lecture 8 - hydrodynamic stretch lecture notes... · lecture 8 flame stretch and lewis number...

6/26/11

Copyright ©2011 by Moshe Matalon. This material is not to be sold, reproduced or distributed without the prior written permission of the owner, M. Matalon. 1

LECTURE 8 Flame stretch and Lewis number effects

1

Hydrodynamic theory

Matalon & Matkowsky (JFM, 1982) Matalon et al. (JFM, 2003)

Heat conduction, species diffusion, viscous dissipation, are resolved on the diffusion scale ~ lf , which constitutes the flame, within which there is a thin reaction zone of thickness ~ β-1 lf .

Asymptotic matching provides jump conditions across the flame and an equation for the flame speed. Results were carried out to O(δ) to retain the influences of a finite flame thickness.

The theory accounts for thermal expansion, temperature-dependent transport, nonunity and distinct Lewis numbers, effects due to stoichiometry and reaction orders and radiative heat losses.

burned

O(δ)

n

unburned

T u

Y u

T b

Y= 0 ω

l f

u

b

!

!

flame zone

Multi-scale approach thin flame δ ≡ lf/L � 1large activation energy β � 1

2

6/26/11


with O(δβ-1) temperature and density corrections in the burned gas neglected

T =TuTb

!"#

$#! =

!u!b

!"#

$#Y = Yu

0

!"#

$#

(F < 0)(F > 0)

burned

unburned

n

ρu

ρb

Vf

F (x, y, z, t) = 0

∇ · v = 0

ρDv

Dt= −∇p+ µ∇2v + ρg

for F < 0 and F > 0

Jump conditions across F = 0Flame speed equation

We consider a two-reactants mixture undergoing combustion described by a one-step overall reaction (reaction order two), with constant transport propertiesλ, µ, ρD. The flame of arbitrary shape propagates in an unconfined environmentunder adiabatic conditions.

σ ≡ ρu/ρu = Tb/Tu

3

[[p+ ρ(v·n)(v·n− Vf )]] = terms ofO(δ)

[[n× (v × n)]] = terms ofO(δ)

[[ρ(v·n− Vf )]] = terms ofO(δ)

Jump conditions across F = 0

Flame speed relation

Sf = SL − LK

burned

unburned

n

ρu

ρb

Vf

F (x, y, z, t) = 0

where, Sf ≡ −Vf + v·n|F=0−

Note that both, the reciprocal of the Reynolds number Re, as well asthe Markstein length are both O(δ).

L (in cm) is the so-called Markstein lengthK the flame stretch rate (1/s), is a measure of the deformation of the flamesurface resulting from its motion and the underlying hydrodynamic strain.

4

6/26/11


Flame Stretch

Consider a set of points on the flame surface which remain on the flame surface but move along it due to the underlying fluid velocity. These points form an element of area A, which is continuously deformed as a result of the motion of the flame and the underlying strain rate. Flame stretch is defined as the fractional area change of a Lagrangian surface element of the flame surface (it is a kinematics concept).

ξ1 ξ2

n

n

e1

e2

P

flame sheet

A(t)

K ≡ 1

A

dA

dt

K = Sfκ+Ks

from kinematic considerations

Within the context of the hydrodynamic theory Sf ≈ SL, so that K = SLκ+Ks

κ and Ks are the curvature and strain rate.

5

Matalon et al., JFM, 2003

Matalon CST, 1993

K = −Vf κ− n ·∇× (v × n)

K = SLκ+Ks

curvature

κ =1

R 1+

1

R 2

1/R1 and 1/R2 are the twoprinciple radii of curvature

κ = −∇ · n

strain rate

E = 12

�∇v + (∇v)T

�

is the rate of strain tensor

Ks = −n ·E · n

normal straining tangential straining

Ks = −(v · n)κ+∇τ · vτ

K = −Vf κ+∇τ · vτ

Note that these expressions are all identical; depending on the flame configura-tion, one or the other may turned out to be easier to use.

Other useful relations:

6

6/26/11


Note that the flame speed equation is a relation that describes the instantaneousshape and location of the flame surface

∂F

∂t+ v∗ ·∇F = (SL − LK) |∇F |

where v∗ is the gas velocity just ahead of the flame surface (i.e., at F = 0−)and K depends on the curvature of the flame surface, as well on the local hy-drodynamic strain rate.

where the velocity v is calculated from the solution of the NS equations, andv∗ determined from the knowledge of the flow field ahead of the flame.

The function F can be seen as a scalar field, whose level surface F (x, t) = const.represents the flame surface. Here the constant is taken to be zero.

This equation is often referred to as the “level set equation”

∂F

∂t+ v∗ ·∇F = SL|∇F |

�1 + L∇· ∇F

|∇F |

�+

L2

�∇F ·

�∇v∗+(∇v∗)T

�·∇F

�

A hybrid Navier-Stokes/front-capturing methodology is required for the numer-ical implementation of the hydrodynamic model.

7 Rastigejev & Matalon, JFM 2006, Creta & Matalon, PCI 2011Peters 2000

1. Spherically expanding flame

RR

RRR

dtdA

A

RA!! 2

4814

2

2

===

=

!!

!

K

!

Ks = "err r= R = "#vr

#r r= R

=$ "1$

2 ˙ R R

% =2R

, Vf = " ˙ R

radial flow v &n = 0

!

K = SL" + K s =2SLR

+# $1#

2 ˙ R R

=2 ˙ R R

!

vr =0 r < R

" #1"

˙ R Rr

$

% &

'

( )

2

r > R

*

+ ,

- ,

LSR !=!

Note: the inwardly propagating spherical flame is negatively stretched (i.e. compressed)

!

K = "Vf# =2 ˙ R R

n R(t)−r = 0

Examples of flame stretch

8

6/26/11


!""

"""

"

2v22)v(lim1lim

v,

2

22

00

2

==#$#+

=##=

#=#=

%#%# rtrrtr

tA

A

trrA

ttK

0,0),0,0,1(),0,v,( ==!== "fVu nv

!2="===dz

zzs eKK

!2)( =""#$%= nvnK

z r

n

razudz

!! =""=

>

v),(2:for

d

Note: the flame in a rear-stagnation point flow is negatively stretched (i.e. compressed)

2. Flame in a stagnation-point Flow Examples of flame stretch

9

Flame shape: (circular cone with sharp apex)

Flow field: v = (-U, 0 , 0) (uniform flow)

for more details about this and other examples, see Matalon, CST 1983

t = (- sin α, 0, - cos α)

U

n

t α

0 r

z

!

K = "U2rsin 2#

The flame is negatively stretched (compressed).

R1 = −r/ cosα and R2 = ∞

c = cotα

F (z, r,ϕ) = c2r2 − z2

3. Bunsen flame Examples of flame stretch

10

6/26/11


(a) the normal propagating motion can expand or contract the flame dependingon the sign of the curvature

(b) normal straining caused by vn can contract or expand the flame dependingon the sign of the curvature

(c) tangential straining, represented by the divergence of the tangential velocityvector, can have a compressing or expanding effect.

K = SL κ��(a)

overall strainKS� �� −(v · n)κ� ��

(b)

+∇τ · vτ� �� (c)

General flame surface

normal straining tangential straining

n

11

The Markstein length

L =

�σ lnσ

σ − 1+

β(Leeff − 1)

2(σ − 1)

� σ

1

ln ξ

ξ − 1dξ

�lf

For a stoichiometric mixture (φ = 0) the effective Lewis number is the aver-age of LeF and LeO. For an off-stoichiometric mixture (φ � 1) the deficientcomponent is more heavily weighted such that for very lean/rich mixtures theeffective Lewis number is practically that of the fuel/oxidizer, respectively.

The effective Lewis number of the mixture Leeff is a weighted average of theindividual Lewis numbers, LeF and LeO, defined as

Leeff =

LeO + (1 + φ)LeF

2 + φlean mixture

LeF + (1 + φ)LeO

1 + φrich mixture

φ =

�β�φ−1 − 1

�lean mixture

β(φ− 1) rich mixture.

where φ > 0 measures thedeviation from stoichiometry

12

6/26/11


Bechtold & Matalon, C&F 2001

•  Leeff < 1 for lean H2/air, and rich hydrocarbon/air mixtures •  Leeff > 1 for rich H2/air, and lean hydrocarbon/air mixtures

(except for CH4 which behaves more like H2)

Llf

=σ lnσ

σ − 1+

β(Leeff − 1)

2(σ − 1)

� σ

1

ln ξ

ξ − 1dξMarkstein number

13

A more general expression for the effective Markstein length can be obtained fora one-step overall chemical reaction but of arbitrary reaction orders, a and b forfuel and oxidizer respectively, and in mixtures for which the viscosity µ, thermalconductivity λ, and diffusivities DF ,DO depend on temperature. Specifically,it is assumed that λ, ρD, µ have the same temperature dependence.

λ(x) is the functional dependence on the dimensionless temperature x=T/Tu.Typically, λ(x)=xα with α = 0.7; constant properties correspond to λ ≡ 1.

L =

�σ

σ − 1

� σ

1

λ(ξ)

ξdξ +

β(Leeff − 1)

2(σ − 1)

� σ

1ln�σ − 1

ξ − 1

�λ(ξ)ξ

dξ

�lf

Leeff =

LeO +ALeF1 +A lean mixture

LeF +ALeO1 +A rich mixture

where the coefficient A > 0, depends on the reaction orders a, b with respect tothe fuel or oxidizer, respectively, and on the equivalence ratio φ. They are givenby

A ≡ G(a, b; φ)bG(a, b−1; φ)

− 1 , G(m,n, z) ≡� ∞

0ςm(ς + z)ne−ς dς ;

with φ > 0 defined earlier (the deviation from stoichiometry).

Note that G(1, 1, z) = 2 + z and G(1, 0, z) = 1, and we recover the previousexpression when a = b = 1.

14

6/26/11


Markstein number relative to the burned gas

The convention we have used is to define the flame speed relative to the gasvelocity f the fresh mixture ahead of the flame. If the flame speed is definedrelative to the gas velocity of the burned gas behind the flame,

Sbf = σSL − σLbK

and the corresponding Markstein number (which could be easily deduced fromknowledge of the jump conditions across the flame) is

Lb =

�1

σ − 1

� σ

1

λ(ξ)

ξdξ +

β(Leeff − 1)

2(σ − 1)

� σ

1ln

�σ − 1

x− 1

�λ(ξ)

ξdx

�lf

Note that the laminar flame speed relative to the burned gas is simply σSL, butLb �= L is not a simple multiplication of L by σ.

15

The Markstein number is uniquely determined only in the asymptotic limit con-sidered here, where the whole flame is treated as a surface that coincides withthe reaction sheet and the flame speed evaluated at this location.

Experimental measurements, on the other hand, are typically taken at a spe-cific reference location inside the flame zone. To make meaningful comparisonsbetween theory and experiments the Markstein number must, therefore, be ad-justed by properly calculating the gas velocity at the chosen reference location.

Tien & Matalon, CNF 1991Bechtold & Matalon, CNF 2001

The magnitude of the Markstein number, in general, drops as a result of thatadjustment; thus for light fuels the corrected Markstein numbers may possiblyhave negative values at lean conditions.

Comment about the Markstein length

16

6/26/11


Flame Speed and Temperature

Sf = SL −�

σ

σ − 1

� σ

1

λ(ξ)

ξdξ +

β(Leeff − 1)

2(σ − 1)

� σ

1ln

�σ − 1

ξ − 1

�λ(ξ)ξ

dξ

�lfK

Tf = Ta − Tu(Leeff − 1)

�� σ

1ln�σ − 1

ξ − 1

�λ(ξ)ξ

dξ

�lfKSL

Note that variations of Tf from the adiabatic temperature are small, on the order O(δβ−1).The expression for Tf is slightly modified at near stoichiometric conditions.

17

• Positive stretch reduces Tf if Le > 1 and increases Tf if Le < 1. Theflame temperature is unaffected by stretch if Le ≈ 1.

• Positive stretch increase or decreases Sf if Le is larger or smaller thana critical, slightly negative Le∗. Roughly speaking we can say, that Sf

increases/decreases with stretch when Le is less/greater than one.

K = 2�Stretch rate

Law , 2007

Experimentally measured temperature for different strain rates. Nitrogen-diluted CH4/air flames at near-stoichiometry (N2/O2 = 5, φ = 0.95). The estimated Leeff ≈ 1 so that T ≈ Ta independent of ε.

d ∼ a+ 1/2�K

K

d

18

6/26/11


Law , 2007

(a)  lean H2 / air mixture LeH2 < 1 rich H2 / air mixture LeO2 > 1

(b)  lean C3H8 / air mixture LeC3H8 > 1 rich C3H8 / air mixture LeO2 < 1

Stretch rate K(1/s)Stretch rate K(1/s)

(a) hydrogen-air mixture (1 atm) (b) propane-air mixture (1 atm)

SL

SL

SL

SL

Calculated flame speed as a function of the stretch rate, for counterflow flames illustrating the distinct dependence of flame speed on stretch and the effect of Lewis number.

19

Outwardly propagating spherical flames

unburned gas

burned

Here δ = lf/R, so that including O(δ) terms provides information about smallerflames (earlier time). One finds

R = σSL

�1− 2σLb lf

R

�

Lb =

�1

σ − 1

� σ

1

λ(ξ)

ξdξ +

β(Leeff − 1)

2(σ − 1)

� σ

1ln

�σ − 1

ξ − 1

�λ(ξ)ξ

dξ

�lf

where Lb �= L is the Markstein length, defined relativeto the gas velocity of the burned gas

Note that Lb changes sign at a critical value Le∗eff . Thus, in lean hydrocar-bon/air mixtures, where Leeff is typically larger than one, the propagation speedof a spherical flame increases as the flame grows larger; the reverse is true forrich hydrocarbon/air mixtures.

20

6/26/11


1.3

1.48

0.78

1.68

Strehlow , 1984

For propane-air mixtures, the transition where the behavior of R changes fromincreasing to decreasing function of R occurs at equivalence ratio φ ≈ 1.3.

Addabbo et al., PCI 2002

Propagation speed vs flame radius

21

The dependence of the flame speed on curvature (via stretch) ensures a smoothflame shape near the tip.

22

6/26/11


Flame propagating in a channel

x

y

burned

unburned

2a U

n

For simplicity, consider a constant density flow and adiabatic walls.

Then the velocity field is given by u(y) = u0(1− y2) and the flame front repre-sented by x = Ut− ϕ(y) where U is the propagation speed.

n =(1,!! y )

1+! y2

Vf =!U1+! y

2, " =

! yy

1+! y2( ) 3/2

Ks =! y

1+! y2

! y! yy

1+! y2SL

Sf = SL − LK

u(y)+U ~ SL 1+! y2using

23

!=y"

!

"#y = 1+#2 $ ˜ u 0 (1$ y 2) $ ˜ U #($1) =#(1) = 0

⇒

!u0 (1! y2 )+ !U

1+! y2

=1!"! yy

1+! y2 level-set equation

Use SL as a unit of speed, a as a unit of length and a/SL a unit of time. TheMarkstein number for σ ≈ 1 is α = 1 + le/2 where le = β(Le− 1).

What happens when we lower le below −2; i.e., when Le is sufficiently low? 24

6/26/11


reaction rate contours

Flame propagating in a channel against an imposed Poiseuille flow - computed

based on a constant density or diffusive-thermal model. Here le = β(Le - 1)

25

Bunsen burner flames – lean/rich propane & methane air mixtures

Le > 1 Le < 1 Le > 1 Le < 1

Law, 200726

6/26/11


Bunsen flame

27

The dependence of the flame speed on curvature (via stretch) ensures a smoothflame shape near the tip (asymptotic solution).

0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

x

y

0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

x

y

Uniform inflow Parabolic inflow

Computations based on a hybrid Navier-Stokes/level-set approachRastigejev & Matalon, JFM 2006

28

6/26/11


Creta & Matalon, PCI 2011

Steady-propagation in a quiescent mixture at a speed 1.15SL, computed forσ = 4 with L > 0, using a hybrid Navier-Stokes/level-set approach.

Flow field through a corrugated flame

29

E N D

30

lecture 8 - hydrodynamic stretch lecture notes... · lecture 8 flame stretch and lewis number...

Documents