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

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

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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)

Paul Ronney

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

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

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

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

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)

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

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

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

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

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

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

Results - video - low Lewis number 8.6% CH4 - 32.0% O2 - 60.0% CO2, horizontal, 12.7 mm cell (Le ≈ 0.7)

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

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

Results - stoichiometric, very thin cell 9.5% CH4 - 90.5% air, horizontal, 3.1 mm cell

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

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

Horizontal propagation Horizontal propagation Horizontal propagation Lewis number effects 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 3.0% C3H8 - 97.0% air Horizontal propagation 12.7 mm cell, Pe = 166

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

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…)

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)

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

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

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

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 ???

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

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

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

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

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

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

Effect of JS parameter … but propane far less impressive

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

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

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!

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

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

Results - wrinkling characteristics Individual images show clearly defined wavelength selection

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

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

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

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

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

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)

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

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

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 ?

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

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…)

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