Theoretical And Numerical Combustion

November 2019
PDF

Download

This document was uploaded by user and they confirmed that they have the permission to share it. If you are author or own the copyright of this book, please report to us by using this DMCA report form. Report DMCA

Overview

Download & View Theoretical And Numerical Combustion as PDF for free.

More details

Words: 233,433
Pages: 487

Preview
Full text

! " #! $ % & '( ! ) ! *! (

+ ( *! . ) & 0. & * ! ! 1 ) & . ! ( 1 ! 1 ( 1 2! & " 3 ) & , *# !

"

, / / , /

- 4 & 5) & * ! ) ! " " % ) " * ! & " ') & 3 " ') & , , % , & , &

& %# 1

& 5 ) . 1 ! & )# & 7 & %! ! 1 8.* & 7) & & 7) & ) & 7) ! & $ ! . & %! ! ! *! . & %! . ! ! $ ! . & %! ! ! 9 & & % ') ! ') ! & : ( ! ! *! &

" " 6 6 " " " "" "" ",/

- - - -" --6 -6 6 6 6 6" 6" 66 66

/ /" /6 "

') ; " : & " 0. " 7) &

&

!

+ & '! +& 0. ) & 1 < & :2=* 2## ( $ # ( :! 2 % 2 # #> & &# . ! " " 1 =* " = " * ! , ;# ! , ;'* % , ( , $ &) # , * % , ; # !

" # !

5 # 1 & > # 1 & 1 %! & 5) # 2 % Æ!> %# ¼ ) # 7 % # :2=* ) & 5) # . ! 1 ? . @ 2

" -

"

" 6 / "

" "/ "/ " " ", "-

, , , ,, - - -/ / /

1 '! 8 $ A'8$B 7 & 1 8! 7 ;! A87;B " ! , 5 ! ! A B - 7 # ¼¼ C¼¼ 6 7 # & / D # .5 .5 ! 3 ;'* ) & + ') :2=* 2%! & .( ! ;'* " * &) # ;'* =* ) & 1 =* =* ! * & # =* ) ! *!# # & =* " 5 & , =* ! # !

$ # !

" + " 5 # " 1! & > < & " *% ) & " 1 ) & 3 " 2 ) ) & 3 " 1 ) # " E ) =* " * ) " # " :2=* ) & " 2 # ( " 7 %! ! " 7) " 7 %! ! " 7 % !

/ /" /, / " 6 / 6 6 6 " " ",

%$"6 ,/ ,/ ,/ , ,-/ - -, 6/ 6 6 6" 6// /

"" 7 % ! ;'* ) & " ; '! 7 " +%! ! " ! "" =* ) & "" * & "" 2 # ) & "" * # &

, + , F & , 5 # , * ! &E , # ) ! &E , E & , =* &E , =* & & 3 , +& &E , +& &E

"

&!'!!

&!'

- + - 2 .# & - ( - 5 - G # - ; # - ; # -" ; #E # -, ; # -- 2 -6 1 G 3 -/ 2 #! ! &) - 2 # & - 2 ( A B # & - 2 ( 7. # & - 2 ! . # & - 2 < & - ; # -" 1 #! # &

/ " , ,

, / / , 6

""

" " "/ " " " ", ,/ , , ," ," ,, ,,6 - -

-

6 + 6 ! 6 1 ! 6 :# =.* ( ! 6 1 ; 0 + A;0+B 6 1 =* 8 #! ' ( 6 1 =* 8 #! =.* ( 6" '# 6 ') 6 2 & %) A*+.B 6 2 .&# & A*+.B 6 * .&# & A8 8B 6 2 &# & A8B 6 2 . A=*HB 6" 2 A2*HB 6 2 ! .# & 6 2 ! # & 6" $ ! & 6" 5! 6" 4 )E ! 6, 2 :! &

-

- * - + # & - 1 A B & - % - 4 ;# ! - + - ;'* # ! - ;'* - ;'* .) !

% ( )

+

-" -" -, -6 6/ 6 // // // / /"

*%

/6 , 6 6 " , 6 " " /

"

1 ) ( # % 1 )! # & ! 2 I ( .

# 1 # ! >

I ? .. @ ! ! ! # ! 1 # 5 3 # $ !I #! 3 ) .)

H ! ) # !

* I Æ ! # %

& ! + #I # # & 2 & #! = .# & ) # # & > & I # !I ! Æ # ( 1 ) ! #! (

1 ! )

AD I ; 4 'I H I I ; H I 8 # I 5 B . A0 8 B

+ I

> ## & I ! # . H # & +

! A B ! :

I 2 I G I 0 8 3# 5J 1 ) (

)

)

1 &# I & # A 0 8 B 1 ! (

A !B 1 ) ! > 1 # & I I ! % : # %I I # ! ! # # !

1 ) #3 >

% ( # & % = * (. . # & 1 3# *% Æ # & > I % . & I ! )

) & + ! ! #% 7 ! ! 1! # ! # I

# 3 I ! # ! ') % ) & I & & #

& %

& > ) 2 ! . ?@ & . #! &

! '! &E 2## %# 2 % A:2=*> :! 2# = * I ;'*> ;# '! * I =*> = * B & #

) & 2 . 3# & I :2=*I ;'* =* + 1 ( ! A ! =* :2=* ;'* B 3

" & 1 & ). ) & # !

)

! % 7 ! 1 #! & % :2=* ) % : % ;'* =*

, &E 1 ! 2 ! =* 7 # :2=* * & #! % ! ! I # &

-

! # 1 # ! #%! !

8 # & )# ! # & :

# # & #

;# '! * &

6 ( ! ! # & 7 ( A;'* =*B A -B ( ! ! < # 1 # ! ! & !

1 I

# # # I (! ) ! % * I . !

,- #.) /01/#

+ 7J( $7: =:*E+=5E$5* / + = 5 !( 1 E '=*''+G1

,- , 234050#3 ; '7 $5: =:* -' 5

1 ( # & ## . ( =.* ( # 1 ( I #! !

! H I D 1 (

# # 1 =.* ( ! . . # ! ( I 3 # K # ) 1 % !> K

AB

# # 1 . # & >

! K I ! % I

#! A ! BI

#

# # & ( # L D # # A # / ! BI % #% # & > (

#.. ) ) # I >

K

K

AB

I K - A B # I K ! ! # * ! . # >

K

AB

AB

( > K

# ) # !>

K

A B

# & I #! ! I 1 # % #! A BI ! A BI #! A B ! A B 1 ! M + I ! # K / # A:! 5 B ## ) / D Æ ! K 6- + I ! A #!B 1 ! # ! > K

L M

A"B

! 3 K A # ! K A K M B B I # % ! K L 1 ( > ! 3 K #! > A B K + I # ( > )I M # . K 6- 1 M A # )

B !> M K M

A,B

)I ! ", 9E# ! M ,- 9E 1 # M M ! )

*

7 # A#E B

7 ! M A9E#B

", "/ -6 -6

//" // / // //// /// //-

/ / /

7 ! M A9E B

,//- 6 / / /

!"

1 A B ! K

# > K

K

A-B

+ I # # # & ! # ! A* 5 I G!

B # A ! # B H I ! ! # 1 # ! # I ! H # ! A/// /// 9EA#DBB # A "/// 9EA#DBB 1 !>

K

A6B

) I 1 3 #! ! A# I B 1 ! % !>

K

K

L M

K

L

M

A/B

#! K # !I # '( ABI A B A6B>

K

K

LM

K

L

L M

M K

AB

7 #! A B ! A B # & I ! )I >

K

K

AB

'( AB ) ! A1B A B + ! # #%! G I ! E & I # !

Scaled molar heat capacity

7

CO2

6 H2O CO 5

4

Ideal diatomic gas

H2 N2 500

1000

1500

2000

2500

Temperature (K)

# # # 3000

Mass heat capacity

2500

H2O

2000

CO

CO2 1500

N2 1000 500

1000

1500

2000

2500

Temperature (K)

$ %&'( # #

"

)

)

* +

) ! # I # ! /// // 9EA# DB # // D /// D ! A B I ! ) I % >

K

K

AB

!> K K

2

1 ! K 1 % !> K

!

Æ L

!

L

AB

!

! ! 1 ! " K 1 ! 3 AD B Æ D ! > Æ K K # I / 4 $ % !> $ K

Æ

K Æ

Æ L !

L !

!

A B

,

6!! . 1 Æ % 1 Æ ) A I B & ! Æ A& B ( ! ## 1 A' ##B> # . # *% A ! B < ! ) 1 & Æ 3 ; % !> ' K

%

K

&

& &

A"B

& K % A B ! Æ 1 ; ' * & ' (! I # I # !

& ! # AG BI % # #! I & ' ## ! ! & 1 5 I (I > ( K

" "

K K % A B % %

A,B

* A # B A &) # B # A' ##I ##B

.! 7

! # # ) >

¼ "

* +

¼¼

"

# K , )

A-B

¼ ! I " " Æ # 7 > ¼¼

¼ " K

¼¼

"

" K /

# K , )

A6B

¼¼

" K "

"

¼

A/B

-

!I ! I -N -N ! ) >

-N K

-N K

"

-N K "

AB

# # *# -N # '( A6BI >

-N K

"

K/

# 1 # # >

K O

AB

¼

O

¼¼

AB

# K 1 # 1! ! # 2 > K .

)

K .

)

AB

') # # # ) . I ) / A (! #! K B 8 ## I !# # # # ! ) + # & I

A * B # . 2 ) . ! 7 # 1 G'7D+= AD B I I # . # I / E 1 # ( >

K

6

)

M0

M

A B

K 1 M ! # # #

# > M M0 ! ! ! # # 1 (

, $

# * *

# &

2 I I > ) I )! ! A9 B ! & I ! ! ! Æ $ ! Æ! > # ! ! (

/

# # & 5 #! A ! E !B> ) Æ # A!! "/ E B 2 # & ! ! # ( ! & ! A B

1.) ' # ! &I 3 # 1 )3 ! #% ( ! 3 & 1 ! % ( A# B # %# A) )B

Stream 1: Fuel

Fuel + Air

Stream 2: oxidizer (a) Premixed flame

(b) Diffusion flame

-. / 0 1 + ) A# BI )3 ) ! ¼ + "¼ " Æ # )3 # ( ! ¼ "¼ 1 L "

(234

A"B

A ) L L BI )3 >

K

¼ " K4 ¼ "

A,B

1 4 1 ( # )

> 5K4

K

+ > 5K4

A-B

N N

A6B

N N ! & )3 1 ( ) # > . 5 6 A ) B # 5 7 A )3 ) B 7 ! + ! E & I # I !! ," # )!# * !I > K

L

4 5

L ,"

A/B

1 ! 4 # ) A5 K B

2

3 45 " 3 45 " 3 45 3 45

4 " 3 6 ! !5 ! 6 3 45

6 77 3 53 3 "! 77

7 7"" 7 757 7 75 7 7

! * + !37 4 ! ! A1 B ) # # ! > # . # ) #%! ! # I 1 ! .

) & ! !

1.) &I )3 ! # A B & !

# ! A )3 )3 BI % % ( > 5K4

AB

1 3 & A )3B G I # ( 5 > 5 K 4

N N

AB

N N & % )3 ! 5 5 !> 5 K 5

N N

AB

N N & # ! ) I )3 ! N K N 5 K 5 ) ! I ( ! #%!

1 ( # # & >

3

L

!

K

L !

L !

8 K

$ L !

8

A B

8 # # ' # ( ) I & % ! > ! ! #! # >- >/ ! # # & ! 2 (I :! # & > # (

! !""

' ' ' (

#!$%&

I # & ! 2 ! ) < > # < ! ! # A; 4 ' B

1 ( A"B # # & A # B> 3

L

K/

!

A"B

1 ( > L 3

!

AA L B B K -N

K ,

A,B

. ! -N 8! % >

K /

-N K /

A-B

1 ! # ! >

9

K

9 9

A B L A 9 B

L

A8

8

B K , A6B

K ! Æ 9 > 9 K 1 * A

# B # 1 ! A6B ! 3 ! & 7!I Æ ! A' ##B % ! > K

&

!

A/B

) * + ,

& Æ ) 1 ( > L 3

K !

!

&

!

AB

L -N

) ;

! !# ! & > # ; ) E &

# & ; # #! # & A ! K /B # > # . ; ! ) ! &

1.4

CO2

1.2 O2 Lewis number

1.0

CH4 H2O

0.8

OH 0.6 0.4 H2 0.2 0.0 -2

-1

0

1

2

3

4

5

x [cm]

8 9: 1 + !!5 , 0 ; ; # * # !

!

7 % # # & 0 !I !> K 1 ( = ( A,B ! A B> ! + !

( A,B K /

-N K / I ( > L 3

K !

!

K/

AB

1 ( ! ( !

( A,B ( A"B ! >

Æ! ) ) G I # + I :G* B 3 ! Æ

'( AB A & ( A& K &B 0 I # I # Æ I ! #

) 1 # >

1 % # ( A"B ! ( 1 A ! B ! # K ! 1 % # *# ( # # ( # ( A"B ! > L A B K -N L 3 ! !

&

!

L

!

A&

& B !

AB

'( AB ! 1 :G* ( ! 1 #

% Æ > I .# ! 1 ## # & 2 Æ ( A B> & K & + I '( AB

+ I ! ! ( AB>

3

L

!

A L B K

!

&

!

L -N

AB

1 ! ! # + ( I (

"

> L 3

K !

!

&

!

K

K /

&

!

A B

2 I ! # %

! ( + I

A=.B I % I 1 & Æ ! I ' ## ! % . ) )

" #! 1 #! ( ( # ) = % ! A'( "BI # A ! !I #! ( B ! (! 8 > &8 K &3

8 8 L 3 !

K

8 L 3

8 !

A"B

*# ( #! AD B>

& K &3

3

L

!

A B K

: !

L

!

A$ B L N L

8A L B

A,B

N A ) I &)BI ! 8 A L B ! 8 1 #! &) : > : K

% L !

A-B

1 &) ) ! ; A% ! B % . #

,

$ # #! !> K L ! ( A"B ! >

& &3

K

& &3

& &3

!

& & K &3 &3

& &3

!

A6B

$ # '( A6B '( A,B # ( >

& K &3

L 3

!

A B K

3

: L A B L N L ! !

8A L B A /B

1 ( #! AB ! # % #! ( 7!# ( A B ! >

3

L

!

$ L !

K

A B

8

*# ( '( A,B # ( >

& K &3

L A B K 3 !

: L $ L N L ! !

8

A B

1 ( ! '( A B A6B>

& &3

K

3

L

!

A B K

&

&3

: !

L

!

L N L

8

A B

P K ! # 1 ) ! ! ! ) #! ! # A M B #! ! &) : A B * # % A K M BI # '( A B # ( A,B >

& & K -N L &3 &3

L L

!

N L

% !

!

8

L

!

A B

-

-N > -N K

1 I K #

A

M -N

A

B

:G* '( A B 1

B 3 >

) ! !>

K

K /

+ I # 3 ! ## -N 1 ( #! ! '( A B A6B>

& K &3

3

!

L

!

A B K -N L L N L !

L $

%

!

!

A "B

8

2 ! # A ? . @ #! 1 B 2# '( A "B A B ( K L >

!

& K &3

L A B K -N L 3 !

L

!

A$ B L N L

%

!

!

A ,B

8 A L B

+ !I ( K L ! # '( A B A B>

& K &3

L 3

!

L 3

!

A B K

-N L

L

!

#

/

%

!

A B L N L

-

!

8 A L B

A -B

# !$& .

& /

.

6

+ A .7 BI ( . *# K !

A ## I ## *

BI >

& K &3

& & L

&3 &3

A 6B

:# '( A B # > &

&

K -N ¼ L &3 &3

L

!

N L

L

% !

L ! !

A"/B

8

= -N ¼ ( -N I ( > N K

M -N

Q

N ¼ K

-N K

-N

M -N

A"B

1 ( 1! ! ! * ! ( ( 2 ( ! I # % >

& K &3

& & L &3 &3

A"B

# '( A "B >

& &3

K -N ¼¼ L

!

!

%

!

A B L $

!

L N L !

A"B

8

-N ¼¼ K -N A( -N (B 1 " 3 #! (

L

L

L

A% B

A

A B L N L

B L

A B L N L

A$ B L N L

M L

8

8

8A L B

8 A L B

B L L N L

8

8

8A L B

B L $ L N L

B L

A

M

K L

K L

8 A L B

L L N L

A

L $ L N L

A

A% B

A$ B L N L

K L

K

'!

+ ) / * ) < * = * 0. 1

) * 1 3 Æ Æ ) = = >

A% B

K

K

L

A% B

! K -N L " K -N L

K

K N L KN L

'#!

K K K L M K K L M L K K L K

K

1

1

* L

*

/

1 #! ( ! % 3 1 #

$

+ &# AH BI & A! 4# # / E # 2 // "// E BI # # % >

1 7 ) & 4# 2 > Æ ( ! 1 ! ! # A BI ! ! ! > L K 3 !

L ! ! !

1 ( ! > K 2

! K ! '

K

3 K 23 '

A"B

K 2

A" B

' A ) 3 B 2 +) # ( + # :! K 2 ' I '( A"B > K !

3

L , ! ! !

A) B

A)B

A""B

A) B

'( A""B I # :! ! & I # ) 1! ) ! & 7 I ## ( K ! K K 23 1 I ! # # & ! # # & > K

A",B

+ #! ( I & &3 ! 3

+ !I # P K A ! B ( # ) ! #

1 ( A"/B >

&

K -N L ¼

&3

!

% !

!

L N L

8

A"-B

%& '( ! % ! # . I # ( K K A & RBI ( A"/B K / ( >

I ( & > & &

K -N ¼ L L &3 &3

% !

!

L

L N L !

I I ( & >

&

K -N ¼ L &3

!

!

%

L N L

8

A"6B

8

A,/B

G -N ¼ -N A'( "B (>

-N K ¼

-N K

-N

M

N K

-

-N

M -N K -N

A,B

-N K / 1 #! ! ( % (>

& K &3

L

!

L 3

!

A B K -N L

A$ B L N L

!

%

!

8 A L B

' '

A,B

& K &3

L

L 3

!

!

L 3

A B K -N L

A B L N L

!

%

!

8A L B

A,B

= I ( I ( ( > K

K A

B K A; B

; K :# '( A "B # # ! ( >

;

& K &3

; ;

L -N L ! !

%

!

L

L N L !

8

A,B

&# I > '( A,BI % :G* A B H # & I ( ) + -N I )I

#I ! + ; !I $$

% '( A,B #

! # & + ! ( # & A -B

! ' !

+ ) ! A )B ) (> ). #! K D ) # ( ! # &4 &3 K & &3 ( A'( B>

& & K -N L L

&3 &3 !

L !

% !

L N L !

!

8

A, B

1 , 3 ( # & 7 %

%

L

K/

* > K

A B

H > %

L

AA L B B K -N

H > %

L

AA L B B K

B L A& N K

7

L

K L

&

L

L

A B K -N

-N K

L

&

8

'#! A B %!

A$ B L N L

L M -N : K %

8A L B

* + 0:> + + * ) !5 = . *

&# I I ! 3 A8 K /BI # ## ( ! % 1 - 7 %

%

L

K/

* > K

A B

H > %

L

AA L B B K -N

H > %

L

AA L B B K

B L A& N K

&

7

L

K L

&

'#! A B %!

L

A B K -N

-N K

0 K -N ¼ L -N ¼ K

A% B

N : K

%

M -N

L

* + # : $ 0

1 ! . & ## ) # A ) )B I ! ( =! # ) & >

+ %# ) I !

+ ! *

7! ! & I ! !

AH B ; & ! ! # & A& !> B

( I # ) & % ) %# G I ! & # ! ! > ! & 1 ! ! & # . )! ! A B H ) I ! ( A B G I ! ! %I ! .! ! # &

# )

,

-

A B 1 1! ( &

. ) & I ( % # 1 - A ) # !

B>

7

3

* K

'#!

-N ¼ K

L 3 !

3

L

L

K/

AB

AA L B B K -N

AB

> !

K -N ¼ L

!

!

%

!

!

AB

-N

1 ( ## # 4# # * #

& ) ! & ! H & !I # '( AB AB & A # 4# B > K K 4#

A"B

- ' ' ' # &.

#& ##& & #0%& ' ( ' 1 . #! & #! & #02&

3 0! ' ( ' 1 . 4 1, 1 ' "% 5 ' 6, ! 7 89 ' ' ' ( ' ' 1

!

AA L B B K -N

K -N ¼ L ! !

% !

!

6

A,B A-B

1 ( # -N A2 B A !I )B ! ) & I ! ! Æ 1! A ! K /B ) # & A# B> A! K /B K I A ! BI A! K /B K 7

STATE 1: fresh gas

6

STATE 2: burnt gas Reaction rate

5

Normalized temperature T / T 1

4 Normalized fuel mass fraction Y / Y F1

3 2 1 0 0.0

Abscissa (x)

x=l

; 1 , . 0 !> + ? > +

1 # & ! ) >

1 ! 3 + & !I ! # ! K / # # !I

(

: ! # < %! 1 ? !@ AH B

I ) ! ! #3# I > & # # & A! K B 2 ! ) &# >

/

& # 1 ! *

' & #I ! ) ! ! ( & 4# A & %) B 1 # > % I ! & 4# #

1 & %# ) & ## . + . + ! ( &

() $ * # '( A"B A-B !

! ! 1! H ! 3 % #I # ! #! ! >

AEE # E B #

. & 1 & E A # &# ! ! # . )!# ! ! # B 1 # 1 I 5:'7+S # # 3 /// # /6 5:'7+S . & % 1 ! . & >

# I % & # % & I ! A( I I I B

: . ' ' ' ' '

0.6

x [cm]

0.6

x [cm]

0.8

0.8

H2 O2 OH H2O O H

1.0

1.0

0.0015

0.0010

0.0005

0.2

0.0000

2e+12

1e+12

0.2

0

5e+11

5e+11

0.4

0.4

0.6

x [cm]

0.6

x [cm]

0.8

1.0

1.0

HO2 H2O2

0.8

1.0

0.8

0.4

0.4

Y Heat release [erg/cm3]

0.6

0.4

0.2

0.0 0.2

0.0 0.2

1000.0

2000.0

3000.0

4000.0

Y

# % & ! # ( ! A6B % #> % # !

I # $ !I (# // I 3# // % ! # #! ) + ! I 3 . & I I # . & ! 3 ! # ! ) I ) % A '( B> . I / # K , ) I ( /// "/// H "/// & ( ) ! Æ 0 # A +BI A ++B

> 477 ( -1 #

-@$AB , . 0 !

Temperature [K]

2400

Maximum flame temperature (K)

2200 2000 1800 1600 1400 1200 1000

0

1

2 3 Equivalence ratio

4

5

: . , 0 !777 2* # * # ! ) . 0

* +*

A ) &BI # )

0 !I + 3 ++ ! A ## *

I D I ## *

B 1 . ( &! > # (! # I )I ! (# & . & ! ? &@ # # ) ) ! ( # # ) ) ! #E & # & A " %

& B 1 # & ( I ) &! + .! # # )!

* ! ! + ( I

# ! . ) & !> ! ! 8#

A B # & (!

1 ) .! * & ) ! I ! # & !> )I L L # 1 #I & # T 0 I . 2 ! # ! !

A ;3 ## *

B ! !

A ) D I 2# B # ) ! ! & # ! 2# ! #. & / ( / * ! ! # L L ! &

* 0 0 #

1 * / 0 @ * A C

1 ) 1 A Æ I !I ÆB ! . # & 1 ! ! ! 3 . A B #

! + & I I ! ! ! A ! ! ) B . & ! ! 2 )I # ! & ( E &

5:'7+S>

! # :+ A>EE!E # B

! # ;

! 1 ! A B + & ) ! # ; / / ! A # B

40

0

Laminar flame speed s L (cm/s)

50

30

20

10 Complex transport Lewis =1 for all species 0 0.6

0.8

1.0 Equivalence Ratio, φ

1.2

1.4

) / . 0 > . 1 9: !

& 0 ; ; # * # !

# # ( #% & 2 !# ! I % # ! Æ ! # ! ) # # #! ) A5 H3 B

!"

1 ) & ) !

! A )I ! & B I . ! ) & ! & %# > !

) 8# ) ( & ( % ! 1 ! )!

) ! &

1 % ( ! & # AH B>

2 # K ! K 1! Æ & K & ; ( ' K % A & B K ' ! ! ' K

! ! # > ) K '( A-B

"¼

"¼¼

A/B

1 3 > K / 1 2 > K . ' 1 -N # # ! '( AB> -N K "

N K "

(

AB

1 -N ( > -N K

M -N K

AM "B K " =

= >

=

K

M

" K "

AB

M

" K "

M

" "

AB

"

" K "¼¼ "¼ # = ! + A ! # B 1 !

% (! 1 % ! > =¼ K

=

K

M

" "

AB

1 -N -N !> -N K =¼ -N

A B

= I = =¼ # * % = > =K

=

¼

K = K

M

" "

A"B

= 1 -N -N > -N K

=-N

A,B

1 ) 1! = = . 3 1 = = ! !

1 -N A/B ! ! )3 1 ! ! & )! > K 0! A B 1 ¼ ¼¼ "¼ K "¼¼ K / " K " A-B K " K " K / I # '( AB AB>

-N K . " ' K > '

A6B

> K . "

A/B

' # ! # A ) ! BI ! ! ! & > . I !

2 " "

!

!

'%& 7 757 "" 6!

3 6 !5 !

'%&' "7!77 65577 6" 77 !7"77

,

6 77 3 53 3 "! 77

7 7"" 7 757 7 75 7 7

8 ! ! " " / * ) * + !37 1 # A K B 1 # . ( A & B>

K K K 4# 4#

K ! !

4#

!

L -N

AB

!

=-N

AB

&

K ! !

%

AB

+ ) I -N #

+ !

2 ) # '( AB AB ! K ! K L A # B 3 ( & # > 4# K 4# A

½ ½

B K =

-N ! K U ½

-N ! K =U

AB A B

½

U 1 ( ) # & '( AB # A 4# B & ! U '( A BI

-

I ! A=U B ! #! &) 4# = # U '( AB A B > A

B K =

K L =

A-B

& '( A-B ) ! ! # >

A BL

M K A BL

A B K

M

M A B

A6B K 6- 2 . '( A6B ! '( AB # >

" K / "

A B K

M

" K "

= K = ¼

A/B

# '( A-B 1 3 >

1 ! A M I B ) &

A) .I ). / B I

1 + ' , 9 ' ' Æ 3 ' , ; #!$6& ' ' 1 .

# & / #00<&

' ' ' 1 ' 1 ' 9 3 , ' , .

/

#006&

6

1 I I I )I A! # ) .B ) ( 1 I

! #. !I ) + 1 & '( A-B 1

#

(!> 1 # 1 ! % > I ! I # 1

1 ! #%! ! # &

Adiabatic flame temperature (K)

Full chemistry and variable Cp One-step chemistry and variable Cp One-step chemistry and constant Cp

2400

2200

2000

1800

1600

1400 0.5

0.6

0.7

0.8

0.9

Equivalence ratio φ

1.0

1.1

1.2

C 0 , 0 377 ( *

1 ! ! # ( I ! 3# #!I # ! # G'7D+= 8 '( A-B + I )I ( 8! 7 ;! A87;B A;! I 8! B ) & 2 )I # " & E & >

7> ! !>

/

7> *#. ! !>

7> *#. ! !>

#

1 7 7 > (. I ) # ! 7 2 # ( I &I ! / D 7 7 = K ""// 9E# K // 9EA#DB + ( // D %

+ & '( AB AB ! % # > K

CK

B K =

A

AB

1 # # / # C # / 3 1 ( C '( AB AB # '( A-B> 4# K ! ! 4#

C K ! !

& ! % C !

L -N

AB

-N

AB

*# ( # ! ; ' K % A &B K # > 4# AC L B K & AC L B ! ! !

AB

A B ( * C L # # I ! '( AB > CL K

A B

1 # ! ) & ! # ! ) ! 1 '' #= !& '' '' ' 1. = !

( > # A ! BI A ! B I ! C 1 A B C # '( A B 1 ( C > C 4# K ! !

% C !

> A L CA

BB

A

CB )

L CA

B

A"B

1 C A B # 2

+ '( A"B ! & > & ! & ! % # ! ( 1 A H H B 1 I #! ) & I A-N '( AB A"BB ) ! H >

'

CB ) / A?A CBCB ! ) A/ ?B ) > -N K > '( A

? K A

B K = A B

(

/ K ?

A,B

A-B

1 ? / ! & ! 1! & 1 K // & A B % ! . &

!

6 4

'%& !!7 34"

64 !"7

#

74" 7 5

7 ! 6

) * # . 0 1 ! C > K

L ?C A ?B

> # ; #00"&&

A6B

300x10

-9

Reduced reaction rate

(1-α)/β exp(−1−β/α)

200

100

0 0.0

1-1/(α+β)

0.5 Reduced temperature θ= (T-T1) / (T2-T1)

8 = $

# 7 4"# 7

! D

1.0

* D

-N ! > C # , -N ? K /, / K -I # / K /I # - # -N # / # -I ) > #! < -N !I & # - ! > & # & I # / A # B -N 3 # C ! ! + I ! A B % A!! / / # / 7 C 7 B ! 3 # - ! C K 1 # ! ! ! AH B> #I ) # - C

/ 1 ) ! '( A,B ) ! )! I ! # ) -N ACB # > C K A? L / B A/B

! / / # ? 1 ) A# ,B> N

-

K

>

? ) / / ?

AB

1 3 .3 ) &

# I I #! A / IB

Reduced reaction rate

α = 0.75 α = 0.75 α = 0.75 α = 0.75

0.2

β=4 β=8 β=13 β=20

0.4 0.6 Reduced temperature Θ = (T-T1) / (T2-T1)

0.8

1.0

8 = $ * D 7 * *

> / : : * ) . $ E * * 0

0

Æ K / A# 6B + ( I (! # C / A BI ! A / IB A Æ K / B 1 % ! #> ÆI # :! I # & I # # / I ! & ) & ! / ! / A/ /BI I I Æ R 2 ! ! 3 # 6 A C K / / B ! A ! .# & ## ! B 3 A C K / B ! 1 ) 3 ! ) A 3 I % 3 / B ! >

! . ! & I

. & 5:'7+S

! E 3 % # 3 E 3 * % ( ! Æ! # . & . ! &

. E ! & # Æ .

Reduced reaction rate

Zone 1: diffusion and convection: thickness = 1 in θ space

Zone 2: diffusion and reaction: thickness = 2/β in θ space

0.0

0.2

0.4 0.6 Reduced temperature θ= (T-T1) / (T2-T1)

0.8

1.0

) ' F . 0

I %# # A7 B 2 ( ) A) B A/ B % # ! . ! & ! 1 ! ! I ! 1 ! . & ! ! & ! ) ! I ! & ! . &

,! $ $ % I ) ) & & 1 ! 1 % ! V I .D D !. 2 ! !

5 ) !) 8!.9 & 7:; 7 ; <8&;=

'( A"B ! % ! # !> @K

4#

%

!

AB

'( A"B ! ## & > C C K @ @

W-

AB

- % AH B !> - K A

CB ) / A?A CBCB

AB

W 3 (! A ?& @B> WK

% > 4#

'( K

%> 4

#

'(

A B

2 > # $ I ! ) & ! W # ? / 1 & A B A%B A> B 1 & > # ! # & + ) A BI %> # 1 Æ !> ! % % K 23 '# # # > WK

> & 4#

A"B

K % ! # &

4 ! ! ( ! '( AB 1 % V I D D AVDB # ! >

W K / /

2 !

H >

W K / / L

/

A? B

A,B

A-B

+ # W // & W # ! ! % A'( A BBI & ! # > 4# K

/

> &

'

A6B

1 # $& '

"

VD A'( A,BB > 4# K

/

)

/ ?

' A > & B L

?

A /B

/

H A'( A-BB

5 !)! ! !>

2 ! 2 ) VD ! ( ! 2 ! ) ) & ! # 1 &

# 6 ## ! C K / C & 700x10

-9

Reduced reaction rate

600

Echekki Ferziger form

500 400 300 Arrhenius form

200 100 0 0.0

0.2

0.4 0.6 Reduced temperature θ= (Τ−Τ1) / (Τ2−Τ1)

0.8

θc = 1- 1/ β

1.0

C '' F . 0 C,

2 # I 2 ! # / ! ' 3# A ' B + ' I 3 C A CB C 6 C '( AB ABI ( A

B> K K 4# A B C 4# K ! !

% C !

L -N !

A B

,

CB AC C B A B K / 1 ( !

-N ! K A

G C 3 '( AB # &

@ A'( ABB> C @

K

C @

W-

A B

W K % A 4# B - K A CB AC C B 2 # ! % K % I & W '( A B 2 # '( AB> C K / I %) ! K @ K / + @ K A 4# B! % K ! Æ Æ & # !> ÆK

% & K 4# 4#

A B

! # &

A -

@ =

G Æ

D D D ! D D D

!

H

C '' F . Æ / Æ

* Æ

1 ! % ## '( A B ! K ! K / A

-N ! K /B>

CK

/

A "B

'Æ

! 6 /I # # > XK

Æ

A L

Æ ' B 4#

C

K A 1

A ,B

' ;99 3 / #! =& #= = & # 0$& , ? ' ' ' Æ. , ' 9 1 ' ? 9 0$

-

C K

A -B

/

8! I % '( A -B A "B

! K / 8! # ! K /I > K / A/

B 4Æ

#

K / A/

B &4

A 6B

#

2 ) ) & > 4# K

/ A/

% B

'

K

A/ A/

A& B' BB'

XK

/ Æ

A"/B

1 ! 3 ) & ## 4# # / 1 # / ! %

-N A > B C 2 ) ' 1 > A 2 B A ' B # & > K > / )A/ ?B A 1 B # % # % ) # '( A"B ? K /, / K - 1 # !

# ) & ' A'( A"/BB 2. A'( A /BB > 4#

A& B

A"B

'

+ I & (

Æ (

Æ 1

& # ! 1 3 C & / ! 1 ) ! K / C K C K / 1 # A ½ A/ B4# & N ! !B 4# ) A ½ '( ABB

1) !

1 # ) & ! ! 7!

1 3 ! ! ! & + & I D & > '

6

. ? 4# I I + !

3 ?

: ? K A B / K ? 4 > Æ K & #

>! ? C K

! 7 /> C K A / B Æ ! 6 /> C K / Æ !@ ? A!B K CA B L !@ !) >!? A!B K L 4 ?C A ?B K L 4 A B & ! ? > -N K A CB AC C B K / A/ B4 & 2 > -N K > > K A4 / B &

#

#

'

#

#

'(

C F . 0 1 )

/ 0 ) * 1 * + !76 1.0

Reduced temperature θ

0.8 θ= θc = 1-1/ β

0.6

0.4

0.2 Arrhenius form Echekki Ferziger model 0.0 -5

-4

-3

-2 -1 Reduced abscissa (x/ δ)

0

1

2

1 : C

% K %

&

&

'

'

A > & B'

( # .F * (

0 0 ) = ' = = ! = $ * ) . * ! : ) Æ # # # G * D > D ! ! *

CB AC C B

A

1

% K %

A

'

CB AC C B

% K %

> '

4 D

A > & B'A L ( B

% K %A B

> '

H

A > & B'

% K %A B

-N '( AB

4#

> '

1

VD

7

/

"

- ' $

7 # ) & ) 1 AH B .! ; # > ¼ "¼ 1 L "

A"B

# !> -N K "¼ >

)

)

)

A"B

¼ H ) ! ( "¼ " 1 ¼ ¼ ( 5 K 4 K A" B A" B 1 %> 1 ) % K % A B 1 + ' ' ! )3 ; I &

! % . ! & A5 7 B A # ( B>

4# K

¼ ) > ) ) ') ') %"¼ A"

"¼ B) )

'

) ) / ) ) / A , B' ) ?

% !> A , B K

½

B)

BL/

5 '

)

A"B

A" B

* B

&I 1 + #I ! ) )

2 ( ) '( A"B ! # % > ! % > >

4# K

" ¼ A" ¼ " ¼ B) > ') ') &

/ ) ) / A , B' ) ?

) )

'

A""B

K I "¼ K K / ! ; I A , B K '( A""B ) A6B

0.7 Numerical solution Experiments Asymptotic solution

0.6

Flame speed (m/s)

0.5 0.4 0.3 0.2 0.1 0.0 0.6

0.8

1.0 Equivalence ratio φ

1.2

1.4

& . +* 1 " ) ! E & # # ) & E & ! # > L L A",B 1 # > > K /- , 0C K " D

/ K /

K -"//

% K //-

K

= K /// D

K /

K / A D B

A"-B A"6B

1 ) > % & 5 K /- # ! & # / ( H ) AY 1 <BI ! #

$ : .F G !

" % " & ! "

-. $ # . # * @ . ' '

" " G /* >

.F 9: >

(

(

# ) # # +*

" $ >

¼ ¼

>

>

" % "

¼ ¼ ¼ $

# !

!

A 2 > )' * '

# !

; /* >

; > #

! ! ! 3 45

+" +

% &

!

!( "

#

' '

$

+

)' * ' $

0 ) . :

. *# ! 2 # I # # ) #

# !

& 1 #3 # ! %) ! # ) H

! & # ( I ! # ) I ! * ! Æ > # ) A ! B 1 ! ! # & K / ## # '( A,B .! > -N K > A

CB

)

)

/ A CB ?A CB

A,/B

1 ? K /, / K - 1 K / / K -I ! K / K / A # -B 1 ) !

Æ

Reduced reaction rate

40x10

n f =0.2 n f =0.4 n f =1

-3

30

20

10

0 0.0

0.2

0.4 0.6 Reduced temperature (T-T1) / (T2-T1)

0.8

1.0

8 = $ * D * * ' >

/ : ' 7

/

0 $

'( A""B ! & # )I ! & # ! I # I & 4# #! !>

4# A B K 4# A B

A,B

1 # Æ ) '( A""B ! # # ! ') ! ) ! > 4# A, B K 4# A , B

(

(

A,B

2 )I K / K " I ! H

! B E &I ) ?+ K / '( A,B ! 7# D

A1 ,B 1 , # ) '( A,B E ! A / I // // DI /- 5 B E & ! 7# D A/ / I // ,// DI /- 5 I 7# B

$ 7 $ ! $ ! - 7 ! "

& *

,

7" 7357 73!6 7 36 ! 3 ! 7

#

#

!7" ,7"76 !5! ,7346 777 ,763 ! 7 ! 7 !5 7 !

. . * + 4 & 2 & 0 $ ('

! # # ) & . . & # . ) A,B 1 ,

"

φ= 1, P=1 atm φ= 1, P=5 atm φ= 1, P=10 atm

0.8 0.6 0.4 0.2

METHANE - AIR

0.0 300

φ= 1, P=1 atm φ= 1, P=5 atm φ= 1, P=10 atm

1.0

Flame speed (m/s)

Flame speed (m/s)

1.0

350 400 450 Fresh gas temperature (K)

0.8 0.6 0.4 0.2

PROPANE - AIR

0.0

500

300

350 400 450 Fresh gas temperature (K)

500

. 1 & 2 # & 0 $ (' #

# $ "% %# # & ( ! ( .

> I & # 3 & 1 ! ! % . ) & * ( % & A ) . %# B ! # + ! # % I ! A BI # I ! A B

* ! + * I & Æ # > ÆK

% & K 4# 4#

K

Æ4# K &

A,B

A,B

K % A BI I % # ( &

! & :! 1 Æ A ? @

,

B ! ! !

& + I !

) A !

!

B 2 ! # % #>

Reduced temperature

(T-T1)/(T-T2)

Æ# K

)

A, B

1

STATE 1: fresh gas

STATE 2: burnt gas

0

δLt

δL o

Abscissa

I1 0 ' . 0 2 A? @ B Æ# ! ! %# C # // /66 A # B Æ# ! #

! > & I # Æ# # A ) # B ! # ?@ # # Æ# # Æ# #

* # I Æ# 1 Æ# ( % & 8# 1 )I 8 # & Æ#, > , Æ# A% B K Æ A% B

A,"B

+ * % A # 5 A'( A,BB

-

BI ! %> , Æ# K Æ

, Æ# K

& 4#

K Æ

A,,B

, 1 Æ# # ! '( A,,B ( & !

% 1 #

& Æ# & I ! ! ! ( '( A-B = I Æ# I 3 Æ ( 1 3 #! ! Æ K Æ# / 1 - 3 % ! !

" "& '# " ( ) ' Æ . 2 '

Æ

D 7 7! 7

; 1 J

!) # # # "" '# " (

I/* ' ; ' @ '

Æ ) Æ

-(

Æ

-(

* Æ * Æ

J #

Æ

'

' 1 F ' ! 1 & ! A1 (234B ! + & # ) ! I ! & 1

> )I # I ! ) Æ# 3 )! %# &

( ! +

. ! & I Q . & I ( ! !

6

& ' " $ I ! ! ## . ) & 2 ! . & & 1 % & % ') & % # * ,

"

1 '

1 & % ! # & . A7 7 ! I 5 B> EK

.

A,-B

. 3

2 & ## . & < # & AH B 1 # ! 2 # ) ! & A 5 B> EK

F F > F L F

A,6B

F F & # # ! & A# "B 1 AF F > B #

& ! ) >

F K FF >

G

!

A-/B

I I E > E K AÆ

B

G !

A-B

1 & # ! & A B , ) % & B 1 & ! ! # ! F >

4- A

F K F L 4 F A-B F & ! F 1 & ! . 4 F & & -

-

* # '( A-B '( A,6B ! > EK

F F > F L F

L 4- A F B

A-B

"/

n

w

R1

p

m Burnt gases

A(t) R2 Fresh gases

I1 0 0 *

I ) > E K AÆ

B

L 4!

!

A-B

1 F ! + & & !>

F K L

A- B

8 F '( A-B #> # # F % ! # & ( !

# ! # A# ,BI F

! ! # '' & 1 ' '' 3 4 ' , , #! "&@ ' , @

' '' , 4 # & .

#! !# &

$

' 4 .

!

!#

$

# #&& # # & #& "&.

#& @

#& @

! @

#! &$

"

$

"

!

A K . K /B>

F K (

A( B

F K (

A( B

K

(

'( A- B 1 4- F '( A-B #> & # & *!I & A# ,BI F A

B> (

K

Burnt gases Fresh gases

Burnt gases

Fresh gases

(a) Cylindrical flame

(b) Spherical flame

-' 0 > # * . * + 3 1 % :G* '( A-B # A 5 B>

F L 4 A F B

E K AF F L FFB >

A-"B

-

F F ## # & F F LF A# "B 1 A FB > F '( A-"B ! & A # ;B F 3 # > F F LF F K A FB > F K F F > F L F

1

A-,B

) :G* '( A-"B>

F

K

F F LF A FB > F K AÆ

K

Æ

F !

!

!

K

B

!

! !

A B K F

A--B

"

I '( A-"B . > EK

F 4 L

-

K F L 4- F

A-6B

+ '( A-6BI & >

FI & . !I I 4 F K 4 A L BI

% I

-

-

" * ! $

2 & ! # # F >

F F K /

A6/B

F 3 > ! ) 1 ! ! # & A ) # # & # /B ! & '( A-B A6/B F F K 4- >

F L 4 F K AF AF F BF B K F

EK

-

A6B

F # # ! A # ;B

" %' $ - ! # 1 ) &

#. ) )! ! < .!=

% ! ! A B &I ! ! # < A# -B 2 I & 3 %) . < 1 & 3 0 !I &

; #0A!& .

# & B# &C #Æ ! ! & B+ ! C +

!

!0 ! ! !

!

+

+ ! ! ! + +

"

3 # 2!I '( A-6B ! & !

> EK

F L 4 F K F F > F L F 4 K

-

-

(

L

A( B (

(

4 K ( 4 K / -

-

A6B

( 4- ! & 2 ! '( A6B F 3 E 3 Burnt gases

Fresh gases

Steady flame front

0 > 0 F & EK

I >

F L 4 F K

-

A( B L ( ( (

4

-

K

(

4

-

K/

A6B

#. ! . 2 & ! # 6 1

& ! ! # & > ! ! # & # & 1 ! &

# 6 . ! # 1 ! & )# ! 3 &I K /I K I K AA! B, /B '( A6B & 3 ! # & 2 & %# ! !

" %' $ # / %# ! & )! ! 1 % A# /B # & A ! ! &B # & # + I & ! & # # ! A% F '( A-6BB # & I # %# ! > K H! I K HB H & 0 F < + # & I + &

!! A< L ! # # # # I & 1 & !I & ! # & # A# /B> ! F K 4- '( A-6B * ! & ! # # & (A3B> ( . K A6B EK . 3

( 3

+ ! & > & . ! # # # & I & >

& & Q &

Flame front

"

Fresh gases (U j1 )

d

Stagnation plane

(a) The single flame: fresh gas against combustion products (steady flow)

Combustion products (U j2 )

Fresh gases (U j1 ) Stagnation plane (b) The twin flames: fresh gas against fresh gas (steady flow)

Flame fronts Fresh gases (U j2 )

Fresh gases Burnt

(c) The spherical flame: unsteady stretched curved flame

gases

. . 0

""

( ' 1 ? @ & ! + ! Æ % & ! * , # % & * , % ! & * ,

& Æ %#

. 1 1 & 4# I ! % # ! & #

# . # ! 1 % ! ! & # 4# # & ; A '( ABB & !

% # # & > 4# K

½

-N !

A6 B

½

1 I & # I ! ) 1 # I ! & % A1 5 I 5 B 3 1 6 # # >

A 1

I1

C I

* 1. * 0: :

1 0 ' # % ! ( I I ! I ! % >

; % & I C K C A# B 2 !

I # C % & >

C C

F K

A6"B

",

Position of isolevel Θ f at time t+dt

FRESH GAS s d n dt

w dt n Position of isolevel Θ f at time t

u dt BURNT GAS

J 0 1 F F 1 ! # !>

C 3

F C K / L

A6,B

F F 1 ! & 4 K ! + C :# C ! CF '( A6,B > F F 4 K K C C3 A6-B

1 ! # & & # ! & 1 !

1 & F> & F

FBF K 4 FF

4- K A F

A66B

!>

FF K C

4- K F F

C 3

LF

C K &C C C &3

A//B

( C K C 1 :G* '( A6-B

A//B ! !

"-

) )I # '( ABI '( A//B ! >

&C ¼ K 4- K C &3 C -N L !

% !

!

A/B

1 ! =* & A "B ; 4 I C + Æ > & # & I #

G I (! ) ( #! & & A5 B

2 4 ! % ! '( A6 B> 4 K

½

-N

½

A/B

+ & 2 4- 4 ( . # C ! 4 (! # # C & A1 /B 1 ! ! ) > 4- AC K /B 1 # &

A 1

I1

I

C

. . )

.

). .

&

+

=

2 9 > 9 >

+ D D

1 ) . . $ 1 ! ## . & 4 # ' &I ) #

"6

& & ! ( 4 # > I & ! 1 4 ( 4 # 1 ! ) # F K F 1 F K F 2 3 ! & 4 # 8 & # ! & ) 6 4 # I & # ! 1 & ! F K A 4 # BF # > 1 4 K F F K 4 # #

8 Displacement speed of isolevel

Normalized speeds

6 Flow velocity 4 Absolute speed of isolevel 2

0

-2 0

20

40 60 80 Distance along normal to flame front

100

120

8 0:# # , 0 . # 1. "

1 & !I ! ( AB + ) I & I & # F ( & AF F BF K AF F BF A/B # !> K

FF K 4

L A L 4 B

K L 4 #

A/B

1 ! L ?C A ?B C ? ! % ! '( AB '( A-B 1 & # & # ! L 4 # ? ? 1 & & > 4- ACB K 4

FF K 4 K 4

#

#

L

?C ?

A/ B

,/

+ & # # & A # # . 4 #B + 4 # A ?B # I ( & A 4 # ! B

! C I 4- # 1 3

C D

I

+D ! #D! #

0 0: * 0 D + 3! # + 3

. $ 1 & & . # ! % + & A BI & ! Æ I )! ! 1 ! # ! & ! A8 I 8 ; I H I 5

I H I B 1 ## I I E ! # & I # > 4- A/B 4 #

'-

K E 4

#

'

4

K E 4 # 4#

A/"B

'- ' 7 # 1 % # AC K /B 1 7 # I # ! '( A/"B > 4- A/B K 4 #

)

EÆ 4 #

4 K 4 #

Æ & % ! '( A ÆK

%

4 #

K

&

4 #

EÆ

4#

)

A/,B

B> A/-B

)- K '- Æ ) K ' Æ 7 ! 1 ) EÆ 4 # D 3 H

,

!I (# )

! EÆ 4 # 77 4 ) ! 7 !I I #. & !I 9

#> )-

K

'- Æ

K

)

L / A'

K K / A' Æ '

B

A L !B !

B

/ K A B

A L !B !

!

!

A/6B

A/B

1 / # ! '( A-B> + #! 1 # 7 # '- ! & ! & & > # 7 # ! # AH B ' # ) & ! A #. !I # . #!I BI & !I ) # & ! # & # ## ! ! A'# . ;B I # 7 # Æ * ( # & > & A8! I 2# I 2# I BI # & A*! ZBI 3 4 & A1 *!BI # & A; *#B # %# 1 ! ( ! ! Æ ! & >

# & I & I &

1 *# & ! ! & A# / B & I ( '( A-6B 3 * # & & 2 ! ! A;! H I ;! H I ;! I ;! H BI A+ 3 I ; 8 D ' '' - 8 1

,

;B A I ## *

B ! # & 2 I 7 ! > & # ! # I ! I ) +! ! # & A)I !

BI & & *# & ! & 3 > F K/ 4 K / AB '( A66B & ! > 4- ACB K

FF

AB

1.0

0.8

0.6

0.4 Reduced temperature Θ

0.2

Velocity 0.0 -1.0

-0.8

-0.6 -0.4 Distance to stagnation plane

-0.2

0.0

8 1 . 0 # ! ! % # ) & A # ) &B H & # & I ! ! ! !> K H! H + ) I % ! ! & I # % Æ 7 !I & # ! 3 # + ) ) & 3

1

'( AB 4- A/B * ( A !I ; *#B> # ! 3 I ) # ! % 3 I + #I ( # # ! ! )!

,

A4# '# I '# ;B 2 # ! H I # # ) # A7 # )B 4 # ! % ! # # A'( A6 BB $ ! (! ! A B #! A1 B

C I

D + D & = 7 . . J K Æ

K K Æ

I1 > . > >

. 0 0: * ) L. M L M : &

*# & % # # A # BI

# & ! A # B 0 !I ; ' K % A &B A% A BB A&B & 2 # & A ! B # 3 ! ! # # & A . %# ! # /B ' K 2 '( A/BI & ' K 2!I # # 1 ! & Adiabatic flame

Consumption speed sc

LeF = 1

Markstein zone sc =sLo 0

Non adiabatic flame Stretch Κ Critical stretch

/ 0 7 ( !> F ( $ '

+ !!7

,

H ' 7 A# BI > ) 7 A/B ) 7 / # I A/,B 4 E ! 1 & ( #%> ! #. & A'# . ;B + I (# ! ! # H & I ) Consumption speed sc

Adiabatic flame

LeF < 1

Markstein zone sc =sLo-Lac Κ

Non adiabatic flame Stretch Κ

0

Critical stretch

/ 0 ( / !> * ( $ ' + !!7

' 6 A# "BI #

& A) 6 /B 1 ! ! # ' (# G I ' # # (# AH B H ! # 2 ! ) # A4 EB & # ! ( 1 (# 4 # Æ 1 ) ) ) . & Consumption speed sc

Adiabatic flame

Markstein zone sc =sLo-Lac Κ

LeF > 1

0

Non adiabatic flame

Stretch Κ Critical stretch

/ 0 ( 0 !> * ( $ ' + !!7

,

&! + I & & I 4 I 4- 4 ! ! # > ) A# ,B ! & A8 !I 5 B 1 4 & 3 1 4- & ) )! & * ! %) ! ! #I ! # + & I 4- ! # / 4 + # ,I # & # ) & & ( & Flame tip: The flame moves at the flow speed u: sa = 0, sd= u, sc=sL . Fresh gas u=sd

Flame front

) 0 >

2 . & ' K A5 B 3 & A# -B> I 4 ( 4 # G I 4- K & ) ( -4 # ! %) -4 # 1 ) >

! # -I & ) # ) # # &> % &

#! ! > & # A # -B 0 & ) I

2 ( A& ! ) BI & # # A! - B ! 3 A5 B , E ! ' 14' ' '4 4 E #E + &

,"

A * # 0 -

* A1 B & ! ! & 1! & # # & ! # 1 &

.> . > >

& = , J K

C I

D + D 7 . . 00 , K K K K

. , ( ! - ) L. M L M : &

1.! 2 & 7 # &# A ! I 2# I B +## & # ) A# 6B & (A3B ! & & + # & & I & ! ! & 1 ! & & ! (A3B (! (A3B 1 & . K J(A3B > . K ( A3 B AB EK . 3

( 3

7 # & ) & & * ( & & (A3B 2 %

,,

$ I & & ## & Æ# I 4 (A3B ! # ( >

+ L 3

!

Fresh gas (state 1)

R

r(t)

AB

1.0

Normalized speeds

Burnt gas (state 2)

AA L B+ B K -N +

R=0 δL0

Reduced temperature

0.8

θ

2

0.6

Yp=Yp

Products mass fraction Y

p

0.4 0.2 0.0 0

1

2

3

4

Normalized radius (R/r(t))

0 1 ( # ( K / ( K !> 3

½

+

K

½

-N +

A B

&) 3 ( K / ( K 1 :G* '( A B ! & & ! >

½

-N + K J(A3B

½

-N + ( K J(A3B + 4

A"B

# '( A,B + ! # A# 6B 0 ;G* '( A B ) # + & I ) !>

½

+ K +

J (A3B

A,B

H# ( ) # ! '( A"B % ) 4 (A3B>

,-

4 K

(A3B 3

A-B

'( A-B 4 + ( 3 ! # > ! # # ! 1 ( 3 ! # 3 & & A(B H # I 7 (A3BI A(B 3 6 (A3BI A(B ! # ! ( ( K / # 1 ( &) # > 3

J (A3B L J (A3B

AB K

K J AB

(A3B

A6B

(A3B 3

A/B

1.0

Normalized velocities u/(dr/dt)

Displacement speed 0.8 Absolute speed 0.6

0.4 Flow speed 0.2

0.0 0

1

2

3 Normalized radius (R/r(t))

4

5

6

: 0 0 F 0 ' 1 & ! AB # A# /B 2 & A K (A3BBI AB + ) A B 3A(A3BB + 4- K 4 > 4- K

(A3B K 4 3

AB

> 4- (

4

,6

.> . > >

+ + 1 2 K K

C I

D + D + ++ 12 + 1 2 K K

IÆ

IÆ

. 0 0 ' F ) L.,

M L M : & 1 & # 3 & 3 1 ' # ) I ! Æ ! 7 # A & B # 2 & & *# '( A BI ! # ) > ) ! ! # A ( K / ( K (A3BB # & A ( K (A3B

B 1 + Æ ( K (A3B L Æ# ( + A(B + #I ! # & 3 >

½

+ K +

J

(A3B L A L BÆ# J(A3B +

AB

H# A# 6B ( ) # ! '( A"B ) 4 (A3B % & > 4 K

Æ# L A L B (A3B

(A3B 3

AB

# & I ( Æ/ # 3 . A-B ! 2# :# # ) AB ! 7 # & Æ A5 B>

& :G* '( AB # 4 ! # 4 ! ')# 7 # ( # # % & # A ! I 5 B

)* + & % )3 ) ! > & I )# # 3 # + ) & I )# )3 ( ! > A % B & 1 & ) & A; I 8# I 5 I 8# B ( & # ! & %# + A ) B ) &I ! > A ! # B )3 A B # )3 3 ! # 1 ) 3 ! & )3 # >

! &I #

! # I )3 ) (! 1 )# )3 > & ! # )# )

1 & # ! ! &I )3 # # A# B + . I & 3 A# BI # ! 3I * -

-

Temperature

YO=YO0

Fuel mass fraction YF

YF=YF0

Oxidizer mass fraction YO T=TO0

T=TF0

Heat release

Diffusion zone

Reaction zone

Abscissa

Diffusion zone

I/ 0

2 & ) ? @ ) & A ,B> & # )3 # )3 2 #!I 3 #%! ! & % 8# # ?# @ & I & ! I ! I ) &

2 & ? @> & # # %! & # ! ! # 2#I ! ) & !

& & A B

+ I )# ! & 1 & # > )#I # ( I ( 1! ! . # G I # Æ! ) & )# !

7! & ! & * A) I & I )# B # & ! > % )# )3Q A & B )# * & ! %! * I # )# I ') ! & * "

Fuel mass fraction Y F at t= t1

-

Temperature at t= t1 Temperature at t= t2 > t1

Fuel mass fraction Y F at t= t2 > t1

Abscissa x

(a) the unsteady unstretched diffusion flame x2

Fuel mass fraction on the axis

Temperature on the axis

Oxidizer jet

Fuel jet axis

x1

(b) the steady stretched diffusion flame

N / 0

!"

* 7! & 3 # A ). B # ! 1 I 4! % & * " & Z!I I &

# #. A) K B >

" ¼

* +

"¼¼

AB

!3 3 & #>

G . ! 7 G . Æ & (

! I

I

-

G . ( . > K

2 '

% AB ! A1 BI )3 AB A B> " 1 L " * AB + "+

1 A1 I I B ( # !> L 3

!

A B K

!

!

L -N

AB

* -N #. -N K "

A'( B>

AB

)3 > -N K 4-N

" "

4 K

A B

4 % '( A,B 1 ! # '( A,B> -N K

=-N

A"B

$ # I ( I )3 > L 3

!

L 3 3

!

L

A B K A B K

!

A B K

! !

!

L -N

A,B

!

L 4-N

A-B

=

!

%

!

-N

A6B

# '( A,B A6B ! I # ! ; A' K % K B ( > K K 4

Q

K K

L =

Q

K K 4

L =

A/B

( I > K L 3

!

A K B K

!

K !

AB

-

K A B # 1 K ! (! L C ) & A B> K K A L CB L = + ! ) & I L C ( ! ! A '( B & K )3 ) G I K #

)# ! 1 K I K K I % '( A/B ( ! 3 1 A % # B

- *

3

* 3

O.F * 3

3

3

-

-

-

3

; * 1 + 3!7

.F .F *

= 3 % !> K

K K

K K

# K , ,

AB

2 E ( > L 3

!

A B K

!

!

AB

! > K K / )3 A1 B 2 (> K K K

AB ) E )3 ') # # ! # 1 >

4 L

K K 4 L

B L K A B L

0 1

0 1

A

0 1

B L

A B

A

0 1

A B

> ' " ' , , '' # ' , 1 ' "&, # # # '

-"

1 % '( A B ! > K 5L

5

L

A"B

( 5 > 5K4

A,B

1 ( & ( )# )3 + A # B ( I # 5 )3 & A B 1 . & ! #> ! ) & 8I ! ) & I !

8

*

O.F *

O.F ) $. 4

7

7

!

7

; # . 4 1 + 3!

Þ ) 1 ) '( A B I )I ) > K 8 A, B

K ,

A-B

' # % I # # >

G . & ) 3 ! 1 > K A, 3B

K A, 3B

A6B

-,

z gradient x2 Iso z surface y3

x1 Tangent plane to iso z surface

y2

x3

8 / 0 P

# -

1 A, 3BI A, 3B '( A6B % 1 G < % ! !I # ( AB A! I !I ! I 3B A I B I B I B B B . A# B + # ( I # # # & A# # B B B # & A# # B A H I 5 B 5! !I ! & ! . !

& A B + . & I ( & & # ' & & > & ! A "B $ ! I ( AB ! >

L 3

L A B 3 !

L 3 !

L

! !

!

K -N

!

A/B

1 ;G* '( A/B ! ) A'( B ( >

K -N L 3

! !

K -N L L

AB

--

L >

LK

!

!

AB

1 ( !>

K -N L L 3

AB

'( AB AB & ( 1! ! ! & > ( I ! # A!B L )# A # B 0 L %I & ( ! & I ) 3I 2 G> K A, 3B

K A, 3B

AB

2 # ) I 3 ! L> & . 1 L A B + .# &) & + ! & ! > & I L A B 2 )# ! L

+ ! $

1 & '( AB AB 3 ) + ! >

$ ! & 1 I # ? ! & @

*! & 1 & !I # & A ! %B + > K A B

K A B

A B

F ' ,

, ,

0 ; 1

4 E , . 7 7 , 7 7 7

-6

1 ! I ! & ) #I ) )! = 3 ! $ # '( A BI '( AB AB > -N K

L

-N K

)#

L

A"B

)#

'( A"B L ! ! L # & . 1 % 3 #

'# $ 7) & . ! & A%# A! , 3B A! , 3BB >

2 )# A B '( AB ) % A! , 3B

! 3

2 & A B & A A BI '( AB A BI '( AB B &

8 & > # ! % K K 4 I K K = L K K 4 = L I 1 I & & ) A BI A B A B * # 3 # & !

. > & )# 2 ! & ) * ! & 2 & A# !I ! B )# 0 I ! # &

3 $ 1 ) ! & # >

6/

Boundary conditions Initial conditions Geometry Flow field

Assumptions on reversibility and chemistry speed

2: Flame structure problem

1: Mixing problem determine the field of the mixture fraction

Scalar dissipation rate field

specify: - species Yk for k = 1 to N - temperature T as functions of the mixture fraction z

z(x i ,t)

T(z)

Yk (z)

Solution of the full problem: T(xi,t), Yk (x i,t)

I : > 0 . ) 4 4 '

'( A !B> # ! %! . & A & B ?(@ A ? . !@B + % # I I ! & A B 1 ! ( !

+ !> # AB ! # A '( AB 3 B %

# %! ! A(B ! 1 %! !I )3 ) 1 % !3 * 1 ?(. @ ? !@ GI A B A B # $ %! I !I I )3 ! ( > +

K A B

A,B

A B ( 1 + I )3 !

! " ! " # " $ ! %

# & ' $ ( ! )$ * " !

&

! "#" ' ( ! "#$#*

! "#$#% ) )

+ # + , " (-)

. ! ' +- /0 , " $

. 1 2

1

1 2

1 2

/3

1 2 "

$ 1 4 " ' - +- /3

$

/

1 2 # 5 # # ! " # "

! +- /3 1 2$ # 6 * " - +- /7

% ( ! ) *

1 2 ( ) $ 1

1

/

" ! $ 1 4 2

TO0 Temperature

TF0 0

YO0

0

1 Oxidizer mass fraction

Fuel mass fraction

z YF0

1

z

( , , # " " ! & 8 # "$

# "

, # - 8-

9 / # : / " $ !

# ;* :

% # 1 2 1 2 ! +- 0$

1 4 1 2 1 4

1

4

$ 1 2 1 1

4 4

/

" # 8 # " ! ' - +- & $

1

4

1

4

1

4

" 1 - - $ " # ! " < # < 22&$ #

( ( ( ( 2 2 ( (

-

- ( - /

- ( - /

- ( - /

- ( - /

* * * * $ $ * *

* %" * %" * %" * %"

* * * * * * * *

$ $ % % % % $ $

. . 0 0 0 0 " 30 " 30

. "0 . 0 *1 0 % .1 " 30 *$ .

*** %. % $0 ."" $"$ %*3 $4

+ +# "#$' +# "#*1 +# "#""# 2 5 $ 6 4$ 6#

# ;* :

0 5 ( ) " ! " " +- / - # " # " " * " $ , # , " ! * 8 # " %

, # # ' " - = # 5 " # > " " ! " #

&

Tad Equilibrium lines Temperature : possible states TO0 TF0

Mixing line 0

1

zst Equilibrium lines

Fuel mass fraction

z YF0

Mixing line

0

zst

1

z

1

z

Mixing line YO0

Equilibrium lines

Oxidizer mass fraction

0

zst

2 7 8

5 9 : ! , #

= # # , # 1 +- +- /$

1 4

4 1 4

4 4

+- " * - $ " ! ! 7 4 ' +- / ; -

$ % & '

( )*

0

4 4 +- % - ? " " @"

- Pure mixing

FUEL z

Fresh mixture T=zT F 0 +(1-z)T O 0 YF= YF0 z Y O = Y O 0 (1- z)

Pure mixing 1-z

OXIDIZER

Combustion Burnt products T=zT F 0 +(1-z)T O 0 + Q YF 0 z /Cp

if z
T=zT F 0 +(1-z)T O 0 + Q YF 0 z st /Cp (1-z)/(1-zs t) if z>zs t

+ 7 8 ;, #

% , # " # # +- //3 % ! - 1 1 , # +- $

1 4

&

6 " " ! +- / " " A +- //3 " # $ 1 4

0

" , # +- &$ , # # " " !

6 # " . ! / # " . 3 -

7

- ! # " " +- / $ . 1 7 . % # ! 1 +- $

1 4

4 1 44 4 4

3

# " - -

Stream 1: Temperature TF0 Fuel mass fraction YF0

Diffusion flame. Maximum temperature Tad

Outlet plane. Mean temperature Tm

Stream 2: Temperature TO0 Oxidizer mass fractionYO0

< 7 # 8 # " . " ! # " . $

1

. . 4 1 4 . 4 . . 4 . 4 4

! $

1

. 1 . 4 . 4

1

. 1 . 4 . 4

- 1

2

3

$

4 4

1 4 4 4 4

1

4 4 4 4 4

/

" - B , # '

" ; - 1 ? 8 , # - 8

Temperatures (K)

2000

1500 Mean burner outlet temperature Maximum temperature in burner 1000

500

0.0

0.5

1.0 1.5 2.0 Global equivalence ratio φg

2.5

3.0

< - 7

# *' %"' " # = Æ 0' *1 0# = *0 $*

> #

$ ' & & + % ( % &

%

& !

( , ' - ( - . & ( ! " * %

' ' ' & /

5 # :8 / 8 - +- $

4 1

" , B - #

: -

@ @ 6 9

! " #

' 8 , # "

# ' # / * "

5 * " C " # , % % # 1 " +- " $ 4 1

"

" # % / #

C

, # Æ # " % # +- $

4 1

&

$ ( ( 0 &( ' 1* 2 & ( & & "* * 3

22

Ignited flame started at t=0

z 1

Pure fuel

Pure oxidizer

x x=0

> 7 8# ' , # ! # " ' 1 2 ! 2 ' 8 $ 1 2 1 0

" > ; $

1

D

4 1 2

7

% # " # " ! " 8 ' , Æ 8 -

$

1

3

, - E 1 / $

4 / 12

" 0 7 $

1

F /

G

$ 1

1

"

/

¾

/

&2

&

% 1 4 1 * 0 ( 4' * ! Æ

2

" 1 @*

,

' $ H * H 1 / , "

1.0

Mass fractions

2000

Fuel Oxidizer

0.8

Temperature

1600 0.6 1200 0.4 800 0.2 400 0.0 0.0

0.2

0.4 0.6 0.8 Mixture fraction z

1.0

0.0

0.2

0.4 0.6 0.8 Mixture fraction z

1.0

- 7 8# =

" ?# = $0 = "#0#

' # 8 # 1 22& # //22 I

1 &222 *JB* / 1 22 JB*BI +- &2 1 H$ / " : * " / # $

1

1 / 1 4 /

&/

# 1 # 1 //3 + # " # . +- 7 # K " " " #

$

5 ' ( " ' * . & & 6 % & 6 0 / 7 & / 6

$ ' ' & 0 ( " ! & &

2/

1.0 Fuel YF Oxidizer Y O Mixture fraction z

0.8

2000

Temperature

1500

0.6 0.4

1000

0.2 500 0.0 -4

-2 0 2 Reduced abscissa η

4

-4

-2 0 2 Reduced abscissa η

4

- 7 8# / 3 ?' " ?# = % # K. 1

·

. 1

K. 1

·

1

1

1

/

¾

&

&

Reaction rate Fuel mass fraction

Abscissa x f-

xf

x f+

+ 7 8 , 5 #

! * $ , "

! " " # ' , # * "

% * ( ' . 8- / /

·

2

! " # Oxidizer

x1

Flame (x1=xf)

x2

Fuel

! > 7 8# , # / , # # # , # # " ' $

1

1

&&

" - 1 2 - # $

4 4 1 4 &0 + 8 8 A - - "

8 $

4 4 1 4

&7

0

( ' "* '

' & % , & & $ " 9*

' " ( ' Æ & 1* ( & ' & " & & , * , . : /

2

5 # , 8 - $

1

&3

4 /! 12 ! !

&

E ! 1

/ $

" +- " +- 7$ 4 1 2 $

1

F /

! G

! 1

"

1 02

/

6 ! # # $

! 1

1

/

/

0

" / ' # # " 1 / ' # # / . $ # K K. 1

· ½

½

. 1

. 1 K

· ½

½

1

½ ½

/

¾

0/

0

" ! # $

! 1

/

1

/ 1 4

0

2& # 2 1 2 2& 2 # ! # "

" 1 1 2& "

2&

' " : // * - , # 8 # " +- 02 $

1/

1

0&

1 2 6 +- 02 0& $

1

/ / 1

/ / 1

00

" # 8 1 2 # $

Scalar dissipation (1/s)

1

/

07

4

10 8 6

Flame position (z=z st )

4 2

Scalar dissipation at z=zst

0 -4

-2

0

2

4

Reduced abscissa eta

!

-

8 " #

& " # / ' 1 &2 - 2

$ % & & & ; : / 7 / 6

20

2 +- 07 "$ 1 2 E 0& 07 " % 8 # " 5 * # " "

- +- 07

! " # % 8 # " : 8 ,

# " 1 2 8 - &3 $

1 03 " $ 1 4 1 2 8

! ' $

1 "$

"

/

2 1

/

D

"

" 1 4 1

0

72

: +- 0 +- 03 $

" " $

" 1

" 4 " 1 2

4

7

7/

' #

27

, # " 2 222 " 22 " 22 2 # " , +- 0$ # # 1 " I " # (- ) +- 7/$ 1

" 1

4

7

' 0 " - , # 8 2 1.0

aeq / a 0.8 0.6 0.4 0.2 0.0 -1

0

1

2

3

4

5

at

& > 7 8

# / ' 8, 5 * # = +# "#1" # # - $

' , #

# &/

' , # "

- L

- # " ,

# 6 0 > "

@ * @

"#0#"

"#0#" B 8 %

"#0#% B 8 *

"#0#* ! 8

¾

@ *

*

*%

½ ½ ·½

B 8

¼

* * % %

)* +! !

A

!

<

' ! " ( !

8 # / ' # ! #

! > 7 8 , C ( # = 8 8 * , D * # = 8 %

@ * @ * *

*

* @ * @

! " # $ % &

/ 8

* * * @ @ *

+

!

!

23

2

! & ' # 6 9

1 2 1 2 1 " > %

1 4

$ # ! "

' , # " 8 < # 7

x2 z

Oxidizer

Fuel

Flame 2e0

Lf

E 7 8# / F % F , 8,# ' < < ! # " ! # " 8 $

# " $ 1 1 #$% % 8 # "$ 1 1 2 1 1

A - 0 $

1 2 1 1 1 1 #$% %

7

8 - , - $

1

7&

" $

$ ( <

9 ( = ( 9' ' & % ( * ( ' !

2

1 2

1 2 1

1 1 2 ¾

¾

(a)

x2

(b)

YO

YF, YO, z T

z

T

T

YF z

YF YO x1

L f

< ' F 7 8 # "#*1# C ' ' ' '###

- 8 = "#0# +- 7& $

1

¼

/

¼

&

&

70

" 1 " < / +- 70 $

77

> # ! +- / 9 < 3 3 # ' 1 ' 1 2 1 +- 77$

' 1

4 1 ( 4

4 ' % , & , < &

73

" ( 1 < E # " < < E (

%

& '

, # # > " 8 L ! @ 6 = 6 9

% ( 8" ) ) $

)1

7

¼

" 1 2 # 1 > " 8L ! " - = 8 # " - $

1

* 1 &

- $

1

32

3

E ) +- 7 $

) 1 1 ) )

3/

- 32 $

1 )

1

1 )

3

) 1 2 1 2

- 3 " $

) ) 1 )

)

3

$ ' "* 4 3. & ' & > ( *

"* %

/

1 $ 1 ) ) )

3&

" +- &3 " 1 4 1 2 & ) 6 9 $

1

+

)

30

> " 8L ! " # A * ( 8" ) ) +- 7 "

" , #

% (' ) = , # ! # +

# , 8 $ ! - /7 ' 8 #

# "

!

% * ' : = # # " " , , # - L *M $

, 1

-

-

$ / & ! "

37

Equilibrium lines Temperature : possible states TO0 TF0

Mixing line 0 Fuel mass fraction

z

1

zst

YF0

Equilibrium lines Mixing line

0

zst

Oxidizer mass fraction YO0

1

z

1

z

Mixing line Equilibrium lines

0

zst

2 7 8 5 >

#

" - - # " L # " - Æ # / # "8 - # L *M " # / # L *M # - # " - L *M = , * * % , ! " * * # ! 5 ! 8 ,

# /2 " " " ! "

! "

; ! # Infinitely fast chemistry Temperature

Finite rate chemistry TO0

possible states with infinitely fast irreversible chemistry

Mixing line TF0 0

1

zst

Fuel mass fraction

Mixing line

z

YF0

Finite rate chemistry

Infinitely fast chemistry 0

1

zst

Oxydizer mass fraction YO0

z

Mixing line Finite rate chemistry Infinitely fast chemistry

0

zst

1

z

+ , 5 5

#

L - ,

# " C 8

" - # " # ! ' +- / /0 * - , # +- 0& - # " - L *M $ $ - N L *M , 8 @ 6 9 N 9

# & ! " &

&

= " +- / /$

. 1 .

L *M $

, 1

.

/

33

3

" ! ! - $ 1 // / 1 / # +-

Infinitely fast chemistry

Scalar dissipation at stoichiometric point ( χ st ) or Flame strain (a) or Inverse Damkohler (Da -1 )

Finite rate chemistry

Quenching

A ' > 2:G 5 5 #

# " L *M . K @ 6 @O

P = / " "

L *M $ , # 8 ! ! . 8 & +- 0 " K * - ; Æ $ Æ * " " ! ! + " - L *M

0

, @" " - L *M 6 9 $

, 1 & "

1 /

/

Q

/ 4

/

2

' 9 - $

1

-

/

" - # # $ - 1 Æ 1 * " , # $ # - , #

% # 0 ! # - 5 - # 5 , - " " L *M

" ? - # #

% & ' +! @" " - 8 @" " " * + : 9 : 6 9 : 9 9

FINITE RATE CHEMISTRY (NON EQUILIBRIUM) Temperature

Temperature

Infinitely fast chemistry limit

Independent of strain

Finite rate chemistry (low strain) Finite rate chemistry (high strain)

IRREVERSIBLE REACTION TO0

TF0

Mixing line 0

TO0 Mixing line 0

1

zst

1

zst

Temperature

Temperature

TF0

Infinitely fast chemistry limit Finite rate chemistry (low strain)

Independent of strain

Finite rate chemistry (high strain)

REVERSIBLE REACTION TO0

TO0 Mixing line 0

zst

TF0

1

Mixing line 0

zst

TF0 1

FAST CHEMISTRY (EQUILIBRIUM)

! 7 8 ' #

7

3

, - # + ,

# # $

C *

:

: , Æ - ,

: *

> / , , , "

#

* Æ > " 8 " - 8 , # #

: / // 8 , # N 22 I 7&2 I 22 , # 8 # L 6 ; " 6 9 9

, # ? , # - > "8 - +- &

5 8 " 6 0 1 2 " $

¼¼

3 4

2 1 1

+ 5 8 # 5 1 $ 0 68 6 1 5 5 $

7 1

1

0.060 1.0 H2 O2 OH H2O

0.6

H O

0.040 Y

Y

0.8

0.4

0.020

0.2 0.0 1.0

0.5

0.0 0.5 x [cm]

0.000 1.0

1.0

Temperature [K]

2e04

Y

HO2 H2O2

1e04

0e+00 1.0

0.5

0.0 0.5 x [cm]

0.5

0.0 0.5 x [cm]

1.0

0.5

0.0 0.5 x [cm]

1.0

3000 2000 1000 0 1.0

1.0

!

> 7 8 , # ! " # H # C = *#0# " 6 8$ 8 6$ 1 * " ? " 2 $

¼¼ 1

1 2

&

C - 7 8 1 0 - 7 $

7 7 4 7 1

4 9

0

" , Æ - , 1 9 7 $

9 1

:. 1

1 ¼¼

1 2

7

/2

:. +- / & 8 B, - " 7 $

7 7 4 7 1

3

8 # // 1 2 "$ 0 0 1 0 5 1 3 +- & ! # $

4 1

4

' 7 $

7 1

/

¾

¾

4

/¾ ¾ /¾ ¾ 4 4 4 4 ¾ ¾ ¾ ¾

22

+- /$

1

7 2 ¾ ¾

/

2

$

7 1

/

¾

¾

4

¾ /¾ /¾ ¾ 4 4 4 4 ¾ ¾ ¾ ¾

2

2/

$

1

/ 2 /

7

¾

¾

¾

¾

2

' - / +- & " - "

/ # $ " # # , C ! - @" , /& " $

8 " // @"

/

0.20 Scalar defined on O element (zO) Scalar defined on O2 and H2 (z1)

Normalized scalars

0.15

0.10

0.05

0.00 0.00

0.05 0.10 0.15 Mixture fraction defined on H element (zH)

0.20

H +# "#44 +# "#*" +# "#** > 7 8 # "#%%#

1.0 Real Hydrogen Real Oxygen Ideal flame structure Hydrogen Ideal flame structure Oxygen

Mass fractions

0.8

0.6

0.4

0.2

0.0 0.00

0.05

0.10

0.15 0.20 Mixture fraction

0.25

0.30

0.35

>

7 8 # "#%% , 8 ! "#%

#

//

- /

- @" - # # +- * ; 1 4 " ' "

- " @"

" ! 7 # 8 D - / - - @" 8 Æ 8

# *

7 /& - - 5 / $ 8 #

5 > " " -

$ , " * - @" # B # " - " /

C L #

-

-

L

@" 1 @" 1

@ ? ?

'

C > 7 8

" * > 8 , % , " , #

' # // , # /0 " # B # , Æ @" >

, , # 9 9

Method 1: 1D flame (full chemistry and transport) a = 300 s Method 2: Equilibrium calculation (with single step reaction) Method 3: Equilibrium calculation (with full chemistry)

4000

Temperature (K)

/

-1

3000

2000

1000

0.0

0.2

0.4 0.6 Mixture fraction (zH)

0.8

1.0

H > 7 8 # "#%%#

Max. Temp. Sens. Coeff. [%]

5

0

-5

-10

-15

p = 1 bar a∞ = 100 s− 1 TF = TOx = 300 K

-20 N2

O2

H

OH O H2 Lewis Number

H 2 O H O2 H 2 O 2

! 8

- 8 ;, C >

& + ! ( ( #

! " # = # " # # 8 " * "

: Æ ' Æ - # " ; # " * $

6 "

: # * # - Æ 6 /

8 # 8 N 8

4 0 ( & % ? ( % % % * & & ' % '

/&

/0

- Æ *

" "

= # " # " 8 # " # * " (# 8 ) ! 5 # # -

<

" # / R 8 # E ? : * E'?: - # " & % L?: 8 0 % @+: " , 7 % - C 6 & # 6 0

$ C "

" * > ! @ 9- ! # Æ E '

" " # " $

" 1 " 4 "

/

$ '' ' ' & ' &

& "* + "* ( ' ' Æ ' ( ' . / 6 36 $

/7

! ; " " - # " " $

;1

" "

% " " # ; 2 # " " 8 # "$ # " ? Æ 8 ' " , # " " " # # " , ! # " I I # 8 > * 8 > * = ==$

>1

= = 1 == =

0

$ ( & '

( & & & 5 & ( & 5

/3

' E (= " ( " # " I > * 8 I $

1

7

>

E $

( 1 ( 1

> 1 1

3

 1 1 ( >

' " # /0 # - 8 % # +- /3 " * $ += # ! = " == " $

+= 1

= > 1 = =

2

% " ! = $

= = - = 1 1 = >

I < $

+ 1

>

D

> + < 1

8>

/

" * 8 ? I ! $ + < 1 1 ( + <

:

/

E ruí(r)/ν uílt / ν

uí(r)

3

4 3

uí 1

1

uíK ηK

lt

r ηK

r

6

lt

: uí(r)/r uíK /η K

r/uí(r)

3

lt/ u í 3

2 2

uí/l t η K /uíK

r ηK

lt

r

ηK

lt

+ ' & ' > #

= E (= = = += 1 == = 8 % * # "$

" , "

" "8* " # < , # /

@ ' & * ! " &

2

- 6 8 # # " / Burnt gases

Turbulent fresh gases

sT

Turbulent flame brush: speed sT thickness δT

! > 8 8, 5# % ( ) # # # " " + * " # % * @ Æ 3 = = " " " # " 8 ' @ Æ ( # " " " # ) " = # '8S S$

4

" - # - * 8 " # # # + 8 " # * Æ "

# * Æ

$

"

& 4

= - 8 , , ' '

* 6 # =

* I ' ;Æ #

.

% < <

Ambiant air

d

U Flame

fuel

Lf

E 8 5 # # ' - & E ( @" N + $

( 1

?

" ? * <

/

Transition

Lf Laminar

Turbulent

Blow-off

Lift-off

; F 8 F & D D, #

Re

! #

# ' " # " E

( @" N + Æ E # " 8 # ' 8 E ( "

# < Æ < # ( # ) ( "8 ,) : # ' " E <

# " 8 $ Æ " 8 , "8 ,

6 L 6L # " $

E'?: E ' ? : *

# " # " E'?: - " - - E 8" - - - - $ " # " " 6C : - - # " , ! # " * ! # < # " E'?: 0

Temperature DNS RANS

LES

Time

= , 2)!' &/)! ;+! 8 #

@+:

" , - - 7 @+: # " ( ) # - * , < # @+: " "8- 0

L?:

" ? 8: * - "

$ ,

$ < # L?: " 0 * 8 " L " * L?: # "

E'?: @+: L?: ! 7 ' - " 8 , " 8 @+: , 8 , " 8 8 % E'?: # " % - 6L 8 # " " $ L?: " E

@+: " * "

E(k)

Modeled in RANS Computed in DNS Computed in LES

Modeled in LES

kc

k

= , # &/)!' ;+!

2)! # >7 , ;+!

> # " E - - * % 6C L?: L?: 8 # - / ! & & & 3 " L?: ! # " 9 ; ; 6C

$ 6C 2 6 ! " $ " # ! " 6C = " L?: 6C - , ! H 2 &2 L?: & 6CT L?: # "

# 3 %

! " / B # * 2 ! - % 6C # " $ " ' -

&

2)! 8 , > 9 # #

/ # = 8 # H 5'

' #

= C < =C< - 8 , &# ? :: ' ' %#

0

, " # # ' 8C ; 6 ; * / " / + 6C - <

# # " " # * @ 8 " L?: 6C @+:

# "

> C I '* C

# " " " = I S I S S C N 9 ' ; 8 - % ( ) # 2 @+: ? 8 # ! # : # " " " @+: " E'?:86C

0.00

0.05

0.10

0.00

0.05

0.10

; + ! - 8 , # 0#4#

, ) # @+: $

@ # "

" '

C # " 6

: 8 " " # " " 6 3 @+: "

7

* " ,

% "

@ " B

@+: $ ! " - , & & 2 @+: E'?: " *

6 " E'?: @+: L?: # !

, &/)!

;+!

2)!

,%+!

> J K > 5

%2 8,' '### > J K > >

&/)! >

- >

-.

> 8, 5 >

> > "2 > > >

5 ' >

C , &/)!' ;+! 2)! #

- % 8

# 9@% B # ! "

# ' # ! 8 " L?: ' Æ # (# ) 6 & # / ? L?:

3

2 1 0 −1 −2 0

2

4

6

8

10

12

; 8

- 8 > 8 # 0#*' , # ! &# ? :: ' > ' %#

2 1 0 −1 −2 0

2

4

6

8

10

12

8 - 8

> 8 ' # 0#**# ! &# ? :: ' ' %#

: # 8 E'?: @+: @+:

/ @+: # && ' / # @+: + # E'?: /22 % ( ) # # " # 8 # ! - * # "

# * % " ! " # 8! ' #

/ , " % - $

' E'?: - @ L E # "

L?: @+:8* - 8 # "

" 8 # " L?: " " @+: 7 : L?: @+:

- 8 8

2

2 1 0 −1 −2 0

2

4

6

8

10

12

;+! 5 - 8

> 8 ' # 0#*%# ! &# ? :: ' ' %#

2 1 0 −1 −2 0

2

4

6

8

10

12

/ 5'

&/)! >

% # 0#*%# / 8 # ! &# ? :: ' #

%&' ; - - E'?: - -

%

/' 00 )

- $

4 1 2

&

6 - 4 4 1

0

4 1

@ 4 .

A ,6 4 A 1 . 4 4 / ,

8 1 0

7

@ A 4 - 3

% # " E

- " " # " " 1 " 4 " A " - & $

4 1 4 4 1 2 4 1

" - " # - # " " "*" 6L $ # " E - # " E # "

"

- " # " Æ 8" = I $

" 1

"

/2

/

'

- " # $

" 1 " 4 "

"

" 1 2

/

A - $

12 4 6 4 1 4

4 1

!

/

4 . 8 1 0 @ 4

,6 A A 1 . 4 / 4 4 , "

//

A 4 -

/

@ A!

/&

6 6 ,6 6 6 6 1 4 1 4 4 ,

/0

- " E - # " + , Æ # " " $

" " E "

'

" " - - * " #8 " " - $

" 1 " 4 "

/7

6 " " " 8

C - E

" " , " " "

&

%

*' ' 0 )

< * " - 8 +- // /&

- E C " * 8 # " 8 > "

> , E *

" A #$ # $

1

9#B

/3

" B 9# : 8 ? ; : " " # $ 8 +- /3 " * # & &3

% & #

Æ8 E , B +- /3 , # $

@ 1 ,

,

/

" , ( ) , Æ ? , # - /& " $

/

1/

" / ,

2

. $ "

5 - 8 " - - , - 0

% 1 # ( " ; - > !

* @ E - " # +- &$

1 B 4 1

4 / 8

/ Æ

" B B 1 " * Æ I * +- * 8 $ 81 /

/

- " B $ " -

- 8- " 8- ' ( ) *

9 * $

""

"9 B 1 < " ""

" 9 $

9 1

/

< N < # " $

& '( "* * & *' > ' $ % ' "* 4 ' ' > + ' %

&

( ) * +,

' - * 8 $

B 1 < 8

&

" 1 22 < 5 - < -. ( )

8

>

% J @ $

B 1

8 >

0

" * 8 > " - $

8 4 B4 8 1 > 4 B4 > 1 5 $

5 1

B 8 4 5 > C B > > 4 5 C 8

7

>8

3

" E ; - $

1 22 D C 1 2 D

C 1 D

1 D

1 /

2

, % 8> > I " ? "8 * " " *$

+ - 8 > Æ "

+- 7 3 E

" E

# "

' " - # " " 6 7

0

N # " - # 8

8 > # "

, * C * > > > : * U

" 8 8 > ! " * 8 > I

!

A 8 > # " : * " ' : C ':C % ; - 8 E " * 8 > 9 " - E E : C E:C C " > * @

% / " ' Æ 8 6 " ! 5$

4

4 5

" . ' " $

. 1 . !½

"

" . 8 " ' 8 .

7

. $

"

1

"

# 4 5# # #

$

1 $

D

D

# 1

# 4 # # #

" 5# # $

5# 1

#

#

Q " % 8Q F8 QG 8 %

DD 4 D 4 % %Q

/

. ; $

" . 1 . 4 4 5 4 4 4 5 4 4 5 4 4 +- Æ " - # - ; !½

" " * + $

+- 6 & 0 ? " # " # " N8 " ?" C % " * ' 1 " $

" 4 . 1 . !½

&

" ! 1 2 1 -

3

%% / 0 # 02 0 ;

6 & 0 . ! " ' " " * Æ " & 8

+- ' 8 # & " 0 #

( !) # " # 8 # E'?: E'?: ' " & # " * # " # : * # E # " @ # * " 6 / 5 # , $ 0 : ! # # '

V 7 " . V . V '

V $ " # - 2 * E'?: " * ! & " # ' ; 6 ' Æ # " - 3 ' " - E'?: $

' D

E'?: * "

y Fresh gases

Limits of turbulent flame brush

A Instantaneous flame front

Burnt gases

C Flame holder

Burnt gases

Transverse cut axis

B

= 8 #

y

y

Laminar profiles

Turbulent profiles

Temperature

Reaction rates

C 5 # 0#*$#

&2

Reduced reaction rates

Laminar rate profile ω(Θ)

Turbulent rate profile ω(Θ)

0 0.0

0.2

0.4 0.6 Reduced temperature θ = (T-T1) / (T2-T1)

0.8

1.0

& 5 8 # # " !

$ - , # # , ? E " 6

8 # " & Æ *

6L * " "

" ' E'?: " 6 7

, " E'?: @+: ' @+: # * " @+: # 5 E'?: # " * #

$ 6

&

t hot T2

T

T1 Time t cold t

( 8 5 #

( % L ? : L?: 8 : # " - & 3 "

L?: , " " # B 8

S S 9 9 N 9

,

# 34

' W - # " " 8 - %

- ' B

# " * , # " # 5 * " ! L?: "

" L?: % L?: 6 8 & # 6 0

&/

, 4 # L?: # " ,

" L?: 8 " E E " " " # " - - * # " 8 " E #

8 , L?: " " $ ) / .

: # " - 6 - C L?: 8 # " 8 # " # "

L?: # " ( ,) "8C

" " # " = # # 8 # " % , * 6

# E ! - A , ' *

# ' "8C " % " 6 88@" 6@ "8 " # " ?

" # "

6 3 % - "

$ 4

+ )* * ' A

' A = ' ( A A * $ 4+) / AA

* % 0 ' * 9 * ' '

&

N% N%% +- &2 L "

"

L?: # "$ # B , # " , 6 = 6@ "8 # " ? "

% # " - " C ? # " L?: # " - : B # - " " 8 * - : # "8 " 8 " * * # 5 # - - N # * X "

Æ C # " E # " # , # * " ! # " 6 9 " " A " ; > 9 @ ; " C 5 * E ! U E - " ' , ! "

&

# " 9 @ # "

, 6 " 8 # " " ; >8 - 8 # " S 6 % Æ B - # " " - ' # - # 8 L?: % " " # " * # " ! % < "8 # " Æ $ N I N # " # " E 6 "8C S J ! -.+ + 01"

; # " 8 " 8 , 8 # " # 8 " 6 " /L # L # &&& " 8 # # " # 8 6 8 # " 8 " 8 - " " (# B ) " # " 8 &/ B

@ 6 9 ' "8 " J * E L E @ E L C 8 # N " 8 %

, " 8 " L?: ; 8 L?:

&&

" 8 L?: " 0

L # " > # - 9 * I 9 # " L - - : = " " : L< L 1

C L?: # " # L?: # " 8 , @ "

" , 8 ? + ? : 8 8 " 9 @ ! " L?: # , 8 L?: # " " " * X # " # # " - E = # " , #

, # 69A 6 62 %

8 # " ' " , # " - 22 69A ' /L L - $ !

' " > 85 #

" " 9 * ; C 8 # "

3D Tanahashi et al 1999 (TSFP)

3D

Rutland and Ferziger 1991 Poinsot, Veynante, Candel 1991

Leonard and Hill 1988 Ashurst, Peters, Smooke 1987 Rutland and TrouvÈ 1990 Cant, Rutland and TrouvÈ 1990 Laverdant and Candel 1989 ElTahry, Rutland, Ferziger 1991 Rutland and Trouve 1993

2D Barr 1990 Katta and Roquemore 1993

No chemistry

TrouvÈ and Poinsot 1994 Rutland and Cant 1994 Zhang and Rutland 1995 Swaminathan and Bilger 1999 Montgomery, Kosaly, Riley 1997

Ashurst and Barr 1983 Osher and Sethian 1987 Ashurst 1987

3D McMurtry and Givi 1989

Ashurst, Kerstein, Kerr, Gibson 1987 Yeung, Pope 1990 Cattolica, Barr, Mansour 1989

Constant density Variable density

One-step chemistry

2D

Constant density Variable density

2D Patnaik and Kailasanath 1988 Katta and Roquemore 1998 Baum et al 1994 Haworth et al 2000 Chen et al 1998 Tanahashi et al 1999

Constant density Variable density

Complex chemistry

&0

NB: Non exhaustive list

+ 2)! 8# " *

" 8 = " * " L S : * L " L?: @ 8 ENC " ! - " ? 8: * - ENC "8 @+: S I S S

, L $

# "

&7

# " I

#

" " 6 " ! ' 0 ! H 1 '0 # " ! 8 # < ! < ' 1 0 H < I $ < ( +- !$ H 6 $

< 0 $

0 ( -

0

( 0

7

- 0 -

E ( E

# 8 " # # * - , 6 0 L?: = 8 8 "

# - " /2 % " # * Æ /2 % # * ! ' 0Æ # Æ 2& 2/ " ! ' /& < "

' $

< ' 0 Æ Æ

3

&3

" , # * ' Æ Æ /& * $ Æ Æ L *M , 1 - - - - " - < 1 - 1 < - 1 Æ E L *M $

< < ( , 1 1 Æ Æ

$ Æ Æ

( ,

0

&2

E 7 L *M &2 " " ' 0 1 222 1 /2 ( 2 7 ( , /&22 &2 E ( 1 222 L *M

/& L *M E- L?: /2 " < Æ ' 022 1 /2 E ( 1 &222 ( 1 /22 L *M ( 1 /22 L *M +- &2 , 1 & " " L?: " /2 E ( 1 < Æ 1 /22 < Æ 1 0 2 " < # * Æ 7 E

< &2 # ? 0 " 8 8 ' - " " /2 %6 " L?: 5 L?: ' #

$

& '

( $

" & ' Æ B' Æ ( / * !

&

1000

RMS velocity / Flame speed

Re = 200

Da = 1

100

DNS domain

Aircraft engines

10

Re = 1

1

Piston engines

1

10

Infinitely thin flames 100

Integral scale / Flame thickness

1000

& 2)!

, 2)!# = " 3 , % 8 # = & % 2:G 0 $# = 8 # Æ $ "# = 5 8 5 Æ $ *# "

< /2 = " L?: ,

$ " & " 9 Æ *

' ( @ ' Æ ! 67 $ ' "

' 4

02

) * + < # " ! " , 7 - " 8 # " " 9 6 ! @ @ C 9 : C I !

5 +& " % @+: 8 , 8 - " - " $

" 1

"

&

" @+: @+: " /$

2 +& / $ # 8 8 1 H

8 1

&/

2 "

" 8 " * 8 , /H " H !

2 /$

1 1

# H

2 "

H/ F 1 /

&

" ! H

2 /$

1 1

0 H

H0 4 4

&

$ &( >= ) 0 >= %( ' *

'

0

F

F

1/∆

1

0

k kc

-∆ / 2

0

x

∆/2

F

−∆ / 2

0

∆/2

x

C 5 # >7 5 L 5 L 5 # ' !$

1

&&

8" $

" 1

"

&0

- " " " 1 " " 8 # " ; - 8 - & 3 $

6 E'?: @+: ! $ " 1 2 - $ " 1 " $ " 1 " 4 " " 1 2 " 1 " - - " " - 8

" " ! ! " S C

,

0/

5 * ) - " - E - &$

4 12 6 4 1 F- 4

1 @ 4

A ,6 A 1 / 4 F 4 ,

"

G

4 .

!

&7

&3

8 1 0

A A

4 -

@ A ! 4 .

&

02

,6 6 6 1 4 ,

0

% - " - $

3 -

3 # # A A & # ' E'?: 8 #

$

@ 1 ,

.

/

1/

0/

0

- 8 8 6 8 6 E * " - @+: +- &7 02 " E'?: - * ?

6 * " ,

5

' 0

< ! # " $ E 1 # # A A L 9 6 ! @ @ C : 9 C I ! # "

" 4

: * 8 A # ; - +- $

Æ 1

4

1

/ 9

0

" $

""

1 % H < "9 "

0

" < % 9 9 +- 0 +- 0 < ! < H$

""

1 % H "9 "

0&

% % 2/ A % # " C : * * "

0

"

8 " " $ 00 1

! //

E(k)

Computed in LES

Modeled in LES

: * " @

k

! # % +- 00 - * " " Æ : * : * %

5

< S S * " / : * % ' ! H @+: ! H # $

$

$

1

# $

1 %

$ $

07

03

0&

" S $

%

&

* $$ '( ) 1

0

" @+: H + E : *

$

Æ Æ

1 1

/ H ""9 "" 9 1 / "" / H$ ""9$"" 9$ 1 /

72 7

"

E(k)

Computed in LES

Modeled in LES

I " $ $

k

M M 0#34 # = : , & 5 5

, ;+! & #

" $

Æ 1 / *

7/

00

- * " ( ) ! Æ

" C %

Æ

" @ C : "

L @ " * @ C @ + L R ? C * +LR?C @ $

H H H 1 22& &

7

" & 1 I $

H 1 F 4

G

"

1 H

7

% ' " L @ C @ " +- 7 8

" $

H H H 1 222 &

7&

" 3

' E'?: @+: # $

1 9#

70

" 9# : E : * S

"*( ( ( + + + " $" / 7 /

5

07

"

< @+: . " , # 7 E'?: & " 7 ? - Æ

- 6 & # 6 0 #

2 . (

# $

. 1 .

!

½

"

77

- - - - % " # " ?" C

' * +- E'?: $

. 1 .

!½

"

4

73

" 1 1 2 " ! 1 - C ?" : < " # #

"

S " E # " " . ' "

= ( > ( 36 *' + ( 3#

03

$

" " " $

$ $ 1 8 %

7

" S 8

E A L?: $ 8 ! L *M ' L *M

! - 5 ! " +- 7 # " , " # * # " * ! LJ * . 8 ' L?: : : E E C ::EEC

$

$

$

. 1 . # 4 . #

&

. # '(

)

'

32

8 +- 32 $

. 1 . # 4 . #

&

. # 4. %(% '(

)

3

* $

. %(% 1 G

'

3/

" G : ' " . # , ! 8 ? , G

5% + # 0

A @+: L?: 0/ 0 $ <

0

# " " - * $ , , ' " " @ - L?: : 0 < 0 H " H @+: ! C 8 # - 8 , <)*

"

$ < <)*

' <)*

1 H " H ( <)*

- <)*

H $

H0 ( " H

3

6 L?: 7 3 %

/2 E " 6 - Æ % #

/2

6 E'?: @+: A E'?: " 8 # "

@+:$ - 8 ' - 8 -

! ,+ @ # B # /2 22 B 2 6 / % * Æ # " 8* " B ! " " # * : # -

#

%

# # 2 ! 0

L *M 2 " * # ' # @ & * # 8 # " -

/ # O 1 $

1

4 . @ 4

7

&

7/

Volume V

Turbulent flame propagating against the mean flow at speed sT

Flame front Cross section A: premixed flow entering at speed sT x1

Fresh gas

Burnt gas

Locally laminar flame front propagating at speed sL

, : # % - 1 " # $

1 ,

. 1

+

. @

&/

" . "

. +- &/ " # & # " # +- // # 6 " EC: # +- > * S '8S X * L 9

" &/$ " , ( ,) - L *M 2 # # $

7

% . # @ $

+

. @ 1 .

&

A +- &/ $

. 1

.

&

Turbulent flame speed (sT)

sL0

Turbulent RMS velocity in the fresh gas (uí) Low turbulence zone s T = a uí

Bending zone

Quenching limit

H 8 , &
'

8 . . " > "

Æ - 0 - C

8 I " 8 " > " * - #

S $ 6 : ' - "

-

7

% # # C " , # > " # # " " # * E * , # " 1 3 E 2 , ! # " ' , # # " # +- /2 " # " # 4 1 4 $ # , #

B 1 3 1 2& B # * 2 # " , 8 (# 8 ) - I 1 I/ * $ 6

I 6 4 I

6I 1

I I 6 I I 6 6 I 4 6 & '( ) & '( ) & '( ) & '( ) -

--

---

&&

-+

" E>: % %% %%% - %N " , # : # " ( ! ) " @" N + " C # ' # " 8 " 8 # 8 " & 5 " # , ! & C # - : # " " " 8 # "$ ( ) ( ) " , ; " # , # ) $ &&

+ "*( / 7

7&

H 5 ,: , 8' ' ' # / > 8> MH , 8 & C # + ! < #

. ( )

EC: ) % " " * J > " ! " " $ " " : EC: < $ ) # - " A ( )

, - & EC: ! $

70

= N N , 8# C / ,: ' 8 > ## N

, # C 9 ' 8 # ' 7 8 C C# & + ! C # < #

Velocity

Velocity ub

uí uíu

uu

uu

Fresh gases

t

t

Velocity

C

A

ub

Burnt gases

B

25 8#

uíb

t

77

- * ) % E'?: " L " E'?: # - - - >8

E'?: & ? E'?: ( ) # " - : & & ' " # " # , " ' , " " &0 S " # " 8 ; # 8 8 " # +- /3 &3 , &7 " # " " # % # #

%

" #

' ! #

# " ; C 8 ;C@ ; 8C 8@ 7 ++% 7%

5 ;C@ $ : && # V " $ V 1 2 V 1

6V V " * " $

6 V 1 Æ V 4 Æ

V

&0

% 2 ( (

'

73

&

# ! 8 , : ' > # B 8 , : ' #

! 5' 2)! 8 H ( , C B > ( 8' L

#

7

" Æ L 8Æ ' 4 1 "

- " $

"1 )

)

" V6V V 1 " 4 "

$

&7

$

" " " " $ V1

V6V V 1 V 1 2 4 V 1 1

&3

1 $ 1 $V V 1 V

&

" $ $

1 ) 4 $ 1

) 4 $

&2

" ) $

1 ) 1 $ 4 - 4 -V

&

" - $

-1

) $

1 )$

&/

" ) $ 6 +- & & $

1 ' $

1

V

4 -V

D

1

) ) $ 4 $ 1

4 -V

4 - V

V

4V )

&

$

&

"

- " " " Æ V - "

32

.

A # +- &3 &

" E V V $

V1

4 -V

4 - V

&&

" +- /7 : & # $

V 1 V

V 1

-V

V

&0

4 -V

E && &0 &3 & - , " V V - & 7

1.0

Reynolds average

0.8

τ

0.6

0 1 2 4 6 10

0.4 0.2 0.0 0.0

0.2

0.4 0.6 Favre (mass-weighted) average

0.8

1.0

& O O

% ' O O *

O +# $#*$# - #

A ;C@ # V $

V 1

V

V

V $ )

1 V

&7

3

0.6

τ

0.5

1 2 4 6 10

0.4 0.3 0.2 0.1 0.0 0.0

0.2

0.4

0.6

0.8

1.0

C

& O & O

% ' O O * O +# $#*3# " ) $ ! 8 $ 8 8 # 8 " ; $ ) # +- &7 5 $

V 1

9#B V

&3

# 8 8 , , &0 @ ; : N

&7 "

N &3 ' 8 N B # ! &2 ' # *" *" " # N IP P = ; "

*" *" ' +- &7

3/

& ' 2 2 # V 2 V 2 V 2 ' 2 # V 2 V V 2 ' & # V " ; " 8 " # V ? # # " # !

- 4

A ;C@ +- & " $

1

D

1 ) )

) )

D

1 $ $

$ $

&

E $

) 4 V $ 4 V V $ ) $ ) V '( ) & '( ) & '( ) &

1

&/2

" E " +- &/2 "

( ) * 8$

1 V 8) 4 V 8$ 8 1 / & '( ) &'()

)

V $ ) & '( )

4 V /

&/

$

" 8 8 * 5 " ; 8 # " 6 : 6 & @ L 8 N8 # ! N &2 % 8 # " EC:

3

x2

Propane + air

x1

A

B

C - 8 8 H # # E

' ;

$

)

2 2

V

V

2 2

E E

$

)

' 2 2

V

' 2 2

V E E

! ' +# $#*1' 8 > # $#*# ( ( > #

H H> 8 # H ' H # #

3

% ; ! " * 8 " EC: +- &/ +- &/ ' 8 / &2 &

$ , + ' : &

' "

- " # 8 8 8 (8 ) "

8 L ; ; ; 9 = '8S ; ; L 9

I " *

" # " # ! * ! ' # "

" # " *

* " *

8 " : &/ : &// E 8 L?: " : &/

% / " Æ 2 "0 C 8 $

" * ! EC: <

3&

' E'?: A EC: * 8O - # # # +- &/ " 8O " # % 8O EC: " " EC: % 8 " 8O - % # A

%

1 0

% 8 " # * Æ () ! I 8 &//

=

" > * " # " " " = +- ! = $

= = - = 1 1 = >

&/

6 # - - 1 Æ , = 1 - =- # B $ , = , #

" 5 " , = # $

*

9 ! " & = 3#( " 9 Æ / & ' 9 Æ & C" > D Æ / 6 $ & "

' * 9 Æ 0 * ! / Æ / Æ = '

0 "( ' &

30

L *M , = " = I ! 8 < = = " # T -

6 " 8 =$

L *M , - 0$ , 1 , < 1

- - < < < 1 1 - - Æ

&/

I ! G I I $

G 1

- - 1 1 1 , - -

Æ

&/&

A +- 0 7 /7 $

G 1

< Æ >

Æ

1

1

Æ

&/0

E ( $

< ( 1 1

$

<

( 1 , G

Æ

&/7

&/3

L *M , ' ,

# " # " * (# ) %

# # 5 " L *M " ,

&

' $

' ' & + % ( % &

? ( & & %

' * * &

% & ' *

37

" 8 ( ) - $

*

$ * ' *

' * ' *

*

= : 8 I

! / 8

C # A L *M I ! < Æ &/$

= G

# * I %

#

# " * ( # ) (# ) " $

$ " " * #

# (" * # ) $ # " * # # 8 * ( # " *) ( # )

- - - G ,

I # *

# # # " * # * " (* # ) ( ! ) ' +- &/0 I (# )

Æ # - "

, - $ ("8 )

# G 1 * " I 8=

33

< Æ &/ 8 9

" *8

$ # * Æ ! * Æ Æ Æ $

= Æ

Æ ( # ) '

& L?:

= Æ I * #

! - "$ I ! T " * Æ I ! " I ! G Æ 1 $

G 1

Æ Æ Æ Æ

# ÆÆ " $

1

Æ

1

Æ

&/

2 G 22

G G # ! ! " " * 8

! (* 8" * # $) L?: # &

=

G G , ! , ? (* # )

& ; L &&

% " 0 8 " " * "

, $

' ! ,

6

I 8= I # I !

G 1 2 G 1 2 G 1

3

D

a=

1

RMS velocity / flame speed (uí/s 0 )

Well stirred reactor Distributed reaction zones

L

Re t= 1

Klim

ov

Wil

lia

lim ms

Ka

it:

=1

Corrugated flamelets 1 Laminar combustion

1

Wrinkled flamelets

Integral length scale / flame thickness

(lt / δ )

C 5 # Æ ) > ( #

Thickened flame

Ka RMS velocity /flame speed (uí/s 0 )

00 = 1

Da= 1 Thickened-wrinkled flame

L

Re t= 1

Klim

1

ov

Wil

s liam

lim

it:

Ka

=1

Wrinkled flamelets Laminar combustion

1

Integral length scale / flame thickness

(lt / δ )

< 5 ( 5 # Æ ) > #

2

BURNT GAS

FRESH GAS

BURNT GAS

FRESH GAS

2)! , : 8 : >, : 8 #

H 5 # H : # ! 5 # = 8 : $ # , 8 > :# 5 # T = 300 K T = 300 K T = 300 K

flamelet preheat zone

flamelet reaction zone

(c)

turbulent flame thickness

T = 2000 K

(b)

turbulent flame thickness

(a)

T = 2000 K

T = 300 K

mean reaction zone

mean preheat zone

T = 2000 K

T = 2000 K

T = 300 K

T = 300 K

turbulent flame thickness

T = 2000 K

T = 2000 K

mean preheat zone mean reaction zone

= 9 2

, " % ? , : 8 , : 8' : >, : 8' : 8#

Æ ! Æ " Æ Æ #$ % & "

""

Æ " " " "

" ' ( "" " $" )

" " "

" " " " " (

" " ( (

" ' " * " +

$" ( "

' $" " " ( " "

"Æ

" "

"

' ( " " " ," ' - . -

" - $" " - " * " " / 0 " " * " ( 1 2 ) 2 ) ( " 3 +" - ) - ) 4 5 6 ' 4 4% 4 47 " "

! " "

"

" "" . $" " "" ( + 8 9 + "" " " * ' : 7/

4

Fresh gases

Burnt gases

A

Instantaneous flame front

B

Laminar strained flamelet (curved and unsteady)

. " "" "Æ $" " " " " " 3 " " ( $" "" ( " "" " ' " "

" $" "" , "" 4 7

" ;" * < . < ="

"

" "" " " ' ( * #$ 4 / 0

: 6>

?

" (" : 6> * #$ 4 5&/ 0

: 6

. > 8 9 ( !" # $

6

" ) 4 7 ? " "" " " ' $" ( 0 Æ + " 0 Æ 0 " . > 8

9 ! (

" ," " ' 4 45 #(

: 6>

" " "

<" " "

*/ " " 3 " " " " "Æ " @ ( ( " " " A" " Æ"

" " / ' : 6>

" " 3 ( " " " " " "" "" ' : 5 Æ "/ "" ( "

" " ) "" " " $"

Curved flamelet

Fresh gases

Burnt gases

Stretched flamelet

2 "

"

Æ " ! " "

%

3 "

3 "" "" " " " " ! ? " " " " ' " ( "" " $" "

' @ ( "" / ( " "" ( ? "" " " ( "

! ) % 4 + " " ' % A @"" "Æ/ $" " 8 9 " ' : & A "" % 4 "" * 8"" 9 " # "" ) $" "

" "" 3 "" " / " ( +

+ " ' : & / 0 "" ""

!

" ( + " / 0 Æ

Æ

: 64

B "

+ " #$ : 47 0 !

$" "

Vortex velocity / flame speed (u’(r)/sL 0 )

Po(r) = 1

:

Ka(r) = 1

u’RMS

Turbulence line u ’ ( r ) 3 /r = ε

Vr(r) = 1

1 Kolmogorov line Re(r) = 1 Vortex size / flame thickness (r/ δ )

η /δ Kolmogorov scale

1

l/ δ Integral scale

( " ( Æ / 0 : 66 Æ Æ " ( "

"

C "

" " " " ( +

Æ . / $" 3 ' : & 0 Æ 0 .

Æ 0 (

89 D Æ ( " ""

" 3 " $" < ( " ( + " " " (

7

$" " "" * ' @ ( " " " $" + ? - < ! " ??" B" '+ E ? ) ? ( = B 1 B ?" ) ? ) + " " " * ! @ ( * " ' : ! " " 0 > ( " $" " " ( * "

" Æ " " Computation domain

Vortex pair Max speed u'(r) Flame front (speed sL )

Symmetry axis r D

!" # ! $" " 1F) "

"

" - "

$" 4 5 6 3 1F) "

1F) " " / E

" @ ( 1F) " $" D

" 1F) ! " " *

"( G #$ %5 / G 0

89 " * E 3 Æ"

5

? " 1F) E " "

) " E " " $" " ) " " " " % - " 4 " + + " " ( ( ' : * ( " " ( <" " >& Æ >>

$ % & ' ( ! ( " '

: 4> : 4 ( + " " $" Æ 0 %& 0 4& ' " : 4> : 4 " H G * " " 0 "

&

0 Æ

$ % & ' (

! 0 >& ( " "

"" $" + " 0 Æ " ( " . $" ( 0 " + " " ! 0 7 $" " ( " " "

' : 4> " ( " ! 0 4> 0 4% " " " " ( 3 $" ( "" ("

! " @ ( " E ' : 44 ( - " 4 "

" @ ( ( '" * $" 4 " " $" 6 % "

" ' : 44/ $" " " $" " 2" ) " " : I ! * "

1F)/

0 >4 J :: Æ

Æ

: 6%

" ""

" Æ " " ( "

) " ( B - " ( ( " " 1F) E K" ( " " 1F) ' : 46

! "" ( " " ' : 44 "

" / ! ( "" 4 ! "" "" $" + $" " "

4>>

! "

) & Æ * ' $ + $ ,

Vortex velocity / flame speed (u’(r)/sL 0 ) u’RMS

4>

Cutoff limit

No effect zone

C

Quenching limit

Kolmogorov line A

D B

1

Vortex size / flame thickness (r/ δ ) 1 Kolmogorov

100 Integral

Cut off

C Distributed reaction zones ? RMS velocity /flame speed (u’/sL 0 ) A Klim

D

ov

Wil

im s l liam

it:

Ka

=1

B Corrugated flamelets

1

Pseudo laminar flames

Wrinkled flamelets

1

100

Integral length scale / flame thickness (l/δ )

!" #

4>4

" " ' ( "" $" + $"

" 4 * L 8 9 $" F

" 1F) " A "

" " "" " ' : 44 ! "" * , ' : 4% Æ " " " $" E , 3 * ! "Æ " "" * < $" " " < "" ""

" < ' : 4%

' : 4 " + ( " " "" " "

" " ( E " $" ' : 4% " 0 >> ' : 6

E "" " " " " "" " " " " B F ) B!F) $" " % : ' ( $" % 44 % 4: *

* "

! " "

" "" ( "

) 4 %/

"

" G (

#$ 4 65 / G 0

! (

: 6:

: 67

" ( " ( > (

4>6

"

! + "

! / ! 0 ! ! !

> " ? "" "

"

" 4 " - " 4 % 6 A

" " " H 0 #! "

! #$ 4 6: "" / HJ! 0

: 65

H ! $" H H 0 > H 0 " " "" ( " / $ $ J 0 > $ $% J $ J $& 0 $ $ $ $% $% $%

$ $ H J H 0 $%$ $ $%

$H H $%

J G

: 6& : 6 : %>

G 0 G ! " " H 3 #$ : %> " #$ % 4: /

$H $H 0 $% $%

: %

$" * H " ," " % : 4 $" " $" / ""

G "" H L " "

" "" "

)" $" "

" (

" $" H " $" / % &'( ) * + , - ) * + ,

4>%

B

" $" "

) % : 6 ""

" "

" "" ,"

$

"

" * " " F ) $" / G "" H ! " "" " " G "

) : 6 4 : 6 5 ?

"" "( "

" #$ % 4& ) : 6 &

"" "" " * % %% ) % : % (

/

0 H ( G 0 G H

J H

: %4

$" " " " H

"" 1 M "

8 9 ' : 4 ( "

" " " * " " " " % : % 3 " $"/ ' % 5 G H #

G H $" " " " "

!" # $ !#%

E ) ) ) " # , A #,A "" "

" B 1 M ' " ! !" : 6 4 ( "" / + " " "" ¼¼ "" " "" H (

/

G 0

¼¼ H

: %6

4>:

" "" "" (

/ 0

(

: %%

"" * % 4 < % *

" " ) ( $" #$ : %% # , A " " ( "" % : 6 ¼¼ ! *

" #$ : %6 $" "" H * "/

¼¼ H 0 HH

H 0 H H 0 H

: %:

" " " H 0 > H 0 " " H 0 H * #,A /

G 0 H H (

: %7

" " $" " $" #,A " 1 "

: %7 " / " ( " #$ : %7 L #,A

" !" ' : 4: #,A !" / #,A " > + " H ( ' % 5 !" " 0 > + ' : 4: " H ) ;" " ) , 3 #,A " !" : 6 4 " " # ,A ( F " . !/'($ !/'0$ "

. !/'/$

. !/'0$ . !/'($

1 2 3## % /(,45 * + 4 * + , 7* !, * $8 ¾½

6 * !, * $ !9 ,:::$ *

4>7

Mean reaction rates

Arrhenius model ω(Θ)

EBU model ω(Θ)

0 0.0

0.2

0.4

0.6

0.8

1.0

Reduced temperature Θ = (T-T1) / (T2-T1)

' - +.( $" "

) : 6 " "

" " ' ( 3F') ? " E #,A " ,

& ' ( "" ( "" ) % 6 ' ( !2 !2 , 2" B + 2" N "" / 0 J)

: %5

) * " "" B?) 8+$"9 / $+ $+ J 0 +

: %& $ $% .9; < !% % $

4>5

"" "

+ + * ' + + +* "

A" "" * $" 2" #( ( " "" #$ : %& " " Æ" $" " 8 " 9

" + $" ," * " +$"

$" " / +$" $" "" $" " " ! 4 5 $" $" . " " " ' $" ( " : % )" " " " " + $" $" "

" " +* $" ) ; ! * +$" E E E 4 ) : 6 7

! ' ) $!')%

" " , ?

- * 55 "; , ?

- , < - < " " ,?- " "" ( " " " "" "" ) : 6 ! " " A "

" " "/

" - " "

H (

" " " " " ' : 47/

&H 0 ) ÆH J " Æ H J , -H

"

: %

"

) " , "

(" ÆH Æ H 1 " H 0 > " " H 0 %

!3 = > $ !

4>&

pdf( Θ )

Θ

0

1

+ " &H H 0 / ) J " J , 0

-H % H 0

: :>

-> 0 - 0 > ,?-

" B 1 M " ) " , ! H 0 > " H 0 #$ : % " /

&H 0 ) ÆH J " Æ H

: :

! + H ' : 45 ! ) : 6

" " ) " B ' (

"" "( "

A" G " /

G 0

H&H G H 0 ,

H-H G H >

: :4

! , " (

) ,?- "

G * "

$" "

" ) : 6 7 $

4>

Θ 1

0 time

, & ) / 0 1 / 0 2

) " $" H/ $ $ $ H J H 0 $ $% $%

$H $%

J G

: :6

, ?

$" H H 0 H H / $ $ $ O H HP J O H HP 0 $ $% $%

$ $H $H OH HP J 4 4H G J G $% $% $% : :% A ,?-

" " " H $" + " H H 0 > $" : :% " / 4 /

$H $H 0 4HG G $% $%

$H $H 4 G 0 4H $% $%

H * /

0

. 4H

: ::

: :7

H 0

HG -HH HG 0 G G -HH

: :5

" " H . 0 . G

. "" ( H + ! $"

4>

( ? , !

" " ( "" / . 0

¼¼ H

: :&

"" ( " ! $" ¼¼

H ) : 6 5 "

" " H 0 > H 0 H 0 H #$ : %:/ G 0 H H : : 4H ( "" (

0 ( #$ : %% (

: %7 ) #,A ,?- #,A #,A < #,A ,?- " " H

! , , - +

$" "

' : 4& " "" E ! " , (

E , * ! ' 4 & !" (

" " 3 "

" <

"

(

"

$" -

G / G 0 G -

: 7>

$" $" " ' : 45 /

- 0 4

H H

: 7

"" (

/

4

T2

T2

T T1

T T1 time

time Fresh gas Limits of turbulent flame brush

A Flame front

u=s d

C

B

T2 T

Burnt gas

T1 time

% 4

" " " ""

$" (

B " " H ' " " #$ : :

/ H J H : 74 - 0 4 J H

$" - " ( $" " " ( "

" ? B

#$ : 7 ""

G #$ : 7> " " / G 0 Æ

: 76

"" Æ " $"

H 0 > H 0 " " ' : 4

44

Θ 1

0 Time

tt

! 3 .45 /

G 0 4 H H Æ (

: 7%

0 ( ! #,A (

Æ 0 Æ #

Æ" (

: 7> " : 6 7 F ,?- " (

G " " "" "

" "

" "" ) : 6 " $" "" "( H B

* + , 3 " "" $"*

$" ! "

" " "

" " " " " " / ? , < E , / G 0 /Q

: 7:

/ "

" " Q "/ 0 0 " "

46

! " "" ( " / ""@ " " Q

Fresh

Fresh

Burnt

Flame

Burnt Flame

LOCAL LEVEL (library): Flamelet consumption : wL

Flame “stretch”

GLOBAL LEVEL: Flame surface density

Mean reaction rate:

Σ

ω = ρ0 w L Σ

' : 6> " " "" " "" " " " / " " ( "

2 ) " " " " / " " $" "

, " 8 9 1 " / " "

$" / / 0 1

: 77

#( C C #$ 7 74

( 1

4%

* / 1 0

&

: 75

" & " 3 1

" / 1

: 7&

* "" ( " Q " (

$"

" # Instantaneous flame front α

mean surface Θ

Ly

.45 3 & 0 & / / 6 , #$ : 7> " / Q0

*

: 5

9 " ? 5 4,,@

+ !/0:$ , ) 3 < 5 ¼ ¼ ¼ + 4,/@

+ 4 ,/@

!/@4$

4:

* " "

" $"

H " H " ' : 6

" " >: #$ : 7 /

H 2 H H 2 J H Q0 0 J H

: 54

2 " / 0

: 56

* " ? / / 0

"

* 0 / 2 G 0 H H : 5%

! 0 ( (

#$ : %7 : : : 7% ! " "" (

: 5% " 3F') " ? " E , ' " Q Æ" "

$" - " "+ $" Q C C ," / " " 2" / Q0

: 5:

" "

"

" " " "" <" 1F) " (

: 6% " "

$ ! # C" $" " Q " " $" " " ? , 1 " " E <

47

E " E C ! E "

" H " /

Q 0 H ÆH H 0 H H 0 H &H

: 57

Æ H 1 Æ " H H 0 H H H 0 H &H * H 0 H ! $" ) % : : " Q " " H H 0 H " &H * ! ( $" " * H 0 H

" * $" " " H " E C $" / $Q $ $ J Q J O * QP 0 $ $% $%

$ $* Æ * * Q J Q $% $%

: 55

* " 3 3 0 H H " " 3 0 $* $% "

" * /

# 0

# H ÆH H

H ÆH H

: 5&

-L) Q $" : 55 " " * * BL) (

" @" " " " " ( #$ 4 &% < 4 ( $" / ? E

" H " " * ," " ( " " * * ( " H

" 3 H " "

45

" $" " $" "" "( " " " A ' 0 J "" " / Q 0 Q J Q

: 5

$ $ $ 0 Æ * * Æ * * J Æ * * $% $% $%

: &>

" " * "" A #$ : 5 : &> " $" : 55 /

$Q $ Q 0 $ Q J Q J Q J $* Q J $ $% $% $%

: &

C" " " " #$ : & " + : 6 Q $" /

$Q $ Q 0 $ J $ $% $%

$Q

$%

J Q J Q

: &4

" 3 (

"" "( " (

"

" ""

" "" ) " " " " " " ) : : 6 ." " " $" " * " " . " " " " BL) #$ : & " " ( $" ! " ? , 1 ' "" 0 ( " " "

" A ! $ ! $

A +

4&

47!+5 '.

' '42 '46 '46 !

4. 4

.

'! ' ! '<2 '<6 '

8

8

8

6 9 8 :

0 8 Ô 8 ¼ 9 8

8

8 8 8

8 ; 8 ; 8

¼¼ ¼¼

8

8

¼¼ ¼¼

9

8 Æ

2= 8

9

8 2

¼

¼

8

2 2 9 Ô

¼

¼

¼ 8

8

2 8

¼

¼

2 8

¼

¼

<

8 3 =>6

$ ? ? ? ? ? ? ? ? ? Æ +3 =>1 ' @ ; Æ ," =:A +3 =22B , '< ? ¼ $4 - ? * 8? 4. & - +3 =>1? 3 !

Q " " H $" Q /

Q 0 $ $Q $ J $ $% $%

$Q

$%

4

Q J ) Q " / ( H

: &6

) " #$ : &6 F

" $"" " $" / G 0

) H " (

: &%

#,A #,A * " #$" : 7: : &4 " $" : 6& : %> " < "" / " " " ( " $" : 7: : &4 3 " " "

) 4 6 ' " "" ( + " ? " " < 1 ," 1" < L" ( " $" "" " F " " Q " " ," C ' " $" " " ? , $"

#$ : :7 " " " : 6 < " " " 1" < L"

"

% E +$" " E E /

: &:

>BC, /( + 4 .9; !% $ = $ A * !, * $ !>BC& * !, * /($ .9; !/'0$

$+ $+ J 0 +

$ $%

44>

* " + 0 + (

" " ? " E (

" " 3 " #$ : &: " +

+ ! $" 0 +

" C C 3 " " +*

" " H " "

" " " 1 " E - , $,% " &H " " " " H H J H / &H H 0 45'566 H H H J H

: &7

' * + /

&H H 0

: &5

" ' ( " ( / H0

H &H H

H 0 ¼¼

H H

&H H

: &&

' " - " / G 0

G H &H H

: &

9 *

*

44

!

" & " &H 0 &H ! ' $" - / - 0 - 0

-H&H H 0

-H&H H

: >

(

(

" " ( - " " ( " ; &R R R / E R ! R J R R ! R J R 0 &R R R R

:

+ /

½ ¾

&R R R R R R 0

: 4

R R R "

" ! $" - (

G / -0

½ ¾

- R R &R R R R R R

: 6

E " * "" * $" " * " ( ( " " " #$ : & : 6 ( K" " G H - R R " Æ" * E E 1+/ " $"

&

3 " ( " ( 3 " " * ' " " B

D !# # E < $ ! D $

444

" "

" " " !

. " *( + " ( " / $" @ * ,?- : 6 : " /

H

" /

H 0 > " H 0 ? " " " " ( " , , " ""/ &H 0

S' J 5 H H 0 H H ' 5 S'S5

' 5 + / ' 5 0

S " * /

H H H

: %

: :

S% 0

: 7

" " " " * ' 5 " " " ¼¼ H H /

'0H

H H H ¼¼

T

50

'

' H

: 5

"" " " "" "

' : 64 A " " H 0 > " H 0 " " ( " > H " (" * " " " D ¼¼ " $"

H # B F ! $

G !$ ! $ "

446

-

0.0

0.2

0.4

0.6

0.8

' > 4> :> >4 > >4

5 > 4> :> >4 6> 4>

H >:> >:> >:> >:> >55 >>

H 7 >4> >4: >>5 >7 >>

1.0

+ " #

¼¼ # * / / ¼¼ +3 =AC / / 3 B!F) $" " H ?" " H $" #$ : :6 H $" H/

$ $ H $ J H 0 $ $% $%

$H $%

4

$H $H J 4HG $% $%

: &

" ) $" $" " H #$ : %> H/ $ $ $ H J H 0 4H $% $ $%

$H G J 4H H $%

:

)" #$ : #$ : & $"

H / " " H 0H $ H $

J

$ 0 $ H $% $%

$H $%

$ J 4H $%

""

$H $%

" "

$H $H 4 H 4 $% $%

$ H $%

"

$H J 4HG $%

: >>

44%

$" "

( ) 7 % 6 A " " / ?" " "" B!F)

" B " "" " "

" / H

0

$H $%

0

H

$ H 7½ $%

$ H $ H 7¾ $% $%

: > : >4

7½ 7¾ "" ) "

H "" "" ( " (

" /

$H $H H 08 08 H $% $% (

: >6

8 " " " " "" "/

G 0 HG 0 H H

G H & H H H H

: >%

" : :5 : : " / ( " "" "

" #$ : > : >4 "

" / "

) 7 % 6 (

" : >6 ( " " H " " " " "

H $" . ( " " " " Æ" - F " E 2" F

" "/ "

; " " " / &R R R &R &R &R : >:

44:

(

" "" ,"

" " "

"

$ ! # # ! ( $" " " ( E 1+ C " " ' "

&R R R $" /

$ & $ & J 0 $ $%

$ ! 0 R & $%

""

$ $ $R $R

$!

$!

$% $%

!

?" (

0 R &

: >7

$ G R R R & $R

<

# ! 0 R $" # " R * $" : >7 " " * " "" " * " ( " R " ( " $" #$ : >7 $" ! $" ( A" " ( (

" Æ" * " / $" " " ,"

D

H !# $ ! I I $

447

" ( $" E $" / ?< " 8" 9 " E E 1+ !" " " ( " Æ" "

. ' , !

" G " " " " "" "( H <

" " "" " 1 ? < < / H 0 H 0

$ H 78 $%

: >5

"" "" 78 "" ) "

" " / #(

: >5 "" ' " #$ : >5 " $""

" " H $" ' " "" " F Æ" "" " " " D " ( " " G ' " #$ : >5 <'1 " " "

" /

$ $ H H 0 $%$ J $ $%

J 78

$ H $%

J G

: >&

#$ : >& ( " " "" " / " " " "" " 78 B!F) L ( , ) - , , , " ( "

"" #$ : >5 "

445

#$ : 5 ) : 6 " " " ' : 7 1 " "" B 1F) " C C E B" < " "" ' ' : 66 "( 1F) "" " H " ?

"" " ' : 6% 1F) " / D 1F) B" "" 0 / " " "" (

: 5 D 1F)
: 5 ""

@ /

$ ¼¼ /¼¼

& !" /

? - * $ ' $ '$? ? & 'D $ & @ & ' ( 3 ,?

- ,?- " #$ : 5 "" "( " 3 " $" " ! ' : 6% "

(

44&

'

$ '$ / ? ? ? ? ? @ ) ! @ & ' ( $% > " * <" " H $ H " " $% > " * "" H$ H C ' "" * "" (

D "" " ( " * "" " " < ,?

- 1F) "

" " "" " "" #$ : 4 C " 8, " 9 * / 9 0

4)

: >

) " " " ) 2 . Æ " "" " " 9 9 ( J < EG% K

(E EG%

44

' ( " : 5 ! : > " " ( 6 F " " "

" C E <" "" "

"

" " <" "" " "

' : 5 ( " "" "" ( " "

" " 3 ,?

- " ( $" "" "( H $" "

$" / $ H $ 1

J

$ H $%

0

11

H

$ $%

$ $%

111

111

$ H $%

H

$& $%

H

$ $%

1:

$H $%

1

1 J

H

$& $%

: >

11 J

G :

" " "( H "

3 #$ : > 3 " " 33 * 333 "" * 3C C " "

C3

" C33 ""

" C333 3U

U " #$ : > Æ" " ( 1 " " " + " " ! 1F) " #$ : > ' : 6: #$ : > " " ' ' : 6:

C333 3U " $% >

" C3 C33 H $ H

46>

$% > U " " H$ H 1.5

1.0

0.5

0

-0.5

-1.0

-3.0

(I) (III) (V) (VII) (IX) unbalanced

0.0

0.2

0.4 0.6 mean progress variable

0.8

(II) (IV) (VI) (VIII) (X)

1.0

@ % /

$ !" & ? - * , ? @

' " : 6: " #$ : > "( " : > " + " " H " " "

3 " "

1F) " + )" " <'1 ( , " 3 ( " ;" L " "" " "" -

" -#) "" / ) : % 7 ' : %& " " -#)

¼¼ *¼¼ ! $

¼¼ ¼¼ ! $ G K ¼¼ ¼¼ !J$ ¼¼ *¼¼ !/,,4$ %

/

46

' , (

! B!F) $" "" (

""

$" - $"

" " ) #,A ,?- <'? ?, : 6

" "" / 0

0 - 0 (

:

D <E,
" " % 4/ 0 0 - " 0 !

: 4

(

#$ % 6 / : 6 0 0 , (

$" " " " " " " /

" "" "" "" ! ' " (

""@ " ! "" " " ! $" ""@ " ( ? ,

$" C

9 5 * * !+"

!/,,'$

" % /(/ ! . !//0$ !. !//L$$ !C9 /($

464

! $" 3F')/ 3 "" F ' ) ? " E " 1F) @ ( E

: 4 6 " ? " ) ? " ) " " +

""

" "" / 0 ) S"

Æ

(

: :

"" B?) "

Æ

) Æ " S" * 1F) /

S" 0 ( J >% J ( J >% J >% 0

Æ

4 0 6

( 4

>

: 7 : 5

Æ " " " . Æ + S" " ' : 67 S F Æ " Æ " Æ "" B " * " #,A ,?- " <'?4 <'?4 : 6 1" ' ( 3F')#,A " " / G 0 S"

Æ

H H (

: &

" #,A "" #,A #(

: & "

" " , -;

0 12 2" " 2 + "" " " "

" " + "" ""

466

1000

Γk

u’ / sL0

100 10 1 0.1 0.01 0.001 10

0

10

1

10

2

10

3

10

4

lt / δL0

," Æ ; +3 =22B Æ ¼ ! 12E 21E 21E 211

* " " 3 Æ " E E " EE " " ( ? " L 2 ' 1" !

" #,A : 6 6 " / G 0

H H

:

"" $"

"" /

$ H $H 0 J

H H $% $%

: 4>

"" ) " ! "" "(

" "" + "" EE ( " #$ : 4> H 0 > * H H #$ : 4> " / $H $ H 0 J H : 4 $% $% " V $" +/ V 0 %

>

: 44

46%

"" " "" /

4

: 46

EE " " #$ : 4> "" /

0 4

: 4%

(

" "" ""

' ( : % "" "" " ( 0 # ( % : 6 (

"" 3F') : 6 B ) % 4 "

""

""

$$

S

¼ ¼ Æ ¼

%¢

4

$$

4

%¢

$$

(

3 3F')

(

4

4

%¢

$$ %¢ S

¼ ¼ Æ ¼

4

¼

¼ Æ

¼

$$

4

¼ %¢ ¼

$$ %¢ S

¼ ¼ ¼ Æ

¼

¼

! %

&F G +3 =26H 0 ? ¼ ¼ 0 5 ! * Æ * Æ ! 2 "" A "" "" D 3F') G ! ¼ + 4$5 K

46:

14 12

8

sT

/

sL0

10

6 4 2 0 0

2

4

u'

/

sL0

6

! !

8

10

¼ ! ?

G ? ," +.( * G ) 0 H'? 0 1'1A Æ 0 H' (

"" Æ " "" Æ " ' : 65

D "

"

" + "" "" " "" " " $" ( " F

EE 3 ( $" 1" "

+ 3

) "" $" + ;

Æ" "" " " ! "

"" < 5 + "" ( /

'( )

<

+ , < 3 " "" " + ;

" " " + ' : 6&

C GB%

! E $

467

*

3 "" " " "

" ' : 6 F " " + 8 9

premixed reactants

u = sT

Flame

) & ) ) ) &

Fresh gases

Hot burnt gases

!

&

Flame

heat fluxes

) 3 5 + 4

F ( "" ( " "" ( : 6 >

" ( $"" L B!F) "

+ " " + ," ( + " " " ( " ' : 6& : 6 ! ( Æ" ' : %> ( "" C ( @ +

465

C ' " " * ;

< + " Q $" 3 " * " " " " !

" " " "

$"" " " " " / #$ : 7:

" $"" ) " #,A ,? ' : %> "" +

( @ # ) & ? & % ? :1 C1 & @ ) " $" Q " + " ' : 6 )" + " ""

+ + " ? Q $" "

"

! " $" Q Q : 6 4& $" " " ?

" G Æ <

A

46&

D " #,A : 6 6 : 6 > + " ( H + + "" H

' : 6 ; F

+ 3 + Æ" " " " " / ' + ( " " 3 " "" " " ( ; ' ( "" ," " "

( " " "" " "" 3 + ; " " + . "

+ < 5 5 6 4 ' 5 :

B!F) "" ( " "

" ? " -#)

& 4 1 B!F) " " 6>

) # , A ) D " " * " ! -#) " + ! Æ" " " ( /

Æ ( " > " -#) + V ' : %

H + "

" " " 3 "

46

" " "

-#) "

E

flame front

LES computational mesh size (∆x)

Fresh gases Θ=0

Burnt gases

Θ=1

' & * Æ 5+ ) I

/ 0 1 / 0 2

Æ" / " * " $" +$" *

"

* + 3 -#) " * "

" "" " " " " E " " : 6 5 " ( -#) " ( " < 7

&

, 567

! * (

B!F) " " ' ( #,A : 6 6 /

G 0 &'& H H

: 4:

&'& " "" / &'&

V

&'& (&'&

: 47

EG% C

GB% ! /(:$ =.% !" > $

4%>

" "" * * + V &'&

" "" (&'& " "" "" (&'& (

$" 3

"/ -#)#,A " " " H * " " " ( '" -M M '" ?M "

/ # , " * B!F) ( + 3 -#) ( "

+ ! "

) % 5 % -#) * +

" "

& 8 " ( ! " ( ," DWB" ' ( . "

Æ (

4 : 4 &/

(

T

Æ

( 0

(

: 45

( " ( 3 " < ( <

Æ " < ' "Æ < " -#) " ' : %4 ) (

" !" " " " $" " ( + @ " (

" " "

" < & < Æ " A" Æ " * " 1 M " ""

#

H H

$ ,E 5 *

+ " * ) 6

4%

Fresh gases LES computational mesh size (∆x)

real flame front

Θ=1

sL Θ=0 thickened flame front

Burnt gases

* *

: 4 4/ 0

0 Æ

: 4&

< < ! "

) : 6 ""

Æ "" < " 1F) ! < ' : %6 ! Æ " = " " Æ " ( 3F') -#) V< Æ 3 " 1" / E( /

(

! #$ : 45 / 0 =

< ( <

= < ( = <

Æ

Æ 0 < Æ

: 4

" ""

&&

9

+$" /

+ "

4%4

!" $

E * 0 = .

Æ ? # *

& * % )

' : %% 3 -#) "

" * + 0 + + *

+ H " -#) +$" / $ + $ + J 0 + $ $%

: 6>

! " 8""9 " #$ : %5/ 0 J) : 6

G > G*

4%6

G < G*

LES mesh size

flame front (G = G*) Fresh gases

sd Burnt gases

) ) 0 )£ )

?

" "" /

V 7 0 V 47 7

: 64

7

) * * 3 ) * 0 3 ( " B!F) " -#) " " ;" * "" " "" #$" : 6> " F

" : 6 % "" * $" " #$ : 6> " " Æ" / " * " " " E 1 +$" " $"

" "" ( "

&

+ , ) 7 ,

! * H

" $"/ $ $ H $ H J 0 $ $% $%

$H $%

J G 0 H

: 66

4%%

$H $%

J G 0 H

: 6%

#$ 4 " H /

$ $ H $ $ H J J 0 H H $ $% $% $%

-2

-1

0

x/∆

1

2

! "

H -#) " F

" 2"

, *

H * < (

#$ % :% * + V " + V ' : %: ! * " " 4 VV < +* +$"

H / H $" " " * ( 1F) ( " 1.0

0.8

0.6

0.4

0.2

0.0

: 6:

G . !/,(($ !/,(4$

* =.% . !/,('$ =.% EG%

Q " " " " "

" X " " " ; ?

H Q 0 X H

#$ : 6% H , /

+% J +3 H=H ) I ) I I 0 H I ( / / I & / & / & 6 II 0 >

.

progress variable

4%:

$" Q X E B!F) ( ! (

, , C $" , . L < Q X ! ' % 6 " ( (

&*

7 ( ) 7

A "(

" #$ % 57/ H 0 H

$ H 78 $%

: 67

L ," " 1F) ' % & , " " "( -#) B!F) : 6 & "" ' : %7 ," " -#) "( B!F) "

$" -#) B!F) ) " ( " " ! " "( * + V ' : %5 " " " ( " " " " " ' : %&/ , C " C "" ( @ + " " * ' : %& '

H * ' : %& B!F) " "" " ( " H " -#) * -#) ( " " 3 " B!F) " * -#) " "" ' : % < "" " " " " ' : > * " "" -#) " " )" " Æ" B!F) " $" " , ! + " "( ' ( 4 % :

" $" / 0 0 84*'*

: 65

4%7

H H

0.20

0.00

0.10

−0.10

−0.20

−0.30 0.0

0.2

0.4

0.6

0.8

1.0

+ ( / ? $ / / / ? % % 0 1'= E % 0 6 % 0 B $-" ) $ ! !" . ?2AA>? K H> ! .

H H

0.15

0.10

0.05

0.00

−0.05 0.0

0.2

0.4

0.6

0.8

1.0

+ ( / ? $ / / / ? % ) I E 6I E :I HI % 0 B' =HB 5+ ) I $ ! , !" . !" H> ! .

4%5

$%: 8 , 2 2

8 $%: 2

5 # ) +"4- , ='C= *!E 3 + 0 1'B=E * =1 L . @

0

"

/

H 0 H H 0 H H H

: 6&

'

" " - " " " H / 0

J H

: 6

" " 0 /

H 0 H

H J H

H J H

>

: %>

% " " H 0 #$ : %> *

" " " "( )" " " (

: 67 " " " " ! " " B

/

0 0

: %

" " " B

4%&

0.020

0.010

0.000

−0.010

−0.020

−0.030 −0.015

−0.005

0.005

0.015

¼¼ /¼¼ $

=H> 6 ? B 21 ¼¼ /¼¼ , 1 . @ $

," ( "

"" ( + ! $" " ( " "" ( " " "" 1 $"

" " "" <

" "

" "" ( Æ" "

"Æ 1 " " 1F) " + "

K"G% G K < =.% < M

4%

, ;67

F" $" 1F) * " "

) % 7 4 ""

"" B!F) #( "

) : 6 7 '

" "" " "" $" *

" 1F)/ .

" Y $" "

1F) " "" " : 4 6 4 !

" * "" " "" : : 6Y 6 . " " * " " " " : : :Y 3 "

" + "" " B!F) -#) 1F) " / )

" " 1F) "

B!F) -#) " 8 9 ' (

"

" " " " " $" " " " " 1F) " D 1F) " " 8 9 " 1F) B!F) -#) ' ( $" "" 1F)

;67

#

B!F) -#) "

" "" "" Æ" ( 1F) " $"

" " 1F) " *

" * $" ( 1F) !

4:>

_ Θ=0

_ Θ=1

Fresh premixed gas + Turbulence

Burnt gas

N

n Instantaneous flame front

Θ=0 Θ=1 x Mean flame front <sc>(x)

Θ(x)

x

!" 1 - / - 2 & . / * "" ( 3 " " ' : :>/

- >/ $" - + " " ( " " "

$" " 1F) ( " " 3

- / * + C " - 4/ $" ( 1F) " / "

4:

" " H - 6/ " "

" " < ( " " * #$ 4 >4/ 0 !

G

: %4

" " + "" + F / 1F) % ( " > " % ' % $" " 1F) " $" / 0 : %6

) " "

" @"" : :

5+@+5 57'-5 +"+4.5+ -@+$-J+! $-" -,-5 ,5+$,"J 5+ -'+ -@+$-J+!

M ! - ! , - ! ,

5 !" D

" " " "" 3

" 3 " " ; ' ( G " " "

" " " / G 0 !

: %%

4:4

F #$ : %% 3 (

$" " " L $"

" " " ""

" "Æ 1F) " " * : : 6 " * ( "" "

" #) ? "" "

" "Æ "" "" ? " * ( ,?- : 6 A" 1F) " "" " " " % 7 6 ! 1F) " " " ' : : "

Θ=0+

sd1

Burnt gases

Burnt gases

Θ=1−

Θ=0+ sd 0

Θ=1− sd

Fresh gases Fresh gases 0<Θ<1

DNS flame (resolved front)

Modeled flame (infinitely thin front)

!" !" & ) *

.

" * * H

" 2 " H 0 H ( " "

4:6

H >&

" F

" ' ( H ( " ! 4 5

"" H 0 > $" Æ" ( 1F)/ " H " " #$ 4 >

$" " " +

3

0 H 0 > H 0 H A "" " ( " " H F

+ " H 3

H " / ' : :4 " " H ( @"" 1F) L ' H 0 >4 " "

" " D H 0 > " " 3

"

Æ" * ( L " " / 0

: %:

8 9 " " ' : :4

7 , ( ;67

, - '

"

" : 6 1F) "

"" ( " " " "" + L E B" " Z E * " "

4:%

! ? $ /£ 0 1'6E ! /£ 0 1'AE ! ? ! ? +3 =2H= < *

" " ' : :>/ 0 > 8 9 "" F" " " " * ! " < " " ( ( - " - " " ' 0 >:

" " < ," -" ' ( "" (" " " " "" ( " "" * " " " 1F) <" " L E B" " Z B" " Z " E B" < ," " 3 " " " ' : :6 " " " - "

" B" " Z < " " - " / " " - " - " " . " "

" ( ! " B" " Z

4::

' & 2

! 5& $ N 5& 02? 5& 026 5& 0 1>

" >

1F) "

( N" < "

! " B!F) : 4 6 * 4 7 " " "" " / " " " "" L " 1F)

' ( L E ? $" Æ" " " " $" " A

"" " (

4:7

L "" " " "Æ */ ""

? " E * " " "" $"* N"

& K" " ( 1F) ' : :>

H 9 *

H " 9

+ - ! , )

/ :! !" & % 5&

!

& ' (

C ' : :% ' : :: H " " H 0 > H 0 - " " E ! * " 0 , " 0 >& ' : :% " "

L ' : :: "

3 - " " 0 >6 ' : ::

" * " " ' : :7 " E

4:5

+ - ! , )

/ :! !" & % 5&

!

& ' (

' " H " " $" "

" $" B ' $" ' (

" : : " : 7 " - ( - - ) K" < . " " " ( 2 ) B

? " " " " $" - " ;" *

"

5+@+5 57'-5 +"+4.5+ -@+$-J+! -'+ -@+$-J+!

! - ! , - ! ,

$ ! 0 / - ! , 0 0 - ! , 0 1

% ! " : : % " " 0 - *

* " "" " ? "

L $" ' ( #$ : %%

4:&

? ? 5& :! !" & ' (

" $" $" " " / " "" " $" ' ( B" " Z " E L E ( " "" " " + " ) " " " " " " ( * " " , , . " "" + "

' : :6 ; + " " . " " + / " - " " " - " - .

" B" " Z " E " B " L E < " " * "" " "" " " L " " + " " "

4:

. ! $" "" ( "" ! " " " " ? ! * ( " ?

" ? " - " $" E

" ? " " , < L E F

"

" 6> " " "" " " * 3 " Q 0 " "" * " "

"" L " " < ? E < 1" : : :

&

<

E " 1F) "

" ) 1F)

"

" " ( " B

" LD ," @ L L # < < @ L ( " '" "

1F) $" ( 1F) ( $" " " 3 ( $" * * * : J @ $" @ "

> >> " Æ " " Æ " "

F" $" " ( E E E

" ," ," # < < "" 3 " "

47>

" " * <EA )

! Æ" " ) "

* " * " L E - ," ( <

' # ' " : :5 ( 1F) " L @D ? @ " 3 "" 0 > C Z Z E " ," "" B " 4& 6> % $" >6: Æ

" 6>> @ ( + 4: * " " " <" ," " + / ' " ( F $" ' " ( ( " " + " ! " (

" " " L ( " < 1F) 1F) "

," " "

! C" ( ( ( 1F) ' ( DL
" " DL
( " " "

$"

47

!" < #7 & A #2A 4 .

& ' ( ? ? 7<

474

' & ( 2 # 3 0 21E 3 0 6A> G < *

$" $" , ) ) ? ' L@D ( " DL ," DL " "" $" > : %>> ' @ L DL " " 5 %: " ;A " ' : :& 3 1F) *

" " " ( #

L

1F) " ( ( L ? " 1F) " " ) ? (

7 , ( ;67 1F) ( " " " "" " + * "" ' 1F)

476

"" (

" ( 6//% ! " N" ! " ! " 1F) " "

' " + " "" 8 9 ) B" ?( C ? "++ = + ( ( ( "

4!5 ? 5 &

E E !" & ? ? - * $ + ' , $ N 1 ; " "" " * + + D ; 1F) "" * " " !

' : : "

" " !

! " " B" " Z !

" $" "

47%

" " " " ; L ; * " " " " . ; 0 "

T ; 0 > " " T ; 0

" " ! 1F) " ' : 7> " +/

" E ! " < B" " Z " " " " " @ ( *

? ( !"

& ? ? - * ' , $ + $ N

* , ( , 3 * "

"" ( * "" " " " "" 3 " " " Q " " ( " $" " Q $" E < E " E C

47:

: 6 7 #$ : 55/ $Q $ Q $ Q $ * Q J J J 0 Q $ $% $% $%

: %7

$" * ) : 6 7 #$ : %7 "" $" * "" * " " ( Q$" " " " #$ : %7 1" < " $" $" " "" " "" 1F) " $" " ! #$ : %7 " 3 " T " (

" $" #$ 4 &/ 0 3 : %5 3 " " " / 0 3 3 ' 4 4> " +/ + " " + 3 "" ( 1F) " " " " " /

0 3

!

: %&

! 3 " "" " " " 3 " 1F) " /

" "" N" T

" * "" 2 ; E T

" * "" < B" " Z T " *

"" " E ' " 3 " " " 3 ( /

3 >4&

: %

477

G ?

? & & E & & 3E $ ? ? !" $ ' , + O

0 N" ' : 7 " 3 "

4> I ) " " " " * * < 2 ; E B" " Z ! "

" ' : 7 + 2 ; E " " (

" ( "" ? " E

< 1F) " " ( ' " $" $"

" "

" ( $" 3 ,?

- #$ : 5 "

$" * , ,

"

" *

" Q * 6 (

$" * " ( " ? B

$"

" 0 84*'* ' : 74

"" " " 1F) (

" >5 " ,?

- "

475

!" ?

& & % . 45 & ' (

" ( " "

$"

-

;67 ,

1F) " " " -#) < B!F) " ! : : * % 5 ) 1F) ( " "" " ' % & *

H $" : 6% ( " ' : 76 3 " * " "(

" B!F)

" "( )" " * #$ : 6: ( 1F) ' : 7% ,?- " " #$ : 54 ,

47&

0.30

0.20

0.10

0.00

−0.10 0.0

0.2

0.4

0.6

0.8

1.0

+ / 3 =2:H $ '$/ ? K ? % '$ / 'P$ / / / $/ &F +3 =2:H - 3 ) $ ! Æ .

0.50

/ 0.25

0.00 0.0

0.2

0.4

0.6

0.8

1.0

+ +3 =2:= / ) 2Æ .

"" ( " " " ' ( " " " ( $" F ( ( ! "" ( " "

" " ? "" ( < : " ( / " ," *

( Æ" " "" ( ' " " L ( " 1" ( "" " * "" " "

" *

) 7 4 " * " "" ( "

) 7 6 B!F) ( ) 7 % ) 7 : -#) $" ' 1F) " ) 7 7

47

45>

! "#$ * ( : = ( ! ( "" ( ' 7 " -#) " " * @ ; 8) 1 9 ( " " " " / + ; * " " ) ( " + ; ( ' 7 " 1 ' 7 4 #( B!F) " ' 7 4 A " ; "

* 7 8 , , ( < ( "" ( ( * " " Æ" " "

" + " ' ( / " (+ " " " " $" @ ""/ " ( " * "" 1" "" ( / " ( " " ( ! " $" "" ""

" ;" * "" ( " 3 " " ," "" ( : 4 " ( /

" " " (+ " " " ( ' ( " @ " $" ?" " " " " ; 3 " (+ + " " ' 7 6 )"

" " Æ" " > K B !>KB$ % G = != > ;%"$

! 5HH H $ #

- C > G # B % ! 5HH H H- $

45

Computational domain

Turbulent diffusion flame brush

Pilot flames

Air

Air

Main fuel: CH4+air Premixed gases for pilot flame

! % Q ! 5 + < ? ? 2AAA

" F ( + ( " " "

(

*

( 3

' " " " * ( ! * " "" ( " "

+ " / "Æ + " " " Æ 5

454

! ! & $-" 6

6 + $ < ? ? 2AAA

Burnt gases entrainment

Ambient air

Ambient air Mean Flame

Air entrainment

Air entrainment

Fuel

- Q $ ) )

456

" < & E + " " " ? "

" ' (

" *

" "" 7 % " " (+

" " " " " " " " " " "/ " $" " (+ " " + " " $" " 7 " + "

? / " +

,

)

)

@ & - ! 5& - =!

" " - R S < ' ) &

$ ) ) - ) &

5 , 21!

) & !

. ) .

( " +

" (+ '+ ' 7 % " " " (

" " "

" " '+

45%

Diffusion flame

Fuel Oxidizer

! % )

' ( E + " 3 " ( " (+ " ( " ' 7 : " ( E

1 " B" 1 C 2 C Premixing zone Fuel

Rich premixed flame

Diffusion flame

Oxidizer Lean premixed flame

! % ) & ' 7 7 * "" D " "

" " *

) & G . + "" ( "

" F

" " " " 1 C ( " ?"[ + ?" "" " +

45:

"" " " " ?"[ + ?" ; " " " " "" L " ?M " < "" " + ( +$" : 6 %

& - " " ) 81 9 ' 7 4 + " / ( " ; ( " + " ' 7 5 $" Æ " $" " ( " ' " " ; " Fuel

Premixed jet

non premixed flame Oxidizer

pilot flame (premixed)

) - )

" . " (+ "Æ + " "

( ' 7 & " " " " ; / " ; "Æ "

" Æ "

!" " 1

" ; ' " " Æ" " ( "

457

mixing layer Fuel

flame auto-ignition point

Oxidizer

δi

) 7 Æ

+ 3 " " + " " D " " + " " + / (a)

Oxidizer Diffusion flame Ignition point

Fuel

(b)

Oxidizer Fuel Burnt

Diffusion flame Ignition point

) ) " ( " " + " / " " * " "" " " * " " " ( 1 * " " " " " + ' 7 3 8"9 ' 7 " (+ " + ( " (+ B" + " (+ ; 3 " (+ ; " " " " + " ' 7

455

" " " " 3 " ( " " + " (+ (+ " "

"" ' * " ' 7 (+

" " " + D " " "Æ "

Oxidizer Fuel

Diffusion flame Immediate ignition

)

BA 3 )

F

" $""

"" ( " "

" (+ ' 7 >

( ) ' " + " ( "Æ " + +

"

' " + " " " "" ( , < 2" ) " " " ; ) " ( ' 7 ." " + + / ( ( ! + " + " " " ( + " + " + " + * ( " + %

- C > G # B ! 5HH H H- $

45&

(a)

(b)

(c)

" & & ) & ' & ; 3 / " + ; " )"

" " / " " " F " $" " ; " + $" * " " "" " ( " " "

*& , ( 3 ! ( ( " + ' 7 4 * " "" ( " ; ; " + "" 1 " "" " + ' 7 6

air propane propane air

& & Q Q

& . @3 N? ? 6111

45

120

100

Air flow rate (g/s)

Extinction 80

60

Lifted flame

40

Anchored flame

20

Instability Laminar rim stabilized flame

0 0

1

2

3

4

5

Propane flow rate (g/s)

B26 & . @3 N? ? 6111

, 2( ? )

? R S B2: )

Q 7 . @3 N? ? 6111

4&>

' " + " ! " E-3' "+ A; ' 7 %

" ; ( " / " " " ; " " ( "" E-3' ( " / ; "" ! ' 7 : ;

89 ) " ; 3 '

; + " + " ' 7 7 -

" ( + 8 9 "" " < ' 7 : 7 7 + / * 8 9 "" ' 7 %

; 8 9 + " " B " " "" ( " " " ' + " A" " " " ( *

' ( ' 7 5 " * " ? "

#1< *

" 7 % %

" " (+ " ! ; " (+ " ( " " *

" " ? * $"

% $ ! "" ( : 4 " *

" " !" ( % : % ," ( Æ" "" ( " ' ( ( D ( ( " Æ" ( ( * / " "

< 6 , "" ( " * + @ ( " 1F)

4&

4 ( R S B2: ) Q . @3 N? ? 6111

4 ( ? R S B2: ) ) & *& . @3 N? ? 6111

4&4

4 $-" + ! ' 4 BHH . @3 N? ? 6111

* +>2 ;67 ) ( " : 4 6 "" " $" " " " " " " " " <" E B ; " " @"" "" ( "

5 % & Flame

Oxidizer

r

Fuel

diagnostic axis

! " ) 6 % " ( *

" "

" D

L @ ( * " "" ! ( @ ( " <" E

4&6

+ 1F) ' 7 & <" E " /

* " "

Æ / Æ 0

0

> **

"

4 .

7

.

> 0 > 0 > " " " Æ !

* - <" E C E / 0 7 4 . + + 1 M " #$ 6 & ! 8 9 * + 0 Æ

(

( + * ' 7 & "

! * " 7 4/ ! / Æ ! / +

@ $

5 Æ 0 2 6 !! 0 6 7 8 8Æ

@ Æ %

¼

¼ % Æ

' ) !" # ? % +3 B6

' @ ( * Æ Æ ( : 4 6 / EN

4&%

! 1 M " ( / 0 0 Æ

Æ

7 6

J Æ Æ L 1 M " ! ( B " * / , 0

0

Æ

Æ

7 %

" 0 Æ '

Æ ( B " "

!" & %

- B61 B62 5 $ '

<" E " " " ( " " * !" ! ' 7 C" " + ' 7 4> * + ( " " " ' 7 4 1 M " ( & '" " 1 M " ) * 1F) " & /

4&:

u’ δi/τ 10000

CASE D

Curvature effects + Unsteady effects

1000

CASE C Quenching Da=

100

Da ext No quenching

Da= 10

Da LFA

3

4

5 6 7 89

0.1 LFA applies

No unsteady effects CASE A

LFA

CASE B Re vort ex =Re crit. 2

Unsteady effects

2

3

4

5 6 7 8

1 Quenching with Unsteady effects

Curvature effects with Unsteady effects

Unsteady effects without quenching

r δi

5 % # $ B6 '

' "Æ 1 M " )

" -'!

3 " "Æ " , ( " " ' 7 4! ) 0 4& 3 , " " " " / ( " " ' 7 4, ' < ) ! 1 M " ( ( "Æ " 89 -'! 3 " " ' 7 4< "

"

"

4&7

. " ( $" " 1 ," ' 7 41 " $" " 1 M " ( / $" 1 M " " 1F)/ 0 >%& < !

< ,

< <

< 1

4 3 6 ? B2> ' - !

B61 ' - 5- , .? %? % ? % ' 3 ! '

4&5

)"

* " ( " * " " " @ (

" -'!

"" ( * " " 3 " ( " " 6 % 6 1 A" * " ( + " "" ( " ( " ) : 4 6 ' : 4% ! " Æ" " * !

* ""

$" 3

* 1 M " ) " ( ""

" " .

*

7

- "" ( " "" + " % 4 ' "

" ( ' 7 44 " / "

*

+ (" ( > > 3 " (+ ( * " " 3 ("

* V> 0 +

+ 3 ("

V> "

> 0 > "

( / ' ( * 0 > > 0 > * ( ' 6 & 6 % %

" !

< 6 6 : 4 " + 1 M " + #$ 6 &5/ + 0

+

.

7 :

4&&

Fuel z=1

z = zst

lt

ld Oxidizer z=0

ηk

lr

' . ("

" > 0 > . " (" #$ 6 44 " "

/

"

.

7 7

" " " " ( "" ( "

("

" > 0 > . / " 7 5 .

. * /

. 0 4 > >

7 &

# > # " > 0 > . " $" "

" !

" "" " " " " " #$ 6 77 /

#

$

. 0 . ( 4 4>

0 . < > 0 .

< > < >

7

0& & E

>

+ " +

! " + ,! Æ

/

¼¼ > . 0 . > &> > 0 . <<>> &> > 0 . >

4&

7 >

. . > > ¼¼

" ¼¼ " &> > > "" F . " "

. " ' ( * " . J " . " (+ > 0 > " " > 0 . > 3 < > 0 > 0 < > 0 0 > ¼¼

> > > "

. " " + / + 1 M " / + 0

.

+

.

7

7 4

! 8 9 B " "

/ B 0 ' " ' (+ D/ < J - A

7 6

! - "

" 1 M " + /

+

7 %

' 0 J - J ! ( 1 M " + + " + 7 6 ," $" K !0,4$ ! ! ! Æ ¼¼ ¼¼ ! !% %

%

$$ % 0'(5

!

¼¼ %

¼¼ & %

!% + ,$

< !% + 4$ G '

% > # ! 0(,$

4>

*

! % $

7 " $

#

0

%" 27 0 % 0 2 " $ 7

R S

'

" $

0 %7

%" # %

" $

!*T

* < ; "" ( "

"" ," "" ( ! " $"

" "" B

* * "" * 3 "" " " "" !

" " "" ( " "" " % 4 ' % * "

.

" + /

+ 0

.

7 :

F * " " B " "" / 0 0 0 +

7 7

( +

* (" / *

>

7 5

"" " ' " 1 M " * 8 9 ' "

3 ! $ 3

! " # ! $! % & & Æ ' # "

# " " " !

! ( ! # ) #!* " ! + # ! " # ! ! !, $! ! ! " ", " " ! ! " & !

) - * #! # # #! & ! . )/% 0 * ! ! " 1 " $! ) ! ! "* + -

-

)2 3*

4 ! " #,# ) * # Æ ! ! !& )5* . # 4 6 ) 2 * ! " / # ! ! 1 ! " ' " # ) # 2 *

Da

D fl = Da

Laminar

Flamelet

U

ea nst

dy

e

ct ffe

s

A LF a

D fl = Da

ext a

Extinction

1

Ret

! " " # ! " ! # $! ! ! ! " " ) ! ' 0 * (

# & " ! # ! , " ! 7 !

" , # " # !& "

! & ! ! " " ! 8#& #! $! " & + 8#& #! ! ! #! " !& ! " & ) 4! 0* 9# & ) * !& " # ( : " " # & "7 # # & ! & " ! );#! ; 6 5 "" < 6 *

$! # ! 1.9 # " ! " " 4! 0 ) 0 * 4 " ! " ! ! & " )/% 2* 4 &# ! &

% 2 0 = # : ! ! 2 0 + ! % : ! ! 2 0 2 0 0 2 0 2 " " : ! " "

4 " & ! # +

> , $! ! =! " ) * > , ! % ) - * > , = ? & % ) - * >0 , 5 " %

! "

> , ' ! " !" ! ) * ) * ' @ ! # , ! " ) ! * ) * ' ) * ) * !#! ! )/% *+ -

A

A

-

) * A

) * A

-

) * A

) *

)2 *

! > >0 : ) , ! ! * & " " ) 2 * $! : % ) /% 0 0 *+

* )

A

A

A

* A )

) * -

A

) * -

) * - B -

)2 B*

)2 *

A C

-

)2 *

)2 *

# ) & " & "* ! " ' " ' ! !#! /% )2 * $! " % !

! = ' ! $! % + " !

C 1 ) * # "

) 0 * $" ) * # # )/% 0 3* )> * 9 )5 ; 5* !& & ,# " " ) 3* ! = # ! C ! Æ " "

( 0 ! ! " "+

0

. # " & # ! : ) * . " !

C ! $! " ! ! # &

" ! ! "

( ! ! " & + . # " & # ! &# : ) * ! #! ) * . " !

C ! $! ) * " $! " " ! &# # $ &# & ! & Æ + ! #! "
)C * # & "+

) *

-

D

) *

)2 0*

C -

C ) *

)2 *

! ) * ! &# % # & & ! - # & % ! ! ) * ! ,"" )* /% )2 0* )2 * ## ! & & " "

+ $! " /% )2 0* .

! & % ) * ) * ) * " !#! " % )4=4 # 2 0 * )/% 2 * " %

# % C + 1.9 & & " ) * & " ) * ! "" ) * $! ! " 2 0 2 0

( ! " % )/% 2 * & & . # C !& " " " $! " "

/% )2 * " )C * $! ! # : ! 2 0 0 : ! 2 0 2 $! & & " ! # ! ! , ! " " % # , % D " & & E ! ! ! !

! " & ! " ! 7 ! )C * " 9&! ! " ! & ! ! ! , % & % ( ! & & " !

) * ) * # /% )2 0* (

! ! C : )C * ) * ! /% )2 * /% )2 * ( ! " ! !

! & & " ! . # # # " # ! ! # " "! !

! ! & & " ! )F& ! BBB & * $" 2 0 ' ! ! !

& & £ '

! "#$

! "#%

! "##

! "#"

( )*! + # $ $ % # & $ '

2

. ! 4! ) * : & " ! ! 5 " " " ) ! * ! !#! ) * ) * # &

/% ) * ) * & " : ! ! % " ) G* ! ) # 2* (

: ! & " &# ) * ) * /% )2 0* +

- ) * ) *

D

- ) * ) *

)2 2*

= ! "

) *) *

-

D

) *) *

)2 G*

$! ! ! ! "" ) * ! $! ! " : ! )E?; 6 4! '* 5 " " ) G* ! "" " ! " % # # " ! ! ! # $! " ) G* ! ! & # + H ) A * ) * ) * ) * )2 3* ) * H )* H )* ! ! ' ) * ! H !& " : " /% ) * ) 2* $! & + -

) *

D

-

)2 *

$ ! , # 4! , % & 9&! ! " " " " ! # - B - ! # B )6 * ( " ! ' ! # # ) *

MEAN VARIABLES USED IN RANS CODE

YO0

G

T(z)

YF0

Y F (z)

TF0

Dynamic pressure

~U ~V ~ W ~z

TO0 Y O (z)

ρ

~ Τ

0

1

zst

z

LOCAL FLAME STRUCTURE

~

z”2 p(z)

TURBULENCE VARIABLES k

ε

~

z

PRESUMED PDF OF z: function of~z and z”2

, & ! ! ! " ! " " " . " # " ! " ! : & " ! # 2 0 ! ! " " 1 " E ! # 2 0 ! 1.9 & # ! &,&# " % $! % ) 6 & =! " * ! ! % : ! &# & !

% % ! & $! ! % # ! ! : " ! ! $! ! & $! % " & " ! 1.9 ) " %

& ! * $! % # " & ) ! % * " & ) ! % * ! ! " ) " % 0 * ! " # " $! # ! "

! '

( ) *+," % '

3

: ( ! ! # ! ! E " & ! ! & " % ! "" ) * )6 '* $! " % /% ) B2* !

. ! & % , )/% 2 * ( 1.9 # # & " % $! "

% & ! " % I " ) BB* ! ) & *+

A

-

A

"

A

)2 B*

!+ -

)2 *

! ! ! : $! ! & ! !# ) # * ( ! /% )2 B* +

-

-

)2 *

( ! !& " ! ! # # ! ! ! ! ; : " " ! : ) /% * ! $! " +

-

-

A0

A

)2 *

% )-" *." '

! ! 1> & ! /% )2 * + - A

)2 0*

! ! ! : ! ! ! " ) :* 9# #

# # # 1.9 /% )2 * + -

-

)2 *

$! : ! ' $" 2

*

-

$ / + 0' 1$$ ¼¼ ¼¼ 0' "12 , $ -

)*! 3

$ .

4 ! !& " ! & " % )2 B*+

$! : 1> /% )2 B* " " # # + -

!½

)2 G*

! ! " & & " ! " ) 0 * !½ " ! " # ! = /% )2 B* # # " # Æ # 1 " $! " #

/% )2 G* # ! " & " ! &

) */0" 1 2 + */*" '

BB

$! " ! # # +

-

)2 3*

!¾

! !¾ " ! " . & /% )2 * ! ! ! " ! & ( ! , ! ! ! ! # ! ! & : $! # ! ! " # - + - "

-"

)2 *

! " $! ! " +

-

)2 0B*

"

$! % 1.9 !+

A

-

!½

A

!¾

"

)2 0*

$! % " & ! /% )2 B* )2 * )2 * . # : ! ! ( ! ! # 2 0 ! " ! ! )! " ! * . # ! : " ! ) # 2 * ! " $! , # # & J K )L ' * $! + : , # " ! ! ); #*

$! & ! ( !

C & & " ! & & " )

MEAN VARIABLES USED IN RANS CODE

B

YF0

T(z)

YO0

Dynamic pressure TO0

U

Y O (z)

Τ

V

0

Yk

W

Y F (z)

TF0 1

zst

z

LOCAL FLAME STRUCTURE

z z”2 p(z) TURBULENCE k

ε

z PRESUMED PDF OF z

, 4

+ 5 +6 * ! >& ! ! "+ ! & # ! ) " * ! ! !& " ) ! * " ! " & ! ) 2 0 * $ ! !& " "

# : ! $! : ! /,;,@ ) * ! ' ! !

$! / 4 & " =# =7# ! /,;,@ )/;@ * " $! " # C ) * ' ) * ) * " # # #! + C -

) A *

) A *

)2 0*

! ( /% )2 0* ! " ! :
B

$! " ! /;@ ) * ! =#

+ /4 # " ! ! " " 7 " " /4 ! " ! /4

" # ! ' " !

= " /% ) 2* # + )2 0* C - # &# #+

#C -

) *

)2 00*

! ) * ! 7 M # ! ", " ! !& # & & ! ! : ! , ! ! , & - ! ( ! # $" +

-

$ ) *

-

Æ ) *

)2 0 *

! $ Æ & ! >& ! , /% )2 00* !

+ #C -

) *

) * -

) *

) *

)2 02*

! ) * - ! - ) * ! "" !& - . " ! " & ) * $! ) * . : " ! # /

)2 02* ! " & ! )2 B* )2 *

! . # : ! % , # ! " " . : ! ! " & " # % ! ! " ) # B* $! , # 2 2 ! ! ! # ' #

Ignition point Fuel

B

Stoichiometric line B

A

Ignited zone

C

Oxidizer Mixing zone

Quenched zone

7 + 4 8

# & " % ! ' ) " ! #! " * ( # 2 2 # ! # & Æ ! : + ! # ) . # 2 2* # )6 ;* % ! )6 4*D . ; 4 !& ! " & . % ! ! ! " $ ) * ! " "

" & & " ) * " ! & # $! ! ! " !#! )4=4* "" " %

$! " # ! " " ! ! + ! , # " ! ! " $! % ' " ! " # ! " ! # 2 )6 * # 2 G ! ! ! JK " & " # $! : ! Æ E"& ! ! ! ! " ! " $! ! " ! ! : ) ) * ) ** ! " & : ! )6 ! 6 ; ! !* ; ! : + & J#K " D # ! $ ! ! ! ) % &, ! !* ! ! " ! " @ # !

B0

Fuel

YF0 TF0

Instantaneous stoichiometric isocontour

Instantaneous flamelet

Stagnation point laminar flame

FUEL

YO0 Oxidizer

OXIDIZER

TO0 YF0

YO0

TF0

TO0

+ ! + & !#! ! ! % ! #

" " ' ( !& : ! ! " ! " @ ! ! # & " ) * ) * /% ) * . ) * ) * " # ! " J " K $! ! " +

) * ) *

)2 0G*

) * ) *

)2 03*

! ! # & "+ -

! # + ) * - ) * )* )2 0* 6"" ! $! ) * # # ,

' % ' $

1

3 & ( # 4 & " '

B

" : ! ) 2 0 * . " % . , " )6 ! 6 * ! # # ! +

)* - Æ ) *

)2 B*

" ! ! # " )/"#

) '* ) * % % &

! ! ' ! & + -

) * -

% ! & )* !+

)% * - % " + % - #

'A

6 *+ )2 *

%

)2 *

)2 *

- ( )/"#

Probability distribution function

6 *

0.0

0.5

1.0

1.5

Scalar dissipation/mean scalar dissipation

2.0

2.5

( χst / χst )

¼¼ $ / 6 2$%9 22"9 ¼¼ :9 $ :;9 0' "%: . ¼¼ $ 2

6 5

% ! " 1 " " ! * " 5 ; % ! ! "" )* : # 2 3 ! # " & %+ % - # & # & ) - (G* ! - B(B2* % - B( & ) % - B( )B(

B2

! "#$" "%! ! ! $&" ! + ! !" '( ! ( 0 9 0 ( ½ (

(

)*!

, + Y O (z, χ st )

MEAN VARIABLES USED IN RANS CODE

T(z, χ st )

YO0

Dynamic pressure

Y F (z, χ st )

TO0

~

U

YF0

~

TF0

V

~

W

~ Τ

ρ

~z

0

1

zst

z

LOCAL FLAME STRUCTURE: function of scalar dissipation

~ z”2 p(z) TURBULENCE VARIABLES k z

ε

~

PRESUMED PDF OF z: ~ function of z and z”2

k,

ε

~

~χ as a function of ~ χ = c~ z”2 ε / k

Model for

and z”2:

PRESUMED PDF OF χst: function of

~st χ

Model for mean stoichiometric scalar dissipation rate

~ χ st as a function of ~ χ and p(z) (use shape function f(z) for χ=χ st f(z) )

) < 8 5+ 4 0' "#; '

BG

( ! # " 2 # ! ) 0 * " + ) ) * - ) * - )2 0* ) ) * !

) ) * -

) *

)2

*

$! " )/% 2 *+ -"

)2 2*

$! # ' , " & & " '

$" 2 2 # 2 $! ! : ! $! 1.9 & & " ) # 2 * = ! " ! ! ) " * & " ) * : " ! 1.9 4 # : ! ) 2 0 * % + : ! !#! ! ! $! !& " & " , & % ) " " * " 2 0

( ! 4 = 4 )4=4* ! & " 8 ; # : " " $! " & & " % # ! & # & & ! - $! ! # & "+

) *

)2 G*

$! ! % )8 ; #*+

; % ! . %

! & $! % & & ! , # % Æ # " # & " ! Æ

B3

. ! "" ) * # $! ! & " & , ) - ,* 9&! " % !& " % # " # & ! " # & ! #!

$! ! Æ & # " % "" ) * /% )2 G* )2 G* ) * /% )2 0G* 9&! ! ! #

" % ! ) * & ) * ) * . # " %

! ) 2 0 2*

. 2 0 0 # ! & & " & ! " " C ( ! % & " & ! &# C $! &

" # !

( " # " # ! " & & " ) * !

2 0 C ) * " ! " $! C ! # # + C -

C ) *) *

)2 3*

$! 7 # , + ) * - ) *) *

)2 *

, ) * #, " # ) 2 0 * $! ' # 2 B $" 2 G $! 1.9 & & $! ! "" " ! ! " )/% 2 3*

! $& ! "%! ! ! $&" ! ! !" '( ! ( 5 0 9 0 ( ( & +

B

"#$"

&

&

&

, 5+ MEAN VARIABLES USED IN RANS CODE

ω k ( z ,χ st)

ρ U V 0

W Yk

ωk

T

ωT

1

zst

z

LOCAL FLAME STRUCTURE: function of stoichiometric scalar dissipation

z z”2 p(z) TURBULENCE VARIABLES k

z PRESUMED PDF OF χst:

ε PRESUMED PDF OF z: function of z and z”2

χ as a function of and z”2: χ = c z”2 ε / k Model for k,

ε

function of

χ st

Model for mean stoichiometric scalar dissipation χ st as a function of χ and p(z) (use shape function f(z) for

χ=χ st f(z) )

* < 8 5+

B

" #! ! E !

# # ! ) ! * NC ) * ! " $! C ! ! ! O " ! NC ) *+ C )* O C - N )2 2B* $! O : )6 F& ! F 8! F *+ )2 2* O - - ) * ! ) - * ! &# ! ! - C ) * ! # &# # ! ! N F& ! F !& ! ! + C -

C ) *) * -

N)*) * O

)2 2*

! ! ! ! " # !+ NC )* NC ) * )2 2* ! ! % & , - )* ! C ) "" ! N * ! " . ! % ! &# " % & " ! 1.9 . ! & " % ! O !& " : " =" ; ! !& & O,% & # $! " % # " )* ' )E*+ O

A

O

-

O

%

A *+O

,

A

,

O

)2 20*

5 (

# ( 6' # ' - /7/ 8 5 " % :9

&

# ) /*."

! + ! # ! " * , , & ! ' , . # /% ) 2* , # & " , - NC , - , ! ! ! Æ $! +O ! /% ) 3* & , ' # " " : $! ) /% )2 20** " ! # # ! ! ! ), O , O* & & " ) O O* $! ! J

! K )=" ;* 1 !& & " ! , "" & # ! # % O % & & # % ! ! )6 F& ! F 8! F F& ! F * $! ' $" 2 3 # 2 ; % ! & # % " % " & $! !& & & #+ , # ! ! " " " " )F = 4 F > $&* # , NC )* % ! # " >& &

% " & # ! !

$! ! & " " + ! , ( ! # % $! + C -

½

¾

(((

C ) ( ( ( * ) ( ( ( * ( ( (

)2 2 * ! ) ( ( ( * ! 7 "" 1 C & " ).! * ! $" 0 $! & # ! ! " " ! ! ! ) C * " ! ! 7 "" ) ( ( ( * ! ! # ! ! $ " +

% & ; ! ' ( 4 ' ' ' 3 & '

! '& # $& ! "%! ! ! $&" ! =& =& ! !" '( ! + 0 < ( & 4 ( & 4 ) & &

"#$" =& 9 =& >

=& & =& > & =& >

, + + MEAN VARIABLES USED IN RANS CODE

ρ

Ω k ( χ st )

~

Ω T (χ st)

U

~

LOCAL FLAME STRUCTURE: function of scalar dissipation

V

~

W

~ ~ ω T = Ω T (χ st) Σ

~

ω k = Ω k ( χ st ) Σ

Yk

~ T

Model for stoichiometric scalar dissipation ~ χst as a function of k and ε

~z TURBULENCE VARIABLES k

FLAME SURFACE DENSITY

Σ

ε

+ < 8 + +

. # & ! ! "" # , #, " & " !

& @ ! ! & Æ ! ! 7 # ! & " ( ! # #! !

! ( ! " % & ! 7 $! % ! " )/% B2* . " !

" % E ! ! ! ! # ) #* F !& " ! ) 6 '* 6 " Æ " #& " # !

" , "

9 " #& " " # " ! # ! " . # # "

" " # !#! "" ! . # ! ! " & # , ! " # )J K*

!

" #

$! 5 / = )5/=* ! & " 8 )8 8 8 8 8 ==! * " , ! " # ( 5/ 5/= & # # ! "# & "# ! " # ' , " $ "% # &&+

! " #

, : $! , "

$ ' 1

" 2 " "

" " ***"

' " " 6' & 5 $

0

(b)

(a)

l x

x

x0

?. @ / 0 3 A 5 B 8 ) # 2 * # # & # ' # # 2 " )J K* $! " & ! # " ' $! ! " # !

! & & ' ) P ! ! 8#& #! P ! 5/ : '* & % - : # # & "

" " " , ", % +

-

A C

)2 2G*

4 ! " /% )2 2G* 5/= & : ! " % ! % 9&! " 7 ! !& " " 9 ! , % ! )/% 2 2G* # !& 5/= " # )==! * " )==! = 4! = =! 4! )= 8 = ! = * " " : Æ & & M" )6 1" * ! ! 5/=

! $ 6"" 1.9 5/ "!

: ) 0 G *+

)# * )# *

-

C )# *

-

)# * )# * ) )# # * #

)2 23*

C )# * ) )# # * #

)2 2*

! ) ! 5/ : . " % " ! ! $! /% )2 2* " +

C

-

C ) * ) )# # * #

-

C ) * Æ ) )# * * ) )# # * #

)2 GB*

! Æ ! , $! + C )# * -

C ) *

Æ ) )# * * ) )# # * #

"# ) # *

C ) * ) # *

! ! "# "" ) # * ! " )Q* $! "" " ! " " & # " % ! ," % 1.9 4 1 ! , ! # , ) 2 0 * " ! : & + H)* A * ) * - ) * )2 G* H)**H) * ! * -

) *

- ) *

) *

)2 G*

. )" * " % " & - ) * " # # & % # +

- ) * -

) *

)2 G*

! . : # ! ! 5/ : " ( " )4 1 * " )4 4 ;! * & !#! # # # ! = & )1& F& ! 1& *

2

! % $! "# , "" " : ! ! ) * "! !

)4 1 ; 8 * $! + -

) * ) *

)2 G0*

( # , & " ! )4! 0 /% 22*+ - ) ) * - ) * )2 G * ! ! & ! " ) * ! " + -

) * ) *

)2 G2*

. # ! ! ! # & "+

)# *

) * ) # * ) # *

)2 GG*

¼ ¼

) *

)2 G3*

( # ! /% )2 GG* +

)# *

) * ) # *

)2 G*

! ) * # " % ) 2* $! ) " & " # *

!#!+ )# * -

) ) * ) # *

G

)2 3B*

. , ! ) # * # ! : . ! % ! ) Q 7 R! ; 8 6 = 4 ;! * . % " !! # & ) ; 8 *+

-

/

A

!"

) *

)2 3*

! ! : & / ! 6 " ! "# " & & " ! "# " !" " !

"

M& !& " " " " # ) 2 * ( ! # ! & # ) 2 2 * 1 #

2 2 $! ! ! #" " ) 2 2 *

& '

. & , " # 9 # & " ; $! #, ! ! ! ! Q1( ! ) ! * ! " # & " , "! )* ) * . )9* " ) * # !# " : $! ! " " : # 2 9 ' # JK ) # 2 0*+ ! : " - : : ! ! ) * )* ! " ! ! ) /% 3* $!

<=% '

1>>&> >

3

C + . %22 D E F 0 ! G (

Flame front

z = zst

Fuel n z=z s t

T

. ω

Oxidizer

cut along normal to the flame front n Statistics over flame cuts: - Tst vs Da . - ∫ ω dn vs Da - Da histogram

. - Integrated reaction rate ∫ ω dn - Temperature at the stoichiometric level Tst - Local Damkohler number Da

*! G ( +

9 ) : 9 6 9 ' .

Extinction in flame / vortex interaction

Extinction in steady stagnation point flames

Limit of LFA domain

Steady laminar flamelet assumption (LFA) holds D a ext

DaLFA

DaSE

Da

Increasing stretch and scalar dissipation rate

5 *!

χf [s ] −1

50

10

5

2

1

1.0

200

0.8

150 2

∫ωhrdn [W/cm ]

(a)

T /Tfast

0.6

Da

Da

LFA

ext

Da

100

initial

0.4 50

0.2 0.0

1.00

0 100.00

10.00 Da/DaSE

C 7 +

¼ :$ 9 :H 5 4 I

*! . *! / . /

9 6 + . 8 6 0 ! EG 9:JJJ9 F (

Æ

B

! # C 0 # ! ! ! ! " "!& ) * C 0) * 9 ! " ! ! # % # )! , J " K* $! " # ! ! ! 9 $! & ! ! " !& # ; ! ! " " 4 6 ) 2 # 2 *+ !" ! ! " : ! ! ! ! ! " ! ) 2 * . # 4 6 ) 2 * ! ! " + - !" - B(0!" 1 # 2 2 " & ) - ( 1 " 1 " - 3* # 2 G " ! #! " & ) - 2(B 1 " 1 " - * χf [s ] −1

50

10

5

2

1

1.0

200

0.8

150 2

∫ωhrdn [W/cm ]

(c) T /Tfast

0.6

100 0.4 50

0.2 0.0

1.00

10.00

0 100.00

Da/DaSE

C 7 + ¼ "2 9 J$ / "1" 0 ! G (

! : ) # 2 2* # ! " & ! # ! ! ! ! " # ! ! !

- ! & + "! # !

! ! # ! " . ! " & ) # 2 G* ! " ! ! & ! ! ! ! " " ! ! & $! ! # & ! # : " ! !& ! " ! ! & !" ! " $! " % ! ! )' ! * $! ! ) # 2 G* & !#! ! " ! % ! " >& & " - B(0!" ! ! ! & : # ! ! " ! !" =& # 2 G ! " ! ! ! ) # 2 2* & !#! ! " : $! " # ! '

' = !& 9 & # # ! " & " ! ) :# # 2 3* $! # ! # " # " ( "! 7 ! " $! "7 & ! # ! &

Fuel

Air

z=1

z=0

T = 300 K

T = 1000 K

Non-reflecting

Non-reflecting

Periodic

Periodic

* 3 5 6 )

5 7

. # # &# ! ! )C * , # ! # & & ! . !

# 2 )* # JK & ! # $ - B( )! = 1 & * # ! & !

C + C - 2

)2 3*

! ! ! & - B(B

; # ) #* C ) * ) * ' , ! ! # $ - B(
) / 4

&

4

2:: 2:# 2:$ 6 4 2%$B 4 2"; 4 22;H9 6 ) ? @ 9 2:$ 6 22%% 0 ! 3 (

Æ

$! # & ! & ! J &K # $ . # 2 ) #!* # & ! $ ' # " ! J &K ) - # $ # * !& # ! $! !& " " = & # " : " 4=4 )4 = 4 2 0 * # 2 0B :#

* . . +6 "1H 6

A

# 2 3 ! : : ! " ! " < ! : ) % " * ! ! ! # & 0 $! ! " E ! ! ! ! : ! !

% ' ! & 7 ! # ! & ! : ! $! ! " ! " & $! :# ! ! # ( ! ! : ! # ! ! ! ' & E ! ! ! ! # ! : ! ! $! = " ! !& "

& " & 8! # # :# $ ! ' 7 ! "

# 2 0 . ! # 2 0 )* ! ! # $ ! & ! & ! ! & ! " ! ! , $! : # : " ! : & ! ) # 2 0 #!* 9 ! ! # ! : && ! : ! ) # 2*

0

Oxidizer (z=0, hot) Computational domain

Fuel (z=1, cold)

+ *! 6 8 A

*! 6 8 .

6

A

& ' F 8! F !& & # ! " & # , # . ,: !# " : 7 ! " # & ! # 2 0 $! : " ! ! , - " " = ) &# # * # # & , ! & " 1 " # 2 00 1 ' $" 2 . ! ! " 1 " ! #

; ! : # " ! )$" 2 *+ ! & " ! # ! " "

K L M A M

2.1

8.0

1.9 6.0

1.7 1.5

4.0

1.3 2.0

1.1 0.9 1.5

3.5

5.5

7.5

9.5

11.5

X

0.0 −0.6

−0.4

−0.2

0.0 y

0.2

0.4

K + 6 4 9 & 9 8 + 6 9 + :: 4 :2% 9 $2: 1:2 M A M

C " ! N ! ! #

! : " 9 # $" 2 - B ! " ! $! " " GB S ! ! ! ! ! 9 ! ! " "! ! ! ! " & ! # 2 00" $! # # " ! & & " $! " ! ! ! ! ! )/% 2 2B* # 2 0 & :

2

&

>

2 :2% :2 :: :2 :2H :2 :$

$2: :# :1 :;

1:2 $: :; $%

) K + 6 9 8 + 2 65 :: 6 M A M

! C ! O ! NC - C O $! % " ! # ! ! ) ! " ' &# #* ! " # ! & " ! !# " $! : # & ! ! # F

8! F 8! F F 8! F !& ! & # ! " % ! O " ! $! " ! ! " ! ! # "

.6 4 & 9 + > M A M =& & >

$ " & ! ! & ! & ! ! M ! ! & ! & Æ ! " ! " !" $!

" " " ! # # M $! !& & # ( " ! # :

Æ ! # " = M !& " " + ! ! ! " & !#! & " ! ( " & " # ! "

BB BB 8 ! " 0BB 2BB 8 " # $! " # & & , ! ! ! ! # & # # # ! " Æ & " # ! & ! ' ! + ! " % ! ! !#! ! ! ! J K ! # ! # ! =& ! # " $! ! + ! % ! " " ! ! " ! . ! ! # , # & ! #! ! $! ! ! ! ' Æ )5 /' *
! " ! ! ! # ! "

G

3

WALLS

Wall quenches flame

Wall modifies turbulence length scales and creates turbulence

Flame heats wall Turbulence changes friction and heat transfer to wall

Flame modifies turbulence FLAME

TURBULENCE Turbulence wrinkles the flame and strains flamelets

+ 6 6

9 + ! ! " ! " ! " ! ! ! ! " . ! & M

" ! ! # ! ! # /& !#! M ! ! " 4 $! !& " 4 # " # G ( ! # # " " "

) 2* ! , ! " ! + $! : ! % ! ! ! ! $! ! ! ! ! ! " ) * + # ) /% ) G** " "

" . ! !

" !" ! ! % ! $!

" ! # M ! ! ! ! ! & . " " ! # ! O ; % ! # & # ! ! = ! # ! & !#! ! & ! " # #

WALLS (1) Wall: - quenches flamelets - decreases flame surface - creates unburnt hydrocarbons

(3) Turbulence model must be modified near walls (k and ε )

(2) Flame touches wall: - large instantaneous heat flux Φ m

(4) Turbulence controls: - Φ : mean heat flux to wall - τ w : mean wall friction

(5) Flame changes turbulence: variable density effects + increased viscosity FLAME

TURBULENCE (6) Turbulence controls: - the flame surface density Σ - the flamelet speed wL

3 6

$! ! : " , " , " $! ! & ! " ,,!, ) # ! ! & 8 4* 1 " & " 4" , # & # " " && # $! "& " ! " #! + ! " !

! ! ! ! " ! /;@ ! )/% ) 02**+ C - "%&

I* I)

)G *

9 , ! # ' " " " ! ! C # : ( # /;@ ) * ! ! # ! ) : * ! ! # ! 0 = ! ! T ! ' 5# ! ! "! % ! # ! " 6 # ! !#! !

# / # # M ! + ! ! ! !

B

! ,,!, )8 4* # # ! ! E ! " # ! =& ,,!, # & " : ! # # "

$! " ! ! "

& 2 $! " ! + " # )! O * )! # *

! "#$ $ ( ) $! : M % ! # )L > # ; ;! < ! ; 8 ! 5 4 # <" * $! " ! ' " + ! 3 " ! ! T !#! ! $! 3 ' " ! ! ! Æ - - ) " * ) * : 6 "+ / -

3

)G *

Æ

Premixed flame front Wall

y Fresh reactants

Burnt gases

6 + 6

$! ! T : "+ T - -

3

'

)G *

! - ! # ! & ! 4 # % ) # G * ; & ! 3 ! ! ! # !

T # & ! ! ! $! ! T " ! J K )! ! - " ) * * + ) -

T

)G 0*

! # ! !U# ! ! ! ! ): " - " ) * ! ! ! # ! " * 9 !

$" G 4! + ! " JK ! ! # " J K & % ! #

*

C6

! Æ "

D

N # N $

+ * +C6

$! M 3 ! ! T ! # ! 4 ! # G 0 ! , # ! ! ) - * ! # ! :
! #! ! # ) - * $! # ! )!#! ! * # ! + ! " ! . M

! % ! #+ 3 ! % ! ) - # G 0* Q ! ! T ! . % ! # ! " # " ! # !

$! % ! # 3 " ! % ! # 6 " / + / - 3 Æ )> # F 5 * . ! % ! # !

' ! ) ! 6 " / " + # ! ! ! % ! # T ! ! # ! 3 + T -

( 3

)G *

! ( ! 9 ! /% )G * ! ! ! " # 1 & # M " ! ! ! " ! ! ! !

WALL

T=T1=TW

t=t1 sc=sLo

t=t3

y(t2)

t=t2 sc<sLo

WALL

WALL

WALL

T=T2 y(t1)

yQ

t=t4

/ + 6 6

. . ' 6

O

/% )G * " ! ! ) % ! #+ ) -

(

/

)G 2*

# ! : Æ - - ) " * # ! " - A M "& & :# # G 0 , " " ! ) # G *+ >U % ! # )>EV*+ ! ! )( - *

# !U % ! # ) # G 0 G * $! ! " ! )< ! ; * )<" 6 6 ; * )> # L F

* 1 ## ! % ! # 6 " / ! ! > & ) ! B(0 ! ! ! ! & " /% )G 2* ! / + # # ! ! ! ! ! % " U! !

4! ) % & * # % ! # )L

# 8 ; ;! ; ;! * >& ! ! ) )> # * ## ! ! " ! ! " ! " $! " " <" " # ! ! U % ! # )

y (wall distance) Cold premixed gas

Flame front

Burnt gas abscissa (a): Head On Quenching (HOQ)

yQ Cold premixed gas

Flame front

Burnt gas (b): Side Wall Quenching (SWQ)

Cold premixed gas

2R

Flame front

Burnt gas

(c): Tube Quenching

. +C6

+6 ! " & 8W W = =!& ' = & " 5 4 # V ! # 6 " / ! ! G ## # & ) " B(2 )/% )G 2** . ! U " " % ! # )< * # & ! # / $" % ! #+ % ! # " Æ )5 F /" L ! * ) # G * $! !,

! " " " & & ! " ! # + ! " " ! ! ! % ! # !

# !#! ! . : ! " #! " # ! # ! ) # G 2* 6 " ! ). > * " ! " )* !

B

(

& '*

. : # G G # ! >EV :# )6 * $!

0

Grid with holes such that Pe = r/d < 50

Flame front

Reactants

3 + ' 6

! + #, & " & $! , ' ! ! ! !D ! " $! ! ! ) # G * . & ' ! ! % ! ! # # G G & ! 6 " / - 3Æ )! ' ! * ! " ) * ) ! ! ' ! # ! ! & ! * !

' ! ) - T ) " ) ** $ ! " ' " ! )Æ ! ! : ! #

- Æ

/% G * $! ! " ! ! I - B( ! < ! ! : ! : & % ! # ! $! JK # # ! # G G $! & " 9 # ! I - B( ! / - (0 ) - B( $! & # # ! )5 F * ! ! # & " /% )G 2* $! ! )! & #* ! % ! # ! )6 * /& # ! # % ! # % ! # " ! S : # ! ! " !

( + '* 4 # M ! ! ! ! ! ! & "+ & !#! ! ! " G !

% ' & Æ & Æ $

1 Flame power ρ 1 sc Cp (T2 -T 1 )

FQ=3.4

F (Heat flux/flame power)

0 0

2

tQ

4

6

8

Time

Peclet number P = y/d

12 Flame moves towards the wall

After quenching, isotherms move away from the wall

PQ =3.4

0 0

2

4

6

8

Time

! I O I

" ! # ! $! Æ ! " # : ! # + = ! ! & :# + # # # EV % ! # ! ! M # ! ! ! , ) # G 3* ! ! & !

2

$! ! & !

! & ! ! & ! , ! M # ! ! ! & ! $ # ! & ! ) # G 3* " 6 ! ! # ! &

I O I
, G ( 9 / 9 08

M & ! #! & ! + & ! ! ! 5 41 !" ! $! & # ! ! & # BB 5 41 $ M6 # ! ! #! >& ! & ! ! ! ) * ! ! ! # $" G ' ! " # )< ! ; * ! # , ) A E ' 6* )6 * ! ! M )6 ; * ! !# M# )F* !

M )> * /& !#! ! & ! # " ! ! ) ## #

G

! ! # % !# M# (

E&

< ! X ;

.!

4!

T

1

%

)

) A6 /

$!

B(

6

) A6 /

9

B(0

6 X ; G

$M , "

9

4

B(

B(0

F

$M6 , "

9

4

0(3

B(

> BBB

$M B "

9

4

B

B(0

0 +56

% "&$ $ /& !#! M & &" ! & # ! ! ! "

! $! " " && ! " G . # G ! ! ! ! ! " ! " !" . & " ! " ! " ! " # # # ! + ! ! ! ! )! # G * !
" ! % #+ $! ! T ! " & ! 6 # ! & ! ! # $! % ! 1.9 $! 9 " ! ! ! T ! ! ! " % ! # T # ! M % ! # ! " " 1.9

! " ! " " ! ! ! 9 % ! # ) G * ! !

3

Instantaneous wall heat flux Φ Phase 3 Flux detector

Fresh gases

Burnt gases

Phase 2

Φm Φ

Phase 1

Phase 1

Instantaneous flame front

Time

) 07 +C6

6

+ ! & ! ! ! T ! &# # ! ; # &# # Æ+ & : & # ! .&# # ! # ,,!, Æ =& # ! ! ! # ! ' " # 2 ) # BB 8 ! ! " # ! GBB 8* # !

" 4 ! " G ! + 4 " ! # ! # % ! #Y ( " "! ! !& ! #! "" % ! # ! ! ! ! $! "

$! ! # ! " "! ' $! ! ! " ' ! , # $! # ! " ! !" ! = ! !& " & F !& ! " " ! Æ ! 9 !& " )6 * ! )9 ; ; .! 1 * & % & ! "!& ! ! ! " "

+* *! +56

6 M . 9 8 + Æ 9

0 ! , (

(

,-& '*

# G B 9 , )6 * > & " & F : ! ! ! ! ! ! : ! #!,! ! # " ! ! ) # * " # ! ! ! $! $! $! ! ! # % ! $! " ! ! ! - (0 $! # ! # & - 2( Æ

% ! & ) # G B *

0B

Maximum distance of flame to wall 80

Flame brush thickness

P = y/δ

100

60

40

Minimum distance of flame to wall

P = PQ = 3.4

20

0 0

2

4

6

t /t F

8

++ *! +56

4 +C6

0 ! , (

F (Heat flux/Flame power)

HOQ 0.3

Head On Quenching (HOQ) Maximum flux to wall

0.2 Mean flux to wall

Minimum flux to wall

0.1

0.0

2

4

6

8

t/tF

+ *! +56

4 6

+ 0 ! , (

6, # 9 ! # ! ! & ! " ! & ! ! " "!& ! ! ! ! # ! ! " ! ! % ! # 3 ! + # G & ! " ! $! 6 " / - (0 " ! & " ! >EV :# ) # G G* $! ! !

! " & ! " !, % ! # )>EV* ) # G * $!, 9 !& " "! ); * & " ).! 1 * $! :# ! ; # G + !

0

Upper wall y

COLD

BURNT

Lower wall

COLD

+ ( *! + 6

6

G .6

+9 9

6 6

! " !

! ! " ! ! ; " " ! # ! : )8 * $! ! ! , )!! & * !& # ! ! ! E ! ! ! ! ! ! ' % ! # E ! ! ! ! & ! ! ! " # ) # G 0* # J #K ! # ! " ' )! # !& " : * $! # ! # & ' " ! ! :# " !! & ! # ! # ! . ! ! ! # ! " !

)& ' " # & ! * & ! = ! T

! ! & # ! ! ! T )" * $! # : " !, 9 ! & ! ! ! !U " U >& ! # ! ! ## ! ! & " ! "

( ,-& ' ' . ( ! ! ! " !" !& # ! "!& " 4" !" ! " ", " )4! 3* " Æ " ' )9* " & ! " ' ! $ # " ' # " ' "! ,! )", # * & # " F 6 ( ! : ! " # " " ! ! ! ,! (

0

+ *! +C6

6 G

6 ( - , . +6 ;:1

! ! ! ,! ( % ! ! # ( ! !

! ! ( ! ,! " # ! ! " " # & " ! " # $! # # !& " ! ! ! " ' $! ! ! !

! ! " ! !

# " )& * " ! ! . 4! 3 ! # " " ! # G ! ! "! & ! ! & ! !" >& ! & ! & ! ,

(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

0

+ . + 8 65

L *! / 4 6

4 6

9 6

' M ,

" + ! " ' "! ! " # # G 2 & ! ! ! ! ! ! ! ! . # " ' ! " " , " ! " & " 1& " ' ! ) # , # * Æ " ! #

(

$ ' '*

9 ! ! " " ( # ! # 1.9 ! # = # # "

+ " !#! ! ,,!, ! $! : !

>& ! # # ! " " " , " "
00

Total reaction rate

2.0

Adiabatic flame holder Cold flame holder

1.8 1.6 1.4 1.2 1.0

0

20

40

60

80

Reduced time (Uo t/l)

+ . +

( 6 +5 M ,

" 7 : . ! " " # ! # 1 9 !& " & " ) JUU!UK # " * E ! & 6 ! ( ! #" & 9 ! " % ! : ! $! ! " )% ! # #* . ! ! ; + ! , & % ! O & !

# G 0 " ! $! ! ! ! # " $! ! : % ! ! & $! % /% ) 3* & ! " $! " % + O

A

O

-

O

%

A * H)

A

Æ

O #

7(

O

O

)G G*

! ! ! # ! ) - ! # - B " #* $! ! 7( 9 7( - B( % ! # ! ! ! ! ! ' % !

? # 2 0" @5;+( ./ '

$

0

9 ! ! " ! ! 8* " ! + -

!

A * 8* - )I

)G 3*

$! % ! " # ). #"# *

(

$ ' '*

= ! ! & M " ! ! ; #

# ! # ! 1.9

" ! !& ! ! % ! " & ! ! # G >& ! " # : " $! ! + ! ,,!, ! & ! ! & ) ! # ' " * ( " ! ! ' ! 0 2 $! & # ! ( ! ! ,, !, ! : " ! ! ! ! ' $! " ! ' # $! " " ! ! " ! ! # 8 4 ! " " E ! ! ! #

$% %%$

( 1.9 " ! " & $! ! % & ! " ! " ! & !& # & 5,,!, & ! $! " ! ! : # " " ! ! !& ! & ! 3 ! ) # G G* ; & )8 4*+ ! # ! ! & # & " % # ! & ! 3 ! "

5 A

& A B 2 C 2 0 ./C" A B C" = ' ((( " ' 5 (((

5 ' ' @ 7"

02

& ! + ' - '

3

- +

)G *

! + ! & * ! ! T ( ! ! ! # 3 & ' T ! % Æ ! ! # ! "
First grid point near wall: velocity u1 , temperature T1

τW

Φ

y1

WALL

+ , 655 56

& % $% %%$

$! ,,!, # & " . ! " , ! #! 3+ - ' + + ' ! & ! $! & ! " ! " + - T) + *+ 3 -

3 3+

-

+ 3 '

-

+

-

' +

-

)' * + T

. " & ! ,,!, + >+ # # " # # >+ " ! >+ # # " # >0+ =! "

' D '

)G B*

0G

> + ! >2+ # >G+ & >3+ D ' " ) , ! * ! " ! " $! & >3 ( " " # ! 9 + 9-

+ '

-

T

)G *

' + '

; & ! ! & ! ! !

# & & & )$" G *

I + N & 2 '2 '2 2

( ( 6

I 6

'

! 6

+ @ # 9 ! " +

- ' A 9

"+

)G *

: # & ' ! Æ - ' -

' ''

- -

''

-

'

)G *

# ! ! & " + ' - '' ) ' * ! ! ' - ) A 9 *

- -

' /,

-

A 9

)G 0*

& ! ! 9 " !! >3 /% )G 0* ! ! ! # !& ! & ! Æ ! ! " ) *+ ' -

- -

/,

-

)G *

03

! # ! ! "!& , ! ! , & ! " ! @ ! ! & : & ! # & $" G 0 )! ! # 8 4*+

P M :2H M ' :2H / $## Q %

,-

P . . :1$ M ! ' :1$ / $2;% Q 1J ! 5 655 56

! ,

+6

4 ! ' ! J& "K ! & , : ! 3 $! " # ! ! ! )J# ' K* # G 3 14

14

+

Velocity (u )

12

12

10

10

8

8

6

6

+

4

Velocity (u )

4 +

2

+

Temperature (T )

2

Temperature (T )

0

0 6 8

2

4

6 8

2

1 10 Wall distance in wall units (y

4

6 8

0 +

)

20 40 60 + Wall distance in wall units (y )

+ /655 56

5 +6
+

-

-

+ :3

)G 2*

! - : + - - B(B : - B(0 $! : )# & # * "& ! # ! ! +

0

" );! ;! * $! )# & # * " " # ! )/% )0 ** ! & +

/ -

+

+

)G G*

:3

# ! / % ! ! : # + -

+ :3

-

-

)G 3*

:3

% $% %%$

9 ( ! ,,!, # ! ' ! + ' ),& , * ! ): : & : * ' ) * ,,!, $! " ,,!, & # " ! # & , ,!, ! !" # ! # ! ! & . !& ! ! ! ! # : & ! & " # ) # G G $" G *+ . # & ! ! ! & ! , + ! ! . ! ! $" G 0 ! & + ! ! T /& !#! ! & " & # ! 3 # ! B(3 ! & +

+

- (00

+ 3 '

A

)G *

! ! + . ! !! 3 # ! B(3 ( /% )G * " " ! - 3 ( 3 # ) ! BBB * ! : # ! ! ! " : >& # + ! ! : # 3 - + 3 ! : $" G 0+

+ 3

- (BG

A ( '

3 # ! (

'

! )G B*

B

0 T " # /% )G B*+ T-

)' *' +

)G *

$! " # ! : # " " # ! & % ! : " ! % " & ! % ! " " + -

+

-

-

+

)G *

:3

2 $! & ! & ! ' - ' + ! T ! & $! #! : & # ! % "& ! , # " & ! & !

! : ! $ ! 1 ! # ! % ! "

4 0' ;:J4 ( ( 0' ;$2 I 9 + N ) 4 ) * ) +6 ' ! :

- ' 655 56

5 & : $! ! " ! : # # ' & # /% )G * " + & !#! # ' + # & ! !

" " /% )G * ( " ! % " ! " & ! " 4! # 5 ) L! 5 4 4 * ! ! ! ,,!, )$" G 2*+ $! ! & ! ! : " ! " # ( # /% )G * ! & " ! # ! & # ! % " + - -

>& # + ! ! : " " 3 - + 3 ' ! ! # & " /% )G B* 0 $! ! T & + # /% )G B*+

T - )' * + $! & # /% )G 2* " # ! " % ) /% 0 G* ! ! ( ! % ! / & # + + / - ' + 3 ! & " ! % " - + ):3 * 2 . % ) # ! % ! : ! *

" ,& !

! : ! $ ! 1 ! # ! %

4 9 9 ½ ( 4 ¾ ) ( ( 0' ;$2 I 9 + N )

' 4

! * ) +6 ' '

! "

! :

- ( / 655 56

5 & : ' % $! & ! & ,,!, ! & ! ! ! ! /% )G * ! ' ( " !

: ( # ! 5 41

B " & ! M # 9 & # ! B( ). #"# * @ ! & ! Æ ! # # : ! ! $ & % " " & ) /% G 0* . ! ! " ,,!, " & $!

! ! " ! " : + ! #!, " # ! " " ! ! ! );! * $! # ! ) " * ! & ! 4 & # 9 " 1 " " # # " !& & # ,,!, ! !

" # 4 ) 8 4 > . #"# . #"# * E ! " > . #"# " !+ ! ! ! ,,!, "

& & " # ! # $! & " + ! ! & ; ! I ! ! ! 3 ! ! "+ -

'

3

; -

'

I -

'

)G *

= # % ! ! ! & " + ! ! ! ! % ! # 3 " " ; " I ! $" G G )>

. #"# * $! ! , ! $" G 0 ! 9 # '

+ P M , + + :2H M + ' :2H / , $## + Q %

,-

+ P . R ! + + :1$ M + ' :1$ / R $2;% + Q 1J

.

! 655 56

+6 &

@ ! ,,!, " & # # $ & : 3 ; ! : # +

-

· .½

'

3 -

'

3 -

'

+ 3 '

-

+ 3

)G 0*

;

·

/½

'

-

' ' +

)G *

! # ! " ! & ! 3 " ! & ' $! I " ! # +

I -

·

A 9

-

9

A 9

-

9

'

)G 2*

# ! : $! # ! # ! #+ -

+

)G G*

-

$ " ! , % " ! %

)0 G*+ / -

!+ -

+

+ '

:3

-

+ '

)G 3*

)G *

:3

4 ! ! )/% )G 3** ! ! : # " ' < $! % " & ! ,,!, ! " " $" G 3+ . # & ! & ! $! & ! " . ! ! $" G G ! & + ! ! T & # # ! B(3 + "+

' +

- (00

+ 3

A

)G B*

! + ! . ! !! # ! B(3 ( /% )G B* " " ! ; - &

0

>& # + ! ! : # - + 3 ! ! : $" )G G*+

I - (BG A (

)G *

0 I T " # /% )G * )G 2*+ T-

' + '

I

)G *

'

$! " # ! : # " " # ! & % " ! % " & ! % ! " " + -

+

-

-

+ '

)G *

:3

2 $! & ! & ! ' - ' + ! T ! & % " # "& ! , # # & & ! & !

! : ! $ ! 1 ! # ! % ! "

4 0' ;124 ( + ( R 0' ;1: I R 9 + N · ½ ½ 4 ) *

) +6 ' ! :

. 6 ' 655 56

& $! ! ! % ,% " ! 4! # 5 #!

' $! # " & # # $! , # , + )!* & ! ; ! # "! " " & # , $! : ! ! " " $! " !& " # )1 #! 6 < * . & ! #+ ! " 1 7 ! #' " #! J # #K " " ! # " ! ! ! ! " ! #' >& ! ! ! ! # " 7 ! ! " " " +

# ) & * ! ! ! & ! & ! D ! ! " ! . ! " " " !& ! #! Æ + " ! ! # !& ! #! " # ! ! ! ! " ! ) 8 8* $! " " " " " ! ! " " " ! " " ! )8 * >& " # ! # " " ! ! = " " ! # " "

& # # ! ! " /&

! " # " " &

; $

2

! " " " # : " + & ! & " & " ! " & =& & #! " !" ! B " !& ! ! ! ! & ! " !" )! " #! ! * . % !

" " " "! # : & ! " ## # ! " " !" , & 9 ! & ! ' ! " + # " ! ! ! ! ! & " " )4! * " " )4 6 ;Z < >7 1 ! =# # [ # 4 4 #!

4 * # # )1 #!* $! ! # ! M $! ! " , # ) 3 * , ! ! ! : $! "7 & "

)5 5 ! ' = ( # 6 8 * " & . & % # ! & 3 3 0 " " 3 ! ! 4 5# / " "

"

'! ( ) $ ( ! : ! # & ! " " $ : " !

/ % . ! , # . " ! # ! & " & $! " # # " /% & ! & % + # 4 : : + >B , " > , ' & ). - B* > , ' & ! )C - B*

G

> , # # " & + & ! & =&

! & # ' )J K* >0 , + & " JK " % JB?+ " 9 ! ! ! " ! " > , & + ! & ) ! & * ! !# ! - B $! # % ! " " ! +

- 0

1

- 1 ¼

)3 *

! ! ) * ! >2 , , + = - B @ ! ! % # +

=

A (= -B

)3 *

A = = -

)3 *

- 1 ¼

)3 0*

! = ! & & . " ) =* ! ) = * + - A

= - = A =

D

D

- A

)3 *

" # = ! & % & # : # &+

A (= - B

=

)3 2*

A - B

)3 G*

$! & " ", 5 ' # /% )3 *+

"

!

"

-

¼

)3 3*

3

! " ! # " + " -

7

-

7

4

)3 *

& " # /% )3 3* & " )

= * Æ " &+

"

A (= - B

=

A - B

)3 B* )3 *

$! % " " " ! , & % + - B "

)3 *

& # ! & % % /% )3 * "

/ ) ( , ) & # # # * ! & : = - )B B * /% )3 B* )3 * +

"

# ! & % +

A

A

-B

-B

-B "

)3 * )3 0*

)3 *

$! /% )3 * ! & # & ) # 3 *+ - 2 ) "* A 2 ) A " *

)3 2*

@ # /% )3 * ! !+ -

2 ) " * 2 ) A "*

"

)3 G*

A + (t-z/c o ) A - (t+z/c o )

z

+ <5 6 $! & " # ! #! ) - A * & # & 2 ! ! - " ! 2 & & # ! - & " !& # + - " $! # " & " && % " ! ! # ! ! & # ( !" ! # " # ) - B( >1 " - 3 B 1 * &

BBB 6 # 0 ; & # & " BBB)" * 2 M ! ! # " $! ! ' & # & + ! ! " " ? : "+ ? -

)3 3*

"

$! ! Æ : ! ! & # & - + -

2 ) " *

)3 *

2 ) A "*

! # /% )3 2* )3 G*+ -

? A

)3 B*

?

$! Æ : # & " ! : )$" 3 * - & ! #! & & ! - A ! + ! 2 ' - ? - ( ! : - )$" 3 * & & - ! 2 ' - B ? - ( !

& # & ! ) - B* !

Æ - ' ? )$" 3 *

E ' $ =;- ¼ " 2 +0 ¼ ¼ " ¼ 2 + C0

2B

) * ! & # ' ! Æ - ! ! ? : )$" 3 0* ( # Æ ! " ! # ! % ); 6 *

G

(

z

+ Æ -

* +

* +

2

:

(1): Infinite duct on the right side

z

:

(2): Infinite duct on the left side p1=0 z

2

:

2

(3): Duct terminating in large vessel

u1=0 z

2

:

(4): Duct terminating on a rigid wall

+ + Æ - 5 / 0 . # ! & & " " #+ - )2 2*

)3 *

= - )=2 2 *

)3 *

2

- )2

*

)3 *

! @ - 2 2 " = 2 & )* # ! " @ # & &

& ! & % " 1= ), ,%* & ! & " "

2

! + 2 " 2 - 23 $! &# & ! % " "+

-

)3 0*

" 2 + -

2 - 2 2

)3 *

! 2 ! 7# 2 ( ! ! &# ! : & + - )2 2 * )3 2* ! # ! 7# ! & ! & % )3 * )3 0* "+

$! +

@2 A " (= 2 - B

)3 G*

@= 2 A 2 - B

)3 3*

2 - " 2

)3 *

! ! & % " ! >!' % + 2 A 2 - B

)3 B*

! ! & " - "
) # 3 * ! J# &K $! # & " ! # & # & # # & # ! " !" ! # !

$! ! ! ! &+ ) 3 * - 2 ) 3 *2 & " # ! # ! ! $! # + # ) 3 0* # & " ) 3 * # M & # ) 3 2* # ) 3 G*

2

y n x

z

K 6 / " ( , ) # 3 * ! & ! >!' %

)3 B*+ 2 A 2 - B )3 * E ! ! & & ! @ # /% )3 3*+ = (2 - B

)3 *

! = ! 6 # & # # ! # ! " ! ! + 2 ) 3 * - 2

)3 *

" # 2 ! >!' % )3 B* ! ! - ! " " +

& & # ! & 2 - 2 & & # ! # & 2 - 2

$! # " ! + ) * - 2

2

A 2

2

)3 0*

! ! & + ) * -

2

"

2

2

2

)3 *

$! !& " : , & # , 3

/!

2

"

$! & " # & # # & " ! ) * ) # 3 * ( ! /% )3 * )3 * # ! & , " + A ) ! * - B )3 2* ! A -

)3 G* y

dz x S(z) z

dV = S(z) dz

/ 6 ( ! % & )&# & ! ! *+ -

!

3 !

D

-

3

!

!

-

!

3 !

)3 3*

$! % " ' B " #+ - A

- A

- A

)3 *

" !# ! + - - $! & ! # ! 3 , " ! & " " !# ! 1 # ! : /% )3 2* )3 G* # # # " ) - B* +

"

A

!

A

)! * - B

)3 0B*

-B

)3 0*

. # ) A A A *

" " ! # ) # 3 0* ( # # ! ' % )3 0B* ! , % )3 0* " !& # # B & ! 7 !#! ! # ! ! +

!

- !

-

)3 0*

20

Section Sj+1

Section Sj

: integration volume

z

0 z-

z+

S ! # ! ! & !#! ! ! ! & ! # " ! ! + ! & ! & . /% )3 0* " ! - B ) # 3 *+ " " & $ ) " ! & ! " * ! 7 )3 0* "+

!

- ! A $

-

)3 0*

Loudspeaker Section Sj+1 Section Sj

S / " * # ! & " # ) # * " !& # & ) # 3 * $! ! ) # 3 2*+ 2 ) 3 * - 2) 3*

)3 00*

! ! ! ' ! # # ! ! 2) 3* " ! & ! # " # 2 ! >!' % )3 B* # &+ 2

A

2 3

A ) * 2 - B

)3 0 *

2

y x

a b

0 z

K 6 - $! ( & ! & & " ! ! ! >!' % " 2) 3* - B )* )3* ! !+

B B

A

-B A

)3 02*

$! ! % + B B

! . & +

-

- .

)3 0G*

A . - -

)3 03*

; " ' & )/% 3 * ) # 3 2*+ B )B* - B )* - B

)B* - )* - B

)3 0*

$! /% )3 0G* ! " )3 0* + B )* - )*

)3* - ). 3*

)3 B*

). * - B

)3 *

! ! & " . + ) * - B

E & ) . * )! J K & "* " . + & & - 1 . - 0 )3 *

! 1 0 & # )1 0 B* ! & # # & )1 0* + & & )3 * 2) 3* - 4 1 0 3

! # ) 3 * - 2 ) 3 *

2

! )1 0* +

& & ) 3 * - 4 1 0 3

2

)3 0*

22

!

- "

& & - )1 * )0 *

)3

*

$! " + " ) % . - &* & # # # ' " ! " ) 3 * # & " /% )3 0* $! 3 ! & & #! 1 ) 0* ! ) 3* 6 . # ! # ! - )1&* )0&* $! & " ) 1 0 * # & ) 1 M 0 #*+ ( & ) )1&* A )0&* * ! # ! 1 0 " +

) 3 * - 4 1

&

& 0 3

2

)3 2*

$! # # ! # ! $! ! ! )B B* ! ! ) 3 * - 2 !

: 3 0 ( 1 0 ' ! ) 3* ! & " # # ! !

E ! ! ! # & ) )1&* A )0&* * # " + - @ $! # ! +

& & ) 3 * - 4 1 0 3 2

)3 G*

! # ! ! # # & ! #! ! $! J,K 4, % ) & " - " - &. " * ! ! #! % , # $! , & " 4 )1 0* # & ! & ' ! " ( ! ! - B +

)4 * - 1

&

A 0

&

)3 3*

$! , % .4 ! & " 4 "+

.4

-

4

"

&

-

"

1

A

0 ¾½

)3 *

2G

Mode (m,n) with a frequency higher than fcm,n propagates in the duct

Mode (m,n) is cut-off when its frequency is smaller than fcm,n

fcm,n

0

Frequency f

, 9 6 57 ' . > #! % & ! . < .4 # ! ! ) # 3 G* 5 % ! ! . .4 , . # : ! ! # ) - 0B M* ! - B( - B( , % $" 3 . % ! # " !" ! # & " # !#! ! " !" ! % ! #! ! 3 B >' ! )B * GBB >' ! ) B* . # )B B* # $! # # !#! ! # % ! 3 B >' " ! #

/

I8 ( 57 ' .

2 2 2 : : 2 : : 2 H%2 :;22 :J22

2: 2$ ( 57 ' .

/(

"

= " !"

! & " . ! "& # ) ! 1 - 0 - B* # !#! ! ! ! # ! " 6 # & # ! " " $! " + C

! #! - ) # 3 3* $!

" C ( ! # ! & & # ) * & # #! ) * . # /% )3 0* )3 * ! ! ! & # A + ) * - 2

2

A 2

2

)3 2B*

! 2 ! ' $

23

Duct 1

Duct

....

z1

j-1

Duct j

Duct j+1

Length = lj - 1

Length = lj

Length = lj + 1

Section = Sj-1

Section = Sj

Section = Sj+1

A j-1 +

A j+

A j+1 +

A j-1 -

A j-

A j+1 -

z2

zj

Interface 1

Duct J

....

z j+1

Interface

(j-1)

zJ

Interface j

z J+1

Interface (J-1)

, 6 ! ! & + 2

) * -

"

2

2

2

)3 2*

. A - 7 " " # ! ! " /% )3 0*+

) * -

!

) *

! ) * - !

) *

)3 2*

2 A 2 - 2 A 2

)3 2*

! )2 2 * - ! )2 2 *

)3 20*

! ! ! ! & A A ! A !#! +

2

2

-

2

2

!

) A H *

) H *

) H *

) A H *

)3 2 *

! H ! " A A A + H - ! ! . #" D ! & ! : )A - * )A - C * +

5

D-

2 5

-D

2 5

2

2

)3 22*

2

$ ! " " " : "! ! # . ! % ) $" 3 * ! ! #! C +

2

2

-

25 - 5 2 5

)3 2G*

/% )3 22* " )3 2G* ! ! ! ' + D A D 5 )3 23* D A D

! + Æ

0 : 1 :4 5 0

T" ·½ # : Q T" # : T" ! " :$ # : T" # : Q T" 2 $

5 "

3 2 ! + Æ

"

$

$

"

$

3 3 % # $ % 2# 3 $ 3 3 3 3$ 3 $ # $ 3$ 3 3 $ 3 0 : 1

! 4

6 4 A6 9 6 4

, 1 1 : . + Æ $ 6 . . 6

& # /% )3 23* & ! # % ! >& # ! , % ! ! & ! ! " )6 6 * $! #" ' $" 3

GB

// . : ! ! " " ! # ! ! ! #! & & $! ! # ! & ! " ! # & " " 2 ) # * # " " & # ! >!' % )3 B*+ 2 A 2 - B

)3 2*

E & & ' ! # " ' )/% 3 3*+ = (2 - B 2 )3 GB* z A

lz

0

ly

y

lx x

) ( E # 2 ) 3 * )! ! & * : ! >!' % ! ! " :# & " : ! 2 )R * # :# ! ! # & ) # 3 * " " " ! # 2 ! 2 ) 3 * - B )* )3*? ) * " # ! >!' % # &+ B B

A

A

? ?

A - B

)3 G*

$! ! % & # ! & " . ! !+ - A . A )3 G*

G

$! " ! # ! + B A B -B

A . - B

? A ? - B

)3 G*

" )' & * ' # + B

) - B * -

3

?

)3 - B . * -

) - B * - B

)3 G0*

$! ! % + B - )*

- ). 3*

? - ) *

)3 G *

! " )3 G0* - 3 - . - + ) * - B D !+

- 0

). . * - B D

&

. - 0.

&

) * - B &

-0

.

)3 G2* )3 GG*

$! ! ! & +

2 ) 3 * - /

0

&

2

) 3 * - 2 ) 3 *

- / 0

0. &

& .

0

3

0.

& .

3

&

)3 G3*

0

&

2

)3 G*

! - " - & )0 * A )0. . * A )0 * $! % ! )0 0. 0 * # & "+ . -

"

&

-

"

0

A

0. .

A

0

½¾

)3 3B*

. " !" # ! #! - & ' . - B( - B( $" 3 0 ' ! % ! : $! & : ! " # BBB 8 7 - ( ! " 30B M $! " !" " # $! # & #! 8 ) # 3 B* & " )E F * ! )1 0 G* ! 1 0 G & ! # G # "+

E / 46 A 46 )E F * - C4 )&4 * G& 8

)3 3*

G

/ /% /! ' I8 : 2 2 #$$ $ 2 2 ! H#% 1 2 2 . :$"; # 2 2 :"J2 2 2 : $::$ : 2 : 8 5 $:%2 : : 2 5 #$#2 ' L

z

2a

+* ( ! C4 ! ; 0 $! & 4 ! C4 )&4 * - B ' $" 3 $! % # & "+ .47 -

3 2 : $

"

4

A

G

½¾ )3 3*

8

2 : $ 2 :$$ $111 2%H" :"J; $;:; 2J;$ $:1% 1:;1

1 1$1H 1;$% #:J$

M ! & ! : " # " ! "& " " ! #! % " & ) ! * # ! % ( " & "& # " BB BBB >' ! " # :

/1

G

2 0 .

$! & !& ! & ! + ,% # # & ) 3 * ! ! ! & #! # ! ! & ! ! # ' ! ! #!,% # & & ) 3 2* ! ! ! &, #! ! ! & ( ! " " ! & ! >!' )8 * . >!' & & % ) !

! : # *+ ! & #! # ! ! & " ! ! ,& # " , ! & # 3 ( & & , : ! #! 8 NECK: length L, surface S, mean radius a

Combustion duct

VOLUME V (any shape)

++ . I 8 $! & # )! ! ! & * , )! # ! !

* , )! ! #* ! % .* # & "+ .* -

"

&

! 8 ,

)3 3*

! 8 #! ! 8 A ( " !

! & $! ! >!' " !" ! ! Æ ' )6 8 * ! ) ! # 3 * ! ! " " ! ! " @ ! : ! >!' % .* ! ! & " ! % ! !"

/3

'

$! # " " " ( ! : 4 ! ! " ! # "

G0

( & # & ! , # $! # 3 2 $! # ! & # & ! ' ! ! # )5 5 ! '* " & ! ! # /% )3 B* )3 *+ A (= - B )3 30* "

=

A - B

)3 3 *

= # /% )3 30* " /% )3 3 * " = # ! # &+

= A "

A ()= * - B

)3 32*

$! # : & "+ -

= A "

. - =

)3 3G*

/% )3 32* ! ! ! # & " ! . +

)3 33*

A (. - B

Boundary A n

Domain V

dA

+ ' HHH $! # " " # & , ) ! , 4 * " " 2 ) # 3 *+

A

. (= 2 - B

)3 3*

G

! - 8 , ! # ! = ! ! 2 /% )3 3* ! ! ! ! # ! # # ! " $! . (0 - = (= /% )3 3* ! " 4 + ! =(= ' )! = - = - B* ! & )= - B " = - B* ! ) - B* . % " " & " ) # 3 * ! # ! ! ! !#! ! " ( ! & " : & # ! &# ! E # & & # ! ! # )! & & & * " # & & ! " ! #!, 4! $! ! & " " )1 1 $! Q 9 *

Constant velocity inlet

Wall

Constant pressure outlet

Wall

Constant velocity inlet

Wall Constant pressure outlet

Wall

+ 0 6

9

4 + 8

9

6 $! ! & ! , # ! ! ! # # &+ ! " !

" : ! # # ! " "

# ! 3 ! # # " " " #! " "

- F/ 1 ' "

!& "

G2

'% ) $ # ! & & % . & & ! % ! ! # ! ! "

'

/

$! # ! 9& , % ! # & & : "+ .

-

.

A =(.

$ 3 ! & + > , ' & ). - B* > , ' & ! )C - B* $! % % & , # +

A (= -B

)3 B*

- A (

)3 *

=

! " ! # % ) 2B* % +

- C A

A + =

&

=

9 ,

(

)3 *

= ! & & ! , /% )3 B* )3 * " " & & % . & ! & ! # % )3 * " ! % - E "

% )*+

)* 7

A (= -

C A + =

&

=

9 ,

(

A

E E

)3 *

$! % )3 * " )*+ =

" A )* - ( 7

)3 0*

GG

: # ! " "+ " - 7 " # ! & & /% )3 * ! &# /% )3 0* & & % ! # ! + " ( )* )* - ) ( *

7

C A + =

&

7

=

9 ,

(

) )E** = + = )3 *

, # & # /% )3 * ! & % )3 * & 3

/

4 '

$ " & & % # + >2 , , ) 3 * >G , #! $! & & " # : + C " " C #! % ) 2* 7 ! ! 7 - " )! ! ! " * . # )8 * ! !

! 1> /% )3 * ! ! ! ! & " # ! : % +

" 7

)*

7

)*

-

C

= + =

)3 2*

. & & & & & & )5> * ! & & , % . ! ! , ! + . - 2+ .

-

.

A

.

- )@ A @*. - @

"

. - @ ) 5 * .

)3 G*

- " =! " 5 - " ! !

. # # " . " " .

G3

/% )3 2* ! "+

" )*

)*

0

C

7 = + =

)3 3*

$! % ' " & : 5 ' # 3 ! - A ! ! )* " /% )3 3* " % ! ! # +

"

- )7 *

C

7 = + =

)3 *

$" 3 2 ! & % , # # ! , # ! = + = + ! " " $! "#! " " ! & " ! ! " ! ! ! 1> ! % ! " $! " "

" $! ' /% )3 * Æ ! #! " " , % " ,

,

*5 +6 +6

4 5 4 5 4 : & 4 5 4 5

( 6 ' 5 +6 / 4 ' 5 ' % & " $! % ! & = : /% )3 * # ! ) # # " & & & & &* ! & %

+ = - )3 BB*

! " $! % " " : # # & & #! /% )3 * $! & ! E

G

"! & ! ! /% )3 * : + )* 7

A (= -

C

)3 B*

$ ',! ) ! * ! % + (= -

C

)3 B*

! C ! 1 # :, /% )3 B* , # # & & & & )4 #! 5> *+

A (= -

7

7 C 7

)3 B*

! C ! ! # & # # & " ! ) * ) # 3 * , " ! /% )3 BB* )3 B* " # # 3 # &+ - )3 B0*

7

/

A

!

)! * -

7 C 7

)3 B *

5 '

$! & #! ! # " # ! ' ! ' ! " . - BB >' ! & #! - - ". " # BBB 8 ! (3 ! ' ) Æ "* ! B ( ! " " JK & #! ! ! " : ! 7 % !#! " & ! & % 4 ! !

# 3 0 ( ! ! ! # " ',! ' - 9 ! ! ! # # - ( # # /% )3 B0* )3 B * - -

# ! !

# + \ ] \! ]

·

·½

·½

·

·½

·½

-

-B

)3 B2*

7 C N 7

)3 BG*

3B

Flame Fresh gas

Burnt gas

Section Sj

Section Sj+1

z

z j + 1 - zj+1 zj+1+

+ S + C ! N

·

·½

·½

! ! " ! ! C

$! : ! ! ! !#! ! + ) * - )

*

)3 B3*

! ! ! ! ! & 7 !#! ! +

! ) * ! ) *

7 C N "

)3 B*

! " & ! ! A A ( ! 7 " " ! ! & /% )3 B* ! ! ! & . " & ! # 3 0 @ # ! 3 G ! & A A A + ) * - 2

! ! & +

) * -

2

"

2

A 2

2

2

2

2

)3 B*

)3 *

! " & ! & " ! ! A

' ' ' 1 & ' ; ! 6 G 1 4 '

3

$! # ! NC " ! + NC - N2

)3 *

. ! A : - # A A A ! 7 )3 B3* )3 B* +

) * - ) * ! ) * - ! ) * A

!

7 C N "

)3 *

2 A 2 - 2 A 2 ! "

)2 2 * -

! "

)3 0*

7

)2 2 *A

"

N

)3 *

! ! & A A ! A "+

2 2

) A H * ) H *

) H * ) A H *

! H # & "+ H -

2 2

A

"

!

1 ¾ 1 ¾

N

)3 2*

N

" ! "

)3 G*

!

/% )3 2* " +

2 2

-

2 2

A 6

)3 3*

!

) A H * ) H *

) H * ) A H *

" 6 !

1 ¾ 1 ¾

N

N

)3 *

$! # ! /% )3 2 * " & "

$! ! & A !

A A $! ! )A A *: !#! 6 /% )3 3*

3

Duct 1

Duct j

Duct j+1

Length = lj - 1

Duct

j-1

Length = lj

Length = lj + 1

Section = Sj-1

Section = Sj

Duct J

Section = Sj+1 Flame

...

A j-1 +

A j+

A j+1 +

A j+1 -

A j-

A j+1 -

z1 z2

zj Interface

Interface 1

...

z j+1 (j-1)

Interface j

zJ

z J+1

Interface (J-1)

+ 5 /! " $! & " ,

! ) # 3 * $ 6 " ! ! , # " #! #" $! #" D94 !& ! !+

2 2

- D9

2 2

A

&

D9 6 A 6

)3 B*

$! D9 # A ! ! D9 # A + D9 - ( ( (

D9 - ( ( (

)3 *

< # ! )3 B* ! A - C # ! " + 2

2

-

25 - 5 2 5

)3 *

! ! ! & ) ,'* ! ) " ! & 2 5 25 A - C * $! # ! ! " ! # + & ! " # ! $! #" ' $" 3 G # ! # 3 ! "

& " " )! 2 5 2 *

3

! + Æ

0 : 1 :4

5 0

·½ T" ·½ ·½

' # : Q T" # : T" : " $ '' # : T" # : Q T"

5 "

"

5 ) 6"

6"

·½ ·½ ·½

5 3 2" ! + Æ

! 4

) ) ) #

& ¾

( ( ( !

$

= * *

& ¾

* + %

=

2" " / : 0

" ) ) ) #

3$

3$

$

"

* *2 * $ %

) ) ) #

$ 3

3

$ $ * & *Q 2$ * %

6 Q 6$

3 3 3$ 3 $ # $

6 4

I 9 6 4

3$ 3 3 $ 3 0 : 1

, 6 5 1 1 : . + Æ $ 6 ! "

" . " "

4 " " 4 D " " "

30

/ 2 ' . # " : 3 B+ -

= A "

)3 *

= # /% )3 B* " # ! ! % )3 BB* " = +

A . -

! -

)7 * 7

C

. - =

)3 0*

$! 1> " ! C $! " ! #
! " ! $! % & "

" : " 1 #! /% )3 0* " & ! " #! ( # & ! ! & , ! " +

8

, A

. (= 2 -

8

,

)3 *

( /% )3 * $ & # ! #! ! " /% )3 * " &# & $! " & ! ! " ) $" 3 3

* # ! & & " & ! + - )2 )*2 *

= - )=2 )*2 *

C - )#C 2 )*2 *

)3 2*

! 2 )* =2 )* #C 2 )* & #
A -

)3 G*

5 ( & 9 2 9 ) +C," =8- 2 & & ¼½ 9 9

3

0 !

7

. #

8

(

9

9 5 5

9 + 6 ! ! ,&# # ! ! "+

-

H, 8

H-

-

0"

2 2 A

= 2 =2 0

)3 3*

$! ,&# & # ! " +

-

) -

) , 8

. -

)2 = 2 *

)3 *

. ! &# +

-

8

!,

! -

-

)7 * 7

C -

)7 * )2 #C 2 * 7

)3 B*

M # + ! ,&# # )& , * ! ,&# & 2 #C 2 ) " ! * " & ! # ! $! #! ! # > " " # ! ! ! " ! # ! ! 2 =2 #C 2 " + 2 )* - /

= 2 - I=

#C 2 - 4

)3 *

! > $! # " "+ >-

) A *

)3 *

$! #! > " & ! " ! " ! ! "

32

' *
" "& $! ! # " ! ! ! " ! ! ! ! " " ! : : # ! " >& #! " # ! ! # ! " " # ! & ! &

/ & $! 1 #! ! ! # ! & ! " " ( # & !! ! # " ! : ! " )6 * 4" " ! & ! ! C ! ! #

!+ )7 * C

)3 * 7

.!#! ! ! ! 1 #! #!

+ ! ! # ! ! E ! &# 1 #! ! # & " /% )3 B*+

)7 * )7 *

)2 #C 2 * C ! 7 7

)3 0*

! " #! ( ! # # &

" ! " " # ! ! ) & ! * ! ! # ! " " " # ! ! )6 # * ( # ' ! ! # " " /% )3 G* ! # ! ! " # # # 3 2 ! ! & " ! " ## - B ) " ! # # ! # * ( ! #! > # & " /% )3 * & ! # + ! # ! ! >& # #+ ! " ", , ! ) ! * ! ! , ! ! # ! #! > " ' $! "

! ) * ! ! # " ! ) * $! # ! 1 #! ! ! " # : " ! " =&

3G

! : ! # ! " ! " " ! ! 1 #! " # & ! " ( ! ! # ! "

! !#! ! 1 #! $! " ! & " # " " # ! ! ( " # & ! ! " 6 6 $! ! " ! " " Non-linear zone: limit-cycle

Pressure oscillations

Linear zone: exponential growth

0

20

40

60

80 Time

100

120

140

+ K 6 $! !# & ! & " ! " # ) 3 0 * ! 0 , ) 3 0 * $! ! ! ! 1 7 " )1 7 1 7* ! ! ! , ) 3 0 0* ( ! :# " $! ! #! #! ! ! ! " ! ' ! ! ! 0 , 3 0 ! ! 4 " !

/

$ '

$! & 3 " #! ! " ! $! : # ' !

! "

!" & " ! $! " ) ! ! ! Æ " ! " *

33

$" 3 G #! ! !# % " " ( ! ! !# "

# 3 G ! ! $! " ! :& + ! &

2 2 2 2 !

Premixed gases Flame holder Section 1: cold gas

Section 2: burnt gas

A1+ A1-

A2+

Outlet

A2a

b

+ 0 5 8 ( ! " ' - ! ! ! ! ! ! # !& " 1#

! # ! ! ! ! # ! !& " 6 : ! ! ) - A *+ ) A * - B ! ! Æ $! ! # + )B * - B ! ! Æ A @ # ! $" 3 G ! " && ) - * $! #" D )% ! ! % * ! 6 +

½ ) A H * ½ ) H * ! D - - ½ ) H * ½ ) A H *

"

$

1 " # ½ ¾½ N % 6 )3 * ! 1½¾ N

½

! H - ) " ! *) " ! * )3 * " & ! & 2

2 2 2 ! # ! " + -

2

2

- -

2 ¾ - 2

)3 2*

$! " N ! " ! , 3 0 ! N " " " ! ! "

/

2 6

3

'

. " # ! ! " N

! # & $" 3 G # N)2 2 2 2 * ! ! ! ! ! " ( # ! # ! N ! + N #! ! ! , " ! ! ; !

N)2 2 2 2 * " ! , !& " ! " " " ! )4 4 * ! ! " " " ! & ! ! )) - ** ! ! ! & ! ! & ! ! " # ! & " ! " ) ! R 6 * C " + $! # " N 7 C N - ! 0 ) * "

)3 G*

! 0 ! ! ( & " ! + 7 NC - ! 02+ )* "

)3 3*

! ! " N+ 7 "

N-

0 "

½ ½ ! 2+ )2 2 *

)3 *

" # N /% )3 * # " )3 2* &, ! ! ! ' $!

& % ! + ) * ) * H ) * )*) A 02+ * - B

)3 0B*

! - " - " & # /% )3 0B* & ! # ( ! # & ! ) 2* " " " $! % ! " " ! ( # !

" >& & # ! ! # ! " 3 0 0

B

/ + . /% )3 0B* " : & ! ! .+ -

" - " - "

! - !

-

)3 0*

$! :# ! #! # # # " 7 ) # 3 3* ( ! H - /% )3 0B* + )* )*02+ - B

)* -

# ! 0 +

02+

)3 0*

A 02+

)* - 02+ ! - " ! ! " #

)3 0*

Premixed gases Flame holder Section 1: fresh gas

Section 2: burnt gas

A1+ A1-

A2+

Outlet

A2-

z=0

z=2a

z=a

+ ) L ! %$ < ! " )0 - B* ! : % - " ) # % '+ ) * - B* # & "+ )* - B - &0

)3 00*

$! ! & #! - - &" - 3 ! ! ! #! $! : ! ! %,& $! & ' " # ! & ! !

! )3 * + 2 - 2 D

A@ 2

2 - @2 -

)3 0 *

¾ ! " #

$ % & ' ! ( ) ' !

" % ) ( *# $ & + )

Pressure amplitude

Velocity amplitude

1/4 wave mode 3/4 wave mode Flame holder

0.00 1/4 wave mode 3/4 wave mode Flame holder

0.0

0.2

0.4 0.6 0.8 Abscissa (z/2a)

1.0

0.00

0.0

0.2

0.4 0.6 Abscissa (z/2a)

0.8

1.0

, ) '! % - ¼ # ./ 0

¼ ¼

¼ ) ' #! $ &/ ' ' + # $ &/ # & & ¼ $ & # # & & 0

) -

' # 1'# 0

'

2)

' #! ' & ' " ' & 3 " ' (

1# ! & !# 4 '

" #! ./ 65 #! ./ ' ! " %

# ./ 0

65

¾

4 ' 65 0

-

2

7 # 1# # ' & '

# '

0

-

-

2

' % ! $ 1# /! " $ % 1# '

/8'! & 7 8/ $ '! #! 0 ) '

-

2

'!#

8 /8'! *# 7 & /8'! # &/ % - ¼ # ./ 0

¼

¼

2

+ &0 ) -

-

'

22

Quarter wave mode

0

To/2

To

Three quarter wave mode

0

2 T1

T1

3 T1

Three quarter and quarter wave modes together

0

To/2

To

"! #$ %&'( 7 *# ) & /8'! / '! 0 -

- -

2

3 ! & ! & /8'! 8/ '! # !# ( ' & & ' 9 0 ( /8'! # & % # 8/ '! ## : # : #8 +!8/ + # ! ' &/ # ' # #0 !# & 4 &/ + # ' ;# ' ' + / 9 8 4

& # 0 4 & % & '

% & 7 # ' + & ! 0

< & # &# ' ! # ' + ' &/ & & & # & &/ & & & '

! & &/ ;

, + Æ /

# %

7 " % ! & & & ' " ' ( % = ' / / %

! & 8 " & &

" ./ & " ' & & 8 ' % , ' Æ & < ! & " % 0 8 " % ># ; > / : ? & @. / 2 3 &

# " & & ! % & A % & 9 ' &

& "B # & 1 & C' D

& *# ! 4 0

(

# '!

< ' = = # ! ' % $& ! % & *# & # ' ! % ' , 9 & & # '8

9 & & #! ! %

'! ! % # &' 9 & E & & # & & ! % & # & " ' 0 " ' # !' C D > # #( 4 ' Æ # " '

" ' 9 Æ ' + &

#

2

Hot recirculated gases

Fresh gases

Vortex formation by hydrodynamic instabilities or acoustic mode

Mixing and convection

Combustion after a ‘delay’

) * " ' ' !

A ! # 4 = 8

1& + & F( 1 = = , F# ! + & " ' # ' # & & # ! ' 8 8 0 '8

F ' ! F &/ &

& & +

" '

'!# & 7 & & " '

&/ & +

0 2 )) ' 9

& ! + # = = ! 4 ' A Æ 8 ' &' 9 & 0

G9 # ' ! &/ + ! ' # & &/ & &/ # ! % ## = = 9 +

& + & ! 8

, % +

! " $& + # Æ8 & + <

%

F # # 4 & " ' , 4 #

" ' : ! >

% # " ' ' # # # " & #+8 # ! ;H 1

+ &/ & ' ! # ' # $ ! + !! # &

# & #

* # ' # 8 0 &/ ! $ & & &/ ' &/ 1 # < Æ ! ' & % & # % ' @ 1 9' 3# # ! '

= & ! : # # ! ! & " ' '! ! I ! & ! " ' ? ; ! & Æ # & & # 7 % *#

' ! ; $ & # I

( # # !0 & !' & *# ! ( & # ' # ! % $& /! # " % # ! &/ * # # !' & *# ' & & ! # ! 2) =( ! Æ 2) "

& & # & = % " 4 ! 2) =( # ! % & * # *# ! & ! & !# ! /( ' ' & *#

#* +

,

- + * . / # 5 #B J )

G # 2) =( 5 #B J )

, * 0 %11 $/ 0 %11 0 +

t=1.32 ms

t=1.55 ms

t=1.78 ms

Normalized values

t=1.09 ms

Inlet flow rate Averaged reaction rate

0.0

0.5

1.0

1.5

2.0

2.5

Time (ms)

+ 2 $/ 0%11 $ 3 4512

3 (' 6 + *# ' *# 2 !' !# + & & # ! % & & % I & ) *# 2 ! # ! # ' " ! *# 2 & # ' ! , ! ' !

# & # # ! 22 # # ' & # % Æ & # " & & ' 0 ! % & # ! & ' # # ! 9 & &9

E & ' 0 & #8 ! % ! & ' @# . / ! ) +

2

!'

$

1

)

22

7'4 8/ + 0 %11 0

))

7 Æ ! # 8 ! ! # $ Æ &/ 4 & $ !# ! 8 & / ' #8 ! # " ' 0 @.

@# . @. / > &

" & * @. %8 ! ! # *# 2 8

0 & @. & & C D ' % # & ! ( B 7 &

& @. & 0

@. ! & " ' & " 'B

: @. '! 0 !

G # @. & #

& % ' % ! & # @. ! & + ' &

& " ' >*F 8 % # & 7 # # # " '

&

#

#

' + @# . ! ) ! # + &' ! 0 % & @. 2 2

4 ' " ' + !! " ' " ' + = = $ " ' + *# & " ' # ' ! ' ' $

)

#

*# #0 & &

&8 > ! & # &9 '!

# @. = '! & + & & &9

x

x

Perturbation

Perturbation

t t=t o

t t=t o

3 8 8 7 / ' @. 0

* + 0 & &9 # 8 *# " ' $ % & & &

&8% 0 & &9 '! *# " ' ' ' $ + % & # " ' '

1 ' !

# @. / & # ' % # & % ' " '

)

Flame

Flame

Outside forcing Acoustics (a) Forced modes

(b) Self excited modes

. * 7 / & & # ! C D # ' & # / & % %# ! + & $ % 9 " & &

#

# > ' # C& D ! # & &# # 7

( &

&/ CD # $& & # # 9 & ' ! & # ' 8 & @. & & & # 8 * &

/ & ! &

7 @. # # & # ' '8+ + @. % &8% % 9 % !# # &/ + !# & &8% % 0 # + ! & % &8% / # # # 0 / < % # & =!# ' &' & 8 % > # & / & & &

)

! & 8 & 0 & ' # > # # & @. I '9 &8% < 8 0 # " ' # F# ' & / & # Æ 9

& & &8% : @! ? & ; & # # ! = '! # % # & / ! !

& @. & & & 8 % & ; & 7## #8 *# I & % & ; *#

, 9# + 0 7 & ' " ' ! ( ! # ' ' 8 @. & " ' # 9 * B 9 " 7## > 2 # 2))))) # )) ! !! & ' & & # ' & # + & & & " ' # & )) ? /! ! & B 7 & " ' & # #! *# " # # ! % 8 & *# &

)

: 9# 0 % '! # 0 I! 2) =( % & " ' # & # ! < @. ' " ' '! " ' # / 2) & " ' & # &/ 2) =( & # & & # *# ) #8 & ! @. ! %0 # 8

! ! !8 I # & " &

' % & 4 & & # @. % & & & 7 ' ! ' " ' & & ) # & & % *# *# ' & @. ' 8

"% 8 % !'# ' ' ! ( ! !# ! ) & @. ) & & ) / %

)2

; 7'4 8/ ;

$ 4<( 0

+ . 0 %11 ;

* 0 %1<

)

* +# # & # " '

# ! ' % * % & ; @. & ' & # +# * *# % I # #

)) ? ! # 2 @. # 4 7 8"# E>:> > ' +% ! 7 8"# %( #

( B" # 7

" ' # &8% 2 ) =( ! ' & & ) # 2) # & ' ! A CH 4 +air

D Exhaust B C

+ * +

Pressure (Atm)

*# ' ! &

! 7 *# ! # ! & & B '! ## & 4 ! #+ *# !

! # & & )2 *# $ ' !

) 1.2 1.1 1.0 0.9

0

20

40

60

80x10

-3

Time (s)

; + +

Velocity (m/s)

)

36 32 28

0

20

40

60

80x10

-3

Time (s)

; + + $ + & &

& *# 2 " '9# # 9 ! # % & 9 & & # & ' ! ' & " ( # 7 #+ & ! + & & $ 2 ) =( ! & 9 ! ! &/ & )) =( ! # " #0 ' ¼ ) B & ! ) B 9 ' / 0

! 9 ! " 0 8 ' ' # ! *# 2 '!# 2 ) =( # I B

! ! " " '9# '!# ! & " 7 ! " & ' 8 ! ' % & ! +

> ' ' F ' *# 8 ' # &/ )) =( + ! &/ & # ! 8 & ' " !

% ' & # # 2 0 & &8 9# & #

)

CH4 + air

0

. +

1.20 Upper wall (C) Lower wall (D) Inlet (A)

Pressure (atm)

1.15 1.10 1.05 1.00 0.95 0.90 0.85 78

79

80 Time (s)

81

82x10

-3

+

0 %'1 +

F!# / & # " '

& !# E! 9 / F E @# . * ! #8 '

@ > & '! / & F E & & ' ' % 0

# & & / = '!

+ & & E 8 # " ' ' & #8 & '#0

G &

" ' @. FE / & '! " & ' E!8 9 $ '! ! ! & # + E '! &

7 @. FE # ! ( ! '! !

8"# #

E

! #8 & # 8 '! '! '## # A8 ! 9 : ' @ '! ' I .! ' '! 8 # & '! ' " ' )

)

* % ' # # ' 8 & 8 ! / A! 9 : ' ' & ' 8 8

: =

; @ " ' &

> ## '! # 8 # " ' * % & 8 / & ! & * !

& # '! !

7 Æ & E!8 9 9 & 8 9# 0 % # '8

! & . / ?

=# =# .#/
L % & E!8 9 / F# & #! & E!8 9 / '8

K &

L

F ( & / 9 '8 # & # '8

& # E!8 9 / CED % & ! ! # & # /

' & / ! &

E!8 9 / 0

$ . ' ! ! 8

! & . E!8 9 / $ ' & 4 '!

#

E % & + & E!8 9 / 8

9' F

& # " ' + & 8# " '

#& ' $ % % & & . E!8 9 / ! % & & 4 " ' " ' 8"# ' 8 ' & 8# " ' 2

& # " ' #! % & & " ' ( & '! " ' * ! % & ! " ' ' 1 ; " '

7 # / & &# & & 4 '!

# % ! & . / ?

.#/ : > .>:> & % .>:> E!8 9 / E!8 9 > : > E>:> ! & & C8 D / & E!8 9 / & ' & # " ' 8 7

! & '!

# ' % & '! ; 8 & . / 2 & E!8 9 / * & #

!

"#

& & ?

.#/ :> * % & ! % ' & & 8 1 9' 1 9' E = '!

! 9 & % 8 + & . / < + / & E!8 9 / #! 2 & '# 8 & % + 4 # % $ + '

& # 0

&

; & 9 ' ! & & ! * % + & # ! & ! ! / &

Æ & '8

:

[email protected]

E7A$.1

E7A$.1

3?.

3?.

E 8#

E 8#

1#

" '

2

2

2-

" '

2

2-

" '

)

-

" '

-

0

= $ .

. 3# F 9' ( E!8 9 E>:> 9 8 # . C! D

+ E!8 9 !

4 4 * % 8 # ! " ' ' 8 / +! C D 0 & .>:> & . / & C! D ? '# # ! , & ' & ! ! ' & " ' & % ! ' + 7 / CD 7 CD ' % ' +% & ! / &

# ! E & ' # % #! & $ & " ' % 8 A ' !# ! / * % +%# ! & 8 & ! " ' ' C D ' / CD &

& " ' .% &

&

! & ' # & 0 % # (

# & ! &# E>:> /0

# / & # ! &

7 Æ

' 0

E & !! ! /8 4 ! 8 48 # # ! &

$ ' 9 ' 8 !! + 4 ' & & !# '! ' 4# '' 4# E '! ## & & & + # 8 4# *# '' 4# & # '! $ ' '! ## ' & ' '! # +

!

$ %& ' (

7 % + ! # & '#

0

= 8 7 # ! % %

& :

= 8 A & # ) 9 ! 5 )

= 8 *9M ' ) ' ! & 4 !

G

" / ! > ' ' ! 0 ! ! ) ! ! !" ! ! ! N" - O # - 5 ! ! ! ! ! ! ! ! ! ! ! !

&

! ! !$ $ % 5

! ! !

&

9 E

' " # ' + 0

" -

& -

-

2

"% & # & % + 0 ! !

#

% '

!$

!

> ( *# & '! ## 0 ! ! ! - & ) ! ! ! !" !

& -

& - & - & - &

! ! ! ! ! N - O N - O - 5 ! ! ! ! ! ! ! ! ! - & - & ! ! ! ! ! ! ! ! ! - & - & ) ! ! ! ! ! ! ! ! ! ! - & - & ! ! ! ! ! ! ! ! !$ -$ & -& $ $ % 5

& 9 E ! ! ! ! 4 & & / !! & & !! ! ! #! 0

& & & & & &

@ - @ - @

@ - @

@ @

@ @ @

! ! ! ! - ! ! ! ! ! ! ! ! ! ! !$

!

Æ ! "# $ %&'

2

x2 OUTLET

INLET

L 5 (u+c)

L 5 (u+c)

L 2 (u)

L 2 (u)

L3

(u)

L3

(u)

L4

(u)

L4

(u)

L 5+k k=1,N (u)

L 5+k k=1,N (u)

L1

L1

(u-c)

x1 = L

(u-c) x1

x3

> * ' @ M & '!

' ! 0 P

P

-

' & #! ! & '! !# #! ! !P ! & ! ' ! ' ! ! ' ! @ M #! 0 @

@

@

@

@

!

! ! ! ! ! ! ! ! ! ! ! ! ! - ! !

2

@

!$

!

& 9 E

)

7 & @ M #! # ( E!8 9 / & 8 ! '! > 8 ## '!

' ! $& ¼ ¼

! '! ¼ ¼ ! # - ) * 0 ! ! - ) ! !

! - @ ) !

7 #! @ & ! & '! : # @ ! & '!

# & E>:> ! # & / $ / # ! ! 0 ! !# !! # ' % / !! /8 /# & &M ' ! @ @ # & ## & & # 8 4

# & ! @ & '! ## & OUTSIDE REGION: use simplified method (boundary layer theory) to evaluate waves entering the CFD domain

INLET COMPUTATIONAL DOMAIN: use full CFD

? 7 ' 4 & # 0 ' & 9 ' @ & # '! ! ' & !

$ + ' + *# * % &8 & ! & & #

& # @ & # # / 2 ) 7 &

& & ! & # '! = '! & & ! & !! & ! & %

G # % ! & # '! & & /

$ '! & & ! % ! & # '! ! 7 % & # '! ! ! '

!

"# ) * & +)*,

% & ! @ & # '! & 8 E!8 9 / = '! & 8 ! / E>:> & ! & '! ! ! 8 %# 8 ! @3F$ 7 @3F$ &

& / ## ! ! C D & & '!

# @3F$ 4 & # & ! $ & ! ! @3F$ 0

! - @ - @ - @ ) !

! - @ - @ )

!

! @ @ ) !

! - @ ) ! ! - @ ) 2 ! !$

- @ ) ! ! %

!! & /8 & * % !! & " ' * # + 0 ! !

@ - @ - @

)

! !

@ - @ - - @

@

@ @ - - @

!* !

!+ !

)

)

)

)

' + " - - * < 0 3 & & @3F$ & ' & # 7 # %

& & @ 0 ! ! ! ! ! ! ! !

@ @ @ @ @-

- @ @

-

@ @ @ - - -

< ! # @3F$ * % # / # @ ) & / $ #

# @ @ & ./

+% ! & '! @ # A ! & '! ! # @3F$ % E!8 9 / ! ! ! : 8 ! ! # & / ! 4! 9 # @3F$ 8 # '! ! @ & ! / 0 % ! # ' ( )* + % + %, - . / ## #0 %' 1

-.

02

%, 3 4 & ' 5 + 67

!

"# %-.- # (

E>:> ! ! & . / & '

+ % *# 7 #! 0

* .>:> 8 # ! / & & / $ % & *#

+ # /

* ! / # @3F$ %

9 ' @ # # '! &8 & 9 ' @ # # # '! * % & *# *# ' # '! @ @3F$

## 0 @ @

#& ! @ & '! !# # *# & 8

!! ! @ & '! # # ! # ! #! /

G # ! / & ' ! & @ & ! ' #! .>:> $ &

& *# !

& # # / ' ./ & @

9 & E>:> G # ! / ' ' & & # '! ## @3F$ ! # & !# CD E ! &

% & / ' @3F$ 9 ' & ' '

/ ' !

! "# %-.- %& ' ( E!8 9 / / . / $ E>:> E!8 9 # . ! .>:> # '

)

Physical boundary conditions ECBC conditions: impose one or more independent variables Ex: Outlet pressure is fixed STEP3

Equations used in code

STEP1 Density x1 momentum

The remaining conservation equations are used on the boundary to time advance density, velocities and species. The amplitude of incoming waves is estimated by the LODI relation

x2 momentum x3 momentum Species (k=1 to N)

Energy equation is cancelled: not needed on the boundary (pressure is fixed)

Energy

LODI relations

STEP2 Using the physical conditions and the LODI equations gives the amplitude variation of incoming waves. For fixed pressure:

L1

= -L 5

=,>, # $ #* * Physical boundary conditions ECBC conditions: impose one or more independent variables Ex: Outlet pressure is fixed

Viscous conditions: impose weak conditions on viscous and diffusion terms. Ex: constant heat flux, normal stresses and species flux along x1

Equations used in code

STEP1 Density x1 momentum

STEP3 The remaining conservation equations are used on the boundary to time advance density, velocities and species. The amplitude of incoming waves is estimated by the LODI relation. The viscous conditions are used to set values to diffusive terms in the conservation equations.

x2 momentum x3 momentum Species (k=1 to N)

Energy equation is cancelled: not needed on the boundary (pressure is fixed)

Energy

LODI relations

STEP2

Using the physical conditions and the LODI equations gives the amplitude variation of incoming waves. For fixed pressure

L1

= -L 5

=,>, = . $ #* *

! ! ## 4 ' ! 8 # ( ! .>:> $ E>:> ! #

&# ! 4 ! / ! !# !8 ! & ! & E>:> & / # *# ' E>:> & ! ' +

& . *# E!8 9 / *#

:.&

:.1

:.'

:.<

# #,>, , 2 2

= . ; #,>, @ = , , , 2 2 = =

; =

7A=

7A=

7A=

+

. ;

; * & ! ' . rel="nofollow">:> 8

$ E>:> & 8 .>:> ! & 9' 3# ( & 8 # " ' ' 3 8

" ' & . / & & ' ' E!8 9 * " ' &

$ #

>1

+

# #,>, ; , = = &

>'

+

& ><

& =?

?

:

= . #,>, @ , , , ½¾ 3 4 = ½ ½ 3 4 ½¿ ½ 3 4 ½ ½ 3 4 ½ & ' = ½¾ 3 4

½ ½ ½¿ ½ 3 4 ½ 3 4 ½ ½ 3 4 & ' = ½¾ 3 4

½ ½ 3 4 ½¿ ½ 3 4 ½ ½ 3 4 ½ & ' = 3 4 3 4 < 4 = B 3 4 3 4 ' & =

; =

+

. ; ; *

$ rel="nofollow"> $8 / & E!8 9 & . / - ' & 0 # 9 ' ' # / ' ! ! & E!8 9 / $ # 2-

## 9' * % $8 ' & . / 3# ! ! & E!8 9 / 0

# F < & ' " ' $8 > $8

#! $8 > $8 & ' '8

& & E!8 9 #! 3# = '! & Æ + $8 8"# & E>:> * ! +% '! ! > $8 $8 /! & 8 0 %

! 8" & '! & @3F$ & = '! & / 40 $8 & ' ! ! ' $8 +% '! ! # * 8 " ' & $8 # E>:> #& ' & & & $8 3" ' : & : : 8 "#

* # E!8 9 / - ! . ## 9' & F 0 # !

& # ' "% # ! ! # * # " ' 4 "% & # % F!$ ! % ! ' ! ! # ( # !! # & # % ( & /

!/ & ' 8 & # 8 / % & E>:> * #

& % # & !! & & & / '! ! & 7 @3F$

! & ! & 4 '! # A & ! & % #& ' @9 & E>:> & # / + > &

&0 & % 8 ' '

" # / ' > # 7 # + &

& & # #! ! Æ & '8

7 # ! & . ( 8 & 4 & & 8 " ' ! % & E!8 9 $ & # # & E>:> & '# 0

7 " ' ' +% ! $8 7 8"# " ' $8 E 8"# " ' : : 7 "# " ' : 7 8 ' 2 E, 7 ' 7,

E *# 0 ) (

* 0

! 0 12 2 # 3 & +, C D " ' ' & ! '

& $ ) > $8 * # 8 " ' - '! # *# 0 @ @ @ @ @ ' & @ !# &

" ' & *! & $ & / E ! ! @ & 4 '!

# 3 & '! @ !# & #! E>:> & ' 0

2

! +% & / ) +% # /

& $ ./ # / / @

7 ! +% @3F$ ## & '# %

& '! @ # 0 @ @

7 +% @3F$ #! & '! @ 0 @ @ - @ @

# / @3F$ 2 ' 0 @ @ @ )

& ./ 0 ! ! ! - & ) ! ! !

' & #! / 0

)

& @ - @ - @ @

@ @ @ 3 - .>:> ! ' 9' 2 -

'! 9 ' ! / & . E!8 9 /

! 0 1 12 +, $ # !

/ & '! ! ' 7 ' > +% !

& " & '! $ % !

! & '! C#D ! , - C#D ' #

8"# & & !

# & '# '! %

0 @ ., ,

@ . - - @ .

@ .

% . & * ( & ' ! & Æ ! & 8"#

0 '! ## ' " # ! P # & & ! ' * # . # ! "#

* ! & . ! & ! # ! ' '! # # ' & " % 2 . Æ ( & &# " '

! 1 12 +. ., 7 > % & '!

' "# ! # !# G # 8"# ' '! ! ! 8 # / * # & 8"# # '8 0 " ' *# 2 ' ! ! * C& 8"#D & ' " '

Q ; & ! '! " # " ' '

½ + 8 +0 &

4 & ½ " '! # ½ , & 8"# & & 9 # 8 1 9' 1 9' ? K! > & 9 8"#

! & E>:> > : * *# +! '! @ @ @ @ @ ! ' @ # $&

+ ½ '8 7 ! ' ! 9 ! @ & # '!

4 ½ ' '0 3 0

+ +% +% & ! ! / 9 & / & # '! @ +% 0 @ / ½

$&

½ " '! # #

9 ! ½ , / ) ./ & " '! ) & 8"# & &

Infinity conditions (imposed pressure) Computational domain

INLET

L5 L2 L3 L4 L 5+k k=1,N

OUTLET

L1

; * * / ' 1 9' ' ! # / 0 / . ( ' % < " ' ( ( & . # ' # & % & % ! @ & @ ./ '0 @ / ½ - @

2

# & !! ' & 8 ' + 9 ! ½ $ ./ & ' * ! " ' #

"% # "% % # ( 7 @ ' & @ #! ./ & / !

! 0 1 12 +., * & # " & '! & $ # .>:>

! & %

" & '! & &

: 7

+% # / @3F$

## & " '! 0 @ @ 7 # #

"% > 4 "% & ! " ' 7 @ ' & @ #! @ @ & / $ %

! 0 # 2 +%4, 7 8 ' E, ! ! E

"% 3

"% % ( 7 ! +% ./ ) 7 # / @3F$ ## & " '! @ @ @ @ @ @ ( ! ( ./ ./ ) > # ! & @ & @ @ @

& / & # & ./

& $ & ./ E & ( "% % ' # / & "% !! !% ! # ! / * 8 & "% 9# ( ! & % ½ 0 !% !

½

% R

½

½

' R (

!/ 0 2 +04, 7 ' ( .>:> 0 ! ' ( ! ( 4 "% # ' ( #

( "% # ' 4 "% # ' % ( 7 ! ( @ @ @ @ ( & ./ ) 3 '! @ !# # ' ' " '! @ # ' ( *# 0

! ' ( / @3F$ ## & " '!

0 @ @ @ & @ @ !! # & 4 "% % & # !

& "% # ' # ! '0 % ) # ) ) ! & % 1# ! $ # & ./ ' ./

! " #$

+ % & ! 8# + *# 7 # 8 8 8 ! # * #8 + 4 # @ 8 %8 G # #8

&

4 & 0 '8

+ E>:> & ' x2

x2

x 2 =l Inlet

Slip walls

x1= l

u1

U1

Outlet

U2

x1

x 2 =-l (a) Axial inlet velocity

(b) Boundary conditions

> @ 0 0 8' 7, $ ) ! ! ( % ! # # + *# 0 )

-

)

' & + ! & 9

$

# E>:> $8 & # ! & " ' ! + & * 4 & & 0 & 8# 0

:0 > : 1 9' & % & ! 1 ! &

$ & / %

& 8"# 0 ! !

! - / ½ ) !

' / ½ E>:> & 0 / .¼ ( : # ! & ./ & ( Æ 8 & / 1 9' 1 9' ! ! & .¼ ) & .¼ )2 ! 4 ' E>:> & % & ! & ! / ½ # / & & ' E>:> % # '! @ ./

: 0 > : E>:> & ' . ) $ & 8"#

:0 * : 8"# E>:> & ' . )

:0 > : "#

½ ' E>:>

& & '# ! (

# & ' & (0 ) ) 1 (2 ))) ( )) 2 *# ) ! & " ' ( & ' ( 7 ( & 2) ' ) ! ! - )2 ' # # & " ' *# ' # : Æ .¼ / )2 ## 1 9' ' '! ' 9 ' & ) $&

" ' &

0 >& * A 5<%

2 & & & '! # ## &

0 >1 . =,>,

2 *# ' & 8"# : $ '! !# 7 #

+

" ' '

E!8 9 / ' & 8"# 8 #

! % 4 & ' , 6/' 5 ! 8 % +

*# # 8"# : ' . )2 $ '!

& ( 2

! " ' / " & . '9 * . )2 ' A & / )) )2 ) 2 ' 0 . )) &# & . ) *# ' ' ! & $ # . . ) # " ' ! & . ! & .¼ ./ ! 1 9'

0 >' . =,>,

2

0 >< =,>, 2 * ! & & "# : *# ) $ + # & ! !

' & # $ ! # # ( < )2 0 (

' )0 # ! "% ( *# ) ' % & & 7 # *# & & & ! & & 8 "# E>:> *# 4 & & / & * ! & " ' 9 : *# ) & " '

! #$ # *# ' & & # & & ' % " ' # " ( & # 4 # # % " ## ' & " ' x2= 2 l Hot gases

Non reflecting boundaries

x1 x 1 =5 l Fresh premixed gases

Laminar flame front

x2=-2l

% ! + *# ! !

(

& + ' *# + " ## & ' '0 # # & " # ' & # ' # 7 # 8 ./ 8 & '#0 ( 3 ) 4 ) 1 02 )

2)

x2= 2 l

U

u1

T

T1

0

1

Y1

T2

x2=-2l (a) Axial inlet velocity

(b) Temperature

(c) Fuel mass fraction

' 3 4 ( ! # ! ./ 1 & # ! 2

>' % & S #! " * )) ! + ( " ' % + # + 7 ) " # & ## ' & # 7 ! +% # $8 / * & : & : 8"# E>:>

8"# , E>:> : & / ) # # !! ! ) & + #! *# E

2

"

' + % ! *# * *# #! #8 ! ! + 7 ! ! ( " '

" # ' & # # " & % # )

>' ;& # ' & : 4 78 # ! + ' *# 2 % 8

# # ' E !# ! & ' 1 & " ' 1 ) ' #

= '! / &

>&

% & #$ $' !/ 5# 2& $ ' 9 ' # '! 9 " ' 0 '!

% & '! $

9 ' # '! 0 '! ! '!# & + C'## D # 4 #+ & $ ' !# : ! # # 4 > 9# '! & A! 9 : ' A! 9 $ &' 9 & # " '

( % & '! " & # ! & + ! 17E '

'## * @. FE + ! ( ! ! ( FE '!

! #+ ! & '! > #8 # ' & '! *# 0

; '! CD '! A! 9 ! # '!# 8 ! & E!8 9 / '!

E '! C/D '! ! '!# & ( # # " ' # ! Physical wave ("p"): group velocity equal to flow speed

Numerical wave ("q"): negative group velocity

Mesh points

+

$ & 8 ! / 5 0 ! ! -5 ) ! !

2

A! 9 : ' ' C/D '! # ! # ! ' & & ! 5 & 4# # $ # 5 #!0 C/D '! ! 7 # & 5 ' $ ' &/ % & C '8D C'## D ' ! '!# / R 8 4 % # ! #! 5 A! 9 : ' $ # &

# & & # ! # ! 5

. , . , * .

C 6 .& .' .&'6' ½

C @ > 2 > $ 8

" ' C/D '! & '! !# "0 !! '! 5 ' '! 5 C/D '!

' ! ! & 0 ' %8

# ! '! # !

" ' E C/D '! # ' ' A! 9 : ' A8 ! 9 ; 0

$ # & # &/ 8 # FE @. ' # & & C/D '! '

E '! G # % '! ' '!

9 ' & : 5 ,

D . * * / ; > ;

7 % *# ' # & # !# '! #

8 & ' # ! & C/D '! % 5 *# ## & '

! 9 : & C/D '! # *# '! ! & # ! / ' '! ! # *# : '! & ) ( ( )2 " '! *# : & "# 7 A! 9 : ' : '! ! " C/D '! & ## ' # 9 *# & C/D '! # " CD '! ## ' , / ! & " & C /D C/ D '! ! ( " & '! # & C/D '! ## & ! 9# & & " & C/D '! CD '! ## ' : = ; @ ' % " ' ! # & &9 & '! +% / & 0 % & # " '! 3! @. FE

0

#

& + !# &

INLET

OUTLET Incident physical wave Speed = u+c, amplitude A1

Reflected physical wave Speed = u-c, amplitude Ap Reflected numerical wave Speed = group velocity of wiggles, amplitude Aq x1 = 0

x1 x1 = L

)

% & *# ## ' " Æ 8 ( #! *# 0 " Æ & '! " Æ & '! & '! 7 / ( & " '! 0 7 / ( & " '! 0

!/ 6 ! & / ! '! / & #! #

'! !# #

& 8 '! # 8"# '89 ' E>:> '

' ! ! & " 0 & & ) *# ' & " ' 0 ! % # " ' ## # 8"# *# ! + & ! % ( ) # & 6 &

8! ! % # ! % 0

)

-

!6 ! !6 !

6

%

-

1

2

! % # 1 ! % ! % & ! ' # &

# & ! & & ( & 7 1 " & ! % ( 0 / & ! 1 *(#

+ ( 0 ½

- % 1

1

2

" ' & & '#0

1 02 ))))

2

! % & ) ) + 1 # 0 , ,

x2= l Non reflecting lateral boundaries Non reflecting outlet

Uniform supersonic inlet flow at speed u0

x 1= l

x 1= - l

x1

Initial vortex

x2=-l

; *

10 )2

0 ))))2

22

$ # & "# E>:> 8 ' & & 0

:0 1& & 8"# 1 9' :0 E 8"# E>:> ' . )2

*# ) ' ! # ! + # : & 0 F #! ! & ' ! ! # ! " ' ' ' ! " ' & $ *# ) ! % # 9' & #! ! # & ! ! ! % ! ' " ' % ! % ))) 7& 0 ! % ! *# ) > : ' ! % ! ' # & ! + +0 ! # ! % C/D '! ' & ' ! " ' '! # # ! " & & ' ! % 7 0 ' & ! ' $ ' # 9' % ! )2 % ! 7 # " '

: &9 '

@ *

>& 9 * * ) 2 ? * 2 * *# ' 8"# E>:> : *# ) , ! % ! 0 ! + ! # ! + & " '! ' 7 0 # ! % # ! % ' % ! ' ) % ! + & ' # ! &

0 A! 9 ; & ! / ' & ! : = &

" '

!

@ *

=,>, >' 9 * * ) 2 ; * C D & &

( ! $ ) #$ % ; " ' ! ' 1 " ' ' 8 ' *#

# ' 8 & &8' 0 # (0 ) " ' 0

P ) )P ) 2

0 ' % 1 1 02 2 < ) ! " ' 5 0 5 0 )

0 8 ' E 8"# ( x2= l INLET: imposed velocities and temperature

OUTLET

x 2 = -l x1

x1 = L ISOTHERMAL NO SLIP WALLS

+ $ 2 ! " ' 5 $& % # & &

! # &

# # 0 ! 5 8 21 2 !

0 0 % ! + & #! 0

! 0

8 !

5 0 0

2

% + ## # / ) ' # # % &

0

8 0

0

2

' % ! % 0

0 ! 5

8 ! 0

)

E ' '

' 4 & ! # # ' #

! # / ' & # & 0

1

" ...!

+

!

2

; / +

>&2 =,>, >' * =,>, ><

& : & 1 9' 8"# E>:> & : ' . )2 & : '

½

$ 8 ' ' # E>:> E, * ; " ' % ! & & ./ 2 & # '! 0 @

! !

! ' & E>:> : 7 !# *# #! ! & " ' & : *# 8"# E>:>

: *# "# E>:> : *# " ' ! ! # ' "# : *# ' ' & : & & )

2 + >& *# + & & ' ' E>:> : : *# & & + 0

4 )) ½ ½ # ! 4 ' #! " ' ))

!

!

,

>' =,>,

' % ! + ./ 2 2 ! #! *# 2 & E>:> : # / # # # # : *# 0 #

# " ' ! + *# 8"# E>:> : ! *# 0

# & # )!! E ! ! # ! + *# + *# E

% / ½ 0 " '! # # "# & : #! & & !8 & '! ! + *# = '8 ! + ! *# ! # & ' +#

2 + =,>, >' 3 4&7

!

2 + =,>, ><

& .E 2 ?2 > 2 )2 C 2 + F 92 &554 ; * . 2 *

; , : 2 + 2 %17.%'' 1 .C 2 C F > 2 ) &5%5 ,

* 1&'.1&% ' .C 2 C2 > 2 )2 8 2 E = F 9 2 E &5%< 9 G ; , : 2 + 2 747.7&1 < .C 2 C2 > 2 ) F 92 H , &5%% ; * ; , : 2 + 2 ('&.('% 7 2 H F E2 + &55' 9

: !

'4'I'&'2 # J 2 ; F 2 , K &55% ; 2

" ; , : 2 + 2 (5'.(55 ( 2 9 F 8 2 , # &5%& . $ &('.&%7 % 2 ) &5%< # $ 8 +. , 5 2 , &55( % & '( '! ! + )2 :=+2 ; &4 2 ,2 # 2 0 F @ 2 ) 144& 9 # G ) && 2 ,2 +2 ; F ) 2 > &55( : . : * + * ! # + 5(1%%& &1 2 )2 @ 2 )2 # 2 0 F +2 ; &55% 9 # * , $ , , ; 2 = 6 -2 J&.%1 &' 2 ? ; &554 C *

. : ,( $ ,

1<7.17'2 , ; 2 -6=. &< 2 ? ; &55' 0 2 %(.&4' &7 2 ? ; F >2 + H &5%' * 11(.17J &J 2 ? ;2 H 2 2 H 2 E F C2 , 8 &5%( = . , 1'<'.1'7' 2

2

&( 2 ? ; F EE2 + &5%5 0 * &(.'( &% 2 ? ;2 + 2 = F 2 E ) &5%( =

. 9 ''5.'(7 &5 2 H ;2 82 E : F 0 2 C E &55( 0 6 * 6 &.1< 14 2 H ;2 82 E : F 0 2 C E &55% #G * &.&7 1& >2 +2 CL 2 )2 2 D2 > 2 +2 , 2 E2 ) 2 >2 ) 2 F $ 2 &55J #* * / - ; , : 2 51' . 5'4 11 >2 F , 2 K M &551 D . . ; , : 2 +. 2 1'&.1'( 1' >2 2 0 2 C K2 , 2 , ) F , 2 K.M 1444 #* ,D6816=1 7<<.7J5 1< >2 + H &554 * &&&.&17 17 > 2 E &5(< E /0 1% % 2 157.'4% 1J >N 2 E F ?2 0 &5J% , 3 ; , : 2 + 2 &J5.&%& 1( >2 K ) F 9 2 K = &5%1 = $ . 444 5 577.5J& 1% >2 E &55< & ' + )2 #

, + 15 >2 E2 +2 ;2 8 2 ) F ) 2 = &55< - ) = 8 6D 6= * 5 &.'1 '4 >2 E2 +2 ; K F 8 2 ) , &55' - ) = 8 6D 6= * 6 ') 17.1.&2 17.1.J '& >2 E2 +2 ; K F ; L 2 ) &55< . 5 , 1<(.1J& '1 > 2 82 E 2 +2 ?2 K2 ,2 2 >2 H2 E2 F + 2 ? &554 : . * 1) 9:0 D8 ; , : 2 + 2 %&(.%1' '' > 2 >2 # 2 0 F +2 ; &555 ) = =D* * J5.%' '< >

2 K E F , 2 = &5%' H '7 > 2 9 9 &57< 4 EC 8 'J > 2 F , 2 ? D &5J' $ ') + '( > 2 ? &5%4 ; . * : ') @ '% > 2 ? &5%% ; * ; , : 2 + 2 <(7.<%%

2

'5 <4 <& <1 <' << <7 <J <( <% <5 74 7& 71 7' 7< 77 7J 7( 7% 75 J4 J& J1

> 2 ? &5%5 ; G 4 /0 &4& > 2 ? &55' , , <'( > 2 # F ) 2 ? K &557 $.

<7(.<J< >2 K &5%J ;

(5.51 >2 K F > 2 K 8 &5%1 0 . * %(.57 >2 K F > 2 K 8 &5%1 8 ! # + %144J' >* 2 C2 )2 2 8 2 = F 9 2 + &5%% // 444 5 (%'.(54 > 2 E F @ 2 ) 1444 9 * @. : 40 7

<<5 . <712 ,:E=# > 2 E2 @ 2 )2 > 2 8 F ; 2 &55% ) =

9 # * " ; , : 2 + 2 5&(. 51( > 2 &5(% /% 2 + )2 - L + E

, . + J > 2 &5%< E 5 2 'J&.'(4 > 2 &5%% ; , 1<7 > 2 F ) 2 E &55% ' # ;#,8=:+ > 2 +2 8 2 2 +2 ; F > 2 ; &551 . ; , : 2 + 2 74'.7&4 > 2 8 F ; 2 &55% ; * " ; , : 2 + 2 5(&. 5(% > 2 )2 C 2 + 8 F C2 O K &55J > 2 E 2

$ . &(J.&5% > 2 + &5(( , 4 /0 ''.7< > 2 +2 , 2 ; F ? 2 K 8 &5%& $ ') + >2 H = , &5%4 ; * :

&&72 @ >2 H = , &554 , / + 4 '&7.''7 >2 H = , F ,2 &55&

HP , / 4 + 4 8 90 1&(.1<4 >2 H = ,2 , 2 E F 92 + &5%5 ;

* : / 0 )

7<&.7J'2 9

2 @ >2 H = , F 92 + &5%J + $ * 17' >2 H = ,2 92 + 2 E2 C F E2 K > &5%& ; * &1(.&<4

2

J' J< J7 JJ J( J% J5 (4 (& (1 (' (< (7 (J (( (% (5 %4 %& %1 %' %< %7

>2 H = ,2 92 + F E2 K > &5%< 0 $ * &<'.&(1 >2 H = , F E2 K > &5(( * 4 4 15& . '&5 >2 H = , F + 2 = &55< 9 : / )

J'.&&'2 + > *2 C2 2 H2 +2 ; F 0 / 2 K &55J 0 . . 1(.<< > *2 C2 +2 ; F 0 / 2 K &55( + * . $ 5 &5&.1&5 > *2 C2 +2 ; F 0 / 2 K 8 &55< . : , $ ,

&7(.&(<2 , ; 2 = 6 - > 2 K F , 2 &5%< ; 5 '<& > 2 K F 9 2 C &5%1 $ ' , - + > 2 K F 8 2 + &5%% : , $ ,

&5.1(2 , ; 2 = 6 - > 2 + F 2 ; # ? &51% )G 55%.&447 > 2 ? F 0 2 0 &5(4 9 <1& > 2 ; ) F DP 2 + K &5(( . 3- ; , : 2 &74' . &7&7 , 2 ? 8 F E 2 &55J 444 . 4 : + 5J.47&J , 2 &557 % 2 ) , 2 2 8 2 , F +2 ; &55J : :

%'.&&12 = : 2 H + 2 ) , 2 2 @ 2 )2 92 02 ) 2 =2 >2 E F +2 ; &55< /;4 < 5.&J , 2 2 @ 2 )2 92 02 E 2 #2 ) 2 = F +2 ; &554 ,

* : 40 0 $

&5.J<2 ? 2 , 2 E2 E 2 #2 ) 2 =2 +2 ;2 @ 2 ) F 92 0 &5%% #* . 145.1'J , 2 E F +2 ; &554 0 $

&.&7 ,2 F >2 H = , &5%% * ; , : 2 + 2 (5&.(55 ,2 2 + 2 > F >2 H = , &554 E * ; , : 2 + 2 %45.%&7 ,2 2 2 , F ;L &554 : , $ ,

1(&.1(52 , ; 2 -6=. , 2 E 82 C 2 ) F 2 &55< ; G

5 , 114.1'J

22

%J %( %% %5 54 5& 51 5' 5< 57 5J 5( 5% 55 &44 &4& &41 &4' &4< &47 &4J &4( &4%

,2 K2 >2 + H F E2 = = &5%5 + * . &4&.&1& , 2 8 ,2 , 2 F H 2 8 &5%5 + , /0 +

1J7( , 2 K 82 # 2 ; F H2 ? &55% ; . * &7.<% , 2 E2 8 2 E F + 2 = 1444 0 6 G = ; , : 2 + 2 , 2 H F

2 : C &5%( : * '7'.'(7 , 2 H F

2 : C &55& ; * (.1J , 2 ? H F ) 2 K &55& = :

! # + 5&41J% , 2 , , F 9 2 > # &5%4 D * 8 # $ &%5.14( , 2 +2 92 , H2 , 2 H F

2 : C &5%% @

* ; , : 2 + 2 ('5.(<7 , 2 , F 8 2 H M &55% ) * ''J.'<% , 2 8 F 92 , H &5%< &1'.&17 ,2 8 F ,2 E ? &5%5 . 8 # $ 1&.<( ,2 + &5%7 ) * , &.75 ,2 + F C2 + &5%' ;

G 5 % 2 1<7.1J' ,2 + F K2 C &5%' + * 5 ,2 +

9&.9&1 ,2 + F 9QL2 &5%< ; 84; 4 < 15&.''% ,2 + F ?2 0 &5%1 #G G *

* 5 17&.1%1 , 2 , ?2 2 ? F 8 2 &5%& $ 3= ; , : 2 + 2 &7%'.&7%5 ,2 , E &55J , , <1( . 745 ,2 D2 )2 02 @ 2 ) F +2 ; 1444 * , &%<'.&%J' ,2 ;2 9 2 F E2 + &55' > . 444 5 &7(<.&7%1 ,2 ? &55( ) Æ . , 4 &<%7 . &<%( ,2 ? F > 2 ? H &555 .

* , 4 (<J . (<%

2

&45 &&4 &&& &&1 &&' &&< &&7 &&J &&( &&% &&5 &14 &1& &11 &1' &1< &17 &1J &1( &1% &15 &'4 &'& &'1 &''

,2 ? F 2 K K &55< $ , 4 1%J% . 1%(4 ,2 ? F 2 K K &55% 75' . J4J , 2 ) C2 )2 2 0 ?2 K2 8 2 E F 9

2 0 &551 @ ,2 9 &57& $ + : 5 4 /> &J'.&(% ,2 9 &571 $ + :: 5 4 /> . , 2 > F +2 ; &55< #G G :.

G ? ; , : 2 + 2 &'%'.&'54 , 2 > F +2 ; &55J G

9 &&&.&'( ) 2 ;2 , 2 ;2 ) 2 =2 2 K , F 2 &55J

G 81 D1 =1 G 1(&.1%( ) 2 = &551 ; 8 . G * &J'.&%& ) 2 = F , 2 &551 ; * . G J(.%7 ) 2 =2 , 2 F E 2 0 # &5%J ; G * 14'.1&( ) 2 =2 C2 @2 ;L 2 2 , 2 E F # 2 # &5%( ,

* : / 0 ) +

75&.J'(2 )2 8 &%&( , )2 8 &%'4 4 , &'J ) > H 2 E2 2 K K2 H2 C F ,2 ? &55% : . * ) &47.&11 ) 2 K # F H 2 K D &5%5 + . .

$ 2 2 '75.'(< ) 2 K # F H 2 K D &554 ; . '7%.'(4 ) *2 2 92 > F + 2 &5%5 D

'5.J4 ) 2 > F ,2 + &554 ; *

* 'J&.'(7 ) K 2 + # F 0 2 8 &55% 9 *

. , 4 115% . 1'&< )2 &5%< . # + %<41'4 ) &5(' 0 5 7 '&<'.&7< ) 2 K ? &5%5 0 * (&.%% )2 + F @ 2 9 &55J ; * . . * * - ; , : 2 + 2 1''.1<4 ) /2 , &55< : / )

'(7 . <(<2

2

&'< ) 2 ) 2 2 ) > F ;2 , &554 ; * G 6 * ; , : 2 + 2 '17.''1 &'7 )/2 + C F 2 ? 8 &5%& # , - + &'J )2 K 02 2 )2 2 ? 92 +2 E F C2 9 + &55< ; * * . 1&'.115 &'( )2 K.E2 B 2 E F > 2 ; &555 ') ):.: ; 1 ( /0 &, 175.1J< &'% )2 K E2 > *2 C F > 2 ; &55J ') < : G + ! # + 5J&5J< &'5 )2 K E2 @ 2 ) F +2 ; &55' * &4&.&&( &<4 )2 02 , 2 + F 9 2 E &55J 9 . 5 &.'J &<& )2 + &5%% G = $ 5 8 4 1<7I1J( &<1 # 2 ; F , 2 K 8 &55J - G * . &%<.141 &<' # 2 ; F 0 / 2 K &55'

* 15'.'7& &<< #G 2 # F + 2 = &5%% G ; , : 2 + 2 J5'.(44 &<7 # 2 0 F 92 , H &55< 0

$ ? ; , : 2 + 2 &'<&.&'<( &<J # ; 2 #2 2 , K F 0 / 2 K &55&

* . &77.&(' &<( #$2 > F E 2 &5(( J15.J7& &<% #2 F C2 @ &55< 4 @ &<5 #/ 2 D 2 C 2 F 9

2 ) &551 : G $ . ; , : 2 + 2 &<J7 &74 0 2 + ?2 0

2 ) F 0 2 0 # &5%< $ ; ; , : 2 + 2 %7.54 &7& 0 2 &5J5 $ : , $

1'&.1JJ2 &71 0 /2 #2 H 2 @ F 92 1444 )G .G

= ; , : 2 + 2 &7' 0 / 2 K &55( 9 : 8)

15 . <(2 9 # + $ . @ &7< 0 / 2 K 8 F + L2 E &55( $ @ @ &77 0 2 02 ) 2 >2 @ 2 ) F , 2 E &55<

$

. * ?

; , : 2 + 2 &1('.&1%& &7J 0 2 02 92 02 @ 2 ) F , 2 &55' D . . * &.1J

2

&7( 0 2 , C2 ?2 8 F 92 , H &555 : 2 (((.(5< &7% 0*2 2 C2 02 E 2 2 2 E F 82 K &551 : , $ ,

<4'.<172 , ; 2 = 6 - &75 0 2 E2 ?2 82 C 2 E2 2 C +2 C 2 ) E2 >2 , ;2 82 H2 C 2 ? , F 92 @ &557 C:.E / . C:.

C:.576447%2 C : &J4 0 2 , F 9RR2 , &55< 9 G / ? ; , : 2 + 2 &17( . &1J< &J& 0 2 , F ER 2 : &557 9

G / 444 5 1''5 &J1 C2 0 &55' . , &1%1.&1%< &J' C 2 E2 EÆ2 2 2 F E2 C &55( D * : @ +

15& . '442 H + &J< C 2 E2 + 2 -2 E2 + F ,2 ? 8 &55& . , 4 &(J4 . &(J7 &J7 C 2 F H 2 &5%% D . * ; , : 2 + 2 JJ7.J(7 &JJ C 2 02 2 E , F H2 D E &551 #G . ; , : 2 + 2 11'.1'4 &J( C 2 F E2 + &557 ; $ * 5 , 1< . '( &J% C 2 F @ 2 9 1444 ;

* 5 11(.1J4 &J5 C 2 E &554 =. # $ 444 5( 1474.147% &(4 C2 @ &5%% ! ' %% % ) N 2 + @: &(& C2 @ &555 ) > R &(1 C2 @ F 2 E &5%( , * * 5 , '1(.'<7 &(' C2 @ F 2 E &5%( #* * .

* 1'.<5 &(< C2 @ F 2 E &55'

* 1<&.17J &(7 C2 F + 2 > &551 + 5 1<(.1(( &(J C2 F B 2 M &55J * , 4 &11<.&1'J &(( C2 + &5%5 E .

, &.&4( &(% C2 + &55< * : / )

<(7.7(12 + &(5 C2 : &5%( +

2

&%4 C 2 02 >2 H F , 2 K M &5%5 , ( 1<& &%& C 2 0 , &55J , * - ; , : 2 + 2 '%& . '%% &%1 C2 O K2 8$2 E B2 9 2 E F ? 2 1444 9 E . * <&.7% &%' C 2 9 &555 @%0 & %2 %% &% ( + )2 - L &%< C 2 D &554 ; * G ; , : 2 + 2 (<'.%'7 &%7 C 2 D 9 F 2 C K &557 : G

* &4(.&&< &%J C 2 H2 9 2 ) C F 2 = &5%< ) ') + &%( C2 > F R2 &5(% : 4 5 4 '<'.'7( &%% C 2 # &5%( ; G ) , # 717.7'7 &%5 8 2 > F C2 ) &5%<

; , : 2 + 2 117.1'1 &54 82 E2 K 2 @ F > 2 ) &55J ; :

%& . &7<2 H + &5& 82 B2 /2 )2 , 2 0 # F @ 2 C &55J : .

- ; , : 2 + 2 1(&(.1(1' &51 8 2 M2 ;2 E F ;2 E H + &55J , * : @0 + 2 4

&752 &5' 8 2 ) K F 2 0 8 &5(1 9$

+.&5<2 = &5< 8 2 ,2 >2 E2 + 2 = F ) 2 8 1444 S &&(.&15 &57 8 2 # F ,2 1444

* = ; , : 2 + 2 &5J 8 2 )2 , 2 >2 +2 ; F >2 1444 =

. '57.<&( &5( 8 2 ) ,2 ) 2 E ,2 + 2 > F >2 K &5%% ; .

G ; , : 2 + 2 7%5.75( &5% 8 2 ) , F +2 ; K &554 ; 9 * * : , $ ,

1%&.15%2 , ; 2 -6=. &55 8 2 ) , F +2 ; K &551 = 9 G

* 5 <47.<'J 144 8 2 C ? &5(5 = 5 , 111.1'( 14& 8 2 K F ; 2 1444 , 0 E * = ; , : 2 + 2

)

141 14' 14< 147 14J 14( 14% 145 1&4 1&& 1&1 1&' 1&< 1&7 1&J 1&( 1&% 1&5 114 11& 111 11' 11< 117

8 2 K 2 D/2 F H 2 8 &5(< ) I 5 <&(.<<< 8 2 K > &5%% $ EC.8 8 2 9 &5%J :. 4 /0) &((.1&( 8 2 9 &5%( = $ $ J7.54 82 E &55% %2 % ' ( + )2 # , + 82 E2 >2 E2 +2 ; F @ 2 ) &55J * : @ $ ') / /

14&.11<2 # ;#,8=:+ 82 E2 @ 2 )2 >2 E F +2 ; &557 * 3 ) &5.&5 8/ 2 K D &5(7 EC.8 8 2 , &5%% 8 $ ! ') K ? 8 2 K D2 ,2 , 0 F > 2 > &5J5 $ 2 K ? F 82 , E F 8 2 + &5%< +

5 'J7 8 2 8 , F 8 2 ? &5<5 )G ! , ; , : 2 + 2 17< . 1JJ 82 ? E2 @ 2 F C 2 &5%J 8 $ 2

G 3 ; , : 2 + 2 &%7'.&%J4 :2 8 C2 9 2 ; F 0 / 2 K 8 &55( 9 . , 4 '%1J . '%'' : /2 F 92 , H &5%1 * *

* 36 ; , : 2 + 2 '1(.''7 : /2 2 E2 H F 92 , H &5%1 #G 2 G2 * 6 * 15'.'4% K2 K &5%J * %&.&&J K2 K2 9

2 K F H2 &5%% : * ; , : 2 + 2 747.7&< K 2 E &551 E. * ! # + 5147%5 K /2 ,2 , 2 > F +2 ; 144& - . K /2 K2 9QL2 2 2 E F 8 2 0 &55( * 5 &<5 I &(& K /2 K2 ?2 2 G2 + C F 2 &55' ; 5 J7.54 K 2 ? F 9 2 > # &5%1 ) TD * T 8 # $ &%5.1&1 K 2 ? + F 9 2 > # &5(1 ; / 1. $ 5 $ # $ '4&

11J K2 ; &55< E ? ;

, : 2 + 2 J%(.(4< 11( H 2 H2 C 2 K 82 D2 # F >2 K + &55& = &&7.&'< 11% H2 2 +2 M H F @ 2 ) C &55% ;. 2

2 2 * <7.J& 115 H2 + E &555 ! 0 $ '! ! ' + )2 - 1'4 H/2 F E 2 0 # &5%J G * J7.%< 1'& H 2 @2 9 2 F B2 @ &55J

* . - ; , : 2 + 2 1<5.17( 1'1 H2 @ F $ 2 ? E &55' G 1(<.1%1 1'' H2 @ F $ 2 ? E &55% ) E K )G 0 - 0 , E 444 5 14<<.147< 1'< H2 ? E F , 2 E # &55' 0 0 # $ EC 8 1'7 H

2 K2 C2 K 02 2 E F E 2 K &5%7 +#E:O 0 . =)%7.%1<42 = 9 1'J H

2 K2 2 0 E F E 2 K &5%5 , .:: 0 , H + C.+ , H =)%5.%445>2 = 9 1'( H

2 K2 ?/2 K F E 2 K &5%' . 2 2 G Æ =)%'.%1452 = 9 1'% H 2 K D2 > 2 ; ;2 >2 + H F /2 K &55< =D* ,D *

<J4.<JJ 1'5 H 2 K D F C2 ) &5%5 #* . 5 , &(1.&51 1<4 H 2 K D2 @ 2 92 H 2 )2 8 2 C 92 C 2 02 )2 K ? F D

2 H &5%& E 444 5 17<.1J1 1<& H 2 E &5%7 8 5 '&.7% 1<1 H 2 &5%% * '5& 1<' H 2 &5%5 9 .# + ::

* '5( . <&' 1<< H 2 &554 9 . + ' E* G .

G 5 <&& . <'7 1<7 H 2 &55& 9 . + J E G * 5 'J& . '5< 1<J H 2 &551 9 + < G (7 . 5J 1<( H 2 2 2 ? F ?2 0 &5%% 0 $ , /0 4 1(1%.1('& 1<% H 2 ; E2 E 2 ) F ?2 ) # &5%J ; . $ 3 ;

, : 2 + 2 &7%'.&7%5

1<5 174 17& 171 17' 17< 177 17J 17( 17% 175 1J4 1J& 1J1 1J' 1J< 1J7 1JJ 1J( 1J% 1J5 1(4 1(& 1(1 1('

H2 K2 E2 + F E 2 &5%( ; 5 &''I&JJ H2 ? &55& , . .

444 5 7<(.77< H 2 9 #2 0 2 2 ,

2 > F 2 K @ &5%1 $ K ? H2 + =2 >2 H = ,2 C

2 ) F 2 > &55% #* &51.14J H2 + =2 2 >2 >2 H = , F 9QL2 &55' 0 * 1(J H 2 M &554 E G * @ '1( . ''< H 2 M F > 2 ? &555 , , 757 . J%( H2 = &5<& ; / 4 :/ '4& H 2 &5(7 D 5 7 '55.<&4 H 2 8.D &5(4 : , 4 1((.15% H2 H H &5%J , $ K ? 9Æ 2 + &5'5 + ' A! 8 , 2 $ 9 2 82 , 2 E2 H 2 ) F > 2 + &55% : >E9

&7' . &(' 9 2 9 F 9 /2 # &5%< ,2 2 ( - B 2 # 9 C . # E: 9 2 > # &55J

:

&5' . &512 H + 9 2 E F , 2 &5%5 , G * * 5 , ,) &'<.&<' 9 2 E F , 2 E &5%% G . * . (5.5J 9 2 E F , 2 E &5%5 G * . (5 92 , H &5%% ) ;

, : 2 + 2 &'%&.&<41 92 , H F 2 , K 1444 2 * , <75.747 92 , H2 B 2 ) 9 F M2 C &5%J + * * 3 ; , : 2 + 2 &<&5.&<1J 9

2 ; ?2 9

2 K2 = 2 ) F 2 ) &55' 9 * HLL *

&<J.&J4 9 8 2 + &55< 2 ! # , + 9 2 &551 , G 5 , &J.<1 9 2 F 82 K &5%% ) = 5 C 17.<'

1(< 1(7 1(J 1(( 1(% 1(5 1%4 1%& 1%1 1%' 1%< 1%7 1%J 1%( 1%% 1%5 154 15& 151 15' 15< 157 15J 15( 15%

9 2 E &554 ' H + 9 2 E &55(

. : 8)

& . 1%2 9 # + $ . @ 9 2 E F E 2 D &55J = . 4 /0 <7 . %1 9 2 > F @ # 2 C &5%( ! $ 1 + 92 + F ?2 0 &5%( + * 1'( 92 + F >2 H = , &5%& , G * 444 5 147.1&' 92 + 2 >2 H = , F E2 K > &5(5 #G * 1%7.'4& 92 + 2 9QL2 F ?2 0 &5%' * . 9 17( 92 + F ?2 0 &5%1 * 1%( 92 + F ?2 0 &5%' * . &.<1 92 + F ?2 0 &55< ; : / )

1.J&2 + 9 9 2 H2 >2 H = , F E2 K > &5%< * &55.1&' 92 &5(< ; G

4 4 &44( 92 F , 2 &5(J G

57.&&( 92 F ?2 0 &55' $ D* - + 9 2 F , 2 K 1444 )

* = ; , : 2 + 2 9QL2 2 D 2 +2 @ /2 F 8 2 0 K &55< #G . 9 G : , $ ,

7.&%2 , ;

9 2 0 , F =2 &5(7 ;

. 2 G &45.&1< 92 K 82 #/ 2 D2 C 2 F 2 0 &554 - $ ; , : 2 + 2 <<&.<<J 92 9 F 2 ) &5%< . * $ 5 '''.'71 92 H 8 1444 D G G =

; , : 2 + 2 92 H 8 F >2 H = , &55% , G G " ; , : 2 + 2 1&J7 . 1&(& 9/2 #2 H

2 K F E 2 K &55& 0 + + 8. C + H =)%(.%1<%2 = 9. E E2 + 2 2 K K F E 2 ? &5%5 #G .

* 5

15(.''1 E 2 > 0 F E 2 > 8 &5(J D 3- ; , : 2 + 2 (&5.(1(

155 E 2 #2 ) 2 #2 +2 ;2 @ 2 )2 92 02 , 2 F # 2 # &5%5 8 ) * > 9 # 2 @ 2 > 2 5%.&&( '44 E / 2 C E F E 2 @ : &55& 0

:@ , , &(J.&%' ; 0/ C @/2 12

<5.7%2 &55& '4& E 2 ; F > 2 &55< * $ <<'.<7( '41 E 2 ;2 2 K E F >2 , ; &555 0 * . *. . + :: ( 77(.7%1 '4' E 2 0 # F > 2 K # &5(( ; . ;

;?.5.+-2 + $ '4< E2 #2 > 2 ; F +2 ; K &55( = * &5% . 11' '47 E2 #2 + ) ,/2 2 > 2 ; F +2 ; K &55(

G * 1<' . 1%1 '4J E2 E F E2 > K &5%1 0 5 1'5 '4( E 2 0 F , 2 K + &55( . * : @ +

&4' . &&<2 H + '4% EE 2 + 2 C 2 ; ,2 H 2 F H 2 H &55' 9 . * , 4 &41' . &4'< '45 EE 2 + 2 E 2 F H 2 &551

. . ; , : 2 + 2 1(& . 1(% '&4 EE2 + F C2 + &5%5 ) * .

* &(&.&%7 '&& E

2 K + F = 2 0 ; E &55( .

: @ +

'4& . '&42 H + '&1 E 2 , F H/2 K 1444 . 4 /0 &.'1 '&' E 2 ,2 9 2 ; F ,2 ? &55J . 5 '7' '&< E 2 , F +2 ; &55& $ * '&&.''1 '&7 E 2 , F

2 H &5%5 E

, +

4 &4' '&J E 2 , F

2 H &55& ; 5 <15.<%< '&( E 2 F H 2 &551

* . ; , : 2 + 2 '&% E 2 2 EE 2 + F H 2 &55' *

: + $ ! ')

%( . '&<2 , - + '&5 E 2 2 EE 2 + 2 H 2 F , 2 K M &55< + =D* . 5 , ,) &J&.&J% '14 E 2 E F H 2 K , &5%4 9 . * &<'.&7<

2

'1& E2 ) &5%< ; + : * 17.'& '11 E 2 8 +2 E 2 2 2 E F H

2 &5%1 ; * * 36 ; , : 2 + 2 &%&.&5J '1' E2 ; &5%4 + . &(7 '1< E 2 H2 ;G2 E 2 2 E )2 G 2 E F 92 E > &55% ,.

* 2 2 * 2 G " ; , : 2 + 2 J5'.(41 '17 E2 + F H2 K &5%1 = 5 '<&.'(( '1J E2 +2 $ 2 H2 ,2 ? F 9

2 &55& .

, 1(<J.1(7( '1( E 2 , K2 HL2 C F 2 K K &55( ) = ; = * , E

8 .D* H &&'.&<< '1% E 2 + &5%& #* * 3= ; , : 2 + 2 55' . &444 '15 E 2 + E F : 2 H - &5J% + - + ''4 E 2 ,2 )2 K 02 2 )2 ) 2 E F 2 E &55% @ * '<1.'7% ''& E 2 > ) &5%4 , 5 7 <'(.<71 ''1 EQ/2 9 F E2 E C &55( : ./ . G. &J.'& ''' ER 2 , E2 > 2 8 F + 2 = &55< + * ? ; ; , : 2 + 2 &455 . &&4J ''< =2 , F ,2 + &5%( #G * J5.(& ''7 = 2 0 &55< , $ 2 ! + )2 #

, 9 ''J = 2 0 &55% ) 5 ,

<&%.<11 ''( = 2 0 ; E F E

2 K + &55( 9 . : 8)

1J< . 1%42 9 # + $ . @ ''% =2 H &55< $ ! ') ) ) + )2 : ''5 =2 H H F # 2 # &551 E* , J4J.J17 '<4 =2 , 1444 + $ + )2 # , + '<& DP> 2 # &5%4 ;

: / ) 4 , + 9 '<1 D 2 &5%( , 5 747.7'& '<' D 2 K F 2 &5(% ; 4 5 4 <&5.<<J '<< D2 # F >2 K + &5%( 8 $ 0 ') #

'<7 '<J '<( '<% '<5 '74 '7& '71 '7' '7< '77 '7J '7( '7% '75 'J4 'J& 'J1 'J' 'J< 'J7 'JJ 'J( 'J%

D2 : &5(J 5 , 17&.1J5 D 2 F 2 K &5%( 0

8.K %(.JJ2 :,# +2 C F H 2 H &55& #G . . ; , : 2 + 2 &J<&.&J<( +2 C F H 2 H &55' ./ 3 444 4 : + 5'.41<& +2 C F H 2 H &55% $ . * " ; , : 2 + 2 (&& . (&( +2 C2 H 2 H2 9 2 H F D2 # &5%% ) ; , : 2 + 2 &7&(.&71J +2 + F ?/2 K &55%

" ; , : 2 + 2 <57.74< + 2 + F ,2 + &5%1 : G * 5 1&5 + 2 = &5%< 9 G . * , '&5.''5 + 2 = &5%J 9

3 ; , : 2 + 2 &1'&.&174 + 2 = &555 ; . . 5 &4( . &'1 + 2 = 1444 , - + + 2 8 &5J7 0 3 ;

, : 2 + 2 &1((.&1%' +2 K2 @ 2 )2 , 2 F +2 ; &55( ) C. $ * : @ +

'1&.''42 H + + &5%& 4 B EC 8 + 2 , ) F E2 + &55%

, '4<&.'4<< + 2 - &555 9 . , 4 ''7.'J1 + 2 - F , 2 K &55J 9

:

1J5 . ''J2 H + +$ 2 K &555 ') .@ + 2 8 F + 2 = &55% . * G G G 1J.<4 + 2 8 F + 2 = &55% - . G " ; , : 2 + 2 &47(.&4J< + L 2 D 8 F 2 ? , &551 8 G *

: + 51.4451 + 2 ;2 ; 2 +2 + 2 = F E2 E &55% * ''7.'7' +2 ; &55J - * . - ; , : 2 + 2 1&5.1'1

'J5 '(4 '(& '(1 '(' '(< '(7 '(J '(( '(% '(5 '%4 '%& '%1 '%' '%< '%7 '%J '%( '%% '%5 '54 '5& '51

+2 ; &55% , 0 * 1(5.1%< +2 ;2 2 ,2 # 2 0 F @ 2 ) &555 9 # 3 ; ) , &.J +2 ; F , 2 &5%% &1&.&7' +2 ;2 , 2 F ;L 2 &55J

. * , 7'&.7(J +2 ;2 # 2 ; F E2 E C &551 * <7.(' +2 ;2 8 2 ) F > *2 C &55' ) . * &&%.&'' +2 ;2 9 , 2 ;2 , 2 F # 2 # &5%J #* Æ * 5 7 1J7.1(% +2 ; F 9 2 &551 > 5 , &4<.&15 +2 ;2 ;L 2 2 @ 2 )2 , 2 F # 2 # &5%( @ * 5 1J7.151 +2 ;2 @ 2 )2 > 2 02 , 2 2 # 2 # F 2 K &5%% :.

; , : 2 + 2 &'J'.&'(4 +2 ;2 @ 2 ) F , 2 &55& S * 5 7J&.J47 +2 ;2 @ 2 )2 M 2 >2 ;L 2 2 2 K.E F , 2 &551

5 , &''&.&'7( +2 ; K2 @ 2 ) F , 2 &554 ) * ; , : 2 + 2 J&'.J&5 + 2 &5%% ; 5 <<7.<J5 + 2 F , 2 ? &5%% ; * ; , : 2 + 2 (%&.(%5 + 2 > &5%7 + , &&5.&51 + 2 > &554 ,

; , : 2 + 2 75&.J&1 + 2 >2 M 2 + H F C2 &5%5 ; , 14&4.14&% +

2 + F >2 E &55( * 2 . $ '1( . '<% +

2 +2 2 E F >2 E &55J 8 6 . - ; , : 2 + 2 1J5'.1(44 + 2 9 &517 : D $ $ ) > > &'J + 2 # ? &5J5 3 ; , : 2 + 2 &4&.&&' +2 &5(& 0 # 2 9 &%(% ; * 8 '&5.'1&

'5' '5< '57 '5J '5( '5% '55 <44 <4& <41 <4' <4< <47 <4J <4( <4% <45 <&4 <&& <&1 <&' <&< <&7 <&J <&( <&%

2 + 82 ; 2 )2 2 K , F , 2 1444 ) 6 * . , 117.1%1 2 K &55J 2 2% ! ' ( (%% % + )2 - L 2 K F @ 2 9 &55J 9#

G , 4 11<% . 1174 2 ? , F + 2 8 , &5(( EC.8 2 0 8 &5J& , + 2 + 9 &%75 =

, <&5.<11 2 + 9 &%75 =/ 2 # G

9 / / 4 ,> ''5 2 + K &5(1 ' 8 + 2 ? 9 F )2 K 0 &55& * * 1<7.17J 2 ? 92 )2 K 02 ) 2 E , F C2 9 + &55' : $ * $ 7%.J5 2 ? 92 )2 K 02 ) 2 E , F G 2 K &551 0

$ * * . ; , : 2 + 2 &J5.&(J 2 ) # F E 2 0 # &57J $ 5 , <7J.<J1 2 > &5%5 E * " ) 1J&&.1J&J 2 ) 8 F 2 K , &5%4 . = 5 , 77.(4 2 ) 8 F 2 K , &5%& > = '1(.''% 2 C F E* 2 E &55& , &7%(.&75( 2 C 2 @ 2 9 F 9QL2 &557 #G , &<<(.&<7< 2 , K F ,2 &55< ; * : , $ ,

(7.5<2 , ; 2 -6=. 2 , K F 0 / 2 K &5%5 : * * " 444 4 : + %5.4&1(2 2 , K F 0 / 2 K &55& . * '<'.'J4 2 , K F ;L 2 &554 + * . 9 . : , $ ,

155.'&42 , ; 2 = 6 - 2 , K F ;L 2 &55' ) * 9 <&.7( 2 + 1444 + $ ') .@ 2 F > 2 &5%% U T ; , : 2 7J5 . 7(( 2 K.E2 M 2 >2 +2 ; F , 2 &55' 9. $ . 'J'.'%& 2 K E &55' * . * * E ; , : 2 + 2 + 5'.4J&

<&5 <14 <1& <11 <1' <1< <17 <1J <1( <1% <15 <'4 <'& <'1 <'' <'< <'7 <'J <'( <'% <'5 <<4 <<& <<1

2 K E F E 2 ; &555 0 *

* + : * 7'(.77J 2 02 + 2 - F >2 # &555 . . , 4 &75J . &J4( 2 2 # 2 C2 82 E M F H 2 8 D &55& ; 5 $ <('.<5& 2 K &5%1 #G 9 * * 36 ; , : 2 + 2 &7<&.&7<% &577 < EC.8 2 ; F +2 ; &555 : 9 # : 4 / <$

('.%<2 , ; 2 = 6 - 2 C F S 2 K &554 ) E

* 15%.'&& 2 H &55J E

- ; , : 2 + 2 %'&.%<J 2 H F + 2 = &5%% #* E . )G 0 1'.<< 2 H2 + 2 = F ?2 0 &55< . <4(.<1( 2 B.2 K2 # F D/2 &554 : * 8 11J.11%

2 : C2 E2 K > F >2 H = , &5%1 ; * 36 ; , : 2 + 2 <1'.<'& 2 @2 E 2 @ F H 2 &55( . . &%' . 1&7 2 ) F B2 # # &5%7 , * 3 5 , $ : %7.&1<% 2 ; F E 2 &55J E

* - ; , : 2 + 2 155.'4J 2 F ) 2 # &55<

. .

1<4.174 2 ) > &5(& E* 3 ; , : 2 + 2 J<5.J7( 2 ) > &5(J ) # .> .- 3- ; , : 2 &J7(.&JJ' 2 #2 2 F 2 &55< 4 /0 1%(.'&5 2 2 82 E F >2 E &55< 1) 366. :/; $ 0 , # >@2 (5'.%44 2 ? &5(% , , &7(.&(J 2 K , &5(( : , 4 (5( 2 ) F + 2 8 &5(& K=0 2 1 # =).=> '(2 - = > 2 = F > 2 ? &555 * 7&5.7<7

)

<<' ;2 , F >2 + &5(% D 5 '('.'55 <<< ; 2 E2 2 ;2 2 E F E 2 ; &555 9 * =D* . E . * 4(,C $ ; , : 2 + 2 744.74' <<7 ; 2 E2 M2 =2 02 E F ;2 E &555 0 81.

* 3 ) >

K# # 2 75.J< <<J ; 2 8 F 9 2 K 9 &5(1 4 C E:; + <<( ; 2 )2 2 + 82 0 2 C2 C 2 K F 2 K , 1444 .

* 6 * = ; , : 2 + 2 <<% ; 2 )2 2 + 82 2 , F , 2 &55% #* .

* * " ; , : 2 + 2 (&5.(1J <<5 ; 2 H ? &5%( ;

5 ( , &.1< <74 ; 2 &55& * : 4 / <$

1('.1%J2 , ; 2 = 6 - <7& ; 2 2 , 2 F )2 K ? &5%% 9 - 4 : %%.4&<5 <71 ; 2 F +2 ; &55< ; $ 5 &.'& <7' ; 2 2 @ 2 )2 >2 H = , F E 2 ; &55< ;

* : , $ ,

57.&1<2 , ; 2 = 6 - <7< ;G2 K E F 2 C &555 #* ) .9 .P@P 2 E '7.71 <77 @ 2 , E F # 2 0 &55% ) #* ) 9 0

" ; , : 2 + 2 7&'.7&5 <7J H 2 # &55J ' 2 + )2 # , + <7( @ H 2 # F @ 2 ) &55(

.

* 33 ) 1&.(2 1&.&1 <7% @ H 2 # F @ 2 ) &55% )

* , 4 1'<( . 1'J% <75 @ H 2 #2 @ 2 ) F , 2 &55J ) . * - ; , : 2 + 2 '7.<1 <J4 @ 2 D &555 0 81.D1 ;6,0)6556<<2 ,#0, <J& @ 2 92 > *2 #2 >2 H = , F H2 ? &557

* 2

, 4 1<5J <J1 @ 2 9 F +2 ; &55% ) = * 4 /0 J77.J51 <J' @ 2 9 F @ 2 ) 1444 :

= ; , : 2 + 2

<J< @ 2 )2 , 2 F E2 K + &5%( , * : ! /

'%J.'5%2 @ <J7 @ 2 )2 )2 K E ) F +2 K &55< #*

* ? ; , : 2 + 2 &1<5.&17J <JJ @ 2 )2 92 02 E 2 # F , 2 &5%5 ,

* " ) 1J1&.1J1J <J( @ 2 )2 +2 K2 )2 K E F E 2 , &55J #*

* - ;

, : 2 + 2 <&'.<14 <J% @ 2 ) F +2 ; &55( #G * 5 %'.&&< <J5 @ 2 ) F +2 ; &55( 9 #

* : 4 / <$

17'.1(<2 , ; 2 = 6 - <(4 @ 2 ) F +2 ; &55( 9 # : 8) + ?

&47.&'72 9

+ $ 2 <(& @ 2 )2 ;L 2 2 >2 H = , F E 2 ; &55( C . * 5 1J'.15' <(1 @ 2 &5%1 : 4%7<<2 + - 2 )

E # <(' @ 2 F > 2 K > &5%1 $ ! $ ( 2 :E

E <(< @ 2 F + 2 # , &5%J = $. 8 $ , @F 2 &.&1 <(7 @ 2 2 , 2 K M F +/2 ? &551 E * * 81. 444 5 '57.<41 <(J @ 2 F E //2 E &55& ; . 5 &.17 <(( HLL2 ; F E2 C &57' ; ; , : 2 + 2 &('.&(( <(% @ 2 2 C 2 F ? 2 , H &5%< -

$ ; , : 2 + 2 (J.%' <(5 ? 2 8 C2 ;2 C2 C2 ) F 0 2 , &55%

. 9# * " ;

, : 2 + 2 %55 . 54( <%4 ? 2 , F ) 2 0 &5%& E D* 8 . 0 0 '&.<' <%& ? 2 , H2 /2 F 9 2 C &5%&

$ %&.55 <%1 ?

2 @ &5&% 5 %<4 <%' ?

2 @ &5&5 5 %& <%< ? 2 0 E &55& 7 ' ') EC.8 <%7 ? 2 : F > *2 C &557 8 $ * 15J.'&4 <%J ?2 0 &5%7 > , <%( ?2 0 &551 ; . (

; , : 2 + 2 &.&%

<%% M 2 @2 D/2 , C2 ; 2 2 C2 ; > F / 2 , C &551 ) * $ , &7&4 <%5 M2 : F ;2 8 &5%< ) ; , : 2 + 2 &%%'.&%51 <54 M2 @ F ,2 0 # , &5%J . $ . &.17 <5& M 2 + H2 C2 F + 2 > &554 '<4.'J7 <51 B 2 D &554 )

* , $ &(%.&%% <5' B 2 F 2 , &557 + * G <<(.<J& <5< B2 2 , 2 2 +2 ;2 ;L 2 F # 2 # &5%J 8 $

* 3 ;

, : 2 + 2 &<1(.&<'<

* 2 11J .G $ 2 'J( )R G 2 &&7 * 2 15& * 2 &%J 2 '75 G 2 &7 2 &' 0P 2 &' ) = * 2 '&( 2 1<5 ) = )= 2 &'1 * 2 1<5 * 2 &%( 2 J #>- 2 14<

2 <1'

2 &J

2 &J

$ 2 54

$

G *

2 '1 G 2 && * 2 &4

2 &44

* 2 &1

* 2 <&4 $2 54 0P

* 2 &' 2 &7 2 J4 *

2 <7

. 2 77 * 2 <%

2 '(4 2 '(%

2 '(7 2 '%< . 2 '(<

2 '7%

* 2 'J4 2 '(5 . 2 'J< 2 %

G 2 57 * 2 '5 2 &<J 0 . 2 &<& 2 &<& 0 2 &%4 2 &'5 $ 2 &(.E.9 2 &(( $ 2 1&4 2 1&< 2 145 2 J G 2 &(( > . 2 5' . 2 ''7 2 <&J 2 <&& ,E, 2 '4( , 0 2 1&( 2 '%5 2 &71 2 &7 * 2 &%(

2 J( 2 J% 2 J% * 2 &(1 /2 1(& * 2 1(' * 2 1'7 2 '<& 2 J&

$ 2 &4( 2 &47 2 75 * 2 '&4 *

* 2 1&< $2 1&7

2 1&1 2 1&7 9 # 9#2 1<' 2 7J 2 7% 2 7J 2 7( 2 7(

* 2 &%7 2 (7 6 * * 2 1%1

* 2 &5J $ G 2 %% G 2 15<

* 2 1&'

* * 2 &51 * . 2 '4'

* 2 '4% * 2 &%( )= * 2 171 2 <4& 6 2 &%1 2 &%1 C . $ * 9 # 9#2 1<& + P2 1&5 =2 14J C 2 &J< 2 &J7

2 <'( 2 'J& 2 < G Æ 2 ( )=2 &77 . 2 ''1 2 'J 2 'J

$2 &% $2 &5 8 / $ . 2 'J& 8 / 2 '(' 8 .) / G. 2 &&& 2 '57 2 &<% :;=0 2 1'1 H/ 2 &%J 2 J 2 &( H.? 2 &%( 2 &57 H 2 &1% H++ * 2 1'1 G 2 %% 2 %% 9 #

2 <44 9 # 9# 2 &'1 2 &J4 * 2 '&' * 2 1'% 2 && 9 2 ( G 2 %< * 2 '' 9 # 2 '&' 9 D ) : 2 <&( 2 'J< .E 2 &71 .E $2 &5 E 2 '4& . 2 &7 2 & 2 1 G 2 15' 2 '44

* 9#2 &J< * 2 %7 G 2 %5 2 &&% 2 5'

2 '512 '5' 2 &1 . 2 '%5 =,>, 2 <1% 2 <1% 2 <1< . 2 <17 . 2 <1J 2 <1( 2 <1< 2 <&5 2 <&4 2 <'J . G 2 <&'

* 2 <4 2 &&% 2 %7 + 2 ''4 + 2 <<' + 2 (

$ * 2 117 2 114 * . 2 '&& 9# * 2 '&< * 2 114

* 2 11&

* 2 <& * 2 14' $ . * 2 11<

2 '77 $ . . 2 '54 $ 2 ''& 6 2 '<4 . 2 ''1 2 ''' . 2 ''1 6 2 ''4 G 2 &&J * * 2 &5J * 2 (<

2

$2 &J * 2 (' )=2 &77 . 2 ''& 2 % 2 '%J

$2 '%< 2 '51 Æ 2 '75

* 2 'J4 )=2 &7J 9#2 &J% 2 54 = 2 &'1 * 2 151

* 2 141 2 &1( 2 &<J 2 && $ . * 2 15% 2 '44 G 2 %% 9#2 &J< 2 &<( . * . 2 <41 G 2 &11

* 2 '1 * 2 '' * 2 '' 2 &J' G 2 (

2 '7% 2 (J 2 (& 2 J G 2 <1 2 7< * 2 5< 2 5' 2 && 2 &4 2 'J 2 J& 2 J 2 75 2 '57 2 &JJ 2 &J

.$ . 2 '51 * 2 'J% .G $ 2 'J( 2 <4( 2 'JJ

2 &(< .G * 2 &55 2 &1( 2 &1( * . 2 &%7 * 2 151 * . 2 &%% + P * 2 &%% 2 &<%

2 &'4 2 &(1 H++ 2 1'1 2 1'& 2 &'& * # .

2 '4& * 2 14< >E9 2 14( # > - 2 14< 2 1&1

2 114

2 14J * 2 11J $2 115 2 &<7 2 &<' * 2 &<< + .H 2 &<7 9#2 &JJ 2 J $ . 2 '7% 2 '(( . # $2 <&4 = . $2 <&4 . 2 &%( 2 <'J B0H * 2 <<

Theoretical And Numerical Combustion

Overview

More details

Related Documents

Theoretical And Numerical Combustion

Combustion

Theoretical

Combustion

Fuels And Combustion

Fuels And Combustion