This document discusses turbulence modeling in computational fluid dynamics (CFD). It contains three main points: 1. Turbulence models used in CFD simulations like RANS and LES are introduced. Important turbulence concepts such as eddies, length scales, and the energy cascade are explained. 2. Reynolds-averaged Navier-Stokes (RANS) equations are presented along with Reynolds stress tensor and turbulent heat flux terms. Common RANS turbulence models and their governing equations are outlined. 3. Large eddy simulation (LES) is described as an alternative to RANS. Filtering operations in LES to separate large and small scales are discussed. Root-mean-square velocities are presented as a

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23 Turbulent velocity scale
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div K div p E u u u e e u u E
t
u U u
v V v
w W w
µ µ+ = +
= +
= +
= +
= +
k
k
k
k
kk
vii
viik
K
vi
11 22 33 12 13 232 . 2 ( 2 2 2 )ij ij
e e e e e e e eµ µ= + + + + +
ije

19. )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.=$ <
= ( '% ? "% ? $& H< 3 L H 6) ) % & # Π3 < %$
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

20. )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

21. )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=

22. 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

23. )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% &
=

24. )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

25. 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

26. 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 ,
_ -*..?= ( 3ΠN WBH $. # *#
)
= :< ( ' 4%
= 4) %$?<M2
SK .%= 3> / 3. 3e' = 3WQ@4I 8H *
Q . J( KW 3.L D[Qk. $ = 3
SK%. B b = 3.
%?QLΠ= 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
= +

27. )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 3ΠN W 4)realizable(B <B0< !H<..3
=. B P J ΠN 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

28. )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

29. )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
=

30. . 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 $ ΠB. 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

31. 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

32. .**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
= = =

33. <?%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
=

34. •( '; <
)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

35. )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
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