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



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( )



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



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



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



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;



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



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 !



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)


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



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%



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



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



-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



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



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



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

#

$%

&

'(



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)



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%



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



-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



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?



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.



-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



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



-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



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%



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



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



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



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



-500

0

500

1000

1500

2000

2500

0.1 1 10 100

DNS

2007 First- Variable Pr

2007 Second- Constant 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



Predictions k & εN: Reτ0= 395; Weτ0= 25; β=0.9, L2=900k+

0

1

2

3

4

5

1 10 100

DNS




k+

y+

!N

+

0

0.05

0.1

0.15

0.2

0.1 1 10 100

DNS




!N+

y+



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*



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

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