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Direct numerical simulation of turbulent,
chemically reacting flows
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Jeffrey Joseph Doom
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Krishnan Mahesh, Advisor
January, 2009
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c Jeffrey Joseph Doom 2009
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Acknowledgments
The person I would like to thank first is my adviser Professor Krishnan Mahesh.
His experience, intelligence, and patience were instrumental. He provided a
great graduate experience for which that I am very grateful.
I am most indebted to my wife, Sarah for her endless support. I would like
to thank my Mother, Eileen Doom and Sarahs Parents, Roger and Deb Haar
for their incredible patience and support. I would like to thank my children,
Kaitlyn and Jonathan. They bring so much joy and happiness to my life. Next,
I thank Dr. Ken Murphy. He was the one that encouraged me to study science
and mathematics. I would also like to thank the professors and students at the
Aerospace Engineering and Mechanics Department for providing an enjoyable
place to work. In particular, I would like to thank Mr. Michael Mattson,
Dr. Suman Muppidi and Dr. Yucheng Hou for their enjoyable conversations.
Finally, I thank my deceased father, Ronald Doom. My father taught me the
most important skills of all, good work ethic and patience.
This work was supported by the AFOSR under grant FA95500410341
and by the Doctoral Dissertation fellowship awarded by the Graduate School
at the University of Minnesota. Computing resources were provided by the
Minnesota Supercomputing Institute, the San Diego Supercomputing Center
and the National Center for Supercomputing Applications.
i
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To my wife, children, and parents
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Abstract
This dissertation: (i) develops a novel numerical method for DNS/LES of com-
pressible, turbulent reacting flows, (ii) performs several validation simulations,
(iii) studies autoignition of a hydrogen vortex ring in air and (iv) studies a
hydrogen/air turbulent diffusion flame.
The numerical method is spatially non-dissipative, implicit and applicable
over a range of Mach numbers. The compressible NavierStokes equations are
rescaled so that the zero Mach number equations are discretely recovered in
the limit of zero Mach number. The dependent variables are colocated in
space, and thermodynamic variables are staggered from velocity in time. The
algorithm discretely conserves kinetic energy in the incompressible, inviscid,
nonreacting limit. The chemical source terms are implicit in time to allow
for stiff chemical mechanisms. The algorithm is readily applicable to complex
chemical mechanisms. Good results are obtained for validation simulations.
The algorithm is used to study autoignition in laminar vortex rings. A
nine species, nineteen reaction mechanism for H2/air combustion proposed
by Mueller et al. [37] is used. Diluted H2 at ambient temperature (300 K)
is injected into hot air. The simulations study the effect of fuel/air ratio,
oxidizer temperature, Lewis number and stroke ratio (ratio of piston stroke
length to diameter). Results show that autoignition occurs in fuel lean, high
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Abstract iv
temperature regions with low scalar dissipation at a most reactive mixture
fraction, MR (Mastorakos et al. [32]). Subsequent evolution of the flame is
not predicted by MR; a most reactive temperature TMR is defined and shown
to predict both the initial autoignition as well as subsequent evolution. For
stroke ratios less than the formation number, ignition in general occurs behind
the vortex ring and propagates into the core. At higher oxidizer temperatures,
ignition is almost instantaneous and occurs along the entire interface between
fuel and oxidizer. For stroke ratios greater than the formation number, ignition
initially occurs behind the leading vortex ring, then occurs along the length
of the trailing column and propagates towards the ring. Lewis number is
seen to affect both the initial ignition as well as subsequent flame evolution
significantly. Nonuniform Lewis number simulations provide faster ignition
and burnout time but a lower maximum temperature. The fuel rich reacting
vortex ring provides the highest maximum temperature and the higher oxidizer
temperature provides the fastest ignition time. The fuel lean reacting vortex
ring has little effect on the flow and behaves similar to a nonreacting vortex
ring.
We then study autoignition of turbulent H2/air diffusion flames using the
Mueller et al. [37] mechanism. Isotropic turbulence is superimposed on an
unstrained diffusion flame where diluted H2 at ambient temperature interacts
with hot air. Both, unity and nonunity Lewis number are studied. The results
are contrasted to the homogeneous mixture problem and laminar diffusion
flames. Results show that autoignition occurs in fuel lean, low vorticity,
high temperature regions with low scalar dissipation around a most reactive
mixture fraction, MR (Mastorakos et al. [32]). However, unlike the laminar
flame where auto-ignition occurs at MR, the turbulent flame autoignites over
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Abstract v
a very broad range of around MR, which cannot completely predict the onset
of ignition. The simulations also study the effects of three-dimensionality. Past
twodimensional simulations (Mastorakos et al. [32]) show that when flame
fronts collide, extinction occurs. However, our three dimensional results show
that when flame fronts collide; they can either increase in intensity, combine
without any appreciable change in intensity or extinguish. This behavior is
due to the threedimensionality of the flow.
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Table of contents
Acknowledgments i
Dedication ii
Abstract iii
Table of contents vi
List of tables ix
List of figures x
Nomenclature xiii
1 Introduction 1
1.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Autoignition of hydrogen vortex ring reacting with hot air . . 3
1.3 Autoignition in turbulent H2/air diffusion flames . . . . . . . 6
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Numerical method 11
2.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Implicit source term . . . . . . . . . . . . . . . . . . . 18
2.2.2 Zero Mach number limit . . . . . . . . . . . . . . . . . 23
3 Validation simulations 25
3.1 Simple chemistry . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.1 Laminar premixed flame . . . . . . . . . . . . . . . . . 26
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TABLE OF CONTENTS viii
A.1.2 Momentum . . . . . . . . . . . . . . . . . . . . . . . . 87
A.1.3 Species . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
A.1.4 Equation of state . . . . . . . . . . . . . . . . . . . . . 88
A.1.5 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B Flamelet modeling 92
Bibliography 97
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List of Tables
3.1 Parameters for laminar diffusion flame with onestep chemistry. 29
3.2 Parameters for turbulent diffusion flame with onestep chemistry. 34
3.3 Parameters for diffusion flame with onestep chemistry. . . . . 35
3.4 Parameters for laminar diffusion flame. . . . . . . . . . . . . . 37
3.5 Simulation parameters for laminar jet flame. . . . . . . . . . . 38
3.6 Simulations parameters for lifted jet flame. . . . . . . . . . . . 39
4.1 Inlet boundary conditions for reacting vortex ring. . . . . . . . 44
4.2 Mueller Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Global results for homogeneous mixture problem. . . . . . . . 51
4.4 Parameters used in reacting vortex ring simulations. . . . . . 51
4.5 Global behavior of reacting vortex ring. . . . . . . . . . . . . . 56
5.1 References variables for laminar H2/air diffusion flame. . . . . 69
ix
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List of Figures
2.1 Storage of variables. . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Variation of acoustic CFL and number of outer loop iterations
with reference Mach number. . . . . . . . . . . . . . . . . . . 22
3.1 Comparison of computed temperature to analytical solution for
a laminar premixed flame and spatial order of accuracy. . . . . 26
3.2 Comparison of asymptotic solution to numerical solution for a
laminar diffusion flame. . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Contour plot of temperature for a jet flame and center line pro-
file of mixture fraction. . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Scalar dissipation rate and reaction rate for turbulent diffusion
flame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5 Isosurface plot of the reaction rate for turbulent diffusion flame
and contour plot of temperature showing entire computational
domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.6 Scatter plot of fuel mass fraction and oxidizer mass fraction. . 34
3.7 Comparison between Chemkin and present algorithm for a wellstirred reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.8 Grid convergence study showing profiles of a major species and
a minor species. . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.9 Time versus maximum temperature and heat release for a lam-
inar diffusion jet flame. . . . . . . . . . . . . . . . . . . . . . . 39
3.10 Contours of a laminar diffusion jet flame. . . . . . . . . . . . . 40
3.11 Contour plots of a lifted jet flame. . . . . . . . . . . . . . . . . 41
4.1 Schematic of reacting vortex ring. . . . . . . . . . . . . . . . . 43
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LIST OF FIGURES xi
4.2 Comparison using onestep chemistry to Hewett and Madnia
and a comparison of a major species & temperature between
Chemkin and present algorithm for a wellstirred reactor using
the Mueller mechanism. . . . . . . . . . . . . . . . . . . . . . 45
4.3 Grid convergence study for a twodimensional laminar reacting
vortex ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Profile of reaction rates for the species H versus time and tem-
poral evolution of temperature with all reactions, and reactions
1, 2, 3, 9, 10 & 11. . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5 Mixture fraction versus chem for varying temperature and mix-
ture fraction versus chem for varying fuel/air ratio for the ho-
mogeneous mixture problem. . . . . . . . . . . . . . . . . . . . 49
4.6 Domain average of rates of reactions 1,2,3,9,10 and 11 for the
species H in our reacting vortex ring. . . . . . . . . . . . . . . 52
4.7 Evolution of autoignition for the reacting vortex ring. . . . . 53
4.8 Comparison between MR and TMR. . . . . . . . . . . . . . . . 54
4.9 Time evolution of heat release. . . . . . . . . . . . . . . . . . . 584.10 Contours of heat release and velocity vectors for nonuniform
and uniform Lewis number. . . . . . . . . . . . . . . . . . . . 59
4.11 Contours of heat release, temperature, and velocity vectors for
L/D = 2 and L/D = 6. . . . . . . . . . . . . . . . . . . . . . . 60
4.12 Time evolution of temperature. . . . . . . . . . . . . . . . . . 62
4.13 Maximum profiles of heat release, temperature and vorticity
magnitude versus time. . . . . . . . . . . . . . . . . . . . . . . 64
5.1 Schematic of initial conditions for the turbulent diffusion flame. 685.2 Comparison between Chemkin and present algorithm for a well
stirred reactor. Also, grid convergence study of laminar diffu-
sion flame. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Scatter plot of the laminar diffusion flame with unity and non
unity Lewis number mixture fractions versus passive scalar. . . 71
5.4 Evolution of mixture fraction versus heat release and temporal
evolution of maximum heat release and temperature for the
laminar diffusion flame. . . . . . . . . . . . . . . . . . . . . . . 72
5.5 Evolution of temperature from a nondimensional time of 0 to 4. 73
5.6 Evolution ofYHO2 from a nondimensional time of 0 to 4. . . . 74
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LIST OF FIGURES xii
5.7 Evolution ofYOH from a nondimensional time of 0 to 4. . . . 74
5.8 Contours of n, T, , , YHO2, and || for the turbulent H2/airdiffusion flame. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.9 Plot of mixture fraction () versus temperature in Kelvin (T). 76
5.10 Contour lines of mixture fraction and temperature superposed
on shaded contours of reaction rate at different instants of time. 78
5.11 Comparison of temperature for unity Lewis number and non
unity Lewis number. . . . . . . . . . . . . . . . . . . . . . . . 79
5.12 Effect of grid resolution for a turbulent diffusion flame. . . . . 80
5.13 Time evolution of flame fronts. . . . . . . . . . . . . . . . . . . 82
B.1 Comparison of a major species & temperature between Chemkin
and flamelet code for a wellstirred reactor. . . . . . . . . . . 95
B.2 Temporal evolution of mixture fraction versus T, , YOH, and
YNO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
B.3 Contours of||, T, YH2O, and YH. . . . . . . . . . . . . . . . 97
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Nomenclature
x, y, z Coordinate axes, coordinates
Density
u, v, w Velocity in the x, y and z directions
Yk Mass fraction of species k
p Pressure
T Temperature
t Time
V Volume
A Area
E Total energy per unit mass
Re Reynolds number
Sc Schmidt numberLe Lewis number
P r Prandtl number
Dynamic viscosity
W Mean molecular weight
xiii
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Nomenclature xiv
Ru Universal gas constant
Dk Diffusion coefficient of the kth species
cp Specific heat at constant pressure
cv Specific heat at constant volume
Qdk Heat of reaction per unit mass
k Mass reaction rate for the kth
speciesn Heat release
Subscripts and Superscript
( )k kth species
( )r Reference variable
( )face Faces of control volume
( )cv Control volume( )d Dimensional value
Abbreviations
DNS Direct Numerical Simulation
LES Largeeddy Simulation
RANS Reynoldsaveraged NavierStokes
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Chapter 1
Introduction
1.1 Numerical method
The direct numerical simulation (DNS) and large-eddy simulation (LES) of
turbulent reacting flows are extremely challenging. Combustion involves a
large number of chemical species, and associated chemical reactions. Different
chemical reactions possess different time-scales, and different reaction zone
thicknesses. Turbulence introduces its own range of length and time-scales.
Furthermore, the nonlinear nature of turbulence can cause errors at the smaller
scales to corrupt the large scales of the solution. Turbulent combustion can
occur at very low Mach numbers (e.g. gas-turbine combustors at low pressures)
or very high Mach numbers (e.g. scramjets). Very low Mach numbers imply
numerical stiffness because of a large difference between the speed of sound
and flow velocities, while very high Mach numbers result in shock waves, and
their attendant problems. Desirable requirements for an algorithm to perform
DNS/LES of turbulent reacting flows are therefore: (i) to handle high Reynoldsnumbers robustly and accurately, without the use of numerical dissipation,
1
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1.1. Numerical method 2
(ii) to handle acoustic stiffness efficiently, (iii) to deal with chemical stiffness,
and combustion-induced large spatial gradients, and (iv) to deal with shock
waves/detonation. This dissertation proposes an approach that deals with
requirements (i) through (iii).
Most DNS/LES of turbulent reacting flows appear to either use the com-
pressible NavierStokes equations with Pade spatial discretization, and explicit
time-advancement (e.g. Poinsot [11], Trouve [50], Lele [27], & Pantano [39]),
or the zero Mach number equations along with a pressure-projection approach
(e.g. Majda & Sethian [31], Montgomery & Riley [36], Rutland & Ferziger [43]
[44], Pember et al [40], and Mahesh et al. [30]). A secondorder, staggered,
implicit compressible algorithm was proposed by Wall et al. [53]. However, the
Pade schemes become unstable at high Reynolds numbers; when explicit time-
advancement is used, they require very small time-step at low Mach numbers,
and for stiff chemical mechanisms. The zero Mach number equations are very
efficient at low Mach numbers because they analytically project acoustic waves
out; also along with implicit time-advancement they can resolve chemical stiff-
ness efficiently. However, due to the complete absence of acoustic effects, they
are not applicable to finite Mach number flows.
The approach proposed in this dissertation is an extension of the algorithm
developed by Hou & Mahesh [23] for compressible flows without chemical re-
action. The Hou & Mahesh algorithm is colocated in space, symmetric in
both space and time, and hence non-dissipative. Motivated by the theoretical
work of Thompson [49], and similar to the non-dimensionalization used by Bijl
& Wesseling [1], and van der Heul et al. [52]. The NavierStokes equations
are non-dimensionalized using an incompressible scaling for pressure, and the
energy equation is interpreted as an equation for the divergence of velocity. A
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1.2. Autoignition of hydrogen vortex ring reacting with hot air 3
pressurecorrection approach is used to constrain the divergence of the velocity
field to satisfy the energy equation. The resulting system of equations ana-
lytically projects acoustic waves out, in the limit of zero Mach number. The
discrete equations are fully implicit, and are constrained to discretely conserve
kinetic energy in the limit of incompressible, inviscid flow. These features
make the algorithm stable and accurate at high Reynolds numbers, and effi-
cient and accurate at very low Mach numbers. Hou & Mahesh [23] show results
for one-dimensional acoustic waves, a periodic shock tube, the incompressible
Taylor problem, and inviscid isotropic turbulence on very coarse grids. We
extend the Hou & Mahesh [23] algorithm to include chemical reaction.
1.2 Autoignition of hydrogen vortex ring re-acting with hot air
The motion of a fluid column of length L through an orifice of diameter D
produces a vortex ring. Nonreacting vortex rings have been studied by many
researchers, and a large volume of work exists (see e.g. the review by Shariff
and Leonard [47]). Less is known about reacting vortex rings. This dissertation
is motivated by the relevance of vortex rings to fuel injection, and uses direct
numerical simulation (DNS) to study the flow generated by a piston pushing
a ring of gaseous fuel into hot oxidizer.
Past work on nonreacting vortex rings (Maxworthy [33], [34]; Didden [12];
Glezer [18]) has studied the process of vortex ring formation, their kinematics,
and the temporal evolution of vortex circulation. A notable finding is that
of Gharib et al [17] who show that a vortex ring achieves its maximum cir-
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1.2. Autoignition of hydrogen vortex ring reacting with hot air 4
culation at the critical stroke ratio (termed formation number) across which
the flow transitions from a toroidal vortex ring to a vortex ring with trailing
column of vorticity. Here, the stroke length ratio is defined as the ratio of the
piston stroke length to the nozzle exit diameter. Sau and Mahesh [46] show
how the stroke length affects entrainment and mixing. They show that en-
trainment is highest at the formation number, and decreases when the trailing
column forms. They explain this behavior by showing that a toroidal vor-
tex ring entrains ambient fluid efficiently, but the trailing column does not.
As the stroke length increases towards the formation number, the vortex ring
remains toroidal, but increases in size and circulation, which increases entrain-
ment. When the stroke ratio exceeds the formation number, the leading vortex
ring remains the same, and the trailing column increases in length. The frac-
tional contribution of the leading vortex ring to overall entrainment therefore
decreases. Since the trailing column does not entrain ambient fluid efficiently,
the overall entrainment drops.
Chen et al. [5][7] have performed experiments on reacting vortex rings of
(pure/ nitrogen diluted) propane and ethane issuing into air. The fuel and oxi-
dizer are initially at the same temperature (300 K) in their experiments, and a
pilot flame is used to initiate combustion. Luminosity measurements are used
to provide data on flame shape and height. The fuel volume and ring circu-
lation are varied in a controlled manner. The nozzle exitvelocity is assumed
tophat (U0), and ring circulation is estimated from hotwire measurements
as = U20 t where t is the duration of the pulse. The fuel volume is esti-
mated as VF = U0tA where A is the nozzle crosssectional area. Chen et al.
also perform simulations of methane/air vortex rings using a steady laminar
flamelet approach, where the chemical species and temperature are functions
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1.2. Autoignition of hydrogen vortex ring reacting with hot air 5
of the conserved scalar and scalar dissipation rate. The speed of the vortex ring
is seen to initially increase, and then decrease at later times. Dilatation due
to heat release is shown to be responsible for this behavior. The experiments
show that as the fuel volume or circulation is changed, the flame structure and
burnout time change significantly. The formation number (which they term
the overfill limit) does not significantly change from the nonreacting case.
Fuel volume which exceeds the overfill limit is seen to decrease heat release
and ring speed. Also, the burnout time and flame shapes are quite different
for ethane and propane under similar conditions although the stoichiometry
and diffusivity are nearly identical. Nitrogen dilution (for propane) decreases
burnout time and flame luminosity, but does not appreciably change flame
structure.
Simulations of reacting vortex rings in a slightly different configuration
were performed by Hewett & Madnia [19] and Safta & Madnia [45]. Hewett
& Madnia solved the axisymmetric compressible NavierStokes equations and
used a onestep Arrhenius mechanism to model combustion of a hydrocarbon
in air. No pilot flame exists; instead the vortex ring of fuel issues into hot
oxidizer. The stroke ratio of the vortex ring is 2 for all cases considered. The
simulations study the effect of oxidizer temperature which is controlled such
that ignition occurs either when the vortex rings forms, or occurs after the
vortex ring has fully formed. They find that at higher oxidizer temperatures,
most of the reaction occurs in front of the vortex ring, while at lower temper-
atures, most of the reaction occurs inside the ring. The heat released during
reaction is shown to affect the ring vorticity and strain rate. Safta & Madnia
[45] consider the same problem, but use a methane-hydrogen mixture as fuel.
They contrast three chemical mechanisms - GRIMech v3.0, and two reduced
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1.3. Autoignition in turbulent H2/air diffusion flames 6
mechanisms with 11 step, 15 species and 12 step, 16 species respectively.
The objective of this dissertation is to study the coupling between the
fluid flow and chemistry with as few simplifying assumptions as possible. We
therefore use direct numerical simulation to study the flow when a round nozzle
injects H2 + N2 fuel at room temperature into air at higher temperatures. The
amount of fuel injected is controlled by varying the stroke length or stroke ratio.
The governing equations are the threedimensional, unsteady, compressible,
NavierStokes equations along with conservation equations for the nine species
described by the Mueller et al. [37] chemical mechanism. The simulations
consider the effects of Lewis number, stroke length ratio and fuel/air mixtures
(equivalence ratio) on autoignition and the flow.
1.3 Autoignition in turbulent H2/air diffu-
sion flames
The autoignition of turbulent diffusion flames is central to applications such
as diesel engines and scramjet engines where fuel is injected into hot oxidizer.
Fuel and oxidizer mix through convection and diffusion, then autoignite due
to the high temperatures of the oxidizer. Direct numerical simulation (DNS) is
used to study a diffusion flame where fuel (hydrogen/nitrogen) reacts with air
(oxygen/nitrogen). For laminar flames, fuel and oxidizer diffuse, then react,
yielding products. In turbulent flames, one might see autoignition, extinction
and even reignition due to the turbulence.
Mahalingam et al. [29] performed DNS of three-dimensional turbulent
nonpremixed flames with finite rate chemistry. They incorporated onestep
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1.3. Autoignition in turbulent H2/air diffusion flames 7
and twostep reaction models for the chemistry. Their results show that the
intermediate species concentration overshoots the experiments. They suggest
that this behavior is due to timedependent strain rates in the turbulent flow.
Montgomery et al. [36] also performed DNS of a three-dimensional turbulent
hydrogenoxygen nonpremixed flame using a reduced mechanism (7species,
tenreactions). The simulations were used to suggest a mixture fraction and
progress variable model for the chemical reactions. Im et al. [24] performed
DNS of twodimensional turbulent nonpremixed hydrogenair flames to study
autoignition. They show that peak values ofHO2 align with maximum scalar
dissipation during ignition. Im et al. also noted that weak and moderate tur-
bulence enhanced autoignition while stronger turbulence delayed ignition.
Echekki & Chen [15] used DNS to study autoignition of a hydrogen/air mix-
ture in two dimensional turbulence. Their results show spatially localized
sites where autoignition begins, which they define as a kernel. These ker-
nels have a build up of radicals at high temperatures, fuellean mixtures, and
low dissipation rates. Mastorakos et al. performed twodimensional simula-
tion of autoignition of laminar and turbulent diffusion flames with onestep
chemistry. They found that ignition always occurs at a welldefined mix-
ture fraction (MR). Hilbert & Thevenin [20] performed similar simulations to
Mastorakos et al. [32] and Im et al. [24]. The difference between Hilbert &
Thevenin [21] and others is that the simulation uses multicomponent diffusion
velocities.
A significant difference between the past work cited above, and our DNS is
that we perform fully three dimensional simulations using finite rate chemistry.
Also, we incorporate the Mueller mechanism for hydrogen/air (nine species
and nineteen reaction) and not a reduced mechanism. The objectives of our
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1.4. Contributions 8
simulations are: (i) to study major differences between two dimensional and
three dimensional simulations of turbulent diffusion flames. In particular,
what happens to the flame front in three dimensions? (ii) How do the ignition
kernels evolve in time? (iii) What is the effect of unity Lewis number and
nonunity Lewis number on ignition, and (iv) How does the mixture fraction
compare to a passive scalar?
1.4 Contributions
The principal contributions of this work are:
We have developed a novel numerical method for DNS/LES of reactingflow. The algorithm is spatially nondissipative, implicit and applicable
over a range of Mach number. It discretely conserves kinetic energy in
the incompressble, invisid, nonreacting limit. The algorithm is readily
applied to complex chemistry.
DNS is used to study autoignition of a hydrogen vortex ring in hot air.The most reactive mixture fraction (Mastorakos et al. [32]) is found to
predict the onset of initial ignition but not the subsequent evolution. A
most reactive temperature TMR is defined and shown to predict both the
initial autoignition as well as subsequent evolution.
The effect of stroke ratio and oxidizer temperature is studied. For strokeratios less than the formation number, ignition in general occurs behind
the vortex ring and propagates into the core. At higher oxidizer tem-
peratures, ignition is almost instantaneous and occurs along the entire
interface between fuel and oxidizer. For stroke ratios greater than the
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1.4. Contributions 9
formation number, ignition initially occurs behind the leading vortex
ring, then occurs along the length of the trailing column and propagates
towards the ring.
Lewis number is shown to affect both the initial ignition as well as sub-sequent flame evolution significantly. Nonuniform Lewis number simu-
lations provide faster ignition and burnout time but a lower maximum
temperature.
The effect of fuel/air ratio was studied. The fuel rich reacting vortexring provides the highest maximum temperature and the higher oxidizer
temperature provides the fastest ignition time. The fuel lean reacting
vortex ring has little effect on the flow and behaves similar to a non
reacting vortex ring.
DNS is used to study a turbulent hydrogen/air diffusion flame. Auto-ignition is shown to occur in fuel lean, high temperature regions with
low scalar dissipation and low vorticity.
Unlike laminar diffusion flames where auto-ignition occurs at MR, the
turbulent flame autoignites over a very broad range of around MR.
The simulations study the effects of three-dimensionality. Twodimensionalsimulations show that when flame fronts collide, extinction occurs (Mas-
torakos et al. [32]). However, our three dimensional results show that
when flame fronts collide; they can either increase in intensity or extin-
guish.
Comparison between unity Lewis number and nonunity Lewis number
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1.4. Contributions 10
simulations show that viscous diffusion is important even in the turbulent
regime.
It is shown that inadequate grid resolution causes autoignition to occurin fuelrich regions at low temperature. This is a non physical effect of
grid resolution.
This dissertation is organized as follows. The numerical method is de-
scribed in chapter 2. Chapter 3 discusses validation cases of varying com-
plexity. Chapter 4 considers autoignition of a hydrogen vortex ring reacting
with hot air. Direct numerical simulation of autoignition in turbulent non
premixed H2/air flames is discussed in chapter 5.
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Chapter 2
Numerical method
This chapter extends the Hou & Mahesh [23] algorithm to include the effects
of chemical reaction. The resulting algorithm treats the chemical source terms
implicitly, solves the species equations in a segregated manner, which allows
easy extension to multiple species and chemical reactions, and reduces to the
zero Mach number equations in the limit of very small Mach number. The
chapter is organized as follows. Section 2.1 describes the non-dimensional gov-
erning equations, and their behavior in the limit of very small Mach number.
The discrete scheme is described in section 2.2. The positioning of variables,
and details of the pressurecorrection approach, discretization of the chemical
source terms are discussed. Also, the behavior of the algorithm in the zero
Mach number limit is illustrated.
2.1 Governing equations
The governing equations are the compressible, reacting Navier-Stokes equation
for an ideal gas:
11
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2.1. Governing equations 12
d
td+
udjxdj
= 0, (2.1)
dYdktd
+Ydk u
dj
xdj=
xdj
dDdk
Ydkxdj
+ dk, (2.2)
dudi
td+
dudi udj
xdj=
pd
xdi+
dij
xdj(2.3)
dEd
td+
dEd + pd
udjxdj
=diju
di
xdj+
xdj
d
cdpP r
Td
xdj
+
Nk=1
Qdkdk (2.4)
pd = dRdTd = dRuWd
Td. (2.5)
where the superscript d denotes the dimensional value. The variables , p, Yk
and ui denote the density, pressure, mass fraction of species k and velocities
respectively. E = cvTd + udi u
di /2 denotes the total energy per unit mass and
ij = dudixdj
+ud
j
xdi 2
3
udk
xdk
ij
is the viscous stress tensor. Ddk, cp and P r
denote the diffusion coefficient of the kth species, specific heat at constant
pressure and the Prandtl number. For the source term, Qdk is the heat of
reaction per unit mass and Qdkdk is the heat release due to combustion for
the kth species. k is the mass reaction rate for the kth species. Ru is the
universal gas constant and Wd is the mean molecular weight of the mixture
defined as: 1Wd
=N
k=1YkWk
. The source term is modeled using the Arrhenius
law in this paper.
The reacting governing equation are non-dimensionalized as follows. Let
r, Yr, L, & Tr denote the reference density, mass fraction, length and tem-perature respectively. The reference velocity, dynamic viscosity and pressure
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2.1. Governing equations 13
are denoted by ur, r and pr respectively. The ratio =cdpcdv
is assumed to
be constant (e.g. Poinsot & Veynante [42]). This yields the following non-
dimensional variables:
=d
r, ui =
udiur
, t =td
L/ur, =
d
r, p =
pd prrur2
, (2.6)
T =
Td
Tr , Mr =
urar =
urRrTr , pr = rRrTr, (2.7)
Yk =YdkYr
, Dk =DdkurL
, k =Ldk
urrYr, Qk =
YrQdk
cp,rTr, (2.8)
Rr =RuWr
, and W =Wd
Wr. (2.9)
Note that pressure is nondimensionalized using an incompressible scaling.
This non-dimensionalization is motivated by Thompson [49], Bijl & Wesseling
[1], Van der Heul et al. [52] and Hou & Mahesh [23]. Therefore, the non-
dimensional governing equations for reacting flows are:
t+
ujxj
= 0, (2.10)
Ykt
+Ykuj
xj=
1
ReSck
xj
Ykxj
+ k, (2.11)
ui
t+
uiuj
xj=
p
xi+
1
Re
ij
xj, (2.12)
M2r
t
p +
12
uiui
+
xj
p +
12
uiui
uj
+
ujxj
=( 1)M2r
Re
ijuixj
+1
RePr
xj
W
T
xj
+
Nk=1
Qkk (2.13)
TW = M2rp + 1. (2.14)
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2.2. Discretization 14
where Sck is the Schmidt number for the kth species. When the reference Mach
number is zero, the non-dimensional reacting governing equations reduce to:
t+
ujxj
= 0, (2.15)
uit
+uiuj
xj= p
xi+
1
Re
ijxj
, (2.16)
Ykt
+ Ykujxj
= 1ReSck
xj
Ykxj
+ k, (2.17)ujxj
=1
RePr
xj
W
T
xj
+
Nk=1
Qkk, (2.18)
T
W= 1. (2.19)
Notice that the divergence of velocity equals the sum of the terms involving
thermal conduction and heat release. If the density is constant and there is
no heat release, the energy equation reduces to the incompressible continuity
equation. In the presence of heat release, the zero Mach number reacting
equations (Majda & Sethian [31]) are obtained. Most projection methods for
the zero Mach number equations project the momentum ui to satisfy the
momentum equation. Here, the velocity is projected to satisfy the energy
equation. The reaction source term can be quite complicated for multiple
species, and is discussed in more detail in section 2.2.1.
2.2 Discretization
Density, pressure, and temperature are staggered in time from velocity by
Hou and Mahesh [23]. The mass fraction of species k are similarly staggered
in time here. The thermodynamic variables and mass fraction of species k are
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2.2. Discretization 15
VNt
VNtVN
t
VNt
VNt
VNt
t+ 12
t+ 12
1
2t+
T1
2t+ , , ,
t u pi Y
k,
Figure 2.1: Storage of variables.
advanced in time from t + 12
to t + 32
, illustrated in figure 1. The variables are
colocated in space, to allow easy application to unstructured grids.
Integrating over the control volume and applying Gausss theorem yields the
discrete governing equations. The discrete continuity and species equations
are
t+ 3
2
cv t+1
2
cv
t+
1
V
faces
t+1facesVt+1N Aface = 0 (2.20)
and
(Sk)t+ 3
2
cv (Sk)t+1
2
cv
t+
1
V
faces
(Sk)t+1faces V
t+1N Aface
=1
ReSck
1
V
faces
1
Skxj
Sk2
xj
t+1faces
NjAface + t+1k . (2.21)
where cv denotes i,j,k and faces denotes summation over all the faces of thecontrol volume. faces and vN denotes the density and normal face velocity at
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2.2. Discretization 16
each face. Note that Sk = Yk. The variables (Sk)faces denotes the species at
the face and Nj is the outward normal vector at the face. The variables V and
Aface denote the volume of the control volume and the area of the face. The
discrete momentum equation is:
(gi)t+1cv (gi)tcv
t
+1
V faces (gi)t+ 1
2
faces Vt+ 1
2
N Aface
= 1V
faces
ptfacesNjAface +1
Re
1
V
faces
(ij)t+ 1
2
face NjAface. (2.22)
Here, gi = ui denotes the momentum in the i direction and ij is the stress
tensor. The discrete energy equation is given by:
M2r
tpcv + 1
2
uiuit+ 1
2
+1
V facespcv + 1
2
uiuit+ 1
2
face vt+ 1
2
N Aface+
1
V
face
vt+ 1
2
N Aface =( 1)M2r
Re
1
V
faces
(ijui)t+ 1
2
face NjAface
+1
RePr
1
V
faces
W
Tt+1
2
N
Aface +
Nk=1
Qkt+ 1
2
k . (2.23)
The algorithm solves each equation separately. This allows one to add multiple
species with relative ease, which allows easy extension to complex chemistry.
Note that the chemical source term is handled implicitly. Details of the implicit
procedure are described in section 2.2.1. The algorithm is a pressure-correction
method. An important feature is that the face-normal velocities are projected
to satisfy the constraint on the divergence which is determined by the energy
equation. At small Mach number, this feature ensures that the velocity field is
discretely divergence free. This is in contrast to most approaches that project
the momentum which is constrained by the continuity equation. A predictor-
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2.2. Discretization 17
corrector approach is used to solve the momentum and energy equation:
pt+3
2,k+1 = pt+
3
2,k + p. (2.24)
The corrector step is the difference between the predictor equation and the
exact equation which is defined as:
gt+1,k+1i,cv = g
i,cv t
4
p
xi
cv
, (2.25)
ut+1,k+1i,cv = u
i,cv t
4t+1cv
p
xi
cv
. (2.26)
Substituting equation (2.26) into the nonlinear term uiui converts kinetic en-
ergy into an equation for p which is:
(uiui)t+1,k+1 =
ui
t
4t+1cv
p
xi
ui
t
4t+1cv
p
xi
= ui u
i t
2t+1ui
p
xi+ O(p2). (2.27)
Equation (2.27) is then substituted into equation (2.23), and yields a discrete
energy equation in terms of p. Note that p is the difference between itera-
tions, which implies that p converges to zero at each time-step. This allows
the high order terms in equation (2.27) to be neglected.
The implementation to solve the following discrete equations are to initial-
ize the outer loop:
t+3
2 ,0 = t+1
2 , ut+1,0i = uti, Tt+3
2 ,0 = Tt+1
2 , vt+1,0N = vtN, St+ 3
2
,0
k = St+ 1
2k .
(2.28)
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2.2. Discretization 18
The next procedure is to advance the continuity equation (2.20), then advance
the species equation (2.21). Once the species is advance, advance the momen-
tum equation (2.22), then obtain vN by interpolation. After obtaining vN, the
next step is to solve the pressure correction equation (2.23). Use the corrector
steps to update pressure (2.24), momentum (2.25) and the velocities (2.26),
then check the convergence for the pressure, momentum, density, and species
between outer loop iterations.
2.2.1 Implicit source term
Consider a chemical system ofN species reacting through M reactions denoted
as (Poinsot & Veynante [42]):
Nk=1
kjk Nk=1
kjk (2.29)
for j = 1, M where k is a symbol for species k.
kj and
kjare the stoichio-
metric coefficients of species k for j reactions. The reaction term is defined
as:
k = Wk
Mj=1
kjQj (2.30)
where
Qj = Kfj
Nk=1
YkWk
kj
forward reaction
KrjNk=1
YkWk
kj
reverse reaction
. (2.31)
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2.2. Discretization 19
Using the empirical Arrhenius law,
Kfj = AfjTj exp
Ej
RT
= AijT
j exp
Taj
T
. (2.32)
Therefore, the source term is:
k = Wk
Mj=1
kj Kfj Nk=1
SkWk
kjWk
Mj=1
kj
Krj
Nk=1
SkWk
kj
. (2.33)
The backward rates Krj are computed through the equilibrium constants:
Krj =Kfj
paRT Nk=1 expS0jR H0jRT (2.34)
where S0j and H0j are entropy and enthalpy, respectively. The discrete form
of equation B.11 is:
t+1k = Wk
Mj=1
kj
Kt+1fj,cv
Nk=1
St+ 3
2
k,cv + St+ 1
2
k,cv
2Wk
kj
WkMj=1
kj
Kt+1rj,cv Nk=1
St+ 32k,cv + St+ 12k,cv2Wk
kj . (2.35)
Substituting equation (2.35) into equation (2.21) for t+1k yields:
(Sk)t+ 3
2
cv (Sk)t+1
2
cv
t+
1
V
faces
(Sk)t+1faces V
t+1N Aface
= 1ReSck
1V
faces
1
Skxj
Sk2
xj
t+1faces
NjAface
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2.2. Discretization 20
+Wk
Mj=1
kj
Kt+1fj,cv N
k=1
St+ 32k,cv + St+ 12k,cv
2Wk
kj
WkMj=1
kj
Kt+1rj,cv N
k=1
St+ 32k,cv + St+ 12k,cv
2Wk
kj
. (2.36)
Equation (2.36) can be represented as:
apSt+ 3
2
cv +nb
anbSt+ 3
2
nb = RHS. (2.37)
where nb are the neighbors of the cv. A parallel, algebraic multi-grid approach
is used to solve the system of algebraic equations. The structured grid interface
of the Hypre library (Lawrence Livermore National Laboratory 2003) is used.
Three different examples of the implicit source term are described below.
If N = 1, M = 1 and only forward reaction is assumed, this implies that
Q1 = A11T1 exp
Ta1
T
Y1W1
11
(2.38)
which implies
1 = W1A1111T1 exp
Ta1
T
Y1W1
11. (2.39)
Note that for a one-step premixed flame:
= BY exp
TaTd
(2.40)
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2.2. Discretization 21
which upon discretization yields:
t+1 = BSt+ 3
2
cv + St+ 1
2
cv
2exp
TaTt+1cv
. (2.41)
Equation (2.41) is the source term used in section 3.1.1.
Consider a two-step reaction from Chen [4] and Mahalingam [29] given by:
A + B I
A + I P. (2.42)
Note that A is the fuel, B is the oxidizer, I is the intermediate step, and P is
the product. The source term for species A is defined as:
k = B1YAYB exp
Ta1
T
+ B2YAYI exp
Ta2
T
. (2.43)
The discrete form is:
k = B1St+ 3
2
A,cv + St+ 1
2
A,cv
2St+1B,cv exp
Ta1
Tt+1cv
+B2
St+ 3
2
A,cv + St+ 1
2
A,cv
2 St+1
I,cv exp Ta2Tt+1cv . (2.44)This implicit source term is used in section 3.1.4. Note that the other species
are handled in a similar manner.
The final example is a nonlinear source term which is used in 3.1.2 and
3.1.3:
k = AF+OYFF YOO expTaT . (2.45)
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2.2. Discretization 22
10-7
10-6
10-5
10-4
10-3
10-2
10-110
-1
100
101
102
103
(a) Mr
CF L
10-7
10-6
10-5
10-4
10-3
10-2
10-12
3
4
5
6
7
8
9
10
(b) Mr
iter
Figure 2.2: Variation of (a) acoustic CFL and (b) number of outer loop itera-tions with Mr. Taylor problem , Reacting problem .
Let F = 2 and O = 1. This implies:
k = A2+1Y2FY1O expTa
T
= AS2FS1O exp
Ta
T
(2.46)
where F is the fuel and O is the oxidizer. The discrete equation for the fuel
species is:
t+1
k
=
ASt+ 3
2
F,cv + St+ 1
2
F,cv
2 (1)
St+1
F,cv(2)
St+1
O,cv
expTa
Tt+1cv . (2.47)
The nonlinear source term is linearized by only solving for term (1) and term
(2) is from previous iterations. The outer loop ensures that the nonlinear
corrections converge to zero.
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2.2. Discretization 23
2.2.2 Zero Mach number limit
The governing equations analytically reduce to the zero Mach number equa-
tions, as shown in section 2.1. Since the dependent variables are spatially
colocated, it appears that the pressureprojection step might result in odd
even decoupling. It is well known that the incompressible equations require
staggering, temporal dissipation, or low order basis functions for pressure, to
avoid oddeven decoupling. However, note that the inner loop in the proposed
algorithm uses nearest neighbors for the pressure equation; it therefore does
not suffer from oddeven decoupling. The tolerance assigned to the outer loop
can in practice, be controlled to obtain low Mach number solutions. However,
a more reliable approach which explicitly introduces temporal biasing is that
proposed by [53], which is now used here.Recall that the pressure gradient term in the momentum equation is defined
as:
p
xi
t+ 12
=1
4
p
xi
t 12
+1
2
p
xi
t+ 12
+1
4
p
xi
t+ 32
. (2.48)
Temporal biasing can be introduced by computing the pressure gradient as:
p
xi
t+ 12
=
1
4
p
xi
t 12
+1
2
p
xi
t+ 12
+
1
4+
p
xi
t+ 32
. (2.49)
Taylor expansion of the pressure at time level t + 12
yields:
1
4
p
xi
t 12
+1
2
p
xi
t+ 12
+
1
4+
p
xi
t+ 32
=
xipt+ 12 + 2tpt
t+ 12 + O t2 . (2.50)
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2.2. Discretization 24
i.e.; the leading order term is first order in time, for a given in equation 2.50.
Alternate biased formulations of the pressure gradient can be used, but as the
Taylor series expansion shows, using of the order of t would be the most
timeaccurate.
Two examples are used to illustrate the low Mach number behavior of the
algorithm. The first example is the nonreacting Taylor problem discussed in
more detail in Hou & Mahesh [23], while the second example corresponds to
a synthetic premixed flame for which an analytical solution is obtained: i.e.
T = sin x + 2, Y = sin x + 2, u = sin x + 2, p =4
3cos x sin x,
= 1/T, = cos x sin x. (2.51)
For the Taylor problem, t was set to 0.001 and for the reacting case t
was 0.0001. Figures 2.2 (a) show the variation of acoustic CFL, and number
of outer loop iterations with Mr respectively. Mr varies from 102 to 105 for
the Taylor problem, and from 103 to 106 for the reacting case. Solutions
were also obtained for Mr equal to zero, but are not shown due to the use of
logarithmic axes. Here, was chosen to be 0.05. Note that the acoustic CFL
varies inversely with Mr; i.e. low Mach numbers can be computed at fixed
timestep. Also, note that the number of outer loop iterations at low Mach
number is only twice that at finite Mach number, and remains constant at
very low Mach numbers.
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Chapter 3
Validation simulations
The algorithm is applied to simulate flames using both simple (3.1) and com-
plex (3.2) chemical mechanisms. Section 3.1.1 considers a onedimensional
steady laminar premixed flame while an unsteady laminar diffusion flame is
considered in section 3.1.2. These two examples illustrate the ability to han-
dle large heat release at low Mach number. Section 3.1.3 considers a steady
twodimensional laminar reacting jet flame with large heat release at low Mach
number. Threedimensional numerical simulations of turbulent nonpremixed
flames with finite rate chemistry are considered in section 3.1.4. This problem
illustrates application to turbulence and finite Mach number. The final ex-
amples in section 3.2 illustrates application to complex chemistry for H2/air
combustion using a 9 species, 19 reaction mechanism from Mueller et al. [37].
The wellstirred reactor problem in section 3.2.1 is used to study chemical
stiffness. A grid convergence study of a laminar diffusion flame is shown in
section 3.2.2. Section 3.2.3 and 3.2.4 discuss simulations of a laminar jet flame
and a lifted jet flame, respectively.
25
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3.1. Simple chemistry 26
-10 -5 0 5 101
2
3
4
5
(a) x
T
10-1
100
10-3
10-2
10-1
(b) dx
Error
Figure 3.1: (a) Comparison of computed temperature to analytical solutionfor a laminar premixed flame. analytic solution, numerical solution.Sc = Re = P r = 1, = 10, = 0.8, Mr = 0.01, (b) spatial order ofaccuracy. Algorithm, second order, Sc = Re = P r = 1, = 0.5, = 2,Mr = 0.01.
3.1 Simple chemistry
3.1.1 Laminar premixed flame
An irreversible, onestep, laminar premixed flame is considered as an example
of a low Mach number reacting flow. The reaction is given by R P where
R is the reactant and P is the product. The reaction source term is that
proposed by Echekki & Ferziger [16]. It approximates the Arrhenius source
term, and is useful for validation, in that it permits analytical solution. The
reaction model is defined as:
=
0 for < c,
( 1)( 1) otherwise .(3.1)
The corresponding analytical solution is given by:
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3.1. Simple chemistry 27
=
(1 1)exp(x) for x 0,1 1 exp [(1 ) x] for x 0 .
(3.2)
Generally, is the order of 10 for hydrocarbon combustion (Echekki & Ferziger
[16]).
Figure 3.1 (a) shows computed results for a premixed flame where = 10,
= 0.8, and the inflow Mach number is equal to 0.01. The Echekki & Ferziger
source term is used. The inflow velocity is set equal to the flame speed, and the
solution is initialized using a hyperbolic tangent profile for the temperature
distribution. The inlet boundary condition for temperature, density, species
and velocity are set to one. Zero derivative boundary conditions are specified
at the outflow. Boundary conditions based on characteristic analysis are not
specified, due the simplicity of the solution at the boundaries, and the use of
sponge boundary conditions at both inflow and outflow boundaries to absorb
acoustic waves that are generated by the initial transient. A cooling term,
(U Uref) is added to the right hand side of the governing equations overthe sponge zone, whose length is 10 percent of the domain. Here U and
Uref denote the vector of conservative variables and the reference solution
respectively. The coefficient is a polynomial function defined as:
(x) = As(x xs)n
(Lx xs)n , (3.3)
where xs and Lx denotes the start of the sponge and the length of the domain.
n and As are equal to three. After a brief transient, the solution settles into
a steady state which is shown in figure 3.1 (a). The computed and analytical
solutions seem to agree well.
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3.1. Simple chemistry 29
one-step reaction defined as:
FYF + OYO PYP (3.4)
where YF, YO and YP are the mass fractions of the fuel, oxidizer and products.
The computed solution is compared to an asymptotic solution developed by
Cuenot & Poinsot [11]. Their analysis is applicable to diffusion flames with
variable density, non-uniform Lewis number, and finite rate chemistry. Cuenot
& Poinsot studied unsteady unstrained, steady strained and unsteady strained
H2O2 flames. Case 2 from their paper corresponds to an unsteady, unstrainedflame, and was chosen for comparison. Table 3.1 lists the conditions for case
2.
Table 3.1: Test condition.
case LeF TF,0 LeO TO,0 F O A Ta Q/cp Re2 1 300 K 1 300 K 8 2 1 108 3600 K 6000 K 10000
Note that LeF and LeO denote the Lewis number for fuel and oxidizer.
TF,0 and TO,0 are the initial temperatures of the fuel and oxidizer. is the
equivalence ratio which is defined as:
= sYF,0YO,0
=OWOFWF
YF,0YO,0
(3.5)
where s is the stoichiometric ratio. The equivalence ratio is used to determine if
the mixture is rich ( > 1), lean ( < 1) or stoichiometric ( = 1). F and O
denote the stoichiometric coefficients of the species. A is the preexponential
factor and Ta is the activation temperature. The Arrhenius reaction rate is
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3.1. Simple chemistry 30
defined by Cuenot & Poinsot as:
k = kWkA
YFWF
F YOWO
Oexp
Ta
T
. (3.6)
Figure 3.2 is a comparison of the computed solution to the asymptotic so-
lution from Cuenot & Poinsot. The asymptotic solution was used to initialize
the temperature, fuel mass fraction, and oxidizer mass fraction. The initial
velocity profile was calculated from the energy equation. The initial pressure
remains uniform in the flame and the initial density is obtained from tem-
perature. Initially the reaction rate is zero because the asymptotic solution
assumes infinitely fast chemistry. However, the simulation involves (fast) finite
rate chemistry. Therefore, the reaction rate rapidly recovers in the simulation
(Cuenot & Poinsot). The inlet boundary condition for temperature, density,
and mass fraction of fuel are set to one. Mass fraction of oxidizer and velocity
are set to zero. Zero derivative boundary conditions are specified at the out-
flow. The initial time was set at ct/L = 20 and the solution was advanced to
a time of ct/L = 64. Again, the sponge boundary conditions were used at
both inflow and outflow boundaries to absorb acoustic waves generated by the
initial transient. The sponge zone is 10 percent of the domain. Also, the flow
Mach number is 0.001 and the grid used 512 points. Reasonable agreement
between computed and asymptotic solution is obtained.
3.1.3 Laminar twodimensional jet flame
A steady two dimensional reacting laminar jet flame from Poinsot & Veynante
[42] is considered. This problem illustrates the ability to handle large heat
release and nearly incompressible flow for a one-step diffusion flame with a
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3.1. Simple chemistry 31
(a) x
y
0 20 40 60 80 100 120 140
0.2
0.4
0.6
0.8
1
(b) x
Figure 3.3: (a) Contour plot of temperature for a jet flame. (b) Center line pro-file of the mixture fraction . asymptotic solution from Poinsot & Veynante[11], numerical solution. Mr = 0.01, Re = 500, Sck = P r = 1.
Damkohler number of 50 106
. From Poinsot & Veynante [42], if one assumes
constant mass flow rate (u = constant), v = w = 0 and D = constant, then
the mixture fraction satisfies:
Fuf
x= FDF
2
x2.
Note that is defined as:
= 2
YF YO + 1
+
1 + 1
(3.7)
where is the equivalence ratio. A similarity solution is obtained:
(x, y) =1
2
xexp
y
2
4x
(3.8)
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3.1. Simple chemistry 33
(a) (b)
Figure 3.4: (a) Scalar dissipation rate and (b) reaction rate at z = 4for turbulent diffusion flame. Notice that scalar dissipation is high where thereaction rate is low. Mr = 0.1, Re = 935, Sck = P r = 1.
(a) (b)
Figure 3.5: (a) Isosurface plot of the reaction rate for turbulent diffusionflame. (b) Contour plot of temperature showing entire computational domain.
Mr = 0.1, Re = 935, Sck = P r = 1.
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3.1. Simple chemistry 35
Table 3.3: Parameters for diffusion flame.
zst A B A (m3 mol1 s1)
0.5 8 0.8 1 1 1012
where k is the wave number, k0 is the wave number at which E(k) is maximum,
and u0 is the rms velocity.
The initial diffusion flame is obtained by asymptotic solution (Cuenot &
Poinsot [11]); table 3 shows the relevant parameters. From table 3, zst is
defined as zst =1
1+where is the equivalence ratio. A and B are the sto-
ichiometric coefficients. A is the preexponential factor. The diffusion flame
is advanced in time to remove any acoustic transients. After the initial tran-
sients are removed, the initial turbulent velocity field is superimposed on the
diffusion flame. The solution is advanced from 0 to 2.0 eddy turnover times
and analyzed at each 0.1 eddy turnover time. Local extinction is observed to
occur in the reaction rate. Holes occur in the reaction zones where pure mix-
ing exists and the hole location corresponds to a high rate of scalar dissipation
rate as predicted by laminar flamelet theory. This is shown in figures 3.4 and
3.5 (a). Figure 3.5 (b) is a contour plot of temperature showing the entire
computational domain. Figure 3.6 shows two-dimensional mass concentration
distribution of fuel mass fraction and oxidizer mass fraction where good agree-
ment is obtained with Chen [3] (figure 2 (a) in their paper). The lower bound
on the species is the location of the flame and the upper bound is influenced
by extinction. The distribution between the lower and upper bounds is the
turbulence induced mixing (Mahalingam [29] & Chen [3]).
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3.2. Complex chemistry 36
(a) time (sec)
YH2 T
0 0.02 0.04 0.06 0.08 0.10
1E-06
2E-06
3E-06
4E-06
(b) time (sec)
YHO2
Figure 3.7: (a) Comparison of a major species & temperature betweenChemkin and present algorithm for a wellstirred reactor. ([YH2 ] Algo-rithm, Chemkin & [T] in kelvin, Algorithm, Chemkin) (b) Illustra-tion of the effect of timestep by comparing the explicit Euler (dt = 1.0e 8)to present implicit source term (dt = 0.001 & dt = 0.0001). implicitdt = 1.0e 3, implicit dt = 1.0d 4 explicit dt = 1.0d 8.
3.2 Complex chemistry
This section illustrates application of the algorithm to complex chemistry.
Combustion of H2 and air is considered, using a 9 species and 19 reaction
mechanism [37].
3.2.1 Well stirred reactor
Figure 3.7 illustrates the ability of the implicit procedure described in section
2.2 to handle chemical stiffness. A perfectlystirred reactor problem is consid-
ered, the timestep is varied, and the results are compared to Chemkin [26]
for accuracy. The implicit treatment of the chemical source terms allows the
time step to be five orders of magnitude higher (1.0e3) than that possibleusing explicit Euler time advancement (1.0e8). The maximum error of the
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3.2. Complex chemistry 37
Table 3.4: Parameters for laminar diffusion flame.
YH2 YO2 YN2,F YN2,O TF TO p u Mr pr Tr0.029 0.233 0.971 0.767 1 4 0 0 0.4 1 atm 300 K
implicit solution (computed using YHO2) compared to the explicit solution was
around 5 percent, for dt = 1.0e
3. For dt = 0.0001, the percent error was
around 0.01 percent.
3.2.2 Laminar diffusion flame
A laminar diffusion flame with complex chemistry was simulated. The param-
eters are listed in table 3.4. Figure 3.8 shows results from a grid convergence
study. Uniform grids ranging from 64 to 2048 points were used to obtain
a gridconverged solution. A nonuniform grid of 128 points that clustered
points in the flame was also used. The resolution was estimated to accurately
compute the instant when the flame was thinnest and the heat release was
highest. Figures 3.8 (a) and (b) show that the solution is stable and qual-
itatively correct even for the 64 points which only has 2 to 3 points in the
flame.
3.2.3 Laminar jet flame
A three dimensional laminar hydrogen/air round jet flame in heated coflow is
considered. The jet Reynolds number is 1000, fuel jet width is 1 cm and other
relevant parameters are listed in table 3.5. The inflow boundary conditions
are shown in table 4.1. The first derivative is set to zero for thermodynamic
variables, species, and velocities at the outflow. At the y and z boundaries, the
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3.2. Complex chemistry 38
(a) x
YH2O
(b) x
YH
Figure 3.8: Grid convergence study showing profiles of a major species anda minor species. (a) H2O (b) H. N=64, N=128 , N=128 nonuniform grid, N=256, N=512, N=1028, N=2048.
Table 3.5: Simulation parameters
YH2 YO2 YN2 T U pFuel 0.029 0 0.971 300 (K) 187 m/s 101325 PaOxidizer 0 0.233 0.767 1200 (K) 75 m/s 101325 Pa
thermodynamic variables, velocities and species are set to a constant. Sponge
boundary conditions are used at the transverse boundaries to absorb acoustic
waves that are generated by the flame. The computational grid is 128 by 64
by 64 and the domain is 30 by 10 by 10. P r is 0.7, is equal to T0.7 and unity
Lewis number (Lek = 1) is assumed.
Figure 3.9 shows the temporal variation of maximum n, T, YOH, YO, YH,
YH2O, YHO2 and YH2O2 . A steady state solution is obtained at 170 units of
time and the final maximum temperature is 2370 K. Figure 3.10 illustrates
the steady state contours of T(K), YOH, YHO2 and YH2O2.
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3.2. Complex chemistry 39
(a) time
n T
(b) time
Figure 3.9: (a) time versus maximum n and T. (b) time versusmaximum species of YOH, YO, YH, YH2O, YHO2 and YH2O2.
Table 3.6: Simulation parameters
YH2 YO2 YN2 T U pFuel 0.11 0 0.89 400 (K) 347 m/s 101325 PaOxidizer 0 0.233 0.767 1100 (K) 100 m/s 101325 Pa
3.2.4 Sandia lifted jet flame
Experiments on a lifted turbulent hydrogen/air slot jet flame in a heated coflow
were performed at Sandia National Laboratories. The jet Reynolds number
is 11,000 and the simulation parameters are listed in table 3.6. The fuel jet
width is 2 mm. We perform preliminary two dimensions simulations under
the same conditions as the Sandia jet flame. The boundary conditions are
similar to that described in section 3.2.3. The computational grid is 1024 by
512 by 2 and the domain is 30 by 14 by 1. P r is 0.7, is equal to T0.7 and
the nonuniform Lewis numbers are from Im et al. [24].
Figure 3.11 (a) is a contour plot of temperature; note the temperature in-
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3.2. Complex chemistry 41
(a) (b)
(c) (d)
Figure 3.11: Contour plots of (a) T(K), (b) YHO2, (c) z and (d) divergence.
crease down stream of the jet as expected for a lifted jet. The precursor to
autoignition is adequate accumulation of the species YHO2. Figure 3.11 (b)
shows the species YHO2 exists down stream in the jet. Figure 3.11 (c) shows the
rollup of vorticity as the jet evolves, while the acoustic waves generated by the
combustion are illustrated in figure 3.11 (d). Note that the acoustic waves ap-
pear to emanate from the ignition regions. Future simulations should consider
threedimensionality, larger domain and nonreflecting boundary conditions
to better represent the acoustic field.
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Chapter 4
Autoignition of a hydrogen
vortex ring reacting with hot air
The objective of this chapter is to study autoignition of fuel injected into
hot air. We therefore use direct numerical simulation to study the flow when
a round nozzle injects H2 + N2 fuel at room temperature into air at higher
temperatures. The amount of fuel injected is controlled by varying the stroke
length or stroke ratio. The governing equations are the threedimensional, un-
steady, compressible, NavierStokes equations along with conservation equa-
tions for the nine species described by the Mueller et al. [37] chemical mecha-
nism. The algorithm proposed in chapter 2 is used. The simulations consider
the effects of Lewis number, stroke length ratio and fuel/air mixtures (equiv-
alence ratio) on autoignition and the flow.
Chapter 4 is organized as follows. Section 4.1 describes relevant details
of the simulation. It provides a brief description of the initial and boundary
conditions. Results from a validation and grid convergence study are shownin section 4.2. The homogeneous mixture problem is discussed in section 4.3
42
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4.1. Problem statement 43
Figure 4.1: Schematic of the problem.
and is used to obtain a chemical time scale and MR. Section 4.4 discusses
results for the reacting vortex ring. A brief summary in section 4.5 concludes
this chapter.
4.1 Problem statement
Figure 4.1 shows a schematic of the problem and illustrates the ring vortex
and the trailing column which forms at large stroke ratios. The dimensional
variables for the reacting vortex ring are chosen such that the equivalence ratio
() is one, and a stoichiometric mixture is obtained where YH2,0 equals 0.02936
and YO2,0 equals 0.233. For fuel lean mixtures, equals 0.25 (YH2 = 0.0074)
and for fuel rich mixtures equals 4 (YH2 = 0.1185). For all simulations, Re is
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4.2. Validation and gridconvergence study 45
-3 -2.5 -2 -1.5 -1 -0.5 00
0.2
0.4
0.6
0.8
1
(a) r
YF
(b) time
T YH2
Figure 4.2: (a) Comparison using onestep chemistry to Hewett and Madnia( [19]). Present, Hewett & Madnia. (b) Comparison of a major species& temperature between Chemkin and present algorithm for a wellstirred re-actor using the Mueller mechanism. ([YH2] Algorithm, Chemkin & [T]
Algorithm, Chemkin).
the sponge zone whose length is 10 percent of the domain. Here U and Uref
denote the vector of conservative variables and the reference solution respec-
tively. The coefficient for the sponge at the outflow, is a polynomial function
defined as:
(x) = As(x xs)n
(Lx xs)n,
where xs and Lx denotes the start of the sponge and the length of the domain.
n and As are equal to three.
4.2 Validation and gridconvergence study
The simulations are first validated using a onestep chemical mechanism, and
compared to the one-step simulations of Hewett & Madnia [19]. Our results
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4.2. Validation and gridconvergence study 46
(a) y
YH
(b) y
YH2O
Figure 4.3: Grid convergence study for a twodimensional laminar reactingvortex ring. (a) YH (b) YH2O. N=64, N=128, Nx=128, Ny=Nz=128nonuniform in y, Nx = 256, Ny=Nz=128 nonuniform in y, N =256, N = 512.
are compared to case 5 in their paper where the temperature ratio is 6:1 and
the stroke length is two. Figure 4.2 (a) shows that good agreement is obtained.
Next, complex chemistry is validated using the homogeneous mixture or
wellstirred reactor problem. We use the 9 species (H2, O2, OH, O, H, H2O,
HO2, H2O2, and N2), 19 reaction Mueller mechanism (Mueller et al. [37])
which is shown in table 4.2. A detailed experiment and simulation of a lam-
inar hydrogen jet diffusion flame was performed by Cheng et al. [8]. Their
work shows that chemical mechanisms from Miller & Bowman [35] , GRIMech
3.0 [48], Mueller et al [37], Maas & Warnatz [28], and O Conaire et al. [38]
compare well with experimental results. Therefore, the Mueller mechanism
should perform adequately for our hydrogen/air reacting vortex ring simula-
tion. Figure 4.2 (b) is a comparison between Chemkin [26] and our algorithm
incorporating the Mueller mechanism. Good agreement is observed for the
homogeneous mixture.
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4.2. Validation and gridconvergence study 47
Table 4.2: Mueller Mechanism: (cm3moleseckcalK)
Afj j EjH2/O2 Chain Reactions
1 H+ O2 O + OH 1.910E14 0.0 16.442 O + H2 H+ OH 5.080E04 2.67 6.293 H2 + OH H2O + H 2.160E08 1.51 3.434 O + H2O OH + OH 2.970E06 2.02 13.4
H2/O2 Dissociation/Recombination Reactions5 H2 + M H+ H+ M 4.580E19 1.40 104.386 O + O + M O2 + M 6.160E15 0.5 07 O + H+ M OH + M 4.710E18 1.0 08 H+ OH + M H2O + M 2.210E22 2.0 0
Formation and Consumption of HO29 H+ O2 + M HO2 + M 3.50E16 0.41 1.1210 HO2 + H H2 + O2 1.660E13 0.0 0.8211 HO2 + H OH + OH 7.080E13 0.0 0.312 HO2 + O OH + O2 3.250E13 0.0 0
13 HO2 + OH H2O + O2 2.890E13 0.0 0.50Formation and Consumption of H2O2
14 HO2 + HO2 H2O2 + O2 4.200E14 0.0 11.9815 H2O2 + M OH + OH + M 1.20E17 0.0 45.516 H2O2 + H H2O + OH 2.410E13 0.0 3.9717 H2O2 + H H2 + HO2 4.820E13 0.0 7.9518 H2O2 + O OH + HO2 9.550E06 2.0 3.9719 H2O2 + OH H2O + HO2 1.000E12 0.0 0
The three dimensional computational grid used in the simulations is based
on a grid refinement study of a twodimensional H2/air reacting vortex ring.
The computational domain was 10 by 10 in x and y where the nozzle is in the
y direction. Uniform grids ranging from 64 to 512 points were used to obtain
a gridconverged solution. Also, a nonuniform grid of 128 points in the y
direction (to cluster points around the nozzle) and 128 uniform points in the
x direction was used. Figures 4.3 (a) and (b) show profiles in the middle of
the reacting vortex ring. Note that grid convergence is obtained for the non
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4.3. Homogeneous problem 48
(a) time
H
(b) time
Tn
Figure 4.4: (a) Rates of reactions 1,2,3,9,10, and 11 for species H. (b) Temporalevolution of temperature with all reactions, and reactions 1, 2, 3, 9, 10 & 11.All reactions: (T) , (n) reactions 1, 2, 3, 9, 10, & 11: (T) ,(n) .
uniform grid of 1282 and uniform 2562 and 5122 grids. To ensure adequate
resolution for the three dimensional reacting vortex ring, a nonuniform 128
grid in y, z directions and 256 in the x direction was chosen, and the size of the
computational domain was 10 by 10 by 10 in the three coordinate directions.
4.3 Homogeneous problem
We first consider the homogeneous mixture or wellstirred reactor problem.
Our objectives are to: (i) obtain an ignition time scale; (ii) obtain an opti-
mum mixture fraction for autoignition (Mastorakos et al. [32]); (iii) identify
which reactions contribute to autoignition and which reactions contribute af-
ter autoignition. Echekki & Chen [15] used DNS to study autoignition of
a hydrogen/air mixture in two dimensional turbulence. They found that theintermediate species O, H, OH, and HO2 play an important role in the auto
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4.3. Homogeneous problem 49
(a)
chem
(b)
chem
Figure 4.5: (a) Mixture fraction versus chem for varying temperature in kelvin. = 1, T = 1100, = 1, T = 1200, = 1, T = 1300, and = 1, T = 1400. (b) Mixture fraction versus chem for varying fuel/air
ratio, = 1/4, T = 1200, = 1, T = 1200, = 4, T = 1200and = 8, T = 1200.
ignition process. In the Mueller mechanism, reactions 1,2,3,9,10 and 11 have
a significant contribution to the minor species O, H, OH, and HO2. Figure
4.4 (a) shows reaction rates of species H due to reactions 1,2,3,9,10 and 11.
Figure 4.4 (b) shows temperature and heat release when all the reactions are
simulated, and when only the autoignition reaction (1,2,3,9,10, and 11) are
simulated. Note that the maximum heat release for the full reaction set andthe autoignition reaction set occurs at about the same time. Up until this
instant of maximum heat release, the autoignition reactions are dominant;
after this instant, the other reactions are also significant. This instant is de-
fined as chem and will be used to obtain the most reactive mixture fraction,
MR.
Mastorakos et al. [32] used onestep chemistry and the homogeneous mix-
ture problem to obtain chem and the mixture fraction most likely to auto
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4.3. Homogeneous problem 50
ignite (MR). The following equations are used to set up the wellstirred
reactor problem in terms of the mixture fraction defined as:
YH2 = YH2,0, YO2 = YO2,0 (1 ) , YN2 = 1 YH2 YO2, (4.3)
T = TO (TO TF) .
For example, if = 0.5, YH2,0 = 0.029, YO2,0 = 0.233, TF = 300 (K) and
TO = 1200 (K), then YH2 , YO2, YN2 , & T would be 0.5, 0.0145, 0.1165, 0.869, &
750 (K), respectively. One can now consider different initial mixture fractions,
obtain the minimum ignition time (chem) and thereby obtain the most reactive
mixture fraction (MR).
We obtain MR for the Mueller mechanism. Figure 4.5 (a) shows mixture
fraction versus chem for initial temperatures that range from 1100 to 1400
Kelvin and figure 4.5 (b) shows mixture fraction versus chem for fuel/air ratios
() that range from 0.25 to 8. Note in figure 4.5 that the minimum chem
corresponds to MR. Figure 4.5 also shows the effect of fuel/air mixture and
temperature on chem. As temperature increases, chem becomes smaller and
MR shifts to the right. As increases, chem becomes smaller and MR shifts
to the left. These results illustrate the balance between adequate fuel and
high enough temperature for autoignition to occur. Table 4.3 lists the most
reactive mixture fraction and corresponding values of the initial temperature,
fuel/air ratio and chem.
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4.4. Results for reacting vortex ring 51
Table 4.3: Global results for homogeneous mixture problem.
T0 MR chem TMR1100 1 0.06 3.4e 4 10521200 1 0.08 1.4e 4 11281300 1 0.09 8.5e 5 12101400 1 0.10 5.6e 5 12901800 1 0.13 1.8e 5 16051200 1/4 0.1 3.5e
4 1110
1200 4 0.06 8.0e 5 11461200 8 0.04 6.4e 5 1164
4.4 Results for reacting vortex ring
Table 4.4 lists the different cases studied. The simulations consider the effect
of Lewis number, stroke length ratio (L/D), fuel/air ratio and oxidizer tem-
perature. The nonuniform Lewis numbers are obtained from Im et al [24].
Two stroke ratios of 2 and 6 are considered. Note that for nonreacting vortex
rings, a stroke ratio of 2 yields a coherent vortex ring, while a stroke ratio of 6
yields a vortex ring followed by a trailing column of vorticity (Sau & Mahesh
[46]). Oxidizer temperatures (TO) of 1200 and 1800 Kelvin are considered. Fi-
nally, fuel/air ratios were chosen such that the equivalence ratios () are equal
to 1/4, 1 and 4 yielding lean, stoichiometic and rich mixtures, respectively.
Table 4.4: Parameters used in reacting vortex ring simulations.
Lewis number (L/D) TOnonuniform 1 2 1200unity 1 2 1200nonuniform 0.25 2 1200nonuniform 4 2 1200nonuniform 1 2 1800
nonuniform 1 6 1200
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4.4. Results for reacting vortex ring 52
time
YH
Figure 4.6: Domain average of rates of reactions 1,2,3,9,10 and 11 for thespecies H. nonuniform Lewis reaction 1, reaction 2, reaction3, reaction 9, reaction 10, reaction 11.
4.4.1 Ignition and flame evolution
We first consider the nonuniform Lewis number simulations at stroke ratio
of 2 and oxidizer temperature of 1200 K. Figure 4.6 shows reactions 1,2,3,9,10
and 11 for the species H. Time is normalized by chem obtained from table
4.3. Comparison to a corresponding plot for the homogeneous mixture (figure
4.4 a) shows qualitative similarities along with some differences. The reaction
rates are seen to peak at time greater than chem for the reacting vortex ring,
reflecting the finite amount of time taken for fuel and oxidizer to be advected
and then diffuse to yield regions of MR. Also, the relative importance of
reactions 9 and 11 is higher for the vortex ring,. The time scale on which
the reaction rates decay to zero following the initial ignition, is longer for the
reacting vortex ring, reflecting the propagation of the flame through the vortex
ring.
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4.4. Results for reacting vortex ring 53
n
(a)
T
(b)
n
T
(c)
n
T
(d)
n
T
(e)
n
T
(f)
n
T
Figure 4.7: Twodimensional slices showing variation in time of versus n.The color contours correspond to n. On each color contour, a colored contourline is plotted for , T and . The color contour of n ranges from 0 to 1.For the contour lines, purple is (contour lines range from 0.04 to 0.12), T isblack (contour lines range from 1120 to 1140 K) and is red (contour linesrange from 0.0 to 0.15). Nondimensional time: (a) 2.5, (b) 3.0, (c) 3.5, (d)4.0, (e) 4.5 and (f) 5.0.
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4.4. Results for reacting vortex ring 54
TMR
MR
LD
= 6
TO = 1800
t : 0.25 0.75 1.25 1.75 2.25 2.75 3.25
TMR
MR
t : 2.5 3.0 3.5 4.0 4.5 5.0 5.5
TMR
MR
LD
= 2
t : 2.5 3.0 3.5 4.0 4.5 5.0 5.5
Figure 4.8: Time evolution of heat release (n), MR and TMR. On each colorcontour of heat release, a contour line is superimposed for and T. Thecontour of n ranges from 0 to 1. The contour lines of range from 0.04 to0.12 (MR) and temperature range from 1092 to 1164 K (TMR) for L/D = 2and L/D = 6. For oxidizer temperature of 1800 K, ranges from 0.10 to 0.18(MR) and temperature ranges from 1590 to 1650 K (TMR).
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4.4. Results for reacting vortex ring 55
Figure 4.7 illustrates the process of autoignition in the vortex ring using
scatter plots of mixture fraction versus heat release at different instances in
time. At each time instant, heat release contours in a radial crosssection are
shown. Superposed on the heat release contours is a colored contour line of
mixture fraction (), temperature (T) and scalar dissipation (), respectively.
Note that the color contours of heat release (n) range from 0 to 1. Mixture
fraction and scalar dissipation are defined as (Poinsot and Veynante [?]):
=
1
1 +
YFY0F
YOY0O
+ 1
, (4.4)
=2
Re
xj
xj
, (4.5)
respectively. For the contour lines, mixture fraction is purple (contour lines
range from 0.04 to 0.12), temperature is black (contour lines range from 1120
to 1140 K) and scalar dissipation is red (contour lines vary over the entire range
of 0.0 to 0.15). Note that the ranges for the contour lines of mixture fraction
and temperature were chosen based on the results of the homogeneous mixture
problem. Recall MR is 0.08 from the homogeneous problem when T0 = 1200
K and = 1 (table 4.3).
Figure 4.7 shows that autoignition occurs in regions of high temperature,
fuel-lean mixtures, and low scalar dissipation where = MR. Ignition occurs
at a MR of 0.08 and propagates towards the stoichiometric side ( = 0.5).
Most of the ignition occurs behind the vortex ring and then moves rapidly
into the core. The contour line of mixture fraction shows that autoignition
occurs in fuel lean regions ( = MR = 0.08). Also, the contour lines of scalar
dissipation show that autoignition occurs in regions of low scalar dissipation.
The superposed contour lines of temperature are chosen to lie around TMR
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4.4. Results for reacting vortex ring 56
which is 1130 K (TMR = 1200 MR [1200 300]). Interestingly, maximumheat release follows TMR at every instant of time in figure 4.7. Figure 4.8 is a
side by side comparison of MR and TMR. This figure shows that the onset of
ignition occurs at MR and TMR. As ignition continues from behind the vortex
ring into the core, MR is in the very fuel lean region and does not follow the
path of ignition. TMR on the other hand, follows the path of ignition all the
way into the core. This suggests that TMR instead of MR might be better
in predicting the evolution of autoignition. This behavior is observed in all
cases simulated (figure 4.8); i.e. in the preignition phase, MR predicts the
onset of ignition fairly well, however TMR predicts both the onset of ignition
as well as its subsequent evolution.
Table 4.5 lists most reactive mixture fraction, burnout time in seconds,
maximum temperature in Kelvin and ignition time in seconds obtained from
all the simulations. Note that the onset of ignition largely depends on fuel/air
ratio and oxidizer temperature. However, in the postignition phase, the evo-
lution of the flame is greatly affected by other parameters. This behavior is
discussed in the follow
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