Hao He- Numerical Simulations of Unsteady Flows in a Pulse Detonation Engine by the Space-Time...

8/3/2019 Hao He- Numerical Simulations of Unsteady Flows in a Pulse Detonation Engine by the Space-Time Conservation El

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NUMERICAL SIMULATIONS OF UNSTEADYFLOWS IN A PULSE DETONATION ENGINE

BY THE SPACE-TIME CONSERVATION ELEMENT

AND SOLUTION ELEMENT METHOD

DISSERTATION

Presented in Partial Fulfillment of the Requirements

for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Hao He, M.S.

*****

The Ohio State University

2006

Dissertation Committee:

Prof. S.-T. (John) Yu, Advisor

Prof. Walter R. LempertProf. Mohammad Samimy

Prof. Ahmet Selamet

Approved by

_________________________

Advisor

Graduate Program in Mechanical Engineering


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Copyright by

Hao He

2006


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ABSTRACT

This dissertation is focused on the development of a numerical framework for

time-accurate solutions of high-speed unsteady flows with specific applications to the

development of the Pulse Detonation Engine (PDE) concept. The numerical method

employed is the space-time Conservation Element and Solution Element (CESE) method,

which is a novel numerical method for time-accurate solutions of nonlinear hyperbolic

equations. As a part of the outcome of the present work, a general-purposed two- and

three-dimensional CESE code has been developed. The code can use both structured and

unstructured meshes composed of tetrahedrons or hexahedrons for three-dimensional

flows and quadrilaterals and triangles for two-dimensional flows. The code is fully

parallelized for large-scaled calculations. Parallel computing is based on domain

decomposition. Message Passing Interface (MPI) is used for data communication

between computer nodes. Extensive pre and post processing codes have been developed

to streamline the numerical calculations, so that the core solver could be directly

connected with commercial grid generators and solution animation software. The code

has been applied to analyze various flow fields related to the PDE concept. First,

numerical results of one-, two- and three-dimensional detonation waves are reported. The

chemical reactions were modeled by a one-step, finite-rate, irreversible global reaction.

The classical Zeldovich, von Neumann, and Doering (ZND) analytical solution was used


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to set up the initial conditions as well as for code validation. In the three-dimensional

calculations, detonations in square, round, and annular tubes at different sizes were

successfully simulated. Salient features of detonation waves were crisply resolved,

including triple points and numerical soot traces on walls. Second, as a promising

detonation initiation means, implosion with shock focusing was investigated. Polygonal

and circular converging shock fronts were simulated. In two-dimensional calculations, we

found a double-implosion mechanism in a successful detonation initiation process. Third,

the plume dynamics of a PDE fueled by propane/air mixtures were studied to support the

prototype development at NASA Glenn Research Center (GRC). Numerical results show

that in each PDE cycle the engine is actively producing thrust forces only in about 6% of

one cycle time period. The rest of the time is occupied by the blow-down and refueling

processes. Since the PDE tube is always open, the processes depend on the flow

conditions outside the PDE tube. In the near-field plume, complex shock/shock and

shock/vortex interactions were found. In the far field, a spherical expansion wave is the

dominant flow feature. This dissertation work is synergy of a very accurate and efficient

CFD method, i.e., the CESE method, and the modern parallel computing technology. The

resultant software is a state-of-the-art numerical framework, ready to be applied to obtain

time-accurate solutions of hyperbolic equations. This approach of applying the CESE

method and parallel computing could point to a new direction for high-fidelity

simulations of complex flow fields of advanced propulsion systems.


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Dedicated to my parents


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ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Professor Sheng-Tao

John Yu, for his patience, guidance and encouragement. My thanks go to Professor

Walter R. Lempert, Professor Mohammad Samimy, and Professor Ahmet Selamet for

serving as the members of my dissertation committee.

I would like to express my appreciation to Dr. Zeng-Chan Zhang and Dr. Moujin

Zhang for the instructive discussions regarding numerical methods and algorithm. I also

wish to thank Bao Wang and Minghao Cai for their help on maintaining the computer

facilities.


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VITA

November, 1973Born in China

July, 1995..B.E., Engineering Mechanics, Tsinghua

University, Beijing, China

May, 2003..M.S., Mechanical Engineering, Wayne State

University, Detroit, USA

1995-1999..Staff Member, Department of Engineering

Mechanics, Tsinghua University, Beijing, China.

1999-2003.Graduate Research Assistant, Department of

Mechanical Engineering, Wayne State University,

Detroit, USA

2003-present..Graduate Research Assistant, Department ofMechanical Engineering, The Ohio State

University

PUBLICATIONS

Bao Wang, Hao He, and S.-T. John Yu, 2005, Direct Calculation of Wave Implosion for

Detonation Initiation in Pulsed Detonation Engines, AIAA Journal, Vol. 43, No. 10, pp.2157-2169.

FIELDS OF STUDY

Major Field: Mechanical Engineering


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TABLE OF CONTENTS

Page

Abstract ....................................................................................................................... iiiDedication ................................................................................................................... iiv

Acknowledgments..........................................................................................................v

Vita....................................................................................................................... vi

List of Tables ............................................................................................................... ix

List of Figures................................................................................................................xNomenclature............................................................................................................ xvii

Chapters:

1. Introduction................................................................................................................1

1.1 Literature Review.............................................................................................11.2 Motivation and Objectives...............................................................................4

1.3 Numerical Methods..........................................................................................6

1.4 Organization.....................................................................................................8

2. The Model Equations.................................................................................................9

2.1 Three-Dimensional Euler Equations................................................................92.2 Euler Equations for Reacting Flows ..............................................................11

2.3 The ZND Solution..........................................................................................14

3. The Space-Time CESE Method...............................................................................27

3.1 Conventional Finite Volume Method ............................................................26

3.2 The CESE Method .........................................................................................32

3.3 The Modified CESE Method .........................................................................403.4 Two-Dimensional Euler Solver .....................................................................47

3.4.1 Conservation Elements and Solution Elements ..................................48

3.4.2 Approximations within a Solution Element........................................51

3.4.3 Evaluation ofum ..................................................................................533.4.4 Evaluation ofumx and umy ....................................................................56


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3.5 Three-Dimensional Euler Solver ...................................................................59

3.5.1 Conservation Elements and Solution Elements ..................................603.5.2 Approximations within a Solution Element........................................64

3.5.3 Evaluation ofum ..................................................................................66

3.5.4 Evaluation ofumx, umy and umz.............................................................69

4. Parallel Computation ...............................................................................................72

4.1 Introduction....................................................................................................724.2 Hardware System...........................................................................................78

4.3 Software and Programming ...........................................................................844.4 System Specifications ....................................................................................88

4.5 Useful Links...................................................................................................99

5. Detonations ............................................................................................................100

5.1 Introduction..................................................................................................1005.2 One-Dimensional Detonations.....................................................................1035.3 Two-Dimensional Detonation Waves..........................................................110

5.4 Three-Dimensional Detonation Waves in a Duct ........................................111

5.5 Detonations in Circular and Annular Tubes ................................................130

6. Implosion and Explosion .......................................................................................146

6.1 Introduction..................................................................................................1456.2 Two-Dimensional Implosion and Explosion ...............................................150

6.3 Three-Dimensional Implosion and Explosion .............................................163

7. Pulse Detonation Engine Aeroacoustics ................................................................172

7.1 Introduction..................................................................................................172

7.2 Initial Condition of Detonation Wave..........................................................178

7.3 Near Field Calculations................................................................................181

7.4 Far Field Calculations..................................................................................1937.5 Concluding Remarks....................................................................................202

8. Conclusions............................................................................................................203

8.1 Achievements and Findings.........................................................................2038.2 Recommendations for Future Work.............................................................207

Bibliography ..............................................................................................................211


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LIST OF TABLES

Table Page

4.1 Parallel computing performance ...................................................................86

4.2 Specifications of the1st

generation cluster in 1999 .......................................90

4.3 Specifications of the 2nd

generation cluster in 2000 .....................................92

4.4 Specifications of the 3rd generation cluster in 2002......................................95

4.5 Specifications of the 4th

generation cluster in 2003......................................97


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LIST OF FIGURES

Figure Page

2.1 A schematic of the ZND detonation wave....................................................15

2.2 Diagram of the Rayleigh lines and Hugoniot curves with different ..........21

2.3 Flow variable profiles of a one-dimensional ZND detonation wave fromthe analytical solution: (a) mass fraction of reactant; (b) pressure; (c)

density, and (d) velocity. Parameters: = 1.2, q0 = 50,E+ = 50, andf=

1.6..................................................................................................................26

3.1 A schematic of the space-time integral.........................................................29

3.2 Space-time meshes by the conventional finite volume methods in (a) one

and (b) two spatial dimensions .....................................................................30

3.3 Space-time geometry of the conventional finite volume methods inE2.......31

3.4 Schematics of the CESE method in one spatial dimension. (a)

Zigzagging SEs. (b) Integration over CE to solve u and ux at the new

time level.......................................................................................................34

3.5 CE+ and CE- for solving u and ux at the new time level in the a- scheme ..38

3.6 Grid points in the x-y plane for the original CESE method..........................39

3.7 Schematics of the modified CESE method in one spatial dimension: (a)

the staggered space-time mesh, (b) SE (j, n), shown as the yellow part,

and CE (j, n)..................................................................................................45

3.8 The space-time mesh in two spatial dimensions: (a) grid points in thex-y

plane, (b) SE and CE for the two-dimensional scheme ................................49

3.9 Spatial translation of the quadrilateral * * * *1 2 3 4A A A A .......................................57

3.10 A schematic of the 3D spatial mesh in thex-y-zspace.................................60


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4.1 A schematic of a Beowulf cluster .................................................................73

4.2 Domain decomposition examples: (a) mesh for a simple three-dimensional domain of a box, (b) tow-dimensional mesh for a

flow-over-cylinder problem, (c) a three-dimensional mesh of an annulus

tube; different colors represent different sub-domains .................................76

4.3 Network performance of two gigabit Ethernet systems with Intels

traditional x86 architecture and CSA, respectively, and a fast Ethernetsystem with single, double and triple fast Ethernet network interface

cards (NICs) installed on each computer node, where the channelbonding technology is used for the multiple card cases. Benchmarked by

netperf 2.1 .....................................................................................................81

4.4 A schematic of channel bonding technology applied on a Fast Ethernet

system with dual network cards used on each node .....................................82

4.5 A schematic of the speedup ..........................................................................86

4.6 The 1st

generation cluster in 1999.................................................................89

4.7 The 2nd

generation cluster in 2000................................................................91

4.8 The 3rd

generation cluster in 2002 ................................................................93

4.9 The 4th

generation cluster in 2003 in OSU ...................................................96

5.1 A schematic of the piston problem .............................................................104

5.2 Dimensionless shock front pressure history in the piston problem for

2.1= , 500 =q , 50+

=E , and (a)f= 2.0, (b)f= 1.6..............................105

5.3 Dimensionless time history of the shock front pressure in the instability

problems for 2.1= , 500 =q , 50=+E , and (a)f= 1.8, (b)f= 1.6,

with the mesh resolution 50 pts/L1/2............................................................107

5.4 The dimensionlessp-v diagram of the detonation for parameters 2.1= ,

500=

q , 50=

+

E andf= 1.8....................................................................108

5.5 Variation of the dimensionless peak pressure with mesh resolution forvarious schemes. The relative mesh spacing is defined as 5/n with n as

the number of mesh nodes for the half-reaction zone. Flow parameters:

2.1= , 500 =q , 50=+E , andf= 1.6.....................................................109


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5.6 Two-dimensional detonation waves: (a) mass fraction of reactant, (b)

pressure, (c) vorticity and (d) temperature. Flow parameters:f= 1.6, =1.2, q0 = 50,E

+= 50....................................................................................111

5.7 Snapshots of pressure contour at different time for a detonation wave

propagating in a square duct: (a) t= 12.0, (b) t= 13.5. The cross sectionsize is 88. The unit length is the half reaction length. Flow parameters:

f= 1.6, = 1.2, q0 = 50,E+

= 50 .................................................................113

5.8 Snapshots of different contour for a detonation wave propagating in asquare duct: (a) temperature, (b) concentration of the reactant. t= 13.5.

Cross section size: 88. Flow parameters:f= 1.6, = 1.2, q0 = 50,E+

=

50.................................................................................................................114

5.9 Snapshots of pressure contour on a cross section near the shock front of

detonation waves propagating in a square duct: (a),(b),(c) rectangular

mode; (d),(e),(f) diagonal mode. Cross section size: 88. Flowparameters:f= 1.6, = 1.2, q0 = 50,E

+= 50..............................................117

5.10 Numerical soot trace on the side wall: (a) rectangular mode, (b) under-

expanded diagonal mode, (c) fully developed diagonal mode with four

heads. Cross section size: 88 for cases (a) and (b), 2020 for case (c).

Flow parameters:f= 1.6, = 1.2, q0 = 50,E+

= 50.....................................119

5.11 Snapshots at different time: (a), (b), (c) pressure iso-surfaces; (d), (e), (f)

pressure contour on a cross section near the wave front. Cross section

size: 4040. Flow parameters:f= 1.6, = 1.2, q0 = 50,E+

= 50 ...............120

5.12 Numerical soot trace on one side wall. Cross section size: 4040. Flow

parameters:f= 1, = 1.2,E+

= 35, (a) q0 = 30; (b) q0 = 40; (c) q0 = 60.....122

5.13 Numerical soot trace on one side wall. Cross section size: 4040. Flow

parameters:f= 1, = 1.2, q0 = 50, (a)E+

= 20; (b)E+

= 40; (c)E+

= 45;

(d)E+

= 50...................................................................................................123

5.14 Numerical soot trace on one side wall. Cross section size: 4040. The

reactants are H2-Air mixtures at: (a) = 0.7; (b) = 0.8; (c) = 1.0; (d)

= 1.4 ........................................................................................................123

5.15 Cell width versus equivalence ratio for detonations in the H2-Air

mixtures. Cell width is normalized based on the minimal

numerical/experimental cell width obtained at the stoichiometric

condition ( = 1.0) .....................................................................................127


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5.16 Numerical soot trace of the two-dimensional calculation of a detonation

propagating in a H2-Air mixture at = 1.4................................................127

5.17 Time history of the pressure peaks at the shock front for two- and

three-dimensional calculations of a CJ detonation wave in a H2-Air

mixture at = 1.4. The peak pressure and time are both dimensionless...129

5.18 The TDW combustor geometry and the set-up...........................................132

5.19 A schematic of the mesh used for the three-dimensional calculations onspinning detonation in a circular tube. The tube radius is 2. The unit

length is the half reaction length.................................................................133

5.20 Calculated soot trace on the tube wall of a single head spinning detonationwave. The tube radius is 2. The unit length is the half reaction length.

Flow parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50 ....................................135

5.21 A schematic of a single-head detonation wave and comparison betweenthe experimental picture and the numerical result. Tube radius: 2. Flow

parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50..............................................135

5.22 A schematic of the single-head spinning detonation in a rotating frame....136

5.23 Pressure iso-surfaces of a spinning detonation at different time. Tube

radius: 2. Flow parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50 ....................138

5.24 Snapshots of pressure contour on the sliced x-zandx-y planes at

different time. Tube radius: 2. Flow parameters:f= 1.2, = 1.2, q0 = 50,E

+= 50. .......................................................................................................139

5.25 Snapshots of pressure contour on the tube wall of a spinning detonation

at different time. Tube radius: 2. Flow parameters:f= 1.2, = 1.2, q0 =

50,E+

= 50 ..................................................................................................140

5.26 Calculated multi-headed detonation wave in a round tube: (a) pressure

iso-surfaces; (b) pressure contour on the slicedy-zplane; (c) pressure

contour on the slicedx-y plane. Tube radius: 6. Flow parameters:f= 1.2,

= 1.2, q0 = 50,E+

= 50..............................................................................142

5.27 Calculated soot trace on the tube wall of a multi-headed detonation wave.

Tube radius: 6. Flow parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50 ...........143


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5.28 A schematic of the mesh for the 3D calculations on detonations in an

annular tube. The external and internal radius is 6 and 3, respectively. The

unit length is the half reaction length..........................................................143

5.29 Pressure iso-surfaces of an annular detonation at different time. External

and internal radius: 6 and 3. Flow parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50 .............................................................................................................144

5.30 Calculated pressure contour on they-zandx-zplanes of the multi-

headed detonation wave in an annular chamber. External and internal

radius: 6 and 3. Flow parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50 ...........145

5.31 Numerical soot trace on the outer and inner tube walls of the multi-headed

detonation wave in an annular chamber. External and internal radius: 6

and 3. Flow parameters:f= 1.2, = 1.2, q0 = 50,E+

= 50..........................145

6.1 A schematic of the mesh for 2D implosion/explosion calculations............151

6.2 Snapshots of pressure contour of a square front implosion at time = 0,

0.1, 0.2, 0.25, 0.3, 0.4, 0.5, 0.65, and 0.8. Domain radius: 1.Circumradius of the square front: 0.707, i.e., edge length 1.0.Pout/Pin=

20. out/in= 20............................................................................................153

6.3 Snapshots of pressure contour of an octagonal front implosion at time =

0, 0.09, 0.18, 0.27, 0.315, 0.36, 0.495, 0.63, and 0.72. Domain radius: 1.

Circumradius of the octagonal front: 0.7.Pout/Pin= 20. out/in= 20 ..........154

6.4 Snapshots of pressure contour of a circular front implosion at time = 0,0.09, 0.18, 0.27, 0.36, 0.45, 0.585, 0.72, 0.90, respectively. Domain

radius: 1. Radius of the circular front: 0.667.Pout/Pin= 20. out/in= 20.....155

6.5 Snapshots of pressure contour of a circular front implosion with

chemical reaction at time = 0, 0.09, 0.18, 0.27, 0.315, 0.36, 0.405, 0.54,0.90, respectively. Domain radius: 1. Radius of the circular front: 0.667.

Pout/Pin= 20. out/in= 20. Detonation parameters:f= 1, = 1.4, q0 = 50,

E+

= 20 ........................................................................................................156

6.6 The evolution of the polygonal shock. In each polygonal section, the

flow field is a planar shock wave entering a converging channel ..............157

6.7 Dimensionless time histories of (a) temperature, and (b) pressure, at the

focal point for the three non-reacting cases.Pout/Pin= 20. out/in= 20. .....161


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6.8 Dimensional time histories of (a) temperature, and (b) pressure, at the

focal point for the imploding reacting flow.Pout/Pin= 20. out/in= 20.Detonation parameters:f= 1, = 1.4, q0 = 50,E

+= 20 ..............................162

6.9 A schematic of the mesh for 3D implosion/explosion problems................164

6.10 Snapshots of pressure contour on the sliced planesx = 2 andy = 0 of an

octagonal front implosion at time = 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35,0.5, 0.75, 1.25. Tube radius: 1. Circumradius of the octagonal front: 0.75.

Pout/Pin= 20. out/in= 20.............................................................................167

6.11 Snapshots of pressure contour on the sliced planesx = 4 andy = 0 of an

octagonal front implosion at time = 0, 0.24, 0.4, 0.56, 0.8, 1.8, 2.8, 4.0.

Tube radius: 2. Circumradius of the octagonal front: 0.3.Pout/Pin= 20.

out/in= 20..................................................................................................169

6.12 Dimensionless time histories of (a) temperature, and (b) pressure, at thefocal point for the three cases with different initial condition and mesh

size. Tube radius: 2.Pout/Pin= 20. out/in= 20............................................171

7.1 A typical PDE cycle....................................................................................173

7.2 A schematic of setting up the initial condition ...........................................179

7.3 A schematic of computational domain for the near-field calculations .......181

7.4 A schematic of the sampling point distribution in the computational

domain for near field calculations. Domain size: 12.5 ft in length, 4.5 ftin radius.......................................................................................................183

7.5 Time history of pressure at the closed end of the PDE tube.......................184

7.6 Three snapshots of pressure contour of a PDE plume at different time of

the initial stage of expansion. Domain size: 12.5 ft in length, 4.5 ft in

radius. t= 0.642, 1.124, and 1.927 ms, respectively...................................186

7.7 Vortex-shock interaction in the near field of a PDE plume........................188

7.8 Time histories of pressure at two checking points located at (a) (5.66,5.875) and (b) (22.64, 5.875).................................................................189

7.9 Time histories of pressure and density at the location (5.66 in, 0.49 ft)

for a three-cycle calculation........................................................................190


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7.10 Calculated wave characteristics at sampling points along L-1, L-2, L-3

and L-4: (a) peak pressure, and (b) the wave speed. The experimental

data is measured on L-1 only......................................................................192

7.11 A schematic of the computational domain and the distribution of

sampling points for far field calculations. Domain size: 57 ft in length,30 ft in radius ..............................................................................................194

7.12 Three snapshots of pressure contour of a PDE plume at different time in

the far field. Domain size: 57 ft in length, 30 ft in radius...........................196

7.13 Time histories of pressure at various locations on the circles at (a) 10 ft

and (b) 50 ft away from the PDE tube exit .................................................198

7.14 Amplitude of pressure fluctuations along the central line at different

distance from the PDE tube exit .................................................................200

7.15 Sound strength (in dB) at different distance from the PDE tube exit .........201


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NOMENCLATURE

Variables and Constants:

Cp specific heat at constant pressure

Cv specific heat at constant volume

D detonation wave velocity

E total energy

E+

activation energy

e specific internal energy

ei sensible internal energy

ec chemical potential

f overdriven factor

f, fm flux component inx direction

gm flux component iny direction

h space-time density flux

h specific enthalpyhf heat of formationhm space-time density flux component

K pre-exponential factor of the Arrhenius kinetics

Kf forward reaction rate

L1/2 half reaction length

M Mach number

Mw molecular weight

n normal vector of space-time surface

P,p pressure

q0 heat release due to chemical reaction

qm flux component inzdirection

R gas constant

Ru universal gas constant

R, r radius


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S surface of space-time volume

s surface element of space-time volume

T temperature

t time

u x-component of velocityu, um conserved flow variables

V space-time volume

v y-component of velocity

v specific volume (1/)

Z mass fraction of the reactant

specific heat ratio

mass fraction of the product

m source term

density

area of space-time surface

equivalence ratio

source term due to chemical reactions

Subscripts:

0 unburned state of gas mixture

in inside value j temporal index of space-time mesh grid

n spatial index of space-time mesh grid

out outside value

ref reference value

CJ Chapman-Jouguet variable

Superscripts:

* dimensionless variable


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CHAPTER 1

INTRODUCTION

1.1 Literature Review

The Pulsed Detonation Engine (PDE) is a novel propulsion concept based on the

use of the high-pressure gases generated by repetitive detonation waves for thrust forces

(Kailasanath, 2001; Kailasanath et al., 1999; Li et al., 2000; Ebrahimi et al., 2000;

Schauer et al., 2001). For a fly regime of Mach 2 to 4, a PDE could improve vehicle

performance and cost effectiveness as compared with conventional air-breathing ramjet

engines. Though not yet implemented on any operational aircraft, there has been much

work done on the PDE related concepts during the past decades.

Among many technical issues about the PDE concept, it was recognized that

successful development of a PDE lies in the development of robust and repeatable

detonation initiation processes with minimum use of sensitized gases, e.g., oxygen, and

energy deposition in the forms of plasma and/or electric arcs. In the past, Nicholls et al.

(1957), Krzycki (1962), and Wortman et al. (1992) had difficulties in achieving reliable


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detonation initiation by using relatively insensitive fuel/air mixtures. Helman et al. (1986),

however, successfully achieved detonation in an ethylene/air mixture by first detonating

an ethylene/oxygen mixture inside a smaller tube, i.e., a pre-detonator or initiator. The

blast wave from the initiator would propagate into the main combustor chamber to

initiate the detonation wave for propulsion. Aided by a similar pre-detonator, Hinkey et al.

(1995) developed a hydrogen-fueled PDE. Nevertheless, reliable detonation initiation

process has been one of major stumbling blocks in the further development of the PDE

concept.

In addition to the detonation initiation process, it is also important to study the

aero acoustics of the PDE plume. It has been envisioned that a PDE would generate thrust

force by releasing a series of detonation waves from the detonation chamber. The

propagating blast wave at a frequency of 30 to 100 Hz would generate an intensive

acoustic field around the PDE. In the near field, theses blast waves could impose

significant vibrations to the fuselage of the vehicle. In the far field, the acoustic signature

of the vehicle could be of concern. Moreover, the developing blast waves in the near field

could also impact the fluid flow inside the detonation chamber. As a result, the PDE

cycle, i.e., purging, fueling, detonation initiation, and blow down processes, could be

influenced.


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As another subject, researchers are now investigating the possibility of developing

a continuous detonation engine by using a spinning detonation wave around an annular

chamber. Since it is a continuous detonation wave, no repetitive detonation initiation

processes, as that in a typical PDE, is needed. A group of Russian scientists at the

Lavrentev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences,

Novosibirsk, have been developing a novel detonation-based annular combustor since

late 50s (Mikhailov and Topchiyan, 1965; Nikolaev and Topchiyan, 1977;

Bykovskii and Mitrofanov, 1980, 2000; Bykovskii and Vedernikov, 1996). They

considered combustion of acetylene-oxygen mixtures in annular chambers with

constriction of the exit cross section. Experimental observation indicated that

conventional turbulent subsonic combustion of the mixture prevailed in a large part of the

annular combustor. However, further investigation showed distinct wave structure

occurred at the leading edge of the reaction zone and the combustion mode was actually

related to a spinning detonation wave along the circumferences of the annulus. The

reaction zone was apparently continuously initiated by the rotating detonation wave with

velocities close to the speed of the sound of the combustion products, i.e., the CJ velocity

of the detonation for the given fuel/air mixture. In the past decades, they reported various

flow geometries and operational conditions of the above-mentioned transverse detonation

waves (TDW) in annular cylindrical chambers. Combustion of both gaseous and

two-phase propellants was considered in their works.


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The most remarkable feature of these combustors is that the axial flow was

subsonic while the detonation waves traveled tangentially in the azimuthal

direction in the annular chamber. It is imperative to understand why the detonation

wave would be confined to spin at a near CJ velocity only in the azimuthal

direction without progressing upstream in the axial direction.

1.2 Motivation and Objectives

To fly higher and faster and even enter the earth orbit, the advanced propulsion

devices are required. Detonation wave could be used in the further development of

advanced propulsion systems. Computational Fluid Dynamics (CFD) has been playing an

important role in aerospace engineering for years. This Ph.D. research is a continuation of

these efforts in applying the state-of-the-art numerical method to modeling detonating

flows in the development of the envisioned PDE propulsion systems. The goals of the

present dissertation are twofold:

(i) By combining a high-fidelity CFD method and the modern parallelcomputing technology, we plan to develop a general numerical

framework to provide high-fidelity numerical solutions of unsteady

flows.

(ii) We plan to use this numerical tool to conduct detailed simulation ofdetonation waves and detonation initiation processes.


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5

Specific contribution by this research work include

(i) The extension of the space-time Conservation Element and SolutionElement (CESE) method for two- and three-dimensional chemically

reacting flows.

(ii) Quadrilateral cells and hexahedral cells will be used for two- andthree-dimensional calculations, respectively.

(iii) The solver supports parallel computing for fine-mesh resolution inlarge spatial domains in both two and three spatial dimensions.

(iv) Thorough code validation by comparing the CFD results with theclassical ZND solutions, and in-depth studies of the aerodynamic

structure of the three-dimensional detonation waves.

(v) Detailed simulation of the PDE plume dynamics for acousticdistribution, which is pertinent to a prototype PDE being tested at the

NASA Glenn Research Center (GRC);

(vi) Study of the detonation initiation process by using theimplosion/explosion mechanism, which will play a key role in an

operational PDE in the future.


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1.3 Numerical Methods

The CESE method is a novel numerical scheme for nonlinear hyperbolic

conservation laws. Developed by Chang (1995) and coworkers (Chang et al., 1999; Yu

and Chang, 1997; Wang and Chang, 1999; Zhang et al., 1999, 2002), the tenet of the

CESE method is a unified treatment of space and time in calculating flux conservation.

Contrast to modern upwind schemes, the method does not use a Riemann solver or a

reconstruction procedure as the building blocks. Nevertheless, the shock capturing

capabilities of the CESE method is superb. In particular, the CESE method is capable of

capturing shocks and acoustic waves simultaneously, while the magnitude of the pressure

jump across the shock wave is several orders in magnitude greater than the averaged

amplitude of pressure fluctuations in the acoustic waves.

In the setting of the CESE method, the space-time domain is divided into

non-overlapping Conservation Elements (CEs), in which the space-time flux conservation

is imposed by integrating the conservation laws in a novel space-time integral form.

Inside a CE, the flow solution could be discontinuous. Since the summation of the

non-overlapping CEs is equal to the whole space-time domain, flux conservation is

enforced locally and globally. To facilitate the integration over CEs, non-overlapping

Solution Elements (SEs) are also defined in the space-time domain. Inside each SE, flow

variables and fluxes are assumed to be continuous and are discretized by a predetermined


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7

function. No flow discontinuity is allowed inside a SE. The flow discontinuity only

occurs across the boundary of neighboring SEs. In general, SEs do not coincide with CEs.

In this work, the flow solution in each SE is discretized by the first-order Taylor series

expansion. Thus, a linear distribution of the flow variables is assumed in the SEs, and the

scheme is second-order accurate in both space and time.

To date, numerous highly accurate solutions have been obtained by the CESE

method, including cavitations (Qin et al., 2001), aero acoustics of supersonic jets (Loh et

al., 1995), turbulent flows with dense sprays (Im et al., 2004), MHD flows (Zhang et al.,

2003), and flows with complex shock systems (Wang et al., 1998). Previously,

applications of the CESE method to detonations were carried out by Park et al. (1999).

Important detonation wave features were reported, including the values of the peak

pressures and the oscillation frequencies of certain one- and two-dimensional detonation

waves. Further investigation of two- and three-dimensional detonations by the CESE

method was reported by Zhang et al. (2001) and He et al. (2001). Recently, Im et al.

(2002) and Wang et al. (2004) applied the CESE method to calculate certain detonation

initiation processes with detailed finite-rate chemistry models. They reported detonation

initiation results by shock reflection in conventional shock tubes, and spherical and

cylindrical implosion/explosion processes. Chemical reactions of H2/O2/Ar and C3H8/air

gas mixtures were simulated by finite-rate models.


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In the present numerical framework, parallel computing is an important part,

because large-sized fine meshes require huge computer resources, which may exceed the

power of a single PC or a workstation. In this research work, parallel computing has been

preformed by using a Beowulf computer cluster. In the numerical method, parallel

computing is realized by domain decomposition. The mesh is first partitioned either

manually or by using software, e.g., METIS. In the calculations, the data transmission

between the computer nodes was done by using Message Passing Interface (MPI).

1.4 Organization

This dissertation is organized as follows: Chapter 2 presents the theoretical model

equations for detonations. Chapter 3 illustrates the CESE method. Chapter 4 introduces

the parallel computing technology and its implementation for the present work. Chapter 5

addresses the studies on detonation waves in one, two, and three spatial dimensions.

Chapter 6 discusses the two- and three-dimensional implosion/explosion processes used

for detonation initiation and transition. Chapter 7 presents the results for PDE plume aero

acoustics simulation. At the end, the effectiveness and limitation of the present research

is discussed and the conclusions are provided.


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CHAPTER 2

THE MODEL EQUATIONS

2.1 Three-Dimensional Euler Equations

The three-dimensional Euler equations for can be expressed in the following vector

form:

0m m m mu f g q

t x y z

+ + + =

, (2.1)

where m =1, 2, 3, 4, 5, indicating the continuity equation, the three momentum equations

along the three Cartesian coordinates, and the energy equation, respectively. um is the

flow variables,fm,gm, qm are the components of the flux vectors:

[ ]m

u

u v

w

E

=

,

2

[ ]

( )

m

u

u p

f uv

uw

E p u

+ = +

, 2[ ]

( )

m

v

uv

g v p

vw

E p v

= + +

,2

[ ]

( )

m

w

wu

q wv

w p

E p w

=

+ +

,

(2.2)

where is the density, u, v, w are the three velocity components, andp is the pressure.


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The specific total energyEis defined as

2 2 2 2 2 21 1( ) ( )

2 ( 1) 2

pE e u v w u v w

-= + + + = + + +

, (2.3)

where e is the specific internal energy, and = Cp/Cv is the specific heat ratio.

To proceed, the above equations are non-dimensionalized based on a reference

state, denoted by a subscript 0. Usually the standard state or the freestream conditions

are taken. A reference velocity is defined as 0RT , which resembles the speed of the

sound in the reference state. Furthermore, a reference length x0 is needed. It is usually

defined as one of the geometric characteristic length, e.g., the diameter of the hole for a

free jet problem. The specific total energy Eand the specific internal energy e are made

dimensionless byRT0.

With the above definitions of the reference state and the space and time scales, the

dimensionless variables (denoted by a bar) are defined as follows:

0

=

,0p

pp =

,0

0

0 p

RT

T

TT

==

,0RT

uu =

,0RT

vv =

,0RT

ww =

,

0x

xx =

,00 / RTx

tt =

,0RT

EE =

. (2.4)

By substituting these dimensionless variables into the governing equations, one obtains a

set of dimensionless Euler equations, which remain unchanged in the form as compared

with Eqs. (2.1)-(2.3).


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2.2 Euler Equations for Reacting Flows

Compared with the original form of the Euler equations, the Euler equations for

reacting flows have an additional equation for chemical reactions. The three-dimensional

version can be expressed in the following vector form:

mmmmm

z

q

y

g

x

f

t

u=

+

+

+

, (2.5)

where m =1, 2, 3, 4, 5 and 6, indicating the continuity equation, the three momentum

equations along the three Cartesian coordinates, the energy equation, and the species

equation, respectively. There is one more term shown in um,fm,gm, qm, andm:

=

Z

Ew

v

u

um

][ ,

+

+

=

uZ

upEuw

uv

pu

u

fm

)(

][

2

,

+

+=

vZ

vpEvw

pv

uv

v

gm

)(

][

2

,

++

=

wZ

wpEpw

wv

wu

w

qm

)(

][2

,

=

0

0

0

0

0

][ m , (2.6)

whereZis the mass fraction of the reactant. The specific total energyEis defined as

2 2 2 2 2 2

0 0

1 1( ) ( )

2 ( 1) 2

pE e Zq u v w Zq u v w

-= + + + + = + + + +

, (2.7)


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where q0 is the heat release due to chemical reactions. To close the equation set, the gas

mixture is assumed polytropic, i.e., (i) the mole number of the reacting gas mixture is a

constant; (ii) the molecular weight of the reacting gas mixture is a constant; (iii) the gas

mixture is ideal, and (iv) the specific heats Cp and Cv of the gas mixture are constants. For

an ideal gas,p = RT, where Tis the temperatureandR = Ru/Mw is the gas constant with

Ru as the universal gas constant and Mw the molecular weight of the gas mixture. To

model the chemical reactions, a one-step irreversible reaction is employed. The source

term in the species equation due to chemical reaction in Eq. (2.6) can be written as

ZTREK u )/exp(+=

, (2.8)

whereKis the Arrhenius coefficient andE+

is the activation energy. The species equation

and the continuity equation will enforce mass conservation of the reactant, the product,

and the reacting gas mixture.

In the present research, the above equations and their two-dimensional version are

used to simulate two- and three-dimensional detonations and implosion/explosion

processes.

To model PDE plumes, the axis-symmetric version of the above equations is

employed:

mmmm

y

g

x

f

t

u=

+

+

, (2.9)


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where the flow variables um, the fluxfm,gm, and the source termm are

=

Z

E

vu

um

][ ,

+

+=

uZ

upE

uvpu

u

fm

)(

][

2

,

+

+=

vZ

vpE

pv

uv

v

gm

)(

][2

,

+

+

=

yvZ

yvpE

yv

yuv

yv

m

/

/)(

/

/

/

][ 2 . (2.10)

To proceed, the above equations are made dimensionless and the state of the

unburned gas is taken as the reference state. As to the reference length, to calculate the

ZND detonations, we let x0 =L1/2, the half- reaction length, defined as the distance from

the shock front to the point where a half of the reactant is consumed by the combustion.

For the PDE plume calculations, we let x0 = R, whereR is the radius of the thruster tube.

The heat release q0 are made dimensionless by RT0. The activation energy E+

is made

dimensionless byRuT0.

Thus, in addition to Eqs. (2.4), we have the following definition of the

dimensionless variables (denoted by a bar):

00 /xRT

KK =

,0TR

EE

u

++ =

,0

0

0RT

qq =

. (2.11)


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By substituting these dimensionless variables into the governing equations, a set of

dimensionless Euler equations is obtained, which remain unchanged in the form as

compared with Eqs. (2.5)-(2.6), or (2.9)-(2.10) for the PDE simulations, except the source

term :

)/exp( TEZK += . (2.12)

Note that the universal gas constantRu in the exponential function has been absorbed into

the dimensionless activation energy+E . Since the derived dimensionless equations are

identical to their dimensional counterparts except Eq. (2.12), the bar for dimensionless

variables will be dropped in the following discussions.

2.3The ZND Solution

Although the classical Zeldovich, von Neumann, and Doering (ZND) solution has

been widely cited, I have not been able to find the complete derivation in the literature.

As a part of this dissertation work, the complete ZND solution has been carefully derived

based on incomplete information and hints provided in the open literature. The successful

use of the classical ZND solution for code validation and as the initial condition is critical

to the present Ph.D. research.

In the ZND model for detonation, we assume that (i) the flow is one dimensional;

(ii) transport effects, i.e., heat conduction, radiation, diffusion, and viscosity, are


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neglected; (iii) the reaction rate is null ahead of the shocks and at a finite rate

immediately behind the shock; (iv) chemical reactions are modeled by a one-step,

irreversible reaction step at a finite rate; and (v) all thermodynamic variables other than

the chemical composition are in local thermodynamic equilibrium. Refer to Fig. 2.1 for a

schematic of the ZND detonation wave.

Figure 2.1: A schematic of the ZND detonation wave.


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Consider the one-dimensional Euler equations for chemically reacting flow in the

dimensional form,

mmm

xf

tu =

+

. (2.13)

In a vector form, the flow variables and fluxes are

=

Z

E

uum

][ ,

+

+=

uZ

upE

pu

u

fm

)(][

2

,

=

0

0

0

][ m , (2.14)

where the total energy 2/2ueE += and internal energy ci eee += .

To simplify the analysis, the polytropic assumptions are invoked (refer to the

discussion right below Eq. (2.7)). Because the gas mixture is ideal, we have Cp = R/(1-)

and Cv = R/(1-) with = Cp/Cv and R = Cp - Cv. Furthermore, )1/( == pvTCe vi

where v = 1/ is the specific volume.

We also assume that the reacting gas mixture is composed of only two species: A

as the reactant and B as the product. Let fBfA hhq =0 where hfA and hfB are the heat of

formation of species A and B. The specific enthalpy of the reacting gas mixture is

00)1()( ZqRTTCZqTChZZhTTCh vpfBfArefp ++=+=++= , (2.15)

where the heat of formation of species B is defined as refpfB TCh = . Therefore, we have

0Zqec = and 0)1/( Zqpve += . We consider a simple one-step irreversible

reaction between the reactant (A) and the product (B): BA fK . According to the law

of mass action,


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[ ] [ ]AKdtAd f=/ , and [ ] [ ]AKdtBd f=/ , (2.16)

where [A], [B] are the mole concentration of species A and B, respectively. According to

the Arrhenius kinetics, )/exp( TREKK uf+

= . The source term in the species equation

due to chemical reaction can be written as

ZTREK u )/exp(+= . (2.17)

To be consistent with the classical ZND model, we reformulate the species equation

based on the mass fraction of the product instead of the reactant. Let = 1-Zbe the

mass fraction of the product, which is frequently referred to as the progress variable since

= 1indicates completion of the chemical reaction. The species equation becomes

=

+

+

TR

EK

x

u

t uexp)1(

)(

. (2.18)

In this case, the definition of internal energy needs further clarification. Let the

specific enthalpy of the gas mixture be

0( ) (1 )p ref fA fB ph C T T h h C T q = + + = . (2.19)

Here the heat of formation of species A is defined as refpfA TCh = . Note that when

formulating in terms of Z, Eq. (2.15) implies hfB = CpTref. As a result, we have

0001

1

qpv

qRTRT

pvqTCe p

=

== . (2.20)

And the total energyEbecomes

21

2

0

uq

pvE +

=

. (2.21)


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We assume that the detonation wave propagates at a constant velocity D, and

u =u-D is the fluid velocity in the shock frame. We then transform the coordinates such

that the spatial origin is on the shock front. The equations of this steady problem in the

new coordinate system become

0)(

=dx

ud , (2.22a)

0)( 2

=+

dx

pud , (2.22b)

[ ] 0)( =+dxupEd , (2.22c)

)1(exp)(

=

+

TR

EK

dx

ud

u

. (2.22d)

For convenience, the carat on the top of velocity u is dropped in the following

discussions.

To proceed, we substitute the continuity equation (2.22a) into the energy equation

(2.22c), and we have

02

2

=

++

upve

dx

d, (2.23)

where v = 1/ is the specific volume. Integrating the continuity (2.22a), momentum

(2.22b), and energy (2.23) equations gives the classical Rayleigh line relation:

vv

ppu

=

0

02)( , orvv

ppvu

=0

022 . (2.24)


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This relation states that the mass flow rate u, which is a constant due to mass

conservation, is equal to the square root of the negative slope of a line in the (p, v) plane

connecting the initial state and an evolving state. Because mass and momentum must

conserve, a process evolving on a (p, v) plane is restricted to occur along a straight line,

i.e., the Rayleigh line. Moreover, because (u)2

is always positive, a steady combustion

process described on a (p, v) plane cannot be simultaneously lower or higher in both

pressure and specific volume.

We then substitute the Rayleigh line relation into the integral form of the energy

equation (2.22c), and we have

)11

)((2

1

0

00

+= ppee , or ))((2

1000 vvppee += . (2.25)

Consider Eq. (2.20) and let = 0 for the unburned state. We then obtain the Hugoniot

curve:

0))((2

1

1000

00 =++

qvvpp

vppv

. (2.26)

Based on a given starting state, the above equation provides a set of hyperbolic curves,

which restricts the evolution of the thermodynamic properties on a (p, v) plane.

Depends on the evolution of the progressive variable , a specific Hugoniot curve could

be identified, and viable solutions must locate at the intersection points of the Rayleigh

line and Hugoniot curves.


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Next, we nondimensionalize the above formulation for further analyses, refer to

Eqs. (2.4) and (2.11). The dimensionless variables, denoted by the superscript *, for the

unburned state, denoted by 0, can be derived as

1*0*

0

*

0 === Tp , 0*

0 Mu = , and )1/(1*

0 = e . (2.27)

Thus, the equation of state changes to

*** Tvp = . (2.28)

The continuity equation becomes

**

0

** Duu == . (2.29)

The dimensionless Rayleigh line relation, i.e., Eq. (2.24), becomes

)1/()1( **20 vpM = , (2.30a)

and the dimensionless Hugoniot curve relation is

02

)1)(1(1

)1( *0

****

=++ qvpvp

. (2.30b)

For convenience, the superscript * and the subscript 0 in M0 denoting the Mach

number of the unburned gas will be dropped in the following discussions. We remark that

in commonplace textbooks for combustion, Eqs. (2.30) would complete the discussions

of the classical Rankine-Hugoniot (R-H) relations. Usually, a graphic illustration of

viable solutions on the (p, v) plane is provided. Refer to Fig. 2.2. However, we would

continue to derive the analytical solutions forp and v based on the use of Eqs. (2.30a-b).


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Figure 2.2: Diagram of the Rayleigh lines and Hugoniot curves with different .

By the given values for the , M, and , two equations of Eqs. (2.30) determine

two unknowns, i.e.,p and v. To proceed, we have the following equation forp from the

Rayleigh line equation, Eq. (2.30a),

2)1(1 Mvp += . (2.31)

Substitute Eq. (2.31) into the Hugoniot curve relations, Eq. (2.30b), and we have

0)1)(2(2

1

1

1)1(0

22222

=++

+ qvvMM

vMvM

. (2.32)

By solving the above second-order polynomial equation forv and simplifying the result,

we have

[ ])(1)1(

12

2

w

M

Mv

++

= , (2.33)


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where

= 01)(q

, (2.34a)

22

22

)1(2

)1(

M

M

=

, (2.34b)

2

2

1

1

Mw

M

=

+. (2.35)

Note that both and w are constants, and () is a function ofonly. To proceed, we

substitute Eq. (2.33) into the Eq. (2.31), and we have the solution forp

[ ])(11

12

w

Mp

++

= . (2.36)

Based on the dimensionless equation of state, Eq. (2.28), we multiply Eq. (2.33) to Eq.

(2.36), to obtain the solution ofTas

[ ]22

22

22

)()()1(1)1(

)1(

wwM

M

pvT +

+

== . (2.37)

Clearly, solution ofp, v, and Tare functions of the progressive variable only. Moreover,

Eqs. (2.33)-(2.37) have two roots denoting the strong and weak detonation solutions

(refer to the two intersections denoted by S and W in Fig. 2.2). For the Chapman-Jouquet

(CJ) detonation, only one solution exists, and we have w() = 0 for= 1. According to

Eq. (2.35), a sonic unburned flow would render w = 0, which would also render

undetermined solution for (). Refer to Eqs. (2.34a-b). Therefore, the reasonable

solution for the CJ detonation is to let (1) = 0, orq0.


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The above discussions show that all thermodynamic variables of the detonating

gas could be calculated as function of the combustion progressive variable . To proceed,

we illustrate the calculation of flow variable inside the reaction zone. We consider the

dimensionless species equation at a steady state:

=

+

*

*

*

*

*exp

)1(

T

E

u

K

dx

d . (2.38)

Refer to Eq. (2.22d) for the dimensional species equation. For convenience, the

superscript * denoting the dimensionless variables will be dropped in the following

discussions. According to Eq. (2.29), u = -Dv in Eq. (2.38), where v can be expressed by

Eq. (2.33).

To proceed, we must first determine the pre-exponential factorKand detonation

speedD before we integrate Eq. (2.38). To calculate K, we use the rate equation, and by

rearranging the Eq. (2.38),

dTEu

dxKK )/exp(1

2/1

0

1

0

+ == . (2.39)

Note that the integration of the progressive variable is performed from 0 to 1/2.

Therefore, the length scalex0 is set to be equal to the half reaction zone length. Equation

(2.39) can be solved by numerical integration. We use Simpsons method. To calculate

the detonation velocityD, we use the definition of the overdriven factorf:


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2)/( CJDDf = . (2.40)

If the CJ shock speed DCJ and the over-driven factor are given, the shock speed D is

determined. To calculate DCJ, we let (1) = 0, and substitute = q0 into Eq. (2.34b) to

obtain

01)2/1(2224 =++ CJCJ MM , (2.41)

where

/)1(2 022 q= . (2.42)

By solving Eq. (2.41), we have

41

21

222

++=CJM . (2.43)

Here, we dropped the root with the negative sign, which is the solution at the deflagration

branch. By using Eqs. (2.27) and (2.29), we have

22

CJCJ MD = . (2.44)

For a given overdriven factorf, D can be calculated by Eq. (2.40). WithKandD

determined, the right hand side of the species equation, Eq. (2.38), is a function ofonly.

Integration of the ODE, i.e., Eq. (2.38) over a spatial domain gives the spatial distribution

of the mass fraction of the combustion product, leading to straightforward calculations of

spatial distribution of all thermodynamic variables in the reaction zone, including p, v,

and T.


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To recap, the input parameters to the above analytical model are , q0,E+,andf,

and the calculation procedure for the reaction zone structure is summarized in the

following.

(i) The detonation velocityD is calculated by Eqs. (2.40), (2.43), and (2.44).(ii) With the known value ofD, Eq. (2.39) is integrated to obtainK.(iii) By integrating Eq. (2.38), the distribution of inside the reaction zone

could be calculated.

(iv) With the given profile, the distribution ofp, v,and Tinside the reactionzone can be calculated by Eqs. (2.33), (2.36), and (2.37), respectively.

A typical ZND solution is shown in Fig. 2.3, including species distributions,

pressure, density, and velocity. The input parameters are = 1.2, q0 = 50,E+

= 50, andf=

1.6.


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(a) (b)

(c) (d)

Figure 2.3: Flow variable profiles of a one-dimensional ZND detonation wave from the

analytical solution: (a) mass fraction of reactant; (b) pressure; (c) density, and (d) velocity.

Parameters: = 1.2, q0 = 50,E+

= 50, andf= 1.6.


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CHAPTER 3

THE SPACE-TIME CESE METHOD

3.1 Conventional Finite Volume Method

Finite volume methods are formulated according to flux balance over a fixed

spatial domain. The conservation laws state that the rate of change of the total amount of

a substance contained in a fixed spatial domain, i.e., the control volume V, is equal to the

flux of that substance across the boundary ofV, denoted as S(V). Consider the differential

form of a conservation law as follows:

0u

t

+ =

f , (3.1)

where u is density of the conserved flow variable, fis the spatial flux vector. By applying

Reynoldss transport theorem to the above equation, one can obtain the integral form as:

( )0

V S VudV d

t

+ =

f s , (3.2)


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where dVis a spatial volume element in V, ds = dn with dand n being the area and the

unit outward normal vector of a surface element on S(V) respectively. By integrating Eq.

(3.2) over the time interval (ts, tf), we have

( )( ) 0f

sf s

t

V V t S V t t t t

udV udV d dt = =

+ = f s . (3.3)

The discretization of Eq. (3.3) is the focus of the finite-volume methods.

Lets consider the one dimensional case first. Let time and space (length) be the

two orthogonal coordinates of a space-time system, i.e., x1= x andx2= t. They constitute

a two-dimensional Euclidean space E2. Define ( , )f uh , then by using the Gauss

divergence theorem, Eq. (3.1) becomes

( )0

S Rd = h s , (3.4)

whereR is a space-time region inE2 and S(R) is the boundary ofR, ds = dn with dand

n being the area and the unit outward normal vector of a surface element on S(R)

respectively. Equation (3.4) states that the total space-time flux h leaving the space-time

volumeR through S(R) vanishes. Refer to Fig. 3.1 for a schematic of Eq. (3.4). Note that

all mathematical operations can be carried out as thoughEN were an ordinary Euclidean

space. For the current caseN= 2.


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r+dr

r

dr

ds

V

S(V)

t

x

Figure 3.1: A schematic of the space-time integral.

Figure 3.2(a) shows the arrangement of conservation cells for the one spatial

dimensional problem in the finite volume methods. Note that conventional finite volume

methods discretize and perform integration on the spatial dimensions only, as shown in

Eq. (3.3). Thus fixed control volume in space is required. Due to this constraint, the shape

of the conservation cells must be rectangular on the x-tplane. Refer to Fig. 3.2(a). The

mesh points are usually placed at the center of the spatial mesh, i.e., on the boundary of

the space-time conservation cells (marked by dots in Fig. 3.2(a)). The conservation cells

must stack up exactly on top of each other; no staggering in time is allowed. This

arrangement results in vertical interfaces extended in time between the neighboring

conservation cells. Across the interface, flow information travels in both directions. As

such, interfacial fluxes generally must be evaluated by interpolating the data from the


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mesh points embedded in these two cells. Determining how this interpolation should be

carried out properly under temporally evolving solution is a difficult problem. Usually an

upwind bias method or a Riemann solver must be employed to calculate the nonlinear

fluxes.

(a)

(b)

Figure 3.2: Space-time meshes by the conventional finite volume methods in (a) one

and (b) two spatial dimensions.


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To proceed, consider a rectangular space-time region R with vertices ABCD on

thex-tplane, as depicted in Fig. 3.3. Then the surface ofR, i.e., S(R) is formed by the line

segmentsAB,BC, CD, andDA. Let t= ts at CD, t= tf atAB,x =xs atBC, andx =xfatDA.

Then because ( , )f u=h , Eq. (3.4) implies

0 f f f f

s s s s f s f f

x x t t

x x t t t t t t x x x x

udx udx fdt fdt = = = =

+ = . (3.5)

Figure 3.3: Space-time geometry of the conventional finite volume methods inE2.

The discretization in muilti-spatial dimensions is performed similarly. Refer to

Fig. 3.2(b) for the two-dimensional case. A conservation cell is a uniform-cross-section

cylinder in the space-time domain. Note that though a circular cross section is shown, the

cross section could be of any shape. Usually its rectangular or triangular for structured


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and unstructured meshes, respectively. Each conservation cell is a cylinder in space-time

with its spatial projection being the control volume in space and its top and bottom faces

representing two constant time levels. Again, because the control volume is a fixed

spatial domain, these conservation cells generally are stacked up exactly on the top of

each other. With this arrangement, the vertical interface that separates any two

neighboring cells will always be sandwiched between two neighboring columns of mesh

points (as shown in Fig. 3.2(a) for one-dimensional case). As such, the difficulty in

evaluating the flux at the vertical interface of two neighboring conservation cells always

exists. As will be shown, with a new space-time discretization and arrangement of

conservation cells adopted in the space-time CESE method, the above difficulty can be

bypassed completely.

3.2 The CESE Method

In contrast to the conventional finite volume methods, the space-time CESE

method treats space and time equally. The integration of Eq. (3.4) is performed on both

space and time dimensions to solve marching variables in the new time levels. Moreover,

the CESE method has separate definitions of Conservation Element (CE) and Solution

Element (SE). SEs are used to discretize the space-time domain, and the conservation

laws are performed over CEs to solve the unknowns. CEs are non-overlapping space-time

domains such that (i) the whole computational domain can be filled by the union of all


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CEs; (ii) flux conservation is enforced over each CE or over a union of several

neighboring CEs; and (iii) inside a CE, flow discontinuity is allowed. SEs are

non-overlapping space-time domains such that (i) an SE does not necessarily coincide

with a CE; (ii) the union of all SEs does not have to fill the whole computational domain;

(iii) flow variables and fluxes could be discontinuous across interfaces of neighboring

SEs; and (iv) within an SE, flow variable and fluxes are assumed continuous, and they

are approximated by the first-order Taylor series expansion in both space and time.

Consider the one dimensional case:

0u f

t x

+ =

. (3.6)

Refer to Fig. 3.4(a), the space-time domain is divided into many non-overlapping

rhombus SEs. Each SE is associated with a grid point (marked by a dot). In the SE, flow

variables are assumed continuous and a Taylor series is used to discretize the equation.

For any (x, t) SE (j, n), u(x, t), f(x, t) and h(x, t), are approximated by *( , ; , )u x t j n ,

*( , ; , )f x t j n , and *( , ; , )t j nh , respectively. Let

*( , ; , ) ( ) ( ) ( ) ( )n n n n j x j j t ju x t j n u u x x u t t + + , (3.7)

where (xj, tn) are the coordinates of the mesh point (j, n) at the center of SE(j, n). Note

that nju , ( )n

x ju , and ( )nt ju , as the numerical analogues of the values of u, ux, ut

respectively, are constant in SE(j, n).


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(a)

(b)

Figure 3.4: Schematics of the CESE method in one spatial dimension. (a) Zigzagging SEs.

(b) Integration over CE to solve u and ux at the new time level.


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Let nj

f and ( ')njf denote the value offand df/du when u assumes the value ofn

ju .

Let

( ) ( ') ( )n n n

x j j x j f f u

, and ( ) ( ') ( )n n n

t j j t j f f u

. (3.8)

Because

f f u

x u x

=

, and

f f u

t u t

=

, (3.9)

( )nx j

f and ( )nt j

f can be considered as the numerical analogues of the value off/x and

f/tat (xj, tn), respectively. As a result, we assume that

*( , ; , ) ( ) ( ) ( ) ( )n n n n j x j j t jf x t j n f f x x f t t + + . (3.10)

Because h = (f, u), we also assume that

* * *( , ; , ) ( ( , ; , ), ( , ; , ))x t j n f x t j n u x t j nh . (3.11)

Note that, by their definitions, njf and ( ')

n

jf are functions ofn

ju ; ( )n

x jf are functions

of nju and ( )

n

x ju ; and ( )n

t jf are functions ofn

ju and ( )n

t ju .

Moreover, we assume that, for any (x, t) SE (j, n), u = *( , ; , )u x t j n and f =

*( , ; , )f x t j n satisfy Eq. (3.6), i.e.,

* *

( , ; , ) ( , ; , ) 0u x t j n f x t j nt x

+ =

. (3.12)

According to Eqs. (3.7) and (3.10), the above equation is equivalent to

( ) ( )n nt j x ju f= . (3.13)


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Since ( )nx jf are functions ofn

ju and ( )nx ju , Eq. (3.13) implies that ( )

n

t ju are also

functions of nj

u and ( )nx ju . Thus, we conclude that the only independent discrete

variables needed to solve aren

ju and ( )n

x ju .

To proceed, let the space-time domain be divided into non-overlapping rectangular

CEs. Refer to Fig. 3.4. Each CE is associated with a mesh point. We then employ local

space-time flux conservation over CEs to solve the unknowns. Figure 3.4(b) shows the CE

and SE associated with mesh points (j, n), and two neighborinh SE associated with mesh

points (j-1, n-1) and (j+1, n-1)on the previous time level. We assume that u* and ux* at mesh

points (j-1, n-1) and (j+1, n-1) are known and used to calculate nju and ( )n

x ju at the new

time level n. Note that the line segments forming the boundary of CE(j, n) belong to the

three SEs respectively. As a result, by imposing flux conservation over the CE, i.e.,

*

( ( , ))0

S CE j nd = h s , (3.14)

one obtains

1/ 2 1/ 2 1/ 2 1/ 2

1/ 2 1/ 2 1/ 2 1/ 2( ) / 2n n n n n

j j j j ju u u s s + += + + , (3.15)

where

2

( / 4)( ) ( / ) ( / 4 )( )n n n n

j x j j t js x u t x f t x f = + + , (3.16)

Given the values of the marching variables at t= tn-1/2

, the value ofujn

can be calculated

explicitly.


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To solve (umx)jn

at point (j, n), central differencing is performed:

( ) [( ) ( ) ] / 2n n n x j x j x ju u u+ = + , (3.17)

where

1/ 2( ) ( ) /( / 2)n n n

x j j ju u u x

= , (3.18)

1/ 2 1/ 2

1/ 2 1/ 2 1/ 2( / 2)( )n n n

j j t ju u t u

= + . (3.19)

For flows with discontinuities, Eq. (3.17) is replaced by a re-weighting procedure to add

artificial damping:

( ) (( ) , ( ) , )n n n x j x j x ju W u u += , (3.20)

where the function Wis defined as

( , , ) x x x xW x xx x

+ + +

+

+=+

. (3.21)

is an adjustable constant, and usually = 1 or 2.

Compared with the conventional finite volume method, there is no fixed spatial

domain constraint in the CESE method. Thus, the SEs could be arranged in a stagging

way. In time marching calculations, the flow variables at neighboring cells leapfrog each

other. Across oblique interfaces of neighboring SEs, discontinuities of u*

(and h*) are

allowed. Flow information propagates only in one direction, i.e., toward the future, through


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each oblique interface. While within the SEs, the unknows (including the flow cariables

and their derivatives) are distributed linearly. Thus, the calculation of the space-time flux

passing these interfaces of CEs is simple and straightforward. No Riemann problem is

encountered for the nonlinear equations.

Note that there is a family of schemes constructed with the CESE method. The

above introduced method is celled a- scheme because of the introduced adjustable

constant . In the a- scheme, ux is not solved by the above mentioned differencing method.

Instead, the two unknowns u and ux are treated equivalent and solved simultaneously by

constructing two CEs within the same three SEs. Its straightforward to construct two

equations to solve two unknowns.

Figure 3.5: CE+ and CE- for solving u and ux at the new time level in the a- scheme.


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To proceed we briefly discuss the extension to two spatial dimensions. For

equations in two spatial dimensions, the SEs and CEs are in a three-dimensional Euclidian

spaceE3. With three unknowns u, ux, and uy now, according to the idea of the a- scheme,

three equations must be provided to solve the new flow variables at the new time step. Thus

triangle meshes is used. Figure 3.6 shows the mesh arrangement for the two spatial

dimensions. Note that the unknowns are located at the centroid of each triangle.

Similarly, tetrahedron meshes are used in three spatial dimensional cases and four

unknowns (u, ux, uy, and uz) at each mesh node are calculated for three spatial dimensional

equations. Details of the original CESE schemes in multiple spatial dimensions please refer

to Wang and Chang (1999).

Figure 3.6: Grid points in thex-y plane for the original CESE method.


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Note that in the present work, the a- scheme is used and extended to solve Euler

equations in two and three spatial dimensions. Hence only one CE is constructed to solve

u and the spatial derivatives are obtained by the differencing method. Therefore, different

with the original mesh type settings, quadrilateral and tetrahedron cells are used in two-

and three- dimensional cases, respectively. See the following sections for details.

3.3 The Modified CESE Method

Consider the one-dimensional Euler equations for reacting flows with source term

for chemical reactions, refer to Eqs. (2.13)-(2.14):

m mm

u f

t x

+ =

, m = 1, 2, 3, 4. (2.13)

Notem is the function ofum, m = 1, 2, 3, 4. Letx1= x,x2= tand ( , )m m mf uh , then by

using the Gauss divergence theorem, we have the similar equation as Eq. (3.4) except the

source term on the right hand side (RHS):

( )m m

S R Rd dR = h s , m = 1, 2, 3, 4. (3.22)

By the same definition of SE and CE as shown in Fig. 3.4, for any (x, t) SE (j, n), um (x,

t), fm (x, t) and hm (x, t), are approximated similarly by*

( , ; , )u x t j n ,*

( , ; , )f x t j n , and

*( , ; , )x t j nh , respectively. By the linear distribution assumption within SEs, we have

* ( , ; , ) ( ) ( ) ( ) ( ) ( )n n n nm m j mx j j mt ju x t j n u u x x u t t + + . (3.23)


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Let ( )nm jf and ,( )n

m l jf denote the value offm and fm/ul, m, l = 1, 2, 3, 4,

respectively, when um assumes the value of ( )n

m ju . Let

4

,

1

( ) ( ) ( )n n nmx j m l j lx jl

f f u=

, m = 1, 2, 3, 4, (3.24a)

and

4

,

1

( ) ( ) ( )n n nmt j m l j lt jl

f f u=

, m = 1, 2, 3, 4. (3.24b)

Because

4

1

m m l

l l

f f u

x u x=

=

, and

4

1

m m l

l l

f f u

t u t=

=

, (3.25)

( )nmx jf and ( )n

mt jf can be considered as the numerical analogues of the value of fm/x

and fm/tat (xj, tn), respectively. As a result, we assume

*

( , ; , ) ( ) ( ) ( ) ( ) ( )n n n n

m m j mx j j mt jf x t j n f f x x f t t = + + , (3.26)

and

* * *( , ; , ) ( ( , ; , ), ( , ; , ))m m mt j n f x t j n u x t j n=h . (3.27)

Note that, by their definitions, for any m = 1, 2, 3, 4, ( )nm jf and ,( )n

m l jf are functions of

( )nm ju ; ( )n

mx jf are functions of ( )nm ju and ( )

n

mx ju ; and ( )nmt jf are functions of ( )

n

m ju

and ( )nmt ju .

Assume that, for any (x, t) SE (j, n), um =* ( , ; , )mu x t j n and fm =

*( , ; , )mf x t j n

satisfy Eq. (2. ), i.e.,

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