Phase FMDF Formulation ................. 110

3.3 Numerical Solution ............................ 112
3.3.1 Consistency of FMDF ...................... 114
3.3.2 Combustor Parameters And Boundary Conditions ....... 116

3.4 Results and Discussion .......................... 117
3.4.1 Effect of Sauter Mean Diameter (SMD) ............. 117
3.4.2 Effect of Spray Angle (a) ..................... 120
3.4.3 Effect of Injection Angle (B) ................... 122
3.4.4 Effect of Mass Loading Ratio (F) ................ 123
3.4.5 Effect of Droplet Injection Velocity (up) ............ 125
3.4.6 Effect of Injection Frequency (2p) ................ 126
3.4.7 Effect of Equivalence Ratio ((25) ................. 128
3.4.8 Effect of Inlet Gas Temperature (Tin) .............. 130
3.4.9 Effect of Inﬂow Turbulence .................... 132

3.5 Summary and Conclusions ........................ 134
3.6 Tables and Figures ............................ 137
3.7 Nomenclature ............................... 178
REFERENCES ................................... 180

vii

1.1

1.2

1.3

2.1

3.1

LIST OF TABLES

Various non—dimensional statistics of the developed initial turbulence

ﬁeld. .................................... 27
Turbulence parameters in various simulations. ............. 27
Flame parameters in various simulations ................. 27

Important parameters of Sandia’s piloted turbulent methane jet ﬂames
D and F ................................... 80

Spray parameters of various tested cases ................. 138

viii

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1.10

LIST OF FIGURES

Computational domain, boundary conditions, and initial vorticity and
ﬂame ﬁelds. ................................

A schematic view of the ﬂame surface, illustrating various methods
employed for the analysis of DNS data. .................

Instantaneous contours of temperature for case I-3 at different times;
(a) t/To = 0.0, (b) t/To = 1.0, (c) t/To = 1.5 and (d) t/To = 2.0.

Instantaneous contours of OH mass fraction for case I—3 at different

times; (a) t/To = 0.0, (b) t/To =1.0, (c) t/To = 1.5 and (d) t/To = 2.0.

Instantaneous contours of various scalars for case I—3 at t/ro = 2.0;
(a) mixture fraction, (b) CO mass fraction, (c) 002 mass fraction and
(d) H mass fraction. ...........................

Instantaneous contours of various ﬂow variables for case I—3 at t / To =
2.0; (a) vorticity and (b) baroclinic torque ................

Two-dimensional energy spectral density function in reacting case I—3

at different times; t/To = 0.0, (dashed—dot line), t/To = 2.0 (solid line).

The conditional probability density function (PDF) of (a) temperature
and (b) strain rate, conditioned on the mixture fraction being between
0.21 < Z < 0.578. The PDFs are calculated from the Eulerian data
at t/To = 0.5 (solid line), t/To = 1.0 (dotted line) and t/ro = 2.0
(dashed-dot line) ..............................

Scatter plots of the temperature in the mixture fraction space for case
I-3 at (a) t/To = 0.25, (b) t/To = 1.0, (c) t/To = 1.5 and (d) t/To = 2.0.
Solid lines show the initial laminar proﬁles ................

Scatter plots of different species mass fractions in the mixture fraction
space for case I-3 at t/ro = 1.0; (a) CO mass fraction, (b) 0H mass
fraction,

(c) 002 mass fraction and (d) H mass fraction. Solid lines show the
initial laminar proﬁles. ..........................

ix

28

29

30

31

32

33

34

35

36

 

1.11 Positions of the ﬂame normal cross—sections on the (a) stoichiomet-
ric mixture fraction iso—surface and (b) temperature contours. Flame

normal A is at node 425, B is at node 1109, BI is at node 1265, C is
at node 1425 and D is at node 1715. .................. 37

1.12 Comparison of various turbulent variables along the ﬂame normal di-
rection with the corresponding laminar values at different locations
and at normalized time of t/To = 2.0 for case I-3; (a) temperature,
(b) OH mass fraction, (c) production rate of OH, and (C1) C0 mass
fraction. Laminar proﬁles; dashed line, turbulent proﬁles along ﬂame
normal A; squared line, B—B’; circled line, C; triangled line; D; plused
line. .................................... 38

1.13 Variations of various turbulence and ﬂame variables along the stoi-
chiometric mixture fraction iso-surface at 15/70 = 2.0 for case I-3; (a)
temperature,

(b) scalar dissipation rate, (c) OH production rate, and (d) tangential
strain rate. ................................ 39

1.14 Variations of (a) total kinetic energy, and (b) pressure-dilatation term
in the kinetic energy equation along the stoichiometric mixture frac-
tion iso—surface at t/To = 2.0 for case I-3 ................. 40

1.15 Variations of (a) vorticity magnitude, and (b) baroclinic term in the
vorticity equation along the stoichiometric mixture fraction iso-surface
at t/To = 2.0 for case I-3 .......................... 41

1.16 The location of Lagrangian particles at different sections on the stoi-
chiometric mixture fraction iso—surface or perpendicular to this surface
for case X—3; (a) t/To = 0.0, (b) t/To = 0.5 and (c) t/To = 1.0. . . . . 42

1.17 Temporal variations of the Lagrangian particle properties; (a) temper-
ature, (b) mixture fraction, and (c) strain rate. Each symbol repre—
sents a particle initially located at different positions in the domain.
Particle A is represented with “0”, particle B with “V”, particle C
with “0”, particle D with “+” and particle E with “I”. Initially the
relative distance of the particles with respect to the stoichiometric
mixture fraction surface is the same. .................. 43

1.18 Temporal variations of the conditional mean temperature conditioned
on the strain rate and mixture fraction; (a) low mixture fraction range,
(b) stoichiometric mixture fraction range, (c) high mixture fraction
range. ((1) Percentage of the difference of the conditional average val-

. ues of the particle temperature. ..................... 44

1.19

1.20

1.21

1.22

1.23

2.1

2.2

2.3

2.4

2.5

Temporal variations of the probability density function (PDF) of the
temperature and the strain rate conditioned on Z = Z stoich at t/To =
2.0. The crosses refer to Eulerian (grid) data, the circles refer to
Lagrangian (particle) data; (a) PDF of temperature, (b) PDF of strain
rate. .................................... 45

The scatter plots of temperature in the mixture fraction space for
different cases at t/ro = 1.5; (a) case I—3, (b) case I-2 and (c) case
L1. ((1) The scatter plot of temperature for low strain rate case, case
VIII-3. Solid lines show the initial laminar proﬁles. .......... 46

The percentage of the difference between the laminar and average
turbulent temperatures at t/To = 1.5 in the mixture fraction domain
for various fuel-air premixing levels. The data are gathered from the
Eulerian grid points. ........................... 47

Instantaneous contours of the CO mass fraction for (a) case I-3 and
(b) case IX-3 at t/ro = 1.5 ......................... 48

Temporal variations of the percentage of grid points in the vicinity of
the ﬂame surface (Z % Zstoz'ch) with temperatures less than 1400 K;
(a) various Reynolds numbers, and (b) various Damkohler numbers.
The data are gathered from the Eulerian grid points. ......... 49

Schematic view of the Sandia’s piloted methane jet ﬂame. Main jet
and pilot jet diameters are 7.2mm and 18.2mm, respectively. . . . . 81

Comparison between Monte Carlo (MC) and ﬁnite difference (FD)
values of the normalized instantaneous ﬁltered temperature; (a) A E =
2A, and 04> = 8,16, (b) AE = A, and 04> = 8,16. MC with Cd) = 8;
thick solid line, FD with 045 = 8; thick dashed line, MC with C¢ = 16;

solid line, FD with C¢> = 16; dashed line. ............. j . . . 82
Speed-up of the parallelized LES / FMDF calculations relative to those
conducted on a 2-processor node. .................... 83
Isosurfaces and isocontours of vorticity magnitude, 9, obtained by
LES/FMDF with the 12-step chemistry model; (a) ﬂame D and (b)
ﬂame F. .................................. 84
Isosurfaces and isocontours of ﬁltered temperature, (T) L, obtained by

LES/FMDF with the 12-step chemistry model; (a) ﬂame D and (b)
ﬂame F. .................................. 85

xi

2.6

2.7

2.8

2.9

2.10

2.11

2.12

The time-averaged (mean) values and the RMS values of the ﬁltered
axial velocity ﬁeld at :r/ D = 15; (a) mean velocity for ﬂame D, (b)
mean velocity for ﬂame F, (c) velocity RMS for ﬂame D and (d) ve-
locity RMS for ﬂame F. Symbols, experimental data; dashed-dot line,
1—step model; dashed line, ﬂamelet model; solid line, 12-step model. .

The time-averaged (mean) temperature and RMS of temperature at
33/ D = 15; (a) mean temperature for ﬂame D, (b) mean temperature
for ﬂame F, (c) RMS of temperature for ﬂame D and (d) RMS of
temperature for ﬂame F. Symbols, experimental data; dashed-dot line,
1-step model; dashed line, ﬂamelet model; solid line, 12-step model.

Proﬁles of various species mass fractions as predicted by LES/ F MDF
with the 12-step model at :r/D = 15; (a) C02 (b) H2, (c) CO, (d)
OH. Circles, measured ﬂame D; squares, measured ﬂame F; dashed
line, computed ﬂame D; solid line, computed ﬂame F ..........

Proﬁles of mean heat release for the ﬂamelet and 12-step models in
ﬂame D (dashed line, ﬂamelet; solid line, 12—step) and in ﬂame F (long
dashed line, ﬂamelet; dashed-dot line, 12—step) at 33/ D = 7.5 ......

Conditional mean temperature and 02, CO, H2 mass fractions for
ﬂame D as predicted by LES / FMDF with 12-step kinetics mechanism
and with C¢ = 10; (a) and (b) show the results, at 23/0 = 7.5; (c)
and ((1) show the results, at :r/ D = 15 (lines, calculations; symbols,
measurements) . ..............................

Conditional mean temperature and Oz, CO, H2 mass fractions for
ﬂame F as predicted by LES/FMDF with 12—step kinetics mechanism
and with C¢ = 10; (a) and (b) show the results, at 23/0 = 7.5; (c)
and ((1) show the results, at :r/ D = 15 (lines, calculations; symbols,
measurements) . ..............................

Scatter plots of temperature vs. mixture fraction at r/ D = 0.83 for
ﬂame D; (a) measurements at :r/D = 7.5 (975 sample points); (b)
measurements at :r/D = 15 (780 sample points); (c) calculations
with LES/FMDF and 12-step mechanism at 32/ D = 7.5 (973 sam-
ple points); (d) calculations with LES/FMDF and 12—step mechanism
at 513/ D = 15 (762 sample points). ...................

xii

86

87

88

89

90

91

2.13

2.14

2.15

2.16

2.17

2.18

Scatter plots of temperature vs. mixture fraction at r/ D = 0.83 for
ﬂame F; (a) measurements at as/D = 7.5 (1550 sample points); (b)
measurements at :r/ D = 15 (976 sample points); (0) calculations
with LES/FMDF and 12-step mechanism at x/ D = 7.5 (1347 sample
points); ((1) calculations with LES/FMDF and 12-step mechanism at
23/ D = 15 (1097 sample points). .....................

Comparison between LES/ F MDF and experimental data in the mix—
ture fraction space for the 12-step model with C¢ = 10 at x/ D = 15;
(a) mean temperature and 02 mass fraction for ﬂame D; (b)mean tem-
perature and 02 mass fraction for ﬂame F; (c) mean CO and H2 mass
fractions for ﬂame D; (d) mean CO and H2 mass fractions for ﬂame
F (lines, calculations; symbols, measurements) ..............

The radial variations of the mean temperature as obtained by

LES / FMDF with 12-step model and different 04> values at x/ D = 15;
(a) ﬂame D and (b) ﬂame F (symbols, measurements; long-dashed line,
C¢ = 6; dashed-dot line, C¢ = 8; solid line, C4; = 10; dashed-dot—dot
line, C¢ = DM 1; dashed line, Cd) = DM 2) ................

Conditional mean and RMS of various ﬂame variables as obtained
by LES/FMDF with the 12-step model and different Cy, values for
ﬂame D; (a) mean and RMS of temperature at :c/ D = 7.5; (b) mean
of CO and H2 mass fractions at x/D = 7.5; (0) mean and RMS of
temperature at :c/ D = 15; (d) mean of CO and H2 mass fractions at
x/ D = 15 (symbols, measurements; long-dashed line, C4; = 6; dashed-
dot line, C¢ = 8; solid line, C¢ = 10; dashed-dot-dot line, C4; = DM 1;
dashed line, C¢ = DM 2). ........................

Conditional mean and RMS of various ﬂame variables as obtained
by LES/FMDF with the 12-step model and different C¢ values for
ﬂame F; (a) mean and RMS of temperature at x/ D = 7.5; (b) mean
of CO and H2 mass fractions at LII/D = 7.5; (c) mean and RMS of
temperature at 113/ D = 15; ((1) mean of CO and H2 mass fractions at
z/ D = 15 (symbols, measurements; long-dashed line, 0¢ = 6; dashed-
dot line, C4; = 8; solid line, Cg; = 10; dashed-dot-dot line, C¢ = DM 1;
dashed line, C4, = DM 2). ........................

Scatter plots of temperature vs. mixture fraction as obtained by

LES / FMDF with the 12-step model and different C¢ values for ﬂame
D at 22/0 = 7.5, and r/D = 0.83; (a) C), = 6 (1102 sample points); (b)
C¢ = 8 (1022 sample points); (c) Cd) = DM1 (1042 sample points);
(d) C), = DM 2 (973 sample points). ..................

xiii

93

94

95

96

97

2.19 Scatter plots of temperature vs. mixture fraction as obtained by
LES/FMDF with the 12-step model and different C¢ values for ﬂame
F m/D = 7.5, and r/D = 0.83; (a) 04> = 6 (1418 sample points); (b)
C43 = 8 (1350 sample points); (c) C4, = DM1 (1376 sample points);
((1) Cd) = DM 2 (1134 sample points) ................... 99

3.1 The Eulerian and the Lagrangian ﬁelds of the two-phase LES / FMDF
methodology. ............................... 139

3.2 Iso—contours of the instantaneous ﬁltered temperature and fuel mass
fraction in the non—reacting axisymmetric dump combustor without
spray as obtained by LES/FDMF; (a) FD values of temperature, (b)

MC values of temperature, (c) FD values of fuel mass fraction, and
(d) MC values of fuel mass fraction .................... 139

 

3.3 Iso—contours of the instantaneous ﬁltered temperature and fuel mass
fraction in the reacting axisymmetric dump combustor without spray
as obtained by LES/FDMF; (a) FD values of temperature, (b) MC
values of temperature, (c) FD values of fuel mass fraction, and ((1)
MC values of fuel mass fraction ...................... 140

3.4 ISO-contours of the instantaneous ﬁltered temperature and fuel mass
fraction in the reacting axisymmetric dump combustor with spray as
obtained by LES / F DMF; (a) FD values of temperature, (b) MC values
of temperature, (c) FD values of fuel mass fraction, and (d) MC values
of fuel mass fraction. ........................... 140

3.5 Radial variations of the instantaneous ﬁltered temperature and fuel
mass fraction at axial locations of :r/D = 2 and 26/ D = 4; (a) Non-
isothermal non-reacting case without spray, (b) N on—isothermal react-
ing case without spray, (c) Non-isothermal reacting case with spray.

The results are obtained by averaging the instantaneous data in the
azimuthal direction and not in time. .................. 141

3.6 Images of the axisymmetric dump combustor; (a) Picture of exper-
imental setup [1] and (b) Illustration of computational domain and
droplet injection. ............................. 142

3.7 ISO-contours of the instantaneous of ﬁltered temperature and fuel mass
fraction as obtained by LES/FDMF with various SMD values; (a)
SMD = 30pm, (b) SMD = 45pm, (c) SMD = 60pm, (d) SMD =
90pm, (e) SMD = 120nm. ........................ 143

xiv

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

Radial variations of various quantities at axial locations of 23/ D = 3
and 33/ D = 6 with various SMD values; (a) Instantaneous ﬁltered
temperature, (b) Instantaneous ﬁltered fuel mass fraction, (0) Root
mean square (RMS) of ﬁltered pressure (solid line, 30pm; dashed line,
45pm; long-dashed line, 60pm; dashed-dot line, 90pm; dashed—dot—dot
line, 120nm). ...............................

Axial variations of integrated droplet number density with various
SMD values (solid line, 30pm; dashed line, 45pm; long-dashed line,
60pm; dashed-dot line, 90pm; dashed—dot-dot line, 120nm). .....

Radial variations of various droplet quantities at axial locations of
32/ D = 3 and :r/ D = 6 with various SMD values; (a) Mean droplet
temperature, (b) Mean droplet axial velocity (solid line, 30pm; dashed
line, 45pm; long-dashed line, 60pm; dashed-dot line, 90pm; dashed-
dot—dot line, 120nm) ............................

ISO-contours of the instantaneous ﬁltered density and fuel mass frac-
tion as obtained by LES / FDMF with various spray angles; (a) a z 0,
(b) Oz 2 20, (c) or z 40, (d) a m 80. ...................

Radial variations of various quantities at axial locations of x/ D = 3
and 113/ D = 6 with various spray angles; (a) Instantaneous ﬁltered
temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c) Root
mean square (RMS) of ﬁltered pressure (solid line, a z 0; dashed line,
a m 20; long-dashed line, a z 40; dashed-dot line, a z 80) .......

Axial variations of integrated droplet number density with various
spray angles (solid line, a z 0; dashed line, a z 20; long-dashed line,
a :21 40; dashed-dot line, a m 80). ....................

Radial variations of various droplet quantities at axial locations of
x/ D = 3 and 23/ D = 6 with various spray angles; (a) Mean droplet
temperature, (b) Mean droplet axial velocity (solid line, a z 0; dashed
line, a m 20; long-dashed line, a m 40; dashed-dot line, oz 3 80). . . .

ISO-contours of the instantaneous ﬁltered density and temperature as
obtained by LES/FDMF with various injection angles; (a) B m 0, (b)
6 w 30, (c) [3 z 45, (d) [3 m 60. .....................

Radial variations of various quantities at axial locations of :c/ D = 3
and x/ D = 6 with various injection angles; (a) Instantaneous ﬁltered
temperature, (b) Instantaneous ﬁltered fuel mass fraction, (0) Root
mean square (RMS) of ﬁltered pressure (solid line, B m 0; dashed line,
6 “~V 30; long-dashed line, 6 z 45; dashed-dot line, 6 z 60). ......

XV

146

147

148

149

150

152

 

3.17

3.18

3.20

3.21

3.22

3.23

3.24

3.25

Axial variations of integrated droplet number density with various
injection angles (solid line, B m 0; dashed line, 6 z 30; long-dashed
line, B m 45; dashed-dot line, B m 60) ...................

Radial variations of various droplet quantities at axial location of
:r/D = 3 with various injection angles; (a) Mean droplet tempera-
ture, (b) Mean droplet axial velocity (solid line, ,6 z 0; dashed line,
ﬂ 2 30; long-dashed line, ,6 z 45; dashed—dot line, 6 z 60). ......

ISO-contours of the instantaneous of ﬁltered temperature and fuel mass
fraction as obtained by LES / FDMF with various droplet mass loading
ratios; (a) F = 0.01, (b) F = 0.055, (c) F = 0.11, (d) F = 0.22, (e)
F = 0.44 ...................................

Radial variations of various quantities at axial locations of x/ D = 3
and :L'/ D = 6 with various droplet mass loading ratios; (a) Instanta—
neous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass frac-
tion, (c) Instantaneous ﬁltered pressure (solid line, F = 0.01; dashed
line, F = 0.055; long-dashed line, F = 0.11; dashed—dot line, F = 0.22;
dashed-dot-dot line, F = 0.44) .......................

Axial variations of integrated droplet number density with various
droplet mass loading ratios (solid line, F = 0.01; dashed line, F =
0.055; long-dashed line, F = 0.11; dashed-dot line, F = 0.22; dashed-
dot-dot line, F = 0.44) ...........................

Radial variations of various droplet quantities at axial location of
23/ D = 3 with various droplet mass loading ratios; (a) Mean droplet
temperature, (b) Mean droplet axial velocity (solid line, F = 0.01;
dashed line, F = 0.055; long-dashed line, F = 0.11; dashed-dot line,
F = 0.22; dashed—dot-dot line, F = 0.44). ................

Iso—contours of the instantaneous of ﬁltered temperature and fuel mass
fraction as obtained by LES/FDMF with various droplet injection
velocities; (a) up = 15m/s, (b) up = 30m/s, (c) up = 60m/s. .....

Axial variations of integrated droplet number density with various
droplet injection velocities (solid line, up = 15m/s; dashed line, up =
30m/s; long-dashed line, up = 60m/s). .................

Radial variations of various quantities at axial locations of 33/ D = 3
and 113/ D = 6 with various droplet injection velocities; (a) Instanta—
neous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass frac-
tion, (c) Instantaneous ﬁltered total kinetic energy (TKE) (solid line,

up = 15m/s; dashed line, up = 30m/s; long-dashed line, up = 60m/s).

xvi

153

154

155

156

157

158

159

159

160

 

3.26

3.27

3.28

3.29

3.30

3.31

3.32

3.33

3.34

ISO-contours of the instantaneous ﬁltered temperature and fuel mass
fraction as obtained by LES / FDMF with various injection frequencies;

(a) 1/2 = 38Hz, (b) 1,!) = 250Hz, (c) w = 500Hz, (d) w = lkHz ..... 161
ISO-surfaces of the instantaneous ﬁltered pressure as obtained by
LES/FDMF with various injection frequencies; (a) ’90 = 38H 2, (b)

1/2 = 250Hz, (c) if = 500Hz, (d) 212 = 1kHz. .............. 162
Radial variations of various quantities at axial locations of x/ D = 3

and 32/ D = 6 with various injection frequencies; (a) Instantaneous
ﬁltered fuel mass fraction, (b) Root mean square (RMS) of ﬁltered
pressure (solid line, 11) = 38H 2:; dashed line, 2/2 = 250Hz; long-dashed
line, 11) = 500Hz; dashed-dot line, 2/2 = 1kHz). ............. 163

ISO-contours of the instantaneous of ﬁltered temperature and fuel mass
fraction as obtained by LES/FDMF with various initial equivalence
ratios; (a) (b = 0.01, (b) cf) = 0.1, (c) (15 = 0.4, (d) ab = 0.7, (e)

(b = 1.00, (f) (15 = 2.00. .......................... 164
ISO—surfaces of the instantaneous vorticity ﬁeld as obtained by

LES/FDMF with various initial equivalence ratios; (a) 45 = 0.01, (b)

925 = 0.1, (c) d) = 0.4, (d) ¢ = 0.7, (e) ¢ = 1.00, (f) 45 = 2.00 ....... 165
Radial variations of various quantities at axial locations of 53/ D = 3

and a: / D = 6 with various initial equivalence ratios; (a) Instantaneous
ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c)
Root mean square (RMS) of ﬁltered pressure (solid line, gb = 0.01;
dashed line, (,2) = 0.1; long-dashed line, gb = 0.4; dashed-dot line,
¢ = 0.7; dashed-dot-dot line, (15 = 0.1; dotted-delta line 45 = 0.2). . . . 166

Radial variations of various droplet quantities at axial location of
:r/ D = 3 with various initial equivalence ratios; (a) Mean droplet
temperature, (b) Total droplet number (solid line, ¢ = 0.01; dashed
line, ¢ = 0.1; long-dashed line, (15 = 0.4; dashed-dot line, d) = 0.7;
dashed-dot-dot line, ab = 0.1; dotted-delta line 43 = 0.2) ......... 167

ISO-contours of the instantaneous of ﬁltered temperature and fuel mass
fraction as obtained by LES/FDMF with various inlet temperatures;
(a) Tin = 3750K, (b) Tm = 4750K, (c) Tin = 5750K, (d) T; =
6750K, (e) Tin = 7750K, (f) Tin = 11000K. .............. 168

ISO-surfaces of the instantaneous vorticity ﬁeld as obtained by

LES/FDMF with various inlet temperatures; (a) Tin = 3750K, (b)

Tin = 4750K, (c) Tm = 5750K, (d) T in = 6750K, (e) T,- = 7750K,

(f) Tin = 1100°K .............................. 169

xvii

3.35

3.36

3.37

3.38

3.39

3.40

3.41

3.42

Radial variations of various quantities at axial locations of m/D =
3 and 22/ D = 6 with various inlet temperatures; (a) Instantaneous
ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c)
Root mean square (RMS) of ﬁltered pressure (solid line, Tm = 3750K;
dashed line, Tin = 4750K; long-dashed line, Tin = 57 50K ; dashed-dot
line, Tin = 6750K; dashed-dot—dot line, Tin = 7750K; dotted—delta
line Tin = 11000K) ............................. 170

Axial variations of integrated droplet number density with various
inlet temperature (solid line, Tm = 3750K; dashed line, Tin = 4750K;
long-dashed line, Tin = 5750K; dashed-dot line, Tm = 6750K; dashed-
dot-dot line, Tm = 7750K; dotted-delta line Tm = 11000K). ..... 171

Radial variations of various droplet quantities at axial location of
x/ D = 3 with various initial carrier gas temperatures; (a) Mean
droplet temperature, (b) Mean droplet axial velocity (solid line, Tin =
3750K; dashed line, Tin = 4750K; long-dashed line, Tin = 5750K;
dashed-dot line, Tin = 6750K; dashed-dot-dot line, Tin = 7750K;
dotted-delta line Tin = 11000K). .................... 172

ISO—contours of the instantaneous of ﬁltered temperature and fuel mass
fraction as obtained by LES/FDMF with various turbulence forcing
amplitudes; (a) emf, (b) 2 x emf, (c) 4 x aref. ............ 173

ISO-surfaces of the instantaneous vorticity ﬁeld as obtained by
LES / FDMF with various turbulence forcing amplitudes; (a) emf, (b)
2 X arefw (C) 4 X Uref- .......................... 174

Radial variations of various quantities at axial locations of 33/ D = 3
and x/ D = 6 with various turbulence forcing amplitudes; (a) Instan—
taneous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass frac-
tion, (c) Root mean square (RMS) of ﬁltered pressure (solid line, emf;
dashed line, 2 X emf; long-dashed line, 4 x Uref)- ........... 175

Axial variations of integrated droplet number density with various
turbulence forcing amplitudes (solid line, aref; dashed line, 2 x emf;
long-dashed line, 4 x Urefl- ....................... 176

Radial variations of various droplet quantities at axial location of
a: / D = 3 with various turbulence forcing amplitudes; (a) Mean droplet
temperature, (b) Mean droplet axial velocity (solid line, emf; dashed
line, 2 X Uref; long-dashed line, 4 x Urefl- ............... 177

xviii

CHAPTER 1

THE STRUCTURE OF
PARTIALLY-PREMIXED METHANE FLAMES
IN HIGH-INTENSITY TURBULENT FLOWS

1 . 1 Introduction

Accurate description of various ﬂame regimes in practical combustion systems is difﬁ-
cult due to strong interactions between turbulence and combustion and the presence
of a wide range of time and length scales. Understandably, the models developed
for combustion systems often assume a certain type of reaction (e.g. slow, fast, pre-
mixed, non—premixed), even though most systems cannot be described based on just
one type of reaction. Among the proposed numerical models for turbulent combus-
tion (see [2—6] for example), the so called “ﬂamelet” models are very popular due
to their relative simplicity [7—12]. In these models, the smallest turbulent motions
(often characterized by the Kolmogorov scale) are assumed to be much larger than
the ﬂame thickness. Thus, the inner structure of the ﬂame is not affected by the
turbulence [13].

Despite its limitations, the direct numerical simulation (DNS) method, has been
widely employed for fundamental understanding of various type of turbulent ﬂames
[4, 12—18]. In many of the reported simulations, the ﬂow is assumed to be two-
dimensional [9, 19, 20]. This is based on the observation that the structure of the

transported scalar is mainly two-dimensional, even though the turbulence itself is

three-dimensional [4, 19]. Along with the simpliﬁcations for the ﬂow ﬁeld, the chem-
istry is also often described by simpler (reduced) reaction models [21] in DNS.

Using DNS with simple chemistry, Poinsot at al. [11] investigated the effects of
partial fuel—air premixing on turbulent ﬂames and have reported signiﬁcant variations
in the local ﬂame burning rate with partial premixing. A similar investigation was
conducted by Baum et al. [9] for premixed hydrogen ﬂames in two-dimensional tur-
bulence using DN S. Also, to better understand the effects of various ﬂow/ combustion
parameters on turbulent premixed ﬂames, Haworth at al. [22, 23] have conducted a
series of simulations of two-dimensional, premixed, turbulent, propane-air ﬂames.
Their DNS results indicate that for the non-unity Lewis number, the local ﬂame
structure correlates more with the local ﬂame curvature than with the tangential
strain rate. Bell et al. [24] studied the effect of turbulence on a premixed methane
ﬂame with a more sophisticated 20-species, 84-reaction mechanism. Consistent with
earlier studies conducted with simpler chemistry model, they have observed that the
turbulence enhances the ﬂame speed with a rate faster than the growth in ﬂame
surface area. Angoshtari [25] studied the strain and curvature effects on premixed
methane ﬂames in high-intensity two-dimensional isotropic turbulence with 1-step,
5-step and 12-step reaction models. It is shown that the 12—step mechanism is able
to accurately predict the tear-offs, local extinction and ﬂame pocket formations in
an intense turbulence ﬁeld.

The characteristics of triple ﬂames in partially-premixed hydrogen-air mixtures,
and the effects of unsteady strain rate on ﬂame propagation in hydrogen-air and
methane-air ﬂames were investigated by Im at al. [26—28]. They indicated that the
triple ﬂame structure in a partially-premixed hydrogen-air ﬂame is similar to that
in a diffusion ﬂame. Maas and Thévenin [20] also used data from two-dimensional
DN S of turbulent non—premixed hydrogen—air ﬂames for assessing the Intrinsic Low

Dimensional Manifold (ILDM) model. The interactions between chemical reaction

and turbulence in homogeneous compressible turbulent ﬂows have been studied by
Jaberi et al. [29—31]. Their results indicate the complexity of combustion effects on
density, pressure, temperature and dilatational velocity ﬁelds.

The main objective of this study is to investigate the local and global behavior
of partially-premixed and non-premixed (initially) laminar methane ﬂames in high-
intensity isotropic turbulent ﬂows. The response of different types of ﬂames to a
high intensity turbulent ﬁeld and the effects of various ﬂow/ﬂame parameters on
the ﬂame extinction and reignition are studied. The chemistry is modeled using a
10-step, 14-species reduced reaction mechanism, which has been proven to be accu-
rate under various conditions [32]. A high-order ﬁnite difference method [33, 34] is
used to integrate the governing equations with periodic and non-reﬂecting boundary
conditions [35, 36] perpendicular and parallel to the ﬂame. The results obtained by

Eulerian [15, 37], and Lagrangian [38, 39] analysis of DNS data are discussed below.

1 .2 Governing Equations

In this section, the governing equations for a compressible reacting ﬂow are presented.
For complete description of these equations the reader is referred to the textbooks
on the subject [4, 40]. The standard equations representing the conservation of mass

and momentum are:
i + 8(Wi)
8t 8583'

 

= 0, (1.1)

8(pui) 6(puiuj) _ Op O’Tij . . _

 

 

where p is the density of mixture, u,- represents the velocity vector, p is the pressure,

rij is the viscous stress tensor, and f,- is the external force in 2th direction. In this

study, the total energy (at) equation is solved,

 

 

N3
a(pet) 8(Wz’et) _ €513 ﬂew-u,- . ' .
at + ax, — 3:5,?“ 5,3. +Qe+p§1lefs,z(uz+Vs,.)I, (1.3)

where Qe is the external energy source or sink, 91]“ is the total stress tensor, and
fs,i is the body force on species 8 in the 2th direction. The mass fraction of reactive

species are described by the following equations:

30%) 3(rmz'Ys) __ 8(stVs,i) . __
at + ax,- — 8% +w3 (3-1,...,N,), (1.4)

 

where Y3 indicates the mass fraction of species 3, (N 3 is the total number of species),
V3,,- is the ith component of diffusion velocity for species 3, and as = W3 2:le 123er
(113,— is the difference between stoichiometric reaction constants as obtained by CHEM-
KIN [41]) is the rate of mass production/ destruction per unit volume for species 3
with molecular weight of W3.

The above system of partial differential equations (Eqs. 1.1 - 1.4) are closed by
the equation of state for an ideal gas. The total energy and species enthalpy are
expressed as:

N3 T
1
et = § :hsrs —p/p+ gum, hs(T) = [T Cps(T)dT+Ah0,s. (1.5)
0

321

In Eq. 1.5, hs is the enthalpy, Cps is the speciﬁc heat, and Ahoﬁ is the enthalpy of for-
mation of species 3. Here, the enthalpy of formation and the speciﬁc heat coefficients
are obtained via tables, polynomial functions and the CHEMKIN thermodynamic
database [41]. The viscosity coefﬁcient, )1 of the mixture in the deﬁnition of the ﬂuid

viscous stress tensor, Tij is calculated based on the viscosity of species [42—44] and

the heat ﬂux vector in Eq. 1.3 is given by:

(H NS
(12' = _/\$ + lethsVs,il- (1-6)
2 3:1

The ﬁrst term in Eq. 1.6 represents the Fourier heat conduction term and the second
term is the heat transport due to “differential diffusion”. The heat transfer due
to radiation and Dufour effects has been neglected. The mixture-averaged thermal
conductivity, A is calculated from the thermal conductivities of individual species.
Computationally, it is very expensive to calculate the complete expression for the
diffusion velocity, V3,,- in Eqs. 1.3 and 1.4. A less intensive, yet sufﬁciently accurate,
model is used in this study. In this model, the diffusion velocity is assumed to be
the opposite of the binary diffusion coefficient which is calculated from the mixture-
averaged formula given by Hirscfelder et al. [45]. The modeled diffusion velocity
does not satisfy the global mass conservation equation. To overcome this problem, a

correction term is added to each species diffusion velocity.

1 .3 Numerical Solution

Equations. 1.1 - 1.4 are integrated using the fourth order compact scheme [33, 34]
with the periodic boundary conditions on the boundaries perpendicular to the ﬂame
and non-reﬂecting outﬂow conditions [35, 36] on the boundaries parallel to the ﬂame
(Figure 1.1). All simulations are conducted with 512 x 512 uniformly distributed
rectangular grid. The turbulence/ ﬂame statistics are monitored to assure the accu—
racy. Some of these statistics are calculated by averaging over all spatial directions,
even though the ﬂow is not homogeneous in one direction. The important statistical
quantities for a 2D turbulent ﬂow are the total energy Q”, the vorticity magnitude

9, the dissipation rate of vorticity r], the (integral and micro) length scales L, l and

the corresponding Reynolds numbers 1261,, Rel deﬁned as
_ .1. . . Q _ l . . _ V 2 1 7
00— 2mm), — ,(w.w.>, n—V(| WI >. m

1 1 1 1 1
bag/773, l=<u9/n)2. ReL=(2av)?L/u. Rez=(2av>?/v. (1.8)

In turbulent combustion simulations, a signiﬁcant part of the computational time is
normally devoted to the calculation of molecular properties like viscosity and thermal
conductivity [21, 25]. Evaluating the diffusion velocities could be also very expensive
as it involves solution of a system of equations of 3N 32 unknowns (N3 is the number of
species) at each grid point and time step. Our approach is to solve for the zeroth—order
(or mixture-averaged) coefficients in the deﬁnition of diffusion velocity V3,,- [45] and
to calculate the mixture viscosity as ,u m 29:31 X sits. The variations of the species
properties (e.g. gas constants) are also approximated by fourth-degree polynomials,
which considerably reduces the computational time. The above approximations have
been tested for turbulent ﬂames and found to yield accurate results. Our analysis
indicate that about 75% of total computational time will be spent for calculating the
diffusion coefﬁcients if these approximations are not made.

The chemical kinetics model is based on a reduced reaction mechanism [46, 47],
which represents the combustion of methane with a 10—step and 14—species reaction.
This mechanism has been reduced from a skeletal mechanism, which itself is derived
from the GRI—Mech 2.11 [48] mechanism. The 10—step mechanism has been widely
tested [32, 46, 47]. Mohammad [32] studied the effects of strain rate, fuel-air premix-
ing, pressure, and preheating temperature on opposed—jet laminar methane-air ﬂames
via detailed and reduced mechanisms. For all of the tested ﬂames, they have found
the predictions of the reduced 10- and 12-step mechanisms to be in good agreement

with those of the detailed GRI mechanisms.

1.4 Results and Discussion

In the simulations considered below, the initial turbulence ﬁeld is generated by
solving the governing equations and boundary conditions with an initially random,
solenoidal, and Gaussian velocity ﬁeld for sufﬁciently long time. The developed
isotropic velocity ﬁeld, described by a set of non-dimensional variables in Table 1.1,
is multiplied by appropriate scale factors and is used as an initial velocity ﬁeld in
all of our reacting simulations. Additionally, the laminar ﬂame data obtained by
low-strain opposed-jet simulations are used for initialization of scalar (temperature
and mass fraction) ﬁelds (Figure 1.1). Tables 1.2 and 1.3 list the initial and ﬁnal
values of some of the important ﬂow/ ﬂame parameters for various simulations con-
sidered in this study. In Table 1.2, u’ represents the root mean square (rms) of the
velocity ﬂuctuations and To is the eddy turn-over time. Also in Table 1.2, 150. is the
Damkbhler number which is the ratio of ﬂow to chemical time scales. Damk6hler
values are relatively low due to high turbulent intensity. The chemical time scale
(deﬁned based on the laminar ﬂame thickness, ﬂame speed and the air viscosity at
ambient temperature) is ﬁxed. 15a is varied by changing the ﬂow time scales. The
effects of initial turbulent intensity, Reynolds number, Damk6hler number, fuel-air
premixing, and ﬂame to turbulence length scale ratio are studied. Partial premixing
of 0% (diffusion ﬂame), 37.5%, and 75% air in the fuel stream are considered. Some
of the ﬂame parameters, such as the ﬂame thickness, are listed in Table 1.3.

The results presented below are obtained by three different types of analysis
(Figure 1.2). Our primary goal is to study the effects of turbulence on the ﬂame
structure and the ﬂame dynamics in a “strong” turbulent ﬁeld. The physical and
compositional ﬂame structures are examined with “ standard Eulerian methods” [15]
and the data at ﬁxed grid points along and perpendicular to the ﬂame. The ﬂame
dynamics is studied by a Lagrangian (particle) method. It is noted here that the

images in the following sections of this dissertation are presented in color.

1.4.1 Physical Flame and Turbulence Structures

In this section, the effect of turbulence on the physical structure of the ﬂame is
studied through examination of “ﬂame variables” like temperature, mixture fraction
and species mass fractions. The effects of combustion on turbulent energy and vor-
ticity are also investigated. Figures 1.3-1.5 show the iso-contours of several ﬂame
variables for case I-3. The temperature contours in Figure 1.3 are consistent with
the (stoichiometric) mixture fraction contours in Figures 1.5 and 1.11, illustrating
the physical ﬂame structure at different times. Evidently, the initial laminar ﬂame
is highly distorted and stretched by the very strong strain ﬁeld to the extent that
the extinction becomes important. Structures similar to edge ﬂames [49] appear on
either side of the ﬂame at locations where extinction is signiﬁcant. Figure 1.4 shows
that the contours of OH mass fraction follow those of the temperature as long as
the ﬂame extinction is not signiﬁcant. However, there are some discontinuities in
the OH mass fraction contours which are not observed in the temperature contours
and are due to (local) ﬂame extinction. It has been argued that the OH radical
represents the ﬂame behavior better than the temperature [25, 50]. It is to be noted
that the ﬂame is affected by an initially isotropic turbulent ﬂow which has no mean
ﬂow convection.

The contours of other major and minor species exhibit similar patterns to those
of temperature and YOH- This is shown in Figure 1.5, where the iso-contours of
the mixture fraction, CO mass fraction, C02 mass fraction and H mass fraction
are considered. Clearly, the patterns in the mixture fraction and carbon monoxide
iso-contours resemble those of temperature. However, there are some differences due
to ﬁnite-rate chemistry and extinction effects. These effects are more pronounced in
the carbon dioxide and hydrogen atom contours. The latter show similar behavior
to that shown for OH radical in Figure 1.4.

To assess the effects of combustion on the turbulence, contours of vorticity and

other ﬂow variables are also considered. Figure 1.6(a) shows the iso—contours of
vorticity at t/To = 2.0. For an isotropic non-reacting ﬂow, there is no preferred
directional dependency. However, as the turbulence is affected more by the ﬂame,
small-scale, high intensity vorticity values are generated near the ﬂame by the “baro-
clinic torque”. It is known that the turbulence is affected by the reaction because (i)
heat of reaction induces volumetric expansion, which opposes the vorticity ﬁeld, (ii)
the temperature dependency of viscosity increases the dissipation of vorticity close to
ﬂame, (iii) vorticity is generated due to the “baroclinic” effect. The latest is very im-
portant in the simulated case considered here. This is evident in Figure 1.6(b), where
it is shown that the high baroclinic and vorticity values are well correlated. The baro—
clinic torque ([—w . (vP x Vp)]/p2) is dependent on vorticity and alignment of the
pressure and density gradients. There seems to be also a good correlation between
the high vorticity values and the ﬂame temperature or the mixture fraction. However,
the total kinetic energy (TKE) is not strongly correlated with the ﬂame. The TKE
mainly changes by the viscous-dissipation and the pressure-dilatation terms [30]. The
viscous-dissipation always transfers energy from the kinetic energy to the internal en—
ergy, but the pressure-dilatation term may transfer energy from the internal energy
to the kinetic energy at some locations. Our results (not shown) also indicate a very
signiﬁcant increase in the strain rate at regions close to the ﬂame, where the vorticity
magnitude is high. The increase in strain rate in turn promotes the scalar dissipation
rate, mixing and reaction. However, it also enhances the ﬁnite-rate chemistry effects

and in some regions even causes ﬂame extinction.

1.4.2 Turbulence and Flame Statistics

Expectedly, with the non-periodic boundary condition in the ﬂame-normal direc-
tion, the ﬂow does not remain homogeneous in that direction. Nevertheless, the

space-averaged statistics provide useful information on the overall ﬂow behavior and

are considered in this section. For example, Figure 1.7 shows that the combustion
increases the ﬂow kinetic energy at small scales (or high wavenumbers of energy spec-
trum), which is understandable since the heat release is conﬁned to narrow reaction
zone. However, our results (not shown here) indicate that the kinetic energy decays
on average at all times, even though the decay rate is affected signiﬁcantly by the
combustion. Similarly, the spectral density of temperature attains higher values at
moderate and high wavenumbers in the reacting case. There are also some effects on
the large-scale ﬂuctuations of temperature and density.

Figure 1.8 shows the conditional PDFs of temperature and strain rate, condi—
tioned on the mixture fraction being between 0.21 < Z < 0.578. The conditional
statistics have been widely employed in the analysis of both non—reacting [51, 52] and
reacting [29, 53—56] turbulent ﬂows. Figure 1.8(a) suggests that there is considerable
difference between laminar and turbulent ﬂames. For the turbulent ﬂame, the PDF
of temperature broadens in time due to ﬁnite—rate chemistry effects and local ﬂame
extinction, a clear indication of strong turbulence/strain rate effect. These results
are consistent with those reported by Sripakagorn et al. [37]. Figure 1.8(b) shows
that the strain ﬁeld is also affected by the combustion. As the turbulence modiﬁes
the laminar ﬂame and the local strain rate increases, the ﬂame exhibits signiﬁcantly
more ﬁnite-rate chemistry effects and even extinction. Nevertheless, the maximum
values of strain rate, scalar dissipation rate, and OH mass fraction continue to in-
crease in time. This is consistent with the formation of separated ﬂame pockets, as
the OH contour plots in Figure 1.4 suggests.

The DN S data indicate that the correlation coeﬂicient between temperature and
scalar dissipation rate decreases as the reaction zone is signiﬁcantly distorted by the
turbulence and the ﬂame experiences extinction. Early in the simulation, Y0 H is well
correlated with the scalar dissipation rate in the thin laminar ﬂame zones, where the

scalar dissipation rate is relatively high. The correlations between the production

10

rates of OH and CO with the scalar dissipation rate are also signiﬁcant at earlier
times. Later on, however, with the signiﬁcant increase in strain and scalar dissipation
rate the ﬂame exhibits signs of extinction and lower values of YOH- This lowers the
overall correlation between the OH mass fraction and the scalar dissipation rate. Our
results (not shown) also indicate lower correlation between the OH mass fraction and
strain rate. The results for conditional averaged strain rate, conditioned on the scalar
dissipation rate, indicate that the strain rate is strongly correlated with the scalar

dissipation rate at high strain rate regions.

1.4.3 Compositional Flame Structure

Scatter plots of variables like temperature in the mixture fraction domain illustrate
the overall state of ﬂame/ turbulence in the “ﬂame coordinate system”. In this coor-
dinate system, the temperature normally peaks somewhere near the stoichiometric
mixture fraction for a ﬂamelet-type combustion and closely follows the laminar pro-
ﬁle depending on the local value of tangential strain rate. However, at locations
where the tangential strain rate is very signiﬁcant, deviations from laminar behavior
is expected.

Figures 1.9 shows the scatter plots of temperature at different times in the mixture
fraction domain for case I-3. The solid lines represent the initial laminar proﬁles.
As the strong turbulent strain ﬁeld affects the ﬂame, the heat is transported away
from the reaction zone with a rate much faster than the rate it is produced by the
reaction. This effectively cools down the ﬂame and leads to low ﬂame temperature
and extinction which are marked with considerable scatter in data and deviation
from laminar ﬂame. In some regions, the locally extinguished ﬂame reignites due to
lower strain rate at later times.

The scatter plots of some of the major and minor species in Figure 1.10 also

indicate signiﬁcant differences between the turbulent and laminar ﬂames. Interest—

11

ingly, the maximum values of YCO2: and Y00 are lower than the corresponding
laminar values, whereas those of Y0 H and YH are substantially higher in the tur-
bulent ﬂame. Such a behavior, termed the “superﬂamelet” behavior [57], has also
been observed in the experimental [58—60] and large eddy simulation (LES) [57]
data. The superﬂamelet behavior can be related to variable strain/ scalar dissipation
ﬁelds as the laminar ﬂame calculations show similar behavior when a step change
in scalar dissipation rate is imposed [61]. Consistent with the LES results of James
and Jaberi [57], the superﬂamelet behavior does not seem to be very signiﬁcant for
YCO (Figure 1.10(a)). This is in contrast to the experimental observations [58—60].
Additionally, the net YCO production rate in the turbulent case is different on the
lean and reach sides of the ﬂame, as the turbulence enhances YC'O production on the
lean side but has the tendency to decrease it on the reach side. Similarly, OH and H
radicals exhibit different behaviors on the rich and lean ﬂame sides (Figure 1.10(b)
and Figure 1.10(d)). Experimental data of Barlow at al. [62] and DNS results of

Mahalingam et al. [63] show similar trends for these species.

1.4.4 Flame Normal Analysis

Despite being useful in the overall assessment of ﬂame/turbulence behavior, the
spaceaveraged statistics contain limited information concerning the local ﬂame and
turbulence structures. To better understand the turbulence effects on the ﬂame and
to directly compare the structures of turbulent and laminar ﬂames, the variations of
ﬂow variables along the ﬂame surface and normal to this surface are considered in
this section. In our ﬂame normal analysis, the ﬂame surface is represented by the
stoichiometric mixture fraction iso-surface (or iso—line in 2D ﬂow). In the ﬂamelet
models, it is assumed that the structure of turbulent ﬂame perpendicular to the
stoichiometric mixture fraction (Zstoz'ch) surface is laminar like, and ﬂame state is

characterized by the stoichiometric scalar dissipation rate (Xstoz'chl- For Xstoz‘ch > Xq

12

(Xq is the quenching limit of the scalar dissipation rate), ﬂame extinction is possible.
Here, we compare the structure of turbulent ﬂame in the direction normal to the
stoichiometric mixture fraction iso—surface with that of the laminar ﬂame. Proﬁles of
temperature, and other turbulent variables along and perpendicular to this surface
are extracted from the DNS data and are compared with the laminar opposed—jet
data. All the results discussed in this section are for case I—3.

Figure 1.11 shows the locations of several ﬂame normal cross-sections along the
stoichiometric mixture fraction surface and on the temperature contours. These
cross-sections are selected in a way that various types of ﬂames are represented. For
instance, sections C and A are located at regions where the ﬂame extinction is either
very signiﬁcant or not important. Additionally, section B-B, is selected for studying
the interactions between ‘two neighboring ﬂames.

Figure 1.12 shows the variations of temperature, OH mass fraction, OH pro-
duction rate, and CO mass fraction in the ﬂame normal direction at sections A,
B—B’, C and D. Laminar results are also shown for comparison. It is observed in Fig-
ure 1.12(a) that the predicted turbulent and laminar temperatures are relatively close
at section A. This suggests that the ﬁnite-rate chemistry effects and ﬂame extinction
are insigniﬁcant at section A. In contrast, the laminar and turbulent temperatures
are very different in the ﬂame normal direction at section C, where the ﬂame is highly
stretched and virtually turned off. At this location, the peak turbulent temperature
is around 400 K.

The temperature proﬁle along the ﬂame normal direction at section B-BI (F ig-
ure 1.12(a)) can be divided into three different regions. The ﬁrst region is the “main”
ﬂame region with the highest temperature around the stoichiometric mixture frac—
tion point. The location of peak temperature in this region is close to that of the
laminar ﬂame, even though the magnitude of temperature is lower. Evidently, the

ﬂame is not strongly affected by the turbulence in this region. As we move along

13

the ﬂame normal direction from B to B], we reach to the second region, where the
temperature is considerably lower. In this region, the mixture is lean as the very low
mixture fraction values indicate. In the third region, where the mixture fraction is
close to stoichiometric and the temperature is normally expected to be comparable
to the adiabatic ﬂame temperature, the temperature is actually very low due to ﬂame
extinction. At section D, the ﬂame still experiences extinction, yet the extent of it
is not as signiﬁcant as it is at sections B—B’ and C. This is also consistent with the
temperature iso—contours in Figure 1.11(b), where it is observed that the ﬂame at
section D is thicker than that at section C. The temperature proﬁle in the fuel rich
side at section D clearly shows the effects of two neighboring ﬂames on each other.
Figures 1.12(b), 1.12(c), 1.12(d) show that the turbulent proﬁles of OH and CO
mass fractions and OH production rate along the ﬂame normal direction at sections
A, B—BI, C and D are different than the laminar ones, particularly where the strain
rate is high. It is noted that the heat is transported away from the ﬂame zone with a
rate much faster than the rate it is produced in high strain rate regions, which cools
down the ﬂame and decreases the ﬂame temperature. Expectedly, this will result in
a decrease in rate of reaction and even local ﬂame extinction. Consistent with the
temperature results in Figure 1.12(a), the OH concentration in turbulent ﬂame is
shown to be higher than those in the laminar ﬂame at section A. This “superﬂamelet”
behavior is due to relatively weak but variable strain rate at this region. In contrast,
Y0 H values are very small for all mixture fraction values at section C, conﬁrming
the ﬂame extinction at this section. This is understandable as the production rate
of OH is virtually zero at section C (Figure 1.12(c)). The proﬁle of YC'O along the
ﬂame normal direction at section C (Figure 1.12(d)) exhibits behavior closer to the
temperature than to YOH- The non-zero (convex shape) values of YOU proﬁle at
section C suggests that there is still some reaction, even though the ﬂame is strongly

strained at this section.

14

Figures 1.13-1.15 show the spatial variation of several turbulent quantities along
the stoichiometric mixture fraction iso—surface. The locations of ﬂame normal cross
sections A, B-B’, C and D are also shown in these ﬁgures. All the reported variables
display oscillatory behavior along the ﬂame surface, consistent with the highly wrin-
kled ﬂame structure shown in Figures 1.3 and 1.4. As the proﬁles of temperature
and stoichiometric scalar dissipation rate, Xstm‘ch in Figures 1.13(a) and (b) show,
at section (or point) A the ﬂame has its maximum temperature and minimum scalar
dissipation rate; indicating a behavior similar to laminar ﬂames. In contrast at point
C, the temperature is the lowest, while the scalar dissipation rate is relatively high.
Also at point C, the OH production rate is nearly zero (Figure 1.13(c)), an indica—
tion of ﬂame extinction at this point. Interestingly, the maximum value of the scalar
dissipation rate occur at point D and not at C. At point C, the scalar dissipation rate
is signiﬁcant because of lower diffusivity coeﬂicient and stronger turbulent straining
in the extinguished ﬂame (Figure 1.11(b)). The ﬂame extinction is also signiﬁcant at
point D. However, the temperature and consequently the molecular diffusivity and
scalar dissipation rate are relatively higher at this section.

Our results (not shown) indicate that along the stoichiometric mixture fraction
surface the vorticity is somewhat uncorrelated with the ﬂame variables like tempera-
ture and OH. However, the scalar dissipation rate is found to be strongly correlated
with the ﬂame temperature, even at locations with noticeable ﬂame extinction [37].
By comparing the strain rate and the scalar dissipation rates along the stoichiometric
surface, one can examine the effect of the strain rate on the scalar dissipation rate
and the ﬂame. However, our analysis for the entire computational domain indicate
that the local values of strain rate and scalar dissipation rate are not very well cor-
related. This suggests that the coupling between the scalar and turbulent ﬁelds is
only signiﬁcant in the “ﬂame region”.

Figure 1.14 shows the spatial variations of the total kinetic energy (TKE) and

15

 

the pressure—dilatation term in the kinetic energy equation along the stoichiomet-
ric mixture fraction surface. Direct numerical simulations of homogeneous reacting
turbulent ﬂows [29—31] have shown that the heat of reaction indirectly affects the
mean turbulent kinetic energy mainly through the pressure—dilatation and viscous-
dissipation terms. It is observed in Figure 1.14(a) that the TKE attains its lowest
and highest values at locations where the temperature is high or low. However, there
is a relatively weak correlation between TKE and ﬂame temperature. On the other
hand, comparison between Figures 1.13(d) and 1.14(a) indicates that the maximum
values of TKE generally coincide with the maximum values of the strain rate except
at point C. At this point, the strain rate is signiﬁcant, but TKE is relatively small.
The low values of TKE are due to several reasons, one being lack of energy trans-
fer from the internal energy by the pressure-dilatation term. In the TKE equation,
the terms that are responsible for production and destruction of TKE have different
positive and negative contributions at different locations. For example, the pressure-
dilatation term has a tendency to alternatively transfer energy between the internal
energy and the kinetic energy [29, 30]. This is evident in Figure 1.14(b), where it is
shown that the variations of pressure-dilatation along the ﬂame surface generally co-
incide with that of TKE. However, as Figure 1.14(b) suggests the pressure-dilatation
is insigniﬁcant at point C. On the other hand, there is a signiﬁcant energy transfer
from the internal energy to the kinetic energy by the pressure-dilatation term at
point D (Figures 1.14(a) and 1.14(b)) which supports the high values of TKE at
this location. The viscous-dissipation term in the TKE equation does not vary so
much except close to point D. At this point, the turbulent scalar ﬂuctuations, and
the ﬂame temperature/ molecular diffusivity coeﬂﬁcient are signiﬁcant (see the scalar
dissipation and temperature plots in Figure 1.13). The high temperature values at
point D is due to heat diffusion from the neighboring ﬂames and transfer of energy

from TKE as well as the heat of reaction.

16

To further examine the effects of combustion on turbulence, the variations of
vorticity magnitude ((2) along the stoichiometric mixture fraction surface are consid-
ered in Figure 1.15(a). The results in this ﬁgure indicates that 9 peaks at locations
where the ﬂame stretching and/ or curvature are signiﬁcant. Evidently, the highest
value of (2 occurs at x/Lm a: 0.8, where a highly distorted ﬂame is located (Fig-
ure 1.11(b)). These results can be better explained by examining the baroclinic term
in the vorticity equation [29, 30] in Fig. 1.15(b). The baroclinic term is composed
of vorticity, pressure gradient and density gradient vectors; both the magnitude and
relative alignment of which are important [30]. In two-dimensional ﬂows, vorticity
has only one component. The vorticity and density proﬁles along the stoichiometric
mixture fraction surface (not shown) suggest that the high negative values of the
baroclinic term are mainly due to negative vorticity, and low density values. Our
results also indicate that along the stoichiometric mixture fraction surface the cosine
of angle between the pressure and density gradient vectors is close to zero wherever
the baroclinic term is signiﬁcant. (Figure 1.15(b)) and is close to one wherever the
baroclinic term is insigniﬁcant. The contributions of the volumetric expansion term
in the vorticity magnitude equation could be negative or positive depending on the
ﬂow being in expansion or in contraction [29, 30]. Our results (not shown) indicate
that the highest vorticity values occur where the volumetric expansion term has its

highest negative value or the ﬂow is in strong contraction.

1.4.5 Lagrangian Flame Analysis

In the Lagrangian method considered in this section, some of the ﬂuid elements are
marked with notional particles that are moved with the local ﬂuid velocity and have

the same local ﬂuid properties. These particles cover several important ﬂame zones

17

[ii

and are transported according to the following equation:

(n)
4X.- _ (n)
dt — v2- , (1.9)

 

where X,- is the displacement vector, v,- represents the ﬂuid velocity interpolated
to the particle position and n is the particle index. Equation 1.9 is integrated via
an explicit ﬁnite difference method and the ﬂow quantities at particle locations are
evaluated by a high order interpolation scheme. The instantaneous and average
(Lagrangian) particle results for cases I-3 and X-3 are discussed below.

Figure 1.16 shows the evolution of a set of Lagrangian particles originally located
along the stoichiometric mixture fraction surface and perpendicular to this surface for
the case X-3. As expected, the particle movement is directly related to the stretching
and bending of the ﬂame by the turbulent strain ﬁeld and is very different at diﬂerent
sections. Also, as expected, particles in the left section (labeled (i) in Figure 1.16(a))
are affected less than those in the middle and right sections ((ii), (iii), (iv)) due to
weaker strain ﬁeld at this location (see Figure 1.3). The results at t/To = 0.5 and
t / To = 1.0 suggest that the ﬂame stretching in the right section (iv) is more than that
in the left section (i), while the ﬂames in the middle sections are strongly distorted
by the turbulence. These observations are consistent with the vorticity and scalar
dissipation rate contours.

Figure 1.17 shows the temporal variations of the turbulent ﬂame temperature,
mixture fraction and strain rate for several (sample) particles in case I-3. Different
than those particles in the previous ﬁgure, these particles are initially located close
to the stoichiometric mixture fraction surface, all with the same distance from this
surface. Expectedly, the behaviors of particles are different as they follow different
trajectories but the temperature of all of them decrease in time due to very low strain

rate, endothermic reactions, and/ or ﬂame extinction. While some particles display

18

strong ﬁnite—rate chemistry effects and extinction, some are weakly affected by the
turbulence. For instance, consider particle “A”, symbolized by “0”. Figures 1.17(c)
and 1.17(a) show that with a sharp increase in the strain rate, the temperature of this
particle decreases between times t/To = 0 and t/TO = 1.1 to a very low value. Our
results (not shown) also indicate a signiﬁcant decline in the OH production rate for
this particle. The temperature decline is mainly caused by high strain rate and ﬁnite
rate chemistry effects. From t/To = 1.1 to t/To = 1.5, the particle moves out of the
ﬂame zone to the fuel rich side as shown in Figure 1.17(b), resulting in an increase in
its mixture fraction value. After t/To = 1.5, particle A again experiences an increase
in temperature and consequently an increase in the viscosity coefficient and a decrease
in the strain rate. As the low OH production rate values suggest, this increase in
temperature is caused by movement of the particle closer to the ﬂame and not the
reignition. The particle B, symbolized by “V”, shows similar behavior before t/ro =
1.5. However, this particle does not move back to the ﬂame zone after t/To = 1.5 and
its temperature continously decreases. In contrast to those particles experiencing
strong strain effects and even extinction, there are others that are not signiﬁcantly
affected by the turbulence. For instance, particle “C” (symbolized by “0” shows high
temperature (Figure 1.17(a)) and OH production rate at all times since it stays in
the ﬂame zone and experiences relatively low strain rates. Interestingly, particle “D”,

77

symbolized by “+ , also keeps its high temperatures before t/To = 1.0. However,
after t/To = 1.0, the temperature of this particle decreases (Figure 1.17(a)) partly
due to high strain rates (Figure 1.17(c)).

Figure 1.17(a) shows that the particle E, symbolized by “I,” experiences both the
extinction and the reignition. In turbulent ﬂames, the local extinction and reignition
processes are primarily controlled by the turbulent strain, mixing and reaction at

small scales. Figure 1.17(c)) shows that the strain rate is initially high for this

particle, leading to sharp decline in temperature before t/To = 0.8. Between times

19

t / To = 0.8 to t / To = 1.1, the temperature of particle E continues to decrease while the
strain rate decreases and the particle moves out of the ﬂame zone. Between t / To = 1.1
and t/To = 1.4, the particle E stays in the fuel rich side, where it experiences high
strain rates (Figure 1.17(c)), accompanied by low OH production rates. Between
t/To = 1.4 and t/To = 1.8, both the temperature and the OH production rate of
particle E increase, suggesting reignition. However, as shown in Figures 1.17(a) and
(b) the temperature of particle E again decreases after t / To = 1.8.

Figure 1.18 shows the conditional average values of the particle temperature for
three different range of initial mixture fraction at different times. These conditional
averages are obtained by averaging the temperature of a group of particles which have
the same initial mixture fraction and (time) integrated strain rates. The averages
are conditioned on the (initial) mixture fraction to study the collective behavior
of particles that are initially at the same location with respect to the ﬂame but
follow diﬂerent paths latter on. Also, the averages are conditioned on the integrated
strain rate instead of instantaneous strain rate to include the turbulence history
effects. The conditional mean temperature in the low mixture fraction (oxidizer)
zone, stoichiometric (ﬂame) zone, and high mixture fraction (fuel) zone are shown in
Figures 1.18(a), (b), and (c), respectively. Evidently, in all cases the conditional mean
temperature decreases. However, the decrease in temperature is more pronounced
when the integrated strain rate is higher. A comparison between these results with
those obtained based on conditioning with respect to instantaneous strain rate (not
shown) suggests that the integrated strain rate represents the effect of turbulence on
the ﬂame better than the instantaneous strain rate.

Figure 1.18(d) shows the difference between average particle temperature when
it is conditioned on the lowest and highest integrated strain rates. The mean tem-
perature difference, represented by A << T >>, is obtained by subtracting the

average particle temperature conditioned on high integrated strain rate from that

20

conditioned on low integrated strain rate for a range of mixture fraction values. The
results for three different mixture fraction ranges are shown. Figure 1.18(d) shows
that for all three mixture fraction ranges considered here, the average temperature
signiﬁcantly decreases at high strain rates due to ﬁnite-rate chemistry effects and
ﬂame extinction. At low strain rates, the ﬂame temperature is generally higher.
Nevertheless, the difference between the average temperatures evaluated at low and
high integrated strain rates is the highest when the initial mixture fraction of the
particles is in the stoichiometric mixture fraction range. The results for the condi-
tional averaged values of OH production rate (not shown) are consistent with those
shown for the temperature.

Figure 1.19 shows the conditional PDFs of the temperature and the strain rate at
t/To = 2.0. The results at other times are similar and are not shown. These PDFs are
calculated from the Eulerian grid points and Lagrangian particle data for Z x Z stoich-
Evidently, the PDFs obtained from the Lagrangian particles agree well with those
obtained from the Eulerian grid points. This suggests the accuracy of Lagrangian
particle analysis. Comparison between the PDFs at t/To = 2.0 with those at earlier
times (not shown) indicate that the initially narrow proﬁle of the conditional PDFs
of temperature of the laminar ﬂame extends toward lower temperature values in
time as a consequence of ﬁnite-rate chemistry/ ﬂame extinction. The ﬂame also has a
signiﬁcant (local) effect on the turbulence as the conditional PDFs of the strain rate
also broadens toward higher strain rate values in time due to combustion generated

turbulence at small scales.

1.4.6 Effects of Various Flow and Flame Parameters

In this section, the dependency of ﬂame and turbulence behaviors on several different
ﬂow and ﬂame parameters are examined by various simulations. Table 1.2 and 1.3

provide the relevant information on the cases considered. Cases II-3, III-3 and VIII-

21

3 have different turbulence intensities and case VIII-3 has been included to show
the ﬂame behavior under laminar like conditions. In cases IV—6 and V—7, turbulent
Reynolds numbers are varied, while the Damkohler number is kept constant. In
cases VI-6 and VII-7, the Damkohler number is changed by modifying the integral
turbulent time scale and turbulent intensity, while the chemical time and Reynolds
number are kept constant. Finally, cases I-1 and L2 involve different premixing levels
of the fuel with air and are considered to investigate the turbulence effects on various
types of ﬂames.

Table 1.3 shows the parameters of three ﬂames with different initial premixing
level. For these ﬂames, the iso-contours of various turbulence and ﬂame variables
and the scatter plots of temperature and species mass fractions are compared. F ig-
ures 1.20(a), (b) and (0) show the scatter plots of temperature in the mixture frac—
tion space for 75%, 37.5% and 0% air-fuel premixing levels, respectively. In all three
cases, the ﬂame temperature is considerably lower than the corresponding laminar
temperature, similarly showing departure from the ﬂamelet behavior in all ﬂames.
In Figure 1.20(d), it is shown that the 75% air-fuel premixed ﬂame does not exhibit
very different behavior than the ﬂamelet behavior when turbulence is relatively weak.
This conﬁrms that the scatters in Figures 1.20(a), (b), (c) are due to strong turbu-
lent strain ﬁeld. The results in Figure 1.20 also indicate that the ﬂame is thinner
for lower premixing level, even though the peak values themselves are approximately
the same in all ﬂames. Moreover, the ﬁnite-rate chemistry effects and extinction do
not seem to be much more signiﬁcant in the non-premixed ﬂame, in comparison to
partially-premixed ﬂames. Like temperature, the scatter plots of OH mass fraction
indicate signiﬁcant local ﬂame extinction for all the three ﬂames and again there is
an “over-prediction” of the laminar OH concentration. This behavior, referred to as
superﬂamelet behavior, has also been observed for the H radical and is consistent

with the experimental and DNS results [57—60]. Laminar calculations conducted

22

F.

with a step function scalar dissipation rate also show superﬂamelet behavior [61].
Further examination of the data indicate that distinct ﬂame pockets with high Y0 H
values are created by the strong turbulence ﬁeld in all three ﬂames. Additionally, our
results indicate that the CO concentration increases when the amount of premixed
air in the fuel stream is increased. However, similar to temperature, the maximum
values of the scalar dissipation rate are found to be nearly the same in all ﬂames.

Figure 1.21 shows the percentage of average temperature difference between tur-
bulent and laminar ﬂames for different levels of fuel-air premixing at t/To = 1.5.
Consistent with the temperature plots shown in Figures 1.9 and 1.20, the maximum
temperature difference, representing the extent of ﬁnite-rate chemistry effects, occurs
close to the stoichiometric mixture fraction point in each ﬂame. Also, the maximum
difference between averaged turbulent and laminar temperatures is higher in the
37.5%-air ﬂame in comparison to the other two. This suggests that the 37.5%-air
ﬂame is experiencing more extinction. In comparison to the 75%-air ﬂame, the dif-
ference between the averaged turbulent and laminar temperatures is smaller in the
0%-air (non-premixed) ﬂame.

The iso-contours of YCO in Figure 1.22 illustrates the physical structure of various
ﬂames as they are inﬂuenced by different ﬂows. The temperature contours exhibit
similar trend and are not shown. The effects of turbulence intensity are shown
in Figure 1.22(a) and (b), where cases I—3 and IX-3 are considered. As expected,
the ﬂame tends to preserve its laminar structure in the lower intensity case IX-3,
whereas it is strongly affected by the higher intensity turbulence case L3, to a degree
that turbulent and laminar ﬂame structures become completely different at some
locations. The YCO proﬁles are smoother in the ﬂame zone in the low intensity case
due to relatively higher temperature and molecular diffusion. The results for cases
I-3, IV-6 and V—7 (not shown) indicate that the peak values of YCO are lower and

the contours of radical species like OH are smoother in the ﬂame region at lower

23

Reynolds numbers. Similarly, the OH concentration values are noticeably lower at
lower Damkohler numbers. The magnitude of the scalar dissipation rate is found to
increase with a decrease in Damkohler number.

In Figure 1.23, the percentage of grid points which are close to stoichiometric
mixture fraction surface and have temperatures below 1400K is shown for ﬂames
with different Reynolds and Damkohler numbers. These results quantify the effects
of Reynolds and Damkohler number on the ﬁnite—rate chemistry and ﬂame extinc-
tion. Evidently, there is more scatter in temperature at lower Reynolds number
(Figure 1.23(a)), which is partly due to increased thermal diffusivity. Also, there
seems to be slightly more scatter in the data for cases with lower Damkbhler number
(Figure 1.23(b)) which can be attributed to relatively higher turbulent intensity. The
temperature iso—contours and the conditional PDFs of the temperature, conditioned
on the mixture fraction being close to its stoichiometric value, exhibit trends similar

to those shown in Figure 1.23.

1.5 Summary and Conclusions

Direct numerical simulations (DN S) are conducted to investigate the interactions of
a methane-air laminar ﬂame with a highly unsteady, and strained two—dimensional
isotropic turbulent ﬂow. The reaction is modeled with a 10-step, 14-species methane-
air reduced mechanism. This mechanism has shown to reproduce the important fea-
tures of laminar methane ﬂames as obtained by the detailed mechanisms for various
strain rates, premixing levels, pressures, and preheating temperatures.

The DNS data are analyzed by both grid-based “Eulerian” and particle-based
“Lagrangian” methods. The results indicate that the local extinction frequently
occurs at high strain and scalar dissipation rate regions. This is accompanied with

the structures similar to edge ﬂames on either side of the extinct ﬂame and pockets

24

 

of isolated burning ﬂames. Expectedly, the ﬂame has a negative overall effect on the
vorticity and kinetic energy due to reduced local Reynolds number and combustion-
induced volumetric (dilatational) ﬂuid motions. However, the ﬂame was found to
generate vorticity close to the reaction zone, mainly due to the baroclinic effects.
The variations of vorticity and baroclinic torque along the ﬂame surface conﬁrm the
vorticity generation by the baroclinic torque.

The strong strain ﬁeld also makes the compositional turbulent ﬂame structure
very different than that of the laminar ﬂame. Close to the stoichiometric mixture
fraction surface, the ﬂame exhibits very different temperatures between unburned
and adiabatic values; demonstrating the signiﬁcance of ﬁnite—rate chemistry effects
and ﬂame extinction. However, close examination of the data along the ﬂame sur-
face indicates that at some points the ﬂame does not experience extinction, even
though the strain and scalar dissipation rates are high. This is partly explained by
considering the vorticity and kinetic energy variations along the stoichiometric mix-
ture fraction (ﬂame) surface. It is observed that in some locations with high scalar
dissipation rate, the pressure—dilatation and viscous—dissipation terms in the kinetic
energy equation transfer energy from the kinetic energy to the internal energy.

The two-way interactions between turbulence and combustion and the dynamical
behavior of the ﬂow / ﬂame variables are shown to be well captured by the Lagrangian
particle analysis. It is observed that the particles experience very different behavior,
depending on their trajectories, even if they are initially at the same position within
the ﬂame. While some particles exhibit complete extinction, some are not virtually
affected by the turbulence. It is concluded that the time integrated Lagrangian strain
rate is a better parameter for characterization of the ﬁnite-rate chemistry effects and
ﬂame extinction than the strain rate itself.

The inﬂuences of (initial) turbulence intensity, Reynolds and Damkohler numbers,

and air-fuel premixing on the ﬂame properties and turbulence are also studied. It

25

is observed that by changing the ﬂame-air premixing, the turbulence is not greatly
aﬂected, even though the ﬂame always experiences extinction in an intense turbulence
ﬁeld. The CO concentration was found to increase and the ﬂame thickness was found
to decrease with an increase in the partial premixing of the fuel with oxidizer. Also,
within the ﬂame zone, the OH concentration was found to increase with an increase

in Reynolds number or Damkbhler number.

26

1.6 Tables and Figures

Table 1.1: Various non-dimensional statistics of the developed initial turbulence ﬁeld.

 

 

(7v

(2

7) l

L

Rel

ReL

 

 

0.359

 

8.94

 

 

2.75 0.06

 

 

0.427

9.71

 

230.32

 

 

 

Table 1.2: Turbulence parameters in various simulations.

 

 

 

 

 

 

 

 

 

 

 

 

 

Case u’ (cm . 3’1) l (cm 7'0 (3) Rel 12a

ti tz' ti ti if tz‘
I — * 2256 0.021 6636-5 293 103 5.046—03
II — * 3000 0.021 4.98e—5 390 135 3.79e—03
III — * 4800 0.021 3.11e-5 624 257 2.37e—03
IV — a: 1641 0.015 6.63e-5 152 90 5.04e-03
V — * 1026 0.010 6636-5 63 46 5.048—03
VI — * 3102 0.015 3.516-5 293 144 2678-03
VII — * 4963 0.010 1.85e5 293 138 1.04e-03
VIII — * 6 0.021 2.49e-2 0.8 2.1 1.89e+00
IX — * 226 0.021 6.63e4 30 24 5.04e—02
X — a: 4800 0.029 4.13e—5 846 357 3.14e—03

 

 

 

 

 

 

 

* Flame conditions/ parameters are different and are provided in Table 1.3.

Table 1.3: Flame parameters in various simulations.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

No. Premixing (%) 6 (cm) 6 / L Turbulence cases considered
at: — 1 0 0.327 2.19 I

at: — 2 37.5 0.250 1.67 I

=I< — 3 75 0.263 1.76 I, II, III,VIII,IX

=0: -— 4 75 0.353 2.36 I

* — 5 75 0.443 2.97 I

a: — 6 75 0.191 1.76 IV, VI

* — 7 75 0.119 1.76 V,VII

* — 8 75 0.349 1.76 X

 

 

 

* Turbulence conditions/ parameters are presented in Table 1.2.

27

 

 

Nonreﬂecting

 

  

 

 

 

 

 

Nonreﬂecting

Figure 1.1: Computational domain, boundary conditions, and initial vorticity and
ﬂame ﬁelds.

 

Eulerian Grid-based Analysis:
Information about various ﬂame and
turbulence properties are gathered

from ﬁxed grid poinls in the domain.

Flame Normal Analysis:
Information about various ﬂame
properties are gathered along normal
direction to the ﬂame.

Lagrangian Particle-based Analysis:
Information about various flame properties
are gathered from lagrangian particles
injected to the ﬂow domain.

 

 

 

 

Figure 1.2: A schematic view of the ﬂame surface, illustrating various methods em-
ployed for the analysis of DNS data.

28

 

 

 

 

 

T T
0.5 2.2E+03 0.5 2.2E+03
(a) 7. (b)

i 1.8E+03 1.8E+03

I >1
:1 0 I.3E+O3 :1 0 1.3E+03

>. >1
8.5E+02 8.5E+02
4.0E+02 4.0E+02

'0 5o o 5 -o.50 0.5
x X/Lx
r T
0.5 2.213403 0.5 2.2E+O3
(c) H (d)

If: 1-8E+03 .. I.8E+03
r50 1-3E+03 {To ‘ 1.3E+03

>u >3
8.5E+02 8.5E+02
4,015+02 4.0E+O2

'0-50 0.5 '0-50 0.5
X/Lx X/Lx

Figure 1.3: Instantaneous contours of temperature for case I—3 at different times;
(a) t/To = 0.0, (b) t/To =1.0, (c) t/To = 1.5 and (d) t/To = 2.0.

29

OH OH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 4.0E-03 0.5 . 4.0E-03
(a) | (h)
f; 3.05—03 , 3.01303
3 ' 2.0503 go 4) 2.0303
3:” >~ 5 ¢ ‘
1.0E-03 , 1.0E-O3
0.0E+00 I.I 0.0E+00
'0 50 0:5 '0'50 0:5
X/Lx X/Lx
OH on
0.5 4.0E-03 0.5 4.013-035
(c) (d) j
3.05-03 ) ( f 3.0E-03
é‘O/p A 54 , 2,013-03 £0; a; ‘ ' v “ 2.0E-03
>3 ‘1 >.
:- _ , 1.0E-03 "S Q“ (I 1.0E-03
. J, .
0.0E+00 0.0E+00
‘0-50 035 '0'5( E5
X/Lx X/Lx

Figure 1.4: Instantaneous contours of OH mass fraction for case I—3 at different
times; (a) t/To = 0.0, (b) t/To 21.0, (c) t/To =1.5 and (d) t/To = 2.0.

30

 

 

 

 

YéLx

v
4?

3

 

 

 

 

 

 

-0.5 ‘
0 0.5
X/Lx
0 e
“ (d)
I
>1
d 0 _' ‘ T
>’ e I
A
-0.5( (is
X/Lx

 

sti-

 

CO
6.5E-02

4.9E-02
3.3E-02
1.7E-02
l. lE-03

Figure 1.5: Instantaneous contours of various scalars for case I—3 at t/To = 2.0;

(a) mixture fraction, (b) CO mass fraction, (c) C02 mass fraction and (d) H mass

fraction.

31

(0

. 9.0E+02
n 3.6E+02

-6.1E+01

    

-4.8E+02
-9.0E+02

 

B

3.2E+04
1 .9E+04
6. 1E+03
-6.9E+03

 

-2.0E+04

 

0

ﬁnd

0.5
X/Lx

Figure 1.6: Instantaneous contours of various ﬂow variables for case I-3 at t / To = 2.0;
(a) vorticity and (b) baroclinic torque.

 

 

(k)

a 10'1 -
10'3 -
10'5 r
10'7 -
10‘9 .

100 101 J [(171102 I 103

E

 

 

‘
1 111111

 

Figure 1.7: Two—dimensional energy spectral density function in reacting case I-3 at
different times; t/To = 0.0, (dashed-dot line), t/ro = 2.0 (solid line).

33

 

l’(<T Izst)

 

 

 

 

'5 1 1 1 1 L 1 1 1 1 1 1 1 1 1 1 1
10 600 1200 1800 2400
T [Kl

 

_
'1‘ -‘
-5 a
10 : -'

3..

p—A
9.
\I
III]!

 

10'8

0‘ if ’500‘ 1000‘ 1500‘ 2000 2500
- -2

St [s 1leO

 

 

 

Figure 1.8: The conditional probability density function (PDF) of (a) temperature
and (b) strain rate, conditioned on the mixture fraction being between

0.21 < Z < 0.578. The PDFS are calculated from the Eulerian data at t/To = 0.5
(solid line), t/To = 1.0 (dotted line) and t/To = 2.0 (dashed-dot line).

34

nAnn

 

 

 

 

 

 

 

 

 

 

 

g £1800:
E E
E '3 1200
a E
E a
600
g 1800 g 1300
E 5
E1200 3 1200
E- I“
g 13 ’
600 600

 

 

 

 

 

 

 

 

0.5
Mixture Fraction

Figure 1.9: Scatter plots of the temperature in the mixture fraction space for case
I—3 at (a) t/To = 0.25, (b) t/To = 1.0, (c) t/To = 1.5 and (d) t/To = 2.0. Solid lines
show the initial laminar proﬁles.

35

 

 

C0 Mass Fraction
.0
i

 

    

 

 

 

0.5
Mixture Fraction

 

0.5
Mixture Fraction

 

 

_o
DI‘
cn
o

a

:(c) ":(d)

  
  
 
  

3::
_.
N

I
_o
_.
{It
' I

I .I

".' 5M
..a 313*;

r} , 1,

:5 .Ju 4..
v.

_O
o
LII

C02 Mass Fractron
.0 .°
1 v r g 1 1 r r a 1 r r r 1 v
H Mass Fraction x 10
O

m L l

0.5
Mixture Fraction

 

 

 

 

 

.9

 

0.5
Mixture Fraction

Figure 1.10: Scatter plots of different species mass fractions in the mixture fraction
space for case I-3 at t/To = 1.0; (a) CO mass fraction, (b) OH mass fraction,

(c) 002 mass fraction and (d) H mass fraction. Solid lines show the initial laminar
proﬁles.

36

 

0.5

 

 

 

.1’ \'\ D
Ir
- \ l .
e” f """ ’ c ,’
‘ """""""" \ ....... .. \. ".5 .
>3 0 l' ‘ j l I if (3}, / __I
[- (nr'.\ \‘1 :l. :’ ‘.\ ..... '1‘ <.
[:j. . '\.v ! I “‘ .\[
uv- ” \
'. .1) .
\ ..... 1| ..... 3
i-
B
1 Stoichiometric Mixture Fraction lsosurface
.0.5 1 1 1 1 I 1 1 1 1
0 0.5 1

X/Lx

 

 

 

 

 

 

 

0 1110.51 11
X/Lx

Figure 1.11: Positions of the ﬂame normal cross-sections on the (a) stoichiometric
mixture fraction iso—surface and (b) temperature contours. Flame normal A is at

node 425, B is at node 1109, B, is at node 1265, C is at. node 1425 and D is at node
1715.

37

Temperature [K]
1'3
8

O‘s
. 8 1

W011 [moles/cm's-s'l]

 

 

 

 

 

1 0:5 5
Mixture Fraction

 

 

 

 

 

0.5
Mixture Fraction

p
O
(I!

OH Mass Fraction x 10

C0 Mass Fraction

_C>
O
A

.o .o
O O
N w

P
O
H

.o .o
O

P

 

E (b)

 

 

 

0.5 l
Mixture Fraction

 

E (d)

 

 

 

 

‘ 0.5 ‘ ’ 1
Mixture Fraction

Figure 1.12: Comparison of various turbulent variables along the ﬂame normal direc-
tion with the corresponding laminar values at different locations and at normalized
time of t/To = 2.0 for case I—3; (a) temperature, (b) OH mass fraction, (0) produc-
tion rate of OH, and (d) CO mass fraction. Laminar proﬁles; dashed line, turbulent

proﬁles along ﬂame normal A; squared line, B-B’; circled line, C; triangled line; D;
plused line.

38

 

2

( )
” a
- <
. o‘ ‘m .
E. _ 1“. 10' ,1 1‘; . H.
4800- ill, I 'mfi I ’18 1
2 ! Lil ' ‘ [1 , '1 j' [m ; 1 1:
a [ i' 5 I 1‘ ‘ ' 1 . 1."
_ 1 _1 1 [ 1). 1 1 1 :1 1
| _ 1 '1 ‘ , 1 i Q ‘.
E1200-,- 1; .1 1
a, ‘ 1' ” 1, ‘1 .' i 58 1
E :1, l [5910 1’ m "'
13 1‘ 1 15115 ,’ 1
1 01 0,1 1
600i m1, W1 1
\I \.,/
P144

 

 

 

“5013“10'00‘ ‘1‘5'00

Node Number

2000‘ ‘

 

1-1 0.04
'1‘ '(C)
'0 I 1
'1' - 1‘
E 0.02: i
9 ’ :3}:
\ ’ 'le
0"[11‘15'

L

WOH [moles

 

0.02 l

I
1
1 .1 1
1 I .1
11 11 ,.
1; Ii 0 ;
[:1 l i1 m1 . D]
,[ , i'f 'il 0 e‘.
l. 1,1 0;: q) 01;,
1'1 1" ;; Q.“ m 01,1
11 <01 m1.
‘1." "‘1' ‘ tuft?“ """1 11:1“? ‘
'1<:'1' :11 1 1.
'1 "1, 1 [ l
11 t \ '1
0 I 11 CG '1:
CI) 1 , g . '5
)3 0 1
1. O
:1 W
I

 

 

 

-0.04

"‘500'1000‘1‘5003000

Node Number

xstoich [8.1]

1500

 

5 .

§ ,

:(b)

 

I
8 11
I:
r I. I 11
L l[ m 1. [i
ll 1,6 1". I!
I[ [1301‘ I[ ,3
<ll W‘m [‘11 51 ill
" 1 "i 5 | [1 1‘11
1 1 . I1' ' 1 _ 1 ~
I.- 10‘ 11111‘ 1"
1""- at), . ,\ [,1». , 11 1
1~ 11111 11. 11:1. .mw 1- 1
'1t-l'11 11'11 1‘ ‘.:‘;,'1
.Ill ‘5 I . - l 0 l] I
1.1.11'111 1‘-°'1:1‘~.1 11
(11‘ 1 l 1 1 0 ‘3 '21 J
’ ‘ tn "
1 l l l l l l

 

 

" 01

. O[
”’1

-. m 1

"l 1 . [1 U

.1 I 8 1 '

1.! 1 ,m 1 8

11 ,1 ‘1 1; 1 11m

i“. <11 ’1 “l !'1

~51 0'01 1 '7“. [1

-ll 1 1 ' 1‘

1: o I

 

 

1000
Node Number

Figure 1.13: Variations of various turbulence and ﬂame variables along the stoichio-
metric mixture fraction iso—surface at t/To = 2.0 for case I-3; (a) temperature,
(b) scalar dissipation rate, (0) OH production rate, and (d) tangential strain rate.

39

 

 

 

5000

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5(3)
11 4000"- 7:11- 9.
= 1‘1 8
F4 0:. '
x ‘ 0!: V3
«”3000“- W13
'?1 " m {E
N " . .1
E ’ .. 8 3‘1 1,
-2111- - .
.- 5163 I: 113110 lI'11‘h‘11.’
- - r: 1 ' o .I :1 :1 1
100011511 11 .‘1'11 ”18,11. 111119131
”1,11 11.. 1101‘ 11115111!
"I! '11-”: IlI‘ilI I“ II Iii! Id: ‘1' 11;” VIII "H
V J 11 J1 . ' 1 1 1 4
O 1000 2000
Node Number
21° 6000_
3: 3(b) ‘En C?
—- 4000- 0' 1 0
'3'? < m g 1 c?)
N O O I
2000- 0 I
E g ,‘Im I i g 1
— 1l 1 11%:[1 l 1111 R11
§ 01111 ’1' l1 ' 11 1
. 1"; 1' 1.
11. U 1
'3 -2000- 33) 1 .
m“ 4000}
a
2
“-1 -6000

1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 1
500 1000 1500 2000
Node Number

Figure 1.14: Variations of (a) total kinetic energy, and (b) pressure-dilatation term
in the kinetic energy equation along the stoichiometric mixture fraction iso-surface
at t/To = 2.0 for case LB.

40

 

 

 

 

 

 

 

 

 

 

 

 

;(a>
1000?
.7. -
o 800 - Q
ﬂ +3
O
1600 - c2
6'. \m I!
- < m o 8 i !
Ci 1 . 0' W m a J 1
200: o o . _i‘-.,.,_ﬂ Hi!
‘i‘ $1 “0]; ""‘xéi’i Nil
:1", it“ {1‘ ' 2% -'J‘. «Jiii‘l U| :i ilk
0“ “II, ~AJ “/11". v . . :7", 1‘: ”Kb"!
100 2000
Node Number
'c _
—* ~(b) a
X 2000 - \ i o
N:_ F m ‘ g
m * 0'
h" " o
- U)
E IOOOE < my: i
H - 8 8 5"! £ .51
.0 r f mi LU) ”if Hill I‘
E 0 t" L“ ""I“""* “ ~ 1: 115$}w y» L,» ,
g ! I1
”“000 i 1600‘ i ' ‘20'00‘
Node Number

Figure 1.15: Variations of (a) vorticity magnitude, and (b) baroclinic term in the
vorticity equation along the stoichiometric mixture fraction iso-surface at t/To = 2.0
for case I-3. ‘

41

 

0.5

 

  

 

 

 

 

 

 

 

 

 

 

(a)
1; i
'5 , i
0:. A —r-— i w - '4':—
>. l l l
I - l
(i) (ii) (iii) (iv)
”(’50 ‘ 0:5 ll
X/LX
0.5
(b)
I x ‘L
go .4 c ~ «,-
) ‘1‘ ;
\ \ 4'
'0-50 ‘ of? l
X/Lx
0.5
»(C)
J * “ V )-
L/ \ ﬁg“? y
2;.
‘050 ‘ 0:5 ‘ ll
X/Lx

5 and (c) t/To =1.0.

42

 

Figure 1.16: The location of Lagrangian particles at different sections on the stoi-
chiometric mixture fraction iso-surface or perpendicular to this surface for case X-3;
(a) t/To = 0.0, (b) t/To = O.

 

 

 

 

 

 

 

 

 

 

 

  

 

 

v ”'1 ..I.I.;;
$1200- '11'
a l ‘: a"
Q 600' . :zzdxovti'
H ; 'vv
0 ‘ o? i‘ 1f5‘ ‘ ‘ i
t/‘Co
0.6»
= , (b)
c '; o“...
:5 0.5 ..000:' 'vvt'O‘a’
E 0.4- ..:=::""“" -.
: Immiﬁaﬁgﬂn O --"-.11
0.3L
E. g
"is 021
2 ‘ ;
040* ‘ofs i ”1:5“ '5
W0
'2‘ 2000
3 , (C)
x 1
11.1500" .00. . ‘
.2. : .’ .
{31000} .fv'f', ,;
9‘ * 1 1V
E soof
’ ........ .1l
m i i 1 l 1 1 l . ~ I
00 0.5 1 1.5 2
t/To

Figure 1.17: Temporal variations of the Lagrangian particle properties; (a) temper-
ature, (b) mixture fraction, and (c) strain rate. Each symbol represents a particle
initially located at different positions in the domain. Particle A is represented with
,particle B with “V”, particle C with “0” ,particle D with “+” and particle E with
“I”. Initially the relative distance of the particles with respect to the stoichiometric

‘6. ”

mixture fraction surface is the same.

43

2400

 

 

 

 

 

 

 

 

 

 

1 (a) 0.2<Z<0.3363
1

E1800

A .

A

{-1

V 1200 -— lSt(t)dt<6.0E4
: ------- 6.054 < lSt(t)dt < 1.255
. - ----------- 1.255 < lSt(t)dt < 1.855
1 1.8E5 <lSlt(t)dt J

6000.5 1 1.5 2
V10
2400

:(c) 0.3717<z<0.5

$31180“L ““=’-:-::_;\—.-I.:.:.:.:.:, ---------

A ~~~~~~

A _________

H .

V 1200 -— lSt(t)dt<6.0E4
: ------- 6.054 < lSt(t)dt < 1.255
- - ----------- 1.255 <lSt(t)dt < 1.8E5
1 - ------------------ 1.8E5 < lSt(t)dt

6000.5 1 1.5 2

Wu

 

 

 

 

 

2400
~(b) 0.3363<Z<0.3717
51800 i """ ~~:.::-~ZZ:.:.I.II: “““““
A Z ____________________________
A y ................
{-1 r
V 1200 -— ISt(t)dt<6.0E4
: ------- 6.0B4 < lSt(t)dt < 1.255
. - ------------ 1.255 < lSt(t)dt < 1.855
-------- .' 1.855 <lSt(t)dt
6000.5 1 1.5 2
We

 

 

 

 

 

. . , O.2<Z<0.3363
5 _: _ ............ 0.3363<Z<0.37l7
_ ...... ; .......... o .37l17<Z<0.5 .
00.5 1 1-5 2
tl’l:0

Figure 1.18: Temporal variations of the conditional mean temperature conditioned
on the strain rate and mixture fraction; (a) low mixture fraction range, (b) stoichio-
metric mixture fraction range, (0) high mixture fraction range. ((1) Percentage of the
difference of the conditional average values of the particle temperature.

44

 

)

st

 

P(<T Iz

 

 

 

‘660LL ‘ i2'oo ‘ ’ 118106 ' ' 2400
T [K]

 

 

 

 

 

d‘ ' 366‘ 1110'0‘0' '11'5'0'0‘ ‘22'0'0'0‘ 12500
St [s' 1X10-

Figure 1.19: Temporal variations of the probability density function (PDF) of the
temperature and the strain rate conditioned on Z = Zstoz‘ch at t/To = 2.0. The
crosses refer to Eulerian (grid) data, the circles refer to Lagrangian (particle) data;
(a) PDF of temperature, (b) PDF of strain rate.

45

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ l _
111800 51800
S E
L

E1200 E31200
S. g .
E E -
0 o t
E- i- .

600 600

(-

£1800” g 1800
E - E
S 1200. E 1200
=- a.
E E
9 o
P » 1-

600 600

0.5 0 0.5 l
Mixture Fraction Mixture Fraction

Figure 1.20: The scatter plots of temperature in the mixture fraction space for dif-
ferent cases at t/To = 1.5; (a) case I-3, (b) case I-2 and (0) case H. (d) The scatter
plot of temperature for low strain rate case, case VIII-3. Solid lines show the initial
laminar proﬁles.

46

 

 

 

 

 

 

24_

A

A

[‘1-

V

l

.J

B

V l

¢\° :1 _ ,,,,,,, k ........ 0% air
-87: _——o——-- 37.5% air
-16 ‘ L4_'0.51,1 I 11

MixtureFracllon

Figure 1.21: The percentage of the difference between the laminar and average tur-
bulent temperatures at t / To = 1.5 in the mixture fraction domain for various fuel—air
premixing levels. The data are gathered from the Eulerian grid points.

47

YCO
n 0.06
7i 45 0.05

0.04

 

0.5

0.03

‘ 0.01

yle

 

 

YCO

H 0.06

1 0.05

 

0.04
0.03
0.01

 

 

 

0.5 '
X/Lx

Figure 1.22: Instantaneous contours of the CO mass fraction for (a) case I-3 and
(b) case IX—3 at t/To = 1.5.

48

 

 

 

 

 

 

A
O

 

= Da=5.04E-03 ..
————-———— Da=2.67E-03
i- ————————— . --------- - Da=1.04E-03_//’/ .

 

'7

OJ
C
I T 1

% of points with T < 1400 K
S 8

 

 

 

O
b

 

Figure 1.23: Temporal variations of the percentage of grid points in the vicinity
of the ﬂame surface (Z % Zstoz‘ch) with temperatures less than 1400 K; (a) various
Reynolds numbers, and (b) various Damkohler numbers. The data are gathered from
the Eulerian grid points.

49

CHAPTER 2

LARGE-EDDY SIMULATIONS OF
TURBULENT METHANE JET FLAMES WITH
FILTERED MASS DENSITY FUNCTION

2.1 Introduction

The well documented experimental data generated by the Sandia National Labora-
tory researchers and their collaborators for turbulent jet ﬂames [58, 59, 64] have been
widely employed for assessment of numerical models. Both Reynolds-Averaged Simu-
lation (RAS) and Large-Eddy Simulation (LES) models have been evaluated [65—68].
For example, Jones and Kakhi [69] used the RAS based probability density function
(PDF) method with linear mean square estimation (LMSE) and Curl’s coalescence-
dispersion (CCD) mixing closures, standard turbulence models and 4—step kinetics
mechanisms to simulate non-piloted non-premixed jet ﬂames at various jet speeds.
Their results indicate that at lower jet speed of 41m/s (ﬂame L), there is no sig-
niﬁcant extinction, but at higher speed of 48m/s (ﬂame B), there are some. In the
extinction-reignition region (15 S m/ D S 35) of the ﬂame B, the predicted tempera-
ture by the LMSE model is shown to be smaller than the experimental values, while
CCD model overpredicts the experimental data at the same region. Muradoglu et
al. [70] have conducted similar jet simulations with a ﬂamelet based chemistry and
interaction by exchange with the mean (IEM) mixing model and have found good

agreement with the low-speed equilibrium ﬂame L data.

50

The assumed PDF method with a reduced kinetics mechanism was used by Lan-
denfeld et al. [66] to simulate the Sandia’s piloted jet ﬂame (ﬂame D). The intrinsic
low-dimensional manifolds (ILDM) algorithm [71, 72] was implemented along with
reaction progress variables in these calculations. Additionally, transport equations
for the mean and variance of mixture fraction and species mass fractions are solved,
together with a Beta PDF for the mixture fraction and a Reynolds stress closure for
the turbulence. Generally, the results were found to compare well with the low-speed
piloted jet ﬂame D data.

Xu and Pope [65] have simulated the piloted jet ﬂames at various jet speeds
with a joint velocity-composition—turbulent frequency RAS/PDF model. The 16
species augmented me—thane mechanism [73, 74] have been implemented with the
in situ adaptive tabulation (ISAT) [75] algorithm for the chemistry. The Euclidian
minimum spanning tree (EMST) model [76], which treats the mixing locally in the
composition space, is used for the mixing model. Overall, the results were found to
be in good agreement with the experimental data.

The models based on LES have also been used for prediction of Sandia’s jet
ﬂames [67, 77]. Different subgrid-scale (SGS) models have been employed. One of
them is based on the ﬁltered density function (FDF) methodology [78]. This method
is the counterpart of the PDF method in RAS [4, 5, 78—81]. The fundamental
advantage of the LES/FDF is that it accounts for the effects of chemical reaction
exactly while calculating the resolved turbulence ﬁeld, thus allowing a more reliable
prediction of turbulent ﬂames. The literature on SGS closures via the FDF method
has been growing at a relatively fast pace since its ﬁrst introduction [78]. The scalar
FDF is considered in Refs. [82—85], the scalar ﬁltered mass density function (FMDF)
in Refs. [86—90], the velocity FDF in Ref. [91] and the velocity-scalar FDF in Ref. [92].
Some recent data on experimental validation of F DF are also available [93].

In the present work, the scalar FMDF methodology is employed as a SGS closure

51

in LES of turbulent methane jet ﬂames. The main objectives are (1) to further assess
the applicability and the extent of validity of the FMDF method for SGS closure of
complex turbulent ﬂames when local ﬂame extinction is important, (2) to investigate
the accuracy of various ways of implementing the mixing model, (3) to study the
effects of chemical kinetics on LES predictions for both near-equilibrium and non-
equilibrium ﬂames. The chemistry is implemented following two different approaches.
In the ﬁrst approach, all reactive species are obtained by direct solution of an appro-
priate kinetics mechanism. In the second approach, only the FMDF of the mixture
fraction is considered; the temperature and species concentrations are obtained from
a set of ﬂamelet tables. The tables are generated by solving the steady-state one-
dimensional opposed jet equations with detailed (GRI) mechanism. Considering the
steady-state ﬂamelet assumption in the second approach, it is expected that the ex-
tinction would not be predicted correctly in this approach. However, it would be
useful to evaluate the overall performance of the FMDF/ﬂamelet method for near-
equilibrium ﬂames and show its limitations in non-equilibrium ﬂames.

The Sandia’s “high speed” piloted turbulent methane jet ﬂames D and F [64]
are considered in this study. Flame D involves limited regions of local extinction,
while ﬂame F tends towards total extinction. The existence of different level of local
extinction in these ﬂames provides a good means of assessing the capabilities of the
models to predict realistic combustion systems. The geometrical conﬁgurations in
these two ﬂames is the same (see Fig. 2.1), but the jet inlet velocity in ﬂame F is
twice of that in ﬂame D. The formulation of LES/FMDF methodology is reviewed
in section 2.2, followed by section 2.3 which describes the consistency issues, the
parallelization algorithm, the implementation of chemistry, and the computational
parameters. Results obtained by the LES / F MDF for ﬂames D and F are discussed

in section 2.4, and the main conclusions are stated in 2.5.

52

2.2 Formulation

In the hybrid Eulerian-Lagrangian LES/FMDF methodology, the ﬁltered velocity
equations are solved together with the joint scalar FMDF equation. These equations
are described in two different sections. The variables are listed in the nomenclature
section. The resolved variables are obtained by application of the spatial ﬁltering

operator [94],
+00

01x, t>>e = / f(x’.t)H(x’,x)dx’. (2.1)

—00
where ’H denotes the ﬁlter function, (f (x, t)) 3 represents the ﬁltered value of the
transport variable f (x, t) and x = (:1:,y,z) denotes the spatial coordinates, with :1:
axis parallel to the jet and y, z perpendicular to :13. In variable density ﬂows it is
more convenient to consider the Favre ﬁltered quantity, (f (x, t)) L =(p f ) g / (p) 3. For

spatially & temporally invariant and localized ﬁlter functions, H(x’, x) E H (x’ — x)

with the properties[94], H (x) = H (—x), and ffooo H (x)dx = 1.

2.2.1 The Filtered Transport Equations

With the application of the ﬁlter function to the transport equations [95] one can

derive the following ﬁltered equations,

5W8 3(p>e<uzf)L

 

 

 

 

 

at + 8% = 0 (2-2)
8<p>ga(tuj>L + a<P>€<ZZf<W>L = _%%€ + 3:32;” __ 19%,]- (2.3)

a “i a a J? 6M?
3(p)ea(t¢ )L+3(p>e(8::f<¢ )L 2 _ (milk 3332' +(psa>g, a=1,2,...,0. (2.4)

In these equations, the scalar (composition and energy) ﬁeld is denoted by (150, E

Ya, a =1,2,...,N3, 450 E h 2 25:11:01qu in which ha 2 hg +f71~gopa(T’)dT'.

53

Equations 2.2-2.4 are closed by the following constitutive relations

N3

0». z <p>eRO<T>L Z 59,,—,>—L- (2.5)
1 a

u- 811- u.
(591M115 (82,33).” 2;:L—§61j%Z—L), <u>L=Pr<k/op>1. (2.6)

 

 

a 3(¢a>L 1 k
J- z —- D , D = —
< 2 >6 (10M >L 33:2, < >L (

 

 

(10>eL6 l” (2'7)

In Eqs. 2.2 — 2.4, the hydrodynamic SGS closure problem is associated with[96] 721- =
(P>e((Uin)L - (Ui)L(Uj)L) and M? = <P>€((ui¢a>L - (Ui>L(¢a)L) denoting the
SOS stresses and the SOS scalar ﬂuxes, respectively. In reacting ﬂows, an additional
model is required for the ﬁltered reaction rates (pSa)g = (p) [(50) L- It will be
shown in the next section that both (p) g and (Sa) L are determined exactly with the
knowledge of the FMDF. For ’12-], the model used by Jaberi et al. [97] is considered:

T..- = -2(p>m (<8.- 5>L — $8.05..) + $01055... (2.8)

where (19,-j) L is the resolved strain rate and E := [(119) Mu?) L — ((119) [jg/((119) L) [I],
Vt = CRAgx/E, u? = u,- — U,- (U, is a reference velocity in the 3:,- direction and the
subscript 3’ denotes ﬁltering at the secondary level with a characteristic size larger
than that at grid level). We have found the performance of this model to be better
than the Smagorinsky type closures [98—100], and we did not even have to implement
the “dynamic” procedure [101, 102] for evaluation of the model coeﬂicients.

For the closure of the subgrid mass ﬂux Mg], the model most often used in LES

of non—reacting ﬂows [103—105] is used

a<¢a>L
3:132' ’

 

Mgr: -—(p)th a=1,2,...,o. (2.9)

54

It must be indicated here that this model is not used directly in the FMDF approach
but modeled FMDF transport equation is constructed to be consistent with it as

discussed below.

2.2.2 The Filtered Mass Density Function (FMDF)

The scalar ﬁltered mass density function (FMDF) is defined as[84, 85, 88, 97]

+00
FL(1/2,x; ,t) E [00 p(x’,t)( [1/J,¢(x',t)]H(x’— x)dx' (2.10)
< 11p. ¢(x. t>1 = 61¢ — «Ax, 01 2 II 610.. - 5.1x, 81 (2.11)

021
where 6 denotes the delta function and 1p denotes the composition domain of the
scalar array ¢(x, t). The term ([¢,¢(x,t)] is the “ﬁne-grained” density [5, 106],
and Eq. 2.10 implies that the F MDF is the mass weighted spatially ﬁltered value
of the ﬁne-grained density. To ensure that the FMDF has all the properties of the
PDF, we will only consider “positive” ﬁlter functions [107] for which all the moments
foooo me(:r)da: exist for m 2 O. The following exact deterministic equation describes

the variations of FMDF in space and time

 

 

 

 

BFL 8<ui>LFL _ 8 (9C 82 a¢a 695,8 A]
7373— + (9135 — (7333—2 <pD3_$i>e _ (Moat/J5 [<pD 3332' 3902' |¢>eFL/p
a , — ,- F a S}, F

In Eq. 2.12, (AIB)g denotes the ﬁltered value of the variable A, “conditioned” on B,
and the hat is used for the quantities which are dependent only on the scalar ﬁeld.
In Eq. 2.12, the last term on the right—hand side represents the effects of chemical
reaction and is in a closed form. The second and the third terms on the right-

hand side are unclosed. They represent the effects of SOS mixing and convection,

55

respectively. The convective ﬂux is modeled here as

3(FL/<p>e)

l<w|¢>e - (“ilLlFL = -(p>eDt 82:,- ,

Dt = Vt/Sct (2.13)

where Vt is the SGS viscosity and is to be determined by hydrodynamic LES and
Set is the SGS Schmidt number. The ﬁrst “moment” of Eq. 2.13 recovers the model
given in Eq. 2.9.

The closure for the SGS mixing can be via any of the ones used in RAS/PDF
methods [108—110]. The simplest one is the LMSE model, [106, 111]

 

 

 

82 a¢a agbﬁ A _ a
810.18% [< Dam, 33:2. l¢>€ FL/p] — 15$; lQmWa - <¢a>LlFLla (2.14)

where Qm(x, t) is the frequency of mixing within the subgrid. This frequency is
modeled as 9m = Cg((D) L + Dﬂ/(Aﬁ), where Cg is the mixing model coefficient
and A H is the characteristic size of the ﬁlter. The mixing model coeﬂicient Cg
represents the velocity-to—scalar timescale ratio [65]. It needs to be either empirically
speciﬁed [65, 112—118] or directly obtained via a dynamic model [89, 90, 119-121].
In the present work, a range of constant empirical values and two different dynamic

models are employed to calculate Cg:

1. In this approach, the LMSE model is used in its standard form, while the
mixing model coefﬁcient Cg is assumed to be constant[112—1l4]. Here,
Cg = 5 is used as the base value for the reference case since it generates
the closest predictions to the experimental data. However, other values of

Cg are also considered.

2. In this approach, the mixing coefficient Cg is computed as a function of

time and space by a dynamic method. The dynamic method is based on

56

the generalized subgrid variance, 0?, = ((031)) L — (¢(a))% and the total

subgrid scalar dissipation rate, 60,. The latter is deﬁned as

a a 68a

(2w)

6a = 2<P>£((D>L+Dt)

 

 

0... = 09102)]. + 00/1185)

where the subscripts in parentheses are excluded from the summation con-
vention. The model for ea in Eq. 2.15 is obtained by assuming that the
dissipation term balances the production term in the generalized subgrid
variance equation citepCJGP98,raman1. By using 9m, 0?, and 60,, the

model coefﬁcient can be computed locally as

A%;V(¢(a))L ' V<¢(a)>L
CO = 2 7

0a

 

a=LZ ”N, (2w)

Evidently, this formulation allows different coefficients for different species,
meaning “Differential diffusion” effects may be included. However, the
computational time needed for the calculation of Cg could become exten-
sive when realistic chemistry models with numerous species are employed.
Therefore, to reduce the computational overhead, a modiﬁed version of

Eq. 2.16 is considered in which mixture fraction variable Z is used,

C =A§.V<Z>L-V<Z>L
“ <z2>L—<z>i '

 

(2N)

In the numerical implementation of the model, the gradient of the favre-
ﬁltered mixture fraction V(Z) L is calculated from the ﬁxed ﬁnite difference
grid points with a purpose of decreasing the numerical noise, while the

subgrid scalar variance is determined from the Monte Carlo particles. An

57

additional statistical averaging over the homogeneous (azimuthal) direc-
tion is also performed to prevent numerical instabilities. In the discussion

below, this dynamic model is referred to as DM1.

3. In this approach, a dynamic method proposed by Raman and Pitsch [90]
is employed for calculating the coefﬁcient Cg. This model is referred to as
DM2. In the DM2 model, the subgrid scale mixing frequency am is again
directly evaluated from the subgrid scalar variance and the dissipation
rate. However, in contrast to DM1, the subgrid scalar variance in DM2 is
obtained by a model. The equations [90] describing the conserved scalar

variance 2772 and dissipation rate X are
2"? = CZAfHVZ - VZ, (2.18)

x = 2(13 + D,)VZ . v2, (2.19)

where the operator “(If)” denotes the favre—ﬁltering operator (. . .) L- Based
on these equations, the scalar mixing timescale 76!) (which is the inverse of

subgrid mixing frequency (1 / 9m» is computed as

CzA2

—~————, (2.20)
2(1) + DT)

where the model coefficient C Z = 2/ Cg is obtained by a dynamic proce-
dure [120].

The IBM model has some limitations [108, 109] which are not discussed here, but
per results obtained in previous studies [5, 78, 109, 110, 122—127], it can be safely
indicated that while the IBM is not quite satisfactory in RAS/PDF, it functions

reasonably well in LES / FMDF, and its numerical implementation is computationally

58

convenient.
With the closures for the SGS convection and mixing, the modeled F MDF trans-

port equation may be written as

 

871;; + W = 5% (p),((D)L + 00% + (2.21)
3 8[5:01le
59—4); l9m(¢a _ <¢a)L)FLl " 87%: °

The transport equations for the SGS moments can be obtained by direct integration
of the above FMDF equation over the composition domain. The equations for the

ﬁrst subgrid Favre moment, ((00,) L is identical with that in Eq. 2.4.

2.3 Numerical Solution

The numerical method used for LES/ F MDF has two components. For the solution
of the hydrodynamic ﬁeld, a high-order accurate ﬁnite difference (FD) method which
has proven effective for LES [4] is employed. The FD discretization procedure is
based on the “compact parameter” scheme [128] which yields up to 6th order spatial
accuracy. All the ﬁnite difference operations for obtaining the ﬁltered values are
performed on ﬁxed and uniform grid points.

The SGS empirical “constants” are C], CR, Set. Based on our experience, in
which a variety of different ﬂows (2D and 3D, constant and variable density, different
chemistry schemes, etc.) are considered, we have determined CI z 0.01, CR z 0.02,
Set z 0.4 — 0.7. Interestingly, the range of some of these values is the same as that
typically used in equivalent models in RAS [79]. The magnitudes of the molecular
parameters are the same as those typically used for hydrocarbon-air ﬂames [129, 130].
For methane (k/cp)L % 2.58 X 10’5 ((T)L/298)0‘7, Pr m 0.75 and 010a is speciﬁed

through polynomial ﬁts as functions of the temperature.

59

2.3.1 The Lagrangian Monte Carlo Method

The most convenient means of solving the F MDF transport equation is via the “La-
grangian Monte Carlo” procedure [5, 131]. The basis of this procedure is the same
as that described in recent papers [70, 84, 88, 91, 97, 132—134]. Therefore, here only
the basic features of the procedure are described. With the Lagrangian procedure,
the FMDF is represented by an ensemble of computational “stochastic elements” (or
“particles”) which are transported in the “physical space” by the combined actions
of large scale convection and diffusion (molecular and subgrid). In addition, trans-
port in the “composition space” occurs due to chemical reaction and SGS mixing.
In doing so, the notional particles evolve via a “stochastic process,” described by the

set of stochastic differential equations (SDES) [135, 136]

dX,-(t) = D,(x(t), t)dt + £(X(t), t)dW,-(t), d¢;'(t) = 72a(¢+, t)dt. (2.22)

where X,- is the Lagrangian position of the particles, D and 8 are known as the
“drift” and “diffusion” coefficients, and W, denotes the Wiener-Levy process [137].
452; denotes the scalar value of the particle with the Lagrangian position vector Xi.
Eq. 3.28 deﬁnes what is known as the general “diffusion” process [136-138]; thus the
PDFs of the stochastic processes (X,(t), ¢3’(t)) are governed by the Fokker-Planck
equation. A comparison between the standard Fokker-Planck equation corresponding
to Eq. 3.28 with the FMDF equation (Eq. 2.21) under consideration identiﬁes the

parameters of Eq. 3.28,

 

1 al<P>e(<D)L + Dtll
Me 3372' ’

 

5 E \/2((D>L + Dt), Di 5 (”ilL + < (223)

Ra E —Qm(¢;— _ <¢a>Ll + §a(¢+)-

60

With this analogy, the F MDF is represented by an ensemble of Monte Carlo particles,
each with a set of scalars (taunt) = ¢a(X(")(t),t) and Lagrangian position vector
X(") . A splitting operation then can be employed in which the transports in physical
and compositional domains are treated separately. The simplest means of simulating
the spatial transport in Eq. 3.28 is via the Euler—Maruyamma approximation [139].
The transfer of information from the ﬁxed ﬁnite difference points to the location of
the Monte Carlo particles are conducted via (fourth and second order) interpolation.
The ﬁltered scalar quantities are calculated by weighted averaging of the particle
values over space with average volumetric size of A E- Ideally, A E has to be very

small.

2.3.2 Consistency

As stated before, the equation governing the ﬁrst subgrid Favre-moment of the scalar
45a obtained from the FMDF equation is identical with the ﬁltered scalar Eq. 2.4,
indicating that the ﬁltered temperature, scalar and density may be alternatively
obtained from the Eulerian, ﬁnite difference (FD) or Lagrangian, Monte Carlo (MC)
solutions. This implies a mathematical consistency between FD and MC parts of the
hybrid scheme. However, due to ﬁnite grid size in FD and limited number of MC
particles, consistency may not be achieved in practice. In the following, the possible
sources of inconsistency are identiﬁed and conditions leading to a consistent solution
are discussed.

Four preliminary simulations with constant values of C4, = 2Cg = 8,16 and
A E = A (A is grid spacing) and AB = 2A have been performed to study the
consistency of LES / F MDF for conditions of Sandia’s piloted jet ﬂames. The reaction
is turned off in these simulations but the variable density / temperature effects are still
important due to strong pilot at inﬂow. The instantaneous temperature proﬁles for

cases with A3 = 2A and A E = A are compared in Fig. 2.2. It is observed that

61

the difference between the MC and FD solutions becomes negligible when A E = A.
Moreover, the mixing model coefﬁcient, which does not appear explicitly in the F avre-
ﬁltered equation, does not seem to have a signiﬁcant effect on the ﬁltered temperature
when A E = 2A, even though the temperature proﬁles seem to be slightly more
diffused for higher Cg (compare results for Cqb = 16 with those for C¢ = 8 in Fig.
2.2a). For the reacting simulations considered in the next section, A E was chosen to
be equal to 2A. In all of these simulations we have found the difference between the
FD and MC solutions to be less than 3 percent for instantaneous temperatures and

even lower for time averaged values.

2.3.3 Parallelization

As mentioned before, in the hybrid LES/FMDF methodology the ﬁltered continu-
ity, momentum and energy equations (Eqs. 2.2-2.4) are solved by a ﬁnite difference
(FD) method over a ﬁxed Eulerian grid system. On the other'hand, all scalars
are obtained from the FMDF by solving its transport equation with a Lagrangian
stochastic Monte Carlo MC method. The employed MC method involves grid-free
particles interacting with the ﬁxed background Eulerian grid. However, there is no
inter-particle interactions, suggesting that MC calculations are potentially efficient
in a parallel environment. However, the particle transport and mixing require in-
teractions with the F D ﬁeld. Moreover, the source term in the FD energy equation
should be calculated by averaging the particle reaction source terms. Consequently,
the MC calculations can not be performed independent from the FD calculations.
To obtain a good statistical representation, a signiﬁcant number of particles are
required within each FD cell. Typically, about ten particles are required for each
FD grid for A E = 2A. A typical hybrid (FD — MC) simulation with ten particles
per cell is about 5 times more expensive than its corresponding FD simulation with

no MC particles. Hence, the usage of “complex” multi-step kinetics mechanisms in

62

these simulations would only be possible by proper application of parallel processing

techniques. Two different parallelization schemes may be used:

1. In the ﬁrst approach, all processors are solving the same FD equations for
the whole domain. Consequently no speed-up is gained in the FD part.
However, since the FD calculations are small fraction of total calculations
in the hybrid scheme for reacting simulations, it is theoretically possible to
achieve relatively high parallel efﬁciencies. In the MC part, particles are
equally divided among the processors. Since the particles are not trans-
fered between the processors, there is no communication load for the MC
calculations and the load distribution is exactly uniform. Inter-processor
communications are only required in the averaging process , where a local
ensemble average value of particles is calculated on each FD cell. Since
each processor is dealing with particles traversing the whole domain, global
communications are required for calculating the summation and broadcast-
ing the outcome back to the processors. For n f number of FD grid points
and up number of processors, the averaging calculations require a sum-
mation of up arrays of size n f and 2 x n f x np times transfer of data
between the processors. On distributed memory architecture, this amount
of data transfer can be prohibitive, since the network speed is normally
much slower than the memory transfer rate. The main drawback is the
linear increase in the number of operations and communications with the

number of processors.

2. The second approach is considerably more complex than the ﬁrst one.
However, it is proven to be scalable and much more efﬁcient. In this ap-
proach, both the FD domain and the MC particles are distributed among

the processors. A limited number of communications are required when the

63

subgroup of particles are located in the boundaries of the FD subdomain
deﬁned for each processor. With this, the communications required for
the averaging process described in the ﬁrst parallelization method is elim—
inated. However, since each processor needs to keep track of the particles
located in its own FD subdomain, particles that exit the subdomain have
to be transferred to its respective processor. For keeping the same load on
every processor, it is necessary to have the same number of particles per
processor at all times. Indeed, our analysis indicate that about one percent
of the particles traverse the boundaries of the subdomains ate each time
step, suggesting very little communication load. Limited communications
are also required for calculating the ensemble averages on the boundary
points. Inter-processor communications are also required for interpolating
a FD quantity on a particle location and for differencing and ﬁltering FD

operations.

To evaluate the difference between the above parallel processing methods, a
benchmarking simulation with 1.2 million FD grid points and 7 million MC particles
was conducted. Single-processor simulations were not possible due to the hardware
structure, thus the speedups reported are relative to a 2—processor simulation. The
speedups obtained for the 4-, 8-, and 16—processor simulations are plotted in Fig.
2.3. For 4- and 8- processor simulations, superlinear speedups are obtained, since
the problem size for each processor becomes smaller and the higher speed cache is
used more efﬁciently. For the 2—, 4-, and 8—processor simulations, only one processor
per node is used. However, for the 16—processor simulation, 2 processors per node
are used, which have to share the memory and network bandwidth, resulting in a
sublinear speedup in this case. It is clear that despite apparent complexity of the

second parallel processing method, it is a very efﬁcient method.

64

2.3.4 Chemistry

In quantitative comparison with laboratory data, the role that the chemical kinetics
model plays may become important. In this work, the chemistry model is based on
(i) non-equilibrium (ﬁnite—rate) and (ii) near-equilibrium models.

In (i), the ﬁnite rate kinetics effects are modeled with a one—step global mecha-
nism [140], or a 12-step reduced mechanism. In (ii), the transport equation of FMDF
for the mixture fraction is solved together with a set of ﬂamelet tables, generated by
laminar opposed jet ﬂame simulations and detailed kinetics model. All other thermo-
chemical variables are constructed from the ﬂamelet data. Additionally, in (i), the

transport equation for the sensible enthalpy (hs = fl?) cp(T')dT’) is solved

5(p)e(hs)L 8<P>£<ui>L<hs>L (9(th 5M?

 

 

 

 

= — _ 0
where
(J2 )3 N L8<Op>L (9333' 7 M2 — <p>th 81:3 (225)

and the term (pSahg)g is obtained from FMDF and Monte Carlo particles. In (ii),
a transport equation for the (E) L = (RT) L is solved (in the ﬂamelet table RT is a
function of mixture fraction). This equation, as derived by multiplying the modeled

FMDF equation by E and integrating over the mixture fraction space, is

3<p)e(E)L 3<p)8(uz')L(E)L _3<J§7)£ _ 3M?

 

 

 

 

at + 33;, = (91,2. 8.2:,- + ([2)ng [(ZG)L - (Z)L(G)Ll
(2.26)

where
G = $1315), (J57) ” yieénagii’; M3 = -<p>2D16<E> (2.27)

6332' O

The reduced mechanism considered in (i) is the 12—step mechanism of Sung et al. [141].

65

This mechanism is developed from the GRI 1.2 detailed mechanism and involves 16
species (H2, H, 02, OH, H20, H02, H202, CH3, CH4, C0, C02, CHZO, C2H2,
C2H4, C2H6, N2). It contains more unsteady intermediates than the conventional
4 and 5-steps mechanisms, and it has proven effective in a range of applications
including auto—ignition, laminar ﬂame propagation [74], and variety of premixed,
non-premixed and partially-premixed laminar opposed jet ﬂames [32]. In all these
applications, the 12—step results were found to be almost indistinguishable from the

GRI results. The detailed mechanism in (ii) is based on GRI—3.

2.3.5 Jet Parameters and Boundary Conditions

Figure 2.1 shows a schematic view of the Sandia’s piloted methane jet ﬂame, and the
coordinate system used in our simulations. For these simulations, a FD mesh with
160 x 161 x 161 grid points was considered for a domain of 16 x 12 x 12 jet diameters
in the x, y, and 2 directions, respectively. The approximate number of MC elements
per each FD cell is 8 and A E = 2A. The main jet composition is 25% CH4 and 75%
air with a Reynolds number of 22400 in Flame D. Flame F has the same parameters,
except the jet speed or Reynolds number, which is doubled. Detailed speciﬁcations
of the ﬂames and measurement methods are available elsewhere [142, 143], and are
not discussed here. However, for convenience the main parameters of ﬂames D and
F are listed in table 2.1.

Non-reﬂecting boundary conditions [144] are considered for the outlet boundary
and zero—gradient conditions are used for the lateral boundaries. For the inﬂow,
non-reﬂecting boundary conditions are used with a prescribed velocity based on ex-
periment. It should be noted that the temperature is not measured at the nozzle
exit, and :13/ d = 1 is the closest distance to the nozzle where such measurements are
conducted. The sensitivity of the model predictions to the uncertainty in the pilot

boundary conditions is an important consideration [145], specially with regards to

66

ﬂame F, which is at a state very close to global extinction. Here, we use the values

close to those suggested by Barlow [142].

2.4 Results and Discussion

The isosurfaces and isocontours of the instantaneous vorticity magnitude, and tem-
perature for ﬂame D and ﬂame F simulations are shown in Figs. 2.4 - 2.5. These
ﬁgures indicate that there is a transition to turbulence at m/ D N 5 in both ﬂames.
Nevertheless, the ﬂow ﬁeld in ﬂame F appears to be more turbulent due to increased
jet speed, and lesser overall heat release effects particularly when 12-step chemistry
is employed. The results (not shown) for ﬂame D indicate that the ﬂow ﬁeld is not
very different when ﬂamelet and 1-step chemistry are used but the turbulence is
stronger when 12-step chemistry is employed. This is due to damping effect of reac-
tion on turbulence, which is more signiﬁcant in our 1-step and ﬂamelet simulations.
The temperature isosurfaces in ﬂame D, as obtained by 1-step and ﬂamelet models
(not shown) seem to be continuous without a noticeable sign of extinction. This is
consistent with the nature of 1-step reaction and ﬂamelet models which do not allow
signiﬁcant or any local extinction. However, the temperature isosurface as obtained
by the 12-step mechanism in ﬂame D (Fig. 2.5a) exhibits discontinuities at :5/ D R: 5
and 33/ D m 15; suggesting limited (local) extinction at some regions of the ﬂame. For
ﬂame F, the local extinction is much more signiﬁcant and clearly visible in Fig. 2.5b,
where the temperature isosurfaces are shown to be severely broken after :c/ D m 5
when 12-step model is used. This is consistent with Fig. 2.4b that shows stronger
turbulent ﬂow in ﬂame F and lesser effect of combustion on turbulence due to en-
hanced local ﬂame extinction. In contrast, the local extinction is not so important
when l-step model is used or is not present when ﬂamelet model is used. These are

also consistent with the results shown below. It is noted here that the images in the

67

following sections of this dissertation are presented in color.

2.4.1 Turbulence and Flame Statistics

Figure 2.6 shows the radial variations of the time-averaged mean axial velocity and
RMS of axial velocity for ﬂames D and F at 55/ D = 15. In this ﬁgure, the LES
predicted Favre-averaged results obtained with ﬂamelet, 1-step and 12—step chemistry
models are compared to Favre-averaged experimental data. The measurement error
is estimated to be below 5% for the mean velocity and about 10% for the RMS [143].
In general, the agreement between the calculated and the measured mean values
is good for both ﬂames for all of the tested chemistry models, suggesting that the
chemistry effects on the mean axial velocity is not signiﬁcant.

For both ﬂames, the predicted RMS values are lower and higher than the exper-
imental values at T/ D R” 1 and x/ D m 0, respectively. The agreement between the
numerical and measured values is better at r/ D > 1. For ﬂame D, the predictions
with 12-step model are higher at r/ D < 1 while the predicted RMS values with 1-
step, ﬂamelet and 12-step models are close at r/ D > 1. This is understandable since
in the jet core region the effect of combustion on turbulence is more signiﬁcant. In
this region, the turbulence damping is less when 12—step model is used. The 1-step
and ﬂamelet models predict higher heat release and mean temperature (Fig. 2.7a).
Similar trend is observed in ﬂame F (Fig. 2.6d). For this ﬂame, the predicted values
by the ﬂamelet model are the lowest due to highest heat release effects. The results
in Fig. 2.6 are consistent with those shown in Figs. 2.5 and 2.7.

The favre—averaged mean values of the temperature for ﬂames D and F are shown
in Figs. 2.7a and 2.7b, respectively. The measured temperatures in ﬂame F are much
lower than those in ﬂame D; at 33/ D = 15, the peak temperature is about 1100K
in ﬂame F compared to 1750K in ﬂame D. Nevertheless, the radial variation of the

mean temperature is observed to be reasonably well predicted by the LES/FMDF

68

when 12-step model is employed. The predicted mean temperatures are higher than
the corresponding experimental values for both ﬂames when 1-step or ﬂamelet model
is used. The 1-step model is based on an irreversible and fast reaction that gener-
ally overpredicts the ﬂame temperature, particularly in the rich side of the ﬂame.
However, the mean temperatures calculated by the ﬂamelet model are higher than
those via 1-step in ﬂame F, since the ﬂamelet model does not allow extinction. These
results show the importance of chemical kinetics model particularly when ﬂame F is
simulated. It is possible to improve the 1-step model predictions by adjusting the
reaction parameters. However, we have decided to use the same original values for
these parameters.

Figs. 2.7c and 2.7d show the predicted and measured values of the RMS of tem-
perature. Overall, the computed results for ﬂames D and F are not very different
than the experimental data in the jet core region when 12-step model is used. The
difference is much more signiﬁcant away from centerline at r/ D > 1.5. This can
be attributed to the differences in the physical structure of turbulence. The results
shown below indicate a better comparison between experiment and LES when tem-
perature RMS is plotted in the mixture fraction domain. The RMS of temperature
is affected by the combined effects of turbulence and heat conduction on one hand,
and the chemical heat release on the other.

The axial variations of the mean temperature, mixture fraction and axial velocity
along the jet centerline (not shown) are found to be in good agreement with the
experimental data for both ﬂames D and F when 12-step model is used. The 1-
step and ﬂamelet model predictions are less accurate, as expected. Also, the mean
temperature and mixture fraction proﬁles as obtained from the ﬁnite difference part
of LES/FMDF are found to be close with those calculated from the Monte—Carlo
part, indicating the insigniﬁcance of the numerical error.

Radial proﬁles of the C02 mass fraction for ﬂames D and F as obtained by the

69

12-step model are shown in Fig. 2.8a. There is again good agreement between the
computed and measured values in both ﬂames, particularly at the jet core region. The
agreement is less away from the centerline. Again, this can be primarily attributed
to the differences in the turbulence structure. The radial proﬁles of the intermediate
species H2, CO and OH in Figs. 2.8b, 2.8c, 2.8d also indicate a good agreement
between LES/FMDF and experiment when 12-step model is employed. The results
for other species (e.g. CH4 and 02) are similar to those shown in Fig. 2.8 and are
not shown. In comparison with the experimental data, the LES/FMDF predictions
with 1-step and ﬂamelet models exhibit considerable differences, specially for ﬂame
F.

The radial variation of the mean heat release in the energy equation at :1: / D = 7.5
as calculated by the ﬂamelet and 12—step models are shown in Fig. 2.9. The results
at 27/ D = 15 exhibit similar trends and are not shown. For ﬂame D, the LES / FMDF
results for 12-step and ﬂamelet models are relatively close which is consistent with
the isosurfaces and statistics of the temperature. However, for ﬂame F, the predicted
heat release with ﬂamelet model is substantially higher than that with the 12-step
model. This is consistent with the mean and the isocontours of temperature in Figs.

2.7b and 2.5b.

2.4.2 Compositional Flame Structure

The measured and computed conditional average and RMS values of various ﬂame
variables conditioned on the mixture fraction, in Figs. 2.10 and 2.11, show the struc-
ture of the piloted jet ﬂames D and F [142]. The numerical results are obtained
by the LES/FMDF with the 12-step chemistry model and Cg = 10. As commonly
done in experimental and computational studies [65, 89, 90, 112—115, 117—121], the
conditional averages and RMS are computed by using the data in a plane perpen—

dicular to the jet axis at several different times. Figs. 2.10a and 2.10c show that for

70

ﬂame D the peak values of conditional mean temperature are slightly overpredicted
by the LES/FMDF at both locations from the nozzle, which is consistent with the
underprediction of 02 mass fraction. The slightly higher predicted values of mean
temperature at both locations for ﬂame D in Figs. 2.10a and 2.10c are consistent
with less scatter in the temperature data shown below in Figs. 2.12c and 2.12d. This
could be due to SGS stress, SGS scalar ﬂux, or mixing models. The conditional RMS
of temperature, as shown in Figs. 2.10a and c, are also slightly underpredicted by
LES/FMDF.

Figs. 2.10b and 2.10d show the conditional mean mass fractions of the CO and
H2 species for ﬂame D. Evidently, the H2 concentration is very well predicted at both
lean and rich sides of the ﬂame away from the stoichiometric location, where it is over
predicted. The predicted CO mass fraction also agree well with the experimental data
on the lean side of the ﬂame, but they tend to be higher than the measured data on the
rich side. For ﬂame D, the reported conditional averages of the temperature, 02, CO
and H2 mass fractions exhibit trends similar to those appeared in the literature [67,
89,90,146,147]

The conditional mean and RMS of the temperature and the 02 mass fraction as
obtained by LES/FMDF with the 12—step kinetics model for ﬂame F are shown in
Figs. 2.11a and 2.11c. At x/D = 7.5, the LES/FMDF results are in good agreement
with the experimental data. However, the conditional average temperature is slightly
overpredicted on the rich side of the ﬂame at z/ D = 15, which is again consistent
with the underestimation of the 02 mass fraction. Interestingly, the (conditional)
RMS values of the temperature are well predicted at :r/ D = 15. The increase in the
magnitude of the temperature RMS in ﬂame F is due to stronger strain ﬁeld in this
ﬂame in comparison to that for ﬂame D. However, the higher level of local extinction
in ﬂame F leads to the conditional mean temperature in this ﬂame to be lower than

that in ﬂame D.

71

The conditional C0 and H2 mass fraction for ﬂame F are shown in Figs. 2.11b
and 2.11d. Again, the experimental results are well predicted for all mixture fraction
values at :r/ D = 7 .5, while there is a slight discrepancy between the experimental
and computational data for Z 2 0.45 at 33/ D = 15. Despite some uncertainty in the
ﬂow conditions, the predictions of the ﬂame F by the LES / FMDF are in overall good
agreement with the experimental data when 12—step chemistry model is employed.
The error in the conditional mean CO mass fraction and temperature on the rich
side of the ﬂame are consistent with those reported by others [65, 113, 115, 148, 149].

The scatter plots of temperature in the mixture fraction space as obtained by
the LES/FMDF with the 12—step mechanism for both ﬂames D and F are compared
with the experimental data in Figs. 2.12 and 2.13. The steady-state results obtained
from low-strain laminar opposed jet ﬂame simulations are also shown for compar-
ison. For ﬂame D, as Figs. 2.12a and 2.12b indicate, there is a limited scatter in
the experimental data at cc/ D = 7.5 and at ct/D = 15, suggesting that the local
extinction is insigniﬁcant [64]. For ﬂame F, there is conSiderable local extinction
and signiﬁcant scatter in the data at x/D = 7.5, and 15 (Figs. 2.13a and 2.13b).
Also at both locations, the LES/FMDF results for ﬂame F are considerably lower
than the laminar ﬂamelet results, while they are close to the laminar ﬂamelet results
for ﬂame D. Overall, the computed temperatures show a reasonably good agreement
with the experimental data at different locations. For ﬂame D, there are some ﬁnite-
rate effects in the experimental data that is not fully captured by the LES / FMDF.
This could be due to SGS turbulence and mixing models. At :c/ D = 7.5, a limited
local extinction is predicted by the LES/FMDF for ﬂame F when 1-step model is
employed, which is consistent with the isosurface contours of the temperature. Our
results (not shown) indicate that the temperatures are signiﬁcantly overpredicted by
the 1-step model in the rich side of the ﬂame.

Fig. 2.14 shows the predicted favre time—averaged mean and RMS of temperature,

72

and the predicted favre mean values of 02, CO and H2 mass fractions in ﬂames D
and F as a function of the mean mixture fraction in comparison to the corresponding
experimental data. These values are obtained by cross—referencing the radial mean
and rms values of the temperature and species mass fractions with the radial mixture
fraction. The computed data are obtained by the LES/FMDF with the 12-step
chemistry model and mixing model constant of Cg = 10.

For ﬂame D, both the peak and shape of scalar and temperature proﬁles for
mean and RMS are well predicted by the LES / FMDF (Fig. 2.14a). At axial location
of :c/D = 15, the RMS of temperature appears to be slightly underpredicted for
(Z): 3 0.125. Considering the highly sensitive and oscillatory behavior of ﬂame
F, the favre-averaged mean and RMS of temperature, and the favre-averaged mean
mass fraction of 02 are actually well predicted by LES/FMDF (Fig. 2.14b), even
though the mean temperature is somewhat underpredicted on the rich side of the
ﬂame. The RMS of temperature for ﬂame F exhibit similar trend to that for ﬂame
D. As shown in Fig. 2.14c and d, the mean mass fractions of CO and H2 are also
in overall good agreement with those of experiment for both ﬂames at 33/ D = 15.
However, there are some discrepancies in the CO proﬁle for ﬂame F at :13/ D = 7.5 on
the rich side (not shown), mainly due to ﬁnite rate effects. This is consistent with
the underpredictions of CO in the rich side of the ﬂame that has been reported in
the literature [146]. Additional error might be resulted from the constant mixing
coefﬁcient in the LMSE mixing model. However, similar trends have been observed
in the calculated mean proﬁles of YCO by Pitsch et al. [90, 146], who used a dynamic
method to compute the IBM mixing model coefficient. This issue is discussed further

in the following section.

73

2.4.3 Subgrid-Scale Mixing

In this section we focus on the sensitivity of the LES/ F MDF calculations to the
mixing model coefﬁcient Cg which is either empirically prescribed or dynamically
evaluated in the simulations considered below. All of the following simulations are
conducted with the 12-step reaction model.

Fig. 2.15a and b show radial proﬁles of the mean (favre-averaged) temperature for
ﬂames D and F as obtained by the LES/FMDF with various Cg. The mean species
mass fractions exhibit similar trends to that of temperature and not shown. For ﬂame
F, a comparison between the results obtained with constant values of Cg = 6, 8, 10
indicates that by increasing Cg the mixing/ combustion enhances and the average
temperature increases, albeit not linearly, which is expected and consistent with the
previous studies [65, 113, 116]. However, the inﬂuences of Cg on the temperature
proﬁles of ﬂame D do not seem to be very signiﬁcant. This is because of the less
sensitivity of ﬂame D, which is close to equilibrium, to the molecular mass and
thermal diffusivity coefﬁcients. The mean temperatures predicted by the dynamic
models DM1 and DM2 are also shown in Fig. 2.15a, b to be close to those obtained by
constant Cg for ﬂame D. However, comparison of the results obtained with DM1 and
DM2 with those of Cg = 6, 8, 10 for ﬂame F indicates that the physical ﬂame structure
is affected more by the mixing model in this ﬂame. The higher mean temperature
values observed with DM2 for ﬂame F is due to higher level of mixing. With DM1,
the predicted mixing is relatively lower than that obtained with constant coefﬁcient
of Cg = 10. Consistently, the predicted temperature RMS by DM1 is found to be
higher than those calculated with Cg = 10. The dynamic model DM2 predicts the
highest RMS temperature. The results for constant mixing coefﬁcients (not shown)
indicate that the mean heat release is signiﬁcantly lower for Cg = 6, 8 in comparison
with Cg = 10 in ﬂame F simulations. Combined with the stronger turbulence in

ﬂame F, these lower heat release values yields substantially lower mean temperature

74

proﬁles (Fig. 2.15b). Also, consistent with the mean temperature proﬁles shown
in the preceding ﬁgures, substantially higher mean heat release values are observed
when DM2 model is employed, particularly for ﬂame F.

The effects of mixing model on the compositional structure of ﬂames D and F
are shown in Figs. 2.16 - 2.17, where the variations of various ﬂame variables in the
mixture fraction domain for different mixing model coefﬁcients are considered. In
Figs. 2.16a and 2.16c, the conditional proﬁles of the mean temperature and RMS
of temperature for ﬂame D with different constant Cg and dynamic models at axial
locations of :13/ D = 7.5 and 93/ D = 15 are compared with the experimental data.
Overall, the mean temperature is well predicted by the LES/FMDF. However, at
a: / D = 7.5, the peak values of the conditional temperature are observed to be slightly
overestimated for all Cg coefficients. The conditional RMS of temperature is shown
to be also well captured with LES / F MDF when Cg = 6, 8 or DM1 is used, while it is
underestimated with the DM2 model at all mixture fraction values. The inaccuracy
of simulations with DM2 model is mainly due to local equilibrium assumption in
the dynamic model as Raman and Pitch [90] suggested. Consistent with our earlier
results and with those obtained by others [65, 113, 116], the RMS of temperature
decreases and the mean temperature increases as Cg increases. The results at a: / D =
15 are somewhat similar to those at :13/ D = 7.5, but more visibly, the conditional
mean temperature are overestimated by the LES/FMDF in the stoichiometric and
the fuel-rich regions. Also, the deviations between the experimental and LES/FMDF
values of the temperature RMS are higher at x/D = 15, mainly due to stronger
turbulence at downstream locations.

Figs. 2.16b and 2.16d show the conditional averages of species C0 and Hg for
ﬂame D at two different downstream locations. At a: / D = 7.5, the predicted values
are generally in agreement with the reported experimental data at all mixture fraction

values for all Cg values. There seems to be a slight overprediction of the experimental

75

data around the stoichiometric point when Cg = 8, 10 or DM2 is used, which is
mainly due to differences in heat release. As expected this effect is more noticeable
at further downstream locations (Fig. 2.16d). The experimental proﬁles of H2 mean
mass fraction at both axial locations are overpredicted by the LES/FMDF in the
stoichiometric region but are well captured at all other mixture fractions by the
simulations.

Figs. 2.17a and 2.17c show the computed and measured conditional mean and
RMS of temperature for ﬂame F at two downstream locations. At 23/ D = 7.5, the
local extinction and conditional mean temperature and the extend of local extinction
present in ﬂame F are well predicted for Cg = 6, 8 and DM1 model. However, the
DM2 model overpredicts the peak temperature, even though the predictions on the
lean and rich sides of the ﬂame are still in fairly good agreement with the experiment.
Also at this location, the RMS ﬂuctuations of the conditional mean temperature are
well predicted for all Cg’s. At further downstream location of x/ D = 15, where
the effects of turbulence is more signiﬁcant, the conditional mean temperature is
underpredicted at stoichiometric and lean side of the ﬂame by the LES / FMDF when
mixing coefficient is constant Cg = 6, 8. The simulations conducted with the dy-
namic models of DM1 and DM2 overpredict the measured mean temperatures for
the mixture fraction values of Z Z 0.25. The poorest agreement is for DM2 model,
mainly a result of higher heat release. At 25/ D = 15, the computed conditional RMS
of temperatures are also in good agreement with the experimental data for Cg = 6, 8
and DM1. With the DM2 model, there is an underprediction of data at lean side of
the ﬂame, possibly due to sensitivity of the ﬂame F to the oscillations in the dynamic
model coefﬁcient. Similar to what have been shown for ﬂame D, the CO and H2 mass
fractions proﬁles agree well with the experimental data for ﬂame F at m/ D = 7.5
as observed in Fig. 2.17b. Only the peak values of these species are overestimated

when DM2 is used, which is consistent with the expository temperature proﬁles in

76

Fig. 2.17a. Similar results have been observed at lower jet speed for the (piloted jet)
ﬂame E by Raman and Pistch [90]. Close examination of the mass fraction proﬁles at
x/ D = 15 (Fig. 2.17d) indicates that the LES/FMDF results with constant mixing
coefﬁcients of Cg = 6, 8 are better than those with DM1 and DM2 models. Both
dynamic models tend to underpredict the rate of change of the mean CO mass frac-
tion proﬁle with the Reynolds number, which is consistent with the results obtained
by others for jet ﬂames [65, 115, 147—149]. Nevertheless, the strong inﬂuences of
the turbulence on the ﬂame and the level of local ﬂame extinction as seen in the
conditional variables seem to be well captured by the LES/ F MDF, not only with
constant mixing coefficients but also with the dynamically computed, composition
dependent, mixing coefficients.

For ﬂame D, the computed time-averaged mean temperature with different Cg
models (not shown) are found to be close to each other, and generally in agreement
with the experimental data. Similarly, the predicted values of mean mass fraction
of CO and Hg for different Cg’s agree well with the experimental data. Again, by
increasing the SGS mixing coefﬁcient Cg the peak mean temperature are found to
increase.

The mean values of the temperature and species mass fractions are more sensitive
to Cg for ﬂame F. However, the measured mean temperature proﬁles and the level
of local ﬂame extinction seem to be well predicted by the simulations when Cg = 10.
There are some minor differences between LES/FMDF and experiment when DM2
is employed. This is consistent with the overprediction of C0 mass fraction by DM2
model in ﬂame F.

The scatter plots of temperature in the mixture fraction domain for various mixing
models and for both ﬂames D and F are shown in Figs. 2.18 - 2.19. These results
obtained, similar to experiment, by collecting the data from the cells that are at

radial distance of z/d = 0.83. Only the results at :c/ D = 7.5 are shown. The trends

77

are similar at different axial locations. The solid lines in Figs. 2.18 - 2.19 show the
temperature for a low-strain (10/3) laminar opposed jet ﬂame proﬁle.

Figs. 2.18 shows the scatter plots of temperature for ﬂame D. In general, the
computed temperature for ﬂame D are in reasonably good agreement with the mea-
surements. Evidently, the number of data points with relatively low temperature is
lowest for case DM2, which is consistent with the overestimation of the peak condi-
tional mean temperature in Fig. 2.16a by this model. Simulations with the dynamic
model DM1 and with constant coefﬁcients of Cg = 6, 8, 10 exhibit a more signiﬁcant
ﬁnite rate chemistry effect, making the prediction of the conditional mean temper-
ature closer to experiment for these cases. The results at x/ D = 15 (not shown)
exhibit similar trends, even though the distribution of the scattered temperature
points is narrower in the fuel-rich side [65, 145].

For ﬂame F, it can be safely stated that the challenging task of representing
the increasing level of local extinction is fairly accomplished by the LES/FMDF
when 12-step model is employed. As shown in Fig. 2.19, the results calculated with
different Cg coefficients are somewhat similar and are in quite good agreement with
the reported experimental data; this is consistent with the accurate predictions of
conditional mean temperatures in Fig. 2.17a. The predictions of the dynamic models
DM1 and DM2 tend to show less number of “low temperature” data points and
consequently higher average temperature, nevertheless, they seem to well capture

the “two banded structure” observed in the experimental data (Fig. 2.13).

2.5 Summary and Conclusions

This paper is concerned with the application and assessment of the ﬁltered mass
density function (FMDF) for large-eddy simulation (LES) of Sandia’s “high speed”

piloted turbulent methane jet ﬂames [64]. The LES/FMDF calculations are con-

78

ducted with both ﬁnite-rate and ﬂamelet-type reaction models. The ﬂamelet model
employs detailed (GRI—3) kinetics mechanism. The ﬁniterate model employs 1-step
and 12—step mechanisms. Various subgrid-scale mixing models are also used and
tested. A scalable algorithm for parallelization of the hybrid (Eulerian-Lagrangian)
LES/FMDF methodology is presented and the consistency of its Monte Carlo - ﬁ-
nite difference solutions were discussed. The parallel algorithm was shown to yield
superlinear speedups with respect to single node calculations. The Favre-averaged
temperatures for the two jet speeds (referred to as ﬂames D and F) were found to
be in fairly good agreement with the experimental data. For the lower speed jet
(ﬂame D), the numerical results are consistent with the experiment exhibiting near-
equilibrium ﬂame structure with limited local extinction. The higher degrees of local
extinction observed in the higher speed jet (ﬂame F) is successfully captured and
reproduced with the LES/FMDF when 12—step chemistry model is employed. As
expected, the ﬂamelet and 12—step chemistry models generate results close to each
other and comparable to experiment for ﬂame D. For ﬂame F, only the 12-step model
predictions are comparable to experiment. For this ﬂame, the higher heat release pre
dicted by the 1-step and ﬂamelet models suppress the effects of turbulence on the
ﬂame. The sensitivity of the calculations to the subgrid mixing model coefﬁcient
Cg is also inspected by performing simulations with different model coefﬁcients and
with two dynamic models. The LES/FMDF results seem to be much more sensitive
to the mixing model (and the way model coefﬁcient is evaluated) in ﬂame F than
in ﬂame D. Even though, the results with different mixing coefﬁcients and models
are not that much different for ﬂame D, the calculations conducted with constant
coefﬁcient of Cg = 10 yield the most accurate overall agreements for both ﬂames.
Further improvements in the predictions might be possible with better mixing and

subgrid stress/ scalar ﬂux closures.

79

2.6 Tables and Figures

Table 2.1: Important parameters of Sandia’s piloted turbulent methane jet ﬂames D

and F.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Flame D Flame F
Rejet 22400 44800
Main jet diameter 7.2 mm 7.2 mm
Pilot jet diameter 18.2 mm 18.2 mm
Main jet bulk velocity 49.6 m/s 99.2 m/s
Main jet peak velocity 63.1 m/s 126.2 m/s
Main jet temperature 300 K 300 K
Pilot bulk velocity 11.4 m/s 22.8 m/s
Pilot temperature 1880 K 1880 K
Pilot mixture fraction 0.27 0.27
Co—ﬂow bulk velocity 0.9 m/s 0.9 m/s

 

80

 

AX

 

 

 

Methane + Air

Coﬂow
Pilot

 

Figure 2.1: Schematic view of the Sandia’s piloted methane jet ﬂame. Main jet and
pilot jet diameters are 7.2mm and 18.2mm, respectively.

81

(T)L

 

 

 

 

 

 

 

 

I ] T I I l
(a) /A\.
- g" x, -
,f/ \
I \
L .” . _
II
I,
_ i 1
// -
/
/
1 l 1 I; l 1
0.5 1 1.5 2
r/D
I I l I I I
b ,I'A‘\
_ ( ) ,/ .\ J
,4: \
ﬂ \
"/7
__ // _,
///
_ I a
-— —]
_ [I], .
I l I l I l
0.5 1 1.5 2
r/D

Figure 2.2: Comparison between Monte Carlo (MC) and ﬁnite difference (FD) values
of the normalized instantaneous ﬁltered temperature; (a) A E = 2A, and Cg = 8, 16,
(b) A3 = A, and Cg = 8,16. MC with Cg = 8; thick solid line, FD with Cg = 8;
thick dashed line, MC with Cg = 16; solid line, FD with Cg = 16; dashed line.

82

 

 

 

 

 

15 -[§—LT3§7I5'1\71I5F] 5 ><
(x6999 “
[- ,.-1“' 4
(139293?”
‘6 ‘///
3‘ 10 _ “0'0?” 4
8 $9_,,-/ Y
x o c
:9? ra\\ {ficien
a ,..u f,"
5 f 19%.? u
L
2 4 6 8 10 e 12 I 14 L 16

no. of nrocesSOrs

Figure 2.3: Speed-up of the parallelized LES/ F MDF calculations relative to those
conducted on a 2—processor node.

83

 

Figure 2.4: Isosurfaces and isocontours of vorticity magnitude, 9, obtained by
LES/FMDF with the 12-step chemistry model; (a) ﬂame D and (b) ﬂame F.

84

 

Figure 2.5: Isosurfaces and isocontours of ﬁltered temperature, (T) L, obtained by
LES/FMDF with the 12-step chemistry model; (a) ﬂame D and (b) ﬂame F.

85

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.6: The time-averaged (mean) values and the RMS values of the ﬁltered
axial velocity ﬁeld at :1: / D = 15; (a) mean velocity for ﬂame D, (b) mean velocity for
ﬂame F, (c) velocity RMS for ﬂame D and (d) velocity RMS for ﬂame F. Symbols,
experimental data; dashed-dot line, 1-step model; dashed line, ﬂamelet model; solid
line, 12-step model.

86

2000
1500:

“ .
[S 1000}

50 "

600

821mm,)

Figure 2.7: The time-averaged (mean) temperature and RMS of temperature at
a: / D = 15; (a) mean temperature for ﬂame D, (b) mean temperature for ﬂame F, (c)
RMS of temperature for ﬂame D and (d) RMS of temperature for ﬂame F. Symbols,
experimental data; dashed-dot line, 1—step model; dashed line, ﬂamelet model; solid
line, 12-step model.

 

I

 

 

 

 

.h
C
O

 

 

 

O ‘3.
A 4 3
r/D
(C) <
. O
I!!! \ .
° 0
.~
* 3
r/D

87

 

 

 

 

 

 

 

 

 

 

 

(b) .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.8: Proﬁles of various species mass fractions as predicted by LES/FMDF
with the 12—step model at III/D = 15; (a) C02 (b) H2, (C) CO, (d) 0H. Circles,
measured ﬂame D; squares, measured ﬂame F; dashed line, computed ﬂame D; solid
line, computed ﬂame F.

88

400 . . T . 1 T

 

 

 

 

 

“ -------------- Flamelet, ﬂame D
l \ -——-——-—- lZ-step, ﬂame D
300 -— ,' “ —————— Flamelet, ﬂame F 7
'l \‘ —— 12-step, ﬂalne F
/\"q I, ‘\
Q) 200 ~ I I
v

100

 

 

 

 

 

Figure 2.9: Proﬁles of mean heat release for the ﬂamelet and 12-step models in
ﬂame D (dashed line, ﬂamelet; solid line, 12—step) and in ﬂame F (long dashed line,
ﬂamelet; dashed-dot line, 12-step) at :17/ D = 7.5.

89

 

 

Cond. Mean Temp.

 

 

 

x/ D = 15
2000 i
6..
E
[1’
t
a
N 1000
E
E
D A.
“M.— _
00 0.5
Mixture Fraction

 

  

 

Cond. Mean Y02

Cond. Mean 1’02

 

 

 

   
   

 

 

  

 

0.08
E (b)
’8
E
r:
§0.04—
”d
8
D
0 0.5
Mixture Fraction
x/D = 15
0.08 . . '
S ((0
>5, .
E
E004
E
l)

 

   

 

 

 

0.08

o
i
Cond. Mean YCO

  

0.08

o
E
Cond. Mean YCO

Figure 2.10: Conditional mean temperature and 02, CO, H2 mass fractions for ﬂame
D as predicted by LES/ F MDF with 12—step kinetics mechanism and with Cg = 10;
(a) and (b) show the results, at x/ D = 7.5; (c) and ((1) show the results, at 55/ D = 15

(lines, calculations; symbols, measurements).

90

N
O
O
O

Cond. Mean Temp.
8
o

 

 

 

 

 

 

 

 

 

 

ois 1O
Mixture Fraction

Cond. Mean 1’02

Cond. Mean Y02

0.08

H2X10

Cond. Mean Y
9
c:
.b

Cond. Megn YHZ x109
E 53 o

O

 

 

 
    

  
  

 

H2
0.5
Mixture Fraction

4

x/D =15

o
E
Cond. Mean YCO

 

 

r

(d)

 

   

 

Mixture Fraction

0.08

 

0.08

o
E
Cond. Mean YCO

p—n
C

Figure 2.11: Conditional mean temperature and 02, 00, H2 mass fractions for ﬂame
F as predicted by LES/FMDF with 12—step kinetics mechanism and with 005 = 10;
(a) and (b) show the results, at z/ D = 7.5; (c) and (d) show the results, at x/ D = 15

(lines, calculations; symbols, measurements).

91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2500 2500
r(a) , (b)
“,3 2000» 8 2000»
t: a
$1500. g 1500
§ 1000 § 1000
E. » EN .
500 500
O 0.5 O 0.5
Mixture Fraction Mixture Fraction
2500 2500 w
L(C) (d)
m 2000— / ‘\ m 2000* .
g r At? g _ ﬂ“...
:3 1500* . = g 1500
5 1000 E 1000
m m
[N » [N
500 500
0 0:5 0 0.5
Mixture Fraction Mixture Fraction

Figure 2.12: Scatter plots of temperature vs. mixture fraction at r/ D = 0.83 for
ﬂame D; (a) measurements at :v/ D = 7.5 (975 sample points); (b) measurements
at z/ D = 15 (780 sample points); (c) calculations with LES/FMDF and 12-step
mechanism at :c/ D = 7.5 (973 sample points); (d) calculations with LES / FMDF and
12-step mechanism at T/ D = 15 (762 sample points).

92

 

Temperature

 

 

 

0.5
Mixture Fraction

 

 

2500

1 000

Temperature

til
o
O

2000 *

g.—
M
O
O

 

Figure 2.13: Scatter plots of temperature vs. mixture fraction at r/ D = 0.83 for
ﬂame F; (a) measurements at x/ D = 7.5 (1550 sample points); (b) measurements
at :c/ D = 15 (976 sample points); (c) calculations with LES/FMDF and 12—step
mechanism at :r/ D = 7.5 (1347 sample points); (d) calculations with LES/FMDF

 

 

0.5
Mixture Fraction

 

2500

 

 

 

 

 

 

 

 

(b)
2 2000 .73.
S "w
g 1500
§ 1000 —
a.
500
o ' 0.5 1
Mixture Fraction
2500 .
(d)
2 2000 ' .5
E 1500
N .
§ 2'.
1000 .. .
500
0 0.5 l
Mixture Fraction

and 12-step mechanism at x/ D = 15 (1097 sample points).

93

 

 

 

 

 

 

 

 

 

 

“0.1

~ 0.05

(%)LXIO
(17%on

 

 

 

 

 

 

 

 

Figure 2.14: Comparison between LES / FMDF and experimental data in the mixture
fraction space for the 12-step model with C¢ = 10 at 27/ D = 15; (a) mean tempera-
ture and 02 mass fraction for ﬂame D; (b)mean temperature and 02 mass fraction
for ﬂame F; (c) mean C0 and H2 mass fractions for ﬂame D; (d) mean CO and H2
mass fractions for ﬂame F (lines, calculations; symbols, measurements).

94

 

2000? (a) T a

 

 

 

 

 

2000

l
e
v

1

p—i
LII
O
O
l
l

 

 

 

 

 

r/D

Figure 2.15: The radial variations of the mean temperature as obtained by
LES/FMDF with 12-step model and different C¢ values at x/D = 15; (a) ﬂame
D and (b) ﬂame F (symbols, measurements; long-dashed line, at = 6; dashed-dot
line, C¢ = 8; solid line, 005 = 10; dashed-dot-dot line, C¢ = DM 1; dashed line,
C¢ = DM 2).

95

 

 

0.08 a . 7 0.06

 
  
      

    

 

 

 

 

 

 

 

   
   

 

 

 

 

 

 

@3000: (3y 7emp. § 0
i 1500 E” i B
2
g 1000 g 0.04 ” 0.03%:
E 500_ rms Temp. ( g 3%
Ida. D
00'” 0.5 g z '1
Mixture Fraction
a a , 0.08 0.06
@1000: (0) Temp. § 0
if 1500 EN >9
. = 2
g 1000 g 004 4003g
E 500 rms Temp. . E E
0 m1 8 1 0 D
00 0.5 1 0.5
Mixture Fraction Mixture Fraction

Figure 2.16: Conditional mean and RMS of various ﬂame variables as obtained by
LES/FMDF with the 12-step model and different C¢ values for ﬂame D; (a) mean
and RMS of temperature at z/ D = 7.5; (b) mean of C0 and H2 mass fractions at
m/D = 7.5; (0) mean and RMS of temperature at :r/D = 15; ((1) mean of CO and
H2 mass fractions at :13/ D = 15 (symbols, measurements; long-dashed line, C¢ = 6;
dashed-dot line, C¢ = 8; solid line, C¢ = 10; dashed-dot-dot line, C}; = DM 1;
dashed line, C¢ = DM 2).

96

N
O
O
O

Cond Mean Temp.

N
O
O
O

500

C ond. Mean Temp.

0b ‘ (f5 ‘ 1

 

1500 ~
1000

 

Ln
O
99

  

 

 

 

1 .‘ “a.
0.5 1
Mixture Fraction

 

1500
1000- _»f

 

 

 

 

Mixture Fraction

2x10

Cond. Mean YH

 

2x10

Cond. Mean YH

 

 

 

 
  

 

05

‘ Mixture Fraction

0.06

O
o
m
Cond. Mean YCO

 

 

 

_ (d?

 

 

 

(is
Mixture Fraction

0.06

 

o
o
m
Cond. Mean YCO

Figure 2.17: Conditional mean and RMS of various ﬂame variables as obtained by
LES/FMDF with the 12-step model and different C¢ values for ﬂame F; (a) mean
and RMS of temperature at 27/ D = 7.5; (b) mean of CO and H2 mass fractions at
z/D = 7 .5; (c) mean and RMS of temperature at x/D = 15; (d) mean of CO and
H2 mass fractions at 33/ D = 15 (symbols, measurements; long-dashed line, Cd, = 6;
dashed-dot line, C¢ = 8; solid line, C4; = 10; dashed-dot-dot line, 045 = DM 1;
dashed line, C4, = DM 2).

97

 

2500

_. N
U1 0
O O
O O

Temperature

1000*

ll:
0
O

 

 

 

0.5
Mixture Fraction

 

1500

1 000

Temperature

U1
0
O

 

0.5
Mixture Fraction

 

 

2500

Temperature
H t—t N
0 UI O
O O O
O O 0

LI!
0
O

 

 

 

O

 

0.5 ]
Mixture Fraction

 

2500

N
O
O
0

Temperature
8 G
O O

. c? . o

VI

0

O
l

 

 

 

C

 

0.5 ’ 1
Mixture Fraction

Figure 2.18: Scatter plots of temperature vs. mixture fraction as obtained by
LES/FMDF with the 12—step model and different C¢ values for ﬂame D at 33/0 = 7.5,
and r/D = 0.83; (a) C0) = 6 (1102 sample points); (b) Cd) = 8 (1022 sample points);
(c) C45 = DIVIl (1042 sample points); (d) 0(1) = 01112 (973 sample points).

98

 

 

 

 

 

 

 

 

2500
(a)
2 2000» m
g ,
EiISOO ‘;i y g; L
§ 1000 ”“4
[N '1 . i
500
0 0.5 1
Mixture Fraction
2500
2 2000
E
E 1500
8.
§ 1000
B4
500
0 0.5
Mixture Fraction

 

 

 

 

 

 

 

 

 

 

(b)
g 2000 gm)
g 1500 f
a. 1* "
§ 1000 ,n/f.” .
5. »~’-‘ 3
500 ‘.
0 0.5 l
Mixture Fraction
2500
2 2000
g ,,
L 1500 5_-
8.. ‘ 3;.
§ 1000 ‘ ”:3,
R \.\\’
500 " .l
0 0.5 l
Mixture Fraction

Figure 2.19: Scatter plots of temperature vs. mixture fraction as obtained by

LES/FMDF with the 12-step model and different Ca; values for ﬂame F a: / D = 7.5,
and r/D = 0.83; (a) Cd, = 6 (1418 sample points); (b) CC, = 8 (1350 sample points);

(c) Cg; = DM1 (1376 sample points); (d) C9, = DMZ (1134 sample points).

99

CHAPTER 3

LARGE-SCALE SIMULATIONS OF
TURBULENT SPRAY COMBUSTION IN A
DUMP COMBUSTOR VIA TWO-PHASE FMDF

3.1 Introduction

Detailed experimental measurements and numerical simulations of multiphase tur-
bulent ﬂows continue to be important but challenging [150—155]. An important
subcategory of two—phase ﬂows is the ﬂow of dispersed droplets in ﬂuids that oc-
curs in spray combustion in various energy conversion and propulsion devices such
as internal combustion engines, gas-turbine combustors, oil-ﬁred industrial furnaces,
liquid-fueled rocket engines, etc. The efficiency and the stability of turbulent com-
bustion and the emissions in such liquid-fuel combustion systems are dependent on
the coupled and complicated effects of various parameters such as the fuel spray
characteristics, the fuel type, the geometry, and the input/ output ﬂow conditions.
It is often very difficult to predict the ﬂow and combustion behaviors in such sys-
tems under various operating conditions. Considering the fact that turbulence itself
has remained as an ”unsolved” problem, one may expect the two-phase turbulent
reacting ﬂows to be almost unpredictable! In fact, with the addition of evaporating
and reacting liquid droplets, the additional physical complexities due to multi-phase
transport including the couplings between the phases, make the analytical/ numerical

description of the problem very difficult. A major challenge is to develop affordable

100

and accurate computational ﬂuid dynamics (CFD) models which can describe the
complicated interactions among turbulence, combustion and spray in a wide range
of time and length scales.

This chapter presents a brief description of a novel two—phase large eddy simu-
lation (LES) model (that we have developed for the numerical simulation of two-
phase turbulent reacting ﬂows based on the two-phase ﬁltered mass density function
(FMDF) methodology. A comprehensive analysis of the turbulent combustion in
a spray-controlled lean, premixed dump combustor is also presented. Among the
theoretical, experimental and numerical studies appeared in the literature for mod-
eling/ understanding of two—phase turbulent reacting ﬂows [156—167], we refer to a
class of Lagrangian statistical methods which dates back to Taylor [168] method of
relating the ﬂuid particle dispersion to the Lagrangian auto-correlation of ﬂuid ele
ments. Taylor’s theory has been the basis of many other theoretical investigations
involving the dispersion of ﬂuid and light tracer particles [169—171].

Experimental studies of spray combustion in turbulent ﬂows may be classiﬁed
into two groups; (i) studies concerning a single droplet evaporation and combus-
tion in turbulent ﬂow conditions, (ii) studies concerning spray development under
conditions comparable to real-life applications. For the latter, numerous studies of
evaporating and combusting sprays have been motivated by a wide variety of prac-
tical applications; see, for example, Ref. [172—193] and references cited therein. As a
geometrically simple problem but relevant to those used in air-breathing propulsion
systems, the spray combustion in a dump combustor is one of the most commonly
employed conﬁgurations in gas turbine and ramjet engine combustors. An example
is the experimental work of El Banhawy et al. [177] that includes a cylindrical com-
bustion chamber equipped with a rotating cup atomizer, producing near-monosized
droplets. The results for this experiment indicate that an increase in the mean droplet

diameter reduces the combustion intensity in the regions away from the centerline

101

for both high and low swirl ﬂames. The results obtained with different degrees of air
swirl, but the same mean droplet diameter, indicates that with the reduction in swirl,
the reaction rate within the initial part of the combustor and close to the center line
decreases. Khalil et al. [178] also investigated the effect of swirl on the ﬂow pattern
and spray combustion air in a cylindrical furnace. They indicated that by increasing
the swirl intensity increases the size and strength of the central recirculation zone,
and there are linear relations between the swirl intensity and both the average and
maximum recirculated mass ﬂow rate. Gupta et al. [188] examined the spray ﬂame
characteristics of kerosene and methanol spray ﬂames in a co-annular, swirl-stabilized
combustor. They found that when the airﬂow ﬁeld and the amount of heat released
by the fuel are similar, the structure and the dynamics will also be similar. Sommer-
feld et al. [182] studied the droplet dispersion and evaporation in a heated turbulent
air stream and the effects they have on the ﬂow in a sudden expansion by measuring
the droplet size distribution, the correlations between droplet size and velocity, and
the droplet mass ﬂux. They found that air velocities and turbulence have signiﬁcant
effect on droplet evaporation rate. Among the other available experimental stud-
ies on spray combustion in dump combustors, the experimental studies of Yu et al.
[189—193] are of particular interest since various combustion and spray parameters
were investigated. They showed that with the pulse injection, synchronized with the
large-scale shedded vortices, the spatial distribution of fuel droplets in the ﬂow ﬁeld,
and therefore the combustion instability cane be controlled. They also showed that
the basic ﬂame characteristics, in particular the ﬂame length and intensity could be
controlled by controlling fuel spray injection.

From a mathematical/ computational point of view, the turbulent spray combus-
tion is generally studied with three different methods [194, 195]: (i) Eulerian-Eulerian
methods [155, 196—205], (ii) Eulerian-Lagrangian methods [151, 153, 196, 206—214],

and (iii) Lagrangian-Lagrangian methods [195, 211, 215, 216]. In the ﬁrst group,

102

the continuum transport equations for both phases are solved. These equations are
somewhat similar and are often obtained by some sort of volume averaging which is
conceptually different than the ensemble averaging in Reynolds-averaged simulation
(RAS) or space averaging in large-eddy simulation (LES). In the second group, the
carrier ﬂuid continuum equations are solved in its ”instantaneous” form in direct
numerical simulation (DN S) or in its ”averaged” form in RAS and LES over a ﬁxed
Eulerian grid system. However, the ”dispersed” phase (droplets) is described by a
set of modeled Lagrangian equations which determine the position, velocity, temper-
ature, and other properties [153, 196, 207, 208] of the dispersed phase. In the third
group, both phases are described in the Lagrangian context.

Eulerian-Eulerian and Eulerian-Lagrangian methods have been widely employed
in DN S, LES or RAS of turbulent spray combustion [217—229]. In comparison, there
have been limited number of simulations conducted based on Lagrangian-Lagrangian
method [195]. One may also consider some of the joint probability density function
(PDF) and the joint mass density function (MDF) models [211, 215] as Lagrangian-
Lagrangian models. Here, we do not discuss the Eulerian-Eulerian and Eulerian-
Lagrangian models, rather we refer the interested reader to review articles and books
on subject [151, 153, 155, 164, 196, 207].

Despite known beneﬁts of LES-based models, specially their ability to capture
the unsteady physical features of complex turbulent ﬂows [155, 196, 208, 230, 231],
they have not widely used for numerical simulations of turbulent spray combus-
tion [228, 232—235]. Among the limited number of LES studies involving turbulent
spray combustion we refer the reader to the papers by Cuenot et al. [198]; Sankaran
et al. [212]; Ying-wen et al. [214]; Ham et al. [228]; Okong’o et al. [236]; Leboissetier
et al. [237]; Patel et al. [238]; Mahesh et al. [239]; Afshari et al. [240].

Cuenot et al. [198] simulated turbulent spray combustion in diﬂerent geometric

conﬁgurations with Eulerian-Eulerian LES models. Then the solver is applied to the

103

Vesta combustor of Turbomeca, composed of 18 main burners ignited by two pilot
fames, which illustrated the capacity of LES to compute complex two-phase reacting
ﬂows in transient regimes. The two—phase reacting LES model employed in Refs. [212,
238] is based on the Eulerian-Lagrangian method. In this model, the carrier gas
equations are solved on ﬁxed Eulerian grid. However, the mixing and reaction are
implemented in one—dimensional (1D) domain via linear eddy model (LEM). The
spray is represented by Lagrangian droplets employing empirical evaporation and
secondary break-up submodels. A similar hybrid Eulerian-Lagrangian approach is
employed in Refs. [228, 239], where the zero Mach number N avier-Stokes equations
are solved on an unstructured grid and the spray is modeled with Lagrangian droplets.
In the studies conducted by Okong’o et al. [236] and Leboissetier et al. [237], the
DNS data for a temporal mixing layer laden with evaporating droplets are used to
assess different subgrid-scale (SGS) models for carrier gas, droplet and evaporated
vapor. It is shown that with the SGS scale-similarity models the predicted droplet
distribution by LES is comparable to that of DNS. However, the mass, momentum
and heat transfer between droplet and carrier gas phases shown to not be accurately
represented by the proposed deterministic closures. The LES models in Ref. [209,
212, 214, 228, 239] have been applied to realistic (complex) gas turbine combustors
with various level of success.

In this work, a new two-phase LES model and a new Lagrangian-Eulerian-Lagran-
gian numerical scheme [240] are developed for two-phase turbulent reacting ﬂows in
complex geometrical conﬁgurations. The model is applied to high Reynolds number
turbulent combustion in a spray-controlled lean premixed dump combustor [191].
The velocity ﬁeld is obtained solving the ﬁltered Eulerian equations with a high-
order compact ﬁnite difference method. The spray model is based on a Lagrangian
mathematical/computational methodology that allows two-way mass, momentum

and energy coupling between phases. The subgrid gas—liquid combustion is modeled

104

with the two-phase ﬁltered mass density function (F MDF ) The new LES/ F MDF is
used for systematic study of the effects of various spray, fuel / air and ﬂow parameters
on turbulence and combustion.

The remainder of this chapter is organized as follows: ﬁrst the governing equa-
tions for velocity, scalar and droplets are presented in Section 2. Followed by the
description of numerical solution procedure in Section 3. The two-phase LES / FMDF
results are presented in Section 4 and the chapter is completed by a summary and

some concluding remarks in Section 5.

3.2 Two-Phase LES / FMDF Model Equations

In this section, the governing equations for two—phase compressible turbulent reacting
ﬂows, as solved in the LES/FMDF model are presented. The two-phase LES/FMDF
model is based on a Lagrangian-Eulerian-Lagrangian mathematical/computational
methodology in which a combined set of strongly coupled Eulerian ﬁnite—difference
(FD) gas-phase, Lagrangian liquid/droplet-phase, and Lagrangian stochastic Monte

Carlo (MC) particle equations are solved.

3.2.1 Eulerian Gas-Phase LES Equations

The carrier phase is considered to be a compressible and Newtonian, and ideal gas
with zero bulk viscosity. The LES equations describing the carrier gas, as obtained
by ﬁltering the Navier-Stokes, energy and scalar equations, are

8_(a_ptﬂ + a<p>aliji>L = <3p>l (3.1)

ul- (9 u; u- a 72" an.
6(p>1a<t >1. + <p>l<azsz< J>L =_aé<):>z + (3362” _BT;+(p)lgi+(Sui)1 (3.2)

 

 

105

8(p>(13[H)L + 3(P)1(IgziL<H)L 2 _%%>1 _ ‘21:: +(SH)1 (3.3)

 

 

a<p>g<t¢a>z + 8<p)1(1g;f<¢a>r = fie—If?” _ 8;? + (1050.); + (33); (3'4)

 

 

where (f (x, t))l and ( f(x, t)) L = (p f>l/ (p); represent the ﬁltered and the Favre-
ﬁltered values of the transport variable f (x, t) and p, u;, P, T and H are the ﬂuid
density, velocity, pressure, temperature, and total enthalpy, respectively. The species’
mass fractions are represented in a side equation with gba E Ya, a = 1,2, ...,Ns.

Equations 3.1-3.4 are closed by the following constitutive relations

 

”3 <¢> >
(m z <p>gR0<T>L Z ——,—;—;—-—é (3.5)
a=1 a
u- 3 U' u
(Ta->1 e um. (518% + 16,—’3 — $5,125?) , <u>L = Pan/Cm (3.6)
«I?» z —<p>g<D>La<§§:L. <D>L = mfgpib (3.7)

In Eqs. 3.5 - 3.7, W0, is the molecular weight of species a, 11 is the dynamic viscosity
and n is the thermal conductivity coefﬁcient.

The SGS closures that appear in the ﬁltered equations include the SGS stress
I‘z-j = (p) l [(uzuj) L — (uz)L(uj)L] , the SGS energy and the scalar ﬂux terms [87, 88,
231, 241] that are usually modeled with similarity and diffusivity type closures [87,
240, 242—244]. Additional models are required for the ﬁltered source/sink terms.
These are described in the next section, where the governing equations for the two—
phase FMDF is presented.

It must be indicated here that diffusivity type closures are not used directly in
the F MDF approach but the modeled FMDF transport equation is constructed to

be consistent with them as discussed below.

106

3.2.2 Lagrangian Liquid / Droplet-Phase Equations

The dispersed droplet ﬁeld equations are derived based on the assumptions that the
droplets are ﬁne and heavy, and the mixture is dilute. Under these conditions, the
droplet collisions are infrequent and one may ignore the droplet-droplet interactions.
Also, the droplets are assumed to be spherical and their movement are governed
by an empirically corrected Stokesian drag force. There are numerous studies on
the transport properties of spherical droplets/ particles in turbulent ﬂows [231, 245—
249]. The density of droplets is considered to be much larger than the density of
the carrier gas such that the dominant force on the droplets is the drag force. In
addition, heat transfer due to radiation is neglected. Consequently, the following
set of governing equations employed in a Lagrangian frame to describe the droplet
displacement vector (X,), the droplet velocity vector (11,-), the droplet temperature

(Td), and the dr0plet mass (md) are used.

 

% 2 vi, (3.8)

id? = $2; (3.9)

% z 90011;:07:dLV (3.10)

93% = md = —md (%) (%) 1n [1 + BM], (3.11)

where F; and Qcom, are the modiﬁed Stokes drag force, and the heat ﬂux to the

surrounding gas, deﬁned as

 

F.- = m. (2) (u: — 22.), (3.12)
Qcmw = 7m (2) (3?;T) (T* — Td). (3.13)

107

The subscript (1 indicates the droplet property, and the asterisk refers to the local ﬂuid
variables which are interpolated to the droplet location. LV = hg), — (C L — Cp,V)Td is
the droplet latent heat of vaporization, 11?, is the enthalpy of formation of the evapo-
rated vapor, C L is the heat capacity of the droplet, and pr is the constant pressure
speciﬁc heat of the evaporated vapor. Also, C is the ratio of the heat capacity of the
droplet CL to constant pressure speciﬁc heat of the gas mixture C; = Ea Yan,a,
where the dependency of the speciﬁc heat of species CW, to temperature is ap-
proximated via high degree polynomials. The gas-phase Prandtl (Pr) and Schmidt
(Sc) numbers, on the other hand, are both set to be constant. Additionally, the
empirically corrected droplet Nusselt number N u = 2 + 0.55212631/ 2P'r1/ 3 and Sher-
wood number Sh = 2 + 0.552Recll/ 2501/ 3, where Red is empirically corrected droplet
Reynolds number, Red = de*|u;‘ — vi] / 11*, deﬁned based on the droplet diameter
Dd and local slip velocity. The mass transfer number BM for evaporating droplets
is calculated as BM = (140,3 — Yv) / (1 — Yvrg) in which the subscript “8” denotes a
droplet surface parameter.

In Eq. 3.12, Td is the droplet time constant,

Td _ p.103
— 18u*’

 

(3.14)

and fl is the empirical correction function to the Stokes drag which is obtained by

the following correlation

1 + 0.0545Red + 0.1Re(11/ 2(1 — 0.03Red)

 

(3.15)

a = 0.09 + 0.077exp(—0.4Red) b = 0.4 + 0.77exp(—0.04Red),

where Reb = p*Ude/u* is based on the blowing velocity. Also, in Eq. 3.13 f2 is the

108

empirical correction factor to the evaporative heat transfer and is deﬁned as

5
f2 = 8,, _a, (3.16)

 

with the normalized evaporation parameter, [3 = —1.5Perr'nd/mp [250].

Finally, the vapor mass fraction at droplet surface is calculated from the surface
molar fraction using the Langmuir-Knudsen evaporation model [251], which takes
into account both equilibrium (subscript eq) and non-equilibrium (subscript neq)

effects:

 

Xneq,s
Y = , 3.17
12,3 Xneq,s + (1 _ Xneq,s)WC/WV ( )

where WC and WV are the molecular weights of the carrier gas and evaporated
liquid vapor, respectively. In Eq. 3.17, the equilibrium and non-equilibrium vapor

mole fractions at droplet surface are deﬁned as

2L
Xneq,s = Xeq,s "' (if ) , (3.18)
: ____.._ __ — — , 3.19
Xeq’s P“ exp { RD/ WV TB,L Td ( )

where Patm is atmospheric pressure, R0 is the universal gas constant and TB, L is the
liquid boiling temperature. The term L K in the deﬁnition of non-equilibrium mole

fraction is the Knudsen layer thickness and is deﬁned as

:1: 2 T R0 W 1/2
LK = 11 ( 7r d.(S'cP/* V)) (3.20)

 

The integrated effects of droplets on the carrier gas mass, momentum, energy
and species mass fraction are expressed through several source/ sink terms (i.e. Sp,

591’ SH, 50,33 in Eqs. 3.1- 3.4). The mass source term, 5,; represents the mass

109

contribution of droplets via evaporation, the momentum source term, Sui represents
the momentum transfer between two phases due to drag force, and the heat source
term, 3 H represents the exchange of the internal and kinetic energy by convective
heat transfer and particle drag. These terms represent the two-way coupling of the

Lagrangian droplets on the Eulerian ﬁeld by localized volume averaging and are

deﬁned as:

 

1 nd
Sp = 7‘7 Zmd, (3.21)
1 nd
Sui = -g[7 2 (E + mdvz’)a (322)
. 21:21;
1 nd v-v-
__ __ . . ' Ll
SE — 6V (Q + szz + md (has + 2 )) , (3.24)

where the summation is taken over all droplets in a volume 6V = 61:3 centered at
each Eulerian (grid) point and has = prTd + h?) is the evaporated vapor enthalpy

at droplet surface.

3.2.3 Two-Phase FMDF Formulation

In conventional LES methods developed for the reacting ﬂows, the ﬁltered equation
for the scalars (i.e. Eq. 3.4) are solved together with the mass, momentum and
energy equations. In the scalar equation, the ﬁltered chemical source/sink terms
are not closed and need modeling. In this study, the subgrid combustion model is
based on the F MDF methodology and the temperature and species mass-fractions

are obtained from the FMDF. The chemical source/ sink terms are also determined

exactly from the FMDF.

110

The scalar F MDF is the joint probability density function of the scalars at
subgrid-level [84, 252—254], deﬁned as:

+00
PL(¢,X; ,t) E [00 p(x',t)€ [1b, ¢(x’,t)] H(x' — x)dx’ (3.25)
s 121:, ¢(x, 1111 = 61¢ — ¢(x, t1] 2 II 61%. — ¢..(x, t>1 (326)

0:1

where 'H and 6 denote the ﬁlter and delta functions, respectively, and E [(1), 1/J(X, 15)] is
the “ﬁne-grained” density [88]. Equation 3.25 indicates that the FMDF is the mass
weighted spatially ﬁltered value of the ﬁne-grained density. To ensure that the F MDF
has all the properties of the PDF, we will only consider “positive” ﬁlter functions [107]
for which all the moments ffooo (L‘mH(.’lI)dl' exist for m _>_ 0.

The variable 1,0 in Eqs. 3.25 and 3.26 represents the composition space of the
scalar (species mass fractions and speciﬁc enthalpy) array ¢(x, t) that is denoted by
(pa 5 Ya, a = 1,2, . . .,N3, a, a h, = 25;, imam 113,0, = [73% op,a(T')dT'. The
deterministic equation describes the variations of FMDF in space and time is derived

from the original (unﬁltered) governing equations and has the following ﬁnal form:

 

ﬁﬂwuélfm = 5%[<p>1<<D>L+Dt> W]

0’ a 0' a A
+ a; 5,7,; 19m (21.. — «lam PL] — 02:1 5,; [SOPL]
8 ¢a<Spltb> (53%)] + (Spl’iplpL

+ £le <p>1 ‘ (p); <p>1

 

(3.27)

In Eq 3.27, (AlB) denotes the ﬁltered value of the variable A, “conditioned” on B,
and the hat is used for the quantities which are dependent only on the scalar ﬁeld.
50 and 9m denote the production rate of species a and the SGS mixing frequency,
respectively. The molecular and SGS diffusivity coefficients are denoted by D and

0,1. The effects of molecular/SGS mixing and SGS convection are modeled with

111

closures similar to those used in RAN S/ PDF methods [88]. The last three terms on

the right-hand side represents the effects of droplets and combustion.

3.3 Numerical Solution

As illustrated in Fig. 3.1, the numerical method used for solving the Eulerian and
Lagrangian equations in the two-phase LES/FMDF methodology has three main
components. For the carrier gas velocity and pressure, a high-order accurate ﬁ-
nite difference (FD) method [4] is employed. The discretization of the ﬁltered Eu—
lerian gas-phase LES equations (Eqs. 3.1-3.4) is based on the “compact parame-
ter” ﬁnite difference scheme [128], which yields up to sixth order spatial accuracies.
The time differencing is based on a third order low storage explicit Runge—Kutta
method [255]. All the ﬁnite difference operations are performed on ﬁxed and uniform
grid points. For the solution of spray and dispersed droplet-phase, the Lagrangian
Eqs. 3.8—3.11, are solved with a point particle method [256]. Time advancement in
the Lagrangian droplet (spray) equations is performed using an explicit second order
accurate Adams-Bashforth scheme. The Eulerian carrier gas-phase variables at the
droplet locations are obtained by a fourth-order Lagrangian interpolation scheme.
Finally, a Lagrangian monte carlo particle method is used for the solution of gas-
phase scalar (species and temperature) FMDF [70, 84, 88, 91, 97, 132—134, 240, 252].
Here only the basic features of the FMDF solution procedure are described. With
the Lagrangian procedure, the F MDF is represented by an ensemble of computa-
tional “stochastic elements” (or “particles”) which are transported in the “physical
space” by the combined actions of large scale convection and diffusion (molecular and
subgrid). In addition, transport in the “composition space” occurs due to chemical

reaction and SGS mixing. In doing so, the notional particles evolve via a “stochastic

112

process,” described by a set of stochastic differential equations (SDEs) [135, 136],

dX,-(t) = D,(x(t), t)dt + £(X(t), t)dW,-(t), dag-(t) = Ra(¢+, t)dt. (3.28)

where X,- is the Lagrangian position of the particles, D and E are known as the
“drift” and “diffusion” coefﬁcients, and W,- denotes the Wiener-Levy process [137].
o; denotes the scalar value of the particle with the Lagrangian position vector Xi.
Eq. 3.28 deﬁnes what is known as the general “diffusion” process [136—138]; thus the
PDFs of the stochastic processes (Xi(t), $30)) are governed by the Fokker-Planck
equation. A comparison between the standard Fokker—Planck equation corresponding
to Eq. 3.28 with the F MDF equation (Eq. 3.27) under consideration identiﬁes the

parameters of Eq. 3.28,

 

1 3l<P)t((D)L + Dtll
p)! 3932' ’

 

5 E W(<D>L + Dt), Di E (uz‘>L + < (3-29)

72.. s —9m(¢>2: — <¢a>L1+ §a(¢+).

With this analogy, the F MDF is represented by an ensemble of Monte Carlo particles,
each with a set of scalars 91551”) (t) = ¢a(X(”)(t),t) and Lagrangian position vector
X("). A splitting operation then can be employed in which the transports in physical
and compositional domains are treated separately. The simplest means of simulating
the spatial transport in Eq. 3.28 is via the Euler-Maruyamma approximation [139].
The transfer of information from the ﬁxed ﬁnite difference points to the location of
the Monte Carlo particles are conducted via (fourth and second order) interpolation.
The ﬁltered scalar quantities are calculated by weighted averaging of the particle
values over space with average volumetric size of AE. Ideally, A E has to be very
small. In this study, the chemical reaction is modeled by a “simple” non-equilibrium

(ﬁnite-rate) one-step global mechanism [140]. However, all the reaction terms in the

113

FMDF equation are calculated exactly with no SGS model.

3.3.1 Consistency of FMDF

As stated before, the equation governing the ﬁrst subgrid Favre-moment of the scalar
(ba as obtained from the FMDF equation (Eq. 3.25) is identical with the ﬁltered scalar
equation (Eq. 3.4). This indicates that the ﬁltered temperature, scalar and density
may be alternatively obtained from the Eulerian, ﬁnite difference (FD) or Lagrangian,
Monte Carlo (MC) solutions, implying a mathematical consistency between FD and
MC parts of the hybrid scheme. However, due to ﬁnite grid size in FD and lim-
ited number of MC particles, absolute consistency may not be achieved in practice.
Nevertheless, various simulations results (i.e. non-reacting non—isothermal, reacting
without spray, reacting with spray) are presented below to demonstrate the consis-
tency and the accuracy of the two—phase FMDF formulation and its Lagrangian MC
method with the conventional LES models and its Eulerian FD method. For this,
the instantaneous ﬁltered temperature and fuel mass fraction variables as calculated

by LES/ FD and FMDF / MC models are compared.

N on—reacting non-isothermal simulations without spray

First we consider non-reacting non-isothermal case without spray. In this case, there
are signiﬁcant density and temperature variations in the combustor between the
inlet temperature 1100K and initial temperature of 300K. However, the pressure
variations remain small at all times.

Fig. 3.2 shows the iso-contours of the instantaneous ﬁltered temperature and fuel
mass fraction at relatively long time as obtained by FD and MC. The FD values
are obtained directly from the ﬁnite-difference grid points and the MC variables are
obtained from the Monte Carlo particles and are interpolated to ﬁnite-difference grid

points for plotting. The results in Fig. 3.2 indicate that the Monte Carlo predictions

114

of the temperature and fuel mass fraction are almost the same as the ﬁnite differ-
ence results. Although the temperature/ density effects on molecular and turbulent
mixing are signiﬁcant, the data indicate very good consistency between FD and MC
suggesting that the ﬁnite difference and Monte Carlo solvers are both reliable and

accurate.

Reacting simulations without spray

The results obtained by FMDF/ MC and LES/ F D for reacting case without spray
are compared in this section. The reaction is for a lean-premixed n-heptane—air
mixture with the equivalence ratio of 0.7, which is modeled with a one-step global
mechanism [257]. The incoming cold fuel-air mixture is ignited by the preheated gas
mixture within the combustor.

As Fig. 3.3 indicates, the predictions of instantaneous temperature and fuel mass
fraction as by FMDF / MC for reacting ﬂow case without spray fairly well agree with
those of LES/ FD. Evidently, the values obtained by the FD and MC methods are
very close at all locations, indicating a good consistency between two methods in the

reacting case without spray.

Reacting simulations with spray

The consistency between LES / FD and FMDF / MC results for reacting ﬂows cannot
be established unless the nonlinear reaction source/ sink terms are calculated from
the FMDF method and used in the conventional LES/ FD equations. The reason
that these nonlinear terms have to be modeled in conventional LES/ FD equations,
while they are closed in the FMDF / MC equations.

Fig. 3.4 shows the comparison between the instantaneous ﬁltered temperature
and fuel mass fraction obtained with the two-phase FMDF/ MC and LES/F D for

reacting case with the spray. The spray is injected towards the centerline jet at the

115

inlet. The turbulent scalar mixing is very well captured by both models. There are
some negligible oscillations at the shear layer in the MC particle predictions, which
is mainly due to the restrictions in the MC particle averaging process and transfer
of Lagrangian data to Eulerian grids as discussed above.

In Fig. 3.5, the radial variations of the instantaneous temperature and mass frac-
tion of the evaporated fuel are plotted at axial locations of x/ D = 2 and x/ D = 4
for all three cases discussed above. Evidently, the Monte-Carlo particle values are
generally in good agreement with the ﬁnite difference ones, even in the case with
reaction and spray. The differences in the case with spray are primarily due to av-
eraging of the Lagrangian droplets that is intentionally conducted in an inconsistent
manner to show the signiﬁcance of the droplet averaging procedure. In this proce—
dure, the droplet source terms are ﬁrst calculated by averaging over ﬁnite-difference
cells and then the averaged values are interpolated to the Monte Carlo particle 10-
cations. A better approach is to ﬁrst interpolate the droplet values to the particle
location then calculate the droplet/source terms in the ﬁnite difference equations
from the particles. It is to be noted that in reacting and/or two-phase ﬂows the
reaction and droplet source/ sink terms in the ﬁnite difference equations are obtained
from the Monte Carlo particles. This is for the testing of the numerical methods
and for showing the consistency. Such information is not available in standard ﬁnite

difference methods.

3.3.2 Combustor Parameters And Boundary Conditions

In Fig. 3.6, a picture of the axisymmetric dump combustor and experimental setup [1]
are shown along with the computational and spray. Numerical simulations are per-
formed on a grid of 121 x 51 x 51 for a domain of 8 x 5 X 5 jet diameters in the x, y,
and 2 directions, respectively. The approximate number of MC particles is 1800000.

The ﬂow Reynolds number, calculated based on the inlet diameter and mean bulk

116

inlet velocity is 38580 in both experiment and LES. The ﬂow Mach number is 0.18.
We tried to keep the spray parameters are close to those suggested by Ken Yu et
al. [191], even though the spray is not well characterized.

For the inﬂow, non-reﬂecting boundary conditions [258] are used with a prescribed
inﬂow gas velocity based on experiment. First zero derivative boundary conditions
are considered for all variables on the combustor wall. For the outlet boundary,

non-reﬂecting boundary conditions are used.

3.4 Results and Discussion

The results of LES/FMDF for two-phase reacting simulations of Ken Yu’s spray-
controlled laboratory-scale lean dump combustor [191] are presented and discussed
in this section. The effects of various combustion and spray parameters similar to
those studied by Ken Yu et al. [191] have been investigated.

Table 3.1 summarizes the spray/ ﬂow parameters for all cases considered. Cases
I-V are to considered to study the effects of the initial Sauter mean diameter (SMD),
spray angle a, spray injection angle 6, liquid fuel to gas mass loading ratio I‘, and
droplet injection velocity up . Case VI is considered to asses the eﬂects of pulsative
spray with different values of droplet injection frequency w on the combustion. Fi-
nally, the effects of ﬂow parameters such as the inﬂow equivalence ratio g1) of the
premixed heptane/ air mixture, inlet temperature Tin and the amplitude of turbulent
ﬂuctuations aref are investigated by considering cases VII-IX. It is noted here that

the images in the following sections of this dissertation are presented in color.

3.4.1 Effect of Sauter Mean Diameter (SMD)

To examine the effect of initial droplet size on turbulent spray combustion, ﬁve dif-

ferent cases with initial Sauter mean diameter (SMD) of 30, 45, 60, 90 and 120 pm

117

are considered. SMD is an important spray parameter that has been commonly con-
sidered in spray studies [259]. The mass loading ratio of droplets, P is kept constant
around 2 0.1 when SMD changes by changing the number of injected droplets. The
other spray and ﬂow parameters are also kept constant as evident in Table 3.1.

Fig. 3.7 shows the iso-contours of instantaneous ﬁltered temperature and ﬁltered
fuel mass fraction as obtained by two-phase LES/FMDF with various Sauter Mean
Diameter. The relative amount of n-heptane fuel distribution in the combustor is
strongly affected by SMD as droplets follow trajectories depending on their initial
size. The droplets are injected at an initial temperature of 298K. For the n-heptane,
the boiling temperature is Tboz’l z 371K. The results in Fig. 3.7 indicate that for
smaller values of SMD most droplets closely follow the ﬂow due to their small in-
ertia. As shown in Fig. 3.7(a), (b), and (c), the small droplets even reach to the
locations close to exit, where they are ﬁnally vaporized. There are high levels of fuel
mass fraction in the center of the combustor, suggesting that most particles com-
pletely evaporate before they reach to exit nozzle. In contrast to small droplets, large
droplets are found to have high enough inertia to cross “high vorticity” and “high
temperature” areas, and reach to the side walls of the combustor. Even though the
life times of larger droplets are more than those of small ones, they quickly evaporate
in the high temperature regions, resulting in an increase in the fuel mass fraction
as shown in Fig. 3.7. The instantaneous ﬁltered temperature clearly indicate the
thermal effects of evaporating droplets on the gas. First, the droplets warm up to
their boiling temperature by the heat transfer from the carrier gas. The evaporation
starts when the dynamic equilibrium between the carrier gas and droplets is reached.
The gas temperature may ﬁnally rise due to the added vapor internal energy and
combustion.

The instantaneous radial values of the ﬁltered temperature, the ﬁltered fuel mass

fraction and the RMS of pressure as obtained by two-phase LES/FMDF are shown

118

in Fig. 3.8 at axial locations of :13/ D = 3 and :r/ D = 6 . Consistent with the former
iso—contour plots, higher values of fuel mass fraction are observed at locations close
to side walls of the combustor for large SMD. Nevertheless, the results for S M D =
45, 60pm show that for r / D < 0.5 as the droplet size decreases the fuel concentration
increases. However, for SM D = 30pm the fuel concentration is relatively low at
nearly all r/ D values and both axial locations, which indicates that a relatively
small number of droplets survive at downstream locations. Fig. 3.8 also shows the
cooling effect of vaporizing droplets. This is especially signiﬁcant for r/ D < 0.5 at
113/D = 3 and for r/D > 0.5 at x/D = 6, where the drop in gas temperature is
evident. It is also quite clear in Fig. 3.8 that for suppressing the pressure instabilities
smaller droplets are more desirable as the radial variations of the RMS of pressure is
higher for larger droplets, consistent with the experimental observations. However,
as the results with SM D = 30pm and SM D = 45pm indicate, injection of small
droplets damps the pressure oscillations due to lack of homogeneous combustion.

Fig. 3.9 shows the axial variations of the integrated droplet number density within
the combustor for various SMD values. At locations close to the spray injectors
and the combustor inlet, the integrated droplet number is approximately the same
for all SMD values with a peak indicating the droplet injection. Consistent with
Fig. 3.7 which shows the injected droplets, the integrated droplet number values
decay to zero at locations closer to inlet for the sprays with S M D = 30, 45am, in
comparison to those with higher SMD of 90 and 120nm. At a: / D m 2, the results for
S M D = 90, 120nm show peak values of droplet number, indicating the accumulation
of unburned droplets closed to the side walls of the combustor (Fig. 3.7). However,
the droplet number is still signiﬁcant at :13/ D m 5 for the spray with S M D = 60pm
since the droplets with this average size are neither large enough to cross the high
velocity main gas stream nor small enough to vaporize earlier.

The simulated values of the mean droplet temperature and the axial velocity

119

versus 7'/ D for various SMD are shown in Fig. 3.10 at axial locations of a: / D = 3 and
x/ D = 6. It is shown in Fig. 3.10a that the mean droplet temperature is generally
below the n—heptane liquid boiling temperature Tbotl as 1.25 for all SMD at both
downstream locations, indicating the accuracy of the droplet evaporation model.
At x/D = 3 and close to combustor axis (r/D < 0.5), the mean temperature is
relatively high as a result of low droplet evaporation for all cases with SMD values
greater than 30pm but decreases with the initial SMD values at locations away from
the combustor axis. The proﬁles of mean droplet axial velocity in Fig. 3.10b indicate
that the axial velocity of the smaller droplets decrease rapidly by the momentum
interactions with the carrier gas. The reason for higher mean droplet axial velocities
at :1:/ D = 3 in the centerline region for cases with S M D = 45, 60, 90, 120nm is the

reduced response to ﬂuid motions of larger droplets.

3.4.2 Effect of Spray Angle (oz)

Another important spray parameter considered in the present study is the spray angle
a. Four different cases with different spray cone angles from 00 (mono-dispersed) to
80° (widely-dispersed) are considered. The initial Sauter mean diameter is set to
45pm in all cases. Sample results are shown below.

Fig. 3.11 shows the effects of spray angle a on the instantaneous iso-surfaces of
density, and iso—contours of fuel mass fraction. Overall, the central region of the
combustor is dominated by high density gas due to relatively small droplet size for
all spray angles. Nevertheless, the density is higher close to the combustor side walls
for wider sprays (Fig. 3.11d). As the spray angle is increased, droplets penetrate less
in the axial direction causing lower fuel mass fraction values within the main fuel
stream. The instantaneous values of C02 mass fraction (not shown) also suggests
that a better droplet dispersion leads to more homogeneous and faster reaction.

Fig. 3.12 shows that for all spray angles the instantaneous ﬁltered temperature

120

are relatively close and the cooling effect of droplets is not very signiﬁcant away
from the combustor centerline. This is primarily due to small initial droplet SMD.
Nevertheless, the radial proﬁles of fuel mass fraction in Figure 3.12b are consistent
with the mass fraction iso—contours, indicating a higher fuel evaporation for smaller
spray angle close combustor centerline (r / D < 0.3). What also shown in Fig. 3.12 is
the radial variations of the RMS of pressure for different values of a. At locations close
to the walls, the RMS of pressure increases as the spray angle increases. However,
for the smaller spray angle of oz z 200, the pressure variations are relatively high
suggesting that other factors are important.

The integrated droplet number density plots in Fig. 3.13 indicate that more un-
burned droplets exist within the combustor for smaller spray angle, which is due to
' non-uniform droplet dispersion. Since the droplets for smaller spray angles mostly
reside within the relatively cold main ﬂow stream, droplet evaporation takes longer
time.

The radial variations of mean droplet temperature and axial velocity for various
0: at :13/ D = 3 and x/D = 6 are shown in Fig. 3.14. The. results indicate that the
mean droplet temperature is higher for a z 200, 400 and 600 compared to that for
oz % 00 at :13/ D = 3 due to higher heat transfer. Similarly, the mean droplet velocities
are higher for wider spray angles, partly because of momentum interactions between
the droplets and the carrier gas. Since the initially large droplets get smaller by
evaporation, which is more signiﬁcant for wider sprays, they get affected more by
the main ﬂow motions. At x/ D = 6, both the mean droplet temperature and axial
velocity attain their highest values for a z 800 close to combustor axis. By getting
away from the combustor axis the shear layer in the regions between the cold and hot
regions, the mean droplet temperature and axial velocity start to increase, suggesting

higher fuel evaporation and combustion at higher spray angles.

121

3.4.3 Effect of Injection Angle (,8)

The spray injection angle 5 is deﬁned as the angle between the spray axis and the
combustor axis. The location of injectors and the angle of injection has direct effect
on the evaporation and burning of the liquid fuel. In this section, LES/FMDF results
obtained for 6 a: 00, 300, 450 and 600 are presented and discussed.

The iso-surfaces and iso—contours of instantaneous ﬁltered gas density and tem-
perature and droplet dispersion for different 6 are shown in Fig. 3.15. As observed
in the ﬁltered density iso—surfaces, the density variations in the combustor is more
signiﬁcant for straight injection or F = 00. This can be related to the interactions of
the droplet with the shear layer and turbulence. Also, for 6 3 0°, the droplet cooling
effect is clearly observed at the points close to combustor side walls. This effect is
less signiﬁcant at other injection angles except at locations close'to the combustor
exit, which can be explained based on the interactions of sprays. As the injection
angle increases, the incoming droplets from two separate sprays disperse less, interact
more and act like a single spray inside the main incoming ”carrier gas stream, which
penetrates farther in the combustor.

Fig. 3.16 shows the radial variations of the instantaneous ﬁltered temperature
and fuel mass fraction at axial locations of .r / D = 3, 6 as obtained by LES/FMDF.
Consistent with the iso—contours of ﬁltered temperature (Fig. 3.15) and the fuel
mass fraction (not shown), the results with ,8 = 30°, 450 and 60" are found not to
be signiﬁcantly different than each other. From the radial variations of fuel mass
fraction, it can be stated that the injection angle increases the fuel concentration
in the main stream (T/ D < 0.5). As one moves outward radially to the sidewalls
(T/ D > 0.5), however, the fuel mass fraction values become higher for 6 = 00. The
radial variations of the RMS of ﬁltered pressure (Fig. 3.16c) indicate that the pressure
variations have signiﬁcantly higher values when ,8 = 00 not only at x/ D = 3 but also

at :13/ D = 6. There are discrepancies among the results obtained with different 6 at

122

T/ D < 0.5.

The axial variations of the integrated number of droplets are presented in Fig. 3.17
for different injection angles. Consistent with the visualized droplet distribution in
Fig. 3.15, there are less number of unburned droplets in the combustor for the case
with the injection angle of 6 = 00. Since all of the droplets are injected only into
the main carrier gas stream, the rate of evaporation of droplets and their integrated
numbers of droplets are pretty much the same for 6 = 300, 450 and 600.

The radial variations of the mean droplet temperature and axial velocity for
various injection angles are shown in Fig. 3.18 at axial location of x/D = 3. Our
results (not shown here) indicate that a small fraction of droplets survive at far

downstream locations when 6 = 450.

3.4.4 Effect of Mass Loading Ratio (I‘)

One of the important parameters that modify the turbulence/ combustion structure
is the mass loading ratio P which is the ratio of liquidfuel mass to carrier gas
mass. To further examine and understand the impact of this parameter on a spray
driven axisymmetric dump combustor, a set of simulations is performed with ﬁve
different mass loading ratios (I‘ = 0.01, 0.055, 0.11, 0.22 and 0.44). Fig. 3.19 shows
the instantaneous iso-contours of the ﬁltered temperature and fuel mass fraction for
different I‘. Here, the mass ﬂow rate of the liquid spray is controlled by multiplying
the droplet source terms in Eqs. 3.1-3.4 with a correction coeﬂicient rather than
changing the actual number of droplets in simulations due to computational cost.
The iso—contours of temperature for very high mass loading ratio (Fig.3.190) indicate
a signiﬁcant cooling effect of the droplets on the gas temperature, along with local
ﬂame extinction close to combustor exit. The absence of sufﬁcient oxygen is another
factor, which prevents the complete combustion and causes local ﬂame extinction.

It is also observed in Fig. 3.19 that the gas temperature diffusion in the central

123

region of the combustor increases with the loading ratio; high temperature values are
observed for r/ D < 0.5. Nevertheless, it seems that the temperature distributions
are not very different for mass loading ratios of F = 0.055, 0.11 and 0.22. For very
low mass loading ratios (Fig.3.19a), lower temperature values are observed in the
“thermal shear layer zone” mainly due to insufﬁcient fuel for combustion and not the
cooling effect of droplets as the fuel mass fraction iso-contours suggest. An interesting
observation is that by increasing the mass loading ratio, the sprays interact more and
the droplet scattering is decreased.

Figs. 3.20a and 3.20b show the radial variations of the instantaneous ﬁltered tem-
perature and fuel mass fraction at different axial locations. The general observation
is that the cooling effect of droplets is signiﬁcant only for F = 0.01 at :r/ D = 3.
For higher values of F, the radial proﬁles of the ﬁltered temperature for different F
are close to each other. At 33/ D = 6, the increase in mass loading ratio decreases
the carrier gas temperature at T/ D > 0.5 due to combustion and added mass for
F = 0.01. Since the relatively cold carrier gas stream gradually mixes and warms
up along the combustor centerline the temperature at r/ D < 0.5 increase in the
case with F = 0.01. However, the reason for the higher ﬁltered temperature values
observed in the cases with F = 0.44 at points r/D < 0.5 could be attributed to
diffusion of the hot combustion products from the outer burning zone to the central
region of the combustor as the vaporized cold fuel partly react and partly pushes the
gas to the center. The effect of dispersed droplet phase on the instantaneous ﬁltered
pressure for different mass loading ratios is shown in Fig. 3.20c. As expected, the
effect of droplets on the instantaneous mean gas pressure is signiﬁcantly different for
different mass loading ratios. An important observation is that by decreasing droplet
mass loading ratios from a very high value to a very low value the mean gas pressure
increases. There seems to be an optimum mass loading ratio for the lowest pressure

oscillations in the combustor.

124

In Fig. 3.21, the axial variations of the integrated droplet number density are
shown for different F. It appears that by increasing the droplet mass loading ratio, the
number of unburned droplets at locations close to combustor inlet and exit increases.
This indicates that the increase of the mass loading ratio decreases the relative rate
of the evaporation of individual fuel droplets, which is due to an increase in fuel
vapor mass fraction, since Yms — Yv determines the rate of evaporation [260].

The droplet statistics are also effected by the two-way coupling effects in a non-
linear way for various F. Fig. 3.22 shows the radial variations of the mean droplet
temperature and axial velocity at axial location of :r/ D = 3. Even though the devia-
tions between the results with low and moderate values of F are not very pronounced,
the LES/FMDF results with F = 0.44 seems to be very different. At :r/ D = 3, the
mean droplet temperature is high, where the mean droplet axial velocity is low due
to the signiﬁcantly increased thermal and momentum couplings between the carrier
gas-phase and the dispersed liquid-phase. Consistent with the results in Fig 3.19,
the droplet results (not shown here) indicate that a few unburned droplets survive

within the relatively cold main ﬂow stream along the centerline F = 0.11, 0.22.

3.4.5 Effect of Droplet Injection Velocity (up)

This section investigates the effects of droplet injection velocity up on both the car-
rier gas and turbulence, combustion and spray. Three different droplet injection
velocities, up = 15, 30 and 60 m/s are considered, while other spray and mean ﬂow
parameters remain as those given in Table 3.1.

Fig. 3.23 presents the iso—contours of the instantaneous ﬁltered temperature and
fuel mass fraction for different up. It is observed that droplets are able to cross the
carrier gas stream as the injection velocity is increased (Fig. 3.23c). As a result of this,
the carrier gas temperature eventually decreases where the droplets are vaporized.

The axial distance beyond which no droplets could be detected is much smaller for

125

higher injection velocities, since the droplets penetrate towards the sidewalls rather
than the central carrier gas stream. This trend is clearly seen in the fuel mass fraction
iso—contours in Fig. 3.23c where the elevated fuel mass fraction values, due to droplet
evaporation, are observed at the upper and the lower sides of the central gas stream.
Comparison of the results for up = 15 m/s and up = 30 m/s indicates that the
droplet evaporation increases as the injection velocity increases since the convective
heat transfer to the droplets is enhanced by the increased relative velocity between
the carrier gas and droplets.

In Fig. 3.24, the axial variations of the integrated droplet number density are
presented for different droplet injection velocities. Overall, the number of unburned
droplets decreases by increasing the injection velocity as argued above. Nevertheless,
at some locations (e.g. x/ D m 2) the number of unburned droplets are found to be
still less for up = 15 m/s in comparison to those for up = 30 m/s.

The radial proﬁles of the instantaneous ﬁltered temperature, fuel mass fraction
and total kinetic energy (TKE) at two different axial locations are shown in Fig. 3.25,
where TKE is deﬁned as ((21,) L(u,) L) / 2. For both axial locations, the ﬁltered tem-
perature is considerably lower for up = 60 m/s than those for up = 15, 30 m/s at
'r / D > 0.5, indicating the signiﬁcance of the cooling effect of droplets. The lower tem-
perature for up = 15 m/s is mainly due to turbulence-droplet thermal interactions
as Fig. 3.23a suggests. A comparison of the corresponding TKE proﬁles shown in
Fig. 3.25c demonstrates that the mean total kinetic energy increases as the droplets
are injected with higher velocities, which is again a consequence of momentum cou—

pling between phases.

3.4.6 Effect of Injection Frequency (11))

Pulsating liquid-fuel sprays are commonly used in active control of the combustion.

Suppressing the pressure oscillations via creating controlled out-of-phase heat release

126

oscillations is the basic idea behind the active control strategies [189]. Ken Yu et
al. [191] have showed that the pressure oscillation amplitude reached the minimum
value when the start of the pulsed fuel injection was synchronized with the inlet
vortex shedding process. They achieved up to a 15—dB reduction in sound pressure
level with phasing of fuel injection and pressure.

In pulsating sprays, the droplet injection frequency 1/2 is deﬁned as the total
number of openings and closings of the fuel actuators in one second. In other words,
the number of times the liquid—fuel is injected per second. Our study is motivated
by the desire to better understand the effects of droplet injection frequency on two-
phase turbulent reactive ﬂows. Simulations with various injection frequencies (z/J =
38H 2 ~ 1kH z) are conducted. For all cases studied, approximately 4200 droplets
are injected when the fuel actuators are opened.

Fig. 3.26 shows a series of 2D plots of the instantaneous ﬁltered temperature
and fuel mass fraction at different frequencies. Expectedly, the extent of spatial
fuel modulation become more evident at higher injection frequencies [189]. This is
attributed to shortening of the time gap between two pulses. Before the droplets
that are concentrated in some locations either evaporate or leave the combustor, the
liquid-fuel actuator are opened for next the injection. The cooling effect of evaporated
droplets is quite clear at these locations, as the temperature and the evaporated fuel
mass fraction contours show. However, it is observed that for low frequencies the
locations with high fuel concentration are rare, but their size is relatively larger.
From the comparison of both temperature and fuel mass fraction images at different
frequencies, it can be concluded that by increasing the droplet injection frequency
the large—scale discontinuities observed at very low frequencies in the main carrier
gas stream disappear.

In Fig. 3.27, the iso—surfaces of the instantaneous mean gas pressure are shown.

Comparison of the results with different injection frequencies indicates that in general

127

by an increase in the injection frequency, the mean gas pressure increases. The results
for 1b = 38H 2, 500Hz show that the mean gas pressure at locations close to inlet
is lower than that at the rest of combustor. Furthermore, signiﬁcant local pressure
drops are observed for i!) = 1kHz. These results suggest that the effect of spray on
the pressure ﬁeld is rather complicated and is strongly dependent on the injection
frequency. A systematic parametric study certainly needed for the selection of the
optimum operation conditions.

The radial variations of the instantaneous ﬁltered fuel mass fraction are shown
in Fig. 3.28 at axial locations of :r/ D = 3, 6. Expectedly, the fuel mass fraction
values are lower for lower injection frequency. Fig. 3.28 shows the effect of pulsating
spray on the mean gas pressure through radial variations of the RMS of pressure.
The pressure ﬂuctuations are lower for 1,!) = lkHz in comparison to those for other
frequencies, indicating that the damping effect of spray is more signiﬁcant at this par-
ticular frequency. As mentioned before, the damping of pulsating sprays is primarily
modiﬁed by creating out-of-phase heat release oscillations. Therefore, as Fig. 3.28
shows the pulsating spray is not that effective in suppressing the pressure oscillations

at other frequencies.

3.4.7 Effect of Equivalence Ratio (gb)

The effect of initial equivalence ratio of the carrier gas on combustion ﬂow variables
are here by changing the initial fuel-air equivalence ratio from d) = 0.01 through to
ab = 2.0, while the other ﬂow and spray parameters are kept constant (Table 3.1).
The instantaneous temperature and fuel mass fraction iso—contours in Fig. 3.29
indicate that the structure of the temperature and turbulence ﬁelds is changed by
altering the equivalence ratio. As the initial equivalence ratio of the carrier gas
decreases, the intensity of large-scale turbulent motions increases. As a result, the

relatively cold incoming main stream become more turbulent, causing locally reduced

128

temperature zones within the hot surrounding ﬂuid. In addition to its effect on tem-
perature, the enhanced turbulent mixing affects the distribution of the heptane mass
fraction within the combustor, also evident in high concentration fuel zones through-
out the combustor. Nevertheless, the spatially averaged values of the temperature
and fuel concentration (not shown here) increases and decreases, respectively, which
is expected since by an increase in fuel-air ratio in a lean combustor the reaction rate
and heat release increase.

Fig. 3.30 illustrates how the initial equivalence ratio of the carrier gas stream
affects the instantaneous vorticity ﬁeld. Evidently, as the initial equivalence ratio
decreases regions with high vorticity values start to appear in the combustor, which
promotes the formation rich detached elements from the tip of the main gas stream.

In Fig. 3.31, the radial variations of the instantaneous ﬁltered temperature, fuel
mass fraction and the RMS of pressure are presented at different axial locations. The
results at 33/ D = 3 show that the centerline temperature is somewhat similar for all
values of qt. Nevertheless, the structure of the temperature ﬁeld appear to be very
different in different fuel-lean and fuel-rich cases. The results with (I) = 0.01, 0.1
and 0.4 at :c/ D = 6 suggest that the high temperature ﬁeld engulfs more cold ﬂuid
from the central stream as 45 increases, which is an indirect effect of equivalence ratio
on the temperature through turbulence. As shown in Fig. 3.31b, the radial proﬁles
of instantaneous ﬁltered fuel mass fraction for ¢ = 0.01 and 0.4 are observed to
have higher values at points close to combustor sidewalls in comparison to those for
(25 = 0.7 —- 2.0. Fig. 3.31(c) shows that the pressure ﬂuctuations are also signiﬁcantly
affected by gb and are the highest when the equivalence ratio is 2.0. However, the
magnitude of the pressure ﬂuctuations does not seem to vary linearly and monoton-
ically with qb, as the RMS for (15 = 0.01 are generally higher compared to those for
¢ = 0.1 — 1.0.

Fig. 3.32a shows the radial variations of mean droplet temperature with different

129

 

values of ab at axial location 113/D = 3. It is observed from that at r/ D < 0.3, the
mean droplet temperature attains higher values as the equivalence ratio increases,
indicating more heat transfer from the carrier gas to the droplets. At 0.3 < r/ D <
0.6, the mean droplet temperature decreases as the equivalence ratio increases since
the relatively cold central main stream becomes thicker as (16 increases. The radial
proﬁles of the droplet dumber density in Fig. 3.32b show that by increasing the initial
equivalence ratio of the incoming premixed gas stream, the droplet evaporation rate
may increase or decrease. By increasing the equivalence ratio of a lean mixture,
heat release increases resulting in more convective heat transfer to the droplets and
faster evaporation on one hand, and smaller difference of 140,3 — Yv and lower droplet

vaporization rate on the other..

3.4.8 Effect of Inlet Gas Temperature (Tm)

In this section, a series of numerical experiments were conducted with different inlet
gas temperatures to quantify the dependence of turbulence/ spray interactions on the
ﬂow preheating. The heated inlet ﬂow simulations were performed at inlet tempera—
tures of 375K, 475K, 575K, 675K, 775K and 1100K. The initial equivalence ratio
of the heptane-air gas mixture was ﬁxed at 915 = 0.7.

Comparison of the instantaneous ﬁltered temperature and fuel mass fraction iso-
contours for various inlet temperatures in Fig. 3.33 indicates that a signiﬁcantly
smaller number of unburned droplets survive at downstream locations, which is ex-
pected. The cooling effect of droplet vaporization is, however, more signiﬁcant in
the results with higher inlet temperature, especially when the inlet gas temperature
is higher than the liquid boiling temperature. The iso—contours of fuel mass fraction
indicates that droplets tend to accumulate more as the inlet temperature decreases.
Additionally, the diffusion of the vaporized fuel increases as the inlet temperature

increases, which results in signiﬁcantly higher fuel concentration close to combustor

130

inlet.

The vorticity ﬁeld and how it is affected by the inlet gas temperature is shown
in Fig. 3.34. Apparently, as the inlet gas temperature decreases regions with high
vorticity values are created, particularly at :r/ D < 4. Compared to the cases with
Tin = 675, 775K, the vorticity values seem to be higher in case with Tin = 1100K
at 517/ D > 5. Since the reaction signiﬁcantly affects the density and pressure dis-
tribution, the higher inlet temperature alters the vorticity generation through the
baroclinic term in the vorticity equation [261].

Fig. 3.35 shows the radial variations of the instantaneous ﬁltered temperature,
fuel mass fraction and the RMS of gas ﬁeld pressure at axial locations of :1: / D = 3, 6.
At both axial locations, the ﬁltered temperature increases as the inlet temperature
increases, which is expected. However, the temperature for the case with Tin = 375K
seems to be higher at :12/ D = 6 close to the combustor centerline. The radial proﬁles
of fuel mass fraction shown in Fig. 3.35b indicate that for relatively lower Tin the
mass fraction of heptane in the main gas stream increases as the inlet temperature
increases due to the droplet vaporization. However, it decreases for highly heated
ﬂows as a result of signiﬁcant fuel consumption in the combustion process. The
radial variations of the pressure RMS in Fig. 3.35c demonstrate variations of pressure
ﬂuctuations with the inlet temperature. It is observed that in general for very high
or very low Tm the pressure ﬂuctuations are pronounced by high heat release or lack
of substantial droplet evaporation.

In Fig. 3.36, the axial variations of the integrated unburned droplet number den-
sity are shown for the cases with different inlet temperatures. Expectedly, the number
of droplets that still survive within the combustor decreases as the inlet temperature
of the carrier gas increases. Some droplets seem to be able to leave the combustor
without evaporation in the simulations with Tin = 375K and 475K, which is not

desirable.

131

The mean temperature and axial velocity values of the droplets are plotted ver-
sus the radial location of the droplets in Fig. 3.37 at axial location of x/D = 3.
Expectedly, the mean droplet temperature increases as the inlet carrier gas temper—
ature increases due to enhanced convective heat transfer. However, for very high
inlet temperatures (e.g. Tm 2 675K) where all of the droplets reach to the boiling
temperature point, the full evaporation of fuel droplets is unavoidable before they
reach to farther downstream locations. Analysis of the mean droplet axial velocity
(Fig. 3.37) for different inlet gas temperatures suggests that as the fuel droplets pass
through the ﬂame and start to evaporate due to convective heat transfer, the axial
velocity of the smaller droplets is decreased due to momentum interactions with the

carrier gas.

3.4.9 Effect of Inﬂow Turbulence

To investigate the effects of the inlet turbulence forcing amplitude on spray com-
bustion and on heat and mass transfer rate of dispersed droplets, simulations with
different magnitudes of ”ref (inlet turbulence forcing amplitude) are conducted in
this section.

Fig. 3.38 shows the iso—contours of instantaneous ﬁltered temperature and fuel
mass fraction for three different turbulence Uref- These iso—contours indicate that as
the forcing amplitude increases, there is more temperature and fuel vapor diffusion
within the combustor. The relatively cold and rich central gas stream is observed to
be totaly broken in the case with strong inlet turbulence (4 x oref), where the forma-
tion of concentrated droplet regions is quite clear. These “isolated droplet islands”
at some locations may cause local ﬂame extinction. The results also show that “the
burn-out” time of droplets becomes shorter as they escape from the relatively cold
central ﬂow stream and are signiﬁcantly exposed to hot gas regions.

In Fig. 3.39, the iso—contours of instantaneous vorticity ﬁeld for different turbu-

132

 

lence forcing are presented. Expectedly, the vortex structures which are primarily
responsible for the altered “circulation” and mixing of fuel and oxidizer seem to be
augmented by increasing the turbulence forcing. Interestingly, the high magnitude
vorticity contours are observed to be in the upper side of the combustor close to spray
injection regions. This suggests that the evaporating droplet mass and energy trans-
fer to the carrier gas can be also considered as mechanisms responsible for vortex
formation.

The radial variations of the instantaneous ﬁltered temperature, fuel mass fraction
and RMS of pressure at :r/D = 3 and 113/D = 6 are shown in Fig. 3.40. It is
observed that at both axial locations the temperature iso-contour values are higher
at r / D < 0.5 for higher turbulence forcing. On the other hand, the gas temperature
is observed to decrease as one moves out radially. This is mainly due to better mixing
of the hot products and cold fuel—rich mixture. This is also quite clear in the proﬁles
of the fuel mass fraction shown in Fig. 3.40b, where the fuel concentration is higher for
higher turbulence forcing at locations close to combustor axis and lower at locations
close to side walls. Thus, by increasing the forcing amplitude more homogeneous
droplet evaporation and combustion is expected. Nevertheless, as shown in the radial
proﬁles of the RMS of mean pressure (Fig. 3.40c), the pressure ﬂuctuations seem
to be much higher for high Uref (4 x Urefl- On the other hand, the results with
moderate turbulence forcing (2 X Uref) indicate that the pressure ﬂuctuations are
better suppressed compared to those in cases with the lowest Uref-

The axial variations of the integrated droplet number density for different tur-
bulence forcing are shown in Fig. 3.41. It is observed that the droplets completely
burn-out earlier as the turbulence forcing increases. Further integration of the droplet
number density proﬁle along the x-axis (not shown) indicates that the highest droplet
evaporation ratio is achieved in the results with a moderate inlet turbulence intensity

(2 x Urefl- Consequently, there seems to be no clear correlation between the droplet

133

 

evaporation rate and the inlet turbulence intensity, even though by increasing Uref
the combustion becomes more homogeneous.

The effects of inlet turbulence on droplets are illustrated in Fig. 3.42 via the
radial variations of the mean droplet temperature and axial velocity at :13/ D = 3.
Both the droplet temperature and velocity are observed to be lower for the highest
simulated inlet turbulent intensity (4 x aref). Our other results (not shown) indicate
that this can be attributed to the droplet size. Due to evaporation delay, larger
droplets survive at this particular location, which tend to preserve more their initial

temperature and velocity.

3.5 Summary and Conclusions

Large eddy simulations (LES) of two—phase turbulent reacting ﬂows in a spray-
controlled axisymmetric dump combustor are conducted to investigate the effects
various spray properties and ambient ﬂow conditions on the droplet and carrier
gas temperature, pressure, and velocity. A hybrid Lagrangian-Eulerian-Lagrangian
methodology is developed and implemented for solving the Lagrangian spray, Eu-
lerian ﬁnite-difference (FD) velocity, and Lagrangian-Monte Carlo (MC) gas scalar
equations via conventional numerical models and a PDF-based two-phase subgrid
combustion model, termed the ﬁltered mass density function (F MDF). Full two-way
mass, momentum and energy coupling between the evaporating droplets and the
carrier lean-premixed heptane-air gas mixture are considered. Different initial values
of droplet Sauter mean diameter, spray and injection angles, liquid fuel to gas mass
loading ratio, droplets injection velocity, pulsative spray injection frequency, carrier
gas equivalence ratio, inlet temperature and turbulence intensity are considered.

The results for non-reacting, reacting without spray and spray-controlled com-

bustion cases show that the LES/ FD and FMDF/ MC are consistent in all tested ﬂow

134

~

- “a

. -
.i. I. .-
J.

44‘

conditions. In a qualitative agreement with experimental observations [262, 263], the
LES/FMDF results indicate that the initial droplet size has a strong inﬂuence on
the ﬂow dynamic and thermal characteristics of the carrier gas and the evaporating
droplets. For example, with an increase in initial droplet size, the rate of evaporation
decreases. The mass, momentum and energy of the evaporating droplets decrease
the mean gas temperature and increase combustion of lean mixture. Consistent with
experimental data [250], smaller droplets closely follow the ﬂow structure, while large
droplets are found to cross high vorticity areas and reach to the wall where they go
through a complicated interaction process.

The results indicate that as the spray angle is increased the penetration of the
droplets decreases. Therefore, droplets are vaporizing at upstream locations, which
yields higher fuel mass fraction and combustion at those locations. It is also observed
that the burn-out ratio of the injected droplets decreases as the spray cone angle de
creases. Similarly, the results with different injection angles indicate that the droplet
penetrate more in the axial direction as the injection angle increases. Except for zero
degree injection angle, where the injection is parallel to the combustor axis, the total
number of vaporized droplets seems to be close to each other for different injection
angles since the droplets are injected into the core carrier gas stream. Nevertheless,
our results (not shown) indicate that the effect of injection angle is more signiﬁcant
when larger droplets are dispersed. Both for large spray and injection angles, the
pressure ﬂuctuations are observed to be signiﬁcantly suppressed.

The effect of droplet mass loading ratio is investigated in several cases, ranging
from highly dilute to highly dense sprays. It is shown that by increasing the droplet
mass loading the rate of evaporation of individual droplets decrease. Local ﬂame
extinctions are also observed for very high mass loading ratios due to signiﬁcant
droplet cooling effect and insuﬂicient oxidizer to sustain the combustion.

By comparing the results with different droplet injection velocities, it is found

135

ff‘R-n

that droplets with higher injection velocities are able to cross the high vorticity
regions and reach to high temperature reacting zones, resulting in an increase in the
droplet evaporation. Furthermore, it is found that the mean total kinetic energy of
the carrier gas and the vorticity are both enhanced when droplets are injected with
higher velocities. Our results (not shown) also indicate that the pressure ﬂuctuations
could be decreased beyond a certain point by injecting faster droplets. The results
with diﬂerent injection frequencies demonstrate that the damping effect of spray is
quite signiﬁcant at a certain high injection frequency. This suggests that by creating
out of-phase heat release oscillations the pressure ﬂuctuations can be well managed.

Investigation of the effects of carrier gas properties on combustion indicates that
the physical structure of mean gas is modiﬁed differently by the spray for different
initial equivalence ratio, inlet temperature and turbulence intensity. As the inlet
equivalence ratio decreases or the inlet turbulence intensity increases, the large-scale
turbulent motions are observed to become more pronounced, resulting in very dif-
ferent spray effects on the combustion. The radial variations of droplet properties
indicate that by increasing the initial equivalence ratio the droplet evaporating rate
tends to increase due to higher heat release. Comparison between the cases with dif-
ferent turbulence intensity indicates that even though the stronger turbulence causes
more homogeneous combustion, there seems to be no signiﬁcant change in the total
number of vaporized droplets.

Our results reveal several important physical effects of different spray/ ﬂow pa—
rameters on two-phase turbulent reacting ﬂows and indicate that the numerical sim-
ulations via hybrid Lagrangian-Eulerian-Lagrangian two-phase LES / FMDF method-
ology is affordable, consistent and reliable for turbulent spray combustion in complex

systems.

136

 

3.6 Tables and Figures

137

 

 

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138

Spray-Control led
_ Dump Combustor

    
 
    

   

_1', r ." l - I
:‘ne ‘ .- .rwr’”.

 

7‘ . - -..::m..é'+-.;'z. (.1;
/ Eulerian Grid -—
f, Fuel Droplets O
U Monte Carlo Particles 0
§ Mass, Momentum, Scalar
5 Terms from Drc plets
3 .
Lu

 

\‘
‘ \
\ s
\ .

 

 

OFuel Droplets . . .

0 Monte Carlo m 2211?...»
Particles ‘ l A] s. . . :
Eulerian Finite memo anon

M t C 1
Difference Grid Operator ﬁning 0

Figure 3.1: The Eulerian and the Lagrangian ﬁelds of the two-phase LES/FMDF
methodology.

 

Finite Difference Monte Carlo

Figure 3.2: ISO-contours of the instantaneous ﬁltered temperature and fuel
mass fraction in the non-reacting axisymmetric dump combustor without
spray as obtained by LES/FDMF; (a) FD values of temperature, (b) MC

values of temperature, (c) FD values of fuel mass fraction, and ((1) MC
values of fuel mass fraction.

139

 

 

 

 

 

 

 

d)

Finite Difference ( Monte Carlo
Figure 3.3: ISO-contours of the instantaneous ﬁltered temperature and
fuel mass fraction in the reacting axisymmetric dump combustor without
spray as obtained by LES / FDMF ; (a) FD values of temperature, (b) MC
values of temperature, (c) FD values of fuel mass fraction, and ((1) MC
values of fuel mass fraction.

 

‘5 _ﬁl’.mt H .
‘5 , a» ’ " .
F 5' ., ., n. '
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(b)
7 “£341:
(C)
Finite Difference Monte Carlo

Figure 3.4: ISO-contours of the instantaneous ﬁltered temperature and fuel
mass fraction in the reacting axisymmetric dump combustor with spray
as obtained by LES / FDMF; (a) FD values of temperature, (b) MC values
of temperature, (c) FD values of fuel mass fraction, and (d) MC values of
fuel mass fraction.

140

s..- .

x/D = 2 x/D = 4 x/D =

 

   

 

  
  
  

 

 

 

 

   

     

«2 6 v 6 g . g
E 5 u. 11..
E e m m
8.2 8.2 53 ' 8 '
g g 2 2
I— 00 0.8 16 1— 0o 0.8 1.6 8 g
r/D r/D Le (44
(a)
0 9 0 9 3 . §0.06
5 E u. u. .
E 6 ' e 6 m m
8.3. 33 53 . {30.03
g g 2 2
E— 0 . H 00 8 16 8 B 0 . 6
r/D r/D “-4 U-a r/D
(b)
010 «>1 ,8; 6
LL. LL.
gs «‘3 § a
g g 2 2
I— . 1.6 l— T’ 7’
r/D Li L:
(C)
Instantaneous Temperature Instantaneous Fuel Mass Fraction

Figure 3.5: Radial variations of the instantaneous ﬁltered temperature
and fuel mass fraction at axial locations of x/ D = 2 and x/D = 4; (a) Non-
isothermal non-reacting case without spray, (b) Non-isothermal reacting
case without spray, (c) Non-isothermal reacting case with spray. The
results are obtained by averaging the instantaneous data in the azimuthal
direction and not in time.

141

I . t Combustion Chamber
gnl er Spray Injectors

    

(a)

 

(b)

Figure 3.6: Images of the axisymmetric dump combustor; (a) Picture of experimental
setup [1] and (b) Illustration of computational domain and droplet injection.

142

 

 

 

 

 

 

 

 

 

 

 

 

Instantaneous Fuel Mass Fraction

 

T= l.60 7.00 YF= 0.00 0.20

Figure 3.7: ISO-contours of the instantaneous of ﬁltered temperature and
fuel mass fraction as obtained by LES / FDMF with various SMD values;
(a) SMD = 30pm, (b) SMD = 45pm, (c) SMD = 60am, ((1) SMD 2 90am,
(8) SMD = 120nm.

143

 

 

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0

A
m
v
9
DJ

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Fuel Mass Frac.
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r/D

 

(C)

Figure 3.8: Radial variations of various quantities at axial locations of
m/D = 3 and 33/0 = 6 with various SMD values; (a) Instantaneous ﬁl-
tered temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c) Root
mean square (RMS) of ﬁltered pressure (solid line, 30pm; dashed line,
45pm; long-dashed line, 60pm; dashed-dot line, 90pm; dashed-dot-dot line,
120nm).

144

 

 

 

 

 

 

Figure 3.9: Axial variations of integrated droplet number density with various SMD
values (solid line, 30pm; dashed line, 45pm; long-dashed line, 60pm; dashed-dot line,
90pm; dashed-dot-dot line, 120nm).

 

145

x/D=3 x/D=6

 

p—s
N
p—a
N

 

 

 

 

 

 

 

Mean Droplet Temp
1
Mean Droplet Temp

A
m
v

.12 -1.2

 

 

 

 

Mean Droplet Vel
8
7I

Mean Droplet Vel
O
O\

 

 

 

     

 

. — 0.8 1.6
r/D r/D
(b)

Figure 3.10: Radial variations of various droplet quantities at axial loca-
tions of x/ D = 3 and m/ D = 6 with various SMD values; (a) Mean droplet
temperature, (b) Mean droplet axial velocity (solid line, 30pm; dashed
line, 45pm; long-dashed line, 60pm; dashed-dot line, 90pm; dashed-dot-dot
line, 120nm).

146

 

Instantaneous Density Instantaneous Fuel Mass Fraction

—t_ _r_
P = 0.15 0.50 YF= 0.00 0.20

Figure 3.11: ISO-contours of the instantaneous ﬁltered density and fuel
mass fraction as obtained by LES / FDMF with various spray angles; (a)
020, (b) (1:820, (c) az40, (d) 0%80.

147

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

10 10
5 £23

c6 c3

33 33

2.5 Q5”

8 E

d) d.)
1—1 1—4 .

0 ‘ 0.8 1.6 0 ‘ 0.8 ‘ 1.6

r/D r/D

(a)
o’ o’
303- EEO-3'
LL. u. .
m m "
$0.2 $0.2
E61 E61
:0 . o .
:1 ~ 5
"5 00 u“ 00

   

A
on
V

 

 

   

 

 

 

 

 

 

C 2 v 2

S S

U) U)

CD U)

9.3 23

0.. 9..

CH CH

O C

CD U)

500 ‘ 018 ‘ 116 500 ‘ 0:8 ‘ 1i6

r/D r/D

(C)

Figure 3.12: Radial variations of various quantities at axial locations of
x/ D = 3 and 93/ D = 6 with various spray angles; (a) Instantaneous ﬁltered
temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c) Root mean
square (RMS) of ﬁltered pressure (solid line, a z 0; dashed line, a as 20;
long-dashed line, a z 40; dashed-dot line, a z 80).

148

 

 

12

 

 

 

 

'2. w» .
1h",
'Z' 4 All“! "- . .2 .
\ ‘ “kiwi: l.‘ {Hi}
Nu»- W...” l," ,
0o 4 8

Figure 3.13: Axial variations of integrated droplet number density with various spray
angles (solid line, a z 0; dashed line, a m 20; long-dashed line, a z 40; dashed—dot
line, a z 80).

 

149

 

x/D=3 x/D=6

 

 

H
N
p—d
N

 

 

 

 

 

Mean Droplet Temp
Mean Droplet Temp.

 

0.8 ‘ 1.6

p—t
C
.9
OO
ﬁ—t
O\

A
93
v
p—t
N

 

1.2

 

  

 

Mean Droplet Vel
2

Mean Droplet Vel.
O
O

 

 

 

 

 

é:
Pr
00
_‘l-
O\
c9
5

. v 0:8 1:6
r/D r/D
(b)

Figure 3.14: Radial variations of various droplet quantities at axial loca-
tions of :r/ D = 3 and 513/ D = 6 with various spray angles; (a) Mean droplet
temperature, (b) Mean droplet axial velocity (solid line, a z 0; dashed
line, a as 20; long-dashed line, a z 40; dashed-dot line, a z 80).

150

 

 

(b)

 

    

4 -1
x/D 6 8 ‘ %D
(d)

Instantaneous Density Instantaneous Temperature

P= 0.15 0.50 T= 1.60 7.00

Figure 3.15: ISO-contours of the instantaneous ﬁltered density and tem-
perature as obtained by LES/FDMF with various injection angles; (a)
(mo, (b) ﬁz30, (c) ﬁz45, (d) ﬁ~60.

151

 

x/D=3 x/D=6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

10 10

E E

s f s

O.) O.)

0., 5- g, 5

E E '

d.) d)

E“ [—

0 ‘ 0.8 ‘ 1.6 o ‘ 018 ‘ 116
r/D r/D

(a)

g o'

H 0.3* a 03»

LR t. LL. »

a 0.2-‘\- a 0.2;

_. 0.1 _. 0.1

(D , d)

s s

L“ 00 L“ 00
(b)

D 2 v 2

S S

a a

Q) G)

H H

9.. 9..

m m

C O

g E

 

 

   

(C)

Figure 3.16: Radial variations of various quantities at axial locations of
as/D = 3 and x/D = 6 with various injection angles; (a) Instantaneous
ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c)
Root mean square (RMS) of ﬁltered pressure (solid line, ﬂ m 0; dashed
line, B m 30; long-dashed line, ﬂ 2 45; dashed-dot line, ﬂ z 60).

152

 

 

 

 

 

 

 

 

Figure 3.17: Axial variations of integrated droplet number density with various in-
jection angles (solid line, B m 0; dashed line, ﬂ z 30; long-dashed line, H w 45;
dashed-dot line, ,6 z 60).

 

 

 

153

x/D=3

 

 

 

 

 

 

 

 

 

9:12
E
Q)
E"
.13
D4
0
H
c:
g
o l
2 1.6
(a)
_.1.2
O)
>
E’ \\'-.
5:06-
a
5
E
00 “ 018 1:6

r/D
(b)
Figure 3.18: Radial variations of various droplet quantities at axial loca-
tion of 27/ D = 3 with various injection angles; (a) Mean droplet tempera-

ture, (b) Mean droplet axial velocity (solid line, 3 m 0; dashed line, ﬂ 2 30;
long-dashed line, ﬂ a: 45; dashed-dot line, ﬂ 2 60).

154

 

 

 

 

 

 

 

 

 

 

 

(d)
err"; _W' "—'— '_ ﬂ
’2 .,. 3"):

' .-'.. whgég. ” ~' ‘ .
. T3.._ “We." .- ”m , - W “
4 ‘ u... .

 

     

.. ”‘3'; 1.; i "I 1
(e)
Instantaneous Temperature Instantaneous Fuel Mass Fraction
T= 1.60 7.00 YF= 0.00 0.20

Figure 3.19: Iso-contours of the instantaneous of ﬁltered temperature and
fuel mass fraction as obtained by LES / FDMF with various droplet mass
loading ratios; (a) I‘ = 0.01, (b) I‘ = 0.055, (c) I‘ = 0.11, (d) P = 0.22, (e)
I‘ = 0.44.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

10 10
E E

8 $3 .

8,5- a ‘- '
{-4 [—< J

0 ‘ 018 ‘ 116 0 018 ‘ 116

r/D r/D

(a)

50.6 C50.6

E E3
LL. N. LL.

3 \‘= a

«s 0.3~ \‘ cc: 0.3%
E 2
T) T)
5: ~ - a

00 0.8 1.6
r/D

(b)

32 32

 

 

 

 

 

 

 

(C)

Figure 3.20: Radial variations of various quantities at axial locations of
rc/ D = 3 and :1:/ D = 6 with various droplet mass loading ratios; (a) Instan-
taneous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass frac-
tion, (c) Instantaneous ﬁltered pressure (solid line, I‘ = 0.01; dashed line,
I‘ = 0.055; long-dashed line, F = 0.11; dashed-dot line, 1" = 0.22; dashed-dot-
dot line, F = 0.44).

156

 

 

 

 

 

 

 

 

Figure 3.21: Axial variations of integrated droplet number density with various
droplet mass loading ratios (solid line, I‘ = 0.01; dashed line, F = 0.055; long-dashed
line, F = 0.11; dashed-dot line, I‘ = 0.22; dashed-dot-dot line, F = 0.44).

 

 

157

x/D=3

 

p—L
N

  

 

 

 

Mean Droplet Temp.

 

 

 

 

 

10 0.8 1.6
r/D
(a)
_, 1.2
G)
>
E5
$0 6
Q
g
E
00 ‘ 6'8 116

(b)

Figure 3.22: Radial variations of various droplet quantities at axial lo-
cation of a: / D = 3 with various droplet mass loading ratios; (a) Mean
droplet temperature, (b) Mean droplet axial velocity (solid line, I‘ = 0.01;
dashed line, I‘ = 0.055; long-dashed line, I‘ = 0.11; dashed-dot line, I‘ = 0.22;
dashed-dot-dot line, I‘ = 0.44).

158

 

 

 

 

 

 

 

 

 

 

Instantaneous Temperature Instantaneous Fuel Mass Fraction

T: 1.60 7.00 Yr: 0.00 0.20

Figure 3.23: ISO—contours of the instantaneous of ﬁltered temperature
and fuel mass fraction as obtained by LES/FDMF with various droplet
injection velocities; (a) up = 15m/s, (b) up = 30m/s, (c) up = 60m/s.

 

 

 

 

 

Figure 3.24: Axial variations of integrated droplet number density with various
droplet injection velocities (solid line, up = 15m/s; dashed line, up = 30m/s; long—
dashed line, up = 60m/s).

159

 

x/D=3 x/D=6

 

p—s
O
.—A
O

 

Temperature
Temperature
LII

 

 

 

 

 

 

 

A
N
v

 

0.3

 

 

 

 

Fuel Mass F rac.

 

Fuel Mass Frac.

 

 

   

A
on
v

 

 

 
 

 

 

 

 

   

 

 

0.8 1.6 00 ’ 018

r/D r/D
(C)

Figure 3.25: Radial variations of various quantities at axial locations of

:13/ D = 3 and x/ D = 6 with various droplet injection velocities; (a) Instan-

taneous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass frac-

tion, (0) Instantaneous ﬁltered total kinetic energy (TKE) (solid line,
up = 15m/s; dashed line, up = 30m/s; long-dashed line, up = 60m/s).

160

 

 

 

 

 

 

 

 

 

rat) 1

 

 

 

40‘"
(d)
Instantaneous Temperature Instantaneous Fuel Mass Fraction
T= 1.60 7.00 YF= 0.00 0.20

Figure 3.26: ISO-contours of the instantaneous ﬁltered temperature and
fuel mass fraction as obtained by LES/FDMF with various injection fre-
quencies; (a) w = 38Hz, (b) 1,11 = 250Hz, (c) 1/2 = 500112, (d) 1/2 = 1kHz.

161

 

P=22 ” 29

Figure 3.27: ISO—surfaces of the instantaneous ﬁltered pressure as obtained by
LES/FDMF with various injection frequencies; (a) I/J = 38H 2, (b) 10 = 250Hz, (c)
II) = 500Hz, (d) 1/2 =1kHz.

162

 

x/D=3 x/D=6

 

.9
o.)
o
w

 

.0
N
/

Fuel Mass Frac
O
'l
o/' I,
/
Fuel Mass F rac
o .o
v— N
l' f '
//

 

 

 

 

 

00 0.8 1.6

A
DD
V

 

 

 

 

 

 

RMS of Pressure

 

RMS of Pressure

 

 

 

(b)

Figure 3.28: Radial variations of various quantities at axial locations of
223/ D = 3 and 33/ D = 6 with various injection frequencies; (a) Instanta-
neous ﬁltered fuel mass fraction, (b) Root mean square (RMS) of ﬁltered
pressure (solid line, 7,!) = 38H 2; dashed line, 10 = 25OH z; long-dashed line,
1/1 = 500Hz; dashed-dot line, «[2 = 1kHz).

163

 

 

 

 

 

 

 

 

 

 

 

     

T= 1.60 7.00 YF= 0.00 0.20

Figure 3.29: Iso-contours of the instantaneous of ﬁltered temperature
and fuel mass fraction as obtained by LES/FDMF with various initial
equivalence ratios; (a) (25 = 0.01, (b) q) = 0.1, (c) 05 = 0.4, (d) (1) = 0.7, (e)
d) = 1.00, (f) 05 = 2.00.

164

 

  

812/94

 

(f)
I-I_
0): 1.00 0.60

Figure 3.30: ISO-surfaces of the instantaneous vorticity ﬁeld as obtained by
LES/FDMF with various initial equivalence ratios; (8.) 05 = 0.01, (b) (b = 0.1, (0)
¢ = 0.4, (d) ,0 = 0.7, (e) ¢ =1.00, (f) ¢ = 2.00.

165

 

 

 

p—s
O
h-d
O

Temperature
Temperature

 

 

 

 

 

 

   

A
93
v

 

 

p
m

.9
N

 

 

 

Fuel Mass Frac.
O

 

Fuel Mass Frac.

 

   

A
on
v

 

 

   

 

 

 

 

 

 

a) _ a)
g 2 .11 g
m (I)
(D U)
52’ £23
1 ’.-
9H --', CH
O J:§.;$"..\ /1 . \ o
m L—/"--"V:./ ........... \H m
E 00 018 116 E 00 018 1'6
r/D r/D

(C)

Figure 3.31: Radial variations of various quantities at axial locations of
33/ D = 3 and z/ D = 6 with various initial equivalence ratios; (a) Instanta-
neous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass fraction,
(c) Root mean square (RMS) of ﬁltered pressure (solid line, 05 = 0.01;
dashed line, ()5 = 0.1; long-dashed line, (15 = 0.4; dashed-dot line, 05 = 0.7;
dashed-dot-dot line, 05 = 0.1; dotted-delta line <25 = 0.2).

166

 

 

x/D=3

 

Mean Droplet Temp.

 

 

 

 

(a)

 

 

 

 

 

(b)

Figure 3.32: Radial variations of various droplet quantities at axial loca-
tion of :r / D = 3 with various initial equivalence ratios; (a) Mean droplet
temperature, (b) Total droplet number (solid line, 0’) = 0.01; dashed line,
05 = 0.1; long-dashed line, 06 = 0.4; dashed-dot line, 05 = 0.7; dashed-dot-dot
line, 05 = 0.1; dotted-delta line 05 = 0.2).

167

 

 

 

 

 

 

 

Instantaneous Temperature Instantaneous Fuel Mass Fraction

T= 1.60 7.00 YF= 0.00 0.20

Figure 3.33: ISO-contours of the instantaneous of ﬁltered temperature and
fuel mass fraction as obtained by LES/FDMF with various inlet temper-
atures; (a) Tin = 3750K, (b) Tm = 4750K, (c) Tin = 5750K, (d) Tin = 6750K,
(e) Tm = 7750K, (f) Tin = 11000K.

168

 

  

'>
'-1
8185‘

 

(f)
I—B_
(D: 1.00 0.60

Figure 3.34: ISO-surfaces of the instantaneous vorticity ﬁeld as obtained by
LES/FDMF with various inlet. temperatures; (a) Tin = 3750K, (b) Tin = 4750K, (c)
Tin, = 5750K, (d) Tin = 6750K, (e) Tin = 7750K, (f) Tin = 11000K.

169

 

 

 

 

x / D =
l 0
o (L.
5 A1“- C-Wﬁi‘FL\sg-m
21%, £.?’/ 7", .......... .-

w I. 0 In" .........
H ,o H!
(D / '/
g. 5 - .-//,:
E -’//-"
o _ 44} I;

E“ 23.1%.?

0 0.8 1.6
r/D
(a)

 

0.3

.o
N

 

Fuel Mass Frac.
O

A
at
v

 

 

 

N

A

 

 

 

RMS of Pressure

(C)

 

x/D=6

 

h—L
O

A

Temperature

bl$'¢'&.¢
'A.I~_¢_an‘h/- ‘1'—

 

oo" /
I
I

/' ,.//

     

‘§
h
‘-.

 

 

 

13

 

.9
o:

.O
N

Fuel Mass Frac.
O

RMS of Pressure

 

 

 

 

r/D

 

y.

 

 

[v«a.Ana-a.-A..A--A..A..A_,A__A ‘6
. "A”

 

 

Figure 3.35: Radial variations of various quantities at axial locations of
x/ D = 3 and 122/D = 6 with various inlet temperatures; (a) Instantaneous
ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass fraction, (c)
Root mean square (RMS) of ﬁltered pressure (solid line, Tin = 3750K;
dashed line, Tin = 4750K; long-dashed line, Tin = 5750K; dashed-dot line,
Tgn = 6750K; dashed-dot-dot line, Tm = 7750K; dotted-delta line Tin =

11000K).

 

 

 

 

Figure 3.36: Axial variations of integrated droplet number density with various inlet
temperature (solid line, Tin = 3750K; dashed line, Tin = 4750K; long-dashed line,
Tin = 5750K; dashed-dot line, Tin = 6750K; dashed-dot-dot line, Tin = 7750K;
dotted-delta line Tin = 11000K).

171

x/D=3

p—a
N

 

 

 

Mean Droplet Temp.

 

A
93
v
p—s
N

 

Mean Droplet Vel.
O

 

 

.AAAAAAAAAAA

08 1.6
r/D

 

 

(b)

Figure 3.37: Radial variations of various droplet quantities at axial lo-
cation of x/ D = 3 with various initial carrier gas temperatures; (a)
Mean droplet temperature, (b) Mean droplet axial velocity (solid line,
Tin = 3750K; dashed line, Tm = 4750K; long-dashed line, Tin = 5750K;
dashed-dot line, Tin = 6750K; dashed-dot-dot line, Tin = 7750K; dotted-
delta line Tin = 11000K).

172

all

 

 

 

5" 1
Ilka.“

w
I
[J
'1‘“
as,”

 

 

 

 

 

 

(C)

Instantaneous Temperature Instantaneous Fuel Mass Fraction
T= 1.60 7.00 YF= 0.00 0.20

Figure 3.38: ISO-contours of the instantaneous of ﬁltered temperature and
fuel mass fraction as obtained by LES/FDMF with various turbulence
forcing amplitudes; (a) 0ref9 (b) 2 x aref, (c) 4 X Uref'

173

 

    

4
x/D 6 819/1)
(C)
_@_
(1):].00 0.60

Figure 3.39: ISO-surfaces of the instantaneous vorticity ﬁeld as obtained by
LES/FDMF with various turbulence forcing amplitudes; (a) arefv (b) 2 x Uref: (c)
4 X ”ref-

174

 

X/D=3 x/D=6

 

 

 

 

 

 

 

 

 

10 10
E E
:6 c6
33‘ t";

g, 5 Q4 5 ,.
E E "
CD Q)
E-' {-4
0 ‘ 0.8 1 1.6 0
r/D

 

A
DD
V

 

 

.O
m
.o
o)

.O
N
.C
N

 

 

 

Fuel Mass Frac.
O

 

   

 

 

Fuel Mass Frac.
O

c?
o
oo
p—A
Ox

00 10.8113

A
U"
v

 

N

 

 

 

 

 

 

 

 

RMS of Pressure
RMS of Pressure

as
or-
00
H)-
ON

 

(C)

Figure 3.40: Radial variations of various quantities at axial locations of
23/ D = 3 and 23/ D = 6 with various turbulence forcing amplitudes; (a)
Instantaneous ﬁltered temperature, (b) Instantaneous ﬁltered fuel mass
fraction, (c) Root mean square (RMS) of ﬁltered pressure (solid line, Uref;
dashed line, 2 x ”ref; long-dashed line, 4 x aref).

175

 

 

 

 

 

Figure 3.41: Axial variations of integrated droplet number density with various tur-
bulence forcing amplitudes (solid line, ”ref; dashed line, 2 X Uref; long-dashed line,

4 X 0ref)-

176

x/D=3

 

 

 

 

Mean Droplet Temp.

1.6

 

A
93
v

 

1.2

Mean Droplet Vel.
0
ON

 

 

 

 

O0 ‘ . 018 116
r/D
(b)

Figure 3.42: Radial variations of various droplet quantities at axial lo-
cation of z/ D = 3 with various turbulence forcing amplitudes; (a) Mean
droplet temperature, (b) Mean droplet axial velocity (solid line, aref;
dashed line, 2 x aref; long-dashed line, 4 x 0ref)~

177

 

3.7

11,2

Nomenclature

: Speciﬁc heat of the mixture at constant pressure
: Constant of the stochastic mixing closure

: Constant in the subgrid scalar stress

: Constant in the subgrid scalar stress

: Molecular diffusion coefﬁcient

: Subgrid diffusion coefﬁcient

: Drift coefficient in the SDE

: Diffusion coefﬁcient in the SDE

: Joint scalars ﬁltered mass density function
: Filter function

: Filter function

: Enthalpy

: Enthalpy of species a

: Enthalpy of formation of species a

: ith component of the ﬂux of scalar a

: Thermal conductivity of the mixture

: Molecular Lewis number

: 71th component of the subgrid scalar flux of species a
: Number of species

: Molecular Prandtl number

: Universal gas constant

: Reynolds number

: SGS Schmidt number

: Production rate of species a

: Strain rate tensor

: Temperature

: Reference temperature

: Subgrid scale stresses

: Time

: ith component of the velocity vector

: ith component of the reference velocity in MKEV closure
: Molecular weight of species a

: Wiener-Levy process

: Position vector

: ith component of the position vector

: Lagrangian position of the particles

: Streamwise coordinate

: Mass fractions of species oz

: Coordinates deﬁning the plane normal to a:
: Mixture fraction

178

 

Greek Symbols:

V : Gradient operator

A : Grid spacing in LES

6 : Dirac delta function

6,3- : Kronecker delta

A E : Ensemble domain width
A H : Grid level ﬁlter width

AH; : Secondary level ﬁlter width

 

u : Molecular viscosity, in = p11

Vt : Subgrid viscosity

(2m : SGS mixing frequency

«,0 : Composition space

(I) : Scalar ﬁeld

$0: : The compositional values of scalar a
Z; : The compositional values of stochastic scalar a

p : Density

0 : Number of scalars, a = N3 + 1

Tij : Molecular stresses

T¢ : Scalar mixing timescale

Symbols

( )g : Filtered value

( ) L : Favre ﬁltered value

( I )L: Conditional Favre ﬁltered value

A : Quantities which depend only on the scalar composition,

2:6. §(¢, x, t) a S(¢(x, t))

00 : Ambient
Subscripts

k : Dummy index
Superscripts

+ : Properties of the stochastic Monte Carlo particles
(n) : Index of the Monte Carlo particles

179

References

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180

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