Theoretical And Numerical Combustion

  • November 2019
  • PDF

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
         

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

           

           

           

           

           

           

           

           

           

           

 +                                    (                 *! .       ) &           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   # !     #!     !       >

A< B K / 

A6B

 < K A,  , ,  , < B          !  1 !    !   = .!    1           !    

 ! )    5:'7+S AD   I D   I D   I   >EE  # 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        &    <  I H     !  &    <   <    <  

        

"

u = (u1 ( x 2 ) , 0 ) x2

n Splitter plate

Flame front

x1

      . >  0      F      <  + # &  I +    &      

    !!  A <  L <  B  1 # &     !    A !   )   !I   !          !  ) !B +                     1  &   %#  A# /B    > !  #   # #    # 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

    







-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 !  C F  '( 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



   









-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



"                               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

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                             !   # "        # "                !  ' E    (=             $

(= 1

== 



"   =               ! =     # " *      = =             <        E          E   $

( 1 (<  1

< 

&

"     22  /222         :   E                              # "               ,                    > !            # "            I        I        #          8                       >   *   8    >           *    =        ==$

>1

=  = 1 == =

0

 $   (   & '       

  (              &      &  &  5    &   (   &          5          

/3

           

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

 1

  

7

>

        E   $

( 1 (  1

  >  1 1  

3

              <    I                                   -   & 0  7$



<  > 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  8 8@ " 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        #   "      ,                " #  *  #  " *     !  L J    *                       .      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        <               I                  EC:     %                     =   ! = 

              *   +- 0$ = 1> &//

=

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

= = - = 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



   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)   <B "    " "     0 6/              "       !"    "  "      (       "     (  "            ( 

 : 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

   !"     ?               

           &         &  %              . 4 5               &     '  (   

"  (          "          "        

  $"

-

;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 :         ;   

   8 9   )        "              ;          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         /   E N

                       

            

     

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*

          

 

 ;



·



 '



 -



     '  '  +

)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       '!    '! '##    #      A 8 ! 9  : '  @   '!   '         I     .!    '    '!       8 #   &      '!      '  " ' )

)

         

*  %     '   #   #    '          8     &   8   !   /  A ! 9  : '   '  &  ' 8    8 

 :  =   

 ;    @  " ' &   



>  ##      '!   #       8     # " '  *  %      & 8      /      &  !  &       *                   !   

                  &  #    '! !      



7   Æ  &       E!8 9     9 & 8      9# 0 %       # '8 

  ! &  . /   ?

  =#   =#   .#/   <I  K &

   L            % &  E!8 9 /   F# &  #!  &       E!8 9 /     '8       



      K &

   L  



F (     &     /     9 '8 # &     # '8 

&  # E!8 9 /   CE D         %   &  !  !      #  &               # /                

              

       '     &         / !  &   

 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 '! 

#         % !  &  . /  ?

  .#/   <I  =     &   . >   :  >   .>:>    &        %  .>:>     E!8 9 /     E!8 9 >   :  >   E>:> ! &    & C  8    D   /   &  E!8 9 /             &     '     &    # " '      8      7  

         !  & '! 

#           '    %  &    '!  ;   8            &   . /    2 &   E!8 9 /   *   & #     

      

!

"# 

   &    & ?

  .#/   <I  &      &     9#  &  C.D      F4 ! & 8 %       .>:>    *  %  &  !       %  '     &     &  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    /      C D    7       C D '  %     ' +%  &   !        /   &

#    ! E          &       '   # %  #!     &    $ &   " '                         %                  8    A '                        !#    !  /       *  % +%#  !        &  8     &   !  " '     ' C  D     '  /  C D     &    

 &    " '  .%   &  

 &  

 !             &     '  #  &          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$     !  # & !#     C D    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  0 2 )

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  : '  A 8 ! 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 0 2 ))))

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

  ;           *        

 1 0 )2

0 ))))2

22

$        #  &    " # E>:>  8  '  &        &   0

 

:0 1&    &   8" #   1  9'   :0 E 8" # E>:>   ' . )2

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

     :     & 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 0 2 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 E 2 + &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 C 2 , 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 8 2 E : F 0  2 C E &55( 0 6          *   6               &.1< 14  2 H ;2 8 2 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 =    $          .        4 44 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%%       // 4 44 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          4 44 .  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 E 2 = = &5%5 +     *     .       &4&.&1& , 2 8 ,2 , 2  F H  2  8 &5%5 +             , /0 +

 1J7( , 2 K 82 # 2 ; F H 2 ? &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 K 2 C &5%' + *              5 ,2 +

  9&.9&1 , 2 + F 9Q L 2  &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 E 2 + &55' >             .      4 44 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 C 2 @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

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

  

    *        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 C 2 @ &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 9  2  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 8  2  H2 C  2 ? , F 9 2 @ &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   /   4 44 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 E 2 + 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 H 2 D E &551 #G                    .        ; , :  2 + 2 11'.1'4 &J( C 2  F E 2 + &557 ;  $              *    5  ,  1< . '( &J% C 2  F @  2 9 1444 ;              

       

*  5     11(.1J4 &J5 C 2 E &554 = .         #  $   4 44 5(   1474.147% &(4 C 2 @ &5%%     !     '   %% % ) N    2 + @: &(& C 2 @ &555    )   > R  &(1 C 2 @ F  2 E &5%( ,  *     *            5  ,  '1(.'<7 &(' C 2 @ F  2 E &5%( #*     *      .

 *       1'.<5 &(< C 2 @ 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 C 2 : &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%< ) ')  +  &%( C 2 > F  R2  &5(% :             4 5 4    '<'.'7( &%% C 2 # &5%( ; G                        )     ,  #   717.7'7 &%5 8 2 > F C 2  ) &5%<          

            ; , :  2 + 2 117.1'1 &54 82 E2 K   2  @ F > 2  ) &55J ;        :         

 %& . &7<2 H    +  &5& 8 2 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 8 2 ? 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     G 2            *     6 *     15'.'4% K 2 K &5%J             *       %&.&&J K 2 K2 9

2 K F H 2  &5%% :      *                 ; , :  2 + 2 747.7&< K  2 E &551 E.           *         !   # +  5147%5 K /2 ,2 , 2 > F + 2 ; 144& -              .               K /2 K2 9Q L 2 2  2 E F 8 2 0 &55(                   *      5     &<5 I &(& K /2 K2 ?2  2 G 2 + 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 D 2 #  F >2 K + &55& =                     &&7.&'< 11% H 2 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 B 2 @ &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  4 44 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                   4 44 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 E 2 + F E 2  &5%( ;                   5    &''I&JJ H2  ? &55& ,              . .  

  4 44 5   7<(.77< H  2 9 #2 0 2  2 ,

2  > F  2 K @ &5%1    $   K ?  H 2 + =2 >2 H = ,2 C

 2 )  F 2 > &55% #*                 &51.14J H 2 + =2 2 >2 >2 H = , F 9Q L 2  &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   *     4 44 5   147.1&' 92 + 2 >2 H = , F E2 K > &5(5 #G                *        1%7.'4& 92 + 2 9Q L 2  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&' 9  2  &5(< ;         G     

  4  4     &44( 9  2  F ,  2  &5(J           G    

       57.&&( 9  2  F ?2 0  &55'      $   D* -   +  9  2  F , 2 K 1444 )

           

 *     =        ; , :  2 + 2    9Q L 2 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 E 2 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' E 2 ; &5%4 +      .         &(7 '1< E  2  H2 ; G2 E 2  2 E )2  G 2  E F 9 2 E > &55% ,.

   *        2   2 * 2   G   "        ; , :  2 + 2 J5'.(41 '17 E 2 + F H2 K &5%1 =            5    '<&.'(( '1J E 2 +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 E 2 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 '<< D 2 #  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%

 

D 2 : &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 4 44 4      : +  5'.41<& + 2 C F H  2 H &55%         $      .  *          "        ; , :  2 + 2 (&& . (&( + 2 C2 H  2 H2 9 2 H F D 2 # &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 E 2 + &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 E 2 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 E 2 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 9Q L 2  &557 #G            ,   &<<(.&<7<  2 , K F , 2  &55< ;       *   : , $   ,

 (7.5<2 ,    ;     2   - 6=.   2 , K F 0 / 2 K &5%5 :      *    *   "  4 44 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 8 2 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 K 2 # 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 H 2 ? &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 E 2 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.   4 44 5    '57.<41 <(J @  2  F E //2 E &55& ;           .     5    &.17 <((  HLL 2 ; F E 2 C &57' ;                     ; , :  2 + 2 &('.&(( <(% @ 2  2 C 2  F ? 2 , H &5%< -            

$            ; , :  2 + 2 (J.%' <(5 ?  2 8 C2 ;2 C2 C 2  ) 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 M 2 @ 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 &(<       U T     *  2 15  )=2 &7' >.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 <<



Related Documents

Combustion
April 2020 18
Theoretical
November 2019 40
Combustion
October 2019 27
Fuels And Combustion
December 2019 9
Fuels And Combustion
December 2019 12