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Fundamentals of Flame Stabilities

Additional Issues - Nonpremixed Flames

Nonpremixed Flame Instabilities

- Some experimental observations

- Cell structures typically observed in

the direction of no flow straining

- Asymptotic analysis showed similar

types of diffusion flame instabilities

(Kim, 1997)

• Cellular instability for Le < 1 Schematic of diffusion flame instability

(Kim, 1997)

• Pulsating instability near

extinction condition for

Le > 1. Hexagonal cellular instability in axisymmetric

methane/air jet flames (Lo Jacono et al., 2003).

Fundamentals of Flame Stabilities

Additional Issues - Edge Flames

Edge Flame Instabilities (Thatcher & Dold, 2001)

- S > 1 for L < L

e c

- Edges continue to propagate even when the trailing

diffusion flame no longer exists (in a temporally

oscillating mode).

=

L L c

c

Fundamentals of Flame Stabilities

Additional Issues - Micro-Combustion

Nonpremixed Combustion in a Microchannel

(Miese et al., 2005)

- Cell structures observed in a narrow

channel subjected to heat loss

- Number of cells depends on mixture and

flow conditions.

Fundamentals of Flame Stabilities

Flame Stretch:

Effects of Aerodynamics on

Flame Propagation

Fundamentals of Flame Stabilities

Flame Stretch - Introduction

In many flame instability studies, it was

essential to consider that flame speed

depends on flow conditions.

- Markstein

• Flame speed depends on curvature

- Diffusive-thermal instability:

• Flame speed changes due to imbalance between

mass and heat transport (Lewis number)

⇒ Formal theoretical description of the net effects of

flow field on flame speed is needed.

Fundamentals of Flame Stabilities

Flame Stretch - Definition

dU

κ =

Karlovitz (1953): dy δ κ

Williams (1975): 1 dA ( )

κ = = T

, Ka Karlovitz Number

A dt S L

Kinematic consideration n

: unit normal vector

V : surface velocity (lab frame)

f

v

: flow velocity (lab frame)

e e

, : tangent vectors on surface

p q

⎡ ⎤

= ⋅ − ⋅

V n v n

S ⎣ ⎦

L f =

G 0

Fundamentals of Flame Stabilities

Flame Stretch - Definition

A. Matalon (1983)

{ }

( )

( ) ( )

κ = − ∇ × × ⋅ − ⋅ ∇ ⋅

v n n V n n

= f

G 0 =

G 0

B. Chung & Law (1984)

( ) ( )

κ = ∇ ⋅ + ⋅ ∇ ⋅

v V n n

t s , t f t ( )

= = + ⋅

v V V V V n n

where ;

s , t t f t f

C. Candel & Poinsot (1990)

( )

κ = − ∇ + ∇ ⋅ + ∇ ⋅

nn

: v v S v

L

( ) ( ) ( )

= + ∇ + ∇ ⋅ = + ∇ ⋅

e e e e : v S n a S n

p p q q L T L

tangential strain curvature

Note that in A & B, curvature effects may be included in the first

term. Fundamentals of Flame Stabilities

Flame Stretch – Physical Implication

Chung & Law (1984) ( ) ( )

κ = ∇ ⋅ + ⋅ ∇ ⋅

v V n n

t s , t f t

(3)

(1) (2)

(1) Flow nonuniformity along the flame surface

( )

= × × ≠ 0 only if flow is oblique to the surface

v n v n

s , t

V

(2) : Flow velocity in the inertial frame

f ⋅ =

V n 0 for stationary flames

f

∇ ⋅ ≠

(3) 0 only for curved flames.

n

t Fundamentals of Flame Stabilities

Flame Stretch – Examples

Examples ( ) ( )

κ = ∇ ⋅ + ⋅ ∇ ⋅

v V n n

t s , t f t

(a) Spherical expanding flame κ

= ∇ ⋅ ≠ ⋅ ≠ ⇒ ≠

0, but 0, 0 0

v n V n

s t t f

,

(b) Steady spherical flame κ

⋅ = ∇ ⋅ = ⇒ =

0, 0 0

V n v

f t s t

,

(c) Steady curved flame in uniform flow

κ

⋅ = ∇ ⋅ ≠ ⇒ ≠

0, but 0 0

V n v

f t s t

,

Fundamentals of Flame Stabilities

Flame Stretch – Further Examples

Steady Stagnation Flow (Hiemenz) u

( ) ( )

κ = ∇ ⋅ + ⋅ ∇ ⋅

v V n n v

t s , t f t

⎛ ⎞

1

a : strain rate (s )

⎜ ⎟

⎛ ⎞

a

= − = ⎧

⎜ ⎟

x , ay , 0 0,

v ⎜ ⎟

0 : 2-D slab

+ =

⎝ ⎠ ⎨

1

k k

⎜ ⎟

,

s t ⎩

⎝ ⎠

1: axisymmetric

= =

n j e i

, t ∂

⎧ ( ) ( )

= 2D

ax a

⎪ ∂

⎪ x

⇒ ⋅ = ∇ ⋅ = ⎨

0,

V n v ∂ ⎛ ⎞

f t 1 1 ( )

⎪ =

2

⎜ ⎟ axisymmetric

ar a

⎪ ∂ ⎝ ⎠

⎩ 2

r r

κ = a

Fundamentals of Flame Stabilities

Flame Stretch – Further Examples n

Spherical Flames ( ) dR

e

( )

κ = ∇ ⋅ + ⋅ ∇ ⋅ = f

p V

v V n n f

e

,

t s t f t dt

q

dR R

= ⋅ = f

0,

v V n f

s , t f dt = =

0 0

( ) ( )

∂ ⋅ ∂ ⋅ ∂ ∂

⎛ ⎞

∂ ∂ e n e n e e

n n

∇ ⋅ = ⋅ + ⋅ = + − ⋅ +

p q p q

⎜ ⎟

n e e n

∂ ∂ ∂ ∂ ∂ ∂

t p q ⎝ ⎠

p q p q p q

1 1

= =

1 1

= + R R

1 2

R R

1 2 >

dR

2 0 for outward prop.

κ = f ⎨ <

⎩ 0 for inward prop.

R dt

f

Fundamentals of Flame Stabilities

Flame Stretch – Further Examples

Bunsen Flame ( ) ( )

κ = ∇ ⋅ + ⋅ ∇ ⋅

v V n n

t s t f t

,

( ) ( )

α α

⋅ = = = −

V n 0, v 0, 0, w , n cos , 0, sin n

f α ∂ ∂

⎡ ⎤

sin ( ) ( )

κ α α α

= − +

rw cos cos w cos

⎢ ⎥ ( )

=

∂ ∂

⎣ ⎦ v 0, 0, w

r r z

α =

If constant (except at the tip)

α

w sin 2 ( )

κ =− < 0

2 R f

Fundamentals of Flame Stabilities

Effects of Flame Stretch - Phenomenology

Key Parameters:

λ ( ) ( )

α = , D deficient , D excess

ρ i j

c p

α ( ) ( )

=

Le Lewis number ; D / D Preferential diffusion

i i j

D

i

Example: Bunsen flame tip

= =

Le 1: no effect D / D 1: no effect

i i j

heat

> <

Le 1: stronger tip D / D 1: stronger tip

i i j

mass

(lean propane/air) (rich hydrogen/air)

< >

Le 1: tip opening / 1: tip opening

D D

i def. i j

(lean hydrogen/air) (rich propane)

excess

Fundamentals of Flame Stabilities

Effects of Flame Stretch for Bunsen Flames

Experimental Observation (Law & Sung, 2000)

rich C H /air lean C H /air rich CH /air lean CH /air

3 8 3 8 4 4

Fundamentals of Flame Stabilities

Effects of Flame Stretch - Counterflow

κ

Example: Counterflow flames ( > 0) Heat: loss

Mass: gain

=

= D D

/ 1: no effect

Le 1: no effect i j

i > <

Le 1: weaker flame D D

/ 1: weaker flame

i i j

(lean propane/air) (rich hydrogen/air)

< >

Le 1: stronger flame D D

/ 1: stronger flame

i i j

(lean hydrogen/air) (rich propane)

Fundamentals of Flame Stabilities

Effects of Flame Stretch in Counterflow

Numerical Results: Flame Temperature vs. Stretch

Fundamentals of Flame Stabilities

Effects of Flame Stretch on Flame Speed

Markstein (1950)

S μ

= − ∇

L f

1 (heuristic curvature effect)

S κ =

, 0

L

Asymptotic Analysis for Low Stretch Flames

S κ

= − + = −

"

L L

1 1 MaKa

S κ =

, 0

L Markstein length δκ

L

= =

Ma Markstein number; Ka Karlovitz number

δ S L

Clavin & Williams (1982), Clavin & Garcia (1983) λ

T T dT

T

γ = =

( ) b

b u

β ; J

− γ

⎛ ⎞

− ( )

λ

Le 1

1 1 T T T

T

= + u

⎜ ⎟ u u

Ma J D

γ γ

⎝ ⎠ ⎛ ⎞ ⎛ ⎞

λ −

2 T T T dT

T

= b

⎜ ⎟ ⎜ ⎟

u b u

ln

D ( )

λ

− −

⎝ ⎠ ⎝ ⎠

T T T T T T

T

u

b u u u

Fundamentals of Flame Stabilities

Effects of Flame Stretch on Flame Speed

Empirical Corrections for Larger Ka

S κ = = +

Faeth et al. , 0

L 1 MaKa

S L

Poinsot et al. ( )

− ∇ ⋅

1 L / S C v

S ( )

κ =

= L , 0 t

L R : radius of curvature

( )

S 1 C L / R

κ =

L , 0

Further ambiguities

- Definition of flame speed

- Definition of flame thickness


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DESCRIZIONE DISPENSA

Materiale didattico per il corso di Combustione del Prof. Mauro Valorani, all'interno del quale sono affrontati i seguenti argomenti: instabilità intrinseca della fiamma; instabilità Saffman - Taylor; analisi di Joulin e Sivashinsky; approssimazione di Darcy; margini di fiamma; effetti dell'aerodinamica sulla propagazione delle fiamme.


DETTAGLI
Esame: Combustione
Corso di laurea: Corso di laurea magistrale in ingegneria aeronautica
SSD:
A.A.: 2007-2008

I contenuti di questa pagina costituiscono rielaborazioni personali del Publisher Atreyu di informazioni apprese con la frequenza delle lezioni di Combustione e studio autonomo di eventuali libri di riferimento in preparazione dell'esame finale o della tesi. Non devono intendersi come materiale ufficiale dell'università La Sapienza - Uniroma1 o del prof Valorani Mauro.

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