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2
Eddy
3
Length scale
4
Vortex stretching
5
Energy cascade
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7
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Reynolds Average Novier Stockes
9
Closed set
10
Correlation
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j i i j
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j
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11
Reynolds stress tensor
12
Turbulent Heat flux
i ju u
ju T
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p j9C u Tix
minla ar turbulent
du
u v
dy
µ= + =
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13
Zero equation model
14
One equation model
15
Two equation model
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t oµ=µ +µ
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16
Rapidly developing flows
17
Recirculating flows
18
Adversed pressure gradient
ml
k
tµ
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19
Crude simulation
20
Coarse mesh
21
Length scales
22
Wall bounded flows
23 Turbulent velocity scale
24 Turbulent length scale
k

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25
Turbulent kinetic energy
26
Viscous dissipation rate of Turbulent kinetic energy
k-L
k-L
k-L
k(t)2 2 21
K= (U +V +W )
2
2 2 21
k= (u +v +w )
2
11 12 13
21 22 23
31 32 33
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e e e
e e e e
e e e
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21 22 23
31 32 33
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1 1
( ) ( )
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ij ij ij
j i j i
U uU u
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k
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)31(
&L ) 8 J @ ) $ <0B# 8) %n ( ' Y[ 1 L ) 4
($ B1 N W @J &B.B 2 ( <W$ M?'(@( = :< L ) "( ')29(
<0B# = :< L ) : 8 &4f -B.
. & % 8 = :< L2 ' ' * ?r N B K<0B# % &=[ Q@(B$
.O W ( '' N B = D "N W T $ = P L ) %% JB.=$ <
= ( '% ? "% ? $& H< 3 L H 6) ) % & # Œ 3 < %$
B " B.
$. 9 : o ' %= > 9 : $0<L 7 !)3-32(p @ 9 : Y '
K<0B#27
B.
)32(
G , . 3? ( RH< 0< )< "9 : o ' %B.3 P P
- 2 ( ! < &% *#%n ; < * p @ % &= :< r O E ^ : W J
% %n% &4 < $ 4f..9 : e' %n ) 3 f J% &K<0B#*$ .
B . $G& $. 0 3 NX W $ <o '! 0<.
*4@ ( #'%* 8•< ? $•<0B#*= >9 :%&%p @*<0B#
;).-.
)33(
*#$6.9 :$= >9 :3 <J @%&**<0B#B.
SK%L2 ')32(*$•( ')33(•P <!.
27
Large scale turbulance
2 .ij ije e=
iv
vi
k-L
L
Lk-L
kL
Lkl
3/2
k
k
l
=
=
Ll
lL
t C lµ =
l

)34(
46 QP.:#2 '09/0J.
== :< L2 ' " <)35($)36(%$0< B ?.
)35(
)36(
% H( L 7 L2 'B
= :<b 0 ;
( r+
r=[ Q-
== :<F P ;+H r
L2 '! N W . V w , N B""""< &.< =:)37(%
. V* X $ $ 4 F &$ <0B# % &? " B.
)37(
" <0B# N< , =B $<0B# ."b 0 . W$% . @(k.
<0B# 8 %n Y[ $ (k K ? &=[ Q r B( ? Fp @
. p @ " B.( ' =)36(%c*# k L ) Y[ $ ($ ( L ) 6 <
( ' Y[)35(..l 2 D 3 f b /J BB ^ '"$ B ^ ' !&
G&B < <0B# 8) %n 0 : G& ^ '.6 QL )
Y[ $ (( ' % ' L 7 L ) 3Z[1.
G ) %= ( % &4 B ; E)38(. B 0<.
2
t
k
C l Cµµ = =
µC
k-LLk
( )
( U) ( ) 2 .t
t ij ij
k
k
div k div grad k E E
t
µ
µ
!
+ = +
2
1 2
( )
( U) ( ) 2 .t
t ij ijdiv div grad C E E C
t k k
µ
µ
!
+ = +
kL
kL
kL
kLkL
µCkNLN1LC2LCk-L
1
2
0.09
1.0
1.30
1.44
1.92
k
C
C
C
µ
!
!
=
=
=
=
=
kNLNkLtµ
Lk
L
kkL
kL/k
L
k-L

)38(
S, ! * 4E =$ . W < $ ! J)38(v Q I - ;& J ! &
G% &! X !F W)'3"2"1=i? ( Di=jB(K< , 0< "01 : 3 !
* & * f " < &
)39(
!F W % 8 z P . 3B$!PD D $ <0B# 8) %n 0 : $
B._ 7<- !F W G 0(d & : 3 4# z P ! & * 8 " B&
: ? @1B.G % 4 ,$ $@ c 3 .B( !F W % &$ B% &
* 8 P* <D &O W T % ' $ % &B.
6--(6E #!#FG# 1 !Fluent
=<= :< L2 ' ")40($)41(%$@ B ?Fluent0<-
.
)40(
)41(
! N W . V w , N B L2 '"""": < &J b N)W G/ *#$
(n%8)( <0B#J N. ; < . *.
(n%8)( <0B#N$%..
2 2
2
3 3
k
i j t ij ij ij
k
ij ij
u
u u S k
x
S E
µ=
=
2 2 2 0i
t ij t t
j
u
S divu
x
µ µ µ= = =
2 2 2
-9(u +v +w )-29k
kL
k-LLk
( ) ( ) ( )t
i k b
i j k j
k
k ku G G
t x x x
µ
µ
!
+ = + + +
2
1 3 2( ) ( ) ( ) ( )t
i k b
i j j
u C G C G C
t x x x k k
µ
µ
!
+ = + + +
µCkNLN1LC2LC
kG=
bG=

K •( '* @(n%8)K<0B#28
BG3; < *29
$**
<0B#B$$ &*#^ D[E1(B<0J$ B@K(-Y[B
*( Kf*y*30
B.
; $v 131
%L 7)42(B.
)42(
W*•E)38()•E4(N-)42(! & -
)43(
$. G r.
7-6I-06J(
I$< JB== >>[<-)8 ,B;N< ,("B$< $&$
L+ ," &S(< E*#.•< ? $•<0B#)%:%. D<0B#W
&? >N:<W&BE:'%** 8
)44(
%vQ $%."9 :B*#B..'%
*.8J•%'%3.K<0B#f<;< ? $=[ QJ.
3.D[E1^*9 :K<0B#32
<0JB.P#$B"S(
&?B•< ? $<0B#0<$<-" B.%N?B)45(B.
28
Turbulent kinetic energy production
29
Mean flow
30
Turbulent flux of fluctuating density
31
Exact relation
32
Turbulent time scale
kG
bG
kG
i
k i j
j
u
G u u
x
=
2
k tG Sµ=
ij ijS= 2S Sij ijS =E
t*
[ ] [ ][ ]
[ ] [ ] [ ]
2
t
t
Velocity Length
or
Velocity Time
=
=
k2 2 21
k= (u +v +w )
2
k
L
[ ] [ ]
2
k = Velocity[ ]
k
= Time
L
kL

)45(
. V@0<_ -• 2%( '.& -#.:$@0<L2 ' ND
( :<3 'B.
7-1-4!# E0 "
• D< K(k4*<0B#0<•E< K(* 8B
)46(
c. V*G3D)c* f:4 @B(
)47(
c= ''Q33
*.0J
)48(
*#r(n%8)K<0B#)D $(%4• 2%)$&*ByK(
•E)49(.#.
)49(
Jf>B"3? (3J%B4*
B"%•)N W$3J%!F W@•)N WB.
?B"$c)46("** 8
)50(
>
33
Local equilibrium
[ ]
2 2
.t
k k
cons Cµ= × =
µCkL
µC
u u
y y%
&
=
2w
u v u&
= =
( )k
p =
( )k u
p u v
y
=
j i
ij i k
k k
u u
p -u u -
x x
j ku u=
u
y
uv
t
u
-9u v µ
y
=
t u y% &
=

)51(
0<c)48(*EU < <
)52(
N-• 2< K(cB< ,<-%. VB3ccG. V
/ &
)3-53(
?B"$@•E)52(E *U < <
)54(
= D*0<w <P• D< K("& 8B.L )" K
:.#B.3:)L 1"P.#.(
=<0<B.#>B%3 ':*.)<=
<L'%"L%G?GO >$L%(3 F ,
,.
L 0 1"o'Q=<PB• B
3 fNz <..K ?$F(* + &=<NF$
.
1-#>B%3 ':6 Q2N 70*" BP#)%L )
)%* W< K(W<B".S(U -• D* W< K(<‚ P3#
<>B$:6 Q3 'J.
2-* W< K(<3 ':N $ ,.( 'X%• 01./
0<B..3y.#)(%3; B)<'.$%; B
3
( ) 2k u u u
p u v u
y y y% %
& &
&
= = =
( )
2
4 4
4
t t
u vu y u
u
y
%
%
& &
&
× = = ' = =
u v
P
.
u v
Cons
%
=
2
t µ
k
* =C
L
2
u v u v
C Cµ µ
% %
= ' =
u v
- 0.3
P
(
µC =0.09
k-LµC
k-L
k-L
k-L
µC
µC
µC
µC =0.09

3-$•E:"= '[j? @k :G*$k := 7P"*$> :
Y%*%$L 01./" $[j%@,% &"*K
:.#%)<')< $% $ ,.T( 'XyZ[1B
%3 f0<.
4-P#* W< K(%y $*<0B#FB".S()%0<B
.* $#6F Q•( '<)%*<0B#.$S(=< "4 @
"*#B"6F Q'-L[?8%$ $".K?#%
Z[1y B=L V#**4 @*#W< J".0<
B.
5-D*%X3 f?88 ,"0<4•( '%'3$[
KL2 'B"? >:&E:N-*6 D& H<%*
3 'J.'$[$•( '( :<%$*4•( '( :<?%3 '
X*$**0<"'4( ' =%!
8-J LM!# &#
=<<W$•EBoussinesq Eddy-Viscosity?" B%o >' $NF
<)N?8/.%FN B,$ $@ T%B*$@L V( ' T" < &3
=.&& -D%034
".':;3=G ,
" BD%p @& -.
* '4•P <!363"=*.0J3=NG ,F $ ,
8%LSwirling$@• DRecirculating<&LG<.JB.
3"=G ,< &% &8N ?8B$%Z E%"! [w <E T<B..
<- B* Bk :o'Q$L W=%F &$%3=$)F3 "=
L 1< J..G/%3"Lr(%n8)K<0B#<r=[ Q
..
34
Over diffusive
µC
k-Lk-L
k-L
µC
µC
kL
µC
k-L
k-L
tµ
k-L

Y[-%% < :=<3=%%* f% WL lT
7..-K3J‡ :)
% (2% &Bo 'Q
% (L8+ ,
L% 8p @X $
2% &%$$% &J $
L$8- f$
*V% (N >X> :J T.
L['<% (X> :J T.
Z[13"B[%%$%Z[1=( ' $%L 1. J€
N&%%=$@(=% &PB.?( 3! & -.- ,.
9-O! P0 Q
'%&%o </=<$Y &&4)F%= 3G ,
_ -*..?= ( 3Œ N WBH $. # *#
)
= :< ( ' 4%
= 4) %$?<M2
SK .%= 3> / 3. 3e' = 3WQ@4I 8H *
Q . J( KW 3.L D[Qk. $ = 3
SK%. B b = 3.
%?QLŒ = 3SK . $ !%*#!( ' .4
! N W T * 4 %G(! ) =.
35
Realizable k-L Model
k-L
k-L
k-L
k-L
k-L
k-L
L
ji
i j t ij
j i
uu 2
-9u u µ - 9kE
x x 3
= +

)55(
' 0<o& -.B !.
)56(
E &( ' ƒf . !)56(&84. .)j .)(.($. +#
( ' .!K &: B p @ G0K /.'K &
)57(
B.G K#& = (8B .)j40 .^ B & -QM 2 L
/ &.+ &3$ XW 3%M # B36
)(@B & / W K.S(4
ND%?XW 3Œ N W 4)realizable(B <B0< !H<..3
=. B P J Œ N W.' L 7_ -M2 *'"H $N?B
. B P ; < * *.= 3%W.&%$= .)*# . <
? ) *= 3
f< & G.
< & G- f f.
V .J(%< & 9 ?' 8
%%8- f $< &
B.
9-1-! E #O! P0 Q
= :< ( ' $ = 3)58(%$ND.
36
Schwarz inequality
2 2
2
3
t
U
u k
x
=
2
t µ
k
µ =C 9
L
2
2 2
2
3
k U
u k C
x
µ=
2
u
1
3.7
3
k U
x Cµ
=
( )
2 2 2
S T S Tu u u u)
µC
k-LµCkL
k-L
k-L
kL

)58(
L2 ')58($=' <oB.Q $= 6)59(#..
)59(
.B= :< ( 'H($ . ?Q= *# 6L$ 0< <
B.
<?%N WŒ[ < K). . 3(B&0 E: K+%.'
& L ) U /<D B & 01 K+: J4.F@B 4f $.( D= ?
$ . ' <0 k : " U /B <B $.=%'
%< = .) $ . B ND NF^ 1 7-%< Bw
<%. * 8.
9-2--A& R%S! $ %0!%4 -
$ )?M2 <==. <L$ 0 34
' ? & / . V"$ . $ G r $%B)60(. B b.
( ) ( )
( ) ( ) 2
1 2 1 3
1 max 0.43, 2
5
j t
k b
j j t j
j t
b
kj j j
ij ij
u kk k
G G
t x x x
u k
C S C C G C
t x x x k
k
C S S S S
µ
µ
!
µ
µ
!
*
*
*
+ = + + +
+ = + + + +
+
+ ,
= = =- .
+/ 0
kGbGk-L
1
2
1.44
0.9
1
1.2
k
C
C
!
!
=
=
=
=
kk-L
k
krealizable-k-L
k-L
realizable-k-L
tµrealizable-k-Lk-LµC
kL

)60(
$ ; < G- f r$ .%B.$)61()B.
)61(
% ; B
= L2 '$< & % e.". % e * L2 ' )B # <% ; B U <D
.
% $ $:X$B '
% L$ :< $ -:$
% &:K< ( I$
2
0
1
2
t
s
ij ij ij ij
ij ij ijk k
ij ij ijk k
k
C
C
kU
A A
U S S
µ
µ
µ
1
1
&
&
=
=
+
= + 2 2
2 = 2
2 = 2
ijUkV0AsA
( )
0
1
3
4.04
6 cos
1
cos 6
3
1
2
s
ij jk ki
ij ij
ji
ij
j i
A
A
W
S S S
W
S
S S S
uu
S
x x
=
=
=
=
=
= +
kL
kL
0
k
n
=0
n
=

. 3? D > L )L [> "* = . % ; B L F@ kB ) 9 <.
<' 1 * J 0<CFDL% &$<-.: * $ O > " 3 . 8 ,
$XP3 % $ $ X 3 N ( w < . D P < $ B$.. < [> † & J
B )6 : "% &k % $ $ X % -$*% &-K<0B# L B *7/8 4 $
$# . = ' = >.
10-SU2V 0 !#
=X $ $ )<' K<0B# =B.< J P ) = 3%*2 % &
P G- f $ Œ B. V ! <D 4 *$ "% &<B % K 8f . : " B =
..* % | "= % & ?G *# &% &. - " % <8 . & ( B.
' $ $ N B z Q 3*!& <' 1 % & &* 8 *# . R 3 B
&.
. : $% &< = "* % ;:N - 7 T % &.9B @J
2 % = 3o 'Q B % &)2 $ ( ) $k / % &(?3 ' $ G/ % $ <B -
*N ' 0< ;: L[?8 ) T % $ . =[ Q r < - <0B# 8) %n ( r &
. V' " = % &cL 1 "J.
* [?8 3 + & = 3* $ 8- f % &G & % &Q $ X "p @ % &)^[j
2% &9 W %J $ % & J SJ $(L V N B = 3 * f ". K<0B# - * k E-.
*&%V=T % &G 2 > %% &. D ! T % (@ #
G o 'Q ? N ( *% &%= "G ,.N ( %$ .) = 3 -2
. 9 D X f f G- f.
kL
kL
kL
k-L
k-L
k-L

11-6(%) ! 3)4 -(
+ ,( G ( ' = K<0B# 4 [ = 337
(RSM)) = .$38
@B.
= % XG , ? K & "*PD % &$ + , G * % &% K 8f
.$ ".8 ; B 3 .E ; </ ( 0 % &* 4%n J <D B
B ) ) .W <0B# 8).O W ( G = :< ( ' " K Y >< L V % *
0< ( G *.
G ,3=% &K<0B#B0<•EEddy-Viscosity*0(% &G
(J%.**; <;).=RSM%•)&4%G
(4•( '( :<4< JB.+ fG(%4•('%" B
'8%(:<<%4•('%ND" B6B"3%3 'O W
X8%(4•('%6•( '( :<B.? ( D4•('%"+ f
! & /(=$( '%0<"! FL 736G(= P)62(
*3R E$G>O W)$:L 1TO W)B..
)62(
PW &##RSM
=% &Eddy-ViscosityL• 2%7<39
"? F40(G(
. &-".-&.3 f"=Eddy-Viscosity'%•0(6( TG
()3zL6 T(B.+ f*.( D 3yQY,
$Z[E1+ ," JKc0<Eddy-Viscosity)8%(J%
37
Reynolds stess model
38
Second order model
39
Attached boundary layer flows
k-L
k-L
2
2ij
ij i j
u u v u w
R u u v u v v w
w u w v w w
= = =
2
u

.**k ; <.( 1B$S(K*)<=% &Eddy-Viscosity
=..
•'=% &0RSM<=1968$/* ( $40
K80 <$$%)
. &(%8%(T#B.3=.3 $K%* + &Second Order Closure$
Second Moment Closure$Second Order Modeling<- BB.*#*'ŒŒ
=% &Eddy-Viscosity<B SJB$c0(&%= PG(‚ :<NDL2 '= :<
0)#8%(& H<%< $B(.#.E &)63(* 8" B
G4 (* :<41
B$3*. '.( D$'%ND3•( '= :<$
.( D'%ND6•( '= :<%3 'XG(*$**B.
)63(
G(%4*$'%G(%4*'%
<)(&!&%3 '= >9 :!&.( D$'%$!&.( D'%"$[S L2 '
ND4•( 'QK@B.S(=RSM=% &wEddy-Viscosity+ ,B"3? (
$J + ,3"=3=&6(Bfo 16<: W%K<0B#F
$$%o >X $$<L)<'B.3=%L Vy + ,
L.' )>$" J*#*L%k E-
$L%G- f42
$* $43
$LF P#44
L.45
B3=
(%#. W--..
40
Donaldson
41
Symetric
42
Swirl flows
43
Rotational flows
44
Free convection flows
45
Buoyant flows
2
2
ij
ij i j
u u v
R u u
v u v
= = =
2
2ij
ij i j
u u v u w
R u u v u v v w
w u w v w w
= = =

<?%P<*B*#.6 TL<0B#L BN?B213
$Tˆ$ $@
214
" B'$[? #:8%1""o( /01: " B4413
8%1<0B#K ?L$ 0N WD[%.** 8B• D< K(
*•<0B#$'%$%4• 01./)*3*•<0B#N W7(:8%1<0B#
L 164B.
)64(
3Y[<-B3y8%(* 8&<D4*•<0B#4 [)'
*• 2%<0B#$%4y./("0<=% &Eddy-Viscosity=% &( ' $%>
$ <$" T[† &J.WN W)‡-2L'%(= )& /
.B.30<=% &Eddy-Viscosity? F &LL BT4 ,$ $@46
$"!
LL B8- f47
$LB • VG48
^[11B.
12--0&!.(( E #X 8# 0!% 43)4(%) !
•( '( :<v 1%G(L )! PD$3= DyQ.#.
3c2( Kf. V$!)E &"!L%<%
y $&"4 xVS,%K<0B#$L2 'k*#!S,%*!B(
K&$%PD)$M ((B.‚ P&%prime$overbar%G•0(
$; <<%<0B#0<! F'
)65(
12-1-.(&% /
•( '(%
)66(
46
Anisotropic
47
Highly swirling flows
48
Stress diriven secondary flows
2
u2
vww
2 2
1 , 0.4 , 0.6u k v k w w k= = =
k-Lk-V
iB
u u u= +
0k
k
u
x
=

•( '; <
)67(
O 0$•( '66$67K ?
)68(
D[& B$.; <$; B!S,%… Q.
12-2-.(1% )2
•( '(%
)69(
•( '; <
)70(
6%•( ' y B<
)71(
,KO<8%B.
O 0•( ' $K ?
)a(
'•Mij& - 8 L 1.B !.
)b(
= D+ fL )N ?8"! &•( '( :<%8%(=RSM.#.
0k
k
u
x
=
0k
k
u
x
=
2
1i i i
k i
k i k k
u u p u
u B
t x x x x
+ = + +
2
1i i i i
ik k
k k i k k
u u u p u
u u B
t x x x x x
+ + = + +
1i i
ii k
i k k
Du p u
u u B
Dt x x x
+ = + +
D
Dt
2
1ii i i i i
ik k k k
k k k k i k k
u u u u u p u
u u u u B
t x x x x x x x
+ + + = + +
2
1j j j ji i
jk k k k
k k k k i k k
u u uu u u p
u u u u B
t x x x x x x x
+ + + = + +
j iu a+u b

)72(
H( L 1%'
( r
= :<b 0 ;+
=[ Q r-
= :<K<0B# G 8 N :< L V E+
=rH+= :<F P ;
( '72( ' GB0@ N* 8&'%G GB & = :<N:< (4
. 2 ( '.)(
12-3-E #3)4(%) !
)73(
*#
F PK<0B#•E $*; <49
(K<0B#•E $G; <50
(K<0B#•E $$%PD51
K< ) &38$G<0B#52
=[ QK<0B#53
49 Advection ( By Mean Flow )
50 Production ( By Mean Strain )
51 Production ( By Body Force )
52 Pressure-Strain Correlation
53 Dissipation
( )
( ) ( )
( ) ( )
j i i ji j i j k
k i j j k
k k k k
i j j ji i
i jk j ik i j j i
k k j i j i
u u uu u u u u u
u u u u u
t x x x x
u u u up p u u
u u u B u B
x x x x x x
+ + + + =
+ + + + + +
ij i jR u u=
ij i jR u u=
ij i jR u u=
ij i jR u u=
ij i jR u u=ij i jR u u=
2 2 2
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