Notes Lecture 13 - cefrc.princeton.edu

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

Summer 2013


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

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


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)

Moshe Matalon

Summer 2013


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.

Moshe Matalon

Summer 2013


[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

Moshe Matalon

Summer 2013


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!

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


  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

Moshe Matalon

  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

Moshe Matalon

Summer 2013


Flames in tubes are generally curved (convex towards the unburned gas)

  Flames in tubes

Uberoi, POF 1959

Moshe Matalon

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

Moshe Matalon

Summer 2013


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).

Moshe Matalon

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 ?

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


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

Moshe Matalon

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

Moshe Matalon

Summer 2013


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.

Moshe Matalon

€

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).

Moshe Matalon

Summer 2013


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

Moshe Matalon

Y T

T Y Le < 1

Thermal diffusivity < Mass diffusivity

Moshe Matalon

Summer 2013


(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

Moshe Matalon

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

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


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

Moshe Matalon

Effect of strain with σ = 5 Effect of gravity with σ = 5

Moshe Matalon

Summer 2013


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)

Moshe Matalon

Summer 2013


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

Moshe Matalon

Summer 2013


Bradley et al. (C&F, 2000)

Iso-octane (ϕ = 1.4 – 1.6)

Methane (ϕ = 1.0)

Moshe Matalon

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.

Moshe Matalon

Summer 2013


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

Moshe Matalon

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


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


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


  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


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


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


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


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

Notes Lecture 13 - cefrc.princeton.edu

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