Turbulent flow of non-Newtonian
liquids through an axisymmetric
sudden expansion
Rob Poole
Department of Engineering,
University of Liverpool
Osborne Reynolds
Seminar
30th April 2003
Introduction
• Osborne Reynolds (1883,1895)
• Newtonian flows - large literature exists
• Non-Newtonian - Few previous studies [Pak
et al (1990)]
– Experimental: flow visualisation
• Aims of this study
– Use of LDA to provide quantitative data
– Investigate effect on reattachment length
Osborne Reynolds
– Database for CFD validation
Seminar
30th April 2003
Experimental rig
Fully
developed
pipe flow
d= 26 mm D=52 mm
R = D2 / d2 = 4
Osborne Reynolds
Seminar
30th April 2003
Working fluids
• Water
• Three concentrations of polyacrylamide (PAA)
– 0.02%, 0.05% and 0.1%
– Shear thinning to various degrees
– Increasing viscoelasticity with concentration
– Large extensional viscosities
– Highly drag reducing
– Optically transparent
Osborne Reynolds
Seminar
30th April 2003
Working fluids cont…
N1
• Rheological data obtained
– Shear viscosity vs shear rate
– First normal stress difference
vs shear stress
Osborne Reynolds
Seminar
30th April 2003
Viscosity (Pa s)
Rheological data
10
2
10
1
μCY  μ 
μ0  μ
1  (
CY
10
 )
0.02% PAA

a n/a
0.05% PAA
0
0.1% PAA
10-1
10
-2
10
-3
10
-3
10
-2
10
-1
10
0
10
1
Shear rate (1/s)
10
2
10
3
Figure 2: Viscosity versus shear rate for 0.02,0.05 and 0.1% of polyacrylamide
(including Carreau-Yasuda fit)
10
4
Osborne Reynolds
Seminar
30th April 2003
First normal stress difference N 1 (Pa)
Rheological data cont …
10
3
0.1% PAA
102
1
10 0
10
10
1
Shear stress (Pa)
Figure 3: First normal stress difference N1
Osborne Reynolds
Seminar
30th April 2003
versus shear stress for 0.1% PAA.
10
2
Estimation of Reynolds number
• Difficulty - no single value for the
viscosity characterises the fluid.
• Method adopted - estimate the
maximum shear rate at ‘inlet’
(x/h=1).
• Example 0.02% PAA
dV
c 
dy
 3000 s 1
Max
Osborne Reynolds
Seminar
30th April 2003
Estimation of Reynolds number
• This shear rate is then used to
obtain a viscosity from the Carreau-Yasuda
model:
μC  2.82 x10-3 Pa.s
• Hence a Reynolds number of
U B h
U B h
Re1 
 26000 Re2 
 22700
C
 CH
Osborne Reynolds
Seminar
30th April 2003
Mean axial velocity profiles
2
2
0
0
1
1
0.5
0.5
0.5
0.5
0
x/h
x/h 0
rr/ /RR
y/h
y/h
1.5
1.5
1
1
1
1
2
2
3
3
4
4
5
5
6
6
2
0.02% PAA
1
1
Water
0
2
0
1.5
0.5
r/ R
1
1
r/ R
y/h
y/h
1.5
0.5
0.5
0.5
0
x/hx/h
0
1
8
9
10
12
8
9
10
12
Figure 5 (b): Mean axial velocity (U/UB) profiles
Figure 5 (b): Mean axial velocity (U/UB) profiles
16
16
20
20
1
Osborne Reynolds
Seminar
30th April 2003
Streamlines
1
1
y/hy/h
1
1
Water
0.5
-0.08<<0
[0.02 steps]
0<0.5 <0.35 [0.05 steps]
0.5
0.5
0
x/h
0
x/h
6
8
XR
12
2
4
Figure 7 2(a):Streamline
pattern
Re=30000
6 for Water
8
XR
12
4
Figure 7 (a):Streamline pattern for Water Re=30000
16
16
0.02% PAA
-0.09<1  <-0.01 [0.02 steps]
0<  <0.3 [0.05 steps]
1
1
1
y/hy/h
0
20
0
20
0.5
0.5
0.5
0
x/h
0
x/h
0.5
2
4
6
8
10
12
Figure 7 2(b):Streamline
pattern
PAA
4
6 for 0.02%
8
10 Re=26000
12
Figure 7 (b):Streamline pattern for 0.02% PAA Re=26000
16
16
Osborne Reynolds
0
Seminar
X
0
30th April 2003
X
R
R
Axial Reynolds stresses (u)
22
00
0.25
0.25
y/h
r/ R
r/ R
1.5
1.5
11
0.5
0.5
0.5
0.5
x/h
x/h
00
11
22
33
44
55
11
66
22
0.02% PAA
Water
00
y/h
R
rr // R
1.5
1.5
11
0.5
0.5
0.5
0.5
00
x/h
x/h
88
99
10
12
10
12
Figure
10
(b):
Axial
turbulence
intensity
(u'
/U
Figure 10 (b): Axial turbulence intensity (u' /UB)B)profiles
profiles
16
16
2020
11
Osborne Reynolds
Seminar
30th April 2003
Radial Reynolds stresses (v)
22
0.25
0.25
00
11
0.5
0.5
rr/ /RR
y/h
y/h
1.5
1.5
0.5
0.5
0
x/h
x/h 0
22
1
1
2
2
3
3
4
4
5
5
6
6
0.02% PAA
11
00
Water
11
0.5
0.5
rr/ /RR
y/h
y/h
1.5
1.5
0.5
0.5
00
x/h
x/h
88
99
10
12
10
12
Figure
12
(b):
Radial
turbulence
intensity
(v'
/U
profiles
Figure 12 (b): Radial turbulence intensity (v' /UB)) profiles
B
16
16
20
20
11
Osborne Reynolds
Seminar
30th April 2003
Mean axial velocity profiles
4
3.5
3
3
y/h
No recirculation
2.5
0.1% PAA
Re  4000
XR32
2
1.5
1
1
1
0.5
0
x/h
1
3
6
12
Figure 9 (a): Mean axial velocity (U/UB) profiles for 0.1% PAA Re= 4000
Osborne Reynolds
Seminar
30th April 2003
Concluding remarks
• Turbulent flow through an axisymmetric sudden
expansion of area expansion ratio (i.e. D2/d2) 4.
• Water and two lowest conc. of PAA - axisymmetric.
• Reattachment lengths were
Water XR 10 step heights
0.02% and 0.05% PAA XR 20 step heights
Osborne Reynolds
Seminar
30th April 2003
Concluding remarks cont…
• Increase in XR caused by modifications to turbulence
structure with large reductions in v and w resulting
in reduced transverse transfer of axial momentum.
• At highest conc. of PAA axisymmetric flow could
not be achieved. This could be due to an elastic
instability or a slight geometric imperfection that is
accentuated by viscoelasticity.
Osborne Reynolds
Seminar
30th April 2003