plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio robert j poole...

Plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio

Robert J PooleDepartment of Engineering, University of Liverpool, UK

Manuel A Alves

CEFT, Faculdade de Engenharia, Universidade do Porto, Portugal

Fernando T PinhoaCEFT, Faculdade de Engenharia, Universidade do Porto, Portugal bUniversidade do Minho, Portugal

Paulo J Oliveira

Departamento de Engenharia Electromecânica, Universidade da Beira Interior, Portugal

AERC 20074th Annual European Rheology Conference

April 12-14, Napoli - Italy

Outline

• Introduction

• Governing equations

• Numerical method / grid dependency issues

• Newtonian results

• UCM simulations: “High” ER followed by “Low” ER

• Conclusions



Introduction

Prevailing view….vortex suppressed by elasticity and totally eliminated at “high” Deborah



Not the whole story (AERC 2006 Poole et al, JNNFM 2007 to appear)

UCM/Oldroyd-B (β = 1/9) simulations, 1:3 expansion ratio, creeping flow

• Maximum obtainable De ≈ 1• Effect of elasticity is to reduce but not eliminate recirculation

• Enhanced pressure drop observed

Why investigate expansion flows of viscoelastic liquids?

Governing equations

1) Mass 0 u

2) Momentum (creeping flow) τ p0

3) Constitutive equation Upper Convected Maxwell model (UCM)

τuuτuuτuττ

TT

t



Essentially phenomenological model• “Simplest” viscoelastic differential model• Capable of capturing qualitative features of many highly-elastic

flows

Numerical method

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

2) Structured, collocated and non-orthogonal meshes

3) Discretization (formally second order)Diffusive terms: central differences (CDS)Convective terms: CUBISTA (Alves et al (2003))

4) Special formulations for cell-face velocities and stresses



Computational domain and meshes

Y

X

ER=D/d

L2= 100dL1= 20d

h

d

D

UB

symmetry axis

d

U.De B



Expansion ratios (ER)

1:1.5

1:2

1:3

1:4

1:8

1:16

1:32Fully-developedinlet velocity and stress profiles

Neumann b.c.s at exit

Low ER

High ER

Representative mesh details

ER = 4 NC DOF (xMIN)/d

M1 15 000 90 000 0.01

M2 60 000 360 000 0.005



ER = 16 NC DOF (xMIN)/d

M1 21 500 129 000 0.01

M2 86 000 516 000 0.005

ER = 1.5 NC DOF (xMIN)/d

M1 14 500 87 000 0.005

M2 58 000 348 000 0.0025

Representative grid dependency and numerical accuracy

ER and fluid XR (= xR / d) XR# %

error

M1 M2

Newtonian ER =1.5 0.3300 0.3298 0.3298 0.02%

Newtonian ER = 2 0.5915 0.5914 0.5913 0.01%

Newtonian ER =4 1.4977 1.4994 1.4999 0.04%

Newtonian ER = 16 6.5603 6.5573 6.5562 0.02%

De = 1.0 ER =1.5 0.3366 0.3426 0.3447 0.59%

De = 1.0 ER =2 0.5528 0.5501 0.5492 0.16%

De = 1.0 ER =4 1.2339 1.2303 1.2291 0.12%

De = 1.0 ER =16 6.2545 6.2490 6.2471 0.03%#denotes extrapolated value using Richardson’s technique



XXXX

XX

X

X

ER - 1

XR

(=x r

/d)

0 10 20 300

2

4

6

8

10

12

14Newtonian M1Newtonian M2XR = 0.4185 (ER - 1) + 0.2635

X

Newtonian simulations: XR variation with ER



d

Linear fit to data for ER 4 (R2=1)

XXXX

XX

X

X

ER - 1

XR

(=x r

/d)

0 10 20 300

2

4

6

8

10

12


X




X

X

X

X

X

X

ER - 1

XR

(=x r

/d)

0 1 2 3 40

0.25

0.5

0.75

1

1.25

1.5

1.75


X

Deviations from linear fit as ER 1

X

X

X

X

XX

X X X

ER - 1

x r/D

0 10 20 300

0.1

0.2

0.3

0.4

0.5

Newtonian M1Newtonian M2XR = 0.4185 (ER - 1) + 0.2635

X




DH

X X X X X X

X X X X X X

X X X X X X X

De

XR

(=x r

/d)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

2

4

6

8

10

12

14

ER = 4

ER = 32

ER = 16

ER = 8

“High” ER viscoelastic : XR variation with De and ER



Δ M1

X M2

Extrapolated

X

X

X

X

X

X

De

XR

(=x r

/d)

0 0.2 0.4 0.6 0.8 1 1.2 1.41.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

ER = 4.0 M1ER = 4.0 M2Extrapolated

X

1:4 expansion ratio



De = 0.0De = 0.2De = 0.4De = 0.6De = 0.8De = 1.0

1:4 expansion ratio (M2)



XX

X

X

XX

X XX

XX

X

X X X X X X X

De

x r/D

0 0.2 0.4 0.6 0.8 1 1.2 1.40.2

0.25

0.3

0.35

0.4

0.45

“High” ER viscoelastic : scaling of XR



XX

X

XX

XX

XXX

X X X X XX

De / ER

x r/D

10-2 10-1 1000.2

0.25

0.3

0.35

0.4

0.45

ER =4ER =8ER =16ER =32

X X X X X X X X X

X X X X X X X

X

X

De

XR

(=x r

/d)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

ER = 1.5

ER = 3

ER = 2

“Low” ER viscoelastic : XR variation with De and ER



X

X

X

X XX

X

X

X

De

XR

(=x r

/d)

0 0.2 0.4 0.6 0.8 10.3

0.31

0.32

0.33

0.34

0.35

ER = 1.5

1:1.5 expansion ratio 1:2 expansion ratio



XX

X

X

X X

X

De

XR

(=x r

/d)

0 0.2 0.4 0.6 0.8 10.5

0.55

0.6

0.65

ER = 2

1:1.5 expansion ratio



De = 0.0De = 0.1De = 0.2De = 0.3De = 0.4De = 0.6De = 0.8De = 1.0 De = 0.0De = 0.1De = 0.2De = 0.3De = 0.4De = 0.6De = 0.8De = 1.0

X X X X X X XX

X

X XX

X X X X

X

X

XX

X

X

XX

X XX

XX

X

X X X X X X X

De

x r/D

0 0.2 0.4 0.6 0.8 1 1.2 1.40.2

0.25

0.3

0.35

0.4

0.45

ER = 4

ER = 32

ER = 16ER = 8

ER = 1.5

ER = 2

ER = 3

“Low” ER viscoelastic : scaling of XR



X X X X XXXX

XX

X X X X

X

XX

X

XX

XX

XXX

X X X X XX

De / ER

x r/D

10-2 10-1 1000.2

0.25

0.3

0.35

0.4

0.45

ER = 4

ER = 32

ER = 16ER = 8

ER = 1.5

ER = 2

ER = 3

Maximum De 1.0?



McKinley et al scaling criterion for onset of purely elastic instabilities:

independent of ER

Streamlines at De = 1 for ER = 4, 8 and 16

critM

21

11

U

Maximum De 1.0?



McKinley scaling criterion for onset of purely elastic instabilities: critM

21

11

U

Streamlines at De = 1 for ER = 4, 8 and 16

XX

Conclusions



For large expansion ratios ( 8)

• Recirculation length normalised with downstream duct height scales with a Deborah number based on bulk velocity at inlet and downstream duct height (De/ER)

For small expansion ratios ( 2)

• XR initially decreases before increasing at a given level of elasticity (De/ER ~ 0.4)

• In range of De for which steady solutions could be obtained XR

decreases with elasticity

Maximum obtainable De is approximately 1.0: independent of ER

De

C

0 0.5 1 1.50

0.2

0.4

0.6

0.8

1

UCM M1UCM M2

UCM M3

OLD B M1OLD B M2

OLD B M3

4

-1.5

Pressure

NEWT

Enhanced pressure drop

4-3.8

Pressure

UCM

De=0.8 w

fdPP

C 2



0.15% polyacrylamide solutionNewtonian

‘2D’ 1: 13.3 Planar Expansion

Townsend and Walters (1993)

Re < 10

De O(1)?



Frame 002 23 Apr 2006 No Data Set

r/d

ii/(

0UB/d)

10-1 10010-1

100

101

-2/3 slopeXXXYYY

Stress variation around sharp corner

Hinch (1993) JnNFM

32r

Stresses around sharp corner go to infinity as:

r



Normal stresses (ER = 3)


x / d

XX/(

0UB/d)

-2 0 2 4 6-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Newt (De=0)

UCM De=0.2

UCM De=0.4

UCM De=0.6

UCM De=0.8

UCM De=1.0


x / d

YY/(

0UB/d)

-2 0 2 4 6-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Newt (De=0)

UCM De=0.2

UCM De=0.4

UCM De=0.6

UCM De=0.8

UCM De=1.0



plane sudden expansion flows of viscoelastic liquids: effect of expansion ratio robert j poole...

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