notes lecture 13 - cefrc.princeton.edu
TRANSCRIPT
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 1
Lecture 13 Flame instabilities
Moshe Matalon
Darrieus-Landau Instability Darrieus (1938) Landau (1944)
They treated the flame as a surface of density discontinuity, with a constant
flame speed and temperature, similar to the leading order hydrodynamic theory,
and examine the stability of a planar flame front.
burned
unburned
n
),,( tzyfx =
⇢u
⇢b
Vf
Moshe Matalon
gvv ρρ +−∇==⋅∇ pDtD,0
€
v × n[ ] = 0
€
ρ(v ⋅n−Vf )[ ] = 0
€
p+ ρv ⋅n(v ⋅n−Vf )[ ] = 0
Sf ≡ v ⋅n−Vf unb. = SLFlame speed relation
Rankine-Hugoniot conditions
Euler equations
2u)1(][
0][)1(][
L
L
Sp
S
ρσ
σ
−−=
=×
−=⋅
nvnv
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Plane flame (in a frame attached to the front)
x =0
unburned
SL
�SL
v = iSL (x < 0)σSL (x > 0)
!"#
$#
p− p0 =0 (x < 0)−(σ −1)ρuSL
2 (x > 0)
!"#
$#
consider first the case with no gravity
We will retain the notation that the top expression in the bracket is for x < 0 and the bottom expression for x > 0
Moshe Matalon
Introduce small disturbances (denoted by prime) and restrict attention to 2D
!ux + !vy = 0ρ !ut + ρuSL !ux + ρ !u !ux + ρ !v !uy = − !pxρ !vt + ρuSL !vx + ρu !v !vx + ρ !v !vy = − !py
!ux + !vy = 0ρ !ut + ρuSL !ux = − !pxρ !vt + ρuSL !vx = − !py
!ux + !vy = 0!pxx + !pyy = 0
ρ !ut + ρuSL !ux = − !px
u = u(x)+ !u (x, y, t) v = !v (x, y, t) p = p(x)+ !p (x, y, t) f = f (y, t)
)
y
x
unburned
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across x = 0
€
[ " u − " v fy ] = 0
[ !v +(u + !u ) fy ]= 0
€
[ " p ] = 0at x = 0-
u + !u − !v fy − ft = SL ft = !u (0− )
[ !u ]= 0[ !v ]= −(σ −1)SL fy
€
[ " p ] = 0
burned
unburned
n
Vf
u
b
ρ
ρ
),,( tzyfx =
n =(1,− fy,− fz )
1+ fy2 + fz
2Vf =
ft1+ fy
2 + fz2
n ~ 1, − fy( ) t ~ fy , 1( )Vf ~ ft
Moshe Matalon
ft = !u (0− )
[ !u ]= 0[ !v ]= −(σ −1)SL fy
€
[ " p ] = 0
Jump conditions across x = 0
and
!ux + !vy = 0!pxx + !pyy = 0
ρ !ut + ρuSL !ux = − !px
These conditions are to be satisfied across the flame front, i.e. across x = f(x,t), but since f << 1 they can be related to values across x = 0 using a Taylor expansion. Since the basic state consists of piecewise constants functions, u’(x =f) ~ u’(0), v’(x =f) ~ v’(0) and p’(x =f) ~ p’(0) .
Solve in the regions x < 0 with ⇢ = ⇢u and x > 0 with ⇢ = ⇢b
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tikyAef
tikyexPp
tikyex
ω
ω
ω
+=
+=
+=
'
)('
)(' Vv y
x
burned
unburned z
A Normal mode analysis
This is the method of normal modes, where it is assumed that an arbitrary disturbance introduced as an I.C., can be expanded in a Fourier integral (or series) over k. Being a linear problem it suffices to examine the response to one Fourier component.
k = |k| =q
k21 + k22
In three-dimensions, disturbances are of the form f 0 ⇠ eik·z+!t, where k =
(k1, k2) is the wave-vector and z = (y, z). The 2D results (in this case) can be
easily extended to 3D if
2⇡/k
k wavenumber� = 2⇡/k wavelength
Moshe Matalon
We are interested in the behavior of the solution as t ! 1.
f = Aeiky+!t = Aei(ky+!I)te!Rt
! = !R + i!I
If the perturbation goes to zero as t ! 1, which is assured when !R < 0 for all
values of k, the basic state (i.e., the planar flame) is stable.
For instability it is su�cient that !R > 0 for one mode (one value of k)
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0at 0][
)1(][0][
=!"
!#
$
=
−−=
=
xP
ASikVU
Lσ
AU ω=− )0(
0
0
u
2
=+
−=+
=−
ikVUPUSU
PkP
x
xxL
xx
ρρω
When substituting into the system of PDEs, the problem reduces to a set ofODEs for the determination of V (x), P (x), A with k real , and ! in general acomplex number.
Moshe Matalon
Dimensional considerations show that
In the present problem the only parameters are �, SL, in addition to k and !.
Solution to the eigenvalue problem exists only if
F(�, SL; k,!) = 0
dispersion relation
! = !(�, SL; k)
! = !DL(�)SLk
The dimensionless factor !DL(�) and, in particular its sign, remain to be deter-
mined.
The associated boundary conditions are homogeneous. It is, therefore a homo-
geneous boundary value problem with homogeneous BCs, for which the trivial
solution U = V = P = A = 0 is a solution. The question is whether there are
other nontrivial solutions. We are therefore faced with an eigenvalue problem
where ! is the eigenvalue.
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[U]= 0[V ]= −ik(σ −1)SLA
[P]= 0
"
#$
%$
at x = 0
AU ω=− )0(
Pxx − k2P = 0
ρωU + ρuSLUx = −PxUx + ikV = 0
ωεωεεε ~,~,~,~ ==== PPUUVV
Pxx − k2 P = 0
ρuSL Ux = − PxUx + ik V = 0
[ U]= 0[ V ]= −ikSLA
[ P]= 0
"
#$
%$
at x = 0
U(0− ) = ω A
It is instructive to examine first, the case for which � � 1 ⌧ 1
Let � � 1 ⌘ ✏
Moshe Matalon
PρuSL
2 =−C1e
kx
−C2e−kx
"
#$
%$
USL
=C1e
kx
C2e−kx
"
#$
%$
VSL
=iC1e
kx
−iC2e−kx
"
#$
%$
Pxx − k2 P = 0
ρuSL Ux = − PxUx + ik V = 0
[ U]= 0[ V ]= −ikSLA
[ P]= 0
"
#$
%$
at x = 0
U(0− ) = ω A C1 +C2 = kA, C1 =C2 =12kA,
ω =1
2kSL
kSL)1(21
−= σω
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y
x burned
unburned
u
0(0+) = u
0(o�) = ft
v
0(0+)� v
0(o�) = �(��1)SLfy
The concentrated vorticity, through the Biot-Savart law induces an axial velocity u proportional to ft that convects segments of the flame intruding towards the burned gas further upstream and those intruding towards the unburned gas further downstream.
the flame front is convected by the flow
the disturbed flame is equivalent to a flat vortex
sheet at x = 0 of strength ⇠ (� � 1)fy that
always increases in time (because !R > 0)
Moshe Matalon
unburned
burned
A different explanation that appears in the literature, uses a mass conservation argument (based on a quasi-steady picture). A streamtube of initial area A must be larger at the flame than further downstream when the sheet is convex towards the unburned gas (f < 0) before expanding back to size A. Upon crossing the flame, the streamtube first converges and then expands to A far away at infinity. An area increase, just ahead of the flame, causes a velocity decrease there. The flame (which is moving at its own speed) will encounter a lower speed and will have to move faster, thus enhancing the initial wrinkle. The “quasi-steady” assumption in this argument is not totally satisfactory, because the perturbed flow field is strongly unsteady!
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L
L
L
L
kS
xkeiCkxeiC
kxeiCSV
xkeCkxeC
kxeCSU
kxeC
kxeC
SP
/ˆ where
/ˆˆ
/ˆ
)1ˆ
(
)ˆ1(
32
1
32
1
2
1
2u
ωω
σωσω
σω
σω
ω
ρ
=
$%
$&
'
−−−−=
$%
$&'
−+−=
$%
$&
'
−−
+−=
0111
10
3
2
1
/ˆˆ/)1(1
1)/ˆ(ˆ1
=!!!
"
#
$$$
%
&
!!!
"
#
$$$
%
&
−−
−−
−+
CCC
σωωσ
σωω
0)1(ˆ2ˆ)1( 2 =−−++ σσωσωσ
Remove the limitation on �, and consider � � 1 = O(1)
ω =−2σ ± 4σ 2 + 4σ (σ 2 −1)
2(σ +1)
kSL!"#
$%& −−+
+= σσσσσ
ω 2311
Plane flames are unconditionally unstable
For � > 1, one of the two roots is always positive
Darrieus-Landau (DL) or hydrodynamic instability. Moshe Matalon
The DL instability has many ramifications in combustion.
Short waves k � 1 grow faster then long waves k ⌧ 1
Thermal expansion is responsible for the instability
For weak thermal expansion ! ⇠ 12 (� � 1)SLk
The hydrodynamic instability appears to contradict the fact that planar flames
have been observed in the laboratory. Note that the result is not valid for short
enough waves, . There may be stabilizing influences of di↵usion at the shortwave
range, i.e. when � ⇠ lf as we shall discuss later. nevertheless, the instability,
which is a result of thermal expansion is always present, and is the dominant
phenomenon in large scale flames, where di↵usion play a limited role.
� � 1 ⌧ 1
!
k
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Formation of cusps and crests Since the flame propagates normal to itself at a constant speed, successive locations of the front can be constructed geometrically using Huygen’s principle. The displacement over a time interval t is constructed from a family of spheres of radius SLt centered on the flame surface. The envelope draws the new location of the flame front. The convex/concave sections of the flame behave differently during the propagation. The convex sections (towards the unburned gas) of the flame expand so that a possible break in the front heals itself. The concave sections contract forming cusps.
x
f(x,t)
0 0.2 0.4 0.6 0.8
-0.6
-0.4
-0.2
0 t = 0.0
0.024
0.048
0.072
0.096
0.120
0.144
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Bunsen burner flames Bunsen flame experiments show that as the flow rate through the burner is reduced the opening angle of the wedge flame widens and the flame height becomes shorter. One might expect that when the incoming velocity U approaches SL the flame will eventually become flat. Instead the flame adopts a different shape with two sharp crests instead of one. The hydrodynamic instability does not permit flames that are too flat.
Uberoi et al., POF 1958
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Flames in tubes are generally curved (convex towards the unburned gas)
Flames in tubes
Uberoi, POF 1959
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Gravity effects
tikyexPgxS
gxpp
tikyexSS
L
L
L
ωρρσ
ρ
ωσ
++$%&
−−−
−=−
++$%&
=
)()1(
)(
b2
u
u0
Viv
The only difference from the previous discussion is in the jump condition for the pressure.
[p]x= f
= −(σ −1)ρuSL2
[p]x= f
= [p]x=0+
∂p
∂x
#
$%&
'( x=0⋅ f +
−(σ −1)ρuSL2 +[P]
x=0eiky+ω t + (ρu− ρb )g f = −(σ −1)ρuSL
2
[P]x=0= −
σ −1
σρugA
y
x
tikyeAf ω+=
burned
unburned
z g > 0
x =0
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Fr = kSL2/g Froude number
€
ω =1
σ +1(σ 3 +σ 2 −σ )SL
2k 2 − (σ 2 −1)gk −σSLk{ }
! ⇠ 1
� + 1
hp�3
+ �2 � � � �iSLk + . . . for k � 1
Influence of gravity dies out as k ! 1, but the short waves remain unstable
! ⇠r
� (��1)g
� + 1
k1/2 � �SL
� + 1
k + . . . for k ⌧ 1
Long waves are stabilized by gravity for downward propagation (g > 0), but
remain unstable for upward propagation (g < 0).
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1. How does the result of Darrieus and Landau reconcile with the fact that planar flames have been observed in the laboratory ?
2. What is the influence of diffusion, which have been neglected in this analysis ?
3. What are the nonlinear consequences of the hydrodynamic instability ?
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ρcpDTDt
−λ∇2T =QBρY exp(−E/RT )
ρDYDt
− (ρD)∇2Y = −BρY exp(−E/RT )
Y T
lf
x 0
planar flame
reactive-diffusive system
Thermodi↵usive instabilities can be studied by filtering out the hydrodynamic
disturbances. This is most easily done by setting ⇢ = const.
kSL!"#
$%& −−+
+= σσσσσ
ω 2311
Hydrodynamic instability
!
k
�
� = 1! ! 0 as � ! 1
Thermodiffusive Instabilities
Moshe Matalon
Introduce dimensionless variables using Dth/SL for length, Dth/S2L for time,
QYu/cp for temperature, and normalize the mass fraction w.r.t. Yu.
DT
Dt�r2T = !
DY
Dt� Le�1r2Y = �!
We will consider 1 � Le�1small, i.e. near equi-di↵usion flames, then the only
parameter is the reduced Lewis number ` = (Le� 1)� where � is the Zel’dovich
number
�
T =Tu + ex (x < 0)Ta (x > 0)
⎧ ⎨ ⎩
H = Ta − (1− Le−1)
xex (x < 0)0 (x > 0)
⎧ ⎨ ⎩
�
Y =1− ex (x < 0)0 (x > 0)
⎧ ⎨ ⎩
The Planar Flame.
Y T lf
lR x 0
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tikyexhh
tikyexTTωψ
ωφ++=
++=
)(
)()s(
)s(
f = Aeiky+!t
)()((
)0(0)0(0)(
22
2
φφψωψψ
φφωφφ
kk
xxk
xxxxx
xxx
−−=+−−
>=<=+−−
A
A
AAx
−=′+′
−=′
−===
+
][][][
][,][0at
21
ψφψφ
ψφ
Linearized equations.
Stability is assured when !R < 0 for all k; the solution is unstable if there exists
a mode for which !R > 0.
This is an eigenvalue problem for !; solutions exist only if the following solv-
ability condition is satisfied F(!, k; `) = 0, or ! = !(`, k). The growth rate !is in general complex, ! = !R + i!I
Moshe Matalon
�
4λ2 (1− λ) + (1− λ)2 − 4k 2( ) = 0
2223
222
2210
)82()121)(82(283219264
++=+++=−++==
kkbkkbkbb
)(41 2k++= ωλb0!
3 + b1!2 + b2! + b3 = 0
bi = bi(`, k)Sivashinsky, CST 1987
Dispersion relation
0)82( 2 >++ k
0)832()2)(128163()2(128 2224 >−++++−+ kk
Conditions for stability:
Using the Routh-Hurwitz criterion
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Conditions for stability:
2 + 8k2 + ` > 0
128(2k)4 + (�3`2 + 16`+ 128)(2k)2 + (32 + 8`� `2) > 0
k
-2 )31(4 +32/3
Stable Unstable Steady cellular
flames
Unstable
!I = 0 !I
6= 0
Pulsating and/or travelling waves
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y
x
tikyeAf ω+=
burned
unburned
x =0
reaction sheet
marginal states correspond to !R = 0
!I = 0 ) f = Aeiky cellular flames
!I 6= 0k = 0 ) f = Aei!It planar pulsating flames
k 6= 0 ) f = Aeik(y+ct) c = !I/ktraveling waves along the flame surface
)
f = Aei(ky+!I)t e!Rt = Aei(ky+!I)t
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Cellular flames
ω
k k*
2−<
2−>
k1
(Le−1)E(Tb −Tu )RTb
2 = − 2The critical Lewis number for cellular flames:
Cellular instability is predicted for � > 2⇡/k⇤. The first unstable wavenumber
will be k1 where k1 = 2n⇡/d where d is the flame dimension (i.e. the size of
the burner or the tube) and n an integer. Experiments, however, show that the
cells have an intrinsic size that is independent of the dimension of the flame.
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€
Dth (N 2) = 0.19 cm2/s
DC3H8−N2 = 0.11 cm2/s
DO2−N2 = 0.22 cm2/sLe ~ 0.86, 1.73{ } ~ −3, 11{ }
The range of Lewis numbers, from rich to lean mixtures
Hydrocarbon-air flames
Hydrogen-air
Dth (N2 ) = 0.19 cm2/s
DH2−N2= 0.61 cm2/s
Le ~ 0.31 ~ −10
Lean H2 –air mixture
The critical Lewis number for the onset of cellular flames seems to agree, in
general, with the experimental observations. For hydrocarbon-air flames cellu-
lar flames are observed on the rich side of stoichiometry (except possibly for
methane), while for hydrogen-air flames on the lean side).
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Goutelas 2004 30
Observations de l'instabilité de Darrieus-Landau
Photos J.Quinard 1990
Saturation non linéaire, taux de croissance forteSaturation non linéaire, taux de croissance faible
� Mécanisme de base de l'instabilité connu depuis presque 50 ans
� Validation expérimentale des taux de croissance fait seulement en 1998-2000– La flamme est instable, donc la flamme plane n'existe pas dans la nature !
–1) Difficulté expérimentale pour préparer une flamme initialement plane–2) Difficulté expérimentale pour contrôler la perturbation initiale
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Y T
T Y Le < 1
Thermal diffusivity < Mass diffusivity
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(Le−1)E(Tb −Tu )RTb
2 ≈10.93The critical Lewis number for pulsating flames:
The critical Lewis number for the onset of oscillations is much larger than one implying that oscillations are likely to be observed in lean mixtures of heavy fuels, or rich mixtures of light fuels. The critical value, however, is quite large and is not likely to be reached in common combustion mixtures. In the presence of heat loss the critical Lewis number is reduced significantly and the instability may be accessible. Indeed it has been observed in porous plug burners when the flame was sufficiently close to the burner.
Planar Pulsating flames
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Combined effects of hydrodynamic and diffusive -thermal Instabilities
The early work of Markstein in the 1950’s was phenomenological in nature, extending the Darrieus-Landau results by assuming a dependence of the flame speed on curvature. The more rigorous results are based on the hydrodynamic asymptotic theory, which includes the influences of diffusion resulting in the internal flame structure assumed thin but of finite thickness, and extends the DL growth rate to include higher order terms in k.
! =
p�3 + �2 � � � �
� + 1| {z }SLk
!DL
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Planar Flames Pelce & Clavin, (JFM, 1982); Matalon & Matkoswky (JFM, 1982); Frankel & Sivashinsky (CST, 1982)
ω =ωDLSLk − l f B1 + β (Leeff −1)B2 +Pr B3[ ] SLk2
DL instability Heat conduction
stabilizing
Viscous effects stabilizing
Species diffusion stabilizing in lean CnHm /air mixtures rich H2 /air mixtures
The coefficients ωDL , B1, B2, B3 > 0 depend only on thermal expansion σ.
Recall Leeff is a weighted average of the fuel and oxidizer Lewis numbers with a heavier weight on the deficient component.
Moshe Matalon
DL
!
k
*effeff LeLe <
*effeff LeLe >
*effeff LeLe > The short wavelength disturbances (λ > λc= 2π/kc) are
stabilized by diffusion Hydrodynamic instability
The short wavelength disturbances are also unstable Hydrodynamic instability is enhanced by diffusion effects Diffusive-thermal instability
*effeff LeLe <
kc
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0 0.5 1 1.5 2
-1-0.75-0.5
-0.250
0.250.5
0.751
1.251.5
1.752
ω
Leeff=0.4 simulationsMatalon et al. JFM03, Leeff=1Leeff=1 simulationsDarrieus - LandauMatalon et al. JFM03, Leeff=0.4
(a)
k
0 0.1 0.2 0.3 0.4 0.5-0.1
0
0.1
0.2
0.3
0.4 Le=0.4 simulationsMatalon et al. JFM03, Le=1Le=1 simulationsDarrieus - LandauMatalon et al. JFM03, Le=0.4
Comparison of linear growth with DNS for H2/air flames; Altanzis et al., JFM(2012); ETH LAV.
Moshe Matalon
k
-2
Stable
Unstable Steady cellular
flames
k = ω DL
l f B1 + β (Leeff − 1)B2 + Pr B3[ ]
neutral stability curves
ω =ω DLSLk − l f B1 + β (Leeff − 1)B2 + Pr B3[ ] SLk2
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Numerical Results
Neutral stability curves in the limit of large activation energy, but with σ = O(1). The asymptotic expression for k << 1 agrees well with these results.
Jackson & Kapila 1984, CST, 41, 191
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Effect of strain with σ = 5 Effect of gravity with σ = 5
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R
r = R(t) [1 + A(t) Sn(θ,φ)]
1AdAdt
=RR
ω −lfRΩ
#
$%
&
'(
growth-rate:
Spherical Flames (R >> lf) Bechtold & Matalon, (C&F, 1987); Addabbo et al. (ProCI, 2002)
1AdAdt
=RRωDL −
lfRB1 +β(Leeff −1) B2 +Pr B3"# $%
&'(
)*+
hydrodynamic diffusion
Moshe Matalon
*effeff LeLe > The amplitude A first decay, but after the flame reaches a certain critical
size Rc it starts growing in time. The instability is hydrodynamic in nature. *effeff LeLe < The amplitude grows in time immediately upon its incept. The flame is
diffusively unstable
A
R(t)
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Growth rate computed for a lean propane/air fame σ = 5.9, β(Leeff -1) = 4.93
Moshe Matalon
Marginal Stability Curves
Peclet number Pe = R/lf
wav
enum
ber
n
n ~ R
Λmax = 2πR/n ~ R
Λmin= 2πR/n ~ const.
Growing modes: Λmin < Λ < Λmax
n = const.
Pecr
Lean propane/air: σ = 5.9, Leeff =1.49
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Bradley et al. (C&F, 2000)
Iso-octane (ϕ = 1.4 – 1.6)
Methane (ϕ = 1.0)
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Nonlinear Effects The linear theory predicts whether disturbances of given wavelength grow or
decay in time. The consequences of an instability needs to be determined from
a nonlinear analysis.
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steady propagation:
Hydrodynamically unstable flame
L = 1 L = 2 L = 4
L = 0.075 corresponding to Le > Le⇤, for � = 4
critical wavelength �c = 1.7
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γ
U/S
L
40 60 80 100 120 140 1601.00
1.05
1.10
1.15
1.20
1.25
~
σ = 5
σ = 4
σ = 3
σ = 5.5subcritical
supercritical
� = (� � 1)L/L
propagationspeedU/S
L
Propagation speed of the hydrodynamically unstable flame
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 25
Steadily-propagating wrinkled flames in quiescent mixtures
U = 1.08 U = 1.15
x = �Ut+ f(y)
� = 80 � = 160
Moshe Matalon
© 1951 Nature Publishing Group
Garside & Jackson, Nature 1951
First observation of a thermodiffusive instability in diffusion flames J.E. Garside and B. Jackson 1951
Formation of a polyhedral flame, previously observed by Smithells & Ingle (1892) and by Smith & Pickering (1929) only in premixed combustion. The photograph is a polyhedral diffusion flame produced when a mixture of H2 diluted in CO2 is burned at atmospheric pressure upon a cylindrical tube of 1.1 cm in diameter. The surface of the flame was formed of five triangular cells in the shape of a polyhedron. By adjusting the flow rate, the flame could be made to rotate about a central vertical axis. Similar polyhedral flames were observed when N2 was used as the diluent gas instead of CO2.
Polyhedral diffusion flame
Polyhedral premixed flame
S. Sohrab Moshe Matalon
Diffusion Flame Instabilities
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 26
N2 diluted H2/O2 diffusion flame at the base of a splitter plate burner
1 cm
Under normal conditions the base of the flame close to the burner partition, was straight. At sufficiently high flow velocity and for when the hydrogen concentration was substantially reduced the base became cellular. Same behavior was observed when the nitrogen in the fuel stream was replaced with Ar but not with He.
Dongworth & Melvin, 1976
Moshe Matalon
Instabilities in di↵usion flames have been typically observed at high flow rates, or
near-extinction conditions. They are predominately di↵usive-thermal in nature
with hydrodynamics playing a secondary role.
The Burke-Schumann solution of complete combustion (corresponding to the
limit D ! 1) is unconditionally stable. Instabilities result only when there is
su�cient leakage of reactants through the reaction sheet with a flame temper-
ature significantly below Ta.
F(!, k; D, LeF, Le
O,�, �) = 0
Damkohler number DLewis numbers Le
F= Dth
DF, Le
O= Dth
DO
mixture strength � =
YF /YO
⌫FWF /⌫OWO
thermal expansion � = ⇢u/⇢f
Stability analysis
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 27
Instabilities develop when the temperature is lowered significantly below the adiabatic temperature and there are significant reactant leakage, i.e., for Dext< D < D* namely at high flow rates or near-extinction conditions.
D may be controlled by varying the flow rate U, or the burning temperature Ta through dilution (nature or amount) or the mixture strength ϕ, or by varying the stretch rate K (tubular flames).
Instabilities are predominantly thermo-diffusive in nature with hydrodynamic effects playing a secondary role; the mode of the instability depends on the Lewis numbers LeF , LeO and on the initial mixture-strength ϕ.
Tf
Flam
e te
mpe
ratu
re
BS limit
D* a T
stable unstable
Dext D Damköhler number
Moshe Matalon
Planar pulsations
oscillatory cells
1
1
LeF
LeO
high-frequency instability
Stationary cells
Cellular flames - more likely to occur when ϕ < 1 (lean conditions) - cell size λ = 0.5 - 2 cm Pulsating flames - more likely to occur when ϕ > 1 (rich conditions) frequency ωI = 1 - 6 Hz
high-frequency instability
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 28
Nitrogen-diluted H2-O2 Diffusion flame - splitter-plate burner
1 cm
Under normal conditions - base remains straight; at high flow rates and at high dilution rates base is cellular. Same behavior when diluted in Ar but not in He.
High flow rate D < D*
in N2 and Ar LF = 0.33 – 0.35 in He LF = 1.02
Dongworth & Melvin, 1976
Moshe Matalon
Pulsations – Jet Flames
Fuel Stream
Oxidizer Stream Pulsations
4% C 3 H 8 96% N 2
65% O 2 35% N 2 1.90 1.09 0.33 No
6% C 3 H 8 94% N 2
40% O 2 60% N 2 1.86 1.05 0.76 No
8% C 3 H 8 92% N 2
30% O 2 70% N 2 1.83 1.03 1.32 Yes
12% C 3 H 8 88% N 2
19% O 2 81% N 2 1.78 1.00 3.04 Yes
24% C 3 H 8 76% N 2
19.5% O 2 80.5% N 2 1.63 0.93 5.56 Yes
42% C 3 H 8 58% N 2
15.5% O 2 84.5% N 2 1.45 0.85 11.24 Yes
80% C 3 H 8 20% N 2
14.5% O 2 85.5% N 2 1.20 0.73 19.23 Yes
100% C 3 H 8 0% N 2
14% O 2 86% N 2 1.10 0.69 23.26 Yes
Füri, Papas & Monkewitz, 2000
LeF LeO �
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 29
Cellular Diffusion Flame H2 & O2 diluted in CO2 (LeF = 0.35, LeO = 0.86, ϕ = 0.24)
9.1 mm
Robert & Monkewitz
�=1
planar flames
extinction
Transition from planar through cellular flames all the way to extinction as the
mixture strength is reduced.
Moshe Matalon
Flame intensity
Flame position
Planar pulsating diffusion flame
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 30
CO2 diluted H2/air tubular diffusion flame Increasing stretch rate from 207 s-1 to 386 s-1
Shopoff, Wang & Pitz 2011 Moshe Matalon
Thermal expansion further destabilizes the cellular flame extends the range of conditions for cellular flames (increases D*)
LeF =LeO = 0.75 , ϕ = 0.5
D fixed σ = 6.1
4.8 4
3.6
wavenumber
grow
th ra
te
Effect of thermal expansion on cellular flames � = ⇢u/⇢f
k
!
Moshe Matalon
Summer 2013
Moshe Matalon University of Illinois at Urbana-Champaign 31
Effect of thermal expansion on pulsating flames
Thermal expansion has a stabilizing influence on planar oscillations; reduces the range of conditions for oscillations (decreases D*)
wave number
D fixed σ = 2.3
2.9
3.5
LeF = 1.5 , LeO = 1.1 , ϕ = 2
grow
th ra
te !
k
� = ⇢u/⇢f
Moshe Matalon