RECENT DEVELOPMENTS AND CHALLENGES IN TURBULENCE …fpinho/pdfs/BSR2008_Turbulence_short4.pdf ·...
Transcript of RECENT DEVELOPMENTS AND CHALLENGES IN TURBULENCE …fpinho/pdfs/BSR2008_Turbulence_short4.pdf ·...
RECENT DEVELOPMENTS AND CHALLENGES INRECENT DEVELOPMENTS AND CHALLENGES IN
TURBULENCE MODELING FOR VISCOELASTICTURBULENCE MODELING FOR VISCOELASTIC
FLUIDSFLUIDS
P. R. ResendeCentro de Estudos de Fenómenos de Transporte, Universidade do Porto, Portugal
F. T. PinhoCentro de Estudos de Fenómenos de Transporte, Universidade do Porto & Universidade doMinho, Portugal
K. KimKorea Institute of Energy Research,Daejeon, Republic of Korea
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
Conference on “Complex Flows of Complex Fluids”17th-19th March 2008University of Liverpool, United Kingdom
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Summary
2
•Brief review of existing RANS models for FENE-P
•Governing equations for FENE-P in RANS/RACE form
(Reynolds decomposition)
•Development of closures for RANS/RACE of FENE-P(2007 (LDR) and 2008 (LDR & HDR) closures)
•Some results (2007 models only)
•Conclusions and future prospects
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Governing equations: turbulent flow & FENE-P
B = B+ b ' where b ' = 0Reynolds decomposition
^- instantaneous quantitiesOverbar or upper-case letters- time-averaged quantities‘ or lower-case letters- fluctuating quantities
3
!Ui
!xi
= 0Continuity (incompressible):
Momentum : !"Ui
"t+ !Uk
"Ui
"xk= #
"p
"xi+$s
"2Ui
"xk"xk+"% ik,p"xk
! ij = 2"sSij + ! ij ,pRheological constitutive equation: FENE-P
! ij ,p ="p
#f Ckk( )Cij $ f L( )% ij&
'()
!"C
ij
"t+ U
k
"Cij
"xk
# Cjk
"Ui
"xk
# Cik
"Uj
"xk
$
%&'
()= # f C
kk( )Cij# f L( )*
ij+, -.
f L( ) =1f Ckk( ) =L2! 3
L2! Ckk
L Cij( ) :
M Uij( )
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Solution of governing equations
4
Direct Numerical Simul.
Sureshkumar, Beris, Handler (1997) PoF, v9, 743Den Toonder, Hulsen, Kuiken, Nieuwstadt (1997) JFM 337, 193Dimitropoulos, Sureshkumar, Beris (1998) JNNFM v79, 433Dimitropoulos, Sureshkumar, Beris, Handler (2001) PoF v13, 1016Angelis, Casciola, Piva (2002) Comput. Physics v31, 495Ptasinski et al (2003) JFM v490, 251Housiadas, Beris (2003) PoF v15, 2369 Zhou, Akhavan (2003) JNNFM v109, 115Stone, Graham (2003) PoF v15, 1247Yu, Kawaguchi (2003) IJHFF v24, 491Vaithianathan, Collins (2003) J. Comput Physics v187, 1Housiadas, Beris (2004) PoF v16, 1581Dubief et al (2004) JFM v514, 271Yu, Li, Kawaguchi (2004) IJHFF v25, 961Dimitropoulos et al (2005) PoF v17, 011705Li, Gupta, Sureshkumar, Khomami (2006) JNNFM v139, 177Li, Sureshkumar, Khomami (2006) JNNFM v140, 23Benzi, Angelis, L’vov, Procaccia, Tiberkevich (2006) JFM v551, 185& others — see recent reviewWhite, Mungal (2008) Ann. Rev Fluid Mech (2008) v40, 235
Physical understanding
Turbulence model development
Too costly for engineering calculations
LES
RANS/RACE
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Time-averaged governing equations: RANS and RACE
5
Closuresrequired
!Ui
!xi
= 0Continuity:
Momentum balance:
!"Ui
"t+ !Uk
"Ui
"xk= #
"p
"xi+$s
"2Ui
"xk"xk#
"
"xk!uiuk( ) +
"% ik,p"xk
! ij , p ="p
#f Ckk( )Cij $ f L( )% ij&' () +
"p
#f Ckk + ckk( )cij
Cij
!
+ uk"cij"xk
# ckj"ui"xk
+ cik"u j
"xk
$
%&
'
() = #
* ij ,p+p
Rheological constitutive equation: FENE-P
Mij CTij NLTij
RACE
! ij = 2"sSij + ! ij ,p
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Existing models for FENE-P: Li et al (2006)
6
DR = 80 1! exp !0.025 We" ! 6.25( )Re"
125
#$%
&'(!0.225)
*+
,
-.
/01
21
341
511! exp !0.0275L( ))* ,-
Reynolds stress
!"Ui
"t+ !Uk
"Ui
"xk= #
"p
"xi+$s
"2Ui
"xk"xk#
"
"xk!uiuk( ) +
"% ik,p"xk
0 equation model(shear stress only)
!uv = "T ,vdU
dy
!T ,v
= "!T ,N
!T ,N
="u#y
! = a DR( )y + b DR( )"# $%
! xy,pdy0
Re!
" = Mxydy0
Re!
" + NLTxydy0
Re!
"IMxy
= a '+ b 'DR + c 'DR2
INLTxy = a"+ b"DR + c"DR2
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Existing models for FENE-P
FENE-P and based on DNS
Leighton, Walker and Stephens (2002) APS meeting ?
- Reynolds stress transport model- Slow pressure-strain redistribution term is modified by polymer (limits energy redistribution)- New term in RS equation: interaction of τ'p,ij & turbulence- New term in Cij equation (NLTij)- Additional diffusive flux terms not modeled
7
Shaqfeh (2006) AIChE Conference
- k-ε v2-f extension model of Durbin (1995)- Simplified model: τp,ij proportional to mean strain (elongation)- Coefficient has laminar and turbulent contribution- Laminar part proportional to ∂U/∂y- Turbulent part proportional to k;- Modifies pressure strain (v2 equation)- One transport equation for Ckk;
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
DNS cases: channel flow
2hu
1,x
2,y
Fully-developed channel flow
8
We!="u
!
2
#0
Re!=hu
!
"0
Re! = 395," = 0.9,L2= 900
We!= 25,DR = 18%
Low Drag Reduction High Drag Reduction
We!= 100,DR = 37%
• 2007 models (Pinho et al JNNFM 2008 & unpublished) - Only LDR
• 2008 model (under develop.)- Recalculated DNS + LDR & HDR
•Closures valid for 1st & higher order turbulence models
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Time- average polymer stress
-200
-150
-100
-50
0
0. 1 10 100
f(Ckk
) C12
f'c'12
f(Ckk
) C12
f'c'12
f(C
kk)
C12;
We=
25;
DR
=18%
y+
We= 25
DR= 18%
We= 100
DR= 37%
9
! ij ,p ="p
#f Ckk( )Cij $ f L( )% ij&' () +
"p
#f Ckk + ckk( )cij
0.00
0.05
0.10
0.15
0.20
0.1 1 101
102
We=25; DR=18%
We=100; DR=37%
rati
o f
'c'1
2/f
C1
2;W
e=
25
; D
R=
18
%
y+
f'c' 12/f(C
kk)C12
f Ckk( )C12 >> f 'c12 Can be neglected !
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Function f (Ckk)
10
0
100
200
300
400
500
600
700
800
1 10 100
Ckk
Ckk
y+
c 'kk
c 'kk
We= 25
DR= 18%
We= 100
DR= 37%
Ckk> c '
kk
2
f Ckk( )bij ! f Ckk( )bij = 0
This will be used frequently
f Ckk( )blmdi ! f Ckk( )blmdi
Function: f Ckk( ) =L2! 3
L2! Ckk + c 'kk( )
0.
1
10
100
101
102
103
f(Ckk)
Ckk
f(Ckk)
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 11
CTij
: originates in advective term, negligible no need for modeling
Mij
Oldroyd derivativeMean flow distortionExact and large
NLTij
: turbulent distortion originates in distortion of Oldroyd derivative- not negligible Must be modeled
DNS: Housiadas et al (2005) Phys Fluids 17, 35106, Li et al (2006 a) JNNFM
Time-average evolution equation for the conformation: RACE
!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
$
%&
'
() + Dij = #
* ij , p+p
Added for stabilityShould be negligible
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 12
Balance of RACE 1xx
-40
-20
0
20
40
60
1 101
102
CTxx
NLTxx
Mxx
Dxx
Sxx
=-!p,xx
CT
xx;
We=
25;
DR
=18%
y+
We=25; DR=18%
DNS: Housiadas et al (2005), Li et al (2006) JNNFM
-40
-20
0
20
40
1 101
102
CTxx
NLTxx
Mxx
Dxx
Sxx
=-!p,xx
CT
xx;
We=
100;
DR
=37%
y+
We=100; DR=37%
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 13
Balance of RACE 2yy
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
1 101
102
CTyy
NLTyy
Myy
Dyy
Syy
CTyy
NLTyy
Myy
Dyy
Syy
CT
yy;
We=
25;
DR
=18
y+
We=25
DR= 18%
We=100
DR= 37%
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
1 101
102
CT
xy;
We=
25;
DR
=18
y+
Cxy
xy
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Approximate equation for NLTij
SIMPLER THAN EXACT EQUATION:
f Cmm( )c
kj
!ui
!xk
+ f Cmm( )c
ik
!uj
!xk
+ Ckj
f Cmm( )
!ui
!xk
+ Cik
f Cmm( )
!uj
!xk
+ "!u
i
!xk
!ckj
!t+!u
j
!xk
!cik
!t
#
$%%
&
'((+
+"!C
kj
!xn
un
!ui
!xk
+!C
ik
!xn
un
!uj
!xk
+! U
nc
kj( )!x
n
!ui
!xk
+! U
nc
ik( )!x
n
!uj
!xk
+ un
!ckj
!xn
!ui
!xk
+ un
!cik
!xn
!uj
!xk
#
$
%%%
&
'
((()
)"!U
k
!xn
cjn
!ui
!xk
+ cin
!uj
!xk
*
+,,
-
.//+!U
j
!xn
ckn
!ui
!xk
+!U
i
!xn
ckn
!uj
!xk
+ Ckn
!uj
!xn
!ui
!xk
+!u
i
!xn
!uj
!xk
*
+,,
-
.//
#
$
%%
&
'
(()
)" Cjn
!uk
!xn
!ui
!xk
+ Cin
!uk
!xn
!uj
!xk
+ cjn
!uk
!xn
!ui
!xk
+ cin
!uk
!xn
!uj
!xk
+ ckn
!uj
!xn
!ui
!xk
+ ckn
!ui
!xn
!uj
!xk
#
$%%
&
'((= 0
14
L Ckj( )!ui
!xk+ L Cik( )
!u j
!xk
L Ckj( )f Cmm( )
!ui
!xk+
L Cik( )f Cmm( )
!u j
!xk
NLTij = ckj!ui
!xk+ cik
!u j
!xk
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
-1000
-500
0
500
1000
1500
1 101
102
f'c'1k!u'
1/!x
k
f'c'2k!u'
2/!x
k
f'c'3k!u'
3/!x
k
f(Ckk
)NLTxx
/2
f(Ckk
)NLTyy
/2
f(Ckk
)NLTzz
/2
y+
We=25, DR= 18%
-2000
0
2000
4000
6000
1 101
102
f'c'1k!u'
1/!x
k
f'c'2k!u'
2/!x
k
f'c'3k!u'
3/!x
k
f(Ckk
)NLTxx
/2
f(Ckk
)NLTyy
/2
f(Ckk
)NLTzz
/2
y+
We=100, DR= 37%
f 'c ' jk!ui!xk
+ f 'c 'ik!u j
!xk" f Cmm( ) ckj
!ui!xk
+ cik!u j
!xk
#
$%
&
'( = f Cmm( )NLTij
15
Simplifications for modeling NLTij
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Previous models for NLTij-- 2007 1st model
16
f Cmm( )NLTij
!= function Sij ,Wij ,Cij ," ij
N,#uiu j
#xk,#Cij
#xk,#NLTij#xn
,Mij ,uiu j
$
%&
'
()
Exact equation is too complex.Alternative model based on:1) Identification of possible dependencies from inspection of exact equation2) Simplicity, but capturing main features3) We=25 (DR=18%)
First model
f Cmm( )NLTij
!= fµ
1
CE3
u"2
#0
2Ckkuiu j +
C$14
#0
uiukWknCnj + u jukWknCni + ukuiWjnCnk( )%
&'
(
)*
CE3= 0.00035;C
!14= 0.00015 fµ1 = 1! exp ! y
+ 26.5( )( )2
2 coefficients 1 damping function
Pinho, Li, Younis, Sureshkumar (2008) JNNFM, in press
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 17
un!u j
!xk" 02) Homogeneous turbulence (negligible turbulent diffusion)
suggests un!cij
!xk" 0CTij = uk
!cij
!xk" 01)
Un
!ckj!xn
!ui!xk
+!cik!xn
!u j
!xk
"
#$
%
&' = 0
un!ckj
!xn
!ui
!xk+ un
!cik
!xn
!u j
!xk" 0
3) Invariance laws
Ckn
!u j
!xn
!ui!xk
+!ui!xn
!u j
!xk
"
#$
%
&' + Cjn
!uk!xn
!ui!xk
+ Cin
!uk!xn
!u j
!xk( C)N
fN24
15
)* +We,0 +-T
Cmm. ij
4) Homogeneous isotropic turbulence
Hinze (1975); Mathieu & Scott (2000)
!ui!xk
!u j
!xl=8
3
k
" f
2# ij# kl $
1
4# ik# jl + # il# jk( )%
&'(
)*
Taylor’s longitudinal micro-scale ! = 20"k
# f
2
Modeling NLTij 1- 2008 model and 2nd 2007 model
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 18
Modeling NLTij 2- 2008 model and 2nd 2007 model5) ad-hoc + symmetry, invariance, permutation, realizability
!Uk
!xncjn
!ui!xk
+ cin!u j
!xk
"
#$
%
&' +
!Uj
!xnckn
!ui!xk
+!Ui
!xnckn
!u j
!xk(
CN3
!Uj
!xk
!Um
!xnCkn
uium
"02SpqSpq
+!Ui
!xk
!Um
!xnCkn
u jum
"02SpqSpq
#
$%%
&
'((
6)Ckj f Cmm( )!ui
!xk+ Cik f Cmm( )
!u j
!xk
! Ckj f Cmm( )"ui
"xk+ Cik f Cmm( )
"u j
"xk= 0
! CN2
Ckj f Cmm( )"Ui
"xk+ Cik f Cmm( )
"Uj
"xk
#
$%
&
'(
7) Decoupling 3rd order correlation
cjn!uk
!xn
!ui
!xk+ cin
!uk
!xn
!u j
!xk+ ckn
!u j
!xn
!ui
!xk+ ckn
!ui
!xn
!u j
!xk"
!CN4
fN1
Cjn
"Uk
"xn
"Ui
"xk+ Cin
"Uk
"xn
"Uj
"xk+ Ckn
"Uj
"xn
"Ui
"xk+ Ckn
"Ui
"xn
"Ui
"xk
#
$%
&
'(
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 19
Closure for NLTij- 2008 model
fN1 = 1! 0.8exp ! y+ 30( )"#
$%2
fN2 = 1! exp ! y+ 25( )"#
$%4
CN1
= 12.7
CN2
= 0.32
CN3
= 0.024
CN4
= 1.11
CN5
= 1.13
f Cmm( )NLTij
!=f Cmm( )
!CN1
Cij f Cmm( )!We"0
# CN2Ckj
$Ui
$xk+ Cik
$Uj
$xk
%
&'
(
)*
+,-
.-
/0-
1-
+CN3
Ckn
202SpqSpq
uium$Uj
$xk
$Um
$xn+ u jum
$Ui
$xk
$Um
$xn
%
&'
(
)*
#CN4fN1 Cjn
$Uk
$xn
$Ui
$xk+ Cin
$Uk
$xn
$Uj
$xk+ Ckn
$Uj
$xn
$Ui
$xk+$Ui
$xn
$Uj
$xk
345
678
%
&'
(
)*
+CN5fN2
4
15
9:2 sWe"0
Cmm; ij
modified term & reprocessed DNS dataWe=25 (LDR) & 100 (HDR)
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
2007 models for NLTij: First versus second model
20
-1000
0
1000
2000
3000
4000
0.1 1 10 100
f(Ckk)*NLTxyf(Ckk)*NLTyyf(Ckk)*NLTiif(Ckk)*NLTxyf(Ckk)*NLTyyf(Ckk)*NLTiif(Ckk)*NLTxyf(Ckk)*NLTyyf(Ckk)*NLTii
y+
DNS
First
Second
We=25, DR=18%
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 21
Performance of the 2008 NLTij closure: xx and yy
-2000
-1000
0
1000
2000
3000
4000
0.1 1 101
102
DNSModelDNSModel
NL
T*
xx
-DN
S;
We=
25
, D
R=
18
y+
We=25, DR=18
We=100, DR=37%
NLTxx
xx
-200
0
200
400
600
800
1000
0.1 1 101
102
DNSModelDNSModel
NL
T*
yy
-DN
S;
We=
25
, D
R=
18
y+
We=25, DR=18
We=100, DR=37%
NLTyy
yy
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
-2000
-1000
0
1000
2000
3000
4000
5000
6000
0.1 1 101
102
DNSModelDNSModel
NL
T*
kk
-DN
S;
We=
25
, D
R=
18
y+
We=25, DR=18
We=100, DR=37%
NLTkk
22
Performance of the 2008 NLTij closure: trace and xy
-200
0
200
400
600
800
1000
0.1 1 101
102
DNSModel
DNS
Model
NL
T*
xy
-DN
S;
We=
25
, D
R=
18
y+
We=25, DR=18
We=100, DR=37%
NLTxy
trace xy
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Modeling the Reynolds stress
23
2) Dissipation of turbulent kinetic energy: !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
#
$%
&
'(= 0
As for Newtonian fluids, most terms in εN are approximated
New term (will be considered in the future)
1) Reynolds stresses: Prandtl-Kolmogorov model (k-ε closure)
!uiu j = 2"T Sij !2
3k#ij
!T =Cµ fµk2
!"N+"
Vwith
Next slide
Major issue: what model to use for Reynolds stresses?
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Transport equation for turbulent kinetic energy
24
!Dk
Dt= "!uiuk
#Ui
#xk" !ui
#k '
dxi"#p 'ui#xi
+$s
#2k
#xi#xi"$s
#ui#xk
#ui#xk
+#% ik , p
'ui
#xk" % ik , p
' #ui#xk
QV
!"V−εΝD
NQNP
k0
-0.20
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
1 101
102
Pk
QN
QV
DN
-!N
-!V
Pk
QN
QV
DN
-!N
-!V
Pk;
We=
25, D
R=
18
y+
We=25
DR=18%
We=100
DR=37% Need to model well εV
Qv is small
Need to modify model of εN
When We increases (DR )
Pk decreasesεN decreasesQv increases in buffer l., but remains smallεV increases in inertial l.
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
-0.06
-0.04
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
40 60 80 100 300
Pk
QN
QV
DN
-!N
-!V
Pk
QN
QV
DN
-!N
-!V
Pk;
We=
25, D
R=
18
y+
We=25
DR=18%
We=100
DR=37%
25
Zoom of balance of k: inertial sub-layer
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Assumptions for viscoelastic stress work: εV
!V "1
#$ ik ,p' %ui
%xk=&p
#'Cik f Cmm + cmm( )
%ui%xk
+ cik f Cmm + cmm( )%ui%xk
(
)*
+
,-
26
-2000
0
2000
4000
6000
8000
10000
1 101
102
Cikf'!u
i/!x
k
f'cik!ui/!x
k
SUM
Cik
*f'
*!u
'i!x
k;
We=
10
0,
DR
=3
7
y+
We=100; DR= 37%
-500
0
500
1000
1500
2000
2500
1 101
102
Cikf'!u
i/!x
k
f'cik!u
i/!x
k
SUM
Cik
*f'
*!u
'i!x
k;
We=
25
, D
R=
18
y+
We=25; DR= 18%
Cik f Cmm + cmm( )!ui
!xk<< cik f Cmm + cmm( )
!ui
!xk
Except in viscous sublayer and buffer, but here is not important!V
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
-2000
0
2000
4000
6000
8000
10000
1 101
102
f'(c'mm)cik!ui/!x
k
f(Cmm)cik!ui/!x
k
f'(c'mm)cik!ui/!x
k
f(Cmm)cik!ui/!x
k
f'c'(
ik)!
u'(
i)/!
x(k
); W
e=
25, D
R=
18
y+
We=25
DR=18%
We=100
DR=37%
27
Further assumptions for viscoelastic stress work: εV
but larger as DR increases
C
!V !O 1( )
at We!0= 25
f 'c 'ik!ui
!xk" C
#V $ f Cmm( )cik
!ui
!xk
This isNLTii
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 28
Viscoelastic stress work model
C!V = 1.27
n = 1.15
We!0= 25" 1.27
We!0= 100" 1.56
!V "#p
$%C
!Vf Cmm( )cik
&ui&xk
= C!V
We'0
25
()*
+,-
n.1 #p
$%f Cmm( )
NLTii
2
Modeled
C!V = 1.076Previous model:
(Weτ0= 25 only)Pinho, Li, Younis, Sureshkumar (2008) JNNFM, in press
!V+Re
"0
( )2
versus
C!V
Re"0
1# $( )
We"0
f Cii( )NLTjj
*
-1000
0
1000
2000
3000
4000
1 101
102
!V
Re"0
2
Model
!V
Re"0
2
Model
EpsV
+R
etau
0**2;
We=
25, D
R=
18
y+
We=25
DR=18%
We=100
DR=37%
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
QV !
"# ik , p'ui
"xk=$p
%"
"xkCik f Cmm + cmm( )ui + cik f Cmm + cmm( )ui&'
()
CFUiik
CUiik
29
Viscoelastic turbulent transport: QV
CFUiik = Cik f Cmm + cmm( )ui
!CFU
2
"
We#0
f Cmm( )Ckn
$uiui
$xn
f Cmm( )CUijk
!= fµ2
25
We"0
#
$%
&
'(
0.53
)C*1uium
+Ckj
+xm+ u jum
+Cik
+xm
#
$%&
'()C*7
!f Cmm( ) ± u j
2Cik ± ui
2Cjk
,-.
/01
,
-.
/
01
C!1= 1.1;C!7
= 0.3
fµ2 = 1! exp ! y+ 26.5( )
-100
-50
0
50
100
150
1 101
102
CFUiik+f(C
mm)CU
iik
ModelCFU
iik+f(C
mm)CU
iik
Model
CUii2/2
y+
We=25
DR=18%
We=100
DR=37%
Closure development followed similar procedures as that for NLTij
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
d
dy!s
dU
dy+ " p,xy # $uv
%
&'
(
)* #
dp
dx= 0Momentum:
! xy,p ="p
#f Ckk( )Cxy
f Ckk( )Cxy = !Cyy
dU
dy+ !NLTxy
f Ckk( )Cxx = 2!Cxy
dU
dy+ !NLTxx +1
f Ckk( )Cyy = !NLTyy +1
f Ckk( )Czz = !NLTzz +1
f Ckk( ) =L2! 3
L2! Cxx + Cyy + Czz( )
Final equations for channel flow: RANS and RACE
30
!"uv = "#TdU
dy
!T =Cµ fµk2
!"N+"
Vwith
Reynolds stress:
fµ = 1! exp!y+
26.5
"
#$
%
&'
(
)*
+
,-2
and
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
0 =d
dy!s +
" fT#T$ k
%
&'(
)*dk
dy
+
,-
.
/0 + Pk 1 " !2 N 1 "DN
+!p
d
dy
f Cmm( )3
CUnny
2
+
,-
.
/0 1!p
f Cmm( )3
NLTnn
2
!N= !!
N+ D
ND
N= 2!s
d k
dy
"
#$%
&'
2
k and ε transport equations: modified Nagano & Hishida
31
Based on Newtonian model of Nagano & Hishida (1984)
!k= 1.1
fT = 1+ 3.5exp ! RT 150( )2"
#$%
Variable Prandtl numbers: Nagano & Shimada (1993), Park and Sung (1995)
0 =d
dy!s +
" fT#T$%
&
'()
*+d !% N
dy
,
-.
/
01 + " f
1C%
1
!% N
k
Pk
"2 " f
2C%
2
% N2
k+ "E + E3 p
E =!s
"#T 1$ fµ( )
d2U
dy2
%&'
()*
2
f1=1 f2 =1! 0.3exp !RT
2( )
C!1= 1.45
!"= 1.3
C!2= 1.90
E! p= 0
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Predictions U+: Reτ0= 395; Weτ0= 25; β=0.9, L2=900
32
NLTij CUijk
Models for
&
changed
0
5
10
15
20
25
30
1 10 100
DNS
Newtonian- Variable Pr2007 First- Variable Pr
2007 Second- constant Pr
2007 Second- Variable Pr
u+
y+
u+
= 2.5 ln y+
+ 5.5
u+
= 11.7 ln y+
-17.0
u+ = y
+
2007 1st model2007 2nd model2007 2nd model
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
-500
0
500
1000
1500
2000
2500
0.1 1 10 100
DNS
2007 First- Variable Pr
2007 Second- Constant Pr
2007 Second- Variable Pr
NLTii
*
y+
Predictions NLTii: Reτ0= 395; Weτ0= 25; β=0.9, L2=900
33
NLTii
*
Models fitted to DNS
Code diverges
Models modified with simulations
Coefficients & reduced by 4
CN1
CN2
Coefficient increased 60%
CN3
CN5
Coefficient reduced 30%
Model 2
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 34
Predictions k & εN: Reτ0= 395; Weτ0= 25; β=0.9, L2=900k+
0
1
2
3
4
5
1 10 100
DNS
2007 First- Variable Pr
2007 Second- Constant Pr
2007 Second- Variable Pr
k+
y+
!N
+
0
0.05
0.1
0.15
0.2
0.1 1 10 100
DNS
2007 First- Variable Pr
2007 Second- Variable Pr
2007 Second- Constant Pr
!N+
y+
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK 35
u 'v '+
Predictions u’v’ & τp,xy: Reτ0= 395; Weτ0= 25; β=0.9, L2=900
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
DNS2007 First- Variable Pr2007 Second- Constant Pr2007 Second- Variable Pr
u'v
'+
y*
!p,xy
+ (zoomed)
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
DNS2007 First- Variable Pr2007 Second- Constant Pr2007 Second- Variable Pr
!p,xy
+
y*
Recent developments and challenges in turbulence modeling of viscoelastic fluids Resende, Pinho, Kim, Younis & Sureshkumar
CEFT-FEUP Centro de Estudos de Fenómenos de Transporte, Universidade do Porto Complex Flows of Complex Fluids, Liverpool, UK
Conclusions and Acknowledgments- Closures for Low DR and High DR
- Closures for NLTij, εV and QV (in fact for εijV and Qij
V)
- Developed simple low Reynolds k-ε model works reasonably well
- Need to incorporate with better Reynolds stress closures:
k-ω, modified k-ε or k-ω, Menter’s SST or Durbin’s v2-f
or RS transport (deficiencies in base model are imp.)
- Need to extend models to Maximum DR, & β & L2
- DNS in other canonical flows required for extension of turb. models
36
Acknowledgments - FundingFundação Calouste Gulbenkian: Project 72259Fundação para a Ciência e TecnologiaProjects SFRH/BSAB/507/2005 ; SFRH/BD/18475/2004Project “Turbopol” POCI/EQU/56342/2004