a turbulence closure for viscoelastic fluids based …fpinho/pdfs/ci103_rhodes2007_turb.pdfa...

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



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

3



-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


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



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



-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



-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



!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+



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



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:



Model for NLTij- First versus second model

12

-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



Viscoelastic turbulent transport: QV

QV !

"# ik ,p'ui

"xk=$p

%"

"xkCik f Cmm + cmm( )ui + cik f Cmm + cmm( )ui&'

()

13

Same simplifications as for !V

Cik f Cmm + cmm( )ui << cik f Cmm + cmm( )ui

cik f Cmm + cmm( )ui ! f Cmm( )cikui

1)

2)

QV=!p

"#

#xkf Cmm( )cikui$% &' Needs modelCUijk ! cjkui



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

14

- Same modelling approach as for second model of NLTij

-20

-10

0

10

20

30

40

0 1 10 100

DNSModel f(Ckk) CU

ii2

QFii2

y+

2 coefficients

1 damping function



0 =d

dy!s +

" fT#T$ k

%&'

()*dk

dy

+

,-

.

/0 + Pk 1 " !2 N 1 "D +!p

d

dy

f Cmm( )3

CUnny

2

+

,-

.

/0 1!p

f Cmm( )3

NLTnn

2

!N= !!

N+D

N DN= 2!s

d k

dy

"

#$

%

&'2

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 !RT2( )

fµ = 1! exp!y+

26.5

"

#$

%

&'

(

)*

+

,-2

k and ! transport equations: modified Nagano & Hishida

15

Based on Newtonian model of Nagano & Hishida (1984)Variable Prandtl numbers: Nagano & Shimada (1993), Park and Sung (1995)

fT =1+ 3.5exp ! RT 150( )2"

#$%&'

!k= 1.1

C!1=1.45;C

!2=1.90;"

!=1.3

!k=1.0

or



0

5

10

15

20

25

30

1 10 100

DNS

Newtonian- ft=1

Newtonian ft=function

First ft=1

First ft=function

First ft=1 No !

V

Second ft=1

Second ft=function

u+

y+

Predictions 1: Re!0= 395; We!0= 25; "=0.9, L2=900

Newtonian log law

Virk’s asymptote

16

NLTij

CUijk

Models for

&

changed



-20

0

20

40

60

80

0.1 1 10 100

CUii2

+

y+


Model fitted to DNS

Modified with simulations Model 1 with ft=1

17

C!1=1.3;C!7

= 0.37

C!1=1.3

2;C!7

=0.37

2

CUijk



-500

0

500

1000

1500

2000

2500

0.1 1 10 100

DNS

First ft=1

First ft=function

First ft=1 No !

V

Second ft=1

Second ft= function

NLTii

*

y+


18

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



NLTyy

*


19

NLTxy

*

0

100

200

300

400

500

0.1 1 10 100

DNS

First ft=1

First ft=function

First ft=1 No !

V

Second ft=1

Second ft=function

NLTyy*

y+

0

100

200

300

400

500

600

1 10 100

DNS

First ft=1

First ft=function

First ft=1 No !

V

Second ft=1

Second ft=function

NLTxy*

y+



!N

+

20


k+

0

1

2

3

4

5

1 10 100

DNS

First ft=1

First ft=function

First ft=1 No !

V

Second ft=1

Second ft=function

k+

y+

0

0.05

0.1

0.15

0.2

0.1 1 10 100

DNS

First ft=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

!N+

y+




21

CyyCxy

-50

-40

-30

-20

-10

0

0.1 1 10 100

DNS

First ft=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

Cxy

y+

0

5

10

15

20

25

0.1 1 10 100

DNS

First ft=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

Cyy

y+




22

CUii2

+Cxx

0

100

200

300

400

500

0.1 1 10 100

DNS

First ft=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

Cxx

y+

-20

-10

0

10

20

30

1 10 100

DNS

First ft=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

CUiiy

+

y+



0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

DNSFirst f

t=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

u'v

'+

y*

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1

DNSFirst f

t=1

First ft

First ft=1 No !

V

Second ft=1

Second ft

"p,xy

+

y*

u 'v '+

!p,xy

+(zoomed)

Reynolds and polymer stresses



Conclusions

24

- Developed simplified k-! model: code and closures are working well

- Viscoelastic turbulent transport not very important at DR= 18%

- Viscoelastic turbulent transport is reasonably well modeled

- Viscoelastic stress power well modeled by NLTij (model 2)

- NLTij required for Cij and !ijV

- NLTij is now fairly good (Model 2)

- Model 2 has viscoelastic turbulence production close to wall

- Isotropic turbulence does not allow a good model

- Deficiencies are related to Newtonian base model: Use better model

- Need to consider anisotropic turbulence: anisotr. k-! and RSM

- Need to consider DNS for HDR & MDR & # & L2



Acknowledgments

Funding

Fundação Calouste GulbenkianProject 72259

Fundação para a Ciência e TecnologiaProject BSAB/507/2005Project Turbopol, nº POCI/EQU/56342/2004

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a turbulence closure for viscoelastic fluids based …fpinho/pdfs/ci103_rhodes2007_turb.pdfa...

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