arXiv:physics/0212057 v1 16 Dec 2002FlameEnhancementandQuenchingin

Fluid

Flows

Natalia

Vlad

imirova

y,Peter

Constan

tinz;Alex

ander

Kiselev

�;Oleg

RuchayskiyyandLeon

idRyzhikz

yASCI/Flash

Center,

TheUniversity

ofChica

go,Chica

go,IL

60637

zDepa

rtmentofMathem

atics,

TheUniversity

ofChica

go,Chica

go,IL

60637and

�Depa

rtmentofMathem

atics,

University

ofWisco

nsin

,Madiso

n,WI53705

(Dated

:January

28,2003)

Weperform

direct

numerical

simulation

s(D

NS)of

anadvected

scalar�eld

which

di�uses

and

reactsaccord

ingto

anonlinear

reactionlaw

.Theobjective

isto

studyhow

thebulk

burningrate

ofthereaction

isa�ected

byan

imposed

ow

.In

particu

lar,weare

interested

incom

parin

gthe

numerical

results

with

recently

pred

ictedanaly

ticalupper

andlow

erbounds.

Wefocuson

reactionenhancem

entandquenchingphenom

enafor

twoclasses

ofimposed

model

ow

swith

di�eren

tge-

ometries:

perio

dic

shear

ow

andcellu

lar ow

.Weare

prim

arilyinterested

inthefast

advection

regime.

We�ndthat

thebulk

burningrate

vin

ashear

ow

satis�es

v�aU+bwhere

Uisthe

typical

ow

velocity

andaisacon

stantdependingon

therelation

ship

betw

eentheoscillation

length

scaleof

the ow

andlam

inar

frontthick

ness.

For

cellular

ow

,weobtain

v�U

1=4.

Wealso

study

am

eextin

ction(quenching)

foran

ignition

-typereaction

lawandcom

pactly

supported

initial

data

forthescalar

�eld

.We�ndthat

inashear

ow

the am

eof

thesize

Wcan

betypically

quenched

bya ow

with

amplitu

deU��W.Thecon

stant�dependson

thegeom

etryof

the ow

andten

ds

toin�nity

ifthe ow

pro�

lehas

aplateau

largerthan

acritical

size.In

acellu

lar ow

,we�ndthat

theadvection

strength

required

forquenchingisU�W

4ifthecell

sizeissm

allerthan

acritical

value.

PACSnumbers:

PACSnumbers:

47.70.Fw,47.27.Te,82.40.Py

I.

INTRODUCTION

Turbulen

tcombustion

inprem

ixed

ow

sis

awidely

studied

topicinboth

scienti�

candindustrialsettin

gs(see

e.g.[14,

39,42]).

Theinterest

inthesubject

isdueto

anim

porta

ntin uence

thatadvectio

ncanhave

onthe

reactionprocess:

both

experim

ental

[13,47]andtheoret-

ical[3,

4,17,18,23,

24,

33,40,50,

55]work

show

sthat

thepropagation

speed

ofthe am

ecanbesig

ni�can

tlyaltered

bythe uid

ow

.Speci�

cally,moderately

inten

selev

elsofturbulen

cehave

theten

dency

toaccelerate

the

amespeed

vbeyondits

laminarvalu

evÆ .

Themech

a-nism

andtheexten

tofthe am

eaccelera

tiondependon

theparticu

larreg

imeof

burning[12].

Thegen

eralrea-

sonfortheenhancem

entisthatthe uid

motio

ndistorts

the amefro

nt,

increasin

gtherea

ctionarea.

Onthe

other

hand,iftheadvection

istoostro

ng,itcanlead

tothe am

eextin

ction.Thecritical

strength

ofadvection

which

leadsto

quenchingdependsontheexten

tof

the

ame,stren

gth

ofreactio

nanddi�usion

,andprop

ertiesofthe ow

.Atthisstag

e,itisunreason

ableto

expect

acom

plete

analy

ticaltheory

describ

ingtheprocess

ofcom

bustion

ina uid

phase.

Indeed

,detailed

modellin

gofthephenom

-enainvolv

essolvingareactio

n-di�usio

nsystem

involv

ing

tempera

ture

(orenergy

)andconcen

tration

sof

reactants

coupled

with

compressib

leNavier-S

tokes

equation

sde-

scribingmotio

nof

themixture

[39,57].

Therefore,

most

ofthestu

dies

inthis�eld

which

seekanalytical

conclu

-sion

suse

heuristic

reasoningorsim

pli�

edmodels,

which

may

approx

imately

describ

ethesystem

incertain

com-

bustio

nregim

es.Som

eofthecom

bustion

regimes

arerelativ

elywell-u

ndersto

od,such

astheso-called

am

eletregim

e,where

amethick

ness

issm

allcompared

tothe

uidvelo

cityscales.

Thegeom

etricoptics

approx

imation

where

theprop

agationof

thefron

tisruled

byHuygen

sprin

cipleisoften

used

asastartin

gpoin

tin

theanaly

sisof

thisregim

e(see

e.g.[34,

38]).Ourgoal

here

isto

studyoneof

themost

widely

used

PDEmodels

ofcom

bustion

,nam

elythescalar

reaction-

di�usion

equation

with

passive

advection

:

@T@t+u�r

T=�r

2T+

1�R(T):

(1)

Here

Tisthenorm

alizedtem

peratu

re,0�

T�

1,uis

the uid

velo

city,which

weassu

meisincom

pressib

le,�

istherm

aldi�usiv

ity,and�isthetypical

reactiontim

e.In

theabsen

ceof

uid

velo

cityEq.(1)

admits

at

prop

-agation

frontwith

laminar

burningvelo

cityof

theord

erof

vÆ � p

�=�andcharacteristic

thick

ness

oftheord

erof

�p��.Themodel(1)

canbederived

fromamore

complete

system

under

assumption

sof

constan

tdensity

andunity

Lew

isnumber

(theratio

ofmaterial

andtem

-peratu

redi�usiv

ity),as

show

n,for

instan

ce,in

[17].The

equation

(1)has

amore

general

applicab

ilitythan

the

geometrical

optics

approx

imation

;moreover,

aswewill

discu

ssbelow

,thegeom

etricaloptics

limitcan

beob-

tained

from(1)

inacertain

param

eterran

ge.Wewillcon

siderreaction

ratesR(T)oftwotypes,

KPP

(Kolm

ogorov,Petrov

skii,

Pisk

unov)[27,

36],andign

i-tion

.TheKPP

typeischaracterized

bythecon

dition

that

thefunction

R(T)ispositiv

eandcon

vexon

thein-

terval0<

T<

1.Thisreaction

typeisused

oftenin

prob

lemson

population

dynam

ics(see

e.g.[5,

26]),but

isrelevan

tin

combustion

modellin

g,forexam

plein

some

autocataly

cticreaction

s[28].

Areaction

termofign

itiontypeischaracterized

bythepresen

ceof

criticalign

itiontem

peratu

re,such

that

thefunction

R(T)isidentically

2

zero below ignition temperature. This type of reactionterm is used widely to model combustion processes (seee.g. [49, 57]), in particular approximating the behavior ofArrhenius-type chemical reactions which vanish rapidlyas temperature approaches zero.Our main goal is to gain insight into the question of

how the geometry and the amplitude of the uid owin uence the combustion process. Our study is partlymotivated by recent analytical work [6, 19, 20, 29, 35]where rigorous bounds on combustion enhancement andquenching are proved. We test the sharpness of results in[6, 19, 20, 29, 35], and in addition derive new predictions.We consider two classes of ows. The �rst is shear ows,a representative of a wider class of ows, called "perco-lating" in [19], which have open streamlines connectingdistant regions of the uid. The second class is cellular ow, where the streamlines are closed and the ow con-sists of isolated cells. For each class of ows, we studyboth ame enhancement and quenching.For ame enhancement study, we consider initial tem-

perature in the form of the laminar front, with T = 1 inthe semi-in�nite region behind the front and T = 0 in thethe semi-in�nite region ahead of the front. Distorted byimposed ow, the ame front propagates as a travellingwave with velocity higher than laminar. The goal of the ame enhancement study is to obtain relations between ame propagation speed v and the properties of the ow,especially for the large advection velocities.In the case of quenching phenomena, we consider ini-

tial temperature to be non-zero in a �nite region. Sincequenching cannot occur for the KPP-type source term[43, 44], we use the ignition-type reaction term. As shownby Kanel, there is a critical size WÆ of initial hot re-gion below which the ame will be extinguished by di�u-sion alone, e.g. with no advection, when the temperaturedrops below the threshold and reaction ceases before the ame establishes a steady travelling wave con�guration[32]. When advection is present, the uid ow stretchesthe initially hot region so that it can be quenched bydi�usion; hot regions of the size much larger than WÆ

can be quenched in this manner. Our goal has been tounderstand how the geometry and amplitude of the owin uence the size of the band W of the initial hot regionthat can be quenched.

II. NUMERICAL SETUP AND METHOD

The simulation is set in two space dimensions, in a ver-tical strip of width L with periodic boundary conditionsin x direction (Fig. 1). In reaction enhancement studies,the initial temperature was set to T = 1 in the lower halfof the domain and to T = 0 in the upper half of the do-main. In the quenching studies the initial temperaturewas set to T = 1 in a horizontal band of width W inthe center of the domain, and to T = 0 elsewhere. Theinterfaces between hot and cold uid were smoothed tomatch the laminar ame thickness.

We consider two types of ows, sinusoidal shear owwith amplitude U and wavelength L, perpendicular tothe initial temperature front(s),

u = U

�0; cos

2�x

L

�; (2)

and cellular ow with amplitude U and wavelength L,

u = U

�sin

2�x

Lcos

2�y

L; � cos

2�x

Lsin

2�y

L

�: (3)

In the quenching simulations, the size of the cell L=2 wasa fraction of W , so that the initial band always containsinteger number of cells.Most of the reaction enhancement computations were

done using KPP reaction rate [27, 36] in the advection-reaction-di�usion equation (1),

R(T ) =1

4T (1 � T ); (4)

with some of the simulations repeated with ignition typereaction,

R(T ) =TÆ

(1� TÆ)2(1� T ); T > TÆ; (5)

where TÆ represents threshold temperature, below whichR(T ) = 0. In quenching studies we use ignition type re-action (5) with threshold temperature TÆ = 0:5. Bothreaction rates (4) and (5) were chosen to exactly match,in the absence of advection, laminar burning velocityvÆ =

p�=� . The corresponding laminar ame thick-

ness is in both cases of the order of Æ =p�� ; for KPP

reaction it is several times wider than for ignition.Equation (1) with reaction rates (4) and (5) has

been solved using a fourth-order explicit �nite di�erencescheme in space and a third-order Adams-Bashforth in-tegration in time. The grid size was chosen so as to ac-curately represent the shear across the reacting region:typically of the order of 12 zones across the ame inter-face for thin fronts and at least 32 zones per period for

FIG. 1: Schematic representation of initial conditions andvelocity �eld. Problem setup, from left to right: reactionenhancement in shear ow; reaction enhancement in cellular ow; quenching in shear ow; quenching in cellular ow. Darktone corresponds to the cold uid (T = 0), light { to the hot uid (T = 1).

3

thick fronts. The computational domain extended a con-siderable distance upstream and downstream from theburning front so that boundary e�ects were negligible.In ame enhancement simulations the overall grid wasremapped following the propagation of the front, therebyallowing for long integration periods | of the order of1000 reaction times � . We found that these long integra-tions were necessary in order to reproduce correctly theasymptotic behavior of the propagation speed in the caseof strong advection.

As a measure of the reaction enhancement we use thebulk burning rate

v(t) =1

L

Z L

0

Z 1

�1

@T (x; y; t)

@tdydx

=1

�L

Z L

0

Z 1

�1

R(T ) dydx: (6)

The second equality in (6) can be justi�ed by integra-tion by parts. The bulk burning rate coincides with thefront velocity in a case where the solution is a travel-ling wave, but provides a more exible measure of com-bustion. Physically, v(t) can be understood as the totalamount of reacted material or the total heat production.In all simulations done for reaction enhancement, v(t)approaches an asymptotic value (for cellular ows oneshould average in time to arrive at this value) and wedenote this asymptotic value v.

We remark that in shear and cellular ows equation(1) admits travelling wave-type solutions, called pulsat-ing fronts (see e.g. [9, 10, 11, 51, 52]). A rigorous stabilitytheory for these solutions exists but is not complete (see[53] for a recent review). Except when quenching occurs,in our simulations we always observed convergence of thesolution to such waves, so the bounds on v also providebounds for the propagation speeds of pulsating fronts.

In the quenching studies, we measure the total amountof burned material per wavelength,

w(t) =1

L

Z L

0

Z 1

�1

T (x; y) dy dx; (7)

which can also be interpreted as the width of the non-perturbed horizontal band with temperature T = 1 sur-rounded by the uid with T = 0. This quantity is relatedto the bulk burning rate per interface, v(t) = _w(t). De-pending on initial conditions and ow parameters, w(t)either approaches an asymptotic value, e.g. v(t)! 0, orincreases with constant rate, that is v(t) ! v (for cellu-lar ow, in the time-averaged sense). In the �rst case wesay that ame quenches; the main objective of quenchingsimulations is to determine under which conditions thishappens.

III. SHEAR FLOW: REACTION

ENHANCEMENT

We carried out simulations with the sinusoidal shear ow with amplitude ow U , and wavelength, L, given byEq. (2). In this section we are interested only in reac-tion enhancement phenomena, and therefore for initialconditions we consider T = 1 in the semi-in�nite domainy < 0, and T = 0 for y > 0. We carried out computa-tions for both KPP and ignition type reactions, but didnot �nd signi�cant di�erences in the qualitative behav-ior. The numerical results presented in this section areobtained with KPP reaction term (4).Of special interest is the dependence of the e�ective

propagation rate v on the velocity amplitude, U , andwavelength, L, which de�nes the characteristic lengthscale of the ow (Fig. 2). For small amplitudes, U � vÆ,our results are in agreement with the quadratic lawv � vÆ + cU2, which goes back to Clavin and Williams[17] for turbulent ow and has been recently proved rig-orously for shear ows in [29]. We did not study thisregime in detail since our main interest is in the strongadvection case.For the amplitudes U � vÆ the results are in good

agreement with the linear law v = aU + b; where thecoeÆcients a and b � vÆ depend on the geometry of the ow. In the situation where the scale of the ow is muchlarger than the reaction length scale, L � Æ, our dataagrees with v = U + vÆ: This law has been proposed in[6] for shear ows which vary slowly compared to thetypical reaction length and rigorously proved in [20] un-der similar assumptions. For any uid ow, the regimeL � Æ is closely related to the so-called geometrical op-tics combustion regime [42], the limit where reaction time

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 8

L/δ = 16

L/δ = 32

L/δ = 64

v=U+vo

FIG. 2: Bulk burning rate (6) as a function of the shear owamplitude for di�erent shear wavelengths.

4

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

laminar front

T = 0.001T = 0.01T = 0.1

T = 0.9T = 0.99T = 0.999

L/δ = 256

FIG. 3: Isotherms within the front in the geometrical opticslimit. Here L=Æ = 256, and U=vÆ = 4.

and length scales approach zero. In the framework of theequation (1) this corresponds to the limit �; � ! 0 while�=� remains constant. Indeed, by rescaling equation (1)with a factor L=Æ in space and time, we �nd that the bulkburning rate v for the original equation (1) is the sameas for equation with modi�ed di�usion and reaction (8)

Tt + u � rT � �Æ

L�T =

L

�R(T ): (8)

As L grows, equation (8) approaches the geometrical op-tics limit.Quite often, front propagation in the thin front and

fast reaction limit is modelled by Hamilton-Jacobi typeequations. One such model is the G-equation

Gt + u � rG = vÆjrGj; (9)

where the front is de�ned by a constant level surface ofthe scalar G (see e.g. [38]). The G-equation describespropagation of the front according to Huygens principle;that is, the front (i) is transported by uid ow, and (ii)propagates normal to itself with the speed vÆ. The lawv = U + vÆ can also be understood from the point ofview of geometrical optics since it is easily derived fromthe G-equation. Recently Majda and Souganidis pointedout that the G-equation does not provide the geometri-cal optics limit of the reaction di�usion equation (8) ina precise sense [25, 37]. However the rigorous boundsderived for the true e�ective equation still give the sameprediction for v in the case of shear ow [37].In the situation where L becomes comparable to Æ,

the coeÆcient proportionality a between velocity ampli-tude and ame propagation rate is no longer equal to

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

0

2

4

6

8

0 2 4 6 8 10 12 14

v/v o

U/vo

L/δ = 1L/δ = 2L/δ = 4L/δ = 8

FIG. 4: Bulk burning rate in high wavenumber sinusoidalshear ow as function of shear amplitude (points), comparedwith isotherm elongation (solid lines) given by Eq. 10.

unity. The rigorous lower bound for v from [35] takesform v � C1U

1

1+C2n, where n = 2�Æ=L: This bound is in

good qualitative agreement with an argument proposedby Abel, Celani, Verni and Vulpiani [1] based on the ef-fective di�usivity for the shear ow. It is well known that,if the problem is considered on suÆciently large time andlength scales, the e�ect of the advection of passive di�u-sive scalar can often be modelled by e�ective di�usivity[7, 8]. The expression of e�ective di�usivity in a strong

shear ow goes back to Taylor [48], �e� = �+ 1

2

�UvÆm

�2�,

where m = 2�Æ=l and l is the typical length scale of the ow. In the presence of reaction, we take l = min(Æ; L);since the advection balances with reaction instead of dif-fusion if L > Æ. This leads to the qualitative predictionv � U if L � Æ and v � UL=Æ if L . Æ. We obtainedgood although not perfect agreement with this predic-tion. This is not surprising given the heuristic derivationof the expression for the e�ective di�usivity and its pos-sible dependence on more subtle geometric properties ofthe ow.Additional understanding of linear dependence v(U )

can be gained from studying the relationship between theburning enhancement and the structure of the front, inparticular level sets of the temperature. Assume that inthe geometrical optics approximation the front is givenby the function y = f(x); then for the travelling waveobeying Huygens principle and propagating with speedv, we have

v = u(x) + vÆp1 + (f 0)2;

where u(x) is pro�le of the shear, u(x) = U cos 2�xL . In

the case where u is a mean zero ow, this leads to theexpression

v =vÆL

Z L

0

p1 + (f 0)2 dx: (10)

Thus, we obtain a well-known fact that the speed of prop-agation is proportional to the area of the front which in

5

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

L/δ = 32T = 0.001T = 0.01T = 0.1

T = 0.9

T = 0.99

T = 0.999

U/vo = 10U/vo = 12U/vo = 14

FIG. 5: Isotherms within the front for cases with L=Æ =32 and U=vÆ = 10; 12; 14. The individual curves have beenrescaled by a factor of (U=vÆ)�1 in the y direction.

geometrical optics limit coincides with a level set of T(see Fig. 3 for a picture of level sets in a situation closeto geometrical optics). It is interesting to test to whatextent this relationship remains true in the situationswhere geometrical optics regime is no longer valid, forexample when L is comparable to Æ. We found that forlarge U , there is still good agreement between the elon-gation factor of the level sets of temperature and com-bustion enhancement (Fig. 4). Moreover, we found thatfor large U the temperature distribution across the frontscales with the shear amplitude (see Fig. 5), providinganother explanation of the linear dependence of the bulkburning rate on the amplitude of the ow. This scalingbehavior can be understood in terms of the approximateself-similarity of equation (1) with respect to the change

of variables ~y = y=ÆU=vÆ

, ~x = x=Æ, ~t = t=� , which gives

@T

@~t+cos

�2�x

L

�@T

@~y=

@2T

@~x2+�vÆU

�2 @2T@~y2

+R(T ): (11)

The only term which now depends on U is the one pro-portional to the second derivative in ~y; and it becomesnegligible in the limit of large U . Indeed, equation (11)without this term is hypoelliptic, and so addition of thesecond derivative term does not constitute a singular per-turbation. That leads to U -independent propagation rate~v � v

U = ~v(L) and to linear proportionality v / U forlarge U=vÆ.We conclude this section by a remark that understand-

ing of the combustion enhancement in a shear ow ap-pears to be useful in some situations where ows withdi�erent structure are involved. In particular, in the

reactive Boussinesq system the ow consists of vorticesmoving along with the reaction front [54]. However inthe frame moving with the front the e�ect of such vor-tical ow is similar to the shear. The prediction for thereaction enhancement in such system based on the re-sults for the shear ows appears to be in agreement withnumerically observed behavior [54].

IV. CELLULAR FLOW: REACTION

ENHANCEMENT

Cellular ows have been studied by many authors (seee.g. [15, 22]), since similar uid motions appear in manyimportant applications; classical examples are the two-dimensional rolls of the Rayleigh-B�enard problem andTaylor vortices in Couette ow.For cellular ow simulations, we use a velocity �eld

given by Eq. (3). The ow is controlled by two param-eters, velocity amplitude U and wavelength L. As insimulations of reaction enhancement by shear ow, weconsider initial conditions with T = 0 in the upper halfof the computational domain (y < 0), and T = 1 in thelower half (y > 0). Most of the results presented in thissection were obtained with KPP reaction term (4); thein uence of the reaction type on the reaction enhance-ment is discussed at the end of this section.There are several regimes, which can be classi�ed ac-

cording to the relations between the characteristic scalespresent in the problem. There are three characteristictime scales: advection, �U = L=U ; reaction, �R = � ; anddi�usion, �D = L2=�. In this paper, we mostly stud-ied two regimes: �U � �R � �D and �R � �U � �D .The �rst is the regime of strong advection; in the sec-ond regime advection can be very strong as well, butis compensated by large cell size L; so that the reac-tive time scale becomes the fastest in the problem. If�D � �R, or equivalently, L � Æ, we have di�usive orsmall cell regime. The remaining regime corresponds to�R � �D � �U ; and therefore U � vÆ. Hence we haveslow advection; this situation is of less interest to us sincev � U + vÆ under very general conditions [19], so the ef-fect of the advection on the propagation speed is minor.In the regime �R � �U � �D our simulations show

good agreement with geometrical optics models. In Fig-ure 7 we show a typical picture of ame in the regimeclose to geometrical optics. The G-equation, (9), whichis closely related to geometrical optics regime, is invari-ant under simultaneous rescaling of time and space bythe same factor. This suggests that for ames approach-ing the geometrical optics limit, one should observe thissimilarity, and indeed we do as shown in Fig. 7. Theearliest prediction of the front propagation speed in acellular ow for large U within the geometrical opticsframework appears to be due to Shy, Ronney, Buckleyand Yakhot [47]. Using heuristic reasoning, they pro-posed that v � U= log(U=vÆ) if one considers front ad-vancing according to Huygens principle. The same law

6

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

III

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

IIIIV

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

IIIIV

V

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

IIIIV

V

VI

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

IIIIV

V

VI

0

1

2

3

0 1 2 3

U/v

o

L/δ

I

II

IIIIV

V

VI

FIG. 6: Burning regimes in cellular ow: (I) �R < �D <�U | slow advection geometrical optics; (II) �R < �U < �D |fast advection geometrical optics; (III) �U < �R < �D |fast advection with radial burning within cells; (IV) �U <�D < �R | fast advection with uniform burning withincells; (V) �D < �U < �R | limited advection in small cells;(VI) �D < �R < �U | slow advection in small cells.

is proposed in [2] based on more detailed analysis. Toillustrate the origin of this law, let us sketch an argu-ment providing the lower bound for v: Let us look at thepropagation of the ame tip along the path ABCDE onthe Fig. 8. Assuming that at every point of ABCDE the ame velocity is given by u(x; y)+ vÆ; and integrating intime, we obtain a lower bound

v

v� �

4

r�UvÆ

�2� 1

log

UvÆ

+

r�UvÆ

�2� 1

! : (12)

This lower bound, when doubled, gives a very good �tto our numerical data and is represented by a solid linein Fig. 9. As one can see in Fig. 7, the tip of the amefollows the path close to ABCDE, but avoiding corners;this may account as a factor for the di�erence in speedcompared with the lower bound. Our results in the ge-ometrical optics regime agree with results of [2]; how-ever we stress that our simulation was carried out for thereaction-di�usion equation (1) using the same numericalscheme in all regimes, while [2] uses a di�erent model ingeometrical optics limit. We remark that in [1, 2] thepossibility of the regime, where �R � �U and v behavesas U3=4, was proposed. We could not de�nitively con-�rm existence of such a regime due to the closeness of(U=vÆ)3=4 and (U=vÆ)= log(U=vÆ) curves in the range oftested parameters (Fig. 9).It should be emphasized that the geometrical optics

regime requires not only the thin front assumption, L�Æ, but also fast reaction in comparison with advection,�R � �U ; in other words, velocity must be limited byU � (L=Æ) vÆ. When this restriction is broken we observe

FIG. 7: Flame in cellular ow with amplitude U=vÆ = 20and period L=Æ = 1024 (upper row) and L=Æ = 512 (lowerrow). Snapshots for the �rst case were taken with time inter-val 24 Æ=vÆ and for the second - with time interval 12 Æ=vÆ.

signi�cant decrease in ame propagation speed comparedto the geometrical optics prediction (Fig. 9). Figure 10further illustrates this point, showing that on a logarith-mic scale, there is a marked change in the slope of (v=vÆ)as a function of U as �R=�U increases. When �U exceeds�R, we observe power-law, v � U1=4, as proposed by Au-doly, Berestycki and Pomeau [6], and con�rmed in [2] in anarrower range of parameters. The measurement of theslope 1=4 was suÆciently precise to distinguish it fromthe v � U1=5 behavior, a lower bound rigorously provedin [35]. The observed v � U1=4 scaling extends to thelimit of cell sizes small in comparison with the laminarthickness, L � Æ. We remark that the laminar frontthickness for the KPP reaction is of the order of 16 Æwhich is large compared to smallest cell size L=2 = 4 Æshown in Fig. 10; limited data available for L=Æ = 4; 2(shown in Fig. 11) also con�rms the v � U1=4 scaling.In the very small cell regime, the v � U1=4 scaling wasrigorously proven in [29] using homogenization approach.

We also studied the dependence of v on the cell sizewhile U=vÆ is �xed. Figure 12 shows changes in the struc-ture of the ame with the increase of the cell size | froma more di�usive front to a front approaching geometricaloptics behavior. The ame propagation speed, normal-ized by (U=vÆ)

1=4 factor, is presented in Fig. 11. As L=Æ

7

B C

DE

A

FIG. 8: Approximate path of the tip of the ame

increases, we see the transition from the �U � �R regime,to the geometrical optics, where ame propagation speedis independent of cell size. For �U � �R the data collapseto a single curve, suggesting the power scaling with L=Æ,with power changing from 1=4 for small L=Æ to 3=4 forlarge L=Æ. The resulting scaling can be summarized as,

v=vÆ � (U=vÆ) = log(U=vÆ); �R � �U � �D; (13)

v=vÆ � (U=vÆ)1=4 (L=Æ)3=4; �U � �R � �D : (14)

v=vÆ � (U=vÆ)1=4 (L=Æ)1=4; �U � �D � �R; (15)

To explain observed the ame propagation speed, letus consider a model based on the e�ective di�usivity,proposed by Audoly, Berestycki and Pomeau [6]. Whenvelocity is high, �U � �R, the sharp temperature gra-dients appear in the narrow boundary layer at cell bor-ders (Fig. 13). The thickness of the boundary layer, h,is determined by the balance of di�usion and advection,�=h2 = U=L, and is much smaller than Æ,

h �p�L=U: (16)

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024L/δ = 512

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024L/δ = 512L/δ = 256

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024L/δ = 512L/δ = 256

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024L/δ = 512L/δ = 256

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024L/δ = 512L/δ = 256

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

v/v o

U/vo

L/δ = 1024L/δ = 512L/δ = 256

FIG. 9: Flame propagation velocity in thin ame regime asfunction of the cellular ow amplitude. Solid line is doubledEq.(12), dashed line is v=vÆ = (U=vÆ)

3=4.

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

L/δ = 1024

L/δ = 512

L/δ = 256

L/δ = 128

L/δ = 64

L/δ = 32

L/δ = 16

L/δ = 8

L/δ = 2048 (prediction)

U1/4

1

2

4

8

16

32

64

128

256

1/256 1/64 1/16 1/4 1 4 16 64

v/v o

τR/τU = (U/vo) / (L/δ)

L/δ = 1024

L/δ = 512

L/δ = 256

L/δ = 128

L/δ = 64

L/δ = 32

L/δ = 16

L/δ = 8

L/δ = 2048 (prediction)

U1/4

FIG. 10: Flame propagation velocity as function of the ratioof laminar burning time to the vortex turnover time. TheL=Æ = 2048 prediction is based on doubled Eq. (12) for geo-metrical optics.

(This argument is a concise, naive version of the con-siderations appearing in the derivation of the e�ectivedi�usivity in cellular ow, [16, 45, 46, 56].)A discrete di�usion equation modelling the original

equation (1) has been suggested in [1, 2]:

@�n@t

=�e�L2

[�n�1 � 2�n + �n+1] +1

�R(�n): (17)

Here �n is the average temperature in the n-th cell; and�e� is e�ective di�usivity

�e� = �L=h: (18)

This leads to the propagation rate,

v =p�e�=� � vÆ

pL=h;

and, taking into account Eq. (16), to the scaling (15). Wefound that the prediction for speed given by (17) agreeswith numerical simulations only if L . Æ:

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

1

2

4

8

16

32

2 4 8 16 32 64 128 256 512 1024

(v/v

o) /

(U/v

o)1/

4

L/δ

U/vo = 512

U/vo = 256

U/vo = 128

U/vo = 64

U/vo = 32

U/vo = 16

U/vo = 8

L1/4

L3/4

FIG. 11: Dependence of the ame propagation velocity onthe size of the vortex.

8

FIG. 12: Flame in a cellular ow with amplitude U=vÆ = 100and period L=Æ = 16; 64; 256 (left to right)

For large cell sizes, L� Æ, the di�usive model (17) nolonger accounts fully for the ame propagation process.The main objection to the model is that (17) assumestemperature is uniform inside the cell, and the reactionterm can be estimated at the average cell temperature.However the numerically observed behavior demonstratesat �rst sharp temperature gradients at the border of thecell, later evolving into ame propagation inside of thecell roughly at the laminar ame speed (see Fig. 13 forthe structure of the front inside cells). Indeed, the cel-lular ow has no eÆcient mechanism for mixing betweenthe streamlines, and the di�usion time scale in that di-rection is of the order L2=� (practically not enhanced)[41]. Therefore the combustion process inside the celltakes time of the order of L=vÆ, rather then Æ=vÆ (whichcorresponds to substituting the average temperature intothe reaction term).

Here, we will modify the e�ective di�usivity model (17)

to account for slower burning inside large cells. As in thecase of small cells, the ame propagation from one cellto another is enhanced because of the high temperaturegradient in the boundary layer with width given by ex-pression (16). The heat coming to the cell through thecell boundary is distributed on the scale of Æ (as opposedto L, in the case of small L). We further notice thatdue to the fast advection, the temperature is essentiallyequal along the streamlines inside the cell (see Fig. 12),which allows the ame to propagate directly from oneboundary layer to another (Fig. 13). That allows us towrite the discrete di�usion equation similar to (17), butreplacing averaging in the cell by the averaging in thestrip of width Æ along the cell border (Æ-layer),

@�n@t

=�e�Æ2

[�n�1 � 2�n + �n+1] +1

�R(�n): (19)

Here �n is the average of the temperature in the Æ-layerof the n-th cell; and �e� is e�ective di�usivity,

�e� = � Æ=h: (20)

The temperature in a laminar front varies on the scaleÆ; and so estimating reaction in a Æ-layer by R(�n) isjusti�ed. Equation (19) does not take into account theheat ux from the Æ-layer to the bulk of the cell. How-ever, since h � Æ; this heat ux does not enter themain balance. In other words, the front described by(19) propagates entirely in the Æ-layers, with the rate

vÆ =p�e�=� � vÆ

pÆ=h: Substituting h from Eq. (16)

we obtain,

vÆ � vÆ (U=vÆ)1=4(L=Æ)�1=4: (21)

The time needed to ignite a new cell, e.g. to warm upa layer of the size of the order Æ in that cell, is equal to�Æ = Æ=vÆ: Once the width of a warmed up layer reachessize of the order of Æ, reaction becomes capable of sustain-ing the temperature. Further propagation of the amecorresponds to the basically laminar front movement in-side the cell, and takes time �cell � L=vÆ.The total bulk burning rate v is of the order of vÆ times

the number of burning cells, which can be estimated asthe ratio of the cell burning time, �cell � L=vÆ, to thetime needed to ignite a cell, �Æ = Æ=vÆ. Therefore thebulk burning rate is obtained by multiplying the numberof burning cells with vÆ; or essentially by normalizing vÆwith a factor of L=Æ,

v � vÆ�cell�Æ

� L

ÆvÆ � vÆ (U=vÆ)

1=4(L=Æ)3=4;

which agrees with numerically observed scaling (14).Finally, we would like to mention the e�ect of the re-

action rate. Similar to the shear ow, we found that thereaction type does not in uence asymptotic scaling lawslike (15) or (14), although there is certainly a di�erencein the constant factors. However, in cellular ows there isan interesting phenomenon which is present for ignition-type but not KPP reactions. The dependence v(U ) is

9

0

0.2

0.4

0.6

0.8

1

-64 -32 0 32 64

T

y/δ

0

0.2

0.4

0.6

0.8

1

-64 -32 0 32 64

T

y/δ

0

0.2

0.4

0.6

0.8

1

-64 -32 0 32 64

T

y/δ

0

0.2

0.4

0.6

0.8

1

-4 0 4

T

y/δ

0

0.2

0.4

0.6

0.8

1

-4 0 4

T

y/δ

0

0.2

0.4

0.6

0.8

1

-4 0 4

T

y/δ

0

0.2

0.4

0.6

0.8

1

-1024 -768 -512 -256 0 256 512 768 1024

T

y/δ

0

0.2

0.4

0.6

0.8

1

-1024 -768 -512 -256 0 256 512 768 1024

T

y/δ

0

0.2

0.4

0.6

0.8

1

-1024 -768 -512 -256 0 256 512 768 1024

T

y/δ

0

0.2

0.4

0.6

0.8

1

128 256 384

T

y/δ

0

0.2

0.4

0.6

0.8

1

128 256 384

T

y/δ

0

0.2

0.4

0.6

0.8

1

128 256 384

T

y/δ

FIG. 13: Temperature pro�le at the middle of the cell (solid) and temperature averaged in x-direction (dashed) for L=Æ =8; U=vÆ = 64 (left) and for L=Æ = 256; U=vÆ = 128 (right). Bottom plots are blow up versions of top plots. Dotted line in thetop plots represents laminar front stretched by factor v=vÆ in y-direction, while in the bottom plots it represents non-modi�edlaminar front.

always monotone increasing in the KPP case, but it mayexhibit a temporary reversal for ignition-type reactions(Fig. 14). This e�ect has been discovered by Kagan andSivashinsky [30] and further studied in [31]. We foundthat this phenomenon is more pronounced when the re-action threshold TÆ is closer to unity, in agreement withthe arguments of [31].

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

KPP

ignition T0 = 3/4

ignition T0 = 1/2

ignition T0 = 1/4

U1/4

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

KPP

ignition T0 = 3/4

ignition T0 = 1/2

ignition T0 = 1/4

U1/4

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

KPP

ignition T0 = 3/4

ignition T0 = 1/2

ignition T0 = 1/4

U1/4

1

2

4

1 2 4 8 16 32 64 128 256 512

v/v o

U/vo

KPP

ignition T0 = 3/4

ignition T0 = 1/2

ignition T0 = 1/4

U1/4

FIG. 14: Flame propagation velocity for di�erent reactions(L=Æ = 8).

V. QUENCHING

In this section we address another e�ect that advec-tion can have on the combustion process | quenching.We say that reaction is quenched if the average tempera-ture goes to zero uniformly with time. Quenching occursin the systems with ignition type reaction when, due todi�usion and advection, temperature drops below the ig-nition threshold everywhere and the integrated reactionrate becomes identically zero.If the size of the region is small enough quenching

can be caused by di�usion alone, e.g. without advec-tion, as shown by Kanel [32]. Kanel considered theone-dimensional reaction-di�usion equation Tt � �Txx =��1R(T ). He found that there exist two critical sizesWÆ � W �

Æ such that if the initial size of the hot region(where T = 1) is smaller than WÆ; reaction quenches,while if the initial size of the hot region is greater thanW �

Æ , two fronts form and propagate in opposite direc-tions. Two di�erent critical sizes are likely an arti-fact of the proof; in our simulations, we always foundWÆ = W �

Æ � Æ and will refer to the single critical size WÆ:In two and three dimensions, the presence of advectionmay lead to stretching of the initial hot spot, thus mak-ing di�usion more eÆcient at cooling, and consequently,at quenching.In our numerical simulations we study quenching un-

der the in uence of advection, in particular in shear andcellular ows. As in previous sections, the prescribed ow velocities are de�ned by Eq. (2) for shear ow andby Eq. (3) for cellular ow. For all simulations we used

10

FIG. 15: Sequence of snapshot of temperature distribution in the shear ow with L=Æ = 4 (top) and in the cellular owwith L=Æ = 4 (bottom). Initial condition was a hot band of the width W=Æ = 6 and W=Æ = 4 for shear and cellular owsimulations respectively. Velocities amplitudes are below critical on the left and above critical on the right (shear: U=vÆ = 13and U=vÆ = 14; cellular: U=vÆ = 600 and U=vÆ = 800). The time is given in units of � .

11

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

w(t

) / δ

t vo/δ

U/vo=15

U/vo=16

U/vo=17

U/vo=18

U/vo=19

U/vo=20

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

w(t

) / δ

t vo/δ

U/vo=15

U/vo=16

U/vo=17

U/vo=18

U/vo=19

U/vo=20

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

w(t

) / δ

t vo/δ

U/vo=15

U/vo=16

U/vo=17

U/vo=18

U/vo=19

U/vo=20

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

w(t

) / δ

t vo/δ

U/vo=15

U/vo=16

U/vo=17

U/vo=18

U/vo=19

U/vo=20

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

w(t

) / δ

t vo/δ

U/vo=15

U/vo=16

U/vo=17

U/vo=18

U/vo=19

U/vo=20

0

20

40

60

80

100

120

140

0 2 4 6 8 10 12

w(t

) / δ

t vo/δ

U/vo=15

U/vo=16

U/vo=17

U/vo=18

U/vo=19

U/vo=20

FIG. 16: Temperature integrated over area 0 < x < L,�1 < y < 1 per wavelength L for di�erent values of shearamplitude U , measured for L=Æ = 6 and W=Æ = 6.

the ignition-type reaction (5) with threshold TÆ = 1=2,since quenching cannot occur for the KPP-type sourceterm [43, 44]. As the initial conditions we use a hori-zontal band of width W with temperature above critical(T = 1) within the band and with temperature belowcritical (T = 0) outside the band.

The typical evolution of the system described above isshown in Fig. 15. We found that the temperature distri-bution in both for shear and cellular ows evolves accord-ing to one of the two possible scenarios, depending on theamplitude of the advection velocity. For lower advectionvelocities, after an initial transient period, the system de-velops solution characterized by a wide, steadily growingburned region between two wrinkled fronts propagatingin opposite directions. These fronts are exactly as de-scribed in the preceding sections with regard to structure,speed, and dependence on the ow properties U and L.For higher advection velocities, the temperature eventu-ally drops below TÆ everywhere, after which no burningoccurs. We denote by Ucr the value of advection veloc-ity which triggers the system between these two scenarios(further we refer to them as burning and quenching), andestablish the relation between Ucr and the initial condi-tions and structure of the ow.

We measure the critical value of velocity amplitudeUcr(L;W ) using the following procedure. For each com-bination of the initial hot band size W and velocity pe-riod L, we execute a number of simulations for di�erentvelocity amplitudes U . For each simulation we measuredthe total amount of burned material per period w(t), de-�ned by Eq. (7), as a function of time (shown in Fig. 16).The burning systems (with higher velocities, where twofront are formed) are characterized in Fig. 16 by constant,non-zero slopes, corresponding to constant reaction rate.These rates are independent of initial conditions, andare equal to double the burning rate v(U;L), since thereare two fronts. The quenched systems are characterizedby evolving to constant w(t). Both formation of steady

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

0

20

40

60

80

100

0 2 4 6 8 10

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

L/δ = 6

L/δ = 8

FIG. 17: The critical amplitude of shear ows with di�erentwavelengths as function of initial width of hot band.

fronts and quenched solutions requires some transitiontime.The summary of results for a sinusoidal shear ow is

given in Fig. 17 and Fig. 18. We found that Ucr scaleslinearly with W (see Fig. 17),

Ucr = �W

�; (22)

as predicted in [20], with the coeÆcient � strongly de-pendent on the wavelength of the advection velocity(Fig. 18). Shear ow is most e�ective at quenching inthe intermediate range of wavelengths, namely when L isof the order of a few reaction lengths Æ. The quenchingmechanism for small and for large L is di�erent; one hasto distinguish between the ability of the ow to stretchthe front over the larger scales and to make the initialhot band uniformly thin.For small L, the rapid spatial variation of the ow

velocity is well approximated by e�ective di�usion. Thee�ective di�usivity for strong shear ow scales as �e� �U2l2=�, where l = min(Æ; L). The characteristic length

0

5

10

15

20

25

0 5 10 15 20

α

L/δ

α = 9 (L/δ)-1 α = (0.25 L/δ)2

0

5

10

15

20

25

0 5 10 15 20

α

L/δ

α = 9 (L/δ)-1 α = (0.25 L/δ)2

0

5

10

15

20

25

0 5 10 15 20

α

L/δ

α = 9 (L/δ)-1 α = (0.25 L/δ)2

FIG. 18: Dependence of the factor � in Eq. (22) on the wave-length of shear ow L. Measurements were taken at W=Æ = 6

12

0

100

200

300

400

0 1 2 3 4

Ucr

/vo

d

FIG. 19: Growth in the value of Ucr for a �xed W as the sizeof at plateau increases.

scale for reaction with this renormalized di�usion behavesas le� � p�e�� and scales linearly with U . Then le� � W

gives Ucr � ÆLW��1 for small L. We remark that in the

limit of small L; the e�ective di�usivity argument can bejusti�ed by a rigorous homogenization procedure [29].For large L, the nature of quenching is related to the

appearance of the (almost) constant regions in the veloc-ity pro�le. We observe the behavior Ucr � L2 for largeL; which can be explained in the following way. In orderfor quenching to occur, the shear ow should stretch theinitial hot region thinner than Kanel's critical length (ofthe order Æ) in time less than the reaction time � , so thatreaction does not have time to compensate cooling byadvection. The stretching is least eÆcient near the tipof the velocity pro�le. At the tip of a sinusoidal pro�le,the di�erence between ow velocities at two points sepa-rated by a distance Æ is U (Æ=L)2. Therefore we obtain a

suÆcient condition for quenching Ucr

�ÆL

�2� � W , which

leads to

Ucr � ��1W

�L

Æ

�2

:

We also examined a degenerate case of shear ows witha plateau in the velocity pro�le. For such ows it hasbeen shown in [20] that quenching does not happen assoon as the size of plateau is larger than certain criticalsize of the order Æ and the size of initial bandwidth Wexceeds WÆ: As expected, we found that Ucr diverged toin�nity as the size of the plateau approached a criticalvalue (Fig. 19), in agreement with results of [20]. Thisphenomenon can be understood in terms of reaction anddi�usion alone: in the region where the pro�le of the ve-locity is at, there is no stretching of the initial hot band.If the size of the hot band is roughly larger than Kanel'scritical sizeWÆ, then reaction can compete with di�usion,there will be no quenching, and eventually propagatingfronts will form.Quenching in cellular ow requires signi�cantly higher

advection amplitudes. For relatively small cell sizes

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

10

100

1000

2 3 4

Ucr

/ v o

W/δ

L/δ = 1

L/δ = 2

L/δ = 4

FIG. 20: The value of Ucr for cellular ows with di�erentperiods.

L . Æ where quenching is possible, we �nd that the crit-ical velocity Ucr satis�es Ucr � W 4 (see Figure 20). No-tice that this correlates with the dependence v � U1=4 forthe speed of advection-enhanced ame. This correlationis not coincidental, and can be explained as follows. Thespeed up law indicates that the size of the region wherereaction happens in a stabilized combustion regime scalesas (U=v0)1=4(L=Æ)1=4Æ for large U: If the width of the ini-tial band of hot material is of the order smaller than(U=vÆ)1=4(L=Æ)1=4Æ, advection carries away the energy ofthe hot material over the larger region faster than reac-tion is able to compensate the falling temperature. Thisleads to quenching.We remark that quenching is impossible if the cell size

is suÆciently large, L & Æ, and W & Æ. The reason issimilar to the at plateau e�ect in the shear ow. Fluidadvection does not provide mixing inside cells in the di-rection perpendicular to the streamlines, and thus if thecell is large enough reaction can sustain itself against dif-fusion. This result has been proved in [21].

VI. CONCLUSIONS

We carried DNS calculations of an advected scalarwhich reacts according to a nonlinear reaction law. Westudied combustion enhancement and quenching phe-nomena in two typical classes of ows, shear and cellular.In a shear ow, we �nd linear dependence v = aU + b ofthe combustion speed v on the amplitude of the ow Uin the strong ow regime. The factor a depends on therelationship between the period of the ow L and typicalreaction length scale Æ; is equal to 1 if L� Æ and tends tozero if L=Æ ! 0: The observed behavior is in agreementwith recent rigorous [19] and numerical [1] results. In acellular ow we studied primarily two regimes character-ized by the relationships �D < �U < �R and �D < �R <�U between di�usion, reaction and advection time scales.We found that combustion speed in the �rst regime is

13

close to predictions of models based on geometrical op-tics limit, v � U= log(U=vÆ): In the second regime wherelarge U dominates, we found v � vÆ(U=vÆ)

1=4(L=Æ)3=4:This agrees with the prediction of the e�ective di�usionmodel [1, 2, 6] in terms of the power of U but has di�erentdependence the cell size L: We proposed an explanationof the observed behavior with a modi�ed e�ective di�u-sion model where enhanced di�usivity is concentrated inthe boundary layers.As opposed to combustion enhancement, quenching

may happen if the reaction term is of ignition type andinitial temperature is higher than critical in a �nite re-gion. If the shear ow velocity pro�le does not have aplateau of suÆciently large size, or the size of the cells incellular ow is not too large, then for any initial hot bandsizeW there exists Ucr(W ) such that for U > Ucr quench-ing takes place. If U < Ucr, two fronts form and propa-gate with the speed of the developed advection-enhanced

front. In the case of shear ow, Ucr depends linearly onW with a factor �(L)=� . Quenching is most eÆcient forthe ows with L on the order of a few typical reactionlengths Æ. For the cellular ows, Ucr scales as W

4. Theresults are in good agreement with theoretical arguments.

VII. ACKNOWLEDGEMENTS

This research is supported in part by the ASCI Flashcenter at the University of Chicago under DOE contractB341495. PC was supported partially by NSF DMS-0202531. AK has been supported by NSF grants DMS-0102554 and DMS-0129470 and Alfred P. Sloan Fellow-ship. LR was supported partially by NSF grant DMS-0203537 and by an Alfred P. Sloan Research Fellowship.

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