Jeff Thesis

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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.

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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

vi


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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

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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

x


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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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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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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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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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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


28/119

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


29/119

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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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


32/119


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-


33/119


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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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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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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+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)


37/119


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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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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2.2.2 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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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:


43/119


=

(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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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


46/119


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


47/119


(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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49/119


(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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51/119


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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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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(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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(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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(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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(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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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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(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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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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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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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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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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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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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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