1. Premixed flame propagation in Hele-Shaw cells: What Darrieus & Landau didn’t tell you
Paul D. Ronney
Dept. of Aerospace & Mechanical Engineering
University of Southern California
Los Angeles, CA USA
National Tsing-Hua University
October 7, 2005

2. University of Southern California
Established 125 years ago this week!
…jointly by a Catholic, a Protestant and a Jew - USC has always been a multi-ethnic, multi-cultural, coeducational university
Today: 32,000 students, 3000 faculty
2 main campuses: University Park and Health Sciences
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3. USC Viterbi School of Engineering
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Andrew Viterbi: co-founder of Qualcomm, co-inventor of CDMA
1900 undergraduates, 3300 graduate students, 165 faculty, 30 degree options
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4. Paul Ronney B.S. Mechanical Engineering, UC Berkeley
M.S. Aeronautics, Caltech
Ph.D. in Aeronautics & Astronautics, MIT
Postdocs: NASA Glenn, Cleveland; US Naval Research Lab, Washington DC
Assistant Professor, Princeton University
Associate/Full Professor, USC
Research interests
Microscale combustion and power generation
(10/4, INER; 10/5 NCKU)
Microgravity combustion and fluid mechanics (10/4, NCU)
Turbulent combustion (10/7, NTHU)
Internal combustion engines
Ignition, flammability, extinction limits of flames (10/3, NCU)
Flame spread over solid fuel beds
Biophysics and biofilms (10/6, NCKU)

5. Paul Ronney

6. Introduction
Models of premixed turbulent combustion don’t agree with experiments nor each other!

7. Introduction - continued...
…whereas in “liquid flame” experiments, ST/SL in 4 different flows is consistent with Yakhot’s model with no adjustable parameters

8. Motivation (continued…)
Why are gaseous flames harder to model & compare (successfully) to experiments?
One reason: self-generated wrinkling due to flame instabilities
Thermal expansion (Darrieus-Landau, DL)
Rayleigh-Taylor (buoyancy-driven, RT)
Viscous fingering (Saffman-Taylor, ST) in Hele-Shaw cells when viscous fluid displaced by less viscous fluid
Diffusive-thermal (DT) (Lewis number)
Needed: simple apparatus for systematic study of DL, RT, ST & DT instabilities & their effects on burning rates

9. Hele-Shaw flow Flow between closely-spaced parallel plates
Momentum eqn. reduces to linear 2-D equation (Darcy’s law)
1000's of references
Practical application to combustion: flame propagation in cylinder crevice volumes

10. Joulin-Sivashinsky (CST, 1994) model
Linear stability analysis of flame propagation in HS cells
Uses Euler-Darcy momentum eqn.
Combined effects of DL, ST, RT & heat loss (but no DT effect - no damping at small l)
Dispersion relation: effects of thermal expansion (), viscosity change across front (F) & buoyancy (G) on relationship between scaled wavelength () and scaled growth rate ()
Characteristic wavelength for ST = (/6)(uUw2/av): smaller scales dominated by DL (no characteristic wavelength)

11. Objectives Measure of premixed flames in Hele-Shaw cells
Propagation rates
Wrinkling characteristics
of premixed flames in Hele-Shaw cells
as a function of
Mixture strength (thus SL) (but density ratio () & viscosity change (fb - fu) don’t vary much over experimentally accessible range of mixtures)
Cell thickness (w)
Propagation direction (upward, downward, horizontal)
Lewis number (vary fuel & inert type)
and compare to JS model predictions

12. Apparatus Aluminum frame sandwiched between Lexan windows
40 cm x 60 cm x 1.27 or or 0.32 cm test section
CH4 & C3H8 fuel, N2 & CO2 diluent - affects Le, Peclet #
Upward, horizontal, downward orientation
Spark ignition (3 locations, ≈ plane initiation)
Exhaust open to ambient pressure at ignition end - flame propagates towards closed end of cell

13. Results - video - “baseline” case
6.8% CH4-air, horizontal, 12.7 mm cell

14. Results - video - upward propagation
6.8% CH4-air, upward, 12.7 mm cell

15. Results - video - downward propagation
6.8% CH4-air, downward, 12.7 mm cell

16. Results - video - high Lewis number
3.0% C3H8-air, horizontal, 12.7 mm cell (Le ≈ 1.7)

17. Results - video - low Lewis number
8.6% CH % O % CO2, horizontal, 12.7 mm cell (Le ≈ 0.7)

18. Results - stoichiometric, baseline thickness
9.5% CH % air, horizontal, 12.7 mm cell

19. Results - stoichiometric, thinner cell
9.5% CH % air, horizontal, 6.3 mm cell

20. Results - stoichiometric, very thin cell
9.5% CH % air, horizontal, 3.1 mm cell

21. Broken flames at very low Pe, Le < 1
6.0% CH4- air, downward, 6.3 mm cell (Pe ≈ 30(!))

22. Results - qualitative Orientation effects
Horizontal propagation - large wavelength wrinkle fills cell
Upward propagation - more pronounced large wrinkle
Downward propagation - globally flat front (buoyancy suppresses large-scale wrinkles); oscillatory modes, transverse waves
Thinner cell: transition to single large “tulip” finger
Consistent with Joulin-Sivashinsky predictions
Large-scale wrinkling observed even at high Le
Broken flames observed near limits for low Le but only rarely & not repeatable
For practical range of conditions, buoyancy & diffusive-thermal effects cannot prevent wrinkling due to viscous fingering and/or thermal expansion
Evidence of preferred wavelengths, but selection mechanism unclear

23. Horizontal propagation Horizontal propagation Horizontal propagation
Lewis number effects
8.6% CH % O % CO2
Horizontal propagation
12.7 mm cell, Pe = 85
6.8% CH % air
Horizontal propagation
12.7 mm cell, Pe = 100
3.0% C3H % air
Horizontal propagation
12.7 mm cell, Pe = 166

24. Results - propagation rates
3-stage propagation
Thermal expansion - most rapid, propagation rate ≈ (u/b)SL
Quasi-steady (slower but still > SL)
Near-end-wall - slowest - large-scale wrinkling suppressed

25. Results - quasi-steady propagation rates
Horizontal, CH4-air (Le ≈ 1)
Quasi-steady propagation rate (ST) always larger than SL - typically ST ≈ 3SL even though u’/SL = 0!
Independent of Pe = SLw/  independent of heat loss
Slightly higher ST/SL for thinner cell despite lower Pe (greater heat loss) (for reasons to be discussed later…)

26. Results - quasi-steady propagation rates
Horizontal, C3H8-air
Very different trend from CH4-air - ST/SL depends significantly on Pe & cell thickness (why? see next slide…)
STILL slightly higher ST/SL for thinner cell despite lower Pe (greater heat loss)

27. Results - quasi-steady propagation rates
C3H8-air (lean): Le ≈ 1.7, lower ST/SL
C3H8-air (rich): Le ≈ 0.9, higher ST/SL (≈ 3), ≈ independent of Pe, similar to CH4-air

28. Results - quasi-steady propagation rates
Horizontal, CH4-O2-CO2 (Le ≈ 0.7)
Similar to CH4-air, no effect of Pe
Slightly higher average ST/SL: 3.5 vs. 3.0, narrow cell again slightly higher

29. Results - quasi-steady propagation rates
Upward, CH4-air (Le ≈ 1)
Higher ST/SL for thicker cell - more buoyancy effect, increases large-scale wrinkling - ≈ no effect of orientation for 1/8” cell
More prevalent at low Pe (low SL) - back to ST/SL ≈ 3 for high Pe

30. Results - quasi-steady propagation rates
Downward, CH4-air (Le ≈ 1)
Higher ST/SL for thinner cell - less buoyancy effect - almost no effect for 1/8” cell
More prevalent at low Pe (low SL) - back to ST/SL ≈ 3 for high Pe
How to correlate ST/SL for varying orientation, SL, w ???

31. Results - quasi-steady propagation rates
Upward, CH4-O2-CO2 (Le ≈ 0.7)
Higher ST/SL for thicker cell - more buoyancy effect, increases large-scale wrinkling - less effect of orientation for 1/8” cell
More prevalent at low Pe (low SL) - back to ST/SL ≈ 4 for high Pe

32. Results - pressure characteristics
Initial pressure rise after ignition
Pressure ≈ constant during quasi-steady phase
Pressure rise higher for faster flames
Slow flame Fast flame

33. Scaling analysis How to estimate “driving force” for flame wrinkling?
Hypothesis: use linear growth rate () of Joulin-Sivashinsky analysis divided by wavenumber (k) (i.e. phase velocity /k) scaled by SL as a dimensionless growth rate
Analogous to a “turbulence intensity”)
Use largest value of growth rate, corresponding to longest half-wavelength mode that fits in cell, i.e., k* = (2/L)/2
(L = width of cell = 39.7 cm)
“Small” L, i.e. L < ST length = (/6)(uUw2/av)
DL dominates - /k = constant
Propagation rate should be independent of L
“Large” L, i.e. L > (/6)(uUw2/av)
ST dominates - /k increases with L
Propagation rate should increase with L
Baseline condition: (6.8% CH4-air, SL = 15.8 cm/s, w = 12.7 mm): ST length = 41 cm > L - little effect of ST

34. Scaling analysis
ST length smaller (thus more important) for slower flames and smaller w - but these conditions will cause flame quenching - how to get smaller ST length without quenching?
ST length = w (/6)(u/av)(1/Pr)Pe for fixed cell width, minimum Pe ≈ 40 set by quenching - easier to get smaller ST length without quenching in thinner cells

36. Effect of JS parameter Results correlate reasonably well with relation
ST/SL ≈ (/kSL)
- suggests dimensionless JS parameter IS the driving force

37. Effect of JS parameter
Very similar for CH4-O2-CO2 mixtures …

38. Effect of JS parameter
… but propane far less impressive

39. Image analysis - flame position
Determine flame position
Video frames digitized, scaled to 256 pixels in x (spanwise) direction
Odd/even video half-frames separated
For each pixel column, flame position in y (propagation) direction (yf) is 1st moment of intensity (I) w.r.t. position, i.e.
Contrast & brightness adjusted to obtain “good” flame trace

40. Flame front lengths
Front length / cell width - measure of wrinkling of flame by instabilities
Relatively constant during test
Higher/lower for upward/downward propagation
Front length / cell width = AT/AL < ST/SL - front length alone cannot account for observed flame acceleration by wrinkling
Curvature in 3rd dimension must account for wrinkling
Assume ST/SL ≈ (AT/AL)(U/SL), where U = speed of curved flame in channel, flat in x-y plane

41. Flame front lengths
Even for horizontally-propagating flames, AT/AL not constant - decreases with increasing Pe - but (inferred) U/SL increases to make (measured) ST/SL constant!

42. Flame front lengths
AT/AL similar with propane - but (inferred) U/SL lower at low Pe to make (measured) ST/SL lower!

43. Flame front lengths
AT/AL correlates reasonably well with JS growth parameter for CH4-air and CH4-O2-CO2
Less satisfying for C3H8-air (high Le)
Expected trend - AT/AL increases as JS parameter increases
… but AT/AL > 1 even when JS parameter < 0

44. Results - wrinkling characteristics
Individual images show clearly defined wavelength selection

45. Results - wrinkling characteristics
…but averaging make them hard to see - 1/2 wave mode dominates spectra…

46. Results - wrinkling characteristics
Because relative amplitudes of modes evolve over time…

47. Results - wrinkling characteristics
Shows up better in terms of amplitude x wavenumber…

48. Wrinkling - different mixture strengths
Modes are very popular for a range of SL…

49. Wrinkling - different cell thicknesses
Characteristic wavelength for ST = 103 cm, 26 cm, 6.4 cm in 12.7, 6.35, 3.2 mm thick cells - for thinner cells, ST dominates DL, more nearly monochromatic behavior (ST has characteristic wavelength, DL doesn’t)
Run 108
9.5% CH4-air
Horizontal propagation
6.35 mm cell

50. Wrinkling - different orientations
Upward = more wrinkling at large scales (RT encouraged); downward = less wrinkling at large scales; smaller scales unaffected (RT dominant at large wavelengths)

51. Wrinkling - different fuel-O2-inerts, same SL
Slightly broader spectrum of disturbances at low Le, less at high Le

52. Conclusions
Flame propagation in quasi-2D Hele-Shaw cells reveals effects of
Thermal expansion - always present
Viscous fingering - narrow channels, high U
Buoyancy - destabilizing/stabilizing at long wavelengths for upward/downward propagation
Lewis number – affects behavior at small wavelengths but propagation rate & large-scale structure unaffected
Heat loss (Peclet number) – little effect, except U affects transition from DL to ST controlled behavior

53. Remark
Most experiments conducted in open flames (Bunsen, counterflow, ...) - gas expansion relaxed in 3rd dimension
… but most practical applications in confined geometries, where unavoidable thermal expansion (DL) & viscous fingering (ST) instabilities cause propagation rates ≈ 3 SL even when heat loss, Lewis number & buoyancy effects are negligible
DL & ST effects may affect propagation rates substantially even when strong turbulence is present - generates wrinkling up to scale of apparatus
(ST/SL)Total = (ST/SL)Turbulence x (ST/SL)ThermalExpansion ?

54. Remark Computational studies suggest similar conclusions
Early times, turbulence dominates
Late times, thermal expansion dominates
H. Boughanem and A. Trouve, 27th Symposium, p. 971.
Initial u'/SL = 4.0 (decaying turbulence); integral-scale Re = 18

55. Future work Examine phase information, mode coupling
Obstacles of specified wavenumber - examine forced response
Linear growth behavior - need to suppress instabilities until specified time / location (e.g. acoustics, Clanet & Searby PRL 1998)
Radial growth from point ignition (Sivashinsky & others…)

56. Thanks to… National Tsing-Hua University
Prof. C. A. Lin, Prof. T. M. Liou
Combustion Institute (Bernard Lewis Lectureship)
NASA (research support)