NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Chapter 2 Turbulent Flow
§ 1 Basic Concept of Turbulence
1,Classification
wall turbulence - friction of wall
free turbulence - interaction of fluid layers
homogeneous turbulence
isotropic turbulence
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CC
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Wuhan 430074,P,R,CHINA
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2,Turbulence average
1) time average
?????
T
Tt
dttx
T
x
0
00 ),(
1lim)( ?? (2-1)
2) space average
????
x
xs
dxtx
x
t
0
00 ),(
1l i m)( ?? (2-2)
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CC
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Wuhan 430074,P,R,CHINA
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3) Probability Density Function
?
?
??
? ???? dPtxtxe )(),(),( 0000
4) Reynolds average
for turbulent quantity ?
??? ??'
where ?? transient value
? ? average value
?’? pulsing value
(2-4)
(2-3)
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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0)( ????? ?????
(2-5)
Assume
x
A
x
A
x
A
x
A
BABABBAAAB
BABA
BABBAABA
BBB
AAA
?
?
?
?
?
?
?
?
?
?
?
?????
??
???????
??
??
'
'')')('(
0''
''
'
'
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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In general,0 ' '? B A Correlation quantity of
turbulence
The above formulae indicate that the average of linear
calculation of turbulent quantity equals linear
calculation of corresponding average of turbulent
values,the average of non-linear calculation of
turbulent quantity equals the sum of non-linear
calculation of corresponding average of turbulent
values and pulsing correlation of these quantities.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Reynolds time average equations
with Reynolds average and assume:
? ? ?
ii
xx ?
?
??
?
?
?
?
?
??
????
??
??
(
0'
?
(2-6)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
?
?
???
?
????? Sv
xxx
v
xt jjjjjj
?
?
??
?
?
?
??
?
??
?
? )'()()()( '
(2-7)
rs
i
j
j
i
i
si
qw
x
v
x
g
x
p
S
hYv
??
?
?
?
?
??
?
?
??
?
,),(,0
,,,1
??
?
?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Note,1) in equation (2-7),the variables are averaged
values
2) the new term comes from the Reynolds average of
non-linear convection term,termed turbulent transport
flux of φ,Just because of this term,the number of un-
known variables is more than the independent equations,
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 2 Turbulent Viscouse Coefficient Models
1,Concept of turbulent viscousity
For two-dimensional boundary layer,
Boussinnesq(1877) proposed
y
uvu
tt ?
???? ??? '' (2-8)
?t - Reynolds stress
u - fluid velocity in main stream
?t - turbulent viscosity coefficient
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
In general,
)(''
3
2
)(
''
j
t
j
ij
j
i
i
j
tij
x
v
k
x
v
x
v
vv
?
?
??
?
?
?
?
?
?
??
?
?
?
??
????
?
),( hY s???
Where k - time averaged value of turbulent kinetic
energy of unit volume of fluid
)(21 2'ivk ? (2-9)
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CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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So,Eq,(2-7) becomes
??
????? S
xx
v
xt jjjj
?
?
??
?
??
?
??
?
? )()()( (2-10)
where
t
t
t ?
? ?
?
?
? ??? - generalized transport
coefficient
?? t- turbulent Prandtl number
?e = ? ? ?t = ??? ? ?t) - effective viscous coefficient
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Note,Turbulent viscous coefficient is only a
assumption,its reliability and applicability
should be proved,
Discussion:
molecular transport,2'v - termed movement
Average free movement distance
Turbulent transportation,k
l
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From dimensional analysis
lkCt 2/1?? ??
(2-10)
For turbulent boundary layer,from similarity of laminar
and turbulent flows,
Prandtl (1925) assumed
y
u
lu
ul
m
mt
?
?
?
?
'
'??
Lm,mixing
length
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
So,
y
ul
mt ?
?? 2?? (2-12)
In general,
j
i
i
j
mt x
v
x
v
l
?
??
?
?
? 2??
(2-13)
or
jj
i
i
jm
i
j
i
i
j
j
i
i
j
mji
xx
v
x
vl
v
x
v
x
v
x
v
x
v
lvv
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
??
?
?
?
??
??
?
2
''
2''
)(
(2-14)
?? ? Ys,h)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
1) Free shear flow
)()( xyxl Em ?? (2-15)
2) Developed turbulent tube flow
42 )/1(06.0)/1(08.014.0/ RyRyRl m ?????
(2-16)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Boundary layer near solid wall
Viscous sublayer
Inertial sublayer
Shear layer
In inertia sublayer
30//
130/Re
??
??
? ???
?
wyy
yu
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Beyond viscous sublayer,
Escudier(1965):
when
?????
????
??
??
m
m
ly
yly
,//
,// (2-17)
y - distance to the wall
? - thickness of boundary layer
? ? 0.41,? ? 0.09
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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In viscous sublayer,
26
)e x p (1
2/12/1
2
?
?
?
?
?
?
?
???
?
?
??
A
A
y
yl
y
u
l
w
m
me f f
?
??
?
???
(2-18)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Discussion on mix length model
Application,1) Boundary layer
2) Jet
Limitation,1)
y
u
t ?
???
2) simple cases.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2,Single equation model
Kolmogorov (1942),Prandtl(1945):
1)
lkCt 2/1?? ??
2) experiment correlation for l
3) differential equation for k
(2-19)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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N-S equations for vi,vj
? ?
? ?
k
jk
j
j
jk
k
j
k
ik
i
i
ik
k
i
x
Tg
x
p
vv
x
v
t
x
Tg
x
p
vv
x
v
t
?
?
???
?
?
??
?
?
?
?
?
?
?
???
?
?
??
?
?
?
?
?
?
????
?
????
)(
)( (2-20)
(2-21)
Where
Tg i ???
Buoyancy force
pT )(
1
?
?? ?
??
Volume expansion
coefficient
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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1) vj × eq.(2-20) + vi× eq.(2-21)→eq,for v i vj
2)vj × eq,for vi + vi × eq,for vj→eq,for v i vj
3)eq,for vivj - eq,for vi vi→eq,for v i’vj’
4)i = j,times 1/2,→ eq,for k
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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)''2/''()()( 2
k
kik
k
k
k x
kvpvv
x
kv
x
k
t ?
???
?
???
?
??
?
? ????
convection diffusion
2)'(''''
k
i
iiki x
v
Tvgvv
?
?
??? ????
Production by
shear force
Production by
buoyancy dissipation
(2-22)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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Modeling ? ?
kT
t
kii
k
i
k
i
i
k
t
k
i
ki
kk
e
kk
t
k
ik
x
T
gTvg
x
v
x
v
x
v
x
v
vv
x
k
x
k
x
k
vpv
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
???
?
?
???
??
?
?
?
??
???
''
)(''
2//''
2
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
From dimensional analysis,define
lkCC
x
v
lklklklk
C
x
v
DD
k
i
t
D
k
i
/)
'
(
///
)
'
(
2/32
2/322/12
2
????
?????
???
????
?
?
?
????
?
?
?
?
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
So,
lkCGGxkxkvxkt Dbk
kk
e
k
k
k
2/3)()()( ?
?
??? ???
?
?
?
??
?
??
?
?
(2-23)
where
kT
t
kb
k
i
k
i
k
i
tk
t
te
x
T
gG
x
v
x
v
x
v
G
lkC
?
?
??
?
?
?
?
?
?
?
?
?
??
?
?
?
?
??
???
?
)(
2/1
CD,C?,l from experiments
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3,Two-equation model (k-??
from ?t = C??k1/2l and ? = CDk3/2/l
?t = C?CD?k2/?
.)( R e2.)(2 eqy n o l d s
x
veqSN
xx
v
k
k
kk
k ??
?
????
?
??
?
? ??
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
)()''()()(
kk
k
k
k
k xx
vxvxt ??? ??? ???? ???? ????????
2
2
)'(2'''2
lk
i
l
k
l
j
k
i
xx
v
x
v
x
v
x
v
??
??
?
?
?
?
?
?? ??
Turbulent
diffusion
Molecular diffusion
Production dissipation
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
For turbulent diffusion,
k
t
k xv ?
??? ?
?
???
?
''
From dimensional analysis
)( 21 ???? CGCkS k ??
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
So,
)()()()( 21 ??????????
?
CGCkxxvxt k
k
e
k
k
k
????? ??? ????
(2-25)
Constants in k-? model
C?CD ?k ?? C1 C2
0.09 1.0 1.3 1.44 1.92
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Application succeeded:
(1) plane jet;
(2) boundary layer alone plane plate;
(3) tube and channel flow;
(4) 2D-3D re-circulating flow
Failed cases:
(1) vortex flow;
(2) buoyancy flow;
(3) gravity layer flow;
(4) boundary layer along curve surface;
(5) low Reynolds flow.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
§ 3 Reynolds Stress Model (RSM)
Shortcoming of k-? model,k,l(?? are scale
quantities and isotropic.
1,Differential Stress Model (DSM)
1) Reynolds Stress Differential Equation
similar to equation for k
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
?? ???? )''()''( jik
k
ji vvvxvvt ??
)]''()''('''[ ji
k
jkjkji
k
vvxvpvvvx ? ???? ??? ???
)''''(
k
i
kj
k
j
ki x
vvv
x
vvv
?
??
?
?? ?
)''(')''(2)''''(
i
j
j
i
k
j
k
i
ijji x
v
x
vp
x
v
x
vTvgTvg
?
??
?
??
?
?
?
???? ???
Dij
Pij
Gij ?ij pij
(2-27)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Dij - diffusion term;
Pij - shear production term;
Gij - buoyancy production term;
?ij - dissipation terms;
Pij - pressure transform term.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2) Second Order Closure
Basic principles:
a) consider physical meaning of each term;
b) dimensional analysis;
c) the term simulated keeps same properties with
original one;
d) three order correlation simulated with gradient ;
e) isotropic dissipation.
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Pij,Gij need not simulated.
Turbulent diffusion (Dalay-Harlow model)
)''(''''''''' ji
l
lksjkiikjkji vvxvv
kCvpvpvvv
?
?????
?????
Dissipation term
???? ij
k
j
k
i
x
v
x
v
3
2)''(2 ?
?
?
?
?
Pressure transport term (Launder-Rotta model)
Pij = Pij,1 + Pij,2 + Pij,3
Pij,1 = -C1(?/k)?(v’iv’j-(2/3)?ijk)
Pij,2 = -C2(Pij-(2/3)?ijGk)
Pij,3 = - C3(Gij-(2/3)?ijGb)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
3) Reynolds stress flux transport equation
Neglect molecular transport
ijijijijij
kijijijji
ji
l
lks
k
ji
k
ji
GPkGC
GPCkvv
k
C
vv
x
vv
k
C
x
vv
x
vv
t
?????
????
?
?
?
?
?
?
?
?
?
?
????
???
?
?
???
3
2
)
3
2
(
)
3
2
()
3
2
''(
)]''(''[)''()''(
3
21
(2-28)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
?????? ?????? ??? ???? bk
l
lks
k
k
k
GGxkvvkCxkvxkt )''()()(
(2-28’)
eq,k is not an independent one,∵ k=1/2(v’i2)
k
CRCGG
k
C
x
vv
k
C
x
v
xt
cfbk
l
lk
k
k
k
2
321 )1)((
)''()()(
?
?
?
?
?
?????
??
?
????
?
?
?
?
?
?
?
?
?
?
(2-29)
Where Rf =-Gb/Gk,Richardson number
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
2
32
1
2
'''
''')''''(
)''('')''()''(
TgC
x
v
TvC
Tv
k
CTg
x
v
Tv
x
T
vv
Tv
x
vv
k
C
x
Tv
x
Tv
t
iT
k
i
kT
iTi
k
i
k
k
ki
i
k
lksT
k
k
k
i
???
?
?
???
?
???
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(2-30)
k
T
Rx
T
Tv
x
T
vv
k
C
x
Tv
x
T
t
k
k
l
lkT
k
k
k
?
?
???
2
2
22
'
1
''2
)
'
''()'()'(
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(2-31)
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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Constants in Reynolds Stress Transport Equation
Model
Cs C1 C2 C3 C1T C2T
0.24 2.2 0.55 0.55 3.0 0.5
C3T R CsT Cc CT C?1 C?2 C?3
0.5 0.8 0.11 0.15 0.13 1.44 1.92 0.8
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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Discussion on DSM
1) Application or advantage:
Fluid flow with vortex,buoyancy,wall effect,etc.
2) Shortcomings:
- complex,11 equations for turbulence;
- 14 constants difficult to determine
- boundary conditions are difficult to be determined
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
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4) Algebraic Stress Model (ASM) (Launder & Rodi)
Basic consideration
- Eq,for stress →Algebraic form
- Keep non-isotropic properties of turbulence.
There are different ways of approximation.
ASM is also called extended k-? model
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
?????? ?????? ??? ???? bk
l
lks
k
k
k
GGxkvvkCxkvxkt )''()()(
k
CRCGG
k
C
x
vv
k
C
x
v
xt
cfbk
l
lk
k
k
k
2
321 )1)((
)''()()(
?
?
?
?
?
?????
??
?
????
?
?
?
?
?
?
?
?
?
?
k
k
iT
k
i
kT
k
ki
T
i
jjii
k
j
kiijji
x
T
Tv
k
RT
TgC
x
v
TvC
x
T
vv
C
k
Tv
TvgTvg
x
v
vv
k
kvv
?
?
??
??
?
?
??
?
?
?
??
?
?
???
''2'
]')1('')1(''[''
''''''()1(
3
2
''
2
2
32
1
?
?
?
??
?
???
NL
CC
National Lab,of Coal Combustion,HUST
Wuhan 430074,P,R,CHINA
E-mail,pcbnlcc@hust.edu.cn
Developing Turbulence Models
Direct Simulation
1) Full Turbulence Simulation FTS
2) Large Eddy Simulation LES
3) Small Eddy Simulation SES
Two-fluid Model