A RANS/RACE A RANS/RACE kk--ωω LOW REYNOLDS NUMBER LOW REYNOLDS NUMBER

TURBULENCE MODEL FOR FENE-P FLUIDSTURBULENCE MODEL FOR FENE-P FLUIDS

P. R. ResendeCentro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia, Universidadedo Porto, Portugal

F. T. PinhoCentro de Estudos de Fenómenos de Transporte, Faculdade de Engenharia, Universidadedo Porto, Portugal

B. A. YounisDep. Civil and Environmental Engineering, University of California, Davis, USA

K. KimDep. Mechanical Engineering,Hanbat National University, Daejeon, South Korea

R. SureshkumarDep. Biomedical and Chemical Engineering, Syracuse University, Syracuse, NY, USA

XVIth International Workshop on Numerical Methods For Non-Newtonian Flows13th-16th June 2010Northampton, MA, USA

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

Drag reduction: motivation

2

Drag reduction in fully-developed channel flow

0

5

10

15

20

25

30

1 10 100

014 9.319.2 15.625.0 19.742.4 27.663.5 33.5100 39153 43.8222 50.5

u+

y+

We!0

DR [%]

u+

= 2.5 ln y+

+ 5.5

u+

= 11.7 ln y+

-17.0

u+ = y

+

Can a k-ω model improve on k-ε ?

Advantages:Valid across all BL (no damping)Better in BL with adverse pres. grad.

Disadvantages:Too sensitive to ω in free stream

Existing models

k-ε: Pinho et al., JNNFM 154 (2008) 89k-ε improved:Resende et a.l,JNNFM(2010), sub.k-ε-v2-f: Iaccarino et al.,JNNFM 165(2010)376

(1st order)

0 order: Li et al., JNNFM 159 (2006) 177

(2nd order)Leighton et al. (2002,2003) APS, ASME

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

DNS cases: channel flow

2hu

1,x

2,y

Fully-developed channel flow

3

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%

DNS test/calibration cases

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 4

Closuresrequired

! 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

!Ui

!xi

= 0Continuity:

Momentum balance:

!"Ui"t

+ !Uk"Ui"xk

= #"p

"xi+$s

"2Ui"xk"xk

#"

"xk!uiuk( ) +

"% ik,p"xk

Reynolds decomposition:Overbar & upper-case: time-averaged quantitiesLower-case: fluctuating quantities

B̂ = B+ b '

New model: Governing Equations

Independent of turbulence model

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 5

Conformation (RACE) equation

!Cij"

+ ! uk#cij#xk

$ ckj#ui#xk

+ cik#u j#xk

%

&'

(

)*

+

,--

.

/00= $ f Ckk( )Cij $ f L( )1 ij+, ./ $ f Ckk + ckk( )cij

CTijMij NLTij

Model for NLTij is identical to that for k-ε (Resende et al. (2010) JNNFM submitted)

f Cmm( )NLTij

!=f Cmm( )

!fN1Cij

f Cmm( )!

" fN2 Ckj#Ui#xk

+ Cik#Uj#xk

$

%&

'

()

*+,

-,

./,

0,

+ fN3Ckn

102SpqSpq

uium#Uj#xk

#Um#xn

+ ujum#Ui#xk

#Um#xn

$

%&

'

() +

1

102SpqSpq

#Uk#xn

#Um#xk

Cjnuium + Cinu jum( )$

%&

'

()

$

%&&

'

())

" fN4 Cjn#Uk#xn

#Ui#xk

+ Cin#Uk#xn

#Uj#xk

+ Ckn#Uj#xn

#Ui#xk

+#Ui#xn

#Uj#xk

234

567

$

%&

'

() + fN5

4

15

8 N

91 sCmm: ij

fNi = f (We!0 , y+)Based on exact equation; explicit model

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

The specific dissipation rate: ω

! !u2

t& t !

l

u;! !

k32

l

1) Estimate of dissipation (large scale)

µT! !

k2

"

Chou (1945)![ ] =

length2

time3

!"uiu j = 2µT Sij !2

3"k# ij

µT= ! kl

How to determine l ? Generally difficult ! Various alternatives

k Transport equation

Prandtl- Kolmogorov closure for Reynolds Stress/Boussinesq app.

6

Kolmogorov (1942)

! !"

k2) Specific dissipation rate:

µT! !

k

"

![ ] =1

timeω is better behaved near walls,

but more sensitive far from walls

! "2#

Ck$ y2

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 7

Reynolds stress closure: eddy viscosity model (k-ε & k-ω)

!uiu j = 2"T Sij !2

3k#ij

Prandtl-Kolmogorov model

!TN= fµ

k

"N

!TP= fµCµ

PfµPCkk

k

"N

!N=

"N

Ckk

New model with Note: C

k= C

µ

Pinho et al. JNNFM (2010) submitted: k-ε

!TN= Cµ fµ

k2

!"N

!T= !

T

N"!

T

P

!TP= Cµ fµCµ

PfµPCkk

k2

!"N

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 8

Transport equation for k

!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

DV

!"V−εΝDNDTPk0

exact exactUnchanged(Newtonian)

PreviousModel

PreviousModel

!"N= !C

kk#

N

Previous model: k-ε context

!"

N= ! !"

N+ D( ) with

and solve equation for !!N

D = 2! sd k

dy

"

#$%

&'

2

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 9

DV=!p"

#

#xkCik f Cmm + cmm( )ui + cik f Cmm + cmm( )ui$%

&'

(!p"

#

#xkf Cmm( )

Cik FU( )i + CU( )ijk2

$

%)

&

'* +!+ p

#2k

#xk2

Cik FU( )i ! fFUCknuiui

!xn fFU = fFU We( )

f Cmm( )CUijk!

= " f#1

uium$Ckj$xm

+ u jum$Cik$xm

%

&'(

)*"f#

7

f Cmm( )!

± u j2Cik ± ui

2Cjk

+,-

./0

f!1 , f!7 = f! We( )Unchanged coefficients & functions;

as in k-ε

!" p ="xy

p

!#

Resende et alJNNFM (2010)sub.

Viscoelastic turbulent diffusion, DV

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 10

Viscoelastic stress work: εV

!V "1

#$ ik ,p' %ui

%xk&'p#(

cik f Cmm + cmm( )%ui%xk

)

*+

,

-.

f 'c 'ik!ui

!xk" f

#V $ f Cmm( )cik

!ui

!xk NLTiif!V = f

!V We( )

Same model as in k-ε(Resende et al (2010) Subm.)

Unchanged

!V = f!V

"p

#$f Cmm( )

NLTii

2

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

0 =d

dy!" p +!s +

# fT$T% k

&'(

)*+dk

dy

,

-.

/

01 + Pk 2 #Ck3

Nk +

!p4

d

dyf Cmm( )

Cnk FU( )n +CUnny2

,

-.

/

01 2!p

f Cmm( )4

NLTnn

2

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)

11

Transport equation of k: final modeled form

New form

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 12

Specific rate of deformation: transport equation

!N=

"N

Cµk

D!N

Dt=1

Cµk

D"N

Dt#!

N

k

Dk

Dt

D!N

Dt= P

!N " #

!N +$

!N + D

!N

T+ D

!N

N+ E

!N

V

!D" N

Dt= C"

1

"kPk +

##xi

$s +$% p + !&T'(

)

*+,

-.#" N

#xi

/

01

2

34 5C"

2

!" 2 +C"

k$s +$% p + !&T( )

#k#xi

#"#xi

+ E" NV

Viscous cross-diffusion (Bredberg et al. 2002)

Dk

Dt= P

k! "

N+#

k+ D

k

T+ D

k

N+ D

k

V! "

V

D!N

Dt= P

!N " #

!N +$

!N + D

!N

T+ D

!N

N+ E

!N

V

Production

Destruction

Redistribution

Turbulentdiffusion

Moleculardiffusion

Viscoelasticinteraction

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 13

Viscoelastic contribution to ω: model

Definition

and modelE

!N

V=1

CµkE

"N

V#!

kD

k

V+!

k"V

Slide 10Slide 9

E!NV " 2#s

#p$ L2 % 3( )

&ui&xm

&

&xk

&

&xmf Cnn( ) f Ĉpp( )cqq' Cik'( )*

+,-

./0

Model of

E!NV " # fDR

! !N2

kC!F1

!V

Ckk$NL2 # 3( )

2

+C!F2 Cii f Ckk( )%& '(2%

&)

'

(*

improved version relative to k-ε of Resende et al (2010), it nowincorporates effects of β & L2

fDR!= fDR

!We

0,",L2( )

E!N

V

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 14

Mean velocity: Reτ0= 395; β=0.9, L2=900

0

5

10

15

20

25

30

100

101

102

We= 0

We= 25

We= 100

DNS- We= 25

DNS- We= 100

We= 0

We= 25

We= 100

u+

y+

u+

= 2.5 ln y+

+ 5.5

u+

= 11.7 ln y+

- 17.0

u+

= y+

k-!

k-"}

}

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 15

Turbulent kinetic energy: Reτ0= 395; β=0.9, L2=900

0

1

2

3

4

5

6

7

1 10 100

DNS- Mansour (We= 0)DNS- We= 25DNS- We= 100We= 0We= 25We= 100We= 0We= 25We= 100

k+

y+

k-!

k-"}}

!T = Cµ fµk2

!"N1# Cµ

PfµPCkk( )

!T = Cµ fµk

"N1#Cµ

PfµPCkk( )

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

Dissipation of k by solvent: Reτ0= 395; β=0.9, L2=900

16

0

0.05

0.1

0.15

0.2

0.25

1 10 100

DNS- Mansour (We=0)DNS- We= 25DNS- We= 100We= 0We= 25We= 100We= 0We= 25We= 100

!+

y+

k-!

k-"}}

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA 17

NLTii: Reτ0= 395; β=0.9, L2=900

-1000

0

1000

2000

3000

4000

5000

1 10 100

DNS- We= 25

DNS- We= 100

We= 0

We= 25

We= 100

We= 0

We= 25

We= 100

NLTii

*

y+

k-!

k-"}}

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

0

100

200

300

400

500

600

700

800

1 10 100

DNS- We=25

DNS- We=100

We=0

We=25

We=100

We=0

We=25

We=100

Cxx

y+

k-!

k-"}}

18

Cxx: Reτ0= 395; β=0.9, L2=900

A RANS/RACE k-ω low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim & Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte XVI IWNMNNF, Northampton, USA

Conclusions, Future Work and Acknowledgments

- k-ω model developed, it works well at Low DR and High DR (50%)

- Closures for elastic terms: similar to corresponding in k-ε

(Resende et al. JNNFM (2010) Submitted)

- Slightly better than k-ε

- More stable (easier convergence)

- Need for 2nd order Reynolds stress closures: deficiency in k

- Need to extend models to Maximum DR, & β & L2

19

Acknowledgments - FundingFundação para a Ciência e TecnologiaProjects PTDC/EQU-FTT/70727/2006 & PTDC/EME-MFE/70186/2006