R. J. Poole and M. P. Escudier Dept. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK, [email protected],[email protected].

R. J. Poole and M. P. EscudierDept. Engineering, Mechanical Engineering, University of LiverpoolLiverpool L69 3GH, UK, [email protected],[email protected]

3D FLOW OF VISCOELASTIC FLUIDS OVER A 3D FLOW OF VISCOELASTIC FLUIDS OVER A

BACKWARD-FACING STEP PRECEDED BY A BACKWARD-FACING STEP PRECEDED BY A

GRADUAL CONTRACTIONGRADUAL CONTRACTION

A. AfonsoCentro de Estudos de Fenómenos de Transporte, DEMEGI Faculdade de Engenharia, Universidade do Porto, Portugal, [email protected]

F. T. PinhoCentro de Estudos de Fenómenos de Transporte, Dep. Eng. MecânicaEscola de Engenharia, Universidade do Minho, Portugal, [email protected]

AERC 200522nd to 24th April 2005 Grenoble, France

3D Viscoelastic flow in smooth contractions European Congress on Rheology

AERC 2005

Flow geometry

Experiments of Poole et al (2004) with solutions of PAA

Area ratio R = d/D = 0.7

Upstream spanwise velocity profiles (x-z plane) at x/h=-8.33 and 0

d = 28mm, h = 6mm,

D = 40mm, w = 80mm

Inlet duct: 120 DH long

(area ratio > 2/3 double backward-facing step )

Aspect ratios A1 = w/h = 13.3

A2 = w/d= 2.86


AERC 2005

Experimental and numerical findings

Figure : Spanwise variation of mean streamwise velocity ( U/UB ) profiles for 0.1% PAA

z / w

U/UB

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

NewtonianCarreau-YasudaPTT

x / h = 0

0.1% PAA

x / h = -8.33

0.1% PAA Re 120

Cat’s ears

Spanwise variation at y/D=0.5GNF

PTT(N2=0)


AERC 2005

Experimental and numerical findings 3

0.1% PAA Re 120Downstream

Figure : Mean streamwise velocity ( U/UB ) profiles for 0.1% PAA

y/D

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

PTTx/h0 1 2 3 4

1

U = 0

XR = 2.3

Newtonian Carreau-Yasuda


AERC 2005

Objective

Cat’s ears: Why?

Shear-thinning:No

Elasticity - : No Ψ1

Qualitative calculation with PTT: parametric investigation

Effect of

Effect of De

Effect of

Effect of Re

Ψ2

ηe

Individual and combined effects


AERC 2005

Governing equations

∂∂tJρui( ) +

∂

∂ξ lρβ lj %u jui( ) = −β li

∂p

∂ξ l+

∂

∂ξ lβ lj %τ ij , p + β ljτ ij ,s( ) + +Jρgi

1) Mass

2) Momentum

3) Constitutive equation

€

1+λε

η pτ kk,p

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟τ ij,p + λ

∂τ ij,p∂t

+∂ ukτ ij,p( )

∂xk

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟=η p

∂ui∂x j

+∂u j∂xi

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟+ λ τ jk,p

∂ui∂xk

+ τ ik,p∂u j∂xk

⎛

⎝ ⎜

⎞

⎠ ⎟−

€

λξ2τ jk,p

∂ui∂xk

+∂uk∂xi

⎛

⎝ ⎜

⎞

⎠ ⎟+ τ ik,p

∂u j∂xk

+∂uk∂x j

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Full PTT (linear stress coefficient)

τ ij = 2η sDij + τ ij , p

Newtonian solvent


AERC 2005

Numerical method: brief description

2) Structured, colocated and non-orthogonal meshes

3) Momentum (ui)

∂∂t

Jρui( )+∂

∂ξlρβlj ˜ u j ˆ u i( )−

∂∂ξl

ηp

Jβljβlj

∂ui∂ξl

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ −

∂ βljτij,s( )

∂ξl=−βli

∂p∂ξl

+∂

∂ξlβlj˜ τ ij( )+Jρgi −

∂∂ξl

ηp

Jβljβlj

∂ui∂ξl

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

solvent polymer

1) Finite volume method (Oliveira et al,1998; Oliveira & Pinho, 1999)

4) Discretization (formally 2nd order)Diffusive terms: central differences (CDS)Advective terms: CUBISTA (deferred correction)

(Alves et al, 2000, 2003)

5) Special formulations for cell-face velocities and stresses


AERC 2005

Computational domain and mesh

5 m (62 DH)

120 h

20 cells

30 cells

102 000 total cells1 020 000 DF


AERC 2005

Inlet flow

x/h=-16

ξ =0.001 ξ =0.2

ε =0.05,λ = 0.1 s,η p = 0.5 Pa.s,η s = 0.05 Pa.s


AERC 2005

Non-dimensional numbers

Re'= ρUbhηs+ηp( )

Reynolds number

De=Ubλp

hDeborah number

Re=ρUbhμc

&γC =Ubh

with and

ε

ξ

Extensional parameter

Slip parameter

Bulk velocity at contraction exit


AERC 2005

Effect of ξ: 1

Re '=5.14

kitten’s ears

Absence of kitten’s ears

Several values of ε

ε =0.15


AERC 2005

Effect of ξ: 2

Re '=5.14

kitten’s ears:high De, high ξ, low ε


AERC 2005

Effect of ξ: 3

Re '=5.14x h=−8.0;−4.0;−2.06;−0.1

kitten’s ears

x/h=-8

x/h=-4

x/h=-2.06

x/h=-0.1

Effect ofinertia


AERC 2005

Effect of εRe '=5.14 ξ =0.2

Closed symbols: kitten’s ears

x/h=-8

x/h=-4

x/h=-2.06

x/h=-0.1

(a) (b)

Effect of De (next slide)


AERC 2005

Effect of De

Re '=5.14 ξ =0.2

ε =0.025ε =0.1

DeDe


AERC 2005

Effect of Re’: 1

ξ =0.2 ε =0.005De=19.6 De=39.2

QuickTime™ and aBMP decompressor

are needed to see this picture.



Re=0.6 Re’=0.48 Re=0.6 Re’=0.43


AERC 2005

Effect of Re’: 2

ξ =0.2 ε =0.005De=19.6 De=39.2

Re=1.7 Re’=1.4 Re=1.7 Re’=1.3






AERC 2005

Effect of Re’: 3

ξ =0.2 ε =0.005De=19.6 De=39.2

Re=3.4 Re’=2.8 Re=3.4 Re’=2.6






AERC 2005

Effect of Re’: 4

ξ =0.2 ε =0.005De=19.6 De=39.2

Re=6.3 Re’=5.2 Re=6.3 Re’=4.7






AERC 2005

Conclusions

•Cat’s ears are qualitatively predicted by PTT (kitten’s ears)

•N2≠ 0 (essential)— high ξ

•Low ε

•High De

•Intermediate Re

•Sometimes enhanced peaks observed at corners

•Low Re: very slim profiles at contraction exit, no peaks

•High Re: flat profiles at contracton exit, no peaks

•Accurate predictions: different transient properties ???

R. J. Poole and M. P. Escudier Dept. Engineering, Mechanical Engineering, University of Liverpool Liverpool L69 3GH, UK, [email protected],[email protected].

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