A k LOW REYNOLDS NUMBER TURBULENCE MODEL FOR …fpinho/pdfs/CI1-AERC2010_Turb_k_omega_v2.pdf · A...

A A kk--!! LOW REYNOLDS NUMBER TURBULENCE LOW REYNOLDS NUMBER TURBULENCE

MODEL FOR TURBULENT CHANNEL FLOW OFMODEL FOR TURBULENT CHANNEL FLOW OF

FENE-P FLUIDSFENE-P FLUIDS

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, 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, Syracyse University, Syracuse, NY, USA

VIth Annual European Rheology Conference

7th-9th April 2010

Göteborg, Sweden

k-! low Re turbulence model for FENE-P fluids Resende, Pinho, Younis, Kim and Sureshkumar

CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 2010, Göteborg, Sweden

Drag reduction: motivation

2

Drag reduction in fully-developed channel flow

Can a k-! model improve on k-" ?

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+

Existing models (1st order)

k-": Pinho et al, JNNFM 154 (2008) 89

k-" improved:Pinho et al (2010) in prep

k-"-v2-f: Iaccarino et al,165(2010)376

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

Disadvantages:Too sensitive to ! in free stream



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


CEFT-FEUP Centro de Estudos de Fenómenos de Transporte AERC 2010, Göteborg, Sweden 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 NLT

ij

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

B = B+ b '

New model: Governing Equations

Independent of turbulence model



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 essentially identical to that for k-", except in some coefficients/ functions

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

#xkCjnuium + 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 s

Cmm: ij

fNi= f (We

!0, y

+)



The specific dissipation rate: !

! !k32

l

1) Estimate of dissipation (large scale)

µT! !

k2

"

Chou (1945)![ ] =length

2

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

6

Kolmogorov (1942)

! !"

k2) Specific dissipation rate:

µT! !

k

"

![ ] =1

time! is better behaved near walls,

but more sensitive far from walls

! "2#

Ck$ y2



Reynolds stress closure: eddy viscosity model (k-" & k-!)

!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 (2010): k-"

!TN= Cµ fµ

k2

!"N

!T= !

T

N"!

T

P

!TP= Cµ fµCµ

PfµPCkk

k2

!"N



Transport equation for k

DV=!p

"#

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

&'

(!p

"#

#xkf Cmm( )

Cik FU( )i+ CU( )

ijk

2

$

%)

&

'*

Cik FU( )i! fFUCkn

uiui

!xn

fFU = fFU We( ) f Cmm( )CUijk

!= " f#

1

uium$Ckj

$xm+ u jum

$Cik

$xm

%

&'(

)*"f#

7

f Cmm( )!

± u j

2Cik ± ui

2Cjk

+,-

./0

f!1 , f!7 = f! We( )

Essentially unchangedCoefficients & functions

!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#"$D

ND

TPk

0

exact exactUnchanged(Newtonian)

PreviousModel

PreviousModel

!"N= !C

kk#

N



Viscoelastic stress work: "V

!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 NLTii

f!V = f

!V We( )

Same model as in k-"

Unchanged

!V = f!V

"p

#$f Cmm( )

NLTii

2



0 =d

dy!p +!s +

" fT#T$ k

%&'

()*dk

dy

+

,-

.

/0 + Pk 1 "Ck2

Nk +

!p

3d

dyf Cmm( )

Cnk FU( )n+CUnny

2

+

,-

.

/0 1!p

f Cmm( )3

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)

10

Transport equation of k: final modeled form

New form



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

.

/0

1

23 4 C"

2

!" 2+ !

C"

k

$s

!+ %T

()*

+,-#k#xi

#"#xi

+ E" N

V

Viscous cross-diffusion (Bredberg et al. 2002)

D!N

Dt= P

!N " #

!N +$

!N + D

!N

T+ D

!N

N+ E

!N

V

Dk

Dt= P

k! "

N+#

k+ D

k

T+ D

k

N+ D

k

V! "

V

Production

Destruction

Redistribution

Turbulentdiffusion

Moleculardiffusion

Viscoelasticinteraction



Viscoelastic contribution to !: model

Definition

and model

E!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 Cpp( )cqq' Cik

'(

)*

+,-

./0

Model of

E!NV " # fDR

! ! N2

kC!F1

!V

! NL2 # 3( )

2

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

%(

&

')

improved version relative to k-", it also incorporates effects of % & L2

fDR!= fDR

!We

0,",L2( )

E!N

V



Mean velocity 1: Re"0= 395; %=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-"}

}



Turbulent kinetic energy: Re"0= 395; %=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( )



Dissipation of k by solvent: Re"0= 395; %=0.9, L2=900

15

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-"}}



NLTii: Re"0= 395; %=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-"}}



Conclusions, Future Work and Acknowledgments

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

- Closure for elastic terms: similar to corresponding in k-"

- Slightly better than k-"

- More stable (easier convergence)

- Need for 2nd order Reynolds stress closures

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

17

Acknowledgments - FundingFundação para a Ciência e TecnologiaProject PTDC/EQU-FTT/70727/2006

Date post:	21-Dec-2018
Category:	Documents
Upload:	truongque
View:	216 times
Download:	0 times

A k LOW REYNOLDS NUMBER TURBULENCE MODEL FOR …fpinho/pdfs/CI1-AERC2010_Turb_k_omega_v2.pdf · A...

Documents