A TURBULENCE CLOSURE FOR VISCOELASTIC A TURBULENCE CLOSURE FOR VISCOELASTIC FLUIDS BASED ON THE FENE-P MODEL FLUIDS BASED ON THE FENE-P MODEL F. T. Pinho Centro de Estudos de Fenómenos de Transporte, FEUP & Universidade do Minho, Portugal P. R. Resende Centro de Estudos de Fenómenos de Transporte, FEUP, Portugal C. F. Li Dep. 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 R. Sureshkumar Dep. Energy, Environmental and Chemical Engineering, Washington University of St. Louis, St Louis, MO, USA B. A. Younis Dep. Civil and Environmental Engineering, University of California, Davis, USA XV th International Workshop on Numerical Methods for Non-Newtonian Flows 6 th -10 th June 2007 Rhodes, Greece
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A TURBULENCE CLOSURE FOR VISCOELASTICA TURBULENCE CLOSURE FOR VISCOELASTIC
FLUIDS BASED ON THE FENE-P MODELFLUIDS BASED ON THE FENE-P MODEL
F. T. PinhoCentro de Estudos de Fenómenos de Transporte, FEUP & Universidade do Minho, Portugal
P. R. ResendeCentro de Estudos de Fenómenos de Transporte, FEUP, 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
R. SureshkumarDep. Energy, Environmental and Chemical Engineering, Washington University of St.
Louis, St Louis, MO, USA
B. A. YounisDep. Civil and Environmental Engineering, University of California, Davis, USA
XVth International Workshop on Numerical Methods for Non-Newtonian Flows
6th-10th June 2007
Rhodes, Greece
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
We! = 25,Re! = 395
" = 0.9,L2= 900
2hu
1,x
2,y
We!="u
!
2
#0
Re!=hu
!
"0
DNS case: LDR- DR= 18%
Fully-developed channel flow
2
B = B+ b ' where b ' = 0
Previous eddy viscosity model- Li et al (2006a) JNNFM v139, 177-189Li et al (2006b) JNNFM v140, 23-40
Remembering Reynolds decomposition
^- instantaneousOverbar or capital letter- time-average‘ or small letter- fluctuations
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
! 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
3
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
-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)]
Cx
yD
TR
(J)
ave
(+ s
ide)
y+
20 times smaller
f Ckk( )C12 >> f 'c12
Time- average polymer stress
! ij ,p ="p
#f Ckk( )Cij $ f L( )%ij&' ()+
"p
#f Ckk + ckk( )cij
f 'cij ! 0
4
! ij ,p ="p
#f Ckk( )Cij $ f L( )%ij&' ()+
"p
#f Ckk + ckk( )cij
? Cij ?
What about
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece 5
!Cij
"
+ ! uk#cij#xk
$ ckj#ui#xk
+ cik#u j
#xk
%
&'
(
)*
+
,--
.
/00= $ f Ckk( )Cij $ f L( )1 ij+, ./
Cij
!
+ uk"cij"xk
# ckj"ui"xk
+ cik"u j
"xk
$
%&
'
() = #
* ij ,p+p
Mij CTij NLT
ij
: turbulent distortion originates in distortion of Oldroyd derivative- not negligible Must be modeled
: originates in advective term, negligible no need for modeling
Exactand large
DNS: Housiadas et al (2005) Phys Fluids 17, 35106, Li et al (2006 a) JNNFM
Time-average equilibrium equation for the conformation
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
Modeling the Reynolds stress
1) Reynolds stresses: Prandtl-Kolmogorov model (k-! closure)
2) Dissipation of turbulent kinetic energy:
6
!uiu j = 2"T Sij !2
3k#ij
!T =Cµ fµk2
!"N+"
Vwith
!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
#
$%
&
'(= 0
As for Newtonian fluids, most terms will be approximated
New term(will be considered in the future)
Next slide
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.1 1 10 100
Pk
QN
QV
DN
!N
!V
Pk
y+
3) Turbulent kinetic energy: its transport equation
7
!Dk
Dt= "!uiuk
#Ui
#xk" !ui
#k '
dxi"#p 'ui#xi
+$s
#2k
#xi#xi"$s
#ui#xk
#ui#xk
+#% ik , p
'ui
#xk" % ik , p
' #ui#xk
QV
!V!"D
N
-Need to model viscoelasticstress work
- Viscoelastic turbulent transport is small
QNP
k0
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
-500
0
500
1000
1500
0.1 1 10 100
Cik f' !u'
i/!x
k
c'ik f' !u'
i/!x
k
SUM
y+
Assumptions for viscoelastic stress work: !V
!V "1
#$ ik ,p' %ui
%xk=&p
#'Cik f Cmm + cmm( )
%ui%xk
+ cik f Cmm + cmm( )%ui%xk
(
)*
+
,-
8
Cik f Cmm + cmm( )!ui
!xk<< cik f Cmm + cmm( )
!ui
!xk
Except in viscous sublayer and buffer, but here is not important
!V
f 'c 'ik!ui
!xk" f Cmm( )cik
!ui
!xk
Also
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece 9
!V "#p
$%f Cmm( )cik
&ui&xk
= C!V
#p
$%f Cmm( )
NLTnn
2
Needs model
C!V = 1.076
Viscoelastic stress work model
!V+Re
"0
( )2
versus
f Ckk( )NLTij*
-500
0
500
1000
1500
2000
2500
0.1 1 101 102
!V
+ Re
"0
2
1.076*f(Cmm)*NLT
nn
+/2
EpsilonVkk/2
y+
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
Key ideas for both models:1) Exact equation2) Assumptions, physical insight and a priori testing of each term3) Test in code and modify values of coefficients
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$14
#0
uiukWknCnj + u jukWknCni + ukuiWjnCnk( )%
&'
(
)*
Modeling NLTij-- first model
10
CE3= 0.00035;C
!14= 0.00015 fµ1 = 1! exp ! y
+ 26.5( )( )2
First model
2 coefficients 1 damping function
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece
Modeling NLTij-- second model
f Cmm( )NLTij
!=f Cmm( )
!CN1
u"2
#$ s
Ckk f Cmm( )%CN2Ckj
&Ui
&xk+Cik
&Uj
&xk
'
()
*
+,
-./
012
+CN3
Ckn
$02SpqSpq
uium&Uj
&xk
&Um
&xn+ u jum
&Ui
&xk
&Um
&xn
'
()
*
+,
% fN1CN4Cjn
&Uk
&xn
&Ui
&xk+Cin
&Uk
&xn
&Uj
&xk+Ckn
&Uj
&xn
&Ui
&xk+&Ui
&xn
&Uj
&xk
345
678
'
()
*
+,
+CN5fN2
4
15
9 N
#$ s
Cmm: ij
CN1= 0.0164;C
N2= 0.32;C
N3= 0.024;C
N4= 1.11;C
N5= 0.045
fN1 = 1! 0.8exp !y+
30
"#$
%&'
(
)*
+
,-
2
fN2 = 1! 0.8exp !y+
25
"#$
%&'
(
)*
+
,-
4
11
uium!ckj
!xm+ uium
!Ckj
!xm" Coef # uium
!Ckj
!xm
4 coefficients:
2 damping functions:
A turbulence closure for viscoelastic fluids based on the FENE-P model
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto XVth IWNMNNF 2007, Rhodes, Greece