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.