Premixed flames in large scale and high intensity turbulent flow

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Premixed flames in large scale and high intensity

turbulent flow

P. Clavin, G. Joulin

To cite this version:

P. Clavin, G. Joulin. Premixed flames in large scale and high intensity turbulent flow. Journal de

Physique Lettres, Edp sciences, 1983, 44 (1), pp.1-12. �10.1051/jphyslet:019830044010100�.

�jpa-00232137�

L-1

Premixed flames

in

_large

scale and

_{high intensity}

turbulent

flow

P. Clavin and G. Joulin

_(*)

Laboratoire de

_Dynamique

et

_{Thermophysique}

des Fluides _(**),

Université de Provence, Centre de _St-Jérôme, 13397 Marseille Cedex _13, France

(Reçu le 11 mai 1982-, revise le 1er octobre, _accepté le 10 novembre ₁₉₈₂₎

Résumé. 2014 La théorie récente de Clavin et Williams _[1] sur les fronts de flammes

_{prémelangées}

dans

les écoulements

_inhomogènes

et instationnaires est

_{généralisée}

au cas où

_{l’amplitude}

de

_déplacement

du front est _grande. Les effets convectifs associés à

_{l’expansion}

du _gaz sont

_pleinement

_pris en _compte dans l’étude de la structure locale du front

_plissé.

On montre

_qu’un

seul scalaire mesurant l’étirement du front contrôle localement la forme et la

_dynamique

de la flamme. Ce scalaire _peut se

_décomposer

en deux termes

_{représentant respectivement}

la contribution de la

_géométrie

de la flamme _(courbure du front avançant avec une vitesse normale

_prescrite)

et celle de

_{l’inhomogénéité}

de l’écoulement

caractérisée _{par le} tenseur du taux des déformations.

Abstract. 2014

The recent

_theory

of Clavin and Williams _[1]

_concerning

the

_premixed

flame fronts in

non uniform as well as

_unsteady

flows is extended to the case of

_{large amplitudes}

of the front

corru-gation.

The convective effects associated with the gas expansion are

_fully

taken into account in the

study

of the flame structure.

_Only

one scalar « the flame stretch » is shown to control the

_shape

and the motion of the front This stretch can be

_split

in two parts, one accounts for the geometry

of the front _{(mean curvature)} and the other for the non

_uniformity

of the flow _(rate of strain _tensor).

LE

JOURNAL

DE

_{PHYSIQUE LETTRBS}

J. _Physique - LETTRES 44 (1983) L-1 - L-12 fer JANVIER 1983, ] Classification Physics Abstracts 47.70 1. Introduction. -

When the flame thickness d

(’"

10- 2

cm at the normal

_conditions)

is smaller than the characteristic

_length

scale A of the _{upstream gas}

_flow,

e =

d/~i

1,

the

dyna-mical

_properties

of a

_premixed

flame are controlled

_by

the

_coupling

of two kinds of mechanisms which can be studied

_{separately [1, 2] :}

a)

Inside the flame thickness the

_temperature

increases from

_{T u}

to

_Tb

and the convective and

diffusive transport

processes of mass and energy control the local structure of the wrinkled front. This structure links the flame

_speed

to the

_shape

of the front and to the local characteristics of the

upstream

gas flow.

b)

But,

in

_addition,

_{hydrodynamical}

effects are

_developed

_upstream

and downstream of the front. When the Mach number is

sufficiently

small

the

_density

_p can be considered to be constant

in the _gas flow outside the flame

_thickness,

but with a different value in the unburnt and burnt

gases,

Pu =1=

pb. This difference

produces

the deflection of the streamlines

through

the tilted front.

(*) Permanent address : Laboratoire

_{d’energetique}

et

_detonique,

L.A. 193 du C.N.R.S., rue

Guillaume-le-Troubadour, 86034 Poitiers Cedex, France.

(**) L.A. no 72 au C.N.R.S.

According

to the

_{incompressible}

fluid

_mechanics,

the flow is modified on a distance of the same order as the wrinkles of the front.

Thus,

the

_{hydrodynamical}

effects make the

_incoming

flow

_depend

on the motion and on the

shape

of the

_{front, but,}

in turn, due to the flame _structure, the flame motion

_depends

on the local characteristics of the flow. The first

study

of this feedback mechanism was carried out

_forty

_years

ago

by

Darrieus

[3]

in 1938 and

by

Landau

[4]

in 1944 but in a crude

_{approximation :}

the structure

of the wrinkled flame was assumed not to be modified from the

_{steady planar}

case. This leads to a

strong

instability

of the front : the flow

_produced

_by

the

_wrinkling

leads to a convective motion

of the front which

_amplifies

the initial wrinkles.

_But,

_clearly,

the overlooked diffusive effects

developed

in the transverse dilation must

_modify

this result at least for short

_{wave-lengths.}

The first

_study

of the wrinkled flame structure was carried out

_twenty

_years later

_by

Barenblatt et al.

_[5]

but in the

_{simplifying approximation}

of the thermal-diffusive model where the convective effects

produced by

the _{difference between Pu and Pb} were

_neglected.

Much

_deeper

_insights

have been

provided by

more recent

_analysis

such as that

presented

in reference

[6]

in which the

_asymptotic

analysis

in

_large

value of the _{activation energy}

_[7]

has been used. But it is

_only

_very

_recently

that a

complete analysis

of the flame structure for small

_amplitudes

of the front

_corrugation

has been carried out

_{[1] by using}

a multiscale

_analysis

based on the

_disparity

of the two scales d and A. The convective effects

_neglected

in

_[5]

and in

_[6]

are

_proved

in

_{[1] to}

be of the same order of

_magnitude

as the diffusion effects but are

_{always stabilizing.}

_Thus,

_according

to the results of Pelce and Clavin

_{[2] concerning}

the limits of

_stability

of

_planar

_fronts,

it turns out that the acceleration of

gravity

and the diffusive effects can overcome the

_{hydrodynamical instability}

mechanism of Darrieus

_[3]

and Landau

_{[4] provided}

that the flame

_propagates

downwards with a slow

_enough

speed.

We _present in this _paper a natural extension of the

study [1]

of the wrinkled flame structure to

the nonlinear case for

_large

_amplitudes

of the front

_corrugation.

But the dimension of the wrinkles is also assumed to be

_large

_compared

to the flame thickness d

and,

as in

_{[ 1 ],}

8 =

~/A ~

1 is used

as a

_perturbation

_parameter

of the

_analysis.

_Furthermore,

the

_{asymptotic expansion}

in

_large

values of the reduced activation is used as in

_[1]

and

_[6].

The

_analysis

does not involve any

assump-tion

_concerning

the

_magnitude

of the _gas

_expansion,

thus,

the convective effects

_{produced by}

the deflection of the streamlines are

_properly

taken into account in the flame structure. The nonlinear

equation

for the local evolution of the front is obtained in terms of the

_upstream

_gas flow as it is modified

_by

the flame. The modification of the flow

_by

the

_{hydrodynamical}

effects described in

b)

can be

_{analytically investigated}

in the linear

_{approximation [2].}

_But,

in the

_general

_case, the

corres-ponding

fluid mechanical

_problem

is non linear and is left to be one of the most difficult in flame

theory.

This

_problem

is not

addressed

here where

_only

the local structure of the flame is

investi-gated

for turbulent flows of

_high

_intensity.

In

_fact,

the

_present

_{study provides only}

the

_boundary

conditions on the flame

front,

considered as a surface of

_{discontinuity moving}

in the turbulent flow.

Moreover,

when the flow

_{inhomogeneities}

are too

_strong,

the front is

_expected

to

_possibly

experience

local

_{quenching [8].}

This mechanism which

_requires

the

_study

of

_higher

orders in the

expansion

in powers of e is not addressed in the _present

_study

which can be used

_only

for

describ-ing

the

dynamical

behaviour of the flamelets far from the

_quenching

limits. The results obtained here are also useful for

_describing

the

_shape

of the front in laminar flows when the

_corrugations

have

_{large amplitude}

as, for

_example,

in the

_tip

of a Bunsen flame or

during

the

propagation

in

tubes.

The results obtained in this paper are _very

_general;

it turns out that

_only

one scalar controls the local

_dynamics

and the

_shape

of the front i.e. the stretch of the surface

_{experienced by}

the front

during

its motion in the _gas flow. This

_quantity

is

_proved

to control the normal

_component

of the front

_velocity

relative to the flow. The relation between these two

_quantities

is linear for small values of s and the

corresponding

coefficient

depends

on the detailed kinetics of the chemical reaction

_sustaining

the heat release. Its

expression

is obtained here for a one

_step

overall chemical

reaction far from the stoichiometric

_composition

and can be

_generalized

to more realistic cases. The role of the flame stretch is more

_easily

understood in the case of a

planar

flame stabilized in a

stagnation

point

flow

_represented

in

_figure

1. The heat

_leaving

the reaction zone

_by

diffusion for

preheating

the

_incoming

_gas is

_equal

to the heat released

_by

the chemical reaction. The rate of this heat

_production

is

_only

controlled

_by

the _temperature of combustion

_resulting

of the

_enthalpy

balance. Inside the

_preheated

zone, the reaction rate can be

_neglected

and the convection balances the diffusion. Due to the flame

stretch,

transverse convection is added to the convection

_produced

in the normal direction

_by

the flame

_speed

both of which take out the heat

_{entering by}

diffusion from the reaction zone.

_Thus,

for a

_given

_temperature

of

combustion,

the flame

_speed

is

expected

to be lowered

_by

a

_positive

stretch.

But,

the overall effect of flame

_stretching

is more difficult

to be

_anticipated

because the

_temperature

of combustion

_Tb

_depends

on the difference between the heat

_diffusivity

and the molecular

_diffusivity.

Fig.

1. - Structure of

a stretched

_planar

flame stabilized in a

_{stagnation point}

flow.

In fact the

_concept

of flame stretch was first introduced a

_long

time _ago, in

1953,

by

Karlo-witz

_[8]

in a

phenomenological description

of the flame

_quenching

in burners. But ten _years

later,

Markstein

[9],

following

the

_{semi-analytical theory}

of Eckhauss

_[10],

assumed that the relevant

_{quantity controlling}

the flame

_speed

was the curvature of the front relative to the «

cur-vature » of the flow _defined as the

_divergence

of the unit flow

_velocity

vector. The

_present

_analysis

clarifies this

_aspect

of the

_question

and reconciles the two

_points

of view of Karlowitz and

Mark-stein. It is shown that the total stretch

_appearing

in the final result can in

fact,

be

expressed,

in the limit e -+

_0,

as the difference between the mean curvature of the front and a scalar built on the « rate of strain tensor » of the

_upstream

flow and on the unit vector

_{perpendicular}

to the front. This scalar differs from that introduced in the

_{phenomenological approach}

of

Markstein ;

it measures in fact the contribution of the flow

_{inhomogeneity}

to the total stretch of the front. As for the curvature of the

front,

it can be

_interpreted

as

_measuring

the stretch

_{produced by}

the natural motion of the front under a constant normal

_{velocity equal}

to the laminar flame

_speed.

Furthermore,

the time

_averaging

of the non linear

_{equation obtained}

for the local evolution of the front

_provides

informations

_concerning

the

_burning

rate of a wrinkled flame stabilized in a turbulent flow. It is found

_that,

even in the

_{high intensity}

_case, the turbulent flame

_speed

is

_given

by

the laminar

_speed

times the _average ratio of the wrinkled flame area to the cross sectional area.

The

_problem

will be formulated in the

_following

section. The

_analysis

_appears in the

subse-quent section and can be omitted in a first

reading.

The reader is referred to references

_[2]

and

_[6]

for a detailed

_presentation

of the two

_techniques

used herein i.e. the

_asymptotic

_expansion

for

large

values of the activation energy and the multiscale method based on e =

d/~1 ~

1.

Never-theless this paper is self contained and all

important

steps

are

presented

here even if

_they

can

be found elsewhere. The results are

presented

and discussed in the last section. 2. Formulation. - The flame model is the

same as in the

_previous

work

[1, 2].

Attention is restricted to exothermic reactions amenable to a

_one-step

_{approximation,}

with the

_degree

of progress describable in terms of the mass fraction

_~A

of a

_single

_{reactant, taken} to be a

limiting

or deficient reactant in the sense that

_{t/J A}

= 0 defines

completion

of the reaction. An Arrhenius

rate is

_adopted

such that as the reduced activation _{energy goes} to

_infinity,

the

_chemistry

is confined

to a

_fluctuating

_surface,

termed the reaction zone whose

_position

is

_employed

to define the

_origin

of a

_moving

coordinate

_system

[6].

With the

density

p

variable,

the mass based thermal

diffusivity

pDth

is assumed to be constant for

_simplicity.

The effect of the

_temperature

_dependence

of

_pDth

have been studied in

_{[ 11 ]}

for small

_amplitudes

of the front

_corrugations

and can

_easily

be

anti-cipated

afterwards in the case of

high amplitudes.

The flame thickness d =

pDth/ Pu

UL and the

transit

_{time tL}

=

d/UL

_are the units of

length

and time

respectively.

UL is the laminar flame

speed

and the

_subscripts

u or - oo refer to the conditions in the fresh mixture. Let

_UT

=

(UT,

0, 0)

and

_{!.f. - (XiX,}

_Y,

_Z,

_T)

_(U

_V

_{W - (0)}

be the uniform and non-uniform

_parts

of the

upstream

flow

_velocity

in the nondimensional form

_{respectively.}

_U-~

is assumed to _vary on nondimensional time and

_length

scales of order

_{1/e; (X,}

Y,

Zg 7) =

(sx,

Ey, sz, st) where x, y,

z, t are the nondimensional coordinates in the

_laboratory

frame. The

_{approximation}

s ~ 1

means that the _space scale of the

_upstream

flow is

_larger

than the flame thickness

_{(d ~}

10-2

_cm).

The

_scaling

of the time is then

_{given by}

the

_Taylor

_hypothesis.

The turbulent flame

speed

UT

is assumed to be of the same order of

_magnitude

as the laminar one _UL,

_UT

=

0( 1 )

in the limit

e-~0.

_But,

in contrast with

_[I],

where the small

_amplitudes

case was studied

(U -00

=

0(8)),

we are

looking

here to the case of

high

turbulent

intensity

when

_{U - 00}

=

0( 1 ).

Let the location of

the flame sheet be defined

_by

x =

0153(y,

z,

t)

in the nondimensional variable. The order of

magni-tude of the

_amplitude

a of the front

_corrugations

can

_easily

be evaluated

_{by assuming}

that the normal

_{burning velocity}

_Un is

_{only weakly}

modified

_by

the

_wrinkling,

_{(Un/UL) -}

1 =

0(E)

such

a _way that one must have

_{3(x/~ - U-}

This

_yields

(I.

= A(Y,

_Z,

_{T) with A}

=

0(1)

in the

Y

/

z Y

E

( ~

~~

( )

limit E -+ 0. The

_moving

coordinates x -

a, n =

~, 5

= z, T = t are introduced.

UT

+

_{i!.- oo(~ ~ ~ ~)}

is the flow

_velocity

outside the flame thickness in the fresh mixture. But due to the heat

_release,

the flow

_velocity

must be modified inside the flame thickness on a

length

scale d.

_{Let U}

=

(U,

V,

W)

be the

_components

of the

velocity

in the

(x,

y,

z)

coordinate

system.

Let r =

p/Pu

be the

density

ratio,

V

=

(V, W)

_represent

the transverse

velocity,

V and

V

denote the

_{gradient involving}

differentiation with

_respect

to x, y, z and

E~,

E yy, E~

respectively.

the

_moving

coordinate _system

_(ç,

_r~,

_{~ ),}

the conservation

equation

for _mass,

_(2),

and for

longitu-dinal and transverse _momentum,

_{(3), (4),}

read

P _{is the nondimensional pressure reduced}

_by

_p.

_uL.

The viscous forces have not been written for

_simplicity

_because,

as it was shown in the

_previous

works for small

_{amplitude corrugations}

[1,2,11],

it is

_anticipated

that the overall effect is neutral in such a _way that the Prandtl numbers

does not _appear in the final result for the

_{dynamical properties}

of the flame fronts.

In the

_{approximation}

of

_vanishingly

small Mach

_number,

the

_{density r}

is related to the

tem-perature

through

the

_perfect

_gas law

_by

0 =

(T - T,~)I(Tf - Tu)

and

y = 1 -

(pf/pj,

(0

0,

y

1),

are the nondimensional

tempe-rature and the

expansion

parameter

respectively.

The

_subscript

f refers to the conditions in the

burned _gases of the

_planar

and adiabatic case. The

_temperature

0 and the nondimensional mass

fraction ~ (~

= 1 in the fresh

mixture)

of the

limiting

_component

obey

the

equations

of _energy

and

_species

conservation. These two

_equations

are

_{coupled by}

a

_production

term with a

tempe-rature

_{dependence given by}

an Arrhenius law and characterized

_by

an activation _{energy E.} When

~

the reduced activation _energy

_{j8 ==}

_{y y}

y-y-

_goes to

_infinity,

the

_production

term is confined to the

kT

f

g y p

surface ~

= 0 and _can be

replaced by

the

following jump

conditions

resulting

of the

quasi steady

and

_planar

character of the reaction zone

_{[ 1, 2, 6] :}

(7)

express the conservation of the

enthalpy through

the reaction zone and

₍₆₎

states that the heat diffusion flux

_leaving

the reaction zone

_along

the normal direction toward the

_preheated

zone is balanced

_by

the chemical

_production

of heat released inside the reaction zone. These relations are valid in the

_{asymptotic limit ~i -~}

oo i.e. when the thickness of the reaction zone shrinks to zero, up to the order

_{1/#, [6]. (6)}

shows that the relative difference between the

tempe-rature of combustion

_Tb

and the adiabatic flame

_temperature

_Tf

must be of order

_{1 /~3}

in the limit

~i

-~ oo,

9 -

1

₌

0(!//?).

In order to

_satisfy

this

_condition,

the Lewis number

_Le

=

DthlDmob

~=0

which is the ratio of the thermal

_diffusivity

and of the molecular

_diffusivity,

must be close to

unity [6], Le -

1 =

0(1 /fl),

When the viscous effects are

_overlooked,

the

_jump

conditions

_concerning

the flow field are

When written in the

_moving

coordinate

_system,

these

_equations

take the

_following

form :

with the

_following

_{boundary conditions, ~-~2013oo:0==0,~=l;~>0:~=0;~-~2013oo:}

0 = 0. A similar

equation (10’)

holds for 0 but where

_Le

is

_{replaced by}

1.

_{(10) emphasizes}

the

order of

_magnitude

in the limit 8 -+ 0 when the time derivative and the _space variation in the

transverse direction are

_anticipated

to be of the same order of

_magnitude

as for the _upstream flow

_{U _ ~.}

The

_system

₍₂₎

₍₃₎

_{(4) (5)}

₍₁₀₎

with the

_jump

conditions

₍₆₎

_{(7) (9)}

is to be solved in

pertur-bation of

s in order to obtain the

_equation

for the evolution of the flame front. The two

_parameters

of

_expansion

_{jS " ~}

and s are

_independent

one comes from the

_chemistry

and the other from the limited _{range of the} wave

_lengths

under

investigation.

The

asymptotic limit

-+ ₊ oo is taken before the limit e -~ 0.

3.

_Analysis.

-

By

identifying

the contributions of the

_upstream

flow

_(referred

_by

the

_subscript

-

oo)

and of the flame thickness

_(referred

_{by )}

_{respectively,}

the solutions of the

_equations

_(2),

(3)

and

₍₄₎

can be written

_{for ~}

0 in the

_following

form :

Noticed that all the

_quantities

with A vary as 0

and ~

on the

length

scale d in the

longitudinal

direction and must _go to zero

_{when ~ -~ 2013}

oo

_(upstream

_boundary

of the flame

_thickness).

Although

the

_{hydrodynamical}

effects described

_{in b ) of § 1,}

are known to make

_{U _ ~}

to be a

functional of the

_position

of the front

A,

_{_U _ ~}

is considered here as a

given quantity

and

only

the flow modification

_{_tl}

inside the flame is to be determined in the

_study

of the flame structure.

According

to

₍₅₎

one has lim r = 1 and the definition

(1)

provide

the

_{expression S- 00 :}

~-~ - 00

in such a _way that the normal

_{burning velocity}

can be

_expressed

as :

This

_quantity

is in fact the main unknown of the

problem

and may be considered as an «

eigen-quantity » determining

the flame motion.

_{S - 00}

will be determined at the dominant order in the limit

_~3

-+ oo _{and up} to

₀₍₆₎

in the limit s -+ 0.

Thus,

the

_{following expansions}

are introduced

At the zeroth

order,

so,

(2)

implies

aSo/a~

= 0 and then

So

= 0.

(10) yields

for ~

0 :

Then the

_jump

condition

_{(7) implies}

_00(~

=

0)

= 1 ₊

0(1/J?)

and

(6) yields :

Recalling (12),

the result

₍₁₅₎

means

_that

at the dominant

_order,

_M~/~L

=

1,

the normal

burning

velocity

is not modified. This is a trivial result

because,

in the limit s -+

_0,

the effect of the

wrinkling

must

_{effectively disappear.}

The

_following

non trivial order will

_give

the first correction. From

₍₁₄₎

and

₍₅₎

one obtains the

_expression

for _’0 and

_{then, (1),}

₍₃₎

and

_{(4) provide}

the

value’of

_{00, Po}

,nd f/o

(16) corresponds

to the usual deflection of the streamlines

_through

the front

_produced

by

the

modification of the normal

_component

of the relative

velocity.

Then,

a

straightforward integration

of the mass balance

_{(2) provides}

S 1 (ç)

to

_give

In order to obtain the modification of the normal

_{burning velocity given by}

_{S - 1,}

it is neces-sary to

_compute

the modification of the

_temperature

of combustion

_{0 1 (~}

=

0).

This _can be

carried out

_by

_integrating

from ~ == 2013 00 to ç

= 0 the order _e of the _sum

of (10)

(10’)

and

₍₂₎

times

(0

+ ~ - 1)

to

_give

-where

_{{ }}

is the same bracket as in

_(17).

The derivation of

₍₁₈₎

has

_required

the use of the

_jump

condition

₍₇₎

with

a8/a~

_1.:=0+

=

O(e2)

and the fact

that,

_as

(Le - 1),

0(~

=

0)

is of

order ~ ~ 1

in the

_{limit ~}

-+ oo. Moreover one can

_compute

the modification of the heat flux

_entering

the

combustion zone

_{by integrating}

_{from ~ = 2013}

oo

_{to ~}

= 0 the difference between

(10’)

times

(1/1 - y)

and

₍₂₎

times

r~-1

= 1 ₊

0(y/l - y)

_to

give

_at the orders

where the

_expansion

in power of e has been used for

_{S _ ~; (A}

+

_6~).

The final result is obtained

_by

The bracket

_{{ }o}

is the same as in

₍₁₇₎

and

_(C/d)

is a non-dimensional coefficient

_{given by}

C as defined

_by

_(21),

is a

length

that we will call « Markstein

_length

_», in

_fact,

such a

length

was introduced for the linear case in the

_{phenomenological approach}

of Markstein

_{[9]. According}

to

₍₁₂₎

and

_(12’),

the result

_(20),

when

_expressed

in terms of the normal

_burning

_speed

_un, takes the

_following

form :

4. Results and discussion. - Turbulent

jlame speed.

-

When the

_{expression ( 12)-( 12’)}

of the -normal

burning velocity

Un introduced in

(22),

the final result

provides

a local

equation

for the

evolution of the

front,

x =

a( y,

z, t) which takes the

following

dimensionalized form :

with

_{(u_ ~,}

_{~-oo) =}

_{(UL U_ ~,}

I

uL V _ ~)

denoting

the non uniform

part

of the

upstream

flow

field.

-

-When the random process is

stationary

as well as

_homogeneous

in transverse directions on time average, the time average of local modifications to structure of laminar flamelets is zero and do not influence the turbulent flame

_speed.

This result

_generalizes

the one obtained for small

amplitudes

of the front

_{corrugation [1]. Only}

the effect of an area increase associated with front

wrinkling

appears; the turbulent flame

speed

is

equal

to the laminar

speed

times the mean area increase :

As it can be

_anticipated

from

_{[ 1 ],}

this result is limited

only

to the two first

orders 8°

and

e1

of the

_expansion

_{in powers} of e.

Flame stretch. -

Following

Karlowitz

[8],

one _may seek to

_interprete

the final result in terms

of flame stretch. In dimensionalized

_form,

₍₂₂₎

reads :

The stretch can be defined

precisely

as the time derivative of the

logarithm

of an

_elementary

area y of the flame front the

_points

of which move in the flame surface with a

_{tangential velocity}

equal

to the

_tangential

_component

of the gas

velocity immediately

ahead of the surface

[17].

The

in the flame surface. The mathematical

_expression

of such a stretch has been

_{given by}

Buck-master

_{[ 12] :}

with

it is worthwhile

_emphasizing

that the local

_quantity

defined

_{by (25)}

is

_{representative}

of the total stretch

_experienced

_by

a surface

_moving

in the flowfield

_(u,

_t~)

with a motion characterized

by

a normal

_velocity

relative to the _gas

_equal

to _un. This total stretch is not

_only

a characteristic of the flow

_{(u, v)}

(

but

_depends

also on _Un. It is clear from

₍₂₅₎

that

~J~"r~

=

0(s),

thus the

_)

p n

( )

L Q

dt

tot

(

dominant order in the limit e -+ 0 of the total stretch is obtained

_{by introducing}

in

₍₂₅₎

the

domi-nant order of _un

_{given by (15),}

_un =

uL(1

+

_O(e)).

_By

_comparison

with

₍₂₄₎

we obtain

Thus,

as

_{anticipated by}

Karlowitz

_{[8], (26)}

_expresses

_that,

at the dominant non trivial order in _e, the total stretch is the

_only

scalar

_controlling

the normal

_{burning velocity.}

This result is not

in

_complete

_agreement with the conclusion of the

_{analysis by}

Buckmaster

_[12]

who

_questioned

the usefulness of a criterion based

_only

on the stretch defined

_by

_(25).

In

_fact,

_following

the

_previous

work of

_{Sivashinsky [13],}

Buckmaster

_[12]

used the « slow

_varying

flame model » defined

by

#

-~

_{oo, 1 2013}

_Le

=

0( 1 )

and _e =

1 /~i.

In this

way instead

of (25),

the « volumial » stretch is found

to be the scalar

_controlling

_un in reference

[12].

This

_quantity

is in fact

_expressed

as the sum of

1 -T-)

and the time derivative of the

_logarithm

of the local flame thickness.

_Furthermore,

B 7

_B

dt / _’0’ g ’

the

_expression

obtained in reference

_{[ 12]}

for the coefficient

C/d

contains

only

the second

_part

of

the

_complete

formula

₍₂₁₎

the first

_part

of which has been

_{proved recently}

to be essential in the

stability analysis

of

_planar

fronts

_{[ 1, 2].}

As was shown in reference

_{[ 14], the joint}

use of 1-

_Le

=

0(1)

and of c =

1/#

lead,

_even in the

planar

case, to time derivative terms

_which,

like the additional

terms

_appearing

in reference

_[12],

are not relevant. As it has been shown in the

_{stability analysis}

of the thermal diffusive model

_(see

for

_example

ref

_[6])

the domain of

_stability

is characterized

byl - Le

_Y =

0(1/~) and thus, the assumption 1/L~=

01 with a

=

0(p)

leads to results

including

e

( I

~)

~ p

I

e

( )

_at

(~)

g

unsteady

effects associated with

_planar

fronts instabilities too for _away from the

_stability

limits for

_concerning

usual

_premixed

flames.

By using

the two

_independent

_limits

-+ oo and e -+

0,

the

_present

analytical study

not

_only

confirms the

_importance

of the flame stretch

_concept

_{anticipated previously by}

_{phenomenological}

considerations but also

_provides

the

_precise

formulation of its influence on

dynamics

of the flame front. For usual values of the gas

expansion (y

>

_0.8)

and of the diffusivities involved in the tradi-tional

hydrocarbon-air

mixtures,

(21) predicts,

in contrast to the thermal diffusive

approxima-tion characterized

_by

y =

0,

_a

positive

value

of C/d

for

any

composition

with a

_{possible exception}

for

_hydrogen

flames

_[2].

Thus,

in _agreement with the

_experimental

_data,

a

_positive

stretch is

predicted

to

_produce

a decrease of the normal

_{burning velocity}

_Un. But the value of

_C/d

_depends

on the kinetics of the reaction and is also sensitive to the heat losses

_[15].

For each

_particular

case,

the

_expression

of

_r.,fd

can be

_easily

obtained

_{by using}

the linear

_{approximation [11,}

_15].

The formula

₍₂₁₎

is valid

_only

for adiabatic flames amenable to a _one-step kinetics

_{approximation}

with mass based

thermal

diffusivities assumed to be constant. But the results

₍₂₃₎

and

₍₂₆₎

are

general

and can be used for _every

_premixed

flames. Front curvature. - It is worthwhile

noticing

that the

_{early semi-analytical}

linear

_theory

of

Eckhauss

_[10]

showed that the flame

_velocity

is influenced

_by

the

_joint

effects of the front

_shape

and of the flow

_{inhomogeneities.}

An extension to the nonlinear case has been

_proposed

_by

Mark-stein

_[9]

who assumed that the relevant

_quantity

could be the difference of the mean curvature

of the front

CR

= -~- + -5-

defined to be

_positive

when the front is convex toward the burnt

B~

~1

i

~2

z

mixture)

and of the « curvature of the

_upstream

flow » defined as the

_divergence

of a unit vector

in the direction of the flow. Such an

_expression

matches

_effectively

the result

(24)

in the linear

approximation,

in this case,

l~R ~ 2013 V2(X

and -

_V.~..~

is the linear

_{approximation}

of the curvature of the flow

as defined

_by

Markstein. .

In fact the exact result

₍₂₄₎

can

_easily

be

_expressed

in terms of

_{1 /R}

to

_{give :}

where n is the unit vector normal to the front and

_Vu

is the « rate of strain tensor » of the

upstream

flow,

_u _ ~ =

(u _ ~,

v

evaluated _at the flame

position.

The second _term of the

r.h.s. of

₍₂₈₎

is

different

from the

_quantity

_{proposed by}

Markstein and can receive a _very

simple

physical interpretation.

The

_quantity

_{V.M-~ 2013}

_n.Vu.~.n

_represents

the relative

_growth

rate,

_1 _d d of an

infinitesimal area y of the surface

_{perpendicular}

to n and convected

_by

the

_{flow!!. - x>’}

a~ Tt)

p ~ Y _ «.

Thus,

as

_{y.!! - 00 =}

_0,

₍₂₈₎

can be written as :

{2013 da

is the stretch that

_the

front would have

_experienced

if it was

_only

convected

_by

the

o~ dt . p Y Y

B

co v

’

upstream

flow _{u- 00.} Thus the two different forms

₍₂₆₎

and

₍₂₉₎

of the same final result can be

easily

Y understood

_{by noticing}

y g that -

uL

is the

stretc 1

da

that

the front would have

expe-R

a

_dt

rienced if it moved in a

_upstream

quiescent

medium with a normal

burning velocity equal

to _{uL :}

I ~ - I

~ ~

I ~

(!. dt = (.!. at

+ (l

at

.

~

dt tot a

d~ ~

dt

B tot B n B conv

The main interest of the forms

₍₂₈₎

and

₍₂₉₎

is to

_separate

in the total

stretch,

the contribution of the

_geometry

of the front from the contribution of the

_{inhomogeneities}

of the

_upstream

flow. Burned _gases. - It is worthwhile to underline that we have chosen in this paper to _express

It is _easy,

_{by using}

_{(9) (16)}

and

₍₁₇₎

to _express the result in term of the downstream flow u+~., =

(u+ ~, v + ~)

just

behind the reaction zone :

-with

with

_M~

=

1

_UL and

uT

_denoting

the

_velocity

of the burned _gas in the laminar and turbulent 1-y

cases

_{respectively.}

A similar

_relationship

was deduced for the

_spherical

_geometry

_by

Frankel and

_{Sivashinsky [16]}

who noticed that

_contrarily

to the value

_{(21 ) of C/d}

obtained

_by

Clavin and Williams

[ 1 ],

Cb/d

given by (31 )

can

_{easily change sign}

for the usual

_{hydrocarbon-air}

flames. When

expressed

in terms of the normal

_{burning velocity}

but relative to the burned _gases,

_un

=

{

uT

₊

M+ooL " a«lat -

~+ooL-Y~ }

(1

+

I Vrx,12)-1/2

the result

(30)

can be written in a similar form

to

_{(26) :}

--Thus a

_positive

flame stretch can

_{produce only}

a decrease of the normal

_{burning velocity}

relative to the fresh mixture but can

_produce

an increase of the normal

_{burning velocity}

relative

to the burned gases. This is

possible

because the modification of the flame structure

_by

the front

wrinkling

involves a

_change

of the normal mass

_flux,

_though

the flame thickness. This result is very

spectacular

in

_spherical

flame

_propagation

in

_quiescent

medium studied in reference

_[16];

in this case, the

corresponding

modification is due to

_unsteady

effects. Each of the two forms

(30)-(32)

and

_(25)-(27)

can be useful

_depending

on the

_{configuration.}

For

_converging

_spherical

_flames,

the fresh mixture is at _{rest, u _ ~} =

0,

and

(23)-(27) provide directly

the front evolution. For

diverging

flames when the

burned

gases are at rest,

(30)-(32) give directly

the solution. In these

cases, the difference between C

and Cb

describes the difference in the

_propagation

between a

converging

and a

diverging

flame. In the

spherical steady

case, one

has 2013==(20132013)

and

~

_a

dt _conv

thus,

’

1 -j-)

= 0. The two forms

₍₂₇₎

and

₍₃₂₎

show that there is no modification of the normal

dt

( )

(

)

’ot

burning

velocity

in this

_[18]

case.

_When,

as for idealized Bunsen

flames,

the

_upstream

flow can be considered as

_{uniform u}

_{= 0,}

₍₂₃₎

_{gives directly}

the

_shape

of the front. But in most of the cases,

as for the

_propagation

in tubes or for the turbulent wrinkled

_flames,

_u and _u are

_unknown;

then the

_complete

solution of the

_{problem requires}

the solution of the

_{incompressible}

fluid mecha-nics

_problem

ahead and behind the front. The relations

₍₂₃₎

and

₍₃₀₎

have to be considered as

two _{of the boundaries conditions necessary} to solve this difficult

_problem.

The other

_jump

References

[1] CLAVIN, P. and WILLIAMS, F. A., J. Fluid. Mech. 116 ₍₁₉₈₂₎ 251.

[2] PELCE, P. and CLAVIN, P., J. Fluid. Mech. 124 (1982) 219.

[3] DARRIEUS, G., 6th International _Congress

_{of Appl.}

Mech. _(Paris) 1945.

[4] LANDAU, L., Acta

_Physicochim.

URSS 19 ₍₁₉₄₄₎ 17.

[5] BARENBLATT, G. I., ZELDOVICH, Y. and ISTRATOV, A. G., Prikl. Mekh. Tekh. Fiz. 2 (1962) 21.

[6] JOULIN, G. and _{CLAVIN, P.,} J. Comb. and Flame 35 (1979) 139.

[7]

BUSH, W. B. and FENDELL, F. E., Comb. Sci. Technol. 1 ₍₁₉₇₀₎ 421.

[8]

KARLOWITZ, B. et al., IV Symposium on Combustion (1953) 613.

[9]

MARKSTEIN, G. H., Non

_{steady flame}

_{propagation (Pergamon Press)} 1964.

[10]

ECKHAUSS, W., J. Fluid. Mech. 10 (1961) 80.

[11]

CLAVIN, P. and GARCIA, P., J. Méc., to _{appear in January} 1983.

[12]

BUCKMASTER, J., Acta Astronautica 6

₍₁₉₇₉₎

741.

[13]

SIVASHINSKY, G. I., Acta Astronautica 3 ₍₁₉₇₆₎ 889.

[14]

JOULIN, G., Existence, stabilité et structuration

_{des flammes prémélangées,}

_Thèse, Poitiers _(1979).

[15] CLAVIN, P. and NICOLI, C., Comb. and Flame, submitted _(1982).

[16]

FRANKEL, M. L. and SIVASHINSKY, G. _I., J. Comb. Sci. Tech. submitted _(1982).

[17]

WILLIAMS, F. A.,

Agard Conference

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