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A FIRST-ORDER TURBULENCE MODEL FORA FIRST-ORDER TURBULENCE MODEL FOR
VISCOELASTIC FENE-P FLUIDSVISCOELASTIC FENE-P FLUIDS
F. T. PinhoCentro de Estudos de Fenómenos de Transporte, FEUP & Universidade do Minho, Portugal
C. F. LiDep. Energy, Environmental and Chemical Engineering, Washington University of St.
Louis, St Louis, MO, USA
Currently at School of Energy and Power Engineering, Zhenjiang, Jiangsu, P. R. China
B. A. YounisDep. Civil and Environmental Engineering, University of California, Davis, USA
R. SureshkumarDep. Energy, Environmental and Chemical Engineering, Washington University of St.
Louis, St Louis, MO, USA
4th Annual European Rheology Conference 2007
12th-14th April 2007
Napoli, Italy
Acknowledgments: FCT, Gulbenkian Foundation
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
10-3
10-2
10-1
103 104 105
0.125% PAA0.2% PAA0.25% CMC0.2% XG0.09%/0.09% CMC/XG
f
Re
Blasius eq.
Virk's MDRA asymptote
- Reduction of shear Reynolds stress (DR)- Increase of normal streamwise Reynolds stress- Dampening of normal radial and tangential Reynolds stress
0
10
20
30
40
50
60
100 101 102 103
0.125% PAA Re=429000.25% CMC Re= 166000.3% CMC Re=43000.09/0.09% CMC/XG Re=45300
u+
y+
Relevance: drag reduction in turbulent pipe flow
Deficit of Reynolds stress
2
Escudier et al JNNFM (1999)
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Time-average governing equations: turbulent flow & FENE-P
!Ui
!xi
= 0Continuity:
Momentum balance:
!"Ui
"t+!Uk
"Ui
"xk=#
"p
"xi+$s
"2Ui
"xk"xk#
"
"xk!uiuk( )+
"% ik,p"xk
! ij = 2"sSij + ! ij ,pRheological constitutive equation: FENE-P
! ij ,p ="p
#f Ckk( )Cij $ f L( )%ij
&'
()
f Ckk( )Cij
+ !"C
ij
"t+U
k
"Cij
"xk
# Cjk
"Ui
"xk
# Cik
"Uj
"xk
$
%&
'
() = f (L)*
ij
!Cij
!t+Uk
!Cij
!xk" C jk
!Ui
!xk" Cik
!U j
!xk
#
$%%
&
'((= Cij
)
= "* ij ,p+p
^- instantaneousOverbar or capital letter- time-average‘ or small letter- fluctuations
3
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
DNS, DR=18% (LDR)
We! = 25,Re! = 395
" = 0.9,L2= 900
2hu
1,x
2,y
We!="u
!
2
#0
Re!=hu
!
"0
DNS case: LDR
Fully-developed channel flow
4
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Reynolds decomposition of conformation tensor
! ij ,p ="p
#f Ckk( )Cij $ f L( )%ij&' ()+
"p
#f Ckk + ckk( )cij f 'cij
Time average polymeric stress
5
B = B+ b ' where b ' = 0
Negligible
f Ckk( ) =L2! 3
L2!Ckk
Function:
-0.035
-0.030
-0.025
-0.020
-0.015
-0.010
-0.0050
0.0
0.1 1 101
102
c'12
/[L2-(C
kk+c'
kk)]
C12
/[L2-(C
kk)]
CxyD
TR
(J)
ave
(+ s
ide)
y+
20 times smaller
f Ckk( )C12 >> f 'c12
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Cij
!
+ uk"cij"xk
# ckj"ui"xk
+ cik"u j
"xk
$
%&
'
() = #
* ij , p+p
!Cij
"
+! uk#cij#xk
$ ckj#ui#xk
+ cik#u j
#xk
%
&''
(
)**
+
,--
.
/00= $ f Ckk( )Cij $ f L( )1ij+, ./
CTij NLTij
6
Time-average conformation tensor equation
Mij
xx xy
-10000
-5000
0
5000
10000
15000
10-1 100 101 102
CTNLTMD
S= - !p
CT
(J)
aver
y+
xx component(a)
-1500
-1000
-500
0
500
1000
10-1 100 101 102
CTNLTMD
S = - !p
CT
(J)
aver
y+
xy component(b)
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy 7
yy
Polymer stress: DNS
! ij ,p"p
= #Cij
$
#uk%cij%xk
+ ckj%ui%xk
+ cik%u j
%xk
&
'((
)
*++
CTij
NLTij: originates in distortion of Oldroyd derivative- not negligible Must be modeled
: originates in advective term, negligible no need for modeling
DNS: Housiadas et al (2005), Li et al (2006) JNNFM
Mij
Exact
-400
-200
0
200
400
600
10-1 100 101 102
CTNLTMD
S= - !p
CT
(J)
aver
y+
yy component(c)
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Modeling the Reynolds stress
1) Reynolds stresses: Prandtl-Kolmogorov model
2) Dissipation of turbulent kinetic energy:
8
!uiu j = 2"T Sij !2
3k#ij
!T =Cµ fµk2
!"N
with
!N
2!s
"ui
"xm
""x
m
# Dui
Dt
$
%&
'
()+ 2!
s
"ui
"xm
""x
m
#uk
"Ui
"xk
$
%&
'
()+ 2!
s
"ui
"xm
""x
m
# "ui
uk
"xk
$
%&
'
()
+2!s
"ui
"xm
""x
m
"p '
"xi
#
$%
&
'() 2*!
s
2"u
i
"xm
""x
m
"2ui
"xk
2
#
$%
&
'() 2!
s
"ui
"xm
""x
m
"+ 'ik , p
"xk
#
$%
&
'(= 0
As for Newtonian fluids, the whole equation will be approximated
New term
Next slide
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.1 1 10 100
Pk
QN
QV
DN
!N
!V
Pk
y+
3) Turbulent kinetic energy: its transport equation
9
!Dk
Dt+ !uiuk
"Ui
"xk= #!ui
"k '
dxi#"p 'ui"xi
+$s
"2k
"xi"xi#$s
"ui"xk
"ui"xk
+"% ik , p
'ui
"xk# % ik , p
' "ui"xk
QV
!V
!"DN
-Need to model viscoelasticstress work
- Viscoelastic turbulent transport is small
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Viscoelastic stress work: !V
!V "1
#$ ik ,p' %ui
%xk=&p
#'Cik f Cmm + cmm( )
%ui%xk
+ cik f Cmm + cmm( )%ui%xk
(
)*
+
,-
Cik f Cmm + cmm( )!ui
!xk< cik f Cmm + cmm( )
!ui
!xkAssumptions & DNS:
!V "#p
$%f Cmm( )cik
&ui&xk
= C!V
#p
$%f Cmm( )
NLTnn
2
Needs modelC
!V = 1.076
(from DNS- next slide)
10
0
50
100
150
200
250
300
350
1 10 100
Ckk
TR
ME
AN
(J)
av
er
y+
ckk2
Ckk> c '
kk
2
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
-2500
-2000
-1500
-1000
-500
0
500
1000
0.1 1 101 102 103
!V (DNS)
-1.7*f(Ckk)*NLT
kk/2
EpsilonVkk/2
y+
(e)
Performance of the viscoelastic work model
!V> 0
Negative sign: different definitions
!V+Re
"0
( )2
versus
f Ckk( )NLTij*
11
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Key ideas:1) Write down exact equation- complex 4 lines long2) Assumptions, physical insight, trial-and-error3) Do a priori testing of each term4) Select appropriate combination and dimensional homogeneity5) Test in code and modify- under investigation
f Cmm( )NLTij
!= function Sij ,Wij ,Cij ,"ij
N,#uiu j
#xk,#Cij
#xk,#NLTij#xn
,Mij ,uiu j
$
%&&
'
())
f Cmm( )NLTij
!= fµ
1
CE3
u"2
#0
2Ckkuiu j +
C$14
#0
uiukWknCnj +u jukWknCni +ukuiWjnCnk( )%
&'
(
)*
Modeling NLTij
12
uium!ckj
!xm+ uium
!Ckj
!xm" Coef # uium
!Ckj
!xm
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
CE3= 0.00035;C
!14= 0.00015 fµ1 = 1! exp ! y
+ 26.5( )( )2
Model for NLTij
-1000
-500
0
500
1000
1500
2000
2500
3000
0.1 1 10 100 1000
f(Ckk)*NLTxy*f(Ckk)*NLTii*Pred f(Ckk)*NLTxy*Pred f(Ckk)*NLTii*f(Ckk)*NLTyy*Pred f(Ckk)*NLTyy*
f(Ckk)*NLTxy
y+
13
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Viscoelastic turbulent transport: QV
QV !
"# ik ,p'ui
"xk=$p
%"
"xkCik f Cmm + cmm( )ui + cik f Cmm + cmm( )ui&'
()
QV=!p
"#
#xkf Cmm( )CUiik
$% &'
Weak coupling between ckk and cij, ui
Ckk> c '
kk
2
f Cmm( ) =L2! 3
L2! Cmm + cmm( )
Cik f Cmm + cmm( )ui < cik f Cmm + cmm( )ui
Neglect of this term is irrelevant because non-neglected term is modeled
Needs model CUijk( )General case
14
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
f Cmm( )CUijk
!= fµ
2
"C#1
uium$Ckj
$xm+ u jum
$Cik
$xm
%
&'
(
)*"
C#7
!f Cmm( ) ± u j
2Cik ± ui
2Cjk
+,-
./0
+
,-
.
/0
C!1=1.3;C!7
= 0.37
fµ2 =1! exp !y+
26.5
"
#$
%
&'
Model for CUijk
-20
-10
0
10
20
30
40
0 1 10 100
QFii2
Pred QFii2
QFii2
y+
15
- Same modelling approach as with NLTij
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
d
dy!s
dU
dy+ " p,xy #$uv
%
&'
(
)*#
dp
dx= 0Momentum:
! xy,p ="p
#f Ckk( )Cxy
f Ckk( )Cxy = !Cyy
dU
dy+!NLTxy
f Ckk( )Cxx = 2!Cxy
dU
dy+ !NLTxx +1
f Ckk( )Cyy = !NLTyy +1
f Ckk( )Czz = !NLTzz +1
f Ckk( ) =L2! 3
L2! Cxx +Cyy +Czz( )
!"uv = "#TdU
dy
!T = Cµ fµk2
!"N
with
Reynolds stress:
Final equations: low Re k-! type model for channel flow
16
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
0 =d
dy!s +
"#T$ k
%
&'
(
)*dk
dy
+
,-
.
/0+Pk 1" !2
N 1"D+!p
d
dy
f Cmm( )3
CUnny
2
+
,-
.
/01!p
f Cmm( )3
NLTnn
2
!N= !!
N+D
N DN= 2!s
d k
dy
"
#$
%
&'2
0 =d
dy!s +
"#T$%
&
'(
)
*+d !% N
dy
,
-.
/
01+" f1C%
1
!% N
k
Pk
"2" f
2C%
2
%N2
k+"E +E3 p
E =!s
"#T 1$ fµ( )
d2U
dy2
%
&'
(
)*2
f1=1
f2 =1! 0.3exp !RT2( )
fµ = 1! exp!y+
26.5
"
#$
%
&'
(
)*
+
,-2
k and ! transport equations: modified Nagano & Hishida
17
based on Newtonian model of Nagano & Hishida (1984)
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
0
5
10
15
20
25
30
1 10 100
u+
la
m
y+
u+
Predictions 1: Re!0= 395; We!0= 25; "=0.9, L2=900
X DNSBlack: Newtonian ! = !
s+!
p
! = !wall
! = !wall ;same !Q;Re"0
= 443
FENE-P simulationsC
!14=1.5 "10
#4;C
E3= 2.98 "10
#4;
Newtonian log law
Virk’s asymptote
C!V = 1.076
18
C!1 =1.3;C!7 = 0.37
C!1 = 0.65;C!7 = 0.18
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Predictions 2: Re!0= 395; We!0= 25; "=0.9, L2=900
-20
0
20
40
60
80
0.1 1 10 100
CUii2
+
y+
Model fitted to DNS
Modified with simulations
19
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
k+
0
1
2
3
4
5
0.1 1 10 100
k+
y+
NLTii
*
-500
0
500
1000
1500
2000
2500
0.1 1 10 100
NLTii*
y+
Predictions 3: Re!0= 395; We!0= 25; "=0.9, L2=900
20
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
NLTyy
*
0
100
200
300
400
500
0.1 1 10 100
NLTyy*
y+
Predictions 4: Re!0= 395; We!0= 25; "=0.9, L2=900
0
5
10
15
20
25
0.1 1 10 100
Cyy
y+
Cyy
21
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Re! = 395;C"1= 0.65;C"7
= 0.18;C#14=1.5 $10%4;C
E3= 2.98 $10%4
0
5
10
15
20
25
1 10 100
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
u+
y+
u+
22
Effect of Weissenberg 1: Re!0= 395; "=0.9, L2=900
k+
0
1
2
3
4
5
0.1 1 10 100
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
k+
y+
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Effect of Weissenberg 2: Re!0= 395; "=0.9, L2=900
23
CUii2
+NLTii
*
-500
0
500
1000
1500
2000
2500
0.1 1 10 100
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
NLTii
*
y+
-10
0
10
20
30
0.1 1 10 100
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
CUii2
+
y+
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Effect of Weissenberg 3: Re!0= 395; "=0.9, L2=900
24
NLTyy
*Cyy
0
5
10
15
20
25
30
35
0.1 1 10 100
DNSNewtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
Cyy
y+
0
100
200
300
400
500
600
0.1 1 10 100
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
NLTyy
*
y+
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Effect of Weissenberg 4: Re!0= 395; "=0.9, L2=900
25
0
100
200
300
400
500
0.1 1 10 100
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
Cxx
y+
Cxx
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
DNS
Newtonian
We!=0
We!=12.5
We!=15
We!=20
We!=25
We!=27.5
u'v'+
y*
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Conclusions
- Developed simplified k-! model: code and closures are working
- Viscoelastic turbulent transport not very important at DR= 18%
- Viscoelastic turbulent transport is reasonably well modeled
- Viscoelastic stress power well modeled by NLTij
- NLTij required for Cij and !ijV
- NLTij closure needs significant improvement
- Need to model viscoelastic turbulence production close to wall
- Deficiencies also related to Newtonian part of model
- Isotropic turbulence does not allow a good model
- Need to consider anisotropic turbulence: anisotr. k-! and RSM
26
A first-order turbulence model for viscoelastic FENE-P fluids
4th AERC- 2007, Napoli, Italy
Conclusions: Models for other constitutive equations
! ij = 2"sSij +"p
#f1Ckk ,L,…( )Cij $ f2 Ckk ,L,…( )%ij&' ()
Constitutive equations can be rewritten as a function of the conformation tensor
Different functions are used
Turbulence models can be modifications of turbulence modelsdeveloped for FENE-P
27
Dr.
Pas
quin
oR
ossa
naP
.le T
ecch
io 8
0801
25 N
apol
ir.p
asqu
ino@
unin
a.it
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e P
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arie
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