Premixed flame propagation in Hele-Shaw cells: What Darrieus

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Premixed flame propagation in Hele-Shaw cells
What Darrieus Landau didnt 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

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University of Southern California

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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)

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Paul Ronney
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Introduction

• Models of premixed turbulent combustion dont
  agree with experiments nor each other!

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Introduction - continued...

• whereas in liquid flame experiments, ST/SL in
  4 different flows is consistent with Yakhots
  model with no adjustable parameters

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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

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Hele-Shaw flow

• Flow between closely-spaced parallel plates
• Momentum eqn. reduces to linear 2-D equation
  (Darcys law)
• 1000's of references
• Practical application to combustion flame
  propagation in cylinder crevice volumes

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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)

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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) dont 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

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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

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Results - video - baseline case

• 6.8 CH4-air, horizontal, 12.7 mm cell

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Results - video - upward propagation

• 6.8 CH4-air, upward, 12.7 mm cell

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Results - video - downward propagation
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Results - video - high Lewis number

• 3.0 C3H8-air, horizontal, 12.7 mm cell (Le 1.7)

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Results - video - low Lewis number

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

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Results - stoichiometric, baseline thickness
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Results - stoichiometric, thinner cell
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Results - stoichiometric, very thin cell
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Broken flames at very low Pe, Le lt 1

• 6.0 CH4- air, downward, 6.3 mm cell (Pe 30(!))

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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

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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
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Results - propagation rates

• 3-stage propagation
• Thermal expansion - most rapid, propagation rate
  (?u/?b)SL
• Quasi-steady (slower but still gt SL)
• Near-end-wall - slowest - large-scale wrinkling
  suppressed

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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)

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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)

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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

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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

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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

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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 ???

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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

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Results - pressure characteristics

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

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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 lt ST length (?/6)(?uUw2/?av)
• DL dominates - ?/k constant
• Propagation rate should be independent of L
• Large L, i.e. L gt (?/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 gt L -
  little effect of ST

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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

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Effect of JS parameter

• Results correlate reasonably well with relation
• ST/SL 1 0.64 (?/kSL)
• - suggests dimensionless JS parameter IS the
  driving force

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Effect of JS parameter

• Very similar for CH4-O2-CO2 mixtures

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Effect of JS parameter

• but propane far less impressive

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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

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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 lt 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

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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!

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Flame front lengths

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

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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 gt 1 even when JS parameter lt 0

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Results - wrinkling characteristics

• Individual images show clearly defined wavelength
  selection

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Results - wrinkling characteristics

• but averaging make them hard to see - 1/2 wave
  mode dominates spectra

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Results - wrinkling characteristics

• Because relative amplitudes of modes evolve over
  time

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Results - wrinkling characteristics

• Shows up better in terms of amplitude x
  wavenumber

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Wrinkling - different mixture strengths

• Modes 3 - 5 are very popular for a range of SL

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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 doesnt)

Run 108 9.5 CH4-air Horizontal propagation 6.35
mm cell
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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)

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Wrinkling - different fuel-O2-inerts, same SL

• Slightly broader spectrum of disturbances at low
  Le, less at high Le

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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

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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 ?

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Remark

• Computational studies suggest similar conclusions
• Early times, turbulence dominates
• Late times, thermal expansion dominates
• H. Boughanem and A. Trouve, 27th Symposium, p.
  971.

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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)

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Thanks to

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