Premixed flame propagation in Hele-Shaw cells: What Darrieus & Landau didn’t tell you

1. Premixed flame propagation in Hele-Shaw cells:What Darrieus & Landau didn’t tell you http://ronney.usc.edu/research Paul D. Ronney Dept. of Aerospace & Mechanical Engineering University of Southern California Los Angeles, CA 90089-1453 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 • USC Trojans football team ranked #1 in USA last 2 years

3. USC Viterbi School of Engineering • Naming gift by Andrew & Erma Viterbi • Andrew Viterbi: co-founder of Qualcomm, co-inventor of CDMA • 1900 undergraduates, 3300 graduate students, 165 faculty, 30 degree options • $135 million external research funding • Distance Education Network (DEN): 900 students in 28 M.S. degree programs; 171 MS degrees awarded in 2005 • More info: http://viterbi.usc.edu

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 • 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 0.635 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% CH4 - 32.0% O2 - 60.0% CO2, horizontal, 12.7 mm cell (Le ≈ 0.7)

18. Results - stoichiometric, baseline thickness 9.5% CH4 - 90.5% air, horizontal, 12.7 mm cell

19. Results - stoichiometric, thinner cell 9.5% CH4 - 90.5% air, horizontal, 6.3 mm cell

20. Results - stoichiometric, very thin cell 9.5% CH4 - 90.5% 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. Lewis number effects 3.0% C3H8 - 97.0% air Horizontal propagation 12.7 mm cell, Pe = 166 8.6% CH4 - 34.4% O2 - 57.0% CO2 Horizontal propagation 12.7 mm cell, Pe = 85 6.8% CH4 - 93.2% air Horizontal propagation 12.7 mm cell, Pe = 100

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

32. 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. Effect of JS parameter • Results correlate reasonably well with relation ST/SL ≈1 + 0.64 (/kSL) - suggests dimensionless JS parameter IS the driving force

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

36. Effect of JS parameter • … but propane far less impressive

37. 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

38. 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

39. 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!

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

41. 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

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

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

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

45. Wrinkling - different mixture strengths • Modes 3 - 5 are very popular for a range of SL…

46. 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

47. 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)

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

49. 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

50. 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 ?