Premixed flame propagation in Hele-Shaw cells:

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
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
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-

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-
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
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-
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-
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
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More info:
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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, S
T
/S
L

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 (

⤬ 癩獣o獩瑹t
change across front (F) & buoyancy (G) on relationship between
scaled wavelength (

⤠慮d 獣sl敤 gro睴w r慴攠(

)


Characteristic wavelength for ST = (

㘩
(

u
Uw
2
/

av
): smaller
scales
dominated by DL (no characteristic wavelength)

Objectives


Measure


Propagation rates


Wrinkling characteristics

of premixed flames in Hele
-
Shaw cells

as a function of


Mixture strength (thus SL) (but density ratio (

)

& 癩獣s獩瑹t
捨慮g攠⡦
b

-

f
u
) 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


CH
4

&

C
3
H
8

fuel,

N
2

&

CO
2

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% CH
4
-
air, horizontal, 12.7 mm cell

Results
-

video
-

upward propagation

6.8% CH
4
-
air, upward, 12.7 mm cell

Results
-

video
-

downward propagation























6.8% CH
4
-
air, downward, 12.7 mm cell


















3.0% C
3
H
8
-
air, horizontal, 12.7 mm cell (Le ≈ 1.7)

Results
-

video
-

high Lewis number

Results
-

video
-

low Lewis number



















8.6% CH
4
-

32.0% O
2
-

60.0% CO
2
, horizontal, 12.7 mm cell (Le ≈ 0.7)

Results
-

stoichiometric, baseline thickness




















9.5% CH
4
-

90.5% air, horizontal, 12.7 mm cell




















9.5% CH
4
-

90.5% air, horizontal, 6.3 mm cell

Results
-

stoichiometric, thinner cell




















9.5% CH
4
-

90.5% air, horizontal, 3.1 mm cell

Results
-

stoichiometric, very thin cell

Broken flames at very low Pe, Le < 1


















6.0% CH
4
-

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

Lewis number effects

8.6% CH
4
-

34.4% O
2

-

57.0% CO
2


Horizontal propagation

12.7 mm cell, Pe = 85

6.8% CH
4

-

93.2% air

Horizontal propagation

12.7 mm cell, Pe = 100

3.0% C
3
H
8

-

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
)S
L


Quasi
-
steady (slower but still >
S
L
)


Near
-
end
-
wall
-

slowest
-

large
-
scale wrinkling suppressed


Results
-

quasi
-
steady propagation rates


Horizontal, CH
4
-
air (Le ≈ 1)


Quasi
-
steady propagation rate (S
T
) always larger than S
L

-

typically S
T

≈ 3S
L

even though u’/S
L

= 0!


Independent of Pe = S
L
w/




independen琠o映hea琠loss


Slightly higher S
T
/S
L

for thinner cell despite lower Pe (greater heat
loss) (for reasons to be discussed later…)



Horizontal, C
3
H
8
-
air


Very different trend from CH
4
-
air
-

S
T
/S
L

depends significantly on Pe &
cell thickness (why? see next slide…)


STILL slightly higher S
T
/S
L

for thinner cell despite lower Pe (greater heat
loss)


Results
-

quasi
-
steady propagation rates


C
3
H
8
-
air (lean): Le ≈ 1.7, lower S
T
/S
L



C
3
H
8
-
air (rich): Le ≈ 0.9, higher S
T
/S
L

(≈ 3), ≈ independent of Pe,
similar to CH
4
-
air



Results
-

quasi
-
steady propagation rates


Horizontal, CH
4
-
O
2
-
CO
2

(Le ≈ 0.7)


Similar to CH
4
-
air, no effect of Pe


Slightly higher average S
T
/S
L
: 3.5 vs. 3.0, narrow cell again slightly
higher


Results
-

quasi
-
steady propagation rates

Results
-

quasi
-
steady propagation rates


Upward, CH
4
-
air (Le ≈ 1)


Higher S
T
/S
L

for thicker cell
-

more buoyancy effect, increases large
-
scale wrinkling
-

≈ no effect of orientation for 1/8” cell


More prevalent at low Pe (low S
L
)
-

back to S
T
/S
L

≈ 3 for high Pe


Results
-

quasi
-
steady propagation rates


Downward, CH
4
-
air (Le ≈ 1)


Higher S
T
/S
L

for thinner cell
-

less buoyancy effect
-

almost no effect
for 1/8” cell


More prevalent at low Pe (low S
L
)
-

back to S
T
/S
L

≈ 3 for high Pe


How to correlate S
T
/S
L

for varying orientation, S
L
, w ???



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⁊ou汩n
-
楶ash楮sky ana汹s楳 d楶楤ed⁢y wavenumber
k) (椮e⸠
phase ve汯c楴y

⽫) sca汥d by⁓
L

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⤯㈠



(L = width of cell = 39.7 cm)


“Small” L, i.e. L < ST length =
(

㘩
(

u
Uw
2
/

av
)

»
DL dominates
-


欠= 捯n獴慮t

»
Propagation rate should be independent of L


“Large” L, i.e. L >
(

㘩
(

u
Uw
2
/

av
)

»
ST dominates
-


欠in捲敡獥猠睩瑨 L

»
Propagation rate should increase with L


Baseline condition: (6.8% CH
4
-
air, S
L

= 15.8 cm/s, w = 12.7
mm): ST length = 41 cm > L
-

little effect of ST


Effect of JS parameter


Results correlate reasonably well with relation


S
T
/S
L

≈

1 + 0.64 (

歓
L
)


-

suggests dimensionless JS parameter IS the driving force

Effect of JS parameter


Very similar for CH
4
-
O
2
-
CO
2

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 (y
f
) 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 = A
T
/A
L

< S
T
/S
L

-

front length alone cannot
account for observed flame acceleration by wrinkling


Curvature in 3rd dimension must account for wrinkling


Assume S
T
/S
L

≈ (A
T
/A
L
)(U/S
L
), where U = speed of curved flame in
channel, flat in x
-
y plane




Flame front lengths


Even for horizontally
-
propagating flames, A
T
/A
L

not constant
-

decreases with increasing Pe
-

but (inferred) U/S
L

increases to
make (measured) S
T
/S
L

constant!




Flame front lengths


A
T
/A
L

similar with propane
-

but (inferred) U/S
L

lower at low Pe to
make (measured) S
T
/S
L

lower!




Flame front lengths


A
T
/A
L

correlates reasonably well with JS growth parameter for
CH
4
-
air and CH
4
-
O
2
-
CO
2


Less satisfying for C
3
H
8
-
air (high Le)


Expected trend
-

A
T
/A
L

increases as JS parameter increases


… but A
T
/A
L

> 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


Shows up better in terms of amplitude x wavenumber…










Wrinkling
-

different mixture strengths


Modes 3
-

5 are very popular for a range of S
L
…










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
-
O
2
-
inerts, same S
L

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

Conclusions


Flame

propagation

in

quasi
-
2
D

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

3
rd

dimension


…

but

most

practical

applications

in

confined

geometries,

where

unavoidable

thermal

expansion

(DL)

&

viscous

fingering

(ST)

instabilities

cause

propagation

rates

≈

3

S
L

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


(S
T
/S
L
)
Total

=

(S
T
/S
L
)
Turbulence

x

(S
T
/S
L
)
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'/S
L

= 4.0 (decaying turbulence); integral
-
scale Re = 18

Thanks to…


National Tsing
-
Hua University


Prof. C. A. Lin, Prof. T. M. Liou


Combustion Institute (Bernard Lewis Lectureship)


NASA (research support)