A FIRST-ORDER TURBULENCE MODEL FOR VISCOELASTIC FENE …fpinho/pdfs/C5-AERC2007v2FENE-P.pdf · A...

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, USACurrently 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 200712th-14th April 2007Napoli, Italy

Acknowledgments:FCT (POCI/EQU/56342/2004);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)



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



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



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)]

Cx

yD

TR

(J)

ave

(+ s

ide)

y+

20 times smaller

f Ckk( )C12 >> f 'c12



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)


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

MijExact

-400

-200

0

200

400

600

10-1 100 101 102

CTNLTMD

S= - !p

CT

(J)

aver

y+

yy component(c)



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



-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

εΝDN

-Need to model viscoelasticstress work

- Viscoelastic turbulent transport is small



Viscoelastic stress work: εV

!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



-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



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



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



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



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



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-ε type model for channel flow

16



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 ε transport equations: modified Nagano & Hishida

17

based on Newtonian model of Nagano & Hishida (1984)



0

5

10

15

20

25

30

1 10 100

u+

lam

y+

u+

Predictions 1: Reτ0= 395; Weτ0= 25; β=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




-20

0

20

40

60

80

0.1 1 10 100

CUii2

+

y+

Model fitted to DNS

Modified with simulations

19



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+


20



NLTyy

*

0

100

200

300

400

500

0.1 1 10 100

NLTyy*

y+


0

5

10

15

20

25

0.1 1 10 100

Cyy

y+

Cyy

21



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: Reτ0= 395; β=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+




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+




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+




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*



Conclusions- Developed simplified k-ε 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 εijV

- 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-ε and RSM

26

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