Autoignition in turbulent
two-phase flows
Giulio Borghesi
Wolfson College
University of Cambridge
A thesis submitted for the degree of
Doctor of Philosophy
Monday, 6th of August, 2012

A Christianne e Giada,
come testimonianza della mia amicizia.
Con affetto,
Giulio

Declaration
This dissertation is the result of my own work and includes nothing which is the
outcome of work done in collaboration except where specifically indicated in the
text. The dissertation contains approximately 46,000 words, 76 figures and 4
tables.
Giulio Borghesi
Hopkinson Laboratory, Cambridge
Monday, 6th of August, 2012

Acknowledgements
The work presented in this dissertation would have not been car-
ried out without the help of many people, to whom I am profoundly
grateful. First, and foremost, I would like to express my most sin-
cere admiration and thanks to my supervisor, Professor Epaminon-
das Mastorakos, for his continuous guidance and support during my
time in Cambridge, both on an academic and on a personal level. I
thank him not only for his many suggestions and advices, but also for
having being very supportive and close during and after my sickness.
I also thank my advisor, Professor Stewart Cant, for some useful dis-
cussions on numerical methods, and for allowing me to use his DNS
code SENGA2. For his technical support, I thank Mr. Peter Benie,
whose computing expertise allowed me to focus on my research work,
without having to worry about the IT infrastructure. Funding from
the European Union through the Marie Curie Project MYPLANET
enabled me to complete this work without financial worry, and I grate-
fully acknowledge the financial contribution received.
I am grateful to Dr. Fernando Biagioli, of Alstom Power Switzerland,
who encouraged me to embrace the challenges of doctoral studies, and
who has always been very supportive of my work. A special mention
goes to Chesterton Rowing Club, and particularly to Mr. Simon Em-
mings, with whom I had the pleasure of discovering that, when done
properly, rowing is actually a wonderful sport.
I have been fortunate enough to meet many special people during the
three years I spent in Cambridge. In particular, I would like to thank
my friends and coworkers Davide Cavaliere, James Kariuki, Andrea
Maffioli, Andrea Pastore and Michael Youtsos for all the good time
we had together, and for having acted as examples and sources of
inspiration for becoming a better person. I would also like to thank
Yuri Wright and Michele Bolla from ETH Zu¨rich, not only for the
many productive discussions we had, but especially for their friend-
ship, and for the nice time that we shared in Switzerland and England.
I own a very special thank to M.D.s Giovanni Grazia, Salvatore Voce
and Giovanni Rosti, and to the Medical Oncology unit of Treviso hos-
pital, Italy, for having taken care of my health during the most delicate
period of my life. During my journey through hospitals, I learnt how
even a single word, if misplaced, may change an individual’s mood
from calm to panic within seconds. No bad thought has crossed my
mind since I put my life in your hands, and there is no word to express
the immense gratitude that I feel towards you. I strongly believe that
I couldn’t have put my trust in better persons.
Last, but certainly not least, I would like to thank my family and two
special persons, Giada Vignali and Christianne Simo˜es da Silveira
Cintra, who overwhelmed me with love and attention when I needed
it the most. I am very fortunate to have people like you in my life,
and there is nothing or nobody that is more precious to me. Thank
you for everything, and please forgive me if my words are not good or
strong enough to express my feelings for you.
Abstract
This dissertation deals with the numerical investigation of the physics
of sprays autoigniting at diesel engine conditions using Direct Nu-
merical Simulations (DNS), and with the modelling of droplet related
effects within the Conditional Moment Closure (CMC) method for
turbulent non-premixed combustion. The dissertation can be split in
four different sections, with the content of each being summarized be-
low.
The first part of the dissertation introduces the equations that govern
the temporal and spatial evolution of a turbulent reacting flow, and
provides an extensive review of the CMC method for both single and
two-phase flows. The problem of modelling droplet related effects in
the CMC transport equations is discussed in detail, and physically-
sound models for the unclosed terms that appear in these equations
and that are affected by the droplet presence are derived.
The second part of the dissertation deals with the application of the
CMC method to the numerical simulation of several n-heptane sprays
igniting at conditions relevant to diesel engine combustion. Droplet-
related terms in the CMC equations were closed with the models
developed in the first part of the dissertation. For all conditions in-
vestigated, CMC could correctly capture the ignition, propagation
and anchoring phases of the spray flame. Inclusion of droplet terms
in the CMC equations had little influence on the numerical predic-
tions, in line with the findings of other authors.
The third part of the dissertation presents a DNS study on the au-
toignition of n-heptane sprays at high pressure / low temperature
conditions. The analysis revealed that spray ignition occurs first in
well-mixed locations with a specific value of the mixture fraction.
Changes in the operating conditions (initial turbulence intensity of
the background gas, global equivalence ratio in the spray region, ini-
tial droplet size distribution) affected spray ignition through changes
in the mixture formation process. For each spray, a characteristic ig-
nition delay time and a characteristic droplet evaporation time could
be defined. The ratio between these time scales was suggested as a
key parameter for controlling the ignition delay of the spray.
The last part of the dissertation exploits the DNS simulations to per-
form an a priori analysis of the applicability of the CMC method to
autoigniting sprays. The study revealed that standard models for the
mixing quantities used in CMC provide poor approximations in two-
phase flows, and are partially responsible for the poor prediction of
the ignition delay time. It was also observed that first-order closure
of the chemical source terms performs poorly during the onset of igni-
tion, suggesting that second-order closures may be more appropriate
for studying spray autoignition problems.
The contribution of the work presented in this dissertation is to pro-
vides a detailed insight into the physics of spray autoignition at diesel
engine conditions, to propose and derive original methods for incorpo-
rating droplet evaporation effects within CMC in a physically-sound
manner, and to assess the applicability and shortcomings of the CMC
method to autoigniting sprays.
Contents
Contents ix
List of Figures xiii
List of Tables xxv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Diesel engine combustion . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Numerical simulation of turbulent reacting flows . . . . . . . . . . 4
1.3.1 Direct Numerical Simulation . . . . . . . . . . . . . . . . . 5
1.3.2 Reynolds-Averaged Navier-Stokes . . . . . . . . . . . . . . 7
1.3.3 Large Eddy Simulation . . . . . . . . . . . . . . . . . . . . 8
1.4 The turbulent combustion problem . . . . . . . . . . . . . . . . . 9
1.5 Objectives and outlook of the dissertation . . . . . . . . . . . . . 12
2 The governing equations of two-phase reacting flows 15
2.1 Mathematical description of two-phase flows . . . . . . . . . . . . 16
2.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Gaseous phase governing equations . . . . . . . . . . . . . 20
2.2.1.1 Mass conservation . . . . . . . . . . . . . . . . . 20
2.2.1.2 Momentum conservation . . . . . . . . . . . . . . 20
2.2.1.3 Energy conservation . . . . . . . . . . . . . . . . 21
2.2.1.4 Species equation . . . . . . . . . . . . . . . . . . 22
2.2.2 Liquid phase governing equations . . . . . . . . . . . . . . 22
2.2.3 Interphase transfer terms . . . . . . . . . . . . . . . . . . . 24
ix
2.2.4 Mixture fraction equation . . . . . . . . . . . . . . . . . . 25
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Conditional Moment Closure for spray combustion 29
3.1 The CMC equations for single phase flows . . . . . . . . . . . . . 30
3.1.1 Derivation of the CMC equations . . . . . . . . . . . . . . 30
3.1.2 Closure of the CMC equations . . . . . . . . . . . . . . . . 33
3.2 The CMC equations for two-phase flows . . . . . . . . . . . . . . 40
3.2.1 Derivation of the CMC equations for sprays . . . . . . . . 42
3.2.2 Modelling droplet effects in CMC . . . . . . . . . . . . . . 44
3.2.2.1 Existing models . . . . . . . . . . . . . . . . . . . 44
3.2.2.2 New methods for estimating P˜ (ξs) and 〈N | ξs〉 . . 48
3.2.2.3 Alternative approach by other authors . . . . . . 51
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4 Simulations of n-heptane sprays autoignition with 2D-CMC 55
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 Experimental configuration . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 CFD setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 CMC setup . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.2.1 Closure of the CMC equations . . . . . . . . . . . 61
4.3.2.2 Numerical methods and boundary conditions . . 62
4.3.2.3 Chemistry . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4.1 Unconditional averages . . . . . . . . . . . . . . . . . . . . 63
4.4.1.1 Spray penetration . . . . . . . . . . . . . . . . . 63
4.4.1.2 Flow structure . . . . . . . . . . . . . . . . . . . 66
4.4.1.3 Influence of droplet terms on ξ˜′′2 predictions . . . 75
4.4.2 Conditional averages . . . . . . . . . . . . . . . . . . . . . 78
4.4.2.1 Autoignition . . . . . . . . . . . . . . . . . . . . 78
4.4.2.2 Flame propagation . . . . . . . . . . . . . . . . . 79
4.4.2.3 Structure of flame base . . . . . . . . . . . . . . . 82
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
x
5 Direct Numerical Simulations of spray autoignition 87
5.1 Introduction and literature review . . . . . . . . . . . . . . . . . . 88
5.2 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Numerical procedure and computational parameters . . . . 92
5.2.2 Chemical mechanism . . . . . . . . . . . . . . . . . . . . . 96
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1 Homogeneous and flamelet calculations . . . . . . . . . . . 97
5.3.2 Main observations . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.2.1 General remarks . . . . . . . . . . . . . . . . . . 101
5.3.2.2 Inert mixing phase . . . . . . . . . . . . . . . . . 104
5.3.2.3 Pre-ignition phase . . . . . . . . . . . . . . . . . 106
5.3.2.4 Ignition phase . . . . . . . . . . . . . . . . . . . . 110
5.3.3 Topology of the autoigniting kernels . . . . . . . . . . . . . 114
5.3.4 Evaporation and mixing . . . . . . . . . . . . . . . . . . . 120
5.3.4.1 General remarks . . . . . . . . . . . . . . . . . . 120
5.3.4.2 Influence of turbulence . . . . . . . . . . . . . . . 122
5.3.4.3 Influence of global equivalence ratio . . . . . . . . 127
5.3.4.4 Influence of initial droplet size distribution . . . . 132
5.3.5 Ignition behavior corresponding to high values of the air
stream temperature . . . . . . . . . . . . . . . . . . . . . . 142
5.3.6 Ignition behavior corresponding to diluted oxygen concen-
tration in air . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6 Validation of the CMC method for spray autoignition 165
6.1 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . 166
6.1.1 0D-CMC equations for two-phase flows . . . . . . . . . . . 166
6.1.2 Closure of the 0D-CMC equations . . . . . . . . . . . . . . 167
6.1.2.1 Modelling strategy A . . . . . . . . . . . . . . . . 167
6.1.2.2 Modelling strategy B . . . . . . . . . . . . . . . . 168
6.1.2.3 Modelling strategy C . . . . . . . . . . . . . . . . 169
6.1.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . 169
6.1.4 Initial and boundary conditions . . . . . . . . . . . . . . . 169
xi
6.1.4.1 Operating conditions investigated . . . . . . . . . 169
6.1.4.2 Initialization of the conditional moments . . . . . 171
6.1.5 Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.2.1 Modelling of conditional scalar dissipation rate . . . . . . . 171
6.2.2 Modelling of chemical source terms . . . . . . . . . . . . . 174
6.2.3 Modelling of mixture fraction PDF . . . . . . . . . . . . . 178
6.2.4 Prediction of the ignition event . . . . . . . . . . . . . . . 180
6.2.4.1 Low temperature cases . . . . . . . . . . . . . . . 180
6.2.4.2 High temperature case . . . . . . . . . . . . . . . 186
6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7 Conclusions 195
7.1 Summary of the main findings . . . . . . . . . . . . . . . . . . . . 195
7.1.1 Prediction of diesel spray autoignition with CMC . . . . . 195
7.1.2 Numerical investigation of autoignition in sprays . . . . . . 197
7.1.3 Applicability of the CMC method for studying spray au-
toignition . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
7.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . 201
A Evaluation of mixture fraction at saturation conditions 205
B Initialization of the conditional temperature in sprays 207
References 211
xii
List of Figures
3.1 Functional dependence of the scalar dissipation rate on mixture
fraction as obtained by using the AMC model [100]. . . . . . . . . 38
4.1 Experimental (left) and computed (right) mean equivalence ratio
fields, plotted at the instant of time preceding ignition for each
of the four oxygen dilutions investigated. Far left: 21 % O2 case.
Left: 15 % O2 case. Right: 12 % O2 case. Far right: 10 % O2 case.
Experimental data from [55]. . . . . . . . . . . . . . . . . . . . . . 64
4.2 Experimental and computed radial profiles of mixture fraction at
different axial positions and at t = 6 ms after start of injection.
Data shown refer to a spray evaporating in an inert environment.
Experimental data from [55]. . . . . . . . . . . . . . . . . . . . . . 65
4.3 Measured and predicted ignition delay time against O2 mole frac-
tion in ambient gas. Experimental data from [54]. . . . . . . . . . 66
4.4 Temporal evolution of flame lift-off height for the different ambient
oxygen concentrations listed in Table 4.2. Experimental data from
[54]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5 From left to right: mean mixture fraction, mixture fraction vari-
ance, mean OH mass fraction and mean temperature fields, plotted
at different time instants. Black line is the stoichiometric mixture
fraction isosurface. Droplet terms in CMC equations included.
Data shown for Case A in Table 4.2. . . . . . . . . . . . . . . . . 70
xiii
4.6 From left to right: mean mixture fraction, mixture fraction vari-
ance, mean OH mass fraction and mean temperature fields, plotted
at different time instants. Black line is the stoichiometric mixture
fraction isosurface. Droplet terms in CMC equations included.
Data shown for Case B in Table 4.2. . . . . . . . . . . . . . . . . 71
4.7 From left to right: mean mixture fraction, mixture fraction vari-
ance, mean OH mass fraction and mean temperature fields, plotted
at different time instants. Black line is the stoichiometric mixture
fraction isosurface. Droplet terms in CMC equations included.
Data shown for Case C in Table 4.2. . . . . . . . . . . . . . . . . 72
4.8 From left to right: mean mixture fraction, mixture fraction vari-
ance, mean OH mass fraction and mean temperature fields, plotted
at different time instants. Black line is the stoichiometric mixture
fraction isosurface. Droplet terms in CMC equations included.
Data shown for Case D in Table 4.2. . . . . . . . . . . . . . . . . 73
4.9 From left to right: mean mixture fraction (left) and mixture frac-
tion variance (central and right) fields, plotted at two time instants
prior to ignition. Black line is the stoichiometric mixture fraction
isosurface. Droplet terms in mixture fraction variance equation ei-
ther neglected (central figure) or modelled (right figure) according
to Equation 3.52. Data shown for Case B in Table 4.2. . . . . . . 74
4.10 From left to right: production terms in ξ˜′′2 transport equation due
to droplets evaporation, due to the spatial gradient of ξ˜, and the
difference between the two, plotted at t = 0.9 ms. Units: kg/(m3 s).
Black line is the stoichiometric mixture fraction isosurface. Data
shown for Case B in Table 4.2. . . . . . . . . . . . . . . . . . . . . 75
4.11 Mixture fraction PDFs plotted at several spatial locations along
the spray axis for the mixing fields shown in Figure 4.9. Black
lines: droplet source terms in mixture fraction variance equation
neglected. Red lines: droplet source terms in mixture fraction
variance equation included. Data shown for Case B in Table 4.2. . 76
xiv
4.12 Temporal evolution of the conditional mean temperature at the
autoignition location (r = 0 mm, z = 42 mm). Data shown for
Case B in Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.13 Temporal evolution of conditional mean temperature and oxygen
mass fraction at selected locations within the flame. Droplet terms
in CMC equations included. Data shown for Case B in Table 4.2. 81
4.14 Source terms in CMC temperature equation at r = 3.0 mm, z =
32.5 mm and selected instants of time. Droplet terms in CMC
equations included. Data shown for Case B in Table 4.2. . . . . . 83
4.15 Source terms in CMC temperature equation at flame anchoring
locations (off and along spray axis) for t=3.0 ms. Data shown for
Case B in Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1 Sketch of the computational domain for the 923 grid. . . . . . . . 95
5.2 Initial droplet size distribution and cumulative liquid volume frac-
tion for Case E in Table 5.1. . . . . . . . . . . . . . . . . . . . . . 96
5.3 Calculations of autoignition time τid,0 in homogeneous n-C7H16 /
air mixtures at p = 24 bar for different values of XO2 and Tair.
Fuel stream temperature was equal to Tf = 450 K for all sets of
conditions investigated. . . . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Left: calculations of autoignition time in unsteady n-C7H16 / air
flamelets with Le = 1 under constant maximum scalar dissipation
N0. Right: same as in left figure, but N0 was normalized by its crit-
ical value, and τid by its reference value. Fuel stream temperature
was equal to Tf = 450 K for all sets of conditions investigated. . . 99
5.5 Calculations of unsteady laminar flamelets for XO2 = 0.21 and
different values of the air stream temperature. Fuel temperature
equal to Tf = 450 K for both cases investigated. Cases referenced
in the figures have the same operating conditions of the flamelets
and are listed in Table 5.1. The maximum scalar dissipation rate
across the flamelet was constant and equal to N0 = 20 s
−1, which
corresponds to a N0/N0,crit of 0.017 for (a) and of 0.155 for (b). . . 100
xv
5.6 Calculation of unsteady laminar flamelet for XO2 = 0.10 and Tair =
1000 K, corresponding to conditions of Case B in Table 5.1. Fuel
temperature equal to Tf = 450 K. The maximum scalar dissipation
rate across the flamelet was constant and equal to N0 = 20 s
−1,
which corresponds to a N0/N0,crit of 0.303. . . . . . . . . . . . . . 101
5.7 Left: temporal evolution of volume-averaged heat release rate for
Cases A, D, E and F in Table 5.1. Right: detailed view of Case
A, showing locations at which species and temperature contour /
scatterplots are given. . . . . . . . . . . . . . . . . . . . . . . . . 102
5.8 Scatterplots of heat release rate, temperature, scalar dissipation
rate and n-C7H16, O2, H, OH, HO2 and CH2O mass fractions plot-
ted against mixture fraction at t = tIN = 0.6 ms in Figure 5.7(b).
Data shown correspond to Case A in Table 5.1. . . . . . . . . . . 105
5.9 Contour plots of temperature, mixture fraction, scalar dissipation
rate, HO2 and CH2O mass fractions and heat release rate corre-
sponding to the axial plane x = 0.5L at t = tIN = 0.6 ms. Data
shown correspond to Case A in Table 5.1. . . . . . . . . . . . . . 106
5.10 Scatterplots of heat release rate, temperature, scalar dissipation
rate and n-C7H16, O2, H, OH, HO2 and CH2O mass fractions plot-
ted against mixture fraction at t = tPI4 = 1.0 ms in Figure 5.7(b).
Data shown correspond to Case A in Table 5.1. . . . . . . . . . . 107
5.11 Contour plots of temperature, mixture fraction, scalar dissipation
rate, HO2 and CH2O mass fractions and heat release rate corre-
sponding to the axial plane x = 0.5L at t = tPI4 = 1.0 ms. Data
shown correspond to Case A in Table 5.1. . . . . . . . . . . . . . 108
5.12 Temporal evolution of conditional mean heat release, temperature,
mixture fraction PDF and selected species mass fractions. Refer-
enced instants of time are given in Figure 5.7(b). Their numer-
ical values are: tPI1 = 0.70 ms, tPI2 = 0.80 ms, tPI3 = 0.90 ms,
tPI4 = 1.00 ms, tPI5 = 1.05 ms. Data shown correspond to Case A
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
xvi
5.13 Temporal evolution of T = 1250 K isosurface. Surface colored
according to local mixture fraction value. Referenced instants of
time are given in Figure 5.7(b). Data shown correspond to Case
A in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.14 Scatterplots of temperature and n-C7H16, O2, H2 mass fractions
plotted against mixture fraction at different instants of time during
ignition. Referenced instants of time are given in Figure 5.7(b).
Data shown correspond to Case A in Table 5.1. . . . . . . . . . . 112
5.15 Scatterplots of HO2, CH2O, C4H8 and CO2 mass fractions plotted
against mixture fraction at different instants of time during igni-
tion. Referenced instants of time are given in Figure 5.7(b). Data
shown correspond to Case A in Table 5.1. . . . . . . . . . . . . . 113
5.16 Conditional heat release rate, plotted at different instants of time
and for different values of the scalar dissipation rate. Referenced
instants of time are given in Figure 5.7(b). Data shown correspond
to Case A in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . 115
5.17 Left: temporal evolution of cross-correlation coefficient between
heat release and scalar dissipation rate. Right: cross-correlation
coefficient, doubly conditioned on mixture fraction and ignition
progress variable, plotted at t = tIG3 = 1.10 ms. Referenced in-
stants of time are given in Figure 5.7(b). Their numerical values
are: tPI2 = 0.80 ms, tPI3 = 0.90 ms, tPI4 = 1.00 ms, tPI5 = 1.05 ms.
Data shown correspond to Case A in Table 5.1 . . . . . . . . . . . 117
5.18 Temporal evolution of ξ = ξMR isosurface. Surface colored accord-
ing to local value of the ratio N/Ncrit,ξMR , with Ncrit,ξMR being the
critical value for ignition of the conditional scalar dissipation rate
at ξMR = 0.2. Gray surfaces in the bottom right figure correspond
to T = 1250 K isosurface. Referenced instants of time are given in
Figure 5.7(b). Data shown correspond to Case A in Table 5.1. . . 118
xvii
5.19 Temporal evolution of conditional OH mass fraction in flow regions
with different levels of scalar dissipation rate. (a): case N ≤ 10 s−1.
(b): case N > 10 s−1. Referenced instants of time are given in
Figure 5.7(b). Their numerical values are: tPI3 = 0.90 ms, tPI4 =
1.00 ms, tPI5 = 1.05 ms. Data shown correspond to Case A in Table
5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.20 Temporal evolution of normalized mean droplet evaporation rate,
liquid temperature increase, droplet diameter and normalized mean
scalar dissipation rate. Data shown correspond to Case A in Table
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.21 Temporal evolution of mean scalar dissipation rate as a function
of the distance from the droplet centre. Data shown correspond to
Case A in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.22 Temporal evolution of (Left) droplet mean evaporation rate and
mean scalar dissipation rate, and (Right) droplet Sauter diameter
and mean liquid temperature, for different intensities of the initial
gas-phase turbulence. Data shown correspond to Cases A and D
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.23 Conditionally-averaged scalar dissipation rate, the conditioning be-
ing on mixture fraction, plotted at different instants of time. Data
shown correspond to Case A (Left) and D (Right) in Table 5.1. . 125
5.24 Temporal evolution of the droplet mean Stokes number and droplet
mean modified Sherwood number for different intensities of the
initial gas-phase turbulence. . . . . . . . . . . . . . . . . . . . . . 126
5.25 Left: temporal evolution of volume-averaged heat release rate for
Case F in Table 5.1, showing locations at which species and tem-
perature contour / conditional means are given. Right: comparison
of mixture fraction PDF at selected instants of time for Cases D
and F in Table 5.1. Numerical values of referenced instants of time
are: tIND = 0.7 ms, tPID = 1.0 ms, tPIF2 = 0.7 ms, tPIF5 = 1.0 ms. . . 128
xviii
5.26 Temporal evolution of conditional mean heat release, temperature
and selected species mass fractions. Referenced instants of time are
given in Figure 5.25. Their numerical values are: tPIF1 = 0.60 ms,
tPIF2 = 0.70 ms, tPIF3 = 0.80 ms, tPIF4 = 0.90 ms, tPIF5 = 1.00 ms.
Data shown correspond to Case F in Table 5.1. . . . . . . . . . . 129
5.27 Temporal evolution of T = 1250 K isosurface. Surface colored
according to local mixture fraction value. Referenced instants of
time are given in Figure 5.25. Data shown correspond to Case F
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.28 Temporal evolution of (Left) droplet mean evaporation rate and
volume-averaged scalar dissipation rate, and (Right) droplet Sauter
mean diameter and liquid mean temperature. Data shown corre-
spond to different initial values of the global equivalence ratio. . . 131
5.29 Left: conditionally-averaged scalar dissipation rate, the condition-
ing being on mixture fraction, plotted at different instants of time.
Data shown correspond to Case F in Table 5.1. Right: conditionally-
averaged scalar dissipation rate for Cases D and F in Table 5.1 at
two selected instants of time. . . . . . . . . . . . . . . . . . . . . . 132
5.30 Temporal evolution of (Left) mean liquid temperature and Sauter
mean diameter, and (Right) residual liquid volume fraction and
mean droplet evaporation rate, for different initial droplet size dis-
tributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.31 Left: droplet size distribution, plotted at different instants of time.
Right: temporal evolution of the number of residual droplets in the
computational domain, normalized by its initial value. Data shown
correspond to Case E (Left) and Cases D and E (Right) in Table
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.32 Left: conditionally-averaged scalar dissipation rate, the condition-
ing being on mixture fraction, plotted at different instants of time.
Data shown correspond to Case E in Table 5.1. Right: conditionally-
averaged scalar dissipation rate for Cases D and E in Table 5.1 at
two selected instants of time. . . . . . . . . . . . . . . . . . . . . . 136
xix
5.33 Left: temporal evolution of volume-averaged heat release rate for
Case E in Table 5.1, showing locations at which species and tem-
perature contour / conditional means are given. Right: comparison
of mixture fraction PDF at selected instants of time for Cases D
and E in Table 5.1. Numerical values of referenced instants of time
are: tIND = 0.7 ms, tPID = 1.0 ms, tPIE1 = 0.6 ms, tPIE2 = 0.9 ms. . 138
5.34 Scatterplots of temperature and n-C7H16, O2, CH2O mass fractions
plotted against mixture fraction at different instants of time during
ignition. Referenced instants of time are given in Figure 5.33. Data
shown correspond to Case E in Table 5.1. . . . . . . . . . . . . . . 140
5.35 Temporal evolution of T = 1250 K isosurface. Surface colored
according to local mixture fraction value. Referenced instants of
time are given in Figure 5.33. Data shown correspond to Case E
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.36 Temporal evolutions of volume-averaged heat release rate for Cases
A and C in Table 5.1. Markers denote locations at which species
and temperature scatterplots are given. . . . . . . . . . . . . . . . 143
5.37 Scatterplots of temperature and O2, CH2O, OH mass fractions
plotted against mixture fraction at different instants of time during
ignition. Referenced instant of time are given in Figure 5.36. Data
shown correspond to Case C in Table 5.1. . . . . . . . . . . . . . . 144
5.38 Temporal evolution of T = 1750 K isosurface. Surface colored
according to local mixture fraction value. Referenced instants of
time are given in Figure 5.36. Data shown correspond to Case C
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.39 Temporal evolution of cross-correlation coefficient between heat
release and scalar dissipation rate. Referenced instants of time are
given in Figure 5.36. Their numerical values are: tPIH1 = 0.180 ms,
tPIH2 = 0.210 ms, tPIH3 = 0.240 ms, tIGH1 = 0.270 ms, tIGH3 =
0.285 ms. Data shown correspond to Case C in Table 5.1. . . . . . 146
5.40 Temporal evolutions of volume-averaged heat release rate for Case
B in Table 5.1. Markers denote locations at which species and
temperature scatterplots are given. . . . . . . . . . . . . . . . . . 148
xx
5.41 Temporal evolution of conditional mean heat release, temperature,
mixture fraction PDF and selected species mass fractions. Refer-
enced instants of time are given in Figure 5.40. Their numerical
values are: tPIB1 = 1.00 ms, tPIB2 = 1.20 ms, tPIB3 = 1.40 ms,
tPIB4 = 1.60 ms, tPIB5 = 1.80 ms. Data shown correspond to Case
B in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.42 Contour plots of mixture fraction (left), CH2O (center) and heat
release rate (right) corresponding to the axial plane x = 0.5L at
selected instants of time. Referenced instants of time are given in
Figure 5.40. Data shown correspond to Case B in Table 5.1. . . . 151
5.43 Scatterplots of temperature and O2, CH2O, OH mass fractions
plotted against mixture fraction at different instants of time during
ignition. Referenced instants of time are given in Figure 5.40. Data
shown correspond to Case B in Table 5.1. . . . . . . . . . . . . . . 153
5.44 Temporal evolution of T = 1200 K isosurface. Surface colored
according to local mixture fraction value. Referenced instants of
time are given in Figure 5.40. Data shown correspond to Case B
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.45 Temporal evolution of ξ = 0.04 isosurface. Surface colored accord-
ing to local temperature value. Data shown correspond to Case B
in Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.46 Temporal evolution of cross-correlation coefficient between heat
release and scalar dissipation rate. Referenced instants of time are
given in Figure 5.40. Their numerical values are: tPIB1 = 1.00 ms,
tPIB4 = 1.60 ms, tPI6 = 2.20 ms, tPI7 = 2.80 ms, tPI8 = 3.40 ms.
Data shown correspond to Case B in Table 5.1 . . . . . . . . . . . 157
5.47 Conditionally-averaged scalar dissipation rate, the conditioning be-
ing on mixture fraction, plotted at different instants of time. Data
shown correspond to Case B in Table 5.1 . . . . . . . . . . . . . . 158
6.1 Comparison between the conditional scalar dissipation rate profiles
as obtained using the AMC model, and by conditionally averaging
the DNS data. Data shown for Case A in Table 6.1. . . . . . . . . 172
xxi
6.2 Comparison between the conditional scalar dissipation rate profiles
as obtained using the AMC model, and by conditionally averaging
the DNS data. Data shown for Case C in Table 6.1. . . . . . . . . 173
6.3 Comparison between the conditional heat release rate term as ob-
tained using the first order closure in CMC, and by conditionally
averaging the DNS data, at two selected instants of time. Data
shown for Case A in Table 6.1. . . . . . . . . . . . . . . . . . . . . 175
6.4 Comparison between the conditional heat release rate term as ob-
tained using the first order closure in CMC, and by conditionally
averaging the DNS data, at two selected instants of time. Data
shown for Case C in Table 6.1. . . . . . . . . . . . . . . . . . . . . 176
6.5 Comparison between the mixture fraction PDF as obtained using
a β distribution, and by processing the DNS data. Data shown for
Case A in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.6 Comparison between the mixture fraction PDF as obtained using
a β distribution, and by processing the DNS data. Data shown for
Case F in Table 6.1. . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.7 Temporal evolution of selected conditional averages at most reac-
tive mixture fraction. Data shown correspond to Case A in Table
6.1 (ξMR = 0.2). The conditional moments were obtained by con-
ditionally averaging the DNS data and from the numerical solution
of the zero-dimensional CMC equations. . . . . . . . . . . . . . . 181
6.8 Scatterplots of temperature against mixture fraction at selected
instants of time early after the start of evaporation. Red line cor-
responds to the conditional temperature profile as obtained with
the use of the standard CMC initialization procedure. Data shown
correspond to Case A in Table 6.1. . . . . . . . . . . . . . . . . . 183
6.9 Temporal evolution of selected conditional averages at most reac-
tive mixture fraction. Data shown correspond to Case F in Table
6.1 (ξMR = 0.185). The conditional moments were obtained by
conditionally averaging the DNS data and from the numerical so-
lution of the zero-dimensional CMC equations. . . . . . . . . . . . 184
xxii
6.10 Temporal evolution of selected conditional averages at most reac-
tive mixture fraction. Data shown correspond to Case C in Table
6.1 (ξMR = 0.045). The conditional moments were obtained by
conditionally averaging the DNS data and from the numerical so-
lution of the zero-dimensional CMC equations. . . . . . . . . . . . 187
6.11 Comparison between standard and improved initialization proce-
dures for the conditional temperature in the CMC method. Data
shown correspond to Case C in Table 6.1. . . . . . . . . . . . . . . 188
6.12 Temporal evolution of selected conditional averages at most reac-
tive mixture fraction. Data shown correspond to Case C in Table
6.1 (ξMR = 0.045). The conditional moments were obtained by
conditionally averaging the DNS data and from the numerical so-
lution of the zero-dimensional CMC equations, with the unclosed
terms modeled according to strategies C (CMC - Standard) and
B (CMC - Improved) described in Section 6.1.2. The conditional
temperature was initialized using either the standard initialization
procedure described in Section 6.1.4.2, or the improved method
detailed in Appendix B. . . . . . . . . . . . . . . . . . . . . . . . 189
A.1 Evaluation of mixture fraction at saturation ξs for droplets evap-
orating in flow regions where substantial chemical reactions have
occurred. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
xxiii

List of Tables
4.1 Ambient and fuel injector conditions. Source: [55]. . . . . . . . . . 59
4.2 Ambient gas composition (in molar fractions), ignition delay time
(in milliseconds) and stoichiometric mixture fraction for the differ-
ent operating conditions investigated. Source: [55]. . . . . . . . . 60
5.1 Turbulence, oxidizer and spray characteristics for the various spray
simulations performed. Fuel temperature and turbulence integral
length scale set equal to Tf = 450 K and lt = L/6 for all cases
investigated. XO2 : oxygen concentration in the oxidizer. Ret: tur-
bulent Reynolds number, Ret = u
′lt/ν. ξst: stoichiometric mixture
fraction. ξMR,0: most reactive mixture fraction of the correspond-
ing homogeneous system (see Section 5.3.1). . . . . . . . . . . . . 93
6.1 Operating conditions, ignition delay time and most reactive mix-
ture fraction of the Direct Numerical Simulations used for validat-
ing the CMC method for autoigniting sprays. . . . . . . . . . . . 170
xxv

Chapter 1
Introduction
1.1 Motivation
Fossil fuels play a key role in our modern society in many contexts, such as
energy supply, transportation and chemical transformation processes. It is esti-
mated that approximately 80 % of the current total primary energy supply is met
through the combustion of either coal of hydrocarbon fuels, with the remaining
20 % being covered with a combination of nuclear power, hydroelectric power and
renewable sources [3]. This scenario has not changed much compared to thirty
years ago, when fossil fuels topped as much as 85 % of the energy demand, and
there are reasons to believe that the situation will not change drastically in the
following decades.
Concerns about the environmental impact of the primary energy sources in
terms of both pollutants and greenhouse gases production have lead governments
to adopt very stringent regulations regarding the emission standards of new ve-
hicles and power plants. Similar concerns about the limited availability and
rising price of conventional fuels have encouraged and motivated research on
fuel-efficient engines. Meeting these requirements has proven to be an extraor-
dinary challenge, and requires a profound understanding of the physics of the
combustion process used to convert the chemical energy of the fuel into thermal
energy. Such an insight view on the fundamental aspects of combustion science
1
has become possible only recently due to a series of advancements in both exper-
imental and computational techniques.
Computational Fluid Dynamics (CFD) is emerging as a powerful and reliable
tool for assisting the design of state-of-the-art combustion chambers. Numerical
simulations of reactive flows are however challenging, due to the difficulties in
describing how turbulence affects the rate at which heat release reactions pro-
ceed. The situation becomes even more complex when the fuel is supplied in
liquid form, due to the appearance of additional complex interacting physical
phenomena, such as the atomization and evaporation of the liquid fuel, or the
exchanges of mass, momentum and energy occurring between the gaseous and
the liquid phase. Spray combustion is at the heart of many practical devices,
including liquid-fueled industrial furnaces, aeronautical gas turbines, and diesel
engines, and thus deep physical understanding and proper modelling of turbu-
lent combustion processes in two-phase flows are mandatory to promote further
development of these applications.
This work deals with the investigation of the physics of ignition and flame
propagation in diesel sprays, and on the subsequent implementation and valida-
tion of terms describing spray-related effects in numerical models for simulating
turbulent non-premixed combustion processes. Although the models derived in
this work are applicable to any turbulent reacting flow in which fuel is supplied
in liquid form, focus is put here on diesel engine combustion, for which a brief
description is provided in the following section.
1.2 Diesel engine combustion
Compression-ignition (CI) engines are the most fuel-efficient engines ever devel-
oped for transportation purposes [28]. This is due to their higher compression
ratio and absence of throttling losses as compared to standard spark-ignition (SI)
engines. Despite their high efficiency, diesel engines are traditionally character-
ized by high levels of nitrogen oxides (NOx) and particulate matter emission,
which are the result of the highly stratified mixture in which combustion occurs.
2
This negative aspect of diesel combustion has prompted considerable theoretical
and experimental efforts aimed at acquiring a profound understanding of the in-
cylinder combustion processes, so to meet current and future emission regulations.
A detailed picture of conventional diesel combustion has been presented in
recent years by Flynn et al. [37]. Their description starts from the instant when
the liquid fuel is injected in the combustion chamber and begins to penetrate
within the bulk gas, leading to air entrainment. As the cold liquid jet is heated
by mixing with the hot oxidizer, a thin sheet of air / fuel vapor mixture is formed
around the jet’s periphery. Further entrainment of hot air is responsible for start-
ing the chemical reactions that are responsible for breaking down the fuel species,
leading to the formation of intermediate species and water vapor. Temperatures
of the order of 1600 − 1700 K are reached as these reactions occur. The lack of
oxygen and the abundance of hydrocarbon fragments such as C2H2, C2H4 and
C3H3 in this region are responsible for the formation of the polycyclic aromatic
hydrocarbons (PAHs) that constitute the inception sites for the formation of
particulate matter in diesel spray flames. At about the same time, a diffusion
flame forms around the jet’s periphery. This flame burns at stoichiometric condi-
tions, for which the temperature of the combustion products is the highest, and
is the main responsible for the production of nitrogen oxides in diesel combustion.
Various methods have been proposed in the engine-combustion community
to address the simultaneous needs of reducing the emissions and increasing the
efficiency of compression-ignition engines. These approaches usually rely on the
principle of generating a charge that is enough well-mixed to allow for either
premixed or partially premixed combustion. One of such methods is known as
Homogeneous Charge Compression Ignition (HCCI) combustion, in which a lean,
homogeneous fuel / air mixture is generated and then compression ignited [28].
Fuel-rich regions in the charge, which are responsible for soot formation, are then
avoided; and the occurrence of combustion in a premixed mode does not lead
to the high flame temperatures that characterize diffusion flames and which are
responsible for NOx emissions. In practical compression-ignition engines, HCCI-
type combustion is achieved by exploiting a technique known as diesel low tem-
3
perature combustion. A straightforward implementation of the HCCI combustion
concept, in fact, is not possible due to the low volatilities of standard diesel fuels,
and the ease with which they autoignite.
HCCI combustion is characterized by the appearance of several burning ker-
nels in the charge long after fuel injection and evaporation are complete. Con-
trolling the ignition delay time of the system becomes therefore fundamental in
achieving the desired levels of emission whilst delivering high thermal efficien-
cies. From a fundamental point of view, this requires a profound understanding
of the physics of turbulent spray autoignition; while, from a modelling point of
view, it implies the need for a turbulent combustion model that is able to capture
the complex interactions existing between evaporation, turbulence and chemistry.
At this point, a reader with little or no experience in the fields of numerical
simulations in general, and turbulent combustion in particular, could raise more
than a question about the significance of turbulent combustion modelling, since
no definition has been provided so far. This conceptual gap will be filled in the
next section, where a short introduction to CFD for turbulent reacting flows will
be given, and the problem of modelling the turbulence-chemistry interaction will
be presented and discussed.
1.3 Numerical simulation of turbulent reacting
flows
The numerical simulation of a turbulent reacting flow consists in using a compu-
tational algorithm to solve a set of partial differential equations that describe the
temporal and spatial evolution of the macroscopic properties of the flow. These
equations will be given in Chapter 2 and are known as the Navier-Stokes equa-
tions. Depending on the application considered, different numerical techniques
for solving the Navier-Stokes equations can be used, the most popular one being
the finite volume method [140]. Finite difference [78, 60] and spectral methods
[49] are also popular choices, especially for those problems where the geometry is
4
simple and high-order differencing schemes are required to minimize the numeri-
cal errors.
The numerical methods mentioned above are based on the discretization of
the governing equations over a computational grid. The grid can be thought
of as an ensemble of finite-size cells that discretize the flow domain. Turbulent
flows are characterized by the presence of a multitude of coherent flow structures
known as eddies, whose dimensions range from a maximum, given by the integral
length scale l0, to a minimum, corresponding to the Kolmogorov length scale
ηK . A numerical simulation in which all the turbulent structures within the
flow are resolved is called Direct Numerical Simulation (DNS), and requires a
computational grid whose cells size is smaller than the Kolmogorov scale. Since
ηK scales with the Reynolds number of the flow Re as [113]:
ηK ∼ l0Re− 34 (1.1)
and since the computational cost of a simulation scales linearly with the num-
ber of cells in the grid, even the simulation of a flow with moderate turbulence
levels would require far more computational resources than those currently af-
fordable. This implies that, in simulating high Reynolds number flows, not all
the flow scales can be directly solved for, and a statistical description should be
sought instead. Two approaches, known as Large Eddy Simulation (or LES) and
Reynolds Averaged Navier Stokes (or RANS), have been widely used for decades
for simulating practical turbulent flows, and are described in the following. Their
presentations is preceded, for completeness, by a short description of DNS.
1.3.1 Direct Numerical Simulation
DNS solves numerically the set of equations that describe a turbulent reacting
flow by resolving all chemical and flow scales. It allows for the most accurate de-
scription of a flow that can be obtained through a numerical simulation, since all
scales of motions are computed explicitly. It is, however, a computationally inten-
sive technique: the grid over which the governing equations are discretized must
be able to resolve not only the Kolmogorov length scale, but also the thickness
5
of the reaction zone, limiting the applicability of DNS to problems characterized
by values of the Reynolds and Damko¨hler numbers of the order of O(102) [145].
The increasing availability of high-performance computing clusters worldwide will
lead to a relaxation of these constraints in the future: simulations of flows with
values of Re up to O(103) are starting to become common, and, in recent years,
a turbulent lifted hydrogen jet flame with a jet Reynolds number of 11,000 was
simulated [159]. For the moment being, however, DNS cannot be used systemati-
cally to investigate practical flows, and should instead be considered as a research
tool that can provide invaluable help in understanding the interactions between
turbulence and chemistry by investigating simplified problems.
Although DNS is commonly perceived as being model free, in simulating a re-
acting flow a choice has to be made concerning the chemical mechanism describing
the combustion chemistry. Detailed mechanisms for the combustion of hydrocar-
bon fuels usually consist of hundreds of species and thousands of reactions [130],
and cannot be used not just in DNS, but also in RANS and LES, due to their ex-
cessive memory requirements. In order to ease the computational burden, either
reduced or global mechanisms are employed when performing a multidimensional
simulation [110]. Reduced mechanisms usually consist of tens of species and
hundreds of elementary reactions; although smaller in size than detailed mecha-
nisms, their use in DNS is still very expensive and, apart from a few exceptions
[98, 99], it has been limited to two-dimensional simulations [23, 146, 158]. Three-
dimensional simulations of reacting flows [86, 137, 22], in fact, still heavily rely on
global kinetic mechanisms, consisting of a handful of species and reactions only.
This choice allows to keep the memory requirements within acceptable limits, as
the larger number of grid points required to discretize a three-dimensional do-
main is compensated by the smaller number of species equations that have to be
solved at each computational node.
DNS of two-phase reacting flows was pioneered by Mashayek, who studied
the evaporation and combustion of fuel droplets in homogeneous shear turbu-
lence [83]. Most of the works that appeared in the literature later on focussed on
the spark-assisted [147, 98, 99] and the spontaneous ignition [148, 149, 125, 126]
6
of n-heptane, which has often been used as a model fuel for diesel, due to its com-
parable cetane number and the availability of many kinetics schemes to describe
its oxidation [130]. Other works investigated the effects of inert droplet evapora-
tion on the mixing properties [154] and the conditional statistics [153] of reacting
mixing layers. DNS was also used to investigate the structure of spray flames
[117, 94], showing the coexistence of premixed and diffusion-like structures. It
should be noted that a proper direct numerical simulation of a two-phase flow
would require a grid resolution small enough to capture the wake region surround-
ing the evaporating droplet. Although several works where the near-droplet field
was resolved have recently appeared in the specialized literature for two-phase
flows [163, 33, 32], the high resolution requirements of these simulations place a
severe limit on the number of droplets that can be tracked with a realistic grid
size, and it cannot be the method of choice when the focus of the study is on
larger scale features of spray-gas interaction and mixing. The works mentioned
above, apart from where differently specified, do not solve for the flow field inside
the droplets or near the droplet wakes; thus, rigorously speaking, they cannot be
considered as proper direct numerical simulations. Despite this nomenclature is
not accurate, it has been frequently used in the specialised literature and has been
widely accepted by the combustion community; for these reasons, while bearing
in mind its limitations, it will be retained in this work.
1.3.2 Reynolds-Averaged Navier-Stokes
RANS is an approach for simulating turbulent flows that solves for a set of trans-
port equations describing the evolution of the mean flow properties. A mean
property has to be interpreted here as the result of an ensemble average of the
instantaneous value of the property over a large number of realizations of the
flow. Mean properties vary over length scales that are several orders of magni-
tude larger than the Kolmogorov length scale. This implies that much coarser
grids than those required by DNS can be used to discretize the computational do-
main, allowing for the numerical simulation of high-Reynolds number flows at an
almost inexpensive computational cost. Additionally, the numerical schemes used
in RANS are often of low order, and thus even the most complicated geometries
7
can be simulated. Despite these advantages, the price to pay when using RANS
based methods is in the loss of the information concerning the turbulent fluctua-
tions occurring in the flow, and in the appearance of unclosed terms on the right
hand side of the governing equations, which stem from the non-linear nature of
the convective terms and represent the influence of turbulent transport processes
on the mean flow properties. Modelling of these quantities represent a classical
problem in the study of turbulence, and some of the most popular approaches
are discussed in every advanced textbook on turbulent combustion, such as [110]
and [20]. Additionally, in the presence of chemical processes occurring within the
flow, the source terms describing the production / destruction rate of each of the
unsteady species that participate in the reactions are also unclosed. Modelling of
the chemical source terms in a turbulent flow is a problem that affects not just
RANS based methods, but LES ones too, and will be addressed in more detail in
Section 1.4.
1.3.3 Large Eddy Simulation
LES is a method that lies half way between RANS and DNS. It attempts at solv-
ing directly for the largest, most-energetic turbulent motions in the flow field,
while the influence of the smallest scales on the larger ones has to be modeled.
The governing equations solved by LES are obtained by applying a spatial filter-
ing operation to the Navier-Stokes equations. Similarly to RANS methods, this
leads to the appearance of unclosed terms on the right hand side of the governing
equations, which again stem from the non-linear nature of the convective fluxes.
These quantities describe how the motion of the resolved scales is influenced by
the presence of the unresolved turbulent eddies. Similarly to their RANS coun-
terparts, they are unclosed and thus proper modelling strategies must be devised.
The interested reader can refer to the book of Pope [113] for a description of some
of the most popular models.
LES has been motivated by the limitations of both DNS and RANS methods.
Differently from DNS, the dynamics of the small scale turbulent structures is not
8
resolved in LES. This greatly reduces the computational power required by a sim-
ulation, since the characteristic cell size of the grid has now to be smaller than just
the smallest resolved motion in the flow, and not the Kolmogorov length scale
anymore. Additionally, being the largest scales those with the highest energy
content [113], modelling of the small scales only has little influence on the correct
resolution of the major features of the flow investigated. This aspect represents a
major difference between LES and RANS, where all turbulent scales are modeled
and accurate capturing of the mean flow features is strongly affected by the choice
of the turbulence model used. Although cheaper than DNS, Large Eddy Simu-
lation is still considerably more expensive than RANS, limiting its widespread
use in industry. This implies that, outside of the academic environment, the use
of LES is justified only in those situations where information on the mean flow
properties is not sufficient anymore, and the unsteady features of the flow need
to be captured to gain a complete understanding of the problem under exami-
nation. Some examples demonstrating the power of LES in capturing complex
phenomena in turbulent reacting flows include the simulation of spark ignition
in combustion chambers for aeronautical gas turbines [142], and the study of the
combustion dynamics of premixed flames in gas turbines for power generation
[17].
1.4 The turbulent combustion problem
The numerical simulation of any turbulent flow with a sufficiently high Reynolds
number requires the use of a RANS or LES technique for the description of
turbulent transport phenomena, due to the excessive computational cost of DNS.
This leads to the appearance of additional terms in the equations that describe the
evolution of the averaged (filtered) fields, which stem from the non-linear nature of
the convective terms. These quantities describe the effects of unresolved turbulent
transport on the resolved fields: they remain unknown and must be evaluated by
mean of a turbulence model. Although a correct description of turbulent stresses
already represents a difficult task for many flow configurations, an even more
complex modelling challenge arises in the presence of chemical reactions. This
is related to the mathematical form of the chemical source terms ω˙α, which are
9
complex functions of both the species concentrations Y and the temperature T .
Being ω˙α strongly non-linear, its expectation, or filtered part in the context of
LES, cannot be expressed using simply:
˜˙ωα 6= ω˙α(Y˜β, T˜ ) (1.2)
since this modelling choice would lead to errors in estimating ˜˙ωα up to sev-
eral orders of magnitude [110]. A multitude of methods have been proposed for
evaluating the averaged (filtered) chemical source term: for the specific case of
non-premixed combustion, a mention is deserved for the Laminar Flamelet model
[106, 107], the Probability Density Function (PDF) method [111] and the Condi-
tional Moment Closure (CMC) [68, 12]. A brief outline of the first two approaches
is given below, while the CMC model will be described in detail in Chapter 3.
The Laminar Flamelet concept for turbulent diffusion flames was introduced
in the framework of RANS during the 1980’s, and extended to LES only in re-
cent years [109]. It views a turbulent diffusion flame as an ensemble of laminar
diffusion flamelets, whose structure can be described using a coordinate-free for-
mulation under the hypothesis of equal diffusivities of all chemical species [106].
The underlying assumption is that, for sufficiently high values of the Damko¨hler
number, heat release reactions are confined in a thin layer in the proximity of the
stoichiometric mixture fraction isosurface: the flame can then be treated locally
as a one-dimensional laminar flame sheet embedded in the turbulent flow, and the
physical coordinate can be replaced by the mixture fraction. A rigorous definition
of the mixture fraction, which will be denoted by the symbol ξ throughout this
work, will be given in Chapter 2; for the moment, it may be thought as an inert
tracer, representing the local amount of mixture that has been generated from
the fuel stream. The coupling between the non-equilibrium chemical calculation
and the flow field is provided by the pressure and a quantity known as the scalar
dissipation rate:
χ = 2D
(
∂ξ
∂xj
)2
(1.3)
10
which accounts for the strain effects of the flow field on the transport of the
reactive scalars in mixture fraction space. In its standard implementation, the
Flamelet model consists of a library of quasi-steady flamelet structures which
are parametrized in terms of the mixture fraction and χ [107]. Mean values of
the reactive scalars are retrieved by evaluating, for each chemical species α, the
following integral:
Y˜α =
∫ 1
0
∫ ∞
0
Yα(η, ζ)P˜η,ζ(η, ζ)dζdη (1.4)
where Yα(η, ζ) represents the quasi-steady flamelet solution at ξ = η and for χ = ζ
corresponding to the α-th species mass fraction, and P˜η,ζ(η, ζ) is the Favre joint
PDF of mixture fraction and scalar dissipation rate. The main limit of this for-
mulation, known as Steady Laminar Flamelet Model (SLFM), lies in the fact that
transient phenomena may not be captured correctly, since only steady flamelets
are considered in the library. Unsteady flamelets equations, with preassigned his-
tories of scalar dissipation rate, were first solved by Mauss et al. [88] to simulate
extinction and reignition in a steady turbulent jet diffusion flame. Their calcu-
lations revealed that flamelets transitioning between steady states exhibit statis-
tics that cannot be reproduced using quasi-steady flamelets only. No mention
was done on the procedure for determining the scalar dissipation rate histories.
This point was addressed by Pitsch et al. [108], who proposed the Representative
Interactive Flamelet (RIF) model. In the RIF method, the instantaneous value
of the scalar dissipation rate to be used in the flamelet equation is evaluated at
each time step, and corresponds to the domain-averaged value of 〈χ| ξs〉. This
last term is the conditional expectation of the scalar dissipation rate evaluated
at the stoichiometric mixture fraction value. A further development of RIF is
represented by the Eulerian Particle Flamelet Model (EPFM), in which differ-
ent regions of the computational domain are assigned to flamelets with different
histories. This approach allows to account for the presence of inhomogeneities
in the flow field, which are neglected in the standard RIF method. Simulations
performed by Barths et al. [7] for a direct injection diesel engine showed that
EPFM yields considerably better predictions than RIF, provided that a sufficient
11
number of unsteady flamelets is chosen.
The transported PDF method was developed in the same period as the lami-
nar flamelet model. It solves for the transport equation of either the joint PDF of
composition and velocity, or the joint PDF of composition only. Chemical source
terms appearing in the governing equations are closed, whereas terms describing
micro-mixing effects need to be modeled. The PDF transport equation, due to its
high dimensionality, must be solved numerically by mean of stochastic methods,
as the use of standard solution techniques would be computationally unafford-
able. One of the most popular approaches is the Lagrangian stochastic particle
method, in which the joint PDF is represented by an ensemble of stochastic par-
ticles, whose evolution is tracked in a Lagrangian framework [111]. More recent
formulations, such as the Stochastic Fields method proposed by Valin˜o [144],
allow to represent the PDF in terms of several Monte Carlo stochastic fields,
which can be computed using an Eulerian grid. Irrespective of the particular
solution procedure chosen, the use of a stochastic approach implies the need for a
high number of samples to obtain a solution which is statistically convergent: this
constraint makes the PDF method one of the most computationally intensive tur-
bulent combustion models available. Recent applications of the transported PDF
approach include simulations of piloted methane flames with increasing levels of
extinction [57], the study of HCCI combustion with particular emphasis on the
effects of turbulence-chemistry interactions on the ignition timing and the emis-
sions [161], and the capturing of micromixing effects in problems representative
of atmospheric chemistry and dispersion [39].
1.5 Objectives and outlook of the dissertation
The objective of the work presented in this dissertation is to provide a better
understanding of the physics of hydrocarbons sprays igniting at conditions rele-
vant to diesel engines through the use of Direct Numerical Simulations, and to
derive, implement and validate a physically-sound modelling of terms describing
droplet related effects within an advanced model for the numerical simulation of
12
turbulent non-premixed combustion, the CMC method.
Closures of the spray related terms appearing in the CMC equations have al-
ready been proposed in the past [65, 121]. These works, however, did not properly
take into account the influence that droplet evaporation has on the mixing field.
Additionally, the term describing droplet evaporation has not been modeled in a
rigorous way, following first principles. This motivates our interest in developing
closures that better reproduce the physics of the problem investigated, and that
incorporate a full description of the influence that the presence of the liquid spray
has on the background gaseous phase.
Concerning the fundamental investigation of autoignition processes in sprays,
there is only a limited number of DNS works on this subject that have appeared in
the literature [148, 149, 125]. These studies focused on high temperature ignition,
for which the chemistry behaves in an Arrhenius way, and no information was
provided for ignition occurring at the low temperature values relevant for HCCI
combustion. The DNS simulations presented in this dissertation aim at filling
this gap by shedding some light on the characterization of the ignition kernels in
sprays at high pressure / low temperature conditions. In particular, our goal is
to verify whether some known results for autoignition occurring in single-phase
flows continue to hold even when the fuel is supplied to the oxidizer in a liquid
form. The DNS data will also serve the additional purpose of validating the CMC
method for two-phase flows.
The dissertation is structured as follows. Chapter 2 provides a brief discussion
on the mathematical description of gaseous flows in the presence of a liquid spray
and introduces the equations governing the temporal and spatial evolution of the
macroscopic properties of these flows. The CMC governing equations for single-
and two-phase flows are then presented and discussed in Chapter 3, together with
a detailed review of the different modelling approaches that have been proposed
for the unclosed terms appearing in these equations. The problem of closing
droplet-related terms in CMC is also introduced, and an original approach for
a physically-sound modelling of these quantities is derived. Chapter 4 presents
13
results from the numerical simulation of several n-heptane sprays, autoigniting at
diesel engine relevant conditions, where the interaction between turbulence and
chemistry was described with the CMC method for two-phase flow presented in
Chapter 3. The scope of these simulations was to study diesel spray combustion
and to validate the proposed modelling of the terms describing spray-related ef-
fects in CMC. A first assessment of the influence of these terms on the numerical
predictions was also attempted. Chapter 5 presents an extensive numerical study
on the physics of spray autoignition at conditions relevant for HCCI and diesel
engine combustion. The investigation was carried out by mean of Direct Nu-
merical Simulations. The influence of several parameters on spray autoignition
was studied, among which the initial strength of the turbulent fluctuations in the
background carrier phase, the initial temperature and oxygen dilution in the oxi-
dizer stream, the initial droplet size distribution of the spray, and the initial value
of the global equivalence ratio in the spray region. These simulations were used
to understand the interaction between evaporation, turbulence and chemistry in
autoigniting sprays and to assess a priori the applicability and shortcomings of
the CMC method to the study of this class of problems. Results obtained from
this analysis are presented in Chapter 6. Finally, Chapter 7 summarizes the main
conclusions from the results presented in this dissertation and identifies areas in
which further research work is needed.
14
Chapter 2
The governing equations of
two-phase reacting flows
The temporal and spatial evolution of a flow field is governed by a system of par-
tial differential equations, known as the Navier-Stokes equations, that represent
conservation of mass, momentum and energy within the fluid system considered.
In the presence of chemical reactions, however, these relations do not provide a
complete description of the flow, and must be supplemented by additional balance
equations for the mass fraction of each of the reacting species. Furthermore, for
certain classes of problems, such as those concerning non-premixed combustion,
it may be useful to introduce additional variables to describe the flow behavior:
the most important ones are the mixture fraction and its variance, as they can
be used to provide a measure of the degree of mixing between the fuel and the
oxidizer streams.
The work presented in this dissertation deals with the numerical simulation of
turbulent reacting sprays. The presence of a disperse, liquid phase within a bulk,
gaseous stream introduces an additional modelling challenge, since it requires a
proper description not only of the evolution of each phase, but also of their mu-
tual interactions. This description strongly depends on the method chosen for
simulating the evolution of the gaseous and the liquid phase; thus, it is important
to have an overview of the different methods available for simulating multi-phase
flows. Knowing the strengths and the weaknesses of each of these methods, in
15
fact, will provide a useful guide for choosing the most convenient approach.
The chapter is organized as follows. The mainstream approaches used for the
simulation of two-phase flows are presented and discussed at first, and a choice
is made for the method to be used in the remainder of the dissertation. Then,
the equations governing the evolution of the gaseous and liquid phases are given.
These equations are those solved by the computational algorithm used for the
direct numerical simulations described in Chapter 5, and represent the starting
point for the derivation of the CMC equations presented in Chapter 3. Derivation
of the RANS equations that are used for the simulations presented in Chapter
4 is also based on these equations. Finally, a new variable that describes the
extent of the mixing between the air and fuel streams in a combustion system,
known as the mixture fraction, is introduced, and its governing equation is derived
under certain simplifying assumptions. This quantity plays a crucial role in the
modelling of turbulent non-premixed combustion, and its importance will become
evident in the next chapters.
2.1 Mathematical description of two-phase flows
Multiphase flows comprise a wide class of problems in fluid mechanics where two
or more phases are present within the same fluid system. A particular example
of a multiphase flow is represented by two-phase flows, where liquid droplets are
dispersed within a gaseous stream. Several approaches have been proposed in
the literature for the numerical simulations of a two-phase flow: in the context of
reacting sprays, however, the choice has often fallen on two formulations, known
as Eulerian-Eulerian and Eulerian-Lagrangian. These approaches both describe
the evolution of the gaseous phase in an Eulerian framework, and differ for the
description of the discrete phase: in the Eulerian-Eulerian method, an Eulerian
viewpoint is adopted, while the Eulerian-Lagrangian method is based on a La-
grangian framework. In both formulations, the variables representing the gaseous
and liquid properties are instantaneous spatial averages taken over a sufficiently
small volume that contains both phases [131, 132, 36]. This averaging proce-
dure implies that both the gas and liquid properties are defined at any point in
16
physical space, regardless of whether that point, at the particular instant of time
considered, is within the gas or the liquid. This procedure allows resolution only
on scales larger than the average spacing between neighboring droplets: hence,
details on the fine scale structure of both the flow and mixing fields are lost. If
a resolution of a two-phase flow up to the gas-liquid interface is required, use of
different approaches, such as the instantaneous two-fluid formulation of Kataoka
[59], should be made. These methods track the position and evolution of the
gas-liquid interface, and have been used to study mixing between the oxidizer
and the fuel vapor under turbulent conditions [33, 163]. Although they provide
a complete description of all fluid scales, their computational cost is extremely
high, limiting the number of droplets that can be affordably computed to less
than a few hundreds. For this reason, these methods will not be considered in
the remainder of this work.
Eulerian-Lagrangian approaches are very popular in the combustion commu-
nity, and have been used in both fundamental [125, 135, 116, 133, 155] and ap-
plied works [52, 4, 56]. In these methods, the droplets constituting the spray
are tracked individually. This requires the solution, for each droplet, of a set of
transport equations describing the temporal evolution of their position, velocity,
temperature and mass. The main advantage of this approach is that following the
particle motions through a Lagrangian tracking scheme appears as the natural,
most intuitive choice. Another advantage is that a wide range of well estab-
lished models exists for the terms describing the interaction between the bulk
and discrete phases [132]. Additionally, the handling of boundary conditions,
such as spray initialization and droplet behavior at solid walls, as well as the
treatment of droplet-droplet collisions and droplet breakup, is less problematic
than in Eulerian-Eulerian methods, and several physically-sound models exist
for describing these phenomena. The main weakness of the Eulerian-Lagrangian
framework lies in its limitation in achieving good parallel computing performance,
due to the difficulties in redistributing the computational load equally among the
computational nodes. It can be shown [118] that, on large grids, only strategies
based on the partitioning of the computational domain will be efficient, since they
minimize the amount of data that have to be exchanged between the processors at
17
each iteration. For single-phase flows, splitting the grid into subdomains having
similar number of cells should allow for a reasonably efficient parallel computa-
tion. This strategy, however, may not be appropriate when applied to two-phase
flows, due to the fact that the droplets are not necessarily distributed homo-
geneously through the domain: situations may then arise in which most of the
droplets are assigned to few processors only, hence creating a strong load imbal-
ance among the nodes. Also, the droplet distribution may not be constant during
the simulation itself [118]; in simulating the reignition sequence of a gas turbine
combustor, for example, droplets are first uniformly located in the combustion
chamber, and then evaporate quickly after ignition, leaving droplets only in the
near injector region. Clearly, more efficient domain partitioning strategies have
to be developed for these applications.
Another limitation of the Eulerian-Lagrangian approach is related to the so
called parcel approximation. Realistic sprays are made up of billions of droplets,
each evolving along a different trajectory. In simulating a practical device, track-
ing of each droplet in the spray is currently unachievable, due to the prohibitive
memory requirements. Instead, the parcel approximation is used. Droplets within
the computational domain are grouped into classes, according to some sorting
criteria such as their initial diameter, velocity, injection location, etc. It is then
assumed that droplets belonging to the same class follow the same evolution: as
a consequence, it is possible to replace droplets belonging to a class by a corre-
sponding average droplet, or parcel. A parcel is characterized not only by the
physical properties of its corresponding droplets, such as the temperature, diam-
eter, or the velocity, but also by the number of droplets it represents. The parcel
approximation allows for simulating a polydisperse spray in a relatively cheap and
straightforward manner, contrary to Eulerian-Eulerian methods, where equations
for each droplet class would have to be solved for on the entire Eulerian grid,
leading to an extremely high computational cost [118]; however, it also affects
the quality of the numerical solution considerably. Nakamura et al. [94] found
that, in simulating a reacting laminar counterflow with air and a n-decane spray
injected from the upper port, and a premixed mixture of air and n-decane from
the lower port, the increase in temperature due to chemical reactions is shifted
18
towards the lower port with increasing droplets / parcel ratio. The droplet charac-
teristic evaporation time was also found to increase when the number of droplets
represented by a parcel increased. The authors linked the observed suppression
of evaporation to the fact that, initially, the mass evaporated from a parcel is
higher than that evaporated from a single droplet: as a consequence, the par-
tial vapor fuel pressure approaches its saturation value faster, inhibiting further
droplet evaporation later on.
Eulerian-Eulerian formulations are less popular than Eulerian-Lagrangian ones.
Differently from Eulerian-Lagrangian methods, the liquid phase is solved for using
an Eulerian framework. These approaches have some interesting characteristics,
such as the possibility of using the same solution algorithm for both the gaseous
and liquid phases, and their suitability for parallel computations [90]. They may
be considerably faster than Eulerian-Lagrangian approaches for simulating com-
plex, large-scale devices, such as a liquid-fueled annular combustion chamber
of a gas turbine, where an extremely large number of computational parcels is
required for representing the liquid phase correctly. However, for simpler con-
figurations, the computational time required by a simulation is comparable, if
not longer, to that required if an Eulerian-Lagrangian method were used [118].
The approach seems also to be limited, at least from a practical point of view,
to the simulation of monodisperse sprays. Computing a polydisperse spray, in
fact, would require solving separate Eulerian transport equations for each of the
droplets classes constituting the spray, which is not currently possible on modern
computers due to the high memory requirements. Last, but not least, there is
no consensus yet regarding the modelling of the unclosed terms that appear in
the equations governing the Eulerian liquid fields [118]. Although promising, it
is clear that Eulerian-Eulerian formulations need further development and vali-
dation, and thus will not be used in this work.
19
2.2 Governing equations
2.2.1 Gaseous phase governing equations
The equations describing the evolution of mass, momentum, energy and species
mass fractions in a gaseous medium containing a disperse, liquid phase are given
below. The interactions between the two phases are treated using the Eulerian-
Lagrangian approach. These equations are those solved in the Direct Numerical
Simulations presented in Chapter 5, with the caveat on the DNS terminology
expressed in Chapter 1, and constitute the starting points for deriving the RANS
equations. They are formally identical to the equations that describe the evo-
lution of a purely gaseous medium, whose derivation can be found in any com-
bustion textbook [75, 76]. The only difference lies in the presence of additional
source terms on their right hand sides, which account for the transfers of mass,
momentum and energy between the two phases.
2.2.1.1 Mass conservation
The equation describing conservation of mass within the flow field is:
∂ρ
∂t
+
∂ρuj
∂xj
= Γm (2.1)
where ρ is the fluid density, uj the j-th velocity component of the flow field,
and Γm is the evaporated liquid mass per unit volume and time. Note that the
Einstein summation convention for repeated roman indices has been used here.
This convention will be adopted in the remainder of the dissertation, except where
differently specified.
2.2.1.2 Momentum conservation
The equation describing conservation of momentum in the i-th direction is:
∂ρui
∂t
+
∂ρuiuj
∂xj
= − ∂p
∂xi
+
∂τij
∂xj
+ ρFi + Γui (2.2)
p is the pressure, while Fi is the i-th component of the body force per unit mass
acting on the flow field. τij is the viscous stress tensor and is responsible for
20
dissipation of momentum into heat due to the molecular viscosity of the fluid.
For a Newtonian fluid, τij can be expressed as:
τij = µ
(
∂ui
∂xj
+
∂uj
∂xi
)
− 2
3
µδij
∂uj
∂xj
where δij is the Kronecker delta, being equal to 1 for i = j, and zero otherwise.
The last term on the right hand side of Equation 2.2 accounts for the change in
momentum of the gas phase due to evaporation of liquid fuel.
2.2.1.3 Energy conservation
The equation describing conservation of total energy is:
∂ρE
∂t
+
∂ρujE
∂xj
= −∂puj
∂xj
+
∂τijui
∂xj
− ∂qj
∂xj
+ ΓE (2.3)
where the internal energy E is:
E =
Ns∑
α=1
Yαhα − p
ρ
+
1
2
umum (2.4)
In presenting Equation 2.3, heat transfer due to radiation has been neglected. hα
is the specific enthalpy of species α, and is defined as:
hα = h
0
α +
∫ T
T0
cp,αdT (2.5)
where h0α is the specific enthalpy of species α at the reference temperature T0,
and cp,α the corresponding specific heat capacity. qj is the j-th component of the
heat flux vector, which is expressed through the Fourier law:
qj = −λ ∂T
∂xj
+ ρ
N∑
α=1
Vα,jYαhα (2.6)
where λ is the heat conductivity coefficient. Vα,j is the j-th component of the
molecular diffusion velocity of species α. ΓE describes the change in internal
energy of the gaseous flow due to evaporation of liquid fuel.
21
2.2.1.4 Species equation
The transport equation for the α-th species mass fraction is:
∂ρYα
∂t
+
∂ρujYα
∂xj
= −∂ρVα,jYα
∂xj
+ ρω˙α + ΓYα (2.7)
ω˙α is the production / destruction rate per unit volume of the α-th species due
to chemical reactions. Vα,j is evaluated according to Fick’s law, modified so to
ensure consistency between the species and continuity equations:
Vα,jYα = −Dα∂Yα
∂xj
+
(
N∑
β=1
Dβ
∂Yβ
∂xj
)
Yα (2.8)
where Dα is the molecular diffusion coefficient of the α-th species. ΓYα is the
source term due to liquid fuel evaporation. It is equal to Γm for the fuel species
(subscript F is used in the following), and zero otherwise.
2.2.2 Liquid phase governing equations
In the Eulerian-Lagragian framework, the disperse phase is treated by track-
ing the evolution of every droplet in the spray. Since the sprays considered in
this work are dilute, droplet-droplet interactions are neglected. The temperature
within each droplet is assumed to be uniform, and no internal flow circulation
is considered: as a consequence, the motion of each droplet is affected by the
aerodynamic drag force only. In order to evaluate the droplet evaporation rate
and the amount of heat exchanged with the gaseous phase, a thin film assumption
(subscript f) is used. Physical properties in the thin film are computed according
to the 1/3 rule. For each droplet d, its position xd, velocity vd, diameter ad and
temperature Td are computed by solving the following system of equations [2]:
dxd
dt
= vd (2.9)
dvd
dt
=
u(xd, t)− vd
τ vd
(2.10)
22
da2d
dt
= −a
2
d
τ pd
(2.11)
and:
dTd
dt
=
1
τTd
[
T (xd, t)− Td −BT,d Lv
WF cp,f
(
Tcrit − Td
Tcrit − Tref
)0.38]
(2.12)
u and T are the gaseous velocity and temperature, evaluated at the droplet
location. Tref is the liquid boiling temperature at the reference pressure pref , Tcrit
the liquid critical temperature, Lv the molar latent heat of evaporation at the
reference temperature and WF the molecular weight of the fuel species. The
Spalding number for mass transfer is given by:
BM,d =
Y sF,d − YF (xd, t)
1− Y sF,d
(2.13)
where Y sF,d is the fuel vapor mass fraction at the droplet surface, evaluated ac-
cording to the Clausius-Clapeyron law. The Spalding number for heat transfer
BT,d is evaluated from BM,d following the procedure described by Abramzon and
Sirignano [2]:
BT,d = (1 +BM,d)
cp,F Shc
cp,fNucLeF − 1 (2.14)
Shc and Nuc are the modified Sherwood and Nusselt numbers. Their evaluation is
also described in [2], and the interested reader is referred there for further details.
LeF is the Lewis number of the fuel species. The relaxation times appearing in
the droplet governing equations are given by [2]:
τ vd =
ρLa
2
d
18µf
(
1 + 1
6
Re
2
3
d
) (2.15)
τ pd =
ρLa
2
d
4Shc
PrLef
µf
1
ln (1 +BM,d)
(2.16)
τTd =
ρLa
2
d
6Shc
PrLef
µf
BT,d
ln (1 +BM,d)
cp,L
cp,f
(2.17)
23
where Red is the droplet Reynolds number, ρL the liquid fuel density, cp,L the
heat capacity of the liquid and Pr the Prandtl number, set equal to 0.7 here.
In the simulations presented in the following chapters, the liquid fuel chosen
was n-heptane, for which pref = 1 bar, Tres = 371.58 K, Tcrit = 540.15 K, Lv =
31.80 kJ/mol and cp,L = 2260.40 J/(Kg K).
2.2.3 Interphase transfer terms
The source terms describing the coupling between the gaseous and the liquid
phases are given by [2]:
Γm = − 1
V
∑
d
αd
dmd
dt
(2.18)
Γui = −
1
V
∑
d
αd
dmdvd,i
dt
(2.19)
ΓE = − 1
V
∑
d
αd
(
cLPmd
T (xd, t)− Td
τTd
+
dmd
dt
hF (Td) +
1
2
dmdv
2
d,i
dt
)
(2.20)
ΓYα = δα,FΓm (2.21)
where the sums are taken over all particles within the computational domain. md
is the mass of droplet d, and V is the volume of the computational cell in which
droplet d is located. The coefficient αd is a function of space. In the RANS sim-
ulations described in Chapter 4, the Particle-Source In Cell (PSI-Cell) approach
[27] is followed, and αd is thus equal to one if droplet d is found within the cell
corresponding to the generic spatial location considered, and zero otherwise. In
the DNS simulations presented in Chapter 5, αd is evaluated according to the
following expression:
αd(x) =

A · exp (−1
2
|x− xd|2
)
if |x− xd| < 2ad,0
0 otherwise
(2.22)
24
where:
A = V
/∫
Vd
exp
(
−1
2
|x− xd|2
)
dV
and Vd is the spherical volume centered at the droplet location and having radius
equal to 2ad,0, where ad,0 is the initial value of the Sauter mean diameter
1 of
the spray that is being simulated. The definition of αd given by Equation 2.22
redistributes the droplet source terms in the neighborhood of the droplet accord-
ing to a distance function that decreases exponentially with increasing distance
from the droplet centre. The distance function has a cut-off length of 2ad,0 to
avoid any influence of the droplet source terms in the far field. This method
of computing the droplet source terms has been used here to avoid the strong
inhomogeneities in the flow field that the PSI-Cell method would generate in the
presence of dilute sprays, and that would lead to resolution problems when high-
order central difference schemes are used for discretizing the governing equations.
This approach is not new, and has already been experimented successfully by
other authors [83, 149].
2.2.4 Mixture fraction equation
In problems involving combustion of two initially non-premixed streams, it is
useful to introduce an auxiliary variable, known as the mixture fraction, that
describes the extent of mixing between the fuel and oxidizer streams. To define
this quantity, we consider the equation describing a generic chemical reaction:
N∑
i=1
ν ′iMi =
N∑
i=1
ν ′′i Mi (2.23)
Here, νi is the stoichiometric coefficient of species Mi. Atomic elements are con-
served under chemical reactions, hence:
N∑
i=1
(ν ′i − ν ′′i )µikWi = 0 (2.24)
1The Sauter mean diameter is defined as the ratio between the volume occupied by the
spray and its corresponding surface area
25
where µik is the mass fraction of the k-th element in species Mi. The mass fraction
of element k is now given by:
Zk =
N∑
i=1
µikYi (2.25)
and is conserved under chemical reaction. This means that, in the hypothesis of
Fickian diffusion and equal molecular diffusivities for all species, Zk satisfies the
following transport equation:
∂ρZk
∂t
+
∂ρujZk
∂xj
=
∂
∂xj
(
ρD
∂Zk
∂xj
)
(2.26)
whereDi = D. Note that Equation 2.26 is only valid for a purely gaseous medium.
Following Bilger [10], we now consider a situation in which two streams, denoted
by subscripts a and b respectively, mix with each other. We can then define the
following new variable:
ξ =
Zk − Zk,b
Zk,a − Zk,b (2.27)
where Zk,a, Zk,b are the values of Zk within the streams. ξ is called the mixture
fraction. It represents the mass fraction of atoms originated from stream a at
any location and time within the flow field, and varies between 0 in stream b,
and 1 in stream a. It is straightforward to demonstrate that, for a purely gaseous
medium, ξ satisfies the following transport equation:
∂ρξ
∂t
+
∂ρujξ
∂xj
=
∂
∂xj
(
ρD
∂ξ
∂xj
)
(2.28)
In a gaseous medium containing a disperse, liquid phase, the mixture fraction is
no longer a conserved scalar, since fuel vapor is continuously generated by the
evaporation of droplets; as a consequence, Equation 2.28 is no longer valid. A
derivation of the mixture fraction transport equation for a two-phase flow can be
found in [29], and reads:
∂ρξ
∂t
+
∂ρujξ
∂xj
=
∂
∂xj
(
ρD
∂ξ
∂xj
)
+ Γm (2.29)
26
Note that Equations 2.28 and 2.29 only differ for the presence of the evaporation
rate term Γm on the right hand side of Equation 2.29. This term accounts for the
generation of fuel vapor by evaporation.
The Favre-averaged form of Equation 2.29 is used in the RANS-CMC simula-
tions described in Chapter 4 to compute the mean mixture fraction field. In the
Direct Numerical Simulations presented in Chapter 5, the instantaneous value of
the mixture fraction is computed from the definition based on the elemental mass
fractions proposed by Bilger [11, 13], and not by solving its transport equation.
The mean mixture fraction is used in Chapter 4 to characterize the mixing be-
tween the air and fuel streams, while the instantaneous mixture fraction is used
in Chapters 5 and 6 for data exploration and to perform the conditional averaging
of the DNS data.
2.3 Summary
The objective of this chapter was to provide an exhaustive presentation of the
methods used for simulating combustion processes in turbulent sprays. The
mainstream approaches for the numerical simulation of flows where a discrete,
liquid-phase is dispersed into a bulk, gaseous phase were reviewed at first. The
discussion focussed especially on the Eulerian-Eulerian and Eulerian-Lagrangian
methods, for which the corresponding strengths and weaknesses were highlighted.
After having identified in the Eulerian-Lagrangian approach the most convenient
framework for the numerical simulation of the dilute sprays that will be studied
in this dissertation, the Navier-Stokes equations were presented. These equations
describe the temporal and spatial evolution of a turbulent reacting flow, and are
those solved in performing the Direct Numerical Simulations presented in Chap-
ter 5. They also represent the starting point for deriving the RANS equations
that are solved in the simulations presented in Chapter 4. Following the presenta-
tion of the Navier-Stokes equations, the equations governing the evolution of the
liquid phase are given, and a description of terms describing inter-phase transfer
processes is provided. Finally, a new quantity that describes mixing between the
air and fuel streams, the mixture fraction, is defined, and its transport equation
27
is derived under certain simplifying assumptions. This quantity plays a central
role in the CMC method for turbulent non-premixed combustion. Additionally,
it will be used in the remainder of this dissertation for data exploration and data
averaging purposes.
28
Chapter 3
Conditional Moment Closure for
spray combustion
The Conditional Moment Closure (CMC) is one of the most advanced models
available for the simulation of turbulent, non-premixed combustion. It has been
proposed independently by Bilger [12] and Klimenko [68], using two different
approaches to derive the equations governing the evolution of the conditional
moments. The CMC equations can be derived in a rigorous manner, with clearly
stated assumptions, and this constitutes one of the strengths of the approach. Al-
though quite recent, extensive reviews of the model are already available [69, 74].
CMC was developed initially for RANS of single-phase flows, and later ex-
tended to both LES [97] and two-phase flow combustion [133, 121, 92]. The main
idea behind the Conditional Moment Closure is to link the fluctuations of the
reactive scalars to those of a variable representative of the combustion process.
For problems in which the fuel and the oxidizer are already well mixed before en-
tering the reaction zone, such a quantity would be the progress variable, while, for
non-premixed situations, the choice falls on the mixture fraction, which describes
the state of the mixing between the fuel and oxidizer streams. The Conditional
Moment Closure then solves for the transport equations of the reactive variables,
conditionally averaged on the value of the quantity that has been assumed to be
representative of the combustion process. If the conditioning variable was cho-
sen adequately, fluctuations of the conditional averages about their mean values
29
are small, and the chemical source terms appearing in the CMC equations can
be closed at first order. Second order closures [87, 103] of the chemical source
terms were also proposed, which may be suitable for capturing flame extinction
/ reignition phenomena in those situations where fluctuations of the conditional
moments are important.
The chapter is organized as follows. The derivation of the CMC equations
for a single-phase flow is presented first, together with a discussion on the dif-
ferent approaches that have been proposed for their closure. Following this, the
Conditional Moment Closure for multiphase flows is introduced. The rigorous
derivation of the two-phase CMC governing equations proposed by Mortensen
and Bilger [92] is presented, and the quantities describing inter-phase transfer
processes are identified. New modelling strategies for these terms are then dis-
cussed. The evaporation of liquid droplets in a gaseous medium is likely to have
a strong impact on the scalar mixing field, which is characterized by the mixture
fraction PDF and the conditional scalar dissipation rate. Modelling of these terms
in the context of a two-phase flow has received little or no attention in the past,
and the final part of the chapter tries to bridge this gap by describing a possible
approach for incorporating droplet effects into standard single-phase models for
the mixing quantities. One should note that only the RANS version of the CMC
model will be considered in this and the following chapters; extension of CMC to
LES of single-phase flows has already been discussed elsewhere [97, 141], while
LES-CMC for sprays is still a novel field of research, with no related study having
appeared in the literature until very recently [143], and represents a fertile ground
for future work.
3.1 The CMC equations for single phase flows
3.1.1 Derivation of the CMC equations
The CMC equations can be derived using either the decomposition method of
Bilger [12], or the joint PDF method of Klimenko [68]. Although different in
nature, these two techniques lead to the same governing equations for the first
30
order conditional moments. In the following, the decomposition method will be
outlined, while the joint PDF method will be described in Section 3.2.
The starting point for deriving the conditional moment transport equations is to
consider the following decomposition of the generic reactive variable Y [12]:
Y (x, t) = Q(ξ(x, t),x, t) + Y ′′(x, t) (3.1)
where Q(η,x, t) is the expectation of the reactive scalar, conditional on the mix-
ture fraction being equal to η. Y refers to either the mass fraction of the α-th
chemical species in the mixture, Yα, or the temperature, YT . Only the transport
equations for the conditional species mass fractions will be derived here. Deriva-
tion of the conditional temperature equation, in fact, is similar, and it will be
omitted. Differentiation of Equation 3.1 leads to:
∂Y
∂t
=
∂Q
∂t
+
∂Q
∂η
∂ξ
∂t
+
∂Y ′′
∂t
(3.2)
and:
∂Y
∂xj
=
∂Q
∂xj
+
∂Q
∂η
∂ξ
∂xj
+
∂Y ′′
∂xj
(3.3)
The diffusion term that appears in the transport equations for the species mass
fractions can be rewritten as:
∂
∂xj
(
ρD
∂Yα
∂xj
)
=
∂
∂xj
(
ρD
∂Qα
∂xj
)
+
∂Qα
∂η
∂
∂xj
(
ρD
∂ξ
∂xj
)
+ρD
(
∂ξ
∂xj
)2
∂2Qα
∂η2
+ ρD
∂ξ
∂xj
∂2Qα
∂xj∂η
+
∂
∂xj
(
ρD
∂Y ′′α
∂xj
)
(3.4)
where use has been made of Equation 3.3. Substituting Equations 3.2, 3.3 and
3.4 into Equation 2.7, and making use of Equation 2.28 yields:
31
ρ
∂Qα
∂t
+ ρuj
∂Qα
∂xj
− ρN ∂
2Qα
∂η2
− ∂
∂xj
(
ρD
∂Qα
∂xj
)
− ρD ∂ξ
∂xj
∂2Qα
∂xj∂η
+ρ
∂Y
′′
α
∂t
+ ρuj
∂Y ′′α
∂xj
− ∂
∂xj
(
ρD
∂Y
′′
α
∂xj
)
= ρω˙α (3.5)
where N = D
(
∂ξ
∂xj
)2
is half the scalar dissipation rate, representing the rate at
which fluctuations of the conditioning scalar ξ are destroyed. Taking the Favre
conditional expectation of Equation 3.5 leads to:
ρη
∂Qα
∂t
+ ρη 〈uj|η〉 ∂Qα
∂xj
− ρη 〈N |η〉 ∂
2Qα
∂η2
= ρη 〈ω˙α|η〉+ eQα + eYα (3.6)
where:
eQα =
〈
∂
∂xj
(
ρD
∂Qα
∂xj
)
+ ρD
∂ξ
∂xj
∂2Qα
∂xj∂η
∣∣∣∣ η〉 (3.7)
eYα = −
〈
ρ
∂Y ′′α
∂t
+ ρuj
∂Y ′′α
∂xj
− ∂
∂xj
(
ρD
∂Y ′′α
∂xj
)∣∣∣∣ η〉 (3.8)
In the above equations, the notation:
〈·| η〉 = 〈·| ξ(x, t) = η〉
indicates an ensemble averaging operation, subject to the fulfillment of the con-
dition on the right of the vertical bar. Equation 3.6 is the unclosed governing
equation for the conditional species mass fractions. In the hypothesis of high
Reynolds number flow, eQα can be neglected, and the following expression can be
used for eYα [12]:
eYαP (η) = −
∂
∂xj
(
ρη
〈
u′′jY
′′
α
∣∣〉P (η)) (3.9)
Here, the mixture fraction PDF P (η) has been introduced. Substituting Equation
3.9 into Equation 3.6, and dividing by the conditionally averaged density, the
32
CMC transport equation for the species mass fractions is obtained:
∂Qα
∂t
+ 〈uj|η〉 ∂Qα
∂xj
= 〈N |η〉 ∂
2Qα
∂η2
+ 〈ω˙α|η〉 − 1
ρP˜ (η)
∂ρP˜ (η)
〈
u′′jY
′′
α
∣∣〉
∂xj
(3.10)
where P˜ (η) = ρηP (η)/ρ is the Favre-averaged mixture fraction PDF. The nota-
tions (˜·) and (·) indicate a Favre and a Reynolds averaging operation respectively.
Details on these operations can be found in [113] and are not repeated here. The
CMC transport equation for the temperature is obtained following a similar ap-
proach, and reads:
∂QT
∂t
+ 〈uj|η〉 ∂QT
∂xj
= 〈N |η〉 ∂
2QT
∂η2
+
〈ω˙H |η〉
ρη 〈cp| η〉 +
1
〈cp| η〉
〈
1
ρ
∂p
∂t
∣∣∣∣ η〉
+ 〈N | η〉
[
1
〈cp| η〉
(
∂ 〈cp| η〉
∂η
+
N∑
α=1
cp,α
∂Qα
∂η
)]
∂QT
∂η
− 1
ρP˜ (η)
∂ρP˜ (η)
〈
u′′jT
′′∣∣〉
∂xj
(3.11)
3.1.2 Closure of the CMC equations
The Conditional Moment Closure yields a set of transport equations which are
unclosed, and hence need proper modelling. Terms to be modeled are the condi-
tional expectation of the flow velocity 〈uj| η〉, the conditional scalar dissipation
rate 〈N | η〉, the Favre-averaged mixture fraction PDF P˜ (η), the conditional tur-
bulent transport terms eYα , eT , the conditional density and pressure terms, ρη,〈
1
ρ
∂p
∂t
∣∣∣ η〉, and the conditional reaction rates 〈ω˙α|η〉, 〈ω˙H |η〉.
The conditional reaction rates are often closed at first order using the condi-
tional mean mass fractions and temperature, e.g. 〈ω˙α(Yα, T )|η〉 = ω˙α(Qα, QT ),
and 〈ω˙H |η〉 = −
∑N
α=1 hα(QT ) 〈 ω˙α| η〉. These closures implicitly assume that fluc-
tuations of the reactive scalars are mostly due to mixture fraction fluctuations: in
other words, for given mixture fraction value, fluctuations of the reactive scalars
around their conditional mean values are supposed to be small. This hypothesis
33
has been validated in several numerical and experimental works, such as in the
piloted methanol flames studied by Masri et al. [84]. Mastorakos and Bilger [87],
however, pointed out that, in simulating autoignition of turbulent mixtures, the
above assumption may constitute a poor approximation, as conditional quanti-
ties often exhibit considerable scatter during the thermal runaway of the system.
Second order closures were thus proposed, based on either a Taylor expansion
of the chemical source term, [65], or on a presumed shape for the conditional
joint PDF of the species mass fractions and temperature, [103]. Prediction of
the ignition delay of a reactive flow generally improves when using second-order
closures, as shown in [103]; however, as the computational cost of the simulations
also increases, it is generally preferred to employ a first-order closure. There
is no restriction on the choice of the chemical scheme used for evaluating the
reaction rates, apart for the memory requirements: in principle, even detailed
mechanisms can be used, although this implies the availability of large compu-
tational resources. Reduced schemes are often employed to relax the numerical
requirements; however, since the CMC equations are solved over a wide range of
mixture fraction and scalar dissipation rate values, they must have been validated
for the operating conditions of interest prior to their use.
Concerning the conditional velocity term 〈uj| η〉, a common modelling ap-
proach is [69]:
〈uj| η〉 = u˜j +
u˜′′j ξ′′
ξ˜′′2
·
(
η − ξ˜
)
(3.12)
which is based on the assumption of a Gaussian joint PDF between uj and ξ.
The unconditional covariance of the velocity and mixture fraction is modelled as:
u˜′′j ξ′′ = −Dt
∂ξ˜
∂xj
(3.13)
whereDt is the turbulent diffusivity, which is evaluated from the turbulence model
used. Li and Bilger [79] investigated the statistics of the conditional transverse
velocity component in a reactive scalar mixing layer, and found that the linear
velocity model given by Equation 3.12 performs well close to the mixing layer
34
axis, and for values of the mixture fraction close to the mean. Departures from
linear behavior were observed for large values of
∣∣∣η − ξ˜∣∣∣ and at locations close to
the mixing layer edge, and these were explained in the light of a theory based
on the turbulent mixing length. The conditional velocity term received further
attention in the work of Mortensen [91], in which it was pointed out that Equation
3.12 is not appropriate for inhomogeneous flows, as it cannot conserve the central
moments of the mixture fraction of order greater than one. Use of a model for
〈uj| η〉 suggested by Pope [111] was thus proposed:
〈uj| η〉 = u˜j − DT
P (η)
∂P (η)
∂xj
(3.14)
In those configurations where the flow is axisymmetric and the conditional mo-
ments exhibit little variations along the cross-stream coordinate, radially-averaged
CMC equations can be solved for, and 〈uj| η〉 can then be modelled as a PDF-
weighted value as [160]:
〈uj| η〉 =
∫ R
0
u˜jP (η)rdr∫ R
0
P (η)rdr
(3.15)
where R is the characteristic radius of the flow. Another approach, investigated
in the context of CMC with LES [141], is to neglect the dependence of the veloc-
ity components on mixture fraction, e.g. 〈uj| η〉 = u˜j. Although the availability
of several alternatives, and despite its limitation, the linear velocity model is still
widely used, due to its stability and its ease of implementation, and it is often
the model of choice in simulating turbulent reacting flows with CMC.
The mixture fraction PDF can be computed through a Monte-Carlo simulation
of the flow. In standard practice, however, this option is too computationally
expensive, and a presumed shape approach for P˜ (η) is preferred. Popular choices
are the clipped-Gaussian and the β-function distributions [69]. These curves are
parametrized in terms of the mean mixture fraction and of its variance; hence,
transport equations for both ξ˜ and ξ˜′′2 have to be solved for in the CFD simulation.
For the case of a single-phase flow, these equations are derived from Equation 2.28
35
and read:
∂ρξ˜
∂t
+
∂ρu˜j
∂xj
=
∂
∂xj
(
ρD
∂ξ˜
∂xj
)
− ∂ρu˜
′′
j ξ
′′
∂xj
(3.16)
and:
∂ρξ˜′′2
∂t
+
∂ρu˜j ξ˜′′2
∂xj
= −∂ρ
˜u′′j (ξ′′2)
′′
∂xj
− 2ρN˜
+
∂
∂xj
(
ρD
∂ξ˜′′2
∂xj
)
− 2ρu˜′′j ξ′′
∂ξ˜
∂xj
(3.17)
Equations 3.16 and 3.17 contain several unclosed terms. Turbulent fluxes are
usually modeled according to the gradient diffusion hypothesis, while a scale-
similarity assumption is adopted for the unconditional scalar dissipation rate:
N˜ = Cχ

k
ξ˜′′2 (3.18)
where Cχ is a model constant, set equal to 2.0 here, and k,  are the turbulent
kinetic energy and its rate of dissipation respectively. These latter quantities are
obtained from the turbulence model used for simulating the flow. Once ξ˜, ξ˜′′2
have been computed, the mixture fraction PDF is reconstructed at each spatial
location according to [102]:
P˜ (η) = γ1δ(η) + (1− γ1 − γ2) G(η)
Ig
+ γ2δ(1− η)
G(η) =
1√
2piξ˜′′2
exp
(
−(η − ξ˜)
2
2ξ˜′′2
)
Ig =
∫ 1
0
G(η)dη; γ1 =
∫ 0
−∞
G(η)dη; γ2 =
∫ ∞
1
G(η)dη (3.19)
for the clipped Gaussian distribution, and:
36
P˜ (η) =
ηr−1 (1− η)s−1
Ib
Ib =
∫ 1
0
ηr−1 (1− η)s−1 dη
r = ξ˜
(
ξ˜
1− ξ˜
ξ˜′′2
− 1
)
; s = r
1− ξ˜
ξ˜
(3.20)
for the β-function distribution [20]. Modelling of the mixture fraction PDF is
currently an active field of research, due to the importance of this term in turbu-
lent non-premixed combustion models. The β-function distribution has been the
model of choice in most CMC related works in recent years, as it combines a good
description of the mixing field with simplicity and stability. There are turbulent
flows, however, for which this model was found unable to predict the correct PDF
shape: one example is the double scalar mixing layer, which attempts to mimic
the mixing field in piloted flames. For this configuration, a model for P˜ (η) based
on the mapping closure concept was proposed [21], and good agreement was found
in an a priori test with DNS data; however, further validation is needed before it
can be applied with confidence to a wider range of problems.
The conditional scalar dissipation rate represents the rate at which mixture
fraction fluctuations are destroyed, and hence the rate at which molecular scalar
mixing is occurring. It is a term of foremost importance not only in CMC, but
also in other models for turbulent non-premixed combustion, such as flamelets
approaches [106] and PDF methods [111]. Modelling of 〈N | η〉 and P˜ (η) are not
independent of each other, as these two quantities are linked together by the PDF
transport equation [69]:
∂ρηP (η)
∂t
+
∂ρη 〈uj| η〉P (η)
∂xj
= −∂
2ρη 〈N | η〉P (η)
∂η2
(3.21)
Equation 3.21 is valid for a single-phase flow, and can be integrated twice in
mixture fraction space to obtain the following expression for the conditional scalar
37
Figure 3.1: Functional dependence of the scalar dissipation rate on mixture frac-
tion as obtained by using the AMC model [100].
dissipation rate [69]:
P˜ (η) 〈N | η〉 = −1
ρ
{
∂ρI˜1(η)
∂t
+
∂
∂xj
(
ρu˜j I˜1(η) + ρ
u˜′′j ξ′′
ξ˜′′2
I˜2(η)
)}
(3.22)
where:
I˜n(η) =
∫ 1
η
(η0 − η)(η0 − ξ˜)n−1P˜ (η0)dη0 (3.23)
〈N | η〉 can be determined from Equation 3.22 once the mixture fraction PDF has
been computed. This procedure was found to provide very good predictions for
the conditional scalar dissipation rate in turbulent flows characterized by inho-
mogeneous mixing between the reactant streams [31]. A second advantage is that
it enforces consistency between modelling of 〈N | η〉 and P˜ (η). The main draw-
backs lie in the strong dependence of the computed conditional scalar dissipation
rate on the model used for the conditional velocity [89], and in the numerical
procedure used for evaluating the terms on the right hand side of Equation 3.22,
which may lead to negative, unphysical values for 〈N | η〉 [72].
38
Errors in evaluating 〈N | η〉 are expected to affect a CMC calculation only
when they occur within the reaction zone [69]. As a consequence, sufficient accu-
racy can be obtained in most cases by using presumed shape models for the con-
ditional scalar dissipation rate, such as the Amplitude Mapping Closure (AMC)
of O’Brien and Jiang [100], or the model proposed by Girimaji [41]. AMC repre-
sents the conditional scalar dissipation rate profile in a counterflow configuration.
It is the model most often used in practical applications, as it provides accurate
predictions of 〈N | η〉 in inhomogeneous flow conditions [63], and consists in the
following functional form for the conditional scalar dissipation rate:
〈N | η〉 = N0G(η) (3.24)
where:
G(η) = exp
(
−2 [erf−1(2η − 1)]2) (3.25)
and:
N0 =
N˜
2
∫ 1
0
G(η)P˜ (η)dη
(3.26)
The function G(η) has a bell-shape profile peaking at 1 for η = 0.5 and is shown
in Figure 3.1. The model of Girimaji [41] is based on the assumption of a β
distribution for the mixture fraction PDF, and was derived by integrating twice
in mixture fraction space the spatially homogeneous form of Equation 3.21. It
reads:
〈N | η〉 = −2
ξ˜
(
1− ξ˜
)
ξ˜′′2
I(η)
P˜ (η)
N˜
2
(3.27)
where:
I(η) =
∫ η
0
{
ξ˜ ln
(
η0 −G1
)
+
(
1− ξ˜
) [
ln
(
1− η0)−G2]} ·
P˜ (η0)(η − η0)dη0 (3.28)
39
and:
G1 =
∫ 1
0
ln(η)P˜ (η)dη, G2 =
∫ 1
0
ln(1− η)P˜ (η)dη (3.29)
The last terms to be modeled in the CMC equations are ρη,
〈
1
ρ
∂p
∂t
∣∣∣ η〉, eYα
and eT . It is usually assumed that pressure is constant in mixture fraction space,
leading to: 〈
1
ρ
∂p
∂t
∣∣∣∣ η〉 = 1ρη ∂p∂t (3.30)
The equation of state can be used to evaluate the conditional density, yielding:
ρη =
p
R 〈T | η〉∑Nα=1 QαWα (3.31)
eYα , eT represent transport due to turbulent fluxes, and have been investigated
in detail in [119]. They depend on the covariance of the conditional velocity and
species mass fractions and temperature respectively, which are usually modelled
as:
〈
u′′jY
′′
α
∣∣ η〉 = −Dt∂Qα
∂xj
,
〈
u′′jT
′′∣∣ η〉 = −Dt∂QT
∂xj
(3.32)
where the turbulent diffusivity Dt is evaluated from the turbulence model used
in the CFD simulation, and is assumed to be constant in mixture fraction space.
3.2 The CMC equations for two-phase flows
CMC has been successfully applied to combustion of several two-phase flows, in-
cluding spray autoignition in either open reactors [67, 152], closed combustion
vessels [151] or Diesel engines [104, 150, 127], and solid particle combustion in
bagasse-fired boilers [121]. To date, however, there has been no consensus on
how to treat the terms describing interphase transfer processes that appear in
the mixture fraction variance and the conditional moments transport equations.
40
Smith et al. [133] were among the first to derive CMC transport equations valid
in the context of a two-phase flow. Their formulation was used by Kim and Huh
[67] to investigate the response of the ignition delay time to changes in the air
temperature of a n-heptane spray injected in an open reactor. One of the main
findings was that inclusion of droplet terms in the combustion model yielded no
noticeable difference for both the ignition delay time and the conditional profiles
of the species mass fractions and temperature.
The n-heptane autoigniting spray problem studied by Kim and Huh was re-
visited by Wright et al. [152] with the aim of investigating in more detail the
flame propagation phase following autoignition. No attempt was made to include
droplet evaporation effects within the CMC and mixture fraction variance equa-
tions; the authors motivated this choice due to a lack of consensus on how to
model these terms, and based on their small effects on the conditional expecta-
tions as reported in previous works [67]. Calculated ignition delays showed very
good agreement with the experimental data set investigated. Effects due to the
initial turbulence strength in the ambient gas were also correctly captured. This
may suggest that, for certain kinds of problems, spray terms in CMC do not play
a key role for the numerical predictions, and can be safely neglected.
A derivation of the CMC equations for two-phase flows was also proposed
by Rogerson et al. [121] in an attempt to study NO and CO emissions from a
bagasse-fired boiler. Their formulation contained additional terms with respect
to the one proposed by Smith et al. [133], which arose from the presence of
a source term due to particle devolatilization in the gas-phase continuity equa-
tion. Although a closure for the particle terms was provided, their effects on the
combustion model were not explored in detail, and hence no conclusion could be
drawn on their importance.
The CMC equations for spray combustion used in the works mentioned above
were derived from transport equations which did not rigorously describe spray
flows. A rigorous derivation of the CMC equations for two-phase flows was pro-
posed only later by Mortensen and Bilger [92] using the joint-PDF method of
41
Klimenko [69] and the instantaneous two-fluid flow formulation of Kataoka [59],
and it is presented in the following section.
3.2.1 Derivation of the CMC equations for sprays
The starting point for deriving the CMC equations for two-phase flows in a rigor-
ous way is to consider the governing equation for the fine-grained mixture fraction
PDF within phase k, ψk = δ(ξk − η), which is given by [92]:
∂ρkψkφk
∂t
+
∂ρkuj,kψkφk
∂xj
=
∂
∂xj
(
ρkDφk
∂ψk
∂xj
)
− ∂
2ρkNkψkφk
∂η2
−∂ρkξkVˆξ,kψk
∂η
+ ρkΠkψk (3.33)
where:
Πk =
∣∣∣∣ ∂f∂xj
∣∣∣∣ δ(f) (uj,i − uj,k) · nj,k, Vˆξ,k = − ∣∣∣∣ ∂f∂xj
∣∣∣∣ δ(f)Vj,k,ξ · nj,k (3.34)
are the volumetric rate of liquid evaporation, and the volumetric diffusion velocity
across the interface between the two phases, evaluated on a per volume basis.
φk is the indicator function, equal to one within phase k, and zero otherwise.
In deriving Equation 3.33, Fickian diffusion has been assumed within phase k.
Taking the expectation of Equation 3.33, the transport equation for the mixture
fraction PDF is derived:
∂ρη 〈θ〉P (η)
∂t
+
∂ρη 〈θ〉 〈uj| η〉P (η)
∂xj
= −∂
2ρη 〈θ〉 〈N | η〉P (η)
∂η2
−
∂ρη
〈
ξVˆξ
∣∣∣ η〉P (η)
∂η
+ ρη 〈Π| η〉P (η) (3.35)
where use has been made of the properties of the fine-grained PDF [69]. For
the ease of reading, the phase subscript has been dropped: this convention will
be kept from now on for simplicity. Also, the molecular diffusion term has been
42
neglected everywhere, except at the phase interface, since we are interested here
in studying high Reynolds number flows. 〈θ〉 is the gas volume fraction and, for
the dilute sprays considered in this work, the assumption of this quantity being
equal to one will be made. Assuming that Fick’s law holds at the phase interface,
and that the liquid phase consists of a single fuel species only, the following
relation between Vˆξ and Π can be derived from the interfacial mass and species
jump relations:
ξVˆξ = Π(1− ξ) (3.36)
Next, the transport equation for the product between Yα and φ is considered.
This reads as [92]:
∂ρYαψφ
∂t
+
∂ρujYαψφ
∂xj
=
∂
∂xj
(
ρDφ
∂Yαψ
∂xj
)
+ 2
∂
∂η
(
ρDφ
∂Yα
∂xj
∂ξ
∂xj
ψ
)
+ρφω˙αψ − ∂
2ρNYαψφ
∂η2
− ∂ρYαξVˆξψ
∂η
+ ρYα(Π + Vˆα)ψ (3.37)
Averaging, making use of Equations 3.35 and 3.36, and applying the primary
closure assumption as in Klimenko and Bilger [69], the transport equation for the
conditional species mass fractions is derived:
∂Qα
∂t
+ 〈uj| η〉 ∂Qα
∂xj
= − 1
ρη 〈θ〉P (η)
∂ρη 〈θ〉P (η) 〈u′′i Y ′′α | η〉
∂xj
+ 〈N | η〉 ∂
2Qα
∂η2
+
[
δαF −Qα − (1− η) ∂Qα
∂η
] 〈Π| η〉
〈θ〉
+ 〈 ω˙α| η〉 − 1
ρη 〈θ〉P (η)
∂ (1− η) ρηP (η) 〈Y ′′α Π′′| η〉
∂η
(3.38)
where δαF is the Kronecker delta, being equal to 1 if the α-th species is the fuel
species, and zero otherwise. The transport equation for the conditional temper-
ature can be derived in a similar manner, and reads [124]:
43
∂QT
∂t
+ 〈uj| η〉 ∂QT
∂xj
= − 1
ρη 〈θ〉P (η)
∂ρη 〈θ〉P (η)
〈
u′′jT
′′∣∣ η〉
∂xj
+ 〈N | η〉
[
1
cpη
(
∂cpη
∂η
+
N∑
α=1
cp,αη
∂Qα
∂η
)
∂QT
∂η
+
∂2QT
∂η2
]
+
1
cpη
〈
1
ρ
∂p
∂t
∣∣∣∣ η〉− 1ρη 〈θ〉P (η) ∂(1− η)ρηP (η) 〈T
′′Π′′| η〉
∂η
+
1
cpη
〈 ω˙H | η〉 −
[
hfg
cpη
+QT − (1− η)∂QT
∂η
] 〈Π| η〉
〈θ〉 (3.39)
Equations 3.38 and 3.39 contain additional terms with respect to the CMC equa-
tions for single-phase flows. These terms represent the effects of droplets evapora-
tion on the gaseous phase, and need to be modeled. Proper modelling strategies
will be outlined in the following section. Additionally, the models described in
Section 3.1.2 for the mixture fraction PDF and the conditional scalar dissipation
rate do not consider the presence of a disperse phase within the bulk gas, and
physically-sound corrections should be implemented to account for this. These
issues will be also addressed in the next section.
3.2.2 Modelling droplet effects in CMC
3.2.2.1 Existing models
The terms describing the interphase transfer processes in the CMC equations
are functions of three unclosed quantities, the conditional covariance between
the evaporation rate and the species mass fraction and temperature, 〈Y ′′Π′′| η〉,
〈T ′′Π′′| η〉, and the conditional evaporation rate, 〈Π| η〉. To the author’s knowl-
edge, no attempt has been made yet to model the conditional covariance terms,
while several approaches have been put forward to provide a closed form for the
conditional evaporation rate. Re`veillon and Vervisch [116] were among the first
to suggest a model for this quantity, which reads as:
〈Π| η〉 = α(d0, d, ρ, φ)ηm (3.40)
44
The function α depends on the local properties of the spray, while the exponent
m is determined dynamically by exploiting the following property of conditional
averages:
Π˜ =
∫ 1
0
〈Π| η〉P (η)dη (3.41)
A second model for the conditional evaporation rate is the one proposed by Sreed-
hara and Huh [135], which prescribes a linear dependence of 〈Π| η〉 on the mixture
fraction:
〈Π| η〉 =

Π˜∫ ξs
ξ˜
(η−ξ˜)P˜ (η)dη (η − ξ˜) for η ≥ ξ˜
0 for η < ξ˜
(3.42)
Comparison of these models against the conditional evaporation rates obtained
from Direct Numerical Simulations of evaporating droplets in isotropic decaying
turbulence was encouraging. In both cases, the mixture fraction values at the
computational grid nodes were used for extracting 〈Π| η〉 from the numerical
data. These, however, differ from the mixture fraction values at the droplet
surfaces, which are those at which evaporation occurs, and that should be used in
evaluating the conditional averages. Additionally, both models provide functional
forms for 〈Π| η〉 which are continuous in mixture fraction space, and thus appear
to ignore the physics of the evaporation process, which occurs at ξs only. This
feature of evaporation was taken into account in the work of Schroll et al. [124],
who proposed a closure for the conditional evaporation rate in which 〈Π| η〉 is
non-zero only at the saturation mixture fraction value ξs,d of each of the droplet
in the computational cell considered:
ρη 〈Π| η〉 =
1
V P˜ (η)
∑
d
m˙dδ(η − ξs,d) (3.43)
Equation 3.43 must be consistent with the model used for the droplet source
term in the gas-phase continuity equation. This is easily verified by evaluating
45
the convolution integral between 〈Π| η〉 and P (η):∫ 1
0
〈Π| η〉P (η)dη = 1
ρV
∑
d
m˙d (3.44)
The term ξs,d in Equation 3.43 is not known, and must be determined. Assuming
the fuel vapor at the generic droplet surface to be saturated, as is commonly
done in spray computations, and knowing the droplet temperature, the satura-
tion pressure ps can be evaluated through integration of the Clausius-Clapeyron
equation between a reference state and the saturation state:
dp
p
=
hfg
R/WF
dT
T 2
(3.45)
with WF the molecular mass of the fuel. Once ps is known, the fuel mass fraction
at the droplet surface, Y sF , can be estimated as:
Y sF =
1
1 +
[(
p
ps
− 1
)
Wox
WF
] (3.46)
where Wox is the molecular mass of the oxidizer gas. The mixture fraction at
saturation can be determined by assuming ξs and Y
s
F to be equal. This assump-
tion does not account for fuel consumption due to chemical reactions. It is exact
for the case of an inert flow, and approximately correct for autoigniting problems
in the preignition phase, and may be used when droplets complete evaporation
prior to entering the flame region. A more advanced method for estimating ξs,
where fuel consumption is taken into account, is described in Appendix A.
Equation 3.43 provides a description of the evaporation process which is phys-
ically sound: it requires, however, an accurate characterization of the fine-scale
mixing structure of the flow, which has to be provided by a suitable model for
the mixture fraction PDF. As discussed in Section 3.1.2, it is standard practice
to make use of presumed shape models for P˜ (η). This requires the solution of
transport equations for ξ˜ and ξ˜′′2, which, for a two-phase flow, can be derived
46
from Equation 2.29 and read as [29]:
∂ρξ˜
∂t
+
∂ρu˜j ξ˜
∂xj
=
∂
∂xj
(
ρD
∂ξ˜
∂xj
)
− ∂ρu˜
′′
j ξ
′′
∂xj
+ ρΠ˜ (3.47)
and:
∂ρξ˜′′2
∂t
+
∂ρu˜j ξ˜′′2
∂xj
= −∂ρ
˜u′′j (ξ′′2)
′′
∂xj
+
∂
∂xj
(
ρD
∂ξ˜′′2
∂xj
)
− 2ρu˜′′j ξ′′
∂ξ˜
∂xj
−2ρN˜ + 2ρ
(
ξ˜Π− ξ˜ Π˜
)
+ ρ
(
ξ˜2Π˜− ξ˜2Π
)
(3.48)
The last term in Equation 3.47, and the last two terms in Equation 3.48 de-
scribe the generation of mixture fraction and mixture fraction fluctuations due
to evaporation of the fuel droplets. These terms are unclosed, and proper mod-
elling strategies are required. Modelling of the mean evaporation rate is done
consistently with the gas-phase continuity equation:
Π˜ =
1
ρV
∑
d
m˙d (3.49)
Concerning the mixture fraction variance equation, several approaches have been
proposed for the modelling of the droplet-related terms. Re`veillon and Vervisch
[116] made use of their single droplet model to evaluate the averages of the prod-
ucts between Π and ξ, ξ2 respectively:
ξ˜Π =
∫ 1
0
η 〈Π| η〉P (η)dη, ξ˜2Π =
∫ 1
0
η2 〈Π| η〉P (η)dη (3.50)
The shortcoming of this approach lies in the model used for the conditional evap-
oration rate, which does not correctly represent the physics of evaporation. By
assuming linear correlations between Π, ξ and their turbulence intensities, Holl-
mann and Gutheil [52] proposed the following closure:
2ρ
(
ξ˜Π− ξ˜ Π˜
)
+ ρ
(
ξ˜2Π˜− ξ˜2Π
)
= ρΠ˜ξ˜′′2
(
1− 2ξ˜
)
ξ˜
(3.51)
47
As for the single droplet model, one may question whether a linear relationship
between the evaporation rate and the mixture fraction is physically sound or not.
The feature of liquid fuel evaporation occurring at ξs only was taken into account
in the model proposed by Demoulin and Borghi [29]:
ξ˜Π =
1
ρV
∑
d
ξs,im˙d, ξ˜2Π =
1
ρV
∑
d
ξ2s,dm˙d (3.52)
Since ξs ≥ ξ˜, and, by definition, ξ ≤ 1, the net effect of evaporation is to increase
the level of mixture fraction fluctuations in the droplet-laden region. It should be
noted, however, that the presence of evaporating droplets in the gaseous flow is
expected to have a strong impact on the mixing field, and, consequently, on the
scalar dissipation rate. In particular, the small scale structures generated during
evaporation leads to an increased dissipation rate. Schroll et al. [125] studied
spray autoignition in homogeneous decaying turbulence by mean of DNS. They
found that the spatially averaged scalar dissipation rate increases with decreas-
ing initial droplet diameter due to the smaller scale structures exhibited by the
mixture fraction field. Other DNS studies have also shown the effect of inert (e.g.
not fuel) droplet evaporation on scalar dissipation [58]. High values of N˜ pro-
mote fast destruction of scalar fluctuations, and it remains unclear whether the
net effect of these source/sink terms results in an increase of ξ˜′′2 or not. Droplet
effects on scalar mixing are not usually taken into account explicitly: N˜ is often
modelled by using Equation 3.18, which was originally proposed for single-phase
flows, and which is based on the concept of energy cascade, where the dissipa-
tion rate is determined by the energy of the large scales and the rate at which
this cascades to the smaller scales. In contrast, in sprays, scalar fluctuations are
generated at the droplet scale too due to the localized evaporation and although
some thoughts have been put forward [26], reliable models for droplet evaporation
effects on scalar dissipation are not available yet.
3.2.2.2 New methods for estimating P˜ (ξs) and 〈N | ξs〉
The mixture fraction PDF obtained from presumed shape models for single-phase
flows cannot be used as it is to model the conditional evaporation rate term, since
48
it may underestimate the amount of fuel vapor generated during evaporation, and
hence overestimate the influence of droplet evaporation on the gaseous phase. A
relatively simple and effective way of dealing with this problem is to provide a
physically-sound correction of the mixture fraction PDF at saturation conditions,
and it is described in the following. This method makes use of the following
mathematical property of probability density functions [112]:
P˜ (ξs) |〈∇ξ| ξs〉| = Σs (3.53)
where Σs is the surface density of the ξ = ξs isosurface, and 〈∇ξ| ξs〉 is the
expectation of the gradient of ξ at the isosurface. Σs is evaluated by assuming
that, in the flow region where evaporation occurs, fuel vapor at ξs exists at the
droplet surface only. Thus:
Σs =
1
V
∑
d|ξ=ξs
4pir2d (3.54)
where the sum is taken over the droplets evaporating at ξ = ξs within the com-
putational cell considered. The expectation of the mixture fraction gradient at
the droplet surface remains unknown, and needs modelling. Assuming spherical
symmetry, and neglecting unsteady terms in the governing equations, an estimate
of |〈∇ξ| ξs〉| for a single droplet is provided by the diffusive balance of the fuel
species mass fraction at the droplet surface [92]:
|〈∇ξ| ξs〉| = m˙ (1− ξs)
4pir2 (ρD)s
(3.55)
In the case of a polydisperse spray, the average value of 〈∇ξ| ξs〉 should be used.
Thus:
|〈∇ξ| ξs〉| = 1
Nd,ξs
∑
d|ξ=ξs
m˙d (1− ξs)
4pir2d (ρD)s
(3.56)
where Nd,ξs is the number of droplets evaporating at ξs within the computational
cell considered. Substituting Equations 3.54 and 3.56 into Equation 3.53, we
49
obtain:
P˜ (ξs) =
Nd,ξs
V
·
∑
d|ξ=ξs 4pir
2
d∑
d|ξ=ξs
m˙d(1−ξs)
4pir2d(ρD)s
(3.57)
which provides the required estimate of the mixture fraction PDF at saturation.
The assumptions made in evaluating Σs and 〈∇ξ| ξs〉 are acceptable when
the difference between ξs and ξ˜, the mean mixture fraction in the computational
cell considered, is large. Under this condition, in fact, the probability of finding
gaseous mixture at saturation conditions away from the droplet surface is low,
and so the use of Equations 3.54 and 3.56 for evaluating the surface density and
the mean mixture fraction gradient at the ξ = ξs isosurface is justified. There is
yet no guideline on how to determine the minimum value of ξs − ξ˜ below which
the estimate provided by Equation 3.57 is no longer valid. Here, the threshold
value of 2.5σ is used, where σ is the mixture fraction r.m.s.. The mixture fraction
PDF is then modeled as:
P˜ (η) =
 P (ξ˜, ξ˜
′′2; η) if η 6= ξs or ξs − ξ˜ ≤ 2.5σ
P˜ (ξs) otherwise
(3.58)
where P (ξ˜, ξ˜′′2; η) represents the distribution obtained using a two-parameter pre-
sumed shape model. Note that P (ξ˜, ξ˜′′2; η) must be rescaled in order to ensure
that P˜ (η) integrates to one.
The considerations made on the modelling of the mixture fraction PDF in two-
phase flows also apply to the conditional scalar dissipation rate. Presumed shape
models are not able to describe the effects on scalar mixing of the small-scale
structures generated by droplet evaporation; as a consequence, the functional
forms for 〈N | η〉 given by standard single-phase flow models must be corrected to
account for the droplet presence. In particular, an increased rate of scalar dissi-
pation is expected in the droplet near field, and hence at mixture fraction values
close to the saturation ones. This is a direct consequence of the steep gradients
in the fuel mass fraction field which arise in the proximity of the evaporating
50
droplets. As seen in evaluating the mixture fraction PDF, an estimate of these
gradients is provided by the fuel species diffusive balance through Equation 3.56.
If we further assume that the conditional average of the square of the gradient is
equal to the conditional mean gradient squared, we obtain:
〈N | ξs〉 ' D 〈∇ξ| ξs〉2 = D
 1
Nd,ξs
∑
d|ξ=ξs
m˙d (1− ξs)
4pir2d (ρD)s
2 (3.59)
As for the mixture fraction PDF, the above expression is valid only when the
difference ξs − ξ˜ is sufficiently large. If this condition is not satisfied, the con-
tribution to 〈N | ξs〉 given by the gaseous mixture away from the droplet surface
tends to become the principal one, and Equation 3.59 would then overestimate
the conditional scalar dissipation rate at saturation. Similarly to the choice made
for the mixture fraction PDF, the correction is applied only when ξs − ξ˜ is lower
than 2.5σ.
3.2.2.3 Alternative approach by other authors
The models described in the previous section provide some characterization of the
impact of evaporation on the mixing field. These effects are accounted for at the
droplet surfaces only, where evaporation occurs, and are neglected elsewhere. Due
to this known shortcoming, it is worth mentioning here an alternative approach for
the modelling of the mixture fraction PDF and the conditional scalar dissipation
rate in two-phase flows. This model was recently proposed by Zoby et al. [163],
and it is based on the asymptotic analysis of the mixing field in the near droplet
region proposed by Klimenko and Bilger [69]:
ξ = ξ∞ + A exp
(−αr2) (3.60)
where A and α are given by:
A =
m˙ (ξs − ξ∞)
4piρDl
, α =
U
4Dl
(3.61)
51
In the above equations, U is the mean axial flow velocity, while l and r are the
axial and transverse distances from the droplet centre. ξ∞ represents the mixture
fraction value far from the droplets. Assuming that ∂ξ
∂r
 ∂ξ
∂l
, one obtains:
N ' 2D
(
∂ξ
∂r
)2
(3.62)
Note that the scalar dissipation rate was defined here as twice the product of the
molecular diffusivity and the square of the mixture fraction gradient. Differenti-
ation of Equation 3.60 then yields:
∂ξ
∂r
= −2Aαr exp (−αr2) (3.63)
This expression can now be substituted into Equation 3.62 to obtain the value
of the scalar dissipation rate as a function of the local mixture fraction and the
axial distance from the droplet centre:
N(ξ, l) = 8Dα
[
ξ2 ln
(
A
ξ
)]
(3.64)
The conditional scalar dissipation rate is finally obtained by integrating Equation
3.64 over the droplet region:
〈N | η〉 = 1
Rc
∫ Rc
0
N(ξ, l)dl =
2Uη2
Rc
[
ln
(
A
η
)
ln (Rc)− ln
2 (Rc)
2
]
(3.65)
where Rc is the inter-droplet spacing. Once 〈N | η〉 is known, P (η) can be deter-
mined from the steady-state form of the mixture fraction PDF transport equa-
tion. Klimenko and Bilger [69] recommended the use of the following relationship
between P (η) and 〈N | η〉:
P (η) 〈N | η〉 = cm˙ (ξs − ξ∞)
ρ
(3.66)
where c is the droplets number density. Thus:
P (η) =
cm˙ (ξs − ξ∞)
2ρUη2
Rc
[
ln
(
A
η
)
ln (Rc)− ln
2(Rc)
2
] (3.67)
52
The models given by Equations 3.65 and 3.67 were compared against Direct
Numerical Simulations of evaporating droplet arrays, in which the inter and the
near droplet fields were fully resolved. Good agreement with numerical results was
found for both the mixture fraction PDF and the conditional scalar dissipation
rate at locations sufficiently far from the droplet. The main limitation of these
models lies in the assumption that the mixing field is characterized by a wake-
like structure at all axial positions away from the droplet. This is not generally
true when locations close to the droplet surface are considered [69], and hence
we may expect inaccuracies in the predicted value of P (η) and 〈N | η〉 at mixture
fraction values close to the saturation ones. Additionally, one may question how
the volume over which the local scalar dissipation rate is integrated should be
chosen for sprays with strong inhomogeneities in droplet density, as this may
have a strong impact on the computed mixing quantities. Despite these remarks,
the approach appears to be promising, and its application to CMC should be
explored.
3.3 Summary
The objective of this chapter was to provide an accurate and up-to-date review
of the Conditional Moment Closure for spray combustion in RANS. Derivation of
the CMC transport equations for single-phase flows was presented first, followed
by a detailed discussion on the modelling of the unclosed terms that appear in
these equations. Then, CMC equations for two-phase flows were discussed. The
rigorous derivation of these equations proposed by Mortensen and Bilger [92]
was presented in order to highlight the quantities describing inter-phase transfer
processes in CMC. Modelling approaches for these terms were subsequently dis-
cussed, and their strengths and limitations were highlighted. The last part of the
chapter dealt with the modelling of the mixture fraction PDF and the conditional
scalar dissipation rate in the presence of evaporating droplets within the gaseous
phase. It was recognized that evaporation strongly affects the mixture formation,
and that models for the quantities characterizing the mixing that were originally
developed for single-phase flows are unable to capture these effects. Physically
sound corrections for these terms were therefore proposed, together with the
53
presentation of an alternative modelling approach that was proposed by other
authors and that recently appeared in the literature.
54
Chapter 4
Simulations of n-heptane sprays
autoignition with 2D-CMC
The Conditional Moment Closure for two-phase flows described in Chapter 3 is
used here to simulate several autoigniting diesel sprays, for which a large body
of experimental data exists [54, 55] to allow for models validation. Both the mix-
ture fraction PDF and the conditional scalar dissipation rate are closed using the
physically-sound models for two-phase flows that have been derived in Section
3.2.2.2. Modelling of the conditional evaporation rate is done according to the
work of Schroll et al. [124]. Scope of these simulations is to study diesel sprays
with CMC, and to attempt a first assessment of the influence of terms describing
droplet related effects in CMC on the numerical predictions of turbulent reacting
flows.
The chapter is structured as follows. First, modelling of diesel spray com-
bustion is briefly reviewed, with particular emphasis on the challenges that it
poses and on the different approaches that have been proposed in the literature
to tackle this problem. Then, the scientific questions that the simulations pre-
sented in this chapter are meant to answer are presented. These are followed up
by a short description of the experimental set-up that has been simulated, and by
a summary of the numerical techniques used to perform the simulations. Results
are then presented. Particular emphasis is given to the influence that the droplet
terms appearing in the CMC equations have on spray autoignition and on the
55
subsequent flame propagation phase. The last section provides a summary of the
main findings, and suggests directions for future work.
4.1 Introduction
The numerical simulation of an autoigniting spray represents the natural bench-
mark for any turbulent combustion model for two-phase flows due to both the
many physical phenomena (ignition, flame propagation, stabilization) that char-
acterize this problem and its importance for many practical devices, such as
liquid-fueled industrial furnaces, diesel engines, and gas turbines. Despite several
modelling approaches for spray combustion have been proposed over the last two
decades, the challenges posed by this problem remain formidable due to the cou-
pling of many complex physical phenomena interacting in a wide range of length
and time scales, such as liquid fuel atomization, interphase exchanges, and the
turbulent combustion itself.
As it has been discussed in Chapters 2 and 3, the presence of evaporating
droplets within the bulk phase introduces extra terms in the equations governing
the evolution of the fluid dynamics quantities, and in the model used for de-
scribing the interaction between turbulence and chemical reactions. These terms,
which are unclosed, describe how the presence of liquid in the flow affects the
bulk gas, and need proper modelling. Most of the work on spray combustion
that appeared in the literature accounted explicitly for the droplet source terms
appearing in the Navier-Stokes and mean mixture fraction equations, while ne-
glected, partly or totally, those associated with the mixture fraction variance and
the turbulent combustion model. Some of these works are reviewed in the fol-
lowing to illustrate the variety of approaches, as well as the difficulties and the
uncertainties, that are encountered in modelling spray combustion.
In simulating split injection and combustion in an automotive DI diesel en-
gine, Barths et al. [7] and Hasse and Peters [48] employed a standard flamelet
model for gaseous combustion. The mixture fraction PDF was modeled using a
β-function, which was parametrized in terms of the mean mixture fraction and its
56
variance. Although the governing equation for ξ˜ included the source term arising
from evaporation of the liquid phase, droplet related effects were neglected in
the mixture fraction variance equation. Flamelet based models were also used
to simulate a turbulent methanol / air spray diffusion flame by Hollman and
Gutheil [52, 53]. Droplet source terms in the mixture fraction variance equa-
tion were accounted for in both works. A laminar gaseous flamelet library was
used in [52], whereas results presented in [53] were based on a laminar counter-
flow spray flamelet library, hence incorporating droplet evaporation effects. The
above turbulent methanol / air spray flame was also studied by Ge and Gutheil
[40] using a coupled PDF / laminar spray flamelet model. Instead of assuming
a β-function shape for the mixture fraction PDF, as it was done in [52, 53], the
joint PDF of the mixture fraction and enthalpy of the gaseous phase was solved
using a Monte Carlo technique. The PDF transport equation explicitly took into
account droplet related effects. Species mass fractions were computed using a
flamelet model based on a library of laminar spray flamelets, as in [53].
Ignition of a diesel spray in a constant volume vessel, burning at high tem-
perature and high pressure conditions, was simulated by Tap and Veynante [139]
using a generalized flame surface density approach. Their model described the
fuel / air mixing process by solving transport equations for the mean mixture frac-
tion and its variance, with droplet related terms accounted for in both equations.
These were treated using the models proposed by Demoulin and Borghi [30]. The
surface-averaged reaction rates, however, were computed using a flamelet model
based on a laminar gaseous flamelet library. Spray autoignition has also been
tackled using the PDF method. Zhu et al. [162] studied a n-heptane spray au-
toigniting in a combustion vessel at p = 37 bar and Tox = 773− 873 K by solving
the joint PDF transport equation of the progress variable and mixture fraction.
A Monte Carlo technique was employed for the numerical solution, and droplet
source terms were accounted for in the PDF equation. Another approach, em-
ployed by Asimov et al. [6], made use of a multi-dimensional CFD framework
based on the flame surface density and extinction concepts to predict the lift-off
height of autoigniting n-heptane sprays at high temperature / high pressure con-
ditions. No detail, however, was given on the treatment of the terms describing
57
interphase transfer processes in the governing equations.
The available literature on the application of the Conditional Moment Closure
to spray combustion has already been reviewed in detail in Chapter 3 and only the
main conclusions are summarized here. It was observed that, despite the variety
of approaches that have been proposed over the last ten years, no consensus has
been reached on how droplet source terms should be modeled, and whether their
inclusion (or exclusion) in the CMC equations affects the numerical predictions.
This is surprising, also considering the popularity that CMC has been gaining
since its introduction due to successful prediction of a wide range of single-phase
turbulent reacting flows, including, among the others, turbulent lifted jet-flames
[63, 105], bluff-body stabilized flames [66], autoigniting methane and methane-
blends mixtures [64, 123], and piloted turbulent methane flames [35]. It is even
more surprising to notice that, as opposed to spray combustion, application of
Conditional Moment Closure to either novel or traditionally challenging areas,
such as LES simulations [95, 96, 38] or the prediction of NOx [122, 5] and soot
[73, 160, 16] emissions from hydrocarbon flames, has already reached an agreed
formulation in the CMC community.
The scope of the simulations presented in this chapter is to use CMC with the
rigorous and physically-sound modelling of the droplet terms described in Chapter
3 for studying diesel sprays. A second objective is to attempt a first assessment of
the influence of these terms on the numerical predictions; in particular, we want
to determine if the assumption of these terms being small and hence negligible,
as suggested by some authors [65, 152], is appropriate. To this end, several
autoigniting n-heptane sprays, burning at conditions relevant to diesel engines,
were simulated using the commercial CFD code STAR-CD coupled to an in-house
CMC solver. Experimental data included the ignition delay time and the lift-off
height for several ambient oxygen concentrations. These were used to test the
capability of CMC of correctly capturing autoignition, flame propagation, and
flame stabilization under a wide range of operating conditions.
58
4.2 Experimental configuration
The experiments modelled in this chapter were performed at Sandia National
Laboratories. Details on the configuration, operating conditions and measure-
ments available can be found in [54, 55]. Additional information regarding the
characterization of the spray are given in [128, 51, 129]. The experiments con-
sisted in the injection and subsequent burning of a n-heptane spray within a
closed combustion vessel at high temperature / high pressure conditions. The
combustion chamber had a cubic shape, with an edge length of 108 mm. The fuel
injector was mounted at the center of a metal side-port, so that the spray was
directed into the center of the chamber. The ambient temperature and density
were kept constant and equal to 1000 K, 14.8 kg/m3 respectively, corresponding
to an ambient pressure of approximately 42.5 bar. A summary of the ambient
and fuel injector conditions is reported in Table 4.1.
Four different ambient oxygen concentrations were tested to simulate the ef-
fect of exhaust gas recirculation (EGR) in diesel engines, and the corresponding
ambient gas compositions are reported in Table 4.2. The fuel was injected for
7 ms, allowing for spray development, autoignition and flame stabilization. The
aim of the study was to investigate the effect of oxygen dilution on the ignition
delay time and flame lift-off height. Data presented in [55, 54] for the ignition
delay time were obtained by ensemble averaging ' 150 − 200 injection events
for each of the operating conditions considered. Concerning the temporal evolu-
tion of the flame lift-off height, 10 injection events were used [55]. This allows
Fuel type: n-heptane
Fuel temperature: 373K
Injection duration: 7 ms
Orifice diameter: 100µm
Orifice pressure drop: 150 MPa
Ambient temperature: 1000 K
Ambient density: 14.8 kg/m3
Initial turbulent velocity fluctuations: 1 m/s
Table 4.1: Ambient and fuel injector conditions. Source: [55].
59
for comparison of the experimental results against data obtained from unsteady
RANS. The low oxygen content of the ambient gas and the long injection dura-
tion constitute the main differences with other experimental work on diesel spray
autoignition, such as that of Wright et al. [151], and motivate the choice of this
set of data for the simulations presented here.
4.3 Numerical methods
4.3.1 CFD setup
The flow field was simulated using the commercial CFD code STAR-CD with a
RANS method. RANS governing equations are assumed to be known, and there-
fore they are not given here: for more details, the interested reader is referred
to any turbulence textbook, such as [113]. A two-dimensional structured grid,
consisting of 109×143 nodes in the axial / radial directions, was used. The mesh
was refined close to the spray axis to better capture the spray development. As
discussed in Chapter 2, an Eulerian-Lagrangian formulation was used to study
the temporal and spatial evolution of the properties of the gaseous and liquid
phases. Turbulence was modeled through the RNG k- model [156], with stan-
dard values for the model constants. The Reitz-Diwakar model [114] was used for
simulating both atomization and secondary spray breakup. The prescribed cone
angle was set equal to 6.9 degrees, according to [54]. The number of droplets
injected per time step was 50. The thermo-physical properties of the droplets
were calculated internally by STAR. Since the sprays investigated in this study
Case O2 CO2 H2O ξst τid
A 21.0 6.11 3.56 0.061 0.42
B 15.0 6.22 3.63 0.0447 0.61
C 12.0 6.28 3.65 0.0361 0.83
D 10.0 6.33 3.67 0.0304 1.06
Table 4.2: Ambient gas composition (in molar fractions), ignition delay time
(in milliseconds) and stoichiometric mixture fraction for the different operating
conditions investigated. Source: [55].
60
are dilute, droplet-droplet interaction and coalescence were ignored. STAR-CD
standard built-in models were employed for evaluating the droplet source term
in the gas-phase momentum equations, as well as for describing droplet turbu-
lent dispersion [1]. Droplet source terms in the continuity and energy equations
were closed according to the evaporation model of Abramzon and Sirignano [2].
Heat transfer to solid boundaries and radiation were both neglected. No trans-
port equation for the species mass fractions had to be solved by the CFD code,
as these were computed by the CMC module. The mean values returned to
STAR-CD were calculated by integrating the conditional averages over the PDF
in mixture fraction space, and were used to determine the mean flow density. The
interfacing between the two codes has been described in detail in [152, 104], and
the interested reader is referred there for further details.
4.3.2 CMC setup
4.3.2.1 Closure of the CMC equations
The numerical simulations presented in this chapter are based on the solution
of the CMC transport equations for two-phase flows derived by Mortensen and
Bilger [92], Equations 3.38 and 3.39. The impact of evaporation on the com-
bustion model was assessed by simulating each of the conditions listed in Table
4.2 twice, with droplet terms being either neglected or included in the govern-
ing equations, and then comparing the corresponding results. Below, a summary
of the modelling choices for the unclosed terms in the CMC equations is presented.
The mixture fraction PDF was modeled according to a β-function distribu-
tion. As discussed in Chapter 3, this requires the solution of transport equations
for the mean mixture fraction and its variance, Equations 3.47 and 3.48. In those
simulations where evaporation effects were accounted for, the droplet source terms
appearing in the ξ˜′′2 equation were modeled according to Equation 3.52. Further-
more, the resulting functional form for P˜ was corrected at ξs following the method
outlined in Section 3.2.2. The conditional evaporation rate was either neglected
or closed according to Equation 3.43. Equation 3.46 was employed to evaluate
61
the fuel mass fraction at saturation for each of the droplets in the computational
domain. ξs was then assumed to be equal to Y
s
F . The improved method for
estimating ξs that is described in Appendix A was also tested for completeness,
but no substantial difference with the approach described above was found. No
attempt was made of modelling the conditional covariances between the evapora-
tion rate and the species mass fractions and temperature, which were neglected.
The conditional scalar dissipation rate was calculated according to the AMC
model. In the simulations that took into account evaporation related effects, the
modeled expression for 〈N | η〉 was further corrected at ξs according to Equation
3.59. Standard single-phase flow models were employed for the remaining terms.
The linear model was used for the conditional velocities. Conditional turbulent
fluxes were closed according to the gradient diffusion hypothesis. A first-order
closure of the chemical source terms was employed.
4.3.2.2 Numerical methods and boundary conditions
A second-order central differencing scheme was used for discretizing the diffusion
terms in the CMC equations, while an upwind scheme was used for the convective
terms. Concerning the domain discretization, 101 nodes in mixture fraction space
were used, clustered around the stoichiometric mixture fraction ηst. In physical
space, 25 by 35 cells were used in radial and axial directions respectively. Full
operator splitting between transport in physical space, transport in conserved
scalar space and chemistry was employed to reduce the number of ODEs that had
to be solved simultaneously. The validity of this technique has been discussed
in detail by Wright et al. [152] and De Paola et al. [104]. The corresponding
sets of discretized equations were integrated with the stiff integrator VODPK
[19]. Absolute and relative solver tolerances were set equal to 10−14 (10−6 for the
temperature equation) and 10−6 respectively for all conditions investigated. Five
internal CMC time steps were computed for each CFD time step, here set equal
to 10−6 s. The conditional expectations for the unsteady species mass fractions
were initialized according to an adiabatic frozen mixing distribution: the mass
fraction of each species was assumed to vary linearly between its values at the
oxidizer (e.g. η = 0) and the fuel (e.g. η = 1) side, whose compositions were
62
given as inputs (cf. Table 4.2). The initial temperature distribution was chosen
so to yield a linear variation of the mixture enthalpy between the oxidizer and fuel
streams. The enthalpy of the fuel stream was evaluated as if the fuel was already
in gaseous form, with its temperature being equal to the initial temperature of
the droplets.
4.3.2.3 Chemistry
Reaction rates were computed using the reduced n-heptane kinetic mechanism
of Liu et al. [80]. The mechanism was derived from a skeletal one consisting of
43 species and 185 reactions. It contains 22 non steady-state species, reacting
according to 18 global steps. The mechanism was validated in [80] in terms of
ignition delays for homogeneous mixtures at different equivalence ratios, temper-
atures and pressures by mean of experimental data from [25]. It has also been
used for non-premixed autoignition problems by Wright et al. [151].
4.4 Results
Results are first presented for the general flow structure. The behavior of the
conditional moments and source terms in the CMC temperature equation is then
analyzed at selected spatial locations and instants of time to shed light on the
ignition, propagation and flame stabilization mechanisms. Assessment of terms
describing evaporation effects in the governing equations is done by comparing
results extracted from simulations in which these quantities were either neglected
or included.
4.4.1 Unconditional averages
4.4.1.1 Spray penetration
Experiments [55] showed that autoignition always occurs at the head of the pen-
etrating spray jet. This observation suggests that an accurate prediction of the
ignition spot location requires an accurate simulation of the spray penetration
63
and mixing. Figure 4.1 compares the measured and simulated equivalence ra-
tio fields at the instant of time preceding ignition for the four different ambient
oxygen concentrations listed in Table 4.2. Experimental data were obtained by
performing Rayleigh scattering measurements in an inert environment (e.g. 0 %
oxygen) and then processing the images with the appropriate (Na/Nf )st ratio to
reconstruct the corresponding mixing field. The mean fields shown in Figure 4.1
are based on an ensemble average of approximately 30-40 instantaneous measure-
ments each. For each of the four oxygen dilutions considered, the equivalence
ratio field as obtained from the numerical simulations was reconstructed from
the corresponding distribution of the mean mixture fraction. This was extracted
from the numerical simulations of the reactive flow. In fact, since the amount of
heat released prior to ignition is low, the flow field is expected to be little affected
by the presence of chemical reactions, and hence the approximation introduced
in evaluating the equivalence ratio fields from the reacting simulations is accept-
able. Spray penetration is overpredicted when the concentration of oxygen in
the ambient gas is the highest, and underpredicted when is the lowest. For all
cases investigated, the computed mixing fields appear to be considerably more
homogeneous than in the experiments. A possible explanation is that the rate at
which scalar mixing is occurring is overpredicted by the turbulence model used
Figure 4.1: Experimental (left) and computed (right) mean equivalence ratio
fields, plotted at the instant of time preceding ignition for each of the four oxygen
dilutions investigated. Far left: 21 % O2 case. Left: 15 % O2 case. Right: 12 % O2
case. Far right: 10 % O2 case. Experimental data from [55].
64
(a) Radial profiles at z = 20 mm (b) Radial profiles at z = 40 mm
Figure 4.2: Experimental and computed radial profiles of mixture fraction at
different axial positions and at t = 6 ms after start of injection. Data shown refer
to a spray evaporating in an inert environment. Experimental data from [55].
in the simulations.
Figure 4.2 shows the computed and measured radial profiles of mean mixture
fraction at z = 20 and 40 mm and t = 6 ms, where z is the axial distance
from the nozzle. Both experimental and numerical data were obtained for a
spray evaporating in an inert environment. The simulations show more diffused
profiles, which are compatible with the notion of faster turbulent mixing. The
choice of the primary break-up model may also play a role in the discrepancy
observed between the experiments and the simulations, as it affects the droplet
size distribution in the resulting spray: sprays containing smaller drops evaporate
faster, leading to a more homogeneous mixture. Although the prediction of the
mixing field is not excellent, no adjustment of either the turbulence or the spray
models used will be sought here. The purpose of the chapter is not to provide
an extremely accurate reproduction of the experimental results, but to assess the
impact of the terms describing evaporation effects in the combustion model on
the numerical simulations of turbulent reacting sprays.
65
4.4.1.2 Flow structure
Figure 4.3 compares the experimental and computed ignition delay times for the
conditions listed in Table 4.2. The experimental ignition delays were determined
from the analysis of the apparent heat-release rate (AHRR) data. The AHRR
signal plotted against time shows small heat release caused by a cool flame, fol-
lowed by rapid high-temperature ignition and premixed burn. The ignition delay
is defined as the beginning of this high-temperature pressure rise [55]. Based
on this definition, τid was evaluated in the numerical simulations as the instant
of time after start of injection at which the unconditional mean temperature
first reaches the value of 1400 K at any point within the computational domain.
Due to the steep rise in T˜ following autoignition, this definition is little affected
by the choice of the threshold value for the mean temperature. The numerical
simulations overpredict the experiments for all ambient oxygen concentrations
considered. The relative error, defined as the difference between computed and
experimental ignition delays, normalized by the experimental ignition delay, in-
creases with increasing oxygen dilution, ranging from 62 % for Case A to 102 % for
Case D. The trend of increasing τid with decreasing ambient oxygen concentration
was correctly reproduced by the numerical simulations, although absolute values
Figure 4.3: Measured and predicted ignition delay time against O2 mole fraction
in ambient gas. Experimental data from [54].
66
could not be captured. This may be due to uncertainties in modelling droplet
primary breakup, as well as in the models employed for closing the turbulent
transport terms and the CMC equations. The choice of the kinetic mechanism
used for computing the reaction rates may also play a role here. Wright et al.
[151] simulated an n-heptane spray autoigniting in a closed combustion vessel
with two-dimensional CMC. The ambient gas temperature and pressure were
equal to 776 K and 80 bar respectively. Reaction rates were computed using two
kinetic mechanisms: the one of Liu et al. [80], which is also the one used in this
work, and the one proposed by Bikas [9]. The computed ignition delay matched
the experimental data when using the Liu et al. [80] chemistry, while the Bikas
mechanism [9] yielded an ignition delay approximately 30 % shorter than in the
experiments. At 50 bar, however, CMC with the Bikas chemistry was found to
give accurate predictions of the ignition delay time [152]. These results suggest
that the choice of the kinetic mechanism used for computing the reaction rates
may strongly affect the predicted τid. Inclusion of the droplet source terms in the
CMC equations leads to a slight increase of the predicted ignition delays, which
reaches a maximum of 9 % for Case B. This is discussed in more detail in Section
4.4.2.1.
Experimental and computed flame lift-off heights Lf are plotted against time
in Figure 4.4. Measured values were obtained by acquiring temporally resolved
images of high-temperature broadband chemiluminescence from several injection
events (' 10), and then identifying, for each image, the high-temperature region
closest to the injector. The resulting data were then ensemble averaged. In the
numerical simulations, the lift-off height was identified as the region closest to
the injector where the mean temperature was found to be above 1400 K, con-
sistently with the procedure adopted for evaluating the ignition delay. As the
axial gradient of T˜ is rather steep, the choice of the threshold value for the mean
temperature affects the evaluation of Lf only slightly. The axial location of the
autoignition spot is overpredicted for Case A, and underpredicted for the other
cases. The final lift-off height is captured with good accuracy, being slightly
underpredicted for the 15 % O2 ambient oxygen concentration case, and slightly
overpredicted for the 10 % O2 one. The experiments show a sharp decrease in
67
(a) 21 % O2 case (b) 15 % O2 case
(c) 12 % O2 case (d) 10 % O2 case
Figure 4.4: Temporal evolution of flame lift-off height for the different ambient
oxygen concentrations listed in Table 4.2. Experimental data from [54].
Lf following autoignition, corresponding to a fast upstream propagation of the
flame and the numerical simulations predict reasonably well the rate of upstream
flame propagation (dLf/dt), although the final time at which the flame stabilises
is overpredicted due to the fact that the ignition time is overpredicted. Wright
et al. [152] simulated the autoignition of a n-heptane spray at high pressure
68
(p = 50 bar) and intermediate oxidizer temperatures (Tair = 783 − 823 K) in an
open reactor, and found that the flame propagation phase following autoignition
is mainly driven by the combined action of preignition reactions and gas volu-
metric expansion following autoignition. In contrast, analysis of source terms in
the CMC temperature equation in the present problem, shown in Section 4.4.2.2,
suggests that a conventional turbulent flame propagation mechanism is active
here. The slower flame propagation rate may be related to errors in estimating
the flow velocity, or the turbulent flame speed, or both. The presence of droplet
related terms in the CMC equations does not have a strong influence on Lf : the
location of the ignition spot remains unaffected, and the flame tip propagates
upstream at almost the same rate. However, due to the longer ignition delay, the
corresponding Lf − t curves are shifted along the horizontal axis.
Temporal evolutions of mean mixture fraction, mixture fraction variance,
mean OH mass fraction and mean temperature fields are shown in Figures 4.5
- 4.8 for the different ambient oxygen dilutions investigated. The stoichiometric
mixture fraction isosurface is also plotted. The mixture fraction variance exhibits
two peaks, one close to the injection point (located at r = 0.0 mm, z = 0.0 mm),
the other along the spray axis, at the location where the mean mixture fraction
reaches its maximum value. As time progresses, the spray expands in both down-
stream and radial directions, and a slight reduction in the peak values of both ξ˜
and ξ˜′′2 is observed. This is probably caused by a reduction in droplet evapora-
tion rate as the gaseous mixture builds up, due to the corresponding decrease in
Spalding number. Mean OH mass fraction and mean temperature fields reveal
that ignition first occurs along the spray axis, in a region where the mixture is
expected to be lean. The experiments also show ignition to occur along the spray
centerline. As the amount of oxygen in the ambient gas decreases, the region
over which the first increase in temperature occurs becomes broader. Following
autoignition, the flame propagates upstream along the ξ˜ = ξst isosurface, and
simultaneously expands towards the centre of the spray. Flame anchoring occurs
along the stoichiometric mixture fraction isosurface, as revealed by the Y˜OH field.
Decreasing the oxygen content in the ambient gas results in a decrease of both
the maximum flame temperature and the maximum values of OH mass fraction
69
(a) t=0.7 ms
(b) t=0.9 ms
(c) t=1.9 ms
Figure 4.5: From left to right: mean mixture fraction, mixture fraction variance,
mean OH mass fraction and mean temperature fields, plotted at different time
instants. Black line is the stoichiometric mixture fraction isosurface. Droplet
terms in CMC equations included. Data shown for Case A in Table 4.2.
70
(a) t=1.09 ms
(b) t=1.40 ms
(c) t=2.40 ms
Figure 4.6: From left to right: mean mixture fraction, mixture fraction variance,
mean OH mass fraction and mean temperature fields, plotted at different time
instants. Black line is the stoichiometric mixture fraction isosurface. Droplet
terms in CMC equations included. Data shown for Case B in Table 4.2.
71
(a) t=1.51 ms
(b) t=1.95 ms
(c) t=3.31 ms
Figure 4.7: From left to right: mean mixture fraction, mixture fraction variance,
mean OH mass fraction and mean temperature fields, plotted at different time
instants. Black line is the stoichiometric mixture fraction isosurface. Droplet
terms in CMC equations included. Data shown for Case C in Table 4.2.
72
(a) t=2.12 ms
(b) t=2.85 ms
(c) t=4.5 ms
Figure 4.8: From left to right: mean mixture fraction, mixture fraction variance,
mean OH mass fraction and mean temperature fields, plotted at different time
instants. Black line is the stoichiometric mixture fraction isosurface. Droplet
terms in CMC equations included. Data shown for Case D in Table 4.2.
73
(a) t=0.40 ms
(b) t=0.90 ms
Figure 4.9: From left to right: mean mixture fraction (left) and mixture fraction
variance (central and right) fields, plotted at two time instants prior to ignition.
Black line is the stoichiometric mixture fraction isosurface. Droplet terms in
mixture fraction variance equation either neglected (central figure) or modelled
(right figure) according to Equation 3.52. Data shown for Case B in Table 4.2.
reached during ignition, propagation and anchoring of the flame; the overall flame
behavior, however, remains unaltered. In the works of Tap and Veynante [139]
and Gopalakrishnan and Abraham [42], who simulated diesel spray autoignition
using a generalized flame surface density approach and a laminar flamelet model
respectively, ignition was observed at the spray edge. The ambient gas tempera-
74
ture and pressure investigated had both values close to the ones considered here.
The different ignition location could have been caused by the absence of physical
space transport terms in both the laminar flamelet model and the generalized
flame surface density model. Wright et al. [152] showed that, when spatially
homogeneous CMC equations are solved for, the predicted ignition location for a
diesel spray shifts from the spray axis to its edge. Such a zero-dimensional CMC
calculation is equivalent to a laminar flamelet model with several flamelets allo-
cated to fixed regions of the computational domain. Each CMC cell then follows
its own evolution, which is determined only by the local scalar dissipation rate.
As a consequence, heat generated early at the spray tip does not get convected
with the spray, which clearly misses an important physical mechanism, and may
lead to erroneous predictions.
4.4.1.3 Influence of droplet terms on ξ˜′′2 predictions
Figure 4.9 compares the mixture fraction variance fields as obtained when droplet
terms are either neglected or included in Equation 3.48. Also shown are the cor-
(a) Source term: S1 (b) Source term: S2 (c) Difference: S1 − S2
Figure 4.10: From left to right: production terms in ξ˜′′2 transport equation due to
droplets evaporation, due to the spatial gradient of ξ˜, and the difference between
the two, plotted at t = 0.9 ms. Units: kg/(m3 s). Black line is the stoichiometric
mixture fraction isosurface. Data shown for Case B in Table 4.2.
75
(a) t=0.40 ms (b) t=0.90 ms
Figure 4.11: Mixture fraction PDFs plotted at several spatial locations along the
spray axis for the mixing fields shown in Figure 4.9. Black lines: droplet source
terms in mixture fraction variance equation neglected. Red lines: droplet source
terms in mixture fraction variance equation included. Data shown for Case B in
Table 4.2.
responding mean mixture fraction fields. The analysis that follows is done for
Case B in Table 4.2. Results shown in the next sections will also refer to this
case, apart from where differently specified, due to the similar behavior that
characterizes all the sprays considered. Two instants of time prior to ignition are
considered. Evaporation is responsible for increasing the level of mixture fraction
fluctuations. The increase is not spatially uniform: it is appreciable close to the
injection point, reaches a maximum along the spray axis at z = 13.5 mm, where
ξ˜ peaks and evaporation is most intense, and finally diminishes, becoming almost
unnoticeable, further downstream, where evaporation is complete. While there
may be a difference of up to 200 % for the peak values, the spatial spreading
of the two fields is similar, and appears not to be affected by the presence of
the droplet terms in the governing equation. The picture looks qualitatively the
same for the two instants of time considered: this is because the fuel has been
injected continuously throughout the experiment, allowing for the establishment
of quasi-steady conditions.
76
The above observations are further substantiated by the spatial fields of the
production terms in the ξ˜′′2 transport equation due to droplet evaporation, S1,
and large-scale gradients of ξ˜, S2, which are shown for t=0.90 ms in Figure 4.10:
S1 = 2ρ¯
(
ξ˜Π− ξ˜ Π˜
)
+ ρ¯
(
ξ˜2Π˜− ξ˜2Π
)
, S2 = 2
µt
Sc
ξ˜′′2
(
∂ξ˜
∂xj
)2
(4.1)
Also shown is the difference between these two fields, S1−S2. The droplet source
term has a localized spatial distribution, being nonzero only in the evaporation
region: there, it outweighs S2 by approximately one order of magnitude, thus
justifying the large differences in the peak values of ξ˜′′2 observed in Figure 4.9.
Elsewhere, the difference between S1 and S2 is negative, indicating that the stan-
dard mechanism of turbulence generation dominates. This explains why, apart
from the regions where droplets evaporate, the mixture fraction variance fields
computed with and without droplet terms in its governing equation look quali-
tatively and quantitatively similar.
The mixture fraction PDFs corresponding to the mixing fields shown in Figure
4.9 are plotted at several locations along the spray axis in Figure 4.11. The
difference in mixture fraction variance results in different shapes of P˜ (η) at the
location where evaporation is the strongest. In particular, since the mixture
fraction variance values are higher when droplet terms are included in Equation
3.48, the corresponding PDF becomes wider. Closer to the injection point, the
difference between the PDFs is smaller, though still appreciable, while further
downstream the two curves appear virtually identical. This is related to the
vanishing difference in mixture fraction variance in the region where droplets
evaporation is complete. The above evidence suggests that evaporative effects
have a strong impact on the prediction of the mixing field, and should be taken
into account in the numerical investigation of spray flows.
77
(a) CMC without droplets terms (b) CMC with droplets terms
Figure 4.12: Temporal evolution of the conditional mean temperature at the
autoignition location (r = 0 mm, z = 42 mm). Data shown for Case B in Table
4.2.
4.4.2 Conditional averages
4.4.2.1 Autoignition
Figure 4.12 shows the temporal evolution of the conditional mean temperature at
the autoignition spot (r = 0 mm, z = 42 mm). The first mixture to ignite occurs
in correspondence of a specific mixture fraction value, ξMR. Following [85], this
is referred to as most reactive mixture fraction. The value of ξMR depends on
the operating conditions considered and, for the present case, is richer than the
stoichiometric one. Once ignition has occurred, heat is rapidly diffused in mix-
ture fraction space and a fully burning solution is recovered shortly after. Similar
results were also obtained in earlier applications of CMC to spray flames [67] and
in DNS of two-phase, reacting flows [125].
The physical behavior depicted above remains unchanged whether or not
droplet related terms are included in the CMC equations. Evaporative cooling is
most effective at high η values, leading to a drop in conditional mean temperature
that reaches a maximum of 57 K at η = 0.355. At leaner mixture compositions,
its main effect consists in a decrease of the most reactive mixture fraction value,
78
which shifts from about 0.15 to about 0.11. This is reasonable: as the conditional
temperature diminishes due to droplet evaporation, chemical reactions at ξMR are
slowed down. Since the oxidizer temperature is higher than the fuel temperature,
the radical pool starts to develop at lower mixture fraction values, for which the
lower fuel concentration is compensated by the faster chemistry. Another effect
of evaporative cooling is a reduction in the value of 〈T | ξst〉 for the fully burning
distribution of around 10 K, which may be partly responsible for the slower rate
of flame propagation during the premixed flame propagation mode.
As it will be discussed in the next sections, the inclusion of droplet source
terms within the CMC equations has a weak effect not only on the autoignition,
but also on the propagation and anchoring mechanisms of the spray flames inves-
tigated in this work. This is a direct consequence of the large separation existing
in mixture fraction space between the values of ξ at which droplet evaporation
occurs, and those at which chemistry becomes important, and arise from the
rapid heating process that liquid droplet undergo when injected in an hot air
environment. A more comprehensive assessment of the influence of droplets re-
lated terms on numerical predictions of autoigniting sprays would require testing
against a wider range of operating conditions than those considered here.
4.4.2.2 Flame propagation
Temporal evolutions of conditional mean temperature and mean oxygen mass
fraction are shown in Figure 4.13 at different locations within the flame region.
Early during the CMC cell history, chemical reactions proceed in a region around
the most reactive mixture fraction. This is clearly shown by the corresponding
decrease in 〈YO2 | η〉. The amount of heat released in this phase, however, is small,
and does not lead to an appreciable increase in 〈T | η〉. At later times, following
flame propagation, the conditional mean temperature suddenly increases at ξst
first, and at neighboring mixture fraction values later on. This is not surprising.
As pointed out in Section 4.4.1.2, a classical premixed flame propagation mode
holds here, with essentially diffusion of heat and species from the ignited regions
to the unignited regions. Hence, the first increase in 〈T | η〉 for regions other than
79
the autoignition spot is expected around stoichiometry, where the turbulent flame
speed peaks. For other mixture fraction values, the rise in temperature will occur
after a longer time, due to the corresponding slower flame propagation speed and
the time needed for the reaction zones to expand across mixture fraction space.
Long after the flame has propagated, when quasi-steady conditions have been
reached, a full burning temperature distribution is established.
Analysis of the contributions to the CMC temperature equation is presented
in Figure 4.14 for location r = 3.0 mm, z = 32.5 mm and two separate instants
of time. At t = 1.0 ms, ignition has not occurred yet, and no significant heat
transfer due to spatial diffusion takes place at the location considered. Chemical
reactions, which peak around ξMR, generate some heat, which is partly diffused
in mixture fraction space, and partly convected away by the mean flow. Spatial
diffusion terms become important at later times, after the flame has ignited and
started to propagate upstream. At t = 1.27 ms, axial and radial diffusion become
comparable in magnitude to the other contributions in the CMC temperature
equation, with both these quantities peaking around ξst. This indicates that the
flame front is moving upstream, although it has not reached yet the spatial lo-
cation under examination here. Chemical reactions are still most intense around
ξMR. The mean flow continues to act as a sink of heat, convecting heat away
to neighboring regions. Pressure work is negligible at both instants of time and
in the whole mixture fraction space. A similar mechanism of flame propagation
was observed in [152] for a n-heptane spray autoigniting at 50 bar in an open
reactor configuration. In particular, spatial transport terms, both convective and
diffusive ones, were found to play an important role in the flame spreading pro-
cess. The existence of a most reactive mixture fraction value, at which spray
autoignition occurs, was also found. Similar conclusions about the existence of a
most reactive mixture fraction in sprays were drawn in [125]. The capturing of
the correct lift-off height and these analogies with previous works suggest that
RANS-CMC is able to capture, at least qualitatively, the mechanisms of flame
autoignition and propagation for the n-heptane sprays examined here, indicating
its suitability for studying combustion under diesel engine conditions.
80
(a) r = 0.0 mm, z = 32.5 mm
(b) r = 3.0 mm, z = 32.5 mm
Figure 4.13: Temporal evolution of conditional mean temperature and oxygen
mass fraction at selected locations within the flame. Droplet terms in CMC
equations included. Data shown for Case B in Table 4.2.
The presence of droplet terms in the CMC equations does not affect appre-
ciably the flame propagation phase. This is due to the almost negligible influence
that these terms have on the full burning temperature distribution, as depicted
in Section 4.4.2.1, and so on the spatial diffusion of heat.
81
4.4.2.3 Structure of flame base
The balance of terms in the CMC temperature equation is shown in Figure 4.15
for the 15 % O2 dilution case. The locations considered are the flame anchoring
points off (r = 3.0 mm, z = 24 mm) and along (r = 0.0 mm, z = 27.5 mm) the
spray axis, 3 ms after start of injection, when quasi-steady conditions have been
attained. Pressure (not shown in Figure 4.15) and spatial diffusion terms are
found to be small in the whole mixture fraction space. Chemistry is the dom-
inant term, and is balanced by diffusion in mixture fraction space and spatial
convection, with its radial component being larger than the axial one off the
spray axis. The contribution of spatial transport terms to the flame stabilization
mechanism appears to be important here. Similar conclusions were found for
lifted jet flames in cold air [63]. However, an important difference between the
lifted jet flame in cold air and the lifted flame in a diesel spray in hot air is that
in the former a convective-diffusive balance sets in at the flame base, while in
the latter a convective-reactive balance appears, with the spatial diffusion terms
being important, but not as important as in the lifted jet flame. Although not
shown here, inclusion of evaporation effects in the governing equation has almost
no effect on the flame anchoring mechanism.
4.5 Conclusions
Autoignition of a n-heptane spray, burning at diesel engine relevant conditions,
was simulated with first-order, two-dimensional CMC with full treatment of
droplet terms in the mixture fraction variance and CMC equations. Numerical
results were assessed by comparison against experimental data for a range of am-
bient oxygen concentrations. Autoignition always occurred along the spray axis,
in agreement with experiments. The resulting flame then propagated upstream
along the stoichiometric mixture fraction isoline, and simultaneously expanded
towards the centre of the jet, until it reached its final position. Experiments
showed the ignition delay time to increase with a reduction in ambient oxygen
concentration. This trend was captured by the numerical simulations, although
82
(a) t=1.00 ms (b) t=1.27 ms
Figure 4.14: Source terms in CMC temperature equation at r = 3.0 mm, z =
32.5 mm and selected instants of time. Droplet terms in CMC equations included.
Data shown for Case B in Table 4.2.
no quantitative agreement with experimental data could be achieved; in partic-
ular, the ignition delay was overpredicted for all oxygen dilutions considered.
It was argued that, among other factors, the kinetic mechanism for n-heptane
combustion used in the simulations presented in this chapter might play a role
in overpredicting the ignition delay. The location of the autoignition spot was
shifted upstream with respect to the experiments for all conditions investigated,
except for the 21 % O2 dilution case, where it occurred downstream of the ex-
perimental location. The final lift-off height was always well predicted and the
average flame propagation rate was reasonably close to the experimental value
for low oxygen dilutions, while it was underpredicted for the 10 % and 12 % di-
lution cases. Analysis of source terms in the CMC temperature equation during
flame expansion revealed that a flame propagation mode holds here associated
with a convective-diffusive balance, while the final stabilization point involves
a convective-reactive balance. The correct capturing of the final lift-off height
and of the three main phases that characterize a reacting diesel sprays (ignition,
flame propagation, flame anchoring) clearly indicate the suitability of CMC for
studying combustion processes occurring in compression-ignition engines.
83
(a) r = 4.0 mm, z = 25.0 mm (b) r = 0.0 mm, z = 27.5 mm
Figure 4.15: Source terms in CMC temperature equation at flame anchoring
locations (off and along spray axis) for t=3.0 ms. Data shown for Case B in
Table 4.2.
Inclusion of droplet terms in the mixture fraction variance and CMC equa-
tions was found to affect the mixing field predictions, especially in those locations
where evaporation was most intense. In particular, an increase of up to 200 % for
the peak value of ξ˜′′2 when droplet terms were included in the mixture fraction
variance equation was observed. This was related to the high values of these
source terms in the droplet-laden regions of the flow, which were found to be
approximately an order of magnitude higher than those associated with the tra-
ditional gas-phase mechanisms responsible for generating turbulent fluctuations.
The ignition delay time was only little affected by the addition of the droplet
source terms in the governing equations: this is because, for the range of con-
ditions investigated, there was a clear separation between the droplet saturation
mixture fraction value, where fuel evaporation occurred, and the stoichiometric
mixture fraction, where most of chemical reactions took place. The most reactive
mixture fraction was observed to shift towards leaner values when droplet terms
were accounted for in the CMC equations. Also, the value of 〈T | ξst〉 was found
to be about 10 K lower with respect to the cases where droplet source terms were
neglected. Both flame propagation and flame anchoring mechanisms were virtu-
ally unaffected by the presence of the droplet terms in the governing equations.
84
This is, again, a consequence of the large difference existing between the values
of ξs and ξst for the conditions investigated in the simulations presented in this
chapter. Future work is needed to assess the influence of the droplet terms ap-
pearing in the CMC equations on the numerical simulations of sprays for a wider
range of operating conditions.
85

Chapter 5
Direct Numerical Simulations of
spray autoignition
Direct Numerical Simulations are used to investigate the physics of spray
autoignition at the high pressure and low temperature values that characterize
HCCI combustion, for which only a few DNS work are available in the literature.
The main purpose of these simulations is to describe the topology of the ignition
kernel. In particular, our aim is to verify if, similarly to autoignition in gaseous
flows, a most reactive mixture fraction can still be identified, and whether the
well-known negative correlation between heat release and scalar dissipation rate
that characterizes single-phase flows [86] also holds when the fuel is supplied in
a liquid form. The simulations presented in this chapter are also meant to ex-
plore the strong dependences existing between evaporation, mixing and ignition
in sprays, and how changes in the operating conditions (e.g. global equivalence
ratio, initial turbulence intensity of the background phase, etc.) may in turn
affect these processes. Another purpose is to provide a large set of data that
can be used to validate the CMC method for autoigniting sprays. To this scope,
several simulations with a zero-dimensional CMC code have been performed, and
the corresponding results are presented in Chapter 6.
The chapter is structured as follows. The problem of spontaneous ignition
in hydrocarbon sprays is reviewed at first, with particular emphasis on existing
works that attempted at describing the structure of the ignition kernel. Then,
87
details on the computational algorithm used for performing the simulations pre-
sented in this chapter are given, together with a description of the operating con-
ditions investigated. Following this, results are presented for the general topology
of the ignition kernel, which was found to be invariant to changes in the spray
properties. Specific effects on fuel / air mixing and ignition delay time that stem
from changes in the operating conditions are then investigated. The chapter ends
with a summary of the main findings, and with the identification of some key
parameters that may allow control of spray autoignition.
5.1 Introduction and literature review
HCCI combustion has been attracting significant attention in recent years due
to its potential for reduced NOx and particulate matter emissions whilst main-
taining the high thermal efficiency typical of compression-ignition diesel engines
[28, 157]. At the low temperature and high pressure conditions corresponding to
HCCI, combustion occurs through volumetric autoignition. Controlling the igni-
tion delay time of the system becomes thus fundamental in order to achieve the
desired levels of pollutant emissions and to avoid excessive rates of heat release,
which would result in engine knock. Achieving this goal requires a profound un-
derstanding of the physics of spray autoignition.
There is a consistent body of numerical and experimental studies available in
the literature dealing with the problem of autoignition in turbulent gaseous flows.
These have recently been the subject of an extensive review by Mastorakos [85].
Since one may expect spray autoignition to share several features of single-phase
flow autoignition, and in order to highlight differences with respect to the purely
gaseous case, it is worth summarizing the key findings of some of these works.
Using two-dimensional Direct Numerical Simulations with a single-step chem-
istry for methane, Mastorakos et al. [86] studied the characteristics of autoignition
in laminar and turbulent shearless mixing layers between cold fuel and hot air
streams. One major conclusion of this work was that autoigniting kernels were
found to develop in those regions characterized by a well-defined mixture fraction,
88
which was named most reactive mixture fraction ξMR and whose numerical value
depended only on the fuel and oxidizer temperatures and the activation energy.
Additionally, it was observed that, among all flow regions where ξ = ξMR, those
that ignited first were characterized by low values of the scalar dissipation rate.
These results have been later confirmed with a wide range of additional DNS re-
sults, including, among others, configurations with two-dimensional domains and
detailed chemistry for hydrogen [34] and three-dimensional domains with single
[136] and four-step [137] n-heptane chemistries.
Concerning the experimental characterization of gaseous flow autoignition,
Gordon et al. [43] performed OH- and CH2O-LIF imaging of methane jets is-
suing into a vitiated coflow. In their work, it was observed that autoignition is
a localized phenomenon and is always preceded by the formation of a pool of
precursor species, among which a dominant role is played by formaldehyde. OH
levels were found to remain low during this phase. As ignition became imminent,
mass fractions of both CH2O and OH reached a maximum; however, at times
following ignition, the peak in formaldehyde was observed to decrease, whilst
the peak in the mole fraction of OH remained approximately constant. It was
concluded that the localized build-up of formaldehyde was an indicator of au-
toignition. Similar conclusions could be inferred from the numerical simulation
[44] of the autoigniting nitrogen / hydrogen jet experiments described in [46]. In
this case, however, the role of the ignition precursor was taken by the HO2 radical.
Autoignition in diesel sprays plays an important role not only relative to the
flame initiation, but also for its anchoring. Although lifted flames are commonly
encountered in many devices of practical interests, there has been so far little
consensus among researchers regarding the dominant stabilization mechanism,
which has often been explained on the basis of either premixed flame or non-
premixed flamelet theories [81]. In a direct numerical simulation of a turbulent
lifted hydrogen jet flame in a heated coflow, Yoo et al. [159] demonstrated that
an important role in stabilizing a lifted flame is played by localized flow autoigni-
tion. Similar conclusions were obtained from the Direct Numerical Simulations
[62] of the diluted hydrogen jets issuing into a heated turbulent coflow studied
89
experimentally by Markides and Mastorakos [82].
Despite the large body of information available on the interaction between
turbulence and autoignition in single-phase flows, little is known on this topic
when the fuel is supplied in a liquid form. A recent investigation carried out on
turbulent dilute methanol sprays issuing in a vitiated coflow [101] has revealed
that single and two-phase flow autoignition exhibits similar features, in particu-
lar with respect to the early formation of formaldehyde in the radical pool and
the importance of kernels in initiating combustion. A marked difference with au-
toignition in gaseous flows was identified with the frequent appearance of double
reaction zones, which were attributed to vaporizing droplets leading to mixing
and the local formation of ignitable mixtures. No information could be inferred
on the nature of the locally evolving ignition kernels and their interaction with
the turbulent flow due to the difficulty of applying advanced laser diagnostic to
spray flows. This limitation is intrinsic to the experimental techniques currently
used in studying combustion and it is not likely to be solved in the near future.
For this reason, Direct Numerical Simulations coupled with complex chemistry
represent today the most powerful tool for unveiling the main physical features
of spray combustion. One should note that the use of detailed chemistry in DNS
has been hindered so far by its enormous computational requirements. Three-
dimensional simulations have often relied on global chemistries to describe the
complex phenomena leading to ignition [137], while the use of reduced and skele-
tal kinetics has been confined to two-dimensional configurations [148, 158]. The
influence of the kinetic scheme on the numerical simulation of self-igniting n-
heptane jets was analyzed by Viggiano [146]. Her study revealed that, depending
on the operating conditions investigated, there may be situations in which igni-
tion is fluid-dynamics controlled, and others in which chemical kinetics play a
dominant role. In the latter case, the use of a detailed scheme is required to
predict ignition properly, while in the former even global chemistries may be suf-
ficient to obtain reliable results.
The suitability of a two-dimensional configuration for studying autoignition
in non-homogeneous mixtures was discussed by Sreedhara and Lakshmisha [137].
90
Their study revealed that the structure of the autoignition sites in non-premixed
mixtures remains the same irrespective of whether the flow is represented in two or
three dimensions. These results were obtained using a four-step global n-heptane
mechanism only, and their validity for more complex chemistries has not been
assessed yet. Differences may be expected, especially in those situations where
the interaction between chemistry and turbulence is strong and a correct rep-
resentation of the turbulent eddies is mandatory. The above discussion justifies
the choice of a three-dimensional configuration with detailed chemistry treatment
for the simulations presented in this chapter. It should be emphasized that no
3D-DNS with detailed chemistry of spray autoignition has been performed in
the past: previous work with complex chemistry DNS was limited to the spark-
assisted ignition of sprays [98, 99].
Several detailed numerical simulations of autoigniting sprays have been per-
formed in recent years. Wang and Rutland [148, 149] investigated the effects of
initial air temperature and equivalence ratio on the ignition of n-heptane fuel
sprays by means of two-dimensional DNS with detailed chemistry. Their study
focused on high temperature ignition at moderate pressure levels, for which chem-
ical reactions exhibit an Arrhenius-like behavior. It was found that an increase in
the air temperature resulted in a faster ignition event, while an opposite behavior
was observed for the global equivalence ratio. This dependency was explained as
a consequence of the increase in evaporated fuel mass, which was in turn responsi-
ble for a more intense cooling of the carrier phase. Ignition was found to occur at
few isolated locations at first, and then to spread out rapidly to the whole mixture.
Schroll et al. [125] used three-dimensional DNS with one-step global chemistry
to analyze the behavior of spray autoignition in mixture fraction space. Their
study revealed the existence of a mixture fraction value ξMR at which autoigni-
tion is more likely to occur. Additional analysis showed that the first spots to
ignite occurred in those regions where ξ ' ξMR and the scalar dissipation rate was
small. These results are fully consistent with the observations made for autoigni-
tion in single-phase flows [86]. A close relationship between spray parameters
(e.g. initial droplet diameter) and mixing was also noted. However, due to the
91
use of a single-step chemistry, nothing could be inferred on the ignition behavior
of sprays in the range of operating conditions relevant to HCCI and diesel engines.
The scope of the simulations presented in this chapter is to provide some
physical insight into the complex interactions between turbulence, chemistry and
fuel / air mixing that characterize hydrocarbon sprays autoigniting at low tem-
peratures and moderate pressure levels, for which no fundamental computational
study exists. More specifically, our aim is to verify whether a most reactive
mixture fraction can still be found for sprays autoigniting in the negative tem-
perature coefficient (NTC) regime, and if the negative correlation between heat
release and scalar dissipation rate that characterizes gaseous flow autoignition
continues to hold for these conditions. We also want to investigate how changes
in the operating conditions, such as oxidizer temperature, oxygen dilution in the
air stream, global equivalence ratio in the droplet-laden region and initial droplet
size distribution of the spray affects mixture formation and the ignition process.
The data obtained from the simulations will also be used to clarify some aspects
concerning the modelling of spray effects in CMC. A discussion on this will be
provided in Chapter 6.
5.2 Mathematical formulation
5.2.1 Numerical procedure and computational parame-
ters
The equations governing the temporal and spatial evolution of the gaseous and
liquid phases have been given in Chapter 2 and will not be repeated here. These
equations were solved numerically using the in-house DNS code Senga2 [98]. First
and second order spatial derivatives were evaluated using a tenth-order explicit
central difference scheme [60]. Periodic boundary conditions were applied in all
spatial directions. Operator splitting between transport in physical space and
chemistry was implemented to ease the constraints on the time-step size. DNS
with operator splitting was studied in detail and validated in [70]. A fourth-order,
low storage, explicit Runge-Kutta scheme [61] was used to advance in time the
92
Case: L [mm] Tair [K] XO2 [−] ad,0 [µm] q [−] k0 [m2s−2] Φ0 [−]
A 2.1 1000 0.21 25 − 0.25 2
B 2.1 1000 0.10 25 − 0.25 2
C 2.1 1350 0.21 25 − 0.25 2
D 2.9 1000 0.21 25 − 0.70 2
E 2.9 1000 0.21 25 3.5 0.70 2
F 2.9 1000 0.21 25 − 0.70 4
Case: Ret [−] G0 [−] τt [ms] τid,0 [ms] τev [ms] ξst [−] ξMR,0 [−]
A 19 9.6 1.52 0.84 0.82 0.062 0.166
B 19 4.8 1.52 1.82 0.82 0.031 0.20
C 11 9.6 1.52 0.13 0.60 0.062 0.03
D 73 15.2 0.75 0.84 0.82 0.062 0.166
E 73 19.6 0.75 0.84 1.60 0.062 0.166
F 73 30.2 0.75 0.84 0.82 0.062 0.166
Table 5.1: Turbulence, oxidizer and spray characteristics for the various spray
simulations performed. Fuel temperature and turbulence integral length scale
set equal to Tf = 450 K and lt = L/6 for all cases investigated. XO2 : oxygen
concentration in the oxidizer. Ret: turbulent Reynolds number, Ret = u
′lt/ν.
ξst: stoichiometric mixture fraction. ξMR,0: most reactive mixture fraction of the
corresponding homogeneous system (see Section 5.3.1).
gas-phase transport equations without chemical source terms. Then, chemical
reactions were computed using the implicit solver VODPK [19]. Unity Lewis
number was assumed for all species. The time-step size ∆t was chosen to be
5 × 10−9 s: this is more than 10 times smaller than the smallest non-chemical
time scale in the simulation, that is the acoustic time scale in a grid cell. The
droplet transport equations were advanced at the beginning of every time step
using the same explicit Runge-Kutta scheme employed for the gas-phase. The
droplet source terms were redistributed to the neighboring grid nodes using the
procedure detailed in Chapter 2.
The cubic domain had a length L of either 2.1 or 2.9 mm. The grid reso-
lution was equal to ∆x = 22.8µm in both geometries, resulting in 92 and 128
computational nodes along each spatial direction. The initial ambient pressure
was p0 = 24 bar, while two different values of the initial ambient gas temperature
93
were investigated, namely Tair = 1000 K and Tair = 1350 K, corresponding to low
and high temperature ignition.
The turbulent velocity field was initialized according to a Batchelor-Townsend
energy spectrum [8] with the integral length scale L11 = L/6. Turbulence was
isotropically decaying, and there was no initial mean flow. The initial turbu-
lent velocity fluctuations were equal to u′0 = 0.25 m/s for the 92
3 grid, and to
u′0 = 0.7 m/s for the 128
3 grid. The initial values of the Kolmogorov length scales
were ηK = 39.7µm and ηK = 27.9µm for the cases with low oxidizer temperature,
and ηK = 51.3µm for the case with high oxidizer temperature, indicating that
gaseous-phase turbulence was well resolved for all configurations investigated.
Fuel droplets were initially distributed in a layer extending from x = 0.3L to
x = 0.7L for the 923 grid, and from x = 0.35L to x = 0.65L for the 1283 grid. A
sketch of the computational domain for the 923 grid is shown in Figure 5.1. The
initial droplet velocity was set equal to that of the background carrier-phase. The
overall equivalence ratio in the droplet-laden region was varied between Φ0 = 2
and Φ0 = 4, corresponding to an initial group combustion number G0 ranging
between 9.6 and 30.2. G0 was evaluated as [24]:
G0 = 1.5LeF
(
1 + 0.276Sc1/3Re
1/2
d,0
)
n
2/3
T
ad,0
ld,0
(5.1)
where nT is the total number of droplets in the droplet-laden region, and Red,0,
ld,0 are the initial droplet Reynolds number, equal to zero here, and the initial
mean inter-droplet distance.
The initial fuel temperature and the initial diameter of the droplets were
equal to Tf = 450 K and ad,0 = 25µm for all cases investigated. Both mono and
polydisperse sprays were studied. In the latter case, the droplet diameters were
initialized according to a Rosin-Rammler distribution [77]:
F (ad,0) = 1− exp
[
−
(ad,0
r
)q]
(5.2)
94
Figure 5.1: Sketch of the computational domain for the 923 grid.
where F (ad) is the cumulative distribution function of the droplet diameter and
represents the relative amount of liquid contained within droplets having diameter
smaller than ad. q is the shape parameter and controls the width of the distri-
bution. The amplitude parameter r was adjusted to obtain the desired value of
the spray Sauter mean diameter. Since the Rosin-Rammler distribution assumes
an infinite range of droplet sizes, the tails of the distribution were not considered
in the spray initialization: more specifically, only values of ad that satisfied the
condition 0.05 < F (ad) < 0.95 were used to assign the droplet diameters. For the
polydisperse case investigated in this chapter, aSMD,0 = 25µm and q = 3.5, re-
sulting in the distribution shown in Figure 5.2. It should be noted that, although
the most probable droplet diameter falls around 20µm, a large part of the liquid
mass is associated with larger drops. There is a considerable presence of small
droplets in the spray (e.g. ad,0 < 20µm) but, due to their corresponding small
size, their contribution to the total liquid mass is almost null.
Details of the various simulations performed can be found in Table 5.1, to-
gether with the corresponding reference times for eddy turnover τt, droplet evap-
oration τev and homogeneous mixture ignition τid,0. The reference evaporation
time was defined as the time needed for a motionless droplet with initial diame-
ter ad,0 to evaporate in a hot, quiescent medium, and it was evaluated by solving
numerically Equations 2.11 and 2.12 with constant conditions in the gas phase,
95
Figure 5.2: Initial droplet size distribution and cumulative liquid volume fraction
for Case E in Table 5.1.
e.g. T = T0 and Yf = 0. In the case of a polydisperse spray, τid,0 was taken to
be the reference evaporation time of the largest drop initialized in the compu-
tational domain. The reference ignition delay time will be defined in Section 5.3.1.
The operating conditions for the simulations presented in this chapter were
chosen so as to obtain comparable values for at least two of the three reference
times, and to investigate the response of spray autoignition to changes in tur-
bulence strength, initial mass loading, oxygen dilution and initial droplet size
distribution. In particular, a comparison between Cases A and B reveals the
effect of dilution; between A and C the differences between high temperature
and low temperature autoignition; between A and D the effect of u′; between D
and E the effect of polydispersity, and between D and F the effect of the overall
equivalence ratio (e.g. initial number of droplets in the spray).
5.2.2 Chemical mechanism
Reaction rates were computed using the reduced n-heptane kinetic mechanism of
Liu et al. [80], which is the same used for the simulations presented in Chapter
4. It consists of 22 non steady-state species, reacting according to 18 global
96
steps. Further information on the mechanism are given in Section 4.3.2.3 and the
interested reader is referred there for further details.
5.3 Results
5.3.1 Homogeneous and flamelet calculations
To better understand the ignition behavior of a turbulent spray, it is useful to
introduce a reference ignition delay time τid,0. In this work, this is taken to be the
minimum among the computed ignition delays of several homogeneous n-C7H16
/ air mixtures corresponding to different values of the mixture fraction, with
their initial temperatures being initialized by assuming inert mixing between the
fuel and oxidizer streams. The value of ξ at which the ignition delay reaches
a minimum is referred to as most reactive mixture fraction ξMR,0 [85]. We dis-
tinguish here between the most reactive mixture fraction of the homogeneous
reactor, ξMR,0, and the one corresponding to the occurrence of ignition in the
DNS simulations, which will be referred to as ξMR in the following. Note that,
in evaluating τid,0, the ignition delay of an homogeneous mixture was taken to be
the time needed for the mixture temperature to reach a value equal to 95 % of
the corresponding burning temperature at that composition. This definition will
be also used to evaluate the ignition delay of the DNS cases investigated in this
chapter. Definitions based on the temporal derivative of the mixture tempera-
ture were also tested for completeness, but no substantial difference with the first
method was found.
Results obtained for the different combinations of ambient oxygen concentra-
tion and air and fuel temperature investigated in the simulations presented in this
chapter are shown in Figure 5.3. Ignition in non-diluted air at high temperature
occurs quickly at mixture fraction values located on the lean side of stoichiometry.
The most reactive mixture fraction for these conditions is equal to ξMR,0 = 0.03,
and the corresponding reference ignition delay is τid,0 = 0.13 ms. A very differ-
ent behavior is observed when the temperature of the air stream is lowered to
Tair = 1000 K; the quickest ignition is now observed at rich mixture compositions,
97
Figure 5.3: Calculations of autoignition time τid,0 in homogeneous n-C7H16 /
air mixtures at p = 24 bar for different values of XO2 and Tair. Fuel stream
temperature was equal to Tf = 450 K for all sets of conditions investigated.
and occurs at considerably later times. For non-diluted air, ξMR,0 = 0.166 and
τid,0 = 0.84 ms. Decreasing the oxygen content in the low-temperature oxidizer
stream leads to an increase in the values of both the most reactive mixture frac-
tion and the ignition delay: for 10 % oxygen dilution, values of ξMR,0 = 0.2 and
τid,0 = 1.82 ms were found.
A second key quantity in the study of a spray autoignition problem is the crit-
ical scalar dissipation rate N0,crit of the laminar flamelet with the same oxygen
dilution, operating pressure and temperatures of the air and fuel streams [85].
This represents the threshold value of N0 above which ignition does not occur,
with N0 being the maximum scalar dissipation rate across the flamelet. N0,crit
can be used to study the influence of mixing and turbulence on ignition. It was
evaluated here by computing several transient flamelets corresponding to differ-
ent values of N0. The functional dependence between N and ξ in the flamelet
equation was modeled according to the AMC model of O’Brien and Jiang [100].
The computed dependences of τid on N0 for the different sets of conditions
investigated in this work are shown in Figure 5.4. For conditions corresponding
to low air temperature and no oxygen dilution, N0,crit was found to be 129 s
−1.
98
(a) τid vs N0 (b) τid/τid,0 vs N0/N0,crit
Figure 5.4: Left: calculations of autoignition time in unsteady n-C7H16 / air
flamelets with Le = 1 under constant maximum scalar dissipation N0. Right:
same as in left figure, but N0 was normalized by its critical value, and τid by its
reference value. Fuel stream temperature was equal to Tf = 450 K for all sets of
conditions investigated.
Increasing Tair leads to a corresponding increase in the critical scalar dissipation
rate, which rises up to 1192 s−1 for Tair = 1350 K. Diluting the oxygen in the air
stream had an opposite effect, with N0,crit decreasing to 66 s
−1 when XO2 = 0.1.
The ignition delay time can be normalized by the reference ignition delay obtained
from the homogeneous reactor calculations. Similarly, the maximum scalar dissi-
pation rate in the flamelet can be normalized by its critical value. The functional
dependence of the normalized ignition delay on the normalized scalar dissipation
rate is also shown in Figure 5.4. It is seen that the sensitivity of the ignition
delay to changes in scalar dissipation increases with increasing air temperature
and increasing oxygen content in the oxidizer stream.
Apart from the ignition time, the transient behavior is also important in order
to shed some light on the ignition and subsequent flame propagation processes.
The maximum scalar dissipation rate in these simulations was kept constant and
equal to 20 s−1, corresponding to a N0/N0,crit ratio ranging between 0.015 for con-
ditions coresponding to Case C, and 0.303 for conditions corresponding to Case
99
B in Table 5.1. Results are shown in Figures 5.5 and 5.6. When Tair = 1350 K
(Figure 5.5(a)), the ignition kernel appears to be very sharp in mixture fraction
space and is located on the lean side of stoichiometry, in accordance with the
homogeneous ignition calculations. Once a burning spot is established, the flame
spreads quickly towards richer mixture composition, and a fully burning solu-
tion is obtained shortly after. If the temperature of the air stream is decreased
(Figure 5.5(b)), the most reactive mixture fraction shifts towards rich mixture
compositions, as predicted by the homogeneous calculations. The ignition kernel
appears to be wider in mixture fraction space with respect to the high tempera-
ture case, and the flame propagation phase is considerably slower and non-even.
In particular, the flame appears to move faster towards leaner mixture composi-
tions. Reducing the oxygen concentration in the oxidizer exacerbates the former
behavior, as it can be seen from Figure 5.6. Although a most reactive mixture
fraction can still be observed, ignition is now occurring in a wider range of mixture
compositions. Once ignition has occurred, the time needed for reaching a fully
(a) Ta = 1350 K (Case C) (b) Ta = 1000 K (Case A)
Figure 5.5: Calculations of unsteady laminar flamelets for XO2 = 0.21 and differ-
ent values of the air stream temperature. Fuel temperature equal to Tf = 450 K
for both cases investigated. Cases referenced in the figures have the same operat-
ing conditions of the flamelets and are listed in Table 5.1. The maximum scalar
dissipation rate across the flamelet was constant and equal to N0 = 20 s
−1, which
corresponds to a N0/N0,crit of 0.017 for (a) and of 0.155 for (b).
100
Figure 5.6: Calculation of unsteady laminar flamelet for XO2 = 0.10 and Tair =
1000 K, corresponding to conditions of Case B in Table 5.1. Fuel temperature
equal to Tf = 450 K. The maximum scalar dissipation rate across the flamelet
was constant and equal to N0 = 20 s
−1, which corresponds to a N0/N0,crit of 0.303.
burning solution is larger than for the undiluted case due to the correspondingly
lower temperature values reached at any ξ value by the burnt mixture, which are
responsible for weaker heat transfer and hence slower flame propagation.
5.3.2 Main observations
5.3.2.1 General remarks
Cases A, D, E and F in Table 5.1 are characterized by the same values of the
initial air stream temperature and oxygen dilution. Their ignition behavior is
similar and it will be described in the following. The analysis below is presented
for Case A, apart from where differently specified. This case is characterized
by low turbulence intensity, initial monodisperse spray and a global equivalence
ratio in the droplet-laden region equal to 2. Differences due to initial droplet
distribution and changes in u′ and Φ0 will be addressed in detail in separate
sections. Effects due to initial air temperature and initial oxygen dilution will
also be treated separately, as ignition at these conditions exhibits very different
characteristics than those observed for the cases mentioned above.
101
(a) Cases A, D, E and F (b) Detailed view of Case A
Figure 5.7: Left: temporal evolution of volume-averaged heat release rate for
Cases A, D, E and F in Table 5.1. Right: detailed view of Case A, showing
locations at which species and temperature contour / scatterplots are given.
The temporal evolutions of the volume-averaged heat release rates for the cases
in Table 5.1 corresponding to low air temperature and standard oxygen concen-
tration in the oxidizer stream are shown in Figure 5.7(a). The time t = 0 ms
corresponds to the instant at which droplets start to evaporate. In order to com-
pare data from simulations with different widths of the droplet-laden layer, and
therefore of the global equivalence ratio in the computational domain ΦV, the
data shown in Figure 5.7 were multiplied by the ratio between ΦV,A and ΦV.
Here, ΦV,A is the global equivalence ratio in the computational domain for the
baseline Case A.
The ignition sequence can be split into three different phases, each corre-
sponding to a peculiar behavior of the mean heat release rate. We distinguish
here between an inert mixing phase (the label IN will be used in the following),
a pre-ignition phase (PI) and an ignition and flame propagation phase (IG). The
inert mixing period (up to about 0.6 ms in Figure 5.7) is dominated by droplet
evaporation and mixing of the resulting fuel vapor with the surrounding hot air.
Heat release reactions are negligible during this phase. Once a reactive mixture
102
has been formed and the radical pool has grown enough in size, exothermic reac-
tions start to be significant. This marks the beginning of the pre-ignition phase,
which is initially characterized by a constant growth in the rate of heat release.
As ignition is approached, the heat release rate stops growing and exhibits either
a plateau or a sudden decrease to a local minimum, depending on the initial
value of the global equivalence ratio in the droplet-laden layer. This is followed
shortly after by its abrupt increase over a much shorter period of time, which
corresponds to the beginning of the ignition and flame propagation phase. This
phase is characterized by the formation of several isolated ignition kernels and
occurs after about 1.05 ms in Figure 5.7(b). These grow in size with time and,
once they reach a critical size, they eventually start to merge, leading shortly to
the combustion of the whole mixture. The present inert phase is conceptually
similar to the conventionally-defined mixture formation phase often discussed in
the diesel autoignition literature, such as in [50]. For the range of parameters
examined here, this phase seems to have the same duration for all cases studied.
The ignition delay time for all cases investigated, apart from those with di-
luted oxygen concentration in the oxidizer stream and high value of the air tem-
perature, is on the order of one millisecond. This is approximately 30 % longer
than τid,0 for the same ambient pressure and air / fuel temperature. Therefore,
consistent with gaseous autoignition DNS [85, 86], the autoignition of inhomo-
geneous systems occurs later than the fastest homogeneous mixture. Turbulence
appears to promote a faster ignition event, consistent with [85, 86]. Less distinct
conclusions can be made for the effects of inhomogeneities in the initial droplet
size distribution. On one hand, the pre-ignition phase appears to be faster for
polydisperse sprays, with the corresponding peak in heat release rate reached at
earlier times. Ignition, however, seems to occur slightly earlier for sprays which
are initially monodisperse. Additionally, higher values of mean heat release rate
are reached when the initial sizes of the droplets are the same.
A more quantitative description of the phenomena occurring in each of the
ignition phases is presented in the following subsections.
103
5.3.2.2 Inert mixing phase
The inert mixing phase is characterized by virtually no fuel consumption, since
the radical pool has not enough time to grow to a significant size. This can be
observed in Figure 5.8, which shows the scatterplots of heat release rate, tem-
perature, scalar dissipation rate and the mass fractions of n-C7H16, O2, CH2O
and of the OH, H and HO2 radicals plotted against mixture fraction at t = tIN.
The instants of time at which details on the solution are given have been marked
in Figure 5.7. We will implicitly refer to this figure in the following, apart from
where differently specified.
Despite the early instant of time chosen and the low turbulence intensity that
characterizes Case A, a considerable scatter in heat release rate and species mass
fractions is observed for any mixture fraction value. This behavior can be bet-
ter understood by plotting the contours of temperature, mixture fraction, scalar
dissipation rate, heat release rate and HO2 and CH2O mass fractions, which are
shown in Figure 5.9 for t = tIN = 0.6 ms. A slice cutting the computational
domain through the x = 0.5L plane was chosen to display the fields. Since
the sprays considered in this work are dilute, the evaporation process exhibits a
spotty nature, with small, cold fuel-rich islands surrounded by the hot ambient
gas. The regions with the steepest gradients in the mixture fraction field are
localized around the droplets, where the fuel vapor is generated. Turbulent dis-
persion may lead to the formation of small clusters of drops, which are usually
characterized by a more uniform mixture fraction distribution and, consequently,
by smaller values of the scalar dissipation rate.
The complicated mixing pattern described above is responsible for the forma-
tion of a strongly inhomogeneous mixture, in which well-mixed spots alternate
with poorly mixed ones. This ultimately leads to the observed scatter in the
heat release rate and species mass fractions. The most reactive regions start to
develop close to either individual droplets or small groups of droplets, where suf-
ficient fuel vapor is found. In particular, reactions are found to be most intense
around ξ = 0.1, which is to be compared with the value ξMR,0 = 0.166 found from
104
Figure 5.8: Scatterplots of heat release rate, temperature, scalar dissipation rate
and n-C7H16, O2, H, OH, HO2 and CH2O mass fractions plotted against mixture
fraction at t = tIN = 0.6 ms in Figure 5.7(b). Data shown correspond to Case A
in Table 5.1.
the homogeneous calculations. The considerable scatter in the temperature data
at this early time is not due to heat release effects but it is rather a consequence of
the droplet heating and evaporation processes. It is evident that, in this so-called
inert phase, OH and H are found in very small mass fractions (10−8 and 10−11
respectively), while HO2 and CH2O are seen at mass fractions of the order of 10
−4.
105
(a) Temperature (K) (b) Mixture fraction (c) Dissipation rate (1/s)
(d) HO2 mass fraction (e) CH2O mass fraction (f) Heat release rate
(
W/m3
)
Figure 5.9: Contour plots of temperature, mixture fraction, scalar dissipation
rate, HO2 and CH2O mass fractions and heat release rate corresponding to the
axial plane x = 0.5L at t = tIN = 0.6 ms. Data shown correspond to Case A in
Table 5.1.
5.3.2.3 Pre-ignition phase
The instant of time at which the volume-averaged heat release rate is non-
negligible anymore marks the beginning of the pre-ignition phase. This is initially
characterized by a slow growth in heat release, culminating in a local maximum
reached at t = tPI4 = 1.0 ms. Scatterplots and contour plots of temperature,
heat release rate and selected species mass fractions at this instant of time are
shown in Figures 5.10 and 5.11. This is done to provide better understanding
of the processes characterizing this phase. Although most of the liquid fuel has
evaporated, the mixture still appears to be highly inhomogeneous. Few rich spots
can be identified: these correspond to the locations of the evaporating droplets.
The most reactive regions are found again in the proximity of either single or
small groups of droplets. According to the droplet group combustion theory of
106
Figure 5.10: Scatterplots of heat release rate, temperature, scalar dissipation rate
and n-C7H16, O2, H, OH, HO2 and CH2O mass fractions plotted against mixture
fraction at t = tPI4 = 1.0 ms in Figure 5.7(b). Data shown correspond to Case A
in Table 5.1.
Chiu and Liu [24], the combustion regime appears to be intermediate between
the single droplet and the internal group combustion modes. The number of
locations at which reactions proceed vigorously has reduced significantly com-
pared to those observed at earlier times. This becomes particularly evident when
the contour plots of formaldehyde in Figures 5.9 and 5.11 are compared. Ex-
107
amination of the scatterplots reveals that exothermic reactions are most intense
around ξ = 0.2. This value of the mixture fraction is higher than the one found
at t = tIN = 0.6 ms and indicates that, as time progresses, the development of
the ignition kernel shifts towards richer mixture compositions. Temporal shifting
of the mixture fraction value at which the heat release rate peaks was already
observed in previous numerical works on autoigniting gaseous flows and is dis-
cussed in [85]. Fuel and oxygen consumption is not negligible anymore, and the
temperature has risen well above its value corresponding to inert mixing between
the air and fuel streams.
As previously discussed, once the volume-averaged heat release rate has reached
a local maximum, it decreases slightly before ignition occurs. To explain this be-
havior, the temporal evolutions of the mixture fraction PDF and of the conditional
averages of the heat release rate, temperature and mass fractions of some selected
(a) Temperature (K) (b) Mixture fraction (c) Dissipation rate (1/s)
(d) HO2 mass fraction (e) CH2O mass fraction (f) Heat release rate
(
W/m3
)
Figure 5.11: Contour plots of temperature, mixture fraction, scalar dissipation
rate, HO2 and CH2O mass fractions and heat release rate corresponding to the
axial plane x = 0.5L at t = tPI4 = 1.0 ms. Data shown correspond to Case A in
Table 5.1.
108
Figure 5.12: Temporal evolution of conditional mean heat release, temperature,
mixture fraction PDF and selected species mass fractions. Referenced instants
of time are given in Figure 5.7(b). Their numerical values are: tPI1 = 0.70 ms,
tPI2 = 0.80 ms, tPI3 = 0.90 ms, tPI4 = 1.00 ms, tPI5 = 1.05 ms. Data shown
correspond to Case A in Table 5.1.
species are shown in Figure 5.12, the conditioning being on the mixture fraction.
As time progresses, the consumption of oxygen and n-heptane becomes more and
more significant. The concentrations of the intermediate and final species, such
as CH2O, CO and CO2 (CO2 is not shown in the figure), increase at all mixture
109
fraction values, with their peak values shifting towards richer mixtures, and an
increment in the conditional temperature is observed. The temporal evolution
of the radical species, e.g. H, OH, HO2 and CH3, is slightly different. At early
times, their conditional mass fractions increase for any mixture fraction value. As
ignition is approached, however, a two-fold behavior is observed. In the mixture
fraction range comprised between 0.05 and 0.15, their concentration decreases,
while it continues to grow for richer mixture conditions. The lack of radicals
inhibits the ignition reactions; thus, it should not be surprising that the condi-
tional heat release rate exhibits a very similar temporal trend, decreasing where
the radicals mass fractions are diminishing, and increasing otherwise. This also
explains the sharp decrease in the mean heat release rate that occurs just before
ignition; rather than indicating a sudden disappearance of the ignition kernels, it
is the result of a slight decrease of the reaction intensity in the moderately rich
mixture pockets. Although small, the effects of this decrease are amplified by
the relative abundance in the flow of regions where ξ ' 0.1, as revealed by the
mixture fraction PDF. Figure 5.12 also shows explicitly the shift of ξMR (defined
as the mixture fraction value at which the heat release rate peaks at a particular
time) from about 0.1 to about 0.2, bracketing the reference value ξMR,0. Ignition
happens quite close to ξMR,0, as it will be discussed next.
5.3.2.4 Ignition phase
The ignition phase begins at the end of the pre-ignition period, when the mean
heat release rate reaches a local minimum, followed by a rapid increase in time
with an exponential-like behavior. This phase is characterized by local ignition
and subsequent flame propagation. The ignition sequence is depicted in Figure
5.13, which shows the temporal evolution of the T = 1250 K isosurface from
t = tIG2 = 1.09 ms to t = tIG5 = 1.15 ms. The isosurface color corresponds to the
local mixture fraction value. Ignition first occurs at few spatial locations where
the mixture is locally rich (ξMR ' 0.2). Note that ξMR is slightly larger than
ξMR,0 (ξMR,0 = 0.166 for conditions corresponding to Case A). It thus appears
that the concept of a most reactive mixture fraction continues to be valid also for
sprays igniting in the NTC regime. As time progresses, the burning spots grow
110
(a) t = tIG2 = 1.09 ms (b) t = tIG3 = 1.10 ms
(c) t = tIG4 = 1.13 ms (d) t = tIG5 = 1.15 ms
Figure 5.13: Temporal evolution of T = 1250 K isosurface. Surface colored ac-
cording to local mixture fraction value. Referenced instants of time are given in
Figure 5.7(b). Data shown correspond to Case A in Table 5.1.
in size, expand toward leaner flow regions and eventually start to merge with
each other. This process is accompanied by the appearance of additional ignition
kernels, which speed up the occurrence of combustion over the entire volume.
Similar spotty nature of the ignition sequence was described by Wang and Rut-
land [148]. The set of conditions investigated in their work, however, differs from
the one considered for most of the simulations presented in this chapter in that
the oxidizer temperature was considerably higher (Tair = 1100− 1300 K) and the
operating pressure was lower (p = 6 bar). Additionally, despite the lower values
of the global equivalence ratio considered (in [148], Φ was varied between 0.5 and
111
(a) t = tIG1 = 1.075 ms
(b) t = tIG3 = 1.100 ms
(c) t = tIG4 = 1.130 ms
(d) t = tIG5 = 1.150 ms
Figure 5.14: Scatterplots of temperature and n-C7H16, O2, H2 mass fractions
plotted against mixture fraction at different instants of time during ignition.
Referenced instants of time are given in Figure 5.7(b). Data shown correspond
to Case A in Table 5.1.
112
(a) t = tIG1 = 1.075 ms
(b) t = tIG3 = 1.100 ms
(c) t = tIG4 = 1.130 ms
(d) t = tIG5 = 1.150 ms
Figure 5.15: Scatterplots of HO2, CH2O, C4H8 and CO2 mass fractions plotted
against mixture fraction at different instants of time during ignition. Referenced
instants of time are given in Figure 5.7(b). Data shown correspond to Case A in
Table 5.1.
113
1.5, while values between 2 and 4 were considered here. Φ is found to increase
the number of ignition spots and its effect on spray ignition will be discussed in
detail in the next sections), ignition was found to occur simultaneously in a larger
number of spatial locations than those shown in Figure 5.13.
The scatterplots of temperature and selected species mass fractions against
ξ during ignition are shown in Figures 5.14 and 5.15. At t = tIG1 = 1.075 ms,
there has been already a considerable consumption of fuel and oxygen around
ξMR = 0.2, and ignition appears to be imminent. Once a flame kernel is formed,
an abrupt change in the mass fractions of all species occurs, which corresponds
to the transition of the mixture from a pre-ignition state to a burning diffusion
flame. The reactants, the products, and some intermediate and radical species,
such as OH, H2 and CH2O (OH is not shown in the figure), undergo a monotonic
transition between these two states. Other species, however, exhibit a more
complex behavior. This is the case of some radicals, such as HO2 and CH3 (CH3
is not shown in the figure), or intermediates like C4H8, whose concentrations first
increase during the transition between the inert and the burning solutions, before
being depleted quickly once a diffusion flame is established. The role of CH2O
as a precursor to ignition in both single and two-phase flows has been discussed
in detail in the experimental works of Gordon et al. [43] and O’Loughlin and
Masri [101]. The behavior of this species in the simulations considered in this
chapter is fully consistent with the experimental findings, confirming the observed
similarity between autoignition in single and two-phase flows. Once it has ignited,
the flame expands towards mixtures that are either leaner or richer than ξMR:
the propagation towards the lean side of stoichiometry, however, appears to be
faster, in line with the observations made for the corresponding laminar transient
flamelet in Section 5.3.1.
5.3.3 Topology of the autoigniting kernels
Two- and three-dimensional DNS studies of single-phase autoigniting flows showed
that a fuel / air mixture always ignites first at those spatial locations where
ξ ' ξMR,0 and the scalar dissipation rate is low [86, 137]. Similar behaviors were
114
(a) t = tPI1 = 0.70 ms (b) t = tPI3 = 0.90 ms
(c) t = tPI4 = 1.00 ms (d) t = tIG3 = 1.10 ms
Figure 5.16: Conditional heat release rate, plotted at different instants of time
and for different values of the scalar dissipation rate. Referenced instants of time
are given in Figure 5.7(b). Data shown correspond to Case A in Table 5.1.
also observed for spray autoignition [125], although the correlation existing be-
tween heat release and scalar dissipation rate was found to be weaker than for
purely gaseous mixtures. The picture emerging from the simulations presented
in this work, while confirming some of observations made previously by other
115
authors, reveals additional details of spray autoignition.
Results presented in Section 5.3.2.4 and analysis of Figure 5.16, which shows
the doubly-conditioned heat release rate at different instant of times, the con-
ditioning being on the mixture fraction and the scalar dissipation rate, clearly
indicate that ignition occurs around ξMR ' 0.2. The correlation existing between
heat release and scalar dissipation rate is stronger than the one observed in [125].
This is indeed confirmed by Figure 5.17, which shows the temporal evolution of
the conditional cross-correlation coefficient 〈σ| η〉 between these two quantities,
the conditioning being again on ξ. Prior to ignition, 〈σ| η〉 is negative in the
range of mixture compositions where the heat release is most intense. Values
down to −0.7 are reached: this is more than twice as low as the minimum of
−0.3 that was found in [125] for spray autoignition, although it is still higher
than the value of −0.9 found for gaseous fuel autoignition [86]. These results
indicate that the scalar dissipation rate is an important controlling parameter for
the ignition of not only gaseous mixtures, but also for diesel sprays, and that the
influence of mixture fraction gradients on τid is somewhat weaker for two- rather
than single-phase flows.
An indirect confirmation of the observed strong dependence between heat
release and scalar dissipation rate is provided by Figure 5.18, which shows the
temporal evolution of the ξ = ξMR isosurface for Case A, colored using the ratio
between the local value of the scalar dissipation rate and the value of the scalar
dissipation rate that inhibits ignition at ξMR, Ncrit,ξMR = N0,critG(ξMR) (ξMR = 0.2
here). At early times, these surfaces are small in size, and are characterized by
values of the scalar dissipation rate which range from moderate to high. As
ignition is approached, most of the liquid fuel has evaporated already, and in-
homogeneities in the mixing field start to be smeared out by diffusive processes.
Some of the ξMR isosurfaces begin to merge with each other, while others disap-
pear as they mix with leaner mixtures. These mixing processes are responsible
for an overall reduction of the scalar dissipation rate, although regions where N is
above the critical value for ignition can still be found even at very late times after
evaporation has started. When ignition occurs, well-mixed conditions have been
116
(a) Temporal evolution of 〈σ| η〉 (b) 〈σ| η, c〉 at t = tIG3 = 1.10 ms
Figure 5.17: Left: temporal evolution of cross-correlation coefficient between heat
release and scalar dissipation rate. Right: cross-correlation coefficient, doubly
conditioned on mixture fraction and ignition progress variable, plotted at t =
tIG3 = 1.10 ms. Referenced instants of time are given in Figure 5.7(b). Their
numerical values are: tPI2 = 0.80 ms, tPI3 = 0.90 ms, tPI4 = 1.00 ms, tPI5 =
1.05 ms. Data shown correspond to Case A in Table 5.1
achieved over the vast majority of the ξMR isosurfaces. The first ignition kernels,
however, do not appear on any of these regions, but only in those locations where
the scalar dissipation rate has been low throughout the whole simulation.
It is interesting to note the existence of an interval of mixture fraction values
in the region of lean to moderately rich mixtures where the correlation coeffi-
cient is strongly positive. The width of this interval grows with time and may
be interpreted as a direct consequence of the behavior, described in the previous
sections, that radical species exhibit close to ignition. This speculation is con-
firmed by Figure 5.19, which shows the temporal evolution of the conditional OH
radical mass fraction in those flow regions where the scalar dissipation rate is be-
low and above the threshold value of 10 s−1. In both situations, a decrease in the
radical mass fractions is observed in the mixture fraction range between 0.05 and
0.15. This reduction, however, is more marked when the local scalar dissipation
117
(a) t = tIN = 0.60 ms (b) t = tPI3 = 0.90 ms
(c) t = tPI4 = 1.00 ms (d) t = tIG3 = 1.10 ms
Figure 5.18: Temporal evolution of ξ = ξMR isosurface. Surface colored according
to local value of the ratio N/Ncrit,ξMR , with Ncrit,ξMR being the critical value for
ignition of the conditional scalar dissipation rate at ξMR = 0.2. Gray surfaces in
the bottom right figure correspond to T = 1250 K isosurface. Referenced instants
of time are given in Figure 5.7(b). Data shown correspond to Case A in Table
5.1.
rate is low, leading to a larger drop in the heat release rate and explaining the
corresponding positive value of the correlation coefficient.
Once combustion has begun, a sudden change in 〈σ| η〉 around the most reac-
tive mixture fraction is observed. Similar behavior was also noted in [125], and it
was linked to the enhanced droplet evaporation induced by the increase in tem-
perature following ignition. This is not the case here, since most of the droplets
118
(a) N ≤ 10 s−1 (b) N > 10 s−1
Figure 5.19: Temporal evolution of conditional OH mass fraction in flow regions
with different levels of scalar dissipation rate. (a): caseN ≤ 10 s−1. (b): caseN >
10 s−1. Referenced instants of time are given in Figure 5.7(b). Their numerical
values are: tPI3 = 0.90 ms, tPI4 = 1.00 ms, tPI5 = 1.05 ms. Data shown correspond
to Case A in Table 5.1
have evaporated completely before ignition, while the surviving ones are boiling
and thus are not affected by changes in temperature of the gas phase. In order
to shed light on this quick variation of 〈σ| η〉, the cross-correlation coefficient has
been plotted in Figure 5.16 at t = 1.1 ms and for different values of the following
ignition progress variable c [99]:
c =
YO2 − YO2,i(η)
YO2,b(η)− YO2,i(η)
(5.3)
where YO2,i, YO2,b are the oxygen concentrations evaluated at the local mixture
fraction value and corresponding to the inert and Burke-Schumann [76] solution
respectively. It is seen that, for low values of the progress variable, the jump in
〈σ| η〉 does not occur. This suggests that the drastic change in the correlation
coefficient after combustion begins is related to the propagation of the flame front
following ignition. Previous work on autoigniting sprays revealed that, once a
burning kernel is formed within the flow, it behaves as a spark for the surrounding
mixture, which is already in a state close to ignition [18]. Turbulence favors the
119
diffusion of heat from the hot, burning spots to the nearly igniting ones, and
thus has a positive effect on the propagation of the reacting zones within the
mixture. This explains the observed jump in 〈σ| η〉: in those flow regions close
to a burning spot, high values of scalar dissipation rate ultimately lead to the
ignition of the surrounding mixture due to heat diffusion, and thus to high values
of the heat release rate and to a positive correlation between these two quantities.
The nearly zero value of 〈σ| η〉 around ξMR after ignition is just a consequence of
the averaging of this quantity over the regions of the flow which are far from the
igniting spots, and thus where a negative correlation between heat release and
scalar dissipation rate still holds, and those regions where the mixture is ignited
by turbulent heat diffusion, for which the heat release and scalar dissipation rate
are positively correlated.
5.3.4 Evaporation and mixing
5.3.4.1 General remarks
The strong influence of the mixing quality on the ignition delay of the spray sug-
gests a detailed investigation of the evaporation process. Figure 5.20 shows the
Figure 5.20: Temporal evolution of normalized mean droplet evaporation rate,
liquid temperature increase, droplet diameter and normalized mean scalar dissi-
pation rate. Data shown correspond to Case A in Table 5.1.
120
Figure 5.21: Temporal evolution of mean scalar dissipation rate as a function of
the distance from the droplet centre. Data shown correspond to Case A in Table
5.1.
temporal evolution of the mean evaporation rate, mean droplet diameter, mean
liquid temperature and mean scalar dissipation rate for Case A in Table 5.1. The
mean liquid temperature T˜l has been defined here as the mass-weighted average
droplet temperature, T˜l =
∑
imd,iTd,i/
∑
imd,i, where the sum is taken over all
droplets in the spray. The data shown have been normalized by their maximum
values; in particular, the mean liquid temperature has been normalized by the
difference between the liquid boiling temperature (Tf,boi = 534.7 K) and the ini-
tial droplet temperature (Td,0 = 450 K). The initial fuel temperature considered
in this work is relatively high and, as a consequence, there is substantial fuel
evaporation even at the beginning of the simulation. As time progresses, the
droplets are heated up and their evaporation rate initially increases. The fuel va-
por produced during this phase starts to mix with the surrounding air and, since
initially the mixing rate is slower than the evaporation rate, the mean scalar dis-
sipation rate increases. The fuel boiling temperature is reached around t = 0.45
ms; at this time, the mean droplet diameter has decreased by 18 % of its initial
value. The evaporation rate reaches its maximum value just an instant of time
earlier, and then it starts to decrease. A similar trend is found for the scalar
dissipation rate, although its maximum value is reached at a slightly later time.
121
As ignition is approached, the droplets become progressively smaller, with some
of them disappearing, and the evaporation rate eventually falls below its initial
value.
Some details on the structure of the mixing field are revealed by Figure 5.21,
which shows the average value of the scalar dissipation rate as a function of the
distance from the droplet centre. This was obtained by evaluating, for each grid
node in the computational domain, the corresponding distance from the closest
droplet in the spray and then averaging the scalar dissipation rate values for
those nodes with similar distances from their closest droplet. Data shown in Fig-
ure 5.21 refer to Case A in Table 5.1, but similar results were also found for the
other configurations investigated in this work. At every instant of time considered
in the analysis, the scalar dissipation rate exhibits the same characteristic profile,
featuring a local maximum in the proximity of the droplet surface, followed by
a sharp decrease to a plateau where its value remains approximately constant.
The peak value of N is found to be a strong function of time, following the same
behavior of the volume-averaged scalar dissipation rate, while the mixing rate in
the plateau stays on the same level of intensity throughout the whole simulation.
Due to the dispersed nature of the sprays studied in this work, the extent of the
plateau initially grows with time, then it remains constant once the fuel vapor
has spread through the inter-droplet regions. Note that the local peak in scalar
dissipation rate occurring away from the droplet surface is an artifact of the point
source approximation used in this work to treat the liquid phase, which does not
allow for resolution of the gas phase in the near-droplet regions.
The above discussion suggests the existence of a strong connection between
evaporation and mixing. In the following, we aim at studying how changes in
the turbulence intensity, spray properties and initial mass loading in the droplet
laden layer affects evaporation and, in turn, mixing and autoignition.
5.3.4.2 Influence of turbulence
Figure 5.22 shows the temporal evolution of the mean droplet evaporation rate,
the volume-averaged scalar dissipation rate, the mean liquid temperature and the
122
mean droplet diameter for Cases A and D in Table 5.1, corresponding to low and
high initial turbulence intensity respectively. In evaluating the volume-averaged
scalar dissipation rate, only the contributions corresponding to ξ ≥ 0.025 were
considered to avoid N˜ being dominated by the flow regions with little or no fuel
content. Turbulence is responsible for faster droplet evaporation, and leads to
initially higher values of scalar dissipation. During the transient droplet heating
phase, the mean droplet evaporation rate is higher for conditions corresponding
to stronger turbulence. As soon as the droplets reach their boiling temperature,
this trend is reversed. Larger mean evaporation rate values are then found for the
weaker turbulence case, due to the corresponding larger mean size of the droplets.
This shift is accompanied by a reversal in the trends for the volume-averaged
scalar dissipation rate, whose values become higher for weaker turbulence condi-
tions. Similar behavior is also found for the conditional scalar dissipation rate,
as shown in Figure 5.23. In view of the strong correlation existing between scalar
dissipation and heat release, this fully explains why the ignition delay time de-
creases as the initial turbulence strength increases.
(a) ˜˙md and N˜ (b) T˜l and aSMD
Figure 5.22: Temporal evolution of (Left) droplet mean evaporation rate and
mean scalar dissipation rate, and (Right) droplet Sauter diameter and mean liquid
temperature, for different intensities of the initial gas-phase turbulence. Data
shown correspond to Cases A and D in Table 5.1.
123
The influence of turbulence on the ignition of gaseous hydrogen / air mixtures
was studied in [15] and [82]. Although the experimental configurations investi-
gated in these works were different, both studies concluded that turbulence has
a delaying effect on ignition. It was suggested in [82] that turbulence could have
been responsible for promoting higher levels of scalar dissipation rate, which in
turn resulted in a longer ignition delay time. In the framework of autoigniting
spray, the correlation existing between ignition and turbulence strength was in-
vestigated by Wright et al. [152], who simulated the n-heptane spray experiments
described in [71] using a RANS-CMC approach. They found that increasing the
turbulence level in the background carrier phase lead to a decrease in the ignition
delay time for all conditions investigated, and this was attributed to the fact that,
for low values of the scalar dissipation rate, the ignition delay time was seen to
decrease with increasing N . The effect of turbulence on ignition in sprays was
also investigated by Wang and Rutland [149], who studied how the ignition delay
time of n-heptane liquid-fuel spray jets is affected by the initial droplet velocity.
An increase in the initial liquid-fuel jet velocity, and hence on the initial shear
between the liquid and the gaseous phase, was found to cause a decrease of the
ignition delay time for those sprays where the characteristic droplet evaporation
time was small compared to the ignition delay time. However, when these two
characteristic times were comparable, increasing the fuel jet velocity was seen to
be responsible for an increase in the ignition delay time. This was attributed to
the corresponding increasing importance of evaporative cooling on the ignition
process. These results are consistent with the dependency between ignition and
turbulence found for the simulations considered in this chapter. Turbulence acts
on ignition through the scalar dissipation rate and, depending on the particular
set of conditions investigated, higher turbulence levels may result in lower val-
ues of scalar dissipation rate during most of the time preceding ignition, hence
leading to a shorter τid, as discussed in [86, 85]. One should note that the com-
plex interaction existing between evaporation and mixing in sprays may lead to
behaviors that are apparently different from those observed in gaseous flows: in
particular, the role of u′ in promoting / delaying ignition may be not the same, as
higher turbulence intensity promotes faster liquid evaporation, which eventually
124
(a) Case A (b) Case D
Figure 5.23: Conditionally-averaged scalar dissipation rate, the conditioning be-
ing on mixture fraction, plotted at different instants of time. Data shown corre-
spond to Case A (Left) and D (Right) in Table 5.1.
results in better mixture formation due to the corresponding longer time avail-
able for mixing processes to take place. In single-phase flows, turbulence seems
to promote higher values of 〈N | η〉 at first; however, when u′ is large, well-mixed
conditions are achieved quickly, and the conditional scalar dissipation rate falls
to zero.
Despite the fact that the effect on the mean evaporation rate is strong, the
initial turbulence level has a weak influence on droplet heating, as it can be seen
from the temporal evolution of the mean liquid temperature (Figure 5.22(b)), and
therefore on the mean value of the fuel mass fraction at saturation. Equations
2.11 and 2.12 then suggest that the observed differences in evaporation rates at
early times, when differences in droplet diameter are still small, must come from
convective-related effects through the droplet modified Sherwood number. This
is confirmed by Figure 5.24, which shows the temporal evolutions of Shc and the
droplet mean Stokes number St for low and high levels of the initial gas-phase
turbulence intensity. St is defined as the ratio of the droplet characteristic re-
laxation time τ vd and the Kolmogorov time scale τη =
√
ν/, and indicates the
125
Figure 5.24: Temporal evolution of the droplet mean Stokes number and droplet
mean modified Sherwood number for different intensities of the initial gas-phase
turbulence.
ability of a droplet to adapt to local variations of the carrier-phase velocity [115].
Since no relative motion between the droplets and the background carrier phase
exists at the beginning of the simulation, and since the initial droplet size is the
same for Cases A and D considered here, the initial droplet relaxation times are
also the same. Differences in St then arise due to the different initial Kolmogorov
time scale, which decreases with increasing turbulence intensity.
The studies available in the literature on the evaporation of droplets in tur-
bulent flows have been recently reviewed by Birouk and Go¨kalp [14]. The ef-
fect of turbulence on the evaporation rate of large hydrocarbon droplets (ad,0 =
O(1) mm) was investigated both in the presence or the absence of a mean convec-
tive flow. In those problems characterized by a zero mean velocity of the carrier
gas, increasing the strength of the turbulent velocity fluctuations always resulted
in an enhancement of the droplet evaporation rate. This was attributed to the
faster transport rate of the fuel vapor available in the proximity of the droplet
surface. The picture that emerges from the simulations presented in this chapter
is consistent with these results. Turbulence can only affect the evaporation rate
of those droplets whose characteristic relaxation time is large compared to the
126
Kolmogorov time scale, so that a non-zero relative motion between the gaseous
and the liquid phase can be established. As discussed in the previous paragraph,
for fixed value of the initial droplet diameter, the stronger the turbulence in-
tensity, the larger the initial Stokes number of the droplet and thus the relative
motion between the phases. This enhances the rate of heat and mass transfer
between the bulk gas and the droplet, resulting in faster liquid fuel evaporation.
As the droplet diameter falls below a critical size, its relaxation time becomes
comparable to τη: St then falls below one, and the droplet start to behave as a
particle tracer of the bulk phase. No relative motion between the phases exists
anymore, and evaporation proceeds as in a laminar configuration with no mean
flow, e.g. Shc = Sh = 2.
5.3.4.3 Influence of global equivalence ratio
The effect of the initial global equivalence ratio on the ignition of n-heptane sprays
was studied numerically by Wang and Rutland [148] with two-dimensional DNS.
Their work focused on spray autoignition under conditions corresponding to low
pressure and high oxidizer temperature. These are different from the conditions
considered in this work, where the air temperature has been kept low and the
pressure high for all cases investigated. It was found that an increase in the value
of Φ0 was responsible for increasing the ignition delay time. This was attributed
essentially to the increased importance of the evaporative cooling of the carrier
gas phase as the initial number of droplets in the computational domain was in-
creased. The effect was rather strong, as a variation in Φ0 from 0.5 to 1.5 lead to
a corresponding increase in τid from 0.91 to 3.30 ms. In order to check whether
a similar trend also holds for ignition at high pressures and low oxidizer tem-
perature, a simulation (Case F) with Φ0 = 4 in the droplet-laden layer was run.
Apart from the value of the global equivalence ratio, the other initial conditions
correspond to those of Case D in Table 5.1. The picture that emerges from these
simulations is rather different than the one observed in [148]. The ignition delay
time now undergoes a very slight decrease as the value of Φ0 is brought from 2
to 4. In addition, differently from the cases with Φ0 = 2, the volume-averaged
heat release rate exhibits a plateau before the ignition event, without any prior
127
(a) Mean heat release rate (b) Mixture fraction PDF
Figure 5.25: Left: temporal evolution of volume-averaged heat release rate for
Case F in Table 5.1, showing locations at which species and temperature contour
/ conditional means are given. Right: comparison of mixture fraction PDF at
selected instants of time for Cases D and F in Table 5.1. Numerical values of
referenced instants of time are: tIND = 0.7 ms, tPID = 1.0 ms, tPIF2 = 0.7 ms,
tPIF5 = 1.0 ms.
sudden drop.
In order to shed some light on the physics of spray autoignition at high Φ, the
numerical solution was analyzed at selected instants of time. These are shown
in Figure 5.25, together with the comparison between the mixture fraction PDF
at two selected instants of time for Cases D and F, which correspond to low
and high values of Φ0 respectively and to the same initial turbulence strength.
The instants of time at which the mixture fraction PDF has been plotted for
Case D were not marked on the corresponding volume-averaged heat release rate
curve: they correspond to the instant at which the heat release rate starts to
rise (t = tIND = 0.7 ms) and to the one at which it reaches a local maximum
before starting to decrease (t = tPID = 1.0 ms). It is evident that increasing the
global equivalence ratio strongly affects the mixture composition, with consider-
able amount of mixture pockets now found at higher ξ values than for the Φ0 = 2
128
Figure 5.26: Temporal evolution of conditional mean heat release, temperature
and selected species mass fractions. Referenced instants of time are given in
Figure 5.25. Their numerical values are: tPIF1 = 0.60 ms, tPIF2 = 0.70 ms, tPIF3 =
0.80 ms, tPIF4 = 0.90 ms, tPIF5 = 1.00 ms. Data shown correspond to Case F in
Table 5.1.
case. Temporal evolutions of the conditional means of the heat release rate, tem-
perature and selected species mass fractions are shown in Figure 5.26. These are
qualitatively similar to those plotted in Figure 5.12, which correspond to Case A
in Table 5.1. As ignition is approached, radical species mass fractions decrease
in value in the mixture fraction range between 0.02 and 0.12. Similar behavior is
found for the heat release rate. The observed plateau in the temporal evolution
of the mean heat release rate should thus be attributed not to a different igni-
tion behavior, but rather to the higher probability of finding very rich mixture
pockets, for which the heat release rate increases monotonically with time, when
Φ0 is increased. These contributions then compensate the lower amounts of heat
release provided by leaner spots, leading to the observed flat behavior of the heat
release rate preceding ignition.
129
(a) t = tIGF2 = 1.03 ms (b) t = tIGF3 = 1.05 ms
(c) t = tIGF4 = 1.07 ms (d) t = tIGF5 = 1.09 ms
Figure 5.27: Temporal evolution of T = 1250 K isosurface. Surface colored ac-
cording to local mixture fraction value. Referenced instants of time are given in
Figure 5.25. Data shown correspond to Case F in Table 5.1.
The differences in mixture composition among the cases studied here also
affect the ignition sequence. This is revealed by Figure 5.27, which shows the
temporal evolution of the T = 1250 K isosurface at four selected instants of time
during ignition for Case F. The spotty nature of the ignition event described for
the cases with Φ0 = 2 is still evident: however, differently from those simula-
tions, ignition of the whole mixture now appears to proceed at a considerably
faster rate. This is again a direct consequence of the abundance in the flow of
mixture pockets with ξ ' ξMR,0. A strong negative correlation between heat re-
lease and scalar dissipation rate is found, similarly to the Φ0 = 2 cases, and it is
130
not shown here.
The strong influence that scalar dissipation rate continues to have on ignition
suggests to seek for eventual changes in the evaporation and mixing processes as
Φ0 is increased. To this end, temporal evolutions of the mean droplet evaporation
rate, volume-averaged scalar dissipation rate, Sauter mean diameter and mean
liquid temperature are plotted in Figure 5.28. The mean properties of the liquid
phase appear to be only slightly affected by the initial concentration of the spray:
the most noticeable difference lies in the longer time needed by the droplets
to reach their boiling temperature when Φ0 is increased. The volume-averaged
scalar dissipation rate, however, is strongly affected by the initial spray density,
particularly as ignition is approached: higher values of the global equivalence ratio
are in fact responsible for higher values of N˜ after t ' 0.4 ms. This result seems to
be in contrast with the fact that the ignition delay time decreases as Φ0 increases.
To solve this apparent contradiction, the temporal evolution of the conditional
scalar dissipation rate has been plotted in Figure 5.29 together with a comparison
of the 〈N | η〉 profiles corresponding to Cases D and F at two different instants
(a) ˜˙m and N˜ (b) aSMD and T˜l
Figure 5.28: Temporal evolution of (Left) droplet mean evaporation rate and
volume-averaged scalar dissipation rate, and (Right) droplet Sauter mean diam-
eter and liquid mean temperature. Data shown correspond to different initial
values of the global equivalence ratio.
131
(a) Case F (b) Cases D and F
Figure 5.29: Left: conditionally-averaged scalar dissipation rate, the conditioning
being on mixture fraction, plotted at different instants of time. Data shown cor-
respond to Case F in Table 5.1. Right: conditionally-averaged scalar dissipation
rate for Cases D and F in Table 5.1 at two selected instants of time.
of time. When plotted in mixture fraction space, no significative difference for
the conditional scalar dissipation rate is observed between the two simulations
considered here. Since 〈N | η〉 tends to peak around ξ = 0.15− 0.25, we conclude
that the observed differences in the temporal evolution of the volume-averaged
scalar dissipation rate are due to the different shapes of the mixture fraction PDF
corresponding to the different values of Φ0 considered here. In fact, the higher
values of N˜ found as the global equivalence ratio is increased are simply the
consequence of the larger relative quantity of very rich mixture pockets found in
denser sprays, which are characterized by more intense mixing processes. These
results also suggest that, as already noted by Schroll et al. [125], N˜ is not a good
indicator for predicting how delayed the ignition of the spray will be compared
to the ignition delay of the corresponding homogeneous reactor, but 〈N | η〉 is.
5.3.4.4 Influence of initial droplet size distribution
The initial droplet size distribution of sprays found in practical applications is
usually not monodisperse. Droplets with different initial sizes heat up and evap-
132
(a) T˜l and aSMD (b) φ and ˜˙m
Figure 5.30: Temporal evolution of (Left) mean liquid temperature and Sauter
mean diameter, and (Right) residual liquid volume fraction and mean droplet
evaporation rate, for different initial droplet size distributions.
orate differently, resulting in a different mixing behavior that may affect ignition.
In order to investigate the influence of spray polydispersity on autoignition, a
simulation (Case E) was run where the initial size of the droplets was assigned ac-
cording to a Rosin-Rammler distribution. The parameters of the Rosin-Rammler
curve were chosen so that the initial Sauter mean diameter of the spray was
aSMD = 25µm and the droplet diameter varied between 5 and 40µm.
Several studies on the dependence of evaporation and combustion of droplets
on their initial diameter are available in the literature. Harstad and Bellan [47]
showed that small and large droplets exhibit very different heating and evapora-
tive behaviors, with the d2 evaporation law [76] retrieved only for small droplets
at low pressures. Ignition of n-heptane droplets with different initial sizes (ad,0
ranging between 10 and 200µm) was studied numerically by Stauch et al. [138].
Similarly to the findings of Harstad and Bellan, they observed that, for the range
of conditions investigated in their study, the droplets did not follow the d2 law.
The ignition delay time was found to be a strong function of the initial droplet
diameter, increasing with increasing ad,0: this dependence, however, varied with
the operating conditions considered (e.g. pressure, initial air temperature). No
133
dependence on the initial fuel temperature was observed, implying that, for fixed
value of the initial droplet diameter, evaporation had a negligible effect on ig-
nition. One should note that this result could be an artifact of the high values
of the oxidizer temperature considered, which resulted in ignition occurring on
the lean side of stoichiometry. For the simulations corresponding to low ambient
gas temperature considered in this chapter, ignition is always observed in regions
where ξ > ξst: since a considerable time may be required to generate a rich mix-
ture, it is expected that the rate at which evaporation proceeds may strongly
affect the time of the first appearance of spots with optimal ignition conditions in
the flow, and hence the ignition delay. The delaying effect of larger initial droplet
size on ignition was also observed experimentally by Gordon and Mastorakos [45]
for atmospheric turbulent dilute diesel sprays, and numerically by Schroll et al.
[125] for n-heptane droplets evaporating and reacting in a turbulent medium. In
the latter work, the decrease of τid with increasing ad,0 was explained as a conse-
quence of the increase in the values of the conditional scalar dissipation rate in
the flow with increasing initial droplet size. An opposite trend for τid with respect
to ad,0 was obtained by Wang and Rutland [149]. In their work, they found that,
as the initial droplet diameter was increased, a corresponding decrease in the
ignition delay time was observed. This result was interpreted as a consequence
of the weaker evaporation rate of larger droplets, which was in turn responsible
for a less intense evaporative cooling of the bulk gaseous phase. In general, these
different trends may be interpreted as a consequence of the differences in the
operating conditions investigated: as noted by Stauch et al. [138], correlations
between quantities differ not only quantitatively, but also qualitatively, for dif-
ferent sets of ambient pressures and oxidizer temperatures.
Given the strong differences observed for the evaporative and ignition be-
haviors of droplets having different sizes, it is expected that the initial presence
of both small and large droplets in the bulk phase will lead to a considerable
deviation from the corresponding behaviors observed for monodisperse sprays.
Differences in the liquid phase behavior can be observed in Figure 5.30, which
shows the temporal evolutions of the mean liquid temperature, the Sauter mean
diameter, the residual liquid volume fraction φ and the mean droplet evaporation
134
(a) Droplet size distribution (b) Normalized droplets number
Figure 5.31: Left: droplet size distribution, plotted at different instants of time.
Right: temporal evolution of the number of residual droplets in the computational
domain, normalized by its initial value. Data shown correspond to Case E (Left)
and Cases D and E (Right) in Table 5.1.
rate for Cases D and E in Table 5.1. These simulations are characterized by the
same operating conditions and initial turbulence properties; the only difference
lies in the different initial droplet size distribution of the liquid phase. φ has
been defined here as the ratio between the instantaneous and initial values of the
volume occupied by the droplets. One can observe that sprays with large initial
droplets are difficult to heat up and, consequently, their evaporation proceeds
slower than for fine sprays. This can be seen from the temporal evolution of the
residual liquid volume fraction. Apart for a short initial transient, during which
the smallest droplets in the polydisperse spray reach their boiling temperature
and evaporate completely, this quantity is always lower for the monodisperse
spray case, indicating a faster evaporation process. The longer time needed for
the liquid phase to heat up results in a shift of the instant at which the mean
droplet evaporation rate peaks. One should note that, when the peak in m˙ is
reached, the liquid mean temperature is lower than its boiling point, differently
from what was observed for the monodisperse cases studied earlier in this chap-
ter. This is because the evaporation rate varies as the square of the droplet
135
(a) Case E (b) Cases D and E
Figure 5.32: Left: conditionally-averaged scalar dissipation rate, the conditioning
being on mixture fraction, plotted at different instants of time. Data shown cor-
respond to Case E in Table 5.1. Right: conditionally-averaged scalar dissipation
rate for Cases D and E in Table 5.1 at two selected instants of time.
diameter and, when the condition Td = Tf,boi is reached (Tf,boi = 534.7 K for the
conditions considered here), the droplets have already shrunk in size enough to
compensate for the corresponding increase in BM,d due to the larger values of Y
s
F,d.
The decrease in Sauter mean diameter for the polydisperse case is considerably
slower than for the monodisperse one (see Figure 5.30). This is a consequence of
the different evaporation behavior of the small and large droplets found in these
sprays. Small droplets heat up quickly and evaporate rapidly while, as it has
already been said, the opposite occurs for large droplets. This implies that, as
time progresses, the behavior of the Sauter mean diameter will be dominated by
the larger droplets, providing an explanation for its slow temporal decrease.
The temporal evolution of the droplet diameter PDF of the spray is shown in
Figure 5.31(a). Early after the start of evaporation, the PDF exhibits a nearly
symmetric shape, with its peak occurring around a = 20µm. As time progresses,
the peak of the distribution moves towards lower diameters as a consequence of
136
the evaporation process. Additionally, the shape of the PDF loses its symmetry
and becomes progressively skewed to the right. This is due to the increasing
difference in characteristic evaporation time of the droplets as their initial diam-
eter is increased. Figure 5.31(b) shows the temporal evolution of the number of
droplets in the computational domain for Cases D and E. In order to allow for
a comparison, this quantity was normalized for each of the cases considered here
using the corresponding initial number of droplets contained in the droplet-laden
layer. Since periodic boundaries were used in all directions in the numerical sim-
ulations, a decrease in the number of droplets indicates that some of them have
evaporated completely. While for a monodisperse spray the number of droplets
remains constant for a considerable amount of time before suddenly undergoing
an abrupt decrease, polydisperse sprays exhibit a continuous and nearly-linear
decrease in the droplet number, which is a direct consequence of the wide range
of droplet lifetimes that characterizes this kind of sprays.
The different evaporation behavior of polydisperse sprays as compared to
monodisperse ones is expected to have a strong impact on the air / fuel mix-
ing. Shown in Figure 5.32(a) is the temporal evolution of the conditional scalar
dissipation rate for Case E. In order to allow for a comparison with monodis-
perse sprays, profiles of 〈N | η〉 for Cases D and E are shown in Figure 5.32(b)
at two selected instant of times. Similarly to what was already observed for the
monodisperse cases considered in this chapter, the conditional scalar dissipation
rate initially increases monotonically with time; the maximum values of 〈N | η〉
reached, however, are now lower. At later times, the observed trend differs from
the one of a monodisperse spray in that the conditional scalar dissipation rate
settles down to an approximately constant level, without decreasing with time
anymore. This different behavior can again be interpreted in the light of the
wide range of initial droplet diameters that characterizes polydisperse sprays.
The higher thermal inertia of large droplets as compared to the smaller ones im-
plies that, early after the start of evaporation, most of the fuel vapor has been
generated from the smaller droplets in the spray, which heat up and reach their
boiling temperature quickly. The combination of fast evaporation, which results
in more time for air and fuel to mix, and the relative small amount of fuel vapor
137
(a) Volume-averaged heat release rate (b) Mixture fraction PDF
Figure 5.33: Left: temporal evolution of volume-averaged heat release rate for
Case E in Table 5.1, showing locations at which species and temperature contour
/ conditional means are given. Right: comparison of mixture fraction PDF at
selected instants of time for Cases D and E in Table 5.1. Numerical values of
referenced instants of time are: tIND = 0.7 ms, tPID = 1.0 ms, tPIE1 = 0.6 ms,
tPIE2 = 0.9 ms.
generated, which is a consequence of the small surface area of the hottest droplets
in the spray, is responsible for the observed initially low values of scalar dissipa-
tion rate. At later times, most of the fuel vapor is provided by those droplets
which were initially the largest and which have been slowly heated up to reach
a temperature close to their boiling point. Since the diameter of these droplets
remains large even at times close to ignition, the amount of fuel vapor generated
through evaporation is substantial. This leads to the formation of rich mixture
pockets that are rather inhomogeneous, yielding the observed high values of the
conditional scalar dissipation rate. This observation is consistent with the find-
ings of Schroll et al. [125], who showed that increasing the initial droplet size of
a monodisperse spray leads to higher values of the conditional scalar dissipation
rate in the whole mixture fraction space.
The ignition sequence of Case E is studied by analyzing the numerical solution
138
at the instants of time given in Figure 5.33. As shown in the previous section,
the mixture distribution and, in particular, the lack or abundance of spots where
ξ ' ξMR,0 may strongly affect the ignition of two-phase flows. It is therefore
interesting to investigate how the mixture fraction PDF changes for sprays with
different initial droplet size distributions. For this reason, Figure 5.33(b) shows
the comparison of the PDF shapes for Cases D and E for two instants of time at
the beginning and the end of the corresponding pre-ignition phase. The instants
of time chosen for plotting the mixture fraction PDF of Case D are those used in
discussing the effects on ignition due to global equivalence ratio, t = tIND = 0.7 ms
and t = tPID = 1.0 ms. The PDF shapes for the two cases studied here is similar,
indicating that fuel distribution is not strongly affected by the initial droplet size
distribution of the spray. Differences in the ignition behavior should thus be solely
attributed to the corresponding differences in the temporal evolution of the scalar
dissipation rate. It should be noted that the shape of the mixture fraction PDF
was found to be strongly affected by the initial value of the droplet diameter in
[125], with its peak shifting towards richer mixture compositions with increasing
a0. The lack of such a dependency here may be attributed to the simultaneous
presence of both small and large droplets in the spray, whose effects on P˜ (η) end
up in compensating each other.
Scatterplots of temperature and selected species mass fractions are shown in
Figure 5.34 and reveals a very interesting feature of polydisperse sprays, as igni-
tion is now occurring simultaneously at two different values of ξ, both of which
are close to ξMR,0. The concept of most reactive mixture fraction thus appears to
hold also for polydisperse sprays, although it may need some modification to ac-
count for the appearance of ignition kernels at more than one location in mixture
fraction space. Another peculiar difference with the monodisperse spray cases
lies in the presence of a considerable amount of fuel pockets that have undergone
little chemical reactions when ignition occurs, as shown by the scatterplots of
n-C7H16 and CH2O mass fractions. These behaviors are again a consequence of
the initial presence of both small and large drops in the spray. Small droplets
evaporate quickly and lead to the formation of well-mixed spots. However, as
time progresses, these regions continue to mix with the surrounding air and may
139
(a) t = tIGE1 = 1.050 ms
(b) t = tIGE2 = 1.070 ms
(c) t = tIGE3 = 1.090 ms
(d) t = tIGE5 = 1.110 ms
Figure 5.34: Scatterplots of temperature and n-C7H16, O2, CH2O mass fractions
plotted against mixture fraction at different instants of time during ignition.
Referenced instants of time are given in Figure 5.33. Data shown correspond to
Case E in Table 5.1.
140
(a) t = tIGE2 = 1.070 ms (b) t = tIGE3 = 1.090 ms
(c) t = tIGE4 = 1.100 ms (d) t = tIGE5 = 1.110 ms
Figure 5.35: Temporal evolution of T = 1250 K isosurface. Surface colored ac-
cording to local mixture fraction value. Referenced instants of time are given in
Figure 5.33. Data shown correspond to Case E in Table 5.1.
become too lean for ignition to occur. Those that remain rich enough during the
whole pre-ignition phase eventually ignite. Their corresponding maximum value
of the mixture fraction depends on the initial size of the droplets present there
and this explains why ignition in polydisperse sprays occurs over a wider range
of mixture fraction values. The evaporation of large droplets initially proceeds
at a very low rate. However, as soon as they have been heated up, it becomes
substantial and, as discussed previously, this leads to the formation of very rich
spots. The delay by which these regions are formed and the strong fluctuations in
the scalar quantities that characterize them explain why a considerable amount
of fuel vapor has undergone little chemical reactions only as ignition is occurring
141
elsewhere in the flow. The similar values of P (ξMR) found for Cases D and E
(Figure 5.33(b). ξMR is the one corresponding to Case D and is equal to 0.2)
together with the higher levels of scalar dissipation associated with Case E (Fig-
ure 5.32(b)) are consistent with the view that ignition can now occur at several
locations in mixture fraction space. Also, the simultaneous presence in the flow of
regions that have ignited and others which have substantially remained unreacted
is compatible with the notion that droplets of different sizes ignite at different
times [138, 125, 45].
The temporal evolution of the T = 1250 K isosurface is shown in Figure 5.35.
Ignition kernels again develop independently at several locations in the gaseous
flow and grow in size with time. However, differently from what was observed for
the monodisperse sprays considered in this chapter (Cases A, D and F), the flame
expansion phase is not accompanied anymore by the appearance of additional
burning spots which aid the ignition of the entire mixture. This is a consequence
of the low availability of flow regions characterized by ideal igniting conditions,
e.g. ξ = ξMR,0 and low scalar dissipation rate.
5.3.5 Ignition behavior corresponding to high values of
the air stream temperature
Ignition at high values of the air stream temperature exhibits several aspects
that differ from low-temperature ignition. Some of these differences are unveiled
by the temporal evolutions of the volume-averaged heat release rate, which are
given in Figure 5.36 for Cases A and C. Since heat release data corresponding
to different operating conditions have different magnitudes, the volume-averaged
heat release rate of Case C was divided by a factor of 10 to allow for a compar-
ison. The ignition delay time decreases as higher values of Tair are considered:
for the case investigated here, τid = 0.28 ms. This is consistent with previous
studies of spray autoignition that appeared in the literature [138, 149, 45], where
an increase in the oxidizer temperature was always found to be responsible for a
corresponding decrease in the ignition delay. The heat release rate now exhibits
a strictly monotonic increasing behavior, without the decrease prior to ignition
142
Figure 5.36: Temporal evolutions of volume-averaged heat release rate for Cases
A and C in Table 5.1. Markers denote locations at which species and temperature
scatterplots are given.
that was observed for the low-temperature cases. One should note that τid is now
more than twice the corresponding reference ignition delay. This large difference
is probably an artifact of the method used for computing τid,0. In fact, the tem-
perature in the homogeneous reactor calculations is initialized by assuming an
inert mixing distribution between the fuel and oxidizer streams. This procedure
does not take into account the fact that, in a spray flow, the gas phase tempera-
ture for a given mixture composition is lower than the one corresponding to inert
mixing due to evaporative cooling effects. Hence, τid,0 refers to a mixture whose
initial temperature is higher than the corresponding one for sprays, thus leading
to a shorter reference ignition delay time.
Scatterplots of temperature and O2, OH and CH2O mass fractions during the
ignition transient are shown in Figure 5.37. The instants of time at which the
scatterplots are given are marked in Figure 5.36. Ignition is seen to occur first
around ξMR = 0.045, which is close to ξMR,0 = 0.03. This value of ξ is lower than
the stoichiometric mixture fraction value (ξst = 0.0625). Ignition occurring on
the lean side of stoichiometry was also observed in [138] for isolated n-heptane
droplets in air at high temperature (Tair = 1200 K). Once a burning kernel is
143
(a) t = tIGH1 = 0.270 ms
(b) t = tIGH3 = 0.285 ms
(c) t = tIGH4 = 0.288 ms
(d) t = tIGH5 = 0.290 ms
Figure 5.37: Scatterplots of temperature and O2, CH2O, OH mass fractions plot-
ted against mixture fraction at different instants of time during ignition. Refer-
enced instant of time are given in Figure 5.36. Data shown correspond to Case
C in Table 5.1.
144
(a) t = tIGH2 = 0.280 ms (b) t = tIGH3 = 0.285 ms
(c) t = tIGH4 = 0.288 ms (d) t = tIGH5 = 0.290 ms
Figure 5.38: Temporal evolution of T = 1750 K isosurface. Surface colored ac-
cording to local mixture fraction value. Referenced instants of time are given in
Figure 5.36. Data shown correspond to Case C in Table 5.1.
established, the flame starts to propagate towards both leaner and richer mix-
ture compositions. The propagation towards the rich side, however, occurs faster.
Similarly to the low temperature cases, the flame propagation phase is accompa-
nied by the appearance of additional burning kernels, which speed up the ignition
of the whole mixture. This spotty nature of the ignition process is shown in Figure
5.38, where the T = 1750 K isosurface has been plotted at four different instants
of time during ignition. Differently from the low-temperature case for the same
turbulence conditions (Case A), it appears that ignition is now dominated by the
continuous appearance of new burning kernels, which do not have enough time to
145
Figure 5.39: Temporal evolution of cross-correlation coefficient between heat
release and scalar dissipation rate. Referenced instants of time are given in
Figure 5.36. Their numerical values are: tPIH1 = 0.180 ms, tPIH2 = 0.210 ms,
tPIH3 = 0.240 ms, tIGH1 = 0.270 ms, tIGH3 = 0.285 ms. Data shown correspond to
Case C in Table 5.1.
grow in size and remain small. As discussed in Section 5.3.2.4, the simultaneous
occurrence of a large number of ignition kernels for conditions corresponding to
high Tair values was already observed by Wang and Rutland [148]. Based on the
results shown here, we may consider it to be a signature feature of high temper-
ature ignition.
The behavior of intermediate and radical species is similar to the one found for
the low temperature cases considered in this chapter and for autoigniting gaseous
flows [43]. As ignition occurs, a sudden increase in the mass fractions of OH and
H (H is not shown in the figure) is observed. The distributions of these species
in mixture fraction space are very narrow and peak around ξst. CH2O and HO2
(not shown), on the other hand, exhibit a characteristic two-fold behavior, peak-
ing during the ignition transient and then being depleted rapidly once a diffusion
flame is established. This is identical to the behavior observed for low values of
the air stream temperature.
146
The cross-correlation coefficient between the heat release rate and the scalar
dissipation rate conditioned on the mixture fraction value is shown in Figure 5.39.
Prior to ignition, a strong negative correlation exists between these quantities in
the proximity of ξMR. Values as low as −0.88 are reached, which are even lower
than those found for the low-temperature cases. Thus, even at high values of the
air stream temperature, ignition is found to occur first at those locations where ξ is
close to ξMR,0 and the value of the scalar dissipation rate is low. After ignition has
occurred, the cross-correlation coefficient becomes approximately equal to zero in
the whole range of mixture fraction values considered here. Similarly to what
has already been observed for the low-temperature cases, this is an artifact of the
propagation of a burning flame from its ignition kernel to the surrounding flow.
Turbulence promotes the flame propagation, and is thus responsible for a positive
correlation between heat release and scalar dissipation rate. This balances out
with the negative correlation which holds for the igniting kernels, yielding an
overall zero value of the cross-correlation coefficient.
5.3.6 Ignition behavior corresponding to diluted oxygen
concentration in air
HCCI engines rely on the principle of dilute premixed or partially premixed com-
bustion to reduce emissions. Dilute mixtures are obtained by either being lean
with the fuel / air equivalence ratio, or through the use of high levels of exhaust
gas recirculation [28]. Given the strong interest in understanding the key factors
controlling ignition at HCCI conditions, a simulation was run with the values of
the air temperature and of the oxygen dilution in the air stream chosen to be com-
parable to those found in an HCCI engine, e.g. Tair = 1000 K and XO2 = 10 %.
This simulation is listed as Case B in Table 5.1.
Temporal evolution of the volume-averaged heat release rate for Case B is
shown in Figure 5.40. Markers indicate those instants of time at which the nu-
merical solution has been investigated further. The ignition sequence can be split
into an inert / pre-ignition phase (denoted by the label PIB) and an ignition and
flame propagation phase (label IGB). Differently from what has been observed for
147
Figure 5.40: Temporal evolutions of volume-averaged heat release rate for Case
B in Table 5.1. Markers denote locations at which species and temperature
scatterplots are given.
the cases with standard air as the bulk phase and the same value of the oxidizer
temperature, the levels of heat release rate reached during the pre-ignition phase
are now considerably lower than those found at ignition, being approximately
one-two orders of magnitude smaller. Early during the pre-ignition phase, the
volume-averaged heat release rate increases monotonically up to t = 1.375 ms,
when it reaches a local maximum. This is followed up by a slight decrease lead-
ing to a local minimum occurring at t = 1.825 ms. In autoigniting sprays with
undiluted air as the oxidizer, the minimum was found to be a precursor to an
abrupt increase of the heat release rate over a short period of time, correspond-
ing to the occurrence of ignition at some locations in the flow. This is not the
case here. After having reached a minimum, the volume-averaged heat release
rate starts to slowly increase with time. Heat release becomes substantial only
at much later times with respect to its local minimum, at t = 4.0 ms. Ignition
occurs at t = 4.18 ms, which is considerably later than the corresponding ignition
delay for the homogeneous reactor, τid,0 = 1.82 ms. The ratio between these char-
acteristic times is equal to 2.297 now, which is almost twice as large as the value
1.273 found for Case D. Despite a change in the steepness of the heat release
rate curve following ignition is still evident, this is smaller than for the undiluted
148
Figure 5.41: Temporal evolution of conditional mean heat release, temperature,
mixture fraction PDF and selected species mass fractions. Referenced instants
of time are given in Figure 5.40. Their numerical values are: tPIB1 = 1.00 ms,
tPIB2 = 1.20 ms, tPIB3 = 1.40 ms, tPIB4 = 1.60 ms, tPIB5 = 1.80 ms. Data shown
correspond to Case B in Table 5.1.
cases, suggesting that the flame propagation phase is now occurring over a longer
period of time. It should be noted that, for Case B, the characteristic droplet
lifetime is much lower than the reference ignition delay time. Thus, mixing may
have a strong influence on ignition, and care should be taken in drawing general
conclusions from the results presented here. This point is discussed in more detail
149
in the following paragraphs.
In order to explain the observed behavior of the volume-averaged heat release
rate early during the pre-ignition phase, temporal evolutions of the conditional
mean heat release rate, temperature, mixture fraction PDF and selected species
mass fractions, the conditioning being on the mixture fraction value, are shown
in Figure 5.41. Referenced instants of time are given in Figure 5.40. One pe-
culiar aspect of Case B is the narrow range of ξ values for which the mixture
fraction PDF is non-zero. The upper limit of this range, ξmax, is seen to de-
crease with time after t = 1.0 ms. This is a consequence of the large difference
existing between the characteristic droplet lifetime (equal to τev = 0.82 ms) and
the ignition delay time (which was found to be τid = 4.15 ms). Early during the
evaporation process, the rate at which fuel vapor is supplied to the bulk phase
through liquid fuel evaporation overcomes the fuel / oxidizer mixing rate, leading
to the formation of an inhomogeneous mixture characterized by the presence of
locally rich spots. The mechanism responsible for supplying the fuel vapor to the
oxidizer, however, lasts only until when droplets are still present in the flow. Af-
ter evaporation of the liquid fuel has completed, mixing processes continue to be
important throughout the whole period of time leading to ignition and are very
effective in stirring the flow, smoothing the inhomogeneities in the mixing field.
As a consequence, the maximum value of the mixture fraction in the flow starts
to decrease rapidly with time after evaporation is complete. At t = 1.80 ms, ξmax
is only equal to 0.08, which is considerably lower than ξMR,0 (ξMR,0 = 0.20 here).
Figure 5.41 shows that, for the range of operating conditions considered here,
reactions at early times during the pre-ignition phase tend to be more intense
around ξ = 0.12− 0.15, where peaks in the heat release rate and the radical and
intermediate species mass fractions (OH and CH2O) are found. At later times,
however, mixing leads to the disappearance of these spots having optimal igni-
tion conditions; additionally, chemical reactions in the range of moderately rich
mixture fraction values (ξ = 0.07 − 0.10) are seen to decrease with time after
t = tPIB2 = 1.20 ms, similarly to what has been observed for the cases with no
oxygen dilution in the bulk gas. These observations are fully consistent with the
observed behavior of the heat release rate during the pre-ignition phase. They
150
(a) t = tPIB1 = 1.00 ms
(b) t = tPIB4 = 1.60 ms
(c) t = tPIB6 = 3.00 ms
(d) t = tIGB1 = 4.15 ms
Figure 5.42: Contour plots of mixture fraction (left), CH2O (center) and heat
release rate (right) corresponding to the axial plane x = 0.5L at selected in-
stants of time. Referenced instants of time are given in Figure 5.40. Data shown
correspond to Case B in Table 5.1.
151
also provide the explanation for the substantial difference existing between the
reference and the actual ignition delay time. The lack of regions in the flow where
ξ = ξMR,0 forces the ignition spots to develop at leaner mixture compositions, for
which the conditions for the occurrence of ignition are less favorable. Thus, the
large difference between the actual and reference ignition delays can be attributed
to the lost of spots with composition equal to ξMR,0 due to mixing.
The physical picture described above is reinforced by the visualization of the
contour plots of mixture fraction, heat release rate and formaldehyde, which are
shown in Figure 5.42 for selected instants of time spanning the whole pre-ignition
phase. A slice cutting the computational domain through the x = 0.5L plane was
chosen to visualize the data. At early times, when droplets are still present within
the flow, the observed behaviors for mixing and chemical reactions are similar to
those described for the undiluted cases in Section 5.3.2.3. Small, fuel-rich pockets,
which correspond approximatively to the droplet locations, are seen to develop at
several locations in the otherwise lean carrier gas. Heat release reactions are found
around these regions; however, differently from the undiluted cases, it now seems
that the most reactive spots are located exclusively around individual droplets.
This is consistent with the low value of the G number that characterizes Case B.
At later times, once the liquid fuel has evaporated completely, a different picture
is observed. The inhomogeneous structure of the mixture fraction field progres-
sively disappears due to mixing of the fuel-containing spots with the surrounding
air, leading to the formation of an approximately homogeneous mixture with few
broad regions whose composition is slightly richer. Analysis of the CH2O and
heat release rate fields reveals that pre-ignition reactions are most intense in the
locations characterized by the highest values of ξ during the whole simulation,
which are also those where ignition first occurs.
The ignition sequence is shown in Figure 5.43. Similarly to the undiluted
cases, ignition is occurring first in those regions characterized by a specific value
of the local mixture fraction, which is equal to ξMR = 0.04 here. One should note
that ξMR is considerably lower than ξMR,0 = 0.2; this is a direct consequence of
the long time available for mixing, which is responsible for generating well-mixed
152
(a) t = tIGB1 = 4.150 ms
(b) t = tIGB2 = 4.225 ms
(c) t = tIGB3 = 4.275 ms
(d) t = tIGB4 = 4.450 ms
Figure 5.43: Scatterplots of temperature and O2, CH2O, OH mass fractions plot-
ted against mixture fraction at different instants of time during ignition. Refer-
enced instants of time are given in Figure 5.40. Data shown correspond to Case
B in Table 5.1.
153
spots whose richest composition is leaner than ξMR,0. After a burning kernel has
been established, the resulting flame starts to propagate in mixture fraction space
and ignites the neighboring flow regions. As previously observed, the flame prop-
agation phase is slower than for the undiluted cases: even after 0.2 ms from the
start of ignition, there is still a considerable amount of flow regions which have
not ignited yet. It should be noted that oxygen dilution in the air stream results
in lower values of the burning flame temperature for a given mixture fraction
value as compared to the undiluted cases. This implies weaker heat fluxes from
the flow regions that are burning to the unreacted ones, providing an explanation
for the observed slower propagation of the flame front.
Temporal evolution of the T = 1200 K isosurface, colored according to the lo-
cal mixture fraction value, is shown in Figure 5.44. The isosurface is seen to exist
already at times prior to ignition: this is because, differently from the undiluted
cases, ignition now occurs more gradually, although a clear, sudden transition
from the non-reacting to the reacting state can still be identified. The nature
of the ignition process is again spotty, with burning kernels developing in just
a limited number of spatial locations in the flow. Once established, these hot
regions grow in size with time as a result of the flame propagation phase. This
process, however, is not accompanied by the simultaneous appearance of addi-
tional ignition kernels. In fact, although Case B was run for a longer time after
the first occurrence of ignition as compared to the other cases investigated in this
chapter, only two independently burning regions could be identified at the end
of the simulation. One could attribute this to the disappearance of the igniting
kernels due to their mixing with leaner flow regions. To investigate this possibil-
ity, the temporal evolution of the ξ = ξMR isosurface (ξMR = 0.04 here) during
the ignition transient has been plotted in Figure 5.45. The isosurface is seen not
to change much in shape and size during ignition, suggesting that the cause for
the small number of burning kernels found is probably related to the lack of flow
regions which remain sufficiently rich during the whole pre-ignition phase.
It is interesting to check whether a negative correlation between heat release
and scalar dissipation rate also holds for ignition under diluted air conditions.
154
(a) t = tIGB1 = 4.150 ms (b) t = tIGB2 = 4.225 ms
(c) t = tIGB3 = 4.275 ms (d) t = tIGB5 = 4.450 ms
Figure 5.44: Temporal evolution of T = 1200 K isosurface. Surface colored ac-
cording to local mixture fraction value. Referenced instants of time are given in
Figure 5.40. Data shown correspond to Case B in Table 5.1.
To this end, the cross-correlation coefficient 〈σ| η〉 between these two quantities
has been plotted in Figure 5.46 at selected instants of time early during the
pre-ignition phase. It should be noted that, once evaporation is complete, the
scalar dissipation rate rapidly falls everywhere in the flow. Behaviors similar to
those observed for the undiluted cases are found. 〈σ| η〉 is positive in the mix-
ture fraction range between 0 − 0.02 and 0.08 − 0.12, then it becomes negative.
Depending on the particular instant of time considered, values between −0.35
and −0.9 are reached. The latter limit is even lower than the minimum value
of 〈σ| η〉 found in the simulations with no oxygen dilution, and comparable to
155
(a) t = tIGB2 = 4.225 ms (b) t = tIGB3 = 4.275 ms
(c) t = tIGB4 = 4.350 ms (d) t = tIGB5 = 4.450 ms
Figure 5.45: Temporal evolution of ξ = 0.04 isosurface. Surface colored according
to local temperature value. Referenced instants of time are given in Figure 5.40.
Data shown correspond to Case B in Table 5.1.
those observed in autoigniting gaseous flows [86]. Around t = tPIB4 = 1.60 ms,
the conditional scalar dissipation rate falls below 1 s−1 for all values of ξ (see
Figure 5.47). The richest mixture found in the flow field at this instant of time
has a mixture fraction value ξ ' 0.12, which is leaner than ξMR,0 (ξMR,0 = 0.2
here), and continues to decrease as ignition is approached. The cross-correlation
coefficient is now positive for all values of ξ. In general, we may conclude that,
early during the pre-ignition phase, when vigorous mixing between air and fuel
vapor is sustained by droplet evaporation, heat release reactions in sprays igniting
under diluted air conditions proceed faster in those regions where good mixing
156
Figure 5.46: Temporal evolution of cross-correlation coefficient between heat re-
lease and scalar dissipation rate. Referenced instants of time are given in Figure
5.40. Their numerical values are: tPIB1 = 1.00 ms, tPIB4 = 1.60 ms, tPI6 = 2.20 ms,
tPI7 = 2.80 ms, tPI8 = 3.40 ms. Data shown correspond to Case B in Table 5.1
conditions are achieved, similarly to what has been observed for spray ignition in
undiluted air. At later times, however, once evaporation is complete, mixing pro-
cesses proceed at such low rates and weak intensities that their effect on ignition
is practically negligible. Burning kernels then develop faster in correspondence
of those spots where the local mixture fraction value has been sufficiently rich
during the whole pre-ignition phase. Although a more detailed analysis is re-
quired to draw definitive conclusions, these results suggest that the ratio between
the reference ignition delay time and the characteristic droplet lifetime plays an
important role in controlling spray ignition, especially for those situations where
evaporation is complete long before the ignition event.
5.4 Conclusions
The physics of n-heptane spray autoignition, burning at conditions relevant to
diesel and HCCI engines, was investigated numerically through Direct Numer-
ical Simulations with complex chemistry. The focus of the study presented in
this chapter was to provide a physical insight into the complex interactions be-
157
Figure 5.47: Conditionally-averaged scalar dissipation rate, the conditioning be-
ing on mixture fraction, plotted at different instants of time. Data shown corre-
spond to Case B in Table 5.1
tween turbulence, chemistry and fuel / air mixing that characterize spontaneous
ignition in sprays, and to verify whether well-known results from the theory of
single-phase flow autoignition continue to hold when the fuel is supplied to the
oxidizer in liquid form.
Three main phases were identified for the low-temperature ignition of monodis-
perse sprays in standard air. These were called inert phase, pre-ignition phase
and ignition and flame propagation phase. The inert phase was characterized by
negligible values of the volume-averaged heat release rate. Droplet evaporation
and mixing of the resulting fuel vapor with the surrounding air were found to be
the dominant processes there. The pre-ignition phase corresponded to the period
of time spanning from the instant at which the mean heat release rate started not
to be negligible anymore to the onset of ignition. Chemical reactions were seen
to develop around either individual or small clusters of droplet, in line with the
expectations from the droplet group combustion theory of Chiu and Liu [24]. The
volume-averaged heat release rate increased slowly at first, before experiencing
a sudden drop prior to ignition; this was explained as a result of the temporal
decrease in the reaction intensity that interested those regions having moderately
158
rich mixture compositions (ξ = 0.05− 0.15). The ignition and flame propagation
phase started after the sudden, exponential-like temporal increase in heat release
rate that accompanied ignition. The ignition process was found to be spotty,
with several burning kernels developing independently from one another which
started to grow up in size and eventually merged together as time progressed.
The ignition delay time was in the order of τid ' 1.0 ms for all cases investigated.
Ignition corresponding to high values of the oxidizer temperature did not show
the presence of an inert mixing-phase; additionally, the heat release rate always
exhibited a monotonically increasing behavior, without the sudden decrease prior
to ignition that characterized the low-temperature cases. The ignition delay time
was considerably shorter than for the low-temperature sprays due to the Arrhe-
nius dependence of the reaction rates on temperature, and it was approximately
equal to τid ' 0.285 ms. Concerning the ignition of sprays under diluted oxygen
conditions, no clear distinction between the inert and pre-ignition phases could be
made, as the corresponding values of the volume-averaged heat release rate were
similar and considerably lower than those occurring at ignition. Also, the tran-
sition between the non-reacting and the reacting states was seen to occur more
gradually than for the undiluted sprays, and the flame propagation phase follow-
ing ignition appeared to be slower due to the lower temperature values reached
by the burnt gases. The ignition delay time was considerably longer than for the
undiluted spray cases, and it was equal to τid = 4.2 ms.
One major conclusion from the study presented in this chapter was that, for
all the monodisperse spray cases investigated, ignition was always found to occur
first in those flow regions where the local mixture fraction was equal to ξMR. The
numerical value of ξMR was close to ξMR,0 of the corresponding homogeneous reac-
tor and depended on the operating conditions considered. Additionally, apart for
one case where the spray was igniting into diluted air, a strong negative correla-
tion (〈σ| ξMR〉 ' −0.7) was found between heat release and scalar dissipation rate
in the range of mixture fraction values around ξMR. For the spray corresponding
to high values of the oxidizer temperature, values down to −0.88 were reached.
This was in contrast with the findings of Schroll et al. [125], who reported only
159
mild negative values for the cross-correlation coefficient between these quantities
(〈σ| ξMR〉 ' −0.3). These results mimic the well-known behaviors observed for
autoignition in single-phase flows [86, 136, 85] and indicate that, for most of the
operating conditions investigated in this chapter, spray ignition proceeds in a
gaseous-like way. The behavior of formaldehyde, which has been identified in
experimental studies as an important precursor of autoignition for hydrocarbon
fuels [43], was found to be similar to the one observed in purely gaseous flows,
increasing during the pre-ignition phase, and being depleted rapidly after ignition
has occurred. These findings are consistent with the measurements of O’Loughlin
and Masri on methanol spray flames [101] and further reinforce the idea that, for
the operating conditions investigated in our simulations, differences existing be-
tween autoignition of gaseous and liquid fuels, if any, are small.
An important discovery made in this chapter was that the concept of most
reactive mixture fraction, although still valid, needs some modification for poly-
disperse spray. Ignition kernels, in fact, were observed at more than one location
in mixture fraction space. The existence of several mixture fraction values at
which the flow ignites was interpreted as a direct consequence of the wide range
of characteristic droplet lifetimes found in polydisperse sprays. The presence of
small droplets in the spray, in particular, leads to early formation of ignitable
mixtures due to the corresponding rapid heating and evaporation of the liquid
fuel. However, as soon as evaporation of the smallest droplets is complete, these
regions continue to mix with the surrounding hot air and may become too lean for
ignition to occur. Those that remain rich enough during the whole pre-ignition
phase, they eventually ignite, and it is expected that the value of ξ at which igni-
tion occurs depends in a complex manner on the ratios between the lifetime of the
small droplets, the ignition delay time and the turbulence time scale. Analysis of
the polydisperse case also revealed that, at the instant of time corresponding to
ignition, several regions in the flow field existed where the fuel and the oxidizer
had barely started to react. These regions were linked to the presence of large
droplets in the flow, whose evaporation initially proceeded at a very slow rate due
to their large thermal inertia and hence corresponding slow heating process. This
implied a slow formation of mixture spots with optimal ignition conditions. Once
160
these regions were formed, the droplet temperature had become high enough to
cause intense liquid evaporation, resulting in high values of the scalar dissipation
rate which were responsible for slowing down, if not interrupting, the pre-ignition
phase. These results were found to be consistent with the numerical simulations
of Stauch et al. [138] and Schroll et al. [125], and with the experimental findings
of Gordon and Mastorakos [45], who all reported an increase in the ignition delay
time of the spray with increasing initial droplet diameter.
Changes in the initial turbulence intensity and global equivalence ratio in the
droplet-laden layer were seen to affect ignition through the temporal evolution of
the conditional scalar dissipation rate. For the monodisperse sprays igniting at
conditions corresponding to standard air and low temperature value studied in
this chapter, 〈N | η〉 was seen to increase at first, reach a maximum, and then to
decrease at later times. A similar behavior was also observed for higher values
of the initial turbulence strength; however, the conditional scalar dissipation rate
was now found to build up over a shorter period of time, and to relax towards
lower values as ignition was approached. The ignition delay was found to become
shorter as u′ was increased, consistently with the observed behavior of 〈N | η〉 and
the strong negative correlation existing between heat release and scalar dissipa-
tion rate. This result was in line with previous experimental [71] and numerical
[149] findings for autoigniting sprays.
The initial value of the global equivalence ratio in the droplet-laden layer did
not affect the ignition delay time appreciably, differently from what has been
observed by other authors [148]. Analysis of the temporal evolution of the con-
ditional scalar dissipation rate profiles revealed that a change in Φ0 did not lead
to a corresponding change in 〈N | η〉 in the inert and pre-ignition phases, thus
explaining the observed insensitivity of τid to the global equivalence ratio. The
ignition sequence was characterized by the development of a larger number of
ignition kernels as compared to the cases with same initial conditions but lower
value of Φ0. This was a consequence of the different shapes of the mixture frac-
tion PDF for these cases, in particular of the larger value of P (ξMR) for the case
with the higher global equivalence ratio, which implied an higher probability of
161
finding regions in the flow with optimal ignition conditions. The change in the
PDF shape was also responsible for a change in the temporal evolution of the
unconditional mean scalar dissipation rate N˜ : in particular, this quantity was
observed to increase with increasing Φ0. The insensitivity of τid to changes in
Φ0 indicates that the average scalar dissipation rate does not provide valuable
information on how delayed the ignition event will be as compared to the ignition
delay of the corresponding homogeneous reactor, in line with the findings of other
authors [125].
Diluting the oxygen concentration in the bulk gas had a retarding effect on the
ignition delay, as expected from the homogeneous reactor calculations. However,
differently from the simulations where standard air was used as oxidizer, the
ratio between τid and τid,0 was large, being equal to 2.297. For the undiluted
case with same initial turbulence strength and bulk gas temperature, a value of
1.273 was found. The reasons for such a large difference have to be searched
in the fact that droplet evaporation is now complete much earlier before the
onset of ignition. In the absence of a source of fuel vapor, mixing processes
become very effective in stirring the flow and smoothing the inhomogeneities in
the mixing field generated by the evaporation process. This shortly leads to the
formation of a flow field where the maximum mixture fraction value is below
ξMR,0, implying that ignition kernels are forced to develop at conditions far from
the optimal ones. Analysis of the cross-correlation coefficient between heat release
and scalar dissipation rate revealed that, early during the pre-ignition phase,
these quantities are negatively correlated, similarly to what has been observed for
sprays igniting in undiluted air. At later times, once evaporation was complete,
the value of the scalar dissipation rate rapidly fell to zero everywhere in the flow,
and 〈σ| η〉 became positive for all mixture fraction values. Burning kernels were
seen to develop faster in correspondence of those flow regions whose composition
remained rich enough, and hence closer to ξMR,0, during the whole pre-ignition
phase. Although further analysis is required to draw definitive conclusions, these
results suggest that the ratio between the reference ignition delay time τid,0 and
the characteristic droplet lifetime τev may be a key parameter for controlling
162
spray ignition, especially for those situations where evaporation is complete early
before a burning kernel has been formed in the flow.
163

Chapter 6
Validation of the CMC method
for spray autoignition
The application of the Conditional Moment Closure to the study of turbulent
reacting flows relies on the use of models that provide a description of the mixing
between the fuel and oxidizer streams. In the RANS-CMC simulations presented
in Chapter 4, it has been implicitly assumed that standard modelling of the con-
ditional scalar dissipation rate and mixture fraction PDF, with physically-sound
corrections to account for evaporation-related phenomena occurring in the near-
droplet field, can provide a fair approximation of these quantities. This approach
is known to present some limitations, as there is a consistent body of work in the
literature showing the deficiencies of conventional models in characterizing the
mixing status of a turbulent flow, not only in the context of sprays [40, 163, 32],
but also for certain gaseous flows with complex mixing patterns [93].
Despite the fact that modelling choices may be irrelevant in the numerical
simulations of steadily burning flames, they are expected to play a decisive role
in capturing complex phenomena such as flame extinction and reignition [38], and
hence deserve careful examination. To this end, the numerical simulations pre-
sented in Chapter 5 are exploited here to check the validity of some of the models
used for closing the CMC equations in the framework of sprays. The assessment
is done by using a zero-dimensional CMC code (e.g. with no spatial transport
terms in the CMC governing equations) to simulate the ignition of some of the
165
turbulent sprays that have been investigated with DNS. Modelling of the unclosed
terms is done based on quantities extracted from the DNS data; as such, the anal-
ysis presented here is performed under the most favorable conditions, since the
information that are required by the CMC sub-models are already available, and
does not need to be determined from the numerical solution of modeled transport
equations.
The chapter is organized as follows. The zero-dimensional CMC equations are
presented first, together with a description of the different modelling strategies
employed for closing these equations, the numerical schemes used in the CMC
code and the initialization of the CMC arrays. Then, assessment of standard
modelling approaches for the mixture fraction PDF and the conditional scalar
dissipation rate is performed, and possible implications for the numerical pre-
diction of spray ignition are discussed. The validity of a first-order closure of
the chemical source terms in the context of autoigniting sprays is also discussed.
Finally, the predicting capabilities of the CMC method for simulating spray au-
toignition are tested by comparing the results obtained from the zero-dimensional
CMC simulations with the DNS data.
6.1 Mathematical formulation
6.1.1 0D-CMC equations for two-phase flows
The 0D-CMC equations for two-phase flows are obtained from the corresponding
multidimensional ones (Equations 3.38 and 3.39) by neglecting those terms that
describe transport in physical space. The governing equation for the conditional
moment of the α-th species mass fraction then becomes:
∂Qα
∂t
= 〈N | η〉 ∂
2Qα
∂η2
+ 〈 ω˙α| η〉+
[
δαF −Qα − (1− η) ∂Qα
∂η
]
〈Π| η〉 (6.1)
Due to the dilute spray assumption, the gas volume fraction 〈θ〉 has been set to
one. Additionally, the term that accounts for the effect of the conditional cross
166
fluctuations between the generic reactive scalar consider (species mass fraction or
temperature) and the evaporation rate has been neglected, as its modelling is at
present unclear. The conditional temperature equation becomes:
∂QT
∂t
= 〈N | η〉
[
1
cpη
(
∂cpη
∂η
+
N∑
α=1
cp,αη
∂Qα
∂η
)
∂QT
∂η
+
∂2QT
∂η2
]
+
1
cpη
〈
1
ρ
∂p
∂t
∣∣∣∣ η〉+ 1cpη 〈 ω˙H | η〉 − 〈Π| η〉
[
hfg
cpη
+QT − (1− η)∂QT
∂η
]
(6.2)
Equations 6.1 and 6.2 were solved numerically using an existing in-house code,
which is described in Section 6.1.3. The code was modified to account for the
contributions due to the presence of the droplet source terms in the CMC equa-
tions.
6.1.2 Closure of the 0D-CMC equations
The different strategies used for modelling the unclosed terms appearing in the
zero-dimensional CMC equations are described below. They have been chosen to
highlight the importance of a proper modelling of the conditional scalar dissipa-
tion rate and mixture fraction PDF, and to assess the impact of droplet terms
on the numerical predictions.
6.1.2.1 Modelling strategy A
Closure of the conditional scalar dissipation rate is obtained using the AMC
model of O’Brien and Jiang [100]. The corresponding functional form for 〈N | η〉
was given in Chapter 3 (see Equations 3.24-3.26) and is repeated below for com-
pleteness:
〈N | η〉 = N0G(η)
with:
G(η) = exp
(
−2 [erf−1(2η − 1)]2) , N0 = χ˜
2
∫ 1
0
G(η)P˜ (η)dη
167
The Favre-averaged mixture fraction PDF is computed using a β-function dis-
tribution. Values of the mean scalar dissipation rate N0, mean mixture fraction
ξ˜ and its variance ξ˜′′ are evaluated every ∆t = 10−5 s using data from the DNS
solutions between x = L/6 and x = 5L/6 (between x = L/4 and 3L/4 for the
simulations corresponding to higher values of the initial turbulence strength, so
to account for the thinner relative width of the droplet laden layer). Averaged
quantities were not computed using data from the entire domain to avoid an
excessive contribution from those regions where air / fuel mixing does not oc-
cur throughout the whole simulation. Pressure work is evaluated by fitting the
pressure curve obtained from the DNS simulations with a third-order spline, and
then computing its temporal derivative. Chemical source terms are closed at first
order. These approaches for evaluating ∂p
∂t
and 〈 ω˙α| η〉 will be used also for the
other modelling strategies considered in this chapter. Droplet source terms are
neglected.
The strategy for closing the 0D-CMC equations depicted here corresponds to
the standard approach used in simulating multi-dimensional turbulent reacting
two-phase flows with CMC. Hence, it will be used as a reference case for assessing
the capability of CMC to provide reliable predictions for autoigniting sprays.
6.1.2.2 Modelling strategy B
The conditional scalar dissipation rate is obtained by conditionally averaging the
DNS data. Since 〈N | η〉 is now assigned a priori and no account is made for
the droplet source terms, no modelling is required for the mixture fraction PDF.
This strategy corresponds to an ideal CMC simulation of a turbulent reacting
two-phase flow where the mixing of the air stream and the fuel vapor is known
and does not need to be reconstructed from averaged quantities. It is expected
that replacing the modeled profile of the conditional scalar dissipation rate with
the exact one will lead to an improvement of the predictive capabilities of CMC
due to the strong influence that 〈N | η〉 has been shown to have on the ignition
process of both single and two-phase flows. No account is done for droplet related
effects.
168
6.1.2.3 Modelling strategy C
Both the conditional scalar dissipation rate and the Favre-averaged mixture frac-
tion PDF are determined from the DNS simulations. These quantities are then
corrected at ξs following the procedure described in Section 3.2.2.2. To avoid an
excessive contribution from those regions of the flow that are never reached by the
fuel vapor, the set of data used for determining P˜ (η) is the same one described in
Section 6.1.2.1 for computing the mean mixture fraction and its variance. Droplet
source terms are accounted for in the CMC equations, with the conditional evap-
oration rate being modeled according to Equation 3.43. This is reported below
for completeness:
〈Π| η〉 = 1
ρV P˜ (η)
∑
d
m˙dδ(η − ξs,d)
6.1.3 Numerical methods
The terms describing diffusion in mixture fraction space were discretized using a
second-order central differencing scheme. A first-order upwind scheme was used
for the convective term appearing in the expression for the droplet source terms.
The corresponding sets of discretized equations were integrated with the stiff
integrator VODPK [19] using the method of lines. Absolute and relative solver
tolerances were set equal to 10−14 (10−6 for the temperature equation) and 10−6
respectively for all conditions investigated. The time step size was 10−6 s for all
simulations.
6.1.4 Initial and boundary conditions
6.1.4.1 Operating conditions investigated
Three different sets of initial conditions have been chosen for the validation of the
CMC method for spray autoignition presented in this chapter. These correspond
to Cases A, C and F in Table 5.1. For the convenience of the reader, a summary
of these conditions is provided in Table 6.1, together with the corresponding ig-
nition delay time and most reactive mixture fraction. Case A is characterized
by low values of the initial turbulence strength and oxidizer temperature. It is
169
Case: L [mm] Tair [K] XO2 [−] Ret [−] τid [ms] ξMR [−] Φ0 [−]
A 2.1 1000 0.21 19 1.09 0.2 2
C 2.1 1350 0.21 11 0.28 0.045 2
F 2.9 1000 0.21 73 1.03 0.185 4
Table 6.1: Operating conditions, ignition delay time and most reactive mixture
fraction of the Direct Numerical Simulations used for validating the CMC method
for autoigniting sprays.
the baseline simulation for assessing the effect of changes in the initial spray con-
ditions. Case C also corresponds to initially weak turbulence levels in the bulk
phase, but the spray is now igniting in an environment characterized by higher
temperature values. Finally, Case F has the same initial air temperature as Case
A, but the initial turbulence intensity is stronger, and the global equivalence ratio
in the droplet-laden region is higher.
Cases C and F have been chosen since they exhibit several features that could
be challenging to model. In particular, the mixture fraction PDF of Case F
has a bimodal character, with a strong hump in correspondence of the most
reactive mixture fraction, which may be difficult to capture with a β-function
distribution. Additionally, this spray is characterized by higher values of the
mean scalar dissipation rate than Case A, although the opposite is true for the
conditional scalar dissipation rate. Since closure of 〈N | η〉 with the Amplitude
Mapping Closure relies on N˜ , this case then represents a probing test also for the
correct modelling of scalar mixing. Regarding Case C, ignition exhibits here an
Arrhenius-like behavior due to the initial high temperature of the oxidizer. This
implies that proper initialization of the conditional moments, in particular of the
conditional temperature, is crucial in correctly predicting the ignition delay time.
This case thus represents a strong candidate for highlighting the validity of the
methods used in two-phase flows CMC for the initialization of the conditional
moments.
170
6.1.4.2 Initialization of the conditional moments
The approach used for initializing the conditional moments is the same described
in Chapter 4 for the RANS-CMC simulations of the autoigniting diesel sprays.
An adiabatic frozen mixing distribution is assumed for the initialization of the
conditional moments, with each species mass fraction varying linearly between its
value at the oxidizer (e.g. η = 0) and the fuel (e.g. η = 1) side. Consistently with
the DNS simulations, the oxidizer stream corresponds to pure air (YO2 = 0.233,
YN2 = 0.767), while the fuel stream consists of n-heptane only (YC7H16 = 1).
The initial temperature distribution is chosen so to yield a linear variation of the
mixture enthalpy between the oxidizer and fuel streams. The enthalpy of the fuel
stream is evaluated as if the fuel is already in gaseous form, with its temperature
being equal to the initial droplet temperature.
6.1.5 Chemistry
Reaction rates were computed with the same reduced n-heptane kinetic mecha-
nism of Liu et al. [80] that was used in the simulations presented in Chapters 4
and 5. The mechanism consists of 22 non steady-state species, which react ac-
cording to 18 global steps, and was derived from a skeletal mechanism consisting
of 43 species and 185 reactions. Additional information on the mechanism can be
found in Chapter 4, and the interested reader is referred there for further details.
6.2 Results
6.2.1 Modelling of conditional scalar dissipation rate
One of the main results obtained from the numerical simulations presented in
Chapter 5 is that the strong negative correlation between heat release and scalar
dissipation rate that has been observed during the ignition of single-phase flows
[86, 85] also holds for sprays. This finding indicates that proper modelling of the
conditional scalar dissipation rate is crucial in capturing the physics of the igni-
tion process with the Conditional Moment Closure. The zero-dimensional CMC
simulations presented in this chapter make use of either the Amplitude Mapping
171
Figure 6.1: Comparison between the conditional scalar dissipation rate profiles
as obtained using the AMC model, and by conditionally averaging the DNS data.
Data shown for Case A in Table 6.1.
Closure proposed by O’Brien and Jiang [100], or the conditionally averaged DNS
data to provide a closed form for 〈N | η〉. AMC has often been the method of
choice for modelling the conditional scalar dissipation rate in multi-dimensional
CMC simulations of both single [103] and two-phase flows [152, 104, 18] due
to its numerical stability and computational efficiency. Its validity for complex
flow configurations [93] or spray applications [163], however, has often been ques-
tioned. It is therefore of interest to check the extent by which the modeled profiles
of the conditional scalar dissipation rate differ from the DNS ones for the dilute
sprays considered in this chapter, and whether these differences are expected to
affect the numerical predictions of the ignition event or not.
Figure 6.1 compares the modeled 〈N | η〉 profiles against those obtained by
conditionally averaging the DNS data at three different instants of time during
the spray evolution. The data shown refers to Case A. The modeled profiles
closely follow the DNS ones for values of the mixture fraction up to 0.1, with
differences in values between corresponding curves being larger at earlier times.
However, when richer mixture compositions are considered, strong deviations are
observed. This was expected: in fact, it is well known that, in a two-phase flow,
the conditional scalar dissipation rate is a function which is not defined anymore
172
over the whole mixture fraction range (e.g. ξ is now comprised between 0 and
ξmax, with ξmax being a function of time and generally lower than one), and that
its peak does not necessarily occur at ξ = 0.5, as predicted by AMC. One way of
dealing with the first issue would be to normalize the mixture fraction by ξmax,
and then apply the AMC model over the normalized mixture fraction space [163].
This procedure would yield a modeled profile of the conditional scalar dissipation
rate that correctly varies between ξ = 0 and ξ = ξmax; however, the maximum of
the 〈N | η〉 curve would now occur at ξmax/2, which does not always correspond
to the situation shown in Figure 6.1. Additionally, the value of ξmax in RANS or
LES simulations is not known, and there is currently no guideline on how to eval-
uate it. Errors in estimating the conditional scalar dissipation rate may have a
strong impact on the prediction of the ignition event when they occur around the
most reactive mixture fraction, which, for Case A, is equal to ξMR = 0.2. Since
AMC overpredicts the value of 〈N | ξMR〉 at all instants of time considered, it is
expected that the use of this model in the zero-dimensional CMC code in place
of the profiles extracted from the DNS data will result in a delayed ignition event.
The above limitation of AMC in providing the right shape of the conditional
Figure 6.2: Comparison between the conditional scalar dissipation rate profiles
as obtained using the AMC model, and by conditionally averaging the DNS data.
Data shown for Case C in Table 6.1.
173
scalar dissipation rate profile has been encountered for all sprays considered in
this chapter. Figure 6.2, for example, compares the modeled and DNS profiles
of the conditional scalar dissipation rate at selected instants of time for Case C.
AMC provides again a fair approximation of the conditional scalar dissipation
rate for low values of the mixture fraction. However, as higher values of ξ are
considered, differences become evident. The situation is exacerbated here by the
fact that, for this high temperature case, ignition occurs a short time after the
start of evaporation. A direct consequence of this early spray ignition is that the
interval of mixture fraction values over which 〈N | η〉 is non-zero is narrow, since
the time available for evaporation and mixture formation was limited. Despite
differences at high values of the mixture fraction remain always large, the dis-
crepancy observed at ξMR (ξMR = 0.045 here) becomes small after t = 0.12 ms.
This difference, however, is non-negligible at earlier instants of time, and it may
affect the correct prediction of the ignition event.
6.2.2 Modelling of chemical source terms
The main strength of the Conditional Moment Closure for non-premixed combus-
tion lies in the hypothesis of the fluctuations of the reactive scalars around their
conditional averages being small. This assumption allows for a first-order closure
of the chemical source terms appearing in the CMC equations [12, 69]. Although
the correlation existing between the reactive scalars and the mixture fraction is
usually strong for steadily burning diffusion flames, this may not be the case in
the presence of transient phenomena, such as ignition or flame extinction [84].
In these situations, it has been suggested by several authors that a second-order
closure of the chemical source terms may be required [134, 103].
Second-order closures of the CMC equations are costly and difficult to im-
plement, since they require not only evaluation of the conditional covariances of
the reactive scalars, but also modelling of the unclosed terms appearing in their
transport equations. So far, their use has been limited to few selected cases,
mainly for demonstrative purposes, whereas most of the applications of CMC
174
(a) t = 0.90 ms (b) t = 1.05 ms
Figure 6.3: Comparison between the conditional heat release rate term as ob-
tained using the first order closure in CMC, and by conditionally averaging the
DNS data, at two selected instants of time. Data shown for Case A in Table 6.1.
to the study of transient phenomena have successfully relied on first-order clo-
sures [152, 142]. These considerations suggest that it may be worth investigating
whether a first-order closure of the conditionally averaged chemical source terms
is adequate or not for simulating spray autoignition.
Figure 6.3, which refers to case A, compares the conditional heat release rate
term as obtained by conditionally averaging the DNS data, and by using the
first-order CMC closure assumption. In the latter case, one has:
〈 ω˙H | η〉 = −
N∑
α=1
〈hαω˙α| η〉 ' −
N∑
α=1
hα(QT )ω˙α(Qβ, QT ) (6.3)
The conditional moments used to evaluate 〈 ω˙H | η〉 from Equation 6.3 are those
obtained from the DNS data and, as such, they are exact. This implies that
the assessment of the first-order closure assumption is done here under the most
favorable conditions, as the evaluation of the conditional moments is not affected
by any of the modelling choices done in closing the CMC equations. Larger dif-
ferences should be expected when the conditional moments obtained from the
175
(a) t = 0.240 ms (b) t = 0.285 ms
Figure 6.4: Comparison between the conditional heat release rate term as ob-
tained using the first order closure in CMC, and by conditionally averaging the
DNS data, at two selected instants of time. Data shown for Case C in Table 6.1.
numerical solution of the CMC equations are used.
Results indicate that the conditional heat release rate term can be safely eval-
uated with Equation 6.3 only at times which are sufficiently far from the onset of
ignition. As ignition becomes imminent, the deviation between the DNS data and
those obtained with the first-order closure assumption becomes large, especially
in correspondence of the most reactive mixture fraction. This is not surprising,
given the spotty nature of the ignition event described in Chapter 5. Since flow
regions with the same value of the mixture fraction will ignite at different times,
the scatter in the conditional moments will be large (see Figures 5.14 and 5.15),
thus violating the conditions for using a first-order closure. One should note that,
close to ignition, the value of 〈 ω˙H | η〉 at the most reactive mixture fraction is un-
derpredicted when using Equation 6.3; thus, it may be expected that, for Case
A, the zero-dimensional CMC simulations will yield a longer ignition delay time
as compared to the DNS data.
Similar conclusions hold for Case C, for which the corresponding modeled and
176
computed forms of the conditional heat release rate are shown in Figure 6.4. It
is clear that, before the onset of ignition, the first-order closure provides again
a good approximation of 〈 ω˙H | η〉. However, during the thermal runaway of the
system, fluctuations in the reactive scalars become large, and Equation 6.3 is no
longer a good model for the conditional heat release rate, with differences be-
tween the modeled and computed profiles of 〈 ω˙H | η〉 reaching values up to one
order of magnitude. It may now be useful to remember that, for Case C, the
most reactive mixture fraction is ξMR = 0.045, which is close to the stoichiomet-
ric value ξst = 0.062. Since the difference between the temperatures of the cold
and burning flamelets solutions at ξMR is very large and close to its maximum
value, fluctuations of the reactive scalars are also large at ξMR; in particular, they
are expected to be larger than those observed for Case A at its most reactive
mixture fraction. This provides an explanation for the much larger difference ob-
served between the modeled and computed forms of the conditional heat release
rate at ξMR as compared to Case A. One final observation is that, differently from
Case A, the first-order closure now overpredicts 〈 ω˙H | η〉 around ξMR; hence, the
CMC simulations are expected to yield a shorter ignition delay as compared to
the DNS data.
These findings are in line with previous studies on the suitability of first-
order CMC for autoignition problems [103, 120]. Richardson et al. [120] showed
that, in an autoignitive lifted hydrogen flame, first-order closure of the chemical
source terms performed poorly in the stabilization region. Second-order closure,
on the other hand, allowed for an excellent approximation of 〈 ω˙α| η〉. It was also
noted that most of the second order contribution could be reconstructed from
the knowledge of a limited number of covariances of the reactive scalars, in line
with the results of De Paola et al. [103]. Although a second-order closure was not
attempted here, results shown in this section and previous findings suggest that it
should be implemented in future applications of the CMC method to autoignitive
flames.
177
6.2.3 Modelling of mixture fraction PDF
The mixture fraction PDF plays a crucial role in the CMC method to retrieve the
unconditional averages of the reactive scalars from the conditional ones. Addi-
tionally, it is employed in some of the models used for closing the unknown terms
appearing in the CMC equations, such as the Amplitude Mapping Closure for the
conditional scalar dissipation rate. It is thus evident that a correct representation
of the mixture fraction PDF is fundamental in providing accurate predictions of
turbulent reactive flows with CMC.
As already discussed in detail in Chapter 3, modelling of the mixture frac-
tion PDF is commonly done by mean of presumed-shape distributions, with the
β-function distribution being one of the most popular choices. The accuracy of
presumed-shape PDFs in describing the mixture composition in sprays has often
been the subject of debate. Ge and Gutheil [40] performed numerical simula-
tions of turbulent reacting sprays using a flamelet model. They showed that
improved predictions were obtained when the mixture fraction PDF, instead of
being modeled, was computed through the numerical solution of its transport
equation, suggesting that presumed shape models may not be adequate for de-
Figure 6.5: Comparison between the mixture fraction PDF as obtained using a β
distribution, and by processing the DNS data. Data shown for Case A in Table
6.1.
178
scribing mixing in two-phase flows. This speculation has been confirmed recently
by Duret et al. [32], who studied evaporation in non-reacting sprays by mean of
Direct Numerical Simulations with full resolution of the near-droplet field. One
of their major findings was that, in a two-phase flow, the β-function distribution
may not be able to retrieve the correct shape of the mixture fraction PDF even
if corrections for the presence of the gas-liquid interfaces are taken into account,
and that a log-normal distribution appears to be a more appropriate choice for
this class of problems.
Despite the concerns raised about its validity, the β-function distribution is
still widely used in CMC for two-phase flows [67, 152, 18]. Hence, it may be
of interest to check whether this model may provide acceptable profiles of the
mixture fraction PDF for the dilute sprays investigated in this chapter. To this
end, a comparison between the computed and modeled PDF profiles is performed
in Figure 6.5 at two selected instants of time preceding ignition. The data shown
refers to Case A. Similarly to the procedure used for evaluating ξ˜ and ξ˜′′2, only
data between x = L/6 and x = 5L/6 were considered to extract the PDF from the
numerical simulations. The results presented here seem the confirm the worries
previously expressed by other authors about the validity of a β-function distri-
bution for describing air / fuel mixing in sprays; in fact, at any instant of time
considered, the mixture fraction PDF exhibits multiple humps, which cannot be
reproduced with a β distribution. Additionally, the presence of regions in the
flow having rich mixture compositions is always underpredicted.
The severe limitations of the β-function distribution in capturing the correct
PDF shape become more evident when sprays with higher initial values of the
global equivalence ratio are considered. As discussed in Chapter 5, increasing
Φ0 from 2 to 4 accentuates the bimodal character of the PDF, leading to the
occurrence of a strong hump around ξ = 0.2. The β-function distribution is not
able to capture this behavior. This is clearly shown in Figure 6.6, where the
comparison between the data extracted from the numerical simulations and the
modeled PDF profiles is presented for Case F at two selected instants of time.
Errors done in determining the mixture fraction PDF may affect the modelling of
179
Figure 6.6: Comparison between the mixture fraction PDF as obtained using a β
distribution, and by processing the DNS data. Data shown for Case F in Table
6.1.
other quantities, such as the conditional scalar dissipation rate, and hence may
ultimately lead to erroneous predictions of the ignition event. From the results
presented in this section, it is clear that more accurate models for this quantity
are needed in the framework of spray combustion.
6.2.4 Prediction of the ignition event
6.2.4.1 Low temperature cases
The temporal evolution of the conditional temperature, and of the conditional
mass fractions of C7H16, O2, CO, OH and CH2O at ξ = ξMR is shown in Figure
6.7. The data set considered here corresponds to Case A in Table 6.1, for which
ξMR = 0.2. The conditional moments were obtained by conditionally averaging
the DNS data, and from the numerical solution of the zero-dimensional CMC
equations, with the unclosed terms being modeled according to strategies A and
B discussed in Section 6.1.2. Results obtained with modelling strategy C, where
droplet evaporation effects are taken into account in the governing equations, are
identical to those corresponding to strategy B. This finding will be discussed in
more detail later in this section.
180
Figure 6.7: Temporal evolution of selected conditional averages at most reactive
mixture fraction. Data shown correspond to Case A in Table 6.1 (ξMR = 0.2).
The conditional moments were obtained by conditionally averaging the DNS data
and from the numerical solution of the zero-dimensional CMC equations, with
the unclosed terms modeled according to the strategies described in Section 6.1.2.
CMC is seen to overpredict the ignition delay time for all the modelling strate-
gies considered. The error in estimating the onset of ignition is larger when AMC
is used to evaluate the conditional scalar dissipation rate, with τid being approx-
imately 20 % longer than in the DNS simulation. This is not surprising, as this
model was found to provide a poor approximation of the conditional scalar dissi-
pation rate at rich mixture compositions, and in particular around ξMR, with its
value being considerably overpredicted. Use of the 〈N | η〉 profiles extracted from
the DNS data leads to a substantial improvement in the CMC results, highlighting
once more the importance of an accurate modelling of this quantity. Discrepancies
with the DNS solution, however, are still in the order of 10 % (τid ' 1.1 τid,DNS),
and they may be interpreted as a consequence of the poor approximation of the
chemical source terms during the ignition transient provided by the first-order
181
closure assumption. The fact that the current modelling of the droplet source
terms in the CMC equations has a negligible influence on the conditional mo-
ments around the stoichiometric mixture fraction may also partly explain the
observed differences.
Another peculiar difference between the DNS results and the CMC simula-
tions lies in the different nature of the ignition event. A two-phase ignition process
can be identified in the CMC simulations, irrespective of the strategy followed for
closing the CMC equations. This is evident from the temporal evolution of tem-
perature and of some radicals and intermediate species, such as OH and CH2O.
In particular, the OH radical exhibits a double peak, with the former being small
and occurring around t = 1.0 ms, while the latter is large and corresponds to the
ignition event. The temperature and the mass fraction of formaldehyde are seen
to grow up to the first OH peak, after which they exhibit a plateau. Later, they
undergo a sudden increase in correspondence of the second OH peak. Following
ignition, both quantities relax to their respective values for a burning flamelet
solution, with CH2O being rapidly depleted as the flame is established. No two-
stage ignition can be seen from the DNS data. Additionally, the ignition event
is considerably less dramatic than in the CMC simulations; the increase in tem-
perature is more gradual, and the mass fractions of the radical and intermediate
species do not reach values that are too high. It is interesting to note how, de-
spite the ignition event occurring earlier in the DNS simulation than in the CMC
ones, fuel and oxidizer consumption starts to be significant at later times in DNS
than in CMC. Since this behavior is observed also when the conditional scalar
dissipation rate is closed using the DNS data, one may attribute it to the poor
approximation of the chemical source terms provided by the first-order closure
assumption during the ignition transient.
The last point raised in the previous paragraph deserves some more attention.
In fact, the analysis presented in Section 6.2.2 revealed that, if instants of time
which are sufficiently far from the ignition event are considered, the first-order
closure assumption is actually a good model for the chemical source terms ap-
pearing in the CMC equations. The early consumption of the reactants cannot
182
(a) t = 0.2 ms (b) t = 0.4 ms
Figure 6.8: Scatterplots of temperature against mixture fraction at selected in-
stants of time early after the start of evaporation. Red line corresponds to the
conditional temperature profile as obtained with the use of the standard CMC
initialization procedure. Data shown correspond to Case A in Table 6.1.
thus be attributed to poor modelling of the reaction rates. Similarly, it is not
a consequence of the mixing model used, as it occurs also when the conditional
scalar dissipation rate is closed with the profiles obtained from the DNS data.
This indicates that proper initialization of the conditional moments and correct
modelling of the droplet source terms play an important role in the successful
simulation of autoigniting sprays with CMC. A confirmation of this is provided
by Figure 6.8, which shows the conditional temperature obtained from the CMC
simulations and by conditionally averaging the DNS data at selected instants
of time early after the start of evaporation. The data shown refers to Case A.
Differences between the curves are evident, highlighting the importance of prop-
erly accounting for the evaporative cooling of the gaseous phase when simulating
the temporal evolution of the conditional temperature. The higher temperatures
found in the CMC simulations early during the spray development are in fact
responsible for the observed early consumption of the fuel and oxidizer species as
compared to the DNS data, since use of higher values of the conditional temper-
ature is responsible for higher values of the reaction rates.
183
Figure 6.9: Temporal evolution of selected conditional averages at most reactive
mixture fraction. Data shown correspond to Case F in Table 6.1 (ξMR = 0.185).
The conditional moments were obtained by conditionally averaging the DNS data
and from the numerical solution of the zero-dimensional CMC equations, with
the unclosed terms modeled according to the strategies described in Section 6.1.2.
As it has been mentioned earlier, including the droplet source terms in the
CMC equations does not affect the numerical results, provided that the same
model for 〈N | η〉 is used. This is a consequence of the fact that evaporation af-
fects the conditional moments at the saturation mixture fraction only. For the
sprays considered here, the value of ξs corresponding to the initial droplet tem-
perature is 0.54, which is higher than the maximum value of the resolved mixture
fraction reached anywhere in the flow during the numerical simulation. This im-
plies that droplet source terms can affect the conditional moments at the mixture
fraction values relevant for the chemical reactions only if an appropriate model
for the conditional scalar dissipation rate is provided. This model must be able
to describe air / fuel mixing not only in the inter droplet space, but also in the
184
near droplet field, that is in the whole mixture fraction range comprised between
0 and ξs. In the numerical simulations presented in Chapter 4, such a model was
provided by the Amplitude Mapping Closure, which, however, it has been shown
to provide poor approximations for sprays. In general, modelling of 〈N | η〉 in the
near droplet field is an extremely challenging task, and it is expected to have a
strong impact not only on the numerical results, but also on the stability of the
computational algorithm. Given also the scarcity of data available for charac-
terizing the mixture fraction PDF and the conditional scalar dissipation rate at
locations close to the droplet surface, more accurate modelling of these quantities
is not attempted here. Instead, we propose a different initialization procedure of
the conditional temperature, which is able to account for the evaporative cooling
of the gaseous phase due to liquid fuel evaporation. This procedure is detailed in
Appendix B, and is applied to the spray igniting under high temperature condi-
tions described in the next section. It is not used for the low-temperature sprays
studied here since it cannot account for the heating of the liquid fuel droplet,
which is substantial when the initial value of the air temperature is low due to
the correspondingly long ignition delay time. This deficiency, in fact, may lead to
inaccurate predictions, and one may then question whether there is any advan-
tage in changing the CMC initialization procedure when the difference between
the initial and final liquid temperatures is large.
Results obtained for Case F are shown in Figure 6.9. These are qualitatively
similar to those obtained for Case A, and so they will not be discussed in detail.
One point that deserves some attention is that both CMC approaches considered
here correctly yield a shorter ignition delay time with respect to Case A. This
is a consequence of the lower values of the conditional scalar dissipation rate at
the most reactive mixture fraction that characterize this spray. The fact that
CMC with modelling strategy A can correctly capture this trend is surprising,
considering that the unconditional scalar dissipation rate, which is used by the
Amplitude Mapping Closure to construct the conditional scalar dissipation rate
profile, is higher for Case F than for Case A. The explanation for this apparent
discrepancy can be found in the different modeled shapes of the mixture fraction
PDFs for these two sprays, and on the properties of the 〈N | η〉 profiles obtained
185
from AMC. Despite the β-function distribution cannot reproduce the hump in
the mixture fraction PDF around ξMR that has been observed for Case F, it still
predicts a longer PDF tail than for Case A. Now, the profile of the conditional
scalar dissipation rate given by AMC is a monotonically increasing function of
η up to 0.5; this implies that the wrongly predicted PDF shape combines in the
AMC model with the wrongly modeled 〈N | η〉 profile to yield a value of N0 that is
lower for Case F than for Case A, hence explaining the shorter ignition delay time
for the former spray. It is interesting to note how two rather approximate models
for the mixing quantities, when combined together, can still be used successfully
to capture the effects of changes in the operating conditions of the system.
6.2.4.2 High temperature case
The analysis presented in Section 6.2.4.1 for Cases A and F in Table 6.1 is re-
peated here for Case C, which corresponds to a spray igniting at high pressure
in a hot environment. Figure 6.10 shows the temporal evolution of selected con-
ditional moments at ξMR (ξMR = 0.045) obtained by conditionally averaging the
DNS data and by solving the zero-dimensional CMC equations, with the unclosed
terms being modeled according to the strategies described in Section 6.1.2. Dif-
ferently from the situation observed for the sprays igniting at low temperature
conditions, CMC now predicts a shorter ignition delay time as compared to the
DNS data. This is particularly evident from the temporal evolution of the mass
fractions of n-heptane, oxygen and formaldehyde; in fact, the reactants are con-
sumed quicker in the CMC simulations, and CH2O builds up at a faster rate.
These behaviors indicate a faster ignition process. Use of the Amplitude Map-
ping Closure for the modelling of the conditional scalar dissipation rate results
in a delayed ignition event as compared to the case where the DNS profiles are
employed. This is consistent with the high levels of scalar dissipation at ξMR
predicted by AMC early during the mixture formation phase, and underlines
once more the importance of a proper modelling of 〈N | η〉 for simulating spray
ignition successfully. τid is found to be approximately equal to 0.65 τid,DNS when
modelling strategy A is used, and to 0.60 τid,DNS when the CMC equations are
closed according to strategies B and C.
186
Figure 6.10: Temporal evolution of selected conditional averages at most reactive
mixture fraction. Data shown correspond to Case C in Table 6.1 (ξMR = 0.045).
The conditional moments were obtained by conditionally averaging the DNS data
and from the numerical solution of the zero-dimensional CMC equations, with
the unclosed terms modeled according to the strategies described in Section 6.1.2.
A peculiar difference between the DNS data and the CMC predictions, which
was also observed for the low-temperature sprays considered in this chapter, lies
in the quick transition between the inert and burning flamelet solutions in the
CMC simulations once ignition has occurred. This is a direct consequence of
the poor approximation of the chemical source terms provided by the first-order
closure during the ignition transient, as discussed in Section 6.2.2. The lack of
a strong discontinuity in the conditional moments extracted from the DNS data
makes it difficult to identify the instant at which the mixture ignites, if only
averaged information on the reactive scalars are available. In the numerical sim-
ulations presented in Chapter 4, the onset of ignition was identified as the instant
of time at which the unconditional temperature first reached a threshold value of
1400 K at any point in the computational domain. Although results presented in
187
(a) Standard CMC initialization (b) Improved CMC initialization
Figure 6.11: Comparison between standard and improved initialization proce-
dures for the conditional temperature in the CMC method. Data shown corre-
spond to Case C in Table 6.1.
Section 6.2.4.1 suggest that this approach can still be used for determining τid in
sprays igniting at low temperature conditions, this is not the case when the initial
oxidizer temperature is high. Data shown in Figure 6.10 indicate that methods
based on the mass fraction of some ignition precursors, such as formaldehyde, are
more appropriate for this class of problems.
The early prediction of the ignition event in CMC cannot be attributed com-
pletely to the first-order closure of the chemical source terms, as this model is
actually quite accurate at times prior to the formation of a burning kernel. Spray
ignition in an hot environment exhibits an Arrhenius-like behavior, and is thus
very sensitive to the initial conditions, in particular with respect to the initial
value of the temperature. The initialization of the conditional moments that
is routinely used in CMC does not account for the evaporative cooling of the
gaseous phase; in fact, this effect should be accounted for by the droplet source
terms appearing in the CMC equations. As it has already been discussed in detail
in the previous sections, accounting for these contributions in CMC raise several
issues, both from the point of view of the modelling of the mixing quantities and
188
Figure 6.12: Temporal evolution of selected conditional averages at most reactive
mixture fraction. Data shown correspond to Case C in Table 6.1 (ξMR = 0.045).
The conditional moments were obtained by conditionally averaging the DNS data
and from the numerical solution of the zero-dimensional CMC equations, with
the unclosed terms modeled according to strategies C (CMC - Standard) and B
(CMC - Improved) described in Section 6.1.2. The conditional temperature was
initialized using either the standard initialization procedure described in Section
6.1.4.2, or the improved method detailed in Appendix B.
of the numerical stability of the computational algorithm. Instead of being intro-
duced explicitly in the governing equations through the droplet source terms, the
evaporative cooling of the gaseous phase is taken into account here by an ad-hoc
initialization of the conditional temperature, which is described in Appendix B.
A comparison between the standard and improved procedures for initializing
the conditional temperature is shown in Figure 6.11. The DNS data used for
assessing the new method proposed in this chapter were extracted at early times
after evaporation has started, at t = 0.05 ms. It is clear that, if droplet source
189
terms in the CMC equations are not accounted for, the standard initialization of
the conditional moments is responsible for values of the conditional temperature
which are too high, and that will eventually lead to early ignition for the high
temperature spray considered here. Using the improved initialization procedure
solves this issue, as the conditional temperature profile now closely follows the
DNS data.
Results obtained when using the zero-dimensional CMC with the new ini-
tialization procedure for the conditional moments, and with the unclosed terms
being modeled according to strategy B, are shown in Figure 6.12. For compari-
son purposes, CMC results obtained with the standard initialization method and
modelling strategy C for the unclosed terms are also plotted. A considerable
improvement in the CMC predictions can be noticed with the use of the new
initialization procedure; however, ignition now occurs more slowly with respect
to the DNS simulation, as revealed by the temporal evolution of the formalde-
hyde mass fraction at ξMR. One explanation for this behavior, apart from the
uncertainties introduced by a first-order closure of the chemical source terms, is
that the present CMC formulation does not account for the heating of the liq-
uid fuel during spray evaporation. In fact, as the liquid phase is heated up, the
enthalpy content of the fuel vapor that is generated from droplet evaporation is
higher, and leads to the formation of mixture spots that are hotter than those
obtained from the evaporation of the initially cold droplets. One evidence of this
phenomenon is given by the scatter in the temperature data obtained from the
DNS simulations that can be observed in Figure 6.8. This implies that, as the
increase in the droplet temperature starts to become significant, the values of the
conditional temperature of the gas phase will be underpredicted, hence leading to
slower chemical reactions and to a delayed ignition event. Despite the heating of
the liquid phase is only moderate for Case C, it is strong enough for yielding the
observed discrepancy between the DNS data and the CMC simulations, due to
the correspondingly strong sensitivity of the reaction rates to changes in temper-
ature. This also justify why the new initialization procedure should not be used
when the heating of the liquid phase is expected to be substantial, as in Cases
A and F. Note that, since the ignition event does not occur abruptly anymore,
190
an exact quantification of the ignition delay time for the CMC simulation with
improved initial conditions has not been attempted.
6.3 Conclusions
The applicability of the CMC method to the numerical simulation of autoignition
in hydrocarbon sprays has been investigated by exploiting the DNS simulations
presented in Chapter 5. Modelling of the unclosed terms that appear in the CMC
equations was discussed at first. Results showed that standard CMC modelling
of the mixture fraction PDF and the conditional scalar dissipation rate, which
is based on the β-function distribution and the Amplitude Mapping Closure of
O’Brien and Jiang [100], is unable to capture several features of mixing in sprays,
in accordance with the findings of other investigators [163, 32]. In particular,
AMC does not provide an accurate estimate of the conditional scalar dissipa-
tion rate at the most reactive mixture fraction, with the value of 〈N | ξMR〉 being
usually overpredicted. Additionally, the modeled profile of the conditional scalar
dissipation rate has non-zero values over the whole mixture fraction space. This
is unphysical, since the maximum value of the mixture fraction in a spray, ξmax,
is usually lower than one (e.g. P (η) = 0 for η > ξmax), and 〈N | η〉 must be zero in
correspondence of those values of η for which P (η) = 0. Concerning the mixture
fraction PDF, the DNS data revealed the presence of several humps which could
not be captured by the β-function distribution. This finding was recognized as a
potential issue for the accuracy of a numerical simulation, since P (η) is used not
only to retrieve the unconditional averages from the conditional ones, but it also
appears in some of the models used for closing the CMC equations.
Examination of the accuracy of the first-order closure of the chemical source
terms usually employed in the CMC method revealed that, at those instants of
time which are sufficiently far from the onset of ignition, this model provides
a very good estimate of the conditional species production / destruction rates.
However, as a burning kernel is established, fluctuations of the reactive scalars
around their conditional means become large, and the use of a first-order closure
191
would lead to severe errors in estimating the chemical source terms. Discrep-
ancies between computed and modeled source terms were found to be larger for
those values of the most reactive mixture fraction that were closer to the stoi-
chiometric value, due to the correspondingly larger fluctuations of the reactive
scalars. These findings clearly indicate the need for higher order closures of the
chemical source terms in studying spray ignition, at least during the transition
of the reactive scalars from their inert to their burning solutions.
The discussion on the validity of some of the models used in closing the CMC
equations was followed by an assessment of the capability of the CMC method to
successfully simulate autoignition in sprays. The assessment was done by running
zero-dimensional CMC simulations of some of the sprays investigated with DNS
in Chapter 5, and then comparing the results obtained from these simulations
against those extracted from the DNS data. Sprays igniting at conditions corre-
sponding to either low or high values of the oxidizer temperature were considered
in the analysis. For the sprays corresponding to low values of the oxidizer tem-
perature, CMC predicted a two-stage ignition process, while single-stage ignition
was observed from the post-processing of the DNS data. Ignition corresponding
to high values of the oxidizer temperature was found to occur in a single-stage
only for both DNS and CMC; however, the onset of ignition always occurred too
early in the CMC simulations. Use of the AMC model for the conditional scalar
dissipation rate in place of the profiles extracted from the DNS data was always
seen to delay the onset of ignition. This was expected, since AMC overpredicts
the conditional scalar dissipation rate at ξMR for all sprays investigated here. For
the low temperature cases, τid was found to be approximately 120 % and 110 %
the ignition delay found in the DNS simulations when modelling strategy A or
B and C were used respectively. These values were equal to 65 % and 60 % for
the high temperature case. The effect of simultaneous changes in the initial tur-
bulence strength and global equivalence ratio of the spray on the ignition delay
time could be correctly captured by CMC, independently of the model used for
the conditional scalar dissipation rate.
192
Inclusion of the droplet source terms in the CMC governing equations did
not affect the numerical results in the range of mixture fraction values that were
relevant for chemical reactions. Proper modelling of these terms, in fact, requires
accurate modelling of both the conditional scalar dissipation rate and the mix-
ture fraction PDF in the near droplet field, that is for mixture fraction values up
to saturation. The models developed in Chapter 3 provide corrections for these
quantities at ξs only; this implies that, if the global model used for the condi-
tional scalar dissipation rate does not provide a description of scalar mixing up
to ξs, then effects due to droplet evaporation will remain confined at the satu-
ration mixture fraction. In general, it was recognized that modelling of mixing
in the near droplet field is an extremely challenging task, and that numerical
results may be strongly affected by the choice of the model used. In order to cope
with these issues, and to provide an easy and reliable method for incorporating
the effects of evaporative cooling in CMC, a new initialization method for the
conditional temperature was proposed. This procedure cannot account for the
heating of the liquid phase, and it can thus be applied only to sprays character-
ized by a short ignition delay time, for which droplet heating is at most moderate.
The new initialization procedure was implemented in the zero-dimensional
CMC code to simulate spray ignition corresponding to high values of the oxidizer
temperature. A considerable improvement in the CMC predictions was obtained,
although the evolution of the ignition kernel was now found to proceed at a slower
rate than in the DNS simulations. This was linked to the inability of the present
CMC method to account for changes in the liquid fuel temperature, which in
turn is responsible for underpredicting the gas-phase temperature, and hence the
species production / destruction rates.
193

Chapter 7
Conclusions
This dissertation presented a detailed analysis of the physics of spray autoignition
and of its modelling with the CMC method. The theoretical development focused
on the description of the scalar mixing field in correspondence of the surface of
the evaporating droplets, and also on the investigation of the complex interac-
tions existing between evaporation, turbulence and mixing in autoigniting sprays
by mean of Direct Numerical Simulations. The former study is essential for the
correct modelling of droplet source terms in CMC, while the latter is important
not only to understand how autoignition in the presence of liquid droplets occurs,
but also to identify current weaknesses of turbulent combustion models in cap-
turing spray autoignition. Applications consisted in the numerical simulations of
several diesel sprays where CMC with full treatment of the droplet source terms
was performed for the first time, and in the assessment of the applicability of the
CMC method to the numerical simulation of spray autoignition. A summary of
the main conclusions of this dissertation is provided in the next section, followed
by suggestions for research topics that should be explored in future works.
7.1 Summary of the main findings
7.1.1 Prediction of diesel spray autoignition with CMC
The CMC method with full treatment of the droplet source terms in the CMC and
mixture fraction variance equations was used in Chapter 4 to study the ignition of
195
several n-heptane sprays at conditions corresponding to diesel engine combustion.
The configuration investigated was a cubic combustion vessel that was operated
at p = 42.5 bar. The initial ambient gas temperature was Tair = 1000 K, and
the initial ambient oxygen molar concentration was varied between XO2 = 10 %
and XO2 = 21 % to simulate the effect of exhaust gas recirculation. The CMC
equations were closed using standard modelling approaches for the mixture frac-
tion PDF and the conditional scalar dissipation rate; these quantities, however,
were corrected according to the models developed in Chapter 3 to account for the
presence of evaporating droplets in the flow. The flow field was computed using
a commercial RANS-CFD solver. Reaction rates were evaluated with a reduced
kinetic mechanism.
For all cases investigated, ignition occurred at the spray tip, in line with the
experimental findings. The resulting flame then propagated upstream along the
stoichiometric mixture fraction isoline, until it reached its final position. This
was located away from the spray axis. The numerical simulations were able to
provide accurate predictions of the final flame lift-off height for all oxygen dilu-
tions considered. CMC was found to overpredict the ignition delay time, with
the difference between the computed and measured values being larger for higher
oxygen dilutions. It was suggested that one of the causes for this discrepancy
could be identified in the chemical mechanism used. The trend of increasing ig-
nition delay time with decreasing oxygen content in the ambient gas, however,
was correctly reproduced.
Analysis of the source terms appearing in the transport equation for the con-
ditional temperature during flame expansion revealed that a flame propagation
mode holds for the diesel sprays that have been investigated. Flame propagation
was associated with a convective-diffusive balance, while the final stabilization
point involved a convective-reactive balance. The correct capturing of the main
features of diesel sprays indicates the suitability of CMC for studying combustion
processes occurring in compression-ignition engines.
196
Inclusion of droplet source terms in the CMC and mixture fraction variance
equations was found to have a negligible influence on the prediction of the ignition
event and subsequent flame propagation. The final lift-off height and the flame
anchoring mechanism also remained unaffected. These results were interpreted
on the light of the large distance in mixture fraction space between the saturation
mixture fraction, where liquid fuel evaporation occurs, and the stoichiometric and
most reactive mixture fractions, where most of the heat release reactions take
place. It was recognized that there may be situations in which the separation
between ξs and ξst remains small, and that a comprehensive assessment of the
influence of droplet terms on the numerical simulation of autoigniting sprays with
CMC should be based on a wider range of operating conditions.
7.1.2 Numerical investigation of autoignition in sprays
The physics of n-heptane spray autoignition at high pressure was studied by
mean of Direct Numerical Simulations in Chapter 5. The configuration investi-
gated consisted of a cubic domain with a droplet-laden layer located in the middle
of the inhomogeneous x direction. The operating pressure was p = 24 bar, and
the ambient gas temperature was either Tair = 1000 K or Tair = 1350 K. The ef-
fects of several changes in the operating conditions of the spray were investigated,
among which the initial turbulence strength, the global equivalence ratio in the
droplet-laden layer, the initial droplet size distribution and the oxygen dilution
in the ambient gas. Reaction rates were computed with the same reduced kinetic
mechanism for n-heptane that was used in the RANS-CMC simulations presented
in Chapter 4.
The ignition sequences of the sprays corresponding to Tair = 1000 K were sim-
ilar, and they could be split into three distinct phases. These were named inert
mixing phase, pre-ignition phase, and ignition and flame propagation phase. The
inert mixing phase was characterized by droplet evaporation and mixing of the
resulting fuel vapor with the oxidizer. Heat release reactions were initially neg-
ligible, and started to become significant only in the pre-ignition phase, during
which a constant growth in the rate of heat release was observed. This eventually
197
reached a local maximum, which was followed shortly after by an abrupt increase,
corresponding to the onset of autoignition. The appearance of a burning kernel
within the flow marked the beginning of the ignition and flame propagation phase.
The nature of the ignition process was spotty, with hot kernels appearing first at
those locations characterized by a well-defined value of the mixture fraction and
low levels of scalar dissipation throughout the whole numerical simulation. The
ignition delay time was approximately equal to 1.0 ms for all cases investigated.
The negative correlation existing between heat release and scalar dissipation rate
was similar to the one observed by other investigators for purely gaseous mix-
tures; this is a novel finding, and suggests that a strong similarity exists between
ignition in single and two-phase flows. A strong negative correlation between
heat release and scalar dissipation was also found for the spray corresponding to
Tair = 1350 K; however, for this case, there was no inert mixing phase, and the
ignition delay time was considerably shorter due to the Arrhenius dependence of
the reaction rates on temperature, and equal to τid = 0.285 ms.
Changes in the initial turbulence strength and global equivalence ratio were
found to affect the ignition process through corresponding changes in the con-
ditional scalar dissipation rate and mixture fraction PDF. In particular, higher
turbulence levels resulted in faster liquid droplet evaporation, which was respon-
sible for initially higher values of the conditional scalar dissipation rate. At
later times, however, when most of the liquid phase had evaporated, turbulence
promoted better mixing between the fuel vapor and the oxidizer, resulting in
lower levels of the conditional scalar dissipation rate and thus in faster ignition.
Increasing the global equivalence ratio in the droplet-laden layer resulted in a
higher probability of finding flow regions with optimal conditions for igniting
(e.g. ξ = ξMR and low values of the scalar dissipation rate); under these condi-
tions, the ignition process was dominated by the appearance of burning kernels
rather than by flame propagation, resulting in overall faster ignition of the entire
spray flow.
Initializing the spray with an inhomogeneous droplet size distribution resulted
in the formation of a strongly inhomogeneous mixing field, where leaner, well-
198
mixed spots alternated with richer ones that remained rather unmixed during
the whole numerical simulation. This was interpreted as a direct consequence of
the different rates at which droplets of initial different size heat up and evaporate.
Differently from the cases where the spray was initially monodisperse, ignition
was seen to occur simultaneously at several locations in mixture fraction space.
This finding is again novel, and it was interpreted as a sign of ignition occurring
first in the locations initially occupied by the small droplets, whose fast evapora-
tion leads to the formation of well-mixed spots that, by the time of ignition, can
reach compositions different from the one corresponding to the minimum value
of τid.
Dilution of the oxygen content in the ambient gas was accompanied by a large
increase in the ignition delay time, with τid being now equal to 4.2 ms. For this
case, the characteristic evaporation time was much shorter than the reference ig-
nition delay time of the corresponding homogeneous system. This was responsible
for the formation of mixture spots in which very good mixing conditions could be
achieved; however, the composition of these regions became lean with respect to
the most reactive mixture fraction of the homogeneous system shortly after evap-
oration was complete. Ignition was found to occur earlier in the locations whose
composition remained rich enough, and hence closer to the theoretical optimal
value ξMR,0, during the whole pre-ignition phase. These findings suggest that the
ratio between the characteristic droplet evaporation time and the characteristic
ignition delay time may play a key role in controlling the onset of ignition in
sprays.
7.1.3 Applicability of the CMC method for studying spray
autoignition
The applicability of the CMC method to the numerical simulation of autoigni-
tion in hydrocarbon sprays has been studied in Chapter 6 by exploiting the DNS
simulations presented in Chapter 5. The validity of some of the most popular
modelling approaches for closing the CMC equations was investigated at first in
199
the context of spray autoignition. Then, results obtained from zero-dimensional
CMC simulations of several autoigniting sprays were compared against the DNS
data to determine whether CMC can successfully predict the ignition event or not.
The analysis revealed that standard models for the mixture fraction PDF and
the conditional scalar dissipation rate, such as the β-function distribution and
the Amplitude Mapping Closure, fail at capturing several features of mixing in
sprays. In particular, AMC was found to overpredict the scalar dissipation at the
most reactive mixture fraction for all conditions investigated, and to yield non-
zero values of 〈N | η〉 also in correspondence of those values of η for which P˜ (η)
was null. The β-function distribution could not capture the multiple humps of the
mixture fraction PDF extracted from the DNS simulations. This was recognized
as a potential source of inaccuracies in the CMC method, given the key role that
this quantity plays in retrieving the unconditional moments from the conditional
ones, and in constructing the submodels used to close the CMC equations. An
a priori analysis of the first-order closure of the chemical source term revealed
that this model is accurate at times which are sufficiently prior to the onset of
ignition. However, as a burning kernel is established, fluctuations of the reactive
scalars around their conditional means become large, and a second-order closure
of the reaction rates should be sought instead.
Zero-dimensional CMC simulations were not able to capture the ignition de-
lay time correctly. For the sprays igniting at conditions corresponding to low
values of the ambient gas temperature, τid was overpredicted, while it was un-
derpredicted for the case where ignition occurred in hot air. The choice of the
model used for the scalar dissipation rate had a strong impact on the numerical
results, highlighting once more the importance of an accurate modelling of this
quantity to obtain reliable predictions. For the low-temperature sprays, CMC
predicted a two-stage ignition event, differently from the DNS simulations, where
single-stage ignition was observed. Additionally, the transition to the burning
solution following ignition occurred abruptly in CMC, while it was more gradual
in DNS.
200
Inclusion of droplet source terms in the CMC equations did not affect the
numerical predictions, since no model was provided to describe scalar mixing in
the near-droplet field. Given the formidable challenge of characterizing the flow
field in the proximity of the droplet surface, and in order to provide a viable way
of incorporating the effects of evaporative cooling of the gaseous phase in CMC, a
new initialization procedure for the conditional temperature was proposed. This
method takes into account the amount of heat that has been transferred from
the gaseous to the liquid phase; however, it cannot account for the heating of the
fuel droplet and, as such, it should only be applied to those problems where the
characteristic ignition delay time is much shorter than the corresponding droplet
evaporation time. The new initialization procedure was applied to the spray
igniting at conditions corresponding to high values of the air temperature. A
considerable improvement with respect to the standard initialization method was
obtained, indicating that droplet evaporation effects must be captured in order
to successfully simulate spray autoignition.
7.2 Suggestions for future work
The work presented in this dissertation has shown some of the capabilities and
limitations of first-order CMC applied to the numerical simulation of autoigniting
sprays. The combination of a RANS method with CMC was able to capture the
main features of diesel spray flames, at least from a qualitative point of view.
Quantitative predictions, however, could not always be obtained, and this was
attributed to poor modelling of the unclosed terms appearing in the CMC equa-
tions.
The conditional scalar dissipation rate was identified as a quantity of fore-
most important for capturing the ignition delay time of the spray. Modelling
of this term in CMC, however, is based on simple closures originally derived for
homogeneous configurations, and is often found to be very inaccurate. A con-
siderable effort should thus be put on the characterization of scalar mixing in
sprays for a wide range of operating conditions. These should include variations
201
in the global equivalence ratio, initial droplet size distribution, initial turbulence
intensity of the bulk phase, and characteristic evaporation time of the droplets.
In fact, strong variations in the behavior of the conditional scalar dissipation rate
are expected as the spray evaporates under different initial conditions. Given
the current limitations of experimental techniques, this analysis should be done
by mean of Direct Numerical Simulations, preferably with full resolution of the
near-droplet field.
First-order closure of the chemical source terms failed at providing a good
approximation of the reaction rates during the onset of ignition. This result im-
plies that use of higher order CMC should be investigated for spray autoignition,
as well as its computational practicality. In particular, strategies for reducing
the computational expense and the modelling complexities of high order closures
of the chemistry terms should be sought. Results presented in this dissertation
showed that, at times which are sufficiently far from the ignition event, the first-
order closure provides a fair approximation of the chemical source terms. This
suggests that solution of the higher order conditional moments may be required
only in a limited interval of time during the ignition event, thus allowing for a
potential reduction in the computational resources required. It is clear, however,
that the feasibility of such an approach, although appealing, should be investi-
gated in more detail.
Effects due to the presence of evaporating droplets in the flow on combus-
tion were found to be strong, especially for ignition occurring at high values of
the oxidizer temperature. Capturing these effects requires not only inclusion of
the droplet source terms in the CMC equations, but also proper modelling of
scalar mixing processes occurring in the near-droplet field. In fact, evaporation
terms have a direct effects on the conditional moments at the saturation mixture
fraction only. This implies that, if the model used for the conditional scalar dis-
sipation rate is unable to describe the mixing field in the proximity of the droplet
surface, then heat release reactions, which occurs at conditions close to the stoi-
chiometric ones, will not be affected by the presence of the evaporating droplets
in the flow. Given the formidable challenge of describing phenomena occurring
202
in the proximity of the droplets, for which the data currently available for mod-
elling purposes is scarce, it could be worth investigating alternative approaches
for incorporating spray effects in CMC. To this scope, an improved method for
initializing the conditional temperature has been described in this dissertation.
Despite this new procedure needs several improvements and can only be applied
to a restricted range of operating conditions, its use has shown a considerable
improvement in the numerical predictions, and it can thus act as a starting point
for the development of better, more comprehensive strategies to account for the
effects of droplet evaporation on the bulk ambient gas.
Finally, concerning the investigation of the physics of spray autoignition, the
analysis presented in this dissertation only considered briefly the effect that a
dilution in the oxygen content of the ambient gas has on the ignition process. Due
to the increasing interest on combustion modes that heavily rely on high dilutions
of the oxidizer to achieve low emissions, such as HCCI or Mild combustion, a more
detailed numerical study on this subject should be performed. In particular, the
complex interactions existing between evaporation, mixing and chemical reactions
in sprays igniting under dilute conditions have to be unveiled, especially for those
situations where the characteristic times of the main phenomena of interest, that
is turbulence, evaporation, and ignition, are similar.
203

Appendix A
Evaluation of mixture fraction at
saturation conditions
The fuel mass fraction at the droplet surface, Y sF , can be determined once the
droplet temperature is known using the Clausius-Clapeyron equation. It reads:
Y sF =
1
1 +
[(
p
ps
− 1
)
Wox
WF
] (A.1)
In the absence of chemical reactions in the flow, ξs and Y
s
F are equal. This is no
longer the case when the droplet evaporates in a medium where fuel consumption
has been substantial, such as in the burnt products of a diffusion flame. If such a
situation arises, ξs can be determined by exploiting the CMC solution. The con-
ditional fuel species mass fraction, in fact, gives a functional dependence between
the mixture fraction and YF . ξs then corresponds to the solution of the following
non-linear equation:
〈YF | ξ〉 = Y sF (A.2)
A graphical sketch of this method is shown in Figure A.1, where ξs is determined
for a non-boiling droplet evaporating in a region filled with burnt gases. This
205
Figure A.1: Evaluation of mixture fraction at saturation ξs for droplets evapo-
rating in flow regions where substantial chemical reactions have occurred.
figure also reveals the existence of a range of values of Y sF for which the assump-
tion ξs = Y
s
F does not clearly hold. The extent of this range depends on how far
chemical reactions have advanced at the location where the droplet is found.
One should note that the dependence between ξs and Y
s
F is not necessarily
univocal. This is again shown in Figure A.1 for the case of evaporation occurring
in a flow region where ignition is occurring. If such a situation arises, ξs is
obtained by computing the solutions of Equation A.2, and then taking the one
with the highest value. In fact, being the sources of the fuel vapor, droplets are
always expected to be found in the regions with the richest mixture composition.
206
Appendix B
Initialization of the conditional
temperature in sprays
The effects of evaporative cooling on the temperature of the gaseous phase can
be taken into account in the CMC method with a proper initialization of the con-
ditional temperature, without having to include the droplet source terms in the
CMC equations. The original initialization procedure described in this appendix
has its starting point in the droplet source term for the gaseous-phase energy
equation. This term was previously given in Equation 2.20, and is reported be-
low for the convenience of the reader:
ΓE = − 1
V
∑
d
αd
(
cLPmd
T (xd, t)− Td
τTd
+
dmd
dt
hF (Td) +
1
2
dmdv
2
d,i
dt
)
(B.1)
This quantity is made up of three contributions. The first two account for the
amount of heat transferred from the gaseous to the liquid phase through convec-
tion / diffusion, and for the change in enthalpy of the mixture due to evaporation
of the liquid fuel. The last term represents the contribution to the internal en-
ergy of the bulk phase due to the kinetic energy of the evaporated fuel; it is
usually several order of magnitude lower than the other quantities appearing in
207
Equation B.1, and can be safely neglected [94]. The expression for ΓE can be
manipulated in such a way that the droplet mass can be rewritten in terms of
its evaporation rate. In fact, starting from the definitions of the characteristic
droplet evaporation and heating times given by Equations 2.16 and 2.17, which
are again reported below for the ease of reading:
τ pd =
ρLa
2
d
4Shc
PrLef
µf
1
ln (1 +Bm,d)
(B.2)
τTd =
ρLa
2
d
6Shc
PrLef
µf
BT,d
ln (1 +Bm,d)
cLP
cFP
(B.3)
it is straightforward to show that:
τTd =
2
3
τ pd
BT,dc
L
P
cFP
(B.4)
Substitution of the above expression into Equation B.1 gives:
ΓE = − 1
V
∑
d
αd
(
2
3
md
τ pd
cFP
T (xd, t)− Td
BT,d
+ m˙dhF (Td)
)
(B.5)
We now turn our attention to Equation 2.11, which describes the temporal evo-
lution of the droplet diameter and is written below for the benefit of the reader:
da2d
dt
= −a
2
d
τ pd
(B.6)
Since md = pia
3
d/8, this equation can be rewritten as:
m˙d = −3
2
md
τ pd
(B.7)
Combining the above expression with Equation B.7 then yields:
ΓE =
1
V
∑
d
αd
dmd
dt
(
cLP
T (xd, t)− Td
BT,d
− hF (Td)
)
(B.8)
208
We now assume spatial homogeneity of the internal energy and mixture fraction
fields. Additionally, we ask that all droplets within the computational domain
have the same temperature and Spalding number for heat transfer. As a conse-
quence of the first hypothesis, the spatial derivatives appearing in the governing
equations for E and ξ vanish. The second assumption allows one to bring the
quantity cLP (T (xd, t)− Td) /BT,d − hF (Td) appearing in Equation B.8 out of the
summation sign. Starting from Equations 2.3 and 2.29, we thus obtain:
dξ
dt
=
1
ρV
∑
d
dm˙d
dt
(B.9)
dE
dt
=
dξ
dt
(
cLP
T (xd, t)− Td
BT,d
− hF (Td)
)
(B.10)
Equation B.10 can now be used to initialize the conditional temperature by elim-
inating the temporal dependence from it and performing an integration between
ξ = 0 and the generic mixture fraction value ξ = η. This would yield the change
in internal energy of the bulk phase corresponding to the formation of an homo-
geneous mixture whose composition is ξ = η. The change in temperature can
then be computed from the knowledge of the specific heat at constant volume of
the gaseous phase. In its practical implementation, the conditional temperature
is initialized by performing step by step integrations between consecutive grid
nodes in mixture fraction space. For example, for the temperature value at the
i+ 1-th node, we have:
Ti+1 = Ti +
∆η
cv
(
cLP
Ti − Td
BT,d(ηi)
− hF (Td)
)
(B.11)
where the value of the conditional temperature at node i was used to compute the
heat transferred between the gaseous and liquid phases. For i = 1, Ti corresponds
to the initial temperature of the ambient gas Tair.
209

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