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Propagation rates, instabilities, and scaling for hydrogen flames

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Propagation rates, instabilities, and scaling for hydrogen flames

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Content PROPAGATION RATES, INSTABILITIES, AND SCALING
FOR HYDROGEN FLAMES
by
Zhenghong Zhou
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
December 2023
Copyright 2023 Zhenghong Zhou

ii
Acknowledgments
I owe deeply to my family, whose unwavering support has been the cornerstone of my
academic journey. Completing my doctoral degree would have been an impossible task without
their encouragement.
I want to extend my sincere gratitude to the esteemed faculty at the University of Southern
California, who have served as academic role models. I am particularly thankful to Professor
Carlos A. Pantano-Rubino, Professor Fokion N. Egolfopoulos, Professor Satwindar S. Sadhal, and
Professor Theodore T. Tsotsis for their roles on my qualify/defense committee. Special
acknowledgment goes to my advisor, Professor Paul D. Ronney, for his invaluable guidance in both
my academic and personal development as a researcher. His mentorship provided me with
opportunities to broaden my horizons, including travel within the United States and to Europe.
To my friends and lab mates in CPL, your camaraderie has been a source of strength and
inspiration. While it is impossible to name everyone who has contributed to technical discussions
and experimental support, I must express my gratitude to Ashkan Davani, Eugene Kong, Si Shen,
Hang Song, Jakrapop (Boom) Wongwiwat, Brandie Rhodes, Yang Shi, Patharapong (Winry)
Bhuripanyo, Mar Battistella, Jui-Yang (Ray) Wang, Shih-Yao (BoB) Huang, Fares Maimani, and
Benjamin Cohen. The challenges we faced, including accidents like fires and explosions, have left
an indelible mark on my life.

iii
Finally, a special thank you to Kaiyuan Wang. Her support during the pandemic and through
various challenges has been invaluable. Her presence has made my life in Los Angeles so much
better.

iv
TABLE OF CONTENTS
Acknowledgments..................................................................................................................... ii
List of Figures.......................................................................................................................... vi
Nomenclature........................................................................................................................... xi
Abstract.................................................................................................................................. xiv
Chapter 1................................................................................................................................... 1
1.1 Background..................................................................................................................... 1
1.1.1 Hydrogen combustion.............................................................................................. 1
1.1.2 Flame propagation in open space and confinement................................................. 3
1.2 Previous Investigations................................................................................................... 6
1.2.1 Flame with edges ................................................................................................ 6
1.2.2 Propagation speed, extinction limit, and scaling of edge-flame ........................11
1.2.3 Lewis number and Diffusive-thermal instability (DTI).................................... 18
1.2.4 Flow in the Hele-Shaw cell............................................................................... 24
1.2.5 Instabilities in the Hele-Shaw cell .................................................................... 26
1.2.6 Propagation speed and scaling of flame fronts in a narrow channel................. 34
1.2.7 Objectives ......................................................................................................... 43
Chapter 2: Methodology ......................................................................................................... 45
2.1 Apparatus and Procedures............................................................................................. 45
2.1.1 The slot-jet counterflow burner ........................................................................ 45
2.1.2 Edge-flame configuration in counterflow burners............................................ 50
2.2 Mixture strength and stoichiometric mixture fraction .................................................. 51
2.3 The Hele-Shaw cell....................................................................................................... 54
Chapter 3: Nonpremixed hydrogen edge-flames.................................................................... 61
3.1 Flame structures............................................................................................................ 61
3.2 Propagation rates........................................................................................................... 65
3.3 Regimes of flame behavior........................................................................................... 70
3.4 Computations of σext and comparison with experiment................................................ 71
3.5 Conclusion .................................................................................................................... 73
Chapter 4: Premixed hydrogen edge-flames........................................................................... 75
4.1 Calculations of strained flames..................................................................................... 75
4.2 Modes of edge-flames................................................................................................... 80
4.3 Regimes of flame behavior........................................................................................... 84
4.4 Edge-flame propagation speed...................................................................................... 90
4.5 Conclusion ........................................................................................................................ 94

v
Chapter 5: Quasi-2D hydrogen flames in the Hele-Shaw cell................................................ 95
5.1 Flame structures............................................................................................................ 95
5.1.1 The effect of instabilities .................................................................................. 97
5.1.2 The effect of cell dimension ........................................................................... 102
5.1.3 The effect of mixture strength..........................................................................113
5.2 The transition from cellular to Sawtooth .....................................................................115
5.3 Propagation rates..........................................................................................................118
5.2.1. The effect of instabilities .................................................................................119
5.2.2. The effect of mixture strength......................................................................... 122
5.2.3. The effect of cell dimension ........................................................................... 124
5.4 JS parameter calculations............................................................................................ 126
5.5 Conclusion .................................................................................................................. 129
Chapter 6: Further work........................................................................................................ 131
6.1 Investigation into scaling parameters.......................................................................... 131
6.2 Flow field simulation .................................................................................................. 131
6.3 New Experiment Apparatus........................................................................................ 132
References............................................................................................................................. 133

vi
List of Figures
Fig. 1. Flame spreading over a fuel-bed. .......................................................................................................................7
Fig. 2. Candle flame burning under microgravity conditions........................................................................................8
Fig. 3. A wrinkled flame front torn by local turbulence................................................................................................9
Fig. 4. Schematic diagrams of conventional flamelet (moderate and high Le), proposed broken...............................10
Fig. 5. “Short length” flame in Clayton et al. [33]. .....................................................................................................16
Fig. 6. False-color direct images of single premixed edge-flames. Reactive mixture flows from the bottom
and cold inert from the top. All flames propagate from left to right, and the height of each image is scaled
with the jet spacing (d) [33].........................................................................................................................................17
Fig. 7. False-color direct images of twin premixed edge-flames. Global strain rate (σ) is shown in each
image. All flames propagate from left to right, and the height of each image is scaled with the jet spacing
(d) [33].........................................................................................................................................................................17
Fig. 8. Thermal and molecular mass diffusions to a curved flame front [31]..............................................................19
Fig. 9. Leeff in H2-O2 flames as a function of H2-O2 ratio [38].....................................................................................20
Fig. 10. cellular structure in hydrocarbon-O2-N2 flames at atmospheric pressure in Markstein (1951) [39]. .............21
Fig. 11. Cellular flames in Kaiser et al. [57]. ..............................................................................................................24
Fig. 12. A propagating flame front in a narrow channel..............................................................................................25
Fig. 13. Diagram of flow and a curved flame front interaction [31]............................................................................27
Fig. 14. Diagram of upward (left) and downward (right) propagating flame and buoyancy effects [69]....................30
Fig. 15. Oil-water interface experiment demonstrates Saffman-Taylor Instability by Tabeling et al. [70].................32
Fig. 16. Diagram of the Saffman-Taylor instability mechanism .................................................................................33
Fig. 17. Effects of ST and RT instabilities on DL instability’s dimensionless growth rate ( ) for a range of
example hydrocarbon-air flames with U = 20cm/s and d = 1.27cm............................................................................37
Fig. 18. Schematic of the experimental apparatus (single premixed configuration) ...................................................47
Fig. 19. Example of effect of edge-flame location on propagation rate in the laboratory reference frame
adopted from [30]........................................................................................................................................................49

vii
Fig. 20. Schematics of the Hele-Shaw experiment apparatus......................................................................................55
Fig. 21. Example of a frame of a video imported into the MATLAB program; the flame is propagating
toward the right, and the red line represents the recognized flame location. Left: flame in a gas mixture with
Leeff > 1; Right: flame in a gas mixture with Leeff < 1..................................................................................................56
Fig. 22. Example of experimentally measured flame position and corresponding pressure in the combustion
chamber. ......................................................................................................................................................................59
Fig. 23. An exemplary plot of wrinkled flame length ratio lflame /w vs. time generated from MATLAB
program........................................................................................................................................................................59
Fig. 24. False-color direct images of edge-flames in mixtures with N = 18. H2-N2 flows from the bottom
upwards, O2-N2 from the top downwards. Slot spacing d = 7.5 mm. White dashed lines indicate the
stagnation plane location. (a) Zst = 0.6, σ = 80s-1
, advancing flat flame (mode I); (b) Zst = 0.4, σ = 80s-1
,
retreating flat flame (mode II); (c) Zst = 0.9, σ = 80s-1
, advancing broken flame (mode III) (d) Zst = 0.70, σ =
110s-1
, stationary broken flame (mode IV); (e) computed reaction rate contours for a mode III flame with
Le=0.33 [49]. ...............................................................................................................................................................62
Fig. 25. False-color direct images of broken-flames. (a) N = 18, Zst = 0.8, σ = 100s-1
; (b) N = 18.5, Zst = 0.8,
σ = 100s-1
; (c) N = 19, Zst = 0.8, σ = 100s-1
; (d) N = 18, Zst = 0.7, σ = 100s-1
; (e) N = 18, Zst = 0.7, σ = 120s-1
;
(f) N = 17, Zst = 0.9, σ = 20s-1
; (g) N = 18, Zst = 0.85, σ = 240s-1
;(h) computed reaction rate contours for a
mode IV flame with Le=0.33 [49]. Images (a) and (f) are advancing (Mode III), and all others are
stationary (Mode IV). ..................................................................................................................................................62
Fig. 26. Broken-flame propagation sequence: Zst = 0.9, σ = 80s-1
, propagating broken-flame (mode III). The
time between frames is 0.183 s....................................................................................................................................63
Fig. 27. Effect of Zst and σ on scaled edge-flame speeds: (a) N=17; (b) N=18. Filled symbols indicate
continuous flat flames and open symbols indicate broken flames...............................................................................64
Fig. 28. Effect of scaled strain rate ε on scaled Uedge: (a) Zst=0.15; (b) Zst=0.5; (c) Zst=0.9. Filled symbols
indicate continuous flat flames (Modes I, advancing and II, retreating), and open symbols indicate broken
flames (Modes III, advancing and IV, stationary). ......................................................................................................67
Fig. 29. Flame response maps in Zst-ε space: (a) N=17; (b) N=18. Recall mode designations: (I) advancing
flat flames, (II) retreating flat flames, (III) advancing flat flames, and (IV) stationary broken-flames.
Vertical dashed lines indicate minimum (pure H2 vs. O2-N2) and maximum (H2-N2 vs. pure O2) values of Zst
attainable. Dashed curves indicate the transition to turbulent structures. ...................................................................68
Fig. 30. Flame response maps in N-ε space: (a) Zst=0.3; (b) Zst=0.8. Dashed curves indicate the transition to
turbulent structures. .....................................................................................................................................................70
Fig. 31. Comparison of measured (solid lines and data points) and computed (dashed curves) values of
extinction strain rate (σext) for varying Zst with N = 17 and N = 18.............................................................................71
Fig. 32. Computed effects of Zst on σext for H2-N2/O2-N2 mixtures with standard transport and artificial
transport causing Lef=Leo≈1, along with results for CH4-N2/O2-N2 (Lef=Leo≈1) and i-CH4-N2/O2-N2
(Lef>Leo≈1) mixtures. ................................................................................................................................................72
Fig. 33. Calculated laminar burning velocity SL for varying adiabatic flame temperature (Tad) and
equivalence ratio (φ). Note that lean mixtures at Tad ≤ 1100 K have SL ≤ 1cm∙s-1 and thus Pe ≤ 10, and
thus could not burn as plane premixed flames in the apparatus utilized......................................................................76
Fig. 34. Calculated (a) extinction strain rates, (b) SL
*
/SL for varying adiabatic flame temperature (Tad) and
equivalence ratio (φ) for twin flames...........................................................................................................................77

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Fig. 35. Calculated (a) extinction strain rates, (b) SL
*
/SL for varying adiabatic flame temperature (Tad) and
equivalence ratio (φ) for single flames. .......................................................................................................................78
Fig. 36. False-color direct images of twin premixed edge-flames propagating or retreating from left to right.
The upper and lower borders of the images represent the locations of the jet exits (spaced 12.5 mm apart.)
White dashed lines indicate the stagnation plane location. (a) φ = 1.5, σ = 50 s-1
, Tad = 1200 K, advancing
flat edge-flame (Mode I); (b) φ = 1.5, σ = 50 s-1
, Tad = 1175 K, retreating flat edge-flame (Mode II); (c) φ =
0.7, σ = 40 s-1
, Tad = 1175 K, advancing wrinkled edge-flame (Mode III); (d) φ = 0.7, σ = 25 s-1
, Tad =
1175 K, advancing wrinkled edge-flame (Mode III); (e) φ = 0.4, σ = 50 s-1
, Tad = 850 K, advancing broken
edge-flame (Mode IV); (f) φ = 0.2, σ = 20 s-1
, Tad = 1175 K, advancing broken-flame (Mode IV). ..........................83
Fig. 37. False-color direct images of single premixed edge-flames propagating from left to right (except (e)
which is stationary.) The upper and lower borders of the images represent the locations of the jet exits.
White dashed lines indicate the stagnation plane location. H2-O2-N2 mixtures flow from the bottom
upwards and N2 from the top downwards. (a) φ = 1.5, σ = 40 s-1
, Tad = 1300 K, advancing flat-flame (Mode
I); (b) φ = 0.8, σ = 40 s-1 Tad = 1300 K, advancing wrinkled edge-flame (Mode III); (c) φ = 0.6, σ = 40 s-1
,
Tad = 1200 K, advancing broken edge-flame (Mode IV); (d) φ = 0.6, σ = 40 s-1
, Tad = 1100 K, advancing
broken edge-flame (Mode IV); (e) φ = 0.6, σ = 78 s-1
, Tad = 1100 K, stationary broken edge-flame (Mode
V); (f) φ = 0.4, σ = 17 s-1
, Tad = 1100 K, continuous short-length moving flame train (Mode VI).............................84
Fig. 38. Maps of modes of flame behavior in φ - ε space for twin edge-flames with (a) Tad = 1175 K and (b)
950 K; Τad - ε space for twin edge-flames with (c) φ = 1.5 and (d) φ = 0.5. Mode designations are: (I)
advancing flat edge-flames, (II) retreating flat edge-flames, (III) advancing wrinkled edge-flames, (IV)
advancing broken edge-flames, (V) stationary edge-flames and (VI) “Short-length” flames. Dashed curves
indicate transition to turbulent structures.....................................................................................................................87
Fig. 39. Maps of modes of flame behavior in φ - ε space for single edge-flames with (a) Tad = 1300 K and
(b) 1100 K; Τad - ε space for twin edge-flames with (c) φ = 1.5 and (d) φ = 0.5. Mode designations are: (I)
advancing flat edge-flames, (II) retreating flat edge-flames, (III) advancing wrinkled edge-flames, (IV)
advancing broken edge-flames, (V) stationary edge-flames and (VI) “Short-length” flames. Dashed curves
indicate the transition to turbulent structures...............................................................................................................88
Fig. 40. Comparison of extinction strain rate (σext) from simulation and experiment..................................................89
Fig. 41. Effect of scaled strain rate e and equivalence ratio φ on twin edge-flame speeds Ũ for fixed
adiabatic flame temperature Tad (upper) or strain rate σ (lower). Filled symbols indicate continuous flames;
open symbols indicate broken flames..........................................................................................................................92
Fig. 42. Effect of scaled strain rate e and equivalence ratio φ on single edge-flame speeds Ũ for fixed
adiabatic flame temperature Tad (upper) or strain rate σ (lower). Filled symbols indicate continuous flames;
open symbols indicate broken flames..........................................................................................................................93
Fig. 43. Effect of adiabatic flame temperature Tad on scaled twin edge-flame speeds Ũ for equivalence ratio
φ = 0.3. Filled symbols indicate continuous flames; open symbols indicate broken flames........................................94
Fig. 44. Two theoretically possible flame modes in the Hele-Shaw cell: (a) a single flame sheet; (b)
individual cylinders. ....................................................................................................................................................96
Fig. 45. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 0.7, d = 1.27 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 1.27 cm, w = 45.72 cm,
horizontal (c)Tad = 1300 K, φ = 0.7, d = 1.27 cm, w = 45.72 cm, upward. All mixtures have Leeff = 0.43..................99
Fig. 46. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300
K, f = 1.0, d = 1.27 cm, w = 45.72 cm, downward (b)Tad = 1300 K, f = 1.0, d = 1.27 cm, w = 45.72 cm,
horizontal (c)Tad = 1300 K, f = 1.0, d = 1.27 cm, w = 45.72 cm, upward. All mixtures have Leeff = 0.73.............100

ix
Fig. 47. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 2.5, d = 1.27 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 1.27 cm, w = 45.72 cm,
horizontal (c)Tad = 1300 K, φ = 2.5, d = 1.27 cm, w = 45.72 cm, upward. All mixtures have Leeff = 1.04................101
Fig. 48. Direct Numerical Simulation (DNS) of a horizontally steady flame in gas mixtures with varying
Leeff values: 1.5 (left), 0.7 (middle), and 0.3 (right), compared with experimental results. The flame front is
delineated by the reaction rate isocontour (depicted in blue), and the calculated streamline pattern
represents the flow field. Courtesy of [81]. ..............................................................................................................102
Fig. 49. DNS simulation of the effect of lateral domain size on the flame front. Courtesy of [81]...........................103
Fig. 50. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 0.7, d = 1.27 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 1.27 cm, w = 11.43 cm,
horizontal (c)Tad = 1300 K, φ = 0.7, d = 1.27 cm, w = 11.43 cm, upward. All mixtures have Leeff = 0.43................105
Fig. 51. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 1.0, d = 1.27 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 1.0, d = 1.27 cm, w = 11.43 cm,
horizontal (c)Tad = 1300 K, φ = 1.0, d = 1.27 cm, w = 11.43 cm, upward. All mixtures have Leeff = 0.73................105
Fig. 52. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 2.5, d = 1.27 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 1.27 cm, w = 11.43 cm,
horizontal (c)Tad = 1300 K, φ = 2.5, d = 1.27 cm, w = 11.43 cm, upward. All mixtures have Leeff = 0.43................106
Fig. 53. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 0.7, d = 0.64 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm,
horizontal (c)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm, upward.................................................................108
Fig. 54. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 0.7, d = 0.64 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm,
horizontal (c)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm, upward.................................................................109
Fig. 55. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 2.5, d = 0.64 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 0.64 cm, w = 45.72 cm,
horizontal (c)Tad = 300 K, φ = 2.5, d = 0.64 cm, w = 45.72 cm, upward...................................................................110
Fig. 56. Unsteady cellular flame structure for Le = 0.5 with simulation for large lateral domain size and
isothermal boundary conditions (left) and experiment for φ = 0.78, Leeff = 0.51, d = 0.32cm, w = 45.72cm,
Tad = 1300K (right). DNS Courtesy of [85]...............................................................................................................111
Fig. 57. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 0.7, d = 0.64 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 11.43 cm,
horizontal (c)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 11.43 cm, upward.................................................................112
Fig. 58. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 1.0, d = 0.64 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 1.0, d = 0.64 cm, w = 11.43 cm,
horizontal (c)Tad = 1300 K, φ = 1.0, d = 0.64 cm, w = 11.43 cm, upward.................................................................112
Fig. 59. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K,
φ = 2.5, d = 0.64cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 0.64cm, w = 11.43 cm,
horizontal (c)Tad = 1300 K, φ = 2.5, d = 0.64cm, w = 11.43 cm, upward..................................................................112
Fig. 60. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1100 K,
φ = 0.1, d = 1.27 cm, w = 45.72 cm, upward (b)Tad = 1100 K, φ = 0.1, d = 0.64 cm, w = 45.72 cm, upward
(c)Tad = 1100 K, φ = 0.1, d = 0.64 cm, w = 11.43 cm, upward (d)Tad = 900 K, φ = 0.1, d = 1.27 cm, w =
45.72 cm, upward (e)Tad = 900 K, φ = 0.1, d = 0.64 cm, w = 45.72 cm, upward (f)Tad = 900 K, φ = 0.1, d =
0.64 cm, w = 11.43 cm, upward. ...............................................................................................................................114

x
Fig. 61. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 900 K,
φ = 0.1, d = 0.32cm, w = 11.43 cm, horizontal (b)Tad = 900 K, φ = 0.1, d = 0.64cm, w = 11.43 cm,
horizontal (c)Tad = 900 K, φ = 0.1, d = 1.27cm, w = 11.43 cm, horizontal (d)Tad = 900 K, φ = 0.1, d =
0.32cm, w = 45.72 cm, horizontal. ............................................................................................................................115
Fig. 62. Transition time and location vs. the equivalence ratio φ and Leeff in a horizontal Hele-Shaw cell
with d = 0.5 cm..........................................................................................................................................................117
Fig. 63. Scaled propagation speed U = ST/SL vs. Pe= SLd/α for H2-O2-N2 mixtures at 1300K: (a)all
mixtures;(b)lean mixtures;(c) rich mixtures. .............................................................................................................120
Fig. 64. Scaled average flame speed U = (ST/SL
*
)(w/lflame) vs. Pe*
= SL
*
d/α for H2-O2-N2 mixtures at 1300K:
(a)all mixtures and cell dimensions;(b)w = 45.72 cm ;(c) w = 11.43 cm...................................................................121
Fig. 65. Effect of adiabatic flame temperature, chamber thickness and propagation direction on flame
propagation speeds ST in wide (a) and narrow (b) cells for ultra-lean H2-O2-N2, Leeff ≈ 0.3....................................123
Fig. 66. Effect of adiabatic flame temperature, chamber thickness and propagation direction on Scaled
average flame speed U = (ST/SL
*
)(w/lflame) in wide (a) and narrow (b) cells for ultra-lean H2-O2-N2, Leeff ≈
0.3..............................................................................................................................................................................124
Fig. 67. Scaled propagation U vs. JS parameter for H2-O2-N2 mixture range: (a)φ < 1.0, Leeff < 0.7; (b) all
mixture tested. ...........................................................................................................................................................127
Fig. 68. Calculated extinction strain rates for varying adiabatic flame temperature Tad and equivalence ratio
φ for twin premixed flames and single flames with hot diluent opposing flow.........................................................128

xi
Nomenclature
α Thermal diffusivity of unburned gas mixtures
β Zel’dovich number
c Specific heat capacity
Cp Constant pressure specific heat
d Hele-Shaw cell thickness; counterflow spacing
D Coefficient of mass diffusivity
Da Damköhler number
δT Flame thickness
Ea Overall activation energy
ε Expansion coefficient; dimensionless strain rate
f Exchange coefficient of momentum
fu Exchange coefficient of momentum of the unburned gas mixture
fb Exchange coefficient of momentum of product gas mixture
fav Average exchange coefficient of momentum
F Dimensionless friction factor
Fr Froude number
g Gravity field parallel to the plate
G Dimensionless gravity factor

xii
h Exchange coefficient of heat loss
Ka Karlovitz number
κ Heat loss coefficent
k Wavenumber = 2π/λ
kRT Rayleigh-Taylor instability’s critical wavenumber = 2π/w
kST Saffman-Taylor instability’s critical wavenumber = 2π/w
L Hele-Shaw cell length
w Hele-Shaw cell width
lflame Two-dimensional wrinkled flame length
Leeff Effective Lewis number
Lec Critical Lewis number characterizes occurring of cellular flame by
DT instability
λ Flame wavelength
λc Critical wavelength characterizes occurring of cellular flame by DT
instability
λST Saffman-Taylor instability wavelength
Λ Dimensionless wavelength
μ Dynamic viscosity of the gas mixture
μu Dynamic viscosity of the fresh unburned gas mixture
μb Dynamic viscosity of the burned gas mixture
μav Average dynamic viscosity of both fresh unburned gas mixture and
burned gas mixture
n Reaction order
ω Growth rate of flame wrinkle
ω̇ Overall reaction rate
Ω Dimensionless growth rate

xiii
Pe Péclet number
P Pressure
ℜ Universal gas constant
ρ Gas mixture density
ρu Unburned gas mixture density
ρb Products mixture density
ST Flame propagation speed
SL Laminar flame speed
SL
* Effective laminar flame speed
SL
′ Modified laminar flame speed
σ Dimensionless growth rate (JS parameter); global strain rate
σext Extinction strain rate
t time
Tad Adiabatic flame temperature
T Temperature
Tu Temperature of the unburned gas mixture
u Two-dimensional velocity field
u and v Velocity components
U Burning velocity normal to the flame front
Ũ Dimensionless edge-flame propagation speed
Uedge Dimensional edge-flame propagation speed
Ujet Velocity of counterflow jets
Usheath Velocity of counterflow shielding flow
Zst Stoichiometric mixture fraction

xiv
Abstract
Hydrogen-air mixture combustion (especially lean) presents unique properties due to its lower
minimum flame temperatures supporting combustion, lower Lewis number (Le, ratio of mixture
thermal diffusivity to fuel mass diffusivity) and lower radiative emission. These properties result
in clean combustion with minimal NO emissions and no byproducts such as CO or CO2.
However, the implementation of hydrogen-air combustion in engines is hampered by a limited
understanding of the complex relationships between flame instabilities, heat losses,
hydrodynamic strain, and flame front curvature.
To develop a more comprehensive understanding of the burning properties of hydrogen-air
mixtures and thereby obtain knowledge that informs the design and operation of energy
conversion devices, properly designed experiments in two laboratory apparatuses were employed
in this study:
1) a counterflow slot-jet burner is used to assess the effects of steady hydrodynamic strain
with minimized thermal expansion on propagation properties for H2-O2-N2 mixtures. The
experiments focus on non-planar flame structures that occur only in low Le mixtures due to the
diffusive-thermal instability and its effects on edge-flame propagation/retreat rates and extinction
limits. The study investigates the impact of flow configurations (nonpremixed vs. premixed),

xv
mixture composition and global strain rate (σ) on flame morphology and propagation rates (Uedge)
were investigated. Both positive and negative Uedge were observed depending on these factors.
Different flame structures were categorized in the form of flame regime maps. Depending on σ
and the dilution level, continuous flames can transition to wrinkled or broken structures, or even
further into isolated, stationary flame islands due to diffusive thermal instability characteristic of
low Le. Such transitions enable flames to survive under conditions where extinction would have
occurred. These findings were in surprisingly good agreement with theoretical predictions.
Moreover, the combined effects of mixture composition and σ were found to correlate well
regarding the effect of a non-dimensional strain rate (ε) on a scaled propagation rate Ũ.
2) a narrow channel, i.e., the Hele-Shaw cell, is used to study the effects of thermal expansion
without hydrodynamic strain (other than that generated by self-induced flow) and its interaction
with other instabilities on H2-O2-N2 flame propagation properties. This is considered especially
relevant to real applications, as most laboratory experiments are conducted in open apparatuses
where thermal expansion is relaxed in the directions parallel to the flame plane. In contrast,
combustion in most practical devices (e.g., IC engines) occurs in very confined geometries where
this relaxation is not possible. The flame propagation rates (ST) and patterns were studied under
different channel thicknesses (d) and widths (w), mixture compositions, and channel orientations
(upward, horizontal, and downward). These conditions mimic those obtained in many practical
devices with high aspect ratio combustion chambers, such as the crevice volume in pistoncylinder engines at the time of ignition. Various flame structures and propagation rates were
observed depending on the interaction between four major instabilities: thermal expansion of the
burned gas (Darrieus-Landau instability, DL); buoyancy effect (Rayleigh-Taylor instability, RT);
viscosity contrast across the flame surface (Saffman-Taylor instability, ST); effective Lewis

xvi
number (Leeff) (diffusive-thermal instability, DT). Lean H2-O2-N2 flames were observed to burn
with broken structures under very low adiabatic temperature conditions where extinction would
have occurred without low Leeff. A non-dimensional JS parameter is employed to scale the
driving forces for flame propagation with wrinkling, and it exhibits a good correlation with Ũ in
the regime of lean mixtures.

1
Chapter 1
Introduction
1.1 Background
1.1.1 Hydrogen combustion
Despite the rapid advancements in Electric Vehicles (EVs), hydrogen combustion remains a
topic of paramount significance for several compelling reasons. First, hydrogen offers a high
energy density by weight, making it an attractive alternative for applications where weight is
critical, such as aviation and maritime transport. Suppose the issues of economical large-scale
hydrogen production, distribution, and on-vehicle storage are resolved. In that case, the most
feasible use of hydrogen for vehicle propulsion will still be internal combustion (IC) engines,
where lean hydrogen-air mixtures with lower flame temperatures and higher gas specific heat
ratios (γ), more than 40% efficiency could be reached. Hydrogen combustion systems can
provide the energy output necessary for these demanding applications without the weight burden
of large battery systems EVs carry. Second, the flexibility of hydrogen as a fuel offers a unique
advantage. Hydrogen can be utilized in various applications, allowing for a multi-faceted
approach to decarbonization. It can be produced from various sources, including renewable

2
energy, and stored for extended periods, providing a feasible option for seasonal energy shifts.
This adaptability sets hydrogen apart as a versatile energy carrier that can be integrated into
existing infrastructure, accelerating the transition to cleaner energy solutions.
The development of hydrogen combustion technologies also complements the rise of EVs by
filling in gaps where EVs may not be the most efficient or practical solution. Together, these
technologies provide a more comprehensive approach to achieving sustainability and energy
security goals. Therefore, hydrogen combustion continues to be a critical area of study and
development, with its unique set of advantages that make it indispensable in the broader context
of global energy solutions.
Recent interest in utilizing fuel-lean or highly diluted mixtures in internal combustion engines
and other power generation systems has surged. In the context of hydrogen, despite a focus on
Proton-Exchange Membrane (PEM) fuel cells in many vehicle designs, internal combustion (IC)
engines are often the more favorable option due to their substantially higher power-to-weight
ratios and negligible efficiency drawbacks compared to fuel cells. Hydrogen fuel effectively
mitigates most environmental issues associated with hydrocarbon-fueled IC engines. Because
lean hydrogen-air mixtures of equivalence ratio of 0.45 (adiabatic flame temperature Tad ≈ 1400
K) produce the same burning velocity (40 cm/s) as stoichiometric hydrocarbon-air mixtures (Tad
≈ 2250 K), hydrogen-powered internal combustion engines can operate with highly fuel-lean
conditions while maintaining low flame temperatures that effectively minimize the generation of
thermal NOx emissions. Additionally, the combustion of hydrogen fuel results in zero emissions
of hydrocarbons, carbon monoxide, carbon dioxide, and “prompt” pollutants.
Hydrogen-air flames have some unique properties compared to hydrocarbon-air flames. For a
given flame temperature (Tf), the characteristic reaction rates of hydrogen-air flames are much

3
higher. As a result, the lean flammability limit and minimum flame temperatures of hydrogen-air
mixtures are much lower than hydrocarbon-air mixtures. Also, because of the low flame
temperature, flame wrinkling via thermal expansion is weaker than in hydrocarbon fuels. The
Lewis number (Le) is about 0.3 for lean H2-air flames, whereas for all hydrocarbon-air flames Le
is close to or larger than unity. Flame fronts in mixtures with such low Le are inherently unstable
and wrinkle spontaneously due to diffusive-thermal instabilities [1] in ways that hydrocarbon-air
flames cannot. Due to their reduced temperatures and lower amounts of combustion products,
lean hydrogen-air flames exhibit significantly lower radiative losses compared to their
hydrocarbon-air counterparts - by an approximate factor of 10 in lean-limit conditions. While a
number of studies conducted in both laboratory settings and internal combustion engines have
indicated that these unique flame characteristics can influence engine performance in unexpected
manners, a comprehensive investigation into this phenomenon has yet to be undertaken.
1.1.2 Flame propagation in open space and confinement
Flame propagation has long been a focal point in the academic discourse of combustion
research. Standard combustion experimental setups—such as counterflow, Bunsen burner, and
fan-stir reactors—facilitate burning in open spaces or away from chamber walls to prevent
flame-wall interactions. As a result, the flame can propagate more uniformly and predictably,
often modeled effectively by theories. Flame instabilities are generally simpler to analyze due to
the lack of geometric constraints, making open-space combustion a good baseline for theoretical
studies and numerical simulations.
While flames in open spaces serve as a useful starting point for understanding basic
combustion principles such as chemical kinetics, fluid dynamics, and mixture diffusion, they
don’t closely resemble real-world combustion scenarios found in f internal combustion engines,

4
gas turbines, and industrial furnaces, where flames exist within compact combustion chambers.
When a flame is in a confined environment, it interacts intimately with the walls or boundaries.
These interactions can manifest in various ways, from flame quenching and stabilization to the
emergence of specific combustion instabilities. Such interactions and their resulting phenomena
aren’t as pronounced, or might even be absent, in more open flames. For instance, the elevated
temperature of the burned gas behind a flame raises the gas’s viscosity. This change in viscosity
across the flame can cause significant wrinkling of the flame front due to friction with chamber
walls in confined spaces [2].
Confinement can lead to enhanced turbulence and mixing due to the proximity of walls,
potentially rendering more efficient combustion processes characterized by reduced emissions.
Yet, with benefits come challenges. Confined flames can sometimes couple with acoustic modes,
resulting in self-sustained oscillations known as thermoacoustic instabilities. These oscillations
can hinder the optimal operation of critical devices, from gas turbines to rocket engines.
In an era where environmental concerns are at the forefront, confined flames offer a unique
emission profile. Understanding and studying these emissions are pivotal as human beings
globally strive for more environmentally friendly combustion solutions. In essence, the study of
confined flames is critical to unlocking safer, more efficient, and greener combustion
technologies for the future. Therefore, understanding combustion in confined spaces is of
paramount importance.
Fundamental insights into combustion science can be derived from studying flames within
simplified geometrical confines such as narrow channels, tubes, or slots. These basic
configurations are particularly relevant as micro-combustion technologies are gaining attention
for their potential in portable power generation, often featuring high aspect ratios to compete

5
with traditional battery systems [3–7]. For instance, consider the combustion chamber of an
internal combustion engine when the piston reaches its top dead center (TDC). This scenario
closely resembles a narrow channel with a high aspect ratio; typically, the combustion chamber
might measure around 1 cm in height and 10 cm in diameter. Understanding the dynamics of
combustion within this confined space becomes paramount. This is particularly true when one
considers the implications for engine emissions, which can be adversely affected by incomplete
combustion and flame quenching. Therefore, the study of flames in constrained geometries not
only advances fundamental understanding but also has direct implications for the design and
operation of practical combustion systems.[8,9].
The experimental framework in this thesis is bifurcated into two key segments, each attending
to specific conditions crucial for an understanding of flame dynamics and for the enhancement of
real-world combustion systems. The first segment focuses on flame propagation in a domain
characterized by steady strain, an absence of turbulence, and minimal influence of thermal
expansion. The second segment, in contrast, addresses scenarios where there is no steady strain,
turbulence is negligible, but the impact of thermal expansion is significant. Consequently, the
factors most responsible for burning velocity enhancement, flame extinction, H2O2 emissions,
etc., can be identified. For instance, in the open-space counterflow experiment, one will
anticipate observing a full spectrum of edge flames—ranging from continuous to broken and
from advancing to retreating. These edge-flame experiments will serve as a predictive tool to
discern which type of edge flame is most likely to occur under specified mixture and property
conditions. Subsequently, the outcomes will be evaluated to see to what extent the (much simpler)
counterflow slot-jet experiments can be used to predict the behavior of turbulent lean H2-air
flames.

6
Additionally, this work would provide a comprehensive database on hydrogen-air flame
behavior that would be useful in other types of combustion systems (e.g., gas turbines) and
develop a better understanding of hydrogen combustion near lean flammability limits that will be
beneficial in reducing fire and explosion hazards associated with the use of hydrogen fuel.
1.2 Previous Investigations
1.2.1 Flame with edges
Flame sheets may become distorted or wrinkled due to instabilities triggered by thermal
expansion, buoyancy, or imbalanced heat and mass diffusion. Additionally, the presence of
boundaries, heat losses, and varying flow fields can give rise to flame edges.
The concept of "edge-flames" began to be recognized and reasonably well understood in the
1990s, even though the flame theory is a subject with a much longer history, rooted in the 1928
work of Burke and Schumann on diffusion flames, followed a few decades later by the work of
Darrieus, Landau, Frank-Kamenetskii, and Zel’dovich on deflagrations. The one-dimensional
flame structures of diffusion and deflagration are central to any discussion of flame physics, and
their universal characteristics are applicable to various flame configurations.
Edge-flames [10,11] are idealized as two-dimensional structures that describe the transition
regions between burning and non-burning portions of flame sheets, which may offer insights into
many types of non-uniform flame phenomena, e.g., flames stabilized near a cold wall or splitter
plate, the leading edge of a flame spreading across a condensed-phase fuel surface, or flames in
highly turbulent flow fields where “holes” in the flame sheet may open or “heal” [12]. Moreover,
edge-flames are essential for premixed flame stabilization in shear layers [13]. Additionally,

7
edge-flame libraries could potentially extend flamelet models of turbulent combustion, which
assume unbroken flame surfaces, to conditions involving local quenching [14].
Figure 1 illustrates a flame advancing over a solid or liquid fuel-bed. The majority of the fuel
flux (which is non-uniform) from the bed is consumed in a nominally 1D diffusion flame.
However, reactions at the bed’s surface temperature are minimal, resulting in a non-reactive
space between the flame and the bed, giving the flame a distinct edge [10].
Figure 2 illustrates a candle flame in the microgravity environment. Due to the absence of
buoyancy-driven currents that would otherwise disturb the flame shape, it assumes a nearhemispherical form with a clearly defined, circular boundary; past this boundary, the air-fuel
mixture is not strong enough to sustain combustion. This showcases how the absence of
gravitational forces can lead to a more symmetric and stable flame configuration.
Fig. 1. Flame spreading over a fuel-bed.

8
Fig. 2. Candle flame burning under microgravity conditions.
Figure 3 depicts a diffusion flame subject to turbulent disturbances. Flame quenching takes
place at specific times and locations where the scalar dissipation rate is sufficiently high. The
scalar dissipation rate gauges the rate at which a scalar property, such as the concentration of a
chemical species in a reactive flow, diminishes due to the effects of turbulence. As a result of this
high scalar dissipation rate, voids or ‘holes’ appear within the flame structure. Each of these
holes possesses its distinct boundary, which in turn acts as an edge for the surrounding onedimensional flame.

9
Fig. 3. A wrinkled flame front torn by local turbulence.
The scalar dissipation rate is intimately related to strain (stretch) and imbalance diffusion due
to the Lewis number Le. Flame strain rate refers to the rate at which a flame’s surface area
changes. This phenomenon can be instigated by the movement of unburned gas surrounding the
flame or due to the flame front’s deformation, which may result from turbulence or other flow
dynamics. As the flame stretches positively, the area available for combustion intensifies, thereby
amplifying the burning process. When subjected to negative stretch, the flame might not only
compress but could also thicken, resulting in a reduced combustion speed. In extreme situations,
this compression can result in local flame quenching or complete extinguishment.
The implications of flame strain are vast and varied. In the realm of turbulent combustion,
flames are exposed to alternating positive and negative strain rates, leading to intricate and often
unpredictable flame behaviors. Especially in premixed flames, the effects of stretch can be very
pronounced due to the flame structure’s high sensitivity to alterations in the velocity of the
unburned mixture and the inherent curvature of the flame itself.

10
Fig. 4. Schematic diagrams of conventional flamelet (moderate and high Le), proposed broken
flamelet (low Le) and distributed combustion (any Le, at very high turbulence levels).
In laboratory settings, flames subject to temporally and spatially uniform hydrodynamic strain
are frequently employed to model the local interactions of flame fronts with turbulent flow fields
[15–17]. This “laminar flamelet” concept presumes that each segment of the flame front will act
as if it were an isolated, steady front experiencing uniform hydrodynamic strain (Figure 4a).
However, the laminar flamelet model cannot apply universally to mixtures with low Le
because even with a uniform mixture in a uniform strain field, diffusive-thermal instability (DTI)
inherent to low Le scenarios can cause a flame front to become non-uniform, potentially akin to
what is illustrated in Figure 4b. The influence of strain on DTI is also substantial, as documented
in the theory works [1,18,19]. This is particularly pertinent for the turbulent combustion of
hydrogen, as the Lewis number has a significant impact on the flame’s wrinkling behavior and its
associated burning rates, as noted in early studies [17,20,21].

11
Computations by several authors [1,19] have shed light on a variety of unique flame
behaviors under strain for low-Le mixtures. At minimal levels of strain, the flame tends to
display mild wrinkling. However, when the strain is sufficiently high, the flame structure
transforms into discontinuous “flame tubes,” which are elongated in the direction of extensional
strain. This transformation allows the flame to withstand levels of strain that would otherwise
lead to its extinction if it were to remain planar or only moderately wrinkled. Essentially, the
strain-induced flame weakening reaches a point where the flame can only persist through the
fortification brought about by its curvature. This behavior is exclusive to low-Le mixtures
because only at low Le is the curved flame more robust than a flat flame. An infinite chain of
traveling tubes/cells is predicted for moderately high strain, whereas two then one isolated
stationary tube(s) are predicted at progressively higher strain. Yet higher strain causes complete
flame extinguishment.
1.2.2 Propagation speed, extinction limit, and scaling of edge-flame
The most significant property of an edge-flame is its propagation speed (Uedge), defined as the
speed of the edge moving from the burned gases toward the unburned gases in the direction
parallel to the flame sheet. Previous theory studies in premixed [19,22–24] and nonpremixed
configurations[25–28] predict that edge-flames may propagate from the burning region into the
unburned region, forming an “ignition front” with Uedge > 0 or retreat from the burning region
into the burned region, forming an “extinction front” with Uedge < 0. Depending on the
environment, edge-flame front could even be stationary with Uedge = 0. Uedge and extinction
limits rely primarily on (1) the mixture strength, which is characterized by the adiabatic flame
temperature Tad; (2) Lewis number (Le, ratio of mixture thermal diffusivity to reactant mass
diffusivity); (3) global strain rate (σ); (4) heat loss due to conduction or radiation.

12
The global strain rate for a counterflow slot-jet is given by equation (1)
(1)
where σ is the global strain rate, Uupper and Ulower are the upper and lower jet exit velocities, ρupper
and ρlower are the corresponding densities of the streams, and d is the nozzle spacing [29]. For the
experiments to be discussed in the following chapters, the two streams will have nearly equal
densities, and thus, the simplified relation
(2)
is appropriate. Compared to the local strain rate, the global strain is typically considered the
more appropriate parameter in a counterflow configuration because the local strain rate will vary
in the direction of flow coming from the jets due to thermal expansion effects. Moreover, local σ
will also vary in the long dimension of the slot jet due to changes in the flow field near the flame
edges. Thus, there does not appear to be any meaningful local σ.
Daou et al.[25] expressed the combined effects of strain and heat losses in terms of a
dimensionless flame thickness ε ≡ β(σα/(2SL
2
))1/2 and a dimensionless heat loss κ ≡ β(α/SL
2
)κ0. In
these expressions, β is the nondimensional activation energy (Zel’dovich number), α is the gas
thermal diffusivity, SL is the laminar burning velocity of a stoichiometric mixture of the fuel and
oxidizer streams, and κ0 is a linear volumetric heat-loss coefficient (units s−1
). For premixed
flames, simulations[19,22] suggested that ε ≡ (σα/(2SL
2
))1/2, while κ is the same as for
nonpremixed flames. κ0 is estimated [30] for slot-jet counterflows as 7.5α /d2
. In fact, dimension
analysis shows that ε2 is a scaled Karlovitz number (ratio of strain rate to chemical reaction rate)
and κ–1/2 is a scaled Péclet number (κ–1 is a ratio of heat generation to heat loss). These theories

13
predict that edge-flames with Le =1 exhibit two extinction limits, in general, with Uedge < 0 for ε
close to these limits and Uedge > 0 away from limits. When ε is high, extinction will happen
because of insufficient residence time. Uedge becomes more negative as ε gets higher. The
condition Uedge→−∞ occurs as σ approaches that of the extinction strain rate (σext) of the uniform
planar flame. When ε is sufficiently low, the second extinction limit is approached due to
conductive and radiative heat losses. For conditions that lead to Uedge < 0, a continuous flame
sheet could exist indefinitely. But suppose a flame edge develops locally in this sheet, such as a
hole caused by a local high strain rate in a turbulent flow or the insertion of a heat sink, the flame
sheet will be completely extinguished by the edge-flame that propagates (retreats) with a
negative propagation rate and results in the hole’s growth. At intermediate ε, theories predict
Uedge is nearly independent of ε, with Uedge/SL ≈ 1 for Le = 1 and larger/smaller Uedge/SL for
smaller/larger Le. These theories assume constant gas density, but in fact, Uedge is strongly
influenced by thermal expansion. By considering the effect of thermal expansion on the
momentum of the gases, Ruetsch et al. [31] predicted that Uedge/SL (in the unburned gas reference
frame far upstream) is accelerated in proportion to (ρu/ρb)
1/2, where ρu and ρb are the unburned
and burned gas densities, respectively. This acceleration has been confirmed experimentally for
nonpremixed hydrocarbon flames [30,32]. This relationship still holds for single premixed
hydrocarbon flames [33] because the thermal expansion-induced flowfield resulting from the
single flame with a single reactive sheet closely resembles nonpremixed flame. However, for
twin premixed flame, Uedge/SL was scaled with density ratio ρu/ρb rather than (ρu/ρb)
1/2 because
the low-density products are partly trapped between the twin-flames, analogous to expanding
spherical flames where mass conservation dictates that flame fronts propagate outward at a speed
of (ρu/ρb)SL rather than SL itself.

14
Prior to Cha’s work[30], few experimental studies have systematically addressed the effects of
environmental parameters on Uedge, such as mixture strength and for nonpremixed particularly
stoichiometric mixture fraction (Zst ≡ 1/(1+νXf /Xo)), which affects the flame position relative to
the stagnation plane in a counterflow. In the definition of Zst above, ν is the stoichiometric
oxygen-to-fuel mass ratio, and Xf and Xo are the mass fractions of fuel and oxygen in the reactant
mixture streams. While pure fuel burning with highly-diluted oxygen (air) corresponds to low Zst
(typically 0.06 for hydrocarbon-air mixtures) is prevalent in traditional nonpremixed combustion,
new fuels and combustion technologies (biofuels, oxyfuel combustion, massive exhaust gas
recirculation, etc.) leads to much broader Zst ranges (up to 0.8 for pure O2 burning with highlydiluted hydrocarbon fuel and up to 0.93 for highly-diluted H2.) In a counterflow, the flame
location moves from the oxidizer side of the stagnation plane to the fuel side as Zst increases,
resulting in significant differences in the reactant temperature/composition/time history, which in
turn substantially affects the burning rates characterized by Uedge or extinction strain rate (σext)
[30,32,34]. This was shown to cause very asymmetric flame sheet properties with respect to Zst
= 0.5 (where the flame sheet resides at the stagnation plane), which was attributed [34] to shifts
in O2 concentration profiles as Zst increases to coincide more closely with peak temperature
locations, leading to increased radical production rates and thus more robust flames.
Cha and Ronney [30] focused primarily on Zst = 0.5 with one data set taken at Zst = 0.2 and
0.8 for CH4/N2 - O2/N2 mixtures for which Lewis numbers of fuel (Lef ) and oxygen (Leo) are
near unity. Studies of Le effects are relevant to systems employing fuels with very high or low
molecular masses, resulting in widely varying Lef, which strongly affects Uedge and extinction
behavior.

15
Song et al. [32] expanded the previous work and systematically studied Zst effects on edgeflame properties for varying σ and Lewis numbers (by changing fuel and diluent type). It was
shown that Uedge and extinction behaviors are coupled with Zst: there are two competing factors.
One is a chemical effect: at low Zst the flame location is on the oxygen side of the stagnation
plane, and radicals produced primarily by the fuel must diffuse upstream to the reaction zone,
whereas at high Zst the flame lies on the fuel side and radicals are readily convected downstream
to the reaction zone, thereby strengthening the flame, consequently, σext increases monotonically
with Zst. The other is a transport effect: at low Zst reaction is O2-limited, whereas at high Zst
reaction is fuel-limited, which leads to a shift in effective Le (Leeff, detailed discussion in the
following section) from that of O2 (Leo) to fuel (Lef) as Zst increases. As a result of the
interactions of Lewis number and chemistry effects, Uedge may increase or decrease with
increasing Zst or have a non-monotonic (U-shaped) behavior. Despite these potentially
complicated interactions, it was found that all observed Uedge vs. Zst trends are consistent with
computed values of extinction strain rate (σext) of these mixtures in a 1D counterflow. Thus, σext
serves as a simple surrogate for predicting edge-flame behavior. These results indicate that the
behavior of highly turbulent nonpremixed flames near extinction (where edge-flames develop
[35]) depends critically on (1) fuel type (through Lewis number and chemical effects), (2) degree
of dilution of both fuel and oxidizer (characterized by Zst), and (3) strain rate.
Clayton et al. [33] examined propagation rates, structures, and extinction limits of single and
twin hydrocarbon premixed edge-flames in the same slot-jet counterflow burner for varying ε, κ
and Le. Most phenomena, including flame shapes, burning intensities, the occurrence of
propagating and retreating edge-flames, propagation speeds relative to SL, high-ε and low-ε
extinction limits, could be interpreted based on theoretical developments via scaled parameters

16
characterizing strain rate (ε) and heat loss (κ). Flame structures for various mixtures and global
strain rates for single/twin premixed edge-flame are shown in Figure 6 and Figure 7 respectively.
Fig. 5. “Short length” flame in Clayton et al. [33].
Figure 5 shows the experimental observation of continuous trains of “short-length” edgeflames at very low ε was not predicted by theory and was hypothesized to be due to thermal
expansion effects not included in the abovementioned theories. No retreating edge-flames were
obtained for single premixed edge-flames; this is consistent with theory [24], which predicts
negative Uedge values exist for twin but not for single premixed flames. Direct Numerical
Simulation (DNS) of edge-flames in the counterflow apparatus with heated flame anchors would
be required to test this hypothesis. Other DNS studies [35] modeling preheated reactants with
appreciable low-temperature chemistry have identified far more varied edge-flame structures
than those observed here; experiments to test these predictions would be relevant to autoignition
in diesel engine combustion.

17
Fig. 6. False-color direct images of single premixed edge-flames. Reactive mixture flows from the bottom and cold
inert from the top. All flames propagate from left to right, and the height of each image is scaled with the jet
spacing (d) [33].
Fig. 7. False-color direct images of twin premixed edge-flames. Global strain rate (σ) is shown in each image. All
flames propagate from left to right, and the height of each image is scaled with the jet spacing (d) [33].

18
1.2.3 Lewis number and Diffusive-thermal instability (DTI)
Diffusive-thermal instability (DTI) is generated when thermal diffusion and molecular
diffusion outside/into a flame sheet are not balanced [36,37]. For a curved or stretched flame
front, heat diffuses from the flame front into unburned gas. In contrast, radicals/molecules
diffuse from unburned gas toward the reaction zone to sustain the flame, as shown in Figure 8.
The bulge of the flame acts like a local source of heat, and a sink of reactant radicals, and the
local flame speeds are affected.
DTI is a small-scale instability that depends on flame structures, and it can be characterized
by the Lewis number Le ≡ α/D. For a certain molecule species, Le is the ratio between the
thermal diffusivity and the mass diffusivity of the bulk mixture. However, for a multi-reactant
system, the estimation of Le for the gas mixture becomes more complicated, and the concept of
effective Lewis number (Leeff) is introduced. A traditional approach of estimating Leeff for a tworeactant system is based on the stoichiometrically limiting reactant: the reactant is readily
consumed in the gas mixture before all other reactants are consumed. Here introduces the
equivalence ratio:
(1)
φ = 1 corresponds to the case that all fuel and oxidizer in the gas mixture are consumed
(stoichiometric); when φ < 1, fuel is the deficient reactant, and Leeff is closer to Lef; when φ > 1,
the oxidizer is the deficient reactant, and Leeff is closer to Leo. For example, a lean H2-O2-N2
mixture will have Leeff less than unity, while a rich H2-O2-N2 mixture will have a Leeff slightly
more than 1.

19
Fig. 8. Thermal and molecular mass diffusions to a curved flame front [31].
For equivalence φ closer to 1, several methods are available to quantify Leeff in multi-gas
mixtures. In this study, the estimate is based on a linear stability analysis by Joulin and Mitani
[38], which considers reaction rates and weights by the abundant reactant. The results vary based
on the reaction order of the global reaction, as shown in Figure 9.

20
Fig. 9. Leeff in H2-O2 flames as a function of H2-O2 ratio [38].
Since the effective Lewis number Leeff depicts the relative strength of thermal and mass
diffusion within the flame, a Leeff value of unity naturally implies that the two types of diffusions
are balanced. Consequently, the local flame temperature is unaffected. When Leeff > 1, thermal
enthalpy diffusion dominates over mass diffusion of the oxidizer, leading the flame front to lose
more heat than it gains in reactant radicals for chemical reactions. Consequently, the local flame
temperature and flame speed decrease. When Leeff <1, mass diffusion is more significant than
thermal diffusion, resulting in a strong influx of reactant radical molecules into the flame front,
thereby increasing both the local flame temperature and flame speed.
In gas mixtures with sufficient low Leeff, DTI can break the flame into small cellular structures
at the most amplified wavelengths. Markstein [39] performed a famous experiment to
demonstrate such cellular structure, as shown in Figure 10. This type of flame is primarily
observed in lean H2-O2-N2 and rich hydrocarbon-O2-N2 gas mixtures, where Leeff is less than
unity due to the high mass diffusivity of hydrogen and the low mass diffusivity of hydrocarbon
correspondingly.
More theoretical and experimental work has shown that both premixed flames[37,39–43] and
nonpremixed flames[44–46] exhibit diffusive-thermal instability (DTI) that leads to cellular
flame structures at low Le of the stoichiometrically limiting reactant (fuel or oxidant). It is also
well established theoretically that flame stretch due to hydrodynamic strain or front curvature
affects these instabilities [1,18,19,27,47–49]. Stretch effects on low-Le DTI are relevant to
turbulent combustion of lean mixtures of light fuels such as hydrogen and methane since the
low-Le DTI is known to affect the wrinkling characteristics and burning rates of turbulent flames
in these fuels [20,21]. Moreover, the strain and front curvature induced by turbulence will affect

21
these wrinkling characteristics. Theory predicts varied effects of flame stretch on low-Le DTI.
The strength of the stretch effect is typically characterized by the Damköhler number (Da~
SL
2
/Σα ~ ε -2
), which indicates the ratio of the stretch rate to a characteristic chemical reaction
rate of the mixture at unstretched conditions.
Fig. 10. cellular structure in hydrocarbon-O2-N2 flames at atmospheric pressure in Markstein (1951) [39].

22
For unstretched (Da = ∞) premixed flames, DTI is predicted to occur for all mixtures with
sufficiently low Le of the stoichiometrically limiting reactant [37]. Sivashinsky et al.[18] and
Buckmaster and Ludford [50] showed that flame stretch due to extensional hydrodynamic strain
could suppress DTI. Buckmaster and Short[1] and Daou and Liñán [19] showed that for
premixed stretched flames in a counterflow straining configuration, instability would occur for
all values of Da down to and below the extinction value for sufficiently low Le. In such cases, at
low Da, which is close to the extinction limit, the flame structure becomes discontinuous,
resulting in “flame tubes” elongated in the direction of extensional strain. These “flame tubes”
enable the flame to survive in the presence of strain that would cause it to extinguish if forced to
remain planar or only moderately wrinkled. Buckmaster and Short [1] predicted an infinite chain
of tubes for moderately low Da, then one or two isolated stationary tube(s) at progressively
lower Da. If Da becomes yet lower, complete flame extinguishment will take place. It is
interesting to note that these structures occur at values of Da that are lower than the value
corresponding to the extinction of plane flames. This behavior is somewhat analogous to
spherically symmetric “flame balls” observed in microgravity experiments [51], where
enhancement of flame temperature at low Le causes the curved flame to survive under conditions
a plane flame could not.
These theoretical predictions might be compared to a few experimental observations on
premixed flames. Ishizuka et al. [52] and Ishizuka and Law [53] examined lean methane-air and
rich propane-air mixtures in a counterflow. Still, these mixtures have Lewis numbers only
slightly less than unity (about 0.95 and 0.80, respectively) and were not found to exhibit the wide
range of structures described above. Indeed, for these mixtures with near-unity Le, the effect of
stretch was shown to suppress cellular instability merely. Liu and Ronney [54] suggest the

23
occurrence of single tube-like flame structures for lean H2-air mixtures (Le ≈ 0.30) at low Da
that is very near blow-off extinction limits in a slot-jet counterflow apparatus. In contrast, no
similar behavior was observed in lean CH4-O2-CO2 mixtures (Le ≈ 0.86); instead, cellular flames
were exhibited at high Da, but were suppressed at lower Da.
Thus, both experiments and theory seem to show that flame-tube behavior occurs only at
sufficiently low Le, i.e., well below the critical value for the occurrence of cellular flames and
lower than those attainable from lean CH4-air, lean CH4-O2- CO2, or rich C3H8-air mixtures.
For nonpremixed flames, the behavior predicted by theory differs from that of premixed
flames. DTI can occur only at low values of Da near extinction conditions because, at higher Da,
the structure of a nonpremixed flame sheet is determined only by mixing considerations without
coupling to chemical reactions. Under such conditions, the flame is unconditionally stable [46].
At sufficiently low Da, reactant leakage through the flame front occurs [55], which causes partial
premixing of reactants. Thus the local heat release rate becomes dependent on chemical reaction
rates in addition to mixing rates.
These models and experiments [45] show that for the DTI to occur, both Da and Le need to be
sufficiently low. The experiments also showed similar results for cases where the driving
mechanism for extinction is heat loss, which becomes crucial at very low σ or high Da, rather
than finite residence time at low Da. Tube-like flames have also been predicted for nonpremixed
flames.
Experimentally, Liu and Ronney’s [54] discovery suggests that a single isolated flame tube is
likely to occur in H2-N2 vs. O2-N2 flames (Lef ≈ 0.30, Leo ≈ 1.0) very near extinction as what
actually happens in the premixed case. Shay and Ronney [56] did not find any such behavior in
CH4-CO2 vs. O2-CO2 (Lef ≈ 0.72, Leo ≈ 0.87). Indeed, Chen et al. [45] found that CH4-CO2 vs.

24
O2-CO2 mixtures were only marginally capable of producing cellular structures in nonpremixed
flames near extinction, even though in premixtures, they readily produce cells for sufficiently
high Da [54]. This observation again indicates the need for sufficiently low Le to observe the
unusual sequence of flame-tube behavior predicted computationally.
Kaiser et al. [57] again examined both premixed and nonpremixed H2-O2-N2 flames in a
counterflow geometry. A wide range of flame structures was observed, particularly close to
extinction limits. These structures result from the interactions of several effects, including
diffusive-thermal instability, flame stretch, conductive heat loss to burner exits, and the shear
layer between the jets and the ambient atmosphere.
Fig. 11. Cellular flames in Kaiser et al. [57].
1.2.4 Flow in the Hele-Shaw cell
A Hele-Shaw cell consists of two large parallel plates separated by a tiny gap. This flow field
inside a Hele-Shaw cell is comparatively straightforward to describe, allowing for simpler
analytical and computational models. As demonstrated in Figure 12, a flame propagating in a

25
Hele-Shaw cell approximates a two-dimensional flow when depth-averaged across the thin
channel, thus rendering it a quasi-2D flow.
Fig. 12. A propagating flame front in a narrow channel
For such flow configurations, a set of simplified two-dimensional single-phase conservation
equations can be employed. When the system is characterized by a low Reynolds number and
when the ratio of the cell thickness and the width d/w is much smaller than 1, the following
equations can be used:
(3)
(4)
Essentially, gas-phase combustion experiments in a Hele-Shaw cell circumvent the issue of
residual film that shows up in liquid non-reactive experiments. However, they introduce other
complexities, most notably related to heat loss. This is a significant concern because heat loss
directly influences critical properties such as density and viscosity ratios across the flame. The
thermal gradients thereby generated can result in a range of instabilities, affecting both flame

26
structure and propagation speed. Moreover, because the flame in a combustion setup is selfsustaining, an additional energy equation is needed to account for the heat released during
combustion.
1.2.5 Instabilities in the Hele-Shaw cell
In narrow channels, several instabilities are expected to manifest and cause flame fronts to
wrinkle, even in laminar flow conditions. These instabilities can arise from various factors, such
as hydrodynamic interactions, thermal gradients, and chemical kinetics. They could potentially
influence the flame structure and its propagation characteristics, making it imperative to account
for these phenomena in both theoretical and practical applications.
Darrieus-Landau instability
Due to the strong exothermic nature of combustion, a large temperature gradient always exists
across the flame front. Under the assumption of ideal gas behavior at constant pressure, the
density of the gas mixture exhibits an inverse relationship with temperature. This leads to a
significant density difference between the unburned and burned gas mixtures, which further
introduces a baroclinic generation of vorticity into the system. This misalignment between
density and pressure gradients at the flame front leads to complex multidimensional interactions
between the flame and the flow.
Figure 13 shows the flow and flame interactions affected by this hydrodynamic instability.
The streamlines across the flame front indicate flow cross-section area variations due to
curvature. Here, the hydrodynamic length scale associated with Darrieus-Landau (DL) instability
is significantly larger than the flame thickness, so the flame is effectively considered as having
zero thickness. This enables the utilization of Bernoulli’s equation

27
(5)
to describe the conservation of momentum across the flame front. In the presence of a tilted
flame element, the continuity of the tangential velocity component requires that the streamlines
bend toward the local normal vector of the flame sheet as they cross it. This leads to a situation
where a stream tube will display an expanded cross-sectional area at the flame if the flame sheet
is convex towards the unburned gas, as depicted in the diagram.
Fig. 13. Diagram of flow and a curved flame front interaction [31].
In the framework of incompressible flow, this increase in area corresponds to a decrease in
fluid velocity. A perturbation in the flame sheet that displaces the flame upstream will
geometrically be convex toward the unburned gas. Such a perturbation encounters a decreased
fluid velocity as it passes through the flame, and this perturbation will be inclined to move even

28
further upstream. Conversely, a perturbation that displaces the flame downstream will cause the
stream tube to contract, leading to an increased fluid velocity and reinforcing the tendency for
the flame to displace further downstream. This mechanism generates multidimensional flame and
flow interactions [58,59].
The growth rate of DL instability does not depend on internal flame structures, and thus the
zero-flame thickness assumption was made. It neither has any characteristic wavelength, as such
curvature effect is independent of the wavelength (λ ≡ 2π/k) of flame wrinkles, where k is the
transversally harmonic wrinkles of wavenumber. Instead, the growth rate of DL instability solely
depends on the density ratio between the burned gas (ρb) and unburned gas (ρu). Characterized by
the thermal expansion coefficient, ε ≡ ρb /ρu, the growth rate of DL is
(6)
or as described by Joulin and Sivashinsky [60] in their linear analysis of a quasi-2D flame,
including only DL instability, the dimensionless growth rate (σ ≡ ωDL/kU) is,
(7)
Because ε is less than unity for all combustion processes, σ is always positive for all wavelengths.
Therefore, a planar premixed flame is unconditionally unstable.
In both two-dimensional and three-dimensional configurations, Darrieus-Landau solutions
predict that a characteristic length scale should fall within the range of flame thickness. However,
empirical observations have shown this scale to be at least two orders of magnitude larger
[39,61–63]. This discrepancy has led to a prevailing belief among researchers that finite flame

29
thickness and transport processes should not be ignored but instead integrated into the models for
a more accurate depiction of real-world phenomena [64].
Rayleigh-Taylor instability
The Rayleigh-Taylor (RT) instability, identified initially in liquid systems due to buoyancy
effects [39,65], has also been observed in combustion scenarios. Experimental measurements [66]
of slow-moving, fuel-lean methane flames influenced by buoyancy in a glass tube have been
found to be consistent with theoretical models describing the ascent of hot air bubbles [67]. This
instability can be characterized by the alignment of the gravitational field with the direction of
flame propagation.
In upward propagating flames, the high-temperature products gas mixture with a small density
stays beneath the low-temperature reactants gas mixture whose density is much larger. This
arrangement is expected to experience instability due to buoyant convection. This effect also
results in flame stretch at the top of the flame front in tube flames, as shown in Figure 14.
In downward-propagating flames, the Rayleigh-Taylor (RT) instability acts in a stabilizing
manner, serving to offset the destabilizing effects brought about by Darrieus-Landau (DL)
instability at larger hydrodynamic wavelengths, as cited in source [68]. In this orientation, the
less dense combustion products are situated above the denser reactant mixture, which inherently
flattens the flame front. Within the context of tube flames, the combustion products near the tube
wall are subject to faster cooling due to thermal losses, thereby generating sinking boundary
layers along the tube wall, a phenomenon depicted in Figure 14.

30
Fig. 14. Diagram of upward (left) and downward (right) propagating flame and buoyancy effects [69].
The Rayleigh-Taylor (RT) instability predominantly operates on larger hydrodynamic scales
rather than smaller diffusive scales and is, therefore, independent of the intricacies of flame
structure. The key parameters governing RT instability include the ratio of densities between the
burned and unburned regions, as well as the gravitational field acting parallel to the direction of
flame propagation. Theoretically, the maximum wavelength of RT instability is constrained by
the dimensions of the experimental combustion chamber, particularly the chamber width (w).
Therefore, the critical wavenumber for RT instability is kRT = 2π/w, which overlaps with the
effective wavelength of DL instability. RT instability will interact with DL instability and affect
the overall dimensionless growth rate (σ ≡ ωDL&RT/kU) of flame instability. Joulin and
Sivashinsky [60] derived this combined growth rate as,
(8)
σ increases for upward propagating flames and decreases for downward propagating flames.

31
Saffman-Taylor instability
The Saffman-Taylor (ST) instability, also known as viscous fingering, was initially identified
in investigations of non-reactive fluid interfaces with differing viscosities [2,70]. Tabeling’s
comprehensive experimental study [70] elucidated the characteristic finger-shaped growth
patterns at an oil-water interface, as shown in Figure 15. This form of instability is observed in
narrow channels where a less-viscous fluid displaces a more-viscous fluid. Consequently, ST
instability is primarily studied and modeled in the Hele-Shaw cell, a narrow channel sandwiched
between two transparent plates [71,72]. Hele-Shaw cell has one of the simplest geometries,
where flow behavior has been studied extensively in Homsy [73].
Indeed, Saffman-Taylor (ST) instability has been extensively studied in the context of nonreactive fluids, but its direct applicability to flame dynamics is constrained by several
fundamental differences. One of these is the absence of surface tension in combustion
phenomena, a factor that plays a significant role in the ST instability in non-reactive interfaces.
The influence of curvature effects on the flame front could be analogized to the role of surface
tension in non-combustive systems. However, these curvature effects are still pending further
investigation.
In the combustion field, ST instability is usually studied on top of DL and RT instabilities in
tube flame [74,75]. Like DL and RT instabilities, ST instability is a hydrodynamic instability that
does not depend on flame structures and has the most effects on larger hydrodynamic scales.
However, due to its friction-driven nature, ST instability is only observed in confined
environments (e.g., in a Hele-Shaw cell) with dependence on its geometry. Consequently, the
instability can be characterized by the dynamic viscosity ratio between products gas mixtures
and reactants gas mixture (µb /µu) and the thickness d of the narrow channel. Kang, Im & Baek

32
[76] also find that ST instability is most prominent at long wavelengths, indicating flames with
high viscosity contrast propagating thin channels (small d) would present the most prominent
effects of ST instability. The hot products gas mixture has a higher viscosity than the cold
reactants gas mixture µb /µu ∝ (Tb /Tu)
0.7, thus creating imbalanced friction force and slight
variation of pressure across the flame.
Fig. 15. Oil-water interface experiment demonstrates Saffman-Taylor Instability by Tabeling et al. [70].
Numerical studies have found it to occur at small Péclet numbers Pe in a channel flame,
significantly impacting flame shape and propagation rate. Kang, Im & Baek [76] shows a
significant velocity increase in the burned gas behind the flame front. Then, the net effect of

33
increased velocity and the friction force of more viscous fluid exceeds the resistance of incoming
fluid (unburned gas). The incoming flow would be deflected by the less viscous fluid (burned
gas), as shown in Figure 16.
Fig. 16. Diagram of the Saffman-Taylor instability mechanism
In contrast to the non-reactive interface between oil and water, a flame front serves as an
exothermic, self-propagating interface. Here, a variation in viscosity is intrinsically linked with a
corresponding density change. Consequently, ST instability is seldom considered in isolation and
is often coupled with DL instability in empirical research. A comprehensive linear analysis,
accounting for all three types of hydrodynamic instabilities—ST, DL, and RT—was put forth by
Joulin and Sivashinsky in 1994 [60].

34
Diffusive-Thermal instability
The growth rate of Diffusive-Thermal Instability (DTI) operates over a considerably narrower
effective wavelength range in comparison to other hydrodynamic instabilities under
consideration [77]. Characteristic wavelength () occurs at a critical Leeff, which is typically
less than 1. At small wavelength compared to , flames favor curvatures and create broken
cellular flames. At larger wavelength but still significantly smaller than the effective wavelength
range of DL instability, DTI can act to stabilize the flame front against DL-induced disturbances
[78]. An in-depth discussion of DTI and its relationship with Leeff is presented in Section 1.2.3.
1.2.6 Propagation speed and scaling of flame fronts in a narrow channel
Joulin-Sivashinsky Parameter
Although many literary works study each individual of the instabilities, investigations,
especially experimental works involving multiple-instability effects on flame behaviors and
propagation velocity, are very limited.
In Joulin and Sivashinsky’s [60] theoretical and numerical study of the aforementioned flame
instabilities, the width and length of the Hele-Shaw cell are much larger than the flame thickness
(). Linear stability analysis was utilized to derive a dispersion relation of a combined
instability growth rate of a quasi-2D flame. It incorporated DL, RT, and ST instabilities but
omitted DT effects by assuming Leeff = 1. They assumed Euler-Darcy flow, which retains the
momentum term yet averaged across the gap, and the flame was treated as an infinitely thin
discontinuity (zero flame thickness assumption) with specified local normal propagation speeds.
The study showed that due to persistent density increases and viscosity decreases across
flame fronts (modified by heat losses through chamber walls), DL and ST instabilities are

35
unconditionally destabilizing, whereas RT effects can be destabilizing or stabilizing for upward
or downward propagation directions, respectively.
JS parameter (σ) is a dimensionless linear growth rate (σ ≡ ω/kU) that characterizes the driven
force of instability wrinkled flames. It is used to scale the flame propagation speed measured (ST)
normalized by a linear flame speed, such as SL, similar to how turbulent intensity (′) is used to
describe turbulent flame propagation speed.
This investigation focuses on flames in flow without any forced turbulence. Therefore, the
combined growth rate of all instabilities exhibited by flame can be seen as an additional driving
force for flame propagation. The non-adiabatic Euler-Darcy equations are,
(9)
(10)
(11)
where u is the 2-D velocity field in a flame propagating against unburned gas mixtures with
stationary Hele-Shaw cell walls. In the above equations, the density () and temperature (T) are
averaged over the thickness of the cell d. g is the gravity field aligned with the x-axis (g < 0 for
downward propagation, g = 0 for horizontal propagation, and g > 0 for upward propagation). In
line with the prior explanation, the flame is modeled as a reactive discontinuity in the flow.
Symbols f and ℎ are the momentum and energy exchange coefficients based on plate temperature,
and these coefficients differ for the unburned and burned regions adjacent to the flame front.
Consequently, the derivation employs distinct f and ℎ values for each of these areas. This analysis
allows separate representations of thermal and transport properties for unburned and burned

36
product gas mixture. The flame is endowed with a known burning velocity U relative to the fresh
unburned medium,
(12)
where D is the front velocity in a fixed frame and n is the unit normal to the front. With ε ≡ρb /ρu;
fav ≡ (fb+fu)/2; Ub = (ρu/ρb)U=U/ε, the final growth rate () of transversally harmonic wrinkles of
wavenumber (k) is represented as
(13)
(14)
defining dimensionless growth rate, dimensionless wavelength, dimensionless friction factor and
dimensionless gravity factor,
The final simplified equation becomes,
(15)
In the equation above, G represents RT instability, while denotes ST instability. The
wavelength associated with F is measured as , where
is the Péclet number. It is noteworthy that Darrieus-Landau (DL) instability lacks a characteristic
wavenumber. In contrast, both RT and ST instabilities do exhibit preferred wavelengths. In the
context of the experimental setup for this study, this preferred wavelength corresponds to the
largest wavelength that can be accommodated within the width (w). Therefore, a characteristic

37
wavenumber, k = 2π/w, is utilized for the analyses conducted in this study. The analytical growth
rates are shown in Figure 17.
Fig. 17. Effects of ST and RT instabilities on DL instability’s dimensionless growth rate ( ) for a range of example
hydrocarbon-air flames with U = 20cm/s and d = 1.27cm.
In order to observe Saffman-Taylor (ST) instability within a Hele-Shaw cell, the ST
wavelength λST must be less than the cell width w. Therefore, for a fixed w, either Pe or d has to
be sufficiently small. While a low flame speed would naturally result in a small Pe, maintaining
a high flame speed and temperature is critical to avoid flame quenching. On the other hand,
reducing d means increasing the effects of heat loss, which can suppress density and viscosity
variation across the flame and thus suppress DL, RT, and ST instabilities. In experiments, ST
instability is mainly observed in fast flames propagating through a cell with the smallest d.
Joulin and Sivashinsky also predicted that DL effects dominate at small wavelengths,
whereas ST effects dominate at longer wavelengths. One important flame instability mechanism
not considered by JS is the diffusive-thermal mechanism (DTI). When Le < Lec < 1, where Lec is
a critical Lewis number, cellular flames emerge. These flames exhibit real growth rates that peak

38
at finite wavenumbers, whereas for Le > Lec, the instability mechanisms serve to discourage the
formation of wrinkles in the flame front. Since this instability is dependent on diffusional effects
that occur on the scale of the flame thickness, it does not affect very long wavelengths. However,
at short wavelengths, it is so dominant that diffusion damps out all wrinkling. This prevents
arbitrarily small wrinkles (which could otherwise be permitted by the DL, ST, and RT
mechanisms) from appearing.
From an experimental standpoint, achieving a perfect 2-D flame is impractical. Inherent thirddimensional curvature introduces another set of small-scale length effects in the d-scale of the
flame propagating in a narrow channel. These effects contribute to the uncertainties and
limitations associated with utilizing parameters derived from the Joulin-Sivashinsky (JS) model.
Therefore, while the JS model serves as a useful theoretical framework, its application in realworld experiments must consider these inherent complexities and deviations from idealized
conditions.
Propagation speed and scaling
Flame propagation speed (ST) in a closed configuration primarily depends on several
parameters:
(1) mixture strength and thus adiabatic flame temperature (Tad).
(2) the effective Lewis number (Leeff).
(3) Fuel and inert gas types.
(4) Propagation directions.
(5) Combustion channel thickness (d) and width (w)

39
It is hypothesized that in a sufficiently long narrow channel with an open ignition end, one
can observe a quasi-steady region of ST. If this scenario is successfully achieved in a Hele-Shaw
cell setup, one can develop scaling parameters to evaluate the effects of ST. The most
straightforward and intuitive scaling parameter to consider first would likely be the laminar
burning speed SL.
Jarosinsky [79] conducted an experimental study of flame propagation in narrow planes but
did not describe front shapes. CPL at USC studied such flames in lean CH4-air (Le ≈ 1.0), C3H8-
air (Le ≈ 1.7), and CH4-O2-CO2 mixtures (Le ≈ 0.7) mixtures in a large Hele-Shaw apparatus.
These experiments indicate that flame wrinkling is an unavoidable phenomenon in these
configurations due to the presence of density and viscosity gradients across the flame front—
both destabilizing factors. Most significantly, for all cases, the measured propagation speed of
the wrinkled flame (ST) is always significantly greater than SL, even though no forced turbulence
is present. The average value of ST/SL is about 2.8 for CH4-air mixtures with Le ≈ 1, which shows
the enhancement of propagation rate due to wrinkling from the instability mechanisms described
above. Very similar results were found for C3H8-air mixtures (Le ≈ 1.7), but with a lower average
ST/SL ≈ 2.0 and for CH4-O2-CO2 mixtures (Le ≈ 0.7) but with a higher average ST/SL ≈ 3.7. Thus,
the Lewis number profoundly impacts the average propagation rates of flames subject to selfgenerated wrinkling, even when such wrinkling occurs in conjunction with hydrodynamic in
addition to diffusive-thermal effects.
A similar conclusion was reached via numerical simulations by Kang et al. [76], who
extended Joulin and Sivashinsky’s analysis by performing numerical simulations with a 2D
compressible reactive Navier-Stokes model. Linear growth rates of DL, ST, and RT instabilities
were computed within the Poiseuille flow assumption. Their results agreed with theoretical and

40
numerical studies by Altantzis et al. [80] regarding DL instability with variances concerning ST
instabilities.
In addition, Fernandez-Galisteo et al. [81] performed an extensive analysis of quasi-isobaric,
quasi-2D flames in the Hele-Shaw cell using average flow properties across the channel gap,
similar to the experimental setup of this research. Their simulations incorporate DL, RT, ST, and
DT instabilities. Although the study uses only one-step chemistry, constant heat capacity, and
variable transport coefficients, it shows very good qualitative and, in some cases, quantitative
agreement with the experiments conducted by CPL.
Another study by Berger et al. [82] finds that for a 2D planar lean hydrogen flame in an
adequately large domain with confinements, the flame speed is essentially independent of the
domain size. Despite the absence of third-dimensional viscosity effects and the use of multi-step
chemistry, this 2-D numerical work, which employed large-scale Direct Numerical Simulations
(DNS), showcased flame shapes influenced by instabilities that are remarkably similar to those
found in Fernandez-Galisteo et al. [81].
For effects on a length scale comparable to flame thickness (d-scale), it has been determined
that heat loss to walls significantly impacts both the flammability and the propagation velocity of
planar flames with a unity effective Lewis number Leeff. For channel width comparable to flame
length (~15), the flames can experience partial extinction due to excessive heat loss [83]. For
flames with Leeff < 1, the high heat loss through the walls leads to fragmented cellular structures
with “dead” spaces both in the central regions and near the walls of the channel [84]. In another
numerical simulation study [85], laminar flames with Leeff > 1 exhibit lower flame velocity in a
narrow channel with isothermal cold walls compared to those in channels with adiabatic walls.
However, laminar flames with Leeff < 1 have unexpectedly higher flame velocity in the channel

41
with isothermal cold walls than with adiabatic walls. This is because increased flame curvature
near the isothermal cold walls creates pockets of higher-temperature product gas behind the
flame.
Although some experiments investigated flames propagating through a narrow channel [86–
89], they nearly all analyze flame extinction and propagation velocities in d-scales (10-30).
Not to mention the absence of experiments to verify those investigations and to look at an overall
effect from all four instabilities on flame behaviors and propagation velocity.
As previously stated, the JS analysis amalgamates the impacts of the three large-scale
instabilities but falls short in addressing the diffusive-thermal (DT) instability. To minimize the
impacts of the issue in this work, in addition to commonly used laminar flame speed SL, attempts
with two methods are made to segregate small length-scale effects from large length-scale
hydrodynamic instabilities. One is to use the “effective laminar burning speed” (SL
*
) based on
extinction strain rate (σext) as utilized in the edge-flame experiments. The other is to decompose
ST into 2D and small-scale components, which leads to a “modified laminar flame speed” (SL
′
).
Wrinkled flames are particularly susceptible to the influence of flame stretch. In scenarios
where flames advance toward the closed end of a channel under constant pressure, it is plausible
to assume that the unburned gas located in front of the flame remains stationary. Under these
circumstances, flame stretch is enhanced by curvature, particularly in gaseous mixtures with
misaligned thermal and mass diffusion rates, which is intimately connected to DT instability.
This study attempts to use stretched flame theories to incorporate the effects of Leeff in the
scaling parameter.

42
The concept of flame stretching and the Karlovitz number is indeed pivotal in the
study of flame dynamics, particularly when analyzing the susceptibility of a flame to various
instabilities and the phenomenon of flame extinction. Ka is a dimensionless parameter defined as
the ratio of the chemical time scale often represented by δT/SL to the physical time scale of a
stretched flame 1/σ, which measures the physical flow time scale of a stretched flame. It is found
that spontaneous flame extinction happens when these two scales are comparable: under strong
stretch, the time of flow can be too short for sufficient chemical reactions, thus extinguishing the
flame.
Under strong stretch, the stretch rate is proportional to laminar flame speed and thermal
diffusivity, σ ~ SL
2
/α. At its limit,
(16)
In premixed flame, SL ∝ √αω̇ . Therefore, by comparing with Equation (16), the stretch rate σ
should scale with the overall reaction rate ω̇ (σ ~ ω̇). For cases near the limit, the extinction
strain rate (σext) is associated with the highest reaction rate. Then, the “Effective burning velocity”
(SL
*
) is defined as,
(17)
This SL
* is a function of Leeff of the gas mixture. Therefore, by scaling ST with SL
*
, Leeff effects
and DTI should be readily accounted for.
A single premixed counterflow configuration with warm diluent is used to estimate the
extinction strain rate (σext). The temperature of the warm diluent is set to 90% of the adiabatic
temperature Tad of the combustible mixture. This configuration best mimics the situation at the

43
flame front in a Hele-Shaw cell, compared to single premixed combustible gas versus cold inert
or twin premixed cold reactants.
On the other hand, according to Damköhler’s second hypothesis, there should also be a factor
that incorporates large-length scale wrinkling caused by DL, RT, and ST by computing the flame
surface area ratio. In this quasi-2D case, the wrinkled flame length ratio lflame/w measured from
experimental videos is expected to catch such large-length scale characteristics. Then, the
decomposition model of ST is shown as,
(18)
In the equation above, ST is a measured quantity, SL
* is a computed value from Chemkin
Pro/Cantera, and U is an “averaged local flame speed” normal to the flame front. In the
meantime, ST /U accounts for the effect of lflame/w. Therefore, Equation (18) can be transformed
into the expression of a scaled “average flame speed.”
(19)
1.2.7 Objectives
The thesis aims to contribute to the understanding of hydrogen combustion by focusing on
two key experimental setups: edge-flame and Hele-Shaw experiments. In the case of edge-flame
experiments, the objective is to identify and elucidate various types of flame structures, such as
flat flames, cellular flames, moving tubes, and stationary tubes. The study seeks to measure
edge-flame propagation rates under well-characterized straining flows and to explore the
influence of factors like mixture strength, equivalence ratio, strain rate, and radiative heat losses.

44
In the context of Hele-Shaw experiments, the aim is to assess how the unique properties of
hydrogen flames impact flame propagation rates and structures in narrow channels. The
questions to be addressed include whether ST /SL values in H2-air mixtures exceed those found in
hydrocarbon mixtures, given the lower temperatures and thermal expansion. The study also
intends to investigate the roles of mixture strength, equivalence ratio, combustion chamber
dimensions (cell thickness and width), and chamber orientation in these results. The influence of
heat losses will also be examined.
Additionally, the study will attempt to define the limits of flame acceleration mechanisms,
such as thermal expansion, viscosity contrast, and diffusive-thermal effects, when external
turbulence is absent. Overall, the work seeks to address several unresolved questions in the field
of hydrogen combustion, with implications for both academic research and practical applications.

45
Chapter 2: Methodology
2.1 Apparatus and Procedures
2.1.1 The slot-jet counterflow burner
This study chooses a slot-jet counterflow apparatus over the most used axisymmetric
counterflowing jet apparatus. A difficulty with the axisymmetric apparatus in the current study
context is that extensional strain is multidirectional, affecting both coordinate directions parallel
to the flame surface. This multidirectional straining complicates the investigation of flame
properties in a controlled strain environment. In contrast, a counterflow slot-jet apparatus
provides extensional strain in the direction orthogonal to the plane of the slots, but there is very
little strain or convection in the direction along the length of the slots. Thus, the flame is strained
only in one of the two coordinates in the plane of the flame. This way, an edge flame nearly
unstrained in the direction of propagation can be obtained. Additionally, computations by Ashurst
et al. [90] have shown that highly strained regions of turbulent flows exhibit a most probable
ratio of strain along the three principal axes in the ratio 0.75:0.25:−1, where positive values
denote extensional strain.
Moreover, DNS studies of premixed flames [91] have shown that the flame surface normal
preferentially aligns with the most compressive strain direction and that the strain in the plane of

46
the flame is statistically much higher in one direction than the orthogonal direction in the plane
of the flame. Thus, highly strained regions, where strain effects are most important, do not
typically exhibit nearly equal rates of extensional strain in the plane of the flame along two of the
principal axes. The counterflow slot-jet configuration provides strain rates in the ratio 1:0:−1,
whereas round jets provide 0.5:0.5:−1. Thus, The slot-jet configuration offers distinct advantages
in representing the straining conditions typically encountered by flames in turbulent flows,
thereby rendering a more realistic evaluation of the stretch environment. This is in contrast to
axisymmetric jets, which may not as realistically simulate the complexity of strain conditions
experienced by flames in such turbulent settings. It should be noted, however, that the laminar
flamelet model cannot apply for mixtures with low Le owing to the inherent diffusive-thermal
instabilities that yield a non-uniform flame front even under conditions of a uniform mixture and
strain field. Additionally, because the convection velocity in the long dimension of the slot is
very small, the propagation speed in the laboratory frame is nearly equal to the propagation
speed relative to the cold unburned gas far ahead of the edge flame (or behind, in the case of
retreating edge flames). This simplifies the interpretation of the experimental data.

47
Fig. 18. Schematic of the experimental apparatus (single premixed configuration)
The counterflow slot-jet burner employed for the experiments consisted of two 0.5×13-cm
central rectangular jets. Equal values of Ujet were employed for the upper and lower streams. On
both sides of these jets, additional 0.5×13-cm slot jets provided N2 sheath flow to prevent the
formation of a secondary nonpremixed flame with ambient air, as mentioned in Kaiser [57]. The
interior of all six jets was filled with steel wool, and the jet exits were fitted with a stainless-steel
honeycomb of 0.7-mm channel width to ensure uniformity of the exit flow. The jets and, thus,
the gases at the jet exits were maintained at room temperature by water cooling. Gas flows were
controlled by commercial mass-flow controllers with accuracy ±1% of full scale (calibrated with
flow calibrators). The sheath flow velocities (Usheath) were matched to those of the central jets,
i.e., Usheath = Ujet, to avoid shear-layer instabilities between the reactive flow and sheath flows.
This choice also produced flames nearly flat and parallel to the jet exit planes. Test experiments

48
showed that Uedge is relatively insensitive to Usheath if Usheath falls in the 0.7~1.3 Ujet range. The
apparatus was enclosed in a ventilated box to minimize potential interference from room drafts at
low Ujet.
Figure 18 shows a schematic of this apparatus. The edge-flames were recorded with highspeed intensified video using a camera sensitive to near-IR emissions near 823 nm, where H2O
has a weak emission band. Uedge and general flame behavior (broken vs. continuous, burning vs.
extinguished) were cross-checked with shadowgraph images, and no differences were found, but
only direct video images are reported here because of the challenges associated with interpreting
shadowgraph images. Because the slot-jet aspect ratio is finite, there is a slight extensional flow
along the slot length, which slightly influences Uedge in the laboratory frame; following prior
work [30,32], this bias was nullified by interpolating Uedge vs. position along the slot to the jet
centerline. To accomplish this, the edge-flame propagation speed in the laboratory frame of
reference (as determined from the video records) was plotted as a function of position (x) along
the slot. A nearly linear relationship between Uedge (x) and x was found, an example of which is
shown in Figure 18. This strongly suggests that the edge-flame propagation speed relative to the
unburned gas is essentially constant and that there is a small, nearly constant velocity gradient
along the slot of magnitude dUedge(x)/dx. Consequently, a linear least-square fit was applied to
the data in the vicinity of the center of the slot and interpolated to determine Uedge at the burner
center where symmetry requires that there is no flow in the propagation direction and thus,
values of Uedge in the laboratory frame and relative to the gas are identical. Note that for the
example shown in Figure 18, dUedge(x)/dx is 51 times smaller than the primary strain rate σ; for
all conditions examined, this ratio was greater than 25 (i.e., comparable to or greater than the slot

49
aspect ratio). Therefore, the velocity gradient in the direction of edge-flame propagation does not
influence the results significantly.
Fig. 19. Example of effect of edge-flame location on propagation rate in the laboratory reference frame adopted
from [30]
In some cases, the flames were broken, leaving islands behind the leading edge. As a practical
matter, in this work, Uedge is defined to be the propagation speed of the leading flame island. The
islands behind the front were practically stationary (Uedge = 0). For retreating flames, there is no
issue because islands were never observed. At jet Reynolds numbers Re = Ujetd/ν > 500,
unsteady flames were observed, indicating apparent transitions to turbulent flow; thus, results for
Re > 500 were discarded.
For conditions resulting in Uedge > 0, an N2 jet was used to “erase” the established flame from
one end to nearly the other and then retracted, enabling the edge-flame to propagate. The range
of conditions for which Uedge < 0 is not directly accessible in a slot-jet apparatus with “bare” slot
ends. To overcome this limitation, electrical wire resistance heaters attached to ceramic spacers

50
were installed at both ends of the slots. These heaters substantially increased the total mixture
enthalpy near the ends of the slot, thereby locally creating a mixture with Uedge > 0 that
“anchored” the weaker unheated mixture away from the ends. The heating wires anchored flames
by enhancing local flame temperature and thus reaction rates at the flame ends under conditions
in which they would retreat without localized heating. Then, the N2 jet was momentarily
introduced at one slot end to separate the flame from its anchoring hot wire, thereby triggering an
extinction wave.
2.1.2 Edge-flame configuration in counterflow burners
As discussed in the last section, the most canonical, readily characterized configuration for
studying edge-flames is plane strain with edges propagating in the third (unstrained) dimension,
which is employed in many theoretical and computational studies [19,22,25] and can be easily
approximated experimentally using counterflowing slot-jets that have a large length-to-width
ratio. Regarding the plane strain setup, three major mixing configurations are commonly used:
nonpremixed, single premixed, and twin premixed flame configurations.
Nonpremixed edge-flame configuration
Diffusion flames are encountered when the fuel and oxidizer are spatially separated and
sufficient energy is available, similar to the conditions for premixed flames. In this scenario,
mixtures of H2-N2 and O2-N2 are designed at specific values of Xo and Xf to achieve the desired
stoichiometric mixture fraction, Zst. The flame occurs in a region known as the mixing layer,
where the fuel and oxidizer are mixed, which can either occur due to molecular diffusion or the
turbulent motion of gas. As with the premixed case, the properties of the reactants and the
surrounding physical factors, such as heat loss, will affect the propagation and evolution of these
flames.

51
In the experiments presented, H2-N2 was issued from the lower jet and O2-N2 from the upper
jet. Reversing these flows had no significant effect on the results, hence buoyancy effects were
insignificant for the conditions tested.
Single premixed flame configuration
Premixed flames occur when the fuel and oxidizer are mixed before combustion. Single
premixed edge-flames feature premixed reactants (fuel+O2+diluent) flowing against a stream of
pure diluent gas. These flames are particularly relevant at lower turbulence intensities (u’), where
wrinkled flame sheets have fresh reactants on one side of the flame sheet and burned products on
the other side. In the experiments presented, H2-O2-N2 was issued from the lower jet and cold N2
was issued from the upper jet. Reversing these flows had no significant effect on the results;
hence, buoyancy effects were insignificant for the conditions tested.
Twin premixed edge-flame configuration
Twin premixed edge-flames are characterized by flows of premixed reactants opposing each
other. These flames become particularly relevant at higher turbulence intensities (u’), where the
possibility of highly folded flame sheets exhibiting back-to-back flames may occur locally.
2.2 Mixture strength and stoichiometric mixture fraction
For nonpremixed edge-flame configurations, mixture strengths (and thus Tad) are controlled
by changing the amount of diluent gas (N, defined as the moles of inert in the H2:O2:N2 = 2:1:N
mixture formed when fuel and oxidant streams are combined in stoichiometric proportions). The
stoichiometric mixture fraction Zst is adjusted by changing the portion of N2 on the H2 side
versus the O2 side. For example, , N = 18 (H2:O2:N2 = 2:1:18), if all the N2 is on the fuel side
(H2:N2 = 2:18, i.e., 10% H2/90% N2, O2/N2 = 1:0 (100% O2)), the stoichiometric mixture fraction

52
Zst = 0.9407 whereas if all the N2 is on the O2 side (H2:N2 = 2:0 (100% H2), O2:N2 = 1:18 (5.26%
O2 / 94.74% N2), the stoichiometric mixture fraction Zst = 0.0074. For these extreme Zst cases
and all intermediate Zst, combining fuel and oxidant streams in stoichiometric proportions results
in premixtures with H2:O2:N2 = 2:1:18, all having the same premixed flame properties such as
Tad = 1068 K. Equal jet exit velocities (Uupper=Ulower) was employed, and since ρupper ≈ ρlower, the
stagnation plane (where the two opposing jets meet) location was essentially at the midplane
between the jets. Table 1 shows the mixtures employed; SL values are extremely low. These
mixtures could not burn as nonadiabatic plane premixed flames in our apparatus (for d = 0.75 cm,
thus Péclet number SLd/α ≈ 16 even for the least-diluted case, which is below the typical
extinction limit criterion Pe = 40) but can exist as nonpremixed flames due to low LeH2 which
enhances flame temperature [92] thus reaction rates.
Table 1 Properties of mixtures tested. SL is calculated using CHEMKIN with Li et al. [93] kinetics. For all cases, Lef
≈0.33 and Leo ≈1.07.
N ρu/ρb SL(cm/s) σext (1/s)
Zst =0.15
σext (1/s)
Zst =0.5
σext (1/s)
Zst =0.9
15 3.75 4.40 176.7 230.3 182.9
16 3.62 2.90 111.2 153.8 95.8
17 3.50 1.78 70.0 102.7 50.2
18 3.39 1.04 44.1 68.5 26.3
19 3.29 0.49 27.7 45.8 13.7
20 3.20 0.35 17.4 30.6 7.2
In the case of premixed flame, to control independently the effective Lewis number Leeff
(which depends largely on the deficient reactant which in turn is strongly affected by the
equivalence ratio φ) and the laminar burning velocity SL (which depends largely on the adiabatic
flame temperature Tad), combustible mixtures were formed with H2:O2 = 2φ:1 and diluted with
N2 to obtain a desired Tad. For example, Tad = 1000 K for both H2:O2:N2 = 1:1:9.55 and 4:1:18.2,

53
yet φ is different (0.5 and 2, respectively) and thus Leeff ≈ LeH2 ≈ 0.3 and Leeff ≈ LeO2 ≈ 1.1,
respectively.) This scheme enables a much more systematic study of Leeff and SL than would
otherwise be possible with H2-air mixtures, where φ and SL cannot be controlled independently.
For mixtures with φ closer to unity, the values of Leeff were estimated using the method
proposed by Joulin and Mitani [38]. Values of H2-O2-N2 composition, Tad, and Leeff for all
mixtures tested are given in the supplemental materials. In this work, most conditions were tested
within 50% to 100% of the full range of mass flow controllers whose accuracy is 0.5% of
reading plus 0.2% of the full range. Assuming 50% of the full range used, for a target σ of 50 s-1
,
σ ± Δσ = 50 × [1± (0.5%+2×0.20%)] = 50 ± 0.45 s-1
; for a mixture of H2:O2:N2 = 2:1:14.78
whose mixture Tad is 1200 K, H2:O2:N2 = 2 ± 0.018: 1 ± 0.009:14.78 ± 0.133. φ and Tad with
largest possible error is about φ = 1 ± 0.02 and Tad = 1200 ± 13.5 K.
To determine the boundaries of the different regimes of flame behaviors observed, two of the
three principal experimental parameters (Tad, σ and φ) were fixed, and one was varied, depending
on which space of parameters (φ − ε or Tad − ε) was to be explored and to which parameter the
boundary was most sensitive. For example, to determine the boundaries in φ − ε space, the
mixture composition was held constant while σ was changed gradually by changing the jet exit
velocity U. Critical transition values of σ were recorded for several different φ values. Similarly,
to change φ while holding Tad and σ constant, the total flow was held constant while changing
the H2/O2 ratio and adjusting N2 to maintain fixed Tad.

54
2.3 The Hele-Shaw cell
Flame propagation in confinement is studied using a Hele-Shaw cell due to its simple
geometry and easy access to video recording. This cell consists of two acrylic plates separated by
a hollow aluminum frame, forming a combustion chamber with dimensions of 106.68 cm in
length (L). Depending on the choice of acrylic insert plates, the chamber’s thickness (d) can be
either 1.27 cm, 0.64 cm, or 0.32 cm, and its width (w) can be 45.72 cm, 22.86 cm, or 11.43 cm.
To achieve a quasi-2D flame, the thickness d must meet specific criteria. It should be sufficiently
large (d > O(10α/SL), for unity Leeff) to permit flame propagation, which ensures that the flame
has enough space to propagate without being quenched by the walls. On the other hand, d should
also be adequately small (d < O(103
α/SL)) to inhibit the onset of instability-driven flame
wrinkling perpendicular to the plates. This constraint minimizes the influence of threedimensional effects that could distort the flame structure and complicate the analysis.
The Hele-Shaw cell rests on two aluminum rails attached to a steel cart frame, and can be
rotated using electric hoists. This setup permits the chamber to adopt multiple orientations so the
flame can propagate in upward, horizontal, downward, and potentially 45-degree angled
directions. Such versatility enables the investigation of gravity or buoyancy effects on flame
behavior.
Four spark electrodes are positioned at one end of the chamber, allowing for the rapid
convergence of four circular flames into a singular flame upon ignition by a capacitive discharge
ignition system. At the ignition end of the cell, an exhaust manifold with a honeycomb flame
straightening design is connected to the atmosphere via a ball valve. This arrangement serves to
inhibit both flame flashback and external flame propagation through the exhaust.

55
The experiment system further includes a separate pressure chamber, a partial pressure mixing
system, the identical high-speed camera used in the edge-flame experiment, a vacuum pump, and
control programs executed via LabVIEW on a desktop computer. A detailed schematic of this
experimental configuration is presented in Figure 20.
Fig. 20. Schematics of the Hele-Shaw experiment apparatus.
At the beginning of each experiment series, the mixing chamber is evacuated to remove any
residual gas that could interfere with subsequent tests. Following the evacuation, the chamber is
filled with the desired gas mixture using the partial pressure method to a few atm. The HeleShaw cell is then initially evacuated and then filled with the mixed gas to a pressure just slightly
above atmospheric (~1.03 atm) to prevent air backflow upon opening the ball valve. Prior to
ignition, the ball valve is opened, allowing combustion to proceed at nearly constant pressure, as
verified by time-resolved chamber pressure measurements. The entire apparatus is housed within

56
a dark enclosure to mitigate the effects of ambient light. This is a crucial consideration for
hydrogen experiments where the flame may exhibit minimal or no coloration depending on the
inert gas used. The high-speed camera records the flame propagation by capturing the nearinferred light emission from water vapor in the hot product gas. Details of each experiment,
including the date, types of mixtures, target and actual mixture fractions, cell dimensions, flame
orientations, target adiabatic flame temperature (Tad), and frame rate, are encoded in the video
file’s name. Afterward, the laminar burning velocity SL and extinction strain rate σext are
calculated based on the recorded actual mixture fractions. For hydrocarbon mixtures, the USC
Mech II chemical kinetic model is employed via Chemkin Pro/Cantera, while the FFCM-1 model
is utilized for hydrogen mixtures.
Fig. 21. Example of a frame of a video imported into the MATLAB program; the flame is propagating toward the
right, and the red line represents the recognized flame location. Left: flame in a gas mixture with Leeff > 1; Right:
flame in a gas mixture with Leeff < 1.
Each experimental video is analyzed through a MATLAB program, which first collects
relevant information about the experiment from the video file’s name and then imports the video
and converts each frame into a two-dimensional matrix of light intensity, as illustrated in Figure

57
21. The estimated position of the flame in each frame is identified by searching for the flame
front, which is assumed to exhibit the highest gradient in illumination contrast.
Background noise is assessed using the first and last frames of the video—specifically, the
frame immediately preceding ignition and the frame following flame extinguishment at the end
wall. Then, each frame is put through a Gaussian filter with a cut-off length scale of chamber
thickness (d) and gamma of 0.7. This process is to distinguish small-wavelength wrinkling
attributed to diffusive-thermal instability (DTI) from large-wavelength wrinkling dominated by
hydrodynamic instability in the 2-D flame length measurement. In a depth-averaged 2-D flame
image, wavelengths smaller than the cell thickness are considered small-scale wrinkles.
Subsequently, variations in luminosity contrast are determined by computing the 2D gradient
of each frame. The MATLAB program employs the ‘edge’ function using the Canny method [94],
which identifies the local maxima of the image gradient calculated via the derivative of a
Gaussian filter. Unlike other methods, such as Sobel and Prewitt [95], the Canny method is able
to detect both strong and weak edges with two thresholds. Weak edges are only included if it is
connected to a strong edge. This approach has been proven to show reduced noise.
However, the resulting gradient mask often exhibits fragmented lines due to the high noise
levels near flame fronts. Detected edges must be dilated using linear structuring elements to find
a smooth outline. To rectify this, detected edges are dilated using linear structuring elements to
achieve a smoother outline. The dilation size is set to the chamber thickness d, ensuring that
wrinkles or gaps of larger wavelengths are preserved. This technique also connects cells spaced
less than the chamber thickness apart, commonly observed in gas mixtures with Leeff less than
one, as depicted in Figure 21(right). Then, the ‘imfill’ and ‘bwareaopen’ functions serve to fill

58
small voids within the mask and eliminate residual noise, yielding a binary image that accurately
represents the flame fronts.
The program recognizes the front edge of the illuminated area in the direction of propagation
as the flame position, shown as a red trace in Figure 21. By summing all pixels behind this flame
edge and normalizing the sum by the cell’s width and length, an average flame position and
combustion chamber pressure versus time plot is generated, as shown in Figure 22. This time
plot reveals that after the initial thermal expansion-dominated transient period, a quasi-steady
state period with a nearly constant slope (thus steady propagation speed) ensues until the flame
nears the opposite end of the Hele-Shaw cell. During this quasi-steady state, the combustion
chamber pressure remains relatively stable, confirming that the unburned gas is not compressed,
and the flame is unaffected by pressure variations. The flame propagation speed ST is estimated
as the average slope of the flame position plot in the quasi-steady state region. This method
ensured that the filter size used in the Gaussian filter does not affect the estimation of flame
propagation speeds. The root-mean-square error (RMSE) is calculated for each ST based on
equation 16, where STn is the slope of each frame, and N is the number of frames within the
quasi-steady state region.
(20)

59
Fig. 22. Example of experimentally measured flame position and corresponding pressure in the combustion chamber.
Fig. 23. An exemplary plot of wrinkled flame length ratio lflame /w vs. time generated from MATLAB program.
An additional metric obtainable from the video is the length of the 2-D flame wrinkled by
instabilities, denoted as lflame. This parameter is derived by measuring the length of the alreadyidentified flame edge during the ST calculation. The estimation uses Vossepoel and Smeulder’s

60
algorithm for perimeter length calculations with 8-connectivity [96]. Then, the results are
normalized using the cell’s width w. As shown in Figure 23, the plot of flame length ratio
(lflame/w) versus time exhibits similar developmental phases as in Figure 22. These phases include
a thermal expansion-dominated region, a quasi-steady state region, and an end-wall region. The
wrinkled flame length ratio for each experiment is estimated as the average value within the
quasi-steady state region. The root-mean-square error (RMSE) for this ratio is computed using
the same method as employed for ST.

61
Chapter 3: Nonpremixed hydrogen edge-flames
3.1 Flame structures
The flame structures of nonpremixed H2-O2-N2 edge-flames in a slot-jet counterflow burner
were investigated under varying conditions, including stoichiometric mixture fraction Zst, global
strain rate σ, mixture strength N, and jet spacing d. Figure 24 shows false-color direct images of
the four types of structures observed, denoted modes I-IV: (I) advancing flat edge-flames, (II)
retreating flat edge-flames, (III) advancing broken edge-flames, and (IV) stationary broken edgeflames. (No retreating broken flames were observed under any condition tested.)
As expected, for Zst less (greater) than 0.5, the flame lies on the O2 (H2) side of the stagnation
plane. This is because the flame tends to localize near the side of the more limited reactant. The
broken flames exhibit a nearly flat morphology, probably because the reaction zone must remain
close to the stoichiometric mixing location, unlike low-Le cellular premixed flames, which may
be highly curved. The broken-flame structures are remarkably similar to those predicted
theoretically (Fig. 24(e)) [49]. For Mode III, individual cells formed behind the leading edgeflame and remained stationary rather than splitting apart after the formation of a continuousflame, as predicted in [49]. A splitting sequence is shown in Fig. 25. While advancing brokenflames (Mode III) would recover after being “erased” by the N2 jet, stationary broken-flame do
not recover after erasure.

62
Fig. 24. False-color direct images of edge-flames in mixtures with N = 18. H2-N2 flows from the bottom upwards,
O2-N2 from the top downwards. Slot spacing d = 7.5 mm. White dashed lines indicate the stagnation plane location.
(a) Zst = 0.6, σ = 80s-1
, advancing flat flame (mode I); (b) Zst = 0.4, σ = 80s-1
, retreating flat flame (mode II); (c) Zst =
0.9, σ = 80s-1
, advancing broken flame (mode III) (d) Zst = 0.70, σ = 110s-1
, stationary broken flame (mode IV); (e)
computed reaction rate contours for a mode III flame with Le=0.33 [49].
Fig. 25. False-color direct images of broken-flames. (a) N = 18, Zst = 0.8, σ = 100s-1
; (b) N = 18.5, Zst = 0.8, σ =
100s-1
; (c) N = 19, Zst = 0.8, σ = 100s-1
; (d) N = 18, Zst = 0.7, σ = 100s-1
; (e) N = 18, Zst = 0.7, σ = 120s-1
; (f) N = 17,

63
Zst = 0.9, σ = 20s-1
; (g) N = 18, Zst = 0.85, σ = 240s-1
;(h) computed reaction rate contours for a mode IV flame with
Le=0.33 [49]. Images (a) and (f) are advancing (Mode III), and all others are stationary (Mode IV).
Fig. 26. Broken-flame propagation sequence: Zst = 0.9, σ = 80s-1
, propagating broken-flame (mode III). The time
between frames is 0.183 s.
Figures 25(a-h) show false-color direct images of broken-flames at high Zst. Figures 25(a-c)
show a sequence with increasing diluent amount N where Zst and σ are held constant; as N
increases, the mixture becomes weaker, and the flame “void fraction” increases. Figures 25(d-e)
show the same behavior with increasing σ where N and Zst are held constant. Figure 25(f) shows
a case with very low σ (near the heat-loss-induced limit) with many irregular cells. Finally,
Figure 25(g) shows a case with very high σ, where the flow, as seen by shadowgraph images (not
shown), is clearly turbulent (which occurs when slot-jet Reynolds numbers exceed 500), but the
cells themselves remain almost stationary. Stationary broken-flame structures (mode IV) are
again remarkably similar to those predicted theoretically (Figure 25(h)) [49].

64
Fig. 27. Effect of Zst and σ on scaled edge-flame speeds: (a) N=17; (b) N=18. Filled symbols indicate continuous
flat flames and open symbols indicate broken flames.
-3
-2
-1
0
1
2
3
0.0 0.2 0.4 0.6 0.8 1.0 Zst
(a)
σ =15/s
σ =30/s
σ =50/s
σ =70/s
σ =90/s
σ =110/s
Ũ
-3
-2
-1
0
1
2
3
0.0 0.2 0.4 0.6 0.8 1.0 Zst
(b)
σ =15/s
σ =20/s
σ =40/s
σ =60/s
σ =70/s
σ =100/s
σ =120/s
Ũ

65
3.2 Propagation rates
It is found in this study that the traditionary scaling for edge-flame propagation speed
Ũ=(Uedge/SL)(ρu/ρb)
1/2 is no longer unsatisfactory for H2-O2-N2 edge-flames because the low LeH2
significantly enhances flame temperatures [92] and thus the reaction rate ω; in fact, for the
mixtures tested (see Table 1) values of SL are so low that they cannot burn as plane premixed
flames in our apparatus (Péclet numbers Pe ≡ SLd/α are below the typical extinction limit
criterion Pe = 40, even for the least-diluted mixture). Park et al. [97] asserted that for weak H2-
O2-N2 flames, the global extinction strain rate σext rather than SL may be used to characterize
global reaction rates. Moreover, Song et al. [32] also showed that σext serves as a simple
surrogate for predicting edge-flame behavior; following this approach, Uedge will scale with
(αω)
1/2~(ασext)
1/2 rather than SL; thus, this work defines Ũ = Uedge/[ασext(ρu/ρb)]1/2. Also, σ is
scaled with σext, leading to a dimensionless strain rate ε ≡ σ/σext. It will be shown that Ũ and ε
are indeed appropriate scalings; one may also note they can be used for Lef ≈ Leo ≈ 1 since, in
that case, σext ~ SL
2
/α ~ ω [15].
Figure 27 shows the effects of Zst and σ on scaled edge-flame speeds for two mixture
strengths. It should be emphasized that for each plot, every point corresponds to the exact same
mixture when fuel and oxidant streams are combined in stoichiometric proportions (thus, every
point on the plot has the same Tad, ρu/ρb and SL as discussed in Introduction), yet Uedge varies
drastically depending on Zst and σ. As with nonpremixed hydrocarbon edge-flames [30], Uedge is
maximum at intermediate σ with heat-loss induced extinction at low σ in addition to the high-σ
extinction limit. Except at very high Zst, Uedge increases with increasing Zst, which is consistent
with the notion of decreasing Leeff as Zst increases (see Introduction). Scaled Uedge values may

66
exceed unity, consistent with the expectation of low Leeff at high Zst. It is somewhat surprising,
however, that upon transition from continuous (filled symbols) to broken (open symbols)
advancing edge-flames, there is no significant change in Uedge values. Figure 28 shows the
effects of scaled strain rate (ε) on scaled Uedge for several dilution levels (N) for three fixed Zst
values. For low (0.15) and intermediate (0.5) Zst, as ε increases Uedge first decreases slowly then
decreases drastically until extinguishment. For high (0.9) Zst, the trend is quite different, though
all cases have a similar maximum scaled Uedge (1.5 ~ 2.5). The reason for this difference is
discussed in the following section. In Figure 27, all curves overlap except for near-extinction
and broken-flame cases, demonstrating that ε and Ũ are proper scaling parameters for strain rate
and edge-flame speed, respectively.
-3
-2
-1
0
1
2
3
0.0 0.3 0.6 ε 0.9 1.2 1.5
(a)
N=15
N=16
N=17
N=18
Ũ

67
Fig. 28. Effect of scaled strain rate ε on scaled Uedge: (a) Zst=0.15; (b) Zst=0.5; (c) Zst=0.9. Filled symbols indicate
continuous flat flames (Modes I, advancing and II, retreating), and open symbols indicate broken flames (Modes III,
advancing and IV, stationary).
-3
-2
-1
0
1
2
3
0.0 0.3 0.6 ε 0.9 1.2 1.5
(b)
N=15
N=16
N=17
N=18
N=19
Ũ
-3
-2
-1
0
1
2
3
0.0 2.0 4.0 ε 6.0 8.0 10.0
(c)
N=15
N=16
N=17
N=18
N=19
N=20
Ũ

68
Fig. 29. Flame response maps in Zst-ε space: (a) N=17; (b) N=18. Recall mode designations: (I) advancing flat
flames, (II) retreating flat flames, (III) advancing flat flames, and (IV) stationary broken-flames. Vertical dashed
0.1
1
10
0.0 0.2 0.4 0.6 0.8 1.0 Zst
(a)
I
II III extinction
IV
extinction
turbulent
ε
0.1
1
10
0.0 0.2 0.4 0.6 0.8 1.0 Zst
(b)
extinction
III
IV
extinction
I
II
turbulent
ε

69
lines indicate minimum (pure H2 vs. O2-N2) and maximum (H2-N2 vs. pure O2) values of Zst attainable. Dashed
curves indicate the transition to turbulent structures.
0.0
0.4
0.8
1.2
1.6
2.0
15 16 17 18 19 20 N
(a)
I
II
extinction
extinction
ε
0
1
2
3
4
5
6
15 16 17 18 19 20 N
(b)
extinction
III
extinction
II
IV
turbulent
ε

70
Fig. 30. Flame response maps in N-ε space: (a) Zst=0.3; (b) Zst=0.8. Dashed curves indicate the transition to
turbulent structures.
3.3 Regimes of flame behavior
Figure 29 shows maps of flame behavior in Zst - ε space for two fixed N values. Note that on
each map, every point produces the same stoichiometric mixture of fuel and oxidant streams.
For all Zst, the two extinction limits at high and low ε are evident. For Zst < 0.6, no broken flames
are observed; only for Zst > 0.6 can Modes I-IV all be observed as ε increases. For N = 17, Zst >
0.85, and N = 18, Zst > 0.78, only broken flames are observed, indicating the dominance of lowLe DTI. For N = 17, Zst < 0.08, and N = 18, Zst < 0.15, no combustion occurs at any ε. Note that
for N=17 at Zst ≈ 0.58, ε ≈ 55 and for N = 18 at Zst ≈ 0.60, ε ≈ 100, a rather remarkable
bifurcation occurs - moving radially outward from these points for small distances in Zst - ε space,
depending on the direction one may encounter any of four of the observed flame structures or
extinction! Figure 30 shows maps of flame behavior in N - ε space for two fixed Zst values.
Again, the low ε and high ε extinction limits are evident, and only for higher Zst are broken
flames observed.

71
Fig. 31. Comparison of measured (solid lines and data points) and computed (dashed curves) values of extinction
strain rate (σext) for varying Zst with N = 17 and N = 18.
3.4 Computations of σext and comparison with experiment
Previous work [30] on Zst influences on nonpremixed edge-flames in hydrocarbon fuels
demonstrated that many aspects of multidimensional edge-flame behavior could be interpreted in
terms of the readily-computed adiabatic 1D counterflow extinction strain rate (σext). Such
computations (Figure 30) for H2-N2/O2-N2 flames were performed with CHEMKIN using the
same gap d as in the experiments. Measured values of σext are similar to but mostly higher than
the computed values. For Zst > 0.6, two sets of experimental data are shown for the “extinction”
of flat flames (transition to broken-flames) and the extinction of broken-flames. Remarkably,
computed values of σext correspond well with the transition to broken-flames observed in
0
50
100
150
200
250
0.0 0.2 0.4 0.6 0.8 1.0
σext (s-1)
Zst
N=17
continuous-ames
N=17
computations
N=18
continuous-ames
N=18
computations
N=17
broken-ames
N=18
broken-ames

72
experiments. Values of σ at the extinction of broken-flames increase rapidly with increasing Zst,
and at Zst > 0.8 could not be observed because of the transition to turbulent flow (see Fig. 29).
Figure 31 again illustrates the bifurcation in flame behavior near Zst ≈ 0.6. This bifurcation in
experiment data results from a one-dimensional to a multi-dimensional structure, whereas the
computations are strictly one-dimensional and thus could not show the bifurcation. Reference
[49] reports computations employing schematic single-step chemistry and predicts that brokenflames exist only when the continuous flame cannot, which is precisely seen experimentally. It is
unclear whether the peak in σext that occurs at Zst ≈ 0.6 is fundamentally related to this
bifurcation or is purely coincidental.
Fig. 32. Computed effects of Zst on σext for H2-N2/O2-N2 mixtures with standard transport and artificial transport
causing Lef=Leo≈1, along with results for CH4-N2/O2-N2 (Lef=Leo≈1) and i-CH4-N2/O2-N2 (Lef>Leo≈1) mixtures.
0
50
100
150
200
250
0.0 0.2 0.4 0.6 0.8 1.0
σext (s-1)
Zst
"Le=1" H2 /O2 /N2 = 2 / 1 / 15
H2 / O2 / N2 = 2 / 1 / 15
CH4 / O2 / N2 = 1 / 2 / 9.5
iC4H10 / O2 / N2 = 1 / 6.5 / 30

73
As discussed in the Introduction, Zst influences on nonpremixed edge-flames in hydrocarbon
fuels are affected by both Lewis numbers of fuel and oxygen and the unique chemistry of these
fuels [32]. To separate Le and chemistry effects for hydrogen, computations of σext were
performed with Lennard-Jones parameters and molecular masses of H2 and H set to those of O2
and O, respectively, to force the Lewis numbers of H2 and O2 to be equal (and both near unity).
Comparisons of H2-N2/O2-N2 flames with standard transport and Lef = Leo≈1 are shown in Fig.
32. With higher Lef, σext decreases 4-5 times lower and becomes almost independent of Zst. By
comparing these data with CH4-N2/O2-N2 flames, which also have Lef=Leo≈1 but the unique
hydrocarbon chemistry that leads to σext monotonically increasing with Zst (also shown in Fig.
32), it can be concluded that (1) the low Le of H2 causes more robust flames with higher σext; (2)
Lef effects, not chemistry effects, are responsible for the non-monotonic effect of Zst on σext in
H2-N2/O2-N2 flames. Figure 32 also shows results for i-C4H10-N2/O2-N2 mixtures where (in
contrast to H2-N2/O2-N2) the high Lef of i-C4H10 causes a decrease in Leeff (from Lef to Leo) as Zst
decreases, and thus non-monotonic behavior in contrast to the monotonic trend with CH4-N2/O2-
N2 mixtures.
3.5 Conclusion
The effects of stoichiometric mixture fraction (Zst) on flame structure and propagation rates
(Uedge) in nonpremixed edge-flames H2-N2/O2-N2 mixtures were studied using a counterflow
slot-jet apparatus. Both advancing and retreating edge-flames were characterized by scaled Uedge
and strain rate (σ). Both high-σ residence-time limit extinction limits and low-σ heat lossinduced extinction limits were observed. As in prior work on hydrocarbon edge-flames [30],
Uedge is neither independent of Zst nor symmetric with respect to Zst = 0.5, but for very different

74
reasons. For hydrocarbons, a chemical effect leads to Uedge monotonically increasing with Zst,
but this effect is absent in H2-N2/O2-N2 flames. Instead, the low Lewis number of H2 and the
shift from oxygen-limited to fuel-limited reaction as Zst increases cause the observed behavior.
Above Zst ≈ 0.6, a bifurcation leads to the possibility of broken (in many cases still nearly planar)
edge-flames, which may be advancing or stationary but were not observed for retreating edges.
The broken flames continue to survive under conditions (high strain or heat loss) where
continuous plane edge-flames cannot. These results indicate that the behavior of turbulent
nonpremixed hydrogen flames and the effects of stoichiometric mixture faction thereupon should
not be anticipated based on extrapolation of Lewis number and strain rate effects for flames in
hydrocarbons or other fuels.

75
Chapter 4: Premixed hydrogen edge-flames
4.1 Calculations of strained flames
Experiments on premixed hydrocarbon/O2/diluent edge-flames [33] have shown that for
single premixed edge-flames, Uedge/SL scales with (ρu/ρb)
1/2 because of the flow field generated by
thermal expansion is similar to that of single premixed flames whereas for twin premixed edgeflames, Uedge/SL scales with ρu /ρb due to the low-density products that are partly trapped between
the twin flames which generates a flow field similar to a flame propagating away from a rigid
wall. The same scalings with ρu/ρb are applied in this work.
Theories of premixed edge-flames [8,9] predict that for adiabatic conditions with Leeff = 1,
constant density and low strain rate, Uedge/SL = 1. Fundamentally, this scaling is appropriate
because for Leeff ≈ 1, Tad is close to that of an adiabatic premixed unstrained continuous flame
and thus the overall reaction rate ω can be estimated as SL
2
/α [15], where α is the unburned gas
thermal diffusivity. However, this scaling of Uedge with SL is unsatisfactory for the H2-O2-N2
flames of interest in this study for two reasons. First, calculated values of SL using CHEMKIN
software and kinetic data from Li et al. [91]. for varying Tad and φ (Fig. 33) show that even for
fixed Tad, SL varies significantly with φ; in particular, for fixed Tad, SL is nearly constant for lean
mixtures but substantially larger (in some cases by more than an order of magnitude) for rich
mixtures. This is due to the decreasing O2 availability in rich mixtures which in turn leads to an

76
increase in the rate of H radical removal via H + O2 + M → HO2 + M. This is not an important
factor for rich hydrocarbon-O2-N2 mixtures because the hydrocarbon molecule itself is a sink of
H atoms whereas obviously H2 cannot act as a sink of H atoms. This trend is less pronounced in
mixtures with higher Tad because there is less impact of H atom loss via H + O2 + M → HO2 + M
compared to chain branching via H + O2 → OH + O at higher temperatures. The second reason is
that, as already mentioned in Chapter 3, for strained flames in lean H2-O2-N2 mixtures the low
Leeff significantly enhances flame temperatures [15] and thus ω.
Fig. 33. Calculated laminar burning velocity SL for varying adiabatic flame temperature (Tad) and equivalence ratio
(φ). Note that lean mixtures at Tad ≤ 1100 K have SL ≤ 1cm∙s-1 and thus Pe ≤ 10, and thus could not burn as plane
premixed flames in the apparatus utilized.

77
Fig. 34. Calculated (a) extinction strain rates, (b) SL
*
/SL for varying adiabatic flame temperature (Tad) and
equivalence ratio (φ) for twin flames

78
Fig. 35. Calculated (a) extinction strain rates, (b) SL
*
/SL for varying adiabatic flame temperature (Tad) and
equivalence ratio (φ) for single flames.

79
Following the successful scaling approach for nonpremixed H2-N2/O2-N2 flames, Uedge will be
scaled by an “effective burning velocity” SL
* = (αω)
1/2∼ (ασext)
1/2 rather than SL. The extinction
strain rate σext must be determined from computations rather than experiments because, for low
Leeff mixtures near extinction limits, planar flames cannot be established experimentally due to
DTI.
Combining the scaling considerations for SL
* and density ratios, this work defines the
dimensionless edge-flame speed Ũ as
 edge
n
u
ext
b
U
U
ρ ασ
ρ
≡      
(21)
where n = ½ for single edge-flames and n = 1 for twin edge-flames. It will be shown that Ũ is an
appropriate scaling parameter in that Ũ is close to unity for most experimental results.
Correspondingly, the strain rate σ is scaled with σext, leading to a dimensionless strain rate ε
defined as
ext
σ
ε
σ ≡ (22)
Using the estimation method described in [98], the characteristic radiative loss rate was
found to be less than 0.5s-1 for all mixtures tested in this work, which is far smaller than the
smallest strain rates examined (5s-1). Consequently, radiative transport is negligible compared to
convective transport in this study.
Figures 34 and 35 show the calculated extinction strain rates σext and the values of the
“effective” SL
* relative to SL; that is, SL
*
/SL = (ασext)
1/2/SL, for twin and single premixed strained
counterflow flames, respectively. It can be seen that for H2-O2-N2 mixtures at low φ, the Lewis

80
number effects cause σext to be much larger than for stoichiometric mixtures and SL
*
/SL to be
much higher than unity. In contrast, for stoichiometric mixtures, SL
*
/SL is very close to unity and
for φ sufficiently larger than unity, σext approaches an asymptotic value of about 0.3. It is worth
noting that while the trends of SL
*
/SL with varying Tad and φ are virtually identical for twin and
single flames (compare Fig. 34(b) and Fig. 35(b)), the trends of SL and σext separately are
somewhat different. This again points to the proposed scaling with SL
* as being appropriate for
the flames examined in this investigation.
4.2 Modes of edge-flames
Contrary to nonpremixed edge-flames, where the dynamics are restricted due to the necessity
for the reaction zone to be situated at the stoichiometric mixing location, premixed flames are not
subject to such limitations. As a result, they can be anticipated to display a broader range of
dynamic behaviors.
Six modes of edge-flame behavior, denoted I-VI, were observed. Which modes are exhibited
depends strongly on the type of edge-flame (twin or single), Tad, φ, and σ, as shown in regime
maps (Figs 36 and 37). Of the three primary experimental variables (Tad, ϕ, and σ), no fixed
value of any one of these three could enable observation of all six modes of behavior merely by
varying the other two. The largest number of modes observed in one plane of (Tad, σ, φ) space is
for twin premixed flames at Tad = 1175 K, where four modes could be observed by varying φ and
σ (Fig. 38(a)). Laminar Mode V and Mode VI do not exist for mixtures with Tad = 1175 K for
any point of σ and φ. After the leading edge-flame propagation, for Modes I-V, the remaining
flame continues to burn steadily, whereas for Mode VI, a periodic train of flamelets is observed
as discussed below.

81
Mode I: advancing flat edge-flame (Figs. 36(a) and 37(a).) Twin advancing flat edgeflames consist of two quasi-planar parallel flames that fold over and connect at the flame edge.
As σ increased or Tad (thus SL or SL
*
) decreased, the spacing between the twin flames decreased,
approaching the stagnation plane and eventually merging. In contrast, single flat advancing edgeflames consist of a single quasi-planar premixed flame trailing an advancing edge. For the
single-flame case, the edge suffers massive heat losses which decreases its propagation speed;
thus, it bends toward the stagnation plane (upward, for our standard configuration with the
reactive mixture fed from the lower jet) where the convection velocity is lower.
Mode II: retreating flat edge-flame (Fig. 36(b).) Retreating edge-flames were only
observed for a very narrow range of conditions near the high-σ extinction limit (Fig 38(a) and
Fig 38(c)) and only for twin flames. These flames are superficially similar in appearance to
advancing flat edge-flames (Mode I). Despite considerable experimental effort, no retreating
single premixed edge-flames could be observed; this is consistent with theory [24], which
predicts that negative Uedge values do not exist for single premixed flames. Also, no retreating
wrinkled or broken flames were observed.
Mode III: advancing wrinkled edge-flame (Figs. 36(c), 36(d), 37(b).) For low-φ (thus low
Leeff) mixtures in both twin and single edge-flame configurations, DTI may cause flat advancing
flames to become wrinkled, particularly near extinction limits. Once the edge-flame propagates
across the slot, the trailing wrinkled flame is nearly steady. The transition from Mode I to Mode
III is gradual, so the transition was defined as when the amplitude of the wrinkles exceeded 10%
of the wavelength of the wrinkles; this definition is arbitrary but not subjective.
Mode IV: advancing broken edge-flame (Figs. 36(e), 36(f), 37(c), 37(d).) For low-φ
mixtures in both twin and single edge-flame configurations near extinction limits, Mode III

82
wrinkled flames transition to a visibly broken structure. As a result of low Leeff, the faster/slower
burning regions of the flame surface (corresponding to higher/lower visible intensity) move
towards/away from the jet exits where the convective velocity is higher/lower. The leading edge
of the propagating flame leaves in its path a train of nearly stationary cellular structures. The
higher reaction rate at the troughs of the cells depletes fuel from the adjacent cusps, leading to
local extinction at the cusps [99]. The lengths of the burning and extinguished regions vary
depending on the equivalence ratio φ and strain rate σ. Despite considerable experimental effort,
no retreating wrinkled or broken flames were found, an observation also reported for
nonpremixed edge-flames in low-Le mixtures. The transition from Mode III to Mode IV is
gradual, so the transition was defined as when the minimum intensity in the nearly extinguished
regions was less than 90% of the intensity in the most intensely burning regions; again, this
definition is arbitrary but not subjective.
Mode V: stationary broken edge-flame (Fig. 37(e).) These were observed for both twin and
single flames only for low-φ mixtures for a very narrow range of conditions near the high-σ
extinction limit (Figs. 38(b), 38(d), 39(b), 39(d).) While Fig. 37(d) (Mode IV) and 37(e) (Mode
V) appear similar, Fig. 37(d) is the result of an advancing edge-flame whereas Fig. 37(e) was
obtained from a continuous flame after increasing σ or reducing Tad and cannot be produced by
the remnant of an advancing edge-flame.
Mode VI: “short-length” edge-flame (Fig. 37(f).) These are distinct from Mode IV flames
in that a continuous train of short flames propagates from one side of the slot jet to the other.
This mode occurs only for single edge-flames and only near the low-σ extinction limit caused by
heat losses. As explained in [33], the leading edge-flame leaves behind a combination of
reactants and products that is too weak (due to reactant depletion and heat losses) to sustain a

83
trailing flame, but once this mixture is swept out of the vicinity of the mixing layer on a time
scale of the order 1/σ, another edge-flame initiated by the heated wire propagates across the slot.
Fig. 36. False-color direct images of twin premixed edge-flames propagating or retreating from left to right. The
upper and lower borders of the images represent the locations of the jet exits (spaced 12.5 mm apart.) White dashed
lines indicate the stagnation plane location. (a) φ = 1.5, σ = 50 s-1
, Tad = 1200 K, advancing flat edge-flame (Mode
I); (b) φ = 1.5, σ = 50 s-1
, Tad = 1175 K, retreating flat edge-flame (Mode II); (c) φ = 0.7, σ = 40 s-1
, Tad = 1175 K,
advancing wrinkled edge-flame (Mode III); (d) φ = 0.7, σ = 25 s-1
, Tad = 1175 K, advancing wrinkled edge-flame
(Mode III); (e) φ = 0.4, σ = 50 s-1
, Tad = 850 K, advancing broken edge-flame (Mode IV); (f) φ = 0.2, σ = 20 s-1
, Tad
= 1175 K, advancing broken-flame (Mode IV).
a
b
c
d
e
f

84
Fig. 37. False-color direct images of single premixed edge-flames propagating from left to right (except (e) which is
stationary.) The upper and lower borders of the images represent the locations of the jet exits. White dashed lines
indicate the stagnation plane location. H2-O2-N2 mixtures flow from the bottom upwards and N2 from the top
downwards. (a) φ = 1.5, σ = 40 s-1
, Tad = 1300 K, advancing flat-flame (Mode I); (b) φ = 0.8, σ = 40 s-1 Tad = 1300
K, advancing wrinkled edge-flame (Mode III); (c) φ = 0.6, σ = 40 s-1
, Tad = 1200 K, advancing broken edge-flame
(Mode IV); (d) φ = 0.6, σ = 40 s-1
, Tad = 1100 K, advancing broken edge-flame (Mode IV); (e) φ = 0.6, σ = 78 s-1
,
Tad = 1100 K, stationary broken edge-flame (Mode V); (f) φ = 0.4, σ = 17 s-1
, Tad = 1100 K, continuous short-length
moving flame train (Mode VI).
4.3 Regimes of flame behavior
To determine the boundaries of the different regimes of flame behaviors observed, two of the
three principal experimental parameters (Tad, σ and φ) were fixed and one was varied, depending
on which space of parameters (φ - ε or Τad - ε) was to be explored and to which parameter the
boundary was most sensitive. For example, to determine the boundaries in φ - ε space, the
mixture composition was held constant while σ was changed gradually by changing the jet exit
velocity U. Critical transition values of σ were recorded for several different φ values. Similarly,
a
b
d
c
f
e

85
to change φ while holding Tad and σ constant, the total flow was held constant while changing
the H2/O2 ratio and adjusting N2 to maintain fixed Tad.
Figures 38(a) and 38(b) show the regimes of flame behavior for twin flames in φ - ε space for
Tad values at 1175 K and 950 K respectively. Figures 38(c) and 38(d) show the regimes in Tad -
ε space for two different φ values at 1.5 and 0.5. Figures 38a-d show the corresponding regime
maps for single flames.
For twin flames, Figure 38(a) shows the presence of 4 modes: Mode I (advancing flat edgeflames) and Mode II (retreating flat edge-flames) only exist for φ > 0.9 (and thus Leeff is close to
or larger than unity) whereas Modes III (advancing wrinkled edge-flame) and IV (advancing
broken edge-flame) reside in the lean side of the regime map. For weaker mixtures (Figure
38(b)), the flames cannot survive without the benefit of temperature enhancement due to
wrinkling caused by DTI and thus only exist at low φ corresponding to low Leeff, and for these
cases, no planar flame structures were observed. In fact, essentially no flat flames exist at φ < 0.7
for either twin and single flames at any value of ε and/or Tad recorded, which agrees with
previously reported in experiment [100] and simulation [101] that the neutral diffusive-thermal
instability boundary lies near φ = 0.6. As Tad is decreased, the range of epsilon where any type
of flame can exist narrows. Modes I and II vanish first at about 1100 K, followed by Mode III at
approximately 875 K. Modes V and IV disappear last, below 850 K, as Fig. 38(c) shows.
For single-flames, regime maps and mode transition process are somewhat similar to twin
flames, but single flames require much higher Tad (by 125 - 150 K) to achieve this similarity.
This is because single flames experience heat losses to both the upstream (reactive) and

86
downstream (cold N2) sides. In contrast, twin flames are nearly adiabatic on the downstream side
due to the back-to-back configuration.
The high-ε extinction limits occur near ε = 1 as expected since ε ≡ σ /σext and thus ε = 1
would correspond to perfect agreement between the computational and experimental values of
σext. An exception to this is for lean single flames at low Tad (Figs. 39(b) and 39(d)) where at the
high-ε extinction limit occurs in the range 2.5 < ε < 10. This observation indicates that these
flames are more robust than the computations predict which is likely due to the fact that these
extremely weak low-Tad single edge-flames only exist due to cellular structures caused by DTI;
the resulting enhancement of local flame temperatures caused by DTI enables flames to exist at
higher ε (thus higher strain rate σ) than would otherwise be possible (a factor that cannot be
properly accounted for with the 1D strained-flame computations.) The author hypothesizes that
the unexpectedly high values of ε (Fig. 40)at extinction (and of Ũ, Fig. 42) may be due to the
additional curvature at the single-flame tip compared to twin flames; 1D simulations can capture
the influence of Leeff but not the 2D curvature effects that become more significant as σ
increases or Tad decreases. Multidimensional simulations would be needed to test this hypothesis.
For conditions exhibiting flat flames at the low-ε extinction limit (essentially, for all φ ≥ 0.7
cases; see Figs. 38(a) and 39(a)), the extinction conditions correspond to jet exit velocities U ≈ SL
and thus the flames are attached to the jet exits, leading to extinction via massive heat losses.
Essentially all φ < 0.7 cases exhibit cellular structures due to DTI and no flat flames exist and
thus the criterion U ≈ SL would not apply. In these cases, the low-ε extinction limit still
corresponds to flames attached to the jet exits, but due to the broken structures, no values of SL
can be assigned or inferred.

87
Fig. 38. Maps of modes of flame behavior in φ - ε space for twin edge-flames with (a) Tad = 1175 K and (b) 950 K;
Τad - ε space for twin edge-flames with (c) φ = 1.5 and (d) φ = 0.5. Mode designations are: (I) advancing flat edgeflames, (II) retreating flat edge-flames, (III) advancing wrinkled edge-flames, (IV) advancing broken edge-flames,
(V) stationary edge-flames and (VI) “Short-length” flames. Dashed curves indicate transition to turbulent structures.

88
Fig. 39. Maps of modes of flame behavior in φ - ε space for single edge-flames with (a) Tad = 1300 K and (b) 1100
K; Τad - ε space for twin edge-flames with (c) φ = 1.5 and (d) φ = 0.5. Mode designations are: (I) advancing flat
edge-flames, (II) retreating flat edge-flames, (III) advancing wrinkled edge-flames, (IV) advancing broken edgeflames, (V) stationary edge-flames and (VI) “Short-length” flames. Dashed curves indicate the transition to turbulent
structures.

89
Fig. 40. Comparison of extinction strain rate (σext) from simulation and experiment
10
100
0.5 1.0 1.5 2.0
σext (s-1)
φ
a
1175K
computation
1175K
experiment
950K
experiment
950K
computation
Twin premixed ame
10
100
0.0 0.5 1.0 1.5
σext (s-1)
φ
b
1300K
computation
1300K
experiment
1100K
experiment
1100K
computation
Single premixed ame

90
4.4 Edge-flame propagation speed
Figures 41 and 42 show measured values of scaled edge-flame speeds Ũ (Eq.(21)) for twin
and single edge flames, respectively. Figures 41(a) and 42(a) show, for one fixed value of
adiabatic flame temperature Tad, the effect of scaled strain rate ε on Ũ for several fixed values of
equivalence ratio φ. Figures 41(b) and 42(b) show, for one fixed dimensional strain rate σ, Ũ as a
function φ for several values of Tad. The values of Tad were selected to illustrate as clearly as
possible all of the flame behaviors observed. With a fixed strain rate σ value of 50 s-1
, 1200 K is
a temperature with all positive Ũ (with 1250 K or even 1300 K, the high strain rate flame will be
in the turbulent regime) for the twin configuration. 1175 K is a temperature that mode II
retreating-flat flame regime, which only happens under very limited conditions: retreating-flat
flame regime is possible for weaker mixtures in theory, but very elusive with fast negative Ũ in
experiment. 850K is a temperature that edge-flame can only burn lean with broken structures.
950K is a temperature that edge-flame can only burn as wrinkled flames.
The plots of Ũ vs. ε are qualitatively similar to those reported previously for nonpremixed H2-
N2/O2-N2 flames in that there are high-ε (strain-induced) and low-ε (heat loss-induced) limits
with a peak value of Ũ of order unity. Single edge-flames exhibit somewhat higher values, and it
would be informative to distinguish whether the data scale more closely with n = 1 or n = ½;
however, the accessible range of density ratios (corresponding to values of 1100 K < Tad < 1300
K) is only ρu /ρb = 3.48 to 4.06. Values of Tad < 1100 K could not be burned and values of Tad >
1300 K resulted in fast-burning flames with required strain rates σ corresponding to values of Ũ
that were in the turbulent flow region. Hence, with such a small range of accessible values of ρu
/ρb, scaling with n = 1 or n = ½ essentially only causes a nearly constant shift in the values of Ũ.

91
A different apparatus with smaller slot-jet widths would be needed to investigate such fastburning mixtures. Nonetheless, these data again demonstrate that SL
*
, not SL, is the more
appropriate scaling parameter because SL
* accounts for the influence of Leeff particularly for small
values of this parameter - note that SL
*
/SL may exceed 50 for low φ and is essentially infinite for
even lower values of Tad than those shown in Figs. 34 and 35, as computations [102] show that
for adiabatic unstrained flames SL decreases without limit as Tad decreases; Figure 41(a) also
shows that some mixtures in the twin-flame configuration have no positive Ũ and thus cannot be
sustained unless the edges of flames are anchored in some manner (in this work via electricallyheated wires.)
Figures 41(b) and 42(b) show that for fixed φ thus fixed Leeff, away from extinction limits, the
proposed scaling of Ũ leads to data nearly collapsing onto a single value, even for widely
varying Tad thus a very wide range of σext. The large drop in Ũ near φ = 1 for mixtures of 1175 K
and 1200 K is due to the sharp change in σext near φ = 1 (Figs. 34(a) and 35(a)). The 850 K and
950 K mixtures at larger φ are very close to the heat loss induced extinction limit, with
correspondingly low values of σext where heat loss from the flame to the slot-jet burner becomes
dominant, causing a deviation from the values where the data would otherwise collapse onto.
Interestingly, the transition from continuous to broken flame structures as Tad decreases for
fixed φ has no significant effect on the scaled edge-flame speed Ũ (Fig. 43).

92
Fig. 41. Effect of scaled strain rate e and equivalence ratio φ on twin edge-flame speeds Ũ for fixed adiabatic flame
temperature Tad (upper) or strain rate σ (lower). Filled symbols indicate continuous flames; open symbols indicate
broken flames.

93
Fig. 42. Effect of scaled strain rate e and equivalence ratio φ on single edge-flame speeds Ũ for fixed adiabatic flame
temperature Tad (upper) or strain rate σ (lower). Filled symbols indicate continuous flames; open symbols indicate
broken flames.

94
Fig. 43. Effect of adiabatic flame temperature Tad on scaled twin edge-flame speeds Ũ for equivalence ratio φ = 0.3.
Filled symbols indicate continuous flames; open symbols indicate broken flames.
4.5 Conclusion
An experimental study of premixed H2-O2-N2 edge-flames exhibited a wide variety of flame
structures depending on configuration (single or twin), mixture strength Tad, Lewis number Leeff
and strain rate σ. A non-dimensional propagation speed Ũ = Uedge/[SL
*
(ρu/ρb)
n
] is shown to scale
Uedge data much more effectively than SL alone. SL
* = (ασext)
1/2 is akin to an effective burning
velocity and provides a means of incorporating Lewis number effects in the estimation of
effective reaction rate in stretched flames. Density ratios with n = 1 and ½ for twin and single
flames, respectively, account for thermal expansion effects.
Some observations of flame structure and propagation rates were qualitatively similar to those
found in studies of nonpremixed H2-N2/O2-N2 flames; however, the premixed flames exhibit
more modes of behavior and more dynamic instabilities. This is because the nominal reaction
zone location of a nonpremixed flame is constrained to lie where the reactant fluxes are in
0
500
1000
1500
2000
2500
3000
3500
800 900 1000 1100 1200
σext (s-1)
Uedge (cm∙s-1)
Tad
φ = 0.3
σ = 50s-1
Twin premixed ame
800 900 1000 1100 1200
3500
3000
2500
2000
1500
1000
500
0
80
70
60
50
40
30
20
10
0

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stoichiometric proportions, i.e., at the mixing layer location, which is nominally flat. Thus, the
cellular instabilities are largely perturbations about this nominally flat, uniform flame sheet. In
contrast, premixed flame locations are not constrained by a particular mixing layer location and
thus may be substantially non-planar.
These observations may contribute to extending the laminar flamelet concept to “broken”
flamelets in H2-O2-N2 mixtures. Flame sheet wrinkling is largely affected by the interaction
between diffusive-thermal instability and turbulence-induced strain [100,101]. For smaller ratios
of turbulence intensity (u’) to laminar flame speed SL where moderately-wrinkled flame
structures occur, local extinction would lead to the formation of edge-flames similar to the single
premixed studied in this work. For larger ratios of u’/SL, highly wrinkled flames substantially
concave to the products will form, resulting in local flame structures similar to the twin premixed
edge-flame configuration studied in this work. In either case, the formation of cellular structures
at low Leeff will significantly affect the propagation speeds and extinction conditions that have
been characterized in this work.
Chapter 5: Quasi-2D hydrogen flames in the Hele-Shaw cell
5.1 Flame structures
Figure 44 shows two theoretical modes possible for flame structures in a quasi-2D cell. The
first is a single flame sheet formed by the merging of individual flame sheets originating from
the electrodes. The second postulation is that a 2D analog (flame circles) could potentially exist

96
in the quasi-cylindrical geometry of a Hele-Shaw cell since steady spherical flame balls can exist
in large chambers [37]. In the context of flame balls, these structures are possible only under
microgravity conditions, where the influence of buoyancy is minimized. In contrast, flame
circles could potentially occur in a horizontally oriented Hele-Shaw cell, where such
gravitational constraints are absent. Unlike flame balls, flame circles are not inherently stable
because Laplace’s equation does not have a steady solution in unbounded, cylindrical geometry,
which becomes singular as r → ∞. The singularity has a very weak logarithmic nature, and thus
one could expect such structures to persist on long time scales [103].
Contrary to this expectation, such behavior was not observed in the majority of experiments
conducted in this study. It can thus be concluded that low-Le flames in a Hele-Shaw cell more
closely resemble propagating flames influenced by diffusive-thermal instabilities rather than
flame circles with minimal propagation effects. Notably, under extreme conditions where the
mixture strength is very weak, with Leeff or φ near extinction thresholds, a few intriguing flame
behaviors were observed, manifesting as “dot/cookie/plate” flame shapes.
Fig. 44. Two theoretically possible flame modes in the Hele-Shaw cell: (a) a single flame sheet; (b) individual
cylinders.

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5.1.1 The effect of instabilities
Flame structures are significantly influenced by the interplay of multiple instabilities,
including Darrieus-Landau (DL), Saffman-Taylor (ST), Rayleigh-Taylor (RT), and diffusivethermal (DT) instabilities. Their combined effects can result in a wide array of flame
morphologies, ranging from simple, planar fronts to complex cellular structures. A
comprehensive understanding of the roles and interactions among these instabilities is
indispensable for accurately predicting flame characteristics and behaviors under varying
operational conditions.
A series of sequential, superimposed images that capture the behavior of H2-O2-N2 flames
under a variety of conditions is presented in this section. These conditions include different
mixture combinations, thereby resulting in varying adiabatic flame temperatures Tad, equivalence
ratios ϕ, and effective Lewis numbers Leeff. Additionally, the figures illustrate flames in cells with
diverse dimensions, specified by their height d and width w, as well as in different orientations of
flame propagation.
Thermal expansion (DL), buoyancy (RT) and viscous contrast (ST)
Figures 45 to 47 depict flame propagation under the conditions of Tad = 1300 K, d = 1.27 cm,
w = 45.72 cm. Subfigures (a), (b), and (c) represent downward, horizontal, and upward
propagation, respectively. The main difference among these figures is the equivalence ratio φ and
thus effective Lewis number Leeff.
Consider Figure 47, which corresponds to φ = 2.5, Leeff = 1.04. In all directions of flame
propagation, the flame front exhibits a smooth cusp shape devoid of intricate small-scale
structures due to unity Leeff and the DL instability. As expected, the downward-propagating flame
front in Figure 47(a) is flattened due to RT instability, as the less dense burned product gas

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overlays the denser unburned gas. Comparatively, the horizontally propagating flame in Figure
47(b) displays more pronounced curved cusp shapes. Figure 47(c) illustrates an upwardpropagating flame; upon ignition, the four waves coming from the electrodes quickly merge into
a single, continuous front. Subsequently, RT instability acts to destabilize this front, transforming
the cusp structure into a larger, more curved front. In this case, the cell width w is about 0.4λST,
so it is not “wide” enough for ST instability to be captured.
Thermal diffusive (DT) instability
Instead of smooth flame fronts, the flames with Leeff = 0.3 in Figure 45 have cellular and
angular “sawtooth” structures. These characteristics are due to the combined effects of smallscale DTI and larger-scale instabilities. The cellular structure is a result of DT instabilities, as
discussed in the Introduction. In such cases, the accelerated mass diffusion rates in the gas
mixture enhance the velocity of the flame peaks, thereby promoting flame curvature effects.
When Leeff falls below a critical threshold, the flame fronts become fragmented.
The numerical study by Fernandez-Galisteo et al. [81] on quasi-2D flame propagation in a
Hele-Shaw cell aligns closely with the conditions of this experimental research. Their results
substantiate the observations made in this study concerning the impact of Leeff on flame
morphologies. For example, Figure 48 reveals remarkable similarities between the simulated and
experimental flame structures in gas mixtures with varying Leeff values. In mixtures where Leeff is
above unity, the flames manifest a cusped configuration, whereas in low Leeff mixtures, the
flames become cellular and angular. Similar patterns have also been reported by Altantzis et al.
[80] in a channel with single-step chemistry, and by Bergera et al. [82] using detailed chemistry
but without confinement and wall friction. Although neither study provides a comprehensive
theoretical explanation for these behaviors, their collective findings affirm that the effects of DT

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and DL instabilities can independently influence the hydrodynamic structure of flame fronts.
These effects do not necessitate the presence of wall viscosity, detailed chemistry, or
confinement. While these observed behaviors certainly merit an in-depth and comprehensive
analysis, such an investigation is beyond the scope of the present experimental study.
Fig. 45. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 0.7,
d = 1.27 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 1.27 cm, w = 45.72 cm, horizontal (c)Tad =
1300 K, φ = 0.7, d = 1.27 cm, w = 45.72 cm, upward. All mixtures have Leeff = 0.43.
Similar to Figure 47, the effect of RT instability is evident when Leeff is 0.3. In the case of a
downward-propagating flame, gravitational forces flatten the flame front, particularly during the
initial stages of propagation. Even as sawtooth structures begin to emerge, the flame front

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remains relatively flat compared to its horizontal (as seen in Fig. 45(b)) and upward (as seen in
Fig. 45(c)) counterparts.
Fig. 46. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, f = 1.0,
d = 1.27 cm, w = 45.72 cm, downward (b)Tad = 1300 K, f = 1.0, d = 1.27 cm, w = 45.72 cm, horizontal (c)Tad =
1300 K, f = 1.0, d = 1.27 cm, w = 45.72 cm, upward. All mixtures have Leeff = 0.73.

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Fig. 47. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 2.5,
d = 1.27 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 1.27 cm, w = 45.72 cm, horizontal (c)Tad =
1300 K, φ = 2.5, d = 1.27 cm, w = 45.72 cm, upward. All mixtures have Leeff = 1.04.

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Fig. 48. Direct Numerical Simulation (DNS) of a horizontally steady flame in gas mixtures with varying Leeff
values: 1.5 (left), 0.7 (middle), and 0.3 (right), compared with experimental results. The flame front is delineated by
the reaction rate isocontour (depicted in blue), and the calculated streamline pattern represents the flow field.
Courtesy of [81].
Figure 46 investigates the case where Leeff is 0.7, a value below unity but potentially around
the critical Lewis number Lec for cellular structures to manifest. Immediately post-ignition,
small-scale cellular structures appear. These cells lack the sawtooth shape and instead present a
remarkably stable flame front with deep troughs. Over time, the size of these cells diminishes
and eventually vanishes, giving way to the sawtooth structure. The transition between these two
states will be further discussed in the following sections. It is noteworthy that as Leeff increases,
the sawtooth structures become less acute and adopt a more rounded appearance.
5.1.2 The effect of cell dimension
Cell width w
The cell lateral width w has a significant impact on ST instability, which is dependent on the
wavenumber k. A variation in cell width could introduce new behaviors associated with ST

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instability. Computational studies on 2D freely propagating flames in periodic domains subject to
DL instabilities [80,104–106] have demonstrated that the overall flame propagation rate is
influenced by the cell width w but becomes independent for sufficiently large values. In this
regime, flames evolve into a configuration characterized by a single cusp-like structure,
commonly referred to as the “tulip flame.”
Fig. 49. DNS simulation of the effect of lateral domain size on the flame front. Courtesy of [81]
The numerical investigation by Fernandez-Galisteo et al. [81] further explored the effect of w,
as Figure 49 shows. They found that below a critical w, DL instability is suppressed by thermal
conduction, leading the flame to flatten after an initial transient period and evolve into steady
planar propagation with ST/SL = 1, whereas for larger w values, the flame develops a single-cusp
structure after initial transients. This is because if k exceeds a neutral-stability wavenumber kc =
2π/wc, which separates the unstable and stable regimes, flame wrinkles are inhibited from
growing, and the flame transitions into a planar configuration. Additionally, in the largest lateral

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domain, they observed sporadic wrinkle formations on the leading crests of the flame. These
wrinkles propagated transversely along the flame surface before disappearing at the trailing
cusps, a phenomenon that aligns well with the observations made in this experimental study.
Furthermore, a smaller lateral domain size could increase the surface area-to-volume ratio,
potentially leading to enhanced heat loss. However, this reduced size could also alter the flow
field and distribution. It is conceivable that the confined space forces the flame fronts to remain
in close proximity, thereby mutually increasing their temperatures. As a result, the flame,
particularly at the center, might propagate more quickly than it would in a larger cell.
Figures 50 to 52 show the flame front morphology in the Hele-Shaw cell under the
conditions of Tad = 1300 K, d = 1.27 cm, w = 11.43 cm. Subfigures (a), (b), and (c) represent
downward, horizontal, and upward propagation, respectively. The main difference among these
figures is the equivalence ratio φ and thus effective Lewis number Leeff.
Consider Figure 52, which corresponds to φ = 2.5, Leeff = 1.04. Similar to its wider
counterpart, in all directions of flame propagation, the flame front exhibits a smooth cusp shape
devoid of intricate small-scale structures due to DL. Notably, in the narrow cells, the effects of
RT and ST instabilities appear to be minimal. The flame in all three orientations initially shows a
slight cusp structure but soon evolves into a stable, mildly curved front.
For flames with Leeff = 0.3 as shown in Figure 50, they display cellular and angular structures
due to the combined effects of small-scale DTI and larger-scale instabilities. Similar to Figure 52,
the influence of ST and RT instabilities is minimal in narrow cells when Leeff is 0.3. Interestingly,
two stable structures are observed for such flames: either a V-shaped sawtooth or a single
inclined front with cellular structures. Despite efforts to synchronize the sparks and tests with
different spark locations, these structures appear to be independent of the initial spark conditions.

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Figure 51 examines the case where Leeff = 0.7, a value below unity but potentially around or
above the critical Lewis number Lec for cellular structures to occur. Unlike in the wide cell, no
post-ignition small-scale cellular structures are observed. This case is most susceptible to RT
instability because the G value in Equation (15) is dominant in this case, thus displaying a
downward-propagating front that is flattened (but still exhibits cellular structures!), a curved
front for horizontal propagation, and an upward-propagating front with even more curvature.
Fig. 50. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 0.7,
d = 1.27 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 1.27 cm, w = 11.43 cm, horizontal (c)Tad =
1300 K, φ = 0.7, d = 1.27 cm, w = 11.43 cm, upward. All mixtures have Leeff = 0.43.
Fig. 51. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 1.0,
d = 1.27 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 1.0, d = 1.27 cm, w = 11.43 cm, horizontal (c)Tad =
1300 K, φ = 1.0, d = 1.27 cm, w = 11.43 cm, upward. All mixtures have Leeff = 0.73.

106
Fig. 52. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 2.5,
d = 1.27 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 1.27 cm, w = 11.43 cm, horizontal (c)Tad =
1300 K, φ = 2.5, d = 1.27 cm, w = 11.43 cm, upward. All mixtures have Leeff = 0.43.
Cell thickness d
The thickness of the Hele-Shaw cell serves as a critical parameter that significantly
influences flame characteristics, including heat loss and ST instability. A thinner cell not only
increases the surface area-to-volume ratio, potentially leading to enhanced heat loss but also
approximates a two-dimensional structure more closely. This closer approximation to a 2D
structure restricts the development of certain three-dimensional flame behaviors, such as DT
instabilities in the height direction or changes in the flow profile. Additionally, the cell thickness
has a direct impact on ST instability. As mentioned in the Introduction, ST instability is most
prominent at long wavelengths, indicating flames with high viscosity contrast propagating thin
channels (small d) would present the most prominent effects of ST instability. Understanding the
intricate interplay between cell thickness, heat loss, and various instabilities is crucial for
accurately characterizing and predicting flame behaviors in confined geometries.
Figures 53 to 55 illustrate flame propagation under the conditions of Tad = 1300 K, d = 0.64
cm, w = 45.72 cm. Subfigures (a), (b), and (c) represent downward, horizontal, and upward
propagation, respectively. The main difference among these figures is the equivalence ratio φ and
thus effective Lewis number Leeff.

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Consider Figure 55, which corresponds to φ = 2.5, Leeff = 1.04. In all directions of flame
propagation, the flame front exhibits a cusp shape devoid of intricate small-scale structures due
to DL. Unlike the case with d = 1.27cm, the downward-propagating flame front in Figure 55(a)
is not flattened due to RT instability. Conversely, after an initial period, the flame front separates
into two flames. The leading edge adheres to the lateral walls and eventually re-connects near the
end wall. In Figure 55(b), the horizontally propagating flame initially advances as a flat front
with minor cusp structures, later transitioning into a large cusp accompanied by trailing minicusp structures. Figure 55(c) depicts an upward-propagating flame, essentially an exaggerated
version of the behavior seen in Figure 55(b).
This observed behavior results from a combination of factors: Leeff number, isothermal wall
conditions, and the interaction of DL and ST instabilities in the flow field. Computational work
by Kurdyumov [85] suggests that for Le < 1, flames may propagate faster in confinement with
isothermal walls than in those with adiabatic walls. This surprising effect is attributed to
increased flame curvature near a cold wall, which leads to elevated post-flame temperatures
despite heat losses to the wall (Fig. 56). In the meantime, numerical investigations by FernandezGalisteo et al. [81] indicate that a wrinkled or elongated flame, often observed in scenarios
involving variable viscosity, results in an increased flame surface area and a corresponding rise
in the outlet gas velocity at the open end. Notably, larger negative velocities occur at the flame
cusps in cases of variable viscosity. There is a small acceleration of the fresh gases ahead of the
flame at the flame crests in the direction of the flame propagation caused by thermal expansion.
Caused by thermal expansion, a small acceleration of the fresh gases occurs ahead of the flame at
the crests in the direction of flame propagation. This fresh gas flow is rapidly diverted in the
opposite direction towards the flame troughs, thereby exerting additional strain on the elongated

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flame front. Such straining in the variable-viscosity context fosters a new hydrodynamic-scale
instability, which, in conjunction with the aforementioned effects, leads to flame separation and
partial extinction in the middle of the cell.
Fig. 53. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 0.7,
d = 0.64 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm, horizontal (c)Tad =
1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm, upward
For flames with Leeff = 0.3 as shown in Figure 54, the sawtooth structures—due to the
combined effects of small-scale DTI and larger-scale instabilities—are maintained but with much
fewer dendrites compared to a thicker cell. In the context of the thinner cell, the orientation has a
reduced impact on flame structures; both horizontal and downward-propagating flames appear

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nearly identical. However, the upward-propagating flame exhibits a more pronounced curvature,
exhibiting a tulip shape and even showing partial extinction at the center of the cell. The flame
near the lateral walls appears brighter than the central region due to the super-adiabatic flame
pockets created by the curvature effect induced by the isothermal wall, as previously discussed.
Fig. 54. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 0.7,
d = 0.64 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm, horizontal (c)Tad =
1300 K, φ = 0.7, d = 0.64 cm, w = 45.72 cm, upward.
Figure 53 focuses on the case where Leeff = 0.7. This mixture, having the stoichiometric
combination, possesses the lowest laminar flame speed SL as well as ST (details to be discussed

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later). This reduced speed results in flatter flame fronts and increases the flame’s susceptibility to
partial extinction.
Fig. 55. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 2.5,
d = 0.64 cm, w = 45.72 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 0.64 cm, w = 45.72 cm, horizontal (c)Tad = 300
K, φ = 2.5, d = 0.64 cm, w = 45.72 cm, upward

111
Fig. 56. Unsteady cellular flame structure for Le = 0.5 with simulation for large lateral domain size and isothermal
boundary conditions (left) and experiment for φ = 0.78, Leeff = 0.51, d = 0.32cm, w = 45.72cm, Tad = 1300K (right).
DNS Courtesy of [85]
Figures 57 to 59 show flame propagation in a narrow cell under the conditions of Tad = 1300
K, d = 0.64 cm, w = 11.43 cm. Subfigures (a), (b), and (c) represent downward, horizontal, and
upward propagation, respectively. The main difference among these figures is the equivalence
ratio φ and thus effective Lewis number Leeff.
Figure 59, which corresponds to φ = 2.5, Leeff = 1.04, is similar to its counterpart with a wider
cell; in all directions of flame propagation, the flame front exhibits a smooth cusp shape. In the
narrow cells, the effects of RT instability appear to be minimal, while ST instability seems to be
more influential, and thus the downward case preserves its cusp shape during propagation.
For flames with Leeff = 0.3, as shown in Figure 57, all three cases appear identical, featuring a
sawtooth structure. This uniformity suggests that DTI may be the dominant factor in these cases.

112
Fig. 57. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 0.7,
d = 0.64 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 0.7, d = 0.64 cm, w = 11.43 cm, horizontal (c)Tad =
1300 K, φ = 0.7, d = 0.64 cm, w = 11.43 cm, upward
Fig. 58. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 1.0,
d = 0.64 cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 1.0, d = 0.64 cm, w = 11.43 cm, horizontal (c)Tad =
1300 K, φ = 1.0, d = 0.64 cm, w = 11.43 cm, upward
Fig. 59. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1300 K, φ = 2.5,
d = 0.64cm, w = 11.43 cm, downward (b)Tad = 1300 K, φ = 2.5, d = 0.64cm, w = 11.43 cm, horizontal (c)Tad = 1300
K, φ = 2.5, d = 0.64cm, w = 11.43 cm, upward

113
Figure 58 examines the case where Leeff = 0.7. a condition most susceptible to both RT and
ST instabilities. The downward-propagating flame front is nearly flat but still exhibits slight
cellular structures, with a slight convexity towards the open duct. The horizontally propagating
flame displays a slightly curved front that bends towards the end wall. The upward-propagating
flame exhibits even more pronounced curvature.
Experiments were also conducted in a cell with a thickness d of 0.32 cm. No qualitative
differences were observed in these tests compared to those in cells with different dimensions.
Specific cases that deviate from this general observation will be discussed in subsequent chapters.
5.1.3 The effect of mixture strength
As introduced in Section 2.2, to control independently the effective Lewis number Leeff,
combustible mixtures were formed with H2:O2 = 2φ:1 and diluted with N2 to obtain the desired
Tad. For example, Tad = 1000 K for both H2:O2:N2 = 1:1:9.55 and 4:1:18.2, yet φ is different (0.5
and 2, respectively. The quantity of the diluent influences Tad, which in turn significantly affects
the laminar flame speed SL and the flame front propagation speed ST. In the presence of
instabilities, varying levels of diluent give rise to distinct flame structures.
Figure 60 compares flame propagation under conditions where φ = 0.1 but diluted to Tad =
1100 K and 900 K, both in an upward propagation orientation. Figures 60(a) and 60(d) reveal
that the flame in the weaker mixture (900 K) is more susceptible to buoyancy effects,
propagating with a more stretched curve front. It should be noted that the flame light intensity at
900 K is actually weaker in the experiment, and the flame front—primarily composed of hot
water vapor product—is thinner than it appears in the image. To enhance visibility, the 900 K
flame was recorded at 10 fps, sometimes even lower, in contrast to the 30 fps used for the 1100 K
mixtures.

114
Figures 60(b) and 60(e) were captured in a thinner cell of d = 0.64 cm. In these instances, the
flame is weaker at the center, a phenomenon previously discussed, which is attributable to the
curvature induced by the isothermal wall. This curvature leads to the formation of hot pockets of
product that trail the flame front, causing it to advance more rapidly. It is also worth noting that
the upward-propagating flame exhibits minimal sawtooth structure. In Figure 60(b), particularly
at the initial stage of propagation and centrally located, separated cells akin to those observed in
edge-flames are present. Figure 60€ shows the emergence of elongated flame tubes, as described
in [107,108].
Figures 60(c) and 60(f) show flames in a narrow cell. While the flame at 1100 K retains its
sawtooth structure, the flame at 900 K propagates with a beautiful, lava lamp-like curved front.
Fig. 60. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 1100 K, φ = 0.1,
d = 1.27 cm, w = 45.72 cm, upward (b)Tad = 1100 K, φ = 0.1, d = 0.64 cm, w = 45.72 cm, upward (c)Tad = 1100 K, φ
= 0.1, d = 0.64 cm, w = 11.43 cm, upward (d)Tad = 900 K, φ = 0.1, d = 1.27 cm, w = 45.72 cm, upward (e)Tad = 900
K, φ = 0.1, d = 0.64 cm, w = 45.72 cm, upward (f)Tad = 900 K, φ = 0.1, d = 0.64 cm, w = 11.43 cm, upward.

115
Fig. 61. Sequential, superimposed images of flame propagation in the Hele-Shaw cell with (a)Tad = 900 K, φ = 0.1, d
= 0.32cm, w = 11.43 cm, horizontal (b)Tad = 900 K, φ = 0.1, d = 0.64cm, w = 11.43 cm, horizontal (c)Tad = 900 K, φ
= 0.1, d = 1.27cm, w = 11.43 cm, horizontal (d)Tad = 900 K, φ = 0.1, d = 0.32cm, w = 45.72 cm, horizontal.
Figure 61 shows more flame front with Tad = 900 K, captured in cells of differing dimensions.
As one moves from Figure 61(a) to 61(c), there is an increase in cell thickness, which is
accompanied by a corresponding increase in the width of individual cells within the cellular
structure. In the thinnest cell, elongated tubes are evident, propagating in alignment with the
cell’s lengthwise direction. Figure 61(d) depicts a flame front in a wider cell, exhibiting cellular
structures that are akin to those observed in edge-flame experiments.
5.2 The transition from cellular to Sawtooth
In this study, a transitional behavior in flame structures is occasionally observed, as
exemplified in Figures 46(a) and 46(c). Immediately following ignition, the flame front

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manifests small-scale cellular structures. These structures are distinct from the sawtooth shape
and exhibit a notably stable front characterized by deep troughs. As time progresses, these
cellular structures diminish in size and eventually disappear, giving way to a transient flat flame
front. This flat front subsequently evolves into the more familiar sawtooth structure.
This transitional behavior is observed only within a narrow range of lean mixtures in the wide
cell, and no such transition was noted in the narrow cell. The transition is more readily observed
in cells with a thickness of d = 1.27 cm. For d = 0.64 cm, the transition still occurs in some
mixtures, while for d = 0.32 cm, no transition was observed. This absence may also be partly
attributed to the partial extinction of the flame at the center of the cell.
Figure 62 shows the temporal and spatial occurrence of this transition. The behavior appears
to be somewhat symmetric with respect to the equivalence ratio φ and thus the Leeff. Whether this
behavior occurs around a critical Leeff, whether this critical number is the same Lec for the
cellular structures to occur, and whether and whether it represents a bifurcation phenomenon
remains a subject for further numerical and experimental investigation.
It is also possible that this observed behavior is a consequence of the developing flow field.
Initially, the flame front bends towards the end wall but eventually has to bend backward due to
faster lateral wall propagation. The small-scale structures induced by DTI are a stabilizing factor,
competing with DL and ST instabilities and ultimately losing out. In mixtures where DTI is
dominant or heat loss is significant, thereby weakening DL and ST, the flame front maintains a
rounded sawtooth shape and no transition is observed. This could also be attributed to the failure
of the z-averaged mixture property assumption due to small-scale 3D structures caused by DTI.

117
Fig. 62. Transition time and location vs. the equivalence ratio φ and Leeff in a horizontal Hele-Shaw cell with d = 0.5
cm.
In some cases, the wavelength of post-ignition cells is about the Saffman-Taylor wavelength.
Moreover, the shape of the troughs between cells resembles the “tulip” structure that is
characteristic of ST instability, whereas for DTI, the gap between cells should have
approximately the same perturbation amplitude as the superadiabatic regions. The Ka upon
ignition is likely high, the hydrodynamic instabilities dominated DTI, and the small-scale

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instabilities only have minimal manifestation on top of the “tulip” flame. Another interesting
point shown in Figure 62 is that the transition time t and transition local x/L almost have identical
profiles, indicating the same ST in the transition stage.
When the transition occurs, it is sometimes accompanied by acoustic waves, leading to
oscillating flame fronts. However, these oscillations do not impact the average propagation speed
of the flame, as this is calculated as a regression value averaged over flame locations, and the
acoustic waves are typically periodic. Furthermore, even with identical mixtures, acoustic waves
are not consistently observed, yet the acquired propagation speed ST values are closely aligned
with or without oscillation.
Lastly, in previous studies conducted in CPL with a shorter cell (about 60 cm in length), such
transition has never been reported. Further investigation into the impact of cell length could
provide valuable insights into the underlying physics of this phenomenon.
5.3 Propagation rates
Sections 5.1 and 5.2 have explored the intricate flame front structures influenced by DL, ST,
RT, and DT instabilities. As these instabilities induce different flame shapes, they also have a
direct impact on flame propagation speeds. This is because flame propagation speeds are
intrinsically linked to the surface area of the flame, which in turn is proportional to the degree of
wrinkling exhibited by the flame front. In this research, the physical propagation speed of the
flame front is denoted as ST. While the term ST is traditionally associated with flame speeds in
forced turbulence environments, the author adopts this notation here to emphasize that selfinduced flame instabilities can similarly lead to flame wrinkling, thereby accelerating
propagation speeds.

119
Experimentally, ST was determined based on video recordings. The fill fraction occupied by
the flame is quantified for each frame, from which ST is inferred. When a transition in flame
structure occurs, in most cases, ST will also change, resulting in two distinct ST values, as
illustrated in Figure 22. Both values are recorded, but the one observed after the transition is used
for subsequent computations, as it represents a more stable propagation speed for the flame. In
cases where the flame is partially extinguished—particularly in lean mixtures with low Lewis
numbers—the flame behavior is unsteady [85]. Flame pockets near the wall propagate more
stably, while those near the center exhibit oscillations. Given this, the forefront of the flame
serves as a more appropriate frame of reference for measuring ST, as it exhibits minimal velocity
oscillations. Therefore, the propagation speed of the flame near the lateral walls is chosen as the
ST value for these specific cases.
5.2.1. The effect of instabilities
Figure 63 presents the scaled propagation speed U = ST/SL as a function of Pe= SLd/α for
both lean and rich H2-O2-N2 mixtures at 1300K for varying cell dimensions. Filled symbols
represent data from wide cells, while hollow symbols denote narrow cells. As anticipated, the
combined effects of DL and ST always serve to destabilize the flame, thereby enhancing its
propagation speed. RT stabilizes/destabilizes the flame during downward/upward propagation,
resulting in the smallest/largest U.
In relation to the influence of DTI, the most obvious observation is the convergence of U
values for all three orientations as the mixture becomes leaner. This occurs because a smaller
Lewis number corresponds to a larger effective burning speed SL
*
, indicating that DTI becomes
the dominant factor affecting flame propagation speed. This phenomenon is clearly shown in
Figure 63.

120
Fig. 63. Scaled propagation speed U = ST/SL vs. Pe= SLd/α for H2-O2-N2 mixtures at 1300K: (a)all mixtures;(b)lean
mixtures;(c) rich mixtures.

121
Fig. 64. Scaled average flame speed U = (ST/SL
*
)(w/lflame) vs. Pe*
= SL
*
d/α for H2-O2-N2 mixtures at 1300K: (a)all
mixtures and cell dimensions;(b)w = 45.72 cm ;(c) w = 11.43 cm.

122
Figure 64 shows the scaled average flame speed U = (ST/SL
*
)(w/lflame) as a function of Pe*
=
SL
*
d/α for both lean and rich H2-O2-N2 mixtures at 1300K for varying cell dimensions. In all
cases, the scaled average flame speed values lie about 2±1, meaning that after taking into
account the effect of instabilities and cell dimensions, the averaged flame speed is always about
twice as much as the computed effective laminar burning velocity SL
* due to the curvature in the
x-z plane (d direction).
5.2.2. The effect of mixture strength
Figure 65 shows the effect of mixture strength on the flame propagation speed. For the
physical speed ST, a clear monotonic trend is observed with increasing Tad, which is consistent
with the Arrhenius law governing reaction rates. Both the orientation and cell dimensions
demonstrate an influence on ST. In general, upward-propagating flames in cells with the largest
dimensions (d and w) exhibit the highest ST. An outlier is observed for the 1200K mixture in a
cell with d = 0.32 cm; this may be attributed to changes in the duct’s surface area, possibly due to
pressure-induced gas inflow, leading to an unusually high flame speed.
In Figure 66, the scaled average flame speed U/SL
* remains a constant value around unity
across varying Tad values, suggesting that appropriate scaling was applied. Note that for the
thinnest cell, d = 0.32 cm, and d = 0.64 cm below 1000K, the U/SL
* value fluctuates. However,
the flame remains sustainable with the help of the DTI-induced cellular structure. In the absence
of these small-scale structures, these flames would be extinguished due to heat loss. It should be
mentioned that the effective burning speed SL
* used for this particular plot was computed based
on the extinction strain rate σext of a twin premixed configuration rather than the single-warm
configuration utilized in other computations due to the unavailability of σext for low adiabatic
temperatures.

123
Fig. 65. Effect of adiabatic flame temperature, chamber thickness and propagation direction on flame propagation
speeds ST in wide (a) and narrow (b) cells for ultra-lean H2-O2-N2, Leeff ≈ 0.3.

124
Fig. 66. Effect of adiabatic flame temperature, chamber thickness and propagation direction on Scaled average
flame speed U = (ST/SL
*
)(w/lflame) in wide (a) and narrow (b) cells for ultra-lean H2-O2-N2, Leeff ≈ 0.3.
5.2.3. The effect of cell dimension
Cell width w
In the case of lean mixtures, where Leeff < 0.7, Figure 63(a) shows that in a thicker cell, the
scaled propagation speed U varies from 3 (in downward propagation in a thin cell with heavy

125
heat loss) to 9 (in upward propagation in a thick cell influenced by RT instability). Conversely, in
a narrower cell, U ranges from approximately 1 to 12. For the rich cases shown in Figure 63(b),
the same conclusion that flame propagates slower in a narrow cell can be drawn.
These observations are consistent with the findings of Fernandez-Galisteo et al. [81], which
suggest that a narrow cell can suppress large-scale instabilities, making DTI the dominant factor,
whose effect is accounted for by the SL
*
. It is worth noting that in the simulation work with Le =1
presented in Figure 49, ST/SL is about one if the cell reaches a critical lateral width value. Future
work will explore smaller lateral widths to investigate whether U can be further reduced.
Figure 63 further illustrates that for the majority of mixtures, the narrow cell effectively
suppresses large-scale instabilities, resulting in closely aligned U values. An exception to this
trend occurs in weak mixtures with a relatively “high” number of Le (still as low as 0.6). In such
cases, buoyancy effects become dominant. It remains an open question whether this is due to RT
instability or a buoyancy-induced flow velocity limit, and further investigation is warranted.
Figures 65 and 66 demonstrate the effect of cell width w rather clearly. In Figure 65(a), the
propagation speed ST spans from 25 cm/s to 100 cm/s, whereas in Figure 65(b), ST range is
reduced to 40 cm/s to 70cm/s. The blue line in the graph, which represents a combined effect of
surface area ratio and Leeff, is about the median value of ST.
Cell thickness d
In terms of flame propagation speed ST, the thickness of the cell plays a significant role,
particularly concerning heat loss, as evidenced in Figures 63 through 66. Thinner cells generally
result in lower average propagation speeds or extinction, especially for mixtures with Lewis
numbers that are not extremely low. However, for super-lean mixtures, cellular structures enable
the formation of super-adiabatic cells or pockets, allowing the flame to persist under conditions

126
where it would otherwise extinguish in a planar configuration. Computational results using
Chemkin indicate that the minimum flame temperature for these cells consistently exceeds 1050
K, even for mixtures with an adiabatic flame temperature Tad of 850 K or lower.
5.4 JS parameter calculations
To identify the driving forces behind the various instabilities, the JS parameter was computed
based on Joulin-Sivashinsky’s was calculated based on linear dispersion relation for DT, ST, and
RT. Rather than utilizing the laminar flame speed SL, the effective burning speed SL
* which was
employed in an effort to account for the influence of small-scale DT instabilities.
Figure 67 shows the relationship between the scaled averaged flame speed U =
(ST/SL
*
)(w/lflame) and the JS parameter σ∗ for H2-O2-N2 mixtures. For lean mixtures (Figure 67(a))
with φ < 1.0, Leeff < 0.7, the scaled averaged flame speed values are about 2±1 for varying σ∗
values, showing satisfactory scaling results: as mentioned in the prior sections, the physical value
of ST could vary from 4 cm/s to 100 cm/s. Also, the interaction between all instabilities together
with cell dimensions further complicates the flame propagation process. When all mixtures are
considered, U = (ST/SL
*
)(w/lflame) does not change by a lot, and the fluctuation is due to (a) the
computation of SL*, which will be discussed below and (b) the value of lflame/w computed from
the edge-detecting Matlab program.
When the Hele-Shaw cell dimensions are reduced, either in width or thickness, heat loss
becomes a significant factor, leading to smaller U values compared to adiabatic conditions.
However, because the extinction strain rate is only available for mixtures with adiabatic no less
than 1100K, further efforts to obtain the SL
* for low-temperature mixtures will be necessary.

127
Fig. 67. Scaled propagation U vs. JS parameter for H2-O2-N2 mixture range: (a)φ < 1.0, Leeff < 0.7; (b) all mixture
tested.
As previously discussed in Section 5.2.2, the use of the extinction strain rate σext from twin
premixed counterflow flames may artificially inflate the calculated U values. Specifically, Figure
67 shows that σext for twin flames remains relatively constant for rich mixtures, which
contradicts the experimental observation that the propagation speed increases with increasing φ.
Furthermore, the twin premixed counterflow flame configuration does not accurately represent
the flame front dynamics within a Hele-Shaw cell. To better model this, cold reactive mixtures

128
versus hot diluent or product in a single premixed configuration should be applied. This
approach more closely mimics the actual conditions in the Hele-Shaw cell, where the flame front
separates cold premixed reactants from hot combustion products.
Fig. 68. Calculated extinction strain rates for varying adiabatic flame temperature Tad and equivalence ratio φ for
twin premixed flames and single flames with hot diluent opposing flow.
The computational results for this method (represented by dashed lines in Figure 68) show a
different trend compared to the twin premixed configuration. For rich mixtures, the propagation
speed increases with the equivalence ratio φ, which would help flatten the U trend. Moreover, the
lean part of the curve exhibits less dramatic changes in value, and thus, the U values for ultralean mixtures would slightly increase. In future work, the temperature of the hot product will be
fine-tuned for each mixture composition because there is no evidence that a universal 10% drop
from the adiabatic temperature is the most accurate estimate.

129
5.5 Conclusion
Flames propagating in narrow channels have been systematically investigated, focusing on
both flame shapes and propagation speeds. These behaviors are influenced by a combination of
Darrieus-Landau (DL), Rayleigh-Taylor (RT), Saffman-Taylor (ST), and diffusive-thermal (DT)
instabilities.
Remarkably, flame propagation speeds can significantly exceed laminar speeds even in the
absence of forced turbulence. This is attributed to self-induced instabilities. Thermal expansion
(DL) is ubiquitous across all flames unless negated by excessive heat loss in slow-burning
scenarios. Buoyancy or RT effects are direction-dependent and more pronounced in thicker
chambers. In contrast, in thinner chambers with fast flames, ST instability becomes significant.
For H2-O2-N2 mixture, DT instabilities are particularly crucial for lean mixtures and become
increasingly dominant as the value of φ further decreases. DT-induced cellular structures enable
flames to sustain conditions that would otherwise lead to extinction in planar flames. Notably, in
the case of upward-propagating flames with low effective Lewis numbers Leeff, the propagation
speed can exceed the laminar flame speed by more than an order of magnitude!
The dimensions of the Hele-Shaw cell also significantly impact both flame front structures
and the propagation speed ST. This study confirms that a narrow cell can suppress large-scale
instabilities, resulting in smoother flame fronts and more uniform propagation speed profiles.
Several novel phenomena have been observed, including flame structure transitions in a
narrow range of lean mixtures in wide cells, elongated flames in very weak mixtures, and
partially extinguished flame pairs. Numerical simulations support some of these findings, others
are still pending further investigation.

130
The JS parameter, based on the effective burning speed SL
*
, has shown promising
performance. It linearly correlates with the scaled propagation speed U across all orientations
and chamber dimensions (d and w), at least for lean mixtures. Alongside the Péclet number, the
JS parameter provides a robust interpretation of the driving forces behind self-induced
instabilities.

131
Chapter 6: Further work
6.1 Investigation into scaling parameters
While the current JS parameter demonstrates good performance, some outliers still require
further investigation. An adjusted JS parameter will be calculated based on the revised extinction
strain rate σext to more closely emulate the flame front conditions observed in this experimental
study.
Additionally, the JS dispersion relationship based on the Euler-Darcy flow may warrant reexamination. This is particularly relevant given that the modern combustion community is
increasingly questioning whether the concept of zero-flame thickness is well-defined. The
ultimate question that arises is whether there exists another parameter that could serve as a more
effective substitute for the JS parameter.
6.2 Flow field simulation
The phenomenon of angular flame fronts has been observed in both computational
simulations and previous experiments conducted by CPL. Despite these observations, the
fundamental theory explaining this behavior remains elusive, and no in-depth scholarly work has
been dedicated to this specific issue. Previous experimental work has indicated that the driving
force behind the angular flame front arises from the interplay between DL and DT instabilities

132
rather than being attributable to RT and ST instabilities or chemical kinetics. Further research is
imperative for a comprehensive understanding of this flame behavior and its growth rate.
Additionally, this study has uncovered new phenomena, including transitions in flame
structure within a narrow equivalence ratio range for lean mixtures in wide cells, the presence of
elongated flames in extremely weak mixtures, and the occurrence of partially extinguished flame
pairs. While numerical simulations support some of these findings, others are still pending
further investigation. Utilizing DNS could provide deeper insights into the underlying physics of
these phenomena.
6.3 New Experiment Apparatus
For accurate flame speed measurements, the flame must propagate under constant pressure
conditions. This ensures that the unburned gas ahead of the flame front remains relatively
stationary, not being pushed by the pressure increase due to the thermal expansion of the burned
products behind the flame. The regulation of such pressure variations is achieved through the
outflow of excess pressure from the combustion products, a process that is dependent on the
cross-sectional area of the exhaust pipe.
In the current experimental setup, the presence of acoustic waves has been observed,
indicating the need for a refined design incorporating pressure sensors for precise pressure
monitoring. Additionally, expanding the dimensions of the experimental cell in length would
provide a longer quasi-steady-state region, thereby enhancing the accuracy of flame speed
measurements. Further investigations could also explore the impact of varying the cell width w
to determine if a truly planar flame can be observed under certain conditions.

133
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Abstract (if available)

Abstract To develop a more comprehensive understanding of the burning properties of hydrogen-air mixtures and thereby obtain knowledge that informs the design and operation of energy conversion devices, this study employs properly designed experiments in two laboratory apparatuses. The first setup is a counterflow slot-jet burner used to assess the effects of steady hydrodynamic strain with minimized thermal expansion on propagation properties for H2-O2-N2 mixtures. The experiments focus on non-planar flame structures that occur only in low Le mixtures due to diffusive-thermal instabilities and their effects on edge-flame propagation/retreat rates and extinction limits. The second apparatus is a narrow channel, i.e., the Hele-Shaw cell, which is used to study the effects of thermal expansion without hydrodynamic strain (other than that generated by self-induced flow) and its interaction with other instabilities on H2-O2-N2 flame propagation properties. Various flame structures and propagation rates were observed depending on the interaction between four major instabilities: thermal expansion of the burned gas (Darrieus-Landau instability, DL); buoyancy effect (Rayleigh-Taylor instability, RT); viscosity contrast across the flame surface (Saffman-Taylor instability, ST); effective Lewis number (Leeff)(diffusive-thermal instability, DT). Flame morphology in both apparatuses is recorded; flame propagation rates were measured and then scaled with proper scaling parameters.

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Asset Metadata

Creator Zhou, Zhenghong (author)

Core Title Propagation rates, instabilities, and scaling for hydrogen flames

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Mechanical Engineering

Degree Conferral Date 2023-12

Publication Date 11/01/2023

Defense Date 10/20/2023

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag flame instabilities,flame propagation,hydrogen,OAI-PMH Harvest,scaling

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ronney, Paul (committee chair), Pantano-Rubino, Carlos (committee member), Tsotsis, Theodore (committee member)

Creator Email zhenghoz@usc.edu,zhouzhenghong.sjtu@gmail.com

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