Transcript Document

Plane sudden expansion flows of viscoelastic liquids:
effect of expansion ratio
Robert J Poole
Department of Engineering, University of Liverpool, UK
Manuel A Alves
CEFT, Faculdade de Engenharia, Universidade do Porto, Portugal
Paulo J Oliveira
Departamento de Engenharia Electromecânica, Universidade da Beira
Interior, Portugal
Fernando T Pinho
aCEFT,
Faculdade de Engenharia, Universidade do Porto, Portugal
do Minho, Portugal
bUniversidade
AERC 2007
4th 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
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Introduction
Why investigate expansion flows of viscoelastic liquids?
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
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Governing equations
 u  0
1) Mass
2) Momentum (creeping flow)
3) Constitutive equation
0   p    τ
Upper Convected Maxwell model (UCM)
τ

   uτ   τ   u  uT    τ  u  uT  τ 
 t

 
Essentially phenomenological model
• “Simplest” viscoelastic differential model
• Capable of capturing qualitative features of many highly-elastic flows
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
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
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Computational domain and meshes
L1= 20d
L2= 100d
Neumann b.c.s at exit
ER=D/d
h
D
d
UB
Expansion ratios
(ER)
1:1.5
1:2
Low ER
1:3
X
symmetry axis
Y
1:4
1:8
Fully-developed
inlet velocity and
stress profiles
De 
 .U B
d
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
1:16
1:32
High ER
Representative mesh details
ER = 1.5
NC
DOF
(xMIN)/d
M1
14 500
87 000
0.005
M2
58 000
348 000
0.0025
NC
DOF
(xMIN)/d
M1
15 000
90 000
0.01
M2
60 000
360 000
0.005
NC
DOF
(xMIN)/d
M1
21 500
129 000
0.01
M2
86 000
516 000
0.005
ER = 4
ER = 16
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Representative grid dependency and numerical
accuracy
ER and fluid
XR#
XR (= xR / d)
% 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
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Newtonian simulations: XR variation with ER
14
Newtonian M1
Newtonian M2
XR = 0.4185 (ER - 1) + 0.2635
X
12
XR (= xr / d)
10
d
8
X
6
4
X
2
X
X
XX
X
0
0
Linear fit to data for
ER  4 (R2=1)
X
10
ER - 1
20
30
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Newtonian simulations: XR variation with ER
14
2
12
Newtonian M1
Newtonian M2
1.75
XR = 0.4185 (ER - 1) + 0.2635
X
1.5
XR (= xr / d)
10
XR (= xr / d)
X
Newtonian M1
Newtonian M2
XR = 0.4185 (ER - 1) + 0.2635
X
1.25
8
X
6
X
1
0.75
X
4
0.5
X
Deviations from
linear fit as ER  1
X
2
X
X
XX
X
0
0
0.25
X
10
ER - 1
20
0X
0
X
30
1
2
ER - 1
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4th Annual European Rheology Conference
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3
4
Newtonian simulations: XR variation with ER
0.5
0.4
X
X
X
X
H
X
xr / D
0.3
X
X
0.2
X
0.1
X
0X
0
10
Newtonian M1
Newtonian M2
XR = 0.4185 (ER - 1) + 0.2635
ER - 1
20
30
AERC 2007
4th Annual European Rheology Conference
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D
“High” ER viscoelastic : XR variation with De and ER
14
12
ER = 32
XR (= xr / d)
10
Δ M1
8
X M2
X
6
4
X
2
X
0
0
X
X
X
X
X
X
X
X
X
X
X
ER = 16
ER = 8
X
0.2
X
0.4
X
0.6
X
0.8
De
X
1
ER = 4
1.2
1.4
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
 Extrapolated
1:4 expansion ratio
1.5X
X
1.45
X
XR (= xr / d)
1.4
1.35
X
1.3
X
1.25
X
1.2
X
1.15
1.1
0
0.2
ER = 4.0 M1
ER = 4.0 M2
Extrapolated
0.4
0.6
0.8
De
1
1.2
AERC 2007
4th Annual European Rheology Conference
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1.4
1:4 expansion ratio (M2)
De = 1.0
0.0
0.2
0.4
0.6
0.8
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
“High” ER viscoelastic : scaling of XR
0.45
0.45
X
X
X
X
X
X
X
X
X
xr / D
X
X
0.35
X
X
X
0.3
0.2
0
0.2
0.4
X X
X XX
X
X
X
X
X
X
X
X
0.3
ER =32
ER =16
ER =8
ER =4
0.25
X
X
X
X
0.35
X
X
0.4
xr / D
X
0.4X
0.25
0.6
0.8
De
1
1.2
1.4
0.2
10
-2
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4th Annual European Rheology Conference
April 12-14, Napoli - Italy
10
-1
De / ER
10
0
“Low” ER viscoelastic : XR variation with De and ER
1.2
X
1
ER = 3
X
XR (= xr / d)
0.8
0.6X
0.4
X
ER = 2
X
X
X
X
X
X
X
X
X
X
X
X
X
X
ER = 1.5
0.2
0
0
0.2
0.4
0.6
De
0.8
1
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
1:1.5 expansion ratio
1:2 expansion ratio
0.65
0.35
X
XR (= xr / d)
XR (= xr / d)
0.34
X
0.33X
X
X
X
0.32
X
X
X
0.6
X
X
X
0.55
X
X
X
0.31
0.3
ER = 1.5
0
0.2
0.4
0.6
0.8
1
X
ER = 2
0.5
0
0.2
De
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
0.4
0.6
De
0.8
1
1:1.5 expansion ratio
De = 1.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
De = 1.0
0.0
0.1
0.2
0.3
0.4
0.6
0.8
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
“Low” ER viscoelastic : scaling of XR
0.45
0.45
X
X
X
X
X
X
X
xr / D
0.35X
X
X
X
ER = 8
X
X
ER = 4
X
0.4
X
ER = 16
ER = 3
X
X
X
0.25
X
0.2
0
X X
0.2
X
X
0.4
X
X
0.6
X
X
ER = 2
X
0.8
De
X
X
ER = 8 X
ER = 16
X X
X XX
X
X
ER = 4 X
X
X
ER = 3
X
X
0.35
X
0.3X
X
ER = 32
ER = 32
X
X
xr / D
X
0.4X
0.3
X
X
ER = 2 X
X
X
X
X XX
0.25
X ER = 1.5
1
1.2
ER = 1.5
1.4
0.2
10
-2
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
X
X
10
-1
De / ER
X X XX
X
X
10
0
Maximum De  1.0?
1
2
McKinley et al scaling criterion for onset of purely elastic instabilities:
 U  11 
     M crit


 independent of ER
Streamlines at De = 1 for ER = 4, 8 and 16
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Maximum De  1.0?
1
2
McKinley scaling criterion for onset of purely elastic instabilities:
 XX
Streamlines at De = 1 for ER = 4, 8 and 16
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
 U  11 
     M crit


Conclusions
For large expansion ratios (  8)
• In range of De for which steady solutions could be obtained XR
decreases with elasticity
• 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)
Maximum obtainable De is approximately 1.0: independent of ER
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Enhanced pressure drop
1
Pressure
UCM M1
UCM M2
UCM M3
OLD B M1
OLD B M2
OLD B M3

0.8
0.6
NEWT


-1.5

C

4

0.4
Pressure

fd
0.2
0

P  P 
C
UCM
De=0.8
2 w
-3.8
0
0.5
De
1
1.5
4
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
‘2D’ 1: 13.3 Planar Expansion
Re < 10
Townsend and Walters (1993)
De O(1)?
Newtonian
0.15% polyacrylamide solution
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Stress variation around sharp corner
Frame 002  23 Apr 2006  No Data Set
ii / (0U B / d)
10
-2/3 slope
XX
XY
YY
1
r
Hinch (1993) JnNFM
10
0
Stresses around
sharp corner go to
infinity as:
r 2 3
10
-1
10-1
100
r/d
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
Normal stresses (ER = 3)
Frame 002  28 Apr 2006  No Data Set
Frame 002  23 Apr 2006  No Data Set
1.4
0.2
1.2
0
1
YY / (0U B / d)
XX / (0U B / d)
-0.2
-0.4
-0.6
Newt (De=0)
UCM De=0.2
UCM De=0.4
UCM De=0.6
UCM De=0.8
UCM De=1.0
-0.8
-1
-1.2
Newt (De=0)
UCM De=0.2
UCM De=0.4
UCM De=0.6
UCM De=0.8
UCM De=1.0
-2
0
2
4
0.8
0.6
0.4
0.2
0
6
-0.2
-2
x/d
AERC 2007
4th Annual European Rheology Conference
April 12-14, Napoli - Italy
0
2
x/d
4
6