XIIIth International Workshop on Numerical …fpinho/pdfs/CI54_Lausanne2003...V. Valtsifer and N. Zvereva, Computer simulation and experimental investigation of rheological behaviour

XIIIth International Workshop on Numerical Methods for non-Newtonian Flows

Hôtel de la Paix, Lausanne, 4-7 June 2003

PROGRAMME Wednesday Afternoon, 4 June 2003 ....................................................................... 2

Novel Numerical Methods I (Chairman : Roland Keunings) .................................... 2 High-Order Numerical Methods (Chairman : Antony Beris) .................................... 2

Thursday Morning, 5 June 2003 .............................................................................. 3 Novel Numerical Methods II (Chairman : Tim Phillips) ............................................ 3 Concentrated Solutions and Melts (Chairman : Mike Graham ) .............................. 3

Thursday Afternoon, 5 June 2003 ........................................................................... 4 Viscoplastic/Inelastic Flows (Chairman : Ian Frigaard)............................................ 4 Suspensions and Mixing (Chairman : Gareth McKinley) ......................................... 4

Friday Morning, 6 June 2003 ................................................................................... 5 Bead-Spring Modelling and Brownian Dynamics (Chairman : Eric Shaqfeh) .......... 5 Turbulent Flows (Chairman : Radhakrishna Sureshkumar) ................................... 5

Friday Afternoon, 6 June 2003 ................................................................................ 6 Free Surface Flows (Chairman : Bamin Khomami)................................................. 6 Stability and Nonlinear Dynamics (Chairman : Peter Monkewitz) ........................... 6

Saturday Morning, 7 June 2003............................................................................... 7 Macroscopic Constitutive Modelling and Benchmark Problems (Chairman : Jay Schieber)................................................................................................................. 7 Theoretical Developments (Chairman : Raj Huilgol) ............................................... 7

1

Wednesday Afternoon, 4 June 2003

Registration 10:00 AM �– 12:00 PM Lunch 12:00 PM �– 1:30 PM

Introduction 1:45 PM �– 2:00 PM Lectures 2:00 PM �– 6:30 PM

Novel Numerical Methods I (Chairman : Roland Keunings)

2:00 PM �– 2:20 PM M. Laso and J. Ramírez. Implicit micro-macro methods.

2:25 PM �– 2:45 PM M. Ellero, M. Kröger and S. Hess. A hybrid method for efficient CONNFFESSIT simulations of fully uncorrelated ensembles of polymers.

2:50 PM �– 3:10 PM C. Le Bris, B. Jourdain and T. Lelièvre. On variance reduction issues in the micro-macro simulations of polymeric fluids.

3:15 PM �– 3:35 PM C. Chauvière and A. Lozinski. Simulation of dilute polymer solutions using a Fokker-Planck equation (comparison between 2D and 3D FENE models).

3:40 PM �– 4:00 PM G. Pan and C. Manke. Simulation of polymer solutions by dissipative particle dynamics.

Break 4:05 PM �– 4:25 PM

High-Order Numerical Methods (Chairman : Antony Beris)

4:25 PM �– 4:45 PM M. I. Gerritsma. Least-squares spectral element methods for non-Newtonian flow.

4:50 PM �– 5:10 PM N. Fiétier. Simulation of viscoelastic fluid flows through contractions and constrictions with spectral and mortar element methods.

5:15 PM �– 5:35 PM R. G. M. van Os and T. N. Phillips. The prediction of complex flows of polymer melts using spectral elements.

5:40 PM �– 6:00 PM T. N. Phillips and K. D. Smith. A spectral element approach to the simulation of viscoelastic flows using Brownian configuration fields.

6:05 PM �– 6:25 PM X. Ma, V. Symeonidis and G.E. Karniadakis. A spectral vanishing viscosity method for stabilizing viscoelastic flows.

Dinner 7:00 PM

2

Thursday Morning, 5 June 2003

Breakfast 7:00 AM �– 8:30 AM Lectures 8:30 AM - 12:10 AM

Novel Numerical Methods II (Chairman : Tim Phillips)

8:30 AM �– 8:50 AM I. J. Keshtiban, F. Belblidia and M. F. Webster. Simulating weakly compressible non-Newtonian flows.

8:55 AM �– 9:15 AM H. K. Rasmussen. The 3D Lagrangian integral method.

9:20 AM �– 9:40 AM A. Lozinski and R. G. Owens. Modelling highly non-homogeneous flows of dilute polymeric solutions using Fokker-Planck-based numerical methods.

Concentrated Solutions and Melts (Chairman : Mike Graham )

9:45 AM �– 10:05 AM J. van Meerveld and H. C. Öttinger. Molecular-based description of polydisperse polymeric liquids.

Break 10:10 AM �– 10:30 AM

10:30 AM �– 10:50 AM P. Wapperom and R. Keunings. Impact of decoupling approximation between tube stretch and orientation in rheometrical and complex flow simulation of entangled linear polymers.

10:55 AM �– 11:15 AM J. Fang, A. Lozinski and R. G. Owens. More realistic kinetic models for concentrated solutions and melts.

11:20 AM �– 11:40 AM J. Schieber. Solving a full chain temporary network model with sliplinks, contour-length fluctuations, chain stretching, and constraint release using Brownian dynamics.

11:45 AM �– 12:05 PM P. K. Bhattacharjee, J. Ravi Prakash and T. Sridhar. Stress relaxation after step extensional strain in an entangled polymer solution.

12:10 PM �– 12:30 PM T.M. Nicholson. Measurement and modelling of polymer melt flow and extrudate swell.

3

Thursday Afternoon, 5 June 2003

Lunch 12:40 PM �– 1:45 PM Lectures 2:00 PM �– 6:10 PM

Viscoplastic/Inelastic Flows (Chairman : Ian Frigaard)

2:00 PM �– 2:20 PM E. Mitsoulis. Flow of viscoplastic fluids through expansions and contractions.

2:25 PM �– 2:45 PM E. Mitsoulis and R. Huilgol. Finite stopping times in Couette and Poiseuille

flows of viscoplastic fluids. 2:50 PM �– 3:10 PM

M. A. Moyers-Gonzalez and I. A. Frigaard. Accurate numerical solution of multiple visco-plastic fluids in ducts.

3:15 PM �– 3:35 PM S. Alexandrov. Frictional effects in viscoplastic flows.

3:40 PM �– 4:00 PM D. Vola. On a numerical strategy to compute non-Newtonian fluids gravity currents.

Break 4:05 PM �– 4:25 PM

4:25 PM �– 4:50 PM

S. Miladinova and G. Lebon. Thin-film flow of a power-law liquid down an inclined plate.

Suspensions and Mixing (Chairman : Gareth McKinley)

4:55 PM �– 5:15 PM V. Legat. Micro-macro modelling of black carbon mixing.

5:20 PM �– 5:40 PM W. R. Hwang, M. A. Hulsen, H. E. H. Meijer. Direct simulations of particle suspensions in viscoelastic fluids in Lees-Edwards sliding bi-periodic frames.

5:45 PM �– 6:05 PM V. Valtsifer and N. Zvereva, Computer simulation and experimental investigation of rheological behaviour of nanoparticles in suspension.

Reception 6:45 PM

Bus departs 7:45 PM

Lake cruise with buffet dinner on the �“Henry Dunant�”

from Ouchy 8:00 PM �– 10:30 PM

4

Friday Morning, 6 June 2003

Breakfast 7:00 AM �– 8:30 AM Lectures 8:30 AM - 12:10 AM

Bead-Spring Modelling and Brownian Dynamics (Chairman : Eric Shaqfeh)

8:30 AM �– 8:50 PM R. Prabhakar and J. Ravi Prakash. Superposition of finite extensibility, hydrodynamic interaction and excluded volume effects in bead-spring chain models for dilute polymer solutions.

8:55 AM �– 9:15 AM P. T. Underhill and P. S. Doyle. On the coarse-graining of polymers into bead-spring chains.

9:20 AM �– 9:40 AM R. Akhavan, Q. Zhou. A multi-mode FENE bead-spring chain model for dilute polymer solutions.

9:45 AM �– 10:05 AM R. M. Jendrejack, J. J. de Pablo and M. D. Graham. DNA dynamics in a microchannel: relaxation, diffusion and cross-stream migration during flow

Break 10:10 AM �– 10:30 AM

Turbulent Flows (Chairman : Radhakrishna Sureshkumar)

10:30 AM �– 10:50 AM V. K. Gupta, R. Sureshkumar and B. Khomami. Numerical simulation of polymer chain dynamics in turbulent channel flow.

10:55 AM �– 11:15 AM K. D. Housiadas and A. N. Beris. Direct numerical simulations of polymer-induced drag reduction in turbulent channel flows.

11:20 AM �– 11:40 AM M. Manhart. A coupled DNS/Monte-Carlo solver for dilute suspensions of small fibres in a Newtonian solvent.

11:45 AM �– 12:05 PM D. O. A. Cruz and F. T. Pinho. A low Reynolds number k model for drag reducing fluids.

5

Friday Afternoon, 6 June 2003

Lunch 12:15 PM �– 1:45 PM Lectures 2:00 PM �– 5:40 PM

Free Surface Flows (Chairman : Bamin Khomami)

2:00 PM �– 2:20 PM G. McKinley. Free surface flows of viscoelastic fluids 2:25 PM �– 2:45 PM

G. Bhatara, E. S. G. Shaqfeh and B. Khomami. A study of a free surface viscoelastic Hele-Shaw cell flow using the finite element method

2:50 PM �– 3:10 PM A. Bonito, M. Laso and M. Picasso. Numerical simulation of 3D non-Newtonian flows with free surfaces.

3:15 PM �– 3:35 PM K. Foteinopoulou, V. Mavrantzas and J. Tsamopoulos. Numerical simulation of bubble growth during filament stretching of pressure-sensitive adhesive materials.

3:40 PM �– 4:00 PM Y. Dimakopoulos and J. Tsamopoulos. Gas-penetration in straight tubes partially or completely occupied by a viscoelastic fluid.

Break 4:05 PM �– 4:25 PM

Stability and Nonlinear Dynamics (Chairman : Peter Monkewitz)

4:25 PM �– 4:45 PM K. Atalik and R. Keunings. On the occurrence of even harmonics in large amplitude oscillatory shear experiments.

4:50 PM �– 5:10 PM B. Sadanandan, K. Arora and R. Sureshkumar. Stability analysis of non-viscometric viscoelastic flows.

5:15 PM �– 5:35 PM M. Sahin and R. G. Owens. Linear stability analysis of the non-Newtonian flow past a confined circular cylinder in a channel.

Workshop Banquet. Bus Departs 6:30 PM

6

Saturday Morning, 7 June 2003

Breakfast 7:00 AM �– 8:55 AM Lectures 8:55 AM �–12:10 AM

Macroscopic Constitutive Modelling and Benchmark Problems (Chairman : Jay Schieber)

8:55 AM �– 9:15 AM G. Mompean, L. Thais and L. Helin. Numerical simulation of viscoelastic flows using algebraic extra-stress models based on differential constitutive equations.

9:20 AM �– 9:40 AM Y. Fan. Boundary layers in the viscoelastic flow around a confined cylinder. 9:45 AM �– 10:05 AM

M. A. Alves, P. J. Oliveira and F. T. Pinho. Flow of PTT fluids through contractions �– effect of contraction ratio.

Break 10:10 AM �– 10:30 AM

Theoretical Developments (Chairman : Raj Huilgol)

10:30 AM �– 10:50 AM X. Xie and M. Pasquali. A convenient way of imposing inflow boundary conditions in two- and three-dimensional viscoelastic flows.

10:55 AM �– 11:15 AM B. Caswell, G. E. Karniadakis and V. Symeonidis. The hole-pressure due to a tube on one wall of a plane channel.

11:20 AM �– 11:40 AM A. R. Davies. Transient decay rates in some common constitutive models of differential and integral type.

11:45 AM �– 12:05 PM M. Renardy. Jet breakup of a Giesekus fluid with inertia.

Lunch 12:15 PM �– 1:45 PM

7

D. O. A. CruzDepartamento de Engenharia Mecânica, Universidade Federal do Pará,Belém, Brasil, [email protected]

A LOW REYNOLDS NUMBER A LOW REYNOLDS NUMBER k-k-!! MODEL MODEL

FOR DRAG REDUCING FLUIDSFOR DRAG REDUCING FLUIDS

F. T. PinhoCentro de Estudos de Fenómenos de Transporte, Faculdade deEngenharia, Universidade do Porto, [email protected],http://www.fe.up.pt/~fpinho

XIIIth International Workshop on Numerical Methods for Non-Newtonian Flows4 - 7th June 2003Hôtel de la Paix, Lausanne, Switzerland

Low Reynolds number k-! model for drag reducing fluids XIIIth International Workshop on Numerical

Methods for Non-Newtonian Flows

Outline

• Introduction

• Proposed model

• Results

• Conclusions

• Future developments



Introduction

• Fluid rheology affects “hydrodynamic” behaviour

• Viscoelastic effects on turbulence are largely unpredicted

• Phenomenological models exist

• Fluid rheology: measured properties

• Flow rate or pressure gradient and geometry

• Turbulent flow characteristics

Turbulence modelling of viscoelasticengineering flows remains a challenge

WE WANT



Introduction - Objective

Objective: Development of a coupled turbulence-constitutive equation closure

• Selection of (simple) constitutive equation

• Reynolds averaged equations

• Modelling new terms and modification of transport equations

• Damping functions for low Reynolds number effects

• Comparisons for parameterisation and behaviour assessment



Proposed model - Rheological constitutive model Route 1

FENE-P:

! ij ="p

#f Akk( )Akk $%ij[ ]

f Akk( )Aij + !"Aij

"t+ uk

"Aij

"xk# A jk

"ui"xk

# Aik

"u j

"xk

$

% &

'

( ) = *ij

f Akk( ) = L2

L2 ! Akk

with

•Viscoelastic model: shear-thinning, viscoelastic, DNS simulations

TOO COMPLEX at this stage:double, triple & quadruple (?) correlations



Proposed model - Rheological constitutive model Route 2

•Simple model: shear-thinning, with assumed relevant features

•Reduced modifications relative to Newtonian turbulence model

•Drag reduction: relevance of strain-hardening extensional

viscosity for constant shear viscosity

•Fluids are usually shear-thinning

•Strain/shear- hardening Trouton ratio

MODIFIED GENERALISED NEWTONIAN FLUID



Proposed model - Modified GNF

! ij = 2µSij

µ = !v "Ke ˙ # 2[ ]p$1

2 % µ = Kv ˙ & 2[ ]n$1

2 Ke ˙ # 2[ ]p$1

2

13!e ˙ " ( )!v ˙ # ( )

= Ke ˙ " 2[ ]p$12

•Effect of only in turbulent flow

•Effect of reduced under low Reynolds number

conditions

˙ !

˙ !

Shear viscositycontribution Extensional

viscositycontribution

via Trouton ratio

n <1

p >1



Proposed model - Transport equations 1

•Momentum

! "Ui"t

+ !Uk"Ui"xk

= # "p"xi

+" 2µ Sik + 2µ'sik # !uiuk( )

"xk

New stressModified stress

Closure by 2-equationmodel: k-!•Turbulent kinetic energy

! DkDt

= "#u j p#x j

" ##x j

12!uiu jui " 2µuisij " 2µ'uiSij " 2µ'uisij

$ % &

' ( ) " 2µsij

2 " 2µ'sij2 "

!2µ'sijSij ! "uiu jSijNew term

Dissipation: !



Proposed model - Transport equations 2

•Rate of dissipation

1) Very complex: as with Newtonian fluids, all terms are modelled

2) There are new terms originating from advection (order of mag.)

3) Weakness of any turbulence model

•Two models tested:

Model A: modified stress and without new stress

Model B: modified stress and new stress (preliminary res.)



Proposed model - Average viscosity (Models A & B)

µ = fvµ h + 1! fv( )"v

!v = Kv ˙ " 2[ ]n#1

2

µ h = Cµ!( )3m(m"1)A28+3m(m"1)A2 2

4m(m"1)A28+3m(m"1)A2 k

6m(m"1)A28+3m(m"1)A2#

8"3(m"1)A2[ ]m8+3m(m"1)A2 B

88+3m(m"1)A2

pure shear viscosity at walls

high Reynolds number contribution(effect of fluctuating " and ! ). .

(derived from order of magnitude and pdf arguments)

Rheological measurements: B, m



Proposed model - New stress (Model B) 1

2µ' sij ?

µ' ! KvKe ˙ " ' p#1 ˙ $ ' n#1

˙ ! ' " ˙ # 'A!

sijsijA!

= SA!

2µ' sij !KvKeA"p#1 S

p+n#2sij

In boundary layers: Pk = !" # !uv $U$y

% 2µ S2

sij !"ui"x j

#uiujL Inviscid estimate of dissipation of k



Proposed model - New stress (Model B) 2

2µ' sij = 1+C0[ ]p+n!2 !1{ }KvKeA"p!1

#$T%U%y

& ' (

) * + 2

2µ

,

-

.

.

.

.

.

/

0

1 1 1 1 1

p+n!22

$T1LC

dUdydUdy

1Lc

= !uR3with and uR

2 = k

exp ! ku"2

# $ %

& ' ( !1

) * +

, - . 1/

to match low and high Reynolds number behaviour 1Lc

= !u"3

1Lc

= !k3/ 2



Proposed models A & B - Pipe flow

•Momentum:

•Reynolds stress:

•k equation:

• ! equation:

•Parameters and functions: Nagano - Hishida (except f!)

1r

ddr

r µ dUdr

! "uv + 2µ' sxr# $ %

& ' (

) * +

, - . !

dp dx

= 0

Model B

Model B

1rddr

r µ !

+ "T#k

$

% &

'

( ) dkdr

*

+ ,

-

. / 0 uv1U

1r0 2µ' sxr

1U1r

0 ˜ 2 + 2" 1 k1r

$ % &

' ( )

2= 0

1rddr

r µ + !"T#$

%

& '

(

) * d ˜ $ dr

+

, -

.

/ 0 + !f1C$1

˜ $ kP 1 !f2C$2

˜ $ 2

k+" "T 11 fµ( ) 22U

2r2%

& ' '

(

) * *

2+ C$3

"T#E"

d ˜ $ drdµ dr

= 0

!"uv = "#T$U$r

% ! "uv = "Cµ fµk2

˜ & $U$r



Proposed model - Damping function

fµ = 1! 1+ 1! n1+ n

y+" # $

% & ' ! 1+n

1!n A+( ) *

+ *

, - *

. * / 1! 1+ p !1

3! py+C

1! p2! p

"

# $ $

%

& ' '

! 3! p p!1 A+(

) *

+ *

,

- *

. *

viscometric contribution extensional contribution

(wall viscosity)

to quantify

•Van Driest’s (1956) philosophy: Stokes second problem

•Shear-thinning and strain-hardening contributions

Van Driest’s parameter

fµ = 1! exp ! y+ A+( )[ ]" 1! exp ! y+ A+( )[ ]Newtonian:



Results - Experimental (Escudier et al, 1999) 1

10-3

10-2

10-1

103 104 105

0.125% PAA0.2% PAA0.25% CMC0.3% CMC0.2% XG0.09/0.09% XG/CMC

f

Re0

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+







•N1 data not used



Results model A- Determination of C - 0.125% PAA

10-3

10-2

10-1

103 104 105

Exp dataC=5; M1C=7; M1C=9; M1C=5; M2C=7; M2C=9; M2M1; p=1M2; p=1

f

Rew

f = 64 / Re

f = 0.316Re!0.25

inelastic shear-thinning

MDRA-Virk

M1: in f! M2: in f!

y +

yw+

Best damping function

Formulation M2

C= 9

Two formulations of

fµ



Results model A - Friction factor - Limiting cases

10-3

10-2

10-1

103 104 105

f

Reg

1.00.8

0.6

0.4

n

10-3

10-2

10-1

103 104 105

f

Rew

p1.01.21.4

1.6

1.8

Virk's MDRA

f=64/Re

Dodge & Metzner, n=1

Ke=2Ke=0.5

Inelastic shear-thinningn<1;p=1

Strain-hardeningn= 1; p>1



Results model A - Friction factor - Other drag reducing fluids 1

10-3

10-2

10-1

103 104 105

Exp dataPred M1Pred M2

f

Rew

f = 0.316Re!0.25

f=64/Re

Dodge & Metzner(1959)

MDRA- Virk

0.25% CMC0.2% XG

10-3

10-2

10-1

103 104 105


f

Rew

f = 0.316Re!0.25

f=64/Re


MDRA- Virk



Results model A - Friction factor - Other drag reducing fluids 2

10-3

10-2

10-1

103 104 105


f

Rew

f = 0.316Re!0.25

f=64/Re


MDRA- Virk

0.3% CMC

10-3

10-2

10-1

103 104 105


f

Rew

f = 0.316Re!0.25

f=64/Re


MDRA-Virk

0.09% CMC/ XG



Results model A - velocity profile - Limiting cases

u+ = yDM+[ ]1/ n

0

5

10

15

20

25

30

35

40

10-1 100 101 102 103

u+

y+DM

n=0.4

0.6

0.8

1.0

Dodge & Metzner log law

0

10

20

30

40

50

60

70

80

100 101 102 103

p=1p=1.2p=1.4p=1.6p=1.8

u+

y+

w

Virk's MDRA

Newtonian log law

Inelastic shear-thinningn<1;p=1

Strain-hardeningn= 1; p>1



Results model A - velocity profile - drag reducing fluids 1

0

5

10

15

20

25

30

35

1 10 100 1000

Exp dataPred M1Pred M2Pred Newtu+=2.5lny+

w+5.5

u+=11.7lny+w-17.0

u+=y+w

u+

y+w

0.09% CMC/ XG (Re=45 300)

0

5

10

15

20

25

30

35

1 10 100 1000

Exp dataPred M1Pred M2u+=2.5lny+

w+5.5

u+=11.7lny+w-17.0

u+=y+w

u+

y+w

0.25% CMC (Re=16 500)



Results model A - velocity profile - drag reducing fluids 2

0

5

10

15

20

25

30

35

1 10 100 1000


w+5.5

u+=11.7lny+w-17.0

u+=y+w

u+

y+w

0.3% CMC (Re=4 300)

0

5

10

15

20

25

30

35

1 10 100 1000


w+5.5

u+=11.7lny+w-17.0

u+=y+w

u+

y+w

0.125% PAA (Re=42 900)



Results model A - Turbulent kinetic energy

0

2

4

6

8

10

12

14

1 10 100 1000

Exp dataPred NewtPred M1Pred M2

k+

y+w0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.0 0.2 0.4 0.6 0.8 1.0

Exp dataPred NewtPred M1Pred M2

k/U2

r/R

0.125% PAA (Re= 42900)



Results model A - Shear stress & damping function

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Pred NewtPred M1Pred M2

uv/u

t2

r/R

0.125% PAA (Re= 42900)

0.0

0.2

0.4

0.6

0.8

1.0

1 10 100 1000

NewtModel 1Model 2

fµ

y+w



Results model B - 1

10-3

10-2

10-1

103 104 105

0.125%PAA p=1Exp dataC=9 C

0=0

C=9 C0=-0.6

C=7 C0=-0.6

C=7 C0=-0.9

C=5 C0=-0.9

f

Rew

0.125% PAA 0.125% PAA - Re!40000

0

10

20

30

40

50

60

10-1 100 101 102 103

Exp. dataC=9 C

0=0

C=9 C0=-0.6

C=5 C0=-0.9

u+

y+

w



Results model B - 2

0

2

4

6

8

10

12

100 101 102 103

Exp. dataC=9 C

0=0

C=9 C0=-0.6

C=5 C0=-0.9

k+

y+w

0.125% PAA - Re!40000

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

2µSij

-uv

2µ'sij

! /"U

!2

r/R

C=9 C0=-0.6



Conclusions

• Coupled turbulence-rheological model derived. Only requires fluid properties as input

• No previous tuning required

• Predicts intense drag reduction (DR) for large p

• Predicts well DR for inelastic shear-thinning fluid

• Predictions of DR for elastic fluids is fair• Improvements required in u+ - y + & k

• New term in momentum and k helps, but more inv. required

Different formulation?



Future developments

• Wall consistent damping functions

• Non-linear k-! for Reynolds stress anisotropy

• Reynolds stress model for Reynolds stress anisotropy

• Adoption of true viscoelastic rheological equation:Oldroyd-B/FENE-CR and FENE-P

• Development of turbulence closure. Use of DNSresults

• k-!, non-linear k-! and Reynolds stress models

SHORT TERM

MEDIUM TERM

D. O. A. CruzDepartamento de Engenharia Mecânica, Universidade Federal do Pará,Belém, Brasil, [email protected]

A LOW REYNOLDS NUMBER A LOW REYNOLDS NUMBER k-k-!! MODEL MODEL

FOR DRAG REDUCING FLUIDSFOR DRAG REDUCING FLUIDS

F. T. PinhoCentro de Estudos de Fenómenos de Transporte, Faculdade deEngenharia, Universidade do Porto, [email protected],http://www.fe.up.pt/~fpinho

XIIIth International Workshop on Numerical Methods for Non-Newtonian Flows4 - 7th June 2003Hôtel de la Paix, Lausanne, Switzerland

ACKNOWLEDGMENTS1) FCT (Portugal):Proj. POCTI/EME/37711/2001; POCTI/EQU/37699/20012) ICCTI (Portugal)- CNPq (Brasil)

Documents

XIIIth International Workshop on Numerical …fpinho/pdfs/CI54_Lausanne2003...V. Valtsifer and N. Zvereva, Computer simulation and experimental investigation of rheological behaviour