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Turbulent flow of non-Newtonian liquids through an axisymmetric
sudden expansion
Rob Poole
Department of Engineering,
University of LiverpoolOsborne Reynolds
Seminar 30th April 2003
Introduction
Osborne Reynolds
Seminar 30th April 2003
• 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– Database for CFD validation
Osborne Reynolds
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 fluidsWorking 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…Working fluids cont…
• Rheological data obtained– Shear viscosity vs shear rate– First normal stress difference
vs shear stress
N1
Osborne Reynolds
Seminar 30th April 2003
Rheological data
Figure 2: Viscosity versus shear rate for 0.02,0.05 and 0.1% of polyacrylamide(including Carreau-Yasuda fit)
Shear rate (1/s)
Vis
cosi
ty(P
as)
10-3 10-2 10-1 100 101 102 103 10410-3
10-2
10-1
100
101
102
0.02% PAA
0.05% PAA
0.1% PAA
anaCY
CY
μμμμ /
0
)(1
Osborne Reynolds
Seminar 30th April 2003
Rheological data cont …
0.1% PAA
Figure 3: First normal stress difference N1 versus shear stress for 0.1% PAA.
Shear stress (Pa)
Fir
stno
rmal
stre
ssdi
ffer
ence
N1
(Pa)
100 101 102101
102
103
Osborne Reynolds
Seminar 30th April 2003
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
13000 sdy
dV
Max
c
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
26000Re1 C
BhU
22700Re2 CH
BhU
• Hence a Reynolds number of
Mean axial velocity profilesy/
h
r/R
0
0.5
1
1.5
2 0
0.5
1
x/h 9 2016108 12
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1
1
1 3 4 6x/h 2 5
Figure 5 (b): Mean axial velocity (U/UB) profiles
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1
1
1 3 4 6x/h 2 5
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1x/h 9 2016108 12
Osborne Reynolds
Seminar 30th April 2003
0.02% PAA
Water
Streamlines
Figure 7 (a):Streamline pattern for Water Re=30000
y/h
0 0
0.5 0.5
1 1
x/h 122 4 6 8 XR 2016
Water
-0.08<<0 [0.02 steps]
0< <0.35 [0.05 steps]
Figure 7 (a):Streamline pattern for Water Re=30000
Figure 7 (b):Streamline pattern for 0.02% PAA Re=26000
y/h
0 0
0.5 0.5
1 1
x/h 162 4 6 8 12 XR10
0.02% PAA
-0.09< <-0.01 [0.02 steps]
0< <0.3 [0.05 steps]
Osborne Reynolds
Seminar 30th April 2003
Axial Reynolds stresses (u)y/
h
r/R
0
0.5
1
1.5
2 0
0.5
1x/h 9 2016108 12
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1
0.25
1 3 4 6x/h 2 5
Figure 10 (b): Axial turbulence intensity (u' /UB) profiles
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1
0.25
1 3 4 6x/h 2 5
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1x/h 9 2016108 12
Osborne Reynolds
Seminar 30th April 2003
0.02% PAA
Water
Radial Reynolds stresses (v)y/
h
r/R
0
0.5
1
1.5
2 0
0.5
1x/h 9 2016108 12
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1
0.25
1 3 4 6x/h 2 5
Figure 12 (b): Radial turbulence intensity (v' /UB) profiles
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1
0.25
1 3 4 6x/h 2 5
y/h
r/R
0
0.5
1
1.5
2 0
0.5
1x/h 9 2016108 12
Osborne Reynolds
Seminar 30th April 2003
0.02% PAA
Water
Osborne Reynolds
Seminar 30th April 2003
y/h
0
0.5
1
1.5
2
2.5
3
3.5
4
1
3
x/h
1
1 6x/h 3 12
0.1% PAA
Re 4000
XR32
Mean axial velocity profiles
No recirculation
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.